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<!DOCTYPE html> <html lang="en"> <head> <meta content="text/html; charset=utf-8" http-equiv="content-type"/> <title>Plug-and-Play image restoration with Stochastic deNOising REgularization</title>  <meta content="width=device-width, initial-scale=1, shrink-to-fit=no" name="viewport"/> <link href="https://cdn.jsdelivr.net/npm/bootstrap@5.3.0/dist/css/bootstrap.min.css" rel="stylesheet" type="text/css"/> <link href="/static/browse/0.3.4/css/ar5iv_0.7.4.min.css" rel="stylesheet" type="text/css"/> <link href="/static/browse/0.3.4/css/latexml_styles.css" rel="stylesheet" type="text/css"/> <script src="https://cdn.jsdelivr.net/npm/bootstrap@5.3.0/dist/js/bootstrap.bundle.min.js"></script> <script src="https://cdnjs.cloudflare.com/ajax/libs/html2canvas/1.3.3/html2canvas.min.js"></script> <script src="/static/browse/0.3.4/js/addons.js"></script> <script src="/static/browse/0.3.4/js/feedbackOverlay.js"></script> <meta content="Machine Learning, ICML" lang="en" name="keywords"/> <base href="/html/2402.01779v2/"/></head> <body> <nav class="ltx_page_navbar"> <nav class="ltx_TOC"> <ol class="ltx_toclist"> <li class="ltx_tocentry ltx_tocentry_section"><a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#S1" title="1 Introduction ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">1 </span>Introduction</span></a></li> <li class="ltx_tocentry ltx_tocentry_section"> <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#S2" title="2 Stochastic deNOising REgularization (SNORE) ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">2 </span>Stochastic deNOising REgularization (SNORE)</span></a> <ol class="ltx_toclist ltx_toclist_section"> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#S2.SS1" title="2.1 Background ‣ 2 Stochastic deNOising REgularization (SNORE) ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">2.1 </span>Background</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"> <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#S2.SS2" title="2.2 SNORE regularization ‣ 2 Stochastic deNOising REgularization (SNORE) ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">2.2 </span>SNORE regularization</span></a> <ol class="ltx_toclist ltx_toclist_subsection"> <li class="ltx_tocentry ltx_tocentry_paragraph"><a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#S2.SS2.SSS0.Px1" title="Interpretation of the SNORE regularization ‣ 2.2 SNORE regularization ‣ 2 Stochastic deNOising REgularization (SNORE) ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_title">Interpretation of the SNORE regularization</span></a></li> <li class="ltx_tocentry ltx_tocentry_paragraph"><a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#S2.SS2.SSS0.Px2" title="Optimization algorithms ‣ 2.2 SNORE regularization ‣ 2 Stochastic deNOising REgularization (SNORE) ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_title">Optimization algorithms</span></a></li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_subsection"> <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#S2.SS3" title="2.3 Related Works ‣ 2 Stochastic deNOising REgularization (SNORE) ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">2.3 </span>Related Works</span></a> <ol class="ltx_toclist ltx_toclist_subsection"> <li class="ltx_tocentry ltx_tocentry_paragraph"><a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#S2.SS3.SSS0.Px1" title="Other Stochastic Plug-and-Play algorithms ‣ 2.3 Related Works ‣ 2 Stochastic deNOising REgularization (SNORE) ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_title">Other Stochastic Plug-and-Play algorithms</span></a></li> <li class="ltx_tocentry ltx_tocentry_paragraph"><a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#S2.SS3.SSS0.Px2" title="Link with diffusion based method ‣ 2.3 Related Works ‣ 2 Stochastic deNOising REgularization (SNORE) ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_title">Link with diffusion based method</span></a></li> </ol> </li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_section"> <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#S3" title="3 Convergence Analysis ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">3 </span>Convergence Analysis</span></a> <ol class="ltx_toclist ltx_toclist_section"> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#S3.SS1" title="3.1 Asymptotics of critical points when 𝜎→0 ‣ 3 Convergence Analysis ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">3.1 </span>Asymptotics of critical points when <math alttext="\sigma\to 0" class="ltx_Math" display="inline"><semantics><mrow><mi>σ</mi><mo stretchy="false">→</mo><mn>0</mn></mrow><annotation-xml encoding="MathML-Content"><apply><ci>→</ci><ci>𝜎</ci><cn type="integer">0</cn></apply></annotation-xml><annotation encoding="application/x-tex">\sigma\to 0</annotation><annotation encoding="application/x-llamapun">italic_σ → 0</annotation></semantics></math></span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#S3.SS2" title="3.2 Unbiased algorithm analysis ‣ 3 Convergence Analysis ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">3.2 </span>Unbiased algorithm analysis</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#S3.SS3" title="3.3 Biased algorithm analysis ‣ 3 Convergence Analysis ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">3.3 </span>Biased algorithm analysis</span></a></li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_section"> <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#S4" title="4 Experiments ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">4 </span>Experiments</span></a> <ol class="ltx_toclist ltx_toclist_section"> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#S4.SS1" title="4.1 Deblurring ‣ 4 Experiments ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">4.1 </span>Deblurring</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#S4.SS2" title="4.2 Inpainting ‣ 4 Experiments ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">4.2 </span>Inpainting</span></a></li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_section"><a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#S5" title="5 Conclusion ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">5 </span>Conclusion</span></a></li> <li class="ltx_tocentry ltx_tocentry_section"><a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#S6" title="6 Impact Statement ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">6 </span>Impact Statement</span></a></li> <li class="ltx_tocentry ltx_tocentry_section"><a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#S7" title="7 Acknowledgements ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">7 </span>Acknowledgements</span></a></li> <li class="ltx_tocentry ltx_tocentry_appendix"><a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#A1" title="Appendix A Reproducibility Statement ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">A </span>Reproducibility Statement</span></a></li> <li class="ltx_tocentry ltx_tocentry_appendix"><a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#A2" title="Appendix B Related works ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">B </span>Related works</span></a></li> <li class="ltx_tocentry ltx_tocentry_appendix"> <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#A3" title="Appendix C SNORE in simple cases ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">C </span>SNORE in simple cases</span></a> <ol class="ltx_toclist ltx_toclist_appendix"> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#A3.SS1" title="C.1 Gaussian Prior ‣ Appendix C SNORE in simple cases ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">C.1 </span>Gaussian Prior</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"> <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#A3.SS2" title="C.2 Gaussian Mixture prior ‣ Appendix C SNORE in simple cases ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">C.2 </span>Gaussian Mixture prior</span></a> <ol class="ltx_toclist ltx_toclist_subsection"> <li class="ltx_tocentry ltx_tocentry_subsubsection"><a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#A3.SS2.SSS1" title="C.2.1 Theoretical considerations ‣ C.2 Gaussian Mixture prior ‣ Appendix C SNORE in simple cases ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">C.2.1 </span>Theoretical considerations</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsubsection"><a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#A3.SS2.SSS2" title="C.2.2 Simulations ‣ C.2 Gaussian Mixture prior ‣ Appendix C SNORE in simple cases ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">C.2.2 </span>Simulations</span></a></li> </ol> </li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_appendix"> <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#A4" title="Appendix D Proofs of section 3 ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">D </span>Proofs of section <span class="ltx_text ltx_ref_tag">3</span></span></a> <ol class="ltx_toclist ltx_toclist_appendix"> <li class="ltx_tocentry ltx_tocentry_subsection"> <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#A4.SS1" title="D.1 Proof of critical Point Analysis ‣ Appendix D Proofs of section 3 ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">D.1 </span>Proof of critical Point Analysis</span></a> <ol class="ltx_toclist ltx_toclist_subsection"> <li class="ltx_tocentry ltx_tocentry_subsubsection"><a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#A4.SS1.SSS1" title="D.1.1 Proof of Proposition 3.1 ‣ D.1 Proof of critical Point Analysis ‣ Appendix D Proofs of section 3 ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">D.1.1 </span>Proof of Proposition <span class="ltx_text ltx_ref_tag">3.1</span></span></a></li> <li class="ltx_tocentry ltx_tocentry_subsubsection"><a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#A4.SS1.SSS2" title="D.1.2 Proof of Proposition 3.2 ‣ D.1 Proof of critical Point Analysis ‣ Appendix D Proofs of section 3 ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">D.1.2 </span>Proof of Proposition <span class="ltx_text ltx_ref_tag">3.2</span></span></a></li> <li class="ltx_tocentry ltx_tocentry_subsubsection"><a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#A4.SS1.SSS3" title="D.1.3 Proof of Proposition 3.3 ‣ D.1 Proof of critical Point Analysis ‣ Appendix D Proofs of section 3 ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">D.1.3 </span>Proof of Proposition <span class="ltx_text ltx_ref_tag">3.3</span></span></a></li> <li class="ltx_tocentry ltx_tocentry_subsubsection"><a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#A4.SS1.SSS4" title="D.1.4 Proof of Proposition 3.4 ‣ D.1 Proof of critical Point Analysis ‣ Appendix D Proofs of section 3 ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">D.1.4 </span>Proof of Proposition <span class="ltx_text ltx_ref_tag">3.4</span></span></a></li> <li class="ltx_tocentry ltx_tocentry_subsubsection"><a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#A4.SS1.SSS5" title="D.1.5 Proof of Proposition 3.5 ‣ D.1 Proof of critical Point Analysis ‣ Appendix D Proofs of section 3 ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">D.1.5 </span>Proof of Proposition <span class="ltx_text ltx_ref_tag">3.5</span></span></a></li> </ol> </li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_appendix"><a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#A5" title="Appendix E Discussion on Assumptions ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">E </span>Discussion on Assumptions</span></a></li> <li class="ltx_tocentry ltx_tocentry_appendix"> <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#A6" title="Appendix F Additional Experiments ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">F </span>Additional Experiments</span></a> <ol class="ltx_toclist ltx_toclist_appendix"> <li class="ltx_tocentry ltx_tocentry_paragraph"><a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#A6.SS0.SSS0.Px1" title="Metrics ‣ Appendix F Additional Experiments ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_title">Metrics</span></a></li> <li class="ltx_tocentry ltx_tocentry_paragraph"><a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#A6.SS0.SSS0.Px2" title="Denoiser ‣ Appendix F Additional Experiments ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_title">Denoiser</span></a></li> <li class="ltx_tocentry ltx_tocentry_paragraph"><a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#A6.SS0.SSS0.Px3" title="RED Prox ‣ Appendix F Additional Experiments ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_title">RED Prox</span></a></li> <li class="ltx_tocentry ltx_tocentry_paragraph"><a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#A6.SS0.SSS0.Px4" title="SNORE Prox ‣ Appendix F Additional Experiments ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_title">SNORE Prox</span></a></li> <li class="ltx_tocentry ltx_tocentry_paragraph"><a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#A6.SS0.SSS0.Px5" title="DiffPIR ‣ Appendix F Additional Experiments ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_title">DiffPIR</span></a></li> <li class="ltx_tocentry ltx_tocentry_paragraph"><a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#A6.SS0.SSS0.Px6" title="PnP SGD ‣ Appendix F Additional Experiments ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_title">PnP SGD</span></a></li> <li class="ltx_tocentry ltx_tocentry_paragraph"><a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#A6.SS0.SSS0.Px7" title="Computational time ‣ Appendix F Additional Experiments ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_title">Computational time</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"> <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#A6.SS1" title="F.1 Deblurring ‣ Appendix F Additional Experiments ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">F.1 </span>Deblurring</span></a> <ol class="ltx_toclist ltx_toclist_subsection"> <li class="ltx_tocentry ltx_tocentry_paragraph"><a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#A6.SS1.SSS0.Px1" title="Parameters setting ‣ F.1 Deblurring ‣ Appendix F Additional Experiments ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_title">Parameters setting</span></a></li> <li class="ltx_tocentry ltx_tocentry_paragraph"><a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#A6.SS1.SSS0.Px2" title="More deblurring results ‣ F.1 Deblurring ‣ Appendix F Additional Experiments ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_title">More deblurring results</span></a></li> <li class="ltx_tocentry ltx_tocentry_paragraph"><a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#A6.SS1.SSS0.Px3" title="Influence of 𝛼_{𝑚-1} ‣ F.1 Deblurring ‣ Appendix F Additional Experiments ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_title">Influence of <math alttext="\alpha_{m-1}" class="ltx_Math" display="inline"><semantics><msub><mi>α</mi><mrow><mi>m</mi><mo>−</mo><mn>1</mn></mrow></msub><annotation-xml encoding="MathML-Content"><apply><csymbol cd="ambiguous">subscript</csymbol><ci>𝛼</ci><apply><minus></minus><ci>𝑚</ci><cn type="integer">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex">\alpha_{m-1}</annotation><annotation encoding="application/x-llamapun">italic_α start_POSTSUBSCRIPT italic_m - 1 end_POSTSUBSCRIPT</annotation></semantics></math></span></a></li> <li class="ltx_tocentry ltx_tocentry_paragraph"><a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#A6.SS1.SSS0.Px4" title="Influence of 𝜎_{𝑚-1} ‣ F.1 Deblurring ‣ Appendix F Additional Experiments ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_title">Influence of <math alttext="\sigma_{m-1}" class="ltx_Math" display="inline"><semantics><msub><mi>σ</mi><mrow><mi>m</mi><mo>−</mo><mn>1</mn></mrow></msub><annotation-xml encoding="MathML-Content"><apply><csymbol cd="ambiguous">subscript</csymbol><ci>𝜎</ci><apply><minus></minus><ci>𝑚</ci><cn type="integer">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex">\sigma_{m-1}</annotation><annotation encoding="application/x-llamapun">italic_σ start_POSTSUBSCRIPT italic_m - 1 end_POSTSUBSCRIPT</annotation></semantics></math></span></a></li> <li class="ltx_tocentry ltx_tocentry_paragraph"><a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#A6.SS1.SSS0.Px5" title="Influence of 𝑚 ‣ F.1 Deblurring ‣ Appendix F Additional Experiments ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_title">Influence of <math alttext="m" class="ltx_Math" display="inline"><semantics><mi>m</mi><annotation-xml encoding="MathML-Content"><ci>𝑚</ci></annotation-xml><annotation encoding="application/x-tex">m</annotation><annotation encoding="application/x-llamapun">italic_m</annotation></semantics></math></span></a></li> <li class="ltx_tocentry ltx_tocentry_paragraph"><a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#A6.SS1.SSS0.Px6" title="On the step-size 𝛿 ‣ F.1 Deblurring ‣ Appendix F Additional Experiments ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_title">On the step-size <math alttext="\delta" class="ltx_Math" display="inline"><semantics><mi>δ</mi><annotation-xml encoding="MathML-Content"><ci>𝛿</ci></annotation-xml><annotation encoding="application/x-tex">\delta</annotation><annotation encoding="application/x-llamapun">italic_δ</annotation></semantics></math></span></a></li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_subsection"> <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#A6.SS2" title="F.2 Uncertainty of SNORE ‣ Appendix F Additional Experiments ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">F.2 </span>Uncertainty of SNORE</span></a> <ol class="ltx_toclist ltx_toclist_subsection"> <li class="ltx_tocentry ltx_tocentry_paragraph"><a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#A6.SS2.SSS0.Px1" title="Seed sensitivity ‣ F.2 Uncertainty of SNORE ‣ Appendix F Additional Experiments ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_title">Seed sensitivity</span></a></li> <li class="ltx_tocentry ltx_tocentry_paragraph"><a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#A6.SS2.SSS0.Px2" title="Initialization sensitivity ‣ F.2 Uncertainty of SNORE ‣ Appendix F Additional Experiments ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_title">Initialization sensitivity</span></a></li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_subsection"> <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#A6.SS3" title="F.3 Inpainting ‣ Appendix F Additional Experiments ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">F.3 </span>Inpainting</span></a> <ol class="ltx_toclist ltx_toclist_subsection"> <li class="ltx_tocentry ltx_tocentry_paragraph"><a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#A6.SS3.SSS0.Px1" title="Parameters setting ‣ F.3 Inpainting ‣ Appendix F Additional Experiments ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_title">Parameters setting</span></a></li> <li class="ltx_tocentry ltx_tocentry_paragraph"><a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#A6.SS3.SSS0.Px2" title="More results ‣ F.3 Inpainting ‣ Appendix F Additional Experiments ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_title">More results</span></a></li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_subsection"> <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#A6.SS4" title="F.4 Super-resolution ‣ Appendix F Additional Experiments ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">F.4 </span>Super-resolution</span></a> <ol class="ltx_toclist ltx_toclist_subsection"> <li class="ltx_tocentry ltx_tocentry_paragraph"><a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#A6.SS4.SSS0.Px1" title="Parameters setting ‣ F.4 Super-resolution ‣ Appendix F Additional Experiments ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_title">Parameters setting</span></a></li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#A6.SS5" title="F.5 SAR despeckling ‣ Appendix F Additional Experiments ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">F.5 </span>SAR despeckling</span></a></li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_appendix"><a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#A7" title="Appendix G SNORE applies the denoiser on its training domain ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">G </span>SNORE applies the denoiser on its training domain</span></a></li> <li class="ltx_tocentry ltx_tocentry_appendix"><a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#A8" title="Appendix H Discussion on the boundedness of (𝐱_𝑘)_{𝑘∈ℕ} ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">H </span>Discussion on the boundedness of <math alttext="({\mathbf{x}}_{k})_{k\in{\mathbb{N}}}" class="ltx_Math" display="inline"><semantics><msub><mrow><mo stretchy="false">(</mo><msub><mi>𝐱</mi><mi>k</mi></msub><mo stretchy="false">)</mo></mrow><mrow><mi>k</mi><mo>∈</mo><mi>ℕ</mi></mrow></msub><annotation-xml encoding="MathML-Content"><apply><csymbol cd="ambiguous">subscript</csymbol><apply><csymbol cd="ambiguous">subscript</csymbol><ci>𝐱</ci><ci>𝑘</ci></apply><apply><in></in><ci>𝑘</ci><ci>ℕ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex">({\mathbf{x}}_{k})_{k\in{\mathbb{N}}}</annotation><annotation encoding="application/x-llamapun">( bold_x start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) start_POSTSUBSCRIPT italic_k ∈ blackboard_N end_POSTSUBSCRIPT</annotation></semantics></math></span></a></li> </ol></nav> </nav> <div class="ltx_page_main"> <div class="ltx_page_content"><div class="section" id="target-section"><div id="license-tr">License: arXiv.org perpetual non-exclusive license</div><div id="watermark-tr">arXiv:2402.01779v2 [eess.IV] 05 Apr 2024</div></div> <article class="ltx_document ltx_pruned_first"> <h1 class="ltx_title ltx_title_document">Plug-and-Play image restoration with Stochastic deNOising REgularization</h1> <div class="ltx_authors"> <span class="ltx_creator ltx_role_author"> <span class="ltx_personname">Marien Renaud </span></span> <span class="ltx_author_before"> </span><span class="ltx_creator ltx_role_author"> <span class="ltx_personname">Jean Prost </span></span> <span class="ltx_author_before"> </span><span class="ltx_creator ltx_role_author"> <span class="ltx_personname">Arthur Leclaire </span></span> <span class="ltx_author_before"> </span><span class="ltx_creator ltx_role_author"> <span class="ltx_personname">Nicolas Papadakis </span></span> </div> <div class="ltx_abstract"> <h6 class="ltx_title ltx_title_abstract">Abstract</h6> <p class="ltx_p" id="id1.id1">Plug-and-Play (PnP) algorithms are a class of iterative algorithms that address image inverse problems by combining a physical model and a deep neural network for regularization. Even if they produce impressive image restoration results, these algorithms rely on a non-standard use of a denoiser on images that are less and less noisy along the iterations, which contrasts with recent algorithms based on Diffusion Models (DM), where the denoiser is applied only on re-noised images. We propose a new PnP framework, called Stochastic deNOising REgularization (SNORE), which applies the denoiser only on images with noise of the adequate level. It is based on an explicit stochastic regularization, which leads to a stochastic gradient descent algorithm to solve ill-posed inverse problems. A convergence analysis of this algorithm and its annealing extension is provided. Experimentally, we prove that SNORE is competitive with respect to state-of-the-art methods on deblurring and inpainting tasks, both quantitatively and qualitatively.</p> </div> <div class="ltx_keywords">Machine Learning, ICML </div> <div class="ltx_para" id="p2"> <br class="ltx_break"/> </div> <section class="ltx_section" id="S1"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">1 </span>Introduction</h2> <div class="ltx_para" id="S1.p1"> <p class="ltx_p" id="S1.p1.2">Many imaging problems can be formulated as <span class="ltx_text ltx_font_italic" id="S1.p1.2.1">inverse problems</span> seeking to recover high-quality images <math alttext="{\mathbf{x}}^{*}" class="ltx_Math" display="inline" id="S1.p1.1.m1.1"><semantics id="S1.p1.1.m1.1a"><msup id="S1.p1.1.m1.1.1" xref="S1.p1.1.m1.1.1.cmml"><mi id="S1.p1.1.m1.1.1.2" xref="S1.p1.1.m1.1.1.2.cmml">𝐱</mi><mo id="S1.p1.1.m1.1.1.3" xref="S1.p1.1.m1.1.1.3.cmml">*</mo></msup><annotation-xml 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id="S1.p1.2.m2.1c">{\mathbf{y}}</annotation><annotation encoding="application/x-llamapun" id="S1.p1.2.m2.1d">bold_y</annotation></semantics></math> by solving a problem of the form</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A8.EGx1"> <tbody id="S1.E1"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle{\mathbf{x}}^{*}\in\operatorname*{arg\,min}_{{\mathbf{x}}\in{% \mathbb{R}}^{d}}{\mathcal{F}({\mathbf{x}},{\mathbf{y}})}+\alpha\mathcal{R}({% \mathbf{x}})," class="ltx_Math" display="inline" id="S1.E1.m1.4"><semantics id="S1.E1.m1.4a"><mrow id="S1.E1.m1.4.4.1" xref="S1.E1.m1.4.4.1.1.cmml"><mrow id="S1.E1.m1.4.4.1.1" xref="S1.E1.m1.4.4.1.1.cmml"><msup id="S1.E1.m1.4.4.1.1.2" xref="S1.E1.m1.4.4.1.1.2.cmml"><mi id="S1.E1.m1.4.4.1.1.2.2" xref="S1.E1.m1.4.4.1.1.2.2.cmml">𝐱</mi><mo id="S1.E1.m1.4.4.1.1.2.3" 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id="S1.E1.m1.4.4.1.1.3.2.2.2.cmml" xref="S1.E1.m1.4.4.1.1.3.2.2.2">ℱ</ci></apply><interval closure="open" id="S1.E1.m1.4.4.1.1.3.2.3.1.cmml" xref="S1.E1.m1.4.4.1.1.3.2.3.2"><ci id="S1.E1.m1.1.1.cmml" xref="S1.E1.m1.1.1">𝐱</ci><ci id="S1.E1.m1.2.2.cmml" xref="S1.E1.m1.2.2">𝐲</ci></interval></apply><apply id="S1.E1.m1.4.4.1.1.3.3.cmml" xref="S1.E1.m1.4.4.1.1.3.3"><times id="S1.E1.m1.4.4.1.1.3.3.1.cmml" xref="S1.E1.m1.4.4.1.1.3.3.1"></times><ci id="S1.E1.m1.4.4.1.1.3.3.2.cmml" xref="S1.E1.m1.4.4.1.1.3.3.2">𝛼</ci><ci id="S1.E1.m1.4.4.1.1.3.3.3.cmml" xref="S1.E1.m1.4.4.1.1.3.3.3">ℛ</ci><ci id="S1.E1.m1.3.3.cmml" xref="S1.E1.m1.3.3">𝐱</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.E1.m1.4c">\displaystyle{\mathbf{x}}^{*}\in\operatorname*{arg\,min}_{{\mathbf{x}}\in{% \mathbb{R}}^{d}}{\mathcal{F}({\mathbf{x}},{\mathbf{y}})}+\alpha\mathcal{R}({% \mathbf{x}}),</annotation><annotation encoding="application/x-llamapun" id="S1.E1.m1.4d">bold_x start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT ∈ start_OPERATOR roman_arg roman_min end_OPERATOR start_POSTSUBSCRIPT bold_x ∈ blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT end_POSTSUBSCRIPT caligraphic_F ( bold_x , bold_y ) + italic_α caligraphic_R ( bold_x ) ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(1)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S1.p1.6">where <math alttext="\mathcal{F}" class="ltx_Math" display="inline" id="S1.p1.3.m1.1"><semantics id="S1.p1.3.m1.1a"><mi class="ltx_font_mathcaligraphic" id="S1.p1.3.m1.1.1" xref="S1.p1.3.m1.1.1.cmml">ℱ</mi><annotation-xml encoding="MathML-Content" id="S1.p1.3.m1.1b"><ci id="S1.p1.3.m1.1.1.cmml" xref="S1.p1.3.m1.1.1">ℱ</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p1.3.m1.1c">\mathcal{F}</annotation><annotation encoding="application/x-llamapun" id="S1.p1.3.m1.1d">caligraphic_F</annotation></semantics></math> measures the fidelity to the degraded observation <math alttext="{\mathbf{y}}" class="ltx_Math" display="inline" id="S1.p1.4.m2.1"><semantics id="S1.p1.4.m2.1a"><mi id="S1.p1.4.m2.1.1" xref="S1.p1.4.m2.1.1.cmml">𝐲</mi><annotation-xml encoding="MathML-Content" id="S1.p1.4.m2.1b"><ci id="S1.p1.4.m2.1.1.cmml" xref="S1.p1.4.m2.1.1">𝐲</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p1.4.m2.1c">{\mathbf{y}}</annotation><annotation encoding="application/x-llamapun" id="S1.p1.4.m2.1d">bold_y</annotation></semantics></math> and <math alttext="\mathcal{R}" class="ltx_Math" display="inline" id="S1.p1.5.m3.1"><semantics id="S1.p1.5.m3.1a"><mi class="ltx_font_mathcaligraphic" id="S1.p1.5.m3.1.1" xref="S1.p1.5.m3.1.1.cmml">ℛ</mi><annotation-xml encoding="MathML-Content" id="S1.p1.5.m3.1b"><ci id="S1.p1.5.m3.1.1.cmml" xref="S1.p1.5.m3.1.1">ℛ</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p1.5.m3.1c">\mathcal{R}</annotation><annotation encoding="application/x-llamapun" id="S1.p1.5.m3.1d">caligraphic_R</annotation></semantics></math> is a regularization term weighted by a parameter <math alttext="\alpha>0" class="ltx_Math" display="inline" id="S1.p1.6.m4.1"><semantics id="S1.p1.6.m4.1a"><mrow id="S1.p1.6.m4.1.1" xref="S1.p1.6.m4.1.1.cmml"><mi id="S1.p1.6.m4.1.1.2" xref="S1.p1.6.m4.1.1.2.cmml">α</mi><mo id="S1.p1.6.m4.1.1.1" xref="S1.p1.6.m4.1.1.1.cmml">></mo><mn id="S1.p1.6.m4.1.1.3" xref="S1.p1.6.m4.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S1.p1.6.m4.1b"><apply id="S1.p1.6.m4.1.1.cmml" xref="S1.p1.6.m4.1.1"><gt id="S1.p1.6.m4.1.1.1.cmml" xref="S1.p1.6.m4.1.1.1"></gt><ci id="S1.p1.6.m4.1.1.2.cmml" xref="S1.p1.6.m4.1.1.2">𝛼</ci><cn id="S1.p1.6.m4.1.1.3.cmml" type="integer" xref="S1.p1.6.m4.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p1.6.m4.1c">\alpha>0</annotation><annotation encoding="application/x-llamapun" id="S1.p1.6.m4.1d">italic_α > 0</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S1.p2"> <p class="ltx_p" id="S1.p2.1">The regularization is crucial to complete the missing information of the observation by bringing prior knowledge on the high-quality image. Choosing a relevant prior <math alttext="\mathcal{R}" class="ltx_Math" display="inline" id="S1.p2.1.m1.1"><semantics id="S1.p2.1.m1.1a"><mi class="ltx_font_mathcaligraphic" id="S1.p2.1.m1.1.1" xref="S1.p2.1.m1.1.1.cmml">ℛ</mi><annotation-xml encoding="MathML-Content" id="S1.p2.1.m1.1b"><ci id="S1.p2.1.m1.1.1.cmml" xref="S1.p2.1.m1.1.1">ℛ</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p2.1.m1.1c">\mathcal{R}</annotation><annotation encoding="application/x-llamapun" id="S1.p2.1.m1.1d">caligraphic_R</annotation></semantics></math> in Problem (<a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#S1.E1" title="1 ‣ 1 Introduction ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_tag">1</span></a>) is a long-standing <cite class="ltx_cite ltx_citemacro_citep">(Rudin et al., <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#bib.bib59" title="">1992</a>; Mallat, <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#bib.bib44" title="">1999</a>; Zoran & Weiss, <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#bib.bib81" title="">2011</a>)</cite> and challenging task and recent approaches explore deep learning techniques that learn a prior from a database of clean images <cite class="ltx_cite ltx_citemacro_citep">(Zhang et al., <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#bib.bib77" title="">2017</a>; Lunz et al., <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#bib.bib41" title="">2018</a>; Laumont et al., <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#bib.bib37" title="">2022b</a>)</cite>.</p> </div> <div class="ltx_para" id="S1.p3"> <p class="ltx_p" id="S1.p3.2">Problem (<a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#S1.E1" title="1 ‣ 1 Introduction ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_tag">1</span></a>) can be addressed with proximal splitting algorithms <cite class="ltx_cite ltx_citemacro_citep">(Boyd et al., <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#bib.bib5" title="">2011</a>)</cite> which are first-order optimization algorithms based on the recursive application of gradient-descent and/or proximal operators of functions <math alttext="\mathcal{F}" class="ltx_Math" display="inline" id="S1.p3.1.m1.1"><semantics id="S1.p3.1.m1.1a"><mi class="ltx_font_mathcaligraphic" id="S1.p3.1.m1.1.1" xref="S1.p3.1.m1.1.1.cmml">ℱ</mi><annotation-xml encoding="MathML-Content" id="S1.p3.1.m1.1b"><ci id="S1.p3.1.m1.1.1.cmml" xref="S1.p3.1.m1.1.1">ℱ</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p3.1.m1.1c">\mathcal{F}</annotation><annotation encoding="application/x-llamapun" id="S1.p3.1.m1.1d">caligraphic_F</annotation></semantics></math> and <math alttext="\mathcal{R}" class="ltx_Math" display="inline" id="S1.p3.2.m2.1"><semantics id="S1.p3.2.m2.1a"><mi class="ltx_font_mathcaligraphic" id="S1.p3.2.m2.1.1" xref="S1.p3.2.m2.1.1.cmml">ℛ</mi><annotation-xml encoding="MathML-Content" id="S1.p3.2.m2.1b"><ci id="S1.p3.2.m2.1.1.cmml" xref="S1.p3.2.m2.1.1">ℛ</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p3.2.m2.1c">\mathcal{R}</annotation><annotation encoding="application/x-llamapun" id="S1.p3.2.m2.1d">caligraphic_R</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S1.p4"> <p class="ltx_p" id="S1.p4.1">The Plug-and-Play (PnP) framework <cite class="ltx_cite ltx_citemacro_citep">(Venkatakrishnan et al., <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#bib.bib75" title="">2013</a>)</cite> consists in replacing, within a proximal splitting algorithm, the proximal step on the regularization <math alttext="\mathcal{R}" class="ltx_Math" display="inline" id="S1.p4.1.m1.1"><semantics id="S1.p4.1.m1.1a"><mi class="ltx_font_mathcaligraphic" id="S1.p4.1.m1.1.1" xref="S1.p4.1.m1.1.1.cmml">ℛ</mi><annotation-xml encoding="MathML-Content" id="S1.p4.1.m1.1b"><ci id="S1.p4.1.m1.1.1.cmml" xref="S1.p4.1.m1.1.1">ℛ</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p4.1.m1.1c">\mathcal{R}</annotation><annotation encoding="application/x-llamapun" id="S1.p4.1.m1.1d">caligraphic_R</annotation></semantics></math> by a denoising operation; and it allows to use implicit regularization priors encoded by pre-trained image denoisers. Similarly, following the Regularization by Denoising (RED) framework <cite class="ltx_cite ltx_citemacro_citep">(Romano et al., <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#bib.bib58" title="">2017</a>)</cite>, a gradient-descent step on the regularization can be substituted by a learned denoiser. It has been observed that plugging a pre-trained state-of-the-art deep denoiser is essential for achieving the best quality results in many imaging contexts <cite class="ltx_cite ltx_citemacro_citep">(Metzler et al., <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#bib.bib48" title="">2018</a>; Ryu et al., <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#bib.bib60" title="">2019</a>; Hurault et al., <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#bib.bib31" title="">2022b</a>; Renaud et al., <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#bib.bib57" title="">2023</a>; Ulondu-Mendes et al., <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#bib.bib74" title="">2023</a>)</cite>.</p> </div> <div class="ltx_para" id="S1.p5"> <p class="ltx_p" id="S1.p5.1">In another line of works, inverse problems solvers based on denoising diffusion models (DDM) <cite class="ltx_cite ltx_citemacro_citep">(Ho et al., <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#bib.bib28" title="">2020</a>; Song et al., <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#bib.bib66" title="">2021b</a>)</cite> have demonstrated their ability to provide high-quality restoration even for severely ill-posed problems where a large amount of information is missing <cite class="ltx_cite ltx_citemacro_citep">(Chung et al., <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#bib.bib12" title="">2023</a>; Song et al., <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#bib.bib63" title="">2023</a>)</cite>. DDM and PnP both rely on deep denoisers to implicitly model the prior distribution, and they decouple prior and data-fidelity terms in order to provide flexible solvers. However, while the theoretical properties of PnP algorithms regarding convergence have been studied in depth <cite class="ltx_cite ltx_citemacro_citep">(Sreehari et al., <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#bib.bib68" title="">2016</a>; Sun et al., <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#bib.bib70" title="">2019b</a>; Gavaskar & Chaudhury, <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#bib.bib25" title="">2019</a>; Ryu et al., <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#bib.bib60" title="">2019</a>; Cohen et al., <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#bib.bib14" title="">2021</a>; Hurault et al., <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#bib.bib30" title="">2022a</a>)</cite>, the theoretical properties of restoration algorithms based on DDM remain poorly understood. Indeed, DDM methods rely on heuristics to approximate the score of the intractable likelihood. To the best of our knowledge, the impact of the approximation error on the generated samples remains to be quantified.</p> </div> <div class="ltx_para" id="S1.p6"> <p class="ltx_p" id="S1.p6.1">In this work, we seek to develop an inverse problem solver that inherits the superior restoration quality of DDM methods, while satisfying the theoretical guarantees of convergence that are met for some PnP algorithms. Our key observation is that, contrary to the PnP framework, in diffusion based methods the denoiser is applied on a <em class="ltx_emph ltx_font_italic" id="S1.p6.1.1">noisy version of the image</em> at each iteration of the sampling process. We postulate that applying the denoiser to noisy images is a main ingredient to the impressive performance of diffusion-based samplers, as it avoids a domain shift relative to the data on which the denoiser was trained.</p> </div> <div class="ltx_para" id="S1.p7"> <p class="ltx_p" id="S1.p7.1">This motivates us to propose SNORE (Stochastic deNOising REgularization), a stochastic PnP algorithm, which differs from classical PnP schemes by injecting noise to the input of the denoiser at each iteration. SNORE minimizes a classical variational objective, where the regularization term is defined as the average value of the smoothed log prior on noisy version of the image of interest, and can be viewed as a relaxed version of the usual negative log-prior.</p> </div> <div class="ltx_para" id="S1.p8"> <p class="ltx_p" id="S1.p8.1"><span class="ltx_text ltx_font_bold" id="S1.p8.1.1">Contributions.</span> <span class="ltx_text ltx_font_bold" id="S1.p8.1.2">(a)</span> We propose a new explicit regularization leading to a novel PnP framework, named <span class="ltx_text ltx_font_italic" id="S1.p8.1.3">Stochastic deNOising Regularization (SNORE)</span>, in which the denoiser is applied on a noisy version of the image at each iteration. <span class="ltx_text ltx_font_bold" id="S1.p8.1.4">(b)</span> We show that SNORE can be optimized by a stochastic gradient-descent algorithm (Algorithm <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#alg2" title="Algorithm 2 ‣ Optimization algorithms ‣ 2.2 SNORE regularization ‣ 2 Stochastic deNOising REgularization (SNORE) ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_tag">2</span></a>). We prove that this algorithm converges with the exact MMSE denoiser (Proposition <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#S3.Thmproposition3" title="Proposition 3.3. ‣ 3.2 Unbiased algorithm analysis ‣ 3 Convergence Analysis ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_tag">3.3</span></a>) and we bound the error with an inexact MMSE denoiser (Proposition <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#S3.Thmproposition5" title="Proposition 3.5. ‣ 3.3 Biased algorithm analysis ‣ 3 Convergence Analysis ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_tag">3.5</span></a>). <span class="ltx_text ltx_font_bold" id="S1.p8.1.5">(c)</span> With a critical point analysis (Proposition <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#S3.Thmproposition2" title="Proposition 3.2. ‣ 3.1 Asymptotics of critical points when 𝜎→0 ‣ 3 Convergence Analysis ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_tag">3.2</span></a>), we motivate the practical use of an annealed algorithm (Algorithm <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#alg3" title="Algorithm 3 ‣ Optimization algorithms ‣ 2.2 SNORE regularization ‣ 2 Stochastic deNOising REgularization (SNORE) ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_tag">3</span></a>). Finally, we demonstrate the efficiency of SNORE to solve inverse problems.</p> </div> </section> <section class="ltx_section" id="S2"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">2 </span>Stochastic deNOising REgularization (SNORE)</h2> <div class="ltx_para" id="S2.p1"> <p class="ltx_p" id="S2.p1.1">In this section, we propose a new stochastic regularization, SNORE (Equation <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#S2.E5" title="5 ‣ 2.2 SNORE regularization ‣ 2 Stochastic deNOising REgularization (SNORE) ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_tag">5</span></a>), that can be used for PnP restoration in such a way that the denoising step now applies to a noisy version of the current image.</p> </div> <div class="ltx_para" id="S2.p2"> <p class="ltx_p" id="S2.p2.1">We first recall the RED regularization (Section <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#S2.SS1" title="2.1 Background ‣ 2 Stochastic deNOising REgularization (SNORE) ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_tag">2.1</span></a>) and the corresponding PnP algorithm (Algorithm <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#alg1" title="Algorithm 1 ‣ 2.1 Background ‣ 2 Stochastic deNOising REgularization (SNORE) ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_tag">1</span></a>). Then we introduce the SNORE regularization leading to Algorithm <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#alg2" title="Algorithm 2 ‣ Optimization algorithms ‣ 2.2 SNORE regularization ‣ 2 Stochastic deNOising REgularization (SNORE) ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_tag">2</span></a>. We also propose an Annealing SNORE algorithm (Algorithm <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#alg3" title="Algorithm 3 ‣ Optimization algorithms ‣ 2.2 SNORE regularization ‣ 2 Stochastic deNOising REgularization (SNORE) ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_tag">3</span></a>) inspired by annealed importance sampling <cite class="ltx_cite ltx_citemacro_citep">(Neal, <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#bib.bib50" title="">1998</a>)</cite>. Finally, we discuss the positioning of our method in relation to existing related works (Section <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#S2.SS3" title="2.3 Related Works ‣ 2 Stochastic deNOising REgularization (SNORE) ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_tag">2.3</span></a>).</p> </div> <section class="ltx_subsection" id="S2.SS1"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">2.1 </span>Background</h3> <div class="ltx_para" id="S2.SS1.p1"> <p class="ltx_p" id="S2.SS1.p1.3"><span class="ltx_text ltx_font_bold" id="S2.SS1.p1.3.1">Bayesian inverse problem</span> An inverse problem formulated as in Problem (<a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#S1.E1" title="1 ‣ 1 Introduction ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_tag">1</span></a>) has a general Bayesian interpretation. From the observation <math alttext="{\mathbf{y}}\in{\mathbb{R}}^{q}" class="ltx_Math" display="inline" id="S2.SS1.p1.1.m1.1"><semantics id="S2.SS1.p1.1.m1.1a"><mrow id="S2.SS1.p1.1.m1.1.1" xref="S2.SS1.p1.1.m1.1.1.cmml"><mi id="S2.SS1.p1.1.m1.1.1.2" xref="S2.SS1.p1.1.m1.1.1.2.cmml">𝐲</mi><mo id="S2.SS1.p1.1.m1.1.1.1" xref="S2.SS1.p1.1.m1.1.1.1.cmml">∈</mo><msup id="S2.SS1.p1.1.m1.1.1.3" xref="S2.SS1.p1.1.m1.1.1.3.cmml"><mi id="S2.SS1.p1.1.m1.1.1.3.2" xref="S2.SS1.p1.1.m1.1.1.3.2.cmml">ℝ</mi><mi id="S2.SS1.p1.1.m1.1.1.3.3" xref="S2.SS1.p1.1.m1.1.1.3.3.cmml">q</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p1.1.m1.1b"><apply id="S2.SS1.p1.1.m1.1.1.cmml" xref="S2.SS1.p1.1.m1.1.1"><in id="S2.SS1.p1.1.m1.1.1.1.cmml" xref="S2.SS1.p1.1.m1.1.1.1"></in><ci id="S2.SS1.p1.1.m1.1.1.2.cmml" xref="S2.SS1.p1.1.m1.1.1.2">𝐲</ci><apply id="S2.SS1.p1.1.m1.1.1.3.cmml" xref="S2.SS1.p1.1.m1.1.1.3"><csymbol cd="ambiguous" id="S2.SS1.p1.1.m1.1.1.3.1.cmml" xref="S2.SS1.p1.1.m1.1.1.3">superscript</csymbol><ci id="S2.SS1.p1.1.m1.1.1.3.2.cmml" xref="S2.SS1.p1.1.m1.1.1.3.2">ℝ</ci><ci id="S2.SS1.p1.1.m1.1.1.3.3.cmml" xref="S2.SS1.p1.1.m1.1.1.3.3">𝑞</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p1.1.m1.1c">{\mathbf{y}}\in{\mathbb{R}}^{q}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p1.1.m1.1d">bold_y ∈ blackboard_R start_POSTSUPERSCRIPT italic_q end_POSTSUPERSCRIPT</annotation></semantics></math> (typically <math alttext="q<d" class="ltx_Math" display="inline" id="S2.SS1.p1.2.m2.1"><semantics id="S2.SS1.p1.2.m2.1a"><mrow id="S2.SS1.p1.2.m2.1.1" xref="S2.SS1.p1.2.m2.1.1.cmml"><mi id="S2.SS1.p1.2.m2.1.1.2" xref="S2.SS1.p1.2.m2.1.1.2.cmml">q</mi><mo id="S2.SS1.p1.2.m2.1.1.1" xref="S2.SS1.p1.2.m2.1.1.1.cmml"><</mo><mi id="S2.SS1.p1.2.m2.1.1.3" xref="S2.SS1.p1.2.m2.1.1.3.cmml">d</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p1.2.m2.1b"><apply 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id="S2.SS1.p1.3.m3.1.1.cmml" xref="S2.SS1.p1.3.m3.1.1"><ci id="S2.SS1.p1.3.m3.1.1.1.cmml" xref="S2.SS1.p1.3.m3.1.1.1">^</ci><ci id="S2.SS1.p1.3.m3.1.1.2.cmml" xref="S2.SS1.p1.3.m3.1.1.2">𝐱</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p1.3.m3.1c">\hat{{\mathbf{x}}}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p1.3.m3.1d">over^ start_ARG bold_x end_ARG</annotation></semantics></math> defined by</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A8.EGx2"> <tbody id="S2.Ex1"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\hat{{\mathbf{x}}}" class="ltx_Math" display="inline" id="S2.Ex1.m1.1"><semantics id="S2.Ex1.m1.1a"><mover accent="true" id="S2.Ex1.m1.1.1" xref="S2.Ex1.m1.1.1.cmml"><mi id="S2.Ex1.m1.1.1.2" xref="S2.Ex1.m1.1.1.2.cmml">𝐱</mi><mo id="S2.Ex1.m1.1.1.1" 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xref="S2.Ex1.m2.2.2.2.1.1.1.1.3">𝐲</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Ex1.m2.2c">\displaystyle=\operatorname*{arg\,max}_{{\mathbf{x}}\in{\mathbb{R}}^{d}}{p({% \mathbf{x}}|{\mathbf{y}})}=\operatorname*{arg\,min}_{{\mathbf{x}}\in{\mathbb{R% }}^{d}}{-\log p({\mathbf{x}}|{\mathbf{y}})}</annotation><annotation encoding="application/x-llamapun" id="S2.Ex1.m2.2d">= start_OPERATOR roman_arg roman_max end_OPERATOR start_POSTSUBSCRIPT bold_x ∈ blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT end_POSTSUBSCRIPT italic_p ( bold_x | bold_y ) = start_OPERATOR roman_arg roman_min end_OPERATOR start_POSTSUBSCRIPT bold_x ∈ blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT end_POSTSUBSCRIPT - roman_log italic_p ( bold_x | bold_y )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="S2.Ex2"><tr class="ltx_equation ltx_eqn_row 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</table> <p class="ltx_p" id="S2.SS1.p1.8">Thus the data-fidelity <math alttext="\mathcal{F}" class="ltx_Math" display="inline" id="S2.SS1.p1.4.m1.1"><semantics id="S2.SS1.p1.4.m1.1a"><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p1.4.m1.1.1" xref="S2.SS1.p1.4.m1.1.1.cmml">ℱ</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.p1.4.m1.1b"><ci id="S2.SS1.p1.4.m1.1.1.cmml" xref="S2.SS1.p1.4.m1.1.1">ℱ</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p1.4.m1.1c">\mathcal{F}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p1.4.m1.1d">caligraphic_F</annotation></semantics></math> is related to the image forward model, and the regularization <math alttext="\mathcal{R}" class="ltx_Math" display="inline" id="S2.SS1.p1.5.m2.1"><semantics id="S2.SS1.p1.5.m2.1a"><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p1.5.m2.1.1" xref="S2.SS1.p1.5.m2.1.1.cmml">ℛ</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.p1.5.m2.1b"><ci id="S2.SS1.p1.5.m2.1.1.cmml" xref="S2.SS1.p1.5.m2.1.1">ℛ</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p1.5.m2.1c">\mathcal{R}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p1.5.m2.1d">caligraphic_R</annotation></semantics></math> reflects the adopted prior model on images (which can be improper). Adding a weighting parameter <math alttext="\alpha>0" class="ltx_Math" display="inline" id="S2.SS1.p1.6.m3.1"><semantics id="S2.SS1.p1.6.m3.1a"><mrow id="S2.SS1.p1.6.m3.1.1" xref="S2.SS1.p1.6.m3.1.1.cmml"><mi id="S2.SS1.p1.6.m3.1.1.2" xref="S2.SS1.p1.6.m3.1.1.2.cmml">α</mi><mo id="S2.SS1.p1.6.m3.1.1.1" xref="S2.SS1.p1.6.m3.1.1.1.cmml">></mo><mn id="S2.SS1.p1.6.m3.1.1.3" xref="S2.SS1.p1.6.m3.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p1.6.m3.1b"><apply id="S2.SS1.p1.6.m3.1.1.cmml" xref="S2.SS1.p1.6.m3.1.1"><gt id="S2.SS1.p1.6.m3.1.1.1.cmml" xref="S2.SS1.p1.6.m3.1.1.1"></gt><ci id="S2.SS1.p1.6.m3.1.1.2.cmml" xref="S2.SS1.p1.6.m3.1.1.2">𝛼</ci><cn id="S2.SS1.p1.6.m3.1.1.3.cmml" type="integer" xref="S2.SS1.p1.6.m3.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p1.6.m3.1c">\alpha>0</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p1.6.m3.1d">italic_α > 0</annotation></semantics></math> (see Problem <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#S1.E1" title="1 ‣ 1 Introduction ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_tag">1</span></a>) is equivalent to adding a temperature parameter on the prior <math alttext="p" class="ltx_Math" display="inline" id="S2.SS1.p1.7.m4.1"><semantics id="S2.SS1.p1.7.m4.1a"><mi id="S2.SS1.p1.7.m4.1.1" xref="S2.SS1.p1.7.m4.1.1.cmml">p</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.p1.7.m4.1b"><ci id="S2.SS1.p1.7.m4.1.1.cmml" xref="S2.SS1.p1.7.m4.1.1">𝑝</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p1.7.m4.1c">p</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p1.7.m4.1d">italic_p</annotation></semantics></math>, which becomes <math alttext="p^{\alpha}" class="ltx_Math" display="inline" id="S2.SS1.p1.8.m5.1"><semantics id="S2.SS1.p1.8.m5.1a"><msup id="S2.SS1.p1.8.m5.1.1" xref="S2.SS1.p1.8.m5.1.1.cmml"><mi id="S2.SS1.p1.8.m5.1.1.2" xref="S2.SS1.p1.8.m5.1.1.2.cmml">p</mi><mi id="S2.SS1.p1.8.m5.1.1.3" xref="S2.SS1.p1.8.m5.1.1.3.cmml">α</mi></msup><annotation-xml encoding="MathML-Content" id="S2.SS1.p1.8.m5.1b"><apply id="S2.SS1.p1.8.m5.1.1.cmml" xref="S2.SS1.p1.8.m5.1.1"><csymbol cd="ambiguous" id="S2.SS1.p1.8.m5.1.1.1.cmml" xref="S2.SS1.p1.8.m5.1.1">superscript</csymbol><ci id="S2.SS1.p1.8.m5.1.1.2.cmml" xref="S2.SS1.p1.8.m5.1.1.2">𝑝</ci><ci id="S2.SS1.p1.8.m5.1.1.3.cmml" xref="S2.SS1.p1.8.m5.1.1.3">𝛼</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p1.8.m5.1c">p^{\alpha}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p1.8.m5.1d">italic_p start_POSTSUPERSCRIPT italic_α end_POSTSUPERSCRIPT</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S2.SS1.p2"> <p class="ltx_p" id="S2.SS1.p2.5"><span class="ltx_text ltx_font_bold" id="S2.SS1.p2.5.1">Data Fidelity</span> The forward model is supposed to have a known form</p> <table class="ltx_equation ltx_eqn_table" id="S2.Ex3"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="{\mathbf{y}}=\mathcal{A}({\mathbf{x}})+{\mathbf{n}}," class="ltx_Math" display="block" id="S2.Ex3.m1.2"><semantics id="S2.Ex3.m1.2a"><mrow id="S2.Ex3.m1.2.2.1" xref="S2.Ex3.m1.2.2.1.1.cmml"><mrow id="S2.Ex3.m1.2.2.1.1" xref="S2.Ex3.m1.2.2.1.1.cmml"><mi id="S2.Ex3.m1.2.2.1.1.2" xref="S2.Ex3.m1.2.2.1.1.2.cmml">𝐲</mi><mo id="S2.Ex3.m1.2.2.1.1.1" xref="S2.Ex3.m1.2.2.1.1.1.cmml">=</mo><mrow id="S2.Ex3.m1.2.2.1.1.3" xref="S2.Ex3.m1.2.2.1.1.3.cmml"><mrow id="S2.Ex3.m1.2.2.1.1.3.2" xref="S2.Ex3.m1.2.2.1.1.3.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.Ex3.m1.2.2.1.1.3.2.2" xref="S2.Ex3.m1.2.2.1.1.3.2.2.cmml">𝒜</mi><mo id="S2.Ex3.m1.2.2.1.1.3.2.1" xref="S2.Ex3.m1.2.2.1.1.3.2.1.cmml">⁢</mo><mrow id="S2.Ex3.m1.2.2.1.1.3.2.3.2" xref="S2.Ex3.m1.2.2.1.1.3.2.cmml"><mo id="S2.Ex3.m1.2.2.1.1.3.2.3.2.1" stretchy="false" xref="S2.Ex3.m1.2.2.1.1.3.2.cmml">(</mo><mi id="S2.Ex3.m1.1.1" xref="S2.Ex3.m1.1.1.cmml">𝐱</mi><mo id="S2.Ex3.m1.2.2.1.1.3.2.3.2.2" stretchy="false" xref="S2.Ex3.m1.2.2.1.1.3.2.cmml">)</mo></mrow></mrow><mo id="S2.Ex3.m1.2.2.1.1.3.1" xref="S2.Ex3.m1.2.2.1.1.3.1.cmml">+</mo><mi id="S2.Ex3.m1.2.2.1.1.3.3" xref="S2.Ex3.m1.2.2.1.1.3.3.cmml">𝐧</mi></mrow></mrow><mo id="S2.Ex3.m1.2.2.1.2" xref="S2.Ex3.m1.2.2.1.1.cmml">,</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.Ex3.m1.2b"><apply id="S2.Ex3.m1.2.2.1.1.cmml" xref="S2.Ex3.m1.2.2.1"><eq id="S2.Ex3.m1.2.2.1.1.1.cmml" xref="S2.Ex3.m1.2.2.1.1.1"></eq><ci id="S2.Ex3.m1.2.2.1.1.2.cmml" xref="S2.Ex3.m1.2.2.1.1.2">𝐲</ci><apply id="S2.Ex3.m1.2.2.1.1.3.cmml" xref="S2.Ex3.m1.2.2.1.1.3"><plus id="S2.Ex3.m1.2.2.1.1.3.1.cmml" xref="S2.Ex3.m1.2.2.1.1.3.1"></plus><apply id="S2.Ex3.m1.2.2.1.1.3.2.cmml" xref="S2.Ex3.m1.2.2.1.1.3.2"><times id="S2.Ex3.m1.2.2.1.1.3.2.1.cmml" xref="S2.Ex3.m1.2.2.1.1.3.2.1"></times><ci id="S2.Ex3.m1.2.2.1.1.3.2.2.cmml" xref="S2.Ex3.m1.2.2.1.1.3.2.2">𝒜</ci><ci id="S2.Ex3.m1.1.1.cmml" xref="S2.Ex3.m1.1.1">𝐱</ci></apply><ci id="S2.Ex3.m1.2.2.1.1.3.3.cmml" xref="S2.Ex3.m1.2.2.1.1.3.3">𝐧</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Ex3.m1.2c">{\mathbf{y}}=\mathcal{A}({\mathbf{x}})+{\mathbf{n}},</annotation><annotation encoding="application/x-llamapun" id="S2.Ex3.m1.2d">bold_y = caligraphic_A ( bold_x ) + bold_n ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S2.SS1.p2.4">with the degradation operator <math alttext="\mathcal{A}:{\mathbb{R}}^{d}\mapsto{\mathbb{R}}^{q}" class="ltx_Math" display="inline" id="S2.SS1.p2.1.m1.1"><semantics id="S2.SS1.p2.1.m1.1a"><mrow id="S2.SS1.p2.1.m1.1.1" xref="S2.SS1.p2.1.m1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p2.1.m1.1.1.2" xref="S2.SS1.p2.1.m1.1.1.2.cmml">𝒜</mi><mo id="S2.SS1.p2.1.m1.1.1.1" lspace="0.278em" rspace="0.278em" xref="S2.SS1.p2.1.m1.1.1.1.cmml">:</mo><mrow id="S2.SS1.p2.1.m1.1.1.3" xref="S2.SS1.p2.1.m1.1.1.3.cmml"><msup id="S2.SS1.p2.1.m1.1.1.3.2" xref="S2.SS1.p2.1.m1.1.1.3.2.cmml"><mi id="S2.SS1.p2.1.m1.1.1.3.2.2" xref="S2.SS1.p2.1.m1.1.1.3.2.2.cmml">ℝ</mi><mi id="S2.SS1.p2.1.m1.1.1.3.2.3" xref="S2.SS1.p2.1.m1.1.1.3.2.3.cmml">d</mi></msup><mo id="S2.SS1.p2.1.m1.1.1.3.1" stretchy="false" xref="S2.SS1.p2.1.m1.1.1.3.1.cmml">↦</mo><msup id="S2.SS1.p2.1.m1.1.1.3.3" xref="S2.SS1.p2.1.m1.1.1.3.3.cmml"><mi id="S2.SS1.p2.1.m1.1.1.3.3.2" xref="S2.SS1.p2.1.m1.1.1.3.3.2.cmml">ℝ</mi><mi id="S2.SS1.p2.1.m1.1.1.3.3.3" xref="S2.SS1.p2.1.m1.1.1.3.3.3.cmml">q</mi></msup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p2.1.m1.1b"><apply id="S2.SS1.p2.1.m1.1.1.cmml" xref="S2.SS1.p2.1.m1.1.1"><ci id="S2.SS1.p2.1.m1.1.1.1.cmml" xref="S2.SS1.p2.1.m1.1.1.1">:</ci><ci id="S2.SS1.p2.1.m1.1.1.2.cmml" xref="S2.SS1.p2.1.m1.1.1.2">𝒜</ci><apply id="S2.SS1.p2.1.m1.1.1.3.cmml" xref="S2.SS1.p2.1.m1.1.1.3"><csymbol cd="latexml" id="S2.SS1.p2.1.m1.1.1.3.1.cmml" xref="S2.SS1.p2.1.m1.1.1.3.1">maps-to</csymbol><apply id="S2.SS1.p2.1.m1.1.1.3.2.cmml" xref="S2.SS1.p2.1.m1.1.1.3.2"><csymbol cd="ambiguous" id="S2.SS1.p2.1.m1.1.1.3.2.1.cmml" xref="S2.SS1.p2.1.m1.1.1.3.2">superscript</csymbol><ci id="S2.SS1.p2.1.m1.1.1.3.2.2.cmml" xref="S2.SS1.p2.1.m1.1.1.3.2.2">ℝ</ci><ci id="S2.SS1.p2.1.m1.1.1.3.2.3.cmml" xref="S2.SS1.p2.1.m1.1.1.3.2.3">𝑑</ci></apply><apply id="S2.SS1.p2.1.m1.1.1.3.3.cmml" xref="S2.SS1.p2.1.m1.1.1.3.3"><csymbol cd="ambiguous" id="S2.SS1.p2.1.m1.1.1.3.3.1.cmml" xref="S2.SS1.p2.1.m1.1.1.3.3">superscript</csymbol><ci id="S2.SS1.p2.1.m1.1.1.3.3.2.cmml" xref="S2.SS1.p2.1.m1.1.1.3.3.2">ℝ</ci><ci id="S2.SS1.p2.1.m1.1.1.3.3.3.cmml" xref="S2.SS1.p2.1.m1.1.1.3.3.3">𝑞</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p2.1.m1.1c">\mathcal{A}:{\mathbb{R}}^{d}\mapsto{\mathbb{R}}^{q}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p2.1.m1.1d">caligraphic_A : blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT ↦ blackboard_R start_POSTSUPERSCRIPT italic_q end_POSTSUPERSCRIPT</annotation></semantics></math>, and the noise <math alttext="{\mathbf{n}}\sim\mathcal{N}(0,\sigma_{{\mathbf{y}}}^{2}\mathbf{I}_{q})" class="ltx_Math" display="inline" id="S2.SS1.p2.2.m2.2"><semantics id="S2.SS1.p2.2.m2.2a"><mrow id="S2.SS1.p2.2.m2.2.2" xref="S2.SS1.p2.2.m2.2.2.cmml"><mi id="S2.SS1.p2.2.m2.2.2.3" xref="S2.SS1.p2.2.m2.2.2.3.cmml">𝐧</mi><mo id="S2.SS1.p2.2.m2.2.2.2" xref="S2.SS1.p2.2.m2.2.2.2.cmml">∼</mo><mrow id="S2.SS1.p2.2.m2.2.2.1" xref="S2.SS1.p2.2.m2.2.2.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p2.2.m2.2.2.1.3" xref="S2.SS1.p2.2.m2.2.2.1.3.cmml">𝒩</mi><mo id="S2.SS1.p2.2.m2.2.2.1.2" xref="S2.SS1.p2.2.m2.2.2.1.2.cmml">⁢</mo><mrow id="S2.SS1.p2.2.m2.2.2.1.1.1" xref="S2.SS1.p2.2.m2.2.2.1.1.2.cmml"><mo id="S2.SS1.p2.2.m2.2.2.1.1.1.2" stretchy="false" xref="S2.SS1.p2.2.m2.2.2.1.1.2.cmml">(</mo><mn id="S2.SS1.p2.2.m2.1.1" xref="S2.SS1.p2.2.m2.1.1.cmml">0</mn><mo id="S2.SS1.p2.2.m2.2.2.1.1.1.3" xref="S2.SS1.p2.2.m2.2.2.1.1.2.cmml">,</mo><mrow id="S2.SS1.p2.2.m2.2.2.1.1.1.1" xref="S2.SS1.p2.2.m2.2.2.1.1.1.1.cmml"><msubsup id="S2.SS1.p2.2.m2.2.2.1.1.1.1.2" xref="S2.SS1.p2.2.m2.2.2.1.1.1.1.2.cmml"><mi id="S2.SS1.p2.2.m2.2.2.1.1.1.1.2.2.2" xref="S2.SS1.p2.2.m2.2.2.1.1.1.1.2.2.2.cmml">σ</mi><mi id="S2.SS1.p2.2.m2.2.2.1.1.1.1.2.2.3" xref="S2.SS1.p2.2.m2.2.2.1.1.1.1.2.2.3.cmml">𝐲</mi><mn id="S2.SS1.p2.2.m2.2.2.1.1.1.1.2.3" xref="S2.SS1.p2.2.m2.2.2.1.1.1.1.2.3.cmml">2</mn></msubsup><mo id="S2.SS1.p2.2.m2.2.2.1.1.1.1.1" xref="S2.SS1.p2.2.m2.2.2.1.1.1.1.1.cmml">⁢</mo><msub id="S2.SS1.p2.2.m2.2.2.1.1.1.1.3" xref="S2.SS1.p2.2.m2.2.2.1.1.1.1.3.cmml"><mi id="S2.SS1.p2.2.m2.2.2.1.1.1.1.3.2" xref="S2.SS1.p2.2.m2.2.2.1.1.1.1.3.2.cmml">𝐈</mi><mi id="S2.SS1.p2.2.m2.2.2.1.1.1.1.3.3" xref="S2.SS1.p2.2.m2.2.2.1.1.1.1.3.3.cmml">q</mi></msub></mrow><mo id="S2.SS1.p2.2.m2.2.2.1.1.1.4" stretchy="false" xref="S2.SS1.p2.2.m2.2.2.1.1.2.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p2.2.m2.2b"><apply id="S2.SS1.p2.2.m2.2.2.cmml" xref="S2.SS1.p2.2.m2.2.2"><csymbol cd="latexml" id="S2.SS1.p2.2.m2.2.2.2.cmml" xref="S2.SS1.p2.2.m2.2.2.2">similar-to</csymbol><ci id="S2.SS1.p2.2.m2.2.2.3.cmml" xref="S2.SS1.p2.2.m2.2.2.3">𝐧</ci><apply id="S2.SS1.p2.2.m2.2.2.1.cmml" xref="S2.SS1.p2.2.m2.2.2.1"><times id="S2.SS1.p2.2.m2.2.2.1.2.cmml" xref="S2.SS1.p2.2.m2.2.2.1.2"></times><ci id="S2.SS1.p2.2.m2.2.2.1.3.cmml" xref="S2.SS1.p2.2.m2.2.2.1.3">𝒩</ci><interval closure="open" id="S2.SS1.p2.2.m2.2.2.1.1.2.cmml" xref="S2.SS1.p2.2.m2.2.2.1.1.1"><cn id="S2.SS1.p2.2.m2.1.1.cmml" type="integer" xref="S2.SS1.p2.2.m2.1.1">0</cn><apply id="S2.SS1.p2.2.m2.2.2.1.1.1.1.cmml" xref="S2.SS1.p2.2.m2.2.2.1.1.1.1"><times id="S2.SS1.p2.2.m2.2.2.1.1.1.1.1.cmml" xref="S2.SS1.p2.2.m2.2.2.1.1.1.1.1"></times><apply id="S2.SS1.p2.2.m2.2.2.1.1.1.1.2.cmml" xref="S2.SS1.p2.2.m2.2.2.1.1.1.1.2"><csymbol cd="ambiguous" id="S2.SS1.p2.2.m2.2.2.1.1.1.1.2.1.cmml" xref="S2.SS1.p2.2.m2.2.2.1.1.1.1.2">superscript</csymbol><apply id="S2.SS1.p2.2.m2.2.2.1.1.1.1.2.2.cmml" xref="S2.SS1.p2.2.m2.2.2.1.1.1.1.2"><csymbol cd="ambiguous" id="S2.SS1.p2.2.m2.2.2.1.1.1.1.2.2.1.cmml" xref="S2.SS1.p2.2.m2.2.2.1.1.1.1.2">subscript</csymbol><ci id="S2.SS1.p2.2.m2.2.2.1.1.1.1.2.2.2.cmml" xref="S2.SS1.p2.2.m2.2.2.1.1.1.1.2.2.2">𝜎</ci><ci id="S2.SS1.p2.2.m2.2.2.1.1.1.1.2.2.3.cmml" xref="S2.SS1.p2.2.m2.2.2.1.1.1.1.2.2.3">𝐲</ci></apply><cn id="S2.SS1.p2.2.m2.2.2.1.1.1.1.2.3.cmml" type="integer" xref="S2.SS1.p2.2.m2.2.2.1.1.1.1.2.3">2</cn></apply><apply id="S2.SS1.p2.2.m2.2.2.1.1.1.1.3.cmml" xref="S2.SS1.p2.2.m2.2.2.1.1.1.1.3"><csymbol cd="ambiguous" id="S2.SS1.p2.2.m2.2.2.1.1.1.1.3.1.cmml" xref="S2.SS1.p2.2.m2.2.2.1.1.1.1.3">subscript</csymbol><ci id="S2.SS1.p2.2.m2.2.2.1.1.1.1.3.2.cmml" xref="S2.SS1.p2.2.m2.2.2.1.1.1.1.3.2">𝐈</ci><ci id="S2.SS1.p2.2.m2.2.2.1.1.1.1.3.3.cmml" xref="S2.SS1.p2.2.m2.2.2.1.1.1.1.3.3">𝑞</ci></apply></apply></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p2.2.m2.2c">{\mathbf{n}}\sim\mathcal{N}(0,\sigma_{{\mathbf{y}}}^{2}\mathbf{I}_{q})</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p2.2.m2.2d">bold_n ∼ caligraphic_N ( 0 , italic_σ start_POSTSUBSCRIPT bold_y end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT bold_I start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT )</annotation></semantics></math> where <math alttext="\sigma_{{\mathbf{y}}}>0" class="ltx_Math" display="inline" id="S2.SS1.p2.3.m3.1"><semantics id="S2.SS1.p2.3.m3.1a"><mrow id="S2.SS1.p2.3.m3.1.1" xref="S2.SS1.p2.3.m3.1.1.cmml"><msub id="S2.SS1.p2.3.m3.1.1.2" xref="S2.SS1.p2.3.m3.1.1.2.cmml"><mi id="S2.SS1.p2.3.m3.1.1.2.2" xref="S2.SS1.p2.3.m3.1.1.2.2.cmml">σ</mi><mi id="S2.SS1.p2.3.m3.1.1.2.3" xref="S2.SS1.p2.3.m3.1.1.2.3.cmml">𝐲</mi></msub><mo id="S2.SS1.p2.3.m3.1.1.1" xref="S2.SS1.p2.3.m3.1.1.1.cmml">></mo><mn id="S2.SS1.p2.3.m3.1.1.3" xref="S2.SS1.p2.3.m3.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p2.3.m3.1b"><apply id="S2.SS1.p2.3.m3.1.1.cmml" xref="S2.SS1.p2.3.m3.1.1"><gt id="S2.SS1.p2.3.m3.1.1.1.cmml" xref="S2.SS1.p2.3.m3.1.1.1"></gt><apply id="S2.SS1.p2.3.m3.1.1.2.cmml" xref="S2.SS1.p2.3.m3.1.1.2"><csymbol cd="ambiguous" id="S2.SS1.p2.3.m3.1.1.2.1.cmml" xref="S2.SS1.p2.3.m3.1.1.2">subscript</csymbol><ci id="S2.SS1.p2.3.m3.1.1.2.2.cmml" xref="S2.SS1.p2.3.m3.1.1.2.2">𝜎</ci><ci id="S2.SS1.p2.3.m3.1.1.2.3.cmml" xref="S2.SS1.p2.3.m3.1.1.2.3">𝐲</ci></apply><cn id="S2.SS1.p2.3.m3.1.1.3.cmml" type="integer" xref="S2.SS1.p2.3.m3.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p2.3.m3.1c">\sigma_{{\mathbf{y}}}>0</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p2.3.m3.1d">italic_σ start_POSTSUBSCRIPT bold_y end_POSTSUBSCRIPT > 0</annotation></semantics></math>. Then <math alttext="\mathcal{F}({\mathbf{x}},{\mathbf{y}})=\frac{1}{\sigma_{{\mathbf{y}}}^{2}}\|% \mathcal{A}({\mathbf{x}})-{\mathbf{y}}\|^{2}" class="ltx_Math" display="inline" id="S2.SS1.p2.4.m4.4"><semantics id="S2.SS1.p2.4.m4.4a"><mrow id="S2.SS1.p2.4.m4.4.4" xref="S2.SS1.p2.4.m4.4.4.cmml"><mrow id="S2.SS1.p2.4.m4.4.4.3" xref="S2.SS1.p2.4.m4.4.4.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p2.4.m4.4.4.3.2" xref="S2.SS1.p2.4.m4.4.4.3.2.cmml">ℱ</mi><mo id="S2.SS1.p2.4.m4.4.4.3.1" xref="S2.SS1.p2.4.m4.4.4.3.1.cmml">⁢</mo><mrow id="S2.SS1.p2.4.m4.4.4.3.3.2" xref="S2.SS1.p2.4.m4.4.4.3.3.1.cmml"><mo id="S2.SS1.p2.4.m4.4.4.3.3.2.1" stretchy="false" xref="S2.SS1.p2.4.m4.4.4.3.3.1.cmml">(</mo><mi id="S2.SS1.p2.4.m4.1.1" xref="S2.SS1.p2.4.m4.1.1.cmml">𝐱</mi><mo id="S2.SS1.p2.4.m4.4.4.3.3.2.2" xref="S2.SS1.p2.4.m4.4.4.3.3.1.cmml">,</mo><mi id="S2.SS1.p2.4.m4.2.2" xref="S2.SS1.p2.4.m4.2.2.cmml">𝐲</mi><mo id="S2.SS1.p2.4.m4.4.4.3.3.2.3" stretchy="false" xref="S2.SS1.p2.4.m4.4.4.3.3.1.cmml">)</mo></mrow></mrow><mo id="S2.SS1.p2.4.m4.4.4.2" xref="S2.SS1.p2.4.m4.4.4.2.cmml">=</mo><mrow id="S2.SS1.p2.4.m4.4.4.1" xref="S2.SS1.p2.4.m4.4.4.1.cmml"><mfrac id="S2.SS1.p2.4.m4.4.4.1.3" xref="S2.SS1.p2.4.m4.4.4.1.3.cmml"><mn id="S2.SS1.p2.4.m4.4.4.1.3.2" xref="S2.SS1.p2.4.m4.4.4.1.3.2.cmml">1</mn><msubsup id="S2.SS1.p2.4.m4.4.4.1.3.3" xref="S2.SS1.p2.4.m4.4.4.1.3.3.cmml"><mi id="S2.SS1.p2.4.m4.4.4.1.3.3.2.2" xref="S2.SS1.p2.4.m4.4.4.1.3.3.2.2.cmml">σ</mi><mi id="S2.SS1.p2.4.m4.4.4.1.3.3.2.3" xref="S2.SS1.p2.4.m4.4.4.1.3.3.2.3.cmml">𝐲</mi><mn id="S2.SS1.p2.4.m4.4.4.1.3.3.3" xref="S2.SS1.p2.4.m4.4.4.1.3.3.3.cmml">2</mn></msubsup></mfrac><mo id="S2.SS1.p2.4.m4.4.4.1.2" xref="S2.SS1.p2.4.m4.4.4.1.2.cmml">⁢</mo><msup id="S2.SS1.p2.4.m4.4.4.1.1" xref="S2.SS1.p2.4.m4.4.4.1.1.cmml"><mrow id="S2.SS1.p2.4.m4.4.4.1.1.1.1" xref="S2.SS1.p2.4.m4.4.4.1.1.1.2.cmml"><mo id="S2.SS1.p2.4.m4.4.4.1.1.1.1.2" stretchy="false" xref="S2.SS1.p2.4.m4.4.4.1.1.1.2.1.cmml">‖</mo><mrow id="S2.SS1.p2.4.m4.4.4.1.1.1.1.1" xref="S2.SS1.p2.4.m4.4.4.1.1.1.1.1.cmml"><mrow id="S2.SS1.p2.4.m4.4.4.1.1.1.1.1.2" xref="S2.SS1.p2.4.m4.4.4.1.1.1.1.1.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p2.4.m4.4.4.1.1.1.1.1.2.2" xref="S2.SS1.p2.4.m4.4.4.1.1.1.1.1.2.2.cmml">𝒜</mi><mo id="S2.SS1.p2.4.m4.4.4.1.1.1.1.1.2.1" xref="S2.SS1.p2.4.m4.4.4.1.1.1.1.1.2.1.cmml">⁢</mo><mrow id="S2.SS1.p2.4.m4.4.4.1.1.1.1.1.2.3.2" xref="S2.SS1.p2.4.m4.4.4.1.1.1.1.1.2.cmml"><mo id="S2.SS1.p2.4.m4.4.4.1.1.1.1.1.2.3.2.1" stretchy="false" xref="S2.SS1.p2.4.m4.4.4.1.1.1.1.1.2.cmml">(</mo><mi id="S2.SS1.p2.4.m4.3.3" xref="S2.SS1.p2.4.m4.3.3.cmml">𝐱</mi><mo id="S2.SS1.p2.4.m4.4.4.1.1.1.1.1.2.3.2.2" stretchy="false" xref="S2.SS1.p2.4.m4.4.4.1.1.1.1.1.2.cmml">)</mo></mrow></mrow><mo id="S2.SS1.p2.4.m4.4.4.1.1.1.1.1.1" xref="S2.SS1.p2.4.m4.4.4.1.1.1.1.1.1.cmml">−</mo><mi id="S2.SS1.p2.4.m4.4.4.1.1.1.1.1.3" xref="S2.SS1.p2.4.m4.4.4.1.1.1.1.1.3.cmml">𝐲</mi></mrow><mo id="S2.SS1.p2.4.m4.4.4.1.1.1.1.3" stretchy="false" xref="S2.SS1.p2.4.m4.4.4.1.1.1.2.1.cmml">‖</mo></mrow><mn id="S2.SS1.p2.4.m4.4.4.1.1.3" xref="S2.SS1.p2.4.m4.4.4.1.1.3.cmml">2</mn></msup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p2.4.m4.4b"><apply id="S2.SS1.p2.4.m4.4.4.cmml" xref="S2.SS1.p2.4.m4.4.4"><eq id="S2.SS1.p2.4.m4.4.4.2.cmml" xref="S2.SS1.p2.4.m4.4.4.2"></eq><apply id="S2.SS1.p2.4.m4.4.4.3.cmml" xref="S2.SS1.p2.4.m4.4.4.3"><times id="S2.SS1.p2.4.m4.4.4.3.1.cmml" xref="S2.SS1.p2.4.m4.4.4.3.1"></times><ci id="S2.SS1.p2.4.m4.4.4.3.2.cmml" xref="S2.SS1.p2.4.m4.4.4.3.2">ℱ</ci><interval closure="open" id="S2.SS1.p2.4.m4.4.4.3.3.1.cmml" xref="S2.SS1.p2.4.m4.4.4.3.3.2"><ci id="S2.SS1.p2.4.m4.1.1.cmml" xref="S2.SS1.p2.4.m4.1.1">𝐱</ci><ci id="S2.SS1.p2.4.m4.2.2.cmml" xref="S2.SS1.p2.4.m4.2.2">𝐲</ci></interval></apply><apply id="S2.SS1.p2.4.m4.4.4.1.cmml" xref="S2.SS1.p2.4.m4.4.4.1"><times id="S2.SS1.p2.4.m4.4.4.1.2.cmml" xref="S2.SS1.p2.4.m4.4.4.1.2"></times><apply id="S2.SS1.p2.4.m4.4.4.1.3.cmml" xref="S2.SS1.p2.4.m4.4.4.1.3"><divide id="S2.SS1.p2.4.m4.4.4.1.3.1.cmml" xref="S2.SS1.p2.4.m4.4.4.1.3"></divide><cn id="S2.SS1.p2.4.m4.4.4.1.3.2.cmml" type="integer" xref="S2.SS1.p2.4.m4.4.4.1.3.2">1</cn><apply id="S2.SS1.p2.4.m4.4.4.1.3.3.cmml" xref="S2.SS1.p2.4.m4.4.4.1.3.3"><csymbol cd="ambiguous" id="S2.SS1.p2.4.m4.4.4.1.3.3.1.cmml" xref="S2.SS1.p2.4.m4.4.4.1.3.3">superscript</csymbol><apply id="S2.SS1.p2.4.m4.4.4.1.3.3.2.cmml" xref="S2.SS1.p2.4.m4.4.4.1.3.3"><csymbol cd="ambiguous" id="S2.SS1.p2.4.m4.4.4.1.3.3.2.1.cmml" xref="S2.SS1.p2.4.m4.4.4.1.3.3">subscript</csymbol><ci id="S2.SS1.p2.4.m4.4.4.1.3.3.2.2.cmml" xref="S2.SS1.p2.4.m4.4.4.1.3.3.2.2">𝜎</ci><ci id="S2.SS1.p2.4.m4.4.4.1.3.3.2.3.cmml" xref="S2.SS1.p2.4.m4.4.4.1.3.3.2.3">𝐲</ci></apply><cn id="S2.SS1.p2.4.m4.4.4.1.3.3.3.cmml" type="integer" xref="S2.SS1.p2.4.m4.4.4.1.3.3.3">2</cn></apply></apply><apply id="S2.SS1.p2.4.m4.4.4.1.1.cmml" xref="S2.SS1.p2.4.m4.4.4.1.1"><csymbol cd="ambiguous" id="S2.SS1.p2.4.m4.4.4.1.1.2.cmml" xref="S2.SS1.p2.4.m4.4.4.1.1">superscript</csymbol><apply id="S2.SS1.p2.4.m4.4.4.1.1.1.2.cmml" xref="S2.SS1.p2.4.m4.4.4.1.1.1.1"><csymbol cd="latexml" id="S2.SS1.p2.4.m4.4.4.1.1.1.2.1.cmml" xref="S2.SS1.p2.4.m4.4.4.1.1.1.1.2">norm</csymbol><apply id="S2.SS1.p2.4.m4.4.4.1.1.1.1.1.cmml" xref="S2.SS1.p2.4.m4.4.4.1.1.1.1.1"><minus id="S2.SS1.p2.4.m4.4.4.1.1.1.1.1.1.cmml" xref="S2.SS1.p2.4.m4.4.4.1.1.1.1.1.1"></minus><apply id="S2.SS1.p2.4.m4.4.4.1.1.1.1.1.2.cmml" xref="S2.SS1.p2.4.m4.4.4.1.1.1.1.1.2"><times id="S2.SS1.p2.4.m4.4.4.1.1.1.1.1.2.1.cmml" xref="S2.SS1.p2.4.m4.4.4.1.1.1.1.1.2.1"></times><ci id="S2.SS1.p2.4.m4.4.4.1.1.1.1.1.2.2.cmml" xref="S2.SS1.p2.4.m4.4.4.1.1.1.1.1.2.2">𝒜</ci><ci id="S2.SS1.p2.4.m4.3.3.cmml" xref="S2.SS1.p2.4.m4.3.3">𝐱</ci></apply><ci id="S2.SS1.p2.4.m4.4.4.1.1.1.1.1.3.cmml" xref="S2.SS1.p2.4.m4.4.4.1.1.1.1.1.3">𝐲</ci></apply></apply><cn id="S2.SS1.p2.4.m4.4.4.1.1.3.cmml" type="integer" xref="S2.SS1.p2.4.m4.4.4.1.1.3">2</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p2.4.m4.4c">\mathcal{F}({\mathbf{x}},{\mathbf{y}})=\frac{1}{\sigma_{{\mathbf{y}}}^{2}}\|% \mathcal{A}({\mathbf{x}})-{\mathbf{y}}\|^{2}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p2.4.m4.4d">caligraphic_F ( bold_x , bold_y ) = divide start_ARG 1 end_ARG start_ARG italic_σ start_POSTSUBSCRIPT bold_y end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG ∥ caligraphic_A ( bold_x ) - bold_y ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT</annotation></semantics></math>. </p> </div> <div class="ltx_para" id="S2.SS1.p3"> <p class="ltx_p" id="S2.SS1.p3.3"><span class="ltx_text ltx_font_bold" id="S2.SS1.p3.3.1">Deep Learning regularization</span> With a Bayesian interpretation, the regularization <math alttext="\mathcal{R}=-\log p" class="ltx_Math" display="inline" id="S2.SS1.p3.1.m1.1"><semantics id="S2.SS1.p3.1.m1.1a"><mrow id="S2.SS1.p3.1.m1.1.1" xref="S2.SS1.p3.1.m1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p3.1.m1.1.1.2" xref="S2.SS1.p3.1.m1.1.1.2.cmml">ℛ</mi><mo id="S2.SS1.p3.1.m1.1.1.1" xref="S2.SS1.p3.1.m1.1.1.1.cmml">=</mo><mrow id="S2.SS1.p3.1.m1.1.1.3" xref="S2.SS1.p3.1.m1.1.1.3.cmml"><mo id="S2.SS1.p3.1.m1.1.1.3a" rspace="0.167em" xref="S2.SS1.p3.1.m1.1.1.3.cmml">−</mo><mrow id="S2.SS1.p3.1.m1.1.1.3.2" xref="S2.SS1.p3.1.m1.1.1.3.2.cmml"><mi id="S2.SS1.p3.1.m1.1.1.3.2.1" xref="S2.SS1.p3.1.m1.1.1.3.2.1.cmml">log</mi><mo id="S2.SS1.p3.1.m1.1.1.3.2a" lspace="0.167em" xref="S2.SS1.p3.1.m1.1.1.3.2.cmml">⁡</mo><mi id="S2.SS1.p3.1.m1.1.1.3.2.2" xref="S2.SS1.p3.1.m1.1.1.3.2.2.cmml">p</mi></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p3.1.m1.1b"><apply id="S2.SS1.p3.1.m1.1.1.cmml" xref="S2.SS1.p3.1.m1.1.1"><eq id="S2.SS1.p3.1.m1.1.1.1.cmml" xref="S2.SS1.p3.1.m1.1.1.1"></eq><ci id="S2.SS1.p3.1.m1.1.1.2.cmml" xref="S2.SS1.p3.1.m1.1.1.2">ℛ</ci><apply id="S2.SS1.p3.1.m1.1.1.3.cmml" xref="S2.SS1.p3.1.m1.1.1.3"><minus id="S2.SS1.p3.1.m1.1.1.3.1.cmml" xref="S2.SS1.p3.1.m1.1.1.3"></minus><apply id="S2.SS1.p3.1.m1.1.1.3.2.cmml" xref="S2.SS1.p3.1.m1.1.1.3.2"><log id="S2.SS1.p3.1.m1.1.1.3.2.1.cmml" xref="S2.SS1.p3.1.m1.1.1.3.2.1"></log><ci id="S2.SS1.p3.1.m1.1.1.3.2.2.cmml" xref="S2.SS1.p3.1.m1.1.1.3.2.2">𝑝</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p3.1.m1.1c">\mathcal{R}=-\log p</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p3.1.m1.1d">caligraphic_R = - roman_log italic_p</annotation></semantics></math> defines a model on the data. Recently, Deep Neural Networks (DNN) have proved their effectiveness in learning a realistic model from a database of observations. The RED framework <cite class="ltx_cite ltx_citemacro_citep">(Romano et al., <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#bib.bib58" title="">2017</a>)</cite> uses the performance of DNN for image restoration. It consists in adopting a prior regularization <math alttext="\mathcal{R}" class="ltx_Math" display="inline" id="S2.SS1.p3.2.m2.1"><semantics id="S2.SS1.p3.2.m2.1a"><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p3.2.m2.1.1" xref="S2.SS1.p3.2.m2.1.1.cmml">ℛ</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.p3.2.m2.1b"><ci id="S2.SS1.p3.2.m2.1.1.cmml" xref="S2.SS1.p3.2.m2.1.1">ℛ</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p3.2.m2.1c">\mathcal{R}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p3.2.m2.1d">caligraphic_R</annotation></semantics></math> whose gradient <math alttext="\nabla\mathcal{R}" class="ltx_Math" display="inline" id="S2.SS1.p3.3.m3.1"><semantics id="S2.SS1.p3.3.m3.1a"><mrow id="S2.SS1.p3.3.m3.1.1" xref="S2.SS1.p3.3.m3.1.1.cmml"><mo id="S2.SS1.p3.3.m3.1.1.1" rspace="0.167em" xref="S2.SS1.p3.3.m3.1.1.1.cmml">∇</mo><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p3.3.m3.1.1.2" xref="S2.SS1.p3.3.m3.1.1.2.cmml">ℛ</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p3.3.m3.1b"><apply id="S2.SS1.p3.3.m3.1.1.cmml" xref="S2.SS1.p3.3.m3.1.1"><ci id="S2.SS1.p3.3.m3.1.1.1.cmml" xref="S2.SS1.p3.3.m3.1.1.1">∇</ci><ci id="S2.SS1.p3.3.m3.1.1.2.cmml" xref="S2.SS1.p3.3.m3.1.1.2">ℛ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p3.3.m3.1c">\nabla\mathcal{R}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p3.3.m3.1d">∇ caligraphic_R</annotation></semantics></math> is given by a pre-trained denoiser. This implicit relation relies on the regularization defined by</p> <table class="ltx_equation ltx_eqn_table" id="S2.E2"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\mathcal{R}({\mathbf{x}})\approx\mathcal{P}_{\sigma}({\mathbf{x}}):=-\log p_{% \sigma}({\mathbf{x}})," class="ltx_Math" display="block" id="S2.E2.m1.4"><semantics id="S2.E2.m1.4a"><mrow id="S2.E2.m1.4.4.1" xref="S2.E2.m1.4.4.1.1.cmml"><mrow id="S2.E2.m1.4.4.1.1" xref="S2.E2.m1.4.4.1.1.cmml"><mrow id="S2.E2.m1.4.4.1.1.2" xref="S2.E2.m1.4.4.1.1.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.E2.m1.4.4.1.1.2.2" xref="S2.E2.m1.4.4.1.1.2.2.cmml">ℛ</mi><mo id="S2.E2.m1.4.4.1.1.2.1" xref="S2.E2.m1.4.4.1.1.2.1.cmml">⁢</mo><mrow id="S2.E2.m1.4.4.1.1.2.3.2" xref="S2.E2.m1.4.4.1.1.2.cmml"><mo id="S2.E2.m1.4.4.1.1.2.3.2.1" stretchy="false" xref="S2.E2.m1.4.4.1.1.2.cmml">(</mo><mi id="S2.E2.m1.1.1" xref="S2.E2.m1.1.1.cmml">𝐱</mi><mo id="S2.E2.m1.4.4.1.1.2.3.2.2" stretchy="false" xref="S2.E2.m1.4.4.1.1.2.cmml">)</mo></mrow></mrow><mo id="S2.E2.m1.4.4.1.1.3" xref="S2.E2.m1.4.4.1.1.3.cmml">≈</mo><mrow id="S2.E2.m1.4.4.1.1.4" xref="S2.E2.m1.4.4.1.1.4.cmml"><msub id="S2.E2.m1.4.4.1.1.4.2" xref="S2.E2.m1.4.4.1.1.4.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.E2.m1.4.4.1.1.4.2.2" xref="S2.E2.m1.4.4.1.1.4.2.2.cmml">𝒫</mi><mi id="S2.E2.m1.4.4.1.1.4.2.3" xref="S2.E2.m1.4.4.1.1.4.2.3.cmml">σ</mi></msub><mo id="S2.E2.m1.4.4.1.1.4.1" xref="S2.E2.m1.4.4.1.1.4.1.cmml">⁢</mo><mrow id="S2.E2.m1.4.4.1.1.4.3.2" xref="S2.E2.m1.4.4.1.1.4.cmml"><mo id="S2.E2.m1.4.4.1.1.4.3.2.1" stretchy="false" xref="S2.E2.m1.4.4.1.1.4.cmml">(</mo><mi id="S2.E2.m1.2.2" xref="S2.E2.m1.2.2.cmml">𝐱</mi><mo id="S2.E2.m1.4.4.1.1.4.3.2.2" rspace="0.278em" stretchy="false" xref="S2.E2.m1.4.4.1.1.4.cmml">)</mo></mrow></mrow><mo id="S2.E2.m1.4.4.1.1.5" rspace="0.278em" xref="S2.E2.m1.4.4.1.1.5.cmml">:=</mo><mrow id="S2.E2.m1.4.4.1.1.6" xref="S2.E2.m1.4.4.1.1.6.cmml"><mo id="S2.E2.m1.4.4.1.1.6a" rspace="0.167em" xref="S2.E2.m1.4.4.1.1.6.cmml">−</mo><mrow id="S2.E2.m1.4.4.1.1.6.2" xref="S2.E2.m1.4.4.1.1.6.2.cmml"><mrow id="S2.E2.m1.4.4.1.1.6.2.2" xref="S2.E2.m1.4.4.1.1.6.2.2.cmml"><mi id="S2.E2.m1.4.4.1.1.6.2.2.1" xref="S2.E2.m1.4.4.1.1.6.2.2.1.cmml">log</mi><mo id="S2.E2.m1.4.4.1.1.6.2.2a" lspace="0.167em" xref="S2.E2.m1.4.4.1.1.6.2.2.cmml">⁡</mo><msub id="S2.E2.m1.4.4.1.1.6.2.2.2" xref="S2.E2.m1.4.4.1.1.6.2.2.2.cmml"><mi id="S2.E2.m1.4.4.1.1.6.2.2.2.2" xref="S2.E2.m1.4.4.1.1.6.2.2.2.2.cmml">p</mi><mi id="S2.E2.m1.4.4.1.1.6.2.2.2.3" xref="S2.E2.m1.4.4.1.1.6.2.2.2.3.cmml">σ</mi></msub></mrow><mo id="S2.E2.m1.4.4.1.1.6.2.1" xref="S2.E2.m1.4.4.1.1.6.2.1.cmml">⁢</mo><mrow id="S2.E2.m1.4.4.1.1.6.2.3.2" xref="S2.E2.m1.4.4.1.1.6.2.cmml"><mo id="S2.E2.m1.4.4.1.1.6.2.3.2.1" stretchy="false" xref="S2.E2.m1.4.4.1.1.6.2.cmml">(</mo><mi id="S2.E2.m1.3.3" xref="S2.E2.m1.3.3.cmml">𝐱</mi><mo id="S2.E2.m1.4.4.1.1.6.2.3.2.2" stretchy="false" xref="S2.E2.m1.4.4.1.1.6.2.cmml">)</mo></mrow></mrow></mrow></mrow><mo id="S2.E2.m1.4.4.1.2" xref="S2.E2.m1.4.4.1.1.cmml">,</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.E2.m1.4b"><apply id="S2.E2.m1.4.4.1.1.cmml" xref="S2.E2.m1.4.4.1"><and id="S2.E2.m1.4.4.1.1a.cmml" xref="S2.E2.m1.4.4.1"></and><apply id="S2.E2.m1.4.4.1.1b.cmml" xref="S2.E2.m1.4.4.1"><approx id="S2.E2.m1.4.4.1.1.3.cmml" xref="S2.E2.m1.4.4.1.1.3"></approx><apply id="S2.E2.m1.4.4.1.1.2.cmml" xref="S2.E2.m1.4.4.1.1.2"><times id="S2.E2.m1.4.4.1.1.2.1.cmml" xref="S2.E2.m1.4.4.1.1.2.1"></times><ci id="S2.E2.m1.4.4.1.1.2.2.cmml" xref="S2.E2.m1.4.4.1.1.2.2">ℛ</ci><ci id="S2.E2.m1.1.1.cmml" xref="S2.E2.m1.1.1">𝐱</ci></apply><apply id="S2.E2.m1.4.4.1.1.4.cmml" xref="S2.E2.m1.4.4.1.1.4"><times id="S2.E2.m1.4.4.1.1.4.1.cmml" xref="S2.E2.m1.4.4.1.1.4.1"></times><apply id="S2.E2.m1.4.4.1.1.4.2.cmml" xref="S2.E2.m1.4.4.1.1.4.2"><csymbol cd="ambiguous" id="S2.E2.m1.4.4.1.1.4.2.1.cmml" xref="S2.E2.m1.4.4.1.1.4.2">subscript</csymbol><ci id="S2.E2.m1.4.4.1.1.4.2.2.cmml" xref="S2.E2.m1.4.4.1.1.4.2.2">𝒫</ci><ci id="S2.E2.m1.4.4.1.1.4.2.3.cmml" xref="S2.E2.m1.4.4.1.1.4.2.3">𝜎</ci></apply><ci id="S2.E2.m1.2.2.cmml" xref="S2.E2.m1.2.2">𝐱</ci></apply></apply><apply id="S2.E2.m1.4.4.1.1c.cmml" xref="S2.E2.m1.4.4.1"><csymbol cd="latexml" id="S2.E2.m1.4.4.1.1.5.cmml" xref="S2.E2.m1.4.4.1.1.5">assign</csymbol><share href="#S2.E2.m1.4.4.1.1.4.cmml" id="S2.E2.m1.4.4.1.1d.cmml" xref="S2.E2.m1.4.4.1"></share><apply id="S2.E2.m1.4.4.1.1.6.cmml" xref="S2.E2.m1.4.4.1.1.6"><minus id="S2.E2.m1.4.4.1.1.6.1.cmml" xref="S2.E2.m1.4.4.1.1.6"></minus><apply id="S2.E2.m1.4.4.1.1.6.2.cmml" xref="S2.E2.m1.4.4.1.1.6.2"><times id="S2.E2.m1.4.4.1.1.6.2.1.cmml" xref="S2.E2.m1.4.4.1.1.6.2.1"></times><apply id="S2.E2.m1.4.4.1.1.6.2.2.cmml" xref="S2.E2.m1.4.4.1.1.6.2.2"><log id="S2.E2.m1.4.4.1.1.6.2.2.1.cmml" xref="S2.E2.m1.4.4.1.1.6.2.2.1"></log><apply id="S2.E2.m1.4.4.1.1.6.2.2.2.cmml" xref="S2.E2.m1.4.4.1.1.6.2.2.2"><csymbol cd="ambiguous" id="S2.E2.m1.4.4.1.1.6.2.2.2.1.cmml" xref="S2.E2.m1.4.4.1.1.6.2.2.2">subscript</csymbol><ci id="S2.E2.m1.4.4.1.1.6.2.2.2.2.cmml" xref="S2.E2.m1.4.4.1.1.6.2.2.2.2">𝑝</ci><ci id="S2.E2.m1.4.4.1.1.6.2.2.2.3.cmml" xref="S2.E2.m1.4.4.1.1.6.2.2.2.3">𝜎</ci></apply></apply><ci id="S2.E2.m1.3.3.cmml" xref="S2.E2.m1.3.3">𝐱</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.E2.m1.4c">\mathcal{R}({\mathbf{x}})\approx\mathcal{P}_{\sigma}({\mathbf{x}}):=-\log p_{% \sigma}({\mathbf{x}}),</annotation><annotation encoding="application/x-llamapun" id="S2.E2.m1.4d">caligraphic_R ( bold_x ) ≈ caligraphic_P start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ( bold_x ) := - roman_log italic_p start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ( bold_x ) ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(2)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S2.SS1.p3.7">where <math alttext="p_{\sigma}" class="ltx_Math" display="inline" id="S2.SS1.p3.4.m1.1"><semantics id="S2.SS1.p3.4.m1.1a"><msub id="S2.SS1.p3.4.m1.1.1" xref="S2.SS1.p3.4.m1.1.1.cmml"><mi id="S2.SS1.p3.4.m1.1.1.2" xref="S2.SS1.p3.4.m1.1.1.2.cmml">p</mi><mi id="S2.SS1.p3.4.m1.1.1.3" xref="S2.SS1.p3.4.m1.1.1.3.cmml">σ</mi></msub><annotation-xml encoding="MathML-Content" id="S2.SS1.p3.4.m1.1b"><apply id="S2.SS1.p3.4.m1.1.1.cmml" xref="S2.SS1.p3.4.m1.1.1"><csymbol cd="ambiguous" id="S2.SS1.p3.4.m1.1.1.1.cmml" xref="S2.SS1.p3.4.m1.1.1">subscript</csymbol><ci id="S2.SS1.p3.4.m1.1.1.2.cmml" xref="S2.SS1.p3.4.m1.1.1.2">𝑝</ci><ci id="S2.SS1.p3.4.m1.1.1.3.cmml" xref="S2.SS1.p3.4.m1.1.1.3">𝜎</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p3.4.m1.1c">p_{\sigma}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p3.4.m1.1d">italic_p start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT</annotation></semantics></math> is the convolution <math alttext="p\ast\mathcal{N}_{\sigma}" class="ltx_Math" display="inline" id="S2.SS1.p3.5.m2.1"><semantics id="S2.SS1.p3.5.m2.1a"><mrow id="S2.SS1.p3.5.m2.1.1" xref="S2.SS1.p3.5.m2.1.1.cmml"><mi id="S2.SS1.p3.5.m2.1.1.2" xref="S2.SS1.p3.5.m2.1.1.2.cmml">p</mi><mo id="S2.SS1.p3.5.m2.1.1.1" lspace="0.222em" rspace="0.222em" xref="S2.SS1.p3.5.m2.1.1.1.cmml">∗</mo><msub id="S2.SS1.p3.5.m2.1.1.3" xref="S2.SS1.p3.5.m2.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p3.5.m2.1.1.3.2" xref="S2.SS1.p3.5.m2.1.1.3.2.cmml">𝒩</mi><mi id="S2.SS1.p3.5.m2.1.1.3.3" xref="S2.SS1.p3.5.m2.1.1.3.3.cmml">σ</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p3.5.m2.1b"><apply id="S2.SS1.p3.5.m2.1.1.cmml" xref="S2.SS1.p3.5.m2.1.1"><ci id="S2.SS1.p3.5.m2.1.1.1.cmml" xref="S2.SS1.p3.5.m2.1.1.1">∗</ci><ci id="S2.SS1.p3.5.m2.1.1.2.cmml" xref="S2.SS1.p3.5.m2.1.1.2">𝑝</ci><apply id="S2.SS1.p3.5.m2.1.1.3.cmml" xref="S2.SS1.p3.5.m2.1.1.3"><csymbol cd="ambiguous" id="S2.SS1.p3.5.m2.1.1.3.1.cmml" xref="S2.SS1.p3.5.m2.1.1.3">subscript</csymbol><ci id="S2.SS1.p3.5.m2.1.1.3.2.cmml" xref="S2.SS1.p3.5.m2.1.1.3.2">𝒩</ci><ci id="S2.SS1.p3.5.m2.1.1.3.3.cmml" xref="S2.SS1.p3.5.m2.1.1.3.3">𝜎</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p3.5.m2.1c">p\ast\mathcal{N}_{\sigma}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p3.5.m2.1d">italic_p ∗ caligraphic_N start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT</annotation></semantics></math> between <math alttext="p" class="ltx_Math" display="inline" id="S2.SS1.p3.6.m3.1"><semantics id="S2.SS1.p3.6.m3.1a"><mi id="S2.SS1.p3.6.m3.1.1" xref="S2.SS1.p3.6.m3.1.1.cmml">p</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.p3.6.m3.1b"><ci id="S2.SS1.p3.6.m3.1.1.cmml" xref="S2.SS1.p3.6.m3.1.1">𝑝</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p3.6.m3.1c">p</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p3.6.m3.1d">italic_p</annotation></semantics></math> and <math alttext="\mathcal{N}_{\sigma}=\mathcal{N}(0,\sigma^{2}\mathbf{I}_{d})" class="ltx_Math" display="inline" id="S2.SS1.p3.7.m4.2"><semantics id="S2.SS1.p3.7.m4.2a"><mrow id="S2.SS1.p3.7.m4.2.2" xref="S2.SS1.p3.7.m4.2.2.cmml"><msub id="S2.SS1.p3.7.m4.2.2.3" xref="S2.SS1.p3.7.m4.2.2.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p3.7.m4.2.2.3.2" xref="S2.SS1.p3.7.m4.2.2.3.2.cmml">𝒩</mi><mi id="S2.SS1.p3.7.m4.2.2.3.3" xref="S2.SS1.p3.7.m4.2.2.3.3.cmml">σ</mi></msub><mo id="S2.SS1.p3.7.m4.2.2.2" xref="S2.SS1.p3.7.m4.2.2.2.cmml">=</mo><mrow id="S2.SS1.p3.7.m4.2.2.1" xref="S2.SS1.p3.7.m4.2.2.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p3.7.m4.2.2.1.3" xref="S2.SS1.p3.7.m4.2.2.1.3.cmml">𝒩</mi><mo id="S2.SS1.p3.7.m4.2.2.1.2" xref="S2.SS1.p3.7.m4.2.2.1.2.cmml">⁢</mo><mrow id="S2.SS1.p3.7.m4.2.2.1.1.1" xref="S2.SS1.p3.7.m4.2.2.1.1.2.cmml"><mo id="S2.SS1.p3.7.m4.2.2.1.1.1.2" stretchy="false" xref="S2.SS1.p3.7.m4.2.2.1.1.2.cmml">(</mo><mn id="S2.SS1.p3.7.m4.1.1" xref="S2.SS1.p3.7.m4.1.1.cmml">0</mn><mo id="S2.SS1.p3.7.m4.2.2.1.1.1.3" xref="S2.SS1.p3.7.m4.2.2.1.1.2.cmml">,</mo><mrow id="S2.SS1.p3.7.m4.2.2.1.1.1.1" xref="S2.SS1.p3.7.m4.2.2.1.1.1.1.cmml"><msup id="S2.SS1.p3.7.m4.2.2.1.1.1.1.2" xref="S2.SS1.p3.7.m4.2.2.1.1.1.1.2.cmml"><mi id="S2.SS1.p3.7.m4.2.2.1.1.1.1.2.2" xref="S2.SS1.p3.7.m4.2.2.1.1.1.1.2.2.cmml">σ</mi><mn id="S2.SS1.p3.7.m4.2.2.1.1.1.1.2.3" xref="S2.SS1.p3.7.m4.2.2.1.1.1.1.2.3.cmml">2</mn></msup><mo id="S2.SS1.p3.7.m4.2.2.1.1.1.1.1" xref="S2.SS1.p3.7.m4.2.2.1.1.1.1.1.cmml">⁢</mo><msub id="S2.SS1.p3.7.m4.2.2.1.1.1.1.3" xref="S2.SS1.p3.7.m4.2.2.1.1.1.1.3.cmml"><mi id="S2.SS1.p3.7.m4.2.2.1.1.1.1.3.2" xref="S2.SS1.p3.7.m4.2.2.1.1.1.1.3.2.cmml">𝐈</mi><mi id="S2.SS1.p3.7.m4.2.2.1.1.1.1.3.3" xref="S2.SS1.p3.7.m4.2.2.1.1.1.1.3.3.cmml">d</mi></msub></mrow><mo id="S2.SS1.p3.7.m4.2.2.1.1.1.4" stretchy="false" xref="S2.SS1.p3.7.m4.2.2.1.1.2.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p3.7.m4.2b"><apply id="S2.SS1.p3.7.m4.2.2.cmml" xref="S2.SS1.p3.7.m4.2.2"><eq id="S2.SS1.p3.7.m4.2.2.2.cmml" xref="S2.SS1.p3.7.m4.2.2.2"></eq><apply id="S2.SS1.p3.7.m4.2.2.3.cmml" xref="S2.SS1.p3.7.m4.2.2.3"><csymbol cd="ambiguous" id="S2.SS1.p3.7.m4.2.2.3.1.cmml" xref="S2.SS1.p3.7.m4.2.2.3">subscript</csymbol><ci id="S2.SS1.p3.7.m4.2.2.3.2.cmml" xref="S2.SS1.p3.7.m4.2.2.3.2">𝒩</ci><ci id="S2.SS1.p3.7.m4.2.2.3.3.cmml" xref="S2.SS1.p3.7.m4.2.2.3.3">𝜎</ci></apply><apply id="S2.SS1.p3.7.m4.2.2.1.cmml" xref="S2.SS1.p3.7.m4.2.2.1"><times id="S2.SS1.p3.7.m4.2.2.1.2.cmml" xref="S2.SS1.p3.7.m4.2.2.1.2"></times><ci id="S2.SS1.p3.7.m4.2.2.1.3.cmml" xref="S2.SS1.p3.7.m4.2.2.1.3">𝒩</ci><interval closure="open" id="S2.SS1.p3.7.m4.2.2.1.1.2.cmml" xref="S2.SS1.p3.7.m4.2.2.1.1.1"><cn id="S2.SS1.p3.7.m4.1.1.cmml" type="integer" xref="S2.SS1.p3.7.m4.1.1">0</cn><apply id="S2.SS1.p3.7.m4.2.2.1.1.1.1.cmml" xref="S2.SS1.p3.7.m4.2.2.1.1.1.1"><times id="S2.SS1.p3.7.m4.2.2.1.1.1.1.1.cmml" xref="S2.SS1.p3.7.m4.2.2.1.1.1.1.1"></times><apply id="S2.SS1.p3.7.m4.2.2.1.1.1.1.2.cmml" xref="S2.SS1.p3.7.m4.2.2.1.1.1.1.2"><csymbol cd="ambiguous" id="S2.SS1.p3.7.m4.2.2.1.1.1.1.2.1.cmml" xref="S2.SS1.p3.7.m4.2.2.1.1.1.1.2">superscript</csymbol><ci id="S2.SS1.p3.7.m4.2.2.1.1.1.1.2.2.cmml" xref="S2.SS1.p3.7.m4.2.2.1.1.1.1.2.2">𝜎</ci><cn id="S2.SS1.p3.7.m4.2.2.1.1.1.1.2.3.cmml" type="integer" xref="S2.SS1.p3.7.m4.2.2.1.1.1.1.2.3">2</cn></apply><apply id="S2.SS1.p3.7.m4.2.2.1.1.1.1.3.cmml" xref="S2.SS1.p3.7.m4.2.2.1.1.1.1.3"><csymbol cd="ambiguous" id="S2.SS1.p3.7.m4.2.2.1.1.1.1.3.1.cmml" xref="S2.SS1.p3.7.m4.2.2.1.1.1.1.3">subscript</csymbol><ci id="S2.SS1.p3.7.m4.2.2.1.1.1.1.3.2.cmml" xref="S2.SS1.p3.7.m4.2.2.1.1.1.1.3.2">𝐈</ci><ci id="S2.SS1.p3.7.m4.2.2.1.1.1.1.3.3.cmml" xref="S2.SS1.p3.7.m4.2.2.1.1.1.1.3.3">𝑑</ci></apply></apply></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p3.7.m4.2c">\mathcal{N}_{\sigma}=\mathcal{N}(0,\sigma^{2}\mathbf{I}_{d})</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p3.7.m4.2d">caligraphic_N start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT = caligraphic_N ( 0 , italic_σ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT bold_I start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT )</annotation></semantics></math>. The relation between denoising and regularization is made explicit with Tweedie’s formula <cite class="ltx_cite ltx_citemacro_citep">(Efron, <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#bib.bib21" title="">2011</a>)</cite>:</p> <table class="ltx_equation ltx_eqn_table" id="S2.E3"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\nabla\mathcal{P}_{\sigma}({\mathbf{x}})=-\frac{1}{\sigma^{2}}\left(D_{\sigma}% ^{\star}({\mathbf{x}})-{\mathbf{x}}\right)," class="ltx_Math" display="block" id="S2.E3.m1.3"><semantics id="S2.E3.m1.3a"><mrow id="S2.E3.m1.3.3.1" xref="S2.E3.m1.3.3.1.1.cmml"><mrow id="S2.E3.m1.3.3.1.1" xref="S2.E3.m1.3.3.1.1.cmml"><mrow id="S2.E3.m1.3.3.1.1.3" xref="S2.E3.m1.3.3.1.1.3.cmml"><mrow id="S2.E3.m1.3.3.1.1.3.2" xref="S2.E3.m1.3.3.1.1.3.2.cmml"><mo id="S2.E3.m1.3.3.1.1.3.2.1" rspace="0.167em" xref="S2.E3.m1.3.3.1.1.3.2.1.cmml">∇</mo><msub id="S2.E3.m1.3.3.1.1.3.2.2" xref="S2.E3.m1.3.3.1.1.3.2.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.E3.m1.3.3.1.1.3.2.2.2" xref="S2.E3.m1.3.3.1.1.3.2.2.2.cmml">𝒫</mi><mi id="S2.E3.m1.3.3.1.1.3.2.2.3" xref="S2.E3.m1.3.3.1.1.3.2.2.3.cmml">σ</mi></msub></mrow><mo id="S2.E3.m1.3.3.1.1.3.1" xref="S2.E3.m1.3.3.1.1.3.1.cmml">⁢</mo><mrow id="S2.E3.m1.3.3.1.1.3.3.2" xref="S2.E3.m1.3.3.1.1.3.cmml"><mo id="S2.E3.m1.3.3.1.1.3.3.2.1" stretchy="false" xref="S2.E3.m1.3.3.1.1.3.cmml">(</mo><mi id="S2.E3.m1.1.1" xref="S2.E3.m1.1.1.cmml">𝐱</mi><mo id="S2.E3.m1.3.3.1.1.3.3.2.2" stretchy="false" xref="S2.E3.m1.3.3.1.1.3.cmml">)</mo></mrow></mrow><mo id="S2.E3.m1.3.3.1.1.2" xref="S2.E3.m1.3.3.1.1.2.cmml">=</mo><mrow id="S2.E3.m1.3.3.1.1.1" xref="S2.E3.m1.3.3.1.1.1.cmml"><mo id="S2.E3.m1.3.3.1.1.1a" xref="S2.E3.m1.3.3.1.1.1.cmml">−</mo><mrow id="S2.E3.m1.3.3.1.1.1.1" xref="S2.E3.m1.3.3.1.1.1.1.cmml"><mfrac id="S2.E3.m1.3.3.1.1.1.1.3" xref="S2.E3.m1.3.3.1.1.1.1.3.cmml"><mn id="S2.E3.m1.3.3.1.1.1.1.3.2" xref="S2.E3.m1.3.3.1.1.1.1.3.2.cmml">1</mn><msup id="S2.E3.m1.3.3.1.1.1.1.3.3" xref="S2.E3.m1.3.3.1.1.1.1.3.3.cmml"><mi id="S2.E3.m1.3.3.1.1.1.1.3.3.2" xref="S2.E3.m1.3.3.1.1.1.1.3.3.2.cmml">σ</mi><mn id="S2.E3.m1.3.3.1.1.1.1.3.3.3" xref="S2.E3.m1.3.3.1.1.1.1.3.3.3.cmml">2</mn></msup></mfrac><mo id="S2.E3.m1.3.3.1.1.1.1.2" xref="S2.E3.m1.3.3.1.1.1.1.2.cmml">⁢</mo><mrow id="S2.E3.m1.3.3.1.1.1.1.1.1" xref="S2.E3.m1.3.3.1.1.1.1.1.1.1.cmml"><mo id="S2.E3.m1.3.3.1.1.1.1.1.1.2" xref="S2.E3.m1.3.3.1.1.1.1.1.1.1.cmml">(</mo><mrow id="S2.E3.m1.3.3.1.1.1.1.1.1.1" xref="S2.E3.m1.3.3.1.1.1.1.1.1.1.cmml"><mrow id="S2.E3.m1.3.3.1.1.1.1.1.1.1.2" xref="S2.E3.m1.3.3.1.1.1.1.1.1.1.2.cmml"><msubsup id="S2.E3.m1.3.3.1.1.1.1.1.1.1.2.2" xref="S2.E3.m1.3.3.1.1.1.1.1.1.1.2.2.cmml"><mi id="S2.E3.m1.3.3.1.1.1.1.1.1.1.2.2.2.2" xref="S2.E3.m1.3.3.1.1.1.1.1.1.1.2.2.2.2.cmml">D</mi><mi id="S2.E3.m1.3.3.1.1.1.1.1.1.1.2.2.2.3" xref="S2.E3.m1.3.3.1.1.1.1.1.1.1.2.2.2.3.cmml">σ</mi><mo id="S2.E3.m1.3.3.1.1.1.1.1.1.1.2.2.3" xref="S2.E3.m1.3.3.1.1.1.1.1.1.1.2.2.3.cmml">⋆</mo></msubsup><mo id="S2.E3.m1.3.3.1.1.1.1.1.1.1.2.1" xref="S2.E3.m1.3.3.1.1.1.1.1.1.1.2.1.cmml">⁢</mo><mrow id="S2.E3.m1.3.3.1.1.1.1.1.1.1.2.3.2" xref="S2.E3.m1.3.3.1.1.1.1.1.1.1.2.cmml"><mo id="S2.E3.m1.3.3.1.1.1.1.1.1.1.2.3.2.1" stretchy="false" xref="S2.E3.m1.3.3.1.1.1.1.1.1.1.2.cmml">(</mo><mi id="S2.E3.m1.2.2" xref="S2.E3.m1.2.2.cmml">𝐱</mi><mo id="S2.E3.m1.3.3.1.1.1.1.1.1.1.2.3.2.2" stretchy="false" xref="S2.E3.m1.3.3.1.1.1.1.1.1.1.2.cmml">)</mo></mrow></mrow><mo id="S2.E3.m1.3.3.1.1.1.1.1.1.1.1" xref="S2.E3.m1.3.3.1.1.1.1.1.1.1.1.cmml">−</mo><mi id="S2.E3.m1.3.3.1.1.1.1.1.1.1.3" xref="S2.E3.m1.3.3.1.1.1.1.1.1.1.3.cmml">𝐱</mi></mrow><mo id="S2.E3.m1.3.3.1.1.1.1.1.1.3" xref="S2.E3.m1.3.3.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow></mrow><mo id="S2.E3.m1.3.3.1.2" xref="S2.E3.m1.3.3.1.1.cmml">,</mo></mrow><annotation-xml 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xref="S2.E3.m1.3.3.1.1.1.1.1.1.1.3">𝐱</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.E3.m1.3c">\nabla\mathcal{P}_{\sigma}({\mathbf{x}})=-\frac{1}{\sigma^{2}}\left(D_{\sigma}% ^{\star}({\mathbf{x}})-{\mathbf{x}}\right),</annotation><annotation encoding="application/x-llamapun" id="S2.E3.m1.3d">∇ caligraphic_P start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ( bold_x ) = - divide start_ARG 1 end_ARG start_ARG italic_σ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG ( italic_D start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT ( bold_x ) - bold_x ) ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(3)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S2.SS1.p3.8">where <math alttext="D_{\sigma}^{\star}" class="ltx_Math" display="inline" id="S2.SS1.p3.8.m1.1"><semantics id="S2.SS1.p3.8.m1.1a"><msubsup id="S2.SS1.p3.8.m1.1.1" xref="S2.SS1.p3.8.m1.1.1.cmml"><mi id="S2.SS1.p3.8.m1.1.1.2.2" xref="S2.SS1.p3.8.m1.1.1.2.2.cmml">D</mi><mi id="S2.SS1.p3.8.m1.1.1.2.3" xref="S2.SS1.p3.8.m1.1.1.2.3.cmml">σ</mi><mo id="S2.SS1.p3.8.m1.1.1.3" xref="S2.SS1.p3.8.m1.1.1.3.cmml">⋆</mo></msubsup><annotation-xml encoding="MathML-Content" id="S2.SS1.p3.8.m1.1b"><apply id="S2.SS1.p3.8.m1.1.1.cmml" xref="S2.SS1.p3.8.m1.1.1"><csymbol cd="ambiguous" id="S2.SS1.p3.8.m1.1.1.1.cmml" xref="S2.SS1.p3.8.m1.1.1">superscript</csymbol><apply id="S2.SS1.p3.8.m1.1.1.2.cmml" xref="S2.SS1.p3.8.m1.1.1"><csymbol cd="ambiguous" id="S2.SS1.p3.8.m1.1.1.2.1.cmml" xref="S2.SS1.p3.8.m1.1.1">subscript</csymbol><ci id="S2.SS1.p3.8.m1.1.1.2.2.cmml" xref="S2.SS1.p3.8.m1.1.1.2.2">𝐷</ci><ci id="S2.SS1.p3.8.m1.1.1.2.3.cmml" xref="S2.SS1.p3.8.m1.1.1.2.3">𝜎</ci></apply><ci id="S2.SS1.p3.8.m1.1.1.3.cmml" xref="S2.SS1.p3.8.m1.1.1.3">⋆</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p3.8.m1.1c">D_{\sigma}^{\star}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p3.8.m1.1d">italic_D start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT</annotation></semantics></math> is the Minimum Mean Square Error (MMSE) denoiser is defined by</p> <table class="ltx_equation ltx_eqn_table" id="S2.E4"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="D_{\sigma}^{\star}({\tilde{\mathbf{x}}}):=\mathbb{E}[{\mathbf{x}}|{\tilde{% \mathbf{x}}}]=\int_{{\mathbb{R}}^{d}}{{\mathbf{x}}p_{{\mathbf{x}}|{\tilde{% \mathbf{x}}}}({\mathbf{x}}|{\tilde{\mathbf{x}}})d{\mathbf{x}}}," class="ltx_Math" display="block" id="S2.E4.m1.2"><semantics id="S2.E4.m1.2a"><mrow id="S2.E4.m1.2.2.1" 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\mathbf{x}}}]=\int_{{\mathbb{R}}^{d}}{{\mathbf{x}}p_{{\mathbf{x}}|{\tilde{% \mathbf{x}}}}({\mathbf{x}}|{\tilde{\mathbf{x}}})d{\mathbf{x}}},</annotation><annotation encoding="application/x-llamapun" id="S2.E4.m1.2d">italic_D start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT ( over~ start_ARG bold_x end_ARG ) := blackboard_E [ bold_x | over~ start_ARG bold_x end_ARG ] = ∫ start_POSTSUBSCRIPT blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT end_POSTSUBSCRIPT bold_x italic_p start_POSTSUBSCRIPT bold_x | over~ start_ARG bold_x end_ARG end_POSTSUBSCRIPT ( bold_x | over~ start_ARG bold_x end_ARG ) italic_d bold_x ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(4)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S2.SS1.p3.9">for <math 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xref="S2.SS1.p3.9.m1.4.4.2.2.1.1.1.1.3.2">𝐈</ci><ci id="S2.SS1.p3.9.m1.4.4.2.2.1.1.1.1.3.3.cmml" xref="S2.SS1.p3.9.m1.4.4.2.2.1.1.1.1.3.3">𝑑</ci></apply></apply></interval></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p3.9.m1.4c">{\tilde{\mathbf{x}}}={\mathbf{x}}+{\boldsymbol{\epsilon}}\text{ with }{\mathbf% {x}}\sim p({\mathbf{x}}),{\boldsymbol{\epsilon}}\sim\mathcal{N}(0,\sigma^{2}% \mathbf{I}_{d})</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p3.9.m1.4d">over~ start_ARG bold_x end_ARG = bold_x + bold_italic_ϵ with bold_x ∼ italic_p ( bold_x ) , bold_italic_ϵ ∼ caligraphic_N ( 0 , italic_σ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT bold_I start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT )</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S2.SS1.p4"> <p class="ltx_p" id="S2.SS1.p4.3">In practice, we do not have access to the exact MMSE denoiser <math alttext="D_{\sigma}^{\star}" class="ltx_Math" display="inline" id="S2.SS1.p4.1.m1.1"><semantics id="S2.SS1.p4.1.m1.1a"><msubsup id="S2.SS1.p4.1.m1.1.1" xref="S2.SS1.p4.1.m1.1.1.cmml"><mi id="S2.SS1.p4.1.m1.1.1.2.2" xref="S2.SS1.p4.1.m1.1.1.2.2.cmml">D</mi><mi id="S2.SS1.p4.1.m1.1.1.2.3" xref="S2.SS1.p4.1.m1.1.1.2.3.cmml">σ</mi><mo id="S2.SS1.p4.1.m1.1.1.3" xref="S2.SS1.p4.1.m1.1.1.3.cmml">⋆</mo></msubsup><annotation-xml encoding="MathML-Content" id="S2.SS1.p4.1.m1.1b"><apply id="S2.SS1.p4.1.m1.1.1.cmml" xref="S2.SS1.p4.1.m1.1.1"><csymbol cd="ambiguous" id="S2.SS1.p4.1.m1.1.1.1.cmml" xref="S2.SS1.p4.1.m1.1.1">superscript</csymbol><apply id="S2.SS1.p4.1.m1.1.1.2.cmml" xref="S2.SS1.p4.1.m1.1.1"><csymbol cd="ambiguous" id="S2.SS1.p4.1.m1.1.1.2.1.cmml" xref="S2.SS1.p4.1.m1.1.1">subscript</csymbol><ci id="S2.SS1.p4.1.m1.1.1.2.2.cmml" xref="S2.SS1.p4.1.m1.1.1.2.2">𝐷</ci><ci id="S2.SS1.p4.1.m1.1.1.2.3.cmml" xref="S2.SS1.p4.1.m1.1.1.2.3">𝜎</ci></apply><ci id="S2.SS1.p4.1.m1.1.1.3.cmml" xref="S2.SS1.p4.1.m1.1.1.3">⋆</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p4.1.m1.1c">D_{\sigma}^{\star}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p4.1.m1.1d">italic_D start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT</annotation></semantics></math>, but only to a deep denoiser <math alttext="D_{\sigma}" class="ltx_Math" display="inline" id="S2.SS1.p4.2.m2.1"><semantics id="S2.SS1.p4.2.m2.1a"><msub id="S2.SS1.p4.2.m2.1.1" xref="S2.SS1.p4.2.m2.1.1.cmml"><mi id="S2.SS1.p4.2.m2.1.1.2" xref="S2.SS1.p4.2.m2.1.1.2.cmml">D</mi><mi id="S2.SS1.p4.2.m2.1.1.3" xref="S2.SS1.p4.2.m2.1.1.3.cmml">σ</mi></msub><annotation-xml encoding="MathML-Content" id="S2.SS1.p4.2.m2.1b"><apply id="S2.SS1.p4.2.m2.1.1.cmml" xref="S2.SS1.p4.2.m2.1.1"><csymbol cd="ambiguous" id="S2.SS1.p4.2.m2.1.1.1.cmml" xref="S2.SS1.p4.2.m2.1.1">subscript</csymbol><ci id="S2.SS1.p4.2.m2.1.1.2.cmml" xref="S2.SS1.p4.2.m2.1.1.2">𝐷</ci><ci id="S2.SS1.p4.2.m2.1.1.3.cmml" xref="S2.SS1.p4.2.m2.1.1.3">𝜎</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p4.2.m2.1c">D_{\sigma}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p4.2.m2.1d">italic_D start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT</annotation></semantics></math> that is trained to approximate the MMSE <math alttext="D_{\sigma}^{\star}" class="ltx_Math" display="inline" id="S2.SS1.p4.3.m3.1"><semantics id="S2.SS1.p4.3.m3.1a"><msubsup id="S2.SS1.p4.3.m3.1.1" xref="S2.SS1.p4.3.m3.1.1.cmml"><mi id="S2.SS1.p4.3.m3.1.1.2.2" xref="S2.SS1.p4.3.m3.1.1.2.2.cmml">D</mi><mi id="S2.SS1.p4.3.m3.1.1.2.3" xref="S2.SS1.p4.3.m3.1.1.2.3.cmml">σ</mi><mo id="S2.SS1.p4.3.m3.1.1.3" xref="S2.SS1.p4.3.m3.1.1.3.cmml">⋆</mo></msubsup><annotation-xml encoding="MathML-Content" id="S2.SS1.p4.3.m3.1b"><apply id="S2.SS1.p4.3.m3.1.1.cmml" xref="S2.SS1.p4.3.m3.1.1"><csymbol cd="ambiguous" id="S2.SS1.p4.3.m3.1.1.1.cmml" xref="S2.SS1.p4.3.m3.1.1">superscript</csymbol><apply id="S2.SS1.p4.3.m3.1.1.2.cmml" xref="S2.SS1.p4.3.m3.1.1"><csymbol cd="ambiguous" id="S2.SS1.p4.3.m3.1.1.2.1.cmml" xref="S2.SS1.p4.3.m3.1.1">subscript</csymbol><ci id="S2.SS1.p4.3.m3.1.1.2.2.cmml" xref="S2.SS1.p4.3.m3.1.1.2.2">𝐷</ci><ci id="S2.SS1.p4.3.m3.1.1.2.3.cmml" xref="S2.SS1.p4.3.m3.1.1.2.3">𝜎</ci></apply><ci id="S2.SS1.p4.3.m3.1.1.3.cmml" xref="S2.SS1.p4.3.m3.1.1.3">⋆</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p4.3.m3.1c">D_{\sigma}^{\star}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p4.3.m3.1d">italic_D start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT</annotation></semantics></math>. Then a gradient descent scheme (Algorithm <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#alg1" title="Algorithm 1 ‣ 2.1 Background ‣ 2 Stochastic deNOising REgularization (SNORE) ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_tag">1</span></a>), as described by <cite class="ltx_cite ltx_citemacro_citet">Reehorst & Schniter (<a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#bib.bib55" title="">2019</a>)</cite>, can be run to obtain an approximate solution of Problem (<a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#S1.E1" title="1 ‣ 1 Introduction ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_tag">1</span></a>).</p> </div> <figure class="ltx_float ltx_float_algorithm ltx_framed_top" id="alg1"> <figcaption class="ltx_caption"><span class="ltx_tag ltx_tag_float"><span class="ltx_text ltx_font_bold" id="alg1.2.1.1">Algorithm 1</span> </span> RED <cite class="ltx_cite ltx_citemacro_citep">(Romano et al., <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#bib.bib58" title="">2017</a>)</cite></figcaption> <div class="ltx_listing ltx_listing" id="alg1.3"> <div class="ltx_listingline" id="alg1.l1"> <span class="ltx_tag ltx_tag_listingline"><span class="ltx_text" id="alg1.l1.1.1.1" style="font-size:80%;">1:</span></span> <span class="ltx_text ltx_font_bold" id="alg1.l1.2">Param.:</span> <math alttext="{\mathbf{x}}_{0}\in{\mathbb{R}}^{d}" class="ltx_Math" display="inline" id="alg1.l1.m1.1"><semantics id="alg1.l1.m1.1a"><mrow id="alg1.l1.m1.1.1" xref="alg1.l1.m1.1.1.cmml"><msub id="alg1.l1.m1.1.1.2" xref="alg1.l1.m1.1.1.2.cmml"><mi id="alg1.l1.m1.1.1.2.2" xref="alg1.l1.m1.1.1.2.2.cmml">𝐱</mi><mn id="alg1.l1.m1.1.1.2.3" xref="alg1.l1.m1.1.1.2.3.cmml">0</mn></msub><mo id="alg1.l1.m1.1.1.1" xref="alg1.l1.m1.1.1.1.cmml">∈</mo><msup id="alg1.l1.m1.1.1.3" xref="alg1.l1.m1.1.1.3.cmml"><mi id="alg1.l1.m1.1.1.3.2" xref="alg1.l1.m1.1.1.3.2.cmml">ℝ</mi><mi id="alg1.l1.m1.1.1.3.3" xref="alg1.l1.m1.1.1.3.3.cmml">d</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="alg1.l1.m1.1b"><apply id="alg1.l1.m1.1.1.cmml" xref="alg1.l1.m1.1.1"><in id="alg1.l1.m1.1.1.1.cmml" xref="alg1.l1.m1.1.1.1"></in><apply id="alg1.l1.m1.1.1.2.cmml" xref="alg1.l1.m1.1.1.2"><csymbol cd="ambiguous" id="alg1.l1.m1.1.1.2.1.cmml" xref="alg1.l1.m1.1.1.2">subscript</csymbol><ci id="alg1.l1.m1.1.1.2.2.cmml" xref="alg1.l1.m1.1.1.2.2">𝐱</ci><cn id="alg1.l1.m1.1.1.2.3.cmml" type="integer" xref="alg1.l1.m1.1.1.2.3">0</cn></apply><apply id="alg1.l1.m1.1.1.3.cmml" xref="alg1.l1.m1.1.1.3"><csymbol cd="ambiguous" id="alg1.l1.m1.1.1.3.1.cmml" xref="alg1.l1.m1.1.1.3">superscript</csymbol><ci id="alg1.l1.m1.1.1.3.2.cmml" xref="alg1.l1.m1.1.1.3.2">ℝ</ci><ci id="alg1.l1.m1.1.1.3.3.cmml" xref="alg1.l1.m1.1.1.3.3">𝑑</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="alg1.l1.m1.1c">{\mathbf{x}}_{0}\in{\mathbb{R}}^{d}</annotation><annotation encoding="application/x-llamapun" id="alg1.l1.m1.1d">bold_x start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ∈ blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT</annotation></semantics></math>, <math alttext="\sigma>0" class="ltx_Math" display="inline" id="alg1.l1.m2.1"><semantics id="alg1.l1.m2.1a"><mrow id="alg1.l1.m2.1.1" xref="alg1.l1.m2.1.1.cmml"><mi id="alg1.l1.m2.1.1.2" xref="alg1.l1.m2.1.1.2.cmml">σ</mi><mo id="alg1.l1.m2.1.1.1" xref="alg1.l1.m2.1.1.1.cmml">></mo><mn id="alg1.l1.m2.1.1.3" xref="alg1.l1.m2.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="alg1.l1.m2.1b"><apply id="alg1.l1.m2.1.1.cmml" xref="alg1.l1.m2.1.1"><gt id="alg1.l1.m2.1.1.1.cmml" xref="alg1.l1.m2.1.1.1"></gt><ci id="alg1.l1.m2.1.1.2.cmml" xref="alg1.l1.m2.1.1.2">𝜎</ci><cn id="alg1.l1.m2.1.1.3.cmml" type="integer" xref="alg1.l1.m2.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="alg1.l1.m2.1c">\sigma>0</annotation><annotation encoding="application/x-llamapun" id="alg1.l1.m2.1d">italic_σ > 0</annotation></semantics></math>, <math alttext="\alpha>0" class="ltx_Math" display="inline" id="alg1.l1.m3.1"><semantics id="alg1.l1.m3.1a"><mrow id="alg1.l1.m3.1.1" xref="alg1.l1.m3.1.1.cmml"><mi id="alg1.l1.m3.1.1.2" xref="alg1.l1.m3.1.1.2.cmml">α</mi><mo id="alg1.l1.m3.1.1.1" xref="alg1.l1.m3.1.1.1.cmml">></mo><mn id="alg1.l1.m3.1.1.3" xref="alg1.l1.m3.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="alg1.l1.m3.1b"><apply id="alg1.l1.m3.1.1.cmml" xref="alg1.l1.m3.1.1"><gt id="alg1.l1.m3.1.1.1.cmml" xref="alg1.l1.m3.1.1.1"></gt><ci id="alg1.l1.m3.1.1.2.cmml" xref="alg1.l1.m3.1.1.2">𝛼</ci><cn id="alg1.l1.m3.1.1.3.cmml" type="integer" xref="alg1.l1.m3.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="alg1.l1.m3.1c">\alpha>0</annotation><annotation encoding="application/x-llamapun" id="alg1.l1.m3.1d">italic_α > 0</annotation></semantics></math>, <math alttext="\delta>0" class="ltx_Math" display="inline" id="alg1.l1.m4.1"><semantics id="alg1.l1.m4.1a"><mrow id="alg1.l1.m4.1.1" xref="alg1.l1.m4.1.1.cmml"><mi id="alg1.l1.m4.1.1.2" xref="alg1.l1.m4.1.1.2.cmml">δ</mi><mo id="alg1.l1.m4.1.1.1" xref="alg1.l1.m4.1.1.1.cmml">></mo><mn id="alg1.l1.m4.1.1.3" xref="alg1.l1.m4.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="alg1.l1.m4.1b"><apply id="alg1.l1.m4.1.1.cmml" xref="alg1.l1.m4.1.1"><gt id="alg1.l1.m4.1.1.1.cmml" xref="alg1.l1.m4.1.1.1"></gt><ci id="alg1.l1.m4.1.1.2.cmml" xref="alg1.l1.m4.1.1.2">𝛿</ci><cn id="alg1.l1.m4.1.1.3.cmml" type="integer" xref="alg1.l1.m4.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="alg1.l1.m4.1c">\delta>0</annotation><annotation encoding="application/x-llamapun" id="alg1.l1.m4.1d">italic_δ > 0</annotation></semantics></math>, <math alttext="N\in{\mathbb{N}}" class="ltx_Math" display="inline" id="alg1.l1.m5.1"><semantics id="alg1.l1.m5.1a"><mrow id="alg1.l1.m5.1.1" xref="alg1.l1.m5.1.1.cmml"><mi id="alg1.l1.m5.1.1.2" xref="alg1.l1.m5.1.1.2.cmml">N</mi><mo id="alg1.l1.m5.1.1.1" xref="alg1.l1.m5.1.1.1.cmml">∈</mo><mi id="alg1.l1.m5.1.1.3" xref="alg1.l1.m5.1.1.3.cmml">ℕ</mi></mrow><annotation-xml encoding="MathML-Content" id="alg1.l1.m5.1b"><apply id="alg1.l1.m5.1.1.cmml" xref="alg1.l1.m5.1.1"><in id="alg1.l1.m5.1.1.1.cmml" xref="alg1.l1.m5.1.1.1"></in><ci id="alg1.l1.m5.1.1.2.cmml" xref="alg1.l1.m5.1.1.2">𝑁</ci><ci id="alg1.l1.m5.1.1.3.cmml" xref="alg1.l1.m5.1.1.3">ℕ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="alg1.l1.m5.1c">N\in{\mathbb{N}}</annotation><annotation encoding="application/x-llamapun" id="alg1.l1.m5.1d">italic_N ∈ blackboard_N</annotation></semantics></math> </div> <div class="ltx_listingline" id="alg1.l2"> <span class="ltx_tag ltx_tag_listingline"><span class="ltx_text" id="alg1.l2.1.1.1" style="font-size:80%;">2:</span></span> <span class="ltx_text ltx_font_bold" id="alg1.l2.2">Input:</span> degraded image <math alttext="{\mathbf{y}}" class="ltx_Math" display="inline" id="alg1.l2.m1.1"><semantics id="alg1.l2.m1.1a"><mi id="alg1.l2.m1.1.1" xref="alg1.l2.m1.1.1.cmml">𝐲</mi><annotation-xml encoding="MathML-Content" id="alg1.l2.m1.1b"><ci id="alg1.l2.m1.1.1.cmml" xref="alg1.l2.m1.1.1">𝐲</ci></annotation-xml><annotation encoding="application/x-tex" id="alg1.l2.m1.1c">{\mathbf{y}}</annotation><annotation encoding="application/x-llamapun" id="alg1.l2.m1.1d">bold_y</annotation></semantics></math> </div> <div class="ltx_listingline" id="alg1.l3"> <span class="ltx_tag ltx_tag_listingline"><span class="ltx_text" id="alg1.l3.1.1.1" style="font-size:80%;">3:</span></span> <span class="ltx_text ltx_font_bold" id="alg1.l3.2">Output:</span> restored image <math alttext="{\mathbf{x}}_{N}" class="ltx_Math" display="inline" id="alg1.l3.m1.1"><semantics id="alg1.l3.m1.1a"><msub id="alg1.l3.m1.1.1" xref="alg1.l3.m1.1.1.cmml"><mi id="alg1.l3.m1.1.1.2" xref="alg1.l3.m1.1.1.2.cmml">𝐱</mi><mi id="alg1.l3.m1.1.1.3" xref="alg1.l3.m1.1.1.3.cmml">N</mi></msub><annotation-xml encoding="MathML-Content" id="alg1.l3.m1.1b"><apply id="alg1.l3.m1.1.1.cmml" xref="alg1.l3.m1.1.1"><csymbol cd="ambiguous" id="alg1.l3.m1.1.1.1.cmml" xref="alg1.l3.m1.1.1">subscript</csymbol><ci id="alg1.l3.m1.1.1.2.cmml" xref="alg1.l3.m1.1.1.2">𝐱</ci><ci id="alg1.l3.m1.1.1.3.cmml" xref="alg1.l3.m1.1.1.3">𝑁</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="alg1.l3.m1.1c">{\mathbf{x}}_{N}</annotation><annotation encoding="application/x-llamapun" id="alg1.l3.m1.1d">bold_x start_POSTSUBSCRIPT italic_N end_POSTSUBSCRIPT</annotation></semantics></math> </div> <div class="ltx_listingline" id="alg1.l4"> <span class="ltx_tag ltx_tag_listingline"><span class="ltx_text" id="alg1.l4.1.1.1" style="font-size:80%;">4:</span></span> <span class="ltx_text ltx_font_bold" id="alg1.l4.2">for</span> <math alttext="k=0,1,\dots,N-1" class="ltx_Math" display="inline" id="alg1.l4.m1.4"><semantics id="alg1.l4.m1.4a"><mrow id="alg1.l4.m1.4.4" xref="alg1.l4.m1.4.4.cmml"><mi id="alg1.l4.m1.4.4.3" xref="alg1.l4.m1.4.4.3.cmml">k</mi><mo id="alg1.l4.m1.4.4.2" xref="alg1.l4.m1.4.4.2.cmml">=</mo><mrow id="alg1.l4.m1.4.4.1.1" xref="alg1.l4.m1.4.4.1.2.cmml"><mn id="alg1.l4.m1.1.1" xref="alg1.l4.m1.1.1.cmml">0</mn><mo id="alg1.l4.m1.4.4.1.1.2" xref="alg1.l4.m1.4.4.1.2.cmml">,</mo><mn id="alg1.l4.m1.2.2" xref="alg1.l4.m1.2.2.cmml">1</mn><mo id="alg1.l4.m1.4.4.1.1.3" xref="alg1.l4.m1.4.4.1.2.cmml">,</mo><mi id="alg1.l4.m1.3.3" mathvariant="normal" xref="alg1.l4.m1.3.3.cmml">…</mi><mo id="alg1.l4.m1.4.4.1.1.4" xref="alg1.l4.m1.4.4.1.2.cmml">,</mo><mrow id="alg1.l4.m1.4.4.1.1.1" xref="alg1.l4.m1.4.4.1.1.1.cmml"><mi id="alg1.l4.m1.4.4.1.1.1.2" xref="alg1.l4.m1.4.4.1.1.1.2.cmml">N</mi><mo id="alg1.l4.m1.4.4.1.1.1.1" xref="alg1.l4.m1.4.4.1.1.1.1.cmml">−</mo><mn id="alg1.l4.m1.4.4.1.1.1.3" xref="alg1.l4.m1.4.4.1.1.1.3.cmml">1</mn></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="alg1.l4.m1.4b"><apply id="alg1.l4.m1.4.4.cmml" xref="alg1.l4.m1.4.4"><eq id="alg1.l4.m1.4.4.2.cmml" xref="alg1.l4.m1.4.4.2"></eq><ci id="alg1.l4.m1.4.4.3.cmml" xref="alg1.l4.m1.4.4.3">𝑘</ci><list id="alg1.l4.m1.4.4.1.2.cmml" xref="alg1.l4.m1.4.4.1.1"><cn id="alg1.l4.m1.1.1.cmml" type="integer" xref="alg1.l4.m1.1.1">0</cn><cn id="alg1.l4.m1.2.2.cmml" type="integer" xref="alg1.l4.m1.2.2">1</cn><ci id="alg1.l4.m1.3.3.cmml" xref="alg1.l4.m1.3.3">…</ci><apply id="alg1.l4.m1.4.4.1.1.1.cmml" xref="alg1.l4.m1.4.4.1.1.1"><minus id="alg1.l4.m1.4.4.1.1.1.1.cmml" xref="alg1.l4.m1.4.4.1.1.1.1"></minus><ci id="alg1.l4.m1.4.4.1.1.1.2.cmml" xref="alg1.l4.m1.4.4.1.1.1.2">𝑁</ci><cn id="alg1.l4.m1.4.4.1.1.1.3.cmml" type="integer" xref="alg1.l4.m1.4.4.1.1.1.3">1</cn></apply></list></apply></annotation-xml><annotation encoding="application/x-tex" id="alg1.l4.m1.4c">k=0,1,\dots,N-1</annotation><annotation encoding="application/x-llamapun" id="alg1.l4.m1.4d">italic_k = 0 , 1 , … , italic_N - 1</annotation></semantics></math> <span class="ltx_text ltx_font_bold" id="alg1.l4.3">do</span> </div> <div class="ltx_listingline" id="alg1.l5"> <span class="ltx_tag ltx_tag_listingline"><span class="ltx_text" id="alg1.l5.1.1.1" style="font-size:80%;">5:</span></span> <math alttext="{\mathbf{x}}_{k+1}\leftarrow{\mathbf{x}}_{k}-\delta\nabla\mathcal{F}({\mathbf{% x}}_{k},{\mathbf{y}})-\frac{\alpha\delta}{\sigma^{2}}\left({\mathbf{x}}_{k}-D_% {\sigma}({\mathbf{x}}_{k})\right)" class="ltx_Math" display="inline" id="alg1.l5.m1.3"><semantics id="alg1.l5.m1.3a"><mrow id="alg1.l5.m1.3.3" xref="alg1.l5.m1.3.3.cmml"><msub id="alg1.l5.m1.3.3.4" xref="alg1.l5.m1.3.3.4.cmml"><mi id="alg1.l5.m1.3.3.4.2" xref="alg1.l5.m1.3.3.4.2.cmml">𝐱</mi><mrow id="alg1.l5.m1.3.3.4.3" xref="alg1.l5.m1.3.3.4.3.cmml"><mi id="alg1.l5.m1.3.3.4.3.2" xref="alg1.l5.m1.3.3.4.3.2.cmml">k</mi><mo id="alg1.l5.m1.3.3.4.3.1" xref="alg1.l5.m1.3.3.4.3.1.cmml">+</mo><mn id="alg1.l5.m1.3.3.4.3.3" xref="alg1.l5.m1.3.3.4.3.3.cmml">1</mn></mrow></msub><mo id="alg1.l5.m1.3.3.3" stretchy="false" xref="alg1.l5.m1.3.3.3.cmml">←</mo><mrow id="alg1.l5.m1.3.3.2" xref="alg1.l5.m1.3.3.2.cmml"><msub id="alg1.l5.m1.3.3.2.4" xref="alg1.l5.m1.3.3.2.4.cmml"><mi id="alg1.l5.m1.3.3.2.4.2" xref="alg1.l5.m1.3.3.2.4.2.cmml">𝐱</mi><mi id="alg1.l5.m1.3.3.2.4.3" xref="alg1.l5.m1.3.3.2.4.3.cmml">k</mi></msub><mo id="alg1.l5.m1.3.3.2.3" xref="alg1.l5.m1.3.3.2.3.cmml">−</mo><mrow id="alg1.l5.m1.2.2.1.1" xref="alg1.l5.m1.2.2.1.1.cmml"><mi id="alg1.l5.m1.2.2.1.1.3" xref="alg1.l5.m1.2.2.1.1.3.cmml">δ</mi><mo id="alg1.l5.m1.2.2.1.1.2" lspace="0.167em" xref="alg1.l5.m1.2.2.1.1.2.cmml">⁢</mo><mrow id="alg1.l5.m1.2.2.1.1.4" xref="alg1.l5.m1.2.2.1.1.4.cmml"><mo id="alg1.l5.m1.2.2.1.1.4.1" rspace="0.167em" xref="alg1.l5.m1.2.2.1.1.4.1.cmml">∇</mo><mi class="ltx_font_mathcaligraphic" id="alg1.l5.m1.2.2.1.1.4.2" xref="alg1.l5.m1.2.2.1.1.4.2.cmml">ℱ</mi></mrow><mo id="alg1.l5.m1.2.2.1.1.2a" xref="alg1.l5.m1.2.2.1.1.2.cmml">⁢</mo><mrow id="alg1.l5.m1.2.2.1.1.1.1" xref="alg1.l5.m1.2.2.1.1.1.2.cmml"><mo id="alg1.l5.m1.2.2.1.1.1.1.2" stretchy="false" xref="alg1.l5.m1.2.2.1.1.1.2.cmml">(</mo><msub id="alg1.l5.m1.2.2.1.1.1.1.1" xref="alg1.l5.m1.2.2.1.1.1.1.1.cmml"><mi id="alg1.l5.m1.2.2.1.1.1.1.1.2" xref="alg1.l5.m1.2.2.1.1.1.1.1.2.cmml">𝐱</mi><mi id="alg1.l5.m1.2.2.1.1.1.1.1.3" xref="alg1.l5.m1.2.2.1.1.1.1.1.3.cmml">k</mi></msub><mo id="alg1.l5.m1.2.2.1.1.1.1.3" xref="alg1.l5.m1.2.2.1.1.1.2.cmml">,</mo><mi id="alg1.l5.m1.1.1" xref="alg1.l5.m1.1.1.cmml">𝐲</mi><mo id="alg1.l5.m1.2.2.1.1.1.1.4" stretchy="false" xref="alg1.l5.m1.2.2.1.1.1.2.cmml">)</mo></mrow></mrow><mo id="alg1.l5.m1.3.3.2.3a" xref="alg1.l5.m1.3.3.2.3.cmml">−</mo><mrow id="alg1.l5.m1.3.3.2.2" xref="alg1.l5.m1.3.3.2.2.cmml"><mfrac id="alg1.l5.m1.3.3.2.2.3" 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xref="alg1.l5.m1.3.3.2.2.1.1.1.1.1.1.1.3">𝑘</ci></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="alg1.l5.m1.3c">{\mathbf{x}}_{k+1}\leftarrow{\mathbf{x}}_{k}-\delta\nabla\mathcal{F}({\mathbf{% x}}_{k},{\mathbf{y}})-\frac{\alpha\delta}{\sigma^{2}}\left({\mathbf{x}}_{k}-D_% {\sigma}({\mathbf{x}}_{k})\right)</annotation><annotation encoding="application/x-llamapun" id="alg1.l5.m1.3d">bold_x start_POSTSUBSCRIPT italic_k + 1 end_POSTSUBSCRIPT ← bold_x start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT - italic_δ ∇ caligraphic_F ( bold_x start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT , bold_y ) - divide start_ARG italic_α italic_δ end_ARG start_ARG italic_σ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG ( bold_x start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT - italic_D start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ( bold_x start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) )</annotation></semantics></math> </div> <div class="ltx_listingline" id="alg1.l6"> <span class="ltx_tag ltx_tag_listingline"><span class="ltx_text" id="alg1.l6.1.1.1" style="font-size:80%;">6:</span></span> <span class="ltx_text ltx_font_bold" id="alg1.l6.2">end</span> <span class="ltx_text ltx_font_bold" id="alg1.l6.3">for</span> </div> </div> </figure> <div class="ltx_para" id="S2.SS1.p5"> <p class="ltx_p" id="S2.SS1.p5.3">Algorithm <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#alg1" title="Algorithm 1 ‣ 2.1 Background ‣ 2 Stochastic deNOising REgularization (SNORE) ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_tag">1</span></a> involves the computation of <math alttext="D_{\sigma}({\mathbf{x}}_{k})" class="ltx_Math" display="inline" id="S2.SS1.p5.1.m1.1"><semantics id="S2.SS1.p5.1.m1.1a"><mrow id="S2.SS1.p5.1.m1.1.1" xref="S2.SS1.p5.1.m1.1.1.cmml"><msub id="S2.SS1.p5.1.m1.1.1.3" xref="S2.SS1.p5.1.m1.1.1.3.cmml"><mi id="S2.SS1.p5.1.m1.1.1.3.2" xref="S2.SS1.p5.1.m1.1.1.3.2.cmml">D</mi><mi id="S2.SS1.p5.1.m1.1.1.3.3" xref="S2.SS1.p5.1.m1.1.1.3.3.cmml">σ</mi></msub><mo id="S2.SS1.p5.1.m1.1.1.2" xref="S2.SS1.p5.1.m1.1.1.2.cmml">⁢</mo><mrow id="S2.SS1.p5.1.m1.1.1.1.1" xref="S2.SS1.p5.1.m1.1.1.1.1.1.cmml"><mo id="S2.SS1.p5.1.m1.1.1.1.1.2" stretchy="false" xref="S2.SS1.p5.1.m1.1.1.1.1.1.cmml">(</mo><msub id="S2.SS1.p5.1.m1.1.1.1.1.1" xref="S2.SS1.p5.1.m1.1.1.1.1.1.cmml"><mi id="S2.SS1.p5.1.m1.1.1.1.1.1.2" xref="S2.SS1.p5.1.m1.1.1.1.1.1.2.cmml">𝐱</mi><mi id="S2.SS1.p5.1.m1.1.1.1.1.1.3" xref="S2.SS1.p5.1.m1.1.1.1.1.1.3.cmml">k</mi></msub><mo id="S2.SS1.p5.1.m1.1.1.1.1.3" stretchy="false" xref="S2.SS1.p5.1.m1.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p5.1.m1.1b"><apply id="S2.SS1.p5.1.m1.1.1.cmml" xref="S2.SS1.p5.1.m1.1.1"><times id="S2.SS1.p5.1.m1.1.1.2.cmml" xref="S2.SS1.p5.1.m1.1.1.2"></times><apply id="S2.SS1.p5.1.m1.1.1.3.cmml" xref="S2.SS1.p5.1.m1.1.1.3"><csymbol cd="ambiguous" 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alttext="{\mathbf{x}}_{k}" class="ltx_Math" display="inline" id="S2.SS1.p5.2.m2.1"><semantics id="S2.SS1.p5.2.m2.1a"><msub id="S2.SS1.p5.2.m2.1.1" xref="S2.SS1.p5.2.m2.1.1.cmml"><mi id="S2.SS1.p5.2.m2.1.1.2" xref="S2.SS1.p5.2.m2.1.1.2.cmml">𝐱</mi><mi id="S2.SS1.p5.2.m2.1.1.3" xref="S2.SS1.p5.2.m2.1.1.3.cmml">k</mi></msub><annotation-xml encoding="MathML-Content" id="S2.SS1.p5.2.m2.1b"><apply id="S2.SS1.p5.2.m2.1.1.cmml" xref="S2.SS1.p5.2.m2.1.1"><csymbol cd="ambiguous" id="S2.SS1.p5.2.m2.1.1.1.cmml" xref="S2.SS1.p5.2.m2.1.1">subscript</csymbol><ci id="S2.SS1.p5.2.m2.1.1.2.cmml" xref="S2.SS1.p5.2.m2.1.1.2">𝐱</ci><ci id="S2.SS1.p5.2.m2.1.1.3.cmml" xref="S2.SS1.p5.2.m2.1.1.3">𝑘</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p5.2.m2.1c">{\mathbf{x}}_{k}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p5.2.m2.1d">bold_x start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT</annotation></semantics></math> that is not necessarily noisy. As a denoiser is trained to denoise images with noise, the application of <math alttext="D_{\sigma}" class="ltx_Math" display="inline" id="S2.SS1.p5.3.m3.1"><semantics id="S2.SS1.p5.3.m3.1a"><msub id="S2.SS1.p5.3.m3.1.1" xref="S2.SS1.p5.3.m3.1.1.cmml"><mi id="S2.SS1.p5.3.m3.1.1.2" xref="S2.SS1.p5.3.m3.1.1.2.cmml">D</mi><mi id="S2.SS1.p5.3.m3.1.1.3" xref="S2.SS1.p5.3.m3.1.1.3.cmml">σ</mi></msub><annotation-xml encoding="MathML-Content" id="S2.SS1.p5.3.m3.1b"><apply id="S2.SS1.p5.3.m3.1.1.cmml" xref="S2.SS1.p5.3.m3.1.1"><csymbol cd="ambiguous" id="S2.SS1.p5.3.m3.1.1.1.cmml" xref="S2.SS1.p5.3.m3.1.1">subscript</csymbol><ci id="S2.SS1.p5.3.m3.1.1.2.cmml" xref="S2.SS1.p5.3.m3.1.1.2">𝐷</ci><ci id="S2.SS1.p5.3.m3.1.1.3.cmml" xref="S2.SS1.p5.3.m3.1.1.3">𝜎</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p5.3.m3.1c">D_{\sigma}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p5.3.m3.1d">italic_D start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT</annotation></semantics></math> to images that are out of the training domain might be irrelevant. To bypass this issue, we propose a new regularization.</p> </div> </section> <section class="ltx_subsection" id="S2.SS2"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">2.2 </span>SNORE regularization</h3> <div class="ltx_para" id="S2.SS2.p1"> <p class="ltx_p" id="S2.SS2.p1.1">We propose the SNORE regularization <math alttext="\mathcal{R}_{\sigma}" class="ltx_Math" display="inline" id="S2.SS2.p1.1.m1.1"><semantics id="S2.SS2.p1.1.m1.1a"><msub id="S2.SS2.p1.1.m1.1.1" xref="S2.SS2.p1.1.m1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS2.p1.1.m1.1.1.2" xref="S2.SS2.p1.1.m1.1.1.2.cmml">ℛ</mi><mi id="S2.SS2.p1.1.m1.1.1.3" xref="S2.SS2.p1.1.m1.1.1.3.cmml">σ</mi></msub><annotation-xml encoding="MathML-Content" id="S2.SS2.p1.1.m1.1b"><apply id="S2.SS2.p1.1.m1.1.1.cmml" xref="S2.SS2.p1.1.m1.1.1"><csymbol cd="ambiguous" id="S2.SS2.p1.1.m1.1.1.1.cmml" xref="S2.SS2.p1.1.m1.1.1">subscript</csymbol><ci id="S2.SS2.p1.1.m1.1.1.2.cmml" xref="S2.SS2.p1.1.m1.1.1.2">ℛ</ci><ci id="S2.SS2.p1.1.m1.1.1.3.cmml" xref="S2.SS2.p1.1.m1.1.1.3">𝜎</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p1.1.m1.1c">\mathcal{R}_{\sigma}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p1.1.m1.1d">caligraphic_R start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT</annotation></semantics></math>, whose gradient applies the MMSE denoiser on noisy images. This new regularization is defined by</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A8.EGx3"> <tbody id="S2.E5"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\mathcal{R}_{\sigma}({\mathbf{x}})" class="ltx_Math" display="inline" id="S2.E5.m1.1"><semantics id="S2.E5.m1.1a"><mrow id="S2.E5.m1.1.2" xref="S2.E5.m1.1.2.cmml"><msub id="S2.E5.m1.1.2.2" xref="S2.E5.m1.1.2.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.E5.m1.1.2.2.2" xref="S2.E5.m1.1.2.2.2.cmml">ℛ</mi><mi id="S2.E5.m1.1.2.2.3" xref="S2.E5.m1.1.2.2.3.cmml">σ</mi></msub><mo id="S2.E5.m1.1.2.1" xref="S2.E5.m1.1.2.1.cmml">⁢</mo><mrow id="S2.E5.m1.1.2.3.2" xref="S2.E5.m1.1.2.cmml"><mo id="S2.E5.m1.1.2.3.2.1" stretchy="false" xref="S2.E5.m1.1.2.cmml">(</mo><mi id="S2.E5.m1.1.1" xref="S2.E5.m1.1.1.cmml">𝐱</mi><mo id="S2.E5.m1.1.2.3.2.2" stretchy="false" xref="S2.E5.m1.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.E5.m1.1b"><apply id="S2.E5.m1.1.2.cmml" xref="S2.E5.m1.1.2"><times id="S2.E5.m1.1.2.1.cmml" xref="S2.E5.m1.1.2.1"></times><apply id="S2.E5.m1.1.2.2.cmml" xref="S2.E5.m1.1.2.2"><csymbol cd="ambiguous" id="S2.E5.m1.1.2.2.1.cmml" xref="S2.E5.m1.1.2.2">subscript</csymbol><ci id="S2.E5.m1.1.2.2.2.cmml" xref="S2.E5.m1.1.2.2.2">ℛ</ci><ci id="S2.E5.m1.1.2.2.3.cmml" xref="S2.E5.m1.1.2.2.3">𝜎</ci></apply><ci id="S2.E5.m1.1.1.cmml" xref="S2.E5.m1.1.1">𝐱</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.E5.m1.1c">\displaystyle\mathcal{R}_{\sigma}({\mathbf{x}})</annotation><annotation encoding="application/x-llamapun" id="S2.E5.m1.1d">caligraphic_R start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ( bold_x )</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle=-\mathbb{E}_{{\tilde{\mathbf{x}}}\sim 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id="S2.E5.m2.3.3.1.1.1.1.1.2.2.1.cmml" xref="S2.E5.m2.3.3.1.1.1.1.1.2.2">subscript</csymbol><ci id="S2.E5.m2.3.3.1.1.1.1.1.2.2.2.cmml" xref="S2.E5.m2.3.3.1.1.1.1.1.2.2.2">𝑝</ci><ci id="S2.E5.m2.3.3.1.1.1.1.1.2.2.3.cmml" xref="S2.E5.m2.3.3.1.1.1.1.1.2.2.3">𝜎</ci></apply></apply><apply id="S2.E5.m2.2.2.cmml" xref="S2.E5.m2.3.3.1.1.1.1.1.3.2"><ci id="S2.E5.m2.2.2.1.cmml" xref="S2.E5.m2.2.2.1">~</ci><ci id="S2.E5.m2.2.2.2.cmml" xref="S2.E5.m2.2.2.2">𝐱</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.E5.m2.3c">\displaystyle=-\mathbb{E}_{{\tilde{\mathbf{x}}}\sim p_{\sigma}({\tilde{\mathbf% {x}}}|{\mathbf{x}})}\left(\log p_{\sigma}({\tilde{\mathbf{x}}})\right)</annotation><annotation encoding="application/x-llamapun" id="S2.E5.m2.3d">= - blackboard_E start_POSTSUBSCRIPT over~ start_ARG bold_x end_ARG ∼ italic_p start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ( over~ start_ARG bold_x end_ARG | bold_x ) end_POSTSUBSCRIPT ( roman_log italic_p start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ( over~ start_ARG bold_x end_ARG ) )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(5)</span></td> </tr></tbody> <tbody id="S2.E6"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\nabla_{{\mathbf{x}}}\mathcal{R}_{\sigma}({\mathbf{x}})" class="ltx_Math" display="inline" id="S2.E6.m1.1"><semantics id="S2.E6.m1.1a"><mrow id="S2.E6.m1.1.2" xref="S2.E6.m1.1.2.cmml"><mrow id="S2.E6.m1.1.2.2" xref="S2.E6.m1.1.2.2.cmml"><msub id="S2.E6.m1.1.2.2.1" xref="S2.E6.m1.1.2.2.1.cmml"><mo id="S2.E6.m1.1.2.2.1.2" xref="S2.E6.m1.1.2.2.1.2.cmml">∇</mo><mi id="S2.E6.m1.1.2.2.1.3" xref="S2.E6.m1.1.2.2.1.3.cmml">𝐱</mi></msub><msub 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xref="S2.E6.m1.1.2.2.1.2">∇</ci><ci id="S2.E6.m1.1.2.2.1.3.cmml" xref="S2.E6.m1.1.2.2.1.3">𝐱</ci></apply><apply id="S2.E6.m1.1.2.2.2.cmml" xref="S2.E6.m1.1.2.2.2"><csymbol cd="ambiguous" id="S2.E6.m1.1.2.2.2.1.cmml" xref="S2.E6.m1.1.2.2.2">subscript</csymbol><ci id="S2.E6.m1.1.2.2.2.2.cmml" xref="S2.E6.m1.1.2.2.2.2">ℛ</ci><ci id="S2.E6.m1.1.2.2.2.3.cmml" xref="S2.E6.m1.1.2.2.2.3">𝜎</ci></apply></apply><ci id="S2.E6.m1.1.1.cmml" xref="S2.E6.m1.1.1">𝐱</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.E6.m1.1c">\displaystyle\nabla_{{\mathbf{x}}}\mathcal{R}_{\sigma}({\mathbf{x}})</annotation><annotation encoding="application/x-llamapun" id="S2.E6.m1.1d">∇ start_POSTSUBSCRIPT bold_x end_POSTSUBSCRIPT caligraphic_R start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ( bold_x )</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle=-\frac{1}{\sigma^{2}}\left(\mathbb{E}_{{\tilde{\mathbf{x}}}\sim p% 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start_ARG italic_σ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG ( blackboard_E start_POSTSUBSCRIPT over~ start_ARG bold_x end_ARG ∼ italic_p start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ( over~ start_ARG bold_x end_ARG | bold_x ) end_POSTSUBSCRIPT ( italic_D start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT ( over~ start_ARG bold_x end_ARG ) ) - bold_x ) .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(6)</span></td> </tr></tbody> </table> </div> <div class="ltx_para" id="S2.SS2.p2"> <p class="ltx_p" id="S2.SS2.p2.4">Minimizing <math alttext="\mathcal{R}_{\sigma}({\mathbf{x}})" class="ltx_Math" display="inline" id="S2.SS2.p2.1.m1.1"><semantics id="S2.SS2.p2.1.m1.1a"><mrow id="S2.SS2.p2.1.m1.1.2" xref="S2.SS2.p2.1.m1.1.2.cmml"><msub id="S2.SS2.p2.1.m1.1.2.2" xref="S2.SS2.p2.1.m1.1.2.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS2.p2.1.m1.1.2.2.2" xref="S2.SS2.p2.1.m1.1.2.2.2.cmml">ℛ</mi><mi id="S2.SS2.p2.1.m1.1.2.2.3" xref="S2.SS2.p2.1.m1.1.2.2.3.cmml">σ</mi></msub><mo id="S2.SS2.p2.1.m1.1.2.1" xref="S2.SS2.p2.1.m1.1.2.1.cmml">⁢</mo><mrow id="S2.SS2.p2.1.m1.1.2.3.2" xref="S2.SS2.p2.1.m1.1.2.cmml"><mo id="S2.SS2.p2.1.m1.1.2.3.2.1" stretchy="false" xref="S2.SS2.p2.1.m1.1.2.cmml">(</mo><mi id="S2.SS2.p2.1.m1.1.1" xref="S2.SS2.p2.1.m1.1.1.cmml">𝐱</mi><mo id="S2.SS2.p2.1.m1.1.2.3.2.2" stretchy="false" xref="S2.SS2.p2.1.m1.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p2.1.m1.1b"><apply id="S2.SS2.p2.1.m1.1.2.cmml" xref="S2.SS2.p2.1.m1.1.2"><times id="S2.SS2.p2.1.m1.1.2.1.cmml" xref="S2.SS2.p2.1.m1.1.2.1"></times><apply id="S2.SS2.p2.1.m1.1.2.2.cmml" xref="S2.SS2.p2.1.m1.1.2.2"><csymbol cd="ambiguous" id="S2.SS2.p2.1.m1.1.2.2.1.cmml" xref="S2.SS2.p2.1.m1.1.2.2">subscript</csymbol><ci id="S2.SS2.p2.1.m1.1.2.2.2.cmml" xref="S2.SS2.p2.1.m1.1.2.2.2">ℛ</ci><ci id="S2.SS2.p2.1.m1.1.2.2.3.cmml" xref="S2.SS2.p2.1.m1.1.2.2.3">𝜎</ci></apply><ci id="S2.SS2.p2.1.m1.1.1.cmml" xref="S2.SS2.p2.1.m1.1.1">𝐱</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p2.1.m1.1c">\mathcal{R}_{\sigma}({\mathbf{x}})</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p2.1.m1.1d">caligraphic_R start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ( bold_x )</annotation></semantics></math> is equivalent to maximizing <math alttext="\mathbb{E}_{{\tilde{\mathbf{x}}}\sim p_{\sigma}({\tilde{\mathbf{x}}}|{\mathbf{% x}})}\left(\log p_{\sigma}({\tilde{\mathbf{x}}})\right)" class="ltx_Math" display="inline" id="S2.SS2.p2.2.m2.3"><semantics id="S2.SS2.p2.2.m2.3a"><mrow id="S2.SS2.p2.2.m2.3.3" xref="S2.SS2.p2.2.m2.3.3.cmml"><msub id="S2.SS2.p2.2.m2.3.3.3" xref="S2.SS2.p2.2.m2.3.3.3.cmml"><mi 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id="S2.SS2.p2.2.m2.2.2.cmml" xref="S2.SS2.p2.2.m2.3.3.1.1.1.3.2"><ci id="S2.SS2.p2.2.m2.2.2.1.cmml" xref="S2.SS2.p2.2.m2.2.2.1">~</ci><ci id="S2.SS2.p2.2.m2.2.2.2.cmml" xref="S2.SS2.p2.2.m2.2.2.2">𝐱</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p2.2.m2.3c">\mathbb{E}_{{\tilde{\mathbf{x}}}\sim p_{\sigma}({\tilde{\mathbf{x}}}|{\mathbf{% x}})}\left(\log p_{\sigma}({\tilde{\mathbf{x}}})\right)</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p2.2.m2.3d">blackboard_E start_POSTSUBSCRIPT over~ start_ARG bold_x end_ARG ∼ italic_p start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ( over~ start_ARG bold_x end_ARG | bold_x ) end_POSTSUBSCRIPT ( roman_log italic_p start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ( over~ start_ARG bold_x end_ARG ) )</annotation></semantics></math>. The last quantity is maximum if noisy versions of <math alttext="{\mathbf{x}}" class="ltx_Math" display="inline" id="S2.SS2.p2.3.m3.1"><semantics id="S2.SS2.p2.3.m3.1a"><mi id="S2.SS2.p2.3.m3.1.1" xref="S2.SS2.p2.3.m3.1.1.cmml">𝐱</mi><annotation-xml encoding="MathML-Content" id="S2.SS2.p2.3.m3.1b"><ci id="S2.SS2.p2.3.m3.1.1.cmml" xref="S2.SS2.p2.3.m3.1.1">𝐱</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p2.3.m3.1c">{\mathbf{x}}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p2.3.m3.1d">bold_x</annotation></semantics></math> are highly probable in the noisy prior distribution, <math alttext="p_{\sigma}({\mathbf{x}})" class="ltx_Math" display="inline" id="S2.SS2.p2.4.m4.1"><semantics id="S2.SS2.p2.4.m4.1a"><mrow id="S2.SS2.p2.4.m4.1.2" xref="S2.SS2.p2.4.m4.1.2.cmml"><msub id="S2.SS2.p2.4.m4.1.2.2" xref="S2.SS2.p2.4.m4.1.2.2.cmml"><mi id="S2.SS2.p2.4.m4.1.2.2.2" xref="S2.SS2.p2.4.m4.1.2.2.2.cmml">p</mi><mi id="S2.SS2.p2.4.m4.1.2.2.3" xref="S2.SS2.p2.4.m4.1.2.2.3.cmml">σ</mi></msub><mo id="S2.SS2.p2.4.m4.1.2.1" xref="S2.SS2.p2.4.m4.1.2.1.cmml">⁢</mo><mrow id="S2.SS2.p2.4.m4.1.2.3.2" xref="S2.SS2.p2.4.m4.1.2.cmml"><mo id="S2.SS2.p2.4.m4.1.2.3.2.1" stretchy="false" xref="S2.SS2.p2.4.m4.1.2.cmml">(</mo><mi id="S2.SS2.p2.4.m4.1.1" xref="S2.SS2.p2.4.m4.1.1.cmml">𝐱</mi><mo id="S2.SS2.p2.4.m4.1.2.3.2.2" stretchy="false" xref="S2.SS2.p2.4.m4.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p2.4.m4.1b"><apply id="S2.SS2.p2.4.m4.1.2.cmml" xref="S2.SS2.p2.4.m4.1.2"><times id="S2.SS2.p2.4.m4.1.2.1.cmml" xref="S2.SS2.p2.4.m4.1.2.1"></times><apply id="S2.SS2.p2.4.m4.1.2.2.cmml" xref="S2.SS2.p2.4.m4.1.2.2"><csymbol cd="ambiguous" id="S2.SS2.p2.4.m4.1.2.2.1.cmml" xref="S2.SS2.p2.4.m4.1.2.2">subscript</csymbol><ci id="S2.SS2.p2.4.m4.1.2.2.2.cmml" xref="S2.SS2.p2.4.m4.1.2.2.2">𝑝</ci><ci id="S2.SS2.p2.4.m4.1.2.2.3.cmml" xref="S2.SS2.p2.4.m4.1.2.2.3">𝜎</ci></apply><ci id="S2.SS2.p2.4.m4.1.1.cmml" xref="S2.SS2.p2.4.m4.1.1">𝐱</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p2.4.m4.1c">p_{\sigma}({\mathbf{x}})</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p2.4.m4.1d">italic_p start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ( bold_x )</annotation></semantics></math>. In other words: <span class="ltx_text ltx_font_italic" id="S2.SS2.p2.4.1">An image looks clean if its noisy versions look as noisy images</span>.</p> </div> <div class="ltx_para" id="S2.SS2.p3"> <p class="ltx_p" id="S2.SS2.p3.1">SNORE regularization can be seen as a relaxation of the classical PnP regularization <math alttext="-\log(p\ast\mathcal{N}_{\sigma})" class="ltx_Math" display="inline" id="S2.SS2.p3.1.m1.2"><semantics id="S2.SS2.p3.1.m1.2a"><mrow id="S2.SS2.p3.1.m1.2.2" xref="S2.SS2.p3.1.m1.2.2.cmml"><mo id="S2.SS2.p3.1.m1.2.2a" rspace="0.167em" xref="S2.SS2.p3.1.m1.2.2.cmml">−</mo><mrow id="S2.SS2.p3.1.m1.2.2.1.1" xref="S2.SS2.p3.1.m1.2.2.1.2.cmml"><mi id="S2.SS2.p3.1.m1.1.1" xref="S2.SS2.p3.1.m1.1.1.cmml">log</mi><mo id="S2.SS2.p3.1.m1.2.2.1.1a" xref="S2.SS2.p3.1.m1.2.2.1.2.cmml">⁡</mo><mrow id="S2.SS2.p3.1.m1.2.2.1.1.1" xref="S2.SS2.p3.1.m1.2.2.1.2.cmml"><mo id="S2.SS2.p3.1.m1.2.2.1.1.1.2" stretchy="false" xref="S2.SS2.p3.1.m1.2.2.1.2.cmml">(</mo><mrow id="S2.SS2.p3.1.m1.2.2.1.1.1.1" xref="S2.SS2.p3.1.m1.2.2.1.1.1.1.cmml"><mi id="S2.SS2.p3.1.m1.2.2.1.1.1.1.2" xref="S2.SS2.p3.1.m1.2.2.1.1.1.1.2.cmml">p</mi><mo id="S2.SS2.p3.1.m1.2.2.1.1.1.1.1" lspace="0.222em" rspace="0.222em" xref="S2.SS2.p3.1.m1.2.2.1.1.1.1.1.cmml">∗</mo><msub id="S2.SS2.p3.1.m1.2.2.1.1.1.1.3" xref="S2.SS2.p3.1.m1.2.2.1.1.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS2.p3.1.m1.2.2.1.1.1.1.3.2" xref="S2.SS2.p3.1.m1.2.2.1.1.1.1.3.2.cmml">𝒩</mi><mi id="S2.SS2.p3.1.m1.2.2.1.1.1.1.3.3" xref="S2.SS2.p3.1.m1.2.2.1.1.1.1.3.3.cmml">σ</mi></msub></mrow><mo id="S2.SS2.p3.1.m1.2.2.1.1.1.3" stretchy="false" xref="S2.SS2.p3.1.m1.2.2.1.2.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p3.1.m1.2b"><apply id="S2.SS2.p3.1.m1.2.2.cmml" xref="S2.SS2.p3.1.m1.2.2"><minus id="S2.SS2.p3.1.m1.2.2.2.cmml" xref="S2.SS2.p3.1.m1.2.2"></minus><apply id="S2.SS2.p3.1.m1.2.2.1.2.cmml" xref="S2.SS2.p3.1.m1.2.2.1.1"><log id="S2.SS2.p3.1.m1.1.1.cmml" xref="S2.SS2.p3.1.m1.1.1"></log><apply id="S2.SS2.p3.1.m1.2.2.1.1.1.1.cmml" xref="S2.SS2.p3.1.m1.2.2.1.1.1.1"><ci id="S2.SS2.p3.1.m1.2.2.1.1.1.1.1.cmml" xref="S2.SS2.p3.1.m1.2.2.1.1.1.1.1">∗</ci><ci id="S2.SS2.p3.1.m1.2.2.1.1.1.1.2.cmml" xref="S2.SS2.p3.1.m1.2.2.1.1.1.1.2">𝑝</ci><apply id="S2.SS2.p3.1.m1.2.2.1.1.1.1.3.cmml" xref="S2.SS2.p3.1.m1.2.2.1.1.1.1.3"><csymbol cd="ambiguous" id="S2.SS2.p3.1.m1.2.2.1.1.1.1.3.1.cmml" xref="S2.SS2.p3.1.m1.2.2.1.1.1.1.3">subscript</csymbol><ci id="S2.SS2.p3.1.m1.2.2.1.1.1.1.3.2.cmml" xref="S2.SS2.p3.1.m1.2.2.1.1.1.1.3.2">𝒩</ci><ci id="S2.SS2.p3.1.m1.2.2.1.1.1.1.3.3.cmml" xref="S2.SS2.p3.1.m1.2.2.1.1.1.1.3.3">𝜎</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p3.1.m1.2c">-\log(p\ast\mathcal{N}_{\sigma})</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p3.1.m1.2d">- roman_log ( italic_p ∗ caligraphic_N start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT )</annotation></semantics></math>, following the idea of <cite class="ltx_cite ltx_citemacro_citet">Scarvelis et al. (<a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#bib.bib61" title="">2023</a>)</cite>.</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A8.EGx4"> <tbody id="S2.E7"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\mathcal{R}_{\sigma}({\mathbf{x}})=-\left(\log(p\ast\mathcal{N}_{% \sigma})\ast\mathcal{N}_{\sigma}\right)({\mathbf{x}})." class="ltx_Math" display="inline" id="S2.E7.m1.4"><semantics id="S2.E7.m1.4a"><mrow id="S2.E7.m1.4.4.1" xref="S2.E7.m1.4.4.1.1.cmml"><mrow id="S2.E7.m1.4.4.1.1" xref="S2.E7.m1.4.4.1.1.cmml"><mrow id="S2.E7.m1.4.4.1.1.3" xref="S2.E7.m1.4.4.1.1.3.cmml"><msub id="S2.E7.m1.4.4.1.1.3.2" xref="S2.E7.m1.4.4.1.1.3.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.E7.m1.4.4.1.1.3.2.2" xref="S2.E7.m1.4.4.1.1.3.2.2.cmml">ℛ</mi><mi id="S2.E7.m1.4.4.1.1.3.2.3" xref="S2.E7.m1.4.4.1.1.3.2.3.cmml">σ</mi></msub><mo id="S2.E7.m1.4.4.1.1.3.1" xref="S2.E7.m1.4.4.1.1.3.1.cmml">⁢</mo><mrow id="S2.E7.m1.4.4.1.1.3.3.2" xref="S2.E7.m1.4.4.1.1.3.cmml"><mo id="S2.E7.m1.4.4.1.1.3.3.2.1" stretchy="false" xref="S2.E7.m1.4.4.1.1.3.cmml">(</mo><mi id="S2.E7.m1.1.1" xref="S2.E7.m1.1.1.cmml">𝐱</mi><mo id="S2.E7.m1.4.4.1.1.3.3.2.2" stretchy="false" xref="S2.E7.m1.4.4.1.1.3.cmml">)</mo></mrow></mrow><mo id="S2.E7.m1.4.4.1.1.2" xref="S2.E7.m1.4.4.1.1.2.cmml">=</mo><mrow id="S2.E7.m1.4.4.1.1.1" xref="S2.E7.m1.4.4.1.1.1.cmml"><mo id="S2.E7.m1.4.4.1.1.1a" xref="S2.E7.m1.4.4.1.1.1.cmml">−</mo><mrow id="S2.E7.m1.4.4.1.1.1.1" xref="S2.E7.m1.4.4.1.1.1.1.cmml"><mrow id="S2.E7.m1.4.4.1.1.1.1.1.1" xref="S2.E7.m1.4.4.1.1.1.1.1.1.1.cmml"><mo id="S2.E7.m1.4.4.1.1.1.1.1.1.2" xref="S2.E7.m1.4.4.1.1.1.1.1.1.1.cmml">(</mo><mrow id="S2.E7.m1.4.4.1.1.1.1.1.1.1" xref="S2.E7.m1.4.4.1.1.1.1.1.1.1.cmml"><mrow id="S2.E7.m1.4.4.1.1.1.1.1.1.1.1.1" 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xref="S2.E7.m1.4.4.1.1.1.1.1.1.1.1.1.1.1.3">subscript</csymbol><ci id="S2.E7.m1.4.4.1.1.1.1.1.1.1.1.1.1.1.3.2.cmml" xref="S2.E7.m1.4.4.1.1.1.1.1.1.1.1.1.1.1.3.2">𝒩</ci><ci id="S2.E7.m1.4.4.1.1.1.1.1.1.1.1.1.1.1.3.3.cmml" xref="S2.E7.m1.4.4.1.1.1.1.1.1.1.1.1.1.1.3.3">𝜎</ci></apply></apply></apply><apply id="S2.E7.m1.4.4.1.1.1.1.1.1.1.3.cmml" xref="S2.E7.m1.4.4.1.1.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S2.E7.m1.4.4.1.1.1.1.1.1.1.3.1.cmml" xref="S2.E7.m1.4.4.1.1.1.1.1.1.1.3">subscript</csymbol><ci id="S2.E7.m1.4.4.1.1.1.1.1.1.1.3.2.cmml" xref="S2.E7.m1.4.4.1.1.1.1.1.1.1.3.2">𝒩</ci><ci id="S2.E7.m1.4.4.1.1.1.1.1.1.1.3.3.cmml" xref="S2.E7.m1.4.4.1.1.1.1.1.1.1.3.3">𝜎</ci></apply></apply><ci id="S2.E7.m1.3.3.cmml" xref="S2.E7.m1.3.3">𝐱</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.E7.m1.4c">\displaystyle\mathcal{R}_{\sigma}({\mathbf{x}})=-\left(\log(p\ast\mathcal{N}_{% \sigma})\ast\mathcal{N}_{\sigma}\right)({\mathbf{x}}).</annotation><annotation encoding="application/x-llamapun" id="S2.E7.m1.4d">caligraphic_R start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ( bold_x ) = - ( roman_log ( italic_p ∗ caligraphic_N start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ) ∗ caligraphic_N start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ) ( bold_x ) .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(7)</span></td> </tr></tbody> </table> </div> <div class="ltx_para" id="S2.SS2.p4"> <p class="ltx_p" id="S2.SS2.p4.8">In Appendix <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#A3" title="Appendix C SNORE in simple cases ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_tag">C</span></a>, we prove that <math alttext="\mathcal{R}_{\sigma}" class="ltx_Math" display="inline" id="S2.SS2.p4.1.m1.1"><semantics id="S2.SS2.p4.1.m1.1a"><msub id="S2.SS2.p4.1.m1.1.1" xref="S2.SS2.p4.1.m1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS2.p4.1.m1.1.1.2" xref="S2.SS2.p4.1.m1.1.1.2.cmml">ℛ</mi><mi id="S2.SS2.p4.1.m1.1.1.3" xref="S2.SS2.p4.1.m1.1.1.3.cmml">σ</mi></msub><annotation-xml encoding="MathML-Content" id="S2.SS2.p4.1.m1.1b"><apply id="S2.SS2.p4.1.m1.1.1.cmml" xref="S2.SS2.p4.1.m1.1.1"><csymbol cd="ambiguous" id="S2.SS2.p4.1.m1.1.1.1.cmml" xref="S2.SS2.p4.1.m1.1.1">subscript</csymbol><ci id="S2.SS2.p4.1.m1.1.1.2.cmml" xref="S2.SS2.p4.1.m1.1.1.2">ℛ</ci><ci id="S2.SS2.p4.1.m1.1.1.3.cmml" xref="S2.SS2.p4.1.m1.1.1.3">𝜎</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p4.1.m1.1c">\mathcal{R}_{\sigma}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p4.1.m1.1d">caligraphic_R start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT</annotation></semantics></math> provides the same minimum than <math alttext="\mathcal{P}_{\sigma}" class="ltx_Math" display="inline" id="S2.SS2.p4.2.m2.1"><semantics id="S2.SS2.p4.2.m2.1a"><msub id="S2.SS2.p4.2.m2.1.1" xref="S2.SS2.p4.2.m2.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS2.p4.2.m2.1.1.2" xref="S2.SS2.p4.2.m2.1.1.2.cmml">𝒫</mi><mi id="S2.SS2.p4.2.m2.1.1.3" xref="S2.SS2.p4.2.m2.1.1.3.cmml">σ</mi></msub><annotation-xml encoding="MathML-Content" id="S2.SS2.p4.2.m2.1b"><apply id="S2.SS2.p4.2.m2.1.1.cmml" xref="S2.SS2.p4.2.m2.1.1"><csymbol cd="ambiguous" id="S2.SS2.p4.2.m2.1.1.1.cmml" xref="S2.SS2.p4.2.m2.1.1">subscript</csymbol><ci id="S2.SS2.p4.2.m2.1.1.2.cmml" xref="S2.SS2.p4.2.m2.1.1.2">𝒫</ci><ci id="S2.SS2.p4.2.m2.1.1.3.cmml" xref="S2.SS2.p4.2.m2.1.1.3">𝜎</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p4.2.m2.1c">\mathcal{P}_{\sigma}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p4.2.m2.1d">caligraphic_P start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT</annotation></semantics></math> if the prior is Gaussian. We also detail the case of Gaussian Mixture prior, with the convergence analysis of <math alttext="\nabla\mathcal{P}_{\sigma}" class="ltx_Math" display="inline" id="S2.SS2.p4.3.m3.1"><semantics id="S2.SS2.p4.3.m3.1a"><mrow id="S2.SS2.p4.3.m3.1.1" xref="S2.SS2.p4.3.m3.1.1.cmml"><mo id="S2.SS2.p4.3.m3.1.1.1" rspace="0.167em" xref="S2.SS2.p4.3.m3.1.1.1.cmml">∇</mo><msub id="S2.SS2.p4.3.m3.1.1.2" xref="S2.SS2.p4.3.m3.1.1.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS2.p4.3.m3.1.1.2.2" xref="S2.SS2.p4.3.m3.1.1.2.2.cmml">𝒫</mi><mi id="S2.SS2.p4.3.m3.1.1.2.3" xref="S2.SS2.p4.3.m3.1.1.2.3.cmml">σ</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p4.3.m3.1b"><apply id="S2.SS2.p4.3.m3.1.1.cmml" xref="S2.SS2.p4.3.m3.1.1"><ci id="S2.SS2.p4.3.m3.1.1.1.cmml" xref="S2.SS2.p4.3.m3.1.1.1">∇</ci><apply id="S2.SS2.p4.3.m3.1.1.2.cmml" xref="S2.SS2.p4.3.m3.1.1.2"><csymbol cd="ambiguous" id="S2.SS2.p4.3.m3.1.1.2.1.cmml" xref="S2.SS2.p4.3.m3.1.1.2">subscript</csymbol><ci id="S2.SS2.p4.3.m3.1.1.2.2.cmml" xref="S2.SS2.p4.3.m3.1.1.2.2">𝒫</ci><ci id="S2.SS2.p4.3.m3.1.1.2.3.cmml" xref="S2.SS2.p4.3.m3.1.1.2.3">𝜎</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p4.3.m3.1c">\nabla\mathcal{P}_{\sigma}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p4.3.m3.1d">∇ caligraphic_P start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="\nabla\mathcal{R}_{\sigma}" class="ltx_Math" display="inline" id="S2.SS2.p4.4.m4.1"><semantics id="S2.SS2.p4.4.m4.1a"><mrow id="S2.SS2.p4.4.m4.1.1" xref="S2.SS2.p4.4.m4.1.1.cmml"><mo id="S2.SS2.p4.4.m4.1.1.1" rspace="0.167em" xref="S2.SS2.p4.4.m4.1.1.1.cmml">∇</mo><msub id="S2.SS2.p4.4.m4.1.1.2" xref="S2.SS2.p4.4.m4.1.1.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS2.p4.4.m4.1.1.2.2" xref="S2.SS2.p4.4.m4.1.1.2.2.cmml">ℛ</mi><mi id="S2.SS2.p4.4.m4.1.1.2.3" xref="S2.SS2.p4.4.m4.1.1.2.3.cmml">σ</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p4.4.m4.1b"><apply id="S2.SS2.p4.4.m4.1.1.cmml" xref="S2.SS2.p4.4.m4.1.1"><ci id="S2.SS2.p4.4.m4.1.1.1.cmml" xref="S2.SS2.p4.4.m4.1.1.1">∇</ci><apply id="S2.SS2.p4.4.m4.1.1.2.cmml" xref="S2.SS2.p4.4.m4.1.1.2"><csymbol cd="ambiguous" id="S2.SS2.p4.4.m4.1.1.2.1.cmml" xref="S2.SS2.p4.4.m4.1.1.2">subscript</csymbol><ci id="S2.SS2.p4.4.m4.1.1.2.2.cmml" xref="S2.SS2.p4.4.m4.1.1.2.2">ℛ</ci><ci id="S2.SS2.p4.4.m4.1.1.2.3.cmml" xref="S2.SS2.p4.4.m4.1.1.2.3">𝜎</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p4.4.m4.1c">\nabla\mathcal{R}_{\sigma}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p4.4.m4.1d">∇ caligraphic_R start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT</annotation></semantics></math> to <math alttext="-\nabla\log p" class="ltx_Math" display="inline" id="S2.SS2.p4.5.m5.1"><semantics id="S2.SS2.p4.5.m5.1a"><mrow id="S2.SS2.p4.5.m5.1.1" xref="S2.SS2.p4.5.m5.1.1.cmml"><mo id="S2.SS2.p4.5.m5.1.1a" rspace="0.167em" xref="S2.SS2.p4.5.m5.1.1.cmml">−</mo><mrow id="S2.SS2.p4.5.m5.1.1.2" xref="S2.SS2.p4.5.m5.1.1.2.cmml"><mrow id="S2.SS2.p4.5.m5.1.1.2.1" xref="S2.SS2.p4.5.m5.1.1.2.1.cmml"><mo id="S2.SS2.p4.5.m5.1.1.2.1.1" rspace="0.167em" xref="S2.SS2.p4.5.m5.1.1.2.1.1.cmml">∇</mo><mi id="S2.SS2.p4.5.m5.1.1.2.1.2" xref="S2.SS2.p4.5.m5.1.1.2.1.2.cmml">log</mi></mrow><mo id="S2.SS2.p4.5.m5.1.1.2a" lspace="0.167em" xref="S2.SS2.p4.5.m5.1.1.2.cmml">⁡</mo><mi id="S2.SS2.p4.5.m5.1.1.2.2" xref="S2.SS2.p4.5.m5.1.1.2.2.cmml">p</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p4.5.m5.1b"><apply id="S2.SS2.p4.5.m5.1.1.cmml" xref="S2.SS2.p4.5.m5.1.1"><minus id="S2.SS2.p4.5.m5.1.1.1.cmml" xref="S2.SS2.p4.5.m5.1.1"></minus><apply id="S2.SS2.p4.5.m5.1.1.2.cmml" xref="S2.SS2.p4.5.m5.1.1.2"><apply id="S2.SS2.p4.5.m5.1.1.2.1.cmml" xref="S2.SS2.p4.5.m5.1.1.2.1"><ci id="S2.SS2.p4.5.m5.1.1.2.1.1.cmml" xref="S2.SS2.p4.5.m5.1.1.2.1.1">∇</ci><log id="S2.SS2.p4.5.m5.1.1.2.1.2.cmml" xref="S2.SS2.p4.5.m5.1.1.2.1.2"></log></apply><ci id="S2.SS2.p4.5.m5.1.1.2.2.cmml" xref="S2.SS2.p4.5.m5.1.1.2.2">𝑝</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p4.5.m5.1c">-\nabla\log p</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p4.5.m5.1d">- ∇ roman_log italic_p</annotation></semantics></math> when <math alttext="\sigma\to 0" class="ltx_Math" display="inline" id="S2.SS2.p4.6.m6.1"><semantics id="S2.SS2.p4.6.m6.1a"><mrow id="S2.SS2.p4.6.m6.1.1" xref="S2.SS2.p4.6.m6.1.1.cmml"><mi id="S2.SS2.p4.6.m6.1.1.2" xref="S2.SS2.p4.6.m6.1.1.2.cmml">σ</mi><mo id="S2.SS2.p4.6.m6.1.1.1" stretchy="false" xref="S2.SS2.p4.6.m6.1.1.1.cmml">→</mo><mn id="S2.SS2.p4.6.m6.1.1.3" xref="S2.SS2.p4.6.m6.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p4.6.m6.1b"><apply id="S2.SS2.p4.6.m6.1.1.cmml" xref="S2.SS2.p4.6.m6.1.1"><ci id="S2.SS2.p4.6.m6.1.1.1.cmml" xref="S2.SS2.p4.6.m6.1.1.1">→</ci><ci id="S2.SS2.p4.6.m6.1.1.2.cmml" xref="S2.SS2.p4.6.m6.1.1.2">𝜎</ci><cn id="S2.SS2.p4.6.m6.1.1.3.cmml" type="integer" xref="S2.SS2.p4.6.m6.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p4.6.m6.1c">\sigma\to 0</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p4.6.m6.1d">italic_σ → 0</annotation></semantics></math> and with a 1D illustration of the difference between <math alttext="\nabla\mathcal{R}_{\sigma}" class="ltx_Math" display="inline" id="S2.SS2.p4.7.m7.1"><semantics id="S2.SS2.p4.7.m7.1a"><mrow id="S2.SS2.p4.7.m7.1.1" xref="S2.SS2.p4.7.m7.1.1.cmml"><mo id="S2.SS2.p4.7.m7.1.1.1" rspace="0.167em" xref="S2.SS2.p4.7.m7.1.1.1.cmml">∇</mo><msub id="S2.SS2.p4.7.m7.1.1.2" xref="S2.SS2.p4.7.m7.1.1.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS2.p4.7.m7.1.1.2.2" xref="S2.SS2.p4.7.m7.1.1.2.2.cmml">ℛ</mi><mi id="S2.SS2.p4.7.m7.1.1.2.3" xref="S2.SS2.p4.7.m7.1.1.2.3.cmml">σ</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p4.7.m7.1b"><apply id="S2.SS2.p4.7.m7.1.1.cmml" xref="S2.SS2.p4.7.m7.1.1"><ci id="S2.SS2.p4.7.m7.1.1.1.cmml" xref="S2.SS2.p4.7.m7.1.1.1">∇</ci><apply id="S2.SS2.p4.7.m7.1.1.2.cmml" xref="S2.SS2.p4.7.m7.1.1.2"><csymbol cd="ambiguous" id="S2.SS2.p4.7.m7.1.1.2.1.cmml" xref="S2.SS2.p4.7.m7.1.1.2">subscript</csymbol><ci id="S2.SS2.p4.7.m7.1.1.2.2.cmml" xref="S2.SS2.p4.7.m7.1.1.2.2">ℛ</ci><ci id="S2.SS2.p4.7.m7.1.1.2.3.cmml" xref="S2.SS2.p4.7.m7.1.1.2.3">𝜎</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p4.7.m7.1c">\nabla\mathcal{R}_{\sigma}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p4.7.m7.1d">∇ caligraphic_R start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="-\nabla\log p_{\sigma}" class="ltx_Math" display="inline" id="S2.SS2.p4.8.m8.1"><semantics id="S2.SS2.p4.8.m8.1a"><mrow id="S2.SS2.p4.8.m8.1.1" xref="S2.SS2.p4.8.m8.1.1.cmml"><mo id="S2.SS2.p4.8.m8.1.1a" rspace="0.167em" xref="S2.SS2.p4.8.m8.1.1.cmml">−</mo><mrow id="S2.SS2.p4.8.m8.1.1.2" xref="S2.SS2.p4.8.m8.1.1.2.cmml"><mrow id="S2.SS2.p4.8.m8.1.1.2.1" xref="S2.SS2.p4.8.m8.1.1.2.1.cmml"><mo id="S2.SS2.p4.8.m8.1.1.2.1.1" rspace="0.167em" xref="S2.SS2.p4.8.m8.1.1.2.1.1.cmml">∇</mo><mi id="S2.SS2.p4.8.m8.1.1.2.1.2" xref="S2.SS2.p4.8.m8.1.1.2.1.2.cmml">log</mi></mrow><mo id="S2.SS2.p4.8.m8.1.1.2a" lspace="0.167em" xref="S2.SS2.p4.8.m8.1.1.2.cmml">⁡</mo><msub id="S2.SS2.p4.8.m8.1.1.2.2" xref="S2.SS2.p4.8.m8.1.1.2.2.cmml"><mi id="S2.SS2.p4.8.m8.1.1.2.2.2" xref="S2.SS2.p4.8.m8.1.1.2.2.2.cmml">p</mi><mi id="S2.SS2.p4.8.m8.1.1.2.2.3" xref="S2.SS2.p4.8.m8.1.1.2.2.3.cmml">σ</mi></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p4.8.m8.1b"><apply id="S2.SS2.p4.8.m8.1.1.cmml" xref="S2.SS2.p4.8.m8.1.1"><minus id="S2.SS2.p4.8.m8.1.1.1.cmml" xref="S2.SS2.p4.8.m8.1.1"></minus><apply id="S2.SS2.p4.8.m8.1.1.2.cmml" xref="S2.SS2.p4.8.m8.1.1.2"><apply id="S2.SS2.p4.8.m8.1.1.2.1.cmml" xref="S2.SS2.p4.8.m8.1.1.2.1"><ci id="S2.SS2.p4.8.m8.1.1.2.1.1.cmml" xref="S2.SS2.p4.8.m8.1.1.2.1.1">∇</ci><log id="S2.SS2.p4.8.m8.1.1.2.1.2.cmml" xref="S2.SS2.p4.8.m8.1.1.2.1.2"></log></apply><apply id="S2.SS2.p4.8.m8.1.1.2.2.cmml" xref="S2.SS2.p4.8.m8.1.1.2.2"><csymbol cd="ambiguous" id="S2.SS2.p4.8.m8.1.1.2.2.1.cmml" xref="S2.SS2.p4.8.m8.1.1.2.2">subscript</csymbol><ci id="S2.SS2.p4.8.m8.1.1.2.2.2.cmml" xref="S2.SS2.p4.8.m8.1.1.2.2.2">𝑝</ci><ci id="S2.SS2.p4.8.m8.1.1.2.2.3.cmml" xref="S2.SS2.p4.8.m8.1.1.2.2.3">𝜎</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p4.8.m8.1c">-\nabla\log p_{\sigma}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p4.8.m8.1d">- ∇ roman_log italic_p start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT</annotation></semantics></math>.</p> </div> <section class="ltx_paragraph" id="S2.SS2.SSS0.Px1"> <h5 class="ltx_title ltx_title_paragraph">Interpretation of the SNORE regularization</h5> <div class="ltx_para" id="S2.SS2.SSS0.Px1.p1"> <p class="ltx_p" id="S2.SS2.SSS0.Px1.p1.1">We first underline that <math alttext="\mathcal{R}_{\sigma}" class="ltx_Math" display="inline" id="S2.SS2.SSS0.Px1.p1.1.m1.1"><semantics id="S2.SS2.SSS0.Px1.p1.1.m1.1a"><msub id="S2.SS2.SSS0.Px1.p1.1.m1.1.1" xref="S2.SS2.SSS0.Px1.p1.1.m1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS2.SSS0.Px1.p1.1.m1.1.1.2" xref="S2.SS2.SSS0.Px1.p1.1.m1.1.1.2.cmml">ℛ</mi><mi id="S2.SS2.SSS0.Px1.p1.1.m1.1.1.3" xref="S2.SS2.SSS0.Px1.p1.1.m1.1.1.3.cmml">σ</mi></msub><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS0.Px1.p1.1.m1.1b"><apply id="S2.SS2.SSS0.Px1.p1.1.m1.1.1.cmml" xref="S2.SS2.SSS0.Px1.p1.1.m1.1.1"><csymbol cd="ambiguous" id="S2.SS2.SSS0.Px1.p1.1.m1.1.1.1.cmml" xref="S2.SS2.SSS0.Px1.p1.1.m1.1.1">subscript</csymbol><ci id="S2.SS2.SSS0.Px1.p1.1.m1.1.1.2.cmml" xref="S2.SS2.SSS0.Px1.p1.1.m1.1.1.2">ℛ</ci><ci id="S2.SS2.SSS0.Px1.p1.1.m1.1.1.3.cmml" xref="S2.SS2.SSS0.Px1.p1.1.m1.1.1.3">𝜎</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS0.Px1.p1.1.m1.1c">\mathcal{R}_{\sigma}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS0.Px1.p1.1.m1.1d">caligraphic_R start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT</annotation></semantics></math> can be re-written as</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A8.EGx5"> <tbody id="S2.Ex4"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\mathcal{R}_{\sigma}({\mathbf{x}})=-\mathbb{E}_{{\tilde{\mathbf{x% }}}\sim p_{\sigma}({\tilde{\mathbf{x}}}|{\mathbf{x}})}\left(\log p_{\sigma}({% \tilde{\mathbf{x}}})\right)" class="ltx_Math" display="inline" id="S2.Ex4.m1.4"><semantics id="S2.Ex4.m1.4a"><mrow id="S2.Ex4.m1.4.4" xref="S2.Ex4.m1.4.4.cmml"><mrow id="S2.Ex4.m1.4.4.3" xref="S2.Ex4.m1.4.4.3.cmml"><msub id="S2.Ex4.m1.4.4.3.2" xref="S2.Ex4.m1.4.4.3.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.Ex4.m1.4.4.3.2.2" xref="S2.Ex4.m1.4.4.3.2.2.cmml">ℛ</mi><mi id="S2.Ex4.m1.4.4.3.2.3" xref="S2.Ex4.m1.4.4.3.2.3.cmml">σ</mi></msub><mo id="S2.Ex4.m1.4.4.3.1" xref="S2.Ex4.m1.4.4.3.1.cmml">⁢</mo><mrow id="S2.Ex4.m1.4.4.3.3.2" xref="S2.Ex4.m1.4.4.3.cmml"><mo id="S2.Ex4.m1.4.4.3.3.2.1" stretchy="false" xref="S2.Ex4.m1.4.4.3.cmml">(</mo><mi id="S2.Ex4.m1.2.2" xref="S2.Ex4.m1.2.2.cmml">𝐱</mi><mo id="S2.Ex4.m1.4.4.3.3.2.2" stretchy="false" xref="S2.Ex4.m1.4.4.3.cmml">)</mo></mrow></mrow><mo id="S2.Ex4.m1.4.4.2" xref="S2.Ex4.m1.4.4.2.cmml">=</mo><mrow id="S2.Ex4.m1.4.4.1" xref="S2.Ex4.m1.4.4.1.cmml"><mo id="S2.Ex4.m1.4.4.1a" xref="S2.Ex4.m1.4.4.1.cmml">−</mo><mrow id="S2.Ex4.m1.4.4.1.1" xref="S2.Ex4.m1.4.4.1.1.cmml"><msub id="S2.Ex4.m1.4.4.1.1.3" xref="S2.Ex4.m1.4.4.1.1.3.cmml"><mi id="S2.Ex4.m1.4.4.1.1.3.2" xref="S2.Ex4.m1.4.4.1.1.3.2.cmml">𝔼</mi><mrow id="S2.Ex4.m1.1.1.1" xref="S2.Ex4.m1.1.1.1.cmml"><mover accent="true" id="S2.Ex4.m1.1.1.1.3" xref="S2.Ex4.m1.1.1.1.3.cmml"><mi id="S2.Ex4.m1.1.1.1.3.2" xref="S2.Ex4.m1.1.1.1.3.2.cmml">𝐱</mi><mo id="S2.Ex4.m1.1.1.1.3.1" xref="S2.Ex4.m1.1.1.1.3.1.cmml">~</mo></mover><mo id="S2.Ex4.m1.1.1.1.2" xref="S2.Ex4.m1.1.1.1.2.cmml">∼</mo><mrow id="S2.Ex4.m1.1.1.1.1" xref="S2.Ex4.m1.1.1.1.1.cmml"><msub id="S2.Ex4.m1.1.1.1.1.3" xref="S2.Ex4.m1.1.1.1.1.3.cmml"><mi id="S2.Ex4.m1.1.1.1.1.3.2" xref="S2.Ex4.m1.1.1.1.1.3.2.cmml">p</mi><mi id="S2.Ex4.m1.1.1.1.1.3.3" xref="S2.Ex4.m1.1.1.1.1.3.3.cmml">σ</mi></msub><mo id="S2.Ex4.m1.1.1.1.1.2" xref="S2.Ex4.m1.1.1.1.1.2.cmml">⁢</mo><mrow id="S2.Ex4.m1.1.1.1.1.1.1" xref="S2.Ex4.m1.1.1.1.1.1.1.1.cmml"><mo id="S2.Ex4.m1.1.1.1.1.1.1.2" stretchy="false" xref="S2.Ex4.m1.1.1.1.1.1.1.1.cmml">(</mo><mrow id="S2.Ex4.m1.1.1.1.1.1.1.1" 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xref="S2.Ex4.m1.4.4.1.1.1.1.1.2.1.cmml">log</mi><mo id="S2.Ex4.m1.4.4.1.1.1.1.1.2a" lspace="0.167em" xref="S2.Ex4.m1.4.4.1.1.1.1.1.2.cmml">⁡</mo><msub id="S2.Ex4.m1.4.4.1.1.1.1.1.2.2" xref="S2.Ex4.m1.4.4.1.1.1.1.1.2.2.cmml"><mi id="S2.Ex4.m1.4.4.1.1.1.1.1.2.2.2" xref="S2.Ex4.m1.4.4.1.1.1.1.1.2.2.2.cmml">p</mi><mi id="S2.Ex4.m1.4.4.1.1.1.1.1.2.2.3" xref="S2.Ex4.m1.4.4.1.1.1.1.1.2.2.3.cmml">σ</mi></msub></mrow><mo id="S2.Ex4.m1.4.4.1.1.1.1.1.1" xref="S2.Ex4.m1.4.4.1.1.1.1.1.1.cmml">⁢</mo><mrow id="S2.Ex4.m1.4.4.1.1.1.1.1.3.2" xref="S2.Ex4.m1.3.3.cmml"><mo id="S2.Ex4.m1.4.4.1.1.1.1.1.3.2.1" stretchy="false" xref="S2.Ex4.m1.3.3.cmml">(</mo><mover accent="true" id="S2.Ex4.m1.3.3" xref="S2.Ex4.m1.3.3.cmml"><mi id="S2.Ex4.m1.3.3.2" xref="S2.Ex4.m1.3.3.2.cmml">𝐱</mi><mo id="S2.Ex4.m1.3.3.1" xref="S2.Ex4.m1.3.3.1.cmml">~</mo></mover><mo id="S2.Ex4.m1.4.4.1.1.1.1.1.3.2.2" stretchy="false" xref="S2.Ex4.m1.3.3.cmml">)</mo></mrow></mrow><mo id="S2.Ex4.m1.4.4.1.1.1.1.3" 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end_ARG ∼ italic_p start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ( over~ start_ARG bold_x end_ARG | bold_x ) end_POSTSUBSCRIPT ( roman_log italic_p start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ( over~ start_ARG bold_x end_ARG ) )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="S2.Ex5"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle=\mathcal{KL}(p_{\sigma}({\tilde{\mathbf{x}}}|{\mathbf{x}})\|p_{% \sigma}({\tilde{\mathbf{x}}}))-\mathbb{E}_{{\tilde{\mathbf{x}}}\sim p_{\sigma}% ({\tilde{\mathbf{x}}}|{\mathbf{x}})}\left(\log p_{\sigma}({\tilde{\mathbf{x}}}% |{\mathbf{x}})\right)" class="ltx_Math" display="inline" id="S2.Ex5.m1.4"><semantics id="S2.Ex5.m1.4a"><mrow id="S2.Ex5.m1.4.4" xref="S2.Ex5.m1.4.4.cmml"><mi id="S2.Ex5.m1.4.4.4" xref="S2.Ex5.m1.4.4.4.cmml"></mi><mo 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p_{\sigma}% ({\tilde{\mathbf{x}}}|{\mathbf{x}})}\left(\log p_{\sigma}({\tilde{\mathbf{x}}}% |{\mathbf{x}})\right)</annotation><annotation encoding="application/x-llamapun" id="S2.Ex5.m1.4d">= caligraphic_K caligraphic_L ( italic_p start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ( over~ start_ARG bold_x end_ARG | bold_x ) ∥ italic_p start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ( over~ start_ARG bold_x end_ARG ) ) - blackboard_E start_POSTSUBSCRIPT over~ start_ARG bold_x end_ARG ∼ italic_p start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ( over~ start_ARG bold_x end_ARG | bold_x ) end_POSTSUBSCRIPT ( roman_log italic_p start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ( over~ start_ARG bold_x end_ARG | bold_x ) )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="S2.E8"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_left 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xref="S2.E8.m1.1.1.2">𝐱</ci></apply></apply></apply></apply><ci id="S2.E8.m1.2.2.1.1.1.3.cmml" xref="S2.E8.m1.2.2.1.1.1.3">𝐶</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.E8.m1.2c">\displaystyle=\mathcal{KL}(p_{\sigma}({\tilde{\mathbf{x}}}|{\mathbf{x}})\|p_{% \sigma}({\tilde{\mathbf{x}}}))+C,</annotation><annotation encoding="application/x-llamapun" id="S2.E8.m1.2d">= caligraphic_K caligraphic_L ( italic_p start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ( over~ start_ARG bold_x end_ARG | bold_x ) ∥ italic_p start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ( over~ start_ARG bold_x end_ARG ) ) + italic_C ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(8)</span></td> </tr></tbody> </table> </div> <div class="ltx_para" id="S2.SS2.SSS0.Px1.p2"> <p class="ltx_p" id="S2.SS2.SSS0.Px1.p2.1">where we introduced the Kullback-Leibler divergence <math alttext="\mathcal{KL}(\mu\|\nu):=\int_{{\mathbb{R}}^{d}}{\log\left(\frac{d\mu}{d\nu}% \right)d\mu}" class="ltx_Math" display="inline" id="S2.SS2.SSS0.Px1.p2.1.m1.3"><semantics id="S2.SS2.SSS0.Px1.p2.1.m1.3a"><mrow id="S2.SS2.SSS0.Px1.p2.1.m1.3.3" xref="S2.SS2.SSS0.Px1.p2.1.m1.3.3.cmml"><mrow id="S2.SS2.SSS0.Px1.p2.1.m1.3.3.1" xref="S2.SS2.SSS0.Px1.p2.1.m1.3.3.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS2.SSS0.Px1.p2.1.m1.3.3.1.3" xref="S2.SS2.SSS0.Px1.p2.1.m1.3.3.1.3.cmml">𝒦</mi><mo id="S2.SS2.SSS0.Px1.p2.1.m1.3.3.1.2" xref="S2.SS2.SSS0.Px1.p2.1.m1.3.3.1.2.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S2.SS2.SSS0.Px1.p2.1.m1.3.3.1.4" xref="S2.SS2.SSS0.Px1.p2.1.m1.3.3.1.4.cmml">ℒ</mi><mo id="S2.SS2.SSS0.Px1.p2.1.m1.3.3.1.2a" xref="S2.SS2.SSS0.Px1.p2.1.m1.3.3.1.2.cmml">⁢</mo><mrow id="S2.SS2.SSS0.Px1.p2.1.m1.3.3.1.1.1" xref="S2.SS2.SSS0.Px1.p2.1.m1.3.3.1.1.1.1.cmml"><mo 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xref="S2.SS2.SSS0.Px1.p2.1.m1.3.3.3.2.3.2">𝜇</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS0.Px1.p2.1.m1.3c">\mathcal{KL}(\mu\|\nu):=\int_{{\mathbb{R}}^{d}}{\log\left(\frac{d\mu}{d\nu}% \right)d\mu}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS0.Px1.p2.1.m1.3d">caligraphic_K caligraphic_L ( italic_μ ∥ italic_ν ) := ∫ start_POSTSUBSCRIPT blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT end_POSTSUBSCRIPT roman_log ( divide start_ARG italic_d italic_μ end_ARG start_ARG italic_d italic_ν end_ARG ) italic_d italic_μ</annotation></semantics></math> and the constant </p> </div> <div class="ltx_para" id="S2.SS2.SSS0.Px1.p3"> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A8.EGx6"> <tbody id="S2.Ex6"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle C=-\mathbb{E}_{{\tilde{\mathbf{x}}}\sim p_{\sigma}({\tilde{% \mathbf{x}}}|{\mathbf{x}})}\left(\log p_{\sigma}({\tilde{\mathbf{x}}}|{\mathbf% {x}})\right)=\frac{d}{2}\left(1+\log\left(2\pi\sigma^{2}\right)\right)." class="ltx_Math" display="inline" id="S2.Ex6.m1.3"><semantics id="S2.Ex6.m1.3a"><mrow id="S2.Ex6.m1.3.3.1" xref="S2.Ex6.m1.3.3.1.1.cmml"><mrow id="S2.Ex6.m1.3.3.1.1" xref="S2.Ex6.m1.3.3.1.1.cmml"><mi id="S2.Ex6.m1.3.3.1.1.4" xref="S2.Ex6.m1.3.3.1.1.4.cmml">C</mi><mo id="S2.Ex6.m1.3.3.1.1.5" xref="S2.Ex6.m1.3.3.1.1.5.cmml">=</mo><mrow id="S2.Ex6.m1.3.3.1.1.1" xref="S2.Ex6.m1.3.3.1.1.1.cmml"><mo id="S2.Ex6.m1.3.3.1.1.1a" xref="S2.Ex6.m1.3.3.1.1.1.cmml">−</mo><mrow id="S2.Ex6.m1.3.3.1.1.1.1" xref="S2.Ex6.m1.3.3.1.1.1.1.cmml"><msub id="S2.Ex6.m1.3.3.1.1.1.1.3" xref="S2.Ex6.m1.3.3.1.1.1.1.3.cmml"><mi id="S2.Ex6.m1.3.3.1.1.1.1.3.2" xref="S2.Ex6.m1.3.3.1.1.1.1.3.2.cmml">𝔼</mi><mrow id="S2.Ex6.m1.1.1.1" xref="S2.Ex6.m1.1.1.1.cmml"><mover accent="true" 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p_{\sigma}({\tilde{\mathbf{x}}}|{\mathbf% {x}})\right)=\frac{d}{2}\left(1+\log\left(2\pi\sigma^{2}\right)\right).</annotation><annotation encoding="application/x-llamapun" id="S2.Ex6.m1.3d">italic_C = - blackboard_E start_POSTSUBSCRIPT over~ start_ARG bold_x end_ARG ∼ italic_p start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ( over~ start_ARG bold_x end_ARG | bold_x ) end_POSTSUBSCRIPT ( roman_log italic_p start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ( over~ start_ARG bold_x end_ARG | bold_x ) ) = divide start_ARG italic_d end_ARG start_ARG 2 end_ARG ( 1 + roman_log ( 2 italic_π italic_σ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) ) .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> </div> <div class="ltx_para" id="S2.SS2.SSS0.Px1.p4"> <p class="ltx_p" id="S2.SS2.SSS0.Px1.p4.6">Hence the potential <math alttext="\mathcal{R}_{\sigma}({\mathbf{x}})" class="ltx_Math" display="inline" 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xref="S2.SS2.SSS0.Px1.p4.2.m2.2.2.1.1.1.1.1.1.1.2.2">𝐱</ci></apply><ci id="S2.SS2.SSS0.Px1.p4.2.m2.2.2.1.1.1.1.1.1.1.3.cmml" xref="S2.SS2.SSS0.Px1.p4.2.m2.2.2.1.1.1.1.1.1.1.3">𝐱</ci></apply></apply><apply id="S2.SS2.SSS0.Px1.p4.2.m2.2.2.1.1.1.3.cmml" xref="S2.SS2.SSS0.Px1.p4.2.m2.2.2.1.1.1.3"><times id="S2.SS2.SSS0.Px1.p4.2.m2.2.2.1.1.1.3.1.cmml" xref="S2.SS2.SSS0.Px1.p4.2.m2.2.2.1.1.1.3.1"></times><apply id="S2.SS2.SSS0.Px1.p4.2.m2.2.2.1.1.1.3.2.cmml" xref="S2.SS2.SSS0.Px1.p4.2.m2.2.2.1.1.1.3.2"><csymbol cd="ambiguous" id="S2.SS2.SSS0.Px1.p4.2.m2.2.2.1.1.1.3.2.1.cmml" xref="S2.SS2.SSS0.Px1.p4.2.m2.2.2.1.1.1.3.2">subscript</csymbol><ci id="S2.SS2.SSS0.Px1.p4.2.m2.2.2.1.1.1.3.2.2.cmml" xref="S2.SS2.SSS0.Px1.p4.2.m2.2.2.1.1.1.3.2.2">𝑝</ci><ci id="S2.SS2.SSS0.Px1.p4.2.m2.2.2.1.1.1.3.2.3.cmml" xref="S2.SS2.SSS0.Px1.p4.2.m2.2.2.1.1.1.3.2.3">𝜎</ci></apply><apply id="S2.SS2.SSS0.Px1.p4.2.m2.1.1.cmml" xref="S2.SS2.SSS0.Px1.p4.2.m2.2.2.1.1.1.3.3.2"><ci id="S2.SS2.SSS0.Px1.p4.2.m2.1.1.1.cmml" xref="S2.SS2.SSS0.Px1.p4.2.m2.1.1.1">~</ci><ci id="S2.SS2.SSS0.Px1.p4.2.m2.1.1.2.cmml" xref="S2.SS2.SSS0.Px1.p4.2.m2.1.1.2">𝐱</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS0.Px1.p4.2.m2.2c">\mathcal{KL}(p_{\sigma}({\tilde{\mathbf{x}}}|{\mathbf{x}})\|p_{\sigma}({\tilde% {\mathbf{x}}}))</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS0.Px1.p4.2.m2.2d">caligraphic_K caligraphic_L ( italic_p start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ( over~ start_ARG bold_x end_ARG | bold_x ) ∥ italic_p start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ( over~ start_ARG bold_x end_ARG ) )</annotation></semantics></math>. This last quantity leads to another interpretation. Minimizing <math alttext="\mathcal{R}_{\sigma}({\mathbf{x}})" class="ltx_Math" display="inline" id="S2.SS2.SSS0.Px1.p4.3.m3.1"><semantics id="S2.SS2.SSS0.Px1.p4.3.m3.1a"><mrow id="S2.SS2.SSS0.Px1.p4.3.m3.1.2" xref="S2.SS2.SSS0.Px1.p4.3.m3.1.2.cmml"><msub id="S2.SS2.SSS0.Px1.p4.3.m3.1.2.2" xref="S2.SS2.SSS0.Px1.p4.3.m3.1.2.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS2.SSS0.Px1.p4.3.m3.1.2.2.2" xref="S2.SS2.SSS0.Px1.p4.3.m3.1.2.2.2.cmml">ℛ</mi><mi id="S2.SS2.SSS0.Px1.p4.3.m3.1.2.2.3" xref="S2.SS2.SSS0.Px1.p4.3.m3.1.2.2.3.cmml">σ</mi></msub><mo id="S2.SS2.SSS0.Px1.p4.3.m3.1.2.1" xref="S2.SS2.SSS0.Px1.p4.3.m3.1.2.1.cmml">⁢</mo><mrow id="S2.SS2.SSS0.Px1.p4.3.m3.1.2.3.2" xref="S2.SS2.SSS0.Px1.p4.3.m3.1.2.cmml"><mo id="S2.SS2.SSS0.Px1.p4.3.m3.1.2.3.2.1" stretchy="false" xref="S2.SS2.SSS0.Px1.p4.3.m3.1.2.cmml">(</mo><mi id="S2.SS2.SSS0.Px1.p4.3.m3.1.1" xref="S2.SS2.SSS0.Px1.p4.3.m3.1.1.cmml">𝐱</mi><mo id="S2.SS2.SSS0.Px1.p4.3.m3.1.2.3.2.2" stretchy="false" xref="S2.SS2.SSS0.Px1.p4.3.m3.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS0.Px1.p4.3.m3.1b"><apply id="S2.SS2.SSS0.Px1.p4.3.m3.1.2.cmml" xref="S2.SS2.SSS0.Px1.p4.3.m3.1.2"><times id="S2.SS2.SSS0.Px1.p4.3.m3.1.2.1.cmml" xref="S2.SS2.SSS0.Px1.p4.3.m3.1.2.1"></times><apply id="S2.SS2.SSS0.Px1.p4.3.m3.1.2.2.cmml" xref="S2.SS2.SSS0.Px1.p4.3.m3.1.2.2"><csymbol cd="ambiguous" id="S2.SS2.SSS0.Px1.p4.3.m3.1.2.2.1.cmml" xref="S2.SS2.SSS0.Px1.p4.3.m3.1.2.2">subscript</csymbol><ci id="S2.SS2.SSS0.Px1.p4.3.m3.1.2.2.2.cmml" xref="S2.SS2.SSS0.Px1.p4.3.m3.1.2.2.2">ℛ</ci><ci id="S2.SS2.SSS0.Px1.p4.3.m3.1.2.2.3.cmml" xref="S2.SS2.SSS0.Px1.p4.3.m3.1.2.2.3">𝜎</ci></apply><ci id="S2.SS2.SSS0.Px1.p4.3.m3.1.1.cmml" xref="S2.SS2.SSS0.Px1.p4.3.m3.1.1">𝐱</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS0.Px1.p4.3.m3.1c">\mathcal{R}_{\sigma}({\mathbf{x}})</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS0.Px1.p4.3.m3.1d">caligraphic_R start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ( bold_x )</annotation></semantics></math> is equivalent to <span class="ltx_text ltx_font_italic" id="S2.SS2.SSS0.Px1.p4.6.3">find the Gaussian mode <math alttext="p_{\sigma}({\tilde{\mathbf{x}}}|{\mathbf{x}})" class="ltx_Math" display="inline" id="S2.SS2.SSS0.Px1.p4.4.1.m1.1"><semantics id="S2.SS2.SSS0.Px1.p4.4.1.m1.1a"><mrow id="S2.SS2.SSS0.Px1.p4.4.1.m1.1.1" xref="S2.SS2.SSS0.Px1.p4.4.1.m1.1.1.cmml"><msub id="S2.SS2.SSS0.Px1.p4.4.1.m1.1.1.3" xref="S2.SS2.SSS0.Px1.p4.4.1.m1.1.1.3.cmml"><mi id="S2.SS2.SSS0.Px1.p4.4.1.m1.1.1.3.2" xref="S2.SS2.SSS0.Px1.p4.4.1.m1.1.1.3.2.cmml">p</mi><mi id="S2.SS2.SSS0.Px1.p4.4.1.m1.1.1.3.3" xref="S2.SS2.SSS0.Px1.p4.4.1.m1.1.1.3.3.cmml">σ</mi></msub><mo id="S2.SS2.SSS0.Px1.p4.4.1.m1.1.1.2" mathvariant="italic" xref="S2.SS2.SSS0.Px1.p4.4.1.m1.1.1.2.cmml">⁢</mo><mrow id="S2.SS2.SSS0.Px1.p4.4.1.m1.1.1.1.1" xref="S2.SS2.SSS0.Px1.p4.4.1.m1.1.1.1.1.1.cmml"><mo 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end_ARG )</annotation></semantics></math> in terms of KL divergence</span>.</p> </div> </section> <section class="ltx_paragraph" id="S2.SS2.SSS0.Px2"> <h5 class="ltx_title ltx_title_paragraph">Optimization algorithms</h5> <div class="ltx_para" id="S2.SS2.SSS0.Px2.p1"> <p class="ltx_p" id="S2.SS2.SSS0.Px2.p1.1">With SNORE regularization, we solve the following optimization problem to restore an image</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A8.EGx7"> <tbody id="S2.E9"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\operatorname*{arg\,min}_{{\mathbf{x}}\in{\mathbb{R}}^{d}}{% \mathcal{J}({\mathbf{x}}):=\mathcal{F}({\mathbf{x}},{\mathbf{y}})+\alpha% \mathcal{R}_{\sigma}({\mathbf{x}})}." class="ltx_Math" display="inline" id="S2.E9.m1.5"><semantics id="S2.E9.m1.5a"><mrow id="S2.E9.m1.5.5.1" xref="S2.E9.m1.5.5.1.1.cmml"><mrow 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start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT end_POSTSUBSCRIPT caligraphic_J ( bold_x ) := caligraphic_F ( bold_x , bold_y ) + italic_α caligraphic_R start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ( bold_x ) .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(9)</span></td> </tr></tbody> </table> </div> <div class="ltx_para" id="S2.SS2.SSS0.Px2.p2"> <p class="ltx_p" id="S2.SS2.SSS0.Px2.p2.1">Due to the formulation of <math alttext="\mathcal{R}_{\sigma}" class="ltx_Math" display="inline" id="S2.SS2.SSS0.Px2.p2.1.m1.1"><semantics id="S2.SS2.SSS0.Px2.p2.1.m1.1a"><msub id="S2.SS2.SSS0.Px2.p2.1.m1.1.1" xref="S2.SS2.SSS0.Px2.p2.1.m1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS2.SSS0.Px2.p2.1.m1.1.1.2" xref="S2.SS2.SSS0.Px2.p2.1.m1.1.1.2.cmml">ℛ</mi><mi id="S2.SS2.SSS0.Px2.p2.1.m1.1.1.3" xref="S2.SS2.SSS0.Px2.p2.1.m1.1.1.3.cmml">σ</mi></msub><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS0.Px2.p2.1.m1.1b"><apply id="S2.SS2.SSS0.Px2.p2.1.m1.1.1.cmml" xref="S2.SS2.SSS0.Px2.p2.1.m1.1.1"><csymbol cd="ambiguous" id="S2.SS2.SSS0.Px2.p2.1.m1.1.1.1.cmml" xref="S2.SS2.SSS0.Px2.p2.1.m1.1.1">subscript</csymbol><ci id="S2.SS2.SSS0.Px2.p2.1.m1.1.1.2.cmml" xref="S2.SS2.SSS0.Px2.p2.1.m1.1.1.2">ℛ</ci><ci id="S2.SS2.SSS0.Px2.p2.1.m1.1.1.3.cmml" xref="S2.SS2.SSS0.Px2.p2.1.m1.1.1.3">𝜎</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS0.Px2.p2.1.m1.1c">\mathcal{R}_{\sigma}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS0.Px2.p2.1.m1.1d">caligraphic_R start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT</annotation></semantics></math> as an expectation, we implement<span class="ltx_note ltx_role_footnote" id="footnote1"><sup class="ltx_note_mark">1</sup><span class="ltx_note_outer"><span class="ltx_note_content"><sup class="ltx_note_mark">1</sup><span class="ltx_tag ltx_tag_note">1</span>Note that a possible stochastic gradient of <math alttext="\mathcal{R}_{\sigma}" class="ltx_Math" display="inline" id="footnote1.m1.1"><semantics id="footnote1.m1.1b"><msub id="footnote1.m1.1.1" xref="footnote1.m1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="footnote1.m1.1.1.2" xref="footnote1.m1.1.1.2.cmml">ℛ</mi><mi id="footnote1.m1.1.1.3" xref="footnote1.m1.1.1.3.cmml">σ</mi></msub><annotation-xml encoding="MathML-Content" id="footnote1.m1.1c"><apply id="footnote1.m1.1.1.cmml" xref="footnote1.m1.1.1"><csymbol cd="ambiguous" id="footnote1.m1.1.1.1.cmml" xref="footnote1.m1.1.1">subscript</csymbol><ci id="footnote1.m1.1.1.2.cmml" xref="footnote1.m1.1.1.2">ℛ</ci><ci id="footnote1.m1.1.1.3.cmml" xref="footnote1.m1.1.1.3">𝜎</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="footnote1.m1.1d">\mathcal{R}_{\sigma}</annotation><annotation encoding="application/x-llamapun" 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id="footnote1.m2.2.2.1.1.1.3.2.3.cmml" xref="footnote1.m2.2.2.1.1.1.3.2.3">𝜎</ci></apply><apply id="footnote1.m2.1.1.cmml" xref="footnote1.m2.2.2.1.1.1.3.3.2"><ci id="footnote1.m2.1.1.1.cmml" xref="footnote1.m2.1.1.1">~</ci><ci id="footnote1.m2.1.1.2.cmml" xref="footnote1.m2.1.1.2">𝐱</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="footnote1.m2.2d">\frac{1}{\sigma^{2}}\left({\tilde{\mathbf{x}}}-D_{\sigma}({\tilde{\mathbf{x}}}% )\right)</annotation><annotation encoding="application/x-llamapun" id="footnote1.m2.2e">divide start_ARG 1 end_ARG start_ARG italic_σ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG ( over~ start_ARG bold_x end_ARG - italic_D start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ( over~ start_ARG bold_x end_ARG ) )</annotation></semantics></math>. We choose to only add noise in the denoiser to reduce the residual noise on the image.</span></span></span> a stochastic gradient descent algorithm (Algorithm <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#alg2" title="Algorithm 2 ‣ Optimization algorithms ‣ 2.2 SNORE regularization ‣ 2 Stochastic deNOising REgularization (SNORE) ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_tag">2</span></a>) to solve Problem (<a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#S2.E9" title="9 ‣ Optimization algorithms ‣ 2.2 SNORE regularization ‣ 2 Stochastic deNOising REgularization (SNORE) ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_tag">9</span></a>).</p> </div> <figure class="ltx_float ltx_float_algorithm ltx_framed_top" id="alg2"> <figcaption class="ltx_caption"><span class="ltx_tag ltx_tag_float"><span class="ltx_text ltx_font_bold" id="alg2.2.1.1">Algorithm 2</span> </span> SNORE</figcaption> <div class="ltx_listing ltx_listing" id="alg2.3"> <div class="ltx_listingline" id="alg2.l1"> <span class="ltx_tag ltx_tag_listingline"><span class="ltx_text" id="alg2.l1.1.1.1" style="font-size:80%;">1:</span></span> <span class="ltx_text ltx_font_bold" id="alg2.l1.2">Param.:</span> init. <math alttext="{\mathbf{x}}_{0}\in{\mathbb{R}}^{d}" class="ltx_Math" display="inline" id="alg2.l1.m1.1"><semantics id="alg2.l1.m1.1a"><mrow id="alg2.l1.m1.1.1" xref="alg2.l1.m1.1.1.cmml"><msub id="alg2.l1.m1.1.1.2" xref="alg2.l1.m1.1.1.2.cmml"><mi id="alg2.l1.m1.1.1.2.2" xref="alg2.l1.m1.1.1.2.2.cmml">𝐱</mi><mn id="alg2.l1.m1.1.1.2.3" xref="alg2.l1.m1.1.1.2.3.cmml">0</mn></msub><mo id="alg2.l1.m1.1.1.1" xref="alg2.l1.m1.1.1.1.cmml">∈</mo><msup id="alg2.l1.m1.1.1.3" xref="alg2.l1.m1.1.1.3.cmml"><mi id="alg2.l1.m1.1.1.3.2" xref="alg2.l1.m1.1.1.3.2.cmml">ℝ</mi><mi id="alg2.l1.m1.1.1.3.3" xref="alg2.l1.m1.1.1.3.3.cmml">d</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="alg2.l1.m1.1b"><apply id="alg2.l1.m1.1.1.cmml" xref="alg2.l1.m1.1.1"><in id="alg2.l1.m1.1.1.1.cmml" xref="alg2.l1.m1.1.1.1"></in><apply id="alg2.l1.m1.1.1.2.cmml" xref="alg2.l1.m1.1.1.2"><csymbol cd="ambiguous" id="alg2.l1.m1.1.1.2.1.cmml" xref="alg2.l1.m1.1.1.2">subscript</csymbol><ci id="alg2.l1.m1.1.1.2.2.cmml" xref="alg2.l1.m1.1.1.2.2">𝐱</ci><cn id="alg2.l1.m1.1.1.2.3.cmml" type="integer" xref="alg2.l1.m1.1.1.2.3">0</cn></apply><apply id="alg2.l1.m1.1.1.3.cmml" xref="alg2.l1.m1.1.1.3"><csymbol cd="ambiguous" id="alg2.l1.m1.1.1.3.1.cmml" xref="alg2.l1.m1.1.1.3">superscript</csymbol><ci id="alg2.l1.m1.1.1.3.2.cmml" xref="alg2.l1.m1.1.1.3.2">ℝ</ci><ci id="alg2.l1.m1.1.1.3.3.cmml" xref="alg2.l1.m1.1.1.3.3">𝑑</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="alg2.l1.m1.1c">{\mathbf{x}}_{0}\in{\mathbb{R}}^{d}</annotation><annotation encoding="application/x-llamapun" id="alg2.l1.m1.1d">bold_x start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ∈ blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT</annotation></semantics></math>, <math alttext="\sigma>0" class="ltx_Math" display="inline" id="alg2.l1.m2.1"><semantics id="alg2.l1.m2.1a"><mrow id="alg2.l1.m2.1.1" xref="alg2.l1.m2.1.1.cmml"><mi id="alg2.l1.m2.1.1.2" xref="alg2.l1.m2.1.1.2.cmml">σ</mi><mo id="alg2.l1.m2.1.1.1" xref="alg2.l1.m2.1.1.1.cmml">></mo><mn id="alg2.l1.m2.1.1.3" xref="alg2.l1.m2.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="alg2.l1.m2.1b"><apply id="alg2.l1.m2.1.1.cmml" xref="alg2.l1.m2.1.1"><gt id="alg2.l1.m2.1.1.1.cmml" xref="alg2.l1.m2.1.1.1"></gt><ci id="alg2.l1.m2.1.1.2.cmml" xref="alg2.l1.m2.1.1.2">𝜎</ci><cn id="alg2.l1.m2.1.1.3.cmml" type="integer" xref="alg2.l1.m2.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="alg2.l1.m2.1c">\sigma>0</annotation><annotation encoding="application/x-llamapun" id="alg2.l1.m2.1d">italic_σ > 0</annotation></semantics></math>, <math alttext="\alpha>0" class="ltx_Math" display="inline" id="alg2.l1.m3.1"><semantics id="alg2.l1.m3.1a"><mrow id="alg2.l1.m3.1.1" xref="alg2.l1.m3.1.1.cmml"><mi id="alg2.l1.m3.1.1.2" xref="alg2.l1.m3.1.1.2.cmml">α</mi><mo id="alg2.l1.m3.1.1.1" xref="alg2.l1.m3.1.1.1.cmml">></mo><mn id="alg2.l1.m3.1.1.3" xref="alg2.l1.m3.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="alg2.l1.m3.1b"><apply id="alg2.l1.m3.1.1.cmml" xref="alg2.l1.m3.1.1"><gt id="alg2.l1.m3.1.1.1.cmml" xref="alg2.l1.m3.1.1.1"></gt><ci id="alg2.l1.m3.1.1.2.cmml" xref="alg2.l1.m3.1.1.2">𝛼</ci><cn id="alg2.l1.m3.1.1.3.cmml" type="integer" xref="alg2.l1.m3.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="alg2.l1.m3.1c">\alpha>0</annotation><annotation encoding="application/x-llamapun" id="alg2.l1.m3.1d">italic_α > 0</annotation></semantics></math>, <math alttext="\delta>0" class="ltx_Math" display="inline" id="alg2.l1.m4.1"><semantics id="alg2.l1.m4.1a"><mrow id="alg2.l1.m4.1.1" xref="alg2.l1.m4.1.1.cmml"><mi id="alg2.l1.m4.1.1.2" xref="alg2.l1.m4.1.1.2.cmml">δ</mi><mo id="alg2.l1.m4.1.1.1" xref="alg2.l1.m4.1.1.1.cmml">></mo><mn id="alg2.l1.m4.1.1.3" xref="alg2.l1.m4.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="alg2.l1.m4.1b"><apply id="alg2.l1.m4.1.1.cmml" xref="alg2.l1.m4.1.1"><gt id="alg2.l1.m4.1.1.1.cmml" xref="alg2.l1.m4.1.1.1"></gt><ci id="alg2.l1.m4.1.1.2.cmml" xref="alg2.l1.m4.1.1.2">𝛿</ci><cn id="alg2.l1.m4.1.1.3.cmml" type="integer" xref="alg2.l1.m4.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="alg2.l1.m4.1c">\delta>0</annotation><annotation encoding="application/x-llamapun" id="alg2.l1.m4.1d">italic_δ > 0</annotation></semantics></math>, <math alttext="N\in{\mathbb{N}}" class="ltx_Math" display="inline" id="alg2.l1.m5.1"><semantics id="alg2.l1.m5.1a"><mrow id="alg2.l1.m5.1.1" xref="alg2.l1.m5.1.1.cmml"><mi id="alg2.l1.m5.1.1.2" xref="alg2.l1.m5.1.1.2.cmml">N</mi><mo id="alg2.l1.m5.1.1.1" xref="alg2.l1.m5.1.1.1.cmml">∈</mo><mi id="alg2.l1.m5.1.1.3" xref="alg2.l1.m5.1.1.3.cmml">ℕ</mi></mrow><annotation-xml encoding="MathML-Content" id="alg2.l1.m5.1b"><apply id="alg2.l1.m5.1.1.cmml" xref="alg2.l1.m5.1.1"><in id="alg2.l1.m5.1.1.1.cmml" xref="alg2.l1.m5.1.1.1"></in><ci id="alg2.l1.m5.1.1.2.cmml" xref="alg2.l1.m5.1.1.2">𝑁</ci><ci id="alg2.l1.m5.1.1.3.cmml" xref="alg2.l1.m5.1.1.3">ℕ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="alg2.l1.m5.1c">N\in{\mathbb{N}}</annotation><annotation encoding="application/x-llamapun" id="alg2.l1.m5.1d">italic_N ∈ blackboard_N</annotation></semantics></math> </div> <div class="ltx_listingline" id="alg2.l2"> <span class="ltx_tag ltx_tag_listingline"><span class="ltx_text" id="alg2.l2.1.1.1" style="font-size:80%;">2:</span></span> <span class="ltx_text ltx_font_bold" id="alg2.l2.2">Input:</span> degraded image <math alttext="{\mathbf{y}}" class="ltx_Math" display="inline" id="alg2.l2.m1.1"><semantics id="alg2.l2.m1.1a"><mi id="alg2.l2.m1.1.1" xref="alg2.l2.m1.1.1.cmml">𝐲</mi><annotation-xml encoding="MathML-Content" id="alg2.l2.m1.1b"><ci id="alg2.l2.m1.1.1.cmml" xref="alg2.l2.m1.1.1">𝐲</ci></annotation-xml><annotation encoding="application/x-tex" id="alg2.l2.m1.1c">{\mathbf{y}}</annotation><annotation encoding="application/x-llamapun" id="alg2.l2.m1.1d">bold_y</annotation></semantics></math> </div> <div class="ltx_listingline" id="alg2.l3"> <span class="ltx_tag ltx_tag_listingline"><span class="ltx_text" id="alg2.l3.1.1.1" style="font-size:80%;">3:</span></span> <span class="ltx_text ltx_font_bold" id="alg2.l3.2">Output:</span> restored image <math alttext="{\mathbf{x}}_{N}" class="ltx_Math" display="inline" id="alg2.l3.m1.1"><semantics id="alg2.l3.m1.1a"><msub id="alg2.l3.m1.1.1" xref="alg2.l3.m1.1.1.cmml"><mi id="alg2.l3.m1.1.1.2" xref="alg2.l3.m1.1.1.2.cmml">𝐱</mi><mi id="alg2.l3.m1.1.1.3" xref="alg2.l3.m1.1.1.3.cmml">N</mi></msub><annotation-xml encoding="MathML-Content" id="alg2.l3.m1.1b"><apply id="alg2.l3.m1.1.1.cmml" xref="alg2.l3.m1.1.1"><csymbol cd="ambiguous" id="alg2.l3.m1.1.1.1.cmml" xref="alg2.l3.m1.1.1">subscript</csymbol><ci id="alg2.l3.m1.1.1.2.cmml" xref="alg2.l3.m1.1.1.2">𝐱</ci><ci id="alg2.l3.m1.1.1.3.cmml" xref="alg2.l3.m1.1.1.3">𝑁</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="alg2.l3.m1.1c">{\mathbf{x}}_{N}</annotation><annotation encoding="application/x-llamapun" id="alg2.l3.m1.1d">bold_x start_POSTSUBSCRIPT italic_N end_POSTSUBSCRIPT</annotation></semantics></math> </div> <div class="ltx_listingline" id="alg2.l4"> <span class="ltx_tag ltx_tag_listingline"><span class="ltx_text" id="alg2.l4.1.1.1" style="font-size:80%;">4:</span></span> <span class="ltx_text ltx_font_bold" id="alg2.l4.2">for</span> <math alttext="k=0,1,\dots,N-1" class="ltx_Math" display="inline" id="alg2.l4.m1.4"><semantics id="alg2.l4.m1.4a"><mrow id="alg2.l4.m1.4.4" xref="alg2.l4.m1.4.4.cmml"><mi id="alg2.l4.m1.4.4.3" xref="alg2.l4.m1.4.4.3.cmml">k</mi><mo id="alg2.l4.m1.4.4.2" xref="alg2.l4.m1.4.4.2.cmml">=</mo><mrow id="alg2.l4.m1.4.4.1.1" xref="alg2.l4.m1.4.4.1.2.cmml"><mn id="alg2.l4.m1.1.1" xref="alg2.l4.m1.1.1.cmml">0</mn><mo id="alg2.l4.m1.4.4.1.1.2" xref="alg2.l4.m1.4.4.1.2.cmml">,</mo><mn id="alg2.l4.m1.2.2" xref="alg2.l4.m1.2.2.cmml">1</mn><mo id="alg2.l4.m1.4.4.1.1.3" xref="alg2.l4.m1.4.4.1.2.cmml">,</mo><mi id="alg2.l4.m1.3.3" mathvariant="normal" xref="alg2.l4.m1.3.3.cmml">…</mi><mo id="alg2.l4.m1.4.4.1.1.4" xref="alg2.l4.m1.4.4.1.2.cmml">,</mo><mrow id="alg2.l4.m1.4.4.1.1.1" xref="alg2.l4.m1.4.4.1.1.1.cmml"><mi id="alg2.l4.m1.4.4.1.1.1.2" xref="alg2.l4.m1.4.4.1.1.1.2.cmml">N</mi><mo id="alg2.l4.m1.4.4.1.1.1.1" xref="alg2.l4.m1.4.4.1.1.1.1.cmml">−</mo><mn id="alg2.l4.m1.4.4.1.1.1.3" xref="alg2.l4.m1.4.4.1.1.1.3.cmml">1</mn></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="alg2.l4.m1.4b"><apply id="alg2.l4.m1.4.4.cmml" xref="alg2.l4.m1.4.4"><eq id="alg2.l4.m1.4.4.2.cmml" xref="alg2.l4.m1.4.4.2"></eq><ci id="alg2.l4.m1.4.4.3.cmml" xref="alg2.l4.m1.4.4.3">𝑘</ci><list id="alg2.l4.m1.4.4.1.2.cmml" xref="alg2.l4.m1.4.4.1.1"><cn id="alg2.l4.m1.1.1.cmml" type="integer" xref="alg2.l4.m1.1.1">0</cn><cn id="alg2.l4.m1.2.2.cmml" type="integer" xref="alg2.l4.m1.2.2">1</cn><ci id="alg2.l4.m1.3.3.cmml" xref="alg2.l4.m1.3.3">…</ci><apply id="alg2.l4.m1.4.4.1.1.1.cmml" xref="alg2.l4.m1.4.4.1.1.1"><minus id="alg2.l4.m1.4.4.1.1.1.1.cmml" xref="alg2.l4.m1.4.4.1.1.1.1"></minus><ci id="alg2.l4.m1.4.4.1.1.1.2.cmml" xref="alg2.l4.m1.4.4.1.1.1.2">𝑁</ci><cn id="alg2.l4.m1.4.4.1.1.1.3.cmml" type="integer" xref="alg2.l4.m1.4.4.1.1.1.3">1</cn></apply></list></apply></annotation-xml><annotation encoding="application/x-tex" id="alg2.l4.m1.4c">k=0,1,\dots,N-1</annotation><annotation encoding="application/x-llamapun" id="alg2.l4.m1.4d">italic_k = 0 , 1 , … , italic_N - 1</annotation></semantics></math> <span class="ltx_text ltx_font_bold" id="alg2.l4.3">do</span> </div> <div class="ltx_listingline" id="alg2.l5"> <span class="ltx_tag ltx_tag_listingline"><span class="ltx_text" id="alg2.l5.1.1.1" style="font-size:80%;">5:</span></span> <math alttext="{\boldsymbol{\epsilon}}\leftarrow\mathcal{N}(0,\mathbf{I}_{d})" class="ltx_Math" display="inline" id="alg2.l5.m1.2"><semantics id="alg2.l5.m1.2a"><mrow id="alg2.l5.m1.2.2" xref="alg2.l5.m1.2.2.cmml"><mi id="alg2.l5.m1.2.2.3" mathvariant="bold-italic" xref="alg2.l5.m1.2.2.3.cmml">ϵ</mi><mo id="alg2.l5.m1.2.2.2" stretchy="false" xref="alg2.l5.m1.2.2.2.cmml">←</mo><mrow id="alg2.l5.m1.2.2.1" xref="alg2.l5.m1.2.2.1.cmml"><mi class="ltx_font_mathcaligraphic" id="alg2.l5.m1.2.2.1.3" xref="alg2.l5.m1.2.2.1.3.cmml">𝒩</mi><mo id="alg2.l5.m1.2.2.1.2" xref="alg2.l5.m1.2.2.1.2.cmml">⁢</mo><mrow id="alg2.l5.m1.2.2.1.1.1" xref="alg2.l5.m1.2.2.1.1.2.cmml"><mo id="alg2.l5.m1.2.2.1.1.1.2" stretchy="false" xref="alg2.l5.m1.2.2.1.1.2.cmml">(</mo><mn id="alg2.l5.m1.1.1" xref="alg2.l5.m1.1.1.cmml">0</mn><mo id="alg2.l5.m1.2.2.1.1.1.3" xref="alg2.l5.m1.2.2.1.1.2.cmml">,</mo><msub id="alg2.l5.m1.2.2.1.1.1.1" xref="alg2.l5.m1.2.2.1.1.1.1.cmml"><mi id="alg2.l5.m1.2.2.1.1.1.1.2" xref="alg2.l5.m1.2.2.1.1.1.1.2.cmml">𝐈</mi><mi id="alg2.l5.m1.2.2.1.1.1.1.3" xref="alg2.l5.m1.2.2.1.1.1.1.3.cmml">d</mi></msub><mo id="alg2.l5.m1.2.2.1.1.1.4" stretchy="false" xref="alg2.l5.m1.2.2.1.1.2.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="alg2.l5.m1.2b"><apply id="alg2.l5.m1.2.2.cmml" xref="alg2.l5.m1.2.2"><ci id="alg2.l5.m1.2.2.2.cmml" xref="alg2.l5.m1.2.2.2">←</ci><ci id="alg2.l5.m1.2.2.3.cmml" xref="alg2.l5.m1.2.2.3">bold-italic-ϵ</ci><apply id="alg2.l5.m1.2.2.1.cmml" xref="alg2.l5.m1.2.2.1"><times id="alg2.l5.m1.2.2.1.2.cmml" xref="alg2.l5.m1.2.2.1.2"></times><ci id="alg2.l5.m1.2.2.1.3.cmml" xref="alg2.l5.m1.2.2.1.3">𝒩</ci><interval closure="open" id="alg2.l5.m1.2.2.1.1.2.cmml" xref="alg2.l5.m1.2.2.1.1.1"><cn id="alg2.l5.m1.1.1.cmml" type="integer" xref="alg2.l5.m1.1.1">0</cn><apply id="alg2.l5.m1.2.2.1.1.1.1.cmml" xref="alg2.l5.m1.2.2.1.1.1.1"><csymbol cd="ambiguous" id="alg2.l5.m1.2.2.1.1.1.1.1.cmml" xref="alg2.l5.m1.2.2.1.1.1.1">subscript</csymbol><ci id="alg2.l5.m1.2.2.1.1.1.1.2.cmml" xref="alg2.l5.m1.2.2.1.1.1.1.2">𝐈</ci><ci id="alg2.l5.m1.2.2.1.1.1.1.3.cmml" xref="alg2.l5.m1.2.2.1.1.1.1.3">𝑑</ci></apply></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="alg2.l5.m1.2c">{\boldsymbol{\epsilon}}\leftarrow\mathcal{N}(0,\mathbf{I}_{d})</annotation><annotation encoding="application/x-llamapun" id="alg2.l5.m1.2d">bold_italic_ϵ ← caligraphic_N ( 0 , bold_I start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT )</annotation></semantics></math> </div> <div class="ltx_listingline" id="alg2.l6"> <span class="ltx_tag ltx_tag_listingline"><span class="ltx_text" id="alg2.l6.1.1.1" style="font-size:80%;">6:</span></span> <math alttext="{\tilde{\mathbf{x}}}_{k}\leftarrow{\mathbf{x}}_{k}+\sigma{\boldsymbol{\epsilon}}" class="ltx_Math" display="inline" id="alg2.l6.m1.1"><semantics id="alg2.l6.m1.1a"><mrow id="alg2.l6.m1.1.1" xref="alg2.l6.m1.1.1.cmml"><msub id="alg2.l6.m1.1.1.2" xref="alg2.l6.m1.1.1.2.cmml"><mover accent="true" id="alg2.l6.m1.1.1.2.2" xref="alg2.l6.m1.1.1.2.2.cmml"><mi id="alg2.l6.m1.1.1.2.2.2" xref="alg2.l6.m1.1.1.2.2.2.cmml">𝐱</mi><mo id="alg2.l6.m1.1.1.2.2.1" xref="alg2.l6.m1.1.1.2.2.1.cmml">~</mo></mover><mi id="alg2.l6.m1.1.1.2.3" xref="alg2.l6.m1.1.1.2.3.cmml">k</mi></msub><mo id="alg2.l6.m1.1.1.1" stretchy="false" xref="alg2.l6.m1.1.1.1.cmml">←</mo><mrow id="alg2.l6.m1.1.1.3" xref="alg2.l6.m1.1.1.3.cmml"><msub id="alg2.l6.m1.1.1.3.2" xref="alg2.l6.m1.1.1.3.2.cmml"><mi id="alg2.l6.m1.1.1.3.2.2" xref="alg2.l6.m1.1.1.3.2.2.cmml">𝐱</mi><mi id="alg2.l6.m1.1.1.3.2.3" xref="alg2.l6.m1.1.1.3.2.3.cmml">k</mi></msub><mo id="alg2.l6.m1.1.1.3.1" xref="alg2.l6.m1.1.1.3.1.cmml">+</mo><mrow id="alg2.l6.m1.1.1.3.3" xref="alg2.l6.m1.1.1.3.3.cmml"><mi id="alg2.l6.m1.1.1.3.3.2" xref="alg2.l6.m1.1.1.3.3.2.cmml">σ</mi><mo id="alg2.l6.m1.1.1.3.3.1" xref="alg2.l6.m1.1.1.3.3.1.cmml">⁢</mo><mi id="alg2.l6.m1.1.1.3.3.3" mathvariant="bold-italic" xref="alg2.l6.m1.1.1.3.3.3.cmml">ϵ</mi></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="alg2.l6.m1.1b"><apply id="alg2.l6.m1.1.1.cmml" xref="alg2.l6.m1.1.1"><ci id="alg2.l6.m1.1.1.1.cmml" xref="alg2.l6.m1.1.1.1">←</ci><apply id="alg2.l6.m1.1.1.2.cmml" xref="alg2.l6.m1.1.1.2"><csymbol cd="ambiguous" id="alg2.l6.m1.1.1.2.1.cmml" xref="alg2.l6.m1.1.1.2">subscript</csymbol><apply id="alg2.l6.m1.1.1.2.2.cmml" xref="alg2.l6.m1.1.1.2.2"><ci id="alg2.l6.m1.1.1.2.2.1.cmml" xref="alg2.l6.m1.1.1.2.2.1">~</ci><ci id="alg2.l6.m1.1.1.2.2.2.cmml" xref="alg2.l6.m1.1.1.2.2.2">𝐱</ci></apply><ci id="alg2.l6.m1.1.1.2.3.cmml" xref="alg2.l6.m1.1.1.2.3">𝑘</ci></apply><apply id="alg2.l6.m1.1.1.3.cmml" xref="alg2.l6.m1.1.1.3"><plus id="alg2.l6.m1.1.1.3.1.cmml" xref="alg2.l6.m1.1.1.3.1"></plus><apply id="alg2.l6.m1.1.1.3.2.cmml" xref="alg2.l6.m1.1.1.3.2"><csymbol cd="ambiguous" id="alg2.l6.m1.1.1.3.2.1.cmml" xref="alg2.l6.m1.1.1.3.2">subscript</csymbol><ci id="alg2.l6.m1.1.1.3.2.2.cmml" xref="alg2.l6.m1.1.1.3.2.2">𝐱</ci><ci id="alg2.l6.m1.1.1.3.2.3.cmml" xref="alg2.l6.m1.1.1.3.2.3">𝑘</ci></apply><apply id="alg2.l6.m1.1.1.3.3.cmml" xref="alg2.l6.m1.1.1.3.3"><times id="alg2.l6.m1.1.1.3.3.1.cmml" xref="alg2.l6.m1.1.1.3.3.1"></times><ci id="alg2.l6.m1.1.1.3.3.2.cmml" xref="alg2.l6.m1.1.1.3.3.2">𝜎</ci><ci id="alg2.l6.m1.1.1.3.3.3.cmml" xref="alg2.l6.m1.1.1.3.3.3">bold-italic-ϵ</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="alg2.l6.m1.1c">{\tilde{\mathbf{x}}}_{k}\leftarrow{\mathbf{x}}_{k}+\sigma{\boldsymbol{\epsilon}}</annotation><annotation encoding="application/x-llamapun" id="alg2.l6.m1.1d">over~ start_ARG bold_x end_ARG start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ← bold_x start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT + italic_σ bold_italic_ϵ</annotation></semantics></math> </div> <div class="ltx_listingline" id="alg2.l7"> <span class="ltx_tag ltx_tag_listingline"><span class="ltx_text" id="alg2.l7.1.1.1" style="font-size:80%;">7:</span></span> <math alttext="{\mathbf{x}}_{k+1}\leftarrow{\mathbf{x}}_{k}-\delta\nabla\mathcal{F}({\mathbf{% x}}_{k},{\mathbf{y}})-\frac{\alpha\delta}{\sigma^{2}}\left({\mathbf{x}}_{k}-D_% {\sigma}({\tilde{\mathbf{x}}}_{k})\right)" class="ltx_Math" display="inline" id="alg2.l7.m1.3"><semantics id="alg2.l7.m1.3a"><mrow id="alg2.l7.m1.3.3" xref="alg2.l7.m1.3.3.cmml"><msub id="alg2.l7.m1.3.3.4" xref="alg2.l7.m1.3.3.4.cmml"><mi id="alg2.l7.m1.3.3.4.2" xref="alg2.l7.m1.3.3.4.2.cmml">𝐱</mi><mrow id="alg2.l7.m1.3.3.4.3" xref="alg2.l7.m1.3.3.4.3.cmml"><mi id="alg2.l7.m1.3.3.4.3.2" xref="alg2.l7.m1.3.3.4.3.2.cmml">k</mi><mo id="alg2.l7.m1.3.3.4.3.1" xref="alg2.l7.m1.3.3.4.3.1.cmml">+</mo><mn id="alg2.l7.m1.3.3.4.3.3" xref="alg2.l7.m1.3.3.4.3.3.cmml">1</mn></mrow></msub><mo id="alg2.l7.m1.3.3.3" stretchy="false" xref="alg2.l7.m1.3.3.3.cmml">←</mo><mrow id="alg2.l7.m1.3.3.2" xref="alg2.l7.m1.3.3.2.cmml"><msub id="alg2.l7.m1.3.3.2.4" xref="alg2.l7.m1.3.3.2.4.cmml"><mi id="alg2.l7.m1.3.3.2.4.2" xref="alg2.l7.m1.3.3.2.4.2.cmml">𝐱</mi><mi id="alg2.l7.m1.3.3.2.4.3" xref="alg2.l7.m1.3.3.2.4.3.cmml">k</mi></msub><mo id="alg2.l7.m1.3.3.2.3" xref="alg2.l7.m1.3.3.2.3.cmml">−</mo><mrow id="alg2.l7.m1.2.2.1.1" xref="alg2.l7.m1.2.2.1.1.cmml"><mi id="alg2.l7.m1.2.2.1.1.3" xref="alg2.l7.m1.2.2.1.1.3.cmml">δ</mi><mo id="alg2.l7.m1.2.2.1.1.2" lspace="0.167em" xref="alg2.l7.m1.2.2.1.1.2.cmml">⁢</mo><mrow id="alg2.l7.m1.2.2.1.1.4" xref="alg2.l7.m1.2.2.1.1.4.cmml"><mo id="alg2.l7.m1.2.2.1.1.4.1" rspace="0.167em" xref="alg2.l7.m1.2.2.1.1.4.1.cmml">∇</mo><mi class="ltx_font_mathcaligraphic" id="alg2.l7.m1.2.2.1.1.4.2" xref="alg2.l7.m1.2.2.1.1.4.2.cmml">ℱ</mi></mrow><mo id="alg2.l7.m1.2.2.1.1.2a" xref="alg2.l7.m1.2.2.1.1.2.cmml">⁢</mo><mrow id="alg2.l7.m1.2.2.1.1.1.1" xref="alg2.l7.m1.2.2.1.1.1.2.cmml"><mo id="alg2.l7.m1.2.2.1.1.1.1.2" stretchy="false" xref="alg2.l7.m1.2.2.1.1.1.2.cmml">(</mo><msub id="alg2.l7.m1.2.2.1.1.1.1.1" xref="alg2.l7.m1.2.2.1.1.1.1.1.cmml"><mi id="alg2.l7.m1.2.2.1.1.1.1.1.2" xref="alg2.l7.m1.2.2.1.1.1.1.1.2.cmml">𝐱</mi><mi id="alg2.l7.m1.2.2.1.1.1.1.1.3" xref="alg2.l7.m1.2.2.1.1.1.1.1.3.cmml">k</mi></msub><mo id="alg2.l7.m1.2.2.1.1.1.1.3" xref="alg2.l7.m1.2.2.1.1.1.2.cmml">,</mo><mi id="alg2.l7.m1.1.1" xref="alg2.l7.m1.1.1.cmml">𝐲</mi><mo id="alg2.l7.m1.2.2.1.1.1.1.4" stretchy="false" xref="alg2.l7.m1.2.2.1.1.1.2.cmml">)</mo></mrow></mrow><mo id="alg2.l7.m1.3.3.2.3a" xref="alg2.l7.m1.3.3.2.3.cmml">−</mo><mrow id="alg2.l7.m1.3.3.2.2" xref="alg2.l7.m1.3.3.2.2.cmml"><mfrac id="alg2.l7.m1.3.3.2.2.3" xref="alg2.l7.m1.3.3.2.2.3.cmml"><mrow id="alg2.l7.m1.3.3.2.2.3.2" xref="alg2.l7.m1.3.3.2.2.3.2.cmml"><mi id="alg2.l7.m1.3.3.2.2.3.2.2" xref="alg2.l7.m1.3.3.2.2.3.2.2.cmml">α</mi><mo id="alg2.l7.m1.3.3.2.2.3.2.1" xref="alg2.l7.m1.3.3.2.2.3.2.1.cmml">⁢</mo><mi id="alg2.l7.m1.3.3.2.2.3.2.3" xref="alg2.l7.m1.3.3.2.2.3.2.3.cmml">δ</mi></mrow><msup id="alg2.l7.m1.3.3.2.2.3.3" xref="alg2.l7.m1.3.3.2.2.3.3.cmml"><mi id="alg2.l7.m1.3.3.2.2.3.3.2" xref="alg2.l7.m1.3.3.2.2.3.3.2.cmml">σ</mi><mn id="alg2.l7.m1.3.3.2.2.3.3.3" xref="alg2.l7.m1.3.3.2.2.3.3.3.cmml">2</mn></msup></mfrac><mo id="alg2.l7.m1.3.3.2.2.2" xref="alg2.l7.m1.3.3.2.2.2.cmml">⁢</mo><mrow id="alg2.l7.m1.3.3.2.2.1.1" xref="alg2.l7.m1.3.3.2.2.1.1.1.cmml"><mo id="alg2.l7.m1.3.3.2.2.1.1.2" xref="alg2.l7.m1.3.3.2.2.1.1.1.cmml">(</mo><mrow id="alg2.l7.m1.3.3.2.2.1.1.1" xref="alg2.l7.m1.3.3.2.2.1.1.1.cmml"><msub id="alg2.l7.m1.3.3.2.2.1.1.1.3" xref="alg2.l7.m1.3.3.2.2.1.1.1.3.cmml"><mi id="alg2.l7.m1.3.3.2.2.1.1.1.3.2" xref="alg2.l7.m1.3.3.2.2.1.1.1.3.2.cmml">𝐱</mi><mi id="alg2.l7.m1.3.3.2.2.1.1.1.3.3" xref="alg2.l7.m1.3.3.2.2.1.1.1.3.3.cmml">k</mi></msub><mo id="alg2.l7.m1.3.3.2.2.1.1.1.2" xref="alg2.l7.m1.3.3.2.2.1.1.1.2.cmml">−</mo><mrow id="alg2.l7.m1.3.3.2.2.1.1.1.1" xref="alg2.l7.m1.3.3.2.2.1.1.1.1.cmml"><msub id="alg2.l7.m1.3.3.2.2.1.1.1.1.3" xref="alg2.l7.m1.3.3.2.2.1.1.1.1.3.cmml"><mi id="alg2.l7.m1.3.3.2.2.1.1.1.1.3.2" xref="alg2.l7.m1.3.3.2.2.1.1.1.1.3.2.cmml">D</mi><mi id="alg2.l7.m1.3.3.2.2.1.1.1.1.3.3" xref="alg2.l7.m1.3.3.2.2.1.1.1.1.3.3.cmml">σ</mi></msub><mo id="alg2.l7.m1.3.3.2.2.1.1.1.1.2" xref="alg2.l7.m1.3.3.2.2.1.1.1.1.2.cmml">⁢</mo><mrow id="alg2.l7.m1.3.3.2.2.1.1.1.1.1.1" xref="alg2.l7.m1.3.3.2.2.1.1.1.1.1.1.1.cmml"><mo id="alg2.l7.m1.3.3.2.2.1.1.1.1.1.1.2" stretchy="false" xref="alg2.l7.m1.3.3.2.2.1.1.1.1.1.1.1.cmml">(</mo><msub id="alg2.l7.m1.3.3.2.2.1.1.1.1.1.1.1" xref="alg2.l7.m1.3.3.2.2.1.1.1.1.1.1.1.cmml"><mover accent="true" id="alg2.l7.m1.3.3.2.2.1.1.1.1.1.1.1.2" xref="alg2.l7.m1.3.3.2.2.1.1.1.1.1.1.1.2.cmml"><mi id="alg2.l7.m1.3.3.2.2.1.1.1.1.1.1.1.2.2" xref="alg2.l7.m1.3.3.2.2.1.1.1.1.1.1.1.2.2.cmml">𝐱</mi><mo id="alg2.l7.m1.3.3.2.2.1.1.1.1.1.1.1.2.1" xref="alg2.l7.m1.3.3.2.2.1.1.1.1.1.1.1.2.1.cmml">~</mo></mover><mi id="alg2.l7.m1.3.3.2.2.1.1.1.1.1.1.1.3" xref="alg2.l7.m1.3.3.2.2.1.1.1.1.1.1.1.3.cmml">k</mi></msub><mo id="alg2.l7.m1.3.3.2.2.1.1.1.1.1.1.3" stretchy="false" xref="alg2.l7.m1.3.3.2.2.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><mo id="alg2.l7.m1.3.3.2.2.1.1.3" xref="alg2.l7.m1.3.3.2.2.1.1.1.cmml">)</mo></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="alg2.l7.m1.3b"><apply id="alg2.l7.m1.3.3.cmml" xref="alg2.l7.m1.3.3"><ci id="alg2.l7.m1.3.3.3.cmml" xref="alg2.l7.m1.3.3.3">←</ci><apply id="alg2.l7.m1.3.3.4.cmml" xref="alg2.l7.m1.3.3.4"><csymbol cd="ambiguous" id="alg2.l7.m1.3.3.4.1.cmml" xref="alg2.l7.m1.3.3.4">subscript</csymbol><ci id="alg2.l7.m1.3.3.4.2.cmml" xref="alg2.l7.m1.3.3.4.2">𝐱</ci><apply id="alg2.l7.m1.3.3.4.3.cmml" xref="alg2.l7.m1.3.3.4.3"><plus id="alg2.l7.m1.3.3.4.3.1.cmml" 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id="alg2.l7.m1.2.2.1.1.4.2.cmml" xref="alg2.l7.m1.2.2.1.1.4.2">ℱ</ci></apply><interval closure="open" id="alg2.l7.m1.2.2.1.1.1.2.cmml" xref="alg2.l7.m1.2.2.1.1.1.1"><apply id="alg2.l7.m1.2.2.1.1.1.1.1.cmml" xref="alg2.l7.m1.2.2.1.1.1.1.1"><csymbol cd="ambiguous" id="alg2.l7.m1.2.2.1.1.1.1.1.1.cmml" xref="alg2.l7.m1.2.2.1.1.1.1.1">subscript</csymbol><ci id="alg2.l7.m1.2.2.1.1.1.1.1.2.cmml" xref="alg2.l7.m1.2.2.1.1.1.1.1.2">𝐱</ci><ci id="alg2.l7.m1.2.2.1.1.1.1.1.3.cmml" xref="alg2.l7.m1.2.2.1.1.1.1.1.3">𝑘</ci></apply><ci id="alg2.l7.m1.1.1.cmml" xref="alg2.l7.m1.1.1">𝐲</ci></interval></apply><apply id="alg2.l7.m1.3.3.2.2.cmml" xref="alg2.l7.m1.3.3.2.2"><times id="alg2.l7.m1.3.3.2.2.2.cmml" xref="alg2.l7.m1.3.3.2.2.2"></times><apply id="alg2.l7.m1.3.3.2.2.3.cmml" xref="alg2.l7.m1.3.3.2.2.3"><divide id="alg2.l7.m1.3.3.2.2.3.1.cmml" xref="alg2.l7.m1.3.3.2.2.3"></divide><apply id="alg2.l7.m1.3.3.2.2.3.2.cmml" xref="alg2.l7.m1.3.3.2.2.3.2"><times id="alg2.l7.m1.3.3.2.2.3.2.1.cmml" 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xref="alg2.l7.m1.3.3.2.2.1.1.1.3.2">𝐱</ci><ci id="alg2.l7.m1.3.3.2.2.1.1.1.3.3.cmml" xref="alg2.l7.m1.3.3.2.2.1.1.1.3.3">𝑘</ci></apply><apply id="alg2.l7.m1.3.3.2.2.1.1.1.1.cmml" xref="alg2.l7.m1.3.3.2.2.1.1.1.1"><times id="alg2.l7.m1.3.3.2.2.1.1.1.1.2.cmml" xref="alg2.l7.m1.3.3.2.2.1.1.1.1.2"></times><apply id="alg2.l7.m1.3.3.2.2.1.1.1.1.3.cmml" xref="alg2.l7.m1.3.3.2.2.1.1.1.1.3"><csymbol cd="ambiguous" id="alg2.l7.m1.3.3.2.2.1.1.1.1.3.1.cmml" xref="alg2.l7.m1.3.3.2.2.1.1.1.1.3">subscript</csymbol><ci id="alg2.l7.m1.3.3.2.2.1.1.1.1.3.2.cmml" xref="alg2.l7.m1.3.3.2.2.1.1.1.1.3.2">𝐷</ci><ci id="alg2.l7.m1.3.3.2.2.1.1.1.1.3.3.cmml" xref="alg2.l7.m1.3.3.2.2.1.1.1.1.3.3">𝜎</ci></apply><apply id="alg2.l7.m1.3.3.2.2.1.1.1.1.1.1.1.cmml" xref="alg2.l7.m1.3.3.2.2.1.1.1.1.1.1"><csymbol cd="ambiguous" id="alg2.l7.m1.3.3.2.2.1.1.1.1.1.1.1.1.cmml" xref="alg2.l7.m1.3.3.2.2.1.1.1.1.1.1">subscript</csymbol><apply id="alg2.l7.m1.3.3.2.2.1.1.1.1.1.1.1.2.cmml" xref="alg2.l7.m1.3.3.2.2.1.1.1.1.1.1.1.2"><ci id="alg2.l7.m1.3.3.2.2.1.1.1.1.1.1.1.2.1.cmml" xref="alg2.l7.m1.3.3.2.2.1.1.1.1.1.1.1.2.1">~</ci><ci id="alg2.l7.m1.3.3.2.2.1.1.1.1.1.1.1.2.2.cmml" xref="alg2.l7.m1.3.3.2.2.1.1.1.1.1.1.1.2.2">𝐱</ci></apply><ci id="alg2.l7.m1.3.3.2.2.1.1.1.1.1.1.1.3.cmml" xref="alg2.l7.m1.3.3.2.2.1.1.1.1.1.1.1.3">𝑘</ci></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="alg2.l7.m1.3c">{\mathbf{x}}_{k+1}\leftarrow{\mathbf{x}}_{k}-\delta\nabla\mathcal{F}({\mathbf{% x}}_{k},{\mathbf{y}})-\frac{\alpha\delta}{\sigma^{2}}\left({\mathbf{x}}_{k}-D_% {\sigma}({\tilde{\mathbf{x}}}_{k})\right)</annotation><annotation encoding="application/x-llamapun" id="alg2.l7.m1.3d">bold_x start_POSTSUBSCRIPT italic_k + 1 end_POSTSUBSCRIPT ← bold_x start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT - italic_δ ∇ caligraphic_F ( bold_x start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT , bold_y ) - divide start_ARG italic_α italic_δ end_ARG start_ARG italic_σ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG ( bold_x start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT - italic_D start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ( over~ start_ARG bold_x end_ARG start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) )</annotation></semantics></math> </div> <div class="ltx_listingline" id="alg2.l8"> <span class="ltx_tag ltx_tag_listingline"><span class="ltx_text" id="alg2.l8.1.1.1" style="font-size:80%;">8:</span></span> <span class="ltx_text ltx_font_bold" id="alg2.l8.2">end</span> <span class="ltx_text ltx_font_bold" id="alg2.l8.3">for</span> </div> </div> </figure> <figure class="ltx_float ltx_float_algorithm ltx_framed_top" id="alg3"> <figcaption class="ltx_caption"><span class="ltx_tag ltx_tag_float"><span class="ltx_text ltx_font_bold" id="alg3.2.1.1">Algorithm 3</span> </span> Annealed SNORE</figcaption> <div class="ltx_listing ltx_listing" id="alg3.3"> <div class="ltx_listingline" id="alg3.l1"> <span class="ltx_tag ltx_tag_listingline"><span class="ltx_text" id="alg3.l1.1.1.1" style="font-size:80%;">1:</span></span> <span class="ltx_text ltx_font_bold" id="alg3.l1.2">Param.:</span> init. <math alttext="{\mathbf{x}}_{0}\in{\mathbb{R}}^{d}" class="ltx_Math" display="inline" id="alg3.l1.m1.1"><semantics id="alg3.l1.m1.1a"><mrow id="alg3.l1.m1.1.1" xref="alg3.l1.m1.1.1.cmml"><msub id="alg3.l1.m1.1.1.2" xref="alg3.l1.m1.1.1.2.cmml"><mi id="alg3.l1.m1.1.1.2.2" xref="alg3.l1.m1.1.1.2.2.cmml">𝐱</mi><mn id="alg3.l1.m1.1.1.2.3" xref="alg3.l1.m1.1.1.2.3.cmml">0</mn></msub><mo id="alg3.l1.m1.1.1.1" xref="alg3.l1.m1.1.1.1.cmml">∈</mo><msup id="alg3.l1.m1.1.1.3" xref="alg3.l1.m1.1.1.3.cmml"><mi id="alg3.l1.m1.1.1.3.2" xref="alg3.l1.m1.1.1.3.2.cmml">ℝ</mi><mi id="alg3.l1.m1.1.1.3.3" xref="alg3.l1.m1.1.1.3.3.cmml">d</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="alg3.l1.m1.1b"><apply id="alg3.l1.m1.1.1.cmml" xref="alg3.l1.m1.1.1"><in id="alg3.l1.m1.1.1.1.cmml" xref="alg3.l1.m1.1.1.1"></in><apply id="alg3.l1.m1.1.1.2.cmml" xref="alg3.l1.m1.1.1.2"><csymbol cd="ambiguous" id="alg3.l1.m1.1.1.2.1.cmml" xref="alg3.l1.m1.1.1.2">subscript</csymbol><ci id="alg3.l1.m1.1.1.2.2.cmml" xref="alg3.l1.m1.1.1.2.2">𝐱</ci><cn id="alg3.l1.m1.1.1.2.3.cmml" type="integer" xref="alg3.l1.m1.1.1.2.3">0</cn></apply><apply id="alg3.l1.m1.1.1.3.cmml" xref="alg3.l1.m1.1.1.3"><csymbol cd="ambiguous" id="alg3.l1.m1.1.1.3.1.cmml" xref="alg3.l1.m1.1.1.3">superscript</csymbol><ci id="alg3.l1.m1.1.1.3.2.cmml" xref="alg3.l1.m1.1.1.3.2">ℝ</ci><ci id="alg3.l1.m1.1.1.3.3.cmml" xref="alg3.l1.m1.1.1.3.3">𝑑</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="alg3.l1.m1.1c">{\mathbf{x}}_{0}\in{\mathbb{R}}^{d}</annotation><annotation encoding="application/x-llamapun" id="alg3.l1.m1.1d">bold_x start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ∈ blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT</annotation></semantics></math>, <math alttext="\delta>0" class="ltx_Math" display="inline" id="alg3.l1.m2.1"><semantics id="alg3.l1.m2.1a"><mrow id="alg3.l1.m2.1.1" xref="alg3.l1.m2.1.1.cmml"><mi id="alg3.l1.m2.1.1.2" xref="alg3.l1.m2.1.1.2.cmml">δ</mi><mo id="alg3.l1.m2.1.1.1" xref="alg3.l1.m2.1.1.1.cmml">></mo><mn id="alg3.l1.m2.1.1.3" xref="alg3.l1.m2.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="alg3.l1.m2.1b"><apply id="alg3.l1.m2.1.1.cmml" xref="alg3.l1.m2.1.1"><gt id="alg3.l1.m2.1.1.1.cmml" xref="alg3.l1.m2.1.1.1"></gt><ci id="alg3.l1.m2.1.1.2.cmml" xref="alg3.l1.m2.1.1.2">𝛿</ci><cn id="alg3.l1.m2.1.1.3.cmml" type="integer" xref="alg3.l1.m2.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="alg3.l1.m2.1c">\delta>0</annotation><annotation encoding="application/x-llamapun" id="alg3.l1.m2.1d">italic_δ > 0</annotation></semantics></math>, annealing schedule <math alttext="m\in{\mathbb{N}}" class="ltx_Math" display="inline" id="alg3.l1.m3.1"><semantics id="alg3.l1.m3.1a"><mrow id="alg3.l1.m3.1.1" xref="alg3.l1.m3.1.1.cmml"><mi id="alg3.l1.m3.1.1.2" xref="alg3.l1.m3.1.1.2.cmml">m</mi><mo id="alg3.l1.m3.1.1.1" xref="alg3.l1.m3.1.1.1.cmml">∈</mo><mi id="alg3.l1.m3.1.1.3" xref="alg3.l1.m3.1.1.3.cmml">ℕ</mi></mrow><annotation-xml encoding="MathML-Content" id="alg3.l1.m3.1b"><apply id="alg3.l1.m3.1.1.cmml" xref="alg3.l1.m3.1.1"><in id="alg3.l1.m3.1.1.1.cmml" xref="alg3.l1.m3.1.1.1"></in><ci id="alg3.l1.m3.1.1.2.cmml" xref="alg3.l1.m3.1.1.2">𝑚</ci><ci id="alg3.l1.m3.1.1.3.cmml" xref="alg3.l1.m3.1.1.3">ℕ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="alg3.l1.m3.1c">m\in{\mathbb{N}}</annotation><annotation encoding="application/x-llamapun" id="alg3.l1.m3.1d">italic_m ∈ blackboard_N</annotation></semantics></math>, <math alttext="\sigma_{0}>\dots>\sigma_{m-1}\approx 0" class="ltx_Math" display="inline" id="alg3.l1.m4.1"><semantics id="alg3.l1.m4.1a"><mrow id="alg3.l1.m4.1.1" xref="alg3.l1.m4.1.1.cmml"><msub id="alg3.l1.m4.1.1.2" xref="alg3.l1.m4.1.1.2.cmml"><mi id="alg3.l1.m4.1.1.2.2" xref="alg3.l1.m4.1.1.2.2.cmml">σ</mi><mn id="alg3.l1.m4.1.1.2.3" xref="alg3.l1.m4.1.1.2.3.cmml">0</mn></msub><mo id="alg3.l1.m4.1.1.3" xref="alg3.l1.m4.1.1.3.cmml">></mo><mi id="alg3.l1.m4.1.1.4" mathvariant="normal" xref="alg3.l1.m4.1.1.4.cmml">⋯</mi><mo id="alg3.l1.m4.1.1.5" xref="alg3.l1.m4.1.1.5.cmml">></mo><msub id="alg3.l1.m4.1.1.6" xref="alg3.l1.m4.1.1.6.cmml"><mi id="alg3.l1.m4.1.1.6.2" xref="alg3.l1.m4.1.1.6.2.cmml">σ</mi><mrow id="alg3.l1.m4.1.1.6.3" xref="alg3.l1.m4.1.1.6.3.cmml"><mi id="alg3.l1.m4.1.1.6.3.2" xref="alg3.l1.m4.1.1.6.3.2.cmml">m</mi><mo id="alg3.l1.m4.1.1.6.3.1" xref="alg3.l1.m4.1.1.6.3.1.cmml">−</mo><mn id="alg3.l1.m4.1.1.6.3.3" xref="alg3.l1.m4.1.1.6.3.3.cmml">1</mn></mrow></msub><mo id="alg3.l1.m4.1.1.7" xref="alg3.l1.m4.1.1.7.cmml">≈</mo><mn id="alg3.l1.m4.1.1.8" xref="alg3.l1.m4.1.1.8.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="alg3.l1.m4.1b"><apply id="alg3.l1.m4.1.1.cmml" xref="alg3.l1.m4.1.1"><and id="alg3.l1.m4.1.1a.cmml" xref="alg3.l1.m4.1.1"></and><apply id="alg3.l1.m4.1.1b.cmml" xref="alg3.l1.m4.1.1"><gt id="alg3.l1.m4.1.1.3.cmml" xref="alg3.l1.m4.1.1.3"></gt><apply id="alg3.l1.m4.1.1.2.cmml" xref="alg3.l1.m4.1.1.2"><csymbol cd="ambiguous" id="alg3.l1.m4.1.1.2.1.cmml" xref="alg3.l1.m4.1.1.2">subscript</csymbol><ci id="alg3.l1.m4.1.1.2.2.cmml" xref="alg3.l1.m4.1.1.2.2">𝜎</ci><cn id="alg3.l1.m4.1.1.2.3.cmml" type="integer" xref="alg3.l1.m4.1.1.2.3">0</cn></apply><ci id="alg3.l1.m4.1.1.4.cmml" xref="alg3.l1.m4.1.1.4">⋯</ci></apply><apply id="alg3.l1.m4.1.1c.cmml" xref="alg3.l1.m4.1.1"><gt id="alg3.l1.m4.1.1.5.cmml" xref="alg3.l1.m4.1.1.5"></gt><share href="#alg3.l1.m4.1.1.4.cmml" id="alg3.l1.m4.1.1d.cmml" xref="alg3.l1.m4.1.1"></share><apply id="alg3.l1.m4.1.1.6.cmml" xref="alg3.l1.m4.1.1.6"><csymbol cd="ambiguous" id="alg3.l1.m4.1.1.6.1.cmml" xref="alg3.l1.m4.1.1.6">subscript</csymbol><ci id="alg3.l1.m4.1.1.6.2.cmml" xref="alg3.l1.m4.1.1.6.2">𝜎</ci><apply id="alg3.l1.m4.1.1.6.3.cmml" xref="alg3.l1.m4.1.1.6.3"><minus id="alg3.l1.m4.1.1.6.3.1.cmml" xref="alg3.l1.m4.1.1.6.3.1"></minus><ci id="alg3.l1.m4.1.1.6.3.2.cmml" xref="alg3.l1.m4.1.1.6.3.2">𝑚</ci><cn id="alg3.l1.m4.1.1.6.3.3.cmml" type="integer" xref="alg3.l1.m4.1.1.6.3.3">1</cn></apply></apply></apply><apply id="alg3.l1.m4.1.1e.cmml" xref="alg3.l1.m4.1.1"><approx id="alg3.l1.m4.1.1.7.cmml" xref="alg3.l1.m4.1.1.7"></approx><share href="#alg3.l1.m4.1.1.6.cmml" id="alg3.l1.m4.1.1f.cmml" xref="alg3.l1.m4.1.1"></share><cn id="alg3.l1.m4.1.1.8.cmml" type="integer" xref="alg3.l1.m4.1.1.8">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="alg3.l1.m4.1c">\sigma_{0}>\dots>\sigma_{m-1}\approx 0</annotation><annotation encoding="application/x-llamapun" id="alg3.l1.m4.1d">italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT > ⋯ > italic_σ start_POSTSUBSCRIPT italic_m - 1 end_POSTSUBSCRIPT ≈ 0</annotation></semantics></math>, <math alttext="\alpha_{0},\dots,\alpha_{m-1}>0" class="ltx_Math" display="inline" id="alg3.l1.m5.3"><semantics id="alg3.l1.m5.3a"><mrow id="alg3.l1.m5.3.3" xref="alg3.l1.m5.3.3.cmml"><mrow id="alg3.l1.m5.3.3.2.2" xref="alg3.l1.m5.3.3.2.3.cmml"><msub id="alg3.l1.m5.2.2.1.1.1" xref="alg3.l1.m5.2.2.1.1.1.cmml"><mi id="alg3.l1.m5.2.2.1.1.1.2" xref="alg3.l1.m5.2.2.1.1.1.2.cmml">α</mi><mn id="alg3.l1.m5.2.2.1.1.1.3" xref="alg3.l1.m5.2.2.1.1.1.3.cmml">0</mn></msub><mo id="alg3.l1.m5.3.3.2.2.3" xref="alg3.l1.m5.3.3.2.3.cmml">,</mo><mi id="alg3.l1.m5.1.1" mathvariant="normal" xref="alg3.l1.m5.1.1.cmml">…</mi><mo id="alg3.l1.m5.3.3.2.2.4" xref="alg3.l1.m5.3.3.2.3.cmml">,</mo><msub id="alg3.l1.m5.3.3.2.2.2" xref="alg3.l1.m5.3.3.2.2.2.cmml"><mi id="alg3.l1.m5.3.3.2.2.2.2" xref="alg3.l1.m5.3.3.2.2.2.2.cmml">α</mi><mrow id="alg3.l1.m5.3.3.2.2.2.3" xref="alg3.l1.m5.3.3.2.2.2.3.cmml"><mi id="alg3.l1.m5.3.3.2.2.2.3.2" xref="alg3.l1.m5.3.3.2.2.2.3.2.cmml">m</mi><mo id="alg3.l1.m5.3.3.2.2.2.3.1" xref="alg3.l1.m5.3.3.2.2.2.3.1.cmml">−</mo><mn id="alg3.l1.m5.3.3.2.2.2.3.3" xref="alg3.l1.m5.3.3.2.2.2.3.3.cmml">1</mn></mrow></msub></mrow><mo id="alg3.l1.m5.3.3.3" xref="alg3.l1.m5.3.3.3.cmml">></mo><mn id="alg3.l1.m5.3.3.4" xref="alg3.l1.m5.3.3.4.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="alg3.l1.m5.3b"><apply id="alg3.l1.m5.3.3.cmml" xref="alg3.l1.m5.3.3"><gt id="alg3.l1.m5.3.3.3.cmml" xref="alg3.l1.m5.3.3.3"></gt><list id="alg3.l1.m5.3.3.2.3.cmml" xref="alg3.l1.m5.3.3.2.2"><apply id="alg3.l1.m5.2.2.1.1.1.cmml" xref="alg3.l1.m5.2.2.1.1.1"><csymbol cd="ambiguous" id="alg3.l1.m5.2.2.1.1.1.1.cmml" xref="alg3.l1.m5.2.2.1.1.1">subscript</csymbol><ci id="alg3.l1.m5.2.2.1.1.1.2.cmml" xref="alg3.l1.m5.2.2.1.1.1.2">𝛼</ci><cn id="alg3.l1.m5.2.2.1.1.1.3.cmml" type="integer" xref="alg3.l1.m5.2.2.1.1.1.3">0</cn></apply><ci id="alg3.l1.m5.1.1.cmml" xref="alg3.l1.m5.1.1">…</ci><apply id="alg3.l1.m5.3.3.2.2.2.cmml" xref="alg3.l1.m5.3.3.2.2.2"><csymbol cd="ambiguous" id="alg3.l1.m5.3.3.2.2.2.1.cmml" xref="alg3.l1.m5.3.3.2.2.2">subscript</csymbol><ci id="alg3.l1.m5.3.3.2.2.2.2.cmml" xref="alg3.l1.m5.3.3.2.2.2.2">𝛼</ci><apply id="alg3.l1.m5.3.3.2.2.2.3.cmml" xref="alg3.l1.m5.3.3.2.2.2.3"><minus id="alg3.l1.m5.3.3.2.2.2.3.1.cmml" xref="alg3.l1.m5.3.3.2.2.2.3.1"></minus><ci id="alg3.l1.m5.3.3.2.2.2.3.2.cmml" xref="alg3.l1.m5.3.3.2.2.2.3.2">𝑚</ci><cn id="alg3.l1.m5.3.3.2.2.2.3.3.cmml" type="integer" xref="alg3.l1.m5.3.3.2.2.2.3.3">1</cn></apply></apply></list><cn id="alg3.l1.m5.3.3.4.cmml" type="integer" xref="alg3.l1.m5.3.3.4">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="alg3.l1.m5.3c">\alpha_{0},\dots,\alpha_{m-1}>0</annotation><annotation encoding="application/x-llamapun" id="alg3.l1.m5.3d">italic_α start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , … , italic_α start_POSTSUBSCRIPT italic_m - 1 end_POSTSUBSCRIPT > 0</annotation></semantics></math>, <math alttext="N_{0},\dots,N_{m-1}\in{\mathbb{N}}" class="ltx_Math" display="inline" id="alg3.l1.m6.3"><semantics id="alg3.l1.m6.3a"><mrow id="alg3.l1.m6.3.3" xref="alg3.l1.m6.3.3.cmml"><mrow id="alg3.l1.m6.3.3.2.2" xref="alg3.l1.m6.3.3.2.3.cmml"><msub id="alg3.l1.m6.2.2.1.1.1" xref="alg3.l1.m6.2.2.1.1.1.cmml"><mi id="alg3.l1.m6.2.2.1.1.1.2" xref="alg3.l1.m6.2.2.1.1.1.2.cmml">N</mi><mn id="alg3.l1.m6.2.2.1.1.1.3" xref="alg3.l1.m6.2.2.1.1.1.3.cmml">0</mn></msub><mo id="alg3.l1.m6.3.3.2.2.3" xref="alg3.l1.m6.3.3.2.3.cmml">,</mo><mi id="alg3.l1.m6.1.1" mathvariant="normal" xref="alg3.l1.m6.1.1.cmml">…</mi><mo id="alg3.l1.m6.3.3.2.2.4" xref="alg3.l1.m6.3.3.2.3.cmml">,</mo><msub id="alg3.l1.m6.3.3.2.2.2" xref="alg3.l1.m6.3.3.2.2.2.cmml"><mi id="alg3.l1.m6.3.3.2.2.2.2" xref="alg3.l1.m6.3.3.2.2.2.2.cmml">N</mi><mrow id="alg3.l1.m6.3.3.2.2.2.3" xref="alg3.l1.m6.3.3.2.2.2.3.cmml"><mi id="alg3.l1.m6.3.3.2.2.2.3.2" xref="alg3.l1.m6.3.3.2.2.2.3.2.cmml">m</mi><mo id="alg3.l1.m6.3.3.2.2.2.3.1" xref="alg3.l1.m6.3.3.2.2.2.3.1.cmml">−</mo><mn id="alg3.l1.m6.3.3.2.2.2.3.3" xref="alg3.l1.m6.3.3.2.2.2.3.3.cmml">1</mn></mrow></msub></mrow><mo id="alg3.l1.m6.3.3.3" xref="alg3.l1.m6.3.3.3.cmml">∈</mo><mi id="alg3.l1.m6.3.3.4" xref="alg3.l1.m6.3.3.4.cmml">ℕ</mi></mrow><annotation-xml encoding="MathML-Content" id="alg3.l1.m6.3b"><apply id="alg3.l1.m6.3.3.cmml" xref="alg3.l1.m6.3.3"><in id="alg3.l1.m6.3.3.3.cmml" xref="alg3.l1.m6.3.3.3"></in><list id="alg3.l1.m6.3.3.2.3.cmml" xref="alg3.l1.m6.3.3.2.2"><apply id="alg3.l1.m6.2.2.1.1.1.cmml" xref="alg3.l1.m6.2.2.1.1.1"><csymbol cd="ambiguous" id="alg3.l1.m6.2.2.1.1.1.1.cmml" xref="alg3.l1.m6.2.2.1.1.1">subscript</csymbol><ci id="alg3.l1.m6.2.2.1.1.1.2.cmml" xref="alg3.l1.m6.2.2.1.1.1.2">𝑁</ci><cn id="alg3.l1.m6.2.2.1.1.1.3.cmml" type="integer" xref="alg3.l1.m6.2.2.1.1.1.3">0</cn></apply><ci id="alg3.l1.m6.1.1.cmml" xref="alg3.l1.m6.1.1">…</ci><apply id="alg3.l1.m6.3.3.2.2.2.cmml" xref="alg3.l1.m6.3.3.2.2.2"><csymbol cd="ambiguous" id="alg3.l1.m6.3.3.2.2.2.1.cmml" xref="alg3.l1.m6.3.3.2.2.2">subscript</csymbol><ci id="alg3.l1.m6.3.3.2.2.2.2.cmml" xref="alg3.l1.m6.3.3.2.2.2.2">𝑁</ci><apply id="alg3.l1.m6.3.3.2.2.2.3.cmml" xref="alg3.l1.m6.3.3.2.2.2.3"><minus id="alg3.l1.m6.3.3.2.2.2.3.1.cmml" xref="alg3.l1.m6.3.3.2.2.2.3.1"></minus><ci id="alg3.l1.m6.3.3.2.2.2.3.2.cmml" xref="alg3.l1.m6.3.3.2.2.2.3.2">𝑚</ci><cn id="alg3.l1.m6.3.3.2.2.2.3.3.cmml" type="integer" xref="alg3.l1.m6.3.3.2.2.2.3.3">1</cn></apply></apply></list><ci id="alg3.l1.m6.3.3.4.cmml" xref="alg3.l1.m6.3.3.4">ℕ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="alg3.l1.m6.3c">N_{0},\dots,N_{m-1}\in{\mathbb{N}}</annotation><annotation encoding="application/x-llamapun" id="alg3.l1.m6.3d">italic_N start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , … , italic_N start_POSTSUBSCRIPT italic_m - 1 end_POSTSUBSCRIPT ∈ blackboard_N</annotation></semantics></math> </div> <div class="ltx_listingline" id="alg3.l2"> <span class="ltx_tag ltx_tag_listingline"><span class="ltx_text" id="alg3.l2.1.1.1" style="font-size:80%;">2:</span></span> <span class="ltx_text ltx_font_bold" id="alg3.l2.2">Input:</span> degraded image <math alttext="{\mathbf{y}}" class="ltx_Math" display="inline" id="alg3.l2.m1.1"><semantics id="alg3.l2.m1.1a"><mi id="alg3.l2.m1.1.1" xref="alg3.l2.m1.1.1.cmml">𝐲</mi><annotation-xml encoding="MathML-Content" id="alg3.l2.m1.1b"><ci id="alg3.l2.m1.1.1.cmml" xref="alg3.l2.m1.1.1">𝐲</ci></annotation-xml><annotation encoding="application/x-tex" id="alg3.l2.m1.1c">{\mathbf{y}}</annotation><annotation encoding="application/x-llamapun" id="alg3.l2.m1.1d">bold_y</annotation></semantics></math> </div> <div class="ltx_listingline" id="alg3.l3"> <span class="ltx_tag ltx_tag_listingline"><span class="ltx_text" id="alg3.l3.1.1.1" style="font-size:80%;">3:</span></span> <span class="ltx_text ltx_font_bold" id="alg3.l3.2">Output:</span> restored image <math alttext="{\mathbf{x}}_{N_{m-1}}" class="ltx_Math" display="inline" id="alg3.l3.m1.1"><semantics id="alg3.l3.m1.1a"><msub id="alg3.l3.m1.1.1" xref="alg3.l3.m1.1.1.cmml"><mi id="alg3.l3.m1.1.1.2" xref="alg3.l3.m1.1.1.2.cmml">𝐱</mi><msub id="alg3.l3.m1.1.1.3" xref="alg3.l3.m1.1.1.3.cmml"><mi id="alg3.l3.m1.1.1.3.2" xref="alg3.l3.m1.1.1.3.2.cmml">N</mi><mrow id="alg3.l3.m1.1.1.3.3" xref="alg3.l3.m1.1.1.3.3.cmml"><mi id="alg3.l3.m1.1.1.3.3.2" xref="alg3.l3.m1.1.1.3.3.2.cmml">m</mi><mo id="alg3.l3.m1.1.1.3.3.1" xref="alg3.l3.m1.1.1.3.3.1.cmml">−</mo><mn id="alg3.l3.m1.1.1.3.3.3" xref="alg3.l3.m1.1.1.3.3.3.cmml">1</mn></mrow></msub></msub><annotation-xml encoding="MathML-Content" id="alg3.l3.m1.1b"><apply id="alg3.l3.m1.1.1.cmml" xref="alg3.l3.m1.1.1"><csymbol cd="ambiguous" id="alg3.l3.m1.1.1.1.cmml" xref="alg3.l3.m1.1.1">subscript</csymbol><ci id="alg3.l3.m1.1.1.2.cmml" xref="alg3.l3.m1.1.1.2">𝐱</ci><apply id="alg3.l3.m1.1.1.3.cmml" xref="alg3.l3.m1.1.1.3"><csymbol cd="ambiguous" id="alg3.l3.m1.1.1.3.1.cmml" xref="alg3.l3.m1.1.1.3">subscript</csymbol><ci id="alg3.l3.m1.1.1.3.2.cmml" xref="alg3.l3.m1.1.1.3.2">𝑁</ci><apply id="alg3.l3.m1.1.1.3.3.cmml" xref="alg3.l3.m1.1.1.3.3"><minus id="alg3.l3.m1.1.1.3.3.1.cmml" xref="alg3.l3.m1.1.1.3.3.1"></minus><ci id="alg3.l3.m1.1.1.3.3.2.cmml" xref="alg3.l3.m1.1.1.3.3.2">𝑚</ci><cn id="alg3.l3.m1.1.1.3.3.3.cmml" type="integer" xref="alg3.l3.m1.1.1.3.3.3">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="alg3.l3.m1.1c">{\mathbf{x}}_{N_{m-1}}</annotation><annotation encoding="application/x-llamapun" id="alg3.l3.m1.1d">bold_x start_POSTSUBSCRIPT italic_N start_POSTSUBSCRIPT italic_m - 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math> </div> <div class="ltx_listingline" id="alg3.l4"> <span class="ltx_tag ltx_tag_listingline"><span class="ltx_text" id="alg3.l4.1.1.1" style="font-size:80%;">4:</span></span> <span class="ltx_text ltx_font_bold" id="alg3.l4.2">for</span> <math alttext="i=0,1,\dots,m-1" class="ltx_Math" display="inline" id="alg3.l4.m1.4"><semantics id="alg3.l4.m1.4a"><mrow id="alg3.l4.m1.4.4" xref="alg3.l4.m1.4.4.cmml"><mi id="alg3.l4.m1.4.4.3" xref="alg3.l4.m1.4.4.3.cmml">i</mi><mo id="alg3.l4.m1.4.4.2" xref="alg3.l4.m1.4.4.2.cmml">=</mo><mrow id="alg3.l4.m1.4.4.1.1" xref="alg3.l4.m1.4.4.1.2.cmml"><mn id="alg3.l4.m1.1.1" xref="alg3.l4.m1.1.1.cmml">0</mn><mo id="alg3.l4.m1.4.4.1.1.2" xref="alg3.l4.m1.4.4.1.2.cmml">,</mo><mn id="alg3.l4.m1.2.2" xref="alg3.l4.m1.2.2.cmml">1</mn><mo id="alg3.l4.m1.4.4.1.1.3" xref="alg3.l4.m1.4.4.1.2.cmml">,</mo><mi id="alg3.l4.m1.3.3" mathvariant="normal" xref="alg3.l4.m1.3.3.cmml">…</mi><mo id="alg3.l4.m1.4.4.1.1.4" xref="alg3.l4.m1.4.4.1.2.cmml">,</mo><mrow id="alg3.l4.m1.4.4.1.1.1" xref="alg3.l4.m1.4.4.1.1.1.cmml"><mi id="alg3.l4.m1.4.4.1.1.1.2" xref="alg3.l4.m1.4.4.1.1.1.2.cmml">m</mi><mo id="alg3.l4.m1.4.4.1.1.1.1" xref="alg3.l4.m1.4.4.1.1.1.1.cmml">−</mo><mn id="alg3.l4.m1.4.4.1.1.1.3" xref="alg3.l4.m1.4.4.1.1.1.3.cmml">1</mn></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="alg3.l4.m1.4b"><apply id="alg3.l4.m1.4.4.cmml" xref="alg3.l4.m1.4.4"><eq id="alg3.l4.m1.4.4.2.cmml" xref="alg3.l4.m1.4.4.2"></eq><ci id="alg3.l4.m1.4.4.3.cmml" xref="alg3.l4.m1.4.4.3">𝑖</ci><list id="alg3.l4.m1.4.4.1.2.cmml" xref="alg3.l4.m1.4.4.1.1"><cn id="alg3.l4.m1.1.1.cmml" type="integer" xref="alg3.l4.m1.1.1">0</cn><cn id="alg3.l4.m1.2.2.cmml" type="integer" xref="alg3.l4.m1.2.2">1</cn><ci id="alg3.l4.m1.3.3.cmml" xref="alg3.l4.m1.3.3">…</ci><apply id="alg3.l4.m1.4.4.1.1.1.cmml" xref="alg3.l4.m1.4.4.1.1.1"><minus id="alg3.l4.m1.4.4.1.1.1.1.cmml" xref="alg3.l4.m1.4.4.1.1.1.1"></minus><ci id="alg3.l4.m1.4.4.1.1.1.2.cmml" xref="alg3.l4.m1.4.4.1.1.1.2">𝑚</ci><cn id="alg3.l4.m1.4.4.1.1.1.3.cmml" type="integer" xref="alg3.l4.m1.4.4.1.1.1.3">1</cn></apply></list></apply></annotation-xml><annotation encoding="application/x-tex" id="alg3.l4.m1.4c">i=0,1,\dots,m-1</annotation><annotation encoding="application/x-llamapun" id="alg3.l4.m1.4d">italic_i = 0 , 1 , … , italic_m - 1</annotation></semantics></math> <span class="ltx_text ltx_font_bold" id="alg3.l4.3">do</span> </div> <div class="ltx_listingline" id="alg3.l5"> <span class="ltx_tag ltx_tag_listingline"><span class="ltx_text" id="alg3.l5.1.1.1" style="font-size:80%;">5:</span></span> <span class="ltx_text ltx_font_bold" id="alg3.l5.2">for</span> <math alttext="k=0,1,\dots,N_{i}-1" class="ltx_Math" display="inline" id="alg3.l5.m1.4"><semantics id="alg3.l5.m1.4a"><mrow id="alg3.l5.m1.4.4" xref="alg3.l5.m1.4.4.cmml"><mi id="alg3.l5.m1.4.4.3" xref="alg3.l5.m1.4.4.3.cmml">k</mi><mo id="alg3.l5.m1.4.4.2" xref="alg3.l5.m1.4.4.2.cmml">=</mo><mrow id="alg3.l5.m1.4.4.1.1" xref="alg3.l5.m1.4.4.1.2.cmml"><mn id="alg3.l5.m1.1.1" xref="alg3.l5.m1.1.1.cmml">0</mn><mo id="alg3.l5.m1.4.4.1.1.2" xref="alg3.l5.m1.4.4.1.2.cmml">,</mo><mn id="alg3.l5.m1.2.2" xref="alg3.l5.m1.2.2.cmml">1</mn><mo id="alg3.l5.m1.4.4.1.1.3" xref="alg3.l5.m1.4.4.1.2.cmml">,</mo><mi id="alg3.l5.m1.3.3" mathvariant="normal" xref="alg3.l5.m1.3.3.cmml">…</mi><mo id="alg3.l5.m1.4.4.1.1.4" xref="alg3.l5.m1.4.4.1.2.cmml">,</mo><mrow id="alg3.l5.m1.4.4.1.1.1" xref="alg3.l5.m1.4.4.1.1.1.cmml"><msub id="alg3.l5.m1.4.4.1.1.1.2" xref="alg3.l5.m1.4.4.1.1.1.2.cmml"><mi id="alg3.l5.m1.4.4.1.1.1.2.2" xref="alg3.l5.m1.4.4.1.1.1.2.2.cmml">N</mi><mi id="alg3.l5.m1.4.4.1.1.1.2.3" xref="alg3.l5.m1.4.4.1.1.1.2.3.cmml">i</mi></msub><mo id="alg3.l5.m1.4.4.1.1.1.1" xref="alg3.l5.m1.4.4.1.1.1.1.cmml">−</mo><mn id="alg3.l5.m1.4.4.1.1.1.3" xref="alg3.l5.m1.4.4.1.1.1.3.cmml">1</mn></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="alg3.l5.m1.4b"><apply id="alg3.l5.m1.4.4.cmml" xref="alg3.l5.m1.4.4"><eq id="alg3.l5.m1.4.4.2.cmml" xref="alg3.l5.m1.4.4.2"></eq><ci id="alg3.l5.m1.4.4.3.cmml" xref="alg3.l5.m1.4.4.3">𝑘</ci><list id="alg3.l5.m1.4.4.1.2.cmml" xref="alg3.l5.m1.4.4.1.1"><cn id="alg3.l5.m1.1.1.cmml" type="integer" xref="alg3.l5.m1.1.1">0</cn><cn id="alg3.l5.m1.2.2.cmml" type="integer" xref="alg3.l5.m1.2.2">1</cn><ci id="alg3.l5.m1.3.3.cmml" xref="alg3.l5.m1.3.3">…</ci><apply id="alg3.l5.m1.4.4.1.1.1.cmml" xref="alg3.l5.m1.4.4.1.1.1"><minus id="alg3.l5.m1.4.4.1.1.1.1.cmml" xref="alg3.l5.m1.4.4.1.1.1.1"></minus><apply id="alg3.l5.m1.4.4.1.1.1.2.cmml" xref="alg3.l5.m1.4.4.1.1.1.2"><csymbol cd="ambiguous" id="alg3.l5.m1.4.4.1.1.1.2.1.cmml" xref="alg3.l5.m1.4.4.1.1.1.2">subscript</csymbol><ci id="alg3.l5.m1.4.4.1.1.1.2.2.cmml" xref="alg3.l5.m1.4.4.1.1.1.2.2">𝑁</ci><ci id="alg3.l5.m1.4.4.1.1.1.2.3.cmml" xref="alg3.l5.m1.4.4.1.1.1.2.3">𝑖</ci></apply><cn id="alg3.l5.m1.4.4.1.1.1.3.cmml" type="integer" xref="alg3.l5.m1.4.4.1.1.1.3">1</cn></apply></list></apply></annotation-xml><annotation encoding="application/x-tex" id="alg3.l5.m1.4c">k=0,1,\dots,N_{i}-1</annotation><annotation encoding="application/x-llamapun" id="alg3.l5.m1.4d">italic_k = 0 , 1 , … , italic_N start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT - 1</annotation></semantics></math> <span class="ltx_text ltx_font_bold" id="alg3.l5.3">do</span> </div> <div class="ltx_listingline" id="alg3.l6"> <span class="ltx_tag ltx_tag_listingline"><span class="ltx_text" id="alg3.l6.1.1.1" style="font-size:80%;">6:</span></span> <math alttext="{\boldsymbol{\epsilon}}\leftarrow\mathcal{N}(0,\mathbf{I}_{d})" class="ltx_Math" display="inline" id="alg3.l6.m1.2"><semantics id="alg3.l6.m1.2a"><mrow id="alg3.l6.m1.2.2" xref="alg3.l6.m1.2.2.cmml"><mi id="alg3.l6.m1.2.2.3" mathvariant="bold-italic" xref="alg3.l6.m1.2.2.3.cmml">ϵ</mi><mo id="alg3.l6.m1.2.2.2" stretchy="false" xref="alg3.l6.m1.2.2.2.cmml">←</mo><mrow id="alg3.l6.m1.2.2.1" xref="alg3.l6.m1.2.2.1.cmml"><mi class="ltx_font_mathcaligraphic" id="alg3.l6.m1.2.2.1.3" xref="alg3.l6.m1.2.2.1.3.cmml">𝒩</mi><mo id="alg3.l6.m1.2.2.1.2" xref="alg3.l6.m1.2.2.1.2.cmml">⁢</mo><mrow id="alg3.l6.m1.2.2.1.1.1" xref="alg3.l6.m1.2.2.1.1.2.cmml"><mo id="alg3.l6.m1.2.2.1.1.1.2" stretchy="false" xref="alg3.l6.m1.2.2.1.1.2.cmml">(</mo><mn id="alg3.l6.m1.1.1" xref="alg3.l6.m1.1.1.cmml">0</mn><mo id="alg3.l6.m1.2.2.1.1.1.3" xref="alg3.l6.m1.2.2.1.1.2.cmml">,</mo><msub id="alg3.l6.m1.2.2.1.1.1.1" xref="alg3.l6.m1.2.2.1.1.1.1.cmml"><mi id="alg3.l6.m1.2.2.1.1.1.1.2" xref="alg3.l6.m1.2.2.1.1.1.1.2.cmml">𝐈</mi><mi id="alg3.l6.m1.2.2.1.1.1.1.3" xref="alg3.l6.m1.2.2.1.1.1.1.3.cmml">d</mi></msub><mo id="alg3.l6.m1.2.2.1.1.1.4" stretchy="false" xref="alg3.l6.m1.2.2.1.1.2.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="alg3.l6.m1.2b"><apply id="alg3.l6.m1.2.2.cmml" xref="alg3.l6.m1.2.2"><ci id="alg3.l6.m1.2.2.2.cmml" xref="alg3.l6.m1.2.2.2">←</ci><ci id="alg3.l6.m1.2.2.3.cmml" xref="alg3.l6.m1.2.2.3">bold-italic-ϵ</ci><apply id="alg3.l6.m1.2.2.1.cmml" xref="alg3.l6.m1.2.2.1"><times id="alg3.l6.m1.2.2.1.2.cmml" xref="alg3.l6.m1.2.2.1.2"></times><ci id="alg3.l6.m1.2.2.1.3.cmml" xref="alg3.l6.m1.2.2.1.3">𝒩</ci><interval closure="open" id="alg3.l6.m1.2.2.1.1.2.cmml" xref="alg3.l6.m1.2.2.1.1.1"><cn id="alg3.l6.m1.1.1.cmml" type="integer" xref="alg3.l6.m1.1.1">0</cn><apply id="alg3.l6.m1.2.2.1.1.1.1.cmml" xref="alg3.l6.m1.2.2.1.1.1.1"><csymbol cd="ambiguous" id="alg3.l6.m1.2.2.1.1.1.1.1.cmml" xref="alg3.l6.m1.2.2.1.1.1.1">subscript</csymbol><ci id="alg3.l6.m1.2.2.1.1.1.1.2.cmml" xref="alg3.l6.m1.2.2.1.1.1.1.2">𝐈</ci><ci id="alg3.l6.m1.2.2.1.1.1.1.3.cmml" xref="alg3.l6.m1.2.2.1.1.1.1.3">𝑑</ci></apply></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="alg3.l6.m1.2c">{\boldsymbol{\epsilon}}\leftarrow\mathcal{N}(0,\mathbf{I}_{d})</annotation><annotation encoding="application/x-llamapun" id="alg3.l6.m1.2d">bold_italic_ϵ ← caligraphic_N ( 0 , bold_I start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT )</annotation></semantics></math> </div> <div class="ltx_listingline" id="alg3.l7"> <span class="ltx_tag ltx_tag_listingline"><span class="ltx_text" id="alg3.l7.1.1.1" style="font-size:80%;">7:</span></span> <math alttext="{\tilde{\mathbf{x}}}_{k}\leftarrow{\mathbf{x}}_{k}+\sigma_{i}{\boldsymbol{% \epsilon}}" class="ltx_Math" display="inline" id="alg3.l7.m1.1"><semantics id="alg3.l7.m1.1a"><mrow id="alg3.l7.m1.1.1" xref="alg3.l7.m1.1.1.cmml"><msub id="alg3.l7.m1.1.1.2" xref="alg3.l7.m1.1.1.2.cmml"><mover accent="true" id="alg3.l7.m1.1.1.2.2" xref="alg3.l7.m1.1.1.2.2.cmml"><mi id="alg3.l7.m1.1.1.2.2.2" xref="alg3.l7.m1.1.1.2.2.2.cmml">𝐱</mi><mo id="alg3.l7.m1.1.1.2.2.1" xref="alg3.l7.m1.1.1.2.2.1.cmml">~</mo></mover><mi id="alg3.l7.m1.1.1.2.3" xref="alg3.l7.m1.1.1.2.3.cmml">k</mi></msub><mo id="alg3.l7.m1.1.1.1" stretchy="false" xref="alg3.l7.m1.1.1.1.cmml">←</mo><mrow id="alg3.l7.m1.1.1.3" xref="alg3.l7.m1.1.1.3.cmml"><msub id="alg3.l7.m1.1.1.3.2" xref="alg3.l7.m1.1.1.3.2.cmml"><mi id="alg3.l7.m1.1.1.3.2.2" xref="alg3.l7.m1.1.1.3.2.2.cmml">𝐱</mi><mi id="alg3.l7.m1.1.1.3.2.3" xref="alg3.l7.m1.1.1.3.2.3.cmml">k</mi></msub><mo id="alg3.l7.m1.1.1.3.1" xref="alg3.l7.m1.1.1.3.1.cmml">+</mo><mrow id="alg3.l7.m1.1.1.3.3" xref="alg3.l7.m1.1.1.3.3.cmml"><msub id="alg3.l7.m1.1.1.3.3.2" xref="alg3.l7.m1.1.1.3.3.2.cmml"><mi id="alg3.l7.m1.1.1.3.3.2.2" xref="alg3.l7.m1.1.1.3.3.2.2.cmml">σ</mi><mi id="alg3.l7.m1.1.1.3.3.2.3" xref="alg3.l7.m1.1.1.3.3.2.3.cmml">i</mi></msub><mo id="alg3.l7.m1.1.1.3.3.1" xref="alg3.l7.m1.1.1.3.3.1.cmml">⁢</mo><mi id="alg3.l7.m1.1.1.3.3.3" mathvariant="bold-italic" xref="alg3.l7.m1.1.1.3.3.3.cmml">ϵ</mi></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="alg3.l7.m1.1b"><apply id="alg3.l7.m1.1.1.cmml" xref="alg3.l7.m1.1.1"><ci id="alg3.l7.m1.1.1.1.cmml" xref="alg3.l7.m1.1.1.1">←</ci><apply id="alg3.l7.m1.1.1.2.cmml" xref="alg3.l7.m1.1.1.2"><csymbol cd="ambiguous" id="alg3.l7.m1.1.1.2.1.cmml" xref="alg3.l7.m1.1.1.2">subscript</csymbol><apply id="alg3.l7.m1.1.1.2.2.cmml" xref="alg3.l7.m1.1.1.2.2"><ci id="alg3.l7.m1.1.1.2.2.1.cmml" xref="alg3.l7.m1.1.1.2.2.1">~</ci><ci id="alg3.l7.m1.1.1.2.2.2.cmml" xref="alg3.l7.m1.1.1.2.2.2">𝐱</ci></apply><ci id="alg3.l7.m1.1.1.2.3.cmml" xref="alg3.l7.m1.1.1.2.3">𝑘</ci></apply><apply id="alg3.l7.m1.1.1.3.cmml" xref="alg3.l7.m1.1.1.3"><plus id="alg3.l7.m1.1.1.3.1.cmml" xref="alg3.l7.m1.1.1.3.1"></plus><apply id="alg3.l7.m1.1.1.3.2.cmml" xref="alg3.l7.m1.1.1.3.2"><csymbol cd="ambiguous" id="alg3.l7.m1.1.1.3.2.1.cmml" xref="alg3.l7.m1.1.1.3.2">subscript</csymbol><ci id="alg3.l7.m1.1.1.3.2.2.cmml" xref="alg3.l7.m1.1.1.3.2.2">𝐱</ci><ci id="alg3.l7.m1.1.1.3.2.3.cmml" xref="alg3.l7.m1.1.1.3.2.3">𝑘</ci></apply><apply id="alg3.l7.m1.1.1.3.3.cmml" xref="alg3.l7.m1.1.1.3.3"><times id="alg3.l7.m1.1.1.3.3.1.cmml" xref="alg3.l7.m1.1.1.3.3.1"></times><apply id="alg3.l7.m1.1.1.3.3.2.cmml" xref="alg3.l7.m1.1.1.3.3.2"><csymbol cd="ambiguous" id="alg3.l7.m1.1.1.3.3.2.1.cmml" xref="alg3.l7.m1.1.1.3.3.2">subscript</csymbol><ci id="alg3.l7.m1.1.1.3.3.2.2.cmml" xref="alg3.l7.m1.1.1.3.3.2.2">𝜎</ci><ci id="alg3.l7.m1.1.1.3.3.2.3.cmml" xref="alg3.l7.m1.1.1.3.3.2.3">𝑖</ci></apply><ci id="alg3.l7.m1.1.1.3.3.3.cmml" xref="alg3.l7.m1.1.1.3.3.3">bold-italic-ϵ</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="alg3.l7.m1.1c">{\tilde{\mathbf{x}}}_{k}\leftarrow{\mathbf{x}}_{k}+\sigma_{i}{\boldsymbol{% \epsilon}}</annotation><annotation encoding="application/x-llamapun" id="alg3.l7.m1.1d">over~ start_ARG bold_x end_ARG start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ← bold_x start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT + italic_σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT bold_italic_ϵ</annotation></semantics></math> </div> <div class="ltx_listingline" id="alg3.l8"> <span class="ltx_tag ltx_tag_listingline"><span class="ltx_text" id="alg3.l8.1.1.1" style="font-size:80%;">8:</span></span> <math alttext="{\mathbf{x}}_{k+1}\leftarrow{\mathbf{x}}_{k}-\delta\nabla\mathcal{F}({\mathbf{% x}}_{k},{\mathbf{y}})-\frac{\alpha_{i}\delta}{\sigma_{i}^{2}}\left({\mathbf{x}% }_{k}-D_{\sigma_{i}}({\tilde{\mathbf{x}}}_{k})\right)" class="ltx_Math" display="inline" id="alg3.l8.m1.3"><semantics id="alg3.l8.m1.3a"><mrow id="alg3.l8.m1.3.3" xref="alg3.l8.m1.3.3.cmml"><msub id="alg3.l8.m1.3.3.4" xref="alg3.l8.m1.3.3.4.cmml"><mi id="alg3.l8.m1.3.3.4.2" xref="alg3.l8.m1.3.3.4.2.cmml">𝐱</mi><mrow id="alg3.l8.m1.3.3.4.3" xref="alg3.l8.m1.3.3.4.3.cmml"><mi id="alg3.l8.m1.3.3.4.3.2" xref="alg3.l8.m1.3.3.4.3.2.cmml">k</mi><mo id="alg3.l8.m1.3.3.4.3.1" xref="alg3.l8.m1.3.3.4.3.1.cmml">+</mo><mn id="alg3.l8.m1.3.3.4.3.3" xref="alg3.l8.m1.3.3.4.3.3.cmml">1</mn></mrow></msub><mo id="alg3.l8.m1.3.3.3" stretchy="false" xref="alg3.l8.m1.3.3.3.cmml">←</mo><mrow id="alg3.l8.m1.3.3.2" xref="alg3.l8.m1.3.3.2.cmml"><msub id="alg3.l8.m1.3.3.2.4" xref="alg3.l8.m1.3.3.2.4.cmml"><mi id="alg3.l8.m1.3.3.2.4.2" xref="alg3.l8.m1.3.3.2.4.2.cmml">𝐱</mi><mi id="alg3.l8.m1.3.3.2.4.3" xref="alg3.l8.m1.3.3.2.4.3.cmml">k</mi></msub><mo id="alg3.l8.m1.3.3.2.3" xref="alg3.l8.m1.3.3.2.3.cmml">−</mo><mrow id="alg3.l8.m1.2.2.1.1" xref="alg3.l8.m1.2.2.1.1.cmml"><mi 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id="alg3.l8.m1.3.3.2.2.1.1.1.1.1.1.1.2.2.cmml" xref="alg3.l8.m1.3.3.2.2.1.1.1.1.1.1.1.2.2">𝐱</ci></apply><ci id="alg3.l8.m1.3.3.2.2.1.1.1.1.1.1.1.3.cmml" xref="alg3.l8.m1.3.3.2.2.1.1.1.1.1.1.1.3">𝑘</ci></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="alg3.l8.m1.3c">{\mathbf{x}}_{k+1}\leftarrow{\mathbf{x}}_{k}-\delta\nabla\mathcal{F}({\mathbf{% x}}_{k},{\mathbf{y}})-\frac{\alpha_{i}\delta}{\sigma_{i}^{2}}\left({\mathbf{x}% }_{k}-D_{\sigma_{i}}({\tilde{\mathbf{x}}}_{k})\right)</annotation><annotation encoding="application/x-llamapun" id="alg3.l8.m1.3d">bold_x start_POSTSUBSCRIPT italic_k + 1 end_POSTSUBSCRIPT ← bold_x start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT - italic_δ ∇ caligraphic_F ( bold_x start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT , bold_y ) - divide start_ARG italic_α start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_δ end_ARG start_ARG italic_σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG ( bold_x start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT - italic_D start_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( over~ start_ARG bold_x end_ARG start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) )</annotation></semantics></math> </div> <div class="ltx_listingline" id="alg3.l9"> <span class="ltx_tag ltx_tag_listingline"><span class="ltx_text" id="alg3.l9.1.1.1" style="font-size:80%;">9:</span></span> <span class="ltx_text ltx_font_bold" id="alg3.l9.2">end</span> <span class="ltx_text ltx_font_bold" id="alg3.l9.3">for</span> </div> <div class="ltx_listingline" id="alg3.l10"> <span class="ltx_tag ltx_tag_listingline"><span class="ltx_text" id="alg3.l10.1.1.1" style="font-size:80%;">10:</span></span> <span class="ltx_text ltx_font_bold" id="alg3.l10.2">end</span> <span class="ltx_text ltx_font_bold" id="alg3.l10.3">for</span> </div> </div> </figure> <div class="ltx_para" id="S2.SS2.SSS0.Px2.p3"> <p class="ltx_p" id="S2.SS2.SSS0.Px2.p3.1">Inspired by annealed importance sampling <cite class="ltx_cite ltx_citemacro_citep">(Neal, <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#bib.bib50" title="">1998</a>)</cite> and the recent use of such a decreasing of <math alttext="\sigma" class="ltx_Math" display="inline" id="S2.SS2.SSS0.Px2.p3.1.m1.1"><semantics id="S2.SS2.SSS0.Px2.p3.1.m1.1a"><mi id="S2.SS2.SSS0.Px2.p3.1.m1.1.1" xref="S2.SS2.SSS0.Px2.p3.1.m1.1.1.cmml">σ</mi><annotation-xml encoding="MathML-Content" id="S2.SS2.SSS0.Px2.p3.1.m1.1b"><ci id="S2.SS2.SSS0.Px2.p3.1.m1.1.1.cmml" xref="S2.SS2.SSS0.Px2.p3.1.m1.1.1">𝜎</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.SSS0.Px2.p3.1.m1.1c">\sigma</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.SSS0.Px2.p3.1.m1.1d">italic_σ</annotation></semantics></math> in diffusion model <cite class="ltx_cite ltx_citemacro_citep">(Sun et al., <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#bib.bib71" title="">2023</a>)</cite>, we also develop an Annealed SNORE Algorithm (Algorithm <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#alg3" title="Algorithm 3 ‣ Optimization algorithms ‣ 2.2 SNORE regularization ‣ 2 Stochastic deNOising REgularization (SNORE) ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_tag">3</span></a>). This algorithm, which proves more efficient in practice, is supported by a critical point analysis (Proposition <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#S3.Thmproposition2" title="Proposition 3.2. ‣ 3.1 Asymptotics of critical points when 𝜎→0 ‣ 3 Convergence Analysis ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_tag">3.2</span></a>).</p> </div> </section> </section> <section class="ltx_subsection" id="S2.SS3"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">2.3 </span>Related Works</h3> <section class="ltx_paragraph" id="S2.SS3.SSS0.Px1"> <h5 class="ltx_title ltx_title_paragraph">Other Stochastic Plug-and-Play algorithms</h5> <div class="ltx_para" id="S2.SS3.SSS0.Px1.p1"> <p class="ltx_p" id="S2.SS3.SSS0.Px1.p1.1">In the existing literature, stochastic versions of Plug-and-Play have already been proposed. Most of these works intend to accelerate the computation by a stochastic mini-batch approximation on the data-fidelity <cite class="ltx_cite ltx_citemacro_citep">(Tang & Davies, <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#bib.bib73" title="">2020</a>)</cite> or the regularization <cite class="ltx_cite ltx_citemacro_citep">(Sun et al., <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#bib.bib69" title="">2019a</a>)</cite>. On the other hand, SNORE does not aim at accelerating PnP algorithms but it proposes a stochastic improvement of PnP by injecting noise inside the classical PnP regularization.</p> </div> <div class="ltx_para" id="S2.SS3.SSS0.Px1.p2"> <p class="ltx_p" id="S2.SS3.SSS0.Px1.p2.1"><cite class="ltx_cite ltx_citemacro_citet">Laumont et al. (<a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#bib.bib37" title="">2022b</a>)</cite> propose to run a stochastic gradient descent algorithm (PnP SGD) with the PnP regularization. Contrary to PnP SGD, SNORE injects the noise inside the denoiser only, and not in the data-fidelity term. Moreover, in SNORE, the standard deviation of the injected noise is fixed (for fixed <math alttext="\sigma" class="ltx_Math" display="inline" id="S2.SS3.SSS0.Px1.p2.1.m1.1"><semantics id="S2.SS3.SSS0.Px1.p2.1.m1.1a"><mi id="S2.SS3.SSS0.Px1.p2.1.m1.1.1" xref="S2.SS3.SSS0.Px1.p2.1.m1.1.1.cmml">σ</mi><annotation-xml encoding="MathML-Content" id="S2.SS3.SSS0.Px1.p2.1.m1.1b"><ci id="S2.SS3.SSS0.Px1.p2.1.m1.1.1.cmml" xref="S2.SS3.SSS0.Px1.p2.1.m1.1.1">𝜎</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.SSS0.Px1.p2.1.m1.1c">\sigma</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.SSS0.Px1.p2.1.m1.1d">italic_σ</annotation></semantics></math>).</p> </div> <div class="ltx_para" id="S2.SS3.SSS0.Px1.p3"> <p class="ltx_p" id="S2.SS3.SSS0.Px1.p3.1">Another line of works target image restoration by sampling the posterior law instead of solving Problem <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#S1.E1" title="1 ‣ 1 Introduction ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_tag">1</span></a>.. This can be done within a PnP framework using the Tweedie formula (Equation <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#S2.E3" title="3 ‣ 2.1 Background ‣ 2 Stochastic deNOising REgularization (SNORE) ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_tag">3</span></a>) to compute a Langevin dynamic <cite class="ltx_cite ltx_citemacro_citep">(Laumont et al., <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#bib.bib36" title="">2022a</a>; Renaud et al., <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#bib.bib57" title="">2023</a>)</cite> or a Gibbs sampling <cite class="ltx_cite ltx_citemacro_citep">(Coeurdoux et al., <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#bib.bib13" title="">2023</a>; Bouman & Buzzard, <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#bib.bib4" title="">2023</a>)</cite>.</p> </div> </section> <section class="ltx_paragraph" id="S2.SS3.SSS0.Px2"> <h5 class="ltx_title ltx_title_paragraph">Link with diffusion based method</h5> <div class="ltx_para" id="S2.SS3.SSS0.Px2.p1"> <p class="ltx_p" id="S2.SS3.SSS0.Px2.p1.2">Denoising diffusion models (DDM) are a class of generative models that can generate images by gradually transforming noise into data with deep denoising networks <cite class="ltx_cite ltx_citemacro_citep">(Ho et al., <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#bib.bib28" title="">2020</a>; Song et al., <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#bib.bib66" title="">2021b</a>)</cite>. A key feature of DDMs is that, for the adequate weighting schedule <math alttext="w_{t}" class="ltx_Math" display="inline" id="S2.SS3.SSS0.Px2.p1.1.m1.1"><semantics id="S2.SS3.SSS0.Px2.p1.1.m1.1a"><msub id="S2.SS3.SSS0.Px2.p1.1.m1.1.1" xref="S2.SS3.SSS0.Px2.p1.1.m1.1.1.cmml"><mi id="S2.SS3.SSS0.Px2.p1.1.m1.1.1.2" xref="S2.SS3.SSS0.Px2.p1.1.m1.1.1.2.cmml">w</mi><mi id="S2.SS3.SSS0.Px2.p1.1.m1.1.1.3" xref="S2.SS3.SSS0.Px2.p1.1.m1.1.1.3.cmml">t</mi></msub><annotation-xml encoding="MathML-Content" id="S2.SS3.SSS0.Px2.p1.1.m1.1b"><apply id="S2.SS3.SSS0.Px2.p1.1.m1.1.1.cmml" xref="S2.SS3.SSS0.Px2.p1.1.m1.1.1"><csymbol cd="ambiguous" id="S2.SS3.SSS0.Px2.p1.1.m1.1.1.1.cmml" xref="S2.SS3.SSS0.Px2.p1.1.m1.1.1">subscript</csymbol><ci id="S2.SS3.SSS0.Px2.p1.1.m1.1.1.2.cmml" xref="S2.SS3.SSS0.Px2.p1.1.m1.1.1.2">𝑤</ci><ci id="S2.SS3.SSS0.Px2.p1.1.m1.1.1.3.cmml" xref="S2.SS3.SSS0.Px2.p1.1.m1.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.SSS0.Px2.p1.1.m1.1c">w_{t}</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.SSS0.Px2.p1.1.m1.1d">italic_w start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT</annotation></semantics></math>, the log-likelihood of the generative model <math alttext="p_{\theta}({\mathbf{x}})" class="ltx_Math" display="inline" id="S2.SS3.SSS0.Px2.p1.2.m2.1"><semantics id="S2.SS3.SSS0.Px2.p1.2.m2.1a"><mrow id="S2.SS3.SSS0.Px2.p1.2.m2.1.2" xref="S2.SS3.SSS0.Px2.p1.2.m2.1.2.cmml"><msub id="S2.SS3.SSS0.Px2.p1.2.m2.1.2.2" xref="S2.SS3.SSS0.Px2.p1.2.m2.1.2.2.cmml"><mi id="S2.SS3.SSS0.Px2.p1.2.m2.1.2.2.2" xref="S2.SS3.SSS0.Px2.p1.2.m2.1.2.2.2.cmml">p</mi><mi id="S2.SS3.SSS0.Px2.p1.2.m2.1.2.2.3" xref="S2.SS3.SSS0.Px2.p1.2.m2.1.2.2.3.cmml">θ</mi></msub><mo id="S2.SS3.SSS0.Px2.p1.2.m2.1.2.1" xref="S2.SS3.SSS0.Px2.p1.2.m2.1.2.1.cmml">⁢</mo><mrow id="S2.SS3.SSS0.Px2.p1.2.m2.1.2.3.2" xref="S2.SS3.SSS0.Px2.p1.2.m2.1.2.cmml"><mo id="S2.SS3.SSS0.Px2.p1.2.m2.1.2.3.2.1" stretchy="false" xref="S2.SS3.SSS0.Px2.p1.2.m2.1.2.cmml">(</mo><mi id="S2.SS3.SSS0.Px2.p1.2.m2.1.1" xref="S2.SS3.SSS0.Px2.p1.2.m2.1.1.cmml">𝐱</mi><mo id="S2.SS3.SSS0.Px2.p1.2.m2.1.2.3.2.2" stretchy="false" xref="S2.SS3.SSS0.Px2.p1.2.m2.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.SSS0.Px2.p1.2.m2.1b"><apply id="S2.SS3.SSS0.Px2.p1.2.m2.1.2.cmml" xref="S2.SS3.SSS0.Px2.p1.2.m2.1.2"><times id="S2.SS3.SSS0.Px2.p1.2.m2.1.2.1.cmml" xref="S2.SS3.SSS0.Px2.p1.2.m2.1.2.1"></times><apply id="S2.SS3.SSS0.Px2.p1.2.m2.1.2.2.cmml" xref="S2.SS3.SSS0.Px2.p1.2.m2.1.2.2"><csymbol cd="ambiguous" id="S2.SS3.SSS0.Px2.p1.2.m2.1.2.2.1.cmml" xref="S2.SS3.SSS0.Px2.p1.2.m2.1.2.2">subscript</csymbol><ci id="S2.SS3.SSS0.Px2.p1.2.m2.1.2.2.2.cmml" xref="S2.SS3.SSS0.Px2.p1.2.m2.1.2.2.2">𝑝</ci><ci id="S2.SS3.SSS0.Px2.p1.2.m2.1.2.2.3.cmml" xref="S2.SS3.SSS0.Px2.p1.2.m2.1.2.2.3">𝜃</ci></apply><ci id="S2.SS3.SSS0.Px2.p1.2.m2.1.1.cmml" xref="S2.SS3.SSS0.Px2.p1.2.m2.1.1">𝐱</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.SSS0.Px2.p1.2.m2.1c">p_{\theta}({\mathbf{x}})</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.SSS0.Px2.p1.2.m2.1d">italic_p start_POSTSUBSCRIPT italic_θ end_POSTSUBSCRIPT ( bold_x )</annotation></semantics></math> is lower-bounded by a (negative) mixture of denoising losses at different noise levels <cite class="ltx_cite ltx_citemacro_citep">(Ho et al., <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#bib.bib28" title="">2020</a>; Song et al., <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#bib.bib65" title="">2021a</a>)</cite>:</p> <table class="ltx_equation ltx_eqn_table" id="S2.E10"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\log 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id="S2.E10.m1.7.8.2.3.2.1" stretchy="false" xref="S2.E10.m1.7.8.2.cmml">(</mo><mi id="S2.E10.m1.7.7" xref="S2.E10.m1.7.7.cmml">𝐱</mi><mo id="S2.E10.m1.7.8.2.3.2.2" stretchy="false" xref="S2.E10.m1.7.8.2.cmml">)</mo></mrow></mrow><mo id="S2.E10.m1.7.8.1" xref="S2.E10.m1.7.8.1.cmml">≥</mo><munder id="S2.E10.m1.7.8.3" xref="S2.E10.m1.7.8.3.cmml"><munder accentunder="true" id="S2.E10.m1.5.5" xref="S2.E10.m1.5.5.cmml"><mrow id="S2.E10.m1.5.5.5" xref="S2.E10.m1.5.5.5.cmml"><mo id="S2.E10.m1.5.5.5a" xref="S2.E10.m1.5.5.5.cmml">−</mo><mrow id="S2.E10.m1.5.5.5.5" xref="S2.E10.m1.5.5.5.5.cmml"><msub id="S2.E10.m1.5.5.5.5.3" xref="S2.E10.m1.5.5.5.5.3.cmml"><mi id="S2.E10.m1.5.5.5.5.3.2" xref="S2.E10.m1.5.5.5.5.3.2.cmml">𝔼</mi><mrow id="S2.E10.m1.3.3.3.3.3" xref="S2.E10.m1.3.3.3.3.3.cmml"><mrow id="S2.E10.m1.3.3.3.3.3.5.2" xref="S2.E10.m1.3.3.3.3.3.5.1.cmml"><mi id="S2.E10.m1.1.1.1.1.1.1" xref="S2.E10.m1.1.1.1.1.1.1.cmml">t</mi><mo id="S2.E10.m1.3.3.3.3.3.5.2.1" 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xref="S2.E10.m1.5.5.5.5.1.1.1.1.3">2</cn></apply></apply></apply></apply></apply></apply><apply id="S2.E10.m1.6.6.1.cmml" xref="S2.E10.m1.6.6.1"><times id="S2.E10.m1.6.6.1.2.cmml" xref="S2.E10.m1.6.6.1.2"></times><ci id="S2.E10.m1.6.6.1.3.cmml" xref="S2.E10.m1.6.6.1.3">ℒ</ci><ci id="S2.E10.m1.6.6.1.1.cmml" xref="S2.E10.m1.6.6.1.1">𝐱</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.E10.m1.7c">\log p_{\theta}({\mathbf{x}})\geq\underbrace{-\mathbb{E}_{t,{\tilde{\mathbf{x}% }}\sim p_{\sigma}({\tilde{\mathbf{x}}}|{\mathbf{x}})}\left[w_{t}\|{\mathbf{x}}% -D_{\sigma_{t}}({\tilde{\mathbf{x}}})\|^{2}\right]}_{\mathcal{L}\left({\mathbf% {x}}\right)}</annotation><annotation encoding="application/x-llamapun" id="S2.E10.m1.7d">roman_log italic_p start_POSTSUBSCRIPT italic_θ end_POSTSUBSCRIPT ( bold_x ) ≥ under⏟ start_ARG - blackboard_E start_POSTSUBSCRIPT italic_t , over~ start_ARG bold_x end_ARG ∼ italic_p start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ( over~ start_ARG bold_x end_ARG | bold_x ) end_POSTSUBSCRIPT [ italic_w start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT ∥ bold_x - italic_D start_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( over~ start_ARG bold_x end_ARG ) ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ] end_ARG start_POSTSUBSCRIPT caligraphic_L ( bold_x ) end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(10)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S2.SS3.SSS0.Px2.p1.5">In order to use DDMs for regularizing inverse problems, several works have proposed to replace the intractable log-likelihood by its lower-bound (<a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#S2.E10" title="10 ‣ Link with diffusion based method ‣ 2.3 Related Works ‣ 2 Stochastic deNOising REgularization (SNORE) ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_tag">10</span></a>) <cite class="ltx_cite ltx_citemacro_citep">(Poole et al., <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#bib.bib54" title="">2022</a>; Wang et al., <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#bib.bib76" title="">2023</a>; Feng & Bouman, <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#bib.bib23" title="">2023</a>; Mardani et al., <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#bib.bib45" title="">2023</a>)</cite>. In particular, the gradient of the lower-bound is:</p> <table class="ltx_equation ltx_eqn_table" id="S2.E11"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\nabla_{{\mathbf{x}}}\mathcal{L}\left({\mathbf{x}}\right)=\mathbb{E}_{t,{% \tilde{\mathbf{x}}}\sim p_{\sigma}({\tilde{\mathbf{x}}}|{\mathbf{x}})}\left[% \frac{w_{t}}{2}J_{D_{\sigma_{t}}}^{\top}\left({\mathbf{x}}-D_{\sigma_{t}}({% \tilde{\mathbf{x}}})\right)^{2}\right]" class="ltx_Math" display="block" id="S2.E11.m1.6"><semantics id="S2.E11.m1.6a"><mrow id="S2.E11.m1.6.6" xref="S2.E11.m1.6.6.cmml"><mrow id="S2.E11.m1.6.6.3" xref="S2.E11.m1.6.6.3.cmml"><mrow id="S2.E11.m1.6.6.3.2" xref="S2.E11.m1.6.6.3.2.cmml"><msub id="S2.E11.m1.6.6.3.2.1" xref="S2.E11.m1.6.6.3.2.1.cmml"><mo id="S2.E11.m1.6.6.3.2.1.2" xref="S2.E11.m1.6.6.3.2.1.2.cmml">∇</mo><mi id="S2.E11.m1.6.6.3.2.1.3" 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\tilde{\mathbf{x}}})\right)^{2}\right]</annotation><annotation encoding="application/x-llamapun" id="S2.E11.m1.6d">∇ start_POSTSUBSCRIPT bold_x end_POSTSUBSCRIPT caligraphic_L ( bold_x ) = blackboard_E start_POSTSUBSCRIPT italic_t , over~ start_ARG bold_x end_ARG ∼ italic_p start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ( over~ start_ARG bold_x end_ARG | bold_x ) end_POSTSUBSCRIPT [ divide start_ARG italic_w start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT end_ARG start_ARG 2 end_ARG italic_J start_POSTSUBSCRIPT italic_D start_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT end_POSTSUBSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ⊤ end_POSTSUPERSCRIPT ( bold_x - italic_D start_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( over~ start_ARG bold_x end_ARG ) ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ]</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(11)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S2.SS3.SSS0.Px2.p1.4">where <math alttext="J_{D_{\sigma_{t}}}" class="ltx_Math" display="inline" id="S2.SS3.SSS0.Px2.p1.3.m1.1"><semantics id="S2.SS3.SSS0.Px2.p1.3.m1.1a"><msub id="S2.SS3.SSS0.Px2.p1.3.m1.1.1" xref="S2.SS3.SSS0.Px2.p1.3.m1.1.1.cmml"><mi id="S2.SS3.SSS0.Px2.p1.3.m1.1.1.2" xref="S2.SS3.SSS0.Px2.p1.3.m1.1.1.2.cmml">J</mi><msub id="S2.SS3.SSS0.Px2.p1.3.m1.1.1.3" xref="S2.SS3.SSS0.Px2.p1.3.m1.1.1.3.cmml"><mi id="S2.SS3.SSS0.Px2.p1.3.m1.1.1.3.2" xref="S2.SS3.SSS0.Px2.p1.3.m1.1.1.3.2.cmml">D</mi><msub id="S2.SS3.SSS0.Px2.p1.3.m1.1.1.3.3" xref="S2.SS3.SSS0.Px2.p1.3.m1.1.1.3.3.cmml"><mi id="S2.SS3.SSS0.Px2.p1.3.m1.1.1.3.3.2" xref="S2.SS3.SSS0.Px2.p1.3.m1.1.1.3.3.2.cmml">σ</mi><mi id="S2.SS3.SSS0.Px2.p1.3.m1.1.1.3.3.3" xref="S2.SS3.SSS0.Px2.p1.3.m1.1.1.3.3.3.cmml">t</mi></msub></msub></msub><annotation-xml encoding="MathML-Content" id="S2.SS3.SSS0.Px2.p1.3.m1.1b"><apply id="S2.SS3.SSS0.Px2.p1.3.m1.1.1.cmml" xref="S2.SS3.SSS0.Px2.p1.3.m1.1.1"><csymbol cd="ambiguous" id="S2.SS3.SSS0.Px2.p1.3.m1.1.1.1.cmml" xref="S2.SS3.SSS0.Px2.p1.3.m1.1.1">subscript</csymbol><ci id="S2.SS3.SSS0.Px2.p1.3.m1.1.1.2.cmml" xref="S2.SS3.SSS0.Px2.p1.3.m1.1.1.2">𝐽</ci><apply id="S2.SS3.SSS0.Px2.p1.3.m1.1.1.3.cmml" xref="S2.SS3.SSS0.Px2.p1.3.m1.1.1.3"><csymbol cd="ambiguous" id="S2.SS3.SSS0.Px2.p1.3.m1.1.1.3.1.cmml" xref="S2.SS3.SSS0.Px2.p1.3.m1.1.1.3">subscript</csymbol><ci id="S2.SS3.SSS0.Px2.p1.3.m1.1.1.3.2.cmml" xref="S2.SS3.SSS0.Px2.p1.3.m1.1.1.3.2">𝐷</ci><apply id="S2.SS3.SSS0.Px2.p1.3.m1.1.1.3.3.cmml" xref="S2.SS3.SSS0.Px2.p1.3.m1.1.1.3.3"><csymbol cd="ambiguous" id="S2.SS3.SSS0.Px2.p1.3.m1.1.1.3.3.1.cmml" xref="S2.SS3.SSS0.Px2.p1.3.m1.1.1.3.3">subscript</csymbol><ci id="S2.SS3.SSS0.Px2.p1.3.m1.1.1.3.3.2.cmml" xref="S2.SS3.SSS0.Px2.p1.3.m1.1.1.3.3.2">𝜎</ci><ci id="S2.SS3.SSS0.Px2.p1.3.m1.1.1.3.3.3.cmml" xref="S2.SS3.SSS0.Px2.p1.3.m1.1.1.3.3.3">𝑡</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.SSS0.Px2.p1.3.m1.1c">J_{D_{\sigma_{t}}}</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.SSS0.Px2.p1.3.m1.1d">italic_J start_POSTSUBSCRIPT italic_D start_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math> is the Jacobian matrix of the denoiser with respect to the input. Hence, the gradient of DDMs lower-bound (<a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#S2.E11" title="11 ‣ Link with diffusion based method ‣ 2.3 Related Works ‣ 2 Stochastic deNOising REgularization (SNORE) ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_tag">11</span></a>) has a similar formulation than the gradient of our regularization function (<a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#S2.E6" title="6 ‣ 2.2 SNORE regularization ‣ 2 Stochastic deNOising REgularization (SNORE) ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_tag">6</span></a>), with the difference that it includes the Jacobian of the denoiser, and it is averaged over multiple noise levels <math alttext="\sigma_{t}" class="ltx_Math" display="inline" id="S2.SS3.SSS0.Px2.p1.4.m2.1"><semantics id="S2.SS3.SSS0.Px2.p1.4.m2.1a"><msub id="S2.SS3.SSS0.Px2.p1.4.m2.1.1" xref="S2.SS3.SSS0.Px2.p1.4.m2.1.1.cmml"><mi id="S2.SS3.SSS0.Px2.p1.4.m2.1.1.2" xref="S2.SS3.SSS0.Px2.p1.4.m2.1.1.2.cmml">σ</mi><mi id="S2.SS3.SSS0.Px2.p1.4.m2.1.1.3" xref="S2.SS3.SSS0.Px2.p1.4.m2.1.1.3.cmml">t</mi></msub><annotation-xml encoding="MathML-Content" id="S2.SS3.SSS0.Px2.p1.4.m2.1b"><apply id="S2.SS3.SSS0.Px2.p1.4.m2.1.1.cmml" xref="S2.SS3.SSS0.Px2.p1.4.m2.1.1"><csymbol cd="ambiguous" id="S2.SS3.SSS0.Px2.p1.4.m2.1.1.1.cmml" xref="S2.SS3.SSS0.Px2.p1.4.m2.1.1">subscript</csymbol><ci id="S2.SS3.SSS0.Px2.p1.4.m2.1.1.2.cmml" xref="S2.SS3.SSS0.Px2.p1.4.m2.1.1.2">𝜎</ci><ci id="S2.SS3.SSS0.Px2.p1.4.m2.1.1.3.cmml" xref="S2.SS3.SSS0.Px2.p1.4.m2.1.1.3">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.SSS0.Px2.p1.4.m2.1c">\sigma_{t}</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.SSS0.Px2.p1.4.m2.1d">italic_σ start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT</annotation></semantics></math>. It has been found that removing the Jacobian in (<a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#S2.E11" title="11 ‣ Link with diffusion based method ‣ 2.3 Related Works ‣ 2 Stochastic deNOising REgularization (SNORE) ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_tag">11</span></a>) yields better results in practice <cite class="ltx_cite ltx_citemacro_citep">(Poole et al., <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#bib.bib54" title="">2022</a>)</cite>. Several works propose theoretical justifications to omit the Jacobian matrix <cite class="ltx_cite ltx_citemacro_citep">(Wang et al., <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#bib.bib76" title="">2023</a>; Mardani et al., <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#bib.bib45" title="">2023</a>)</cite>, by assuming that the denoising network provides the exact score of some prior function. Although we rely on a similar assumption, our theoretical analysis in Section <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#S3.SS3" title="3.3 Biased algorithm analysis ‣ 3 Convergence Analysis ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_tag">3.3</span></a> also covers the case of an imperfect denoiser.</p> </div> <div class="ltx_para" id="S2.SS3.SSS0.Px2.p2"> <p class="ltx_p" id="S2.SS3.SSS0.Px2.p2.1">A different approach for solving inverse problem with a DDM prior is to guide the generative process of an unconditional DDM to generate images consistent with an observation <math alttext="{\mathbf{y}}" class="ltx_Math" display="inline" id="S2.SS3.SSS0.Px2.p2.1.m1.1"><semantics id="S2.SS3.SSS0.Px2.p2.1.m1.1a"><mi id="S2.SS3.SSS0.Px2.p2.1.m1.1.1" xref="S2.SS3.SSS0.Px2.p2.1.m1.1.1.cmml">𝐲</mi><annotation-xml encoding="MathML-Content" id="S2.SS3.SSS0.Px2.p2.1.m1.1b"><ci id="S2.SS3.SSS0.Px2.p2.1.m1.1.1.cmml" xref="S2.SS3.SSS0.Px2.p2.1.m1.1.1">𝐲</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.SSS0.Px2.p2.1.m1.1c">{\mathbf{y}}</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.SSS0.Px2.p2.1.m1.1d">bold_y</annotation></semantics></math> <cite class="ltx_cite ltx_citemacro_citep">(Song et al., <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#bib.bib67" title="">2022</a>; Kawar et al., <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#bib.bib34" title="">2022</a>; Chung et al., <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#bib.bib11" title="">2022</a>, <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#bib.bib12" title="">2023</a>; Song et al., <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#bib.bib63" title="">2023</a>; Luther & Seung, <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#bib.bib42" title="">2023</a>; Zhu et al., <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#bib.bib80" title="">2023</a>)</cite>. Notice that <cite class="ltx_cite ltx_citemacro_citet">Luther & Seung (<a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#bib.bib42" title="">2023</a>)</cite> proposed an algorithm similar to Algorithm <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#alg3" title="Algorithm 3 ‣ Optimization algorithms ‣ 2.2 SNORE regularization ‣ 2 Stochastic deNOising REgularization (SNORE) ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_tag">3</span></a> but do not provide strong theoretical motivation or analysis. Such reverse diffusion processes involve a gradual decrease of the strength of the denoising network, analogous to our annealing procedure. However, those approaches aim at sampling from the posterior distribution of the inverse problem, whereas we adopt a (stochastic) optimization perspective. Despite their impressive practical results, DDM guided rely on heuristics to approximate the intractable likelihood model on noisy data. The impact of the approximation error on the distribution of generated samples remains to be quantified.</p> </div> </section> </section> </section> <section class="ltx_section" id="S3"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">3 </span>Convergence Analysis</h2> <div class="ltx_para" id="S3.p1"> <p class="ltx_p" id="S3.p1.3">In this section, we provide a theoretical analysis of our regularization SNORE and a convergence analysis of the associated algorithm. Problem (<a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#S2.E9" title="9 ‣ Optimization algorithms ‣ 2.2 SNORE regularization ‣ 2 Stochastic deNOising REgularization (SNORE) ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_tag">9</span></a>) is non-convex due to our regularization. Hence, in the best-case scenario, one can only expect a convergence of the algorithm towards a critical point of the target functional <math alttext="\mathcal{J}" class="ltx_Math" display="inline" id="S3.p1.1.m1.1"><semantics id="S3.p1.1.m1.1a"><mi class="ltx_font_mathcaligraphic" id="S3.p1.1.m1.1.1" xref="S3.p1.1.m1.1.1.cmml">𝒥</mi><annotation-xml encoding="MathML-Content" id="S3.p1.1.m1.1b"><ci id="S3.p1.1.m1.1.1.cmml" xref="S3.p1.1.m1.1.1">𝒥</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.p1.1.m1.1c">\mathcal{J}</annotation><annotation encoding="application/x-llamapun" id="S3.p1.1.m1.1d">caligraphic_J</annotation></semantics></math>, defined in relation (<a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#S2.E9" title="9 ‣ Optimization algorithms ‣ 2.2 SNORE regularization ‣ 2 Stochastic deNOising REgularization (SNORE) ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_tag">9</span></a>). Note that all global and local minima of <math alttext="\mathcal{J}" class="ltx_Math" display="inline" id="S3.p1.2.m2.1"><semantics id="S3.p1.2.m2.1a"><mi class="ltx_font_mathcaligraphic" id="S3.p1.2.m2.1.1" xref="S3.p1.2.m2.1.1.cmml">𝒥</mi><annotation-xml encoding="MathML-Content" id="S3.p1.2.m2.1b"><ci id="S3.p1.2.m2.1.1.cmml" xref="S3.p1.2.m2.1.1">𝒥</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.p1.2.m2.1c">\mathcal{J}</annotation><annotation encoding="application/x-llamapun" id="S3.p1.2.m2.1d">caligraphic_J</annotation></semantics></math> are critical points of <math alttext="\mathcal{J}" class="ltx_Math" display="inline" id="S3.p1.3.m3.1"><semantics id="S3.p1.3.m3.1a"><mi class="ltx_font_mathcaligraphic" id="S3.p1.3.m3.1.1" xref="S3.p1.3.m3.1.1.cmml">𝒥</mi><annotation-xml encoding="MathML-Content" id="S3.p1.3.m3.1b"><ci id="S3.p1.3.m3.1.1.cmml" xref="S3.p1.3.m3.1.1">𝒥</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.p1.3.m3.1c">\mathcal{J}</annotation><annotation encoding="application/x-llamapun" id="S3.p1.3.m3.1d">caligraphic_J</annotation></semantics></math>. Based on the existing literature <cite class="ltx_cite ltx_citemacro_citep">(Tadic & Doucet, <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#bib.bib72" title="">2017</a>; Laumont et al., <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#bib.bib37" title="">2022b</a>)</cite>, we analyze our stochastic gradient descent in this challenging non-convex context.</p> </div> <div class="ltx_para" id="S3.p2"> <p class="ltx_p" id="S3.p2.5">We first show (Section <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#S3.SS1" title="3.1 Asymptotics of critical points when 𝜎→0 ‣ 3 Convergence Analysis ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_tag">3.1</span></a>) that our regularization <math alttext="\mathcal{R}_{\sigma}" class="ltx_Math" display="inline" id="S3.p2.1.m1.1"><semantics id="S3.p2.1.m1.1a"><msub id="S3.p2.1.m1.1.1" xref="S3.p2.1.m1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.p2.1.m1.1.1.2" xref="S3.p2.1.m1.1.1.2.cmml">ℛ</mi><mi id="S3.p2.1.m1.1.1.3" xref="S3.p2.1.m1.1.1.3.cmml">σ</mi></msub><annotation-xml encoding="MathML-Content" id="S3.p2.1.m1.1b"><apply id="S3.p2.1.m1.1.1.cmml" xref="S3.p2.1.m1.1.1"><csymbol cd="ambiguous" id="S3.p2.1.m1.1.1.1.cmml" xref="S3.p2.1.m1.1.1">subscript</csymbol><ci id="S3.p2.1.m1.1.1.2.cmml" xref="S3.p2.1.m1.1.1.2">ℛ</ci><ci id="S3.p2.1.m1.1.1.3.cmml" xref="S3.p2.1.m1.1.1.3">𝜎</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p2.1.m1.1c">\mathcal{R}_{\sigma}</annotation><annotation encoding="application/x-llamapun" id="S3.p2.1.m1.1d">caligraphic_R start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT</annotation></semantics></math> is a relevant approximation of the ideal regularization <math alttext="-\log p" class="ltx_Math" display="inline" id="S3.p2.2.m2.1"><semantics id="S3.p2.2.m2.1a"><mrow id="S3.p2.2.m2.1.1" xref="S3.p2.2.m2.1.1.cmml"><mo id="S3.p2.2.m2.1.1a" rspace="0.167em" xref="S3.p2.2.m2.1.1.cmml">−</mo><mrow id="S3.p2.2.m2.1.1.2" xref="S3.p2.2.m2.1.1.2.cmml"><mi id="S3.p2.2.m2.1.1.2.1" xref="S3.p2.2.m2.1.1.2.1.cmml">log</mi><mo id="S3.p2.2.m2.1.1.2a" lspace="0.167em" xref="S3.p2.2.m2.1.1.2.cmml">⁡</mo><mi id="S3.p2.2.m2.1.1.2.2" xref="S3.p2.2.m2.1.1.2.2.cmml">p</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.p2.2.m2.1b"><apply id="S3.p2.2.m2.1.1.cmml" xref="S3.p2.2.m2.1.1"><minus id="S3.p2.2.m2.1.1.1.cmml" xref="S3.p2.2.m2.1.1"></minus><apply id="S3.p2.2.m2.1.1.2.cmml" xref="S3.p2.2.m2.1.1.2"><log id="S3.p2.2.m2.1.1.2.1.cmml" xref="S3.p2.2.m2.1.1.2.1"></log><ci id="S3.p2.2.m2.1.1.2.2.cmml" xref="S3.p2.2.m2.1.1.2.2">𝑝</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p2.2.m2.1c">-\log p</annotation><annotation encoding="application/x-llamapun" id="S3.p2.2.m2.1d">- roman_log italic_p</annotation></semantics></math>. Then we analyze the asymptotic behavior of the critical points of Problem (<a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#S2.E9" title="9 ‣ Optimization algorithms ‣ 2.2 SNORE regularization ‣ 2 Stochastic deNOising REgularization (SNORE) ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_tag">9</span></a>) which motivates the annealing Algorithm <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#alg3" title="Algorithm 3 ‣ Optimization algorithms ‣ 2.2 SNORE regularization ‣ 2 Stochastic deNOising REgularization (SNORE) ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_tag">3</span></a>. Next we prove, at fixed <math alttext="\sigma" class="ltx_Math" display="inline" id="S3.p2.3.m3.1"><semantics id="S3.p2.3.m3.1a"><mi id="S3.p2.3.m3.1.1" xref="S3.p2.3.m3.1.1.cmml">σ</mi><annotation-xml encoding="MathML-Content" id="S3.p2.3.m3.1b"><ci id="S3.p2.3.m3.1.1.cmml" xref="S3.p2.3.m3.1.1">𝜎</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.p2.3.m3.1c">\sigma</annotation><annotation encoding="application/x-llamapun" id="S3.p2.3.m3.1d">italic_σ</annotation></semantics></math>, the convergence of Algorithm <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#alg2" title="Algorithm 2 ‣ Optimization algorithms ‣ 2.2 SNORE regularization ‣ 2 Stochastic deNOising REgularization (SNORE) ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_tag">2</span></a> to a critical point of Problem (<a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#S2.E9" title="9 ‣ Optimization algorithms ‣ 2.2 SNORE regularization ‣ 2 Stochastic deNOising REgularization (SNORE) ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_tag">9</span></a>) in the case of using the exact MMSE denoiser <math alttext="D_{\sigma}^{\star}" class="ltx_Math" display="inline" id="S3.p2.4.m4.1"><semantics id="S3.p2.4.m4.1a"><msubsup id="S3.p2.4.m4.1.1" xref="S3.p2.4.m4.1.1.cmml"><mi id="S3.p2.4.m4.1.1.2.2" xref="S3.p2.4.m4.1.1.2.2.cmml">D</mi><mi id="S3.p2.4.m4.1.1.2.3" xref="S3.p2.4.m4.1.1.2.3.cmml">σ</mi><mo id="S3.p2.4.m4.1.1.3" xref="S3.p2.4.m4.1.1.3.cmml">⋆</mo></msubsup><annotation-xml encoding="MathML-Content" id="S3.p2.4.m4.1b"><apply id="S3.p2.4.m4.1.1.cmml" xref="S3.p2.4.m4.1.1"><csymbol cd="ambiguous" id="S3.p2.4.m4.1.1.1.cmml" xref="S3.p2.4.m4.1.1">superscript</csymbol><apply id="S3.p2.4.m4.1.1.2.cmml" xref="S3.p2.4.m4.1.1"><csymbol cd="ambiguous" id="S3.p2.4.m4.1.1.2.1.cmml" xref="S3.p2.4.m4.1.1">subscript</csymbol><ci id="S3.p2.4.m4.1.1.2.2.cmml" xref="S3.p2.4.m4.1.1.2.2">𝐷</ci><ci id="S3.p2.4.m4.1.1.2.3.cmml" xref="S3.p2.4.m4.1.1.2.3">𝜎</ci></apply><ci id="S3.p2.4.m4.1.1.3.cmml" xref="S3.p2.4.m4.1.1.3">⋆</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p2.4.m4.1c">D_{\sigma}^{\star}</annotation><annotation encoding="application/x-llamapun" id="S3.p2.4.m4.1d">italic_D start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT</annotation></semantics></math> (Section <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#S3.SS2" title="3.2 Unbiased algorithm analysis ‣ 3 Convergence Analysis ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_tag">3.2</span></a>). In Section <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#S3.SS3" title="3.3 Biased algorithm analysis ‣ 3 Convergence Analysis ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_tag">3.3</span></a>, we quantify the error of Algorithm <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#alg2" title="Algorithm 2 ‣ Optimization algorithms ‣ 2.2 SNORE regularization ‣ 2 Stochastic deNOising REgularization (SNORE) ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_tag">2</span></a> with an inexact denoiser <math alttext="D_{\sigma}" class="ltx_Math" display="inline" id="S3.p2.5.m5.1"><semantics id="S3.p2.5.m5.1a"><msub id="S3.p2.5.m5.1.1" xref="S3.p2.5.m5.1.1.cmml"><mi id="S3.p2.5.m5.1.1.2" xref="S3.p2.5.m5.1.1.2.cmml">D</mi><mi id="S3.p2.5.m5.1.1.3" xref="S3.p2.5.m5.1.1.3.cmml">σ</mi></msub><annotation-xml encoding="MathML-Content" id="S3.p2.5.m5.1b"><apply id="S3.p2.5.m5.1.1.cmml" xref="S3.p2.5.m5.1.1"><csymbol cd="ambiguous" id="S3.p2.5.m5.1.1.1.cmml" xref="S3.p2.5.m5.1.1">subscript</csymbol><ci id="S3.p2.5.m5.1.1.2.cmml" xref="S3.p2.5.m5.1.1.2">𝐷</ci><ci id="S3.p2.5.m5.1.1.3.cmml" xref="S3.p2.5.m5.1.1.3">𝜎</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p2.5.m5.1c">D_{\sigma}</annotation><annotation encoding="application/x-llamapun" id="S3.p2.5.m5.1d">italic_D start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT</annotation></semantics></math>. Proofs can be found in Appendix <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#A4" title="Appendix D Proofs of section 3 ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_tag">D</span></a>. A discussion on technical assumptions is given in Appendix <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#A5" title="Appendix E Discussion on Assumptions ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_tag">E</span></a>.</p> </div> <section class="ltx_subsection" id="S3.SS1"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">3.1 </span>Asymptotics of critical points when <math alttext="\sigma\to 0" class="ltx_Math" display="inline" id="S3.SS1.1.m1.1"><semantics id="S3.SS1.1.m1.1b"><mrow id="S3.SS1.1.m1.1.1" xref="S3.SS1.1.m1.1.1.cmml"><mi id="S3.SS1.1.m1.1.1.2" xref="S3.SS1.1.m1.1.1.2.cmml">σ</mi><mo id="S3.SS1.1.m1.1.1.1" stretchy="false" xref="S3.SS1.1.m1.1.1.1.cmml">→</mo><mn id="S3.SS1.1.m1.1.1.3" xref="S3.SS1.1.m1.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.1.m1.1c"><apply id="S3.SS1.1.m1.1.1.cmml" xref="S3.SS1.1.m1.1.1"><ci id="S3.SS1.1.m1.1.1.1.cmml" xref="S3.SS1.1.m1.1.1.1">→</ci><ci id="S3.SS1.1.m1.1.1.2.cmml" xref="S3.SS1.1.m1.1.1.2">𝜎</ci><cn id="S3.SS1.1.m1.1.1.3.cmml" type="integer" xref="S3.SS1.1.m1.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.1.m1.1d">\sigma\to 0</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.1.m1.1e">italic_σ → 0</annotation></semantics></math> </h3> <div class="ltx_para" id="S3.SS1.p1"> <p class="ltx_p" id="S3.SS1.p1.4">Inspired by <cite class="ltx_cite ltx_citemacro_citet">Laumont et al. (<a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#bib.bib37" title="">2022b</a>, Proposition 1)</cite>, we first demonstrate that <math alttext="\nabla\mathcal{R}_{\sigma}" class="ltx_Math" display="inline" id="S3.SS1.p1.1.m1.1"><semantics id="S3.SS1.p1.1.m1.1a"><mrow id="S3.SS1.p1.1.m1.1.1" xref="S3.SS1.p1.1.m1.1.1.cmml"><mo id="S3.SS1.p1.1.m1.1.1.1" rspace="0.167em" xref="S3.SS1.p1.1.m1.1.1.1.cmml">∇</mo><msub id="S3.SS1.p1.1.m1.1.1.2" xref="S3.SS1.p1.1.m1.1.1.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.SS1.p1.1.m1.1.1.2.2" xref="S3.SS1.p1.1.m1.1.1.2.2.cmml">ℛ</mi><mi id="S3.SS1.p1.1.m1.1.1.2.3" xref="S3.SS1.p1.1.m1.1.1.2.3.cmml">σ</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p1.1.m1.1b"><apply id="S3.SS1.p1.1.m1.1.1.cmml" xref="S3.SS1.p1.1.m1.1.1"><ci id="S3.SS1.p1.1.m1.1.1.1.cmml" xref="S3.SS1.p1.1.m1.1.1.1">∇</ci><apply id="S3.SS1.p1.1.m1.1.1.2.cmml" xref="S3.SS1.p1.1.m1.1.1.2"><csymbol cd="ambiguous" id="S3.SS1.p1.1.m1.1.1.2.1.cmml" xref="S3.SS1.p1.1.m1.1.1.2">subscript</csymbol><ci id="S3.SS1.p1.1.m1.1.1.2.2.cmml" xref="S3.SS1.p1.1.m1.1.1.2.2">ℛ</ci><ci id="S3.SS1.p1.1.m1.1.1.2.3.cmml" xref="S3.SS1.p1.1.m1.1.1.2.3">𝜎</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p1.1.m1.1c">\nabla\mathcal{R}_{\sigma}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p1.1.m1.1d">∇ caligraphic_R start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT</annotation></semantics></math> converges uniformly to <math alttext="-\nabla\log p" class="ltx_Math" display="inline" id="S3.SS1.p1.2.m2.1"><semantics id="S3.SS1.p1.2.m2.1a"><mrow id="S3.SS1.p1.2.m2.1.1" xref="S3.SS1.p1.2.m2.1.1.cmml"><mo id="S3.SS1.p1.2.m2.1.1a" rspace="0.167em" xref="S3.SS1.p1.2.m2.1.1.cmml">−</mo><mrow id="S3.SS1.p1.2.m2.1.1.2" xref="S3.SS1.p1.2.m2.1.1.2.cmml"><mrow id="S3.SS1.p1.2.m2.1.1.2.1" xref="S3.SS1.p1.2.m2.1.1.2.1.cmml"><mo id="S3.SS1.p1.2.m2.1.1.2.1.1" rspace="0.167em" xref="S3.SS1.p1.2.m2.1.1.2.1.1.cmml">∇</mo><mi id="S3.SS1.p1.2.m2.1.1.2.1.2" xref="S3.SS1.p1.2.m2.1.1.2.1.2.cmml">log</mi></mrow><mo id="S3.SS1.p1.2.m2.1.1.2a" lspace="0.167em" xref="S3.SS1.p1.2.m2.1.1.2.cmml">⁡</mo><mi id="S3.SS1.p1.2.m2.1.1.2.2" xref="S3.SS1.p1.2.m2.1.1.2.2.cmml">p</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p1.2.m2.1b"><apply id="S3.SS1.p1.2.m2.1.1.cmml" xref="S3.SS1.p1.2.m2.1.1"><minus id="S3.SS1.p1.2.m2.1.1.1.cmml" xref="S3.SS1.p1.2.m2.1.1"></minus><apply id="S3.SS1.p1.2.m2.1.1.2.cmml" xref="S3.SS1.p1.2.m2.1.1.2"><apply id="S3.SS1.p1.2.m2.1.1.2.1.cmml" xref="S3.SS1.p1.2.m2.1.1.2.1"><ci id="S3.SS1.p1.2.m2.1.1.2.1.1.cmml" xref="S3.SS1.p1.2.m2.1.1.2.1.1">∇</ci><log id="S3.SS1.p1.2.m2.1.1.2.1.2.cmml" xref="S3.SS1.p1.2.m2.1.1.2.1.2"></log></apply><ci id="S3.SS1.p1.2.m2.1.1.2.2.cmml" xref="S3.SS1.p1.2.m2.1.1.2.2">𝑝</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p1.2.m2.1c">-\nabla\log p</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p1.2.m2.1d">- ∇ roman_log italic_p</annotation></semantics></math> on every compact when <math alttext="\sigma\rightarrow 0" class="ltx_Math" display="inline" id="S3.SS1.p1.3.m3.1"><semantics id="S3.SS1.p1.3.m3.1a"><mrow id="S3.SS1.p1.3.m3.1.1" xref="S3.SS1.p1.3.m3.1.1.cmml"><mi id="S3.SS1.p1.3.m3.1.1.2" xref="S3.SS1.p1.3.m3.1.1.2.cmml">σ</mi><mo id="S3.SS1.p1.3.m3.1.1.1" stretchy="false" xref="S3.SS1.p1.3.m3.1.1.1.cmml">→</mo><mn id="S3.SS1.p1.3.m3.1.1.3" xref="S3.SS1.p1.3.m3.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p1.3.m3.1b"><apply id="S3.SS1.p1.3.m3.1.1.cmml" xref="S3.SS1.p1.3.m3.1.1"><ci id="S3.SS1.p1.3.m3.1.1.1.cmml" xref="S3.SS1.p1.3.m3.1.1.1">→</ci><ci id="S3.SS1.p1.3.m3.1.1.2.cmml" xref="S3.SS1.p1.3.m3.1.1.2">𝜎</ci><cn id="S3.SS1.p1.3.m3.1.1.3.cmml" type="integer" xref="S3.SS1.p1.3.m3.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p1.3.m3.1c">\sigma\rightarrow 0</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p1.3.m3.1d">italic_σ → 0</annotation></semantics></math>. This type of result requires technical assumptions, such as <math alttext="\nabla\log p" class="ltx_Math" display="inline" id="S3.SS1.p1.4.m4.1"><semantics id="S3.SS1.p1.4.m4.1a"><mrow id="S3.SS1.p1.4.m4.1.1" xref="S3.SS1.p1.4.m4.1.1.cmml"><mrow id="S3.SS1.p1.4.m4.1.1.1" xref="S3.SS1.p1.4.m4.1.1.1.cmml"><mo id="S3.SS1.p1.4.m4.1.1.1.1" rspace="0.167em" xref="S3.SS1.p1.4.m4.1.1.1.1.cmml">∇</mo><mi id="S3.SS1.p1.4.m4.1.1.1.2" xref="S3.SS1.p1.4.m4.1.1.1.2.cmml">log</mi></mrow><mo id="S3.SS1.p1.4.m4.1.1a" lspace="0.167em" xref="S3.SS1.p1.4.m4.1.1.cmml">⁡</mo><mi id="S3.SS1.p1.4.m4.1.1.2" xref="S3.SS1.p1.4.m4.1.1.2.cmml">p</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p1.4.m4.1b"><apply id="S3.SS1.p1.4.m4.1.1.cmml" xref="S3.SS1.p1.4.m4.1.1"><apply id="S3.SS1.p1.4.m4.1.1.1.cmml" xref="S3.SS1.p1.4.m4.1.1.1"><ci id="S3.SS1.p1.4.m4.1.1.1.1.cmml" xref="S3.SS1.p1.4.m4.1.1.1.1">∇</ci><log id="S3.SS1.p1.4.m4.1.1.1.2.cmml" xref="S3.SS1.p1.4.m4.1.1.1.2"></log></apply><ci id="S3.SS1.p1.4.m4.1.1.2.cmml" xref="S3.SS1.p1.4.m4.1.1.2">𝑝</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p1.4.m4.1c">\nabla\log p</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p1.4.m4.1d">∇ roman_log italic_p</annotation></semantics></math> to be defined everywhere and smooth. </p> </div> <div class="ltx_theorem ltx_theorem_assumption" id="S3.Thmtheorem1"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S3.Thmtheorem1.1.1.1">Assumption 3.1</span></span><span class="ltx_text ltx_font_bold" id="S3.Thmtheorem1.2.2">.</span> </h6> <div class="ltx_para" id="S3.Thmtheorem1.p1"> <p class="ltx_p" id="S3.Thmtheorem1.p1.6"><span class="ltx_text ltx_font_bold" id="S3.Thmtheorem1.p1.6.1">(a)</span> The prior distribution <math alttext="p\in\mathrm{C}^{1}({\mathbb{R}}^{d},]0,+\infty[)" class="ltx_math_unparsed" display="inline" id="S3.Thmtheorem1.p1.1.m1.1"><semantics id="S3.Thmtheorem1.p1.1.m1.1a"><mrow id="S3.Thmtheorem1.p1.1.m1.1b"><mi id="S3.Thmtheorem1.p1.1.m1.1.1">p</mi><mo id="S3.Thmtheorem1.p1.1.m1.1.2">∈</mo><msup id="S3.Thmtheorem1.p1.1.m1.1.3"><mi id="S3.Thmtheorem1.p1.1.m1.1.3.2" mathvariant="normal">C</mi><mn id="S3.Thmtheorem1.p1.1.m1.1.3.3">1</mn></msup><mrow id="S3.Thmtheorem1.p1.1.m1.1.4"><mo id="S3.Thmtheorem1.p1.1.m1.1.4.1" stretchy="false">(</mo><msup id="S3.Thmtheorem1.p1.1.m1.1.4.2"><mi id="S3.Thmtheorem1.p1.1.m1.1.4.2.2">ℝ</mi><mi id="S3.Thmtheorem1.p1.1.m1.1.4.2.3">d</mi></msup><mo id="S3.Thmtheorem1.p1.1.m1.1.4.3">,</mo><mo id="S3.Thmtheorem1.p1.1.m1.1.4.4" stretchy="false">]</mo></mrow><mn id="S3.Thmtheorem1.p1.1.m1.1.5">0</mn><mo id="S3.Thmtheorem1.p1.1.m1.1.6" rspace="0em">,</mo><mo id="S3.Thmtheorem1.p1.1.m1.1.7" lspace="0em">+</mo><mi id="S3.Thmtheorem1.p1.1.m1.1.8" mathvariant="normal">∞</mi><mrow id="S3.Thmtheorem1.p1.1.m1.1.9"><mo id="S3.Thmtheorem1.p1.1.m1.1.9.1" stretchy="false">[</mo><mo id="S3.Thmtheorem1.p1.1.m1.1.9.2" stretchy="false">)</mo></mrow></mrow><annotation encoding="application/x-tex" id="S3.Thmtheorem1.p1.1.m1.1c">p\in\mathrm{C}^{1}({\mathbb{R}}^{d},]0,+\infty[)</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem1.p1.1.m1.1d">italic_p ∈ roman_C start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT , ] 0 , + ∞ [ )</annotation></semantics></math> with <math alttext="\|p\|_{\infty}+\|\nabla p\|_{\infty}<+\infty" class="ltx_Math" display="inline" id="S3.Thmtheorem1.p1.2.m2.2"><semantics id="S3.Thmtheorem1.p1.2.m2.2a"><mrow id="S3.Thmtheorem1.p1.2.m2.2.2" xref="S3.Thmtheorem1.p1.2.m2.2.2.cmml"><mrow id="S3.Thmtheorem1.p1.2.m2.2.2.1" xref="S3.Thmtheorem1.p1.2.m2.2.2.1.cmml"><msub id="S3.Thmtheorem1.p1.2.m2.2.2.1.3" xref="S3.Thmtheorem1.p1.2.m2.2.2.1.3.cmml"><mrow id="S3.Thmtheorem1.p1.2.m2.2.2.1.3.2.2" xref="S3.Thmtheorem1.p1.2.m2.2.2.1.3.2.1.cmml"><mo id="S3.Thmtheorem1.p1.2.m2.2.2.1.3.2.2.1" stretchy="false" xref="S3.Thmtheorem1.p1.2.m2.2.2.1.3.2.1.1.cmml">‖</mo><mi id="S3.Thmtheorem1.p1.2.m2.1.1" xref="S3.Thmtheorem1.p1.2.m2.1.1.cmml">p</mi><mo id="S3.Thmtheorem1.p1.2.m2.2.2.1.3.2.2.2" stretchy="false" xref="S3.Thmtheorem1.p1.2.m2.2.2.1.3.2.1.1.cmml">‖</mo></mrow><mi id="S3.Thmtheorem1.p1.2.m2.2.2.1.3.3" mathvariant="normal" xref="S3.Thmtheorem1.p1.2.m2.2.2.1.3.3.cmml">∞</mi></msub><mo id="S3.Thmtheorem1.p1.2.m2.2.2.1.2" xref="S3.Thmtheorem1.p1.2.m2.2.2.1.2.cmml">+</mo><msub id="S3.Thmtheorem1.p1.2.m2.2.2.1.1" xref="S3.Thmtheorem1.p1.2.m2.2.2.1.1.cmml"><mrow id="S3.Thmtheorem1.p1.2.m2.2.2.1.1.1.1" xref="S3.Thmtheorem1.p1.2.m2.2.2.1.1.1.2.cmml"><mo id="S3.Thmtheorem1.p1.2.m2.2.2.1.1.1.1.2" stretchy="false" xref="S3.Thmtheorem1.p1.2.m2.2.2.1.1.1.2.1.cmml">‖</mo><mrow id="S3.Thmtheorem1.p1.2.m2.2.2.1.1.1.1.1" xref="S3.Thmtheorem1.p1.2.m2.2.2.1.1.1.1.1.cmml"><mo id="S3.Thmtheorem1.p1.2.m2.2.2.1.1.1.1.1.1" rspace="0.167em" 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xref="S3.Thmtheorem1.p1.2.m2.2.2.3.2"></infinity></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem1.p1.2.m2.2c">\|p\|_{\infty}+\|\nabla p\|_{\infty}<+\infty</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem1.p1.2.m2.2d">∥ italic_p ∥ start_POSTSUBSCRIPT ∞ end_POSTSUBSCRIPT + ∥ ∇ italic_p ∥ start_POSTSUBSCRIPT ∞ end_POSTSUBSCRIPT < + ∞</annotation></semantics></math>. <span class="ltx_text ltx_font_bold" id="S3.Thmtheorem1.p1.6.2">(b)</span> The prior score is sub-polynomial, there exist <math alttext="A\in{\mathbb{R}}^{+}" class="ltx_Math" display="inline" id="S3.Thmtheorem1.p1.3.m3.1"><semantics id="S3.Thmtheorem1.p1.3.m3.1a"><mrow id="S3.Thmtheorem1.p1.3.m3.1.1" xref="S3.Thmtheorem1.p1.3.m3.1.1.cmml"><mi id="S3.Thmtheorem1.p1.3.m3.1.1.2" xref="S3.Thmtheorem1.p1.3.m3.1.1.2.cmml">A</mi><mo id="S3.Thmtheorem1.p1.3.m3.1.1.1" xref="S3.Thmtheorem1.p1.3.m3.1.1.1.cmml">∈</mo><msup id="S3.Thmtheorem1.p1.3.m3.1.1.3" xref="S3.Thmtheorem1.p1.3.m3.1.1.3.cmml"><mi id="S3.Thmtheorem1.p1.3.m3.1.1.3.2" xref="S3.Thmtheorem1.p1.3.m3.1.1.3.2.cmml">ℝ</mi><mo id="S3.Thmtheorem1.p1.3.m3.1.1.3.3" xref="S3.Thmtheorem1.p1.3.m3.1.1.3.3.cmml">+</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem1.p1.3.m3.1b"><apply id="S3.Thmtheorem1.p1.3.m3.1.1.cmml" xref="S3.Thmtheorem1.p1.3.m3.1.1"><in id="S3.Thmtheorem1.p1.3.m3.1.1.1.cmml" xref="S3.Thmtheorem1.p1.3.m3.1.1.1"></in><ci id="S3.Thmtheorem1.p1.3.m3.1.1.2.cmml" xref="S3.Thmtheorem1.p1.3.m3.1.1.2">𝐴</ci><apply id="S3.Thmtheorem1.p1.3.m3.1.1.3.cmml" xref="S3.Thmtheorem1.p1.3.m3.1.1.3"><csymbol cd="ambiguous" id="S3.Thmtheorem1.p1.3.m3.1.1.3.1.cmml" xref="S3.Thmtheorem1.p1.3.m3.1.1.3">superscript</csymbol><ci id="S3.Thmtheorem1.p1.3.m3.1.1.3.2.cmml" xref="S3.Thmtheorem1.p1.3.m3.1.1.3.2">ℝ</ci><plus id="S3.Thmtheorem1.p1.3.m3.1.1.3.3.cmml" xref="S3.Thmtheorem1.p1.3.m3.1.1.3.3"></plus></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem1.p1.3.m3.1c">A\in{\mathbb{R}}^{+}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem1.p1.3.m3.1d">italic_A ∈ blackboard_R start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT</annotation></semantics></math> and <math alttext="q\in{\mathbb{N}}" class="ltx_Math" display="inline" id="S3.Thmtheorem1.p1.4.m4.1"><semantics id="S3.Thmtheorem1.p1.4.m4.1a"><mrow id="S3.Thmtheorem1.p1.4.m4.1.1" xref="S3.Thmtheorem1.p1.4.m4.1.1.cmml"><mi id="S3.Thmtheorem1.p1.4.m4.1.1.2" xref="S3.Thmtheorem1.p1.4.m4.1.1.2.cmml">q</mi><mo id="S3.Thmtheorem1.p1.4.m4.1.1.1" xref="S3.Thmtheorem1.p1.4.m4.1.1.1.cmml">∈</mo><mi id="S3.Thmtheorem1.p1.4.m4.1.1.3" xref="S3.Thmtheorem1.p1.4.m4.1.1.3.cmml">ℕ</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem1.p1.4.m4.1b"><apply id="S3.Thmtheorem1.p1.4.m4.1.1.cmml" xref="S3.Thmtheorem1.p1.4.m4.1.1"><in id="S3.Thmtheorem1.p1.4.m4.1.1.1.cmml" xref="S3.Thmtheorem1.p1.4.m4.1.1.1"></in><ci 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xref="S3.Thmtheorem1.p1.5.m5.1.1.3">superscript</csymbol><ci id="S3.Thmtheorem1.p1.5.m5.1.1.3.2.cmml" xref="S3.Thmtheorem1.p1.5.m5.1.1.3.2">ℝ</ci><ci id="S3.Thmtheorem1.p1.5.m5.1.1.3.3.cmml" xref="S3.Thmtheorem1.p1.5.m5.1.1.3.3">𝑑</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem1.p1.5.m5.1c">\forall{\mathbf{x}}\in{\mathbb{R}}^{d}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem1.p1.5.m5.1d">∀ bold_x ∈ blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT</annotation></semantics></math>, <math alttext="\|\nabla\log p({\mathbf{x}})\|\leq A(1+\|{\mathbf{x}}\|^{q})" class="ltx_Math" display="inline" id="S3.Thmtheorem1.p1.6.m6.4"><semantics id="S3.Thmtheorem1.p1.6.m6.4a"><mrow id="S3.Thmtheorem1.p1.6.m6.4.4" xref="S3.Thmtheorem1.p1.6.m6.4.4.cmml"><mrow id="S3.Thmtheorem1.p1.6.m6.3.3.1.1" xref="S3.Thmtheorem1.p1.6.m6.3.3.1.2.cmml"><mo id="S3.Thmtheorem1.p1.6.m6.3.3.1.1.2" stretchy="false" 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id="S3.Thmtheorem1.p1.6.m6.4.4.2.1.1.3" stretchy="false" xref="S3.Thmtheorem1.p1.6.m6.4.4.2.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem1.p1.6.m6.4b"><apply id="S3.Thmtheorem1.p1.6.m6.4.4.cmml" xref="S3.Thmtheorem1.p1.6.m6.4.4"><leq id="S3.Thmtheorem1.p1.6.m6.4.4.3.cmml" xref="S3.Thmtheorem1.p1.6.m6.4.4.3"></leq><apply id="S3.Thmtheorem1.p1.6.m6.3.3.1.2.cmml" xref="S3.Thmtheorem1.p1.6.m6.3.3.1.1"><csymbol cd="latexml" id="S3.Thmtheorem1.p1.6.m6.3.3.1.2.1.cmml" xref="S3.Thmtheorem1.p1.6.m6.3.3.1.1.2">norm</csymbol><apply id="S3.Thmtheorem1.p1.6.m6.3.3.1.1.1.cmml" xref="S3.Thmtheorem1.p1.6.m6.3.3.1.1.1"><times id="S3.Thmtheorem1.p1.6.m6.3.3.1.1.1.1.cmml" xref="S3.Thmtheorem1.p1.6.m6.3.3.1.1.1.1"></times><apply id="S3.Thmtheorem1.p1.6.m6.3.3.1.1.1.2.cmml" xref="S3.Thmtheorem1.p1.6.m6.3.3.1.1.1.2"><apply id="S3.Thmtheorem1.p1.6.m6.3.3.1.1.1.2.1.cmml" xref="S3.Thmtheorem1.p1.6.m6.3.3.1.1.1.2.1"><ci id="S3.Thmtheorem1.p1.6.m6.3.3.1.1.1.2.1.1.cmml" xref="S3.Thmtheorem1.p1.6.m6.3.3.1.1.1.2.1.1">∇</ci><log id="S3.Thmtheorem1.p1.6.m6.3.3.1.1.1.2.1.2.cmml" xref="S3.Thmtheorem1.p1.6.m6.3.3.1.1.1.2.1.2"></log></apply><ci id="S3.Thmtheorem1.p1.6.m6.3.3.1.1.1.2.2.cmml" xref="S3.Thmtheorem1.p1.6.m6.3.3.1.1.1.2.2">𝑝</ci></apply><ci id="S3.Thmtheorem1.p1.6.m6.1.1.cmml" xref="S3.Thmtheorem1.p1.6.m6.1.1">𝐱</ci></apply></apply><apply id="S3.Thmtheorem1.p1.6.m6.4.4.2.cmml" xref="S3.Thmtheorem1.p1.6.m6.4.4.2"><times id="S3.Thmtheorem1.p1.6.m6.4.4.2.2.cmml" xref="S3.Thmtheorem1.p1.6.m6.4.4.2.2"></times><ci id="S3.Thmtheorem1.p1.6.m6.4.4.2.3.cmml" xref="S3.Thmtheorem1.p1.6.m6.4.4.2.3">𝐴</ci><apply id="S3.Thmtheorem1.p1.6.m6.4.4.2.1.1.1.cmml" xref="S3.Thmtheorem1.p1.6.m6.4.4.2.1.1"><plus id="S3.Thmtheorem1.p1.6.m6.4.4.2.1.1.1.1.cmml" xref="S3.Thmtheorem1.p1.6.m6.4.4.2.1.1.1.1"></plus><cn id="S3.Thmtheorem1.p1.6.m6.4.4.2.1.1.1.2.cmml" type="integer" xref="S3.Thmtheorem1.p1.6.m6.4.4.2.1.1.1.2">1</cn><apply id="S3.Thmtheorem1.p1.6.m6.4.4.2.1.1.1.3.cmml" xref="S3.Thmtheorem1.p1.6.m6.4.4.2.1.1.1.3"><csymbol cd="ambiguous" id="S3.Thmtheorem1.p1.6.m6.4.4.2.1.1.1.3.1.cmml" xref="S3.Thmtheorem1.p1.6.m6.4.4.2.1.1.1.3">superscript</csymbol><apply id="S3.Thmtheorem1.p1.6.m6.4.4.2.1.1.1.3.2.1.cmml" xref="S3.Thmtheorem1.p1.6.m6.4.4.2.1.1.1.3.2.2"><csymbol cd="latexml" id="S3.Thmtheorem1.p1.6.m6.4.4.2.1.1.1.3.2.1.1.cmml" xref="S3.Thmtheorem1.p1.6.m6.4.4.2.1.1.1.3.2.2.1">norm</csymbol><ci id="S3.Thmtheorem1.p1.6.m6.2.2.cmml" xref="S3.Thmtheorem1.p1.6.m6.2.2">𝐱</ci></apply><ci id="S3.Thmtheorem1.p1.6.m6.4.4.2.1.1.1.3.3.cmml" xref="S3.Thmtheorem1.p1.6.m6.4.4.2.1.1.1.3.3">𝑞</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem1.p1.6.m6.4c">\|\nabla\log p({\mathbf{x}})\|\leq A(1+\|{\mathbf{x}}\|^{q})</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem1.p1.6.m6.4d">∥ ∇ roman_log italic_p ( bold_x ) ∥ ≤ italic_A ( 1 + ∥ bold_x ∥ start_POSTSUPERSCRIPT italic_q end_POSTSUPERSCRIPT )</annotation></semantics></math>.</p> </div> </div> <div class="ltx_para" id="S3.SS1.p2"> <p class="ltx_p" id="S3.SS1.p2.1">Assumption <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#S3.Thmtheorem1" title="Assumption 3.1. ‣ 3.1 Asymptotics of critical points when 𝜎→0 ‣ 3 Convergence Analysis ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_tag">3.1</span></a><span class="ltx_text ltx_font_bold" id="S3.SS1.p2.1.1">(a)</span> ensures that the prior is smooth, non-degenerate and Lipschitz.</p> </div> <div class="ltx_theorem ltx_theorem_assumption" id="S3.Thmtheorem2"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S3.Thmtheorem2.1.1.1">Assumption 3.2</span></span><span class="ltx_text ltx_font_bold" id="S3.Thmtheorem2.2.2">.</span> </h6> <div class="ltx_para" id="S3.Thmtheorem2.p1"> <p class="ltx_p" id="S3.Thmtheorem2.p1.5">The noisy prior score is sub-polynomial, there exist <math alttext="B\in{\mathbb{R}}^{+}" class="ltx_Math" display="inline" id="S3.Thmtheorem2.p1.1.m1.1"><semantics id="S3.Thmtheorem2.p1.1.m1.1a"><mrow id="S3.Thmtheorem2.p1.1.m1.1.1" xref="S3.Thmtheorem2.p1.1.m1.1.1.cmml"><mi id="S3.Thmtheorem2.p1.1.m1.1.1.2" xref="S3.Thmtheorem2.p1.1.m1.1.1.2.cmml">B</mi><mo id="S3.Thmtheorem2.p1.1.m1.1.1.1" xref="S3.Thmtheorem2.p1.1.m1.1.1.1.cmml">∈</mo><msup id="S3.Thmtheorem2.p1.1.m1.1.1.3" xref="S3.Thmtheorem2.p1.1.m1.1.1.3.cmml"><mi id="S3.Thmtheorem2.p1.1.m1.1.1.3.2" xref="S3.Thmtheorem2.p1.1.m1.1.1.3.2.cmml">ℝ</mi><mo id="S3.Thmtheorem2.p1.1.m1.1.1.3.3" xref="S3.Thmtheorem2.p1.1.m1.1.1.3.3.cmml">+</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem2.p1.1.m1.1b"><apply id="S3.Thmtheorem2.p1.1.m1.1.1.cmml" xref="S3.Thmtheorem2.p1.1.m1.1.1"><in id="S3.Thmtheorem2.p1.1.m1.1.1.1.cmml" xref="S3.Thmtheorem2.p1.1.m1.1.1.1"></in><ci id="S3.Thmtheorem2.p1.1.m1.1.1.2.cmml" xref="S3.Thmtheorem2.p1.1.m1.1.1.2">𝐵</ci><apply id="S3.Thmtheorem2.p1.1.m1.1.1.3.cmml" xref="S3.Thmtheorem2.p1.1.m1.1.1.3"><csymbol cd="ambiguous" id="S3.Thmtheorem2.p1.1.m1.1.1.3.1.cmml" xref="S3.Thmtheorem2.p1.1.m1.1.1.3">superscript</csymbol><ci id="S3.Thmtheorem2.p1.1.m1.1.1.3.2.cmml" xref="S3.Thmtheorem2.p1.1.m1.1.1.3.2">ℝ</ci><plus id="S3.Thmtheorem2.p1.1.m1.1.1.3.3.cmml" xref="S3.Thmtheorem2.p1.1.m1.1.1.3.3"></plus></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem2.p1.1.m1.1c">B\in{\mathbb{R}}^{+}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem2.p1.1.m1.1d">italic_B ∈ blackboard_R start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT</annotation></semantics></math>, <math alttext="\beta\in{\mathbb{R}}" class="ltx_Math" display="inline" id="S3.Thmtheorem2.p1.2.m2.1"><semantics id="S3.Thmtheorem2.p1.2.m2.1a"><mrow id="S3.Thmtheorem2.p1.2.m2.1.1" xref="S3.Thmtheorem2.p1.2.m2.1.1.cmml"><mi id="S3.Thmtheorem2.p1.2.m2.1.1.2" xref="S3.Thmtheorem2.p1.2.m2.1.1.2.cmml">β</mi><mo id="S3.Thmtheorem2.p1.2.m2.1.1.1" xref="S3.Thmtheorem2.p1.2.m2.1.1.1.cmml">∈</mo><mi id="S3.Thmtheorem2.p1.2.m2.1.1.3" xref="S3.Thmtheorem2.p1.2.m2.1.1.3.cmml">ℝ</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem2.p1.2.m2.1b"><apply id="S3.Thmtheorem2.p1.2.m2.1.1.cmml" xref="S3.Thmtheorem2.p1.2.m2.1.1"><in id="S3.Thmtheorem2.p1.2.m2.1.1.1.cmml" xref="S3.Thmtheorem2.p1.2.m2.1.1.1"></in><ci id="S3.Thmtheorem2.p1.2.m2.1.1.2.cmml" xref="S3.Thmtheorem2.p1.2.m2.1.1.2">𝛽</ci><ci id="S3.Thmtheorem2.p1.2.m2.1.1.3.cmml" xref="S3.Thmtheorem2.p1.2.m2.1.1.3">ℝ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem2.p1.2.m2.1c">\beta\in{\mathbb{R}}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem2.p1.2.m2.1d">italic_β ∈ blackboard_R</annotation></semantics></math> and <math alttext="r\in{\mathbb{N}}" class="ltx_Math" display="inline" id="S3.Thmtheorem2.p1.3.m3.1"><semantics id="S3.Thmtheorem2.p1.3.m3.1a"><mrow id="S3.Thmtheorem2.p1.3.m3.1.1" xref="S3.Thmtheorem2.p1.3.m3.1.1.cmml"><mi id="S3.Thmtheorem2.p1.3.m3.1.1.2" xref="S3.Thmtheorem2.p1.3.m3.1.1.2.cmml">r</mi><mo id="S3.Thmtheorem2.p1.3.m3.1.1.1" xref="S3.Thmtheorem2.p1.3.m3.1.1.1.cmml">∈</mo><mi id="S3.Thmtheorem2.p1.3.m3.1.1.3" xref="S3.Thmtheorem2.p1.3.m3.1.1.3.cmml">ℕ</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem2.p1.3.m3.1b"><apply id="S3.Thmtheorem2.p1.3.m3.1.1.cmml" xref="S3.Thmtheorem2.p1.3.m3.1.1"><in id="S3.Thmtheorem2.p1.3.m3.1.1.1.cmml" xref="S3.Thmtheorem2.p1.3.m3.1.1.1"></in><ci id="S3.Thmtheorem2.p1.3.m3.1.1.2.cmml" xref="S3.Thmtheorem2.p1.3.m3.1.1.2">𝑟</ci><ci id="S3.Thmtheorem2.p1.3.m3.1.1.3.cmml" xref="S3.Thmtheorem2.p1.3.m3.1.1.3">ℕ</ci></apply></annotation-xml><annotation encoding="application/x-tex" 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xref="S3.Thmtheorem2.p1.5.m5.4.4.2.1.1.1.1"></plus><cn id="S3.Thmtheorem2.p1.5.m5.4.4.2.1.1.1.2.cmml" type="integer" xref="S3.Thmtheorem2.p1.5.m5.4.4.2.1.1.1.2">1</cn><apply id="S3.Thmtheorem2.p1.5.m5.4.4.2.1.1.1.3.cmml" xref="S3.Thmtheorem2.p1.5.m5.4.4.2.1.1.1.3"><csymbol cd="ambiguous" id="S3.Thmtheorem2.p1.5.m5.4.4.2.1.1.1.3.1.cmml" xref="S3.Thmtheorem2.p1.5.m5.4.4.2.1.1.1.3">superscript</csymbol><apply id="S3.Thmtheorem2.p1.5.m5.4.4.2.1.1.1.3.2.1.cmml" xref="S3.Thmtheorem2.p1.5.m5.4.4.2.1.1.1.3.2.2"><csymbol cd="latexml" id="S3.Thmtheorem2.p1.5.m5.4.4.2.1.1.1.3.2.1.1.cmml" xref="S3.Thmtheorem2.p1.5.m5.4.4.2.1.1.1.3.2.2.1">norm</csymbol><ci id="S3.Thmtheorem2.p1.5.m5.2.2.cmml" xref="S3.Thmtheorem2.p1.5.m5.2.2">𝐱</ci></apply><ci id="S3.Thmtheorem2.p1.5.m5.4.4.2.1.1.1.3.3.cmml" xref="S3.Thmtheorem2.p1.5.m5.4.4.2.1.1.1.3.3">𝑟</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem2.p1.5.m5.4c">\|\nabla\log p_{\sigma}({\mathbf{x}})\|\leq B\sigma^{\beta}(1+\|{\mathbf{x}}\|% ^{r})</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem2.p1.5.m5.4d">∥ ∇ roman_log italic_p start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ( bold_x ) ∥ ≤ italic_B italic_σ start_POSTSUPERSCRIPT italic_β end_POSTSUPERSCRIPT ( 1 + ∥ bold_x ∥ start_POSTSUPERSCRIPT italic_r end_POSTSUPERSCRIPT )</annotation></semantics></math>.</p> </div> </div> <div class="ltx_para" id="S3.SS1.p3"> <p class="ltx_p" id="S3.SS1.p3.9">Under the so-called manifold hypothesis <cite class="ltx_cite ltx_citemacro_citep">(De Bortoli, <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#bib.bib17" title="">2023</a>)</cite> (see Assumption <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#S3.Thmtheorem5" title="Assumption 3.5 (Manifold hypothesis). ‣ 3.2 Unbiased algorithm analysis ‣ 3 Convergence Analysis ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_tag">3.5</span></a>), Assumption <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#S3.Thmtheorem2" title="Assumption 3.2. ‣ 3.1 Asymptotics of critical points when 𝜎→0 ‣ 3 Convergence Analysis ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_tag">3.2</span></a> is verified with <math alttext="r=1" class="ltx_Math" display="inline" id="S3.SS1.p3.1.m1.1"><semantics id="S3.SS1.p3.1.m1.1a"><mrow id="S3.SS1.p3.1.m1.1.1" xref="S3.SS1.p3.1.m1.1.1.cmml"><mi id="S3.SS1.p3.1.m1.1.1.2" xref="S3.SS1.p3.1.m1.1.1.2.cmml">r</mi><mo id="S3.SS1.p3.1.m1.1.1.1" xref="S3.SS1.p3.1.m1.1.1.1.cmml">=</mo><mn id="S3.SS1.p3.1.m1.1.1.3" xref="S3.SS1.p3.1.m1.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p3.1.m1.1b"><apply id="S3.SS1.p3.1.m1.1.1.cmml" xref="S3.SS1.p3.1.m1.1.1"><eq id="S3.SS1.p3.1.m1.1.1.1.cmml" xref="S3.SS1.p3.1.m1.1.1.1"></eq><ci id="S3.SS1.p3.1.m1.1.1.2.cmml" xref="S3.SS1.p3.1.m1.1.1.2">𝑟</ci><cn id="S3.SS1.p3.1.m1.1.1.3.cmml" type="integer" xref="S3.SS1.p3.1.m1.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p3.1.m1.1c">r=1</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p3.1.m1.1d">italic_r = 1</annotation></semantics></math> and <math alttext="\beta=-2" class="ltx_Math" display="inline" id="S3.SS1.p3.2.m2.1"><semantics id="S3.SS1.p3.2.m2.1a"><mrow id="S3.SS1.p3.2.m2.1.1" xref="S3.SS1.p3.2.m2.1.1.cmml"><mi id="S3.SS1.p3.2.m2.1.1.2" xref="S3.SS1.p3.2.m2.1.1.2.cmml">β</mi><mo id="S3.SS1.p3.2.m2.1.1.1" xref="S3.SS1.p3.2.m2.1.1.1.cmml">=</mo><mrow id="S3.SS1.p3.2.m2.1.1.3" xref="S3.SS1.p3.2.m2.1.1.3.cmml"><mo id="S3.SS1.p3.2.m2.1.1.3a" xref="S3.SS1.p3.2.m2.1.1.3.cmml">−</mo><mn id="S3.SS1.p3.2.m2.1.1.3.2" xref="S3.SS1.p3.2.m2.1.1.3.2.cmml">2</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p3.2.m2.1b"><apply id="S3.SS1.p3.2.m2.1.1.cmml" xref="S3.SS1.p3.2.m2.1.1"><eq id="S3.SS1.p3.2.m2.1.1.1.cmml" xref="S3.SS1.p3.2.m2.1.1.1"></eq><ci id="S3.SS1.p3.2.m2.1.1.2.cmml" xref="S3.SS1.p3.2.m2.1.1.2">𝛽</ci><apply id="S3.SS1.p3.2.m2.1.1.3.cmml" xref="S3.SS1.p3.2.m2.1.1.3"><minus id="S3.SS1.p3.2.m2.1.1.3.1.cmml" xref="S3.SS1.p3.2.m2.1.1.3"></minus><cn id="S3.SS1.p3.2.m2.1.1.3.2.cmml" type="integer" xref="S3.SS1.p3.2.m2.1.1.3.2">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p3.2.m2.1c">\beta=-2</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p3.2.m2.1d">italic_β = - 2</annotation></semantics></math>. Assumption <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#S3.Thmtheorem2" title="Assumption 3.2. ‣ 3.1 Asymptotics of critical points when 𝜎→0 ‣ 3 Convergence Analysis ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_tag">3.2</span></a> has also been proved with <math alttext="r=1" class="ltx_Math" display="inline" id="S3.SS1.p3.3.m3.1"><semantics id="S3.SS1.p3.3.m3.1a"><mrow id="S3.SS1.p3.3.m3.1.1" xref="S3.SS1.p3.3.m3.1.1.cmml"><mi id="S3.SS1.p3.3.m3.1.1.2" xref="S3.SS1.p3.3.m3.1.1.2.cmml">r</mi><mo id="S3.SS1.p3.3.m3.1.1.1" xref="S3.SS1.p3.3.m3.1.1.1.cmml">=</mo><mn id="S3.SS1.p3.3.m3.1.1.3" xref="S3.SS1.p3.3.m3.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p3.3.m3.1b"><apply id="S3.SS1.p3.3.m3.1.1.cmml" xref="S3.SS1.p3.3.m3.1.1"><eq id="S3.SS1.p3.3.m3.1.1.1.cmml" xref="S3.SS1.p3.3.m3.1.1.1"></eq><ci id="S3.SS1.p3.3.m3.1.1.2.cmml" xref="S3.SS1.p3.3.m3.1.1.2">𝑟</ci><cn id="S3.SS1.p3.3.m3.1.1.3.cmml" type="integer" xref="S3.SS1.p3.3.m3.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p3.3.m3.1c">r=1</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p3.3.m3.1d">italic_r = 1</annotation></semantics></math> and <math alttext="\beta=0" class="ltx_Math" display="inline" id="S3.SS1.p3.4.m4.1"><semantics id="S3.SS1.p3.4.m4.1a"><mrow id="S3.SS1.p3.4.m4.1.1" xref="S3.SS1.p3.4.m4.1.1.cmml"><mi id="S3.SS1.p3.4.m4.1.1.2" xref="S3.SS1.p3.4.m4.1.1.2.cmml">β</mi><mo id="S3.SS1.p3.4.m4.1.1.1" xref="S3.SS1.p3.4.m4.1.1.1.cmml">=</mo><mn id="S3.SS1.p3.4.m4.1.1.3" xref="S3.SS1.p3.4.m4.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p3.4.m4.1b"><apply id="S3.SS1.p3.4.m4.1.1.cmml" xref="S3.SS1.p3.4.m4.1.1"><eq id="S3.SS1.p3.4.m4.1.1.1.cmml" xref="S3.SS1.p3.4.m4.1.1.1"></eq><ci id="S3.SS1.p3.4.m4.1.1.2.cmml" xref="S3.SS1.p3.4.m4.1.1.2">𝛽</ci><cn id="S3.SS1.p3.4.m4.1.1.3.cmml" type="integer" xref="S3.SS1.p3.4.m4.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p3.4.m4.1c">\beta=0</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p3.4.m4.1d">italic_β = 0</annotation></semantics></math> in <cite class="ltx_cite ltx_citemacro_citep">(De Bortoli et al., <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#bib.bib18" title="">2023</a>)</cite>, under the Assumption <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#S3.Thmtheorem1" title="Assumption 3.1. ‣ 3.1 Asymptotics of critical points when 𝜎→0 ‣ 3 Convergence Analysis ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_tag">3.1</span></a><span class="ltx_text ltx_font_bold" id="S3.SS1.p3.9.1">(b)</span> with <math alttext="q=1" class="ltx_Math" display="inline" id="S3.SS1.p3.5.m5.1"><semantics id="S3.SS1.p3.5.m5.1a"><mrow id="S3.SS1.p3.5.m5.1.1" xref="S3.SS1.p3.5.m5.1.1.cmml"><mi id="S3.SS1.p3.5.m5.1.1.2" xref="S3.SS1.p3.5.m5.1.1.2.cmml">q</mi><mo id="S3.SS1.p3.5.m5.1.1.1" xref="S3.SS1.p3.5.m5.1.1.1.cmml">=</mo><mn id="S3.SS1.p3.5.m5.1.1.3" xref="S3.SS1.p3.5.m5.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p3.5.m5.1b"><apply id="S3.SS1.p3.5.m5.1.1.cmml" xref="S3.SS1.p3.5.m5.1.1"><eq id="S3.SS1.p3.5.m5.1.1.1.cmml" xref="S3.SS1.p3.5.m5.1.1.1"></eq><ci id="S3.SS1.p3.5.m5.1.1.2.cmml" xref="S3.SS1.p3.5.m5.1.1.2">𝑞</ci><cn id="S3.SS1.p3.5.m5.1.1.3.cmml" type="integer" xref="S3.SS1.p3.5.m5.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p3.5.m5.1c">q=1</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p3.5.m5.1d">italic_q = 1</annotation></semantics></math> and the fact that there exist <math alttext="m_{0}>0" class="ltx_Math" display="inline" id="S3.SS1.p3.6.m6.1"><semantics id="S3.SS1.p3.6.m6.1a"><mrow id="S3.SS1.p3.6.m6.1.1" xref="S3.SS1.p3.6.m6.1.1.cmml"><msub id="S3.SS1.p3.6.m6.1.1.2" xref="S3.SS1.p3.6.m6.1.1.2.cmml"><mi id="S3.SS1.p3.6.m6.1.1.2.2" xref="S3.SS1.p3.6.m6.1.1.2.2.cmml">m</mi><mn id="S3.SS1.p3.6.m6.1.1.2.3" xref="S3.SS1.p3.6.m6.1.1.2.3.cmml">0</mn></msub><mo id="S3.SS1.p3.6.m6.1.1.1" xref="S3.SS1.p3.6.m6.1.1.1.cmml">></mo><mn id="S3.SS1.p3.6.m6.1.1.3" xref="S3.SS1.p3.6.m6.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p3.6.m6.1b"><apply id="S3.SS1.p3.6.m6.1.1.cmml" xref="S3.SS1.p3.6.m6.1.1"><gt id="S3.SS1.p3.6.m6.1.1.1.cmml" xref="S3.SS1.p3.6.m6.1.1.1"></gt><apply id="S3.SS1.p3.6.m6.1.1.2.cmml" xref="S3.SS1.p3.6.m6.1.1.2"><csymbol cd="ambiguous" id="S3.SS1.p3.6.m6.1.1.2.1.cmml" xref="S3.SS1.p3.6.m6.1.1.2">subscript</csymbol><ci id="S3.SS1.p3.6.m6.1.1.2.2.cmml" xref="S3.SS1.p3.6.m6.1.1.2.2">𝑚</ci><cn id="S3.SS1.p3.6.m6.1.1.2.3.cmml" type="integer" xref="S3.SS1.p3.6.m6.1.1.2.3">0</cn></apply><cn id="S3.SS1.p3.6.m6.1.1.3.cmml" type="integer" xref="S3.SS1.p3.6.m6.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p3.6.m6.1c">m_{0}>0</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p3.6.m6.1d">italic_m start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT > 0</annotation></semantics></math> and <math alttext="d_{0}\geq 0" class="ltx_Math" display="inline" id="S3.SS1.p3.7.m7.1"><semantics id="S3.SS1.p3.7.m7.1a"><mrow id="S3.SS1.p3.7.m7.1.1" xref="S3.SS1.p3.7.m7.1.1.cmml"><msub id="S3.SS1.p3.7.m7.1.1.2" xref="S3.SS1.p3.7.m7.1.1.2.cmml"><mi id="S3.SS1.p3.7.m7.1.1.2.2" xref="S3.SS1.p3.7.m7.1.1.2.2.cmml">d</mi><mn id="S3.SS1.p3.7.m7.1.1.2.3" xref="S3.SS1.p3.7.m7.1.1.2.3.cmml">0</mn></msub><mo id="S3.SS1.p3.7.m7.1.1.1" xref="S3.SS1.p3.7.m7.1.1.1.cmml">≥</mo><mn id="S3.SS1.p3.7.m7.1.1.3" xref="S3.SS1.p3.7.m7.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p3.7.m7.1b"><apply id="S3.SS1.p3.7.m7.1.1.cmml" xref="S3.SS1.p3.7.m7.1.1"><geq id="S3.SS1.p3.7.m7.1.1.1.cmml" xref="S3.SS1.p3.7.m7.1.1.1"></geq><apply id="S3.SS1.p3.7.m7.1.1.2.cmml" xref="S3.SS1.p3.7.m7.1.1.2"><csymbol cd="ambiguous" id="S3.SS1.p3.7.m7.1.1.2.1.cmml" xref="S3.SS1.p3.7.m7.1.1.2">subscript</csymbol><ci id="S3.SS1.p3.7.m7.1.1.2.2.cmml" xref="S3.SS1.p3.7.m7.1.1.2.2">𝑑</ci><cn id="S3.SS1.p3.7.m7.1.1.2.3.cmml" type="integer" xref="S3.SS1.p3.7.m7.1.1.2.3">0</cn></apply><cn id="S3.SS1.p3.7.m7.1.1.3.cmml" type="integer" xref="S3.SS1.p3.7.m7.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p3.7.m7.1c">d_{0}\geq 0</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p3.7.m7.1d">italic_d start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ≥ 0</annotation></semantics></math> such that <math alttext="\forall{\mathbf{x}}\in{\mathbb{R}}^{d}" class="ltx_Math" display="inline" id="S3.SS1.p3.8.m8.1"><semantics id="S3.SS1.p3.8.m8.1a"><mrow id="S3.SS1.p3.8.m8.1.1" xref="S3.SS1.p3.8.m8.1.1.cmml"><mrow id="S3.SS1.p3.8.m8.1.1.2" xref="S3.SS1.p3.8.m8.1.1.2.cmml"><mo id="S3.SS1.p3.8.m8.1.1.2.1" rspace="0.167em" xref="S3.SS1.p3.8.m8.1.1.2.1.cmml">∀</mo><mi id="S3.SS1.p3.8.m8.1.1.2.2" xref="S3.SS1.p3.8.m8.1.1.2.2.cmml">𝐱</mi></mrow><mo id="S3.SS1.p3.8.m8.1.1.1" xref="S3.SS1.p3.8.m8.1.1.1.cmml">∈</mo><msup id="S3.SS1.p3.8.m8.1.1.3" xref="S3.SS1.p3.8.m8.1.1.3.cmml"><mi id="S3.SS1.p3.8.m8.1.1.3.2" xref="S3.SS1.p3.8.m8.1.1.3.2.cmml">ℝ</mi><mi id="S3.SS1.p3.8.m8.1.1.3.3" xref="S3.SS1.p3.8.m8.1.1.3.3.cmml">d</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p3.8.m8.1b"><apply id="S3.SS1.p3.8.m8.1.1.cmml" xref="S3.SS1.p3.8.m8.1.1"><in id="S3.SS1.p3.8.m8.1.1.1.cmml" xref="S3.SS1.p3.8.m8.1.1.1"></in><apply id="S3.SS1.p3.8.m8.1.1.2.cmml" xref="S3.SS1.p3.8.m8.1.1.2"><csymbol cd="latexml" id="S3.SS1.p3.8.m8.1.1.2.1.cmml" xref="S3.SS1.p3.8.m8.1.1.2.1">for-all</csymbol><ci id="S3.SS1.p3.8.m8.1.1.2.2.cmml" xref="S3.SS1.p3.8.m8.1.1.2.2">𝐱</ci></apply><apply id="S3.SS1.p3.8.m8.1.1.3.cmml" xref="S3.SS1.p3.8.m8.1.1.3"><csymbol cd="ambiguous" id="S3.SS1.p3.8.m8.1.1.3.1.cmml" xref="S3.SS1.p3.8.m8.1.1.3">superscript</csymbol><ci id="S3.SS1.p3.8.m8.1.1.3.2.cmml" xref="S3.SS1.p3.8.m8.1.1.3.2">ℝ</ci><ci id="S3.SS1.p3.8.m8.1.1.3.3.cmml" xref="S3.SS1.p3.8.m8.1.1.3.3">𝑑</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p3.8.m8.1c">\forall{\mathbf{x}}\in{\mathbb{R}}^{d}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p3.8.m8.1d">∀ bold_x ∈ blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT</annotation></semantics></math>, <math alttext="\langle\nabla\log p({\mathbf{x}}),{\mathbf{x}}\rangle\leq-m_{0}\|{\mathbf{x}}% \|^{2}+d_{0}\|{\mathbf{x}}\|" class="ltx_Math" display="inline" id="S3.SS1.p3.9.m9.5"><semantics id="S3.SS1.p3.9.m9.5a"><mrow id="S3.SS1.p3.9.m9.5.5" xref="S3.SS1.p3.9.m9.5.5.cmml"><mrow id="S3.SS1.p3.9.m9.5.5.1.1" xref="S3.SS1.p3.9.m9.5.5.1.2.cmml"><mo id="S3.SS1.p3.9.m9.5.5.1.1.2" stretchy="false" xref="S3.SS1.p3.9.m9.5.5.1.2.cmml">⟨</mo><mrow id="S3.SS1.p3.9.m9.5.5.1.1.1" xref="S3.SS1.p3.9.m9.5.5.1.1.1.cmml"><mrow id="S3.SS1.p3.9.m9.5.5.1.1.1.2" xref="S3.SS1.p3.9.m9.5.5.1.1.1.2.cmml"><mrow id="S3.SS1.p3.9.m9.5.5.1.1.1.2.1" xref="S3.SS1.p3.9.m9.5.5.1.1.1.2.1.cmml"><mo id="S3.SS1.p3.9.m9.5.5.1.1.1.2.1.1" rspace="0.167em" xref="S3.SS1.p3.9.m9.5.5.1.1.1.2.1.1.cmml">∇</mo><mi id="S3.SS1.p3.9.m9.5.5.1.1.1.2.1.2" xref="S3.SS1.p3.9.m9.5.5.1.1.1.2.1.2.cmml">log</mi></mrow><mo id="S3.SS1.p3.9.m9.5.5.1.1.1.2a" lspace="0.167em" xref="S3.SS1.p3.9.m9.5.5.1.1.1.2.cmml">⁡</mo><mi id="S3.SS1.p3.9.m9.5.5.1.1.1.2.2" xref="S3.SS1.p3.9.m9.5.5.1.1.1.2.2.cmml">p</mi></mrow><mo id="S3.SS1.p3.9.m9.5.5.1.1.1.1" xref="S3.SS1.p3.9.m9.5.5.1.1.1.1.cmml">⁢</mo><mrow id="S3.SS1.p3.9.m9.5.5.1.1.1.3.2" xref="S3.SS1.p3.9.m9.5.5.1.1.1.cmml"><mo id="S3.SS1.p3.9.m9.5.5.1.1.1.3.2.1" stretchy="false" xref="S3.SS1.p3.9.m9.5.5.1.1.1.cmml">(</mo><mi id="S3.SS1.p3.9.m9.1.1" xref="S3.SS1.p3.9.m9.1.1.cmml">𝐱</mi><mo id="S3.SS1.p3.9.m9.5.5.1.1.1.3.2.2" stretchy="false" xref="S3.SS1.p3.9.m9.5.5.1.1.1.cmml">)</mo></mrow></mrow><mo id="S3.SS1.p3.9.m9.5.5.1.1.3" xref="S3.SS1.p3.9.m9.5.5.1.2.cmml">,</mo><mi id="S3.SS1.p3.9.m9.2.2" xref="S3.SS1.p3.9.m9.2.2.cmml">𝐱</mi><mo id="S3.SS1.p3.9.m9.5.5.1.1.4" stretchy="false" xref="S3.SS1.p3.9.m9.5.5.1.2.cmml">⟩</mo></mrow><mo id="S3.SS1.p3.9.m9.5.5.2" xref="S3.SS1.p3.9.m9.5.5.2.cmml">≤</mo><mrow id="S3.SS1.p3.9.m9.5.5.3" xref="S3.SS1.p3.9.m9.5.5.3.cmml"><mrow id="S3.SS1.p3.9.m9.5.5.3.2" xref="S3.SS1.p3.9.m9.5.5.3.2.cmml"><mo id="S3.SS1.p3.9.m9.5.5.3.2a" xref="S3.SS1.p3.9.m9.5.5.3.2.cmml">−</mo><mrow id="S3.SS1.p3.9.m9.5.5.3.2.2" xref="S3.SS1.p3.9.m9.5.5.3.2.2.cmml"><msub id="S3.SS1.p3.9.m9.5.5.3.2.2.2" xref="S3.SS1.p3.9.m9.5.5.3.2.2.2.cmml"><mi id="S3.SS1.p3.9.m9.5.5.3.2.2.2.2" xref="S3.SS1.p3.9.m9.5.5.3.2.2.2.2.cmml">m</mi><mn id="S3.SS1.p3.9.m9.5.5.3.2.2.2.3" xref="S3.SS1.p3.9.m9.5.5.3.2.2.2.3.cmml">0</mn></msub><mo id="S3.SS1.p3.9.m9.5.5.3.2.2.1" xref="S3.SS1.p3.9.m9.5.5.3.2.2.1.cmml">⁢</mo><msup id="S3.SS1.p3.9.m9.5.5.3.2.2.3" xref="S3.SS1.p3.9.m9.5.5.3.2.2.3.cmml"><mrow id="S3.SS1.p3.9.m9.5.5.3.2.2.3.2.2" xref="S3.SS1.p3.9.m9.5.5.3.2.2.3.2.1.cmml"><mo id="S3.SS1.p3.9.m9.5.5.3.2.2.3.2.2.1" stretchy="false" xref="S3.SS1.p3.9.m9.5.5.3.2.2.3.2.1.1.cmml">‖</mo><mi id="S3.SS1.p3.9.m9.3.3" xref="S3.SS1.p3.9.m9.3.3.cmml">𝐱</mi><mo id="S3.SS1.p3.9.m9.5.5.3.2.2.3.2.2.2" stretchy="false" xref="S3.SS1.p3.9.m9.5.5.3.2.2.3.2.1.1.cmml">‖</mo></mrow><mn id="S3.SS1.p3.9.m9.5.5.3.2.2.3.3" xref="S3.SS1.p3.9.m9.5.5.3.2.2.3.3.cmml">2</mn></msup></mrow></mrow><mo id="S3.SS1.p3.9.m9.5.5.3.1" xref="S3.SS1.p3.9.m9.5.5.3.1.cmml">+</mo><mrow id="S3.SS1.p3.9.m9.5.5.3.3" xref="S3.SS1.p3.9.m9.5.5.3.3.cmml"><msub id="S3.SS1.p3.9.m9.5.5.3.3.2" xref="S3.SS1.p3.9.m9.5.5.3.3.2.cmml"><mi id="S3.SS1.p3.9.m9.5.5.3.3.2.2" xref="S3.SS1.p3.9.m9.5.5.3.3.2.2.cmml">d</mi><mn id="S3.SS1.p3.9.m9.5.5.3.3.2.3" xref="S3.SS1.p3.9.m9.5.5.3.3.2.3.cmml">0</mn></msub><mo id="S3.SS1.p3.9.m9.5.5.3.3.1" xref="S3.SS1.p3.9.m9.5.5.3.3.1.cmml">⁢</mo><mrow id="S3.SS1.p3.9.m9.5.5.3.3.3.2" xref="S3.SS1.p3.9.m9.5.5.3.3.3.1.cmml"><mo id="S3.SS1.p3.9.m9.5.5.3.3.3.2.1" stretchy="false" xref="S3.SS1.p3.9.m9.5.5.3.3.3.1.1.cmml">‖</mo><mi id="S3.SS1.p3.9.m9.4.4" xref="S3.SS1.p3.9.m9.4.4.cmml">𝐱</mi><mo id="S3.SS1.p3.9.m9.5.5.3.3.3.2.2" stretchy="false" xref="S3.SS1.p3.9.m9.5.5.3.3.3.1.1.cmml">‖</mo></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p3.9.m9.5b"><apply id="S3.SS1.p3.9.m9.5.5.cmml" xref="S3.SS1.p3.9.m9.5.5"><leq id="S3.SS1.p3.9.m9.5.5.2.cmml" xref="S3.SS1.p3.9.m9.5.5.2"></leq><list id="S3.SS1.p3.9.m9.5.5.1.2.cmml" xref="S3.SS1.p3.9.m9.5.5.1.1"><apply id="S3.SS1.p3.9.m9.5.5.1.1.1.cmml" xref="S3.SS1.p3.9.m9.5.5.1.1.1"><times id="S3.SS1.p3.9.m9.5.5.1.1.1.1.cmml" xref="S3.SS1.p3.9.m9.5.5.1.1.1.1"></times><apply id="S3.SS1.p3.9.m9.5.5.1.1.1.2.cmml" xref="S3.SS1.p3.9.m9.5.5.1.1.1.2"><apply id="S3.SS1.p3.9.m9.5.5.1.1.1.2.1.cmml" xref="S3.SS1.p3.9.m9.5.5.1.1.1.2.1"><ci id="S3.SS1.p3.9.m9.5.5.1.1.1.2.1.1.cmml" xref="S3.SS1.p3.9.m9.5.5.1.1.1.2.1.1">∇</ci><log id="S3.SS1.p3.9.m9.5.5.1.1.1.2.1.2.cmml" xref="S3.SS1.p3.9.m9.5.5.1.1.1.2.1.2"></log></apply><ci id="S3.SS1.p3.9.m9.5.5.1.1.1.2.2.cmml" xref="S3.SS1.p3.9.m9.5.5.1.1.1.2.2">𝑝</ci></apply><ci id="S3.SS1.p3.9.m9.1.1.cmml" xref="S3.SS1.p3.9.m9.1.1">𝐱</ci></apply><ci id="S3.SS1.p3.9.m9.2.2.cmml" xref="S3.SS1.p3.9.m9.2.2">𝐱</ci></list><apply id="S3.SS1.p3.9.m9.5.5.3.cmml" xref="S3.SS1.p3.9.m9.5.5.3"><plus id="S3.SS1.p3.9.m9.5.5.3.1.cmml" xref="S3.SS1.p3.9.m9.5.5.3.1"></plus><apply id="S3.SS1.p3.9.m9.5.5.3.2.cmml" xref="S3.SS1.p3.9.m9.5.5.3.2"><minus id="S3.SS1.p3.9.m9.5.5.3.2.1.cmml" xref="S3.SS1.p3.9.m9.5.5.3.2"></minus><apply id="S3.SS1.p3.9.m9.5.5.3.2.2.cmml" xref="S3.SS1.p3.9.m9.5.5.3.2.2"><times id="S3.SS1.p3.9.m9.5.5.3.2.2.1.cmml" xref="S3.SS1.p3.9.m9.5.5.3.2.2.1"></times><apply 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id="S3.SS1.p3.9.m9.5.5.3.3.cmml" xref="S3.SS1.p3.9.m9.5.5.3.3"><times id="S3.SS1.p3.9.m9.5.5.3.3.1.cmml" xref="S3.SS1.p3.9.m9.5.5.3.3.1"></times><apply id="S3.SS1.p3.9.m9.5.5.3.3.2.cmml" xref="S3.SS1.p3.9.m9.5.5.3.3.2"><csymbol cd="ambiguous" id="S3.SS1.p3.9.m9.5.5.3.3.2.1.cmml" xref="S3.SS1.p3.9.m9.5.5.3.3.2">subscript</csymbol><ci id="S3.SS1.p3.9.m9.5.5.3.3.2.2.cmml" xref="S3.SS1.p3.9.m9.5.5.3.3.2.2">𝑑</ci><cn id="S3.SS1.p3.9.m9.5.5.3.3.2.3.cmml" type="integer" xref="S3.SS1.p3.9.m9.5.5.3.3.2.3">0</cn></apply><apply id="S3.SS1.p3.9.m9.5.5.3.3.3.1.cmml" xref="S3.SS1.p3.9.m9.5.5.3.3.3.2"><csymbol cd="latexml" id="S3.SS1.p3.9.m9.5.5.3.3.3.1.1.cmml" xref="S3.SS1.p3.9.m9.5.5.3.3.3.2.1">norm</csymbol><ci id="S3.SS1.p3.9.m9.4.4.cmml" xref="S3.SS1.p3.9.m9.4.4">𝐱</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p3.9.m9.5c">\langle\nabla\log p({\mathbf{x}}),{\mathbf{x}}\rangle\leq-m_{0}\|{\mathbf{x}}% \|^{2}+d_{0}\|{\mathbf{x}}\|</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p3.9.m9.5d">⟨ ∇ roman_log italic_p ( bold_x ) , bold_x ⟩ ≤ - italic_m start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ∥ bold_x ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + italic_d start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ∥ bold_x ∥</annotation></semantics></math>.</p> </div> <div class="ltx_theorem ltx_theorem_proposition" id="S3.Thmproposition1"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S3.Thmproposition1.1.1.1">Proposition 3.1</span></span><span class="ltx_text ltx_font_bold" id="S3.Thmproposition1.2.2">.</span> </h6> <div class="ltx_para" id="S3.Thmproposition1.p1"> <p class="ltx_p" id="S3.Thmproposition1.p1.5"><span class="ltx_text ltx_font_italic" id="S3.Thmproposition1.p1.5.5">Under Assumptions <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#S3.Thmtheorem1" title="Assumption 3.1. ‣ 3.1 Asymptotics of critical points when 𝜎→0 ‣ 3 Convergence Analysis ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_tag">3.1</span></a>-<a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#S3.Thmtheorem2" title="Assumption 3.2. ‣ 3.1 Asymptotics of critical points when 𝜎→0 ‣ 3 Convergence Analysis ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_tag">3.2</span></a>, for <math alttext="\mathbf{K}" class="ltx_Math" display="inline" id="S3.Thmproposition1.p1.1.1.m1.1"><semantics id="S3.Thmproposition1.p1.1.1.m1.1a"><mi id="S3.Thmproposition1.p1.1.1.m1.1.1" xref="S3.Thmproposition1.p1.1.1.m1.1.1.cmml">𝐊</mi><annotation-xml encoding="MathML-Content" id="S3.Thmproposition1.p1.1.1.m1.1b"><ci id="S3.Thmproposition1.p1.1.1.m1.1.1.cmml" xref="S3.Thmproposition1.p1.1.1.m1.1.1">𝐊</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmproposition1.p1.1.1.m1.1c">\mathbf{K}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmproposition1.p1.1.1.m1.1d">bold_K</annotation></semantics></math> a compact of <math alttext="{\mathbb{R}}^{d}" class="ltx_Math" display="inline" id="S3.Thmproposition1.p1.2.2.m2.1"><semantics id="S3.Thmproposition1.p1.2.2.m2.1a"><msup id="S3.Thmproposition1.p1.2.2.m2.1.1" xref="S3.Thmproposition1.p1.2.2.m2.1.1.cmml"><mi id="S3.Thmproposition1.p1.2.2.m2.1.1.2" xref="S3.Thmproposition1.p1.2.2.m2.1.1.2.cmml">ℝ</mi><mi id="S3.Thmproposition1.p1.2.2.m2.1.1.3" xref="S3.Thmproposition1.p1.2.2.m2.1.1.3.cmml">d</mi></msup><annotation-xml encoding="MathML-Content" id="S3.Thmproposition1.p1.2.2.m2.1b"><apply id="S3.Thmproposition1.p1.2.2.m2.1.1.cmml" xref="S3.Thmproposition1.p1.2.2.m2.1.1"><csymbol cd="ambiguous" id="S3.Thmproposition1.p1.2.2.m2.1.1.1.cmml" xref="S3.Thmproposition1.p1.2.2.m2.1.1">superscript</csymbol><ci id="S3.Thmproposition1.p1.2.2.m2.1.1.2.cmml" xref="S3.Thmproposition1.p1.2.2.m2.1.1.2">ℝ</ci><ci id="S3.Thmproposition1.p1.2.2.m2.1.1.3.cmml" xref="S3.Thmproposition1.p1.2.2.m2.1.1.3">𝑑</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmproposition1.p1.2.2.m2.1c">{\mathbb{R}}^{d}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmproposition1.p1.2.2.m2.1d">blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT</annotation></semantics></math>, <math alttext="\nabla\mathcal{R}_{\sigma}" class="ltx_Math" display="inline" id="S3.Thmproposition1.p1.3.3.m3.1"><semantics id="S3.Thmproposition1.p1.3.3.m3.1a"><mrow id="S3.Thmproposition1.p1.3.3.m3.1.1" xref="S3.Thmproposition1.p1.3.3.m3.1.1.cmml"><mo id="S3.Thmproposition1.p1.3.3.m3.1.1.1" mathvariant="normal" rspace="0.167em" xref="S3.Thmproposition1.p1.3.3.m3.1.1.1.cmml">∇</mo><msub id="S3.Thmproposition1.p1.3.3.m3.1.1.2" xref="S3.Thmproposition1.p1.3.3.m3.1.1.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.Thmproposition1.p1.3.3.m3.1.1.2.2" xref="S3.Thmproposition1.p1.3.3.m3.1.1.2.2.cmml">ℛ</mi><mi id="S3.Thmproposition1.p1.3.3.m3.1.1.2.3" xref="S3.Thmproposition1.p1.3.3.m3.1.1.2.3.cmml">σ</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmproposition1.p1.3.3.m3.1b"><apply id="S3.Thmproposition1.p1.3.3.m3.1.1.cmml" xref="S3.Thmproposition1.p1.3.3.m3.1.1"><ci id="S3.Thmproposition1.p1.3.3.m3.1.1.1.cmml" xref="S3.Thmproposition1.p1.3.3.m3.1.1.1">normal-∇</ci><apply id="S3.Thmproposition1.p1.3.3.m3.1.1.2.cmml" xref="S3.Thmproposition1.p1.3.3.m3.1.1.2"><csymbol cd="ambiguous" id="S3.Thmproposition1.p1.3.3.m3.1.1.2.1.cmml" xref="S3.Thmproposition1.p1.3.3.m3.1.1.2">subscript</csymbol><ci id="S3.Thmproposition1.p1.3.3.m3.1.1.2.2.cmml" xref="S3.Thmproposition1.p1.3.3.m3.1.1.2.2">ℛ</ci><ci id="S3.Thmproposition1.p1.3.3.m3.1.1.2.3.cmml" xref="S3.Thmproposition1.p1.3.3.m3.1.1.2.3">𝜎</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmproposition1.p1.3.3.m3.1c">\nabla\mathcal{R}_{\sigma}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmproposition1.p1.3.3.m3.1d">∇ caligraphic_R start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT</annotation></semantics></math> converges uniformly to <math alttext="-\nabla\log p" class="ltx_Math" display="inline" id="S3.Thmproposition1.p1.4.4.m4.1"><semantics id="S3.Thmproposition1.p1.4.4.m4.1a"><mrow id="S3.Thmproposition1.p1.4.4.m4.1.1" xref="S3.Thmproposition1.p1.4.4.m4.1.1.cmml"><mo id="S3.Thmproposition1.p1.4.4.m4.1.1a" mathvariant="normal" rspace="0.167em" xref="S3.Thmproposition1.p1.4.4.m4.1.1.cmml">−</mo><mrow id="S3.Thmproposition1.p1.4.4.m4.1.1.2" xref="S3.Thmproposition1.p1.4.4.m4.1.1.2.cmml"><mrow id="S3.Thmproposition1.p1.4.4.m4.1.1.2.1" xref="S3.Thmproposition1.p1.4.4.m4.1.1.2.1.cmml"><mo id="S3.Thmproposition1.p1.4.4.m4.1.1.2.1.1" mathvariant="normal" rspace="0.167em" xref="S3.Thmproposition1.p1.4.4.m4.1.1.2.1.1.cmml">∇</mo><mi id="S3.Thmproposition1.p1.4.4.m4.1.1.2.1.2" mathvariant="normal" xref="S3.Thmproposition1.p1.4.4.m4.1.1.2.1.2.cmml">log</mi></mrow><mo id="S3.Thmproposition1.p1.4.4.m4.1.1.2a" lspace="0.167em" mathvariant="italic" xref="S3.Thmproposition1.p1.4.4.m4.1.1.2.cmml">⁡</mo><mi id="S3.Thmproposition1.p1.4.4.m4.1.1.2.2" xref="S3.Thmproposition1.p1.4.4.m4.1.1.2.2.cmml">p</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmproposition1.p1.4.4.m4.1b"><apply id="S3.Thmproposition1.p1.4.4.m4.1.1.cmml" xref="S3.Thmproposition1.p1.4.4.m4.1.1"><minus id="S3.Thmproposition1.p1.4.4.m4.1.1.1.cmml" xref="S3.Thmproposition1.p1.4.4.m4.1.1"></minus><apply id="S3.Thmproposition1.p1.4.4.m4.1.1.2.cmml" xref="S3.Thmproposition1.p1.4.4.m4.1.1.2"><apply id="S3.Thmproposition1.p1.4.4.m4.1.1.2.1.cmml" xref="S3.Thmproposition1.p1.4.4.m4.1.1.2.1"><ci id="S3.Thmproposition1.p1.4.4.m4.1.1.2.1.1.cmml" xref="S3.Thmproposition1.p1.4.4.m4.1.1.2.1.1">normal-∇</ci><log id="S3.Thmproposition1.p1.4.4.m4.1.1.2.1.2.cmml" xref="S3.Thmproposition1.p1.4.4.m4.1.1.2.1.2"></log></apply><ci id="S3.Thmproposition1.p1.4.4.m4.1.1.2.2.cmml" xref="S3.Thmproposition1.p1.4.4.m4.1.1.2.2">𝑝</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmproposition1.p1.4.4.m4.1c">-\nabla\log p</annotation><annotation encoding="application/x-llamapun" id="S3.Thmproposition1.p1.4.4.m4.1d">- ∇ roman_log italic_p</annotation></semantics></math> on <math alttext="\mathbf{K}" class="ltx_Math" display="inline" id="S3.Thmproposition1.p1.5.5.m5.1"><semantics id="S3.Thmproposition1.p1.5.5.m5.1a"><mi id="S3.Thmproposition1.p1.5.5.m5.1.1" xref="S3.Thmproposition1.p1.5.5.m5.1.1.cmml">𝐊</mi><annotation-xml encoding="MathML-Content" id="S3.Thmproposition1.p1.5.5.m5.1b"><ci id="S3.Thmproposition1.p1.5.5.m5.1.1.cmml" xref="S3.Thmproposition1.p1.5.5.m5.1.1">𝐊</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmproposition1.p1.5.5.m5.1c">\mathbf{K}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmproposition1.p1.5.5.m5.1d">bold_K</annotation></semantics></math>,</span></p> <table class="ltx_equation ltx_eqn_table" id="S3.Ex7"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\lim_{\sigma\to 0}\sup_{\mathbf{K}}\|\nabla\mathcal{R}_{\sigma}+\nabla\log p\|% =0." class="ltx_Math" display="block" id="S3.Ex7.m1.1"><semantics id="S3.Ex7.m1.1a"><mrow id="S3.Ex7.m1.1.1.1" xref="S3.Ex7.m1.1.1.1.1.cmml"><mrow id="S3.Ex7.m1.1.1.1.1" xref="S3.Ex7.m1.1.1.1.1.cmml"><mrow id="S3.Ex7.m1.1.1.1.1.1" xref="S3.Ex7.m1.1.1.1.1.1.cmml"><munder id="S3.Ex7.m1.1.1.1.1.1.2" xref="S3.Ex7.m1.1.1.1.1.1.2.cmml"><mo id="S3.Ex7.m1.1.1.1.1.1.2.2" movablelimits="false" xref="S3.Ex7.m1.1.1.1.1.1.2.2.cmml">lim</mo><mrow id="S3.Ex7.m1.1.1.1.1.1.2.3" xref="S3.Ex7.m1.1.1.1.1.1.2.3.cmml"><mi id="S3.Ex7.m1.1.1.1.1.1.2.3.2" xref="S3.Ex7.m1.1.1.1.1.1.2.3.2.cmml">σ</mi><mo id="S3.Ex7.m1.1.1.1.1.1.2.3.1" stretchy="false" xref="S3.Ex7.m1.1.1.1.1.1.2.3.1.cmml">→</mo><mn id="S3.Ex7.m1.1.1.1.1.1.2.3.3" xref="S3.Ex7.m1.1.1.1.1.1.2.3.3.cmml">0</mn></mrow></munder><mrow id="S3.Ex7.m1.1.1.1.1.1.1" xref="S3.Ex7.m1.1.1.1.1.1.1.cmml"><munder id="S3.Ex7.m1.1.1.1.1.1.1.2" xref="S3.Ex7.m1.1.1.1.1.1.1.2.cmml"><mo id="S3.Ex7.m1.1.1.1.1.1.1.2.2" lspace="0.167em" movablelimits="false" rspace="0em" xref="S3.Ex7.m1.1.1.1.1.1.1.2.2.cmml">sup</mo><mi id="S3.Ex7.m1.1.1.1.1.1.1.2.3" xref="S3.Ex7.m1.1.1.1.1.1.1.2.3.cmml">𝐊</mi></munder><mrow id="S3.Ex7.m1.1.1.1.1.1.1.1.1" xref="S3.Ex7.m1.1.1.1.1.1.1.1.2.cmml"><mo id="S3.Ex7.m1.1.1.1.1.1.1.1.1.2" stretchy="false" xref="S3.Ex7.m1.1.1.1.1.1.1.1.2.1.cmml">‖</mo><mrow id="S3.Ex7.m1.1.1.1.1.1.1.1.1.1" xref="S3.Ex7.m1.1.1.1.1.1.1.1.1.1.cmml"><mrow id="S3.Ex7.m1.1.1.1.1.1.1.1.1.1.2" xref="S3.Ex7.m1.1.1.1.1.1.1.1.1.1.2.cmml"><mo id="S3.Ex7.m1.1.1.1.1.1.1.1.1.1.2.1" rspace="0.167em" xref="S3.Ex7.m1.1.1.1.1.1.1.1.1.1.2.1.cmml">∇</mo><msub id="S3.Ex7.m1.1.1.1.1.1.1.1.1.1.2.2" xref="S3.Ex7.m1.1.1.1.1.1.1.1.1.1.2.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.Ex7.m1.1.1.1.1.1.1.1.1.1.2.2.2" xref="S3.Ex7.m1.1.1.1.1.1.1.1.1.1.2.2.2.cmml">ℛ</mi><mi id="S3.Ex7.m1.1.1.1.1.1.1.1.1.1.2.2.3" xref="S3.Ex7.m1.1.1.1.1.1.1.1.1.1.2.2.3.cmml">σ</mi></msub></mrow><mo id="S3.Ex7.m1.1.1.1.1.1.1.1.1.1.1" xref="S3.Ex7.m1.1.1.1.1.1.1.1.1.1.1.cmml">+</mo><mrow id="S3.Ex7.m1.1.1.1.1.1.1.1.1.1.3" xref="S3.Ex7.m1.1.1.1.1.1.1.1.1.1.3.cmml"><mrow id="S3.Ex7.m1.1.1.1.1.1.1.1.1.1.3.1" xref="S3.Ex7.m1.1.1.1.1.1.1.1.1.1.3.1.cmml"><mo id="S3.Ex7.m1.1.1.1.1.1.1.1.1.1.3.1.1" rspace="0.167em" xref="S3.Ex7.m1.1.1.1.1.1.1.1.1.1.3.1.1.cmml">∇</mo><mi id="S3.Ex7.m1.1.1.1.1.1.1.1.1.1.3.1.2" xref="S3.Ex7.m1.1.1.1.1.1.1.1.1.1.3.1.2.cmml">log</mi></mrow><mo id="S3.Ex7.m1.1.1.1.1.1.1.1.1.1.3a" lspace="0.167em" xref="S3.Ex7.m1.1.1.1.1.1.1.1.1.1.3.cmml">⁡</mo><mi id="S3.Ex7.m1.1.1.1.1.1.1.1.1.1.3.2" xref="S3.Ex7.m1.1.1.1.1.1.1.1.1.1.3.2.cmml">p</mi></mrow></mrow><mo id="S3.Ex7.m1.1.1.1.1.1.1.1.1.3" stretchy="false" xref="S3.Ex7.m1.1.1.1.1.1.1.1.2.1.cmml">‖</mo></mrow></mrow></mrow><mo id="S3.Ex7.m1.1.1.1.1.2" xref="S3.Ex7.m1.1.1.1.1.2.cmml">=</mo><mn id="S3.Ex7.m1.1.1.1.1.3" xref="S3.Ex7.m1.1.1.1.1.3.cmml">0</mn></mrow><mo id="S3.Ex7.m1.1.1.1.2" lspace="0em" xref="S3.Ex7.m1.1.1.1.1.cmml">.</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.Ex7.m1.1b"><apply id="S3.Ex7.m1.1.1.1.1.cmml" xref="S3.Ex7.m1.1.1.1"><eq id="S3.Ex7.m1.1.1.1.1.2.cmml" xref="S3.Ex7.m1.1.1.1.1.2"></eq><apply id="S3.Ex7.m1.1.1.1.1.1.cmml" xref="S3.Ex7.m1.1.1.1.1.1"><apply id="S3.Ex7.m1.1.1.1.1.1.2.cmml" xref="S3.Ex7.m1.1.1.1.1.1.2"><csymbol cd="ambiguous" id="S3.Ex7.m1.1.1.1.1.1.2.1.cmml" xref="S3.Ex7.m1.1.1.1.1.1.2">subscript</csymbol><limit id="S3.Ex7.m1.1.1.1.1.1.2.2.cmml" 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xref="S3.Ex7.m1.1.1.1.1.1.1.1.1.2">norm</csymbol><apply id="S3.Ex7.m1.1.1.1.1.1.1.1.1.1.cmml" xref="S3.Ex7.m1.1.1.1.1.1.1.1.1.1"><plus id="S3.Ex7.m1.1.1.1.1.1.1.1.1.1.1.cmml" xref="S3.Ex7.m1.1.1.1.1.1.1.1.1.1.1"></plus><apply id="S3.Ex7.m1.1.1.1.1.1.1.1.1.1.2.cmml" xref="S3.Ex7.m1.1.1.1.1.1.1.1.1.1.2"><ci id="S3.Ex7.m1.1.1.1.1.1.1.1.1.1.2.1.cmml" xref="S3.Ex7.m1.1.1.1.1.1.1.1.1.1.2.1">∇</ci><apply id="S3.Ex7.m1.1.1.1.1.1.1.1.1.1.2.2.cmml" xref="S3.Ex7.m1.1.1.1.1.1.1.1.1.1.2.2"><csymbol cd="ambiguous" id="S3.Ex7.m1.1.1.1.1.1.1.1.1.1.2.2.1.cmml" xref="S3.Ex7.m1.1.1.1.1.1.1.1.1.1.2.2">subscript</csymbol><ci id="S3.Ex7.m1.1.1.1.1.1.1.1.1.1.2.2.2.cmml" xref="S3.Ex7.m1.1.1.1.1.1.1.1.1.1.2.2.2">ℛ</ci><ci id="S3.Ex7.m1.1.1.1.1.1.1.1.1.1.2.2.3.cmml" xref="S3.Ex7.m1.1.1.1.1.1.1.1.1.1.2.2.3">𝜎</ci></apply></apply><apply id="S3.Ex7.m1.1.1.1.1.1.1.1.1.1.3.cmml" xref="S3.Ex7.m1.1.1.1.1.1.1.1.1.1.3"><apply id="S3.Ex7.m1.1.1.1.1.1.1.1.1.1.3.1.cmml" xref="S3.Ex7.m1.1.1.1.1.1.1.1.1.1.3.1"><ci id="S3.Ex7.m1.1.1.1.1.1.1.1.1.1.3.1.1.cmml" xref="S3.Ex7.m1.1.1.1.1.1.1.1.1.1.3.1.1">∇</ci><log id="S3.Ex7.m1.1.1.1.1.1.1.1.1.1.3.1.2.cmml" xref="S3.Ex7.m1.1.1.1.1.1.1.1.1.1.3.1.2"></log></apply><ci id="S3.Ex7.m1.1.1.1.1.1.1.1.1.1.3.2.cmml" xref="S3.Ex7.m1.1.1.1.1.1.1.1.1.1.3.2">𝑝</ci></apply></apply></apply></apply></apply><cn id="S3.Ex7.m1.1.1.1.1.3.cmml" type="integer" xref="S3.Ex7.m1.1.1.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Ex7.m1.1c">\lim_{\sigma\to 0}\sup_{\mathbf{K}}\|\nabla\mathcal{R}_{\sigma}+\nabla\log p\|% =0.</annotation><annotation encoding="application/x-llamapun" id="S3.Ex7.m1.1d">roman_lim start_POSTSUBSCRIPT italic_σ → 0 end_POSTSUBSCRIPT roman_sup start_POSTSUBSCRIPT bold_K end_POSTSUBSCRIPT ∥ ∇ caligraphic_R start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT + ∇ roman_log italic_p ∥ = 0 .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> </div> </div> <div class="ltx_para" id="S3.SS1.p4"> <p class="ltx_p" id="S3.SS1.p4.12">Proposition <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#S3.Thmproposition1" title="Proposition 3.1. ‣ 3.1 Asymptotics of critical points when 𝜎→0 ‣ 3 Convergence Analysis ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_tag">3.1</span></a> proves that our score <math alttext="\nabla\mathcal{R}_{\sigma}" class="ltx_Math" display="inline" id="S3.SS1.p4.1.m1.1"><semantics id="S3.SS1.p4.1.m1.1a"><mrow id="S3.SS1.p4.1.m1.1.1" xref="S3.SS1.p4.1.m1.1.1.cmml"><mo id="S3.SS1.p4.1.m1.1.1.1" rspace="0.167em" xref="S3.SS1.p4.1.m1.1.1.1.cmml">∇</mo><msub id="S3.SS1.p4.1.m1.1.1.2" xref="S3.SS1.p4.1.m1.1.1.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.SS1.p4.1.m1.1.1.2.2" xref="S3.SS1.p4.1.m1.1.1.2.2.cmml">ℛ</mi><mi id="S3.SS1.p4.1.m1.1.1.2.3" xref="S3.SS1.p4.1.m1.1.1.2.3.cmml">σ</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p4.1.m1.1b"><apply id="S3.SS1.p4.1.m1.1.1.cmml" xref="S3.SS1.p4.1.m1.1.1"><ci id="S3.SS1.p4.1.m1.1.1.1.cmml" xref="S3.SS1.p4.1.m1.1.1.1">∇</ci><apply id="S3.SS1.p4.1.m1.1.1.2.cmml" xref="S3.SS1.p4.1.m1.1.1.2"><csymbol cd="ambiguous" id="S3.SS1.p4.1.m1.1.1.2.1.cmml" xref="S3.SS1.p4.1.m1.1.1.2">subscript</csymbol><ci id="S3.SS1.p4.1.m1.1.1.2.2.cmml" xref="S3.SS1.p4.1.m1.1.1.2.2">ℛ</ci><ci id="S3.SS1.p4.1.m1.1.1.2.3.cmml" xref="S3.SS1.p4.1.m1.1.1.2.3">𝜎</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p4.1.m1.1c">\nabla\mathcal{R}_{\sigma}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p4.1.m1.1d">∇ caligraphic_R start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT</annotation></semantics></math> is close to the ideal score <math alttext="-\nabla\log p" class="ltx_Math" display="inline" id="S3.SS1.p4.2.m2.1"><semantics id="S3.SS1.p4.2.m2.1a"><mrow id="S3.SS1.p4.2.m2.1.1" xref="S3.SS1.p4.2.m2.1.1.cmml"><mo id="S3.SS1.p4.2.m2.1.1a" rspace="0.167em" xref="S3.SS1.p4.2.m2.1.1.cmml">−</mo><mrow id="S3.SS1.p4.2.m2.1.1.2" xref="S3.SS1.p4.2.m2.1.1.2.cmml"><mrow id="S3.SS1.p4.2.m2.1.1.2.1" xref="S3.SS1.p4.2.m2.1.1.2.1.cmml"><mo id="S3.SS1.p4.2.m2.1.1.2.1.1" rspace="0.167em" xref="S3.SS1.p4.2.m2.1.1.2.1.1.cmml">∇</mo><mi id="S3.SS1.p4.2.m2.1.1.2.1.2" xref="S3.SS1.p4.2.m2.1.1.2.1.2.cmml">log</mi></mrow><mo id="S3.SS1.p4.2.m2.1.1.2a" lspace="0.167em" xref="S3.SS1.p4.2.m2.1.1.2.cmml">⁡</mo><mi id="S3.SS1.p4.2.m2.1.1.2.2" xref="S3.SS1.p4.2.m2.1.1.2.2.cmml">p</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p4.2.m2.1b"><apply id="S3.SS1.p4.2.m2.1.1.cmml" xref="S3.SS1.p4.2.m2.1.1"><minus id="S3.SS1.p4.2.m2.1.1.1.cmml" xref="S3.SS1.p4.2.m2.1.1"></minus><apply id="S3.SS1.p4.2.m2.1.1.2.cmml" xref="S3.SS1.p4.2.m2.1.1.2"><apply id="S3.SS1.p4.2.m2.1.1.2.1.cmml" xref="S3.SS1.p4.2.m2.1.1.2.1"><ci id="S3.SS1.p4.2.m2.1.1.2.1.1.cmml" xref="S3.SS1.p4.2.m2.1.1.2.1.1">∇</ci><log id="S3.SS1.p4.2.m2.1.1.2.1.2.cmml" xref="S3.SS1.p4.2.m2.1.1.2.1.2"></log></apply><ci id="S3.SS1.p4.2.m2.1.1.2.2.cmml" xref="S3.SS1.p4.2.m2.1.1.2.2">𝑝</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p4.2.m2.1c">-\nabla\log p</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p4.2.m2.1d">- ∇ roman_log italic_p</annotation></semantics></math> when <math alttext="\sigma\to 0" class="ltx_Math" display="inline" id="S3.SS1.p4.3.m3.1"><semantics id="S3.SS1.p4.3.m3.1a"><mrow id="S3.SS1.p4.3.m3.1.1" xref="S3.SS1.p4.3.m3.1.1.cmml"><mi id="S3.SS1.p4.3.m3.1.1.2" xref="S3.SS1.p4.3.m3.1.1.2.cmml">σ</mi><mo id="S3.SS1.p4.3.m3.1.1.1" stretchy="false" xref="S3.SS1.p4.3.m3.1.1.1.cmml">→</mo><mn id="S3.SS1.p4.3.m3.1.1.3" xref="S3.SS1.p4.3.m3.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p4.3.m3.1b"><apply id="S3.SS1.p4.3.m3.1.1.cmml" xref="S3.SS1.p4.3.m3.1.1"><ci id="S3.SS1.p4.3.m3.1.1.1.cmml" xref="S3.SS1.p4.3.m3.1.1.1">→</ci><ci id="S3.SS1.p4.3.m3.1.1.2.cmml" xref="S3.SS1.p4.3.m3.1.1.2">𝜎</ci><cn id="S3.SS1.p4.3.m3.1.1.3.cmml" type="integer" xref="S3.SS1.p4.3.m3.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p4.3.m3.1c">\sigma\to 0</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p4.3.m3.1d">italic_σ → 0</annotation></semantics></math>. With this uniform approximation result, we are now able to study the behavior of the critical points of our optimization problem when <math alttext="\sigma\to 0" class="ltx_Math" display="inline" id="S3.SS1.p4.4.m4.1"><semantics id="S3.SS1.p4.4.m4.1a"><mrow id="S3.SS1.p4.4.m4.1.1" xref="S3.SS1.p4.4.m4.1.1.cmml"><mi id="S3.SS1.p4.4.m4.1.1.2" xref="S3.SS1.p4.4.m4.1.1.2.cmml">σ</mi><mo id="S3.SS1.p4.4.m4.1.1.1" stretchy="false" xref="S3.SS1.p4.4.m4.1.1.1.cmml">→</mo><mn id="S3.SS1.p4.4.m4.1.1.3" xref="S3.SS1.p4.4.m4.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p4.4.m4.1b"><apply id="S3.SS1.p4.4.m4.1.1.cmml" xref="S3.SS1.p4.4.m4.1.1"><ci id="S3.SS1.p4.4.m4.1.1.1.cmml" xref="S3.SS1.p4.4.m4.1.1.1">→</ci><ci id="S3.SS1.p4.4.m4.1.1.2.cmml" xref="S3.SS1.p4.4.m4.1.1.2">𝜎</ci><cn id="S3.SS1.p4.4.m4.1.1.3.cmml" type="integer" xref="S3.SS1.p4.4.m4.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p4.4.m4.1c">\sigma\to 0</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p4.4.m4.1d">italic_σ → 0</annotation></semantics></math>. For <math alttext="\mathbf{K}" class="ltx_Math" display="inline" id="S3.SS1.p4.5.m5.1"><semantics id="S3.SS1.p4.5.m5.1a"><mi id="S3.SS1.p4.5.m5.1.1" xref="S3.SS1.p4.5.m5.1.1.cmml">𝐊</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.p4.5.m5.1b"><ci id="S3.SS1.p4.5.m5.1.1.cmml" xref="S3.SS1.p4.5.m5.1.1">𝐊</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p4.5.m5.1c">\mathbf{K}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p4.5.m5.1d">bold_K</annotation></semantics></math> a compact of <math alttext="{\mathbb{R}}^{d}" class="ltx_Math" display="inline" id="S3.SS1.p4.6.m6.1"><semantics id="S3.SS1.p4.6.m6.1a"><msup id="S3.SS1.p4.6.m6.1.1" xref="S3.SS1.p4.6.m6.1.1.cmml"><mi id="S3.SS1.p4.6.m6.1.1.2" xref="S3.SS1.p4.6.m6.1.1.2.cmml">ℝ</mi><mi id="S3.SS1.p4.6.m6.1.1.3" xref="S3.SS1.p4.6.m6.1.1.3.cmml">d</mi></msup><annotation-xml encoding="MathML-Content" id="S3.SS1.p4.6.m6.1b"><apply id="S3.SS1.p4.6.m6.1.1.cmml" xref="S3.SS1.p4.6.m6.1.1"><csymbol cd="ambiguous" id="S3.SS1.p4.6.m6.1.1.1.cmml" xref="S3.SS1.p4.6.m6.1.1">superscript</csymbol><ci id="S3.SS1.p4.6.m6.1.1.2.cmml" xref="S3.SS1.p4.6.m6.1.1.2">ℝ</ci><ci id="S3.SS1.p4.6.m6.1.1.3.cmml" xref="S3.SS1.p4.6.m6.1.1.3">𝑑</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p4.6.m6.1c">{\mathbb{R}}^{d}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p4.6.m6.1d">blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT</annotation></semantics></math>, we define <math alttext="\mathbf{S}_{\sigma,\mathbf{K}}=\{{\mathbf{x}}\in\mathbf{K}|\nabla\mathcal{F}({% \mathbf{x}},{\mathbf{y}})+\alpha\nabla\mathcal{R}_{\sigma}({\mathbf{x}})=0\}" class="ltx_Math" display="inline" id="S3.SS1.p4.7.m7.7"><semantics id="S3.SS1.p4.7.m7.7a"><mrow id="S3.SS1.p4.7.m7.7.7" xref="S3.SS1.p4.7.m7.7.7.cmml"><msub id="S3.SS1.p4.7.m7.7.7.4" xref="S3.SS1.p4.7.m7.7.7.4.cmml"><mi id="S3.SS1.p4.7.m7.7.7.4.2" xref="S3.SS1.p4.7.m7.7.7.4.2.cmml">𝐒</mi><mrow id="S3.SS1.p4.7.m7.2.2.2.4" xref="S3.SS1.p4.7.m7.2.2.2.3.cmml"><mi id="S3.SS1.p4.7.m7.1.1.1.1" xref="S3.SS1.p4.7.m7.1.1.1.1.cmml">σ</mi><mo id="S3.SS1.p4.7.m7.2.2.2.4.1" xref="S3.SS1.p4.7.m7.2.2.2.3.cmml">,</mo><mi id="S3.SS1.p4.7.m7.2.2.2.2" xref="S3.SS1.p4.7.m7.2.2.2.2.cmml">𝐊</mi></mrow></msub><mo id="S3.SS1.p4.7.m7.7.7.3" xref="S3.SS1.p4.7.m7.7.7.3.cmml">=</mo><mrow id="S3.SS1.p4.7.m7.7.7.2.2" xref="S3.SS1.p4.7.m7.7.7.2.3.cmml"><mo id="S3.SS1.p4.7.m7.7.7.2.2.3" stretchy="false" xref="S3.SS1.p4.7.m7.7.7.2.3.1.cmml">{</mo><mrow id="S3.SS1.p4.7.m7.6.6.1.1.1" xref="S3.SS1.p4.7.m7.6.6.1.1.1.cmml"><mi id="S3.SS1.p4.7.m7.6.6.1.1.1.2" xref="S3.SS1.p4.7.m7.6.6.1.1.1.2.cmml">𝐱</mi><mo id="S3.SS1.p4.7.m7.6.6.1.1.1.1" xref="S3.SS1.p4.7.m7.6.6.1.1.1.1.cmml">∈</mo><mi id="S3.SS1.p4.7.m7.6.6.1.1.1.3" xref="S3.SS1.p4.7.m7.6.6.1.1.1.3.cmml">𝐊</mi></mrow><mo id="S3.SS1.p4.7.m7.7.7.2.2.4" lspace="0em" rspace="0.167em" xref="S3.SS1.p4.7.m7.7.7.2.3.1.cmml">|</mo><mrow id="S3.SS1.p4.7.m7.7.7.2.2.2" xref="S3.SS1.p4.7.m7.7.7.2.2.2.cmml"><mrow id="S3.SS1.p4.7.m7.7.7.2.2.2.2" xref="S3.SS1.p4.7.m7.7.7.2.2.2.2.cmml"><mrow id="S3.SS1.p4.7.m7.7.7.2.2.2.2.2" xref="S3.SS1.p4.7.m7.7.7.2.2.2.2.2.cmml"><mrow id="S3.SS1.p4.7.m7.7.7.2.2.2.2.2.2" xref="S3.SS1.p4.7.m7.7.7.2.2.2.2.2.2.cmml"><mo id="S3.SS1.p4.7.m7.7.7.2.2.2.2.2.2.1" rspace="0.167em" xref="S3.SS1.p4.7.m7.7.7.2.2.2.2.2.2.1.cmml">∇</mo><mi class="ltx_font_mathcaligraphic" id="S3.SS1.p4.7.m7.7.7.2.2.2.2.2.2.2" xref="S3.SS1.p4.7.m7.7.7.2.2.2.2.2.2.2.cmml">ℱ</mi></mrow><mo id="S3.SS1.p4.7.m7.7.7.2.2.2.2.2.1" xref="S3.SS1.p4.7.m7.7.7.2.2.2.2.2.1.cmml">⁢</mo><mrow id="S3.SS1.p4.7.m7.7.7.2.2.2.2.2.3.2" xref="S3.SS1.p4.7.m7.7.7.2.2.2.2.2.3.1.cmml"><mo id="S3.SS1.p4.7.m7.7.7.2.2.2.2.2.3.2.1" stretchy="false" xref="S3.SS1.p4.7.m7.7.7.2.2.2.2.2.3.1.cmml">(</mo><mi id="S3.SS1.p4.7.m7.3.3" xref="S3.SS1.p4.7.m7.3.3.cmml">𝐱</mi><mo id="S3.SS1.p4.7.m7.7.7.2.2.2.2.2.3.2.2" xref="S3.SS1.p4.7.m7.7.7.2.2.2.2.2.3.1.cmml">,</mo><mi id="S3.SS1.p4.7.m7.4.4" xref="S3.SS1.p4.7.m7.4.4.cmml">𝐲</mi><mo id="S3.SS1.p4.7.m7.7.7.2.2.2.2.2.3.2.3" stretchy="false" xref="S3.SS1.p4.7.m7.7.7.2.2.2.2.2.3.1.cmml">)</mo></mrow></mrow><mo id="S3.SS1.p4.7.m7.7.7.2.2.2.2.1" xref="S3.SS1.p4.7.m7.7.7.2.2.2.2.1.cmml">+</mo><mrow id="S3.SS1.p4.7.m7.7.7.2.2.2.2.3" xref="S3.SS1.p4.7.m7.7.7.2.2.2.2.3.cmml"><mi id="S3.SS1.p4.7.m7.7.7.2.2.2.2.3.2" xref="S3.SS1.p4.7.m7.7.7.2.2.2.2.3.2.cmml">α</mi><mo id="S3.SS1.p4.7.m7.7.7.2.2.2.2.3.1" lspace="0.167em" xref="S3.SS1.p4.7.m7.7.7.2.2.2.2.3.1.cmml">⁢</mo><mrow id="S3.SS1.p4.7.m7.7.7.2.2.2.2.3.3" xref="S3.SS1.p4.7.m7.7.7.2.2.2.2.3.3.cmml"><mo id="S3.SS1.p4.7.m7.7.7.2.2.2.2.3.3.1" rspace="0.167em" xref="S3.SS1.p4.7.m7.7.7.2.2.2.2.3.3.1.cmml">∇</mo><msub id="S3.SS1.p4.7.m7.7.7.2.2.2.2.3.3.2" xref="S3.SS1.p4.7.m7.7.7.2.2.2.2.3.3.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.SS1.p4.7.m7.7.7.2.2.2.2.3.3.2.2" xref="S3.SS1.p4.7.m7.7.7.2.2.2.2.3.3.2.2.cmml">ℛ</mi><mi id="S3.SS1.p4.7.m7.7.7.2.2.2.2.3.3.2.3" xref="S3.SS1.p4.7.m7.7.7.2.2.2.2.3.3.2.3.cmml">σ</mi></msub></mrow><mo id="S3.SS1.p4.7.m7.7.7.2.2.2.2.3.1a" xref="S3.SS1.p4.7.m7.7.7.2.2.2.2.3.1.cmml">⁢</mo><mrow id="S3.SS1.p4.7.m7.7.7.2.2.2.2.3.4.2" xref="S3.SS1.p4.7.m7.7.7.2.2.2.2.3.cmml"><mo id="S3.SS1.p4.7.m7.7.7.2.2.2.2.3.4.2.1" stretchy="false" xref="S3.SS1.p4.7.m7.7.7.2.2.2.2.3.cmml">(</mo><mi id="S3.SS1.p4.7.m7.5.5" xref="S3.SS1.p4.7.m7.5.5.cmml">𝐱</mi><mo id="S3.SS1.p4.7.m7.7.7.2.2.2.2.3.4.2.2" stretchy="false" xref="S3.SS1.p4.7.m7.7.7.2.2.2.2.3.cmml">)</mo></mrow></mrow></mrow><mo id="S3.SS1.p4.7.m7.7.7.2.2.2.1" xref="S3.SS1.p4.7.m7.7.7.2.2.2.1.cmml">=</mo><mn id="S3.SS1.p4.7.m7.7.7.2.2.2.3" xref="S3.SS1.p4.7.m7.7.7.2.2.2.3.cmml">0</mn></mrow><mo id="S3.SS1.p4.7.m7.7.7.2.2.5" stretchy="false" xref="S3.SS1.p4.7.m7.7.7.2.3.1.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p4.7.m7.7b"><apply id="S3.SS1.p4.7.m7.7.7.cmml" xref="S3.SS1.p4.7.m7.7.7"><eq id="S3.SS1.p4.7.m7.7.7.3.cmml" xref="S3.SS1.p4.7.m7.7.7.3"></eq><apply id="S3.SS1.p4.7.m7.7.7.4.cmml" xref="S3.SS1.p4.7.m7.7.7.4"><csymbol cd="ambiguous" id="S3.SS1.p4.7.m7.7.7.4.1.cmml" xref="S3.SS1.p4.7.m7.7.7.4">subscript</csymbol><ci id="S3.SS1.p4.7.m7.7.7.4.2.cmml" xref="S3.SS1.p4.7.m7.7.7.4.2">𝐒</ci><list id="S3.SS1.p4.7.m7.2.2.2.3.cmml" xref="S3.SS1.p4.7.m7.2.2.2.4"><ci id="S3.SS1.p4.7.m7.1.1.1.1.cmml" xref="S3.SS1.p4.7.m7.1.1.1.1">𝜎</ci><ci id="S3.SS1.p4.7.m7.2.2.2.2.cmml" xref="S3.SS1.p4.7.m7.2.2.2.2">𝐊</ci></list></apply><apply id="S3.SS1.p4.7.m7.7.7.2.3.cmml" xref="S3.SS1.p4.7.m7.7.7.2.2"><csymbol cd="latexml" id="S3.SS1.p4.7.m7.7.7.2.3.1.cmml" xref="S3.SS1.p4.7.m7.7.7.2.2.3">conditional-set</csymbol><apply id="S3.SS1.p4.7.m7.6.6.1.1.1.cmml" xref="S3.SS1.p4.7.m7.6.6.1.1.1"><in id="S3.SS1.p4.7.m7.6.6.1.1.1.1.cmml" xref="S3.SS1.p4.7.m7.6.6.1.1.1.1"></in><ci id="S3.SS1.p4.7.m7.6.6.1.1.1.2.cmml" xref="S3.SS1.p4.7.m7.6.6.1.1.1.2">𝐱</ci><ci id="S3.SS1.p4.7.m7.6.6.1.1.1.3.cmml" xref="S3.SS1.p4.7.m7.6.6.1.1.1.3">𝐊</ci></apply><apply id="S3.SS1.p4.7.m7.7.7.2.2.2.cmml" xref="S3.SS1.p4.7.m7.7.7.2.2.2"><eq id="S3.SS1.p4.7.m7.7.7.2.2.2.1.cmml" xref="S3.SS1.p4.7.m7.7.7.2.2.2.1"></eq><apply id="S3.SS1.p4.7.m7.7.7.2.2.2.2.cmml" xref="S3.SS1.p4.7.m7.7.7.2.2.2.2"><plus id="S3.SS1.p4.7.m7.7.7.2.2.2.2.1.cmml" xref="S3.SS1.p4.7.m7.7.7.2.2.2.2.1"></plus><apply id="S3.SS1.p4.7.m7.7.7.2.2.2.2.2.cmml" xref="S3.SS1.p4.7.m7.7.7.2.2.2.2.2"><times id="S3.SS1.p4.7.m7.7.7.2.2.2.2.2.1.cmml" xref="S3.SS1.p4.7.m7.7.7.2.2.2.2.2.1"></times><apply id="S3.SS1.p4.7.m7.7.7.2.2.2.2.2.2.cmml" xref="S3.SS1.p4.7.m7.7.7.2.2.2.2.2.2"><ci id="S3.SS1.p4.7.m7.7.7.2.2.2.2.2.2.1.cmml" xref="S3.SS1.p4.7.m7.7.7.2.2.2.2.2.2.1">∇</ci><ci id="S3.SS1.p4.7.m7.7.7.2.2.2.2.2.2.2.cmml" xref="S3.SS1.p4.7.m7.7.7.2.2.2.2.2.2.2">ℱ</ci></apply><interval closure="open" id="S3.SS1.p4.7.m7.7.7.2.2.2.2.2.3.1.cmml" xref="S3.SS1.p4.7.m7.7.7.2.2.2.2.2.3.2"><ci id="S3.SS1.p4.7.m7.3.3.cmml" xref="S3.SS1.p4.7.m7.3.3">𝐱</ci><ci id="S3.SS1.p4.7.m7.4.4.cmml" xref="S3.SS1.p4.7.m7.4.4">𝐲</ci></interval></apply><apply id="S3.SS1.p4.7.m7.7.7.2.2.2.2.3.cmml" xref="S3.SS1.p4.7.m7.7.7.2.2.2.2.3"><times id="S3.SS1.p4.7.m7.7.7.2.2.2.2.3.1.cmml" xref="S3.SS1.p4.7.m7.7.7.2.2.2.2.3.1"></times><ci id="S3.SS1.p4.7.m7.7.7.2.2.2.2.3.2.cmml" xref="S3.SS1.p4.7.m7.7.7.2.2.2.2.3.2">𝛼</ci><apply id="S3.SS1.p4.7.m7.7.7.2.2.2.2.3.3.cmml" xref="S3.SS1.p4.7.m7.7.7.2.2.2.2.3.3"><ci id="S3.SS1.p4.7.m7.7.7.2.2.2.2.3.3.1.cmml" xref="S3.SS1.p4.7.m7.7.7.2.2.2.2.3.3.1">∇</ci><apply id="S3.SS1.p4.7.m7.7.7.2.2.2.2.3.3.2.cmml" xref="S3.SS1.p4.7.m7.7.7.2.2.2.2.3.3.2"><csymbol cd="ambiguous" id="S3.SS1.p4.7.m7.7.7.2.2.2.2.3.3.2.1.cmml" xref="S3.SS1.p4.7.m7.7.7.2.2.2.2.3.3.2">subscript</csymbol><ci id="S3.SS1.p4.7.m7.7.7.2.2.2.2.3.3.2.2.cmml" xref="S3.SS1.p4.7.m7.7.7.2.2.2.2.3.3.2.2">ℛ</ci><ci id="S3.SS1.p4.7.m7.7.7.2.2.2.2.3.3.2.3.cmml" xref="S3.SS1.p4.7.m7.7.7.2.2.2.2.3.3.2.3">𝜎</ci></apply></apply><ci id="S3.SS1.p4.7.m7.5.5.cmml" xref="S3.SS1.p4.7.m7.5.5">𝐱</ci></apply></apply><cn id="S3.SS1.p4.7.m7.7.7.2.2.2.3.cmml" type="integer" xref="S3.SS1.p4.7.m7.7.7.2.2.2.3">0</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p4.7.m7.7c">\mathbf{S}_{\sigma,\mathbf{K}}=\{{\mathbf{x}}\in\mathbf{K}|\nabla\mathcal{F}({% \mathbf{x}},{\mathbf{y}})+\alpha\nabla\mathcal{R}_{\sigma}({\mathbf{x}})=0\}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p4.7.m7.7d">bold_S start_POSTSUBSCRIPT italic_σ , bold_K end_POSTSUBSCRIPT = { bold_x ∈ bold_K | ∇ caligraphic_F ( bold_x , bold_y ) + italic_α ∇ caligraphic_R start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ( bold_x ) = 0 }</annotation></semantics></math>, the set of critical points of Problem (<a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#S2.E9" title="9 ‣ Optimization algorithms ‣ 2.2 SNORE regularization ‣ 2 Stochastic deNOising REgularization (SNORE) ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_tag">9</span></a>) in <math alttext="\mathbf{K}" class="ltx_Math" display="inline" id="S3.SS1.p4.8.m8.1"><semantics id="S3.SS1.p4.8.m8.1a"><mi id="S3.SS1.p4.8.m8.1.1" xref="S3.SS1.p4.8.m8.1.1.cmml">𝐊</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.p4.8.m8.1b"><ci id="S3.SS1.p4.8.m8.1.1.cmml" xref="S3.SS1.p4.8.m8.1.1">𝐊</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p4.8.m8.1c">\mathbf{K}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p4.8.m8.1d">bold_K</annotation></semantics></math>. In order to study the behavior of the critical points set <math alttext="\mathbf{S}_{\sigma,\mathbf{K}}" class="ltx_Math" display="inline" id="S3.SS1.p4.9.m9.2"><semantics id="S3.SS1.p4.9.m9.2a"><msub id="S3.SS1.p4.9.m9.2.3" xref="S3.SS1.p4.9.m9.2.3.cmml"><mi id="S3.SS1.p4.9.m9.2.3.2" xref="S3.SS1.p4.9.m9.2.3.2.cmml">𝐒</mi><mrow id="S3.SS1.p4.9.m9.2.2.2.4" xref="S3.SS1.p4.9.m9.2.2.2.3.cmml"><mi id="S3.SS1.p4.9.m9.1.1.1.1" xref="S3.SS1.p4.9.m9.1.1.1.1.cmml">σ</mi><mo id="S3.SS1.p4.9.m9.2.2.2.4.1" xref="S3.SS1.p4.9.m9.2.2.2.3.cmml">,</mo><mi id="S3.SS1.p4.9.m9.2.2.2.2" xref="S3.SS1.p4.9.m9.2.2.2.2.cmml">𝐊</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="S3.SS1.p4.9.m9.2b"><apply id="S3.SS1.p4.9.m9.2.3.cmml" xref="S3.SS1.p4.9.m9.2.3"><csymbol cd="ambiguous" id="S3.SS1.p4.9.m9.2.3.1.cmml" xref="S3.SS1.p4.9.m9.2.3">subscript</csymbol><ci id="S3.SS1.p4.9.m9.2.3.2.cmml" xref="S3.SS1.p4.9.m9.2.3.2">𝐒</ci><list id="S3.SS1.p4.9.m9.2.2.2.3.cmml" xref="S3.SS1.p4.9.m9.2.2.2.4"><ci id="S3.SS1.p4.9.m9.1.1.1.1.cmml" xref="S3.SS1.p4.9.m9.1.1.1.1">𝜎</ci><ci id="S3.SS1.p4.9.m9.2.2.2.2.cmml" xref="S3.SS1.p4.9.m9.2.2.2.2">𝐊</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p4.9.m9.2c">\mathbf{S}_{\sigma,\mathbf{K}}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p4.9.m9.2d">bold_S start_POSTSUBSCRIPT italic_σ , bold_K end_POSTSUBSCRIPT</annotation></semantics></math> when <math alttext="\sigma\to 0" class="ltx_Math" display="inline" id="S3.SS1.p4.10.m10.1"><semantics id="S3.SS1.p4.10.m10.1a"><mrow id="S3.SS1.p4.10.m10.1.1" xref="S3.SS1.p4.10.m10.1.1.cmml"><mi id="S3.SS1.p4.10.m10.1.1.2" xref="S3.SS1.p4.10.m10.1.1.2.cmml">σ</mi><mo id="S3.SS1.p4.10.m10.1.1.1" stretchy="false" xref="S3.SS1.p4.10.m10.1.1.1.cmml">→</mo><mn id="S3.SS1.p4.10.m10.1.1.3" xref="S3.SS1.p4.10.m10.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p4.10.m10.1b"><apply id="S3.SS1.p4.10.m10.1.1.cmml" xref="S3.SS1.p4.10.m10.1.1"><ci id="S3.SS1.p4.10.m10.1.1.1.cmml" xref="S3.SS1.p4.10.m10.1.1.1">→</ci><ci id="S3.SS1.p4.10.m10.1.1.2.cmml" xref="S3.SS1.p4.10.m10.1.1.2">𝜎</ci><cn id="S3.SS1.p4.10.m10.1.1.3.cmml" type="integer" xref="S3.SS1.p4.10.m10.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p4.10.m10.1c">\sigma\to 0</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p4.10.m10.1d">italic_σ → 0</annotation></semantics></math>, we define below the notion of limit for sets <math alttext="\mathbf{S}_{\sigma,\mathbf{K}}" class="ltx_Math" display="inline" id="S3.SS1.p4.11.m11.2"><semantics id="S3.SS1.p4.11.m11.2a"><msub id="S3.SS1.p4.11.m11.2.3" xref="S3.SS1.p4.11.m11.2.3.cmml"><mi id="S3.SS1.p4.11.m11.2.3.2" xref="S3.SS1.p4.11.m11.2.3.2.cmml">𝐒</mi><mrow id="S3.SS1.p4.11.m11.2.2.2.4" xref="S3.SS1.p4.11.m11.2.2.2.3.cmml"><mi id="S3.SS1.p4.11.m11.1.1.1.1" xref="S3.SS1.p4.11.m11.1.1.1.1.cmml">σ</mi><mo id="S3.SS1.p4.11.m11.2.2.2.4.1" xref="S3.SS1.p4.11.m11.2.2.2.3.cmml">,</mo><mi id="S3.SS1.p4.11.m11.2.2.2.2" xref="S3.SS1.p4.11.m11.2.2.2.2.cmml">𝐊</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="S3.SS1.p4.11.m11.2b"><apply id="S3.SS1.p4.11.m11.2.3.cmml" xref="S3.SS1.p4.11.m11.2.3"><csymbol cd="ambiguous" id="S3.SS1.p4.11.m11.2.3.1.cmml" xref="S3.SS1.p4.11.m11.2.3">subscript</csymbol><ci id="S3.SS1.p4.11.m11.2.3.2.cmml" xref="S3.SS1.p4.11.m11.2.3.2">𝐒</ci><list id="S3.SS1.p4.11.m11.2.2.2.3.cmml" xref="S3.SS1.p4.11.m11.2.2.2.4"><ci id="S3.SS1.p4.11.m11.1.1.1.1.cmml" xref="S3.SS1.p4.11.m11.1.1.1.1">𝜎</ci><ci id="S3.SS1.p4.11.m11.2.2.2.2.cmml" xref="S3.SS1.p4.11.m11.2.2.2.2">𝐊</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p4.11.m11.2c">\mathbf{S}_{\sigma,\mathbf{K}}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p4.11.m11.2d">bold_S start_POSTSUBSCRIPT italic_σ , bold_K end_POSTSUBSCRIPT</annotation></semantics></math> when <math alttext="\sigma\to 0" class="ltx_Math" display="inline" id="S3.SS1.p4.12.m12.1"><semantics id="S3.SS1.p4.12.m12.1a"><mrow id="S3.SS1.p4.12.m12.1.1" xref="S3.SS1.p4.12.m12.1.1.cmml"><mi id="S3.SS1.p4.12.m12.1.1.2" xref="S3.SS1.p4.12.m12.1.1.2.cmml">σ</mi><mo id="S3.SS1.p4.12.m12.1.1.1" stretchy="false" xref="S3.SS1.p4.12.m12.1.1.1.cmml">→</mo><mn id="S3.SS1.p4.12.m12.1.1.3" xref="S3.SS1.p4.12.m12.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p4.12.m12.1b"><apply id="S3.SS1.p4.12.m12.1.1.cmml" xref="S3.SS1.p4.12.m12.1.1"><ci id="S3.SS1.p4.12.m12.1.1.1.cmml" xref="S3.SS1.p4.12.m12.1.1.1">→</ci><ci id="S3.SS1.p4.12.m12.1.1.2.cmml" xref="S3.SS1.p4.12.m12.1.1.2">𝜎</ci><cn id="S3.SS1.p4.12.m12.1.1.3.cmml" type="integer" xref="S3.SS1.p4.12.m12.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p4.12.m12.1c">\sigma\to 0</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p4.12.m12.1d">italic_σ → 0</annotation></semantics></math>. To do so, we first introduce cluster points of sets.</p> </div> <div class="ltx_theorem ltx_theorem_definition" id="Thmdefinitionx1"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="Thmdefinitionx1.1.1.1">Definition</span></span><span class="ltx_text ltx_font_bold" id="Thmdefinitionx1.2.2">.</span> </h6> <div class="ltx_para" id="Thmdefinitionx1.p1"> <p class="ltx_p" id="Thmdefinitionx1.p1.9">For a sequence of sets <math alttext="(\mathbf{S}_{k})_{k\in{\mathbb{N}}}\in\left({\mathbb{R}}^{d}\right)^{{\mathbb{% N}}}" class="ltx_Math" display="inline" id="Thmdefinitionx1.p1.1.m1.2"><semantics id="Thmdefinitionx1.p1.1.m1.2a"><mrow id="Thmdefinitionx1.p1.1.m1.2.2" xref="Thmdefinitionx1.p1.1.m1.2.2.cmml"><msub id="Thmdefinitionx1.p1.1.m1.1.1.1" xref="Thmdefinitionx1.p1.1.m1.1.1.1.cmml"><mrow id="Thmdefinitionx1.p1.1.m1.1.1.1.1.1" xref="Thmdefinitionx1.p1.1.m1.1.1.1.1.1.1.cmml"><mo id="Thmdefinitionx1.p1.1.m1.1.1.1.1.1.2" stretchy="false" xref="Thmdefinitionx1.p1.1.m1.1.1.1.1.1.1.cmml">(</mo><msub id="Thmdefinitionx1.p1.1.m1.1.1.1.1.1.1" xref="Thmdefinitionx1.p1.1.m1.1.1.1.1.1.1.cmml"><mi id="Thmdefinitionx1.p1.1.m1.1.1.1.1.1.1.2" xref="Thmdefinitionx1.p1.1.m1.1.1.1.1.1.1.2.cmml">𝐒</mi><mi id="Thmdefinitionx1.p1.1.m1.1.1.1.1.1.1.3" xref="Thmdefinitionx1.p1.1.m1.1.1.1.1.1.1.3.cmml">k</mi></msub><mo id="Thmdefinitionx1.p1.1.m1.1.1.1.1.1.3" stretchy="false" xref="Thmdefinitionx1.p1.1.m1.1.1.1.1.1.1.cmml">)</mo></mrow><mrow id="Thmdefinitionx1.p1.1.m1.1.1.1.3" xref="Thmdefinitionx1.p1.1.m1.1.1.1.3.cmml"><mi id="Thmdefinitionx1.p1.1.m1.1.1.1.3.2" xref="Thmdefinitionx1.p1.1.m1.1.1.1.3.2.cmml">k</mi><mo id="Thmdefinitionx1.p1.1.m1.1.1.1.3.1" xref="Thmdefinitionx1.p1.1.m1.1.1.1.3.1.cmml">∈</mo><mi id="Thmdefinitionx1.p1.1.m1.1.1.1.3.3" xref="Thmdefinitionx1.p1.1.m1.1.1.1.3.3.cmml">ℕ</mi></mrow></msub><mo id="Thmdefinitionx1.p1.1.m1.2.2.3" xref="Thmdefinitionx1.p1.1.m1.2.2.3.cmml">∈</mo><msup id="Thmdefinitionx1.p1.1.m1.2.2.2" xref="Thmdefinitionx1.p1.1.m1.2.2.2.cmml"><mrow id="Thmdefinitionx1.p1.1.m1.2.2.2.1.1" xref="Thmdefinitionx1.p1.1.m1.2.2.2.1.1.1.cmml"><mo id="Thmdefinitionx1.p1.1.m1.2.2.2.1.1.2" xref="Thmdefinitionx1.p1.1.m1.2.2.2.1.1.1.cmml">(</mo><msup id="Thmdefinitionx1.p1.1.m1.2.2.2.1.1.1" xref="Thmdefinitionx1.p1.1.m1.2.2.2.1.1.1.cmml"><mi id="Thmdefinitionx1.p1.1.m1.2.2.2.1.1.1.2" xref="Thmdefinitionx1.p1.1.m1.2.2.2.1.1.1.2.cmml">ℝ</mi><mi id="Thmdefinitionx1.p1.1.m1.2.2.2.1.1.1.3" xref="Thmdefinitionx1.p1.1.m1.2.2.2.1.1.1.3.cmml">d</mi></msup><mo id="Thmdefinitionx1.p1.1.m1.2.2.2.1.1.3" xref="Thmdefinitionx1.p1.1.m1.2.2.2.1.1.1.cmml">)</mo></mrow><mi id="Thmdefinitionx1.p1.1.m1.2.2.2.3" xref="Thmdefinitionx1.p1.1.m1.2.2.2.3.cmml">ℕ</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="Thmdefinitionx1.p1.1.m1.2b"><apply id="Thmdefinitionx1.p1.1.m1.2.2.cmml" xref="Thmdefinitionx1.p1.1.m1.2.2"><in id="Thmdefinitionx1.p1.1.m1.2.2.3.cmml" xref="Thmdefinitionx1.p1.1.m1.2.2.3"></in><apply id="Thmdefinitionx1.p1.1.m1.1.1.1.cmml" xref="Thmdefinitionx1.p1.1.m1.1.1.1"><csymbol cd="ambiguous" id="Thmdefinitionx1.p1.1.m1.1.1.1.2.cmml" xref="Thmdefinitionx1.p1.1.m1.1.1.1">subscript</csymbol><apply id="Thmdefinitionx1.p1.1.m1.1.1.1.1.1.1.cmml" xref="Thmdefinitionx1.p1.1.m1.1.1.1.1.1"><csymbol cd="ambiguous" id="Thmdefinitionx1.p1.1.m1.1.1.1.1.1.1.1.cmml" xref="Thmdefinitionx1.p1.1.m1.1.1.1.1.1">subscript</csymbol><ci id="Thmdefinitionx1.p1.1.m1.1.1.1.1.1.1.2.cmml" xref="Thmdefinitionx1.p1.1.m1.1.1.1.1.1.1.2">𝐒</ci><ci id="Thmdefinitionx1.p1.1.m1.1.1.1.1.1.1.3.cmml" xref="Thmdefinitionx1.p1.1.m1.1.1.1.1.1.1.3">𝑘</ci></apply><apply id="Thmdefinitionx1.p1.1.m1.1.1.1.3.cmml" xref="Thmdefinitionx1.p1.1.m1.1.1.1.3"><in id="Thmdefinitionx1.p1.1.m1.1.1.1.3.1.cmml" xref="Thmdefinitionx1.p1.1.m1.1.1.1.3.1"></in><ci id="Thmdefinitionx1.p1.1.m1.1.1.1.3.2.cmml" xref="Thmdefinitionx1.p1.1.m1.1.1.1.3.2">𝑘</ci><ci id="Thmdefinitionx1.p1.1.m1.1.1.1.3.3.cmml" xref="Thmdefinitionx1.p1.1.m1.1.1.1.3.3">ℕ</ci></apply></apply><apply id="Thmdefinitionx1.p1.1.m1.2.2.2.cmml" xref="Thmdefinitionx1.p1.1.m1.2.2.2"><csymbol cd="ambiguous" id="Thmdefinitionx1.p1.1.m1.2.2.2.2.cmml" xref="Thmdefinitionx1.p1.1.m1.2.2.2">superscript</csymbol><apply id="Thmdefinitionx1.p1.1.m1.2.2.2.1.1.1.cmml" xref="Thmdefinitionx1.p1.1.m1.2.2.2.1.1"><csymbol cd="ambiguous" id="Thmdefinitionx1.p1.1.m1.2.2.2.1.1.1.1.cmml" xref="Thmdefinitionx1.p1.1.m1.2.2.2.1.1">superscript</csymbol><ci id="Thmdefinitionx1.p1.1.m1.2.2.2.1.1.1.2.cmml" xref="Thmdefinitionx1.p1.1.m1.2.2.2.1.1.1.2">ℝ</ci><ci id="Thmdefinitionx1.p1.1.m1.2.2.2.1.1.1.3.cmml" xref="Thmdefinitionx1.p1.1.m1.2.2.2.1.1.1.3">𝑑</ci></apply><ci id="Thmdefinitionx1.p1.1.m1.2.2.2.3.cmml" xref="Thmdefinitionx1.p1.1.m1.2.2.2.3">ℕ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmdefinitionx1.p1.1.m1.2c">(\mathbf{S}_{k})_{k\in{\mathbb{N}}}\in\left({\mathbb{R}}^{d}\right)^{{\mathbb{% N}}}</annotation><annotation encoding="application/x-llamapun" id="Thmdefinitionx1.p1.1.m1.2d">( bold_S start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) start_POSTSUBSCRIPT italic_k ∈ blackboard_N end_POSTSUBSCRIPT ∈ ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT blackboard_N end_POSTSUPERSCRIPT</annotation></semantics></math>, <math alttext="{\mathbf{z}}" class="ltx_Math" display="inline" id="Thmdefinitionx1.p1.2.m2.1"><semantics id="Thmdefinitionx1.p1.2.m2.1a"><mi id="Thmdefinitionx1.p1.2.m2.1.1" xref="Thmdefinitionx1.p1.2.m2.1.1.cmml">𝐳</mi><annotation-xml encoding="MathML-Content" id="Thmdefinitionx1.p1.2.m2.1b"><ci id="Thmdefinitionx1.p1.2.m2.1.1.cmml" xref="Thmdefinitionx1.p1.2.m2.1.1">𝐳</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmdefinitionx1.p1.2.m2.1c">{\mathbf{z}}</annotation><annotation encoding="application/x-llamapun" id="Thmdefinitionx1.p1.2.m2.1d">bold_z</annotation></semantics></math> is called a cluster point of these sets if any neighborhood of <math alttext="{\mathbf{z}}" class="ltx_Math" display="inline" id="Thmdefinitionx1.p1.3.m3.1"><semantics id="Thmdefinitionx1.p1.3.m3.1a"><mi id="Thmdefinitionx1.p1.3.m3.1.1" xref="Thmdefinitionx1.p1.3.m3.1.1.cmml">𝐳</mi><annotation-xml encoding="MathML-Content" id="Thmdefinitionx1.p1.3.m3.1b"><ci id="Thmdefinitionx1.p1.3.m3.1.1.cmml" xref="Thmdefinitionx1.p1.3.m3.1.1">𝐳</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmdefinitionx1.p1.3.m3.1c">{\mathbf{z}}</annotation><annotation encoding="application/x-llamapun" id="Thmdefinitionx1.p1.3.m3.1d">bold_z</annotation></semantics></math> is visited infinitely often by <math alttext="(\mathbf{S}_{k})" class="ltx_Math" display="inline" id="Thmdefinitionx1.p1.4.m4.1"><semantics id="Thmdefinitionx1.p1.4.m4.1a"><mrow id="Thmdefinitionx1.p1.4.m4.1.1.1" xref="Thmdefinitionx1.p1.4.m4.1.1.1.1.cmml"><mo id="Thmdefinitionx1.p1.4.m4.1.1.1.2" stretchy="false" xref="Thmdefinitionx1.p1.4.m4.1.1.1.1.cmml">(</mo><msub id="Thmdefinitionx1.p1.4.m4.1.1.1.1" xref="Thmdefinitionx1.p1.4.m4.1.1.1.1.cmml"><mi id="Thmdefinitionx1.p1.4.m4.1.1.1.1.2" xref="Thmdefinitionx1.p1.4.m4.1.1.1.1.2.cmml">𝐒</mi><mi id="Thmdefinitionx1.p1.4.m4.1.1.1.1.3" xref="Thmdefinitionx1.p1.4.m4.1.1.1.1.3.cmml">k</mi></msub><mo id="Thmdefinitionx1.p1.4.m4.1.1.1.3" stretchy="false" xref="Thmdefinitionx1.p1.4.m4.1.1.1.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="Thmdefinitionx1.p1.4.m4.1b"><apply id="Thmdefinitionx1.p1.4.m4.1.1.1.1.cmml" xref="Thmdefinitionx1.p1.4.m4.1.1.1"><csymbol cd="ambiguous" id="Thmdefinitionx1.p1.4.m4.1.1.1.1.1.cmml" xref="Thmdefinitionx1.p1.4.m4.1.1.1">subscript</csymbol><ci id="Thmdefinitionx1.p1.4.m4.1.1.1.1.2.cmml" xref="Thmdefinitionx1.p1.4.m4.1.1.1.1.2">𝐒</ci><ci id="Thmdefinitionx1.p1.4.m4.1.1.1.1.3.cmml" xref="Thmdefinitionx1.p1.4.m4.1.1.1.1.3">𝑘</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmdefinitionx1.p1.4.m4.1c">(\mathbf{S}_{k})</annotation><annotation encoding="application/x-llamapun" id="Thmdefinitionx1.p1.4.m4.1d">( bold_S start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT )</annotation></semantics></math>, i.e. <math alttext="\forall\epsilon>0" class="ltx_Math" display="inline" id="Thmdefinitionx1.p1.5.m5.1"><semantics id="Thmdefinitionx1.p1.5.m5.1a"><mrow id="Thmdefinitionx1.p1.5.m5.1.1" xref="Thmdefinitionx1.p1.5.m5.1.1.cmml"><mrow id="Thmdefinitionx1.p1.5.m5.1.1.2" xref="Thmdefinitionx1.p1.5.m5.1.1.2.cmml"><mo id="Thmdefinitionx1.p1.5.m5.1.1.2.1" rspace="0.167em" xref="Thmdefinitionx1.p1.5.m5.1.1.2.1.cmml">∀</mo><mi id="Thmdefinitionx1.p1.5.m5.1.1.2.2" xref="Thmdefinitionx1.p1.5.m5.1.1.2.2.cmml">ϵ</mi></mrow><mo id="Thmdefinitionx1.p1.5.m5.1.1.1" xref="Thmdefinitionx1.p1.5.m5.1.1.1.cmml">></mo><mn id="Thmdefinitionx1.p1.5.m5.1.1.3" xref="Thmdefinitionx1.p1.5.m5.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="Thmdefinitionx1.p1.5.m5.1b"><apply id="Thmdefinitionx1.p1.5.m5.1.1.cmml" xref="Thmdefinitionx1.p1.5.m5.1.1"><gt id="Thmdefinitionx1.p1.5.m5.1.1.1.cmml" xref="Thmdefinitionx1.p1.5.m5.1.1.1"></gt><apply id="Thmdefinitionx1.p1.5.m5.1.1.2.cmml" xref="Thmdefinitionx1.p1.5.m5.1.1.2"><csymbol cd="latexml" id="Thmdefinitionx1.p1.5.m5.1.1.2.1.cmml" xref="Thmdefinitionx1.p1.5.m5.1.1.2.1">for-all</csymbol><ci id="Thmdefinitionx1.p1.5.m5.1.1.2.2.cmml" xref="Thmdefinitionx1.p1.5.m5.1.1.2.2">italic-ϵ</ci></apply><cn id="Thmdefinitionx1.p1.5.m5.1.1.3.cmml" type="integer" xref="Thmdefinitionx1.p1.5.m5.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmdefinitionx1.p1.5.m5.1c">\forall\epsilon>0</annotation><annotation encoding="application/x-llamapun" id="Thmdefinitionx1.p1.5.m5.1d">∀ italic_ϵ > 0</annotation></semantics></math>, <math alttext="\forall k_{0}\in{\mathbb{N}}" class="ltx_Math" display="inline" id="Thmdefinitionx1.p1.6.m6.1"><semantics id="Thmdefinitionx1.p1.6.m6.1a"><mrow id="Thmdefinitionx1.p1.6.m6.1.1" xref="Thmdefinitionx1.p1.6.m6.1.1.cmml"><mrow id="Thmdefinitionx1.p1.6.m6.1.1.2" xref="Thmdefinitionx1.p1.6.m6.1.1.2.cmml"><mo id="Thmdefinitionx1.p1.6.m6.1.1.2.1" rspace="0.167em" xref="Thmdefinitionx1.p1.6.m6.1.1.2.1.cmml">∀</mo><msub id="Thmdefinitionx1.p1.6.m6.1.1.2.2" xref="Thmdefinitionx1.p1.6.m6.1.1.2.2.cmml"><mi id="Thmdefinitionx1.p1.6.m6.1.1.2.2.2" xref="Thmdefinitionx1.p1.6.m6.1.1.2.2.2.cmml">k</mi><mn id="Thmdefinitionx1.p1.6.m6.1.1.2.2.3" xref="Thmdefinitionx1.p1.6.m6.1.1.2.2.3.cmml">0</mn></msub></mrow><mo id="Thmdefinitionx1.p1.6.m6.1.1.1" xref="Thmdefinitionx1.p1.6.m6.1.1.1.cmml">∈</mo><mi id="Thmdefinitionx1.p1.6.m6.1.1.3" xref="Thmdefinitionx1.p1.6.m6.1.1.3.cmml">ℕ</mi></mrow><annotation-xml encoding="MathML-Content" id="Thmdefinitionx1.p1.6.m6.1b"><apply id="Thmdefinitionx1.p1.6.m6.1.1.cmml" xref="Thmdefinitionx1.p1.6.m6.1.1"><in id="Thmdefinitionx1.p1.6.m6.1.1.1.cmml" xref="Thmdefinitionx1.p1.6.m6.1.1.1"></in><apply id="Thmdefinitionx1.p1.6.m6.1.1.2.cmml" xref="Thmdefinitionx1.p1.6.m6.1.1.2"><csymbol cd="latexml" id="Thmdefinitionx1.p1.6.m6.1.1.2.1.cmml" xref="Thmdefinitionx1.p1.6.m6.1.1.2.1">for-all</csymbol><apply id="Thmdefinitionx1.p1.6.m6.1.1.2.2.cmml" xref="Thmdefinitionx1.p1.6.m6.1.1.2.2"><csymbol cd="ambiguous" id="Thmdefinitionx1.p1.6.m6.1.1.2.2.1.cmml" xref="Thmdefinitionx1.p1.6.m6.1.1.2.2">subscript</csymbol><ci id="Thmdefinitionx1.p1.6.m6.1.1.2.2.2.cmml" xref="Thmdefinitionx1.p1.6.m6.1.1.2.2.2">𝑘</ci><cn id="Thmdefinitionx1.p1.6.m6.1.1.2.2.3.cmml" type="integer" xref="Thmdefinitionx1.p1.6.m6.1.1.2.2.3">0</cn></apply></apply><ci id="Thmdefinitionx1.p1.6.m6.1.1.3.cmml" xref="Thmdefinitionx1.p1.6.m6.1.1.3">ℕ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmdefinitionx1.p1.6.m6.1c">\forall k_{0}\in{\mathbb{N}}</annotation><annotation encoding="application/x-llamapun" id="Thmdefinitionx1.p1.6.m6.1d">∀ italic_k start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ∈ blackboard_N</annotation></semantics></math>, there exist <math alttext="k\geq k_{0}" class="ltx_Math" display="inline" id="Thmdefinitionx1.p1.7.m7.1"><semantics id="Thmdefinitionx1.p1.7.m7.1a"><mrow id="Thmdefinitionx1.p1.7.m7.1.1" xref="Thmdefinitionx1.p1.7.m7.1.1.cmml"><mi id="Thmdefinitionx1.p1.7.m7.1.1.2" xref="Thmdefinitionx1.p1.7.m7.1.1.2.cmml">k</mi><mo id="Thmdefinitionx1.p1.7.m7.1.1.1" xref="Thmdefinitionx1.p1.7.m7.1.1.1.cmml">≥</mo><msub id="Thmdefinitionx1.p1.7.m7.1.1.3" xref="Thmdefinitionx1.p1.7.m7.1.1.3.cmml"><mi id="Thmdefinitionx1.p1.7.m7.1.1.3.2" xref="Thmdefinitionx1.p1.7.m7.1.1.3.2.cmml">k</mi><mn id="Thmdefinitionx1.p1.7.m7.1.1.3.3" xref="Thmdefinitionx1.p1.7.m7.1.1.3.3.cmml">0</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="Thmdefinitionx1.p1.7.m7.1b"><apply id="Thmdefinitionx1.p1.7.m7.1.1.cmml" xref="Thmdefinitionx1.p1.7.m7.1.1"><geq id="Thmdefinitionx1.p1.7.m7.1.1.1.cmml" xref="Thmdefinitionx1.p1.7.m7.1.1.1"></geq><ci id="Thmdefinitionx1.p1.7.m7.1.1.2.cmml" xref="Thmdefinitionx1.p1.7.m7.1.1.2">𝑘</ci><apply id="Thmdefinitionx1.p1.7.m7.1.1.3.cmml" xref="Thmdefinitionx1.p1.7.m7.1.1.3"><csymbol cd="ambiguous" id="Thmdefinitionx1.p1.7.m7.1.1.3.1.cmml" xref="Thmdefinitionx1.p1.7.m7.1.1.3">subscript</csymbol><ci id="Thmdefinitionx1.p1.7.m7.1.1.3.2.cmml" xref="Thmdefinitionx1.p1.7.m7.1.1.3.2">𝑘</ci><cn id="Thmdefinitionx1.p1.7.m7.1.1.3.3.cmml" type="integer" xref="Thmdefinitionx1.p1.7.m7.1.1.3.3">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmdefinitionx1.p1.7.m7.1c">k\geq k_{0}</annotation><annotation encoding="application/x-llamapun" id="Thmdefinitionx1.p1.7.m7.1d">italic_k ≥ italic_k start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math>, and <math alttext="{\mathbf{x}}_{k}\in\mathbf{S}_{k}" class="ltx_Math" display="inline" id="Thmdefinitionx1.p1.8.m8.1"><semantics id="Thmdefinitionx1.p1.8.m8.1a"><mrow id="Thmdefinitionx1.p1.8.m8.1.1" xref="Thmdefinitionx1.p1.8.m8.1.1.cmml"><msub id="Thmdefinitionx1.p1.8.m8.1.1.2" xref="Thmdefinitionx1.p1.8.m8.1.1.2.cmml"><mi id="Thmdefinitionx1.p1.8.m8.1.1.2.2" xref="Thmdefinitionx1.p1.8.m8.1.1.2.2.cmml">𝐱</mi><mi id="Thmdefinitionx1.p1.8.m8.1.1.2.3" xref="Thmdefinitionx1.p1.8.m8.1.1.2.3.cmml">k</mi></msub><mo id="Thmdefinitionx1.p1.8.m8.1.1.1" xref="Thmdefinitionx1.p1.8.m8.1.1.1.cmml">∈</mo><msub id="Thmdefinitionx1.p1.8.m8.1.1.3" xref="Thmdefinitionx1.p1.8.m8.1.1.3.cmml"><mi id="Thmdefinitionx1.p1.8.m8.1.1.3.2" xref="Thmdefinitionx1.p1.8.m8.1.1.3.2.cmml">𝐒</mi><mi id="Thmdefinitionx1.p1.8.m8.1.1.3.3" xref="Thmdefinitionx1.p1.8.m8.1.1.3.3.cmml">k</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="Thmdefinitionx1.p1.8.m8.1b"><apply id="Thmdefinitionx1.p1.8.m8.1.1.cmml" xref="Thmdefinitionx1.p1.8.m8.1.1"><in id="Thmdefinitionx1.p1.8.m8.1.1.1.cmml" xref="Thmdefinitionx1.p1.8.m8.1.1.1"></in><apply id="Thmdefinitionx1.p1.8.m8.1.1.2.cmml" xref="Thmdefinitionx1.p1.8.m8.1.1.2"><csymbol cd="ambiguous" id="Thmdefinitionx1.p1.8.m8.1.1.2.1.cmml" xref="Thmdefinitionx1.p1.8.m8.1.1.2">subscript</csymbol><ci id="Thmdefinitionx1.p1.8.m8.1.1.2.2.cmml" xref="Thmdefinitionx1.p1.8.m8.1.1.2.2">𝐱</ci><ci id="Thmdefinitionx1.p1.8.m8.1.1.2.3.cmml" xref="Thmdefinitionx1.p1.8.m8.1.1.2.3">𝑘</ci></apply><apply id="Thmdefinitionx1.p1.8.m8.1.1.3.cmml" xref="Thmdefinitionx1.p1.8.m8.1.1.3"><csymbol cd="ambiguous" id="Thmdefinitionx1.p1.8.m8.1.1.3.1.cmml" xref="Thmdefinitionx1.p1.8.m8.1.1.3">subscript</csymbol><ci id="Thmdefinitionx1.p1.8.m8.1.1.3.2.cmml" xref="Thmdefinitionx1.p1.8.m8.1.1.3.2">𝐒</ci><ci id="Thmdefinitionx1.p1.8.m8.1.1.3.3.cmml" xref="Thmdefinitionx1.p1.8.m8.1.1.3.3">𝑘</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmdefinitionx1.p1.8.m8.1c">{\mathbf{x}}_{k}\in\mathbf{S}_{k}</annotation><annotation encoding="application/x-llamapun" id="Thmdefinitionx1.p1.8.m8.1d">bold_x start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ∈ bold_S start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT</annotation></semantics></math> such that <math alttext="\|{\mathbf{x}}_{k}-{\mathbf{z}}\|\leq\epsilon" class="ltx_Math" display="inline" id="Thmdefinitionx1.p1.9.m9.1"><semantics id="Thmdefinitionx1.p1.9.m9.1a"><mrow id="Thmdefinitionx1.p1.9.m9.1.1" xref="Thmdefinitionx1.p1.9.m9.1.1.cmml"><mrow id="Thmdefinitionx1.p1.9.m9.1.1.1.1" xref="Thmdefinitionx1.p1.9.m9.1.1.1.2.cmml"><mo id="Thmdefinitionx1.p1.9.m9.1.1.1.1.2" stretchy="false" xref="Thmdefinitionx1.p1.9.m9.1.1.1.2.1.cmml">‖</mo><mrow id="Thmdefinitionx1.p1.9.m9.1.1.1.1.1" xref="Thmdefinitionx1.p1.9.m9.1.1.1.1.1.cmml"><msub id="Thmdefinitionx1.p1.9.m9.1.1.1.1.1.2" xref="Thmdefinitionx1.p1.9.m9.1.1.1.1.1.2.cmml"><mi id="Thmdefinitionx1.p1.9.m9.1.1.1.1.1.2.2" xref="Thmdefinitionx1.p1.9.m9.1.1.1.1.1.2.2.cmml">𝐱</mi><mi id="Thmdefinitionx1.p1.9.m9.1.1.1.1.1.2.3" xref="Thmdefinitionx1.p1.9.m9.1.1.1.1.1.2.3.cmml">k</mi></msub><mo id="Thmdefinitionx1.p1.9.m9.1.1.1.1.1.1" xref="Thmdefinitionx1.p1.9.m9.1.1.1.1.1.1.cmml">−</mo><mi id="Thmdefinitionx1.p1.9.m9.1.1.1.1.1.3" xref="Thmdefinitionx1.p1.9.m9.1.1.1.1.1.3.cmml">𝐳</mi></mrow><mo id="Thmdefinitionx1.p1.9.m9.1.1.1.1.3" stretchy="false" xref="Thmdefinitionx1.p1.9.m9.1.1.1.2.1.cmml">‖</mo></mrow><mo id="Thmdefinitionx1.p1.9.m9.1.1.2" xref="Thmdefinitionx1.p1.9.m9.1.1.2.cmml">≤</mo><mi id="Thmdefinitionx1.p1.9.m9.1.1.3" xref="Thmdefinitionx1.p1.9.m9.1.1.3.cmml">ϵ</mi></mrow><annotation-xml encoding="MathML-Content" id="Thmdefinitionx1.p1.9.m9.1b"><apply id="Thmdefinitionx1.p1.9.m9.1.1.cmml" xref="Thmdefinitionx1.p1.9.m9.1.1"><leq id="Thmdefinitionx1.p1.9.m9.1.1.2.cmml" xref="Thmdefinitionx1.p1.9.m9.1.1.2"></leq><apply id="Thmdefinitionx1.p1.9.m9.1.1.1.2.cmml" xref="Thmdefinitionx1.p1.9.m9.1.1.1.1"><csymbol cd="latexml" id="Thmdefinitionx1.p1.9.m9.1.1.1.2.1.cmml" xref="Thmdefinitionx1.p1.9.m9.1.1.1.1.2">norm</csymbol><apply id="Thmdefinitionx1.p1.9.m9.1.1.1.1.1.cmml" xref="Thmdefinitionx1.p1.9.m9.1.1.1.1.1"><minus id="Thmdefinitionx1.p1.9.m9.1.1.1.1.1.1.cmml" xref="Thmdefinitionx1.p1.9.m9.1.1.1.1.1.1"></minus><apply id="Thmdefinitionx1.p1.9.m9.1.1.1.1.1.2.cmml" xref="Thmdefinitionx1.p1.9.m9.1.1.1.1.1.2"><csymbol cd="ambiguous" id="Thmdefinitionx1.p1.9.m9.1.1.1.1.1.2.1.cmml" xref="Thmdefinitionx1.p1.9.m9.1.1.1.1.1.2">subscript</csymbol><ci id="Thmdefinitionx1.p1.9.m9.1.1.1.1.1.2.2.cmml" xref="Thmdefinitionx1.p1.9.m9.1.1.1.1.1.2.2">𝐱</ci><ci id="Thmdefinitionx1.p1.9.m9.1.1.1.1.1.2.3.cmml" xref="Thmdefinitionx1.p1.9.m9.1.1.1.1.1.2.3">𝑘</ci></apply><ci id="Thmdefinitionx1.p1.9.m9.1.1.1.1.1.3.cmml" xref="Thmdefinitionx1.p1.9.m9.1.1.1.1.1.3">𝐳</ci></apply></apply><ci id="Thmdefinitionx1.p1.9.m9.1.1.3.cmml" xref="Thmdefinitionx1.p1.9.m9.1.1.3">italic-ϵ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmdefinitionx1.p1.9.m9.1c">\|{\mathbf{x}}_{k}-{\mathbf{z}}\|\leq\epsilon</annotation><annotation encoding="application/x-llamapun" id="Thmdefinitionx1.p1.9.m9.1d">∥ bold_x start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT - bold_z ∥ ≤ italic_ϵ</annotation></semantics></math>.</p> </div> </div> <div class="ltx_para" id="S3.SS1.p5"> <p class="ltx_p" id="S3.SS1.p5.11">We now apply this definition of cluster points of sets for a decreasing sequence of <math alttext="\sigma>0" class="ltx_Math" display="inline" id="S3.SS1.p5.1.m1.1"><semantics id="S3.SS1.p5.1.m1.1a"><mrow id="S3.SS1.p5.1.m1.1.1" xref="S3.SS1.p5.1.m1.1.1.cmml"><mi id="S3.SS1.p5.1.m1.1.1.2" xref="S3.SS1.p5.1.m1.1.1.2.cmml">σ</mi><mo id="S3.SS1.p5.1.m1.1.1.1" xref="S3.SS1.p5.1.m1.1.1.1.cmml">></mo><mn id="S3.SS1.p5.1.m1.1.1.3" xref="S3.SS1.p5.1.m1.1.1.3.cmml">0</mn></mrow><annotation-xml 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xref="S3.SS1.p5.2.m2.4.4.2.2.2.2"><csymbol cd="ambiguous" id="S3.SS1.p5.2.m2.4.4.2.2.2.3a.cmml" xref="S3.SS1.p5.2.m2.4.4.2.2.2.2.3">formulae-sequence</csymbol><apply id="S3.SS1.p5.2.m2.4.4.2.2.2.1.1.cmml" xref="S3.SS1.p5.2.m2.4.4.2.2.2.1.1"><in id="S3.SS1.p5.2.m2.4.4.2.2.2.1.1.1.cmml" xref="S3.SS1.p5.2.m2.4.4.2.2.2.1.1.1"></in><apply id="S3.SS1.p5.2.m2.4.4.2.2.2.1.1.2.cmml" xref="S3.SS1.p5.2.m2.4.4.2.2.2.1.1.2"><csymbol cd="latexml" id="S3.SS1.p5.2.m2.4.4.2.2.2.1.1.2.1.cmml" xref="S3.SS1.p5.2.m2.4.4.2.2.2.1.1.2.1">for-all</csymbol><ci id="S3.SS1.p5.2.m2.4.4.2.2.2.1.1.2.2.cmml" xref="S3.SS1.p5.2.m2.4.4.2.2.2.1.1.2.2">𝑛</ci></apply><ci id="S3.SS1.p5.2.m2.4.4.2.2.2.1.1.3.cmml" xref="S3.SS1.p5.2.m2.4.4.2.2.2.1.1.3">ℕ</ci></apply><apply id="S3.SS1.p5.2.m2.4.4.2.2.2.2.2.cmml" xref="S3.SS1.p5.2.m2.4.4.2.2.2.2.2"><gt id="S3.SS1.p5.2.m2.4.4.2.2.2.2.2.2.cmml" xref="S3.SS1.p5.2.m2.4.4.2.2.2.2.2.2"></gt><apply id="S3.SS1.p5.2.m2.4.4.2.2.2.2.2.3.cmml" xref="S3.SS1.p5.2.m2.4.4.2.2.2.2.2.3"><csymbol cd="ambiguous" id="S3.SS1.p5.2.m2.4.4.2.2.2.2.2.3.1.cmml" xref="S3.SS1.p5.2.m2.4.4.2.2.2.2.2.3">subscript</csymbol><ci id="S3.SS1.p5.2.m2.4.4.2.2.2.2.2.3.2.cmml" xref="S3.SS1.p5.2.m2.4.4.2.2.2.2.2.3.2">𝜎</ci><ci id="S3.SS1.p5.2.m2.4.4.2.2.2.2.2.3.3.cmml" xref="S3.SS1.p5.2.m2.4.4.2.2.2.2.2.3.3">𝑛</ci></apply><list id="S3.SS1.p5.2.m2.4.4.2.2.2.2.2.1.2.cmml" xref="S3.SS1.p5.2.m2.4.4.2.2.2.2.2.1.1"><cn id="S3.SS1.p5.2.m2.2.2.cmml" type="integer" xref="S3.SS1.p5.2.m2.2.2">0</cn><apply id="S3.SS1.p5.2.m2.4.4.2.2.2.2.2.1.1.1.cmml" xref="S3.SS1.p5.2.m2.4.4.2.2.2.2.2.1.1.1"><times id="S3.SS1.p5.2.m2.4.4.2.2.2.2.2.1.1.1.1.cmml" xref="S3.SS1.p5.2.m2.4.4.2.2.2.2.2.1.1.1.1"></times><apply id="S3.SS1.p5.2.m2.4.4.2.2.2.2.2.1.1.1.2.cmml" xref="S3.SS1.p5.2.m2.4.4.2.2.2.2.2.1.1.1.2"><csymbol cd="ambiguous" id="S3.SS1.p5.2.m2.4.4.2.2.2.2.2.1.1.1.2.1.cmml" xref="S3.SS1.p5.2.m2.4.4.2.2.2.2.2.1.1.1.2">subscript</csymbol><ci id="S3.SS1.p5.2.m2.4.4.2.2.2.2.2.1.1.1.2.2.cmml" xref="S3.SS1.p5.2.m2.4.4.2.2.2.2.2.1.1.1.2.2">𝜎</ci><ci id="S3.SS1.p5.2.m2.4.4.2.2.2.2.2.1.1.1.2.3.cmml" xref="S3.SS1.p5.2.m2.4.4.2.2.2.2.2.1.1.1.2.3">𝑛</ci></apply><ci id="S3.SS1.p5.2.m2.4.4.2.2.2.2.2.1.1.1.3a.cmml" xref="S3.SS1.p5.2.m2.4.4.2.2.2.2.2.1.1.1.3"><mtext id="S3.SS1.p5.2.m2.4.4.2.2.2.2.2.1.1.1.3.cmml" xref="S3.SS1.p5.2.m2.4.4.2.2.2.2.2.1.1.1.3"> decreases to </mtext></ci><cn id="S3.SS1.p5.2.m2.4.4.2.2.2.2.2.1.1.1.4.cmml" type="integer" xref="S3.SS1.p5.2.m2.4.4.2.2.2.2.2.1.1.1.4">0</cn></apply></list></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p5.2.m2.4c">\mathbf{E}=\{(\sigma_{n})_{n\in{\mathbb{N}}}\in({\mathbb{R}})^{{\mathbb{N}}}|% \forall n\in{\mathbb{N}},\sigma_{n}>0,\sigma_{n}\text{ decreases to }0\}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p5.2.m2.4d">bold_E = { ( italic_σ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ) start_POSTSUBSCRIPT italic_n ∈ blackboard_N end_POSTSUBSCRIPT ∈ ( blackboard_R ) start_POSTSUPERSCRIPT blackboard_N end_POSTSUPERSCRIPT | ∀ italic_n ∈ blackboard_N , italic_σ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT > 0 , italic_σ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT decreases to 0 }</annotation></semantics></math>, and for <math alttext="\boldsymbol{\sigma}=(\sigma_{n})_{n\in{\mathbb{N}}}\in\mathbf{E}" class="ltx_Math" display="inline" id="S3.SS1.p5.3.m3.1"><semantics id="S3.SS1.p5.3.m3.1a"><mrow id="S3.SS1.p5.3.m3.1.1" xref="S3.SS1.p5.3.m3.1.1.cmml"><mi id="S3.SS1.p5.3.m3.1.1.3" xref="S3.SS1.p5.3.m3.1.1.3.cmml">𝝈</mi><mo id="S3.SS1.p5.3.m3.1.1.4" xref="S3.SS1.p5.3.m3.1.1.4.cmml">=</mo><msub id="S3.SS1.p5.3.m3.1.1.1" xref="S3.SS1.p5.3.m3.1.1.1.cmml"><mrow id="S3.SS1.p5.3.m3.1.1.1.1.1" xref="S3.SS1.p5.3.m3.1.1.1.1.1.1.cmml"><mo id="S3.SS1.p5.3.m3.1.1.1.1.1.2" stretchy="false" xref="S3.SS1.p5.3.m3.1.1.1.1.1.1.cmml">(</mo><msub id="S3.SS1.p5.3.m3.1.1.1.1.1.1" xref="S3.SS1.p5.3.m3.1.1.1.1.1.1.cmml"><mi id="S3.SS1.p5.3.m3.1.1.1.1.1.1.2" xref="S3.SS1.p5.3.m3.1.1.1.1.1.1.2.cmml">σ</mi><mi id="S3.SS1.p5.3.m3.1.1.1.1.1.1.3" xref="S3.SS1.p5.3.m3.1.1.1.1.1.1.3.cmml">n</mi></msub><mo id="S3.SS1.p5.3.m3.1.1.1.1.1.3" stretchy="false" xref="S3.SS1.p5.3.m3.1.1.1.1.1.1.cmml">)</mo></mrow><mrow id="S3.SS1.p5.3.m3.1.1.1.3" xref="S3.SS1.p5.3.m3.1.1.1.3.cmml"><mi id="S3.SS1.p5.3.m3.1.1.1.3.2" xref="S3.SS1.p5.3.m3.1.1.1.3.2.cmml">n</mi><mo id="S3.SS1.p5.3.m3.1.1.1.3.1" xref="S3.SS1.p5.3.m3.1.1.1.3.1.cmml">∈</mo><mi id="S3.SS1.p5.3.m3.1.1.1.3.3" xref="S3.SS1.p5.3.m3.1.1.1.3.3.cmml">ℕ</mi></mrow></msub><mo id="S3.SS1.p5.3.m3.1.1.5" xref="S3.SS1.p5.3.m3.1.1.5.cmml">∈</mo><mi id="S3.SS1.p5.3.m3.1.1.6" xref="S3.SS1.p5.3.m3.1.1.6.cmml">𝐄</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p5.3.m3.1b"><apply id="S3.SS1.p5.3.m3.1.1.cmml" xref="S3.SS1.p5.3.m3.1.1"><and id="S3.SS1.p5.3.m3.1.1a.cmml" xref="S3.SS1.p5.3.m3.1.1"></and><apply id="S3.SS1.p5.3.m3.1.1b.cmml" xref="S3.SS1.p5.3.m3.1.1"><eq id="S3.SS1.p5.3.m3.1.1.4.cmml" xref="S3.SS1.p5.3.m3.1.1.4"></eq><ci id="S3.SS1.p5.3.m3.1.1.3.cmml" xref="S3.SS1.p5.3.m3.1.1.3">𝝈</ci><apply id="S3.SS1.p5.3.m3.1.1.1.cmml" xref="S3.SS1.p5.3.m3.1.1.1"><csymbol cd="ambiguous" id="S3.SS1.p5.3.m3.1.1.1.2.cmml" xref="S3.SS1.p5.3.m3.1.1.1">subscript</csymbol><apply id="S3.SS1.p5.3.m3.1.1.1.1.1.1.cmml" xref="S3.SS1.p5.3.m3.1.1.1.1.1"><csymbol cd="ambiguous" id="S3.SS1.p5.3.m3.1.1.1.1.1.1.1.cmml" xref="S3.SS1.p5.3.m3.1.1.1.1.1">subscript</csymbol><ci id="S3.SS1.p5.3.m3.1.1.1.1.1.1.2.cmml" xref="S3.SS1.p5.3.m3.1.1.1.1.1.1.2">𝜎</ci><ci id="S3.SS1.p5.3.m3.1.1.1.1.1.1.3.cmml" xref="S3.SS1.p5.3.m3.1.1.1.1.1.1.3">𝑛</ci></apply><apply id="S3.SS1.p5.3.m3.1.1.1.3.cmml" xref="S3.SS1.p5.3.m3.1.1.1.3"><in id="S3.SS1.p5.3.m3.1.1.1.3.1.cmml" xref="S3.SS1.p5.3.m3.1.1.1.3.1"></in><ci id="S3.SS1.p5.3.m3.1.1.1.3.2.cmml" xref="S3.SS1.p5.3.m3.1.1.1.3.2">𝑛</ci><ci id="S3.SS1.p5.3.m3.1.1.1.3.3.cmml" xref="S3.SS1.p5.3.m3.1.1.1.3.3">ℕ</ci></apply></apply></apply><apply id="S3.SS1.p5.3.m3.1.1c.cmml" xref="S3.SS1.p5.3.m3.1.1"><in id="S3.SS1.p5.3.m3.1.1.5.cmml" xref="S3.SS1.p5.3.m3.1.1.5"></in><share href="#S3.SS1.p5.3.m3.1.1.1.cmml" id="S3.SS1.p5.3.m3.1.1d.cmml" xref="S3.SS1.p5.3.m3.1.1"></share><ci id="S3.SS1.p5.3.m3.1.1.6.cmml" xref="S3.SS1.p5.3.m3.1.1.6">𝐄</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p5.3.m3.1c">\boldsymbol{\sigma}=(\sigma_{n})_{n\in{\mathbb{N}}}\in\mathbf{E}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p5.3.m3.1d">bold_italic_σ = ( italic_σ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ) start_POSTSUBSCRIPT italic_n ∈ blackboard_N end_POSTSUBSCRIPT ∈ bold_E</annotation></semantics></math>, we define the cluster points of the sequence of set <math alttext="(\mathbf{S}_{\sigma_{n},\mathbf{K}})_{n\in{\mathbb{N}}}" class="ltx_Math" display="inline" id="S3.SS1.p5.4.m4.3"><semantics id="S3.SS1.p5.4.m4.3a"><msub id="S3.SS1.p5.4.m4.3.3" xref="S3.SS1.p5.4.m4.3.3.cmml"><mrow id="S3.SS1.p5.4.m4.3.3.1.1" xref="S3.SS1.p5.4.m4.3.3.1.1.1.cmml"><mo id="S3.SS1.p5.4.m4.3.3.1.1.2" stretchy="false" xref="S3.SS1.p5.4.m4.3.3.1.1.1.cmml">(</mo><msub id="S3.SS1.p5.4.m4.3.3.1.1.1" xref="S3.SS1.p5.4.m4.3.3.1.1.1.cmml"><mi id="S3.SS1.p5.4.m4.3.3.1.1.1.2" xref="S3.SS1.p5.4.m4.3.3.1.1.1.2.cmml">𝐒</mi><mrow id="S3.SS1.p5.4.m4.2.2.2.2" xref="S3.SS1.p5.4.m4.2.2.2.3.cmml"><msub id="S3.SS1.p5.4.m4.2.2.2.2.1" xref="S3.SS1.p5.4.m4.2.2.2.2.1.cmml"><mi id="S3.SS1.p5.4.m4.2.2.2.2.1.2" xref="S3.SS1.p5.4.m4.2.2.2.2.1.2.cmml">σ</mi><mi id="S3.SS1.p5.4.m4.2.2.2.2.1.3" xref="S3.SS1.p5.4.m4.2.2.2.2.1.3.cmml">n</mi></msub><mo id="S3.SS1.p5.4.m4.2.2.2.2.2" xref="S3.SS1.p5.4.m4.2.2.2.3.cmml">,</mo><mi id="S3.SS1.p5.4.m4.1.1.1.1" xref="S3.SS1.p5.4.m4.1.1.1.1.cmml">𝐊</mi></mrow></msub><mo id="S3.SS1.p5.4.m4.3.3.1.1.3" stretchy="false" xref="S3.SS1.p5.4.m4.3.3.1.1.1.cmml">)</mo></mrow><mrow id="S3.SS1.p5.4.m4.3.3.3" xref="S3.SS1.p5.4.m4.3.3.3.cmml"><mi id="S3.SS1.p5.4.m4.3.3.3.2" xref="S3.SS1.p5.4.m4.3.3.3.2.cmml">n</mi><mo id="S3.SS1.p5.4.m4.3.3.3.1" xref="S3.SS1.p5.4.m4.3.3.3.1.cmml">∈</mo><mi id="S3.SS1.p5.4.m4.3.3.3.3" xref="S3.SS1.p5.4.m4.3.3.3.3.cmml">ℕ</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="S3.SS1.p5.4.m4.3b"><apply id="S3.SS1.p5.4.m4.3.3.cmml" xref="S3.SS1.p5.4.m4.3.3"><csymbol cd="ambiguous" id="S3.SS1.p5.4.m4.3.3.2.cmml" xref="S3.SS1.p5.4.m4.3.3">subscript</csymbol><apply id="S3.SS1.p5.4.m4.3.3.1.1.1.cmml" xref="S3.SS1.p5.4.m4.3.3.1.1"><csymbol cd="ambiguous" id="S3.SS1.p5.4.m4.3.3.1.1.1.1.cmml" xref="S3.SS1.p5.4.m4.3.3.1.1">subscript</csymbol><ci id="S3.SS1.p5.4.m4.3.3.1.1.1.2.cmml" xref="S3.SS1.p5.4.m4.3.3.1.1.1.2">𝐒</ci><list id="S3.SS1.p5.4.m4.2.2.2.3.cmml" xref="S3.SS1.p5.4.m4.2.2.2.2"><apply id="S3.SS1.p5.4.m4.2.2.2.2.1.cmml" xref="S3.SS1.p5.4.m4.2.2.2.2.1"><csymbol cd="ambiguous" id="S3.SS1.p5.4.m4.2.2.2.2.1.1.cmml" xref="S3.SS1.p5.4.m4.2.2.2.2.1">subscript</csymbol><ci id="S3.SS1.p5.4.m4.2.2.2.2.1.2.cmml" xref="S3.SS1.p5.4.m4.2.2.2.2.1.2">𝜎</ci><ci id="S3.SS1.p5.4.m4.2.2.2.2.1.3.cmml" xref="S3.SS1.p5.4.m4.2.2.2.2.1.3">𝑛</ci></apply><ci id="S3.SS1.p5.4.m4.1.1.1.1.cmml" xref="S3.SS1.p5.4.m4.1.1.1.1">𝐊</ci></list></apply><apply id="S3.SS1.p5.4.m4.3.3.3.cmml" xref="S3.SS1.p5.4.m4.3.3.3"><in id="S3.SS1.p5.4.m4.3.3.3.1.cmml" xref="S3.SS1.p5.4.m4.3.3.3.1"></in><ci id="S3.SS1.p5.4.m4.3.3.3.2.cmml" xref="S3.SS1.p5.4.m4.3.3.3.2">𝑛</ci><ci id="S3.SS1.p5.4.m4.3.3.3.3.cmml" xref="S3.SS1.p5.4.m4.3.3.3.3">ℕ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p5.4.m4.3c">(\mathbf{S}_{\sigma_{n},\mathbf{K}})_{n\in{\mathbb{N}}}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p5.4.m4.3d">( bold_S start_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT , bold_K end_POSTSUBSCRIPT ) start_POSTSUBSCRIPT italic_n ∈ blackboard_N end_POSTSUBSCRIPT</annotation></semantics></math> by <math alttext="\mathbf{S}_{\boldsymbol{\sigma},\mathbf{K}}=\{{\mathbf{x}}\in\mathbf{K}|% \forall\epsilon>0,\forall m\in{\mathbb{N}},\exists n\geq m,{\mathbf{z}}_{n}\in% \mathbf{S}_{\sigma_{n},\mathbf{K}},\|{\mathbf{x}}-{\mathbf{z}}_{n}\|\leq\epsilon\}" class="ltx_Math" display="inline" id="S3.SS1.p5.5.m5.6"><semantics id="S3.SS1.p5.5.m5.6a"><mrow id="S3.SS1.p5.5.m5.6.6" xref="S3.SS1.p5.5.m5.6.6.cmml"><msub id="S3.SS1.p5.5.m5.6.6.4" xref="S3.SS1.p5.5.m5.6.6.4.cmml"><mi id="S3.SS1.p5.5.m5.6.6.4.2" xref="S3.SS1.p5.5.m5.6.6.4.2.cmml">𝐒</mi><mrow id="S3.SS1.p5.5.m5.2.2.2.4" xref="S3.SS1.p5.5.m5.2.2.2.3.cmml"><mi id="S3.SS1.p5.5.m5.1.1.1.1" xref="S3.SS1.p5.5.m5.1.1.1.1.cmml">𝝈</mi><mo id="S3.SS1.p5.5.m5.2.2.2.4.1" xref="S3.SS1.p5.5.m5.2.2.2.3.cmml">,</mo><mi id="S3.SS1.p5.5.m5.2.2.2.2" xref="S3.SS1.p5.5.m5.2.2.2.2.cmml">𝐊</mi></mrow></msub><mo id="S3.SS1.p5.5.m5.6.6.3" xref="S3.SS1.p5.5.m5.6.6.3.cmml">=</mo><mrow id="S3.SS1.p5.5.m5.6.6.2.2" xref="S3.SS1.p5.5.m5.6.6.2.3.cmml"><mo id="S3.SS1.p5.5.m5.6.6.2.2.3" stretchy="false" xref="S3.SS1.p5.5.m5.6.6.2.3.1.cmml">{</mo><mrow id="S3.SS1.p5.5.m5.5.5.1.1.1" xref="S3.SS1.p5.5.m5.5.5.1.1.1.cmml"><mi id="S3.SS1.p5.5.m5.5.5.1.1.1.2" xref="S3.SS1.p5.5.m5.5.5.1.1.1.2.cmml">𝐱</mi><mo id="S3.SS1.p5.5.m5.5.5.1.1.1.1" xref="S3.SS1.p5.5.m5.5.5.1.1.1.1.cmml">∈</mo><mi id="S3.SS1.p5.5.m5.5.5.1.1.1.3" xref="S3.SS1.p5.5.m5.5.5.1.1.1.3.cmml">𝐊</mi></mrow><mo id="S3.SS1.p5.5.m5.6.6.2.2.4" lspace="0em" rspace="0.167em" xref="S3.SS1.p5.5.m5.6.6.2.3.1.cmml">|</mo><mrow id="S3.SS1.p5.5.m5.6.6.2.2.2.2" xref="S3.SS1.p5.5.m5.6.6.2.2.2.3.cmml"><mrow id="S3.SS1.p5.5.m5.6.6.2.2.2.1.1" xref="S3.SS1.p5.5.m5.6.6.2.2.2.1.1.cmml"><mrow id="S3.SS1.p5.5.m5.6.6.2.2.2.1.1.2" xref="S3.SS1.p5.5.m5.6.6.2.2.2.1.1.2.cmml"><mo id="S3.SS1.p5.5.m5.6.6.2.2.2.1.1.2.1" rspace="0.167em" xref="S3.SS1.p5.5.m5.6.6.2.2.2.1.1.2.1.cmml">∀</mo><mi id="S3.SS1.p5.5.m5.6.6.2.2.2.1.1.2.2" xref="S3.SS1.p5.5.m5.6.6.2.2.2.1.1.2.2.cmml">ϵ</mi></mrow><mo id="S3.SS1.p5.5.m5.6.6.2.2.2.1.1.1" xref="S3.SS1.p5.5.m5.6.6.2.2.2.1.1.1.cmml">></mo><mn 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xref="S3.SS1.p5.5.m5.6.6.2.2.2.2.2.2.2.2.2.2.2.1.1.1.3.3">𝑛</ci></apply></apply></apply><ci id="S3.SS1.p5.5.m5.6.6.2.2.2.2.2.2.2.2.2.2.2.3.cmml" xref="S3.SS1.p5.5.m5.6.6.2.2.2.2.2.2.2.2.2.2.2.3">italic-ϵ</ci></apply></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p5.5.m5.6c">\mathbf{S}_{\boldsymbol{\sigma},\mathbf{K}}=\{{\mathbf{x}}\in\mathbf{K}|% \forall\epsilon>0,\forall m\in{\mathbb{N}},\exists n\geq m,{\mathbf{z}}_{n}\in% \mathbf{S}_{\sigma_{n},\mathbf{K}},\|{\mathbf{x}}-{\mathbf{z}}_{n}\|\leq\epsilon\}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p5.5.m5.6d">bold_S start_POSTSUBSCRIPT bold_italic_σ , bold_K end_POSTSUBSCRIPT = { bold_x ∈ bold_K | ∀ italic_ϵ > 0 , ∀ italic_m ∈ blackboard_N , ∃ italic_n ≥ italic_m , bold_z start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ∈ bold_S start_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT , bold_K end_POSTSUBSCRIPT , ∥ bold_x - bold_z start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ∥ ≤ italic_ϵ }</annotation></semantics></math>. Finally, we can define a limit of sets <math alttext="\mathbf{S}_{\sigma,\mathbf{K}}" class="ltx_Math" display="inline" id="S3.SS1.p5.6.m6.2"><semantics id="S3.SS1.p5.6.m6.2a"><msub id="S3.SS1.p5.6.m6.2.3" xref="S3.SS1.p5.6.m6.2.3.cmml"><mi id="S3.SS1.p5.6.m6.2.3.2" xref="S3.SS1.p5.6.m6.2.3.2.cmml">𝐒</mi><mrow id="S3.SS1.p5.6.m6.2.2.2.4" xref="S3.SS1.p5.6.m6.2.2.2.3.cmml"><mi id="S3.SS1.p5.6.m6.1.1.1.1" xref="S3.SS1.p5.6.m6.1.1.1.1.cmml">σ</mi><mo id="S3.SS1.p5.6.m6.2.2.2.4.1" xref="S3.SS1.p5.6.m6.2.2.2.3.cmml">,</mo><mi id="S3.SS1.p5.6.m6.2.2.2.2" xref="S3.SS1.p5.6.m6.2.2.2.2.cmml">𝐊</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="S3.SS1.p5.6.m6.2b"><apply id="S3.SS1.p5.6.m6.2.3.cmml" xref="S3.SS1.p5.6.m6.2.3"><csymbol cd="ambiguous" id="S3.SS1.p5.6.m6.2.3.1.cmml" xref="S3.SS1.p5.6.m6.2.3">subscript</csymbol><ci id="S3.SS1.p5.6.m6.2.3.2.cmml" xref="S3.SS1.p5.6.m6.2.3.2">𝐒</ci><list id="S3.SS1.p5.6.m6.2.2.2.3.cmml" xref="S3.SS1.p5.6.m6.2.2.2.4"><ci id="S3.SS1.p5.6.m6.1.1.1.1.cmml" xref="S3.SS1.p5.6.m6.1.1.1.1">𝜎</ci><ci id="S3.SS1.p5.6.m6.2.2.2.2.cmml" xref="S3.SS1.p5.6.m6.2.2.2.2">𝐊</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p5.6.m6.2c">\mathbf{S}_{\sigma,\mathbf{K}}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p5.6.m6.2d">bold_S start_POSTSUBSCRIPT italic_σ , bold_K end_POSTSUBSCRIPT</annotation></semantics></math> for a continuous <math alttext="\sigma\to 0" class="ltx_Math" display="inline" id="S3.SS1.p5.7.m7.1"><semantics id="S3.SS1.p5.7.m7.1a"><mrow id="S3.SS1.p5.7.m7.1.1" xref="S3.SS1.p5.7.m7.1.1.cmml"><mi id="S3.SS1.p5.7.m7.1.1.2" xref="S3.SS1.p5.7.m7.1.1.2.cmml">σ</mi><mo id="S3.SS1.p5.7.m7.1.1.1" stretchy="false" xref="S3.SS1.p5.7.m7.1.1.1.cmml">→</mo><mn id="S3.SS1.p5.7.m7.1.1.3" xref="S3.SS1.p5.7.m7.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p5.7.m7.1b"><apply id="S3.SS1.p5.7.m7.1.1.cmml" xref="S3.SS1.p5.7.m7.1.1"><ci id="S3.SS1.p5.7.m7.1.1.1.cmml" xref="S3.SS1.p5.7.m7.1.1.1">→</ci><ci id="S3.SS1.p5.7.m7.1.1.2.cmml" xref="S3.SS1.p5.7.m7.1.1.2">𝜎</ci><cn id="S3.SS1.p5.7.m7.1.1.3.cmml" type="integer" xref="S3.SS1.p5.7.m7.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p5.7.m7.1c">\sigma\to 0</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p5.7.m7.1d">italic_σ → 0</annotation></semantics></math> with <math alttext="\mathbf{S}_{\mathbf{K}}=\cup_{\boldsymbol{\sigma}\in\mathbf{E}}{\mathbf{S}_{% \boldsymbol{\sigma},\mathbf{K}}}" class="ltx_Math" display="inline" id="S3.SS1.p5.8.m8.2"><semantics id="S3.SS1.p5.8.m8.2a"><mrow id="S3.SS1.p5.8.m8.2.3" xref="S3.SS1.p5.8.m8.2.3.cmml"><msub id="S3.SS1.p5.8.m8.2.3.2" xref="S3.SS1.p5.8.m8.2.3.2.cmml"><mi id="S3.SS1.p5.8.m8.2.3.2.2" xref="S3.SS1.p5.8.m8.2.3.2.2.cmml">𝐒</mi><mi id="S3.SS1.p5.8.m8.2.3.2.3" xref="S3.SS1.p5.8.m8.2.3.2.3.cmml">𝐊</mi></msub><mo id="S3.SS1.p5.8.m8.2.3.1" rspace="0em" xref="S3.SS1.p5.8.m8.2.3.1.cmml">=</mo><mrow id="S3.SS1.p5.8.m8.2.3.3" xref="S3.SS1.p5.8.m8.2.3.3.cmml"><msub id="S3.SS1.p5.8.m8.2.3.3.1" xref="S3.SS1.p5.8.m8.2.3.3.1.cmml"><mo id="S3.SS1.p5.8.m8.2.3.3.1.2" lspace="0em" xref="S3.SS1.p5.8.m8.2.3.3.1.2.cmml">∪</mo><mrow id="S3.SS1.p5.8.m8.2.3.3.1.3" xref="S3.SS1.p5.8.m8.2.3.3.1.3.cmml"><mi id="S3.SS1.p5.8.m8.2.3.3.1.3.2" xref="S3.SS1.p5.8.m8.2.3.3.1.3.2.cmml">𝝈</mi><mo id="S3.SS1.p5.8.m8.2.3.3.1.3.1" xref="S3.SS1.p5.8.m8.2.3.3.1.3.1.cmml">∈</mo><mi id="S3.SS1.p5.8.m8.2.3.3.1.3.3" xref="S3.SS1.p5.8.m8.2.3.3.1.3.3.cmml">𝐄</mi></mrow></msub><msub id="S3.SS1.p5.8.m8.2.3.3.2" xref="S3.SS1.p5.8.m8.2.3.3.2.cmml"><mi id="S3.SS1.p5.8.m8.2.3.3.2.2" xref="S3.SS1.p5.8.m8.2.3.3.2.2.cmml">𝐒</mi><mrow id="S3.SS1.p5.8.m8.2.2.2.4" xref="S3.SS1.p5.8.m8.2.2.2.3.cmml"><mi id="S3.SS1.p5.8.m8.1.1.1.1" xref="S3.SS1.p5.8.m8.1.1.1.1.cmml">𝝈</mi><mo id="S3.SS1.p5.8.m8.2.2.2.4.1" xref="S3.SS1.p5.8.m8.2.2.2.3.cmml">,</mo><mi id="S3.SS1.p5.8.m8.2.2.2.2" xref="S3.SS1.p5.8.m8.2.2.2.2.cmml">𝐊</mi></mrow></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p5.8.m8.2b"><apply id="S3.SS1.p5.8.m8.2.3.cmml" xref="S3.SS1.p5.8.m8.2.3"><eq id="S3.SS1.p5.8.m8.2.3.1.cmml" xref="S3.SS1.p5.8.m8.2.3.1"></eq><apply id="S3.SS1.p5.8.m8.2.3.2.cmml" xref="S3.SS1.p5.8.m8.2.3.2"><csymbol cd="ambiguous" id="S3.SS1.p5.8.m8.2.3.2.1.cmml" xref="S3.SS1.p5.8.m8.2.3.2">subscript</csymbol><ci id="S3.SS1.p5.8.m8.2.3.2.2.cmml" xref="S3.SS1.p5.8.m8.2.3.2.2">𝐒</ci><ci id="S3.SS1.p5.8.m8.2.3.2.3.cmml" xref="S3.SS1.p5.8.m8.2.3.2.3">𝐊</ci></apply><apply id="S3.SS1.p5.8.m8.2.3.3.cmml" xref="S3.SS1.p5.8.m8.2.3.3"><apply id="S3.SS1.p5.8.m8.2.3.3.1.cmml" xref="S3.SS1.p5.8.m8.2.3.3.1"><csymbol cd="ambiguous" id="S3.SS1.p5.8.m8.2.3.3.1.1.cmml" xref="S3.SS1.p5.8.m8.2.3.3.1">subscript</csymbol><union id="S3.SS1.p5.8.m8.2.3.3.1.2.cmml" xref="S3.SS1.p5.8.m8.2.3.3.1.2"></union><apply id="S3.SS1.p5.8.m8.2.3.3.1.3.cmml" xref="S3.SS1.p5.8.m8.2.3.3.1.3"><in id="S3.SS1.p5.8.m8.2.3.3.1.3.1.cmml" xref="S3.SS1.p5.8.m8.2.3.3.1.3.1"></in><ci id="S3.SS1.p5.8.m8.2.3.3.1.3.2.cmml" xref="S3.SS1.p5.8.m8.2.3.3.1.3.2">𝝈</ci><ci id="S3.SS1.p5.8.m8.2.3.3.1.3.3.cmml" xref="S3.SS1.p5.8.m8.2.3.3.1.3.3">𝐄</ci></apply></apply><apply id="S3.SS1.p5.8.m8.2.3.3.2.cmml" xref="S3.SS1.p5.8.m8.2.3.3.2"><csymbol cd="ambiguous" id="S3.SS1.p5.8.m8.2.3.3.2.1.cmml" xref="S3.SS1.p5.8.m8.2.3.3.2">subscript</csymbol><ci id="S3.SS1.p5.8.m8.2.3.3.2.2.cmml" xref="S3.SS1.p5.8.m8.2.3.3.2.2">𝐒</ci><list id="S3.SS1.p5.8.m8.2.2.2.3.cmml" xref="S3.SS1.p5.8.m8.2.2.2.4"><ci id="S3.SS1.p5.8.m8.1.1.1.1.cmml" xref="S3.SS1.p5.8.m8.1.1.1.1">𝝈</ci><ci id="S3.SS1.p5.8.m8.2.2.2.2.cmml" xref="S3.SS1.p5.8.m8.2.2.2.2">𝐊</ci></list></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p5.8.m8.2c">\mathbf{S}_{\mathbf{K}}=\cup_{\boldsymbol{\sigma}\in\mathbf{E}}{\mathbf{S}_{% \boldsymbol{\sigma},\mathbf{K}}}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p5.8.m8.2d">bold_S start_POSTSUBSCRIPT bold_K end_POSTSUBSCRIPT = ∪ start_POSTSUBSCRIPT bold_italic_σ ∈ bold_E end_POSTSUBSCRIPT bold_S start_POSTSUBSCRIPT bold_italic_σ , bold_K end_POSTSUBSCRIPT</annotation></semantics></math>. Our target points are the critical points of Problem (<a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#S1.E1" title="1 ‣ 1 Introduction ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_tag">1</span></a>), <math alttext="\mathbf{S}^{\star}_{\mathbf{K}}=\{{\mathbf{x}}\in\mathbf{K}|\nabla\mathcal{F}(% {\mathbf{x}})+\alpha\nabla\mathcal{R}({\mathbf{x}})=0\}" class="ltx_Math" display="inline" id="S3.SS1.p5.9.m9.4"><semantics id="S3.SS1.p5.9.m9.4a"><mrow id="S3.SS1.p5.9.m9.4.4" xref="S3.SS1.p5.9.m9.4.4.cmml"><msubsup id="S3.SS1.p5.9.m9.4.4.4" xref="S3.SS1.p5.9.m9.4.4.4.cmml"><mi id="S3.SS1.p5.9.m9.4.4.4.2.2" xref="S3.SS1.p5.9.m9.4.4.4.2.2.cmml">𝐒</mi><mi id="S3.SS1.p5.9.m9.4.4.4.3" xref="S3.SS1.p5.9.m9.4.4.4.3.cmml">𝐊</mi><mo id="S3.SS1.p5.9.m9.4.4.4.2.3" xref="S3.SS1.p5.9.m9.4.4.4.2.3.cmml">⋆</mo></msubsup><mo id="S3.SS1.p5.9.m9.4.4.3" xref="S3.SS1.p5.9.m9.4.4.3.cmml">=</mo><mrow id="S3.SS1.p5.9.m9.4.4.2.2" xref="S3.SS1.p5.9.m9.4.4.2.3.cmml"><mo id="S3.SS1.p5.9.m9.4.4.2.2.3" stretchy="false" xref="S3.SS1.p5.9.m9.4.4.2.3.1.cmml">{</mo><mrow id="S3.SS1.p5.9.m9.3.3.1.1.1" xref="S3.SS1.p5.9.m9.3.3.1.1.1.cmml"><mi id="S3.SS1.p5.9.m9.3.3.1.1.1.2" xref="S3.SS1.p5.9.m9.3.3.1.1.1.2.cmml">𝐱</mi><mo id="S3.SS1.p5.9.m9.3.3.1.1.1.1" xref="S3.SS1.p5.9.m9.3.3.1.1.1.1.cmml">∈</mo><mi id="S3.SS1.p5.9.m9.3.3.1.1.1.3" xref="S3.SS1.p5.9.m9.3.3.1.1.1.3.cmml">𝐊</mi></mrow><mo id="S3.SS1.p5.9.m9.4.4.2.2.4" lspace="0em" rspace="0.167em" xref="S3.SS1.p5.9.m9.4.4.2.3.1.cmml">|</mo><mrow id="S3.SS1.p5.9.m9.4.4.2.2.2" xref="S3.SS1.p5.9.m9.4.4.2.2.2.cmml"><mrow id="S3.SS1.p5.9.m9.4.4.2.2.2.2" xref="S3.SS1.p5.9.m9.4.4.2.2.2.2.cmml"><mrow id="S3.SS1.p5.9.m9.4.4.2.2.2.2.2" xref="S3.SS1.p5.9.m9.4.4.2.2.2.2.2.cmml"><mrow id="S3.SS1.p5.9.m9.4.4.2.2.2.2.2.2" xref="S3.SS1.p5.9.m9.4.4.2.2.2.2.2.2.cmml"><mo id="S3.SS1.p5.9.m9.4.4.2.2.2.2.2.2.1" rspace="0.167em" xref="S3.SS1.p5.9.m9.4.4.2.2.2.2.2.2.1.cmml">∇</mo><mi class="ltx_font_mathcaligraphic" id="S3.SS1.p5.9.m9.4.4.2.2.2.2.2.2.2" xref="S3.SS1.p5.9.m9.4.4.2.2.2.2.2.2.2.cmml">ℱ</mi></mrow><mo id="S3.SS1.p5.9.m9.4.4.2.2.2.2.2.1" xref="S3.SS1.p5.9.m9.4.4.2.2.2.2.2.1.cmml">⁢</mo><mrow id="S3.SS1.p5.9.m9.4.4.2.2.2.2.2.3.2" xref="S3.SS1.p5.9.m9.4.4.2.2.2.2.2.cmml"><mo id="S3.SS1.p5.9.m9.4.4.2.2.2.2.2.3.2.1" stretchy="false" xref="S3.SS1.p5.9.m9.4.4.2.2.2.2.2.cmml">(</mo><mi id="S3.SS1.p5.9.m9.1.1" xref="S3.SS1.p5.9.m9.1.1.cmml">𝐱</mi><mo id="S3.SS1.p5.9.m9.4.4.2.2.2.2.2.3.2.2" stretchy="false" xref="S3.SS1.p5.9.m9.4.4.2.2.2.2.2.cmml">)</mo></mrow></mrow><mo id="S3.SS1.p5.9.m9.4.4.2.2.2.2.1" xref="S3.SS1.p5.9.m9.4.4.2.2.2.2.1.cmml">+</mo><mrow id="S3.SS1.p5.9.m9.4.4.2.2.2.2.3" xref="S3.SS1.p5.9.m9.4.4.2.2.2.2.3.cmml"><mi id="S3.SS1.p5.9.m9.4.4.2.2.2.2.3.2" xref="S3.SS1.p5.9.m9.4.4.2.2.2.2.3.2.cmml">α</mi><mo id="S3.SS1.p5.9.m9.4.4.2.2.2.2.3.1" lspace="0.167em" xref="S3.SS1.p5.9.m9.4.4.2.2.2.2.3.1.cmml">⁢</mo><mrow id="S3.SS1.p5.9.m9.4.4.2.2.2.2.3.3" xref="S3.SS1.p5.9.m9.4.4.2.2.2.2.3.3.cmml"><mo id="S3.SS1.p5.9.m9.4.4.2.2.2.2.3.3.1" rspace="0.167em" xref="S3.SS1.p5.9.m9.4.4.2.2.2.2.3.3.1.cmml">∇</mo><mi class="ltx_font_mathcaligraphic" id="S3.SS1.p5.9.m9.4.4.2.2.2.2.3.3.2" xref="S3.SS1.p5.9.m9.4.4.2.2.2.2.3.3.2.cmml">ℛ</mi></mrow><mo id="S3.SS1.p5.9.m9.4.4.2.2.2.2.3.1a" xref="S3.SS1.p5.9.m9.4.4.2.2.2.2.3.1.cmml">⁢</mo><mrow id="S3.SS1.p5.9.m9.4.4.2.2.2.2.3.4.2" xref="S3.SS1.p5.9.m9.4.4.2.2.2.2.3.cmml"><mo id="S3.SS1.p5.9.m9.4.4.2.2.2.2.3.4.2.1" stretchy="false" xref="S3.SS1.p5.9.m9.4.4.2.2.2.2.3.cmml">(</mo><mi id="S3.SS1.p5.9.m9.2.2" xref="S3.SS1.p5.9.m9.2.2.cmml">𝐱</mi><mo id="S3.SS1.p5.9.m9.4.4.2.2.2.2.3.4.2.2" stretchy="false" xref="S3.SS1.p5.9.m9.4.4.2.2.2.2.3.cmml">)</mo></mrow></mrow></mrow><mo id="S3.SS1.p5.9.m9.4.4.2.2.2.1" xref="S3.SS1.p5.9.m9.4.4.2.2.2.1.cmml">=</mo><mn id="S3.SS1.p5.9.m9.4.4.2.2.2.3" xref="S3.SS1.p5.9.m9.4.4.2.2.2.3.cmml">0</mn></mrow><mo id="S3.SS1.p5.9.m9.4.4.2.2.5" stretchy="false" xref="S3.SS1.p5.9.m9.4.4.2.3.1.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p5.9.m9.4b"><apply id="S3.SS1.p5.9.m9.4.4.cmml" xref="S3.SS1.p5.9.m9.4.4"><eq id="S3.SS1.p5.9.m9.4.4.3.cmml" xref="S3.SS1.p5.9.m9.4.4.3"></eq><apply id="S3.SS1.p5.9.m9.4.4.4.cmml" xref="S3.SS1.p5.9.m9.4.4.4"><csymbol cd="ambiguous" id="S3.SS1.p5.9.m9.4.4.4.1.cmml" xref="S3.SS1.p5.9.m9.4.4.4">subscript</csymbol><apply id="S3.SS1.p5.9.m9.4.4.4.2.cmml" xref="S3.SS1.p5.9.m9.4.4.4"><csymbol cd="ambiguous" id="S3.SS1.p5.9.m9.4.4.4.2.1.cmml" xref="S3.SS1.p5.9.m9.4.4.4">superscript</csymbol><ci id="S3.SS1.p5.9.m9.4.4.4.2.2.cmml" xref="S3.SS1.p5.9.m9.4.4.4.2.2">𝐒</ci><ci id="S3.SS1.p5.9.m9.4.4.4.2.3.cmml" xref="S3.SS1.p5.9.m9.4.4.4.2.3">⋆</ci></apply><ci id="S3.SS1.p5.9.m9.4.4.4.3.cmml" xref="S3.SS1.p5.9.m9.4.4.4.3">𝐊</ci></apply><apply id="S3.SS1.p5.9.m9.4.4.2.3.cmml" xref="S3.SS1.p5.9.m9.4.4.2.2"><csymbol cd="latexml" id="S3.SS1.p5.9.m9.4.4.2.3.1.cmml" xref="S3.SS1.p5.9.m9.4.4.2.2.3">conditional-set</csymbol><apply id="S3.SS1.p5.9.m9.3.3.1.1.1.cmml" xref="S3.SS1.p5.9.m9.3.3.1.1.1"><in id="S3.SS1.p5.9.m9.3.3.1.1.1.1.cmml" xref="S3.SS1.p5.9.m9.3.3.1.1.1.1"></in><ci id="S3.SS1.p5.9.m9.3.3.1.1.1.2.cmml" xref="S3.SS1.p5.9.m9.3.3.1.1.1.2">𝐱</ci><ci id="S3.SS1.p5.9.m9.3.3.1.1.1.3.cmml" xref="S3.SS1.p5.9.m9.3.3.1.1.1.3">𝐊</ci></apply><apply id="S3.SS1.p5.9.m9.4.4.2.2.2.cmml" xref="S3.SS1.p5.9.m9.4.4.2.2.2"><eq id="S3.SS1.p5.9.m9.4.4.2.2.2.1.cmml" xref="S3.SS1.p5.9.m9.4.4.2.2.2.1"></eq><apply id="S3.SS1.p5.9.m9.4.4.2.2.2.2.cmml" xref="S3.SS1.p5.9.m9.4.4.2.2.2.2"><plus id="S3.SS1.p5.9.m9.4.4.2.2.2.2.1.cmml" xref="S3.SS1.p5.9.m9.4.4.2.2.2.2.1"></plus><apply id="S3.SS1.p5.9.m9.4.4.2.2.2.2.2.cmml" xref="S3.SS1.p5.9.m9.4.4.2.2.2.2.2"><times id="S3.SS1.p5.9.m9.4.4.2.2.2.2.2.1.cmml" xref="S3.SS1.p5.9.m9.4.4.2.2.2.2.2.1"></times><apply id="S3.SS1.p5.9.m9.4.4.2.2.2.2.2.2.cmml" xref="S3.SS1.p5.9.m9.4.4.2.2.2.2.2.2"><ci id="S3.SS1.p5.9.m9.4.4.2.2.2.2.2.2.1.cmml" xref="S3.SS1.p5.9.m9.4.4.2.2.2.2.2.2.1">∇</ci><ci id="S3.SS1.p5.9.m9.4.4.2.2.2.2.2.2.2.cmml" xref="S3.SS1.p5.9.m9.4.4.2.2.2.2.2.2.2">ℱ</ci></apply><ci id="S3.SS1.p5.9.m9.1.1.cmml" xref="S3.SS1.p5.9.m9.1.1">𝐱</ci></apply><apply id="S3.SS1.p5.9.m9.4.4.2.2.2.2.3.cmml" xref="S3.SS1.p5.9.m9.4.4.2.2.2.2.3"><times id="S3.SS1.p5.9.m9.4.4.2.2.2.2.3.1.cmml" xref="S3.SS1.p5.9.m9.4.4.2.2.2.2.3.1"></times><ci id="S3.SS1.p5.9.m9.4.4.2.2.2.2.3.2.cmml" xref="S3.SS1.p5.9.m9.4.4.2.2.2.2.3.2">𝛼</ci><apply id="S3.SS1.p5.9.m9.4.4.2.2.2.2.3.3.cmml" xref="S3.SS1.p5.9.m9.4.4.2.2.2.2.3.3"><ci id="S3.SS1.p5.9.m9.4.4.2.2.2.2.3.3.1.cmml" xref="S3.SS1.p5.9.m9.4.4.2.2.2.2.3.3.1">∇</ci><ci id="S3.SS1.p5.9.m9.4.4.2.2.2.2.3.3.2.cmml" xref="S3.SS1.p5.9.m9.4.4.2.2.2.2.3.3.2">ℛ</ci></apply><ci id="S3.SS1.p5.9.m9.2.2.cmml" xref="S3.SS1.p5.9.m9.2.2">𝐱</ci></apply></apply><cn id="S3.SS1.p5.9.m9.4.4.2.2.2.3.cmml" type="integer" xref="S3.SS1.p5.9.m9.4.4.2.2.2.3">0</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p5.9.m9.4c">\mathbf{S}^{\star}_{\mathbf{K}}=\{{\mathbf{x}}\in\mathbf{K}|\nabla\mathcal{F}(% {\mathbf{x}})+\alpha\nabla\mathcal{R}({\mathbf{x}})=0\}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p5.9.m9.4d">bold_S start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT bold_K end_POSTSUBSCRIPT = { bold_x ∈ bold_K | ∇ caligraphic_F ( bold_x ) + italic_α ∇ caligraphic_R ( bold_x ) = 0 }</annotation></semantics></math> where <math alttext="\mathcal{R}=-\log p" class="ltx_Math" display="inline" id="S3.SS1.p5.10.m10.1"><semantics id="S3.SS1.p5.10.m10.1a"><mrow id="S3.SS1.p5.10.m10.1.1" xref="S3.SS1.p5.10.m10.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.SS1.p5.10.m10.1.1.2" xref="S3.SS1.p5.10.m10.1.1.2.cmml">ℛ</mi><mo id="S3.SS1.p5.10.m10.1.1.1" xref="S3.SS1.p5.10.m10.1.1.1.cmml">=</mo><mrow id="S3.SS1.p5.10.m10.1.1.3" xref="S3.SS1.p5.10.m10.1.1.3.cmml"><mo id="S3.SS1.p5.10.m10.1.1.3a" rspace="0.167em" xref="S3.SS1.p5.10.m10.1.1.3.cmml">−</mo><mrow id="S3.SS1.p5.10.m10.1.1.3.2" xref="S3.SS1.p5.10.m10.1.1.3.2.cmml"><mi id="S3.SS1.p5.10.m10.1.1.3.2.1" xref="S3.SS1.p5.10.m10.1.1.3.2.1.cmml">log</mi><mo id="S3.SS1.p5.10.m10.1.1.3.2a" lspace="0.167em" xref="S3.SS1.p5.10.m10.1.1.3.2.cmml">⁡</mo><mi id="S3.SS1.p5.10.m10.1.1.3.2.2" xref="S3.SS1.p5.10.m10.1.1.3.2.2.cmml">p</mi></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p5.10.m10.1b"><apply id="S3.SS1.p5.10.m10.1.1.cmml" xref="S3.SS1.p5.10.m10.1.1"><eq id="S3.SS1.p5.10.m10.1.1.1.cmml" xref="S3.SS1.p5.10.m10.1.1.1"></eq><ci id="S3.SS1.p5.10.m10.1.1.2.cmml" xref="S3.SS1.p5.10.m10.1.1.2">ℛ</ci><apply id="S3.SS1.p5.10.m10.1.1.3.cmml" xref="S3.SS1.p5.10.m10.1.1.3"><minus id="S3.SS1.p5.10.m10.1.1.3.1.cmml" xref="S3.SS1.p5.10.m10.1.1.3"></minus><apply id="S3.SS1.p5.10.m10.1.1.3.2.cmml" xref="S3.SS1.p5.10.m10.1.1.3.2"><log id="S3.SS1.p5.10.m10.1.1.3.2.1.cmml" xref="S3.SS1.p5.10.m10.1.1.3.2.1"></log><ci id="S3.SS1.p5.10.m10.1.1.3.2.2.cmml" xref="S3.SS1.p5.10.m10.1.1.3.2.2">𝑝</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p5.10.m10.1c">\mathcal{R}=-\log p</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p5.10.m10.1d">caligraphic_R = - roman_log italic_p</annotation></semantics></math>. The following proposition finally establishes that the limit of set <math alttext="\mathbf{S}_{\sigma,\mathbf{K}}" class="ltx_Math" display="inline" id="S3.SS1.p5.11.m11.2"><semantics id="S3.SS1.p5.11.m11.2a"><msub id="S3.SS1.p5.11.m11.2.3" xref="S3.SS1.p5.11.m11.2.3.cmml"><mi id="S3.SS1.p5.11.m11.2.3.2" xref="S3.SS1.p5.11.m11.2.3.2.cmml">𝐒</mi><mrow id="S3.SS1.p5.11.m11.2.2.2.4" xref="S3.SS1.p5.11.m11.2.2.2.3.cmml"><mi id="S3.SS1.p5.11.m11.1.1.1.1" xref="S3.SS1.p5.11.m11.1.1.1.1.cmml">σ</mi><mo id="S3.SS1.p5.11.m11.2.2.2.4.1" xref="S3.SS1.p5.11.m11.2.2.2.3.cmml">,</mo><mi id="S3.SS1.p5.11.m11.2.2.2.2" xref="S3.SS1.p5.11.m11.2.2.2.2.cmml">𝐊</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="S3.SS1.p5.11.m11.2b"><apply id="S3.SS1.p5.11.m11.2.3.cmml" xref="S3.SS1.p5.11.m11.2.3"><csymbol cd="ambiguous" id="S3.SS1.p5.11.m11.2.3.1.cmml" xref="S3.SS1.p5.11.m11.2.3">subscript</csymbol><ci id="S3.SS1.p5.11.m11.2.3.2.cmml" xref="S3.SS1.p5.11.m11.2.3.2">𝐒</ci><list id="S3.SS1.p5.11.m11.2.2.2.3.cmml" xref="S3.SS1.p5.11.m11.2.2.2.4"><ci id="S3.SS1.p5.11.m11.1.1.1.1.cmml" xref="S3.SS1.p5.11.m11.1.1.1.1">𝜎</ci><ci id="S3.SS1.p5.11.m11.2.2.2.2.cmml" xref="S3.SS1.p5.11.m11.2.2.2.2">𝐊</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p5.11.m11.2c">\mathbf{S}_{\sigma,\mathbf{K}}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p5.11.m11.2d">bold_S start_POSTSUBSCRIPT italic_σ , bold_K end_POSTSUBSCRIPT</annotation></semantics></math> (in the sense of cluster point) is included in the targeted points.</p> </div> <div class="ltx_theorem ltx_theorem_proposition" id="S3.Thmproposition2"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S3.Thmproposition2.1.1.1">Proposition 3.2</span></span><span class="ltx_text ltx_font_bold" id="S3.Thmproposition2.2.2">.</span> </h6> <div class="ltx_para" id="S3.Thmproposition2.p1"> <p class="ltx_p" id="S3.Thmproposition2.p1.2"><span class="ltx_text ltx_font_italic" id="S3.Thmproposition2.p1.2.2">Under Assumptions <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#S3.Thmtheorem1" title="Assumption 3.1. ‣ 3.1 Asymptotics of critical points when 𝜎→0 ‣ 3 Convergence Analysis ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_tag">3.1</span></a>-<a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#S3.Thmtheorem2" title="Assumption 3.2. ‣ 3.1 Asymptotics of critical points when 𝜎→0 ‣ 3 Convergence Analysis ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_tag">3.2</span></a>, for <math alttext="\mathbf{K}" class="ltx_Math" display="inline" id="S3.Thmproposition2.p1.1.1.m1.1"><semantics id="S3.Thmproposition2.p1.1.1.m1.1a"><mi id="S3.Thmproposition2.p1.1.1.m1.1.1" xref="S3.Thmproposition2.p1.1.1.m1.1.1.cmml">𝐊</mi><annotation-xml encoding="MathML-Content" id="S3.Thmproposition2.p1.1.1.m1.1b"><ci id="S3.Thmproposition2.p1.1.1.m1.1.1.cmml" xref="S3.Thmproposition2.p1.1.1.m1.1.1">𝐊</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmproposition2.p1.1.1.m1.1c">\mathbf{K}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmproposition2.p1.1.1.m1.1d">bold_K</annotation></semantics></math> a compact subset of <math alttext="{\mathbb{R}}^{d}" class="ltx_Math" display="inline" id="S3.Thmproposition2.p1.2.2.m2.1"><semantics id="S3.Thmproposition2.p1.2.2.m2.1a"><msup id="S3.Thmproposition2.p1.2.2.m2.1.1" xref="S3.Thmproposition2.p1.2.2.m2.1.1.cmml"><mi id="S3.Thmproposition2.p1.2.2.m2.1.1.2" xref="S3.Thmproposition2.p1.2.2.m2.1.1.2.cmml">ℝ</mi><mi id="S3.Thmproposition2.p1.2.2.m2.1.1.3" xref="S3.Thmproposition2.p1.2.2.m2.1.1.3.cmml">d</mi></msup><annotation-xml encoding="MathML-Content" id="S3.Thmproposition2.p1.2.2.m2.1b"><apply id="S3.Thmproposition2.p1.2.2.m2.1.1.cmml" xref="S3.Thmproposition2.p1.2.2.m2.1.1"><csymbol cd="ambiguous" id="S3.Thmproposition2.p1.2.2.m2.1.1.1.cmml" xref="S3.Thmproposition2.p1.2.2.m2.1.1">superscript</csymbol><ci id="S3.Thmproposition2.p1.2.2.m2.1.1.2.cmml" xref="S3.Thmproposition2.p1.2.2.m2.1.1.2">ℝ</ci><ci id="S3.Thmproposition2.p1.2.2.m2.1.1.3.cmml" xref="S3.Thmproposition2.p1.2.2.m2.1.1.3">𝑑</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmproposition2.p1.2.2.m2.1c">{\mathbb{R}}^{d}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmproposition2.p1.2.2.m2.1d">blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT</annotation></semantics></math>, we have</span></p> <table class="ltx_equation ltx_eqn_table" id="S3.Ex8"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\mathbf{S}_{\mathbf{K}}\subseteq\mathbf{S}^{\star}_{\mathbf{K}}." class="ltx_Math" display="block" id="S3.Ex8.m1.1"><semantics id="S3.Ex8.m1.1a"><mrow id="S3.Ex8.m1.1.1.1" xref="S3.Ex8.m1.1.1.1.1.cmml"><mrow id="S3.Ex8.m1.1.1.1.1" xref="S3.Ex8.m1.1.1.1.1.cmml"><msub id="S3.Ex8.m1.1.1.1.1.2" xref="S3.Ex8.m1.1.1.1.1.2.cmml"><mi id="S3.Ex8.m1.1.1.1.1.2.2" xref="S3.Ex8.m1.1.1.1.1.2.2.cmml">𝐒</mi><mi id="S3.Ex8.m1.1.1.1.1.2.3" xref="S3.Ex8.m1.1.1.1.1.2.3.cmml">𝐊</mi></msub><mo id="S3.Ex8.m1.1.1.1.1.1" xref="S3.Ex8.m1.1.1.1.1.1.cmml">⊆</mo><msubsup id="S3.Ex8.m1.1.1.1.1.3" xref="S3.Ex8.m1.1.1.1.1.3.cmml"><mi id="S3.Ex8.m1.1.1.1.1.3.2.2" xref="S3.Ex8.m1.1.1.1.1.3.2.2.cmml">𝐒</mi><mi id="S3.Ex8.m1.1.1.1.1.3.3" xref="S3.Ex8.m1.1.1.1.1.3.3.cmml">𝐊</mi><mo id="S3.Ex8.m1.1.1.1.1.3.2.3" xref="S3.Ex8.m1.1.1.1.1.3.2.3.cmml">⋆</mo></msubsup></mrow><mo id="S3.Ex8.m1.1.1.1.2" lspace="0em" xref="S3.Ex8.m1.1.1.1.1.cmml">.</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.Ex8.m1.1b"><apply id="S3.Ex8.m1.1.1.1.1.cmml" xref="S3.Ex8.m1.1.1.1"><subset id="S3.Ex8.m1.1.1.1.1.1.cmml" xref="S3.Ex8.m1.1.1.1.1.1"></subset><apply id="S3.Ex8.m1.1.1.1.1.2.cmml" xref="S3.Ex8.m1.1.1.1.1.2"><csymbol cd="ambiguous" id="S3.Ex8.m1.1.1.1.1.2.1.cmml" xref="S3.Ex8.m1.1.1.1.1.2">subscript</csymbol><ci id="S3.Ex8.m1.1.1.1.1.2.2.cmml" xref="S3.Ex8.m1.1.1.1.1.2.2">𝐒</ci><ci id="S3.Ex8.m1.1.1.1.1.2.3.cmml" xref="S3.Ex8.m1.1.1.1.1.2.3">𝐊</ci></apply><apply id="S3.Ex8.m1.1.1.1.1.3.cmml" xref="S3.Ex8.m1.1.1.1.1.3"><csymbol cd="ambiguous" id="S3.Ex8.m1.1.1.1.1.3.1.cmml" xref="S3.Ex8.m1.1.1.1.1.3">subscript</csymbol><apply id="S3.Ex8.m1.1.1.1.1.3.2.cmml" xref="S3.Ex8.m1.1.1.1.1.3"><csymbol cd="ambiguous" id="S3.Ex8.m1.1.1.1.1.3.2.1.cmml" xref="S3.Ex8.m1.1.1.1.1.3">superscript</csymbol><ci id="S3.Ex8.m1.1.1.1.1.3.2.2.cmml" xref="S3.Ex8.m1.1.1.1.1.3.2.2">𝐒</ci><ci id="S3.Ex8.m1.1.1.1.1.3.2.3.cmml" xref="S3.Ex8.m1.1.1.1.1.3.2.3">⋆</ci></apply><ci id="S3.Ex8.m1.1.1.1.1.3.3.cmml" xref="S3.Ex8.m1.1.1.1.1.3.3">𝐊</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Ex8.m1.1c">\mathbf{S}_{\mathbf{K}}\subseteq\mathbf{S}^{\star}_{\mathbf{K}}.</annotation><annotation encoding="application/x-llamapun" id="S3.Ex8.m1.1d">bold_S start_POSTSUBSCRIPT bold_K end_POSTSUBSCRIPT ⊆ bold_S start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT bold_K end_POSTSUBSCRIPT .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> </div> </div> <div class="ltx_para" id="S3.SS1.p6"> <p class="ltx_p" id="S3.SS1.p6.5">Proposition <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#S3.Thmproposition2" title="Proposition 3.2. ‣ 3.1 Asymptotics of critical points when 𝜎→0 ‣ 3 Convergence Analysis ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_tag">3.2</span></a> means that a sequence of computed critical points with <math alttext="\sigma>0" class="ltx_Math" display="inline" id="S3.SS1.p6.1.m1.1"><semantics id="S3.SS1.p6.1.m1.1a"><mrow id="S3.SS1.p6.1.m1.1.1" xref="S3.SS1.p6.1.m1.1.1.cmml"><mi id="S3.SS1.p6.1.m1.1.1.2" xref="S3.SS1.p6.1.m1.1.1.2.cmml">σ</mi><mo id="S3.SS1.p6.1.m1.1.1.1" xref="S3.SS1.p6.1.m1.1.1.1.cmml">></mo><mn id="S3.SS1.p6.1.m1.1.1.3" xref="S3.SS1.p6.1.m1.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p6.1.m1.1b"><apply id="S3.SS1.p6.1.m1.1.1.cmml" xref="S3.SS1.p6.1.m1.1.1"><gt id="S3.SS1.p6.1.m1.1.1.1.cmml" xref="S3.SS1.p6.1.m1.1.1.1"></gt><ci id="S3.SS1.p6.1.m1.1.1.2.cmml" xref="S3.SS1.p6.1.m1.1.1.2">𝜎</ci><cn id="S3.SS1.p6.1.m1.1.1.3.cmml" type="integer" xref="S3.SS1.p6.1.m1.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p6.1.m1.1c">\sigma>0</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p6.1.m1.1d">italic_σ > 0</annotation></semantics></math> has all its cluster points in <math alttext="\mathbf{S}^{\star}_{\mathbf{K}}" class="ltx_Math" display="inline" id="S3.SS1.p6.2.m2.1"><semantics id="S3.SS1.p6.2.m2.1a"><msubsup id="S3.SS1.p6.2.m2.1.1" xref="S3.SS1.p6.2.m2.1.1.cmml"><mi id="S3.SS1.p6.2.m2.1.1.2.2" xref="S3.SS1.p6.2.m2.1.1.2.2.cmml">𝐒</mi><mi id="S3.SS1.p6.2.m2.1.1.3" xref="S3.SS1.p6.2.m2.1.1.3.cmml">𝐊</mi><mo id="S3.SS1.p6.2.m2.1.1.2.3" xref="S3.SS1.p6.2.m2.1.1.2.3.cmml">⋆</mo></msubsup><annotation-xml encoding="MathML-Content" id="S3.SS1.p6.2.m2.1b"><apply id="S3.SS1.p6.2.m2.1.1.cmml" xref="S3.SS1.p6.2.m2.1.1"><csymbol cd="ambiguous" id="S3.SS1.p6.2.m2.1.1.1.cmml" xref="S3.SS1.p6.2.m2.1.1">subscript</csymbol><apply id="S3.SS1.p6.2.m2.1.1.2.cmml" xref="S3.SS1.p6.2.m2.1.1"><csymbol cd="ambiguous" id="S3.SS1.p6.2.m2.1.1.2.1.cmml" xref="S3.SS1.p6.2.m2.1.1">superscript</csymbol><ci id="S3.SS1.p6.2.m2.1.1.2.2.cmml" xref="S3.SS1.p6.2.m2.1.1.2.2">𝐒</ci><ci id="S3.SS1.p6.2.m2.1.1.2.3.cmml" xref="S3.SS1.p6.2.m2.1.1.2.3">⋆</ci></apply><ci id="S3.SS1.p6.2.m2.1.1.3.cmml" xref="S3.SS1.p6.2.m2.1.1.3">𝐊</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p6.2.m2.1c">\mathbf{S}^{\star}_{\mathbf{K}}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p6.2.m2.1d">bold_S start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT bold_K end_POSTSUBSCRIPT</annotation></semantics></math>, the set of critical points of the ideal optimization problem (<a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#S1.E1" title="1 ‣ 1 Introduction ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_tag">1</span></a>). This result suggests that the annealed algorithm may converge to a point of <math alttext="\mathbf{S}^{\star}_{\mathbf{K}}" class="ltx_Math" display="inline" id="S3.SS1.p6.3.m3.1"><semantics id="S3.SS1.p6.3.m3.1a"><msubsup id="S3.SS1.p6.3.m3.1.1" xref="S3.SS1.p6.3.m3.1.1.cmml"><mi id="S3.SS1.p6.3.m3.1.1.2.2" xref="S3.SS1.p6.3.m3.1.1.2.2.cmml">𝐒</mi><mi id="S3.SS1.p6.3.m3.1.1.3" xref="S3.SS1.p6.3.m3.1.1.3.cmml">𝐊</mi><mo id="S3.SS1.p6.3.m3.1.1.2.3" xref="S3.SS1.p6.3.m3.1.1.2.3.cmml">⋆</mo></msubsup><annotation-xml encoding="MathML-Content" id="S3.SS1.p6.3.m3.1b"><apply id="S3.SS1.p6.3.m3.1.1.cmml" xref="S3.SS1.p6.3.m3.1.1"><csymbol cd="ambiguous" id="S3.SS1.p6.3.m3.1.1.1.cmml" xref="S3.SS1.p6.3.m3.1.1">subscript</csymbol><apply id="S3.SS1.p6.3.m3.1.1.2.cmml" xref="S3.SS1.p6.3.m3.1.1"><csymbol cd="ambiguous" id="S3.SS1.p6.3.m3.1.1.2.1.cmml" xref="S3.SS1.p6.3.m3.1.1">superscript</csymbol><ci id="S3.SS1.p6.3.m3.1.1.2.2.cmml" xref="S3.SS1.p6.3.m3.1.1.2.2">𝐒</ci><ci id="S3.SS1.p6.3.m3.1.1.2.3.cmml" xref="S3.SS1.p6.3.m3.1.1.2.3">⋆</ci></apply><ci id="S3.SS1.p6.3.m3.1.1.3.cmml" xref="S3.SS1.p6.3.m3.1.1.3">𝐊</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p6.3.m3.1c">\mathbf{S}^{\star}_{\mathbf{K}}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p6.3.m3.1d">bold_S start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT bold_K end_POSTSUBSCRIPT</annotation></semantics></math>. In fact annealing <cite class="ltx_cite ltx_citemacro_citep">(Neal, <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#bib.bib50" title="">1998</a>)</cite> consists in successively approximating critical points of <math alttext="\mathbf{S}_{\sigma_{i},\mathbf{K}}" class="ltx_Math" display="inline" id="S3.SS1.p6.4.m4.2"><semantics id="S3.SS1.p6.4.m4.2a"><msub id="S3.SS1.p6.4.m4.2.3" xref="S3.SS1.p6.4.m4.2.3.cmml"><mi id="S3.SS1.p6.4.m4.2.3.2" xref="S3.SS1.p6.4.m4.2.3.2.cmml">𝐒</mi><mrow id="S3.SS1.p6.4.m4.2.2.2.2" xref="S3.SS1.p6.4.m4.2.2.2.3.cmml"><msub id="S3.SS1.p6.4.m4.2.2.2.2.1" xref="S3.SS1.p6.4.m4.2.2.2.2.1.cmml"><mi id="S3.SS1.p6.4.m4.2.2.2.2.1.2" xref="S3.SS1.p6.4.m4.2.2.2.2.1.2.cmml">σ</mi><mi id="S3.SS1.p6.4.m4.2.2.2.2.1.3" xref="S3.SS1.p6.4.m4.2.2.2.2.1.3.cmml">i</mi></msub><mo id="S3.SS1.p6.4.m4.2.2.2.2.2" xref="S3.SS1.p6.4.m4.2.2.2.3.cmml">,</mo><mi id="S3.SS1.p6.4.m4.1.1.1.1" xref="S3.SS1.p6.4.m4.1.1.1.1.cmml">𝐊</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="S3.SS1.p6.4.m4.2b"><apply id="S3.SS1.p6.4.m4.2.3.cmml" xref="S3.SS1.p6.4.m4.2.3"><csymbol cd="ambiguous" id="S3.SS1.p6.4.m4.2.3.1.cmml" xref="S3.SS1.p6.4.m4.2.3">subscript</csymbol><ci id="S3.SS1.p6.4.m4.2.3.2.cmml" xref="S3.SS1.p6.4.m4.2.3.2">𝐒</ci><list id="S3.SS1.p6.4.m4.2.2.2.3.cmml" xref="S3.SS1.p6.4.m4.2.2.2.2"><apply id="S3.SS1.p6.4.m4.2.2.2.2.1.cmml" xref="S3.SS1.p6.4.m4.2.2.2.2.1"><csymbol cd="ambiguous" id="S3.SS1.p6.4.m4.2.2.2.2.1.1.cmml" xref="S3.SS1.p6.4.m4.2.2.2.2.1">subscript</csymbol><ci id="S3.SS1.p6.4.m4.2.2.2.2.1.2.cmml" xref="S3.SS1.p6.4.m4.2.2.2.2.1.2">𝜎</ci><ci id="S3.SS1.p6.4.m4.2.2.2.2.1.3.cmml" xref="S3.SS1.p6.4.m4.2.2.2.2.1.3">𝑖</ci></apply><ci id="S3.SS1.p6.4.m4.1.1.1.1.cmml" xref="S3.SS1.p6.4.m4.1.1.1.1">𝐊</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p6.4.m4.2c">\mathbf{S}_{\sigma_{i},\mathbf{K}}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p6.4.m4.2d">bold_S start_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , bold_K end_POSTSUBSCRIPT</annotation></semantics></math> for a decreasing sequence <math alttext="\sigma_{0}>\dots>\sigma_{m-1}\approx 0" class="ltx_Math" display="inline" id="S3.SS1.p6.5.m5.1"><semantics id="S3.SS1.p6.5.m5.1a"><mrow id="S3.SS1.p6.5.m5.1.1" xref="S3.SS1.p6.5.m5.1.1.cmml"><msub id="S3.SS1.p6.5.m5.1.1.2" xref="S3.SS1.p6.5.m5.1.1.2.cmml"><mi id="S3.SS1.p6.5.m5.1.1.2.2" xref="S3.SS1.p6.5.m5.1.1.2.2.cmml">σ</mi><mn id="S3.SS1.p6.5.m5.1.1.2.3" xref="S3.SS1.p6.5.m5.1.1.2.3.cmml">0</mn></msub><mo id="S3.SS1.p6.5.m5.1.1.3" xref="S3.SS1.p6.5.m5.1.1.3.cmml">></mo><mi id="S3.SS1.p6.5.m5.1.1.4" mathvariant="normal" xref="S3.SS1.p6.5.m5.1.1.4.cmml">⋯</mi><mo id="S3.SS1.p6.5.m5.1.1.5" xref="S3.SS1.p6.5.m5.1.1.5.cmml">></mo><msub id="S3.SS1.p6.5.m5.1.1.6" xref="S3.SS1.p6.5.m5.1.1.6.cmml"><mi id="S3.SS1.p6.5.m5.1.1.6.2" xref="S3.SS1.p6.5.m5.1.1.6.2.cmml">σ</mi><mrow id="S3.SS1.p6.5.m5.1.1.6.3" xref="S3.SS1.p6.5.m5.1.1.6.3.cmml"><mi id="S3.SS1.p6.5.m5.1.1.6.3.2" xref="S3.SS1.p6.5.m5.1.1.6.3.2.cmml">m</mi><mo id="S3.SS1.p6.5.m5.1.1.6.3.1" xref="S3.SS1.p6.5.m5.1.1.6.3.1.cmml">−</mo><mn id="S3.SS1.p6.5.m5.1.1.6.3.3" xref="S3.SS1.p6.5.m5.1.1.6.3.3.cmml">1</mn></mrow></msub><mo id="S3.SS1.p6.5.m5.1.1.7" xref="S3.SS1.p6.5.m5.1.1.7.cmml">≈</mo><mn id="S3.SS1.p6.5.m5.1.1.8" xref="S3.SS1.p6.5.m5.1.1.8.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p6.5.m5.1b"><apply id="S3.SS1.p6.5.m5.1.1.cmml" xref="S3.SS1.p6.5.m5.1.1"><and id="S3.SS1.p6.5.m5.1.1a.cmml" xref="S3.SS1.p6.5.m5.1.1"></and><apply id="S3.SS1.p6.5.m5.1.1b.cmml" xref="S3.SS1.p6.5.m5.1.1"><gt id="S3.SS1.p6.5.m5.1.1.3.cmml" xref="S3.SS1.p6.5.m5.1.1.3"></gt><apply id="S3.SS1.p6.5.m5.1.1.2.cmml" xref="S3.SS1.p6.5.m5.1.1.2"><csymbol cd="ambiguous" id="S3.SS1.p6.5.m5.1.1.2.1.cmml" xref="S3.SS1.p6.5.m5.1.1.2">subscript</csymbol><ci id="S3.SS1.p6.5.m5.1.1.2.2.cmml" xref="S3.SS1.p6.5.m5.1.1.2.2">𝜎</ci><cn id="S3.SS1.p6.5.m5.1.1.2.3.cmml" type="integer" xref="S3.SS1.p6.5.m5.1.1.2.3">0</cn></apply><ci id="S3.SS1.p6.5.m5.1.1.4.cmml" xref="S3.SS1.p6.5.m5.1.1.4">⋯</ci></apply><apply id="S3.SS1.p6.5.m5.1.1c.cmml" xref="S3.SS1.p6.5.m5.1.1"><gt id="S3.SS1.p6.5.m5.1.1.5.cmml" xref="S3.SS1.p6.5.m5.1.1.5"></gt><share href="#S3.SS1.p6.5.m5.1.1.4.cmml" id="S3.SS1.p6.5.m5.1.1d.cmml" xref="S3.SS1.p6.5.m5.1.1"></share><apply id="S3.SS1.p6.5.m5.1.1.6.cmml" xref="S3.SS1.p6.5.m5.1.1.6"><csymbol cd="ambiguous" id="S3.SS1.p6.5.m5.1.1.6.1.cmml" xref="S3.SS1.p6.5.m5.1.1.6">subscript</csymbol><ci id="S3.SS1.p6.5.m5.1.1.6.2.cmml" xref="S3.SS1.p6.5.m5.1.1.6.2">𝜎</ci><apply id="S3.SS1.p6.5.m5.1.1.6.3.cmml" xref="S3.SS1.p6.5.m5.1.1.6.3"><minus id="S3.SS1.p6.5.m5.1.1.6.3.1.cmml" xref="S3.SS1.p6.5.m5.1.1.6.3.1"></minus><ci id="S3.SS1.p6.5.m5.1.1.6.3.2.cmml" xref="S3.SS1.p6.5.m5.1.1.6.3.2">𝑚</ci><cn id="S3.SS1.p6.5.m5.1.1.6.3.3.cmml" type="integer" xref="S3.SS1.p6.5.m5.1.1.6.3.3">1</cn></apply></apply></apply><apply id="S3.SS1.p6.5.m5.1.1e.cmml" xref="S3.SS1.p6.5.m5.1.1"><approx id="S3.SS1.p6.5.m5.1.1.7.cmml" xref="S3.SS1.p6.5.m5.1.1.7"></approx><share href="#S3.SS1.p6.5.m5.1.1.6.cmml" id="S3.SS1.p6.5.m5.1.1f.cmml" xref="S3.SS1.p6.5.m5.1.1"></share><cn id="S3.SS1.p6.5.m5.1.1.8.cmml" type="integer" xref="S3.SS1.p6.5.m5.1.1.8">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p6.5.m5.1c">\sigma_{0}>\dots>\sigma_{m-1}\approx 0</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p6.5.m5.1d">italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT > ⋯ > italic_σ start_POSTSUBSCRIPT italic_m - 1 end_POSTSUBSCRIPT ≈ 0</annotation></semantics></math>. Proposition <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#S3.Thmproposition2" title="Proposition 3.2. ‣ 3.1 Asymptotics of critical points when 𝜎→0 ‣ 3 Convergence Analysis ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_tag">3.2</span></a> thus motivates Algorithm <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#alg3" title="Algorithm 3 ‣ Optimization algorithms ‣ 2.2 SNORE regularization ‣ 2 Stochastic deNOising REgularization (SNORE) ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_tag">3</span></a>, which will be proved efficient in practice.</p> </div> </section> <section class="ltx_subsection" id="S3.SS2"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">3.2 </span>Unbiased algorithm analysis</h3> <div class="ltx_para" id="S3.SS2.p1"> <p class="ltx_p" id="S3.SS2.p1.1">In this section, we prove the convergence of the SNORE Algorithm <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#alg2" title="Algorithm 2 ‣ Optimization algorithms ‣ 2.2 SNORE regularization ‣ 2 Stochastic deNOising REgularization (SNORE) ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_tag">2</span></a> run with the exact MMSE denoiser <math alttext="D_{\sigma}^{\star}" class="ltx_Math" display="inline" id="S3.SS2.p1.1.m1.1"><semantics id="S3.SS2.p1.1.m1.1a"><msubsup id="S3.SS2.p1.1.m1.1.1" xref="S3.SS2.p1.1.m1.1.1.cmml"><mi id="S3.SS2.p1.1.m1.1.1.2.2" xref="S3.SS2.p1.1.m1.1.1.2.2.cmml">D</mi><mi id="S3.SS2.p1.1.m1.1.1.2.3" xref="S3.SS2.p1.1.m1.1.1.2.3.cmml">σ</mi><mo id="S3.SS2.p1.1.m1.1.1.3" xref="S3.SS2.p1.1.m1.1.1.3.cmml">⋆</mo></msubsup><annotation-xml encoding="MathML-Content" id="S3.SS2.p1.1.m1.1b"><apply id="S3.SS2.p1.1.m1.1.1.cmml" xref="S3.SS2.p1.1.m1.1.1"><csymbol cd="ambiguous" id="S3.SS2.p1.1.m1.1.1.1.cmml" xref="S3.SS2.p1.1.m1.1.1">superscript</csymbol><apply id="S3.SS2.p1.1.m1.1.1.2.cmml" xref="S3.SS2.p1.1.m1.1.1"><csymbol cd="ambiguous" id="S3.SS2.p1.1.m1.1.1.2.1.cmml" xref="S3.SS2.p1.1.m1.1.1">subscript</csymbol><ci id="S3.SS2.p1.1.m1.1.1.2.2.cmml" xref="S3.SS2.p1.1.m1.1.1.2.2">𝐷</ci><ci id="S3.SS2.p1.1.m1.1.1.2.3.cmml" xref="S3.SS2.p1.1.m1.1.1.2.3">𝜎</ci></apply><ci id="S3.SS2.p1.1.m1.1.1.3.cmml" xref="S3.SS2.p1.1.m1.1.1.3">⋆</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p1.1.m1.1c">D_{\sigma}^{\star}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p1.1.m1.1d">italic_D start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT</annotation></semantics></math>. With this denoiser, an iteration of the algorithm is computed by</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A8.EGx8"> <tbody id="S3.E12"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle{\mathbf{x}}_{k+1}={\mathbf{x}}_{k}-\delta_{k}\nabla\mathcal{F}({% \mathbf{x}}_{k},{\mathbf{y}})-\alpha\delta_{k}\nabla\log p_{\sigma}({\mathbf{x% }}_{k}+\zeta_{k})," class="ltx_Math" display="inline" id="S3.E12.m1.2"><semantics id="S3.E12.m1.2a"><mrow id="S3.E12.m1.2.2.1" xref="S3.E12.m1.2.2.1.1.cmml"><mrow id="S3.E12.m1.2.2.1.1" xref="S3.E12.m1.2.2.1.1.cmml"><msub id="S3.E12.m1.2.2.1.1.4" xref="S3.E12.m1.2.2.1.1.4.cmml"><mi id="S3.E12.m1.2.2.1.1.4.2" xref="S3.E12.m1.2.2.1.1.4.2.cmml">𝐱</mi><mrow id="S3.E12.m1.2.2.1.1.4.3" xref="S3.E12.m1.2.2.1.1.4.3.cmml"><mi id="S3.E12.m1.2.2.1.1.4.3.2" xref="S3.E12.m1.2.2.1.1.4.3.2.cmml">k</mi><mo 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xref="S3.E12.m1.2.2.1.1.2.2.5.1.2.cmml">log</mi></mrow><mo id="S3.E12.m1.2.2.1.1.2.2.5a" lspace="0.167em" xref="S3.E12.m1.2.2.1.1.2.2.5.cmml">⁡</mo><msub id="S3.E12.m1.2.2.1.1.2.2.5.2" xref="S3.E12.m1.2.2.1.1.2.2.5.2.cmml"><mi id="S3.E12.m1.2.2.1.1.2.2.5.2.2" xref="S3.E12.m1.2.2.1.1.2.2.5.2.2.cmml">p</mi><mi id="S3.E12.m1.2.2.1.1.2.2.5.2.3" xref="S3.E12.m1.2.2.1.1.2.2.5.2.3.cmml">σ</mi></msub></mrow><mo id="S3.E12.m1.2.2.1.1.2.2.2b" xref="S3.E12.m1.2.2.1.1.2.2.2.cmml">⁢</mo><mrow id="S3.E12.m1.2.2.1.1.2.2.1.1" xref="S3.E12.m1.2.2.1.1.2.2.1.1.1.cmml"><mo id="S3.E12.m1.2.2.1.1.2.2.1.1.2" stretchy="false" xref="S3.E12.m1.2.2.1.1.2.2.1.1.1.cmml">(</mo><mrow id="S3.E12.m1.2.2.1.1.2.2.1.1.1" xref="S3.E12.m1.2.2.1.1.2.2.1.1.1.cmml"><msub id="S3.E12.m1.2.2.1.1.2.2.1.1.1.2" xref="S3.E12.m1.2.2.1.1.2.2.1.1.1.2.cmml"><mi id="S3.E12.m1.2.2.1.1.2.2.1.1.1.2.2" xref="S3.E12.m1.2.2.1.1.2.2.1.1.1.2.2.cmml">𝐱</mi><mi id="S3.E12.m1.2.2.1.1.2.2.1.1.1.2.3" 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xref="S3.E12.m1.2.2.1.1.1.1.1.1.1.2">𝐱</ci><ci id="S3.E12.m1.2.2.1.1.1.1.1.1.1.3.cmml" xref="S3.E12.m1.2.2.1.1.1.1.1.1.1.3">𝑘</ci></apply><ci id="S3.E12.m1.1.1.cmml" xref="S3.E12.m1.1.1">𝐲</ci></interval></apply><apply id="S3.E12.m1.2.2.1.1.2.2.cmml" xref="S3.E12.m1.2.2.1.1.2.2"><times id="S3.E12.m1.2.2.1.1.2.2.2.cmml" xref="S3.E12.m1.2.2.1.1.2.2.2"></times><ci id="S3.E12.m1.2.2.1.1.2.2.3.cmml" xref="S3.E12.m1.2.2.1.1.2.2.3">𝛼</ci><apply id="S3.E12.m1.2.2.1.1.2.2.4.cmml" xref="S3.E12.m1.2.2.1.1.2.2.4"><csymbol cd="ambiguous" id="S3.E12.m1.2.2.1.1.2.2.4.1.cmml" xref="S3.E12.m1.2.2.1.1.2.2.4">subscript</csymbol><ci id="S3.E12.m1.2.2.1.1.2.2.4.2.cmml" xref="S3.E12.m1.2.2.1.1.2.2.4.2">𝛿</ci><ci id="S3.E12.m1.2.2.1.1.2.2.4.3.cmml" xref="S3.E12.m1.2.2.1.1.2.2.4.3">𝑘</ci></apply><apply id="S3.E12.m1.2.2.1.1.2.2.5.cmml" xref="S3.E12.m1.2.2.1.1.2.2.5"><apply id="S3.E12.m1.2.2.1.1.2.2.5.1.cmml" xref="S3.E12.m1.2.2.1.1.2.2.5.1"><ci id="S3.E12.m1.2.2.1.1.2.2.5.1.1.cmml" xref="S3.E12.m1.2.2.1.1.2.2.5.1.1">∇</ci><log id="S3.E12.m1.2.2.1.1.2.2.5.1.2.cmml" xref="S3.E12.m1.2.2.1.1.2.2.5.1.2"></log></apply><apply id="S3.E12.m1.2.2.1.1.2.2.5.2.cmml" xref="S3.E12.m1.2.2.1.1.2.2.5.2"><csymbol cd="ambiguous" id="S3.E12.m1.2.2.1.1.2.2.5.2.1.cmml" xref="S3.E12.m1.2.2.1.1.2.2.5.2">subscript</csymbol><ci id="S3.E12.m1.2.2.1.1.2.2.5.2.2.cmml" xref="S3.E12.m1.2.2.1.1.2.2.5.2.2">𝑝</ci><ci id="S3.E12.m1.2.2.1.1.2.2.5.2.3.cmml" xref="S3.E12.m1.2.2.1.1.2.2.5.2.3">𝜎</ci></apply></apply><apply id="S3.E12.m1.2.2.1.1.2.2.1.1.1.cmml" xref="S3.E12.m1.2.2.1.1.2.2.1.1"><plus id="S3.E12.m1.2.2.1.1.2.2.1.1.1.1.cmml" xref="S3.E12.m1.2.2.1.1.2.2.1.1.1.1"></plus><apply id="S3.E12.m1.2.2.1.1.2.2.1.1.1.2.cmml" xref="S3.E12.m1.2.2.1.1.2.2.1.1.1.2"><csymbol cd="ambiguous" id="S3.E12.m1.2.2.1.1.2.2.1.1.1.2.1.cmml" xref="S3.E12.m1.2.2.1.1.2.2.1.1.1.2">subscript</csymbol><ci id="S3.E12.m1.2.2.1.1.2.2.1.1.1.2.2.cmml" xref="S3.E12.m1.2.2.1.1.2.2.1.1.1.2.2">𝐱</ci><ci id="S3.E12.m1.2.2.1.1.2.2.1.1.1.2.3.cmml" xref="S3.E12.m1.2.2.1.1.2.2.1.1.1.2.3">𝑘</ci></apply><apply id="S3.E12.m1.2.2.1.1.2.2.1.1.1.3.cmml" xref="S3.E12.m1.2.2.1.1.2.2.1.1.1.3"><csymbol cd="ambiguous" id="S3.E12.m1.2.2.1.1.2.2.1.1.1.3.1.cmml" xref="S3.E12.m1.2.2.1.1.2.2.1.1.1.3">subscript</csymbol><ci id="S3.E12.m1.2.2.1.1.2.2.1.1.1.3.2.cmml" xref="S3.E12.m1.2.2.1.1.2.2.1.1.1.3.2">𝜁</ci><ci id="S3.E12.m1.2.2.1.1.2.2.1.1.1.3.3.cmml" xref="S3.E12.m1.2.2.1.1.2.2.1.1.1.3.3">𝑘</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.E12.m1.2c">\displaystyle{\mathbf{x}}_{k+1}={\mathbf{x}}_{k}-\delta_{k}\nabla\mathcal{F}({% \mathbf{x}}_{k},{\mathbf{y}})-\alpha\delta_{k}\nabla\log p_{\sigma}({\mathbf{x% }}_{k}+\zeta_{k}),</annotation><annotation encoding="application/x-llamapun" id="S3.E12.m1.2d">bold_x start_POSTSUBSCRIPT italic_k + 1 end_POSTSUBSCRIPT = bold_x start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT - italic_δ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ∇ caligraphic_F ( bold_x start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT , bold_y ) - italic_α italic_δ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ∇ roman_log italic_p start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ( bold_x start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT + italic_ζ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(12)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S3.SS2.p1.3">with <math alttext="\zeta_{k}\sim\mathcal{N}(0,\sigma^{2}\mathbf{I}_{d})" class="ltx_Math" display="inline" id="S3.SS2.p1.2.m1.2"><semantics id="S3.SS2.p1.2.m1.2a"><mrow id="S3.SS2.p1.2.m1.2.2" xref="S3.SS2.p1.2.m1.2.2.cmml"><msub id="S3.SS2.p1.2.m1.2.2.3" xref="S3.SS2.p1.2.m1.2.2.3.cmml"><mi id="S3.SS2.p1.2.m1.2.2.3.2" xref="S3.SS2.p1.2.m1.2.2.3.2.cmml">ζ</mi><mi id="S3.SS2.p1.2.m1.2.2.3.3" xref="S3.SS2.p1.2.m1.2.2.3.3.cmml">k</mi></msub><mo id="S3.SS2.p1.2.m1.2.2.2" xref="S3.SS2.p1.2.m1.2.2.2.cmml">∼</mo><mrow id="S3.SS2.p1.2.m1.2.2.1" xref="S3.SS2.p1.2.m1.2.2.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.SS2.p1.2.m1.2.2.1.3" xref="S3.SS2.p1.2.m1.2.2.1.3.cmml">𝒩</mi><mo id="S3.SS2.p1.2.m1.2.2.1.2" xref="S3.SS2.p1.2.m1.2.2.1.2.cmml">⁢</mo><mrow id="S3.SS2.p1.2.m1.2.2.1.1.1" xref="S3.SS2.p1.2.m1.2.2.1.1.2.cmml"><mo id="S3.SS2.p1.2.m1.2.2.1.1.1.2" stretchy="false" xref="S3.SS2.p1.2.m1.2.2.1.1.2.cmml">(</mo><mn id="S3.SS2.p1.2.m1.1.1" xref="S3.SS2.p1.2.m1.1.1.cmml">0</mn><mo id="S3.SS2.p1.2.m1.2.2.1.1.1.3" xref="S3.SS2.p1.2.m1.2.2.1.1.2.cmml">,</mo><mrow id="S3.SS2.p1.2.m1.2.2.1.1.1.1" xref="S3.SS2.p1.2.m1.2.2.1.1.1.1.cmml"><msup id="S3.SS2.p1.2.m1.2.2.1.1.1.1.2" xref="S3.SS2.p1.2.m1.2.2.1.1.1.1.2.cmml"><mi id="S3.SS2.p1.2.m1.2.2.1.1.1.1.2.2" xref="S3.SS2.p1.2.m1.2.2.1.1.1.1.2.2.cmml">σ</mi><mn id="S3.SS2.p1.2.m1.2.2.1.1.1.1.2.3" xref="S3.SS2.p1.2.m1.2.2.1.1.1.1.2.3.cmml">2</mn></msup><mo id="S3.SS2.p1.2.m1.2.2.1.1.1.1.1" xref="S3.SS2.p1.2.m1.2.2.1.1.1.1.1.cmml">⁢</mo><msub id="S3.SS2.p1.2.m1.2.2.1.1.1.1.3" xref="S3.SS2.p1.2.m1.2.2.1.1.1.1.3.cmml"><mi id="S3.SS2.p1.2.m1.2.2.1.1.1.1.3.2" xref="S3.SS2.p1.2.m1.2.2.1.1.1.1.3.2.cmml">𝐈</mi><mi id="S3.SS2.p1.2.m1.2.2.1.1.1.1.3.3" xref="S3.SS2.p1.2.m1.2.2.1.1.1.1.3.3.cmml">d</mi></msub></mrow><mo id="S3.SS2.p1.2.m1.2.2.1.1.1.4" stretchy="false" xref="S3.SS2.p1.2.m1.2.2.1.1.2.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.p1.2.m1.2b"><apply id="S3.SS2.p1.2.m1.2.2.cmml" xref="S3.SS2.p1.2.m1.2.2"><csymbol cd="latexml" id="S3.SS2.p1.2.m1.2.2.2.cmml" xref="S3.SS2.p1.2.m1.2.2.2">similar-to</csymbol><apply id="S3.SS2.p1.2.m1.2.2.3.cmml" xref="S3.SS2.p1.2.m1.2.2.3"><csymbol cd="ambiguous" id="S3.SS2.p1.2.m1.2.2.3.1.cmml" xref="S3.SS2.p1.2.m1.2.2.3">subscript</csymbol><ci id="S3.SS2.p1.2.m1.2.2.3.2.cmml" xref="S3.SS2.p1.2.m1.2.2.3.2">𝜁</ci><ci id="S3.SS2.p1.2.m1.2.2.3.3.cmml" xref="S3.SS2.p1.2.m1.2.2.3.3">𝑘</ci></apply><apply id="S3.SS2.p1.2.m1.2.2.1.cmml" xref="S3.SS2.p1.2.m1.2.2.1"><times id="S3.SS2.p1.2.m1.2.2.1.2.cmml" xref="S3.SS2.p1.2.m1.2.2.1.2"></times><ci id="S3.SS2.p1.2.m1.2.2.1.3.cmml" xref="S3.SS2.p1.2.m1.2.2.1.3">𝒩</ci><interval closure="open" id="S3.SS2.p1.2.m1.2.2.1.1.2.cmml" xref="S3.SS2.p1.2.m1.2.2.1.1.1"><cn id="S3.SS2.p1.2.m1.1.1.cmml" type="integer" xref="S3.SS2.p1.2.m1.1.1">0</cn><apply id="S3.SS2.p1.2.m1.2.2.1.1.1.1.cmml" xref="S3.SS2.p1.2.m1.2.2.1.1.1.1"><times id="S3.SS2.p1.2.m1.2.2.1.1.1.1.1.cmml" xref="S3.SS2.p1.2.m1.2.2.1.1.1.1.1"></times><apply id="S3.SS2.p1.2.m1.2.2.1.1.1.1.2.cmml" xref="S3.SS2.p1.2.m1.2.2.1.1.1.1.2"><csymbol cd="ambiguous" id="S3.SS2.p1.2.m1.2.2.1.1.1.1.2.1.cmml" xref="S3.SS2.p1.2.m1.2.2.1.1.1.1.2">superscript</csymbol><ci id="S3.SS2.p1.2.m1.2.2.1.1.1.1.2.2.cmml" xref="S3.SS2.p1.2.m1.2.2.1.1.1.1.2.2">𝜎</ci><cn id="S3.SS2.p1.2.m1.2.2.1.1.1.1.2.3.cmml" type="integer" xref="S3.SS2.p1.2.m1.2.2.1.1.1.1.2.3">2</cn></apply><apply id="S3.SS2.p1.2.m1.2.2.1.1.1.1.3.cmml" xref="S3.SS2.p1.2.m1.2.2.1.1.1.1.3"><csymbol cd="ambiguous" id="S3.SS2.p1.2.m1.2.2.1.1.1.1.3.1.cmml" xref="S3.SS2.p1.2.m1.2.2.1.1.1.1.3">subscript</csymbol><ci id="S3.SS2.p1.2.m1.2.2.1.1.1.1.3.2.cmml" xref="S3.SS2.p1.2.m1.2.2.1.1.1.1.3.2">𝐈</ci><ci id="S3.SS2.p1.2.m1.2.2.1.1.1.1.3.3.cmml" xref="S3.SS2.p1.2.m1.2.2.1.1.1.1.3.3">𝑑</ci></apply></apply></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p1.2.m1.2c">\zeta_{k}\sim\mathcal{N}(0,\sigma^{2}\mathbf{I}_{d})</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p1.2.m1.2d">italic_ζ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ∼ caligraphic_N ( 0 , italic_σ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT bold_I start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT )</annotation></semantics></math> and <math alttext="(\delta_{k})_{k\in{\mathbb{N}}}\in{\left({\mathbb{R}}^{+}\right)}^{{\mathbb{N}}}" class="ltx_Math" display="inline" id="S3.SS2.p1.3.m2.2"><semantics id="S3.SS2.p1.3.m2.2a"><mrow id="S3.SS2.p1.3.m2.2.2" xref="S3.SS2.p1.3.m2.2.2.cmml"><msub id="S3.SS2.p1.3.m2.1.1.1" xref="S3.SS2.p1.3.m2.1.1.1.cmml"><mrow id="S3.SS2.p1.3.m2.1.1.1.1.1" xref="S3.SS2.p1.3.m2.1.1.1.1.1.1.cmml"><mo id="S3.SS2.p1.3.m2.1.1.1.1.1.2" stretchy="false" xref="S3.SS2.p1.3.m2.1.1.1.1.1.1.cmml">(</mo><msub id="S3.SS2.p1.3.m2.1.1.1.1.1.1" xref="S3.SS2.p1.3.m2.1.1.1.1.1.1.cmml"><mi id="S3.SS2.p1.3.m2.1.1.1.1.1.1.2" xref="S3.SS2.p1.3.m2.1.1.1.1.1.1.2.cmml">δ</mi><mi id="S3.SS2.p1.3.m2.1.1.1.1.1.1.3" xref="S3.SS2.p1.3.m2.1.1.1.1.1.1.3.cmml">k</mi></msub><mo id="S3.SS2.p1.3.m2.1.1.1.1.1.3" stretchy="false" xref="S3.SS2.p1.3.m2.1.1.1.1.1.1.cmml">)</mo></mrow><mrow id="S3.SS2.p1.3.m2.1.1.1.3" xref="S3.SS2.p1.3.m2.1.1.1.3.cmml"><mi id="S3.SS2.p1.3.m2.1.1.1.3.2" xref="S3.SS2.p1.3.m2.1.1.1.3.2.cmml">k</mi><mo id="S3.SS2.p1.3.m2.1.1.1.3.1" xref="S3.SS2.p1.3.m2.1.1.1.3.1.cmml">∈</mo><mi id="S3.SS2.p1.3.m2.1.1.1.3.3" xref="S3.SS2.p1.3.m2.1.1.1.3.3.cmml">ℕ</mi></mrow></msub><mo id="S3.SS2.p1.3.m2.2.2.3" xref="S3.SS2.p1.3.m2.2.2.3.cmml">∈</mo><msup id="S3.SS2.p1.3.m2.2.2.2" xref="S3.SS2.p1.3.m2.2.2.2.cmml"><mrow id="S3.SS2.p1.3.m2.2.2.2.1.1" xref="S3.SS2.p1.3.m2.2.2.2.1.1.1.cmml"><mo id="S3.SS2.p1.3.m2.2.2.2.1.1.2" xref="S3.SS2.p1.3.m2.2.2.2.1.1.1.cmml">(</mo><msup id="S3.SS2.p1.3.m2.2.2.2.1.1.1" xref="S3.SS2.p1.3.m2.2.2.2.1.1.1.cmml"><mi id="S3.SS2.p1.3.m2.2.2.2.1.1.1.2" xref="S3.SS2.p1.3.m2.2.2.2.1.1.1.2.cmml">ℝ</mi><mo id="S3.SS2.p1.3.m2.2.2.2.1.1.1.3" xref="S3.SS2.p1.3.m2.2.2.2.1.1.1.3.cmml">+</mo></msup><mo id="S3.SS2.p1.3.m2.2.2.2.1.1.3" xref="S3.SS2.p1.3.m2.2.2.2.1.1.1.cmml">)</mo></mrow><mi id="S3.SS2.p1.3.m2.2.2.2.3" xref="S3.SS2.p1.3.m2.2.2.2.3.cmml">ℕ</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.p1.3.m2.2b"><apply id="S3.SS2.p1.3.m2.2.2.cmml" xref="S3.SS2.p1.3.m2.2.2"><in id="S3.SS2.p1.3.m2.2.2.3.cmml" xref="S3.SS2.p1.3.m2.2.2.3"></in><apply id="S3.SS2.p1.3.m2.1.1.1.cmml" xref="S3.SS2.p1.3.m2.1.1.1"><csymbol cd="ambiguous" id="S3.SS2.p1.3.m2.1.1.1.2.cmml" xref="S3.SS2.p1.3.m2.1.1.1">subscript</csymbol><apply id="S3.SS2.p1.3.m2.1.1.1.1.1.1.cmml" xref="S3.SS2.p1.3.m2.1.1.1.1.1"><csymbol cd="ambiguous" id="S3.SS2.p1.3.m2.1.1.1.1.1.1.1.cmml" xref="S3.SS2.p1.3.m2.1.1.1.1.1">subscript</csymbol><ci id="S3.SS2.p1.3.m2.1.1.1.1.1.1.2.cmml" xref="S3.SS2.p1.3.m2.1.1.1.1.1.1.2">𝛿</ci><ci id="S3.SS2.p1.3.m2.1.1.1.1.1.1.3.cmml" xref="S3.SS2.p1.3.m2.1.1.1.1.1.1.3">𝑘</ci></apply><apply id="S3.SS2.p1.3.m2.1.1.1.3.cmml" xref="S3.SS2.p1.3.m2.1.1.1.3"><in id="S3.SS2.p1.3.m2.1.1.1.3.1.cmml" xref="S3.SS2.p1.3.m2.1.1.1.3.1"></in><ci id="S3.SS2.p1.3.m2.1.1.1.3.2.cmml" xref="S3.SS2.p1.3.m2.1.1.1.3.2">𝑘</ci><ci id="S3.SS2.p1.3.m2.1.1.1.3.3.cmml" xref="S3.SS2.p1.3.m2.1.1.1.3.3">ℕ</ci></apply></apply><apply id="S3.SS2.p1.3.m2.2.2.2.cmml" xref="S3.SS2.p1.3.m2.2.2.2"><csymbol cd="ambiguous" id="S3.SS2.p1.3.m2.2.2.2.2.cmml" xref="S3.SS2.p1.3.m2.2.2.2">superscript</csymbol><apply id="S3.SS2.p1.3.m2.2.2.2.1.1.1.cmml" xref="S3.SS2.p1.3.m2.2.2.2.1.1"><csymbol cd="ambiguous" id="S3.SS2.p1.3.m2.2.2.2.1.1.1.1.cmml" xref="S3.SS2.p1.3.m2.2.2.2.1.1">superscript</csymbol><ci id="S3.SS2.p1.3.m2.2.2.2.1.1.1.2.cmml" xref="S3.SS2.p1.3.m2.2.2.2.1.1.1.2">ℝ</ci><plus id="S3.SS2.p1.3.m2.2.2.2.1.1.1.3.cmml" xref="S3.SS2.p1.3.m2.2.2.2.1.1.1.3"></plus></apply><ci id="S3.SS2.p1.3.m2.2.2.2.3.cmml" xref="S3.SS2.p1.3.m2.2.2.2.3">ℕ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p1.3.m2.2c">(\delta_{k})_{k\in{\mathbb{N}}}\in{\left({\mathbb{R}}^{+}\right)}^{{\mathbb{N}}}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p1.3.m2.2d">( italic_δ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) start_POSTSUBSCRIPT italic_k ∈ blackboard_N end_POSTSUBSCRIPT ∈ ( blackboard_R start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT blackboard_N end_POSTSUPERSCRIPT</annotation></semantics></math> the decreasing sequence of step-sizes. Algorithm <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#alg2" title="Algorithm 2 ‣ Optimization algorithms ‣ 2.2 SNORE regularization ‣ 2 Stochastic deNOising REgularization (SNORE) ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_tag">2</span></a> is a stochastic descent algorithm that solves Problem (<a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#S2.E9" title="9 ‣ Optimization algorithms ‣ 2.2 SNORE regularization ‣ 2 Stochastic deNOising REgularization (SNORE) ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_tag">9</span></a>).</p> </div> <div class="ltx_para" id="S3.SS2.p2"> <p class="ltx_p" id="S3.SS2.p2.1">One can note that the stochastic gradient estimation is unbiased. Indeed, by defining</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A8.EGx9"> <tbody id="S3.Ex9"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle f({\mathbf{x}},\zeta)=\nabla\mathcal{F}({\mathbf{x}},{\mathbf{y}% })+\alpha\nabla\log p_{\sigma}({\mathbf{x}}+\zeta)," class="ltx_Math" display="inline" id="S3.Ex9.m1.5"><semantics id="S3.Ex9.m1.5a"><mrow id="S3.Ex9.m1.5.5.1" xref="S3.Ex9.m1.5.5.1.1.cmml"><mrow id="S3.Ex9.m1.5.5.1.1" xref="S3.Ex9.m1.5.5.1.1.cmml"><mrow id="S3.Ex9.m1.5.5.1.1.3" xref="S3.Ex9.m1.5.5.1.1.3.cmml"><mi id="S3.Ex9.m1.5.5.1.1.3.2" xref="S3.Ex9.m1.5.5.1.1.3.2.cmml">f</mi><mo id="S3.Ex9.m1.5.5.1.1.3.1" xref="S3.Ex9.m1.5.5.1.1.3.1.cmml">⁢</mo><mrow id="S3.Ex9.m1.5.5.1.1.3.3.2" xref="S3.Ex9.m1.5.5.1.1.3.3.1.cmml"><mo id="S3.Ex9.m1.5.5.1.1.3.3.2.1" stretchy="false" xref="S3.Ex9.m1.5.5.1.1.3.3.1.cmml">(</mo><mi id="S3.Ex9.m1.1.1" xref="S3.Ex9.m1.1.1.cmml">𝐱</mi><mo id="S3.Ex9.m1.5.5.1.1.3.3.2.2" xref="S3.Ex9.m1.5.5.1.1.3.3.1.cmml">,</mo><mi id="S3.Ex9.m1.2.2" xref="S3.Ex9.m1.2.2.cmml">ζ</mi><mo id="S3.Ex9.m1.5.5.1.1.3.3.2.3" stretchy="false" xref="S3.Ex9.m1.5.5.1.1.3.3.1.cmml">)</mo></mrow></mrow><mo id="S3.Ex9.m1.5.5.1.1.2" xref="S3.Ex9.m1.5.5.1.1.2.cmml">=</mo><mrow id="S3.Ex9.m1.5.5.1.1.1" xref="S3.Ex9.m1.5.5.1.1.1.cmml"><mrow id="S3.Ex9.m1.5.5.1.1.1.3" xref="S3.Ex9.m1.5.5.1.1.1.3.cmml"><mrow id="S3.Ex9.m1.5.5.1.1.1.3.2" xref="S3.Ex9.m1.5.5.1.1.1.3.2.cmml"><mo id="S3.Ex9.m1.5.5.1.1.1.3.2.1" rspace="0.167em" xref="S3.Ex9.m1.5.5.1.1.1.3.2.1.cmml">∇</mo><mi class="ltx_font_mathcaligraphic" id="S3.Ex9.m1.5.5.1.1.1.3.2.2" xref="S3.Ex9.m1.5.5.1.1.1.3.2.2.cmml">ℱ</mi></mrow><mo id="S3.Ex9.m1.5.5.1.1.1.3.1" xref="S3.Ex9.m1.5.5.1.1.1.3.1.cmml">⁢</mo><mrow id="S3.Ex9.m1.5.5.1.1.1.3.3.2" xref="S3.Ex9.m1.5.5.1.1.1.3.3.1.cmml"><mo id="S3.Ex9.m1.5.5.1.1.1.3.3.2.1" stretchy="false" 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})+\alpha\nabla\log p_{\sigma}({\mathbf{x}}+\zeta),</annotation><annotation encoding="application/x-llamapun" id="S3.Ex9.m1.5d">italic_f ( bold_x , italic_ζ ) = ∇ caligraphic_F ( bold_x , bold_y ) + italic_α ∇ roman_log italic_p start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ( bold_x + italic_ζ ) ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S3.SS2.p2.2">we verify that</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A8.EGx10"> <tbody id="S3.Ex10"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\mathbb{E}_{\zeta\sim\mathcal{N}(0,\sigma^{2}\mathbf{I}_{d})}{% \left(f({\mathbf{x}},\zeta)\right)}=\nabla\mathcal{F}({\mathbf{x}},{\mathbf{y}% })+\alpha\nabla\mathcal{R}_{\sigma}({\mathbf{x}})." class="ltx_Math" display="inline" 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xref="S3.Ex10.m1.1.1.1.1.cmml">0</mn><mo id="S3.Ex10.m1.2.2.2.2.1.1.3" xref="S3.Ex10.m1.2.2.2.2.1.2.cmml">,</mo><mrow id="S3.Ex10.m1.2.2.2.2.1.1.1" xref="S3.Ex10.m1.2.2.2.2.1.1.1.cmml"><msup id="S3.Ex10.m1.2.2.2.2.1.1.1.2" xref="S3.Ex10.m1.2.2.2.2.1.1.1.2.cmml"><mi id="S3.Ex10.m1.2.2.2.2.1.1.1.2.2" xref="S3.Ex10.m1.2.2.2.2.1.1.1.2.2.cmml">σ</mi><mn id="S3.Ex10.m1.2.2.2.2.1.1.1.2.3" xref="S3.Ex10.m1.2.2.2.2.1.1.1.2.3.cmml">2</mn></msup><mo id="S3.Ex10.m1.2.2.2.2.1.1.1.1" xref="S3.Ex10.m1.2.2.2.2.1.1.1.1.cmml">⁢</mo><msub id="S3.Ex10.m1.2.2.2.2.1.1.1.3" xref="S3.Ex10.m1.2.2.2.2.1.1.1.3.cmml"><mi id="S3.Ex10.m1.2.2.2.2.1.1.1.3.2" xref="S3.Ex10.m1.2.2.2.2.1.1.1.3.2.cmml">𝐈</mi><mi id="S3.Ex10.m1.2.2.2.2.1.1.1.3.3" xref="S3.Ex10.m1.2.2.2.2.1.1.1.3.3.cmml">d</mi></msub></mrow><mo id="S3.Ex10.m1.2.2.2.2.1.1.4" stretchy="false" xref="S3.Ex10.m1.2.2.2.2.1.2.cmml">)</mo></mrow></mrow></mrow></msub><mo id="S3.Ex10.m1.8.8.1.1.1.2" xref="S3.Ex10.m1.8.8.1.1.1.2.cmml">⁢</mo><mrow 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xref="S3.Ex10.m1.8.8.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S3.Ex10.m1.8.8.1.1.2" xref="S3.Ex10.m1.8.8.1.1.2.cmml">=</mo><mrow id="S3.Ex10.m1.8.8.1.1.3" xref="S3.Ex10.m1.8.8.1.1.3.cmml"><mrow id="S3.Ex10.m1.8.8.1.1.3.2" xref="S3.Ex10.m1.8.8.1.1.3.2.cmml"><mrow id="S3.Ex10.m1.8.8.1.1.3.2.2" xref="S3.Ex10.m1.8.8.1.1.3.2.2.cmml"><mo id="S3.Ex10.m1.8.8.1.1.3.2.2.1" rspace="0.167em" xref="S3.Ex10.m1.8.8.1.1.3.2.2.1.cmml">∇</mo><mi class="ltx_font_mathcaligraphic" id="S3.Ex10.m1.8.8.1.1.3.2.2.2" xref="S3.Ex10.m1.8.8.1.1.3.2.2.2.cmml">ℱ</mi></mrow><mo id="S3.Ex10.m1.8.8.1.1.3.2.1" xref="S3.Ex10.m1.8.8.1.1.3.2.1.cmml">⁢</mo><mrow id="S3.Ex10.m1.8.8.1.1.3.2.3.2" xref="S3.Ex10.m1.8.8.1.1.3.2.3.1.cmml"><mo id="S3.Ex10.m1.8.8.1.1.3.2.3.2.1" stretchy="false" xref="S3.Ex10.m1.8.8.1.1.3.2.3.1.cmml">(</mo><mi id="S3.Ex10.m1.5.5" xref="S3.Ex10.m1.5.5.cmml">𝐱</mi><mo id="S3.Ex10.m1.8.8.1.1.3.2.3.2.2" xref="S3.Ex10.m1.8.8.1.1.3.2.3.1.cmml">,</mo><mi id="S3.Ex10.m1.6.6" 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xref="S3.Ex10.m1.8.8.1.1.3.3"><times id="S3.Ex10.m1.8.8.1.1.3.3.1.cmml" xref="S3.Ex10.m1.8.8.1.1.3.3.1"></times><ci id="S3.Ex10.m1.8.8.1.1.3.3.2.cmml" xref="S3.Ex10.m1.8.8.1.1.3.3.2">𝛼</ci><apply id="S3.Ex10.m1.8.8.1.1.3.3.3.cmml" xref="S3.Ex10.m1.8.8.1.1.3.3.3"><ci id="S3.Ex10.m1.8.8.1.1.3.3.3.1.cmml" xref="S3.Ex10.m1.8.8.1.1.3.3.3.1">∇</ci><apply id="S3.Ex10.m1.8.8.1.1.3.3.3.2.cmml" xref="S3.Ex10.m1.8.8.1.1.3.3.3.2"><csymbol cd="ambiguous" id="S3.Ex10.m1.8.8.1.1.3.3.3.2.1.cmml" xref="S3.Ex10.m1.8.8.1.1.3.3.3.2">subscript</csymbol><ci id="S3.Ex10.m1.8.8.1.1.3.3.3.2.2.cmml" xref="S3.Ex10.m1.8.8.1.1.3.3.3.2.2">ℛ</ci><ci id="S3.Ex10.m1.8.8.1.1.3.3.3.2.3.cmml" xref="S3.Ex10.m1.8.8.1.1.3.3.3.2.3">𝜎</ci></apply></apply><ci id="S3.Ex10.m1.7.7.cmml" xref="S3.Ex10.m1.7.7">𝐱</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Ex10.m1.8c">\displaystyle\mathbb{E}_{\zeta\sim\mathcal{N}(0,\sigma^{2}\mathbf{I}_{d})}{% \left(f({\mathbf{x}},\zeta)\right)}=\nabla\mathcal{F}({\mathbf{x}},{\mathbf{y}% })+\alpha\nabla\mathcal{R}_{\sigma}({\mathbf{x}}).</annotation><annotation encoding="application/x-llamapun" id="S3.Ex10.m1.8d">blackboard_E start_POSTSUBSCRIPT italic_ζ ∼ caligraphic_N ( 0 , italic_σ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT bold_I start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT ) end_POSTSUBSCRIPT ( italic_f ( bold_x , italic_ζ ) ) = ∇ caligraphic_F ( bold_x , bold_y ) + italic_α ∇ caligraphic_R start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ( bold_x ) .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> </div> <div class="ltx_para" id="S3.SS2.p3"> <p class="ltx_p" id="S3.SS2.p3.1">We make our convergence analysis based on previous studies on stochastic gradient algorithm <cite class="ltx_cite ltx_citemacro_citep">(Benaïm, <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#bib.bib2" title="">1999</a>, Corollary 6.7)</cite>, <cite class="ltx_cite ltx_citemacro_citep">(Metivier & Priouret, <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#bib.bib47" title="">1984</a>, Section II-D)</cite> or <cite class="ltx_cite ltx_citemacro_citep">(Tadic & Doucet, <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#bib.bib72" title="">2017</a>)</cite>. An assumption on the step-size decrease is required to ensure convergence.</p> </div> <div class="ltx_theorem ltx_theorem_assumption" id="S3.Thmtheorem3"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S3.Thmtheorem3.1.1.1">Assumption 3.3</span></span><span class="ltx_text ltx_font_bold" id="S3.Thmtheorem3.2.2">.</span> </h6> <div class="ltx_para" id="S3.Thmtheorem3.p1"> <p class="ltx_p" id="S3.Thmtheorem3.p1.2"><span class="ltx_text ltx_font_italic" id="S3.Thmtheorem3.p1.2.2">The step-size decreases to zero but not too fast: <math alttext="\sum_{k=0}^{+\infty}{\delta_{k}}=+\infty" class="ltx_Math" display="inline" id="S3.Thmtheorem3.p1.1.1.m1.1"><semantics id="S3.Thmtheorem3.p1.1.1.m1.1a"><mrow id="S3.Thmtheorem3.p1.1.1.m1.1.1" xref="S3.Thmtheorem3.p1.1.1.m1.1.1.cmml"><mrow id="S3.Thmtheorem3.p1.1.1.m1.1.1.2" xref="S3.Thmtheorem3.p1.1.1.m1.1.1.2.cmml"><msubsup id="S3.Thmtheorem3.p1.1.1.m1.1.1.2.1" 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xref="S3.Thmtheorem3.p1.1.1.m1.1.1.3"></plus><infinity id="S3.Thmtheorem3.p1.1.1.m1.1.1.3.2.cmml" xref="S3.Thmtheorem3.p1.1.1.m1.1.1.3.2"></infinity></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem3.p1.1.1.m1.1c">\sum_{k=0}^{+\infty}{\delta_{k}}=+\infty</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem3.p1.1.1.m1.1d">∑ start_POSTSUBSCRIPT italic_k = 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT + ∞ end_POSTSUPERSCRIPT italic_δ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT = + ∞</annotation></semantics></math> and <math alttext="\sum_{k=0}^{+\infty}{\delta_{k}^{2}}<+\infty" class="ltx_Math" display="inline" id="S3.Thmtheorem3.p1.2.2.m2.1"><semantics id="S3.Thmtheorem3.p1.2.2.m2.1a"><mrow id="S3.Thmtheorem3.p1.2.2.m2.1.1" xref="S3.Thmtheorem3.p1.2.2.m2.1.1.cmml"><mrow id="S3.Thmtheorem3.p1.2.2.m2.1.1.2" xref="S3.Thmtheorem3.p1.2.2.m2.1.1.2.cmml"><msubsup id="S3.Thmtheorem3.p1.2.2.m2.1.1.2.1" 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id="S3.Thmtheorem3.p1.2.2.m2.1.1.2.1.2.3.2.cmml" xref="S3.Thmtheorem3.p1.2.2.m2.1.1.2.1.2.3.2">𝑘</ci><cn id="S3.Thmtheorem3.p1.2.2.m2.1.1.2.1.2.3.3.cmml" type="integer" xref="S3.Thmtheorem3.p1.2.2.m2.1.1.2.1.2.3.3">0</cn></apply></apply><apply id="S3.Thmtheorem3.p1.2.2.m2.1.1.2.1.3.cmml" xref="S3.Thmtheorem3.p1.2.2.m2.1.1.2.1.3"><plus id="S3.Thmtheorem3.p1.2.2.m2.1.1.2.1.3.1.cmml" xref="S3.Thmtheorem3.p1.2.2.m2.1.1.2.1.3"></plus><infinity id="S3.Thmtheorem3.p1.2.2.m2.1.1.2.1.3.2.cmml" xref="S3.Thmtheorem3.p1.2.2.m2.1.1.2.1.3.2"></infinity></apply></apply><apply id="S3.Thmtheorem3.p1.2.2.m2.1.1.2.2.cmml" xref="S3.Thmtheorem3.p1.2.2.m2.1.1.2.2"><csymbol cd="ambiguous" id="S3.Thmtheorem3.p1.2.2.m2.1.1.2.2.1.cmml" xref="S3.Thmtheorem3.p1.2.2.m2.1.1.2.2">superscript</csymbol><apply id="S3.Thmtheorem3.p1.2.2.m2.1.1.2.2.2.cmml" xref="S3.Thmtheorem3.p1.2.2.m2.1.1.2.2"><csymbol cd="ambiguous" id="S3.Thmtheorem3.p1.2.2.m2.1.1.2.2.2.1.cmml" 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∞ end_POSTSUPERSCRIPT italic_δ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT < + ∞</annotation></semantics></math>.</span></p> </div> </div> <div class="ltx_para" id="S3.SS2.p4"> <p class="ltx_p" id="S3.SS2.p4.2">Assumption <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#S3.Thmtheorem3" title="Assumption 3.3. ‣ 3.2 Unbiased algorithm analysis ‣ 3 Convergence Analysis ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_tag">3.3</span></a> guides the choice of the step-size rule to ensure convergence, for instance <math alttext="\delta_{k}=\frac{\delta}{k^{a}}" class="ltx_Math" display="inline" id="S3.SS2.p4.1.m1.1"><semantics id="S3.SS2.p4.1.m1.1a"><mrow id="S3.SS2.p4.1.m1.1.1" xref="S3.SS2.p4.1.m1.1.1.cmml"><msub id="S3.SS2.p4.1.m1.1.1.2" xref="S3.SS2.p4.1.m1.1.1.2.cmml"><mi id="S3.SS2.p4.1.m1.1.1.2.2" xref="S3.SS2.p4.1.m1.1.1.2.2.cmml">δ</mi><mi id="S3.SS2.p4.1.m1.1.1.2.3" xref="S3.SS2.p4.1.m1.1.1.2.3.cmml">k</mi></msub><mo id="S3.SS2.p4.1.m1.1.1.1" xref="S3.SS2.p4.1.m1.1.1.1.cmml">=</mo><mfrac id="S3.SS2.p4.1.m1.1.1.3" xref="S3.SS2.p4.1.m1.1.1.3.cmml"><mi id="S3.SS2.p4.1.m1.1.1.3.2" xref="S3.SS2.p4.1.m1.1.1.3.2.cmml">δ</mi><msup id="S3.SS2.p4.1.m1.1.1.3.3" xref="S3.SS2.p4.1.m1.1.1.3.3.cmml"><mi id="S3.SS2.p4.1.m1.1.1.3.3.2" xref="S3.SS2.p4.1.m1.1.1.3.3.2.cmml">k</mi><mi id="S3.SS2.p4.1.m1.1.1.3.3.3" xref="S3.SS2.p4.1.m1.1.1.3.3.3.cmml">a</mi></msup></mfrac></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.p4.1.m1.1b"><apply id="S3.SS2.p4.1.m1.1.1.cmml" xref="S3.SS2.p4.1.m1.1.1"><eq id="S3.SS2.p4.1.m1.1.1.1.cmml" xref="S3.SS2.p4.1.m1.1.1.1"></eq><apply id="S3.SS2.p4.1.m1.1.1.2.cmml" xref="S3.SS2.p4.1.m1.1.1.2"><csymbol cd="ambiguous" id="S3.SS2.p4.1.m1.1.1.2.1.cmml" xref="S3.SS2.p4.1.m1.1.1.2">subscript</csymbol><ci id="S3.SS2.p4.1.m1.1.1.2.2.cmml" xref="S3.SS2.p4.1.m1.1.1.2.2">𝛿</ci><ci id="S3.SS2.p4.1.m1.1.1.2.3.cmml" xref="S3.SS2.p4.1.m1.1.1.2.3">𝑘</ci></apply><apply id="S3.SS2.p4.1.m1.1.1.3.cmml" xref="S3.SS2.p4.1.m1.1.1.3"><divide id="S3.SS2.p4.1.m1.1.1.3.1.cmml" xref="S3.SS2.p4.1.m1.1.1.3"></divide><ci id="S3.SS2.p4.1.m1.1.1.3.2.cmml" xref="S3.SS2.p4.1.m1.1.1.3.2">𝛿</ci><apply id="S3.SS2.p4.1.m1.1.1.3.3.cmml" xref="S3.SS2.p4.1.m1.1.1.3.3"><csymbol cd="ambiguous" id="S3.SS2.p4.1.m1.1.1.3.3.1.cmml" xref="S3.SS2.p4.1.m1.1.1.3.3">superscript</csymbol><ci id="S3.SS2.p4.1.m1.1.1.3.3.2.cmml" xref="S3.SS2.p4.1.m1.1.1.3.3.2">𝑘</ci><ci id="S3.SS2.p4.1.m1.1.1.3.3.3.cmml" xref="S3.SS2.p4.1.m1.1.1.3.3.3">𝑎</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p4.1.m1.1c">\delta_{k}=\frac{\delta}{k^{a}}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p4.1.m1.1d">italic_δ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT = divide start_ARG italic_δ end_ARG start_ARG italic_k start_POSTSUPERSCRIPT italic_a end_POSTSUPERSCRIPT end_ARG</annotation></semantics></math> with <math alttext="a\in]\frac{1}{2},1]" class="ltx_math_unparsed" display="inline" id="S3.SS2.p4.2.m2.1"><semantics id="S3.SS2.p4.2.m2.1a"><mrow id="S3.SS2.p4.2.m2.1b"><mrow id="S3.SS2.p4.2.m2.1.1"><mi id="S3.SS2.p4.2.m2.1.1.1">a</mi><mo id="S3.SS2.p4.2.m2.1.1.2" rspace="0em">∈</mo><mo id="S3.SS2.p4.2.m2.1.1.3" stretchy="false">]</mo></mrow><mfrac id="S3.SS2.p4.2.m2.1.2"><mn id="S3.SS2.p4.2.m2.1.2.2">1</mn><mn id="S3.SS2.p4.2.m2.1.2.3">2</mn></mfrac><mo id="S3.SS2.p4.2.m2.1.3">,</mo><mn id="S3.SS2.p4.2.m2.1.4">1</mn><mo id="S3.SS2.p4.2.m2.1.5" stretchy="false">]</mo></mrow><annotation encoding="application/x-tex" id="S3.SS2.p4.2.m2.1c">a\in]\frac{1}{2},1]</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p4.2.m2.1d">italic_a ∈ ] divide start_ARG 1 end_ARG start_ARG 2 end_ARG , 1 ]</annotation></semantics></math>.</p> </div> <div class="ltx_theorem ltx_theorem_assumption" id="S3.Thmtheorem4"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S3.Thmtheorem4.1.1.1">Assumption 3.4</span></span><span class="ltx_text ltx_font_bold" id="S3.Thmtheorem4.2.2">.</span> </h6> <div class="ltx_para" id="S3.Thmtheorem4.p1"> <p class="ltx_p" id="S3.Thmtheorem4.p1.2"><span class="ltx_text ltx_font_italic" id="S3.Thmtheorem4.p1.2.2">The data-fidelity term <math alttext="\mathcal{F}_{y}:{\mathbf{x}}\in{\mathbb{R}}^{d}\mapsto\mathcal{F}({\mathbf{x}}% ,{\mathbf{y}})" class="ltx_Math" display="inline" id="S3.Thmtheorem4.p1.1.1.m1.2"><semantics id="S3.Thmtheorem4.p1.1.1.m1.2a"><mrow id="S3.Thmtheorem4.p1.1.1.m1.2.3" xref="S3.Thmtheorem4.p1.1.1.m1.2.3.cmml"><msub id="S3.Thmtheorem4.p1.1.1.m1.2.3.2" xref="S3.Thmtheorem4.p1.1.1.m1.2.3.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.Thmtheorem4.p1.1.1.m1.2.3.2.2" xref="S3.Thmtheorem4.p1.1.1.m1.2.3.2.2.cmml">ℱ</mi><mi id="S3.Thmtheorem4.p1.1.1.m1.2.3.2.3" 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end_POSTSUBSCRIPT : bold_x ∈ blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT ↦ caligraphic_F ( bold_x , bold_y )</annotation></semantics></math> is <math alttext="\mathcal{C}^{\infty}" class="ltx_Math" display="inline" id="S3.Thmtheorem4.p1.2.2.m2.1"><semantics id="S3.Thmtheorem4.p1.2.2.m2.1a"><msup id="S3.Thmtheorem4.p1.2.2.m2.1.1" xref="S3.Thmtheorem4.p1.2.2.m2.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.Thmtheorem4.p1.2.2.m2.1.1.2" xref="S3.Thmtheorem4.p1.2.2.m2.1.1.2.cmml">𝒞</mi><mi id="S3.Thmtheorem4.p1.2.2.m2.1.1.3" mathvariant="normal" xref="S3.Thmtheorem4.p1.2.2.m2.1.1.3.cmml">∞</mi></msup><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem4.p1.2.2.m2.1b"><apply id="S3.Thmtheorem4.p1.2.2.m2.1.1.cmml" xref="S3.Thmtheorem4.p1.2.2.m2.1.1"><csymbol cd="ambiguous" id="S3.Thmtheorem4.p1.2.2.m2.1.1.1.cmml" xref="S3.Thmtheorem4.p1.2.2.m2.1.1">superscript</csymbol><ci id="S3.Thmtheorem4.p1.2.2.m2.1.1.2.cmml" 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encoding="application/x-tex" id="S3.SS2.p5.1.m1.3c">\mathcal{F}({\mathbf{x}},{\mathbf{y}})=\frac{1}{2\sigma_{{\mathbf{y}}}^{2}}\|{% \mathbf{y}}-\mathcal{A}{\mathbf{x}}\|^{2}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p5.1.m1.3d">caligraphic_F ( bold_x , bold_y ) = divide start_ARG 1 end_ARG start_ARG 2 italic_σ start_POSTSUBSCRIPT bold_y end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG ∥ bold_y - caligraphic_A bold_x ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT</annotation></semantics></math> associated to a linear inverse problem with additive white Gaussian noise.</p> </div> <div class="ltx_para" id="S3.SS2.p6"> <p class="ltx_p" id="S3.SS2.p6.1">We define the set of realizations where the sequence is bounded in the compact <math alttext="\mathbf{K}" class="ltx_Math" display="inline" id="S3.SS2.p6.1.m1.1"><semantics id="S3.SS2.p6.1.m1.1a"><mi id="S3.SS2.p6.1.m1.1.1" xref="S3.SS2.p6.1.m1.1.1.cmml">𝐊</mi><annotation-xml 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start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ∈ bold_K } ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S3.SS2.p6.5">and the distance of a point to a set by <math alttext="d({\mathbf{x}},\mathbf{S})=\inf_{{\mathbf{y}}\in\mathbf{S}}{\|{\mathbf{x}}-{% \mathbf{y}}\|}" class="ltx_Math" display="inline" id="S3.SS2.p6.2.m1.3"><semantics id="S3.SS2.p6.2.m1.3a"><mrow id="S3.SS2.p6.2.m1.3.3" xref="S3.SS2.p6.2.m1.3.3.cmml"><mrow id="S3.SS2.p6.2.m1.3.3.3" xref="S3.SS2.p6.2.m1.3.3.3.cmml"><mi id="S3.SS2.p6.2.m1.3.3.3.2" xref="S3.SS2.p6.2.m1.3.3.3.2.cmml">d</mi><mo id="S3.SS2.p6.2.m1.3.3.3.1" xref="S3.SS2.p6.2.m1.3.3.3.1.cmml">⁢</mo><mrow id="S3.SS2.p6.2.m1.3.3.3.3.2" xref="S3.SS2.p6.2.m1.3.3.3.3.1.cmml"><mo id="S3.SS2.p6.2.m1.3.3.3.3.2.1" stretchy="false" xref="S3.SS2.p6.2.m1.3.3.3.3.1.cmml">(</mo><mi id="S3.SS2.p6.2.m1.1.1" xref="S3.SS2.p6.2.m1.1.1.cmml">𝐱</mi><mo id="S3.SS2.p6.2.m1.3.3.3.3.2.2" 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xref="S3.SS2.p6.2.m1.3.3.1.1.1.2">norm</csymbol><apply id="S3.SS2.p6.2.m1.3.3.1.1.1.1.cmml" xref="S3.SS2.p6.2.m1.3.3.1.1.1.1"><minus id="S3.SS2.p6.2.m1.3.3.1.1.1.1.1.cmml" xref="S3.SS2.p6.2.m1.3.3.1.1.1.1.1"></minus><ci id="S3.SS2.p6.2.m1.3.3.1.1.1.1.2.cmml" xref="S3.SS2.p6.2.m1.3.3.1.1.1.1.2">𝐱</ci><ci id="S3.SS2.p6.2.m1.3.3.1.1.1.1.3.cmml" xref="S3.SS2.p6.2.m1.3.3.1.1.1.1.3">𝐲</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p6.2.m1.3c">d({\mathbf{x}},\mathbf{S})=\inf_{{\mathbf{y}}\in\mathbf{S}}{\|{\mathbf{x}}-{% \mathbf{y}}\|}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p6.2.m1.3d">italic_d ( bold_x , bold_S ) = roman_inf start_POSTSUBSCRIPT bold_y ∈ bold_S end_POSTSUBSCRIPT ∥ bold_x - bold_y ∥</annotation></semantics></math>, with <math alttext="{\mathbf{x}}\in{\mathbb{R}}^{d}" class="ltx_Math" display="inline" id="S3.SS2.p6.3.m2.1"><semantics id="S3.SS2.p6.3.m2.1a"><mrow id="S3.SS2.p6.3.m2.1.1" 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xref="S3.SS2.p6.3.m2.1.1.3.3">𝑑</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p6.3.m2.1c">{\mathbf{x}}\in{\mathbb{R}}^{d}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p6.3.m2.1d">bold_x ∈ blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT</annotation></semantics></math> and <math alttext="\mathbf{S}\subset{\mathbb{R}}^{d}" class="ltx_Math" display="inline" id="S3.SS2.p6.4.m3.1"><semantics id="S3.SS2.p6.4.m3.1a"><mrow id="S3.SS2.p6.4.m3.1.1" xref="S3.SS2.p6.4.m3.1.1.cmml"><mi id="S3.SS2.p6.4.m3.1.1.2" xref="S3.SS2.p6.4.m3.1.1.2.cmml">𝐒</mi><mo id="S3.SS2.p6.4.m3.1.1.1" xref="S3.SS2.p6.4.m3.1.1.1.cmml">⊂</mo><msup id="S3.SS2.p6.4.m3.1.1.3" xref="S3.SS2.p6.4.m3.1.1.3.cmml"><mi id="S3.SS2.p6.4.m3.1.1.3.2" xref="S3.SS2.p6.4.m3.1.1.3.2.cmml">ℝ</mi><mi id="S3.SS2.p6.4.m3.1.1.3.3" xref="S3.SS2.p6.4.m3.1.1.3.3.cmml">d</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.p6.4.m3.1b"><apply id="S3.SS2.p6.4.m3.1.1.cmml" xref="S3.SS2.p6.4.m3.1.1"><subset id="S3.SS2.p6.4.m3.1.1.1.cmml" xref="S3.SS2.p6.4.m3.1.1.1"></subset><ci id="S3.SS2.p6.4.m3.1.1.2.cmml" xref="S3.SS2.p6.4.m3.1.1.2">𝐒</ci><apply id="S3.SS2.p6.4.m3.1.1.3.cmml" xref="S3.SS2.p6.4.m3.1.1.3"><csymbol cd="ambiguous" id="S3.SS2.p6.4.m3.1.1.3.1.cmml" xref="S3.SS2.p6.4.m3.1.1.3">superscript</csymbol><ci id="S3.SS2.p6.4.m3.1.1.3.2.cmml" xref="S3.SS2.p6.4.m3.1.1.3.2">ℝ</ci><ci id="S3.SS2.p6.4.m3.1.1.3.3.cmml" xref="S3.SS2.p6.4.m3.1.1.3.3">𝑑</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p6.4.m3.1c">\mathbf{S}\subset{\mathbb{R}}^{d}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p6.4.m3.1d">bold_S ⊂ blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT</annotation></semantics></math>. The fact that we restrict to realizations in <math alttext="\Lambda_{\mathbf{K}}" class="ltx_Math" display="inline" id="S3.SS2.p6.5.m4.1"><semantics id="S3.SS2.p6.5.m4.1a"><msub id="S3.SS2.p6.5.m4.1.1" xref="S3.SS2.p6.5.m4.1.1.cmml"><mi id="S3.SS2.p6.5.m4.1.1.2" mathvariant="normal" xref="S3.SS2.p6.5.m4.1.1.2.cmml">Λ</mi><mi id="S3.SS2.p6.5.m4.1.1.3" xref="S3.SS2.p6.5.m4.1.1.3.cmml">𝐊</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS2.p6.5.m4.1b"><apply id="S3.SS2.p6.5.m4.1.1.cmml" xref="S3.SS2.p6.5.m4.1.1"><csymbol cd="ambiguous" id="S3.SS2.p6.5.m4.1.1.1.cmml" xref="S3.SS2.p6.5.m4.1.1">subscript</csymbol><ci id="S3.SS2.p6.5.m4.1.1.2.cmml" xref="S3.SS2.p6.5.m4.1.1.2">Λ</ci><ci id="S3.SS2.p6.5.m4.1.1.3.cmml" xref="S3.SS2.p6.5.m4.1.1.3">𝐊</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p6.5.m4.1c">\Lambda_{\mathbf{K}}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p6.5.m4.1d">roman_Λ start_POSTSUBSCRIPT bold_K end_POSTSUBSCRIPT</annotation></semantics></math> will be referred to as the ”boundedness assumption”.</p> </div> <div class="ltx_theorem ltx_theorem_proposition" id="S3.Thmproposition3"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S3.Thmproposition3.1.1.1">Proposition 3.3</span></span><span class="ltx_text ltx_font_bold" id="S3.Thmproposition3.2.2">.</span> </h6> <div class="ltx_para" id="S3.Thmproposition3.p1"> <p class="ltx_p" id="S3.Thmproposition3.p1.1"><span class="ltx_text ltx_font_italic" id="S3.Thmproposition3.p1.1.1">Under Assumptions <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#S3.Thmtheorem2" title="Assumption 3.2. ‣ 3.1 Asymptotics of critical points when 𝜎→0 ‣ 3 Convergence Analysis ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_tag">3.2</span></a>-<a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#S3.Thmtheorem4" title="Assumption 3.4. ‣ 3.2 Unbiased algorithm analysis ‣ 3 Convergence Analysis ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_tag">3.4</span></a>, almost surely on <math alttext="\Lambda_{\mathbf{K}}" class="ltx_Math" display="inline" id="S3.Thmproposition3.p1.1.1.m1.1"><semantics id="S3.Thmproposition3.p1.1.1.m1.1a"><msub id="S3.Thmproposition3.p1.1.1.m1.1.1" xref="S3.Thmproposition3.p1.1.1.m1.1.1.cmml"><mi id="S3.Thmproposition3.p1.1.1.m1.1.1.2" mathvariant="normal" xref="S3.Thmproposition3.p1.1.1.m1.1.1.2.cmml">Λ</mi><mi id="S3.Thmproposition3.p1.1.1.m1.1.1.3" xref="S3.Thmproposition3.p1.1.1.m1.1.1.3.cmml">𝐊</mi></msub><annotation-xml encoding="MathML-Content" id="S3.Thmproposition3.p1.1.1.m1.1b"><apply id="S3.Thmproposition3.p1.1.1.m1.1.1.cmml" xref="S3.Thmproposition3.p1.1.1.m1.1.1"><csymbol cd="ambiguous" id="S3.Thmproposition3.p1.1.1.m1.1.1.1.cmml" xref="S3.Thmproposition3.p1.1.1.m1.1.1">subscript</csymbol><ci id="S3.Thmproposition3.p1.1.1.m1.1.1.2.cmml" xref="S3.Thmproposition3.p1.1.1.m1.1.1.2">normal-Λ</ci><ci id="S3.Thmproposition3.p1.1.1.m1.1.1.3.cmml" xref="S3.Thmproposition3.p1.1.1.m1.1.1.3">𝐊</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmproposition3.p1.1.1.m1.1c">\Lambda_{\mathbf{K}}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmproposition3.p1.1.1.m1.1d">roman_Λ start_POSTSUBSCRIPT bold_K end_POSTSUBSCRIPT</annotation></semantics></math>, we have</span></p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A8.EGx11"> <tbody id="S3.Ex12"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\lim_{k\to+\infty}{d({\mathbf{x}}_{k},\mathbf{S_{\sigma,\mathbf{K% }}})}=0," class="ltx_Math" display="inline" id="S3.Ex12.m1.3"><semantics id="S3.Ex12.m1.3a"><mrow id="S3.Ex12.m1.3.3.1" xref="S3.Ex12.m1.3.3.1.1.cmml"><mrow id="S3.Ex12.m1.3.3.1.1" xref="S3.Ex12.m1.3.3.1.1.cmml"><mrow id="S3.Ex12.m1.3.3.1.1.2" xref="S3.Ex12.m1.3.3.1.1.2.cmml"><munder id="S3.Ex12.m1.3.3.1.1.2.3" xref="S3.Ex12.m1.3.3.1.1.2.3.cmml"><mo id="S3.Ex12.m1.3.3.1.1.2.3.2" movablelimits="false" xref="S3.Ex12.m1.3.3.1.1.2.3.2.cmml">lim</mo><mrow id="S3.Ex12.m1.3.3.1.1.2.3.3" xref="S3.Ex12.m1.3.3.1.1.2.3.3.cmml"><mi id="S3.Ex12.m1.3.3.1.1.2.3.3.2" xref="S3.Ex12.m1.3.3.1.1.2.3.3.2.cmml">k</mi><mo id="S3.Ex12.m1.3.3.1.1.2.3.3.1" stretchy="false" xref="S3.Ex12.m1.3.3.1.1.2.3.3.1.cmml">→</mo><mrow id="S3.Ex12.m1.3.3.1.1.2.3.3.3" xref="S3.Ex12.m1.3.3.1.1.2.3.3.3.cmml"><mo id="S3.Ex12.m1.3.3.1.1.2.3.3.3a" xref="S3.Ex12.m1.3.3.1.1.2.3.3.3.cmml">+</mo><mi id="S3.Ex12.m1.3.3.1.1.2.3.3.3.2" mathvariant="normal" xref="S3.Ex12.m1.3.3.1.1.2.3.3.3.2.cmml">∞</mi></mrow></mrow></munder><mrow id="S3.Ex12.m1.3.3.1.1.2.2" xref="S3.Ex12.m1.3.3.1.1.2.2.cmml"><mi id="S3.Ex12.m1.3.3.1.1.2.2.4" xref="S3.Ex12.m1.3.3.1.1.2.2.4.cmml">d</mi><mo id="S3.Ex12.m1.3.3.1.1.2.2.3" xref="S3.Ex12.m1.3.3.1.1.2.2.3.cmml">⁢</mo><mrow id="S3.Ex12.m1.3.3.1.1.2.2.2.2" xref="S3.Ex12.m1.3.3.1.1.2.2.2.3.cmml"><mo id="S3.Ex12.m1.3.3.1.1.2.2.2.2.3" stretchy="false" xref="S3.Ex12.m1.3.3.1.1.2.2.2.3.cmml">(</mo><msub id="S3.Ex12.m1.3.3.1.1.1.1.1.1.1" xref="S3.Ex12.m1.3.3.1.1.1.1.1.1.1.cmml"><mi id="S3.Ex12.m1.3.3.1.1.1.1.1.1.1.2" xref="S3.Ex12.m1.3.3.1.1.1.1.1.1.1.2.cmml">𝐱</mi><mi id="S3.Ex12.m1.3.3.1.1.1.1.1.1.1.3" xref="S3.Ex12.m1.3.3.1.1.1.1.1.1.1.3.cmml">k</mi></msub><mo id="S3.Ex12.m1.3.3.1.1.2.2.2.2.4" xref="S3.Ex12.m1.3.3.1.1.2.2.2.3.cmml">,</mo><msub id="S3.Ex12.m1.3.3.1.1.2.2.2.2.2" xref="S3.Ex12.m1.3.3.1.1.2.2.2.2.2.cmml"><mi id="S3.Ex12.m1.3.3.1.1.2.2.2.2.2.2" xref="S3.Ex12.m1.3.3.1.1.2.2.2.2.2.2.cmml">𝐒</mi><mrow id="S3.Ex12.m1.2.2.2.4" xref="S3.Ex12.m1.2.2.2.3.cmml"><mi id="S3.Ex12.m1.1.1.1.1" xref="S3.Ex12.m1.1.1.1.1.cmml">σ</mi><mo id="S3.Ex12.m1.2.2.2.4.1" xref="S3.Ex12.m1.2.2.2.3.cmml">,</mo><mi id="S3.Ex12.m1.2.2.2.2" xref="S3.Ex12.m1.2.2.2.2.cmml">𝐊</mi></mrow></msub><mo id="S3.Ex12.m1.3.3.1.1.2.2.2.2.5" stretchy="false" xref="S3.Ex12.m1.3.3.1.1.2.2.2.3.cmml">)</mo></mrow></mrow></mrow><mo id="S3.Ex12.m1.3.3.1.1.3" xref="S3.Ex12.m1.3.3.1.1.3.cmml">=</mo><mn id="S3.Ex12.m1.3.3.1.1.4" xref="S3.Ex12.m1.3.3.1.1.4.cmml">0</mn></mrow><mo id="S3.Ex12.m1.3.3.1.2" xref="S3.Ex12.m1.3.3.1.1.cmml">,</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.Ex12.m1.3b"><apply id="S3.Ex12.m1.3.3.1.1.cmml" xref="S3.Ex12.m1.3.3.1"><eq id="S3.Ex12.m1.3.3.1.1.3.cmml" xref="S3.Ex12.m1.3.3.1.1.3"></eq><apply id="S3.Ex12.m1.3.3.1.1.2.cmml" xref="S3.Ex12.m1.3.3.1.1.2"><apply id="S3.Ex12.m1.3.3.1.1.2.3.cmml" xref="S3.Ex12.m1.3.3.1.1.2.3"><csymbol cd="ambiguous" id="S3.Ex12.m1.3.3.1.1.2.3.1.cmml" xref="S3.Ex12.m1.3.3.1.1.2.3">subscript</csymbol><limit id="S3.Ex12.m1.3.3.1.1.2.3.2.cmml" xref="S3.Ex12.m1.3.3.1.1.2.3.2"></limit><apply id="S3.Ex12.m1.3.3.1.1.2.3.3.cmml" xref="S3.Ex12.m1.3.3.1.1.2.3.3"><ci id="S3.Ex12.m1.3.3.1.1.2.3.3.1.cmml" xref="S3.Ex12.m1.3.3.1.1.2.3.3.1">→</ci><ci id="S3.Ex12.m1.3.3.1.1.2.3.3.2.cmml" xref="S3.Ex12.m1.3.3.1.1.2.3.3.2">𝑘</ci><apply id="S3.Ex12.m1.3.3.1.1.2.3.3.3.cmml" xref="S3.Ex12.m1.3.3.1.1.2.3.3.3"><plus id="S3.Ex12.m1.3.3.1.1.2.3.3.3.1.cmml" xref="S3.Ex12.m1.3.3.1.1.2.3.3.3"></plus><infinity id="S3.Ex12.m1.3.3.1.1.2.3.3.3.2.cmml" xref="S3.Ex12.m1.3.3.1.1.2.3.3.3.2"></infinity></apply></apply></apply><apply id="S3.Ex12.m1.3.3.1.1.2.2.cmml" xref="S3.Ex12.m1.3.3.1.1.2.2"><times id="S3.Ex12.m1.3.3.1.1.2.2.3.cmml" xref="S3.Ex12.m1.3.3.1.1.2.2.3"></times><ci id="S3.Ex12.m1.3.3.1.1.2.2.4.cmml" xref="S3.Ex12.m1.3.3.1.1.2.2.4">𝑑</ci><interval closure="open" id="S3.Ex12.m1.3.3.1.1.2.2.2.3.cmml" xref="S3.Ex12.m1.3.3.1.1.2.2.2.2"><apply id="S3.Ex12.m1.3.3.1.1.1.1.1.1.1.cmml" xref="S3.Ex12.m1.3.3.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S3.Ex12.m1.3.3.1.1.1.1.1.1.1.1.cmml" xref="S3.Ex12.m1.3.3.1.1.1.1.1.1.1">subscript</csymbol><ci id="S3.Ex12.m1.3.3.1.1.1.1.1.1.1.2.cmml" xref="S3.Ex12.m1.3.3.1.1.1.1.1.1.1.2">𝐱</ci><ci id="S3.Ex12.m1.3.3.1.1.1.1.1.1.1.3.cmml" xref="S3.Ex12.m1.3.3.1.1.1.1.1.1.1.3">𝑘</ci></apply><apply id="S3.Ex12.m1.3.3.1.1.2.2.2.2.2.cmml" xref="S3.Ex12.m1.3.3.1.1.2.2.2.2.2"><csymbol cd="ambiguous" id="S3.Ex12.m1.3.3.1.1.2.2.2.2.2.1.cmml" xref="S3.Ex12.m1.3.3.1.1.2.2.2.2.2">subscript</csymbol><ci id="S3.Ex12.m1.3.3.1.1.2.2.2.2.2.2.cmml" xref="S3.Ex12.m1.3.3.1.1.2.2.2.2.2.2">𝐒</ci><list id="S3.Ex12.m1.2.2.2.3.cmml" xref="S3.Ex12.m1.2.2.2.4"><ci id="S3.Ex12.m1.1.1.1.1.cmml" xref="S3.Ex12.m1.1.1.1.1">𝜎</ci><ci id="S3.Ex12.m1.2.2.2.2.cmml" xref="S3.Ex12.m1.2.2.2.2">𝐊</ci></list></apply></interval></apply></apply><cn id="S3.Ex12.m1.3.3.1.1.4.cmml" type="integer" xref="S3.Ex12.m1.3.3.1.1.4">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Ex12.m1.3c">\displaystyle\lim_{k\to+\infty}{d({\mathbf{x}}_{k},\mathbf{S_{\sigma,\mathbf{K% }}})}=0,</annotation><annotation encoding="application/x-llamapun" id="S3.Ex12.m1.3d">roman_lim start_POSTSUBSCRIPT italic_k → + ∞ end_POSTSUBSCRIPT italic_d ( bold_x start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT , bold_S start_POSTSUBSCRIPT italic_σ , bold_K end_POSTSUBSCRIPT ) = 0 ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="S3.Ex13"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\lim_{k\to+\infty}{\|\nabla\mathcal{J}({\mathbf{x}}_{k})\|}=0," class="ltx_Math" display="inline" id="S3.Ex13.m1.1"><semantics id="S3.Ex13.m1.1a"><mrow id="S3.Ex13.m1.1.1.1" xref="S3.Ex13.m1.1.1.1.1.cmml"><mrow id="S3.Ex13.m1.1.1.1.1" xref="S3.Ex13.m1.1.1.1.1.cmml"><mrow id="S3.Ex13.m1.1.1.1.1.1" xref="S3.Ex13.m1.1.1.1.1.1.cmml"><munder id="S3.Ex13.m1.1.1.1.1.1.2" xref="S3.Ex13.m1.1.1.1.1.1.2.cmml"><mo id="S3.Ex13.m1.1.1.1.1.1.2.2" movablelimits="false" xref="S3.Ex13.m1.1.1.1.1.1.2.2.cmml">lim</mo><mrow id="S3.Ex13.m1.1.1.1.1.1.2.3" xref="S3.Ex13.m1.1.1.1.1.1.2.3.cmml"><mi id="S3.Ex13.m1.1.1.1.1.1.2.3.2" xref="S3.Ex13.m1.1.1.1.1.1.2.3.2.cmml">k</mi><mo id="S3.Ex13.m1.1.1.1.1.1.2.3.1" stretchy="false" xref="S3.Ex13.m1.1.1.1.1.1.2.3.1.cmml">→</mo><mrow id="S3.Ex13.m1.1.1.1.1.1.2.3.3" xref="S3.Ex13.m1.1.1.1.1.1.2.3.3.cmml"><mo id="S3.Ex13.m1.1.1.1.1.1.2.3.3a" xref="S3.Ex13.m1.1.1.1.1.1.2.3.3.cmml">+</mo><mi id="S3.Ex13.m1.1.1.1.1.1.2.3.3.2" mathvariant="normal" xref="S3.Ex13.m1.1.1.1.1.1.2.3.3.2.cmml">∞</mi></mrow></mrow></munder><mrow id="S3.Ex13.m1.1.1.1.1.1.1.1" xref="S3.Ex13.m1.1.1.1.1.1.1.2.cmml"><mo id="S3.Ex13.m1.1.1.1.1.1.1.1.2" lspace="0em" stretchy="false" xref="S3.Ex13.m1.1.1.1.1.1.1.2.1.cmml">‖</mo><mrow id="S3.Ex13.m1.1.1.1.1.1.1.1.1" xref="S3.Ex13.m1.1.1.1.1.1.1.1.1.cmml"><mrow id="S3.Ex13.m1.1.1.1.1.1.1.1.1.3" xref="S3.Ex13.m1.1.1.1.1.1.1.1.1.3.cmml"><mo id="S3.Ex13.m1.1.1.1.1.1.1.1.1.3.1" rspace="0.167em" xref="S3.Ex13.m1.1.1.1.1.1.1.1.1.3.1.cmml">∇</mo><mi class="ltx_font_mathcaligraphic" id="S3.Ex13.m1.1.1.1.1.1.1.1.1.3.2" xref="S3.Ex13.m1.1.1.1.1.1.1.1.1.3.2.cmml">𝒥</mi></mrow><mo id="S3.Ex13.m1.1.1.1.1.1.1.1.1.2" xref="S3.Ex13.m1.1.1.1.1.1.1.1.1.2.cmml">⁢</mo><mrow id="S3.Ex13.m1.1.1.1.1.1.1.1.1.1.1" xref="S3.Ex13.m1.1.1.1.1.1.1.1.1.1.1.1.cmml"><mo id="S3.Ex13.m1.1.1.1.1.1.1.1.1.1.1.2" stretchy="false" xref="S3.Ex13.m1.1.1.1.1.1.1.1.1.1.1.1.cmml">(</mo><msub id="S3.Ex13.m1.1.1.1.1.1.1.1.1.1.1.1" xref="S3.Ex13.m1.1.1.1.1.1.1.1.1.1.1.1.cmml"><mi id="S3.Ex13.m1.1.1.1.1.1.1.1.1.1.1.1.2" xref="S3.Ex13.m1.1.1.1.1.1.1.1.1.1.1.1.2.cmml">𝐱</mi><mi id="S3.Ex13.m1.1.1.1.1.1.1.1.1.1.1.1.3" xref="S3.Ex13.m1.1.1.1.1.1.1.1.1.1.1.1.3.cmml">k</mi></msub><mo id="S3.Ex13.m1.1.1.1.1.1.1.1.1.1.1.3" stretchy="false" xref="S3.Ex13.m1.1.1.1.1.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S3.Ex13.m1.1.1.1.1.1.1.1.3" stretchy="false" xref="S3.Ex13.m1.1.1.1.1.1.1.2.1.cmml">‖</mo></mrow></mrow><mo id="S3.Ex13.m1.1.1.1.1.2" xref="S3.Ex13.m1.1.1.1.1.2.cmml">=</mo><mn id="S3.Ex13.m1.1.1.1.1.3" xref="S3.Ex13.m1.1.1.1.1.3.cmml">0</mn></mrow><mo id="S3.Ex13.m1.1.1.1.2" xref="S3.Ex13.m1.1.1.1.1.cmml">,</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.Ex13.m1.1b"><apply id="S3.Ex13.m1.1.1.1.1.cmml" xref="S3.Ex13.m1.1.1.1"><eq id="S3.Ex13.m1.1.1.1.1.2.cmml" xref="S3.Ex13.m1.1.1.1.1.2"></eq><apply id="S3.Ex13.m1.1.1.1.1.1.cmml" xref="S3.Ex13.m1.1.1.1.1.1"><apply id="S3.Ex13.m1.1.1.1.1.1.2.cmml" xref="S3.Ex13.m1.1.1.1.1.1.2"><csymbol cd="ambiguous" id="S3.Ex13.m1.1.1.1.1.1.2.1.cmml" xref="S3.Ex13.m1.1.1.1.1.1.2">subscript</csymbol><limit id="S3.Ex13.m1.1.1.1.1.1.2.2.cmml" xref="S3.Ex13.m1.1.1.1.1.1.2.2"></limit><apply id="S3.Ex13.m1.1.1.1.1.1.2.3.cmml" xref="S3.Ex13.m1.1.1.1.1.1.2.3"><ci 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id="S3.Ex13.m1.1.1.1.1.1.1.1.1.3.2.cmml" xref="S3.Ex13.m1.1.1.1.1.1.1.1.1.3.2">𝒥</ci></apply><apply id="S3.Ex13.m1.1.1.1.1.1.1.1.1.1.1.1.cmml" xref="S3.Ex13.m1.1.1.1.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S3.Ex13.m1.1.1.1.1.1.1.1.1.1.1.1.1.cmml" xref="S3.Ex13.m1.1.1.1.1.1.1.1.1.1.1">subscript</csymbol><ci id="S3.Ex13.m1.1.1.1.1.1.1.1.1.1.1.1.2.cmml" xref="S3.Ex13.m1.1.1.1.1.1.1.1.1.1.1.1.2">𝐱</ci><ci id="S3.Ex13.m1.1.1.1.1.1.1.1.1.1.1.1.3.cmml" xref="S3.Ex13.m1.1.1.1.1.1.1.1.1.1.1.1.3">𝑘</ci></apply></apply></apply></apply><cn id="S3.Ex13.m1.1.1.1.1.3.cmml" type="integer" xref="S3.Ex13.m1.1.1.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Ex13.m1.1c">\displaystyle\lim_{k\to+\infty}{\|\nabla\mathcal{J}({\mathbf{x}}_{k})\|}=0,</annotation><annotation encoding="application/x-llamapun" id="S3.Ex13.m1.1d">roman_lim start_POSTSUBSCRIPT italic_k → + ∞ end_POSTSUBSCRIPT ∥ ∇ caligraphic_J ( bold_x start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) ∥ = 0 ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S3.Thmproposition3.p1.3"><span class="ltx_text ltx_font_italic" id="S3.Thmproposition3.p1.3.2">and <math alttext="(\mathcal{J}({\mathbf{x}}_{k}))_{k\in{\mathbb{N}}}" class="ltx_Math" display="inline" id="S3.Thmproposition3.p1.2.1.m1.1"><semantics id="S3.Thmproposition3.p1.2.1.m1.1a"><msub id="S3.Thmproposition3.p1.2.1.m1.1.1" xref="S3.Thmproposition3.p1.2.1.m1.1.1.cmml"><mrow id="S3.Thmproposition3.p1.2.1.m1.1.1.1.1" xref="S3.Thmproposition3.p1.2.1.m1.1.1.1.1.1.cmml"><mo id="S3.Thmproposition3.p1.2.1.m1.1.1.1.1.2" mathvariant="normal" stretchy="false" xref="S3.Thmproposition3.p1.2.1.m1.1.1.1.1.1.cmml">(</mo><mrow id="S3.Thmproposition3.p1.2.1.m1.1.1.1.1.1" xref="S3.Thmproposition3.p1.2.1.m1.1.1.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.Thmproposition3.p1.2.1.m1.1.1.1.1.1.3" xref="S3.Thmproposition3.p1.2.1.m1.1.1.1.1.1.3.cmml">𝒥</mi><mo id="S3.Thmproposition3.p1.2.1.m1.1.1.1.1.1.2" mathvariant="italic" xref="S3.Thmproposition3.p1.2.1.m1.1.1.1.1.1.2.cmml">⁢</mo><mrow id="S3.Thmproposition3.p1.2.1.m1.1.1.1.1.1.1.1" xref="S3.Thmproposition3.p1.2.1.m1.1.1.1.1.1.1.1.1.cmml"><mo id="S3.Thmproposition3.p1.2.1.m1.1.1.1.1.1.1.1.2" mathvariant="normal" stretchy="false" xref="S3.Thmproposition3.p1.2.1.m1.1.1.1.1.1.1.1.1.cmml">(</mo><msub id="S3.Thmproposition3.p1.2.1.m1.1.1.1.1.1.1.1.1" xref="S3.Thmproposition3.p1.2.1.m1.1.1.1.1.1.1.1.1.cmml"><mi id="S3.Thmproposition3.p1.2.1.m1.1.1.1.1.1.1.1.1.2" xref="S3.Thmproposition3.p1.2.1.m1.1.1.1.1.1.1.1.1.2.cmml">𝐱</mi><mi id="S3.Thmproposition3.p1.2.1.m1.1.1.1.1.1.1.1.1.3" xref="S3.Thmproposition3.p1.2.1.m1.1.1.1.1.1.1.1.1.3.cmml">k</mi></msub><mo id="S3.Thmproposition3.p1.2.1.m1.1.1.1.1.1.1.1.3" mathvariant="normal" stretchy="false" xref="S3.Thmproposition3.p1.2.1.m1.1.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S3.Thmproposition3.p1.2.1.m1.1.1.1.1.3" mathvariant="normal" stretchy="false" xref="S3.Thmproposition3.p1.2.1.m1.1.1.1.1.1.cmml">)</mo></mrow><mrow id="S3.Thmproposition3.p1.2.1.m1.1.1.3" xref="S3.Thmproposition3.p1.2.1.m1.1.1.3.cmml"><mi id="S3.Thmproposition3.p1.2.1.m1.1.1.3.2" xref="S3.Thmproposition3.p1.2.1.m1.1.1.3.2.cmml">k</mi><mo id="S3.Thmproposition3.p1.2.1.m1.1.1.3.1" mathvariant="normal" xref="S3.Thmproposition3.p1.2.1.m1.1.1.3.1.cmml">∈</mo><mi id="S3.Thmproposition3.p1.2.1.m1.1.1.3.3" xref="S3.Thmproposition3.p1.2.1.m1.1.1.3.3.cmml">ℕ</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="S3.Thmproposition3.p1.2.1.m1.1b"><apply id="S3.Thmproposition3.p1.2.1.m1.1.1.cmml" xref="S3.Thmproposition3.p1.2.1.m1.1.1"><csymbol cd="ambiguous" id="S3.Thmproposition3.p1.2.1.m1.1.1.2.cmml" xref="S3.Thmproposition3.p1.2.1.m1.1.1">subscript</csymbol><apply id="S3.Thmproposition3.p1.2.1.m1.1.1.1.1.1.cmml" xref="S3.Thmproposition3.p1.2.1.m1.1.1.1.1"><times id="S3.Thmproposition3.p1.2.1.m1.1.1.1.1.1.2.cmml" xref="S3.Thmproposition3.p1.2.1.m1.1.1.1.1.1.2"></times><ci id="S3.Thmproposition3.p1.2.1.m1.1.1.1.1.1.3.cmml" xref="S3.Thmproposition3.p1.2.1.m1.1.1.1.1.1.3">𝒥</ci><apply id="S3.Thmproposition3.p1.2.1.m1.1.1.1.1.1.1.1.1.cmml" xref="S3.Thmproposition3.p1.2.1.m1.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S3.Thmproposition3.p1.2.1.m1.1.1.1.1.1.1.1.1.1.cmml" xref="S3.Thmproposition3.p1.2.1.m1.1.1.1.1.1.1.1">subscript</csymbol><ci id="S3.Thmproposition3.p1.2.1.m1.1.1.1.1.1.1.1.1.2.cmml" xref="S3.Thmproposition3.p1.2.1.m1.1.1.1.1.1.1.1.1.2">𝐱</ci><ci id="S3.Thmproposition3.p1.2.1.m1.1.1.1.1.1.1.1.1.3.cmml" xref="S3.Thmproposition3.p1.2.1.m1.1.1.1.1.1.1.1.1.3">𝑘</ci></apply></apply><apply id="S3.Thmproposition3.p1.2.1.m1.1.1.3.cmml" xref="S3.Thmproposition3.p1.2.1.m1.1.1.3"><in id="S3.Thmproposition3.p1.2.1.m1.1.1.3.1.cmml" xref="S3.Thmproposition3.p1.2.1.m1.1.1.3.1"></in><ci id="S3.Thmproposition3.p1.2.1.m1.1.1.3.2.cmml" xref="S3.Thmproposition3.p1.2.1.m1.1.1.3.2">𝑘</ci><ci id="S3.Thmproposition3.p1.2.1.m1.1.1.3.3.cmml" xref="S3.Thmproposition3.p1.2.1.m1.1.1.3.3">ℕ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmproposition3.p1.2.1.m1.1c">(\mathcal{J}({\mathbf{x}}_{k}))_{k\in{\mathbb{N}}}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmproposition3.p1.2.1.m1.1d">( caligraphic_J ( bold_x start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) ) start_POSTSUBSCRIPT italic_k ∈ blackboard_N end_POSTSUBSCRIPT</annotation></semantics></math> converges to a value of <math alttext="\mathcal{J}(\mathbf{S_{\sigma,\mathbf{K}}})" class="ltx_Math" display="inline" id="S3.Thmproposition3.p1.3.2.m2.3"><semantics id="S3.Thmproposition3.p1.3.2.m2.3a"><mrow id="S3.Thmproposition3.p1.3.2.m2.3.3" xref="S3.Thmproposition3.p1.3.2.m2.3.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.Thmproposition3.p1.3.2.m2.3.3.3" xref="S3.Thmproposition3.p1.3.2.m2.3.3.3.cmml">𝒥</mi><mo id="S3.Thmproposition3.p1.3.2.m2.3.3.2" mathvariant="italic" xref="S3.Thmproposition3.p1.3.2.m2.3.3.2.cmml">⁢</mo><mrow id="S3.Thmproposition3.p1.3.2.m2.3.3.1.1" xref="S3.Thmproposition3.p1.3.2.m2.3.3.1.1.1.cmml"><mo id="S3.Thmproposition3.p1.3.2.m2.3.3.1.1.2" mathvariant="normal" stretchy="false" xref="S3.Thmproposition3.p1.3.2.m2.3.3.1.1.1.cmml">(</mo><msub id="S3.Thmproposition3.p1.3.2.m2.3.3.1.1.1" xref="S3.Thmproposition3.p1.3.2.m2.3.3.1.1.1.cmml"><mi id="S3.Thmproposition3.p1.3.2.m2.3.3.1.1.1.2" xref="S3.Thmproposition3.p1.3.2.m2.3.3.1.1.1.2.cmml">𝐒</mi><mrow id="S3.Thmproposition3.p1.3.2.m2.2.2.2.4" xref="S3.Thmproposition3.p1.3.2.m2.2.2.2.3.cmml"><mi id="S3.Thmproposition3.p1.3.2.m2.1.1.1.1" xref="S3.Thmproposition3.p1.3.2.m2.1.1.1.1.cmml">σ</mi><mo id="S3.Thmproposition3.p1.3.2.m2.2.2.2.4.1" mathvariant="normal" xref="S3.Thmproposition3.p1.3.2.m2.2.2.2.3.cmml">,</mo><mi id="S3.Thmproposition3.p1.3.2.m2.2.2.2.2" xref="S3.Thmproposition3.p1.3.2.m2.2.2.2.2.cmml">𝐊</mi></mrow></msub><mo id="S3.Thmproposition3.p1.3.2.m2.3.3.1.1.3" mathvariant="normal" stretchy="false" xref="S3.Thmproposition3.p1.3.2.m2.3.3.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmproposition3.p1.3.2.m2.3b"><apply id="S3.Thmproposition3.p1.3.2.m2.3.3.cmml" xref="S3.Thmproposition3.p1.3.2.m2.3.3"><times id="S3.Thmproposition3.p1.3.2.m2.3.3.2.cmml" xref="S3.Thmproposition3.p1.3.2.m2.3.3.2"></times><ci id="S3.Thmproposition3.p1.3.2.m2.3.3.3.cmml" xref="S3.Thmproposition3.p1.3.2.m2.3.3.3">𝒥</ci><apply id="S3.Thmproposition3.p1.3.2.m2.3.3.1.1.1.cmml" xref="S3.Thmproposition3.p1.3.2.m2.3.3.1.1"><csymbol cd="ambiguous" id="S3.Thmproposition3.p1.3.2.m2.3.3.1.1.1.1.cmml" xref="S3.Thmproposition3.p1.3.2.m2.3.3.1.1">subscript</csymbol><ci id="S3.Thmproposition3.p1.3.2.m2.3.3.1.1.1.2.cmml" xref="S3.Thmproposition3.p1.3.2.m2.3.3.1.1.1.2">𝐒</ci><list id="S3.Thmproposition3.p1.3.2.m2.2.2.2.3.cmml" xref="S3.Thmproposition3.p1.3.2.m2.2.2.2.4"><ci id="S3.Thmproposition3.p1.3.2.m2.1.1.1.1.cmml" xref="S3.Thmproposition3.p1.3.2.m2.1.1.1.1">𝜎</ci><ci id="S3.Thmproposition3.p1.3.2.m2.2.2.2.2.cmml" xref="S3.Thmproposition3.p1.3.2.m2.2.2.2.2">𝐊</ci></list></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmproposition3.p1.3.2.m2.3c">\mathcal{J}(\mathbf{S_{\sigma,\mathbf{K}}})</annotation><annotation encoding="application/x-llamapun" id="S3.Thmproposition3.p1.3.2.m2.3d">caligraphic_J ( bold_S start_POSTSUBSCRIPT italic_σ , bold_K end_POSTSUBSCRIPT )</annotation></semantics></math>.</span></p> </div> </div> <div class="ltx_para" id="S3.SS2.p7"> <p class="ltx_p" id="S3.SS2.p7.3">Proposition <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#S3.Thmproposition3" title="Proposition 3.3. ‣ 3.2 Unbiased algorithm analysis ‣ 3 Convergence Analysis ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_tag">3.3</span></a> proves that Algorithm <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#alg2" title="Algorithm 2 ‣ Optimization algorithms ‣ 2.2 SNORE regularization ‣ 2 Stochastic deNOising REgularization (SNORE) ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_tag">2</span></a> run with the exact MMSE denoiser (<a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#S2.E4" title="4 ‣ 2.1 Background ‣ 2 Stochastic deNOising REgularization (SNORE) ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_tag">4</span></a>) converges to the set of critical points of Problem (<a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#S2.E9" title="9 ‣ Optimization algorithms ‣ 2.2 SNORE regularization ‣ 2 Stochastic deNOising REgularization (SNORE) ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_tag">9</span></a>). This is a weak convergence in the sense that this does not give <em class="ltx_emph ltx_font_italic" id="S3.SS2.p7.3.1">a priori</em> that there is <math alttext="{\mathbf{x}}^{\star}\in\mathbf{S_{\sigma,\mathbf{K}}}" class="ltx_Math" display="inline" id="S3.SS2.p7.1.m1.2"><semantics id="S3.SS2.p7.1.m1.2a"><mrow id="S3.SS2.p7.1.m1.2.3" xref="S3.SS2.p7.1.m1.2.3.cmml"><msup id="S3.SS2.p7.1.m1.2.3.2" xref="S3.SS2.p7.1.m1.2.3.2.cmml"><mi id="S3.SS2.p7.1.m1.2.3.2.2" xref="S3.SS2.p7.1.m1.2.3.2.2.cmml">𝐱</mi><mo id="S3.SS2.p7.1.m1.2.3.2.3" xref="S3.SS2.p7.1.m1.2.3.2.3.cmml">⋆</mo></msup><mo id="S3.SS2.p7.1.m1.2.3.1" xref="S3.SS2.p7.1.m1.2.3.1.cmml">∈</mo><msub id="S3.SS2.p7.1.m1.2.3.3" xref="S3.SS2.p7.1.m1.2.3.3.cmml"><mi id="S3.SS2.p7.1.m1.2.3.3.2" xref="S3.SS2.p7.1.m1.2.3.3.2.cmml">𝐒</mi><mrow id="S3.SS2.p7.1.m1.2.2.2.4" xref="S3.SS2.p7.1.m1.2.2.2.3.cmml"><mi id="S3.SS2.p7.1.m1.1.1.1.1" xref="S3.SS2.p7.1.m1.1.1.1.1.cmml">σ</mi><mo id="S3.SS2.p7.1.m1.2.2.2.4.1" xref="S3.SS2.p7.1.m1.2.2.2.3.cmml">,</mo><mi id="S3.SS2.p7.1.m1.2.2.2.2" xref="S3.SS2.p7.1.m1.2.2.2.2.cmml">𝐊</mi></mrow></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.p7.1.m1.2b"><apply id="S3.SS2.p7.1.m1.2.3.cmml" xref="S3.SS2.p7.1.m1.2.3"><in id="S3.SS2.p7.1.m1.2.3.1.cmml" xref="S3.SS2.p7.1.m1.2.3.1"></in><apply id="S3.SS2.p7.1.m1.2.3.2.cmml" xref="S3.SS2.p7.1.m1.2.3.2"><csymbol cd="ambiguous" id="S3.SS2.p7.1.m1.2.3.2.1.cmml" xref="S3.SS2.p7.1.m1.2.3.2">superscript</csymbol><ci id="S3.SS2.p7.1.m1.2.3.2.2.cmml" xref="S3.SS2.p7.1.m1.2.3.2.2">𝐱</ci><ci id="S3.SS2.p7.1.m1.2.3.2.3.cmml" xref="S3.SS2.p7.1.m1.2.3.2.3">⋆</ci></apply><apply id="S3.SS2.p7.1.m1.2.3.3.cmml" xref="S3.SS2.p7.1.m1.2.3.3"><csymbol cd="ambiguous" id="S3.SS2.p7.1.m1.2.3.3.1.cmml" xref="S3.SS2.p7.1.m1.2.3.3">subscript</csymbol><ci id="S3.SS2.p7.1.m1.2.3.3.2.cmml" xref="S3.SS2.p7.1.m1.2.3.3.2">𝐒</ci><list id="S3.SS2.p7.1.m1.2.2.2.3.cmml" xref="S3.SS2.p7.1.m1.2.2.2.4"><ci id="S3.SS2.p7.1.m1.1.1.1.1.cmml" xref="S3.SS2.p7.1.m1.1.1.1.1">𝜎</ci><ci id="S3.SS2.p7.1.m1.2.2.2.2.cmml" xref="S3.SS2.p7.1.m1.2.2.2.2">𝐊</ci></list></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p7.1.m1.2c">{\mathbf{x}}^{\star}\in\mathbf{S_{\sigma,\mathbf{K}}}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p7.1.m1.2d">bold_x start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT ∈ bold_S start_POSTSUBSCRIPT italic_σ , bold_K end_POSTSUBSCRIPT</annotation></semantics></math> such that <math alttext="\lim_{k\to+\infty}{\|{\mathbf{x}}_{k}-{\mathbf{x}}^{\star}\|}=0" class="ltx_Math" display="inline" id="S3.SS2.p7.2.m2.1"><semantics id="S3.SS2.p7.2.m2.1a"><mrow id="S3.SS2.p7.2.m2.1.1" xref="S3.SS2.p7.2.m2.1.1.cmml"><mrow id="S3.SS2.p7.2.m2.1.1.1" xref="S3.SS2.p7.2.m2.1.1.1.cmml"><msub id="S3.SS2.p7.2.m2.1.1.1.2" xref="S3.SS2.p7.2.m2.1.1.1.2.cmml"><mo id="S3.SS2.p7.2.m2.1.1.1.2.2" xref="S3.SS2.p7.2.m2.1.1.1.2.2.cmml">lim</mo><mrow id="S3.SS2.p7.2.m2.1.1.1.2.3" xref="S3.SS2.p7.2.m2.1.1.1.2.3.cmml"><mi id="S3.SS2.p7.2.m2.1.1.1.2.3.2" xref="S3.SS2.p7.2.m2.1.1.1.2.3.2.cmml">k</mi><mo id="S3.SS2.p7.2.m2.1.1.1.2.3.1" stretchy="false" xref="S3.SS2.p7.2.m2.1.1.1.2.3.1.cmml">→</mo><mrow id="S3.SS2.p7.2.m2.1.1.1.2.3.3" xref="S3.SS2.p7.2.m2.1.1.1.2.3.3.cmml"><mo id="S3.SS2.p7.2.m2.1.1.1.2.3.3a" xref="S3.SS2.p7.2.m2.1.1.1.2.3.3.cmml">+</mo><mi id="S3.SS2.p7.2.m2.1.1.1.2.3.3.2" mathvariant="normal" xref="S3.SS2.p7.2.m2.1.1.1.2.3.3.2.cmml">∞</mi></mrow></mrow></msub><mrow id="S3.SS2.p7.2.m2.1.1.1.1.1" xref="S3.SS2.p7.2.m2.1.1.1.1.2.cmml"><mo id="S3.SS2.p7.2.m2.1.1.1.1.1.2" lspace="0em" stretchy="false" xref="S3.SS2.p7.2.m2.1.1.1.1.2.1.cmml">‖</mo><mrow id="S3.SS2.p7.2.m2.1.1.1.1.1.1" xref="S3.SS2.p7.2.m2.1.1.1.1.1.1.cmml"><msub id="S3.SS2.p7.2.m2.1.1.1.1.1.1.2" xref="S3.SS2.p7.2.m2.1.1.1.1.1.1.2.cmml"><mi id="S3.SS2.p7.2.m2.1.1.1.1.1.1.2.2" xref="S3.SS2.p7.2.m2.1.1.1.1.1.1.2.2.cmml">𝐱</mi><mi id="S3.SS2.p7.2.m2.1.1.1.1.1.1.2.3" xref="S3.SS2.p7.2.m2.1.1.1.1.1.1.2.3.cmml">k</mi></msub><mo id="S3.SS2.p7.2.m2.1.1.1.1.1.1.1" xref="S3.SS2.p7.2.m2.1.1.1.1.1.1.1.cmml">−</mo><msup id="S3.SS2.p7.2.m2.1.1.1.1.1.1.3" xref="S3.SS2.p7.2.m2.1.1.1.1.1.1.3.cmml"><mi id="S3.SS2.p7.2.m2.1.1.1.1.1.1.3.2" xref="S3.SS2.p7.2.m2.1.1.1.1.1.1.3.2.cmml">𝐱</mi><mo id="S3.SS2.p7.2.m2.1.1.1.1.1.1.3.3" xref="S3.SS2.p7.2.m2.1.1.1.1.1.1.3.3.cmml">⋆</mo></msup></mrow><mo id="S3.SS2.p7.2.m2.1.1.1.1.1.3" stretchy="false" xref="S3.SS2.p7.2.m2.1.1.1.1.2.1.cmml">‖</mo></mrow></mrow><mo id="S3.SS2.p7.2.m2.1.1.2" xref="S3.SS2.p7.2.m2.1.1.2.cmml">=</mo><mn id="S3.SS2.p7.2.m2.1.1.3" xref="S3.SS2.p7.2.m2.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.p7.2.m2.1b"><apply id="S3.SS2.p7.2.m2.1.1.cmml" xref="S3.SS2.p7.2.m2.1.1"><eq id="S3.SS2.p7.2.m2.1.1.2.cmml" xref="S3.SS2.p7.2.m2.1.1.2"></eq><apply id="S3.SS2.p7.2.m2.1.1.1.cmml" xref="S3.SS2.p7.2.m2.1.1.1"><apply id="S3.SS2.p7.2.m2.1.1.1.2.cmml" xref="S3.SS2.p7.2.m2.1.1.1.2"><csymbol cd="ambiguous" id="S3.SS2.p7.2.m2.1.1.1.2.1.cmml" xref="S3.SS2.p7.2.m2.1.1.1.2">subscript</csymbol><limit id="S3.SS2.p7.2.m2.1.1.1.2.2.cmml" xref="S3.SS2.p7.2.m2.1.1.1.2.2"></limit><apply id="S3.SS2.p7.2.m2.1.1.1.2.3.cmml" xref="S3.SS2.p7.2.m2.1.1.1.2.3"><ci id="S3.SS2.p7.2.m2.1.1.1.2.3.1.cmml" xref="S3.SS2.p7.2.m2.1.1.1.2.3.1">→</ci><ci id="S3.SS2.p7.2.m2.1.1.1.2.3.2.cmml" xref="S3.SS2.p7.2.m2.1.1.1.2.3.2">𝑘</ci><apply id="S3.SS2.p7.2.m2.1.1.1.2.3.3.cmml" xref="S3.SS2.p7.2.m2.1.1.1.2.3.3"><plus id="S3.SS2.p7.2.m2.1.1.1.2.3.3.1.cmml" xref="S3.SS2.p7.2.m2.1.1.1.2.3.3"></plus><infinity id="S3.SS2.p7.2.m2.1.1.1.2.3.3.2.cmml" xref="S3.SS2.p7.2.m2.1.1.1.2.3.3.2"></infinity></apply></apply></apply><apply id="S3.SS2.p7.2.m2.1.1.1.1.2.cmml" xref="S3.SS2.p7.2.m2.1.1.1.1.1"><csymbol cd="latexml" id="S3.SS2.p7.2.m2.1.1.1.1.2.1.cmml" xref="S3.SS2.p7.2.m2.1.1.1.1.1.2">norm</csymbol><apply id="S3.SS2.p7.2.m2.1.1.1.1.1.1.cmml" xref="S3.SS2.p7.2.m2.1.1.1.1.1.1"><minus id="S3.SS2.p7.2.m2.1.1.1.1.1.1.1.cmml" xref="S3.SS2.p7.2.m2.1.1.1.1.1.1.1"></minus><apply id="S3.SS2.p7.2.m2.1.1.1.1.1.1.2.cmml" xref="S3.SS2.p7.2.m2.1.1.1.1.1.1.2"><csymbol cd="ambiguous" id="S3.SS2.p7.2.m2.1.1.1.1.1.1.2.1.cmml" xref="S3.SS2.p7.2.m2.1.1.1.1.1.1.2">subscript</csymbol><ci id="S3.SS2.p7.2.m2.1.1.1.1.1.1.2.2.cmml" xref="S3.SS2.p7.2.m2.1.1.1.1.1.1.2.2">𝐱</ci><ci id="S3.SS2.p7.2.m2.1.1.1.1.1.1.2.3.cmml" xref="S3.SS2.p7.2.m2.1.1.1.1.1.1.2.3">𝑘</ci></apply><apply id="S3.SS2.p7.2.m2.1.1.1.1.1.1.3.cmml" xref="S3.SS2.p7.2.m2.1.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S3.SS2.p7.2.m2.1.1.1.1.1.1.3.1.cmml" xref="S3.SS2.p7.2.m2.1.1.1.1.1.1.3">superscript</csymbol><ci id="S3.SS2.p7.2.m2.1.1.1.1.1.1.3.2.cmml" xref="S3.SS2.p7.2.m2.1.1.1.1.1.1.3.2">𝐱</ci><ci id="S3.SS2.p7.2.m2.1.1.1.1.1.1.3.3.cmml" xref="S3.SS2.p7.2.m2.1.1.1.1.1.1.3.3">⋆</ci></apply></apply></apply></apply><cn id="S3.SS2.p7.2.m2.1.1.3.cmml" type="integer" xref="S3.SS2.p7.2.m2.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p7.2.m2.1c">\lim_{k\to+\infty}{\|{\mathbf{x}}_{k}-{\mathbf{x}}^{\star}\|}=0</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p7.2.m2.1d">roman_lim start_POSTSUBSCRIPT italic_k → + ∞ end_POSTSUBSCRIPT ∥ bold_x start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT - bold_x start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT ∥ = 0</annotation></semantics></math>. Assuming that the sequence is bounded in <math alttext="\Lambda_{\mathbf{K}}" class="ltx_Math" display="inline" id="S3.SS2.p7.3.m3.1"><semantics id="S3.SS2.p7.3.m3.1a"><msub id="S3.SS2.p7.3.m3.1.1" xref="S3.SS2.p7.3.m3.1.1.cmml"><mi id="S3.SS2.p7.3.m3.1.1.2" mathvariant="normal" xref="S3.SS2.p7.3.m3.1.1.2.cmml">Λ</mi><mi id="S3.SS2.p7.3.m3.1.1.3" xref="S3.SS2.p7.3.m3.1.1.3.cmml">𝐊</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS2.p7.3.m3.1b"><apply id="S3.SS2.p7.3.m3.1.1.cmml" xref="S3.SS2.p7.3.m3.1.1"><csymbol cd="ambiguous" id="S3.SS2.p7.3.m3.1.1.1.cmml" xref="S3.SS2.p7.3.m3.1.1">subscript</csymbol><ci id="S3.SS2.p7.3.m3.1.1.2.cmml" xref="S3.SS2.p7.3.m3.1.1.2">Λ</ci><ci id="S3.SS2.p7.3.m3.1.1.3.cmml" xref="S3.SS2.p7.3.m3.1.1.3">𝐊</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p7.3.m3.1c">\Lambda_{\mathbf{K}}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p7.3.m3.1d">roman_Λ start_POSTSUBSCRIPT bold_K end_POSTSUBSCRIPT</annotation></semantics></math> is standard in the stochastic gradient descent analysis <cite class="ltx_cite ltx_citemacro_citep">(Benaïm, <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#bib.bib3" title="">2006</a>; Castera et al., <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#bib.bib7" title="">2021</a>)</cite>. We discuss this assumption in Appendix <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#A8" title="Appendix H Discussion on the boundedness of (𝐱_𝑘)_{𝑘∈ℕ} ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_tag">H</span></a>.</p> </div> <div class="ltx_para" id="S3.SS2.p8"> <p class="ltx_p" id="S3.SS2.p8.1">In the previous result, we do not assume that the prior <math alttext="p" class="ltx_Math" display="inline" id="S3.SS2.p8.1.m1.1"><semantics id="S3.SS2.p8.1.m1.1a"><mi id="S3.SS2.p8.1.m1.1.1" xref="S3.SS2.p8.1.m1.1.1.cmml">p</mi><annotation-xml encoding="MathML-Content" id="S3.SS2.p8.1.m1.1b"><ci id="S3.SS2.p8.1.m1.1.1.cmml" xref="S3.SS2.p8.1.m1.1.1">𝑝</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p8.1.m1.1c">p</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p8.1.m1.1d">italic_p</annotation></semantics></math> is smooth but we make Assumption <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#S3.Thmtheorem2" title="Assumption 3.2. ‣ 3.1 Asymptotics of critical points when 𝜎→0 ‣ 3 Convergence Analysis ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_tag">3.2</span></a> of a subpolynomial noisy score. This assumption is difficult to verify for a general prior distribution but can be verified in the case of the so-called manifold hypothesis <cite class="ltx_cite ltx_citemacro_citep">(De Bortoli, <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#bib.bib17" title="">2023</a>)</cite>.</p> </div> <div class="ltx_theorem ltx_theorem_assumption" id="S3.Thmtheorem5"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S3.Thmtheorem5.1.1.1">Assumption 3.5</span></span><span class="ltx_text ltx_font_bold" id="S3.Thmtheorem5.2.2"> </span>(Manifold hypothesis)<span class="ltx_text ltx_font_bold" id="S3.Thmtheorem5.3.3">.</span> </h6> <div class="ltx_para" id="S3.Thmtheorem5.p1"> <p class="ltx_p" id="S3.Thmtheorem5.p1.2">The prior <math alttext="p" class="ltx_Math" display="inline" id="S3.Thmtheorem5.p1.1.m1.1"><semantics id="S3.Thmtheorem5.p1.1.m1.1a"><mi id="S3.Thmtheorem5.p1.1.m1.1.1" xref="S3.Thmtheorem5.p1.1.m1.1.1.cmml">p</mi><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem5.p1.1.m1.1b"><ci id="S3.Thmtheorem5.p1.1.m1.1.1.cmml" xref="S3.Thmtheorem5.p1.1.m1.1.1">𝑝</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem5.p1.1.m1.1c">p</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem5.p1.1.m1.1d">italic_p</annotation></semantics></math> is supported on a compact set <math alttext="\mathcal{M}" class="ltx_Math" display="inline" id="S3.Thmtheorem5.p1.2.m2.1"><semantics id="S3.Thmtheorem5.p1.2.m2.1a"><mi class="ltx_font_mathcaligraphic" id="S3.Thmtheorem5.p1.2.m2.1.1" xref="S3.Thmtheorem5.p1.2.m2.1.1.cmml">ℳ</mi><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem5.p1.2.m2.1b"><ci id="S3.Thmtheorem5.p1.2.m2.1.1.cmml" xref="S3.Thmtheorem5.p1.2.m2.1.1">ℳ</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem5.p1.2.m2.1c">\mathcal{M}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem5.p1.2.m2.1d">caligraphic_M</annotation></semantics></math>.</p> </div> </div> <div class="ltx_para" id="S3.SS2.p9"> <p class="ltx_p" id="S3.SS2.p9.1">Assumption <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#S3.Thmtheorem5" title="Assumption 3.5 (Manifold hypothesis). ‣ 3.2 Unbiased algorithm analysis ‣ 3 Convergence Analysis ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_tag">3.5</span></a> is typically true for an image distribution with bounded pixel values.</p> </div> <div class="ltx_theorem ltx_theorem_proposition" id="S3.Thmproposition4"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S3.Thmproposition4.1.1.1">Proposition 3.4</span></span><span class="ltx_text ltx_font_bold" id="S3.Thmproposition4.2.2">.</span> </h6> <div class="ltx_para" id="S3.Thmproposition4.p1"> <p class="ltx_p" id="S3.Thmproposition4.p1.1"><span class="ltx_text ltx_font_italic" id="S3.Thmproposition4.p1.1.1">Under Assumptions <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#S3.Thmtheorem3" title="Assumption 3.3. ‣ 3.2 Unbiased algorithm analysis ‣ 3 Convergence Analysis ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_tag">3.3</span></a>-<a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#S3.Thmtheorem5" title="Assumption 3.5 (Manifold hypothesis). ‣ 3.2 Unbiased algorithm analysis ‣ 3 Convergence Analysis ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_tag">3.5</span></a>, almost surely on <math alttext="\Lambda_{\mathbf{K}}" class="ltx_Math" display="inline" id="S3.Thmproposition4.p1.1.1.m1.1"><semantics id="S3.Thmproposition4.p1.1.1.m1.1a"><msub id="S3.Thmproposition4.p1.1.1.m1.1.1" xref="S3.Thmproposition4.p1.1.1.m1.1.1.cmml"><mi id="S3.Thmproposition4.p1.1.1.m1.1.1.2" mathvariant="normal" xref="S3.Thmproposition4.p1.1.1.m1.1.1.2.cmml">Λ</mi><mi id="S3.Thmproposition4.p1.1.1.m1.1.1.3" xref="S3.Thmproposition4.p1.1.1.m1.1.1.3.cmml">𝐊</mi></msub><annotation-xml encoding="MathML-Content" id="S3.Thmproposition4.p1.1.1.m1.1b"><apply id="S3.Thmproposition4.p1.1.1.m1.1.1.cmml" xref="S3.Thmproposition4.p1.1.1.m1.1.1"><csymbol cd="ambiguous" id="S3.Thmproposition4.p1.1.1.m1.1.1.1.cmml" xref="S3.Thmproposition4.p1.1.1.m1.1.1">subscript</csymbol><ci id="S3.Thmproposition4.p1.1.1.m1.1.1.2.cmml" xref="S3.Thmproposition4.p1.1.1.m1.1.1.2">normal-Λ</ci><ci id="S3.Thmproposition4.p1.1.1.m1.1.1.3.cmml" xref="S3.Thmproposition4.p1.1.1.m1.1.1.3">𝐊</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmproposition4.p1.1.1.m1.1c">\Lambda_{\mathbf{K}}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmproposition4.p1.1.1.m1.1d">roman_Λ start_POSTSUBSCRIPT bold_K end_POSTSUBSCRIPT</annotation></semantics></math>, we have</span></p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A8.EGx12"> <tbody id="S3.Ex14"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\lim_{k\to+\infty}{d({\mathbf{x}}_{k},\mathbf{S_{\sigma,\mathbf{K% }}})}=0," class="ltx_Math" display="inline" id="S3.Ex14.m1.3"><semantics id="S3.Ex14.m1.3a"><mrow id="S3.Ex14.m1.3.3.1" xref="S3.Ex14.m1.3.3.1.1.cmml"><mrow id="S3.Ex14.m1.3.3.1.1" xref="S3.Ex14.m1.3.3.1.1.cmml"><mrow id="S3.Ex14.m1.3.3.1.1.2" xref="S3.Ex14.m1.3.3.1.1.2.cmml"><munder id="S3.Ex14.m1.3.3.1.1.2.3" xref="S3.Ex14.m1.3.3.1.1.2.3.cmml"><mo id="S3.Ex14.m1.3.3.1.1.2.3.2" movablelimits="false" xref="S3.Ex14.m1.3.3.1.1.2.3.2.cmml">lim</mo><mrow id="S3.Ex14.m1.3.3.1.1.2.3.3" xref="S3.Ex14.m1.3.3.1.1.2.3.3.cmml"><mi id="S3.Ex14.m1.3.3.1.1.2.3.3.2" xref="S3.Ex14.m1.3.3.1.1.2.3.3.2.cmml">k</mi><mo id="S3.Ex14.m1.3.3.1.1.2.3.3.1" stretchy="false" xref="S3.Ex14.m1.3.3.1.1.2.3.3.1.cmml">→</mo><mrow id="S3.Ex14.m1.3.3.1.1.2.3.3.3" xref="S3.Ex14.m1.3.3.1.1.2.3.3.3.cmml"><mo id="S3.Ex14.m1.3.3.1.1.2.3.3.3a" xref="S3.Ex14.m1.3.3.1.1.2.3.3.3.cmml">+</mo><mi id="S3.Ex14.m1.3.3.1.1.2.3.3.3.2" mathvariant="normal" xref="S3.Ex14.m1.3.3.1.1.2.3.3.3.2.cmml">∞</mi></mrow></mrow></munder><mrow id="S3.Ex14.m1.3.3.1.1.2.2" xref="S3.Ex14.m1.3.3.1.1.2.2.cmml"><mi id="S3.Ex14.m1.3.3.1.1.2.2.4" xref="S3.Ex14.m1.3.3.1.1.2.2.4.cmml">d</mi><mo id="S3.Ex14.m1.3.3.1.1.2.2.3" xref="S3.Ex14.m1.3.3.1.1.2.2.3.cmml">⁢</mo><mrow id="S3.Ex14.m1.3.3.1.1.2.2.2.2" xref="S3.Ex14.m1.3.3.1.1.2.2.2.3.cmml"><mo id="S3.Ex14.m1.3.3.1.1.2.2.2.2.3" stretchy="false" xref="S3.Ex14.m1.3.3.1.1.2.2.2.3.cmml">(</mo><msub id="S3.Ex14.m1.3.3.1.1.1.1.1.1.1" xref="S3.Ex14.m1.3.3.1.1.1.1.1.1.1.cmml"><mi id="S3.Ex14.m1.3.3.1.1.1.1.1.1.1.2" xref="S3.Ex14.m1.3.3.1.1.1.1.1.1.1.2.cmml">𝐱</mi><mi id="S3.Ex14.m1.3.3.1.1.1.1.1.1.1.3" xref="S3.Ex14.m1.3.3.1.1.1.1.1.1.1.3.cmml">k</mi></msub><mo id="S3.Ex14.m1.3.3.1.1.2.2.2.2.4" xref="S3.Ex14.m1.3.3.1.1.2.2.2.3.cmml">,</mo><msub id="S3.Ex14.m1.3.3.1.1.2.2.2.2.2" xref="S3.Ex14.m1.3.3.1.1.2.2.2.2.2.cmml"><mi id="S3.Ex14.m1.3.3.1.1.2.2.2.2.2.2" xref="S3.Ex14.m1.3.3.1.1.2.2.2.2.2.2.cmml">𝐒</mi><mrow id="S3.Ex14.m1.2.2.2.4" xref="S3.Ex14.m1.2.2.2.3.cmml"><mi id="S3.Ex14.m1.1.1.1.1" xref="S3.Ex14.m1.1.1.1.1.cmml">σ</mi><mo id="S3.Ex14.m1.2.2.2.4.1" xref="S3.Ex14.m1.2.2.2.3.cmml">,</mo><mi id="S3.Ex14.m1.2.2.2.2" xref="S3.Ex14.m1.2.2.2.2.cmml">𝐊</mi></mrow></msub><mo id="S3.Ex14.m1.3.3.1.1.2.2.2.2.5" stretchy="false" xref="S3.Ex14.m1.3.3.1.1.2.2.2.3.cmml">)</mo></mrow></mrow></mrow><mo id="S3.Ex14.m1.3.3.1.1.3" xref="S3.Ex14.m1.3.3.1.1.3.cmml">=</mo><mn id="S3.Ex14.m1.3.3.1.1.4" xref="S3.Ex14.m1.3.3.1.1.4.cmml">0</mn></mrow><mo id="S3.Ex14.m1.3.3.1.2" xref="S3.Ex14.m1.3.3.1.1.cmml">,</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.Ex14.m1.3b"><apply id="S3.Ex14.m1.3.3.1.1.cmml" xref="S3.Ex14.m1.3.3.1"><eq id="S3.Ex14.m1.3.3.1.1.3.cmml" xref="S3.Ex14.m1.3.3.1.1.3"></eq><apply id="S3.Ex14.m1.3.3.1.1.2.cmml" xref="S3.Ex14.m1.3.3.1.1.2"><apply id="S3.Ex14.m1.3.3.1.1.2.3.cmml" xref="S3.Ex14.m1.3.3.1.1.2.3"><csymbol cd="ambiguous" id="S3.Ex14.m1.3.3.1.1.2.3.1.cmml" xref="S3.Ex14.m1.3.3.1.1.2.3">subscript</csymbol><limit id="S3.Ex14.m1.3.3.1.1.2.3.2.cmml" xref="S3.Ex14.m1.3.3.1.1.2.3.2"></limit><apply id="S3.Ex14.m1.3.3.1.1.2.3.3.cmml" xref="S3.Ex14.m1.3.3.1.1.2.3.3"><ci id="S3.Ex14.m1.3.3.1.1.2.3.3.1.cmml" xref="S3.Ex14.m1.3.3.1.1.2.3.3.1">→</ci><ci id="S3.Ex14.m1.3.3.1.1.2.3.3.2.cmml" xref="S3.Ex14.m1.3.3.1.1.2.3.3.2">𝑘</ci><apply id="S3.Ex14.m1.3.3.1.1.2.3.3.3.cmml" xref="S3.Ex14.m1.3.3.1.1.2.3.3.3"><plus id="S3.Ex14.m1.3.3.1.1.2.3.3.3.1.cmml" xref="S3.Ex14.m1.3.3.1.1.2.3.3.3"></plus><infinity id="S3.Ex14.m1.3.3.1.1.2.3.3.3.2.cmml" xref="S3.Ex14.m1.3.3.1.1.2.3.3.3.2"></infinity></apply></apply></apply><apply id="S3.Ex14.m1.3.3.1.1.2.2.cmml" xref="S3.Ex14.m1.3.3.1.1.2.2"><times id="S3.Ex14.m1.3.3.1.1.2.2.3.cmml" xref="S3.Ex14.m1.3.3.1.1.2.2.3"></times><ci id="S3.Ex14.m1.3.3.1.1.2.2.4.cmml" xref="S3.Ex14.m1.3.3.1.1.2.2.4">𝑑</ci><interval closure="open" id="S3.Ex14.m1.3.3.1.1.2.2.2.3.cmml" xref="S3.Ex14.m1.3.3.1.1.2.2.2.2"><apply id="S3.Ex14.m1.3.3.1.1.1.1.1.1.1.cmml" xref="S3.Ex14.m1.3.3.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S3.Ex14.m1.3.3.1.1.1.1.1.1.1.1.cmml" xref="S3.Ex14.m1.3.3.1.1.1.1.1.1.1">subscript</csymbol><ci id="S3.Ex14.m1.3.3.1.1.1.1.1.1.1.2.cmml" xref="S3.Ex14.m1.3.3.1.1.1.1.1.1.1.2">𝐱</ci><ci id="S3.Ex14.m1.3.3.1.1.1.1.1.1.1.3.cmml" xref="S3.Ex14.m1.3.3.1.1.1.1.1.1.1.3">𝑘</ci></apply><apply id="S3.Ex14.m1.3.3.1.1.2.2.2.2.2.cmml" xref="S3.Ex14.m1.3.3.1.1.2.2.2.2.2"><csymbol cd="ambiguous" id="S3.Ex14.m1.3.3.1.1.2.2.2.2.2.1.cmml" xref="S3.Ex14.m1.3.3.1.1.2.2.2.2.2">subscript</csymbol><ci id="S3.Ex14.m1.3.3.1.1.2.2.2.2.2.2.cmml" xref="S3.Ex14.m1.3.3.1.1.2.2.2.2.2.2">𝐒</ci><list id="S3.Ex14.m1.2.2.2.3.cmml" xref="S3.Ex14.m1.2.2.2.4"><ci id="S3.Ex14.m1.1.1.1.1.cmml" xref="S3.Ex14.m1.1.1.1.1">𝜎</ci><ci id="S3.Ex14.m1.2.2.2.2.cmml" xref="S3.Ex14.m1.2.2.2.2">𝐊</ci></list></apply></interval></apply></apply><cn id="S3.Ex14.m1.3.3.1.1.4.cmml" type="integer" xref="S3.Ex14.m1.3.3.1.1.4">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Ex14.m1.3c">\displaystyle\lim_{k\to+\infty}{d({\mathbf{x}}_{k},\mathbf{S_{\sigma,\mathbf{K% }}})}=0,</annotation><annotation encoding="application/x-llamapun" id="S3.Ex14.m1.3d">roman_lim start_POSTSUBSCRIPT italic_k → + ∞ end_POSTSUBSCRIPT italic_d ( bold_x start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT , bold_S start_POSTSUBSCRIPT italic_σ , bold_K end_POSTSUBSCRIPT ) = 0 ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="S3.Ex15"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\lim_{k\to+\infty}{\|\nabla\mathcal{J}({\mathbf{x}}_{k})\|}=0," class="ltx_Math" display="inline" id="S3.Ex15.m1.1"><semantics id="S3.Ex15.m1.1a"><mrow id="S3.Ex15.m1.1.1.1" xref="S3.Ex15.m1.1.1.1.1.cmml"><mrow id="S3.Ex15.m1.1.1.1.1" xref="S3.Ex15.m1.1.1.1.1.cmml"><mrow id="S3.Ex15.m1.1.1.1.1.1" xref="S3.Ex15.m1.1.1.1.1.1.cmml"><munder id="S3.Ex15.m1.1.1.1.1.1.2" xref="S3.Ex15.m1.1.1.1.1.1.2.cmml"><mo id="S3.Ex15.m1.1.1.1.1.1.2.2" movablelimits="false" xref="S3.Ex15.m1.1.1.1.1.1.2.2.cmml">lim</mo><mrow id="S3.Ex15.m1.1.1.1.1.1.2.3" xref="S3.Ex15.m1.1.1.1.1.1.2.3.cmml"><mi id="S3.Ex15.m1.1.1.1.1.1.2.3.2" xref="S3.Ex15.m1.1.1.1.1.1.2.3.2.cmml">k</mi><mo id="S3.Ex15.m1.1.1.1.1.1.2.3.1" stretchy="false" xref="S3.Ex15.m1.1.1.1.1.1.2.3.1.cmml">→</mo><mrow id="S3.Ex15.m1.1.1.1.1.1.2.3.3" xref="S3.Ex15.m1.1.1.1.1.1.2.3.3.cmml"><mo id="S3.Ex15.m1.1.1.1.1.1.2.3.3a" xref="S3.Ex15.m1.1.1.1.1.1.2.3.3.cmml">+</mo><mi id="S3.Ex15.m1.1.1.1.1.1.2.3.3.2" mathvariant="normal" xref="S3.Ex15.m1.1.1.1.1.1.2.3.3.2.cmml">∞</mi></mrow></mrow></munder><mrow id="S3.Ex15.m1.1.1.1.1.1.1.1" xref="S3.Ex15.m1.1.1.1.1.1.1.2.cmml"><mo id="S3.Ex15.m1.1.1.1.1.1.1.1.2" lspace="0em" stretchy="false" xref="S3.Ex15.m1.1.1.1.1.1.1.2.1.cmml">‖</mo><mrow id="S3.Ex15.m1.1.1.1.1.1.1.1.1" xref="S3.Ex15.m1.1.1.1.1.1.1.1.1.cmml"><mrow id="S3.Ex15.m1.1.1.1.1.1.1.1.1.3" xref="S3.Ex15.m1.1.1.1.1.1.1.1.1.3.cmml"><mo id="S3.Ex15.m1.1.1.1.1.1.1.1.1.3.1" rspace="0.167em" xref="S3.Ex15.m1.1.1.1.1.1.1.1.1.3.1.cmml">∇</mo><mi class="ltx_font_mathcaligraphic" id="S3.Ex15.m1.1.1.1.1.1.1.1.1.3.2" xref="S3.Ex15.m1.1.1.1.1.1.1.1.1.3.2.cmml">𝒥</mi></mrow><mo id="S3.Ex15.m1.1.1.1.1.1.1.1.1.2" xref="S3.Ex15.m1.1.1.1.1.1.1.1.1.2.cmml">⁢</mo><mrow 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xref="S3.Ex15.m1.1.1.1.1.1.1.1.1.1.1.1.3">𝑘</ci></apply></apply></apply></apply><cn id="S3.Ex15.m1.1.1.1.1.3.cmml" type="integer" xref="S3.Ex15.m1.1.1.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Ex15.m1.1c">\displaystyle\lim_{k\to+\infty}{\|\nabla\mathcal{J}({\mathbf{x}}_{k})\|}=0,</annotation><annotation encoding="application/x-llamapun" id="S3.Ex15.m1.1d">roman_lim start_POSTSUBSCRIPT italic_k → + ∞ end_POSTSUBSCRIPT ∥ ∇ caligraphic_J ( bold_x start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) ∥ = 0 ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S3.Thmproposition4.p1.3"><span class="ltx_text ltx_font_italic" id="S3.Thmproposition4.p1.3.2">and <math alttext="(\mathcal{J}({\mathbf{x}}_{k}))_{k\in{\mathbb{N}}}" class="ltx_Math" display="inline" id="S3.Thmproposition4.p1.2.1.m1.1"><semantics id="S3.Thmproposition4.p1.2.1.m1.1a"><msub id="S3.Thmproposition4.p1.2.1.m1.1.1" xref="S3.Thmproposition4.p1.2.1.m1.1.1.cmml"><mrow id="S3.Thmproposition4.p1.2.1.m1.1.1.1.1" xref="S3.Thmproposition4.p1.2.1.m1.1.1.1.1.1.cmml"><mo id="S3.Thmproposition4.p1.2.1.m1.1.1.1.1.2" mathvariant="normal" stretchy="false" xref="S3.Thmproposition4.p1.2.1.m1.1.1.1.1.1.cmml">(</mo><mrow id="S3.Thmproposition4.p1.2.1.m1.1.1.1.1.1" xref="S3.Thmproposition4.p1.2.1.m1.1.1.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.Thmproposition4.p1.2.1.m1.1.1.1.1.1.3" xref="S3.Thmproposition4.p1.2.1.m1.1.1.1.1.1.3.cmml">𝒥</mi><mo id="S3.Thmproposition4.p1.2.1.m1.1.1.1.1.1.2" mathvariant="italic" xref="S3.Thmproposition4.p1.2.1.m1.1.1.1.1.1.2.cmml">⁢</mo><mrow id="S3.Thmproposition4.p1.2.1.m1.1.1.1.1.1.1.1" xref="S3.Thmproposition4.p1.2.1.m1.1.1.1.1.1.1.1.1.cmml"><mo id="S3.Thmproposition4.p1.2.1.m1.1.1.1.1.1.1.1.2" mathvariant="normal" stretchy="false" xref="S3.Thmproposition4.p1.2.1.m1.1.1.1.1.1.1.1.1.cmml">(</mo><msub 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id="S3.Thmproposition4.p1.2.1.m1.1.1.3.3" xref="S3.Thmproposition4.p1.2.1.m1.1.1.3.3.cmml">ℕ</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="S3.Thmproposition4.p1.2.1.m1.1b"><apply id="S3.Thmproposition4.p1.2.1.m1.1.1.cmml" xref="S3.Thmproposition4.p1.2.1.m1.1.1"><csymbol cd="ambiguous" id="S3.Thmproposition4.p1.2.1.m1.1.1.2.cmml" xref="S3.Thmproposition4.p1.2.1.m1.1.1">subscript</csymbol><apply id="S3.Thmproposition4.p1.2.1.m1.1.1.1.1.1.cmml" xref="S3.Thmproposition4.p1.2.1.m1.1.1.1.1"><times id="S3.Thmproposition4.p1.2.1.m1.1.1.1.1.1.2.cmml" xref="S3.Thmproposition4.p1.2.1.m1.1.1.1.1.1.2"></times><ci id="S3.Thmproposition4.p1.2.1.m1.1.1.1.1.1.3.cmml" xref="S3.Thmproposition4.p1.2.1.m1.1.1.1.1.1.3">𝒥</ci><apply id="S3.Thmproposition4.p1.2.1.m1.1.1.1.1.1.1.1.1.cmml" xref="S3.Thmproposition4.p1.2.1.m1.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S3.Thmproposition4.p1.2.1.m1.1.1.1.1.1.1.1.1.1.cmml" xref="S3.Thmproposition4.p1.2.1.m1.1.1.1.1.1.1.1">subscript</csymbol><ci id="S3.Thmproposition4.p1.2.1.m1.1.1.1.1.1.1.1.1.2.cmml" xref="S3.Thmproposition4.p1.2.1.m1.1.1.1.1.1.1.1.1.2">𝐱</ci><ci id="S3.Thmproposition4.p1.2.1.m1.1.1.1.1.1.1.1.1.3.cmml" xref="S3.Thmproposition4.p1.2.1.m1.1.1.1.1.1.1.1.1.3">𝑘</ci></apply></apply><apply id="S3.Thmproposition4.p1.2.1.m1.1.1.3.cmml" xref="S3.Thmproposition4.p1.2.1.m1.1.1.3"><in id="S3.Thmproposition4.p1.2.1.m1.1.1.3.1.cmml" xref="S3.Thmproposition4.p1.2.1.m1.1.1.3.1"></in><ci id="S3.Thmproposition4.p1.2.1.m1.1.1.3.2.cmml" xref="S3.Thmproposition4.p1.2.1.m1.1.1.3.2">𝑘</ci><ci id="S3.Thmproposition4.p1.2.1.m1.1.1.3.3.cmml" xref="S3.Thmproposition4.p1.2.1.m1.1.1.3.3">ℕ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmproposition4.p1.2.1.m1.1c">(\mathcal{J}({\mathbf{x}}_{k}))_{k\in{\mathbb{N}}}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmproposition4.p1.2.1.m1.1d">( caligraphic_J ( bold_x start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) ) start_POSTSUBSCRIPT italic_k ∈ blackboard_N end_POSTSUBSCRIPT</annotation></semantics></math> converges to a value of <math alttext="\mathcal{J}(\mathbf{S_{\sigma,\mathbf{K}}})" class="ltx_Math" display="inline" id="S3.Thmproposition4.p1.3.2.m2.3"><semantics id="S3.Thmproposition4.p1.3.2.m2.3a"><mrow id="S3.Thmproposition4.p1.3.2.m2.3.3" xref="S3.Thmproposition4.p1.3.2.m2.3.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.Thmproposition4.p1.3.2.m2.3.3.3" xref="S3.Thmproposition4.p1.3.2.m2.3.3.3.cmml">𝒥</mi><mo id="S3.Thmproposition4.p1.3.2.m2.3.3.2" mathvariant="italic" xref="S3.Thmproposition4.p1.3.2.m2.3.3.2.cmml">⁢</mo><mrow id="S3.Thmproposition4.p1.3.2.m2.3.3.1.1" xref="S3.Thmproposition4.p1.3.2.m2.3.3.1.1.1.cmml"><mo id="S3.Thmproposition4.p1.3.2.m2.3.3.1.1.2" mathvariant="normal" stretchy="false" xref="S3.Thmproposition4.p1.3.2.m2.3.3.1.1.1.cmml">(</mo><msub id="S3.Thmproposition4.p1.3.2.m2.3.3.1.1.1" xref="S3.Thmproposition4.p1.3.2.m2.3.3.1.1.1.cmml"><mi id="S3.Thmproposition4.p1.3.2.m2.3.3.1.1.1.2" xref="S3.Thmproposition4.p1.3.2.m2.3.3.1.1.1.2.cmml">𝐒</mi><mrow id="S3.Thmproposition4.p1.3.2.m2.2.2.2.4" xref="S3.Thmproposition4.p1.3.2.m2.2.2.2.3.cmml"><mi id="S3.Thmproposition4.p1.3.2.m2.1.1.1.1" xref="S3.Thmproposition4.p1.3.2.m2.1.1.1.1.cmml">σ</mi><mo id="S3.Thmproposition4.p1.3.2.m2.2.2.2.4.1" mathvariant="normal" xref="S3.Thmproposition4.p1.3.2.m2.2.2.2.3.cmml">,</mo><mi id="S3.Thmproposition4.p1.3.2.m2.2.2.2.2" xref="S3.Thmproposition4.p1.3.2.m2.2.2.2.2.cmml">𝐊</mi></mrow></msub><mo id="S3.Thmproposition4.p1.3.2.m2.3.3.1.1.3" mathvariant="normal" stretchy="false" xref="S3.Thmproposition4.p1.3.2.m2.3.3.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmproposition4.p1.3.2.m2.3b"><apply id="S3.Thmproposition4.p1.3.2.m2.3.3.cmml" xref="S3.Thmproposition4.p1.3.2.m2.3.3"><times id="S3.Thmproposition4.p1.3.2.m2.3.3.2.cmml" xref="S3.Thmproposition4.p1.3.2.m2.3.3.2"></times><ci id="S3.Thmproposition4.p1.3.2.m2.3.3.3.cmml" xref="S3.Thmproposition4.p1.3.2.m2.3.3.3">𝒥</ci><apply id="S3.Thmproposition4.p1.3.2.m2.3.3.1.1.1.cmml" xref="S3.Thmproposition4.p1.3.2.m2.3.3.1.1"><csymbol cd="ambiguous" id="S3.Thmproposition4.p1.3.2.m2.3.3.1.1.1.1.cmml" xref="S3.Thmproposition4.p1.3.2.m2.3.3.1.1">subscript</csymbol><ci id="S3.Thmproposition4.p1.3.2.m2.3.3.1.1.1.2.cmml" xref="S3.Thmproposition4.p1.3.2.m2.3.3.1.1.1.2">𝐒</ci><list id="S3.Thmproposition4.p1.3.2.m2.2.2.2.3.cmml" xref="S3.Thmproposition4.p1.3.2.m2.2.2.2.4"><ci id="S3.Thmproposition4.p1.3.2.m2.1.1.1.1.cmml" xref="S3.Thmproposition4.p1.3.2.m2.1.1.1.1">𝜎</ci><ci id="S3.Thmproposition4.p1.3.2.m2.2.2.2.2.cmml" xref="S3.Thmproposition4.p1.3.2.m2.2.2.2.2">𝐊</ci></list></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmproposition4.p1.3.2.m2.3c">\mathcal{J}(\mathbf{S_{\sigma,\mathbf{K}}})</annotation><annotation encoding="application/x-llamapun" id="S3.Thmproposition4.p1.3.2.m2.3d">caligraphic_J ( bold_S start_POSTSUBSCRIPT italic_σ , bold_K end_POSTSUBSCRIPT )</annotation></semantics></math>.</span></p> </div> </div> <div class="ltx_para" id="S3.SS2.p10"> <p class="ltx_p" id="S3.SS2.p10.1">Proposition <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#S3.Thmproposition4" title="Proposition 3.4. ‣ 3.2 Unbiased algorithm analysis ‣ 3 Convergence Analysis ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_tag">3.4</span></a> establishes the convergence of the SNORE algorithm to the critical points of Problem (<a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#S2.E9" title="9 ‣ Optimization algorithms ‣ 2.2 SNORE regularization ‣ 2 Stochastic deNOising REgularization (SNORE) ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_tag">9</span></a>), under the three mild Assumptions <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#S3.Thmtheorem3" title="Assumption 3.3. ‣ 3.2 Unbiased algorithm analysis ‣ 3 Convergence Analysis ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_tag">3.3</span></a>-<a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#S3.Thmtheorem5" title="Assumption 3.5 (Manifold hypothesis). ‣ 3.2 Unbiased algorithm analysis ‣ 3 Convergence Analysis ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_tag">3.5</span></a>.</p> </div> </section> <section class="ltx_subsection" id="S3.SS3"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">3.3 </span>Biased algorithm analysis</h3> <div class="ltx_para" id="S3.SS3.p1"> <p class="ltx_p" id="S3.SS3.p1.2">We now quantify the error of the SNORE algorithm (Algorithm <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#alg2" title="Algorithm 2 ‣ Optimization algorithms ‣ 2.2 SNORE regularization ‣ 2 Stochastic deNOising REgularization (SNORE) ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_tag">2</span></a>) run with an inexact MMSE denoiser <math alttext="D_{\sigma}\approx D_{\sigma}^{\star}" class="ltx_Math" display="inline" id="S3.SS3.p1.1.m1.1"><semantics id="S3.SS3.p1.1.m1.1a"><mrow id="S3.SS3.p1.1.m1.1.1" xref="S3.SS3.p1.1.m1.1.1.cmml"><msub id="S3.SS3.p1.1.m1.1.1.2" xref="S3.SS3.p1.1.m1.1.1.2.cmml"><mi id="S3.SS3.p1.1.m1.1.1.2.2" xref="S3.SS3.p1.1.m1.1.1.2.2.cmml">D</mi><mi id="S3.SS3.p1.1.m1.1.1.2.3" xref="S3.SS3.p1.1.m1.1.1.2.3.cmml">σ</mi></msub><mo id="S3.SS3.p1.1.m1.1.1.1" xref="S3.SS3.p1.1.m1.1.1.1.cmml">≈</mo><msubsup id="S3.SS3.p1.1.m1.1.1.3" xref="S3.SS3.p1.1.m1.1.1.3.cmml"><mi id="S3.SS3.p1.1.m1.1.1.3.2.2" xref="S3.SS3.p1.1.m1.1.1.3.2.2.cmml">D</mi><mi id="S3.SS3.p1.1.m1.1.1.3.2.3" xref="S3.SS3.p1.1.m1.1.1.3.2.3.cmml">σ</mi><mo id="S3.SS3.p1.1.m1.1.1.3.3" xref="S3.SS3.p1.1.m1.1.1.3.3.cmml">⋆</mo></msubsup></mrow><annotation-xml encoding="MathML-Content" id="S3.SS3.p1.1.m1.1b"><apply id="S3.SS3.p1.1.m1.1.1.cmml" xref="S3.SS3.p1.1.m1.1.1"><approx id="S3.SS3.p1.1.m1.1.1.1.cmml" xref="S3.SS3.p1.1.m1.1.1.1"></approx><apply id="S3.SS3.p1.1.m1.1.1.2.cmml" xref="S3.SS3.p1.1.m1.1.1.2"><csymbol cd="ambiguous" id="S3.SS3.p1.1.m1.1.1.2.1.cmml" xref="S3.SS3.p1.1.m1.1.1.2">subscript</csymbol><ci id="S3.SS3.p1.1.m1.1.1.2.2.cmml" xref="S3.SS3.p1.1.m1.1.1.2.2">𝐷</ci><ci id="S3.SS3.p1.1.m1.1.1.2.3.cmml" xref="S3.SS3.p1.1.m1.1.1.2.3">𝜎</ci></apply><apply id="S3.SS3.p1.1.m1.1.1.3.cmml" xref="S3.SS3.p1.1.m1.1.1.3"><csymbol cd="ambiguous" id="S3.SS3.p1.1.m1.1.1.3.1.cmml" xref="S3.SS3.p1.1.m1.1.1.3">superscript</csymbol><apply id="S3.SS3.p1.1.m1.1.1.3.2.cmml" xref="S3.SS3.p1.1.m1.1.1.3"><csymbol cd="ambiguous" id="S3.SS3.p1.1.m1.1.1.3.2.1.cmml" xref="S3.SS3.p1.1.m1.1.1.3">subscript</csymbol><ci id="S3.SS3.p1.1.m1.1.1.3.2.2.cmml" xref="S3.SS3.p1.1.m1.1.1.3.2.2">𝐷</ci><ci id="S3.SS3.p1.1.m1.1.1.3.2.3.cmml" xref="S3.SS3.p1.1.m1.1.1.3.2.3">𝜎</ci></apply><ci id="S3.SS3.p1.1.m1.1.1.3.3.cmml" xref="S3.SS3.p1.1.m1.1.1.3.3">⋆</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p1.1.m1.1c">D_{\sigma}\approx D_{\sigma}^{\star}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p1.1.m1.1d">italic_D start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ≈ italic_D start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT</annotation></semantics></math>. Such a study is crucial as the algorithm is run in practice with a learned denoiser <math alttext="D_{\sigma}" class="ltx_Math" display="inline" id="S3.SS3.p1.2.m2.1"><semantics id="S3.SS3.p1.2.m2.1a"><msub id="S3.SS3.p1.2.m2.1.1" xref="S3.SS3.p1.2.m2.1.1.cmml"><mi id="S3.SS3.p1.2.m2.1.1.2" xref="S3.SS3.p1.2.m2.1.1.2.cmml">D</mi><mi id="S3.SS3.p1.2.m2.1.1.3" xref="S3.SS3.p1.2.m2.1.1.3.cmml">σ</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS3.p1.2.m2.1b"><apply id="S3.SS3.p1.2.m2.1.1.cmml" xref="S3.SS3.p1.2.m2.1.1"><csymbol cd="ambiguous" id="S3.SS3.p1.2.m2.1.1.1.cmml" xref="S3.SS3.p1.2.m2.1.1">subscript</csymbol><ci id="S3.SS3.p1.2.m2.1.1.2.cmml" xref="S3.SS3.p1.2.m2.1.1.2">𝐷</ci><ci id="S3.SS3.p1.2.m2.1.1.3.cmml" xref="S3.SS3.p1.2.m2.1.1.3">𝜎</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p1.2.m2.1c">D_{\sigma}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p1.2.m2.1d">italic_D start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT</annotation></semantics></math> which is not exact. With this denoiser, an iteration of Algorithm <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#alg2" title="Algorithm 2 ‣ Optimization algorithms ‣ 2.2 SNORE regularization ‣ 2 Stochastic deNOising REgularization (SNORE) ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_tag">2</span></a> is computed by</p> <table class="ltx_equation ltx_eqn_table" id="S3.E13"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="{\mathbf{x}}_{k+1}={\mathbf{x}}_{k}-\delta_{k}\nabla\mathcal{F}({\mathbf{x}}_{% k},{\mathbf{y}})-\frac{\alpha\delta_{k}}{\sigma^{2}}\left({\mathbf{x}}_{k}-D_{% \sigma}({\mathbf{x}}_{k}+\zeta_{k})\right)." class="ltx_Math" display="block" id="S3.E13.m1.2"><semantics id="S3.E13.m1.2a"><mrow id="S3.E13.m1.2.2.1" xref="S3.E13.m1.2.2.1.1.cmml"><mrow id="S3.E13.m1.2.2.1.1" 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start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ( bold_x start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT + italic_ζ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) ) .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(13)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S3.SS3.p1.5">It can be rewritten as</p> <table class="ltx_equation ltx_eqn_table" id="S3.E14"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="{\mathbf{x}}_{k+1}={\mathbf{x}}_{k}-\delta_{k}(\nabla\mathcal{J}(x_{k})+\xi_{k% })," class="ltx_Math" display="block" id="S3.E14.m1.1"><semantics id="S3.E14.m1.1a"><mrow id="S3.E14.m1.1.1.1" xref="S3.E14.m1.1.1.1.1.cmml"><mrow id="S3.E14.m1.1.1.1.1" 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xref="S3.E14.m1.1.1.1.1.1.1.1.1.1.1.2.cmml">⁢</mo><mrow id="S3.E14.m1.1.1.1.1.1.1.1.1.1.1.1.1" xref="S3.E14.m1.1.1.1.1.1.1.1.1.1.1.1.1.1.cmml"><mo id="S3.E14.m1.1.1.1.1.1.1.1.1.1.1.1.1.2" stretchy="false" xref="S3.E14.m1.1.1.1.1.1.1.1.1.1.1.1.1.1.cmml">(</mo><msub id="S3.E14.m1.1.1.1.1.1.1.1.1.1.1.1.1.1" xref="S3.E14.m1.1.1.1.1.1.1.1.1.1.1.1.1.1.cmml"><mi id="S3.E14.m1.1.1.1.1.1.1.1.1.1.1.1.1.1.2" xref="S3.E14.m1.1.1.1.1.1.1.1.1.1.1.1.1.1.2.cmml">x</mi><mi id="S3.E14.m1.1.1.1.1.1.1.1.1.1.1.1.1.1.3" xref="S3.E14.m1.1.1.1.1.1.1.1.1.1.1.1.1.1.3.cmml">k</mi></msub><mo id="S3.E14.m1.1.1.1.1.1.1.1.1.1.1.1.1.3" stretchy="false" xref="S3.E14.m1.1.1.1.1.1.1.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S3.E14.m1.1.1.1.1.1.1.1.1.1.2" xref="S3.E14.m1.1.1.1.1.1.1.1.1.1.2.cmml">+</mo><msub id="S3.E14.m1.1.1.1.1.1.1.1.1.1.3" xref="S3.E14.m1.1.1.1.1.1.1.1.1.1.3.cmml"><mi id="S3.E14.m1.1.1.1.1.1.1.1.1.1.3.2" xref="S3.E14.m1.1.1.1.1.1.1.1.1.1.3.2.cmml">ξ</mi><mi id="S3.E14.m1.1.1.1.1.1.1.1.1.1.3.3" xref="S3.E14.m1.1.1.1.1.1.1.1.1.1.3.3.cmml">k</mi></msub></mrow><mo id="S3.E14.m1.1.1.1.1.1.1.1.1.3" stretchy="false" xref="S3.E14.m1.1.1.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow></mrow><mo id="S3.E14.m1.1.1.1.2" xref="S3.E14.m1.1.1.1.1.cmml">,</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.E14.m1.1b"><apply id="S3.E14.m1.1.1.1.1.cmml" xref="S3.E14.m1.1.1.1"><eq id="S3.E14.m1.1.1.1.1.2.cmml" xref="S3.E14.m1.1.1.1.1.2"></eq><apply id="S3.E14.m1.1.1.1.1.3.cmml" xref="S3.E14.m1.1.1.1.1.3"><csymbol cd="ambiguous" id="S3.E14.m1.1.1.1.1.3.1.cmml" xref="S3.E14.m1.1.1.1.1.3">subscript</csymbol><ci id="S3.E14.m1.1.1.1.1.3.2.cmml" xref="S3.E14.m1.1.1.1.1.3.2">𝐱</ci><apply id="S3.E14.m1.1.1.1.1.3.3.cmml" xref="S3.E14.m1.1.1.1.1.3.3"><plus id="S3.E14.m1.1.1.1.1.3.3.1.cmml" xref="S3.E14.m1.1.1.1.1.3.3.1"></plus><ci id="S3.E14.m1.1.1.1.1.3.3.2.cmml" xref="S3.E14.m1.1.1.1.1.3.3.2">𝑘</ci><cn id="S3.E14.m1.1.1.1.1.3.3.3.cmml" type="integer" 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end_POSTSUBSCRIPT ( ∇ caligraphic_J ( italic_x start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) + italic_ξ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(14)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S3.SS3.p1.6">where, by using Equation (<a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#S2.E6" title="6 ‣ 2.2 SNORE regularization ‣ 2 Stochastic deNOising REgularization (SNORE) ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_tag">6</span></a>), the gradient perturbation writes as</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A8.EGx13"> <tbody id="S3.Ex16"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\xi_{k}" class="ltx_Math" display="inline" id="S3.Ex16.m1.1"><semantics id="S3.Ex16.m1.1a"><msub id="S3.Ex16.m1.1.1" xref="S3.Ex16.m1.1.1.cmml"><mi id="S3.Ex16.m1.1.1.2" xref="S3.Ex16.m1.1.1.2.cmml">ξ</mi><mi id="S3.Ex16.m1.1.1.3" xref="S3.Ex16.m1.1.1.3.cmml">k</mi></msub><annotation-xml encoding="MathML-Content" id="S3.Ex16.m1.1b"><apply id="S3.Ex16.m1.1.1.cmml" xref="S3.Ex16.m1.1.1"><csymbol cd="ambiguous" id="S3.Ex16.m1.1.1.1.cmml" xref="S3.Ex16.m1.1.1">subscript</csymbol><ci id="S3.Ex16.m1.1.1.2.cmml" xref="S3.Ex16.m1.1.1.2">𝜉</ci><ci id="S3.Ex16.m1.1.1.3.cmml" xref="S3.Ex16.m1.1.1.3">𝑘</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Ex16.m1.1c">\displaystyle\xi_{k}</annotation><annotation encoding="application/x-llamapun" id="S3.Ex16.m1.1d">italic_ξ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT</annotation></semantics></math></td> <td 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id="S3.Ex16.m2.4.4.3.3.1.1.1.3.cmml" xref="S3.Ex16.m2.4.4.3.3.1.1.1.3">𝑘</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Ex16.m2.4c">\displaystyle=\nabla\mathcal{F}({\mathbf{x}}_{k},{\mathbf{y}})+\frac{\alpha}{% \sigma^{2}}\left({\mathbf{x}}_{k}-D_{\sigma}({\mathbf{x}}_{k}+\zeta_{k})\right% )-\nabla\mathcal{J}(x_{k})</annotation><annotation encoding="application/x-llamapun" id="S3.Ex16.m2.4d">= ∇ caligraphic_F ( bold_x start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT , bold_y ) + divide start_ARG italic_α end_ARG start_ARG italic_σ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG ( bold_x start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT - italic_D start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ( bold_x start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT + italic_ζ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) ) - ∇ caligraphic_J ( italic_x start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT )</annotation></semantics></math></td> <td 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xref="S3.Ex17.m1.1.1.1.1.1.1.1.1.2.1.1.1.2">𝑥</ci><ci id="S3.Ex17.m1.1.1.1.1.1.1.1.1.2.1.1.1.3.cmml" xref="S3.Ex17.m1.1.1.1.1.1.1.1.1.2.1.1.1.3">𝑘</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Ex17.m1.1c">\displaystyle=\frac{\alpha}{\sigma^{2}}\left(D_{\sigma}({\mathbf{x}}_{k}+\zeta% _{k})-D_{\sigma}^{*}(x_{k})\right).</annotation><annotation encoding="application/x-llamapun" id="S3.Ex17.m1.1d">= divide start_ARG italic_α end_ARG start_ARG italic_σ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG ( italic_D start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ( bold_x start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT + italic_ζ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) - italic_D start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT ( italic_x start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) ) .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S3.SS3.p1.4">This stochastic shift <math alttext="\xi_{k}" class="ltx_Math" display="inline" id="S3.SS3.p1.3.m1.1"><semantics id="S3.SS3.p1.3.m1.1a"><msub id="S3.SS3.p1.3.m1.1.1" xref="S3.SS3.p1.3.m1.1.1.cmml"><mi id="S3.SS3.p1.3.m1.1.1.2" xref="S3.SS3.p1.3.m1.1.1.2.cmml">ξ</mi><mi id="S3.SS3.p1.3.m1.1.1.3" xref="S3.SS3.p1.3.m1.1.1.3.cmml">k</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS3.p1.3.m1.1b"><apply id="S3.SS3.p1.3.m1.1.1.cmml" xref="S3.SS3.p1.3.m1.1.1"><csymbol cd="ambiguous" id="S3.SS3.p1.3.m1.1.1.1.cmml" xref="S3.SS3.p1.3.m1.1.1">subscript</csymbol><ci id="S3.SS3.p1.3.m1.1.1.2.cmml" xref="S3.SS3.p1.3.m1.1.1.2">𝜉</ci><ci id="S3.SS3.p1.3.m1.1.1.3.cmml" xref="S3.SS3.p1.3.m1.1.1.3">𝑘</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p1.3.m1.1c">\xi_{k}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p1.3.m1.1d">italic_ξ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT</annotation></semantics></math> is in general biased, i.e. <math alttext="{\mathbb{E}(\xi_{k})\neq 0}" class="ltx_Math" display="inline" id="S3.SS3.p1.4.m2.1"><semantics id="S3.SS3.p1.4.m2.1a"><mrow id="S3.SS3.p1.4.m2.1.1" xref="S3.SS3.p1.4.m2.1.1.cmml"><mrow id="S3.SS3.p1.4.m2.1.1.1" xref="S3.SS3.p1.4.m2.1.1.1.cmml"><mi id="S3.SS3.p1.4.m2.1.1.1.3" xref="S3.SS3.p1.4.m2.1.1.1.3.cmml">𝔼</mi><mo id="S3.SS3.p1.4.m2.1.1.1.2" xref="S3.SS3.p1.4.m2.1.1.1.2.cmml">⁢</mo><mrow id="S3.SS3.p1.4.m2.1.1.1.1.1" xref="S3.SS3.p1.4.m2.1.1.1.1.1.1.cmml"><mo id="S3.SS3.p1.4.m2.1.1.1.1.1.2" stretchy="false" xref="S3.SS3.p1.4.m2.1.1.1.1.1.1.cmml">(</mo><msub id="S3.SS3.p1.4.m2.1.1.1.1.1.1" xref="S3.SS3.p1.4.m2.1.1.1.1.1.1.cmml"><mi id="S3.SS3.p1.4.m2.1.1.1.1.1.1.2" xref="S3.SS3.p1.4.m2.1.1.1.1.1.1.2.cmml">ξ</mi><mi id="S3.SS3.p1.4.m2.1.1.1.1.1.1.3" xref="S3.SS3.p1.4.m2.1.1.1.1.1.1.3.cmml">k</mi></msub><mo id="S3.SS3.p1.4.m2.1.1.1.1.1.3" stretchy="false" xref="S3.SS3.p1.4.m2.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S3.SS3.p1.4.m2.1.1.2" xref="S3.SS3.p1.4.m2.1.1.2.cmml">≠</mo><mn id="S3.SS3.p1.4.m2.1.1.3" xref="S3.SS3.p1.4.m2.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.SS3.p1.4.m2.1b"><apply id="S3.SS3.p1.4.m2.1.1.cmml" xref="S3.SS3.p1.4.m2.1.1"><neq id="S3.SS3.p1.4.m2.1.1.2.cmml" xref="S3.SS3.p1.4.m2.1.1.2"></neq><apply id="S3.SS3.p1.4.m2.1.1.1.cmml" xref="S3.SS3.p1.4.m2.1.1.1"><times id="S3.SS3.p1.4.m2.1.1.1.2.cmml" xref="S3.SS3.p1.4.m2.1.1.1.2"></times><ci id="S3.SS3.p1.4.m2.1.1.1.3.cmml" xref="S3.SS3.p1.4.m2.1.1.1.3">𝔼</ci><apply id="S3.SS3.p1.4.m2.1.1.1.1.1.1.cmml" xref="S3.SS3.p1.4.m2.1.1.1.1.1"><csymbol cd="ambiguous" id="S3.SS3.p1.4.m2.1.1.1.1.1.1.1.cmml" xref="S3.SS3.p1.4.m2.1.1.1.1.1">subscript</csymbol><ci id="S3.SS3.p1.4.m2.1.1.1.1.1.1.2.cmml" xref="S3.SS3.p1.4.m2.1.1.1.1.1.1.2">𝜉</ci><ci id="S3.SS3.p1.4.m2.1.1.1.1.1.1.3.cmml" xref="S3.SS3.p1.4.m2.1.1.1.1.1.1.3">𝑘</ci></apply></apply><cn id="S3.SS3.p1.4.m2.1.1.3.cmml" type="integer" xref="S3.SS3.p1.4.m2.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p1.4.m2.1c">{\mathbb{E}(\xi_{k})\neq 0}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p1.4.m2.1d">blackboard_E ( italic_ξ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) ≠ 0</annotation></semantics></math>. </p> </div> <div class="ltx_theorem ltx_theorem_assumption" id="S3.Thmtheorem6"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S3.Thmtheorem6.1.1.1">Assumption 3.6</span></span><span class="ltx_text ltx_font_bold" id="S3.Thmtheorem6.2.2">.</span> </h6> <div class="ltx_para" id="S3.Thmtheorem6.p1"> <p class="ltx_p" id="S3.Thmtheorem6.p1.7">The learned denoiser <math alttext="D_{\sigma}" class="ltx_Math" display="inline" id="S3.Thmtheorem6.p1.1.m1.1"><semantics id="S3.Thmtheorem6.p1.1.m1.1a"><msub id="S3.Thmtheorem6.p1.1.m1.1.1" xref="S3.Thmtheorem6.p1.1.m1.1.1.cmml"><mi id="S3.Thmtheorem6.p1.1.m1.1.1.2" xref="S3.Thmtheorem6.p1.1.m1.1.1.2.cmml">D</mi><mi id="S3.Thmtheorem6.p1.1.m1.1.1.3" xref="S3.Thmtheorem6.p1.1.m1.1.1.3.cmml">σ</mi></msub><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem6.p1.1.m1.1b"><apply id="S3.Thmtheorem6.p1.1.m1.1.1.cmml" xref="S3.Thmtheorem6.p1.1.m1.1.1"><csymbol cd="ambiguous" id="S3.Thmtheorem6.p1.1.m1.1.1.1.cmml" xref="S3.Thmtheorem6.p1.1.m1.1.1">subscript</csymbol><ci id="S3.Thmtheorem6.p1.1.m1.1.1.2.cmml" xref="S3.Thmtheorem6.p1.1.m1.1.1.2">𝐷</ci><ci id="S3.Thmtheorem6.p1.1.m1.1.1.3.cmml" xref="S3.Thmtheorem6.p1.1.m1.1.1.3">𝜎</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem6.p1.1.m1.1c">D_{\sigma}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem6.p1.1.m1.1d">italic_D start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT</annotation></semantics></math> is <math alttext="\mathcal{C}^{\infty}" class="ltx_Math" display="inline" id="S3.Thmtheorem6.p1.2.m2.1"><semantics id="S3.Thmtheorem6.p1.2.m2.1a"><msup id="S3.Thmtheorem6.p1.2.m2.1.1" xref="S3.Thmtheorem6.p1.2.m2.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.Thmtheorem6.p1.2.m2.1.1.2" xref="S3.Thmtheorem6.p1.2.m2.1.1.2.cmml">𝒞</mi><mi id="S3.Thmtheorem6.p1.2.m2.1.1.3" mathvariant="normal" xref="S3.Thmtheorem6.p1.2.m2.1.1.3.cmml">∞</mi></msup><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem6.p1.2.m2.1b"><apply id="S3.Thmtheorem6.p1.2.m2.1.1.cmml" xref="S3.Thmtheorem6.p1.2.m2.1.1"><csymbol cd="ambiguous" id="S3.Thmtheorem6.p1.2.m2.1.1.1.cmml" xref="S3.Thmtheorem6.p1.2.m2.1.1">superscript</csymbol><ci id="S3.Thmtheorem6.p1.2.m2.1.1.2.cmml" xref="S3.Thmtheorem6.p1.2.m2.1.1.2">𝒞</ci><infinity id="S3.Thmtheorem6.p1.2.m2.1.1.3.cmml" xref="S3.Thmtheorem6.p1.2.m2.1.1.3"></infinity></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem6.p1.2.m2.1c">\mathcal{C}^{\infty}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem6.p1.2.m2.1d">caligraphic_C start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT</annotation></semantics></math> and is a bounded approximation of the exact MMSE denoiser <math alttext="D_{\sigma}^{\star}" class="ltx_Math" display="inline" id="S3.Thmtheorem6.p1.3.m3.1"><semantics id="S3.Thmtheorem6.p1.3.m3.1a"><msubsup id="S3.Thmtheorem6.p1.3.m3.1.1" xref="S3.Thmtheorem6.p1.3.m3.1.1.cmml"><mi id="S3.Thmtheorem6.p1.3.m3.1.1.2.2" xref="S3.Thmtheorem6.p1.3.m3.1.1.2.2.cmml">D</mi><mi id="S3.Thmtheorem6.p1.3.m3.1.1.2.3" xref="S3.Thmtheorem6.p1.3.m3.1.1.2.3.cmml">σ</mi><mo id="S3.Thmtheorem6.p1.3.m3.1.1.3" xref="S3.Thmtheorem6.p1.3.m3.1.1.3.cmml">⋆</mo></msubsup><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem6.p1.3.m3.1b"><apply id="S3.Thmtheorem6.p1.3.m3.1.1.cmml" xref="S3.Thmtheorem6.p1.3.m3.1.1"><csymbol cd="ambiguous" id="S3.Thmtheorem6.p1.3.m3.1.1.1.cmml" xref="S3.Thmtheorem6.p1.3.m3.1.1">superscript</csymbol><apply id="S3.Thmtheorem6.p1.3.m3.1.1.2.cmml" xref="S3.Thmtheorem6.p1.3.m3.1.1"><csymbol cd="ambiguous" id="S3.Thmtheorem6.p1.3.m3.1.1.2.1.cmml" xref="S3.Thmtheorem6.p1.3.m3.1.1">subscript</csymbol><ci id="S3.Thmtheorem6.p1.3.m3.1.1.2.2.cmml" xref="S3.Thmtheorem6.p1.3.m3.1.1.2.2">𝐷</ci><ci id="S3.Thmtheorem6.p1.3.m3.1.1.2.3.cmml" xref="S3.Thmtheorem6.p1.3.m3.1.1.2.3">𝜎</ci></apply><ci id="S3.Thmtheorem6.p1.3.m3.1.1.3.cmml" xref="S3.Thmtheorem6.p1.3.m3.1.1.3">⋆</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem6.p1.3.m3.1c">D_{\sigma}^{\star}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem6.p1.3.m3.1d">italic_D start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT</annotation></semantics></math>, <math alttext="\forall R>0" class="ltx_Math" display="inline" id="S3.Thmtheorem6.p1.4.m4.1"><semantics id="S3.Thmtheorem6.p1.4.m4.1a"><mrow id="S3.Thmtheorem6.p1.4.m4.1.1" xref="S3.Thmtheorem6.p1.4.m4.1.1.cmml"><mrow id="S3.Thmtheorem6.p1.4.m4.1.1.2" xref="S3.Thmtheorem6.p1.4.m4.1.1.2.cmml"><mo id="S3.Thmtheorem6.p1.4.m4.1.1.2.1" rspace="0.167em" xref="S3.Thmtheorem6.p1.4.m4.1.1.2.1.cmml">∀</mo><mi id="S3.Thmtheorem6.p1.4.m4.1.1.2.2" xref="S3.Thmtheorem6.p1.4.m4.1.1.2.2.cmml">R</mi></mrow><mo id="S3.Thmtheorem6.p1.4.m4.1.1.1" xref="S3.Thmtheorem6.p1.4.m4.1.1.1.cmml">></mo><mn id="S3.Thmtheorem6.p1.4.m4.1.1.3" xref="S3.Thmtheorem6.p1.4.m4.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem6.p1.4.m4.1b"><apply id="S3.Thmtheorem6.p1.4.m4.1.1.cmml" xref="S3.Thmtheorem6.p1.4.m4.1.1"><gt id="S3.Thmtheorem6.p1.4.m4.1.1.1.cmml" xref="S3.Thmtheorem6.p1.4.m4.1.1.1"></gt><apply id="S3.Thmtheorem6.p1.4.m4.1.1.2.cmml" xref="S3.Thmtheorem6.p1.4.m4.1.1.2"><csymbol cd="latexml" id="S3.Thmtheorem6.p1.4.m4.1.1.2.1.cmml" xref="S3.Thmtheorem6.p1.4.m4.1.1.2.1">for-all</csymbol><ci id="S3.Thmtheorem6.p1.4.m4.1.1.2.2.cmml" xref="S3.Thmtheorem6.p1.4.m4.1.1.2.2">𝑅</ci></apply><cn id="S3.Thmtheorem6.p1.4.m4.1.1.3.cmml" type="integer" xref="S3.Thmtheorem6.p1.4.m4.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem6.p1.4.m4.1c">\forall R>0</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem6.p1.4.m4.1d">∀ italic_R > 0</annotation></semantics></math>, there exists <math alttext="M(R)>0" class="ltx_Math" display="inline" id="S3.Thmtheorem6.p1.5.m5.1"><semantics id="S3.Thmtheorem6.p1.5.m5.1a"><mrow id="S3.Thmtheorem6.p1.5.m5.1.2" xref="S3.Thmtheorem6.p1.5.m5.1.2.cmml"><mrow id="S3.Thmtheorem6.p1.5.m5.1.2.2" xref="S3.Thmtheorem6.p1.5.m5.1.2.2.cmml"><mi id="S3.Thmtheorem6.p1.5.m5.1.2.2.2" xref="S3.Thmtheorem6.p1.5.m5.1.2.2.2.cmml">M</mi><mo id="S3.Thmtheorem6.p1.5.m5.1.2.2.1" xref="S3.Thmtheorem6.p1.5.m5.1.2.2.1.cmml">⁢</mo><mrow id="S3.Thmtheorem6.p1.5.m5.1.2.2.3.2" xref="S3.Thmtheorem6.p1.5.m5.1.2.2.cmml"><mo id="S3.Thmtheorem6.p1.5.m5.1.2.2.3.2.1" stretchy="false" xref="S3.Thmtheorem6.p1.5.m5.1.2.2.cmml">(</mo><mi id="S3.Thmtheorem6.p1.5.m5.1.1" xref="S3.Thmtheorem6.p1.5.m5.1.1.cmml">R</mi><mo id="S3.Thmtheorem6.p1.5.m5.1.2.2.3.2.2" stretchy="false" xref="S3.Thmtheorem6.p1.5.m5.1.2.2.cmml">)</mo></mrow></mrow><mo id="S3.Thmtheorem6.p1.5.m5.1.2.1" xref="S3.Thmtheorem6.p1.5.m5.1.2.1.cmml">></mo><mn id="S3.Thmtheorem6.p1.5.m5.1.2.3" xref="S3.Thmtheorem6.p1.5.m5.1.2.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem6.p1.5.m5.1b"><apply id="S3.Thmtheorem6.p1.5.m5.1.2.cmml" xref="S3.Thmtheorem6.p1.5.m5.1.2"><gt id="S3.Thmtheorem6.p1.5.m5.1.2.1.cmml" xref="S3.Thmtheorem6.p1.5.m5.1.2.1"></gt><apply id="S3.Thmtheorem6.p1.5.m5.1.2.2.cmml" xref="S3.Thmtheorem6.p1.5.m5.1.2.2"><times id="S3.Thmtheorem6.p1.5.m5.1.2.2.1.cmml" xref="S3.Thmtheorem6.p1.5.m5.1.2.2.1"></times><ci id="S3.Thmtheorem6.p1.5.m5.1.2.2.2.cmml" xref="S3.Thmtheorem6.p1.5.m5.1.2.2.2">𝑀</ci><ci id="S3.Thmtheorem6.p1.5.m5.1.1.cmml" xref="S3.Thmtheorem6.p1.5.m5.1.1">𝑅</ci></apply><cn id="S3.Thmtheorem6.p1.5.m5.1.2.3.cmml" type="integer" xref="S3.Thmtheorem6.p1.5.m5.1.2.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem6.p1.5.m5.1c">M(R)>0</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem6.p1.5.m5.1d">italic_M ( italic_R ) > 0</annotation></semantics></math>, such that <math alttext="\forall{\mathbf{x}}\in\mathcal{B}(0,R)" class="ltx_Math" display="inline" id="S3.Thmtheorem6.p1.6.m6.2"><semantics id="S3.Thmtheorem6.p1.6.m6.2a"><mrow id="S3.Thmtheorem6.p1.6.m6.2.3" xref="S3.Thmtheorem6.p1.6.m6.2.3.cmml"><mrow id="S3.Thmtheorem6.p1.6.m6.2.3.2" xref="S3.Thmtheorem6.p1.6.m6.2.3.2.cmml"><mo id="S3.Thmtheorem6.p1.6.m6.2.3.2.1" rspace="0.167em" xref="S3.Thmtheorem6.p1.6.m6.2.3.2.1.cmml">∀</mo><mi id="S3.Thmtheorem6.p1.6.m6.2.3.2.2" xref="S3.Thmtheorem6.p1.6.m6.2.3.2.2.cmml">𝐱</mi></mrow><mo id="S3.Thmtheorem6.p1.6.m6.2.3.1" xref="S3.Thmtheorem6.p1.6.m6.2.3.1.cmml">∈</mo><mrow id="S3.Thmtheorem6.p1.6.m6.2.3.3" xref="S3.Thmtheorem6.p1.6.m6.2.3.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.Thmtheorem6.p1.6.m6.2.3.3.2" xref="S3.Thmtheorem6.p1.6.m6.2.3.3.2.cmml">ℬ</mi><mo id="S3.Thmtheorem6.p1.6.m6.2.3.3.1" xref="S3.Thmtheorem6.p1.6.m6.2.3.3.1.cmml">⁢</mo><mrow id="S3.Thmtheorem6.p1.6.m6.2.3.3.3.2" xref="S3.Thmtheorem6.p1.6.m6.2.3.3.3.1.cmml"><mo id="S3.Thmtheorem6.p1.6.m6.2.3.3.3.2.1" stretchy="false" xref="S3.Thmtheorem6.p1.6.m6.2.3.3.3.1.cmml">(</mo><mn id="S3.Thmtheorem6.p1.6.m6.1.1" xref="S3.Thmtheorem6.p1.6.m6.1.1.cmml">0</mn><mo id="S3.Thmtheorem6.p1.6.m6.2.3.3.3.2.2" xref="S3.Thmtheorem6.p1.6.m6.2.3.3.3.1.cmml">,</mo><mi id="S3.Thmtheorem6.p1.6.m6.2.2" xref="S3.Thmtheorem6.p1.6.m6.2.2.cmml">R</mi><mo id="S3.Thmtheorem6.p1.6.m6.2.3.3.3.2.3" stretchy="false" xref="S3.Thmtheorem6.p1.6.m6.2.3.3.3.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem6.p1.6.m6.2b"><apply id="S3.Thmtheorem6.p1.6.m6.2.3.cmml" xref="S3.Thmtheorem6.p1.6.m6.2.3"><in id="S3.Thmtheorem6.p1.6.m6.2.3.1.cmml" xref="S3.Thmtheorem6.p1.6.m6.2.3.1"></in><apply id="S3.Thmtheorem6.p1.6.m6.2.3.2.cmml" xref="S3.Thmtheorem6.p1.6.m6.2.3.2"><csymbol cd="latexml" id="S3.Thmtheorem6.p1.6.m6.2.3.2.1.cmml" xref="S3.Thmtheorem6.p1.6.m6.2.3.2.1">for-all</csymbol><ci id="S3.Thmtheorem6.p1.6.m6.2.3.2.2.cmml" xref="S3.Thmtheorem6.p1.6.m6.2.3.2.2">𝐱</ci></apply><apply id="S3.Thmtheorem6.p1.6.m6.2.3.3.cmml" xref="S3.Thmtheorem6.p1.6.m6.2.3.3"><times id="S3.Thmtheorem6.p1.6.m6.2.3.3.1.cmml" xref="S3.Thmtheorem6.p1.6.m6.2.3.3.1"></times><ci id="S3.Thmtheorem6.p1.6.m6.2.3.3.2.cmml" xref="S3.Thmtheorem6.p1.6.m6.2.3.3.2">ℬ</ci><interval closure="open" id="S3.Thmtheorem6.p1.6.m6.2.3.3.3.1.cmml" xref="S3.Thmtheorem6.p1.6.m6.2.3.3.3.2"><cn id="S3.Thmtheorem6.p1.6.m6.1.1.cmml" type="integer" xref="S3.Thmtheorem6.p1.6.m6.1.1">0</cn><ci id="S3.Thmtheorem6.p1.6.m6.2.2.cmml" xref="S3.Thmtheorem6.p1.6.m6.2.2">𝑅</ci></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem6.p1.6.m6.2c">\forall{\mathbf{x}}\in\mathcal{B}(0,R)</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem6.p1.6.m6.2d">∀ bold_x ∈ caligraphic_B ( 0 , italic_R )</annotation></semantics></math>, <math alttext="\|D_{\sigma}({\mathbf{x}})-D_{\sigma}^{\star}({\mathbf{x}})\|\leq M(R)" class="ltx_Math" display="inline" id="S3.Thmtheorem6.p1.7.m7.4"><semantics id="S3.Thmtheorem6.p1.7.m7.4a"><mrow id="S3.Thmtheorem6.p1.7.m7.4.4" xref="S3.Thmtheorem6.p1.7.m7.4.4.cmml"><mrow id="S3.Thmtheorem6.p1.7.m7.4.4.1.1" xref="S3.Thmtheorem6.p1.7.m7.4.4.1.2.cmml"><mo id="S3.Thmtheorem6.p1.7.m7.4.4.1.1.2" stretchy="false" xref="S3.Thmtheorem6.p1.7.m7.4.4.1.2.1.cmml">‖</mo><mrow id="S3.Thmtheorem6.p1.7.m7.4.4.1.1.1" xref="S3.Thmtheorem6.p1.7.m7.4.4.1.1.1.cmml"><mrow id="S3.Thmtheorem6.p1.7.m7.4.4.1.1.1.2" xref="S3.Thmtheorem6.p1.7.m7.4.4.1.1.1.2.cmml"><msub id="S3.Thmtheorem6.p1.7.m7.4.4.1.1.1.2.2" xref="S3.Thmtheorem6.p1.7.m7.4.4.1.1.1.2.2.cmml"><mi id="S3.Thmtheorem6.p1.7.m7.4.4.1.1.1.2.2.2" xref="S3.Thmtheorem6.p1.7.m7.4.4.1.1.1.2.2.2.cmml">D</mi><mi id="S3.Thmtheorem6.p1.7.m7.4.4.1.1.1.2.2.3" xref="S3.Thmtheorem6.p1.7.m7.4.4.1.1.1.2.2.3.cmml">σ</mi></msub><mo id="S3.Thmtheorem6.p1.7.m7.4.4.1.1.1.2.1" xref="S3.Thmtheorem6.p1.7.m7.4.4.1.1.1.2.1.cmml">⁢</mo><mrow id="S3.Thmtheorem6.p1.7.m7.4.4.1.1.1.2.3.2" xref="S3.Thmtheorem6.p1.7.m7.4.4.1.1.1.2.cmml"><mo id="S3.Thmtheorem6.p1.7.m7.4.4.1.1.1.2.3.2.1" stretchy="false" xref="S3.Thmtheorem6.p1.7.m7.4.4.1.1.1.2.cmml">(</mo><mi id="S3.Thmtheorem6.p1.7.m7.1.1" xref="S3.Thmtheorem6.p1.7.m7.1.1.cmml">𝐱</mi><mo id="S3.Thmtheorem6.p1.7.m7.4.4.1.1.1.2.3.2.2" stretchy="false" xref="S3.Thmtheorem6.p1.7.m7.4.4.1.1.1.2.cmml">)</mo></mrow></mrow><mo id="S3.Thmtheorem6.p1.7.m7.4.4.1.1.1.1" xref="S3.Thmtheorem6.p1.7.m7.4.4.1.1.1.1.cmml">−</mo><mrow id="S3.Thmtheorem6.p1.7.m7.4.4.1.1.1.3" xref="S3.Thmtheorem6.p1.7.m7.4.4.1.1.1.3.cmml"><msubsup id="S3.Thmtheorem6.p1.7.m7.4.4.1.1.1.3.2" xref="S3.Thmtheorem6.p1.7.m7.4.4.1.1.1.3.2.cmml"><mi id="S3.Thmtheorem6.p1.7.m7.4.4.1.1.1.3.2.2.2" xref="S3.Thmtheorem6.p1.7.m7.4.4.1.1.1.3.2.2.2.cmml">D</mi><mi id="S3.Thmtheorem6.p1.7.m7.4.4.1.1.1.3.2.2.3" xref="S3.Thmtheorem6.p1.7.m7.4.4.1.1.1.3.2.2.3.cmml">σ</mi><mo id="S3.Thmtheorem6.p1.7.m7.4.4.1.1.1.3.2.3" xref="S3.Thmtheorem6.p1.7.m7.4.4.1.1.1.3.2.3.cmml">⋆</mo></msubsup><mo id="S3.Thmtheorem6.p1.7.m7.4.4.1.1.1.3.1" xref="S3.Thmtheorem6.p1.7.m7.4.4.1.1.1.3.1.cmml">⁢</mo><mrow id="S3.Thmtheorem6.p1.7.m7.4.4.1.1.1.3.3.2" xref="S3.Thmtheorem6.p1.7.m7.4.4.1.1.1.3.cmml"><mo id="S3.Thmtheorem6.p1.7.m7.4.4.1.1.1.3.3.2.1" stretchy="false" xref="S3.Thmtheorem6.p1.7.m7.4.4.1.1.1.3.cmml">(</mo><mi id="S3.Thmtheorem6.p1.7.m7.2.2" xref="S3.Thmtheorem6.p1.7.m7.2.2.cmml">𝐱</mi><mo id="S3.Thmtheorem6.p1.7.m7.4.4.1.1.1.3.3.2.2" stretchy="false" xref="S3.Thmtheorem6.p1.7.m7.4.4.1.1.1.3.cmml">)</mo></mrow></mrow></mrow><mo id="S3.Thmtheorem6.p1.7.m7.4.4.1.1.3" stretchy="false" xref="S3.Thmtheorem6.p1.7.m7.4.4.1.2.1.cmml">‖</mo></mrow><mo id="S3.Thmtheorem6.p1.7.m7.4.4.2" xref="S3.Thmtheorem6.p1.7.m7.4.4.2.cmml">≤</mo><mrow id="S3.Thmtheorem6.p1.7.m7.4.4.3" xref="S3.Thmtheorem6.p1.7.m7.4.4.3.cmml"><mi id="S3.Thmtheorem6.p1.7.m7.4.4.3.2" xref="S3.Thmtheorem6.p1.7.m7.4.4.3.2.cmml">M</mi><mo id="S3.Thmtheorem6.p1.7.m7.4.4.3.1" xref="S3.Thmtheorem6.p1.7.m7.4.4.3.1.cmml">⁢</mo><mrow id="S3.Thmtheorem6.p1.7.m7.4.4.3.3.2" xref="S3.Thmtheorem6.p1.7.m7.4.4.3.cmml"><mo id="S3.Thmtheorem6.p1.7.m7.4.4.3.3.2.1" stretchy="false" xref="S3.Thmtheorem6.p1.7.m7.4.4.3.cmml">(</mo><mi id="S3.Thmtheorem6.p1.7.m7.3.3" xref="S3.Thmtheorem6.p1.7.m7.3.3.cmml">R</mi><mo id="S3.Thmtheorem6.p1.7.m7.4.4.3.3.2.2" stretchy="false" xref="S3.Thmtheorem6.p1.7.m7.4.4.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem6.p1.7.m7.4b"><apply id="S3.Thmtheorem6.p1.7.m7.4.4.cmml" xref="S3.Thmtheorem6.p1.7.m7.4.4"><leq id="S3.Thmtheorem6.p1.7.m7.4.4.2.cmml" xref="S3.Thmtheorem6.p1.7.m7.4.4.2"></leq><apply id="S3.Thmtheorem6.p1.7.m7.4.4.1.2.cmml" xref="S3.Thmtheorem6.p1.7.m7.4.4.1.1"><csymbol cd="latexml" id="S3.Thmtheorem6.p1.7.m7.4.4.1.2.1.cmml" xref="S3.Thmtheorem6.p1.7.m7.4.4.1.1.2">norm</csymbol><apply id="S3.Thmtheorem6.p1.7.m7.4.4.1.1.1.cmml" xref="S3.Thmtheorem6.p1.7.m7.4.4.1.1.1"><minus id="S3.Thmtheorem6.p1.7.m7.4.4.1.1.1.1.cmml" xref="S3.Thmtheorem6.p1.7.m7.4.4.1.1.1.1"></minus><apply id="S3.Thmtheorem6.p1.7.m7.4.4.1.1.1.2.cmml" xref="S3.Thmtheorem6.p1.7.m7.4.4.1.1.1.2"><times id="S3.Thmtheorem6.p1.7.m7.4.4.1.1.1.2.1.cmml" xref="S3.Thmtheorem6.p1.7.m7.4.4.1.1.1.2.1"></times><apply id="S3.Thmtheorem6.p1.7.m7.4.4.1.1.1.2.2.cmml" xref="S3.Thmtheorem6.p1.7.m7.4.4.1.1.1.2.2"><csymbol cd="ambiguous" id="S3.Thmtheorem6.p1.7.m7.4.4.1.1.1.2.2.1.cmml" xref="S3.Thmtheorem6.p1.7.m7.4.4.1.1.1.2.2">subscript</csymbol><ci id="S3.Thmtheorem6.p1.7.m7.4.4.1.1.1.2.2.2.cmml" xref="S3.Thmtheorem6.p1.7.m7.4.4.1.1.1.2.2.2">𝐷</ci><ci id="S3.Thmtheorem6.p1.7.m7.4.4.1.1.1.2.2.3.cmml" xref="S3.Thmtheorem6.p1.7.m7.4.4.1.1.1.2.2.3">𝜎</ci></apply><ci id="S3.Thmtheorem6.p1.7.m7.1.1.cmml" xref="S3.Thmtheorem6.p1.7.m7.1.1">𝐱</ci></apply><apply id="S3.Thmtheorem6.p1.7.m7.4.4.1.1.1.3.cmml" xref="S3.Thmtheorem6.p1.7.m7.4.4.1.1.1.3"><times id="S3.Thmtheorem6.p1.7.m7.4.4.1.1.1.3.1.cmml" xref="S3.Thmtheorem6.p1.7.m7.4.4.1.1.1.3.1"></times><apply id="S3.Thmtheorem6.p1.7.m7.4.4.1.1.1.3.2.cmml" xref="S3.Thmtheorem6.p1.7.m7.4.4.1.1.1.3.2"><csymbol cd="ambiguous" id="S3.Thmtheorem6.p1.7.m7.4.4.1.1.1.3.2.1.cmml" xref="S3.Thmtheorem6.p1.7.m7.4.4.1.1.1.3.2">superscript</csymbol><apply id="S3.Thmtheorem6.p1.7.m7.4.4.1.1.1.3.2.2.cmml" xref="S3.Thmtheorem6.p1.7.m7.4.4.1.1.1.3.2"><csymbol cd="ambiguous" id="S3.Thmtheorem6.p1.7.m7.4.4.1.1.1.3.2.2.1.cmml" xref="S3.Thmtheorem6.p1.7.m7.4.4.1.1.1.3.2">subscript</csymbol><ci id="S3.Thmtheorem6.p1.7.m7.4.4.1.1.1.3.2.2.2.cmml" xref="S3.Thmtheorem6.p1.7.m7.4.4.1.1.1.3.2.2.2">𝐷</ci><ci id="S3.Thmtheorem6.p1.7.m7.4.4.1.1.1.3.2.2.3.cmml" xref="S3.Thmtheorem6.p1.7.m7.4.4.1.1.1.3.2.2.3">𝜎</ci></apply><ci id="S3.Thmtheorem6.p1.7.m7.4.4.1.1.1.3.2.3.cmml" xref="S3.Thmtheorem6.p1.7.m7.4.4.1.1.1.3.2.3">⋆</ci></apply><ci id="S3.Thmtheorem6.p1.7.m7.2.2.cmml" xref="S3.Thmtheorem6.p1.7.m7.2.2">𝐱</ci></apply></apply></apply><apply id="S3.Thmtheorem6.p1.7.m7.4.4.3.cmml" xref="S3.Thmtheorem6.p1.7.m7.4.4.3"><times id="S3.Thmtheorem6.p1.7.m7.4.4.3.1.cmml" xref="S3.Thmtheorem6.p1.7.m7.4.4.3.1"></times><ci id="S3.Thmtheorem6.p1.7.m7.4.4.3.2.cmml" xref="S3.Thmtheorem6.p1.7.m7.4.4.3.2">𝑀</ci><ci id="S3.Thmtheorem6.p1.7.m7.3.3.cmml" xref="S3.Thmtheorem6.p1.7.m7.3.3">𝑅</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem6.p1.7.m7.4c">\|D_{\sigma}({\mathbf{x}})-D_{\sigma}^{\star}({\mathbf{x}})\|\leq M(R)</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem6.p1.7.m7.4d">∥ italic_D start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ( bold_x ) - italic_D start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT ( bold_x ) ∥ ≤ italic_M ( italic_R )</annotation></semantics></math>.</p> </div> </div> <div class="ltx_para" id="S3.SS3.p2"> <p class="ltx_p" id="S3.SS3.p2.2">Assumption <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#S3.Thmtheorem6" title="Assumption 3.6. ‣ 3.3 Biased algorithm analysis ‣ 3 Convergence Analysis ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_tag">3.6</span></a> is also made in <cite class="ltx_cite ltx_citemacro_citep">(Laumont et al., <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#bib.bib37" title="">2022b</a>)</cite>, and it can be ensured if the denoiser is learned with a specific loss <cite class="ltx_cite ltx_citemacro_citep">(Laumont et al., <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#bib.bib36" title="">2022a</a>)</cite>. Moreover, if the activation functions of the denoiser are <math alttext="\mathcal{C}^{\infty}" class="ltx_Math" display="inline" id="S3.SS3.p2.1.m1.1"><semantics id="S3.SS3.p2.1.m1.1a"><msup id="S3.SS3.p2.1.m1.1.1" xref="S3.SS3.p2.1.m1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.SS3.p2.1.m1.1.1.2" xref="S3.SS3.p2.1.m1.1.1.2.cmml">𝒞</mi><mi id="S3.SS3.p2.1.m1.1.1.3" mathvariant="normal" xref="S3.SS3.p2.1.m1.1.1.3.cmml">∞</mi></msup><annotation-xml encoding="MathML-Content" id="S3.SS3.p2.1.m1.1b"><apply id="S3.SS3.p2.1.m1.1.1.cmml" xref="S3.SS3.p2.1.m1.1.1"><csymbol cd="ambiguous" id="S3.SS3.p2.1.m1.1.1.1.cmml" xref="S3.SS3.p2.1.m1.1.1">superscript</csymbol><ci id="S3.SS3.p2.1.m1.1.1.2.cmml" xref="S3.SS3.p2.1.m1.1.1.2">𝒞</ci><infinity id="S3.SS3.p2.1.m1.1.1.3.cmml" xref="S3.SS3.p2.1.m1.1.1.3"></infinity></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p2.1.m1.1c">\mathcal{C}^{\infty}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p2.1.m1.1d">caligraphic_C start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT</annotation></semantics></math> (in our case ELU), then the denoiser is <math alttext="\mathcal{C}^{\infty}" class="ltx_Math" display="inline" id="S3.SS3.p2.2.m2.1"><semantics id="S3.SS3.p2.2.m2.1a"><msup id="S3.SS3.p2.2.m2.1.1" xref="S3.SS3.p2.2.m2.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.SS3.p2.2.m2.1.1.2" xref="S3.SS3.p2.2.m2.1.1.2.cmml">𝒞</mi><mi id="S3.SS3.p2.2.m2.1.1.3" mathvariant="normal" xref="S3.SS3.p2.2.m2.1.1.3.cmml">∞</mi></msup><annotation-xml encoding="MathML-Content" id="S3.SS3.p2.2.m2.1b"><apply id="S3.SS3.p2.2.m2.1.1.cmml" xref="S3.SS3.p2.2.m2.1.1"><csymbol cd="ambiguous" id="S3.SS3.p2.2.m2.1.1.1.cmml" xref="S3.SS3.p2.2.m2.1.1">superscript</csymbol><ci id="S3.SS3.p2.2.m2.1.1.2.cmml" xref="S3.SS3.p2.2.m2.1.1.2">𝒞</ci><infinity id="S3.SS3.p2.2.m2.1.1.3.cmml" xref="S3.SS3.p2.2.m2.1.1.3"></infinity></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p2.2.m2.1c">\mathcal{C}^{\infty}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p2.2.m2.1d">caligraphic_C start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT</annotation></semantics></math>.</p> </div> <div class="ltx_theorem ltx_theorem_assumption" id="S3.Thmtheorem7"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S3.Thmtheorem7.1.1.1">Assumption 3.7</span></span><span class="ltx_text ltx_font_bold" id="S3.Thmtheorem7.2.2">.</span> </h6> <div class="ltx_para" id="S3.Thmtheorem7.p1"> <p class="ltx_p" id="S3.Thmtheorem7.p1.4">The exact MMSE denoiser and the learned denoiser are sublinear, there exists <math alttext="C\geq 0" class="ltx_Math" display="inline" id="S3.Thmtheorem7.p1.1.m1.1"><semantics id="S3.Thmtheorem7.p1.1.m1.1a"><mrow id="S3.Thmtheorem7.p1.1.m1.1.1" xref="S3.Thmtheorem7.p1.1.m1.1.1.cmml"><mi id="S3.Thmtheorem7.p1.1.m1.1.1.2" xref="S3.Thmtheorem7.p1.1.m1.1.1.2.cmml">C</mi><mo id="S3.Thmtheorem7.p1.1.m1.1.1.1" xref="S3.Thmtheorem7.p1.1.m1.1.1.1.cmml">≥</mo><mn id="S3.Thmtheorem7.p1.1.m1.1.1.3" xref="S3.Thmtheorem7.p1.1.m1.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem7.p1.1.m1.1b"><apply id="S3.Thmtheorem7.p1.1.m1.1.1.cmml" xref="S3.Thmtheorem7.p1.1.m1.1.1"><geq id="S3.Thmtheorem7.p1.1.m1.1.1.1.cmml" xref="S3.Thmtheorem7.p1.1.m1.1.1.1"></geq><ci id="S3.Thmtheorem7.p1.1.m1.1.1.2.cmml" xref="S3.Thmtheorem7.p1.1.m1.1.1.2">𝐶</ci><cn id="S3.Thmtheorem7.p1.1.m1.1.1.3.cmml" type="integer" xref="S3.Thmtheorem7.p1.1.m1.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem7.p1.1.m1.1c">C\geq 0</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem7.p1.1.m1.1d">italic_C ≥ 0</annotation></semantics></math> such that <math alttext="\forall{\mathbf{x}}\in{\mathbb{R}}^{d}" class="ltx_Math" display="inline" id="S3.Thmtheorem7.p1.2.m2.1"><semantics id="S3.Thmtheorem7.p1.2.m2.1a"><mrow id="S3.Thmtheorem7.p1.2.m2.1.1" xref="S3.Thmtheorem7.p1.2.m2.1.1.cmml"><mrow id="S3.Thmtheorem7.p1.2.m2.1.1.2" xref="S3.Thmtheorem7.p1.2.m2.1.1.2.cmml"><mo id="S3.Thmtheorem7.p1.2.m2.1.1.2.1" rspace="0.167em" xref="S3.Thmtheorem7.p1.2.m2.1.1.2.1.cmml">∀</mo><mi id="S3.Thmtheorem7.p1.2.m2.1.1.2.2" xref="S3.Thmtheorem7.p1.2.m2.1.1.2.2.cmml">𝐱</mi></mrow><mo id="S3.Thmtheorem7.p1.2.m2.1.1.1" xref="S3.Thmtheorem7.p1.2.m2.1.1.1.cmml">∈</mo><msup id="S3.Thmtheorem7.p1.2.m2.1.1.3" xref="S3.Thmtheorem7.p1.2.m2.1.1.3.cmml"><mi id="S3.Thmtheorem7.p1.2.m2.1.1.3.2" xref="S3.Thmtheorem7.p1.2.m2.1.1.3.2.cmml">ℝ</mi><mi id="S3.Thmtheorem7.p1.2.m2.1.1.3.3" xref="S3.Thmtheorem7.p1.2.m2.1.1.3.3.cmml">d</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem7.p1.2.m2.1b"><apply id="S3.Thmtheorem7.p1.2.m2.1.1.cmml" xref="S3.Thmtheorem7.p1.2.m2.1.1"><in id="S3.Thmtheorem7.p1.2.m2.1.1.1.cmml" xref="S3.Thmtheorem7.p1.2.m2.1.1.1"></in><apply id="S3.Thmtheorem7.p1.2.m2.1.1.2.cmml" xref="S3.Thmtheorem7.p1.2.m2.1.1.2"><csymbol cd="latexml" id="S3.Thmtheorem7.p1.2.m2.1.1.2.1.cmml" xref="S3.Thmtheorem7.p1.2.m2.1.1.2.1">for-all</csymbol><ci id="S3.Thmtheorem7.p1.2.m2.1.1.2.2.cmml" xref="S3.Thmtheorem7.p1.2.m2.1.1.2.2">𝐱</ci></apply><apply id="S3.Thmtheorem7.p1.2.m2.1.1.3.cmml" xref="S3.Thmtheorem7.p1.2.m2.1.1.3"><csymbol cd="ambiguous" id="S3.Thmtheorem7.p1.2.m2.1.1.3.1.cmml" xref="S3.Thmtheorem7.p1.2.m2.1.1.3">superscript</csymbol><ci id="S3.Thmtheorem7.p1.2.m2.1.1.3.2.cmml" xref="S3.Thmtheorem7.p1.2.m2.1.1.3.2">ℝ</ci><ci id="S3.Thmtheorem7.p1.2.m2.1.1.3.3.cmml" xref="S3.Thmtheorem7.p1.2.m2.1.1.3.3">𝑑</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem7.p1.2.m2.1c">\forall{\mathbf{x}}\in{\mathbb{R}}^{d}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem7.p1.2.m2.1d">∀ bold_x ∈ blackboard_R start_POSTSUPERSCRIPT 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cd="ambiguous" id="S3.Thmtheorem7.p1.4.m4.3.3.1.1.1.2.1.cmml" xref="S3.Thmtheorem7.p1.4.m4.3.3.1.1.1.2">subscript</csymbol><ci id="S3.Thmtheorem7.p1.4.m4.3.3.1.1.1.2.2.cmml" xref="S3.Thmtheorem7.p1.4.m4.3.3.1.1.1.2.2">𝐷</ci><ci id="S3.Thmtheorem7.p1.4.m4.3.3.1.1.1.2.3.cmml" xref="S3.Thmtheorem7.p1.4.m4.3.3.1.1.1.2.3">𝜎</ci></apply><ci id="S3.Thmtheorem7.p1.4.m4.1.1.cmml" xref="S3.Thmtheorem7.p1.4.m4.1.1">𝐱</ci></apply></apply><apply id="S3.Thmtheorem7.p1.4.m4.3.3.3.cmml" xref="S3.Thmtheorem7.p1.4.m4.3.3.3"><plus id="S3.Thmtheorem7.p1.4.m4.3.3.3.1.cmml" xref="S3.Thmtheorem7.p1.4.m4.3.3.3.1"></plus><apply id="S3.Thmtheorem7.p1.4.m4.3.3.3.2.1.cmml" xref="S3.Thmtheorem7.p1.4.m4.3.3.3.2.2"><csymbol cd="latexml" id="S3.Thmtheorem7.p1.4.m4.3.3.3.2.1.1.cmml" xref="S3.Thmtheorem7.p1.4.m4.3.3.3.2.2.1">norm</csymbol><ci id="S3.Thmtheorem7.p1.4.m4.2.2.cmml" xref="S3.Thmtheorem7.p1.4.m4.2.2">𝐱</ci></apply><apply id="S3.Thmtheorem7.p1.4.m4.3.3.3.3.cmml" xref="S3.Thmtheorem7.p1.4.m4.3.3.3.3"><times id="S3.Thmtheorem7.p1.4.m4.3.3.3.3.1.cmml" xref="S3.Thmtheorem7.p1.4.m4.3.3.3.3.1"></times><ci id="S3.Thmtheorem7.p1.4.m4.3.3.3.3.2.cmml" xref="S3.Thmtheorem7.p1.4.m4.3.3.3.3.2">𝐶</ci><ci id="S3.Thmtheorem7.p1.4.m4.3.3.3.3.3.cmml" xref="S3.Thmtheorem7.p1.4.m4.3.3.3.3.3">𝜎</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem7.p1.4.m4.3c">\|D_{\sigma}({\mathbf{x}})\|\leq\|{\mathbf{x}}\|+C\sigma</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem7.p1.4.m4.3d">∥ italic_D start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ( bold_x ) ∥ ≤ ∥ bold_x ∥ + italic_C italic_σ</annotation></semantics></math>.</p> </div> </div> <div class="ltx_para" id="S3.SS3.p3"> <p class="ltx_p" id="S3.SS3.p3.1">Assumption <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#S3.Thmtheorem7" title="Assumption 3.7. ‣ 3.3 Biased algorithm analysis ‣ 3 Convergence Analysis ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_tag">3.7</span></a> is the stable condition on the denoiser <math alttext="D_{\sigma}" class="ltx_Math" display="inline" id="S3.SS3.p3.1.m1.1"><semantics id="S3.SS3.p3.1.m1.1a"><msub id="S3.SS3.p3.1.m1.1.1" xref="S3.SS3.p3.1.m1.1.1.cmml"><mi id="S3.SS3.p3.1.m1.1.1.2" xref="S3.SS3.p3.1.m1.1.1.2.cmml">D</mi><mi id="S3.SS3.p3.1.m1.1.1.3" xref="S3.SS3.p3.1.m1.1.1.3.cmml">σ</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS3.p3.1.m1.1b"><apply id="S3.SS3.p3.1.m1.1.1.cmml" xref="S3.SS3.p3.1.m1.1.1"><csymbol cd="ambiguous" id="S3.SS3.p3.1.m1.1.1.1.cmml" xref="S3.SS3.p3.1.m1.1.1">subscript</csymbol><ci id="S3.SS3.p3.1.m1.1.1.2.cmml" xref="S3.SS3.p3.1.m1.1.1.2">𝐷</ci><ci id="S3.SS3.p3.1.m1.1.1.3.cmml" xref="S3.SS3.p3.1.m1.1.1.3">𝜎</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p3.1.m1.1c">D_{\sigma}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p3.1.m1.1d">italic_D start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT</annotation></semantics></math> in the sense that it is bounded in norm. As a practical example, a bounded denoiser <cite class="ltx_cite ltx_citemacro_citep">(Chan et al., <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#bib.bib9" title="">2016</a>, Definition 1)</cite> verifies Assumption <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#S3.Thmtheorem7" title="Assumption 3.7. ‣ 3.3 Biased algorithm analysis ‣ 3 Convergence Analysis ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_tag">3.7</span></a>.</p> </div> <div class="ltx_theorem ltx_theorem_proposition" id="S3.Thmproposition5"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S3.Thmproposition5.1.1.1">Proposition 3.5</span></span><span class="ltx_text ltx_font_bold" id="S3.Thmproposition5.2.2">.</span> </h6> <div class="ltx_para" id="S3.Thmproposition5.p1"> <p class="ltx_p" id="S3.Thmproposition5.p1.4"><span class="ltx_text ltx_font_italic" id="S3.Thmproposition5.p1.4.4">Under Assumptions <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#S3.Thmtheorem3" title="Assumption 3.3. ‣ 3.2 Unbiased algorithm analysis ‣ 3 Convergence Analysis ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_tag">3.3</span></a>, <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#S3.Thmtheorem4" title="Assumption 3.4. ‣ 3.2 Unbiased algorithm analysis ‣ 3 Convergence Analysis ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_tag">3.4</span></a>, <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#S3.Thmtheorem6" title="Assumption 3.6. ‣ 3.3 Biased algorithm analysis ‣ 3 Convergence Analysis ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_tag">3.6</span></a>, <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#S3.Thmtheorem7" title="Assumption 3.7. ‣ 3.3 Biased algorithm analysis ‣ 3 Convergence Analysis ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_tag">3.7</span></a>, for <math alttext="R>0" class="ltx_Math" display="inline" id="S3.Thmproposition5.p1.1.1.m1.1"><semantics id="S3.Thmproposition5.p1.1.1.m1.1a"><mrow id="S3.Thmproposition5.p1.1.1.m1.1.1" xref="S3.Thmproposition5.p1.1.1.m1.1.1.cmml"><mi id="S3.Thmproposition5.p1.1.1.m1.1.1.2" xref="S3.Thmproposition5.p1.1.1.m1.1.1.2.cmml">R</mi><mo id="S3.Thmproposition5.p1.1.1.m1.1.1.1" mathvariant="normal" xref="S3.Thmproposition5.p1.1.1.m1.1.1.1.cmml">></mo><mn id="S3.Thmproposition5.p1.1.1.m1.1.1.3" mathvariant="normal" xref="S3.Thmproposition5.p1.1.1.m1.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmproposition5.p1.1.1.m1.1b"><apply id="S3.Thmproposition5.p1.1.1.m1.1.1.cmml" xref="S3.Thmproposition5.p1.1.1.m1.1.1"><gt id="S3.Thmproposition5.p1.1.1.m1.1.1.1.cmml" xref="S3.Thmproposition5.p1.1.1.m1.1.1.1"></gt><ci id="S3.Thmproposition5.p1.1.1.m1.1.1.2.cmml" xref="S3.Thmproposition5.p1.1.1.m1.1.1.2">𝑅</ci><cn id="S3.Thmproposition5.p1.1.1.m1.1.1.3.cmml" type="integer" xref="S3.Thmproposition5.p1.1.1.m1.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmproposition5.p1.1.1.m1.1c">R>0</annotation><annotation encoding="application/x-llamapun" id="S3.Thmproposition5.p1.1.1.m1.1d">italic_R > 0</annotation></semantics></math> and <math alttext="\mathbf{K}\subseteq\mathcal{B}(0,R)" class="ltx_Math" display="inline" id="S3.Thmproposition5.p1.2.2.m2.2"><semantics id="S3.Thmproposition5.p1.2.2.m2.2a"><mrow id="S3.Thmproposition5.p1.2.2.m2.2.3" xref="S3.Thmproposition5.p1.2.2.m2.2.3.cmml"><mi id="S3.Thmproposition5.p1.2.2.m2.2.3.2" xref="S3.Thmproposition5.p1.2.2.m2.2.3.2.cmml">𝐊</mi><mo id="S3.Thmproposition5.p1.2.2.m2.2.3.1" mathvariant="normal" xref="S3.Thmproposition5.p1.2.2.m2.2.3.1.cmml">⊆</mo><mrow id="S3.Thmproposition5.p1.2.2.m2.2.3.3" xref="S3.Thmproposition5.p1.2.2.m2.2.3.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.Thmproposition5.p1.2.2.m2.2.3.3.2" xref="S3.Thmproposition5.p1.2.2.m2.2.3.3.2.cmml">ℬ</mi><mo id="S3.Thmproposition5.p1.2.2.m2.2.3.3.1" mathvariant="italic" xref="S3.Thmproposition5.p1.2.2.m2.2.3.3.1.cmml">⁢</mo><mrow id="S3.Thmproposition5.p1.2.2.m2.2.3.3.3.2" xref="S3.Thmproposition5.p1.2.2.m2.2.3.3.3.1.cmml"><mo id="S3.Thmproposition5.p1.2.2.m2.2.3.3.3.2.1" mathvariant="normal" stretchy="false" xref="S3.Thmproposition5.p1.2.2.m2.2.3.3.3.1.cmml">(</mo><mn id="S3.Thmproposition5.p1.2.2.m2.1.1" mathvariant="normal" xref="S3.Thmproposition5.p1.2.2.m2.1.1.cmml">0</mn><mo id="S3.Thmproposition5.p1.2.2.m2.2.3.3.3.2.2" mathvariant="normal" xref="S3.Thmproposition5.p1.2.2.m2.2.3.3.3.1.cmml">,</mo><mi id="S3.Thmproposition5.p1.2.2.m2.2.2" xref="S3.Thmproposition5.p1.2.2.m2.2.2.cmml">R</mi><mo id="S3.Thmproposition5.p1.2.2.m2.2.3.3.3.2.3" mathvariant="normal" stretchy="false" xref="S3.Thmproposition5.p1.2.2.m2.2.3.3.3.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmproposition5.p1.2.2.m2.2b"><apply id="S3.Thmproposition5.p1.2.2.m2.2.3.cmml" xref="S3.Thmproposition5.p1.2.2.m2.2.3"><subset id="S3.Thmproposition5.p1.2.2.m2.2.3.1.cmml" xref="S3.Thmproposition5.p1.2.2.m2.2.3.1"></subset><ci id="S3.Thmproposition5.p1.2.2.m2.2.3.2.cmml" xref="S3.Thmproposition5.p1.2.2.m2.2.3.2">𝐊</ci><apply id="S3.Thmproposition5.p1.2.2.m2.2.3.3.cmml" xref="S3.Thmproposition5.p1.2.2.m2.2.3.3"><times id="S3.Thmproposition5.p1.2.2.m2.2.3.3.1.cmml" xref="S3.Thmproposition5.p1.2.2.m2.2.3.3.1"></times><ci id="S3.Thmproposition5.p1.2.2.m2.2.3.3.2.cmml" xref="S3.Thmproposition5.p1.2.2.m2.2.3.3.2">ℬ</ci><interval closure="open" id="S3.Thmproposition5.p1.2.2.m2.2.3.3.3.1.cmml" xref="S3.Thmproposition5.p1.2.2.m2.2.3.3.3.2"><cn id="S3.Thmproposition5.p1.2.2.m2.1.1.cmml" type="integer" xref="S3.Thmproposition5.p1.2.2.m2.1.1">0</cn><ci id="S3.Thmproposition5.p1.2.2.m2.2.2.cmml" xref="S3.Thmproposition5.p1.2.2.m2.2.2">𝑅</ci></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmproposition5.p1.2.2.m2.2c">\mathbf{K}\subseteq\mathcal{B}(0,R)</annotation><annotation encoding="application/x-llamapun" id="S3.Thmproposition5.p1.2.2.m2.2d">bold_K ⊆ caligraphic_B ( 0 , italic_R )</annotation></semantics></math> compact, almost surely on <math alttext="\Lambda_{\mathbf{K}}" class="ltx_Math" display="inline" id="S3.Thmproposition5.p1.3.3.m3.1"><semantics id="S3.Thmproposition5.p1.3.3.m3.1a"><msub id="S3.Thmproposition5.p1.3.3.m3.1.1" xref="S3.Thmproposition5.p1.3.3.m3.1.1.cmml"><mi id="S3.Thmproposition5.p1.3.3.m3.1.1.2" mathvariant="normal" xref="S3.Thmproposition5.p1.3.3.m3.1.1.2.cmml">Λ</mi><mi id="S3.Thmproposition5.p1.3.3.m3.1.1.3" xref="S3.Thmproposition5.p1.3.3.m3.1.1.3.cmml">𝐊</mi></msub><annotation-xml encoding="MathML-Content" id="S3.Thmproposition5.p1.3.3.m3.1b"><apply id="S3.Thmproposition5.p1.3.3.m3.1.1.cmml" xref="S3.Thmproposition5.p1.3.3.m3.1.1"><csymbol cd="ambiguous" id="S3.Thmproposition5.p1.3.3.m3.1.1.1.cmml" xref="S3.Thmproposition5.p1.3.3.m3.1.1">subscript</csymbol><ci id="S3.Thmproposition5.p1.3.3.m3.1.1.2.cmml" xref="S3.Thmproposition5.p1.3.3.m3.1.1.2">normal-Λ</ci><ci id="S3.Thmproposition5.p1.3.3.m3.1.1.3.cmml" xref="S3.Thmproposition5.p1.3.3.m3.1.1.3">𝐊</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmproposition5.p1.3.3.m3.1c">\Lambda_{\mathbf{K}}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmproposition5.p1.3.3.m3.1d">roman_Λ start_POSTSUBSCRIPT bold_K end_POSTSUBSCRIPT</annotation></semantics></math>, there exists <math alttext="M_{\mathbf{K}}\in]0,+\infty[" class="ltx_math_unparsed" display="inline" id="S3.Thmproposition5.p1.4.4.m4.1"><semantics id="S3.Thmproposition5.p1.4.4.m4.1a"><mrow id="S3.Thmproposition5.p1.4.4.m4.1b"><msub id="S3.Thmproposition5.p1.4.4.m4.1.1"><mi id="S3.Thmproposition5.p1.4.4.m4.1.1.2">M</mi><mi id="S3.Thmproposition5.p1.4.4.m4.1.1.3">𝐊</mi></msub><mo id="S3.Thmproposition5.p1.4.4.m4.1.2" mathvariant="normal" rspace="0em">∈</mo><mo id="S3.Thmproposition5.p1.4.4.m4.1.3" mathvariant="normal" stretchy="false">]</mo><mn id="S3.Thmproposition5.p1.4.4.m4.1.4" mathvariant="normal">0</mn><mo id="S3.Thmproposition5.p1.4.4.m4.1.5" mathvariant="normal" rspace="0em">,</mo><mo id="S3.Thmproposition5.p1.4.4.m4.1.6" lspace="0em" mathvariant="normal">+</mo><mi id="S3.Thmproposition5.p1.4.4.m4.1.7" mathvariant="normal">∞</mi><mo id="S3.Thmproposition5.p1.4.4.m4.1.8" mathvariant="normal" stretchy="false">[</mo></mrow><annotation encoding="application/x-tex" id="S3.Thmproposition5.p1.4.4.m4.1c">M_{\mathbf{K}}\in]0,+\infty[</annotation><annotation encoding="application/x-llamapun" id="S3.Thmproposition5.p1.4.4.m4.1d">italic_M start_POSTSUBSCRIPT bold_K end_POSTSUBSCRIPT ∈ ] 0 , + ∞ [</annotation></semantics></math> such that</span></p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A8.EGx14"> <tbody id="S3.Ex18"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\limsup_{k\to+\infty}\|\nabla\mathcal{J}({\mathbf{x}}_{k})\|" class="ltx_Math" display="inline" id="S3.Ex18.m1.1"><semantics id="S3.Ex18.m1.1a"><mrow id="S3.Ex18.m1.1.1" xref="S3.Ex18.m1.1.1.cmml"><munder id="S3.Ex18.m1.1.1.2" xref="S3.Ex18.m1.1.1.2.cmml"><mo id="S3.Ex18.m1.1.1.2.2" movablelimits="false" xref="S3.Ex18.m1.1.1.2.2.cmml">lim sup</mo><mrow id="S3.Ex18.m1.1.1.2.3" xref="S3.Ex18.m1.1.1.2.3.cmml"><mi id="S3.Ex18.m1.1.1.2.3.2" xref="S3.Ex18.m1.1.1.2.3.2.cmml">k</mi><mo id="S3.Ex18.m1.1.1.2.3.1" stretchy="false" xref="S3.Ex18.m1.1.1.2.3.1.cmml">→</mo><mrow id="S3.Ex18.m1.1.1.2.3.3" xref="S3.Ex18.m1.1.1.2.3.3.cmml"><mo id="S3.Ex18.m1.1.1.2.3.3a" xref="S3.Ex18.m1.1.1.2.3.3.cmml">+</mo><mi id="S3.Ex18.m1.1.1.2.3.3.2" mathvariant="normal" xref="S3.Ex18.m1.1.1.2.3.3.2.cmml">∞</mi></mrow></mrow></munder><mrow id="S3.Ex18.m1.1.1.1.1" xref="S3.Ex18.m1.1.1.1.2.cmml"><mo id="S3.Ex18.m1.1.1.1.1.2" lspace="0em" stretchy="false" xref="S3.Ex18.m1.1.1.1.2.1.cmml">‖</mo><mrow id="S3.Ex18.m1.1.1.1.1.1" xref="S3.Ex18.m1.1.1.1.1.1.cmml"><mrow id="S3.Ex18.m1.1.1.1.1.1.3" xref="S3.Ex18.m1.1.1.1.1.1.3.cmml"><mo id="S3.Ex18.m1.1.1.1.1.1.3.1" rspace="0.167em" xref="S3.Ex18.m1.1.1.1.1.1.3.1.cmml">∇</mo><mi class="ltx_font_mathcaligraphic" id="S3.Ex18.m1.1.1.1.1.1.3.2" xref="S3.Ex18.m1.1.1.1.1.1.3.2.cmml">𝒥</mi></mrow><mo id="S3.Ex18.m1.1.1.1.1.1.2" xref="S3.Ex18.m1.1.1.1.1.1.2.cmml">⁢</mo><mrow id="S3.Ex18.m1.1.1.1.1.1.1.1" xref="S3.Ex18.m1.1.1.1.1.1.1.1.1.cmml"><mo id="S3.Ex18.m1.1.1.1.1.1.1.1.2" stretchy="false" xref="S3.Ex18.m1.1.1.1.1.1.1.1.1.cmml">(</mo><msub id="S3.Ex18.m1.1.1.1.1.1.1.1.1" xref="S3.Ex18.m1.1.1.1.1.1.1.1.1.cmml"><mi id="S3.Ex18.m1.1.1.1.1.1.1.1.1.2" xref="S3.Ex18.m1.1.1.1.1.1.1.1.1.2.cmml">𝐱</mi><mi id="S3.Ex18.m1.1.1.1.1.1.1.1.1.3" xref="S3.Ex18.m1.1.1.1.1.1.1.1.1.3.cmml">k</mi></msub><mo id="S3.Ex18.m1.1.1.1.1.1.1.1.3" stretchy="false" xref="S3.Ex18.m1.1.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S3.Ex18.m1.1.1.1.1.3" stretchy="false" xref="S3.Ex18.m1.1.1.1.2.1.cmml">‖</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Ex18.m1.1b"><apply id="S3.Ex18.m1.1.1.cmml" xref="S3.Ex18.m1.1.1"><apply id="S3.Ex18.m1.1.1.2.cmml" xref="S3.Ex18.m1.1.1.2"><csymbol cd="ambiguous" id="S3.Ex18.m1.1.1.2.1.cmml" xref="S3.Ex18.m1.1.1.2">subscript</csymbol><csymbol cd="latexml" id="S3.Ex18.m1.1.1.2.2.cmml" xref="S3.Ex18.m1.1.1.2.2">limit-supremum</csymbol><apply id="S3.Ex18.m1.1.1.2.3.cmml" xref="S3.Ex18.m1.1.1.2.3"><ci id="S3.Ex18.m1.1.1.2.3.1.cmml" xref="S3.Ex18.m1.1.1.2.3.1">→</ci><ci id="S3.Ex18.m1.1.1.2.3.2.cmml" xref="S3.Ex18.m1.1.1.2.3.2">𝑘</ci><apply id="S3.Ex18.m1.1.1.2.3.3.cmml" xref="S3.Ex18.m1.1.1.2.3.3"><plus id="S3.Ex18.m1.1.1.2.3.3.1.cmml" xref="S3.Ex18.m1.1.1.2.3.3"></plus><infinity id="S3.Ex18.m1.1.1.2.3.3.2.cmml" xref="S3.Ex18.m1.1.1.2.3.3.2"></infinity></apply></apply></apply><apply id="S3.Ex18.m1.1.1.1.2.cmml" xref="S3.Ex18.m1.1.1.1.1"><csymbol cd="latexml" id="S3.Ex18.m1.1.1.1.2.1.cmml" xref="S3.Ex18.m1.1.1.1.1.2">norm</csymbol><apply id="S3.Ex18.m1.1.1.1.1.1.cmml" xref="S3.Ex18.m1.1.1.1.1.1"><times id="S3.Ex18.m1.1.1.1.1.1.2.cmml" xref="S3.Ex18.m1.1.1.1.1.1.2"></times><apply id="S3.Ex18.m1.1.1.1.1.1.3.cmml" xref="S3.Ex18.m1.1.1.1.1.1.3"><ci id="S3.Ex18.m1.1.1.1.1.1.3.1.cmml" xref="S3.Ex18.m1.1.1.1.1.1.3.1">∇</ci><ci id="S3.Ex18.m1.1.1.1.1.1.3.2.cmml" xref="S3.Ex18.m1.1.1.1.1.1.3.2">𝒥</ci></apply><apply id="S3.Ex18.m1.1.1.1.1.1.1.1.1.cmml" xref="S3.Ex18.m1.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S3.Ex18.m1.1.1.1.1.1.1.1.1.1.cmml" xref="S3.Ex18.m1.1.1.1.1.1.1.1">subscript</csymbol><ci id="S3.Ex18.m1.1.1.1.1.1.1.1.1.2.cmml" xref="S3.Ex18.m1.1.1.1.1.1.1.1.1.2">𝐱</ci><ci id="S3.Ex18.m1.1.1.1.1.1.1.1.1.3.cmml" xref="S3.Ex18.m1.1.1.1.1.1.1.1.1.3">𝑘</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Ex18.m1.1c">\displaystyle\limsup_{k\to+\infty}\|\nabla\mathcal{J}({\mathbf{x}}_{k})\|</annotation><annotation encoding="application/x-llamapun" id="S3.Ex18.m1.1d">lim sup start_POSTSUBSCRIPT italic_k → + ∞ end_POSTSUBSCRIPT ∥ ∇ caligraphic_J ( bold_x start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) ∥</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\leq M_{\mathbf{K}}\eta^{\frac{1}{2}}," class="ltx_Math" display="inline" id="S3.Ex18.m2.1"><semantics id="S3.Ex18.m2.1a"><mrow id="S3.Ex18.m2.1.1.1" xref="S3.Ex18.m2.1.1.1.1.cmml"><mrow id="S3.Ex18.m2.1.1.1.1" xref="S3.Ex18.m2.1.1.1.1.cmml"><mi id="S3.Ex18.m2.1.1.1.1.2" xref="S3.Ex18.m2.1.1.1.1.2.cmml"></mi><mo id="S3.Ex18.m2.1.1.1.1.1" xref="S3.Ex18.m2.1.1.1.1.1.cmml">≤</mo><mrow id="S3.Ex18.m2.1.1.1.1.3" xref="S3.Ex18.m2.1.1.1.1.3.cmml"><msub id="S3.Ex18.m2.1.1.1.1.3.2" xref="S3.Ex18.m2.1.1.1.1.3.2.cmml"><mi id="S3.Ex18.m2.1.1.1.1.3.2.2" xref="S3.Ex18.m2.1.1.1.1.3.2.2.cmml">M</mi><mi id="S3.Ex18.m2.1.1.1.1.3.2.3" xref="S3.Ex18.m2.1.1.1.1.3.2.3.cmml">𝐊</mi></msub><mo id="S3.Ex18.m2.1.1.1.1.3.1" xref="S3.Ex18.m2.1.1.1.1.3.1.cmml">⁢</mo><msup id="S3.Ex18.m2.1.1.1.1.3.3" xref="S3.Ex18.m2.1.1.1.1.3.3.cmml"><mi id="S3.Ex18.m2.1.1.1.1.3.3.2" xref="S3.Ex18.m2.1.1.1.1.3.3.2.cmml">η</mi><mfrac id="S3.Ex18.m2.1.1.1.1.3.3.3" xref="S3.Ex18.m2.1.1.1.1.3.3.3.cmml"><mn id="S3.Ex18.m2.1.1.1.1.3.3.3.2" xref="S3.Ex18.m2.1.1.1.1.3.3.3.2.cmml">1</mn><mn id="S3.Ex18.m2.1.1.1.1.3.3.3.3" xref="S3.Ex18.m2.1.1.1.1.3.3.3.3.cmml">2</mn></mfrac></msup></mrow></mrow><mo id="S3.Ex18.m2.1.1.1.2" xref="S3.Ex18.m2.1.1.1.1.cmml">,</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.Ex18.m2.1b"><apply id="S3.Ex18.m2.1.1.1.1.cmml" xref="S3.Ex18.m2.1.1.1"><leq id="S3.Ex18.m2.1.1.1.1.1.cmml" xref="S3.Ex18.m2.1.1.1.1.1"></leq><csymbol cd="latexml" id="S3.Ex18.m2.1.1.1.1.2.cmml" xref="S3.Ex18.m2.1.1.1.1.2">absent</csymbol><apply id="S3.Ex18.m2.1.1.1.1.3.cmml" xref="S3.Ex18.m2.1.1.1.1.3"><times id="S3.Ex18.m2.1.1.1.1.3.1.cmml" xref="S3.Ex18.m2.1.1.1.1.3.1"></times><apply id="S3.Ex18.m2.1.1.1.1.3.2.cmml" xref="S3.Ex18.m2.1.1.1.1.3.2"><csymbol cd="ambiguous" id="S3.Ex18.m2.1.1.1.1.3.2.1.cmml" xref="S3.Ex18.m2.1.1.1.1.3.2">subscript</csymbol><ci id="S3.Ex18.m2.1.1.1.1.3.2.2.cmml" xref="S3.Ex18.m2.1.1.1.1.3.2.2">𝑀</ci><ci id="S3.Ex18.m2.1.1.1.1.3.2.3.cmml" xref="S3.Ex18.m2.1.1.1.1.3.2.3">𝐊</ci></apply><apply id="S3.Ex18.m2.1.1.1.1.3.3.cmml" xref="S3.Ex18.m2.1.1.1.1.3.3"><csymbol cd="ambiguous" id="S3.Ex18.m2.1.1.1.1.3.3.1.cmml" xref="S3.Ex18.m2.1.1.1.1.3.3">superscript</csymbol><ci id="S3.Ex18.m2.1.1.1.1.3.3.2.cmml" xref="S3.Ex18.m2.1.1.1.1.3.3.2">𝜂</ci><apply id="S3.Ex18.m2.1.1.1.1.3.3.3.cmml" xref="S3.Ex18.m2.1.1.1.1.3.3.3"><divide id="S3.Ex18.m2.1.1.1.1.3.3.3.1.cmml" xref="S3.Ex18.m2.1.1.1.1.3.3.3"></divide><cn id="S3.Ex18.m2.1.1.1.1.3.3.3.2.cmml" type="integer" xref="S3.Ex18.m2.1.1.1.1.3.3.3.2">1</cn><cn id="S3.Ex18.m2.1.1.1.1.3.3.3.3.cmml" type="integer" xref="S3.Ex18.m2.1.1.1.1.3.3.3.3">2</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Ex18.m2.1c">\displaystyle\leq M_{\mathbf{K}}\eta^{\frac{1}{2}},</annotation><annotation encoding="application/x-llamapun" id="S3.Ex18.m2.1d">≤ italic_M start_POSTSUBSCRIPT bold_K end_POSTSUBSCRIPT italic_η start_POSTSUPERSCRIPT divide start_ARG 1 end_ARG start_ARG 2 end_ARG end_POSTSUPERSCRIPT ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="S3.Ex19"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\limsup_{k\to+\infty}\mathcal{J}({\mathbf{x}}_{k})-\liminf_{k\to+% \infty}\mathcal{J}({\mathbf{x}}_{k})" class="ltx_Math" display="inline" id="S3.Ex19.m1.2"><semantics id="S3.Ex19.m1.2a"><mrow id="S3.Ex19.m1.2.2" xref="S3.Ex19.m1.2.2.cmml"><mrow id="S3.Ex19.m1.1.1.1" xref="S3.Ex19.m1.1.1.1.cmml"><munder id="S3.Ex19.m1.1.1.1.2" xref="S3.Ex19.m1.1.1.1.2.cmml"><mo id="S3.Ex19.m1.1.1.1.2.2" movablelimits="false" xref="S3.Ex19.m1.1.1.1.2.2.cmml">lim sup</mo><mrow id="S3.Ex19.m1.1.1.1.2.3" xref="S3.Ex19.m1.1.1.1.2.3.cmml"><mi id="S3.Ex19.m1.1.1.1.2.3.2" xref="S3.Ex19.m1.1.1.1.2.3.2.cmml">k</mi><mo id="S3.Ex19.m1.1.1.1.2.3.1" stretchy="false" xref="S3.Ex19.m1.1.1.1.2.3.1.cmml">→</mo><mrow id="S3.Ex19.m1.1.1.1.2.3.3" xref="S3.Ex19.m1.1.1.1.2.3.3.cmml"><mo id="S3.Ex19.m1.1.1.1.2.3.3a" xref="S3.Ex19.m1.1.1.1.2.3.3.cmml">+</mo><mi id="S3.Ex19.m1.1.1.1.2.3.3.2" mathvariant="normal" xref="S3.Ex19.m1.1.1.1.2.3.3.2.cmml">∞</mi></mrow></mrow></munder><mrow id="S3.Ex19.m1.1.1.1.1" xref="S3.Ex19.m1.1.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.Ex19.m1.1.1.1.1.3" xref="S3.Ex19.m1.1.1.1.1.3.cmml">𝒥</mi><mo id="S3.Ex19.m1.1.1.1.1.2" xref="S3.Ex19.m1.1.1.1.1.2.cmml">⁢</mo><mrow id="S3.Ex19.m1.1.1.1.1.1.1" xref="S3.Ex19.m1.1.1.1.1.1.1.1.cmml"><mo id="S3.Ex19.m1.1.1.1.1.1.1.2" stretchy="false" xref="S3.Ex19.m1.1.1.1.1.1.1.1.cmml">(</mo><msub id="S3.Ex19.m1.1.1.1.1.1.1.1" xref="S3.Ex19.m1.1.1.1.1.1.1.1.cmml"><mi id="S3.Ex19.m1.1.1.1.1.1.1.1.2" xref="S3.Ex19.m1.1.1.1.1.1.1.1.2.cmml">𝐱</mi><mi id="S3.Ex19.m1.1.1.1.1.1.1.1.3" xref="S3.Ex19.m1.1.1.1.1.1.1.1.3.cmml">k</mi></msub><mo id="S3.Ex19.m1.1.1.1.1.1.1.3" stretchy="false" xref="S3.Ex19.m1.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><mo id="S3.Ex19.m1.2.2.3" xref="S3.Ex19.m1.2.2.3.cmml">−</mo><mrow id="S3.Ex19.m1.2.2.2" xref="S3.Ex19.m1.2.2.2.cmml"><munder id="S3.Ex19.m1.2.2.2.2" xref="S3.Ex19.m1.2.2.2.2.cmml"><mo id="S3.Ex19.m1.2.2.2.2.2" lspace="0em" movablelimits="false" rspace="0.167em" xref="S3.Ex19.m1.2.2.2.2.2.cmml">lim inf</mo><mrow id="S3.Ex19.m1.2.2.2.2.3" xref="S3.Ex19.m1.2.2.2.2.3.cmml"><mi id="S3.Ex19.m1.2.2.2.2.3.2" xref="S3.Ex19.m1.2.2.2.2.3.2.cmml">k</mi><mo id="S3.Ex19.m1.2.2.2.2.3.1" stretchy="false" xref="S3.Ex19.m1.2.2.2.2.3.1.cmml">→</mo><mrow id="S3.Ex19.m1.2.2.2.2.3.3" xref="S3.Ex19.m1.2.2.2.2.3.3.cmml"><mo id="S3.Ex19.m1.2.2.2.2.3.3a" xref="S3.Ex19.m1.2.2.2.2.3.3.cmml">+</mo><mi id="S3.Ex19.m1.2.2.2.2.3.3.2" mathvariant="normal" xref="S3.Ex19.m1.2.2.2.2.3.3.2.cmml">∞</mi></mrow></mrow></munder><mrow id="S3.Ex19.m1.2.2.2.1" xref="S3.Ex19.m1.2.2.2.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.Ex19.m1.2.2.2.1.3" xref="S3.Ex19.m1.2.2.2.1.3.cmml">𝒥</mi><mo id="S3.Ex19.m1.2.2.2.1.2" xref="S3.Ex19.m1.2.2.2.1.2.cmml">⁢</mo><mrow id="S3.Ex19.m1.2.2.2.1.1.1" xref="S3.Ex19.m1.2.2.2.1.1.1.1.cmml"><mo id="S3.Ex19.m1.2.2.2.1.1.1.2" stretchy="false" xref="S3.Ex19.m1.2.2.2.1.1.1.1.cmml">(</mo><msub id="S3.Ex19.m1.2.2.2.1.1.1.1" xref="S3.Ex19.m1.2.2.2.1.1.1.1.cmml"><mi id="S3.Ex19.m1.2.2.2.1.1.1.1.2" xref="S3.Ex19.m1.2.2.2.1.1.1.1.2.cmml">𝐱</mi><mi id="S3.Ex19.m1.2.2.2.1.1.1.1.3" xref="S3.Ex19.m1.2.2.2.1.1.1.1.3.cmml">k</mi></msub><mo id="S3.Ex19.m1.2.2.2.1.1.1.3" stretchy="false" xref="S3.Ex19.m1.2.2.2.1.1.1.1.cmml">)</mo></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Ex19.m1.2b"><apply id="S3.Ex19.m1.2.2.cmml" xref="S3.Ex19.m1.2.2"><minus id="S3.Ex19.m1.2.2.3.cmml" xref="S3.Ex19.m1.2.2.3"></minus><apply id="S3.Ex19.m1.1.1.1.cmml" xref="S3.Ex19.m1.1.1.1"><apply id="S3.Ex19.m1.1.1.1.2.cmml" xref="S3.Ex19.m1.1.1.1.2"><csymbol cd="ambiguous" id="S3.Ex19.m1.1.1.1.2.1.cmml" xref="S3.Ex19.m1.1.1.1.2">subscript</csymbol><csymbol cd="latexml" id="S3.Ex19.m1.1.1.1.2.2.cmml" xref="S3.Ex19.m1.1.1.1.2.2">limit-supremum</csymbol><apply id="S3.Ex19.m1.1.1.1.2.3.cmml" xref="S3.Ex19.m1.1.1.1.2.3"><ci id="S3.Ex19.m1.1.1.1.2.3.1.cmml" xref="S3.Ex19.m1.1.1.1.2.3.1">→</ci><ci id="S3.Ex19.m1.1.1.1.2.3.2.cmml" xref="S3.Ex19.m1.1.1.1.2.3.2">𝑘</ci><apply id="S3.Ex19.m1.1.1.1.2.3.3.cmml" xref="S3.Ex19.m1.1.1.1.2.3.3"><plus id="S3.Ex19.m1.1.1.1.2.3.3.1.cmml" xref="S3.Ex19.m1.1.1.1.2.3.3"></plus><infinity id="S3.Ex19.m1.1.1.1.2.3.3.2.cmml" xref="S3.Ex19.m1.1.1.1.2.3.3.2"></infinity></apply></apply></apply><apply id="S3.Ex19.m1.1.1.1.1.cmml" xref="S3.Ex19.m1.1.1.1.1"><times id="S3.Ex19.m1.1.1.1.1.2.cmml" xref="S3.Ex19.m1.1.1.1.1.2"></times><ci id="S3.Ex19.m1.1.1.1.1.3.cmml" xref="S3.Ex19.m1.1.1.1.1.3">𝒥</ci><apply id="S3.Ex19.m1.1.1.1.1.1.1.1.cmml" xref="S3.Ex19.m1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S3.Ex19.m1.1.1.1.1.1.1.1.1.cmml" xref="S3.Ex19.m1.1.1.1.1.1.1">subscript</csymbol><ci id="S3.Ex19.m1.1.1.1.1.1.1.1.2.cmml" xref="S3.Ex19.m1.1.1.1.1.1.1.1.2">𝐱</ci><ci id="S3.Ex19.m1.1.1.1.1.1.1.1.3.cmml" xref="S3.Ex19.m1.1.1.1.1.1.1.1.3">𝑘</ci></apply></apply></apply><apply id="S3.Ex19.m1.2.2.2.cmml" xref="S3.Ex19.m1.2.2.2"><apply id="S3.Ex19.m1.2.2.2.2.cmml" xref="S3.Ex19.m1.2.2.2.2"><csymbol cd="ambiguous" id="S3.Ex19.m1.2.2.2.2.1.cmml" xref="S3.Ex19.m1.2.2.2.2">subscript</csymbol><csymbol cd="latexml" id="S3.Ex19.m1.2.2.2.2.2.cmml" xref="S3.Ex19.m1.2.2.2.2.2">limit-infimum</csymbol><apply id="S3.Ex19.m1.2.2.2.2.3.cmml" xref="S3.Ex19.m1.2.2.2.2.3"><ci id="S3.Ex19.m1.2.2.2.2.3.1.cmml" xref="S3.Ex19.m1.2.2.2.2.3.1">→</ci><ci id="S3.Ex19.m1.2.2.2.2.3.2.cmml" xref="S3.Ex19.m1.2.2.2.2.3.2">𝑘</ci><apply id="S3.Ex19.m1.2.2.2.2.3.3.cmml" xref="S3.Ex19.m1.2.2.2.2.3.3"><plus id="S3.Ex19.m1.2.2.2.2.3.3.1.cmml" xref="S3.Ex19.m1.2.2.2.2.3.3"></plus><infinity id="S3.Ex19.m1.2.2.2.2.3.3.2.cmml" xref="S3.Ex19.m1.2.2.2.2.3.3.2"></infinity></apply></apply></apply><apply id="S3.Ex19.m1.2.2.2.1.cmml" xref="S3.Ex19.m1.2.2.2.1"><times id="S3.Ex19.m1.2.2.2.1.2.cmml" xref="S3.Ex19.m1.2.2.2.1.2"></times><ci id="S3.Ex19.m1.2.2.2.1.3.cmml" xref="S3.Ex19.m1.2.2.2.1.3">𝒥</ci><apply id="S3.Ex19.m1.2.2.2.1.1.1.1.cmml" xref="S3.Ex19.m1.2.2.2.1.1.1"><csymbol cd="ambiguous" id="S3.Ex19.m1.2.2.2.1.1.1.1.1.cmml" xref="S3.Ex19.m1.2.2.2.1.1.1">subscript</csymbol><ci id="S3.Ex19.m1.2.2.2.1.1.1.1.2.cmml" xref="S3.Ex19.m1.2.2.2.1.1.1.1.2">𝐱</ci><ci id="S3.Ex19.m1.2.2.2.1.1.1.1.3.cmml" xref="S3.Ex19.m1.2.2.2.1.1.1.1.3">𝑘</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Ex19.m1.2c">\displaystyle\limsup_{k\to+\infty}\mathcal{J}({\mathbf{x}}_{k})-\liminf_{k\to+% \infty}\mathcal{J}({\mathbf{x}}_{k})</annotation><annotation encoding="application/x-llamapun" id="S3.Ex19.m1.2d">lim sup start_POSTSUBSCRIPT italic_k → + ∞ end_POSTSUBSCRIPT caligraphic_J ( bold_x start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) - lim inf start_POSTSUBSCRIPT italic_k → + ∞ end_POSTSUBSCRIPT caligraphic_J ( bold_x start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT )</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\leq M_{\mathbf{K}}\eta," class="ltx_Math" display="inline" id="S3.Ex19.m2.1"><semantics id="S3.Ex19.m2.1a"><mrow id="S3.Ex19.m2.1.1.1" xref="S3.Ex19.m2.1.1.1.1.cmml"><mrow id="S3.Ex19.m2.1.1.1.1" xref="S3.Ex19.m2.1.1.1.1.cmml"><mi id="S3.Ex19.m2.1.1.1.1.2" xref="S3.Ex19.m2.1.1.1.1.2.cmml"></mi><mo id="S3.Ex19.m2.1.1.1.1.1" xref="S3.Ex19.m2.1.1.1.1.1.cmml">≤</mo><mrow id="S3.Ex19.m2.1.1.1.1.3" xref="S3.Ex19.m2.1.1.1.1.3.cmml"><msub id="S3.Ex19.m2.1.1.1.1.3.2" xref="S3.Ex19.m2.1.1.1.1.3.2.cmml"><mi id="S3.Ex19.m2.1.1.1.1.3.2.2" xref="S3.Ex19.m2.1.1.1.1.3.2.2.cmml">M</mi><mi id="S3.Ex19.m2.1.1.1.1.3.2.3" xref="S3.Ex19.m2.1.1.1.1.3.2.3.cmml">𝐊</mi></msub><mo id="S3.Ex19.m2.1.1.1.1.3.1" xref="S3.Ex19.m2.1.1.1.1.3.1.cmml">⁢</mo><mi id="S3.Ex19.m2.1.1.1.1.3.3" xref="S3.Ex19.m2.1.1.1.1.3.3.cmml">η</mi></mrow></mrow><mo id="S3.Ex19.m2.1.1.1.2" xref="S3.Ex19.m2.1.1.1.1.cmml">,</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.Ex19.m2.1b"><apply id="S3.Ex19.m2.1.1.1.1.cmml" xref="S3.Ex19.m2.1.1.1"><leq id="S3.Ex19.m2.1.1.1.1.1.cmml" xref="S3.Ex19.m2.1.1.1.1.1"></leq><csymbol cd="latexml" id="S3.Ex19.m2.1.1.1.1.2.cmml" xref="S3.Ex19.m2.1.1.1.1.2">absent</csymbol><apply id="S3.Ex19.m2.1.1.1.1.3.cmml" xref="S3.Ex19.m2.1.1.1.1.3"><times id="S3.Ex19.m2.1.1.1.1.3.1.cmml" xref="S3.Ex19.m2.1.1.1.1.3.1"></times><apply id="S3.Ex19.m2.1.1.1.1.3.2.cmml" xref="S3.Ex19.m2.1.1.1.1.3.2"><csymbol cd="ambiguous" id="S3.Ex19.m2.1.1.1.1.3.2.1.cmml" xref="S3.Ex19.m2.1.1.1.1.3.2">subscript</csymbol><ci id="S3.Ex19.m2.1.1.1.1.3.2.2.cmml" xref="S3.Ex19.m2.1.1.1.1.3.2.2">𝑀</ci><ci id="S3.Ex19.m2.1.1.1.1.3.2.3.cmml" xref="S3.Ex19.m2.1.1.1.1.3.2.3">𝐊</ci></apply><ci id="S3.Ex19.m2.1.1.1.1.3.3.cmml" xref="S3.Ex19.m2.1.1.1.1.3.3">𝜂</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Ex19.m2.1c">\displaystyle\leq M_{\mathbf{K}}\eta,</annotation><annotation encoding="application/x-llamapun" id="S3.Ex19.m2.1d">≤ italic_M start_POSTSUBSCRIPT bold_K end_POSTSUBSCRIPT italic_η ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S3.Thmproposition5.p1.5"><span class="ltx_text ltx_font_italic" id="S3.Thmproposition5.p1.5.1">with the bias <math alttext="\eta=\underset{k\to+\infty}{\limsup}{\|\mathbb{E}(\xi_{k})\|}\underset{\sigma% \to 0}{\leq}\frac{\alpha}{\sigma^{2}}M(R)+o(\sigma)" class="ltx_Math" display="inline" id="S3.Thmproposition5.p1.5.1.m1.3"><semantics id="S3.Thmproposition5.p1.5.1.m1.3a"><mrow id="S3.Thmproposition5.p1.5.1.m1.3.3" xref="S3.Thmproposition5.p1.5.1.m1.3.3.cmml"><mi id="S3.Thmproposition5.p1.5.1.m1.3.3.3" xref="S3.Thmproposition5.p1.5.1.m1.3.3.3.cmml">η</mi><mo id="S3.Thmproposition5.p1.5.1.m1.3.3.2" mathvariant="normal" rspace="0.1389em" xref="S3.Thmproposition5.p1.5.1.m1.3.3.2.cmml">=</mo><mrow id="S3.Thmproposition5.p1.5.1.m1.3.3.1" xref="S3.Thmproposition5.p1.5.1.m1.3.3.1.cmml"><mrow id="S3.Thmproposition5.p1.5.1.m1.3.3.1.1" 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xref="S3.Thmproposition5.p1.5.1.m1.3.3.1.1.6">𝑀</ci><ci id="S3.Thmproposition5.p1.5.1.m1.1.1.cmml" xref="S3.Thmproposition5.p1.5.1.m1.1.1">𝑅</ci></apply><apply id="S3.Thmproposition5.p1.5.1.m1.3.3.1.3.cmml" xref="S3.Thmproposition5.p1.5.1.m1.3.3.1.3"><times id="S3.Thmproposition5.p1.5.1.m1.3.3.1.3.1.cmml" xref="S3.Thmproposition5.p1.5.1.m1.3.3.1.3.1"></times><ci id="S3.Thmproposition5.p1.5.1.m1.3.3.1.3.2.cmml" xref="S3.Thmproposition5.p1.5.1.m1.3.3.1.3.2">𝑜</ci><ci id="S3.Thmproposition5.p1.5.1.m1.2.2.cmml" xref="S3.Thmproposition5.p1.5.1.m1.2.2">𝜎</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmproposition5.p1.5.1.m1.3c">\eta=\underset{k\to+\infty}{\limsup}{\|\mathbb{E}(\xi_{k})\|}\underset{\sigma% \to 0}{\leq}\frac{\alpha}{\sigma^{2}}M(R)+o(\sigma)</annotation><annotation encoding="application/x-llamapun" id="S3.Thmproposition5.p1.5.1.m1.3d">italic_η = start_UNDERACCENT italic_k → + ∞ end_UNDERACCENT start_ARG lim sup end_ARG ∥ blackboard_E ( italic_ξ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) ∥ start_UNDERACCENT italic_σ → 0 end_UNDERACCENT start_ARG ≤ end_ARG divide start_ARG italic_α end_ARG start_ARG italic_σ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG italic_M ( italic_R ) + italic_o ( italic_σ )</annotation></semantics></math>.</span></p> </div> </div> <div class="ltx_para" id="S3.SS3.p4"> <p class="ltx_p" id="S3.SS3.p4.3">The denoiser bias <math alttext="\eta" class="ltx_Math" display="inline" id="S3.SS3.p4.1.m1.1"><semantics id="S3.SS3.p4.1.m1.1a"><mi id="S3.SS3.p4.1.m1.1.1" xref="S3.SS3.p4.1.m1.1.1.cmml">η</mi><annotation-xml encoding="MathML-Content" id="S3.SS3.p4.1.m1.1b"><ci id="S3.SS3.p4.1.m1.1.1.cmml" xref="S3.SS3.p4.1.m1.1.1">𝜂</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p4.1.m1.1c">\eta</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p4.1.m1.1d">italic_η</annotation></semantics></math> has a similar bound than in <cite class="ltx_cite ltx_citemacro_citep">(Laumont et al., <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#bib.bib37" title="">2022b</a>, Proposition 3)</cite>. If the denoiser is well trained, <math alttext="M(R)\approx 0" class="ltx_Math" display="inline" id="S3.SS3.p4.2.m2.1"><semantics id="S3.SS3.p4.2.m2.1a"><mrow id="S3.SS3.p4.2.m2.1.2" xref="S3.SS3.p4.2.m2.1.2.cmml"><mrow id="S3.SS3.p4.2.m2.1.2.2" xref="S3.SS3.p4.2.m2.1.2.2.cmml"><mi id="S3.SS3.p4.2.m2.1.2.2.2" xref="S3.SS3.p4.2.m2.1.2.2.2.cmml">M</mi><mo id="S3.SS3.p4.2.m2.1.2.2.1" xref="S3.SS3.p4.2.m2.1.2.2.1.cmml">⁢</mo><mrow id="S3.SS3.p4.2.m2.1.2.2.3.2" xref="S3.SS3.p4.2.m2.1.2.2.cmml"><mo id="S3.SS3.p4.2.m2.1.2.2.3.2.1" stretchy="false" xref="S3.SS3.p4.2.m2.1.2.2.cmml">(</mo><mi id="S3.SS3.p4.2.m2.1.1" xref="S3.SS3.p4.2.m2.1.1.cmml">R</mi><mo id="S3.SS3.p4.2.m2.1.2.2.3.2.2" stretchy="false" xref="S3.SS3.p4.2.m2.1.2.2.cmml">)</mo></mrow></mrow><mo id="S3.SS3.p4.2.m2.1.2.1" xref="S3.SS3.p4.2.m2.1.2.1.cmml">≈</mo><mn id="S3.SS3.p4.2.m2.1.2.3" xref="S3.SS3.p4.2.m2.1.2.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.SS3.p4.2.m2.1b"><apply id="S3.SS3.p4.2.m2.1.2.cmml" xref="S3.SS3.p4.2.m2.1.2"><approx id="S3.SS3.p4.2.m2.1.2.1.cmml" xref="S3.SS3.p4.2.m2.1.2.1"></approx><apply id="S3.SS3.p4.2.m2.1.2.2.cmml" xref="S3.SS3.p4.2.m2.1.2.2"><times id="S3.SS3.p4.2.m2.1.2.2.1.cmml" xref="S3.SS3.p4.2.m2.1.2.2.1"></times><ci id="S3.SS3.p4.2.m2.1.2.2.2.cmml" xref="S3.SS3.p4.2.m2.1.2.2.2">𝑀</ci><ci id="S3.SS3.p4.2.m2.1.1.cmml" xref="S3.SS3.p4.2.m2.1.1">𝑅</ci></apply><cn id="S3.SS3.p4.2.m2.1.2.3.cmml" type="integer" xref="S3.SS3.p4.2.m2.1.2.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p4.2.m2.1c">M(R)\approx 0</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p4.2.m2.1d">italic_M ( italic_R ) ≈ 0</annotation></semantics></math>, i.e. the denoiser bias is small. Proposition <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#S3.Thmproposition5" title="Proposition 3.5. ‣ 3.3 Biased algorithm analysis ‣ 3 Convergence Analysis ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_tag">3.5</span></a> proves that the smaller is the denoiser bias, the closer the sequence <math alttext="(x_{k})_{k\in\mathbf{N}}" class="ltx_Math" display="inline" id="S3.SS3.p4.3.m3.1"><semantics id="S3.SS3.p4.3.m3.1a"><msub id="S3.SS3.p4.3.m3.1.1" xref="S3.SS3.p4.3.m3.1.1.cmml"><mrow id="S3.SS3.p4.3.m3.1.1.1.1" xref="S3.SS3.p4.3.m3.1.1.1.1.1.cmml"><mo id="S3.SS3.p4.3.m3.1.1.1.1.2" stretchy="false" xref="S3.SS3.p4.3.m3.1.1.1.1.1.cmml">(</mo><msub id="S3.SS3.p4.3.m3.1.1.1.1.1" xref="S3.SS3.p4.3.m3.1.1.1.1.1.cmml"><mi id="S3.SS3.p4.3.m3.1.1.1.1.1.2" xref="S3.SS3.p4.3.m3.1.1.1.1.1.2.cmml">x</mi><mi id="S3.SS3.p4.3.m3.1.1.1.1.1.3" xref="S3.SS3.p4.3.m3.1.1.1.1.1.3.cmml">k</mi></msub><mo id="S3.SS3.p4.3.m3.1.1.1.1.3" stretchy="false" xref="S3.SS3.p4.3.m3.1.1.1.1.1.cmml">)</mo></mrow><mrow id="S3.SS3.p4.3.m3.1.1.3" xref="S3.SS3.p4.3.m3.1.1.3.cmml"><mi id="S3.SS3.p4.3.m3.1.1.3.2" xref="S3.SS3.p4.3.m3.1.1.3.2.cmml">k</mi><mo id="S3.SS3.p4.3.m3.1.1.3.1" xref="S3.SS3.p4.3.m3.1.1.3.1.cmml">∈</mo><mi id="S3.SS3.p4.3.m3.1.1.3.3" xref="S3.SS3.p4.3.m3.1.1.3.3.cmml">𝐍</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="S3.SS3.p4.3.m3.1b"><apply id="S3.SS3.p4.3.m3.1.1.cmml" xref="S3.SS3.p4.3.m3.1.1"><csymbol cd="ambiguous" id="S3.SS3.p4.3.m3.1.1.2.cmml" xref="S3.SS3.p4.3.m3.1.1">subscript</csymbol><apply id="S3.SS3.p4.3.m3.1.1.1.1.1.cmml" xref="S3.SS3.p4.3.m3.1.1.1.1"><csymbol cd="ambiguous" id="S3.SS3.p4.3.m3.1.1.1.1.1.1.cmml" xref="S3.SS3.p4.3.m3.1.1.1.1">subscript</csymbol><ci id="S3.SS3.p4.3.m3.1.1.1.1.1.2.cmml" xref="S3.SS3.p4.3.m3.1.1.1.1.1.2">𝑥</ci><ci id="S3.SS3.p4.3.m3.1.1.1.1.1.3.cmml" xref="S3.SS3.p4.3.m3.1.1.1.1.1.3">𝑘</ci></apply><apply id="S3.SS3.p4.3.m3.1.1.3.cmml" xref="S3.SS3.p4.3.m3.1.1.3"><in id="S3.SS3.p4.3.m3.1.1.3.1.cmml" xref="S3.SS3.p4.3.m3.1.1.3.1"></in><ci id="S3.SS3.p4.3.m3.1.1.3.2.cmml" xref="S3.SS3.p4.3.m3.1.1.3.2">𝑘</ci><ci id="S3.SS3.p4.3.m3.1.1.3.3.cmml" xref="S3.SS3.p4.3.m3.1.1.3.3">𝐍</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p4.3.m3.1c">(x_{k})_{k\in\mathbf{N}}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p4.3.m3.1d">( italic_x start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) start_POSTSUBSCRIPT italic_k ∈ bold_N end_POSTSUBSCRIPT</annotation></semantics></math> is to critical points of Problem (<a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#S2.E9" title="9 ‣ Optimization algorithms ‣ 2.2 SNORE regularization ‣ 2 Stochastic deNOising REgularization (SNORE) ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_tag">9</span></a>) (in terms of gradient norm). This statement generalizes the convergence result of Proposition <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#S3.Thmproposition3" title="Proposition 3.3. ‣ 3.2 Unbiased algorithm analysis ‣ 3 Convergence Analysis ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_tag">3.3</span></a> to the biased case.</p> </div> <figure class="ltx_figure" id="S3.F1"> <div class="ltx_flex_figure"> <div class="ltx_flex_cell ltx_flex_size_1ltx_flex_size_many"> <figure class="ltx_figure ltx_flex_size_10 ltx_flex_size_many ltx_align_center" id="S3.F1.1"><img alt="Refer to caption" class="ltx_graphics ltx_img_square" height="598" id="S3.F1.1.g1" src="extracted/5518834/images/kernel_0.png" width="598"/> </figure> </div> <div class="ltx_flex_cell ltx_flex_size_1ltx_flex_size_many"> <figure class="ltx_figure ltx_flex_size_10 ltx_flex_size_many ltx_align_center" id="S3.F1.2"><img alt="Refer to caption" class="ltx_graphics ltx_img_square" height="598" id="S3.F1.2.g1" src="extracted/5518834/images/kernel_1.png" width="598"/> </figure> </div> <div class="ltx_flex_cell ltx_flex_size_1ltx_flex_size_many"> <figure class="ltx_figure ltx_flex_size_10 ltx_flex_size_many ltx_align_center" id="S3.F1.3"><img alt="Refer to caption" class="ltx_graphics ltx_img_square" height="598" id="S3.F1.3.g1" src="extracted/5518834/images/kernel_2.png" width="598"/> </figure> </div> <div class="ltx_flex_cell ltx_flex_size_1ltx_flex_size_many"> <figure class="ltx_figure ltx_flex_size_10 ltx_flex_size_many ltx_align_center" id="S3.F1.4"><img alt="Refer to caption" class="ltx_graphics ltx_img_square" height="598" id="S3.F1.4.g1" src="extracted/5518834/images/kernel_3.png" width="598"/> </figure> </div> <div class="ltx_flex_cell ltx_flex_size_1ltx_flex_size_many"> <figure class="ltx_figure ltx_flex_size_10 ltx_flex_size_many ltx_align_center" id="S3.F1.5"><img alt="Refer to caption" class="ltx_graphics ltx_img_square" height="598" id="S3.F1.5.g1" src="extracted/5518834/images/kernel_4.png" width="598"/> </figure> </div> <div class="ltx_flex_cell ltx_flex_size_1ltx_flex_size_many"> <figure class="ltx_figure ltx_flex_size_10 ltx_flex_size_many ltx_align_center" id="S3.F1.6"><img alt="Refer to caption" class="ltx_graphics ltx_img_square" height="598" id="S3.F1.6.g1" src="extracted/5518834/images/kernel_5.png" width="598"/> </figure> </div> <div class="ltx_flex_cell ltx_flex_size_1ltx_flex_size_many"> <figure class="ltx_figure ltx_flex_size_10 ltx_flex_size_many ltx_align_center" id="S3.F1.7"><img alt="Refer to caption" class="ltx_graphics ltx_img_square" height="598" id="S3.F1.7.g1" src="extracted/5518834/images/kernel_6.png" width="598"/> </figure> </div> <div class="ltx_flex_cell ltx_flex_size_1ltx_flex_size_many"> <figure class="ltx_figure ltx_flex_size_10 ltx_flex_size_many ltx_align_center" id="S3.F1.8"><img alt="Refer to caption" class="ltx_graphics ltx_img_square" height="598" id="S3.F1.8.g1" src="extracted/5518834/images/kernel_7.png" width="598"/> </figure> </div> <div class="ltx_flex_cell ltx_flex_size_1ltx_flex_size_many"> <figure class="ltx_figure ltx_flex_size_10 ltx_flex_size_many ltx_align_center" id="S3.F1.9"><img alt="Refer to caption" class="ltx_graphics ltx_img_square" height="598" id="S3.F1.9.g1" src="extracted/5518834/images/kernel_8.png" width="598"/> </figure> </div> <div class="ltx_flex_cell ltx_flex_size_1ltx_flex_size_many"> <figure class="ltx_figure ltx_flex_size_10 ltx_flex_size_many ltx_align_center" id="S3.F1.10"><img alt="Refer to caption" class="ltx_graphics ltx_img_square" height="598" id="S3.F1.10.g1" src="extracted/5518834/images/kernel_9.png" width="598"/> </figure> </div> </div> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure"><span class="ltx_text" id="S3.F1.16.3.1" style="font-size:90%;">Figure 1</span>: </span><span class="ltx_text" id="S3.F1.14.2" style="font-size:90%;">Kernels used for deblurring. As in <cite class="ltx_cite ltx_citemacro_citep">(Zhang et al., <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#bib.bib77" title="">2017</a>, <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#bib.bib78" title="">2021</a>; Pesquet et al., <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#bib.bib53" title="">2021</a>; Hurault et al., <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#bib.bib30" title="">2022a</a>)</cite> we test the different methods on 8 real-world camera shake kernels proposed in <cite class="ltx_cite ltx_citemacro_citep">(Levin et al., <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#bib.bib39" title="">2009</a>)</cite> and on the uniform <math alttext="9\times 9" class="ltx_Math" display="inline" id="S3.F1.13.1.m1.1"><semantics id="S3.F1.13.1.m1.1b"><mrow id="S3.F1.13.1.m1.1.1" xref="S3.F1.13.1.m1.1.1.cmml"><mn id="S3.F1.13.1.m1.1.1.2" xref="S3.F1.13.1.m1.1.1.2.cmml">9</mn><mo id="S3.F1.13.1.m1.1.1.1" lspace="0.222em" rspace="0.222em" xref="S3.F1.13.1.m1.1.1.1.cmml">×</mo><mn id="S3.F1.13.1.m1.1.1.3" xref="S3.F1.13.1.m1.1.1.3.cmml">9</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.F1.13.1.m1.1c"><apply id="S3.F1.13.1.m1.1.1.cmml" xref="S3.F1.13.1.m1.1.1"><times id="S3.F1.13.1.m1.1.1.1.cmml" xref="S3.F1.13.1.m1.1.1.1"></times><cn id="S3.F1.13.1.m1.1.1.2.cmml" type="integer" xref="S3.F1.13.1.m1.1.1.2">9</cn><cn id="S3.F1.13.1.m1.1.1.3.cmml" type="integer" xref="S3.F1.13.1.m1.1.1.3">9</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.F1.13.1.m1.1d">9\times 9</annotation><annotation encoding="application/x-llamapun" id="S3.F1.13.1.m1.1e">9 × 9</annotation></semantics></math> kernel and the <math alttext="25\times 25" class="ltx_Math" display="inline" id="S3.F1.14.2.m2.1"><semantics id="S3.F1.14.2.m2.1b"><mrow id="S3.F1.14.2.m2.1.1" xref="S3.F1.14.2.m2.1.1.cmml"><mn id="S3.F1.14.2.m2.1.1.2" xref="S3.F1.14.2.m2.1.1.2.cmml">25</mn><mo id="S3.F1.14.2.m2.1.1.1" lspace="0.222em" rspace="0.222em" xref="S3.F1.14.2.m2.1.1.1.cmml">×</mo><mn id="S3.F1.14.2.m2.1.1.3" xref="S3.F1.14.2.m2.1.1.3.cmml">25</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.F1.14.2.m2.1c"><apply id="S3.F1.14.2.m2.1.1.cmml" xref="S3.F1.14.2.m2.1.1"><times id="S3.F1.14.2.m2.1.1.1.cmml" xref="S3.F1.14.2.m2.1.1.1"></times><cn id="S3.F1.14.2.m2.1.1.2.cmml" type="integer" xref="S3.F1.14.2.m2.1.1.2">25</cn><cn id="S3.F1.14.2.m2.1.1.3.cmml" type="integer" xref="S3.F1.14.2.m2.1.1.3">25</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.F1.14.2.m2.1d">25\times 25</annotation><annotation encoding="application/x-llamapun" id="S3.F1.14.2.m2.1e">25 × 25</annotation></semantics></math> Gaussian kernel with standard deviation 1.6 proposed in <cite class="ltx_cite ltx_citemacro_citep">(Romano et al., <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#bib.bib58" title="">2017</a>)</cite>.</span></figcaption> </figure> <figure class="ltx_figure" id="S3.F2"><img alt="Refer to caption" class="ltx_graphics ltx_centering ltx_img_landscape" height="209" id="S3.F2.g1" src="extracted/5518834/images/Results_restoration_deblurring_all_methods.png" width="598"/> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure"><span class="ltx_text" id="S3.F2.9.4.1" style="font-size:90%;">Figure 2</span>: </span><span class="ltx_text" id="S3.F2.6.3" style="font-size:90%;">Deblurring with various methods of a motion blur kernel with input noise level <math alttext="\sigma_{{\mathbf{y}}}=10/255" class="ltx_Math" display="inline" id="S3.F2.4.1.m1.1"><semantics id="S3.F2.4.1.m1.1b"><mrow id="S3.F2.4.1.m1.1.1" xref="S3.F2.4.1.m1.1.1.cmml"><msub id="S3.F2.4.1.m1.1.1.2" xref="S3.F2.4.1.m1.1.1.2.cmml"><mi id="S3.F2.4.1.m1.1.1.2.2" xref="S3.F2.4.1.m1.1.1.2.2.cmml">σ</mi><mi id="S3.F2.4.1.m1.1.1.2.3" xref="S3.F2.4.1.m1.1.1.2.3.cmml">𝐲</mi></msub><mo id="S3.F2.4.1.m1.1.1.1" xref="S3.F2.4.1.m1.1.1.1.cmml">=</mo><mrow id="S3.F2.4.1.m1.1.1.3" xref="S3.F2.4.1.m1.1.1.3.cmml"><mn id="S3.F2.4.1.m1.1.1.3.2" xref="S3.F2.4.1.m1.1.1.3.2.cmml">10</mn><mo id="S3.F2.4.1.m1.1.1.3.1" xref="S3.F2.4.1.m1.1.1.3.1.cmml">/</mo><mn id="S3.F2.4.1.m1.1.1.3.3" xref="S3.F2.4.1.m1.1.1.3.3.cmml">255</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.F2.4.1.m1.1c"><apply id="S3.F2.4.1.m1.1.1.cmml" xref="S3.F2.4.1.m1.1.1"><eq id="S3.F2.4.1.m1.1.1.1.cmml" xref="S3.F2.4.1.m1.1.1.1"></eq><apply id="S3.F2.4.1.m1.1.1.2.cmml" xref="S3.F2.4.1.m1.1.1.2"><csymbol cd="ambiguous" id="S3.F2.4.1.m1.1.1.2.1.cmml" xref="S3.F2.4.1.m1.1.1.2">subscript</csymbol><ci id="S3.F2.4.1.m1.1.1.2.2.cmml" xref="S3.F2.4.1.m1.1.1.2.2">𝜎</ci><ci id="S3.F2.4.1.m1.1.1.2.3.cmml" xref="S3.F2.4.1.m1.1.1.2.3">𝐲</ci></apply><apply id="S3.F2.4.1.m1.1.1.3.cmml" xref="S3.F2.4.1.m1.1.1.3"><divide id="S3.F2.4.1.m1.1.1.3.1.cmml" xref="S3.F2.4.1.m1.1.1.3.1"></divide><cn id="S3.F2.4.1.m1.1.1.3.2.cmml" type="integer" xref="S3.F2.4.1.m1.1.1.3.2">10</cn><cn id="S3.F2.4.1.m1.1.1.3.3.cmml" type="integer" xref="S3.F2.4.1.m1.1.1.3.3">255</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.F2.4.1.m1.1d">\sigma_{{\mathbf{y}}}=10/255</annotation><annotation encoding="application/x-llamapun" id="S3.F2.4.1.m1.1e">italic_σ start_POSTSUBSCRIPT bold_y end_POSTSUBSCRIPT = 10 / 255</annotation></semantics></math> with a GS-denoiser trained on natural images. Note that Ann-SNORE produces a better perceptual reconstruction (BRISQUE). <span class="ltx_text ltx_font_italic" id="S3.F2.6.3.1">Bottom-Right:</span> Decrease of the optimized function <math alttext="\mathcal{J}" class="ltx_Math" display="inline" id="S3.F2.5.2.m2.1"><semantics id="S3.F2.5.2.m2.1b"><mi class="ltx_font_mathcaligraphic" id="S3.F2.5.2.m2.1.1" xref="S3.F2.5.2.m2.1.1.cmml">𝒥</mi><annotation-xml encoding="MathML-Content" id="S3.F2.5.2.m2.1c"><ci id="S3.F2.5.2.m2.1.1.cmml" xref="S3.F2.5.2.m2.1.1">𝒥</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.F2.5.2.m2.1d">\mathcal{J}</annotation><annotation encoding="application/x-llamapun" id="S3.F2.5.2.m2.1e">caligraphic_J</annotation></semantics></math> (Equation (<a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#S4.E16" title="16 ‣ 4 Experiments ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_tag">16</span></a>)) along the stochastic gradient descent with the last parameters <math alttext="(\alpha_{m-1},\sigma_{m-1})" class="ltx_Math" display="inline" id="S3.F2.6.3.m3.2"><semantics id="S3.F2.6.3.m3.2b"><mrow id="S3.F2.6.3.m3.2.2.2" xref="S3.F2.6.3.m3.2.2.3.cmml"><mo id="S3.F2.6.3.m3.2.2.2.3" stretchy="false" xref="S3.F2.6.3.m3.2.2.3.cmml">(</mo><msub id="S3.F2.6.3.m3.1.1.1.1" xref="S3.F2.6.3.m3.1.1.1.1.cmml"><mi id="S3.F2.6.3.m3.1.1.1.1.2" xref="S3.F2.6.3.m3.1.1.1.1.2.cmml">α</mi><mrow id="S3.F2.6.3.m3.1.1.1.1.3" xref="S3.F2.6.3.m3.1.1.1.1.3.cmml"><mi id="S3.F2.6.3.m3.1.1.1.1.3.2" xref="S3.F2.6.3.m3.1.1.1.1.3.2.cmml">m</mi><mo id="S3.F2.6.3.m3.1.1.1.1.3.1" xref="S3.F2.6.3.m3.1.1.1.1.3.1.cmml">−</mo><mn id="S3.F2.6.3.m3.1.1.1.1.3.3" xref="S3.F2.6.3.m3.1.1.1.1.3.3.cmml">1</mn></mrow></msub><mo id="S3.F2.6.3.m3.2.2.2.4" xref="S3.F2.6.3.m3.2.2.3.cmml">,</mo><msub id="S3.F2.6.3.m3.2.2.2.2" xref="S3.F2.6.3.m3.2.2.2.2.cmml"><mi id="S3.F2.6.3.m3.2.2.2.2.2" xref="S3.F2.6.3.m3.2.2.2.2.2.cmml">σ</mi><mrow id="S3.F2.6.3.m3.2.2.2.2.3" xref="S3.F2.6.3.m3.2.2.2.2.3.cmml"><mi id="S3.F2.6.3.m3.2.2.2.2.3.2" xref="S3.F2.6.3.m3.2.2.2.2.3.2.cmml">m</mi><mo id="S3.F2.6.3.m3.2.2.2.2.3.1" xref="S3.F2.6.3.m3.2.2.2.2.3.1.cmml">−</mo><mn id="S3.F2.6.3.m3.2.2.2.2.3.3" xref="S3.F2.6.3.m3.2.2.2.2.3.3.cmml">1</mn></mrow></msub><mo id="S3.F2.6.3.m3.2.2.2.5" stretchy="false" xref="S3.F2.6.3.m3.2.2.3.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.F2.6.3.m3.2c"><interval closure="open" id="S3.F2.6.3.m3.2.2.3.cmml" xref="S3.F2.6.3.m3.2.2.2"><apply id="S3.F2.6.3.m3.1.1.1.1.cmml" xref="S3.F2.6.3.m3.1.1.1.1"><csymbol cd="ambiguous" id="S3.F2.6.3.m3.1.1.1.1.1.cmml" xref="S3.F2.6.3.m3.1.1.1.1">subscript</csymbol><ci id="S3.F2.6.3.m3.1.1.1.1.2.cmml" xref="S3.F2.6.3.m3.1.1.1.1.2">𝛼</ci><apply id="S3.F2.6.3.m3.1.1.1.1.3.cmml" xref="S3.F2.6.3.m3.1.1.1.1.3"><minus id="S3.F2.6.3.m3.1.1.1.1.3.1.cmml" xref="S3.F2.6.3.m3.1.1.1.1.3.1"></minus><ci id="S3.F2.6.3.m3.1.1.1.1.3.2.cmml" xref="S3.F2.6.3.m3.1.1.1.1.3.2">𝑚</ci><cn id="S3.F2.6.3.m3.1.1.1.1.3.3.cmml" type="integer" xref="S3.F2.6.3.m3.1.1.1.1.3.3">1</cn></apply></apply><apply id="S3.F2.6.3.m3.2.2.2.2.cmml" xref="S3.F2.6.3.m3.2.2.2.2"><csymbol cd="ambiguous" id="S3.F2.6.3.m3.2.2.2.2.1.cmml" xref="S3.F2.6.3.m3.2.2.2.2">subscript</csymbol><ci id="S3.F2.6.3.m3.2.2.2.2.2.cmml" xref="S3.F2.6.3.m3.2.2.2.2.2">𝜎</ci><apply id="S3.F2.6.3.m3.2.2.2.2.3.cmml" xref="S3.F2.6.3.m3.2.2.2.2.3"><minus id="S3.F2.6.3.m3.2.2.2.2.3.1.cmml" xref="S3.F2.6.3.m3.2.2.2.2.3.1"></minus><ci id="S3.F2.6.3.m3.2.2.2.2.3.2.cmml" xref="S3.F2.6.3.m3.2.2.2.2.3.2">𝑚</ci><cn id="S3.F2.6.3.m3.2.2.2.2.3.3.cmml" type="integer" xref="S3.F2.6.3.m3.2.2.2.2.3.3">1</cn></apply></apply></interval></annotation-xml><annotation encoding="application/x-tex" id="S3.F2.6.3.m3.2d">(\alpha_{m-1},\sigma_{m-1})</annotation><annotation encoding="application/x-llamapun" id="S3.F2.6.3.m3.2e">( italic_α start_POSTSUBSCRIPT italic_m - 1 end_POSTSUBSCRIPT , italic_σ start_POSTSUBSCRIPT italic_m - 1 end_POSTSUBSCRIPT )</annotation></semantics></math> of the annealing procedure. </span></figcaption> </figure> </section> </section> <section class="ltx_section" id="S4"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">4 </span>Experiments</h2> <div class="ltx_para" id="S4.p1"> <p class="ltx_p" id="S4.p1.1">In this section, we show the performance of Annealing SNORE (Ann-SNORE) algorithm for image inverse problems, including deblurring and inpainting. Ann-SNORE is compared to several state-of-the-art image restoration methods. In Appendix <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#A6" title="Appendix F Additional Experiments ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_tag">F</span></a> we give more details on our experiments and other results on various images and various inverse problems including deblurring, inpainting, super-resolution and despeckling. A study of Ann-SNORE sensitivity to its parameters and the randomness of the algorithm is also provided.</p> </div> <div class="ltx_para" id="S4.p2"> <p class="ltx_p" id="S4.p2.3">In our experiment, we use a Gradient-Step denoiser <cite class="ltx_cite ltx_citemacro_citep">(Hurault et al., <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#bib.bib30" title="">2022a</a>)</cite> of the form</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A8.EGx15"> <tbody id="S4.E15"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle D_{\sigma}={{\operatorname{Id}}}-\nabla g_{\sigma}," class="ltx_Math" display="inline" id="S4.E15.m1.1"><semantics id="S4.E15.m1.1a"><mrow id="S4.E15.m1.1.1.1" xref="S4.E15.m1.1.1.1.1.cmml"><mrow id="S4.E15.m1.1.1.1.1" xref="S4.E15.m1.1.1.1.1.cmml"><msub id="S4.E15.m1.1.1.1.1.2" xref="S4.E15.m1.1.1.1.1.2.cmml"><mi id="S4.E15.m1.1.1.1.1.2.2" xref="S4.E15.m1.1.1.1.1.2.2.cmml">D</mi><mi id="S4.E15.m1.1.1.1.1.2.3" xref="S4.E15.m1.1.1.1.1.2.3.cmml">σ</mi></msub><mo id="S4.E15.m1.1.1.1.1.1" xref="S4.E15.m1.1.1.1.1.1.cmml">=</mo><mrow id="S4.E15.m1.1.1.1.1.3" xref="S4.E15.m1.1.1.1.1.3.cmml"><mi id="S4.E15.m1.1.1.1.1.3.2" xref="S4.E15.m1.1.1.1.1.3.2.cmml">Id</mi><mo id="S4.E15.m1.1.1.1.1.3.1" lspace="0em" xref="S4.E15.m1.1.1.1.1.3.1.cmml">−</mo><mrow id="S4.E15.m1.1.1.1.1.3.3" xref="S4.E15.m1.1.1.1.1.3.3.cmml"><mo id="S4.E15.m1.1.1.1.1.3.3.1" rspace="0.167em" xref="S4.E15.m1.1.1.1.1.3.3.1.cmml">∇</mo><msub id="S4.E15.m1.1.1.1.1.3.3.2" xref="S4.E15.m1.1.1.1.1.3.3.2.cmml"><mi id="S4.E15.m1.1.1.1.1.3.3.2.2" xref="S4.E15.m1.1.1.1.1.3.3.2.2.cmml">g</mi><mi id="S4.E15.m1.1.1.1.1.3.3.2.3" xref="S4.E15.m1.1.1.1.1.3.3.2.3.cmml">σ</mi></msub></mrow></mrow></mrow><mo id="S4.E15.m1.1.1.1.2" xref="S4.E15.m1.1.1.1.1.cmml">,</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.E15.m1.1b"><apply id="S4.E15.m1.1.1.1.1.cmml" xref="S4.E15.m1.1.1.1"><eq id="S4.E15.m1.1.1.1.1.1.cmml" xref="S4.E15.m1.1.1.1.1.1"></eq><apply id="S4.E15.m1.1.1.1.1.2.cmml" xref="S4.E15.m1.1.1.1.1.2"><csymbol cd="ambiguous" id="S4.E15.m1.1.1.1.1.2.1.cmml" xref="S4.E15.m1.1.1.1.1.2">subscript</csymbol><ci id="S4.E15.m1.1.1.1.1.2.2.cmml" xref="S4.E15.m1.1.1.1.1.2.2">𝐷</ci><ci id="S4.E15.m1.1.1.1.1.2.3.cmml" xref="S4.E15.m1.1.1.1.1.2.3">𝜎</ci></apply><apply id="S4.E15.m1.1.1.1.1.3.cmml" xref="S4.E15.m1.1.1.1.1.3"><minus id="S4.E15.m1.1.1.1.1.3.1.cmml" xref="S4.E15.m1.1.1.1.1.3.1"></minus><ci id="S4.E15.m1.1.1.1.1.3.2.cmml" xref="S4.E15.m1.1.1.1.1.3.2">Id</ci><apply id="S4.E15.m1.1.1.1.1.3.3.cmml" xref="S4.E15.m1.1.1.1.1.3.3"><ci id="S4.E15.m1.1.1.1.1.3.3.1.cmml" xref="S4.E15.m1.1.1.1.1.3.3.1">∇</ci><apply id="S4.E15.m1.1.1.1.1.3.3.2.cmml" xref="S4.E15.m1.1.1.1.1.3.3.2"><csymbol cd="ambiguous" id="S4.E15.m1.1.1.1.1.3.3.2.1.cmml" xref="S4.E15.m1.1.1.1.1.3.3.2">subscript</csymbol><ci id="S4.E15.m1.1.1.1.1.3.3.2.2.cmml" xref="S4.E15.m1.1.1.1.1.3.3.2.2">𝑔</ci><ci id="S4.E15.m1.1.1.1.1.3.3.2.3.cmml" xref="S4.E15.m1.1.1.1.1.3.3.2.3">𝜎</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.E15.m1.1c">\displaystyle D_{\sigma}={{\operatorname{Id}}}-\nabla g_{\sigma},</annotation><annotation encoding="application/x-llamapun" id="S4.E15.m1.1d">italic_D start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT = roman_Id - ∇ italic_g start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(15)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S4.p2.2">where <math alttext="g_{\sigma}" class="ltx_Math" display="inline" id="S4.p2.1.m1.1"><semantics id="S4.p2.1.m1.1a"><msub id="S4.p2.1.m1.1.1" xref="S4.p2.1.m1.1.1.cmml"><mi id="S4.p2.1.m1.1.1.2" xref="S4.p2.1.m1.1.1.2.cmml">g</mi><mi id="S4.p2.1.m1.1.1.3" xref="S4.p2.1.m1.1.1.3.cmml">σ</mi></msub><annotation-xml encoding="MathML-Content" id="S4.p2.1.m1.1b"><apply id="S4.p2.1.m1.1.1.cmml" xref="S4.p2.1.m1.1.1"><csymbol cd="ambiguous" id="S4.p2.1.m1.1.1.1.cmml" xref="S4.p2.1.m1.1.1">subscript</csymbol><ci id="S4.p2.1.m1.1.1.2.cmml" xref="S4.p2.1.m1.1.1.2">𝑔</ci><ci id="S4.p2.1.m1.1.1.3.cmml" xref="S4.p2.1.m1.1.1.3">𝜎</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p2.1.m1.1c">g_{\sigma}</annotation><annotation encoding="application/x-llamapun" id="S4.p2.1.m1.1d">italic_g start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT</annotation></semantics></math> is a learned neural network. With this gradient-step denoiser, <cite class="ltx_cite ltx_citemacro_citet">Hurault et al. (<a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#bib.bib30" title="">2022a</a>)</cite> demonstrated that RED converges to a critical point of an explicit objective function of the form <math alttext="\mathcal{F}({\mathbf{x}},{\mathbf{y}})+\alpha g_{\sigma}\left({\mathbf{x}}\right)" class="ltx_Math" display="inline" id="S4.p2.2.m2.3"><semantics id="S4.p2.2.m2.3a"><mrow id="S4.p2.2.m2.3.4" xref="S4.p2.2.m2.3.4.cmml"><mrow id="S4.p2.2.m2.3.4.2" xref="S4.p2.2.m2.3.4.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.p2.2.m2.3.4.2.2" xref="S4.p2.2.m2.3.4.2.2.cmml">ℱ</mi><mo id="S4.p2.2.m2.3.4.2.1" xref="S4.p2.2.m2.3.4.2.1.cmml">⁢</mo><mrow id="S4.p2.2.m2.3.4.2.3.2" xref="S4.p2.2.m2.3.4.2.3.1.cmml"><mo id="S4.p2.2.m2.3.4.2.3.2.1" stretchy="false" xref="S4.p2.2.m2.3.4.2.3.1.cmml">(</mo><mi id="S4.p2.2.m2.1.1" xref="S4.p2.2.m2.1.1.cmml">𝐱</mi><mo id="S4.p2.2.m2.3.4.2.3.2.2" xref="S4.p2.2.m2.3.4.2.3.1.cmml">,</mo><mi id="S4.p2.2.m2.2.2" xref="S4.p2.2.m2.2.2.cmml">𝐲</mi><mo id="S4.p2.2.m2.3.4.2.3.2.3" stretchy="false" xref="S4.p2.2.m2.3.4.2.3.1.cmml">)</mo></mrow></mrow><mo id="S4.p2.2.m2.3.4.1" xref="S4.p2.2.m2.3.4.1.cmml">+</mo><mrow id="S4.p2.2.m2.3.4.3" xref="S4.p2.2.m2.3.4.3.cmml"><mi id="S4.p2.2.m2.3.4.3.2" xref="S4.p2.2.m2.3.4.3.2.cmml">α</mi><mo id="S4.p2.2.m2.3.4.3.1" xref="S4.p2.2.m2.3.4.3.1.cmml">⁢</mo><msub id="S4.p2.2.m2.3.4.3.3" xref="S4.p2.2.m2.3.4.3.3.cmml"><mi id="S4.p2.2.m2.3.4.3.3.2" xref="S4.p2.2.m2.3.4.3.3.2.cmml">g</mi><mi id="S4.p2.2.m2.3.4.3.3.3" xref="S4.p2.2.m2.3.4.3.3.3.cmml">σ</mi></msub><mo id="S4.p2.2.m2.3.4.3.1a" xref="S4.p2.2.m2.3.4.3.1.cmml">⁢</mo><mrow id="S4.p2.2.m2.3.4.3.4.2" xref="S4.p2.2.m2.3.4.3.cmml"><mo id="S4.p2.2.m2.3.4.3.4.2.1" xref="S4.p2.2.m2.3.4.3.cmml">(</mo><mi id="S4.p2.2.m2.3.3" xref="S4.p2.2.m2.3.3.cmml">𝐱</mi><mo id="S4.p2.2.m2.3.4.3.4.2.2" xref="S4.p2.2.m2.3.4.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.p2.2.m2.3b"><apply id="S4.p2.2.m2.3.4.cmml" xref="S4.p2.2.m2.3.4"><plus id="S4.p2.2.m2.3.4.1.cmml" xref="S4.p2.2.m2.3.4.1"></plus><apply id="S4.p2.2.m2.3.4.2.cmml" xref="S4.p2.2.m2.3.4.2"><times id="S4.p2.2.m2.3.4.2.1.cmml" xref="S4.p2.2.m2.3.4.2.1"></times><ci id="S4.p2.2.m2.3.4.2.2.cmml" xref="S4.p2.2.m2.3.4.2.2">ℱ</ci><interval closure="open" id="S4.p2.2.m2.3.4.2.3.1.cmml" xref="S4.p2.2.m2.3.4.2.3.2"><ci id="S4.p2.2.m2.1.1.cmml" xref="S4.p2.2.m2.1.1">𝐱</ci><ci id="S4.p2.2.m2.2.2.cmml" xref="S4.p2.2.m2.2.2">𝐲</ci></interval></apply><apply id="S4.p2.2.m2.3.4.3.cmml" xref="S4.p2.2.m2.3.4.3"><times id="S4.p2.2.m2.3.4.3.1.cmml" xref="S4.p2.2.m2.3.4.3.1"></times><ci id="S4.p2.2.m2.3.4.3.2.cmml" xref="S4.p2.2.m2.3.4.3.2">𝛼</ci><apply id="S4.p2.2.m2.3.4.3.3.cmml" xref="S4.p2.2.m2.3.4.3.3"><csymbol cd="ambiguous" id="S4.p2.2.m2.3.4.3.3.1.cmml" xref="S4.p2.2.m2.3.4.3.3">subscript</csymbol><ci id="S4.p2.2.m2.3.4.3.3.2.cmml" xref="S4.p2.2.m2.3.4.3.3.2">𝑔</ci><ci id="S4.p2.2.m2.3.4.3.3.3.cmml" xref="S4.p2.2.m2.3.4.3.3.3">𝜎</ci></apply><ci id="S4.p2.2.m2.3.3.cmml" xref="S4.p2.2.m2.3.3">𝐱</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p2.2.m2.3c">\mathcal{F}({\mathbf{x}},{\mathbf{y}})+\alpha g_{\sigma}\left({\mathbf{x}}\right)</annotation><annotation encoding="application/x-llamapun" id="S4.p2.2.m2.3d">caligraphic_F ( bold_x , bold_y ) + italic_α italic_g start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ( bold_x )</annotation></semantics></math>. Furthermore, it is established that this objective function decreases throughout the algorithm. Using the Gradient-Step denoiser in the SNORE Algorithm <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#alg2" title="Algorithm 2 ‣ Optimization algorithms ‣ 2.2 SNORE regularization ‣ 2 Stochastic deNOising REgularization (SNORE) ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_tag">2</span></a> yields a stochastic gradient descent that minimizes the objective</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A8.EGx16"> <tbody id="S4.E16"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\operatorname*{arg\,min}_{{\mathbf{x}}\in{\mathbb{R}}^{d}}{% \mathcal{J}({\mathbf{x}})}:={\mathbb{E}_{{\tilde{\mathbf{x}}}\sim p_{\sigma}({% \tilde{\mathbf{x}}}|{\mathbf{x}})}\left(\mathcal{F}({\mathbf{x}},{\mathbf{y}})% +\frac{\alpha}{\sigma^{2}}g_{\sigma}({\tilde{\mathbf{x}}})\right)}." class="ltx_Math" display="inline" id="S4.E16.m1.6"><semantics id="S4.E16.m1.6a"><mrow id="S4.E16.m1.6.6.1" xref="S4.E16.m1.6.6.1.1.cmml"><mrow id="S4.E16.m1.6.6.1.1" xref="S4.E16.m1.6.6.1.1.cmml"><mrow id="S4.E16.m1.6.6.1.1.3" xref="S4.E16.m1.6.6.1.1.3.cmml"><mrow id="S4.E16.m1.6.6.1.1.3.2" xref="S4.E16.m1.6.6.1.1.3.2.cmml"><munder id="S4.E16.m1.6.6.1.1.3.2.1" xref="S4.E16.m1.6.6.1.1.3.2.1.cmml"><mrow id="S4.E16.m1.6.6.1.1.3.2.1.2" xref="S4.E16.m1.6.6.1.1.3.2.1.2.cmml"><mi id="S4.E16.m1.6.6.1.1.3.2.1.2.2" xref="S4.E16.m1.6.6.1.1.3.2.1.2.2.cmml">arg</mi><mo id="S4.E16.m1.6.6.1.1.3.2.1.2.1" lspace="0.170em" xref="S4.E16.m1.6.6.1.1.3.2.1.2.1.cmml">⁢</mo><mi id="S4.E16.m1.6.6.1.1.3.2.1.2.3" xref="S4.E16.m1.6.6.1.1.3.2.1.2.3.cmml">min</mi></mrow><mrow id="S4.E16.m1.6.6.1.1.3.2.1.3" xref="S4.E16.m1.6.6.1.1.3.2.1.3.cmml"><mi id="S4.E16.m1.6.6.1.1.3.2.1.3.2" xref="S4.E16.m1.6.6.1.1.3.2.1.3.2.cmml">𝐱</mi><mo 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rspace="0.278em" xref="S4.E16.m1.6.6.1.1.2.cmml">:=</mo><mrow id="S4.E16.m1.6.6.1.1.1" xref="S4.E16.m1.6.6.1.1.1.cmml"><msub id="S4.E16.m1.6.6.1.1.1.3" xref="S4.E16.m1.6.6.1.1.1.3.cmml"><mi id="S4.E16.m1.6.6.1.1.1.3.2" xref="S4.E16.m1.6.6.1.1.1.3.2.cmml">𝔼</mi><mrow id="S4.E16.m1.1.1.1" xref="S4.E16.m1.1.1.1.cmml"><mover accent="true" id="S4.E16.m1.1.1.1.3" xref="S4.E16.m1.1.1.1.3.cmml"><mi id="S4.E16.m1.1.1.1.3.2" xref="S4.E16.m1.1.1.1.3.2.cmml">𝐱</mi><mo id="S4.E16.m1.1.1.1.3.1" xref="S4.E16.m1.1.1.1.3.1.cmml">~</mo></mover><mo id="S4.E16.m1.1.1.1.2" xref="S4.E16.m1.1.1.1.2.cmml">∼</mo><mrow id="S4.E16.m1.1.1.1.1" xref="S4.E16.m1.1.1.1.1.cmml"><msub id="S4.E16.m1.1.1.1.1.3" xref="S4.E16.m1.1.1.1.1.3.cmml"><mi id="S4.E16.m1.1.1.1.1.3.2" xref="S4.E16.m1.1.1.1.1.3.2.cmml">p</mi><mi id="S4.E16.m1.1.1.1.1.3.3" xref="S4.E16.m1.1.1.1.1.3.3.cmml">σ</mi></msub><mo id="S4.E16.m1.1.1.1.1.2" xref="S4.E16.m1.1.1.1.1.2.cmml">⁢</mo><mrow id="S4.E16.m1.1.1.1.1.1.1" 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xref="S4.E16.m1.6.6.1.1.1.1.1.1.3.3">subscript</csymbol><ci id="S4.E16.m1.6.6.1.1.1.1.1.1.3.3.2.cmml" xref="S4.E16.m1.6.6.1.1.1.1.1.1.3.3.2">𝑔</ci><ci id="S4.E16.m1.6.6.1.1.1.1.1.1.3.3.3.cmml" xref="S4.E16.m1.6.6.1.1.1.1.1.1.3.3.3">𝜎</ci></apply><apply id="S4.E16.m1.5.5.cmml" xref="S4.E16.m1.6.6.1.1.1.1.1.1.3.4.2"><ci id="S4.E16.m1.5.5.1.cmml" xref="S4.E16.m1.5.5.1">~</ci><ci id="S4.E16.m1.5.5.2.cmml" xref="S4.E16.m1.5.5.2">𝐱</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.E16.m1.6c">\displaystyle\operatorname*{arg\,min}_{{\mathbf{x}}\in{\mathbb{R}}^{d}}{% \mathcal{J}({\mathbf{x}})}:={\mathbb{E}_{{\tilde{\mathbf{x}}}\sim p_{\sigma}({% \tilde{\mathbf{x}}}|{\mathbf{x}})}\left(\mathcal{F}({\mathbf{x}},{\mathbf{y}})% +\frac{\alpha}{\sigma^{2}}g_{\sigma}({\tilde{\mathbf{x}}})\right)}.</annotation><annotation encoding="application/x-llamapun" id="S4.E16.m1.6d">start_OPERATOR roman_arg roman_min end_OPERATOR start_POSTSUBSCRIPT bold_x ∈ blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT end_POSTSUBSCRIPT caligraphic_J ( bold_x ) := blackboard_E start_POSTSUBSCRIPT over~ start_ARG bold_x end_ARG ∼ italic_p start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ( over~ start_ARG bold_x end_ARG | bold_x ) end_POSTSUBSCRIPT ( caligraphic_F ( bold_x , bold_y ) + divide start_ARG italic_α end_ARG start_ARG italic_σ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG italic_g start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ( over~ start_ARG bold_x end_ARG ) ) .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(16)</span></td> </tr></tbody> </table> </div> <section class="ltx_subsection" id="S4.SS1"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">4.1 </span>Deblurring</h3> <div class="ltx_para" id="S4.SS1.p1"> <p class="ltx_p" id="S4.SS1.p1.5">For image deblurring, the degradation operator is a convolution performed with circular boundary conditions. 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end_POSTSUPERSCRIPT</annotation></semantics></math>, where <math alttext="{\mathbf{A}}={\mathbf{F}}{\mathbf{\Lambda}}{\mathbf{F}}^{*}" class="ltx_Math" display="inline" id="S4.SS1.p1.2.m2.1"><semantics id="S4.SS1.p1.2.m2.1a"><mrow id="S4.SS1.p1.2.m2.1.1" xref="S4.SS1.p1.2.m2.1.1.cmml"><mi id="S4.SS1.p1.2.m2.1.1.2" xref="S4.SS1.p1.2.m2.1.1.2.cmml">𝐀</mi><mo id="S4.SS1.p1.2.m2.1.1.1" xref="S4.SS1.p1.2.m2.1.1.1.cmml">=</mo><mrow id="S4.SS1.p1.2.m2.1.1.3" xref="S4.SS1.p1.2.m2.1.1.3.cmml"><mi id="S4.SS1.p1.2.m2.1.1.3.2" xref="S4.SS1.p1.2.m2.1.1.3.2.cmml">𝐅</mi><mo id="S4.SS1.p1.2.m2.1.1.3.1" xref="S4.SS1.p1.2.m2.1.1.3.1.cmml">⁢</mo><mi id="S4.SS1.p1.2.m2.1.1.3.3" xref="S4.SS1.p1.2.m2.1.1.3.3.cmml">𝚲</mi><mo id="S4.SS1.p1.2.m2.1.1.3.1a" xref="S4.SS1.p1.2.m2.1.1.3.1.cmml">⁢</mo><msup id="S4.SS1.p1.2.m2.1.1.3.4" xref="S4.SS1.p1.2.m2.1.1.3.4.cmml"><mi id="S4.SS1.p1.2.m2.1.1.3.4.2" xref="S4.SS1.p1.2.m2.1.1.3.4.2.cmml">𝐅</mi><mo id="S4.SS1.p1.2.m2.1.1.3.4.3" xref="S4.SS1.p1.2.m2.1.1.3.4.3.cmml">*</mo></msup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p1.2.m2.1b"><apply id="S4.SS1.p1.2.m2.1.1.cmml" xref="S4.SS1.p1.2.m2.1.1"><eq id="S4.SS1.p1.2.m2.1.1.1.cmml" xref="S4.SS1.p1.2.m2.1.1.1"></eq><ci id="S4.SS1.p1.2.m2.1.1.2.cmml" xref="S4.SS1.p1.2.m2.1.1.2">𝐀</ci><apply id="S4.SS1.p1.2.m2.1.1.3.cmml" xref="S4.SS1.p1.2.m2.1.1.3"><times id="S4.SS1.p1.2.m2.1.1.3.1.cmml" xref="S4.SS1.p1.2.m2.1.1.3.1"></times><ci id="S4.SS1.p1.2.m2.1.1.3.2.cmml" xref="S4.SS1.p1.2.m2.1.1.3.2">𝐅</ci><ci id="S4.SS1.p1.2.m2.1.1.3.3.cmml" xref="S4.SS1.p1.2.m2.1.1.3.3">𝚲</ci><apply id="S4.SS1.p1.2.m2.1.1.3.4.cmml" xref="S4.SS1.p1.2.m2.1.1.3.4"><csymbol cd="ambiguous" id="S4.SS1.p1.2.m2.1.1.3.4.1.cmml" xref="S4.SS1.p1.2.m2.1.1.3.4">superscript</csymbol><ci id="S4.SS1.p1.2.m2.1.1.3.4.2.cmml" xref="S4.SS1.p1.2.m2.1.1.3.4.2">𝐅</ci><times id="S4.SS1.p1.2.m2.1.1.3.4.3.cmml" xref="S4.SS1.p1.2.m2.1.1.3.4.3"></times></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p1.2.m2.1c">{\mathbf{A}}={\mathbf{F}}{\mathbf{\Lambda}}{\mathbf{F}}^{*}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p1.2.m2.1d">bold_A = bold_F bold_Λ bold_F start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT</annotation></semantics></math>, <math alttext="{\mathbf{F}}" class="ltx_Math" display="inline" id="S4.SS1.p1.3.m3.1"><semantics id="S4.SS1.p1.3.m3.1a"><mi id="S4.SS1.p1.3.m3.1.1" xref="S4.SS1.p1.3.m3.1.1.cmml">𝐅</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.p1.3.m3.1b"><ci id="S4.SS1.p1.3.m3.1.1.cmml" xref="S4.SS1.p1.3.m3.1.1">𝐅</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p1.3.m3.1c">{\mathbf{F}}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p1.3.m3.1d">bold_F</annotation></semantics></math> is the orthogonal matrix of the discrete Fourier transform (and <math alttext="{\mathbf{F}}^{*}" class="ltx_Math" display="inline" id="S4.SS1.p1.4.m4.1"><semantics id="S4.SS1.p1.4.m4.1a"><msup id="S4.SS1.p1.4.m4.1.1" xref="S4.SS1.p1.4.m4.1.1.cmml"><mi id="S4.SS1.p1.4.m4.1.1.2" xref="S4.SS1.p1.4.m4.1.1.2.cmml">𝐅</mi><mo id="S4.SS1.p1.4.m4.1.1.3" xref="S4.SS1.p1.4.m4.1.1.3.cmml">*</mo></msup><annotation-xml encoding="MathML-Content" id="S4.SS1.p1.4.m4.1b"><apply id="S4.SS1.p1.4.m4.1.1.cmml" xref="S4.SS1.p1.4.m4.1.1"><csymbol cd="ambiguous" id="S4.SS1.p1.4.m4.1.1.1.cmml" xref="S4.SS1.p1.4.m4.1.1">superscript</csymbol><ci id="S4.SS1.p1.4.m4.1.1.2.cmml" xref="S4.SS1.p1.4.m4.1.1.2">𝐅</ci><times id="S4.SS1.p1.4.m4.1.1.3.cmml" xref="S4.SS1.p1.4.m4.1.1.3"></times></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p1.4.m4.1c">{\mathbf{F}}^{*}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p1.4.m4.1d">bold_F start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT</annotation></semantics></math> its inverse) and <math alttext="{\mathbf{\Lambda}}" class="ltx_Math" display="inline" id="S4.SS1.p1.5.m5.1"><semantics id="S4.SS1.p1.5.m5.1a"><mi id="S4.SS1.p1.5.m5.1.1" xref="S4.SS1.p1.5.m5.1.1.cmml">𝚲</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.p1.5.m5.1b"><ci id="S4.SS1.p1.5.m5.1.1.cmml" xref="S4.SS1.p1.5.m5.1.1">𝚲</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p1.5.m5.1c">{\mathbf{\Lambda}}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p1.5.m5.1d">bold_Λ</annotation></semantics></math> a diagonal matrix.</p> </div> <div class="ltx_para" id="S4.SS1.p2"> <p class="ltx_p" id="S4.SS1.p2.14">We make a gradual annealing, by keeping <math alttext="\sigma" class="ltx_Math" display="inline" id="S4.SS1.p2.1.m1.1"><semantics id="S4.SS1.p2.1.m1.1a"><mi id="S4.SS1.p2.1.m1.1.1" xref="S4.SS1.p2.1.m1.1.1.cmml">σ</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.p2.1.m1.1b"><ci id="S4.SS1.p2.1.m1.1.1.cmml" xref="S4.SS1.p2.1.m1.1.1">𝜎</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p2.1.m1.1c">\sigma</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p2.1.m1.1d">italic_σ</annotation></semantics></math> and <math alttext="\alpha" class="ltx_Math" display="inline" id="S4.SS1.p2.2.m2.1"><semantics id="S4.SS1.p2.2.m2.1a"><mi id="S4.SS1.p2.2.m2.1.1" xref="S4.SS1.p2.2.m2.1.1.cmml">α</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.p2.2.m2.1b"><ci id="S4.SS1.p2.2.m2.1.1.cmml" xref="S4.SS1.p2.2.m2.1.1">𝛼</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p2.2.m2.1c">\alpha</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p2.2.m2.1d">italic_α</annotation></semantics></math> fixed for some iterations, to efficiently minimize Problem (<a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#S4.E16" title="16 ‣ 4 Experiments ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_tag">16</span></a>). We set <math alttext="1500" class="ltx_Math" display="inline" id="S4.SS1.p2.3.m3.1"><semantics id="S4.SS1.p2.3.m3.1a"><mn id="S4.SS1.p2.3.m3.1.1" xref="S4.SS1.p2.3.m3.1.1.cmml">1500</mn><annotation-xml encoding="MathML-Content" id="S4.SS1.p2.3.m3.1b"><cn id="S4.SS1.p2.3.m3.1.1.cmml" type="integer" xref="S4.SS1.p2.3.m3.1.1">1500</cn></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p2.3.m3.1c">1500</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p2.3.m3.1d">1500</annotation></semantics></math> iterations and <math alttext="m=16" class="ltx_Math" display="inline" id="S4.SS1.p2.4.m4.1"><semantics id="S4.SS1.p2.4.m4.1a"><mrow id="S4.SS1.p2.4.m4.1.1" xref="S4.SS1.p2.4.m4.1.1.cmml"><mi id="S4.SS1.p2.4.m4.1.1.2" xref="S4.SS1.p2.4.m4.1.1.2.cmml">m</mi><mo id="S4.SS1.p2.4.m4.1.1.1" xref="S4.SS1.p2.4.m4.1.1.1.cmml">=</mo><mn id="S4.SS1.p2.4.m4.1.1.3" xref="S4.SS1.p2.4.m4.1.1.3.cmml">16</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p2.4.m4.1b"><apply id="S4.SS1.p2.4.m4.1.1.cmml" xref="S4.SS1.p2.4.m4.1.1"><eq id="S4.SS1.p2.4.m4.1.1.1.cmml" xref="S4.SS1.p2.4.m4.1.1.1"></eq><ci id="S4.SS1.p2.4.m4.1.1.2.cmml" xref="S4.SS1.p2.4.m4.1.1.2">𝑚</ci><cn id="S4.SS1.p2.4.m4.1.1.3.cmml" type="integer" xref="S4.SS1.p2.4.m4.1.1.3">16</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p2.4.m4.1c">m=16</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p2.4.m4.1d">italic_m = 16</annotation></semantics></math> annealing levels. To ensure convergence, we run <math alttext="300" class="ltx_Math" display="inline" id="S4.SS1.p2.5.m5.1"><semantics id="S4.SS1.p2.5.m5.1a"><mn id="S4.SS1.p2.5.m5.1.1" xref="S4.SS1.p2.5.m5.1.1.cmml">300</mn><annotation-xml encoding="MathML-Content" id="S4.SS1.p2.5.m5.1b"><cn id="S4.SS1.p2.5.m5.1.1.cmml" type="integer" xref="S4.SS1.p2.5.m5.1.1">300</cn></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p2.5.m5.1c">300</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p2.5.m5.1d">300</annotation></semantics></math> iterations with the last parameters <math alttext="\sigma_{m-1},\alpha_{m-1}" class="ltx_Math" display="inline" id="S4.SS1.p2.6.m6.2"><semantics id="S4.SS1.p2.6.m6.2a"><mrow id="S4.SS1.p2.6.m6.2.2.2" xref="S4.SS1.p2.6.m6.2.2.3.cmml"><msub id="S4.SS1.p2.6.m6.1.1.1.1" xref="S4.SS1.p2.6.m6.1.1.1.1.cmml"><mi id="S4.SS1.p2.6.m6.1.1.1.1.2" xref="S4.SS1.p2.6.m6.1.1.1.1.2.cmml">σ</mi><mrow id="S4.SS1.p2.6.m6.1.1.1.1.3" xref="S4.SS1.p2.6.m6.1.1.1.1.3.cmml"><mi id="S4.SS1.p2.6.m6.1.1.1.1.3.2" xref="S4.SS1.p2.6.m6.1.1.1.1.3.2.cmml">m</mi><mo id="S4.SS1.p2.6.m6.1.1.1.1.3.1" xref="S4.SS1.p2.6.m6.1.1.1.1.3.1.cmml">−</mo><mn id="S4.SS1.p2.6.m6.1.1.1.1.3.3" xref="S4.SS1.p2.6.m6.1.1.1.1.3.3.cmml">1</mn></mrow></msub><mo id="S4.SS1.p2.6.m6.2.2.2.3" xref="S4.SS1.p2.6.m6.2.2.3.cmml">,</mo><msub id="S4.SS1.p2.6.m6.2.2.2.2" xref="S4.SS1.p2.6.m6.2.2.2.2.cmml"><mi id="S4.SS1.p2.6.m6.2.2.2.2.2" xref="S4.SS1.p2.6.m6.2.2.2.2.2.cmml">α</mi><mrow id="S4.SS1.p2.6.m6.2.2.2.2.3" xref="S4.SS1.p2.6.m6.2.2.2.2.3.cmml"><mi id="S4.SS1.p2.6.m6.2.2.2.2.3.2" xref="S4.SS1.p2.6.m6.2.2.2.2.3.2.cmml">m</mi><mo id="S4.SS1.p2.6.m6.2.2.2.2.3.1" xref="S4.SS1.p2.6.m6.2.2.2.2.3.1.cmml">−</mo><mn id="S4.SS1.p2.6.m6.2.2.2.2.3.3" xref="S4.SS1.p2.6.m6.2.2.2.2.3.3.cmml">1</mn></mrow></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p2.6.m6.2b"><list id="S4.SS1.p2.6.m6.2.2.3.cmml" xref="S4.SS1.p2.6.m6.2.2.2"><apply id="S4.SS1.p2.6.m6.1.1.1.1.cmml" xref="S4.SS1.p2.6.m6.1.1.1.1"><csymbol cd="ambiguous" id="S4.SS1.p2.6.m6.1.1.1.1.1.cmml" xref="S4.SS1.p2.6.m6.1.1.1.1">subscript</csymbol><ci id="S4.SS1.p2.6.m6.1.1.1.1.2.cmml" xref="S4.SS1.p2.6.m6.1.1.1.1.2">𝜎</ci><apply id="S4.SS1.p2.6.m6.1.1.1.1.3.cmml" xref="S4.SS1.p2.6.m6.1.1.1.1.3"><minus id="S4.SS1.p2.6.m6.1.1.1.1.3.1.cmml" xref="S4.SS1.p2.6.m6.1.1.1.1.3.1"></minus><ci id="S4.SS1.p2.6.m6.1.1.1.1.3.2.cmml" xref="S4.SS1.p2.6.m6.1.1.1.1.3.2">𝑚</ci><cn id="S4.SS1.p2.6.m6.1.1.1.1.3.3.cmml" type="integer" xref="S4.SS1.p2.6.m6.1.1.1.1.3.3">1</cn></apply></apply><apply id="S4.SS1.p2.6.m6.2.2.2.2.cmml" xref="S4.SS1.p2.6.m6.2.2.2.2"><csymbol cd="ambiguous" id="S4.SS1.p2.6.m6.2.2.2.2.1.cmml" xref="S4.SS1.p2.6.m6.2.2.2.2">subscript</csymbol><ci id="S4.SS1.p2.6.m6.2.2.2.2.2.cmml" xref="S4.SS1.p2.6.m6.2.2.2.2.2">𝛼</ci><apply id="S4.SS1.p2.6.m6.2.2.2.2.3.cmml" xref="S4.SS1.p2.6.m6.2.2.2.2.3"><minus id="S4.SS1.p2.6.m6.2.2.2.2.3.1.cmml" xref="S4.SS1.p2.6.m6.2.2.2.2.3.1"></minus><ci id="S4.SS1.p2.6.m6.2.2.2.2.3.2.cmml" xref="S4.SS1.p2.6.m6.2.2.2.2.3.2">𝑚</ci><cn id="S4.SS1.p2.6.m6.2.2.2.2.3.3.cmml" type="integer" xref="S4.SS1.p2.6.m6.2.2.2.2.3.3">1</cn></apply></apply></list></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p2.6.m6.2c">\sigma_{m-1},\alpha_{m-1}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p2.6.m6.2d">italic_σ start_POSTSUBSCRIPT italic_m - 1 end_POSTSUBSCRIPT , italic_α start_POSTSUBSCRIPT italic_m - 1 end_POSTSUBSCRIPT</annotation></semantics></math>. For all input noise levels <math alttext="\sigma_{{\mathbf{y}}}" class="ltx_Math" display="inline" id="S4.SS1.p2.7.m7.1"><semantics id="S4.SS1.p2.7.m7.1a"><msub id="S4.SS1.p2.7.m7.1.1" xref="S4.SS1.p2.7.m7.1.1.cmml"><mi id="S4.SS1.p2.7.m7.1.1.2" xref="S4.SS1.p2.7.m7.1.1.2.cmml">σ</mi><mi id="S4.SS1.p2.7.m7.1.1.3" xref="S4.SS1.p2.7.m7.1.1.3.cmml">𝐲</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS1.p2.7.m7.1b"><apply id="S4.SS1.p2.7.m7.1.1.cmml" xref="S4.SS1.p2.7.m7.1.1"><csymbol cd="ambiguous" id="S4.SS1.p2.7.m7.1.1.1.cmml" xref="S4.SS1.p2.7.m7.1.1">subscript</csymbol><ci id="S4.SS1.p2.7.m7.1.1.2.cmml" xref="S4.SS1.p2.7.m7.1.1.2">𝜎</ci><ci id="S4.SS1.p2.7.m7.1.1.3.cmml" xref="S4.SS1.p2.7.m7.1.1.3">𝐲</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p2.7.m7.1c">\sigma_{{\mathbf{y}}}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p2.7.m7.1d">italic_σ start_POSTSUBSCRIPT bold_y end_POSTSUBSCRIPT</annotation></semantics></math>, we set <math alttext="\sigma_{0}=1.8\sigma_{{\mathbf{y}}}" class="ltx_Math" display="inline" id="S4.SS1.p2.8.m8.1"><semantics id="S4.SS1.p2.8.m8.1a"><mrow id="S4.SS1.p2.8.m8.1.1" xref="S4.SS1.p2.8.m8.1.1.cmml"><msub id="S4.SS1.p2.8.m8.1.1.2" xref="S4.SS1.p2.8.m8.1.1.2.cmml"><mi id="S4.SS1.p2.8.m8.1.1.2.2" xref="S4.SS1.p2.8.m8.1.1.2.2.cmml">σ</mi><mn id="S4.SS1.p2.8.m8.1.1.2.3" xref="S4.SS1.p2.8.m8.1.1.2.3.cmml">0</mn></msub><mo id="S4.SS1.p2.8.m8.1.1.1" xref="S4.SS1.p2.8.m8.1.1.1.cmml">=</mo><mrow id="S4.SS1.p2.8.m8.1.1.3" xref="S4.SS1.p2.8.m8.1.1.3.cmml"><mn id="S4.SS1.p2.8.m8.1.1.3.2" xref="S4.SS1.p2.8.m8.1.1.3.2.cmml">1.8</mn><mo id="S4.SS1.p2.8.m8.1.1.3.1" xref="S4.SS1.p2.8.m8.1.1.3.1.cmml">⁢</mo><msub id="S4.SS1.p2.8.m8.1.1.3.3" xref="S4.SS1.p2.8.m8.1.1.3.3.cmml"><mi id="S4.SS1.p2.8.m8.1.1.3.3.2" xref="S4.SS1.p2.8.m8.1.1.3.3.2.cmml">σ</mi><mi id="S4.SS1.p2.8.m8.1.1.3.3.3" xref="S4.SS1.p2.8.m8.1.1.3.3.3.cmml">𝐲</mi></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p2.8.m8.1b"><apply id="S4.SS1.p2.8.m8.1.1.cmml" xref="S4.SS1.p2.8.m8.1.1"><eq id="S4.SS1.p2.8.m8.1.1.1.cmml" xref="S4.SS1.p2.8.m8.1.1.1"></eq><apply id="S4.SS1.p2.8.m8.1.1.2.cmml" xref="S4.SS1.p2.8.m8.1.1.2"><csymbol cd="ambiguous" id="S4.SS1.p2.8.m8.1.1.2.1.cmml" xref="S4.SS1.p2.8.m8.1.1.2">subscript</csymbol><ci id="S4.SS1.p2.8.m8.1.1.2.2.cmml" xref="S4.SS1.p2.8.m8.1.1.2.2">𝜎</ci><cn id="S4.SS1.p2.8.m8.1.1.2.3.cmml" type="integer" xref="S4.SS1.p2.8.m8.1.1.2.3">0</cn></apply><apply id="S4.SS1.p2.8.m8.1.1.3.cmml" xref="S4.SS1.p2.8.m8.1.1.3"><times id="S4.SS1.p2.8.m8.1.1.3.1.cmml" xref="S4.SS1.p2.8.m8.1.1.3.1"></times><cn id="S4.SS1.p2.8.m8.1.1.3.2.cmml" type="float" xref="S4.SS1.p2.8.m8.1.1.3.2">1.8</cn><apply id="S4.SS1.p2.8.m8.1.1.3.3.cmml" xref="S4.SS1.p2.8.m8.1.1.3.3"><csymbol cd="ambiguous" id="S4.SS1.p2.8.m8.1.1.3.3.1.cmml" xref="S4.SS1.p2.8.m8.1.1.3.3">subscript</csymbol><ci id="S4.SS1.p2.8.m8.1.1.3.3.2.cmml" xref="S4.SS1.p2.8.m8.1.1.3.3.2">𝜎</ci><ci id="S4.SS1.p2.8.m8.1.1.3.3.3.cmml" xref="S4.SS1.p2.8.m8.1.1.3.3.3">𝐲</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p2.8.m8.1c">\sigma_{0}=1.8\sigma_{{\mathbf{y}}}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p2.8.m8.1d">italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT = 1.8 italic_σ start_POSTSUBSCRIPT bold_y end_POSTSUBSCRIPT</annotation></semantics></math>, <math alttext="\sigma_{m-1}=0.5\sigma_{{\mathbf{y}}}" class="ltx_Math" display="inline" id="S4.SS1.p2.9.m9.1"><semantics id="S4.SS1.p2.9.m9.1a"><mrow id="S4.SS1.p2.9.m9.1.1" xref="S4.SS1.p2.9.m9.1.1.cmml"><msub id="S4.SS1.p2.9.m9.1.1.2" xref="S4.SS1.p2.9.m9.1.1.2.cmml"><mi id="S4.SS1.p2.9.m9.1.1.2.2" xref="S4.SS1.p2.9.m9.1.1.2.2.cmml">σ</mi><mrow id="S4.SS1.p2.9.m9.1.1.2.3" xref="S4.SS1.p2.9.m9.1.1.2.3.cmml"><mi id="S4.SS1.p2.9.m9.1.1.2.3.2" xref="S4.SS1.p2.9.m9.1.1.2.3.2.cmml">m</mi><mo id="S4.SS1.p2.9.m9.1.1.2.3.1" xref="S4.SS1.p2.9.m9.1.1.2.3.1.cmml">−</mo><mn id="S4.SS1.p2.9.m9.1.1.2.3.3" xref="S4.SS1.p2.9.m9.1.1.2.3.3.cmml">1</mn></mrow></msub><mo id="S4.SS1.p2.9.m9.1.1.1" xref="S4.SS1.p2.9.m9.1.1.1.cmml">=</mo><mrow id="S4.SS1.p2.9.m9.1.1.3" xref="S4.SS1.p2.9.m9.1.1.3.cmml"><mn id="S4.SS1.p2.9.m9.1.1.3.2" xref="S4.SS1.p2.9.m9.1.1.3.2.cmml">0.5</mn><mo id="S4.SS1.p2.9.m9.1.1.3.1" xref="S4.SS1.p2.9.m9.1.1.3.1.cmml">⁢</mo><msub id="S4.SS1.p2.9.m9.1.1.3.3" xref="S4.SS1.p2.9.m9.1.1.3.3.cmml"><mi id="S4.SS1.p2.9.m9.1.1.3.3.2" xref="S4.SS1.p2.9.m9.1.1.3.3.2.cmml">σ</mi><mi id="S4.SS1.p2.9.m9.1.1.3.3.3" xref="S4.SS1.p2.9.m9.1.1.3.3.3.cmml">𝐲</mi></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p2.9.m9.1b"><apply id="S4.SS1.p2.9.m9.1.1.cmml" xref="S4.SS1.p2.9.m9.1.1"><eq id="S4.SS1.p2.9.m9.1.1.1.cmml" xref="S4.SS1.p2.9.m9.1.1.1"></eq><apply id="S4.SS1.p2.9.m9.1.1.2.cmml" xref="S4.SS1.p2.9.m9.1.1.2"><csymbol cd="ambiguous" id="S4.SS1.p2.9.m9.1.1.2.1.cmml" xref="S4.SS1.p2.9.m9.1.1.2">subscript</csymbol><ci id="S4.SS1.p2.9.m9.1.1.2.2.cmml" xref="S4.SS1.p2.9.m9.1.1.2.2">𝜎</ci><apply id="S4.SS1.p2.9.m9.1.1.2.3.cmml" xref="S4.SS1.p2.9.m9.1.1.2.3"><minus id="S4.SS1.p2.9.m9.1.1.2.3.1.cmml" xref="S4.SS1.p2.9.m9.1.1.2.3.1"></minus><ci id="S4.SS1.p2.9.m9.1.1.2.3.2.cmml" xref="S4.SS1.p2.9.m9.1.1.2.3.2">𝑚</ci><cn id="S4.SS1.p2.9.m9.1.1.2.3.3.cmml" type="integer" xref="S4.SS1.p2.9.m9.1.1.2.3.3">1</cn></apply></apply><apply id="S4.SS1.p2.9.m9.1.1.3.cmml" xref="S4.SS1.p2.9.m9.1.1.3"><times id="S4.SS1.p2.9.m9.1.1.3.1.cmml" xref="S4.SS1.p2.9.m9.1.1.3.1"></times><cn id="S4.SS1.p2.9.m9.1.1.3.2.cmml" type="float" xref="S4.SS1.p2.9.m9.1.1.3.2">0.5</cn><apply id="S4.SS1.p2.9.m9.1.1.3.3.cmml" xref="S4.SS1.p2.9.m9.1.1.3.3"><csymbol cd="ambiguous" id="S4.SS1.p2.9.m9.1.1.3.3.1.cmml" xref="S4.SS1.p2.9.m9.1.1.3.3">subscript</csymbol><ci id="S4.SS1.p2.9.m9.1.1.3.3.2.cmml" xref="S4.SS1.p2.9.m9.1.1.3.3.2">𝜎</ci><ci id="S4.SS1.p2.9.m9.1.1.3.3.3.cmml" xref="S4.SS1.p2.9.m9.1.1.3.3.3">𝐲</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p2.9.m9.1c">\sigma_{m-1}=0.5\sigma_{{\mathbf{y}}}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p2.9.m9.1d">italic_σ start_POSTSUBSCRIPT italic_m - 1 end_POSTSUBSCRIPT = 0.5 italic_σ start_POSTSUBSCRIPT bold_y end_POSTSUBSCRIPT</annotation></semantics></math>, <math alttext="\alpha_{0}=0.1\sigma_{0}^{2}\sigma_{{\mathbf{y}}}^{-2}" class="ltx_Math" display="inline" id="S4.SS1.p2.10.m10.1"><semantics id="S4.SS1.p2.10.m10.1a"><mrow id="S4.SS1.p2.10.m10.1.1" xref="S4.SS1.p2.10.m10.1.1.cmml"><msub id="S4.SS1.p2.10.m10.1.1.2" xref="S4.SS1.p2.10.m10.1.1.2.cmml"><mi id="S4.SS1.p2.10.m10.1.1.2.2" xref="S4.SS1.p2.10.m10.1.1.2.2.cmml">α</mi><mn id="S4.SS1.p2.10.m10.1.1.2.3" xref="S4.SS1.p2.10.m10.1.1.2.3.cmml">0</mn></msub><mo id="S4.SS1.p2.10.m10.1.1.1" xref="S4.SS1.p2.10.m10.1.1.1.cmml">=</mo><mrow id="S4.SS1.p2.10.m10.1.1.3" xref="S4.SS1.p2.10.m10.1.1.3.cmml"><mn id="S4.SS1.p2.10.m10.1.1.3.2" xref="S4.SS1.p2.10.m10.1.1.3.2.cmml">0.1</mn><mo id="S4.SS1.p2.10.m10.1.1.3.1" xref="S4.SS1.p2.10.m10.1.1.3.1.cmml">⁢</mo><msubsup id="S4.SS1.p2.10.m10.1.1.3.3" xref="S4.SS1.p2.10.m10.1.1.3.3.cmml"><mi id="S4.SS1.p2.10.m10.1.1.3.3.2.2" xref="S4.SS1.p2.10.m10.1.1.3.3.2.2.cmml">σ</mi><mn id="S4.SS1.p2.10.m10.1.1.3.3.2.3" xref="S4.SS1.p2.10.m10.1.1.3.3.2.3.cmml">0</mn><mn id="S4.SS1.p2.10.m10.1.1.3.3.3" xref="S4.SS1.p2.10.m10.1.1.3.3.3.cmml">2</mn></msubsup><mo id="S4.SS1.p2.10.m10.1.1.3.1a" xref="S4.SS1.p2.10.m10.1.1.3.1.cmml">⁢</mo><msubsup id="S4.SS1.p2.10.m10.1.1.3.4" xref="S4.SS1.p2.10.m10.1.1.3.4.cmml"><mi id="S4.SS1.p2.10.m10.1.1.3.4.2.2" xref="S4.SS1.p2.10.m10.1.1.3.4.2.2.cmml">σ</mi><mi id="S4.SS1.p2.10.m10.1.1.3.4.2.3" xref="S4.SS1.p2.10.m10.1.1.3.4.2.3.cmml">𝐲</mi><mrow id="S4.SS1.p2.10.m10.1.1.3.4.3" xref="S4.SS1.p2.10.m10.1.1.3.4.3.cmml"><mo id="S4.SS1.p2.10.m10.1.1.3.4.3a" xref="S4.SS1.p2.10.m10.1.1.3.4.3.cmml">−</mo><mn id="S4.SS1.p2.10.m10.1.1.3.4.3.2" xref="S4.SS1.p2.10.m10.1.1.3.4.3.2.cmml">2</mn></mrow></msubsup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p2.10.m10.1b"><apply id="S4.SS1.p2.10.m10.1.1.cmml" xref="S4.SS1.p2.10.m10.1.1"><eq id="S4.SS1.p2.10.m10.1.1.1.cmml" xref="S4.SS1.p2.10.m10.1.1.1"></eq><apply id="S4.SS1.p2.10.m10.1.1.2.cmml" xref="S4.SS1.p2.10.m10.1.1.2"><csymbol cd="ambiguous" id="S4.SS1.p2.10.m10.1.1.2.1.cmml" xref="S4.SS1.p2.10.m10.1.1.2">subscript</csymbol><ci id="S4.SS1.p2.10.m10.1.1.2.2.cmml" xref="S4.SS1.p2.10.m10.1.1.2.2">𝛼</ci><cn id="S4.SS1.p2.10.m10.1.1.2.3.cmml" type="integer" xref="S4.SS1.p2.10.m10.1.1.2.3">0</cn></apply><apply id="S4.SS1.p2.10.m10.1.1.3.cmml" xref="S4.SS1.p2.10.m10.1.1.3"><times id="S4.SS1.p2.10.m10.1.1.3.1.cmml" xref="S4.SS1.p2.10.m10.1.1.3.1"></times><cn id="S4.SS1.p2.10.m10.1.1.3.2.cmml" type="float" xref="S4.SS1.p2.10.m10.1.1.3.2">0.1</cn><apply id="S4.SS1.p2.10.m10.1.1.3.3.cmml" xref="S4.SS1.p2.10.m10.1.1.3.3"><csymbol cd="ambiguous" id="S4.SS1.p2.10.m10.1.1.3.3.1.cmml" xref="S4.SS1.p2.10.m10.1.1.3.3">superscript</csymbol><apply id="S4.SS1.p2.10.m10.1.1.3.3.2.cmml" xref="S4.SS1.p2.10.m10.1.1.3.3"><csymbol cd="ambiguous" id="S4.SS1.p2.10.m10.1.1.3.3.2.1.cmml" xref="S4.SS1.p2.10.m10.1.1.3.3">subscript</csymbol><ci id="S4.SS1.p2.10.m10.1.1.3.3.2.2.cmml" xref="S4.SS1.p2.10.m10.1.1.3.3.2.2">𝜎</ci><cn id="S4.SS1.p2.10.m10.1.1.3.3.2.3.cmml" type="integer" xref="S4.SS1.p2.10.m10.1.1.3.3.2.3">0</cn></apply><cn id="S4.SS1.p2.10.m10.1.1.3.3.3.cmml" type="integer" xref="S4.SS1.p2.10.m10.1.1.3.3.3">2</cn></apply><apply id="S4.SS1.p2.10.m10.1.1.3.4.cmml" xref="S4.SS1.p2.10.m10.1.1.3.4"><csymbol cd="ambiguous" id="S4.SS1.p2.10.m10.1.1.3.4.1.cmml" xref="S4.SS1.p2.10.m10.1.1.3.4">superscript</csymbol><apply id="S4.SS1.p2.10.m10.1.1.3.4.2.cmml" xref="S4.SS1.p2.10.m10.1.1.3.4"><csymbol cd="ambiguous" id="S4.SS1.p2.10.m10.1.1.3.4.2.1.cmml" xref="S4.SS1.p2.10.m10.1.1.3.4">subscript</csymbol><ci id="S4.SS1.p2.10.m10.1.1.3.4.2.2.cmml" xref="S4.SS1.p2.10.m10.1.1.3.4.2.2">𝜎</ci><ci id="S4.SS1.p2.10.m10.1.1.3.4.2.3.cmml" xref="S4.SS1.p2.10.m10.1.1.3.4.2.3">𝐲</ci></apply><apply id="S4.SS1.p2.10.m10.1.1.3.4.3.cmml" xref="S4.SS1.p2.10.m10.1.1.3.4.3"><minus id="S4.SS1.p2.10.m10.1.1.3.4.3.1.cmml" xref="S4.SS1.p2.10.m10.1.1.3.4.3"></minus><cn id="S4.SS1.p2.10.m10.1.1.3.4.3.2.cmml" type="integer" xref="S4.SS1.p2.10.m10.1.1.3.4.3.2">2</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p2.10.m10.1c">\alpha_{0}=0.1\sigma_{0}^{2}\sigma_{{\mathbf{y}}}^{-2}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p2.10.m10.1d">italic_α start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT = 0.1 italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_σ start_POSTSUBSCRIPT bold_y end_POSTSUBSCRIPT start_POSTSUPERSCRIPT - 2 end_POSTSUPERSCRIPT</annotation></semantics></math> and <math alttext="\alpha_{m-1}=\sigma_{m-1}^{2}\sigma_{{\mathbf{y}}}^{-2}" class="ltx_Math" display="inline" id="S4.SS1.p2.11.m11.1"><semantics id="S4.SS1.p2.11.m11.1a"><mrow id="S4.SS1.p2.11.m11.1.1" xref="S4.SS1.p2.11.m11.1.1.cmml"><msub id="S4.SS1.p2.11.m11.1.1.2" xref="S4.SS1.p2.11.m11.1.1.2.cmml"><mi id="S4.SS1.p2.11.m11.1.1.2.2" xref="S4.SS1.p2.11.m11.1.1.2.2.cmml">α</mi><mrow id="S4.SS1.p2.11.m11.1.1.2.3" xref="S4.SS1.p2.11.m11.1.1.2.3.cmml"><mi id="S4.SS1.p2.11.m11.1.1.2.3.2" xref="S4.SS1.p2.11.m11.1.1.2.3.2.cmml">m</mi><mo id="S4.SS1.p2.11.m11.1.1.2.3.1" xref="S4.SS1.p2.11.m11.1.1.2.3.1.cmml">−</mo><mn id="S4.SS1.p2.11.m11.1.1.2.3.3" xref="S4.SS1.p2.11.m11.1.1.2.3.3.cmml">1</mn></mrow></msub><mo id="S4.SS1.p2.11.m11.1.1.1" xref="S4.SS1.p2.11.m11.1.1.1.cmml">=</mo><mrow id="S4.SS1.p2.11.m11.1.1.3" xref="S4.SS1.p2.11.m11.1.1.3.cmml"><msubsup id="S4.SS1.p2.11.m11.1.1.3.2" xref="S4.SS1.p2.11.m11.1.1.3.2.cmml"><mi id="S4.SS1.p2.11.m11.1.1.3.2.2.2" xref="S4.SS1.p2.11.m11.1.1.3.2.2.2.cmml">σ</mi><mrow id="S4.SS1.p2.11.m11.1.1.3.2.2.3" xref="S4.SS1.p2.11.m11.1.1.3.2.2.3.cmml"><mi id="S4.SS1.p2.11.m11.1.1.3.2.2.3.2" xref="S4.SS1.p2.11.m11.1.1.3.2.2.3.2.cmml">m</mi><mo id="S4.SS1.p2.11.m11.1.1.3.2.2.3.1" xref="S4.SS1.p2.11.m11.1.1.3.2.2.3.1.cmml">−</mo><mn id="S4.SS1.p2.11.m11.1.1.3.2.2.3.3" xref="S4.SS1.p2.11.m11.1.1.3.2.2.3.3.cmml">1</mn></mrow><mn id="S4.SS1.p2.11.m11.1.1.3.2.3" xref="S4.SS1.p2.11.m11.1.1.3.2.3.cmml">2</mn></msubsup><mo id="S4.SS1.p2.11.m11.1.1.3.1" xref="S4.SS1.p2.11.m11.1.1.3.1.cmml">⁢</mo><msubsup id="S4.SS1.p2.11.m11.1.1.3.3" xref="S4.SS1.p2.11.m11.1.1.3.3.cmml"><mi id="S4.SS1.p2.11.m11.1.1.3.3.2.2" xref="S4.SS1.p2.11.m11.1.1.3.3.2.2.cmml">σ</mi><mi id="S4.SS1.p2.11.m11.1.1.3.3.2.3" xref="S4.SS1.p2.11.m11.1.1.3.3.2.3.cmml">𝐲</mi><mrow id="S4.SS1.p2.11.m11.1.1.3.3.3" xref="S4.SS1.p2.11.m11.1.1.3.3.3.cmml"><mo id="S4.SS1.p2.11.m11.1.1.3.3.3a" xref="S4.SS1.p2.11.m11.1.1.3.3.3.cmml">−</mo><mn id="S4.SS1.p2.11.m11.1.1.3.3.3.2" xref="S4.SS1.p2.11.m11.1.1.3.3.3.2.cmml">2</mn></mrow></msubsup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p2.11.m11.1b"><apply id="S4.SS1.p2.11.m11.1.1.cmml" xref="S4.SS1.p2.11.m11.1.1"><eq id="S4.SS1.p2.11.m11.1.1.1.cmml" xref="S4.SS1.p2.11.m11.1.1.1"></eq><apply id="S4.SS1.p2.11.m11.1.1.2.cmml" xref="S4.SS1.p2.11.m11.1.1.2"><csymbol cd="ambiguous" id="S4.SS1.p2.11.m11.1.1.2.1.cmml" xref="S4.SS1.p2.11.m11.1.1.2">subscript</csymbol><ci id="S4.SS1.p2.11.m11.1.1.2.2.cmml" xref="S4.SS1.p2.11.m11.1.1.2.2">𝛼</ci><apply id="S4.SS1.p2.11.m11.1.1.2.3.cmml" xref="S4.SS1.p2.11.m11.1.1.2.3"><minus id="S4.SS1.p2.11.m11.1.1.2.3.1.cmml" xref="S4.SS1.p2.11.m11.1.1.2.3.1"></minus><ci id="S4.SS1.p2.11.m11.1.1.2.3.2.cmml" xref="S4.SS1.p2.11.m11.1.1.2.3.2">𝑚</ci><cn id="S4.SS1.p2.11.m11.1.1.2.3.3.cmml" type="integer" xref="S4.SS1.p2.11.m11.1.1.2.3.3">1</cn></apply></apply><apply id="S4.SS1.p2.11.m11.1.1.3.cmml" xref="S4.SS1.p2.11.m11.1.1.3"><times id="S4.SS1.p2.11.m11.1.1.3.1.cmml" xref="S4.SS1.p2.11.m11.1.1.3.1"></times><apply id="S4.SS1.p2.11.m11.1.1.3.2.cmml" xref="S4.SS1.p2.11.m11.1.1.3.2"><csymbol cd="ambiguous" id="S4.SS1.p2.11.m11.1.1.3.2.1.cmml" xref="S4.SS1.p2.11.m11.1.1.3.2">superscript</csymbol><apply id="S4.SS1.p2.11.m11.1.1.3.2.2.cmml" xref="S4.SS1.p2.11.m11.1.1.3.2"><csymbol cd="ambiguous" id="S4.SS1.p2.11.m11.1.1.3.2.2.1.cmml" xref="S4.SS1.p2.11.m11.1.1.3.2">subscript</csymbol><ci id="S4.SS1.p2.11.m11.1.1.3.2.2.2.cmml" xref="S4.SS1.p2.11.m11.1.1.3.2.2.2">𝜎</ci><apply id="S4.SS1.p2.11.m11.1.1.3.2.2.3.cmml" xref="S4.SS1.p2.11.m11.1.1.3.2.2.3"><minus id="S4.SS1.p2.11.m11.1.1.3.2.2.3.1.cmml" xref="S4.SS1.p2.11.m11.1.1.3.2.2.3.1"></minus><ci id="S4.SS1.p2.11.m11.1.1.3.2.2.3.2.cmml" xref="S4.SS1.p2.11.m11.1.1.3.2.2.3.2">𝑚</ci><cn id="S4.SS1.p2.11.m11.1.1.3.2.2.3.3.cmml" type="integer" xref="S4.SS1.p2.11.m11.1.1.3.2.2.3.3">1</cn></apply></apply><cn id="S4.SS1.p2.11.m11.1.1.3.2.3.cmml" type="integer" xref="S4.SS1.p2.11.m11.1.1.3.2.3">2</cn></apply><apply id="S4.SS1.p2.11.m11.1.1.3.3.cmml" xref="S4.SS1.p2.11.m11.1.1.3.3"><csymbol cd="ambiguous" id="S4.SS1.p2.11.m11.1.1.3.3.1.cmml" xref="S4.SS1.p2.11.m11.1.1.3.3">superscript</csymbol><apply id="S4.SS1.p2.11.m11.1.1.3.3.2.cmml" xref="S4.SS1.p2.11.m11.1.1.3.3"><csymbol cd="ambiguous" id="S4.SS1.p2.11.m11.1.1.3.3.2.1.cmml" xref="S4.SS1.p2.11.m11.1.1.3.3">subscript</csymbol><ci id="S4.SS1.p2.11.m11.1.1.3.3.2.2.cmml" xref="S4.SS1.p2.11.m11.1.1.3.3.2.2">𝜎</ci><ci id="S4.SS1.p2.11.m11.1.1.3.3.2.3.cmml" xref="S4.SS1.p2.11.m11.1.1.3.3.2.3">𝐲</ci></apply><apply id="S4.SS1.p2.11.m11.1.1.3.3.3.cmml" xref="S4.SS1.p2.11.m11.1.1.3.3.3"><minus id="S4.SS1.p2.11.m11.1.1.3.3.3.1.cmml" xref="S4.SS1.p2.11.m11.1.1.3.3.3"></minus><cn id="S4.SS1.p2.11.m11.1.1.3.3.3.2.cmml" type="integer" xref="S4.SS1.p2.11.m11.1.1.3.3.3.2">2</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p2.11.m11.1c">\alpha_{m-1}=\sigma_{m-1}^{2}\sigma_{{\mathbf{y}}}^{-2}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p2.11.m11.1d">italic_α start_POSTSUBSCRIPT italic_m - 1 end_POSTSUBSCRIPT = italic_σ start_POSTSUBSCRIPT italic_m - 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_σ start_POSTSUBSCRIPT bold_y end_POSTSUBSCRIPT start_POSTSUPERSCRIPT - 2 end_POSTSUPERSCRIPT</annotation></semantics></math>. We initialize with the observation <math alttext="{\mathbf{x}}_{0}={\mathbf{y}}" class="ltx_Math" display="inline" id="S4.SS1.p2.12.m12.1"><semantics id="S4.SS1.p2.12.m12.1a"><mrow id="S4.SS1.p2.12.m12.1.1" xref="S4.SS1.p2.12.m12.1.1.cmml"><msub id="S4.SS1.p2.12.m12.1.1.2" xref="S4.SS1.p2.12.m12.1.1.2.cmml"><mi id="S4.SS1.p2.12.m12.1.1.2.2" xref="S4.SS1.p2.12.m12.1.1.2.2.cmml">𝐱</mi><mn id="S4.SS1.p2.12.m12.1.1.2.3" xref="S4.SS1.p2.12.m12.1.1.2.3.cmml">0</mn></msub><mo id="S4.SS1.p2.12.m12.1.1.1" xref="S4.SS1.p2.12.m12.1.1.1.cmml">=</mo><mi id="S4.SS1.p2.12.m12.1.1.3" xref="S4.SS1.p2.12.m12.1.1.3.cmml">𝐲</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p2.12.m12.1b"><apply id="S4.SS1.p2.12.m12.1.1.cmml" xref="S4.SS1.p2.12.m12.1.1"><eq id="S4.SS1.p2.12.m12.1.1.1.cmml" xref="S4.SS1.p2.12.m12.1.1.1"></eq><apply id="S4.SS1.p2.12.m12.1.1.2.cmml" xref="S4.SS1.p2.12.m12.1.1.2"><csymbol cd="ambiguous" id="S4.SS1.p2.12.m12.1.1.2.1.cmml" xref="S4.SS1.p2.12.m12.1.1.2">subscript</csymbol><ci id="S4.SS1.p2.12.m12.1.1.2.2.cmml" xref="S4.SS1.p2.12.m12.1.1.2.2">𝐱</ci><cn id="S4.SS1.p2.12.m12.1.1.2.3.cmml" type="integer" xref="S4.SS1.p2.12.m12.1.1.2.3">0</cn></apply><ci id="S4.SS1.p2.12.m12.1.1.3.cmml" xref="S4.SS1.p2.12.m12.1.1.3">𝐲</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p2.12.m12.1c">{\mathbf{x}}_{0}={\mathbf{y}}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p2.12.m12.1d">bold_x start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT = bold_y</annotation></semantics></math> and use a fixed step-size, <math alttext="\delta_{k}=\delta=0.1" class="ltx_Math" display="inline" id="S4.SS1.p2.13.m13.1"><semantics id="S4.SS1.p2.13.m13.1a"><mrow id="S4.SS1.p2.13.m13.1.1" xref="S4.SS1.p2.13.m13.1.1.cmml"><msub id="S4.SS1.p2.13.m13.1.1.2" xref="S4.SS1.p2.13.m13.1.1.2.cmml"><mi id="S4.SS1.p2.13.m13.1.1.2.2" xref="S4.SS1.p2.13.m13.1.1.2.2.cmml">δ</mi><mi id="S4.SS1.p2.13.m13.1.1.2.3" xref="S4.SS1.p2.13.m13.1.1.2.3.cmml">k</mi></msub><mo id="S4.SS1.p2.13.m13.1.1.3" xref="S4.SS1.p2.13.m13.1.1.3.cmml">=</mo><mi id="S4.SS1.p2.13.m13.1.1.4" xref="S4.SS1.p2.13.m13.1.1.4.cmml">δ</mi><mo id="S4.SS1.p2.13.m13.1.1.5" xref="S4.SS1.p2.13.m13.1.1.5.cmml">=</mo><mn id="S4.SS1.p2.13.m13.1.1.6" xref="S4.SS1.p2.13.m13.1.1.6.cmml">0.1</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p2.13.m13.1b"><apply id="S4.SS1.p2.13.m13.1.1.cmml" xref="S4.SS1.p2.13.m13.1.1"><and id="S4.SS1.p2.13.m13.1.1a.cmml" xref="S4.SS1.p2.13.m13.1.1"></and><apply id="S4.SS1.p2.13.m13.1.1b.cmml" xref="S4.SS1.p2.13.m13.1.1"><eq id="S4.SS1.p2.13.m13.1.1.3.cmml" xref="S4.SS1.p2.13.m13.1.1.3"></eq><apply id="S4.SS1.p2.13.m13.1.1.2.cmml" xref="S4.SS1.p2.13.m13.1.1.2"><csymbol cd="ambiguous" id="S4.SS1.p2.13.m13.1.1.2.1.cmml" xref="S4.SS1.p2.13.m13.1.1.2">subscript</csymbol><ci id="S4.SS1.p2.13.m13.1.1.2.2.cmml" xref="S4.SS1.p2.13.m13.1.1.2.2">𝛿</ci><ci id="S4.SS1.p2.13.m13.1.1.2.3.cmml" xref="S4.SS1.p2.13.m13.1.1.2.3">𝑘</ci></apply><ci id="S4.SS1.p2.13.m13.1.1.4.cmml" xref="S4.SS1.p2.13.m13.1.1.4">𝛿</ci></apply><apply id="S4.SS1.p2.13.m13.1.1c.cmml" xref="S4.SS1.p2.13.m13.1.1"><eq id="S4.SS1.p2.13.m13.1.1.5.cmml" xref="S4.SS1.p2.13.m13.1.1.5"></eq><share href="#S4.SS1.p2.13.m13.1.1.4.cmml" id="S4.SS1.p2.13.m13.1.1d.cmml" xref="S4.SS1.p2.13.m13.1.1"></share><cn id="S4.SS1.p2.13.m13.1.1.6.cmml" type="float" xref="S4.SS1.p2.13.m13.1.1.6">0.1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p2.13.m13.1c">\delta_{k}=\delta=0.1</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p2.13.m13.1d">italic_δ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT = italic_δ = 0.1</annotation></semantics></math>, as we observe that a decreasing <math alttext="\delta_{k}" class="ltx_Math" display="inline" id="S4.SS1.p2.14.m14.1"><semantics id="S4.SS1.p2.14.m14.1a"><msub id="S4.SS1.p2.14.m14.1.1" xref="S4.SS1.p2.14.m14.1.1.cmml"><mi id="S4.SS1.p2.14.m14.1.1.2" xref="S4.SS1.p2.14.m14.1.1.2.cmml">δ</mi><mi id="S4.SS1.p2.14.m14.1.1.3" xref="S4.SS1.p2.14.m14.1.1.3.cmml">k</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS1.p2.14.m14.1b"><apply id="S4.SS1.p2.14.m14.1.1.cmml" xref="S4.SS1.p2.14.m14.1.1"><csymbol cd="ambiguous" id="S4.SS1.p2.14.m14.1.1.1.cmml" xref="S4.SS1.p2.14.m14.1.1">subscript</csymbol><ci id="S4.SS1.p2.14.m14.1.1.2.cmml" xref="S4.SS1.p2.14.m14.1.1.2">𝛿</ci><ci id="S4.SS1.p2.14.m14.1.1.3.cmml" xref="S4.SS1.p2.14.m14.1.1.3">𝑘</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p2.14.m14.1c">\delta_{k}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p2.14.m14.1d">italic_δ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT</annotation></semantics></math> leads to a slower convergence, as notice by <cite class="ltx_cite ltx_citemacro_citet">Laumont et al. (<a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#bib.bib37" title="">2022b</a>)</cite>.</p> </div> <figure class="ltx_figure" id="S4.F3"> <p class="ltx_p" id="S4.F3.1"><span class="ltx_text" id="S4.F3.1.1" style="width:433.6pt;"><img alt="Refer to caption" class="ltx_graphics ltx_img_landscape" height="133" id="S4.F3.1.1.g1" src="extracted/5518834/images/Results_restoration_inpainting_all_methods.png" width="604"/></span></p> <figcaption class="ltx_caption"><span class="ltx_tag ltx_tag_figure"><span class="ltx_text" id="S4.F3.5.2.1" style="font-size:90%;">Figure 3</span>: </span><span class="ltx_text" id="S4.F3.3.1" style="font-size:90%;">Inpainting with various methods on a random mask (with a proportion <math alttext="p=0.5" class="ltx_Math" display="inline" id="S4.F3.3.1.m1.1"><semantics id="S4.F3.3.1.m1.1b"><mrow id="S4.F3.3.1.m1.1.1" xref="S4.F3.3.1.m1.1.1.cmml"><mi id="S4.F3.3.1.m1.1.1.2" xref="S4.F3.3.1.m1.1.1.2.cmml">p</mi><mo id="S4.F3.3.1.m1.1.1.1" xref="S4.F3.3.1.m1.1.1.1.cmml">=</mo><mn id="S4.F3.3.1.m1.1.1.3" xref="S4.F3.3.1.m1.1.1.3.cmml">0.5</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.F3.3.1.m1.1c"><apply id="S4.F3.3.1.m1.1.1.cmml" xref="S4.F3.3.1.m1.1.1"><eq id="S4.F3.3.1.m1.1.1.1.cmml" xref="S4.F3.3.1.m1.1.1.1"></eq><ci id="S4.F3.3.1.m1.1.1.2.cmml" xref="S4.F3.3.1.m1.1.1.2">𝑝</ci><cn id="S4.F3.3.1.m1.1.1.3.cmml" type="float" xref="S4.F3.3.1.m1.1.1.3">0.5</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.F3.3.1.m1.1d">p=0.5</annotation><annotation encoding="application/x-llamapun" id="S4.F3.3.1.m1.1e">italic_p = 0.5</annotation></semantics></math> of masked pixels) with a GS-denoiser trained on natural images. One can observe here the ability of Ann-SNORE to recover both sharp structures and textural content.</span></figcaption> </figure> <div class="ltx_para" id="S4.SS1.p3"> <p class="ltx_p" id="S4.SS1.p3.4">We compare in Table <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#S4.T1" title="Table 1 ‣ 4.1 Deblurring ‣ 4 Experiments ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_tag">1</span></a> our method to RED restoration algorithm of <cite class="ltx_cite ltx_citemacro_citep">(Romano et al., <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#bib.bib58" title="">2017</a>)</cite> (see Algorithm <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#alg1" title="Algorithm 1 ‣ 2.1 Background ‣ 2 Stochastic deNOising REgularization (SNORE) ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_tag">1</span></a>) with a gradient-descent step on the data-fidelity, “RED Prox” method <cite class="ltx_cite ltx_citemacro_citep">(Romano et al., <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#bib.bib58" title="">2017</a>)</cite> with a proximal-descent step on the data-fidelity (see Algorithm <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#alg4" title="Algorithm 4 ‣ RED Prox ‣ Appendix F Additional Experiments ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_tag">4</span></a>), DiffPIR <cite class="ltx_cite ltx_citemacro_citep">(Zhu et al., <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#bib.bib80" title="">2023</a>)</cite> and PnP SGD <cite class="ltx_cite ltx_citemacro_citep">(Laumont et al., <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#bib.bib37" title="">2022b</a>)</cite>. As <cite class="ltx_cite ltx_citemacro_citep">(Hurault et al., <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#bib.bib30" title="">2022a</a>)</cite>, we evaluate each method on <math alttext="10" class="ltx_Math" display="inline" id="S4.SS1.p3.1.m1.1"><semantics id="S4.SS1.p3.1.m1.1a"><mn id="S4.SS1.p3.1.m1.1.1" xref="S4.SS1.p3.1.m1.1.1.cmml">10</mn><annotation-xml encoding="MathML-Content" id="S4.SS1.p3.1.m1.1b"><cn id="S4.SS1.p3.1.m1.1.1.cmml" type="integer" xref="S4.SS1.p3.1.m1.1.1">10</cn></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p3.1.m1.1c">10</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p3.1.m1.1d">10</annotation></semantics></math> images from CBSD68 <cite class="ltx_cite ltx_citemacro_citep">(Martin et al., <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#bib.bib46" title="">2001</a>)</cite> and <math alttext="10" class="ltx_Math" display="inline" id="S4.SS1.p3.2.m2.1"><semantics id="S4.SS1.p3.2.m2.1a"><mn id="S4.SS1.p3.2.m2.1.1" xref="S4.SS1.p3.2.m2.1.1.cmml">10</mn><annotation-xml encoding="MathML-Content" id="S4.SS1.p3.2.m2.1b"><cn id="S4.SS1.p3.2.m2.1.1.cmml" type="integer" xref="S4.SS1.p3.2.m2.1.1">10</cn></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p3.2.m2.1c">10</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p3.2.m2.1d">10</annotation></semantics></math> blur kernels presented in Figure <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#S3.F1" title="Figure 1 ‣ 3.3 Biased algorithm analysis ‣ 3 Convergence Analysis ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_tag">1</span></a>. The same denoiser trained by <cite class="ltx_cite ltx_citemacro_citep">(Hurault et al., <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#bib.bib30" title="">2022a</a>)</cite> on natural images is used for all methods to ensure a fair comparison. We compare two variants of the Ann-SNORE algorithm, with gradient step on the data-fidelity (Algorithm <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#alg3" title="Algorithm 3 ‣ Optimization algorithms ‣ 2.2 SNORE regularization ‣ 2 Stochastic deNOising REgularization (SNORE) ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_tag">3</span></a>) or with a proximal step (Algorithm <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#alg5" title="Algorithm 5 ‣ SNORE Prox ‣ Appendix F Additional Experiments ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_tag">5</span></a> in Appendix <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#A6" title="Appendix F Additional Experiments ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_tag">F</span></a>). DiffPIR is used with <math alttext="t_{\text{start}}<t_{\text{trained}}" class="ltx_Math" display="inline" id="S4.SS1.p3.3.m3.1"><semantics id="S4.SS1.p3.3.m3.1a"><mrow id="S4.SS1.p3.3.m3.1.1" xref="S4.SS1.p3.3.m3.1.1.cmml"><msub id="S4.SS1.p3.3.m3.1.1.2" xref="S4.SS1.p3.3.m3.1.1.2.cmml"><mi id="S4.SS1.p3.3.m3.1.1.2.2" xref="S4.SS1.p3.3.m3.1.1.2.2.cmml">t</mi><mtext id="S4.SS1.p3.3.m3.1.1.2.3" xref="S4.SS1.p3.3.m3.1.1.2.3a.cmml">start</mtext></msub><mo id="S4.SS1.p3.3.m3.1.1.1" xref="S4.SS1.p3.3.m3.1.1.1.cmml"><</mo><msub id="S4.SS1.p3.3.m3.1.1.3" xref="S4.SS1.p3.3.m3.1.1.3.cmml"><mi id="S4.SS1.p3.3.m3.1.1.3.2" xref="S4.SS1.p3.3.m3.1.1.3.2.cmml">t</mi><mtext id="S4.SS1.p3.3.m3.1.1.3.3" xref="S4.SS1.p3.3.m3.1.1.3.3a.cmml">trained</mtext></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p3.3.m3.1b"><apply id="S4.SS1.p3.3.m3.1.1.cmml" xref="S4.SS1.p3.3.m3.1.1"><lt id="S4.SS1.p3.3.m3.1.1.1.cmml" xref="S4.SS1.p3.3.m3.1.1.1"></lt><apply id="S4.SS1.p3.3.m3.1.1.2.cmml" xref="S4.SS1.p3.3.m3.1.1.2"><csymbol cd="ambiguous" id="S4.SS1.p3.3.m3.1.1.2.1.cmml" xref="S4.SS1.p3.3.m3.1.1.2">subscript</csymbol><ci id="S4.SS1.p3.3.m3.1.1.2.2.cmml" xref="S4.SS1.p3.3.m3.1.1.2.2">𝑡</ci><ci id="S4.SS1.p3.3.m3.1.1.2.3a.cmml" xref="S4.SS1.p3.3.m3.1.1.2.3"><mtext id="S4.SS1.p3.3.m3.1.1.2.3.cmml" mathsize="70%" xref="S4.SS1.p3.3.m3.1.1.2.3">start</mtext></ci></apply><apply id="S4.SS1.p3.3.m3.1.1.3.cmml" xref="S4.SS1.p3.3.m3.1.1.3"><csymbol cd="ambiguous" id="S4.SS1.p3.3.m3.1.1.3.1.cmml" xref="S4.SS1.p3.3.m3.1.1.3">subscript</csymbol><ci id="S4.SS1.p3.3.m3.1.1.3.2.cmml" xref="S4.SS1.p3.3.m3.1.1.3.2">𝑡</ci><ci id="S4.SS1.p3.3.m3.1.1.3.3a.cmml" xref="S4.SS1.p3.3.m3.1.1.3.3"><mtext id="S4.SS1.p3.3.m3.1.1.3.3.cmml" mathsize="70%" xref="S4.SS1.p3.3.m3.1.1.3.3">trained</mtext></ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p3.3.m3.1c">t_{\text{start}}<t_{\text{trained}}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p3.3.m3.1d">italic_t start_POSTSUBSCRIPT start end_POSTSUBSCRIPT < italic_t start_POSTSUBSCRIPT trained end_POSTSUBSCRIPT</annotation></semantics></math> as suggested by the authors <cite class="ltx_cite ltx_citemacro_citep">(Zhu et al., <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#bib.bib80" title="">2023</a>)</cite>. However, this diffusion method does not outperform in our experiments because our denoiser is not trained to tackle highly-noised images. An extra-parameter <math alttext="\beta>0" class="ltx_Math" display="inline" id="S4.SS1.p3.4.m4.1"><semantics id="S4.SS1.p3.4.m4.1a"><mrow id="S4.SS1.p3.4.m4.1.1" xref="S4.SS1.p3.4.m4.1.1.cmml"><mi id="S4.SS1.p3.4.m4.1.1.2" xref="S4.SS1.p3.4.m4.1.1.2.cmml">β</mi><mo id="S4.SS1.p3.4.m4.1.1.1" xref="S4.SS1.p3.4.m4.1.1.1.cmml">></mo><mn id="S4.SS1.p3.4.m4.1.1.3" xref="S4.SS1.p3.4.m4.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p3.4.m4.1b"><apply id="S4.SS1.p3.4.m4.1.1.cmml" xref="S4.SS1.p3.4.m4.1.1"><gt id="S4.SS1.p3.4.m4.1.1.1.cmml" xref="S4.SS1.p3.4.m4.1.1.1"></gt><ci id="S4.SS1.p3.4.m4.1.1.2.cmml" xref="S4.SS1.p3.4.m4.1.1.2">𝛽</ci><cn id="S4.SS1.p3.4.m4.1.1.3.cmml" type="integer" xref="S4.SS1.p3.4.m4.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p3.4.m4.1c">\beta>0</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p3.4.m4.1d">italic_β > 0</annotation></semantics></math> is added to PnP SGD algorithm to increase the algorithm performances. On Table <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#S4.T1" title="Table 1 ‣ 4.1 Deblurring ‣ 4 Experiments ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_tag">1</span></a>, results are presented with distorsion metrics (PSNR, SSIM) and perceptual metrics with reference (LPIPS) and without reference (BRISQUE).</p> </div> <div class="ltx_para" id="S4.SS1.p4"> <p class="ltx_p" id="S4.SS1.p4.1">On Table <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#S4.T1" title="Table 1 ‣ 4.1 Deblurring ‣ 4 Experiments ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_tag">1</span></a>, we observe that Ann-SNORE has similar performance than other state-of-the-art methods. If we compare Ann-SNORE to RED Prox, which performs the best in terms of distortion, we observe that Ann-SNORE performs favorably in terms of perceptual metrics. However, Ann-SNORE remains slower than other methods for deblurring as it requires a sufficient number of annealing levels for the restored images to have a high visual quality (see discussion in Appendix <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#A6" title="Appendix F Additional Experiments ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_tag">F</span></a>).</p> </div> <div class="ltx_para" id="S4.SS1.p5"> <p class="ltx_p" id="S4.SS1.p5.1">On Figure <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#S3.F2" title="Figure 2 ‣ 3.3 Biased algorithm analysis ‣ 3 Convergence Analysis ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_tag">2</span></a>, we provide a qualitative comparison. Note that the Ann-SNORE algorithm provides a more realistic result than RED Prox with equivalent quantitative score (PSNR). The global decreasing behavior of the function <math alttext="\mathcal{J}" class="ltx_Math" display="inline" id="S4.SS1.p5.1.m1.1"><semantics id="S4.SS1.p5.1.m1.1a"><mi class="ltx_font_mathcaligraphic" id="S4.SS1.p5.1.m1.1.1" xref="S4.SS1.p5.1.m1.1.1.cmml">𝒥</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.p5.1.m1.1b"><ci id="S4.SS1.p5.1.m1.1.1.cmml" xref="S4.SS1.p5.1.m1.1.1">𝒥</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p5.1.m1.1c">\mathcal{J}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p5.1.m1.1d">caligraphic_J</annotation></semantics></math> (<a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#S4.E16" title="16 ‣ 4 Experiments ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_tag">16</span></a>) empirically confirms that our algorithm minimizes this function.</p> </div> <figure class="ltx_table" id="S4.T1"> <div class="ltx_inline-block ltx_align_center ltx_transformed_outer" id="S4.T1.4" style="width:433.6pt;height:536.1pt;vertical-align:-0.0pt;"><span class="ltx_transformed_inner" style="transform:translate(78.5pt,-97.1pt) scale(1.56761275071198,1.56761275071198) ;"> <table class="ltx_tabular ltx_guessed_headers ltx_align_middle" id="S4.T1.4.4"> <thead class="ltx_thead"> <tr class="ltx_tr" id="S4.T1.4.4.4"> <th class="ltx_td ltx_align_center ltx_th ltx_th_column" id="S4.T1.4.4.4.5">Noise level</th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column" id="S4.T1.4.4.4.6">Method</th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column" id="S4.T1.1.1.1.1">PSNR<math alttext="\uparrow" class="ltx_Math" display="inline" id="S4.T1.1.1.1.1.m1.1"><semantics id="S4.T1.1.1.1.1.m1.1a"><mo id="S4.T1.1.1.1.1.m1.1.1" stretchy="false" xref="S4.T1.1.1.1.1.m1.1.1.cmml">↑</mo><annotation-xml encoding="MathML-Content" id="S4.T1.1.1.1.1.m1.1b"><ci id="S4.T1.1.1.1.1.m1.1.1.cmml" xref="S4.T1.1.1.1.1.m1.1.1">↑</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.T1.1.1.1.1.m1.1c">\uparrow</annotation><annotation encoding="application/x-llamapun" id="S4.T1.1.1.1.1.m1.1d">↑</annotation></semantics></math> </th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column" id="S4.T1.2.2.2.2">SSIM<math alttext="\uparrow" class="ltx_Math" display="inline" id="S4.T1.2.2.2.2.m1.1"><semantics id="S4.T1.2.2.2.2.m1.1a"><mo id="S4.T1.2.2.2.2.m1.1.1" stretchy="false" xref="S4.T1.2.2.2.2.m1.1.1.cmml">↑</mo><annotation-xml encoding="MathML-Content" id="S4.T1.2.2.2.2.m1.1b"><ci id="S4.T1.2.2.2.2.m1.1.1.cmml" xref="S4.T1.2.2.2.2.m1.1.1">↑</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.T1.2.2.2.2.m1.1c">\uparrow</annotation><annotation encoding="application/x-llamapun" id="S4.T1.2.2.2.2.m1.1d">↑</annotation></semantics></math> </th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column" id="S4.T1.3.3.3.3">LPIPS<math alttext="\downarrow" class="ltx_Math" display="inline" id="S4.T1.3.3.3.3.m1.1"><semantics id="S4.T1.3.3.3.3.m1.1a"><mo id="S4.T1.3.3.3.3.m1.1.1" stretchy="false" xref="S4.T1.3.3.3.3.m1.1.1.cmml">↓</mo><annotation-xml encoding="MathML-Content" id="S4.T1.3.3.3.3.m1.1b"><ci id="S4.T1.3.3.3.3.m1.1.1.cmml" xref="S4.T1.3.3.3.3.m1.1.1">↓</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.T1.3.3.3.3.m1.1c">\downarrow</annotation><annotation encoding="application/x-llamapun" id="S4.T1.3.3.3.3.m1.1d">↓</annotation></semantics></math> </th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column" id="S4.T1.4.4.4.4">BRISQUE<math alttext="\downarrow" class="ltx_Math" display="inline" id="S4.T1.4.4.4.4.m1.1"><semantics id="S4.T1.4.4.4.4.m1.1a"><mo id="S4.T1.4.4.4.4.m1.1.1" stretchy="false" xref="S4.T1.4.4.4.4.m1.1.1.cmml">↓</mo><annotation-xml encoding="MathML-Content" id="S4.T1.4.4.4.4.m1.1b"><ci id="S4.T1.4.4.4.4.m1.1.1.cmml" xref="S4.T1.4.4.4.4.m1.1.1">↓</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.T1.4.4.4.4.m1.1c">\downarrow</annotation><annotation encoding="application/x-llamapun" id="S4.T1.4.4.4.4.m1.1d">↓</annotation></semantics></math> </th> </tr> </thead> <tbody class="ltx_tbody"> <tr class="ltx_tr" id="S4.T1.4.4.5.1"> <td class="ltx_td ltx_border_t" id="S4.T1.4.4.5.1.1"></td> <td class="ltx_td ltx_align_center ltx_border_t" id="S4.T1.4.4.5.1.2">RED</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S4.T1.4.4.5.1.3">29.82</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S4.T1.4.4.5.1.4">0.84</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S4.T1.4.4.5.1.5">0.17</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S4.T1.4.4.5.1.6"><span class="ltx_text ltx_framed_underline" id="S4.T1.4.4.5.1.6.1">21.09</span></td> </tr> <tr class="ltx_tr" id="S4.T1.4.4.6.2"> <td class="ltx_td" id="S4.T1.4.4.6.2.1"></td> <td class="ltx_td ltx_align_center" id="S4.T1.4.4.6.2.2">RED Prox</td> <td class="ltx_td ltx_align_center" id="S4.T1.4.4.6.2.3"><span class="ltx_text ltx_font_bold" id="S4.T1.4.4.6.2.3.1">30.64</span></td> <td class="ltx_td ltx_align_center" id="S4.T1.4.4.6.2.4"><span class="ltx_text ltx_font_bold" id="S4.T1.4.4.6.2.4.1">0.87</span></td> <td class="ltx_td ltx_align_center" id="S4.T1.4.4.6.2.5"><span class="ltx_text ltx_font_bold" id="S4.T1.4.4.6.2.5.1">0.15</span></td> <td class="ltx_td ltx_align_center" id="S4.T1.4.4.6.2.6">45.27</td> </tr> <tr class="ltx_tr" id="S4.T1.4.4.7.3"> <td class="ltx_td ltx_align_center" id="S4.T1.4.4.7.3.1">5/255</td> <td class="ltx_td ltx_align_center" id="S4.T1.4.4.7.3.2">Ann-SNORE</td> <td class="ltx_td ltx_align_center" id="S4.T1.4.4.7.3.3"><span class="ltx_text ltx_framed_underline" id="S4.T1.4.4.7.3.3.1">29.92</span></td> <td class="ltx_td ltx_align_center" id="S4.T1.4.4.7.3.4"><span class="ltx_text ltx_framed_underline" id="S4.T1.4.4.7.3.4.1">0.85</span></td> <td class="ltx_td ltx_align_center" id="S4.T1.4.4.7.3.5">0.17</td> <td class="ltx_td ltx_align_center" id="S4.T1.4.4.7.3.6">28.24</td> </tr> <tr class="ltx_tr" id="S4.T1.4.4.8.4"> <td class="ltx_td" id="S4.T1.4.4.8.4.1"></td> <td class="ltx_td ltx_align_center" id="S4.T1.4.4.8.4.2">Ann-SNORE Prox</td> <td class="ltx_td ltx_align_center" id="S4.T1.4.4.8.4.3"><span class="ltx_text ltx_framed_underline" id="S4.T1.4.4.8.4.3.1">29.92</span></td> <td class="ltx_td ltx_align_center" id="S4.T1.4.4.8.4.4"><span class="ltx_text ltx_framed_underline" id="S4.T1.4.4.8.4.4.1">0.85</span></td> <td class="ltx_td ltx_align_center" id="S4.T1.4.4.8.4.5">0.17</td> <td class="ltx_td ltx_align_center" id="S4.T1.4.4.8.4.6">28.25</td> </tr> <tr class="ltx_tr" id="S4.T1.4.4.9.5"> <td class="ltx_td" id="S4.T1.4.4.9.5.1"></td> <td class="ltx_td ltx_align_center" id="S4.T1.4.4.9.5.2">DiffPIR</td> <td class="ltx_td ltx_align_center" id="S4.T1.4.4.9.5.3">28.55</td> <td class="ltx_td ltx_align_center" id="S4.T1.4.4.9.5.4">0.76</td> <td class="ltx_td ltx_align_center" id="S4.T1.4.4.9.5.5"><span class="ltx_text ltx_framed_underline" id="S4.T1.4.4.9.5.5.1">0.16</span></td> <td class="ltx_td ltx_align_center" id="S4.T1.4.4.9.5.6"><span class="ltx_text ltx_font_bold" id="S4.T1.4.4.9.5.6.1">17.87</span></td> </tr> <tr class="ltx_tr" id="S4.T1.4.4.10.6"> <td class="ltx_td" id="S4.T1.4.4.10.6.1"></td> <td class="ltx_td ltx_align_center" id="S4.T1.4.4.10.6.2">PnP SGD</td> <td class="ltx_td ltx_align_center" id="S4.T1.4.4.10.6.3">29.37</td> <td class="ltx_td ltx_align_center" id="S4.T1.4.4.10.6.4">0.83</td> <td class="ltx_td ltx_align_center" id="S4.T1.4.4.10.6.5">0.20</td> <td class="ltx_td ltx_align_center" id="S4.T1.4.4.10.6.6">26.44</td> </tr> <tr class="ltx_tr" id="S4.T1.4.4.11.7"> <td class="ltx_td ltx_border_t" id="S4.T1.4.4.11.7.1"></td> <td class="ltx_td ltx_align_center ltx_border_t" id="S4.T1.4.4.11.7.2">RED</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S4.T1.4.4.11.7.3">27.18</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S4.T1.4.4.11.7.4">0.72</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S4.T1.4.4.11.7.5">0.25</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S4.T1.4.4.11.7.6"><span class="ltx_text ltx_font_bold" id="S4.T1.4.4.11.7.6.1">20.16</span></td> </tr> <tr class="ltx_tr" id="S4.T1.4.4.12.8"> <td class="ltx_td" id="S4.T1.4.4.12.8.1"></td> <td class="ltx_td ltx_align_center" id="S4.T1.4.4.12.8.2">RED Prox</td> <td class="ltx_td ltx_align_center" id="S4.T1.4.4.12.8.3"><span class="ltx_text ltx_font_bold" id="S4.T1.4.4.12.8.3.1">28.50</span></td> <td class="ltx_td ltx_align_center" id="S4.T1.4.4.12.8.4"><span class="ltx_text ltx_font_bold" id="S4.T1.4.4.12.8.4.1">0.80</span></td> <td class="ltx_td ltx_align_center" id="S4.T1.4.4.12.8.5"><span class="ltx_text ltx_font_bold" id="S4.T1.4.4.12.8.5.1">0.23</span></td> <td class="ltx_td ltx_align_center" id="S4.T1.4.4.12.8.6">51.41</td> </tr> <tr class="ltx_tr" id="S4.T1.4.4.13.9"> <td class="ltx_td ltx_align_center" id="S4.T1.4.4.13.9.1">10/255</td> <td class="ltx_td ltx_align_center" id="S4.T1.4.4.13.9.2">Ann-SNORE</td> <td class="ltx_td ltx_align_center" id="S4.T1.4.4.13.9.3"><span class="ltx_text ltx_framed_underline" id="S4.T1.4.4.13.9.3.1">27.91</span></td> <td class="ltx_td ltx_align_center" id="S4.T1.4.4.13.9.4"><span class="ltx_text ltx_framed_underline" id="S4.T1.4.4.13.9.4.1">0.78</span></td> <td class="ltx_td ltx_align_center" id="S4.T1.4.4.13.9.5"><span class="ltx_text ltx_framed_underline" id="S4.T1.4.4.13.9.5.1">0.24</span></td> <td class="ltx_td ltx_align_center" id="S4.T1.4.4.13.9.6">27.89</td> </tr> <tr class="ltx_tr" id="S4.T1.4.4.14.10"> <td class="ltx_td" id="S4.T1.4.4.14.10.1"></td> <td class="ltx_td ltx_align_center" id="S4.T1.4.4.14.10.2">Ann-SNORE Prox</td> <td class="ltx_td ltx_align_center" id="S4.T1.4.4.14.10.3"><span class="ltx_text ltx_framed_underline" id="S4.T1.4.4.14.10.3.1">27.91</span></td> <td class="ltx_td ltx_align_center" id="S4.T1.4.4.14.10.4"><span class="ltx_text ltx_framed_underline" id="S4.T1.4.4.14.10.4.1">0.78</span></td> <td class="ltx_td ltx_align_center" id="S4.T1.4.4.14.10.5"><span class="ltx_text ltx_font_bold" id="S4.T1.4.4.14.10.5.1">0.23</span></td> <td class="ltx_td ltx_align_center" id="S4.T1.4.4.14.10.6">27.97</td> </tr> <tr class="ltx_tr" id="S4.T1.4.4.15.11"> <td class="ltx_td" id="S4.T1.4.4.15.11.1"></td> <td class="ltx_td ltx_align_center" id="S4.T1.4.4.15.11.2">DiffPIR</td> <td class="ltx_td ltx_align_center" id="S4.T1.4.4.15.11.3">27.47</td> <td class="ltx_td ltx_align_center" id="S4.T1.4.4.15.11.4">0.74</td> <td class="ltx_td ltx_align_center" id="S4.T1.4.4.15.11.5"><span class="ltx_text ltx_framed_underline" id="S4.T1.4.4.15.11.5.1">0.24</span></td> <td class="ltx_td ltx_align_center" id="S4.T1.4.4.15.11.6"><span class="ltx_text ltx_framed_underline" id="S4.T1.4.4.15.11.6.1">21.12</span></td> </tr> <tr class="ltx_tr" id="S4.T1.4.4.16.12"> <td class="ltx_td" id="S4.T1.4.4.16.12.1"></td> <td class="ltx_td ltx_align_center" id="S4.T1.4.4.16.12.2">PnP SGD</td> <td class="ltx_td ltx_align_center" id="S4.T1.4.4.16.12.3">27.61</td> <td class="ltx_td ltx_align_center" id="S4.T1.4.4.16.12.4">0.75</td> <td class="ltx_td ltx_align_center" id="S4.T1.4.4.16.12.5">0.26</td> <td class="ltx_td ltx_align_center" id="S4.T1.4.4.16.12.6">25.74</td> </tr> <tr class="ltx_tr" id="S4.T1.4.4.17.13"> <td class="ltx_td ltx_border_t" id="S4.T1.4.4.17.13.1"></td> <td class="ltx_td ltx_align_center ltx_border_t" id="S4.T1.4.4.17.13.2">RED</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S4.T1.4.4.17.13.3">24.03</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S4.T1.4.4.17.13.4">0.54</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S4.T1.4.4.17.13.5">0.43</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S4.T1.4.4.17.13.6"><span class="ltx_text ltx_font_bold" id="S4.T1.4.4.17.13.6.1">20.75</span></td> </tr> <tr class="ltx_tr" id="S4.T1.4.4.18.14"> <td class="ltx_td" id="S4.T1.4.4.18.14.1"></td> <td class="ltx_td ltx_align_center" id="S4.T1.4.4.18.14.2">RED Prox</td> <td class="ltx_td ltx_align_center" id="S4.T1.4.4.18.14.3"><span class="ltx_text ltx_font_bold" id="S4.T1.4.4.18.14.3.1">26.31</span></td> <td class="ltx_td ltx_align_center" id="S4.T1.4.4.18.14.4"><span class="ltx_text ltx_font_bold" id="S4.T1.4.4.18.14.4.1">0.71</span></td> <td class="ltx_td ltx_align_center" id="S4.T1.4.4.18.14.5"><span class="ltx_text ltx_font_bold" id="S4.T1.4.4.18.14.5.1">0.31</span></td> <td class="ltx_td ltx_align_center" id="S4.T1.4.4.18.14.6">55.48</td> </tr> <tr class="ltx_tr" id="S4.T1.4.4.19.15"> <td class="ltx_td ltx_align_center" id="S4.T1.4.4.19.15.1">20/255</td> <td class="ltx_td ltx_align_center" id="S4.T1.4.4.19.15.2">Ann-SNORE</td> <td class="ltx_td ltx_align_center" id="S4.T1.4.4.19.15.3">25.61</td> <td class="ltx_td ltx_align_center" id="S4.T1.4.4.19.15.4">0.66</td> <td class="ltx_td ltx_align_center" id="S4.T1.4.4.19.15.5"><span class="ltx_text ltx_framed_underline" id="S4.T1.4.4.19.15.5.1">0.32</span></td> <td class="ltx_td ltx_align_center" id="S4.T1.4.4.19.15.6">28.88</td> </tr> <tr class="ltx_tr" id="S4.T1.4.4.20.16"> <td class="ltx_td" id="S4.T1.4.4.20.16.1"></td> <td class="ltx_td ltx_align_center" id="S4.T1.4.4.20.16.2">Ann-SNORE Prox</td> <td class="ltx_td ltx_align_center" id="S4.T1.4.4.20.16.3">25.61</td> <td class="ltx_td ltx_align_center" id="S4.T1.4.4.20.16.4">0.66</td> <td class="ltx_td ltx_align_center" id="S4.T1.4.4.20.16.5"><span class="ltx_text ltx_framed_underline" id="S4.T1.4.4.20.16.5.1">0.32</span></td> <td class="ltx_td ltx_align_center" id="S4.T1.4.4.20.16.6">29.00</td> </tr> <tr class="ltx_tr" id="S4.T1.4.4.21.17"> <td class="ltx_td" id="S4.T1.4.4.21.17.1"></td> <td class="ltx_td ltx_align_center" id="S4.T1.4.4.21.17.2">DiffPIR</td> <td class="ltx_td ltx_align_center" id="S4.T1.4.4.21.17.3"><span class="ltx_text ltx_framed_underline" id="S4.T1.4.4.21.17.3.1">26.05</span></td> <td class="ltx_td ltx_align_center" id="S4.T1.4.4.21.17.4"><span class="ltx_text ltx_framed_underline" id="S4.T1.4.4.21.17.4.1">0.69</span></td> <td class="ltx_td ltx_align_center" id="S4.T1.4.4.21.17.5">0.33</td> <td class="ltx_td ltx_align_center" id="S4.T1.4.4.21.17.6">33.57</td> </tr> <tr class="ltx_tr" id="S4.T1.4.4.22.18"> <td class="ltx_td ltx_border_b" id="S4.T1.4.4.22.18.1"></td> <td class="ltx_td ltx_align_center ltx_border_b" id="S4.T1.4.4.22.18.2">PnP SGD</td> <td class="ltx_td ltx_align_center ltx_border_b" id="S4.T1.4.4.22.18.3">25.50</td> <td class="ltx_td ltx_align_center ltx_border_b" id="S4.T1.4.4.22.18.4">0.63</td> <td class="ltx_td ltx_align_center ltx_border_b" id="S4.T1.4.4.22.18.5">0.33</td> <td class="ltx_td ltx_align_center ltx_border_b" id="S4.T1.4.4.22.18.6"><span class="ltx_text ltx_framed_underline" id="S4.T1.4.4.22.18.6.1">22.67</span></td> </tr> </tbody> </table> </span></div> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_table"><span class="ltx_text" id="S4.T1.10.3.1" style="font-size:90%;">Table 1</span>: </span><span class="ltx_text" id="S4.T1.8.2" style="font-size:90%;">Quantitative comparisons of image deblurring methods on CBSD10 with <math alttext="10" class="ltx_Math" display="inline" id="S4.T1.7.1.m1.1"><semantics id="S4.T1.7.1.m1.1b"><mn id="S4.T1.7.1.m1.1.1" xref="S4.T1.7.1.m1.1.1.cmml">10</mn><annotation-xml encoding="MathML-Content" id="S4.T1.7.1.m1.1c"><cn id="S4.T1.7.1.m1.1.1.cmml" type="integer" xref="S4.T1.7.1.m1.1.1">10</cn></annotation-xml><annotation encoding="application/x-tex" id="S4.T1.7.1.m1.1d">10</annotation><annotation encoding="application/x-llamapun" id="S4.T1.7.1.m1.1e">10</annotation></semantics></math> different blur kernels (see Figure <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#S3.F1" title="Figure 1 ‣ 3.3 Biased algorithm analysis ‣ 3 Convergence Analysis ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_tag">1</span></a>) and three different level of noise <math alttext="{\sigma_{{\mathbf{y}}}\in\{5,10,20\}/255}" class="ltx_Math" display="inline" id="S4.T1.8.2.m2.3"><semantics id="S4.T1.8.2.m2.3b"><mrow id="S4.T1.8.2.m2.3.4" xref="S4.T1.8.2.m2.3.4.cmml"><msub id="S4.T1.8.2.m2.3.4.2" xref="S4.T1.8.2.m2.3.4.2.cmml"><mi id="S4.T1.8.2.m2.3.4.2.2" xref="S4.T1.8.2.m2.3.4.2.2.cmml">σ</mi><mi id="S4.T1.8.2.m2.3.4.2.3" xref="S4.T1.8.2.m2.3.4.2.3.cmml">𝐲</mi></msub><mo id="S4.T1.8.2.m2.3.4.1" xref="S4.T1.8.2.m2.3.4.1.cmml">∈</mo><mrow id="S4.T1.8.2.m2.3.4.3" xref="S4.T1.8.2.m2.3.4.3.cmml"><mrow id="S4.T1.8.2.m2.3.4.3.2.2" xref="S4.T1.8.2.m2.3.4.3.2.1.cmml"><mo id="S4.T1.8.2.m2.3.4.3.2.2.1" stretchy="false" xref="S4.T1.8.2.m2.3.4.3.2.1.cmml">{</mo><mn id="S4.T1.8.2.m2.1.1" xref="S4.T1.8.2.m2.1.1.cmml">5</mn><mo id="S4.T1.8.2.m2.3.4.3.2.2.2" xref="S4.T1.8.2.m2.3.4.3.2.1.cmml">,</mo><mn id="S4.T1.8.2.m2.2.2" xref="S4.T1.8.2.m2.2.2.cmml">10</mn><mo id="S4.T1.8.2.m2.3.4.3.2.2.3" xref="S4.T1.8.2.m2.3.4.3.2.1.cmml">,</mo><mn id="S4.T1.8.2.m2.3.3" xref="S4.T1.8.2.m2.3.3.cmml">20</mn><mo id="S4.T1.8.2.m2.3.4.3.2.2.4" stretchy="false" xref="S4.T1.8.2.m2.3.4.3.2.1.cmml">}</mo></mrow><mo id="S4.T1.8.2.m2.3.4.3.1" xref="S4.T1.8.2.m2.3.4.3.1.cmml">/</mo><mn id="S4.T1.8.2.m2.3.4.3.3" xref="S4.T1.8.2.m2.3.4.3.3.cmml">255</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.T1.8.2.m2.3c"><apply id="S4.T1.8.2.m2.3.4.cmml" xref="S4.T1.8.2.m2.3.4"><in id="S4.T1.8.2.m2.3.4.1.cmml" xref="S4.T1.8.2.m2.3.4.1"></in><apply id="S4.T1.8.2.m2.3.4.2.cmml" xref="S4.T1.8.2.m2.3.4.2"><csymbol cd="ambiguous" id="S4.T1.8.2.m2.3.4.2.1.cmml" xref="S4.T1.8.2.m2.3.4.2">subscript</csymbol><ci id="S4.T1.8.2.m2.3.4.2.2.cmml" xref="S4.T1.8.2.m2.3.4.2.2">𝜎</ci><ci id="S4.T1.8.2.m2.3.4.2.3.cmml" xref="S4.T1.8.2.m2.3.4.2.3">𝐲</ci></apply><apply id="S4.T1.8.2.m2.3.4.3.cmml" xref="S4.T1.8.2.m2.3.4.3"><divide id="S4.T1.8.2.m2.3.4.3.1.cmml" xref="S4.T1.8.2.m2.3.4.3.1"></divide><set id="S4.T1.8.2.m2.3.4.3.2.1.cmml" xref="S4.T1.8.2.m2.3.4.3.2.2"><cn id="S4.T1.8.2.m2.1.1.cmml" type="integer" xref="S4.T1.8.2.m2.1.1">5</cn><cn id="S4.T1.8.2.m2.2.2.cmml" type="integer" xref="S4.T1.8.2.m2.2.2">10</cn><cn id="S4.T1.8.2.m2.3.3.cmml" type="integer" xref="S4.T1.8.2.m2.3.3">20</cn></set><cn id="S4.T1.8.2.m2.3.4.3.3.cmml" type="integer" xref="S4.T1.8.2.m2.3.4.3.3">255</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.T1.8.2.m2.3d">{\sigma_{{\mathbf{y}}}\in\{5,10,20\}/255}</annotation><annotation encoding="application/x-llamapun" id="S4.T1.8.2.m2.3e">italic_σ start_POSTSUBSCRIPT bold_y end_POSTSUBSCRIPT ∈ { 5 , 10 , 20 } / 255</annotation></semantics></math>. Best and second best results are respectively displayed in bold and underlined.</span></figcaption> </figure> </section> <section class="ltx_subsection" id="S4.SS2"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">4.2 </span>Inpainting</h3> <div class="ltx_para" id="S4.SS2.p1"> <p class="ltx_p" id="S4.SS2.p1.6">For image inpainting, the degradation operator <math alttext="{\mathbf{A}}" class="ltx_Math" display="inline" id="S4.SS2.p1.1.m1.1"><semantics id="S4.SS2.p1.1.m1.1a"><mi id="S4.SS2.p1.1.m1.1.1" xref="S4.SS2.p1.1.m1.1.1.cmml">𝐀</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.p1.1.m1.1b"><ci id="S4.SS2.p1.1.m1.1.1.cmml" xref="S4.SS2.p1.1.m1.1.1">𝐀</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p1.1.m1.1c">{\mathbf{A}}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p1.1.m1.1d">bold_A</annotation></semantics></math> is a diagonal matrix with coefficient in <math alttext="\{0,1\}" class="ltx_Math" display="inline" id="S4.SS2.p1.2.m2.2"><semantics id="S4.SS2.p1.2.m2.2a"><mrow id="S4.SS2.p1.2.m2.2.3.2" xref="S4.SS2.p1.2.m2.2.3.1.cmml"><mo id="S4.SS2.p1.2.m2.2.3.2.1" stretchy="false" xref="S4.SS2.p1.2.m2.2.3.1.cmml">{</mo><mn id="S4.SS2.p1.2.m2.1.1" xref="S4.SS2.p1.2.m2.1.1.cmml">0</mn><mo id="S4.SS2.p1.2.m2.2.3.2.2" xref="S4.SS2.p1.2.m2.2.3.1.cmml">,</mo><mn id="S4.SS2.p1.2.m2.2.2" xref="S4.SS2.p1.2.m2.2.2.cmml">1</mn><mo id="S4.SS2.p1.2.m2.2.3.2.3" stretchy="false" xref="S4.SS2.p1.2.m2.2.3.1.cmml">}</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p1.2.m2.2b"><set id="S4.SS2.p1.2.m2.2.3.1.cmml" xref="S4.SS2.p1.2.m2.2.3.2"><cn id="S4.SS2.p1.2.m2.1.1.cmml" type="integer" xref="S4.SS2.p1.2.m2.1.1">0</cn><cn id="S4.SS2.p1.2.m2.2.2.cmml" type="integer" xref="S4.SS2.p1.2.m2.2.2">1</cn></set></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p1.2.m2.2c">\{0,1\}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p1.2.m2.2d">{ 0 , 1 }</annotation></semantics></math>. No noise is added to the degraded observation, <math alttext="{\mathbf{y}}={\mathbf{A}}{\mathbf{x}}" class="ltx_Math" display="inline" id="S4.SS2.p1.3.m3.1"><semantics id="S4.SS2.p1.3.m3.1a"><mrow id="S4.SS2.p1.3.m3.1.1" xref="S4.SS2.p1.3.m3.1.1.cmml"><mi id="S4.SS2.p1.3.m3.1.1.2" xref="S4.SS2.p1.3.m3.1.1.2.cmml">𝐲</mi><mo id="S4.SS2.p1.3.m3.1.1.1" xref="S4.SS2.p1.3.m3.1.1.1.cmml">=</mo><mi id="S4.SS2.p1.3.m3.1.1.3" xref="S4.SS2.p1.3.m3.1.1.3.cmml">𝐀𝐱</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p1.3.m3.1b"><apply id="S4.SS2.p1.3.m3.1.1.cmml" xref="S4.SS2.p1.3.m3.1.1"><eq id="S4.SS2.p1.3.m3.1.1.1.cmml" xref="S4.SS2.p1.3.m3.1.1.1"></eq><ci id="S4.SS2.p1.3.m3.1.1.2.cmml" xref="S4.SS2.p1.3.m3.1.1.2">𝐲</ci><ci id="S4.SS2.p1.3.m3.1.1.3.cmml" xref="S4.SS2.p1.3.m3.1.1.3">𝐀𝐱</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p1.3.m3.1c">{\mathbf{y}}={\mathbf{A}}{\mathbf{x}}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p1.3.m3.1d">bold_y = bold_Ax</annotation></semantics></math>. The proximal operator of <math alttext="\mathcal{F}" class="ltx_Math" display="inline" id="S4.SS2.p1.4.m4.1"><semantics id="S4.SS2.p1.4.m4.1a"><mi class="ltx_font_mathcaligraphic" id="S4.SS2.p1.4.m4.1.1" xref="S4.SS2.p1.4.m4.1.1.cmml">ℱ</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.p1.4.m4.1b"><ci id="S4.SS2.p1.4.m4.1.1.cmml" xref="S4.SS2.p1.4.m4.1.1">ℱ</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p1.4.m4.1c">\mathcal{F}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p1.4.m4.1d">caligraphic_F</annotation></semantics></math> is the orthogonal projection which imposes observed pixels values. This strict condition is relaxed in RED (Algorithm <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#alg1" title="Algorithm 1 ‣ 2.1 Background ‣ 2 Stochastic deNOising REgularization (SNORE) ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_tag">1</span></a>) and Ann-SNORE (Algorithm <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#alg3" title="Algorithm 3 ‣ Optimization algorithms ‣ 2.2 SNORE regularization ‣ 2 Stochastic deNOising REgularization (SNORE) ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_tag">3</span></a>) by taking <math alttext="\nabla\mathcal{F}({\mathbf{x}},{\mathbf{y}})={\mathbf{A}}({\mathbf{x}}-{% \mathbf{y}})" class="ltx_Math" display="inline" id="S4.SS2.p1.5.m5.3"><semantics id="S4.SS2.p1.5.m5.3a"><mrow id="S4.SS2.p1.5.m5.3.3" xref="S4.SS2.p1.5.m5.3.3.cmml"><mrow id="S4.SS2.p1.5.m5.3.3.3" xref="S4.SS2.p1.5.m5.3.3.3.cmml"><mrow id="S4.SS2.p1.5.m5.3.3.3.2" xref="S4.SS2.p1.5.m5.3.3.3.2.cmml"><mo id="S4.SS2.p1.5.m5.3.3.3.2.1" rspace="0.167em" xref="S4.SS2.p1.5.m5.3.3.3.2.1.cmml">∇</mo><mi class="ltx_font_mathcaligraphic" id="S4.SS2.p1.5.m5.3.3.3.2.2" xref="S4.SS2.p1.5.m5.3.3.3.2.2.cmml">ℱ</mi></mrow><mo id="S4.SS2.p1.5.m5.3.3.3.1" xref="S4.SS2.p1.5.m5.3.3.3.1.cmml">⁢</mo><mrow id="S4.SS2.p1.5.m5.3.3.3.3.2" xref="S4.SS2.p1.5.m5.3.3.3.3.1.cmml"><mo id="S4.SS2.p1.5.m5.3.3.3.3.2.1" stretchy="false" xref="S4.SS2.p1.5.m5.3.3.3.3.1.cmml">(</mo><mi id="S4.SS2.p1.5.m5.1.1" xref="S4.SS2.p1.5.m5.1.1.cmml">𝐱</mi><mo id="S4.SS2.p1.5.m5.3.3.3.3.2.2" xref="S4.SS2.p1.5.m5.3.3.3.3.1.cmml">,</mo><mi id="S4.SS2.p1.5.m5.2.2" xref="S4.SS2.p1.5.m5.2.2.cmml">𝐲</mi><mo id="S4.SS2.p1.5.m5.3.3.3.3.2.3" stretchy="false" xref="S4.SS2.p1.5.m5.3.3.3.3.1.cmml">)</mo></mrow></mrow><mo id="S4.SS2.p1.5.m5.3.3.2" xref="S4.SS2.p1.5.m5.3.3.2.cmml">=</mo><mrow id="S4.SS2.p1.5.m5.3.3.1" xref="S4.SS2.p1.5.m5.3.3.1.cmml"><mi id="S4.SS2.p1.5.m5.3.3.1.3" xref="S4.SS2.p1.5.m5.3.3.1.3.cmml">𝐀</mi><mo id="S4.SS2.p1.5.m5.3.3.1.2" xref="S4.SS2.p1.5.m5.3.3.1.2.cmml">⁢</mo><mrow id="S4.SS2.p1.5.m5.3.3.1.1.1" xref="S4.SS2.p1.5.m5.3.3.1.1.1.1.cmml"><mo id="S4.SS2.p1.5.m5.3.3.1.1.1.2" stretchy="false" xref="S4.SS2.p1.5.m5.3.3.1.1.1.1.cmml">(</mo><mrow id="S4.SS2.p1.5.m5.3.3.1.1.1.1" xref="S4.SS2.p1.5.m5.3.3.1.1.1.1.cmml"><mi id="S4.SS2.p1.5.m5.3.3.1.1.1.1.2" xref="S4.SS2.p1.5.m5.3.3.1.1.1.1.2.cmml">𝐱</mi><mo id="S4.SS2.p1.5.m5.3.3.1.1.1.1.1" xref="S4.SS2.p1.5.m5.3.3.1.1.1.1.1.cmml">−</mo><mi id="S4.SS2.p1.5.m5.3.3.1.1.1.1.3" xref="S4.SS2.p1.5.m5.3.3.1.1.1.1.3.cmml">𝐲</mi></mrow><mo id="S4.SS2.p1.5.m5.3.3.1.1.1.3" stretchy="false" xref="S4.SS2.p1.5.m5.3.3.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p1.5.m5.3b"><apply id="S4.SS2.p1.5.m5.3.3.cmml" xref="S4.SS2.p1.5.m5.3.3"><eq id="S4.SS2.p1.5.m5.3.3.2.cmml" xref="S4.SS2.p1.5.m5.3.3.2"></eq><apply id="S4.SS2.p1.5.m5.3.3.3.cmml" xref="S4.SS2.p1.5.m5.3.3.3"><times id="S4.SS2.p1.5.m5.3.3.3.1.cmml" xref="S4.SS2.p1.5.m5.3.3.3.1"></times><apply id="S4.SS2.p1.5.m5.3.3.3.2.cmml" xref="S4.SS2.p1.5.m5.3.3.3.2"><ci id="S4.SS2.p1.5.m5.3.3.3.2.1.cmml" xref="S4.SS2.p1.5.m5.3.3.3.2.1">∇</ci><ci id="S4.SS2.p1.5.m5.3.3.3.2.2.cmml" xref="S4.SS2.p1.5.m5.3.3.3.2.2">ℱ</ci></apply><interval closure="open" id="S4.SS2.p1.5.m5.3.3.3.3.1.cmml" xref="S4.SS2.p1.5.m5.3.3.3.3.2"><ci id="S4.SS2.p1.5.m5.1.1.cmml" xref="S4.SS2.p1.5.m5.1.1">𝐱</ci><ci id="S4.SS2.p1.5.m5.2.2.cmml" xref="S4.SS2.p1.5.m5.2.2">𝐲</ci></interval></apply><apply id="S4.SS2.p1.5.m5.3.3.1.cmml" xref="S4.SS2.p1.5.m5.3.3.1"><times id="S4.SS2.p1.5.m5.3.3.1.2.cmml" xref="S4.SS2.p1.5.m5.3.3.1.2"></times><ci id="S4.SS2.p1.5.m5.3.3.1.3.cmml" xref="S4.SS2.p1.5.m5.3.3.1.3">𝐀</ci><apply id="S4.SS2.p1.5.m5.3.3.1.1.1.1.cmml" xref="S4.SS2.p1.5.m5.3.3.1.1.1"><minus id="S4.SS2.p1.5.m5.3.3.1.1.1.1.1.cmml" xref="S4.SS2.p1.5.m5.3.3.1.1.1.1.1"></minus><ci id="S4.SS2.p1.5.m5.3.3.1.1.1.1.2.cmml" xref="S4.SS2.p1.5.m5.3.3.1.1.1.1.2">𝐱</ci><ci id="S4.SS2.p1.5.m5.3.3.1.1.1.1.3.cmml" xref="S4.SS2.p1.5.m5.3.3.1.1.1.1.3">𝐲</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p1.5.m5.3c">\nabla\mathcal{F}({\mathbf{x}},{\mathbf{y}})={\mathbf{A}}({\mathbf{x}}-{% \mathbf{y}})</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p1.5.m5.3d">∇ caligraphic_F ( bold_x , bold_y ) = bold_A ( bold_x - bold_y )</annotation></semantics></math>. We focus on random mask inpainting with a proportion of masked pixels <math alttext="p=0.5" class="ltx_Math" display="inline" id="S4.SS2.p1.6.m6.1"><semantics id="S4.SS2.p1.6.m6.1a"><mrow id="S4.SS2.p1.6.m6.1.1" xref="S4.SS2.p1.6.m6.1.1.cmml"><mi id="S4.SS2.p1.6.m6.1.1.2" xref="S4.SS2.p1.6.m6.1.1.2.cmml">p</mi><mo id="S4.SS2.p1.6.m6.1.1.1" xref="S4.SS2.p1.6.m6.1.1.1.cmml">=</mo><mn id="S4.SS2.p1.6.m6.1.1.3" xref="S4.SS2.p1.6.m6.1.1.3.cmml">0.5</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p1.6.m6.1b"><apply id="S4.SS2.p1.6.m6.1.1.cmml" xref="S4.SS2.p1.6.m6.1.1"><eq id="S4.SS2.p1.6.m6.1.1.1.cmml" xref="S4.SS2.p1.6.m6.1.1.1"></eq><ci id="S4.SS2.p1.6.m6.1.1.2.cmml" xref="S4.SS2.p1.6.m6.1.1.2">𝑝</ci><cn id="S4.SS2.p1.6.m6.1.1.3.cmml" type="float" xref="S4.SS2.p1.6.m6.1.1.3">0.5</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p1.6.m6.1c">p=0.5</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p1.6.m6.1d">italic_p = 0.5</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S4.SS2.p2"> <p class="ltx_p" id="S4.SS2.p2.7">For Ann-SNORE, we keep the same annealing scheme than for deblurring with <math alttext="500" class="ltx_Math" display="inline" id="S4.SS2.p2.1.m1.1"><semantics id="S4.SS2.p2.1.m1.1a"><mn id="S4.SS2.p2.1.m1.1.1" xref="S4.SS2.p2.1.m1.1.1.cmml">500</mn><annotation-xml encoding="MathML-Content" id="S4.SS2.p2.1.m1.1b"><cn id="S4.SS2.p2.1.m1.1.1.cmml" type="integer" xref="S4.SS2.p2.1.m1.1.1">500</cn></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p2.1.m1.1c">500</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p2.1.m1.1d">500</annotation></semantics></math> iterations. We set <math alttext="\sigma_{0}=50/255" class="ltx_Math" display="inline" id="S4.SS2.p2.2.m2.1"><semantics id="S4.SS2.p2.2.m2.1a"><mrow id="S4.SS2.p2.2.m2.1.1" xref="S4.SS2.p2.2.m2.1.1.cmml"><msub id="S4.SS2.p2.2.m2.1.1.2" xref="S4.SS2.p2.2.m2.1.1.2.cmml"><mi id="S4.SS2.p2.2.m2.1.1.2.2" xref="S4.SS2.p2.2.m2.1.1.2.2.cmml">σ</mi><mn id="S4.SS2.p2.2.m2.1.1.2.3" xref="S4.SS2.p2.2.m2.1.1.2.3.cmml">0</mn></msub><mo id="S4.SS2.p2.2.m2.1.1.1" xref="S4.SS2.p2.2.m2.1.1.1.cmml">=</mo><mrow id="S4.SS2.p2.2.m2.1.1.3" xref="S4.SS2.p2.2.m2.1.1.3.cmml"><mn id="S4.SS2.p2.2.m2.1.1.3.2" xref="S4.SS2.p2.2.m2.1.1.3.2.cmml">50</mn><mo id="S4.SS2.p2.2.m2.1.1.3.1" xref="S4.SS2.p2.2.m2.1.1.3.1.cmml">/</mo><mn id="S4.SS2.p2.2.m2.1.1.3.3" xref="S4.SS2.p2.2.m2.1.1.3.3.cmml">255</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p2.2.m2.1b"><apply id="S4.SS2.p2.2.m2.1.1.cmml" xref="S4.SS2.p2.2.m2.1.1"><eq id="S4.SS2.p2.2.m2.1.1.1.cmml" xref="S4.SS2.p2.2.m2.1.1.1"></eq><apply id="S4.SS2.p2.2.m2.1.1.2.cmml" xref="S4.SS2.p2.2.m2.1.1.2"><csymbol cd="ambiguous" id="S4.SS2.p2.2.m2.1.1.2.1.cmml" xref="S4.SS2.p2.2.m2.1.1.2">subscript</csymbol><ci id="S4.SS2.p2.2.m2.1.1.2.2.cmml" xref="S4.SS2.p2.2.m2.1.1.2.2">𝜎</ci><cn id="S4.SS2.p2.2.m2.1.1.2.3.cmml" type="integer" xref="S4.SS2.p2.2.m2.1.1.2.3">0</cn></apply><apply id="S4.SS2.p2.2.m2.1.1.3.cmml" xref="S4.SS2.p2.2.m2.1.1.3"><divide id="S4.SS2.p2.2.m2.1.1.3.1.cmml" xref="S4.SS2.p2.2.m2.1.1.3.1"></divide><cn id="S4.SS2.p2.2.m2.1.1.3.2.cmml" type="integer" xref="S4.SS2.p2.2.m2.1.1.3.2">50</cn><cn id="S4.SS2.p2.2.m2.1.1.3.3.cmml" type="integer" xref="S4.SS2.p2.2.m2.1.1.3.3">255</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p2.2.m2.1c">\sigma_{0}=50/255</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p2.2.m2.1d">italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT = 50 / 255</annotation></semantics></math>, <math alttext="\sigma_{m-1}=5/255" class="ltx_Math" display="inline" id="S4.SS2.p2.3.m3.1"><semantics id="S4.SS2.p2.3.m3.1a"><mrow id="S4.SS2.p2.3.m3.1.1" xref="S4.SS2.p2.3.m3.1.1.cmml"><msub id="S4.SS2.p2.3.m3.1.1.2" xref="S4.SS2.p2.3.m3.1.1.2.cmml"><mi id="S4.SS2.p2.3.m3.1.1.2.2" xref="S4.SS2.p2.3.m3.1.1.2.2.cmml">σ</mi><mrow id="S4.SS2.p2.3.m3.1.1.2.3" xref="S4.SS2.p2.3.m3.1.1.2.3.cmml"><mi id="S4.SS2.p2.3.m3.1.1.2.3.2" xref="S4.SS2.p2.3.m3.1.1.2.3.2.cmml">m</mi><mo id="S4.SS2.p2.3.m3.1.1.2.3.1" xref="S4.SS2.p2.3.m3.1.1.2.3.1.cmml">−</mo><mn id="S4.SS2.p2.3.m3.1.1.2.3.3" xref="S4.SS2.p2.3.m3.1.1.2.3.3.cmml">1</mn></mrow></msub><mo id="S4.SS2.p2.3.m3.1.1.1" xref="S4.SS2.p2.3.m3.1.1.1.cmml">=</mo><mrow id="S4.SS2.p2.3.m3.1.1.3" xref="S4.SS2.p2.3.m3.1.1.3.cmml"><mn id="S4.SS2.p2.3.m3.1.1.3.2" xref="S4.SS2.p2.3.m3.1.1.3.2.cmml">5</mn><mo id="S4.SS2.p2.3.m3.1.1.3.1" xref="S4.SS2.p2.3.m3.1.1.3.1.cmml">/</mo><mn id="S4.SS2.p2.3.m3.1.1.3.3" xref="S4.SS2.p2.3.m3.1.1.3.3.cmml">255</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p2.3.m3.1b"><apply id="S4.SS2.p2.3.m3.1.1.cmml" xref="S4.SS2.p2.3.m3.1.1"><eq id="S4.SS2.p2.3.m3.1.1.1.cmml" xref="S4.SS2.p2.3.m3.1.1.1"></eq><apply id="S4.SS2.p2.3.m3.1.1.2.cmml" xref="S4.SS2.p2.3.m3.1.1.2"><csymbol cd="ambiguous" id="S4.SS2.p2.3.m3.1.1.2.1.cmml" xref="S4.SS2.p2.3.m3.1.1.2">subscript</csymbol><ci id="S4.SS2.p2.3.m3.1.1.2.2.cmml" xref="S4.SS2.p2.3.m3.1.1.2.2">𝜎</ci><apply id="S4.SS2.p2.3.m3.1.1.2.3.cmml" xref="S4.SS2.p2.3.m3.1.1.2.3"><minus id="S4.SS2.p2.3.m3.1.1.2.3.1.cmml" xref="S4.SS2.p2.3.m3.1.1.2.3.1"></minus><ci id="S4.SS2.p2.3.m3.1.1.2.3.2.cmml" xref="S4.SS2.p2.3.m3.1.1.2.3.2">𝑚</ci><cn id="S4.SS2.p2.3.m3.1.1.2.3.3.cmml" type="integer" xref="S4.SS2.p2.3.m3.1.1.2.3.3">1</cn></apply></apply><apply id="S4.SS2.p2.3.m3.1.1.3.cmml" xref="S4.SS2.p2.3.m3.1.1.3"><divide id="S4.SS2.p2.3.m3.1.1.3.1.cmml" xref="S4.SS2.p2.3.m3.1.1.3.1"></divide><cn id="S4.SS2.p2.3.m3.1.1.3.2.cmml" type="integer" xref="S4.SS2.p2.3.m3.1.1.3.2">5</cn><cn id="S4.SS2.p2.3.m3.1.1.3.3.cmml" type="integer" xref="S4.SS2.p2.3.m3.1.1.3.3">255</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p2.3.m3.1c">\sigma_{m-1}=5/255</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p2.3.m3.1d">italic_σ start_POSTSUBSCRIPT italic_m - 1 end_POSTSUBSCRIPT = 5 / 255</annotation></semantics></math>, <math alttext="\alpha_{0}=\alpha_{m-1}=0.15" class="ltx_Math" display="inline" id="S4.SS2.p2.4.m4.1"><semantics id="S4.SS2.p2.4.m4.1a"><mrow id="S4.SS2.p2.4.m4.1.1" xref="S4.SS2.p2.4.m4.1.1.cmml"><msub id="S4.SS2.p2.4.m4.1.1.2" xref="S4.SS2.p2.4.m4.1.1.2.cmml"><mi id="S4.SS2.p2.4.m4.1.1.2.2" xref="S4.SS2.p2.4.m4.1.1.2.2.cmml">α</mi><mn id="S4.SS2.p2.4.m4.1.1.2.3" xref="S4.SS2.p2.4.m4.1.1.2.3.cmml">0</mn></msub><mo id="S4.SS2.p2.4.m4.1.1.3" xref="S4.SS2.p2.4.m4.1.1.3.cmml">=</mo><msub id="S4.SS2.p2.4.m4.1.1.4" xref="S4.SS2.p2.4.m4.1.1.4.cmml"><mi id="S4.SS2.p2.4.m4.1.1.4.2" xref="S4.SS2.p2.4.m4.1.1.4.2.cmml">α</mi><mrow id="S4.SS2.p2.4.m4.1.1.4.3" xref="S4.SS2.p2.4.m4.1.1.4.3.cmml"><mi id="S4.SS2.p2.4.m4.1.1.4.3.2" xref="S4.SS2.p2.4.m4.1.1.4.3.2.cmml">m</mi><mo id="S4.SS2.p2.4.m4.1.1.4.3.1" xref="S4.SS2.p2.4.m4.1.1.4.3.1.cmml">−</mo><mn id="S4.SS2.p2.4.m4.1.1.4.3.3" xref="S4.SS2.p2.4.m4.1.1.4.3.3.cmml">1</mn></mrow></msub><mo id="S4.SS2.p2.4.m4.1.1.5" xref="S4.SS2.p2.4.m4.1.1.5.cmml">=</mo><mn id="S4.SS2.p2.4.m4.1.1.6" xref="S4.SS2.p2.4.m4.1.1.6.cmml">0.15</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p2.4.m4.1b"><apply id="S4.SS2.p2.4.m4.1.1.cmml" xref="S4.SS2.p2.4.m4.1.1"><and id="S4.SS2.p2.4.m4.1.1a.cmml" xref="S4.SS2.p2.4.m4.1.1"></and><apply id="S4.SS2.p2.4.m4.1.1b.cmml" xref="S4.SS2.p2.4.m4.1.1"><eq id="S4.SS2.p2.4.m4.1.1.3.cmml" xref="S4.SS2.p2.4.m4.1.1.3"></eq><apply id="S4.SS2.p2.4.m4.1.1.2.cmml" xref="S4.SS2.p2.4.m4.1.1.2"><csymbol cd="ambiguous" id="S4.SS2.p2.4.m4.1.1.2.1.cmml" xref="S4.SS2.p2.4.m4.1.1.2">subscript</csymbol><ci id="S4.SS2.p2.4.m4.1.1.2.2.cmml" xref="S4.SS2.p2.4.m4.1.1.2.2">𝛼</ci><cn id="S4.SS2.p2.4.m4.1.1.2.3.cmml" type="integer" xref="S4.SS2.p2.4.m4.1.1.2.3">0</cn></apply><apply id="S4.SS2.p2.4.m4.1.1.4.cmml" xref="S4.SS2.p2.4.m4.1.1.4"><csymbol cd="ambiguous" id="S4.SS2.p2.4.m4.1.1.4.1.cmml" xref="S4.SS2.p2.4.m4.1.1.4">subscript</csymbol><ci id="S4.SS2.p2.4.m4.1.1.4.2.cmml" xref="S4.SS2.p2.4.m4.1.1.4.2">𝛼</ci><apply id="S4.SS2.p2.4.m4.1.1.4.3.cmml" xref="S4.SS2.p2.4.m4.1.1.4.3"><minus id="S4.SS2.p2.4.m4.1.1.4.3.1.cmml" xref="S4.SS2.p2.4.m4.1.1.4.3.1"></minus><ci id="S4.SS2.p2.4.m4.1.1.4.3.2.cmml" xref="S4.SS2.p2.4.m4.1.1.4.3.2">𝑚</ci><cn id="S4.SS2.p2.4.m4.1.1.4.3.3.cmml" type="integer" xref="S4.SS2.p2.4.m4.1.1.4.3.3">1</cn></apply></apply></apply><apply id="S4.SS2.p2.4.m4.1.1c.cmml" xref="S4.SS2.p2.4.m4.1.1"><eq id="S4.SS2.p2.4.m4.1.1.5.cmml" xref="S4.SS2.p2.4.m4.1.1.5"></eq><share href="#S4.SS2.p2.4.m4.1.1.4.cmml" id="S4.SS2.p2.4.m4.1.1d.cmml" xref="S4.SS2.p2.4.m4.1.1"></share><cn id="S4.SS2.p2.4.m4.1.1.6.cmml" type="float" xref="S4.SS2.p2.4.m4.1.1.6">0.15</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p2.4.m4.1c">\alpha_{0}=\alpha_{m-1}=0.15</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p2.4.m4.1d">italic_α start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT = italic_α start_POSTSUBSCRIPT italic_m - 1 end_POSTSUBSCRIPT = 0.15</annotation></semantics></math>. The initialization is done with a modified version of the observation <math alttext="{\mathbf{y}}" class="ltx_Math" display="inline" id="S4.SS2.p2.5.m5.1"><semantics id="S4.SS2.p2.5.m5.1a"><mi id="S4.SS2.p2.5.m5.1.1" xref="S4.SS2.p2.5.m5.1.1.cmml">𝐲</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.p2.5.m5.1b"><ci id="S4.SS2.p2.5.m5.1.1.cmml" xref="S4.SS2.p2.5.m5.1.1">𝐲</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p2.5.m5.1c">{\mathbf{y}}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p2.5.m5.1d">bold_y</annotation></semantics></math>, where the masked pixels are set to the <math alttext="0.5" class="ltx_Math" display="inline" id="S4.SS2.p2.6.m6.1"><semantics id="S4.SS2.p2.6.m6.1a"><mn id="S4.SS2.p2.6.m6.1.1" xref="S4.SS2.p2.6.m6.1.1.cmml">0.5</mn><annotation-xml encoding="MathML-Content" id="S4.SS2.p2.6.m6.1b"><cn id="S4.SS2.p2.6.m6.1.1.cmml" type="float" xref="S4.SS2.p2.6.m6.1.1">0.5</cn></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p2.6.m6.1c">0.5</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p2.6.m6.1d">0.5</annotation></semantics></math> value. We set a fixed step-size <math alttext="\delta_{k}=\delta=0.5" class="ltx_Math" display="inline" id="S4.SS2.p2.7.m7.1"><semantics id="S4.SS2.p2.7.m7.1a"><mrow id="S4.SS2.p2.7.m7.1.1" xref="S4.SS2.p2.7.m7.1.1.cmml"><msub id="S4.SS2.p2.7.m7.1.1.2" xref="S4.SS2.p2.7.m7.1.1.2.cmml"><mi id="S4.SS2.p2.7.m7.1.1.2.2" xref="S4.SS2.p2.7.m7.1.1.2.2.cmml">δ</mi><mi id="S4.SS2.p2.7.m7.1.1.2.3" xref="S4.SS2.p2.7.m7.1.1.2.3.cmml">k</mi></msub><mo id="S4.SS2.p2.7.m7.1.1.3" xref="S4.SS2.p2.7.m7.1.1.3.cmml">=</mo><mi id="S4.SS2.p2.7.m7.1.1.4" xref="S4.SS2.p2.7.m7.1.1.4.cmml">δ</mi><mo id="S4.SS2.p2.7.m7.1.1.5" xref="S4.SS2.p2.7.m7.1.1.5.cmml">=</mo><mn id="S4.SS2.p2.7.m7.1.1.6" xref="S4.SS2.p2.7.m7.1.1.6.cmml">0.5</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p2.7.m7.1b"><apply id="S4.SS2.p2.7.m7.1.1.cmml" xref="S4.SS2.p2.7.m7.1.1"><and id="S4.SS2.p2.7.m7.1.1a.cmml" xref="S4.SS2.p2.7.m7.1.1"></and><apply id="S4.SS2.p2.7.m7.1.1b.cmml" xref="S4.SS2.p2.7.m7.1.1"><eq id="S4.SS2.p2.7.m7.1.1.3.cmml" xref="S4.SS2.p2.7.m7.1.1.3"></eq><apply id="S4.SS2.p2.7.m7.1.1.2.cmml" xref="S4.SS2.p2.7.m7.1.1.2"><csymbol cd="ambiguous" id="S4.SS2.p2.7.m7.1.1.2.1.cmml" xref="S4.SS2.p2.7.m7.1.1.2">subscript</csymbol><ci id="S4.SS2.p2.7.m7.1.1.2.2.cmml" xref="S4.SS2.p2.7.m7.1.1.2.2">𝛿</ci><ci id="S4.SS2.p2.7.m7.1.1.2.3.cmml" xref="S4.SS2.p2.7.m7.1.1.2.3">𝑘</ci></apply><ci id="S4.SS2.p2.7.m7.1.1.4.cmml" xref="S4.SS2.p2.7.m7.1.1.4">𝛿</ci></apply><apply id="S4.SS2.p2.7.m7.1.1c.cmml" xref="S4.SS2.p2.7.m7.1.1"><eq id="S4.SS2.p2.7.m7.1.1.5.cmml" xref="S4.SS2.p2.7.m7.1.1.5"></eq><share href="#S4.SS2.p2.7.m7.1.1.4.cmml" id="S4.SS2.p2.7.m7.1.1d.cmml" xref="S4.SS2.p2.7.m7.1.1"></share><cn id="S4.SS2.p2.7.m7.1.1.6.cmml" type="float" xref="S4.SS2.p2.7.m7.1.1.6">0.5</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p2.7.m7.1c">\delta_{k}=\delta=0.5</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p2.7.m7.1d">italic_δ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT = italic_δ = 0.5</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S4.SS2.p3"> <p class="ltx_p" id="S4.SS2.p3.1">On Table <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#S4.T2" title="Table 2 ‣ 4.2 Inpainting ‣ 4 Experiments ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_tag">2</span></a>, we compare Ann-SNORE to other methods for the inpainting task with <math alttext="p=0.5" class="ltx_Math" display="inline" id="S4.SS2.p3.1.m1.1"><semantics id="S4.SS2.p3.1.m1.1a"><mrow id="S4.SS2.p3.1.m1.1.1" xref="S4.SS2.p3.1.m1.1.1.cmml"><mi id="S4.SS2.p3.1.m1.1.1.2" xref="S4.SS2.p3.1.m1.1.1.2.cmml">p</mi><mo id="S4.SS2.p3.1.m1.1.1.1" xref="S4.SS2.p3.1.m1.1.1.1.cmml">=</mo><mn id="S4.SS2.p3.1.m1.1.1.3" xref="S4.SS2.p3.1.m1.1.1.3.cmml">0.5</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p3.1.m1.1b"><apply id="S4.SS2.p3.1.m1.1.1.cmml" xref="S4.SS2.p3.1.m1.1.1"><eq id="S4.SS2.p3.1.m1.1.1.1.cmml" xref="S4.SS2.p3.1.m1.1.1.1"></eq><ci id="S4.SS2.p3.1.m1.1.1.2.cmml" xref="S4.SS2.p3.1.m1.1.1.2">𝑝</ci><cn id="S4.SS2.p3.1.m1.1.1.3.cmml" type="float" xref="S4.SS2.p3.1.m1.1.1.3">0.5</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p3.1.m1.1c">p=0.5</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p3.1.m1.1d">italic_p = 0.5</annotation></semantics></math> on CBSD68. A qualitative comparison is given in Figure <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#S4.F3" title="Figure 3 ‣ 4.1 Deblurring ‣ 4 Experiments ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_tag">3</span></a>. We observe that Ann-SNORE outperforms other approaches with better distortion and perceptual scores (PSNR, SSIM, LPIPS). Visually, we observe that Ann-SNORE succeeds to restore textures. Other qualitative results are given in Appendix <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#A6" title="Appendix F Additional Experiments ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_tag">F</span></a>. Note that each method (except DiffPIR) is run with the same number of iterations. Hence, contrary to the deblurring task, inpainting with Ann-SNORE does not involve any additional computational load with respect to RED methods. </p> </div> <figure class="ltx_table" id="S4.T2"> <div class="ltx_inline-block ltx_align_center ltx_transformed_outer" id="S4.T2.4" style="width:433.6pt;height:203.5pt;vertical-align:-0.0pt;"><span class="ltx_transformed_inner" style="transform:translate(101.7pt,-47.7pt) scale(1.88393499014468,1.88393499014468) ;"> <table class="ltx_tabular ltx_guessed_headers ltx_align_middle" id="S4.T2.4.4"> <thead class="ltx_thead"> <tr class="ltx_tr" id="S4.T2.4.4.4"> <th class="ltx_td ltx_align_center ltx_th ltx_th_column" id="S4.T2.4.4.4.5">Method</th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column" id="S4.T2.1.1.1.1">PSNR<math alttext="\uparrow" class="ltx_Math" display="inline" id="S4.T2.1.1.1.1.m1.1"><semantics id="S4.T2.1.1.1.1.m1.1a"><mo id="S4.T2.1.1.1.1.m1.1.1" stretchy="false" xref="S4.T2.1.1.1.1.m1.1.1.cmml">↑</mo><annotation-xml encoding="MathML-Content" id="S4.T2.1.1.1.1.m1.1b"><ci id="S4.T2.1.1.1.1.m1.1.1.cmml" xref="S4.T2.1.1.1.1.m1.1.1">↑</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.T2.1.1.1.1.m1.1c">\uparrow</annotation><annotation encoding="application/x-llamapun" id="S4.T2.1.1.1.1.m1.1d">↑</annotation></semantics></math> </th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column" id="S4.T2.2.2.2.2">SSIM<math alttext="\uparrow" class="ltx_Math" display="inline" id="S4.T2.2.2.2.2.m1.1"><semantics id="S4.T2.2.2.2.2.m1.1a"><mo id="S4.T2.2.2.2.2.m1.1.1" stretchy="false" xref="S4.T2.2.2.2.2.m1.1.1.cmml">↑</mo><annotation-xml encoding="MathML-Content" id="S4.T2.2.2.2.2.m1.1b"><ci id="S4.T2.2.2.2.2.m1.1.1.cmml" xref="S4.T2.2.2.2.2.m1.1.1">↑</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.T2.2.2.2.2.m1.1c">\uparrow</annotation><annotation encoding="application/x-llamapun" id="S4.T2.2.2.2.2.m1.1d">↑</annotation></semantics></math> </th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column" id="S4.T2.3.3.3.3">LPIPS<math alttext="\downarrow" class="ltx_Math" display="inline" id="S4.T2.3.3.3.3.m1.1"><semantics id="S4.T2.3.3.3.3.m1.1a"><mo id="S4.T2.3.3.3.3.m1.1.1" stretchy="false" xref="S4.T2.3.3.3.3.m1.1.1.cmml">↓</mo><annotation-xml encoding="MathML-Content" id="S4.T2.3.3.3.3.m1.1b"><ci id="S4.T2.3.3.3.3.m1.1.1.cmml" xref="S4.T2.3.3.3.3.m1.1.1">↓</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.T2.3.3.3.3.m1.1c">\downarrow</annotation><annotation encoding="application/x-llamapun" id="S4.T2.3.3.3.3.m1.1d">↓</annotation></semantics></math> </th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column" id="S4.T2.4.4.4.4">BRISQUE<math alttext="\downarrow" class="ltx_Math" display="inline" id="S4.T2.4.4.4.4.m1.1"><semantics id="S4.T2.4.4.4.4.m1.1a"><mo id="S4.T2.4.4.4.4.m1.1.1" stretchy="false" xref="S4.T2.4.4.4.4.m1.1.1.cmml">↓</mo><annotation-xml encoding="MathML-Content" id="S4.T2.4.4.4.4.m1.1b"><ci id="S4.T2.4.4.4.4.m1.1.1.cmml" xref="S4.T2.4.4.4.4.m1.1.1">↓</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.T2.4.4.4.4.m1.1c">\downarrow</annotation><annotation encoding="application/x-llamapun" id="S4.T2.4.4.4.4.m1.1d">↓</annotation></semantics></math> </th> </tr> </thead> <tbody class="ltx_tbody"> <tr class="ltx_tr" id="S4.T2.4.4.5.1"> <td class="ltx_td ltx_align_center ltx_border_t" id="S4.T2.4.4.5.1.1">RED</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S4.T2.4.4.5.1.2">31.26</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S4.T2.4.4.5.1.3">0.91</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S4.T2.4.4.5.1.4"><span class="ltx_text ltx_framed_underline" id="S4.T2.4.4.5.1.4.1">0.07</span></td> <td class="ltx_td ltx_align_center ltx_border_t" id="S4.T2.4.4.5.1.5">17.13</td> </tr> <tr class="ltx_tr" id="S4.T2.4.4.6.2"> <td class="ltx_td ltx_align_center" id="S4.T2.4.4.6.2.1">RED Prox</td> <td class="ltx_td ltx_align_center" id="S4.T2.4.4.6.2.2">30.31</td> <td class="ltx_td ltx_align_center" id="S4.T2.4.4.6.2.3">0.89</td> <td class="ltx_td ltx_align_center" id="S4.T2.4.4.6.2.4">0.12</td> <td class="ltx_td ltx_align_center" id="S4.T2.4.4.6.2.5">38.29</td> </tr> <tr class="ltx_tr" id="S4.T2.4.4.7.3"> <td class="ltx_td ltx_align_center" id="S4.T2.4.4.7.3.1">Ann-SNORE</td> <td class="ltx_td ltx_align_center" id="S4.T2.4.4.7.3.2"><span class="ltx_text ltx_framed_underline" id="S4.T2.4.4.7.3.2.1">31.65</span></td> <td class="ltx_td ltx_align_center" id="S4.T2.4.4.7.3.3"><span class="ltx_text ltx_framed_underline" id="S4.T2.4.4.7.3.3.1">0.92</span></td> <td class="ltx_td ltx_align_center" id="S4.T2.4.4.7.3.4"><span class="ltx_text ltx_font_bold" id="S4.T2.4.4.7.3.4.1">0.04</span></td> <td class="ltx_td ltx_align_center" id="S4.T2.4.4.7.3.5"><span class="ltx_text ltx_framed_underline" id="S4.T2.4.4.7.3.5.1">7.10</span></td> </tr> <tr class="ltx_tr" id="S4.T2.4.4.8.4"> <td class="ltx_td ltx_align_center" id="S4.T2.4.4.8.4.1">Ann-SNORE Prox</td> <td class="ltx_td ltx_align_center" id="S4.T2.4.4.8.4.2"><span class="ltx_text ltx_font_bold" id="S4.T2.4.4.8.4.2.1">31.94</span></td> <td class="ltx_td ltx_align_center" id="S4.T2.4.4.8.4.3"><span class="ltx_text ltx_font_bold" id="S4.T2.4.4.8.4.3.1">0.93</span></td> <td class="ltx_td ltx_align_center" id="S4.T2.4.4.8.4.4"><span class="ltx_text ltx_font_bold" id="S4.T2.4.4.8.4.4.1">0.04</span></td> <td class="ltx_td ltx_align_center" id="S4.T2.4.4.8.4.5">8.35</td> </tr> <tr class="ltx_tr" id="S4.T2.4.4.9.5"> <td class="ltx_td ltx_align_center ltx_border_b" id="S4.T2.4.4.9.5.1">DiffPIR</td> <td class="ltx_td ltx_align_center ltx_border_b" id="S4.T2.4.4.9.5.2">29.57</td> <td class="ltx_td ltx_align_center ltx_border_b" id="S4.T2.4.4.9.5.3">0.87</td> <td class="ltx_td ltx_align_center ltx_border_b" id="S4.T2.4.4.9.5.4"><span class="ltx_text ltx_framed_underline" id="S4.T2.4.4.9.5.4.1">0.07</span></td> <td class="ltx_td ltx_align_center ltx_border_b" id="S4.T2.4.4.9.5.5"><span class="ltx_text ltx_font_bold" id="S4.T2.4.4.9.5.5.1">4.17</span></td> </tr> </tbody> </table> </span></div> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_table"><span class="ltx_text" id="S4.T2.8.2.1" style="font-size:90%;">Table 2</span>: </span><span class="ltx_text" id="S4.T2.6.1" style="font-size:90%;">Inpainting result for random missing pixel with probability <math alttext="p=0.5" class="ltx_Math" display="inline" id="S4.T2.6.1.m1.1"><semantics id="S4.T2.6.1.m1.1b"><mrow id="S4.T2.6.1.m1.1.1" xref="S4.T2.6.1.m1.1.1.cmml"><mi id="S4.T2.6.1.m1.1.1.2" xref="S4.T2.6.1.m1.1.1.2.cmml">p</mi><mo id="S4.T2.6.1.m1.1.1.1" xref="S4.T2.6.1.m1.1.1.1.cmml">=</mo><mn id="S4.T2.6.1.m1.1.1.3" xref="S4.T2.6.1.m1.1.1.3.cmml">0.5</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.T2.6.1.m1.1c"><apply id="S4.T2.6.1.m1.1.1.cmml" xref="S4.T2.6.1.m1.1.1"><eq id="S4.T2.6.1.m1.1.1.1.cmml" xref="S4.T2.6.1.m1.1.1.1"></eq><ci id="S4.T2.6.1.m1.1.1.2.cmml" xref="S4.T2.6.1.m1.1.1.2">𝑝</ci><cn id="S4.T2.6.1.m1.1.1.3.cmml" type="float" xref="S4.T2.6.1.m1.1.1.3">0.5</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.T2.6.1.m1.1d">p=0.5</annotation><annotation encoding="application/x-llamapun" id="S4.T2.6.1.m1.1e">italic_p = 0.5</annotation></semantics></math> on CBSD68 dataset. Best and second best results are respectively displayed in bold and underlined.</span></figcaption> </figure> </section> </section> <section class="ltx_section" id="S5"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">5 </span>Conclusion</h2> <div class="ltx_para" id="S5.p1"> <p class="ltx_p" id="S5.p1.1">In this work, we introduce a stochastic denoising regularization (SNORE) for solving imaging inverse problems within the PnP framework. This regularization realizes the heuristic idea that <span class="ltx_text ltx_font_italic" id="S5.p1.1.1">an image looks clean if its noisy versions look as noisy images (with the same noise level)</span>. Solving inverse problems with this regularization can be addressed with a provably-convergent stochastic optimization algorithm. The algorithm differs from standard PnP in the fact that the regularization step consists in denoising noised images, thus avoiding a distribution-shift from the data on which the denoiser is trained. It also draws a connection with recent diffusion-based approaches, which also involve noising-denoising steps, but included in a different global scheme that relates to backward diffusion. Experiments conducted on ill-posed inverse problems (deblurring, inpainting) show that SNORE attains state-of-the-art image restoration performance (in terms of full-reference and no-reference quality measures), at the expense of a computational cost which is, for now, larger than competing methods for deblurring. It would be interesting to determine whether the computational cost could be reduced by relying on other stochastic gradient-descent algorithms (e.g. ADAM <cite class="ltx_cite ltx_citemacro_citep">(Kingma & Ba, <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#bib.bib35" title="">2014</a>)</cite>, INNA <cite class="ltx_cite ltx_citemacro_citep">(Castera et al., <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#bib.bib7" title="">2021</a>)</cite>), for which theoretical convergence has not been shown yet for functionals including regularizations such as RED or SNORE.</p> </div> </section> <section class="ltx_section" id="S6"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">6 </span>Impact Statement</h2> <div class="ltx_para" id="S6.p1"> <p class="ltx_p" id="S6.p1.1">The work presented in this paper addresses the highly ill-posed problem of restoring missing information within an image. This sensitivity to errors is particularly pronounced in scenarios involving post-processing algorithms such as segmentation, detection, or classification applied to the reconstructed image, where errors in the reconstruction process may propagate into erroneous decision-making based on the image data. This concern is particularly critical in the context of medical images. SNORE, functioning as a stochastic process reliant on a learned denoiser, inherently produces random fluctuations in its output. In Section <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#A6.SS2" title="F.2 Uncertainty of SNORE ‣ Appendix F Additional Experiments ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_tag">F.2</span></a>, we provide an analysis of SNORE uncertainty to its random seed and its initialization. These experiments suggest a robustness of our method. Subsequent research effort should focus on quantifying the errors associated with SNORE in order to confirm its utility as a reliable reconstruction algorithm.</p> </div> </section> <section class="ltx_section" id="S7"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">7 </span>Acknowledgements</h2> <div class="ltx_para" id="S7.p1"> <p class="ltx_p" id="S7.p1.1">This study has been carried out with financial support from the French Direction Générale de l’Armement and the French Research Agency through the PostProdLEAP project (ANR-19-CE23-0027-01). Experiments presented in this paper were carried out using the PlaFRIM experimental testbed, supported by Inria, CNRS (LABRI and IMB), Universite de Bordeaux, Bordeaux INP and Conseil Regional d’Aquitaine (see https://www.plafrim.fr). 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In Supplement <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#A1" title="Appendix A Reproducibility Statement ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_tag">A</span></a>, we provide a reproducibility statement for this work. In Supplement <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#A2" title="Appendix B Related works ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_tag">B</span></a>, we present in more details some related works. In Supplement <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#A3" title="Appendix C SNORE in simple cases ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_tag">C</span></a>, SNORE regularization is detailed in simple cases such as Gaussian or Gaussian Mixture priors. In Supplement <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#A4" title="Appendix D Proofs of section 3 ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_tag">D</span></a>, we provide the proofs of the different propositions presented in Section <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#S3" title="3 Convergence Analysis ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_tag">3</span></a>. In Supplement <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#A5" title="Appendix E Discussion on Assumptions ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_tag">E</span></a>, we give more explanations about our technical assumptions. In Supplement <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#A6" title="Appendix F Additional Experiments ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_tag">F</span></a>, we provide more details on our experimental setting and present additional numerical results. In Supplement <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#A8" title="Appendix H Discussion on the boundedness of (𝐱_𝑘)_{𝑘∈ℕ} ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_tag">H</span></a>, we discuss the boundedness hypothesis and perspectives of generalization.</p> </div> </section> <section class="ltx_appendix" id="A1"> <h2 class="ltx_title ltx_title_appendix"> <span class="ltx_tag ltx_tag_appendix">Appendix A </span>Reproducibility Statement</h2> <div class="ltx_para" id="A1.p1"> <p class="ltx_p" id="A1.p1.1">Anonymous source code is given in supplementary material. It contains a README.md file that explains step by step how to run the algorithm and replicate the results of the paper. Moreover, the pseudocode of SNORE algorithm is given in Algorithm <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#alg3" title="Algorithm 3 ‣ Optimization algorithms ‣ 2.2 SNORE regularization ‣ 2 Stochastic deNOising REgularization (SNORE) ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_tag">3</span></a> and every comparing methods pseudocodes are given in Appendix <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#A6" title="Appendix F Additional Experiments ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_tag">F</span></a>. All parameters setting is detailed in Appendix <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#A6" title="Appendix F Additional Experiments ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_tag">F</span></a>. The used datasets and the denoiser weights <cite class="ltx_cite ltx_citemacro_citep">(Hurault et al., <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#bib.bib30" title="">2022a</a>)</cite> are given in the supplementary materials. Theoretical results presented in Section <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#S3" title="3 Convergence Analysis ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_tag">3</span></a> are proved in the appendices.</p> </div> </section> <section class="ltx_appendix" id="A2"> <h2 class="ltx_title ltx_title_appendix"> <span class="ltx_tag ltx_tag_appendix">Appendix B </span>Related works</h2> <div class="ltx_para" id="A2.p1"> <p class="ltx_p" id="A2.p1.1">First of all, the convergence analysis of RED (Algorithm <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#alg1" title="Algorithm 1 ‣ 2.1 Background ‣ 2 Stochastic deNOising REgularization (SNORE) ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_tag">1</span></a>) and RED Prox (Algorithm <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#alg4" title="Algorithm 4 ‣ RED Prox ‣ Appendix F Additional Experiments ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_tag">4</span></a>) algorithms have been made in the literature <cite class="ltx_cite ltx_citemacro_citep">(Fermanian et al., <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#bib.bib24" title="">2023</a>; Hurault et al., <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#bib.bib30" title="">2022a</a>)</cite>. However, to our knowledge, no convergence analysis have be developed for DiffPIR <cite class="ltx_cite ltx_citemacro_citep">(Zhu et al., <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#bib.bib80" title="">2023</a>)</cite>.</p> </div> <div class="ltx_para" id="A2.p2"> <p class="ltx_p" id="A2.p2.1">In the existing literature, some stochastic versions of Plug-and-Play have already been proposed.</p> </div> <div class="ltx_para" id="A2.p3"> <p class="ltx_p" id="A2.p3.3"><cite class="ltx_cite ltx_citemacro_citet">Tang & Davies (<a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#bib.bib73" title="">2020</a>)</cite> propose to accelerate the computation of PnP-ADMM algorithms especially when the data-fidelity is heavy to compute by using a mini-batch approximation of the data-fidelity. 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id="A2.p3.1.m1.3.3.2.2.1.1.1.1.3.1.cmml" xref="A2.p3.1.m1.3.3.2.2.1.1.1.1.3">subscript</csymbol><ci id="A2.p3.1.m1.3.3.2.2.1.1.1.1.3.2.cmml" xref="A2.p3.1.m1.3.3.2.2.1.1.1.1.3.2">𝐷</ci><ci id="A2.p3.1.m1.3.3.2.2.1.1.1.1.3.3.cmml" xref="A2.p3.1.m1.3.3.2.2.1.1.1.1.3.3">𝜎</ci></apply><apply id="A2.p3.1.m1.3.3.2.2.1.1.1.1.1.1.1.cmml" xref="A2.p3.1.m1.3.3.2.2.1.1.1.1.1.1"><csymbol cd="ambiguous" id="A2.p3.1.m1.3.3.2.2.1.1.1.1.1.1.1.1.cmml" xref="A2.p3.1.m1.3.3.2.2.1.1.1.1.1.1">subscript</csymbol><ci id="A2.p3.1.m1.3.3.2.2.1.1.1.1.1.1.1.2.cmml" xref="A2.p3.1.m1.3.3.2.2.1.1.1.1.1.1.1.2">𝐱</ci><ci id="A2.p3.1.m1.3.3.2.2.1.1.1.1.1.1.1.3.cmml" xref="A2.p3.1.m1.3.3.2.2.1.1.1.1.1.1.1.3">𝑘</ci></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.p3.1.m1.3c">{\mathbf{x}}_{k+1}={\mathbf{x}}_{k}-\delta_{k}\tilde{\nabla}\mathcal{F}({% \mathbf{x}}_{k},{\mathbf{y}})-\frac{\alpha\delta_{k}}{\sigma^{2}}\left({% \mathbf{x}}_{k}-D_{\sigma}({\mathbf{x}}_{k})\right)</annotation><annotation encoding="application/x-llamapun" id="A2.p3.1.m1.3d">bold_x start_POSTSUBSCRIPT italic_k + 1 end_POSTSUBSCRIPT = bold_x start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT - italic_δ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT over~ start_ARG ∇ end_ARG caligraphic_F ( bold_x start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT , bold_y ) - divide start_ARG italic_α italic_δ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_ARG start_ARG italic_σ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG ( bold_x start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT - italic_D start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ( bold_x start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) )</annotation></semantics></math> with <math alttext="\tilde{\nabla}\mathcal{F}" class="ltx_Math" display="inline" id="A2.p3.2.m2.1"><semantics id="A2.p3.2.m2.1a"><mrow id="A2.p3.2.m2.1.1" xref="A2.p3.2.m2.1.1.cmml"><mover accent="true" id="A2.p3.2.m2.1.1.2" xref="A2.p3.2.m2.1.1.2.cmml"><mo id="A2.p3.2.m2.1.1.2.2" xref="A2.p3.2.m2.1.1.2.2.cmml">∇</mo><mo id="A2.p3.2.m2.1.1.2.1" xref="A2.p3.2.m2.1.1.2.1.cmml">~</mo></mover><mo id="A2.p3.2.m2.1.1.1" lspace="0.167em" xref="A2.p3.2.m2.1.1.1.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="A2.p3.2.m2.1.1.3" xref="A2.p3.2.m2.1.1.3.cmml">ℱ</mi></mrow><annotation-xml encoding="MathML-Content" id="A2.p3.2.m2.1b"><apply id="A2.p3.2.m2.1.1.cmml" xref="A2.p3.2.m2.1.1"><times id="A2.p3.2.m2.1.1.1.cmml" xref="A2.p3.2.m2.1.1.1"></times><apply id="A2.p3.2.m2.1.1.2.cmml" xref="A2.p3.2.m2.1.1.2"><ci id="A2.p3.2.m2.1.1.2.1.cmml" xref="A2.p3.2.m2.1.1.2.1">~</ci><ci id="A2.p3.2.m2.1.1.2.2.cmml" xref="A2.p3.2.m2.1.1.2.2">∇</ci></apply><ci id="A2.p3.2.m2.1.1.3.cmml" xref="A2.p3.2.m2.1.1.3">ℱ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.p3.2.m2.1c">\tilde{\nabla}\mathcal{F}</annotation><annotation encoding="application/x-llamapun" id="A2.p3.2.m2.1d">over~ start_ARG ∇ end_ARG caligraphic_F</annotation></semantics></math> a random batch approximation of <math alttext="\nabla\mathcal{F}" class="ltx_Math" display="inline" id="A2.p3.3.m3.1"><semantics id="A2.p3.3.m3.1a"><mrow id="A2.p3.3.m3.1.1" xref="A2.p3.3.m3.1.1.cmml"><mo id="A2.p3.3.m3.1.1.1" rspace="0.167em" xref="A2.p3.3.m3.1.1.1.cmml">∇</mo><mi class="ltx_font_mathcaligraphic" id="A2.p3.3.m3.1.1.2" xref="A2.p3.3.m3.1.1.2.cmml">ℱ</mi></mrow><annotation-xml encoding="MathML-Content" id="A2.p3.3.m3.1b"><apply id="A2.p3.3.m3.1.1.cmml" xref="A2.p3.3.m3.1.1"><ci id="A2.p3.3.m3.1.1.1.cmml" xref="A2.p3.3.m3.1.1.1">∇</ci><ci id="A2.p3.3.m3.1.1.2.cmml" xref="A2.p3.3.m3.1.1.2">ℱ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.p3.3.m3.1c">\nabla\mathcal{F}</annotation><annotation encoding="application/x-llamapun" id="A2.p3.3.m3.1d">∇ caligraphic_F</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="A2.p4"> <p class="ltx_p" id="A2.p4.7"><cite class="ltx_cite ltx_citemacro_citet">Sun et al. (<a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#bib.bib69" title="">2019a</a>)</cite> propose a similar idea based on a batch of random indices, which are optimized at each step. Take <math alttext="U_{i}\in{\mathbb{R}}^{n\times n_{i}}" class="ltx_Math" display="inline" id="A2.p4.1.m1.1"><semantics id="A2.p4.1.m1.1a"><mrow id="A2.p4.1.m1.1.1" xref="A2.p4.1.m1.1.1.cmml"><msub id="A2.p4.1.m1.1.1.2" xref="A2.p4.1.m1.1.1.2.cmml"><mi id="A2.p4.1.m1.1.1.2.2" xref="A2.p4.1.m1.1.1.2.2.cmml">U</mi><mi id="A2.p4.1.m1.1.1.2.3" xref="A2.p4.1.m1.1.1.2.3.cmml">i</mi></msub><mo id="A2.p4.1.m1.1.1.1" xref="A2.p4.1.m1.1.1.1.cmml">∈</mo><msup id="A2.p4.1.m1.1.1.3" xref="A2.p4.1.m1.1.1.3.cmml"><mi id="A2.p4.1.m1.1.1.3.2" xref="A2.p4.1.m1.1.1.3.2.cmml">ℝ</mi><mrow id="A2.p4.1.m1.1.1.3.3" xref="A2.p4.1.m1.1.1.3.3.cmml"><mi id="A2.p4.1.m1.1.1.3.3.2" xref="A2.p4.1.m1.1.1.3.3.2.cmml">n</mi><mo id="A2.p4.1.m1.1.1.3.3.1" lspace="0.222em" rspace="0.222em" xref="A2.p4.1.m1.1.1.3.3.1.cmml">×</mo><msub id="A2.p4.1.m1.1.1.3.3.3" xref="A2.p4.1.m1.1.1.3.3.3.cmml"><mi id="A2.p4.1.m1.1.1.3.3.3.2" xref="A2.p4.1.m1.1.1.3.3.3.2.cmml">n</mi><mi id="A2.p4.1.m1.1.1.3.3.3.3" xref="A2.p4.1.m1.1.1.3.3.3.3.cmml">i</mi></msub></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="A2.p4.1.m1.1b"><apply id="A2.p4.1.m1.1.1.cmml" xref="A2.p4.1.m1.1.1"><in id="A2.p4.1.m1.1.1.1.cmml" xref="A2.p4.1.m1.1.1.1"></in><apply id="A2.p4.1.m1.1.1.2.cmml" xref="A2.p4.1.m1.1.1.2"><csymbol cd="ambiguous" id="A2.p4.1.m1.1.1.2.1.cmml" xref="A2.p4.1.m1.1.1.2">subscript</csymbol><ci id="A2.p4.1.m1.1.1.2.2.cmml" xref="A2.p4.1.m1.1.1.2.2">𝑈</ci><ci id="A2.p4.1.m1.1.1.2.3.cmml" xref="A2.p4.1.m1.1.1.2.3">𝑖</ci></apply><apply id="A2.p4.1.m1.1.1.3.cmml" xref="A2.p4.1.m1.1.1.3"><csymbol cd="ambiguous" id="A2.p4.1.m1.1.1.3.1.cmml" xref="A2.p4.1.m1.1.1.3">superscript</csymbol><ci id="A2.p4.1.m1.1.1.3.2.cmml" xref="A2.p4.1.m1.1.1.3.2">ℝ</ci><apply id="A2.p4.1.m1.1.1.3.3.cmml" xref="A2.p4.1.m1.1.1.3.3"><times id="A2.p4.1.m1.1.1.3.3.1.cmml" xref="A2.p4.1.m1.1.1.3.3.1"></times><ci id="A2.p4.1.m1.1.1.3.3.2.cmml" xref="A2.p4.1.m1.1.1.3.3.2">𝑛</ci><apply id="A2.p4.1.m1.1.1.3.3.3.cmml" xref="A2.p4.1.m1.1.1.3.3.3"><csymbol cd="ambiguous" id="A2.p4.1.m1.1.1.3.3.3.1.cmml" xref="A2.p4.1.m1.1.1.3.3.3">subscript</csymbol><ci id="A2.p4.1.m1.1.1.3.3.3.2.cmml" xref="A2.p4.1.m1.1.1.3.3.3.2">𝑛</ci><ci id="A2.p4.1.m1.1.1.3.3.3.3.cmml" xref="A2.p4.1.m1.1.1.3.3.3.3">𝑖</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.p4.1.m1.1c">U_{i}\in{\mathbb{R}}^{n\times n_{i}}</annotation><annotation encoding="application/x-llamapun" id="A2.p4.1.m1.1d">italic_U start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∈ blackboard_R start_POSTSUPERSCRIPT italic_n × italic_n start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUPERSCRIPT</annotation></semantics></math> such that <math alttext="\sum_{i=1}^{b}n_{i}=n" class="ltx_Math" display="inline" id="A2.p4.2.m2.1"><semantics id="A2.p4.2.m2.1a"><mrow id="A2.p4.2.m2.1.1" xref="A2.p4.2.m2.1.1.cmml"><mrow id="A2.p4.2.m2.1.1.2" xref="A2.p4.2.m2.1.1.2.cmml"><msubsup id="A2.p4.2.m2.1.1.2.1" xref="A2.p4.2.m2.1.1.2.1.cmml"><mo id="A2.p4.2.m2.1.1.2.1.2.2" xref="A2.p4.2.m2.1.1.2.1.2.2.cmml">∑</mo><mrow id="A2.p4.2.m2.1.1.2.1.2.3" xref="A2.p4.2.m2.1.1.2.1.2.3.cmml"><mi id="A2.p4.2.m2.1.1.2.1.2.3.2" xref="A2.p4.2.m2.1.1.2.1.2.3.2.cmml">i</mi><mo id="A2.p4.2.m2.1.1.2.1.2.3.1" xref="A2.p4.2.m2.1.1.2.1.2.3.1.cmml">=</mo><mn id="A2.p4.2.m2.1.1.2.1.2.3.3" xref="A2.p4.2.m2.1.1.2.1.2.3.3.cmml">1</mn></mrow><mi id="A2.p4.2.m2.1.1.2.1.3" xref="A2.p4.2.m2.1.1.2.1.3.cmml">b</mi></msubsup><msub id="A2.p4.2.m2.1.1.2.2" xref="A2.p4.2.m2.1.1.2.2.cmml"><mi id="A2.p4.2.m2.1.1.2.2.2" xref="A2.p4.2.m2.1.1.2.2.2.cmml">n</mi><mi id="A2.p4.2.m2.1.1.2.2.3" xref="A2.p4.2.m2.1.1.2.2.3.cmml">i</mi></msub></mrow><mo id="A2.p4.2.m2.1.1.1" xref="A2.p4.2.m2.1.1.1.cmml">=</mo><mi id="A2.p4.2.m2.1.1.3" xref="A2.p4.2.m2.1.1.3.cmml">n</mi></mrow><annotation-xml encoding="MathML-Content" id="A2.p4.2.m2.1b"><apply id="A2.p4.2.m2.1.1.cmml" xref="A2.p4.2.m2.1.1"><eq id="A2.p4.2.m2.1.1.1.cmml" xref="A2.p4.2.m2.1.1.1"></eq><apply id="A2.p4.2.m2.1.1.2.cmml" xref="A2.p4.2.m2.1.1.2"><apply id="A2.p4.2.m2.1.1.2.1.cmml" xref="A2.p4.2.m2.1.1.2.1"><csymbol cd="ambiguous" id="A2.p4.2.m2.1.1.2.1.1.cmml" xref="A2.p4.2.m2.1.1.2.1">superscript</csymbol><apply id="A2.p4.2.m2.1.1.2.1.2.cmml" xref="A2.p4.2.m2.1.1.2.1"><csymbol cd="ambiguous" id="A2.p4.2.m2.1.1.2.1.2.1.cmml" xref="A2.p4.2.m2.1.1.2.1">subscript</csymbol><sum id="A2.p4.2.m2.1.1.2.1.2.2.cmml" xref="A2.p4.2.m2.1.1.2.1.2.2"></sum><apply id="A2.p4.2.m2.1.1.2.1.2.3.cmml" xref="A2.p4.2.m2.1.1.2.1.2.3"><eq id="A2.p4.2.m2.1.1.2.1.2.3.1.cmml" xref="A2.p4.2.m2.1.1.2.1.2.3.1"></eq><ci id="A2.p4.2.m2.1.1.2.1.2.3.2.cmml" xref="A2.p4.2.m2.1.1.2.1.2.3.2">𝑖</ci><cn id="A2.p4.2.m2.1.1.2.1.2.3.3.cmml" type="integer" xref="A2.p4.2.m2.1.1.2.1.2.3.3">1</cn></apply></apply><ci id="A2.p4.2.m2.1.1.2.1.3.cmml" xref="A2.p4.2.m2.1.1.2.1.3">𝑏</ci></apply><apply id="A2.p4.2.m2.1.1.2.2.cmml" xref="A2.p4.2.m2.1.1.2.2"><csymbol cd="ambiguous" id="A2.p4.2.m2.1.1.2.2.1.cmml" xref="A2.p4.2.m2.1.1.2.2">subscript</csymbol><ci id="A2.p4.2.m2.1.1.2.2.2.cmml" xref="A2.p4.2.m2.1.1.2.2.2">𝑛</ci><ci id="A2.p4.2.m2.1.1.2.2.3.cmml" xref="A2.p4.2.m2.1.1.2.2.3">𝑖</ci></apply></apply><ci id="A2.p4.2.m2.1.1.3.cmml" xref="A2.p4.2.m2.1.1.3">𝑛</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.p4.2.m2.1c">\sum_{i=1}^{b}n_{i}=n</annotation><annotation encoding="application/x-llamapun" id="A2.p4.2.m2.1d">∑ start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_b end_POSTSUPERSCRIPT italic_n start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = italic_n</annotation></semantics></math> and <math alttext="\sum_{i=1}^{b}U_{i}U_{i}^{T}=\mathbf{I}_{n}" class="ltx_Math" display="inline" id="A2.p4.3.m3.1"><semantics id="A2.p4.3.m3.1a"><mrow id="A2.p4.3.m3.1.1" xref="A2.p4.3.m3.1.1.cmml"><mrow id="A2.p4.3.m3.1.1.2" xref="A2.p4.3.m3.1.1.2.cmml"><msubsup id="A2.p4.3.m3.1.1.2.1" xref="A2.p4.3.m3.1.1.2.1.cmml"><mo id="A2.p4.3.m3.1.1.2.1.2.2" xref="A2.p4.3.m3.1.1.2.1.2.2.cmml">∑</mo><mrow id="A2.p4.3.m3.1.1.2.1.2.3" xref="A2.p4.3.m3.1.1.2.1.2.3.cmml"><mi id="A2.p4.3.m3.1.1.2.1.2.3.2" xref="A2.p4.3.m3.1.1.2.1.2.3.2.cmml">i</mi><mo id="A2.p4.3.m3.1.1.2.1.2.3.1" xref="A2.p4.3.m3.1.1.2.1.2.3.1.cmml">=</mo><mn id="A2.p4.3.m3.1.1.2.1.2.3.3" xref="A2.p4.3.m3.1.1.2.1.2.3.3.cmml">1</mn></mrow><mi id="A2.p4.3.m3.1.1.2.1.3" xref="A2.p4.3.m3.1.1.2.1.3.cmml">b</mi></msubsup><mrow id="A2.p4.3.m3.1.1.2.2" xref="A2.p4.3.m3.1.1.2.2.cmml"><msub id="A2.p4.3.m3.1.1.2.2.2" xref="A2.p4.3.m3.1.1.2.2.2.cmml"><mi id="A2.p4.3.m3.1.1.2.2.2.2" xref="A2.p4.3.m3.1.1.2.2.2.2.cmml">U</mi><mi id="A2.p4.3.m3.1.1.2.2.2.3" xref="A2.p4.3.m3.1.1.2.2.2.3.cmml">i</mi></msub><mo id="A2.p4.3.m3.1.1.2.2.1" xref="A2.p4.3.m3.1.1.2.2.1.cmml">⁢</mo><msubsup id="A2.p4.3.m3.1.1.2.2.3" xref="A2.p4.3.m3.1.1.2.2.3.cmml"><mi id="A2.p4.3.m3.1.1.2.2.3.2.2" xref="A2.p4.3.m3.1.1.2.2.3.2.2.cmml">U</mi><mi id="A2.p4.3.m3.1.1.2.2.3.2.3" xref="A2.p4.3.m3.1.1.2.2.3.2.3.cmml">i</mi><mi id="A2.p4.3.m3.1.1.2.2.3.3" xref="A2.p4.3.m3.1.1.2.2.3.3.cmml">T</mi></msubsup></mrow></mrow><mo id="A2.p4.3.m3.1.1.1" xref="A2.p4.3.m3.1.1.1.cmml">=</mo><msub id="A2.p4.3.m3.1.1.3" xref="A2.p4.3.m3.1.1.3.cmml"><mi id="A2.p4.3.m3.1.1.3.2" xref="A2.p4.3.m3.1.1.3.2.cmml">𝐈</mi><mi id="A2.p4.3.m3.1.1.3.3" xref="A2.p4.3.m3.1.1.3.3.cmml">n</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="A2.p4.3.m3.1b"><apply id="A2.p4.3.m3.1.1.cmml" xref="A2.p4.3.m3.1.1"><eq id="A2.p4.3.m3.1.1.1.cmml" xref="A2.p4.3.m3.1.1.1"></eq><apply id="A2.p4.3.m3.1.1.2.cmml" xref="A2.p4.3.m3.1.1.2"><apply id="A2.p4.3.m3.1.1.2.1.cmml" xref="A2.p4.3.m3.1.1.2.1"><csymbol cd="ambiguous" id="A2.p4.3.m3.1.1.2.1.1.cmml" xref="A2.p4.3.m3.1.1.2.1">superscript</csymbol><apply id="A2.p4.3.m3.1.1.2.1.2.cmml" xref="A2.p4.3.m3.1.1.2.1"><csymbol cd="ambiguous" id="A2.p4.3.m3.1.1.2.1.2.1.cmml" xref="A2.p4.3.m3.1.1.2.1">subscript</csymbol><sum id="A2.p4.3.m3.1.1.2.1.2.2.cmml" xref="A2.p4.3.m3.1.1.2.1.2.2"></sum><apply id="A2.p4.3.m3.1.1.2.1.2.3.cmml" xref="A2.p4.3.m3.1.1.2.1.2.3"><eq id="A2.p4.3.m3.1.1.2.1.2.3.1.cmml" xref="A2.p4.3.m3.1.1.2.1.2.3.1"></eq><ci id="A2.p4.3.m3.1.1.2.1.2.3.2.cmml" xref="A2.p4.3.m3.1.1.2.1.2.3.2">𝑖</ci><cn id="A2.p4.3.m3.1.1.2.1.2.3.3.cmml" type="integer" xref="A2.p4.3.m3.1.1.2.1.2.3.3">1</cn></apply></apply><ci id="A2.p4.3.m3.1.1.2.1.3.cmml" xref="A2.p4.3.m3.1.1.2.1.3">𝑏</ci></apply><apply id="A2.p4.3.m3.1.1.2.2.cmml" xref="A2.p4.3.m3.1.1.2.2"><times id="A2.p4.3.m3.1.1.2.2.1.cmml" xref="A2.p4.3.m3.1.1.2.2.1"></times><apply id="A2.p4.3.m3.1.1.2.2.2.cmml" xref="A2.p4.3.m3.1.1.2.2.2"><csymbol cd="ambiguous" id="A2.p4.3.m3.1.1.2.2.2.1.cmml" xref="A2.p4.3.m3.1.1.2.2.2">subscript</csymbol><ci id="A2.p4.3.m3.1.1.2.2.2.2.cmml" xref="A2.p4.3.m3.1.1.2.2.2.2">𝑈</ci><ci id="A2.p4.3.m3.1.1.2.2.2.3.cmml" xref="A2.p4.3.m3.1.1.2.2.2.3">𝑖</ci></apply><apply id="A2.p4.3.m3.1.1.2.2.3.cmml" xref="A2.p4.3.m3.1.1.2.2.3"><csymbol cd="ambiguous" id="A2.p4.3.m3.1.1.2.2.3.1.cmml" xref="A2.p4.3.m3.1.1.2.2.3">superscript</csymbol><apply id="A2.p4.3.m3.1.1.2.2.3.2.cmml" xref="A2.p4.3.m3.1.1.2.2.3"><csymbol cd="ambiguous" id="A2.p4.3.m3.1.1.2.2.3.2.1.cmml" xref="A2.p4.3.m3.1.1.2.2.3">subscript</csymbol><ci id="A2.p4.3.m3.1.1.2.2.3.2.2.cmml" xref="A2.p4.3.m3.1.1.2.2.3.2.2">𝑈</ci><ci id="A2.p4.3.m3.1.1.2.2.3.2.3.cmml" xref="A2.p4.3.m3.1.1.2.2.3.2.3">𝑖</ci></apply><ci id="A2.p4.3.m3.1.1.2.2.3.3.cmml" xref="A2.p4.3.m3.1.1.2.2.3.3">𝑇</ci></apply></apply></apply><apply id="A2.p4.3.m3.1.1.3.cmml" xref="A2.p4.3.m3.1.1.3"><csymbol cd="ambiguous" id="A2.p4.3.m3.1.1.3.1.cmml" xref="A2.p4.3.m3.1.1.3">subscript</csymbol><ci id="A2.p4.3.m3.1.1.3.2.cmml" xref="A2.p4.3.m3.1.1.3.2">𝐈</ci><ci id="A2.p4.3.m3.1.1.3.3.cmml" xref="A2.p4.3.m3.1.1.3.3">𝑛</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.p4.3.m3.1c">\sum_{i=1}^{b}U_{i}U_{i}^{T}=\mathbf{I}_{n}</annotation><annotation encoding="application/x-llamapun" id="A2.p4.3.m3.1d">∑ start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_b end_POSTSUPERSCRIPT italic_U start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_U start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT = bold_I start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT</annotation></semantics></math> then <math alttext="x_{k+1}=x_{k}-\delta U_{i_{k}}^{T}\nabla\mathcal{J}(x_{k})" class="ltx_Math" display="inline" id="A2.p4.4.m4.1"><semantics id="A2.p4.4.m4.1a"><mrow id="A2.p4.4.m4.1.1" xref="A2.p4.4.m4.1.1.cmml"><msub id="A2.p4.4.m4.1.1.3" xref="A2.p4.4.m4.1.1.3.cmml"><mi id="A2.p4.4.m4.1.1.3.2" xref="A2.p4.4.m4.1.1.3.2.cmml">x</mi><mrow id="A2.p4.4.m4.1.1.3.3" xref="A2.p4.4.m4.1.1.3.3.cmml"><mi id="A2.p4.4.m4.1.1.3.3.2" xref="A2.p4.4.m4.1.1.3.3.2.cmml">k</mi><mo id="A2.p4.4.m4.1.1.3.3.1" xref="A2.p4.4.m4.1.1.3.3.1.cmml">+</mo><mn id="A2.p4.4.m4.1.1.3.3.3" xref="A2.p4.4.m4.1.1.3.3.3.cmml">1</mn></mrow></msub><mo id="A2.p4.4.m4.1.1.2" xref="A2.p4.4.m4.1.1.2.cmml">=</mo><mrow id="A2.p4.4.m4.1.1.1" xref="A2.p4.4.m4.1.1.1.cmml"><msub id="A2.p4.4.m4.1.1.1.3" xref="A2.p4.4.m4.1.1.1.3.cmml"><mi id="A2.p4.4.m4.1.1.1.3.2" xref="A2.p4.4.m4.1.1.1.3.2.cmml">x</mi><mi id="A2.p4.4.m4.1.1.1.3.3" xref="A2.p4.4.m4.1.1.1.3.3.cmml">k</mi></msub><mo id="A2.p4.4.m4.1.1.1.2" xref="A2.p4.4.m4.1.1.1.2.cmml">−</mo><mrow id="A2.p4.4.m4.1.1.1.1" xref="A2.p4.4.m4.1.1.1.1.cmml"><mi id="A2.p4.4.m4.1.1.1.1.3" xref="A2.p4.4.m4.1.1.1.1.3.cmml">δ</mi><mo id="A2.p4.4.m4.1.1.1.1.2" xref="A2.p4.4.m4.1.1.1.1.2.cmml">⁢</mo><msubsup id="A2.p4.4.m4.1.1.1.1.4" xref="A2.p4.4.m4.1.1.1.1.4.cmml"><mi id="A2.p4.4.m4.1.1.1.1.4.2.2" xref="A2.p4.4.m4.1.1.1.1.4.2.2.cmml">U</mi><msub id="A2.p4.4.m4.1.1.1.1.4.2.3" xref="A2.p4.4.m4.1.1.1.1.4.2.3.cmml"><mi id="A2.p4.4.m4.1.1.1.1.4.2.3.2" 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xref="A2.p4.4.m4.1.1.1.2"></minus><apply id="A2.p4.4.m4.1.1.1.3.cmml" xref="A2.p4.4.m4.1.1.1.3"><csymbol cd="ambiguous" id="A2.p4.4.m4.1.1.1.3.1.cmml" xref="A2.p4.4.m4.1.1.1.3">subscript</csymbol><ci id="A2.p4.4.m4.1.1.1.3.2.cmml" xref="A2.p4.4.m4.1.1.1.3.2">𝑥</ci><ci id="A2.p4.4.m4.1.1.1.3.3.cmml" xref="A2.p4.4.m4.1.1.1.3.3">𝑘</ci></apply><apply id="A2.p4.4.m4.1.1.1.1.cmml" xref="A2.p4.4.m4.1.1.1.1"><times id="A2.p4.4.m4.1.1.1.1.2.cmml" xref="A2.p4.4.m4.1.1.1.1.2"></times><ci id="A2.p4.4.m4.1.1.1.1.3.cmml" xref="A2.p4.4.m4.1.1.1.1.3">𝛿</ci><apply id="A2.p4.4.m4.1.1.1.1.4.cmml" xref="A2.p4.4.m4.1.1.1.1.4"><csymbol cd="ambiguous" id="A2.p4.4.m4.1.1.1.1.4.1.cmml" xref="A2.p4.4.m4.1.1.1.1.4">superscript</csymbol><apply id="A2.p4.4.m4.1.1.1.1.4.2.cmml" xref="A2.p4.4.m4.1.1.1.1.4"><csymbol cd="ambiguous" id="A2.p4.4.m4.1.1.1.1.4.2.1.cmml" xref="A2.p4.4.m4.1.1.1.1.4">subscript</csymbol><ci id="A2.p4.4.m4.1.1.1.1.4.2.2.cmml" xref="A2.p4.4.m4.1.1.1.1.4.2.2">𝑈</ci><apply 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xref="A2.p4.4.m4.1.1.1.1.1.1.1.3">𝑘</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.p4.4.m4.1c">x_{k+1}=x_{k}-\delta U_{i_{k}}^{T}\nabla\mathcal{J}(x_{k})</annotation><annotation encoding="application/x-llamapun" id="A2.p4.4.m4.1d">italic_x start_POSTSUBSCRIPT italic_k + 1 end_POSTSUBSCRIPT = italic_x start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT - italic_δ italic_U start_POSTSUBSCRIPT italic_i start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT ∇ caligraphic_J ( italic_x start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT )</annotation></semantics></math>, with <math alttext="i_{k}" class="ltx_Math" display="inline" id="A2.p4.5.m5.1"><semantics id="A2.p4.5.m5.1a"><msub id="A2.p4.5.m5.1.1" xref="A2.p4.5.m5.1.1.cmml"><mi id="A2.p4.5.m5.1.1.2" xref="A2.p4.5.m5.1.1.2.cmml">i</mi><mi id="A2.p4.5.m5.1.1.3" xref="A2.p4.5.m5.1.1.3.cmml">k</mi></msub><annotation-xml encoding="MathML-Content" id="A2.p4.5.m5.1b"><apply id="A2.p4.5.m5.1.1.cmml" xref="A2.p4.5.m5.1.1"><csymbol cd="ambiguous" id="A2.p4.5.m5.1.1.1.cmml" xref="A2.p4.5.m5.1.1">subscript</csymbol><ci id="A2.p4.5.m5.1.1.2.cmml" xref="A2.p4.5.m5.1.1.2">𝑖</ci><ci id="A2.p4.5.m5.1.1.3.cmml" xref="A2.p4.5.m5.1.1.3">𝑘</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.p4.5.m5.1c">i_{k}</annotation><annotation encoding="application/x-llamapun" id="A2.p4.5.m5.1d">italic_i start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT</annotation></semantics></math> a random index in <math alttext="[1,b]" class="ltx_Math" display="inline" id="A2.p4.6.m6.2"><semantics id="A2.p4.6.m6.2a"><mrow id="A2.p4.6.m6.2.3.2" xref="A2.p4.6.m6.2.3.1.cmml"><mo id="A2.p4.6.m6.2.3.2.1" stretchy="false" xref="A2.p4.6.m6.2.3.1.cmml">[</mo><mn id="A2.p4.6.m6.1.1" xref="A2.p4.6.m6.1.1.cmml">1</mn><mo id="A2.p4.6.m6.2.3.2.2" xref="A2.p4.6.m6.2.3.1.cmml">,</mo><mi id="A2.p4.6.m6.2.2" xref="A2.p4.6.m6.2.2.cmml">b</mi><mo id="A2.p4.6.m6.2.3.2.3" stretchy="false" xref="A2.p4.6.m6.2.3.1.cmml">]</mo></mrow><annotation-xml encoding="MathML-Content" id="A2.p4.6.m6.2b"><interval closure="closed" id="A2.p4.6.m6.2.3.1.cmml" xref="A2.p4.6.m6.2.3.2"><cn id="A2.p4.6.m6.1.1.cmml" type="integer" xref="A2.p4.6.m6.1.1">1</cn><ci id="A2.p4.6.m6.2.2.cmml" xref="A2.p4.6.m6.2.2">𝑏</ci></interval></annotation-xml><annotation encoding="application/x-tex" id="A2.p4.6.m6.2c">[1,b]</annotation><annotation encoding="application/x-llamapun" id="A2.p4.6.m6.2d">[ 1 , italic_b ]</annotation></semantics></math> and <math alttext="\mathcal{J}=\mathcal{F}+\alpha\mathcal{R}" class="ltx_Math" display="inline" id="A2.p4.7.m7.1"><semantics id="A2.p4.7.m7.1a"><mrow id="A2.p4.7.m7.1.1" xref="A2.p4.7.m7.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="A2.p4.7.m7.1.1.2" xref="A2.p4.7.m7.1.1.2.cmml">𝒥</mi><mo id="A2.p4.7.m7.1.1.1" xref="A2.p4.7.m7.1.1.1.cmml">=</mo><mrow id="A2.p4.7.m7.1.1.3" xref="A2.p4.7.m7.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="A2.p4.7.m7.1.1.3.2" xref="A2.p4.7.m7.1.1.3.2.cmml">ℱ</mi><mo id="A2.p4.7.m7.1.1.3.1" xref="A2.p4.7.m7.1.1.3.1.cmml">+</mo><mrow id="A2.p4.7.m7.1.1.3.3" xref="A2.p4.7.m7.1.1.3.3.cmml"><mi id="A2.p4.7.m7.1.1.3.3.2" xref="A2.p4.7.m7.1.1.3.3.2.cmml">α</mi><mo id="A2.p4.7.m7.1.1.3.3.1" xref="A2.p4.7.m7.1.1.3.3.1.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="A2.p4.7.m7.1.1.3.3.3" xref="A2.p4.7.m7.1.1.3.3.3.cmml">ℛ</mi></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="A2.p4.7.m7.1b"><apply id="A2.p4.7.m7.1.1.cmml" xref="A2.p4.7.m7.1.1"><eq id="A2.p4.7.m7.1.1.1.cmml" xref="A2.p4.7.m7.1.1.1"></eq><ci id="A2.p4.7.m7.1.1.2.cmml" xref="A2.p4.7.m7.1.1.2">𝒥</ci><apply id="A2.p4.7.m7.1.1.3.cmml" xref="A2.p4.7.m7.1.1.3"><plus id="A2.p4.7.m7.1.1.3.1.cmml" xref="A2.p4.7.m7.1.1.3.1"></plus><ci id="A2.p4.7.m7.1.1.3.2.cmml" xref="A2.p4.7.m7.1.1.3.2">ℱ</ci><apply id="A2.p4.7.m7.1.1.3.3.cmml" xref="A2.p4.7.m7.1.1.3.3"><times id="A2.p4.7.m7.1.1.3.3.1.cmml" xref="A2.p4.7.m7.1.1.3.3.1"></times><ci id="A2.p4.7.m7.1.1.3.3.2.cmml" xref="A2.p4.7.m7.1.1.3.3.2">𝛼</ci><ci id="A2.p4.7.m7.1.1.3.3.3.cmml" xref="A2.p4.7.m7.1.1.3.3.3">ℛ</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.p4.7.m7.1c">\mathcal{J}=\mathcal{F}+\alpha\mathcal{R}</annotation><annotation encoding="application/x-llamapun" id="A2.p4.7.m7.1d">caligraphic_J = caligraphic_F + italic_α caligraphic_R</annotation></semantics></math> the functional to minimize. On can remark that the two previous works propose an acceleration of PnP based on random batches but no noise is added inside the process. SNORE does not aim at accelerating PnP algorithms but it rather proposes a stochastic improvement of PnP by injecting noise inside the classical PnP regularization.</p> </div> <div class="ltx_para" id="A2.p5"> <p class="ltx_p" id="A2.p5.4"><cite class="ltx_cite ltx_citemacro_citet">Laumont et al. (<a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#bib.bib37" title="">2022b</a>)</cite> propose to run a stochastic gradient descent (SGD) algorithm with the PnP regularization. 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end_POSTSUBSCRIPT - italic_D start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ( bold_x start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) ) + italic_δ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT italic_z start_POSTSUBSCRIPT italic_k + 1 end_POSTSUBSCRIPT</annotation></semantics></math> with <math alttext="z_{k+1}\sim\mathcal{N}(0,\mathbf{I}_{d})" class="ltx_Math" display="inline" id="A2.p5.2.m2.2"><semantics id="A2.p5.2.m2.2a"><mrow id="A2.p5.2.m2.2.2" xref="A2.p5.2.m2.2.2.cmml"><msub id="A2.p5.2.m2.2.2.3" xref="A2.p5.2.m2.2.2.3.cmml"><mi id="A2.p5.2.m2.2.2.3.2" xref="A2.p5.2.m2.2.2.3.2.cmml">z</mi><mrow id="A2.p5.2.m2.2.2.3.3" xref="A2.p5.2.m2.2.2.3.3.cmml"><mi id="A2.p5.2.m2.2.2.3.3.2" xref="A2.p5.2.m2.2.2.3.3.2.cmml">k</mi><mo id="A2.p5.2.m2.2.2.3.3.1" xref="A2.p5.2.m2.2.2.3.3.1.cmml">+</mo><mn id="A2.p5.2.m2.2.2.3.3.3" xref="A2.p5.2.m2.2.2.3.3.3.cmml">1</mn></mrow></msub><mo id="A2.p5.2.m2.2.2.2" xref="A2.p5.2.m2.2.2.2.cmml">∼</mo><mrow id="A2.p5.2.m2.2.2.1" 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cd="ambiguous" id="A2.p5.2.m2.2.2.1.1.1.1.1.cmml" xref="A2.p5.2.m2.2.2.1.1.1.1">subscript</csymbol><ci id="A2.p5.2.m2.2.2.1.1.1.1.2.cmml" xref="A2.p5.2.m2.2.2.1.1.1.1.2">𝐈</ci><ci id="A2.p5.2.m2.2.2.1.1.1.1.3.cmml" xref="A2.p5.2.m2.2.2.1.1.1.1.3">𝑑</ci></apply></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.p5.2.m2.2c">z_{k+1}\sim\mathcal{N}(0,\mathbf{I}_{d})</annotation><annotation encoding="application/x-llamapun" id="A2.p5.2.m2.2d">italic_z start_POSTSUBSCRIPT italic_k + 1 end_POSTSUBSCRIPT ∼ caligraphic_N ( 0 , bold_I start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT )</annotation></semantics></math>. The step-size decreases with time so the additive noise standard deviation also reduces with time. Unlike the above mentioned methods, PnP SGD is not an acceleration of RED <cite class="ltx_cite ltx_citemacro_citep">(Romano et al., <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#bib.bib58" title="">2017</a>)</cite> but an another type of algorithm to minimize the same objective function, <math alttext="\mathcal{F}(\cdot,{\mathbf{y}})+\alpha\mathcal{P}_{\sigma}" class="ltx_Math" display="inline" id="A2.p5.3.m3.2"><semantics id="A2.p5.3.m3.2a"><mrow id="A2.p5.3.m3.2.3" xref="A2.p5.3.m3.2.3.cmml"><mrow id="A2.p5.3.m3.2.3.2" xref="A2.p5.3.m3.2.3.2.cmml"><mi class="ltx_font_mathcaligraphic" id="A2.p5.3.m3.2.3.2.2" xref="A2.p5.3.m3.2.3.2.2.cmml">ℱ</mi><mo id="A2.p5.3.m3.2.3.2.1" xref="A2.p5.3.m3.2.3.2.1.cmml">⁢</mo><mrow id="A2.p5.3.m3.2.3.2.3.2" xref="A2.p5.3.m3.2.3.2.3.1.cmml"><mo id="A2.p5.3.m3.2.3.2.3.2.1" stretchy="false" xref="A2.p5.3.m3.2.3.2.3.1.cmml">(</mo><mo id="A2.p5.3.m3.1.1" lspace="0em" rspace="0em" xref="A2.p5.3.m3.1.1.cmml">⋅</mo><mo id="A2.p5.3.m3.2.3.2.3.2.2" xref="A2.p5.3.m3.2.3.2.3.1.cmml">,</mo><mi id="A2.p5.3.m3.2.2" xref="A2.p5.3.m3.2.2.cmml">𝐲</mi><mo id="A2.p5.3.m3.2.3.2.3.2.3" stretchy="false" xref="A2.p5.3.m3.2.3.2.3.1.cmml">)</mo></mrow></mrow><mo id="A2.p5.3.m3.2.3.1" xref="A2.p5.3.m3.2.3.1.cmml">+</mo><mrow id="A2.p5.3.m3.2.3.3" xref="A2.p5.3.m3.2.3.3.cmml"><mi id="A2.p5.3.m3.2.3.3.2" xref="A2.p5.3.m3.2.3.3.2.cmml">α</mi><mo id="A2.p5.3.m3.2.3.3.1" xref="A2.p5.3.m3.2.3.3.1.cmml">⁢</mo><msub id="A2.p5.3.m3.2.3.3.3" xref="A2.p5.3.m3.2.3.3.3.cmml"><mi class="ltx_font_mathcaligraphic" id="A2.p5.3.m3.2.3.3.3.2" xref="A2.p5.3.m3.2.3.3.3.2.cmml">𝒫</mi><mi id="A2.p5.3.m3.2.3.3.3.3" xref="A2.p5.3.m3.2.3.3.3.3.cmml">σ</mi></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="A2.p5.3.m3.2b"><apply id="A2.p5.3.m3.2.3.cmml" xref="A2.p5.3.m3.2.3"><plus id="A2.p5.3.m3.2.3.1.cmml" xref="A2.p5.3.m3.2.3.1"></plus><apply id="A2.p5.3.m3.2.3.2.cmml" xref="A2.p5.3.m3.2.3.2"><times id="A2.p5.3.m3.2.3.2.1.cmml" xref="A2.p5.3.m3.2.3.2.1"></times><ci id="A2.p5.3.m3.2.3.2.2.cmml" xref="A2.p5.3.m3.2.3.2.2">ℱ</ci><interval closure="open" id="A2.p5.3.m3.2.3.2.3.1.cmml" xref="A2.p5.3.m3.2.3.2.3.2"><ci id="A2.p5.3.m3.1.1.cmml" xref="A2.p5.3.m3.1.1">⋅</ci><ci id="A2.p5.3.m3.2.2.cmml" xref="A2.p5.3.m3.2.2">𝐲</ci></interval></apply><apply id="A2.p5.3.m3.2.3.3.cmml" xref="A2.p5.3.m3.2.3.3"><times id="A2.p5.3.m3.2.3.3.1.cmml" xref="A2.p5.3.m3.2.3.3.1"></times><ci id="A2.p5.3.m3.2.3.3.2.cmml" xref="A2.p5.3.m3.2.3.3.2">𝛼</ci><apply id="A2.p5.3.m3.2.3.3.3.cmml" xref="A2.p5.3.m3.2.3.3.3"><csymbol cd="ambiguous" id="A2.p5.3.m3.2.3.3.3.1.cmml" xref="A2.p5.3.m3.2.3.3.3">subscript</csymbol><ci id="A2.p5.3.m3.2.3.3.3.2.cmml" xref="A2.p5.3.m3.2.3.3.3.2">𝒫</ci><ci id="A2.p5.3.m3.2.3.3.3.3.cmml" xref="A2.p5.3.m3.2.3.3.3.3">𝜎</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.p5.3.m3.2c">\mathcal{F}(\cdot,{\mathbf{y}})+\alpha\mathcal{P}_{\sigma}</annotation><annotation encoding="application/x-llamapun" id="A2.p5.3.m3.2d">caligraphic_F ( ⋅ , bold_y ) + italic_α caligraphic_P start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT</annotation></semantics></math>. SNORE has two main differences with this algorithm. First the noise is only injected inside the denoiser, and never in the data-fidelity term. Second, the standard deviation of the injected noise is fixed during the time (for fixed <math alttext="\sigma" class="ltx_Math" display="inline" id="A2.p5.4.m4.1"><semantics id="A2.p5.4.m4.1a"><mi id="A2.p5.4.m4.1.1" xref="A2.p5.4.m4.1.1.cmml">σ</mi><annotation-xml encoding="MathML-Content" id="A2.p5.4.m4.1b"><ci id="A2.p5.4.m4.1.1.cmml" xref="A2.p5.4.m4.1.1">𝜎</ci></annotation-xml><annotation encoding="application/x-tex" id="A2.p5.4.m4.1c">\sigma</annotation><annotation encoding="application/x-llamapun" id="A2.p5.4.m4.1d">italic_σ</annotation></semantics></math>). <cite class="ltx_cite ltx_citemacro_citet">Luther & Seung (<a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#bib.bib42" title="">2023</a>)</cite> proposed an algorithm similar to Algorithm <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#alg3" title="Algorithm 3 ‣ Optimization algorithms ‣ 2.2 SNORE regularization ‣ 2 Stochastic deNOising REgularization (SNORE) ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_tag">3</span></a> run with a blind denoiser but do not provide strong theoretical motivation or analysis.</p> </div> <div class="ltx_para" id="A2.p6"> <p class="ltx_p" id="A2.p6.16">The authors of <cite class="ltx_cite ltx_citemacro_citet">Kadkhodaie & Simoncelli (<a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#bib.bib33" title="">2021</a>)</cite> propose a method to solve linear inverse problems based on a modification of the Monte Carlo Markov Chain (MCMC) of Langevin Algorithm <cite class="ltx_cite ltx_citemacro_citep">(Song & Ermon, <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#bib.bib64" title="">2020</a>)</cite>. 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cd="ambiguous" id="A2.p6.1.m1.1.1.1.4.2.1.cmml" xref="A2.p6.1.m1.1.1.1.4.2">subscript</csymbol><ci id="A2.p6.1.m1.1.1.1.4.2.2.cmml" xref="A2.p6.1.m1.1.1.1.4.2.2">𝛾</ci><ci id="A2.p6.1.m1.1.1.1.4.2.3.cmml" xref="A2.p6.1.m1.1.1.1.4.2.3">𝑘</ci></apply><apply id="A2.p6.1.m1.1.1.1.4.3.cmml" xref="A2.p6.1.m1.1.1.1.4.3"><csymbol cd="ambiguous" id="A2.p6.1.m1.1.1.1.4.3.1.cmml" xref="A2.p6.1.m1.1.1.1.4.3">subscript</csymbol><ci id="A2.p6.1.m1.1.1.1.4.3.2.cmml" xref="A2.p6.1.m1.1.1.1.4.3.2">𝐳</ci><ci id="A2.p6.1.m1.1.1.1.4.3.3.cmml" xref="A2.p6.1.m1.1.1.1.4.3.3">𝑘</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.p6.1.m1.1c">\mathbf{x}_{k+1}=\mathbf{x}_{k}+\delta_{k}\left(\left(\operatorname{Id}-A^{T}A% \right)\left(D(\mathbf{x}_{k})-\mathbf{x}_{k}\right)+A^{T}\left(\mathbf{y}-A% \mathbf{x}_{k}\right)\right)+\gamma_{k}\mathbf{z}_{k}</annotation><annotation encoding="application/x-llamapun" id="A2.p6.1.m1.1d">bold_x start_POSTSUBSCRIPT italic_k + 1 end_POSTSUBSCRIPT = bold_x start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT + italic_δ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ( ( roman_Id - italic_A start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT italic_A ) ( italic_D ( bold_x start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) - bold_x start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) + italic_A start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT ( bold_y - italic_A bold_x start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) ) + italic_γ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT bold_z start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT</annotation></semantics></math>, with <math alttext="\delta_{k}=\frac{\delta_{0}k}{1+\delta_{0}(k-1)}" class="ltx_Math" display="inline" id="A2.p6.2.m2.1"><semantics id="A2.p6.2.m2.1a"><mrow id="A2.p6.2.m2.1.2" xref="A2.p6.2.m2.1.2.cmml"><msub id="A2.p6.2.m2.1.2.2" xref="A2.p6.2.m2.1.2.2.cmml"><mi id="A2.p6.2.m2.1.2.2.2" xref="A2.p6.2.m2.1.2.2.2.cmml">δ</mi><mi id="A2.p6.2.m2.1.2.2.3" xref="A2.p6.2.m2.1.2.2.3.cmml">k</mi></msub><mo id="A2.p6.2.m2.1.2.1" xref="A2.p6.2.m2.1.2.1.cmml">=</mo><mfrac id="A2.p6.2.m2.1.1" xref="A2.p6.2.m2.1.1.cmml"><mrow id="A2.p6.2.m2.1.1.3" xref="A2.p6.2.m2.1.1.3.cmml"><msub id="A2.p6.2.m2.1.1.3.2" xref="A2.p6.2.m2.1.1.3.2.cmml"><mi id="A2.p6.2.m2.1.1.3.2.2" xref="A2.p6.2.m2.1.1.3.2.2.cmml">δ</mi><mn id="A2.p6.2.m2.1.1.3.2.3" xref="A2.p6.2.m2.1.1.3.2.3.cmml">0</mn></msub><mo id="A2.p6.2.m2.1.1.3.1" xref="A2.p6.2.m2.1.1.3.1.cmml">⁢</mo><mi id="A2.p6.2.m2.1.1.3.3" xref="A2.p6.2.m2.1.1.3.3.cmml">k</mi></mrow><mrow id="A2.p6.2.m2.1.1.1" xref="A2.p6.2.m2.1.1.1.cmml"><mn id="A2.p6.2.m2.1.1.1.3" xref="A2.p6.2.m2.1.1.1.3.cmml">1</mn><mo id="A2.p6.2.m2.1.1.1.2" xref="A2.p6.2.m2.1.1.1.2.cmml">+</mo><mrow id="A2.p6.2.m2.1.1.1.1" xref="A2.p6.2.m2.1.1.1.1.cmml"><msub id="A2.p6.2.m2.1.1.1.1.3" xref="A2.p6.2.m2.1.1.1.1.3.cmml"><mi id="A2.p6.2.m2.1.1.1.1.3.2" xref="A2.p6.2.m2.1.1.1.1.3.2.cmml">δ</mi><mn id="A2.p6.2.m2.1.1.1.1.3.3" xref="A2.p6.2.m2.1.1.1.1.3.3.cmml">0</mn></msub><mo id="A2.p6.2.m2.1.1.1.1.2" xref="A2.p6.2.m2.1.1.1.1.2.cmml">⁢</mo><mrow id="A2.p6.2.m2.1.1.1.1.1.1" xref="A2.p6.2.m2.1.1.1.1.1.1.1.cmml"><mo id="A2.p6.2.m2.1.1.1.1.1.1.2" stretchy="false" xref="A2.p6.2.m2.1.1.1.1.1.1.1.cmml">(</mo><mrow id="A2.p6.2.m2.1.1.1.1.1.1.1" xref="A2.p6.2.m2.1.1.1.1.1.1.1.cmml"><mi id="A2.p6.2.m2.1.1.1.1.1.1.1.2" xref="A2.p6.2.m2.1.1.1.1.1.1.1.2.cmml">k</mi><mo id="A2.p6.2.m2.1.1.1.1.1.1.1.1" xref="A2.p6.2.m2.1.1.1.1.1.1.1.1.cmml">−</mo><mn id="A2.p6.2.m2.1.1.1.1.1.1.1.3" xref="A2.p6.2.m2.1.1.1.1.1.1.1.3.cmml">1</mn></mrow><mo id="A2.p6.2.m2.1.1.1.1.1.1.3" stretchy="false" xref="A2.p6.2.m2.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow></mfrac></mrow><annotation-xml encoding="MathML-Content" id="A2.p6.2.m2.1b"><apply id="A2.p6.2.m2.1.2.cmml" xref="A2.p6.2.m2.1.2"><eq id="A2.p6.2.m2.1.2.1.cmml" xref="A2.p6.2.m2.1.2.1"></eq><apply id="A2.p6.2.m2.1.2.2.cmml" xref="A2.p6.2.m2.1.2.2"><csymbol cd="ambiguous" id="A2.p6.2.m2.1.2.2.1.cmml" xref="A2.p6.2.m2.1.2.2">subscript</csymbol><ci id="A2.p6.2.m2.1.2.2.2.cmml" xref="A2.p6.2.m2.1.2.2.2">𝛿</ci><ci id="A2.p6.2.m2.1.2.2.3.cmml" xref="A2.p6.2.m2.1.2.2.3">𝑘</ci></apply><apply id="A2.p6.2.m2.1.1.cmml" xref="A2.p6.2.m2.1.1"><divide id="A2.p6.2.m2.1.1.2.cmml" xref="A2.p6.2.m2.1.1"></divide><apply id="A2.p6.2.m2.1.1.3.cmml" xref="A2.p6.2.m2.1.1.3"><times id="A2.p6.2.m2.1.1.3.1.cmml" xref="A2.p6.2.m2.1.1.3.1"></times><apply id="A2.p6.2.m2.1.1.3.2.cmml" xref="A2.p6.2.m2.1.1.3.2"><csymbol cd="ambiguous" id="A2.p6.2.m2.1.1.3.2.1.cmml" xref="A2.p6.2.m2.1.1.3.2">subscript</csymbol><ci id="A2.p6.2.m2.1.1.3.2.2.cmml" xref="A2.p6.2.m2.1.1.3.2.2">𝛿</ci><cn id="A2.p6.2.m2.1.1.3.2.3.cmml" type="integer" xref="A2.p6.2.m2.1.1.3.2.3">0</cn></apply><ci id="A2.p6.2.m2.1.1.3.3.cmml" xref="A2.p6.2.m2.1.1.3.3">𝑘</ci></apply><apply id="A2.p6.2.m2.1.1.1.cmml" xref="A2.p6.2.m2.1.1.1"><plus id="A2.p6.2.m2.1.1.1.2.cmml" xref="A2.p6.2.m2.1.1.1.2"></plus><cn id="A2.p6.2.m2.1.1.1.3.cmml" type="integer" xref="A2.p6.2.m2.1.1.1.3">1</cn><apply id="A2.p6.2.m2.1.1.1.1.cmml" xref="A2.p6.2.m2.1.1.1.1"><times id="A2.p6.2.m2.1.1.1.1.2.cmml" xref="A2.p6.2.m2.1.1.1.1.2"></times><apply id="A2.p6.2.m2.1.1.1.1.3.cmml" xref="A2.p6.2.m2.1.1.1.1.3"><csymbol cd="ambiguous" id="A2.p6.2.m2.1.1.1.1.3.1.cmml" xref="A2.p6.2.m2.1.1.1.1.3">subscript</csymbol><ci id="A2.p6.2.m2.1.1.1.1.3.2.cmml" xref="A2.p6.2.m2.1.1.1.1.3.2">𝛿</ci><cn id="A2.p6.2.m2.1.1.1.1.3.3.cmml" type="integer" xref="A2.p6.2.m2.1.1.1.1.3.3">0</cn></apply><apply id="A2.p6.2.m2.1.1.1.1.1.1.1.cmml" xref="A2.p6.2.m2.1.1.1.1.1.1"><minus id="A2.p6.2.m2.1.1.1.1.1.1.1.1.cmml" xref="A2.p6.2.m2.1.1.1.1.1.1.1.1"></minus><ci id="A2.p6.2.m2.1.1.1.1.1.1.1.2.cmml" xref="A2.p6.2.m2.1.1.1.1.1.1.1.2">𝑘</ci><cn id="A2.p6.2.m2.1.1.1.1.1.1.1.3.cmml" type="integer" xref="A2.p6.2.m2.1.1.1.1.1.1.1.3">1</cn></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.p6.2.m2.1c">\delta_{k}=\frac{\delta_{0}k}{1+\delta_{0}(k-1)}</annotation><annotation encoding="application/x-llamapun" id="A2.p6.2.m2.1d">italic_δ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT = divide start_ARG italic_δ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT italic_k end_ARG start_ARG 1 + italic_δ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( italic_k - 1 ) end_ARG</annotation></semantics></math> the step-size, <math alttext="D" class="ltx_Math" display="inline" id="A2.p6.3.m3.1"><semantics id="A2.p6.3.m3.1a"><mi id="A2.p6.3.m3.1.1" xref="A2.p6.3.m3.1.1.cmml">D</mi><annotation-xml encoding="MathML-Content" id="A2.p6.3.m3.1b"><ci id="A2.p6.3.m3.1.1.cmml" xref="A2.p6.3.m3.1.1">𝐷</ci></annotation-xml><annotation encoding="application/x-tex" id="A2.p6.3.m3.1c">D</annotation><annotation encoding="application/x-llamapun" id="A2.p6.3.m3.1d">italic_D</annotation></semantics></math> a blind denoiser, <math alttext="\mathbf{z}_{k}\sim\mathcal{N}(0,\operatorname{Id})" class="ltx_Math" display="inline" id="A2.p6.4.m4.2"><semantics id="A2.p6.4.m4.2a"><mrow id="A2.p6.4.m4.2.3" xref="A2.p6.4.m4.2.3.cmml"><msub id="A2.p6.4.m4.2.3.2" xref="A2.p6.4.m4.2.3.2.cmml"><mi id="A2.p6.4.m4.2.3.2.2" xref="A2.p6.4.m4.2.3.2.2.cmml">𝐳</mi><mi id="A2.p6.4.m4.2.3.2.3" xref="A2.p6.4.m4.2.3.2.3.cmml">k</mi></msub><mo id="A2.p6.4.m4.2.3.1" xref="A2.p6.4.m4.2.3.1.cmml">∼</mo><mrow id="A2.p6.4.m4.2.3.3" xref="A2.p6.4.m4.2.3.3.cmml"><mi class="ltx_font_mathcaligraphic" id="A2.p6.4.m4.2.3.3.2" xref="A2.p6.4.m4.2.3.3.2.cmml">𝒩</mi><mo id="A2.p6.4.m4.2.3.3.1" xref="A2.p6.4.m4.2.3.3.1.cmml">⁢</mo><mrow id="A2.p6.4.m4.2.3.3.3.2" xref="A2.p6.4.m4.2.3.3.3.1.cmml"><mo id="A2.p6.4.m4.2.3.3.3.2.1" stretchy="false" xref="A2.p6.4.m4.2.3.3.3.1.cmml">(</mo><mn id="A2.p6.4.m4.1.1" xref="A2.p6.4.m4.1.1.cmml">0</mn><mo id="A2.p6.4.m4.2.3.3.3.2.2" xref="A2.p6.4.m4.2.3.3.3.1.cmml">,</mo><mi id="A2.p6.4.m4.2.2" xref="A2.p6.4.m4.2.2.cmml">Id</mi><mo id="A2.p6.4.m4.2.3.3.3.2.3" stretchy="false" xref="A2.p6.4.m4.2.3.3.3.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="A2.p6.4.m4.2b"><apply id="A2.p6.4.m4.2.3.cmml" xref="A2.p6.4.m4.2.3"><csymbol cd="latexml" id="A2.p6.4.m4.2.3.1.cmml" xref="A2.p6.4.m4.2.3.1">similar-to</csymbol><apply id="A2.p6.4.m4.2.3.2.cmml" xref="A2.p6.4.m4.2.3.2"><csymbol cd="ambiguous" id="A2.p6.4.m4.2.3.2.1.cmml" xref="A2.p6.4.m4.2.3.2">subscript</csymbol><ci id="A2.p6.4.m4.2.3.2.2.cmml" xref="A2.p6.4.m4.2.3.2.2">𝐳</ci><ci id="A2.p6.4.m4.2.3.2.3.cmml" xref="A2.p6.4.m4.2.3.2.3">𝑘</ci></apply><apply id="A2.p6.4.m4.2.3.3.cmml" xref="A2.p6.4.m4.2.3.3"><times id="A2.p6.4.m4.2.3.3.1.cmml" xref="A2.p6.4.m4.2.3.3.1"></times><ci id="A2.p6.4.m4.2.3.3.2.cmml" xref="A2.p6.4.m4.2.3.3.2">𝒩</ci><interval closure="open" id="A2.p6.4.m4.2.3.3.3.1.cmml" xref="A2.p6.4.m4.2.3.3.3.2"><cn id="A2.p6.4.m4.1.1.cmml" type="integer" xref="A2.p6.4.m4.1.1">0</cn><ci id="A2.p6.4.m4.2.2.cmml" xref="A2.p6.4.m4.2.2">Id</ci></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.p6.4.m4.2c">\mathbf{z}_{k}\sim\mathcal{N}(0,\operatorname{Id})</annotation><annotation encoding="application/x-llamapun" id="A2.p6.4.m4.2d">bold_z start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ∼ caligraphic_N ( 0 , roman_Id )</annotation></semantics></math>, <math alttext="\gamma_{k}=\left(\left(1-\beta\delta_{k}\right)^{2}-\left(1-\delta_{k}\right)^% {2}\right)\sigma_{k}" class="ltx_Math" display="inline" id="A2.p6.5.m5.1"><semantics id="A2.p6.5.m5.1a"><mrow id="A2.p6.5.m5.1.1" xref="A2.p6.5.m5.1.1.cmml"><msub id="A2.p6.5.m5.1.1.3" xref="A2.p6.5.m5.1.1.3.cmml"><mi id="A2.p6.5.m5.1.1.3.2" xref="A2.p6.5.m5.1.1.3.2.cmml">γ</mi><mi id="A2.p6.5.m5.1.1.3.3" xref="A2.p6.5.m5.1.1.3.3.cmml">k</mi></msub><mo id="A2.p6.5.m5.1.1.2" xref="A2.p6.5.m5.1.1.2.cmml">=</mo><mrow id="A2.p6.5.m5.1.1.1" xref="A2.p6.5.m5.1.1.1.cmml"><mrow id="A2.p6.5.m5.1.1.1.1.1" 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id="A2.p6.5.m5.1b"><apply id="A2.p6.5.m5.1.1.cmml" xref="A2.p6.5.m5.1.1"><eq id="A2.p6.5.m5.1.1.2.cmml" xref="A2.p6.5.m5.1.1.2"></eq><apply id="A2.p6.5.m5.1.1.3.cmml" xref="A2.p6.5.m5.1.1.3"><csymbol cd="ambiguous" id="A2.p6.5.m5.1.1.3.1.cmml" xref="A2.p6.5.m5.1.1.3">subscript</csymbol><ci id="A2.p6.5.m5.1.1.3.2.cmml" xref="A2.p6.5.m5.1.1.3.2">𝛾</ci><ci id="A2.p6.5.m5.1.1.3.3.cmml" xref="A2.p6.5.m5.1.1.3.3">𝑘</ci></apply><apply id="A2.p6.5.m5.1.1.1.cmml" xref="A2.p6.5.m5.1.1.1"><times id="A2.p6.5.m5.1.1.1.2.cmml" xref="A2.p6.5.m5.1.1.1.2"></times><apply id="A2.p6.5.m5.1.1.1.1.1.1.cmml" xref="A2.p6.5.m5.1.1.1.1.1"><minus id="A2.p6.5.m5.1.1.1.1.1.1.3.cmml" xref="A2.p6.5.m5.1.1.1.1.1.1.3"></minus><apply id="A2.p6.5.m5.1.1.1.1.1.1.1.cmml" xref="A2.p6.5.m5.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="A2.p6.5.m5.1.1.1.1.1.1.1.2.cmml" xref="A2.p6.5.m5.1.1.1.1.1.1.1">superscript</csymbol><apply id="A2.p6.5.m5.1.1.1.1.1.1.1.1.1.1.cmml" xref="A2.p6.5.m5.1.1.1.1.1.1.1.1.1"><minus 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xref="A2.p6.5.m5.1.1.1.1.1.1.1.3">2</cn></apply><apply id="A2.p6.5.m5.1.1.1.1.1.1.2.cmml" xref="A2.p6.5.m5.1.1.1.1.1.1.2"><csymbol cd="ambiguous" id="A2.p6.5.m5.1.1.1.1.1.1.2.2.cmml" xref="A2.p6.5.m5.1.1.1.1.1.1.2">superscript</csymbol><apply id="A2.p6.5.m5.1.1.1.1.1.1.2.1.1.1.cmml" xref="A2.p6.5.m5.1.1.1.1.1.1.2.1.1"><minus id="A2.p6.5.m5.1.1.1.1.1.1.2.1.1.1.1.cmml" xref="A2.p6.5.m5.1.1.1.1.1.1.2.1.1.1.1"></minus><cn id="A2.p6.5.m5.1.1.1.1.1.1.2.1.1.1.2.cmml" type="integer" xref="A2.p6.5.m5.1.1.1.1.1.1.2.1.1.1.2">1</cn><apply id="A2.p6.5.m5.1.1.1.1.1.1.2.1.1.1.3.cmml" xref="A2.p6.5.m5.1.1.1.1.1.1.2.1.1.1.3"><csymbol cd="ambiguous" id="A2.p6.5.m5.1.1.1.1.1.1.2.1.1.1.3.1.cmml" xref="A2.p6.5.m5.1.1.1.1.1.1.2.1.1.1.3">subscript</csymbol><ci id="A2.p6.5.m5.1.1.1.1.1.1.2.1.1.1.3.2.cmml" xref="A2.p6.5.m5.1.1.1.1.1.1.2.1.1.1.3.2">𝛿</ci><ci id="A2.p6.5.m5.1.1.1.1.1.1.2.1.1.1.3.3.cmml" xref="A2.p6.5.m5.1.1.1.1.1.1.2.1.1.1.3.3">𝑘</ci></apply></apply><cn id="A2.p6.5.m5.1.1.1.1.1.1.2.3.cmml" type="integer" xref="A2.p6.5.m5.1.1.1.1.1.1.2.3">2</cn></apply></apply><apply id="A2.p6.5.m5.1.1.1.3.cmml" xref="A2.p6.5.m5.1.1.1.3"><csymbol cd="ambiguous" id="A2.p6.5.m5.1.1.1.3.1.cmml" xref="A2.p6.5.m5.1.1.1.3">subscript</csymbol><ci id="A2.p6.5.m5.1.1.1.3.2.cmml" xref="A2.p6.5.m5.1.1.1.3.2">𝜎</ci><ci id="A2.p6.5.m5.1.1.1.3.3.cmml" xref="A2.p6.5.m5.1.1.1.3.3">𝑘</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.p6.5.m5.1c">\gamma_{k}=\left(\left(1-\beta\delta_{k}\right)^{2}-\left(1-\delta_{k}\right)^% {2}\right)\sigma_{k}</annotation><annotation encoding="application/x-llamapun" id="A2.p6.5.m5.1d">italic_γ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT = ( ( 1 - italic_β italic_δ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT - ( 1 - italic_δ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) italic_σ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT</annotation></semantics></math>, <math alttext="\beta\in[0,1]" class="ltx_Math" display="inline" id="A2.p6.6.m6.2"><semantics id="A2.p6.6.m6.2a"><mrow id="A2.p6.6.m6.2.3" xref="A2.p6.6.m6.2.3.cmml"><mi id="A2.p6.6.m6.2.3.2" xref="A2.p6.6.m6.2.3.2.cmml">β</mi><mo id="A2.p6.6.m6.2.3.1" xref="A2.p6.6.m6.2.3.1.cmml">∈</mo><mrow id="A2.p6.6.m6.2.3.3.2" xref="A2.p6.6.m6.2.3.3.1.cmml"><mo id="A2.p6.6.m6.2.3.3.2.1" stretchy="false" xref="A2.p6.6.m6.2.3.3.1.cmml">[</mo><mn id="A2.p6.6.m6.1.1" xref="A2.p6.6.m6.1.1.cmml">0</mn><mo id="A2.p6.6.m6.2.3.3.2.2" xref="A2.p6.6.m6.2.3.3.1.cmml">,</mo><mn id="A2.p6.6.m6.2.2" xref="A2.p6.6.m6.2.2.cmml">1</mn><mo id="A2.p6.6.m6.2.3.3.2.3" stretchy="false" xref="A2.p6.6.m6.2.3.3.1.cmml">]</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A2.p6.6.m6.2b"><apply id="A2.p6.6.m6.2.3.cmml" xref="A2.p6.6.m6.2.3"><in id="A2.p6.6.m6.2.3.1.cmml" xref="A2.p6.6.m6.2.3.1"></in><ci id="A2.p6.6.m6.2.3.2.cmml" xref="A2.p6.6.m6.2.3.2">𝛽</ci><interval closure="closed" id="A2.p6.6.m6.2.3.3.1.cmml" xref="A2.p6.6.m6.2.3.3.2"><cn id="A2.p6.6.m6.1.1.cmml" type="integer" xref="A2.p6.6.m6.1.1">0</cn><cn id="A2.p6.6.m6.2.2.cmml" type="integer" xref="A2.p6.6.m6.2.2">1</cn></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.p6.6.m6.2c">\beta\in[0,1]</annotation><annotation encoding="application/x-llamapun" id="A2.p6.6.m6.2d">italic_β ∈ [ 0 , 1 ]</annotation></semantics></math> and <math alttext="\sigma_{k}" class="ltx_Math" display="inline" id="A2.p6.7.m7.1"><semantics id="A2.p6.7.m7.1a"><msub id="A2.p6.7.m7.1.1" xref="A2.p6.7.m7.1.1.cmml"><mi id="A2.p6.7.m7.1.1.2" xref="A2.p6.7.m7.1.1.2.cmml">σ</mi><mi id="A2.p6.7.m7.1.1.3" xref="A2.p6.7.m7.1.1.3.cmml">k</mi></msub><annotation-xml encoding="MathML-Content" id="A2.p6.7.m7.1b"><apply id="A2.p6.7.m7.1.1.cmml" xref="A2.p6.7.m7.1.1"><csymbol cd="ambiguous" id="A2.p6.7.m7.1.1.1.cmml" xref="A2.p6.7.m7.1.1">subscript</csymbol><ci id="A2.p6.7.m7.1.1.2.cmml" xref="A2.p6.7.m7.1.1.2">𝜎</ci><ci id="A2.p6.7.m7.1.1.3.cmml" xref="A2.p6.7.m7.1.1.3">𝑘</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.p6.7.m7.1c">\sigma_{k}</annotation><annotation encoding="application/x-llamapun" id="A2.p6.7.m7.1d">italic_σ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT</annotation></semantics></math> an estimation of the noise level of <math alttext="\mathbf{x}_{k}" class="ltx_Math" display="inline" id="A2.p6.8.m8.1"><semantics id="A2.p6.8.m8.1a"><msub id="A2.p6.8.m8.1.1" xref="A2.p6.8.m8.1.1.cmml"><mi id="A2.p6.8.m8.1.1.2" xref="A2.p6.8.m8.1.1.2.cmml">𝐱</mi><mi id="A2.p6.8.m8.1.1.3" xref="A2.p6.8.m8.1.1.3.cmml">k</mi></msub><annotation-xml encoding="MathML-Content" id="A2.p6.8.m8.1b"><apply id="A2.p6.8.m8.1.1.cmml" xref="A2.p6.8.m8.1.1"><csymbol cd="ambiguous" id="A2.p6.8.m8.1.1.1.cmml" xref="A2.p6.8.m8.1.1">subscript</csymbol><ci id="A2.p6.8.m8.1.1.2.cmml" xref="A2.p6.8.m8.1.1.2">𝐱</ci><ci id="A2.p6.8.m8.1.1.3.cmml" xref="A2.p6.8.m8.1.1.3">𝑘</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.p6.8.m8.1c">\mathbf{x}_{k}</annotation><annotation encoding="application/x-llamapun" id="A2.p6.8.m8.1d">bold_x start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT</annotation></semantics></math>. In this method, a blind denoiser <math alttext="D" class="ltx_Math" display="inline" id="A2.p6.9.m9.1"><semantics id="A2.p6.9.m9.1a"><mi id="A2.p6.9.m9.1.1" xref="A2.p6.9.m9.1.1.cmml">D</mi><annotation-xml encoding="MathML-Content" id="A2.p6.9.m9.1b"><ci id="A2.p6.9.m9.1.1.cmml" xref="A2.p6.9.m9.1.1">𝐷</ci></annotation-xml><annotation encoding="application/x-tex" id="A2.p6.9.m9.1c">D</annotation><annotation encoding="application/x-llamapun" id="A2.p6.9.m9.1d">italic_D</annotation></semantics></math> is used instead of a denoiser <math alttext="D_{\sigma}" class="ltx_Math" display="inline" id="A2.p6.10.m10.1"><semantics id="A2.p6.10.m10.1a"><msub id="A2.p6.10.m10.1.1" xref="A2.p6.10.m10.1.1.cmml"><mi id="A2.p6.10.m10.1.1.2" xref="A2.p6.10.m10.1.1.2.cmml">D</mi><mi id="A2.p6.10.m10.1.1.3" xref="A2.p6.10.m10.1.1.3.cmml">σ</mi></msub><annotation-xml encoding="MathML-Content" id="A2.p6.10.m10.1b"><apply id="A2.p6.10.m10.1.1.cmml" xref="A2.p6.10.m10.1.1"><csymbol cd="ambiguous" id="A2.p6.10.m10.1.1.1.cmml" xref="A2.p6.10.m10.1.1">subscript</csymbol><ci id="A2.p6.10.m10.1.1.2.cmml" xref="A2.p6.10.m10.1.1.2">𝐷</ci><ci id="A2.p6.10.m10.1.1.3.cmml" xref="A2.p6.10.m10.1.1.3">𝜎</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.p6.10.m10.1c">D_{\sigma}</annotation><annotation encoding="application/x-llamapun" id="A2.p6.10.m10.1d">italic_D start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT</annotation></semantics></math> which takes the noise level as an input and a non-exact estimation of the noise level of <math alttext="\mathbf{x}_{k}" class="ltx_Math" display="inline" id="A2.p6.11.m11.1"><semantics id="A2.p6.11.m11.1a"><msub id="A2.p6.11.m11.1.1" xref="A2.p6.11.m11.1.1.cmml"><mi id="A2.p6.11.m11.1.1.2" xref="A2.p6.11.m11.1.1.2.cmml">𝐱</mi><mi id="A2.p6.11.m11.1.1.3" xref="A2.p6.11.m11.1.1.3.cmml">k</mi></msub><annotation-xml encoding="MathML-Content" id="A2.p6.11.m11.1b"><apply id="A2.p6.11.m11.1.1.cmml" xref="A2.p6.11.m11.1.1"><csymbol cd="ambiguous" id="A2.p6.11.m11.1.1.1.cmml" xref="A2.p6.11.m11.1.1">subscript</csymbol><ci id="A2.p6.11.m11.1.1.2.cmml" xref="A2.p6.11.m11.1.1.2">𝐱</ci><ci id="A2.p6.11.m11.1.1.3.cmml" xref="A2.p6.11.m11.1.1.3">𝑘</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.p6.11.m11.1c">\mathbf{x}_{k}</annotation><annotation encoding="application/x-llamapun" id="A2.p6.11.m11.1d">bold_x start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT</annotation></semantics></math> is made at each iteration. Moreover the parameter <math alttext="\beta" class="ltx_Math" display="inline" id="A2.p6.12.m12.1"><semantics id="A2.p6.12.m12.1a"><mi id="A2.p6.12.m12.1.1" xref="A2.p6.12.m12.1.1.cmml">β</mi><annotation-xml encoding="MathML-Content" id="A2.p6.12.m12.1b"><ci id="A2.p6.12.m12.1.1.cmml" xref="A2.p6.12.m12.1.1">𝛽</ci></annotation-xml><annotation encoding="application/x-tex" id="A2.p6.12.m12.1c">\beta</annotation><annotation encoding="application/x-llamapun" id="A2.p6.12.m12.1d">italic_β</annotation></semantics></math> is chosen to make the added noise level <math alttext="\gamma_{k}" class="ltx_Math" display="inline" id="A2.p6.13.m13.1"><semantics id="A2.p6.13.m13.1a"><msub id="A2.p6.13.m13.1.1" xref="A2.p6.13.m13.1.1.cmml"><mi id="A2.p6.13.m13.1.1.2" xref="A2.p6.13.m13.1.1.2.cmml">γ</mi><mi id="A2.p6.13.m13.1.1.3" xref="A2.p6.13.m13.1.1.3.cmml">k</mi></msub><annotation-xml encoding="MathML-Content" id="A2.p6.13.m13.1b"><apply id="A2.p6.13.m13.1.1.cmml" xref="A2.p6.13.m13.1.1"><csymbol cd="ambiguous" id="A2.p6.13.m13.1.1.1.cmml" xref="A2.p6.13.m13.1.1">subscript</csymbol><ci id="A2.p6.13.m13.1.1.2.cmml" xref="A2.p6.13.m13.1.1.2">𝛾</ci><ci id="A2.p6.13.m13.1.1.3.cmml" xref="A2.p6.13.m13.1.1.3">𝑘</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.p6.13.m13.1c">\gamma_{k}</annotation><annotation encoding="application/x-llamapun" id="A2.p6.13.m13.1d">italic_γ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT</annotation></semantics></math> decrease through the iterations, compared to <math alttext="\sigma" class="ltx_Math" display="inline" id="A2.p6.14.m14.1"><semantics id="A2.p6.14.m14.1a"><mi id="A2.p6.14.m14.1.1" xref="A2.p6.14.m14.1.1.cmml">σ</mi><annotation-xml encoding="MathML-Content" id="A2.p6.14.m14.1b"><ci id="A2.p6.14.m14.1.1.cmml" xref="A2.p6.14.m14.1.1">𝜎</ci></annotation-xml><annotation encoding="application/x-tex" id="A2.p6.14.m14.1c">\sigma</annotation><annotation encoding="application/x-llamapun" id="A2.p6.14.m14.1d">italic_σ</annotation></semantics></math> that is adapted to the denoiser <math alttext="D_{\sigma}" class="ltx_Math" display="inline" id="A2.p6.15.m15.1"><semantics id="A2.p6.15.m15.1a"><msub id="A2.p6.15.m15.1.1" xref="A2.p6.15.m15.1.1.cmml"><mi id="A2.p6.15.m15.1.1.2" xref="A2.p6.15.m15.1.1.2.cmml">D</mi><mi id="A2.p6.15.m15.1.1.3" xref="A2.p6.15.m15.1.1.3.cmml">σ</mi></msub><annotation-xml encoding="MathML-Content" id="A2.p6.15.m15.1b"><apply id="A2.p6.15.m15.1.1.cmml" xref="A2.p6.15.m15.1.1"><csymbol cd="ambiguous" id="A2.p6.15.m15.1.1.1.cmml" xref="A2.p6.15.m15.1.1">subscript</csymbol><ci id="A2.p6.15.m15.1.1.2.cmml" xref="A2.p6.15.m15.1.1.2">𝐷</ci><ci id="A2.p6.15.m15.1.1.3.cmml" xref="A2.p6.15.m15.1.1.3">𝜎</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.p6.15.m15.1c">D_{\sigma}</annotation><annotation encoding="application/x-llamapun" id="A2.p6.15.m15.1d">italic_D start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT</annotation></semantics></math> in SNORE. This noise <math alttext="\gamma_{k}\mathbf{z}_{k}" class="ltx_Math" display="inline" id="A2.p6.16.m16.1"><semantics id="A2.p6.16.m16.1a"><mrow id="A2.p6.16.m16.1.1" xref="A2.p6.16.m16.1.1.cmml"><msub id="A2.p6.16.m16.1.1.2" xref="A2.p6.16.m16.1.1.2.cmml"><mi id="A2.p6.16.m16.1.1.2.2" xref="A2.p6.16.m16.1.1.2.2.cmml">γ</mi><mi id="A2.p6.16.m16.1.1.2.3" xref="A2.p6.16.m16.1.1.2.3.cmml">k</mi></msub><mo id="A2.p6.16.m16.1.1.1" xref="A2.p6.16.m16.1.1.1.cmml">⁢</mo><msub id="A2.p6.16.m16.1.1.3" xref="A2.p6.16.m16.1.1.3.cmml"><mi id="A2.p6.16.m16.1.1.3.2" xref="A2.p6.16.m16.1.1.3.2.cmml">𝐳</mi><mi id="A2.p6.16.m16.1.1.3.3" xref="A2.p6.16.m16.1.1.3.3.cmml">k</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="A2.p6.16.m16.1b"><apply id="A2.p6.16.m16.1.1.cmml" xref="A2.p6.16.m16.1.1"><times id="A2.p6.16.m16.1.1.1.cmml" xref="A2.p6.16.m16.1.1.1"></times><apply id="A2.p6.16.m16.1.1.2.cmml" xref="A2.p6.16.m16.1.1.2"><csymbol cd="ambiguous" id="A2.p6.16.m16.1.1.2.1.cmml" xref="A2.p6.16.m16.1.1.2">subscript</csymbol><ci id="A2.p6.16.m16.1.1.2.2.cmml" xref="A2.p6.16.m16.1.1.2.2">𝛾</ci><ci id="A2.p6.16.m16.1.1.2.3.cmml" xref="A2.p6.16.m16.1.1.2.3">𝑘</ci></apply><apply id="A2.p6.16.m16.1.1.3.cmml" xref="A2.p6.16.m16.1.1.3"><csymbol cd="ambiguous" id="A2.p6.16.m16.1.1.3.1.cmml" xref="A2.p6.16.m16.1.1.3">subscript</csymbol><ci id="A2.p6.16.m16.1.1.3.2.cmml" xref="A2.p6.16.m16.1.1.3.2">𝐳</ci><ci id="A2.p6.16.m16.1.1.3.3.cmml" xref="A2.p6.16.m16.1.1.3.3">𝑘</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.p6.16.m16.1c">\gamma_{k}\mathbf{z}_{k}</annotation><annotation encoding="application/x-llamapun" id="A2.p6.16.m16.1d">italic_γ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT bold_z start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT</annotation></semantics></math> is added to the entire image and not only to the regularization such as in SNORE. Finally, the authors of <cite class="ltx_cite ltx_citemacro_citet">Kadkhodaie & Simoncelli (<a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#bib.bib33" title="">2021</a>)</cite> do not provide a theoretical analysis of the proposed algorithm, and depending on the noise schedule, the algorithm may perform posterior sampling or stochastic optimization.</p> </div> <div class="ltx_para" id="A2.p7"> <p class="ltx_p" id="A2.p7.1">In section <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#S2.SS3" title="2.3 Related Works ‣ 2 Stochastic deNOising REgularization (SNORE) ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_tag">2.3</span></a>, we separate stochastic version of PnP and Diffusion Model (DM) for clarity, but this separation is not strict. We recall that PnP is using a denoiser inside an optimization algorithm and DM generate data by simulating a reverse diffusion process. However, many recent works do not fit exactly inside these categories. <cite class="ltx_cite ltx_citemacro_citet">Jalal et al. (<a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#bib.bib32" title="">2021</a>); Sun et al. (<a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#bib.bib71" title="">2023</a>)</cite> develop a restoration method by running a Langevin dynamics (instead of a reverse diffusion process) with a score-matching network. Other posterior sampling algorithms can be simulated with Langevin Dynamics run with a PnP approximation of the score <cite class="ltx_cite ltx_citemacro_citep">(Laumont et al., <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#bib.bib36" title="">2022a</a>; Renaud et al., <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#bib.bib57" title="">2023</a>)</cite>. Another line of work, it to run Gibbs sampling <cite class="ltx_cite ltx_citemacro_citep">(Coeurdoux et al., <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#bib.bib13" title="">2023</a>; Bouman & Buzzard, <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#bib.bib4" title="">2023</a>)</cite> with a diffusion model approximation of the score. Our new regularization SNORE is part of this field between PnP and DM.</p> </div> </section> <section class="ltx_appendix" id="A3"> <h2 class="ltx_title ltx_title_appendix"> <span class="ltx_tag ltx_tag_appendix">Appendix C </span>SNORE in simple cases</h2> <div class="ltx_para" id="A3.p1"> <p class="ltx_p" id="A3.p1.2">In this section, we detail the behavior of the SNORE regularization in simple cases (Gaussian prior and Gaussian Mixture prior) in order to develop our intuition on this regularization. We observe that this regularization is equivalent to the PnP regularization with a Gaussian prior. Then, in the Gaussian Mixture case, we demonstrate that the gradient of the SNORE regularization converges to the gradient of the ideal regularization <math alttext="-\nabla\log p" class="ltx_Math" display="inline" id="A3.p1.1.m1.1"><semantics id="A3.p1.1.m1.1a"><mrow id="A3.p1.1.m1.1.1" xref="A3.p1.1.m1.1.1.cmml"><mo id="A3.p1.1.m1.1.1a" rspace="0.167em" xref="A3.p1.1.m1.1.1.cmml">−</mo><mrow id="A3.p1.1.m1.1.1.2" xref="A3.p1.1.m1.1.1.2.cmml"><mrow id="A3.p1.1.m1.1.1.2.1" xref="A3.p1.1.m1.1.1.2.1.cmml"><mo id="A3.p1.1.m1.1.1.2.1.1" rspace="0.167em" xref="A3.p1.1.m1.1.1.2.1.1.cmml">∇</mo><mi id="A3.p1.1.m1.1.1.2.1.2" xref="A3.p1.1.m1.1.1.2.1.2.cmml">log</mi></mrow><mo id="A3.p1.1.m1.1.1.2a" lspace="0.167em" xref="A3.p1.1.m1.1.1.2.cmml">⁡</mo><mi id="A3.p1.1.m1.1.1.2.2" xref="A3.p1.1.m1.1.1.2.2.cmml">p</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="A3.p1.1.m1.1b"><apply id="A3.p1.1.m1.1.1.cmml" xref="A3.p1.1.m1.1.1"><minus id="A3.p1.1.m1.1.1.1.cmml" xref="A3.p1.1.m1.1.1"></minus><apply id="A3.p1.1.m1.1.1.2.cmml" xref="A3.p1.1.m1.1.1.2"><apply id="A3.p1.1.m1.1.1.2.1.cmml" xref="A3.p1.1.m1.1.1.2.1"><ci id="A3.p1.1.m1.1.1.2.1.1.cmml" xref="A3.p1.1.m1.1.1.2.1.1">∇</ci><log id="A3.p1.1.m1.1.1.2.1.2.cmml" xref="A3.p1.1.m1.1.1.2.1.2"></log></apply><ci id="A3.p1.1.m1.1.1.2.2.cmml" xref="A3.p1.1.m1.1.1.2.2">𝑝</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.p1.1.m1.1c">-\nabla\log p</annotation><annotation encoding="application/x-llamapun" id="A3.p1.1.m1.1d">- ∇ roman_log italic_p</annotation></semantics></math> on every compact when <math alttext="\sigma\to 0" class="ltx_Math" display="inline" id="A3.p1.2.m2.1"><semantics id="A3.p1.2.m2.1a"><mrow id="A3.p1.2.m2.1.1" xref="A3.p1.2.m2.1.1.cmml"><mi id="A3.p1.2.m2.1.1.2" xref="A3.p1.2.m2.1.1.2.cmml">σ</mi><mo id="A3.p1.2.m2.1.1.1" stretchy="false" xref="A3.p1.2.m2.1.1.1.cmml">→</mo><mn id="A3.p1.2.m2.1.1.3" xref="A3.p1.2.m2.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="A3.p1.2.m2.1b"><apply id="A3.p1.2.m2.1.1.cmml" xref="A3.p1.2.m2.1.1"><ci id="A3.p1.2.m2.1.1.1.cmml" xref="A3.p1.2.m2.1.1.1">→</ci><ci id="A3.p1.2.m2.1.1.2.cmml" xref="A3.p1.2.m2.1.1.2">𝜎</ci><cn id="A3.p1.2.m2.1.1.3.cmml" type="integer" xref="A3.p1.2.m2.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.p1.2.m2.1c">\sigma\to 0</annotation><annotation encoding="application/x-llamapun" id="A3.p1.2.m2.1d">italic_σ → 0</annotation></semantics></math> and we exhibit the speed of convergence. Finally, we simulate SNORE regularization for 1D distribution and compare it to the PnP regularization.</p> </div> <section class="ltx_subsection" id="A3.SS1"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">C.1 </span>Gaussian Prior</h3> <div class="ltx_para" id="A3.SS1.p1"> <p class="ltx_p" id="A3.SS1.p1.1">In order to understand the behavior of Algorithm <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#alg2" title="Algorithm 2 ‣ Optimization algorithms ‣ 2.2 SNORE regularization ‣ 2 Stochastic deNOising REgularization (SNORE) ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_tag">2</span></a> compared to other algorithms (such as Algorithm <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#alg1" title="Algorithm 1 ‣ 2.1 Background ‣ 2 Stochastic deNOising REgularization (SNORE) ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_tag">1</span></a>) we make the computation in a very simple case where the prior is a Gaussian distribution.</p> </div> <div class="ltx_para" id="A3.SS1.p2"> <p class="ltx_p" id="A3.SS1.p2.2">We suppose that <math alttext="p({\mathbf{x}})=\mathcal{N}({\mathbf{x}};{\mathbf{\mu}},{\mathbf{\Sigma}})" class="ltx_Math" display="inline" id="A3.SS1.p2.1.m1.4"><semantics id="A3.SS1.p2.1.m1.4a"><mrow id="A3.SS1.p2.1.m1.4.5" xref="A3.SS1.p2.1.m1.4.5.cmml"><mrow id="A3.SS1.p2.1.m1.4.5.2" xref="A3.SS1.p2.1.m1.4.5.2.cmml"><mi id="A3.SS1.p2.1.m1.4.5.2.2" xref="A3.SS1.p2.1.m1.4.5.2.2.cmml">p</mi><mo id="A3.SS1.p2.1.m1.4.5.2.1" xref="A3.SS1.p2.1.m1.4.5.2.1.cmml">⁢</mo><mrow id="A3.SS1.p2.1.m1.4.5.2.3.2" xref="A3.SS1.p2.1.m1.4.5.2.cmml"><mo id="A3.SS1.p2.1.m1.4.5.2.3.2.1" stretchy="false" xref="A3.SS1.p2.1.m1.4.5.2.cmml">(</mo><mi id="A3.SS1.p2.1.m1.1.1" xref="A3.SS1.p2.1.m1.1.1.cmml">𝐱</mi><mo id="A3.SS1.p2.1.m1.4.5.2.3.2.2" stretchy="false" xref="A3.SS1.p2.1.m1.4.5.2.cmml">)</mo></mrow></mrow><mo id="A3.SS1.p2.1.m1.4.5.1" xref="A3.SS1.p2.1.m1.4.5.1.cmml">=</mo><mrow id="A3.SS1.p2.1.m1.4.5.3" xref="A3.SS1.p2.1.m1.4.5.3.cmml"><mi class="ltx_font_mathcaligraphic" id="A3.SS1.p2.1.m1.4.5.3.2" xref="A3.SS1.p2.1.m1.4.5.3.2.cmml">𝒩</mi><mo id="A3.SS1.p2.1.m1.4.5.3.1" xref="A3.SS1.p2.1.m1.4.5.3.1.cmml">⁢</mo><mrow id="A3.SS1.p2.1.m1.4.5.3.3.2" xref="A3.SS1.p2.1.m1.4.5.3.3.1.cmml"><mo id="A3.SS1.p2.1.m1.4.5.3.3.2.1" stretchy="false" xref="A3.SS1.p2.1.m1.4.5.3.3.1.cmml">(</mo><mi id="A3.SS1.p2.1.m1.2.2" xref="A3.SS1.p2.1.m1.2.2.cmml">𝐱</mi><mo id="A3.SS1.p2.1.m1.4.5.3.3.2.2" xref="A3.SS1.p2.1.m1.4.5.3.3.1.cmml">;</mo><mi id="A3.SS1.p2.1.m1.3.3" xref="A3.SS1.p2.1.m1.3.3.cmml">μ</mi><mo id="A3.SS1.p2.1.m1.4.5.3.3.2.3" xref="A3.SS1.p2.1.m1.4.5.3.3.1.cmml">,</mo><mi id="A3.SS1.p2.1.m1.4.4" xref="A3.SS1.p2.1.m1.4.4.cmml">𝚺</mi><mo id="A3.SS1.p2.1.m1.4.5.3.3.2.4" stretchy="false" xref="A3.SS1.p2.1.m1.4.5.3.3.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="A3.SS1.p2.1.m1.4b"><apply id="A3.SS1.p2.1.m1.4.5.cmml" xref="A3.SS1.p2.1.m1.4.5"><eq id="A3.SS1.p2.1.m1.4.5.1.cmml" xref="A3.SS1.p2.1.m1.4.5.1"></eq><apply id="A3.SS1.p2.1.m1.4.5.2.cmml" xref="A3.SS1.p2.1.m1.4.5.2"><times id="A3.SS1.p2.1.m1.4.5.2.1.cmml" xref="A3.SS1.p2.1.m1.4.5.2.1"></times><ci id="A3.SS1.p2.1.m1.4.5.2.2.cmml" xref="A3.SS1.p2.1.m1.4.5.2.2">𝑝</ci><ci id="A3.SS1.p2.1.m1.1.1.cmml" xref="A3.SS1.p2.1.m1.1.1">𝐱</ci></apply><apply id="A3.SS1.p2.1.m1.4.5.3.cmml" xref="A3.SS1.p2.1.m1.4.5.3"><times id="A3.SS1.p2.1.m1.4.5.3.1.cmml" xref="A3.SS1.p2.1.m1.4.5.3.1"></times><ci id="A3.SS1.p2.1.m1.4.5.3.2.cmml" xref="A3.SS1.p2.1.m1.4.5.3.2">𝒩</ci><list id="A3.SS1.p2.1.m1.4.5.3.3.1.cmml" xref="A3.SS1.p2.1.m1.4.5.3.3.2"><ci id="A3.SS1.p2.1.m1.2.2.cmml" xref="A3.SS1.p2.1.m1.2.2">𝐱</ci><ci id="A3.SS1.p2.1.m1.3.3.cmml" xref="A3.SS1.p2.1.m1.3.3">𝜇</ci><ci id="A3.SS1.p2.1.m1.4.4.cmml" xref="A3.SS1.p2.1.m1.4.4">𝚺</ci></list></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.SS1.p2.1.m1.4c">p({\mathbf{x}})=\mathcal{N}({\mathbf{x}};{\mathbf{\mu}},{\mathbf{\Sigma}})</annotation><annotation encoding="application/x-llamapun" id="A3.SS1.p2.1.m1.4d">italic_p ( bold_x ) = caligraphic_N ( bold_x ; italic_μ , bold_Σ )</annotation></semantics></math>. Then <math alttext="p_{\sigma}({\tilde{\mathbf{x}}})=\mathcal{N}({\tilde{\mathbf{x}}};{\mathbf{\mu% }},{\mathbf{\Sigma}}+\sigma^{2}\mathbf{I})" class="ltx_Math" display="inline" id="A3.SS1.p2.2.m2.4"><semantics id="A3.SS1.p2.2.m2.4a"><mrow id="A3.SS1.p2.2.m2.4.4" xref="A3.SS1.p2.2.m2.4.4.cmml"><mrow id="A3.SS1.p2.2.m2.4.4.3" xref="A3.SS1.p2.2.m2.4.4.3.cmml"><msub id="A3.SS1.p2.2.m2.4.4.3.2" xref="A3.SS1.p2.2.m2.4.4.3.2.cmml"><mi id="A3.SS1.p2.2.m2.4.4.3.2.2" xref="A3.SS1.p2.2.m2.4.4.3.2.2.cmml">p</mi><mi id="A3.SS1.p2.2.m2.4.4.3.2.3" xref="A3.SS1.p2.2.m2.4.4.3.2.3.cmml">σ</mi></msub><mo id="A3.SS1.p2.2.m2.4.4.3.1" xref="A3.SS1.p2.2.m2.4.4.3.1.cmml">⁢</mo><mrow id="A3.SS1.p2.2.m2.4.4.3.3.2" xref="A3.SS1.p2.2.m2.1.1.cmml"><mo id="A3.SS1.p2.2.m2.4.4.3.3.2.1" stretchy="false" xref="A3.SS1.p2.2.m2.1.1.cmml">(</mo><mover accent="true" id="A3.SS1.p2.2.m2.1.1" xref="A3.SS1.p2.2.m2.1.1.cmml"><mi id="A3.SS1.p2.2.m2.1.1.2" xref="A3.SS1.p2.2.m2.1.1.2.cmml">𝐱</mi><mo id="A3.SS1.p2.2.m2.1.1.1" 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id="A3.SS1.p2.2.m2.4.4.1.1.1.1.3.2.cmml" xref="A3.SS1.p2.2.m2.4.4.1.1.1.1.3.2"><csymbol cd="ambiguous" id="A3.SS1.p2.2.m2.4.4.1.1.1.1.3.2.1.cmml" xref="A3.SS1.p2.2.m2.4.4.1.1.1.1.3.2">superscript</csymbol><ci id="A3.SS1.p2.2.m2.4.4.1.1.1.1.3.2.2.cmml" xref="A3.SS1.p2.2.m2.4.4.1.1.1.1.3.2.2">𝜎</ci><cn id="A3.SS1.p2.2.m2.4.4.1.1.1.1.3.2.3.cmml" type="integer" xref="A3.SS1.p2.2.m2.4.4.1.1.1.1.3.2.3">2</cn></apply><ci id="A3.SS1.p2.2.m2.4.4.1.1.1.1.3.3.cmml" xref="A3.SS1.p2.2.m2.4.4.1.1.1.1.3.3">𝐈</ci></apply></apply></list></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.SS1.p2.2.m2.4c">p_{\sigma}({\tilde{\mathbf{x}}})=\mathcal{N}({\tilde{\mathbf{x}}};{\mathbf{\mu% }},{\mathbf{\Sigma}}+\sigma^{2}\mathbf{I})</annotation><annotation encoding="application/x-llamapun" id="A3.SS1.p2.2.m2.4d">italic_p start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ( over~ start_ARG bold_x end_ARG ) = caligraphic_N ( over~ start_ARG bold_x end_ARG ; italic_μ , bold_Σ + italic_σ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT bold_I )</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="A3.SS1.p3"> <p class="ltx_p" id="A3.SS1.p3.1">The score of these distributions has the following expression</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A8.EGx17"> <tbody id="A3.Ex20"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle-\nabla\log p({\mathbf{x}})" class="ltx_Math" display="inline" id="A3.Ex20.m1.1"><semantics id="A3.Ex20.m1.1a"><mrow id="A3.Ex20.m1.1.2" xref="A3.Ex20.m1.1.2.cmml"><mo id="A3.Ex20.m1.1.2a" rspace="0.167em" xref="A3.Ex20.m1.1.2.cmml">−</mo><mrow id="A3.Ex20.m1.1.2.2" xref="A3.Ex20.m1.1.2.2.cmml"><mrow id="A3.Ex20.m1.1.2.2.2" xref="A3.Ex20.m1.1.2.2.2.cmml"><mrow id="A3.Ex20.m1.1.2.2.2.1" xref="A3.Ex20.m1.1.2.2.2.1.cmml"><mo id="A3.Ex20.m1.1.2.2.2.1.1" rspace="0.167em" 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xref="A3.Ex20.m1.1.2.2.2"><apply id="A3.Ex20.m1.1.2.2.2.1.cmml" xref="A3.Ex20.m1.1.2.2.2.1"><ci id="A3.Ex20.m1.1.2.2.2.1.1.cmml" xref="A3.Ex20.m1.1.2.2.2.1.1">∇</ci><log id="A3.Ex20.m1.1.2.2.2.1.2.cmml" xref="A3.Ex20.m1.1.2.2.2.1.2"></log></apply><ci id="A3.Ex20.m1.1.2.2.2.2.cmml" xref="A3.Ex20.m1.1.2.2.2.2">𝑝</ci></apply><ci id="A3.Ex20.m1.1.1.cmml" xref="A3.Ex20.m1.1.1">𝐱</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.Ex20.m1.1c">\displaystyle-\nabla\log p({\mathbf{x}})</annotation><annotation encoding="application/x-llamapun" id="A3.Ex20.m1.1d">- ∇ roman_log italic_p ( bold_x )</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle={\mathbf{\Sigma}}^{-1}({\mathbf{x}}-{\mathbf{\mu}})" class="ltx_Math" display="inline" id="A3.Ex20.m2.1"><semantics id="A3.Ex20.m2.1a"><mrow id="A3.Ex20.m2.1.1" xref="A3.Ex20.m2.1.1.cmml"><mi id="A3.Ex20.m2.1.1.3" xref="A3.Ex20.m2.1.1.3.cmml"></mi><mo 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id="A3.Ex20.m2.1.1.1.1.1.3" stretchy="false" xref="A3.Ex20.m2.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="A3.Ex20.m2.1b"><apply id="A3.Ex20.m2.1.1.cmml" xref="A3.Ex20.m2.1.1"><eq id="A3.Ex20.m2.1.1.2.cmml" xref="A3.Ex20.m2.1.1.2"></eq><csymbol cd="latexml" id="A3.Ex20.m2.1.1.3.cmml" xref="A3.Ex20.m2.1.1.3">absent</csymbol><apply id="A3.Ex20.m2.1.1.1.cmml" xref="A3.Ex20.m2.1.1.1"><times id="A3.Ex20.m2.1.1.1.2.cmml" xref="A3.Ex20.m2.1.1.1.2"></times><apply id="A3.Ex20.m2.1.1.1.3.cmml" xref="A3.Ex20.m2.1.1.1.3"><csymbol cd="ambiguous" id="A3.Ex20.m2.1.1.1.3.1.cmml" xref="A3.Ex20.m2.1.1.1.3">superscript</csymbol><ci id="A3.Ex20.m2.1.1.1.3.2.cmml" xref="A3.Ex20.m2.1.1.1.3.2">𝚺</ci><apply id="A3.Ex20.m2.1.1.1.3.3.cmml" xref="A3.Ex20.m2.1.1.1.3.3"><minus id="A3.Ex20.m2.1.1.1.3.3.1.cmml" xref="A3.Ex20.m2.1.1.1.3.3"></minus><cn id="A3.Ex20.m2.1.1.1.3.3.2.cmml" type="integer" xref="A3.Ex20.m2.1.1.1.3.3.2">1</cn></apply></apply><apply id="A3.Ex20.m2.1.1.1.1.1.1.cmml" xref="A3.Ex20.m2.1.1.1.1.1"><minus id="A3.Ex20.m2.1.1.1.1.1.1.1.cmml" xref="A3.Ex20.m2.1.1.1.1.1.1.1"></minus><ci id="A3.Ex20.m2.1.1.1.1.1.1.2.cmml" xref="A3.Ex20.m2.1.1.1.1.1.1.2">𝐱</ci><ci id="A3.Ex20.m2.1.1.1.1.1.1.3.cmml" xref="A3.Ex20.m2.1.1.1.1.1.1.3">𝜇</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.Ex20.m2.1c">\displaystyle={\mathbf{\Sigma}}^{-1}({\mathbf{x}}-{\mathbf{\mu}})</annotation><annotation encoding="application/x-llamapun" id="A3.Ex20.m2.1d">= bold_Σ start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ( bold_x - italic_μ )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="A3.Ex21"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle-\nabla\log p_{\sigma}({\tilde{\mathbf{x}}})" class="ltx_Math" 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xref="A3.Ex21.m2.1.1.1.1.2.2.1.1.3">𝜇</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.Ex21.m2.1c">\displaystyle=({\mathbf{\Sigma}}+\sigma^{2}\mathbf{I}_{d})^{-1}({\tilde{% \mathbf{x}}}-{\mathbf{\mu}}).</annotation><annotation encoding="application/x-llamapun" id="A3.Ex21.m2.1d">= ( bold_Σ + italic_σ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT bold_I start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ( over~ start_ARG bold_x end_ARG - italic_μ ) .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> </div> <div class="ltx_para" id="A3.SS1.p4"> <p class="ltx_p" id="A3.SS1.p4.1">Moreover, the SNORE regularization defined in Equation (<a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#S2.E5" title="5 ‣ 2.2 SNORE regularization ‣ 2 Stochastic deNOising REgularization (SNORE) ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_tag">5</span></a>) can be computed in closed form</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A8.EGx18"> <tbody id="A3.Ex22"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\mathcal{R}_{\sigma}({\mathbf{x}})=-\mathbb{E}_{{\tilde{\mathbf{x% }}}\sim p_{\sigma}({\tilde{\mathbf{x}}}|{\mathbf{x}})}\left(\log p_{\sigma}({% \tilde{\mathbf{x}}})\right)=\frac{1}{2}\mathbb{E}_{{\tilde{\mathbf{x}}}\sim p_% {\sigma}({\tilde{\mathbf{x}}}|{\mathbf{x}})}\left(({\tilde{\mathbf{x}}}-{% \mathbf{\mu}})^{T}({\mathbf{\Sigma}}+\sigma^{2}\mathbf{I}_{d})^{-1}({\tilde{% \mathbf{x}}}-{\mathbf{\mu}})\right)" class="ltx_Math" display="inline" id="A3.Ex22.m1.6"><semantics id="A3.Ex22.m1.6a"><mrow id="A3.Ex22.m1.6.6" xref="A3.Ex22.m1.6.6.cmml"><mrow id="A3.Ex22.m1.6.6.4" 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id="A3.Ex22.m1.6.6.2.1.1.1.2.1.1.1.2" xref="A3.Ex22.m1.6.6.2.1.1.1.2.1.1.1.2.cmml">𝚺</mi><mo id="A3.Ex22.m1.6.6.2.1.1.1.2.1.1.1.1" xref="A3.Ex22.m1.6.6.2.1.1.1.2.1.1.1.1.cmml">+</mo><mrow id="A3.Ex22.m1.6.6.2.1.1.1.2.1.1.1.3" xref="A3.Ex22.m1.6.6.2.1.1.1.2.1.1.1.3.cmml"><msup id="A3.Ex22.m1.6.6.2.1.1.1.2.1.1.1.3.2" xref="A3.Ex22.m1.6.6.2.1.1.1.2.1.1.1.3.2.cmml"><mi id="A3.Ex22.m1.6.6.2.1.1.1.2.1.1.1.3.2.2" xref="A3.Ex22.m1.6.6.2.1.1.1.2.1.1.1.3.2.2.cmml">σ</mi><mn id="A3.Ex22.m1.6.6.2.1.1.1.2.1.1.1.3.2.3" xref="A3.Ex22.m1.6.6.2.1.1.1.2.1.1.1.3.2.3.cmml">2</mn></msup><mo id="A3.Ex22.m1.6.6.2.1.1.1.2.1.1.1.3.1" xref="A3.Ex22.m1.6.6.2.1.1.1.2.1.1.1.3.1.cmml">⁢</mo><msub id="A3.Ex22.m1.6.6.2.1.1.1.2.1.1.1.3.3" xref="A3.Ex22.m1.6.6.2.1.1.1.2.1.1.1.3.3.cmml"><mi id="A3.Ex22.m1.6.6.2.1.1.1.2.1.1.1.3.3.2" xref="A3.Ex22.m1.6.6.2.1.1.1.2.1.1.1.3.3.2.cmml">𝐈</mi><mi id="A3.Ex22.m1.6.6.2.1.1.1.2.1.1.1.3.3.3" xref="A3.Ex22.m1.6.6.2.1.1.1.2.1.1.1.3.3.3.cmml">d</mi></msub></mrow></mrow><mo 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id="A3.Ex22.m1.6c">\displaystyle\mathcal{R}_{\sigma}({\mathbf{x}})=-\mathbb{E}_{{\tilde{\mathbf{x% }}}\sim p_{\sigma}({\tilde{\mathbf{x}}}|{\mathbf{x}})}\left(\log p_{\sigma}({% \tilde{\mathbf{x}}})\right)=\frac{1}{2}\mathbb{E}_{{\tilde{\mathbf{x}}}\sim p_% {\sigma}({\tilde{\mathbf{x}}}|{\mathbf{x}})}\left(({\tilde{\mathbf{x}}}-{% \mathbf{\mu}})^{T}({\mathbf{\Sigma}}+\sigma^{2}\mathbf{I}_{d})^{-1}({\tilde{% \mathbf{x}}}-{\mathbf{\mu}})\right)</annotation><annotation encoding="application/x-llamapun" id="A3.Ex22.m1.6d">caligraphic_R start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ( bold_x ) = - blackboard_E start_POSTSUBSCRIPT over~ start_ARG bold_x end_ARG ∼ italic_p start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ( over~ start_ARG bold_x end_ARG | bold_x ) end_POSTSUBSCRIPT ( roman_log italic_p start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ( over~ start_ARG bold_x end_ARG ) ) = divide start_ARG 1 end_ARG start_ARG 2 end_ARG blackboard_E start_POSTSUBSCRIPT over~ start_ARG bold_x end_ARG ∼ italic_p start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ( over~ start_ARG bold_x end_ARG | bold_x ) end_POSTSUBSCRIPT ( ( over~ start_ARG bold_x end_ARG - italic_μ ) start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT ( bold_Σ + italic_σ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT bold_I start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ( over~ start_ARG bold_x end_ARG - italic_μ ) )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="A3.Ex23"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle=\frac{1}{2}\int_{{\mathbb{R}}^{d}}({\tilde{\mathbf{x}}}-{\mathbf% {\mu}})^{T}({\mathbf{\Sigma}}+\sigma^{2}\mathbf{I}_{d})^{-1}({\tilde{\mathbf{x% }}}-{\mathbf{\mu}})\mathcal{N}({\tilde{\mathbf{x}}};{\mathbf{x}},\sigma^{2}% 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}}}-{\mathbf{\mu}})\mathcal{N}({\tilde{\mathbf{x}}};{\mathbf{x}},\sigma^{2}% \mathbf{I}_{d})d{\tilde{\mathbf{x}}}</annotation><annotation encoding="application/x-llamapun" id="A3.Ex23.m1.6d">= divide start_ARG 1 end_ARG start_ARG 2 end_ARG ∫ start_POSTSUBSCRIPT blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ( over~ start_ARG bold_x end_ARG - italic_μ ) start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT ( bold_Σ + italic_σ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT bold_I start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ( over~ start_ARG bold_x end_ARG - italic_μ ) caligraphic_N ( over~ start_ARG bold_x end_ARG ; bold_x , italic_σ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT bold_I start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT ) italic_d over~ start_ARG bold_x end_ARG</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="A3.Ex24"><tr 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}-{\mathbf{x}})\mathcal{N}({\tilde{\mathbf{x}}};{\mathbf{x}},\sigma^{2}\mathbf% {I}_{d})d{\tilde{\mathbf{x}}}+\frac{1}{2}\int_{{\mathbb{R}}^{d}}({\mathbf{x}}-% {\mathbf{\mu}})^{T}({\mathbf{\Sigma}}+\sigma^{2}\mathbf{I}_{d})^{-1}({\mathbf{% x}}-{\mathbf{\mu}})\mathcal{N}({\tilde{\mathbf{x}}};{\mathbf{x}},\sigma^{2}% \mathbf{I}_{d})d{\tilde{\mathbf{x}}}</annotation><annotation encoding="application/x-llamapun" id="A3.Ex24.m1.12d">= divide start_ARG 1 end_ARG start_ARG 2 end_ARG ∫ start_POSTSUBSCRIPT blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ( over~ start_ARG bold_x end_ARG - bold_x ) start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT ( bold_Σ + italic_σ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT bold_I start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ( over~ start_ARG bold_x end_ARG - bold_x ) caligraphic_N ( over~ start_ARG bold_x end_ARG ; bold_x , italic_σ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT bold_I start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT ) italic_d over~ start_ARG bold_x end_ARG + divide start_ARG 1 end_ARG start_ARG 2 end_ARG ∫ start_POSTSUBSCRIPT blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ( bold_x - italic_μ ) start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT ( bold_Σ + italic_σ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT bold_I start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ( bold_x - italic_μ ) caligraphic_N ( over~ start_ARG bold_x end_ARG ; bold_x , italic_σ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT bold_I start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT ) italic_d over~ start_ARG bold_x end_ARG</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="A3.Ex25"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_left 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xref="A3.Ex25.m1.6.6.4.4.7.2.2">𝐱</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.Ex25.m1.6c">\displaystyle+\int_{{\mathbb{R}}^{d}}({\tilde{\mathbf{x}}}-{\mathbf{x}})^{T}({% \mathbf{\Sigma}}+\sigma^{2}\mathbf{I}_{d})^{-1}({\mathbf{x}}-{\mathbf{\mu}})% \mathcal{N}({\tilde{\mathbf{x}}};{\mathbf{x}},\sigma^{2}\mathbf{I}_{d})d{% \tilde{\mathbf{x}}}</annotation><annotation encoding="application/x-llamapun" id="A3.Ex25.m1.6d">+ ∫ start_POSTSUBSCRIPT blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ( over~ start_ARG bold_x end_ARG - bold_x ) start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT ( bold_Σ + italic_σ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT bold_I start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ( bold_x - italic_μ ) caligraphic_N ( over~ start_ARG bold_x end_ARG ; bold_x , italic_σ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT bold_I start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT ) italic_d over~ start_ARG bold_x end_ARG</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="A3.Ex26"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle=\sigma^{2}Tr\left(({\mathbf{\Sigma}}+\sigma^{2}\mathbf{I}_{d})^{% -1}\right)+\frac{1}{2}({\mathbf{x}}-{\mathbf{\mu}})^{T}({\mathbf{\Sigma}}+% \sigma^{2}\mathbf{I}_{d})^{-1}({\mathbf{x}}-{\mathbf{\mu}})." class="ltx_Math" display="inline" id="A3.Ex26.m1.1"><semantics id="A3.Ex26.m1.1a"><mrow id="A3.Ex26.m1.1.1.1" xref="A3.Ex26.m1.1.1.1.1.cmml"><mrow id="A3.Ex26.m1.1.1.1.1" xref="A3.Ex26.m1.1.1.1.1.cmml"><mi id="A3.Ex26.m1.1.1.1.1.6" xref="A3.Ex26.m1.1.1.1.1.6.cmml"></mi><mo id="A3.Ex26.m1.1.1.1.1.5" xref="A3.Ex26.m1.1.1.1.1.5.cmml">=</mo><mrow id="A3.Ex26.m1.1.1.1.1.4" 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xref="A3.Ex26.m1.1.1.1.1.3.3.2.3"><minus id="A3.Ex26.m1.1.1.1.1.3.3.2.3.1.cmml" xref="A3.Ex26.m1.1.1.1.1.3.3.2.3"></minus><cn id="A3.Ex26.m1.1.1.1.1.3.3.2.3.2.cmml" type="integer" xref="A3.Ex26.m1.1.1.1.1.3.3.2.3.2">1</cn></apply></apply><apply id="A3.Ex26.m1.1.1.1.1.4.4.3.1.1.cmml" xref="A3.Ex26.m1.1.1.1.1.4.4.3.1"><minus id="A3.Ex26.m1.1.1.1.1.4.4.3.1.1.1.cmml" xref="A3.Ex26.m1.1.1.1.1.4.4.3.1.1.1"></minus><ci id="A3.Ex26.m1.1.1.1.1.4.4.3.1.1.2.cmml" xref="A3.Ex26.m1.1.1.1.1.4.4.3.1.1.2">𝐱</ci><ci id="A3.Ex26.m1.1.1.1.1.4.4.3.1.1.3.cmml" xref="A3.Ex26.m1.1.1.1.1.4.4.3.1.1.3">𝜇</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.Ex26.m1.1c">\displaystyle=\sigma^{2}Tr\left(({\mathbf{\Sigma}}+\sigma^{2}\mathbf{I}_{d})^{% -1}\right)+\frac{1}{2}({\mathbf{x}}-{\mathbf{\mu}})^{T}({\mathbf{\Sigma}}+% \sigma^{2}\mathbf{I}_{d})^{-1}({\mathbf{x}}-{\mathbf{\mu}}).</annotation><annotation encoding="application/x-llamapun" id="A3.Ex26.m1.1d">= italic_σ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_T italic_r ( ( bold_Σ + italic_σ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT bold_I start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ) + divide start_ARG 1 end_ARG start_ARG 2 end_ARG ( bold_x - italic_μ ) start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT ( bold_Σ + italic_σ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT bold_I start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ( bold_x - italic_μ ) .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> </div> <div class="ltx_para" id="A3.SS1.p5"> <p class="ltx_p" id="A3.SS1.p5.2">The gradient of the SNORE regularization is thus</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A8.EGx19"> <tbody id="A3.Ex27"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\nabla\mathcal{R}_{\sigma}({\mathbf{x}})=({\mathbf{\Sigma}}+% \sigma^{2}\mathbf{I}_{d})^{-1}({\mathbf{x}}-{\mathbf{\mu}})," class="ltx_Math" display="inline" id="A3.Ex27.m1.2"><semantics id="A3.Ex27.m1.2a"><mrow id="A3.Ex27.m1.2.2.1" xref="A3.Ex27.m1.2.2.1.1.cmml"><mrow id="A3.Ex27.m1.2.2.1.1" xref="A3.Ex27.m1.2.2.1.1.cmml"><mrow id="A3.Ex27.m1.2.2.1.1.4" xref="A3.Ex27.m1.2.2.1.1.4.cmml"><mrow id="A3.Ex27.m1.2.2.1.1.4.2" xref="A3.Ex27.m1.2.2.1.1.4.2.cmml"><mo id="A3.Ex27.m1.2.2.1.1.4.2.1" rspace="0.167em" xref="A3.Ex27.m1.2.2.1.1.4.2.1.cmml">∇</mo><msub id="A3.Ex27.m1.2.2.1.1.4.2.2" xref="A3.Ex27.m1.2.2.1.1.4.2.2.cmml"><mi class="ltx_font_mathcaligraphic" id="A3.Ex27.m1.2.2.1.1.4.2.2.2" xref="A3.Ex27.m1.2.2.1.1.4.2.2.2.cmml">ℛ</mi><mi id="A3.Ex27.m1.2.2.1.1.4.2.2.3" xref="A3.Ex27.m1.2.2.1.1.4.2.2.3.cmml">σ</mi></msub></mrow><mo id="A3.Ex27.m1.2.2.1.1.4.1" xref="A3.Ex27.m1.2.2.1.1.4.1.cmml">⁢</mo><mrow id="A3.Ex27.m1.2.2.1.1.4.3.2" xref="A3.Ex27.m1.2.2.1.1.4.cmml"><mo id="A3.Ex27.m1.2.2.1.1.4.3.2.1" stretchy="false" xref="A3.Ex27.m1.2.2.1.1.4.cmml">(</mo><mi id="A3.Ex27.m1.1.1" xref="A3.Ex27.m1.1.1.cmml">𝐱</mi><mo id="A3.Ex27.m1.2.2.1.1.4.3.2.2" stretchy="false" xref="A3.Ex27.m1.2.2.1.1.4.cmml">)</mo></mrow></mrow><mo id="A3.Ex27.m1.2.2.1.1.3" xref="A3.Ex27.m1.2.2.1.1.3.cmml">=</mo><mrow id="A3.Ex27.m1.2.2.1.1.2" xref="A3.Ex27.m1.2.2.1.1.2.cmml"><msup id="A3.Ex27.m1.2.2.1.1.1.1" xref="A3.Ex27.m1.2.2.1.1.1.1.cmml"><mrow id="A3.Ex27.m1.2.2.1.1.1.1.1.1" xref="A3.Ex27.m1.2.2.1.1.1.1.1.1.1.cmml"><mo id="A3.Ex27.m1.2.2.1.1.1.1.1.1.2" stretchy="false" xref="A3.Ex27.m1.2.2.1.1.1.1.1.1.1.cmml">(</mo><mrow id="A3.Ex27.m1.2.2.1.1.1.1.1.1.1" xref="A3.Ex27.m1.2.2.1.1.1.1.1.1.1.cmml"><mi id="A3.Ex27.m1.2.2.1.1.1.1.1.1.1.2" xref="A3.Ex27.m1.2.2.1.1.1.1.1.1.1.2.cmml">𝚺</mi><mo id="A3.Ex27.m1.2.2.1.1.1.1.1.1.1.1" xref="A3.Ex27.m1.2.2.1.1.1.1.1.1.1.1.cmml">+</mo><mrow id="A3.Ex27.m1.2.2.1.1.1.1.1.1.1.3" xref="A3.Ex27.m1.2.2.1.1.1.1.1.1.1.3.cmml"><msup id="A3.Ex27.m1.2.2.1.1.1.1.1.1.1.3.2" xref="A3.Ex27.m1.2.2.1.1.1.1.1.1.1.3.2.cmml"><mi id="A3.Ex27.m1.2.2.1.1.1.1.1.1.1.3.2.2" xref="A3.Ex27.m1.2.2.1.1.1.1.1.1.1.3.2.2.cmml">σ</mi><mn id="A3.Ex27.m1.2.2.1.1.1.1.1.1.1.3.2.3" xref="A3.Ex27.m1.2.2.1.1.1.1.1.1.1.3.2.3.cmml">2</mn></msup><mo id="A3.Ex27.m1.2.2.1.1.1.1.1.1.1.3.1" xref="A3.Ex27.m1.2.2.1.1.1.1.1.1.1.3.1.cmml">⁢</mo><msub id="A3.Ex27.m1.2.2.1.1.1.1.1.1.1.3.3" xref="A3.Ex27.m1.2.2.1.1.1.1.1.1.1.3.3.cmml"><mi id="A3.Ex27.m1.2.2.1.1.1.1.1.1.1.3.3.2" xref="A3.Ex27.m1.2.2.1.1.1.1.1.1.1.3.3.2.cmml">𝐈</mi><mi id="A3.Ex27.m1.2.2.1.1.1.1.1.1.1.3.3.3" xref="A3.Ex27.m1.2.2.1.1.1.1.1.1.1.3.3.3.cmml">d</mi></msub></mrow></mrow><mo id="A3.Ex27.m1.2.2.1.1.1.1.1.1.3" stretchy="false" xref="A3.Ex27.m1.2.2.1.1.1.1.1.1.1.cmml">)</mo></mrow><mrow id="A3.Ex27.m1.2.2.1.1.1.1.3" xref="A3.Ex27.m1.2.2.1.1.1.1.3.cmml"><mo id="A3.Ex27.m1.2.2.1.1.1.1.3a" xref="A3.Ex27.m1.2.2.1.1.1.1.3.cmml">−</mo><mn id="A3.Ex27.m1.2.2.1.1.1.1.3.2" xref="A3.Ex27.m1.2.2.1.1.1.1.3.2.cmml">1</mn></mrow></msup><mo id="A3.Ex27.m1.2.2.1.1.2.3" xref="A3.Ex27.m1.2.2.1.1.2.3.cmml">⁢</mo><mrow id="A3.Ex27.m1.2.2.1.1.2.2.1" xref="A3.Ex27.m1.2.2.1.1.2.2.1.1.cmml"><mo id="A3.Ex27.m1.2.2.1.1.2.2.1.2" stretchy="false" xref="A3.Ex27.m1.2.2.1.1.2.2.1.1.cmml">(</mo><mrow id="A3.Ex27.m1.2.2.1.1.2.2.1.1" xref="A3.Ex27.m1.2.2.1.1.2.2.1.1.cmml"><mi id="A3.Ex27.m1.2.2.1.1.2.2.1.1.2" xref="A3.Ex27.m1.2.2.1.1.2.2.1.1.2.cmml">𝐱</mi><mo id="A3.Ex27.m1.2.2.1.1.2.2.1.1.1" xref="A3.Ex27.m1.2.2.1.1.2.2.1.1.1.cmml">−</mo><mi id="A3.Ex27.m1.2.2.1.1.2.2.1.1.3" xref="A3.Ex27.m1.2.2.1.1.2.2.1.1.3.cmml">μ</mi></mrow><mo id="A3.Ex27.m1.2.2.1.1.2.2.1.3" stretchy="false" xref="A3.Ex27.m1.2.2.1.1.2.2.1.1.cmml">)</mo></mrow></mrow></mrow><mo id="A3.Ex27.m1.2.2.1.2" xref="A3.Ex27.m1.2.2.1.1.cmml">,</mo></mrow><annotation-xml encoding="MathML-Content" id="A3.Ex27.m1.2b"><apply id="A3.Ex27.m1.2.2.1.1.cmml" xref="A3.Ex27.m1.2.2.1"><eq id="A3.Ex27.m1.2.2.1.1.3.cmml" xref="A3.Ex27.m1.2.2.1.1.3"></eq><apply id="A3.Ex27.m1.2.2.1.1.4.cmml" xref="A3.Ex27.m1.2.2.1.1.4"><times id="A3.Ex27.m1.2.2.1.1.4.1.cmml" xref="A3.Ex27.m1.2.2.1.1.4.1"></times><apply id="A3.Ex27.m1.2.2.1.1.4.2.cmml" xref="A3.Ex27.m1.2.2.1.1.4.2"><ci id="A3.Ex27.m1.2.2.1.1.4.2.1.cmml" xref="A3.Ex27.m1.2.2.1.1.4.2.1">∇</ci><apply id="A3.Ex27.m1.2.2.1.1.4.2.2.cmml" xref="A3.Ex27.m1.2.2.1.1.4.2.2"><csymbol cd="ambiguous" id="A3.Ex27.m1.2.2.1.1.4.2.2.1.cmml" xref="A3.Ex27.m1.2.2.1.1.4.2.2">subscript</csymbol><ci id="A3.Ex27.m1.2.2.1.1.4.2.2.2.cmml" xref="A3.Ex27.m1.2.2.1.1.4.2.2.2">ℛ</ci><ci id="A3.Ex27.m1.2.2.1.1.4.2.2.3.cmml" xref="A3.Ex27.m1.2.2.1.1.4.2.2.3">𝜎</ci></apply></apply><ci id="A3.Ex27.m1.1.1.cmml" xref="A3.Ex27.m1.1.1">𝐱</ci></apply><apply id="A3.Ex27.m1.2.2.1.1.2.cmml" 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xref="A3.Ex27.m1.2.2.1.1.1.1.1.1.1.3.2.2">𝜎</ci><cn id="A3.Ex27.m1.2.2.1.1.1.1.1.1.1.3.2.3.cmml" type="integer" xref="A3.Ex27.m1.2.2.1.1.1.1.1.1.1.3.2.3">2</cn></apply><apply id="A3.Ex27.m1.2.2.1.1.1.1.1.1.1.3.3.cmml" xref="A3.Ex27.m1.2.2.1.1.1.1.1.1.1.3.3"><csymbol cd="ambiguous" id="A3.Ex27.m1.2.2.1.1.1.1.1.1.1.3.3.1.cmml" xref="A3.Ex27.m1.2.2.1.1.1.1.1.1.1.3.3">subscript</csymbol><ci id="A3.Ex27.m1.2.2.1.1.1.1.1.1.1.3.3.2.cmml" xref="A3.Ex27.m1.2.2.1.1.1.1.1.1.1.3.3.2">𝐈</ci><ci id="A3.Ex27.m1.2.2.1.1.1.1.1.1.1.3.3.3.cmml" xref="A3.Ex27.m1.2.2.1.1.1.1.1.1.1.3.3.3">𝑑</ci></apply></apply></apply><apply id="A3.Ex27.m1.2.2.1.1.1.1.3.cmml" xref="A3.Ex27.m1.2.2.1.1.1.1.3"><minus id="A3.Ex27.m1.2.2.1.1.1.1.3.1.cmml" xref="A3.Ex27.m1.2.2.1.1.1.1.3"></minus><cn id="A3.Ex27.m1.2.2.1.1.1.1.3.2.cmml" type="integer" xref="A3.Ex27.m1.2.2.1.1.1.1.3.2">1</cn></apply></apply><apply id="A3.Ex27.m1.2.2.1.1.2.2.1.1.cmml" xref="A3.Ex27.m1.2.2.1.1.2.2.1"><minus id="A3.Ex27.m1.2.2.1.1.2.2.1.1.1.cmml" xref="A3.Ex27.m1.2.2.1.1.2.2.1.1.1"></minus><ci id="A3.Ex27.m1.2.2.1.1.2.2.1.1.2.cmml" xref="A3.Ex27.m1.2.2.1.1.2.2.1.1.2">𝐱</ci><ci id="A3.Ex27.m1.2.2.1.1.2.2.1.1.3.cmml" xref="A3.Ex27.m1.2.2.1.1.2.2.1.1.3">𝜇</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.Ex27.m1.2c">\displaystyle\nabla\mathcal{R}_{\sigma}({\mathbf{x}})=({\mathbf{\Sigma}}+% \sigma^{2}\mathbf{I}_{d})^{-1}({\mathbf{x}}-{\mathbf{\mu}}),</annotation><annotation encoding="application/x-llamapun" id="A3.Ex27.m1.2d">∇ caligraphic_R start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ( bold_x ) = ( bold_Σ + italic_σ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT bold_I start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ( bold_x - italic_μ ) ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="A3.SS1.p5.1">and we have that with a Gaussian prior, <math 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id="A3.SS1.p5.1.m1.3c">\nabla\mathcal{R}_{\sigma}({\mathbf{x}})=\nabla\mathcal{P}_{\sigma}({\mathbf{x% }})=-\nabla\log p_{\sigma}({\mathbf{x}})</annotation><annotation encoding="application/x-llamapun" id="A3.SS1.p5.1.m1.3d">∇ caligraphic_R start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ( bold_x ) = ∇ caligraphic_P start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ( bold_x ) = - ∇ roman_log italic_p start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ( bold_x )</annotation></semantics></math>, which makes SNORE equivalent to traditional PnP.</p> </div> </section> <section class="ltx_subsection" id="A3.SS2"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">C.2 </span>Gaussian Mixture prior</h3> <div class="ltx_para" id="A3.SS2.p1"> <p class="ltx_p" id="A3.SS2.p1.5">In this section, we study the behavior of the SNORE regularization in the case of a Gaussian Mixture prior. First, we remember the convergence of <math alttext="-\nabla\log p_{\sigma}" class="ltx_Math" display="inline" id="A3.SS2.p1.1.m1.1"><semantics id="A3.SS2.p1.1.m1.1a"><mrow id="A3.SS2.p1.1.m1.1.1" xref="A3.SS2.p1.1.m1.1.1.cmml"><mo id="A3.SS2.p1.1.m1.1.1a" rspace="0.167em" xref="A3.SS2.p1.1.m1.1.1.cmml">−</mo><mrow id="A3.SS2.p1.1.m1.1.1.2" xref="A3.SS2.p1.1.m1.1.1.2.cmml"><mrow id="A3.SS2.p1.1.m1.1.1.2.1" xref="A3.SS2.p1.1.m1.1.1.2.1.cmml"><mo id="A3.SS2.p1.1.m1.1.1.2.1.1" rspace="0.167em" xref="A3.SS2.p1.1.m1.1.1.2.1.1.cmml">∇</mo><mi id="A3.SS2.p1.1.m1.1.1.2.1.2" xref="A3.SS2.p1.1.m1.1.1.2.1.2.cmml">log</mi></mrow><mo id="A3.SS2.p1.1.m1.1.1.2a" lspace="0.167em" xref="A3.SS2.p1.1.m1.1.1.2.cmml">⁡</mo><msub id="A3.SS2.p1.1.m1.1.1.2.2" xref="A3.SS2.p1.1.m1.1.1.2.2.cmml"><mi id="A3.SS2.p1.1.m1.1.1.2.2.2" xref="A3.SS2.p1.1.m1.1.1.2.2.2.cmml">p</mi><mi id="A3.SS2.p1.1.m1.1.1.2.2.3" xref="A3.SS2.p1.1.m1.1.1.2.2.3.cmml">σ</mi></msub></mrow></mrow><annotation-xml encoding="MathML-Content" 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roman_log italic_p start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT</annotation></semantics></math> to <math alttext="-\nabla\log p" class="ltx_Math" display="inline" id="A3.SS2.p1.2.m2.1"><semantics id="A3.SS2.p1.2.m2.1a"><mrow id="A3.SS2.p1.2.m2.1.1" xref="A3.SS2.p1.2.m2.1.1.cmml"><mo id="A3.SS2.p1.2.m2.1.1a" rspace="0.167em" xref="A3.SS2.p1.2.m2.1.1.cmml">−</mo><mrow id="A3.SS2.p1.2.m2.1.1.2" xref="A3.SS2.p1.2.m2.1.1.2.cmml"><mrow id="A3.SS2.p1.2.m2.1.1.2.1" xref="A3.SS2.p1.2.m2.1.1.2.1.cmml"><mo id="A3.SS2.p1.2.m2.1.1.2.1.1" rspace="0.167em" xref="A3.SS2.p1.2.m2.1.1.2.1.1.cmml">∇</mo><mi id="A3.SS2.p1.2.m2.1.1.2.1.2" xref="A3.SS2.p1.2.m2.1.1.2.1.2.cmml">log</mi></mrow><mo id="A3.SS2.p1.2.m2.1.1.2a" lspace="0.167em" xref="A3.SS2.p1.2.m2.1.1.2.cmml">⁡</mo><mi id="A3.SS2.p1.2.m2.1.1.2.2" xref="A3.SS2.p1.2.m2.1.1.2.2.cmml">p</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="A3.SS2.p1.2.m2.1b"><apply id="A3.SS2.p1.2.m2.1.1.cmml" xref="A3.SS2.p1.2.m2.1.1"><minus id="A3.SS2.p1.2.m2.1.1.1.cmml" xref="A3.SS2.p1.2.m2.1.1"></minus><apply id="A3.SS2.p1.2.m2.1.1.2.cmml" xref="A3.SS2.p1.2.m2.1.1.2"><apply id="A3.SS2.p1.2.m2.1.1.2.1.cmml" xref="A3.SS2.p1.2.m2.1.1.2.1"><ci id="A3.SS2.p1.2.m2.1.1.2.1.1.cmml" xref="A3.SS2.p1.2.m2.1.1.2.1.1">∇</ci><log id="A3.SS2.p1.2.m2.1.1.2.1.2.cmml" xref="A3.SS2.p1.2.m2.1.1.2.1.2"></log></apply><ci id="A3.SS2.p1.2.m2.1.1.2.2.cmml" xref="A3.SS2.p1.2.m2.1.1.2.2">𝑝</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.SS2.p1.2.m2.1c">-\nabla\log p</annotation><annotation encoding="application/x-llamapun" id="A3.SS2.p1.2.m2.1d">- ∇ roman_log italic_p</annotation></semantics></math> and we compute the speed of convergence of <math alttext="\nabla\mathcal{R}_{\sigma}" class="ltx_Math" display="inline" id="A3.SS2.p1.3.m3.1"><semantics id="A3.SS2.p1.3.m3.1a"><mrow id="A3.SS2.p1.3.m3.1.1" xref="A3.SS2.p1.3.m3.1.1.cmml"><mo id="A3.SS2.p1.3.m3.1.1.1" rspace="0.167em" xref="A3.SS2.p1.3.m3.1.1.1.cmml">∇</mo><msub id="A3.SS2.p1.3.m3.1.1.2" xref="A3.SS2.p1.3.m3.1.1.2.cmml"><mi class="ltx_font_mathcaligraphic" id="A3.SS2.p1.3.m3.1.1.2.2" xref="A3.SS2.p1.3.m3.1.1.2.2.cmml">ℛ</mi><mi id="A3.SS2.p1.3.m3.1.1.2.3" xref="A3.SS2.p1.3.m3.1.1.2.3.cmml">σ</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="A3.SS2.p1.3.m3.1b"><apply id="A3.SS2.p1.3.m3.1.1.cmml" xref="A3.SS2.p1.3.m3.1.1"><ci id="A3.SS2.p1.3.m3.1.1.1.cmml" xref="A3.SS2.p1.3.m3.1.1.1">∇</ci><apply id="A3.SS2.p1.3.m3.1.1.2.cmml" xref="A3.SS2.p1.3.m3.1.1.2"><csymbol cd="ambiguous" id="A3.SS2.p1.3.m3.1.1.2.1.cmml" xref="A3.SS2.p1.3.m3.1.1.2">subscript</csymbol><ci id="A3.SS2.p1.3.m3.1.1.2.2.cmml" xref="A3.SS2.p1.3.m3.1.1.2.2">ℛ</ci><ci id="A3.SS2.p1.3.m3.1.1.2.3.cmml" xref="A3.SS2.p1.3.m3.1.1.2.3">𝜎</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.SS2.p1.3.m3.1c">\nabla\mathcal{R}_{\sigma}</annotation><annotation encoding="application/x-llamapun" id="A3.SS2.p1.3.m3.1d">∇ caligraphic_R start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT</annotation></semantics></math> to <math alttext="-\nabla\log p" class="ltx_Math" display="inline" id="A3.SS2.p1.4.m4.1"><semantics id="A3.SS2.p1.4.m4.1a"><mrow id="A3.SS2.p1.4.m4.1.1" xref="A3.SS2.p1.4.m4.1.1.cmml"><mo id="A3.SS2.p1.4.m4.1.1a" rspace="0.167em" xref="A3.SS2.p1.4.m4.1.1.cmml">−</mo><mrow id="A3.SS2.p1.4.m4.1.1.2" xref="A3.SS2.p1.4.m4.1.1.2.cmml"><mrow id="A3.SS2.p1.4.m4.1.1.2.1" xref="A3.SS2.p1.4.m4.1.1.2.1.cmml"><mo id="A3.SS2.p1.4.m4.1.1.2.1.1" rspace="0.167em" xref="A3.SS2.p1.4.m4.1.1.2.1.1.cmml">∇</mo><mi id="A3.SS2.p1.4.m4.1.1.2.1.2" xref="A3.SS2.p1.4.m4.1.1.2.1.2.cmml">log</mi></mrow><mo id="A3.SS2.p1.4.m4.1.1.2a" lspace="0.167em" xref="A3.SS2.p1.4.m4.1.1.2.cmml">⁡</mo><mi id="A3.SS2.p1.4.m4.1.1.2.2" xref="A3.SS2.p1.4.m4.1.1.2.2.cmml">p</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="A3.SS2.p1.4.m4.1b"><apply id="A3.SS2.p1.4.m4.1.1.cmml" xref="A3.SS2.p1.4.m4.1.1"><minus id="A3.SS2.p1.4.m4.1.1.1.cmml" xref="A3.SS2.p1.4.m4.1.1"></minus><apply id="A3.SS2.p1.4.m4.1.1.2.cmml" xref="A3.SS2.p1.4.m4.1.1.2"><apply id="A3.SS2.p1.4.m4.1.1.2.1.cmml" xref="A3.SS2.p1.4.m4.1.1.2.1"><ci id="A3.SS2.p1.4.m4.1.1.2.1.1.cmml" xref="A3.SS2.p1.4.m4.1.1.2.1.1">∇</ci><log id="A3.SS2.p1.4.m4.1.1.2.1.2.cmml" xref="A3.SS2.p1.4.m4.1.1.2.1.2"></log></apply><ci id="A3.SS2.p1.4.m4.1.1.2.2.cmml" xref="A3.SS2.p1.4.m4.1.1.2.2">𝑝</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.SS2.p1.4.m4.1c">-\nabla\log p</annotation><annotation encoding="application/x-llamapun" id="A3.SS2.p1.4.m4.1d">- ∇ roman_log italic_p</annotation></semantics></math> when <math alttext="\sigma\to 0" class="ltx_Math" display="inline" id="A3.SS2.p1.5.m5.1"><semantics id="A3.SS2.p1.5.m5.1a"><mrow id="A3.SS2.p1.5.m5.1.1" xref="A3.SS2.p1.5.m5.1.1.cmml"><mi id="A3.SS2.p1.5.m5.1.1.2" xref="A3.SS2.p1.5.m5.1.1.2.cmml">σ</mi><mo id="A3.SS2.p1.5.m5.1.1.1" stretchy="false" xref="A3.SS2.p1.5.m5.1.1.1.cmml">→</mo><mn id="A3.SS2.p1.5.m5.1.1.3" xref="A3.SS2.p1.5.m5.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="A3.SS2.p1.5.m5.1b"><apply id="A3.SS2.p1.5.m5.1.1.cmml" xref="A3.SS2.p1.5.m5.1.1"><ci id="A3.SS2.p1.5.m5.1.1.1.cmml" xref="A3.SS2.p1.5.m5.1.1.1">→</ci><ci id="A3.SS2.p1.5.m5.1.1.2.cmml" xref="A3.SS2.p1.5.m5.1.1.2">𝜎</ci><cn id="A3.SS2.p1.5.m5.1.1.3.cmml" type="integer" xref="A3.SS2.p1.5.m5.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.SS2.p1.5.m5.1c">\sigma\to 0</annotation><annotation encoding="application/x-llamapun" id="A3.SS2.p1.5.m5.1d">italic_σ → 0</annotation></semantics></math>. Then, simulations are run in 1D to give more intuition.</p> </div> <div class="ltx_para" id="A3.SS2.p2"> <p class="ltx_p" id="A3.SS2.p2.1">Let us suppose that the prior <math alttext="p" class="ltx_Math" display="inline" id="A3.SS2.p2.1.m1.1"><semantics id="A3.SS2.p2.1.m1.1a"><mi id="A3.SS2.p2.1.m1.1.1" xref="A3.SS2.p2.1.m1.1.1.cmml">p</mi><annotation-xml encoding="MathML-Content" id="A3.SS2.p2.1.m1.1b"><ci id="A3.SS2.p2.1.m1.1.1.cmml" xref="A3.SS2.p2.1.m1.1.1">𝑝</ci></annotation-xml><annotation encoding="application/x-tex" id="A3.SS2.p2.1.m1.1c">p</annotation><annotation encoding="application/x-llamapun" id="A3.SS2.p2.1.m1.1d">italic_p</annotation></semantics></math> is a Gaussian Mixture Model,</p> <table class="ltx_equation ltx_eqn_table" id="A3.Ex28"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="p({\mathbf{x}})=\sum_{i=1}^{p}{\pi_{i}\mathcal{N}({\mathbf{x}};{\mathbf{\mu}}_% {i},{\mathbf{\Sigma}}_{i})}," class="ltx_Math" display="block" id="A3.Ex28.m1.3"><semantics id="A3.Ex28.m1.3a"><mrow id="A3.Ex28.m1.3.3.1" xref="A3.Ex28.m1.3.3.1.1.cmml"><mrow id="A3.Ex28.m1.3.3.1.1" xref="A3.Ex28.m1.3.3.1.1.cmml"><mrow id="A3.Ex28.m1.3.3.1.1.4" xref="A3.Ex28.m1.3.3.1.1.4.cmml"><mi id="A3.Ex28.m1.3.3.1.1.4.2" xref="A3.Ex28.m1.3.3.1.1.4.2.cmml">p</mi><mo id="A3.Ex28.m1.3.3.1.1.4.1" xref="A3.Ex28.m1.3.3.1.1.4.1.cmml">⁢</mo><mrow id="A3.Ex28.m1.3.3.1.1.4.3.2" xref="A3.Ex28.m1.3.3.1.1.4.cmml"><mo id="A3.Ex28.m1.3.3.1.1.4.3.2.1" stretchy="false" xref="A3.Ex28.m1.3.3.1.1.4.cmml">(</mo><mi id="A3.Ex28.m1.1.1" xref="A3.Ex28.m1.1.1.cmml">𝐱</mi><mo id="A3.Ex28.m1.3.3.1.1.4.3.2.2" stretchy="false" xref="A3.Ex28.m1.3.3.1.1.4.cmml">)</mo></mrow></mrow><mo id="A3.Ex28.m1.3.3.1.1.3" rspace="0.111em" xref="A3.Ex28.m1.3.3.1.1.3.cmml">=</mo><mrow id="A3.Ex28.m1.3.3.1.1.2" xref="A3.Ex28.m1.3.3.1.1.2.cmml"><munderover id="A3.Ex28.m1.3.3.1.1.2.3" xref="A3.Ex28.m1.3.3.1.1.2.3.cmml"><mo id="A3.Ex28.m1.3.3.1.1.2.3.2.2" movablelimits="false" xref="A3.Ex28.m1.3.3.1.1.2.3.2.2.cmml">∑</mo><mrow id="A3.Ex28.m1.3.3.1.1.2.3.2.3" xref="A3.Ex28.m1.3.3.1.1.2.3.2.3.cmml"><mi id="A3.Ex28.m1.3.3.1.1.2.3.2.3.2" xref="A3.Ex28.m1.3.3.1.1.2.3.2.3.2.cmml">i</mi><mo id="A3.Ex28.m1.3.3.1.1.2.3.2.3.1" xref="A3.Ex28.m1.3.3.1.1.2.3.2.3.1.cmml">=</mo><mn id="A3.Ex28.m1.3.3.1.1.2.3.2.3.3" xref="A3.Ex28.m1.3.3.1.1.2.3.2.3.3.cmml">1</mn></mrow><mi id="A3.Ex28.m1.3.3.1.1.2.3.3" xref="A3.Ex28.m1.3.3.1.1.2.3.3.cmml">p</mi></munderover><mrow id="A3.Ex28.m1.3.3.1.1.2.2" xref="A3.Ex28.m1.3.3.1.1.2.2.cmml"><msub id="A3.Ex28.m1.3.3.1.1.2.2.4" xref="A3.Ex28.m1.3.3.1.1.2.2.4.cmml"><mi id="A3.Ex28.m1.3.3.1.1.2.2.4.2" xref="A3.Ex28.m1.3.3.1.1.2.2.4.2.cmml">π</mi><mi id="A3.Ex28.m1.3.3.1.1.2.2.4.3" xref="A3.Ex28.m1.3.3.1.1.2.2.4.3.cmml">i</mi></msub><mo id="A3.Ex28.m1.3.3.1.1.2.2.3" xref="A3.Ex28.m1.3.3.1.1.2.2.3.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="A3.Ex28.m1.3.3.1.1.2.2.5" xref="A3.Ex28.m1.3.3.1.1.2.2.5.cmml">𝒩</mi><mo id="A3.Ex28.m1.3.3.1.1.2.2.3a" xref="A3.Ex28.m1.3.3.1.1.2.2.3.cmml">⁢</mo><mrow id="A3.Ex28.m1.3.3.1.1.2.2.2.2" xref="A3.Ex28.m1.3.3.1.1.2.2.2.3.cmml"><mo id="A3.Ex28.m1.3.3.1.1.2.2.2.2.3" stretchy="false" xref="A3.Ex28.m1.3.3.1.1.2.2.2.3.cmml">(</mo><mi id="A3.Ex28.m1.2.2" xref="A3.Ex28.m1.2.2.cmml">𝐱</mi><mo id="A3.Ex28.m1.3.3.1.1.2.2.2.2.4" xref="A3.Ex28.m1.3.3.1.1.2.2.2.3.cmml">;</mo><msub id="A3.Ex28.m1.3.3.1.1.1.1.1.1.1" xref="A3.Ex28.m1.3.3.1.1.1.1.1.1.1.cmml"><mi id="A3.Ex28.m1.3.3.1.1.1.1.1.1.1.2" xref="A3.Ex28.m1.3.3.1.1.1.1.1.1.1.2.cmml">μ</mi><mi id="A3.Ex28.m1.3.3.1.1.1.1.1.1.1.3" xref="A3.Ex28.m1.3.3.1.1.1.1.1.1.1.3.cmml">i</mi></msub><mo id="A3.Ex28.m1.3.3.1.1.2.2.2.2.5" xref="A3.Ex28.m1.3.3.1.1.2.2.2.3.cmml">,</mo><msub id="A3.Ex28.m1.3.3.1.1.2.2.2.2.2" xref="A3.Ex28.m1.3.3.1.1.2.2.2.2.2.cmml"><mi 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xref="A3.Ex28.m1.3.3.1.1.2.2.2.2.2"><csymbol cd="ambiguous" id="A3.Ex28.m1.3.3.1.1.2.2.2.2.2.1.cmml" xref="A3.Ex28.m1.3.3.1.1.2.2.2.2.2">subscript</csymbol><ci id="A3.Ex28.m1.3.3.1.1.2.2.2.2.2.2.cmml" xref="A3.Ex28.m1.3.3.1.1.2.2.2.2.2.2">𝚺</ci><ci id="A3.Ex28.m1.3.3.1.1.2.2.2.2.2.3.cmml" xref="A3.Ex28.m1.3.3.1.1.2.2.2.2.2.3">𝑖</ci></apply></list></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.Ex28.m1.3c">p({\mathbf{x}})=\sum_{i=1}^{p}{\pi_{i}\mathcal{N}({\mathbf{x}};{\mathbf{\mu}}_% {i},{\mathbf{\Sigma}}_{i})},</annotation><annotation encoding="application/x-llamapun" id="A3.Ex28.m1.3d">italic_p ( bold_x ) = ∑ start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT italic_π start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT caligraphic_N ( bold_x ; italic_μ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , bold_Σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="A3.SS2.p2.3">with <math alttext="\pi_{i}\geq 0" class="ltx_Math" display="inline" id="A3.SS2.p2.2.m1.1"><semantics id="A3.SS2.p2.2.m1.1a"><mrow id="A3.SS2.p2.2.m1.1.1" xref="A3.SS2.p2.2.m1.1.1.cmml"><msub id="A3.SS2.p2.2.m1.1.1.2" xref="A3.SS2.p2.2.m1.1.1.2.cmml"><mi id="A3.SS2.p2.2.m1.1.1.2.2" xref="A3.SS2.p2.2.m1.1.1.2.2.cmml">π</mi><mi id="A3.SS2.p2.2.m1.1.1.2.3" xref="A3.SS2.p2.2.m1.1.1.2.3.cmml">i</mi></msub><mo id="A3.SS2.p2.2.m1.1.1.1" xref="A3.SS2.p2.2.m1.1.1.1.cmml">≥</mo><mn id="A3.SS2.p2.2.m1.1.1.3" xref="A3.SS2.p2.2.m1.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="A3.SS2.p2.2.m1.1b"><apply id="A3.SS2.p2.2.m1.1.1.cmml" xref="A3.SS2.p2.2.m1.1.1"><geq id="A3.SS2.p2.2.m1.1.1.1.cmml" xref="A3.SS2.p2.2.m1.1.1.1"></geq><apply id="A3.SS2.p2.2.m1.1.1.2.cmml" xref="A3.SS2.p2.2.m1.1.1.2"><csymbol cd="ambiguous" id="A3.SS2.p2.2.m1.1.1.2.1.cmml" xref="A3.SS2.p2.2.m1.1.1.2">subscript</csymbol><ci id="A3.SS2.p2.2.m1.1.1.2.2.cmml" xref="A3.SS2.p2.2.m1.1.1.2.2">𝜋</ci><ci id="A3.SS2.p2.2.m1.1.1.2.3.cmml" xref="A3.SS2.p2.2.m1.1.1.2.3">𝑖</ci></apply><cn id="A3.SS2.p2.2.m1.1.1.3.cmml" type="integer" xref="A3.SS2.p2.2.m1.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.SS2.p2.2.m1.1c">\pi_{i}\geq 0</annotation><annotation encoding="application/x-llamapun" id="A3.SS2.p2.2.m1.1d">italic_π start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ≥ 0</annotation></semantics></math> and <math alttext="\sum_{i=1}^{p}\pi_{i}=1" class="ltx_Math" display="inline" id="A3.SS2.p2.3.m2.1"><semantics id="A3.SS2.p2.3.m2.1a"><mrow id="A3.SS2.p2.3.m2.1.1" xref="A3.SS2.p2.3.m2.1.1.cmml"><mrow id="A3.SS2.p2.3.m2.1.1.2" xref="A3.SS2.p2.3.m2.1.1.2.cmml"><msubsup id="A3.SS2.p2.3.m2.1.1.2.1" xref="A3.SS2.p2.3.m2.1.1.2.1.cmml"><mo id="A3.SS2.p2.3.m2.1.1.2.1.2.2" xref="A3.SS2.p2.3.m2.1.1.2.1.2.2.cmml">∑</mo><mrow 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xref="A3.SS2.p2.3.m2.1.1.1"></eq><apply id="A3.SS2.p2.3.m2.1.1.2.cmml" xref="A3.SS2.p2.3.m2.1.1.2"><apply id="A3.SS2.p2.3.m2.1.1.2.1.cmml" xref="A3.SS2.p2.3.m2.1.1.2.1"><csymbol cd="ambiguous" id="A3.SS2.p2.3.m2.1.1.2.1.1.cmml" xref="A3.SS2.p2.3.m2.1.1.2.1">superscript</csymbol><apply id="A3.SS2.p2.3.m2.1.1.2.1.2.cmml" xref="A3.SS2.p2.3.m2.1.1.2.1"><csymbol cd="ambiguous" id="A3.SS2.p2.3.m2.1.1.2.1.2.1.cmml" xref="A3.SS2.p2.3.m2.1.1.2.1">subscript</csymbol><sum id="A3.SS2.p2.3.m2.1.1.2.1.2.2.cmml" xref="A3.SS2.p2.3.m2.1.1.2.1.2.2"></sum><apply id="A3.SS2.p2.3.m2.1.1.2.1.2.3.cmml" xref="A3.SS2.p2.3.m2.1.1.2.1.2.3"><eq id="A3.SS2.p2.3.m2.1.1.2.1.2.3.1.cmml" xref="A3.SS2.p2.3.m2.1.1.2.1.2.3.1"></eq><ci id="A3.SS2.p2.3.m2.1.1.2.1.2.3.2.cmml" xref="A3.SS2.p2.3.m2.1.1.2.1.2.3.2">𝑖</ci><cn id="A3.SS2.p2.3.m2.1.1.2.1.2.3.3.cmml" type="integer" xref="A3.SS2.p2.3.m2.1.1.2.1.2.3.3">1</cn></apply></apply><ci id="A3.SS2.p2.3.m2.1.1.2.1.3.cmml" xref="A3.SS2.p2.3.m2.1.1.2.1.3">𝑝</ci></apply><apply id="A3.SS2.p2.3.m2.1.1.2.2.cmml" xref="A3.SS2.p2.3.m2.1.1.2.2"><csymbol cd="ambiguous" id="A3.SS2.p2.3.m2.1.1.2.2.1.cmml" xref="A3.SS2.p2.3.m2.1.1.2.2">subscript</csymbol><ci id="A3.SS2.p2.3.m2.1.1.2.2.2.cmml" xref="A3.SS2.p2.3.m2.1.1.2.2.2">𝜋</ci><ci id="A3.SS2.p2.3.m2.1.1.2.2.3.cmml" xref="A3.SS2.p2.3.m2.1.1.2.2.3">𝑖</ci></apply></apply><cn id="A3.SS2.p2.3.m2.1.1.3.cmml" type="integer" xref="A3.SS2.p2.3.m2.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.SS2.p2.3.m2.1c">\sum_{i=1}^{p}\pi_{i}=1</annotation><annotation encoding="application/x-llamapun" id="A3.SS2.p2.3.m2.1d">∑ start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT italic_π start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = 1</annotation></semantics></math>.</p> </div> <section class="ltx_subsubsection" id="A3.SS2.SSS1"> <h4 class="ltx_title ltx_title_subsubsection"> <span class="ltx_tag ltx_tag_subsubsection">C.2.1 </span>Theoretical considerations</h4> <div class="ltx_para" id="A3.SS2.SSS1.p1"> <p class="ltx_p" id="A3.SS2.SSS1.p1.1">The scores of the prior distribution and the noisy prior distributions have the closed forms </p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A8.EGx20"> <tbody id="A3.Ex29"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\nabla\log p({\mathbf{x}})" class="ltx_Math" display="inline" id="A3.Ex29.m1.1"><semantics id="A3.Ex29.m1.1a"><mrow id="A3.Ex29.m1.1.2" xref="A3.Ex29.m1.1.2.cmml"><mrow id="A3.Ex29.m1.1.2.2" xref="A3.Ex29.m1.1.2.2.cmml"><mrow id="A3.Ex29.m1.1.2.2.1" xref="A3.Ex29.m1.1.2.2.1.cmml"><mo id="A3.Ex29.m1.1.2.2.1.1" rspace="0.167em" xref="A3.Ex29.m1.1.2.2.1.1.cmml">∇</mo><mi id="A3.Ex29.m1.1.2.2.1.2" xref="A3.Ex29.m1.1.2.2.1.2.cmml">log</mi></mrow><mo id="A3.Ex29.m1.1.2.2a" lspace="0.167em" xref="A3.Ex29.m1.1.2.2.cmml">⁡</mo><mi id="A3.Ex29.m1.1.2.2.2" xref="A3.Ex29.m1.1.2.2.2.cmml">p</mi></mrow><mo id="A3.Ex29.m1.1.2.1" xref="A3.Ex29.m1.1.2.1.cmml">⁢</mo><mrow id="A3.Ex29.m1.1.2.3.2" xref="A3.Ex29.m1.1.2.cmml"><mo id="A3.Ex29.m1.1.2.3.2.1" stretchy="false" xref="A3.Ex29.m1.1.2.cmml">(</mo><mi id="A3.Ex29.m1.1.1" xref="A3.Ex29.m1.1.1.cmml">𝐱</mi><mo id="A3.Ex29.m1.1.2.3.2.2" stretchy="false" xref="A3.Ex29.m1.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A3.Ex29.m1.1b"><apply id="A3.Ex29.m1.1.2.cmml" xref="A3.Ex29.m1.1.2"><times id="A3.Ex29.m1.1.2.1.cmml" xref="A3.Ex29.m1.1.2.1"></times><apply id="A3.Ex29.m1.1.2.2.cmml" xref="A3.Ex29.m1.1.2.2"><apply id="A3.Ex29.m1.1.2.2.1.cmml" xref="A3.Ex29.m1.1.2.2.1"><ci id="A3.Ex29.m1.1.2.2.1.1.cmml" xref="A3.Ex29.m1.1.2.2.1.1">∇</ci><log id="A3.Ex29.m1.1.2.2.1.2.cmml" xref="A3.Ex29.m1.1.2.2.1.2"></log></apply><ci id="A3.Ex29.m1.1.2.2.2.cmml" xref="A3.Ex29.m1.1.2.2.2">𝑝</ci></apply><ci id="A3.Ex29.m1.1.1.cmml" xref="A3.Ex29.m1.1.1">𝐱</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.Ex29.m1.1c">\displaystyle\nabla\log p({\mathbf{x}})</annotation><annotation encoding="application/x-llamapun" id="A3.Ex29.m1.1d">∇ roman_log italic_p ( bold_x )</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle=-\frac{\sum_{i=1}^{p}{\pi_{i}{\mathbf{\Sigma}}_{i}^{-1}({\mathbf% {x}}-{\mathbf{\mu}}_{i})\mathcal{N}({\mathbf{x}};{\mathbf{\mu}}_{i},{\mathbf{% \Sigma}}_{i})}}{\sum_{i=1}^{p}{\pi_{i}\mathcal{N}({\mathbf{x}};{\mathbf{\mu}}_% {i},{\mathbf{\Sigma}}_{i})}}" class="ltx_Math" display="inline" id="A3.Ex29.m2.7"><semantics id="A3.Ex29.m2.7a"><mrow id="A3.Ex29.m2.7.8" xref="A3.Ex29.m2.7.8.cmml"><mi id="A3.Ex29.m2.7.8.2" xref="A3.Ex29.m2.7.8.2.cmml"></mi><mo id="A3.Ex29.m2.7.8.1" xref="A3.Ex29.m2.7.8.1.cmml">=</mo><mrow id="A3.Ex29.m2.7.8.3" xref="A3.Ex29.m2.7.8.3.cmml"><mo id="A3.Ex29.m2.7.8.3a" 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id="A3.Ex29.m2.7.7.7.3.2.2.2.cmml" xref="A3.Ex29.m2.7.7.7.3.2.2.2"><csymbol cd="ambiguous" id="A3.Ex29.m2.7.7.7.3.2.2.2.1.cmml" xref="A3.Ex29.m2.7.7.7.3.2.2.2">subscript</csymbol><ci id="A3.Ex29.m2.7.7.7.3.2.2.2.2.cmml" xref="A3.Ex29.m2.7.7.7.3.2.2.2.2">𝚺</ci><ci id="A3.Ex29.m2.7.7.7.3.2.2.2.3.cmml" xref="A3.Ex29.m2.7.7.7.3.2.2.2.3">𝑖</ci></apply></list></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.Ex29.m2.7c">\displaystyle=-\frac{\sum_{i=1}^{p}{\pi_{i}{\mathbf{\Sigma}}_{i}^{-1}({\mathbf% {x}}-{\mathbf{\mu}}_{i})\mathcal{N}({\mathbf{x}};{\mathbf{\mu}}_{i},{\mathbf{% \Sigma}}_{i})}}{\sum_{i=1}^{p}{\pi_{i}\mathcal{N}({\mathbf{x}};{\mathbf{\mu}}_% {i},{\mathbf{\Sigma}}_{i})}}</annotation><annotation encoding="application/x-llamapun" id="A3.Ex29.m2.7d">= - divide start_ARG ∑ start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT italic_π start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT bold_Σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ( bold_x - italic_μ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) caligraphic_N ( bold_x ; italic_μ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , bold_Σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) end_ARG start_ARG ∑ start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT italic_π start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT caligraphic_N ( bold_x ; italic_μ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , bold_Σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) end_ARG</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="A3.E17"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\nabla\log p_{\sigma}({\mathbf{x}})" class="ltx_Math" display="inline" id="A3.E17.m1.1"><semantics id="A3.E17.m1.1a"><mrow id="A3.E17.m1.1.2" xref="A3.E17.m1.1.2.cmml"><mrow id="A3.E17.m1.1.2.2" xref="A3.E17.m1.1.2.2.cmml"><mrow id="A3.E17.m1.1.2.2.1" xref="A3.E17.m1.1.2.2.1.cmml"><mo id="A3.E17.m1.1.2.2.1.1" rspace="0.167em" xref="A3.E17.m1.1.2.2.1.1.cmml">∇</mo><mi id="A3.E17.m1.1.2.2.1.2" xref="A3.E17.m1.1.2.2.1.2.cmml">log</mi></mrow><mo id="A3.E17.m1.1.2.2a" lspace="0.167em" xref="A3.E17.m1.1.2.2.cmml">⁡</mo><msub id="A3.E17.m1.1.2.2.2" xref="A3.E17.m1.1.2.2.2.cmml"><mi id="A3.E17.m1.1.2.2.2.2" xref="A3.E17.m1.1.2.2.2.2.cmml">p</mi><mi id="A3.E17.m1.1.2.2.2.3" xref="A3.E17.m1.1.2.2.2.3.cmml">σ</mi></msub></mrow><mo id="A3.E17.m1.1.2.1" xref="A3.E17.m1.1.2.1.cmml">⁢</mo><mrow id="A3.E17.m1.1.2.3.2" xref="A3.E17.m1.1.2.cmml"><mo id="A3.E17.m1.1.2.3.2.1" stretchy="false" xref="A3.E17.m1.1.2.cmml">(</mo><mi id="A3.E17.m1.1.1" xref="A3.E17.m1.1.1.cmml">𝐱</mi><mo id="A3.E17.m1.1.2.3.2.2" stretchy="false" xref="A3.E17.m1.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A3.E17.m1.1b"><apply id="A3.E17.m1.1.2.cmml" xref="A3.E17.m1.1.2"><times id="A3.E17.m1.1.2.1.cmml" xref="A3.E17.m1.1.2.1"></times><apply id="A3.E17.m1.1.2.2.cmml" xref="A3.E17.m1.1.2.2"><apply id="A3.E17.m1.1.2.2.1.cmml" xref="A3.E17.m1.1.2.2.1"><ci id="A3.E17.m1.1.2.2.1.1.cmml" xref="A3.E17.m1.1.2.2.1.1">∇</ci><log id="A3.E17.m1.1.2.2.1.2.cmml" xref="A3.E17.m1.1.2.2.1.2"></log></apply><apply id="A3.E17.m1.1.2.2.2.cmml" xref="A3.E17.m1.1.2.2.2"><csymbol cd="ambiguous" id="A3.E17.m1.1.2.2.2.1.cmml" xref="A3.E17.m1.1.2.2.2">subscript</csymbol><ci id="A3.E17.m1.1.2.2.2.2.cmml" xref="A3.E17.m1.1.2.2.2.2">𝑝</ci><ci id="A3.E17.m1.1.2.2.2.3.cmml" xref="A3.E17.m1.1.2.2.2.3">𝜎</ci></apply></apply><ci id="A3.E17.m1.1.1.cmml" xref="A3.E17.m1.1.1">𝐱</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.E17.m1.1c">\displaystyle\nabla\log p_{\sigma}({\mathbf{x}})</annotation><annotation encoding="application/x-llamapun" id="A3.E17.m1.1d">∇ roman_log italic_p start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ( bold_x )</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle=-\frac{\sum_{i=1}^{p}{\pi_{i}({\mathbf{\Sigma}}_{i}+\sigma^{2}% \mathbf{I}_{d})^{-1}({\mathbf{x}}-{\mathbf{\mu}}_{i})\mathcal{N}({\mathbf{x}};% {\mathbf{\mu}}_{i},{\mathbf{\Sigma}}_{i}+\sigma^{2}\mathbf{I}_{d})}}{\sum_{i=1% }^{p}{\pi_{i}\mathcal{N}({\mathbf{x}};{\mathbf{\mu}}_{i},{\mathbf{\Sigma}}_{i}% +\sigma^{2}\mathbf{I}_{d})}}." class="ltx_Math" display="inline" id="A3.E17.m2.9"><semantics id="A3.E17.m2.9a"><mrow id="A3.E17.m2.9.9.1" xref="A3.E17.m2.9.9.1.1.cmml"><mrow id="A3.E17.m2.9.9.1.1" xref="A3.E17.m2.9.9.1.1.cmml"><mi id="A3.E17.m2.9.9.1.1.2" xref="A3.E17.m2.9.9.1.1.2.cmml"></mi><mo id="A3.E17.m2.9.9.1.1.1" xref="A3.E17.m2.9.9.1.1.1.cmml">=</mo><mrow id="A3.E17.m2.9.9.1.1.3" xref="A3.E17.m2.9.9.1.1.3.cmml"><mo id="A3.E17.m2.9.9.1.1.3a" 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\mathbf{I}_{d})^{-1}({\mathbf{x}}-{\mathbf{\mu}}_{i})\mathcal{N}({\mathbf{x}};% {\mathbf{\mu}}_{i},{\mathbf{\Sigma}}_{i}+\sigma^{2}\mathbf{I}_{d})}}{\sum_{i=1% }^{p}{\pi_{i}\mathcal{N}({\mathbf{x}};{\mathbf{\mu}}_{i},{\mathbf{\Sigma}}_{i}% +\sigma^{2}\mathbf{I}_{d})}}.</annotation><annotation encoding="application/x-llamapun" id="A3.E17.m2.9d">= - divide start_ARG ∑ start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT italic_π start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( bold_Σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT + italic_σ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT bold_I start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ( bold_x - italic_μ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) caligraphic_N ( bold_x ; italic_μ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , bold_Σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT + italic_σ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT bold_I start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT ) end_ARG start_ARG ∑ start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT italic_π start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT caligraphic_N ( bold_x ; italic_μ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , bold_Σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT + italic_σ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT bold_I start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT ) end_ARG .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(17)</span></td> </tr></tbody> </table> </div> <div class="ltx_para" id="A3.SS2.SSS1.p2"> <p class="ltx_p" id="A3.SS2.SSS1.p2.1">First we study the behavior of this noisy prior distribution score when <math alttext="\sigma\to 0" class="ltx_Math" display="inline" id="A3.SS2.SSS1.p2.1.m1.1"><semantics id="A3.SS2.SSS1.p2.1.m1.1a"><mrow id="A3.SS2.SSS1.p2.1.m1.1.1" xref="A3.SS2.SSS1.p2.1.m1.1.1.cmml"><mi id="A3.SS2.SSS1.p2.1.m1.1.1.2" xref="A3.SS2.SSS1.p2.1.m1.1.1.2.cmml">σ</mi><mo id="A3.SS2.SSS1.p2.1.m1.1.1.1" stretchy="false" xref="A3.SS2.SSS1.p2.1.m1.1.1.1.cmml">→</mo><mn id="A3.SS2.SSS1.p2.1.m1.1.1.3" xref="A3.SS2.SSS1.p2.1.m1.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="A3.SS2.SSS1.p2.1.m1.1b"><apply id="A3.SS2.SSS1.p2.1.m1.1.1.cmml" xref="A3.SS2.SSS1.p2.1.m1.1.1"><ci id="A3.SS2.SSS1.p2.1.m1.1.1.1.cmml" xref="A3.SS2.SSS1.p2.1.m1.1.1.1">→</ci><ci id="A3.SS2.SSS1.p2.1.m1.1.1.2.cmml" xref="A3.SS2.SSS1.p2.1.m1.1.1.2">𝜎</ci><cn id="A3.SS2.SSS1.p2.1.m1.1.1.3.cmml" type="integer" xref="A3.SS2.SSS1.p2.1.m1.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.SS2.SSS1.p2.1.m1.1c">\sigma\to 0</annotation><annotation encoding="application/x-llamapun" id="A3.SS2.SSS1.p2.1.m1.1d">italic_σ → 0</annotation></semantics></math>.</p> </div> <div class="ltx_theorem ltx_theorem_proposition" id="A3.Thmproposition1"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="A3.Thmproposition1.1.1.1">Proposition C.1</span></span><span class="ltx_text ltx_font_bold" id="A3.Thmproposition1.2.2">.</span> </h6> <div class="ltx_para" id="A3.Thmproposition1.p1"> <p class="ltx_p" id="A3.Thmproposition1.p1.5"><span class="ltx_text ltx_font_italic" id="A3.Thmproposition1.p1.5.5">If the prior <math alttext="p" class="ltx_Math" display="inline" id="A3.Thmproposition1.p1.1.1.m1.1"><semantics id="A3.Thmproposition1.p1.1.1.m1.1a"><mi id="A3.Thmproposition1.p1.1.1.m1.1.1" xref="A3.Thmproposition1.p1.1.1.m1.1.1.cmml">p</mi><annotation-xml encoding="MathML-Content" id="A3.Thmproposition1.p1.1.1.m1.1b"><ci id="A3.Thmproposition1.p1.1.1.m1.1.1.cmml" xref="A3.Thmproposition1.p1.1.1.m1.1.1">𝑝</ci></annotation-xml><annotation encoding="application/x-tex" id="A3.Thmproposition1.p1.1.1.m1.1c">p</annotation><annotation encoding="application/x-llamapun" id="A3.Thmproposition1.p1.1.1.m1.1d">italic_p</annotation></semantics></math> is a Gaussian Mixture Model, then there exist <math alttext="a_{p}\in{\mathbb{R}}^{+}" class="ltx_Math" display="inline" id="A3.Thmproposition1.p1.2.2.m2.1"><semantics id="A3.Thmproposition1.p1.2.2.m2.1a"><mrow id="A3.Thmproposition1.p1.2.2.m2.1.1" xref="A3.Thmproposition1.p1.2.2.m2.1.1.cmml"><msub id="A3.Thmproposition1.p1.2.2.m2.1.1.2" xref="A3.Thmproposition1.p1.2.2.m2.1.1.2.cmml"><mi id="A3.Thmproposition1.p1.2.2.m2.1.1.2.2" xref="A3.Thmproposition1.p1.2.2.m2.1.1.2.2.cmml">a</mi><mi id="A3.Thmproposition1.p1.2.2.m2.1.1.2.3" xref="A3.Thmproposition1.p1.2.2.m2.1.1.2.3.cmml">p</mi></msub><mo id="A3.Thmproposition1.p1.2.2.m2.1.1.1" mathvariant="normal" xref="A3.Thmproposition1.p1.2.2.m2.1.1.1.cmml">∈</mo><msup id="A3.Thmproposition1.p1.2.2.m2.1.1.3" xref="A3.Thmproposition1.p1.2.2.m2.1.1.3.cmml"><mi id="A3.Thmproposition1.p1.2.2.m2.1.1.3.2" xref="A3.Thmproposition1.p1.2.2.m2.1.1.3.2.cmml">ℝ</mi><mo id="A3.Thmproposition1.p1.2.2.m2.1.1.3.3" mathvariant="normal" xref="A3.Thmproposition1.p1.2.2.m2.1.1.3.3.cmml">+</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="A3.Thmproposition1.p1.2.2.m2.1b"><apply id="A3.Thmproposition1.p1.2.2.m2.1.1.cmml" xref="A3.Thmproposition1.p1.2.2.m2.1.1"><in id="A3.Thmproposition1.p1.2.2.m2.1.1.1.cmml" xref="A3.Thmproposition1.p1.2.2.m2.1.1.1"></in><apply id="A3.Thmproposition1.p1.2.2.m2.1.1.2.cmml" xref="A3.Thmproposition1.p1.2.2.m2.1.1.2"><csymbol cd="ambiguous" id="A3.Thmproposition1.p1.2.2.m2.1.1.2.1.cmml" xref="A3.Thmproposition1.p1.2.2.m2.1.1.2">subscript</csymbol><ci id="A3.Thmproposition1.p1.2.2.m2.1.1.2.2.cmml" xref="A3.Thmproposition1.p1.2.2.m2.1.1.2.2">𝑎</ci><ci id="A3.Thmproposition1.p1.2.2.m2.1.1.2.3.cmml" xref="A3.Thmproposition1.p1.2.2.m2.1.1.2.3">𝑝</ci></apply><apply 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xref="A3.Thmproposition1.p1.3.3.m3.1.1.cmml"><msub id="A3.Thmproposition1.p1.3.3.m3.1.1.2" xref="A3.Thmproposition1.p1.3.3.m3.1.1.2.cmml"><mi id="A3.Thmproposition1.p1.3.3.m3.1.1.2.2" xref="A3.Thmproposition1.p1.3.3.m3.1.1.2.2.cmml">α</mi><mn id="A3.Thmproposition1.p1.3.3.m3.1.1.2.3" mathvariant="normal" xref="A3.Thmproposition1.p1.3.3.m3.1.1.2.3.cmml">0</mn></msub><mo id="A3.Thmproposition1.p1.3.3.m3.1.1.1" mathvariant="normal" xref="A3.Thmproposition1.p1.3.3.m3.1.1.1.cmml">></mo><mn id="A3.Thmproposition1.p1.3.3.m3.1.1.3" mathvariant="normal" xref="A3.Thmproposition1.p1.3.3.m3.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="A3.Thmproposition1.p1.3.3.m3.1b"><apply id="A3.Thmproposition1.p1.3.3.m3.1.1.cmml" xref="A3.Thmproposition1.p1.3.3.m3.1.1"><gt id="A3.Thmproposition1.p1.3.3.m3.1.1.1.cmml" xref="A3.Thmproposition1.p1.3.3.m3.1.1.1"></gt><apply id="A3.Thmproposition1.p1.3.3.m3.1.1.2.cmml" xref="A3.Thmproposition1.p1.3.3.m3.1.1.2"><csymbol cd="ambiguous" 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id="A3.Thmproposition1.p1.4.4.m4.1.1.1.cmml" xref="A3.Thmproposition1.p1.4.4.m4.1.1.1"></in><apply id="A3.Thmproposition1.p1.4.4.m4.1.1.2.cmml" xref="A3.Thmproposition1.p1.4.4.m4.1.1.2"><csymbol cd="latexml" id="A3.Thmproposition1.p1.4.4.m4.1.1.2.1.cmml" xref="A3.Thmproposition1.p1.4.4.m4.1.1.2.1">for-all</csymbol><ci id="A3.Thmproposition1.p1.4.4.m4.1.1.2.2.cmml" xref="A3.Thmproposition1.p1.4.4.m4.1.1.2.2">𝐱</ci></apply><apply id="A3.Thmproposition1.p1.4.4.m4.1.1.3.cmml" xref="A3.Thmproposition1.p1.4.4.m4.1.1.3"><csymbol cd="ambiguous" id="A3.Thmproposition1.p1.4.4.m4.1.1.3.1.cmml" xref="A3.Thmproposition1.p1.4.4.m4.1.1.3">superscript</csymbol><ci id="A3.Thmproposition1.p1.4.4.m4.1.1.3.2.cmml" xref="A3.Thmproposition1.p1.4.4.m4.1.1.3.2">ℝ</ci><ci id="A3.Thmproposition1.p1.4.4.m4.1.1.3.3.cmml" xref="A3.Thmproposition1.p1.4.4.m4.1.1.3.3">𝑑</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.Thmproposition1.p1.4.4.m4.1c">\forall{\mathbf{x}}\in{\mathbb{R}}^{d}</annotation><annotation encoding="application/x-llamapun" id="A3.Thmproposition1.p1.4.4.m4.1d">∀ bold_x ∈ blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT</annotation></semantics></math>, <math alttext="\forall\sigma\in]0,\sigma_{0}]" class="ltx_math_unparsed" display="inline" id="A3.Thmproposition1.p1.5.5.m5.1"><semantics id="A3.Thmproposition1.p1.5.5.m5.1a"><mrow id="A3.Thmproposition1.p1.5.5.m5.1b"><mrow id="A3.Thmproposition1.p1.5.5.m5.1.1"><mo id="A3.Thmproposition1.p1.5.5.m5.1.1.1" mathvariant="normal" rspace="0.167em">∀</mo><mi id="A3.Thmproposition1.p1.5.5.m5.1.1.2">σ</mi><mo id="A3.Thmproposition1.p1.5.5.m5.1.1.3" mathvariant="normal" rspace="0em">∈</mo><mo id="A3.Thmproposition1.p1.5.5.m5.1.1.4" mathvariant="normal" stretchy="false">]</mo></mrow><mn id="A3.Thmproposition1.p1.5.5.m5.1.2" mathvariant="normal">0</mn><mo id="A3.Thmproposition1.p1.5.5.m5.1.3" mathvariant="normal">,</mo><msub id="A3.Thmproposition1.p1.5.5.m5.1.4"><mi id="A3.Thmproposition1.p1.5.5.m5.1.4.2">σ</mi><mn id="A3.Thmproposition1.p1.5.5.m5.1.4.3" mathvariant="normal">0</mn></msub><mo id="A3.Thmproposition1.p1.5.5.m5.1.5" mathvariant="normal" stretchy="false">]</mo></mrow><annotation encoding="application/x-tex" id="A3.Thmproposition1.p1.5.5.m5.1c">\forall\sigma\in]0,\sigma_{0}]</annotation><annotation encoding="application/x-llamapun" id="A3.Thmproposition1.p1.5.5.m5.1d">∀ italic_σ ∈ ] 0 , italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ]</annotation></semantics></math>:</span></p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A8.EGx21"> <tbody id="A3.E18"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\|\nabla\log p_{\sigma}({\mathbf{x}})-\nabla\log p({\mathbf{x}})% \|\leq\sigma^{2}a_{p}(\|{\mathbf{x}}\|^{3}+1)." class="ltx_Math" 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\|\leq\sigma^{2}a_{p}(\|{\mathbf{x}}\|^{3}+1).</annotation><annotation encoding="application/x-llamapun" id="A3.E18.m1.4d">∥ ∇ roman_log italic_p start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ( bold_x ) - ∇ roman_log italic_p ( bold_x ) ∥ ≤ italic_σ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_a start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT ( ∥ bold_x ∥ start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT + 1 ) .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(18)</span></td> </tr></tbody> </table> </div> </div> <div class="ltx_proof" id="A3.SS2.SSS1.6"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof.</h6> <div class="ltx_para" id="A3.SS2.SSS1.1.p1"> <p class="ltx_p" id="A3.SS2.SSS1.1.p1.3">We make a series expansion of <math alttext="\nabla\log p_{\sigma}({\mathbf{x}})" class="ltx_Math" display="inline" id="A3.SS2.SSS1.1.p1.1.m1.1"><semantics id="A3.SS2.SSS1.1.p1.1.m1.1a"><mrow id="A3.SS2.SSS1.1.p1.1.m1.1.2" xref="A3.SS2.SSS1.1.p1.1.m1.1.2.cmml"><mrow id="A3.SS2.SSS1.1.p1.1.m1.1.2.2" xref="A3.SS2.SSS1.1.p1.1.m1.1.2.2.cmml"><mrow id="A3.SS2.SSS1.1.p1.1.m1.1.2.2.1" xref="A3.SS2.SSS1.1.p1.1.m1.1.2.2.1.cmml"><mo id="A3.SS2.SSS1.1.p1.1.m1.1.2.2.1.1" rspace="0.167em" xref="A3.SS2.SSS1.1.p1.1.m1.1.2.2.1.1.cmml">∇</mo><mi id="A3.SS2.SSS1.1.p1.1.m1.1.2.2.1.2" xref="A3.SS2.SSS1.1.p1.1.m1.1.2.2.1.2.cmml">log</mi></mrow><mo id="A3.SS2.SSS1.1.p1.1.m1.1.2.2a" lspace="0.167em" xref="A3.SS2.SSS1.1.p1.1.m1.1.2.2.cmml">⁡</mo><msub id="A3.SS2.SSS1.1.p1.1.m1.1.2.2.2" xref="A3.SS2.SSS1.1.p1.1.m1.1.2.2.2.cmml"><mi id="A3.SS2.SSS1.1.p1.1.m1.1.2.2.2.2" xref="A3.SS2.SSS1.1.p1.1.m1.1.2.2.2.2.cmml">p</mi><mi id="A3.SS2.SSS1.1.p1.1.m1.1.2.2.2.3" xref="A3.SS2.SSS1.1.p1.1.m1.1.2.2.2.3.cmml">σ</mi></msub></mrow><mo id="A3.SS2.SSS1.1.p1.1.m1.1.2.1" xref="A3.SS2.SSS1.1.p1.1.m1.1.2.1.cmml">⁢</mo><mrow id="A3.SS2.SSS1.1.p1.1.m1.1.2.3.2" xref="A3.SS2.SSS1.1.p1.1.m1.1.2.cmml"><mo id="A3.SS2.SSS1.1.p1.1.m1.1.2.3.2.1" stretchy="false" xref="A3.SS2.SSS1.1.p1.1.m1.1.2.cmml">(</mo><mi id="A3.SS2.SSS1.1.p1.1.m1.1.1" xref="A3.SS2.SSS1.1.p1.1.m1.1.1.cmml">𝐱</mi><mo id="A3.SS2.SSS1.1.p1.1.m1.1.2.3.2.2" stretchy="false" xref="A3.SS2.SSS1.1.p1.1.m1.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A3.SS2.SSS1.1.p1.1.m1.1b"><apply id="A3.SS2.SSS1.1.p1.1.m1.1.2.cmml" xref="A3.SS2.SSS1.1.p1.1.m1.1.2"><times id="A3.SS2.SSS1.1.p1.1.m1.1.2.1.cmml" xref="A3.SS2.SSS1.1.p1.1.m1.1.2.1"></times><apply id="A3.SS2.SSS1.1.p1.1.m1.1.2.2.cmml" xref="A3.SS2.SSS1.1.p1.1.m1.1.2.2"><apply id="A3.SS2.SSS1.1.p1.1.m1.1.2.2.1.cmml" xref="A3.SS2.SSS1.1.p1.1.m1.1.2.2.1"><ci id="A3.SS2.SSS1.1.p1.1.m1.1.2.2.1.1.cmml" xref="A3.SS2.SSS1.1.p1.1.m1.1.2.2.1.1">∇</ci><log id="A3.SS2.SSS1.1.p1.1.m1.1.2.2.1.2.cmml" xref="A3.SS2.SSS1.1.p1.1.m1.1.2.2.1.2"></log></apply><apply id="A3.SS2.SSS1.1.p1.1.m1.1.2.2.2.cmml" xref="A3.SS2.SSS1.1.p1.1.m1.1.2.2.2"><csymbol cd="ambiguous" id="A3.SS2.SSS1.1.p1.1.m1.1.2.2.2.1.cmml" xref="A3.SS2.SSS1.1.p1.1.m1.1.2.2.2">subscript</csymbol><ci id="A3.SS2.SSS1.1.p1.1.m1.1.2.2.2.2.cmml" xref="A3.SS2.SSS1.1.p1.1.m1.1.2.2.2.2">𝑝</ci><ci id="A3.SS2.SSS1.1.p1.1.m1.1.2.2.2.3.cmml" xref="A3.SS2.SSS1.1.p1.1.m1.1.2.2.2.3">𝜎</ci></apply></apply><ci id="A3.SS2.SSS1.1.p1.1.m1.1.1.cmml" xref="A3.SS2.SSS1.1.p1.1.m1.1.1">𝐱</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.SS2.SSS1.1.p1.1.m1.1c">\nabla\log p_{\sigma}({\mathbf{x}})</annotation><annotation encoding="application/x-llamapun" id="A3.SS2.SSS1.1.p1.1.m1.1d">∇ roman_log italic_p start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ( bold_x )</annotation></semantics></math> when <math alttext="\sigma" class="ltx_Math" display="inline" id="A3.SS2.SSS1.1.p1.2.m2.1"><semantics id="A3.SS2.SSS1.1.p1.2.m2.1a"><mi id="A3.SS2.SSS1.1.p1.2.m2.1.1" xref="A3.SS2.SSS1.1.p1.2.m2.1.1.cmml">σ</mi><annotation-xml encoding="MathML-Content" id="A3.SS2.SSS1.1.p1.2.m2.1b"><ci id="A3.SS2.SSS1.1.p1.2.m2.1.1.cmml" xref="A3.SS2.SSS1.1.p1.2.m2.1.1">𝜎</ci></annotation-xml><annotation encoding="application/x-tex" id="A3.SS2.SSS1.1.p1.2.m2.1c">\sigma</annotation><annotation encoding="application/x-llamapun" id="A3.SS2.SSS1.1.p1.2.m2.1d">italic_σ</annotation></semantics></math> goes to <math alttext="0" class="ltx_Math" display="inline" id="A3.SS2.SSS1.1.p1.3.m3.1"><semantics id="A3.SS2.SSS1.1.p1.3.m3.1a"><mn id="A3.SS2.SSS1.1.p1.3.m3.1.1" xref="A3.SS2.SSS1.1.p1.3.m3.1.1.cmml">0</mn><annotation-xml encoding="MathML-Content" id="A3.SS2.SSS1.1.p1.3.m3.1b"><cn id="A3.SS2.SSS1.1.p1.3.m3.1.1.cmml" type="integer" xref="A3.SS2.SSS1.1.p1.3.m3.1.1">0</cn></annotation-xml></semantics></math>. Recall that</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A8.EGx22"> <tbody id="A3.Ex30"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\mathcal{N}({\mathbf{x}};{\mathbf{\mu}}_{i},{\mathbf{\Sigma}}_{i}% +\sigma^{2}\mathbf{I}_{d})=\frac{1}{\sqrt{\det(2\pi({\mathbf{\Sigma}}_{i}+% \sigma^{2}\mathbf{I}_{d}))}}\exp{\left(-\frac{1}{2}({\mathbf{x}}-{\mathbf{\mu}% }_{i})^{T}({\mathbf{\Sigma}}_{i}+\sigma^{2}\mathbf{I}_{d})^{-1}({\mathbf{x}}-{% \mathbf{\mu}}_{i})\right)}." class="ltx_Math" display="inline" id="A3.Ex30.m1.4"><semantics id="A3.Ex30.m1.4a"><mrow id="A3.Ex30.m1.4.4.1" xref="A3.Ex30.m1.4.4.1.1.cmml"><mrow id="A3.Ex30.m1.4.4.1.1" xref="A3.Ex30.m1.4.4.1.1.cmml"><mrow id="A3.Ex30.m1.4.4.1.1.2" xref="A3.Ex30.m1.4.4.1.1.2.cmml"><mi class="ltx_font_mathcaligraphic" id="A3.Ex30.m1.4.4.1.1.2.4" xref="A3.Ex30.m1.4.4.1.1.2.4.cmml">𝒩</mi><mo id="A3.Ex30.m1.4.4.1.1.2.3" xref="A3.Ex30.m1.4.4.1.1.2.3.cmml">⁢</mo><mrow id="A3.Ex30.m1.4.4.1.1.2.2.2" xref="A3.Ex30.m1.4.4.1.1.2.2.3.cmml"><mo id="A3.Ex30.m1.4.4.1.1.2.2.2.3" stretchy="false" xref="A3.Ex30.m1.4.4.1.1.2.2.3.cmml">(</mo><mi id="A3.Ex30.m1.2.2" xref="A3.Ex30.m1.2.2.cmml">𝐱</mi><mo id="A3.Ex30.m1.4.4.1.1.2.2.2.4" xref="A3.Ex30.m1.4.4.1.1.2.2.3.cmml">;</mo><msub id="A3.Ex30.m1.4.4.1.1.1.1.1.1" xref="A3.Ex30.m1.4.4.1.1.1.1.1.1.cmml"><mi id="A3.Ex30.m1.4.4.1.1.1.1.1.1.2" xref="A3.Ex30.m1.4.4.1.1.1.1.1.1.2.cmml">μ</mi><mi id="A3.Ex30.m1.4.4.1.1.1.1.1.1.3" xref="A3.Ex30.m1.4.4.1.1.1.1.1.1.3.cmml">i</mi></msub><mo id="A3.Ex30.m1.4.4.1.1.2.2.2.5" xref="A3.Ex30.m1.4.4.1.1.2.2.3.cmml">,</mo><mrow id="A3.Ex30.m1.4.4.1.1.2.2.2.2" xref="A3.Ex30.m1.4.4.1.1.2.2.2.2.cmml"><msub id="A3.Ex30.m1.4.4.1.1.2.2.2.2.2" xref="A3.Ex30.m1.4.4.1.1.2.2.2.2.2.cmml"><mi id="A3.Ex30.m1.4.4.1.1.2.2.2.2.2.2" xref="A3.Ex30.m1.4.4.1.1.2.2.2.2.2.2.cmml">𝚺</mi><mi id="A3.Ex30.m1.4.4.1.1.2.2.2.2.2.3" xref="A3.Ex30.m1.4.4.1.1.2.2.2.2.2.3.cmml">i</mi></msub><mo id="A3.Ex30.m1.4.4.1.1.2.2.2.2.1" xref="A3.Ex30.m1.4.4.1.1.2.2.2.2.1.cmml">+</mo><mrow id="A3.Ex30.m1.4.4.1.1.2.2.2.2.3" xref="A3.Ex30.m1.4.4.1.1.2.2.2.2.3.cmml"><msup id="A3.Ex30.m1.4.4.1.1.2.2.2.2.3.2" xref="A3.Ex30.m1.4.4.1.1.2.2.2.2.3.2.cmml"><mi id="A3.Ex30.m1.4.4.1.1.2.2.2.2.3.2.2" xref="A3.Ex30.m1.4.4.1.1.2.2.2.2.3.2.2.cmml">σ</mi><mn id="A3.Ex30.m1.4.4.1.1.2.2.2.2.3.2.3" xref="A3.Ex30.m1.4.4.1.1.2.2.2.2.3.2.3.cmml">2</mn></msup><mo id="A3.Ex30.m1.4.4.1.1.2.2.2.2.3.1" xref="A3.Ex30.m1.4.4.1.1.2.2.2.2.3.1.cmml">⁢</mo><msub id="A3.Ex30.m1.4.4.1.1.2.2.2.2.3.3" xref="A3.Ex30.m1.4.4.1.1.2.2.2.2.3.3.cmml"><mi id="A3.Ex30.m1.4.4.1.1.2.2.2.2.3.3.2" xref="A3.Ex30.m1.4.4.1.1.2.2.2.2.3.3.2.cmml">𝐈</mi><mi id="A3.Ex30.m1.4.4.1.1.2.2.2.2.3.3.3" xref="A3.Ex30.m1.4.4.1.1.2.2.2.2.3.3.3.cmml">d</mi></msub></mrow></mrow><mo id="A3.Ex30.m1.4.4.1.1.2.2.2.6" stretchy="false" xref="A3.Ex30.m1.4.4.1.1.2.2.3.cmml">)</mo></mrow></mrow><mo id="A3.Ex30.m1.4.4.1.1.4" xref="A3.Ex30.m1.4.4.1.1.4.cmml">=</mo><mrow id="A3.Ex30.m1.4.4.1.1.3" xref="A3.Ex30.m1.4.4.1.1.3.cmml"><mstyle displaystyle="true" id="A3.Ex30.m1.1.1" xref="A3.Ex30.m1.1.1.cmml"><mfrac id="A3.Ex30.m1.1.1a" xref="A3.Ex30.m1.1.1.cmml"><mn id="A3.Ex30.m1.1.1.3" xref="A3.Ex30.m1.1.1.3.cmml">1</mn><msqrt id="A3.Ex30.m1.1.1.1" xref="A3.Ex30.m1.1.1.1.cmml"><mrow id="A3.Ex30.m1.1.1.1.1.1" xref="A3.Ex30.m1.1.1.1.1.1.cmml"><mo id="A3.Ex30.m1.1.1.1.1.1.2" rspace="0em" xref="A3.Ex30.m1.1.1.1.1.1.2.cmml">det</mo><mrow id="A3.Ex30.m1.1.1.1.1.1.1.1" xref="A3.Ex30.m1.1.1.1.1.1.1.1.1.cmml"><mo id="A3.Ex30.m1.1.1.1.1.1.1.1.2" stretchy="false" xref="A3.Ex30.m1.1.1.1.1.1.1.1.1.cmml">(</mo><mrow id="A3.Ex30.m1.1.1.1.1.1.1.1.1" xref="A3.Ex30.m1.1.1.1.1.1.1.1.1.cmml"><mn id="A3.Ex30.m1.1.1.1.1.1.1.1.1.3" xref="A3.Ex30.m1.1.1.1.1.1.1.1.1.3.cmml">2</mn><mo id="A3.Ex30.m1.1.1.1.1.1.1.1.1.2" xref="A3.Ex30.m1.1.1.1.1.1.1.1.1.2.cmml">⁢</mo><mi id="A3.Ex30.m1.1.1.1.1.1.1.1.1.4" xref="A3.Ex30.m1.1.1.1.1.1.1.1.1.4.cmml">π</mi><mo id="A3.Ex30.m1.1.1.1.1.1.1.1.1.2a" xref="A3.Ex30.m1.1.1.1.1.1.1.1.1.2.cmml">⁢</mo><mrow id="A3.Ex30.m1.1.1.1.1.1.1.1.1.1.1" xref="A3.Ex30.m1.1.1.1.1.1.1.1.1.1.1.1.cmml"><mo id="A3.Ex30.m1.1.1.1.1.1.1.1.1.1.1.2" stretchy="false" xref="A3.Ex30.m1.1.1.1.1.1.1.1.1.1.1.1.cmml">(</mo><mrow id="A3.Ex30.m1.1.1.1.1.1.1.1.1.1.1.1" xref="A3.Ex30.m1.1.1.1.1.1.1.1.1.1.1.1.cmml"><msub id="A3.Ex30.m1.1.1.1.1.1.1.1.1.1.1.1.2" xref="A3.Ex30.m1.1.1.1.1.1.1.1.1.1.1.1.2.cmml"><mi id="A3.Ex30.m1.1.1.1.1.1.1.1.1.1.1.1.2.2" xref="A3.Ex30.m1.1.1.1.1.1.1.1.1.1.1.1.2.2.cmml">𝚺</mi><mi id="A3.Ex30.m1.1.1.1.1.1.1.1.1.1.1.1.2.3" xref="A3.Ex30.m1.1.1.1.1.1.1.1.1.1.1.1.2.3.cmml">i</mi></msub><mo id="A3.Ex30.m1.1.1.1.1.1.1.1.1.1.1.1.1" xref="A3.Ex30.m1.1.1.1.1.1.1.1.1.1.1.1.1.cmml">+</mo><mrow id="A3.Ex30.m1.1.1.1.1.1.1.1.1.1.1.1.3" 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id="A3.Ex30.m1.4.4.1.1.3.1.1.1.1.3.3.1.1.3.3.cmml" xref="A3.Ex30.m1.4.4.1.1.3.1.1.1.1.3.3.1.1.3.3">𝑖</ci></apply></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.Ex30.m1.4c">\displaystyle\mathcal{N}({\mathbf{x}};{\mathbf{\mu}}_{i},{\mathbf{\Sigma}}_{i}% +\sigma^{2}\mathbf{I}_{d})=\frac{1}{\sqrt{\det(2\pi({\mathbf{\Sigma}}_{i}+% \sigma^{2}\mathbf{I}_{d}))}}\exp{\left(-\frac{1}{2}({\mathbf{x}}-{\mathbf{\mu}% }_{i})^{T}({\mathbf{\Sigma}}_{i}+\sigma^{2}\mathbf{I}_{d})^{-1}({\mathbf{x}}-{% \mathbf{\mu}}_{i})\right)}.</annotation><annotation encoding="application/x-llamapun" id="A3.Ex30.m1.4d">caligraphic_N ( bold_x ; italic_μ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , bold_Σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT + italic_σ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT bold_I start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT ) = divide start_ARG 1 end_ARG start_ARG square-root start_ARG roman_det ( 2 italic_π ( bold_Σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT + italic_σ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT bold_I start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT ) ) end_ARG end_ARG roman_exp ( - divide start_ARG 1 end_ARG start_ARG 2 end_ARG ( bold_x - italic_μ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT ( bold_Σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT + italic_σ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT bold_I start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ( bold_x - italic_μ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) ) .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="A3.SS2.SSS1.1.p1.7">It is known that <math alttext="\det{({\mathbf{\Sigma}}_{i}+\sigma^{2}\mathbf{I}_{d})}=\det{({\mathbf{\Sigma}}% _{i})}+\sigma^{2}\text{Tr}{({\operatorname{Com}}({\mathbf{\Sigma}}_{i})^{T})}+% \mathcal{O}(\sigma^{4})" class="ltx_math_unparsed" display="inline" id="A3.SS2.SSS1.1.p1.4.m1.1"><semantics id="A3.SS2.SSS1.1.p1.4.m1.1a"><mrow id="A3.SS2.SSS1.1.p1.4.m1.1b"><mo id="A3.SS2.SSS1.1.p1.4.m1.1.2" rspace="0em">det</mo><mrow id="A3.SS2.SSS1.1.p1.4.m1.1.3"><mo id="A3.SS2.SSS1.1.p1.4.m1.1.3.1" stretchy="false">(</mo><msub id="A3.SS2.SSS1.1.p1.4.m1.1.3.2"><mi id="A3.SS2.SSS1.1.p1.4.m1.1.3.2.2">𝚺</mi><mi id="A3.SS2.SSS1.1.p1.4.m1.1.3.2.3">i</mi></msub><mo id="A3.SS2.SSS1.1.p1.4.m1.1.3.3">+</mo><msup id="A3.SS2.SSS1.1.p1.4.m1.1.3.4"><mi id="A3.SS2.SSS1.1.p1.4.m1.1.3.4.2">σ</mi><mn id="A3.SS2.SSS1.1.p1.4.m1.1.3.4.3">2</mn></msup><msub id="A3.SS2.SSS1.1.p1.4.m1.1.3.5"><mi id="A3.SS2.SSS1.1.p1.4.m1.1.3.5.2">𝐈</mi><mi id="A3.SS2.SSS1.1.p1.4.m1.1.3.5.3">d</mi></msub><mo id="A3.SS2.SSS1.1.p1.4.m1.1.3.6" stretchy="false">)</mo></mrow><mo id="A3.SS2.SSS1.1.p1.4.m1.1.4" rspace="0.1389em">=</mo><mo id="A3.SS2.SSS1.1.p1.4.m1.1.5" lspace="0.1389em" rspace="0em">det</mo><mrow id="A3.SS2.SSS1.1.p1.4.m1.1.6"><mo id="A3.SS2.SSS1.1.p1.4.m1.1.6.1" stretchy="false">(</mo><msub id="A3.SS2.SSS1.1.p1.4.m1.1.6.2"><mi id="A3.SS2.SSS1.1.p1.4.m1.1.6.2.2">𝚺</mi><mi id="A3.SS2.SSS1.1.p1.4.m1.1.6.2.3">i</mi></msub><mo id="A3.SS2.SSS1.1.p1.4.m1.1.6.3" stretchy="false">)</mo></mrow><mo id="A3.SS2.SSS1.1.p1.4.m1.1.7">+</mo><msup id="A3.SS2.SSS1.1.p1.4.m1.1.8"><mi id="A3.SS2.SSS1.1.p1.4.m1.1.8.2">σ</mi><mn id="A3.SS2.SSS1.1.p1.4.m1.1.8.3">2</mn></msup><mtext id="A3.SS2.SSS1.1.p1.4.m1.1.9">Tr</mtext><mrow id="A3.SS2.SSS1.1.p1.4.m1.1.10"><mo id="A3.SS2.SSS1.1.p1.4.m1.1.10.1" stretchy="false">(</mo><mi id="A3.SS2.SSS1.1.p1.4.m1.1.1">Com</mi><msup id="A3.SS2.SSS1.1.p1.4.m1.1.10.2"><mrow id="A3.SS2.SSS1.1.p1.4.m1.1.10.2.2"><mo id="A3.SS2.SSS1.1.p1.4.m1.1.10.2.2.1" stretchy="false">(</mo><msub id="A3.SS2.SSS1.1.p1.4.m1.1.10.2.2.2"><mi id="A3.SS2.SSS1.1.p1.4.m1.1.10.2.2.2.2">𝚺</mi><mi id="A3.SS2.SSS1.1.p1.4.m1.1.10.2.2.2.3">i</mi></msub><mo id="A3.SS2.SSS1.1.p1.4.m1.1.10.2.2.3" stretchy="false">)</mo></mrow><mi id="A3.SS2.SSS1.1.p1.4.m1.1.10.2.3">T</mi></msup><mo id="A3.SS2.SSS1.1.p1.4.m1.1.10.3" stretchy="false">)</mo></mrow><mo id="A3.SS2.SSS1.1.p1.4.m1.1.11">+</mo><mi class="ltx_font_mathcaligraphic" id="A3.SS2.SSS1.1.p1.4.m1.1.12">𝒪</mi><mrow id="A3.SS2.SSS1.1.p1.4.m1.1.13"><mo id="A3.SS2.SSS1.1.p1.4.m1.1.13.1" stretchy="false">(</mo><msup id="A3.SS2.SSS1.1.p1.4.m1.1.13.2"><mi id="A3.SS2.SSS1.1.p1.4.m1.1.13.2.2">σ</mi><mn id="A3.SS2.SSS1.1.p1.4.m1.1.13.2.3">4</mn></msup><mo id="A3.SS2.SSS1.1.p1.4.m1.1.13.3" stretchy="false">)</mo></mrow></mrow><annotation encoding="application/x-tex" id="A3.SS2.SSS1.1.p1.4.m1.1c">\det{({\mathbf{\Sigma}}_{i}+\sigma^{2}\mathbf{I}_{d})}=\det{({\mathbf{\Sigma}}% _{i})}+\sigma^{2}\text{Tr}{({\operatorname{Com}}({\mathbf{\Sigma}}_{i})^{T})}+% \mathcal{O}(\sigma^{4})</annotation><annotation encoding="application/x-llamapun" id="A3.SS2.SSS1.1.p1.4.m1.1d">roman_det ( bold_Σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT + italic_σ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT bold_I start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT ) = roman_det ( bold_Σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) + italic_σ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT Tr ( roman_Com ( bold_Σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT ) + caligraphic_O ( italic_σ start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT )</annotation></semantics></math>, with <span class="ltx_text ltx_markedasmath" id="A3.SS2.SSS1.1.p1.7.1">Tr</span> the trace of the matrix and <math alttext="{\operatorname{Com}}({\mathbf{\Sigma}}_{i})" class="ltx_Math" display="inline" id="A3.SS2.SSS1.1.p1.6.m3.2"><semantics id="A3.SS2.SSS1.1.p1.6.m3.2a"><mrow id="A3.SS2.SSS1.1.p1.6.m3.2.2.1" xref="A3.SS2.SSS1.1.p1.6.m3.2.2.2.cmml"><mi id="A3.SS2.SSS1.1.p1.6.m3.1.1" xref="A3.SS2.SSS1.1.p1.6.m3.1.1.cmml">Com</mi><mo id="A3.SS2.SSS1.1.p1.6.m3.2.2.1a" xref="A3.SS2.SSS1.1.p1.6.m3.2.2.2.cmml">⁡</mo><mrow id="A3.SS2.SSS1.1.p1.6.m3.2.2.1.1" xref="A3.SS2.SSS1.1.p1.6.m3.2.2.2.cmml"><mo id="A3.SS2.SSS1.1.p1.6.m3.2.2.1.1.2" stretchy="false" xref="A3.SS2.SSS1.1.p1.6.m3.2.2.2.cmml">(</mo><msub id="A3.SS2.SSS1.1.p1.6.m3.2.2.1.1.1" xref="A3.SS2.SSS1.1.p1.6.m3.2.2.1.1.1.cmml"><mi id="A3.SS2.SSS1.1.p1.6.m3.2.2.1.1.1.2" xref="A3.SS2.SSS1.1.p1.6.m3.2.2.1.1.1.2.cmml">𝚺</mi><mi id="A3.SS2.SSS1.1.p1.6.m3.2.2.1.1.1.3" xref="A3.SS2.SSS1.1.p1.6.m3.2.2.1.1.1.3.cmml">i</mi></msub><mo id="A3.SS2.SSS1.1.p1.6.m3.2.2.1.1.3" stretchy="false" xref="A3.SS2.SSS1.1.p1.6.m3.2.2.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A3.SS2.SSS1.1.p1.6.m3.2b"><apply id="A3.SS2.SSS1.1.p1.6.m3.2.2.2.cmml" xref="A3.SS2.SSS1.1.p1.6.m3.2.2.1"><ci id="A3.SS2.SSS1.1.p1.6.m3.1.1.cmml" xref="A3.SS2.SSS1.1.p1.6.m3.1.1">Com</ci><apply id="A3.SS2.SSS1.1.p1.6.m3.2.2.1.1.1.cmml" xref="A3.SS2.SSS1.1.p1.6.m3.2.2.1.1.1"><csymbol cd="ambiguous" id="A3.SS2.SSS1.1.p1.6.m3.2.2.1.1.1.1.cmml" xref="A3.SS2.SSS1.1.p1.6.m3.2.2.1.1.1">subscript</csymbol><ci id="A3.SS2.SSS1.1.p1.6.m3.2.2.1.1.1.2.cmml" xref="A3.SS2.SSS1.1.p1.6.m3.2.2.1.1.1.2">𝚺</ci><ci id="A3.SS2.SSS1.1.p1.6.m3.2.2.1.1.1.3.cmml" xref="A3.SS2.SSS1.1.p1.6.m3.2.2.1.1.1.3">𝑖</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.SS2.SSS1.1.p1.6.m3.2c">{\operatorname{Com}}({\mathbf{\Sigma}}_{i})</annotation><annotation encoding="application/x-llamapun" id="A3.SS2.SSS1.1.p1.6.m3.2d">roman_Com ( bold_Σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT )</annotation></semantics></math> the comatrix of the matrix <math alttext="{\mathbf{\Sigma}}_{i}" class="ltx_Math" display="inline" id="A3.SS2.SSS1.1.p1.7.m4.1"><semantics id="A3.SS2.SSS1.1.p1.7.m4.1a"><msub id="A3.SS2.SSS1.1.p1.7.m4.1.1" xref="A3.SS2.SSS1.1.p1.7.m4.1.1.cmml"><mi id="A3.SS2.SSS1.1.p1.7.m4.1.1.2" xref="A3.SS2.SSS1.1.p1.7.m4.1.1.2.cmml">𝚺</mi><mi id="A3.SS2.SSS1.1.p1.7.m4.1.1.3" xref="A3.SS2.SSS1.1.p1.7.m4.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="A3.SS2.SSS1.1.p1.7.m4.1b"><apply id="A3.SS2.SSS1.1.p1.7.m4.1.1.cmml" xref="A3.SS2.SSS1.1.p1.7.m4.1.1"><csymbol cd="ambiguous" id="A3.SS2.SSS1.1.p1.7.m4.1.1.1.cmml" xref="A3.SS2.SSS1.1.p1.7.m4.1.1">subscript</csymbol><ci id="A3.SS2.SSS1.1.p1.7.m4.1.1.2.cmml" xref="A3.SS2.SSS1.1.p1.7.m4.1.1.2">𝚺</ci><ci id="A3.SS2.SSS1.1.p1.7.m4.1.1.3.cmml" xref="A3.SS2.SSS1.1.p1.7.m4.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.SS2.SSS1.1.p1.7.m4.1c">{\mathbf{\Sigma}}_{i}</annotation><annotation encoding="application/x-llamapun" id="A3.SS2.SSS1.1.p1.7.m4.1d">bold_Σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math>. Then we have</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A8.EGx23"> <tbody id="A3.Ex31"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\mathcal{N}({\mathbf{x}};{\mathbf{\mu}}_{i},{\mathbf{\Sigma}}_{i}% +\sigma^{2}\mathbf{I}_{d})=\frac{1}{\sqrt{\det(2\pi({\mathbf{\Sigma}}_{i}+% \sigma^{2}\mathbf{I}_{d}))}}\exp{\left(-\frac{1}{2}({\mathbf{x}}-{\mathbf{\mu}% }_{i})^{T}({\mathbf{\Sigma}}_{i}+\sigma^{2}\mathbf{I}_{d})^{-1}({\mathbf{x}}-{% \mathbf{\mu}}_{i})\right)}" class="ltx_Math" display="inline" id="A3.Ex31.m1.6"><semantics id="A3.Ex31.m1.6a"><mrow id="A3.Ex31.m1.6.6" xref="A3.Ex31.m1.6.6.cmml"><mrow id="A3.Ex31.m1.5.5.2" xref="A3.Ex31.m1.5.5.2.cmml"><mi class="ltx_font_mathcaligraphic" id="A3.Ex31.m1.5.5.2.4" xref="A3.Ex31.m1.5.5.2.4.cmml">𝒩</mi><mo id="A3.Ex31.m1.5.5.2.3" xref="A3.Ex31.m1.5.5.2.3.cmml">⁢</mo><mrow id="A3.Ex31.m1.5.5.2.2.2" xref="A3.Ex31.m1.5.5.2.2.3.cmml"><mo id="A3.Ex31.m1.5.5.2.2.2.3" stretchy="false" xref="A3.Ex31.m1.5.5.2.2.3.cmml">(</mo><mi id="A3.Ex31.m1.2.2" xref="A3.Ex31.m1.2.2.cmml">𝐱</mi><mo id="A3.Ex31.m1.5.5.2.2.2.4" xref="A3.Ex31.m1.5.5.2.2.3.cmml">;</mo><msub id="A3.Ex31.m1.4.4.1.1.1.1" xref="A3.Ex31.m1.4.4.1.1.1.1.cmml"><mi id="A3.Ex31.m1.4.4.1.1.1.1.2" xref="A3.Ex31.m1.4.4.1.1.1.1.2.cmml">μ</mi><mi id="A3.Ex31.m1.4.4.1.1.1.1.3" xref="A3.Ex31.m1.4.4.1.1.1.1.3.cmml">i</mi></msub><mo id="A3.Ex31.m1.5.5.2.2.2.5" xref="A3.Ex31.m1.5.5.2.2.3.cmml">,</mo><mrow id="A3.Ex31.m1.5.5.2.2.2.2" xref="A3.Ex31.m1.5.5.2.2.2.2.cmml"><msub id="A3.Ex31.m1.5.5.2.2.2.2.2" xref="A3.Ex31.m1.5.5.2.2.2.2.2.cmml"><mi id="A3.Ex31.m1.5.5.2.2.2.2.2.2" xref="A3.Ex31.m1.5.5.2.2.2.2.2.2.cmml">𝚺</mi><mi id="A3.Ex31.m1.5.5.2.2.2.2.2.3" xref="A3.Ex31.m1.5.5.2.2.2.2.2.3.cmml">i</mi></msub><mo id="A3.Ex31.m1.5.5.2.2.2.2.1" xref="A3.Ex31.m1.5.5.2.2.2.2.1.cmml">+</mo><mrow id="A3.Ex31.m1.5.5.2.2.2.2.3" xref="A3.Ex31.m1.5.5.2.2.2.2.3.cmml"><msup id="A3.Ex31.m1.5.5.2.2.2.2.3.2" xref="A3.Ex31.m1.5.5.2.2.2.2.3.2.cmml"><mi id="A3.Ex31.m1.5.5.2.2.2.2.3.2.2" xref="A3.Ex31.m1.5.5.2.2.2.2.3.2.2.cmml">σ</mi><mn id="A3.Ex31.m1.5.5.2.2.2.2.3.2.3" xref="A3.Ex31.m1.5.5.2.2.2.2.3.2.3.cmml">2</mn></msup><mo id="A3.Ex31.m1.5.5.2.2.2.2.3.1" xref="A3.Ex31.m1.5.5.2.2.2.2.3.1.cmml">⁢</mo><msub id="A3.Ex31.m1.5.5.2.2.2.2.3.3" xref="A3.Ex31.m1.5.5.2.2.2.2.3.3.cmml"><mi id="A3.Ex31.m1.5.5.2.2.2.2.3.3.2" xref="A3.Ex31.m1.5.5.2.2.2.2.3.3.2.cmml">𝐈</mi><mi id="A3.Ex31.m1.5.5.2.2.2.2.3.3.3" xref="A3.Ex31.m1.5.5.2.2.2.2.3.3.3.cmml">d</mi></msub></mrow></mrow><mo id="A3.Ex31.m1.5.5.2.2.2.6" stretchy="false" xref="A3.Ex31.m1.5.5.2.2.3.cmml">)</mo></mrow></mrow><mo id="A3.Ex31.m1.6.6.4" xref="A3.Ex31.m1.6.6.4.cmml">=</mo><mrow id="A3.Ex31.m1.6.6.3" xref="A3.Ex31.m1.6.6.3.cmml"><mstyle displaystyle="true" id="A3.Ex31.m1.1.1" xref="A3.Ex31.m1.1.1.cmml"><mfrac id="A3.Ex31.m1.1.1a" xref="A3.Ex31.m1.1.1.cmml"><mn 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id="A3.Ex31.m1.6.6.3.1.1.1.1.3.3.1.1.2.cmml" xref="A3.Ex31.m1.6.6.3.1.1.1.1.3.3.1.1.2">𝐱</ci><apply id="A3.Ex31.m1.6.6.3.1.1.1.1.3.3.1.1.3.cmml" xref="A3.Ex31.m1.6.6.3.1.1.1.1.3.3.1.1.3"><csymbol cd="ambiguous" id="A3.Ex31.m1.6.6.3.1.1.1.1.3.3.1.1.3.1.cmml" xref="A3.Ex31.m1.6.6.3.1.1.1.1.3.3.1.1.3">subscript</csymbol><ci id="A3.Ex31.m1.6.6.3.1.1.1.1.3.3.1.1.3.2.cmml" xref="A3.Ex31.m1.6.6.3.1.1.1.1.3.3.1.1.3.2">𝜇</ci><ci id="A3.Ex31.m1.6.6.3.1.1.1.1.3.3.1.1.3.3.cmml" xref="A3.Ex31.m1.6.6.3.1.1.1.1.3.3.1.1.3.3">𝑖</ci></apply></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.Ex31.m1.6c">\displaystyle\mathcal{N}({\mathbf{x}};{\mathbf{\mu}}_{i},{\mathbf{\Sigma}}_{i}% +\sigma^{2}\mathbf{I}_{d})=\frac{1}{\sqrt{\det(2\pi({\mathbf{\Sigma}}_{i}+% \sigma^{2}\mathbf{I}_{d}))}}\exp{\left(-\frac{1}{2}({\mathbf{x}}-{\mathbf{\mu}% }_{i})^{T}({\mathbf{\Sigma}}_{i}+\sigma^{2}\mathbf{I}_{d})^{-1}({\mathbf{x}}-{% \mathbf{\mu}}_{i})\right)}</annotation><annotation encoding="application/x-llamapun" id="A3.Ex31.m1.6d">caligraphic_N ( bold_x ; italic_μ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , bold_Σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT + italic_σ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT bold_I start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT ) = divide start_ARG 1 end_ARG start_ARG square-root start_ARG roman_det ( 2 italic_π ( bold_Σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT + italic_σ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT bold_I start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT ) ) end_ARG end_ARG roman_exp ( - divide start_ARG 1 end_ARG start_ARG 2 end_ARG ( bold_x - italic_μ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT ( bold_Σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT + italic_σ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT bold_I start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ( bold_x - italic_μ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="A3.Ex32"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle=\frac{1}{\sqrt{\det{(2\pi{\mathbf{\Sigma}}_{i})}}}\left(1-\sigma% ^{2}\frac{\text{Tr}{({\operatorname{Com}}({\mathbf{\Sigma}}_{i}))}}{2\det{({% \mathbf{\Sigma}}_{i})}}+\mathcal{O}(\sigma^{4})\right)" class="ltx_Math" display="inline" id="A3.Ex32.m1.5"><semantics id="A3.Ex32.m1.5a"><mrow id="A3.Ex32.m1.5.5" xref="A3.Ex32.m1.5.5.cmml"><mi id="A3.Ex32.m1.5.5.3" xref="A3.Ex32.m1.5.5.3.cmml"></mi><mo id="A3.Ex32.m1.5.5.2" xref="A3.Ex32.m1.5.5.2.cmml">=</mo><mrow id="A3.Ex32.m1.5.5.1" xref="A3.Ex32.m1.5.5.1.cmml"><mstyle displaystyle="true" id="A3.Ex32.m1.1.1" xref="A3.Ex32.m1.1.1.cmml"><mfrac 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id="A3.Ex33.m1.2c">\displaystyle\exp{\left(-\frac{1}{2}({\mathbf{x}}-{\mathbf{\mu}}_{i})^{T}({% \mathbf{\Sigma}}_{i})^{-1}({\mathbf{x}}-{\mathbf{\mu}}_{i})+\frac{\sigma^{2}}{% 2}({\mathbf{x}}-{\mathbf{\mu}}_{i})^{T}({\mathbf{\Sigma}}_{i})^{-2}({\mathbf{x% }}-{\mathbf{\mu}}_{i})+\mathcal{O}(\sigma^{4})\right)}</annotation><annotation encoding="application/x-llamapun" id="A3.Ex33.m1.2d">roman_exp ( - divide start_ARG 1 end_ARG start_ARG 2 end_ARG ( bold_x - italic_μ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT ( bold_Σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ( bold_x - italic_μ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) + divide start_ARG italic_σ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG 2 end_ARG ( bold_x - italic_μ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT ( bold_Σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT - 2 end_POSTSUPERSCRIPT ( bold_x - italic_μ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) + caligraphic_O ( italic_σ start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT ) )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="A3.Ex34"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle=\mathcal{N}({\mathbf{x}};{\mathbf{\mu}}_{i},{\mathbf{\Sigma}}_{i% })\left(1-\sigma^{2}\frac{\text{Tr}{({\operatorname{Com}}({\mathbf{\Sigma}}_{i% }))}}{2\det{({\mathbf{\Sigma}}_{i})}}+\mathcal{O}(\sigma^{4})\right)\left(1+% \frac{\sigma^{2}}{2}({\mathbf{x}}-{\mathbf{\mu}}_{i})^{T}({\mathbf{\Sigma}}_{i% })^{-2}({\mathbf{x}}-{\mathbf{\mu}}_{i})+\mathcal{O}(\sigma^{4})\right)" class="ltx_Math" display="inline" id="A3.Ex34.m1.8"><semantics id="A3.Ex34.m1.8a"><mrow 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encoding="application/x-tex" id="A3.Ex34.m1.8c">\displaystyle=\mathcal{N}({\mathbf{x}};{\mathbf{\mu}}_{i},{\mathbf{\Sigma}}_{i% })\left(1-\sigma^{2}\frac{\text{Tr}{({\operatorname{Com}}({\mathbf{\Sigma}}_{i% }))}}{2\det{({\mathbf{\Sigma}}_{i})}}+\mathcal{O}(\sigma^{4})\right)\left(1+% \frac{\sigma^{2}}{2}({\mathbf{x}}-{\mathbf{\mu}}_{i})^{T}({\mathbf{\Sigma}}_{i% })^{-2}({\mathbf{x}}-{\mathbf{\mu}}_{i})+\mathcal{O}(\sigma^{4})\right)</annotation><annotation encoding="application/x-llamapun" id="A3.Ex34.m1.8d">= caligraphic_N ( bold_x ; italic_μ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , bold_Σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) ( 1 - italic_σ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT divide start_ARG Tr ( roman_Com ( bold_Σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) ) end_ARG start_ARG 2 roman_det ( bold_Σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) end_ARG + caligraphic_O ( italic_σ start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT ) ) ( 1 + divide start_ARG italic_σ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG 2 end_ARG ( bold_x - italic_μ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT ( bold_Σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT - 2 end_POSTSUPERSCRIPT ( bold_x - italic_μ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) + caligraphic_O ( italic_σ start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT ) )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="A3.Ex35"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle=\mathcal{N}({\mathbf{x}};{\mathbf{\mu}}_{i},{\mathbf{\Sigma}}_{i% })\left(1+\frac{\sigma^{2}}{2}\left(({\mathbf{x}}-{\mathbf{\mu}}_{i})^{T}({% \mathbf{\Sigma}}_{i})^{-2}({\mathbf{x}}-{\mathbf{\mu}}_{i})-\frac{\text{Tr}{({% \operatorname{Com}}({\mathbf{\Sigma}}_{i}))}}{\det{({\mathbf{\Sigma}}_{i})}}% \right)+\mathcal{O}(\sigma^{4})\right)." class="ltx_Math" display="inline" id="A3.Ex35.m1.5"><semantics id="A3.Ex35.m1.5a"><mrow id="A3.Ex35.m1.5.5.1" xref="A3.Ex35.m1.5.5.1.1.cmml"><mrow id="A3.Ex35.m1.5.5.1.1" xref="A3.Ex35.m1.5.5.1.1.cmml"><mi id="A3.Ex35.m1.5.5.1.1.5" xref="A3.Ex35.m1.5.5.1.1.5.cmml"></mi><mo id="A3.Ex35.m1.5.5.1.1.4" xref="A3.Ex35.m1.5.5.1.1.4.cmml">=</mo><mrow id="A3.Ex35.m1.5.5.1.1.3" xref="A3.Ex35.m1.5.5.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="A3.Ex35.m1.5.5.1.1.3.5" xref="A3.Ex35.m1.5.5.1.1.3.5.cmml">𝒩</mi><mo id="A3.Ex35.m1.5.5.1.1.3.4" xref="A3.Ex35.m1.5.5.1.1.3.4.cmml">⁢</mo><mrow id="A3.Ex35.m1.5.5.1.1.2.2.2" xref="A3.Ex35.m1.5.5.1.1.2.2.3.cmml"><mo id="A3.Ex35.m1.5.5.1.1.2.2.2.3" stretchy="false" xref="A3.Ex35.m1.5.5.1.1.2.2.3.cmml">(</mo><mi id="A3.Ex35.m1.4.4" xref="A3.Ex35.m1.4.4.cmml">𝐱</mi><mo id="A3.Ex35.m1.5.5.1.1.2.2.2.4" 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id="A3.Ex35.m1.5.5.1.1.3.3.1.1.1.1.1.1.2.2.1.1" xref="A3.Ex35.m1.5.5.1.1.3.3.1.1.1.1.1.1.2.2.1.1.1.cmml"><mo id="A3.Ex35.m1.5.5.1.1.3.3.1.1.1.1.1.1.2.2.1.1.2" stretchy="false" xref="A3.Ex35.m1.5.5.1.1.3.3.1.1.1.1.1.1.2.2.1.1.1.cmml">(</mo><msub id="A3.Ex35.m1.5.5.1.1.3.3.1.1.1.1.1.1.2.2.1.1.1" xref="A3.Ex35.m1.5.5.1.1.3.3.1.1.1.1.1.1.2.2.1.1.1.cmml"><mi id="A3.Ex35.m1.5.5.1.1.3.3.1.1.1.1.1.1.2.2.1.1.1.2" xref="A3.Ex35.m1.5.5.1.1.3.3.1.1.1.1.1.1.2.2.1.1.1.2.cmml">𝚺</mi><mi id="A3.Ex35.m1.5.5.1.1.3.3.1.1.1.1.1.1.2.2.1.1.1.3" xref="A3.Ex35.m1.5.5.1.1.3.3.1.1.1.1.1.1.2.2.1.1.1.3.cmml">i</mi></msub><mo id="A3.Ex35.m1.5.5.1.1.3.3.1.1.1.1.1.1.2.2.1.1.3" stretchy="false" xref="A3.Ex35.m1.5.5.1.1.3.3.1.1.1.1.1.1.2.2.1.1.1.cmml">)</mo></mrow><mrow id="A3.Ex35.m1.5.5.1.1.3.3.1.1.1.1.1.1.2.2.3" xref="A3.Ex35.m1.5.5.1.1.3.3.1.1.1.1.1.1.2.2.3.cmml"><mo id="A3.Ex35.m1.5.5.1.1.3.3.1.1.1.1.1.1.2.2.3a" xref="A3.Ex35.m1.5.5.1.1.3.3.1.1.1.1.1.1.2.2.3.cmml">−</mo><mn 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id="A3.Ex35.m1.5.5.1.1.3.3.1.1.2.1.1.1.1.cmml" xref="A3.Ex35.m1.5.5.1.1.3.3.1.1.2.1.1">superscript</csymbol><ci id="A3.Ex35.m1.5.5.1.1.3.3.1.1.2.1.1.1.2.cmml" xref="A3.Ex35.m1.5.5.1.1.3.3.1.1.2.1.1.1.2">𝜎</ci><cn id="A3.Ex35.m1.5.5.1.1.3.3.1.1.2.1.1.1.3.cmml" type="integer" xref="A3.Ex35.m1.5.5.1.1.3.3.1.1.2.1.1.1.3">4</cn></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.Ex35.m1.5c">\displaystyle=\mathcal{N}({\mathbf{x}};{\mathbf{\mu}}_{i},{\mathbf{\Sigma}}_{i% })\left(1+\frac{\sigma^{2}}{2}\left(({\mathbf{x}}-{\mathbf{\mu}}_{i})^{T}({% \mathbf{\Sigma}}_{i})^{-2}({\mathbf{x}}-{\mathbf{\mu}}_{i})-\frac{\text{Tr}{({% \operatorname{Com}}({\mathbf{\Sigma}}_{i}))}}{\det{({\mathbf{\Sigma}}_{i})}}% \right)+\mathcal{O}(\sigma^{4})\right).</annotation><annotation encoding="application/x-llamapun" id="A3.Ex35.m1.5d">= caligraphic_N ( bold_x ; italic_μ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , bold_Σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) ( 1 + divide start_ARG italic_σ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG 2 end_ARG ( ( bold_x - italic_μ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT ( bold_Σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT - 2 end_POSTSUPERSCRIPT ( bold_x - italic_μ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) - divide start_ARG Tr ( roman_Com ( bold_Σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) ) end_ARG start_ARG roman_det ( bold_Σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) end_ARG ) + caligraphic_O ( italic_σ start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT ) ) .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="A3.SS2.SSS1.1.p1.8">We define <math alttext="u_{i}({\mathbf{x}})=\frac{1}{2}\left(({\mathbf{x}}-{\mathbf{\mu}}_{i})^{T}({% \mathbf{\Sigma}}_{i})^{-2}({\mathbf{x}}-{\mathbf{\mu}}_{i})-\frac{\text{Tr}{({% \operatorname{Com}}({\mathbf{\Sigma}}_{i}))}}{\det{({\mathbf{\Sigma}}_{i})}}\right)" class="ltx_Math" display="inline" id="A3.SS2.SSS1.1.p1.8.m1.5"><semantics id="A3.SS2.SSS1.1.p1.8.m1.5a"><mrow id="A3.SS2.SSS1.1.p1.8.m1.5.5" xref="A3.SS2.SSS1.1.p1.8.m1.5.5.cmml"><mrow id="A3.SS2.SSS1.1.p1.8.m1.5.5.3" xref="A3.SS2.SSS1.1.p1.8.m1.5.5.3.cmml"><msub id="A3.SS2.SSS1.1.p1.8.m1.5.5.3.2" xref="A3.SS2.SSS1.1.p1.8.m1.5.5.3.2.cmml"><mi id="A3.SS2.SSS1.1.p1.8.m1.5.5.3.2.2" xref="A3.SS2.SSS1.1.p1.8.m1.5.5.3.2.2.cmml">u</mi><mi id="A3.SS2.SSS1.1.p1.8.m1.5.5.3.2.3" xref="A3.SS2.SSS1.1.p1.8.m1.5.5.3.2.3.cmml">i</mi></msub><mo id="A3.SS2.SSS1.1.p1.8.m1.5.5.3.1" xref="A3.SS2.SSS1.1.p1.8.m1.5.5.3.1.cmml">⁢</mo><mrow id="A3.SS2.SSS1.1.p1.8.m1.5.5.3.3.2" xref="A3.SS2.SSS1.1.p1.8.m1.5.5.3.cmml"><mo id="A3.SS2.SSS1.1.p1.8.m1.5.5.3.3.2.1" stretchy="false" xref="A3.SS2.SSS1.1.p1.8.m1.5.5.3.cmml">(</mo><mi 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id="A3.SS2.SSS1.1.p1.8.m1.5.5.1.1.1.1.3" xref="A3.SS2.SSS1.1.p1.8.m1.5.5.1.1.1.1.3.cmml"><msup id="A3.SS2.SSS1.1.p1.8.m1.5.5.1.1.1.1.1.1" xref="A3.SS2.SSS1.1.p1.8.m1.5.5.1.1.1.1.1.1.cmml"><mrow id="A3.SS2.SSS1.1.p1.8.m1.5.5.1.1.1.1.1.1.1.1" xref="A3.SS2.SSS1.1.p1.8.m1.5.5.1.1.1.1.1.1.1.1.1.cmml"><mo id="A3.SS2.SSS1.1.p1.8.m1.5.5.1.1.1.1.1.1.1.1.2" stretchy="false" xref="A3.SS2.SSS1.1.p1.8.m1.5.5.1.1.1.1.1.1.1.1.1.cmml">(</mo><mrow id="A3.SS2.SSS1.1.p1.8.m1.5.5.1.1.1.1.1.1.1.1.1" xref="A3.SS2.SSS1.1.p1.8.m1.5.5.1.1.1.1.1.1.1.1.1.cmml"><mi id="A3.SS2.SSS1.1.p1.8.m1.5.5.1.1.1.1.1.1.1.1.1.2" xref="A3.SS2.SSS1.1.p1.8.m1.5.5.1.1.1.1.1.1.1.1.1.2.cmml">𝐱</mi><mo id="A3.SS2.SSS1.1.p1.8.m1.5.5.1.1.1.1.1.1.1.1.1.1" xref="A3.SS2.SSS1.1.p1.8.m1.5.5.1.1.1.1.1.1.1.1.1.1.cmml">−</mo><msub id="A3.SS2.SSS1.1.p1.8.m1.5.5.1.1.1.1.1.1.1.1.1.3" xref="A3.SS2.SSS1.1.p1.8.m1.5.5.1.1.1.1.1.1.1.1.1.3.cmml"><mi id="A3.SS2.SSS1.1.p1.8.m1.5.5.1.1.1.1.1.1.1.1.1.3.2" 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xref="A3.SS2.SSS1.1.p1.8.m1.5.5.1.1.1.1.2.2.1.1.1.cmml"><mi id="A3.SS2.SSS1.1.p1.8.m1.5.5.1.1.1.1.2.2.1.1.1.2" xref="A3.SS2.SSS1.1.p1.8.m1.5.5.1.1.1.1.2.2.1.1.1.2.cmml">𝚺</mi><mi id="A3.SS2.SSS1.1.p1.8.m1.5.5.1.1.1.1.2.2.1.1.1.3" xref="A3.SS2.SSS1.1.p1.8.m1.5.5.1.1.1.1.2.2.1.1.1.3.cmml">i</mi></msub><mo id="A3.SS2.SSS1.1.p1.8.m1.5.5.1.1.1.1.2.2.1.1.3" stretchy="false" xref="A3.SS2.SSS1.1.p1.8.m1.5.5.1.1.1.1.2.2.1.1.1.cmml">)</mo></mrow><mrow id="A3.SS2.SSS1.1.p1.8.m1.5.5.1.1.1.1.2.2.3" xref="A3.SS2.SSS1.1.p1.8.m1.5.5.1.1.1.1.2.2.3.cmml"><mo id="A3.SS2.SSS1.1.p1.8.m1.5.5.1.1.1.1.2.2.3a" xref="A3.SS2.SSS1.1.p1.8.m1.5.5.1.1.1.1.2.2.3.cmml">−</mo><mn id="A3.SS2.SSS1.1.p1.8.m1.5.5.1.1.1.1.2.2.3.2" xref="A3.SS2.SSS1.1.p1.8.m1.5.5.1.1.1.1.2.2.3.2.cmml">2</mn></mrow></msup><mo id="A3.SS2.SSS1.1.p1.8.m1.5.5.1.1.1.1.3.4a" xref="A3.SS2.SSS1.1.p1.8.m1.5.5.1.1.1.1.3.4.cmml">⁢</mo><mrow id="A3.SS2.SSS1.1.p1.8.m1.5.5.1.1.1.1.3.3.1" xref="A3.SS2.SSS1.1.p1.8.m1.5.5.1.1.1.1.3.3.1.1.cmml"><mo 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id="A3.SS2.SSS1.1.p1.8.m1.5c">u_{i}({\mathbf{x}})=\frac{1}{2}\left(({\mathbf{x}}-{\mathbf{\mu}}_{i})^{T}({% \mathbf{\Sigma}}_{i})^{-2}({\mathbf{x}}-{\mathbf{\mu}}_{i})-\frac{\text{Tr}{({% \operatorname{Com}}({\mathbf{\Sigma}}_{i}))}}{\det{({\mathbf{\Sigma}}_{i})}}\right)</annotation><annotation encoding="application/x-llamapun" id="A3.SS2.SSS1.1.p1.8.m1.5d">italic_u start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( bold_x ) = divide start_ARG 1 end_ARG start_ARG 2 end_ARG ( ( bold_x - italic_μ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT ( bold_Σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT - 2 end_POSTSUPERSCRIPT ( bold_x - italic_μ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) - divide start_ARG Tr ( roman_Com ( bold_Σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) ) end_ARG start_ARG roman_det ( bold_Σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) end_ARG )</annotation></semantics></math>. For the score approximation, we get</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A8.EGx24"> <tbody id="A3.Ex36"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\nabla\log p_{\sigma}({\mathbf{x}})=-\frac{\sum_{i=1}^{p}{\pi_{i}% ({\mathbf{\Sigma}}_{i}+\sigma^{2}\mathbf{I}_{d})^{-1}({\mathbf{x}}-{\mathbf{% \mu}}_{i})\mathcal{N}({\mathbf{x}};{\mathbf{\mu}}_{i},{\mathbf{\Sigma}}_{i}+% \sigma^{2}\mathbf{I}_{d})}}{\sum_{i=1}^{p}{\pi_{i}\mathcal{N}({\mathbf{x}};{% \mathbf{\mu}}_{i},{\mathbf{\Sigma}}_{i}+\sigma^{2}\mathbf{I}_{d})}}" class="ltx_Math" display="inline" id="A3.Ex36.m1.9"><semantics id="A3.Ex36.m1.9a"><mrow id="A3.Ex36.m1.9.10" xref="A3.Ex36.m1.9.10.cmml"><mrow id="A3.Ex36.m1.9.10.2" xref="A3.Ex36.m1.9.10.2.cmml"><mrow id="A3.Ex36.m1.9.10.2.2" xref="A3.Ex36.m1.9.10.2.2.cmml"><mrow id="A3.Ex36.m1.9.10.2.2.1" xref="A3.Ex36.m1.9.10.2.2.1.cmml"><mo id="A3.Ex36.m1.9.10.2.2.1.1" rspace="0.167em" xref="A3.Ex36.m1.9.10.2.2.1.1.cmml">∇</mo><mi id="A3.Ex36.m1.9.10.2.2.1.2" xref="A3.Ex36.m1.9.10.2.2.1.2.cmml">log</mi></mrow><mo id="A3.Ex36.m1.9.10.2.2a" lspace="0.167em" xref="A3.Ex36.m1.9.10.2.2.cmml">⁡</mo><msub id="A3.Ex36.m1.9.10.2.2.2" xref="A3.Ex36.m1.9.10.2.2.2.cmml"><mi id="A3.Ex36.m1.9.10.2.2.2.2" xref="A3.Ex36.m1.9.10.2.2.2.2.cmml">p</mi><mi id="A3.Ex36.m1.9.10.2.2.2.3" xref="A3.Ex36.m1.9.10.2.2.2.3.cmml">σ</mi></msub></mrow><mo id="A3.Ex36.m1.9.10.2.1" xref="A3.Ex36.m1.9.10.2.1.cmml">⁢</mo><mrow id="A3.Ex36.m1.9.10.2.3.2" xref="A3.Ex36.m1.9.10.2.cmml"><mo id="A3.Ex36.m1.9.10.2.3.2.1" stretchy="false" xref="A3.Ex36.m1.9.10.2.cmml">(</mo><mi id="A3.Ex36.m1.9.9" xref="A3.Ex36.m1.9.9.cmml">𝐱</mi><mo id="A3.Ex36.m1.9.10.2.3.2.2" stretchy="false" xref="A3.Ex36.m1.9.10.2.cmml">)</mo></mrow></mrow><mo id="A3.Ex36.m1.9.10.1" xref="A3.Ex36.m1.9.10.1.cmml">=</mo><mrow id="A3.Ex36.m1.9.10.3" xref="A3.Ex36.m1.9.10.3.cmml"><mo id="A3.Ex36.m1.9.10.3a" xref="A3.Ex36.m1.9.10.3.cmml">−</mo><mstyle displaystyle="true" id="A3.Ex36.m1.8.8" xref="A3.Ex36.m1.8.8.cmml"><mfrac id="A3.Ex36.m1.8.8a" xref="A3.Ex36.m1.8.8.cmml"><mrow id="A3.Ex36.m1.5.5.5" xref="A3.Ex36.m1.5.5.5.cmml"><msubsup id="A3.Ex36.m1.5.5.5.6" xref="A3.Ex36.m1.5.5.5.6.cmml"><mo id="A3.Ex36.m1.5.5.5.6.2.2" xref="A3.Ex36.m1.5.5.5.6.2.2.cmml">∑</mo><mrow id="A3.Ex36.m1.5.5.5.6.2.3" xref="A3.Ex36.m1.5.5.5.6.2.3.cmml"><mi id="A3.Ex36.m1.5.5.5.6.2.3.2" xref="A3.Ex36.m1.5.5.5.6.2.3.2.cmml">i</mi><mo id="A3.Ex36.m1.5.5.5.6.2.3.1" xref="A3.Ex36.m1.5.5.5.6.2.3.1.cmml">=</mo><mn id="A3.Ex36.m1.5.5.5.6.2.3.3" xref="A3.Ex36.m1.5.5.5.6.2.3.3.cmml">1</mn></mrow><mi id="A3.Ex36.m1.5.5.5.6.3" xref="A3.Ex36.m1.5.5.5.6.3.cmml">p</mi></msubsup><mrow id="A3.Ex36.m1.5.5.5.5" xref="A3.Ex36.m1.5.5.5.5.cmml"><msub id="A3.Ex36.m1.5.5.5.5.6" xref="A3.Ex36.m1.5.5.5.5.6.cmml"><mi id="A3.Ex36.m1.5.5.5.5.6.2" 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xref="A3.Ex36.m1.8.8.8.3.2.2.2.3.3.3">𝑑</ci></apply></apply></apply></list></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.Ex36.m1.9c">\displaystyle\nabla\log p_{\sigma}({\mathbf{x}})=-\frac{\sum_{i=1}^{p}{\pi_{i}% ({\mathbf{\Sigma}}_{i}+\sigma^{2}\mathbf{I}_{d})^{-1}({\mathbf{x}}-{\mathbf{% \mu}}_{i})\mathcal{N}({\mathbf{x}};{\mathbf{\mu}}_{i},{\mathbf{\Sigma}}_{i}+% \sigma^{2}\mathbf{I}_{d})}}{\sum_{i=1}^{p}{\pi_{i}\mathcal{N}({\mathbf{x}};{% \mathbf{\mu}}_{i},{\mathbf{\Sigma}}_{i}+\sigma^{2}\mathbf{I}_{d})}}</annotation><annotation encoding="application/x-llamapun" id="A3.Ex36.m1.9d">∇ roman_log italic_p start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ( bold_x ) = - divide start_ARG ∑ start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT italic_π start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( bold_Σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT + italic_σ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT bold_I start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ( bold_x - italic_μ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) caligraphic_N ( bold_x ; italic_μ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , bold_Σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT + italic_σ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT bold_I start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT ) end_ARG start_ARG ∑ start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT italic_π start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT caligraphic_N ( bold_x ; italic_μ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , bold_Σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT + italic_σ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT bold_I start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT ) end_ARG</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="A3.Ex37"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle=-\frac{\sum_{i=1}^{p}{\pi_{i}({\mathbf{\Sigma}}_{i}^{-1}+\sigma^% {2}{\mathbf{\Sigma}}_{i}^{-2}+\mathcal{O}(\sigma^{4}))({\mathbf{x}}-{\mathbf{% \mu}}_{i})\mathcal{N}({\mathbf{x}};{\mathbf{\mu}}_{i},{\mathbf{\Sigma}}_{i})% \left(1+\sigma^{2}u_{i}({\mathbf{x}})+\mathcal{O}(\sigma^{4})\right)}}{\sum_{i% =1}^{p}{\pi_{i}\mathcal{N}({\mathbf{x}};{\mathbf{\mu}}_{i},{\mathbf{\Sigma}}_{% i})\left(1+\sigma^{2}u_{i}({\mathbf{x}})+\mathcal{O}(\sigma^{4})\right)}}" class="ltx_Math" display="inline" id="A3.Ex37.m1.12"><semantics id="A3.Ex37.m1.12a"><mrow id="A3.Ex37.m1.12.13" xref="A3.Ex37.m1.12.13.cmml"><mi id="A3.Ex37.m1.12.13.2" xref="A3.Ex37.m1.12.13.2.cmml"></mi><mo id="A3.Ex37.m1.12.13.1" xref="A3.Ex37.m1.12.13.1.cmml">=</mo><mrow id="A3.Ex37.m1.12.13.3" xref="A3.Ex37.m1.12.13.3.cmml"><mo 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id="A3.Ex37.m1.7.7.7.7.7.3" xref="A3.Ex37.m1.7.7.7.7.7.3.cmml">i</mi></msub><mo id="A3.Ex37.m1.7.7.7.7.6" xref="A3.Ex37.m1.7.7.7.7.6.cmml">⁢</mo><mrow id="A3.Ex37.m1.3.3.3.3.1.1" xref="A3.Ex37.m1.3.3.3.3.1.1.1.cmml"><mo id="A3.Ex37.m1.3.3.3.3.1.1.2" stretchy="false" xref="A3.Ex37.m1.3.3.3.3.1.1.1.cmml">(</mo><mrow id="A3.Ex37.m1.3.3.3.3.1.1.1" xref="A3.Ex37.m1.3.3.3.3.1.1.1.cmml"><msubsup id="A3.Ex37.m1.3.3.3.3.1.1.1.3" xref="A3.Ex37.m1.3.3.3.3.1.1.1.3.cmml"><mi id="A3.Ex37.m1.3.3.3.3.1.1.1.3.2.2" xref="A3.Ex37.m1.3.3.3.3.1.1.1.3.2.2.cmml">𝚺</mi><mi id="A3.Ex37.m1.3.3.3.3.1.1.1.3.2.3" xref="A3.Ex37.m1.3.3.3.3.1.1.1.3.2.3.cmml">i</mi><mrow id="A3.Ex37.m1.3.3.3.3.1.1.1.3.3" xref="A3.Ex37.m1.3.3.3.3.1.1.1.3.3.cmml"><mo id="A3.Ex37.m1.3.3.3.3.1.1.1.3.3a" xref="A3.Ex37.m1.3.3.3.3.1.1.1.3.3.cmml">−</mo><mn id="A3.Ex37.m1.3.3.3.3.1.1.1.3.3.2" xref="A3.Ex37.m1.3.3.3.3.1.1.1.3.3.2.cmml">1</mn></mrow></msubsup><mo id="A3.Ex37.m1.3.3.3.3.1.1.1.2" xref="A3.Ex37.m1.3.3.3.3.1.1.1.2.cmml">+</mo><mrow id="A3.Ex37.m1.3.3.3.3.1.1.1.4" xref="A3.Ex37.m1.3.3.3.3.1.1.1.4.cmml"><msup id="A3.Ex37.m1.3.3.3.3.1.1.1.4.2" xref="A3.Ex37.m1.3.3.3.3.1.1.1.4.2.cmml"><mi id="A3.Ex37.m1.3.3.3.3.1.1.1.4.2.2" xref="A3.Ex37.m1.3.3.3.3.1.1.1.4.2.2.cmml">σ</mi><mn id="A3.Ex37.m1.3.3.3.3.1.1.1.4.2.3" xref="A3.Ex37.m1.3.3.3.3.1.1.1.4.2.3.cmml">2</mn></msup><mo id="A3.Ex37.m1.3.3.3.3.1.1.1.4.1" xref="A3.Ex37.m1.3.3.3.3.1.1.1.4.1.cmml">⁢</mo><msubsup id="A3.Ex37.m1.3.3.3.3.1.1.1.4.3" xref="A3.Ex37.m1.3.3.3.3.1.1.1.4.3.cmml"><mi id="A3.Ex37.m1.3.3.3.3.1.1.1.4.3.2.2" xref="A3.Ex37.m1.3.3.3.3.1.1.1.4.3.2.2.cmml">𝚺</mi><mi id="A3.Ex37.m1.3.3.3.3.1.1.1.4.3.2.3" xref="A3.Ex37.m1.3.3.3.3.1.1.1.4.3.2.3.cmml">i</mi><mrow id="A3.Ex37.m1.3.3.3.3.1.1.1.4.3.3" xref="A3.Ex37.m1.3.3.3.3.1.1.1.4.3.3.cmml"><mo id="A3.Ex37.m1.3.3.3.3.1.1.1.4.3.3a" xref="A3.Ex37.m1.3.3.3.3.1.1.1.4.3.3.cmml">−</mo><mn id="A3.Ex37.m1.3.3.3.3.1.1.1.4.3.3.2" xref="A3.Ex37.m1.3.3.3.3.1.1.1.4.3.3.2.cmml">2</mn></mrow></msubsup></mrow><mo id="A3.Ex37.m1.3.3.3.3.1.1.1.2a" xref="A3.Ex37.m1.3.3.3.3.1.1.1.2.cmml">+</mo><mrow id="A3.Ex37.m1.3.3.3.3.1.1.1.1" xref="A3.Ex37.m1.3.3.3.3.1.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="A3.Ex37.m1.3.3.3.3.1.1.1.1.3" xref="A3.Ex37.m1.3.3.3.3.1.1.1.1.3.cmml">𝒪</mi><mo id="A3.Ex37.m1.3.3.3.3.1.1.1.1.2" xref="A3.Ex37.m1.3.3.3.3.1.1.1.1.2.cmml">⁢</mo><mrow id="A3.Ex37.m1.3.3.3.3.1.1.1.1.1.1" xref="A3.Ex37.m1.3.3.3.3.1.1.1.1.1.1.1.cmml"><mo id="A3.Ex37.m1.3.3.3.3.1.1.1.1.1.1.2" stretchy="false" xref="A3.Ex37.m1.3.3.3.3.1.1.1.1.1.1.1.cmml">(</mo><msup id="A3.Ex37.m1.3.3.3.3.1.1.1.1.1.1.1" xref="A3.Ex37.m1.3.3.3.3.1.1.1.1.1.1.1.cmml"><mi id="A3.Ex37.m1.3.3.3.3.1.1.1.1.1.1.1.2" xref="A3.Ex37.m1.3.3.3.3.1.1.1.1.1.1.1.2.cmml">σ</mi><mn id="A3.Ex37.m1.3.3.3.3.1.1.1.1.1.1.1.3" xref="A3.Ex37.m1.3.3.3.3.1.1.1.1.1.1.1.3.cmml">4</mn></msup><mo id="A3.Ex37.m1.3.3.3.3.1.1.1.1.1.1.3" stretchy="false" xref="A3.Ex37.m1.3.3.3.3.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><mo id="A3.Ex37.m1.3.3.3.3.1.1.3" stretchy="false" xref="A3.Ex37.m1.3.3.3.3.1.1.1.cmml">)</mo></mrow><mo id="A3.Ex37.m1.7.7.7.7.6a" xref="A3.Ex37.m1.7.7.7.7.6.cmml">⁢</mo><mrow id="A3.Ex37.m1.4.4.4.4.2.1" xref="A3.Ex37.m1.4.4.4.4.2.1.1.cmml"><mo id="A3.Ex37.m1.4.4.4.4.2.1.2" stretchy="false" xref="A3.Ex37.m1.4.4.4.4.2.1.1.cmml">(</mo><mrow id="A3.Ex37.m1.4.4.4.4.2.1.1" xref="A3.Ex37.m1.4.4.4.4.2.1.1.cmml"><mi id="A3.Ex37.m1.4.4.4.4.2.1.1.2" xref="A3.Ex37.m1.4.4.4.4.2.1.1.2.cmml">𝐱</mi><mo id="A3.Ex37.m1.4.4.4.4.2.1.1.1" xref="A3.Ex37.m1.4.4.4.4.2.1.1.1.cmml">−</mo><msub id="A3.Ex37.m1.4.4.4.4.2.1.1.3" xref="A3.Ex37.m1.4.4.4.4.2.1.1.3.cmml"><mi id="A3.Ex37.m1.4.4.4.4.2.1.1.3.2" xref="A3.Ex37.m1.4.4.4.4.2.1.1.3.2.cmml">μ</mi><mi id="A3.Ex37.m1.4.4.4.4.2.1.1.3.3" xref="A3.Ex37.m1.4.4.4.4.2.1.1.3.3.cmml">i</mi></msub></mrow><mo id="A3.Ex37.m1.4.4.4.4.2.1.3" stretchy="false" xref="A3.Ex37.m1.4.4.4.4.2.1.1.cmml">)</mo></mrow><mo id="A3.Ex37.m1.7.7.7.7.6b" xref="A3.Ex37.m1.7.7.7.7.6.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="A3.Ex37.m1.7.7.7.7.8" xref="A3.Ex37.m1.7.7.7.7.8.cmml">𝒩</mi><mo id="A3.Ex37.m1.7.7.7.7.6c" xref="A3.Ex37.m1.7.7.7.7.6.cmml">⁢</mo><mrow id="A3.Ex37.m1.6.6.6.6.4.2" xref="A3.Ex37.m1.6.6.6.6.4.3.cmml"><mo id="A3.Ex37.m1.6.6.6.6.4.2.3" stretchy="false" xref="A3.Ex37.m1.6.6.6.6.4.3.cmml">(</mo><mi id="A3.Ex37.m1.1.1.1.1" xref="A3.Ex37.m1.1.1.1.1.cmml">𝐱</mi><mo id="A3.Ex37.m1.6.6.6.6.4.2.4" xref="A3.Ex37.m1.6.6.6.6.4.3.cmml">;</mo><msub id="A3.Ex37.m1.5.5.5.5.3.1.1" xref="A3.Ex37.m1.5.5.5.5.3.1.1.cmml"><mi id="A3.Ex37.m1.5.5.5.5.3.1.1.2" xref="A3.Ex37.m1.5.5.5.5.3.1.1.2.cmml">μ</mi><mi id="A3.Ex37.m1.5.5.5.5.3.1.1.3" xref="A3.Ex37.m1.5.5.5.5.3.1.1.3.cmml">i</mi></msub><mo id="A3.Ex37.m1.6.6.6.6.4.2.5" xref="A3.Ex37.m1.6.6.6.6.4.3.cmml">,</mo><msub id="A3.Ex37.m1.6.6.6.6.4.2.2" xref="A3.Ex37.m1.6.6.6.6.4.2.2.cmml"><mi id="A3.Ex37.m1.6.6.6.6.4.2.2.2" xref="A3.Ex37.m1.6.6.6.6.4.2.2.2.cmml">𝚺</mi><mi id="A3.Ex37.m1.6.6.6.6.4.2.2.3" xref="A3.Ex37.m1.6.6.6.6.4.2.2.3.cmml">i</mi></msub><mo id="A3.Ex37.m1.6.6.6.6.4.2.6" stretchy="false" xref="A3.Ex37.m1.6.6.6.6.4.3.cmml">)</mo></mrow><mo id="A3.Ex37.m1.7.7.7.7.6d" xref="A3.Ex37.m1.7.7.7.7.6.cmml">⁢</mo><mrow id="A3.Ex37.m1.7.7.7.7.5.1" xref="A3.Ex37.m1.7.7.7.7.5.1.1.cmml"><mo id="A3.Ex37.m1.7.7.7.7.5.1.2" xref="A3.Ex37.m1.7.7.7.7.5.1.1.cmml">(</mo><mrow id="A3.Ex37.m1.7.7.7.7.5.1.1" xref="A3.Ex37.m1.7.7.7.7.5.1.1.cmml"><mn id="A3.Ex37.m1.7.7.7.7.5.1.1.3" xref="A3.Ex37.m1.7.7.7.7.5.1.1.3.cmml">1</mn><mo id="A3.Ex37.m1.7.7.7.7.5.1.1.2" xref="A3.Ex37.m1.7.7.7.7.5.1.1.2.cmml">+</mo><mrow id="A3.Ex37.m1.7.7.7.7.5.1.1.4" xref="A3.Ex37.m1.7.7.7.7.5.1.1.4.cmml"><msup id="A3.Ex37.m1.7.7.7.7.5.1.1.4.2" xref="A3.Ex37.m1.7.7.7.7.5.1.1.4.2.cmml"><mi id="A3.Ex37.m1.7.7.7.7.5.1.1.4.2.2" xref="A3.Ex37.m1.7.7.7.7.5.1.1.4.2.2.cmml">σ</mi><mn id="A3.Ex37.m1.7.7.7.7.5.1.1.4.2.3" xref="A3.Ex37.m1.7.7.7.7.5.1.1.4.2.3.cmml">2</mn></msup><mo id="A3.Ex37.m1.7.7.7.7.5.1.1.4.1" xref="A3.Ex37.m1.7.7.7.7.5.1.1.4.1.cmml">⁢</mo><msub id="A3.Ex37.m1.7.7.7.7.5.1.1.4.3" xref="A3.Ex37.m1.7.7.7.7.5.1.1.4.3.cmml"><mi id="A3.Ex37.m1.7.7.7.7.5.1.1.4.3.2" xref="A3.Ex37.m1.7.7.7.7.5.1.1.4.3.2.cmml">u</mi><mi id="A3.Ex37.m1.7.7.7.7.5.1.1.4.3.3" xref="A3.Ex37.m1.7.7.7.7.5.1.1.4.3.3.cmml">i</mi></msub><mo id="A3.Ex37.m1.7.7.7.7.5.1.1.4.1a" xref="A3.Ex37.m1.7.7.7.7.5.1.1.4.1.cmml">⁢</mo><mrow id="A3.Ex37.m1.7.7.7.7.5.1.1.4.4.2" xref="A3.Ex37.m1.7.7.7.7.5.1.1.4.cmml"><mo id="A3.Ex37.m1.7.7.7.7.5.1.1.4.4.2.1" stretchy="false" xref="A3.Ex37.m1.7.7.7.7.5.1.1.4.cmml">(</mo><mi id="A3.Ex37.m1.2.2.2.2" xref="A3.Ex37.m1.2.2.2.2.cmml">𝐱</mi><mo id="A3.Ex37.m1.7.7.7.7.5.1.1.4.4.2.2" stretchy="false" xref="A3.Ex37.m1.7.7.7.7.5.1.1.4.cmml">)</mo></mrow></mrow><mo id="A3.Ex37.m1.7.7.7.7.5.1.1.2a" xref="A3.Ex37.m1.7.7.7.7.5.1.1.2.cmml">+</mo><mrow id="A3.Ex37.m1.7.7.7.7.5.1.1.1" xref="A3.Ex37.m1.7.7.7.7.5.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="A3.Ex37.m1.7.7.7.7.5.1.1.1.3" xref="A3.Ex37.m1.7.7.7.7.5.1.1.1.3.cmml">𝒪</mi><mo id="A3.Ex37.m1.7.7.7.7.5.1.1.1.2" xref="A3.Ex37.m1.7.7.7.7.5.1.1.1.2.cmml">⁢</mo><mrow id="A3.Ex37.m1.7.7.7.7.5.1.1.1.1.1" xref="A3.Ex37.m1.7.7.7.7.5.1.1.1.1.1.1.cmml"><mo id="A3.Ex37.m1.7.7.7.7.5.1.1.1.1.1.2" stretchy="false" xref="A3.Ex37.m1.7.7.7.7.5.1.1.1.1.1.1.cmml">(</mo><msup id="A3.Ex37.m1.7.7.7.7.5.1.1.1.1.1.1" xref="A3.Ex37.m1.7.7.7.7.5.1.1.1.1.1.1.cmml"><mi id="A3.Ex37.m1.7.7.7.7.5.1.1.1.1.1.1.2" xref="A3.Ex37.m1.7.7.7.7.5.1.1.1.1.1.1.2.cmml">σ</mi><mn id="A3.Ex37.m1.7.7.7.7.5.1.1.1.1.1.1.3" xref="A3.Ex37.m1.7.7.7.7.5.1.1.1.1.1.1.3.cmml">4</mn></msup><mo id="A3.Ex37.m1.7.7.7.7.5.1.1.1.1.1.3" stretchy="false" xref="A3.Ex37.m1.7.7.7.7.5.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><mo id="A3.Ex37.m1.7.7.7.7.5.1.3" xref="A3.Ex37.m1.7.7.7.7.5.1.1.cmml">)</mo></mrow></mrow></mrow><mrow id="A3.Ex37.m1.12.12.12" xref="A3.Ex37.m1.12.12.12.cmml"><msubsup id="A3.Ex37.m1.12.12.12.6" xref="A3.Ex37.m1.12.12.12.6.cmml"><mo id="A3.Ex37.m1.12.12.12.6.2.2" xref="A3.Ex37.m1.12.12.12.6.2.2.cmml">∑</mo><mrow id="A3.Ex37.m1.12.12.12.6.2.3" xref="A3.Ex37.m1.12.12.12.6.2.3.cmml"><mi id="A3.Ex37.m1.12.12.12.6.2.3.2" xref="A3.Ex37.m1.12.12.12.6.2.3.2.cmml">i</mi><mo id="A3.Ex37.m1.12.12.12.6.2.3.1" xref="A3.Ex37.m1.12.12.12.6.2.3.1.cmml">=</mo><mn id="A3.Ex37.m1.12.12.12.6.2.3.3" xref="A3.Ex37.m1.12.12.12.6.2.3.3.cmml">1</mn></mrow><mi id="A3.Ex37.m1.12.12.12.6.3" xref="A3.Ex37.m1.12.12.12.6.3.cmml">p</mi></msubsup><mrow id="A3.Ex37.m1.12.12.12.5" xref="A3.Ex37.m1.12.12.12.5.cmml"><msub id="A3.Ex37.m1.12.12.12.5.5" xref="A3.Ex37.m1.12.12.12.5.5.cmml"><mi id="A3.Ex37.m1.12.12.12.5.5.2" xref="A3.Ex37.m1.12.12.12.5.5.2.cmml">π</mi><mi id="A3.Ex37.m1.12.12.12.5.5.3" xref="A3.Ex37.m1.12.12.12.5.5.3.cmml">i</mi></msub><mo id="A3.Ex37.m1.12.12.12.5.4" xref="A3.Ex37.m1.12.12.12.5.4.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="A3.Ex37.m1.12.12.12.5.6" 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\mu}}_{i})\mathcal{N}({\mathbf{x}};{\mathbf{\mu}}_{i},{\mathbf{\Sigma}}_{i})% \left(1+\sigma^{2}u_{i}({\mathbf{x}})+\mathcal{O}(\sigma^{4})\right)}}{\sum_{i% =1}^{p}{\pi_{i}\mathcal{N}({\mathbf{x}};{\mathbf{\mu}}_{i},{\mathbf{\Sigma}}_{% i})\left(1+\sigma^{2}u_{i}({\mathbf{x}})+\mathcal{O}(\sigma^{4})\right)}}</annotation><annotation encoding="application/x-llamapun" id="A3.Ex37.m1.12d">= - divide start_ARG ∑ start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT italic_π start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( bold_Σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT + italic_σ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT bold_Σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT - 2 end_POSTSUPERSCRIPT + caligraphic_O ( italic_σ start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT ) ) ( bold_x - italic_μ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) caligraphic_N ( bold_x ; italic_μ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , bold_Σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) ( 1 + italic_σ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_u start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( bold_x ) + caligraphic_O ( italic_σ start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT ) ) end_ARG start_ARG ∑ start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT italic_π start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT caligraphic_N ( bold_x ; italic_μ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , bold_Σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) ( 1 + italic_σ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_u start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( bold_x ) + caligraphic_O ( italic_σ start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT ) ) end_ARG</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="A3.Ex38"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle=-\sigma^{2}\left(\frac{\sum_{i=1}^{p}{\pi_{i}\left({\mathbf{% \Sigma}}_{i}^{-1}({\mathbf{x}}-{\mathbf{\mu}}_{i})u_{i}({\mathbf{x}})+{\mathbf% {\Sigma}}_{i}^{-2}({\mathbf{x}}-{\mathbf{\mu}}_{i})\right)\mathcal{N}({\mathbf% {x}};{\mathbf{\mu}}_{i},{\mathbf{\Sigma}}_{i})}}{\sum_{i=1}^{p}{\pi_{i}% \mathcal{N}({\mathbf{x}};{\mathbf{\mu}}_{i},{\mathbf{\Sigma}}_{i})}}\right)" class="ltx_Math" display="inline" id="A3.Ex38.m1.8"><semantics id="A3.Ex38.m1.8a"><mrow id="A3.Ex38.m1.8.9" xref="A3.Ex38.m1.8.9.cmml"><mi id="A3.Ex38.m1.8.9.2" xref="A3.Ex38.m1.8.9.2.cmml"></mi><mo id="A3.Ex38.m1.8.9.1" xref="A3.Ex38.m1.8.9.1.cmml">=</mo><mrow id="A3.Ex38.m1.8.9.3" xref="A3.Ex38.m1.8.9.3.cmml"><mo id="A3.Ex38.m1.8.9.3a" xref="A3.Ex38.m1.8.9.3.cmml">−</mo><mrow id="A3.Ex38.m1.8.9.3.2" xref="A3.Ex38.m1.8.9.3.2.cmml"><msup 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id="A3.Ex38.m1.8.8.8.3.2.2.2.cmml" xref="A3.Ex38.m1.8.8.8.3.2.2.2"><csymbol cd="ambiguous" id="A3.Ex38.m1.8.8.8.3.2.2.2.1.cmml" xref="A3.Ex38.m1.8.8.8.3.2.2.2">subscript</csymbol><ci id="A3.Ex38.m1.8.8.8.3.2.2.2.2.cmml" xref="A3.Ex38.m1.8.8.8.3.2.2.2.2">𝚺</ci><ci id="A3.Ex38.m1.8.8.8.3.2.2.2.3.cmml" xref="A3.Ex38.m1.8.8.8.3.2.2.2.3">𝑖</ci></apply></list></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.Ex38.m1.8c">\displaystyle=-\sigma^{2}\left(\frac{\sum_{i=1}^{p}{\pi_{i}\left({\mathbf{% \Sigma}}_{i}^{-1}({\mathbf{x}}-{\mathbf{\mu}}_{i})u_{i}({\mathbf{x}})+{\mathbf% {\Sigma}}_{i}^{-2}({\mathbf{x}}-{\mathbf{\mu}}_{i})\right)\mathcal{N}({\mathbf% {x}};{\mathbf{\mu}}_{i},{\mathbf{\Sigma}}_{i})}}{\sum_{i=1}^{p}{\pi_{i}% \mathcal{N}({\mathbf{x}};{\mathbf{\mu}}_{i},{\mathbf{\Sigma}}_{i})}}\right)</annotation><annotation encoding="application/x-llamapun" id="A3.Ex38.m1.8d">= - italic_σ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( divide start_ARG ∑ start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT italic_π start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( bold_Σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ( bold_x - italic_μ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) italic_u start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( bold_x ) + bold_Σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT - 2 end_POSTSUPERSCRIPT ( bold_x - italic_μ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) ) caligraphic_N ( bold_x ; italic_μ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , bold_Σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) end_ARG start_ARG ∑ start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT italic_π start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT caligraphic_N ( bold_x ; italic_μ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , bold_Σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) end_ARG )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="A3.Ex39"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\qquad\qquad\qquad-\frac{\nabla p({\mathbf{x}})}{p({\mathbf{x}})% \left(1+\frac{\sigma^{2}}{p({\mathbf{x}})}\sum_{i=1}^{p}{\pi_{i}\mathcal{N}({% \mathbf{x}};{\mathbf{\mu}}_{i},{\mathbf{\Sigma}}_{i})u_{i}({\mathbf{x}})}% \right)}+\mathcal{O}(\sigma^{4})" class="ltx_Math" display="inline" id="A3.Ex39.m1.7"><semantics id="A3.Ex39.m1.7a"><mrow id="A3.Ex39.m1.7.7" xref="A3.Ex39.m1.7.7.cmml"><mrow id="A3.Ex39.m1.7.7.3" xref="A3.Ex39.m1.7.7.3.cmml"><mo id="A3.Ex39.m1.7.7.3a" xref="A3.Ex39.m1.7.7.3.cmml">−</mo><mstyle displaystyle="true" id="A3.Ex39.m1.6.6" xref="A3.Ex39.m1.6.6.cmml"><mfrac id="A3.Ex39.m1.6.6a" xref="A3.Ex39.m1.6.6.cmml"><mrow id="A3.Ex39.m1.1.1.1" xref="A3.Ex39.m1.1.1.1.cmml"><mrow id="A3.Ex39.m1.1.1.1.3" xref="A3.Ex39.m1.1.1.1.3.cmml"><mo id="A3.Ex39.m1.1.1.1.3.1" rspace="0.167em" xref="A3.Ex39.m1.1.1.1.3.1.cmml">∇</mo><mi id="A3.Ex39.m1.1.1.1.3.2" xref="A3.Ex39.m1.1.1.1.3.2.cmml">p</mi></mrow><mo id="A3.Ex39.m1.1.1.1.2" xref="A3.Ex39.m1.1.1.1.2.cmml">⁢</mo><mrow id="A3.Ex39.m1.1.1.1.4.2" xref="A3.Ex39.m1.1.1.1.cmml"><mo id="A3.Ex39.m1.1.1.1.4.2.1" stretchy="false" xref="A3.Ex39.m1.1.1.1.cmml">(</mo><mi id="A3.Ex39.m1.1.1.1.1" xref="A3.Ex39.m1.1.1.1.1.cmml">𝐱</mi><mo id="A3.Ex39.m1.1.1.1.4.2.2" stretchy="false" xref="A3.Ex39.m1.1.1.1.cmml">)</mo></mrow></mrow><mrow id="A3.Ex39.m1.6.6.6" xref="A3.Ex39.m1.6.6.6.cmml"><mi id="A3.Ex39.m1.6.6.6.7" xref="A3.Ex39.m1.6.6.6.7.cmml">p</mi><mo id="A3.Ex39.m1.6.6.6.6" xref="A3.Ex39.m1.6.6.6.6.cmml">⁢</mo><mrow id="A3.Ex39.m1.6.6.6.8.2" xref="A3.Ex39.m1.6.6.6.cmml"><mo id="A3.Ex39.m1.6.6.6.8.2.1" stretchy="false" 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italic_p ( bold_x ) end_ARG ∑ start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT italic_π start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT caligraphic_N ( bold_x ; italic_μ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , bold_Σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) italic_u start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( bold_x ) ) end_ARG + caligraphic_O ( italic_σ start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="A3.Ex40"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle=\nabla\log p({\mathbf{x}})-\sigma^{2}c_{p}({\mathbf{x}})+% \mathcal{O}(\sigma^{4})," class="ltx_Math" display="inline" id="A3.Ex40.m1.3"><semantics id="A3.Ex40.m1.3a"><mrow id="A3.Ex40.m1.3.3.1" 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id="A3.Ex40.m1.3.3.1.1.1.1.2.cmml" xref="A3.Ex40.m1.3.3.1.1.1.1.2"></times><ci id="A3.Ex40.m1.3.3.1.1.1.1.3.cmml" xref="A3.Ex40.m1.3.3.1.1.1.1.3">𝒪</ci><apply id="A3.Ex40.m1.3.3.1.1.1.1.1.1.1.cmml" xref="A3.Ex40.m1.3.3.1.1.1.1.1.1"><csymbol cd="ambiguous" id="A3.Ex40.m1.3.3.1.1.1.1.1.1.1.1.cmml" xref="A3.Ex40.m1.3.3.1.1.1.1.1.1">superscript</csymbol><ci id="A3.Ex40.m1.3.3.1.1.1.1.1.1.1.2.cmml" xref="A3.Ex40.m1.3.3.1.1.1.1.1.1.1.2">𝜎</ci><cn id="A3.Ex40.m1.3.3.1.1.1.1.1.1.1.3.cmml" type="integer" xref="A3.Ex40.m1.3.3.1.1.1.1.1.1.1.3">4</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.Ex40.m1.3c">\displaystyle=\nabla\log p({\mathbf{x}})-\sigma^{2}c_{p}({\mathbf{x}})+% \mathcal{O}(\sigma^{4}),</annotation><annotation encoding="application/x-llamapun" id="A3.Ex40.m1.3d">= ∇ roman_log italic_p ( bold_x ) - italic_σ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_c start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT ( bold_x ) + caligraphic_O ( italic_σ start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT ) ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="A3.SS2.SSS1.1.p1.9">with</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A8.EGx25"> <tbody id="A3.Ex41"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle c_{p}({\mathbf{x}})=\frac{\sum_{i=1}^{p}{\pi_{i}\left({\mathbf{% \Sigma}}_{i}^{-1}({\mathbf{x}}-{\mathbf{\mu}}_{i})u_{i}({\mathbf{x}})+{\mathbf% {\Sigma}}_{i}^{-2}({\mathbf{x}}-{\mathbf{\mu}}_{i})\right)\mathcal{N}({\mathbf% {x}};{\mathbf{\mu}}_{i},{\mathbf{\Sigma}}_{i})}}{p({\mathbf{x}})}-\frac{\nabla p% ({\mathbf{x}})}{p^{2}({\mathbf{x}})}\sum_{i=1}^{p}{\pi_{i}\mathcal{N}({\mathbf% {x}};{\mathbf{\mu}}_{i},{\mathbf{\Sigma}}_{i})u_{i}({\mathbf{x}})}." class="ltx_Math" display="inline" 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xref="A3.Ex41.m1.3.3.3.3.1.1.1.1.1.1.1.cmml">)</mo></mrow><mo id="A3.Ex41.m1.3.3.3.3.1.1.1.1.2a" xref="A3.Ex41.m1.3.3.3.3.1.1.1.1.2.cmml">⁢</mo><msub id="A3.Ex41.m1.3.3.3.3.1.1.1.1.4" xref="A3.Ex41.m1.3.3.3.3.1.1.1.1.4.cmml"><mi id="A3.Ex41.m1.3.3.3.3.1.1.1.1.4.2" xref="A3.Ex41.m1.3.3.3.3.1.1.1.1.4.2.cmml">u</mi><mi id="A3.Ex41.m1.3.3.3.3.1.1.1.1.4.3" xref="A3.Ex41.m1.3.3.3.3.1.1.1.1.4.3.cmml">i</mi></msub><mo id="A3.Ex41.m1.3.3.3.3.1.1.1.1.2b" xref="A3.Ex41.m1.3.3.3.3.1.1.1.1.2.cmml">⁢</mo><mrow id="A3.Ex41.m1.3.3.3.3.1.1.1.1.5.2" xref="A3.Ex41.m1.3.3.3.3.1.1.1.1.cmml"><mo id="A3.Ex41.m1.3.3.3.3.1.1.1.1.5.2.1" stretchy="false" xref="A3.Ex41.m1.3.3.3.3.1.1.1.1.cmml">(</mo><mi id="A3.Ex41.m1.1.1.1.1" xref="A3.Ex41.m1.1.1.1.1.cmml">𝐱</mi><mo id="A3.Ex41.m1.3.3.3.3.1.1.1.1.5.2.2" stretchy="false" xref="A3.Ex41.m1.3.3.3.3.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="A3.Ex41.m1.3.3.3.3.1.1.1.3" xref="A3.Ex41.m1.3.3.3.3.1.1.1.3.cmml">+</mo><mrow id="A3.Ex41.m1.3.3.3.3.1.1.1.2" xref="A3.Ex41.m1.3.3.3.3.1.1.1.2.cmml"><msubsup id="A3.Ex41.m1.3.3.3.3.1.1.1.2.3" xref="A3.Ex41.m1.3.3.3.3.1.1.1.2.3.cmml"><mi id="A3.Ex41.m1.3.3.3.3.1.1.1.2.3.2.2" xref="A3.Ex41.m1.3.3.3.3.1.1.1.2.3.2.2.cmml">𝚺</mi><mi id="A3.Ex41.m1.3.3.3.3.1.1.1.2.3.2.3" xref="A3.Ex41.m1.3.3.3.3.1.1.1.2.3.2.3.cmml">i</mi><mrow id="A3.Ex41.m1.3.3.3.3.1.1.1.2.3.3" xref="A3.Ex41.m1.3.3.3.3.1.1.1.2.3.3.cmml"><mo id="A3.Ex41.m1.3.3.3.3.1.1.1.2.3.3a" xref="A3.Ex41.m1.3.3.3.3.1.1.1.2.3.3.cmml">−</mo><mn id="A3.Ex41.m1.3.3.3.3.1.1.1.2.3.3.2" xref="A3.Ex41.m1.3.3.3.3.1.1.1.2.3.3.2.cmml">2</mn></mrow></msubsup><mo id="A3.Ex41.m1.3.3.3.3.1.1.1.2.2" xref="A3.Ex41.m1.3.3.3.3.1.1.1.2.2.cmml">⁢</mo><mrow id="A3.Ex41.m1.3.3.3.3.1.1.1.2.1.1" xref="A3.Ex41.m1.3.3.3.3.1.1.1.2.1.1.1.cmml"><mo id="A3.Ex41.m1.3.3.3.3.1.1.1.2.1.1.2" stretchy="false" xref="A3.Ex41.m1.3.3.3.3.1.1.1.2.1.1.1.cmml">(</mo><mrow id="A3.Ex41.m1.3.3.3.3.1.1.1.2.1.1.1" xref="A3.Ex41.m1.3.3.3.3.1.1.1.2.1.1.1.cmml"><mi id="A3.Ex41.m1.3.3.3.3.1.1.1.2.1.1.1.2" xref="A3.Ex41.m1.3.3.3.3.1.1.1.2.1.1.1.2.cmml">𝐱</mi><mo id="A3.Ex41.m1.3.3.3.3.1.1.1.2.1.1.1.1" xref="A3.Ex41.m1.3.3.3.3.1.1.1.2.1.1.1.1.cmml">−</mo><msub id="A3.Ex41.m1.3.3.3.3.1.1.1.2.1.1.1.3" xref="A3.Ex41.m1.3.3.3.3.1.1.1.2.1.1.1.3.cmml"><mi id="A3.Ex41.m1.3.3.3.3.1.1.1.2.1.1.1.3.2" xref="A3.Ex41.m1.3.3.3.3.1.1.1.2.1.1.1.3.2.cmml">μ</mi><mi id="A3.Ex41.m1.3.3.3.3.1.1.1.2.1.1.1.3.3" xref="A3.Ex41.m1.3.3.3.3.1.1.1.2.1.1.1.3.3.cmml">i</mi></msub></mrow><mo id="A3.Ex41.m1.3.3.3.3.1.1.1.2.1.1.3" stretchy="false" xref="A3.Ex41.m1.3.3.3.3.1.1.1.2.1.1.1.cmml">)</mo></mrow></mrow></mrow><mo id="A3.Ex41.m1.3.3.3.3.1.1.3" xref="A3.Ex41.m1.3.3.3.3.1.1.1.cmml">)</mo></mrow><mo id="A3.Ex41.m1.5.5.5.5.4a" xref="A3.Ex41.m1.5.5.5.5.4.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="A3.Ex41.m1.5.5.5.5.6" xref="A3.Ex41.m1.5.5.5.5.6.cmml">𝒩</mi><mo id="A3.Ex41.m1.5.5.5.5.4b" xref="A3.Ex41.m1.5.5.5.5.4.cmml">⁢</mo><mrow id="A3.Ex41.m1.5.5.5.5.3.2" xref="A3.Ex41.m1.5.5.5.5.3.3.cmml"><mo id="A3.Ex41.m1.5.5.5.5.3.2.3" stretchy="false" xref="A3.Ex41.m1.5.5.5.5.3.3.cmml">(</mo><mi id="A3.Ex41.m1.2.2.2.2" xref="A3.Ex41.m1.2.2.2.2.cmml">𝐱</mi><mo id="A3.Ex41.m1.5.5.5.5.3.2.4" xref="A3.Ex41.m1.5.5.5.5.3.3.cmml">;</mo><msub id="A3.Ex41.m1.4.4.4.4.2.1.1" xref="A3.Ex41.m1.4.4.4.4.2.1.1.cmml"><mi id="A3.Ex41.m1.4.4.4.4.2.1.1.2" xref="A3.Ex41.m1.4.4.4.4.2.1.1.2.cmml">μ</mi><mi id="A3.Ex41.m1.4.4.4.4.2.1.1.3" xref="A3.Ex41.m1.4.4.4.4.2.1.1.3.cmml">i</mi></msub><mo id="A3.Ex41.m1.5.5.5.5.3.2.5" xref="A3.Ex41.m1.5.5.5.5.3.3.cmml">,</mo><msub id="A3.Ex41.m1.5.5.5.5.3.2.2" xref="A3.Ex41.m1.5.5.5.5.3.2.2.cmml"><mi id="A3.Ex41.m1.5.5.5.5.3.2.2.2" xref="A3.Ex41.m1.5.5.5.5.3.2.2.2.cmml">𝚺</mi><mi id="A3.Ex41.m1.5.5.5.5.3.2.2.3" xref="A3.Ex41.m1.5.5.5.5.3.2.2.3.cmml">i</mi></msub><mo id="A3.Ex41.m1.5.5.5.5.3.2.6" stretchy="false" xref="A3.Ex41.m1.5.5.5.5.3.3.cmml">)</mo></mrow></mrow></mrow><mrow id="A3.Ex41.m1.6.6.6" xref="A3.Ex41.m1.6.6.6.cmml"><mi id="A3.Ex41.m1.6.6.6.3" xref="A3.Ex41.m1.6.6.6.3.cmml">p</mi><mo id="A3.Ex41.m1.6.6.6.2" xref="A3.Ex41.m1.6.6.6.2.cmml">⁢</mo><mrow id="A3.Ex41.m1.6.6.6.4.2" xref="A3.Ex41.m1.6.6.6.cmml"><mo id="A3.Ex41.m1.6.6.6.4.2.1" stretchy="false" xref="A3.Ex41.m1.6.6.6.cmml">(</mo><mi id="A3.Ex41.m1.6.6.6.1" xref="A3.Ex41.m1.6.6.6.1.cmml">𝐱</mi><mo id="A3.Ex41.m1.6.6.6.4.2.2" stretchy="false" xref="A3.Ex41.m1.6.6.6.cmml">)</mo></mrow></mrow></mfrac></mstyle><mo id="A3.Ex41.m1.12.12.1.1.2.3" xref="A3.Ex41.m1.12.12.1.1.2.3.cmml">−</mo><mrow id="A3.Ex41.m1.12.12.1.1.2.2" xref="A3.Ex41.m1.12.12.1.1.2.2.cmml"><mstyle displaystyle="true" id="A3.Ex41.m1.8.8" xref="A3.Ex41.m1.8.8.cmml"><mfrac id="A3.Ex41.m1.8.8a" xref="A3.Ex41.m1.8.8.cmml"><mrow id="A3.Ex41.m1.7.7.1" xref="A3.Ex41.m1.7.7.1.cmml"><mrow id="A3.Ex41.m1.7.7.1.3" xref="A3.Ex41.m1.7.7.1.3.cmml"><mo id="A3.Ex41.m1.7.7.1.3.1" rspace="0.167em" xref="A3.Ex41.m1.7.7.1.3.1.cmml">∇</mo><mi id="A3.Ex41.m1.7.7.1.3.2" xref="A3.Ex41.m1.7.7.1.3.2.cmml">p</mi></mrow><mo id="A3.Ex41.m1.7.7.1.2" xref="A3.Ex41.m1.7.7.1.2.cmml">⁢</mo><mrow id="A3.Ex41.m1.7.7.1.4.2" xref="A3.Ex41.m1.7.7.1.cmml"><mo id="A3.Ex41.m1.7.7.1.4.2.1" stretchy="false" xref="A3.Ex41.m1.7.7.1.cmml">(</mo><mi id="A3.Ex41.m1.7.7.1.1" xref="A3.Ex41.m1.7.7.1.1.cmml">𝐱</mi><mo id="A3.Ex41.m1.7.7.1.4.2.2" stretchy="false" xref="A3.Ex41.m1.7.7.1.cmml">)</mo></mrow></mrow><mrow id="A3.Ex41.m1.8.8.2" xref="A3.Ex41.m1.8.8.2.cmml"><msup id="A3.Ex41.m1.8.8.2.3" xref="A3.Ex41.m1.8.8.2.3.cmml"><mi id="A3.Ex41.m1.8.8.2.3.2" xref="A3.Ex41.m1.8.8.2.3.2.cmml">p</mi><mn id="A3.Ex41.m1.8.8.2.3.3" xref="A3.Ex41.m1.8.8.2.3.3.cmml">2</mn></msup><mo id="A3.Ex41.m1.8.8.2.2" xref="A3.Ex41.m1.8.8.2.2.cmml">⁢</mo><mrow id="A3.Ex41.m1.8.8.2.4.2" xref="A3.Ex41.m1.8.8.2.cmml"><mo id="A3.Ex41.m1.8.8.2.4.2.1" stretchy="false" xref="A3.Ex41.m1.8.8.2.cmml">(</mo><mi id="A3.Ex41.m1.8.8.2.1" xref="A3.Ex41.m1.8.8.2.1.cmml">𝐱</mi><mo id="A3.Ex41.m1.8.8.2.4.2.2" stretchy="false" xref="A3.Ex41.m1.8.8.2.cmml">)</mo></mrow></mrow></mfrac></mstyle><mo id="A3.Ex41.m1.12.12.1.1.2.2.3" xref="A3.Ex41.m1.12.12.1.1.2.2.3.cmml">⁢</mo><mrow id="A3.Ex41.m1.12.12.1.1.2.2.2" xref="A3.Ex41.m1.12.12.1.1.2.2.2.cmml"><mstyle displaystyle="true" id="A3.Ex41.m1.12.12.1.1.2.2.2.3" xref="A3.Ex41.m1.12.12.1.1.2.2.2.3.cmml"><munderover id="A3.Ex41.m1.12.12.1.1.2.2.2.3a" xref="A3.Ex41.m1.12.12.1.1.2.2.2.3.cmml"><mo id="A3.Ex41.m1.12.12.1.1.2.2.2.3.2.2" movablelimits="false" xref="A3.Ex41.m1.12.12.1.1.2.2.2.3.2.2.cmml">∑</mo><mrow id="A3.Ex41.m1.12.12.1.1.2.2.2.3.2.3" xref="A3.Ex41.m1.12.12.1.1.2.2.2.3.2.3.cmml"><mi id="A3.Ex41.m1.12.12.1.1.2.2.2.3.2.3.2" xref="A3.Ex41.m1.12.12.1.1.2.2.2.3.2.3.2.cmml">i</mi><mo id="A3.Ex41.m1.12.12.1.1.2.2.2.3.2.3.1" xref="A3.Ex41.m1.12.12.1.1.2.2.2.3.2.3.1.cmml">=</mo><mn id="A3.Ex41.m1.12.12.1.1.2.2.2.3.2.3.3" xref="A3.Ex41.m1.12.12.1.1.2.2.2.3.2.3.3.cmml">1</mn></mrow><mi id="A3.Ex41.m1.12.12.1.1.2.2.2.3.3" 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{x}};{\mathbf{\mu}}_{i},{\mathbf{\Sigma}}_{i})u_{i}({\mathbf{x}})}.</annotation><annotation encoding="application/x-llamapun" id="A3.Ex41.m1.12d">italic_c start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT ( bold_x ) = divide start_ARG ∑ start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT italic_π start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( bold_Σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ( bold_x - italic_μ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) italic_u start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( bold_x ) + bold_Σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT - 2 end_POSTSUPERSCRIPT ( bold_x - italic_μ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) ) caligraphic_N ( bold_x ; italic_μ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , bold_Σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) end_ARG start_ARG italic_p ( bold_x ) end_ARG - divide start_ARG ∇ italic_p ( bold_x ) end_ARG start_ARG italic_p start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( bold_x ) end_ARG ∑ start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT italic_π start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT caligraphic_N ( bold_x ; italic_μ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , bold_Σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) italic_u start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( bold_x ) .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> </div> <div class="ltx_para" id="A3.SS2.SSS1.2.p2"> <p class="ltx_p" id="A3.SS2.SSS1.2.p2.1">This demonstrates the following inequality</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A8.EGx26"> <tbody id="A3.E19"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right 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xref="A3.E19.m1.4.4.1.1.1.1.1.1.2.cmml">σ</mi><mn id="A3.E19.m1.4.4.1.1.1.1.1.1.3" xref="A3.E19.m1.4.4.1.1.1.1.1.1.3.cmml">4</mn></msup><mo id="A3.E19.m1.4.4.1.1.1.1.1.3" stretchy="false" xref="A3.E19.m1.4.4.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><mo id="A3.E19.m1.4.4.1.2" xref="A3.E19.m1.4.4.1.1.cmml">,</mo></mrow><annotation-xml encoding="MathML-Content" id="A3.E19.m1.4b"><apply id="A3.E19.m1.4.4.1.1.cmml" xref="A3.E19.m1.4.4.1"><plus id="A3.E19.m1.4.4.1.1.2.cmml" xref="A3.E19.m1.4.4.1.1.2"></plus><apply id="A3.E19.m1.4.4.1.1.3.cmml" xref="A3.E19.m1.4.4.1.1.3"><minus id="A3.E19.m1.4.4.1.1.3.1.cmml" xref="A3.E19.m1.4.4.1.1.3.1"></minus><apply id="A3.E19.m1.4.4.1.1.3.2.cmml" xref="A3.E19.m1.4.4.1.1.3.2"><times id="A3.E19.m1.4.4.1.1.3.2.1.cmml" xref="A3.E19.m1.4.4.1.1.3.2.1"></times><apply id="A3.E19.m1.4.4.1.1.3.2.2.cmml" xref="A3.E19.m1.4.4.1.1.3.2.2"><apply id="A3.E19.m1.4.4.1.1.3.2.2.1.cmml" xref="A3.E19.m1.4.4.1.1.3.2.2.1"><ci id="A3.E19.m1.4.4.1.1.3.2.2.1.1.cmml" 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id="A3.E19.m1.4.4.1.1.3.3.2.3.cmml" type="integer" xref="A3.E19.m1.4.4.1.1.3.3.2.3">2</cn></apply><apply id="A3.E19.m1.4.4.1.1.3.3.3.cmml" xref="A3.E19.m1.4.4.1.1.3.3.3"><csymbol cd="ambiguous" id="A3.E19.m1.4.4.1.1.3.3.3.1.cmml" xref="A3.E19.m1.4.4.1.1.3.3.3">subscript</csymbol><ci id="A3.E19.m1.4.4.1.1.3.3.3.2.cmml" xref="A3.E19.m1.4.4.1.1.3.3.3.2">𝑐</ci><ci id="A3.E19.m1.4.4.1.1.3.3.3.3.cmml" xref="A3.E19.m1.4.4.1.1.3.3.3.3">𝑝</ci></apply><ci id="A3.E19.m1.3.3.cmml" xref="A3.E19.m1.3.3">𝐱</ci></apply></apply><apply id="A3.E19.m1.4.4.1.1.1.cmml" xref="A3.E19.m1.4.4.1.1.1"><times id="A3.E19.m1.4.4.1.1.1.2.cmml" xref="A3.E19.m1.4.4.1.1.1.2"></times><ci id="A3.E19.m1.4.4.1.1.1.3.cmml" xref="A3.E19.m1.4.4.1.1.1.3">𝒪</ci><apply id="A3.E19.m1.4.4.1.1.1.1.1.1.cmml" xref="A3.E19.m1.4.4.1.1.1.1.1"><csymbol cd="ambiguous" id="A3.E19.m1.4.4.1.1.1.1.1.1.1.cmml" xref="A3.E19.m1.4.4.1.1.1.1.1">superscript</csymbol><ci id="A3.E19.m1.4.4.1.1.1.1.1.1.2.cmml" xref="A3.E19.m1.4.4.1.1.1.1.1.1.2">𝜎</ci><cn id="A3.E19.m1.4.4.1.1.1.1.1.1.3.cmml" type="integer" xref="A3.E19.m1.4.4.1.1.1.1.1.1.3">4</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.E19.m1.4c">\displaystyle\nabla\log p_{\sigma}({\mathbf{x}})\underset{\sigma\to 0}{=}% \nabla\log p({\mathbf{x}})-\sigma^{2}c_{p}({\mathbf{x}})+\mathcal{O}(\sigma^{4% }),</annotation><annotation encoding="application/x-llamapun" id="A3.E19.m1.4d">∇ roman_log italic_p start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ( bold_x ) start_UNDERACCENT italic_σ → 0 end_UNDERACCENT start_ARG = end_ARG ∇ roman_log italic_p ( bold_x ) - italic_σ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_c start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT ( bold_x ) + caligraphic_O ( italic_σ start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT ) ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(19)</span></td> </tr></tbody> </table> </div> <div class="ltx_para" id="A3.SS2.SSS1.3.p3"> <p class="ltx_p" id="A3.SS2.SSS1.3.p3.6">from which we can deduce a pointwise convergence of <math alttext="\nabla\log p_{\sigma}" class="ltx_Math" display="inline" id="A3.SS2.SSS1.3.p3.1.m1.1"><semantics id="A3.SS2.SSS1.3.p3.1.m1.1a"><mrow id="A3.SS2.SSS1.3.p3.1.m1.1.1" xref="A3.SS2.SSS1.3.p3.1.m1.1.1.cmml"><mrow id="A3.SS2.SSS1.3.p3.1.m1.1.1.1" xref="A3.SS2.SSS1.3.p3.1.m1.1.1.1.cmml"><mo id="A3.SS2.SSS1.3.p3.1.m1.1.1.1.1" rspace="0.167em" xref="A3.SS2.SSS1.3.p3.1.m1.1.1.1.1.cmml">∇</mo><mi id="A3.SS2.SSS1.3.p3.1.m1.1.1.1.2" xref="A3.SS2.SSS1.3.p3.1.m1.1.1.1.2.cmml">log</mi></mrow><mo id="A3.SS2.SSS1.3.p3.1.m1.1.1a" lspace="0.167em" xref="A3.SS2.SSS1.3.p3.1.m1.1.1.cmml">⁡</mo><msub id="A3.SS2.SSS1.3.p3.1.m1.1.1.2" xref="A3.SS2.SSS1.3.p3.1.m1.1.1.2.cmml"><mi id="A3.SS2.SSS1.3.p3.1.m1.1.1.2.2" xref="A3.SS2.SSS1.3.p3.1.m1.1.1.2.2.cmml">p</mi><mi id="A3.SS2.SSS1.3.p3.1.m1.1.1.2.3" xref="A3.SS2.SSS1.3.p3.1.m1.1.1.2.3.cmml">σ</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="A3.SS2.SSS1.3.p3.1.m1.1b"><apply id="A3.SS2.SSS1.3.p3.1.m1.1.1.cmml" xref="A3.SS2.SSS1.3.p3.1.m1.1.1"><apply id="A3.SS2.SSS1.3.p3.1.m1.1.1.1.cmml" xref="A3.SS2.SSS1.3.p3.1.m1.1.1.1"><ci id="A3.SS2.SSS1.3.p3.1.m1.1.1.1.1.cmml" xref="A3.SS2.SSS1.3.p3.1.m1.1.1.1.1">∇</ci><log id="A3.SS2.SSS1.3.p3.1.m1.1.1.1.2.cmml" xref="A3.SS2.SSS1.3.p3.1.m1.1.1.1.2"></log></apply><apply id="A3.SS2.SSS1.3.p3.1.m1.1.1.2.cmml" xref="A3.SS2.SSS1.3.p3.1.m1.1.1.2"><csymbol cd="ambiguous" id="A3.SS2.SSS1.3.p3.1.m1.1.1.2.1.cmml" xref="A3.SS2.SSS1.3.p3.1.m1.1.1.2">subscript</csymbol><ci id="A3.SS2.SSS1.3.p3.1.m1.1.1.2.2.cmml" xref="A3.SS2.SSS1.3.p3.1.m1.1.1.2.2">𝑝</ci><ci id="A3.SS2.SSS1.3.p3.1.m1.1.1.2.3.cmml" xref="A3.SS2.SSS1.3.p3.1.m1.1.1.2.3">𝜎</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.SS2.SSS1.3.p3.1.m1.1c">\nabla\log p_{\sigma}</annotation><annotation encoding="application/x-llamapun" id="A3.SS2.SSS1.3.p3.1.m1.1d">∇ roman_log italic_p start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT</annotation></semantics></math> to <math alttext="\nabla\log p" class="ltx_Math" display="inline" id="A3.SS2.SSS1.3.p3.2.m2.1"><semantics id="A3.SS2.SSS1.3.p3.2.m2.1a"><mrow id="A3.SS2.SSS1.3.p3.2.m2.1.1" xref="A3.SS2.SSS1.3.p3.2.m2.1.1.cmml"><mrow id="A3.SS2.SSS1.3.p3.2.m2.1.1.1" xref="A3.SS2.SSS1.3.p3.2.m2.1.1.1.cmml"><mo id="A3.SS2.SSS1.3.p3.2.m2.1.1.1.1" rspace="0.167em" xref="A3.SS2.SSS1.3.p3.2.m2.1.1.1.1.cmml">∇</mo><mi id="A3.SS2.SSS1.3.p3.2.m2.1.1.1.2" xref="A3.SS2.SSS1.3.p3.2.m2.1.1.1.2.cmml">log</mi></mrow><mo id="A3.SS2.SSS1.3.p3.2.m2.1.1a" lspace="0.167em" xref="A3.SS2.SSS1.3.p3.2.m2.1.1.cmml">⁡</mo><mi id="A3.SS2.SSS1.3.p3.2.m2.1.1.2" xref="A3.SS2.SSS1.3.p3.2.m2.1.1.2.cmml">p</mi></mrow><annotation-xml encoding="MathML-Content" id="A3.SS2.SSS1.3.p3.2.m2.1b"><apply id="A3.SS2.SSS1.3.p3.2.m2.1.1.cmml" xref="A3.SS2.SSS1.3.p3.2.m2.1.1"><apply id="A3.SS2.SSS1.3.p3.2.m2.1.1.1.cmml" xref="A3.SS2.SSS1.3.p3.2.m2.1.1.1"><ci id="A3.SS2.SSS1.3.p3.2.m2.1.1.1.1.cmml" xref="A3.SS2.SSS1.3.p3.2.m2.1.1.1.1">∇</ci><log id="A3.SS2.SSS1.3.p3.2.m2.1.1.1.2.cmml" xref="A3.SS2.SSS1.3.p3.2.m2.1.1.1.2"></log></apply><ci id="A3.SS2.SSS1.3.p3.2.m2.1.1.2.cmml" xref="A3.SS2.SSS1.3.p3.2.m2.1.1.2">𝑝</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.SS2.SSS1.3.p3.2.m2.1c">\nabla\log p</annotation><annotation encoding="application/x-llamapun" id="A3.SS2.SSS1.3.p3.2.m2.1d">∇ roman_log italic_p</annotation></semantics></math> with speed <math alttext="\sigma^{2}" class="ltx_Math" display="inline" id="A3.SS2.SSS1.3.p3.3.m3.1"><semantics id="A3.SS2.SSS1.3.p3.3.m3.1a"><msup id="A3.SS2.SSS1.3.p3.3.m3.1.1" xref="A3.SS2.SSS1.3.p3.3.m3.1.1.cmml"><mi id="A3.SS2.SSS1.3.p3.3.m3.1.1.2" xref="A3.SS2.SSS1.3.p3.3.m3.1.1.2.cmml">σ</mi><mn id="A3.SS2.SSS1.3.p3.3.m3.1.1.3" xref="A3.SS2.SSS1.3.p3.3.m3.1.1.3.cmml">2</mn></msup><annotation-xml encoding="MathML-Content" id="A3.SS2.SSS1.3.p3.3.m3.1b"><apply id="A3.SS2.SSS1.3.p3.3.m3.1.1.cmml" xref="A3.SS2.SSS1.3.p3.3.m3.1.1"><csymbol cd="ambiguous" id="A3.SS2.SSS1.3.p3.3.m3.1.1.1.cmml" xref="A3.SS2.SSS1.3.p3.3.m3.1.1">superscript</csymbol><ci id="A3.SS2.SSS1.3.p3.3.m3.1.1.2.cmml" xref="A3.SS2.SSS1.3.p3.3.m3.1.1.2">𝜎</ci><cn id="A3.SS2.SSS1.3.p3.3.m3.1.1.3.cmml" type="integer" xref="A3.SS2.SSS1.3.p3.3.m3.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.SS2.SSS1.3.p3.3.m3.1c">\sigma^{2}</annotation><annotation encoding="application/x-llamapun" id="A3.SS2.SSS1.3.p3.3.m3.1d">italic_σ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT</annotation></semantics></math>. Indeed, there exist <math alttext="{c_{p}:{\mathbb{R}}^{d}\mapsto{\mathbb{R}}}" class="ltx_Math" display="inline" id="A3.SS2.SSS1.3.p3.4.m4.1"><semantics id="A3.SS2.SSS1.3.p3.4.m4.1a"><mrow id="A3.SS2.SSS1.3.p3.4.m4.1.1" xref="A3.SS2.SSS1.3.p3.4.m4.1.1.cmml"><msub id="A3.SS2.SSS1.3.p3.4.m4.1.1.2" xref="A3.SS2.SSS1.3.p3.4.m4.1.1.2.cmml"><mi id="A3.SS2.SSS1.3.p3.4.m4.1.1.2.2" xref="A3.SS2.SSS1.3.p3.4.m4.1.1.2.2.cmml">c</mi><mi id="A3.SS2.SSS1.3.p3.4.m4.1.1.2.3" xref="A3.SS2.SSS1.3.p3.4.m4.1.1.2.3.cmml">p</mi></msub><mo id="A3.SS2.SSS1.3.p3.4.m4.1.1.1" lspace="0.278em" rspace="0.278em" xref="A3.SS2.SSS1.3.p3.4.m4.1.1.1.cmml">:</mo><mrow id="A3.SS2.SSS1.3.p3.4.m4.1.1.3" xref="A3.SS2.SSS1.3.p3.4.m4.1.1.3.cmml"><msup id="A3.SS2.SSS1.3.p3.4.m4.1.1.3.2" xref="A3.SS2.SSS1.3.p3.4.m4.1.1.3.2.cmml"><mi id="A3.SS2.SSS1.3.p3.4.m4.1.1.3.2.2" xref="A3.SS2.SSS1.3.p3.4.m4.1.1.3.2.2.cmml">ℝ</mi><mi id="A3.SS2.SSS1.3.p3.4.m4.1.1.3.2.3" xref="A3.SS2.SSS1.3.p3.4.m4.1.1.3.2.3.cmml">d</mi></msup><mo id="A3.SS2.SSS1.3.p3.4.m4.1.1.3.1" stretchy="false" xref="A3.SS2.SSS1.3.p3.4.m4.1.1.3.1.cmml">↦</mo><mi id="A3.SS2.SSS1.3.p3.4.m4.1.1.3.3" xref="A3.SS2.SSS1.3.p3.4.m4.1.1.3.3.cmml">ℝ</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="A3.SS2.SSS1.3.p3.4.m4.1b"><apply id="A3.SS2.SSS1.3.p3.4.m4.1.1.cmml" xref="A3.SS2.SSS1.3.p3.4.m4.1.1"><ci id="A3.SS2.SSS1.3.p3.4.m4.1.1.1.cmml" xref="A3.SS2.SSS1.3.p3.4.m4.1.1.1">:</ci><apply id="A3.SS2.SSS1.3.p3.4.m4.1.1.2.cmml" xref="A3.SS2.SSS1.3.p3.4.m4.1.1.2"><csymbol cd="ambiguous" id="A3.SS2.SSS1.3.p3.4.m4.1.1.2.1.cmml" xref="A3.SS2.SSS1.3.p3.4.m4.1.1.2">subscript</csymbol><ci id="A3.SS2.SSS1.3.p3.4.m4.1.1.2.2.cmml" xref="A3.SS2.SSS1.3.p3.4.m4.1.1.2.2">𝑐</ci><ci id="A3.SS2.SSS1.3.p3.4.m4.1.1.2.3.cmml" xref="A3.SS2.SSS1.3.p3.4.m4.1.1.2.3">𝑝</ci></apply><apply id="A3.SS2.SSS1.3.p3.4.m4.1.1.3.cmml" xref="A3.SS2.SSS1.3.p3.4.m4.1.1.3"><csymbol cd="latexml" id="A3.SS2.SSS1.3.p3.4.m4.1.1.3.1.cmml" xref="A3.SS2.SSS1.3.p3.4.m4.1.1.3.1">maps-to</csymbol><apply id="A3.SS2.SSS1.3.p3.4.m4.1.1.3.2.cmml" xref="A3.SS2.SSS1.3.p3.4.m4.1.1.3.2"><csymbol cd="ambiguous" id="A3.SS2.SSS1.3.p3.4.m4.1.1.3.2.1.cmml" xref="A3.SS2.SSS1.3.p3.4.m4.1.1.3.2">superscript</csymbol><ci id="A3.SS2.SSS1.3.p3.4.m4.1.1.3.2.2.cmml" xref="A3.SS2.SSS1.3.p3.4.m4.1.1.3.2.2">ℝ</ci><ci id="A3.SS2.SSS1.3.p3.4.m4.1.1.3.2.3.cmml" xref="A3.SS2.SSS1.3.p3.4.m4.1.1.3.2.3">𝑑</ci></apply><ci id="A3.SS2.SSS1.3.p3.4.m4.1.1.3.3.cmml" xref="A3.SS2.SSS1.3.p3.4.m4.1.1.3.3">ℝ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.SS2.SSS1.3.p3.4.m4.1c">{c_{p}:{\mathbb{R}}^{d}\mapsto{\mathbb{R}}}</annotation><annotation encoding="application/x-llamapun" id="A3.SS2.SSS1.3.p3.4.m4.1d">italic_c start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT : blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT ↦ blackboard_R</annotation></semantics></math> and <math alttext="\sigma_{0}>0" class="ltx_Math" display="inline" id="A3.SS2.SSS1.3.p3.5.m5.1"><semantics id="A3.SS2.SSS1.3.p3.5.m5.1a"><mrow id="A3.SS2.SSS1.3.p3.5.m5.1.1" xref="A3.SS2.SSS1.3.p3.5.m5.1.1.cmml"><msub id="A3.SS2.SSS1.3.p3.5.m5.1.1.2" xref="A3.SS2.SSS1.3.p3.5.m5.1.1.2.cmml"><mi id="A3.SS2.SSS1.3.p3.5.m5.1.1.2.2" xref="A3.SS2.SSS1.3.p3.5.m5.1.1.2.2.cmml">σ</mi><mn id="A3.SS2.SSS1.3.p3.5.m5.1.1.2.3" xref="A3.SS2.SSS1.3.p3.5.m5.1.1.2.3.cmml">0</mn></msub><mo id="A3.SS2.SSS1.3.p3.5.m5.1.1.1" xref="A3.SS2.SSS1.3.p3.5.m5.1.1.1.cmml">></mo><mn id="A3.SS2.SSS1.3.p3.5.m5.1.1.3" xref="A3.SS2.SSS1.3.p3.5.m5.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="A3.SS2.SSS1.3.p3.5.m5.1b"><apply id="A3.SS2.SSS1.3.p3.5.m5.1.1.cmml" xref="A3.SS2.SSS1.3.p3.5.m5.1.1"><gt id="A3.SS2.SSS1.3.p3.5.m5.1.1.1.cmml" xref="A3.SS2.SSS1.3.p3.5.m5.1.1.1"></gt><apply id="A3.SS2.SSS1.3.p3.5.m5.1.1.2.cmml" xref="A3.SS2.SSS1.3.p3.5.m5.1.1.2"><csymbol cd="ambiguous" id="A3.SS2.SSS1.3.p3.5.m5.1.1.2.1.cmml" xref="A3.SS2.SSS1.3.p3.5.m5.1.1.2">subscript</csymbol><ci id="A3.SS2.SSS1.3.p3.5.m5.1.1.2.2.cmml" xref="A3.SS2.SSS1.3.p3.5.m5.1.1.2.2">𝜎</ci><cn id="A3.SS2.SSS1.3.p3.5.m5.1.1.2.3.cmml" type="integer" xref="A3.SS2.SSS1.3.p3.5.m5.1.1.2.3">0</cn></apply><cn id="A3.SS2.SSS1.3.p3.5.m5.1.1.3.cmml" type="integer" xref="A3.SS2.SSS1.3.p3.5.m5.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.SS2.SSS1.3.p3.5.m5.1c">\sigma_{0}>0</annotation><annotation encoding="application/x-llamapun" id="A3.SS2.SSS1.3.p3.5.m5.1d">italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT > 0</annotation></semantics></math>, such that for <math alttext="\sigma\in]0,\sigma_{0}]" class="ltx_math_unparsed" display="inline" id="A3.SS2.SSS1.3.p3.6.m6.1"><semantics id="A3.SS2.SSS1.3.p3.6.m6.1a"><mrow id="A3.SS2.SSS1.3.p3.6.m6.1b"><mrow id="A3.SS2.SSS1.3.p3.6.m6.1.1"><mi id="A3.SS2.SSS1.3.p3.6.m6.1.1.1">σ</mi><mo id="A3.SS2.SSS1.3.p3.6.m6.1.1.2" rspace="0em">∈</mo><mo id="A3.SS2.SSS1.3.p3.6.m6.1.1.3" stretchy="false">]</mo></mrow><mn id="A3.SS2.SSS1.3.p3.6.m6.1.2">0</mn><mo id="A3.SS2.SSS1.3.p3.6.m6.1.3">,</mo><msub id="A3.SS2.SSS1.3.p3.6.m6.1.4"><mi id="A3.SS2.SSS1.3.p3.6.m6.1.4.2">σ</mi><mn id="A3.SS2.SSS1.3.p3.6.m6.1.4.3">0</mn></msub><mo id="A3.SS2.SSS1.3.p3.6.m6.1.5" stretchy="false">]</mo></mrow><annotation encoding="application/x-tex" id="A3.SS2.SSS1.3.p3.6.m6.1c">\sigma\in]0,\sigma_{0}]</annotation><annotation encoding="application/x-llamapun" id="A3.SS2.SSS1.3.p3.6.m6.1d">italic_σ ∈ ] 0 , italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ]</annotation></semantics></math>,</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A8.EGx27"> <tbody id="A3.E20"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\|\nabla\log p_{\sigma}({\mathbf{x}})-\nabla\log p({\mathbf{x}})% \|\leq 2\sigma^{2}\|c_{p}({\mathbf{x}})\|," class="ltx_Math" display="inline" id="A3.E20.m1.4"><semantics id="A3.E20.m1.4a"><mrow id="A3.E20.m1.4.4.1" xref="A3.E20.m1.4.4.1.1.cmml"><mrow id="A3.E20.m1.4.4.1.1" xref="A3.E20.m1.4.4.1.1.cmml"><mrow id="A3.E20.m1.4.4.1.1.1.1" xref="A3.E20.m1.4.4.1.1.1.2.cmml"><mo id="A3.E20.m1.4.4.1.1.1.1.2" stretchy="false" xref="A3.E20.m1.4.4.1.1.1.2.1.cmml">‖</mo><mrow id="A3.E20.m1.4.4.1.1.1.1.1" xref="A3.E20.m1.4.4.1.1.1.1.1.cmml"><mrow id="A3.E20.m1.4.4.1.1.1.1.1.2" xref="A3.E20.m1.4.4.1.1.1.1.1.2.cmml"><mrow id="A3.E20.m1.4.4.1.1.1.1.1.2.2" xref="A3.E20.m1.4.4.1.1.1.1.1.2.2.cmml"><mrow id="A3.E20.m1.4.4.1.1.1.1.1.2.2.1" xref="A3.E20.m1.4.4.1.1.1.1.1.2.2.1.cmml"><mo id="A3.E20.m1.4.4.1.1.1.1.1.2.2.1.1" rspace="0.167em" xref="A3.E20.m1.4.4.1.1.1.1.1.2.2.1.1.cmml">∇</mo><mi id="A3.E20.m1.4.4.1.1.1.1.1.2.2.1.2" xref="A3.E20.m1.4.4.1.1.1.1.1.2.2.1.2.cmml">log</mi></mrow><mo id="A3.E20.m1.4.4.1.1.1.1.1.2.2a" lspace="0.167em" xref="A3.E20.m1.4.4.1.1.1.1.1.2.2.cmml">⁡</mo><msub id="A3.E20.m1.4.4.1.1.1.1.1.2.2.2" xref="A3.E20.m1.4.4.1.1.1.1.1.2.2.2.cmml"><mi id="A3.E20.m1.4.4.1.1.1.1.1.2.2.2.2" xref="A3.E20.m1.4.4.1.1.1.1.1.2.2.2.2.cmml">p</mi><mi id="A3.E20.m1.4.4.1.1.1.1.1.2.2.2.3" xref="A3.E20.m1.4.4.1.1.1.1.1.2.2.2.3.cmml">σ</mi></msub></mrow><mo id="A3.E20.m1.4.4.1.1.1.1.1.2.1" xref="A3.E20.m1.4.4.1.1.1.1.1.2.1.cmml">⁢</mo><mrow id="A3.E20.m1.4.4.1.1.1.1.1.2.3.2" xref="A3.E20.m1.4.4.1.1.1.1.1.2.cmml"><mo id="A3.E20.m1.4.4.1.1.1.1.1.2.3.2.1" stretchy="false" xref="A3.E20.m1.4.4.1.1.1.1.1.2.cmml">(</mo><mi id="A3.E20.m1.1.1" xref="A3.E20.m1.1.1.cmml">𝐱</mi><mo id="A3.E20.m1.4.4.1.1.1.1.1.2.3.2.2" stretchy="false" xref="A3.E20.m1.4.4.1.1.1.1.1.2.cmml">)</mo></mrow></mrow><mo id="A3.E20.m1.4.4.1.1.1.1.1.1" xref="A3.E20.m1.4.4.1.1.1.1.1.1.cmml">−</mo><mrow id="A3.E20.m1.4.4.1.1.1.1.1.3" xref="A3.E20.m1.4.4.1.1.1.1.1.3.cmml"><mrow id="A3.E20.m1.4.4.1.1.1.1.1.3.2" xref="A3.E20.m1.4.4.1.1.1.1.1.3.2.cmml"><mrow 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,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(20)</span></td> </tr></tbody> </table> </div> <div class="ltx_para" id="A3.SS2.SSS1.4.p4"> <p class="ltx_p" id="A3.SS2.SSS1.4.p4.1">Next, one can remark that <math alttext="\forall j\in[1,p]" class="ltx_Math" display="inline" id="A3.SS2.SSS1.4.p4.1.m1.2"><semantics id="A3.SS2.SSS1.4.p4.1.m1.2a"><mrow id="A3.SS2.SSS1.4.p4.1.m1.2.3" xref="A3.SS2.SSS1.4.p4.1.m1.2.3.cmml"><mrow id="A3.SS2.SSS1.4.p4.1.m1.2.3.2" xref="A3.SS2.SSS1.4.p4.1.m1.2.3.2.cmml"><mo id="A3.SS2.SSS1.4.p4.1.m1.2.3.2.1" rspace="0.167em" xref="A3.SS2.SSS1.4.p4.1.m1.2.3.2.1.cmml">∀</mo><mi id="A3.SS2.SSS1.4.p4.1.m1.2.3.2.2" xref="A3.SS2.SSS1.4.p4.1.m1.2.3.2.2.cmml">j</mi></mrow><mo id="A3.SS2.SSS1.4.p4.1.m1.2.3.1" xref="A3.SS2.SSS1.4.p4.1.m1.2.3.1.cmml">∈</mo><mrow id="A3.SS2.SSS1.4.p4.1.m1.2.3.3.2" xref="A3.SS2.SSS1.4.p4.1.m1.2.3.3.1.cmml"><mo id="A3.SS2.SSS1.4.p4.1.m1.2.3.3.2.1" stretchy="false" xref="A3.SS2.SSS1.4.p4.1.m1.2.3.3.1.cmml">[</mo><mn id="A3.SS2.SSS1.4.p4.1.m1.1.1" xref="A3.SS2.SSS1.4.p4.1.m1.1.1.cmml">1</mn><mo id="A3.SS2.SSS1.4.p4.1.m1.2.3.3.2.2" xref="A3.SS2.SSS1.4.p4.1.m1.2.3.3.1.cmml">,</mo><mi id="A3.SS2.SSS1.4.p4.1.m1.2.2" xref="A3.SS2.SSS1.4.p4.1.m1.2.2.cmml">p</mi><mo id="A3.SS2.SSS1.4.p4.1.m1.2.3.3.2.3" stretchy="false" xref="A3.SS2.SSS1.4.p4.1.m1.2.3.3.1.cmml">]</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A3.SS2.SSS1.4.p4.1.m1.2b"><apply id="A3.SS2.SSS1.4.p4.1.m1.2.3.cmml" xref="A3.SS2.SSS1.4.p4.1.m1.2.3"><in id="A3.SS2.SSS1.4.p4.1.m1.2.3.1.cmml" xref="A3.SS2.SSS1.4.p4.1.m1.2.3.1"></in><apply id="A3.SS2.SSS1.4.p4.1.m1.2.3.2.cmml" xref="A3.SS2.SSS1.4.p4.1.m1.2.3.2"><csymbol cd="latexml" id="A3.SS2.SSS1.4.p4.1.m1.2.3.2.1.cmml" xref="A3.SS2.SSS1.4.p4.1.m1.2.3.2.1">for-all</csymbol><ci id="A3.SS2.SSS1.4.p4.1.m1.2.3.2.2.cmml" xref="A3.SS2.SSS1.4.p4.1.m1.2.3.2.2">𝑗</ci></apply><interval closure="closed" id="A3.SS2.SSS1.4.p4.1.m1.2.3.3.1.cmml" xref="A3.SS2.SSS1.4.p4.1.m1.2.3.3.2"><cn id="A3.SS2.SSS1.4.p4.1.m1.1.1.cmml" type="integer" xref="A3.SS2.SSS1.4.p4.1.m1.1.1">1</cn><ci id="A3.SS2.SSS1.4.p4.1.m1.2.2.cmml" xref="A3.SS2.SSS1.4.p4.1.m1.2.2">𝑝</ci></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.SS2.SSS1.4.p4.1.m1.2c">\forall j\in[1,p]</annotation><annotation encoding="application/x-llamapun" id="A3.SS2.SSS1.4.p4.1.m1.2d">∀ italic_j ∈ [ 1 , italic_p ]</annotation></semantics></math>,</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A8.EGx28"> <tbody id="A3.Ex42"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\frac{\pi_{j}\mathcal{N}({\mathbf{x}};{\mathbf{\mu}}_{j},{\mathbf% {\Sigma}}_{j})}{p({\mathbf{x}})}=\frac{\pi_{j}\mathcal{N}({\mathbf{x}};{% \mathbf{\mu}}_{j},{\mathbf{\Sigma}}_{j})}{\sum_{i=1}^{p}{\pi_{i}\mathcal{N}({% \mathbf{x}};{\mathbf{\mu}}_{i},{\mathbf{\Sigma}}_{i})}}=\frac{1}{1+\frac{\sum_% {i\neq j}{\pi_{i}\mathcal{N}({\mathbf{x}};{\mathbf{\mu}}_{i},{\mathbf{\Sigma}}% _{i})}}{\pi_{j}\mathcal{N}({\mathbf{x}};{\mathbf{\mu}}_{j},{\mathbf{\Sigma}}_{% j})}}\leq 1." class="ltx_Math" display="inline" id="A3.Ex42.m1.17"><semantics id="A3.Ex42.m1.17a"><mrow id="A3.Ex42.m1.17.17.1" xref="A3.Ex42.m1.17.17.1.1.cmml"><mrow id="A3.Ex42.m1.17.17.1.1" xref="A3.Ex42.m1.17.17.1.1.cmml"><mstyle displaystyle="true" id="A3.Ex42.m1.4.4" xref="A3.Ex42.m1.4.4.cmml"><mfrac id="A3.Ex42.m1.4.4a" xref="A3.Ex42.m1.4.4.cmml"><mrow id="A3.Ex42.m1.3.3.3" xref="A3.Ex42.m1.3.3.3.cmml"><msub id="A3.Ex42.m1.3.3.3.5" xref="A3.Ex42.m1.3.3.3.5.cmml"><mi id="A3.Ex42.m1.3.3.3.5.2" xref="A3.Ex42.m1.3.3.3.5.2.cmml">π</mi><mi id="A3.Ex42.m1.3.3.3.5.3" xref="A3.Ex42.m1.3.3.3.5.3.cmml">j</mi></msub><mo id="A3.Ex42.m1.3.3.3.4" xref="A3.Ex42.m1.3.3.3.4.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="A3.Ex42.m1.3.3.3.6" xref="A3.Ex42.m1.3.3.3.6.cmml">𝒩</mi><mo id="A3.Ex42.m1.3.3.3.4a" xref="A3.Ex42.m1.3.3.3.4.cmml">⁢</mo><mrow id="A3.Ex42.m1.3.3.3.3.2" xref="A3.Ex42.m1.3.3.3.3.3.cmml"><mo id="A3.Ex42.m1.3.3.3.3.2.3" stretchy="false" xref="A3.Ex42.m1.3.3.3.3.3.cmml">(</mo><mi id="A3.Ex42.m1.1.1.1.1" xref="A3.Ex42.m1.1.1.1.1.cmml">𝐱</mi><mo id="A3.Ex42.m1.3.3.3.3.2.4" xref="A3.Ex42.m1.3.3.3.3.3.cmml">;</mo><msub id="A3.Ex42.m1.2.2.2.2.1.1" xref="A3.Ex42.m1.2.2.2.2.1.1.cmml"><mi id="A3.Ex42.m1.2.2.2.2.1.1.2" xref="A3.Ex42.m1.2.2.2.2.1.1.2.cmml">μ</mi><mi id="A3.Ex42.m1.2.2.2.2.1.1.3" xref="A3.Ex42.m1.2.2.2.2.1.1.3.cmml">j</mi></msub><mo id="A3.Ex42.m1.3.3.3.3.2.5" xref="A3.Ex42.m1.3.3.3.3.3.cmml">,</mo><msub id="A3.Ex42.m1.3.3.3.3.2.2" xref="A3.Ex42.m1.3.3.3.3.2.2.cmml"><mi id="A3.Ex42.m1.3.3.3.3.2.2.2" xref="A3.Ex42.m1.3.3.3.3.2.2.2.cmml">𝚺</mi><mi id="A3.Ex42.m1.3.3.3.3.2.2.3" xref="A3.Ex42.m1.3.3.3.3.2.2.3.cmml">j</mi></msub><mo id="A3.Ex42.m1.3.3.3.3.2.6" stretchy="false" xref="A3.Ex42.m1.3.3.3.3.3.cmml">)</mo></mrow></mrow><mrow id="A3.Ex42.m1.4.4.4" xref="A3.Ex42.m1.4.4.4.cmml"><mi id="A3.Ex42.m1.4.4.4.3" xref="A3.Ex42.m1.4.4.4.3.cmml">p</mi><mo id="A3.Ex42.m1.4.4.4.2" xref="A3.Ex42.m1.4.4.4.2.cmml">⁢</mo><mrow id="A3.Ex42.m1.4.4.4.4.2" xref="A3.Ex42.m1.4.4.4.cmml"><mo id="A3.Ex42.m1.4.4.4.4.2.1" stretchy="false" xref="A3.Ex42.m1.4.4.4.cmml">(</mo><mi id="A3.Ex42.m1.4.4.4.1" xref="A3.Ex42.m1.4.4.4.1.cmml">𝐱</mi><mo id="A3.Ex42.m1.4.4.4.4.2.2" stretchy="false" xref="A3.Ex42.m1.4.4.4.cmml">)</mo></mrow></mrow></mfrac></mstyle><mo id="A3.Ex42.m1.17.17.1.1.2" xref="A3.Ex42.m1.17.17.1.1.2.cmml">=</mo><mstyle displaystyle="true" id="A3.Ex42.m1.10.10" xref="A3.Ex42.m1.10.10.cmml"><mfrac id="A3.Ex42.m1.10.10a" xref="A3.Ex42.m1.10.10.cmml"><mrow id="A3.Ex42.m1.7.7.3" xref="A3.Ex42.m1.7.7.3.cmml"><msub id="A3.Ex42.m1.7.7.3.5" xref="A3.Ex42.m1.7.7.3.5.cmml"><mi id="A3.Ex42.m1.7.7.3.5.2" xref="A3.Ex42.m1.7.7.3.5.2.cmml">π</mi><mi id="A3.Ex42.m1.7.7.3.5.3" xref="A3.Ex42.m1.7.7.3.5.3.cmml">j</mi></msub><mo id="A3.Ex42.m1.7.7.3.4" xref="A3.Ex42.m1.7.7.3.4.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="A3.Ex42.m1.7.7.3.6" xref="A3.Ex42.m1.7.7.3.6.cmml">𝒩</mi><mo id="A3.Ex42.m1.7.7.3.4a" xref="A3.Ex42.m1.7.7.3.4.cmml">⁢</mo><mrow id="A3.Ex42.m1.7.7.3.3.2" xref="A3.Ex42.m1.7.7.3.3.3.cmml"><mo id="A3.Ex42.m1.7.7.3.3.2.3" stretchy="false" xref="A3.Ex42.m1.7.7.3.3.3.cmml">(</mo><mi id="A3.Ex42.m1.5.5.1.1" xref="A3.Ex42.m1.5.5.1.1.cmml">𝐱</mi><mo id="A3.Ex42.m1.7.7.3.3.2.4" xref="A3.Ex42.m1.7.7.3.3.3.cmml">;</mo><msub id="A3.Ex42.m1.6.6.2.2.1.1" xref="A3.Ex42.m1.6.6.2.2.1.1.cmml"><mi id="A3.Ex42.m1.6.6.2.2.1.1.2" xref="A3.Ex42.m1.6.6.2.2.1.1.2.cmml">μ</mi><mi id="A3.Ex42.m1.6.6.2.2.1.1.3" xref="A3.Ex42.m1.6.6.2.2.1.1.3.cmml">j</mi></msub><mo id="A3.Ex42.m1.7.7.3.3.2.5" xref="A3.Ex42.m1.7.7.3.3.3.cmml">,</mo><msub id="A3.Ex42.m1.7.7.3.3.2.2" xref="A3.Ex42.m1.7.7.3.3.2.2.cmml"><mi id="A3.Ex42.m1.7.7.3.3.2.2.2" xref="A3.Ex42.m1.7.7.3.3.2.2.2.cmml">𝚺</mi><mi id="A3.Ex42.m1.7.7.3.3.2.2.3" xref="A3.Ex42.m1.7.7.3.3.2.2.3.cmml">j</mi></msub><mo id="A3.Ex42.m1.7.7.3.3.2.6" stretchy="false" xref="A3.Ex42.m1.7.7.3.3.3.cmml">)</mo></mrow></mrow><mrow id="A3.Ex42.m1.10.10.6" xref="A3.Ex42.m1.10.10.6.cmml"><msubsup id="A3.Ex42.m1.10.10.6.4" xref="A3.Ex42.m1.10.10.6.4.cmml"><mo id="A3.Ex42.m1.10.10.6.4.2.2" xref="A3.Ex42.m1.10.10.6.4.2.2.cmml">∑</mo><mrow id="A3.Ex42.m1.10.10.6.4.2.3" xref="A3.Ex42.m1.10.10.6.4.2.3.cmml"><mi id="A3.Ex42.m1.10.10.6.4.2.3.2" xref="A3.Ex42.m1.10.10.6.4.2.3.2.cmml">i</mi><mo id="A3.Ex42.m1.10.10.6.4.2.3.1" xref="A3.Ex42.m1.10.10.6.4.2.3.1.cmml">=</mo><mn id="A3.Ex42.m1.10.10.6.4.2.3.3" xref="A3.Ex42.m1.10.10.6.4.2.3.3.cmml">1</mn></mrow><mi id="A3.Ex42.m1.10.10.6.4.3" xref="A3.Ex42.m1.10.10.6.4.3.cmml">p</mi></msubsup><mrow id="A3.Ex42.m1.10.10.6.3" xref="A3.Ex42.m1.10.10.6.3.cmml"><msub id="A3.Ex42.m1.10.10.6.3.4" xref="A3.Ex42.m1.10.10.6.3.4.cmml"><mi id="A3.Ex42.m1.10.10.6.3.4.2" xref="A3.Ex42.m1.10.10.6.3.4.2.cmml">π</mi><mi id="A3.Ex42.m1.10.10.6.3.4.3" xref="A3.Ex42.m1.10.10.6.3.4.3.cmml">i</mi></msub><mo id="A3.Ex42.m1.10.10.6.3.3" xref="A3.Ex42.m1.10.10.6.3.3.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="A3.Ex42.m1.10.10.6.3.5" xref="A3.Ex42.m1.10.10.6.3.5.cmml">𝒩</mi><mo id="A3.Ex42.m1.10.10.6.3.3a" xref="A3.Ex42.m1.10.10.6.3.3.cmml">⁢</mo><mrow id="A3.Ex42.m1.10.10.6.3.2.2" xref="A3.Ex42.m1.10.10.6.3.2.3.cmml"><mo id="A3.Ex42.m1.10.10.6.3.2.2.3" stretchy="false" xref="A3.Ex42.m1.10.10.6.3.2.3.cmml">(</mo><mi id="A3.Ex42.m1.8.8.4.1" xref="A3.Ex42.m1.8.8.4.1.cmml">𝐱</mi><mo id="A3.Ex42.m1.10.10.6.3.2.2.4" xref="A3.Ex42.m1.10.10.6.3.2.3.cmml">;</mo><msub id="A3.Ex42.m1.9.9.5.2.1.1.1" xref="A3.Ex42.m1.9.9.5.2.1.1.1.cmml"><mi id="A3.Ex42.m1.9.9.5.2.1.1.1.2" xref="A3.Ex42.m1.9.9.5.2.1.1.1.2.cmml">μ</mi><mi id="A3.Ex42.m1.9.9.5.2.1.1.1.3" xref="A3.Ex42.m1.9.9.5.2.1.1.1.3.cmml">i</mi></msub><mo id="A3.Ex42.m1.10.10.6.3.2.2.5" xref="A3.Ex42.m1.10.10.6.3.2.3.cmml">,</mo><msub id="A3.Ex42.m1.10.10.6.3.2.2.2" xref="A3.Ex42.m1.10.10.6.3.2.2.2.cmml"><mi id="A3.Ex42.m1.10.10.6.3.2.2.2.2" xref="A3.Ex42.m1.10.10.6.3.2.2.2.2.cmml">𝚺</mi><mi id="A3.Ex42.m1.10.10.6.3.2.2.2.3" xref="A3.Ex42.m1.10.10.6.3.2.2.2.3.cmml">i</mi></msub><mo id="A3.Ex42.m1.10.10.6.3.2.2.6" stretchy="false" xref="A3.Ex42.m1.10.10.6.3.2.3.cmml">)</mo></mrow></mrow></mrow></mfrac></mstyle><mo id="A3.Ex42.m1.17.17.1.1.3" xref="A3.Ex42.m1.17.17.1.1.3.cmml">=</mo><mstyle displaystyle="true" id="A3.Ex42.m1.16.16" xref="A3.Ex42.m1.16.16.cmml"><mfrac id="A3.Ex42.m1.16.16a" xref="A3.Ex42.m1.16.16.cmml"><mn id="A3.Ex42.m1.16.16.8" xref="A3.Ex42.m1.16.16.8.cmml">1</mn><mrow id="A3.Ex42.m1.16.16.6" xref="A3.Ex42.m1.16.16.6.cmml"><mn id="A3.Ex42.m1.16.16.6.8" 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xref="A3.Ex42.m1.16.16.6.6.6.3.2.2"><csymbol cd="ambiguous" id="A3.Ex42.m1.16.16.6.6.6.3.2.2.1.cmml" xref="A3.Ex42.m1.16.16.6.6.6.3.2.2">subscript</csymbol><ci id="A3.Ex42.m1.16.16.6.6.6.3.2.2.2.cmml" xref="A3.Ex42.m1.16.16.6.6.6.3.2.2.2">𝚺</ci><ci id="A3.Ex42.m1.16.16.6.6.6.3.2.2.3.cmml" xref="A3.Ex42.m1.16.16.6.6.6.3.2.2.3">𝑗</ci></apply></list></apply></apply></apply></apply></apply><apply id="A3.Ex42.m1.17.17.1.1e.cmml" xref="A3.Ex42.m1.17.17.1"><leq id="A3.Ex42.m1.17.17.1.1.4.cmml" xref="A3.Ex42.m1.17.17.1.1.4"></leq><share href="#A3.Ex42.m1.16.16.cmml" id="A3.Ex42.m1.17.17.1.1f.cmml" xref="A3.Ex42.m1.17.17.1"></share><cn id="A3.Ex42.m1.17.17.1.1.5.cmml" type="integer" xref="A3.Ex42.m1.17.17.1.1.5">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.Ex42.m1.17c">\displaystyle\frac{\pi_{j}\mathcal{N}({\mathbf{x}};{\mathbf{\mu}}_{j},{\mathbf% {\Sigma}}_{j})}{p({\mathbf{x}})}=\frac{\pi_{j}\mathcal{N}({\mathbf{x}};{% \mathbf{\mu}}_{j},{\mathbf{\Sigma}}_{j})}{\sum_{i=1}^{p}{\pi_{i}\mathcal{N}({% \mathbf{x}};{\mathbf{\mu}}_{i},{\mathbf{\Sigma}}_{i})}}=\frac{1}{1+\frac{\sum_% {i\neq j}{\pi_{i}\mathcal{N}({\mathbf{x}};{\mathbf{\mu}}_{i},{\mathbf{\Sigma}}% _{i})}}{\pi_{j}\mathcal{N}({\mathbf{x}};{\mathbf{\mu}}_{j},{\mathbf{\Sigma}}_{% j})}}\leq 1.</annotation><annotation encoding="application/x-llamapun" id="A3.Ex42.m1.17d">divide start_ARG italic_π start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT caligraphic_N ( bold_x ; italic_μ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT , bold_Σ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ) end_ARG start_ARG italic_p ( bold_x ) end_ARG = divide start_ARG italic_π start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT caligraphic_N ( bold_x ; italic_μ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT , bold_Σ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ) end_ARG start_ARG ∑ start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT italic_π start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT caligraphic_N ( bold_x ; italic_μ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , bold_Σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) end_ARG = divide start_ARG 1 end_ARG start_ARG 1 + divide start_ARG ∑ start_POSTSUBSCRIPT italic_i ≠ italic_j end_POSTSUBSCRIPT italic_π start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT caligraphic_N ( bold_x ; italic_μ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , bold_Σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) end_ARG start_ARG italic_π start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT caligraphic_N ( bold_x ; italic_μ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT , bold_Σ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ) end_ARG end_ARG ≤ 1 .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> </div> <div class="ltx_para" id="A3.SS2.SSS1.5.p5"> <p class="ltx_p" id="A3.SS2.SSS1.5.p5.1">As a consequence, 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id="A3.SS2.SSS1.5.p5.1.m1.1.1.2.1.cmml" xref="A3.SS2.SSS1.5.p5.1.m1.1.1.2">subscript</csymbol><ci id="A3.SS2.SSS1.5.p5.1.m1.1.1.2.2.cmml" xref="A3.SS2.SSS1.5.p5.1.m1.1.1.2.2">𝛼</ci><ci id="A3.SS2.SSS1.5.p5.1.m1.1.1.2.3.cmml" xref="A3.SS2.SSS1.5.p5.1.m1.1.1.2.3">𝑝</ci></apply><cn id="A3.SS2.SSS1.5.p5.1.m1.1.1.3.cmml" type="integer" xref="A3.SS2.SSS1.5.p5.1.m1.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.SS2.SSS1.5.p5.1.m1.1c">\alpha_{p}\geq 0</annotation><annotation encoding="application/x-llamapun" id="A3.SS2.SSS1.5.p5.1.m1.1d">italic_α start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT ≥ 0</annotation></semantics></math> such that</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A8.EGx29"> <tbody id="A3.E21"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math 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xref="A3.E21.m1.3.3.1.1.1.cmml">)</mo></mrow></mrow><mo id="A3.E21.m1.3.3.1.1.3" stretchy="false" xref="A3.E21.m1.3.3.1.2.1.cmml">‖</mo></mrow><mo id="A3.E21.m1.4.4.3" xref="A3.E21.m1.4.4.3.cmml">≤</mo><mrow id="A3.E21.m1.4.4.2" xref="A3.E21.m1.4.4.2.cmml"><msub id="A3.E21.m1.4.4.2.3" xref="A3.E21.m1.4.4.2.3.cmml"><mi id="A3.E21.m1.4.4.2.3.2" xref="A3.E21.m1.4.4.2.3.2.cmml">α</mi><mi id="A3.E21.m1.4.4.2.3.3" xref="A3.E21.m1.4.4.2.3.3.cmml">p</mi></msub><mo id="A3.E21.m1.4.4.2.2" xref="A3.E21.m1.4.4.2.2.cmml">⁢</mo><mrow id="A3.E21.m1.4.4.2.1.1" xref="A3.E21.m1.4.4.2.1.1.1.cmml"><mo id="A3.E21.m1.4.4.2.1.1.2" stretchy="false" xref="A3.E21.m1.4.4.2.1.1.1.cmml">(</mo><mrow id="A3.E21.m1.4.4.2.1.1.1" xref="A3.E21.m1.4.4.2.1.1.1.cmml"><msup id="A3.E21.m1.4.4.2.1.1.1.2" xref="A3.E21.m1.4.4.2.1.1.1.2.cmml"><mrow id="A3.E21.m1.4.4.2.1.1.1.2.2.2" xref="A3.E21.m1.4.4.2.1.1.1.2.2.1.cmml"><mo id="A3.E21.m1.4.4.2.1.1.1.2.2.2.1" stretchy="false" xref="A3.E21.m1.4.4.2.1.1.1.2.2.1.1.cmml">‖</mo><mi id="A3.E21.m1.2.2" xref="A3.E21.m1.2.2.cmml">𝐱</mi><mo id="A3.E21.m1.4.4.2.1.1.1.2.2.2.2" stretchy="false" xref="A3.E21.m1.4.4.2.1.1.1.2.2.1.1.cmml">‖</mo></mrow><mn id="A3.E21.m1.4.4.2.1.1.1.2.3" xref="A3.E21.m1.4.4.2.1.1.1.2.3.cmml">3</mn></msup><mo id="A3.E21.m1.4.4.2.1.1.1.1" xref="A3.E21.m1.4.4.2.1.1.1.1.cmml">+</mo><mn id="A3.E21.m1.4.4.2.1.1.1.3" xref="A3.E21.m1.4.4.2.1.1.1.3.cmml">1</mn></mrow><mo id="A3.E21.m1.4.4.2.1.1.3" stretchy="false" xref="A3.E21.m1.4.4.2.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="A3.E21.m1.4b"><apply id="A3.E21.m1.4.4.cmml" xref="A3.E21.m1.4.4"><leq id="A3.E21.m1.4.4.3.cmml" xref="A3.E21.m1.4.4.3"></leq><apply id="A3.E21.m1.3.3.1.2.cmml" xref="A3.E21.m1.3.3.1.1"><csymbol cd="latexml" id="A3.E21.m1.3.3.1.2.1.cmml" xref="A3.E21.m1.3.3.1.1.2">norm</csymbol><apply id="A3.E21.m1.3.3.1.1.1.cmml" xref="A3.E21.m1.3.3.1.1.1"><times id="A3.E21.m1.3.3.1.1.1.1.cmml" xref="A3.E21.m1.3.3.1.1.1.1"></times><apply id="A3.E21.m1.3.3.1.1.1.2.cmml" xref="A3.E21.m1.3.3.1.1.1.2"><csymbol cd="ambiguous" id="A3.E21.m1.3.3.1.1.1.2.1.cmml" xref="A3.E21.m1.3.3.1.1.1.2">subscript</csymbol><ci id="A3.E21.m1.3.3.1.1.1.2.2.cmml" xref="A3.E21.m1.3.3.1.1.1.2.2">𝑐</ci><ci id="A3.E21.m1.3.3.1.1.1.2.3.cmml" xref="A3.E21.m1.3.3.1.1.1.2.3">𝑝</ci></apply><ci id="A3.E21.m1.1.1.cmml" xref="A3.E21.m1.1.1">𝐱</ci></apply></apply><apply id="A3.E21.m1.4.4.2.cmml" xref="A3.E21.m1.4.4.2"><times id="A3.E21.m1.4.4.2.2.cmml" xref="A3.E21.m1.4.4.2.2"></times><apply id="A3.E21.m1.4.4.2.3.cmml" xref="A3.E21.m1.4.4.2.3"><csymbol cd="ambiguous" id="A3.E21.m1.4.4.2.3.1.cmml" xref="A3.E21.m1.4.4.2.3">subscript</csymbol><ci id="A3.E21.m1.4.4.2.3.2.cmml" xref="A3.E21.m1.4.4.2.3.2">𝛼</ci><ci id="A3.E21.m1.4.4.2.3.3.cmml" xref="A3.E21.m1.4.4.2.3.3">𝑝</ci></apply><apply id="A3.E21.m1.4.4.2.1.1.1.cmml" xref="A3.E21.m1.4.4.2.1.1"><plus id="A3.E21.m1.4.4.2.1.1.1.1.cmml" xref="A3.E21.m1.4.4.2.1.1.1.1"></plus><apply id="A3.E21.m1.4.4.2.1.1.1.2.cmml" xref="A3.E21.m1.4.4.2.1.1.1.2"><csymbol cd="ambiguous" id="A3.E21.m1.4.4.2.1.1.1.2.1.cmml" xref="A3.E21.m1.4.4.2.1.1.1.2">superscript</csymbol><apply id="A3.E21.m1.4.4.2.1.1.1.2.2.1.cmml" xref="A3.E21.m1.4.4.2.1.1.1.2.2.2"><csymbol cd="latexml" id="A3.E21.m1.4.4.2.1.1.1.2.2.1.1.cmml" xref="A3.E21.m1.4.4.2.1.1.1.2.2.2.1">norm</csymbol><ci id="A3.E21.m1.2.2.cmml" xref="A3.E21.m1.2.2">𝐱</ci></apply><cn id="A3.E21.m1.4.4.2.1.1.1.2.3.cmml" type="integer" xref="A3.E21.m1.4.4.2.1.1.1.2.3">3</cn></apply><cn id="A3.E21.m1.4.4.2.1.1.1.3.cmml" type="integer" xref="A3.E21.m1.4.4.2.1.1.1.3">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.E21.m1.4c">\displaystyle\|c_{p}({\mathbf{x}})\|\leq\alpha_{p}(\|{\mathbf{x}}\|^{3}+1)</annotation><annotation encoding="application/x-llamapun" id="A3.E21.m1.4d">∥ italic_c start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT ( bold_x ) ∥ ≤ italic_α start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT ( ∥ bold_x ∥ start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT + 1 )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(21)</span></td> </tr></tbody> </table> </div> <div class="ltx_para" id="A3.SS2.SSS1.6.p6"> <p class="ltx_p" id="A3.SS2.SSS1.6.p6.4">By combining (<a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#A3.E20" title="20 ‣ Proof. ‣ C.2.1 Theoretical considerations ‣ C.2 Gaussian Mixture prior ‣ Appendix C SNORE in simple cases ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_tag">20</span></a>) and (<a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#A3.E21" title="21 ‣ Proof. ‣ C.2.1 Theoretical considerations ‣ C.2 Gaussian Mixture prior ‣ Appendix C SNORE in simple cases ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_tag">21</span></a>), we obtain that there exist <math alttext="a_{p}" class="ltx_Math" display="inline" id="A3.SS2.SSS1.6.p6.1.m1.1"><semantics id="A3.SS2.SSS1.6.p6.1.m1.1a"><msub id="A3.SS2.SSS1.6.p6.1.m1.1.1" xref="A3.SS2.SSS1.6.p6.1.m1.1.1.cmml"><mi id="A3.SS2.SSS1.6.p6.1.m1.1.1.2" xref="A3.SS2.SSS1.6.p6.1.m1.1.1.2.cmml">a</mi><mi id="A3.SS2.SSS1.6.p6.1.m1.1.1.3" xref="A3.SS2.SSS1.6.p6.1.m1.1.1.3.cmml">p</mi></msub><annotation-xml encoding="MathML-Content" id="A3.SS2.SSS1.6.p6.1.m1.1b"><apply id="A3.SS2.SSS1.6.p6.1.m1.1.1.cmml" xref="A3.SS2.SSS1.6.p6.1.m1.1.1"><csymbol cd="ambiguous" id="A3.SS2.SSS1.6.p6.1.m1.1.1.1.cmml" xref="A3.SS2.SSS1.6.p6.1.m1.1.1">subscript</csymbol><ci id="A3.SS2.SSS1.6.p6.1.m1.1.1.2.cmml" xref="A3.SS2.SSS1.6.p6.1.m1.1.1.2">𝑎</ci><ci id="A3.SS2.SSS1.6.p6.1.m1.1.1.3.cmml" xref="A3.SS2.SSS1.6.p6.1.m1.1.1.3">𝑝</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.SS2.SSS1.6.p6.1.m1.1c">a_{p}</annotation><annotation encoding="application/x-llamapun" id="A3.SS2.SSS1.6.p6.1.m1.1d">italic_a start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="\alpha_{0}>0" class="ltx_Math" display="inline" id="A3.SS2.SSS1.6.p6.2.m2.1"><semantics id="A3.SS2.SSS1.6.p6.2.m2.1a"><mrow id="A3.SS2.SSS1.6.p6.2.m2.1.1" xref="A3.SS2.SSS1.6.p6.2.m2.1.1.cmml"><msub id="A3.SS2.SSS1.6.p6.2.m2.1.1.2" xref="A3.SS2.SSS1.6.p6.2.m2.1.1.2.cmml"><mi id="A3.SS2.SSS1.6.p6.2.m2.1.1.2.2" xref="A3.SS2.SSS1.6.p6.2.m2.1.1.2.2.cmml">α</mi><mn id="A3.SS2.SSS1.6.p6.2.m2.1.1.2.3" xref="A3.SS2.SSS1.6.p6.2.m2.1.1.2.3.cmml">0</mn></msub><mo id="A3.SS2.SSS1.6.p6.2.m2.1.1.1" xref="A3.SS2.SSS1.6.p6.2.m2.1.1.1.cmml">></mo><mn id="A3.SS2.SSS1.6.p6.2.m2.1.1.3" xref="A3.SS2.SSS1.6.p6.2.m2.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="A3.SS2.SSS1.6.p6.2.m2.1b"><apply id="A3.SS2.SSS1.6.p6.2.m2.1.1.cmml" xref="A3.SS2.SSS1.6.p6.2.m2.1.1"><gt id="A3.SS2.SSS1.6.p6.2.m2.1.1.1.cmml" xref="A3.SS2.SSS1.6.p6.2.m2.1.1.1"></gt><apply id="A3.SS2.SSS1.6.p6.2.m2.1.1.2.cmml" xref="A3.SS2.SSS1.6.p6.2.m2.1.1.2"><csymbol cd="ambiguous" id="A3.SS2.SSS1.6.p6.2.m2.1.1.2.1.cmml" xref="A3.SS2.SSS1.6.p6.2.m2.1.1.2">subscript</csymbol><ci id="A3.SS2.SSS1.6.p6.2.m2.1.1.2.2.cmml" xref="A3.SS2.SSS1.6.p6.2.m2.1.1.2.2">𝛼</ci><cn id="A3.SS2.SSS1.6.p6.2.m2.1.1.2.3.cmml" type="integer" xref="A3.SS2.SSS1.6.p6.2.m2.1.1.2.3">0</cn></apply><cn id="A3.SS2.SSS1.6.p6.2.m2.1.1.3.cmml" type="integer" xref="A3.SS2.SSS1.6.p6.2.m2.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.SS2.SSS1.6.p6.2.m2.1c">\alpha_{0}>0</annotation><annotation encoding="application/x-llamapun" id="A3.SS2.SSS1.6.p6.2.m2.1d">italic_α start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT > 0</annotation></semantics></math>, such that <math alttext="\forall{\mathbf{x}}\in{\mathbb{R}}^{d}" class="ltx_Math" display="inline" id="A3.SS2.SSS1.6.p6.3.m3.1"><semantics id="A3.SS2.SSS1.6.p6.3.m3.1a"><mrow id="A3.SS2.SSS1.6.p6.3.m3.1.1" xref="A3.SS2.SSS1.6.p6.3.m3.1.1.cmml"><mrow id="A3.SS2.SSS1.6.p6.3.m3.1.1.2" xref="A3.SS2.SSS1.6.p6.3.m3.1.1.2.cmml"><mo id="A3.SS2.SSS1.6.p6.3.m3.1.1.2.1" rspace="0.167em" xref="A3.SS2.SSS1.6.p6.3.m3.1.1.2.1.cmml">∀</mo><mi id="A3.SS2.SSS1.6.p6.3.m3.1.1.2.2" xref="A3.SS2.SSS1.6.p6.3.m3.1.1.2.2.cmml">𝐱</mi></mrow><mo id="A3.SS2.SSS1.6.p6.3.m3.1.1.1" xref="A3.SS2.SSS1.6.p6.3.m3.1.1.1.cmml">∈</mo><msup id="A3.SS2.SSS1.6.p6.3.m3.1.1.3" xref="A3.SS2.SSS1.6.p6.3.m3.1.1.3.cmml"><mi id="A3.SS2.SSS1.6.p6.3.m3.1.1.3.2" xref="A3.SS2.SSS1.6.p6.3.m3.1.1.3.2.cmml">ℝ</mi><mi id="A3.SS2.SSS1.6.p6.3.m3.1.1.3.3" xref="A3.SS2.SSS1.6.p6.3.m3.1.1.3.3.cmml">d</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="A3.SS2.SSS1.6.p6.3.m3.1b"><apply id="A3.SS2.SSS1.6.p6.3.m3.1.1.cmml" xref="A3.SS2.SSS1.6.p6.3.m3.1.1"><in id="A3.SS2.SSS1.6.p6.3.m3.1.1.1.cmml" xref="A3.SS2.SSS1.6.p6.3.m3.1.1.1"></in><apply id="A3.SS2.SSS1.6.p6.3.m3.1.1.2.cmml" xref="A3.SS2.SSS1.6.p6.3.m3.1.1.2"><csymbol cd="latexml" id="A3.SS2.SSS1.6.p6.3.m3.1.1.2.1.cmml" xref="A3.SS2.SSS1.6.p6.3.m3.1.1.2.1">for-all</csymbol><ci id="A3.SS2.SSS1.6.p6.3.m3.1.1.2.2.cmml" xref="A3.SS2.SSS1.6.p6.3.m3.1.1.2.2">𝐱</ci></apply><apply id="A3.SS2.SSS1.6.p6.3.m3.1.1.3.cmml" xref="A3.SS2.SSS1.6.p6.3.m3.1.1.3"><csymbol cd="ambiguous" id="A3.SS2.SSS1.6.p6.3.m3.1.1.3.1.cmml" xref="A3.SS2.SSS1.6.p6.3.m3.1.1.3">superscript</csymbol><ci id="A3.SS2.SSS1.6.p6.3.m3.1.1.3.2.cmml" xref="A3.SS2.SSS1.6.p6.3.m3.1.1.3.2">ℝ</ci><ci id="A3.SS2.SSS1.6.p6.3.m3.1.1.3.3.cmml" xref="A3.SS2.SSS1.6.p6.3.m3.1.1.3.3">𝑑</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.SS2.SSS1.6.p6.3.m3.1c">\forall{\mathbf{x}}\in{\mathbb{R}}^{d}</annotation><annotation encoding="application/x-llamapun" id="A3.SS2.SSS1.6.p6.3.m3.1d">∀ bold_x ∈ blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT</annotation></semantics></math>, <math alttext="\forall\sigma\in]0,\sigma_{0}]" class="ltx_math_unparsed" display="inline" id="A3.SS2.SSS1.6.p6.4.m4.1"><semantics id="A3.SS2.SSS1.6.p6.4.m4.1a"><mrow id="A3.SS2.SSS1.6.p6.4.m4.1b"><mrow id="A3.SS2.SSS1.6.p6.4.m4.1.1"><mo id="A3.SS2.SSS1.6.p6.4.m4.1.1.1" rspace="0.167em">∀</mo><mi id="A3.SS2.SSS1.6.p6.4.m4.1.1.2">σ</mi><mo id="A3.SS2.SSS1.6.p6.4.m4.1.1.3" rspace="0em">∈</mo><mo id="A3.SS2.SSS1.6.p6.4.m4.1.1.4" stretchy="false">]</mo></mrow><mn id="A3.SS2.SSS1.6.p6.4.m4.1.2">0</mn><mo id="A3.SS2.SSS1.6.p6.4.m4.1.3">,</mo><msub id="A3.SS2.SSS1.6.p6.4.m4.1.4"><mi id="A3.SS2.SSS1.6.p6.4.m4.1.4.2">σ</mi><mn id="A3.SS2.SSS1.6.p6.4.m4.1.4.3">0</mn></msub><mo id="A3.SS2.SSS1.6.p6.4.m4.1.5" stretchy="false">]</mo></mrow><annotation encoding="application/x-tex" id="A3.SS2.SSS1.6.p6.4.m4.1c">\forall\sigma\in]0,\sigma_{0}]</annotation><annotation encoding="application/x-llamapun" id="A3.SS2.SSS1.6.p6.4.m4.1d">∀ italic_σ ∈ ] 0 , italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ]</annotation></semantics></math>,</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A8.EGx30"> <tbody id="A3.E22"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\|\nabla\log p_{\sigma}({\mathbf{x}})-\nabla\log p({\mathbf{x}})% \|\leq\sigma^{2}a_{p}(\|{\mathbf{x}}\|^{3}+1)." class="ltx_Math" display="inline" id="A3.E22.m1.4"><semantics id="A3.E22.m1.4a"><mrow id="A3.E22.m1.4.4.1" xref="A3.E22.m1.4.4.1.1.cmml"><mrow id="A3.E22.m1.4.4.1.1" xref="A3.E22.m1.4.4.1.1.cmml"><mrow id="A3.E22.m1.4.4.1.1.1.1" xref="A3.E22.m1.4.4.1.1.1.2.cmml"><mo id="A3.E22.m1.4.4.1.1.1.1.2" stretchy="false" xref="A3.E22.m1.4.4.1.1.1.2.1.cmml">‖</mo><mrow id="A3.E22.m1.4.4.1.1.1.1.1" xref="A3.E22.m1.4.4.1.1.1.1.1.cmml"><mrow id="A3.E22.m1.4.4.1.1.1.1.1.2" xref="A3.E22.m1.4.4.1.1.1.1.1.2.cmml"><mrow 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.</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(22)</span></td> </tr></tbody> </table> <p class="ltx_p" id="A3.SS2.SSS1.6.p6.5">∎</p> </div> </div> <div class="ltx_para" id="A3.SS2.SSS1.p3"> <p class="ltx_p" id="A3.SS2.SSS1.p3.1">A similar work can be done to evaluate the approximation done by our regularization <math alttext="\mathcal{R}_{\sigma}" class="ltx_Math" display="inline" id="A3.SS2.SSS1.p3.1.m1.1"><semantics id="A3.SS2.SSS1.p3.1.m1.1a"><msub id="A3.SS2.SSS1.p3.1.m1.1.1" xref="A3.SS2.SSS1.p3.1.m1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="A3.SS2.SSS1.p3.1.m1.1.1.2" xref="A3.SS2.SSS1.p3.1.m1.1.1.2.cmml">ℛ</mi><mi id="A3.SS2.SSS1.p3.1.m1.1.1.3" xref="A3.SS2.SSS1.p3.1.m1.1.1.3.cmml">σ</mi></msub><annotation-xml encoding="MathML-Content" id="A3.SS2.SSS1.p3.1.m1.1b"><apply id="A3.SS2.SSS1.p3.1.m1.1.1.cmml" xref="A3.SS2.SSS1.p3.1.m1.1.1"><csymbol cd="ambiguous" id="A3.SS2.SSS1.p3.1.m1.1.1.1.cmml" xref="A3.SS2.SSS1.p3.1.m1.1.1">subscript</csymbol><ci id="A3.SS2.SSS1.p3.1.m1.1.1.2.cmml" xref="A3.SS2.SSS1.p3.1.m1.1.1.2">ℛ</ci><ci id="A3.SS2.SSS1.p3.1.m1.1.1.3.cmml" xref="A3.SS2.SSS1.p3.1.m1.1.1.3">𝜎</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.SS2.SSS1.p3.1.m1.1c">\mathcal{R}_{\sigma}</annotation><annotation encoding="application/x-llamapun" id="A3.SS2.SSS1.p3.1.m1.1d">caligraphic_R start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT</annotation></semantics></math>.</p> </div> <div class="ltx_theorem ltx_theorem_proposition" id="A3.Thmproposition2"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="A3.Thmproposition2.1.1.1">Proposition C.2</span></span><span class="ltx_text ltx_font_bold" id="A3.Thmproposition2.2.2">.</span> </h6> <div class="ltx_para" id="A3.Thmproposition2.p1"> <p class="ltx_p" id="A3.Thmproposition2.p1.5"><span class="ltx_text ltx_font_italic" id="A3.Thmproposition2.p1.5.5">If the prior <math alttext="p" class="ltx_Math" display="inline" id="A3.Thmproposition2.p1.1.1.m1.1"><semantics id="A3.Thmproposition2.p1.1.1.m1.1a"><mi id="A3.Thmproposition2.p1.1.1.m1.1.1" xref="A3.Thmproposition2.p1.1.1.m1.1.1.cmml">p</mi><annotation-xml encoding="MathML-Content" id="A3.Thmproposition2.p1.1.1.m1.1b"><ci id="A3.Thmproposition2.p1.1.1.m1.1.1.cmml" xref="A3.Thmproposition2.p1.1.1.m1.1.1">𝑝</ci></annotation-xml><annotation encoding="application/x-tex" id="A3.Thmproposition2.p1.1.1.m1.1c">p</annotation><annotation encoding="application/x-llamapun" id="A3.Thmproposition2.p1.1.1.m1.1d">italic_p</annotation></semantics></math> is a Gaussian Mixture Model, then there exist <math alttext="\sigma_{0}>0" class="ltx_Math" display="inline" id="A3.Thmproposition2.p1.2.2.m2.1"><semantics id="A3.Thmproposition2.p1.2.2.m2.1a"><mrow id="A3.Thmproposition2.p1.2.2.m2.1.1" xref="A3.Thmproposition2.p1.2.2.m2.1.1.cmml"><msub id="A3.Thmproposition2.p1.2.2.m2.1.1.2" xref="A3.Thmproposition2.p1.2.2.m2.1.1.2.cmml"><mi id="A3.Thmproposition2.p1.2.2.m2.1.1.2.2" xref="A3.Thmproposition2.p1.2.2.m2.1.1.2.2.cmml">σ</mi><mn id="A3.Thmproposition2.p1.2.2.m2.1.1.2.3" mathvariant="normal" xref="A3.Thmproposition2.p1.2.2.m2.1.1.2.3.cmml">0</mn></msub><mo id="A3.Thmproposition2.p1.2.2.m2.1.1.1" mathvariant="normal" xref="A3.Thmproposition2.p1.2.2.m2.1.1.1.cmml">></mo><mn id="A3.Thmproposition2.p1.2.2.m2.1.1.3" mathvariant="normal" xref="A3.Thmproposition2.p1.2.2.m2.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="A3.Thmproposition2.p1.2.2.m2.1b"><apply id="A3.Thmproposition2.p1.2.2.m2.1.1.cmml" xref="A3.Thmproposition2.p1.2.2.m2.1.1"><gt id="A3.Thmproposition2.p1.2.2.m2.1.1.1.cmml" xref="A3.Thmproposition2.p1.2.2.m2.1.1.1"></gt><apply id="A3.Thmproposition2.p1.2.2.m2.1.1.2.cmml" xref="A3.Thmproposition2.p1.2.2.m2.1.1.2"><csymbol cd="ambiguous" id="A3.Thmproposition2.p1.2.2.m2.1.1.2.1.cmml" xref="A3.Thmproposition2.p1.2.2.m2.1.1.2">subscript</csymbol><ci id="A3.Thmproposition2.p1.2.2.m2.1.1.2.2.cmml" xref="A3.Thmproposition2.p1.2.2.m2.1.1.2.2">𝜎</ci><cn id="A3.Thmproposition2.p1.2.2.m2.1.1.2.3.cmml" type="integer" xref="A3.Thmproposition2.p1.2.2.m2.1.1.2.3">0</cn></apply><cn id="A3.Thmproposition2.p1.2.2.m2.1.1.3.cmml" type="integer" xref="A3.Thmproposition2.p1.2.2.m2.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.Thmproposition2.p1.2.2.m2.1c">\sigma_{0}>0</annotation><annotation encoding="application/x-llamapun" id="A3.Thmproposition2.p1.2.2.m2.1d">italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT > 0</annotation></semantics></math> and <math alttext="e_{p}\in{\mathbb{R}}^{+}" class="ltx_Math" display="inline" id="A3.Thmproposition2.p1.3.3.m3.1"><semantics id="A3.Thmproposition2.p1.3.3.m3.1a"><mrow id="A3.Thmproposition2.p1.3.3.m3.1.1" xref="A3.Thmproposition2.p1.3.3.m3.1.1.cmml"><msub id="A3.Thmproposition2.p1.3.3.m3.1.1.2" xref="A3.Thmproposition2.p1.3.3.m3.1.1.2.cmml"><mi id="A3.Thmproposition2.p1.3.3.m3.1.1.2.2" xref="A3.Thmproposition2.p1.3.3.m3.1.1.2.2.cmml">e</mi><mi id="A3.Thmproposition2.p1.3.3.m3.1.1.2.3" xref="A3.Thmproposition2.p1.3.3.m3.1.1.2.3.cmml">p</mi></msub><mo id="A3.Thmproposition2.p1.3.3.m3.1.1.1" mathvariant="normal" xref="A3.Thmproposition2.p1.3.3.m3.1.1.1.cmml">∈</mo><msup id="A3.Thmproposition2.p1.3.3.m3.1.1.3" xref="A3.Thmproposition2.p1.3.3.m3.1.1.3.cmml"><mi id="A3.Thmproposition2.p1.3.3.m3.1.1.3.2" xref="A3.Thmproposition2.p1.3.3.m3.1.1.3.2.cmml">ℝ</mi><mo id="A3.Thmproposition2.p1.3.3.m3.1.1.3.3" mathvariant="normal" xref="A3.Thmproposition2.p1.3.3.m3.1.1.3.3.cmml">+</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="A3.Thmproposition2.p1.3.3.m3.1b"><apply id="A3.Thmproposition2.p1.3.3.m3.1.1.cmml" xref="A3.Thmproposition2.p1.3.3.m3.1.1"><in id="A3.Thmproposition2.p1.3.3.m3.1.1.1.cmml" xref="A3.Thmproposition2.p1.3.3.m3.1.1.1"></in><apply id="A3.Thmproposition2.p1.3.3.m3.1.1.2.cmml" xref="A3.Thmproposition2.p1.3.3.m3.1.1.2"><csymbol cd="ambiguous" id="A3.Thmproposition2.p1.3.3.m3.1.1.2.1.cmml" xref="A3.Thmproposition2.p1.3.3.m3.1.1.2">subscript</csymbol><ci id="A3.Thmproposition2.p1.3.3.m3.1.1.2.2.cmml" xref="A3.Thmproposition2.p1.3.3.m3.1.1.2.2">𝑒</ci><ci id="A3.Thmproposition2.p1.3.3.m3.1.1.2.3.cmml" xref="A3.Thmproposition2.p1.3.3.m3.1.1.2.3">𝑝</ci></apply><apply id="A3.Thmproposition2.p1.3.3.m3.1.1.3.cmml" xref="A3.Thmproposition2.p1.3.3.m3.1.1.3"><csymbol cd="ambiguous" id="A3.Thmproposition2.p1.3.3.m3.1.1.3.1.cmml" xref="A3.Thmproposition2.p1.3.3.m3.1.1.3">superscript</csymbol><ci id="A3.Thmproposition2.p1.3.3.m3.1.1.3.2.cmml" xref="A3.Thmproposition2.p1.3.3.m3.1.1.3.2">ℝ</ci><plus id="A3.Thmproposition2.p1.3.3.m3.1.1.3.3.cmml" xref="A3.Thmproposition2.p1.3.3.m3.1.1.3.3"></plus></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.Thmproposition2.p1.3.3.m3.1c">e_{p}\in{\mathbb{R}}^{+}</annotation><annotation encoding="application/x-llamapun" id="A3.Thmproposition2.p1.3.3.m3.1d">italic_e start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT ∈ blackboard_R start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT</annotation></semantics></math> such that <math alttext="\forall{\mathbf{x}}\in{\mathbb{R}}^{d}" class="ltx_Math" display="inline" id="A3.Thmproposition2.p1.4.4.m4.1"><semantics id="A3.Thmproposition2.p1.4.4.m4.1a"><mrow id="A3.Thmproposition2.p1.4.4.m4.1.1" xref="A3.Thmproposition2.p1.4.4.m4.1.1.cmml"><mrow id="A3.Thmproposition2.p1.4.4.m4.1.1.2" xref="A3.Thmproposition2.p1.4.4.m4.1.1.2.cmml"><mo id="A3.Thmproposition2.p1.4.4.m4.1.1.2.1" mathvariant="normal" rspace="0.167em" xref="A3.Thmproposition2.p1.4.4.m4.1.1.2.1.cmml">∀</mo><mi id="A3.Thmproposition2.p1.4.4.m4.1.1.2.2" xref="A3.Thmproposition2.p1.4.4.m4.1.1.2.2.cmml">𝐱</mi></mrow><mo id="A3.Thmproposition2.p1.4.4.m4.1.1.1" mathvariant="normal" xref="A3.Thmproposition2.p1.4.4.m4.1.1.1.cmml">∈</mo><msup id="A3.Thmproposition2.p1.4.4.m4.1.1.3" xref="A3.Thmproposition2.p1.4.4.m4.1.1.3.cmml"><mi id="A3.Thmproposition2.p1.4.4.m4.1.1.3.2" xref="A3.Thmproposition2.p1.4.4.m4.1.1.3.2.cmml">ℝ</mi><mi id="A3.Thmproposition2.p1.4.4.m4.1.1.3.3" xref="A3.Thmproposition2.p1.4.4.m4.1.1.3.3.cmml">d</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="A3.Thmproposition2.p1.4.4.m4.1b"><apply id="A3.Thmproposition2.p1.4.4.m4.1.1.cmml" xref="A3.Thmproposition2.p1.4.4.m4.1.1"><in id="A3.Thmproposition2.p1.4.4.m4.1.1.1.cmml" xref="A3.Thmproposition2.p1.4.4.m4.1.1.1"></in><apply id="A3.Thmproposition2.p1.4.4.m4.1.1.2.cmml" xref="A3.Thmproposition2.p1.4.4.m4.1.1.2"><csymbol cd="latexml" id="A3.Thmproposition2.p1.4.4.m4.1.1.2.1.cmml" xref="A3.Thmproposition2.p1.4.4.m4.1.1.2.1">for-all</csymbol><ci id="A3.Thmproposition2.p1.4.4.m4.1.1.2.2.cmml" xref="A3.Thmproposition2.p1.4.4.m4.1.1.2.2">𝐱</ci></apply><apply id="A3.Thmproposition2.p1.4.4.m4.1.1.3.cmml" xref="A3.Thmproposition2.p1.4.4.m4.1.1.3"><csymbol cd="ambiguous" id="A3.Thmproposition2.p1.4.4.m4.1.1.3.1.cmml" xref="A3.Thmproposition2.p1.4.4.m4.1.1.3">superscript</csymbol><ci id="A3.Thmproposition2.p1.4.4.m4.1.1.3.2.cmml" xref="A3.Thmproposition2.p1.4.4.m4.1.1.3.2">ℝ</ci><ci id="A3.Thmproposition2.p1.4.4.m4.1.1.3.3.cmml" xref="A3.Thmproposition2.p1.4.4.m4.1.1.3.3">𝑑</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.Thmproposition2.p1.4.4.m4.1c">\forall{\mathbf{x}}\in{\mathbb{R}}^{d}</annotation><annotation encoding="application/x-llamapun" id="A3.Thmproposition2.p1.4.4.m4.1d">∀ bold_x ∈ blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT</annotation></semantics></math> and <math alttext="\sigma\in]0,\sigma_{0}]" class="ltx_math_unparsed" display="inline" id="A3.Thmproposition2.p1.5.5.m5.1"><semantics id="A3.Thmproposition2.p1.5.5.m5.1a"><mrow id="A3.Thmproposition2.p1.5.5.m5.1b"><mrow id="A3.Thmproposition2.p1.5.5.m5.1.1"><mi id="A3.Thmproposition2.p1.5.5.m5.1.1.1">σ</mi><mo id="A3.Thmproposition2.p1.5.5.m5.1.1.2" mathvariant="normal" rspace="0em">∈</mo><mo id="A3.Thmproposition2.p1.5.5.m5.1.1.3" mathvariant="normal" stretchy="false">]</mo></mrow><mn id="A3.Thmproposition2.p1.5.5.m5.1.2" mathvariant="normal">0</mn><mo id="A3.Thmproposition2.p1.5.5.m5.1.3" mathvariant="normal">,</mo><msub id="A3.Thmproposition2.p1.5.5.m5.1.4"><mi id="A3.Thmproposition2.p1.5.5.m5.1.4.2">σ</mi><mn id="A3.Thmproposition2.p1.5.5.m5.1.4.3" mathvariant="normal">0</mn></msub><mo id="A3.Thmproposition2.p1.5.5.m5.1.5" mathvariant="normal" stretchy="false">]</mo></mrow><annotation encoding="application/x-tex" id="A3.Thmproposition2.p1.5.5.m5.1c">\sigma\in]0,\sigma_{0}]</annotation><annotation encoding="application/x-llamapun" id="A3.Thmproposition2.p1.5.5.m5.1d">italic_σ ∈ ] 0 , italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ]</annotation></semantics></math>, </span></p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A8.EGx31"> <tbody id="A3.E23"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\|\nabla\mathcal{R}_{\sigma}({\mathbf{x}})+\nabla\log p({\mathbf{% x}})\|\leq\sigma e_{p}(\|{\mathbf{x}}\|^{2}+1)." class="ltx_Math" display="inline" id="A3.E23.m1.4"><semantics id="A3.E23.m1.4a"><mrow id="A3.E23.m1.4.4.1" xref="A3.E23.m1.4.4.1.1.cmml"><mrow id="A3.E23.m1.4.4.1.1" xref="A3.E23.m1.4.4.1.1.cmml"><mrow id="A3.E23.m1.4.4.1.1.1.1" xref="A3.E23.m1.4.4.1.1.1.2.cmml"><mo id="A3.E23.m1.4.4.1.1.1.1.2" stretchy="false" xref="A3.E23.m1.4.4.1.1.1.2.1.cmml">‖</mo><mrow id="A3.E23.m1.4.4.1.1.1.1.1" xref="A3.E23.m1.4.4.1.1.1.1.1.cmml"><mrow id="A3.E23.m1.4.4.1.1.1.1.1.2" xref="A3.E23.m1.4.4.1.1.1.1.1.2.cmml"><mrow id="A3.E23.m1.4.4.1.1.1.1.1.2.2" xref="A3.E23.m1.4.4.1.1.1.1.1.2.2.cmml"><mo id="A3.E23.m1.4.4.1.1.1.1.1.2.2.1" rspace="0.167em" xref="A3.E23.m1.4.4.1.1.1.1.1.2.2.1.cmml">∇</mo><msub id="A3.E23.m1.4.4.1.1.1.1.1.2.2.2" xref="A3.E23.m1.4.4.1.1.1.1.1.2.2.2.cmml"><mi class="ltx_font_mathcaligraphic" id="A3.E23.m1.4.4.1.1.1.1.1.2.2.2.2" xref="A3.E23.m1.4.4.1.1.1.1.1.2.2.2.2.cmml">ℛ</mi><mi id="A3.E23.m1.4.4.1.1.1.1.1.2.2.2.3" xref="A3.E23.m1.4.4.1.1.1.1.1.2.2.2.3.cmml">σ</mi></msub></mrow><mo id="A3.E23.m1.4.4.1.1.1.1.1.2.1" xref="A3.E23.m1.4.4.1.1.1.1.1.2.1.cmml">⁢</mo><mrow id="A3.E23.m1.4.4.1.1.1.1.1.2.3.2" xref="A3.E23.m1.4.4.1.1.1.1.1.2.cmml"><mo id="A3.E23.m1.4.4.1.1.1.1.1.2.3.2.1" stretchy="false" xref="A3.E23.m1.4.4.1.1.1.1.1.2.cmml">(</mo><mi id="A3.E23.m1.1.1" xref="A3.E23.m1.1.1.cmml">𝐱</mi><mo id="A3.E23.m1.4.4.1.1.1.1.1.2.3.2.2" stretchy="false" xref="A3.E23.m1.4.4.1.1.1.1.1.2.cmml">)</mo></mrow></mrow><mo id="A3.E23.m1.4.4.1.1.1.1.1.1" xref="A3.E23.m1.4.4.1.1.1.1.1.1.cmml">+</mo><mrow id="A3.E23.m1.4.4.1.1.1.1.1.3" xref="A3.E23.m1.4.4.1.1.1.1.1.3.cmml"><mrow id="A3.E23.m1.4.4.1.1.1.1.1.3.2" xref="A3.E23.m1.4.4.1.1.1.1.1.3.2.cmml"><mrow id="A3.E23.m1.4.4.1.1.1.1.1.3.2.1" xref="A3.E23.m1.4.4.1.1.1.1.1.3.2.1.cmml"><mo id="A3.E23.m1.4.4.1.1.1.1.1.3.2.1.1" rspace="0.167em" xref="A3.E23.m1.4.4.1.1.1.1.1.3.2.1.1.cmml">∇</mo><mi id="A3.E23.m1.4.4.1.1.1.1.1.3.2.1.2" xref="A3.E23.m1.4.4.1.1.1.1.1.3.2.1.2.cmml">log</mi></mrow><mo id="A3.E23.m1.4.4.1.1.1.1.1.3.2a" lspace="0.167em" xref="A3.E23.m1.4.4.1.1.1.1.1.3.2.cmml">⁡</mo><mi id="A3.E23.m1.4.4.1.1.1.1.1.3.2.2" xref="A3.E23.m1.4.4.1.1.1.1.1.3.2.2.cmml">p</mi></mrow><mo id="A3.E23.m1.4.4.1.1.1.1.1.3.1" xref="A3.E23.m1.4.4.1.1.1.1.1.3.1.cmml">⁢</mo><mrow id="A3.E23.m1.4.4.1.1.1.1.1.3.3.2" xref="A3.E23.m1.4.4.1.1.1.1.1.3.cmml"><mo id="A3.E23.m1.4.4.1.1.1.1.1.3.3.2.1" stretchy="false" 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id="A3.E23.m1.2.2.cmml" xref="A3.E23.m1.2.2">𝐱</ci></apply></apply></apply><apply id="A3.E23.m1.4.4.1.1.2.cmml" xref="A3.E23.m1.4.4.1.1.2"><times id="A3.E23.m1.4.4.1.1.2.2.cmml" xref="A3.E23.m1.4.4.1.1.2.2"></times><ci id="A3.E23.m1.4.4.1.1.2.3.cmml" xref="A3.E23.m1.4.4.1.1.2.3">𝜎</ci><apply id="A3.E23.m1.4.4.1.1.2.4.cmml" xref="A3.E23.m1.4.4.1.1.2.4"><csymbol cd="ambiguous" id="A3.E23.m1.4.4.1.1.2.4.1.cmml" xref="A3.E23.m1.4.4.1.1.2.4">subscript</csymbol><ci id="A3.E23.m1.4.4.1.1.2.4.2.cmml" xref="A3.E23.m1.4.4.1.1.2.4.2">𝑒</ci><ci id="A3.E23.m1.4.4.1.1.2.4.3.cmml" xref="A3.E23.m1.4.4.1.1.2.4.3">𝑝</ci></apply><apply id="A3.E23.m1.4.4.1.1.2.1.1.1.cmml" xref="A3.E23.m1.4.4.1.1.2.1.1"><plus id="A3.E23.m1.4.4.1.1.2.1.1.1.1.cmml" xref="A3.E23.m1.4.4.1.1.2.1.1.1.1"></plus><apply id="A3.E23.m1.4.4.1.1.2.1.1.1.2.cmml" xref="A3.E23.m1.4.4.1.1.2.1.1.1.2"><csymbol cd="ambiguous" id="A3.E23.m1.4.4.1.1.2.1.1.1.2.1.cmml" xref="A3.E23.m1.4.4.1.1.2.1.1.1.2">superscript</csymbol><apply id="A3.E23.m1.4.4.1.1.2.1.1.1.2.2.1.cmml" xref="A3.E23.m1.4.4.1.1.2.1.1.1.2.2.2"><csymbol cd="latexml" id="A3.E23.m1.4.4.1.1.2.1.1.1.2.2.1.1.cmml" xref="A3.E23.m1.4.4.1.1.2.1.1.1.2.2.2.1">norm</csymbol><ci id="A3.E23.m1.3.3.cmml" xref="A3.E23.m1.3.3">𝐱</ci></apply><cn id="A3.E23.m1.4.4.1.1.2.1.1.1.2.3.cmml" type="integer" xref="A3.E23.m1.4.4.1.1.2.1.1.1.2.3">2</cn></apply><cn id="A3.E23.m1.4.4.1.1.2.1.1.1.3.cmml" type="integer" xref="A3.E23.m1.4.4.1.1.2.1.1.1.3">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.E23.m1.4c">\displaystyle\|\nabla\mathcal{R}_{\sigma}({\mathbf{x}})+\nabla\log p({\mathbf{% x}})\|\leq\sigma e_{p}(\|{\mathbf{x}}\|^{2}+1).</annotation><annotation encoding="application/x-llamapun" id="A3.E23.m1.4d">∥ ∇ caligraphic_R start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ( bold_x ) + ∇ roman_log italic_p ( bold_x ) ∥ ≤ italic_σ italic_e start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT ( ∥ bold_x ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + 1 ) .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(23)</span></td> </tr></tbody> </table> </div> </div> <div class="ltx_proof" id="A3.SS2.SSS1.12"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof.</h6> <div class="ltx_para" id="A3.SS2.SSS1.7.p1"> <p class="ltx_p" id="A3.SS2.SSS1.7.p1.4">To emphasize the dependence of <math alttext="\mathcal{R}" class="ltx_Math" display="inline" id="A3.SS2.SSS1.7.p1.1.m1.1"><semantics id="A3.SS2.SSS1.7.p1.1.m1.1a"><mi class="ltx_font_mathcaligraphic" id="A3.SS2.SSS1.7.p1.1.m1.1.1" xref="A3.SS2.SSS1.7.p1.1.m1.1.1.cmml">ℛ</mi><annotation-xml encoding="MathML-Content" id="A3.SS2.SSS1.7.p1.1.m1.1b"><ci id="A3.SS2.SSS1.7.p1.1.m1.1.1.cmml" xref="A3.SS2.SSS1.7.p1.1.m1.1.1">ℛ</ci></annotation-xml><annotation encoding="application/x-tex" id="A3.SS2.SSS1.7.p1.1.m1.1c">\mathcal{R}</annotation><annotation encoding="application/x-llamapun" id="A3.SS2.SSS1.7.p1.1.m1.1d">caligraphic_R</annotation></semantics></math> in <math alttext="\sigma" class="ltx_Math" display="inline" id="A3.SS2.SSS1.7.p1.2.m2.1"><semantics id="A3.SS2.SSS1.7.p1.2.m2.1a"><mi id="A3.SS2.SSS1.7.p1.2.m2.1.1" xref="A3.SS2.SSS1.7.p1.2.m2.1.1.cmml">σ</mi><annotation-xml encoding="MathML-Content" id="A3.SS2.SSS1.7.p1.2.m2.1b"><ci id="A3.SS2.SSS1.7.p1.2.m2.1.1.cmml" xref="A3.SS2.SSS1.7.p1.2.m2.1.1">𝜎</ci></annotation-xml><annotation encoding="application/x-tex" id="A3.SS2.SSS1.7.p1.2.m2.1c">\sigma</annotation><annotation encoding="application/x-llamapun" id="A3.SS2.SSS1.7.p1.2.m2.1d">italic_σ</annotation></semantics></math>, we will denote it as <math alttext="\mathcal{R}_{\sigma}" class="ltx_Math" display="inline" id="A3.SS2.SSS1.7.p1.3.m3.1"><semantics id="A3.SS2.SSS1.7.p1.3.m3.1a"><msub id="A3.SS2.SSS1.7.p1.3.m3.1.1" xref="A3.SS2.SSS1.7.p1.3.m3.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="A3.SS2.SSS1.7.p1.3.m3.1.1.2" xref="A3.SS2.SSS1.7.p1.3.m3.1.1.2.cmml">ℛ</mi><mi id="A3.SS2.SSS1.7.p1.3.m3.1.1.3" xref="A3.SS2.SSS1.7.p1.3.m3.1.1.3.cmml">σ</mi></msub><annotation-xml encoding="MathML-Content" id="A3.SS2.SSS1.7.p1.3.m3.1b"><apply id="A3.SS2.SSS1.7.p1.3.m3.1.1.cmml" xref="A3.SS2.SSS1.7.p1.3.m3.1.1"><csymbol cd="ambiguous" id="A3.SS2.SSS1.7.p1.3.m3.1.1.1.cmml" xref="A3.SS2.SSS1.7.p1.3.m3.1.1">subscript</csymbol><ci id="A3.SS2.SSS1.7.p1.3.m3.1.1.2.cmml" xref="A3.SS2.SSS1.7.p1.3.m3.1.1.2">ℛ</ci><ci id="A3.SS2.SSS1.7.p1.3.m3.1.1.3.cmml" xref="A3.SS2.SSS1.7.p1.3.m3.1.1.3">𝜎</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.SS2.SSS1.7.p1.3.m3.1c">\mathcal{R}_{\sigma}</annotation><annotation encoding="application/x-llamapun" id="A3.SS2.SSS1.7.p1.3.m3.1d">caligraphic_R start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT</annotation></semantics></math> in the following computations. The behavior of our regularization gradient <math alttext="\nabla\mathcal{R}_{\sigma}({\mathbf{x}})=-\mathbb{E}_{{\tilde{\mathbf{x}}}\sim p% _{\sigma}({\tilde{\mathbf{x}}}|{\mathbf{x}})}\left(\nabla\log p_{\sigma}({% \tilde{\mathbf{x}}})\right)" class="ltx_Math" display="inline" id="A3.SS2.SSS1.7.p1.4.m4.4"><semantics id="A3.SS2.SSS1.7.p1.4.m4.4a"><mrow id="A3.SS2.SSS1.7.p1.4.m4.4.4" xref="A3.SS2.SSS1.7.p1.4.m4.4.4.cmml"><mrow id="A3.SS2.SSS1.7.p1.4.m4.4.4.3" xref="A3.SS2.SSS1.7.p1.4.m4.4.4.3.cmml"><mrow id="A3.SS2.SSS1.7.p1.4.m4.4.4.3.2" xref="A3.SS2.SSS1.7.p1.4.m4.4.4.3.2.cmml"><mo id="A3.SS2.SSS1.7.p1.4.m4.4.4.3.2.1" rspace="0.167em" xref="A3.SS2.SSS1.7.p1.4.m4.4.4.3.2.1.cmml">∇</mo><msub id="A3.SS2.SSS1.7.p1.4.m4.4.4.3.2.2" xref="A3.SS2.SSS1.7.p1.4.m4.4.4.3.2.2.cmml"><mi class="ltx_font_mathcaligraphic" id="A3.SS2.SSS1.7.p1.4.m4.4.4.3.2.2.2" xref="A3.SS2.SSS1.7.p1.4.m4.4.4.3.2.2.2.cmml">ℛ</mi><mi id="A3.SS2.SSS1.7.p1.4.m4.4.4.3.2.2.3" 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xref="A3.SS2.SSS1.7.p1.4.m4.4.4.1.1.1.1.1.1"></times><apply id="A3.SS2.SSS1.7.p1.4.m4.4.4.1.1.1.1.1.2.cmml" xref="A3.SS2.SSS1.7.p1.4.m4.4.4.1.1.1.1.1.2"><apply id="A3.SS2.SSS1.7.p1.4.m4.4.4.1.1.1.1.1.2.1.cmml" xref="A3.SS2.SSS1.7.p1.4.m4.4.4.1.1.1.1.1.2.1"><ci id="A3.SS2.SSS1.7.p1.4.m4.4.4.1.1.1.1.1.2.1.1.cmml" xref="A3.SS2.SSS1.7.p1.4.m4.4.4.1.1.1.1.1.2.1.1">∇</ci><log id="A3.SS2.SSS1.7.p1.4.m4.4.4.1.1.1.1.1.2.1.2.cmml" xref="A3.SS2.SSS1.7.p1.4.m4.4.4.1.1.1.1.1.2.1.2"></log></apply><apply id="A3.SS2.SSS1.7.p1.4.m4.4.4.1.1.1.1.1.2.2.cmml" xref="A3.SS2.SSS1.7.p1.4.m4.4.4.1.1.1.1.1.2.2"><csymbol cd="ambiguous" id="A3.SS2.SSS1.7.p1.4.m4.4.4.1.1.1.1.1.2.2.1.cmml" xref="A3.SS2.SSS1.7.p1.4.m4.4.4.1.1.1.1.1.2.2">subscript</csymbol><ci id="A3.SS2.SSS1.7.p1.4.m4.4.4.1.1.1.1.1.2.2.2.cmml" xref="A3.SS2.SSS1.7.p1.4.m4.4.4.1.1.1.1.1.2.2.2">𝑝</ci><ci id="A3.SS2.SSS1.7.p1.4.m4.4.4.1.1.1.1.1.2.2.3.cmml" xref="A3.SS2.SSS1.7.p1.4.m4.4.4.1.1.1.1.1.2.2.3">𝜎</ci></apply></apply><apply id="A3.SS2.SSS1.7.p1.4.m4.3.3.cmml" xref="A3.SS2.SSS1.7.p1.4.m4.4.4.1.1.1.1.1.3.2"><ci id="A3.SS2.SSS1.7.p1.4.m4.3.3.1.cmml" xref="A3.SS2.SSS1.7.p1.4.m4.3.3.1">~</ci><ci id="A3.SS2.SSS1.7.p1.4.m4.3.3.2.cmml" xref="A3.SS2.SSS1.7.p1.4.m4.3.3.2">𝐱</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.SS2.SSS1.7.p1.4.m4.4c">\nabla\mathcal{R}_{\sigma}({\mathbf{x}})=-\mathbb{E}_{{\tilde{\mathbf{x}}}\sim p% _{\sigma}({\tilde{\mathbf{x}}}|{\mathbf{x}})}\left(\nabla\log p_{\sigma}({% \tilde{\mathbf{x}}})\right)</annotation><annotation encoding="application/x-llamapun" id="A3.SS2.SSS1.7.p1.4.m4.4d">∇ caligraphic_R start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ( bold_x ) = - blackboard_E start_POSTSUBSCRIPT over~ start_ARG bold_x end_ARG ∼ italic_p start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ( over~ start_ARG bold_x end_ARG | bold_x ) end_POSTSUBSCRIPT ( ∇ roman_log italic_p start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ( over~ start_ARG bold_x end_ARG ) )</annotation></semantics></math> is</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A8.EGx32"> <tbody id="A3.Ex43"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\|\nabla\mathcal{R}_{\sigma}({\mathbf{x}})+\nabla\log p({\mathbf{% x}})\|" class="ltx_Math" display="inline" id="A3.Ex43.m1.3"><semantics id="A3.Ex43.m1.3a"><mrow id="A3.Ex43.m1.3.3.1" xref="A3.Ex43.m1.3.3.2.cmml"><mo id="A3.Ex43.m1.3.3.1.2" stretchy="false" xref="A3.Ex43.m1.3.3.2.1.cmml">‖</mo><mrow id="A3.Ex43.m1.3.3.1.1" xref="A3.Ex43.m1.3.3.1.1.cmml"><mrow id="A3.Ex43.m1.3.3.1.1.2" xref="A3.Ex43.m1.3.3.1.1.2.cmml"><mrow id="A3.Ex43.m1.3.3.1.1.2.2" xref="A3.Ex43.m1.3.3.1.1.2.2.cmml"><mo id="A3.Ex43.m1.3.3.1.1.2.2.1" rspace="0.167em" xref="A3.Ex43.m1.3.3.1.1.2.2.1.cmml">∇</mo><msub id="A3.Ex43.m1.3.3.1.1.2.2.2" xref="A3.Ex43.m1.3.3.1.1.2.2.2.cmml"><mi class="ltx_font_mathcaligraphic" id="A3.Ex43.m1.3.3.1.1.2.2.2.2" xref="A3.Ex43.m1.3.3.1.1.2.2.2.2.cmml">ℛ</mi><mi id="A3.Ex43.m1.3.3.1.1.2.2.2.3" xref="A3.Ex43.m1.3.3.1.1.2.2.2.3.cmml">σ</mi></msub></mrow><mo id="A3.Ex43.m1.3.3.1.1.2.1" xref="A3.Ex43.m1.3.3.1.1.2.1.cmml">⁢</mo><mrow id="A3.Ex43.m1.3.3.1.1.2.3.2" xref="A3.Ex43.m1.3.3.1.1.2.cmml"><mo id="A3.Ex43.m1.3.3.1.1.2.3.2.1" stretchy="false" xref="A3.Ex43.m1.3.3.1.1.2.cmml">(</mo><mi id="A3.Ex43.m1.1.1" xref="A3.Ex43.m1.1.1.cmml">𝐱</mi><mo id="A3.Ex43.m1.3.3.1.1.2.3.2.2" stretchy="false" xref="A3.Ex43.m1.3.3.1.1.2.cmml">)</mo></mrow></mrow><mo id="A3.Ex43.m1.3.3.1.1.1" xref="A3.Ex43.m1.3.3.1.1.1.cmml">+</mo><mrow id="A3.Ex43.m1.3.3.1.1.3" xref="A3.Ex43.m1.3.3.1.1.3.cmml"><mrow id="A3.Ex43.m1.3.3.1.1.3.2" xref="A3.Ex43.m1.3.3.1.1.3.2.cmml"><mrow id="A3.Ex43.m1.3.3.1.1.3.2.1" xref="A3.Ex43.m1.3.3.1.1.3.2.1.cmml"><mo id="A3.Ex43.m1.3.3.1.1.3.2.1.1" rspace="0.167em" 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xref="A3.Ex43.m1.3.3.1.1.3.1"></times><apply id="A3.Ex43.m1.3.3.1.1.3.2.cmml" xref="A3.Ex43.m1.3.3.1.1.3.2"><apply id="A3.Ex43.m1.3.3.1.1.3.2.1.cmml" xref="A3.Ex43.m1.3.3.1.1.3.2.1"><ci id="A3.Ex43.m1.3.3.1.1.3.2.1.1.cmml" xref="A3.Ex43.m1.3.3.1.1.3.2.1.1">∇</ci><log id="A3.Ex43.m1.3.3.1.1.3.2.1.2.cmml" xref="A3.Ex43.m1.3.3.1.1.3.2.1.2"></log></apply><ci id="A3.Ex43.m1.3.3.1.1.3.2.2.cmml" xref="A3.Ex43.m1.3.3.1.1.3.2.2">𝑝</ci></apply><ci id="A3.Ex43.m1.2.2.cmml" xref="A3.Ex43.m1.2.2">𝐱</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.Ex43.m1.3c">\displaystyle\|\nabla\mathcal{R}_{\sigma}({\mathbf{x}})+\nabla\log p({\mathbf{% x}})\|</annotation><annotation encoding="application/x-llamapun" id="A3.Ex43.m1.3d">∥ ∇ caligraphic_R start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ( bold_x ) + ∇ roman_log italic_p ( bold_x ) ∥</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math 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id="A3.Ex43.m2.2.2.2.cmml" xref="A3.Ex43.m2.2.2.2">𝐱</ci></apply></apply></apply><apply id="A3.Ex43.m2.4.4.1.1.1.3.cmml" xref="A3.Ex43.m2.4.4.1.1.1.3"><times id="A3.Ex43.m2.4.4.1.1.1.3.1.cmml" xref="A3.Ex43.m2.4.4.1.1.1.3.1"></times><apply id="A3.Ex43.m2.4.4.1.1.1.3.2.cmml" xref="A3.Ex43.m2.4.4.1.1.1.3.2"><apply id="A3.Ex43.m2.4.4.1.1.1.3.2.1.cmml" xref="A3.Ex43.m2.4.4.1.1.1.3.2.1"><ci id="A3.Ex43.m2.4.4.1.1.1.3.2.1.1.cmml" xref="A3.Ex43.m2.4.4.1.1.1.3.2.1.1">∇</ci><log id="A3.Ex43.m2.4.4.1.1.1.3.2.1.2.cmml" xref="A3.Ex43.m2.4.4.1.1.1.3.2.1.2"></log></apply><ci id="A3.Ex43.m2.4.4.1.1.1.3.2.2.cmml" xref="A3.Ex43.m2.4.4.1.1.1.3.2.2">𝑝</ci></apply><ci id="A3.Ex43.m2.3.3.cmml" xref="A3.Ex43.m2.3.3">𝐱</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.Ex43.m2.4c">\displaystyle=\|\mathbb{E}_{{\tilde{\mathbf{x}}}\sim p_{\sigma}({\tilde{% \mathbf{x}}}|{\mathbf{x}})}\left(\nabla\log p_{\sigma}({\tilde{\mathbf{x}}})% \right)-\nabla\log p({\mathbf{x}})\|</annotation><annotation encoding="application/x-llamapun" id="A3.Ex43.m2.4d">= ∥ blackboard_E start_POSTSUBSCRIPT over~ start_ARG bold_x end_ARG ∼ italic_p start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ( over~ start_ARG bold_x end_ARG | bold_x ) end_POSTSUBSCRIPT ( ∇ roman_log italic_p start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ( over~ start_ARG bold_x end_ARG ) ) - ∇ roman_log italic_p ( bold_x ) ∥</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="A3.Ex44"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_eqn_cell"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\leq\|\mathbb{E}_{{\tilde{\mathbf{x}}}\sim p_{\sigma}({\tilde{% \mathbf{x}}}|{\mathbf{x}})}\left(\nabla\log p_{\sigma}({\tilde{\mathbf{x}}})-% \nabla\log p({\tilde{\mathbf{x}}})\right)\|+\|\mathbb{E}_{{\tilde{\mathbf{x}}}% \sim p_{\sigma}({\tilde{\mathbf{x}}}|{\mathbf{x}})}\left(\nabla\log p({\tilde{% \mathbf{x}}})\right)-\nabla\log p({\mathbf{x}})\|" class="ltx_Math" display="inline" id="A3.Ex44.m1.8"><semantics id="A3.Ex44.m1.8a"><mrow id="A3.Ex44.m1.8.8" xref="A3.Ex44.m1.8.8.cmml"><mi id="A3.Ex44.m1.8.8.4" xref="A3.Ex44.m1.8.8.4.cmml"></mi><mo id="A3.Ex44.m1.8.8.3" xref="A3.Ex44.m1.8.8.3.cmml">≤</mo><mrow id="A3.Ex44.m1.8.8.2" xref="A3.Ex44.m1.8.8.2.cmml"><mrow id="A3.Ex44.m1.7.7.1.1.1" xref="A3.Ex44.m1.7.7.1.1.2.cmml"><mo id="A3.Ex44.m1.7.7.1.1.1.2" stretchy="false" xref="A3.Ex44.m1.7.7.1.1.2.1.cmml">‖</mo><mrow id="A3.Ex44.m1.7.7.1.1.1.1" xref="A3.Ex44.m1.7.7.1.1.1.1.cmml"><msub id="A3.Ex44.m1.7.7.1.1.1.1.3" xref="A3.Ex44.m1.7.7.1.1.1.1.3.cmml"><mi id="A3.Ex44.m1.7.7.1.1.1.1.3.2" xref="A3.Ex44.m1.7.7.1.1.1.1.3.2.cmml">𝔼</mi><mrow id="A3.Ex44.m1.1.1.1" xref="A3.Ex44.m1.1.1.1.cmml"><mover accent="true" id="A3.Ex44.m1.1.1.1.3" xref="A3.Ex44.m1.1.1.1.3.cmml"><mi id="A3.Ex44.m1.1.1.1.3.2" xref="A3.Ex44.m1.1.1.1.3.2.cmml">𝐱</mi><mo id="A3.Ex44.m1.1.1.1.3.1" xref="A3.Ex44.m1.1.1.1.3.1.cmml">~</mo></mover><mo id="A3.Ex44.m1.1.1.1.2" xref="A3.Ex44.m1.1.1.1.2.cmml">∼</mo><mrow id="A3.Ex44.m1.1.1.1.1" xref="A3.Ex44.m1.1.1.1.1.cmml"><msub id="A3.Ex44.m1.1.1.1.1.3" xref="A3.Ex44.m1.1.1.1.1.3.cmml"><mi id="A3.Ex44.m1.1.1.1.1.3.2" xref="A3.Ex44.m1.1.1.1.1.3.2.cmml">p</mi><mi id="A3.Ex44.m1.1.1.1.1.3.3" xref="A3.Ex44.m1.1.1.1.1.3.3.cmml">σ</mi></msub><mo id="A3.Ex44.m1.1.1.1.1.2" xref="A3.Ex44.m1.1.1.1.1.2.cmml">⁢</mo><mrow id="A3.Ex44.m1.1.1.1.1.1.1" xref="A3.Ex44.m1.1.1.1.1.1.1.1.cmml"><mo id="A3.Ex44.m1.1.1.1.1.1.1.2" stretchy="false" xref="A3.Ex44.m1.1.1.1.1.1.1.1.cmml">(</mo><mrow id="A3.Ex44.m1.1.1.1.1.1.1.1" xref="A3.Ex44.m1.1.1.1.1.1.1.1.cmml"><mover accent="true" id="A3.Ex44.m1.1.1.1.1.1.1.1.2" xref="A3.Ex44.m1.1.1.1.1.1.1.1.2.cmml"><mi id="A3.Ex44.m1.1.1.1.1.1.1.1.2.2" xref="A3.Ex44.m1.1.1.1.1.1.1.1.2.2.cmml">𝐱</mi><mo id="A3.Ex44.m1.1.1.1.1.1.1.1.2.1" xref="A3.Ex44.m1.1.1.1.1.1.1.1.2.1.cmml">~</mo></mover><mo fence="false" id="A3.Ex44.m1.1.1.1.1.1.1.1.1" xref="A3.Ex44.m1.1.1.1.1.1.1.1.1.cmml">|</mo><mi id="A3.Ex44.m1.1.1.1.1.1.1.1.3" xref="A3.Ex44.m1.1.1.1.1.1.1.1.3.cmml">𝐱</mi></mrow><mo id="A3.Ex44.m1.1.1.1.1.1.1.3" stretchy="false" xref="A3.Ex44.m1.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow></msub><mo id="A3.Ex44.m1.7.7.1.1.1.1.2" xref="A3.Ex44.m1.7.7.1.1.1.1.2.cmml">⁢</mo><mrow id="A3.Ex44.m1.7.7.1.1.1.1.1.1" xref="A3.Ex44.m1.7.7.1.1.1.1.1.1.1.cmml"><mo id="A3.Ex44.m1.7.7.1.1.1.1.1.1.2" xref="A3.Ex44.m1.7.7.1.1.1.1.1.1.1.cmml">(</mo><mrow id="A3.Ex44.m1.7.7.1.1.1.1.1.1.1" xref="A3.Ex44.m1.7.7.1.1.1.1.1.1.1.cmml"><mrow id="A3.Ex44.m1.7.7.1.1.1.1.1.1.1.2" xref="A3.Ex44.m1.7.7.1.1.1.1.1.1.1.2.cmml"><mrow id="A3.Ex44.m1.7.7.1.1.1.1.1.1.1.2.2" xref="A3.Ex44.m1.7.7.1.1.1.1.1.1.1.2.2.cmml"><mrow id="A3.Ex44.m1.7.7.1.1.1.1.1.1.1.2.2.1" xref="A3.Ex44.m1.7.7.1.1.1.1.1.1.1.2.2.1.cmml"><mo id="A3.Ex44.m1.7.7.1.1.1.1.1.1.1.2.2.1.1" rspace="0.167em" xref="A3.Ex44.m1.7.7.1.1.1.1.1.1.1.2.2.1.1.cmml">∇</mo><mi id="A3.Ex44.m1.7.7.1.1.1.1.1.1.1.2.2.1.2" xref="A3.Ex44.m1.7.7.1.1.1.1.1.1.1.2.2.1.2.cmml">log</mi></mrow><mo id="A3.Ex44.m1.7.7.1.1.1.1.1.1.1.2.2a" lspace="0.167em" xref="A3.Ex44.m1.7.7.1.1.1.1.1.1.1.2.2.cmml">⁡</mo><msub id="A3.Ex44.m1.7.7.1.1.1.1.1.1.1.2.2.2" xref="A3.Ex44.m1.7.7.1.1.1.1.1.1.1.2.2.2.cmml"><mi id="A3.Ex44.m1.7.7.1.1.1.1.1.1.1.2.2.2.2" xref="A3.Ex44.m1.7.7.1.1.1.1.1.1.1.2.2.2.2.cmml">p</mi><mi id="A3.Ex44.m1.7.7.1.1.1.1.1.1.1.2.2.2.3" xref="A3.Ex44.m1.7.7.1.1.1.1.1.1.1.2.2.2.3.cmml">σ</mi></msub></mrow><mo id="A3.Ex44.m1.7.7.1.1.1.1.1.1.1.2.1" xref="A3.Ex44.m1.7.7.1.1.1.1.1.1.1.2.1.cmml">⁢</mo><mrow id="A3.Ex44.m1.7.7.1.1.1.1.1.1.1.2.3.2" xref="A3.Ex44.m1.3.3.cmml"><mo id="A3.Ex44.m1.7.7.1.1.1.1.1.1.1.2.3.2.1" stretchy="false" xref="A3.Ex44.m1.3.3.cmml">(</mo><mover accent="true" id="A3.Ex44.m1.3.3" xref="A3.Ex44.m1.3.3.cmml"><mi id="A3.Ex44.m1.3.3.2" xref="A3.Ex44.m1.3.3.2.cmml">𝐱</mi><mo id="A3.Ex44.m1.3.3.1" xref="A3.Ex44.m1.3.3.1.cmml">~</mo></mover><mo id="A3.Ex44.m1.7.7.1.1.1.1.1.1.1.2.3.2.2" stretchy="false" xref="A3.Ex44.m1.3.3.cmml">)</mo></mrow></mrow><mo id="A3.Ex44.m1.7.7.1.1.1.1.1.1.1.1" xref="A3.Ex44.m1.7.7.1.1.1.1.1.1.1.1.cmml">−</mo><mrow id="A3.Ex44.m1.7.7.1.1.1.1.1.1.1.3" xref="A3.Ex44.m1.7.7.1.1.1.1.1.1.1.3.cmml"><mrow id="A3.Ex44.m1.7.7.1.1.1.1.1.1.1.3.2" xref="A3.Ex44.m1.7.7.1.1.1.1.1.1.1.3.2.cmml"><mrow id="A3.Ex44.m1.7.7.1.1.1.1.1.1.1.3.2.1" xref="A3.Ex44.m1.7.7.1.1.1.1.1.1.1.3.2.1.cmml"><mo id="A3.Ex44.m1.7.7.1.1.1.1.1.1.1.3.2.1.1" rspace="0.167em" xref="A3.Ex44.m1.7.7.1.1.1.1.1.1.1.3.2.1.1.cmml">∇</mo><mi id="A3.Ex44.m1.7.7.1.1.1.1.1.1.1.3.2.1.2" xref="A3.Ex44.m1.7.7.1.1.1.1.1.1.1.3.2.1.2.cmml">log</mi></mrow><mo id="A3.Ex44.m1.7.7.1.1.1.1.1.1.1.3.2a" lspace="0.167em" xref="A3.Ex44.m1.7.7.1.1.1.1.1.1.1.3.2.cmml">⁡</mo><mi id="A3.Ex44.m1.7.7.1.1.1.1.1.1.1.3.2.2" xref="A3.Ex44.m1.7.7.1.1.1.1.1.1.1.3.2.2.cmml">p</mi></mrow><mo id="A3.Ex44.m1.7.7.1.1.1.1.1.1.1.3.1" xref="A3.Ex44.m1.7.7.1.1.1.1.1.1.1.3.1.cmml">⁢</mo><mrow id="A3.Ex44.m1.7.7.1.1.1.1.1.1.1.3.3.2" xref="A3.Ex44.m1.4.4.cmml"><mo id="A3.Ex44.m1.7.7.1.1.1.1.1.1.1.3.3.2.1" stretchy="false" xref="A3.Ex44.m1.4.4.cmml">(</mo><mover accent="true" id="A3.Ex44.m1.4.4" xref="A3.Ex44.m1.4.4.cmml"><mi id="A3.Ex44.m1.4.4.2" xref="A3.Ex44.m1.4.4.2.cmml">𝐱</mi><mo id="A3.Ex44.m1.4.4.1" xref="A3.Ex44.m1.4.4.1.cmml">~</mo></mover><mo id="A3.Ex44.m1.7.7.1.1.1.1.1.1.1.3.3.2.2" stretchy="false" xref="A3.Ex44.m1.4.4.cmml">)</mo></mrow></mrow></mrow><mo id="A3.Ex44.m1.7.7.1.1.1.1.1.1.3" xref="A3.Ex44.m1.7.7.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="A3.Ex44.m1.7.7.1.1.1.3" stretchy="false" xref="A3.Ex44.m1.7.7.1.1.2.1.cmml">‖</mo></mrow><mo id="A3.Ex44.m1.8.8.2.3" xref="A3.Ex44.m1.8.8.2.3.cmml">+</mo><mrow id="A3.Ex44.m1.8.8.2.2.1" xref="A3.Ex44.m1.8.8.2.2.2.cmml"><mo id="A3.Ex44.m1.8.8.2.2.1.2" stretchy="false" xref="A3.Ex44.m1.8.8.2.2.2.1.cmml">‖</mo><mrow id="A3.Ex44.m1.8.8.2.2.1.1" xref="A3.Ex44.m1.8.8.2.2.1.1.cmml"><mrow id="A3.Ex44.m1.8.8.2.2.1.1.1" xref="A3.Ex44.m1.8.8.2.2.1.1.1.cmml"><msub id="A3.Ex44.m1.8.8.2.2.1.1.1.3" xref="A3.Ex44.m1.8.8.2.2.1.1.1.3.cmml"><mi id="A3.Ex44.m1.8.8.2.2.1.1.1.3.2" xref="A3.Ex44.m1.8.8.2.2.1.1.1.3.2.cmml">𝔼</mi><mrow id="A3.Ex44.m1.2.2.1" xref="A3.Ex44.m1.2.2.1.cmml"><mover accent="true" id="A3.Ex44.m1.2.2.1.3" xref="A3.Ex44.m1.2.2.1.3.cmml"><mi id="A3.Ex44.m1.2.2.1.3.2" xref="A3.Ex44.m1.2.2.1.3.2.cmml">𝐱</mi><mo id="A3.Ex44.m1.2.2.1.3.1" xref="A3.Ex44.m1.2.2.1.3.1.cmml">~</mo></mover><mo id="A3.Ex44.m1.2.2.1.2" xref="A3.Ex44.m1.2.2.1.2.cmml">∼</mo><mrow id="A3.Ex44.m1.2.2.1.1" xref="A3.Ex44.m1.2.2.1.1.cmml"><msub id="A3.Ex44.m1.2.2.1.1.3" xref="A3.Ex44.m1.2.2.1.1.3.cmml"><mi id="A3.Ex44.m1.2.2.1.1.3.2" xref="A3.Ex44.m1.2.2.1.1.3.2.cmml">p</mi><mi id="A3.Ex44.m1.2.2.1.1.3.3" xref="A3.Ex44.m1.2.2.1.1.3.3.cmml">σ</mi></msub><mo id="A3.Ex44.m1.2.2.1.1.2" xref="A3.Ex44.m1.2.2.1.1.2.cmml">⁢</mo><mrow id="A3.Ex44.m1.2.2.1.1.1.1" xref="A3.Ex44.m1.2.2.1.1.1.1.1.cmml"><mo id="A3.Ex44.m1.2.2.1.1.1.1.2" stretchy="false" xref="A3.Ex44.m1.2.2.1.1.1.1.1.cmml">(</mo><mrow id="A3.Ex44.m1.2.2.1.1.1.1.1" xref="A3.Ex44.m1.2.2.1.1.1.1.1.cmml"><mover accent="true" id="A3.Ex44.m1.2.2.1.1.1.1.1.2" xref="A3.Ex44.m1.2.2.1.1.1.1.1.2.cmml"><mi id="A3.Ex44.m1.2.2.1.1.1.1.1.2.2" xref="A3.Ex44.m1.2.2.1.1.1.1.1.2.2.cmml">𝐱</mi><mo id="A3.Ex44.m1.2.2.1.1.1.1.1.2.1" xref="A3.Ex44.m1.2.2.1.1.1.1.1.2.1.cmml">~</mo></mover><mo fence="false" id="A3.Ex44.m1.2.2.1.1.1.1.1.1" xref="A3.Ex44.m1.2.2.1.1.1.1.1.1.cmml">|</mo><mi id="A3.Ex44.m1.2.2.1.1.1.1.1.3" xref="A3.Ex44.m1.2.2.1.1.1.1.1.3.cmml">𝐱</mi></mrow><mo id="A3.Ex44.m1.2.2.1.1.1.1.3" stretchy="false" xref="A3.Ex44.m1.2.2.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow></msub><mo id="A3.Ex44.m1.8.8.2.2.1.1.1.2" xref="A3.Ex44.m1.8.8.2.2.1.1.1.2.cmml">⁢</mo><mrow id="A3.Ex44.m1.8.8.2.2.1.1.1.1.1" xref="A3.Ex44.m1.8.8.2.2.1.1.1.1.1.1.cmml"><mo id="A3.Ex44.m1.8.8.2.2.1.1.1.1.1.2" xref="A3.Ex44.m1.8.8.2.2.1.1.1.1.1.1.cmml">(</mo><mrow id="A3.Ex44.m1.8.8.2.2.1.1.1.1.1.1" xref="A3.Ex44.m1.8.8.2.2.1.1.1.1.1.1.cmml"><mrow id="A3.Ex44.m1.8.8.2.2.1.1.1.1.1.1.2" xref="A3.Ex44.m1.8.8.2.2.1.1.1.1.1.1.2.cmml"><mrow id="A3.Ex44.m1.8.8.2.2.1.1.1.1.1.1.2.1" xref="A3.Ex44.m1.8.8.2.2.1.1.1.1.1.1.2.1.cmml"><mo id="A3.Ex44.m1.8.8.2.2.1.1.1.1.1.1.2.1.1" rspace="0.167em" xref="A3.Ex44.m1.8.8.2.2.1.1.1.1.1.1.2.1.1.cmml">∇</mo><mi id="A3.Ex44.m1.8.8.2.2.1.1.1.1.1.1.2.1.2" xref="A3.Ex44.m1.8.8.2.2.1.1.1.1.1.1.2.1.2.cmml">log</mi></mrow><mo id="A3.Ex44.m1.8.8.2.2.1.1.1.1.1.1.2a" lspace="0.167em" xref="A3.Ex44.m1.8.8.2.2.1.1.1.1.1.1.2.cmml">⁡</mo><mi id="A3.Ex44.m1.8.8.2.2.1.1.1.1.1.1.2.2" xref="A3.Ex44.m1.8.8.2.2.1.1.1.1.1.1.2.2.cmml">p</mi></mrow><mo id="A3.Ex44.m1.8.8.2.2.1.1.1.1.1.1.1" xref="A3.Ex44.m1.8.8.2.2.1.1.1.1.1.1.1.cmml">⁢</mo><mrow id="A3.Ex44.m1.8.8.2.2.1.1.1.1.1.1.3.2" xref="A3.Ex44.m1.5.5.cmml"><mo 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xref="A3.Ex44.m1.8.8.2.2.1.1.3.2.1.2.cmml">log</mi></mrow><mo id="A3.Ex44.m1.8.8.2.2.1.1.3.2a" lspace="0.167em" xref="A3.Ex44.m1.8.8.2.2.1.1.3.2.cmml">⁡</mo><mi id="A3.Ex44.m1.8.8.2.2.1.1.3.2.2" xref="A3.Ex44.m1.8.8.2.2.1.1.3.2.2.cmml">p</mi></mrow><mo id="A3.Ex44.m1.8.8.2.2.1.1.3.1" xref="A3.Ex44.m1.8.8.2.2.1.1.3.1.cmml">⁢</mo><mrow id="A3.Ex44.m1.8.8.2.2.1.1.3.3.2" xref="A3.Ex44.m1.8.8.2.2.1.1.3.cmml"><mo id="A3.Ex44.m1.8.8.2.2.1.1.3.3.2.1" stretchy="false" xref="A3.Ex44.m1.8.8.2.2.1.1.3.cmml">(</mo><mi id="A3.Ex44.m1.6.6" xref="A3.Ex44.m1.6.6.cmml">𝐱</mi><mo id="A3.Ex44.m1.8.8.2.2.1.1.3.3.2.2" stretchy="false" xref="A3.Ex44.m1.8.8.2.2.1.1.3.cmml">)</mo></mrow></mrow></mrow><mo id="A3.Ex44.m1.8.8.2.2.1.3" stretchy="false" xref="A3.Ex44.m1.8.8.2.2.2.1.cmml">‖</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="A3.Ex44.m1.8b"><apply id="A3.Ex44.m1.8.8.cmml" xref="A3.Ex44.m1.8.8"><leq id="A3.Ex44.m1.8.8.3.cmml" xref="A3.Ex44.m1.8.8.3"></leq><csymbol cd="latexml" 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xref="A3.Ex44.m1.8.8.2.2.1.1.3.2.1.1">∇</ci><log id="A3.Ex44.m1.8.8.2.2.1.1.3.2.1.2.cmml" xref="A3.Ex44.m1.8.8.2.2.1.1.3.2.1.2"></log></apply><ci id="A3.Ex44.m1.8.8.2.2.1.1.3.2.2.cmml" xref="A3.Ex44.m1.8.8.2.2.1.1.3.2.2">𝑝</ci></apply><ci id="A3.Ex44.m1.6.6.cmml" xref="A3.Ex44.m1.6.6">𝐱</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.Ex44.m1.8c">\displaystyle\leq\|\mathbb{E}_{{\tilde{\mathbf{x}}}\sim p_{\sigma}({\tilde{% \mathbf{x}}}|{\mathbf{x}})}\left(\nabla\log p_{\sigma}({\tilde{\mathbf{x}}})-% \nabla\log p({\tilde{\mathbf{x}}})\right)\|+\|\mathbb{E}_{{\tilde{\mathbf{x}}}% \sim p_{\sigma}({\tilde{\mathbf{x}}}|{\mathbf{x}})}\left(\nabla\log p({\tilde{% \mathbf{x}}})\right)-\nabla\log p({\mathbf{x}})\|</annotation><annotation encoding="application/x-llamapun" id="A3.Ex44.m1.8d">≤ ∥ blackboard_E start_POSTSUBSCRIPT over~ start_ARG bold_x end_ARG ∼ italic_p start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ( over~ start_ARG bold_x end_ARG | bold_x ) end_POSTSUBSCRIPT ( ∇ roman_log italic_p start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ( over~ start_ARG bold_x end_ARG ) - ∇ roman_log italic_p ( over~ start_ARG bold_x end_ARG ) ) ∥ + ∥ blackboard_E start_POSTSUBSCRIPT over~ start_ARG bold_x end_ARG ∼ italic_p start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ( over~ start_ARG bold_x end_ARG | bold_x ) end_POSTSUBSCRIPT ( ∇ roman_log italic_p ( over~ start_ARG bold_x end_ARG ) ) - ∇ roman_log italic_p ( bold_x ) ∥</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="A3.Ex45"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_eqn_cell"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\leq\sigma^{2}\mathbb{E}_{{\tilde{\mathbf{x}}}\sim p_{\sigma}({% \tilde{\mathbf{x}}}|{\mathbf{x}})}\left(c_{p}({\tilde{\mathbf{x}}})\right)+\|% \mathbb{E}_{{\tilde{\mathbf{x}}}\sim p_{\sigma}({\tilde{\mathbf{x}}}|{\mathbf{% x}})}\left(\nabla\log p({\tilde{\mathbf{x}}})\right)-\nabla\log p({\mathbf{x}})\|" class="ltx_Math" display="inline" id="A3.Ex45.m1.7"><semantics id="A3.Ex45.m1.7a"><mrow id="A3.Ex45.m1.7.7" xref="A3.Ex45.m1.7.7.cmml"><mi id="A3.Ex45.m1.7.7.4" xref="A3.Ex45.m1.7.7.4.cmml"></mi><mo id="A3.Ex45.m1.7.7.3" xref="A3.Ex45.m1.7.7.3.cmml">≤</mo><mrow id="A3.Ex45.m1.7.7.2" xref="A3.Ex45.m1.7.7.2.cmml"><mrow id="A3.Ex45.m1.6.6.1.1" xref="A3.Ex45.m1.6.6.1.1.cmml"><msup id="A3.Ex45.m1.6.6.1.1.3" xref="A3.Ex45.m1.6.6.1.1.3.cmml"><mi id="A3.Ex45.m1.6.6.1.1.3.2" xref="A3.Ex45.m1.6.6.1.1.3.2.cmml">σ</mi><mn id="A3.Ex45.m1.6.6.1.1.3.3" xref="A3.Ex45.m1.6.6.1.1.3.3.cmml">2</mn></msup><mo id="A3.Ex45.m1.6.6.1.1.2" xref="A3.Ex45.m1.6.6.1.1.2.cmml">⁢</mo><msub id="A3.Ex45.m1.6.6.1.1.4" xref="A3.Ex45.m1.6.6.1.1.4.cmml"><mi id="A3.Ex45.m1.6.6.1.1.4.2" xref="A3.Ex45.m1.6.6.1.1.4.2.cmml">𝔼</mi><mrow id="A3.Ex45.m1.1.1.1" 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\tilde{\mathbf{x}}}|{\mathbf{x}})}\left(c_{p}({\tilde{\mathbf{x}}})\right)+\|% \mathbb{E}_{{\tilde{\mathbf{x}}}\sim p_{\sigma}({\tilde{\mathbf{x}}}|{\mathbf{% x}})}\left(\nabla\log p({\tilde{\mathbf{x}}})\right)-\nabla\log p({\mathbf{x}})\|</annotation><annotation encoding="application/x-llamapun" id="A3.Ex45.m1.7d">≤ italic_σ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT blackboard_E start_POSTSUBSCRIPT over~ start_ARG bold_x end_ARG ∼ italic_p start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ( over~ start_ARG bold_x end_ARG | bold_x ) end_POSTSUBSCRIPT ( italic_c start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT ( over~ start_ARG bold_x end_ARG ) ) + ∥ blackboard_E start_POSTSUBSCRIPT over~ start_ARG bold_x end_ARG ∼ italic_p start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ( over~ start_ARG bold_x end_ARG | bold_x ) end_POSTSUBSCRIPT ( ∇ roman_log italic_p ( over~ start_ARG bold_x end_ARG ) ) - ∇ roman_log italic_p ( bold_x ) ∥</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="A3.Ex46"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_eqn_cell"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\leq\sigma^{2}a_{p}\mathbb{E}_{{\tilde{\mathbf{x}}}\sim p_{\sigma% }({\tilde{\mathbf{x}}}|{\mathbf{x}})}\left(\|{\tilde{\mathbf{x}}}\|^{3}+1% \right)+\|\mathbb{E}_{{\tilde{\mathbf{x}}}\sim p_{\sigma}({\tilde{\mathbf{x}}}% |{\mathbf{x}})}\left(\nabla\log p({\tilde{\mathbf{x}}})\right)-\nabla\log p({% \mathbf{x}})\|." class="ltx_Math" display="inline" id="A3.Ex46.m1.6"><semantics id="A3.Ex46.m1.6a"><mrow id="A3.Ex46.m1.6.6.1" xref="A3.Ex46.m1.6.6.1.1.cmml"><mrow id="A3.Ex46.m1.6.6.1.1" xref="A3.Ex46.m1.6.6.1.1.cmml"><mi id="A3.Ex46.m1.6.6.1.1.4" xref="A3.Ex46.m1.6.6.1.1.4.cmml"></mi><mo id="A3.Ex46.m1.6.6.1.1.3" xref="A3.Ex46.m1.6.6.1.1.3.cmml">≤</mo><mrow id="A3.Ex46.m1.6.6.1.1.2" 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xref="A3.Ex46.m1.6.6.1.1.2.2.1.1.3.2.1.1">∇</ci><log id="A3.Ex46.m1.6.6.1.1.2.2.1.1.3.2.1.2.cmml" xref="A3.Ex46.m1.6.6.1.1.2.2.1.1.3.2.1.2"></log></apply><ci id="A3.Ex46.m1.6.6.1.1.2.2.1.1.3.2.2.cmml" xref="A3.Ex46.m1.6.6.1.1.2.2.1.1.3.2.2">𝑝</ci></apply><ci id="A3.Ex46.m1.5.5.cmml" xref="A3.Ex46.m1.5.5">𝐱</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.Ex46.m1.6c">\displaystyle\leq\sigma^{2}a_{p}\mathbb{E}_{{\tilde{\mathbf{x}}}\sim p_{\sigma% }({\tilde{\mathbf{x}}}|{\mathbf{x}})}\left(\|{\tilde{\mathbf{x}}}\|^{3}+1% \right)+\|\mathbb{E}_{{\tilde{\mathbf{x}}}\sim p_{\sigma}({\tilde{\mathbf{x}}}% |{\mathbf{x}})}\left(\nabla\log p({\tilde{\mathbf{x}}})\right)-\nabla\log p({% \mathbf{x}})\|.</annotation><annotation encoding="application/x-llamapun" id="A3.Ex46.m1.6d">≤ italic_σ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_a start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT blackboard_E start_POSTSUBSCRIPT over~ start_ARG bold_x end_ARG ∼ italic_p start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ( over~ start_ARG bold_x end_ARG | bold_x ) end_POSTSUBSCRIPT ( ∥ over~ start_ARG bold_x end_ARG ∥ start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT + 1 ) + ∥ blackboard_E start_POSTSUBSCRIPT over~ start_ARG bold_x end_ARG ∼ italic_p start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ( over~ start_ARG bold_x end_ARG | bold_x ) end_POSTSUBSCRIPT ( ∇ roman_log italic_p ( over~ start_ARG bold_x end_ARG ) ) - ∇ roman_log italic_p ( bold_x ) ∥ .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> </div> <div class="ltx_para" id="A3.SS2.SSS1.8.p2"> <p class="ltx_p" id="A3.SS2.SSS1.8.p2.3">By using that for <math alttext="a,b\in{\mathbb{R}}^{+}" class="ltx_Math" display="inline" id="A3.SS2.SSS1.8.p2.1.m1.2"><semantics id="A3.SS2.SSS1.8.p2.1.m1.2a"><mrow id="A3.SS2.SSS1.8.p2.1.m1.2.3" xref="A3.SS2.SSS1.8.p2.1.m1.2.3.cmml"><mrow id="A3.SS2.SSS1.8.p2.1.m1.2.3.2.2" 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xref="A3.SS2.SSS1.8.p2.2.m2.2.2.2.1.1.1.2"><csymbol cd="ambiguous" id="A3.SS2.SSS1.8.p2.2.m2.2.2.2.1.1.1.2.1.cmml" xref="A3.SS2.SSS1.8.p2.2.m2.2.2.2.1.1.1.2">superscript</csymbol><ci id="A3.SS2.SSS1.8.p2.2.m2.2.2.2.1.1.1.2.2.cmml" xref="A3.SS2.SSS1.8.p2.2.m2.2.2.2.1.1.1.2.2">𝑎</ci><cn id="A3.SS2.SSS1.8.p2.2.m2.2.2.2.1.1.1.2.3.cmml" type="integer" xref="A3.SS2.SSS1.8.p2.2.m2.2.2.2.1.1.1.2.3">3</cn></apply><apply id="A3.SS2.SSS1.8.p2.2.m2.2.2.2.1.1.1.3.cmml" xref="A3.SS2.SSS1.8.p2.2.m2.2.2.2.1.1.1.3"><csymbol cd="ambiguous" id="A3.SS2.SSS1.8.p2.2.m2.2.2.2.1.1.1.3.1.cmml" xref="A3.SS2.SSS1.8.p2.2.m2.2.2.2.1.1.1.3">superscript</csymbol><ci id="A3.SS2.SSS1.8.p2.2.m2.2.2.2.1.1.1.3.2.cmml" xref="A3.SS2.SSS1.8.p2.2.m2.2.2.2.1.1.1.3.2">𝑏</ci><cn id="A3.SS2.SSS1.8.p2.2.m2.2.2.2.1.1.1.3.3.cmml" type="integer" xref="A3.SS2.SSS1.8.p2.2.m2.2.2.2.1.1.1.3.3">3</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.SS2.SSS1.8.p2.2.m2.2c">(a+b)^{3}\leq 4(a^{3}+b^{3})</annotation><annotation encoding="application/x-llamapun" id="A3.SS2.SSS1.8.p2.2.m2.2d">( italic_a + italic_b ) start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ≤ 4 ( italic_a start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT + italic_b start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT )</annotation></semantics></math>, we get the following inequalities <math alttext="\mathbb{E}_{{\tilde{\mathbf{x}}}\sim p_{\sigma}({\tilde{\mathbf{x}}}|{\mathbf{% x}})}\left(\|{\tilde{\mathbf{x}}}\|^{3}\right)\leq\mathbb{E}_{{\tilde{\mathbf{% x}}}\sim p_{\sigma}({\tilde{\mathbf{x}}}|{\mathbf{x}})}\left(\left(\|{\tilde{% \mathbf{x}}}-{\mathbf{x}}\|+\|{\mathbf{x}}\|\right)^{3}\right)\leq 4\mathbb{E}% _{{\tilde{\mathbf{x}}}\sim p_{\sigma}({\tilde{\mathbf{x}}}|{\mathbf{x}})}\left% (\|{\tilde{\mathbf{x}}}-{\mathbf{x}}\|^{3}+\|{\mathbf{x}}\|^{3}\right)=8d% \sigma^{3}\sqrt{\frac{2}{\pi}}+4\|{\mathbf{x}}\|^{3}" class="ltx_Math" display="inline" id="A3.SS2.SSS1.8.p2.3.m3.10"><semantics 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id="A3.SS2.SSS1.8.p2.3.m3.1.1.1.1.3.2" xref="A3.SS2.SSS1.8.p2.3.m3.1.1.1.1.3.2.cmml">p</mi><mi id="A3.SS2.SSS1.8.p2.3.m3.1.1.1.1.3.3" xref="A3.SS2.SSS1.8.p2.3.m3.1.1.1.1.3.3.cmml">σ</mi></msub><mo id="A3.SS2.SSS1.8.p2.3.m3.1.1.1.1.2" xref="A3.SS2.SSS1.8.p2.3.m3.1.1.1.1.2.cmml">⁢</mo><mrow id="A3.SS2.SSS1.8.p2.3.m3.1.1.1.1.1.1" xref="A3.SS2.SSS1.8.p2.3.m3.1.1.1.1.1.1.1.cmml"><mo id="A3.SS2.SSS1.8.p2.3.m3.1.1.1.1.1.1.2" stretchy="false" xref="A3.SS2.SSS1.8.p2.3.m3.1.1.1.1.1.1.1.cmml">(</mo><mrow id="A3.SS2.SSS1.8.p2.3.m3.1.1.1.1.1.1.1" xref="A3.SS2.SSS1.8.p2.3.m3.1.1.1.1.1.1.1.cmml"><mover accent="true" id="A3.SS2.SSS1.8.p2.3.m3.1.1.1.1.1.1.1.2" xref="A3.SS2.SSS1.8.p2.3.m3.1.1.1.1.1.1.1.2.cmml"><mi id="A3.SS2.SSS1.8.p2.3.m3.1.1.1.1.1.1.1.2.2" xref="A3.SS2.SSS1.8.p2.3.m3.1.1.1.1.1.1.1.2.2.cmml">𝐱</mi><mo id="A3.SS2.SSS1.8.p2.3.m3.1.1.1.1.1.1.1.2.1" xref="A3.SS2.SSS1.8.p2.3.m3.1.1.1.1.1.1.1.2.1.cmml">~</mo></mover><mo fence="false" id="A3.SS2.SSS1.8.p2.3.m3.1.1.1.1.1.1.1.1" xref="A3.SS2.SSS1.8.p2.3.m3.1.1.1.1.1.1.1.1.cmml">|</mo><mi id="A3.SS2.SSS1.8.p2.3.m3.1.1.1.1.1.1.1.3" xref="A3.SS2.SSS1.8.p2.3.m3.1.1.1.1.1.1.1.3.cmml">𝐱</mi></mrow><mo id="A3.SS2.SSS1.8.p2.3.m3.1.1.1.1.1.1.3" stretchy="false" xref="A3.SS2.SSS1.8.p2.3.m3.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow></msub><mo id="A3.SS2.SSS1.8.p2.3.m3.8.8.1.2" xref="A3.SS2.SSS1.8.p2.3.m3.8.8.1.2.cmml">⁢</mo><mrow id="A3.SS2.SSS1.8.p2.3.m3.8.8.1.1.1" xref="A3.SS2.SSS1.8.p2.3.m3.8.8.1.1.1.1.cmml"><mo id="A3.SS2.SSS1.8.p2.3.m3.8.8.1.1.1.2" xref="A3.SS2.SSS1.8.p2.3.m3.8.8.1.1.1.1.cmml">(</mo><msup id="A3.SS2.SSS1.8.p2.3.m3.8.8.1.1.1.1" xref="A3.SS2.SSS1.8.p2.3.m3.8.8.1.1.1.1.cmml"><mrow id="A3.SS2.SSS1.8.p2.3.m3.8.8.1.1.1.1.2.2" xref="A3.SS2.SSS1.8.p2.3.m3.8.8.1.1.1.1.2.1.cmml"><mo id="A3.SS2.SSS1.8.p2.3.m3.8.8.1.1.1.1.2.2.1" stretchy="false" xref="A3.SS2.SSS1.8.p2.3.m3.8.8.1.1.1.1.2.1.1.cmml">‖</mo><mover accent="true" id="A3.SS2.SSS1.8.p2.3.m3.4.4" xref="A3.SS2.SSS1.8.p2.3.m3.4.4.cmml"><mi id="A3.SS2.SSS1.8.p2.3.m3.4.4.2" xref="A3.SS2.SSS1.8.p2.3.m3.4.4.2.cmml">𝐱</mi><mo id="A3.SS2.SSS1.8.p2.3.m3.4.4.1" xref="A3.SS2.SSS1.8.p2.3.m3.4.4.1.cmml">~</mo></mover><mo id="A3.SS2.SSS1.8.p2.3.m3.8.8.1.1.1.1.2.2.2" stretchy="false" xref="A3.SS2.SSS1.8.p2.3.m3.8.8.1.1.1.1.2.1.1.cmml">‖</mo></mrow><mn id="A3.SS2.SSS1.8.p2.3.m3.8.8.1.1.1.1.3" xref="A3.SS2.SSS1.8.p2.3.m3.8.8.1.1.1.1.3.cmml">3</mn></msup><mo id="A3.SS2.SSS1.8.p2.3.m3.8.8.1.1.1.3" xref="A3.SS2.SSS1.8.p2.3.m3.8.8.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="A3.SS2.SSS1.8.p2.3.m3.10.10.5" xref="A3.SS2.SSS1.8.p2.3.m3.10.10.5.cmml">≤</mo><mrow id="A3.SS2.SSS1.8.p2.3.m3.9.9.2" xref="A3.SS2.SSS1.8.p2.3.m3.9.9.2.cmml"><msub id="A3.SS2.SSS1.8.p2.3.m3.9.9.2.3" xref="A3.SS2.SSS1.8.p2.3.m3.9.9.2.3.cmml"><mi id="A3.SS2.SSS1.8.p2.3.m3.9.9.2.3.2" xref="A3.SS2.SSS1.8.p2.3.m3.9.9.2.3.2.cmml">𝔼</mi><mrow id="A3.SS2.SSS1.8.p2.3.m3.2.2.1" xref="A3.SS2.SSS1.8.p2.3.m3.2.2.1.cmml"><mover accent="true" id="A3.SS2.SSS1.8.p2.3.m3.2.2.1.3" xref="A3.SS2.SSS1.8.p2.3.m3.2.2.1.3.cmml"><mi id="A3.SS2.SSS1.8.p2.3.m3.2.2.1.3.2" xref="A3.SS2.SSS1.8.p2.3.m3.2.2.1.3.2.cmml">𝐱</mi><mo id="A3.SS2.SSS1.8.p2.3.m3.2.2.1.3.1" xref="A3.SS2.SSS1.8.p2.3.m3.2.2.1.3.1.cmml">~</mo></mover><mo id="A3.SS2.SSS1.8.p2.3.m3.2.2.1.2" xref="A3.SS2.SSS1.8.p2.3.m3.2.2.1.2.cmml">∼</mo><mrow id="A3.SS2.SSS1.8.p2.3.m3.2.2.1.1" xref="A3.SS2.SSS1.8.p2.3.m3.2.2.1.1.cmml"><msub id="A3.SS2.SSS1.8.p2.3.m3.2.2.1.1.3" xref="A3.SS2.SSS1.8.p2.3.m3.2.2.1.1.3.cmml"><mi id="A3.SS2.SSS1.8.p2.3.m3.2.2.1.1.3.2" xref="A3.SS2.SSS1.8.p2.3.m3.2.2.1.1.3.2.cmml">p</mi><mi id="A3.SS2.SSS1.8.p2.3.m3.2.2.1.1.3.3" xref="A3.SS2.SSS1.8.p2.3.m3.2.2.1.1.3.3.cmml">σ</mi></msub><mo id="A3.SS2.SSS1.8.p2.3.m3.2.2.1.1.2" xref="A3.SS2.SSS1.8.p2.3.m3.2.2.1.1.2.cmml">⁢</mo><mrow id="A3.SS2.SSS1.8.p2.3.m3.2.2.1.1.1.1" xref="A3.SS2.SSS1.8.p2.3.m3.2.2.1.1.1.1.1.cmml"><mo id="A3.SS2.SSS1.8.p2.3.m3.2.2.1.1.1.1.2" stretchy="false" xref="A3.SS2.SSS1.8.p2.3.m3.2.2.1.1.1.1.1.cmml">(</mo><mrow id="A3.SS2.SSS1.8.p2.3.m3.2.2.1.1.1.1.1" xref="A3.SS2.SSS1.8.p2.3.m3.2.2.1.1.1.1.1.cmml"><mover accent="true" id="A3.SS2.SSS1.8.p2.3.m3.2.2.1.1.1.1.1.2" xref="A3.SS2.SSS1.8.p2.3.m3.2.2.1.1.1.1.1.2.cmml"><mi id="A3.SS2.SSS1.8.p2.3.m3.2.2.1.1.1.1.1.2.2" xref="A3.SS2.SSS1.8.p2.3.m3.2.2.1.1.1.1.1.2.2.cmml">𝐱</mi><mo id="A3.SS2.SSS1.8.p2.3.m3.2.2.1.1.1.1.1.2.1" xref="A3.SS2.SSS1.8.p2.3.m3.2.2.1.1.1.1.1.2.1.cmml">~</mo></mover><mo fence="false" id="A3.SS2.SSS1.8.p2.3.m3.2.2.1.1.1.1.1.1" xref="A3.SS2.SSS1.8.p2.3.m3.2.2.1.1.1.1.1.1.cmml">|</mo><mi id="A3.SS2.SSS1.8.p2.3.m3.2.2.1.1.1.1.1.3" xref="A3.SS2.SSS1.8.p2.3.m3.2.2.1.1.1.1.1.3.cmml">𝐱</mi></mrow><mo id="A3.SS2.SSS1.8.p2.3.m3.2.2.1.1.1.1.3" stretchy="false" xref="A3.SS2.SSS1.8.p2.3.m3.2.2.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow></msub><mo id="A3.SS2.SSS1.8.p2.3.m3.9.9.2.2" xref="A3.SS2.SSS1.8.p2.3.m3.9.9.2.2.cmml">⁢</mo><mrow id="A3.SS2.SSS1.8.p2.3.m3.9.9.2.1.1" xref="A3.SS2.SSS1.8.p2.3.m3.9.9.2.1.1.1.cmml"><mo id="A3.SS2.SSS1.8.p2.3.m3.9.9.2.1.1.2" xref="A3.SS2.SSS1.8.p2.3.m3.9.9.2.1.1.1.cmml">(</mo><msup id="A3.SS2.SSS1.8.p2.3.m3.9.9.2.1.1.1" xref="A3.SS2.SSS1.8.p2.3.m3.9.9.2.1.1.1.cmml"><mrow id="A3.SS2.SSS1.8.p2.3.m3.9.9.2.1.1.1.1.1" xref="A3.SS2.SSS1.8.p2.3.m3.9.9.2.1.1.1.1.1.1.cmml"><mo id="A3.SS2.SSS1.8.p2.3.m3.9.9.2.1.1.1.1.1.2" xref="A3.SS2.SSS1.8.p2.3.m3.9.9.2.1.1.1.1.1.1.cmml">(</mo><mrow id="A3.SS2.SSS1.8.p2.3.m3.9.9.2.1.1.1.1.1.1" xref="A3.SS2.SSS1.8.p2.3.m3.9.9.2.1.1.1.1.1.1.cmml"><mrow id="A3.SS2.SSS1.8.p2.3.m3.9.9.2.1.1.1.1.1.1.1.1" xref="A3.SS2.SSS1.8.p2.3.m3.9.9.2.1.1.1.1.1.1.1.2.cmml"><mo id="A3.SS2.SSS1.8.p2.3.m3.9.9.2.1.1.1.1.1.1.1.1.2" stretchy="false" xref="A3.SS2.SSS1.8.p2.3.m3.9.9.2.1.1.1.1.1.1.1.2.1.cmml">‖</mo><mrow id="A3.SS2.SSS1.8.p2.3.m3.9.9.2.1.1.1.1.1.1.1.1.1" xref="A3.SS2.SSS1.8.p2.3.m3.9.9.2.1.1.1.1.1.1.1.1.1.cmml"><mover accent="true" id="A3.SS2.SSS1.8.p2.3.m3.9.9.2.1.1.1.1.1.1.1.1.1.2" xref="A3.SS2.SSS1.8.p2.3.m3.9.9.2.1.1.1.1.1.1.1.1.1.2.cmml"><mi id="A3.SS2.SSS1.8.p2.3.m3.9.9.2.1.1.1.1.1.1.1.1.1.2.2" xref="A3.SS2.SSS1.8.p2.3.m3.9.9.2.1.1.1.1.1.1.1.1.1.2.2.cmml">𝐱</mi><mo id="A3.SS2.SSS1.8.p2.3.m3.9.9.2.1.1.1.1.1.1.1.1.1.2.1" xref="A3.SS2.SSS1.8.p2.3.m3.9.9.2.1.1.1.1.1.1.1.1.1.2.1.cmml">~</mo></mover><mo id="A3.SS2.SSS1.8.p2.3.m3.9.9.2.1.1.1.1.1.1.1.1.1.1" xref="A3.SS2.SSS1.8.p2.3.m3.9.9.2.1.1.1.1.1.1.1.1.1.1.cmml">−</mo><mi id="A3.SS2.SSS1.8.p2.3.m3.9.9.2.1.1.1.1.1.1.1.1.1.3" xref="A3.SS2.SSS1.8.p2.3.m3.9.9.2.1.1.1.1.1.1.1.1.1.3.cmml">𝐱</mi></mrow><mo id="A3.SS2.SSS1.8.p2.3.m3.9.9.2.1.1.1.1.1.1.1.1.3" stretchy="false" xref="A3.SS2.SSS1.8.p2.3.m3.9.9.2.1.1.1.1.1.1.1.2.1.cmml">‖</mo></mrow><mo id="A3.SS2.SSS1.8.p2.3.m3.9.9.2.1.1.1.1.1.1.2" xref="A3.SS2.SSS1.8.p2.3.m3.9.9.2.1.1.1.1.1.1.2.cmml">+</mo><mrow id="A3.SS2.SSS1.8.p2.3.m3.9.9.2.1.1.1.1.1.1.3.2" xref="A3.SS2.SSS1.8.p2.3.m3.9.9.2.1.1.1.1.1.1.3.1.cmml"><mo id="A3.SS2.SSS1.8.p2.3.m3.9.9.2.1.1.1.1.1.1.3.2.1" stretchy="false" xref="A3.SS2.SSS1.8.p2.3.m3.9.9.2.1.1.1.1.1.1.3.1.1.cmml">‖</mo><mi id="A3.SS2.SSS1.8.p2.3.m3.5.5" xref="A3.SS2.SSS1.8.p2.3.m3.5.5.cmml">𝐱</mi><mo id="A3.SS2.SSS1.8.p2.3.m3.9.9.2.1.1.1.1.1.1.3.2.2" stretchy="false" xref="A3.SS2.SSS1.8.p2.3.m3.9.9.2.1.1.1.1.1.1.3.1.1.cmml">‖</mo></mrow></mrow><mo id="A3.SS2.SSS1.8.p2.3.m3.9.9.2.1.1.1.1.1.3" xref="A3.SS2.SSS1.8.p2.3.m3.9.9.2.1.1.1.1.1.1.cmml">)</mo></mrow><mn id="A3.SS2.SSS1.8.p2.3.m3.9.9.2.1.1.1.3" xref="A3.SS2.SSS1.8.p2.3.m3.9.9.2.1.1.1.3.cmml">3</mn></msup><mo id="A3.SS2.SSS1.8.p2.3.m3.9.9.2.1.1.3" xref="A3.SS2.SSS1.8.p2.3.m3.9.9.2.1.1.1.cmml">)</mo></mrow></mrow><mo id="A3.SS2.SSS1.8.p2.3.m3.10.10.6" xref="A3.SS2.SSS1.8.p2.3.m3.10.10.6.cmml">≤</mo><mrow id="A3.SS2.SSS1.8.p2.3.m3.10.10.3" xref="A3.SS2.SSS1.8.p2.3.m3.10.10.3.cmml"><mn id="A3.SS2.SSS1.8.p2.3.m3.10.10.3.3" xref="A3.SS2.SSS1.8.p2.3.m3.10.10.3.3.cmml">4</mn><mo id="A3.SS2.SSS1.8.p2.3.m3.10.10.3.2" xref="A3.SS2.SSS1.8.p2.3.m3.10.10.3.2.cmml">⁢</mo><msub 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xref="A3.SS2.SSS1.8.p2.3.m3.7.7">𝐱</ci></apply><cn id="A3.SS2.SSS1.8.p2.3.m3.10.10.8.3.3.3.cmml" type="integer" xref="A3.SS2.SSS1.8.p2.3.m3.10.10.8.3.3.3">3</cn></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.SS2.SSS1.8.p2.3.m3.10c">\mathbb{E}_{{\tilde{\mathbf{x}}}\sim p_{\sigma}({\tilde{\mathbf{x}}}|{\mathbf{% x}})}\left(\|{\tilde{\mathbf{x}}}\|^{3}\right)\leq\mathbb{E}_{{\tilde{\mathbf{% x}}}\sim p_{\sigma}({\tilde{\mathbf{x}}}|{\mathbf{x}})}\left(\left(\|{\tilde{% \mathbf{x}}}-{\mathbf{x}}\|+\|{\mathbf{x}}\|\right)^{3}\right)\leq 4\mathbb{E}% _{{\tilde{\mathbf{x}}}\sim p_{\sigma}({\tilde{\mathbf{x}}}|{\mathbf{x}})}\left% (\|{\tilde{\mathbf{x}}}-{\mathbf{x}}\|^{3}+\|{\mathbf{x}}\|^{3}\right)=8d% \sigma^{3}\sqrt{\frac{2}{\pi}}+4\|{\mathbf{x}}\|^{3}</annotation><annotation encoding="application/x-llamapun" id="A3.SS2.SSS1.8.p2.3.m3.10d">blackboard_E start_POSTSUBSCRIPT over~ start_ARG bold_x end_ARG ∼ italic_p start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ( over~ start_ARG bold_x end_ARG | bold_x ) end_POSTSUBSCRIPT ( ∥ over~ start_ARG bold_x end_ARG ∥ start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≤ blackboard_E start_POSTSUBSCRIPT over~ start_ARG bold_x end_ARG ∼ italic_p start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ( over~ start_ARG bold_x end_ARG | bold_x ) end_POSTSUBSCRIPT ( ( ∥ over~ start_ARG bold_x end_ARG - bold_x ∥ + ∥ bold_x ∥ ) start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) ≤ 4 blackboard_E start_POSTSUBSCRIPT over~ start_ARG bold_x end_ARG ∼ italic_p start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ( over~ start_ARG bold_x end_ARG | bold_x ) end_POSTSUBSCRIPT ( ∥ over~ start_ARG bold_x end_ARG - bold_x ∥ start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT + ∥ bold_x ∥ start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) = 8 italic_d italic_σ start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT square-root start_ARG divide start_ARG 2 end_ARG start_ARG italic_π end_ARG end_ARG + 4 ∥ bold_x ∥ start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT</annotation></semantics></math>. This leads to the following inequality</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A8.EGx33"> <tbody id="A3.E24"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\|\nabla\mathcal{R}_{\sigma}({\mathbf{x}})+\nabla\log p({\mathbf{% x}})\|" class="ltx_Math" display="inline" id="A3.E24.m1.3"><semantics id="A3.E24.m1.3a"><mrow id="A3.E24.m1.3.3.1" xref="A3.E24.m1.3.3.2.cmml"><mo id="A3.E24.m1.3.3.1.2" stretchy="false" xref="A3.E24.m1.3.3.2.1.cmml">‖</mo><mrow id="A3.E24.m1.3.3.1.1" xref="A3.E24.m1.3.3.1.1.cmml"><mrow id="A3.E24.m1.3.3.1.1.2" xref="A3.E24.m1.3.3.1.1.2.cmml"><mrow id="A3.E24.m1.3.3.1.1.2.2" xref="A3.E24.m1.3.3.1.1.2.2.cmml"><mo id="A3.E24.m1.3.3.1.1.2.2.1" rspace="0.167em" xref="A3.E24.m1.3.3.1.1.2.2.1.cmml">∇</mo><msub id="A3.E24.m1.3.3.1.1.2.2.2" xref="A3.E24.m1.3.3.1.1.2.2.2.cmml"><mi 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id="A3.E24.m1.3.3.1.1.3.2.1.cmml" xref="A3.E24.m1.3.3.1.1.3.2.1"><ci id="A3.E24.m1.3.3.1.1.3.2.1.1.cmml" xref="A3.E24.m1.3.3.1.1.3.2.1.1">∇</ci><log id="A3.E24.m1.3.3.1.1.3.2.1.2.cmml" xref="A3.E24.m1.3.3.1.1.3.2.1.2"></log></apply><ci id="A3.E24.m1.3.3.1.1.3.2.2.cmml" xref="A3.E24.m1.3.3.1.1.3.2.2">𝑝</ci></apply><ci id="A3.E24.m1.2.2.cmml" xref="A3.E24.m1.2.2">𝐱</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.E24.m1.3c">\displaystyle\|\nabla\mathcal{R}_{\sigma}({\mathbf{x}})+\nabla\log p({\mathbf{% x}})\|</annotation><annotation encoding="application/x-llamapun" id="A3.E24.m1.3d">∥ ∇ caligraphic_R start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ( bold_x ) + ∇ roman_log italic_p ( bold_x ) ∥</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\leq\sigma^{2}a_{p}\left(8d\sigma^{3}\sqrt{\frac{2}{\pi}}+4\|{% \mathbf{x}}\|^{3}+1\right)+\|\mathbb{E}_{{\tilde{\mathbf{x}}}\sim p_{\sigma}({% 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xref="A3.E24.m2.5.5.1.1.2.2.1.1.3.2.1.2"></log></apply><ci id="A3.E24.m2.5.5.1.1.2.2.1.1.3.2.2.cmml" xref="A3.E24.m2.5.5.1.1.2.2.1.1.3.2.2">𝑝</ci></apply><ci id="A3.E24.m2.4.4.cmml" xref="A3.E24.m2.4.4">𝐱</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.E24.m2.5c">\displaystyle\leq\sigma^{2}a_{p}\left(8d\sigma^{3}\sqrt{\frac{2}{\pi}}+4\|{% \mathbf{x}}\|^{3}+1\right)+\|\mathbb{E}_{{\tilde{\mathbf{x}}}\sim p_{\sigma}({% \tilde{\mathbf{x}}}|{\mathbf{x}})}\left(\nabla\log p({\tilde{\mathbf{x}}})% \right)-\nabla\log p({\mathbf{x}})\|.</annotation><annotation encoding="application/x-llamapun" id="A3.E24.m2.5d">≤ italic_σ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_a start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT ( 8 italic_d italic_σ start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT square-root start_ARG divide start_ARG 2 end_ARG start_ARG italic_π end_ARG end_ARG + 4 ∥ bold_x ∥ start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT + 1 ) + ∥ blackboard_E start_POSTSUBSCRIPT over~ start_ARG bold_x end_ARG ∼ italic_p start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ( over~ start_ARG bold_x end_ARG | bold_x ) end_POSTSUBSCRIPT ( ∇ roman_log italic_p ( over~ start_ARG bold_x end_ARG ) ) - ∇ roman_log italic_p ( bold_x ) ∥ .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(24)</span></td> </tr></tbody> </table> </div> <div class="ltx_para" id="A3.SS2.SSS1.9.p3"> <p class="ltx_p" id="A3.SS2.SSS1.9.p3.5">As <math alttext="\nabla\log p" class="ltx_Math" display="inline" id="A3.SS2.SSS1.9.p3.1.m1.1"><semantics id="A3.SS2.SSS1.9.p3.1.m1.1a"><mrow id="A3.SS2.SSS1.9.p3.1.m1.1.1" xref="A3.SS2.SSS1.9.p3.1.m1.1.1.cmml"><mrow id="A3.SS2.SSS1.9.p3.1.m1.1.1.1" xref="A3.SS2.SSS1.9.p3.1.m1.1.1.1.cmml"><mo id="A3.SS2.SSS1.9.p3.1.m1.1.1.1.1" rspace="0.167em" xref="A3.SS2.SSS1.9.p3.1.m1.1.1.1.1.cmml">∇</mo><mi id="A3.SS2.SSS1.9.p3.1.m1.1.1.1.2" xref="A3.SS2.SSS1.9.p3.1.m1.1.1.1.2.cmml">log</mi></mrow><mo id="A3.SS2.SSS1.9.p3.1.m1.1.1a" lspace="0.167em" xref="A3.SS2.SSS1.9.p3.1.m1.1.1.cmml">⁡</mo><mi id="A3.SS2.SSS1.9.p3.1.m1.1.1.2" xref="A3.SS2.SSS1.9.p3.1.m1.1.1.2.cmml">p</mi></mrow><annotation-xml encoding="MathML-Content" id="A3.SS2.SSS1.9.p3.1.m1.1b"><apply id="A3.SS2.SSS1.9.p3.1.m1.1.1.cmml" xref="A3.SS2.SSS1.9.p3.1.m1.1.1"><apply id="A3.SS2.SSS1.9.p3.1.m1.1.1.1.cmml" xref="A3.SS2.SSS1.9.p3.1.m1.1.1.1"><ci id="A3.SS2.SSS1.9.p3.1.m1.1.1.1.1.cmml" xref="A3.SS2.SSS1.9.p3.1.m1.1.1.1.1">∇</ci><log id="A3.SS2.SSS1.9.p3.1.m1.1.1.1.2.cmml" xref="A3.SS2.SSS1.9.p3.1.m1.1.1.1.2"></log></apply><ci id="A3.SS2.SSS1.9.p3.1.m1.1.1.2.cmml" xref="A3.SS2.SSS1.9.p3.1.m1.1.1.2">𝑝</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.SS2.SSS1.9.p3.1.m1.1c">\nabla\log p</annotation><annotation encoding="application/x-llamapun" id="A3.SS2.SSS1.9.p3.1.m1.1d">∇ roman_log italic_p</annotation></semantics></math> is <math alttext="\mathcal{C}^{2}" class="ltx_Math" display="inline" id="A3.SS2.SSS1.9.p3.2.m2.1"><semantics id="A3.SS2.SSS1.9.p3.2.m2.1a"><msup id="A3.SS2.SSS1.9.p3.2.m2.1.1" xref="A3.SS2.SSS1.9.p3.2.m2.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="A3.SS2.SSS1.9.p3.2.m2.1.1.2" xref="A3.SS2.SSS1.9.p3.2.m2.1.1.2.cmml">𝒞</mi><mn id="A3.SS2.SSS1.9.p3.2.m2.1.1.3" xref="A3.SS2.SSS1.9.p3.2.m2.1.1.3.cmml">2</mn></msup><annotation-xml encoding="MathML-Content" id="A3.SS2.SSS1.9.p3.2.m2.1b"><apply id="A3.SS2.SSS1.9.p3.2.m2.1.1.cmml" xref="A3.SS2.SSS1.9.p3.2.m2.1.1"><csymbol cd="ambiguous" id="A3.SS2.SSS1.9.p3.2.m2.1.1.1.cmml" xref="A3.SS2.SSS1.9.p3.2.m2.1.1">superscript</csymbol><ci id="A3.SS2.SSS1.9.p3.2.m2.1.1.2.cmml" xref="A3.SS2.SSS1.9.p3.2.m2.1.1.2">𝒞</ci><cn id="A3.SS2.SSS1.9.p3.2.m2.1.1.3.cmml" type="integer" xref="A3.SS2.SSS1.9.p3.2.m2.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.SS2.SSS1.9.p3.2.m2.1c">\mathcal{C}^{2}</annotation><annotation encoding="application/x-llamapun" id="A3.SS2.SSS1.9.p3.2.m2.1d">caligraphic_C start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT</annotation></semantics></math> in <math alttext="{\mathbf{x}}" class="ltx_Math" display="inline" id="A3.SS2.SSS1.9.p3.3.m3.1"><semantics id="A3.SS2.SSS1.9.p3.3.m3.1a"><mi id="A3.SS2.SSS1.9.p3.3.m3.1.1" xref="A3.SS2.SSS1.9.p3.3.m3.1.1.cmml">𝐱</mi><annotation-xml encoding="MathML-Content" id="A3.SS2.SSS1.9.p3.3.m3.1b"><ci id="A3.SS2.SSS1.9.p3.3.m3.1.1.cmml" xref="A3.SS2.SSS1.9.p3.3.m3.1.1">𝐱</ci></annotation-xml><annotation encoding="application/x-tex" id="A3.SS2.SSS1.9.p3.3.m3.1c">{\mathbf{x}}</annotation><annotation encoding="application/x-llamapun" id="A3.SS2.SSS1.9.p3.3.m3.1d">bold_x</annotation></semantics></math>, by the Taylor theorem, there exists <math alttext="r>0" class="ltx_Math" display="inline" id="A3.SS2.SSS1.9.p3.4.m4.1"><semantics id="A3.SS2.SSS1.9.p3.4.m4.1a"><mrow id="A3.SS2.SSS1.9.p3.4.m4.1.1" xref="A3.SS2.SSS1.9.p3.4.m4.1.1.cmml"><mi id="A3.SS2.SSS1.9.p3.4.m4.1.1.2" xref="A3.SS2.SSS1.9.p3.4.m4.1.1.2.cmml">r</mi><mo id="A3.SS2.SSS1.9.p3.4.m4.1.1.1" xref="A3.SS2.SSS1.9.p3.4.m4.1.1.1.cmml">></mo><mn id="A3.SS2.SSS1.9.p3.4.m4.1.1.3" xref="A3.SS2.SSS1.9.p3.4.m4.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="A3.SS2.SSS1.9.p3.4.m4.1b"><apply id="A3.SS2.SSS1.9.p3.4.m4.1.1.cmml" xref="A3.SS2.SSS1.9.p3.4.m4.1.1"><gt id="A3.SS2.SSS1.9.p3.4.m4.1.1.1.cmml" xref="A3.SS2.SSS1.9.p3.4.m4.1.1.1"></gt><ci id="A3.SS2.SSS1.9.p3.4.m4.1.1.2.cmml" xref="A3.SS2.SSS1.9.p3.4.m4.1.1.2">𝑟</ci><cn id="A3.SS2.SSS1.9.p3.4.m4.1.1.3.cmml" type="integer" xref="A3.SS2.SSS1.9.p3.4.m4.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.SS2.SSS1.9.p3.4.m4.1c">r>0</annotation><annotation encoding="application/x-llamapun" id="A3.SS2.SSS1.9.p3.4.m4.1d">italic_r > 0</annotation></semantics></math> such that <math alttext="\forall{\mathbf{y}},{\mathbf{z}}\in\mathcal{B}({\mathbf{x}},r)" class="ltx_Math" display="inline" id="A3.SS2.SSS1.9.p3.5.m5.4"><semantics id="A3.SS2.SSS1.9.p3.5.m5.4a"><mrow id="A3.SS2.SSS1.9.p3.5.m5.4.4" xref="A3.SS2.SSS1.9.p3.5.m5.4.4.cmml"><mrow id="A3.SS2.SSS1.9.p3.5.m5.4.4.1.1" xref="A3.SS2.SSS1.9.p3.5.m5.4.4.1.2.cmml"><mrow id="A3.SS2.SSS1.9.p3.5.m5.4.4.1.1.1" xref="A3.SS2.SSS1.9.p3.5.m5.4.4.1.1.1.cmml"><mo id="A3.SS2.SSS1.9.p3.5.m5.4.4.1.1.1.1" rspace="0.167em" xref="A3.SS2.SSS1.9.p3.5.m5.4.4.1.1.1.1.cmml">∀</mo><mi id="A3.SS2.SSS1.9.p3.5.m5.4.4.1.1.1.2" xref="A3.SS2.SSS1.9.p3.5.m5.4.4.1.1.1.2.cmml">𝐲</mi></mrow><mo id="A3.SS2.SSS1.9.p3.5.m5.4.4.1.1.2" xref="A3.SS2.SSS1.9.p3.5.m5.4.4.1.2.cmml">,</mo><mi id="A3.SS2.SSS1.9.p3.5.m5.3.3" xref="A3.SS2.SSS1.9.p3.5.m5.3.3.cmml">𝐳</mi></mrow><mo id="A3.SS2.SSS1.9.p3.5.m5.4.4.2" xref="A3.SS2.SSS1.9.p3.5.m5.4.4.2.cmml">∈</mo><mrow id="A3.SS2.SSS1.9.p3.5.m5.4.4.3" xref="A3.SS2.SSS1.9.p3.5.m5.4.4.3.cmml"><mi class="ltx_font_mathcaligraphic" id="A3.SS2.SSS1.9.p3.5.m5.4.4.3.2" xref="A3.SS2.SSS1.9.p3.5.m5.4.4.3.2.cmml">ℬ</mi><mo id="A3.SS2.SSS1.9.p3.5.m5.4.4.3.1" xref="A3.SS2.SSS1.9.p3.5.m5.4.4.3.1.cmml">⁢</mo><mrow id="A3.SS2.SSS1.9.p3.5.m5.4.4.3.3.2" xref="A3.SS2.SSS1.9.p3.5.m5.4.4.3.3.1.cmml"><mo id="A3.SS2.SSS1.9.p3.5.m5.4.4.3.3.2.1" stretchy="false" xref="A3.SS2.SSS1.9.p3.5.m5.4.4.3.3.1.cmml">(</mo><mi id="A3.SS2.SSS1.9.p3.5.m5.1.1" xref="A3.SS2.SSS1.9.p3.5.m5.1.1.cmml">𝐱</mi><mo id="A3.SS2.SSS1.9.p3.5.m5.4.4.3.3.2.2" xref="A3.SS2.SSS1.9.p3.5.m5.4.4.3.3.1.cmml">,</mo><mi id="A3.SS2.SSS1.9.p3.5.m5.2.2" xref="A3.SS2.SSS1.9.p3.5.m5.2.2.cmml">r</mi><mo id="A3.SS2.SSS1.9.p3.5.m5.4.4.3.3.2.3" stretchy="false" xref="A3.SS2.SSS1.9.p3.5.m5.4.4.3.3.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="A3.SS2.SSS1.9.p3.5.m5.4b"><apply id="A3.SS2.SSS1.9.p3.5.m5.4.4.cmml" xref="A3.SS2.SSS1.9.p3.5.m5.4.4"><in id="A3.SS2.SSS1.9.p3.5.m5.4.4.2.cmml" xref="A3.SS2.SSS1.9.p3.5.m5.4.4.2"></in><list id="A3.SS2.SSS1.9.p3.5.m5.4.4.1.2.cmml" xref="A3.SS2.SSS1.9.p3.5.m5.4.4.1.1"><apply id="A3.SS2.SSS1.9.p3.5.m5.4.4.1.1.1.cmml" xref="A3.SS2.SSS1.9.p3.5.m5.4.4.1.1.1"><csymbol cd="latexml" id="A3.SS2.SSS1.9.p3.5.m5.4.4.1.1.1.1.cmml" xref="A3.SS2.SSS1.9.p3.5.m5.4.4.1.1.1.1">for-all</csymbol><ci id="A3.SS2.SSS1.9.p3.5.m5.4.4.1.1.1.2.cmml" xref="A3.SS2.SSS1.9.p3.5.m5.4.4.1.1.1.2">𝐲</ci></apply><ci id="A3.SS2.SSS1.9.p3.5.m5.3.3.cmml" xref="A3.SS2.SSS1.9.p3.5.m5.3.3">𝐳</ci></list><apply id="A3.SS2.SSS1.9.p3.5.m5.4.4.3.cmml" xref="A3.SS2.SSS1.9.p3.5.m5.4.4.3"><times id="A3.SS2.SSS1.9.p3.5.m5.4.4.3.1.cmml" xref="A3.SS2.SSS1.9.p3.5.m5.4.4.3.1"></times><ci id="A3.SS2.SSS1.9.p3.5.m5.4.4.3.2.cmml" xref="A3.SS2.SSS1.9.p3.5.m5.4.4.3.2">ℬ</ci><interval closure="open" id="A3.SS2.SSS1.9.p3.5.m5.4.4.3.3.1.cmml" xref="A3.SS2.SSS1.9.p3.5.m5.4.4.3.3.2"><ci id="A3.SS2.SSS1.9.p3.5.m5.1.1.cmml" xref="A3.SS2.SSS1.9.p3.5.m5.1.1">𝐱</ci><ci id="A3.SS2.SSS1.9.p3.5.m5.2.2.cmml" xref="A3.SS2.SSS1.9.p3.5.m5.2.2">𝑟</ci></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.SS2.SSS1.9.p3.5.m5.4c">\forall{\mathbf{y}},{\mathbf{z}}\in\mathcal{B}({\mathbf{x}},r)</annotation><annotation encoding="application/x-llamapun" id="A3.SS2.SSS1.9.p3.5.m5.4d">∀ bold_y , bold_z ∈ caligraphic_B ( bold_x , italic_r )</annotation></semantics></math></p> <table class="ltx_equation ltx_eqn_table" id="A3.Ex48"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\|\nabla\log p({\mathbf{y}})-\nabla\log p({\mathbf{z}})\|\leq M_{{\mathbf{x}}}% \|{\mathbf{y}}-{\mathbf{z}}\|," class="ltx_Math" display="block" id="A3.Ex48.m1.3"><semantics id="A3.Ex48.m1.3a"><mrow id="A3.Ex48.m1.3.3.1" xref="A3.Ex48.m1.3.3.1.1.cmml"><mrow id="A3.Ex48.m1.3.3.1.1" xref="A3.Ex48.m1.3.3.1.1.cmml"><mrow id="A3.Ex48.m1.3.3.1.1.1.1" xref="A3.Ex48.m1.3.3.1.1.1.2.cmml"><mo id="A3.Ex48.m1.3.3.1.1.1.1.2" stretchy="false" xref="A3.Ex48.m1.3.3.1.1.1.2.1.cmml">‖</mo><mrow id="A3.Ex48.m1.3.3.1.1.1.1.1" xref="A3.Ex48.m1.3.3.1.1.1.1.1.cmml"><mrow id="A3.Ex48.m1.3.3.1.1.1.1.1.2" xref="A3.Ex48.m1.3.3.1.1.1.1.1.2.cmml"><mrow id="A3.Ex48.m1.3.3.1.1.1.1.1.2.2" xref="A3.Ex48.m1.3.3.1.1.1.1.1.2.2.cmml"><mrow id="A3.Ex48.m1.3.3.1.1.1.1.1.2.2.1" xref="A3.Ex48.m1.3.3.1.1.1.1.1.2.2.1.cmml"><mo id="A3.Ex48.m1.3.3.1.1.1.1.1.2.2.1.1" rspace="0.167em" xref="A3.Ex48.m1.3.3.1.1.1.1.1.2.2.1.1.cmml">∇</mo><mi id="A3.Ex48.m1.3.3.1.1.1.1.1.2.2.1.2" xref="A3.Ex48.m1.3.3.1.1.1.1.1.2.2.1.2.cmml">log</mi></mrow><mo id="A3.Ex48.m1.3.3.1.1.1.1.1.2.2a" lspace="0.167em" xref="A3.Ex48.m1.3.3.1.1.1.1.1.2.2.cmml">⁡</mo><mi id="A3.Ex48.m1.3.3.1.1.1.1.1.2.2.2" xref="A3.Ex48.m1.3.3.1.1.1.1.1.2.2.2.cmml">p</mi></mrow><mo id="A3.Ex48.m1.3.3.1.1.1.1.1.2.1" xref="A3.Ex48.m1.3.3.1.1.1.1.1.2.1.cmml">⁢</mo><mrow id="A3.Ex48.m1.3.3.1.1.1.1.1.2.3.2" xref="A3.Ex48.m1.3.3.1.1.1.1.1.2.cmml"><mo id="A3.Ex48.m1.3.3.1.1.1.1.1.2.3.2.1" stretchy="false" xref="A3.Ex48.m1.3.3.1.1.1.1.1.2.cmml">(</mo><mi id="A3.Ex48.m1.1.1" xref="A3.Ex48.m1.1.1.cmml">𝐲</mi><mo id="A3.Ex48.m1.3.3.1.1.1.1.1.2.3.2.2" stretchy="false" xref="A3.Ex48.m1.3.3.1.1.1.1.1.2.cmml">)</mo></mrow></mrow><mo id="A3.Ex48.m1.3.3.1.1.1.1.1.1" xref="A3.Ex48.m1.3.3.1.1.1.1.1.1.cmml">−</mo><mrow id="A3.Ex48.m1.3.3.1.1.1.1.1.3" xref="A3.Ex48.m1.3.3.1.1.1.1.1.3.cmml"><mrow id="A3.Ex48.m1.3.3.1.1.1.1.1.3.2" xref="A3.Ex48.m1.3.3.1.1.1.1.1.3.2.cmml"><mrow id="A3.Ex48.m1.3.3.1.1.1.1.1.3.2.1" xref="A3.Ex48.m1.3.3.1.1.1.1.1.3.2.1.cmml"><mo id="A3.Ex48.m1.3.3.1.1.1.1.1.3.2.1.1" rspace="0.167em" xref="A3.Ex48.m1.3.3.1.1.1.1.1.3.2.1.1.cmml">∇</mo><mi id="A3.Ex48.m1.3.3.1.1.1.1.1.3.2.1.2" xref="A3.Ex48.m1.3.3.1.1.1.1.1.3.2.1.2.cmml">log</mi></mrow><mo id="A3.Ex48.m1.3.3.1.1.1.1.1.3.2a" lspace="0.167em" xref="A3.Ex48.m1.3.3.1.1.1.1.1.3.2.cmml">⁡</mo><mi id="A3.Ex48.m1.3.3.1.1.1.1.1.3.2.2" xref="A3.Ex48.m1.3.3.1.1.1.1.1.3.2.2.cmml">p</mi></mrow><mo id="A3.Ex48.m1.3.3.1.1.1.1.1.3.1" xref="A3.Ex48.m1.3.3.1.1.1.1.1.3.1.cmml">⁢</mo><mrow id="A3.Ex48.m1.3.3.1.1.1.1.1.3.3.2" xref="A3.Ex48.m1.3.3.1.1.1.1.1.3.cmml"><mo id="A3.Ex48.m1.3.3.1.1.1.1.1.3.3.2.1" stretchy="false" xref="A3.Ex48.m1.3.3.1.1.1.1.1.3.cmml">(</mo><mi id="A3.Ex48.m1.2.2" xref="A3.Ex48.m1.2.2.cmml">𝐳</mi><mo id="A3.Ex48.m1.3.3.1.1.1.1.1.3.3.2.2" stretchy="false" xref="A3.Ex48.m1.3.3.1.1.1.1.1.3.cmml">)</mo></mrow></mrow></mrow><mo id="A3.Ex48.m1.3.3.1.1.1.1.3" stretchy="false" xref="A3.Ex48.m1.3.3.1.1.1.2.1.cmml">‖</mo></mrow><mo id="A3.Ex48.m1.3.3.1.1.3" xref="A3.Ex48.m1.3.3.1.1.3.cmml">≤</mo><mrow id="A3.Ex48.m1.3.3.1.1.2" xref="A3.Ex48.m1.3.3.1.1.2.cmml"><msub id="A3.Ex48.m1.3.3.1.1.2.3" xref="A3.Ex48.m1.3.3.1.1.2.3.cmml"><mi id="A3.Ex48.m1.3.3.1.1.2.3.2" xref="A3.Ex48.m1.3.3.1.1.2.3.2.cmml">M</mi><mi id="A3.Ex48.m1.3.3.1.1.2.3.3" xref="A3.Ex48.m1.3.3.1.1.2.3.3.cmml">𝐱</mi></msub><mo id="A3.Ex48.m1.3.3.1.1.2.2" xref="A3.Ex48.m1.3.3.1.1.2.2.cmml">⁢</mo><mrow id="A3.Ex48.m1.3.3.1.1.2.1.1" xref="A3.Ex48.m1.3.3.1.1.2.1.2.cmml"><mo id="A3.Ex48.m1.3.3.1.1.2.1.1.2" stretchy="false" xref="A3.Ex48.m1.3.3.1.1.2.1.2.1.cmml">‖</mo><mrow id="A3.Ex48.m1.3.3.1.1.2.1.1.1" xref="A3.Ex48.m1.3.3.1.1.2.1.1.1.cmml"><mi id="A3.Ex48.m1.3.3.1.1.2.1.1.1.2" xref="A3.Ex48.m1.3.3.1.1.2.1.1.1.2.cmml">𝐲</mi><mo id="A3.Ex48.m1.3.3.1.1.2.1.1.1.1" xref="A3.Ex48.m1.3.3.1.1.2.1.1.1.1.cmml">−</mo><mi id="A3.Ex48.m1.3.3.1.1.2.1.1.1.3" xref="A3.Ex48.m1.3.3.1.1.2.1.1.1.3.cmml">𝐳</mi></mrow><mo id="A3.Ex48.m1.3.3.1.1.2.1.1.3" stretchy="false" xref="A3.Ex48.m1.3.3.1.1.2.1.2.1.cmml">‖</mo></mrow></mrow></mrow><mo id="A3.Ex48.m1.3.3.1.2" xref="A3.Ex48.m1.3.3.1.1.cmml">,</mo></mrow><annotation-xml encoding="MathML-Content" id="A3.Ex48.m1.3b"><apply id="A3.Ex48.m1.3.3.1.1.cmml" xref="A3.Ex48.m1.3.3.1"><leq id="A3.Ex48.m1.3.3.1.1.3.cmml" xref="A3.Ex48.m1.3.3.1.1.3"></leq><apply id="A3.Ex48.m1.3.3.1.1.1.2.cmml" xref="A3.Ex48.m1.3.3.1.1.1.1"><csymbol cd="latexml" id="A3.Ex48.m1.3.3.1.1.1.2.1.cmml" xref="A3.Ex48.m1.3.3.1.1.1.1.2">norm</csymbol><apply id="A3.Ex48.m1.3.3.1.1.1.1.1.cmml" xref="A3.Ex48.m1.3.3.1.1.1.1.1"><minus id="A3.Ex48.m1.3.3.1.1.1.1.1.1.cmml" xref="A3.Ex48.m1.3.3.1.1.1.1.1.1"></minus><apply id="A3.Ex48.m1.3.3.1.1.1.1.1.2.cmml" xref="A3.Ex48.m1.3.3.1.1.1.1.1.2"><times id="A3.Ex48.m1.3.3.1.1.1.1.1.2.1.cmml" xref="A3.Ex48.m1.3.3.1.1.1.1.1.2.1"></times><apply id="A3.Ex48.m1.3.3.1.1.1.1.1.2.2.cmml" xref="A3.Ex48.m1.3.3.1.1.1.1.1.2.2"><apply id="A3.Ex48.m1.3.3.1.1.1.1.1.2.2.1.cmml" xref="A3.Ex48.m1.3.3.1.1.1.1.1.2.2.1"><ci id="A3.Ex48.m1.3.3.1.1.1.1.1.2.2.1.1.cmml" xref="A3.Ex48.m1.3.3.1.1.1.1.1.2.2.1.1">∇</ci><log id="A3.Ex48.m1.3.3.1.1.1.1.1.2.2.1.2.cmml" xref="A3.Ex48.m1.3.3.1.1.1.1.1.2.2.1.2"></log></apply><ci id="A3.Ex48.m1.3.3.1.1.1.1.1.2.2.2.cmml" xref="A3.Ex48.m1.3.3.1.1.1.1.1.2.2.2">𝑝</ci></apply><ci id="A3.Ex48.m1.1.1.cmml" xref="A3.Ex48.m1.1.1">𝐲</ci></apply><apply id="A3.Ex48.m1.3.3.1.1.1.1.1.3.cmml" xref="A3.Ex48.m1.3.3.1.1.1.1.1.3"><times id="A3.Ex48.m1.3.3.1.1.1.1.1.3.1.cmml" xref="A3.Ex48.m1.3.3.1.1.1.1.1.3.1"></times><apply id="A3.Ex48.m1.3.3.1.1.1.1.1.3.2.cmml" xref="A3.Ex48.m1.3.3.1.1.1.1.1.3.2"><apply id="A3.Ex48.m1.3.3.1.1.1.1.1.3.2.1.cmml" xref="A3.Ex48.m1.3.3.1.1.1.1.1.3.2.1"><ci id="A3.Ex48.m1.3.3.1.1.1.1.1.3.2.1.1.cmml" xref="A3.Ex48.m1.3.3.1.1.1.1.1.3.2.1.1">∇</ci><log id="A3.Ex48.m1.3.3.1.1.1.1.1.3.2.1.2.cmml" xref="A3.Ex48.m1.3.3.1.1.1.1.1.3.2.1.2"></log></apply><ci id="A3.Ex48.m1.3.3.1.1.1.1.1.3.2.2.cmml" xref="A3.Ex48.m1.3.3.1.1.1.1.1.3.2.2">𝑝</ci></apply><ci id="A3.Ex48.m1.2.2.cmml" xref="A3.Ex48.m1.2.2">𝐳</ci></apply></apply></apply><apply id="A3.Ex48.m1.3.3.1.1.2.cmml" xref="A3.Ex48.m1.3.3.1.1.2"><times id="A3.Ex48.m1.3.3.1.1.2.2.cmml" xref="A3.Ex48.m1.3.3.1.1.2.2"></times><apply id="A3.Ex48.m1.3.3.1.1.2.3.cmml" xref="A3.Ex48.m1.3.3.1.1.2.3"><csymbol cd="ambiguous" id="A3.Ex48.m1.3.3.1.1.2.3.1.cmml" xref="A3.Ex48.m1.3.3.1.1.2.3">subscript</csymbol><ci id="A3.Ex48.m1.3.3.1.1.2.3.2.cmml" xref="A3.Ex48.m1.3.3.1.1.2.3.2">𝑀</ci><ci id="A3.Ex48.m1.3.3.1.1.2.3.3.cmml" xref="A3.Ex48.m1.3.3.1.1.2.3.3">𝐱</ci></apply><apply id="A3.Ex48.m1.3.3.1.1.2.1.2.cmml" xref="A3.Ex48.m1.3.3.1.1.2.1.1"><csymbol cd="latexml" id="A3.Ex48.m1.3.3.1.1.2.1.2.1.cmml" xref="A3.Ex48.m1.3.3.1.1.2.1.1.2">norm</csymbol><apply id="A3.Ex48.m1.3.3.1.1.2.1.1.1.cmml" xref="A3.Ex48.m1.3.3.1.1.2.1.1.1"><minus id="A3.Ex48.m1.3.3.1.1.2.1.1.1.1.cmml" xref="A3.Ex48.m1.3.3.1.1.2.1.1.1.1"></minus><ci id="A3.Ex48.m1.3.3.1.1.2.1.1.1.2.cmml" xref="A3.Ex48.m1.3.3.1.1.2.1.1.1.2">𝐲</ci><ci id="A3.Ex48.m1.3.3.1.1.2.1.1.1.3.cmml" xref="A3.Ex48.m1.3.3.1.1.2.1.1.1.3">𝐳</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.Ex48.m1.3c">\|\nabla\log p({\mathbf{y}})-\nabla\log p({\mathbf{z}})\|\leq M_{{\mathbf{x}}}% \|{\mathbf{y}}-{\mathbf{z}}\|,</annotation><annotation encoding="application/x-llamapun" id="A3.Ex48.m1.3d">∥ ∇ roman_log italic_p ( bold_y ) - ∇ roman_log italic_p ( bold_z ) ∥ ≤ italic_M start_POSTSUBSCRIPT bold_x end_POSTSUBSCRIPT ∥ bold_y - bold_z ∥ ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="A3.SS2.SSS1.9.p3.7">with <math alttext="M_{{\mathbf{x}}}=2\|\nabla^{2}\log p({\mathbf{x}})\|" class="ltx_Math" display="inline" id="A3.SS2.SSS1.9.p3.6.m1.2"><semantics id="A3.SS2.SSS1.9.p3.6.m1.2a"><mrow id="A3.SS2.SSS1.9.p3.6.m1.2.2" xref="A3.SS2.SSS1.9.p3.6.m1.2.2.cmml"><msub id="A3.SS2.SSS1.9.p3.6.m1.2.2.3" xref="A3.SS2.SSS1.9.p3.6.m1.2.2.3.cmml"><mi id="A3.SS2.SSS1.9.p3.6.m1.2.2.3.2" xref="A3.SS2.SSS1.9.p3.6.m1.2.2.3.2.cmml">M</mi><mi id="A3.SS2.SSS1.9.p3.6.m1.2.2.3.3" xref="A3.SS2.SSS1.9.p3.6.m1.2.2.3.3.cmml">𝐱</mi></msub><mo id="A3.SS2.SSS1.9.p3.6.m1.2.2.2" xref="A3.SS2.SSS1.9.p3.6.m1.2.2.2.cmml">=</mo><mrow id="A3.SS2.SSS1.9.p3.6.m1.2.2.1" xref="A3.SS2.SSS1.9.p3.6.m1.2.2.1.cmml"><mn id="A3.SS2.SSS1.9.p3.6.m1.2.2.1.3" xref="A3.SS2.SSS1.9.p3.6.m1.2.2.1.3.cmml">2</mn><mo id="A3.SS2.SSS1.9.p3.6.m1.2.2.1.2" xref="A3.SS2.SSS1.9.p3.6.m1.2.2.1.2.cmml">⁢</mo><mrow id="A3.SS2.SSS1.9.p3.6.m1.2.2.1.1.1" xref="A3.SS2.SSS1.9.p3.6.m1.2.2.1.1.2.cmml"><mo id="A3.SS2.SSS1.9.p3.6.m1.2.2.1.1.1.2" stretchy="false" xref="A3.SS2.SSS1.9.p3.6.m1.2.2.1.1.2.1.cmml">‖</mo><mrow id="A3.SS2.SSS1.9.p3.6.m1.2.2.1.1.1.1" xref="A3.SS2.SSS1.9.p3.6.m1.2.2.1.1.1.1.cmml"><mrow id="A3.SS2.SSS1.9.p3.6.m1.2.2.1.1.1.1.2" xref="A3.SS2.SSS1.9.p3.6.m1.2.2.1.1.1.1.2.cmml"><mrow id="A3.SS2.SSS1.9.p3.6.m1.2.2.1.1.1.1.2.1" xref="A3.SS2.SSS1.9.p3.6.m1.2.2.1.1.1.1.2.1.cmml"><msup id="A3.SS2.SSS1.9.p3.6.m1.2.2.1.1.1.1.2.1.1" xref="A3.SS2.SSS1.9.p3.6.m1.2.2.1.1.1.1.2.1.1.cmml"><mo id="A3.SS2.SSS1.9.p3.6.m1.2.2.1.1.1.1.2.1.1.2" rspace="0.167em" 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id="A3.SS2.SSS1.9.p3.6.m1.2.2.1.1.1.1.2.1.1.2.cmml" xref="A3.SS2.SSS1.9.p3.6.m1.2.2.1.1.1.1.2.1.1.2">∇</ci><cn id="A3.SS2.SSS1.9.p3.6.m1.2.2.1.1.1.1.2.1.1.3.cmml" type="integer" xref="A3.SS2.SSS1.9.p3.6.m1.2.2.1.1.1.1.2.1.1.3">2</cn></apply><log id="A3.SS2.SSS1.9.p3.6.m1.2.2.1.1.1.1.2.1.2.cmml" xref="A3.SS2.SSS1.9.p3.6.m1.2.2.1.1.1.1.2.1.2"></log></apply><ci id="A3.SS2.SSS1.9.p3.6.m1.2.2.1.1.1.1.2.2.cmml" xref="A3.SS2.SSS1.9.p3.6.m1.2.2.1.1.1.1.2.2">𝑝</ci></apply><ci id="A3.SS2.SSS1.9.p3.6.m1.1.1.cmml" xref="A3.SS2.SSS1.9.p3.6.m1.1.1">𝐱</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.SS2.SSS1.9.p3.6.m1.2c">M_{{\mathbf{x}}}=2\|\nabla^{2}\log p({\mathbf{x}})\|</annotation><annotation encoding="application/x-llamapun" id="A3.SS2.SSS1.9.p3.6.m1.2d">italic_M start_POSTSUBSCRIPT bold_x end_POSTSUBSCRIPT = 2 ∥ ∇ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT roman_log italic_p ( bold_x ) ∥</annotation></semantics></math> and <math 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id="A3.SS2.SSS1.9.p3.7.m2.3.3.1.1.1.3.3.cmml" xref="A3.SS2.SSS1.9.p3.7.m2.3.3.1.1.1.3.3">𝑑</ci></apply></apply><apply id="A3.SS2.SSS1.9.p3.7.m2.4.4.2.2.2.cmml" xref="A3.SS2.SSS1.9.p3.7.m2.4.4.2.2.2"><leq id="A3.SS2.SSS1.9.p3.7.m2.4.4.2.2.2.2.cmml" xref="A3.SS2.SSS1.9.p3.7.m2.4.4.2.2.2.2"></leq><apply id="A3.SS2.SSS1.9.p3.7.m2.4.4.2.2.2.1.2.cmml" xref="A3.SS2.SSS1.9.p3.7.m2.4.4.2.2.2.1.1"><csymbol cd="latexml" id="A3.SS2.SSS1.9.p3.7.m2.4.4.2.2.2.1.2.1.cmml" xref="A3.SS2.SSS1.9.p3.7.m2.4.4.2.2.2.1.1.2">norm</csymbol><apply id="A3.SS2.SSS1.9.p3.7.m2.4.4.2.2.2.1.1.1.cmml" xref="A3.SS2.SSS1.9.p3.7.m2.4.4.2.2.2.1.1.1"><minus id="A3.SS2.SSS1.9.p3.7.m2.4.4.2.2.2.1.1.1.1.cmml" xref="A3.SS2.SSS1.9.p3.7.m2.4.4.2.2.2.1.1.1.1"></minus><ci id="A3.SS2.SSS1.9.p3.7.m2.4.4.2.2.2.1.1.1.2.cmml" xref="A3.SS2.SSS1.9.p3.7.m2.4.4.2.2.2.1.1.1.2">𝐲</ci><ci id="A3.SS2.SSS1.9.p3.7.m2.4.4.2.2.2.1.1.1.3.cmml" xref="A3.SS2.SSS1.9.p3.7.m2.4.4.2.2.2.1.1.1.3">𝐱</ci></apply></apply><ci id="A3.SS2.SSS1.9.p3.7.m2.4.4.2.2.2.3.cmml" xref="A3.SS2.SSS1.9.p3.7.m2.4.4.2.2.2.3">𝑟</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.SS2.SSS1.9.p3.7.m2.4c">\mathcal{B}({\mathbf{x}},r)=\{{\mathbf{y}}\in{\mathbb{R}}^{d}|\|{\mathbf{y}}-{% \mathbf{x}}\|\leq r\}</annotation><annotation encoding="application/x-llamapun" id="A3.SS2.SSS1.9.p3.7.m2.4d">caligraphic_B ( bold_x , italic_r ) = { bold_y ∈ blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT | ∥ bold_y - bold_x ∥ ≤ italic_r }</annotation></semantics></math>. </p> </div> <div class="ltx_para" id="A3.SS2.SSS1.10.p4"> <p class="ltx_p" id="A3.SS2.SSS1.10.p4.11">So we have</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A8.EGx34"> <tbody id="A3.Ex49"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\|\mathbb{E}_{{\tilde{\mathbf{x}}}\sim p_{\sigma}({\tilde{\mathbf% {x}}}|{\mathbf{x}})}\left(\nabla\log p({\tilde{\mathbf{x}}})\right)-\nabla\log p% ({\mathbf{x}})\|=\|\int_{{\mathbb{R}}^{d}}{\nabla\log p({\tilde{\mathbf{x}}})% \mathcal{N}({\tilde{\mathbf{x}}};{\mathbf{x}},\sigma^{2}\mathbf{I}_{d})d{% \tilde{\mathbf{x}}}}-\nabla\log p({\mathbf{x}})\|" class="ltx_Math" display="inline" id="A3.Ex49.m1.9"><semantics id="A3.Ex49.m1.9a"><mrow id="A3.Ex49.m1.9.9" xref="A3.Ex49.m1.9.9.cmml"><mrow id="A3.Ex49.m1.8.8.1.1" xref="A3.Ex49.m1.8.8.1.2.cmml"><mo id="A3.Ex49.m1.8.8.1.1.2" stretchy="false" xref="A3.Ex49.m1.8.8.1.2.1.cmml">‖</mo><mrow id="A3.Ex49.m1.8.8.1.1.1" xref="A3.Ex49.m1.8.8.1.1.1.cmml"><mrow id="A3.Ex49.m1.8.8.1.1.1.1" xref="A3.Ex49.m1.8.8.1.1.1.1.cmml"><msub id="A3.Ex49.m1.8.8.1.1.1.1.3" xref="A3.Ex49.m1.8.8.1.1.1.1.3.cmml"><mi id="A3.Ex49.m1.8.8.1.1.1.1.3.2" xref="A3.Ex49.m1.8.8.1.1.1.1.3.2.cmml">𝔼</mi><mrow id="A3.Ex49.m1.1.1.1" 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start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ( over~ start_ARG bold_x end_ARG | bold_x ) end_POSTSUBSCRIPT ( ∇ roman_log italic_p ( over~ start_ARG bold_x end_ARG ) ) - ∇ roman_log italic_p ( bold_x ) ∥ = ∥ ∫ start_POSTSUBSCRIPT blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ∇ roman_log italic_p ( over~ start_ARG bold_x end_ARG ) caligraphic_N ( over~ start_ARG bold_x end_ARG ; bold_x , italic_σ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT bold_I start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT ) italic_d over~ start_ARG bold_x end_ARG - ∇ roman_log italic_p ( bold_x ) ∥</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="A3.Ex50"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\leq\|\int_{\mathcal{B}({\mathbf{x}},r)}{(\nabla\log p({\tilde{% 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xref="A3.Ex50.m1.13.13.1.1.1.1.1.1.1.1.3.cmml">)</mo></mrow></mrow></mrow><mo id="A3.Ex50.m1.13.13.1.1.1.1.1.1.1.3" stretchy="false" xref="A3.Ex50.m1.13.13.1.1.1.1.1.1.1.1.cmml">)</mo></mrow><mo id="A3.Ex50.m1.13.13.1.1.1.1.2.3" xref="A3.Ex50.m1.13.13.1.1.1.1.2.3.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="A3.Ex50.m1.13.13.1.1.1.1.2.4" xref="A3.Ex50.m1.13.13.1.1.1.1.2.4.cmml">𝒩</mi><mo id="A3.Ex50.m1.13.13.1.1.1.1.2.3a" xref="A3.Ex50.m1.13.13.1.1.1.1.2.3.cmml">⁢</mo><mrow id="A3.Ex50.m1.13.13.1.1.1.1.2.2.1" xref="A3.Ex50.m1.13.13.1.1.1.1.2.2.2.cmml"><mo id="A3.Ex50.m1.13.13.1.1.1.1.2.2.1.2" stretchy="false" xref="A3.Ex50.m1.13.13.1.1.1.1.2.2.2.cmml">(</mo><mover accent="true" id="A3.Ex50.m1.7.7" xref="A3.Ex50.m1.7.7.cmml"><mi id="A3.Ex50.m1.7.7.2" xref="A3.Ex50.m1.7.7.2.cmml">𝐱</mi><mo id="A3.Ex50.m1.7.7.1" xref="A3.Ex50.m1.7.7.1.cmml">~</mo></mover><mo id="A3.Ex50.m1.13.13.1.1.1.1.2.2.1.3" xref="A3.Ex50.m1.13.13.1.1.1.1.2.2.2.cmml">;</mo><mi id="A3.Ex50.m1.8.8" xref="A3.Ex50.m1.8.8.cmml">𝐱</mi><mo id="A3.Ex50.m1.13.13.1.1.1.1.2.2.1.4" xref="A3.Ex50.m1.13.13.1.1.1.1.2.2.2.cmml">,</mo><mrow id="A3.Ex50.m1.13.13.1.1.1.1.2.2.1.1" xref="A3.Ex50.m1.13.13.1.1.1.1.2.2.1.1.cmml"><msup id="A3.Ex50.m1.13.13.1.1.1.1.2.2.1.1.2" xref="A3.Ex50.m1.13.13.1.1.1.1.2.2.1.1.2.cmml"><mi id="A3.Ex50.m1.13.13.1.1.1.1.2.2.1.1.2.2" xref="A3.Ex50.m1.13.13.1.1.1.1.2.2.1.1.2.2.cmml">σ</mi><mn id="A3.Ex50.m1.13.13.1.1.1.1.2.2.1.1.2.3" xref="A3.Ex50.m1.13.13.1.1.1.1.2.2.1.1.2.3.cmml">2</mn></msup><mo id="A3.Ex50.m1.13.13.1.1.1.1.2.2.1.1.1" xref="A3.Ex50.m1.13.13.1.1.1.1.2.2.1.1.1.cmml">⁢</mo><msub id="A3.Ex50.m1.13.13.1.1.1.1.2.2.1.1.3" xref="A3.Ex50.m1.13.13.1.1.1.1.2.2.1.1.3.cmml"><mi id="A3.Ex50.m1.13.13.1.1.1.1.2.2.1.1.3.2" xref="A3.Ex50.m1.13.13.1.1.1.1.2.2.1.1.3.2.cmml">𝐈</mi><mi id="A3.Ex50.m1.13.13.1.1.1.1.2.2.1.1.3.3" xref="A3.Ex50.m1.13.13.1.1.1.1.2.2.1.1.3.3.cmml">d</mi></msub></mrow><mo id="A3.Ex50.m1.13.13.1.1.1.1.2.2.1.5" stretchy="false" xref="A3.Ex50.m1.13.13.1.1.1.1.2.2.2.cmml">)</mo></mrow><mo id="A3.Ex50.m1.13.13.1.1.1.1.2.3b" lspace="0em" xref="A3.Ex50.m1.13.13.1.1.1.1.2.3.cmml">⁢</mo><mrow id="A3.Ex50.m1.13.13.1.1.1.1.2.5" xref="A3.Ex50.m1.13.13.1.1.1.1.2.5.cmml"><mo id="A3.Ex50.m1.13.13.1.1.1.1.2.5.1" rspace="0em" xref="A3.Ex50.m1.13.13.1.1.1.1.2.5.1.cmml">𝑑</mo><mover accent="true" id="A3.Ex50.m1.13.13.1.1.1.1.2.5.2" xref="A3.Ex50.m1.13.13.1.1.1.1.2.5.2.cmml"><mi id="A3.Ex50.m1.13.13.1.1.1.1.2.5.2.2" xref="A3.Ex50.m1.13.13.1.1.1.1.2.5.2.2.cmml">𝐱</mi><mo id="A3.Ex50.m1.13.13.1.1.1.1.2.5.2.1" xref="A3.Ex50.m1.13.13.1.1.1.1.2.5.2.1.cmml">~</mo></mover></mrow></mrow></mrow><mo id="A3.Ex50.m1.13.13.1.1.1.3" stretchy="false" xref="A3.Ex50.m1.13.13.1.1.2.1.cmml">‖</mo></mrow><mo id="A3.Ex50.m1.14.14.2.3" xref="A3.Ex50.m1.14.14.2.3.cmml">+</mo><mrow id="A3.Ex50.m1.14.14.2.2.1" xref="A3.Ex50.m1.14.14.2.2.2.cmml"><mo id="A3.Ex50.m1.14.14.2.2.1.2" stretchy="false" xref="A3.Ex50.m1.14.14.2.2.2.1.cmml">‖</mo><mrow id="A3.Ex50.m1.14.14.2.2.1.1" xref="A3.Ex50.m1.14.14.2.2.1.1.cmml"><mstyle displaystyle="true" id="A3.Ex50.m1.14.14.2.2.1.1.3" xref="A3.Ex50.m1.14.14.2.2.1.1.3.cmml"><msub id="A3.Ex50.m1.14.14.2.2.1.1.3a" xref="A3.Ex50.m1.14.14.2.2.1.1.3.cmml"><mo id="A3.Ex50.m1.14.14.2.2.1.1.3.2" xref="A3.Ex50.m1.14.14.2.2.1.1.3.2.cmml">∫</mo><mrow id="A3.Ex50.m1.4.4.2" xref="A3.Ex50.m1.4.4.2.cmml"><msup id="A3.Ex50.m1.4.4.2.4" xref="A3.Ex50.m1.4.4.2.4.cmml"><mi id="A3.Ex50.m1.4.4.2.4.2" xref="A3.Ex50.m1.4.4.2.4.2.cmml">ℝ</mi><mi id="A3.Ex50.m1.4.4.2.4.3" xref="A3.Ex50.m1.4.4.2.4.3.cmml">d</mi></msup><mo id="A3.Ex50.m1.4.4.2.3" xref="A3.Ex50.m1.4.4.2.3.cmml">∖</mo><mrow id="A3.Ex50.m1.4.4.2.5" xref="A3.Ex50.m1.4.4.2.5.cmml"><mi class="ltx_font_mathcaligraphic" id="A3.Ex50.m1.4.4.2.5.2" xref="A3.Ex50.m1.4.4.2.5.2.cmml">ℬ</mi><mo id="A3.Ex50.m1.4.4.2.5.1" xref="A3.Ex50.m1.4.4.2.5.1.cmml">⁢</mo><mrow id="A3.Ex50.m1.4.4.2.5.3.2" xref="A3.Ex50.m1.4.4.2.5.3.1.cmml"><mo id="A3.Ex50.m1.4.4.2.5.3.2.1" stretchy="false" xref="A3.Ex50.m1.4.4.2.5.3.1.cmml">(</mo><mi id="A3.Ex50.m1.3.3.1.1" xref="A3.Ex50.m1.3.3.1.1.cmml">𝐱</mi><mo id="A3.Ex50.m1.4.4.2.5.3.2.2" xref="A3.Ex50.m1.4.4.2.5.3.1.cmml">,</mo><mi id="A3.Ex50.m1.4.4.2.2" xref="A3.Ex50.m1.4.4.2.2.cmml">r</mi><mo id="A3.Ex50.m1.4.4.2.5.3.2.3" stretchy="false" xref="A3.Ex50.m1.4.4.2.5.3.1.cmml">)</mo></mrow></mrow></mrow></msub></mstyle><mrow id="A3.Ex50.m1.14.14.2.2.1.1.2" xref="A3.Ex50.m1.14.14.2.2.1.1.2.cmml"><mrow id="A3.Ex50.m1.14.14.2.2.1.1.1.1.1" xref="A3.Ex50.m1.14.14.2.2.1.1.1.1.1.1.cmml"><mo id="A3.Ex50.m1.14.14.2.2.1.1.1.1.1.2" stretchy="false" xref="A3.Ex50.m1.14.14.2.2.1.1.1.1.1.1.cmml">(</mo><mrow id="A3.Ex50.m1.14.14.2.2.1.1.1.1.1.1" xref="A3.Ex50.m1.14.14.2.2.1.1.1.1.1.1.cmml"><mrow id="A3.Ex50.m1.14.14.2.2.1.1.1.1.1.1.2" xref="A3.Ex50.m1.14.14.2.2.1.1.1.1.1.1.2.cmml"><mrow id="A3.Ex50.m1.14.14.2.2.1.1.1.1.1.1.2.2" xref="A3.Ex50.m1.14.14.2.2.1.1.1.1.1.1.2.2.cmml"><mrow id="A3.Ex50.m1.14.14.2.2.1.1.1.1.1.1.2.2.1" xref="A3.Ex50.m1.14.14.2.2.1.1.1.1.1.1.2.2.1.cmml"><mo id="A3.Ex50.m1.14.14.2.2.1.1.1.1.1.1.2.2.1.1" rspace="0.167em" xref="A3.Ex50.m1.14.14.2.2.1.1.1.1.1.1.2.2.1.1.cmml">∇</mo><mi id="A3.Ex50.m1.14.14.2.2.1.1.1.1.1.1.2.2.1.2" xref="A3.Ex50.m1.14.14.2.2.1.1.1.1.1.1.2.2.1.2.cmml">log</mi></mrow><mo id="A3.Ex50.m1.14.14.2.2.1.1.1.1.1.1.2.2a" lspace="0.167em" xref="A3.Ex50.m1.14.14.2.2.1.1.1.1.1.1.2.2.cmml">⁡</mo><mi id="A3.Ex50.m1.14.14.2.2.1.1.1.1.1.1.2.2.2" xref="A3.Ex50.m1.14.14.2.2.1.1.1.1.1.1.2.2.2.cmml">p</mi></mrow><mo id="A3.Ex50.m1.14.14.2.2.1.1.1.1.1.1.2.1" xref="A3.Ex50.m1.14.14.2.2.1.1.1.1.1.1.2.1.cmml">⁢</mo><mrow id="A3.Ex50.m1.14.14.2.2.1.1.1.1.1.1.2.3.2" xref="A3.Ex50.m1.9.9.cmml"><mo id="A3.Ex50.m1.14.14.2.2.1.1.1.1.1.1.2.3.2.1" stretchy="false" xref="A3.Ex50.m1.9.9.cmml">(</mo><mover accent="true" id="A3.Ex50.m1.9.9" xref="A3.Ex50.m1.9.9.cmml"><mi id="A3.Ex50.m1.9.9.2" xref="A3.Ex50.m1.9.9.2.cmml">𝐱</mi><mo id="A3.Ex50.m1.9.9.1" xref="A3.Ex50.m1.9.9.1.cmml">~</mo></mover><mo id="A3.Ex50.m1.14.14.2.2.1.1.1.1.1.1.2.3.2.2" stretchy="false" xref="A3.Ex50.m1.9.9.cmml">)</mo></mrow></mrow><mo id="A3.Ex50.m1.14.14.2.2.1.1.1.1.1.1.1" xref="A3.Ex50.m1.14.14.2.2.1.1.1.1.1.1.1.cmml">−</mo><mrow id="A3.Ex50.m1.14.14.2.2.1.1.1.1.1.1.3" xref="A3.Ex50.m1.14.14.2.2.1.1.1.1.1.1.3.cmml"><mrow id="A3.Ex50.m1.14.14.2.2.1.1.1.1.1.1.3.2" xref="A3.Ex50.m1.14.14.2.2.1.1.1.1.1.1.3.2.cmml"><mrow id="A3.Ex50.m1.14.14.2.2.1.1.1.1.1.1.3.2.1" xref="A3.Ex50.m1.14.14.2.2.1.1.1.1.1.1.3.2.1.cmml"><mo id="A3.Ex50.m1.14.14.2.2.1.1.1.1.1.1.3.2.1.1" rspace="0.167em" xref="A3.Ex50.m1.14.14.2.2.1.1.1.1.1.1.3.2.1.1.cmml">∇</mo><mi id="A3.Ex50.m1.14.14.2.2.1.1.1.1.1.1.3.2.1.2" xref="A3.Ex50.m1.14.14.2.2.1.1.1.1.1.1.3.2.1.2.cmml">log</mi></mrow><mo id="A3.Ex50.m1.14.14.2.2.1.1.1.1.1.1.3.2a" lspace="0.167em" xref="A3.Ex50.m1.14.14.2.2.1.1.1.1.1.1.3.2.cmml">⁡</mo><mi id="A3.Ex50.m1.14.14.2.2.1.1.1.1.1.1.3.2.2" 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\mathbf{x}},\sigma^{2}\mathbf{I}_{d})d{\tilde{\mathbf{x}}}}\|+\|\int_{{\mathbb% {R}}^{d}\setminus\mathcal{B}({\mathbf{x}},r)}{(\nabla\log p({\tilde{\mathbf{x}% }})-\nabla\log p({\mathbf{x}}))\mathcal{N}({\tilde{\mathbf{x}}};{\mathbf{x}},% \sigma^{2}\mathbf{I}_{d})d{\tilde{\mathbf{x}}}}\|</annotation><annotation encoding="application/x-llamapun" id="A3.Ex50.m1.14d">≤ ∥ ∫ start_POSTSUBSCRIPT caligraphic_B ( bold_x , italic_r ) end_POSTSUBSCRIPT ( ∇ roman_log italic_p ( over~ start_ARG bold_x end_ARG ) - ∇ roman_log italic_p ( bold_x ) ) caligraphic_N ( over~ start_ARG bold_x end_ARG ; bold_x , italic_σ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT bold_I start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT ) italic_d over~ start_ARG bold_x end_ARG ∥ + ∥ ∫ start_POSTSUBSCRIPT blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT ∖ caligraphic_B ( bold_x , italic_r ) end_POSTSUBSCRIPT ( ∇ roman_log italic_p ( over~ start_ARG bold_x end_ARG ) - ∇ roman_log italic_p ( bold_x ) ) caligraphic_N ( over~ start_ARG bold_x end_ARG ; bold_x , italic_σ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT bold_I start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT ) italic_d over~ start_ARG bold_x end_ARG ∥</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="A3.Ex51"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\leq M_{{\mathbf{x}}}\int_{\mathcal{B}({\mathbf{x}},r)}{\|{\tilde% {\mathbf{x}}}-{\mathbf{x}}\|\mathcal{N}({\tilde{\mathbf{x}}};{\mathbf{x}},% \sigma^{2}\mathbf{I}_{d})d{\tilde{\mathbf{x}}}}+\|\int_{{\mathbb{R}}^{d}% \setminus\mathcal{B}({\mathbf{x}},r)}{(\nabla\log p({\tilde{\mathbf{x}}})-% \nabla\log p({\mathbf{x}}))\mathcal{N}({\tilde{\mathbf{x}}};{\mathbf{x}},% \sigma^{2}\mathbf{I}_{d})d{\tilde{\mathbf{x}}}}\|" class="ltx_Math" display="inline" 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id="A3.Ex51.m1.13.13.3.3.1.1.2.2.1.1.3.cmml" xref="A3.Ex51.m1.13.13.3.3.1.1.2.2.1.1.3"><csymbol cd="ambiguous" id="A3.Ex51.m1.13.13.3.3.1.1.2.2.1.1.3.1.cmml" xref="A3.Ex51.m1.13.13.3.3.1.1.2.2.1.1.3">subscript</csymbol><ci id="A3.Ex51.m1.13.13.3.3.1.1.2.2.1.1.3.2.cmml" xref="A3.Ex51.m1.13.13.3.3.1.1.2.2.1.1.3.2">𝐈</ci><ci id="A3.Ex51.m1.13.13.3.3.1.1.2.2.1.1.3.3.cmml" xref="A3.Ex51.m1.13.13.3.3.1.1.2.2.1.1.3.3">𝑑</ci></apply></apply></list><apply id="A3.Ex51.m1.13.13.3.3.1.1.2.5.cmml" xref="A3.Ex51.m1.13.13.3.3.1.1.2.5"><csymbol cd="latexml" id="A3.Ex51.m1.13.13.3.3.1.1.2.5.1.cmml" xref="A3.Ex51.m1.13.13.3.3.1.1.2.5.1">differential-d</csymbol><apply id="A3.Ex51.m1.13.13.3.3.1.1.2.5.2.cmml" xref="A3.Ex51.m1.13.13.3.3.1.1.2.5.2"><ci id="A3.Ex51.m1.13.13.3.3.1.1.2.5.2.1.cmml" xref="A3.Ex51.m1.13.13.3.3.1.1.2.5.2.1">~</ci><ci id="A3.Ex51.m1.13.13.3.3.1.1.2.5.2.2.cmml" xref="A3.Ex51.m1.13.13.3.3.1.1.2.5.2.2">𝐱</ci></apply></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.Ex51.m1.13c">\displaystyle\leq M_{{\mathbf{x}}}\int_{\mathcal{B}({\mathbf{x}},r)}{\|{\tilde% {\mathbf{x}}}-{\mathbf{x}}\|\mathcal{N}({\tilde{\mathbf{x}}};{\mathbf{x}},% \sigma^{2}\mathbf{I}_{d})d{\tilde{\mathbf{x}}}}+\|\int_{{\mathbb{R}}^{d}% \setminus\mathcal{B}({\mathbf{x}},r)}{(\nabla\log p({\tilde{\mathbf{x}}})-% \nabla\log p({\mathbf{x}}))\mathcal{N}({\tilde{\mathbf{x}}};{\mathbf{x}},% \sigma^{2}\mathbf{I}_{d})d{\tilde{\mathbf{x}}}}\|</annotation><annotation encoding="application/x-llamapun" id="A3.Ex51.m1.13d">≤ italic_M start_POSTSUBSCRIPT bold_x end_POSTSUBSCRIPT ∫ start_POSTSUBSCRIPT caligraphic_B ( bold_x , italic_r ) end_POSTSUBSCRIPT ∥ over~ start_ARG bold_x end_ARG - bold_x ∥ caligraphic_N ( over~ start_ARG bold_x end_ARG ; bold_x , italic_σ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT bold_I start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT ) italic_d over~ start_ARG bold_x end_ARG + ∥ ∫ start_POSTSUBSCRIPT blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT ∖ caligraphic_B ( bold_x , italic_r ) end_POSTSUBSCRIPT ( ∇ roman_log italic_p ( over~ start_ARG bold_x end_ARG ) - ∇ roman_log italic_p ( bold_x ) ) caligraphic_N ( over~ start_ARG bold_x end_ARG ; bold_x , italic_σ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT bold_I start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT ) italic_d over~ start_ARG bold_x end_ARG ∥</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="A3.E25"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\leq M_{{\mathbf{x}}}\sigma\int_{{\mathbb{R}}^{d}}{\|{\tilde{% \mathbf{x}}}\|\mathcal{N}({\tilde{\mathbf{x}}};0,\mathbf{I}_{d})d{\tilde{% \mathbf{x}}}}+\|\int_{{\mathbb{R}}^{d}\setminus\mathcal{B}({\mathbf{x}},r)}{(% \nabla\log p({\tilde{\mathbf{x}}})-\nabla\log p({\mathbf{x}}))\mathcal{N}({% \tilde{\mathbf{x}}};{\mathbf{x}},\sigma^{2}\mathbf{I}_{d})d{\tilde{\mathbf{x}}% }}\|." class="ltx_Math" display="inline" id="A3.E25.m1.10"><semantics id="A3.E25.m1.10a"><mrow id="A3.E25.m1.10.10.1" xref="A3.E25.m1.10.10.1.1.cmml"><mrow id="A3.E25.m1.10.10.1.1" xref="A3.E25.m1.10.10.1.1.cmml"><mi id="A3.E25.m1.10.10.1.1.4" xref="A3.E25.m1.10.10.1.1.4.cmml"></mi><mo id="A3.E25.m1.10.10.1.1.3" xref="A3.E25.m1.10.10.1.1.3.cmml">≤</mo><mrow id="A3.E25.m1.10.10.1.1.2" xref="A3.E25.m1.10.10.1.1.2.cmml"><mrow id="A3.E25.m1.10.10.1.1.1.1" xref="A3.E25.m1.10.10.1.1.1.1.cmml"><msub id="A3.E25.m1.10.10.1.1.1.1.3" xref="A3.E25.m1.10.10.1.1.1.1.3.cmml"><mi id="A3.E25.m1.10.10.1.1.1.1.3.2" xref="A3.E25.m1.10.10.1.1.1.1.3.2.cmml">M</mi><mi id="A3.E25.m1.10.10.1.1.1.1.3.3" xref="A3.E25.m1.10.10.1.1.1.1.3.3.cmml">𝐱</mi></msub><mo 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id="A3.E25.m1.10.10.1.1.2.2.1.1.2.2.1.1.2.3.cmml" type="integer" xref="A3.E25.m1.10.10.1.1.2.2.1.1.2.2.1.1.2.3">2</cn></apply><apply id="A3.E25.m1.10.10.1.1.2.2.1.1.2.2.1.1.3.cmml" xref="A3.E25.m1.10.10.1.1.2.2.1.1.2.2.1.1.3"><csymbol cd="ambiguous" id="A3.E25.m1.10.10.1.1.2.2.1.1.2.2.1.1.3.1.cmml" xref="A3.E25.m1.10.10.1.1.2.2.1.1.2.2.1.1.3">subscript</csymbol><ci id="A3.E25.m1.10.10.1.1.2.2.1.1.2.2.1.1.3.2.cmml" xref="A3.E25.m1.10.10.1.1.2.2.1.1.2.2.1.1.3.2">𝐈</ci><ci id="A3.E25.m1.10.10.1.1.2.2.1.1.2.2.1.1.3.3.cmml" xref="A3.E25.m1.10.10.1.1.2.2.1.1.2.2.1.1.3.3">𝑑</ci></apply></apply></list><apply id="A3.E25.m1.10.10.1.1.2.2.1.1.2.5.cmml" xref="A3.E25.m1.10.10.1.1.2.2.1.1.2.5"><csymbol cd="latexml" id="A3.E25.m1.10.10.1.1.2.2.1.1.2.5.1.cmml" xref="A3.E25.m1.10.10.1.1.2.2.1.1.2.5.1">differential-d</csymbol><apply id="A3.E25.m1.10.10.1.1.2.2.1.1.2.5.2.cmml" xref="A3.E25.m1.10.10.1.1.2.2.1.1.2.5.2"><ci id="A3.E25.m1.10.10.1.1.2.2.1.1.2.5.2.1.cmml" xref="A3.E25.m1.10.10.1.1.2.2.1.1.2.5.2.1">~</ci><ci id="A3.E25.m1.10.10.1.1.2.2.1.1.2.5.2.2.cmml" xref="A3.E25.m1.10.10.1.1.2.2.1.1.2.5.2.2">𝐱</ci></apply></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.E25.m1.10c">\displaystyle\leq M_{{\mathbf{x}}}\sigma\int_{{\mathbb{R}}^{d}}{\|{\tilde{% \mathbf{x}}}\|\mathcal{N}({\tilde{\mathbf{x}}};0,\mathbf{I}_{d})d{\tilde{% \mathbf{x}}}}+\|\int_{{\mathbb{R}}^{d}\setminus\mathcal{B}({\mathbf{x}},r)}{(% \nabla\log p({\tilde{\mathbf{x}}})-\nabla\log p({\mathbf{x}}))\mathcal{N}({% \tilde{\mathbf{x}}};{\mathbf{x}},\sigma^{2}\mathbf{I}_{d})d{\tilde{\mathbf{x}}% }}\|.</annotation><annotation encoding="application/x-llamapun" id="A3.E25.m1.10d">≤ italic_M start_POSTSUBSCRIPT bold_x end_POSTSUBSCRIPT italic_σ ∫ start_POSTSUBSCRIPT blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ∥ over~ start_ARG bold_x end_ARG ∥ caligraphic_N ( over~ start_ARG bold_x end_ARG ; 0 , bold_I start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT ) italic_d over~ start_ARG bold_x end_ARG + ∥ ∫ start_POSTSUBSCRIPT blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT ∖ caligraphic_B ( bold_x , italic_r ) end_POSTSUBSCRIPT ( ∇ roman_log italic_p ( over~ start_ARG bold_x end_ARG ) - ∇ roman_log italic_p ( bold_x ) ) caligraphic_N ( over~ start_ARG bold_x end_ARG ; bold_x , italic_σ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT bold_I start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT ) italic_d over~ start_ARG bold_x end_ARG ∥ .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(25)</span></td> </tr></tbody> </table> <p class="ltx_p" id="A3.SS2.SSS1.10.p4.5">One can notice that <math alttext="\nabla\log p" class="ltx_Math" display="inline" id="A3.SS2.SSS1.10.p4.1.m1.1"><semantics id="A3.SS2.SSS1.10.p4.1.m1.1a"><mrow id="A3.SS2.SSS1.10.p4.1.m1.1.1" xref="A3.SS2.SSS1.10.p4.1.m1.1.1.cmml"><mrow id="A3.SS2.SSS1.10.p4.1.m1.1.1.1" xref="A3.SS2.SSS1.10.p4.1.m1.1.1.1.cmml"><mo id="A3.SS2.SSS1.10.p4.1.m1.1.1.1.1" rspace="0.167em" xref="A3.SS2.SSS1.10.p4.1.m1.1.1.1.1.cmml">∇</mo><mi id="A3.SS2.SSS1.10.p4.1.m1.1.1.1.2" xref="A3.SS2.SSS1.10.p4.1.m1.1.1.1.2.cmml">log</mi></mrow><mo id="A3.SS2.SSS1.10.p4.1.m1.1.1a" lspace="0.167em" xref="A3.SS2.SSS1.10.p4.1.m1.1.1.cmml">⁡</mo><mi id="A3.SS2.SSS1.10.p4.1.m1.1.1.2" xref="A3.SS2.SSS1.10.p4.1.m1.1.1.2.cmml">p</mi></mrow><annotation-xml encoding="MathML-Content" id="A3.SS2.SSS1.10.p4.1.m1.1b"><apply id="A3.SS2.SSS1.10.p4.1.m1.1.1.cmml" xref="A3.SS2.SSS1.10.p4.1.m1.1.1"><apply id="A3.SS2.SSS1.10.p4.1.m1.1.1.1.cmml" xref="A3.SS2.SSS1.10.p4.1.m1.1.1.1"><ci id="A3.SS2.SSS1.10.p4.1.m1.1.1.1.1.cmml" xref="A3.SS2.SSS1.10.p4.1.m1.1.1.1.1">∇</ci><log id="A3.SS2.SSS1.10.p4.1.m1.1.1.1.2.cmml" xref="A3.SS2.SSS1.10.p4.1.m1.1.1.1.2"></log></apply><ci id="A3.SS2.SSS1.10.p4.1.m1.1.1.2.cmml" xref="A3.SS2.SSS1.10.p4.1.m1.1.1.2">𝑝</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.SS2.SSS1.10.p4.1.m1.1c">\nabla\log p</annotation><annotation encoding="application/x-llamapun" id="A3.SS2.SSS1.10.p4.1.m1.1d">∇ roman_log italic_p</annotation></semantics></math> is sub-linear, so there exists <math alttext="b_{p}\geq 0" class="ltx_Math" display="inline" id="A3.SS2.SSS1.10.p4.2.m2.1"><semantics id="A3.SS2.SSS1.10.p4.2.m2.1a"><mrow id="A3.SS2.SSS1.10.p4.2.m2.1.1" xref="A3.SS2.SSS1.10.p4.2.m2.1.1.cmml"><msub id="A3.SS2.SSS1.10.p4.2.m2.1.1.2" xref="A3.SS2.SSS1.10.p4.2.m2.1.1.2.cmml"><mi id="A3.SS2.SSS1.10.p4.2.m2.1.1.2.2" xref="A3.SS2.SSS1.10.p4.2.m2.1.1.2.2.cmml">b</mi><mi id="A3.SS2.SSS1.10.p4.2.m2.1.1.2.3" xref="A3.SS2.SSS1.10.p4.2.m2.1.1.2.3.cmml">p</mi></msub><mo id="A3.SS2.SSS1.10.p4.2.m2.1.1.1" xref="A3.SS2.SSS1.10.p4.2.m2.1.1.1.cmml">≥</mo><mn id="A3.SS2.SSS1.10.p4.2.m2.1.1.3" xref="A3.SS2.SSS1.10.p4.2.m2.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="A3.SS2.SSS1.10.p4.2.m2.1b"><apply id="A3.SS2.SSS1.10.p4.2.m2.1.1.cmml" xref="A3.SS2.SSS1.10.p4.2.m2.1.1"><geq id="A3.SS2.SSS1.10.p4.2.m2.1.1.1.cmml" xref="A3.SS2.SSS1.10.p4.2.m2.1.1.1"></geq><apply id="A3.SS2.SSS1.10.p4.2.m2.1.1.2.cmml" xref="A3.SS2.SSS1.10.p4.2.m2.1.1.2"><csymbol cd="ambiguous" id="A3.SS2.SSS1.10.p4.2.m2.1.1.2.1.cmml" xref="A3.SS2.SSS1.10.p4.2.m2.1.1.2">subscript</csymbol><ci id="A3.SS2.SSS1.10.p4.2.m2.1.1.2.2.cmml" xref="A3.SS2.SSS1.10.p4.2.m2.1.1.2.2">𝑏</ci><ci id="A3.SS2.SSS1.10.p4.2.m2.1.1.2.3.cmml" xref="A3.SS2.SSS1.10.p4.2.m2.1.1.2.3">𝑝</ci></apply><cn id="A3.SS2.SSS1.10.p4.2.m2.1.1.3.cmml" type="integer" xref="A3.SS2.SSS1.10.p4.2.m2.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.SS2.SSS1.10.p4.2.m2.1c">b_{p}\geq 0</annotation><annotation encoding="application/x-llamapun" id="A3.SS2.SSS1.10.p4.2.m2.1d">italic_b start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT ≥ 0</annotation></semantics></math> such that <math alttext="\forall{\mathbf{x}}\in{\mathbb{R}}^{d}" class="ltx_Math" display="inline" id="A3.SS2.SSS1.10.p4.3.m3.1"><semantics id="A3.SS2.SSS1.10.p4.3.m3.1a"><mrow id="A3.SS2.SSS1.10.p4.3.m3.1.1" xref="A3.SS2.SSS1.10.p4.3.m3.1.1.cmml"><mrow id="A3.SS2.SSS1.10.p4.3.m3.1.1.2" xref="A3.SS2.SSS1.10.p4.3.m3.1.1.2.cmml"><mo id="A3.SS2.SSS1.10.p4.3.m3.1.1.2.1" rspace="0.167em" xref="A3.SS2.SSS1.10.p4.3.m3.1.1.2.1.cmml">∀</mo><mi id="A3.SS2.SSS1.10.p4.3.m3.1.1.2.2" xref="A3.SS2.SSS1.10.p4.3.m3.1.1.2.2.cmml">𝐱</mi></mrow><mo id="A3.SS2.SSS1.10.p4.3.m3.1.1.1" xref="A3.SS2.SSS1.10.p4.3.m3.1.1.1.cmml">∈</mo><msup id="A3.SS2.SSS1.10.p4.3.m3.1.1.3" xref="A3.SS2.SSS1.10.p4.3.m3.1.1.3.cmml"><mi id="A3.SS2.SSS1.10.p4.3.m3.1.1.3.2" xref="A3.SS2.SSS1.10.p4.3.m3.1.1.3.2.cmml">ℝ</mi><mi id="A3.SS2.SSS1.10.p4.3.m3.1.1.3.3" xref="A3.SS2.SSS1.10.p4.3.m3.1.1.3.3.cmml">d</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="A3.SS2.SSS1.10.p4.3.m3.1b"><apply id="A3.SS2.SSS1.10.p4.3.m3.1.1.cmml" xref="A3.SS2.SSS1.10.p4.3.m3.1.1"><in id="A3.SS2.SSS1.10.p4.3.m3.1.1.1.cmml" xref="A3.SS2.SSS1.10.p4.3.m3.1.1.1"></in><apply id="A3.SS2.SSS1.10.p4.3.m3.1.1.2.cmml" xref="A3.SS2.SSS1.10.p4.3.m3.1.1.2"><csymbol cd="latexml" id="A3.SS2.SSS1.10.p4.3.m3.1.1.2.1.cmml" xref="A3.SS2.SSS1.10.p4.3.m3.1.1.2.1">for-all</csymbol><ci id="A3.SS2.SSS1.10.p4.3.m3.1.1.2.2.cmml" xref="A3.SS2.SSS1.10.p4.3.m3.1.1.2.2">𝐱</ci></apply><apply id="A3.SS2.SSS1.10.p4.3.m3.1.1.3.cmml" xref="A3.SS2.SSS1.10.p4.3.m3.1.1.3"><csymbol cd="ambiguous" id="A3.SS2.SSS1.10.p4.3.m3.1.1.3.1.cmml" xref="A3.SS2.SSS1.10.p4.3.m3.1.1.3">superscript</csymbol><ci id="A3.SS2.SSS1.10.p4.3.m3.1.1.3.2.cmml" xref="A3.SS2.SSS1.10.p4.3.m3.1.1.3.2">ℝ</ci><ci id="A3.SS2.SSS1.10.p4.3.m3.1.1.3.3.cmml" xref="A3.SS2.SSS1.10.p4.3.m3.1.1.3.3">𝑑</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.SS2.SSS1.10.p4.3.m3.1c">\forall{\mathbf{x}}\in{\mathbb{R}}^{d}</annotation><annotation encoding="application/x-llamapun" id="A3.SS2.SSS1.10.p4.3.m3.1d">∀ bold_x ∈ blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT</annotation></semantics></math>, <math alttext="\|\nabla\log p({\mathbf{x}})\|\leq b_{p}(\|{\mathbf{x}}\|+1)" class="ltx_Math" display="inline" id="A3.SS2.SSS1.10.p4.4.m4.4"><semantics id="A3.SS2.SSS1.10.p4.4.m4.4a"><mrow id="A3.SS2.SSS1.10.p4.4.m4.4.4" xref="A3.SS2.SSS1.10.p4.4.m4.4.4.cmml"><mrow id="A3.SS2.SSS1.10.p4.4.m4.3.3.1.1" xref="A3.SS2.SSS1.10.p4.4.m4.3.3.1.2.cmml"><mo id="A3.SS2.SSS1.10.p4.4.m4.3.3.1.1.2" stretchy="false" xref="A3.SS2.SSS1.10.p4.4.m4.3.3.1.2.1.cmml">‖</mo><mrow id="A3.SS2.SSS1.10.p4.4.m4.3.3.1.1.1" xref="A3.SS2.SSS1.10.p4.4.m4.3.3.1.1.1.cmml"><mrow id="A3.SS2.SSS1.10.p4.4.m4.3.3.1.1.1.2" xref="A3.SS2.SSS1.10.p4.4.m4.3.3.1.1.1.2.cmml"><mrow id="A3.SS2.SSS1.10.p4.4.m4.3.3.1.1.1.2.1" xref="A3.SS2.SSS1.10.p4.4.m4.3.3.1.1.1.2.1.cmml"><mo id="A3.SS2.SSS1.10.p4.4.m4.3.3.1.1.1.2.1.1" rspace="0.167em" xref="A3.SS2.SSS1.10.p4.4.m4.3.3.1.1.1.2.1.1.cmml">∇</mo><mi id="A3.SS2.SSS1.10.p4.4.m4.3.3.1.1.1.2.1.2" xref="A3.SS2.SSS1.10.p4.4.m4.3.3.1.1.1.2.1.2.cmml">log</mi></mrow><mo id="A3.SS2.SSS1.10.p4.4.m4.3.3.1.1.1.2a" lspace="0.167em" xref="A3.SS2.SSS1.10.p4.4.m4.3.3.1.1.1.2.cmml">⁡</mo><mi id="A3.SS2.SSS1.10.p4.4.m4.3.3.1.1.1.2.2" xref="A3.SS2.SSS1.10.p4.4.m4.3.3.1.1.1.2.2.cmml">p</mi></mrow><mo id="A3.SS2.SSS1.10.p4.4.m4.3.3.1.1.1.1" xref="A3.SS2.SSS1.10.p4.4.m4.3.3.1.1.1.1.cmml">⁢</mo><mrow id="A3.SS2.SSS1.10.p4.4.m4.3.3.1.1.1.3.2" xref="A3.SS2.SSS1.10.p4.4.m4.3.3.1.1.1.cmml"><mo id="A3.SS2.SSS1.10.p4.4.m4.3.3.1.1.1.3.2.1" stretchy="false" xref="A3.SS2.SSS1.10.p4.4.m4.3.3.1.1.1.cmml">(</mo><mi id="A3.SS2.SSS1.10.p4.4.m4.1.1" xref="A3.SS2.SSS1.10.p4.4.m4.1.1.cmml">𝐱</mi><mo id="A3.SS2.SSS1.10.p4.4.m4.3.3.1.1.1.3.2.2" stretchy="false" xref="A3.SS2.SSS1.10.p4.4.m4.3.3.1.1.1.cmml">)</mo></mrow></mrow><mo id="A3.SS2.SSS1.10.p4.4.m4.3.3.1.1.3" stretchy="false" xref="A3.SS2.SSS1.10.p4.4.m4.3.3.1.2.1.cmml">‖</mo></mrow><mo id="A3.SS2.SSS1.10.p4.4.m4.4.4.3" xref="A3.SS2.SSS1.10.p4.4.m4.4.4.3.cmml">≤</mo><mrow id="A3.SS2.SSS1.10.p4.4.m4.4.4.2" xref="A3.SS2.SSS1.10.p4.4.m4.4.4.2.cmml"><msub id="A3.SS2.SSS1.10.p4.4.m4.4.4.2.3" xref="A3.SS2.SSS1.10.p4.4.m4.4.4.2.3.cmml"><mi id="A3.SS2.SSS1.10.p4.4.m4.4.4.2.3.2" xref="A3.SS2.SSS1.10.p4.4.m4.4.4.2.3.2.cmml">b</mi><mi id="A3.SS2.SSS1.10.p4.4.m4.4.4.2.3.3" xref="A3.SS2.SSS1.10.p4.4.m4.4.4.2.3.3.cmml">p</mi></msub><mo id="A3.SS2.SSS1.10.p4.4.m4.4.4.2.2" xref="A3.SS2.SSS1.10.p4.4.m4.4.4.2.2.cmml">⁢</mo><mrow id="A3.SS2.SSS1.10.p4.4.m4.4.4.2.1.1" xref="A3.SS2.SSS1.10.p4.4.m4.4.4.2.1.1.1.cmml"><mo id="A3.SS2.SSS1.10.p4.4.m4.4.4.2.1.1.2" stretchy="false" xref="A3.SS2.SSS1.10.p4.4.m4.4.4.2.1.1.1.cmml">(</mo><mrow id="A3.SS2.SSS1.10.p4.4.m4.4.4.2.1.1.1" xref="A3.SS2.SSS1.10.p4.4.m4.4.4.2.1.1.1.cmml"><mrow id="A3.SS2.SSS1.10.p4.4.m4.4.4.2.1.1.1.2.2" xref="A3.SS2.SSS1.10.p4.4.m4.4.4.2.1.1.1.2.1.cmml"><mo id="A3.SS2.SSS1.10.p4.4.m4.4.4.2.1.1.1.2.2.1" stretchy="false" xref="A3.SS2.SSS1.10.p4.4.m4.4.4.2.1.1.1.2.1.1.cmml">‖</mo><mi id="A3.SS2.SSS1.10.p4.4.m4.2.2" xref="A3.SS2.SSS1.10.p4.4.m4.2.2.cmml">𝐱</mi><mo id="A3.SS2.SSS1.10.p4.4.m4.4.4.2.1.1.1.2.2.2" stretchy="false" xref="A3.SS2.SSS1.10.p4.4.m4.4.4.2.1.1.1.2.1.1.cmml">‖</mo></mrow><mo id="A3.SS2.SSS1.10.p4.4.m4.4.4.2.1.1.1.1" xref="A3.SS2.SSS1.10.p4.4.m4.4.4.2.1.1.1.1.cmml">+</mo><mn id="A3.SS2.SSS1.10.p4.4.m4.4.4.2.1.1.1.3" xref="A3.SS2.SSS1.10.p4.4.m4.4.4.2.1.1.1.3.cmml">1</mn></mrow><mo id="A3.SS2.SSS1.10.p4.4.m4.4.4.2.1.1.3" stretchy="false" xref="A3.SS2.SSS1.10.p4.4.m4.4.4.2.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="A3.SS2.SSS1.10.p4.4.m4.4b"><apply id="A3.SS2.SSS1.10.p4.4.m4.4.4.cmml" xref="A3.SS2.SSS1.10.p4.4.m4.4.4"><leq id="A3.SS2.SSS1.10.p4.4.m4.4.4.3.cmml" xref="A3.SS2.SSS1.10.p4.4.m4.4.4.3"></leq><apply id="A3.SS2.SSS1.10.p4.4.m4.3.3.1.2.cmml" xref="A3.SS2.SSS1.10.p4.4.m4.3.3.1.1"><csymbol cd="latexml" id="A3.SS2.SSS1.10.p4.4.m4.3.3.1.2.1.cmml" xref="A3.SS2.SSS1.10.p4.4.m4.3.3.1.1.2">norm</csymbol><apply id="A3.SS2.SSS1.10.p4.4.m4.3.3.1.1.1.cmml" xref="A3.SS2.SSS1.10.p4.4.m4.3.3.1.1.1"><times id="A3.SS2.SSS1.10.p4.4.m4.3.3.1.1.1.1.cmml" xref="A3.SS2.SSS1.10.p4.4.m4.3.3.1.1.1.1"></times><apply id="A3.SS2.SSS1.10.p4.4.m4.3.3.1.1.1.2.cmml" xref="A3.SS2.SSS1.10.p4.4.m4.3.3.1.1.1.2"><apply id="A3.SS2.SSS1.10.p4.4.m4.3.3.1.1.1.2.1.cmml" xref="A3.SS2.SSS1.10.p4.4.m4.3.3.1.1.1.2.1"><ci id="A3.SS2.SSS1.10.p4.4.m4.3.3.1.1.1.2.1.1.cmml" xref="A3.SS2.SSS1.10.p4.4.m4.3.3.1.1.1.2.1.1">∇</ci><log id="A3.SS2.SSS1.10.p4.4.m4.3.3.1.1.1.2.1.2.cmml" xref="A3.SS2.SSS1.10.p4.4.m4.3.3.1.1.1.2.1.2"></log></apply><ci id="A3.SS2.SSS1.10.p4.4.m4.3.3.1.1.1.2.2.cmml" xref="A3.SS2.SSS1.10.p4.4.m4.3.3.1.1.1.2.2">𝑝</ci></apply><ci id="A3.SS2.SSS1.10.p4.4.m4.1.1.cmml" xref="A3.SS2.SSS1.10.p4.4.m4.1.1">𝐱</ci></apply></apply><apply id="A3.SS2.SSS1.10.p4.4.m4.4.4.2.cmml" xref="A3.SS2.SSS1.10.p4.4.m4.4.4.2"><times id="A3.SS2.SSS1.10.p4.4.m4.4.4.2.2.cmml" xref="A3.SS2.SSS1.10.p4.4.m4.4.4.2.2"></times><apply id="A3.SS2.SSS1.10.p4.4.m4.4.4.2.3.cmml" xref="A3.SS2.SSS1.10.p4.4.m4.4.4.2.3"><csymbol cd="ambiguous" id="A3.SS2.SSS1.10.p4.4.m4.4.4.2.3.1.cmml" xref="A3.SS2.SSS1.10.p4.4.m4.4.4.2.3">subscript</csymbol><ci id="A3.SS2.SSS1.10.p4.4.m4.4.4.2.3.2.cmml" xref="A3.SS2.SSS1.10.p4.4.m4.4.4.2.3.2">𝑏</ci><ci id="A3.SS2.SSS1.10.p4.4.m4.4.4.2.3.3.cmml" xref="A3.SS2.SSS1.10.p4.4.m4.4.4.2.3.3">𝑝</ci></apply><apply id="A3.SS2.SSS1.10.p4.4.m4.4.4.2.1.1.1.cmml" xref="A3.SS2.SSS1.10.p4.4.m4.4.4.2.1.1"><plus id="A3.SS2.SSS1.10.p4.4.m4.4.4.2.1.1.1.1.cmml" xref="A3.SS2.SSS1.10.p4.4.m4.4.4.2.1.1.1.1"></plus><apply id="A3.SS2.SSS1.10.p4.4.m4.4.4.2.1.1.1.2.1.cmml" xref="A3.SS2.SSS1.10.p4.4.m4.4.4.2.1.1.1.2.2"><csymbol cd="latexml" id="A3.SS2.SSS1.10.p4.4.m4.4.4.2.1.1.1.2.1.1.cmml" xref="A3.SS2.SSS1.10.p4.4.m4.4.4.2.1.1.1.2.2.1">norm</csymbol><ci id="A3.SS2.SSS1.10.p4.4.m4.2.2.cmml" xref="A3.SS2.SSS1.10.p4.4.m4.2.2">𝐱</ci></apply><cn id="A3.SS2.SSS1.10.p4.4.m4.4.4.2.1.1.1.3.cmml" type="integer" xref="A3.SS2.SSS1.10.p4.4.m4.4.4.2.1.1.1.3">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.SS2.SSS1.10.p4.4.m4.4c">\|\nabla\log p({\mathbf{x}})\|\leq b_{p}(\|{\mathbf{x}}\|+1)</annotation><annotation encoding="application/x-llamapun" id="A3.SS2.SSS1.10.p4.4.m4.4d">∥ ∇ roman_log italic_p ( bold_x ) ∥ ≤ italic_b start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT ( ∥ bold_x ∥ + 1 )</annotation></semantics></math>. 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id="A3.SS2.SSS1.10.p4.5.m5.8c">\int_{{\mathbb{R}}^{d}}{\|{\tilde{\mathbf{x}}}\|\mathcal{N}({\tilde{\mathbf{x}% }};0,\mathbf{I}_{d})d{\tilde{\mathbf{x}}}}\leq\sqrt{\int_{{\mathbb{R}}^{d}}{\|% {\tilde{\mathbf{x}}}\|^{2}\mathcal{N}({\tilde{\mathbf{x}}};0,\mathbf{I}_{d})d{% \tilde{\mathbf{x}}}}}=\sqrt{d}</annotation><annotation encoding="application/x-llamapun" id="A3.SS2.SSS1.10.p4.5.m5.8d">∫ start_POSTSUBSCRIPT blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ∥ over~ start_ARG bold_x end_ARG ∥ caligraphic_N ( over~ start_ARG bold_x end_ARG ; 0 , bold_I start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT ) italic_d over~ start_ARG bold_x end_ARG ≤ square-root start_ARG ∫ start_POSTSUBSCRIPT blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ∥ over~ start_ARG bold_x end_ARG ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT caligraphic_N ( over~ start_ARG bold_x end_ARG ; 0 , bold_I start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT ) italic_d over~ start_ARG bold_x end_ARG end_ARG = square-root start_ARG italic_d end_ARG</annotation></semantics></math>. Combining this property with relation (<a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#A3.E25" title="25 ‣ Proof. ‣ C.2.1 Theoretical considerations ‣ C.2 Gaussian Mixture prior ‣ Appendix C SNORE in simple cases ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_tag">25</span></a>), we get</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A8.EGx35"> <tbody id="A3.Ex52"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\|\mathbb{E}_{{\tilde{\mathbf{x}}}\sim p_{\sigma}({\tilde{\mathbf% {x}}}|{\mathbf{x}})}\left(\nabla\log p({\tilde{\mathbf{x}}})\right)-\nabla\log p% ({\mathbf{x}})\|" class="ltx_Math" display="inline" id="A3.Ex52.m1.4"><semantics id="A3.Ex52.m1.4a"><mrow id="A3.Ex52.m1.4.4.1" xref="A3.Ex52.m1.4.4.2.cmml"><mo id="A3.Ex52.m1.4.4.1.2" stretchy="false" xref="A3.Ex52.m1.4.4.2.1.cmml">‖</mo><mrow id="A3.Ex52.m1.4.4.1.1" xref="A3.Ex52.m1.4.4.1.1.cmml"><mrow id="A3.Ex52.m1.4.4.1.1.1" xref="A3.Ex52.m1.4.4.1.1.1.cmml"><msub id="A3.Ex52.m1.4.4.1.1.1.3" xref="A3.Ex52.m1.4.4.1.1.1.3.cmml"><mi id="A3.Ex52.m1.4.4.1.1.1.3.2" xref="A3.Ex52.m1.4.4.1.1.1.3.2.cmml">𝔼</mi><mrow id="A3.Ex52.m1.1.1.1" xref="A3.Ex52.m1.1.1.1.cmml"><mover accent="true" id="A3.Ex52.m1.1.1.1.3" xref="A3.Ex52.m1.1.1.1.3.cmml"><mi id="A3.Ex52.m1.1.1.1.3.2" xref="A3.Ex52.m1.1.1.1.3.2.cmml">𝐱</mi><mo id="A3.Ex52.m1.1.1.1.3.1" xref="A3.Ex52.m1.1.1.1.3.1.cmml">~</mo></mover><mo id="A3.Ex52.m1.1.1.1.2" xref="A3.Ex52.m1.1.1.1.2.cmml">∼</mo><mrow id="A3.Ex52.m1.1.1.1.1" xref="A3.Ex52.m1.1.1.1.1.cmml"><msub id="A3.Ex52.m1.1.1.1.1.3" xref="A3.Ex52.m1.1.1.1.1.3.cmml"><mi id="A3.Ex52.m1.1.1.1.1.3.2" xref="A3.Ex52.m1.1.1.1.1.3.2.cmml">p</mi><mi id="A3.Ex52.m1.1.1.1.1.3.3" xref="A3.Ex52.m1.1.1.1.1.3.3.cmml">σ</mi></msub><mo id="A3.Ex52.m1.1.1.1.1.2" 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xref="A3.Ex52.m1.4.4.1.1.3.2.1.2"></log></apply><ci id="A3.Ex52.m1.4.4.1.1.3.2.2.cmml" xref="A3.Ex52.m1.4.4.1.1.3.2.2">𝑝</ci></apply><ci id="A3.Ex52.m1.3.3.cmml" xref="A3.Ex52.m1.3.3">𝐱</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.Ex52.m1.4c">\displaystyle\|\mathbb{E}_{{\tilde{\mathbf{x}}}\sim p_{\sigma}({\tilde{\mathbf% {x}}}|{\mathbf{x}})}\left(\nabla\log p({\tilde{\mathbf{x}}})\right)-\nabla\log p% ({\mathbf{x}})\|</annotation><annotation encoding="application/x-llamapun" id="A3.Ex52.m1.4d">∥ blackboard_E start_POSTSUBSCRIPT over~ start_ARG bold_x end_ARG ∼ italic_p start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ( over~ start_ARG bold_x end_ARG | bold_x ) end_POSTSUBSCRIPT ( ∇ roman_log italic_p ( over~ start_ARG bold_x end_ARG ) ) - ∇ roman_log italic_p ( bold_x ) ∥</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\leq M_{{\mathbf{x}}}\sqrt{d}\sigma+\int_{{\mathbb{R}}^{d}% \setminus\mathcal{B}({\mathbf{x}},r)}{(b_{p}\|{\tilde{\mathbf{x}}}\|+b_{p}\|{% \mathbf{x}}\|+2)\mathcal{N}({\tilde{\mathbf{x}}};{\mathbf{x}},\sigma^{2}% \mathbf{I}_{d})d{\tilde{\mathbf{x}}}}" class="ltx_Math" display="inline" id="A3.Ex52.m2.8"><semantics id="A3.Ex52.m2.8a"><mrow id="A3.Ex52.m2.8.8" xref="A3.Ex52.m2.8.8.cmml"><mi id="A3.Ex52.m2.8.8.4" xref="A3.Ex52.m2.8.8.4.cmml"></mi><mo id="A3.Ex52.m2.8.8.3" xref="A3.Ex52.m2.8.8.3.cmml">≤</mo><mrow id="A3.Ex52.m2.8.8.2" xref="A3.Ex52.m2.8.8.2.cmml"><mrow id="A3.Ex52.m2.8.8.2.4" xref="A3.Ex52.m2.8.8.2.4.cmml"><msub id="A3.Ex52.m2.8.8.2.4.2" xref="A3.Ex52.m2.8.8.2.4.2.cmml"><mi id="A3.Ex52.m2.8.8.2.4.2.2" xref="A3.Ex52.m2.8.8.2.4.2.2.cmml">M</mi><mi id="A3.Ex52.m2.8.8.2.4.2.3" xref="A3.Ex52.m2.8.8.2.4.2.3.cmml">𝐱</mi></msub><mo id="A3.Ex52.m2.8.8.2.4.1" xref="A3.Ex52.m2.8.8.2.4.1.cmml">⁢</mo><msqrt id="A3.Ex52.m2.8.8.2.4.3" xref="A3.Ex52.m2.8.8.2.4.3.cmml"><mi id="A3.Ex52.m2.8.8.2.4.3.2" 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id="A3.Ex52.m2.8c">\displaystyle\leq M_{{\mathbf{x}}}\sqrt{d}\sigma+\int_{{\mathbb{R}}^{d}% \setminus\mathcal{B}({\mathbf{x}},r)}{(b_{p}\|{\tilde{\mathbf{x}}}\|+b_{p}\|{% \mathbf{x}}\|+2)\mathcal{N}({\tilde{\mathbf{x}}};{\mathbf{x}},\sigma^{2}% \mathbf{I}_{d})d{\tilde{\mathbf{x}}}}</annotation><annotation encoding="application/x-llamapun" id="A3.Ex52.m2.8d">≤ italic_M start_POSTSUBSCRIPT bold_x end_POSTSUBSCRIPT square-root start_ARG italic_d end_ARG italic_σ + ∫ start_POSTSUBSCRIPT blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT ∖ caligraphic_B ( bold_x , italic_r ) end_POSTSUBSCRIPT ( italic_b start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT ∥ over~ start_ARG bold_x end_ARG ∥ + italic_b start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT ∥ bold_x ∥ + 2 ) caligraphic_N ( over~ start_ARG bold_x end_ARG ; bold_x , italic_σ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT bold_I start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT ) italic_d over~ start_ARG bold_x 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xref="A3.Ex53.m1.7.7.1.1.2.2.2.2.1.1.3.3">𝑑</ci></apply></apply></list><apply id="A3.Ex53.m1.7.7.1.1.2.2.2.5.cmml" xref="A3.Ex53.m1.7.7.1.1.2.2.2.5"><csymbol cd="latexml" id="A3.Ex53.m1.7.7.1.1.2.2.2.5.1.cmml" xref="A3.Ex53.m1.7.7.1.1.2.2.2.5.1">differential-d</csymbol><apply id="A3.Ex53.m1.7.7.1.1.2.2.2.5.2.cmml" xref="A3.Ex53.m1.7.7.1.1.2.2.2.5.2"><ci id="A3.Ex53.m1.7.7.1.1.2.2.2.5.2.1.cmml" xref="A3.Ex53.m1.7.7.1.1.2.2.2.5.2.1">~</ci><ci id="A3.Ex53.m1.7.7.1.1.2.2.2.5.2.2.cmml" xref="A3.Ex53.m1.7.7.1.1.2.2.2.5.2.2">𝐱</ci></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.Ex53.m1.7c">\displaystyle\leq M_{{\mathbf{x}}}\sqrt{d}\sigma+\int_{{\mathbb{R}}^{d}% \setminus\mathcal{B}({\mathbf{x}},r)}{(b_{p}\|{\tilde{\mathbf{x}}}\|+b_{p}\|{% \mathbf{x}}\|+2)\mathcal{N}({\tilde{\mathbf{x}}};{\mathbf{x}},\sigma^{2}% \mathbf{I}_{d})d{\tilde{\mathbf{x}}}}.</annotation><annotation encoding="application/x-llamapun" id="A3.Ex53.m1.7d">≤ italic_M start_POSTSUBSCRIPT bold_x end_POSTSUBSCRIPT square-root start_ARG italic_d end_ARG italic_σ + ∫ start_POSTSUBSCRIPT blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT ∖ caligraphic_B ( bold_x , italic_r ) end_POSTSUBSCRIPT ( italic_b start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT ∥ over~ start_ARG bold_x end_ARG ∥ + italic_b start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT ∥ bold_x ∥ + 2 ) caligraphic_N ( over~ start_ARG bold_x end_ARG ; bold_x , italic_σ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT bold_I start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT ) italic_d over~ start_ARG bold_x end_ARG .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="A3.SS2.SSS1.10.p4.8">Moreover on the set <math alttext="{\mathbb{R}}^{d}\setminus\mathcal{B}({\mathbf{x}},r)" class="ltx_Math" display="inline" id="A3.SS2.SSS1.10.p4.6.m1.2"><semantics id="A3.SS2.SSS1.10.p4.6.m1.2a"><mrow 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This leads to <math alttext="\mathcal{N}({\tilde{\mathbf{x}}};{\mathbf{x}},\sigma^{2}\mathbf{I}_{d})\leq 2^% {\frac{d}{2}}\exp{(-\frac{r^{2}}{2\sigma^{2}})}\mathcal{N}({\tilde{\mathbf{x}}% };{\mathbf{x}},2\sigma^{2}\mathbf{I}_{d})" class="ltx_Math" display="inline" id="A3.SS2.SSS1.10.p4.8.m3.8"><semantics id="A3.SS2.SSS1.10.p4.8.m3.8a"><mrow id="A3.SS2.SSS1.10.p4.8.m3.8.8" xref="A3.SS2.SSS1.10.p4.8.m3.8.8.cmml"><mrow id="A3.SS2.SSS1.10.p4.8.m3.6.6.1" xref="A3.SS2.SSS1.10.p4.8.m3.6.6.1.cmml"><mi class="ltx_font_mathcaligraphic" id="A3.SS2.SSS1.10.p4.8.m3.6.6.1.3" xref="A3.SS2.SSS1.10.p4.8.m3.6.6.1.3.cmml">𝒩</mi><mo id="A3.SS2.SSS1.10.p4.8.m3.6.6.1.2" xref="A3.SS2.SSS1.10.p4.8.m3.6.6.1.2.cmml">⁢</mo><mrow id="A3.SS2.SSS1.10.p4.8.m3.6.6.1.1.1" xref="A3.SS2.SSS1.10.p4.8.m3.6.6.1.1.2.cmml"><mo id="A3.SS2.SSS1.10.p4.8.m3.6.6.1.1.1.2" stretchy="false" xref="A3.SS2.SSS1.10.p4.8.m3.6.6.1.1.2.cmml">(</mo><mover accent="true" id="A3.SS2.SSS1.10.p4.8.m3.1.1" 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id="A3.SS2.SSS1.10.p4.8.m3.8.8.3.2.1.1.4.1.cmml" xref="A3.SS2.SSS1.10.p4.8.m3.8.8.3.2.1.1.4">subscript</csymbol><ci id="A3.SS2.SSS1.10.p4.8.m3.8.8.3.2.1.1.4.2.cmml" xref="A3.SS2.SSS1.10.p4.8.m3.8.8.3.2.1.1.4.2">𝐈</ci><ci id="A3.SS2.SSS1.10.p4.8.m3.8.8.3.2.1.1.4.3.cmml" xref="A3.SS2.SSS1.10.p4.8.m3.8.8.3.2.1.1.4.3">𝑑</ci></apply></apply></list></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.SS2.SSS1.10.p4.8.m3.8c">\mathcal{N}({\tilde{\mathbf{x}}};{\mathbf{x}},\sigma^{2}\mathbf{I}_{d})\leq 2^% {\frac{d}{2}}\exp{(-\frac{r^{2}}{2\sigma^{2}})}\mathcal{N}({\tilde{\mathbf{x}}% };{\mathbf{x}},2\sigma^{2}\mathbf{I}_{d})</annotation><annotation encoding="application/x-llamapun" id="A3.SS2.SSS1.10.p4.8.m3.8d">caligraphic_N ( over~ start_ARG bold_x end_ARG ; bold_x , italic_σ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT bold_I start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT ) ≤ 2 start_POSTSUPERSCRIPT divide start_ARG italic_d end_ARG start_ARG 2 end_ARG end_POSTSUPERSCRIPT roman_exp ( - divide start_ARG italic_r start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG 2 italic_σ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG ) caligraphic_N ( over~ start_ARG bold_x end_ARG ; bold_x , 2 italic_σ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT bold_I start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT )</annotation></semantics></math>. Injecting this relation into the previous computation, we obtain</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A8.EGx36"> <tbody id="A3.Ex54"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\|\mathbb{E}_{{\tilde{\mathbf{x}}}\sim p_{\sigma}({\tilde{\mathbf% {x}}}|{\mathbf{x}})}\left(\nabla\log p({\tilde{\mathbf{x}}})\right)-\nabla\log p% ({\mathbf{x}})\|" class="ltx_Math" display="inline" id="A3.Ex54.m1.4"><semantics id="A3.Ex54.m1.4a"><mrow id="A3.Ex54.m1.4.4.1" xref="A3.Ex54.m1.4.4.2.cmml"><mo id="A3.Ex54.m1.4.4.1.2" stretchy="false" xref="A3.Ex54.m1.4.4.2.1.cmml">‖</mo><mrow id="A3.Ex54.m1.4.4.1.1" xref="A3.Ex54.m1.4.4.1.1.cmml"><mrow id="A3.Ex54.m1.4.4.1.1.1" xref="A3.Ex54.m1.4.4.1.1.1.cmml"><msub id="A3.Ex54.m1.4.4.1.1.1.3" xref="A3.Ex54.m1.4.4.1.1.1.3.cmml"><mi id="A3.Ex54.m1.4.4.1.1.1.3.2" 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id="A3.Ex54.m1.4c">\displaystyle\|\mathbb{E}_{{\tilde{\mathbf{x}}}\sim p_{\sigma}({\tilde{\mathbf% {x}}}|{\mathbf{x}})}\left(\nabla\log p({\tilde{\mathbf{x}}})\right)-\nabla\log p% ({\mathbf{x}})\|</annotation><annotation encoding="application/x-llamapun" id="A3.Ex54.m1.4d">∥ blackboard_E start_POSTSUBSCRIPT over~ start_ARG bold_x end_ARG ∼ italic_p start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ( over~ start_ARG bold_x end_ARG | bold_x ) end_POSTSUBSCRIPT ( ∇ roman_log italic_p ( over~ start_ARG bold_x end_ARG ) ) - ∇ roman_log italic_p ( bold_x ) ∥</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\leq M_{{\mathbf{x}}}\sqrt{d}\sigma+2^{\frac{d}{2}}\exp{(-\frac{r% ^{2}}{2\sigma^{2}})}\int_{{\mathbb{R}}^{d}\setminus\mathcal{B}({\mathbf{x}},r)% }{(b_{p}\|{\tilde{\mathbf{x}}}\|+b_{p}\|{\mathbf{x}}\|+2)\mathcal{N}({\tilde{% \mathbf{x}}};{\mathbf{x}},2\sigma^{2}\mathbf{I}_{d})d{\tilde{\mathbf{x}}}}" class="ltx_Math" 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id="A3.Ex54.m2.10.10.3.3.3.2.2.1.1.4.3.cmml" xref="A3.Ex54.m2.10.10.3.3.3.2.2.1.1.4.3">𝑑</ci></apply></apply></list><apply id="A3.Ex54.m2.10.10.3.3.3.2.5.cmml" xref="A3.Ex54.m2.10.10.3.3.3.2.5"><csymbol cd="latexml" id="A3.Ex54.m2.10.10.3.3.3.2.5.1.cmml" xref="A3.Ex54.m2.10.10.3.3.3.2.5.1">differential-d</csymbol><apply id="A3.Ex54.m2.10.10.3.3.3.2.5.2.cmml" xref="A3.Ex54.m2.10.10.3.3.3.2.5.2"><ci id="A3.Ex54.m2.10.10.3.3.3.2.5.2.1.cmml" xref="A3.Ex54.m2.10.10.3.3.3.2.5.2.1">~</ci><ci id="A3.Ex54.m2.10.10.3.3.3.2.5.2.2.cmml" xref="A3.Ex54.m2.10.10.3.3.3.2.5.2.2">𝐱</ci></apply></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.Ex54.m2.10c">\displaystyle\leq M_{{\mathbf{x}}}\sqrt{d}\sigma+2^{\frac{d}{2}}\exp{(-\frac{r% ^{2}}{2\sigma^{2}})}\int_{{\mathbb{R}}^{d}\setminus\mathcal{B}({\mathbf{x}},r)% }{(b_{p}\|{\tilde{\mathbf{x}}}\|+b_{p}\|{\mathbf{x}}\|+2)\mathcal{N}({\tilde{% \mathbf{x}}};{\mathbf{x}},2\sigma^{2}\mathbf{I}_{d})d{\tilde{\mathbf{x}}}}</annotation><annotation encoding="application/x-llamapun" id="A3.Ex54.m2.10d">≤ italic_M start_POSTSUBSCRIPT bold_x end_POSTSUBSCRIPT square-root start_ARG italic_d end_ARG italic_σ + 2 start_POSTSUPERSCRIPT divide start_ARG italic_d end_ARG start_ARG 2 end_ARG end_POSTSUPERSCRIPT roman_exp ( - divide start_ARG italic_r start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG 2 italic_σ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG ) ∫ start_POSTSUBSCRIPT blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT ∖ caligraphic_B ( bold_x , italic_r ) end_POSTSUBSCRIPT ( italic_b start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT ∥ over~ start_ARG bold_x end_ARG ∥ + italic_b start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT ∥ bold_x ∥ + 2 ) caligraphic_N ( over~ start_ARG bold_x end_ARG ; bold_x , 2 italic_σ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT bold_I start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT ) italic_d over~ start_ARG bold_x end_ARG</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="A3.Ex55"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_eqn_cell"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\leq M_{{\mathbf{x}}}\sqrt{d}\sigma+2^{\frac{d}{2}}\exp{(-\frac{r% ^{2}}{2\sigma^{2}})}\int_{{\mathbb{R}}^{d}}{(b_{p}\|{\tilde{\mathbf{x}}}\|+b_{% p}\|{\mathbf{x}}\|+2)\mathcal{N}({\tilde{\mathbf{x}}};{\mathbf{x}},2\sigma^{2}% \mathbf{I}_{d})d{\tilde{\mathbf{x}}}}" class="ltx_Math" display="inline" id="A3.Ex55.m1.8"><semantics id="A3.Ex55.m1.8a"><mrow id="A3.Ex55.m1.8.8" xref="A3.Ex55.m1.8.8.cmml"><mi id="A3.Ex55.m1.8.8.5" xref="A3.Ex55.m1.8.8.5.cmml"></mi><mo id="A3.Ex55.m1.8.8.4" xref="A3.Ex55.m1.8.8.4.cmml">≤</mo><mrow id="A3.Ex55.m1.8.8.3" xref="A3.Ex55.m1.8.8.3.cmml"><mrow id="A3.Ex55.m1.8.8.3.5" xref="A3.Ex55.m1.8.8.3.5.cmml"><msub id="A3.Ex55.m1.8.8.3.5.2" xref="A3.Ex55.m1.8.8.3.5.2.cmml"><mi id="A3.Ex55.m1.8.8.3.5.2.2" xref="A3.Ex55.m1.8.8.3.5.2.2.cmml">M</mi><mi id="A3.Ex55.m1.8.8.3.5.2.3" xref="A3.Ex55.m1.8.8.3.5.2.3.cmml">𝐱</mi></msub><mo id="A3.Ex55.m1.8.8.3.5.1" xref="A3.Ex55.m1.8.8.3.5.1.cmml">⁢</mo><msqrt id="A3.Ex55.m1.8.8.3.5.3" xref="A3.Ex55.m1.8.8.3.5.3.cmml"><mi id="A3.Ex55.m1.8.8.3.5.3.2" xref="A3.Ex55.m1.8.8.3.5.3.2.cmml">d</mi></msqrt><mo id="A3.Ex55.m1.8.8.3.5.1a" xref="A3.Ex55.m1.8.8.3.5.1.cmml">⁢</mo><mi id="A3.Ex55.m1.8.8.3.5.4" xref="A3.Ex55.m1.8.8.3.5.4.cmml">σ</mi></mrow><mo id="A3.Ex55.m1.8.8.3.4" xref="A3.Ex55.m1.8.8.3.4.cmml">+</mo><mrow id="A3.Ex55.m1.8.8.3.3" xref="A3.Ex55.m1.8.8.3.3.cmml"><msup id="A3.Ex55.m1.8.8.3.3.5" xref="A3.Ex55.m1.8.8.3.3.5.cmml"><mn id="A3.Ex55.m1.8.8.3.3.5.2" xref="A3.Ex55.m1.8.8.3.3.5.2.cmml">2</mn><mfrac id="A3.Ex55.m1.8.8.3.3.5.3" xref="A3.Ex55.m1.8.8.3.3.5.3.cmml"><mi id="A3.Ex55.m1.8.8.3.3.5.3.2" xref="A3.Ex55.m1.8.8.3.3.5.3.2.cmml">d</mi><mn id="A3.Ex55.m1.8.8.3.3.5.3.3" xref="A3.Ex55.m1.8.8.3.3.5.3.3.cmml">2</mn></mfrac></msup><mo id="A3.Ex55.m1.8.8.3.3.4" lspace="0.167em" xref="A3.Ex55.m1.8.8.3.3.4.cmml">⁢</mo><mrow id="A3.Ex55.m1.6.6.1.1.1.1" xref="A3.Ex55.m1.6.6.1.1.1.2.cmml"><mi id="A3.Ex55.m1.1.1" xref="A3.Ex55.m1.1.1.cmml">exp</mi><mo id="A3.Ex55.m1.6.6.1.1.1.1a" xref="A3.Ex55.m1.6.6.1.1.1.2.cmml">⁡</mo><mrow id="A3.Ex55.m1.6.6.1.1.1.1.1" xref="A3.Ex55.m1.6.6.1.1.1.2.cmml"><mo id="A3.Ex55.m1.6.6.1.1.1.1.1.2" stretchy="false" xref="A3.Ex55.m1.6.6.1.1.1.2.cmml">(</mo><mrow id="A3.Ex55.m1.6.6.1.1.1.1.1.1" xref="A3.Ex55.m1.6.6.1.1.1.1.1.1.cmml"><mo id="A3.Ex55.m1.6.6.1.1.1.1.1.1a" xref="A3.Ex55.m1.6.6.1.1.1.1.1.1.cmml">−</mo><mstyle displaystyle="true" id="A3.Ex55.m1.6.6.1.1.1.1.1.1.2" xref="A3.Ex55.m1.6.6.1.1.1.1.1.1.2.cmml"><mfrac id="A3.Ex55.m1.6.6.1.1.1.1.1.1.2a" xref="A3.Ex55.m1.6.6.1.1.1.1.1.1.2.cmml"><msup id="A3.Ex55.m1.6.6.1.1.1.1.1.1.2.2" xref="A3.Ex55.m1.6.6.1.1.1.1.1.1.2.2.cmml"><mi id="A3.Ex55.m1.6.6.1.1.1.1.1.1.2.2.2" xref="A3.Ex55.m1.6.6.1.1.1.1.1.1.2.2.2.cmml">r</mi><mn id="A3.Ex55.m1.6.6.1.1.1.1.1.1.2.2.3" xref="A3.Ex55.m1.6.6.1.1.1.1.1.1.2.2.3.cmml">2</mn></msup><mrow id="A3.Ex55.m1.6.6.1.1.1.1.1.1.2.3" xref="A3.Ex55.m1.6.6.1.1.1.1.1.1.2.3.cmml"><mn id="A3.Ex55.m1.6.6.1.1.1.1.1.1.2.3.2" xref="A3.Ex55.m1.6.6.1.1.1.1.1.1.2.3.2.cmml">2</mn><mo id="A3.Ex55.m1.6.6.1.1.1.1.1.1.2.3.1" xref="A3.Ex55.m1.6.6.1.1.1.1.1.1.2.3.1.cmml">⁢</mo><msup id="A3.Ex55.m1.6.6.1.1.1.1.1.1.2.3.3" xref="A3.Ex55.m1.6.6.1.1.1.1.1.1.2.3.3.cmml"><mi id="A3.Ex55.m1.6.6.1.1.1.1.1.1.2.3.3.2" xref="A3.Ex55.m1.6.6.1.1.1.1.1.1.2.3.3.2.cmml">σ</mi><mn id="A3.Ex55.m1.6.6.1.1.1.1.1.1.2.3.3.3" xref="A3.Ex55.m1.6.6.1.1.1.1.1.1.2.3.3.3.cmml">2</mn></msup></mrow></mfrac></mstyle></mrow><mo id="A3.Ex55.m1.6.6.1.1.1.1.1.3" stretchy="false" xref="A3.Ex55.m1.6.6.1.1.1.2.cmml">)</mo></mrow></mrow><mo id="A3.Ex55.m1.8.8.3.3.4a" 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id="A3.Ex55.m1.8.8.3.3.3.2.5.cmml" xref="A3.Ex55.m1.8.8.3.3.3.2.5"><csymbol cd="latexml" id="A3.Ex55.m1.8.8.3.3.3.2.5.1.cmml" xref="A3.Ex55.m1.8.8.3.3.3.2.5.1">differential-d</csymbol><apply id="A3.Ex55.m1.8.8.3.3.3.2.5.2.cmml" xref="A3.Ex55.m1.8.8.3.3.3.2.5.2"><ci id="A3.Ex55.m1.8.8.3.3.3.2.5.2.1.cmml" xref="A3.Ex55.m1.8.8.3.3.3.2.5.2.1">~</ci><ci id="A3.Ex55.m1.8.8.3.3.3.2.5.2.2.cmml" xref="A3.Ex55.m1.8.8.3.3.3.2.5.2.2">𝐱</ci></apply></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.Ex55.m1.8c">\displaystyle\leq M_{{\mathbf{x}}}\sqrt{d}\sigma+2^{\frac{d}{2}}\exp{(-\frac{r% ^{2}}{2\sigma^{2}})}\int_{{\mathbb{R}}^{d}}{(b_{p}\|{\tilde{\mathbf{x}}}\|+b_{% p}\|{\mathbf{x}}\|+2)\mathcal{N}({\tilde{\mathbf{x}}};{\mathbf{x}},2\sigma^{2}% \mathbf{I}_{d})d{\tilde{\mathbf{x}}}}</annotation><annotation encoding="application/x-llamapun" id="A3.Ex55.m1.8d">≤ italic_M start_POSTSUBSCRIPT bold_x end_POSTSUBSCRIPT square-root start_ARG italic_d end_ARG italic_σ + 2 start_POSTSUPERSCRIPT divide start_ARG italic_d end_ARG start_ARG 2 end_ARG end_POSTSUPERSCRIPT roman_exp ( - divide start_ARG italic_r start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG 2 italic_σ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG ) ∫ start_POSTSUBSCRIPT blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ( italic_b start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT ∥ over~ start_ARG bold_x end_ARG ∥ + italic_b start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT ∥ bold_x ∥ + 2 ) caligraphic_N ( over~ start_ARG bold_x end_ARG ; bold_x , 2 italic_σ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT bold_I start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT ) italic_d over~ start_ARG bold_x end_ARG</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="A3.Ex56"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_eqn_cell"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\leq M_{{\mathbf{x}}}\sqrt{d}\sigma+2^{\frac{d}{2}}\exp{(-\frac{r% ^{2}}{2\sigma^{2}})}(b_{p}\sqrt{\|{\mathbf{x}}\|^{2}+b_{p}\sqrt{d}\sigma^{2}}+% b_{p}\|{\mathbf{x}}\|+2)" class="ltx_Math" display="inline" id="A3.Ex56.m1.5"><semantics id="A3.Ex56.m1.5a"><mrow id="A3.Ex56.m1.5.5" xref="A3.Ex56.m1.5.5.cmml"><mi id="A3.Ex56.m1.5.5.4" xref="A3.Ex56.m1.5.5.4.cmml"></mi><mo id="A3.Ex56.m1.5.5.3" xref="A3.Ex56.m1.5.5.3.cmml">≤</mo><mrow id="A3.Ex56.m1.5.5.2" xref="A3.Ex56.m1.5.5.2.cmml"><mrow id="A3.Ex56.m1.5.5.2.4" xref="A3.Ex56.m1.5.5.2.4.cmml"><msub id="A3.Ex56.m1.5.5.2.4.2" xref="A3.Ex56.m1.5.5.2.4.2.cmml"><mi id="A3.Ex56.m1.5.5.2.4.2.2" xref="A3.Ex56.m1.5.5.2.4.2.2.cmml">M</mi><mi id="A3.Ex56.m1.5.5.2.4.2.3" xref="A3.Ex56.m1.5.5.2.4.2.3.cmml">𝐱</mi></msub><mo id="A3.Ex56.m1.5.5.2.4.1" xref="A3.Ex56.m1.5.5.2.4.1.cmml">⁢</mo><msqrt 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xref="A3.Ex56.m1.3.3">𝐱</ci></apply></apply><cn id="A3.Ex56.m1.5.5.2.2.2.1.1.4.cmml" type="integer" xref="A3.Ex56.m1.5.5.2.2.2.1.1.4">2</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.Ex56.m1.5c">\displaystyle\leq M_{{\mathbf{x}}}\sqrt{d}\sigma+2^{\frac{d}{2}}\exp{(-\frac{r% ^{2}}{2\sigma^{2}})}(b_{p}\sqrt{\|{\mathbf{x}}\|^{2}+b_{p}\sqrt{d}\sigma^{2}}+% b_{p}\|{\mathbf{x}}\|+2)</annotation><annotation encoding="application/x-llamapun" id="A3.Ex56.m1.5d">≤ italic_M start_POSTSUBSCRIPT bold_x end_POSTSUBSCRIPT square-root start_ARG italic_d end_ARG italic_σ + 2 start_POSTSUPERSCRIPT divide start_ARG italic_d end_ARG start_ARG 2 end_ARG end_POSTSUPERSCRIPT roman_exp ( - divide start_ARG italic_r start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG 2 italic_σ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG ) ( italic_b start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT square-root start_ARG ∥ bold_x ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + italic_b start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT square-root start_ARG italic_d end_ARG italic_σ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG + italic_b start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT ∥ bold_x ∥ + 2 )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="A3.Ex57"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_eqn_cell"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\leq M_{{\mathbf{x}}}\sqrt{d}\sigma+\mathcal{O}(\sigma^{4})" class="ltx_Math" display="inline" id="A3.Ex57.m1.1"><semantics id="A3.Ex57.m1.1a"><mrow id="A3.Ex57.m1.1.1" xref="A3.Ex57.m1.1.1.cmml"><mi id="A3.Ex57.m1.1.1.3" xref="A3.Ex57.m1.1.1.3.cmml"></mi><mo id="A3.Ex57.m1.1.1.2" xref="A3.Ex57.m1.1.1.2.cmml">≤</mo><mrow id="A3.Ex57.m1.1.1.1" xref="A3.Ex57.m1.1.1.1.cmml"><mrow id="A3.Ex57.m1.1.1.1.3" xref="A3.Ex57.m1.1.1.1.3.cmml"><msub id="A3.Ex57.m1.1.1.1.3.2" xref="A3.Ex57.m1.1.1.1.3.2.cmml"><mi id="A3.Ex57.m1.1.1.1.3.2.2" xref="A3.Ex57.m1.1.1.1.3.2.2.cmml">M</mi><mi id="A3.Ex57.m1.1.1.1.3.2.3" xref="A3.Ex57.m1.1.1.1.3.2.3.cmml">𝐱</mi></msub><mo id="A3.Ex57.m1.1.1.1.3.1" xref="A3.Ex57.m1.1.1.1.3.1.cmml">⁢</mo><msqrt id="A3.Ex57.m1.1.1.1.3.3" xref="A3.Ex57.m1.1.1.1.3.3.cmml"><mi id="A3.Ex57.m1.1.1.1.3.3.2" xref="A3.Ex57.m1.1.1.1.3.3.2.cmml">d</mi></msqrt><mo id="A3.Ex57.m1.1.1.1.3.1a" xref="A3.Ex57.m1.1.1.1.3.1.cmml">⁢</mo><mi id="A3.Ex57.m1.1.1.1.3.4" xref="A3.Ex57.m1.1.1.1.3.4.cmml">σ</mi></mrow><mo id="A3.Ex57.m1.1.1.1.2" xref="A3.Ex57.m1.1.1.1.2.cmml">+</mo><mrow id="A3.Ex57.m1.1.1.1.1" xref="A3.Ex57.m1.1.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="A3.Ex57.m1.1.1.1.1.3" xref="A3.Ex57.m1.1.1.1.1.3.cmml">𝒪</mi><mo id="A3.Ex57.m1.1.1.1.1.2" xref="A3.Ex57.m1.1.1.1.1.2.cmml">⁢</mo><mrow id="A3.Ex57.m1.1.1.1.1.1.1" 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xref="A3.Ex57.m1.1.1.1.3.1"></times><apply id="A3.Ex57.m1.1.1.1.3.2.cmml" xref="A3.Ex57.m1.1.1.1.3.2"><csymbol cd="ambiguous" id="A3.Ex57.m1.1.1.1.3.2.1.cmml" xref="A3.Ex57.m1.1.1.1.3.2">subscript</csymbol><ci id="A3.Ex57.m1.1.1.1.3.2.2.cmml" xref="A3.Ex57.m1.1.1.1.3.2.2">𝑀</ci><ci id="A3.Ex57.m1.1.1.1.3.2.3.cmml" xref="A3.Ex57.m1.1.1.1.3.2.3">𝐱</ci></apply><apply id="A3.Ex57.m1.1.1.1.3.3.cmml" xref="A3.Ex57.m1.1.1.1.3.3"><root id="A3.Ex57.m1.1.1.1.3.3a.cmml" xref="A3.Ex57.m1.1.1.1.3.3"></root><ci id="A3.Ex57.m1.1.1.1.3.3.2.cmml" xref="A3.Ex57.m1.1.1.1.3.3.2">𝑑</ci></apply><ci id="A3.Ex57.m1.1.1.1.3.4.cmml" xref="A3.Ex57.m1.1.1.1.3.4">𝜎</ci></apply><apply id="A3.Ex57.m1.1.1.1.1.cmml" xref="A3.Ex57.m1.1.1.1.1"><times id="A3.Ex57.m1.1.1.1.1.2.cmml" xref="A3.Ex57.m1.1.1.1.1.2"></times><ci id="A3.Ex57.m1.1.1.1.1.3.cmml" xref="A3.Ex57.m1.1.1.1.1.3">𝒪</ci><apply id="A3.Ex57.m1.1.1.1.1.1.1.1.cmml" xref="A3.Ex57.m1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="A3.Ex57.m1.1.1.1.1.1.1.1.1.cmml" xref="A3.Ex57.m1.1.1.1.1.1.1">superscript</csymbol><ci id="A3.Ex57.m1.1.1.1.1.1.1.1.2.cmml" xref="A3.Ex57.m1.1.1.1.1.1.1.1.2">𝜎</ci><cn id="A3.Ex57.m1.1.1.1.1.1.1.1.3.cmml" type="integer" xref="A3.Ex57.m1.1.1.1.1.1.1.1.3">4</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.Ex57.m1.1c">\displaystyle\leq M_{{\mathbf{x}}}\sqrt{d}\sigma+\mathcal{O}(\sigma^{4})</annotation><annotation encoding="application/x-llamapun" id="A3.Ex57.m1.1d">≤ italic_M start_POSTSUBSCRIPT bold_x end_POSTSUBSCRIPT square-root start_ARG italic_d end_ARG italic_σ + caligraphic_O ( italic_σ start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="A3.Ex58"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_eqn_cell"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\leq e_{p}(\|{\mathbf{x}}\|^{2}+1)\sqrt{d}\sigma+\mathcal{O}(% \sigma^{4})." class="ltx_Math" display="inline" id="A3.Ex58.m1.2"><semantics id="A3.Ex58.m1.2a"><mrow id="A3.Ex58.m1.2.2.1" xref="A3.Ex58.m1.2.2.1.1.cmml"><mrow id="A3.Ex58.m1.2.2.1.1" xref="A3.Ex58.m1.2.2.1.1.cmml"><mi id="A3.Ex58.m1.2.2.1.1.4" xref="A3.Ex58.m1.2.2.1.1.4.cmml"></mi><mo id="A3.Ex58.m1.2.2.1.1.3" xref="A3.Ex58.m1.2.2.1.1.3.cmml">≤</mo><mrow id="A3.Ex58.m1.2.2.1.1.2" xref="A3.Ex58.m1.2.2.1.1.2.cmml"><mrow id="A3.Ex58.m1.2.2.1.1.1.1" xref="A3.Ex58.m1.2.2.1.1.1.1.cmml"><msub id="A3.Ex58.m1.2.2.1.1.1.1.3" xref="A3.Ex58.m1.2.2.1.1.1.1.3.cmml"><mi id="A3.Ex58.m1.2.2.1.1.1.1.3.2" xref="A3.Ex58.m1.2.2.1.1.1.1.3.2.cmml">e</mi><mi id="A3.Ex58.m1.2.2.1.1.1.1.3.3" xref="A3.Ex58.m1.2.2.1.1.1.1.3.3.cmml">p</mi></msub><mo id="A3.Ex58.m1.2.2.1.1.1.1.2" xref="A3.Ex58.m1.2.2.1.1.1.1.2.cmml">⁢</mo><mrow id="A3.Ex58.m1.2.2.1.1.1.1.1.1" xref="A3.Ex58.m1.2.2.1.1.1.1.1.1.1.cmml"><mo 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xref="A3.Ex58.m1.2.2.1.1.1.1.1.1.1.cmml">)</mo></mrow><mo id="A3.Ex58.m1.2.2.1.1.1.1.2a" xref="A3.Ex58.m1.2.2.1.1.1.1.2.cmml">⁢</mo><msqrt id="A3.Ex58.m1.2.2.1.1.1.1.4" xref="A3.Ex58.m1.2.2.1.1.1.1.4.cmml"><mi id="A3.Ex58.m1.2.2.1.1.1.1.4.2" xref="A3.Ex58.m1.2.2.1.1.1.1.4.2.cmml">d</mi></msqrt><mo id="A3.Ex58.m1.2.2.1.1.1.1.2b" xref="A3.Ex58.m1.2.2.1.1.1.1.2.cmml">⁢</mo><mi id="A3.Ex58.m1.2.2.1.1.1.1.5" xref="A3.Ex58.m1.2.2.1.1.1.1.5.cmml">σ</mi></mrow><mo id="A3.Ex58.m1.2.2.1.1.2.3" xref="A3.Ex58.m1.2.2.1.1.2.3.cmml">+</mo><mrow id="A3.Ex58.m1.2.2.1.1.2.2" xref="A3.Ex58.m1.2.2.1.1.2.2.cmml"><mi class="ltx_font_mathcaligraphic" id="A3.Ex58.m1.2.2.1.1.2.2.3" xref="A3.Ex58.m1.2.2.1.1.2.2.3.cmml">𝒪</mi><mo id="A3.Ex58.m1.2.2.1.1.2.2.2" xref="A3.Ex58.m1.2.2.1.1.2.2.2.cmml">⁢</mo><mrow id="A3.Ex58.m1.2.2.1.1.2.2.1.1" xref="A3.Ex58.m1.2.2.1.1.2.2.1.1.1.cmml"><mo id="A3.Ex58.m1.2.2.1.1.2.2.1.1.2" stretchy="false" xref="A3.Ex58.m1.2.2.1.1.2.2.1.1.1.cmml">(</mo><msup id="A3.Ex58.m1.2.2.1.1.2.2.1.1.1" xref="A3.Ex58.m1.2.2.1.1.2.2.1.1.1.cmml"><mi id="A3.Ex58.m1.2.2.1.1.2.2.1.1.1.2" xref="A3.Ex58.m1.2.2.1.1.2.2.1.1.1.2.cmml">σ</mi><mn id="A3.Ex58.m1.2.2.1.1.2.2.1.1.1.3" xref="A3.Ex58.m1.2.2.1.1.2.2.1.1.1.3.cmml">4</mn></msup><mo id="A3.Ex58.m1.2.2.1.1.2.2.1.1.3" stretchy="false" xref="A3.Ex58.m1.2.2.1.1.2.2.1.1.1.cmml">)</mo></mrow></mrow></mrow></mrow><mo id="A3.Ex58.m1.2.2.1.2" lspace="0em" xref="A3.Ex58.m1.2.2.1.1.cmml">.</mo></mrow><annotation-xml encoding="MathML-Content" id="A3.Ex58.m1.2b"><apply id="A3.Ex58.m1.2.2.1.1.cmml" xref="A3.Ex58.m1.2.2.1"><leq id="A3.Ex58.m1.2.2.1.1.3.cmml" xref="A3.Ex58.m1.2.2.1.1.3"></leq><csymbol cd="latexml" id="A3.Ex58.m1.2.2.1.1.4.cmml" xref="A3.Ex58.m1.2.2.1.1.4">absent</csymbol><apply id="A3.Ex58.m1.2.2.1.1.2.cmml" xref="A3.Ex58.m1.2.2.1.1.2"><plus id="A3.Ex58.m1.2.2.1.1.2.3.cmml" xref="A3.Ex58.m1.2.2.1.1.2.3"></plus><apply id="A3.Ex58.m1.2.2.1.1.1.1.cmml" xref="A3.Ex58.m1.2.2.1.1.1.1"><times id="A3.Ex58.m1.2.2.1.1.1.1.2.cmml" xref="A3.Ex58.m1.2.2.1.1.1.1.2"></times><apply id="A3.Ex58.m1.2.2.1.1.1.1.3.cmml" xref="A3.Ex58.m1.2.2.1.1.1.1.3"><csymbol cd="ambiguous" id="A3.Ex58.m1.2.2.1.1.1.1.3.1.cmml" xref="A3.Ex58.m1.2.2.1.1.1.1.3">subscript</csymbol><ci id="A3.Ex58.m1.2.2.1.1.1.1.3.2.cmml" xref="A3.Ex58.m1.2.2.1.1.1.1.3.2">𝑒</ci><ci id="A3.Ex58.m1.2.2.1.1.1.1.3.3.cmml" xref="A3.Ex58.m1.2.2.1.1.1.1.3.3">𝑝</ci></apply><apply id="A3.Ex58.m1.2.2.1.1.1.1.1.1.1.cmml" xref="A3.Ex58.m1.2.2.1.1.1.1.1.1"><plus id="A3.Ex58.m1.2.2.1.1.1.1.1.1.1.1.cmml" xref="A3.Ex58.m1.2.2.1.1.1.1.1.1.1.1"></plus><apply id="A3.Ex58.m1.2.2.1.1.1.1.1.1.1.2.cmml" xref="A3.Ex58.m1.2.2.1.1.1.1.1.1.1.2"><csymbol cd="ambiguous" id="A3.Ex58.m1.2.2.1.1.1.1.1.1.1.2.1.cmml" xref="A3.Ex58.m1.2.2.1.1.1.1.1.1.1.2">superscript</csymbol><apply id="A3.Ex58.m1.2.2.1.1.1.1.1.1.1.2.2.1.cmml" xref="A3.Ex58.m1.2.2.1.1.1.1.1.1.1.2.2.2"><csymbol cd="latexml" id="A3.Ex58.m1.2.2.1.1.1.1.1.1.1.2.2.1.1.cmml" xref="A3.Ex58.m1.2.2.1.1.1.1.1.1.1.2.2.2.1">norm</csymbol><ci id="A3.Ex58.m1.1.1.cmml" xref="A3.Ex58.m1.1.1">𝐱</ci></apply><cn id="A3.Ex58.m1.2.2.1.1.1.1.1.1.1.2.3.cmml" type="integer" xref="A3.Ex58.m1.2.2.1.1.1.1.1.1.1.2.3">2</cn></apply><cn id="A3.Ex58.m1.2.2.1.1.1.1.1.1.1.3.cmml" type="integer" xref="A3.Ex58.m1.2.2.1.1.1.1.1.1.1.3">1</cn></apply><apply id="A3.Ex58.m1.2.2.1.1.1.1.4.cmml" xref="A3.Ex58.m1.2.2.1.1.1.1.4"><root id="A3.Ex58.m1.2.2.1.1.1.1.4a.cmml" xref="A3.Ex58.m1.2.2.1.1.1.1.4"></root><ci id="A3.Ex58.m1.2.2.1.1.1.1.4.2.cmml" xref="A3.Ex58.m1.2.2.1.1.1.1.4.2">𝑑</ci></apply><ci id="A3.Ex58.m1.2.2.1.1.1.1.5.cmml" xref="A3.Ex58.m1.2.2.1.1.1.1.5">𝜎</ci></apply><apply id="A3.Ex58.m1.2.2.1.1.2.2.cmml" xref="A3.Ex58.m1.2.2.1.1.2.2"><times id="A3.Ex58.m1.2.2.1.1.2.2.2.cmml" xref="A3.Ex58.m1.2.2.1.1.2.2.2"></times><ci id="A3.Ex58.m1.2.2.1.1.2.2.3.cmml" xref="A3.Ex58.m1.2.2.1.1.2.2.3">𝒪</ci><apply id="A3.Ex58.m1.2.2.1.1.2.2.1.1.1.cmml" xref="A3.Ex58.m1.2.2.1.1.2.2.1.1"><csymbol cd="ambiguous" id="A3.Ex58.m1.2.2.1.1.2.2.1.1.1.1.cmml" xref="A3.Ex58.m1.2.2.1.1.2.2.1.1">superscript</csymbol><ci id="A3.Ex58.m1.2.2.1.1.2.2.1.1.1.2.cmml" xref="A3.Ex58.m1.2.2.1.1.2.2.1.1.1.2">𝜎</ci><cn id="A3.Ex58.m1.2.2.1.1.2.2.1.1.1.3.cmml" type="integer" xref="A3.Ex58.m1.2.2.1.1.2.2.1.1.1.3">4</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.Ex58.m1.2c">\displaystyle\leq e_{p}(\|{\mathbf{x}}\|^{2}+1)\sqrt{d}\sigma+\mathcal{O}(% \sigma^{4}).</annotation><annotation encoding="application/x-llamapun" id="A3.Ex58.m1.2d">≤ italic_e start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT ( ∥ bold_x ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + 1 ) square-root start_ARG italic_d end_ARG italic_σ + caligraphic_O ( italic_σ start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT ) .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="A3.SS2.SSS1.10.p4.10">The last inequality holds because there exists <math alttext="e_{p}\in{\mathbb{R}}^{+}" class="ltx_Math" display="inline" id="A3.SS2.SSS1.10.p4.9.m1.1"><semantics id="A3.SS2.SSS1.10.p4.9.m1.1a"><mrow id="A3.SS2.SSS1.10.p4.9.m1.1.1" xref="A3.SS2.SSS1.10.p4.9.m1.1.1.cmml"><msub id="A3.SS2.SSS1.10.p4.9.m1.1.1.2" xref="A3.SS2.SSS1.10.p4.9.m1.1.1.2.cmml"><mi id="A3.SS2.SSS1.10.p4.9.m1.1.1.2.2" xref="A3.SS2.SSS1.10.p4.9.m1.1.1.2.2.cmml">e</mi><mi id="A3.SS2.SSS1.10.p4.9.m1.1.1.2.3" xref="A3.SS2.SSS1.10.p4.9.m1.1.1.2.3.cmml">p</mi></msub><mo id="A3.SS2.SSS1.10.p4.9.m1.1.1.1" xref="A3.SS2.SSS1.10.p4.9.m1.1.1.1.cmml">∈</mo><msup id="A3.SS2.SSS1.10.p4.9.m1.1.1.3" xref="A3.SS2.SSS1.10.p4.9.m1.1.1.3.cmml"><mi id="A3.SS2.SSS1.10.p4.9.m1.1.1.3.2" xref="A3.SS2.SSS1.10.p4.9.m1.1.1.3.2.cmml">ℝ</mi><mo id="A3.SS2.SSS1.10.p4.9.m1.1.1.3.3" xref="A3.SS2.SSS1.10.p4.9.m1.1.1.3.3.cmml">+</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="A3.SS2.SSS1.10.p4.9.m1.1b"><apply id="A3.SS2.SSS1.10.p4.9.m1.1.1.cmml" 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id="A3.SS2.SSS1.10.p4.9.m1.1c">e_{p}\in{\mathbb{R}}^{+}</annotation><annotation encoding="application/x-llamapun" id="A3.SS2.SSS1.10.p4.9.m1.1d">italic_e start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT ∈ blackboard_R start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT</annotation></semantics></math> such that <math alttext="M_{{\mathbf{x}}}=2\|\nabla^{2}\log p({\mathbf{x}})\|\leq e_{p}(\|{\mathbf{x}}% \|^{2}+1)" class="ltx_Math" display="inline" id="A3.SS2.SSS1.10.p4.10.m2.4"><semantics id="A3.SS2.SSS1.10.p4.10.m2.4a"><mrow id="A3.SS2.SSS1.10.p4.10.m2.4.4" xref="A3.SS2.SSS1.10.p4.10.m2.4.4.cmml"><msub id="A3.SS2.SSS1.10.p4.10.m2.4.4.4" xref="A3.SS2.SSS1.10.p4.10.m2.4.4.4.cmml"><mi id="A3.SS2.SSS1.10.p4.10.m2.4.4.4.2" xref="A3.SS2.SSS1.10.p4.10.m2.4.4.4.2.cmml">M</mi><mi id="A3.SS2.SSS1.10.p4.10.m2.4.4.4.3" xref="A3.SS2.SSS1.10.p4.10.m2.4.4.4.3.cmml">𝐱</mi></msub><mo id="A3.SS2.SSS1.10.p4.10.m2.4.4.5" xref="A3.SS2.SSS1.10.p4.10.m2.4.4.5.cmml">=</mo><mrow id="A3.SS2.SSS1.10.p4.10.m2.3.3.1" 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id="A3.SS2.SSS1.10.p4.10.m2.4d">italic_M start_POSTSUBSCRIPT bold_x end_POSTSUBSCRIPT = 2 ∥ ∇ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT roman_log italic_p ( bold_x ) ∥ ≤ italic_e start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT ( ∥ bold_x ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + 1 )</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="A3.SS2.SSS1.11.p5"> <p class="ltx_p" id="A3.SS2.SSS1.11.p5.2">Combining the above inequalities with Equation (<a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#A3.E24" title="24 ‣ Proof. ‣ C.2.1 Theoretical considerations ‣ C.2 Gaussian Mixture prior ‣ Appendix C SNORE in simple cases ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_tag">24</span></a>), we have that</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A8.EGx37"> <tbody id="A3.Ex59"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\|\nabla\mathcal{R}_{\sigma}({\mathbf{x}})+\nabla\log p({\mathbf{% x}})\|" class="ltx_Math" display="inline" id="A3.Ex59.m1.3"><semantics id="A3.Ex59.m1.3a"><mrow id="A3.Ex59.m1.3.3.1" xref="A3.Ex59.m1.3.3.2.cmml"><mo id="A3.Ex59.m1.3.3.1.2" stretchy="false" xref="A3.Ex59.m1.3.3.2.1.cmml">‖</mo><mrow id="A3.Ex59.m1.3.3.1.1" xref="A3.Ex59.m1.3.3.1.1.cmml"><mrow id="A3.Ex59.m1.3.3.1.1.2" xref="A3.Ex59.m1.3.3.1.1.2.cmml"><mrow id="A3.Ex59.m1.3.3.1.1.2.2" xref="A3.Ex59.m1.3.3.1.1.2.2.cmml"><mo id="A3.Ex59.m1.3.3.1.1.2.2.1" rspace="0.167em" xref="A3.Ex59.m1.3.3.1.1.2.2.1.cmml">∇</mo><msub id="A3.Ex59.m1.3.3.1.1.2.2.2" xref="A3.Ex59.m1.3.3.1.1.2.2.2.cmml"><mi class="ltx_font_mathcaligraphic" id="A3.Ex59.m1.3.3.1.1.2.2.2.2" xref="A3.Ex59.m1.3.3.1.1.2.2.2.2.cmml">ℛ</mi><mi id="A3.Ex59.m1.3.3.1.1.2.2.2.3" xref="A3.Ex59.m1.3.3.1.1.2.2.2.3.cmml">σ</mi></msub></mrow><mo 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xref="A3.Ex59.m1.3.3.1.1.2.1"></times><apply id="A3.Ex59.m1.3.3.1.1.2.2.cmml" xref="A3.Ex59.m1.3.3.1.1.2.2"><ci id="A3.Ex59.m1.3.3.1.1.2.2.1.cmml" xref="A3.Ex59.m1.3.3.1.1.2.2.1">∇</ci><apply id="A3.Ex59.m1.3.3.1.1.2.2.2.cmml" xref="A3.Ex59.m1.3.3.1.1.2.2.2"><csymbol cd="ambiguous" id="A3.Ex59.m1.3.3.1.1.2.2.2.1.cmml" xref="A3.Ex59.m1.3.3.1.1.2.2.2">subscript</csymbol><ci id="A3.Ex59.m1.3.3.1.1.2.2.2.2.cmml" xref="A3.Ex59.m1.3.3.1.1.2.2.2.2">ℛ</ci><ci id="A3.Ex59.m1.3.3.1.1.2.2.2.3.cmml" xref="A3.Ex59.m1.3.3.1.1.2.2.2.3">𝜎</ci></apply></apply><ci id="A3.Ex59.m1.1.1.cmml" xref="A3.Ex59.m1.1.1">𝐱</ci></apply><apply id="A3.Ex59.m1.3.3.1.1.3.cmml" xref="A3.Ex59.m1.3.3.1.1.3"><times id="A3.Ex59.m1.3.3.1.1.3.1.cmml" xref="A3.Ex59.m1.3.3.1.1.3.1"></times><apply id="A3.Ex59.m1.3.3.1.1.3.2.cmml" xref="A3.Ex59.m1.3.3.1.1.3.2"><apply id="A3.Ex59.m1.3.3.1.1.3.2.1.cmml" xref="A3.Ex59.m1.3.3.1.1.3.2.1"><ci id="A3.Ex59.m1.3.3.1.1.3.2.1.1.cmml" xref="A3.Ex59.m1.3.3.1.1.3.2.1.1">∇</ci><log id="A3.Ex59.m1.3.3.1.1.3.2.1.2.cmml" xref="A3.Ex59.m1.3.3.1.1.3.2.1.2"></log></apply><ci id="A3.Ex59.m1.3.3.1.1.3.2.2.cmml" xref="A3.Ex59.m1.3.3.1.1.3.2.2">𝑝</ci></apply><ci id="A3.Ex59.m1.2.2.cmml" xref="A3.Ex59.m1.2.2">𝐱</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.Ex59.m1.3c">\displaystyle\|\nabla\mathcal{R}_{\sigma}({\mathbf{x}})+\nabla\log p({\mathbf{% x}})\|</annotation><annotation encoding="application/x-llamapun" id="A3.Ex59.m1.3d">∥ ∇ caligraphic_R start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ( bold_x ) + ∇ roman_log italic_p ( bold_x ) ∥</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\leq\sigma^{2}a_{p}\left(8d\sigma^{3}\sqrt{\frac{2}{\pi}}+4\|{% \mathbf{x}}\|^{3}+1\right)+e_{p}(\|{\mathbf{x}}\|^{2}+1)\sqrt{d}\sigma+% \mathcal{O}(\sigma^{4})" class="ltx_Math" display="inline" id="A3.Ex59.m2.5"><semantics id="A3.Ex59.m2.5a"><mrow id="A3.Ex59.m2.5.5" 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xref="A3.Ex59.m2.5.5.3.3.1.1">superscript</csymbol><ci id="A3.Ex59.m2.5.5.3.3.1.1.1.2.cmml" xref="A3.Ex59.m2.5.5.3.3.1.1.1.2">𝜎</ci><cn id="A3.Ex59.m2.5.5.3.3.1.1.1.3.cmml" type="integer" xref="A3.Ex59.m2.5.5.3.3.1.1.1.3">4</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.Ex59.m2.5c">\displaystyle\leq\sigma^{2}a_{p}\left(8d\sigma^{3}\sqrt{\frac{2}{\pi}}+4\|{% \mathbf{x}}\|^{3}+1\right)+e_{p}(\|{\mathbf{x}}\|^{2}+1)\sqrt{d}\sigma+% \mathcal{O}(\sigma^{4})</annotation><annotation encoding="application/x-llamapun" id="A3.Ex59.m2.5d">≤ italic_σ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_a start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT ( 8 italic_d italic_σ start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT square-root start_ARG divide start_ARG 2 end_ARG start_ARG italic_π end_ARG end_ARG + 4 ∥ bold_x ∥ start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT + 1 ) + italic_e start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT ( ∥ bold_x ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + 1 ) square-root start_ARG italic_d end_ARG italic_σ + caligraphic_O ( italic_σ start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="A3.Ex60"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_eqn_cell"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\leq\sigma\tilde{e}_{p}(\|{\mathbf{x}}\|^{2}+1)+\sigma^{2}a_{p}(4% \|{\mathbf{x}}\|^{3}+1)+\mathcal{O}(\sigma^{4})," class="ltx_Math" display="inline" id="A3.Ex60.m1.3"><semantics id="A3.Ex60.m1.3a"><mrow id="A3.Ex60.m1.3.3.1" xref="A3.Ex60.m1.3.3.1.1.cmml"><mrow id="A3.Ex60.m1.3.3.1.1" xref="A3.Ex60.m1.3.3.1.1.cmml"><mi id="A3.Ex60.m1.3.3.1.1.5" xref="A3.Ex60.m1.3.3.1.1.5.cmml"></mi><mo id="A3.Ex60.m1.3.3.1.1.4" xref="A3.Ex60.m1.3.3.1.1.4.cmml">≤</mo><mrow 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xref="A3.Ex60.m1.3.3.1.1.3.3.1.1.1.2">𝜎</ci><cn id="A3.Ex60.m1.3.3.1.1.3.3.1.1.1.3.cmml" type="integer" xref="A3.Ex60.m1.3.3.1.1.3.3.1.1.1.3">4</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.Ex60.m1.3c">\displaystyle\leq\sigma\tilde{e}_{p}(\|{\mathbf{x}}\|^{2}+1)+\sigma^{2}a_{p}(4% \|{\mathbf{x}}\|^{3}+1)+\mathcal{O}(\sigma^{4}),</annotation><annotation encoding="application/x-llamapun" id="A3.Ex60.m1.3d">≤ italic_σ over~ start_ARG italic_e end_ARG start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT ( ∥ bold_x ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + 1 ) + italic_σ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_a start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT ( 4 ∥ bold_x ∥ start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT + 1 ) + caligraphic_O ( italic_σ start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT ) ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="A3.SS2.SSS1.11.p5.1">with <math alttext="\tilde{e}_{p}=e_{p}\sqrt{d}" class="ltx_Math" display="inline" id="A3.SS2.SSS1.11.p5.1.m1.1"><semantics id="A3.SS2.SSS1.11.p5.1.m1.1a"><mrow id="A3.SS2.SSS1.11.p5.1.m1.1.1" xref="A3.SS2.SSS1.11.p5.1.m1.1.1.cmml"><msub id="A3.SS2.SSS1.11.p5.1.m1.1.1.2" xref="A3.SS2.SSS1.11.p5.1.m1.1.1.2.cmml"><mover accent="true" id="A3.SS2.SSS1.11.p5.1.m1.1.1.2.2" xref="A3.SS2.SSS1.11.p5.1.m1.1.1.2.2.cmml"><mi id="A3.SS2.SSS1.11.p5.1.m1.1.1.2.2.2" xref="A3.SS2.SSS1.11.p5.1.m1.1.1.2.2.2.cmml">e</mi><mo id="A3.SS2.SSS1.11.p5.1.m1.1.1.2.2.1" xref="A3.SS2.SSS1.11.p5.1.m1.1.1.2.2.1.cmml">~</mo></mover><mi id="A3.SS2.SSS1.11.p5.1.m1.1.1.2.3" xref="A3.SS2.SSS1.11.p5.1.m1.1.1.2.3.cmml">p</mi></msub><mo id="A3.SS2.SSS1.11.p5.1.m1.1.1.1" xref="A3.SS2.SSS1.11.p5.1.m1.1.1.1.cmml">=</mo><mrow id="A3.SS2.SSS1.11.p5.1.m1.1.1.3" xref="A3.SS2.SSS1.11.p5.1.m1.1.1.3.cmml"><msub id="A3.SS2.SSS1.11.p5.1.m1.1.1.3.2" xref="A3.SS2.SSS1.11.p5.1.m1.1.1.3.2.cmml"><mi id="A3.SS2.SSS1.11.p5.1.m1.1.1.3.2.2" xref="A3.SS2.SSS1.11.p5.1.m1.1.1.3.2.2.cmml">e</mi><mi id="A3.SS2.SSS1.11.p5.1.m1.1.1.3.2.3" xref="A3.SS2.SSS1.11.p5.1.m1.1.1.3.2.3.cmml">p</mi></msub><mo id="A3.SS2.SSS1.11.p5.1.m1.1.1.3.1" xref="A3.SS2.SSS1.11.p5.1.m1.1.1.3.1.cmml">⁢</mo><msqrt id="A3.SS2.SSS1.11.p5.1.m1.1.1.3.3" xref="A3.SS2.SSS1.11.p5.1.m1.1.1.3.3.cmml"><mi id="A3.SS2.SSS1.11.p5.1.m1.1.1.3.3.2" xref="A3.SS2.SSS1.11.p5.1.m1.1.1.3.3.2.cmml">d</mi></msqrt></mrow></mrow><annotation-xml encoding="MathML-Content" id="A3.SS2.SSS1.11.p5.1.m1.1b"><apply id="A3.SS2.SSS1.11.p5.1.m1.1.1.cmml" xref="A3.SS2.SSS1.11.p5.1.m1.1.1"><eq id="A3.SS2.SSS1.11.p5.1.m1.1.1.1.cmml" xref="A3.SS2.SSS1.11.p5.1.m1.1.1.1"></eq><apply id="A3.SS2.SSS1.11.p5.1.m1.1.1.2.cmml" xref="A3.SS2.SSS1.11.p5.1.m1.1.1.2"><csymbol cd="ambiguous" id="A3.SS2.SSS1.11.p5.1.m1.1.1.2.1.cmml" xref="A3.SS2.SSS1.11.p5.1.m1.1.1.2">subscript</csymbol><apply id="A3.SS2.SSS1.11.p5.1.m1.1.1.2.2.cmml" xref="A3.SS2.SSS1.11.p5.1.m1.1.1.2.2"><ci id="A3.SS2.SSS1.11.p5.1.m1.1.1.2.2.1.cmml" xref="A3.SS2.SSS1.11.p5.1.m1.1.1.2.2.1">~</ci><ci id="A3.SS2.SSS1.11.p5.1.m1.1.1.2.2.2.cmml" xref="A3.SS2.SSS1.11.p5.1.m1.1.1.2.2.2">𝑒</ci></apply><ci id="A3.SS2.SSS1.11.p5.1.m1.1.1.2.3.cmml" xref="A3.SS2.SSS1.11.p5.1.m1.1.1.2.3">𝑝</ci></apply><apply id="A3.SS2.SSS1.11.p5.1.m1.1.1.3.cmml" xref="A3.SS2.SSS1.11.p5.1.m1.1.1.3"><times id="A3.SS2.SSS1.11.p5.1.m1.1.1.3.1.cmml" xref="A3.SS2.SSS1.11.p5.1.m1.1.1.3.1"></times><apply id="A3.SS2.SSS1.11.p5.1.m1.1.1.3.2.cmml" xref="A3.SS2.SSS1.11.p5.1.m1.1.1.3.2"><csymbol cd="ambiguous" id="A3.SS2.SSS1.11.p5.1.m1.1.1.3.2.1.cmml" xref="A3.SS2.SSS1.11.p5.1.m1.1.1.3.2">subscript</csymbol><ci id="A3.SS2.SSS1.11.p5.1.m1.1.1.3.2.2.cmml" xref="A3.SS2.SSS1.11.p5.1.m1.1.1.3.2.2">𝑒</ci><ci id="A3.SS2.SSS1.11.p5.1.m1.1.1.3.2.3.cmml" xref="A3.SS2.SSS1.11.p5.1.m1.1.1.3.2.3">𝑝</ci></apply><apply id="A3.SS2.SSS1.11.p5.1.m1.1.1.3.3.cmml" xref="A3.SS2.SSS1.11.p5.1.m1.1.1.3.3"><root id="A3.SS2.SSS1.11.p5.1.m1.1.1.3.3a.cmml" xref="A3.SS2.SSS1.11.p5.1.m1.1.1.3.3"></root><ci id="A3.SS2.SSS1.11.p5.1.m1.1.1.3.3.2.cmml" xref="A3.SS2.SSS1.11.p5.1.m1.1.1.3.3.2">𝑑</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.SS2.SSS1.11.p5.1.m1.1c">\tilde{e}_{p}=e_{p}\sqrt{d}</annotation><annotation encoding="application/x-llamapun" id="A3.SS2.SSS1.11.p5.1.m1.1d">over~ start_ARG italic_e end_ARG start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT = italic_e start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT square-root start_ARG italic_d end_ARG</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="A3.SS2.SSS1.12.p6"> <p class="ltx_p" id="A3.SS2.SSS1.12.p6.5">By defining <math alttext="d_{p}=2\tilde{e}_{p}" class="ltx_Math" display="inline" id="A3.SS2.SSS1.12.p6.1.m1.1"><semantics id="A3.SS2.SSS1.12.p6.1.m1.1a"><mrow id="A3.SS2.SSS1.12.p6.1.m1.1.1" xref="A3.SS2.SSS1.12.p6.1.m1.1.1.cmml"><msub id="A3.SS2.SSS1.12.p6.1.m1.1.1.2" xref="A3.SS2.SSS1.12.p6.1.m1.1.1.2.cmml"><mi id="A3.SS2.SSS1.12.p6.1.m1.1.1.2.2" xref="A3.SS2.SSS1.12.p6.1.m1.1.1.2.2.cmml">d</mi><mi id="A3.SS2.SSS1.12.p6.1.m1.1.1.2.3" xref="A3.SS2.SSS1.12.p6.1.m1.1.1.2.3.cmml">p</mi></msub><mo id="A3.SS2.SSS1.12.p6.1.m1.1.1.1" xref="A3.SS2.SSS1.12.p6.1.m1.1.1.1.cmml">=</mo><mrow id="A3.SS2.SSS1.12.p6.1.m1.1.1.3" xref="A3.SS2.SSS1.12.p6.1.m1.1.1.3.cmml"><mn id="A3.SS2.SSS1.12.p6.1.m1.1.1.3.2" xref="A3.SS2.SSS1.12.p6.1.m1.1.1.3.2.cmml">2</mn><mo id="A3.SS2.SSS1.12.p6.1.m1.1.1.3.1" xref="A3.SS2.SSS1.12.p6.1.m1.1.1.3.1.cmml">⁢</mo><msub id="A3.SS2.SSS1.12.p6.1.m1.1.1.3.3" xref="A3.SS2.SSS1.12.p6.1.m1.1.1.3.3.cmml"><mover accent="true" id="A3.SS2.SSS1.12.p6.1.m1.1.1.3.3.2" xref="A3.SS2.SSS1.12.p6.1.m1.1.1.3.3.2.cmml"><mi id="A3.SS2.SSS1.12.p6.1.m1.1.1.3.3.2.2" xref="A3.SS2.SSS1.12.p6.1.m1.1.1.3.3.2.2.cmml">e</mi><mo id="A3.SS2.SSS1.12.p6.1.m1.1.1.3.3.2.1" xref="A3.SS2.SSS1.12.p6.1.m1.1.1.3.3.2.1.cmml">~</mo></mover><mi id="A3.SS2.SSS1.12.p6.1.m1.1.1.3.3.3" xref="A3.SS2.SSS1.12.p6.1.m1.1.1.3.3.3.cmml">p</mi></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="A3.SS2.SSS1.12.p6.1.m1.1b"><apply id="A3.SS2.SSS1.12.p6.1.m1.1.1.cmml" xref="A3.SS2.SSS1.12.p6.1.m1.1.1"><eq id="A3.SS2.SSS1.12.p6.1.m1.1.1.1.cmml" xref="A3.SS2.SSS1.12.p6.1.m1.1.1.1"></eq><apply id="A3.SS2.SSS1.12.p6.1.m1.1.1.2.cmml" xref="A3.SS2.SSS1.12.p6.1.m1.1.1.2"><csymbol cd="ambiguous" id="A3.SS2.SSS1.12.p6.1.m1.1.1.2.1.cmml" xref="A3.SS2.SSS1.12.p6.1.m1.1.1.2">subscript</csymbol><ci id="A3.SS2.SSS1.12.p6.1.m1.1.1.2.2.cmml" xref="A3.SS2.SSS1.12.p6.1.m1.1.1.2.2">𝑑</ci><ci id="A3.SS2.SSS1.12.p6.1.m1.1.1.2.3.cmml" xref="A3.SS2.SSS1.12.p6.1.m1.1.1.2.3">𝑝</ci></apply><apply id="A3.SS2.SSS1.12.p6.1.m1.1.1.3.cmml" xref="A3.SS2.SSS1.12.p6.1.m1.1.1.3"><times id="A3.SS2.SSS1.12.p6.1.m1.1.1.3.1.cmml" xref="A3.SS2.SSS1.12.p6.1.m1.1.1.3.1"></times><cn id="A3.SS2.SSS1.12.p6.1.m1.1.1.3.2.cmml" type="integer" xref="A3.SS2.SSS1.12.p6.1.m1.1.1.3.2">2</cn><apply id="A3.SS2.SSS1.12.p6.1.m1.1.1.3.3.cmml" xref="A3.SS2.SSS1.12.p6.1.m1.1.1.3.3"><csymbol cd="ambiguous" id="A3.SS2.SSS1.12.p6.1.m1.1.1.3.3.1.cmml" xref="A3.SS2.SSS1.12.p6.1.m1.1.1.3.3">subscript</csymbol><apply id="A3.SS2.SSS1.12.p6.1.m1.1.1.3.3.2.cmml" xref="A3.SS2.SSS1.12.p6.1.m1.1.1.3.3.2"><ci id="A3.SS2.SSS1.12.p6.1.m1.1.1.3.3.2.1.cmml" xref="A3.SS2.SSS1.12.p6.1.m1.1.1.3.3.2.1">~</ci><ci id="A3.SS2.SSS1.12.p6.1.m1.1.1.3.3.2.2.cmml" xref="A3.SS2.SSS1.12.p6.1.m1.1.1.3.3.2.2">𝑒</ci></apply><ci id="A3.SS2.SSS1.12.p6.1.m1.1.1.3.3.3.cmml" xref="A3.SS2.SSS1.12.p6.1.m1.1.1.3.3.3">𝑝</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.SS2.SSS1.12.p6.1.m1.1c">d_{p}=2\tilde{e}_{p}</annotation><annotation encoding="application/x-llamapun" id="A3.SS2.SSS1.12.p6.1.m1.1d">italic_d start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT = 2 over~ start_ARG italic_e end_ARG start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT</annotation></semantics></math>, we have demonstrated that there exist <math alttext="\sigma_{0}>0" class="ltx_Math" display="inline" id="A3.SS2.SSS1.12.p6.2.m2.1"><semantics id="A3.SS2.SSS1.12.p6.2.m2.1a"><mrow id="A3.SS2.SSS1.12.p6.2.m2.1.1" xref="A3.SS2.SSS1.12.p6.2.m2.1.1.cmml"><msub id="A3.SS2.SSS1.12.p6.2.m2.1.1.2" xref="A3.SS2.SSS1.12.p6.2.m2.1.1.2.cmml"><mi id="A3.SS2.SSS1.12.p6.2.m2.1.1.2.2" xref="A3.SS2.SSS1.12.p6.2.m2.1.1.2.2.cmml">σ</mi><mn id="A3.SS2.SSS1.12.p6.2.m2.1.1.2.3" xref="A3.SS2.SSS1.12.p6.2.m2.1.1.2.3.cmml">0</mn></msub><mo id="A3.SS2.SSS1.12.p6.2.m2.1.1.1" xref="A3.SS2.SSS1.12.p6.2.m2.1.1.1.cmml">></mo><mn id="A3.SS2.SSS1.12.p6.2.m2.1.1.3" xref="A3.SS2.SSS1.12.p6.2.m2.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="A3.SS2.SSS1.12.p6.2.m2.1b"><apply id="A3.SS2.SSS1.12.p6.2.m2.1.1.cmml" xref="A3.SS2.SSS1.12.p6.2.m2.1.1"><gt id="A3.SS2.SSS1.12.p6.2.m2.1.1.1.cmml" xref="A3.SS2.SSS1.12.p6.2.m2.1.1.1"></gt><apply id="A3.SS2.SSS1.12.p6.2.m2.1.1.2.cmml" xref="A3.SS2.SSS1.12.p6.2.m2.1.1.2"><csymbol cd="ambiguous" id="A3.SS2.SSS1.12.p6.2.m2.1.1.2.1.cmml" xref="A3.SS2.SSS1.12.p6.2.m2.1.1.2">subscript</csymbol><ci id="A3.SS2.SSS1.12.p6.2.m2.1.1.2.2.cmml" xref="A3.SS2.SSS1.12.p6.2.m2.1.1.2.2">𝜎</ci><cn id="A3.SS2.SSS1.12.p6.2.m2.1.1.2.3.cmml" type="integer" xref="A3.SS2.SSS1.12.p6.2.m2.1.1.2.3">0</cn></apply><cn id="A3.SS2.SSS1.12.p6.2.m2.1.1.3.cmml" type="integer" xref="A3.SS2.SSS1.12.p6.2.m2.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.SS2.SSS1.12.p6.2.m2.1c">\sigma_{0}>0</annotation><annotation encoding="application/x-llamapun" id="A3.SS2.SSS1.12.p6.2.m2.1d">italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT > 0</annotation></semantics></math>, <math alttext="d_{p}\geq 0" class="ltx_Math" display="inline" id="A3.SS2.SSS1.12.p6.3.m3.1"><semantics id="A3.SS2.SSS1.12.p6.3.m3.1a"><mrow id="A3.SS2.SSS1.12.p6.3.m3.1.1" xref="A3.SS2.SSS1.12.p6.3.m3.1.1.cmml"><msub id="A3.SS2.SSS1.12.p6.3.m3.1.1.2" xref="A3.SS2.SSS1.12.p6.3.m3.1.1.2.cmml"><mi id="A3.SS2.SSS1.12.p6.3.m3.1.1.2.2" xref="A3.SS2.SSS1.12.p6.3.m3.1.1.2.2.cmml">d</mi><mi id="A3.SS2.SSS1.12.p6.3.m3.1.1.2.3" xref="A3.SS2.SSS1.12.p6.3.m3.1.1.2.3.cmml">p</mi></msub><mo id="A3.SS2.SSS1.12.p6.3.m3.1.1.1" xref="A3.SS2.SSS1.12.p6.3.m3.1.1.1.cmml">≥</mo><mn id="A3.SS2.SSS1.12.p6.3.m3.1.1.3" xref="A3.SS2.SSS1.12.p6.3.m3.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="A3.SS2.SSS1.12.p6.3.m3.1b"><apply id="A3.SS2.SSS1.12.p6.3.m3.1.1.cmml" xref="A3.SS2.SSS1.12.p6.3.m3.1.1"><geq id="A3.SS2.SSS1.12.p6.3.m3.1.1.1.cmml" xref="A3.SS2.SSS1.12.p6.3.m3.1.1.1"></geq><apply id="A3.SS2.SSS1.12.p6.3.m3.1.1.2.cmml" xref="A3.SS2.SSS1.12.p6.3.m3.1.1.2"><csymbol cd="ambiguous" id="A3.SS2.SSS1.12.p6.3.m3.1.1.2.1.cmml" xref="A3.SS2.SSS1.12.p6.3.m3.1.1.2">subscript</csymbol><ci id="A3.SS2.SSS1.12.p6.3.m3.1.1.2.2.cmml" xref="A3.SS2.SSS1.12.p6.3.m3.1.1.2.2">𝑑</ci><ci id="A3.SS2.SSS1.12.p6.3.m3.1.1.2.3.cmml" xref="A3.SS2.SSS1.12.p6.3.m3.1.1.2.3">𝑝</ci></apply><cn id="A3.SS2.SSS1.12.p6.3.m3.1.1.3.cmml" type="integer" xref="A3.SS2.SSS1.12.p6.3.m3.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.SS2.SSS1.12.p6.3.m3.1c">d_{p}\geq 0</annotation><annotation encoding="application/x-llamapun" id="A3.SS2.SSS1.12.p6.3.m3.1d">italic_d start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT ≥ 0</annotation></semantics></math> such that <math alttext="\forall{\mathbf{x}}\in{\mathbb{R}}^{d}" class="ltx_Math" display="inline" id="A3.SS2.SSS1.12.p6.4.m4.1"><semantics id="A3.SS2.SSS1.12.p6.4.m4.1a"><mrow id="A3.SS2.SSS1.12.p6.4.m4.1.1" xref="A3.SS2.SSS1.12.p6.4.m4.1.1.cmml"><mrow id="A3.SS2.SSS1.12.p6.4.m4.1.1.2" xref="A3.SS2.SSS1.12.p6.4.m4.1.1.2.cmml"><mo id="A3.SS2.SSS1.12.p6.4.m4.1.1.2.1" rspace="0.167em" xref="A3.SS2.SSS1.12.p6.4.m4.1.1.2.1.cmml">∀</mo><mi id="A3.SS2.SSS1.12.p6.4.m4.1.1.2.2" xref="A3.SS2.SSS1.12.p6.4.m4.1.1.2.2.cmml">𝐱</mi></mrow><mo id="A3.SS2.SSS1.12.p6.4.m4.1.1.1" xref="A3.SS2.SSS1.12.p6.4.m4.1.1.1.cmml">∈</mo><msup id="A3.SS2.SSS1.12.p6.4.m4.1.1.3" xref="A3.SS2.SSS1.12.p6.4.m4.1.1.3.cmml"><mi id="A3.SS2.SSS1.12.p6.4.m4.1.1.3.2" xref="A3.SS2.SSS1.12.p6.4.m4.1.1.3.2.cmml">ℝ</mi><mi id="A3.SS2.SSS1.12.p6.4.m4.1.1.3.3" xref="A3.SS2.SSS1.12.p6.4.m4.1.1.3.3.cmml">d</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="A3.SS2.SSS1.12.p6.4.m4.1b"><apply id="A3.SS2.SSS1.12.p6.4.m4.1.1.cmml" xref="A3.SS2.SSS1.12.p6.4.m4.1.1"><in id="A3.SS2.SSS1.12.p6.4.m4.1.1.1.cmml" xref="A3.SS2.SSS1.12.p6.4.m4.1.1.1"></in><apply id="A3.SS2.SSS1.12.p6.4.m4.1.1.2.cmml" xref="A3.SS2.SSS1.12.p6.4.m4.1.1.2"><csymbol cd="latexml" id="A3.SS2.SSS1.12.p6.4.m4.1.1.2.1.cmml" xref="A3.SS2.SSS1.12.p6.4.m4.1.1.2.1">for-all</csymbol><ci id="A3.SS2.SSS1.12.p6.4.m4.1.1.2.2.cmml" xref="A3.SS2.SSS1.12.p6.4.m4.1.1.2.2">𝐱</ci></apply><apply id="A3.SS2.SSS1.12.p6.4.m4.1.1.3.cmml" xref="A3.SS2.SSS1.12.p6.4.m4.1.1.3"><csymbol cd="ambiguous" id="A3.SS2.SSS1.12.p6.4.m4.1.1.3.1.cmml" xref="A3.SS2.SSS1.12.p6.4.m4.1.1.3">superscript</csymbol><ci id="A3.SS2.SSS1.12.p6.4.m4.1.1.3.2.cmml" xref="A3.SS2.SSS1.12.p6.4.m4.1.1.3.2">ℝ</ci><ci id="A3.SS2.SSS1.12.p6.4.m4.1.1.3.3.cmml" xref="A3.SS2.SSS1.12.p6.4.m4.1.1.3.3">𝑑</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.SS2.SSS1.12.p6.4.m4.1c">\forall{\mathbf{x}}\in{\mathbb{R}}^{d}</annotation><annotation encoding="application/x-llamapun" id="A3.SS2.SSS1.12.p6.4.m4.1d">∀ bold_x ∈ blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT</annotation></semantics></math> and <math alttext="\forall\sigma\in]0,\sigma_{0}]" class="ltx_math_unparsed" display="inline" id="A3.SS2.SSS1.12.p6.5.m5.1"><semantics id="A3.SS2.SSS1.12.p6.5.m5.1a"><mrow id="A3.SS2.SSS1.12.p6.5.m5.1b"><mrow id="A3.SS2.SSS1.12.p6.5.m5.1.1"><mo id="A3.SS2.SSS1.12.p6.5.m5.1.1.1" rspace="0.167em">∀</mo><mi id="A3.SS2.SSS1.12.p6.5.m5.1.1.2">σ</mi><mo id="A3.SS2.SSS1.12.p6.5.m5.1.1.3" rspace="0em">∈</mo><mo id="A3.SS2.SSS1.12.p6.5.m5.1.1.4" stretchy="false">]</mo></mrow><mn id="A3.SS2.SSS1.12.p6.5.m5.1.2">0</mn><mo id="A3.SS2.SSS1.12.p6.5.m5.1.3">,</mo><msub id="A3.SS2.SSS1.12.p6.5.m5.1.4"><mi id="A3.SS2.SSS1.12.p6.5.m5.1.4.2">σ</mi><mn id="A3.SS2.SSS1.12.p6.5.m5.1.4.3">0</mn></msub><mo id="A3.SS2.SSS1.12.p6.5.m5.1.5" stretchy="false">]</mo></mrow><annotation encoding="application/x-tex" id="A3.SS2.SSS1.12.p6.5.m5.1c">\forall\sigma\in]0,\sigma_{0}]</annotation><annotation encoding="application/x-llamapun" id="A3.SS2.SSS1.12.p6.5.m5.1d">∀ italic_σ ∈ ] 0 , italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ]</annotation></semantics></math>,</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A8.EGx38"> <tbody id="A3.E26"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\|\nabla\mathcal{R}_{\sigma}({\mathbf{x}})+\nabla\log p({\mathbf{% x}})\|\leq\sigma d_{p}(\|{\mathbf{x}}\|^{2}+1)." class="ltx_Math" display="inline" id="A3.E26.m1.4"><semantics id="A3.E26.m1.4a"><mrow id="A3.E26.m1.4.4.1" xref="A3.E26.m1.4.4.1.1.cmml"><mrow id="A3.E26.m1.4.4.1.1" xref="A3.E26.m1.4.4.1.1.cmml"><mrow id="A3.E26.m1.4.4.1.1.1.1" xref="A3.E26.m1.4.4.1.1.1.2.cmml"><mo id="A3.E26.m1.4.4.1.1.1.1.2" stretchy="false" xref="A3.E26.m1.4.4.1.1.1.2.1.cmml">‖</mo><mrow id="A3.E26.m1.4.4.1.1.1.1.1" xref="A3.E26.m1.4.4.1.1.1.1.1.cmml"><mrow id="A3.E26.m1.4.4.1.1.1.1.1.2" xref="A3.E26.m1.4.4.1.1.1.1.1.2.cmml"><mrow id="A3.E26.m1.4.4.1.1.1.1.1.2.2" xref="A3.E26.m1.4.4.1.1.1.1.1.2.2.cmml"><mo id="A3.E26.m1.4.4.1.1.1.1.1.2.2.1" rspace="0.167em" xref="A3.E26.m1.4.4.1.1.1.1.1.2.2.1.cmml">∇</mo><msub id="A3.E26.m1.4.4.1.1.1.1.1.2.2.2" xref="A3.E26.m1.4.4.1.1.1.1.1.2.2.2.cmml"><mi class="ltx_font_mathcaligraphic" id="A3.E26.m1.4.4.1.1.1.1.1.2.2.2.2" xref="A3.E26.m1.4.4.1.1.1.1.1.2.2.2.2.cmml">ℛ</mi><mi id="A3.E26.m1.4.4.1.1.1.1.1.2.2.2.3" xref="A3.E26.m1.4.4.1.1.1.1.1.2.2.2.3.cmml">σ</mi></msub></mrow><mo id="A3.E26.m1.4.4.1.1.1.1.1.2.1" xref="A3.E26.m1.4.4.1.1.1.1.1.2.1.cmml">⁢</mo><mrow id="A3.E26.m1.4.4.1.1.1.1.1.2.3.2" xref="A3.E26.m1.4.4.1.1.1.1.1.2.cmml"><mo id="A3.E26.m1.4.4.1.1.1.1.1.2.3.2.1" stretchy="false" xref="A3.E26.m1.4.4.1.1.1.1.1.2.cmml">(</mo><mi id="A3.E26.m1.1.1" xref="A3.E26.m1.1.1.cmml">𝐱</mi><mo id="A3.E26.m1.4.4.1.1.1.1.1.2.3.2.2" stretchy="false" xref="A3.E26.m1.4.4.1.1.1.1.1.2.cmml">)</mo></mrow></mrow><mo id="A3.E26.m1.4.4.1.1.1.1.1.1" xref="A3.E26.m1.4.4.1.1.1.1.1.1.cmml">+</mo><mrow id="A3.E26.m1.4.4.1.1.1.1.1.3" xref="A3.E26.m1.4.4.1.1.1.1.1.3.cmml"><mrow id="A3.E26.m1.4.4.1.1.1.1.1.3.2" xref="A3.E26.m1.4.4.1.1.1.1.1.3.2.cmml"><mrow id="A3.E26.m1.4.4.1.1.1.1.1.3.2.1" xref="A3.E26.m1.4.4.1.1.1.1.1.3.2.1.cmml"><mo id="A3.E26.m1.4.4.1.1.1.1.1.3.2.1.1" rspace="0.167em" xref="A3.E26.m1.4.4.1.1.1.1.1.3.2.1.1.cmml">∇</mo><mi id="A3.E26.m1.4.4.1.1.1.1.1.3.2.1.2" xref="A3.E26.m1.4.4.1.1.1.1.1.3.2.1.2.cmml">log</mi></mrow><mo id="A3.E26.m1.4.4.1.1.1.1.1.3.2a" lspace="0.167em" xref="A3.E26.m1.4.4.1.1.1.1.1.3.2.cmml">⁡</mo><mi id="A3.E26.m1.4.4.1.1.1.1.1.3.2.2" xref="A3.E26.m1.4.4.1.1.1.1.1.3.2.2.cmml">p</mi></mrow><mo id="A3.E26.m1.4.4.1.1.1.1.1.3.1" xref="A3.E26.m1.4.4.1.1.1.1.1.3.1.cmml">⁢</mo><mrow id="A3.E26.m1.4.4.1.1.1.1.1.3.3.2" xref="A3.E26.m1.4.4.1.1.1.1.1.3.cmml"><mo id="A3.E26.m1.4.4.1.1.1.1.1.3.3.2.1" stretchy="false" xref="A3.E26.m1.4.4.1.1.1.1.1.3.cmml">(</mo><mi id="A3.E26.m1.2.2" xref="A3.E26.m1.2.2.cmml">𝐱</mi><mo id="A3.E26.m1.4.4.1.1.1.1.1.3.3.2.2" stretchy="false" xref="A3.E26.m1.4.4.1.1.1.1.1.3.cmml">)</mo></mrow></mrow></mrow><mo id="A3.E26.m1.4.4.1.1.1.1.3" stretchy="false" xref="A3.E26.m1.4.4.1.1.1.2.1.cmml">‖</mo></mrow><mo id="A3.E26.m1.4.4.1.1.3" xref="A3.E26.m1.4.4.1.1.3.cmml">≤</mo><mrow id="A3.E26.m1.4.4.1.1.2" xref="A3.E26.m1.4.4.1.1.2.cmml"><mi id="A3.E26.m1.4.4.1.1.2.3" xref="A3.E26.m1.4.4.1.1.2.3.cmml">σ</mi><mo id="A3.E26.m1.4.4.1.1.2.2" xref="A3.E26.m1.4.4.1.1.2.2.cmml">⁢</mo><msub id="A3.E26.m1.4.4.1.1.2.4" xref="A3.E26.m1.4.4.1.1.2.4.cmml"><mi id="A3.E26.m1.4.4.1.1.2.4.2" xref="A3.E26.m1.4.4.1.1.2.4.2.cmml">d</mi><mi id="A3.E26.m1.4.4.1.1.2.4.3" xref="A3.E26.m1.4.4.1.1.2.4.3.cmml">p</mi></msub><mo id="A3.E26.m1.4.4.1.1.2.2a" xref="A3.E26.m1.4.4.1.1.2.2.cmml">⁢</mo><mrow id="A3.E26.m1.4.4.1.1.2.1.1" xref="A3.E26.m1.4.4.1.1.2.1.1.1.cmml"><mo id="A3.E26.m1.4.4.1.1.2.1.1.2" stretchy="false" xref="A3.E26.m1.4.4.1.1.2.1.1.1.cmml">(</mo><mrow id="A3.E26.m1.4.4.1.1.2.1.1.1" xref="A3.E26.m1.4.4.1.1.2.1.1.1.cmml"><msup id="A3.E26.m1.4.4.1.1.2.1.1.1.2" xref="A3.E26.m1.4.4.1.1.2.1.1.1.2.cmml"><mrow id="A3.E26.m1.4.4.1.1.2.1.1.1.2.2.2" xref="A3.E26.m1.4.4.1.1.2.1.1.1.2.2.1.cmml"><mo id="A3.E26.m1.4.4.1.1.2.1.1.1.2.2.2.1" stretchy="false" xref="A3.E26.m1.4.4.1.1.2.1.1.1.2.2.1.1.cmml">‖</mo><mi id="A3.E26.m1.3.3" xref="A3.E26.m1.3.3.cmml">𝐱</mi><mo id="A3.E26.m1.4.4.1.1.2.1.1.1.2.2.2.2" stretchy="false" xref="A3.E26.m1.4.4.1.1.2.1.1.1.2.2.1.1.cmml">‖</mo></mrow><mn id="A3.E26.m1.4.4.1.1.2.1.1.1.2.3" xref="A3.E26.m1.4.4.1.1.2.1.1.1.2.3.cmml">2</mn></msup><mo id="A3.E26.m1.4.4.1.1.2.1.1.1.1" xref="A3.E26.m1.4.4.1.1.2.1.1.1.1.cmml">+</mo><mn id="A3.E26.m1.4.4.1.1.2.1.1.1.3" xref="A3.E26.m1.4.4.1.1.2.1.1.1.3.cmml">1</mn></mrow><mo id="A3.E26.m1.4.4.1.1.2.1.1.3" stretchy="false" xref="A3.E26.m1.4.4.1.1.2.1.1.1.cmml">)</mo></mrow></mrow></mrow><mo id="A3.E26.m1.4.4.1.2" lspace="0em" xref="A3.E26.m1.4.4.1.1.cmml">.</mo></mrow><annotation-xml encoding="MathML-Content" id="A3.E26.m1.4b"><apply id="A3.E26.m1.4.4.1.1.cmml" xref="A3.E26.m1.4.4.1"><leq id="A3.E26.m1.4.4.1.1.3.cmml" xref="A3.E26.m1.4.4.1.1.3"></leq><apply id="A3.E26.m1.4.4.1.1.1.2.cmml" xref="A3.E26.m1.4.4.1.1.1.1"><csymbol cd="latexml" id="A3.E26.m1.4.4.1.1.1.2.1.cmml" xref="A3.E26.m1.4.4.1.1.1.1.2">norm</csymbol><apply 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xref="A3.E26.m1.4.4.1.1.2.4">subscript</csymbol><ci id="A3.E26.m1.4.4.1.1.2.4.2.cmml" xref="A3.E26.m1.4.4.1.1.2.4.2">𝑑</ci><ci id="A3.E26.m1.4.4.1.1.2.4.3.cmml" xref="A3.E26.m1.4.4.1.1.2.4.3">𝑝</ci></apply><apply id="A3.E26.m1.4.4.1.1.2.1.1.1.cmml" xref="A3.E26.m1.4.4.1.1.2.1.1"><plus id="A3.E26.m1.4.4.1.1.2.1.1.1.1.cmml" xref="A3.E26.m1.4.4.1.1.2.1.1.1.1"></plus><apply id="A3.E26.m1.4.4.1.1.2.1.1.1.2.cmml" xref="A3.E26.m1.4.4.1.1.2.1.1.1.2"><csymbol cd="ambiguous" id="A3.E26.m1.4.4.1.1.2.1.1.1.2.1.cmml" xref="A3.E26.m1.4.4.1.1.2.1.1.1.2">superscript</csymbol><apply id="A3.E26.m1.4.4.1.1.2.1.1.1.2.2.1.cmml" xref="A3.E26.m1.4.4.1.1.2.1.1.1.2.2.2"><csymbol cd="latexml" id="A3.E26.m1.4.4.1.1.2.1.1.1.2.2.1.1.cmml" xref="A3.E26.m1.4.4.1.1.2.1.1.1.2.2.2.1">norm</csymbol><ci id="A3.E26.m1.3.3.cmml" xref="A3.E26.m1.3.3">𝐱</ci></apply><cn id="A3.E26.m1.4.4.1.1.2.1.1.1.2.3.cmml" type="integer" xref="A3.E26.m1.4.4.1.1.2.1.1.1.2.3">2</cn></apply><cn id="A3.E26.m1.4.4.1.1.2.1.1.1.3.cmml" type="integer" xref="A3.E26.m1.4.4.1.1.2.1.1.1.3">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.E26.m1.4c">\displaystyle\|\nabla\mathcal{R}_{\sigma}({\mathbf{x}})+\nabla\log p({\mathbf{% x}})\|\leq\sigma d_{p}(\|{\mathbf{x}}\|^{2}+1).</annotation><annotation encoding="application/x-llamapun" id="A3.E26.m1.4d">∥ ∇ caligraphic_R start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ( bold_x ) + ∇ roman_log italic_p ( bold_x ) ∥ ≤ italic_σ italic_d start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT ( ∥ bold_x ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + 1 ) .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(26)</span></td> </tr></tbody> </table> <p class="ltx_p" id="A3.SS2.SSS1.12.p6.6">∎</p> </div> </div> <div class="ltx_para" id="A3.SS2.SSS1.p4"> <p class="ltx_p" id="A3.SS2.SSS1.p4.1">Proposition <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#A3.Thmproposition2" title="Proposition C.2. ‣ C.2.1 Theoretical considerations ‣ C.2 Gaussian Mixture prior ‣ Appendix C SNORE in simple cases ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_tag">C.2</span></a> shows that, in a case of a GMM prior, our regularization approximates the prior score with a pointwise speed of <math alttext="\sigma" class="ltx_Math" display="inline" id="A3.SS2.SSS1.p4.1.m1.1"><semantics id="A3.SS2.SSS1.p4.1.m1.1a"><mi id="A3.SS2.SSS1.p4.1.m1.1.1" xref="A3.SS2.SSS1.p4.1.m1.1.1.cmml">σ</mi><annotation-xml encoding="MathML-Content" id="A3.SS2.SSS1.p4.1.m1.1b"><ci id="A3.SS2.SSS1.p4.1.m1.1.1.cmml" xref="A3.SS2.SSS1.p4.1.m1.1.1">𝜎</ci></annotation-xml><annotation encoding="application/x-tex" id="A3.SS2.SSS1.p4.1.m1.1c">\sigma</annotation><annotation encoding="application/x-llamapun" id="A3.SS2.SSS1.p4.1.m1.1d">italic_σ</annotation></semantics></math>. This speed of approximation is slower than for the traditional PnP (Proposition <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#A3.Thmproposition1" title="Proposition C.1. ‣ C.2.1 Theoretical considerations ‣ C.2 Gaussian Mixture prior ‣ Appendix C SNORE in simple cases ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_tag">C.1</span></a>).</p> </div> </section> <section class="ltx_subsubsection" id="A3.SS2.SSS2"> <h4 class="ltx_title ltx_title_subsubsection"> <span class="ltx_tag ltx_tag_subsubsection">C.2.2 </span>Simulations</h4> <div class="ltx_para" id="A3.SS2.SSS2.p1"> <p class="ltx_p" id="A3.SS2.SSS2.p1.3">In Figure <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#A3.F4" title="Figure 4 ‣ C.2.2 Simulations ‣ C.2 Gaussian Mixture prior ‣ Appendix C SNORE in simple cases ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_tag">4</span></a>, we display a non-trivial Gaussian Mixture distribution in 1D with three Gaussians. Using Equation (<a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#A3.Ex29" title="C.2.1 Theoretical considerations ‣ C.2 Gaussian Mixture prior ‣ Appendix C SNORE in simple cases ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_tag">C.2.1</span></a>), we display <math alttext="-\log p_{\sigma}" class="ltx_Math" display="inline" id="A3.SS2.SSS2.p1.1.m1.1"><semantics id="A3.SS2.SSS2.p1.1.m1.1a"><mrow id="A3.SS2.SSS2.p1.1.m1.1.1" xref="A3.SS2.SSS2.p1.1.m1.1.1.cmml"><mo id="A3.SS2.SSS2.p1.1.m1.1.1a" rspace="0.167em" xref="A3.SS2.SSS2.p1.1.m1.1.1.cmml">−</mo><mrow id="A3.SS2.SSS2.p1.1.m1.1.1.2" xref="A3.SS2.SSS2.p1.1.m1.1.1.2.cmml"><mi id="A3.SS2.SSS2.p1.1.m1.1.1.2.1" xref="A3.SS2.SSS2.p1.1.m1.1.1.2.1.cmml">log</mi><mo id="A3.SS2.SSS2.p1.1.m1.1.1.2a" lspace="0.167em" xref="A3.SS2.SSS2.p1.1.m1.1.1.2.cmml">⁡</mo><msub id="A3.SS2.SSS2.p1.1.m1.1.1.2.2" xref="A3.SS2.SSS2.p1.1.m1.1.1.2.2.cmml"><mi id="A3.SS2.SSS2.p1.1.m1.1.1.2.2.2" xref="A3.SS2.SSS2.p1.1.m1.1.1.2.2.2.cmml">p</mi><mi id="A3.SS2.SSS2.p1.1.m1.1.1.2.2.3" xref="A3.SS2.SSS2.p1.1.m1.1.1.2.2.3.cmml">σ</mi></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="A3.SS2.SSS2.p1.1.m1.1b"><apply id="A3.SS2.SSS2.p1.1.m1.1.1.cmml" xref="A3.SS2.SSS2.p1.1.m1.1.1"><minus id="A3.SS2.SSS2.p1.1.m1.1.1.1.cmml" xref="A3.SS2.SSS2.p1.1.m1.1.1"></minus><apply id="A3.SS2.SSS2.p1.1.m1.1.1.2.cmml" xref="A3.SS2.SSS2.p1.1.m1.1.1.2"><log id="A3.SS2.SSS2.p1.1.m1.1.1.2.1.cmml" xref="A3.SS2.SSS2.p1.1.m1.1.1.2.1"></log><apply id="A3.SS2.SSS2.p1.1.m1.1.1.2.2.cmml" xref="A3.SS2.SSS2.p1.1.m1.1.1.2.2"><csymbol cd="ambiguous" id="A3.SS2.SSS2.p1.1.m1.1.1.2.2.1.cmml" xref="A3.SS2.SSS2.p1.1.m1.1.1.2.2">subscript</csymbol><ci id="A3.SS2.SSS2.p1.1.m1.1.1.2.2.2.cmml" xref="A3.SS2.SSS2.p1.1.m1.1.1.2.2.2">𝑝</ci><ci id="A3.SS2.SSS2.p1.1.m1.1.1.2.2.3.cmml" xref="A3.SS2.SSS2.p1.1.m1.1.1.2.2.3">𝜎</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.SS2.SSS2.p1.1.m1.1c">-\log p_{\sigma}</annotation><annotation encoding="application/x-llamapun" id="A3.SS2.SSS2.p1.1.m1.1d">- roman_log italic_p start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT</annotation></semantics></math>. In order to show <math alttext="\mathcal{R}_{\sigma}" class="ltx_Math" display="inline" id="A3.SS2.SSS2.p1.2.m2.1"><semantics id="A3.SS2.SSS2.p1.2.m2.1a"><msub id="A3.SS2.SSS2.p1.2.m2.1.1" xref="A3.SS2.SSS2.p1.2.m2.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="A3.SS2.SSS2.p1.2.m2.1.1.2" xref="A3.SS2.SSS2.p1.2.m2.1.1.2.cmml">ℛ</mi><mi id="A3.SS2.SSS2.p1.2.m2.1.1.3" xref="A3.SS2.SSS2.p1.2.m2.1.1.3.cmml">σ</mi></msub><annotation-xml encoding="MathML-Content" id="A3.SS2.SSS2.p1.2.m2.1b"><apply id="A3.SS2.SSS2.p1.2.m2.1.1.cmml" xref="A3.SS2.SSS2.p1.2.m2.1.1"><csymbol cd="ambiguous" id="A3.SS2.SSS2.p1.2.m2.1.1.1.cmml" xref="A3.SS2.SSS2.p1.2.m2.1.1">subscript</csymbol><ci id="A3.SS2.SSS2.p1.2.m2.1.1.2.cmml" xref="A3.SS2.SSS2.p1.2.m2.1.1.2">ℛ</ci><ci id="A3.SS2.SSS2.p1.2.m2.1.1.3.cmml" xref="A3.SS2.SSS2.p1.2.m2.1.1.3">𝜎</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.SS2.SSS2.p1.2.m2.1c">\mathcal{R}_{\sigma}</annotation><annotation encoding="application/x-llamapun" id="A3.SS2.SSS2.p1.2.m2.1d">caligraphic_R start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT</annotation></semantics></math>, we use Equation (<a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#S2.E5" title="5 ‣ 2.2 SNORE regularization ‣ 2 Stochastic deNOising REgularization (SNORE) ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_tag">5</span></a>) and the Euler method to approximate the integration on <math alttext="{\tilde{\mathbf{x}}}" class="ltx_Math" display="inline" id="A3.SS2.SSS2.p1.3.m3.1"><semantics id="A3.SS2.SSS2.p1.3.m3.1a"><mover accent="true" id="A3.SS2.SSS2.p1.3.m3.1.1" xref="A3.SS2.SSS2.p1.3.m3.1.1.cmml"><mi id="A3.SS2.SSS2.p1.3.m3.1.1.2" xref="A3.SS2.SSS2.p1.3.m3.1.1.2.cmml">𝐱</mi><mo id="A3.SS2.SSS2.p1.3.m3.1.1.1" xref="A3.SS2.SSS2.p1.3.m3.1.1.1.cmml">~</mo></mover><annotation-xml encoding="MathML-Content" id="A3.SS2.SSS2.p1.3.m3.1b"><apply id="A3.SS2.SSS2.p1.3.m3.1.1.cmml" xref="A3.SS2.SSS2.p1.3.m3.1.1"><ci id="A3.SS2.SSS2.p1.3.m3.1.1.1.cmml" xref="A3.SS2.SSS2.p1.3.m3.1.1.1">~</ci><ci id="A3.SS2.SSS2.p1.3.m3.1.1.2.cmml" xref="A3.SS2.SSS2.p1.3.m3.1.1.2">𝐱</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.SS2.SSS2.p1.3.m3.1c">{\tilde{\mathbf{x}}}</annotation><annotation encoding="application/x-llamapun" id="A3.SS2.SSS2.p1.3.m3.1d">over~ start_ARG bold_x end_ARG</annotation></semantics></math>.</p> </div> <figure class="ltx_figure" id="A3.F4"><img alt="Refer to caption" class="ltx_graphics ltx_centering ltx_img_landscape" height="117" id="A3.F4.g1" src="extracted/5518834/images/regularization_values.png" width="598"/> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure"><span class="ltx_text" id="A3.F4.12.5.1" style="font-size:90%;">Figure 4</span>: </span><span class="ltx_text ltx_font_italic" id="A3.F4.8.4" style="font-size:90%;">Leftmost:<span class="ltx_text ltx_font_upright" id="A3.F4.5.1.1"> Score of the prior <math alttext="p" class="ltx_Math" display="inline" id="A3.F4.5.1.1.m1.1"><semantics id="A3.F4.5.1.1.m1.1b"><mi id="A3.F4.5.1.1.m1.1.1" xref="A3.F4.5.1.1.m1.1.1.cmml">p</mi><annotation-xml encoding="MathML-Content" id="A3.F4.5.1.1.m1.1c"><ci id="A3.F4.5.1.1.m1.1.1.cmml" xref="A3.F4.5.1.1.m1.1.1">𝑝</ci></annotation-xml><annotation encoding="application/x-tex" id="A3.F4.5.1.1.m1.1d">p</annotation><annotation encoding="application/x-llamapun" id="A3.F4.5.1.1.m1.1e">italic_p</annotation></semantics></math>. </span>Rightmost:<span class="ltx_text ltx_font_upright" id="A3.F4.8.4.4"> Values of <math alttext="-\log p_{\sigma}" class="ltx_Math" display="inline" id="A3.F4.6.2.2.m1.1"><semantics id="A3.F4.6.2.2.m1.1b"><mrow id="A3.F4.6.2.2.m1.1.1" xref="A3.F4.6.2.2.m1.1.1.cmml"><mo id="A3.F4.6.2.2.m1.1.1b" mathvariant="normal" rspace="0.167em" xref="A3.F4.6.2.2.m1.1.1.cmml">−</mo><mrow id="A3.F4.6.2.2.m1.1.1.2" xref="A3.F4.6.2.2.m1.1.1.2.cmml"><mi id="A3.F4.6.2.2.m1.1.1.2.1" mathvariant="normal" xref="A3.F4.6.2.2.m1.1.1.2.1.cmml">log</mi><mo id="A3.F4.6.2.2.m1.1.1.2b" lspace="0.167em" mathvariant="normal" xref="A3.F4.6.2.2.m1.1.1.2.cmml">⁡</mo><msub id="A3.F4.6.2.2.m1.1.1.2.2" xref="A3.F4.6.2.2.m1.1.1.2.2.cmml"><mi id="A3.F4.6.2.2.m1.1.1.2.2.2" xref="A3.F4.6.2.2.m1.1.1.2.2.2.cmml">p</mi><mi id="A3.F4.6.2.2.m1.1.1.2.2.3" xref="A3.F4.6.2.2.m1.1.1.2.2.3.cmml">σ</mi></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="A3.F4.6.2.2.m1.1c"><apply id="A3.F4.6.2.2.m1.1.1.cmml" xref="A3.F4.6.2.2.m1.1.1"><minus id="A3.F4.6.2.2.m1.1.1.1.cmml" xref="A3.F4.6.2.2.m1.1.1"></minus><apply id="A3.F4.6.2.2.m1.1.1.2.cmml" xref="A3.F4.6.2.2.m1.1.1.2"><log id="A3.F4.6.2.2.m1.1.1.2.1.cmml" xref="A3.F4.6.2.2.m1.1.1.2.1"></log><apply id="A3.F4.6.2.2.m1.1.1.2.2.cmml" xref="A3.F4.6.2.2.m1.1.1.2.2"><csymbol cd="ambiguous" id="A3.F4.6.2.2.m1.1.1.2.2.1.cmml" xref="A3.F4.6.2.2.m1.1.1.2.2">subscript</csymbol><ci id="A3.F4.6.2.2.m1.1.1.2.2.2.cmml" xref="A3.F4.6.2.2.m1.1.1.2.2.2">𝑝</ci><ci id="A3.F4.6.2.2.m1.1.1.2.2.3.cmml" xref="A3.F4.6.2.2.m1.1.1.2.2.3">𝜎</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.F4.6.2.2.m1.1d">-\log p_{\sigma}</annotation><annotation encoding="application/x-llamapun" id="A3.F4.6.2.2.m1.1e">- roman_log italic_p start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="\mathcal{R}_{\sigma}" class="ltx_Math" display="inline" id="A3.F4.7.3.3.m2.1"><semantics id="A3.F4.7.3.3.m2.1b"><msub id="A3.F4.7.3.3.m2.1.1" xref="A3.F4.7.3.3.m2.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="A3.F4.7.3.3.m2.1.1.2" xref="A3.F4.7.3.3.m2.1.1.2.cmml">ℛ</mi><mi id="A3.F4.7.3.3.m2.1.1.3" xref="A3.F4.7.3.3.m2.1.1.3.cmml">σ</mi></msub><annotation-xml encoding="MathML-Content" id="A3.F4.7.3.3.m2.1c"><apply id="A3.F4.7.3.3.m2.1.1.cmml" xref="A3.F4.7.3.3.m2.1.1"><csymbol cd="ambiguous" id="A3.F4.7.3.3.m2.1.1.1.cmml" xref="A3.F4.7.3.3.m2.1.1">subscript</csymbol><ci id="A3.F4.7.3.3.m2.1.1.2.cmml" xref="A3.F4.7.3.3.m2.1.1.2">ℛ</ci><ci id="A3.F4.7.3.3.m2.1.1.3.cmml" xref="A3.F4.7.3.3.m2.1.1.3">𝜎</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.F4.7.3.3.m2.1d">\mathcal{R}_{\sigma}</annotation><annotation encoding="application/x-llamapun" id="A3.F4.7.3.3.m2.1e">caligraphic_R start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT</annotation></semantics></math> for five values of <math alttext="\sigma>0" class="ltx_Math" display="inline" id="A3.F4.8.4.4.m3.1"><semantics id="A3.F4.8.4.4.m3.1b"><mrow id="A3.F4.8.4.4.m3.1.1" xref="A3.F4.8.4.4.m3.1.1.cmml"><mi id="A3.F4.8.4.4.m3.1.1.2" xref="A3.F4.8.4.4.m3.1.1.2.cmml">σ</mi><mo id="A3.F4.8.4.4.m3.1.1.1" mathvariant="normal" xref="A3.F4.8.4.4.m3.1.1.1.cmml">></mo><mn id="A3.F4.8.4.4.m3.1.1.3" mathvariant="normal" xref="A3.F4.8.4.4.m3.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="A3.F4.8.4.4.m3.1c"><apply id="A3.F4.8.4.4.m3.1.1.cmml" xref="A3.F4.8.4.4.m3.1.1"><gt id="A3.F4.8.4.4.m3.1.1.1.cmml" xref="A3.F4.8.4.4.m3.1.1.1"></gt><ci id="A3.F4.8.4.4.m3.1.1.2.cmml" xref="A3.F4.8.4.4.m3.1.1.2">𝜎</ci><cn id="A3.F4.8.4.4.m3.1.1.3.cmml" type="integer" xref="A3.F4.8.4.4.m3.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.F4.8.4.4.m3.1d">\sigma>0</annotation><annotation encoding="application/x-llamapun" id="A3.F4.8.4.4.m3.1e">italic_σ > 0</annotation></semantics></math>.</span></span></figcaption> </figure> <div class="ltx_para" id="A3.SS2.SSS2.p2"> <p class="ltx_p" id="A3.SS2.SSS2.p2.10">First, we observe that both <math alttext="-\log p_{\sigma}" class="ltx_Math" display="inline" id="A3.SS2.SSS2.p2.1.m1.1"><semantics id="A3.SS2.SSS2.p2.1.m1.1a"><mrow id="A3.SS2.SSS2.p2.1.m1.1.1" xref="A3.SS2.SSS2.p2.1.m1.1.1.cmml"><mo id="A3.SS2.SSS2.p2.1.m1.1.1a" rspace="0.167em" xref="A3.SS2.SSS2.p2.1.m1.1.1.cmml">−</mo><mrow id="A3.SS2.SSS2.p2.1.m1.1.1.2" xref="A3.SS2.SSS2.p2.1.m1.1.1.2.cmml"><mi id="A3.SS2.SSS2.p2.1.m1.1.1.2.1" xref="A3.SS2.SSS2.p2.1.m1.1.1.2.1.cmml">log</mi><mo id="A3.SS2.SSS2.p2.1.m1.1.1.2a" lspace="0.167em" xref="A3.SS2.SSS2.p2.1.m1.1.1.2.cmml">⁡</mo><msub id="A3.SS2.SSS2.p2.1.m1.1.1.2.2" xref="A3.SS2.SSS2.p2.1.m1.1.1.2.2.cmml"><mi id="A3.SS2.SSS2.p2.1.m1.1.1.2.2.2" xref="A3.SS2.SSS2.p2.1.m1.1.1.2.2.2.cmml">p</mi><mi id="A3.SS2.SSS2.p2.1.m1.1.1.2.2.3" xref="A3.SS2.SSS2.p2.1.m1.1.1.2.2.3.cmml">σ</mi></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="A3.SS2.SSS2.p2.1.m1.1b"><apply id="A3.SS2.SSS2.p2.1.m1.1.1.cmml" xref="A3.SS2.SSS2.p2.1.m1.1.1"><minus id="A3.SS2.SSS2.p2.1.m1.1.1.1.cmml" xref="A3.SS2.SSS2.p2.1.m1.1.1"></minus><apply id="A3.SS2.SSS2.p2.1.m1.1.1.2.cmml" xref="A3.SS2.SSS2.p2.1.m1.1.1.2"><log id="A3.SS2.SSS2.p2.1.m1.1.1.2.1.cmml" xref="A3.SS2.SSS2.p2.1.m1.1.1.2.1"></log><apply id="A3.SS2.SSS2.p2.1.m1.1.1.2.2.cmml" xref="A3.SS2.SSS2.p2.1.m1.1.1.2.2"><csymbol cd="ambiguous" id="A3.SS2.SSS2.p2.1.m1.1.1.2.2.1.cmml" xref="A3.SS2.SSS2.p2.1.m1.1.1.2.2">subscript</csymbol><ci id="A3.SS2.SSS2.p2.1.m1.1.1.2.2.2.cmml" xref="A3.SS2.SSS2.p2.1.m1.1.1.2.2.2">𝑝</ci><ci id="A3.SS2.SSS2.p2.1.m1.1.1.2.2.3.cmml" xref="A3.SS2.SSS2.p2.1.m1.1.1.2.2.3">𝜎</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.SS2.SSS2.p2.1.m1.1c">-\log p_{\sigma}</annotation><annotation encoding="application/x-llamapun" id="A3.SS2.SSS2.p2.1.m1.1d">- roman_log italic_p start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="\mathcal{R}_{\sigma}" class="ltx_Math" display="inline" id="A3.SS2.SSS2.p2.2.m2.1"><semantics id="A3.SS2.SSS2.p2.2.m2.1a"><msub id="A3.SS2.SSS2.p2.2.m2.1.1" xref="A3.SS2.SSS2.p2.2.m2.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="A3.SS2.SSS2.p2.2.m2.1.1.2" xref="A3.SS2.SSS2.p2.2.m2.1.1.2.cmml">ℛ</mi><mi id="A3.SS2.SSS2.p2.2.m2.1.1.3" xref="A3.SS2.SSS2.p2.2.m2.1.1.3.cmml">σ</mi></msub><annotation-xml encoding="MathML-Content" id="A3.SS2.SSS2.p2.2.m2.1b"><apply id="A3.SS2.SSS2.p2.2.m2.1.1.cmml" xref="A3.SS2.SSS2.p2.2.m2.1.1"><csymbol cd="ambiguous" id="A3.SS2.SSS2.p2.2.m2.1.1.1.cmml" xref="A3.SS2.SSS2.p2.2.m2.1.1">subscript</csymbol><ci id="A3.SS2.SSS2.p2.2.m2.1.1.2.cmml" xref="A3.SS2.SSS2.p2.2.m2.1.1.2">ℛ</ci><ci id="A3.SS2.SSS2.p2.2.m2.1.1.3.cmml" xref="A3.SS2.SSS2.p2.2.m2.1.1.3">𝜎</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.SS2.SSS2.p2.2.m2.1c">\mathcal{R}_{\sigma}</annotation><annotation encoding="application/x-llamapun" id="A3.SS2.SSS2.p2.2.m2.1d">caligraphic_R start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT</annotation></semantics></math> converge to <math alttext="-\log p" class="ltx_Math" display="inline" id="A3.SS2.SSS2.p2.3.m3.1"><semantics id="A3.SS2.SSS2.p2.3.m3.1a"><mrow id="A3.SS2.SSS2.p2.3.m3.1.1" xref="A3.SS2.SSS2.p2.3.m3.1.1.cmml"><mo id="A3.SS2.SSS2.p2.3.m3.1.1a" rspace="0.167em" xref="A3.SS2.SSS2.p2.3.m3.1.1.cmml">−</mo><mrow id="A3.SS2.SSS2.p2.3.m3.1.1.2" xref="A3.SS2.SSS2.p2.3.m3.1.1.2.cmml"><mi id="A3.SS2.SSS2.p2.3.m3.1.1.2.1" xref="A3.SS2.SSS2.p2.3.m3.1.1.2.1.cmml">log</mi><mo id="A3.SS2.SSS2.p2.3.m3.1.1.2a" lspace="0.167em" xref="A3.SS2.SSS2.p2.3.m3.1.1.2.cmml">⁡</mo><mi id="A3.SS2.SSS2.p2.3.m3.1.1.2.2" xref="A3.SS2.SSS2.p2.3.m3.1.1.2.2.cmml">p</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="A3.SS2.SSS2.p2.3.m3.1b"><apply id="A3.SS2.SSS2.p2.3.m3.1.1.cmml" xref="A3.SS2.SSS2.p2.3.m3.1.1"><minus id="A3.SS2.SSS2.p2.3.m3.1.1.1.cmml" xref="A3.SS2.SSS2.p2.3.m3.1.1"></minus><apply id="A3.SS2.SSS2.p2.3.m3.1.1.2.cmml" xref="A3.SS2.SSS2.p2.3.m3.1.1.2"><log id="A3.SS2.SSS2.p2.3.m3.1.1.2.1.cmml" xref="A3.SS2.SSS2.p2.3.m3.1.1.2.1"></log><ci id="A3.SS2.SSS2.p2.3.m3.1.1.2.2.cmml" xref="A3.SS2.SSS2.p2.3.m3.1.1.2.2">𝑝</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.SS2.SSS2.p2.3.m3.1c">-\log p</annotation><annotation encoding="application/x-llamapun" id="A3.SS2.SSS2.p2.3.m3.1d">- roman_log italic_p</annotation></semantics></math> when <math alttext="\sigma\to 0" class="ltx_Math" display="inline" id="A3.SS2.SSS2.p2.4.m4.1"><semantics id="A3.SS2.SSS2.p2.4.m4.1a"><mrow id="A3.SS2.SSS2.p2.4.m4.1.1" xref="A3.SS2.SSS2.p2.4.m4.1.1.cmml"><mi id="A3.SS2.SSS2.p2.4.m4.1.1.2" xref="A3.SS2.SSS2.p2.4.m4.1.1.2.cmml">σ</mi><mo id="A3.SS2.SSS2.p2.4.m4.1.1.1" stretchy="false" xref="A3.SS2.SSS2.p2.4.m4.1.1.1.cmml">→</mo><mn id="A3.SS2.SSS2.p2.4.m4.1.1.3" xref="A3.SS2.SSS2.p2.4.m4.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="A3.SS2.SSS2.p2.4.m4.1b"><apply id="A3.SS2.SSS2.p2.4.m4.1.1.cmml" xref="A3.SS2.SSS2.p2.4.m4.1.1"><ci id="A3.SS2.SSS2.p2.4.m4.1.1.1.cmml" xref="A3.SS2.SSS2.p2.4.m4.1.1.1">→</ci><ci id="A3.SS2.SSS2.p2.4.m4.1.1.2.cmml" xref="A3.SS2.SSS2.p2.4.m4.1.1.2">𝜎</ci><cn id="A3.SS2.SSS2.p2.4.m4.1.1.3.cmml" type="integer" xref="A3.SS2.SSS2.p2.4.m4.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.SS2.SSS2.p2.4.m4.1c">\sigma\to 0</annotation><annotation encoding="application/x-llamapun" id="A3.SS2.SSS2.p2.4.m4.1d">italic_σ → 0</annotation></semantics></math>. We also see that <math alttext="\mathcal{R}_{\sigma}" class="ltx_Math" display="inline" id="A3.SS2.SSS2.p2.5.m5.1"><semantics id="A3.SS2.SSS2.p2.5.m5.1a"><msub id="A3.SS2.SSS2.p2.5.m5.1.1" xref="A3.SS2.SSS2.p2.5.m5.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="A3.SS2.SSS2.p2.5.m5.1.1.2" xref="A3.SS2.SSS2.p2.5.m5.1.1.2.cmml">ℛ</mi><mi id="A3.SS2.SSS2.p2.5.m5.1.1.3" xref="A3.SS2.SSS2.p2.5.m5.1.1.3.cmml">σ</mi></msub><annotation-xml encoding="MathML-Content" id="A3.SS2.SSS2.p2.5.m5.1b"><apply id="A3.SS2.SSS2.p2.5.m5.1.1.cmml" xref="A3.SS2.SSS2.p2.5.m5.1.1"><csymbol cd="ambiguous" id="A3.SS2.SSS2.p2.5.m5.1.1.1.cmml" xref="A3.SS2.SSS2.p2.5.m5.1.1">subscript</csymbol><ci id="A3.SS2.SSS2.p2.5.m5.1.1.2.cmml" xref="A3.SS2.SSS2.p2.5.m5.1.1.2">ℛ</ci><ci id="A3.SS2.SSS2.p2.5.m5.1.1.3.cmml" xref="A3.SS2.SSS2.p2.5.m5.1.1.3">𝜎</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.SS2.SSS2.p2.5.m5.1c">\mathcal{R}_{\sigma}</annotation><annotation encoding="application/x-llamapun" id="A3.SS2.SSS2.p2.5.m5.1d">caligraphic_R start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT</annotation></semantics></math> seems to converge to <math alttext="-\log p" class="ltx_Math" display="inline" id="A3.SS2.SSS2.p2.6.m6.1"><semantics id="A3.SS2.SSS2.p2.6.m6.1a"><mrow id="A3.SS2.SSS2.p2.6.m6.1.1" xref="A3.SS2.SSS2.p2.6.m6.1.1.cmml"><mo id="A3.SS2.SSS2.p2.6.m6.1.1a" rspace="0.167em" xref="A3.SS2.SSS2.p2.6.m6.1.1.cmml">−</mo><mrow id="A3.SS2.SSS2.p2.6.m6.1.1.2" xref="A3.SS2.SSS2.p2.6.m6.1.1.2.cmml"><mi id="A3.SS2.SSS2.p2.6.m6.1.1.2.1" xref="A3.SS2.SSS2.p2.6.m6.1.1.2.1.cmml">log</mi><mo id="A3.SS2.SSS2.p2.6.m6.1.1.2a" lspace="0.167em" xref="A3.SS2.SSS2.p2.6.m6.1.1.2.cmml">⁡</mo><mi id="A3.SS2.SSS2.p2.6.m6.1.1.2.2" xref="A3.SS2.SSS2.p2.6.m6.1.1.2.2.cmml">p</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="A3.SS2.SSS2.p2.6.m6.1b"><apply id="A3.SS2.SSS2.p2.6.m6.1.1.cmml" xref="A3.SS2.SSS2.p2.6.m6.1.1"><minus id="A3.SS2.SSS2.p2.6.m6.1.1.1.cmml" xref="A3.SS2.SSS2.p2.6.m6.1.1"></minus><apply id="A3.SS2.SSS2.p2.6.m6.1.1.2.cmml" xref="A3.SS2.SSS2.p2.6.m6.1.1.2"><log id="A3.SS2.SSS2.p2.6.m6.1.1.2.1.cmml" xref="A3.SS2.SSS2.p2.6.m6.1.1.2.1"></log><ci id="A3.SS2.SSS2.p2.6.m6.1.1.2.2.cmml" xref="A3.SS2.SSS2.p2.6.m6.1.1.2.2">𝑝</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.SS2.SSS2.p2.6.m6.1c">-\log p</annotation><annotation encoding="application/x-llamapun" id="A3.SS2.SSS2.p2.6.m6.1d">- roman_log italic_p</annotation></semantics></math> more slowly that <math alttext="-\log p_{\sigma}" class="ltx_Math" display="inline" id="A3.SS2.SSS2.p2.7.m7.1"><semantics id="A3.SS2.SSS2.p2.7.m7.1a"><mrow id="A3.SS2.SSS2.p2.7.m7.1.1" xref="A3.SS2.SSS2.p2.7.m7.1.1.cmml"><mo id="A3.SS2.SSS2.p2.7.m7.1.1a" rspace="0.167em" xref="A3.SS2.SSS2.p2.7.m7.1.1.cmml">−</mo><mrow id="A3.SS2.SSS2.p2.7.m7.1.1.2" xref="A3.SS2.SSS2.p2.7.m7.1.1.2.cmml"><mi id="A3.SS2.SSS2.p2.7.m7.1.1.2.1" xref="A3.SS2.SSS2.p2.7.m7.1.1.2.1.cmml">log</mi><mo id="A3.SS2.SSS2.p2.7.m7.1.1.2a" lspace="0.167em" xref="A3.SS2.SSS2.p2.7.m7.1.1.2.cmml">⁡</mo><msub id="A3.SS2.SSS2.p2.7.m7.1.1.2.2" xref="A3.SS2.SSS2.p2.7.m7.1.1.2.2.cmml"><mi id="A3.SS2.SSS2.p2.7.m7.1.1.2.2.2" xref="A3.SS2.SSS2.p2.7.m7.1.1.2.2.2.cmml">p</mi><mi id="A3.SS2.SSS2.p2.7.m7.1.1.2.2.3" xref="A3.SS2.SSS2.p2.7.m7.1.1.2.2.3.cmml">σ</mi></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="A3.SS2.SSS2.p2.7.m7.1b"><apply id="A3.SS2.SSS2.p2.7.m7.1.1.cmml" xref="A3.SS2.SSS2.p2.7.m7.1.1"><minus id="A3.SS2.SSS2.p2.7.m7.1.1.1.cmml" xref="A3.SS2.SSS2.p2.7.m7.1.1"></minus><apply id="A3.SS2.SSS2.p2.7.m7.1.1.2.cmml" xref="A3.SS2.SSS2.p2.7.m7.1.1.2"><log id="A3.SS2.SSS2.p2.7.m7.1.1.2.1.cmml" xref="A3.SS2.SSS2.p2.7.m7.1.1.2.1"></log><apply id="A3.SS2.SSS2.p2.7.m7.1.1.2.2.cmml" xref="A3.SS2.SSS2.p2.7.m7.1.1.2.2"><csymbol cd="ambiguous" id="A3.SS2.SSS2.p2.7.m7.1.1.2.2.1.cmml" xref="A3.SS2.SSS2.p2.7.m7.1.1.2.2">subscript</csymbol><ci id="A3.SS2.SSS2.p2.7.m7.1.1.2.2.2.cmml" xref="A3.SS2.SSS2.p2.7.m7.1.1.2.2.2">𝑝</ci><ci id="A3.SS2.SSS2.p2.7.m7.1.1.2.2.3.cmml" xref="A3.SS2.SSS2.p2.7.m7.1.1.2.2.3">𝜎</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.SS2.SSS2.p2.7.m7.1c">-\log p_{\sigma}</annotation><annotation encoding="application/x-llamapun" id="A3.SS2.SSS2.p2.7.m7.1d">- roman_log italic_p start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT</annotation></semantics></math> as suggested by Proposition <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#A3.Thmproposition1" title="Proposition C.1. ‣ C.2.1 Theoretical considerations ‣ C.2 Gaussian Mixture prior ‣ Appendix C SNORE in simple cases ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_tag">C.1</span></a> and Proposition <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#A3.Thmproposition2" title="Proposition C.2. ‣ C.2.1 Theoretical considerations ‣ C.2 Gaussian Mixture prior ‣ Appendix C SNORE in simple cases ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_tag">C.2</span></a>. Finally, <math alttext="\mathcal{R}_{\sigma}" class="ltx_Math" display="inline" id="A3.SS2.SSS2.p2.8.m8.1"><semantics id="A3.SS2.SSS2.p2.8.m8.1a"><msub id="A3.SS2.SSS2.p2.8.m8.1.1" xref="A3.SS2.SSS2.p2.8.m8.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="A3.SS2.SSS2.p2.8.m8.1.1.2" xref="A3.SS2.SSS2.p2.8.m8.1.1.2.cmml">ℛ</mi><mi id="A3.SS2.SSS2.p2.8.m8.1.1.3" xref="A3.SS2.SSS2.p2.8.m8.1.1.3.cmml">σ</mi></msub><annotation-xml encoding="MathML-Content" id="A3.SS2.SSS2.p2.8.m8.1b"><apply id="A3.SS2.SSS2.p2.8.m8.1.1.cmml" xref="A3.SS2.SSS2.p2.8.m8.1.1"><csymbol cd="ambiguous" id="A3.SS2.SSS2.p2.8.m8.1.1.1.cmml" xref="A3.SS2.SSS2.p2.8.m8.1.1">subscript</csymbol><ci id="A3.SS2.SSS2.p2.8.m8.1.1.2.cmml" xref="A3.SS2.SSS2.p2.8.m8.1.1.2">ℛ</ci><ci id="A3.SS2.SSS2.p2.8.m8.1.1.3.cmml" xref="A3.SS2.SSS2.p2.8.m8.1.1.3">𝜎</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.SS2.SSS2.p2.8.m8.1c">\mathcal{R}_{\sigma}</annotation><annotation encoding="application/x-llamapun" id="A3.SS2.SSS2.p2.8.m8.1d">caligraphic_R start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT</annotation></semantics></math> seems to be ”more convex” so easier to minimize and it has a similar minimum that <math alttext="-\log p_{\sigma}" class="ltx_Math" display="inline" id="A3.SS2.SSS2.p2.9.m9.1"><semantics id="A3.SS2.SSS2.p2.9.m9.1a"><mrow id="A3.SS2.SSS2.p2.9.m9.1.1" xref="A3.SS2.SSS2.p2.9.m9.1.1.cmml"><mo id="A3.SS2.SSS2.p2.9.m9.1.1a" rspace="0.167em" xref="A3.SS2.SSS2.p2.9.m9.1.1.cmml">−</mo><mrow id="A3.SS2.SSS2.p2.9.m9.1.1.2" xref="A3.SS2.SSS2.p2.9.m9.1.1.2.cmml"><mi id="A3.SS2.SSS2.p2.9.m9.1.1.2.1" xref="A3.SS2.SSS2.p2.9.m9.1.1.2.1.cmml">log</mi><mo id="A3.SS2.SSS2.p2.9.m9.1.1.2a" lspace="0.167em" xref="A3.SS2.SSS2.p2.9.m9.1.1.2.cmml">⁡</mo><msub id="A3.SS2.SSS2.p2.9.m9.1.1.2.2" xref="A3.SS2.SSS2.p2.9.m9.1.1.2.2.cmml"><mi id="A3.SS2.SSS2.p2.9.m9.1.1.2.2.2" xref="A3.SS2.SSS2.p2.9.m9.1.1.2.2.2.cmml">p</mi><mi id="A3.SS2.SSS2.p2.9.m9.1.1.2.2.3" xref="A3.SS2.SSS2.p2.9.m9.1.1.2.2.3.cmml">σ</mi></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="A3.SS2.SSS2.p2.9.m9.1b"><apply id="A3.SS2.SSS2.p2.9.m9.1.1.cmml" xref="A3.SS2.SSS2.p2.9.m9.1.1"><minus id="A3.SS2.SSS2.p2.9.m9.1.1.1.cmml" xref="A3.SS2.SSS2.p2.9.m9.1.1"></minus><apply id="A3.SS2.SSS2.p2.9.m9.1.1.2.cmml" xref="A3.SS2.SSS2.p2.9.m9.1.1.2"><log id="A3.SS2.SSS2.p2.9.m9.1.1.2.1.cmml" xref="A3.SS2.SSS2.p2.9.m9.1.1.2.1"></log><apply id="A3.SS2.SSS2.p2.9.m9.1.1.2.2.cmml" xref="A3.SS2.SSS2.p2.9.m9.1.1.2.2"><csymbol cd="ambiguous" id="A3.SS2.SSS2.p2.9.m9.1.1.2.2.1.cmml" xref="A3.SS2.SSS2.p2.9.m9.1.1.2.2">subscript</csymbol><ci id="A3.SS2.SSS2.p2.9.m9.1.1.2.2.2.cmml" xref="A3.SS2.SSS2.p2.9.m9.1.1.2.2.2">𝑝</ci><ci id="A3.SS2.SSS2.p2.9.m9.1.1.2.2.3.cmml" xref="A3.SS2.SSS2.p2.9.m9.1.1.2.2.3">𝜎</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.SS2.SSS2.p2.9.m9.1c">-\log p_{\sigma}</annotation><annotation encoding="application/x-llamapun" id="A3.SS2.SSS2.p2.9.m9.1d">- roman_log italic_p start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT</annotation></semantics></math>. This simulation suggests that <math alttext="\mathcal{R}_{\sigma}" class="ltx_Math" display="inline" id="A3.SS2.SSS2.p2.10.m10.1"><semantics id="A3.SS2.SSS2.p2.10.m10.1a"><msub id="A3.SS2.SSS2.p2.10.m10.1.1" xref="A3.SS2.SSS2.p2.10.m10.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="A3.SS2.SSS2.p2.10.m10.1.1.2" xref="A3.SS2.SSS2.p2.10.m10.1.1.2.cmml">ℛ</mi><mi id="A3.SS2.SSS2.p2.10.m10.1.1.3" xref="A3.SS2.SSS2.p2.10.m10.1.1.3.cmml">σ</mi></msub><annotation-xml encoding="MathML-Content" id="A3.SS2.SSS2.p2.10.m10.1b"><apply id="A3.SS2.SSS2.p2.10.m10.1.1.cmml" xref="A3.SS2.SSS2.p2.10.m10.1.1"><csymbol cd="ambiguous" id="A3.SS2.SSS2.p2.10.m10.1.1.1.cmml" xref="A3.SS2.SSS2.p2.10.m10.1.1">subscript</csymbol><ci id="A3.SS2.SSS2.p2.10.m10.1.1.2.cmml" xref="A3.SS2.SSS2.p2.10.m10.1.1.2">ℛ</ci><ci id="A3.SS2.SSS2.p2.10.m10.1.1.3.cmml" xref="A3.SS2.SSS2.p2.10.m10.1.1.3">𝜎</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.SS2.SSS2.p2.10.m10.1c">\mathcal{R}_{\sigma}</annotation><annotation encoding="application/x-llamapun" id="A3.SS2.SSS2.p2.10.m10.1d">caligraphic_R start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT</annotation></semantics></math> is an easier potential to minimize.</p> </div> </section> </section> </section> <section class="ltx_appendix" id="A4"> <h2 class="ltx_title ltx_title_appendix"> <span class="ltx_tag ltx_tag_appendix">Appendix D </span>Proofs of section <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#S3" title="3 Convergence Analysis ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_tag">3</span></a> </h2> <section class="ltx_subsection" id="A4.SS1"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">D.1 </span>Proof of critical Point Analysis</h3> <div class="ltx_para" id="A4.SS1.p1"> <p class="ltx_p" id="A4.SS1.p1.1">First, we recall a part of the proof of <cite class="ltx_cite ltx_citemacro_citep">(Laumont et al., <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#bib.bib37" title="">2022b</a>, Proposition 1)</cite> where it has been proved under some assumptions on the prior distribution that the posterior score approximation converges uniformly to the posterior score on every compact of <math alttext="{\mathbb{R}}^{d}" class="ltx_Math" display="inline" id="A4.SS1.p1.1.m1.1"><semantics id="A4.SS1.p1.1.m1.1a"><msup id="A4.SS1.p1.1.m1.1.1" xref="A4.SS1.p1.1.m1.1.1.cmml"><mi id="A4.SS1.p1.1.m1.1.1.2" xref="A4.SS1.p1.1.m1.1.1.2.cmml">ℝ</mi><mi id="A4.SS1.p1.1.m1.1.1.3" xref="A4.SS1.p1.1.m1.1.1.3.cmml">d</mi></msup><annotation-xml encoding="MathML-Content" id="A4.SS1.p1.1.m1.1b"><apply id="A4.SS1.p1.1.m1.1.1.cmml" xref="A4.SS1.p1.1.m1.1.1"><csymbol cd="ambiguous" id="A4.SS1.p1.1.m1.1.1.1.cmml" xref="A4.SS1.p1.1.m1.1.1">superscript</csymbol><ci id="A4.SS1.p1.1.m1.1.1.2.cmml" xref="A4.SS1.p1.1.m1.1.1.2">ℝ</ci><ci id="A4.SS1.p1.1.m1.1.1.3.cmml" xref="A4.SS1.p1.1.m1.1.1.3">𝑑</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.SS1.p1.1.m1.1c">{\mathbb{R}}^{d}</annotation><annotation encoding="application/x-llamapun" id="A4.SS1.p1.1.m1.1d">blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT</annotation></semantics></math>.</p> </div> <div class="ltx_theorem ltx_theorem_proposition" id="A4.Thmproposition1"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="A4.Thmproposition1.1.1.1">Proposition D.1</span></span><span class="ltx_text ltx_font_bold" id="A4.Thmproposition1.2.2">.</span> </h6> <div class="ltx_para" id="A4.Thmproposition1.p1"> <p class="ltx_p" id="A4.Thmproposition1.p1.5"><cite class="ltx_cite ltx_citemacro_citep"><span class="ltx_text ltx_font_italic" id="A4.Thmproposition1.p1.5.6.1">(</span>Laumont et al.<span class="ltx_text ltx_font_italic" id="A4.Thmproposition1.p1.5.7.2.1.1">, </span><a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#bib.bib37" title="">2022b</a><span class="ltx_text ltx_font_italic" id="A4.Thmproposition1.p1.5.8.3">, Proposition 1)</span></cite><span class="ltx_text ltx_font_italic" id="A4.Thmproposition1.p1.5.5"> Under Assumption <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#S3.Thmtheorem1" title="Assumption 3.1. ‣ 3.1 Asymptotics of critical points when 𝜎→0 ‣ 3 Convergence Analysis ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_tag">3.1</span></a>, for <math alttext="\mathbf{K}" class="ltx_Math" display="inline" id="A4.Thmproposition1.p1.1.1.m1.1"><semantics id="A4.Thmproposition1.p1.1.1.m1.1a"><mi id="A4.Thmproposition1.p1.1.1.m1.1.1" xref="A4.Thmproposition1.p1.1.1.m1.1.1.cmml">𝐊</mi><annotation-xml encoding="MathML-Content" id="A4.Thmproposition1.p1.1.1.m1.1b"><ci id="A4.Thmproposition1.p1.1.1.m1.1.1.cmml" xref="A4.Thmproposition1.p1.1.1.m1.1.1">𝐊</ci></annotation-xml><annotation encoding="application/x-tex" id="A4.Thmproposition1.p1.1.1.m1.1c">\mathbf{K}</annotation><annotation encoding="application/x-llamapun" id="A4.Thmproposition1.p1.1.1.m1.1d">bold_K</annotation></semantics></math> a compact of <math alttext="{\mathbb{R}}^{d}" class="ltx_Math" display="inline" id="A4.Thmproposition1.p1.2.2.m2.1"><semantics id="A4.Thmproposition1.p1.2.2.m2.1a"><msup id="A4.Thmproposition1.p1.2.2.m2.1.1" xref="A4.Thmproposition1.p1.2.2.m2.1.1.cmml"><mi id="A4.Thmproposition1.p1.2.2.m2.1.1.2" xref="A4.Thmproposition1.p1.2.2.m2.1.1.2.cmml">ℝ</mi><mi id="A4.Thmproposition1.p1.2.2.m2.1.1.3" xref="A4.Thmproposition1.p1.2.2.m2.1.1.3.cmml">d</mi></msup><annotation-xml encoding="MathML-Content" id="A4.Thmproposition1.p1.2.2.m2.1b"><apply id="A4.Thmproposition1.p1.2.2.m2.1.1.cmml" xref="A4.Thmproposition1.p1.2.2.m2.1.1"><csymbol cd="ambiguous" id="A4.Thmproposition1.p1.2.2.m2.1.1.1.cmml" xref="A4.Thmproposition1.p1.2.2.m2.1.1">superscript</csymbol><ci id="A4.Thmproposition1.p1.2.2.m2.1.1.2.cmml" xref="A4.Thmproposition1.p1.2.2.m2.1.1.2">ℝ</ci><ci id="A4.Thmproposition1.p1.2.2.m2.1.1.3.cmml" xref="A4.Thmproposition1.p1.2.2.m2.1.1.3">𝑑</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.Thmproposition1.p1.2.2.m2.1c">{\mathbb{R}}^{d}</annotation><annotation encoding="application/x-llamapun" id="A4.Thmproposition1.p1.2.2.m2.1d">blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT</annotation></semantics></math>, <math alttext="\nabla\log p_{\sigma}(\cdot|{\mathbf{y}})" class="ltx_math_unparsed" display="inline" id="A4.Thmproposition1.p1.3.3.m3.1"><semantics id="A4.Thmproposition1.p1.3.3.m3.1a"><mrow id="A4.Thmproposition1.p1.3.3.m3.1b"><mo id="A4.Thmproposition1.p1.3.3.m3.1.1" mathvariant="normal" rspace="0.167em">∇</mo><mi id="A4.Thmproposition1.p1.3.3.m3.1.2" mathvariant="normal">log</mi><msub id="A4.Thmproposition1.p1.3.3.m3.1.3"><mi id="A4.Thmproposition1.p1.3.3.m3.1.3.2">p</mi><mi id="A4.Thmproposition1.p1.3.3.m3.1.3.3">σ</mi></msub><mrow id="A4.Thmproposition1.p1.3.3.m3.1.4"><mo id="A4.Thmproposition1.p1.3.3.m3.1.4.1" mathvariant="normal" stretchy="false">(</mo><mo id="A4.Thmproposition1.p1.3.3.m3.1.4.2" lspace="0em" mathvariant="normal" rspace="0em">⋅</mo><mo fence="false" id="A4.Thmproposition1.p1.3.3.m3.1.4.3" mathvariant="normal" rspace="0.167em" stretchy="false">|</mo><mi id="A4.Thmproposition1.p1.3.3.m3.1.4.4">𝐲</mi><mo id="A4.Thmproposition1.p1.3.3.m3.1.4.5" mathvariant="normal" stretchy="false">)</mo></mrow></mrow><annotation encoding="application/x-tex" id="A4.Thmproposition1.p1.3.3.m3.1c">\nabla\log p_{\sigma}(\cdot|{\mathbf{y}})</annotation><annotation encoding="application/x-llamapun" id="A4.Thmproposition1.p1.3.3.m3.1d">∇ roman_log italic_p start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ( ⋅ | bold_y )</annotation></semantics></math> converges uniformly to <math alttext="\nabla\log p(\cdot|{\mathbf{y}})" class="ltx_math_unparsed" display="inline" id="A4.Thmproposition1.p1.4.4.m4.1"><semantics id="A4.Thmproposition1.p1.4.4.m4.1a"><mrow id="A4.Thmproposition1.p1.4.4.m4.1b"><mo id="A4.Thmproposition1.p1.4.4.m4.1.1" mathvariant="normal" rspace="0.167em">∇</mo><mi id="A4.Thmproposition1.p1.4.4.m4.1.2" mathvariant="normal">log</mi><mi id="A4.Thmproposition1.p1.4.4.m4.1.3">p</mi><mrow id="A4.Thmproposition1.p1.4.4.m4.1.4"><mo id="A4.Thmproposition1.p1.4.4.m4.1.4.1" mathvariant="normal" stretchy="false">(</mo><mo id="A4.Thmproposition1.p1.4.4.m4.1.4.2" lspace="0em" mathvariant="normal" rspace="0em">⋅</mo><mo fence="false" id="A4.Thmproposition1.p1.4.4.m4.1.4.3" mathvariant="normal" rspace="0.167em" stretchy="false">|</mo><mi id="A4.Thmproposition1.p1.4.4.m4.1.4.4">𝐲</mi><mo id="A4.Thmproposition1.p1.4.4.m4.1.4.5" mathvariant="normal" stretchy="false">)</mo></mrow></mrow><annotation encoding="application/x-tex" id="A4.Thmproposition1.p1.4.4.m4.1c">\nabla\log p(\cdot|{\mathbf{y}})</annotation><annotation encoding="application/x-llamapun" id="A4.Thmproposition1.p1.4.4.m4.1d">∇ roman_log italic_p ( ⋅ | bold_y )</annotation></semantics></math> on <math alttext="\mathbf{K}" class="ltx_Math" display="inline" id="A4.Thmproposition1.p1.5.5.m5.1"><semantics id="A4.Thmproposition1.p1.5.5.m5.1a"><mi id="A4.Thmproposition1.p1.5.5.m5.1.1" xref="A4.Thmproposition1.p1.5.5.m5.1.1.cmml">𝐊</mi><annotation-xml encoding="MathML-Content" id="A4.Thmproposition1.p1.5.5.m5.1b"><ci id="A4.Thmproposition1.p1.5.5.m5.1.1.cmml" xref="A4.Thmproposition1.p1.5.5.m5.1.1">𝐊</ci></annotation-xml><annotation encoding="application/x-tex" id="A4.Thmproposition1.p1.5.5.m5.1c">\mathbf{K}</annotation><annotation encoding="application/x-llamapun" id="A4.Thmproposition1.p1.5.5.m5.1d">bold_K</annotation></semantics></math>,</span></p> <table class="ltx_equation ltx_eqn_table" id="A4.Ex61"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\lim_{\sigma\to 0}\sup_{\mathbf{K}}\|\nabla\log p_{\sigma}(\cdot|{\mathbf{y}})% -\nabla\log p(\cdot|{\mathbf{y}})\|=0." class="ltx_math_unparsed" display="block" id="A4.Ex61.m1.1"><semantics id="A4.Ex61.m1.1a"><mrow id="A4.Ex61.m1.1b"><munder id="A4.Ex61.m1.1.1"><mo id="A4.Ex61.m1.1.1.2" movablelimits="false">lim</mo><mrow id="A4.Ex61.m1.1.1.3"><mi id="A4.Ex61.m1.1.1.3.2">σ</mi><mo id="A4.Ex61.m1.1.1.3.1" stretchy="false">→</mo><mn id="A4.Ex61.m1.1.1.3.3">0</mn></mrow></munder><munder id="A4.Ex61.m1.1.2"><mo id="A4.Ex61.m1.1.2.2" lspace="0.167em" movablelimits="false" rspace="0em">sup</mo><mi id="A4.Ex61.m1.1.2.3">𝐊</mi></munder><mo id="A4.Ex61.m1.1.3" lspace="0em" rspace="0.167em">∥</mo><mo id="A4.Ex61.m1.1.4" rspace="0.167em">∇</mo><mi id="A4.Ex61.m1.1.5">log</mi><msub id="A4.Ex61.m1.1.6"><mi id="A4.Ex61.m1.1.6.2">p</mi><mi id="A4.Ex61.m1.1.6.3">σ</mi></msub><mrow id="A4.Ex61.m1.1.7"><mo id="A4.Ex61.m1.1.7.1" stretchy="false">(</mo><mo id="A4.Ex61.m1.1.7.2" lspace="0em" rspace="0em">⋅</mo><mo fence="false" id="A4.Ex61.m1.1.7.3" rspace="0.167em" stretchy="false">|</mo><mi id="A4.Ex61.m1.1.7.4">𝐲</mi><mo id="A4.Ex61.m1.1.7.5" stretchy="false">)</mo></mrow><mo id="A4.Ex61.m1.1.8">−</mo><mo id="A4.Ex61.m1.1.9" rspace="0.167em">∇</mo><mi id="A4.Ex61.m1.1.10">log</mi><mi id="A4.Ex61.m1.1.11">p</mi><mrow id="A4.Ex61.m1.1.12"><mo id="A4.Ex61.m1.1.12.1" stretchy="false">(</mo><mo id="A4.Ex61.m1.1.12.2" lspace="0em" rspace="0em">⋅</mo><mo fence="false" id="A4.Ex61.m1.1.12.3" rspace="0.167em" stretchy="false">|</mo><mi id="A4.Ex61.m1.1.12.4">𝐲</mi><mo id="A4.Ex61.m1.1.12.5" stretchy="false">)</mo></mrow><mo id="A4.Ex61.m1.1.13" lspace="0em" rspace="0.0835em">∥</mo><mo id="A4.Ex61.m1.1.14" lspace="0.0835em">=</mo><mn id="A4.Ex61.m1.1.15">0</mn><mo id="A4.Ex61.m1.1.16" lspace="0em">.</mo></mrow><annotation encoding="application/x-tex" id="A4.Ex61.m1.1c">\lim_{\sigma\to 0}\sup_{\mathbf{K}}\|\nabla\log p_{\sigma}(\cdot|{\mathbf{y}})% -\nabla\log p(\cdot|{\mathbf{y}})\|=0.</annotation><annotation encoding="application/x-llamapun" id="A4.Ex61.m1.1d">roman_lim start_POSTSUBSCRIPT italic_σ → 0 end_POSTSUBSCRIPT roman_sup start_POSTSUBSCRIPT bold_K end_POSTSUBSCRIPT ∥ ∇ roman_log italic_p start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ( ⋅ | bold_y ) - ∇ roman_log italic_p ( ⋅ | bold_y ) ∥ = 0 .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> </div> </div> <div class="ltx_proof" id="A4.SS1.8"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof.</h6> <div class="ltx_para" id="A4.SS1.1.p1"> <p class="ltx_p" id="A4.SS1.1.p1.2">By the Bayes’ theorem, <math alttext="\nabla\log p_{\sigma}(\cdot|{\mathbf{y}})=\nabla\log p({\mathbf{y}}|\cdot)+% \nabla\log p_{\sigma}" class="ltx_math_unparsed" display="inline" id="A4.SS1.1.p1.1.m1.1"><semantics id="A4.SS1.1.p1.1.m1.1a"><mrow id="A4.SS1.1.p1.1.m1.1b"><mo id="A4.SS1.1.p1.1.m1.1.1" rspace="0.167em">∇</mo><mi id="A4.SS1.1.p1.1.m1.1.2">log</mi><msub id="A4.SS1.1.p1.1.m1.1.3"><mi id="A4.SS1.1.p1.1.m1.1.3.2">p</mi><mi id="A4.SS1.1.p1.1.m1.1.3.3">σ</mi></msub><mrow id="A4.SS1.1.p1.1.m1.1.4"><mo id="A4.SS1.1.p1.1.m1.1.4.1" stretchy="false">(</mo><mo id="A4.SS1.1.p1.1.m1.1.4.2" lspace="0em" rspace="0em">⋅</mo><mo fence="false" id="A4.SS1.1.p1.1.m1.1.4.3" rspace="0.167em" stretchy="false">|</mo><mi id="A4.SS1.1.p1.1.m1.1.4.4">𝐲</mi><mo id="A4.SS1.1.p1.1.m1.1.4.5" stretchy="false">)</mo></mrow><mo id="A4.SS1.1.p1.1.m1.1.5">=</mo><mo id="A4.SS1.1.p1.1.m1.1.6" rspace="0.167em">∇</mo><mi id="A4.SS1.1.p1.1.m1.1.7">log</mi><mi id="A4.SS1.1.p1.1.m1.1.8">p</mi><mrow id="A4.SS1.1.p1.1.m1.1.9"><mo id="A4.SS1.1.p1.1.m1.1.9.1" stretchy="false">(</mo><mi id="A4.SS1.1.p1.1.m1.1.9.2">𝐲</mi><mo fence="false" id="A4.SS1.1.p1.1.m1.1.9.3" stretchy="false">|</mo><mo id="A4.SS1.1.p1.1.m1.1.9.4" lspace="0em" rspace="0em">⋅</mo><mo id="A4.SS1.1.p1.1.m1.1.9.5" stretchy="false">)</mo></mrow><mo id="A4.SS1.1.p1.1.m1.1.10">+</mo><mo id="A4.SS1.1.p1.1.m1.1.11" rspace="0.167em">∇</mo><mi id="A4.SS1.1.p1.1.m1.1.12">log</mi><msub id="A4.SS1.1.p1.1.m1.1.13"><mi id="A4.SS1.1.p1.1.m1.1.13.2">p</mi><mi id="A4.SS1.1.p1.1.m1.1.13.3">σ</mi></msub></mrow><annotation encoding="application/x-tex" id="A4.SS1.1.p1.1.m1.1c">\nabla\log p_{\sigma}(\cdot|{\mathbf{y}})=\nabla\log p({\mathbf{y}}|\cdot)+% \nabla\log p_{\sigma}</annotation><annotation encoding="application/x-llamapun" id="A4.SS1.1.p1.1.m1.1d">∇ roman_log italic_p start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ( ⋅ | bold_y ) = ∇ roman_log italic_p ( bold_y | ⋅ ) + ∇ roman_log italic_p start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT</annotation></semantics></math>. As a consequence, Proposition <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#A4.Thmproposition1" title="Proposition D.1. ‣ D.1 Proof of critical Point Analysis ‣ Appendix D Proofs of section 3 ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_tag">D.1</span></a> is equivalent to show that the score converges uniformly on <math alttext="\mathbf{K}" class="ltx_Math" display="inline" id="A4.SS1.1.p1.2.m2.1"><semantics id="A4.SS1.1.p1.2.m2.1a"><mi id="A4.SS1.1.p1.2.m2.1.1" xref="A4.SS1.1.p1.2.m2.1.1.cmml">𝐊</mi><annotation-xml encoding="MathML-Content" id="A4.SS1.1.p1.2.m2.1b"><ci id="A4.SS1.1.p1.2.m2.1.1.cmml" xref="A4.SS1.1.p1.2.m2.1.1">𝐊</ci></annotation-xml><annotation encoding="application/x-tex" id="A4.SS1.1.p1.2.m2.1c">\mathbf{K}</annotation><annotation encoding="application/x-llamapun" id="A4.SS1.1.p1.2.m2.1d">bold_K</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="A4.SS1.2.p2"> <p class="ltx_p" id="A4.SS1.2.p2.4">For <math alttext="f\in\mathrm{C}({\mathbb{R}}^{d},{\mathbb{R}}^{p})" class="ltx_Math" display="inline" id="A4.SS1.2.p2.1.m1.2"><semantics id="A4.SS1.2.p2.1.m1.2a"><mrow id="A4.SS1.2.p2.1.m1.2.2" xref="A4.SS1.2.p2.1.m1.2.2.cmml"><mi id="A4.SS1.2.p2.1.m1.2.2.4" xref="A4.SS1.2.p2.1.m1.2.2.4.cmml">f</mi><mo id="A4.SS1.2.p2.1.m1.2.2.3" xref="A4.SS1.2.p2.1.m1.2.2.3.cmml">∈</mo><mrow id="A4.SS1.2.p2.1.m1.2.2.2" xref="A4.SS1.2.p2.1.m1.2.2.2.cmml"><mi id="A4.SS1.2.p2.1.m1.2.2.2.4" mathvariant="normal" xref="A4.SS1.2.p2.1.m1.2.2.2.4.cmml">C</mi><mo id="A4.SS1.2.p2.1.m1.2.2.2.3" xref="A4.SS1.2.p2.1.m1.2.2.2.3.cmml">⁢</mo><mrow id="A4.SS1.2.p2.1.m1.2.2.2.2.2" xref="A4.SS1.2.p2.1.m1.2.2.2.2.3.cmml"><mo id="A4.SS1.2.p2.1.m1.2.2.2.2.2.3" stretchy="false" xref="A4.SS1.2.p2.1.m1.2.2.2.2.3.cmml">(</mo><msup id="A4.SS1.2.p2.1.m1.1.1.1.1.1.1" xref="A4.SS1.2.p2.1.m1.1.1.1.1.1.1.cmml"><mi id="A4.SS1.2.p2.1.m1.1.1.1.1.1.1.2" xref="A4.SS1.2.p2.1.m1.1.1.1.1.1.1.2.cmml">ℝ</mi><mi id="A4.SS1.2.p2.1.m1.1.1.1.1.1.1.3" xref="A4.SS1.2.p2.1.m1.1.1.1.1.1.1.3.cmml">d</mi></msup><mo id="A4.SS1.2.p2.1.m1.2.2.2.2.2.4" xref="A4.SS1.2.p2.1.m1.2.2.2.2.3.cmml">,</mo><msup id="A4.SS1.2.p2.1.m1.2.2.2.2.2.2" xref="A4.SS1.2.p2.1.m1.2.2.2.2.2.2.cmml"><mi id="A4.SS1.2.p2.1.m1.2.2.2.2.2.2.2" xref="A4.SS1.2.p2.1.m1.2.2.2.2.2.2.2.cmml">ℝ</mi><mi id="A4.SS1.2.p2.1.m1.2.2.2.2.2.2.3" xref="A4.SS1.2.p2.1.m1.2.2.2.2.2.2.3.cmml">p</mi></msup><mo id="A4.SS1.2.p2.1.m1.2.2.2.2.2.5" stretchy="false" xref="A4.SS1.2.p2.1.m1.2.2.2.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="A4.SS1.2.p2.1.m1.2b"><apply id="A4.SS1.2.p2.1.m1.2.2.cmml" xref="A4.SS1.2.p2.1.m1.2.2"><in id="A4.SS1.2.p2.1.m1.2.2.3.cmml" xref="A4.SS1.2.p2.1.m1.2.2.3"></in><ci id="A4.SS1.2.p2.1.m1.2.2.4.cmml" xref="A4.SS1.2.p2.1.m1.2.2.4">𝑓</ci><apply id="A4.SS1.2.p2.1.m1.2.2.2.cmml" xref="A4.SS1.2.p2.1.m1.2.2.2"><times id="A4.SS1.2.p2.1.m1.2.2.2.3.cmml" xref="A4.SS1.2.p2.1.m1.2.2.2.3"></times><ci id="A4.SS1.2.p2.1.m1.2.2.2.4.cmml" xref="A4.SS1.2.p2.1.m1.2.2.2.4">C</ci><interval closure="open" id="A4.SS1.2.p2.1.m1.2.2.2.2.3.cmml" xref="A4.SS1.2.p2.1.m1.2.2.2.2.2"><apply id="A4.SS1.2.p2.1.m1.1.1.1.1.1.1.cmml" xref="A4.SS1.2.p2.1.m1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="A4.SS1.2.p2.1.m1.1.1.1.1.1.1.1.cmml" xref="A4.SS1.2.p2.1.m1.1.1.1.1.1.1">superscript</csymbol><ci id="A4.SS1.2.p2.1.m1.1.1.1.1.1.1.2.cmml" xref="A4.SS1.2.p2.1.m1.1.1.1.1.1.1.2">ℝ</ci><ci id="A4.SS1.2.p2.1.m1.1.1.1.1.1.1.3.cmml" xref="A4.SS1.2.p2.1.m1.1.1.1.1.1.1.3">𝑑</ci></apply><apply id="A4.SS1.2.p2.1.m1.2.2.2.2.2.2.cmml" xref="A4.SS1.2.p2.1.m1.2.2.2.2.2.2"><csymbol cd="ambiguous" id="A4.SS1.2.p2.1.m1.2.2.2.2.2.2.1.cmml" xref="A4.SS1.2.p2.1.m1.2.2.2.2.2.2">superscript</csymbol><ci id="A4.SS1.2.p2.1.m1.2.2.2.2.2.2.2.cmml" xref="A4.SS1.2.p2.1.m1.2.2.2.2.2.2.2">ℝ</ci><ci id="A4.SS1.2.p2.1.m1.2.2.2.2.2.2.3.cmml" xref="A4.SS1.2.p2.1.m1.2.2.2.2.2.2.3">𝑝</ci></apply></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.SS1.2.p2.1.m1.2c">f\in\mathrm{C}({\mathbb{R}}^{d},{\mathbb{R}}^{p})</annotation><annotation encoding="application/x-llamapun" id="A4.SS1.2.p2.1.m1.2d">italic_f ∈ roman_C ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT , blackboard_R start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT )</annotation></semantics></math>, with <math alttext="p\in{\mathbb{N}}" class="ltx_Math" display="inline" id="A4.SS1.2.p2.2.m2.1"><semantics id="A4.SS1.2.p2.2.m2.1a"><mrow id="A4.SS1.2.p2.2.m2.1.1" xref="A4.SS1.2.p2.2.m2.1.1.cmml"><mi id="A4.SS1.2.p2.2.m2.1.1.2" xref="A4.SS1.2.p2.2.m2.1.1.2.cmml">p</mi><mo id="A4.SS1.2.p2.2.m2.1.1.1" xref="A4.SS1.2.p2.2.m2.1.1.1.cmml">∈</mo><mi id="A4.SS1.2.p2.2.m2.1.1.3" xref="A4.SS1.2.p2.2.m2.1.1.3.cmml">ℕ</mi></mrow><annotation-xml encoding="MathML-Content" id="A4.SS1.2.p2.2.m2.1b"><apply id="A4.SS1.2.p2.2.m2.1.1.cmml" xref="A4.SS1.2.p2.2.m2.1.1"><in id="A4.SS1.2.p2.2.m2.1.1.1.cmml" xref="A4.SS1.2.p2.2.m2.1.1.1"></in><ci id="A4.SS1.2.p2.2.m2.1.1.2.cmml" xref="A4.SS1.2.p2.2.m2.1.1.2">𝑝</ci><ci id="A4.SS1.2.p2.2.m2.1.1.3.cmml" xref="A4.SS1.2.p2.2.m2.1.1.3">ℕ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.SS1.2.p2.2.m2.1c">p\in{\mathbb{N}}</annotation><annotation encoding="application/x-llamapun" id="A4.SS1.2.p2.2.m2.1d">italic_p ∈ blackboard_N</annotation></semantics></math> and <math alttext="\|f\|_{\infty}<+\infty" class="ltx_Math" display="inline" id="A4.SS1.2.p2.3.m3.1"><semantics id="A4.SS1.2.p2.3.m3.1a"><mrow id="A4.SS1.2.p2.3.m3.1.2" xref="A4.SS1.2.p2.3.m3.1.2.cmml"><msub id="A4.SS1.2.p2.3.m3.1.2.2" xref="A4.SS1.2.p2.3.m3.1.2.2.cmml"><mrow id="A4.SS1.2.p2.3.m3.1.2.2.2.2" xref="A4.SS1.2.p2.3.m3.1.2.2.2.1.cmml"><mo id="A4.SS1.2.p2.3.m3.1.2.2.2.2.1" stretchy="false" xref="A4.SS1.2.p2.3.m3.1.2.2.2.1.1.cmml">‖</mo><mi id="A4.SS1.2.p2.3.m3.1.1" xref="A4.SS1.2.p2.3.m3.1.1.cmml">f</mi><mo id="A4.SS1.2.p2.3.m3.1.2.2.2.2.2" stretchy="false" xref="A4.SS1.2.p2.3.m3.1.2.2.2.1.1.cmml">‖</mo></mrow><mi id="A4.SS1.2.p2.3.m3.1.2.2.3" mathvariant="normal" xref="A4.SS1.2.p2.3.m3.1.2.2.3.cmml">∞</mi></msub><mo id="A4.SS1.2.p2.3.m3.1.2.1" xref="A4.SS1.2.p2.3.m3.1.2.1.cmml"><</mo><mrow id="A4.SS1.2.p2.3.m3.1.2.3" xref="A4.SS1.2.p2.3.m3.1.2.3.cmml"><mo id="A4.SS1.2.p2.3.m3.1.2.3a" xref="A4.SS1.2.p2.3.m3.1.2.3.cmml">+</mo><mi id="A4.SS1.2.p2.3.m3.1.2.3.2" mathvariant="normal" xref="A4.SS1.2.p2.3.m3.1.2.3.2.cmml">∞</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="A4.SS1.2.p2.3.m3.1b"><apply id="A4.SS1.2.p2.3.m3.1.2.cmml" xref="A4.SS1.2.p2.3.m3.1.2"><lt id="A4.SS1.2.p2.3.m3.1.2.1.cmml" xref="A4.SS1.2.p2.3.m3.1.2.1"></lt><apply id="A4.SS1.2.p2.3.m3.1.2.2.cmml" xref="A4.SS1.2.p2.3.m3.1.2.2"><csymbol cd="ambiguous" id="A4.SS1.2.p2.3.m3.1.2.2.1.cmml" xref="A4.SS1.2.p2.3.m3.1.2.2">subscript</csymbol><apply id="A4.SS1.2.p2.3.m3.1.2.2.2.1.cmml" xref="A4.SS1.2.p2.3.m3.1.2.2.2.2"><csymbol cd="latexml" id="A4.SS1.2.p2.3.m3.1.2.2.2.1.1.cmml" xref="A4.SS1.2.p2.3.m3.1.2.2.2.2.1">norm</csymbol><ci id="A4.SS1.2.p2.3.m3.1.1.cmml" xref="A4.SS1.2.p2.3.m3.1.1">𝑓</ci></apply><infinity id="A4.SS1.2.p2.3.m3.1.2.2.3.cmml" xref="A4.SS1.2.p2.3.m3.1.2.2.3"></infinity></apply><apply id="A4.SS1.2.p2.3.m3.1.2.3.cmml" xref="A4.SS1.2.p2.3.m3.1.2.3"><plus id="A4.SS1.2.p2.3.m3.1.2.3.1.cmml" xref="A4.SS1.2.p2.3.m3.1.2.3"></plus><infinity id="A4.SS1.2.p2.3.m3.1.2.3.2.cmml" xref="A4.SS1.2.p2.3.m3.1.2.3.2"></infinity></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.SS1.2.p2.3.m3.1c">\|f\|_{\infty}<+\infty</annotation><annotation encoding="application/x-llamapun" id="A4.SS1.2.p2.3.m3.1d">∥ italic_f ∥ start_POSTSUBSCRIPT ∞ end_POSTSUBSCRIPT < + ∞</annotation></semantics></math>, we define for <math alttext="{\mathbf{x}}\in{\mathbb{R}}^{d}" class="ltx_Math" display="inline" id="A4.SS1.2.p2.4.m4.1"><semantics id="A4.SS1.2.p2.4.m4.1a"><mrow id="A4.SS1.2.p2.4.m4.1.1" xref="A4.SS1.2.p2.4.m4.1.1.cmml"><mi id="A4.SS1.2.p2.4.m4.1.1.2" xref="A4.SS1.2.p2.4.m4.1.1.2.cmml">𝐱</mi><mo id="A4.SS1.2.p2.4.m4.1.1.1" xref="A4.SS1.2.p2.4.m4.1.1.1.cmml">∈</mo><msup id="A4.SS1.2.p2.4.m4.1.1.3" xref="A4.SS1.2.p2.4.m4.1.1.3.cmml"><mi id="A4.SS1.2.p2.4.m4.1.1.3.2" xref="A4.SS1.2.p2.4.m4.1.1.3.2.cmml">ℝ</mi><mi id="A4.SS1.2.p2.4.m4.1.1.3.3" xref="A4.SS1.2.p2.4.m4.1.1.3.3.cmml">d</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="A4.SS1.2.p2.4.m4.1b"><apply id="A4.SS1.2.p2.4.m4.1.1.cmml" xref="A4.SS1.2.p2.4.m4.1.1"><in id="A4.SS1.2.p2.4.m4.1.1.1.cmml" xref="A4.SS1.2.p2.4.m4.1.1.1"></in><ci id="A4.SS1.2.p2.4.m4.1.1.2.cmml" xref="A4.SS1.2.p2.4.m4.1.1.2">𝐱</ci><apply id="A4.SS1.2.p2.4.m4.1.1.3.cmml" xref="A4.SS1.2.p2.4.m4.1.1.3"><csymbol cd="ambiguous" id="A4.SS1.2.p2.4.m4.1.1.3.1.cmml" xref="A4.SS1.2.p2.4.m4.1.1.3">superscript</csymbol><ci id="A4.SS1.2.p2.4.m4.1.1.3.2.cmml" xref="A4.SS1.2.p2.4.m4.1.1.3.2">ℝ</ci><ci id="A4.SS1.2.p2.4.m4.1.1.3.3.cmml" xref="A4.SS1.2.p2.4.m4.1.1.3.3">𝑑</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.SS1.2.p2.4.m4.1c">{\mathbf{x}}\in{\mathbb{R}}^{d}</annotation><annotation encoding="application/x-llamapun" id="A4.SS1.2.p2.4.m4.1d">bold_x ∈ blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT</annotation></semantics></math></p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A8.EGx39"> <tbody id="A4.Ex62"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle f_{\sigma}({\mathbf{x}})=\left(f\ast\mathcal{N}_{\sigma}\right)(% {\mathbf{x}})=\int_{{\mathbb{R}}^{d}}{f({\tilde{\mathbf{x}}}-{\tilde{\mathbf{x% }}})\mathcal{N}({\mathbf{x}};0,\sigma^{2}\mathbf{I}_{d})d{\tilde{\mathbf{x}}}}." class="ltx_Math" display="inline" id="A4.Ex62.m1.5"><semantics id="A4.Ex62.m1.5a"><mrow id="A4.Ex62.m1.5.5.1" xref="A4.Ex62.m1.5.5.1.1.cmml"><mrow id="A4.Ex62.m1.5.5.1.1" xref="A4.Ex62.m1.5.5.1.1.cmml"><mrow id="A4.Ex62.m1.5.5.1.1.5" xref="A4.Ex62.m1.5.5.1.1.5.cmml"><msub id="A4.Ex62.m1.5.5.1.1.5.2" xref="A4.Ex62.m1.5.5.1.1.5.2.cmml"><mi id="A4.Ex62.m1.5.5.1.1.5.2.2" xref="A4.Ex62.m1.5.5.1.1.5.2.2.cmml">f</mi><mi id="A4.Ex62.m1.5.5.1.1.5.2.3" xref="A4.Ex62.m1.5.5.1.1.5.2.3.cmml">σ</mi></msub><mo id="A4.Ex62.m1.5.5.1.1.5.1" xref="A4.Ex62.m1.5.5.1.1.5.1.cmml">⁢</mo><mrow id="A4.Ex62.m1.5.5.1.1.5.3.2" xref="A4.Ex62.m1.5.5.1.1.5.cmml"><mo id="A4.Ex62.m1.5.5.1.1.5.3.2.1" stretchy="false" xref="A4.Ex62.m1.5.5.1.1.5.cmml">(</mo><mi id="A4.Ex62.m1.1.1" xref="A4.Ex62.m1.1.1.cmml">𝐱</mi><mo id="A4.Ex62.m1.5.5.1.1.5.3.2.2" stretchy="false" xref="A4.Ex62.m1.5.5.1.1.5.cmml">)</mo></mrow></mrow><mo id="A4.Ex62.m1.5.5.1.1.6" xref="A4.Ex62.m1.5.5.1.1.6.cmml">=</mo><mrow id="A4.Ex62.m1.5.5.1.1.1" xref="A4.Ex62.m1.5.5.1.1.1.cmml"><mrow id="A4.Ex62.m1.5.5.1.1.1.1.1" xref="A4.Ex62.m1.5.5.1.1.1.1.1.1.cmml"><mo id="A4.Ex62.m1.5.5.1.1.1.1.1.2" xref="A4.Ex62.m1.5.5.1.1.1.1.1.1.cmml">(</mo><mrow id="A4.Ex62.m1.5.5.1.1.1.1.1.1" xref="A4.Ex62.m1.5.5.1.1.1.1.1.1.cmml"><mi id="A4.Ex62.m1.5.5.1.1.1.1.1.1.2" xref="A4.Ex62.m1.5.5.1.1.1.1.1.1.2.cmml">f</mi><mo id="A4.Ex62.m1.5.5.1.1.1.1.1.1.1" lspace="0.222em" rspace="0.222em" xref="A4.Ex62.m1.5.5.1.1.1.1.1.1.1.cmml">∗</mo><msub id="A4.Ex62.m1.5.5.1.1.1.1.1.1.3" xref="A4.Ex62.m1.5.5.1.1.1.1.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="A4.Ex62.m1.5.5.1.1.1.1.1.1.3.2" xref="A4.Ex62.m1.5.5.1.1.1.1.1.1.3.2.cmml">𝒩</mi><mi id="A4.Ex62.m1.5.5.1.1.1.1.1.1.3.3" xref="A4.Ex62.m1.5.5.1.1.1.1.1.1.3.3.cmml">σ</mi></msub></mrow><mo id="A4.Ex62.m1.5.5.1.1.1.1.1.3" xref="A4.Ex62.m1.5.5.1.1.1.1.1.1.cmml">)</mo></mrow><mo id="A4.Ex62.m1.5.5.1.1.1.2" xref="A4.Ex62.m1.5.5.1.1.1.2.cmml">⁢</mo><mrow id="A4.Ex62.m1.5.5.1.1.1.3.2" 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f_{\sigma}({\mathbf{x}})=\left(f\ast\mathcal{N}_{\sigma}\right)(% {\mathbf{x}})=\int_{{\mathbb{R}}^{d}}{f({\tilde{\mathbf{x}}}-{\tilde{\mathbf{x% }}})\mathcal{N}({\mathbf{x}};0,\sigma^{2}\mathbf{I}_{d})d{\tilde{\mathbf{x}}}}.</annotation><annotation encoding="application/x-llamapun" id="A4.Ex62.m1.5d">italic_f start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ( bold_x ) = ( italic_f ∗ caligraphic_N start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ) ( bold_x ) = ∫ start_POSTSUBSCRIPT blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT end_POSTSUBSCRIPT italic_f ( over~ start_ARG bold_x end_ARG - over~ start_ARG bold_x end_ARG ) caligraphic_N ( bold_x ; 0 , italic_σ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT bold_I start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT ) italic_d over~ start_ARG bold_x end_ARG .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="A4.SS1.2.p2.6">For <math alttext="\epsilon>0" class="ltx_Math" display="inline" id="A4.SS1.2.p2.5.m1.1"><semantics id="A4.SS1.2.p2.5.m1.1a"><mrow id="A4.SS1.2.p2.5.m1.1.1" xref="A4.SS1.2.p2.5.m1.1.1.cmml"><mi id="A4.SS1.2.p2.5.m1.1.1.2" xref="A4.SS1.2.p2.5.m1.1.1.2.cmml">ϵ</mi><mo id="A4.SS1.2.p2.5.m1.1.1.1" xref="A4.SS1.2.p2.5.m1.1.1.1.cmml">></mo><mn id="A4.SS1.2.p2.5.m1.1.1.3" xref="A4.SS1.2.p2.5.m1.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="A4.SS1.2.p2.5.m1.1b"><apply id="A4.SS1.2.p2.5.m1.1.1.cmml" xref="A4.SS1.2.p2.5.m1.1.1"><gt id="A4.SS1.2.p2.5.m1.1.1.1.cmml" xref="A4.SS1.2.p2.5.m1.1.1.1"></gt><ci id="A4.SS1.2.p2.5.m1.1.1.2.cmml" xref="A4.SS1.2.p2.5.m1.1.1.2">italic-ϵ</ci><cn id="A4.SS1.2.p2.5.m1.1.1.3.cmml" type="integer" xref="A4.SS1.2.p2.5.m1.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.SS1.2.p2.5.m1.1c">\epsilon>0</annotation><annotation encoding="application/x-llamapun" id="A4.SS1.2.p2.5.m1.1d">italic_ϵ > 0</annotation></semantics></math>, there exists <math alttext="R\in{\mathbb{R}}^{+}" class="ltx_Math" display="inline" id="A4.SS1.2.p2.6.m2.1"><semantics id="A4.SS1.2.p2.6.m2.1a"><mrow id="A4.SS1.2.p2.6.m2.1.1" xref="A4.SS1.2.p2.6.m2.1.1.cmml"><mi id="A4.SS1.2.p2.6.m2.1.1.2" xref="A4.SS1.2.p2.6.m2.1.1.2.cmml">R</mi><mo id="A4.SS1.2.p2.6.m2.1.1.1" xref="A4.SS1.2.p2.6.m2.1.1.1.cmml">∈</mo><msup id="A4.SS1.2.p2.6.m2.1.1.3" xref="A4.SS1.2.p2.6.m2.1.1.3.cmml"><mi id="A4.SS1.2.p2.6.m2.1.1.3.2" xref="A4.SS1.2.p2.6.m2.1.1.3.2.cmml">ℝ</mi><mo id="A4.SS1.2.p2.6.m2.1.1.3.3" xref="A4.SS1.2.p2.6.m2.1.1.3.3.cmml">+</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="A4.SS1.2.p2.6.m2.1b"><apply id="A4.SS1.2.p2.6.m2.1.1.cmml" xref="A4.SS1.2.p2.6.m2.1.1"><in id="A4.SS1.2.p2.6.m2.1.1.1.cmml" xref="A4.SS1.2.p2.6.m2.1.1.1"></in><ci id="A4.SS1.2.p2.6.m2.1.1.2.cmml" xref="A4.SS1.2.p2.6.m2.1.1.2">𝑅</ci><apply id="A4.SS1.2.p2.6.m2.1.1.3.cmml" xref="A4.SS1.2.p2.6.m2.1.1.3"><csymbol cd="ambiguous" id="A4.SS1.2.p2.6.m2.1.1.3.1.cmml" xref="A4.SS1.2.p2.6.m2.1.1.3">superscript</csymbol><ci id="A4.SS1.2.p2.6.m2.1.1.3.2.cmml" xref="A4.SS1.2.p2.6.m2.1.1.3.2">ℝ</ci><plus id="A4.SS1.2.p2.6.m2.1.1.3.3.cmml" xref="A4.SS1.2.p2.6.m2.1.1.3.3"></plus></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.SS1.2.p2.6.m2.1c">R\in{\mathbb{R}}^{+}</annotation><annotation encoding="application/x-llamapun" id="A4.SS1.2.p2.6.m2.1d">italic_R ∈ blackboard_R start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT</annotation></semantics></math> such that</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A8.EGx40"> <tbody id="A4.Ex63"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\int_{{\mathbb{R}}^{d}\setminus\mathcal{B}(0,R)}{\|f({\mathbf{x}}% -\sigma{\tilde{\mathbf{x}}})-f({\mathbf{x}})\|\mathcal{N}({\tilde{\mathbf{x}}}% ;0,\mathbf{I}_{d})d{\tilde{\mathbf{x}}}}\leq 2\|f\|_{\infty}\int_{{\mathbb{R}}% ^{d}\setminus\mathcal{B}(0,R)}{\mathcal{N}({\tilde{\mathbf{x}}};0,\mathbf{I}_{% d})d{\tilde{\mathbf{x}}}}\leq\frac{\epsilon}{2}." class="ltx_Math" display="inline" id="A4.Ex63.m1.11"><semantics id="A4.Ex63.m1.11a"><mrow id="A4.Ex63.m1.11.11.1" xref="A4.Ex63.m1.11.11.1.1.cmml"><mrow id="A4.Ex63.m1.11.11.1.1" xref="A4.Ex63.m1.11.11.1.1.cmml"><mrow id="A4.Ex63.m1.11.11.1.1.2" xref="A4.Ex63.m1.11.11.1.1.2.cmml"><mstyle displaystyle="true" id="A4.Ex63.m1.11.11.1.1.2.3" xref="A4.Ex63.m1.11.11.1.1.2.3.cmml"><msub id="A4.Ex63.m1.11.11.1.1.2.3a" xref="A4.Ex63.m1.11.11.1.1.2.3.cmml"><mo id="A4.Ex63.m1.11.11.1.1.2.3.2" xref="A4.Ex63.m1.11.11.1.1.2.3.2.cmml">∫</mo><mrow id="A4.Ex63.m1.2.2.2" xref="A4.Ex63.m1.2.2.2.cmml"><msup id="A4.Ex63.m1.2.2.2.4" xref="A4.Ex63.m1.2.2.2.4.cmml"><mi id="A4.Ex63.m1.2.2.2.4.2" xref="A4.Ex63.m1.2.2.2.4.2.cmml">ℝ</mi><mi id="A4.Ex63.m1.2.2.2.4.3" xref="A4.Ex63.m1.2.2.2.4.3.cmml">d</mi></msup><mo id="A4.Ex63.m1.2.2.2.3" xref="A4.Ex63.m1.2.2.2.3.cmml">∖</mo><mrow id="A4.Ex63.m1.2.2.2.5" xref="A4.Ex63.m1.2.2.2.5.cmml"><mi class="ltx_font_mathcaligraphic" id="A4.Ex63.m1.2.2.2.5.2" xref="A4.Ex63.m1.2.2.2.5.2.cmml">ℬ</mi><mo id="A4.Ex63.m1.2.2.2.5.1" xref="A4.Ex63.m1.2.2.2.5.1.cmml">⁢</mo><mrow id="A4.Ex63.m1.2.2.2.5.3.2" xref="A4.Ex63.m1.2.2.2.5.3.1.cmml"><mo id="A4.Ex63.m1.2.2.2.5.3.2.1" stretchy="false" xref="A4.Ex63.m1.2.2.2.5.3.1.cmml">(</mo><mn id="A4.Ex63.m1.1.1.1.1" xref="A4.Ex63.m1.1.1.1.1.cmml">0</mn><mo id="A4.Ex63.m1.2.2.2.5.3.2.2" xref="A4.Ex63.m1.2.2.2.5.3.1.cmml">,</mo><mi id="A4.Ex63.m1.2.2.2.2" xref="A4.Ex63.m1.2.2.2.2.cmml">R</mi><mo id="A4.Ex63.m1.2.2.2.5.3.2.3" stretchy="false" xref="A4.Ex63.m1.2.2.2.5.3.1.cmml">)</mo></mrow></mrow></mrow></msub></mstyle><mrow id="A4.Ex63.m1.11.11.1.1.2.2" xref="A4.Ex63.m1.11.11.1.1.2.2.cmml"><mrow id="A4.Ex63.m1.11.11.1.1.1.1.1.1" xref="A4.Ex63.m1.11.11.1.1.1.1.1.2.cmml"><mo id="A4.Ex63.m1.11.11.1.1.1.1.1.1.2" stretchy="false" xref="A4.Ex63.m1.11.11.1.1.1.1.1.2.1.cmml">‖</mo><mrow id="A4.Ex63.m1.11.11.1.1.1.1.1.1.1" xref="A4.Ex63.m1.11.11.1.1.1.1.1.1.1.cmml"><mrow id="A4.Ex63.m1.11.11.1.1.1.1.1.1.1.1" xref="A4.Ex63.m1.11.11.1.1.1.1.1.1.1.1.cmml"><mi id="A4.Ex63.m1.11.11.1.1.1.1.1.1.1.1.3" xref="A4.Ex63.m1.11.11.1.1.1.1.1.1.1.1.3.cmml">f</mi><mo id="A4.Ex63.m1.11.11.1.1.1.1.1.1.1.1.2" xref="A4.Ex63.m1.11.11.1.1.1.1.1.1.1.1.2.cmml">⁢</mo><mrow id="A4.Ex63.m1.11.11.1.1.1.1.1.1.1.1.1.1" xref="A4.Ex63.m1.11.11.1.1.1.1.1.1.1.1.1.1.1.cmml"><mo id="A4.Ex63.m1.11.11.1.1.1.1.1.1.1.1.1.1.2" stretchy="false" xref="A4.Ex63.m1.11.11.1.1.1.1.1.1.1.1.1.1.1.cmml">(</mo><mrow id="A4.Ex63.m1.11.11.1.1.1.1.1.1.1.1.1.1.1" xref="A4.Ex63.m1.11.11.1.1.1.1.1.1.1.1.1.1.1.cmml"><mi id="A4.Ex63.m1.11.11.1.1.1.1.1.1.1.1.1.1.1.2" xref="A4.Ex63.m1.11.11.1.1.1.1.1.1.1.1.1.1.1.2.cmml">𝐱</mi><mo id="A4.Ex63.m1.11.11.1.1.1.1.1.1.1.1.1.1.1.1" xref="A4.Ex63.m1.11.11.1.1.1.1.1.1.1.1.1.1.1.1.cmml">−</mo><mrow id="A4.Ex63.m1.11.11.1.1.1.1.1.1.1.1.1.1.1.3" xref="A4.Ex63.m1.11.11.1.1.1.1.1.1.1.1.1.1.1.3.cmml"><mi id="A4.Ex63.m1.11.11.1.1.1.1.1.1.1.1.1.1.1.3.2" xref="A4.Ex63.m1.11.11.1.1.1.1.1.1.1.1.1.1.1.3.2.cmml">σ</mi><mo id="A4.Ex63.m1.11.11.1.1.1.1.1.1.1.1.1.1.1.3.1" xref="A4.Ex63.m1.11.11.1.1.1.1.1.1.1.1.1.1.1.3.1.cmml">⁢</mo><mover accent="true" id="A4.Ex63.m1.11.11.1.1.1.1.1.1.1.1.1.1.1.3.3" xref="A4.Ex63.m1.11.11.1.1.1.1.1.1.1.1.1.1.1.3.3.cmml"><mi id="A4.Ex63.m1.11.11.1.1.1.1.1.1.1.1.1.1.1.3.3.2" xref="A4.Ex63.m1.11.11.1.1.1.1.1.1.1.1.1.1.1.3.3.2.cmml">𝐱</mi><mo id="A4.Ex63.m1.11.11.1.1.1.1.1.1.1.1.1.1.1.3.3.1" xref="A4.Ex63.m1.11.11.1.1.1.1.1.1.1.1.1.1.1.3.3.1.cmml">~</mo></mover></mrow></mrow><mo id="A4.Ex63.m1.11.11.1.1.1.1.1.1.1.1.1.1.3" stretchy="false" xref="A4.Ex63.m1.11.11.1.1.1.1.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="A4.Ex63.m1.11.11.1.1.1.1.1.1.1.2" xref="A4.Ex63.m1.11.11.1.1.1.1.1.1.1.2.cmml">−</mo><mrow id="A4.Ex63.m1.11.11.1.1.1.1.1.1.1.3" xref="A4.Ex63.m1.11.11.1.1.1.1.1.1.1.3.cmml"><mi id="A4.Ex63.m1.11.11.1.1.1.1.1.1.1.3.2" xref="A4.Ex63.m1.11.11.1.1.1.1.1.1.1.3.2.cmml">f</mi><mo id="A4.Ex63.m1.11.11.1.1.1.1.1.1.1.3.1" xref="A4.Ex63.m1.11.11.1.1.1.1.1.1.1.3.1.cmml">⁢</mo><mrow id="A4.Ex63.m1.11.11.1.1.1.1.1.1.1.3.3.2" xref="A4.Ex63.m1.11.11.1.1.1.1.1.1.1.3.cmml"><mo id="A4.Ex63.m1.11.11.1.1.1.1.1.1.1.3.3.2.1" stretchy="false" xref="A4.Ex63.m1.11.11.1.1.1.1.1.1.1.3.cmml">(</mo><mi id="A4.Ex63.m1.5.5" xref="A4.Ex63.m1.5.5.cmml">𝐱</mi><mo id="A4.Ex63.m1.11.11.1.1.1.1.1.1.1.3.3.2.2" stretchy="false" xref="A4.Ex63.m1.11.11.1.1.1.1.1.1.1.3.cmml">)</mo></mrow></mrow></mrow><mo id="A4.Ex63.m1.11.11.1.1.1.1.1.1.3" stretchy="false" xref="A4.Ex63.m1.11.11.1.1.1.1.1.2.1.cmml">‖</mo></mrow><mo id="A4.Ex63.m1.11.11.1.1.2.2.3" xref="A4.Ex63.m1.11.11.1.1.2.2.3.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="A4.Ex63.m1.11.11.1.1.2.2.4" xref="A4.Ex63.m1.11.11.1.1.2.2.4.cmml">𝒩</mi><mo id="A4.Ex63.m1.11.11.1.1.2.2.3a" xref="A4.Ex63.m1.11.11.1.1.2.2.3.cmml">⁢</mo><mrow id="A4.Ex63.m1.11.11.1.1.2.2.2.1" xref="A4.Ex63.m1.11.11.1.1.2.2.2.2.cmml"><mo id="A4.Ex63.m1.11.11.1.1.2.2.2.1.2" stretchy="false" xref="A4.Ex63.m1.11.11.1.1.2.2.2.2.cmml">(</mo><mover accent="true" id="A4.Ex63.m1.6.6" xref="A4.Ex63.m1.6.6.cmml"><mi id="A4.Ex63.m1.6.6.2" xref="A4.Ex63.m1.6.6.2.cmml">𝐱</mi><mo id="A4.Ex63.m1.6.6.1" xref="A4.Ex63.m1.6.6.1.cmml">~</mo></mover><mo id="A4.Ex63.m1.11.11.1.1.2.2.2.1.3" xref="A4.Ex63.m1.11.11.1.1.2.2.2.2.cmml">;</mo><mn id="A4.Ex63.m1.7.7" xref="A4.Ex63.m1.7.7.cmml">0</mn><mo id="A4.Ex63.m1.11.11.1.1.2.2.2.1.4" xref="A4.Ex63.m1.11.11.1.1.2.2.2.2.cmml">,</mo><msub id="A4.Ex63.m1.11.11.1.1.2.2.2.1.1" xref="A4.Ex63.m1.11.11.1.1.2.2.2.1.1.cmml"><mi id="A4.Ex63.m1.11.11.1.1.2.2.2.1.1.2" xref="A4.Ex63.m1.11.11.1.1.2.2.2.1.1.2.cmml">𝐈</mi><mi id="A4.Ex63.m1.11.11.1.1.2.2.2.1.1.3" xref="A4.Ex63.m1.11.11.1.1.2.2.2.1.1.3.cmml">d</mi></msub><mo id="A4.Ex63.m1.11.11.1.1.2.2.2.1.5" stretchy="false" xref="A4.Ex63.m1.11.11.1.1.2.2.2.2.cmml">)</mo></mrow><mo id="A4.Ex63.m1.11.11.1.1.2.2.3b" lspace="0em" xref="A4.Ex63.m1.11.11.1.1.2.2.3.cmml">⁢</mo><mrow id="A4.Ex63.m1.11.11.1.1.2.2.5" xref="A4.Ex63.m1.11.11.1.1.2.2.5.cmml"><mo id="A4.Ex63.m1.11.11.1.1.2.2.5.1" rspace="0em" xref="A4.Ex63.m1.11.11.1.1.2.2.5.1.cmml">𝑑</mo><mover accent="true" id="A4.Ex63.m1.11.11.1.1.2.2.5.2" xref="A4.Ex63.m1.11.11.1.1.2.2.5.2.cmml"><mi id="A4.Ex63.m1.11.11.1.1.2.2.5.2.2" xref="A4.Ex63.m1.11.11.1.1.2.2.5.2.2.cmml">𝐱</mi><mo id="A4.Ex63.m1.11.11.1.1.2.2.5.2.1" xref="A4.Ex63.m1.11.11.1.1.2.2.5.2.1.cmml">~</mo></mover></mrow></mrow></mrow><mo id="A4.Ex63.m1.11.11.1.1.5" xref="A4.Ex63.m1.11.11.1.1.5.cmml">≤</mo><mrow id="A4.Ex63.m1.11.11.1.1.3" xref="A4.Ex63.m1.11.11.1.1.3.cmml"><mn id="A4.Ex63.m1.11.11.1.1.3.3" xref="A4.Ex63.m1.11.11.1.1.3.3.cmml">2</mn><mo id="A4.Ex63.m1.11.11.1.1.3.2" 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id="A4.Ex63.m1.11.11.1.1.3.1.1.2.cmml" xref="A4.Ex63.m1.11.11.1.1.3.1.1.2"></times><ci id="A4.Ex63.m1.11.11.1.1.3.1.1.3.cmml" xref="A4.Ex63.m1.11.11.1.1.3.1.1.3">𝒩</ci><list id="A4.Ex63.m1.11.11.1.1.3.1.1.1.2.cmml" xref="A4.Ex63.m1.11.11.1.1.3.1.1.1.1"><apply id="A4.Ex63.m1.9.9.cmml" xref="A4.Ex63.m1.9.9"><ci id="A4.Ex63.m1.9.9.1.cmml" xref="A4.Ex63.m1.9.9.1">~</ci><ci id="A4.Ex63.m1.9.9.2.cmml" xref="A4.Ex63.m1.9.9.2">𝐱</ci></apply><cn id="A4.Ex63.m1.10.10.cmml" type="integer" xref="A4.Ex63.m1.10.10">0</cn><apply id="A4.Ex63.m1.11.11.1.1.3.1.1.1.1.1.cmml" xref="A4.Ex63.m1.11.11.1.1.3.1.1.1.1.1"><csymbol cd="ambiguous" id="A4.Ex63.m1.11.11.1.1.3.1.1.1.1.1.1.cmml" xref="A4.Ex63.m1.11.11.1.1.3.1.1.1.1.1">subscript</csymbol><ci id="A4.Ex63.m1.11.11.1.1.3.1.1.1.1.1.2.cmml" xref="A4.Ex63.m1.11.11.1.1.3.1.1.1.1.1.2">𝐈</ci><ci id="A4.Ex63.m1.11.11.1.1.3.1.1.1.1.1.3.cmml" xref="A4.Ex63.m1.11.11.1.1.3.1.1.1.1.1.3">𝑑</ci></apply></list><apply id="A4.Ex63.m1.11.11.1.1.3.1.1.4.cmml" 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id="A4.Ex63.m1.11.11.1.1.7.3.cmml" type="integer" xref="A4.Ex63.m1.11.11.1.1.7.3">2</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.Ex63.m1.11c">\displaystyle\int_{{\mathbb{R}}^{d}\setminus\mathcal{B}(0,R)}{\|f({\mathbf{x}}% -\sigma{\tilde{\mathbf{x}}})-f({\mathbf{x}})\|\mathcal{N}({\tilde{\mathbf{x}}}% ;0,\mathbf{I}_{d})d{\tilde{\mathbf{x}}}}\leq 2\|f\|_{\infty}\int_{{\mathbb{R}}% ^{d}\setminus\mathcal{B}(0,R)}{\mathcal{N}({\tilde{\mathbf{x}}};0,\mathbf{I}_{% d})d{\tilde{\mathbf{x}}}}\leq\frac{\epsilon}{2}.</annotation><annotation encoding="application/x-llamapun" id="A4.Ex63.m1.11d">∫ start_POSTSUBSCRIPT blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT ∖ caligraphic_B ( 0 , italic_R ) end_POSTSUBSCRIPT ∥ italic_f ( bold_x - italic_σ over~ start_ARG bold_x end_ARG ) - italic_f ( bold_x ) ∥ caligraphic_N ( over~ start_ARG bold_x end_ARG ; 0 , bold_I start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT ) italic_d over~ start_ARG bold_x end_ARG ≤ 2 ∥ italic_f ∥ start_POSTSUBSCRIPT ∞ end_POSTSUBSCRIPT ∫ start_POSTSUBSCRIPT blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT ∖ caligraphic_B ( 0 , italic_R ) end_POSTSUBSCRIPT caligraphic_N ( over~ start_ARG bold_x end_ARG ; 0 , bold_I start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT ) italic_d over~ start_ARG bold_x end_ARG ≤ divide start_ARG italic_ϵ end_ARG start_ARG 2 end_ARG .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="A4.SS1.2.p2.10">Then <math alttext="\tilde{\mathbf{K}}=\mathbf{K}+\mathcal{B}(0,R)" class="ltx_Math" display="inline" id="A4.SS1.2.p2.7.m1.2"><semantics id="A4.SS1.2.p2.7.m1.2a"><mrow id="A4.SS1.2.p2.7.m1.2.3" xref="A4.SS1.2.p2.7.m1.2.3.cmml"><mover accent="true" id="A4.SS1.2.p2.7.m1.2.3.2" xref="A4.SS1.2.p2.7.m1.2.3.2.cmml"><mi id="A4.SS1.2.p2.7.m1.2.3.2.2" xref="A4.SS1.2.p2.7.m1.2.3.2.2.cmml">𝐊</mi><mo id="A4.SS1.2.p2.7.m1.2.3.2.1" xref="A4.SS1.2.p2.7.m1.2.3.2.1.cmml">~</mo></mover><mo id="A4.SS1.2.p2.7.m1.2.3.1" xref="A4.SS1.2.p2.7.m1.2.3.1.cmml">=</mo><mrow id="A4.SS1.2.p2.7.m1.2.3.3" xref="A4.SS1.2.p2.7.m1.2.3.3.cmml"><mi id="A4.SS1.2.p2.7.m1.2.3.3.2" xref="A4.SS1.2.p2.7.m1.2.3.3.2.cmml">𝐊</mi><mo id="A4.SS1.2.p2.7.m1.2.3.3.1" xref="A4.SS1.2.p2.7.m1.2.3.3.1.cmml">+</mo><mrow id="A4.SS1.2.p2.7.m1.2.3.3.3" xref="A4.SS1.2.p2.7.m1.2.3.3.3.cmml"><mi class="ltx_font_mathcaligraphic" id="A4.SS1.2.p2.7.m1.2.3.3.3.2" xref="A4.SS1.2.p2.7.m1.2.3.3.3.2.cmml">ℬ</mi><mo id="A4.SS1.2.p2.7.m1.2.3.3.3.1" xref="A4.SS1.2.p2.7.m1.2.3.3.3.1.cmml">⁢</mo><mrow id="A4.SS1.2.p2.7.m1.2.3.3.3.3.2" xref="A4.SS1.2.p2.7.m1.2.3.3.3.3.1.cmml"><mo id="A4.SS1.2.p2.7.m1.2.3.3.3.3.2.1" stretchy="false" xref="A4.SS1.2.p2.7.m1.2.3.3.3.3.1.cmml">(</mo><mn id="A4.SS1.2.p2.7.m1.1.1" xref="A4.SS1.2.p2.7.m1.1.1.cmml">0</mn><mo id="A4.SS1.2.p2.7.m1.2.3.3.3.3.2.2" xref="A4.SS1.2.p2.7.m1.2.3.3.3.3.1.cmml">,</mo><mi id="A4.SS1.2.p2.7.m1.2.2" xref="A4.SS1.2.p2.7.m1.2.2.cmml">R</mi><mo id="A4.SS1.2.p2.7.m1.2.3.3.3.3.2.3" stretchy="false" xref="A4.SS1.2.p2.7.m1.2.3.3.3.3.1.cmml">)</mo></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="A4.SS1.2.p2.7.m1.2b"><apply id="A4.SS1.2.p2.7.m1.2.3.cmml" xref="A4.SS1.2.p2.7.m1.2.3"><eq id="A4.SS1.2.p2.7.m1.2.3.1.cmml" xref="A4.SS1.2.p2.7.m1.2.3.1"></eq><apply id="A4.SS1.2.p2.7.m1.2.3.2.cmml" xref="A4.SS1.2.p2.7.m1.2.3.2"><ci id="A4.SS1.2.p2.7.m1.2.3.2.1.cmml" xref="A4.SS1.2.p2.7.m1.2.3.2.1">~</ci><ci id="A4.SS1.2.p2.7.m1.2.3.2.2.cmml" xref="A4.SS1.2.p2.7.m1.2.3.2.2">𝐊</ci></apply><apply id="A4.SS1.2.p2.7.m1.2.3.3.cmml" xref="A4.SS1.2.p2.7.m1.2.3.3"><plus id="A4.SS1.2.p2.7.m1.2.3.3.1.cmml" xref="A4.SS1.2.p2.7.m1.2.3.3.1"></plus><ci id="A4.SS1.2.p2.7.m1.2.3.3.2.cmml" xref="A4.SS1.2.p2.7.m1.2.3.3.2">𝐊</ci><apply id="A4.SS1.2.p2.7.m1.2.3.3.3.cmml" xref="A4.SS1.2.p2.7.m1.2.3.3.3"><times id="A4.SS1.2.p2.7.m1.2.3.3.3.1.cmml" xref="A4.SS1.2.p2.7.m1.2.3.3.3.1"></times><ci id="A4.SS1.2.p2.7.m1.2.3.3.3.2.cmml" xref="A4.SS1.2.p2.7.m1.2.3.3.3.2">ℬ</ci><interval closure="open" id="A4.SS1.2.p2.7.m1.2.3.3.3.3.1.cmml" xref="A4.SS1.2.p2.7.m1.2.3.3.3.3.2"><cn id="A4.SS1.2.p2.7.m1.1.1.cmml" type="integer" xref="A4.SS1.2.p2.7.m1.1.1">0</cn><ci id="A4.SS1.2.p2.7.m1.2.2.cmml" xref="A4.SS1.2.p2.7.m1.2.2">𝑅</ci></interval></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.SS1.2.p2.7.m1.2c">\tilde{\mathbf{K}}=\mathbf{K}+\mathcal{B}(0,R)</annotation><annotation encoding="application/x-llamapun" id="A4.SS1.2.p2.7.m1.2d">over~ start_ARG bold_K end_ARG = bold_K + caligraphic_B ( 0 , italic_R )</annotation></semantics></math> (Minkowski sum) is compact, so <math alttext="f" class="ltx_Math" display="inline" id="A4.SS1.2.p2.8.m2.1"><semantics id="A4.SS1.2.p2.8.m2.1a"><mi id="A4.SS1.2.p2.8.m2.1.1" xref="A4.SS1.2.p2.8.m2.1.1.cmml">f</mi><annotation-xml encoding="MathML-Content" id="A4.SS1.2.p2.8.m2.1b"><ci id="A4.SS1.2.p2.8.m2.1.1.cmml" xref="A4.SS1.2.p2.8.m2.1.1">𝑓</ci></annotation-xml><annotation encoding="application/x-tex" id="A4.SS1.2.p2.8.m2.1c">f</annotation><annotation encoding="application/x-llamapun" id="A4.SS1.2.p2.8.m2.1d">italic_f</annotation></semantics></math> is uniformly continuous on <math alttext="\tilde{\mathbf{K}}" class="ltx_Math" display="inline" id="A4.SS1.2.p2.9.m3.1"><semantics id="A4.SS1.2.p2.9.m3.1a"><mover accent="true" id="A4.SS1.2.p2.9.m3.1.1" xref="A4.SS1.2.p2.9.m3.1.1.cmml"><mi id="A4.SS1.2.p2.9.m3.1.1.2" xref="A4.SS1.2.p2.9.m3.1.1.2.cmml">𝐊</mi><mo id="A4.SS1.2.p2.9.m3.1.1.1" xref="A4.SS1.2.p2.9.m3.1.1.1.cmml">~</mo></mover><annotation-xml encoding="MathML-Content" id="A4.SS1.2.p2.9.m3.1b"><apply id="A4.SS1.2.p2.9.m3.1.1.cmml" xref="A4.SS1.2.p2.9.m3.1.1"><ci id="A4.SS1.2.p2.9.m3.1.1.1.cmml" xref="A4.SS1.2.p2.9.m3.1.1.1">~</ci><ci id="A4.SS1.2.p2.9.m3.1.1.2.cmml" xref="A4.SS1.2.p2.9.m3.1.1.2">𝐊</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.SS1.2.p2.9.m3.1c">\tilde{\mathbf{K}}</annotation><annotation encoding="application/x-llamapun" id="A4.SS1.2.p2.9.m3.1d">over~ start_ARG bold_K end_ARG</annotation></semantics></math>. There exists <math alttext="\xi>0" class="ltx_Math" display="inline" id="A4.SS1.2.p2.10.m4.1"><semantics id="A4.SS1.2.p2.10.m4.1a"><mrow id="A4.SS1.2.p2.10.m4.1.1" xref="A4.SS1.2.p2.10.m4.1.1.cmml"><mi id="A4.SS1.2.p2.10.m4.1.1.2" xref="A4.SS1.2.p2.10.m4.1.1.2.cmml">ξ</mi><mo id="A4.SS1.2.p2.10.m4.1.1.1" xref="A4.SS1.2.p2.10.m4.1.1.1.cmml">></mo><mn id="A4.SS1.2.p2.10.m4.1.1.3" xref="A4.SS1.2.p2.10.m4.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="A4.SS1.2.p2.10.m4.1b"><apply id="A4.SS1.2.p2.10.m4.1.1.cmml" xref="A4.SS1.2.p2.10.m4.1.1"><gt id="A4.SS1.2.p2.10.m4.1.1.1.cmml" xref="A4.SS1.2.p2.10.m4.1.1.1"></gt><ci id="A4.SS1.2.p2.10.m4.1.1.2.cmml" xref="A4.SS1.2.p2.10.m4.1.1.2">𝜉</ci><cn id="A4.SS1.2.p2.10.m4.1.1.3.cmml" type="integer" xref="A4.SS1.2.p2.10.m4.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.SS1.2.p2.10.m4.1c">\xi>0</annotation><annotation encoding="application/x-llamapun" id="A4.SS1.2.p2.10.m4.1d">italic_ξ > 0</annotation></semantics></math> such that</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A8.EGx41"> <tbody id="A4.Ex64"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\forall\sigma\in[0,\xi],\forall{\tilde{\mathbf{x}}}\in\mathcal{B}% (0,R),\forall{\mathbf{x}}\in\mathbf{K},\quad\|f({\mathbf{x}}-\sigma{\tilde{% \mathbf{x}}})-f({\mathbf{x}})\|\leq\frac{\epsilon}{2}." class="ltx_Math" display="inline" id="A4.Ex64.m1.6"><semantics id="A4.Ex64.m1.6a"><mrow id="A4.Ex64.m1.6.6.1"><mrow id="A4.Ex64.m1.6.6.1.1.2" xref="A4.Ex64.m1.6.6.1.1.3.cmml"><mrow id="A4.Ex64.m1.6.6.1.1.1.1" xref="A4.Ex64.m1.6.6.1.1.1.1.cmml"><mrow id="A4.Ex64.m1.6.6.1.1.1.1.2" xref="A4.Ex64.m1.6.6.1.1.1.1.2.cmml"><mo id="A4.Ex64.m1.6.6.1.1.1.1.2.1" rspace="0.167em" xref="A4.Ex64.m1.6.6.1.1.1.1.2.1.cmml">∀</mo><mi id="A4.Ex64.m1.6.6.1.1.1.1.2.2" xref="A4.Ex64.m1.6.6.1.1.1.1.2.2.cmml">σ</mi></mrow><mo id="A4.Ex64.m1.6.6.1.1.1.1.1" xref="A4.Ex64.m1.6.6.1.1.1.1.1.cmml">∈</mo><mrow id="A4.Ex64.m1.6.6.1.1.1.1.3.2" xref="A4.Ex64.m1.6.6.1.1.1.1.3.1.cmml"><mo id="A4.Ex64.m1.6.6.1.1.1.1.3.2.1" stretchy="false" xref="A4.Ex64.m1.6.6.1.1.1.1.3.1.cmml">[</mo><mn id="A4.Ex64.m1.1.1" xref="A4.Ex64.m1.1.1.cmml">0</mn><mo id="A4.Ex64.m1.6.6.1.1.1.1.3.2.2" xref="A4.Ex64.m1.6.6.1.1.1.1.3.1.cmml">,</mo><mi id="A4.Ex64.m1.2.2" xref="A4.Ex64.m1.2.2.cmml">ξ</mi><mo id="A4.Ex64.m1.6.6.1.1.1.1.3.2.3" stretchy="false" xref="A4.Ex64.m1.6.6.1.1.1.1.3.1.cmml">]</mo></mrow></mrow><mo id="A4.Ex64.m1.6.6.1.1.2.3" xref="A4.Ex64.m1.6.6.1.1.3a.cmml">,</mo><mrow id="A4.Ex64.m1.6.6.1.1.2.2.2" xref="A4.Ex64.m1.6.6.1.1.2.2.3.cmml"><mrow id="A4.Ex64.m1.6.6.1.1.2.2.1.1" xref="A4.Ex64.m1.6.6.1.1.2.2.1.1.cmml"><mrow id="A4.Ex64.m1.6.6.1.1.2.2.1.1.2" xref="A4.Ex64.m1.6.6.1.1.2.2.1.1.2.cmml"><mo id="A4.Ex64.m1.6.6.1.1.2.2.1.1.2.1" rspace="0.167em" xref="A4.Ex64.m1.6.6.1.1.2.2.1.1.2.1.cmml">∀</mo><mover accent="true" id="A4.Ex64.m1.6.6.1.1.2.2.1.1.2.2" xref="A4.Ex64.m1.6.6.1.1.2.2.1.1.2.2.cmml"><mi id="A4.Ex64.m1.6.6.1.1.2.2.1.1.2.2.2" xref="A4.Ex64.m1.6.6.1.1.2.2.1.1.2.2.2.cmml">𝐱</mi><mo id="A4.Ex64.m1.6.6.1.1.2.2.1.1.2.2.1" xref="A4.Ex64.m1.6.6.1.1.2.2.1.1.2.2.1.cmml">~</mo></mover></mrow><mo id="A4.Ex64.m1.6.6.1.1.2.2.1.1.1" xref="A4.Ex64.m1.6.6.1.1.2.2.1.1.1.cmml">∈</mo><mrow id="A4.Ex64.m1.6.6.1.1.2.2.1.1.3" xref="A4.Ex64.m1.6.6.1.1.2.2.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="A4.Ex64.m1.6.6.1.1.2.2.1.1.3.2" xref="A4.Ex64.m1.6.6.1.1.2.2.1.1.3.2.cmml">ℬ</mi><mo id="A4.Ex64.m1.6.6.1.1.2.2.1.1.3.1" xref="A4.Ex64.m1.6.6.1.1.2.2.1.1.3.1.cmml">⁢</mo><mrow id="A4.Ex64.m1.6.6.1.1.2.2.1.1.3.3.2" xref="A4.Ex64.m1.6.6.1.1.2.2.1.1.3.3.1.cmml"><mo id="A4.Ex64.m1.6.6.1.1.2.2.1.1.3.3.2.1" stretchy="false" xref="A4.Ex64.m1.6.6.1.1.2.2.1.1.3.3.1.cmml">(</mo><mn id="A4.Ex64.m1.3.3" xref="A4.Ex64.m1.3.3.cmml">0</mn><mo id="A4.Ex64.m1.6.6.1.1.2.2.1.1.3.3.2.2" xref="A4.Ex64.m1.6.6.1.1.2.2.1.1.3.3.1.cmml">,</mo><mi id="A4.Ex64.m1.4.4" xref="A4.Ex64.m1.4.4.cmml">R</mi><mo id="A4.Ex64.m1.6.6.1.1.2.2.1.1.3.3.2.3" stretchy="false" xref="A4.Ex64.m1.6.6.1.1.2.2.1.1.3.3.1.cmml">)</mo></mrow></mrow></mrow><mo id="A4.Ex64.m1.6.6.1.1.2.2.2.3" xref="A4.Ex64.m1.6.6.1.1.2.2.3a.cmml">,</mo><mrow id="A4.Ex64.m1.6.6.1.1.2.2.2.2.2" xref="A4.Ex64.m1.6.6.1.1.2.2.2.2.3.cmml"><mrow id="A4.Ex64.m1.6.6.1.1.2.2.2.2.1.1" xref="A4.Ex64.m1.6.6.1.1.2.2.2.2.1.1.cmml"><mrow id="A4.Ex64.m1.6.6.1.1.2.2.2.2.1.1.2" xref="A4.Ex64.m1.6.6.1.1.2.2.2.2.1.1.2.cmml"><mo id="A4.Ex64.m1.6.6.1.1.2.2.2.2.1.1.2.1" rspace="0.167em" xref="A4.Ex64.m1.6.6.1.1.2.2.2.2.1.1.2.1.cmml">∀</mo><mi id="A4.Ex64.m1.6.6.1.1.2.2.2.2.1.1.2.2" xref="A4.Ex64.m1.6.6.1.1.2.2.2.2.1.1.2.2.cmml">𝐱</mi></mrow><mo id="A4.Ex64.m1.6.6.1.1.2.2.2.2.1.1.1" xref="A4.Ex64.m1.6.6.1.1.2.2.2.2.1.1.1.cmml">∈</mo><mi id="A4.Ex64.m1.6.6.1.1.2.2.2.2.1.1.3" xref="A4.Ex64.m1.6.6.1.1.2.2.2.2.1.1.3.cmml">𝐊</mi></mrow><mo id="A4.Ex64.m1.6.6.1.1.2.2.2.2.2.3" rspace="1.167em" xref="A4.Ex64.m1.6.6.1.1.2.2.2.2.3a.cmml">,</mo><mrow id="A4.Ex64.m1.6.6.1.1.2.2.2.2.2.2" xref="A4.Ex64.m1.6.6.1.1.2.2.2.2.2.2.cmml"><mrow id="A4.Ex64.m1.6.6.1.1.2.2.2.2.2.2.1.1" xref="A4.Ex64.m1.6.6.1.1.2.2.2.2.2.2.1.2.cmml"><mo id="A4.Ex64.m1.6.6.1.1.2.2.2.2.2.2.1.1.2" stretchy="false" xref="A4.Ex64.m1.6.6.1.1.2.2.2.2.2.2.1.2.1.cmml">‖</mo><mrow id="A4.Ex64.m1.6.6.1.1.2.2.2.2.2.2.1.1.1" xref="A4.Ex64.m1.6.6.1.1.2.2.2.2.2.2.1.1.1.cmml"><mrow id="A4.Ex64.m1.6.6.1.1.2.2.2.2.2.2.1.1.1.1" xref="A4.Ex64.m1.6.6.1.1.2.2.2.2.2.2.1.1.1.1.cmml"><mi id="A4.Ex64.m1.6.6.1.1.2.2.2.2.2.2.1.1.1.1.3" xref="A4.Ex64.m1.6.6.1.1.2.2.2.2.2.2.1.1.1.1.3.cmml">f</mi><mo id="A4.Ex64.m1.6.6.1.1.2.2.2.2.2.2.1.1.1.1.2" xref="A4.Ex64.m1.6.6.1.1.2.2.2.2.2.2.1.1.1.1.2.cmml">⁢</mo><mrow id="A4.Ex64.m1.6.6.1.1.2.2.2.2.2.2.1.1.1.1.1.1" xref="A4.Ex64.m1.6.6.1.1.2.2.2.2.2.2.1.1.1.1.1.1.1.cmml"><mo id="A4.Ex64.m1.6.6.1.1.2.2.2.2.2.2.1.1.1.1.1.1.2" stretchy="false" xref="A4.Ex64.m1.6.6.1.1.2.2.2.2.2.2.1.1.1.1.1.1.1.cmml">(</mo><mrow id="A4.Ex64.m1.6.6.1.1.2.2.2.2.2.2.1.1.1.1.1.1.1" xref="A4.Ex64.m1.6.6.1.1.2.2.2.2.2.2.1.1.1.1.1.1.1.cmml"><mi id="A4.Ex64.m1.6.6.1.1.2.2.2.2.2.2.1.1.1.1.1.1.1.2" xref="A4.Ex64.m1.6.6.1.1.2.2.2.2.2.2.1.1.1.1.1.1.1.2.cmml">𝐱</mi><mo id="A4.Ex64.m1.6.6.1.1.2.2.2.2.2.2.1.1.1.1.1.1.1.1" xref="A4.Ex64.m1.6.6.1.1.2.2.2.2.2.2.1.1.1.1.1.1.1.1.cmml">−</mo><mrow id="A4.Ex64.m1.6.6.1.1.2.2.2.2.2.2.1.1.1.1.1.1.1.3" xref="A4.Ex64.m1.6.6.1.1.2.2.2.2.2.2.1.1.1.1.1.1.1.3.cmml"><mi id="A4.Ex64.m1.6.6.1.1.2.2.2.2.2.2.1.1.1.1.1.1.1.3.2" xref="A4.Ex64.m1.6.6.1.1.2.2.2.2.2.2.1.1.1.1.1.1.1.3.2.cmml">σ</mi><mo id="A4.Ex64.m1.6.6.1.1.2.2.2.2.2.2.1.1.1.1.1.1.1.3.1" xref="A4.Ex64.m1.6.6.1.1.2.2.2.2.2.2.1.1.1.1.1.1.1.3.1.cmml">⁢</mo><mover accent="true" id="A4.Ex64.m1.6.6.1.1.2.2.2.2.2.2.1.1.1.1.1.1.1.3.3" xref="A4.Ex64.m1.6.6.1.1.2.2.2.2.2.2.1.1.1.1.1.1.1.3.3.cmml"><mi id="A4.Ex64.m1.6.6.1.1.2.2.2.2.2.2.1.1.1.1.1.1.1.3.3.2" xref="A4.Ex64.m1.6.6.1.1.2.2.2.2.2.2.1.1.1.1.1.1.1.3.3.2.cmml">𝐱</mi><mo id="A4.Ex64.m1.6.6.1.1.2.2.2.2.2.2.1.1.1.1.1.1.1.3.3.1" xref="A4.Ex64.m1.6.6.1.1.2.2.2.2.2.2.1.1.1.1.1.1.1.3.3.1.cmml">~</mo></mover></mrow></mrow><mo id="A4.Ex64.m1.6.6.1.1.2.2.2.2.2.2.1.1.1.1.1.1.3" stretchy="false" xref="A4.Ex64.m1.6.6.1.1.2.2.2.2.2.2.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="A4.Ex64.m1.6.6.1.1.2.2.2.2.2.2.1.1.1.2" xref="A4.Ex64.m1.6.6.1.1.2.2.2.2.2.2.1.1.1.2.cmml">−</mo><mrow id="A4.Ex64.m1.6.6.1.1.2.2.2.2.2.2.1.1.1.3" xref="A4.Ex64.m1.6.6.1.1.2.2.2.2.2.2.1.1.1.3.cmml"><mi id="A4.Ex64.m1.6.6.1.1.2.2.2.2.2.2.1.1.1.3.2" xref="A4.Ex64.m1.6.6.1.1.2.2.2.2.2.2.1.1.1.3.2.cmml">f</mi><mo id="A4.Ex64.m1.6.6.1.1.2.2.2.2.2.2.1.1.1.3.1" xref="A4.Ex64.m1.6.6.1.1.2.2.2.2.2.2.1.1.1.3.1.cmml">⁢</mo><mrow id="A4.Ex64.m1.6.6.1.1.2.2.2.2.2.2.1.1.1.3.3.2" xref="A4.Ex64.m1.6.6.1.1.2.2.2.2.2.2.1.1.1.3.cmml"><mo 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xref="A4.Ex64.m1.6.6.1.1.2.2.2.2.2.2.1.1.1.1.1.1.1.3.3"><ci id="A4.Ex64.m1.6.6.1.1.2.2.2.2.2.2.1.1.1.1.1.1.1.3.3.1.cmml" xref="A4.Ex64.m1.6.6.1.1.2.2.2.2.2.2.1.1.1.1.1.1.1.3.3.1">~</ci><ci id="A4.Ex64.m1.6.6.1.1.2.2.2.2.2.2.1.1.1.1.1.1.1.3.3.2.cmml" xref="A4.Ex64.m1.6.6.1.1.2.2.2.2.2.2.1.1.1.1.1.1.1.3.3.2">𝐱</ci></apply></apply></apply></apply><apply id="A4.Ex64.m1.6.6.1.1.2.2.2.2.2.2.1.1.1.3.cmml" xref="A4.Ex64.m1.6.6.1.1.2.2.2.2.2.2.1.1.1.3"><times id="A4.Ex64.m1.6.6.1.1.2.2.2.2.2.2.1.1.1.3.1.cmml" xref="A4.Ex64.m1.6.6.1.1.2.2.2.2.2.2.1.1.1.3.1"></times><ci id="A4.Ex64.m1.6.6.1.1.2.2.2.2.2.2.1.1.1.3.2.cmml" xref="A4.Ex64.m1.6.6.1.1.2.2.2.2.2.2.1.1.1.3.2">𝑓</ci><ci id="A4.Ex64.m1.5.5.cmml" xref="A4.Ex64.m1.5.5">𝐱</ci></apply></apply></apply><apply id="A4.Ex64.m1.6.6.1.1.2.2.2.2.2.2.3.cmml" xref="A4.Ex64.m1.6.6.1.1.2.2.2.2.2.2.3"><divide id="A4.Ex64.m1.6.6.1.1.2.2.2.2.2.2.3.1.cmml" xref="A4.Ex64.m1.6.6.1.1.2.2.2.2.2.2.3"></divide><ci id="A4.Ex64.m1.6.6.1.1.2.2.2.2.2.2.3.2.cmml" xref="A4.Ex64.m1.6.6.1.1.2.2.2.2.2.2.3.2">italic-ϵ</ci><cn id="A4.Ex64.m1.6.6.1.1.2.2.2.2.2.2.3.3.cmml" type="integer" xref="A4.Ex64.m1.6.6.1.1.2.2.2.2.2.2.3.3">2</cn></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.Ex64.m1.6c">\displaystyle\forall\sigma\in[0,\xi],\forall{\tilde{\mathbf{x}}}\in\mathcal{B}% (0,R),\forall{\mathbf{x}}\in\mathbf{K},\quad\|f({\mathbf{x}}-\sigma{\tilde{% \mathbf{x}}})-f({\mathbf{x}})\|\leq\frac{\epsilon}{2}.</annotation><annotation encoding="application/x-llamapun" id="A4.Ex64.m1.6d">∀ italic_σ ∈ [ 0 , italic_ξ ] , ∀ over~ start_ARG bold_x end_ARG ∈ caligraphic_B ( 0 , italic_R ) , ∀ bold_x ∈ bold_K , ∥ italic_f ( bold_x - italic_σ over~ start_ARG bold_x end_ARG ) - italic_f ( bold_x ) ∥ ≤ divide start_ARG italic_ϵ end_ARG start_ARG 2 end_ARG .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> </div> <div class="ltx_para" id="A4.SS1.3.p3"> <p class="ltx_p" id="A4.SS1.3.p3.1">We can deduce for <math alttext="{\mathbf{x}}\in\mathbf{K},\sigma\in[0,\xi]" class="ltx_Math" display="inline" id="A4.SS1.3.p3.1.m1.4"><semantics id="A4.SS1.3.p3.1.m1.4a"><mrow id="A4.SS1.3.p3.1.m1.4.4.2" xref="A4.SS1.3.p3.1.m1.4.4.3.cmml"><mrow id="A4.SS1.3.p3.1.m1.3.3.1.1" xref="A4.SS1.3.p3.1.m1.3.3.1.1.cmml"><mi id="A4.SS1.3.p3.1.m1.3.3.1.1.2" xref="A4.SS1.3.p3.1.m1.3.3.1.1.2.cmml">𝐱</mi><mo id="A4.SS1.3.p3.1.m1.3.3.1.1.1" xref="A4.SS1.3.p3.1.m1.3.3.1.1.1.cmml">∈</mo><mi id="A4.SS1.3.p3.1.m1.3.3.1.1.3" xref="A4.SS1.3.p3.1.m1.3.3.1.1.3.cmml">𝐊</mi></mrow><mo id="A4.SS1.3.p3.1.m1.4.4.2.3" xref="A4.SS1.3.p3.1.m1.4.4.3a.cmml">,</mo><mrow id="A4.SS1.3.p3.1.m1.4.4.2.2" xref="A4.SS1.3.p3.1.m1.4.4.2.2.cmml"><mi id="A4.SS1.3.p3.1.m1.4.4.2.2.2" xref="A4.SS1.3.p3.1.m1.4.4.2.2.2.cmml">σ</mi><mo id="A4.SS1.3.p3.1.m1.4.4.2.2.1" xref="A4.SS1.3.p3.1.m1.4.4.2.2.1.cmml">∈</mo><mrow id="A4.SS1.3.p3.1.m1.4.4.2.2.3.2" xref="A4.SS1.3.p3.1.m1.4.4.2.2.3.1.cmml"><mo id="A4.SS1.3.p3.1.m1.4.4.2.2.3.2.1" stretchy="false" xref="A4.SS1.3.p3.1.m1.4.4.2.2.3.1.cmml">[</mo><mn id="A4.SS1.3.p3.1.m1.1.1" xref="A4.SS1.3.p3.1.m1.1.1.cmml">0</mn><mo id="A4.SS1.3.p3.1.m1.4.4.2.2.3.2.2" xref="A4.SS1.3.p3.1.m1.4.4.2.2.3.1.cmml">,</mo><mi id="A4.SS1.3.p3.1.m1.2.2" xref="A4.SS1.3.p3.1.m1.2.2.cmml">ξ</mi><mo id="A4.SS1.3.p3.1.m1.4.4.2.2.3.2.3" stretchy="false" xref="A4.SS1.3.p3.1.m1.4.4.2.2.3.1.cmml">]</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="A4.SS1.3.p3.1.m1.4b"><apply id="A4.SS1.3.p3.1.m1.4.4.3.cmml" xref="A4.SS1.3.p3.1.m1.4.4.2"><csymbol cd="ambiguous" id="A4.SS1.3.p3.1.m1.4.4.3a.cmml" xref="A4.SS1.3.p3.1.m1.4.4.2.3">formulae-sequence</csymbol><apply id="A4.SS1.3.p3.1.m1.3.3.1.1.cmml" xref="A4.SS1.3.p3.1.m1.3.3.1.1"><in id="A4.SS1.3.p3.1.m1.3.3.1.1.1.cmml" xref="A4.SS1.3.p3.1.m1.3.3.1.1.1"></in><ci id="A4.SS1.3.p3.1.m1.3.3.1.1.2.cmml" xref="A4.SS1.3.p3.1.m1.3.3.1.1.2">𝐱</ci><ci id="A4.SS1.3.p3.1.m1.3.3.1.1.3.cmml" xref="A4.SS1.3.p3.1.m1.3.3.1.1.3">𝐊</ci></apply><apply id="A4.SS1.3.p3.1.m1.4.4.2.2.cmml" xref="A4.SS1.3.p3.1.m1.4.4.2.2"><in id="A4.SS1.3.p3.1.m1.4.4.2.2.1.cmml" xref="A4.SS1.3.p3.1.m1.4.4.2.2.1"></in><ci id="A4.SS1.3.p3.1.m1.4.4.2.2.2.cmml" xref="A4.SS1.3.p3.1.m1.4.4.2.2.2">𝜎</ci><interval closure="closed" id="A4.SS1.3.p3.1.m1.4.4.2.2.3.1.cmml" xref="A4.SS1.3.p3.1.m1.4.4.2.2.3.2"><cn id="A4.SS1.3.p3.1.m1.1.1.cmml" type="integer" xref="A4.SS1.3.p3.1.m1.1.1">0</cn><ci id="A4.SS1.3.p3.1.m1.2.2.cmml" xref="A4.SS1.3.p3.1.m1.2.2">𝜉</ci></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.SS1.3.p3.1.m1.4c">{\mathbf{x}}\in\mathbf{K},\sigma\in[0,\xi]</annotation><annotation encoding="application/x-llamapun" id="A4.SS1.3.p3.1.m1.4d">bold_x ∈ bold_K , italic_σ ∈ [ 0 , italic_ξ ]</annotation></semantics></math> that</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A8.EGx42"> <tbody id="A4.Ex65"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\|f_{\sigma}({\mathbf{x}})-f({\mathbf{x}})\|" class="ltx_Math" display="inline" id="A4.Ex65.m1.3"><semantics id="A4.Ex65.m1.3a"><mrow id="A4.Ex65.m1.3.3.1" xref="A4.Ex65.m1.3.3.2.cmml"><mo id="A4.Ex65.m1.3.3.1.2" stretchy="false" xref="A4.Ex65.m1.3.3.2.1.cmml">‖</mo><mrow id="A4.Ex65.m1.3.3.1.1" xref="A4.Ex65.m1.3.3.1.1.cmml"><mrow id="A4.Ex65.m1.3.3.1.1.2" xref="A4.Ex65.m1.3.3.1.1.2.cmml"><msub id="A4.Ex65.m1.3.3.1.1.2.2" xref="A4.Ex65.m1.3.3.1.1.2.2.cmml"><mi id="A4.Ex65.m1.3.3.1.1.2.2.2" xref="A4.Ex65.m1.3.3.1.1.2.2.2.cmml">f</mi><mi id="A4.Ex65.m1.3.3.1.1.2.2.3" xref="A4.Ex65.m1.3.3.1.1.2.2.3.cmml">σ</mi></msub><mo id="A4.Ex65.m1.3.3.1.1.2.1" xref="A4.Ex65.m1.3.3.1.1.2.1.cmml">⁢</mo><mrow id="A4.Ex65.m1.3.3.1.1.2.3.2" xref="A4.Ex65.m1.3.3.1.1.2.cmml"><mo id="A4.Ex65.m1.3.3.1.1.2.3.2.1" stretchy="false" xref="A4.Ex65.m1.3.3.1.1.2.cmml">(</mo><mi id="A4.Ex65.m1.1.1" xref="A4.Ex65.m1.1.1.cmml">𝐱</mi><mo id="A4.Ex65.m1.3.3.1.1.2.3.2.2" stretchy="false" xref="A4.Ex65.m1.3.3.1.1.2.cmml">)</mo></mrow></mrow><mo id="A4.Ex65.m1.3.3.1.1.1" xref="A4.Ex65.m1.3.3.1.1.1.cmml">−</mo><mrow id="A4.Ex65.m1.3.3.1.1.3" xref="A4.Ex65.m1.3.3.1.1.3.cmml"><mi id="A4.Ex65.m1.3.3.1.1.3.2" xref="A4.Ex65.m1.3.3.1.1.3.2.cmml">f</mi><mo id="A4.Ex65.m1.3.3.1.1.3.1" xref="A4.Ex65.m1.3.3.1.1.3.1.cmml">⁢</mo><mrow id="A4.Ex65.m1.3.3.1.1.3.3.2" xref="A4.Ex65.m1.3.3.1.1.3.cmml"><mo id="A4.Ex65.m1.3.3.1.1.3.3.2.1" stretchy="false" xref="A4.Ex65.m1.3.3.1.1.3.cmml">(</mo><mi id="A4.Ex65.m1.2.2" xref="A4.Ex65.m1.2.2.cmml">𝐱</mi><mo id="A4.Ex65.m1.3.3.1.1.3.3.2.2" stretchy="false" xref="A4.Ex65.m1.3.3.1.1.3.cmml">)</mo></mrow></mrow></mrow><mo id="A4.Ex65.m1.3.3.1.3" stretchy="false" xref="A4.Ex65.m1.3.3.2.1.cmml">‖</mo></mrow><annotation-xml encoding="MathML-Content" id="A4.Ex65.m1.3b"><apply id="A4.Ex65.m1.3.3.2.cmml" xref="A4.Ex65.m1.3.3.1"><csymbol cd="latexml" id="A4.Ex65.m1.3.3.2.1.cmml" xref="A4.Ex65.m1.3.3.1.2">norm</csymbol><apply id="A4.Ex65.m1.3.3.1.1.cmml" xref="A4.Ex65.m1.3.3.1.1"><minus id="A4.Ex65.m1.3.3.1.1.1.cmml" xref="A4.Ex65.m1.3.3.1.1.1"></minus><apply id="A4.Ex65.m1.3.3.1.1.2.cmml" xref="A4.Ex65.m1.3.3.1.1.2"><times id="A4.Ex65.m1.3.3.1.1.2.1.cmml" xref="A4.Ex65.m1.3.3.1.1.2.1"></times><apply id="A4.Ex65.m1.3.3.1.1.2.2.cmml" xref="A4.Ex65.m1.3.3.1.1.2.2"><csymbol cd="ambiguous" id="A4.Ex65.m1.3.3.1.1.2.2.1.cmml" xref="A4.Ex65.m1.3.3.1.1.2.2">subscript</csymbol><ci id="A4.Ex65.m1.3.3.1.1.2.2.2.cmml" xref="A4.Ex65.m1.3.3.1.1.2.2.2">𝑓</ci><ci id="A4.Ex65.m1.3.3.1.1.2.2.3.cmml" xref="A4.Ex65.m1.3.3.1.1.2.2.3">𝜎</ci></apply><ci id="A4.Ex65.m1.1.1.cmml" xref="A4.Ex65.m1.1.1">𝐱</ci></apply><apply id="A4.Ex65.m1.3.3.1.1.3.cmml" xref="A4.Ex65.m1.3.3.1.1.3"><times id="A4.Ex65.m1.3.3.1.1.3.1.cmml" xref="A4.Ex65.m1.3.3.1.1.3.1"></times><ci id="A4.Ex65.m1.3.3.1.1.3.2.cmml" xref="A4.Ex65.m1.3.3.1.1.3.2">𝑓</ci><ci id="A4.Ex65.m1.2.2.cmml" xref="A4.Ex65.m1.2.2">𝐱</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.Ex65.m1.3c">\displaystyle\|f_{\sigma}({\mathbf{x}})-f({\mathbf{x}})\|</annotation><annotation encoding="application/x-llamapun" id="A4.Ex65.m1.3d">∥ italic_f start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ( bold_x ) - italic_f ( bold_x ) ∥</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\leq\int_{{\mathbb{R}}^{d}}{\|f({\mathbf{x}}-{\tilde{\mathbf{x}}}% )-f({\mathbf{x}})\|\mathcal{N}({\tilde{\mathbf{x}}};0,\sigma^{2}\mathbf{I}_{d}% )d{\tilde{\mathbf{x}}}}" class="ltx_Math" display="inline" id="A4.Ex65.m2.5"><semantics id="A4.Ex65.m2.5a"><mrow id="A4.Ex65.m2.5.5" xref="A4.Ex65.m2.5.5.cmml"><mi id="A4.Ex65.m2.5.5.4" xref="A4.Ex65.m2.5.5.4.cmml"></mi><mo id="A4.Ex65.m2.5.5.3" xref="A4.Ex65.m2.5.5.3.cmml">≤</mo><mrow id="A4.Ex65.m2.5.5.2" xref="A4.Ex65.m2.5.5.2.cmml"><mstyle displaystyle="true" id="A4.Ex65.m2.5.5.2.3" xref="A4.Ex65.m2.5.5.2.3.cmml"><msub id="A4.Ex65.m2.5.5.2.3a" xref="A4.Ex65.m2.5.5.2.3.cmml"><mo id="A4.Ex65.m2.5.5.2.3.2" xref="A4.Ex65.m2.5.5.2.3.2.cmml">∫</mo><msup id="A4.Ex65.m2.5.5.2.3.3" xref="A4.Ex65.m2.5.5.2.3.3.cmml"><mi id="A4.Ex65.m2.5.5.2.3.3.2" xref="A4.Ex65.m2.5.5.2.3.3.2.cmml">ℝ</mi><mi id="A4.Ex65.m2.5.5.2.3.3.3" xref="A4.Ex65.m2.5.5.2.3.3.3.cmml">d</mi></msup></msub></mstyle><mrow id="A4.Ex65.m2.5.5.2.2" xref="A4.Ex65.m2.5.5.2.2.cmml"><mrow id="A4.Ex65.m2.4.4.1.1.1.1" xref="A4.Ex65.m2.4.4.1.1.1.2.cmml"><mo id="A4.Ex65.m2.4.4.1.1.1.1.2" stretchy="false" xref="A4.Ex65.m2.4.4.1.1.1.2.1.cmml">‖</mo><mrow id="A4.Ex65.m2.4.4.1.1.1.1.1" xref="A4.Ex65.m2.4.4.1.1.1.1.1.cmml"><mrow id="A4.Ex65.m2.4.4.1.1.1.1.1.1" xref="A4.Ex65.m2.4.4.1.1.1.1.1.1.cmml"><mi id="A4.Ex65.m2.4.4.1.1.1.1.1.1.3" xref="A4.Ex65.m2.4.4.1.1.1.1.1.1.3.cmml">f</mi><mo id="A4.Ex65.m2.4.4.1.1.1.1.1.1.2" xref="A4.Ex65.m2.4.4.1.1.1.1.1.1.2.cmml">⁢</mo><mrow id="A4.Ex65.m2.4.4.1.1.1.1.1.1.1.1" xref="A4.Ex65.m2.4.4.1.1.1.1.1.1.1.1.1.cmml"><mo id="A4.Ex65.m2.4.4.1.1.1.1.1.1.1.1.2" stretchy="false" xref="A4.Ex65.m2.4.4.1.1.1.1.1.1.1.1.1.cmml">(</mo><mrow id="A4.Ex65.m2.4.4.1.1.1.1.1.1.1.1.1" xref="A4.Ex65.m2.4.4.1.1.1.1.1.1.1.1.1.cmml"><mi id="A4.Ex65.m2.4.4.1.1.1.1.1.1.1.1.1.2" xref="A4.Ex65.m2.4.4.1.1.1.1.1.1.1.1.1.2.cmml">𝐱</mi><mo id="A4.Ex65.m2.4.4.1.1.1.1.1.1.1.1.1.1" xref="A4.Ex65.m2.4.4.1.1.1.1.1.1.1.1.1.1.cmml">−</mo><mover accent="true" id="A4.Ex65.m2.4.4.1.1.1.1.1.1.1.1.1.3" xref="A4.Ex65.m2.4.4.1.1.1.1.1.1.1.1.1.3.cmml"><mi id="A4.Ex65.m2.4.4.1.1.1.1.1.1.1.1.1.3.2" xref="A4.Ex65.m2.4.4.1.1.1.1.1.1.1.1.1.3.2.cmml">𝐱</mi><mo id="A4.Ex65.m2.4.4.1.1.1.1.1.1.1.1.1.3.1" xref="A4.Ex65.m2.4.4.1.1.1.1.1.1.1.1.1.3.1.cmml">~</mo></mover></mrow><mo id="A4.Ex65.m2.4.4.1.1.1.1.1.1.1.1.3" stretchy="false" xref="A4.Ex65.m2.4.4.1.1.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo 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id="A4.Ex66.m1.5c">\displaystyle\leq\int_{{\mathbb{R}}^{d}}{\|f({\mathbf{x}}-\sigma{\tilde{% \mathbf{x}}})-f({\mathbf{x}})\|\mathcal{N}({\tilde{\mathbf{x}}};0,\mathbf{I}_{% d})d{\tilde{\mathbf{x}}}}</annotation><annotation encoding="application/x-llamapun" id="A4.Ex66.m1.5d">≤ ∫ start_POSTSUBSCRIPT blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ∥ italic_f ( bold_x - italic_σ over~ start_ARG bold_x end_ARG ) - italic_f ( bold_x ) ∥ caligraphic_N ( over~ start_ARG bold_x end_ARG ; 0 , bold_I start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT ) italic_d over~ start_ARG bold_x end_ARG</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="A4.Ex67"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_eqn_cell"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math 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id="A4.Ex67.m1.11.11.1.1.1.1.1.1" xref="A4.Ex67.m1.11.11.1.1.1.1.1.1.cmml"><mrow id="A4.Ex67.m1.11.11.1.1.1.1.1.1.1" xref="A4.Ex67.m1.11.11.1.1.1.1.1.1.1.cmml"><mi id="A4.Ex67.m1.11.11.1.1.1.1.1.1.1.3" xref="A4.Ex67.m1.11.11.1.1.1.1.1.1.1.3.cmml">f</mi><mo id="A4.Ex67.m1.11.11.1.1.1.1.1.1.1.2" xref="A4.Ex67.m1.11.11.1.1.1.1.1.1.1.2.cmml">⁢</mo><mrow id="A4.Ex67.m1.11.11.1.1.1.1.1.1.1.1.1" xref="A4.Ex67.m1.11.11.1.1.1.1.1.1.1.1.1.1.cmml"><mo id="A4.Ex67.m1.11.11.1.1.1.1.1.1.1.1.1.2" stretchy="false" xref="A4.Ex67.m1.11.11.1.1.1.1.1.1.1.1.1.1.cmml">(</mo><mrow id="A4.Ex67.m1.11.11.1.1.1.1.1.1.1.1.1.1" xref="A4.Ex67.m1.11.11.1.1.1.1.1.1.1.1.1.1.cmml"><mi id="A4.Ex67.m1.11.11.1.1.1.1.1.1.1.1.1.1.2" xref="A4.Ex67.m1.11.11.1.1.1.1.1.1.1.1.1.1.2.cmml">𝐱</mi><mo id="A4.Ex67.m1.11.11.1.1.1.1.1.1.1.1.1.1.1" xref="A4.Ex67.m1.11.11.1.1.1.1.1.1.1.1.1.1.1.cmml">−</mo><mrow id="A4.Ex67.m1.11.11.1.1.1.1.1.1.1.1.1.1.3" xref="A4.Ex67.m1.11.11.1.1.1.1.1.1.1.1.1.1.3.cmml"><mi id="A4.Ex67.m1.11.11.1.1.1.1.1.1.1.1.1.1.3.2" xref="A4.Ex67.m1.11.11.1.1.1.1.1.1.1.1.1.1.3.2.cmml">σ</mi><mo id="A4.Ex67.m1.11.11.1.1.1.1.1.1.1.1.1.1.3.1" xref="A4.Ex67.m1.11.11.1.1.1.1.1.1.1.1.1.1.3.1.cmml">⁢</mo><mover accent="true" id="A4.Ex67.m1.11.11.1.1.1.1.1.1.1.1.1.1.3.3" xref="A4.Ex67.m1.11.11.1.1.1.1.1.1.1.1.1.1.3.3.cmml"><mi id="A4.Ex67.m1.11.11.1.1.1.1.1.1.1.1.1.1.3.3.2" xref="A4.Ex67.m1.11.11.1.1.1.1.1.1.1.1.1.1.3.3.2.cmml">𝐱</mi><mo id="A4.Ex67.m1.11.11.1.1.1.1.1.1.1.1.1.1.3.3.1" xref="A4.Ex67.m1.11.11.1.1.1.1.1.1.1.1.1.1.3.3.1.cmml">~</mo></mover></mrow></mrow><mo id="A4.Ex67.m1.11.11.1.1.1.1.1.1.1.1.1.3" stretchy="false" xref="A4.Ex67.m1.11.11.1.1.1.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="A4.Ex67.m1.11.11.1.1.1.1.1.1.2" xref="A4.Ex67.m1.11.11.1.1.1.1.1.1.2.cmml">−</mo><mrow id="A4.Ex67.m1.11.11.1.1.1.1.1.1.3" xref="A4.Ex67.m1.11.11.1.1.1.1.1.1.3.cmml"><mi id="A4.Ex67.m1.11.11.1.1.1.1.1.1.3.2" xref="A4.Ex67.m1.11.11.1.1.1.1.1.1.3.2.cmml">f</mi><mo id="A4.Ex67.m1.11.11.1.1.1.1.1.1.3.1" xref="A4.Ex67.m1.11.11.1.1.1.1.1.1.3.1.cmml">⁢</mo><mrow id="A4.Ex67.m1.11.11.1.1.1.1.1.1.3.3.2" xref="A4.Ex67.m1.11.11.1.1.1.1.1.1.3.cmml"><mo id="A4.Ex67.m1.11.11.1.1.1.1.1.1.3.3.2.1" stretchy="false" xref="A4.Ex67.m1.11.11.1.1.1.1.1.1.3.cmml">(</mo><mi id="A4.Ex67.m1.5.5" xref="A4.Ex67.m1.5.5.cmml">𝐱</mi><mo id="A4.Ex67.m1.11.11.1.1.1.1.1.1.3.3.2.2" stretchy="false" xref="A4.Ex67.m1.11.11.1.1.1.1.1.1.3.cmml">)</mo></mrow></mrow></mrow><mo id="A4.Ex67.m1.11.11.1.1.1.1.1.3" stretchy="false" xref="A4.Ex67.m1.11.11.1.1.1.1.2.1.cmml">‖</mo></mrow><mo id="A4.Ex67.m1.12.12.2.2.2.3" xref="A4.Ex67.m1.12.12.2.2.2.3.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="A4.Ex67.m1.12.12.2.2.2.4" xref="A4.Ex67.m1.12.12.2.2.2.4.cmml">𝒩</mi><mo id="A4.Ex67.m1.12.12.2.2.2.3a" xref="A4.Ex67.m1.12.12.2.2.2.3.cmml">⁢</mo><mrow id="A4.Ex67.m1.12.12.2.2.2.2.1" xref="A4.Ex67.m1.12.12.2.2.2.2.2.cmml"><mo id="A4.Ex67.m1.12.12.2.2.2.2.1.2" stretchy="false" xref="A4.Ex67.m1.12.12.2.2.2.2.2.cmml">(</mo><mover accent="true" id="A4.Ex67.m1.6.6" xref="A4.Ex67.m1.6.6.cmml"><mi id="A4.Ex67.m1.6.6.2" xref="A4.Ex67.m1.6.6.2.cmml">𝐱</mi><mo id="A4.Ex67.m1.6.6.1" xref="A4.Ex67.m1.6.6.1.cmml">~</mo></mover><mo id="A4.Ex67.m1.12.12.2.2.2.2.1.3" xref="A4.Ex67.m1.12.12.2.2.2.2.2.cmml">;</mo><mn id="A4.Ex67.m1.7.7" xref="A4.Ex67.m1.7.7.cmml">0</mn><mo id="A4.Ex67.m1.12.12.2.2.2.2.1.4" xref="A4.Ex67.m1.12.12.2.2.2.2.2.cmml">,</mo><msub id="A4.Ex67.m1.12.12.2.2.2.2.1.1" xref="A4.Ex67.m1.12.12.2.2.2.2.1.1.cmml"><mi id="A4.Ex67.m1.12.12.2.2.2.2.1.1.2" xref="A4.Ex67.m1.12.12.2.2.2.2.1.1.2.cmml">𝐈</mi><mi id="A4.Ex67.m1.12.12.2.2.2.2.1.1.3" xref="A4.Ex67.m1.12.12.2.2.2.2.1.1.3.cmml">d</mi></msub><mo id="A4.Ex67.m1.12.12.2.2.2.2.1.5" stretchy="false" xref="A4.Ex67.m1.12.12.2.2.2.2.2.cmml">)</mo></mrow><mo id="A4.Ex67.m1.12.12.2.2.2.3b" lspace="0em" xref="A4.Ex67.m1.12.12.2.2.2.3.cmml">⁢</mo><mrow id="A4.Ex67.m1.12.12.2.2.2.5" xref="A4.Ex67.m1.12.12.2.2.2.5.cmml"><mo id="A4.Ex67.m1.12.12.2.2.2.5.1" rspace="0em" xref="A4.Ex67.m1.12.12.2.2.2.5.1.cmml">𝑑</mo><mover accent="true" id="A4.Ex67.m1.12.12.2.2.2.5.2" xref="A4.Ex67.m1.12.12.2.2.2.5.2.cmml"><mi id="A4.Ex67.m1.12.12.2.2.2.5.2.2" xref="A4.Ex67.m1.12.12.2.2.2.5.2.2.cmml">𝐱</mi><mo id="A4.Ex67.m1.12.12.2.2.2.5.2.1" xref="A4.Ex67.m1.12.12.2.2.2.5.2.1.cmml">~</mo></mover></mrow></mrow></mrow><mo id="A4.Ex67.m1.14.14.4.5" xref="A4.Ex67.m1.14.14.4.5.cmml">+</mo><mrow id="A4.Ex67.m1.14.14.4.4" xref="A4.Ex67.m1.14.14.4.4.cmml"><mstyle displaystyle="true" id="A4.Ex67.m1.14.14.4.4.3" xref="A4.Ex67.m1.14.14.4.4.3.cmml"><msub id="A4.Ex67.m1.14.14.4.4.3a" xref="A4.Ex67.m1.14.14.4.4.3.cmml"><mo id="A4.Ex67.m1.14.14.4.4.3.2" xref="A4.Ex67.m1.14.14.4.4.3.2.cmml">∫</mo><mrow id="A4.Ex67.m1.4.4.2" xref="A4.Ex67.m1.4.4.2.cmml"><msup id="A4.Ex67.m1.4.4.2.4" xref="A4.Ex67.m1.4.4.2.4.cmml"><mi id="A4.Ex67.m1.4.4.2.4.2" xref="A4.Ex67.m1.4.4.2.4.2.cmml">ℝ</mi><mi id="A4.Ex67.m1.4.4.2.4.3" xref="A4.Ex67.m1.4.4.2.4.3.cmml">d</mi></msup><mo id="A4.Ex67.m1.4.4.2.3" xref="A4.Ex67.m1.4.4.2.3.cmml">∖</mo><mrow id="A4.Ex67.m1.4.4.2.5" xref="A4.Ex67.m1.4.4.2.5.cmml"><mi class="ltx_font_mathcaligraphic" id="A4.Ex67.m1.4.4.2.5.2" xref="A4.Ex67.m1.4.4.2.5.2.cmml">ℬ</mi><mo id="A4.Ex67.m1.4.4.2.5.1" xref="A4.Ex67.m1.4.4.2.5.1.cmml">⁢</mo><mrow id="A4.Ex67.m1.4.4.2.5.3.2" xref="A4.Ex67.m1.4.4.2.5.3.1.cmml"><mo id="A4.Ex67.m1.4.4.2.5.3.2.1" stretchy="false" xref="A4.Ex67.m1.4.4.2.5.3.1.cmml">(</mo><mn id="A4.Ex67.m1.3.3.1.1" xref="A4.Ex67.m1.3.3.1.1.cmml">0</mn><mo id="A4.Ex67.m1.4.4.2.5.3.2.2" xref="A4.Ex67.m1.4.4.2.5.3.1.cmml">,</mo><mi id="A4.Ex67.m1.4.4.2.2" xref="A4.Ex67.m1.4.4.2.2.cmml">R</mi><mo id="A4.Ex67.m1.4.4.2.5.3.2.3" stretchy="false" xref="A4.Ex67.m1.4.4.2.5.3.1.cmml">)</mo></mrow></mrow></mrow></msub></mstyle><mrow id="A4.Ex67.m1.14.14.4.4.2" xref="A4.Ex67.m1.14.14.4.4.2.cmml"><mrow id="A4.Ex67.m1.13.13.3.3.1.1.1" xref="A4.Ex67.m1.13.13.3.3.1.1.2.cmml"><mo id="A4.Ex67.m1.13.13.3.3.1.1.1.2" stretchy="false" xref="A4.Ex67.m1.13.13.3.3.1.1.2.1.cmml">‖</mo><mrow id="A4.Ex67.m1.13.13.3.3.1.1.1.1" xref="A4.Ex67.m1.13.13.3.3.1.1.1.1.cmml"><mrow id="A4.Ex67.m1.13.13.3.3.1.1.1.1.1" xref="A4.Ex67.m1.13.13.3.3.1.1.1.1.1.cmml"><mi id="A4.Ex67.m1.13.13.3.3.1.1.1.1.1.3" xref="A4.Ex67.m1.13.13.3.3.1.1.1.1.1.3.cmml">f</mi><mo id="A4.Ex67.m1.13.13.3.3.1.1.1.1.1.2" xref="A4.Ex67.m1.13.13.3.3.1.1.1.1.1.2.cmml">⁢</mo><mrow id="A4.Ex67.m1.13.13.3.3.1.1.1.1.1.1.1" xref="A4.Ex67.m1.13.13.3.3.1.1.1.1.1.1.1.1.cmml"><mo id="A4.Ex67.m1.13.13.3.3.1.1.1.1.1.1.1.2" stretchy="false" xref="A4.Ex67.m1.13.13.3.3.1.1.1.1.1.1.1.1.cmml">(</mo><mrow id="A4.Ex67.m1.13.13.3.3.1.1.1.1.1.1.1.1" xref="A4.Ex67.m1.13.13.3.3.1.1.1.1.1.1.1.1.cmml"><mi id="A4.Ex67.m1.13.13.3.3.1.1.1.1.1.1.1.1.2" xref="A4.Ex67.m1.13.13.3.3.1.1.1.1.1.1.1.1.2.cmml">𝐱</mi><mo id="A4.Ex67.m1.13.13.3.3.1.1.1.1.1.1.1.1.1" xref="A4.Ex67.m1.13.13.3.3.1.1.1.1.1.1.1.1.1.cmml">−</mo><mrow id="A4.Ex67.m1.13.13.3.3.1.1.1.1.1.1.1.1.3" xref="A4.Ex67.m1.13.13.3.3.1.1.1.1.1.1.1.1.3.cmml"><mi id="A4.Ex67.m1.13.13.3.3.1.1.1.1.1.1.1.1.3.2" xref="A4.Ex67.m1.13.13.3.3.1.1.1.1.1.1.1.1.3.2.cmml">σ</mi><mo id="A4.Ex67.m1.13.13.3.3.1.1.1.1.1.1.1.1.3.1" xref="A4.Ex67.m1.13.13.3.3.1.1.1.1.1.1.1.1.3.1.cmml">⁢</mo><mover accent="true" id="A4.Ex67.m1.13.13.3.3.1.1.1.1.1.1.1.1.3.3" xref="A4.Ex67.m1.13.13.3.3.1.1.1.1.1.1.1.1.3.3.cmml"><mi id="A4.Ex67.m1.13.13.3.3.1.1.1.1.1.1.1.1.3.3.2" xref="A4.Ex67.m1.13.13.3.3.1.1.1.1.1.1.1.1.3.3.2.cmml">𝐱</mi><mo id="A4.Ex67.m1.13.13.3.3.1.1.1.1.1.1.1.1.3.3.1" xref="A4.Ex67.m1.13.13.3.3.1.1.1.1.1.1.1.1.3.3.1.cmml">~</mo></mover></mrow></mrow><mo id="A4.Ex67.m1.13.13.3.3.1.1.1.1.1.1.1.3" stretchy="false" xref="A4.Ex67.m1.13.13.3.3.1.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="A4.Ex67.m1.13.13.3.3.1.1.1.1.2" xref="A4.Ex67.m1.13.13.3.3.1.1.1.1.2.cmml">−</mo><mrow 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id="A4.Ex67.m1.9.9.1.cmml" xref="A4.Ex67.m1.9.9.1">~</ci><ci id="A4.Ex67.m1.9.9.2.cmml" xref="A4.Ex67.m1.9.9.2">𝐱</ci></apply><cn id="A4.Ex67.m1.10.10.cmml" type="integer" xref="A4.Ex67.m1.10.10">0</cn><apply id="A4.Ex67.m1.14.14.4.4.2.2.1.1.cmml" xref="A4.Ex67.m1.14.14.4.4.2.2.1.1"><csymbol cd="ambiguous" id="A4.Ex67.m1.14.14.4.4.2.2.1.1.1.cmml" xref="A4.Ex67.m1.14.14.4.4.2.2.1.1">subscript</csymbol><ci id="A4.Ex67.m1.14.14.4.4.2.2.1.1.2.cmml" xref="A4.Ex67.m1.14.14.4.4.2.2.1.1.2">𝐈</ci><ci id="A4.Ex67.m1.14.14.4.4.2.2.1.1.3.cmml" xref="A4.Ex67.m1.14.14.4.4.2.2.1.1.3">𝑑</ci></apply></list><apply id="A4.Ex67.m1.14.14.4.4.2.5.cmml" xref="A4.Ex67.m1.14.14.4.4.2.5"><csymbol cd="latexml" id="A4.Ex67.m1.14.14.4.4.2.5.1.cmml" xref="A4.Ex67.m1.14.14.4.4.2.5.1">differential-d</csymbol><apply id="A4.Ex67.m1.14.14.4.4.2.5.2.cmml" xref="A4.Ex67.m1.14.14.4.4.2.5.2"><ci id="A4.Ex67.m1.14.14.4.4.2.5.2.1.cmml" xref="A4.Ex67.m1.14.14.4.4.2.5.2.1">~</ci><ci id="A4.Ex67.m1.14.14.4.4.2.5.2.2.cmml" xref="A4.Ex67.m1.14.14.4.4.2.5.2.2">𝐱</ci></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.Ex67.m1.14c">\displaystyle\leq\int_{\mathcal{B}(0,R)}{\|f({\mathbf{x}}-\sigma{\tilde{% \mathbf{x}}})-f({\mathbf{x}})\|\mathcal{N}({\tilde{\mathbf{x}}};0,\mathbf{I}_{% d})d{\tilde{\mathbf{x}}}}+\int_{{\mathbb{R}}^{d}\setminus\mathcal{B}(0,R)}{\|f% ({\mathbf{x}}-\sigma{\tilde{\mathbf{x}}})-f({\mathbf{x}})\|\mathcal{N}({\tilde% {\mathbf{x}}};0,\mathbf{I}_{d})d{\tilde{\mathbf{x}}}}</annotation><annotation encoding="application/x-llamapun" id="A4.Ex67.m1.14d">≤ ∫ start_POSTSUBSCRIPT caligraphic_B ( 0 , italic_R ) end_POSTSUBSCRIPT ∥ italic_f ( bold_x - italic_σ over~ start_ARG bold_x end_ARG ) - italic_f ( bold_x ) ∥ caligraphic_N ( over~ start_ARG bold_x end_ARG ; 0 , bold_I start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT ) italic_d over~ start_ARG bold_x end_ARG + ∫ start_POSTSUBSCRIPT blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT ∖ caligraphic_B ( 0 , italic_R ) end_POSTSUBSCRIPT ∥ italic_f ( bold_x - italic_σ over~ start_ARG bold_x end_ARG ) - italic_f ( bold_x ) ∥ caligraphic_N ( over~ start_ARG bold_x end_ARG ; 0 , bold_I start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT ) italic_d over~ start_ARG bold_x end_ARG</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="A4.Ex68"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_eqn_cell"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\leq\epsilon." class="ltx_Math" display="inline" id="A4.Ex68.m1.1"><semantics id="A4.Ex68.m1.1a"><mrow id="A4.Ex68.m1.1.1.1" xref="A4.Ex68.m1.1.1.1.1.cmml"><mrow id="A4.Ex68.m1.1.1.1.1" xref="A4.Ex68.m1.1.1.1.1.cmml"><mi id="A4.Ex68.m1.1.1.1.1.2" xref="A4.Ex68.m1.1.1.1.1.2.cmml"></mi><mo id="A4.Ex68.m1.1.1.1.1.1" xref="A4.Ex68.m1.1.1.1.1.1.cmml">≤</mo><mi id="A4.Ex68.m1.1.1.1.1.3" xref="A4.Ex68.m1.1.1.1.1.3.cmml">ϵ</mi></mrow><mo id="A4.Ex68.m1.1.1.1.2" lspace="0em" xref="A4.Ex68.m1.1.1.1.1.cmml">.</mo></mrow><annotation-xml encoding="MathML-Content" id="A4.Ex68.m1.1b"><apply id="A4.Ex68.m1.1.1.1.1.cmml" xref="A4.Ex68.m1.1.1.1"><leq id="A4.Ex68.m1.1.1.1.1.1.cmml" xref="A4.Ex68.m1.1.1.1.1.1"></leq><csymbol cd="latexml" id="A4.Ex68.m1.1.1.1.1.2.cmml" xref="A4.Ex68.m1.1.1.1.1.2">absent</csymbol><ci id="A4.Ex68.m1.1.1.1.1.3.cmml" xref="A4.Ex68.m1.1.1.1.1.3">italic-ϵ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.Ex68.m1.1c">\displaystyle\leq\epsilon.</annotation><annotation encoding="application/x-llamapun" id="A4.Ex68.m1.1d">≤ italic_ϵ .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> </div> <div class="ltx_para" id="A4.SS1.4.p4"> <p class="ltx_p" id="A4.SS1.4.p4.6">So we have the uniform convergence of <math alttext="f_{\sigma}" class="ltx_Math" display="inline" id="A4.SS1.4.p4.1.m1.1"><semantics id="A4.SS1.4.p4.1.m1.1a"><msub id="A4.SS1.4.p4.1.m1.1.1" xref="A4.SS1.4.p4.1.m1.1.1.cmml"><mi id="A4.SS1.4.p4.1.m1.1.1.2" xref="A4.SS1.4.p4.1.m1.1.1.2.cmml">f</mi><mi id="A4.SS1.4.p4.1.m1.1.1.3" xref="A4.SS1.4.p4.1.m1.1.1.3.cmml">σ</mi></msub><annotation-xml encoding="MathML-Content" id="A4.SS1.4.p4.1.m1.1b"><apply id="A4.SS1.4.p4.1.m1.1.1.cmml" xref="A4.SS1.4.p4.1.m1.1.1"><csymbol cd="ambiguous" id="A4.SS1.4.p4.1.m1.1.1.1.cmml" xref="A4.SS1.4.p4.1.m1.1.1">subscript</csymbol><ci id="A4.SS1.4.p4.1.m1.1.1.2.cmml" xref="A4.SS1.4.p4.1.m1.1.1.2">𝑓</ci><ci id="A4.SS1.4.p4.1.m1.1.1.3.cmml" xref="A4.SS1.4.p4.1.m1.1.1.3">𝜎</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.SS1.4.p4.1.m1.1c">f_{\sigma}</annotation><annotation encoding="application/x-llamapun" id="A4.SS1.4.p4.1.m1.1d">italic_f start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT</annotation></semantics></math> to <math alttext="f" class="ltx_Math" display="inline" id="A4.SS1.4.p4.2.m2.1"><semantics id="A4.SS1.4.p4.2.m2.1a"><mi id="A4.SS1.4.p4.2.m2.1.1" xref="A4.SS1.4.p4.2.m2.1.1.cmml">f</mi><annotation-xml encoding="MathML-Content" id="A4.SS1.4.p4.2.m2.1b"><ci id="A4.SS1.4.p4.2.m2.1.1.cmml" xref="A4.SS1.4.p4.2.m2.1.1">𝑓</ci></annotation-xml><annotation encoding="application/x-tex" id="A4.SS1.4.p4.2.m2.1c">f</annotation><annotation encoding="application/x-llamapun" id="A4.SS1.4.p4.2.m2.1d">italic_f</annotation></semantics></math> on <math alttext="\mathbf{K}" class="ltx_Math" display="inline" id="A4.SS1.4.p4.3.m3.1"><semantics id="A4.SS1.4.p4.3.m3.1a"><mi id="A4.SS1.4.p4.3.m3.1.1" xref="A4.SS1.4.p4.3.m3.1.1.cmml">𝐊</mi><annotation-xml encoding="MathML-Content" id="A4.SS1.4.p4.3.m3.1b"><ci id="A4.SS1.4.p4.3.m3.1.1.cmml" xref="A4.SS1.4.p4.3.m3.1.1">𝐊</ci></annotation-xml><annotation encoding="application/x-tex" id="A4.SS1.4.p4.3.m3.1c">\mathbf{K}</annotation><annotation encoding="application/x-llamapun" id="A4.SS1.4.p4.3.m3.1d">bold_K</annotation></semantics></math>: <math alttext="\forall\epsilon>0" class="ltx_Math" display="inline" id="A4.SS1.4.p4.4.m4.1"><semantics id="A4.SS1.4.p4.4.m4.1a"><mrow id="A4.SS1.4.p4.4.m4.1.1" xref="A4.SS1.4.p4.4.m4.1.1.cmml"><mrow id="A4.SS1.4.p4.4.m4.1.1.2" xref="A4.SS1.4.p4.4.m4.1.1.2.cmml"><mo id="A4.SS1.4.p4.4.m4.1.1.2.1" rspace="0.167em" xref="A4.SS1.4.p4.4.m4.1.1.2.1.cmml">∀</mo><mi id="A4.SS1.4.p4.4.m4.1.1.2.2" xref="A4.SS1.4.p4.4.m4.1.1.2.2.cmml">ϵ</mi></mrow><mo id="A4.SS1.4.p4.4.m4.1.1.1" xref="A4.SS1.4.p4.4.m4.1.1.1.cmml">></mo><mn id="A4.SS1.4.p4.4.m4.1.1.3" xref="A4.SS1.4.p4.4.m4.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="A4.SS1.4.p4.4.m4.1b"><apply id="A4.SS1.4.p4.4.m4.1.1.cmml" xref="A4.SS1.4.p4.4.m4.1.1"><gt id="A4.SS1.4.p4.4.m4.1.1.1.cmml" xref="A4.SS1.4.p4.4.m4.1.1.1"></gt><apply id="A4.SS1.4.p4.4.m4.1.1.2.cmml" xref="A4.SS1.4.p4.4.m4.1.1.2"><csymbol cd="latexml" id="A4.SS1.4.p4.4.m4.1.1.2.1.cmml" xref="A4.SS1.4.p4.4.m4.1.1.2.1">for-all</csymbol><ci id="A4.SS1.4.p4.4.m4.1.1.2.2.cmml" xref="A4.SS1.4.p4.4.m4.1.1.2.2">italic-ϵ</ci></apply><cn id="A4.SS1.4.p4.4.m4.1.1.3.cmml" type="integer" xref="A4.SS1.4.p4.4.m4.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.SS1.4.p4.4.m4.1c">\forall\epsilon>0</annotation><annotation encoding="application/x-llamapun" id="A4.SS1.4.p4.4.m4.1d">∀ italic_ϵ > 0</annotation></semantics></math>, there exists <math alttext="\xi>0" class="ltx_Math" display="inline" id="A4.SS1.4.p4.5.m5.1"><semantics id="A4.SS1.4.p4.5.m5.1a"><mrow id="A4.SS1.4.p4.5.m5.1.1" xref="A4.SS1.4.p4.5.m5.1.1.cmml"><mi id="A4.SS1.4.p4.5.m5.1.1.2" xref="A4.SS1.4.p4.5.m5.1.1.2.cmml">ξ</mi><mo id="A4.SS1.4.p4.5.m5.1.1.1" xref="A4.SS1.4.p4.5.m5.1.1.1.cmml">></mo><mn id="A4.SS1.4.p4.5.m5.1.1.3" xref="A4.SS1.4.p4.5.m5.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="A4.SS1.4.p4.5.m5.1b"><apply id="A4.SS1.4.p4.5.m5.1.1.cmml" xref="A4.SS1.4.p4.5.m5.1.1"><gt id="A4.SS1.4.p4.5.m5.1.1.1.cmml" xref="A4.SS1.4.p4.5.m5.1.1.1"></gt><ci id="A4.SS1.4.p4.5.m5.1.1.2.cmml" xref="A4.SS1.4.p4.5.m5.1.1.2">𝜉</ci><cn id="A4.SS1.4.p4.5.m5.1.1.3.cmml" type="integer" xref="A4.SS1.4.p4.5.m5.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.SS1.4.p4.5.m5.1c">\xi>0</annotation><annotation encoding="application/x-llamapun" id="A4.SS1.4.p4.5.m5.1d">italic_ξ > 0</annotation></semantics></math> such that <math alttext="\forall\sigma\in[0,\xi]" class="ltx_Math" display="inline" id="A4.SS1.4.p4.6.m6.2"><semantics id="A4.SS1.4.p4.6.m6.2a"><mrow id="A4.SS1.4.p4.6.m6.2.3" xref="A4.SS1.4.p4.6.m6.2.3.cmml"><mrow id="A4.SS1.4.p4.6.m6.2.3.2" xref="A4.SS1.4.p4.6.m6.2.3.2.cmml"><mo id="A4.SS1.4.p4.6.m6.2.3.2.1" rspace="0.167em" xref="A4.SS1.4.p4.6.m6.2.3.2.1.cmml">∀</mo><mi id="A4.SS1.4.p4.6.m6.2.3.2.2" xref="A4.SS1.4.p4.6.m6.2.3.2.2.cmml">σ</mi></mrow><mo id="A4.SS1.4.p4.6.m6.2.3.1" xref="A4.SS1.4.p4.6.m6.2.3.1.cmml">∈</mo><mrow id="A4.SS1.4.p4.6.m6.2.3.3.2" xref="A4.SS1.4.p4.6.m6.2.3.3.1.cmml"><mo id="A4.SS1.4.p4.6.m6.2.3.3.2.1" stretchy="false" xref="A4.SS1.4.p4.6.m6.2.3.3.1.cmml">[</mo><mn id="A4.SS1.4.p4.6.m6.1.1" xref="A4.SS1.4.p4.6.m6.1.1.cmml">0</mn><mo id="A4.SS1.4.p4.6.m6.2.3.3.2.2" xref="A4.SS1.4.p4.6.m6.2.3.3.1.cmml">,</mo><mi id="A4.SS1.4.p4.6.m6.2.2" xref="A4.SS1.4.p4.6.m6.2.2.cmml">ξ</mi><mo id="A4.SS1.4.p4.6.m6.2.3.3.2.3" stretchy="false" xref="A4.SS1.4.p4.6.m6.2.3.3.1.cmml">]</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A4.SS1.4.p4.6.m6.2b"><apply id="A4.SS1.4.p4.6.m6.2.3.cmml" xref="A4.SS1.4.p4.6.m6.2.3"><in id="A4.SS1.4.p4.6.m6.2.3.1.cmml" xref="A4.SS1.4.p4.6.m6.2.3.1"></in><apply id="A4.SS1.4.p4.6.m6.2.3.2.cmml" xref="A4.SS1.4.p4.6.m6.2.3.2"><csymbol cd="latexml" id="A4.SS1.4.p4.6.m6.2.3.2.1.cmml" xref="A4.SS1.4.p4.6.m6.2.3.2.1">for-all</csymbol><ci id="A4.SS1.4.p4.6.m6.2.3.2.2.cmml" xref="A4.SS1.4.p4.6.m6.2.3.2.2">𝜎</ci></apply><interval closure="closed" id="A4.SS1.4.p4.6.m6.2.3.3.1.cmml" xref="A4.SS1.4.p4.6.m6.2.3.3.2"><cn id="A4.SS1.4.p4.6.m6.1.1.cmml" type="integer" xref="A4.SS1.4.p4.6.m6.1.1">0</cn><ci id="A4.SS1.4.p4.6.m6.2.2.cmml" xref="A4.SS1.4.p4.6.m6.2.2">𝜉</ci></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.SS1.4.p4.6.m6.2c">\forall\sigma\in[0,\xi]</annotation><annotation encoding="application/x-llamapun" id="A4.SS1.4.p4.6.m6.2d">∀ italic_σ ∈ [ 0 , italic_ξ ]</annotation></semantics></math></p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A8.EGx43"> <tbody id="A4.Ex69"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\sup_{{\mathbf{x}}\in\mathbf{K}}\|f_{\sigma}({\mathbf{x}})-f({% \mathbf{x}})\|" class="ltx_Math" display="inline" id="A4.Ex69.m1.3"><semantics id="A4.Ex69.m1.3a"><mrow id="A4.Ex69.m1.3.3" xref="A4.Ex69.m1.3.3.cmml"><munder id="A4.Ex69.m1.3.3.2" xref="A4.Ex69.m1.3.3.2.cmml"><mo id="A4.Ex69.m1.3.3.2.2" movablelimits="false" xref="A4.Ex69.m1.3.3.2.2.cmml">sup</mo><mrow id="A4.Ex69.m1.3.3.2.3" xref="A4.Ex69.m1.3.3.2.3.cmml"><mi id="A4.Ex69.m1.3.3.2.3.2" xref="A4.Ex69.m1.3.3.2.3.2.cmml">𝐱</mi><mo id="A4.Ex69.m1.3.3.2.3.1" xref="A4.Ex69.m1.3.3.2.3.1.cmml">∈</mo><mi id="A4.Ex69.m1.3.3.2.3.3" xref="A4.Ex69.m1.3.3.2.3.3.cmml">𝐊</mi></mrow></munder><mrow id="A4.Ex69.m1.3.3.1.1" xref="A4.Ex69.m1.3.3.1.2.cmml"><mo id="A4.Ex69.m1.3.3.1.1.2" lspace="0em" stretchy="false" xref="A4.Ex69.m1.3.3.1.2.1.cmml">‖</mo><mrow id="A4.Ex69.m1.3.3.1.1.1" xref="A4.Ex69.m1.3.3.1.1.1.cmml"><mrow id="A4.Ex69.m1.3.3.1.1.1.2" xref="A4.Ex69.m1.3.3.1.1.1.2.cmml"><msub id="A4.Ex69.m1.3.3.1.1.1.2.2" xref="A4.Ex69.m1.3.3.1.1.1.2.2.cmml"><mi id="A4.Ex69.m1.3.3.1.1.1.2.2.2" xref="A4.Ex69.m1.3.3.1.1.1.2.2.2.cmml">f</mi><mi id="A4.Ex69.m1.3.3.1.1.1.2.2.3" xref="A4.Ex69.m1.3.3.1.1.1.2.2.3.cmml">σ</mi></msub><mo id="A4.Ex69.m1.3.3.1.1.1.2.1" xref="A4.Ex69.m1.3.3.1.1.1.2.1.cmml">⁢</mo><mrow id="A4.Ex69.m1.3.3.1.1.1.2.3.2" xref="A4.Ex69.m1.3.3.1.1.1.2.cmml"><mo id="A4.Ex69.m1.3.3.1.1.1.2.3.2.1" stretchy="false" xref="A4.Ex69.m1.3.3.1.1.1.2.cmml">(</mo><mi id="A4.Ex69.m1.1.1" xref="A4.Ex69.m1.1.1.cmml">𝐱</mi><mo id="A4.Ex69.m1.3.3.1.1.1.2.3.2.2" stretchy="false" xref="A4.Ex69.m1.3.3.1.1.1.2.cmml">)</mo></mrow></mrow><mo id="A4.Ex69.m1.3.3.1.1.1.1" xref="A4.Ex69.m1.3.3.1.1.1.1.cmml">−</mo><mrow id="A4.Ex69.m1.3.3.1.1.1.3" xref="A4.Ex69.m1.3.3.1.1.1.3.cmml"><mi id="A4.Ex69.m1.3.3.1.1.1.3.2" xref="A4.Ex69.m1.3.3.1.1.1.3.2.cmml">f</mi><mo id="A4.Ex69.m1.3.3.1.1.1.3.1" xref="A4.Ex69.m1.3.3.1.1.1.3.1.cmml">⁢</mo><mrow id="A4.Ex69.m1.3.3.1.1.1.3.3.2" xref="A4.Ex69.m1.3.3.1.1.1.3.cmml"><mo id="A4.Ex69.m1.3.3.1.1.1.3.3.2.1" stretchy="false" xref="A4.Ex69.m1.3.3.1.1.1.3.cmml">(</mo><mi id="A4.Ex69.m1.2.2" xref="A4.Ex69.m1.2.2.cmml">𝐱</mi><mo id="A4.Ex69.m1.3.3.1.1.1.3.3.2.2" stretchy="false" xref="A4.Ex69.m1.3.3.1.1.1.3.cmml">)</mo></mrow></mrow></mrow><mo id="A4.Ex69.m1.3.3.1.1.3" stretchy="false" xref="A4.Ex69.m1.3.3.1.2.1.cmml">‖</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A4.Ex69.m1.3b"><apply id="A4.Ex69.m1.3.3.cmml" xref="A4.Ex69.m1.3.3"><apply id="A4.Ex69.m1.3.3.2.cmml" xref="A4.Ex69.m1.3.3.2"><csymbol cd="ambiguous" id="A4.Ex69.m1.3.3.2.1.cmml" xref="A4.Ex69.m1.3.3.2">subscript</csymbol><csymbol cd="latexml" id="A4.Ex69.m1.3.3.2.2.cmml" xref="A4.Ex69.m1.3.3.2.2">supremum</csymbol><apply id="A4.Ex69.m1.3.3.2.3.cmml" xref="A4.Ex69.m1.3.3.2.3"><in id="A4.Ex69.m1.3.3.2.3.1.cmml" xref="A4.Ex69.m1.3.3.2.3.1"></in><ci id="A4.Ex69.m1.3.3.2.3.2.cmml" xref="A4.Ex69.m1.3.3.2.3.2">𝐱</ci><ci id="A4.Ex69.m1.3.3.2.3.3.cmml" xref="A4.Ex69.m1.3.3.2.3.3">𝐊</ci></apply></apply><apply id="A4.Ex69.m1.3.3.1.2.cmml" xref="A4.Ex69.m1.3.3.1.1"><csymbol cd="latexml" id="A4.Ex69.m1.3.3.1.2.1.cmml" xref="A4.Ex69.m1.3.3.1.1.2">norm</csymbol><apply id="A4.Ex69.m1.3.3.1.1.1.cmml" xref="A4.Ex69.m1.3.3.1.1.1"><minus id="A4.Ex69.m1.3.3.1.1.1.1.cmml" xref="A4.Ex69.m1.3.3.1.1.1.1"></minus><apply id="A4.Ex69.m1.3.3.1.1.1.2.cmml" xref="A4.Ex69.m1.3.3.1.1.1.2"><times id="A4.Ex69.m1.3.3.1.1.1.2.1.cmml" xref="A4.Ex69.m1.3.3.1.1.1.2.1"></times><apply id="A4.Ex69.m1.3.3.1.1.1.2.2.cmml" xref="A4.Ex69.m1.3.3.1.1.1.2.2"><csymbol cd="ambiguous" id="A4.Ex69.m1.3.3.1.1.1.2.2.1.cmml" xref="A4.Ex69.m1.3.3.1.1.1.2.2">subscript</csymbol><ci id="A4.Ex69.m1.3.3.1.1.1.2.2.2.cmml" xref="A4.Ex69.m1.3.3.1.1.1.2.2.2">𝑓</ci><ci id="A4.Ex69.m1.3.3.1.1.1.2.2.3.cmml" xref="A4.Ex69.m1.3.3.1.1.1.2.2.3">𝜎</ci></apply><ci id="A4.Ex69.m1.1.1.cmml" xref="A4.Ex69.m1.1.1">𝐱</ci></apply><apply id="A4.Ex69.m1.3.3.1.1.1.3.cmml" xref="A4.Ex69.m1.3.3.1.1.1.3"><times id="A4.Ex69.m1.3.3.1.1.1.3.1.cmml" xref="A4.Ex69.m1.3.3.1.1.1.3.1"></times><ci id="A4.Ex69.m1.3.3.1.1.1.3.2.cmml" xref="A4.Ex69.m1.3.3.1.1.1.3.2">𝑓</ci><ci id="A4.Ex69.m1.2.2.cmml" xref="A4.Ex69.m1.2.2">𝐱</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.Ex69.m1.3c">\displaystyle\sup_{{\mathbf{x}}\in\mathbf{K}}\|f_{\sigma}({\mathbf{x}})-f({% \mathbf{x}})\|</annotation><annotation encoding="application/x-llamapun" id="A4.Ex69.m1.3d">roman_sup start_POSTSUBSCRIPT bold_x ∈ bold_K end_POSTSUBSCRIPT ∥ italic_f start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ( bold_x ) - italic_f ( bold_x ) ∥</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\leq\epsilon." class="ltx_Math" display="inline" id="A4.Ex69.m2.1"><semantics id="A4.Ex69.m2.1a"><mrow id="A4.Ex69.m2.1.1.1" xref="A4.Ex69.m2.1.1.1.1.cmml"><mrow id="A4.Ex69.m2.1.1.1.1" xref="A4.Ex69.m2.1.1.1.1.cmml"><mi id="A4.Ex69.m2.1.1.1.1.2" xref="A4.Ex69.m2.1.1.1.1.2.cmml"></mi><mo id="A4.Ex69.m2.1.1.1.1.1" xref="A4.Ex69.m2.1.1.1.1.1.cmml">≤</mo><mi id="A4.Ex69.m2.1.1.1.1.3" xref="A4.Ex69.m2.1.1.1.1.3.cmml">ϵ</mi></mrow><mo id="A4.Ex69.m2.1.1.1.2" lspace="0em" xref="A4.Ex69.m2.1.1.1.1.cmml">.</mo></mrow><annotation-xml encoding="MathML-Content" id="A4.Ex69.m2.1b"><apply id="A4.Ex69.m2.1.1.1.1.cmml" xref="A4.Ex69.m2.1.1.1"><leq id="A4.Ex69.m2.1.1.1.1.1.cmml" xref="A4.Ex69.m2.1.1.1.1.1"></leq><csymbol cd="latexml" id="A4.Ex69.m2.1.1.1.1.2.cmml" xref="A4.Ex69.m2.1.1.1.1.2">absent</csymbol><ci id="A4.Ex69.m2.1.1.1.1.3.cmml" xref="A4.Ex69.m2.1.1.1.1.3">italic-ϵ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.Ex69.m2.1c">\displaystyle\leq\epsilon.</annotation><annotation encoding="application/x-llamapun" id="A4.Ex69.m2.1d">≤ italic_ϵ .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> </div> <div class="ltx_para" id="A4.SS1.5.p5"> <p class="ltx_p" id="A4.SS1.5.p5.3">Applying this result with <math alttext="f=p" class="ltx_Math" display="inline" id="A4.SS1.5.p5.1.m1.1"><semantics id="A4.SS1.5.p5.1.m1.1a"><mrow id="A4.SS1.5.p5.1.m1.1.1" xref="A4.SS1.5.p5.1.m1.1.1.cmml"><mi id="A4.SS1.5.p5.1.m1.1.1.2" xref="A4.SS1.5.p5.1.m1.1.1.2.cmml">f</mi><mo id="A4.SS1.5.p5.1.m1.1.1.1" xref="A4.SS1.5.p5.1.m1.1.1.1.cmml">=</mo><mi id="A4.SS1.5.p5.1.m1.1.1.3" xref="A4.SS1.5.p5.1.m1.1.1.3.cmml">p</mi></mrow><annotation-xml encoding="MathML-Content" id="A4.SS1.5.p5.1.m1.1b"><apply id="A4.SS1.5.p5.1.m1.1.1.cmml" xref="A4.SS1.5.p5.1.m1.1.1"><eq id="A4.SS1.5.p5.1.m1.1.1.1.cmml" xref="A4.SS1.5.p5.1.m1.1.1.1"></eq><ci id="A4.SS1.5.p5.1.m1.1.1.2.cmml" xref="A4.SS1.5.p5.1.m1.1.1.2">𝑓</ci><ci id="A4.SS1.5.p5.1.m1.1.1.3.cmml" xref="A4.SS1.5.p5.1.m1.1.1.3">𝑝</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.SS1.5.p5.1.m1.1c">f=p</annotation><annotation encoding="application/x-llamapun" id="A4.SS1.5.p5.1.m1.1d">italic_f = italic_p</annotation></semantics></math> and <math alttext="f=\nabla p" class="ltx_Math" display="inline" id="A4.SS1.5.p5.2.m2.1"><semantics id="A4.SS1.5.p5.2.m2.1a"><mrow id="A4.SS1.5.p5.2.m2.1.1" xref="A4.SS1.5.p5.2.m2.1.1.cmml"><mi id="A4.SS1.5.p5.2.m2.1.1.2" xref="A4.SS1.5.p5.2.m2.1.1.2.cmml">f</mi><mo id="A4.SS1.5.p5.2.m2.1.1.1" xref="A4.SS1.5.p5.2.m2.1.1.1.cmml">=</mo><mrow id="A4.SS1.5.p5.2.m2.1.1.3" xref="A4.SS1.5.p5.2.m2.1.1.3.cmml"><mo id="A4.SS1.5.p5.2.m2.1.1.3.1" rspace="0.167em" xref="A4.SS1.5.p5.2.m2.1.1.3.1.cmml">∇</mo><mi id="A4.SS1.5.p5.2.m2.1.1.3.2" xref="A4.SS1.5.p5.2.m2.1.1.3.2.cmml">p</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="A4.SS1.5.p5.2.m2.1b"><apply id="A4.SS1.5.p5.2.m2.1.1.cmml" xref="A4.SS1.5.p5.2.m2.1.1"><eq id="A4.SS1.5.p5.2.m2.1.1.1.cmml" xref="A4.SS1.5.p5.2.m2.1.1.1"></eq><ci id="A4.SS1.5.p5.2.m2.1.1.2.cmml" xref="A4.SS1.5.p5.2.m2.1.1.2">𝑓</ci><apply id="A4.SS1.5.p5.2.m2.1.1.3.cmml" xref="A4.SS1.5.p5.2.m2.1.1.3"><ci id="A4.SS1.5.p5.2.m2.1.1.3.1.cmml" xref="A4.SS1.5.p5.2.m2.1.1.3.1">∇</ci><ci id="A4.SS1.5.p5.2.m2.1.1.3.2.cmml" xref="A4.SS1.5.p5.2.m2.1.1.3.2">𝑝</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.SS1.5.p5.2.m2.1c">f=\nabla p</annotation><annotation encoding="application/x-llamapun" id="A4.SS1.5.p5.2.m2.1d">italic_f = ∇ italic_p</annotation></semantics></math> (because <math alttext="\nabla p_{\sigma}=\nabla(p\ast\mathcal{N}_{\sigma})=(\nabla p)\ast\mathcal{N}_% {\sigma}" class="ltx_Math" display="inline" id="A4.SS1.5.p5.3.m3.3"><semantics id="A4.SS1.5.p5.3.m3.3a"><mrow id="A4.SS1.5.p5.3.m3.3.3" xref="A4.SS1.5.p5.3.m3.3.3.cmml"><mrow id="A4.SS1.5.p5.3.m3.3.3.4" xref="A4.SS1.5.p5.3.m3.3.3.4.cmml"><mo id="A4.SS1.5.p5.3.m3.3.3.4.1" rspace="0.167em" xref="A4.SS1.5.p5.3.m3.3.3.4.1.cmml">∇</mo><msub id="A4.SS1.5.p5.3.m3.3.3.4.2" xref="A4.SS1.5.p5.3.m3.3.3.4.2.cmml"><mi id="A4.SS1.5.p5.3.m3.3.3.4.2.2" xref="A4.SS1.5.p5.3.m3.3.3.4.2.2.cmml">p</mi><mi id="A4.SS1.5.p5.3.m3.3.3.4.2.3" xref="A4.SS1.5.p5.3.m3.3.3.4.2.3.cmml">σ</mi></msub></mrow><mo id="A4.SS1.5.p5.3.m3.3.3.5" xref="A4.SS1.5.p5.3.m3.3.3.5.cmml">=</mo><mrow id="A4.SS1.5.p5.3.m3.2.2.1.1" xref="A4.SS1.5.p5.3.m3.2.2.1.2.cmml"><mo id="A4.SS1.5.p5.3.m3.1.1" xref="A4.SS1.5.p5.3.m3.1.1.cmml">∇</mo><mrow id="A4.SS1.5.p5.3.m3.2.2.1.1.1" xref="A4.SS1.5.p5.3.m3.2.2.1.2.cmml"><mo id="A4.SS1.5.p5.3.m3.2.2.1.1.1.2" stretchy="false" xref="A4.SS1.5.p5.3.m3.2.2.1.2.cmml">(</mo><mrow id="A4.SS1.5.p5.3.m3.2.2.1.1.1.1" xref="A4.SS1.5.p5.3.m3.2.2.1.1.1.1.cmml"><mi id="A4.SS1.5.p5.3.m3.2.2.1.1.1.1.2" xref="A4.SS1.5.p5.3.m3.2.2.1.1.1.1.2.cmml">p</mi><mo id="A4.SS1.5.p5.3.m3.2.2.1.1.1.1.1" lspace="0.222em" rspace="0.222em" xref="A4.SS1.5.p5.3.m3.2.2.1.1.1.1.1.cmml">∗</mo><msub id="A4.SS1.5.p5.3.m3.2.2.1.1.1.1.3" xref="A4.SS1.5.p5.3.m3.2.2.1.1.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="A4.SS1.5.p5.3.m3.2.2.1.1.1.1.3.2" xref="A4.SS1.5.p5.3.m3.2.2.1.1.1.1.3.2.cmml">𝒩</mi><mi id="A4.SS1.5.p5.3.m3.2.2.1.1.1.1.3.3" xref="A4.SS1.5.p5.3.m3.2.2.1.1.1.1.3.3.cmml">σ</mi></msub></mrow><mo id="A4.SS1.5.p5.3.m3.2.2.1.1.1.3" stretchy="false" xref="A4.SS1.5.p5.3.m3.2.2.1.2.cmml">)</mo></mrow></mrow><mo id="A4.SS1.5.p5.3.m3.3.3.6" xref="A4.SS1.5.p5.3.m3.3.3.6.cmml">=</mo><mrow id="A4.SS1.5.p5.3.m3.3.3.2" xref="A4.SS1.5.p5.3.m3.3.3.2.cmml"><mrow id="A4.SS1.5.p5.3.m3.3.3.2.1.1" xref="A4.SS1.5.p5.3.m3.3.3.2.1.1.1.cmml"><mo id="A4.SS1.5.p5.3.m3.3.3.2.1.1.2" stretchy="false" xref="A4.SS1.5.p5.3.m3.3.3.2.1.1.1.cmml">(</mo><mrow id="A4.SS1.5.p5.3.m3.3.3.2.1.1.1" xref="A4.SS1.5.p5.3.m3.3.3.2.1.1.1.cmml"><mo id="A4.SS1.5.p5.3.m3.3.3.2.1.1.1.1" rspace="0.167em" xref="A4.SS1.5.p5.3.m3.3.3.2.1.1.1.1.cmml">∇</mo><mi id="A4.SS1.5.p5.3.m3.3.3.2.1.1.1.2" xref="A4.SS1.5.p5.3.m3.3.3.2.1.1.1.2.cmml">p</mi></mrow><mo id="A4.SS1.5.p5.3.m3.3.3.2.1.1.3" rspace="0.055em" stretchy="false" xref="A4.SS1.5.p5.3.m3.3.3.2.1.1.1.cmml">)</mo></mrow><mo id="A4.SS1.5.p5.3.m3.3.3.2.2" rspace="0.222em" xref="A4.SS1.5.p5.3.m3.3.3.2.2.cmml">∗</mo><msub id="A4.SS1.5.p5.3.m3.3.3.2.3" xref="A4.SS1.5.p5.3.m3.3.3.2.3.cmml"><mi class="ltx_font_mathcaligraphic" id="A4.SS1.5.p5.3.m3.3.3.2.3.2" xref="A4.SS1.5.p5.3.m3.3.3.2.3.2.cmml">𝒩</mi><mi id="A4.SS1.5.p5.3.m3.3.3.2.3.3" xref="A4.SS1.5.p5.3.m3.3.3.2.3.3.cmml">σ</mi></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="A4.SS1.5.p5.3.m3.3b"><apply id="A4.SS1.5.p5.3.m3.3.3.cmml" xref="A4.SS1.5.p5.3.m3.3.3"><and id="A4.SS1.5.p5.3.m3.3.3a.cmml" xref="A4.SS1.5.p5.3.m3.3.3"></and><apply id="A4.SS1.5.p5.3.m3.3.3b.cmml" xref="A4.SS1.5.p5.3.m3.3.3"><eq id="A4.SS1.5.p5.3.m3.3.3.5.cmml" xref="A4.SS1.5.p5.3.m3.3.3.5"></eq><apply id="A4.SS1.5.p5.3.m3.3.3.4.cmml" xref="A4.SS1.5.p5.3.m3.3.3.4"><ci id="A4.SS1.5.p5.3.m3.3.3.4.1.cmml" xref="A4.SS1.5.p5.3.m3.3.3.4.1">∇</ci><apply id="A4.SS1.5.p5.3.m3.3.3.4.2.cmml" xref="A4.SS1.5.p5.3.m3.3.3.4.2"><csymbol cd="ambiguous" id="A4.SS1.5.p5.3.m3.3.3.4.2.1.cmml" xref="A4.SS1.5.p5.3.m3.3.3.4.2">subscript</csymbol><ci id="A4.SS1.5.p5.3.m3.3.3.4.2.2.cmml" xref="A4.SS1.5.p5.3.m3.3.3.4.2.2">𝑝</ci><ci id="A4.SS1.5.p5.3.m3.3.3.4.2.3.cmml" xref="A4.SS1.5.p5.3.m3.3.3.4.2.3">𝜎</ci></apply></apply><apply id="A4.SS1.5.p5.3.m3.2.2.1.2.cmml" xref="A4.SS1.5.p5.3.m3.2.2.1.1"><ci id="A4.SS1.5.p5.3.m3.1.1.cmml" xref="A4.SS1.5.p5.3.m3.1.1">∇</ci><apply id="A4.SS1.5.p5.3.m3.2.2.1.1.1.1.cmml" xref="A4.SS1.5.p5.3.m3.2.2.1.1.1.1"><ci id="A4.SS1.5.p5.3.m3.2.2.1.1.1.1.1.cmml" xref="A4.SS1.5.p5.3.m3.2.2.1.1.1.1.1">∗</ci><ci id="A4.SS1.5.p5.3.m3.2.2.1.1.1.1.2.cmml" xref="A4.SS1.5.p5.3.m3.2.2.1.1.1.1.2">𝑝</ci><apply id="A4.SS1.5.p5.3.m3.2.2.1.1.1.1.3.cmml" xref="A4.SS1.5.p5.3.m3.2.2.1.1.1.1.3"><csymbol cd="ambiguous" id="A4.SS1.5.p5.3.m3.2.2.1.1.1.1.3.1.cmml" xref="A4.SS1.5.p5.3.m3.2.2.1.1.1.1.3">subscript</csymbol><ci id="A4.SS1.5.p5.3.m3.2.2.1.1.1.1.3.2.cmml" xref="A4.SS1.5.p5.3.m3.2.2.1.1.1.1.3.2">𝒩</ci><ci id="A4.SS1.5.p5.3.m3.2.2.1.1.1.1.3.3.cmml" xref="A4.SS1.5.p5.3.m3.2.2.1.1.1.1.3.3">𝜎</ci></apply></apply></apply></apply><apply id="A4.SS1.5.p5.3.m3.3.3c.cmml" xref="A4.SS1.5.p5.3.m3.3.3"><eq id="A4.SS1.5.p5.3.m3.3.3.6.cmml" xref="A4.SS1.5.p5.3.m3.3.3.6"></eq><share href="#A4.SS1.5.p5.3.m3.2.2.1.cmml" id="A4.SS1.5.p5.3.m3.3.3d.cmml" xref="A4.SS1.5.p5.3.m3.3.3"></share><apply id="A4.SS1.5.p5.3.m3.3.3.2.cmml" xref="A4.SS1.5.p5.3.m3.3.3.2"><ci id="A4.SS1.5.p5.3.m3.3.3.2.2.cmml" xref="A4.SS1.5.p5.3.m3.3.3.2.2">∗</ci><apply id="A4.SS1.5.p5.3.m3.3.3.2.1.1.1.cmml" xref="A4.SS1.5.p5.3.m3.3.3.2.1.1"><ci id="A4.SS1.5.p5.3.m3.3.3.2.1.1.1.1.cmml" xref="A4.SS1.5.p5.3.m3.3.3.2.1.1.1.1">∇</ci><ci id="A4.SS1.5.p5.3.m3.3.3.2.1.1.1.2.cmml" xref="A4.SS1.5.p5.3.m3.3.3.2.1.1.1.2">𝑝</ci></apply><apply id="A4.SS1.5.p5.3.m3.3.3.2.3.cmml" xref="A4.SS1.5.p5.3.m3.3.3.2.3"><csymbol cd="ambiguous" id="A4.SS1.5.p5.3.m3.3.3.2.3.1.cmml" xref="A4.SS1.5.p5.3.m3.3.3.2.3">subscript</csymbol><ci id="A4.SS1.5.p5.3.m3.3.3.2.3.2.cmml" xref="A4.SS1.5.p5.3.m3.3.3.2.3.2">𝒩</ci><ci id="A4.SS1.5.p5.3.m3.3.3.2.3.3.cmml" xref="A4.SS1.5.p5.3.m3.3.3.2.3.3">𝜎</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.SS1.5.p5.3.m3.3c">\nabla p_{\sigma}=\nabla(p\ast\mathcal{N}_{\sigma})=(\nabla p)\ast\mathcal{N}_% {\sigma}</annotation><annotation encoding="application/x-llamapun" id="A4.SS1.5.p5.3.m3.3d">∇ italic_p start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT = ∇ ( italic_p ∗ caligraphic_N start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ) = ( ∇ italic_p ) ∗ caligraphic_N start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT</annotation></semantics></math>, we finally get</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A8.EGx44"> <tbody id="A4.Ex70"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\sup_{\mathbf{K}}\|\nabla\log p_{\sigma}-\nabla\log p\|" class="ltx_Math" display="inline" id="A4.Ex70.m1.1"><semantics id="A4.Ex70.m1.1a"><mrow id="A4.Ex70.m1.1.1" xref="A4.Ex70.m1.1.1.cmml"><munder id="A4.Ex70.m1.1.1.2" xref="A4.Ex70.m1.1.1.2.cmml"><mo id="A4.Ex70.m1.1.1.2.2" movablelimits="false" xref="A4.Ex70.m1.1.1.2.2.cmml">sup</mo><mi id="A4.Ex70.m1.1.1.2.3" xref="A4.Ex70.m1.1.1.2.3.cmml">𝐊</mi></munder><mrow id="A4.Ex70.m1.1.1.1.1" xref="A4.Ex70.m1.1.1.1.2.cmml"><mo id="A4.Ex70.m1.1.1.1.1.2" lspace="0em" stretchy="false" xref="A4.Ex70.m1.1.1.1.2.1.cmml">‖</mo><mrow id="A4.Ex70.m1.1.1.1.1.1" xref="A4.Ex70.m1.1.1.1.1.1.cmml"><mrow id="A4.Ex70.m1.1.1.1.1.1.2" xref="A4.Ex70.m1.1.1.1.1.1.2.cmml"><mrow id="A4.Ex70.m1.1.1.1.1.1.2.1" xref="A4.Ex70.m1.1.1.1.1.1.2.1.cmml"><mo id="A4.Ex70.m1.1.1.1.1.1.2.1.1" rspace="0.167em" xref="A4.Ex70.m1.1.1.1.1.1.2.1.1.cmml">∇</mo><mi id="A4.Ex70.m1.1.1.1.1.1.2.1.2" xref="A4.Ex70.m1.1.1.1.1.1.2.1.2.cmml">log</mi></mrow><mo id="A4.Ex70.m1.1.1.1.1.1.2a" lspace="0.167em" xref="A4.Ex70.m1.1.1.1.1.1.2.cmml">⁡</mo><msub id="A4.Ex70.m1.1.1.1.1.1.2.2" xref="A4.Ex70.m1.1.1.1.1.1.2.2.cmml"><mi id="A4.Ex70.m1.1.1.1.1.1.2.2.2" xref="A4.Ex70.m1.1.1.1.1.1.2.2.2.cmml">p</mi><mi id="A4.Ex70.m1.1.1.1.1.1.2.2.3" xref="A4.Ex70.m1.1.1.1.1.1.2.2.3.cmml">σ</mi></msub></mrow><mo id="A4.Ex70.m1.1.1.1.1.1.1" xref="A4.Ex70.m1.1.1.1.1.1.1.cmml">−</mo><mrow id="A4.Ex70.m1.1.1.1.1.1.3" xref="A4.Ex70.m1.1.1.1.1.1.3.cmml"><mrow id="A4.Ex70.m1.1.1.1.1.1.3.1" xref="A4.Ex70.m1.1.1.1.1.1.3.1.cmml"><mo id="A4.Ex70.m1.1.1.1.1.1.3.1.1" rspace="0.167em" xref="A4.Ex70.m1.1.1.1.1.1.3.1.1.cmml">∇</mo><mi id="A4.Ex70.m1.1.1.1.1.1.3.1.2" xref="A4.Ex70.m1.1.1.1.1.1.3.1.2.cmml">log</mi></mrow><mo id="A4.Ex70.m1.1.1.1.1.1.3a" lspace="0.167em" xref="A4.Ex70.m1.1.1.1.1.1.3.cmml">⁡</mo><mi id="A4.Ex70.m1.1.1.1.1.1.3.2" xref="A4.Ex70.m1.1.1.1.1.1.3.2.cmml">p</mi></mrow></mrow><mo id="A4.Ex70.m1.1.1.1.1.3" stretchy="false" xref="A4.Ex70.m1.1.1.1.2.1.cmml">‖</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A4.Ex70.m1.1b"><apply id="A4.Ex70.m1.1.1.cmml" xref="A4.Ex70.m1.1.1"><apply id="A4.Ex70.m1.1.1.2.cmml" xref="A4.Ex70.m1.1.1.2"><csymbol cd="ambiguous" id="A4.Ex70.m1.1.1.2.1.cmml" xref="A4.Ex70.m1.1.1.2">subscript</csymbol><csymbol cd="latexml" id="A4.Ex70.m1.1.1.2.2.cmml" xref="A4.Ex70.m1.1.1.2.2">supremum</csymbol><ci id="A4.Ex70.m1.1.1.2.3.cmml" xref="A4.Ex70.m1.1.1.2.3">𝐊</ci></apply><apply id="A4.Ex70.m1.1.1.1.2.cmml" xref="A4.Ex70.m1.1.1.1.1"><csymbol cd="latexml" id="A4.Ex70.m1.1.1.1.2.1.cmml" xref="A4.Ex70.m1.1.1.1.1.2">norm</csymbol><apply id="A4.Ex70.m1.1.1.1.1.1.cmml" xref="A4.Ex70.m1.1.1.1.1.1"><minus id="A4.Ex70.m1.1.1.1.1.1.1.cmml" xref="A4.Ex70.m1.1.1.1.1.1.1"></minus><apply id="A4.Ex70.m1.1.1.1.1.1.2.cmml" xref="A4.Ex70.m1.1.1.1.1.1.2"><apply id="A4.Ex70.m1.1.1.1.1.1.2.1.cmml" xref="A4.Ex70.m1.1.1.1.1.1.2.1"><ci id="A4.Ex70.m1.1.1.1.1.1.2.1.1.cmml" xref="A4.Ex70.m1.1.1.1.1.1.2.1.1">∇</ci><log id="A4.Ex70.m1.1.1.1.1.1.2.1.2.cmml" xref="A4.Ex70.m1.1.1.1.1.1.2.1.2"></log></apply><apply id="A4.Ex70.m1.1.1.1.1.1.2.2.cmml" xref="A4.Ex70.m1.1.1.1.1.1.2.2"><csymbol cd="ambiguous" id="A4.Ex70.m1.1.1.1.1.1.2.2.1.cmml" xref="A4.Ex70.m1.1.1.1.1.1.2.2">subscript</csymbol><ci id="A4.Ex70.m1.1.1.1.1.1.2.2.2.cmml" xref="A4.Ex70.m1.1.1.1.1.1.2.2.2">𝑝</ci><ci id="A4.Ex70.m1.1.1.1.1.1.2.2.3.cmml" xref="A4.Ex70.m1.1.1.1.1.1.2.2.3">𝜎</ci></apply></apply><apply id="A4.Ex70.m1.1.1.1.1.1.3.cmml" xref="A4.Ex70.m1.1.1.1.1.1.3"><apply id="A4.Ex70.m1.1.1.1.1.1.3.1.cmml" xref="A4.Ex70.m1.1.1.1.1.1.3.1"><ci id="A4.Ex70.m1.1.1.1.1.1.3.1.1.cmml" xref="A4.Ex70.m1.1.1.1.1.1.3.1.1">∇</ci><log id="A4.Ex70.m1.1.1.1.1.1.3.1.2.cmml" xref="A4.Ex70.m1.1.1.1.1.1.3.1.2"></log></apply><ci id="A4.Ex70.m1.1.1.1.1.1.3.2.cmml" xref="A4.Ex70.m1.1.1.1.1.1.3.2">𝑝</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.Ex70.m1.1c">\displaystyle\sup_{\mathbf{K}}\|\nabla\log p_{\sigma}-\nabla\log p\|</annotation><annotation encoding="application/x-llamapun" 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start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ) end_ARG start_ARG italic_p start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT italic_p end_ARG ∥ .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="A4.SS1.5.p5.10">We define <math alttext="m_{p}=\min\{\inf_{{\mathbf{x}}\in\mathbf{K}}{p({\mathbf{x}})}\}>0" class="ltx_Math" display="inline" id="A4.SS1.5.p5.4.m1.3"><semantics id="A4.SS1.5.p5.4.m1.3a"><mrow id="A4.SS1.5.p5.4.m1.3.3" xref="A4.SS1.5.p5.4.m1.3.3.cmml"><msub id="A4.SS1.5.p5.4.m1.3.3.3" xref="A4.SS1.5.p5.4.m1.3.3.3.cmml"><mi id="A4.SS1.5.p5.4.m1.3.3.3.2" xref="A4.SS1.5.p5.4.m1.3.3.3.2.cmml">m</mi><mi id="A4.SS1.5.p5.4.m1.3.3.3.3" xref="A4.SS1.5.p5.4.m1.3.3.3.3.cmml">p</mi></msub><mo id="A4.SS1.5.p5.4.m1.3.3.4" xref="A4.SS1.5.p5.4.m1.3.3.4.cmml">=</mo><mrow id="A4.SS1.5.p5.4.m1.3.3.1.1" xref="A4.SS1.5.p5.4.m1.3.3.1.2.cmml"><mi id="A4.SS1.5.p5.4.m1.2.2" xref="A4.SS1.5.p5.4.m1.2.2.cmml">min</mi><mo 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xref="A4.SS1.5.p5.4.m1.3.3.1.1.1.1.1">subscript</csymbol><csymbol cd="latexml" id="A4.SS1.5.p5.4.m1.3.3.1.1.1.1.1.2.cmml" xref="A4.SS1.5.p5.4.m1.3.3.1.1.1.1.1.2">infimum</csymbol><apply id="A4.SS1.5.p5.4.m1.3.3.1.1.1.1.1.3.cmml" xref="A4.SS1.5.p5.4.m1.3.3.1.1.1.1.1.3"><in id="A4.SS1.5.p5.4.m1.3.3.1.1.1.1.1.3.1.cmml" xref="A4.SS1.5.p5.4.m1.3.3.1.1.1.1.1.3.1"></in><ci id="A4.SS1.5.p5.4.m1.3.3.1.1.1.1.1.3.2.cmml" xref="A4.SS1.5.p5.4.m1.3.3.1.1.1.1.1.3.2">𝐱</ci><ci id="A4.SS1.5.p5.4.m1.3.3.1.1.1.1.1.3.3.cmml" xref="A4.SS1.5.p5.4.m1.3.3.1.1.1.1.1.3.3">𝐊</ci></apply></apply><apply id="A4.SS1.5.p5.4.m1.3.3.1.1.1.1.2.cmml" xref="A4.SS1.5.p5.4.m1.3.3.1.1.1.1.2"><times id="A4.SS1.5.p5.4.m1.3.3.1.1.1.1.2.1.cmml" xref="A4.SS1.5.p5.4.m1.3.3.1.1.1.1.2.1"></times><ci id="A4.SS1.5.p5.4.m1.3.3.1.1.1.1.2.2.cmml" xref="A4.SS1.5.p5.4.m1.3.3.1.1.1.1.2.2">𝑝</ci><ci id="A4.SS1.5.p5.4.m1.1.1.cmml" xref="A4.SS1.5.p5.4.m1.1.1">𝐱</ci></apply></apply></apply></apply><apply id="A4.SS1.5.p5.4.m1.3.3c.cmml" xref="A4.SS1.5.p5.4.m1.3.3"><gt id="A4.SS1.5.p5.4.m1.3.3.5.cmml" xref="A4.SS1.5.p5.4.m1.3.3.5"></gt><share href="#A4.SS1.5.p5.4.m1.3.3.1.cmml" id="A4.SS1.5.p5.4.m1.3.3d.cmml" xref="A4.SS1.5.p5.4.m1.3.3"></share><cn id="A4.SS1.5.p5.4.m1.3.3.6.cmml" type="integer" xref="A4.SS1.5.p5.4.m1.3.3.6">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.SS1.5.p5.4.m1.3c">m_{p}=\min\{\inf_{{\mathbf{x}}\in\mathbf{K}}{p({\mathbf{x}})}\}>0</annotation><annotation encoding="application/x-llamapun" id="A4.SS1.5.p5.4.m1.3d">italic_m start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT = roman_min { roman_inf start_POSTSUBSCRIPT bold_x ∈ bold_K end_POSTSUBSCRIPT italic_p ( bold_x ) } > 0</annotation></semantics></math>, because <math alttext="\sup_{\mathbf{K}}\|p-p_{\sigma}\|\to 0" class="ltx_Math" display="inline" id="A4.SS1.5.p5.5.m2.1"><semantics id="A4.SS1.5.p5.5.m2.1a"><mrow id="A4.SS1.5.p5.5.m2.1.1" xref="A4.SS1.5.p5.5.m2.1.1.cmml"><mrow id="A4.SS1.5.p5.5.m2.1.1.1" xref="A4.SS1.5.p5.5.m2.1.1.1.cmml"><msub id="A4.SS1.5.p5.5.m2.1.1.1.2" xref="A4.SS1.5.p5.5.m2.1.1.1.2.cmml"><mo id="A4.SS1.5.p5.5.m2.1.1.1.2.2" xref="A4.SS1.5.p5.5.m2.1.1.1.2.2.cmml">sup</mo><mi id="A4.SS1.5.p5.5.m2.1.1.1.2.3" xref="A4.SS1.5.p5.5.m2.1.1.1.2.3.cmml">𝐊</mi></msub><mrow id="A4.SS1.5.p5.5.m2.1.1.1.1.1" xref="A4.SS1.5.p5.5.m2.1.1.1.1.2.cmml"><mo id="A4.SS1.5.p5.5.m2.1.1.1.1.1.2" lspace="0em" stretchy="false" xref="A4.SS1.5.p5.5.m2.1.1.1.1.2.1.cmml">‖</mo><mrow id="A4.SS1.5.p5.5.m2.1.1.1.1.1.1" xref="A4.SS1.5.p5.5.m2.1.1.1.1.1.1.cmml"><mi id="A4.SS1.5.p5.5.m2.1.1.1.1.1.1.2" xref="A4.SS1.5.p5.5.m2.1.1.1.1.1.1.2.cmml">p</mi><mo id="A4.SS1.5.p5.5.m2.1.1.1.1.1.1.1" xref="A4.SS1.5.p5.5.m2.1.1.1.1.1.1.1.cmml">−</mo><msub id="A4.SS1.5.p5.5.m2.1.1.1.1.1.1.3" xref="A4.SS1.5.p5.5.m2.1.1.1.1.1.1.3.cmml"><mi id="A4.SS1.5.p5.5.m2.1.1.1.1.1.1.3.2" xref="A4.SS1.5.p5.5.m2.1.1.1.1.1.1.3.2.cmml">p</mi><mi id="A4.SS1.5.p5.5.m2.1.1.1.1.1.1.3.3" xref="A4.SS1.5.p5.5.m2.1.1.1.1.1.1.3.3.cmml">σ</mi></msub></mrow><mo id="A4.SS1.5.p5.5.m2.1.1.1.1.1.3" stretchy="false" xref="A4.SS1.5.p5.5.m2.1.1.1.1.2.1.cmml">‖</mo></mrow></mrow><mo id="A4.SS1.5.p5.5.m2.1.1.2" stretchy="false" xref="A4.SS1.5.p5.5.m2.1.1.2.cmml">→</mo><mn id="A4.SS1.5.p5.5.m2.1.1.3" xref="A4.SS1.5.p5.5.m2.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="A4.SS1.5.p5.5.m2.1b"><apply id="A4.SS1.5.p5.5.m2.1.1.cmml" xref="A4.SS1.5.p5.5.m2.1.1"><ci id="A4.SS1.5.p5.5.m2.1.1.2.cmml" xref="A4.SS1.5.p5.5.m2.1.1.2">→</ci><apply id="A4.SS1.5.p5.5.m2.1.1.1.cmml" xref="A4.SS1.5.p5.5.m2.1.1.1"><apply id="A4.SS1.5.p5.5.m2.1.1.1.2.cmml" xref="A4.SS1.5.p5.5.m2.1.1.1.2"><csymbol cd="ambiguous" id="A4.SS1.5.p5.5.m2.1.1.1.2.1.cmml" xref="A4.SS1.5.p5.5.m2.1.1.1.2">subscript</csymbol><csymbol cd="latexml" id="A4.SS1.5.p5.5.m2.1.1.1.2.2.cmml" xref="A4.SS1.5.p5.5.m2.1.1.1.2.2">supremum</csymbol><ci id="A4.SS1.5.p5.5.m2.1.1.1.2.3.cmml" xref="A4.SS1.5.p5.5.m2.1.1.1.2.3">𝐊</ci></apply><apply id="A4.SS1.5.p5.5.m2.1.1.1.1.2.cmml" xref="A4.SS1.5.p5.5.m2.1.1.1.1.1"><csymbol cd="latexml" id="A4.SS1.5.p5.5.m2.1.1.1.1.2.1.cmml" xref="A4.SS1.5.p5.5.m2.1.1.1.1.1.2">norm</csymbol><apply id="A4.SS1.5.p5.5.m2.1.1.1.1.1.1.cmml" xref="A4.SS1.5.p5.5.m2.1.1.1.1.1.1"><minus id="A4.SS1.5.p5.5.m2.1.1.1.1.1.1.1.cmml" xref="A4.SS1.5.p5.5.m2.1.1.1.1.1.1.1"></minus><ci id="A4.SS1.5.p5.5.m2.1.1.1.1.1.1.2.cmml" xref="A4.SS1.5.p5.5.m2.1.1.1.1.1.1.2">𝑝</ci><apply id="A4.SS1.5.p5.5.m2.1.1.1.1.1.1.3.cmml" xref="A4.SS1.5.p5.5.m2.1.1.1.1.1.1.3"><csymbol cd="ambiguous" id="A4.SS1.5.p5.5.m2.1.1.1.1.1.1.3.1.cmml" xref="A4.SS1.5.p5.5.m2.1.1.1.1.1.1.3">subscript</csymbol><ci id="A4.SS1.5.p5.5.m2.1.1.1.1.1.1.3.2.cmml" xref="A4.SS1.5.p5.5.m2.1.1.1.1.1.1.3.2">𝑝</ci><ci id="A4.SS1.5.p5.5.m2.1.1.1.1.1.1.3.3.cmml" xref="A4.SS1.5.p5.5.m2.1.1.1.1.1.1.3.3">𝜎</ci></apply></apply></apply></apply><cn id="A4.SS1.5.p5.5.m2.1.1.3.cmml" type="integer" xref="A4.SS1.5.p5.5.m2.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.SS1.5.p5.5.m2.1c">\sup_{\mathbf{K}}\|p-p_{\sigma}\|\to 0</annotation><annotation encoding="application/x-llamapun" id="A4.SS1.5.p5.5.m2.1d">roman_sup start_POSTSUBSCRIPT bold_K end_POSTSUBSCRIPT ∥ italic_p - italic_p start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ∥ → 0</annotation></semantics></math>, there exists <math alttext="\sigma_{1}>0" class="ltx_Math" display="inline" id="A4.SS1.5.p5.6.m3.1"><semantics id="A4.SS1.5.p5.6.m3.1a"><mrow id="A4.SS1.5.p5.6.m3.1.1" xref="A4.SS1.5.p5.6.m3.1.1.cmml"><msub id="A4.SS1.5.p5.6.m3.1.1.2" xref="A4.SS1.5.p5.6.m3.1.1.2.cmml"><mi id="A4.SS1.5.p5.6.m3.1.1.2.2" xref="A4.SS1.5.p5.6.m3.1.1.2.2.cmml">σ</mi><mn id="A4.SS1.5.p5.6.m3.1.1.2.3" xref="A4.SS1.5.p5.6.m3.1.1.2.3.cmml">1</mn></msub><mo id="A4.SS1.5.p5.6.m3.1.1.1" xref="A4.SS1.5.p5.6.m3.1.1.1.cmml">></mo><mn id="A4.SS1.5.p5.6.m3.1.1.3" xref="A4.SS1.5.p5.6.m3.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="A4.SS1.5.p5.6.m3.1b"><apply id="A4.SS1.5.p5.6.m3.1.1.cmml" xref="A4.SS1.5.p5.6.m3.1.1"><gt id="A4.SS1.5.p5.6.m3.1.1.1.cmml" xref="A4.SS1.5.p5.6.m3.1.1.1"></gt><apply id="A4.SS1.5.p5.6.m3.1.1.2.cmml" xref="A4.SS1.5.p5.6.m3.1.1.2"><csymbol cd="ambiguous" id="A4.SS1.5.p5.6.m3.1.1.2.1.cmml" xref="A4.SS1.5.p5.6.m3.1.1.2">subscript</csymbol><ci id="A4.SS1.5.p5.6.m3.1.1.2.2.cmml" xref="A4.SS1.5.p5.6.m3.1.1.2.2">𝜎</ci><cn id="A4.SS1.5.p5.6.m3.1.1.2.3.cmml" type="integer" xref="A4.SS1.5.p5.6.m3.1.1.2.3">1</cn></apply><cn id="A4.SS1.5.p5.6.m3.1.1.3.cmml" type="integer" xref="A4.SS1.5.p5.6.m3.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.SS1.5.p5.6.m3.1c">\sigma_{1}>0</annotation><annotation encoding="application/x-llamapun" id="A4.SS1.5.p5.6.m3.1d">italic_σ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT > 0</annotation></semantics></math> such that for <math alttext="0<\sigma\leq\sigma_{1}" class="ltx_Math" display="inline" id="A4.SS1.5.p5.7.m4.1"><semantics id="A4.SS1.5.p5.7.m4.1a"><mrow id="A4.SS1.5.p5.7.m4.1.1" xref="A4.SS1.5.p5.7.m4.1.1.cmml"><mn id="A4.SS1.5.p5.7.m4.1.1.2" xref="A4.SS1.5.p5.7.m4.1.1.2.cmml">0</mn><mo id="A4.SS1.5.p5.7.m4.1.1.3" xref="A4.SS1.5.p5.7.m4.1.1.3.cmml"><</mo><mi id="A4.SS1.5.p5.7.m4.1.1.4" xref="A4.SS1.5.p5.7.m4.1.1.4.cmml">σ</mi><mo id="A4.SS1.5.p5.7.m4.1.1.5" xref="A4.SS1.5.p5.7.m4.1.1.5.cmml">≤</mo><msub id="A4.SS1.5.p5.7.m4.1.1.6" xref="A4.SS1.5.p5.7.m4.1.1.6.cmml"><mi id="A4.SS1.5.p5.7.m4.1.1.6.2" xref="A4.SS1.5.p5.7.m4.1.1.6.2.cmml">σ</mi><mn id="A4.SS1.5.p5.7.m4.1.1.6.3" xref="A4.SS1.5.p5.7.m4.1.1.6.3.cmml">1</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="A4.SS1.5.p5.7.m4.1b"><apply id="A4.SS1.5.p5.7.m4.1.1.cmml" xref="A4.SS1.5.p5.7.m4.1.1"><and id="A4.SS1.5.p5.7.m4.1.1a.cmml" xref="A4.SS1.5.p5.7.m4.1.1"></and><apply id="A4.SS1.5.p5.7.m4.1.1b.cmml" xref="A4.SS1.5.p5.7.m4.1.1"><lt id="A4.SS1.5.p5.7.m4.1.1.3.cmml" xref="A4.SS1.5.p5.7.m4.1.1.3"></lt><cn id="A4.SS1.5.p5.7.m4.1.1.2.cmml" type="integer" xref="A4.SS1.5.p5.7.m4.1.1.2">0</cn><ci id="A4.SS1.5.p5.7.m4.1.1.4.cmml" xref="A4.SS1.5.p5.7.m4.1.1.4">𝜎</ci></apply><apply id="A4.SS1.5.p5.7.m4.1.1c.cmml" xref="A4.SS1.5.p5.7.m4.1.1"><leq id="A4.SS1.5.p5.7.m4.1.1.5.cmml" xref="A4.SS1.5.p5.7.m4.1.1.5"></leq><share href="#A4.SS1.5.p5.7.m4.1.1.4.cmml" id="A4.SS1.5.p5.7.m4.1.1d.cmml" xref="A4.SS1.5.p5.7.m4.1.1"></share><apply id="A4.SS1.5.p5.7.m4.1.1.6.cmml" xref="A4.SS1.5.p5.7.m4.1.1.6"><csymbol cd="ambiguous" id="A4.SS1.5.p5.7.m4.1.1.6.1.cmml" xref="A4.SS1.5.p5.7.m4.1.1.6">subscript</csymbol><ci id="A4.SS1.5.p5.7.m4.1.1.6.2.cmml" xref="A4.SS1.5.p5.7.m4.1.1.6.2">𝜎</ci><cn id="A4.SS1.5.p5.7.m4.1.1.6.3.cmml" type="integer" xref="A4.SS1.5.p5.7.m4.1.1.6.3">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.SS1.5.p5.7.m4.1c">0<\sigma\leq\sigma_{1}</annotation><annotation encoding="application/x-llamapun" id="A4.SS1.5.p5.7.m4.1d">0 < italic_σ ≤ italic_σ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math>, <math alttext="\forall{\mathbf{x}}\in\mathbf{K}" class="ltx_Math" display="inline" id="A4.SS1.5.p5.8.m5.1"><semantics id="A4.SS1.5.p5.8.m5.1a"><mrow id="A4.SS1.5.p5.8.m5.1.1" xref="A4.SS1.5.p5.8.m5.1.1.cmml"><mrow id="A4.SS1.5.p5.8.m5.1.1.2" xref="A4.SS1.5.p5.8.m5.1.1.2.cmml"><mo id="A4.SS1.5.p5.8.m5.1.1.2.1" rspace="0.167em" xref="A4.SS1.5.p5.8.m5.1.1.2.1.cmml">∀</mo><mi id="A4.SS1.5.p5.8.m5.1.1.2.2" xref="A4.SS1.5.p5.8.m5.1.1.2.2.cmml">𝐱</mi></mrow><mo id="A4.SS1.5.p5.8.m5.1.1.1" xref="A4.SS1.5.p5.8.m5.1.1.1.cmml">∈</mo><mi id="A4.SS1.5.p5.8.m5.1.1.3" xref="A4.SS1.5.p5.8.m5.1.1.3.cmml">𝐊</mi></mrow><annotation-xml encoding="MathML-Content" id="A4.SS1.5.p5.8.m5.1b"><apply id="A4.SS1.5.p5.8.m5.1.1.cmml" xref="A4.SS1.5.p5.8.m5.1.1"><in id="A4.SS1.5.p5.8.m5.1.1.1.cmml" xref="A4.SS1.5.p5.8.m5.1.1.1"></in><apply id="A4.SS1.5.p5.8.m5.1.1.2.cmml" xref="A4.SS1.5.p5.8.m5.1.1.2"><csymbol cd="latexml" id="A4.SS1.5.p5.8.m5.1.1.2.1.cmml" xref="A4.SS1.5.p5.8.m5.1.1.2.1">for-all</csymbol><ci id="A4.SS1.5.p5.8.m5.1.1.2.2.cmml" xref="A4.SS1.5.p5.8.m5.1.1.2.2">𝐱</ci></apply><ci id="A4.SS1.5.p5.8.m5.1.1.3.cmml" xref="A4.SS1.5.p5.8.m5.1.1.3">𝐊</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.SS1.5.p5.8.m5.1c">\forall{\mathbf{x}}\in\mathbf{K}</annotation><annotation encoding="application/x-llamapun" id="A4.SS1.5.p5.8.m5.1d">∀ bold_x ∈ bold_K</annotation></semantics></math>, <math alttext="p_{\sigma}(x)>\frac{m_{p}}{2}" class="ltx_Math" display="inline" id="A4.SS1.5.p5.9.m6.1"><semantics id="A4.SS1.5.p5.9.m6.1a"><mrow id="A4.SS1.5.p5.9.m6.1.2" xref="A4.SS1.5.p5.9.m6.1.2.cmml"><mrow id="A4.SS1.5.p5.9.m6.1.2.2" xref="A4.SS1.5.p5.9.m6.1.2.2.cmml"><msub id="A4.SS1.5.p5.9.m6.1.2.2.2" xref="A4.SS1.5.p5.9.m6.1.2.2.2.cmml"><mi id="A4.SS1.5.p5.9.m6.1.2.2.2.2" xref="A4.SS1.5.p5.9.m6.1.2.2.2.2.cmml">p</mi><mi id="A4.SS1.5.p5.9.m6.1.2.2.2.3" xref="A4.SS1.5.p5.9.m6.1.2.2.2.3.cmml">σ</mi></msub><mo id="A4.SS1.5.p5.9.m6.1.2.2.1" xref="A4.SS1.5.p5.9.m6.1.2.2.1.cmml">⁢</mo><mrow id="A4.SS1.5.p5.9.m6.1.2.2.3.2" xref="A4.SS1.5.p5.9.m6.1.2.2.cmml"><mo id="A4.SS1.5.p5.9.m6.1.2.2.3.2.1" stretchy="false" xref="A4.SS1.5.p5.9.m6.1.2.2.cmml">(</mo><mi id="A4.SS1.5.p5.9.m6.1.1" xref="A4.SS1.5.p5.9.m6.1.1.cmml">x</mi><mo id="A4.SS1.5.p5.9.m6.1.2.2.3.2.2" stretchy="false" xref="A4.SS1.5.p5.9.m6.1.2.2.cmml">)</mo></mrow></mrow><mo id="A4.SS1.5.p5.9.m6.1.2.1" xref="A4.SS1.5.p5.9.m6.1.2.1.cmml">></mo><mfrac id="A4.SS1.5.p5.9.m6.1.2.3" xref="A4.SS1.5.p5.9.m6.1.2.3.cmml"><msub id="A4.SS1.5.p5.9.m6.1.2.3.2" xref="A4.SS1.5.p5.9.m6.1.2.3.2.cmml"><mi id="A4.SS1.5.p5.9.m6.1.2.3.2.2" xref="A4.SS1.5.p5.9.m6.1.2.3.2.2.cmml">m</mi><mi id="A4.SS1.5.p5.9.m6.1.2.3.2.3" xref="A4.SS1.5.p5.9.m6.1.2.3.2.3.cmml">p</mi></msub><mn id="A4.SS1.5.p5.9.m6.1.2.3.3" xref="A4.SS1.5.p5.9.m6.1.2.3.3.cmml">2</mn></mfrac></mrow><annotation-xml encoding="MathML-Content" id="A4.SS1.5.p5.9.m6.1b"><apply id="A4.SS1.5.p5.9.m6.1.2.cmml" xref="A4.SS1.5.p5.9.m6.1.2"><gt id="A4.SS1.5.p5.9.m6.1.2.1.cmml" xref="A4.SS1.5.p5.9.m6.1.2.1"></gt><apply id="A4.SS1.5.p5.9.m6.1.2.2.cmml" xref="A4.SS1.5.p5.9.m6.1.2.2"><times id="A4.SS1.5.p5.9.m6.1.2.2.1.cmml" xref="A4.SS1.5.p5.9.m6.1.2.2.1"></times><apply id="A4.SS1.5.p5.9.m6.1.2.2.2.cmml" xref="A4.SS1.5.p5.9.m6.1.2.2.2"><csymbol cd="ambiguous" id="A4.SS1.5.p5.9.m6.1.2.2.2.1.cmml" xref="A4.SS1.5.p5.9.m6.1.2.2.2">subscript</csymbol><ci id="A4.SS1.5.p5.9.m6.1.2.2.2.2.cmml" xref="A4.SS1.5.p5.9.m6.1.2.2.2.2">𝑝</ci><ci id="A4.SS1.5.p5.9.m6.1.2.2.2.3.cmml" xref="A4.SS1.5.p5.9.m6.1.2.2.2.3">𝜎</ci></apply><ci id="A4.SS1.5.p5.9.m6.1.1.cmml" xref="A4.SS1.5.p5.9.m6.1.1">𝑥</ci></apply><apply id="A4.SS1.5.p5.9.m6.1.2.3.cmml" xref="A4.SS1.5.p5.9.m6.1.2.3"><divide id="A4.SS1.5.p5.9.m6.1.2.3.1.cmml" xref="A4.SS1.5.p5.9.m6.1.2.3"></divide><apply id="A4.SS1.5.p5.9.m6.1.2.3.2.cmml" xref="A4.SS1.5.p5.9.m6.1.2.3.2"><csymbol cd="ambiguous" id="A4.SS1.5.p5.9.m6.1.2.3.2.1.cmml" xref="A4.SS1.5.p5.9.m6.1.2.3.2">subscript</csymbol><ci id="A4.SS1.5.p5.9.m6.1.2.3.2.2.cmml" xref="A4.SS1.5.p5.9.m6.1.2.3.2.2">𝑚</ci><ci id="A4.SS1.5.p5.9.m6.1.2.3.2.3.cmml" xref="A4.SS1.5.p5.9.m6.1.2.3.2.3">𝑝</ci></apply><cn id="A4.SS1.5.p5.9.m6.1.2.3.3.cmml" type="integer" xref="A4.SS1.5.p5.9.m6.1.2.3.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.SS1.5.p5.9.m6.1c">p_{\sigma}(x)>\frac{m_{p}}{2}</annotation><annotation encoding="application/x-llamapun" id="A4.SS1.5.p5.9.m6.1d">italic_p start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ( italic_x ) > divide start_ARG italic_m start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT end_ARG start_ARG 2 end_ARG</annotation></semantics></math>. Thus for <math alttext="0<\sigma\leq\sigma_{1}" class="ltx_Math" display="inline" id="A4.SS1.5.p5.10.m7.1"><semantics id="A4.SS1.5.p5.10.m7.1a"><mrow id="A4.SS1.5.p5.10.m7.1.1" xref="A4.SS1.5.p5.10.m7.1.1.cmml"><mn id="A4.SS1.5.p5.10.m7.1.1.2" xref="A4.SS1.5.p5.10.m7.1.1.2.cmml">0</mn><mo id="A4.SS1.5.p5.10.m7.1.1.3" xref="A4.SS1.5.p5.10.m7.1.1.3.cmml"><</mo><mi id="A4.SS1.5.p5.10.m7.1.1.4" xref="A4.SS1.5.p5.10.m7.1.1.4.cmml">σ</mi><mo id="A4.SS1.5.p5.10.m7.1.1.5" xref="A4.SS1.5.p5.10.m7.1.1.5.cmml">≤</mo><msub id="A4.SS1.5.p5.10.m7.1.1.6" xref="A4.SS1.5.p5.10.m7.1.1.6.cmml"><mi id="A4.SS1.5.p5.10.m7.1.1.6.2" xref="A4.SS1.5.p5.10.m7.1.1.6.2.cmml">σ</mi><mn id="A4.SS1.5.p5.10.m7.1.1.6.3" xref="A4.SS1.5.p5.10.m7.1.1.6.3.cmml">1</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="A4.SS1.5.p5.10.m7.1b"><apply id="A4.SS1.5.p5.10.m7.1.1.cmml" xref="A4.SS1.5.p5.10.m7.1.1"><and id="A4.SS1.5.p5.10.m7.1.1a.cmml" xref="A4.SS1.5.p5.10.m7.1.1"></and><apply id="A4.SS1.5.p5.10.m7.1.1b.cmml" xref="A4.SS1.5.p5.10.m7.1.1"><lt id="A4.SS1.5.p5.10.m7.1.1.3.cmml" xref="A4.SS1.5.p5.10.m7.1.1.3"></lt><cn id="A4.SS1.5.p5.10.m7.1.1.2.cmml" type="integer" xref="A4.SS1.5.p5.10.m7.1.1.2">0</cn><ci id="A4.SS1.5.p5.10.m7.1.1.4.cmml" xref="A4.SS1.5.p5.10.m7.1.1.4">𝜎</ci></apply><apply id="A4.SS1.5.p5.10.m7.1.1c.cmml" xref="A4.SS1.5.p5.10.m7.1.1"><leq id="A4.SS1.5.p5.10.m7.1.1.5.cmml" xref="A4.SS1.5.p5.10.m7.1.1.5"></leq><share href="#A4.SS1.5.p5.10.m7.1.1.4.cmml" id="A4.SS1.5.p5.10.m7.1.1d.cmml" xref="A4.SS1.5.p5.10.m7.1.1"></share><apply id="A4.SS1.5.p5.10.m7.1.1.6.cmml" xref="A4.SS1.5.p5.10.m7.1.1.6"><csymbol cd="ambiguous" id="A4.SS1.5.p5.10.m7.1.1.6.1.cmml" xref="A4.SS1.5.p5.10.m7.1.1.6">subscript</csymbol><ci id="A4.SS1.5.p5.10.m7.1.1.6.2.cmml" xref="A4.SS1.5.p5.10.m7.1.1.6.2">𝜎</ci><cn id="A4.SS1.5.p5.10.m7.1.1.6.3.cmml" type="integer" xref="A4.SS1.5.p5.10.m7.1.1.6.3">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.SS1.5.p5.10.m7.1c">0<\sigma\leq\sigma_{1}</annotation><annotation encoding="application/x-llamapun" id="A4.SS1.5.p5.10.m7.1d">0 < italic_σ ≤ italic_σ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math>,</p> </div> <div class="ltx_para" id="A4.SS1.6.p6"> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A8.EGx45"> <tbody id="A4.Ex71"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\sup_{\mathbf{K}}\|\nabla\log p_{\sigma}-\nabla\log p\|" class="ltx_Math" display="inline" id="A4.Ex71.m1.1"><semantics id="A4.Ex71.m1.1a"><mrow id="A4.Ex71.m1.1.1" xref="A4.Ex71.m1.1.1.cmml"><munder id="A4.Ex71.m1.1.1.2" xref="A4.Ex71.m1.1.1.2.cmml"><mo id="A4.Ex71.m1.1.1.2.2" movablelimits="false" xref="A4.Ex71.m1.1.1.2.2.cmml">sup</mo><mi id="A4.Ex71.m1.1.1.2.3" xref="A4.Ex71.m1.1.1.2.3.cmml">𝐊</mi></munder><mrow id="A4.Ex71.m1.1.1.1.1" xref="A4.Ex71.m1.1.1.1.2.cmml"><mo id="A4.Ex71.m1.1.1.1.1.2" lspace="0em" stretchy="false" xref="A4.Ex71.m1.1.1.1.2.1.cmml">‖</mo><mrow id="A4.Ex71.m1.1.1.1.1.1" xref="A4.Ex71.m1.1.1.1.1.1.cmml"><mrow id="A4.Ex71.m1.1.1.1.1.1.2" xref="A4.Ex71.m1.1.1.1.1.1.2.cmml"><mrow id="A4.Ex71.m1.1.1.1.1.1.2.1" xref="A4.Ex71.m1.1.1.1.1.1.2.1.cmml"><mo id="A4.Ex71.m1.1.1.1.1.1.2.1.1" rspace="0.167em" xref="A4.Ex71.m1.1.1.1.1.1.2.1.1.cmml">∇</mo><mi id="A4.Ex71.m1.1.1.1.1.1.2.1.2" xref="A4.Ex71.m1.1.1.1.1.1.2.1.2.cmml">log</mi></mrow><mo id="A4.Ex71.m1.1.1.1.1.1.2a" lspace="0.167em" xref="A4.Ex71.m1.1.1.1.1.1.2.cmml">⁡</mo><msub id="A4.Ex71.m1.1.1.1.1.1.2.2" xref="A4.Ex71.m1.1.1.1.1.1.2.2.cmml"><mi id="A4.Ex71.m1.1.1.1.1.1.2.2.2" xref="A4.Ex71.m1.1.1.1.1.1.2.2.2.cmml">p</mi><mi id="A4.Ex71.m1.1.1.1.1.1.2.2.3" xref="A4.Ex71.m1.1.1.1.1.1.2.2.3.cmml">σ</mi></msub></mrow><mo id="A4.Ex71.m1.1.1.1.1.1.1" xref="A4.Ex71.m1.1.1.1.1.1.1.cmml">−</mo><mrow id="A4.Ex71.m1.1.1.1.1.1.3" xref="A4.Ex71.m1.1.1.1.1.1.3.cmml"><mrow id="A4.Ex71.m1.1.1.1.1.1.3.1" xref="A4.Ex71.m1.1.1.1.1.1.3.1.cmml"><mo id="A4.Ex71.m1.1.1.1.1.1.3.1.1" rspace="0.167em" xref="A4.Ex71.m1.1.1.1.1.1.3.1.1.cmml">∇</mo><mi id="A4.Ex71.m1.1.1.1.1.1.3.1.2" xref="A4.Ex71.m1.1.1.1.1.1.3.1.2.cmml">log</mi></mrow><mo id="A4.Ex71.m1.1.1.1.1.1.3a" lspace="0.167em" xref="A4.Ex71.m1.1.1.1.1.1.3.cmml">⁡</mo><mi id="A4.Ex71.m1.1.1.1.1.1.3.2" xref="A4.Ex71.m1.1.1.1.1.1.3.2.cmml">p</mi></mrow></mrow><mo id="A4.Ex71.m1.1.1.1.1.3" stretchy="false" xref="A4.Ex71.m1.1.1.1.2.1.cmml">‖</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A4.Ex71.m1.1b"><apply id="A4.Ex71.m1.1.1.cmml" xref="A4.Ex71.m1.1.1"><apply id="A4.Ex71.m1.1.1.2.cmml" xref="A4.Ex71.m1.1.1.2"><csymbol cd="ambiguous" id="A4.Ex71.m1.1.1.2.1.cmml" xref="A4.Ex71.m1.1.1.2">subscript</csymbol><csymbol cd="latexml" id="A4.Ex71.m1.1.1.2.2.cmml" xref="A4.Ex71.m1.1.1.2.2">supremum</csymbol><ci id="A4.Ex71.m1.1.1.2.3.cmml" xref="A4.Ex71.m1.1.1.2.3">𝐊</ci></apply><apply id="A4.Ex71.m1.1.1.1.2.cmml" xref="A4.Ex71.m1.1.1.1.1"><csymbol cd="latexml" id="A4.Ex71.m1.1.1.1.2.1.cmml" xref="A4.Ex71.m1.1.1.1.1.2">norm</csymbol><apply id="A4.Ex71.m1.1.1.1.1.1.cmml" xref="A4.Ex71.m1.1.1.1.1.1"><minus id="A4.Ex71.m1.1.1.1.1.1.1.cmml" xref="A4.Ex71.m1.1.1.1.1.1.1"></minus><apply id="A4.Ex71.m1.1.1.1.1.1.2.cmml" xref="A4.Ex71.m1.1.1.1.1.1.2"><apply id="A4.Ex71.m1.1.1.1.1.1.2.1.cmml" xref="A4.Ex71.m1.1.1.1.1.1.2.1"><ci id="A4.Ex71.m1.1.1.1.1.1.2.1.1.cmml" xref="A4.Ex71.m1.1.1.1.1.1.2.1.1">∇</ci><log id="A4.Ex71.m1.1.1.1.1.1.2.1.2.cmml" xref="A4.Ex71.m1.1.1.1.1.1.2.1.2"></log></apply><apply id="A4.Ex71.m1.1.1.1.1.1.2.2.cmml" xref="A4.Ex71.m1.1.1.1.1.1.2.2"><csymbol cd="ambiguous" id="A4.Ex71.m1.1.1.1.1.1.2.2.1.cmml" xref="A4.Ex71.m1.1.1.1.1.1.2.2">subscript</csymbol><ci id="A4.Ex71.m1.1.1.1.1.1.2.2.2.cmml" xref="A4.Ex71.m1.1.1.1.1.1.2.2.2">𝑝</ci><ci id="A4.Ex71.m1.1.1.1.1.1.2.2.3.cmml" xref="A4.Ex71.m1.1.1.1.1.1.2.2.3">𝜎</ci></apply></apply><apply id="A4.Ex71.m1.1.1.1.1.1.3.cmml" xref="A4.Ex71.m1.1.1.1.1.1.3"><apply id="A4.Ex71.m1.1.1.1.1.1.3.1.cmml" xref="A4.Ex71.m1.1.1.1.1.1.3.1"><ci id="A4.Ex71.m1.1.1.1.1.1.3.1.1.cmml" xref="A4.Ex71.m1.1.1.1.1.1.3.1.1">∇</ci><log id="A4.Ex71.m1.1.1.1.1.1.3.1.2.cmml" xref="A4.Ex71.m1.1.1.1.1.1.3.1.2"></log></apply><ci id="A4.Ex71.m1.1.1.1.1.1.3.2.cmml" xref="A4.Ex71.m1.1.1.1.1.1.3.2">𝑝</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.Ex71.m1.1c">\displaystyle\sup_{\mathbf{K}}\|\nabla\log p_{\sigma}-\nabla\log p\|</annotation><annotation encoding="application/x-llamapun" id="A4.Ex71.m1.1d">roman_sup start_POSTSUBSCRIPT bold_K end_POSTSUBSCRIPT ∥ ∇ roman_log italic_p start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT - ∇ roman_log italic_p ∥</annotation></semantics></math></td> <td class="ltx_td 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0</annotation><annotation encoding="application/x-llamapun" id="A4.Ex71.m2.4d">≤ divide start_ARG 2 ( roman_sup start_POSTSUBSCRIPT bold_K end_POSTSUBSCRIPT ∥ ∇ italic_p start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT - ∇ italic_p ∥ ) ∥ italic_p ∥ start_POSTSUBSCRIPT ∞ end_POSTSUBSCRIPT + ∥ ∇ italic_p ∥ start_POSTSUBSCRIPT ∞ end_POSTSUBSCRIPT ( roman_sup start_POSTSUBSCRIPT bold_K end_POSTSUBSCRIPT ∥ italic_p - italic_p start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ∥ ) end_ARG start_ARG italic_m start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG → 0</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> </div> <div class="ltx_para" id="A4.SS1.7.p7"> <p class="ltx_p" id="A4.SS1.7.p7.1">Thus we have prove Propositon <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#A4.Thmproposition1" title="Proposition D.1. ‣ D.1 Proof of critical Point Analysis ‣ Appendix D Proofs of section 3 ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_tag">D.1</span></a>.</p> </div> <div class="ltx_para" id="A4.SS1.8.p8"> <p class="ltx_p" id="A4.SS1.8.p8.1">∎</p> </div> </div> <section class="ltx_subsubsection" id="A4.SS1.SSS1"> <h4 class="ltx_title ltx_title_subsubsection"> <span class="ltx_tag ltx_tag_subsubsection">D.1.1 </span>Proof of Proposition <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#S3.Thmproposition1" title="Proposition 3.1. ‣ 3.1 Asymptotics of critical points when 𝜎→0 ‣ 3 Convergence Analysis ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_tag">3.1</span></a> </h4> <div class="ltx_para" id="A4.SS1.SSS1.p1"> <p class="ltx_p" id="A4.SS1.SSS1.p1.4">We define <math alttext="\mathbf{L}=\mathbf{K}+\mathcal{B}(0,1)" class="ltx_Math" display="inline" id="A4.SS1.SSS1.p1.1.m1.2"><semantics id="A4.SS1.SSS1.p1.1.m1.2a"><mrow id="A4.SS1.SSS1.p1.1.m1.2.3" xref="A4.SS1.SSS1.p1.1.m1.2.3.cmml"><mi id="A4.SS1.SSS1.p1.1.m1.2.3.2" xref="A4.SS1.SSS1.p1.1.m1.2.3.2.cmml">𝐋</mi><mo id="A4.SS1.SSS1.p1.1.m1.2.3.1" xref="A4.SS1.SSS1.p1.1.m1.2.3.1.cmml">=</mo><mrow id="A4.SS1.SSS1.p1.1.m1.2.3.3" xref="A4.SS1.SSS1.p1.1.m1.2.3.3.cmml"><mi id="A4.SS1.SSS1.p1.1.m1.2.3.3.2" xref="A4.SS1.SSS1.p1.1.m1.2.3.3.2.cmml">𝐊</mi><mo id="A4.SS1.SSS1.p1.1.m1.2.3.3.1" xref="A4.SS1.SSS1.p1.1.m1.2.3.3.1.cmml">+</mo><mrow id="A4.SS1.SSS1.p1.1.m1.2.3.3.3" xref="A4.SS1.SSS1.p1.1.m1.2.3.3.3.cmml"><mi class="ltx_font_mathcaligraphic" id="A4.SS1.SSS1.p1.1.m1.2.3.3.3.2" xref="A4.SS1.SSS1.p1.1.m1.2.3.3.3.2.cmml">ℬ</mi><mo id="A4.SS1.SSS1.p1.1.m1.2.3.3.3.1" xref="A4.SS1.SSS1.p1.1.m1.2.3.3.3.1.cmml">⁢</mo><mrow id="A4.SS1.SSS1.p1.1.m1.2.3.3.3.3.2" xref="A4.SS1.SSS1.p1.1.m1.2.3.3.3.3.1.cmml"><mo id="A4.SS1.SSS1.p1.1.m1.2.3.3.3.3.2.1" stretchy="false" xref="A4.SS1.SSS1.p1.1.m1.2.3.3.3.3.1.cmml">(</mo><mn id="A4.SS1.SSS1.p1.1.m1.1.1" xref="A4.SS1.SSS1.p1.1.m1.1.1.cmml">0</mn><mo id="A4.SS1.SSS1.p1.1.m1.2.3.3.3.3.2.2" xref="A4.SS1.SSS1.p1.1.m1.2.3.3.3.3.1.cmml">,</mo><mn id="A4.SS1.SSS1.p1.1.m1.2.2" xref="A4.SS1.SSS1.p1.1.m1.2.2.cmml">1</mn><mo id="A4.SS1.SSS1.p1.1.m1.2.3.3.3.3.2.3" stretchy="false" xref="A4.SS1.SSS1.p1.1.m1.2.3.3.3.3.1.cmml">)</mo></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="A4.SS1.SSS1.p1.1.m1.2b"><apply id="A4.SS1.SSS1.p1.1.m1.2.3.cmml" xref="A4.SS1.SSS1.p1.1.m1.2.3"><eq id="A4.SS1.SSS1.p1.1.m1.2.3.1.cmml" xref="A4.SS1.SSS1.p1.1.m1.2.3.1"></eq><ci id="A4.SS1.SSS1.p1.1.m1.2.3.2.cmml" xref="A4.SS1.SSS1.p1.1.m1.2.3.2">𝐋</ci><apply id="A4.SS1.SSS1.p1.1.m1.2.3.3.cmml" xref="A4.SS1.SSS1.p1.1.m1.2.3.3"><plus id="A4.SS1.SSS1.p1.1.m1.2.3.3.1.cmml" xref="A4.SS1.SSS1.p1.1.m1.2.3.3.1"></plus><ci id="A4.SS1.SSS1.p1.1.m1.2.3.3.2.cmml" xref="A4.SS1.SSS1.p1.1.m1.2.3.3.2">𝐊</ci><apply id="A4.SS1.SSS1.p1.1.m1.2.3.3.3.cmml" xref="A4.SS1.SSS1.p1.1.m1.2.3.3.3"><times id="A4.SS1.SSS1.p1.1.m1.2.3.3.3.1.cmml" xref="A4.SS1.SSS1.p1.1.m1.2.3.3.3.1"></times><ci id="A4.SS1.SSS1.p1.1.m1.2.3.3.3.2.cmml" xref="A4.SS1.SSS1.p1.1.m1.2.3.3.3.2">ℬ</ci><interval closure="open" id="A4.SS1.SSS1.p1.1.m1.2.3.3.3.3.1.cmml" xref="A4.SS1.SSS1.p1.1.m1.2.3.3.3.3.2"><cn id="A4.SS1.SSS1.p1.1.m1.1.1.cmml" type="integer" xref="A4.SS1.SSS1.p1.1.m1.1.1">0</cn><cn id="A4.SS1.SSS1.p1.1.m1.2.2.cmml" type="integer" xref="A4.SS1.SSS1.p1.1.m1.2.2">1</cn></interval></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.SS1.SSS1.p1.1.m1.2c">\mathbf{L}=\mathbf{K}+\mathcal{B}(0,1)</annotation><annotation encoding="application/x-llamapun" id="A4.SS1.SSS1.p1.1.m1.2d">bold_L = bold_K + caligraphic_B ( 0 , 1 )</annotation></semantics></math>. By Proposition <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#A4.Thmproposition1" title="Proposition D.1. ‣ D.1 Proof of critical Point Analysis ‣ Appendix D Proofs of section 3 ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_tag">D.1</span></a>, <math alttext="\nabla\log p_{\sigma}" class="ltx_Math" display="inline" id="A4.SS1.SSS1.p1.2.m2.1"><semantics id="A4.SS1.SSS1.p1.2.m2.1a"><mrow id="A4.SS1.SSS1.p1.2.m2.1.1" xref="A4.SS1.SSS1.p1.2.m2.1.1.cmml"><mrow id="A4.SS1.SSS1.p1.2.m2.1.1.1" xref="A4.SS1.SSS1.p1.2.m2.1.1.1.cmml"><mo id="A4.SS1.SSS1.p1.2.m2.1.1.1.1" rspace="0.167em" xref="A4.SS1.SSS1.p1.2.m2.1.1.1.1.cmml">∇</mo><mi id="A4.SS1.SSS1.p1.2.m2.1.1.1.2" xref="A4.SS1.SSS1.p1.2.m2.1.1.1.2.cmml">log</mi></mrow><mo id="A4.SS1.SSS1.p1.2.m2.1.1a" lspace="0.167em" xref="A4.SS1.SSS1.p1.2.m2.1.1.cmml">⁡</mo><msub id="A4.SS1.SSS1.p1.2.m2.1.1.2" xref="A4.SS1.SSS1.p1.2.m2.1.1.2.cmml"><mi id="A4.SS1.SSS1.p1.2.m2.1.1.2.2" xref="A4.SS1.SSS1.p1.2.m2.1.1.2.2.cmml">p</mi><mi id="A4.SS1.SSS1.p1.2.m2.1.1.2.3" xref="A4.SS1.SSS1.p1.2.m2.1.1.2.3.cmml">σ</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="A4.SS1.SSS1.p1.2.m2.1b"><apply id="A4.SS1.SSS1.p1.2.m2.1.1.cmml" xref="A4.SS1.SSS1.p1.2.m2.1.1"><apply id="A4.SS1.SSS1.p1.2.m2.1.1.1.cmml" xref="A4.SS1.SSS1.p1.2.m2.1.1.1"><ci id="A4.SS1.SSS1.p1.2.m2.1.1.1.1.cmml" xref="A4.SS1.SSS1.p1.2.m2.1.1.1.1">∇</ci><log id="A4.SS1.SSS1.p1.2.m2.1.1.1.2.cmml" xref="A4.SS1.SSS1.p1.2.m2.1.1.1.2"></log></apply><apply id="A4.SS1.SSS1.p1.2.m2.1.1.2.cmml" xref="A4.SS1.SSS1.p1.2.m2.1.1.2"><csymbol cd="ambiguous" id="A4.SS1.SSS1.p1.2.m2.1.1.2.1.cmml" xref="A4.SS1.SSS1.p1.2.m2.1.1.2">subscript</csymbol><ci id="A4.SS1.SSS1.p1.2.m2.1.1.2.2.cmml" xref="A4.SS1.SSS1.p1.2.m2.1.1.2.2">𝑝</ci><ci id="A4.SS1.SSS1.p1.2.m2.1.1.2.3.cmml" xref="A4.SS1.SSS1.p1.2.m2.1.1.2.3">𝜎</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.SS1.SSS1.p1.2.m2.1c">\nabla\log p_{\sigma}</annotation><annotation encoding="application/x-llamapun" id="A4.SS1.SSS1.p1.2.m2.1d">∇ roman_log italic_p start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT</annotation></semantics></math> converges uniformly to <math alttext="\nabla\log p" class="ltx_Math" display="inline" id="A4.SS1.SSS1.p1.3.m3.1"><semantics id="A4.SS1.SSS1.p1.3.m3.1a"><mrow id="A4.SS1.SSS1.p1.3.m3.1.1" xref="A4.SS1.SSS1.p1.3.m3.1.1.cmml"><mrow id="A4.SS1.SSS1.p1.3.m3.1.1.1" xref="A4.SS1.SSS1.p1.3.m3.1.1.1.cmml"><mo id="A4.SS1.SSS1.p1.3.m3.1.1.1.1" rspace="0.167em" xref="A4.SS1.SSS1.p1.3.m3.1.1.1.1.cmml">∇</mo><mi id="A4.SS1.SSS1.p1.3.m3.1.1.1.2" xref="A4.SS1.SSS1.p1.3.m3.1.1.1.2.cmml">log</mi></mrow><mo id="A4.SS1.SSS1.p1.3.m3.1.1a" lspace="0.167em" xref="A4.SS1.SSS1.p1.3.m3.1.1.cmml">⁡</mo><mi id="A4.SS1.SSS1.p1.3.m3.1.1.2" xref="A4.SS1.SSS1.p1.3.m3.1.1.2.cmml">p</mi></mrow><annotation-xml encoding="MathML-Content" id="A4.SS1.SSS1.p1.3.m3.1b"><apply id="A4.SS1.SSS1.p1.3.m3.1.1.cmml" xref="A4.SS1.SSS1.p1.3.m3.1.1"><apply id="A4.SS1.SSS1.p1.3.m3.1.1.1.cmml" xref="A4.SS1.SSS1.p1.3.m3.1.1.1"><ci id="A4.SS1.SSS1.p1.3.m3.1.1.1.1.cmml" xref="A4.SS1.SSS1.p1.3.m3.1.1.1.1">∇</ci><log id="A4.SS1.SSS1.p1.3.m3.1.1.1.2.cmml" xref="A4.SS1.SSS1.p1.3.m3.1.1.1.2"></log></apply><ci id="A4.SS1.SSS1.p1.3.m3.1.1.2.cmml" xref="A4.SS1.SSS1.p1.3.m3.1.1.2">𝑝</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.SS1.SSS1.p1.3.m3.1c">\nabla\log p</annotation><annotation encoding="application/x-llamapun" id="A4.SS1.SSS1.p1.3.m3.1d">∇ roman_log italic_p</annotation></semantics></math> on <math alttext="\mathbf{L}" class="ltx_Math" display="inline" id="A4.SS1.SSS1.p1.4.m4.1"><semantics id="A4.SS1.SSS1.p1.4.m4.1a"><mi id="A4.SS1.SSS1.p1.4.m4.1.1" xref="A4.SS1.SSS1.p1.4.m4.1.1.cmml">𝐋</mi><annotation-xml encoding="MathML-Content" id="A4.SS1.SSS1.p1.4.m4.1b"><ci id="A4.SS1.SSS1.p1.4.m4.1.1.cmml" xref="A4.SS1.SSS1.p1.4.m4.1.1">𝐋</ci></annotation-xml><annotation encoding="application/x-tex" id="A4.SS1.SSS1.p1.4.m4.1c">\mathbf{L}</annotation><annotation encoding="application/x-llamapun" id="A4.SS1.SSS1.p1.4.m4.1d">bold_L</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="A4.SS1.SSS1.p2"> <p class="ltx_p" id="A4.SS1.SSS1.p2.1">For <math alttext="\epsilon>0,{\mathbf{x}}\in\mathbf{K}" class="ltx_Math" display="inline" id="A4.SS1.SSS1.p2.1.m1.2"><semantics id="A4.SS1.SSS1.p2.1.m1.2a"><mrow id="A4.SS1.SSS1.p2.1.m1.2.2.2" xref="A4.SS1.SSS1.p2.1.m1.2.2.3.cmml"><mrow id="A4.SS1.SSS1.p2.1.m1.1.1.1.1" xref="A4.SS1.SSS1.p2.1.m1.1.1.1.1.cmml"><mi id="A4.SS1.SSS1.p2.1.m1.1.1.1.1.2" xref="A4.SS1.SSS1.p2.1.m1.1.1.1.1.2.cmml">ϵ</mi><mo id="A4.SS1.SSS1.p2.1.m1.1.1.1.1.1" xref="A4.SS1.SSS1.p2.1.m1.1.1.1.1.1.cmml">></mo><mn id="A4.SS1.SSS1.p2.1.m1.1.1.1.1.3" xref="A4.SS1.SSS1.p2.1.m1.1.1.1.1.3.cmml">0</mn></mrow><mo id="A4.SS1.SSS1.p2.1.m1.2.2.2.3" xref="A4.SS1.SSS1.p2.1.m1.2.2.3a.cmml">,</mo><mrow id="A4.SS1.SSS1.p2.1.m1.2.2.2.2" xref="A4.SS1.SSS1.p2.1.m1.2.2.2.2.cmml"><mi id="A4.SS1.SSS1.p2.1.m1.2.2.2.2.2" xref="A4.SS1.SSS1.p2.1.m1.2.2.2.2.2.cmml">𝐱</mi><mo id="A4.SS1.SSS1.p2.1.m1.2.2.2.2.1" xref="A4.SS1.SSS1.p2.1.m1.2.2.2.2.1.cmml">∈</mo><mi id="A4.SS1.SSS1.p2.1.m1.2.2.2.2.3" xref="A4.SS1.SSS1.p2.1.m1.2.2.2.2.3.cmml">𝐊</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="A4.SS1.SSS1.p2.1.m1.2b"><apply id="A4.SS1.SSS1.p2.1.m1.2.2.3.cmml" xref="A4.SS1.SSS1.p2.1.m1.2.2.2"><csymbol cd="ambiguous" id="A4.SS1.SSS1.p2.1.m1.2.2.3a.cmml" xref="A4.SS1.SSS1.p2.1.m1.2.2.2.3">formulae-sequence</csymbol><apply id="A4.SS1.SSS1.p2.1.m1.1.1.1.1.cmml" xref="A4.SS1.SSS1.p2.1.m1.1.1.1.1"><gt id="A4.SS1.SSS1.p2.1.m1.1.1.1.1.1.cmml" xref="A4.SS1.SSS1.p2.1.m1.1.1.1.1.1"></gt><ci id="A4.SS1.SSS1.p2.1.m1.1.1.1.1.2.cmml" xref="A4.SS1.SSS1.p2.1.m1.1.1.1.1.2">italic-ϵ</ci><cn id="A4.SS1.SSS1.p2.1.m1.1.1.1.1.3.cmml" type="integer" xref="A4.SS1.SSS1.p2.1.m1.1.1.1.1.3">0</cn></apply><apply id="A4.SS1.SSS1.p2.1.m1.2.2.2.2.cmml" xref="A4.SS1.SSS1.p2.1.m1.2.2.2.2"><in id="A4.SS1.SSS1.p2.1.m1.2.2.2.2.1.cmml" xref="A4.SS1.SSS1.p2.1.m1.2.2.2.2.1"></in><ci id="A4.SS1.SSS1.p2.1.m1.2.2.2.2.2.cmml" xref="A4.SS1.SSS1.p2.1.m1.2.2.2.2.2">𝐱</ci><ci id="A4.SS1.SSS1.p2.1.m1.2.2.2.2.3.cmml" xref="A4.SS1.SSS1.p2.1.m1.2.2.2.2.3">𝐊</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.SS1.SSS1.p2.1.m1.2c">\epsilon>0,{\mathbf{x}}\in\mathbf{K}</annotation><annotation encoding="application/x-llamapun" id="A4.SS1.SSS1.p2.1.m1.2d">italic_ϵ > 0 , bold_x ∈ bold_K</annotation></semantics></math>,</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A8.EGx46"> <tbody id="A4.Ex72"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\|-\mathcal{R}_{\sigma}({\mathbf{x}})-\nabla\log p({\mathbf{x}})% \|=\|\int_{{\mathbb{R}}^{d}}{\nabla\log p_{\sigma}({\mathbf{x}}+\zeta)\mathcal% {N}(\zeta;0,\sigma^{2}\mathbf{I}_{d})}d\zeta-\nabla\log p({\mathbf{x}})\|" class="ltx_Math" display="inline" id="A4.Ex72.m1.7"><semantics id="A4.Ex72.m1.7a"><mrow id="A4.Ex72.m1.7.7" xref="A4.Ex72.m1.7.7.cmml"><mrow id="A4.Ex72.m1.6.6.1.1" xref="A4.Ex72.m1.6.6.1.2.cmml"><mo id="A4.Ex72.m1.6.6.1.1.2" stretchy="false" xref="A4.Ex72.m1.6.6.1.2.1.cmml">‖</mo><mrow id="A4.Ex72.m1.6.6.1.1.1" xref="A4.Ex72.m1.6.6.1.1.1.cmml"><mrow id="A4.Ex72.m1.6.6.1.1.1.2" xref="A4.Ex72.m1.6.6.1.1.1.2.cmml"><mo id="A4.Ex72.m1.6.6.1.1.1.2a" xref="A4.Ex72.m1.6.6.1.1.1.2.cmml">−</mo><mrow id="A4.Ex72.m1.6.6.1.1.1.2.2" xref="A4.Ex72.m1.6.6.1.1.1.2.2.cmml"><msub id="A4.Ex72.m1.6.6.1.1.1.2.2.2" xref="A4.Ex72.m1.6.6.1.1.1.2.2.2.cmml"><mi class="ltx_font_mathcaligraphic" id="A4.Ex72.m1.6.6.1.1.1.2.2.2.2" xref="A4.Ex72.m1.6.6.1.1.1.2.2.2.2.cmml">ℛ</mi><mi id="A4.Ex72.m1.6.6.1.1.1.2.2.2.3" xref="A4.Ex72.m1.6.6.1.1.1.2.2.2.3.cmml">σ</mi></msub><mo id="A4.Ex72.m1.6.6.1.1.1.2.2.1" xref="A4.Ex72.m1.6.6.1.1.1.2.2.1.cmml">⁢</mo><mrow id="A4.Ex72.m1.6.6.1.1.1.2.2.3.2" xref="A4.Ex72.m1.6.6.1.1.1.2.2.cmml"><mo id="A4.Ex72.m1.6.6.1.1.1.2.2.3.2.1" stretchy="false" xref="A4.Ex72.m1.6.6.1.1.1.2.2.cmml">(</mo><mi id="A4.Ex72.m1.1.1" xref="A4.Ex72.m1.1.1.cmml">𝐱</mi><mo id="A4.Ex72.m1.6.6.1.1.1.2.2.3.2.2" stretchy="false" xref="A4.Ex72.m1.6.6.1.1.1.2.2.cmml">)</mo></mrow></mrow></mrow><mo id="A4.Ex72.m1.6.6.1.1.1.1" xref="A4.Ex72.m1.6.6.1.1.1.1.cmml">−</mo><mrow id="A4.Ex72.m1.6.6.1.1.1.3" xref="A4.Ex72.m1.6.6.1.1.1.3.cmml"><mrow id="A4.Ex72.m1.6.6.1.1.1.3.2" xref="A4.Ex72.m1.6.6.1.1.1.3.2.cmml"><mrow id="A4.Ex72.m1.6.6.1.1.1.3.2.1" xref="A4.Ex72.m1.6.6.1.1.1.3.2.1.cmml"><mo id="A4.Ex72.m1.6.6.1.1.1.3.2.1.1" rspace="0.167em" xref="A4.Ex72.m1.6.6.1.1.1.3.2.1.1.cmml">∇</mo><mi 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end_POSTSUPERSCRIPT bold_I start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT ) italic_d italic_ζ - ∇ roman_log italic_p ( bold_x ) ∥</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="A4.Ex73"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\leq\int_{{\mathbb{R}}^{d}}{\|\nabla\log p_{\sigma}({\mathbf{x}}+% \zeta)-\nabla\log p({\mathbf{x}})\|\mathcal{N}(\zeta;0,\sigma^{2}\mathbf{I}_{d% })}d\zeta" class="ltx_Math" display="inline" id="A4.Ex73.m1.5"><semantics id="A4.Ex73.m1.5a"><mrow id="A4.Ex73.m1.5.5" xref="A4.Ex73.m1.5.5.cmml"><mi id="A4.Ex73.m1.5.5.4" xref="A4.Ex73.m1.5.5.4.cmml"></mi><mo id="A4.Ex73.m1.5.5.3" xref="A4.Ex73.m1.5.5.3.cmml">≤</mo><mrow id="A4.Ex73.m1.5.5.2" xref="A4.Ex73.m1.5.5.2.cmml"><mstyle displaystyle="true" id="A4.Ex73.m1.5.5.2.3" 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xref="A4.Ex74.m1.9.9.4.4.2.2.1.1.3.3">𝑑</ci></apply></apply></list><apply id="A4.Ex74.m1.9.9.4.4.2.5.cmml" xref="A4.Ex74.m1.9.9.4.4.2.5"><csymbol cd="latexml" id="A4.Ex74.m1.9.9.4.4.2.5.1.cmml" xref="A4.Ex74.m1.9.9.4.4.2.5.1">differential-d</csymbol><ci id="A4.Ex74.m1.9.9.4.4.2.5.2.cmml" xref="A4.Ex74.m1.9.9.4.4.2.5.2">𝜁</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.Ex74.m1.9c">\displaystyle\leq\int_{{\mathbb{R}}^{d}}{\|\nabla\log p_{\sigma}({\mathbf{x}}+% \zeta)-\nabla\log p({\mathbf{x}}+\zeta)\|\mathcal{N}(\zeta;0,\sigma^{2}\mathbf% {I}_{d})}d\zeta+\int_{{\mathbb{R}}^{d}}{\|\nabla\log p({\mathbf{x}}+\zeta)-% \nabla\log p({\mathbf{x}})\|\mathcal{N}(\zeta;0,\sigma^{2}\mathbf{I}_{d})}d\zeta</annotation><annotation encoding="application/x-llamapun" id="A4.Ex74.m1.9d">≤ ∫ start_POSTSUBSCRIPT blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ∥ ∇ roman_log italic_p start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ( bold_x + italic_ζ ) - ∇ roman_log italic_p ( bold_x + italic_ζ ) ∥ caligraphic_N ( italic_ζ ; 0 , italic_σ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT bold_I start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT ) italic_d italic_ζ + ∫ start_POSTSUBSCRIPT blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ∥ ∇ roman_log italic_p ( bold_x + italic_ζ ) - ∇ roman_log italic_p ( bold_x ) ∥ caligraphic_N ( italic_ζ ; 0 , italic_σ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT bold_I start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT ) italic_d italic_ζ</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="A4.Ex75"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\leq\int_{\mathcal{B}(0,1)}{\|\nabla\log p_{\sigma}-\nabla\log p% 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id="A4.Ex75.m1.14.14.4.4.2.5.1.cmml" xref="A4.Ex75.m1.14.14.4.4.2.5.1">differential-d</csymbol><ci id="A4.Ex75.m1.14.14.4.4.2.5.2.cmml" xref="A4.Ex75.m1.14.14.4.4.2.5.2">𝜁</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.Ex75.m1.14c">\displaystyle\leq\int_{\mathcal{B}(0,1)}{\|\nabla\log p_{\sigma}-\nabla\log p% \|_{\infty,\mathbf{L}}\mathcal{N}(\zeta;0,\sigma^{2}\mathbf{I}_{d})}d\zeta+% \int_{{\mathbb{R}}^{d}\setminus\mathcal{B}(0,1)}{\left(\|\nabla\log p_{\sigma}% ({\mathbf{x}}+\zeta)\|+\|\nabla\log p({\mathbf{x}}+\zeta)\|\right)\mathcal{N}(% \zeta;0,\sigma^{2}\mathbf{I}_{d})}d\zeta</annotation><annotation encoding="application/x-llamapun" id="A4.Ex75.m1.14d">≤ ∫ start_POSTSUBSCRIPT caligraphic_B ( 0 , 1 ) end_POSTSUBSCRIPT ∥ ∇ roman_log italic_p start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT - ∇ roman_log italic_p ∥ start_POSTSUBSCRIPT ∞ , bold_L end_POSTSUBSCRIPT caligraphic_N ( italic_ζ ; 0 , italic_σ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT bold_I start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT ) italic_d italic_ζ + ∫ start_POSTSUBSCRIPT blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT ∖ caligraphic_B ( 0 , 1 ) end_POSTSUBSCRIPT ( ∥ ∇ roman_log italic_p start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ( bold_x + italic_ζ ) ∥ + ∥ ∇ roman_log italic_p ( bold_x + italic_ζ ) ∥ ) caligraphic_N ( italic_ζ ; 0 , italic_σ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT bold_I start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT ) italic_d italic_ζ</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="A4.Ex76"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle+\int_{{\mathbb{R}}^{d}}{\|\nabla\log p({\mathbf{x}}+\zeta)-% \nabla\log p({\mathbf{x}})\|\mathcal{N}(\zeta;0,\sigma^{2}\mathbf{I}_{d})}d\zeta." 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xref="A4.Ex76.m1.4.4.1.1.2.2.2.1.1.1"></times><apply id="A4.Ex76.m1.4.4.1.1.2.2.2.1.1.2.cmml" xref="A4.Ex76.m1.4.4.1.1.2.2.2.1.1.2"><csymbol cd="ambiguous" id="A4.Ex76.m1.4.4.1.1.2.2.2.1.1.2.1.cmml" xref="A4.Ex76.m1.4.4.1.1.2.2.2.1.1.2">superscript</csymbol><ci id="A4.Ex76.m1.4.4.1.1.2.2.2.1.1.2.2.cmml" xref="A4.Ex76.m1.4.4.1.1.2.2.2.1.1.2.2">𝜎</ci><cn id="A4.Ex76.m1.4.4.1.1.2.2.2.1.1.2.3.cmml" type="integer" xref="A4.Ex76.m1.4.4.1.1.2.2.2.1.1.2.3">2</cn></apply><apply id="A4.Ex76.m1.4.4.1.1.2.2.2.1.1.3.cmml" xref="A4.Ex76.m1.4.4.1.1.2.2.2.1.1.3"><csymbol cd="ambiguous" id="A4.Ex76.m1.4.4.1.1.2.2.2.1.1.3.1.cmml" xref="A4.Ex76.m1.4.4.1.1.2.2.2.1.1.3">subscript</csymbol><ci id="A4.Ex76.m1.4.4.1.1.2.2.2.1.1.3.2.cmml" xref="A4.Ex76.m1.4.4.1.1.2.2.2.1.1.3.2">𝐈</ci><ci id="A4.Ex76.m1.4.4.1.1.2.2.2.1.1.3.3.cmml" xref="A4.Ex76.m1.4.4.1.1.2.2.2.1.1.3.3">𝑑</ci></apply></apply></list><apply id="A4.Ex76.m1.4.4.1.1.2.2.5.cmml" xref="A4.Ex76.m1.4.4.1.1.2.2.5"><csymbol cd="latexml" id="A4.Ex76.m1.4.4.1.1.2.2.5.1.cmml" xref="A4.Ex76.m1.4.4.1.1.2.2.5.1">differential-d</csymbol><ci id="A4.Ex76.m1.4.4.1.1.2.2.5.2.cmml" xref="A4.Ex76.m1.4.4.1.1.2.2.5.2">𝜁</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.Ex76.m1.4c">\displaystyle+\int_{{\mathbb{R}}^{d}}{\|\nabla\log p({\mathbf{x}}+\zeta)-% \nabla\log p({\mathbf{x}})\|\mathcal{N}(\zeta;0,\sigma^{2}\mathbf{I}_{d})}d\zeta.</annotation><annotation encoding="application/x-llamapun" id="A4.Ex76.m1.4d">+ ∫ start_POSTSUBSCRIPT blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ∥ ∇ roman_log italic_p ( bold_x + italic_ζ ) - ∇ roman_log italic_p ( bold_x ) ∥ caligraphic_N ( italic_ζ ; 0 , italic_σ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT bold_I start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT ) italic_d italic_ζ .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="A4.SS1.SSS1.p2.9">Because <math alttext="\mathbf{L}" class="ltx_Math" display="inline" id="A4.SS1.SSS1.p2.2.m1.1"><semantics id="A4.SS1.SSS1.p2.2.m1.1a"><mi id="A4.SS1.SSS1.p2.2.m1.1.1" xref="A4.SS1.SSS1.p2.2.m1.1.1.cmml">𝐋</mi><annotation-xml encoding="MathML-Content" id="A4.SS1.SSS1.p2.2.m1.1b"><ci id="A4.SS1.SSS1.p2.2.m1.1.1.cmml" xref="A4.SS1.SSS1.p2.2.m1.1.1">𝐋</ci></annotation-xml><annotation encoding="application/x-tex" id="A4.SS1.SSS1.p2.2.m1.1c">\mathbf{L}</annotation><annotation encoding="application/x-llamapun" id="A4.SS1.SSS1.p2.2.m1.1d">bold_L</annotation></semantics></math> is compact and <math alttext="\nabla\log p" class="ltx_Math" display="inline" id="A4.SS1.SSS1.p2.3.m2.1"><semantics id="A4.SS1.SSS1.p2.3.m2.1a"><mrow id="A4.SS1.SSS1.p2.3.m2.1.1" xref="A4.SS1.SSS1.p2.3.m2.1.1.cmml"><mrow id="A4.SS1.SSS1.p2.3.m2.1.1.1" xref="A4.SS1.SSS1.p2.3.m2.1.1.1.cmml"><mo id="A4.SS1.SSS1.p2.3.m2.1.1.1.1" rspace="0.167em" xref="A4.SS1.SSS1.p2.3.m2.1.1.1.1.cmml">∇</mo><mi id="A4.SS1.SSS1.p2.3.m2.1.1.1.2" xref="A4.SS1.SSS1.p2.3.m2.1.1.1.2.cmml">log</mi></mrow><mo id="A4.SS1.SSS1.p2.3.m2.1.1a" lspace="0.167em" xref="A4.SS1.SSS1.p2.3.m2.1.1.cmml">⁡</mo><mi id="A4.SS1.SSS1.p2.3.m2.1.1.2" xref="A4.SS1.SSS1.p2.3.m2.1.1.2.cmml">p</mi></mrow><annotation-xml encoding="MathML-Content" id="A4.SS1.SSS1.p2.3.m2.1b"><apply id="A4.SS1.SSS1.p2.3.m2.1.1.cmml" xref="A4.SS1.SSS1.p2.3.m2.1.1"><apply id="A4.SS1.SSS1.p2.3.m2.1.1.1.cmml" xref="A4.SS1.SSS1.p2.3.m2.1.1.1"><ci id="A4.SS1.SSS1.p2.3.m2.1.1.1.1.cmml" xref="A4.SS1.SSS1.p2.3.m2.1.1.1.1">∇</ci><log id="A4.SS1.SSS1.p2.3.m2.1.1.1.2.cmml" xref="A4.SS1.SSS1.p2.3.m2.1.1.1.2"></log></apply><ci id="A4.SS1.SSS1.p2.3.m2.1.1.2.cmml" xref="A4.SS1.SSS1.p2.3.m2.1.1.2">𝑝</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.SS1.SSS1.p2.3.m2.1c">\nabla\log p</annotation><annotation encoding="application/x-llamapun" id="A4.SS1.SSS1.p2.3.m2.1d">∇ roman_log italic_p</annotation></semantics></math> is continuous on <math alttext="\bar{\mathbf{K}}" class="ltx_Math" display="inline" id="A4.SS1.SSS1.p2.4.m3.1"><semantics id="A4.SS1.SSS1.p2.4.m3.1a"><mover accent="true" id="A4.SS1.SSS1.p2.4.m3.1.1" xref="A4.SS1.SSS1.p2.4.m3.1.1.cmml"><mi id="A4.SS1.SSS1.p2.4.m3.1.1.2" xref="A4.SS1.SSS1.p2.4.m3.1.1.2.cmml">𝐊</mi><mo id="A4.SS1.SSS1.p2.4.m3.1.1.1" xref="A4.SS1.SSS1.p2.4.m3.1.1.1.cmml">¯</mo></mover><annotation-xml encoding="MathML-Content" id="A4.SS1.SSS1.p2.4.m3.1b"><apply id="A4.SS1.SSS1.p2.4.m3.1.1.cmml" xref="A4.SS1.SSS1.p2.4.m3.1.1"><ci id="A4.SS1.SSS1.p2.4.m3.1.1.1.cmml" xref="A4.SS1.SSS1.p2.4.m3.1.1.1">¯</ci><ci id="A4.SS1.SSS1.p2.4.m3.1.1.2.cmml" xref="A4.SS1.SSS1.p2.4.m3.1.1.2">𝐊</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.SS1.SSS1.p2.4.m3.1c">\bar{\mathbf{K}}</annotation><annotation encoding="application/x-llamapun" id="A4.SS1.SSS1.p2.4.m3.1d">over¯ start_ARG bold_K end_ARG</annotation></semantics></math>, <math alttext="\nabla\log p" class="ltx_Math" display="inline" id="A4.SS1.SSS1.p2.5.m4.1"><semantics id="A4.SS1.SSS1.p2.5.m4.1a"><mrow id="A4.SS1.SSS1.p2.5.m4.1.1" xref="A4.SS1.SSS1.p2.5.m4.1.1.cmml"><mrow id="A4.SS1.SSS1.p2.5.m4.1.1.1" xref="A4.SS1.SSS1.p2.5.m4.1.1.1.cmml"><mo id="A4.SS1.SSS1.p2.5.m4.1.1.1.1" rspace="0.167em" xref="A4.SS1.SSS1.p2.5.m4.1.1.1.1.cmml">∇</mo><mi id="A4.SS1.SSS1.p2.5.m4.1.1.1.2" xref="A4.SS1.SSS1.p2.5.m4.1.1.1.2.cmml">log</mi></mrow><mo id="A4.SS1.SSS1.p2.5.m4.1.1a" lspace="0.167em" xref="A4.SS1.SSS1.p2.5.m4.1.1.cmml">⁡</mo><mi id="A4.SS1.SSS1.p2.5.m4.1.1.2" xref="A4.SS1.SSS1.p2.5.m4.1.1.2.cmml">p</mi></mrow><annotation-xml encoding="MathML-Content" id="A4.SS1.SSS1.p2.5.m4.1b"><apply id="A4.SS1.SSS1.p2.5.m4.1.1.cmml" xref="A4.SS1.SSS1.p2.5.m4.1.1"><apply id="A4.SS1.SSS1.p2.5.m4.1.1.1.cmml" xref="A4.SS1.SSS1.p2.5.m4.1.1.1"><ci id="A4.SS1.SSS1.p2.5.m4.1.1.1.1.cmml" xref="A4.SS1.SSS1.p2.5.m4.1.1.1.1">∇</ci><log id="A4.SS1.SSS1.p2.5.m4.1.1.1.2.cmml" xref="A4.SS1.SSS1.p2.5.m4.1.1.1.2"></log></apply><ci id="A4.SS1.SSS1.p2.5.m4.1.1.2.cmml" xref="A4.SS1.SSS1.p2.5.m4.1.1.2">𝑝</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.SS1.SSS1.p2.5.m4.1c">\nabla\log p</annotation><annotation encoding="application/x-llamapun" id="A4.SS1.SSS1.p2.5.m4.1d">∇ roman_log italic_p</annotation></semantics></math> is uniformly continuous on <math alttext="\mathbf{L}" class="ltx_Math" display="inline" id="A4.SS1.SSS1.p2.6.m5.1"><semantics id="A4.SS1.SSS1.p2.6.m5.1a"><mi id="A4.SS1.SSS1.p2.6.m5.1.1" xref="A4.SS1.SSS1.p2.6.m5.1.1.cmml">𝐋</mi><annotation-xml encoding="MathML-Content" id="A4.SS1.SSS1.p2.6.m5.1b"><ci id="A4.SS1.SSS1.p2.6.m5.1.1.cmml" xref="A4.SS1.SSS1.p2.6.m5.1.1">𝐋</ci></annotation-xml><annotation encoding="application/x-tex" id="A4.SS1.SSS1.p2.6.m5.1c">\mathbf{L}</annotation><annotation encoding="application/x-llamapun" id="A4.SS1.SSS1.p2.6.m5.1d">bold_L</annotation></semantics></math>. So there exists <math alttext="1\geq\mu>0" class="ltx_Math" display="inline" id="A4.SS1.SSS1.p2.7.m6.1"><semantics id="A4.SS1.SSS1.p2.7.m6.1a"><mrow id="A4.SS1.SSS1.p2.7.m6.1.1" xref="A4.SS1.SSS1.p2.7.m6.1.1.cmml"><mn id="A4.SS1.SSS1.p2.7.m6.1.1.2" xref="A4.SS1.SSS1.p2.7.m6.1.1.2.cmml">1</mn><mo id="A4.SS1.SSS1.p2.7.m6.1.1.3" xref="A4.SS1.SSS1.p2.7.m6.1.1.3.cmml">≥</mo><mi id="A4.SS1.SSS1.p2.7.m6.1.1.4" xref="A4.SS1.SSS1.p2.7.m6.1.1.4.cmml">μ</mi><mo id="A4.SS1.SSS1.p2.7.m6.1.1.5" xref="A4.SS1.SSS1.p2.7.m6.1.1.5.cmml">></mo><mn id="A4.SS1.SSS1.p2.7.m6.1.1.6" xref="A4.SS1.SSS1.p2.7.m6.1.1.6.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="A4.SS1.SSS1.p2.7.m6.1b"><apply id="A4.SS1.SSS1.p2.7.m6.1.1.cmml" xref="A4.SS1.SSS1.p2.7.m6.1.1"><and id="A4.SS1.SSS1.p2.7.m6.1.1a.cmml" xref="A4.SS1.SSS1.p2.7.m6.1.1"></and><apply id="A4.SS1.SSS1.p2.7.m6.1.1b.cmml" xref="A4.SS1.SSS1.p2.7.m6.1.1"><geq id="A4.SS1.SSS1.p2.7.m6.1.1.3.cmml" xref="A4.SS1.SSS1.p2.7.m6.1.1.3"></geq><cn id="A4.SS1.SSS1.p2.7.m6.1.1.2.cmml" type="integer" xref="A4.SS1.SSS1.p2.7.m6.1.1.2">1</cn><ci id="A4.SS1.SSS1.p2.7.m6.1.1.4.cmml" xref="A4.SS1.SSS1.p2.7.m6.1.1.4">𝜇</ci></apply><apply id="A4.SS1.SSS1.p2.7.m6.1.1c.cmml" xref="A4.SS1.SSS1.p2.7.m6.1.1"><gt id="A4.SS1.SSS1.p2.7.m6.1.1.5.cmml" xref="A4.SS1.SSS1.p2.7.m6.1.1.5"></gt><share href="#A4.SS1.SSS1.p2.7.m6.1.1.4.cmml" id="A4.SS1.SSS1.p2.7.m6.1.1d.cmml" xref="A4.SS1.SSS1.p2.7.m6.1.1"></share><cn id="A4.SS1.SSS1.p2.7.m6.1.1.6.cmml" type="integer" xref="A4.SS1.SSS1.p2.7.m6.1.1.6">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.SS1.SSS1.p2.7.m6.1c">1\geq\mu>0</annotation><annotation encoding="application/x-llamapun" id="A4.SS1.SSS1.p2.7.m6.1d">1 ≥ italic_μ > 0</annotation></semantics></math> such that <math alttext="\forall{\mathbf{x}},{\mathbf{y}}\in\mathbf{L}" class="ltx_Math" display="inline" id="A4.SS1.SSS1.p2.8.m7.2"><semantics id="A4.SS1.SSS1.p2.8.m7.2a"><mrow id="A4.SS1.SSS1.p2.8.m7.2.2" xref="A4.SS1.SSS1.p2.8.m7.2.2.cmml"><mrow id="A4.SS1.SSS1.p2.8.m7.2.2.1.1" xref="A4.SS1.SSS1.p2.8.m7.2.2.1.2.cmml"><mrow id="A4.SS1.SSS1.p2.8.m7.2.2.1.1.1" xref="A4.SS1.SSS1.p2.8.m7.2.2.1.1.1.cmml"><mo id="A4.SS1.SSS1.p2.8.m7.2.2.1.1.1.1" rspace="0.167em" xref="A4.SS1.SSS1.p2.8.m7.2.2.1.1.1.1.cmml">∀</mo><mi id="A4.SS1.SSS1.p2.8.m7.2.2.1.1.1.2" xref="A4.SS1.SSS1.p2.8.m7.2.2.1.1.1.2.cmml">𝐱</mi></mrow><mo id="A4.SS1.SSS1.p2.8.m7.2.2.1.1.2" xref="A4.SS1.SSS1.p2.8.m7.2.2.1.2.cmml">,</mo><mi id="A4.SS1.SSS1.p2.8.m7.1.1" xref="A4.SS1.SSS1.p2.8.m7.1.1.cmml">𝐲</mi></mrow><mo id="A4.SS1.SSS1.p2.8.m7.2.2.2" xref="A4.SS1.SSS1.p2.8.m7.2.2.2.cmml">∈</mo><mi id="A4.SS1.SSS1.p2.8.m7.2.2.3" xref="A4.SS1.SSS1.p2.8.m7.2.2.3.cmml">𝐋</mi></mrow><annotation-xml encoding="MathML-Content" id="A4.SS1.SSS1.p2.8.m7.2b"><apply id="A4.SS1.SSS1.p2.8.m7.2.2.cmml" xref="A4.SS1.SSS1.p2.8.m7.2.2"><in id="A4.SS1.SSS1.p2.8.m7.2.2.2.cmml" xref="A4.SS1.SSS1.p2.8.m7.2.2.2"></in><list id="A4.SS1.SSS1.p2.8.m7.2.2.1.2.cmml" xref="A4.SS1.SSS1.p2.8.m7.2.2.1.1"><apply id="A4.SS1.SSS1.p2.8.m7.2.2.1.1.1.cmml" xref="A4.SS1.SSS1.p2.8.m7.2.2.1.1.1"><csymbol cd="latexml" id="A4.SS1.SSS1.p2.8.m7.2.2.1.1.1.1.cmml" xref="A4.SS1.SSS1.p2.8.m7.2.2.1.1.1.1">for-all</csymbol><ci id="A4.SS1.SSS1.p2.8.m7.2.2.1.1.1.2.cmml" xref="A4.SS1.SSS1.p2.8.m7.2.2.1.1.1.2">𝐱</ci></apply><ci id="A4.SS1.SSS1.p2.8.m7.1.1.cmml" xref="A4.SS1.SSS1.p2.8.m7.1.1">𝐲</ci></list><ci id="A4.SS1.SSS1.p2.8.m7.2.2.3.cmml" xref="A4.SS1.SSS1.p2.8.m7.2.2.3">𝐋</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.SS1.SSS1.p2.8.m7.2c">\forall{\mathbf{x}},{\mathbf{y}}\in\mathbf{L}</annotation><annotation encoding="application/x-llamapun" id="A4.SS1.SSS1.p2.8.m7.2d">∀ bold_x , bold_y ∈ bold_L</annotation></semantics></math>, if <math alttext="\|{\mathbf{x}}-{\mathbf{y}}\|\leq\mu" class="ltx_Math" display="inline" id="A4.SS1.SSS1.p2.9.m8.1"><semantics id="A4.SS1.SSS1.p2.9.m8.1a"><mrow id="A4.SS1.SSS1.p2.9.m8.1.1" xref="A4.SS1.SSS1.p2.9.m8.1.1.cmml"><mrow id="A4.SS1.SSS1.p2.9.m8.1.1.1.1" xref="A4.SS1.SSS1.p2.9.m8.1.1.1.2.cmml"><mo id="A4.SS1.SSS1.p2.9.m8.1.1.1.1.2" stretchy="false" xref="A4.SS1.SSS1.p2.9.m8.1.1.1.2.1.cmml">‖</mo><mrow id="A4.SS1.SSS1.p2.9.m8.1.1.1.1.1" xref="A4.SS1.SSS1.p2.9.m8.1.1.1.1.1.cmml"><mi id="A4.SS1.SSS1.p2.9.m8.1.1.1.1.1.2" xref="A4.SS1.SSS1.p2.9.m8.1.1.1.1.1.2.cmml">𝐱</mi><mo id="A4.SS1.SSS1.p2.9.m8.1.1.1.1.1.1" xref="A4.SS1.SSS1.p2.9.m8.1.1.1.1.1.1.cmml">−</mo><mi id="A4.SS1.SSS1.p2.9.m8.1.1.1.1.1.3" xref="A4.SS1.SSS1.p2.9.m8.1.1.1.1.1.3.cmml">𝐲</mi></mrow><mo id="A4.SS1.SSS1.p2.9.m8.1.1.1.1.3" stretchy="false" xref="A4.SS1.SSS1.p2.9.m8.1.1.1.2.1.cmml">‖</mo></mrow><mo id="A4.SS1.SSS1.p2.9.m8.1.1.2" xref="A4.SS1.SSS1.p2.9.m8.1.1.2.cmml">≤</mo><mi id="A4.SS1.SSS1.p2.9.m8.1.1.3" xref="A4.SS1.SSS1.p2.9.m8.1.1.3.cmml">μ</mi></mrow><annotation-xml encoding="MathML-Content" id="A4.SS1.SSS1.p2.9.m8.1b"><apply id="A4.SS1.SSS1.p2.9.m8.1.1.cmml" xref="A4.SS1.SSS1.p2.9.m8.1.1"><leq id="A4.SS1.SSS1.p2.9.m8.1.1.2.cmml" xref="A4.SS1.SSS1.p2.9.m8.1.1.2"></leq><apply id="A4.SS1.SSS1.p2.9.m8.1.1.1.2.cmml" xref="A4.SS1.SSS1.p2.9.m8.1.1.1.1"><csymbol cd="latexml" id="A4.SS1.SSS1.p2.9.m8.1.1.1.2.1.cmml" xref="A4.SS1.SSS1.p2.9.m8.1.1.1.1.2">norm</csymbol><apply id="A4.SS1.SSS1.p2.9.m8.1.1.1.1.1.cmml" xref="A4.SS1.SSS1.p2.9.m8.1.1.1.1.1"><minus id="A4.SS1.SSS1.p2.9.m8.1.1.1.1.1.1.cmml" xref="A4.SS1.SSS1.p2.9.m8.1.1.1.1.1.1"></minus><ci id="A4.SS1.SSS1.p2.9.m8.1.1.1.1.1.2.cmml" xref="A4.SS1.SSS1.p2.9.m8.1.1.1.1.1.2">𝐱</ci><ci id="A4.SS1.SSS1.p2.9.m8.1.1.1.1.1.3.cmml" xref="A4.SS1.SSS1.p2.9.m8.1.1.1.1.1.3">𝐲</ci></apply></apply><ci id="A4.SS1.SSS1.p2.9.m8.1.1.3.cmml" xref="A4.SS1.SSS1.p2.9.m8.1.1.3">𝜇</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.SS1.SSS1.p2.9.m8.1c">\|{\mathbf{x}}-{\mathbf{y}}\|\leq\mu</annotation><annotation encoding="application/x-llamapun" id="A4.SS1.SSS1.p2.9.m8.1d">∥ bold_x - bold_y ∥ ≤ italic_μ</annotation></semantics></math>,</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A8.EGx47"> <tbody id="A4.E27"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\|\nabla\log p({\mathbf{x}})-\nabla\log p({\mathbf{y}})\|\leq\epsilon." class="ltx_Math" display="inline" id="A4.E27.m1.3"><semantics id="A4.E27.m1.3a"><mrow id="A4.E27.m1.3.3.1" xref="A4.E27.m1.3.3.1.1.cmml"><mrow id="A4.E27.m1.3.3.1.1" xref="A4.E27.m1.3.3.1.1.cmml"><mrow id="A4.E27.m1.3.3.1.1.1.1" xref="A4.E27.m1.3.3.1.1.1.2.cmml"><mo id="A4.E27.m1.3.3.1.1.1.1.2" stretchy="false" xref="A4.E27.m1.3.3.1.1.1.2.1.cmml">‖</mo><mrow id="A4.E27.m1.3.3.1.1.1.1.1" xref="A4.E27.m1.3.3.1.1.1.1.1.cmml"><mrow id="A4.E27.m1.3.3.1.1.1.1.1.2" xref="A4.E27.m1.3.3.1.1.1.1.1.2.cmml"><mrow id="A4.E27.m1.3.3.1.1.1.1.1.2.2" xref="A4.E27.m1.3.3.1.1.1.1.1.2.2.cmml"><mrow id="A4.E27.m1.3.3.1.1.1.1.1.2.2.1" xref="A4.E27.m1.3.3.1.1.1.1.1.2.2.1.cmml"><mo id="A4.E27.m1.3.3.1.1.1.1.1.2.2.1.1" rspace="0.167em" xref="A4.E27.m1.3.3.1.1.1.1.1.2.2.1.1.cmml">∇</mo><mi id="A4.E27.m1.3.3.1.1.1.1.1.2.2.1.2" xref="A4.E27.m1.3.3.1.1.1.1.1.2.2.1.2.cmml">log</mi></mrow><mo id="A4.E27.m1.3.3.1.1.1.1.1.2.2a" lspace="0.167em" xref="A4.E27.m1.3.3.1.1.1.1.1.2.2.cmml">⁡</mo><mi id="A4.E27.m1.3.3.1.1.1.1.1.2.2.2" xref="A4.E27.m1.3.3.1.1.1.1.1.2.2.2.cmml">p</mi></mrow><mo id="A4.E27.m1.3.3.1.1.1.1.1.2.1" xref="A4.E27.m1.3.3.1.1.1.1.1.2.1.cmml">⁢</mo><mrow id="A4.E27.m1.3.3.1.1.1.1.1.2.3.2" xref="A4.E27.m1.3.3.1.1.1.1.1.2.cmml"><mo id="A4.E27.m1.3.3.1.1.1.1.1.2.3.2.1" stretchy="false" xref="A4.E27.m1.3.3.1.1.1.1.1.2.cmml">(</mo><mi id="A4.E27.m1.1.1" xref="A4.E27.m1.1.1.cmml">𝐱</mi><mo id="A4.E27.m1.3.3.1.1.1.1.1.2.3.2.2" stretchy="false" xref="A4.E27.m1.3.3.1.1.1.1.1.2.cmml">)</mo></mrow></mrow><mo id="A4.E27.m1.3.3.1.1.1.1.1.1" xref="A4.E27.m1.3.3.1.1.1.1.1.1.cmml">−</mo><mrow id="A4.E27.m1.3.3.1.1.1.1.1.3" xref="A4.E27.m1.3.3.1.1.1.1.1.3.cmml"><mrow id="A4.E27.m1.3.3.1.1.1.1.1.3.2" xref="A4.E27.m1.3.3.1.1.1.1.1.3.2.cmml"><mrow id="A4.E27.m1.3.3.1.1.1.1.1.3.2.1" xref="A4.E27.m1.3.3.1.1.1.1.1.3.2.1.cmml"><mo id="A4.E27.m1.3.3.1.1.1.1.1.3.2.1.1" rspace="0.167em" xref="A4.E27.m1.3.3.1.1.1.1.1.3.2.1.1.cmml">∇</mo><mi id="A4.E27.m1.3.3.1.1.1.1.1.3.2.1.2" xref="A4.E27.m1.3.3.1.1.1.1.1.3.2.1.2.cmml">log</mi></mrow><mo id="A4.E27.m1.3.3.1.1.1.1.1.3.2a" lspace="0.167em" xref="A4.E27.m1.3.3.1.1.1.1.1.3.2.cmml">⁡</mo><mi id="A4.E27.m1.3.3.1.1.1.1.1.3.2.2" xref="A4.E27.m1.3.3.1.1.1.1.1.3.2.2.cmml">p</mi></mrow><mo id="A4.E27.m1.3.3.1.1.1.1.1.3.1" xref="A4.E27.m1.3.3.1.1.1.1.1.3.1.cmml">⁢</mo><mrow id="A4.E27.m1.3.3.1.1.1.1.1.3.3.2" xref="A4.E27.m1.3.3.1.1.1.1.1.3.cmml"><mo id="A4.E27.m1.3.3.1.1.1.1.1.3.3.2.1" stretchy="false" xref="A4.E27.m1.3.3.1.1.1.1.1.3.cmml">(</mo><mi id="A4.E27.m1.2.2" xref="A4.E27.m1.2.2.cmml">𝐲</mi><mo id="A4.E27.m1.3.3.1.1.1.1.1.3.3.2.2" stretchy="false" xref="A4.E27.m1.3.3.1.1.1.1.1.3.cmml">)</mo></mrow></mrow></mrow><mo id="A4.E27.m1.3.3.1.1.1.1.3" stretchy="false" xref="A4.E27.m1.3.3.1.1.1.2.1.cmml">‖</mo></mrow><mo id="A4.E27.m1.3.3.1.1.2" xref="A4.E27.m1.3.3.1.1.2.cmml">≤</mo><mi id="A4.E27.m1.3.3.1.1.3" xref="A4.E27.m1.3.3.1.1.3.cmml">ϵ</mi></mrow><mo id="A4.E27.m1.3.3.1.2" lspace="0em" xref="A4.E27.m1.3.3.1.1.cmml">.</mo></mrow><annotation-xml encoding="MathML-Content" id="A4.E27.m1.3b"><apply id="A4.E27.m1.3.3.1.1.cmml" xref="A4.E27.m1.3.3.1"><leq id="A4.E27.m1.3.3.1.1.2.cmml" xref="A4.E27.m1.3.3.1.1.2"></leq><apply id="A4.E27.m1.3.3.1.1.1.2.cmml" xref="A4.E27.m1.3.3.1.1.1.1"><csymbol cd="latexml" id="A4.E27.m1.3.3.1.1.1.2.1.cmml" xref="A4.E27.m1.3.3.1.1.1.1.2">norm</csymbol><apply id="A4.E27.m1.3.3.1.1.1.1.1.cmml" xref="A4.E27.m1.3.3.1.1.1.1.1"><minus id="A4.E27.m1.3.3.1.1.1.1.1.1.cmml" xref="A4.E27.m1.3.3.1.1.1.1.1.1"></minus><apply id="A4.E27.m1.3.3.1.1.1.1.1.2.cmml" xref="A4.E27.m1.3.3.1.1.1.1.1.2"><times 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xref="A4.E27.m1.3.3.1.1.1.1.1.3.2.1.1">∇</ci><log id="A4.E27.m1.3.3.1.1.1.1.1.3.2.1.2.cmml" xref="A4.E27.m1.3.3.1.1.1.1.1.3.2.1.2"></log></apply><ci id="A4.E27.m1.3.3.1.1.1.1.1.3.2.2.cmml" xref="A4.E27.m1.3.3.1.1.1.1.1.3.2.2">𝑝</ci></apply><ci id="A4.E27.m1.2.2.cmml" xref="A4.E27.m1.2.2">𝐲</ci></apply></apply></apply><ci id="A4.E27.m1.3.3.1.1.3.cmml" xref="A4.E27.m1.3.3.1.1.3">italic-ϵ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.E27.m1.3c">\displaystyle\|\nabla\log p({\mathbf{x}})-\nabla\log p({\mathbf{y}})\|\leq\epsilon.</annotation><annotation encoding="application/x-llamapun" id="A4.E27.m1.3d">∥ ∇ roman_log italic_p ( bold_x ) - ∇ roman_log italic_p ( bold_y ) ∥ ≤ italic_ϵ .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(27)</span></td> </tr></tbody> </table> </div> <div class="ltx_para" id="A4.SS1.SSS1.p3"> <p class="ltx_p" id="A4.SS1.SSS1.p3.26">Then by using Equation (<a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#A4.E27" title="27 ‣ D.1.1 Proof of Proposition 3.1 ‣ D.1 Proof of critical Point Analysis ‣ Appendix D Proofs of section 3 ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_tag">27</span></a>), Assumptions <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#S3.Thmtheorem1" title="Assumption 3.1. ‣ 3.1 Asymptotics of critical points when 𝜎→0 ‣ 3 Convergence Analysis ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_tag">3.1</span></a><span class="ltx_text ltx_font_bold" id="A4.SS1.SSS1.p3.26.1">(b)</span> and Assumptions <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#S3.Thmtheorem2" title="Assumption 3.2. ‣ 3.1 Asymptotics of critical points when 𝜎→0 ‣ 3 Convergence Analysis ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_tag">3.2</span></a>, we have</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A8.EGx48"> <tbody id="A4.Ex77"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\|\nabla\mathcal{R}_{\sigma}({\mathbf{x}})+\nabla\log p({\mathbf{% x}})\|" class="ltx_Math" display="inline" id="A4.Ex77.m1.3"><semantics id="A4.Ex77.m1.3a"><mrow id="A4.Ex77.m1.3.3.1" xref="A4.Ex77.m1.3.3.2.cmml"><mo id="A4.Ex77.m1.3.3.1.2" stretchy="false" xref="A4.Ex77.m1.3.3.2.1.cmml">‖</mo><mrow id="A4.Ex77.m1.3.3.1.1" xref="A4.Ex77.m1.3.3.1.1.cmml"><mrow id="A4.Ex77.m1.3.3.1.1.2" xref="A4.Ex77.m1.3.3.1.1.2.cmml"><mrow id="A4.Ex77.m1.3.3.1.1.2.2" xref="A4.Ex77.m1.3.3.1.1.2.2.cmml"><mo id="A4.Ex77.m1.3.3.1.1.2.2.1" rspace="0.167em" xref="A4.Ex77.m1.3.3.1.1.2.2.1.cmml">∇</mo><msub id="A4.Ex77.m1.3.3.1.1.2.2.2" xref="A4.Ex77.m1.3.3.1.1.2.2.2.cmml"><mi class="ltx_font_mathcaligraphic" id="A4.Ex77.m1.3.3.1.1.2.2.2.2" xref="A4.Ex77.m1.3.3.1.1.2.2.2.2.cmml">ℛ</mi><mi id="A4.Ex77.m1.3.3.1.1.2.2.2.3" xref="A4.Ex77.m1.3.3.1.1.2.2.2.3.cmml">σ</mi></msub></mrow><mo id="A4.Ex77.m1.3.3.1.1.2.1" xref="A4.Ex77.m1.3.3.1.1.2.1.cmml">⁢</mo><mrow id="A4.Ex77.m1.3.3.1.1.2.3.2" xref="A4.Ex77.m1.3.3.1.1.2.cmml"><mo id="A4.Ex77.m1.3.3.1.1.2.3.2.1" stretchy="false" xref="A4.Ex77.m1.3.3.1.1.2.cmml">(</mo><mi id="A4.Ex77.m1.1.1" xref="A4.Ex77.m1.1.1.cmml">𝐱</mi><mo id="A4.Ex77.m1.3.3.1.1.2.3.2.2" stretchy="false" xref="A4.Ex77.m1.3.3.1.1.2.cmml">)</mo></mrow></mrow><mo id="A4.Ex77.m1.3.3.1.1.1" xref="A4.Ex77.m1.3.3.1.1.1.cmml">+</mo><mrow id="A4.Ex77.m1.3.3.1.1.3" xref="A4.Ex77.m1.3.3.1.1.3.cmml"><mrow id="A4.Ex77.m1.3.3.1.1.3.2" xref="A4.Ex77.m1.3.3.1.1.3.2.cmml"><mrow id="A4.Ex77.m1.3.3.1.1.3.2.1" xref="A4.Ex77.m1.3.3.1.1.3.2.1.cmml"><mo id="A4.Ex77.m1.3.3.1.1.3.2.1.1" rspace="0.167em" xref="A4.Ex77.m1.3.3.1.1.3.2.1.1.cmml">∇</mo><mi id="A4.Ex77.m1.3.3.1.1.3.2.1.2" xref="A4.Ex77.m1.3.3.1.1.3.2.1.2.cmml">log</mi></mrow><mo id="A4.Ex77.m1.3.3.1.1.3.2a" lspace="0.167em" xref="A4.Ex77.m1.3.3.1.1.3.2.cmml">⁡</mo><mi id="A4.Ex77.m1.3.3.1.1.3.2.2" xref="A4.Ex77.m1.3.3.1.1.3.2.2.cmml">p</mi></mrow><mo id="A4.Ex77.m1.3.3.1.1.3.1" xref="A4.Ex77.m1.3.3.1.1.3.1.cmml">⁢</mo><mrow id="A4.Ex77.m1.3.3.1.1.3.3.2" xref="A4.Ex77.m1.3.3.1.1.3.cmml"><mo id="A4.Ex77.m1.3.3.1.1.3.3.2.1" stretchy="false" xref="A4.Ex77.m1.3.3.1.1.3.cmml">(</mo><mi id="A4.Ex77.m1.2.2" xref="A4.Ex77.m1.2.2.cmml">𝐱</mi><mo id="A4.Ex77.m1.3.3.1.1.3.3.2.2" stretchy="false" xref="A4.Ex77.m1.3.3.1.1.3.cmml">)</mo></mrow></mrow></mrow><mo id="A4.Ex77.m1.3.3.1.3" stretchy="false" xref="A4.Ex77.m1.3.3.2.1.cmml">‖</mo></mrow><annotation-xml encoding="MathML-Content" id="A4.Ex77.m1.3b"><apply id="A4.Ex77.m1.3.3.2.cmml" 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id="A4.Ex77.m1.3.3.1.1.3.cmml" xref="A4.Ex77.m1.3.3.1.1.3"><times id="A4.Ex77.m1.3.3.1.1.3.1.cmml" xref="A4.Ex77.m1.3.3.1.1.3.1"></times><apply id="A4.Ex77.m1.3.3.1.1.3.2.cmml" xref="A4.Ex77.m1.3.3.1.1.3.2"><apply id="A4.Ex77.m1.3.3.1.1.3.2.1.cmml" xref="A4.Ex77.m1.3.3.1.1.3.2.1"><ci id="A4.Ex77.m1.3.3.1.1.3.2.1.1.cmml" xref="A4.Ex77.m1.3.3.1.1.3.2.1.1">∇</ci><log id="A4.Ex77.m1.3.3.1.1.3.2.1.2.cmml" xref="A4.Ex77.m1.3.3.1.1.3.2.1.2"></log></apply><ci id="A4.Ex77.m1.3.3.1.1.3.2.2.cmml" xref="A4.Ex77.m1.3.3.1.1.3.2.2">𝑝</ci></apply><ci id="A4.Ex77.m1.2.2.cmml" xref="A4.Ex77.m1.2.2">𝐱</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.Ex77.m1.3c">\displaystyle\|\nabla\mathcal{R}_{\sigma}({\mathbf{x}})+\nabla\log p({\mathbf{% x}})\|</annotation><annotation encoding="application/x-llamapun" id="A4.Ex77.m1.3d">∥ ∇ caligraphic_R start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ( bold_x ) + ∇ roman_log italic_p ( bold_x ) ∥</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="A4.Ex78"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\leq(\sup_{\mathbf{L}}\|\nabla\log p_{\sigma}-\nabla\log p\|)\int% _{\mathcal{B}(0,1)}{\mathcal{N}(\zeta;0,\sigma^{2}\mathbf{I}_{d})}d\zeta+\int_% {{\mathbb{R}}^{d}\setminus\mathcal{B}(0,1)}{\left(B\sigma^{\beta}(1+\|{\mathbf% {x}}+\zeta\|^{r})+A(1+\|{\mathbf{x}}+\zeta\|^{q})\right)\mathcal{N}(\zeta;0,% \sigma^{2}\mathbf{I}_{d})}d\zeta" class="ltx_Math" display="inline" id="A4.Ex78.m1.12"><semantics id="A4.Ex78.m1.12a"><mrow id="A4.Ex78.m1.12.12" xref="A4.Ex78.m1.12.12.cmml"><mi id="A4.Ex78.m1.12.12.6" xref="A4.Ex78.m1.12.12.6.cmml"></mi><mo id="A4.Ex78.m1.12.12.5" xref="A4.Ex78.m1.12.12.5.cmml">≤</mo><mrow id="A4.Ex78.m1.12.12.4" xref="A4.Ex78.m1.12.12.4.cmml"><mrow id="A4.Ex78.m1.10.10.2.2" xref="A4.Ex78.m1.10.10.2.2.cmml"><mrow id="A4.Ex78.m1.9.9.1.1.1.1" xref="A4.Ex78.m1.9.9.1.1.1.1.1.cmml"><mo id="A4.Ex78.m1.9.9.1.1.1.1.2" stretchy="false" xref="A4.Ex78.m1.9.9.1.1.1.1.1.cmml">(</mo><mrow id="A4.Ex78.m1.9.9.1.1.1.1.1" xref="A4.Ex78.m1.9.9.1.1.1.1.1.cmml"><munder id="A4.Ex78.m1.9.9.1.1.1.1.1.2" xref="A4.Ex78.m1.9.9.1.1.1.1.1.2.cmml"><mo id="A4.Ex78.m1.9.9.1.1.1.1.1.2.2" lspace="0em" movablelimits="false" rspace="0em" xref="A4.Ex78.m1.9.9.1.1.1.1.1.2.2.cmml">sup</mo><mi id="A4.Ex78.m1.9.9.1.1.1.1.1.2.3" xref="A4.Ex78.m1.9.9.1.1.1.1.1.2.3.cmml">𝐋</mi></munder><mrow id="A4.Ex78.m1.9.9.1.1.1.1.1.1.1" xref="A4.Ex78.m1.9.9.1.1.1.1.1.1.2.cmml"><mo id="A4.Ex78.m1.9.9.1.1.1.1.1.1.1.2" stretchy="false" xref="A4.Ex78.m1.9.9.1.1.1.1.1.1.2.1.cmml">‖</mo><mrow id="A4.Ex78.m1.9.9.1.1.1.1.1.1.1.1" xref="A4.Ex78.m1.9.9.1.1.1.1.1.1.1.1.cmml"><mrow id="A4.Ex78.m1.9.9.1.1.1.1.1.1.1.1.2" xref="A4.Ex78.m1.9.9.1.1.1.1.1.1.1.1.2.cmml"><mrow id="A4.Ex78.m1.9.9.1.1.1.1.1.1.1.1.2.1" xref="A4.Ex78.m1.9.9.1.1.1.1.1.1.1.1.2.1.cmml"><mo id="A4.Ex78.m1.9.9.1.1.1.1.1.1.1.1.2.1.1" rspace="0.167em" xref="A4.Ex78.m1.9.9.1.1.1.1.1.1.1.1.2.1.1.cmml">∇</mo><mi id="A4.Ex78.m1.9.9.1.1.1.1.1.1.1.1.2.1.2" xref="A4.Ex78.m1.9.9.1.1.1.1.1.1.1.1.2.1.2.cmml">log</mi></mrow><mo id="A4.Ex78.m1.9.9.1.1.1.1.1.1.1.1.2a" lspace="0.167em" xref="A4.Ex78.m1.9.9.1.1.1.1.1.1.1.1.2.cmml">⁡</mo><msub id="A4.Ex78.m1.9.9.1.1.1.1.1.1.1.1.2.2" xref="A4.Ex78.m1.9.9.1.1.1.1.1.1.1.1.2.2.cmml"><mi id="A4.Ex78.m1.9.9.1.1.1.1.1.1.1.1.2.2.2" xref="A4.Ex78.m1.9.9.1.1.1.1.1.1.1.1.2.2.2.cmml">p</mi><mi id="A4.Ex78.m1.9.9.1.1.1.1.1.1.1.1.2.2.3" xref="A4.Ex78.m1.9.9.1.1.1.1.1.1.1.1.2.2.3.cmml">σ</mi></msub></mrow><mo id="A4.Ex78.m1.9.9.1.1.1.1.1.1.1.1.1" xref="A4.Ex78.m1.9.9.1.1.1.1.1.1.1.1.1.cmml">−</mo><mrow id="A4.Ex78.m1.9.9.1.1.1.1.1.1.1.1.3" xref="A4.Ex78.m1.9.9.1.1.1.1.1.1.1.1.3.cmml"><mrow id="A4.Ex78.m1.9.9.1.1.1.1.1.1.1.1.3.1" xref="A4.Ex78.m1.9.9.1.1.1.1.1.1.1.1.3.1.cmml"><mo id="A4.Ex78.m1.9.9.1.1.1.1.1.1.1.1.3.1.1" rspace="0.167em" xref="A4.Ex78.m1.9.9.1.1.1.1.1.1.1.1.3.1.1.cmml">∇</mo><mi id="A4.Ex78.m1.9.9.1.1.1.1.1.1.1.1.3.1.2" xref="A4.Ex78.m1.9.9.1.1.1.1.1.1.1.1.3.1.2.cmml">log</mi></mrow><mo id="A4.Ex78.m1.9.9.1.1.1.1.1.1.1.1.3a" lspace="0.167em" xref="A4.Ex78.m1.9.9.1.1.1.1.1.1.1.1.3.cmml">⁡</mo><mi id="A4.Ex78.m1.9.9.1.1.1.1.1.1.1.1.3.2" xref="A4.Ex78.m1.9.9.1.1.1.1.1.1.1.1.3.2.cmml">p</mi></mrow></mrow><mo id="A4.Ex78.m1.9.9.1.1.1.1.1.1.1.3" stretchy="false" xref="A4.Ex78.m1.9.9.1.1.1.1.1.1.2.1.cmml">‖</mo></mrow></mrow><mo id="A4.Ex78.m1.9.9.1.1.1.1.3" stretchy="false" xref="A4.Ex78.m1.9.9.1.1.1.1.1.cmml">)</mo></mrow><mo id="A4.Ex78.m1.10.10.2.2.3" xref="A4.Ex78.m1.10.10.2.2.3.cmml">⁢</mo><mrow id="A4.Ex78.m1.10.10.2.2.2" xref="A4.Ex78.m1.10.10.2.2.2.cmml"><mstyle displaystyle="true" id="A4.Ex78.m1.10.10.2.2.2.2" xref="A4.Ex78.m1.10.10.2.2.2.2.cmml"><msub id="A4.Ex78.m1.10.10.2.2.2.2a" xref="A4.Ex78.m1.10.10.2.2.2.2.cmml"><mo id="A4.Ex78.m1.10.10.2.2.2.2.2" xref="A4.Ex78.m1.10.10.2.2.2.2.2.cmml">∫</mo><mrow id="A4.Ex78.m1.2.2.2" xref="A4.Ex78.m1.2.2.2.cmml"><mi class="ltx_font_mathcaligraphic" id="A4.Ex78.m1.2.2.2.4" xref="A4.Ex78.m1.2.2.2.4.cmml">ℬ</mi><mo id="A4.Ex78.m1.2.2.2.3" xref="A4.Ex78.m1.2.2.2.3.cmml">⁢</mo><mrow id="A4.Ex78.m1.2.2.2.5.2" xref="A4.Ex78.m1.2.2.2.5.1.cmml"><mo id="A4.Ex78.m1.2.2.2.5.2.1" stretchy="false" xref="A4.Ex78.m1.2.2.2.5.1.cmml">(</mo><mn id="A4.Ex78.m1.1.1.1.1" xref="A4.Ex78.m1.1.1.1.1.cmml">0</mn><mo id="A4.Ex78.m1.2.2.2.5.2.2" xref="A4.Ex78.m1.2.2.2.5.1.cmml">,</mo><mn id="A4.Ex78.m1.2.2.2.2" xref="A4.Ex78.m1.2.2.2.2.cmml">1</mn><mo id="A4.Ex78.m1.2.2.2.5.2.3" stretchy="false" xref="A4.Ex78.m1.2.2.2.5.1.cmml">)</mo></mrow></mrow></msub></mstyle><mrow id="A4.Ex78.m1.10.10.2.2.2.1" xref="A4.Ex78.m1.10.10.2.2.2.1.cmml"><mi class="ltx_font_mathcaligraphic" id="A4.Ex78.m1.10.10.2.2.2.1.3" xref="A4.Ex78.m1.10.10.2.2.2.1.3.cmml">𝒩</mi><mo id="A4.Ex78.m1.10.10.2.2.2.1.2" xref="A4.Ex78.m1.10.10.2.2.2.1.2.cmml">⁢</mo><mrow id="A4.Ex78.m1.10.10.2.2.2.1.1.1" xref="A4.Ex78.m1.10.10.2.2.2.1.1.2.cmml"><mo id="A4.Ex78.m1.10.10.2.2.2.1.1.1.2" stretchy="false" xref="A4.Ex78.m1.10.10.2.2.2.1.1.2.cmml">(</mo><mi id="A4.Ex78.m1.5.5" xref="A4.Ex78.m1.5.5.cmml">ζ</mi><mo id="A4.Ex78.m1.10.10.2.2.2.1.1.1.3" xref="A4.Ex78.m1.10.10.2.2.2.1.1.2.cmml">;</mo><mn id="A4.Ex78.m1.6.6" xref="A4.Ex78.m1.6.6.cmml">0</mn><mo id="A4.Ex78.m1.10.10.2.2.2.1.1.1.4" xref="A4.Ex78.m1.10.10.2.2.2.1.1.2.cmml">,</mo><mrow id="A4.Ex78.m1.10.10.2.2.2.1.1.1.1" xref="A4.Ex78.m1.10.10.2.2.2.1.1.1.1.cmml"><msup id="A4.Ex78.m1.10.10.2.2.2.1.1.1.1.2" xref="A4.Ex78.m1.10.10.2.2.2.1.1.1.1.2.cmml"><mi id="A4.Ex78.m1.10.10.2.2.2.1.1.1.1.2.2" xref="A4.Ex78.m1.10.10.2.2.2.1.1.1.1.2.2.cmml">σ</mi><mn id="A4.Ex78.m1.10.10.2.2.2.1.1.1.1.2.3" xref="A4.Ex78.m1.10.10.2.2.2.1.1.1.1.2.3.cmml">2</mn></msup><mo id="A4.Ex78.m1.10.10.2.2.2.1.1.1.1.1" xref="A4.Ex78.m1.10.10.2.2.2.1.1.1.1.1.cmml">⁢</mo><msub id="A4.Ex78.m1.10.10.2.2.2.1.1.1.1.3" xref="A4.Ex78.m1.10.10.2.2.2.1.1.1.1.3.cmml"><mi id="A4.Ex78.m1.10.10.2.2.2.1.1.1.1.3.2" xref="A4.Ex78.m1.10.10.2.2.2.1.1.1.1.3.2.cmml">𝐈</mi><mi id="A4.Ex78.m1.10.10.2.2.2.1.1.1.1.3.3" xref="A4.Ex78.m1.10.10.2.2.2.1.1.1.1.3.3.cmml">d</mi></msub></mrow><mo id="A4.Ex78.m1.10.10.2.2.2.1.1.1.5" stretchy="false" xref="A4.Ex78.m1.10.10.2.2.2.1.1.2.cmml">)</mo></mrow><mo id="A4.Ex78.m1.10.10.2.2.2.1.2a" lspace="0em" xref="A4.Ex78.m1.10.10.2.2.2.1.2.cmml">⁢</mo><mrow id="A4.Ex78.m1.10.10.2.2.2.1.4" xref="A4.Ex78.m1.10.10.2.2.2.1.4.cmml"><mo id="A4.Ex78.m1.10.10.2.2.2.1.4.1" rspace="0em" xref="A4.Ex78.m1.10.10.2.2.2.1.4.1.cmml">𝑑</mo><mi id="A4.Ex78.m1.10.10.2.2.2.1.4.2" xref="A4.Ex78.m1.10.10.2.2.2.1.4.2.cmml">ζ</mi></mrow></mrow></mrow></mrow><mo id="A4.Ex78.m1.12.12.4.5" xref="A4.Ex78.m1.12.12.4.5.cmml">+</mo><mrow id="A4.Ex78.m1.12.12.4.4" xref="A4.Ex78.m1.12.12.4.4.cmml"><mstyle displaystyle="true" id="A4.Ex78.m1.12.12.4.4.3" xref="A4.Ex78.m1.12.12.4.4.3.cmml"><msub id="A4.Ex78.m1.12.12.4.4.3a" xref="A4.Ex78.m1.12.12.4.4.3.cmml"><mo id="A4.Ex78.m1.12.12.4.4.3.2" xref="A4.Ex78.m1.12.12.4.4.3.2.cmml">∫</mo><mrow id="A4.Ex78.m1.4.4.2" xref="A4.Ex78.m1.4.4.2.cmml"><msup id="A4.Ex78.m1.4.4.2.4" xref="A4.Ex78.m1.4.4.2.4.cmml"><mi id="A4.Ex78.m1.4.4.2.4.2" xref="A4.Ex78.m1.4.4.2.4.2.cmml">ℝ</mi><mi id="A4.Ex78.m1.4.4.2.4.3" xref="A4.Ex78.m1.4.4.2.4.3.cmml">d</mi></msup><mo id="A4.Ex78.m1.4.4.2.3" xref="A4.Ex78.m1.4.4.2.3.cmml">∖</mo><mrow id="A4.Ex78.m1.4.4.2.5" xref="A4.Ex78.m1.4.4.2.5.cmml"><mi class="ltx_font_mathcaligraphic" id="A4.Ex78.m1.4.4.2.5.2" xref="A4.Ex78.m1.4.4.2.5.2.cmml">ℬ</mi><mo id="A4.Ex78.m1.4.4.2.5.1" xref="A4.Ex78.m1.4.4.2.5.1.cmml">⁢</mo><mrow id="A4.Ex78.m1.4.4.2.5.3.2" xref="A4.Ex78.m1.4.4.2.5.3.1.cmml"><mo id="A4.Ex78.m1.4.4.2.5.3.2.1" stretchy="false" xref="A4.Ex78.m1.4.4.2.5.3.1.cmml">(</mo><mn id="A4.Ex78.m1.3.3.1.1" xref="A4.Ex78.m1.3.3.1.1.cmml">0</mn><mo id="A4.Ex78.m1.4.4.2.5.3.2.2" xref="A4.Ex78.m1.4.4.2.5.3.1.cmml">,</mo><mn id="A4.Ex78.m1.4.4.2.2" xref="A4.Ex78.m1.4.4.2.2.cmml">1</mn><mo id="A4.Ex78.m1.4.4.2.5.3.2.3" stretchy="false" xref="A4.Ex78.m1.4.4.2.5.3.1.cmml">)</mo></mrow></mrow></mrow></msub></mstyle><mrow id="A4.Ex78.m1.12.12.4.4.2" xref="A4.Ex78.m1.12.12.4.4.2.cmml"><mrow id="A4.Ex78.m1.11.11.3.3.1.1.1" xref="A4.Ex78.m1.11.11.3.3.1.1.1.1.cmml"><mo id="A4.Ex78.m1.11.11.3.3.1.1.1.2" xref="A4.Ex78.m1.11.11.3.3.1.1.1.1.cmml">(</mo><mrow id="A4.Ex78.m1.11.11.3.3.1.1.1.1" xref="A4.Ex78.m1.11.11.3.3.1.1.1.1.cmml"><mrow id="A4.Ex78.m1.11.11.3.3.1.1.1.1.1" xref="A4.Ex78.m1.11.11.3.3.1.1.1.1.1.cmml"><mi id="A4.Ex78.m1.11.11.3.3.1.1.1.1.1.3" xref="A4.Ex78.m1.11.11.3.3.1.1.1.1.1.3.cmml">B</mi><mo id="A4.Ex78.m1.11.11.3.3.1.1.1.1.1.2" xref="A4.Ex78.m1.11.11.3.3.1.1.1.1.1.2.cmml">⁢</mo><msup id="A4.Ex78.m1.11.11.3.3.1.1.1.1.1.4" xref="A4.Ex78.m1.11.11.3.3.1.1.1.1.1.4.cmml"><mi 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xref="A4.Ex78.m1.12.12.4.4.2.5.2">𝜁</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.Ex78.m1.12c">\displaystyle\leq(\sup_{\mathbf{L}}\|\nabla\log p_{\sigma}-\nabla\log p\|)\int% _{\mathcal{B}(0,1)}{\mathcal{N}(\zeta;0,\sigma^{2}\mathbf{I}_{d})}d\zeta+\int_% {{\mathbb{R}}^{d}\setminus\mathcal{B}(0,1)}{\left(B\sigma^{\beta}(1+\|{\mathbf% {x}}+\zeta\|^{r})+A(1+\|{\mathbf{x}}+\zeta\|^{q})\right)\mathcal{N}(\zeta;0,% \sigma^{2}\mathbf{I}_{d})}d\zeta</annotation><annotation encoding="application/x-llamapun" id="A4.Ex78.m1.12d">≤ ( roman_sup start_POSTSUBSCRIPT bold_L end_POSTSUBSCRIPT ∥ ∇ roman_log italic_p start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT - ∇ roman_log italic_p ∥ ) ∫ start_POSTSUBSCRIPT caligraphic_B ( 0 , 1 ) end_POSTSUBSCRIPT caligraphic_N ( italic_ζ ; 0 , italic_σ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT bold_I start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT ) italic_d italic_ζ + ∫ start_POSTSUBSCRIPT blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT ∖ caligraphic_B ( 0 , 1 ) end_POSTSUBSCRIPT ( italic_B italic_σ start_POSTSUPERSCRIPT italic_β end_POSTSUPERSCRIPT ( 1 + ∥ bold_x + italic_ζ ∥ start_POSTSUPERSCRIPT italic_r end_POSTSUPERSCRIPT ) + italic_A ( 1 + ∥ bold_x + italic_ζ ∥ start_POSTSUPERSCRIPT italic_q end_POSTSUPERSCRIPT ) ) caligraphic_N ( italic_ζ ; 0 , italic_σ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT bold_I start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT ) italic_d italic_ζ</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="A4.Ex79"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle+\epsilon\int_{\mathcal{B}(0,\mu)}{\mathcal{N}(\zeta;0,\sigma^{2}% \mathbf{I}_{d})}d\zeta+\int_{{\mathbb{R}}^{d}\setminus\mathcal{B}(0,\mu)}{(A(1% +\|{\mathbf{x}}+\zeta\|^{q})+A(1+\|{\mathbf{x}}\|^{q}))\mathcal{N}(\zeta;0,% \sigma^{2}\mathbf{I}_{d})}d\zeta" class="ltx_Math" display="inline" id="A4.Ex79.m1.12"><semantics id="A4.Ex79.m1.12a"><mrow id="A4.Ex79.m1.12.12" xref="A4.Ex79.m1.12.12.cmml"><mrow id="A4.Ex79.m1.10.10.1" xref="A4.Ex79.m1.10.10.1.cmml"><mo id="A4.Ex79.m1.10.10.1a" xref="A4.Ex79.m1.10.10.1.cmml">+</mo><mrow id="A4.Ex79.m1.10.10.1.1" xref="A4.Ex79.m1.10.10.1.1.cmml"><mi id="A4.Ex79.m1.10.10.1.1.3" xref="A4.Ex79.m1.10.10.1.1.3.cmml">ϵ</mi><mo id="A4.Ex79.m1.10.10.1.1.2" xref="A4.Ex79.m1.10.10.1.1.2.cmml">⁢</mo><mrow id="A4.Ex79.m1.10.10.1.1.1" xref="A4.Ex79.m1.10.10.1.1.1.cmml"><mstyle displaystyle="true" id="A4.Ex79.m1.10.10.1.1.1.2" xref="A4.Ex79.m1.10.10.1.1.1.2.cmml"><msub id="A4.Ex79.m1.10.10.1.1.1.2a" xref="A4.Ex79.m1.10.10.1.1.1.2.cmml"><mo id="A4.Ex79.m1.10.10.1.1.1.2.2" xref="A4.Ex79.m1.10.10.1.1.1.2.2.cmml">∫</mo><mrow id="A4.Ex79.m1.2.2.2" xref="A4.Ex79.m1.2.2.2.cmml"><mi class="ltx_font_mathcaligraphic" id="A4.Ex79.m1.2.2.2.4" xref="A4.Ex79.m1.2.2.2.4.cmml">ℬ</mi><mo id="A4.Ex79.m1.2.2.2.3" xref="A4.Ex79.m1.2.2.2.3.cmml">⁢</mo><mrow id="A4.Ex79.m1.2.2.2.5.2" xref="A4.Ex79.m1.2.2.2.5.1.cmml"><mo id="A4.Ex79.m1.2.2.2.5.2.1" stretchy="false" xref="A4.Ex79.m1.2.2.2.5.1.cmml">(</mo><mn id="A4.Ex79.m1.1.1.1.1" xref="A4.Ex79.m1.1.1.1.1.cmml">0</mn><mo id="A4.Ex79.m1.2.2.2.5.2.2" xref="A4.Ex79.m1.2.2.2.5.1.cmml">,</mo><mi id="A4.Ex79.m1.2.2.2.2" xref="A4.Ex79.m1.2.2.2.2.cmml">μ</mi><mo id="A4.Ex79.m1.2.2.2.5.2.3" stretchy="false" xref="A4.Ex79.m1.2.2.2.5.1.cmml">)</mo></mrow></mrow></msub></mstyle><mrow id="A4.Ex79.m1.10.10.1.1.1.1" xref="A4.Ex79.m1.10.10.1.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="A4.Ex79.m1.10.10.1.1.1.1.3" xref="A4.Ex79.m1.10.10.1.1.1.1.3.cmml">𝒩</mi><mo id="A4.Ex79.m1.10.10.1.1.1.1.2" xref="A4.Ex79.m1.10.10.1.1.1.1.2.cmml">⁢</mo><mrow id="A4.Ex79.m1.10.10.1.1.1.1.1.1" xref="A4.Ex79.m1.10.10.1.1.1.1.1.2.cmml"><mo id="A4.Ex79.m1.10.10.1.1.1.1.1.1.2" stretchy="false" xref="A4.Ex79.m1.10.10.1.1.1.1.1.2.cmml">(</mo><mi id="A4.Ex79.m1.5.5" xref="A4.Ex79.m1.5.5.cmml">ζ</mi><mo id="A4.Ex79.m1.10.10.1.1.1.1.1.1.3" xref="A4.Ex79.m1.10.10.1.1.1.1.1.2.cmml">;</mo><mn id="A4.Ex79.m1.6.6" xref="A4.Ex79.m1.6.6.cmml">0</mn><mo id="A4.Ex79.m1.10.10.1.1.1.1.1.1.4" xref="A4.Ex79.m1.10.10.1.1.1.1.1.2.cmml">,</mo><mrow id="A4.Ex79.m1.10.10.1.1.1.1.1.1.1" xref="A4.Ex79.m1.10.10.1.1.1.1.1.1.1.cmml"><msup id="A4.Ex79.m1.10.10.1.1.1.1.1.1.1.2" xref="A4.Ex79.m1.10.10.1.1.1.1.1.1.1.2.cmml"><mi id="A4.Ex79.m1.10.10.1.1.1.1.1.1.1.2.2" xref="A4.Ex79.m1.10.10.1.1.1.1.1.1.1.2.2.cmml">σ</mi><mn id="A4.Ex79.m1.10.10.1.1.1.1.1.1.1.2.3" xref="A4.Ex79.m1.10.10.1.1.1.1.1.1.1.2.3.cmml">2</mn></msup><mo id="A4.Ex79.m1.10.10.1.1.1.1.1.1.1.1" xref="A4.Ex79.m1.10.10.1.1.1.1.1.1.1.1.cmml">⁢</mo><msub id="A4.Ex79.m1.10.10.1.1.1.1.1.1.1.3" xref="A4.Ex79.m1.10.10.1.1.1.1.1.1.1.3.cmml"><mi id="A4.Ex79.m1.10.10.1.1.1.1.1.1.1.3.2" xref="A4.Ex79.m1.10.10.1.1.1.1.1.1.1.3.2.cmml">𝐈</mi><mi id="A4.Ex79.m1.10.10.1.1.1.1.1.1.1.3.3" xref="A4.Ex79.m1.10.10.1.1.1.1.1.1.1.3.3.cmml">d</mi></msub></mrow><mo id="A4.Ex79.m1.10.10.1.1.1.1.1.1.5" stretchy="false" xref="A4.Ex79.m1.10.10.1.1.1.1.1.2.cmml">)</mo></mrow><mo id="A4.Ex79.m1.10.10.1.1.1.1.2a" lspace="0em" xref="A4.Ex79.m1.10.10.1.1.1.1.2.cmml">⁢</mo><mrow id="A4.Ex79.m1.10.10.1.1.1.1.4" xref="A4.Ex79.m1.10.10.1.1.1.1.4.cmml"><mo id="A4.Ex79.m1.10.10.1.1.1.1.4.1" rspace="0em" xref="A4.Ex79.m1.10.10.1.1.1.1.4.1.cmml">𝑑</mo><mi id="A4.Ex79.m1.10.10.1.1.1.1.4.2" xref="A4.Ex79.m1.10.10.1.1.1.1.4.2.cmml">ζ</mi></mrow></mrow></mrow></mrow></mrow><mo id="A4.Ex79.m1.12.12.4" xref="A4.Ex79.m1.12.12.4.cmml">+</mo><mrow id="A4.Ex79.m1.12.12.3" xref="A4.Ex79.m1.12.12.3.cmml"><mstyle displaystyle="true" id="A4.Ex79.m1.12.12.3.3" xref="A4.Ex79.m1.12.12.3.3.cmml"><msub id="A4.Ex79.m1.12.12.3.3a" xref="A4.Ex79.m1.12.12.3.3.cmml"><mo id="A4.Ex79.m1.12.12.3.3.2" xref="A4.Ex79.m1.12.12.3.3.2.cmml">∫</mo><mrow id="A4.Ex79.m1.4.4.2" xref="A4.Ex79.m1.4.4.2.cmml"><msup id="A4.Ex79.m1.4.4.2.4" xref="A4.Ex79.m1.4.4.2.4.cmml"><mi id="A4.Ex79.m1.4.4.2.4.2" xref="A4.Ex79.m1.4.4.2.4.2.cmml">ℝ</mi><mi id="A4.Ex79.m1.4.4.2.4.3" xref="A4.Ex79.m1.4.4.2.4.3.cmml">d</mi></msup><mo id="A4.Ex79.m1.4.4.2.3" xref="A4.Ex79.m1.4.4.2.3.cmml">∖</mo><mrow id="A4.Ex79.m1.4.4.2.5" xref="A4.Ex79.m1.4.4.2.5.cmml"><mi class="ltx_font_mathcaligraphic" id="A4.Ex79.m1.4.4.2.5.2" xref="A4.Ex79.m1.4.4.2.5.2.cmml">ℬ</mi><mo id="A4.Ex79.m1.4.4.2.5.1" xref="A4.Ex79.m1.4.4.2.5.1.cmml">⁢</mo><mrow id="A4.Ex79.m1.4.4.2.5.3.2" xref="A4.Ex79.m1.4.4.2.5.3.1.cmml"><mo id="A4.Ex79.m1.4.4.2.5.3.2.1" stretchy="false" xref="A4.Ex79.m1.4.4.2.5.3.1.cmml">(</mo><mn id="A4.Ex79.m1.3.3.1.1" xref="A4.Ex79.m1.3.3.1.1.cmml">0</mn><mo id="A4.Ex79.m1.4.4.2.5.3.2.2" xref="A4.Ex79.m1.4.4.2.5.3.1.cmml">,</mo><mi id="A4.Ex79.m1.4.4.2.2" xref="A4.Ex79.m1.4.4.2.2.cmml">μ</mi><mo id="A4.Ex79.m1.4.4.2.5.3.2.3" 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id="A4.Ex79.m1.8.8.cmml" xref="A4.Ex79.m1.8.8">𝜁</ci><cn id="A4.Ex79.m1.9.9.cmml" type="integer" xref="A4.Ex79.m1.9.9">0</cn><apply id="A4.Ex79.m1.12.12.3.2.2.1.1.cmml" xref="A4.Ex79.m1.12.12.3.2.2.1.1"><times id="A4.Ex79.m1.12.12.3.2.2.1.1.1.cmml" xref="A4.Ex79.m1.12.12.3.2.2.1.1.1"></times><apply id="A4.Ex79.m1.12.12.3.2.2.1.1.2.cmml" xref="A4.Ex79.m1.12.12.3.2.2.1.1.2"><csymbol cd="ambiguous" id="A4.Ex79.m1.12.12.3.2.2.1.1.2.1.cmml" xref="A4.Ex79.m1.12.12.3.2.2.1.1.2">superscript</csymbol><ci id="A4.Ex79.m1.12.12.3.2.2.1.1.2.2.cmml" xref="A4.Ex79.m1.12.12.3.2.2.1.1.2.2">𝜎</ci><cn id="A4.Ex79.m1.12.12.3.2.2.1.1.2.3.cmml" type="integer" xref="A4.Ex79.m1.12.12.3.2.2.1.1.2.3">2</cn></apply><apply id="A4.Ex79.m1.12.12.3.2.2.1.1.3.cmml" xref="A4.Ex79.m1.12.12.3.2.2.1.1.3"><csymbol cd="ambiguous" id="A4.Ex79.m1.12.12.3.2.2.1.1.3.1.cmml" xref="A4.Ex79.m1.12.12.3.2.2.1.1.3">subscript</csymbol><ci id="A4.Ex79.m1.12.12.3.2.2.1.1.3.2.cmml" xref="A4.Ex79.m1.12.12.3.2.2.1.1.3.2">𝐈</ci><ci id="A4.Ex79.m1.12.12.3.2.2.1.1.3.3.cmml" xref="A4.Ex79.m1.12.12.3.2.2.1.1.3.3">𝑑</ci></apply></apply></list><apply id="A4.Ex79.m1.12.12.3.2.5.cmml" xref="A4.Ex79.m1.12.12.3.2.5"><csymbol cd="latexml" id="A4.Ex79.m1.12.12.3.2.5.1.cmml" xref="A4.Ex79.m1.12.12.3.2.5.1">differential-d</csymbol><ci id="A4.Ex79.m1.12.12.3.2.5.2.cmml" xref="A4.Ex79.m1.12.12.3.2.5.2">𝜁</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.Ex79.m1.12c">\displaystyle+\epsilon\int_{\mathcal{B}(0,\mu)}{\mathcal{N}(\zeta;0,\sigma^{2}% \mathbf{I}_{d})}d\zeta+\int_{{\mathbb{R}}^{d}\setminus\mathcal{B}(0,\mu)}{(A(1% +\|{\mathbf{x}}+\zeta\|^{q})+A(1+\|{\mathbf{x}}\|^{q}))\mathcal{N}(\zeta;0,% \sigma^{2}\mathbf{I}_{d})}d\zeta</annotation><annotation encoding="application/x-llamapun" id="A4.Ex79.m1.12d">+ italic_ϵ ∫ start_POSTSUBSCRIPT caligraphic_B ( 0 , italic_μ ) end_POSTSUBSCRIPT caligraphic_N ( italic_ζ ; 0 , italic_σ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT bold_I start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT ) italic_d italic_ζ + ∫ start_POSTSUBSCRIPT blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT ∖ caligraphic_B ( 0 , italic_μ ) end_POSTSUBSCRIPT ( italic_A ( 1 + ∥ bold_x + italic_ζ ∥ start_POSTSUPERSCRIPT italic_q end_POSTSUPERSCRIPT ) + italic_A ( 1 + ∥ bold_x ∥ start_POSTSUPERSCRIPT italic_q end_POSTSUPERSCRIPT ) ) caligraphic_N ( italic_ζ ; 0 , italic_σ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT bold_I start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT ) italic_d italic_ζ</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="A4.Ex80"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\leq\sup_{\mathbf{L}}(\|\nabla\log p_{\sigma}-\nabla\log p\|)+% \epsilon+\int_{{\mathbb{R}}^{d}\setminus\mathcal{B}(0,\mu)}{\left(B\sigma^{% 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id="A4.Ex80.m1.6.6.1.1.2.2.1.1.1.1.2.3" xref="A4.Ex80.m1.6.6.1.1.2.2.1.1.1.1.2.3.cmml">2</mn><mo id="A4.Ex80.m1.6.6.1.1.2.2.1.1.1.1.2.2" xref="A4.Ex80.m1.6.6.1.1.2.2.1.1.1.1.2.2.cmml">⁢</mo><mi id="A4.Ex80.m1.6.6.1.1.2.2.1.1.1.1.2.4" xref="A4.Ex80.m1.6.6.1.1.2.2.1.1.1.1.2.4.cmml">A</mi><mo id="A4.Ex80.m1.6.6.1.1.2.2.1.1.1.1.2.2a" xref="A4.Ex80.m1.6.6.1.1.2.2.1.1.1.1.2.2.cmml">⁢</mo><mrow id="A4.Ex80.m1.6.6.1.1.2.2.1.1.1.1.2.1.1" xref="A4.Ex80.m1.6.6.1.1.2.2.1.1.1.1.2.1.1.1.cmml"><mo id="A4.Ex80.m1.6.6.1.1.2.2.1.1.1.1.2.1.1.2" stretchy="false" xref="A4.Ex80.m1.6.6.1.1.2.2.1.1.1.1.2.1.1.1.cmml">(</mo><mrow id="A4.Ex80.m1.6.6.1.1.2.2.1.1.1.1.2.1.1.1" xref="A4.Ex80.m1.6.6.1.1.2.2.1.1.1.1.2.1.1.1.cmml"><mn id="A4.Ex80.m1.6.6.1.1.2.2.1.1.1.1.2.1.1.1.3" xref="A4.Ex80.m1.6.6.1.1.2.2.1.1.1.1.2.1.1.1.3.cmml">1</mn><mo id="A4.Ex80.m1.6.6.1.1.2.2.1.1.1.1.2.1.1.1.2" xref="A4.Ex80.m1.6.6.1.1.2.2.1.1.1.1.2.1.1.1.2.cmml">+</mo><msup id="A4.Ex80.m1.6.6.1.1.2.2.1.1.1.1.2.1.1.1.1" 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encoding="application/x-tex" id="A4.Ex80.m1.6c">\displaystyle\leq\sup_{\mathbf{L}}(\|\nabla\log p_{\sigma}-\nabla\log p\|)+% \epsilon+\int_{{\mathbb{R}}^{d}\setminus\mathcal{B}(0,\mu)}{\left(B\sigma^{% \beta}(1+\|{\mathbf{x}}+\zeta\|^{r})+2A(1+\|{\mathbf{x}}+\zeta\|^{q})+A(1+\|{% \mathbf{x}}\|^{q})\right)\mathcal{N}(\zeta;0,\sigma^{2}\mathbf{I}_{d})}d\zeta,</annotation><annotation encoding="application/x-llamapun" id="A4.Ex80.m1.6d">≤ roman_sup start_POSTSUBSCRIPT bold_L end_POSTSUBSCRIPT ( ∥ ∇ roman_log italic_p start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT - ∇ roman_log italic_p ∥ ) + italic_ϵ + ∫ start_POSTSUBSCRIPT blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT ∖ caligraphic_B ( 0 , italic_μ ) end_POSTSUBSCRIPT ( italic_B italic_σ start_POSTSUPERSCRIPT italic_β end_POSTSUPERSCRIPT ( 1 + ∥ bold_x + italic_ζ ∥ start_POSTSUPERSCRIPT italic_r end_POSTSUPERSCRIPT ) + 2 italic_A ( 1 + ∥ bold_x + italic_ζ ∥ start_POSTSUPERSCRIPT italic_q end_POSTSUPERSCRIPT ) + italic_A ( 1 + ∥ bold_x ∥ start_POSTSUPERSCRIPT italic_q end_POSTSUPERSCRIPT ) ) caligraphic_N ( italic_ζ ; 0 , italic_σ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT bold_I start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT ) italic_d italic_ζ ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="A4.SS1.SSS1.p3.7">where in the last inequality we have used that <math alttext="\mu\leq 1" class="ltx_Math" display="inline" id="A4.SS1.SSS1.p3.1.m1.1"><semantics id="A4.SS1.SSS1.p3.1.m1.1a"><mrow id="A4.SS1.SSS1.p3.1.m1.1.1" xref="A4.SS1.SSS1.p3.1.m1.1.1.cmml"><mi id="A4.SS1.SSS1.p3.1.m1.1.1.2" xref="A4.SS1.SSS1.p3.1.m1.1.1.2.cmml">μ</mi><mo id="A4.SS1.SSS1.p3.1.m1.1.1.1" xref="A4.SS1.SSS1.p3.1.m1.1.1.1.cmml">≤</mo><mn id="A4.SS1.SSS1.p3.1.m1.1.1.3" xref="A4.SS1.SSS1.p3.1.m1.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="A4.SS1.SSS1.p3.1.m1.1b"><apply id="A4.SS1.SSS1.p3.1.m1.1.1.cmml" xref="A4.SS1.SSS1.p3.1.m1.1.1"><leq id="A4.SS1.SSS1.p3.1.m1.1.1.1.cmml" xref="A4.SS1.SSS1.p3.1.m1.1.1.1"></leq><ci id="A4.SS1.SSS1.p3.1.m1.1.1.2.cmml" xref="A4.SS1.SSS1.p3.1.m1.1.1.2">𝜇</ci><cn id="A4.SS1.SSS1.p3.1.m1.1.1.3.cmml" type="integer" xref="A4.SS1.SSS1.p3.1.m1.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.SS1.SSS1.p3.1.m1.1c">\mu\leq 1</annotation><annotation encoding="application/x-llamapun" id="A4.SS1.SSS1.p3.1.m1.1d">italic_μ ≤ 1</annotation></semantics></math>. 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id="A4.SS1.SSS1.p3.2.m2.4.4.5" xref="A4.SS1.SSS1.p3.2.m2.4.4.5.cmml">≤</mo><mrow id="A4.SS1.SSS1.p3.2.m2.4.4.6" xref="A4.SS1.SSS1.p3.2.m2.4.4.6.cmml"><mi id="A4.SS1.SSS1.p3.2.m2.4.4.6.2" xref="A4.SS1.SSS1.p3.2.m2.4.4.6.2.cmml">R</mi><mo id="A4.SS1.SSS1.p3.2.m2.4.4.6.1" xref="A4.SS1.SSS1.p3.2.m2.4.4.6.1.cmml">+</mo><mrow id="A4.SS1.SSS1.p3.2.m2.4.4.6.3.2" xref="A4.SS1.SSS1.p3.2.m2.4.4.6.3.1.cmml"><mo id="A4.SS1.SSS1.p3.2.m2.4.4.6.3.2.1" stretchy="false" xref="A4.SS1.SSS1.p3.2.m2.4.4.6.3.1.1.cmml">‖</mo><mi id="A4.SS1.SSS1.p3.2.m2.3.3" xref="A4.SS1.SSS1.p3.2.m2.3.3.cmml">ζ</mi><mo id="A4.SS1.SSS1.p3.2.m2.4.4.6.3.2.2" stretchy="false" xref="A4.SS1.SSS1.p3.2.m2.4.4.6.3.1.1.cmml">‖</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="A4.SS1.SSS1.p3.2.m2.4b"><apply id="A4.SS1.SSS1.p3.2.m2.4.4.cmml" xref="A4.SS1.SSS1.p3.2.m2.4.4"><and id="A4.SS1.SSS1.p3.2.m2.4.4a.cmml" xref="A4.SS1.SSS1.p3.2.m2.4.4"></and><apply id="A4.SS1.SSS1.p3.2.m2.4.4b.cmml" 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xref="A4.SS1.SSS1.p3.2.m2.4.4.4.2.2.1">norm</csymbol><ci id="A4.SS1.SSS1.p3.2.m2.1.1.cmml" xref="A4.SS1.SSS1.p3.2.m2.1.1">𝐱</ci></apply><apply id="A4.SS1.SSS1.p3.2.m2.4.4.4.3.1.cmml" xref="A4.SS1.SSS1.p3.2.m2.4.4.4.3.2"><csymbol cd="latexml" id="A4.SS1.SSS1.p3.2.m2.4.4.4.3.1.1.cmml" xref="A4.SS1.SSS1.p3.2.m2.4.4.4.3.2.1">norm</csymbol><ci id="A4.SS1.SSS1.p3.2.m2.2.2.cmml" xref="A4.SS1.SSS1.p3.2.m2.2.2">𝜁</ci></apply></apply></apply><apply id="A4.SS1.SSS1.p3.2.m2.4.4c.cmml" xref="A4.SS1.SSS1.p3.2.m2.4.4"><leq id="A4.SS1.SSS1.p3.2.m2.4.4.5.cmml" xref="A4.SS1.SSS1.p3.2.m2.4.4.5"></leq><share href="#A4.SS1.SSS1.p3.2.m2.4.4.4.cmml" id="A4.SS1.SSS1.p3.2.m2.4.4d.cmml" xref="A4.SS1.SSS1.p3.2.m2.4.4"></share><apply id="A4.SS1.SSS1.p3.2.m2.4.4.6.cmml" xref="A4.SS1.SSS1.p3.2.m2.4.4.6"><plus id="A4.SS1.SSS1.p3.2.m2.4.4.6.1.cmml" xref="A4.SS1.SSS1.p3.2.m2.4.4.6.1"></plus><ci id="A4.SS1.SSS1.p3.2.m2.4.4.6.2.cmml" xref="A4.SS1.SSS1.p3.2.m2.4.4.6.2">𝑅</ci><apply id="A4.SS1.SSS1.p3.2.m2.4.4.6.3.1.cmml" xref="A4.SS1.SSS1.p3.2.m2.4.4.6.3.2"><csymbol cd="latexml" id="A4.SS1.SSS1.p3.2.m2.4.4.6.3.1.1.cmml" xref="A4.SS1.SSS1.p3.2.m2.4.4.6.3.2.1">norm</csymbol><ci id="A4.SS1.SSS1.p3.2.m2.3.3.cmml" xref="A4.SS1.SSS1.p3.2.m2.3.3">𝜁</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.SS1.SSS1.p3.2.m2.4c">\|{\mathbf{x}}+\zeta\|\leq\|{\mathbf{x}}\|+\|\zeta\|\leq R+\|\zeta\|</annotation><annotation encoding="application/x-llamapun" id="A4.SS1.SSS1.p3.2.m2.4d">∥ bold_x + italic_ζ ∥ ≤ ∥ bold_x ∥ + ∥ italic_ζ ∥ ≤ italic_R + ∥ italic_ζ ∥</annotation></semantics></math>, where <math alttext="R=\sup_{{\mathbf{x}}\in\mathbf{K}}{\|{\mathbf{x}}\|}<+\infty" class="ltx_Math" display="inline" id="A4.SS1.SSS1.p3.3.m3.1"><semantics id="A4.SS1.SSS1.p3.3.m3.1a"><mrow id="A4.SS1.SSS1.p3.3.m3.1.2" xref="A4.SS1.SSS1.p3.3.m3.1.2.cmml"><mi id="A4.SS1.SSS1.p3.3.m3.1.2.2" xref="A4.SS1.SSS1.p3.3.m3.1.2.2.cmml">R</mi><mo id="A4.SS1.SSS1.p3.3.m3.1.2.3" rspace="0.1389em" xref="A4.SS1.SSS1.p3.3.m3.1.2.3.cmml">=</mo><mrow id="A4.SS1.SSS1.p3.3.m3.1.2.4" xref="A4.SS1.SSS1.p3.3.m3.1.2.4.cmml"><msub id="A4.SS1.SSS1.p3.3.m3.1.2.4.1" xref="A4.SS1.SSS1.p3.3.m3.1.2.4.1.cmml"><mo id="A4.SS1.SSS1.p3.3.m3.1.2.4.1.2" lspace="0.1389em" rspace="0em" xref="A4.SS1.SSS1.p3.3.m3.1.2.4.1.2.cmml">sup</mo><mrow id="A4.SS1.SSS1.p3.3.m3.1.2.4.1.3" xref="A4.SS1.SSS1.p3.3.m3.1.2.4.1.3.cmml"><mi id="A4.SS1.SSS1.p3.3.m3.1.2.4.1.3.2" xref="A4.SS1.SSS1.p3.3.m3.1.2.4.1.3.2.cmml">𝐱</mi><mo id="A4.SS1.SSS1.p3.3.m3.1.2.4.1.3.1" xref="A4.SS1.SSS1.p3.3.m3.1.2.4.1.3.1.cmml">∈</mo><mi id="A4.SS1.SSS1.p3.3.m3.1.2.4.1.3.3" xref="A4.SS1.SSS1.p3.3.m3.1.2.4.1.3.3.cmml">𝐊</mi></mrow></msub><mrow id="A4.SS1.SSS1.p3.3.m3.1.2.4.2.2" xref="A4.SS1.SSS1.p3.3.m3.1.2.4.2.1.cmml"><mo id="A4.SS1.SSS1.p3.3.m3.1.2.4.2.2.1" stretchy="false" xref="A4.SS1.SSS1.p3.3.m3.1.2.4.2.1.1.cmml">‖</mo><mi id="A4.SS1.SSS1.p3.3.m3.1.1" xref="A4.SS1.SSS1.p3.3.m3.1.1.cmml">𝐱</mi><mo id="A4.SS1.SSS1.p3.3.m3.1.2.4.2.2.2" stretchy="false" xref="A4.SS1.SSS1.p3.3.m3.1.2.4.2.1.1.cmml">‖</mo></mrow></mrow><mo id="A4.SS1.SSS1.p3.3.m3.1.2.5" xref="A4.SS1.SSS1.p3.3.m3.1.2.5.cmml"><</mo><mrow id="A4.SS1.SSS1.p3.3.m3.1.2.6" xref="A4.SS1.SSS1.p3.3.m3.1.2.6.cmml"><mo id="A4.SS1.SSS1.p3.3.m3.1.2.6a" xref="A4.SS1.SSS1.p3.3.m3.1.2.6.cmml">+</mo><mi id="A4.SS1.SSS1.p3.3.m3.1.2.6.2" mathvariant="normal" xref="A4.SS1.SSS1.p3.3.m3.1.2.6.2.cmml">∞</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="A4.SS1.SSS1.p3.3.m3.1b"><apply id="A4.SS1.SSS1.p3.3.m3.1.2.cmml" xref="A4.SS1.SSS1.p3.3.m3.1.2"><and id="A4.SS1.SSS1.p3.3.m3.1.2a.cmml" xref="A4.SS1.SSS1.p3.3.m3.1.2"></and><apply id="A4.SS1.SSS1.p3.3.m3.1.2b.cmml" xref="A4.SS1.SSS1.p3.3.m3.1.2"><eq id="A4.SS1.SSS1.p3.3.m3.1.2.3.cmml" xref="A4.SS1.SSS1.p3.3.m3.1.2.3"></eq><ci id="A4.SS1.SSS1.p3.3.m3.1.2.2.cmml" xref="A4.SS1.SSS1.p3.3.m3.1.2.2">𝑅</ci><apply id="A4.SS1.SSS1.p3.3.m3.1.2.4.cmml" xref="A4.SS1.SSS1.p3.3.m3.1.2.4"><apply id="A4.SS1.SSS1.p3.3.m3.1.2.4.1.cmml" xref="A4.SS1.SSS1.p3.3.m3.1.2.4.1"><csymbol cd="ambiguous" id="A4.SS1.SSS1.p3.3.m3.1.2.4.1.1.cmml" xref="A4.SS1.SSS1.p3.3.m3.1.2.4.1">subscript</csymbol><csymbol cd="latexml" id="A4.SS1.SSS1.p3.3.m3.1.2.4.1.2.cmml" xref="A4.SS1.SSS1.p3.3.m3.1.2.4.1.2">supremum</csymbol><apply id="A4.SS1.SSS1.p3.3.m3.1.2.4.1.3.cmml" xref="A4.SS1.SSS1.p3.3.m3.1.2.4.1.3"><in id="A4.SS1.SSS1.p3.3.m3.1.2.4.1.3.1.cmml" xref="A4.SS1.SSS1.p3.3.m3.1.2.4.1.3.1"></in><ci id="A4.SS1.SSS1.p3.3.m3.1.2.4.1.3.2.cmml" xref="A4.SS1.SSS1.p3.3.m3.1.2.4.1.3.2">𝐱</ci><ci id="A4.SS1.SSS1.p3.3.m3.1.2.4.1.3.3.cmml" xref="A4.SS1.SSS1.p3.3.m3.1.2.4.1.3.3">𝐊</ci></apply></apply><apply id="A4.SS1.SSS1.p3.3.m3.1.2.4.2.1.cmml" xref="A4.SS1.SSS1.p3.3.m3.1.2.4.2.2"><csymbol cd="latexml" id="A4.SS1.SSS1.p3.3.m3.1.2.4.2.1.1.cmml" xref="A4.SS1.SSS1.p3.3.m3.1.2.4.2.2.1">norm</csymbol><ci id="A4.SS1.SSS1.p3.3.m3.1.1.cmml" xref="A4.SS1.SSS1.p3.3.m3.1.1">𝐱</ci></apply></apply></apply><apply id="A4.SS1.SSS1.p3.3.m3.1.2c.cmml" xref="A4.SS1.SSS1.p3.3.m3.1.2"><lt id="A4.SS1.SSS1.p3.3.m3.1.2.5.cmml" xref="A4.SS1.SSS1.p3.3.m3.1.2.5"></lt><share href="#A4.SS1.SSS1.p3.3.m3.1.2.4.cmml" id="A4.SS1.SSS1.p3.3.m3.1.2d.cmml" xref="A4.SS1.SSS1.p3.3.m3.1.2"></share><apply id="A4.SS1.SSS1.p3.3.m3.1.2.6.cmml" xref="A4.SS1.SSS1.p3.3.m3.1.2.6"><plus id="A4.SS1.SSS1.p3.3.m3.1.2.6.1.cmml" xref="A4.SS1.SSS1.p3.3.m3.1.2.6"></plus><infinity id="A4.SS1.SSS1.p3.3.m3.1.2.6.2.cmml" xref="A4.SS1.SSS1.p3.3.m3.1.2.6.2"></infinity></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.SS1.SSS1.p3.3.m3.1c">R=\sup_{{\mathbf{x}}\in\mathbf{K}}{\|{\mathbf{x}}\|}<+\infty</annotation><annotation encoding="application/x-llamapun" id="A4.SS1.SSS1.p3.3.m3.1d">italic_R = roman_sup start_POSTSUBSCRIPT bold_x ∈ bold_K end_POSTSUBSCRIPT ∥ bold_x ∥ < + ∞</annotation></semantics></math> because <math alttext="\mathbf{K}" class="ltx_Math" display="inline" id="A4.SS1.SSS1.p3.4.m4.1"><semantics id="A4.SS1.SSS1.p3.4.m4.1a"><mi id="A4.SS1.SSS1.p3.4.m4.1.1" xref="A4.SS1.SSS1.p3.4.m4.1.1.cmml">𝐊</mi><annotation-xml encoding="MathML-Content" id="A4.SS1.SSS1.p3.4.m4.1b"><ci id="A4.SS1.SSS1.p3.4.m4.1.1.cmml" xref="A4.SS1.SSS1.p3.4.m4.1.1">𝐊</ci></annotation-xml><annotation encoding="application/x-tex" id="A4.SS1.SSS1.p3.4.m4.1c">\mathbf{K}</annotation><annotation encoding="application/x-llamapun" id="A4.SS1.SSS1.p3.4.m4.1d">bold_K</annotation></semantics></math> is compact. Moreover on the set <math alttext="{\mathbb{R}}^{d}\setminus\mathcal{B}(0,\mu)" class="ltx_Math" display="inline" id="A4.SS1.SSS1.p3.5.m5.2"><semantics id="A4.SS1.SSS1.p3.5.m5.2a"><mrow id="A4.SS1.SSS1.p3.5.m5.2.3" xref="A4.SS1.SSS1.p3.5.m5.2.3.cmml"><msup id="A4.SS1.SSS1.p3.5.m5.2.3.2" xref="A4.SS1.SSS1.p3.5.m5.2.3.2.cmml"><mi id="A4.SS1.SSS1.p3.5.m5.2.3.2.2" xref="A4.SS1.SSS1.p3.5.m5.2.3.2.2.cmml">ℝ</mi><mi id="A4.SS1.SSS1.p3.5.m5.2.3.2.3" xref="A4.SS1.SSS1.p3.5.m5.2.3.2.3.cmml">d</mi></msup><mo id="A4.SS1.SSS1.p3.5.m5.2.3.1" xref="A4.SS1.SSS1.p3.5.m5.2.3.1.cmml">∖</mo><mrow id="A4.SS1.SSS1.p3.5.m5.2.3.3" xref="A4.SS1.SSS1.p3.5.m5.2.3.3.cmml"><mi class="ltx_font_mathcaligraphic" id="A4.SS1.SSS1.p3.5.m5.2.3.3.2" xref="A4.SS1.SSS1.p3.5.m5.2.3.3.2.cmml">ℬ</mi><mo id="A4.SS1.SSS1.p3.5.m5.2.3.3.1" xref="A4.SS1.SSS1.p3.5.m5.2.3.3.1.cmml">⁢</mo><mrow id="A4.SS1.SSS1.p3.5.m5.2.3.3.3.2" xref="A4.SS1.SSS1.p3.5.m5.2.3.3.3.1.cmml"><mo id="A4.SS1.SSS1.p3.5.m5.2.3.3.3.2.1" stretchy="false" xref="A4.SS1.SSS1.p3.5.m5.2.3.3.3.1.cmml">(</mo><mn id="A4.SS1.SSS1.p3.5.m5.1.1" xref="A4.SS1.SSS1.p3.5.m5.1.1.cmml">0</mn><mo id="A4.SS1.SSS1.p3.5.m5.2.3.3.3.2.2" xref="A4.SS1.SSS1.p3.5.m5.2.3.3.3.1.cmml">,</mo><mi id="A4.SS1.SSS1.p3.5.m5.2.2" xref="A4.SS1.SSS1.p3.5.m5.2.2.cmml">μ</mi><mo id="A4.SS1.SSS1.p3.5.m5.2.3.3.3.2.3" stretchy="false" xref="A4.SS1.SSS1.p3.5.m5.2.3.3.3.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="A4.SS1.SSS1.p3.5.m5.2b"><apply id="A4.SS1.SSS1.p3.5.m5.2.3.cmml" xref="A4.SS1.SSS1.p3.5.m5.2.3"><setdiff id="A4.SS1.SSS1.p3.5.m5.2.3.1.cmml" xref="A4.SS1.SSS1.p3.5.m5.2.3.1"></setdiff><apply id="A4.SS1.SSS1.p3.5.m5.2.3.2.cmml" xref="A4.SS1.SSS1.p3.5.m5.2.3.2"><csymbol cd="ambiguous" id="A4.SS1.SSS1.p3.5.m5.2.3.2.1.cmml" xref="A4.SS1.SSS1.p3.5.m5.2.3.2">superscript</csymbol><ci id="A4.SS1.SSS1.p3.5.m5.2.3.2.2.cmml" xref="A4.SS1.SSS1.p3.5.m5.2.3.2.2">ℝ</ci><ci id="A4.SS1.SSS1.p3.5.m5.2.3.2.3.cmml" xref="A4.SS1.SSS1.p3.5.m5.2.3.2.3">𝑑</ci></apply><apply id="A4.SS1.SSS1.p3.5.m5.2.3.3.cmml" xref="A4.SS1.SSS1.p3.5.m5.2.3.3"><times id="A4.SS1.SSS1.p3.5.m5.2.3.3.1.cmml" xref="A4.SS1.SSS1.p3.5.m5.2.3.3.1"></times><ci id="A4.SS1.SSS1.p3.5.m5.2.3.3.2.cmml" xref="A4.SS1.SSS1.p3.5.m5.2.3.3.2">ℬ</ci><interval closure="open" id="A4.SS1.SSS1.p3.5.m5.2.3.3.3.1.cmml" xref="A4.SS1.SSS1.p3.5.m5.2.3.3.3.2"><cn id="A4.SS1.SSS1.p3.5.m5.1.1.cmml" type="integer" xref="A4.SS1.SSS1.p3.5.m5.1.1">0</cn><ci id="A4.SS1.SSS1.p3.5.m5.2.2.cmml" xref="A4.SS1.SSS1.p3.5.m5.2.2">𝜇</ci></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.SS1.SSS1.p3.5.m5.2c">{\mathbb{R}}^{d}\setminus\mathcal{B}(0,\mu)</annotation><annotation encoding="application/x-llamapun" id="A4.SS1.SSS1.p3.5.m5.2d">blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT ∖ caligraphic_B ( 0 , italic_μ )</annotation></semantics></math>, the inequality <math 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xref="A4.SS1.SSS1.p3.6.m6.1.1.1.3.1.1.cmml">‖</mo></mrow><mn id="A4.SS1.SSS1.p3.6.m6.1.1.1.4" xref="A4.SS1.SSS1.p3.6.m6.1.1.1.4.cmml">2</mn></msup><mn id="A4.SS1.SSS1.p3.6.m6.1.1.3" xref="A4.SS1.SSS1.p3.6.m6.1.1.3.cmml">2</mn></mfrac></mrow></mrow><annotation-xml encoding="MathML-Content" id="A4.SS1.SSS1.p3.6.m6.2b"><apply id="A4.SS1.SSS1.p3.6.m6.2.3.cmml" xref="A4.SS1.SSS1.p3.6.m6.2.3"><geq id="A4.SS1.SSS1.p3.6.m6.2.3.1.cmml" xref="A4.SS1.SSS1.p3.6.m6.2.3.1"></geq><apply id="A4.SS1.SSS1.p3.6.m6.2.3.2.cmml" xref="A4.SS1.SSS1.p3.6.m6.2.3.2"><csymbol cd="ambiguous" id="A4.SS1.SSS1.p3.6.m6.2.3.2.1.cmml" xref="A4.SS1.SSS1.p3.6.m6.2.3.2">superscript</csymbol><apply id="A4.SS1.SSS1.p3.6.m6.2.3.2.2.1.cmml" xref="A4.SS1.SSS1.p3.6.m6.2.3.2.2.2"><csymbol cd="latexml" id="A4.SS1.SSS1.p3.6.m6.2.3.2.2.1.1.cmml" xref="A4.SS1.SSS1.p3.6.m6.2.3.2.2.2.1">norm</csymbol><ci id="A4.SS1.SSS1.p3.6.m6.2.2.cmml" xref="A4.SS1.SSS1.p3.6.m6.2.2">𝜁</ci></apply><cn id="A4.SS1.SSS1.p3.6.m6.2.3.2.3.cmml" type="integer" xref="A4.SS1.SSS1.p3.6.m6.2.3.2.3">2</cn></apply><apply id="A4.SS1.SSS1.p3.6.m6.2.3.3.cmml" xref="A4.SS1.SSS1.p3.6.m6.2.3.3"><plus id="A4.SS1.SSS1.p3.6.m6.2.3.3.1.cmml" xref="A4.SS1.SSS1.p3.6.m6.2.3.3.1"></plus><apply id="A4.SS1.SSS1.p3.6.m6.2.3.3.2.cmml" xref="A4.SS1.SSS1.p3.6.m6.2.3.3.2"><divide id="A4.SS1.SSS1.p3.6.m6.2.3.3.2.1.cmml" xref="A4.SS1.SSS1.p3.6.m6.2.3.3.2"></divide><apply id="A4.SS1.SSS1.p3.6.m6.2.3.3.2.2.cmml" xref="A4.SS1.SSS1.p3.6.m6.2.3.3.2.2"><csymbol cd="ambiguous" id="A4.SS1.SSS1.p3.6.m6.2.3.3.2.2.1.cmml" xref="A4.SS1.SSS1.p3.6.m6.2.3.3.2.2">superscript</csymbol><ci id="A4.SS1.SSS1.p3.6.m6.2.3.3.2.2.2.cmml" xref="A4.SS1.SSS1.p3.6.m6.2.3.3.2.2.2">𝜇</ci><cn id="A4.SS1.SSS1.p3.6.m6.2.3.3.2.2.3.cmml" type="integer" xref="A4.SS1.SSS1.p3.6.m6.2.3.3.2.2.3">2</cn></apply><cn id="A4.SS1.SSS1.p3.6.m6.2.3.3.2.3.cmml" type="integer" xref="A4.SS1.SSS1.p3.6.m6.2.3.3.2.3">2</cn></apply><apply id="A4.SS1.SSS1.p3.6.m6.1.1.cmml" xref="A4.SS1.SSS1.p3.6.m6.1.1"><divide id="A4.SS1.SSS1.p3.6.m6.1.1.2.cmml" xref="A4.SS1.SSS1.p3.6.m6.1.1"></divide><apply id="A4.SS1.SSS1.p3.6.m6.1.1.1.cmml" xref="A4.SS1.SSS1.p3.6.m6.1.1.1"><csymbol cd="ambiguous" id="A4.SS1.SSS1.p3.6.m6.1.1.1.2.cmml" xref="A4.SS1.SSS1.p3.6.m6.1.1.1">superscript</csymbol><apply id="A4.SS1.SSS1.p3.6.m6.1.1.1.3.1.cmml" xref="A4.SS1.SSS1.p3.6.m6.1.1.1.3.2"><csymbol cd="latexml" id="A4.SS1.SSS1.p3.6.m6.1.1.1.3.1.1.cmml" xref="A4.SS1.SSS1.p3.6.m6.1.1.1.3.2.1">norm</csymbol><ci id="A4.SS1.SSS1.p3.6.m6.1.1.1.1.cmml" xref="A4.SS1.SSS1.p3.6.m6.1.1.1.1">𝜁</ci></apply><cn id="A4.SS1.SSS1.p3.6.m6.1.1.1.4.cmml" type="integer" xref="A4.SS1.SSS1.p3.6.m6.1.1.1.4">2</cn></apply><cn id="A4.SS1.SSS1.p3.6.m6.1.1.3.cmml" type="integer" xref="A4.SS1.SSS1.p3.6.m6.1.1.3">2</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.SS1.SSS1.p3.6.m6.2c">\|\zeta\|^{2}\geq\frac{\mu^{2}}{2}+\frac{\|\zeta\|^{2}}{2}</annotation><annotation encoding="application/x-llamapun" id="A4.SS1.SSS1.p3.6.m6.2d">∥ italic_ζ ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≥ divide start_ARG italic_μ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG 2 end_ARG + divide start_ARG ∥ italic_ζ ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG 2 end_ARG</annotation></semantics></math> holds. This leads to <math alttext="\mathcal{N}(\zeta;0,\sigma^{2}\mathbf{I}_{d})\leq 2^{\frac{d}{2}}\exp{(-\frac{% \mu^{2}}{2\sigma^{2}})}\mathcal{N}(\zeta;0,2\sigma^{2}\mathbf{I}_{d})" class="ltx_Math" display="inline" id="A4.SS1.SSS1.p3.7.m7.8"><semantics id="A4.SS1.SSS1.p3.7.m7.8a"><mrow id="A4.SS1.SSS1.p3.7.m7.8.8" xref="A4.SS1.SSS1.p3.7.m7.8.8.cmml"><mrow id="A4.SS1.SSS1.p3.7.m7.6.6.1" xref="A4.SS1.SSS1.p3.7.m7.6.6.1.cmml"><mi class="ltx_font_mathcaligraphic" id="A4.SS1.SSS1.p3.7.m7.6.6.1.3" xref="A4.SS1.SSS1.p3.7.m7.6.6.1.3.cmml">𝒩</mi><mo id="A4.SS1.SSS1.p3.7.m7.6.6.1.2" xref="A4.SS1.SSS1.p3.7.m7.6.6.1.2.cmml">⁢</mo><mrow id="A4.SS1.SSS1.p3.7.m7.6.6.1.1.1" xref="A4.SS1.SSS1.p3.7.m7.6.6.1.1.2.cmml"><mo id="A4.SS1.SSS1.p3.7.m7.6.6.1.1.1.2" stretchy="false" xref="A4.SS1.SSS1.p3.7.m7.6.6.1.1.2.cmml">(</mo><mi id="A4.SS1.SSS1.p3.7.m7.1.1" xref="A4.SS1.SSS1.p3.7.m7.1.1.cmml">ζ</mi><mo id="A4.SS1.SSS1.p3.7.m7.6.6.1.1.1.3" xref="A4.SS1.SSS1.p3.7.m7.6.6.1.1.2.cmml">;</mo><mn id="A4.SS1.SSS1.p3.7.m7.2.2" xref="A4.SS1.SSS1.p3.7.m7.2.2.cmml">0</mn><mo id="A4.SS1.SSS1.p3.7.m7.6.6.1.1.1.4" xref="A4.SS1.SSS1.p3.7.m7.6.6.1.1.2.cmml">,</mo><mrow id="A4.SS1.SSS1.p3.7.m7.6.6.1.1.1.1" xref="A4.SS1.SSS1.p3.7.m7.6.6.1.1.1.1.cmml"><msup id="A4.SS1.SSS1.p3.7.m7.6.6.1.1.1.1.2" xref="A4.SS1.SSS1.p3.7.m7.6.6.1.1.1.1.2.cmml"><mi id="A4.SS1.SSS1.p3.7.m7.6.6.1.1.1.1.2.2" xref="A4.SS1.SSS1.p3.7.m7.6.6.1.1.1.1.2.2.cmml">σ</mi><mn id="A4.SS1.SSS1.p3.7.m7.6.6.1.1.1.1.2.3" xref="A4.SS1.SSS1.p3.7.m7.6.6.1.1.1.1.2.3.cmml">2</mn></msup><mo id="A4.SS1.SSS1.p3.7.m7.6.6.1.1.1.1.1" xref="A4.SS1.SSS1.p3.7.m7.6.6.1.1.1.1.1.cmml">⁢</mo><msub id="A4.SS1.SSS1.p3.7.m7.6.6.1.1.1.1.3" xref="A4.SS1.SSS1.p3.7.m7.6.6.1.1.1.1.3.cmml"><mi id="A4.SS1.SSS1.p3.7.m7.6.6.1.1.1.1.3.2" xref="A4.SS1.SSS1.p3.7.m7.6.6.1.1.1.1.3.2.cmml">𝐈</mi><mi id="A4.SS1.SSS1.p3.7.m7.6.6.1.1.1.1.3.3" xref="A4.SS1.SSS1.p3.7.m7.6.6.1.1.1.1.3.3.cmml">d</mi></msub></mrow><mo id="A4.SS1.SSS1.p3.7.m7.6.6.1.1.1.5" stretchy="false" 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id="A4.SS1.SSS1.p3.7.m7.8c">\mathcal{N}(\zeta;0,\sigma^{2}\mathbf{I}_{d})\leq 2^{\frac{d}{2}}\exp{(-\frac{% \mu^{2}}{2\sigma^{2}})}\mathcal{N}(\zeta;0,2\sigma^{2}\mathbf{I}_{d})</annotation><annotation encoding="application/x-llamapun" id="A4.SS1.SSS1.p3.7.m7.8d">caligraphic_N ( italic_ζ ; 0 , italic_σ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT bold_I start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT ) ≤ 2 start_POSTSUPERSCRIPT divide start_ARG italic_d end_ARG start_ARG 2 end_ARG end_POSTSUPERSCRIPT roman_exp ( - divide start_ARG italic_μ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG 2 italic_σ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG ) caligraphic_N ( italic_ζ ; 0 , 2 italic_σ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT bold_I start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT )</annotation></semantics></math>; and then to the following inequality</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A8.EGx49"> <tbody id="A4.Ex81"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\|\nabla\mathcal{R}_{\sigma}({\mathbf{x}})+\nabla\log p({\mathbf{% x}})\|\leq\|\nabla\log p_{\sigma}-\nabla\log p\|_{\infty,\mathbf{L}}+\epsilon" class="ltx_Math" display="inline" id="A4.Ex81.m1.6"><semantics id="A4.Ex81.m1.6a"><mrow id="A4.Ex81.m1.6.6" xref="A4.Ex81.m1.6.6.cmml"><mrow id="A4.Ex81.m1.5.5.1.1" xref="A4.Ex81.m1.5.5.1.2.cmml"><mo id="A4.Ex81.m1.5.5.1.1.2" stretchy="false" xref="A4.Ex81.m1.5.5.1.2.1.cmml">‖</mo><mrow id="A4.Ex81.m1.5.5.1.1.1" xref="A4.Ex81.m1.5.5.1.1.1.cmml"><mrow id="A4.Ex81.m1.5.5.1.1.1.2" xref="A4.Ex81.m1.5.5.1.1.1.2.cmml"><mrow id="A4.Ex81.m1.5.5.1.1.1.2.2" xref="A4.Ex81.m1.5.5.1.1.1.2.2.cmml"><mo id="A4.Ex81.m1.5.5.1.1.1.2.2.1" rspace="0.167em" xref="A4.Ex81.m1.5.5.1.1.1.2.2.1.cmml">∇</mo><msub id="A4.Ex81.m1.5.5.1.1.1.2.2.2" xref="A4.Ex81.m1.5.5.1.1.1.2.2.2.cmml"><mi class="ltx_font_mathcaligraphic" id="A4.Ex81.m1.5.5.1.1.1.2.2.2.2" xref="A4.Ex81.m1.5.5.1.1.1.2.2.2.2.cmml">ℛ</mi><mi id="A4.Ex81.m1.5.5.1.1.1.2.2.2.3" xref="A4.Ex81.m1.5.5.1.1.1.2.2.2.3.cmml">σ</mi></msub></mrow><mo id="A4.Ex81.m1.5.5.1.1.1.2.1" xref="A4.Ex81.m1.5.5.1.1.1.2.1.cmml">⁢</mo><mrow id="A4.Ex81.m1.5.5.1.1.1.2.3.2" xref="A4.Ex81.m1.5.5.1.1.1.2.cmml"><mo id="A4.Ex81.m1.5.5.1.1.1.2.3.2.1" stretchy="false" xref="A4.Ex81.m1.5.5.1.1.1.2.cmml">(</mo><mi id="A4.Ex81.m1.3.3" xref="A4.Ex81.m1.3.3.cmml">𝐱</mi><mo id="A4.Ex81.m1.5.5.1.1.1.2.3.2.2" stretchy="false" xref="A4.Ex81.m1.5.5.1.1.1.2.cmml">)</mo></mrow></mrow><mo id="A4.Ex81.m1.5.5.1.1.1.1" xref="A4.Ex81.m1.5.5.1.1.1.1.cmml">+</mo><mrow id="A4.Ex81.m1.5.5.1.1.1.3" xref="A4.Ex81.m1.5.5.1.1.1.3.cmml"><mrow id="A4.Ex81.m1.5.5.1.1.1.3.2" xref="A4.Ex81.m1.5.5.1.1.1.3.2.cmml"><mrow id="A4.Ex81.m1.5.5.1.1.1.3.2.1" xref="A4.Ex81.m1.5.5.1.1.1.3.2.1.cmml"><mo id="A4.Ex81.m1.5.5.1.1.1.3.2.1.1" rspace="0.167em" 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^{\beta}(1+(R+\|\zeta\|)^{r})+2A(1+(R+\|\zeta\|)^{q})+A(1+R^{q})\right)2^{% \frac{d}{2}}\exp{(-\frac{\mu^{2}}{2\sigma^{2}})}\mathcal{N}(\zeta;0,2\sigma^{2% }\mathbf{I}_{d})d\zeta}</annotation><annotation encoding="application/x-llamapun" id="A4.Ex82.m1.10d">+ ∫ start_POSTSUBSCRIPT blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT ∖ caligraphic_B ( 0 , italic_μ ) end_POSTSUBSCRIPT ( italic_B italic_σ start_POSTSUPERSCRIPT italic_β end_POSTSUPERSCRIPT ( 1 + ( italic_R + ∥ italic_ζ ∥ ) start_POSTSUPERSCRIPT italic_r end_POSTSUPERSCRIPT ) + 2 italic_A ( 1 + ( italic_R + ∥ italic_ζ ∥ ) start_POSTSUPERSCRIPT italic_q end_POSTSUPERSCRIPT ) + italic_A ( 1 + italic_R start_POSTSUPERSCRIPT italic_q end_POSTSUPERSCRIPT ) ) 2 start_POSTSUPERSCRIPT divide start_ARG italic_d end_ARG start_ARG 2 end_ARG end_POSTSUPERSCRIPT roman_exp ( - divide start_ARG italic_μ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG 2 italic_σ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG ) caligraphic_N ( italic_ζ ; 0 , 2 italic_σ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT bold_I start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT ) italic_d italic_ζ</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="A4.Ex83"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\leq\sup_{\mathbf{L}}(\|\nabla\log p_{\sigma}-\nabla\log p\|)+% \epsilon+(\sigma^{\beta}C_{r,B,R}+C_{q,A,R})\exp{(-\frac{\mu^{2}}{2\sigma^{2}}% )}," class="ltx_Math" display="inline" id="A4.Ex83.m1.8"><semantics id="A4.Ex83.m1.8a"><mrow id="A4.Ex83.m1.8.8.1" xref="A4.Ex83.m1.8.8.1.1.cmml"><mrow id="A4.Ex83.m1.8.8.1.1" xref="A4.Ex83.m1.8.8.1.1.cmml"><mi id="A4.Ex83.m1.8.8.1.1.5" xref="A4.Ex83.m1.8.8.1.1.5.cmml"></mi><mo id="A4.Ex83.m1.8.8.1.1.4" rspace="0.1389em" xref="A4.Ex83.m1.8.8.1.1.4.cmml">≤</mo><mrow 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xref="A4.Ex83.m1.8.8.1.1.3.3.2.1.1.1.2.3.3.3">2</cn></apply></apply></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.Ex83.m1.8c">\displaystyle\leq\sup_{\mathbf{L}}(\|\nabla\log p_{\sigma}-\nabla\log p\|)+% \epsilon+(\sigma^{\beta}C_{r,B,R}+C_{q,A,R})\exp{(-\frac{\mu^{2}}{2\sigma^{2}}% )},</annotation><annotation encoding="application/x-llamapun" id="A4.Ex83.m1.8d">≤ roman_sup start_POSTSUBSCRIPT bold_L end_POSTSUBSCRIPT ( ∥ ∇ roman_log italic_p start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT - ∇ roman_log italic_p ∥ ) + italic_ϵ + ( italic_σ start_POSTSUPERSCRIPT italic_β end_POSTSUPERSCRIPT italic_C start_POSTSUBSCRIPT italic_r , italic_B , italic_R end_POSTSUBSCRIPT + italic_C start_POSTSUBSCRIPT italic_q , italic_A , italic_R end_POSTSUBSCRIPT ) roman_exp ( - divide start_ARG italic_μ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG 2 italic_σ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG ) ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="A4.SS1.SSS1.p3.25">with <math alttext="C_{r,B,R}" class="ltx_Math" display="inline" id="A4.SS1.SSS1.p3.8.m1.3"><semantics id="A4.SS1.SSS1.p3.8.m1.3a"><msub id="A4.SS1.SSS1.p3.8.m1.3.4" xref="A4.SS1.SSS1.p3.8.m1.3.4.cmml"><mi id="A4.SS1.SSS1.p3.8.m1.3.4.2" xref="A4.SS1.SSS1.p3.8.m1.3.4.2.cmml">C</mi><mrow id="A4.SS1.SSS1.p3.8.m1.3.3.3.5" xref="A4.SS1.SSS1.p3.8.m1.3.3.3.4.cmml"><mi id="A4.SS1.SSS1.p3.8.m1.1.1.1.1" xref="A4.SS1.SSS1.p3.8.m1.1.1.1.1.cmml">r</mi><mo id="A4.SS1.SSS1.p3.8.m1.3.3.3.5.1" xref="A4.SS1.SSS1.p3.8.m1.3.3.3.4.cmml">,</mo><mi id="A4.SS1.SSS1.p3.8.m1.2.2.2.2" xref="A4.SS1.SSS1.p3.8.m1.2.2.2.2.cmml">B</mi><mo id="A4.SS1.SSS1.p3.8.m1.3.3.3.5.2" xref="A4.SS1.SSS1.p3.8.m1.3.3.3.4.cmml">,</mo><mi id="A4.SS1.SSS1.p3.8.m1.3.3.3.3" xref="A4.SS1.SSS1.p3.8.m1.3.3.3.3.cmml">R</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="A4.SS1.SSS1.p3.8.m1.3b"><apply id="A4.SS1.SSS1.p3.8.m1.3.4.cmml" xref="A4.SS1.SSS1.p3.8.m1.3.4"><csymbol cd="ambiguous" id="A4.SS1.SSS1.p3.8.m1.3.4.1.cmml" xref="A4.SS1.SSS1.p3.8.m1.3.4">subscript</csymbol><ci id="A4.SS1.SSS1.p3.8.m1.3.4.2.cmml" xref="A4.SS1.SSS1.p3.8.m1.3.4.2">𝐶</ci><list id="A4.SS1.SSS1.p3.8.m1.3.3.3.4.cmml" xref="A4.SS1.SSS1.p3.8.m1.3.3.3.5"><ci id="A4.SS1.SSS1.p3.8.m1.1.1.1.1.cmml" xref="A4.SS1.SSS1.p3.8.m1.1.1.1.1">𝑟</ci><ci id="A4.SS1.SSS1.p3.8.m1.2.2.2.2.cmml" xref="A4.SS1.SSS1.p3.8.m1.2.2.2.2">𝐵</ci><ci id="A4.SS1.SSS1.p3.8.m1.3.3.3.3.cmml" xref="A4.SS1.SSS1.p3.8.m1.3.3.3.3">𝑅</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.SS1.SSS1.p3.8.m1.3c">C_{r,B,R}</annotation><annotation encoding="application/x-llamapun" id="A4.SS1.SSS1.p3.8.m1.3d">italic_C start_POSTSUBSCRIPT italic_r , italic_B , italic_R end_POSTSUBSCRIPT</annotation></semantics></math> a constant depending on <math alttext="r" class="ltx_Math" display="inline" 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id="A4.SS1.SSS1.p3.10.m3.1d">italic_B</annotation></semantics></math> and <math alttext="R" class="ltx_Math" display="inline" id="A4.SS1.SSS1.p3.11.m4.1"><semantics id="A4.SS1.SSS1.p3.11.m4.1a"><mi id="A4.SS1.SSS1.p3.11.m4.1.1" xref="A4.SS1.SSS1.p3.11.m4.1.1.cmml">R</mi><annotation-xml encoding="MathML-Content" id="A4.SS1.SSS1.p3.11.m4.1b"><ci id="A4.SS1.SSS1.p3.11.m4.1.1.cmml" xref="A4.SS1.SSS1.p3.11.m4.1.1">𝑅</ci></annotation-xml><annotation encoding="application/x-tex" id="A4.SS1.SSS1.p3.11.m4.1c">R</annotation><annotation encoding="application/x-llamapun" id="A4.SS1.SSS1.p3.11.m4.1d">italic_R</annotation></semantics></math>; and <math alttext="C_{q,A,R}" class="ltx_Math" display="inline" id="A4.SS1.SSS1.p3.12.m5.3"><semantics id="A4.SS1.SSS1.p3.12.m5.3a"><msub id="A4.SS1.SSS1.p3.12.m5.3.4" xref="A4.SS1.SSS1.p3.12.m5.3.4.cmml"><mi id="A4.SS1.SSS1.p3.12.m5.3.4.2" xref="A4.SS1.SSS1.p3.12.m5.3.4.2.cmml">C</mi><mrow id="A4.SS1.SSS1.p3.12.m5.3.3.3.5" xref="A4.SS1.SSS1.p3.12.m5.3.3.3.4.cmml"><mi id="A4.SS1.SSS1.p3.12.m5.1.1.1.1" xref="A4.SS1.SSS1.p3.12.m5.1.1.1.1.cmml">q</mi><mo id="A4.SS1.SSS1.p3.12.m5.3.3.3.5.1" xref="A4.SS1.SSS1.p3.12.m5.3.3.3.4.cmml">,</mo><mi id="A4.SS1.SSS1.p3.12.m5.2.2.2.2" xref="A4.SS1.SSS1.p3.12.m5.2.2.2.2.cmml">A</mi><mo id="A4.SS1.SSS1.p3.12.m5.3.3.3.5.2" xref="A4.SS1.SSS1.p3.12.m5.3.3.3.4.cmml">,</mo><mi id="A4.SS1.SSS1.p3.12.m5.3.3.3.3" xref="A4.SS1.SSS1.p3.12.m5.3.3.3.3.cmml">R</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="A4.SS1.SSS1.p3.12.m5.3b"><apply id="A4.SS1.SSS1.p3.12.m5.3.4.cmml" xref="A4.SS1.SSS1.p3.12.m5.3.4"><csymbol cd="ambiguous" id="A4.SS1.SSS1.p3.12.m5.3.4.1.cmml" xref="A4.SS1.SSS1.p3.12.m5.3.4">subscript</csymbol><ci id="A4.SS1.SSS1.p3.12.m5.3.4.2.cmml" xref="A4.SS1.SSS1.p3.12.m5.3.4.2">𝐶</ci><list id="A4.SS1.SSS1.p3.12.m5.3.3.3.4.cmml" xref="A4.SS1.SSS1.p3.12.m5.3.3.3.5"><ci id="A4.SS1.SSS1.p3.12.m5.1.1.1.1.cmml" xref="A4.SS1.SSS1.p3.12.m5.1.1.1.1">𝑞</ci><ci id="A4.SS1.SSS1.p3.12.m5.2.2.2.2.cmml" xref="A4.SS1.SSS1.p3.12.m5.2.2.2.2">𝐴</ci><ci id="A4.SS1.SSS1.p3.12.m5.3.3.3.3.cmml" xref="A4.SS1.SSS1.p3.12.m5.3.3.3.3">𝑅</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.SS1.SSS1.p3.12.m5.3c">C_{q,A,R}</annotation><annotation encoding="application/x-llamapun" id="A4.SS1.SSS1.p3.12.m5.3d">italic_C start_POSTSUBSCRIPT italic_q , italic_A , italic_R end_POSTSUBSCRIPT</annotation></semantics></math> a constant depending on <math alttext="q" class="ltx_Math" display="inline" id="A4.SS1.SSS1.p3.13.m6.1"><semantics id="A4.SS1.SSS1.p3.13.m6.1a"><mi id="A4.SS1.SSS1.p3.13.m6.1.1" xref="A4.SS1.SSS1.p3.13.m6.1.1.cmml">q</mi><annotation-xml encoding="MathML-Content" id="A4.SS1.SSS1.p3.13.m6.1b"><ci id="A4.SS1.SSS1.p3.13.m6.1.1.cmml" xref="A4.SS1.SSS1.p3.13.m6.1.1">𝑞</ci></annotation-xml><annotation encoding="application/x-tex" id="A4.SS1.SSS1.p3.13.m6.1c">q</annotation><annotation encoding="application/x-llamapun" id="A4.SS1.SSS1.p3.13.m6.1d">italic_q</annotation></semantics></math>, <math alttext="A" class="ltx_Math" display="inline" id="A4.SS1.SSS1.p3.14.m7.1"><semantics id="A4.SS1.SSS1.p3.14.m7.1a"><mi id="A4.SS1.SSS1.p3.14.m7.1.1" xref="A4.SS1.SSS1.p3.14.m7.1.1.cmml">A</mi><annotation-xml encoding="MathML-Content" id="A4.SS1.SSS1.p3.14.m7.1b"><ci id="A4.SS1.SSS1.p3.14.m7.1.1.cmml" xref="A4.SS1.SSS1.p3.14.m7.1.1">𝐴</ci></annotation-xml><annotation encoding="application/x-tex" id="A4.SS1.SSS1.p3.14.m7.1c">A</annotation><annotation encoding="application/x-llamapun" id="A4.SS1.SSS1.p3.14.m7.1d">italic_A</annotation></semantics></math> and <math alttext="R" class="ltx_Math" display="inline" id="A4.SS1.SSS1.p3.15.m8.1"><semantics id="A4.SS1.SSS1.p3.15.m8.1a"><mi id="A4.SS1.SSS1.p3.15.m8.1.1" xref="A4.SS1.SSS1.p3.15.m8.1.1.cmml">R</mi><annotation-xml encoding="MathML-Content" id="A4.SS1.SSS1.p3.15.m8.1b"><ci id="A4.SS1.SSS1.p3.15.m8.1.1.cmml" xref="A4.SS1.SSS1.p3.15.m8.1.1">𝑅</ci></annotation-xml><annotation encoding="application/x-tex" id="A4.SS1.SSS1.p3.15.m8.1c">R</annotation><annotation encoding="application/x-llamapun" id="A4.SS1.SSS1.p3.15.m8.1d">italic_R</annotation></semantics></math>. By the uniform convergence of <math alttext="\nabla\log p_{\sigma}" class="ltx_Math" display="inline" id="A4.SS1.SSS1.p3.16.m9.1"><semantics id="A4.SS1.SSS1.p3.16.m9.1a"><mrow id="A4.SS1.SSS1.p3.16.m9.1.1" xref="A4.SS1.SSS1.p3.16.m9.1.1.cmml"><mrow id="A4.SS1.SSS1.p3.16.m9.1.1.1" xref="A4.SS1.SSS1.p3.16.m9.1.1.1.cmml"><mo id="A4.SS1.SSS1.p3.16.m9.1.1.1.1" rspace="0.167em" xref="A4.SS1.SSS1.p3.16.m9.1.1.1.1.cmml">∇</mo><mi id="A4.SS1.SSS1.p3.16.m9.1.1.1.2" xref="A4.SS1.SSS1.p3.16.m9.1.1.1.2.cmml">log</mi></mrow><mo id="A4.SS1.SSS1.p3.16.m9.1.1a" lspace="0.167em" xref="A4.SS1.SSS1.p3.16.m9.1.1.cmml">⁡</mo><msub id="A4.SS1.SSS1.p3.16.m9.1.1.2" xref="A4.SS1.SSS1.p3.16.m9.1.1.2.cmml"><mi id="A4.SS1.SSS1.p3.16.m9.1.1.2.2" xref="A4.SS1.SSS1.p3.16.m9.1.1.2.2.cmml">p</mi><mi id="A4.SS1.SSS1.p3.16.m9.1.1.2.3" xref="A4.SS1.SSS1.p3.16.m9.1.1.2.3.cmml">σ</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="A4.SS1.SSS1.p3.16.m9.1b"><apply id="A4.SS1.SSS1.p3.16.m9.1.1.cmml" xref="A4.SS1.SSS1.p3.16.m9.1.1"><apply id="A4.SS1.SSS1.p3.16.m9.1.1.1.cmml" xref="A4.SS1.SSS1.p3.16.m9.1.1.1"><ci id="A4.SS1.SSS1.p3.16.m9.1.1.1.1.cmml" xref="A4.SS1.SSS1.p3.16.m9.1.1.1.1">∇</ci><log id="A4.SS1.SSS1.p3.16.m9.1.1.1.2.cmml" xref="A4.SS1.SSS1.p3.16.m9.1.1.1.2"></log></apply><apply id="A4.SS1.SSS1.p3.16.m9.1.1.2.cmml" xref="A4.SS1.SSS1.p3.16.m9.1.1.2"><csymbol cd="ambiguous" id="A4.SS1.SSS1.p3.16.m9.1.1.2.1.cmml" xref="A4.SS1.SSS1.p3.16.m9.1.1.2">subscript</csymbol><ci id="A4.SS1.SSS1.p3.16.m9.1.1.2.2.cmml" xref="A4.SS1.SSS1.p3.16.m9.1.1.2.2">𝑝</ci><ci id="A4.SS1.SSS1.p3.16.m9.1.1.2.3.cmml" xref="A4.SS1.SSS1.p3.16.m9.1.1.2.3">𝜎</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.SS1.SSS1.p3.16.m9.1c">\nabla\log p_{\sigma}</annotation><annotation encoding="application/x-llamapun" id="A4.SS1.SSS1.p3.16.m9.1d">∇ roman_log italic_p start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT</annotation></semantics></math> to <math alttext="\nabla\log p" class="ltx_Math" display="inline" id="A4.SS1.SSS1.p3.17.m10.1"><semantics id="A4.SS1.SSS1.p3.17.m10.1a"><mrow id="A4.SS1.SSS1.p3.17.m10.1.1" xref="A4.SS1.SSS1.p3.17.m10.1.1.cmml"><mrow id="A4.SS1.SSS1.p3.17.m10.1.1.1" xref="A4.SS1.SSS1.p3.17.m10.1.1.1.cmml"><mo id="A4.SS1.SSS1.p3.17.m10.1.1.1.1" rspace="0.167em" xref="A4.SS1.SSS1.p3.17.m10.1.1.1.1.cmml">∇</mo><mi id="A4.SS1.SSS1.p3.17.m10.1.1.1.2" xref="A4.SS1.SSS1.p3.17.m10.1.1.1.2.cmml">log</mi></mrow><mo id="A4.SS1.SSS1.p3.17.m10.1.1a" lspace="0.167em" xref="A4.SS1.SSS1.p3.17.m10.1.1.cmml">⁡</mo><mi id="A4.SS1.SSS1.p3.17.m10.1.1.2" xref="A4.SS1.SSS1.p3.17.m10.1.1.2.cmml">p</mi></mrow><annotation-xml encoding="MathML-Content" id="A4.SS1.SSS1.p3.17.m10.1b"><apply id="A4.SS1.SSS1.p3.17.m10.1.1.cmml" xref="A4.SS1.SSS1.p3.17.m10.1.1"><apply id="A4.SS1.SSS1.p3.17.m10.1.1.1.cmml" xref="A4.SS1.SSS1.p3.17.m10.1.1.1"><ci id="A4.SS1.SSS1.p3.17.m10.1.1.1.1.cmml" xref="A4.SS1.SSS1.p3.17.m10.1.1.1.1">∇</ci><log id="A4.SS1.SSS1.p3.17.m10.1.1.1.2.cmml" xref="A4.SS1.SSS1.p3.17.m10.1.1.1.2"></log></apply><ci id="A4.SS1.SSS1.p3.17.m10.1.1.2.cmml" xref="A4.SS1.SSS1.p3.17.m10.1.1.2">𝑝</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.SS1.SSS1.p3.17.m10.1c">\nabla\log p</annotation><annotation encoding="application/x-llamapun" id="A4.SS1.SSS1.p3.17.m10.1d">∇ roman_log italic_p</annotation></semantics></math> on <math alttext="\mathbf{L}" class="ltx_Math" display="inline" id="A4.SS1.SSS1.p3.18.m11.1"><semantics id="A4.SS1.SSS1.p3.18.m11.1a"><mi id="A4.SS1.SSS1.p3.18.m11.1.1" xref="A4.SS1.SSS1.p3.18.m11.1.1.cmml">𝐋</mi><annotation-xml encoding="MathML-Content" id="A4.SS1.SSS1.p3.18.m11.1b"><ci id="A4.SS1.SSS1.p3.18.m11.1.1.cmml" xref="A4.SS1.SSS1.p3.18.m11.1.1">𝐋</ci></annotation-xml><annotation encoding="application/x-tex" id="A4.SS1.SSS1.p3.18.m11.1c">\mathbf{L}</annotation><annotation encoding="application/x-llamapun" id="A4.SS1.SSS1.p3.18.m11.1d">bold_L</annotation></semantics></math>, there exists <math alttext="\sigma_{0}>0" class="ltx_Math" display="inline" id="A4.SS1.SSS1.p3.19.m12.1"><semantics id="A4.SS1.SSS1.p3.19.m12.1a"><mrow id="A4.SS1.SSS1.p3.19.m12.1.1" xref="A4.SS1.SSS1.p3.19.m12.1.1.cmml"><msub id="A4.SS1.SSS1.p3.19.m12.1.1.2" xref="A4.SS1.SSS1.p3.19.m12.1.1.2.cmml"><mi id="A4.SS1.SSS1.p3.19.m12.1.1.2.2" xref="A4.SS1.SSS1.p3.19.m12.1.1.2.2.cmml">σ</mi><mn id="A4.SS1.SSS1.p3.19.m12.1.1.2.3" xref="A4.SS1.SSS1.p3.19.m12.1.1.2.3.cmml">0</mn></msub><mo id="A4.SS1.SSS1.p3.19.m12.1.1.1" xref="A4.SS1.SSS1.p3.19.m12.1.1.1.cmml">></mo><mn id="A4.SS1.SSS1.p3.19.m12.1.1.3" xref="A4.SS1.SSS1.p3.19.m12.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="A4.SS1.SSS1.p3.19.m12.1b"><apply id="A4.SS1.SSS1.p3.19.m12.1.1.cmml" xref="A4.SS1.SSS1.p3.19.m12.1.1"><gt id="A4.SS1.SSS1.p3.19.m12.1.1.1.cmml" xref="A4.SS1.SSS1.p3.19.m12.1.1.1"></gt><apply id="A4.SS1.SSS1.p3.19.m12.1.1.2.cmml" xref="A4.SS1.SSS1.p3.19.m12.1.1.2"><csymbol cd="ambiguous" id="A4.SS1.SSS1.p3.19.m12.1.1.2.1.cmml" xref="A4.SS1.SSS1.p3.19.m12.1.1.2">subscript</csymbol><ci id="A4.SS1.SSS1.p3.19.m12.1.1.2.2.cmml" xref="A4.SS1.SSS1.p3.19.m12.1.1.2.2">𝜎</ci><cn id="A4.SS1.SSS1.p3.19.m12.1.1.2.3.cmml" type="integer" xref="A4.SS1.SSS1.p3.19.m12.1.1.2.3">0</cn></apply><cn id="A4.SS1.SSS1.p3.19.m12.1.1.3.cmml" type="integer" xref="A4.SS1.SSS1.p3.19.m12.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.SS1.SSS1.p3.19.m12.1c">\sigma_{0}>0</annotation><annotation encoding="application/x-llamapun" id="A4.SS1.SSS1.p3.19.m12.1d">italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT > 0</annotation></semantics></math> such that <math alttext="\forall\sigma\leq\sigma_{0}" class="ltx_Math" display="inline" id="A4.SS1.SSS1.p3.20.m13.1"><semantics id="A4.SS1.SSS1.p3.20.m13.1a"><mrow id="A4.SS1.SSS1.p3.20.m13.1.1" xref="A4.SS1.SSS1.p3.20.m13.1.1.cmml"><mrow id="A4.SS1.SSS1.p3.20.m13.1.1.2" xref="A4.SS1.SSS1.p3.20.m13.1.1.2.cmml"><mo id="A4.SS1.SSS1.p3.20.m13.1.1.2.1" rspace="0.167em" xref="A4.SS1.SSS1.p3.20.m13.1.1.2.1.cmml">∀</mo><mi id="A4.SS1.SSS1.p3.20.m13.1.1.2.2" xref="A4.SS1.SSS1.p3.20.m13.1.1.2.2.cmml">σ</mi></mrow><mo id="A4.SS1.SSS1.p3.20.m13.1.1.1" xref="A4.SS1.SSS1.p3.20.m13.1.1.1.cmml">≤</mo><msub id="A4.SS1.SSS1.p3.20.m13.1.1.3" xref="A4.SS1.SSS1.p3.20.m13.1.1.3.cmml"><mi id="A4.SS1.SSS1.p3.20.m13.1.1.3.2" xref="A4.SS1.SSS1.p3.20.m13.1.1.3.2.cmml">σ</mi><mn id="A4.SS1.SSS1.p3.20.m13.1.1.3.3" xref="A4.SS1.SSS1.p3.20.m13.1.1.3.3.cmml">0</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="A4.SS1.SSS1.p3.20.m13.1b"><apply id="A4.SS1.SSS1.p3.20.m13.1.1.cmml" xref="A4.SS1.SSS1.p3.20.m13.1.1"><leq id="A4.SS1.SSS1.p3.20.m13.1.1.1.cmml" xref="A4.SS1.SSS1.p3.20.m13.1.1.1"></leq><apply id="A4.SS1.SSS1.p3.20.m13.1.1.2.cmml" xref="A4.SS1.SSS1.p3.20.m13.1.1.2"><csymbol cd="latexml" id="A4.SS1.SSS1.p3.20.m13.1.1.2.1.cmml" xref="A4.SS1.SSS1.p3.20.m13.1.1.2.1">for-all</csymbol><ci id="A4.SS1.SSS1.p3.20.m13.1.1.2.2.cmml" xref="A4.SS1.SSS1.p3.20.m13.1.1.2.2">𝜎</ci></apply><apply id="A4.SS1.SSS1.p3.20.m13.1.1.3.cmml" xref="A4.SS1.SSS1.p3.20.m13.1.1.3"><csymbol cd="ambiguous" id="A4.SS1.SSS1.p3.20.m13.1.1.3.1.cmml" xref="A4.SS1.SSS1.p3.20.m13.1.1.3">subscript</csymbol><ci id="A4.SS1.SSS1.p3.20.m13.1.1.3.2.cmml" xref="A4.SS1.SSS1.p3.20.m13.1.1.3.2">𝜎</ci><cn id="A4.SS1.SSS1.p3.20.m13.1.1.3.3.cmml" type="integer" xref="A4.SS1.SSS1.p3.20.m13.1.1.3.3">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.SS1.SSS1.p3.20.m13.1c">\forall\sigma\leq\sigma_{0}</annotation><annotation encoding="application/x-llamapun" id="A4.SS1.SSS1.p3.20.m13.1d">∀ italic_σ ≤ italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math>, <math alttext="\sup_{\mathbf{L}}\|\nabla\log p_{\sigma}-\nabla\log p\|\leq\epsilon" class="ltx_Math" display="inline" id="A4.SS1.SSS1.p3.21.m14.1"><semantics id="A4.SS1.SSS1.p3.21.m14.1a"><mrow id="A4.SS1.SSS1.p3.21.m14.1.1" xref="A4.SS1.SSS1.p3.21.m14.1.1.cmml"><mrow id="A4.SS1.SSS1.p3.21.m14.1.1.1" xref="A4.SS1.SSS1.p3.21.m14.1.1.1.cmml"><msub id="A4.SS1.SSS1.p3.21.m14.1.1.1.2" xref="A4.SS1.SSS1.p3.21.m14.1.1.1.2.cmml"><mo id="A4.SS1.SSS1.p3.21.m14.1.1.1.2.2" xref="A4.SS1.SSS1.p3.21.m14.1.1.1.2.2.cmml">sup</mo><mi id="A4.SS1.SSS1.p3.21.m14.1.1.1.2.3" xref="A4.SS1.SSS1.p3.21.m14.1.1.1.2.3.cmml">𝐋</mi></msub><mrow id="A4.SS1.SSS1.p3.21.m14.1.1.1.1.1" xref="A4.SS1.SSS1.p3.21.m14.1.1.1.1.2.cmml"><mo id="A4.SS1.SSS1.p3.21.m14.1.1.1.1.1.2" lspace="0em" stretchy="false" xref="A4.SS1.SSS1.p3.21.m14.1.1.1.1.2.1.cmml">‖</mo><mrow id="A4.SS1.SSS1.p3.21.m14.1.1.1.1.1.1" xref="A4.SS1.SSS1.p3.21.m14.1.1.1.1.1.1.cmml"><mrow id="A4.SS1.SSS1.p3.21.m14.1.1.1.1.1.1.2" xref="A4.SS1.SSS1.p3.21.m14.1.1.1.1.1.1.2.cmml"><mrow id="A4.SS1.SSS1.p3.21.m14.1.1.1.1.1.1.2.1" xref="A4.SS1.SSS1.p3.21.m14.1.1.1.1.1.1.2.1.cmml"><mo id="A4.SS1.SSS1.p3.21.m14.1.1.1.1.1.1.2.1.1" rspace="0.167em" xref="A4.SS1.SSS1.p3.21.m14.1.1.1.1.1.1.2.1.1.cmml">∇</mo><mi id="A4.SS1.SSS1.p3.21.m14.1.1.1.1.1.1.2.1.2" xref="A4.SS1.SSS1.p3.21.m14.1.1.1.1.1.1.2.1.2.cmml">log</mi></mrow><mo id="A4.SS1.SSS1.p3.21.m14.1.1.1.1.1.1.2a" lspace="0.167em" xref="A4.SS1.SSS1.p3.21.m14.1.1.1.1.1.1.2.cmml">⁡</mo><msub id="A4.SS1.SSS1.p3.21.m14.1.1.1.1.1.1.2.2" xref="A4.SS1.SSS1.p3.21.m14.1.1.1.1.1.1.2.2.cmml"><mi id="A4.SS1.SSS1.p3.21.m14.1.1.1.1.1.1.2.2.2" xref="A4.SS1.SSS1.p3.21.m14.1.1.1.1.1.1.2.2.2.cmml">p</mi><mi id="A4.SS1.SSS1.p3.21.m14.1.1.1.1.1.1.2.2.3" xref="A4.SS1.SSS1.p3.21.m14.1.1.1.1.1.1.2.2.3.cmml">σ</mi></msub></mrow><mo id="A4.SS1.SSS1.p3.21.m14.1.1.1.1.1.1.1" xref="A4.SS1.SSS1.p3.21.m14.1.1.1.1.1.1.1.cmml">−</mo><mrow id="A4.SS1.SSS1.p3.21.m14.1.1.1.1.1.1.3" xref="A4.SS1.SSS1.p3.21.m14.1.1.1.1.1.1.3.cmml"><mrow id="A4.SS1.SSS1.p3.21.m14.1.1.1.1.1.1.3.1" xref="A4.SS1.SSS1.p3.21.m14.1.1.1.1.1.1.3.1.cmml"><mo id="A4.SS1.SSS1.p3.21.m14.1.1.1.1.1.1.3.1.1" rspace="0.167em" xref="A4.SS1.SSS1.p3.21.m14.1.1.1.1.1.1.3.1.1.cmml">∇</mo><mi id="A4.SS1.SSS1.p3.21.m14.1.1.1.1.1.1.3.1.2" xref="A4.SS1.SSS1.p3.21.m14.1.1.1.1.1.1.3.1.2.cmml">log</mi></mrow><mo id="A4.SS1.SSS1.p3.21.m14.1.1.1.1.1.1.3a" lspace="0.167em" xref="A4.SS1.SSS1.p3.21.m14.1.1.1.1.1.1.3.cmml">⁡</mo><mi id="A4.SS1.SSS1.p3.21.m14.1.1.1.1.1.1.3.2" xref="A4.SS1.SSS1.p3.21.m14.1.1.1.1.1.1.3.2.cmml">p</mi></mrow></mrow><mo id="A4.SS1.SSS1.p3.21.m14.1.1.1.1.1.3" stretchy="false" xref="A4.SS1.SSS1.p3.21.m14.1.1.1.1.2.1.cmml">‖</mo></mrow></mrow><mo id="A4.SS1.SSS1.p3.21.m14.1.1.2" xref="A4.SS1.SSS1.p3.21.m14.1.1.2.cmml">≤</mo><mi id="A4.SS1.SSS1.p3.21.m14.1.1.3" xref="A4.SS1.SSS1.p3.21.m14.1.1.3.cmml">ϵ</mi></mrow><annotation-xml encoding="MathML-Content" id="A4.SS1.SSS1.p3.21.m14.1b"><apply id="A4.SS1.SSS1.p3.21.m14.1.1.cmml" xref="A4.SS1.SSS1.p3.21.m14.1.1"><leq id="A4.SS1.SSS1.p3.21.m14.1.1.2.cmml" xref="A4.SS1.SSS1.p3.21.m14.1.1.2"></leq><apply id="A4.SS1.SSS1.p3.21.m14.1.1.1.cmml" xref="A4.SS1.SSS1.p3.21.m14.1.1.1"><apply id="A4.SS1.SSS1.p3.21.m14.1.1.1.2.cmml" xref="A4.SS1.SSS1.p3.21.m14.1.1.1.2"><csymbol cd="ambiguous" id="A4.SS1.SSS1.p3.21.m14.1.1.1.2.1.cmml" xref="A4.SS1.SSS1.p3.21.m14.1.1.1.2">subscript</csymbol><csymbol cd="latexml" id="A4.SS1.SSS1.p3.21.m14.1.1.1.2.2.cmml" xref="A4.SS1.SSS1.p3.21.m14.1.1.1.2.2">supremum</csymbol><ci id="A4.SS1.SSS1.p3.21.m14.1.1.1.2.3.cmml" xref="A4.SS1.SSS1.p3.21.m14.1.1.1.2.3">𝐋</ci></apply><apply id="A4.SS1.SSS1.p3.21.m14.1.1.1.1.2.cmml" xref="A4.SS1.SSS1.p3.21.m14.1.1.1.1.1"><csymbol cd="latexml" id="A4.SS1.SSS1.p3.21.m14.1.1.1.1.2.1.cmml" xref="A4.SS1.SSS1.p3.21.m14.1.1.1.1.1.2">norm</csymbol><apply id="A4.SS1.SSS1.p3.21.m14.1.1.1.1.1.1.cmml" xref="A4.SS1.SSS1.p3.21.m14.1.1.1.1.1.1"><minus id="A4.SS1.SSS1.p3.21.m14.1.1.1.1.1.1.1.cmml" xref="A4.SS1.SSS1.p3.21.m14.1.1.1.1.1.1.1"></minus><apply id="A4.SS1.SSS1.p3.21.m14.1.1.1.1.1.1.2.cmml" xref="A4.SS1.SSS1.p3.21.m14.1.1.1.1.1.1.2"><apply id="A4.SS1.SSS1.p3.21.m14.1.1.1.1.1.1.2.1.cmml" xref="A4.SS1.SSS1.p3.21.m14.1.1.1.1.1.1.2.1"><ci id="A4.SS1.SSS1.p3.21.m14.1.1.1.1.1.1.2.1.1.cmml" xref="A4.SS1.SSS1.p3.21.m14.1.1.1.1.1.1.2.1.1">∇</ci><log id="A4.SS1.SSS1.p3.21.m14.1.1.1.1.1.1.2.1.2.cmml" xref="A4.SS1.SSS1.p3.21.m14.1.1.1.1.1.1.2.1.2"></log></apply><apply id="A4.SS1.SSS1.p3.21.m14.1.1.1.1.1.1.2.2.cmml" xref="A4.SS1.SSS1.p3.21.m14.1.1.1.1.1.1.2.2"><csymbol cd="ambiguous" id="A4.SS1.SSS1.p3.21.m14.1.1.1.1.1.1.2.2.1.cmml" xref="A4.SS1.SSS1.p3.21.m14.1.1.1.1.1.1.2.2">subscript</csymbol><ci id="A4.SS1.SSS1.p3.21.m14.1.1.1.1.1.1.2.2.2.cmml" xref="A4.SS1.SSS1.p3.21.m14.1.1.1.1.1.1.2.2.2">𝑝</ci><ci id="A4.SS1.SSS1.p3.21.m14.1.1.1.1.1.1.2.2.3.cmml" xref="A4.SS1.SSS1.p3.21.m14.1.1.1.1.1.1.2.2.3">𝜎</ci></apply></apply><apply id="A4.SS1.SSS1.p3.21.m14.1.1.1.1.1.1.3.cmml" xref="A4.SS1.SSS1.p3.21.m14.1.1.1.1.1.1.3"><apply id="A4.SS1.SSS1.p3.21.m14.1.1.1.1.1.1.3.1.cmml" xref="A4.SS1.SSS1.p3.21.m14.1.1.1.1.1.1.3.1"><ci id="A4.SS1.SSS1.p3.21.m14.1.1.1.1.1.1.3.1.1.cmml" xref="A4.SS1.SSS1.p3.21.m14.1.1.1.1.1.1.3.1.1">∇</ci><log id="A4.SS1.SSS1.p3.21.m14.1.1.1.1.1.1.3.1.2.cmml" xref="A4.SS1.SSS1.p3.21.m14.1.1.1.1.1.1.3.1.2"></log></apply><ci id="A4.SS1.SSS1.p3.21.m14.1.1.1.1.1.1.3.2.cmml" xref="A4.SS1.SSS1.p3.21.m14.1.1.1.1.1.1.3.2">𝑝</ci></apply></apply></apply></apply><ci id="A4.SS1.SSS1.p3.21.m14.1.1.3.cmml" xref="A4.SS1.SSS1.p3.21.m14.1.1.3">italic-ϵ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.SS1.SSS1.p3.21.m14.1c">\sup_{\mathbf{L}}\|\nabla\log p_{\sigma}-\nabla\log p\|\leq\epsilon</annotation><annotation encoding="application/x-llamapun" id="A4.SS1.SSS1.p3.21.m14.1d">roman_sup start_POSTSUBSCRIPT bold_L end_POSTSUBSCRIPT ∥ ∇ roman_log italic_p start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT - ∇ roman_log italic_p ∥ ≤ italic_ϵ</annotation></semantics></math>. Then with the polynomial-exponential behavior, there exists, <math alttext="\sigma_{1}>0" class="ltx_Math" display="inline" id="A4.SS1.SSS1.p3.22.m15.1"><semantics id="A4.SS1.SSS1.p3.22.m15.1a"><mrow id="A4.SS1.SSS1.p3.22.m15.1.1" xref="A4.SS1.SSS1.p3.22.m15.1.1.cmml"><msub id="A4.SS1.SSS1.p3.22.m15.1.1.2" xref="A4.SS1.SSS1.p3.22.m15.1.1.2.cmml"><mi id="A4.SS1.SSS1.p3.22.m15.1.1.2.2" xref="A4.SS1.SSS1.p3.22.m15.1.1.2.2.cmml">σ</mi><mn id="A4.SS1.SSS1.p3.22.m15.1.1.2.3" xref="A4.SS1.SSS1.p3.22.m15.1.1.2.3.cmml">1</mn></msub><mo id="A4.SS1.SSS1.p3.22.m15.1.1.1" xref="A4.SS1.SSS1.p3.22.m15.1.1.1.cmml">></mo><mn id="A4.SS1.SSS1.p3.22.m15.1.1.3" xref="A4.SS1.SSS1.p3.22.m15.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="A4.SS1.SSS1.p3.22.m15.1b"><apply id="A4.SS1.SSS1.p3.22.m15.1.1.cmml" xref="A4.SS1.SSS1.p3.22.m15.1.1"><gt id="A4.SS1.SSS1.p3.22.m15.1.1.1.cmml" xref="A4.SS1.SSS1.p3.22.m15.1.1.1"></gt><apply id="A4.SS1.SSS1.p3.22.m15.1.1.2.cmml" xref="A4.SS1.SSS1.p3.22.m15.1.1.2"><csymbol cd="ambiguous" id="A4.SS1.SSS1.p3.22.m15.1.1.2.1.cmml" xref="A4.SS1.SSS1.p3.22.m15.1.1.2">subscript</csymbol><ci id="A4.SS1.SSS1.p3.22.m15.1.1.2.2.cmml" xref="A4.SS1.SSS1.p3.22.m15.1.1.2.2">𝜎</ci><cn id="A4.SS1.SSS1.p3.22.m15.1.1.2.3.cmml" type="integer" xref="A4.SS1.SSS1.p3.22.m15.1.1.2.3">1</cn></apply><cn id="A4.SS1.SSS1.p3.22.m15.1.1.3.cmml" type="integer" xref="A4.SS1.SSS1.p3.22.m15.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.SS1.SSS1.p3.22.m15.1c">\sigma_{1}>0</annotation><annotation encoding="application/x-llamapun" id="A4.SS1.SSS1.p3.22.m15.1d">italic_σ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT > 0</annotation></semantics></math>, such that <math alttext="\forall\sigma\leq\sigma_{1}" class="ltx_Math" display="inline" id="A4.SS1.SSS1.p3.23.m16.1"><semantics id="A4.SS1.SSS1.p3.23.m16.1a"><mrow id="A4.SS1.SSS1.p3.23.m16.1.1" xref="A4.SS1.SSS1.p3.23.m16.1.1.cmml"><mrow id="A4.SS1.SSS1.p3.23.m16.1.1.2" xref="A4.SS1.SSS1.p3.23.m16.1.1.2.cmml"><mo id="A4.SS1.SSS1.p3.23.m16.1.1.2.1" rspace="0.167em" xref="A4.SS1.SSS1.p3.23.m16.1.1.2.1.cmml">∀</mo><mi id="A4.SS1.SSS1.p3.23.m16.1.1.2.2" xref="A4.SS1.SSS1.p3.23.m16.1.1.2.2.cmml">σ</mi></mrow><mo id="A4.SS1.SSS1.p3.23.m16.1.1.1" xref="A4.SS1.SSS1.p3.23.m16.1.1.1.cmml">≤</mo><msub id="A4.SS1.SSS1.p3.23.m16.1.1.3" xref="A4.SS1.SSS1.p3.23.m16.1.1.3.cmml"><mi id="A4.SS1.SSS1.p3.23.m16.1.1.3.2" xref="A4.SS1.SSS1.p3.23.m16.1.1.3.2.cmml">σ</mi><mn id="A4.SS1.SSS1.p3.23.m16.1.1.3.3" xref="A4.SS1.SSS1.p3.23.m16.1.1.3.3.cmml">1</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="A4.SS1.SSS1.p3.23.m16.1b"><apply id="A4.SS1.SSS1.p3.23.m16.1.1.cmml" xref="A4.SS1.SSS1.p3.23.m16.1.1"><leq id="A4.SS1.SSS1.p3.23.m16.1.1.1.cmml" xref="A4.SS1.SSS1.p3.23.m16.1.1.1"></leq><apply id="A4.SS1.SSS1.p3.23.m16.1.1.2.cmml" xref="A4.SS1.SSS1.p3.23.m16.1.1.2"><csymbol cd="latexml" id="A4.SS1.SSS1.p3.23.m16.1.1.2.1.cmml" 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id="A4.SS1.SSS1.p3.24.m17.9c">(\sigma^{\beta}C_{r,B,R}+C_{q,A,R})\exp{(-\frac{\mu^{2}}{2\sigma^{2}})}\leq\epsilon</annotation><annotation encoding="application/x-llamapun" id="A4.SS1.SSS1.p3.24.m17.9d">( italic_σ start_POSTSUPERSCRIPT italic_β end_POSTSUPERSCRIPT italic_C start_POSTSUBSCRIPT italic_r , italic_B , italic_R end_POSTSUBSCRIPT + italic_C start_POSTSUBSCRIPT italic_q , italic_A , italic_R end_POSTSUBSCRIPT ) roman_exp ( - divide start_ARG italic_μ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG 2 italic_σ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG ) ≤ italic_ϵ</annotation></semantics></math>. Finally, for <math alttext="\sigma\leq\min(\sigma_{0},\sigma_{1})" class="ltx_Math" display="inline" id="A4.SS1.SSS1.p3.25.m18.3"><semantics id="A4.SS1.SSS1.p3.25.m18.3a"><mrow id="A4.SS1.SSS1.p3.25.m18.3.3" xref="A4.SS1.SSS1.p3.25.m18.3.3.cmml"><mi id="A4.SS1.SSS1.p3.25.m18.3.3.4" xref="A4.SS1.SSS1.p3.25.m18.3.3.4.cmml">σ</mi><mo id="A4.SS1.SSS1.p3.25.m18.3.3.3" xref="A4.SS1.SSS1.p3.25.m18.3.3.3.cmml">≤</mo><mrow id="A4.SS1.SSS1.p3.25.m18.3.3.2.2" xref="A4.SS1.SSS1.p3.25.m18.3.3.2.3.cmml"><mi id="A4.SS1.SSS1.p3.25.m18.1.1" xref="A4.SS1.SSS1.p3.25.m18.1.1.cmml">min</mi><mo id="A4.SS1.SSS1.p3.25.m18.3.3.2.2a" xref="A4.SS1.SSS1.p3.25.m18.3.3.2.3.cmml">⁡</mo><mrow id="A4.SS1.SSS1.p3.25.m18.3.3.2.2.2" xref="A4.SS1.SSS1.p3.25.m18.3.3.2.3.cmml"><mo id="A4.SS1.SSS1.p3.25.m18.3.3.2.2.2.3" stretchy="false" xref="A4.SS1.SSS1.p3.25.m18.3.3.2.3.cmml">(</mo><msub id="A4.SS1.SSS1.p3.25.m18.2.2.1.1.1.1" xref="A4.SS1.SSS1.p3.25.m18.2.2.1.1.1.1.cmml"><mi id="A4.SS1.SSS1.p3.25.m18.2.2.1.1.1.1.2" xref="A4.SS1.SSS1.p3.25.m18.2.2.1.1.1.1.2.cmml">σ</mi><mn id="A4.SS1.SSS1.p3.25.m18.2.2.1.1.1.1.3" xref="A4.SS1.SSS1.p3.25.m18.2.2.1.1.1.1.3.cmml">0</mn></msub><mo id="A4.SS1.SSS1.p3.25.m18.3.3.2.2.2.4" xref="A4.SS1.SSS1.p3.25.m18.3.3.2.3.cmml">,</mo><msub id="A4.SS1.SSS1.p3.25.m18.3.3.2.2.2.2" xref="A4.SS1.SSS1.p3.25.m18.3.3.2.2.2.2.cmml"><mi id="A4.SS1.SSS1.p3.25.m18.3.3.2.2.2.2.2" xref="A4.SS1.SSS1.p3.25.m18.3.3.2.2.2.2.2.cmml">σ</mi><mn id="A4.SS1.SSS1.p3.25.m18.3.3.2.2.2.2.3" xref="A4.SS1.SSS1.p3.25.m18.3.3.2.2.2.2.3.cmml">1</mn></msub><mo id="A4.SS1.SSS1.p3.25.m18.3.3.2.2.2.5" stretchy="false" xref="A4.SS1.SSS1.p3.25.m18.3.3.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="A4.SS1.SSS1.p3.25.m18.3b"><apply id="A4.SS1.SSS1.p3.25.m18.3.3.cmml" xref="A4.SS1.SSS1.p3.25.m18.3.3"><leq id="A4.SS1.SSS1.p3.25.m18.3.3.3.cmml" xref="A4.SS1.SSS1.p3.25.m18.3.3.3"></leq><ci id="A4.SS1.SSS1.p3.25.m18.3.3.4.cmml" xref="A4.SS1.SSS1.p3.25.m18.3.3.4">𝜎</ci><apply id="A4.SS1.SSS1.p3.25.m18.3.3.2.3.cmml" xref="A4.SS1.SSS1.p3.25.m18.3.3.2.2"><min id="A4.SS1.SSS1.p3.25.m18.1.1.cmml" xref="A4.SS1.SSS1.p3.25.m18.1.1"></min><apply id="A4.SS1.SSS1.p3.25.m18.2.2.1.1.1.1.cmml" xref="A4.SS1.SSS1.p3.25.m18.2.2.1.1.1.1"><csymbol cd="ambiguous" id="A4.SS1.SSS1.p3.25.m18.2.2.1.1.1.1.1.cmml" xref="A4.SS1.SSS1.p3.25.m18.2.2.1.1.1.1">subscript</csymbol><ci id="A4.SS1.SSS1.p3.25.m18.2.2.1.1.1.1.2.cmml" xref="A4.SS1.SSS1.p3.25.m18.2.2.1.1.1.1.2">𝜎</ci><cn id="A4.SS1.SSS1.p3.25.m18.2.2.1.1.1.1.3.cmml" type="integer" xref="A4.SS1.SSS1.p3.25.m18.2.2.1.1.1.1.3">0</cn></apply><apply id="A4.SS1.SSS1.p3.25.m18.3.3.2.2.2.2.cmml" xref="A4.SS1.SSS1.p3.25.m18.3.3.2.2.2.2"><csymbol cd="ambiguous" id="A4.SS1.SSS1.p3.25.m18.3.3.2.2.2.2.1.cmml" xref="A4.SS1.SSS1.p3.25.m18.3.3.2.2.2.2">subscript</csymbol><ci id="A4.SS1.SSS1.p3.25.m18.3.3.2.2.2.2.2.cmml" xref="A4.SS1.SSS1.p3.25.m18.3.3.2.2.2.2.2">𝜎</ci><cn id="A4.SS1.SSS1.p3.25.m18.3.3.2.2.2.2.3.cmml" type="integer" xref="A4.SS1.SSS1.p3.25.m18.3.3.2.2.2.2.3">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.SS1.SSS1.p3.25.m18.3c">\sigma\leq\min(\sigma_{0},\sigma_{1})</annotation><annotation encoding="application/x-llamapun" id="A4.SS1.SSS1.p3.25.m18.3d">italic_σ ≤ roman_min ( italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_σ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT )</annotation></semantics></math>,</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A8.EGx50"> <tbody id="A4.Ex84"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\|\nabla\mathcal{R}_{\sigma}({\mathbf{x}})+\nabla\log p({\mathbf{% x}})\|\leq 3\epsilon." class="ltx_Math" display="inline" id="A4.Ex84.m1.3"><semantics id="A4.Ex84.m1.3a"><mrow id="A4.Ex84.m1.3.3.1" xref="A4.Ex84.m1.3.3.1.1.cmml"><mrow id="A4.Ex84.m1.3.3.1.1" xref="A4.Ex84.m1.3.3.1.1.cmml"><mrow id="A4.Ex84.m1.3.3.1.1.1.1" xref="A4.Ex84.m1.3.3.1.1.1.2.cmml"><mo id="A4.Ex84.m1.3.3.1.1.1.1.2" stretchy="false" xref="A4.Ex84.m1.3.3.1.1.1.2.1.cmml">‖</mo><mrow id="A4.Ex84.m1.3.3.1.1.1.1.1" xref="A4.Ex84.m1.3.3.1.1.1.1.1.cmml"><mrow id="A4.Ex84.m1.3.3.1.1.1.1.1.2" xref="A4.Ex84.m1.3.3.1.1.1.1.1.2.cmml"><mrow id="A4.Ex84.m1.3.3.1.1.1.1.1.2.2" xref="A4.Ex84.m1.3.3.1.1.1.1.1.2.2.cmml"><mo id="A4.Ex84.m1.3.3.1.1.1.1.1.2.2.1" rspace="0.167em" xref="A4.Ex84.m1.3.3.1.1.1.1.1.2.2.1.cmml">∇</mo><msub id="A4.Ex84.m1.3.3.1.1.1.1.1.2.2.2" xref="A4.Ex84.m1.3.3.1.1.1.1.1.2.2.2.cmml"><mi class="ltx_font_mathcaligraphic" id="A4.Ex84.m1.3.3.1.1.1.1.1.2.2.2.2" xref="A4.Ex84.m1.3.3.1.1.1.1.1.2.2.2.2.cmml">ℛ</mi><mi id="A4.Ex84.m1.3.3.1.1.1.1.1.2.2.2.3" xref="A4.Ex84.m1.3.3.1.1.1.1.1.2.2.2.3.cmml">σ</mi></msub></mrow><mo id="A4.Ex84.m1.3.3.1.1.1.1.1.2.1" xref="A4.Ex84.m1.3.3.1.1.1.1.1.2.1.cmml">⁢</mo><mrow id="A4.Ex84.m1.3.3.1.1.1.1.1.2.3.2" 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xref="A4.Ex84.m1.3.3.1.1.1.1.1.3.2.2.cmml">p</mi></mrow><mo id="A4.Ex84.m1.3.3.1.1.1.1.1.3.1" xref="A4.Ex84.m1.3.3.1.1.1.1.1.3.1.cmml">⁢</mo><mrow id="A4.Ex84.m1.3.3.1.1.1.1.1.3.3.2" xref="A4.Ex84.m1.3.3.1.1.1.1.1.3.cmml"><mo id="A4.Ex84.m1.3.3.1.1.1.1.1.3.3.2.1" stretchy="false" xref="A4.Ex84.m1.3.3.1.1.1.1.1.3.cmml">(</mo><mi id="A4.Ex84.m1.2.2" xref="A4.Ex84.m1.2.2.cmml">𝐱</mi><mo id="A4.Ex84.m1.3.3.1.1.1.1.1.3.3.2.2" stretchy="false" xref="A4.Ex84.m1.3.3.1.1.1.1.1.3.cmml">)</mo></mrow></mrow></mrow><mo id="A4.Ex84.m1.3.3.1.1.1.1.3" stretchy="false" xref="A4.Ex84.m1.3.3.1.1.1.2.1.cmml">‖</mo></mrow><mo id="A4.Ex84.m1.3.3.1.1.2" xref="A4.Ex84.m1.3.3.1.1.2.cmml">≤</mo><mrow id="A4.Ex84.m1.3.3.1.1.3" xref="A4.Ex84.m1.3.3.1.1.3.cmml"><mn id="A4.Ex84.m1.3.3.1.1.3.2" xref="A4.Ex84.m1.3.3.1.1.3.2.cmml">3</mn><mo id="A4.Ex84.m1.3.3.1.1.3.1" xref="A4.Ex84.m1.3.3.1.1.3.1.cmml">⁢</mo><mi id="A4.Ex84.m1.3.3.1.1.3.3" xref="A4.Ex84.m1.3.3.1.1.3.3.cmml">ϵ</mi></mrow></mrow><mo id="A4.Ex84.m1.3.3.1.2" lspace="0em" xref="A4.Ex84.m1.3.3.1.1.cmml">.</mo></mrow><annotation-xml encoding="MathML-Content" id="A4.Ex84.m1.3b"><apply id="A4.Ex84.m1.3.3.1.1.cmml" xref="A4.Ex84.m1.3.3.1"><leq id="A4.Ex84.m1.3.3.1.1.2.cmml" xref="A4.Ex84.m1.3.3.1.1.2"></leq><apply id="A4.Ex84.m1.3.3.1.1.1.2.cmml" xref="A4.Ex84.m1.3.3.1.1.1.1"><csymbol cd="latexml" id="A4.Ex84.m1.3.3.1.1.1.2.1.cmml" xref="A4.Ex84.m1.3.3.1.1.1.1.2">norm</csymbol><apply id="A4.Ex84.m1.3.3.1.1.1.1.1.cmml" xref="A4.Ex84.m1.3.3.1.1.1.1.1"><plus id="A4.Ex84.m1.3.3.1.1.1.1.1.1.cmml" xref="A4.Ex84.m1.3.3.1.1.1.1.1.1"></plus><apply id="A4.Ex84.m1.3.3.1.1.1.1.1.2.cmml" xref="A4.Ex84.m1.3.3.1.1.1.1.1.2"><times id="A4.Ex84.m1.3.3.1.1.1.1.1.2.1.cmml" xref="A4.Ex84.m1.3.3.1.1.1.1.1.2.1"></times><apply id="A4.Ex84.m1.3.3.1.1.1.1.1.2.2.cmml" xref="A4.Ex84.m1.3.3.1.1.1.1.1.2.2"><ci id="A4.Ex84.m1.3.3.1.1.1.1.1.2.2.1.cmml" xref="A4.Ex84.m1.3.3.1.1.1.1.1.2.2.1">∇</ci><apply id="A4.Ex84.m1.3.3.1.1.1.1.1.2.2.2.cmml" 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id="A4.Ex84.m1.3.3.1.1.1.1.1.3.2.2.cmml" xref="A4.Ex84.m1.3.3.1.1.1.1.1.3.2.2">𝑝</ci></apply><ci id="A4.Ex84.m1.2.2.cmml" xref="A4.Ex84.m1.2.2">𝐱</ci></apply></apply></apply><apply id="A4.Ex84.m1.3.3.1.1.3.cmml" xref="A4.Ex84.m1.3.3.1.1.3"><times id="A4.Ex84.m1.3.3.1.1.3.1.cmml" xref="A4.Ex84.m1.3.3.1.1.3.1"></times><cn id="A4.Ex84.m1.3.3.1.1.3.2.cmml" type="integer" xref="A4.Ex84.m1.3.3.1.1.3.2">3</cn><ci id="A4.Ex84.m1.3.3.1.1.3.3.cmml" xref="A4.Ex84.m1.3.3.1.1.3.3">italic-ϵ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.Ex84.m1.3c">\displaystyle\|\nabla\mathcal{R}_{\sigma}({\mathbf{x}})+\nabla\log p({\mathbf{% x}})\|\leq 3\epsilon.</annotation><annotation encoding="application/x-llamapun" id="A4.Ex84.m1.3d">∥ ∇ caligraphic_R start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ( bold_x ) + ∇ roman_log italic_p ( bold_x ) ∥ ≤ 3 italic_ϵ .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> </div> </section> <section class="ltx_subsubsection" id="A4.SS1.SSS2"> <h4 class="ltx_title ltx_title_subsubsection"> <span class="ltx_tag ltx_tag_subsubsection">D.1.2 </span>Proof of Proposition <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#S3.Thmproposition2" title="Proposition 3.2. ‣ 3.1 Asymptotics of critical points when 𝜎→0 ‣ 3 Convergence Analysis ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_tag">3.2</span></a> </h4> <div class="ltx_para" id="A4.SS1.SSS2.p1"> <p class="ltx_p" id="A4.SS1.SSS2.p1.18">We will follow the same structure as the proof of <cite class="ltx_cite ltx_citemacro_citep">(Laumont et al., <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#bib.bib37" title="">2022b</a>, Proposition 1)</cite> and use our Proposition <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#S3.Thmproposition1" title="Proposition 3.1. ‣ 3.1 Asymptotics of critical points when 𝜎→0 ‣ 3 Convergence Analysis ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_tag">3.1</span></a>. For <math alttext="{\mathbf{x}}\in\mathbf{S}_{\mathbf{K}}" class="ltx_Math" display="inline" id="A4.SS1.SSS2.p1.1.m1.1"><semantics id="A4.SS1.SSS2.p1.1.m1.1a"><mrow id="A4.SS1.SSS2.p1.1.m1.1.1" xref="A4.SS1.SSS2.p1.1.m1.1.1.cmml"><mi id="A4.SS1.SSS2.p1.1.m1.1.1.2" xref="A4.SS1.SSS2.p1.1.m1.1.1.2.cmml">𝐱</mi><mo id="A4.SS1.SSS2.p1.1.m1.1.1.1" xref="A4.SS1.SSS2.p1.1.m1.1.1.1.cmml">∈</mo><msub id="A4.SS1.SSS2.p1.1.m1.1.1.3" xref="A4.SS1.SSS2.p1.1.m1.1.1.3.cmml"><mi id="A4.SS1.SSS2.p1.1.m1.1.1.3.2" xref="A4.SS1.SSS2.p1.1.m1.1.1.3.2.cmml">𝐒</mi><mi id="A4.SS1.SSS2.p1.1.m1.1.1.3.3" xref="A4.SS1.SSS2.p1.1.m1.1.1.3.3.cmml">𝐊</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="A4.SS1.SSS2.p1.1.m1.1b"><apply id="A4.SS1.SSS2.p1.1.m1.1.1.cmml" xref="A4.SS1.SSS2.p1.1.m1.1.1"><in id="A4.SS1.SSS2.p1.1.m1.1.1.1.cmml" xref="A4.SS1.SSS2.p1.1.m1.1.1.1"></in><ci id="A4.SS1.SSS2.p1.1.m1.1.1.2.cmml" xref="A4.SS1.SSS2.p1.1.m1.1.1.2">𝐱</ci><apply id="A4.SS1.SSS2.p1.1.m1.1.1.3.cmml" xref="A4.SS1.SSS2.p1.1.m1.1.1.3"><csymbol cd="ambiguous" id="A4.SS1.SSS2.p1.1.m1.1.1.3.1.cmml" xref="A4.SS1.SSS2.p1.1.m1.1.1.3">subscript</csymbol><ci id="A4.SS1.SSS2.p1.1.m1.1.1.3.2.cmml" xref="A4.SS1.SSS2.p1.1.m1.1.1.3.2">𝐒</ci><ci id="A4.SS1.SSS2.p1.1.m1.1.1.3.3.cmml" xref="A4.SS1.SSS2.p1.1.m1.1.1.3.3">𝐊</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.SS1.SSS2.p1.1.m1.1c">{\mathbf{x}}\in\mathbf{S}_{\mathbf{K}}</annotation><annotation encoding="application/x-llamapun" id="A4.SS1.SSS2.p1.1.m1.1d">bold_x ∈ bold_S start_POSTSUBSCRIPT bold_K end_POSTSUBSCRIPT</annotation></semantics></math>, by definition there exist <math alttext="(\sigma_{n})_{n\in{\mathbb{N}}}\in\mathbf{E}" class="ltx_Math" display="inline" id="A4.SS1.SSS2.p1.2.m2.1"><semantics id="A4.SS1.SSS2.p1.2.m2.1a"><mrow id="A4.SS1.SSS2.p1.2.m2.1.1" xref="A4.SS1.SSS2.p1.2.m2.1.1.cmml"><msub id="A4.SS1.SSS2.p1.2.m2.1.1.1" xref="A4.SS1.SSS2.p1.2.m2.1.1.1.cmml"><mrow id="A4.SS1.SSS2.p1.2.m2.1.1.1.1.1" xref="A4.SS1.SSS2.p1.2.m2.1.1.1.1.1.1.cmml"><mo id="A4.SS1.SSS2.p1.2.m2.1.1.1.1.1.2" stretchy="false" xref="A4.SS1.SSS2.p1.2.m2.1.1.1.1.1.1.cmml">(</mo><msub id="A4.SS1.SSS2.p1.2.m2.1.1.1.1.1.1" xref="A4.SS1.SSS2.p1.2.m2.1.1.1.1.1.1.cmml"><mi id="A4.SS1.SSS2.p1.2.m2.1.1.1.1.1.1.2" xref="A4.SS1.SSS2.p1.2.m2.1.1.1.1.1.1.2.cmml">σ</mi><mi id="A4.SS1.SSS2.p1.2.m2.1.1.1.1.1.1.3" xref="A4.SS1.SSS2.p1.2.m2.1.1.1.1.1.1.3.cmml">n</mi></msub><mo id="A4.SS1.SSS2.p1.2.m2.1.1.1.1.1.3" stretchy="false" xref="A4.SS1.SSS2.p1.2.m2.1.1.1.1.1.1.cmml">)</mo></mrow><mrow id="A4.SS1.SSS2.p1.2.m2.1.1.1.3" xref="A4.SS1.SSS2.p1.2.m2.1.1.1.3.cmml"><mi id="A4.SS1.SSS2.p1.2.m2.1.1.1.3.2" xref="A4.SS1.SSS2.p1.2.m2.1.1.1.3.2.cmml">n</mi><mo id="A4.SS1.SSS2.p1.2.m2.1.1.1.3.1" xref="A4.SS1.SSS2.p1.2.m2.1.1.1.3.1.cmml">∈</mo><mi id="A4.SS1.SSS2.p1.2.m2.1.1.1.3.3" xref="A4.SS1.SSS2.p1.2.m2.1.1.1.3.3.cmml">ℕ</mi></mrow></msub><mo id="A4.SS1.SSS2.p1.2.m2.1.1.2" xref="A4.SS1.SSS2.p1.2.m2.1.1.2.cmml">∈</mo><mi id="A4.SS1.SSS2.p1.2.m2.1.1.3" xref="A4.SS1.SSS2.p1.2.m2.1.1.3.cmml">𝐄</mi></mrow><annotation-xml encoding="MathML-Content" id="A4.SS1.SSS2.p1.2.m2.1b"><apply id="A4.SS1.SSS2.p1.2.m2.1.1.cmml" xref="A4.SS1.SSS2.p1.2.m2.1.1"><in id="A4.SS1.SSS2.p1.2.m2.1.1.2.cmml" xref="A4.SS1.SSS2.p1.2.m2.1.1.2"></in><apply id="A4.SS1.SSS2.p1.2.m2.1.1.1.cmml" xref="A4.SS1.SSS2.p1.2.m2.1.1.1"><csymbol cd="ambiguous" id="A4.SS1.SSS2.p1.2.m2.1.1.1.2.cmml" xref="A4.SS1.SSS2.p1.2.m2.1.1.1">subscript</csymbol><apply id="A4.SS1.SSS2.p1.2.m2.1.1.1.1.1.1.cmml" xref="A4.SS1.SSS2.p1.2.m2.1.1.1.1.1"><csymbol cd="ambiguous" id="A4.SS1.SSS2.p1.2.m2.1.1.1.1.1.1.1.cmml" xref="A4.SS1.SSS2.p1.2.m2.1.1.1.1.1">subscript</csymbol><ci id="A4.SS1.SSS2.p1.2.m2.1.1.1.1.1.1.2.cmml" xref="A4.SS1.SSS2.p1.2.m2.1.1.1.1.1.1.2">𝜎</ci><ci id="A4.SS1.SSS2.p1.2.m2.1.1.1.1.1.1.3.cmml" xref="A4.SS1.SSS2.p1.2.m2.1.1.1.1.1.1.3">𝑛</ci></apply><apply id="A4.SS1.SSS2.p1.2.m2.1.1.1.3.cmml" xref="A4.SS1.SSS2.p1.2.m2.1.1.1.3"><in id="A4.SS1.SSS2.p1.2.m2.1.1.1.3.1.cmml" xref="A4.SS1.SSS2.p1.2.m2.1.1.1.3.1"></in><ci id="A4.SS1.SSS2.p1.2.m2.1.1.1.3.2.cmml" xref="A4.SS1.SSS2.p1.2.m2.1.1.1.3.2">𝑛</ci><ci id="A4.SS1.SSS2.p1.2.m2.1.1.1.3.3.cmml" xref="A4.SS1.SSS2.p1.2.m2.1.1.1.3.3">ℕ</ci></apply></apply><ci id="A4.SS1.SSS2.p1.2.m2.1.1.3.cmml" xref="A4.SS1.SSS2.p1.2.m2.1.1.3">𝐄</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.SS1.SSS2.p1.2.m2.1c">(\sigma_{n})_{n\in{\mathbb{N}}}\in\mathbf{E}</annotation><annotation encoding="application/x-llamapun" id="A4.SS1.SSS2.p1.2.m2.1d">( italic_σ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ) start_POSTSUBSCRIPT italic_n ∈ blackboard_N end_POSTSUBSCRIPT ∈ bold_E</annotation></semantics></math>, <math alttext="{\mathbf{x}}_{n}\in\mathbf{S}_{\sigma_{n},\mathbf{K}}" class="ltx_Math" display="inline" id="A4.SS1.SSS2.p1.3.m3.2"><semantics id="A4.SS1.SSS2.p1.3.m3.2a"><mrow id="A4.SS1.SSS2.p1.3.m3.2.3" xref="A4.SS1.SSS2.p1.3.m3.2.3.cmml"><msub id="A4.SS1.SSS2.p1.3.m3.2.3.2" xref="A4.SS1.SSS2.p1.3.m3.2.3.2.cmml"><mi id="A4.SS1.SSS2.p1.3.m3.2.3.2.2" xref="A4.SS1.SSS2.p1.3.m3.2.3.2.2.cmml">𝐱</mi><mi id="A4.SS1.SSS2.p1.3.m3.2.3.2.3" xref="A4.SS1.SSS2.p1.3.m3.2.3.2.3.cmml">n</mi></msub><mo id="A4.SS1.SSS2.p1.3.m3.2.3.1" xref="A4.SS1.SSS2.p1.3.m3.2.3.1.cmml">∈</mo><msub id="A4.SS1.SSS2.p1.3.m3.2.3.3" xref="A4.SS1.SSS2.p1.3.m3.2.3.3.cmml"><mi id="A4.SS1.SSS2.p1.3.m3.2.3.3.2" xref="A4.SS1.SSS2.p1.3.m3.2.3.3.2.cmml">𝐒</mi><mrow id="A4.SS1.SSS2.p1.3.m3.2.2.2.2" xref="A4.SS1.SSS2.p1.3.m3.2.2.2.3.cmml"><msub id="A4.SS1.SSS2.p1.3.m3.2.2.2.2.1" xref="A4.SS1.SSS2.p1.3.m3.2.2.2.2.1.cmml"><mi id="A4.SS1.SSS2.p1.3.m3.2.2.2.2.1.2" xref="A4.SS1.SSS2.p1.3.m3.2.2.2.2.1.2.cmml">σ</mi><mi id="A4.SS1.SSS2.p1.3.m3.2.2.2.2.1.3" xref="A4.SS1.SSS2.p1.3.m3.2.2.2.2.1.3.cmml">n</mi></msub><mo id="A4.SS1.SSS2.p1.3.m3.2.2.2.2.2" xref="A4.SS1.SSS2.p1.3.m3.2.2.2.3.cmml">,</mo><mi id="A4.SS1.SSS2.p1.3.m3.1.1.1.1" xref="A4.SS1.SSS2.p1.3.m3.1.1.1.1.cmml">𝐊</mi></mrow></msub></mrow><annotation-xml encoding="MathML-Content" id="A4.SS1.SSS2.p1.3.m3.2b"><apply id="A4.SS1.SSS2.p1.3.m3.2.3.cmml" xref="A4.SS1.SSS2.p1.3.m3.2.3"><in id="A4.SS1.SSS2.p1.3.m3.2.3.1.cmml" xref="A4.SS1.SSS2.p1.3.m3.2.3.1"></in><apply id="A4.SS1.SSS2.p1.3.m3.2.3.2.cmml" xref="A4.SS1.SSS2.p1.3.m3.2.3.2"><csymbol cd="ambiguous" id="A4.SS1.SSS2.p1.3.m3.2.3.2.1.cmml" xref="A4.SS1.SSS2.p1.3.m3.2.3.2">subscript</csymbol><ci id="A4.SS1.SSS2.p1.3.m3.2.3.2.2.cmml" xref="A4.SS1.SSS2.p1.3.m3.2.3.2.2">𝐱</ci><ci id="A4.SS1.SSS2.p1.3.m3.2.3.2.3.cmml" xref="A4.SS1.SSS2.p1.3.m3.2.3.2.3">𝑛</ci></apply><apply id="A4.SS1.SSS2.p1.3.m3.2.3.3.cmml" xref="A4.SS1.SSS2.p1.3.m3.2.3.3"><csymbol cd="ambiguous" id="A4.SS1.SSS2.p1.3.m3.2.3.3.1.cmml" xref="A4.SS1.SSS2.p1.3.m3.2.3.3">subscript</csymbol><ci id="A4.SS1.SSS2.p1.3.m3.2.3.3.2.cmml" xref="A4.SS1.SSS2.p1.3.m3.2.3.3.2">𝐒</ci><list id="A4.SS1.SSS2.p1.3.m3.2.2.2.3.cmml" xref="A4.SS1.SSS2.p1.3.m3.2.2.2.2"><apply id="A4.SS1.SSS2.p1.3.m3.2.2.2.2.1.cmml" xref="A4.SS1.SSS2.p1.3.m3.2.2.2.2.1"><csymbol cd="ambiguous" id="A4.SS1.SSS2.p1.3.m3.2.2.2.2.1.1.cmml" xref="A4.SS1.SSS2.p1.3.m3.2.2.2.2.1">subscript</csymbol><ci id="A4.SS1.SSS2.p1.3.m3.2.2.2.2.1.2.cmml" xref="A4.SS1.SSS2.p1.3.m3.2.2.2.2.1.2">𝜎</ci><ci id="A4.SS1.SSS2.p1.3.m3.2.2.2.2.1.3.cmml" xref="A4.SS1.SSS2.p1.3.m3.2.2.2.2.1.3">𝑛</ci></apply><ci id="A4.SS1.SSS2.p1.3.m3.1.1.1.1.cmml" xref="A4.SS1.SSS2.p1.3.m3.1.1.1.1">𝐊</ci></list></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.SS1.SSS2.p1.3.m3.2c">{\mathbf{x}}_{n}\in\mathbf{S}_{\sigma_{n},\mathbf{K}}</annotation><annotation encoding="application/x-llamapun" id="A4.SS1.SSS2.p1.3.m3.2d">bold_x start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ∈ bold_S start_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT , bold_K end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="\phi:{\mathbb{N}}\to{\mathbb{N}}" class="ltx_Math" display="inline" id="A4.SS1.SSS2.p1.4.m4.1"><semantics id="A4.SS1.SSS2.p1.4.m4.1a"><mrow id="A4.SS1.SSS2.p1.4.m4.1.1" xref="A4.SS1.SSS2.p1.4.m4.1.1.cmml"><mi id="A4.SS1.SSS2.p1.4.m4.1.1.2" xref="A4.SS1.SSS2.p1.4.m4.1.1.2.cmml">ϕ</mi><mo id="A4.SS1.SSS2.p1.4.m4.1.1.1" lspace="0.278em" rspace="0.278em" xref="A4.SS1.SSS2.p1.4.m4.1.1.1.cmml">:</mo><mrow id="A4.SS1.SSS2.p1.4.m4.1.1.3" xref="A4.SS1.SSS2.p1.4.m4.1.1.3.cmml"><mi id="A4.SS1.SSS2.p1.4.m4.1.1.3.2" xref="A4.SS1.SSS2.p1.4.m4.1.1.3.2.cmml">ℕ</mi><mo id="A4.SS1.SSS2.p1.4.m4.1.1.3.1" stretchy="false" xref="A4.SS1.SSS2.p1.4.m4.1.1.3.1.cmml">→</mo><mi id="A4.SS1.SSS2.p1.4.m4.1.1.3.3" xref="A4.SS1.SSS2.p1.4.m4.1.1.3.3.cmml">ℕ</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="A4.SS1.SSS2.p1.4.m4.1b"><apply id="A4.SS1.SSS2.p1.4.m4.1.1.cmml" xref="A4.SS1.SSS2.p1.4.m4.1.1"><ci id="A4.SS1.SSS2.p1.4.m4.1.1.1.cmml" xref="A4.SS1.SSS2.p1.4.m4.1.1.1">:</ci><ci id="A4.SS1.SSS2.p1.4.m4.1.1.2.cmml" xref="A4.SS1.SSS2.p1.4.m4.1.1.2">italic-ϕ</ci><apply id="A4.SS1.SSS2.p1.4.m4.1.1.3.cmml" xref="A4.SS1.SSS2.p1.4.m4.1.1.3"><ci id="A4.SS1.SSS2.p1.4.m4.1.1.3.1.cmml" xref="A4.SS1.SSS2.p1.4.m4.1.1.3.1">→</ci><ci id="A4.SS1.SSS2.p1.4.m4.1.1.3.2.cmml" xref="A4.SS1.SSS2.p1.4.m4.1.1.3.2">ℕ</ci><ci id="A4.SS1.SSS2.p1.4.m4.1.1.3.3.cmml" xref="A4.SS1.SSS2.p1.4.m4.1.1.3.3">ℕ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.SS1.SSS2.p1.4.m4.1c">\phi:{\mathbb{N}}\to{\mathbb{N}}</annotation><annotation encoding="application/x-llamapun" id="A4.SS1.SSS2.p1.4.m4.1d">italic_ϕ : blackboard_N → blackboard_N</annotation></semantics></math> strictly increasing such that <math alttext="{\mathbf{x}}_{\phi(n)}\to{\mathbf{x}}" class="ltx_Math" display="inline" id="A4.SS1.SSS2.p1.5.m5.1"><semantics id="A4.SS1.SSS2.p1.5.m5.1a"><mrow id="A4.SS1.SSS2.p1.5.m5.1.2" xref="A4.SS1.SSS2.p1.5.m5.1.2.cmml"><msub id="A4.SS1.SSS2.p1.5.m5.1.2.2" xref="A4.SS1.SSS2.p1.5.m5.1.2.2.cmml"><mi id="A4.SS1.SSS2.p1.5.m5.1.2.2.2" xref="A4.SS1.SSS2.p1.5.m5.1.2.2.2.cmml">𝐱</mi><mrow id="A4.SS1.SSS2.p1.5.m5.1.1.1" xref="A4.SS1.SSS2.p1.5.m5.1.1.1.cmml"><mi id="A4.SS1.SSS2.p1.5.m5.1.1.1.3" xref="A4.SS1.SSS2.p1.5.m5.1.1.1.3.cmml">ϕ</mi><mo id="A4.SS1.SSS2.p1.5.m5.1.1.1.2" xref="A4.SS1.SSS2.p1.5.m5.1.1.1.2.cmml">⁢</mo><mrow id="A4.SS1.SSS2.p1.5.m5.1.1.1.4.2" xref="A4.SS1.SSS2.p1.5.m5.1.1.1.cmml"><mo id="A4.SS1.SSS2.p1.5.m5.1.1.1.4.2.1" stretchy="false" xref="A4.SS1.SSS2.p1.5.m5.1.1.1.cmml">(</mo><mi id="A4.SS1.SSS2.p1.5.m5.1.1.1.1" xref="A4.SS1.SSS2.p1.5.m5.1.1.1.1.cmml">n</mi><mo id="A4.SS1.SSS2.p1.5.m5.1.1.1.4.2.2" stretchy="false" xref="A4.SS1.SSS2.p1.5.m5.1.1.1.cmml">)</mo></mrow></mrow></msub><mo id="A4.SS1.SSS2.p1.5.m5.1.2.1" stretchy="false" xref="A4.SS1.SSS2.p1.5.m5.1.2.1.cmml">→</mo><mi id="A4.SS1.SSS2.p1.5.m5.1.2.3" xref="A4.SS1.SSS2.p1.5.m5.1.2.3.cmml">𝐱</mi></mrow><annotation-xml encoding="MathML-Content" id="A4.SS1.SSS2.p1.5.m5.1b"><apply id="A4.SS1.SSS2.p1.5.m5.1.2.cmml" xref="A4.SS1.SSS2.p1.5.m5.1.2"><ci id="A4.SS1.SSS2.p1.5.m5.1.2.1.cmml" xref="A4.SS1.SSS2.p1.5.m5.1.2.1">→</ci><apply id="A4.SS1.SSS2.p1.5.m5.1.2.2.cmml" xref="A4.SS1.SSS2.p1.5.m5.1.2.2"><csymbol cd="ambiguous" id="A4.SS1.SSS2.p1.5.m5.1.2.2.1.cmml" xref="A4.SS1.SSS2.p1.5.m5.1.2.2">subscript</csymbol><ci id="A4.SS1.SSS2.p1.5.m5.1.2.2.2.cmml" xref="A4.SS1.SSS2.p1.5.m5.1.2.2.2">𝐱</ci><apply id="A4.SS1.SSS2.p1.5.m5.1.1.1.cmml" xref="A4.SS1.SSS2.p1.5.m5.1.1.1"><times id="A4.SS1.SSS2.p1.5.m5.1.1.1.2.cmml" xref="A4.SS1.SSS2.p1.5.m5.1.1.1.2"></times><ci id="A4.SS1.SSS2.p1.5.m5.1.1.1.3.cmml" xref="A4.SS1.SSS2.p1.5.m5.1.1.1.3">italic-ϕ</ci><ci id="A4.SS1.SSS2.p1.5.m5.1.1.1.1.cmml" xref="A4.SS1.SSS2.p1.5.m5.1.1.1.1">𝑛</ci></apply></apply><ci id="A4.SS1.SSS2.p1.5.m5.1.2.3.cmml" xref="A4.SS1.SSS2.p1.5.m5.1.2.3">𝐱</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.SS1.SSS2.p1.5.m5.1c">{\mathbf{x}}_{\phi(n)}\to{\mathbf{x}}</annotation><annotation encoding="application/x-llamapun" id="A4.SS1.SSS2.p1.5.m5.1d">bold_x start_POSTSUBSCRIPT italic_ϕ ( italic_n ) end_POSTSUBSCRIPT → bold_x</annotation></semantics></math>. By Proposition <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#S3.Thmproposition1" title="Proposition 3.1. ‣ 3.1 Asymptotics of critical points when 𝜎→0 ‣ 3 Convergence Analysis ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_tag">3.1</span></a>, <math alttext="\nabla\mathcal{R}_{\sigma}" class="ltx_Math" display="inline" id="A4.SS1.SSS2.p1.6.m6.1"><semantics id="A4.SS1.SSS2.p1.6.m6.1a"><mrow id="A4.SS1.SSS2.p1.6.m6.1.1" xref="A4.SS1.SSS2.p1.6.m6.1.1.cmml"><mo id="A4.SS1.SSS2.p1.6.m6.1.1.1" rspace="0.167em" xref="A4.SS1.SSS2.p1.6.m6.1.1.1.cmml">∇</mo><msub id="A4.SS1.SSS2.p1.6.m6.1.1.2" xref="A4.SS1.SSS2.p1.6.m6.1.1.2.cmml"><mi class="ltx_font_mathcaligraphic" id="A4.SS1.SSS2.p1.6.m6.1.1.2.2" xref="A4.SS1.SSS2.p1.6.m6.1.1.2.2.cmml">ℛ</mi><mi id="A4.SS1.SSS2.p1.6.m6.1.1.2.3" xref="A4.SS1.SSS2.p1.6.m6.1.1.2.3.cmml">σ</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="A4.SS1.SSS2.p1.6.m6.1b"><apply id="A4.SS1.SSS2.p1.6.m6.1.1.cmml" xref="A4.SS1.SSS2.p1.6.m6.1.1"><ci id="A4.SS1.SSS2.p1.6.m6.1.1.1.cmml" xref="A4.SS1.SSS2.p1.6.m6.1.1.1">∇</ci><apply id="A4.SS1.SSS2.p1.6.m6.1.1.2.cmml" xref="A4.SS1.SSS2.p1.6.m6.1.1.2"><csymbol cd="ambiguous" id="A4.SS1.SSS2.p1.6.m6.1.1.2.1.cmml" xref="A4.SS1.SSS2.p1.6.m6.1.1.2">subscript</csymbol><ci id="A4.SS1.SSS2.p1.6.m6.1.1.2.2.cmml" xref="A4.SS1.SSS2.p1.6.m6.1.1.2.2">ℛ</ci><ci id="A4.SS1.SSS2.p1.6.m6.1.1.2.3.cmml" xref="A4.SS1.SSS2.p1.6.m6.1.1.2.3">𝜎</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.SS1.SSS2.p1.6.m6.1c">\nabla\mathcal{R}_{\sigma}</annotation><annotation encoding="application/x-llamapun" id="A4.SS1.SSS2.p1.6.m6.1d">∇ caligraphic_R start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT</annotation></semantics></math> converges uniformly to <math alttext="-\nabla\log p" class="ltx_Math" display="inline" id="A4.SS1.SSS2.p1.7.m7.1"><semantics id="A4.SS1.SSS2.p1.7.m7.1a"><mrow id="A4.SS1.SSS2.p1.7.m7.1.1" xref="A4.SS1.SSS2.p1.7.m7.1.1.cmml"><mo id="A4.SS1.SSS2.p1.7.m7.1.1a" rspace="0.167em" xref="A4.SS1.SSS2.p1.7.m7.1.1.cmml">−</mo><mrow id="A4.SS1.SSS2.p1.7.m7.1.1.2" xref="A4.SS1.SSS2.p1.7.m7.1.1.2.cmml"><mrow id="A4.SS1.SSS2.p1.7.m7.1.1.2.1" xref="A4.SS1.SSS2.p1.7.m7.1.1.2.1.cmml"><mo id="A4.SS1.SSS2.p1.7.m7.1.1.2.1.1" rspace="0.167em" xref="A4.SS1.SSS2.p1.7.m7.1.1.2.1.1.cmml">∇</mo><mi id="A4.SS1.SSS2.p1.7.m7.1.1.2.1.2" xref="A4.SS1.SSS2.p1.7.m7.1.1.2.1.2.cmml">log</mi></mrow><mo id="A4.SS1.SSS2.p1.7.m7.1.1.2a" lspace="0.167em" xref="A4.SS1.SSS2.p1.7.m7.1.1.2.cmml">⁡</mo><mi id="A4.SS1.SSS2.p1.7.m7.1.1.2.2" xref="A4.SS1.SSS2.p1.7.m7.1.1.2.2.cmml">p</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="A4.SS1.SSS2.p1.7.m7.1b"><apply id="A4.SS1.SSS2.p1.7.m7.1.1.cmml" xref="A4.SS1.SSS2.p1.7.m7.1.1"><minus id="A4.SS1.SSS2.p1.7.m7.1.1.1.cmml" xref="A4.SS1.SSS2.p1.7.m7.1.1"></minus><apply id="A4.SS1.SSS2.p1.7.m7.1.1.2.cmml" xref="A4.SS1.SSS2.p1.7.m7.1.1.2"><apply id="A4.SS1.SSS2.p1.7.m7.1.1.2.1.cmml" xref="A4.SS1.SSS2.p1.7.m7.1.1.2.1"><ci id="A4.SS1.SSS2.p1.7.m7.1.1.2.1.1.cmml" xref="A4.SS1.SSS2.p1.7.m7.1.1.2.1.1">∇</ci><log id="A4.SS1.SSS2.p1.7.m7.1.1.2.1.2.cmml" xref="A4.SS1.SSS2.p1.7.m7.1.1.2.1.2"></log></apply><ci id="A4.SS1.SSS2.p1.7.m7.1.1.2.2.cmml" xref="A4.SS1.SSS2.p1.7.m7.1.1.2.2">𝑝</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.SS1.SSS2.p1.7.m7.1c">-\nabla\log p</annotation><annotation encoding="application/x-llamapun" id="A4.SS1.SSS2.p1.7.m7.1d">- ∇ roman_log italic_p</annotation></semantics></math> on <math alttext="\mathbf{K}" class="ltx_Math" display="inline" id="A4.SS1.SSS2.p1.8.m8.1"><semantics id="A4.SS1.SSS2.p1.8.m8.1a"><mi id="A4.SS1.SSS2.p1.8.m8.1.1" xref="A4.SS1.SSS2.p1.8.m8.1.1.cmml">𝐊</mi><annotation-xml encoding="MathML-Content" id="A4.SS1.SSS2.p1.8.m8.1b"><ci id="A4.SS1.SSS2.p1.8.m8.1.1.cmml" xref="A4.SS1.SSS2.p1.8.m8.1.1">𝐊</ci></annotation-xml><annotation encoding="application/x-tex" id="A4.SS1.SSS2.p1.8.m8.1c">\mathbf{K}</annotation><annotation encoding="application/x-llamapun" id="A4.SS1.SSS2.p1.8.m8.1d">bold_K</annotation></semantics></math>. So <math alttext="\nabla\mathcal{F}(\cdot,{\mathbf{y}})+\alpha\nabla\mathcal{R}_{\sigma}" class="ltx_Math" display="inline" id="A4.SS1.SSS2.p1.9.m9.2"><semantics id="A4.SS1.SSS2.p1.9.m9.2a"><mrow id="A4.SS1.SSS2.p1.9.m9.2.3" xref="A4.SS1.SSS2.p1.9.m9.2.3.cmml"><mrow id="A4.SS1.SSS2.p1.9.m9.2.3.2" xref="A4.SS1.SSS2.p1.9.m9.2.3.2.cmml"><mrow id="A4.SS1.SSS2.p1.9.m9.2.3.2.2" xref="A4.SS1.SSS2.p1.9.m9.2.3.2.2.cmml"><mo id="A4.SS1.SSS2.p1.9.m9.2.3.2.2.1" rspace="0.167em" xref="A4.SS1.SSS2.p1.9.m9.2.3.2.2.1.cmml">∇</mo><mi class="ltx_font_mathcaligraphic" id="A4.SS1.SSS2.p1.9.m9.2.3.2.2.2" xref="A4.SS1.SSS2.p1.9.m9.2.3.2.2.2.cmml">ℱ</mi></mrow><mo id="A4.SS1.SSS2.p1.9.m9.2.3.2.1" xref="A4.SS1.SSS2.p1.9.m9.2.3.2.1.cmml">⁢</mo><mrow id="A4.SS1.SSS2.p1.9.m9.2.3.2.3.2" xref="A4.SS1.SSS2.p1.9.m9.2.3.2.3.1.cmml"><mo id="A4.SS1.SSS2.p1.9.m9.2.3.2.3.2.1" stretchy="false" xref="A4.SS1.SSS2.p1.9.m9.2.3.2.3.1.cmml">(</mo><mo id="A4.SS1.SSS2.p1.9.m9.1.1" lspace="0em" rspace="0em" xref="A4.SS1.SSS2.p1.9.m9.1.1.cmml">⋅</mo><mo id="A4.SS1.SSS2.p1.9.m9.2.3.2.3.2.2" xref="A4.SS1.SSS2.p1.9.m9.2.3.2.3.1.cmml">,</mo><mi id="A4.SS1.SSS2.p1.9.m9.2.2" xref="A4.SS1.SSS2.p1.9.m9.2.2.cmml">𝐲</mi><mo id="A4.SS1.SSS2.p1.9.m9.2.3.2.3.2.3" stretchy="false" xref="A4.SS1.SSS2.p1.9.m9.2.3.2.3.1.cmml">)</mo></mrow></mrow><mo id="A4.SS1.SSS2.p1.9.m9.2.3.1" xref="A4.SS1.SSS2.p1.9.m9.2.3.1.cmml">+</mo><mrow id="A4.SS1.SSS2.p1.9.m9.2.3.3" xref="A4.SS1.SSS2.p1.9.m9.2.3.3.cmml"><mi id="A4.SS1.SSS2.p1.9.m9.2.3.3.2" xref="A4.SS1.SSS2.p1.9.m9.2.3.3.2.cmml">α</mi><mo id="A4.SS1.SSS2.p1.9.m9.2.3.3.1" lspace="0.167em" xref="A4.SS1.SSS2.p1.9.m9.2.3.3.1.cmml">⁢</mo><mrow id="A4.SS1.SSS2.p1.9.m9.2.3.3.3" xref="A4.SS1.SSS2.p1.9.m9.2.3.3.3.cmml"><mo id="A4.SS1.SSS2.p1.9.m9.2.3.3.3.1" rspace="0.167em" xref="A4.SS1.SSS2.p1.9.m9.2.3.3.3.1.cmml">∇</mo><msub id="A4.SS1.SSS2.p1.9.m9.2.3.3.3.2" xref="A4.SS1.SSS2.p1.9.m9.2.3.3.3.2.cmml"><mi class="ltx_font_mathcaligraphic" id="A4.SS1.SSS2.p1.9.m9.2.3.3.3.2.2" xref="A4.SS1.SSS2.p1.9.m9.2.3.3.3.2.2.cmml">ℛ</mi><mi id="A4.SS1.SSS2.p1.9.m9.2.3.3.3.2.3" xref="A4.SS1.SSS2.p1.9.m9.2.3.3.3.2.3.cmml">σ</mi></msub></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="A4.SS1.SSS2.p1.9.m9.2b"><apply id="A4.SS1.SSS2.p1.9.m9.2.3.cmml" xref="A4.SS1.SSS2.p1.9.m9.2.3"><plus id="A4.SS1.SSS2.p1.9.m9.2.3.1.cmml" xref="A4.SS1.SSS2.p1.9.m9.2.3.1"></plus><apply id="A4.SS1.SSS2.p1.9.m9.2.3.2.cmml" xref="A4.SS1.SSS2.p1.9.m9.2.3.2"><times id="A4.SS1.SSS2.p1.9.m9.2.3.2.1.cmml" xref="A4.SS1.SSS2.p1.9.m9.2.3.2.1"></times><apply id="A4.SS1.SSS2.p1.9.m9.2.3.2.2.cmml" xref="A4.SS1.SSS2.p1.9.m9.2.3.2.2"><ci id="A4.SS1.SSS2.p1.9.m9.2.3.2.2.1.cmml" xref="A4.SS1.SSS2.p1.9.m9.2.3.2.2.1">∇</ci><ci id="A4.SS1.SSS2.p1.9.m9.2.3.2.2.2.cmml" xref="A4.SS1.SSS2.p1.9.m9.2.3.2.2.2">ℱ</ci></apply><interval closure="open" id="A4.SS1.SSS2.p1.9.m9.2.3.2.3.1.cmml" xref="A4.SS1.SSS2.p1.9.m9.2.3.2.3.2"><ci id="A4.SS1.SSS2.p1.9.m9.1.1.cmml" xref="A4.SS1.SSS2.p1.9.m9.1.1">⋅</ci><ci id="A4.SS1.SSS2.p1.9.m9.2.2.cmml" xref="A4.SS1.SSS2.p1.9.m9.2.2">𝐲</ci></interval></apply><apply id="A4.SS1.SSS2.p1.9.m9.2.3.3.cmml" xref="A4.SS1.SSS2.p1.9.m9.2.3.3"><times id="A4.SS1.SSS2.p1.9.m9.2.3.3.1.cmml" xref="A4.SS1.SSS2.p1.9.m9.2.3.3.1"></times><ci id="A4.SS1.SSS2.p1.9.m9.2.3.3.2.cmml" xref="A4.SS1.SSS2.p1.9.m9.2.3.3.2">𝛼</ci><apply id="A4.SS1.SSS2.p1.9.m9.2.3.3.3.cmml" xref="A4.SS1.SSS2.p1.9.m9.2.3.3.3"><ci id="A4.SS1.SSS2.p1.9.m9.2.3.3.3.1.cmml" xref="A4.SS1.SSS2.p1.9.m9.2.3.3.3.1">∇</ci><apply id="A4.SS1.SSS2.p1.9.m9.2.3.3.3.2.cmml" xref="A4.SS1.SSS2.p1.9.m9.2.3.3.3.2"><csymbol cd="ambiguous" id="A4.SS1.SSS2.p1.9.m9.2.3.3.3.2.1.cmml" xref="A4.SS1.SSS2.p1.9.m9.2.3.3.3.2">subscript</csymbol><ci id="A4.SS1.SSS2.p1.9.m9.2.3.3.3.2.2.cmml" xref="A4.SS1.SSS2.p1.9.m9.2.3.3.3.2.2">ℛ</ci><ci id="A4.SS1.SSS2.p1.9.m9.2.3.3.3.2.3.cmml" xref="A4.SS1.SSS2.p1.9.m9.2.3.3.3.2.3">𝜎</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.SS1.SSS2.p1.9.m9.2c">\nabla\mathcal{F}(\cdot,{\mathbf{y}})+\alpha\nabla\mathcal{R}_{\sigma}</annotation><annotation encoding="application/x-llamapun" id="A4.SS1.SSS2.p1.9.m9.2d">∇ caligraphic_F ( ⋅ , bold_y ) + italic_α ∇ caligraphic_R start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT</annotation></semantics></math> converges uniformly to <math alttext="\nabla\mathcal{F}(\cdot,{\mathbf{y}})-\alpha\nabla\log p=\nabla\mathcal{J}" class="ltx_Math" display="inline" id="A4.SS1.SSS2.p1.10.m10.2"><semantics id="A4.SS1.SSS2.p1.10.m10.2a"><mrow id="A4.SS1.SSS2.p1.10.m10.2.3" xref="A4.SS1.SSS2.p1.10.m10.2.3.cmml"><mrow id="A4.SS1.SSS2.p1.10.m10.2.3.2" xref="A4.SS1.SSS2.p1.10.m10.2.3.2.cmml"><mrow id="A4.SS1.SSS2.p1.10.m10.2.3.2.2" xref="A4.SS1.SSS2.p1.10.m10.2.3.2.2.cmml"><mrow id="A4.SS1.SSS2.p1.10.m10.2.3.2.2.2" xref="A4.SS1.SSS2.p1.10.m10.2.3.2.2.2.cmml"><mo id="A4.SS1.SSS2.p1.10.m10.2.3.2.2.2.1" rspace="0.167em" xref="A4.SS1.SSS2.p1.10.m10.2.3.2.2.2.1.cmml">∇</mo><mi class="ltx_font_mathcaligraphic" id="A4.SS1.SSS2.p1.10.m10.2.3.2.2.2.2" xref="A4.SS1.SSS2.p1.10.m10.2.3.2.2.2.2.cmml">ℱ</mi></mrow><mo id="A4.SS1.SSS2.p1.10.m10.2.3.2.2.1" xref="A4.SS1.SSS2.p1.10.m10.2.3.2.2.1.cmml">⁢</mo><mrow id="A4.SS1.SSS2.p1.10.m10.2.3.2.2.3.2" xref="A4.SS1.SSS2.p1.10.m10.2.3.2.2.3.1.cmml"><mo id="A4.SS1.SSS2.p1.10.m10.2.3.2.2.3.2.1" stretchy="false" xref="A4.SS1.SSS2.p1.10.m10.2.3.2.2.3.1.cmml">(</mo><mo id="A4.SS1.SSS2.p1.10.m10.1.1" lspace="0em" rspace="0em" xref="A4.SS1.SSS2.p1.10.m10.1.1.cmml">⋅</mo><mo id="A4.SS1.SSS2.p1.10.m10.2.3.2.2.3.2.2" xref="A4.SS1.SSS2.p1.10.m10.2.3.2.2.3.1.cmml">,</mo><mi id="A4.SS1.SSS2.p1.10.m10.2.2" xref="A4.SS1.SSS2.p1.10.m10.2.2.cmml">𝐲</mi><mo id="A4.SS1.SSS2.p1.10.m10.2.3.2.2.3.2.3" stretchy="false" xref="A4.SS1.SSS2.p1.10.m10.2.3.2.2.3.1.cmml">)</mo></mrow></mrow><mo id="A4.SS1.SSS2.p1.10.m10.2.3.2.1" xref="A4.SS1.SSS2.p1.10.m10.2.3.2.1.cmml">−</mo><mrow id="A4.SS1.SSS2.p1.10.m10.2.3.2.3" xref="A4.SS1.SSS2.p1.10.m10.2.3.2.3.cmml"><mi id="A4.SS1.SSS2.p1.10.m10.2.3.2.3.2" xref="A4.SS1.SSS2.p1.10.m10.2.3.2.3.2.cmml">α</mi><mo id="A4.SS1.SSS2.p1.10.m10.2.3.2.3.1" lspace="0.167em" xref="A4.SS1.SSS2.p1.10.m10.2.3.2.3.1.cmml">⁢</mo><mrow id="A4.SS1.SSS2.p1.10.m10.2.3.2.3.3" xref="A4.SS1.SSS2.p1.10.m10.2.3.2.3.3.cmml"><mrow id="A4.SS1.SSS2.p1.10.m10.2.3.2.3.3.1" xref="A4.SS1.SSS2.p1.10.m10.2.3.2.3.3.1.cmml"><mo id="A4.SS1.SSS2.p1.10.m10.2.3.2.3.3.1.1" rspace="0.167em" xref="A4.SS1.SSS2.p1.10.m10.2.3.2.3.3.1.1.cmml">∇</mo><mi id="A4.SS1.SSS2.p1.10.m10.2.3.2.3.3.1.2" xref="A4.SS1.SSS2.p1.10.m10.2.3.2.3.3.1.2.cmml">log</mi></mrow><mo id="A4.SS1.SSS2.p1.10.m10.2.3.2.3.3a" lspace="0.167em" xref="A4.SS1.SSS2.p1.10.m10.2.3.2.3.3.cmml">⁡</mo><mi id="A4.SS1.SSS2.p1.10.m10.2.3.2.3.3.2" xref="A4.SS1.SSS2.p1.10.m10.2.3.2.3.3.2.cmml">p</mi></mrow></mrow></mrow><mo id="A4.SS1.SSS2.p1.10.m10.2.3.1" xref="A4.SS1.SSS2.p1.10.m10.2.3.1.cmml">=</mo><mrow id="A4.SS1.SSS2.p1.10.m10.2.3.3" xref="A4.SS1.SSS2.p1.10.m10.2.3.3.cmml"><mo id="A4.SS1.SSS2.p1.10.m10.2.3.3.1" rspace="0.167em" xref="A4.SS1.SSS2.p1.10.m10.2.3.3.1.cmml">∇</mo><mi class="ltx_font_mathcaligraphic" id="A4.SS1.SSS2.p1.10.m10.2.3.3.2" xref="A4.SS1.SSS2.p1.10.m10.2.3.3.2.cmml">𝒥</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="A4.SS1.SSS2.p1.10.m10.2b"><apply id="A4.SS1.SSS2.p1.10.m10.2.3.cmml" xref="A4.SS1.SSS2.p1.10.m10.2.3"><eq id="A4.SS1.SSS2.p1.10.m10.2.3.1.cmml" xref="A4.SS1.SSS2.p1.10.m10.2.3.1"></eq><apply id="A4.SS1.SSS2.p1.10.m10.2.3.2.cmml" xref="A4.SS1.SSS2.p1.10.m10.2.3.2"><minus id="A4.SS1.SSS2.p1.10.m10.2.3.2.1.cmml" xref="A4.SS1.SSS2.p1.10.m10.2.3.2.1"></minus><apply id="A4.SS1.SSS2.p1.10.m10.2.3.2.2.cmml" xref="A4.SS1.SSS2.p1.10.m10.2.3.2.2"><times id="A4.SS1.SSS2.p1.10.m10.2.3.2.2.1.cmml" xref="A4.SS1.SSS2.p1.10.m10.2.3.2.2.1"></times><apply id="A4.SS1.SSS2.p1.10.m10.2.3.2.2.2.cmml" xref="A4.SS1.SSS2.p1.10.m10.2.3.2.2.2"><ci id="A4.SS1.SSS2.p1.10.m10.2.3.2.2.2.1.cmml" xref="A4.SS1.SSS2.p1.10.m10.2.3.2.2.2.1">∇</ci><ci id="A4.SS1.SSS2.p1.10.m10.2.3.2.2.2.2.cmml" xref="A4.SS1.SSS2.p1.10.m10.2.3.2.2.2.2">ℱ</ci></apply><interval closure="open" id="A4.SS1.SSS2.p1.10.m10.2.3.2.2.3.1.cmml" xref="A4.SS1.SSS2.p1.10.m10.2.3.2.2.3.2"><ci id="A4.SS1.SSS2.p1.10.m10.1.1.cmml" xref="A4.SS1.SSS2.p1.10.m10.1.1">⋅</ci><ci id="A4.SS1.SSS2.p1.10.m10.2.2.cmml" xref="A4.SS1.SSS2.p1.10.m10.2.2">𝐲</ci></interval></apply><apply id="A4.SS1.SSS2.p1.10.m10.2.3.2.3.cmml" xref="A4.SS1.SSS2.p1.10.m10.2.3.2.3"><times id="A4.SS1.SSS2.p1.10.m10.2.3.2.3.1.cmml" xref="A4.SS1.SSS2.p1.10.m10.2.3.2.3.1"></times><ci id="A4.SS1.SSS2.p1.10.m10.2.3.2.3.2.cmml" xref="A4.SS1.SSS2.p1.10.m10.2.3.2.3.2">𝛼</ci><apply id="A4.SS1.SSS2.p1.10.m10.2.3.2.3.3.cmml" xref="A4.SS1.SSS2.p1.10.m10.2.3.2.3.3"><apply id="A4.SS1.SSS2.p1.10.m10.2.3.2.3.3.1.cmml" xref="A4.SS1.SSS2.p1.10.m10.2.3.2.3.3.1"><ci id="A4.SS1.SSS2.p1.10.m10.2.3.2.3.3.1.1.cmml" xref="A4.SS1.SSS2.p1.10.m10.2.3.2.3.3.1.1">∇</ci><log id="A4.SS1.SSS2.p1.10.m10.2.3.2.3.3.1.2.cmml" xref="A4.SS1.SSS2.p1.10.m10.2.3.2.3.3.1.2"></log></apply><ci id="A4.SS1.SSS2.p1.10.m10.2.3.2.3.3.2.cmml" xref="A4.SS1.SSS2.p1.10.m10.2.3.2.3.3.2">𝑝</ci></apply></apply></apply><apply id="A4.SS1.SSS2.p1.10.m10.2.3.3.cmml" xref="A4.SS1.SSS2.p1.10.m10.2.3.3"><ci id="A4.SS1.SSS2.p1.10.m10.2.3.3.1.cmml" xref="A4.SS1.SSS2.p1.10.m10.2.3.3.1">∇</ci><ci id="A4.SS1.SSS2.p1.10.m10.2.3.3.2.cmml" xref="A4.SS1.SSS2.p1.10.m10.2.3.3.2">𝒥</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.SS1.SSS2.p1.10.m10.2c">\nabla\mathcal{F}(\cdot,{\mathbf{y}})-\alpha\nabla\log p=\nabla\mathcal{J}</annotation><annotation encoding="application/x-llamapun" id="A4.SS1.SSS2.p1.10.m10.2d">∇ caligraphic_F ( ⋅ , bold_y ) - italic_α ∇ roman_log italic_p = ∇ caligraphic_J</annotation></semantics></math> on <math alttext="\mathbf{K}" class="ltx_Math" display="inline" id="A4.SS1.SSS2.p1.11.m11.1"><semantics id="A4.SS1.SSS2.p1.11.m11.1a"><mi id="A4.SS1.SSS2.p1.11.m11.1.1" xref="A4.SS1.SSS2.p1.11.m11.1.1.cmml">𝐊</mi><annotation-xml encoding="MathML-Content" id="A4.SS1.SSS2.p1.11.m11.1b"><ci id="A4.SS1.SSS2.p1.11.m11.1.1.cmml" xref="A4.SS1.SSS2.p1.11.m11.1.1">𝐊</ci></annotation-xml><annotation encoding="application/x-tex" id="A4.SS1.SSS2.p1.11.m11.1c">\mathbf{K}</annotation><annotation encoding="application/x-llamapun" id="A4.SS1.SSS2.p1.11.m11.1d">bold_K</annotation></semantics></math>. Because <math alttext="\forall n\in{\mathbb{N}},{\mathbf{x}}_{\phi(n)}\in\mathbf{K}" class="ltx_Math" display="inline" id="A4.SS1.SSS2.p1.12.m12.3"><semantics id="A4.SS1.SSS2.p1.12.m12.3a"><mrow id="A4.SS1.SSS2.p1.12.m12.3.3.2" xref="A4.SS1.SSS2.p1.12.m12.3.3.3.cmml"><mrow id="A4.SS1.SSS2.p1.12.m12.2.2.1.1" xref="A4.SS1.SSS2.p1.12.m12.2.2.1.1.cmml"><mrow id="A4.SS1.SSS2.p1.12.m12.2.2.1.1.2" xref="A4.SS1.SSS2.p1.12.m12.2.2.1.1.2.cmml"><mo id="A4.SS1.SSS2.p1.12.m12.2.2.1.1.2.1" rspace="0.167em" xref="A4.SS1.SSS2.p1.12.m12.2.2.1.1.2.1.cmml">∀</mo><mi id="A4.SS1.SSS2.p1.12.m12.2.2.1.1.2.2" xref="A4.SS1.SSS2.p1.12.m12.2.2.1.1.2.2.cmml">n</mi></mrow><mo id="A4.SS1.SSS2.p1.12.m12.2.2.1.1.1" xref="A4.SS1.SSS2.p1.12.m12.2.2.1.1.1.cmml">∈</mo><mi id="A4.SS1.SSS2.p1.12.m12.2.2.1.1.3" xref="A4.SS1.SSS2.p1.12.m12.2.2.1.1.3.cmml">ℕ</mi></mrow><mo id="A4.SS1.SSS2.p1.12.m12.3.3.2.3" xref="A4.SS1.SSS2.p1.12.m12.3.3.3a.cmml">,</mo><mrow id="A4.SS1.SSS2.p1.12.m12.3.3.2.2" xref="A4.SS1.SSS2.p1.12.m12.3.3.2.2.cmml"><msub id="A4.SS1.SSS2.p1.12.m12.3.3.2.2.2" xref="A4.SS1.SSS2.p1.12.m12.3.3.2.2.2.cmml"><mi id="A4.SS1.SSS2.p1.12.m12.3.3.2.2.2.2" xref="A4.SS1.SSS2.p1.12.m12.3.3.2.2.2.2.cmml">𝐱</mi><mrow id="A4.SS1.SSS2.p1.12.m12.1.1.1" xref="A4.SS1.SSS2.p1.12.m12.1.1.1.cmml"><mi id="A4.SS1.SSS2.p1.12.m12.1.1.1.3" xref="A4.SS1.SSS2.p1.12.m12.1.1.1.3.cmml">ϕ</mi><mo id="A4.SS1.SSS2.p1.12.m12.1.1.1.2" xref="A4.SS1.SSS2.p1.12.m12.1.1.1.2.cmml">⁢</mo><mrow id="A4.SS1.SSS2.p1.12.m12.1.1.1.4.2" xref="A4.SS1.SSS2.p1.12.m12.1.1.1.cmml"><mo id="A4.SS1.SSS2.p1.12.m12.1.1.1.4.2.1" stretchy="false" xref="A4.SS1.SSS2.p1.12.m12.1.1.1.cmml">(</mo><mi id="A4.SS1.SSS2.p1.12.m12.1.1.1.1" xref="A4.SS1.SSS2.p1.12.m12.1.1.1.1.cmml">n</mi><mo id="A4.SS1.SSS2.p1.12.m12.1.1.1.4.2.2" stretchy="false" xref="A4.SS1.SSS2.p1.12.m12.1.1.1.cmml">)</mo></mrow></mrow></msub><mo id="A4.SS1.SSS2.p1.12.m12.3.3.2.2.1" xref="A4.SS1.SSS2.p1.12.m12.3.3.2.2.1.cmml">∈</mo><mi id="A4.SS1.SSS2.p1.12.m12.3.3.2.2.3" xref="A4.SS1.SSS2.p1.12.m12.3.3.2.2.3.cmml">𝐊</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="A4.SS1.SSS2.p1.12.m12.3b"><apply id="A4.SS1.SSS2.p1.12.m12.3.3.3.cmml" xref="A4.SS1.SSS2.p1.12.m12.3.3.2"><csymbol cd="ambiguous" id="A4.SS1.SSS2.p1.12.m12.3.3.3a.cmml" xref="A4.SS1.SSS2.p1.12.m12.3.3.2.3">formulae-sequence</csymbol><apply id="A4.SS1.SSS2.p1.12.m12.2.2.1.1.cmml" xref="A4.SS1.SSS2.p1.12.m12.2.2.1.1"><in id="A4.SS1.SSS2.p1.12.m12.2.2.1.1.1.cmml" xref="A4.SS1.SSS2.p1.12.m12.2.2.1.1.1"></in><apply id="A4.SS1.SSS2.p1.12.m12.2.2.1.1.2.cmml" xref="A4.SS1.SSS2.p1.12.m12.2.2.1.1.2"><csymbol cd="latexml" id="A4.SS1.SSS2.p1.12.m12.2.2.1.1.2.1.cmml" xref="A4.SS1.SSS2.p1.12.m12.2.2.1.1.2.1">for-all</csymbol><ci id="A4.SS1.SSS2.p1.12.m12.2.2.1.1.2.2.cmml" xref="A4.SS1.SSS2.p1.12.m12.2.2.1.1.2.2">𝑛</ci></apply><ci id="A4.SS1.SSS2.p1.12.m12.2.2.1.1.3.cmml" xref="A4.SS1.SSS2.p1.12.m12.2.2.1.1.3">ℕ</ci></apply><apply id="A4.SS1.SSS2.p1.12.m12.3.3.2.2.cmml" xref="A4.SS1.SSS2.p1.12.m12.3.3.2.2"><in id="A4.SS1.SSS2.p1.12.m12.3.3.2.2.1.cmml" xref="A4.SS1.SSS2.p1.12.m12.3.3.2.2.1"></in><apply id="A4.SS1.SSS2.p1.12.m12.3.3.2.2.2.cmml" xref="A4.SS1.SSS2.p1.12.m12.3.3.2.2.2"><csymbol cd="ambiguous" id="A4.SS1.SSS2.p1.12.m12.3.3.2.2.2.1.cmml" xref="A4.SS1.SSS2.p1.12.m12.3.3.2.2.2">subscript</csymbol><ci id="A4.SS1.SSS2.p1.12.m12.3.3.2.2.2.2.cmml" xref="A4.SS1.SSS2.p1.12.m12.3.3.2.2.2.2">𝐱</ci><apply id="A4.SS1.SSS2.p1.12.m12.1.1.1.cmml" xref="A4.SS1.SSS2.p1.12.m12.1.1.1"><times id="A4.SS1.SSS2.p1.12.m12.1.1.1.2.cmml" xref="A4.SS1.SSS2.p1.12.m12.1.1.1.2"></times><ci id="A4.SS1.SSS2.p1.12.m12.1.1.1.3.cmml" xref="A4.SS1.SSS2.p1.12.m12.1.1.1.3">italic-ϕ</ci><ci id="A4.SS1.SSS2.p1.12.m12.1.1.1.1.cmml" xref="A4.SS1.SSS2.p1.12.m12.1.1.1.1">𝑛</ci></apply></apply><ci id="A4.SS1.SSS2.p1.12.m12.3.3.2.2.3.cmml" xref="A4.SS1.SSS2.p1.12.m12.3.3.2.2.3">𝐊</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.SS1.SSS2.p1.12.m12.3c">\forall n\in{\mathbb{N}},{\mathbf{x}}_{\phi(n)}\in\mathbf{K}</annotation><annotation encoding="application/x-llamapun" id="A4.SS1.SSS2.p1.12.m12.3d">∀ italic_n ∈ blackboard_N , bold_x start_POSTSUBSCRIPT italic_ϕ ( italic_n ) end_POSTSUBSCRIPT ∈ bold_K</annotation></semantics></math>, we have that <math alttext="\nabla\mathcal{F}({\mathbf{x}}_{\phi(n)},{\mathbf{y}})+\alpha\nabla\mathcal{R}% _{\sigma_{\phi(n)}}({\mathbf{x}}_{\phi(n)})\to\nabla\mathcal{J}({\mathbf{x}})" class="ltx_Math" display="inline" id="A4.SS1.SSS2.p1.13.m13.7"><semantics id="A4.SS1.SSS2.p1.13.m13.7a"><mrow id="A4.SS1.SSS2.p1.13.m13.7.7" xref="A4.SS1.SSS2.p1.13.m13.7.7.cmml"><mrow id="A4.SS1.SSS2.p1.13.m13.7.7.2" xref="A4.SS1.SSS2.p1.13.m13.7.7.2.cmml"><mrow id="A4.SS1.SSS2.p1.13.m13.6.6.1.1" xref="A4.SS1.SSS2.p1.13.m13.6.6.1.1.cmml"><mrow id="A4.SS1.SSS2.p1.13.m13.6.6.1.1.3" xref="A4.SS1.SSS2.p1.13.m13.6.6.1.1.3.cmml"><mo id="A4.SS1.SSS2.p1.13.m13.6.6.1.1.3.1" rspace="0.167em" xref="A4.SS1.SSS2.p1.13.m13.6.6.1.1.3.1.cmml">∇</mo><mi class="ltx_font_mathcaligraphic" id="A4.SS1.SSS2.p1.13.m13.6.6.1.1.3.2" xref="A4.SS1.SSS2.p1.13.m13.6.6.1.1.3.2.cmml">ℱ</mi></mrow><mo id="A4.SS1.SSS2.p1.13.m13.6.6.1.1.2" xref="A4.SS1.SSS2.p1.13.m13.6.6.1.1.2.cmml">⁢</mo><mrow id="A4.SS1.SSS2.p1.13.m13.6.6.1.1.1.1" xref="A4.SS1.SSS2.p1.13.m13.6.6.1.1.1.2.cmml"><mo id="A4.SS1.SSS2.p1.13.m13.6.6.1.1.1.1.2" stretchy="false" xref="A4.SS1.SSS2.p1.13.m13.6.6.1.1.1.2.cmml">(</mo><msub id="A4.SS1.SSS2.p1.13.m13.6.6.1.1.1.1.1" xref="A4.SS1.SSS2.p1.13.m13.6.6.1.1.1.1.1.cmml"><mi id="A4.SS1.SSS2.p1.13.m13.6.6.1.1.1.1.1.2" xref="A4.SS1.SSS2.p1.13.m13.6.6.1.1.1.1.1.2.cmml">𝐱</mi><mrow id="A4.SS1.SSS2.p1.13.m13.1.1.1" xref="A4.SS1.SSS2.p1.13.m13.1.1.1.cmml"><mi id="A4.SS1.SSS2.p1.13.m13.1.1.1.3" xref="A4.SS1.SSS2.p1.13.m13.1.1.1.3.cmml">ϕ</mi><mo id="A4.SS1.SSS2.p1.13.m13.1.1.1.2" xref="A4.SS1.SSS2.p1.13.m13.1.1.1.2.cmml">⁢</mo><mrow id="A4.SS1.SSS2.p1.13.m13.1.1.1.4.2" xref="A4.SS1.SSS2.p1.13.m13.1.1.1.cmml"><mo id="A4.SS1.SSS2.p1.13.m13.1.1.1.4.2.1" stretchy="false" xref="A4.SS1.SSS2.p1.13.m13.1.1.1.cmml">(</mo><mi id="A4.SS1.SSS2.p1.13.m13.1.1.1.1" xref="A4.SS1.SSS2.p1.13.m13.1.1.1.1.cmml">n</mi><mo id="A4.SS1.SSS2.p1.13.m13.1.1.1.4.2.2" stretchy="false" xref="A4.SS1.SSS2.p1.13.m13.1.1.1.cmml">)</mo></mrow></mrow></msub><mo id="A4.SS1.SSS2.p1.13.m13.6.6.1.1.1.1.3" xref="A4.SS1.SSS2.p1.13.m13.6.6.1.1.1.2.cmml">,</mo><mi id="A4.SS1.SSS2.p1.13.m13.4.4" xref="A4.SS1.SSS2.p1.13.m13.4.4.cmml">𝐲</mi><mo id="A4.SS1.SSS2.p1.13.m13.6.6.1.1.1.1.4" stretchy="false" xref="A4.SS1.SSS2.p1.13.m13.6.6.1.1.1.2.cmml">)</mo></mrow></mrow><mo id="A4.SS1.SSS2.p1.13.m13.7.7.2.3" xref="A4.SS1.SSS2.p1.13.m13.7.7.2.3.cmml">+</mo><mrow id="A4.SS1.SSS2.p1.13.m13.7.7.2.2" xref="A4.SS1.SSS2.p1.13.m13.7.7.2.2.cmml"><mi id="A4.SS1.SSS2.p1.13.m13.7.7.2.2.3" xref="A4.SS1.SSS2.p1.13.m13.7.7.2.2.3.cmml">α</mi><mo id="A4.SS1.SSS2.p1.13.m13.7.7.2.2.2" lspace="0.167em" xref="A4.SS1.SSS2.p1.13.m13.7.7.2.2.2.cmml">⁢</mo><mrow id="A4.SS1.SSS2.p1.13.m13.7.7.2.2.4" xref="A4.SS1.SSS2.p1.13.m13.7.7.2.2.4.cmml"><mo id="A4.SS1.SSS2.p1.13.m13.7.7.2.2.4.1" rspace="0.167em" xref="A4.SS1.SSS2.p1.13.m13.7.7.2.2.4.1.cmml">∇</mo><msub id="A4.SS1.SSS2.p1.13.m13.7.7.2.2.4.2" xref="A4.SS1.SSS2.p1.13.m13.7.7.2.2.4.2.cmml"><mi class="ltx_font_mathcaligraphic" id="A4.SS1.SSS2.p1.13.m13.7.7.2.2.4.2.2" xref="A4.SS1.SSS2.p1.13.m13.7.7.2.2.4.2.2.cmml">ℛ</mi><msub id="A4.SS1.SSS2.p1.13.m13.2.2.1" xref="A4.SS1.SSS2.p1.13.m13.2.2.1.cmml"><mi id="A4.SS1.SSS2.p1.13.m13.2.2.1.3" xref="A4.SS1.SSS2.p1.13.m13.2.2.1.3.cmml">σ</mi><mrow id="A4.SS1.SSS2.p1.13.m13.2.2.1.1.1" xref="A4.SS1.SSS2.p1.13.m13.2.2.1.1.1.cmml"><mi id="A4.SS1.SSS2.p1.13.m13.2.2.1.1.1.3" xref="A4.SS1.SSS2.p1.13.m13.2.2.1.1.1.3.cmml">ϕ</mi><mo id="A4.SS1.SSS2.p1.13.m13.2.2.1.1.1.2" xref="A4.SS1.SSS2.p1.13.m13.2.2.1.1.1.2.cmml">⁢</mo><mrow id="A4.SS1.SSS2.p1.13.m13.2.2.1.1.1.4.2" xref="A4.SS1.SSS2.p1.13.m13.2.2.1.1.1.cmml"><mo id="A4.SS1.SSS2.p1.13.m13.2.2.1.1.1.4.2.1" stretchy="false" xref="A4.SS1.SSS2.p1.13.m13.2.2.1.1.1.cmml">(</mo><mi id="A4.SS1.SSS2.p1.13.m13.2.2.1.1.1.1" xref="A4.SS1.SSS2.p1.13.m13.2.2.1.1.1.1.cmml">n</mi><mo id="A4.SS1.SSS2.p1.13.m13.2.2.1.1.1.4.2.2" stretchy="false" xref="A4.SS1.SSS2.p1.13.m13.2.2.1.1.1.cmml">)</mo></mrow></mrow></msub></msub></mrow><mo id="A4.SS1.SSS2.p1.13.m13.7.7.2.2.2a" xref="A4.SS1.SSS2.p1.13.m13.7.7.2.2.2.cmml">⁢</mo><mrow id="A4.SS1.SSS2.p1.13.m13.7.7.2.2.1.1" xref="A4.SS1.SSS2.p1.13.m13.7.7.2.2.1.1.1.cmml"><mo id="A4.SS1.SSS2.p1.13.m13.7.7.2.2.1.1.2" stretchy="false" xref="A4.SS1.SSS2.p1.13.m13.7.7.2.2.1.1.1.cmml">(</mo><msub id="A4.SS1.SSS2.p1.13.m13.7.7.2.2.1.1.1" xref="A4.SS1.SSS2.p1.13.m13.7.7.2.2.1.1.1.cmml"><mi id="A4.SS1.SSS2.p1.13.m13.7.7.2.2.1.1.1.2" xref="A4.SS1.SSS2.p1.13.m13.7.7.2.2.1.1.1.2.cmml">𝐱</mi><mrow id="A4.SS1.SSS2.p1.13.m13.3.3.1" xref="A4.SS1.SSS2.p1.13.m13.3.3.1.cmml"><mi id="A4.SS1.SSS2.p1.13.m13.3.3.1.3" xref="A4.SS1.SSS2.p1.13.m13.3.3.1.3.cmml">ϕ</mi><mo id="A4.SS1.SSS2.p1.13.m13.3.3.1.2" xref="A4.SS1.SSS2.p1.13.m13.3.3.1.2.cmml">⁢</mo><mrow id="A4.SS1.SSS2.p1.13.m13.3.3.1.4.2" xref="A4.SS1.SSS2.p1.13.m13.3.3.1.cmml"><mo id="A4.SS1.SSS2.p1.13.m13.3.3.1.4.2.1" stretchy="false" xref="A4.SS1.SSS2.p1.13.m13.3.3.1.cmml">(</mo><mi id="A4.SS1.SSS2.p1.13.m13.3.3.1.1" xref="A4.SS1.SSS2.p1.13.m13.3.3.1.1.cmml">n</mi><mo id="A4.SS1.SSS2.p1.13.m13.3.3.1.4.2.2" stretchy="false" xref="A4.SS1.SSS2.p1.13.m13.3.3.1.cmml">)</mo></mrow></mrow></msub><mo id="A4.SS1.SSS2.p1.13.m13.7.7.2.2.1.1.3" stretchy="false" xref="A4.SS1.SSS2.p1.13.m13.7.7.2.2.1.1.1.cmml">)</mo></mrow></mrow></mrow><mo id="A4.SS1.SSS2.p1.13.m13.7.7.3" stretchy="false" xref="A4.SS1.SSS2.p1.13.m13.7.7.3.cmml">→</mo><mrow id="A4.SS1.SSS2.p1.13.m13.7.7.4" xref="A4.SS1.SSS2.p1.13.m13.7.7.4.cmml"><mrow id="A4.SS1.SSS2.p1.13.m13.7.7.4.2" xref="A4.SS1.SSS2.p1.13.m13.7.7.4.2.cmml"><mo id="A4.SS1.SSS2.p1.13.m13.7.7.4.2.1" rspace="0.167em" xref="A4.SS1.SSS2.p1.13.m13.7.7.4.2.1.cmml">∇</mo><mi class="ltx_font_mathcaligraphic" id="A4.SS1.SSS2.p1.13.m13.7.7.4.2.2" xref="A4.SS1.SSS2.p1.13.m13.7.7.4.2.2.cmml">𝒥</mi></mrow><mo id="A4.SS1.SSS2.p1.13.m13.7.7.4.1" xref="A4.SS1.SSS2.p1.13.m13.7.7.4.1.cmml">⁢</mo><mrow id="A4.SS1.SSS2.p1.13.m13.7.7.4.3.2" xref="A4.SS1.SSS2.p1.13.m13.7.7.4.cmml"><mo id="A4.SS1.SSS2.p1.13.m13.7.7.4.3.2.1" stretchy="false" xref="A4.SS1.SSS2.p1.13.m13.7.7.4.cmml">(</mo><mi id="A4.SS1.SSS2.p1.13.m13.5.5" xref="A4.SS1.SSS2.p1.13.m13.5.5.cmml">𝐱</mi><mo id="A4.SS1.SSS2.p1.13.m13.7.7.4.3.2.2" stretchy="false" xref="A4.SS1.SSS2.p1.13.m13.7.7.4.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="A4.SS1.SSS2.p1.13.m13.7b"><apply id="A4.SS1.SSS2.p1.13.m13.7.7.cmml" xref="A4.SS1.SSS2.p1.13.m13.7.7"><ci id="A4.SS1.SSS2.p1.13.m13.7.7.3.cmml" xref="A4.SS1.SSS2.p1.13.m13.7.7.3">→</ci><apply id="A4.SS1.SSS2.p1.13.m13.7.7.2.cmml" xref="A4.SS1.SSS2.p1.13.m13.7.7.2"><plus id="A4.SS1.SSS2.p1.13.m13.7.7.2.3.cmml" xref="A4.SS1.SSS2.p1.13.m13.7.7.2.3"></plus><apply id="A4.SS1.SSS2.p1.13.m13.6.6.1.1.cmml" xref="A4.SS1.SSS2.p1.13.m13.6.6.1.1"><times id="A4.SS1.SSS2.p1.13.m13.6.6.1.1.2.cmml" xref="A4.SS1.SSS2.p1.13.m13.6.6.1.1.2"></times><apply id="A4.SS1.SSS2.p1.13.m13.6.6.1.1.3.cmml" xref="A4.SS1.SSS2.p1.13.m13.6.6.1.1.3"><ci id="A4.SS1.SSS2.p1.13.m13.6.6.1.1.3.1.cmml" xref="A4.SS1.SSS2.p1.13.m13.6.6.1.1.3.1">∇</ci><ci id="A4.SS1.SSS2.p1.13.m13.6.6.1.1.3.2.cmml" xref="A4.SS1.SSS2.p1.13.m13.6.6.1.1.3.2">ℱ</ci></apply><interval closure="open" id="A4.SS1.SSS2.p1.13.m13.6.6.1.1.1.2.cmml" xref="A4.SS1.SSS2.p1.13.m13.6.6.1.1.1.1"><apply id="A4.SS1.SSS2.p1.13.m13.6.6.1.1.1.1.1.cmml" xref="A4.SS1.SSS2.p1.13.m13.6.6.1.1.1.1.1"><csymbol cd="ambiguous" id="A4.SS1.SSS2.p1.13.m13.6.6.1.1.1.1.1.1.cmml" xref="A4.SS1.SSS2.p1.13.m13.6.6.1.1.1.1.1">subscript</csymbol><ci id="A4.SS1.SSS2.p1.13.m13.6.6.1.1.1.1.1.2.cmml" xref="A4.SS1.SSS2.p1.13.m13.6.6.1.1.1.1.1.2">𝐱</ci><apply id="A4.SS1.SSS2.p1.13.m13.1.1.1.cmml" xref="A4.SS1.SSS2.p1.13.m13.1.1.1"><times id="A4.SS1.SSS2.p1.13.m13.1.1.1.2.cmml" xref="A4.SS1.SSS2.p1.13.m13.1.1.1.2"></times><ci id="A4.SS1.SSS2.p1.13.m13.1.1.1.3.cmml" xref="A4.SS1.SSS2.p1.13.m13.1.1.1.3">italic-ϕ</ci><ci id="A4.SS1.SSS2.p1.13.m13.1.1.1.1.cmml" xref="A4.SS1.SSS2.p1.13.m13.1.1.1.1">𝑛</ci></apply></apply><ci id="A4.SS1.SSS2.p1.13.m13.4.4.cmml" xref="A4.SS1.SSS2.p1.13.m13.4.4">𝐲</ci></interval></apply><apply id="A4.SS1.SSS2.p1.13.m13.7.7.2.2.cmml" xref="A4.SS1.SSS2.p1.13.m13.7.7.2.2"><times id="A4.SS1.SSS2.p1.13.m13.7.7.2.2.2.cmml" xref="A4.SS1.SSS2.p1.13.m13.7.7.2.2.2"></times><ci id="A4.SS1.SSS2.p1.13.m13.7.7.2.2.3.cmml" xref="A4.SS1.SSS2.p1.13.m13.7.7.2.2.3">𝛼</ci><apply id="A4.SS1.SSS2.p1.13.m13.7.7.2.2.4.cmml" xref="A4.SS1.SSS2.p1.13.m13.7.7.2.2.4"><ci id="A4.SS1.SSS2.p1.13.m13.7.7.2.2.4.1.cmml" xref="A4.SS1.SSS2.p1.13.m13.7.7.2.2.4.1">∇</ci><apply id="A4.SS1.SSS2.p1.13.m13.7.7.2.2.4.2.cmml" xref="A4.SS1.SSS2.p1.13.m13.7.7.2.2.4.2"><csymbol cd="ambiguous" id="A4.SS1.SSS2.p1.13.m13.7.7.2.2.4.2.1.cmml" xref="A4.SS1.SSS2.p1.13.m13.7.7.2.2.4.2">subscript</csymbol><ci id="A4.SS1.SSS2.p1.13.m13.7.7.2.2.4.2.2.cmml" xref="A4.SS1.SSS2.p1.13.m13.7.7.2.2.4.2.2">ℛ</ci><apply id="A4.SS1.SSS2.p1.13.m13.2.2.1.cmml" xref="A4.SS1.SSS2.p1.13.m13.2.2.1"><csymbol cd="ambiguous" id="A4.SS1.SSS2.p1.13.m13.2.2.1.2.cmml" xref="A4.SS1.SSS2.p1.13.m13.2.2.1">subscript</csymbol><ci id="A4.SS1.SSS2.p1.13.m13.2.2.1.3.cmml" xref="A4.SS1.SSS2.p1.13.m13.2.2.1.3">𝜎</ci><apply id="A4.SS1.SSS2.p1.13.m13.2.2.1.1.1.cmml" xref="A4.SS1.SSS2.p1.13.m13.2.2.1.1.1"><times id="A4.SS1.SSS2.p1.13.m13.2.2.1.1.1.2.cmml" xref="A4.SS1.SSS2.p1.13.m13.2.2.1.1.1.2"></times><ci id="A4.SS1.SSS2.p1.13.m13.2.2.1.1.1.3.cmml" xref="A4.SS1.SSS2.p1.13.m13.2.2.1.1.1.3">italic-ϕ</ci><ci id="A4.SS1.SSS2.p1.13.m13.2.2.1.1.1.1.cmml" xref="A4.SS1.SSS2.p1.13.m13.2.2.1.1.1.1">𝑛</ci></apply></apply></apply></apply><apply id="A4.SS1.SSS2.p1.13.m13.7.7.2.2.1.1.1.cmml" xref="A4.SS1.SSS2.p1.13.m13.7.7.2.2.1.1"><csymbol cd="ambiguous" id="A4.SS1.SSS2.p1.13.m13.7.7.2.2.1.1.1.1.cmml" xref="A4.SS1.SSS2.p1.13.m13.7.7.2.2.1.1">subscript</csymbol><ci id="A4.SS1.SSS2.p1.13.m13.7.7.2.2.1.1.1.2.cmml" xref="A4.SS1.SSS2.p1.13.m13.7.7.2.2.1.1.1.2">𝐱</ci><apply id="A4.SS1.SSS2.p1.13.m13.3.3.1.cmml" xref="A4.SS1.SSS2.p1.13.m13.3.3.1"><times id="A4.SS1.SSS2.p1.13.m13.3.3.1.2.cmml" xref="A4.SS1.SSS2.p1.13.m13.3.3.1.2"></times><ci id="A4.SS1.SSS2.p1.13.m13.3.3.1.3.cmml" xref="A4.SS1.SSS2.p1.13.m13.3.3.1.3">italic-ϕ</ci><ci id="A4.SS1.SSS2.p1.13.m13.3.3.1.1.cmml" xref="A4.SS1.SSS2.p1.13.m13.3.3.1.1">𝑛</ci></apply></apply></apply></apply><apply id="A4.SS1.SSS2.p1.13.m13.7.7.4.cmml" xref="A4.SS1.SSS2.p1.13.m13.7.7.4"><times id="A4.SS1.SSS2.p1.13.m13.7.7.4.1.cmml" xref="A4.SS1.SSS2.p1.13.m13.7.7.4.1"></times><apply id="A4.SS1.SSS2.p1.13.m13.7.7.4.2.cmml" xref="A4.SS1.SSS2.p1.13.m13.7.7.4.2"><ci id="A4.SS1.SSS2.p1.13.m13.7.7.4.2.1.cmml" xref="A4.SS1.SSS2.p1.13.m13.7.7.4.2.1">∇</ci><ci id="A4.SS1.SSS2.p1.13.m13.7.7.4.2.2.cmml" xref="A4.SS1.SSS2.p1.13.m13.7.7.4.2.2">𝒥</ci></apply><ci id="A4.SS1.SSS2.p1.13.m13.5.5.cmml" xref="A4.SS1.SSS2.p1.13.m13.5.5">𝐱</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.SS1.SSS2.p1.13.m13.7c">\nabla\mathcal{F}({\mathbf{x}}_{\phi(n)},{\mathbf{y}})+\alpha\nabla\mathcal{R}% _{\sigma_{\phi(n)}}({\mathbf{x}}_{\phi(n)})\to\nabla\mathcal{J}({\mathbf{x}})</annotation><annotation encoding="application/x-llamapun" id="A4.SS1.SSS2.p1.13.m13.7d">∇ caligraphic_F ( bold_x start_POSTSUBSCRIPT italic_ϕ ( italic_n ) end_POSTSUBSCRIPT , bold_y ) + italic_α ∇ caligraphic_R start_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT italic_ϕ ( italic_n ) end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( bold_x start_POSTSUBSCRIPT italic_ϕ ( italic_n ) end_POSTSUBSCRIPT ) → ∇ caligraphic_J ( bold_x )</annotation></semantics></math>. On the other hand, as <math alttext="\forall n\in{\mathbb{N}}" class="ltx_Math" display="inline" id="A4.SS1.SSS2.p1.14.m14.1"><semantics id="A4.SS1.SSS2.p1.14.m14.1a"><mrow id="A4.SS1.SSS2.p1.14.m14.1.1" xref="A4.SS1.SSS2.p1.14.m14.1.1.cmml"><mrow id="A4.SS1.SSS2.p1.14.m14.1.1.2" xref="A4.SS1.SSS2.p1.14.m14.1.1.2.cmml"><mo id="A4.SS1.SSS2.p1.14.m14.1.1.2.1" rspace="0.167em" xref="A4.SS1.SSS2.p1.14.m14.1.1.2.1.cmml">∀</mo><mi id="A4.SS1.SSS2.p1.14.m14.1.1.2.2" xref="A4.SS1.SSS2.p1.14.m14.1.1.2.2.cmml">n</mi></mrow><mo id="A4.SS1.SSS2.p1.14.m14.1.1.1" xref="A4.SS1.SSS2.p1.14.m14.1.1.1.cmml">∈</mo><mi id="A4.SS1.SSS2.p1.14.m14.1.1.3" xref="A4.SS1.SSS2.p1.14.m14.1.1.3.cmml">ℕ</mi></mrow><annotation-xml encoding="MathML-Content" id="A4.SS1.SSS2.p1.14.m14.1b"><apply id="A4.SS1.SSS2.p1.14.m14.1.1.cmml" xref="A4.SS1.SSS2.p1.14.m14.1.1"><in id="A4.SS1.SSS2.p1.14.m14.1.1.1.cmml" xref="A4.SS1.SSS2.p1.14.m14.1.1.1"></in><apply id="A4.SS1.SSS2.p1.14.m14.1.1.2.cmml" xref="A4.SS1.SSS2.p1.14.m14.1.1.2"><csymbol cd="latexml" id="A4.SS1.SSS2.p1.14.m14.1.1.2.1.cmml" xref="A4.SS1.SSS2.p1.14.m14.1.1.2.1">for-all</csymbol><ci id="A4.SS1.SSS2.p1.14.m14.1.1.2.2.cmml" xref="A4.SS1.SSS2.p1.14.m14.1.1.2.2">𝑛</ci></apply><ci id="A4.SS1.SSS2.p1.14.m14.1.1.3.cmml" xref="A4.SS1.SSS2.p1.14.m14.1.1.3">ℕ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.SS1.SSS2.p1.14.m14.1c">\forall n\in{\mathbb{N}}</annotation><annotation encoding="application/x-llamapun" id="A4.SS1.SSS2.p1.14.m14.1d">∀ italic_n ∈ blackboard_N</annotation></semantics></math>, because <math alttext="{\mathbf{x}}_{\phi(n)}\in\mathbf{S}_{\sigma_{\phi(n)},\mathbf{K}}" class="ltx_Math" display="inline" id="A4.SS1.SSS2.p1.15.m15.4"><semantics id="A4.SS1.SSS2.p1.15.m15.4a"><mrow id="A4.SS1.SSS2.p1.15.m15.4.5" xref="A4.SS1.SSS2.p1.15.m15.4.5.cmml"><msub id="A4.SS1.SSS2.p1.15.m15.4.5.2" xref="A4.SS1.SSS2.p1.15.m15.4.5.2.cmml"><mi id="A4.SS1.SSS2.p1.15.m15.4.5.2.2" xref="A4.SS1.SSS2.p1.15.m15.4.5.2.2.cmml">𝐱</mi><mrow id="A4.SS1.SSS2.p1.15.m15.1.1.1" xref="A4.SS1.SSS2.p1.15.m15.1.1.1.cmml"><mi id="A4.SS1.SSS2.p1.15.m15.1.1.1.3" xref="A4.SS1.SSS2.p1.15.m15.1.1.1.3.cmml">ϕ</mi><mo id="A4.SS1.SSS2.p1.15.m15.1.1.1.2" xref="A4.SS1.SSS2.p1.15.m15.1.1.1.2.cmml">⁢</mo><mrow id="A4.SS1.SSS2.p1.15.m15.1.1.1.4.2" xref="A4.SS1.SSS2.p1.15.m15.1.1.1.cmml"><mo id="A4.SS1.SSS2.p1.15.m15.1.1.1.4.2.1" stretchy="false" xref="A4.SS1.SSS2.p1.15.m15.1.1.1.cmml">(</mo><mi id="A4.SS1.SSS2.p1.15.m15.1.1.1.1" xref="A4.SS1.SSS2.p1.15.m15.1.1.1.1.cmml">n</mi><mo id="A4.SS1.SSS2.p1.15.m15.1.1.1.4.2.2" stretchy="false" xref="A4.SS1.SSS2.p1.15.m15.1.1.1.cmml">)</mo></mrow></mrow></msub><mo id="A4.SS1.SSS2.p1.15.m15.4.5.1" xref="A4.SS1.SSS2.p1.15.m15.4.5.1.cmml">∈</mo><msub id="A4.SS1.SSS2.p1.15.m15.4.5.3" xref="A4.SS1.SSS2.p1.15.m15.4.5.3.cmml"><mi id="A4.SS1.SSS2.p1.15.m15.4.5.3.2" xref="A4.SS1.SSS2.p1.15.m15.4.5.3.2.cmml">𝐒</mi><mrow id="A4.SS1.SSS2.p1.15.m15.4.4.3.3" xref="A4.SS1.SSS2.p1.15.m15.4.4.3.4.cmml"><msub id="A4.SS1.SSS2.p1.15.m15.4.4.3.3.1" xref="A4.SS1.SSS2.p1.15.m15.4.4.3.3.1.cmml"><mi id="A4.SS1.SSS2.p1.15.m15.4.4.3.3.1.2" xref="A4.SS1.SSS2.p1.15.m15.4.4.3.3.1.2.cmml">σ</mi><mrow id="A4.SS1.SSS2.p1.15.m15.2.2.1.1.1" xref="A4.SS1.SSS2.p1.15.m15.2.2.1.1.1.cmml"><mi id="A4.SS1.SSS2.p1.15.m15.2.2.1.1.1.3" xref="A4.SS1.SSS2.p1.15.m15.2.2.1.1.1.3.cmml">ϕ</mi><mo id="A4.SS1.SSS2.p1.15.m15.2.2.1.1.1.2" xref="A4.SS1.SSS2.p1.15.m15.2.2.1.1.1.2.cmml">⁢</mo><mrow id="A4.SS1.SSS2.p1.15.m15.2.2.1.1.1.4.2" xref="A4.SS1.SSS2.p1.15.m15.2.2.1.1.1.cmml"><mo id="A4.SS1.SSS2.p1.15.m15.2.2.1.1.1.4.2.1" stretchy="false" xref="A4.SS1.SSS2.p1.15.m15.2.2.1.1.1.cmml">(</mo><mi id="A4.SS1.SSS2.p1.15.m15.2.2.1.1.1.1" xref="A4.SS1.SSS2.p1.15.m15.2.2.1.1.1.1.cmml">n</mi><mo id="A4.SS1.SSS2.p1.15.m15.2.2.1.1.1.4.2.2" stretchy="false" xref="A4.SS1.SSS2.p1.15.m15.2.2.1.1.1.cmml">)</mo></mrow></mrow></msub><mo id="A4.SS1.SSS2.p1.15.m15.4.4.3.3.2" xref="A4.SS1.SSS2.p1.15.m15.4.4.3.4.cmml">,</mo><mi id="A4.SS1.SSS2.p1.15.m15.3.3.2.2" xref="A4.SS1.SSS2.p1.15.m15.3.3.2.2.cmml">𝐊</mi></mrow></msub></mrow><annotation-xml encoding="MathML-Content" id="A4.SS1.SSS2.p1.15.m15.4b"><apply id="A4.SS1.SSS2.p1.15.m15.4.5.cmml" xref="A4.SS1.SSS2.p1.15.m15.4.5"><in id="A4.SS1.SSS2.p1.15.m15.4.5.1.cmml" xref="A4.SS1.SSS2.p1.15.m15.4.5.1"></in><apply id="A4.SS1.SSS2.p1.15.m15.4.5.2.cmml" xref="A4.SS1.SSS2.p1.15.m15.4.5.2"><csymbol cd="ambiguous" id="A4.SS1.SSS2.p1.15.m15.4.5.2.1.cmml" xref="A4.SS1.SSS2.p1.15.m15.4.5.2">subscript</csymbol><ci id="A4.SS1.SSS2.p1.15.m15.4.5.2.2.cmml" xref="A4.SS1.SSS2.p1.15.m15.4.5.2.2">𝐱</ci><apply id="A4.SS1.SSS2.p1.15.m15.1.1.1.cmml" xref="A4.SS1.SSS2.p1.15.m15.1.1.1"><times id="A4.SS1.SSS2.p1.15.m15.1.1.1.2.cmml" xref="A4.SS1.SSS2.p1.15.m15.1.1.1.2"></times><ci id="A4.SS1.SSS2.p1.15.m15.1.1.1.3.cmml" xref="A4.SS1.SSS2.p1.15.m15.1.1.1.3">italic-ϕ</ci><ci id="A4.SS1.SSS2.p1.15.m15.1.1.1.1.cmml" xref="A4.SS1.SSS2.p1.15.m15.1.1.1.1">𝑛</ci></apply></apply><apply id="A4.SS1.SSS2.p1.15.m15.4.5.3.cmml" xref="A4.SS1.SSS2.p1.15.m15.4.5.3"><csymbol cd="ambiguous" id="A4.SS1.SSS2.p1.15.m15.4.5.3.1.cmml" xref="A4.SS1.SSS2.p1.15.m15.4.5.3">subscript</csymbol><ci id="A4.SS1.SSS2.p1.15.m15.4.5.3.2.cmml" xref="A4.SS1.SSS2.p1.15.m15.4.5.3.2">𝐒</ci><list id="A4.SS1.SSS2.p1.15.m15.4.4.3.4.cmml" xref="A4.SS1.SSS2.p1.15.m15.4.4.3.3"><apply id="A4.SS1.SSS2.p1.15.m15.4.4.3.3.1.cmml" xref="A4.SS1.SSS2.p1.15.m15.4.4.3.3.1"><csymbol cd="ambiguous" id="A4.SS1.SSS2.p1.15.m15.4.4.3.3.1.1.cmml" xref="A4.SS1.SSS2.p1.15.m15.4.4.3.3.1">subscript</csymbol><ci id="A4.SS1.SSS2.p1.15.m15.4.4.3.3.1.2.cmml" xref="A4.SS1.SSS2.p1.15.m15.4.4.3.3.1.2">𝜎</ci><apply id="A4.SS1.SSS2.p1.15.m15.2.2.1.1.1.cmml" xref="A4.SS1.SSS2.p1.15.m15.2.2.1.1.1"><times id="A4.SS1.SSS2.p1.15.m15.2.2.1.1.1.2.cmml" xref="A4.SS1.SSS2.p1.15.m15.2.2.1.1.1.2"></times><ci id="A4.SS1.SSS2.p1.15.m15.2.2.1.1.1.3.cmml" xref="A4.SS1.SSS2.p1.15.m15.2.2.1.1.1.3">italic-ϕ</ci><ci id="A4.SS1.SSS2.p1.15.m15.2.2.1.1.1.1.cmml" xref="A4.SS1.SSS2.p1.15.m15.2.2.1.1.1.1">𝑛</ci></apply></apply><ci id="A4.SS1.SSS2.p1.15.m15.3.3.2.2.cmml" xref="A4.SS1.SSS2.p1.15.m15.3.3.2.2">𝐊</ci></list></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.SS1.SSS2.p1.15.m15.4c">{\mathbf{x}}_{\phi(n)}\in\mathbf{S}_{\sigma_{\phi(n)},\mathbf{K}}</annotation><annotation encoding="application/x-llamapun" id="A4.SS1.SSS2.p1.15.m15.4d">bold_x start_POSTSUBSCRIPT italic_ϕ ( italic_n ) end_POSTSUBSCRIPT ∈ bold_S start_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT italic_ϕ ( italic_n ) end_POSTSUBSCRIPT , bold_K end_POSTSUBSCRIPT</annotation></semantics></math>, <math alttext="\nabla\mathcal{F}({\mathbf{x}}_{\phi(n)},{\mathbf{y}})+\alpha\nabla\mathcal{R}% _{\sigma_{\phi(n)}}({\mathbf{x}}_{\phi(n)})=0" class="ltx_Math" display="inline" id="A4.SS1.SSS2.p1.16.m16.6"><semantics id="A4.SS1.SSS2.p1.16.m16.6a"><mrow id="A4.SS1.SSS2.p1.16.m16.6.6" xref="A4.SS1.SSS2.p1.16.m16.6.6.cmml"><mrow id="A4.SS1.SSS2.p1.16.m16.6.6.2" xref="A4.SS1.SSS2.p1.16.m16.6.6.2.cmml"><mrow id="A4.SS1.SSS2.p1.16.m16.5.5.1.1" xref="A4.SS1.SSS2.p1.16.m16.5.5.1.1.cmml"><mrow id="A4.SS1.SSS2.p1.16.m16.5.5.1.1.3" xref="A4.SS1.SSS2.p1.16.m16.5.5.1.1.3.cmml"><mo id="A4.SS1.SSS2.p1.16.m16.5.5.1.1.3.1" rspace="0.167em" xref="A4.SS1.SSS2.p1.16.m16.5.5.1.1.3.1.cmml">∇</mo><mi class="ltx_font_mathcaligraphic" id="A4.SS1.SSS2.p1.16.m16.5.5.1.1.3.2" xref="A4.SS1.SSS2.p1.16.m16.5.5.1.1.3.2.cmml">ℱ</mi></mrow><mo id="A4.SS1.SSS2.p1.16.m16.5.5.1.1.2" xref="A4.SS1.SSS2.p1.16.m16.5.5.1.1.2.cmml">⁢</mo><mrow id="A4.SS1.SSS2.p1.16.m16.5.5.1.1.1.1" xref="A4.SS1.SSS2.p1.16.m16.5.5.1.1.1.2.cmml"><mo id="A4.SS1.SSS2.p1.16.m16.5.5.1.1.1.1.2" stretchy="false" xref="A4.SS1.SSS2.p1.16.m16.5.5.1.1.1.2.cmml">(</mo><msub id="A4.SS1.SSS2.p1.16.m16.5.5.1.1.1.1.1" xref="A4.SS1.SSS2.p1.16.m16.5.5.1.1.1.1.1.cmml"><mi id="A4.SS1.SSS2.p1.16.m16.5.5.1.1.1.1.1.2" xref="A4.SS1.SSS2.p1.16.m16.5.5.1.1.1.1.1.2.cmml">𝐱</mi><mrow id="A4.SS1.SSS2.p1.16.m16.1.1.1" xref="A4.SS1.SSS2.p1.16.m16.1.1.1.cmml"><mi id="A4.SS1.SSS2.p1.16.m16.1.1.1.3" xref="A4.SS1.SSS2.p1.16.m16.1.1.1.3.cmml">ϕ</mi><mo id="A4.SS1.SSS2.p1.16.m16.1.1.1.2" xref="A4.SS1.SSS2.p1.16.m16.1.1.1.2.cmml">⁢</mo><mrow id="A4.SS1.SSS2.p1.16.m16.1.1.1.4.2" xref="A4.SS1.SSS2.p1.16.m16.1.1.1.cmml"><mo id="A4.SS1.SSS2.p1.16.m16.1.1.1.4.2.1" stretchy="false" xref="A4.SS1.SSS2.p1.16.m16.1.1.1.cmml">(</mo><mi id="A4.SS1.SSS2.p1.16.m16.1.1.1.1" xref="A4.SS1.SSS2.p1.16.m16.1.1.1.1.cmml">n</mi><mo id="A4.SS1.SSS2.p1.16.m16.1.1.1.4.2.2" stretchy="false" xref="A4.SS1.SSS2.p1.16.m16.1.1.1.cmml">)</mo></mrow></mrow></msub><mo id="A4.SS1.SSS2.p1.16.m16.5.5.1.1.1.1.3" xref="A4.SS1.SSS2.p1.16.m16.5.5.1.1.1.2.cmml">,</mo><mi id="A4.SS1.SSS2.p1.16.m16.4.4" xref="A4.SS1.SSS2.p1.16.m16.4.4.cmml">𝐲</mi><mo id="A4.SS1.SSS2.p1.16.m16.5.5.1.1.1.1.4" stretchy="false" xref="A4.SS1.SSS2.p1.16.m16.5.5.1.1.1.2.cmml">)</mo></mrow></mrow><mo id="A4.SS1.SSS2.p1.16.m16.6.6.2.3" xref="A4.SS1.SSS2.p1.16.m16.6.6.2.3.cmml">+</mo><mrow id="A4.SS1.SSS2.p1.16.m16.6.6.2.2" xref="A4.SS1.SSS2.p1.16.m16.6.6.2.2.cmml"><mi id="A4.SS1.SSS2.p1.16.m16.6.6.2.2.3" xref="A4.SS1.SSS2.p1.16.m16.6.6.2.2.3.cmml">α</mi><mo id="A4.SS1.SSS2.p1.16.m16.6.6.2.2.2" lspace="0.167em" xref="A4.SS1.SSS2.p1.16.m16.6.6.2.2.2.cmml">⁢</mo><mrow id="A4.SS1.SSS2.p1.16.m16.6.6.2.2.4" xref="A4.SS1.SSS2.p1.16.m16.6.6.2.2.4.cmml"><mo id="A4.SS1.SSS2.p1.16.m16.6.6.2.2.4.1" rspace="0.167em" xref="A4.SS1.SSS2.p1.16.m16.6.6.2.2.4.1.cmml">∇</mo><msub id="A4.SS1.SSS2.p1.16.m16.6.6.2.2.4.2" xref="A4.SS1.SSS2.p1.16.m16.6.6.2.2.4.2.cmml"><mi class="ltx_font_mathcaligraphic" id="A4.SS1.SSS2.p1.16.m16.6.6.2.2.4.2.2" xref="A4.SS1.SSS2.p1.16.m16.6.6.2.2.4.2.2.cmml">ℛ</mi><msub id="A4.SS1.SSS2.p1.16.m16.2.2.1" xref="A4.SS1.SSS2.p1.16.m16.2.2.1.cmml"><mi id="A4.SS1.SSS2.p1.16.m16.2.2.1.3" xref="A4.SS1.SSS2.p1.16.m16.2.2.1.3.cmml">σ</mi><mrow id="A4.SS1.SSS2.p1.16.m16.2.2.1.1.1" xref="A4.SS1.SSS2.p1.16.m16.2.2.1.1.1.cmml"><mi id="A4.SS1.SSS2.p1.16.m16.2.2.1.1.1.3" xref="A4.SS1.SSS2.p1.16.m16.2.2.1.1.1.3.cmml">ϕ</mi><mo id="A4.SS1.SSS2.p1.16.m16.2.2.1.1.1.2" xref="A4.SS1.SSS2.p1.16.m16.2.2.1.1.1.2.cmml">⁢</mo><mrow id="A4.SS1.SSS2.p1.16.m16.2.2.1.1.1.4.2" xref="A4.SS1.SSS2.p1.16.m16.2.2.1.1.1.cmml"><mo id="A4.SS1.SSS2.p1.16.m16.2.2.1.1.1.4.2.1" stretchy="false" xref="A4.SS1.SSS2.p1.16.m16.2.2.1.1.1.cmml">(</mo><mi id="A4.SS1.SSS2.p1.16.m16.2.2.1.1.1.1" xref="A4.SS1.SSS2.p1.16.m16.2.2.1.1.1.1.cmml">n</mi><mo id="A4.SS1.SSS2.p1.16.m16.2.2.1.1.1.4.2.2" stretchy="false" xref="A4.SS1.SSS2.p1.16.m16.2.2.1.1.1.cmml">)</mo></mrow></mrow></msub></msub></mrow><mo id="A4.SS1.SSS2.p1.16.m16.6.6.2.2.2a" xref="A4.SS1.SSS2.p1.16.m16.6.6.2.2.2.cmml">⁢</mo><mrow id="A4.SS1.SSS2.p1.16.m16.6.6.2.2.1.1" xref="A4.SS1.SSS2.p1.16.m16.6.6.2.2.1.1.1.cmml"><mo id="A4.SS1.SSS2.p1.16.m16.6.6.2.2.1.1.2" stretchy="false" xref="A4.SS1.SSS2.p1.16.m16.6.6.2.2.1.1.1.cmml">(</mo><msub id="A4.SS1.SSS2.p1.16.m16.6.6.2.2.1.1.1" xref="A4.SS1.SSS2.p1.16.m16.6.6.2.2.1.1.1.cmml"><mi id="A4.SS1.SSS2.p1.16.m16.6.6.2.2.1.1.1.2" xref="A4.SS1.SSS2.p1.16.m16.6.6.2.2.1.1.1.2.cmml">𝐱</mi><mrow id="A4.SS1.SSS2.p1.16.m16.3.3.1" xref="A4.SS1.SSS2.p1.16.m16.3.3.1.cmml"><mi id="A4.SS1.SSS2.p1.16.m16.3.3.1.3" xref="A4.SS1.SSS2.p1.16.m16.3.3.1.3.cmml">ϕ</mi><mo id="A4.SS1.SSS2.p1.16.m16.3.3.1.2" xref="A4.SS1.SSS2.p1.16.m16.3.3.1.2.cmml">⁢</mo><mrow id="A4.SS1.SSS2.p1.16.m16.3.3.1.4.2" xref="A4.SS1.SSS2.p1.16.m16.3.3.1.cmml"><mo id="A4.SS1.SSS2.p1.16.m16.3.3.1.4.2.1" stretchy="false" xref="A4.SS1.SSS2.p1.16.m16.3.3.1.cmml">(</mo><mi id="A4.SS1.SSS2.p1.16.m16.3.3.1.1" xref="A4.SS1.SSS2.p1.16.m16.3.3.1.1.cmml">n</mi><mo id="A4.SS1.SSS2.p1.16.m16.3.3.1.4.2.2" stretchy="false" xref="A4.SS1.SSS2.p1.16.m16.3.3.1.cmml">)</mo></mrow></mrow></msub><mo id="A4.SS1.SSS2.p1.16.m16.6.6.2.2.1.1.3" stretchy="false" xref="A4.SS1.SSS2.p1.16.m16.6.6.2.2.1.1.1.cmml">)</mo></mrow></mrow></mrow><mo id="A4.SS1.SSS2.p1.16.m16.6.6.3" xref="A4.SS1.SSS2.p1.16.m16.6.6.3.cmml">=</mo><mn id="A4.SS1.SSS2.p1.16.m16.6.6.4" xref="A4.SS1.SSS2.p1.16.m16.6.6.4.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="A4.SS1.SSS2.p1.16.m16.6b"><apply id="A4.SS1.SSS2.p1.16.m16.6.6.cmml" xref="A4.SS1.SSS2.p1.16.m16.6.6"><eq id="A4.SS1.SSS2.p1.16.m16.6.6.3.cmml" xref="A4.SS1.SSS2.p1.16.m16.6.6.3"></eq><apply id="A4.SS1.SSS2.p1.16.m16.6.6.2.cmml" xref="A4.SS1.SSS2.p1.16.m16.6.6.2"><plus id="A4.SS1.SSS2.p1.16.m16.6.6.2.3.cmml" xref="A4.SS1.SSS2.p1.16.m16.6.6.2.3"></plus><apply id="A4.SS1.SSS2.p1.16.m16.5.5.1.1.cmml" xref="A4.SS1.SSS2.p1.16.m16.5.5.1.1"><times id="A4.SS1.SSS2.p1.16.m16.5.5.1.1.2.cmml" xref="A4.SS1.SSS2.p1.16.m16.5.5.1.1.2"></times><apply id="A4.SS1.SSS2.p1.16.m16.5.5.1.1.3.cmml" xref="A4.SS1.SSS2.p1.16.m16.5.5.1.1.3"><ci id="A4.SS1.SSS2.p1.16.m16.5.5.1.1.3.1.cmml" xref="A4.SS1.SSS2.p1.16.m16.5.5.1.1.3.1">∇</ci><ci id="A4.SS1.SSS2.p1.16.m16.5.5.1.1.3.2.cmml" xref="A4.SS1.SSS2.p1.16.m16.5.5.1.1.3.2">ℱ</ci></apply><interval closure="open" id="A4.SS1.SSS2.p1.16.m16.5.5.1.1.1.2.cmml" xref="A4.SS1.SSS2.p1.16.m16.5.5.1.1.1.1"><apply id="A4.SS1.SSS2.p1.16.m16.5.5.1.1.1.1.1.cmml" xref="A4.SS1.SSS2.p1.16.m16.5.5.1.1.1.1.1"><csymbol cd="ambiguous" id="A4.SS1.SSS2.p1.16.m16.5.5.1.1.1.1.1.1.cmml" xref="A4.SS1.SSS2.p1.16.m16.5.5.1.1.1.1.1">subscript</csymbol><ci id="A4.SS1.SSS2.p1.16.m16.5.5.1.1.1.1.1.2.cmml" xref="A4.SS1.SSS2.p1.16.m16.5.5.1.1.1.1.1.2">𝐱</ci><apply id="A4.SS1.SSS2.p1.16.m16.1.1.1.cmml" xref="A4.SS1.SSS2.p1.16.m16.1.1.1"><times id="A4.SS1.SSS2.p1.16.m16.1.1.1.2.cmml" xref="A4.SS1.SSS2.p1.16.m16.1.1.1.2"></times><ci id="A4.SS1.SSS2.p1.16.m16.1.1.1.3.cmml" xref="A4.SS1.SSS2.p1.16.m16.1.1.1.3">italic-ϕ</ci><ci id="A4.SS1.SSS2.p1.16.m16.1.1.1.1.cmml" xref="A4.SS1.SSS2.p1.16.m16.1.1.1.1">𝑛</ci></apply></apply><ci id="A4.SS1.SSS2.p1.16.m16.4.4.cmml" xref="A4.SS1.SSS2.p1.16.m16.4.4">𝐲</ci></interval></apply><apply id="A4.SS1.SSS2.p1.16.m16.6.6.2.2.cmml" xref="A4.SS1.SSS2.p1.16.m16.6.6.2.2"><times id="A4.SS1.SSS2.p1.16.m16.6.6.2.2.2.cmml" xref="A4.SS1.SSS2.p1.16.m16.6.6.2.2.2"></times><ci id="A4.SS1.SSS2.p1.16.m16.6.6.2.2.3.cmml" xref="A4.SS1.SSS2.p1.16.m16.6.6.2.2.3">𝛼</ci><apply id="A4.SS1.SSS2.p1.16.m16.6.6.2.2.4.cmml" xref="A4.SS1.SSS2.p1.16.m16.6.6.2.2.4"><ci id="A4.SS1.SSS2.p1.16.m16.6.6.2.2.4.1.cmml" xref="A4.SS1.SSS2.p1.16.m16.6.6.2.2.4.1">∇</ci><apply id="A4.SS1.SSS2.p1.16.m16.6.6.2.2.4.2.cmml" xref="A4.SS1.SSS2.p1.16.m16.6.6.2.2.4.2"><csymbol cd="ambiguous" id="A4.SS1.SSS2.p1.16.m16.6.6.2.2.4.2.1.cmml" xref="A4.SS1.SSS2.p1.16.m16.6.6.2.2.4.2">subscript</csymbol><ci id="A4.SS1.SSS2.p1.16.m16.6.6.2.2.4.2.2.cmml" xref="A4.SS1.SSS2.p1.16.m16.6.6.2.2.4.2.2">ℛ</ci><apply id="A4.SS1.SSS2.p1.16.m16.2.2.1.cmml" xref="A4.SS1.SSS2.p1.16.m16.2.2.1"><csymbol cd="ambiguous" id="A4.SS1.SSS2.p1.16.m16.2.2.1.2.cmml" xref="A4.SS1.SSS2.p1.16.m16.2.2.1">subscript</csymbol><ci id="A4.SS1.SSS2.p1.16.m16.2.2.1.3.cmml" xref="A4.SS1.SSS2.p1.16.m16.2.2.1.3">𝜎</ci><apply id="A4.SS1.SSS2.p1.16.m16.2.2.1.1.1.cmml" xref="A4.SS1.SSS2.p1.16.m16.2.2.1.1.1"><times id="A4.SS1.SSS2.p1.16.m16.2.2.1.1.1.2.cmml" xref="A4.SS1.SSS2.p1.16.m16.2.2.1.1.1.2"></times><ci id="A4.SS1.SSS2.p1.16.m16.2.2.1.1.1.3.cmml" xref="A4.SS1.SSS2.p1.16.m16.2.2.1.1.1.3">italic-ϕ</ci><ci id="A4.SS1.SSS2.p1.16.m16.2.2.1.1.1.1.cmml" xref="A4.SS1.SSS2.p1.16.m16.2.2.1.1.1.1">𝑛</ci></apply></apply></apply></apply><apply id="A4.SS1.SSS2.p1.16.m16.6.6.2.2.1.1.1.cmml" xref="A4.SS1.SSS2.p1.16.m16.6.6.2.2.1.1"><csymbol cd="ambiguous" id="A4.SS1.SSS2.p1.16.m16.6.6.2.2.1.1.1.1.cmml" xref="A4.SS1.SSS2.p1.16.m16.6.6.2.2.1.1">subscript</csymbol><ci id="A4.SS1.SSS2.p1.16.m16.6.6.2.2.1.1.1.2.cmml" xref="A4.SS1.SSS2.p1.16.m16.6.6.2.2.1.1.1.2">𝐱</ci><apply id="A4.SS1.SSS2.p1.16.m16.3.3.1.cmml" xref="A4.SS1.SSS2.p1.16.m16.3.3.1"><times id="A4.SS1.SSS2.p1.16.m16.3.3.1.2.cmml" xref="A4.SS1.SSS2.p1.16.m16.3.3.1.2"></times><ci id="A4.SS1.SSS2.p1.16.m16.3.3.1.3.cmml" xref="A4.SS1.SSS2.p1.16.m16.3.3.1.3">italic-ϕ</ci><ci id="A4.SS1.SSS2.p1.16.m16.3.3.1.1.cmml" xref="A4.SS1.SSS2.p1.16.m16.3.3.1.1">𝑛</ci></apply></apply></apply></apply><cn id="A4.SS1.SSS2.p1.16.m16.6.6.4.cmml" type="integer" xref="A4.SS1.SSS2.p1.16.m16.6.6.4">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.SS1.SSS2.p1.16.m16.6c">\nabla\mathcal{F}({\mathbf{x}}_{\phi(n)},{\mathbf{y}})+\alpha\nabla\mathcal{R}% _{\sigma_{\phi(n)}}({\mathbf{x}}_{\phi(n)})=0</annotation><annotation encoding="application/x-llamapun" id="A4.SS1.SSS2.p1.16.m16.6d">∇ caligraphic_F ( bold_x start_POSTSUBSCRIPT italic_ϕ ( italic_n ) end_POSTSUBSCRIPT , bold_y ) + italic_α ∇ caligraphic_R start_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT italic_ϕ ( italic_n ) end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( bold_x start_POSTSUBSCRIPT italic_ϕ ( italic_n ) end_POSTSUBSCRIPT ) = 0</annotation></semantics></math>, we have <math alttext="\nabla\mathcal{J}({\mathbf{x}})=0" class="ltx_Math" display="inline" id="A4.SS1.SSS2.p1.17.m17.1"><semantics id="A4.SS1.SSS2.p1.17.m17.1a"><mrow id="A4.SS1.SSS2.p1.17.m17.1.2" xref="A4.SS1.SSS2.p1.17.m17.1.2.cmml"><mrow id="A4.SS1.SSS2.p1.17.m17.1.2.2" xref="A4.SS1.SSS2.p1.17.m17.1.2.2.cmml"><mrow id="A4.SS1.SSS2.p1.17.m17.1.2.2.2" xref="A4.SS1.SSS2.p1.17.m17.1.2.2.2.cmml"><mo id="A4.SS1.SSS2.p1.17.m17.1.2.2.2.1" rspace="0.167em" xref="A4.SS1.SSS2.p1.17.m17.1.2.2.2.1.cmml">∇</mo><mi class="ltx_font_mathcaligraphic" id="A4.SS1.SSS2.p1.17.m17.1.2.2.2.2" xref="A4.SS1.SSS2.p1.17.m17.1.2.2.2.2.cmml">𝒥</mi></mrow><mo id="A4.SS1.SSS2.p1.17.m17.1.2.2.1" xref="A4.SS1.SSS2.p1.17.m17.1.2.2.1.cmml">⁢</mo><mrow id="A4.SS1.SSS2.p1.17.m17.1.2.2.3.2" xref="A4.SS1.SSS2.p1.17.m17.1.2.2.cmml"><mo id="A4.SS1.SSS2.p1.17.m17.1.2.2.3.2.1" stretchy="false" xref="A4.SS1.SSS2.p1.17.m17.1.2.2.cmml">(</mo><mi id="A4.SS1.SSS2.p1.17.m17.1.1" xref="A4.SS1.SSS2.p1.17.m17.1.1.cmml">𝐱</mi><mo id="A4.SS1.SSS2.p1.17.m17.1.2.2.3.2.2" stretchy="false" xref="A4.SS1.SSS2.p1.17.m17.1.2.2.cmml">)</mo></mrow></mrow><mo id="A4.SS1.SSS2.p1.17.m17.1.2.1" xref="A4.SS1.SSS2.p1.17.m17.1.2.1.cmml">=</mo><mn id="A4.SS1.SSS2.p1.17.m17.1.2.3" xref="A4.SS1.SSS2.p1.17.m17.1.2.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="A4.SS1.SSS2.p1.17.m17.1b"><apply id="A4.SS1.SSS2.p1.17.m17.1.2.cmml" xref="A4.SS1.SSS2.p1.17.m17.1.2"><eq id="A4.SS1.SSS2.p1.17.m17.1.2.1.cmml" xref="A4.SS1.SSS2.p1.17.m17.1.2.1"></eq><apply id="A4.SS1.SSS2.p1.17.m17.1.2.2.cmml" xref="A4.SS1.SSS2.p1.17.m17.1.2.2"><times id="A4.SS1.SSS2.p1.17.m17.1.2.2.1.cmml" xref="A4.SS1.SSS2.p1.17.m17.1.2.2.1"></times><apply id="A4.SS1.SSS2.p1.17.m17.1.2.2.2.cmml" xref="A4.SS1.SSS2.p1.17.m17.1.2.2.2"><ci id="A4.SS1.SSS2.p1.17.m17.1.2.2.2.1.cmml" xref="A4.SS1.SSS2.p1.17.m17.1.2.2.2.1">∇</ci><ci id="A4.SS1.SSS2.p1.17.m17.1.2.2.2.2.cmml" xref="A4.SS1.SSS2.p1.17.m17.1.2.2.2.2">𝒥</ci></apply><ci id="A4.SS1.SSS2.p1.17.m17.1.1.cmml" xref="A4.SS1.SSS2.p1.17.m17.1.1">𝐱</ci></apply><cn id="A4.SS1.SSS2.p1.17.m17.1.2.3.cmml" type="integer" xref="A4.SS1.SSS2.p1.17.m17.1.2.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.SS1.SSS2.p1.17.m17.1c">\nabla\mathcal{J}({\mathbf{x}})=0</annotation><annotation encoding="application/x-llamapun" id="A4.SS1.SSS2.p1.17.m17.1d">∇ caligraphic_J ( bold_x ) = 0</annotation></semantics></math>, which shows that <math alttext="{\mathbf{x}}\in\mathbf{S}^{\star}_{\mathbf{K}}" class="ltx_Math" display="inline" id="A4.SS1.SSS2.p1.18.m18.1"><semantics id="A4.SS1.SSS2.p1.18.m18.1a"><mrow id="A4.SS1.SSS2.p1.18.m18.1.1" xref="A4.SS1.SSS2.p1.18.m18.1.1.cmml"><mi id="A4.SS1.SSS2.p1.18.m18.1.1.2" xref="A4.SS1.SSS2.p1.18.m18.1.1.2.cmml">𝐱</mi><mo id="A4.SS1.SSS2.p1.18.m18.1.1.1" xref="A4.SS1.SSS2.p1.18.m18.1.1.1.cmml">∈</mo><msubsup id="A4.SS1.SSS2.p1.18.m18.1.1.3" xref="A4.SS1.SSS2.p1.18.m18.1.1.3.cmml"><mi id="A4.SS1.SSS2.p1.18.m18.1.1.3.2.2" xref="A4.SS1.SSS2.p1.18.m18.1.1.3.2.2.cmml">𝐒</mi><mi id="A4.SS1.SSS2.p1.18.m18.1.1.3.3" xref="A4.SS1.SSS2.p1.18.m18.1.1.3.3.cmml">𝐊</mi><mo id="A4.SS1.SSS2.p1.18.m18.1.1.3.2.3" xref="A4.SS1.SSS2.p1.18.m18.1.1.3.2.3.cmml">⋆</mo></msubsup></mrow><annotation-xml encoding="MathML-Content" id="A4.SS1.SSS2.p1.18.m18.1b"><apply id="A4.SS1.SSS2.p1.18.m18.1.1.cmml" xref="A4.SS1.SSS2.p1.18.m18.1.1"><in id="A4.SS1.SSS2.p1.18.m18.1.1.1.cmml" xref="A4.SS1.SSS2.p1.18.m18.1.1.1"></in><ci id="A4.SS1.SSS2.p1.18.m18.1.1.2.cmml" xref="A4.SS1.SSS2.p1.18.m18.1.1.2">𝐱</ci><apply id="A4.SS1.SSS2.p1.18.m18.1.1.3.cmml" xref="A4.SS1.SSS2.p1.18.m18.1.1.3"><csymbol cd="ambiguous" id="A4.SS1.SSS2.p1.18.m18.1.1.3.1.cmml" xref="A4.SS1.SSS2.p1.18.m18.1.1.3">subscript</csymbol><apply id="A4.SS1.SSS2.p1.18.m18.1.1.3.2.cmml" xref="A4.SS1.SSS2.p1.18.m18.1.1.3"><csymbol cd="ambiguous" id="A4.SS1.SSS2.p1.18.m18.1.1.3.2.1.cmml" xref="A4.SS1.SSS2.p1.18.m18.1.1.3">superscript</csymbol><ci id="A4.SS1.SSS2.p1.18.m18.1.1.3.2.2.cmml" xref="A4.SS1.SSS2.p1.18.m18.1.1.3.2.2">𝐒</ci><ci id="A4.SS1.SSS2.p1.18.m18.1.1.3.2.3.cmml" xref="A4.SS1.SSS2.p1.18.m18.1.1.3.2.3">⋆</ci></apply><ci id="A4.SS1.SSS2.p1.18.m18.1.1.3.3.cmml" xref="A4.SS1.SSS2.p1.18.m18.1.1.3.3">𝐊</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.SS1.SSS2.p1.18.m18.1c">{\mathbf{x}}\in\mathbf{S}^{\star}_{\mathbf{K}}</annotation><annotation encoding="application/x-llamapun" id="A4.SS1.SSS2.p1.18.m18.1d">bold_x ∈ bold_S start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT bold_K end_POSTSUBSCRIPT</annotation></semantics></math>.</p> </div> </section> <section class="ltx_subsubsection" id="A4.SS1.SSS3"> <h4 class="ltx_title ltx_title_subsubsection"> <span class="ltx_tag ltx_tag_subsubsection">D.1.3 </span>Proof of Proposition <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#S3.Thmproposition3" title="Proposition 3.3. ‣ 3.2 Unbiased algorithm analysis ‣ 3 Convergence Analysis ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_tag">3.3</span></a> </h4> <div class="ltx_para" id="A4.SS1.SSS3.p1"> <p class="ltx_p" id="A4.SS1.SSS3.p1.3">We will apply the result <cite class="ltx_cite ltx_citemacro_citep">(Tadic & Doucet, <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#bib.bib72" title="">2017</a>, Theorem 2.1, (ii))</cite>. The <cite class="ltx_cite ltx_citemacro_citep">(Tadic & Doucet, <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#bib.bib72" title="">2017</a>, Assumption 2.1)</cite> is verified by Assumption <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#S3.Thmtheorem3" title="Assumption 3.3. ‣ 3.2 Unbiased algorithm analysis ‣ 3 Convergence Analysis ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_tag">3.3</span></a>. First we define <math alttext="\xi_{k}=\nabla\mathcal{F}({\mathbf{x}}_{k},{\mathbf{y}})+\alpha\nabla\log p_{% \sigma}({\mathbf{x}}_{k}+\zeta_{k})-\nabla\mathcal{J}_{\sigma}(x_{k})=\alpha(% \nabla\log p_{\sigma}({\mathbf{x}}_{k}+\zeta_{k})-\nabla\log p({\mathbf{x}}_{k% }))" class="ltx_Math" display="inline" id="A4.SS1.SSS3.p1.1.m1.5"><semantics id="A4.SS1.SSS3.p1.1.m1.5a"><mrow id="A4.SS1.SSS3.p1.1.m1.5.5" xref="A4.SS1.SSS3.p1.1.m1.5.5.cmml"><msub id="A4.SS1.SSS3.p1.1.m1.5.5.6" xref="A4.SS1.SSS3.p1.1.m1.5.5.6.cmml"><mi id="A4.SS1.SSS3.p1.1.m1.5.5.6.2" xref="A4.SS1.SSS3.p1.1.m1.5.5.6.2.cmml">ξ</mi><mi id="A4.SS1.SSS3.p1.1.m1.5.5.6.3" xref="A4.SS1.SSS3.p1.1.m1.5.5.6.3.cmml">k</mi></msub><mo id="A4.SS1.SSS3.p1.1.m1.5.5.7" xref="A4.SS1.SSS3.p1.1.m1.5.5.7.cmml">=</mo><mrow id="A4.SS1.SSS3.p1.1.m1.4.4.3" xref="A4.SS1.SSS3.p1.1.m1.4.4.3.cmml"><mrow id="A4.SS1.SSS3.p1.1.m1.3.3.2.2" xref="A4.SS1.SSS3.p1.1.m1.3.3.2.2.cmml"><mrow id="A4.SS1.SSS3.p1.1.m1.2.2.1.1.1" xref="A4.SS1.SSS3.p1.1.m1.2.2.1.1.1.cmml"><mrow id="A4.SS1.SSS3.p1.1.m1.2.2.1.1.1.3" xref="A4.SS1.SSS3.p1.1.m1.2.2.1.1.1.3.cmml"><mo id="A4.SS1.SSS3.p1.1.m1.2.2.1.1.1.3.1" rspace="0.167em" xref="A4.SS1.SSS3.p1.1.m1.2.2.1.1.1.3.1.cmml">∇</mo><mi class="ltx_font_mathcaligraphic" id="A4.SS1.SSS3.p1.1.m1.2.2.1.1.1.3.2" xref="A4.SS1.SSS3.p1.1.m1.2.2.1.1.1.3.2.cmml">ℱ</mi></mrow><mo id="A4.SS1.SSS3.p1.1.m1.2.2.1.1.1.2" xref="A4.SS1.SSS3.p1.1.m1.2.2.1.1.1.2.cmml">⁢</mo><mrow id="A4.SS1.SSS3.p1.1.m1.2.2.1.1.1.1.1" xref="A4.SS1.SSS3.p1.1.m1.2.2.1.1.1.1.2.cmml"><mo id="A4.SS1.SSS3.p1.1.m1.2.2.1.1.1.1.1.2" stretchy="false" xref="A4.SS1.SSS3.p1.1.m1.2.2.1.1.1.1.2.cmml">(</mo><msub id="A4.SS1.SSS3.p1.1.m1.2.2.1.1.1.1.1.1" xref="A4.SS1.SSS3.p1.1.m1.2.2.1.1.1.1.1.1.cmml"><mi id="A4.SS1.SSS3.p1.1.m1.2.2.1.1.1.1.1.1.2" xref="A4.SS1.SSS3.p1.1.m1.2.2.1.1.1.1.1.1.2.cmml">𝐱</mi><mi id="A4.SS1.SSS3.p1.1.m1.2.2.1.1.1.1.1.1.3" xref="A4.SS1.SSS3.p1.1.m1.2.2.1.1.1.1.1.1.3.cmml">k</mi></msub><mo id="A4.SS1.SSS3.p1.1.m1.2.2.1.1.1.1.1.3" xref="A4.SS1.SSS3.p1.1.m1.2.2.1.1.1.1.2.cmml">,</mo><mi id="A4.SS1.SSS3.p1.1.m1.1.1" xref="A4.SS1.SSS3.p1.1.m1.1.1.cmml">𝐲</mi><mo id="A4.SS1.SSS3.p1.1.m1.2.2.1.1.1.1.1.4" stretchy="false" xref="A4.SS1.SSS3.p1.1.m1.2.2.1.1.1.1.2.cmml">)</mo></mrow></mrow><mo id="A4.SS1.SSS3.p1.1.m1.3.3.2.2.3" xref="A4.SS1.SSS3.p1.1.m1.3.3.2.2.3.cmml">+</mo><mrow id="A4.SS1.SSS3.p1.1.m1.3.3.2.2.2" xref="A4.SS1.SSS3.p1.1.m1.3.3.2.2.2.cmml"><mi id="A4.SS1.SSS3.p1.1.m1.3.3.2.2.2.3" xref="A4.SS1.SSS3.p1.1.m1.3.3.2.2.2.3.cmml">α</mi><mo id="A4.SS1.SSS3.p1.1.m1.3.3.2.2.2.2" lspace="0.167em" xref="A4.SS1.SSS3.p1.1.m1.3.3.2.2.2.2.cmml">⁢</mo><mrow id="A4.SS1.SSS3.p1.1.m1.3.3.2.2.2.4" xref="A4.SS1.SSS3.p1.1.m1.3.3.2.2.2.4.cmml"><mrow id="A4.SS1.SSS3.p1.1.m1.3.3.2.2.2.4.1" xref="A4.SS1.SSS3.p1.1.m1.3.3.2.2.2.4.1.cmml"><mo id="A4.SS1.SSS3.p1.1.m1.3.3.2.2.2.4.1.1" rspace="0.167em" xref="A4.SS1.SSS3.p1.1.m1.3.3.2.2.2.4.1.1.cmml">∇</mo><mi 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id="A4.SS1.SSS3.p1.1.m1.5.5.4.1.1.1.2.cmml" xref="A4.SS1.SSS3.p1.1.m1.5.5.4.1.1.1.2"><times id="A4.SS1.SSS3.p1.1.m1.5.5.4.1.1.1.2.2.cmml" xref="A4.SS1.SSS3.p1.1.m1.5.5.4.1.1.1.2.2"></times><apply id="A4.SS1.SSS3.p1.1.m1.5.5.4.1.1.1.2.3.cmml" xref="A4.SS1.SSS3.p1.1.m1.5.5.4.1.1.1.2.3"><apply id="A4.SS1.SSS3.p1.1.m1.5.5.4.1.1.1.2.3.1.cmml" xref="A4.SS1.SSS3.p1.1.m1.5.5.4.1.1.1.2.3.1"><ci id="A4.SS1.SSS3.p1.1.m1.5.5.4.1.1.1.2.3.1.1.cmml" xref="A4.SS1.SSS3.p1.1.m1.5.5.4.1.1.1.2.3.1.1">∇</ci><log id="A4.SS1.SSS3.p1.1.m1.5.5.4.1.1.1.2.3.1.2.cmml" xref="A4.SS1.SSS3.p1.1.m1.5.5.4.1.1.1.2.3.1.2"></log></apply><ci id="A4.SS1.SSS3.p1.1.m1.5.5.4.1.1.1.2.3.2.cmml" xref="A4.SS1.SSS3.p1.1.m1.5.5.4.1.1.1.2.3.2">𝑝</ci></apply><apply id="A4.SS1.SSS3.p1.1.m1.5.5.4.1.1.1.2.1.1.1.cmml" xref="A4.SS1.SSS3.p1.1.m1.5.5.4.1.1.1.2.1.1"><csymbol cd="ambiguous" id="A4.SS1.SSS3.p1.1.m1.5.5.4.1.1.1.2.1.1.1.1.cmml" xref="A4.SS1.SSS3.p1.1.m1.5.5.4.1.1.1.2.1.1">subscript</csymbol><ci id="A4.SS1.SSS3.p1.1.m1.5.5.4.1.1.1.2.1.1.1.2.cmml" xref="A4.SS1.SSS3.p1.1.m1.5.5.4.1.1.1.2.1.1.1.2">𝐱</ci><ci id="A4.SS1.SSS3.p1.1.m1.5.5.4.1.1.1.2.1.1.1.3.cmml" xref="A4.SS1.SSS3.p1.1.m1.5.5.4.1.1.1.2.1.1.1.3">𝑘</ci></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.SS1.SSS3.p1.1.m1.5c">\xi_{k}=\nabla\mathcal{F}({\mathbf{x}}_{k},{\mathbf{y}})+\alpha\nabla\log p_{% \sigma}({\mathbf{x}}_{k}+\zeta_{k})-\nabla\mathcal{J}_{\sigma}(x_{k})=\alpha(% \nabla\log p_{\sigma}({\mathbf{x}}_{k}+\zeta_{k})-\nabla\log p({\mathbf{x}}_{k% }))</annotation><annotation encoding="application/x-llamapun" id="A4.SS1.SSS3.p1.1.m1.5d">italic_ξ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT = ∇ caligraphic_F ( bold_x start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT , bold_y ) + italic_α ∇ roman_log italic_p start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ( bold_x start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT + italic_ζ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) - ∇ caligraphic_J start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ( italic_x start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) = italic_α ( ∇ roman_log italic_p start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ( bold_x start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT + italic_ζ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) - ∇ roman_log italic_p ( bold_x start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) )</annotation></semantics></math>, with <math alttext="\alpha>0,\zeta_{k}\sim\mathcal{N}(0,\sigma^{2}\mathbf{I}_{d})" class="ltx_Math" display="inline" id="A4.SS1.SSS3.p1.2.m2.3"><semantics id="A4.SS1.SSS3.p1.2.m2.3a"><mrow id="A4.SS1.SSS3.p1.2.m2.3.3.2" xref="A4.SS1.SSS3.p1.2.m2.3.3.3.cmml"><mrow id="A4.SS1.SSS3.p1.2.m2.2.2.1.1" xref="A4.SS1.SSS3.p1.2.m2.2.2.1.1.cmml"><mi id="A4.SS1.SSS3.p1.2.m2.2.2.1.1.2" xref="A4.SS1.SSS3.p1.2.m2.2.2.1.1.2.cmml">α</mi><mo id="A4.SS1.SSS3.p1.2.m2.2.2.1.1.1" xref="A4.SS1.SSS3.p1.2.m2.2.2.1.1.1.cmml">></mo><mn id="A4.SS1.SSS3.p1.2.m2.2.2.1.1.3" xref="A4.SS1.SSS3.p1.2.m2.2.2.1.1.3.cmml">0</mn></mrow><mo id="A4.SS1.SSS3.p1.2.m2.3.3.2.3" xref="A4.SS1.SSS3.p1.2.m2.3.3.3a.cmml">,</mo><mrow id="A4.SS1.SSS3.p1.2.m2.3.3.2.2" xref="A4.SS1.SSS3.p1.2.m2.3.3.2.2.cmml"><msub id="A4.SS1.SSS3.p1.2.m2.3.3.2.2.3" xref="A4.SS1.SSS3.p1.2.m2.3.3.2.2.3.cmml"><mi id="A4.SS1.SSS3.p1.2.m2.3.3.2.2.3.2" xref="A4.SS1.SSS3.p1.2.m2.3.3.2.2.3.2.cmml">ζ</mi><mi id="A4.SS1.SSS3.p1.2.m2.3.3.2.2.3.3" xref="A4.SS1.SSS3.p1.2.m2.3.3.2.2.3.3.cmml">k</mi></msub><mo id="A4.SS1.SSS3.p1.2.m2.3.3.2.2.2" xref="A4.SS1.SSS3.p1.2.m2.3.3.2.2.2.cmml">∼</mo><mrow id="A4.SS1.SSS3.p1.2.m2.3.3.2.2.1" xref="A4.SS1.SSS3.p1.2.m2.3.3.2.2.1.cmml"><mi class="ltx_font_mathcaligraphic" id="A4.SS1.SSS3.p1.2.m2.3.3.2.2.1.3" xref="A4.SS1.SSS3.p1.2.m2.3.3.2.2.1.3.cmml">𝒩</mi><mo id="A4.SS1.SSS3.p1.2.m2.3.3.2.2.1.2" xref="A4.SS1.SSS3.p1.2.m2.3.3.2.2.1.2.cmml">⁢</mo><mrow id="A4.SS1.SSS3.p1.2.m2.3.3.2.2.1.1.1" xref="A4.SS1.SSS3.p1.2.m2.3.3.2.2.1.1.2.cmml"><mo id="A4.SS1.SSS3.p1.2.m2.3.3.2.2.1.1.1.2" stretchy="false" xref="A4.SS1.SSS3.p1.2.m2.3.3.2.2.1.1.2.cmml">(</mo><mn id="A4.SS1.SSS3.p1.2.m2.1.1" xref="A4.SS1.SSS3.p1.2.m2.1.1.cmml">0</mn><mo id="A4.SS1.SSS3.p1.2.m2.3.3.2.2.1.1.1.3" xref="A4.SS1.SSS3.p1.2.m2.3.3.2.2.1.1.2.cmml">,</mo><mrow id="A4.SS1.SSS3.p1.2.m2.3.3.2.2.1.1.1.1" xref="A4.SS1.SSS3.p1.2.m2.3.3.2.2.1.1.1.1.cmml"><msup id="A4.SS1.SSS3.p1.2.m2.3.3.2.2.1.1.1.1.2" xref="A4.SS1.SSS3.p1.2.m2.3.3.2.2.1.1.1.1.2.cmml"><mi id="A4.SS1.SSS3.p1.2.m2.3.3.2.2.1.1.1.1.2.2" xref="A4.SS1.SSS3.p1.2.m2.3.3.2.2.1.1.1.1.2.2.cmml">σ</mi><mn id="A4.SS1.SSS3.p1.2.m2.3.3.2.2.1.1.1.1.2.3" xref="A4.SS1.SSS3.p1.2.m2.3.3.2.2.1.1.1.1.2.3.cmml">2</mn></msup><mo id="A4.SS1.SSS3.p1.2.m2.3.3.2.2.1.1.1.1.1" xref="A4.SS1.SSS3.p1.2.m2.3.3.2.2.1.1.1.1.1.cmml">⁢</mo><msub id="A4.SS1.SSS3.p1.2.m2.3.3.2.2.1.1.1.1.3" xref="A4.SS1.SSS3.p1.2.m2.3.3.2.2.1.1.1.1.3.cmml"><mi id="A4.SS1.SSS3.p1.2.m2.3.3.2.2.1.1.1.1.3.2" xref="A4.SS1.SSS3.p1.2.m2.3.3.2.2.1.1.1.1.3.2.cmml">𝐈</mi><mi id="A4.SS1.SSS3.p1.2.m2.3.3.2.2.1.1.1.1.3.3" xref="A4.SS1.SSS3.p1.2.m2.3.3.2.2.1.1.1.1.3.3.cmml">d</mi></msub></mrow><mo id="A4.SS1.SSS3.p1.2.m2.3.3.2.2.1.1.1.4" stretchy="false" xref="A4.SS1.SSS3.p1.2.m2.3.3.2.2.1.1.2.cmml">)</mo></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="A4.SS1.SSS3.p1.2.m2.3b"><apply id="A4.SS1.SSS3.p1.2.m2.3.3.3.cmml" xref="A4.SS1.SSS3.p1.2.m2.3.3.2"><csymbol cd="ambiguous" id="A4.SS1.SSS3.p1.2.m2.3.3.3a.cmml" xref="A4.SS1.SSS3.p1.2.m2.3.3.2.3">formulae-sequence</csymbol><apply id="A4.SS1.SSS3.p1.2.m2.2.2.1.1.cmml" xref="A4.SS1.SSS3.p1.2.m2.2.2.1.1"><gt id="A4.SS1.SSS3.p1.2.m2.2.2.1.1.1.cmml" xref="A4.SS1.SSS3.p1.2.m2.2.2.1.1.1"></gt><ci id="A4.SS1.SSS3.p1.2.m2.2.2.1.1.2.cmml" xref="A4.SS1.SSS3.p1.2.m2.2.2.1.1.2">𝛼</ci><cn id="A4.SS1.SSS3.p1.2.m2.2.2.1.1.3.cmml" type="integer" xref="A4.SS1.SSS3.p1.2.m2.2.2.1.1.3">0</cn></apply><apply id="A4.SS1.SSS3.p1.2.m2.3.3.2.2.cmml" xref="A4.SS1.SSS3.p1.2.m2.3.3.2.2"><csymbol cd="latexml" id="A4.SS1.SSS3.p1.2.m2.3.3.2.2.2.cmml" xref="A4.SS1.SSS3.p1.2.m2.3.3.2.2.2">similar-to</csymbol><apply id="A4.SS1.SSS3.p1.2.m2.3.3.2.2.3.cmml" xref="A4.SS1.SSS3.p1.2.m2.3.3.2.2.3"><csymbol cd="ambiguous" id="A4.SS1.SSS3.p1.2.m2.3.3.2.2.3.1.cmml" xref="A4.SS1.SSS3.p1.2.m2.3.3.2.2.3">subscript</csymbol><ci id="A4.SS1.SSS3.p1.2.m2.3.3.2.2.3.2.cmml" xref="A4.SS1.SSS3.p1.2.m2.3.3.2.2.3.2">𝜁</ci><ci id="A4.SS1.SSS3.p1.2.m2.3.3.2.2.3.3.cmml" xref="A4.SS1.SSS3.p1.2.m2.3.3.2.2.3.3">𝑘</ci></apply><apply id="A4.SS1.SSS3.p1.2.m2.3.3.2.2.1.cmml" xref="A4.SS1.SSS3.p1.2.m2.3.3.2.2.1"><times id="A4.SS1.SSS3.p1.2.m2.3.3.2.2.1.2.cmml" xref="A4.SS1.SSS3.p1.2.m2.3.3.2.2.1.2"></times><ci id="A4.SS1.SSS3.p1.2.m2.3.3.2.2.1.3.cmml" xref="A4.SS1.SSS3.p1.2.m2.3.3.2.2.1.3">𝒩</ci><interval closure="open" id="A4.SS1.SSS3.p1.2.m2.3.3.2.2.1.1.2.cmml" xref="A4.SS1.SSS3.p1.2.m2.3.3.2.2.1.1.1"><cn id="A4.SS1.SSS3.p1.2.m2.1.1.cmml" type="integer" xref="A4.SS1.SSS3.p1.2.m2.1.1">0</cn><apply id="A4.SS1.SSS3.p1.2.m2.3.3.2.2.1.1.1.1.cmml" xref="A4.SS1.SSS3.p1.2.m2.3.3.2.2.1.1.1.1"><times id="A4.SS1.SSS3.p1.2.m2.3.3.2.2.1.1.1.1.1.cmml" xref="A4.SS1.SSS3.p1.2.m2.3.3.2.2.1.1.1.1.1"></times><apply id="A4.SS1.SSS3.p1.2.m2.3.3.2.2.1.1.1.1.2.cmml" xref="A4.SS1.SSS3.p1.2.m2.3.3.2.2.1.1.1.1.2"><csymbol cd="ambiguous" id="A4.SS1.SSS3.p1.2.m2.3.3.2.2.1.1.1.1.2.1.cmml" xref="A4.SS1.SSS3.p1.2.m2.3.3.2.2.1.1.1.1.2">superscript</csymbol><ci id="A4.SS1.SSS3.p1.2.m2.3.3.2.2.1.1.1.1.2.2.cmml" xref="A4.SS1.SSS3.p1.2.m2.3.3.2.2.1.1.1.1.2.2">𝜎</ci><cn id="A4.SS1.SSS3.p1.2.m2.3.3.2.2.1.1.1.1.2.3.cmml" type="integer" xref="A4.SS1.SSS3.p1.2.m2.3.3.2.2.1.1.1.1.2.3">2</cn></apply><apply id="A4.SS1.SSS3.p1.2.m2.3.3.2.2.1.1.1.1.3.cmml" xref="A4.SS1.SSS3.p1.2.m2.3.3.2.2.1.1.1.1.3"><csymbol cd="ambiguous" id="A4.SS1.SSS3.p1.2.m2.3.3.2.2.1.1.1.1.3.1.cmml" xref="A4.SS1.SSS3.p1.2.m2.3.3.2.2.1.1.1.1.3">subscript</csymbol><ci id="A4.SS1.SSS3.p1.2.m2.3.3.2.2.1.1.1.1.3.2.cmml" xref="A4.SS1.SSS3.p1.2.m2.3.3.2.2.1.1.1.1.3.2">𝐈</ci><ci id="A4.SS1.SSS3.p1.2.m2.3.3.2.2.1.1.1.1.3.3.cmml" xref="A4.SS1.SSS3.p1.2.m2.3.3.2.2.1.1.1.1.3.3">𝑑</ci></apply></apply></interval></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.SS1.SSS3.p1.2.m2.3c">\alpha>0,\zeta_{k}\sim\mathcal{N}(0,\sigma^{2}\mathbf{I}_{d})</annotation><annotation encoding="application/x-llamapun" id="A4.SS1.SSS3.p1.2.m2.3d">italic_α > 0 , italic_ζ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ∼ caligraphic_N ( 0 , italic_σ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT bold_I start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT )</annotation></semantics></math>. By definition we have <math alttext="\mathbb{E}_{\zeta_{k}}(\xi_{k})=0" class="ltx_Math" display="inline" id="A4.SS1.SSS3.p1.3.m3.1"><semantics id="A4.SS1.SSS3.p1.3.m3.1a"><mrow id="A4.SS1.SSS3.p1.3.m3.1.1" xref="A4.SS1.SSS3.p1.3.m3.1.1.cmml"><mrow id="A4.SS1.SSS3.p1.3.m3.1.1.1" xref="A4.SS1.SSS3.p1.3.m3.1.1.1.cmml"><msub id="A4.SS1.SSS3.p1.3.m3.1.1.1.3" xref="A4.SS1.SSS3.p1.3.m3.1.1.1.3.cmml"><mi id="A4.SS1.SSS3.p1.3.m3.1.1.1.3.2" xref="A4.SS1.SSS3.p1.3.m3.1.1.1.3.2.cmml">𝔼</mi><msub id="A4.SS1.SSS3.p1.3.m3.1.1.1.3.3" xref="A4.SS1.SSS3.p1.3.m3.1.1.1.3.3.cmml"><mi id="A4.SS1.SSS3.p1.3.m3.1.1.1.3.3.2" xref="A4.SS1.SSS3.p1.3.m3.1.1.1.3.3.2.cmml">ζ</mi><mi id="A4.SS1.SSS3.p1.3.m3.1.1.1.3.3.3" xref="A4.SS1.SSS3.p1.3.m3.1.1.1.3.3.3.cmml">k</mi></msub></msub><mo id="A4.SS1.SSS3.p1.3.m3.1.1.1.2" xref="A4.SS1.SSS3.p1.3.m3.1.1.1.2.cmml">⁢</mo><mrow id="A4.SS1.SSS3.p1.3.m3.1.1.1.1.1" xref="A4.SS1.SSS3.p1.3.m3.1.1.1.1.1.1.cmml"><mo id="A4.SS1.SSS3.p1.3.m3.1.1.1.1.1.2" stretchy="false" xref="A4.SS1.SSS3.p1.3.m3.1.1.1.1.1.1.cmml">(</mo><msub id="A4.SS1.SSS3.p1.3.m3.1.1.1.1.1.1" xref="A4.SS1.SSS3.p1.3.m3.1.1.1.1.1.1.cmml"><mi id="A4.SS1.SSS3.p1.3.m3.1.1.1.1.1.1.2" xref="A4.SS1.SSS3.p1.3.m3.1.1.1.1.1.1.2.cmml">ξ</mi><mi id="A4.SS1.SSS3.p1.3.m3.1.1.1.1.1.1.3" xref="A4.SS1.SSS3.p1.3.m3.1.1.1.1.1.1.3.cmml">k</mi></msub><mo id="A4.SS1.SSS3.p1.3.m3.1.1.1.1.1.3" stretchy="false" xref="A4.SS1.SSS3.p1.3.m3.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="A4.SS1.SSS3.p1.3.m3.1.1.2" xref="A4.SS1.SSS3.p1.3.m3.1.1.2.cmml">=</mo><mn id="A4.SS1.SSS3.p1.3.m3.1.1.3" xref="A4.SS1.SSS3.p1.3.m3.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="A4.SS1.SSS3.p1.3.m3.1b"><apply id="A4.SS1.SSS3.p1.3.m3.1.1.cmml" xref="A4.SS1.SSS3.p1.3.m3.1.1"><eq id="A4.SS1.SSS3.p1.3.m3.1.1.2.cmml" xref="A4.SS1.SSS3.p1.3.m3.1.1.2"></eq><apply id="A4.SS1.SSS3.p1.3.m3.1.1.1.cmml" xref="A4.SS1.SSS3.p1.3.m3.1.1.1"><times id="A4.SS1.SSS3.p1.3.m3.1.1.1.2.cmml" xref="A4.SS1.SSS3.p1.3.m3.1.1.1.2"></times><apply id="A4.SS1.SSS3.p1.3.m3.1.1.1.3.cmml" xref="A4.SS1.SSS3.p1.3.m3.1.1.1.3"><csymbol cd="ambiguous" id="A4.SS1.SSS3.p1.3.m3.1.1.1.3.1.cmml" xref="A4.SS1.SSS3.p1.3.m3.1.1.1.3">subscript</csymbol><ci id="A4.SS1.SSS3.p1.3.m3.1.1.1.3.2.cmml" xref="A4.SS1.SSS3.p1.3.m3.1.1.1.3.2">𝔼</ci><apply id="A4.SS1.SSS3.p1.3.m3.1.1.1.3.3.cmml" xref="A4.SS1.SSS3.p1.3.m3.1.1.1.3.3"><csymbol cd="ambiguous" id="A4.SS1.SSS3.p1.3.m3.1.1.1.3.3.1.cmml" xref="A4.SS1.SSS3.p1.3.m3.1.1.1.3.3">subscript</csymbol><ci id="A4.SS1.SSS3.p1.3.m3.1.1.1.3.3.2.cmml" xref="A4.SS1.SSS3.p1.3.m3.1.1.1.3.3.2">𝜁</ci><ci id="A4.SS1.SSS3.p1.3.m3.1.1.1.3.3.3.cmml" xref="A4.SS1.SSS3.p1.3.m3.1.1.1.3.3.3">𝑘</ci></apply></apply><apply id="A4.SS1.SSS3.p1.3.m3.1.1.1.1.1.1.cmml" xref="A4.SS1.SSS3.p1.3.m3.1.1.1.1.1"><csymbol cd="ambiguous" id="A4.SS1.SSS3.p1.3.m3.1.1.1.1.1.1.1.cmml" xref="A4.SS1.SSS3.p1.3.m3.1.1.1.1.1">subscript</csymbol><ci id="A4.SS1.SSS3.p1.3.m3.1.1.1.1.1.1.2.cmml" xref="A4.SS1.SSS3.p1.3.m3.1.1.1.1.1.1.2">𝜉</ci><ci id="A4.SS1.SSS3.p1.3.m3.1.1.1.1.1.1.3.cmml" xref="A4.SS1.SSS3.p1.3.m3.1.1.1.1.1.1.3">𝑘</ci></apply></apply><cn id="A4.SS1.SSS3.p1.3.m3.1.1.3.cmml" type="integer" xref="A4.SS1.SSS3.p1.3.m3.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.SS1.SSS3.p1.3.m3.1c">\mathbb{E}_{\zeta_{k}}(\xi_{k})=0</annotation><annotation encoding="application/x-llamapun" id="A4.SS1.SSS3.p1.3.m3.1d">blackboard_E start_POSTSUBSCRIPT italic_ζ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_ξ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) = 0</annotation></semantics></math> and</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A8.EGx51"> <tbody id="A4.Ex85"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math 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xref="A4.Ex85.m1.1.1.1.1.1.1.1.1.1.1.1.1.1.2">𝑥</ci><ci id="A4.Ex85.m1.1.1.1.1.1.1.1.1.1.1.1.1.1.3.cmml" xref="A4.Ex85.m1.1.1.1.1.1.1.1.1.1.1.1.1.1.3">𝑘</ci></apply></apply><apply id="A4.Ex85.m1.1.1.1.1.1.1.1.1.1.3.cmml" xref="A4.Ex85.m1.1.1.1.1.1.1.1.1.1.3"><csymbol cd="ambiguous" id="A4.Ex85.m1.1.1.1.1.1.1.1.1.1.3.1.cmml" xref="A4.Ex85.m1.1.1.1.1.1.1.1.1.1.3">subscript</csymbol><ci id="A4.Ex85.m1.1.1.1.1.1.1.1.1.1.3.2.cmml" xref="A4.Ex85.m1.1.1.1.1.1.1.1.1.1.3.2">𝜉</ci><ci id="A4.Ex85.m1.1.1.1.1.1.1.1.1.1.3.3.cmml" xref="A4.Ex85.m1.1.1.1.1.1.1.1.1.1.3.3">𝑘</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.Ex85.m1.1c">\displaystyle{\mathbf{x}}_{k+1}={\mathbf{x}}_{k}-\delta_{k}(\nabla\mathcal{J}_% {\sigma}(x_{k})+\xi_{k}).</annotation><annotation encoding="application/x-llamapun" id="A4.Ex85.m1.1d">bold_x start_POSTSUBSCRIPT italic_k + 1 end_POSTSUBSCRIPT = bold_x start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT - italic_δ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ( ∇ caligraphic_J start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ( italic_x start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) + italic_ξ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> </div> <div class="ltx_para" id="A4.SS1.SSS3.p2"> <p class="ltx_p" id="A4.SS1.SSS3.p2.1">We have, by using Assumption <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#S3.Thmtheorem2" title="Assumption 3.2. ‣ 3.1 Asymptotics of critical points when 𝜎→0 ‣ 3 Convergence Analysis ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_tag">3.2</span></a> and <math alttext="R_{\mathbf{K}}=\sup_{{\mathbf{x}}\in\mathbf{K}}{\|{\mathbf{x}}\|}" class="ltx_Math" display="inline" id="A4.SS1.SSS3.p2.1.m1.1"><semantics id="A4.SS1.SSS3.p2.1.m1.1a"><mrow id="A4.SS1.SSS3.p2.1.m1.1.2" 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xref="A4.SS1.SSS3.p2.1.m1.1.2.3.2.2.1">norm</csymbol><ci id="A4.SS1.SSS3.p2.1.m1.1.1.cmml" xref="A4.SS1.SSS3.p2.1.m1.1.1">𝐱</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.SS1.SSS3.p2.1.m1.1c">R_{\mathbf{K}}=\sup_{{\mathbf{x}}\in\mathbf{K}}{\|{\mathbf{x}}\|}</annotation><annotation encoding="application/x-llamapun" id="A4.SS1.SSS3.p2.1.m1.1d">italic_R start_POSTSUBSCRIPT bold_K end_POSTSUBSCRIPT = roman_sup start_POSTSUBSCRIPT bold_x ∈ bold_K end_POSTSUBSCRIPT ∥ bold_x ∥</annotation></semantics></math></p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A8.EGx52"> <tbody id="A4.Ex86"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\mathbb{E}(\|\xi_{k}\|^{2}|{\mathbf{x}}_{k})" class="ltx_Math" display="inline" id="A4.Ex86.m1.1"><semantics id="A4.Ex86.m1.1a"><mrow id="A4.Ex86.m1.1.1" xref="A4.Ex86.m1.1.1.cmml"><mi id="A4.Ex86.m1.1.1.3" xref="A4.Ex86.m1.1.1.3.cmml">𝔼</mi><mo id="A4.Ex86.m1.1.1.2" xref="A4.Ex86.m1.1.1.2.cmml">⁢</mo><mrow id="A4.Ex86.m1.1.1.1.1" xref="A4.Ex86.m1.1.1.1.1.1.cmml"><mo id="A4.Ex86.m1.1.1.1.1.2" stretchy="false" xref="A4.Ex86.m1.1.1.1.1.1.cmml">(</mo><mrow id="A4.Ex86.m1.1.1.1.1.1" xref="A4.Ex86.m1.1.1.1.1.1.cmml"><msup id="A4.Ex86.m1.1.1.1.1.1.1" xref="A4.Ex86.m1.1.1.1.1.1.1.cmml"><mrow id="A4.Ex86.m1.1.1.1.1.1.1.1.1" xref="A4.Ex86.m1.1.1.1.1.1.1.1.2.cmml"><mo id="A4.Ex86.m1.1.1.1.1.1.1.1.1.2" stretchy="false" xref="A4.Ex86.m1.1.1.1.1.1.1.1.2.1.cmml">‖</mo><msub id="A4.Ex86.m1.1.1.1.1.1.1.1.1.1" xref="A4.Ex86.m1.1.1.1.1.1.1.1.1.1.cmml"><mi id="A4.Ex86.m1.1.1.1.1.1.1.1.1.1.2" xref="A4.Ex86.m1.1.1.1.1.1.1.1.1.1.2.cmml">ξ</mi><mi id="A4.Ex86.m1.1.1.1.1.1.1.1.1.1.3" xref="A4.Ex86.m1.1.1.1.1.1.1.1.1.1.3.cmml">k</mi></msub><mo id="A4.Ex86.m1.1.1.1.1.1.1.1.1.3" stretchy="false" xref="A4.Ex86.m1.1.1.1.1.1.1.1.2.1.cmml">‖</mo></mrow><mn 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id="A4.Ex86.m1.1.1.1.1.1.3.2.cmml" xref="A4.Ex86.m1.1.1.1.1.1.3.2">𝐱</ci><ci id="A4.Ex86.m1.1.1.1.1.1.3.3.cmml" xref="A4.Ex86.m1.1.1.1.1.1.3.3">𝑘</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.Ex86.m1.1c">\displaystyle\mathbb{E}(\|\xi_{k}\|^{2}|{\mathbf{x}}_{k})</annotation><annotation encoding="application/x-llamapun" id="A4.Ex86.m1.1d">blackboard_E ( ∥ italic_ξ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT | bold_x start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT )</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle=\alpha^{2}\mathbb{E}(\|\nabla\log p_{\sigma}({\mathbf{x}}_{k}+% \zeta_{k})-\mathbb{E}_{\zeta\sim\mathcal{N}(0,\sigma^{2}\mathbf{I}_{d})}\left(% \nabla\log p_{\sigma}({\mathbf{x}}_{k}+\zeta)\right)\|^{2}|{\mathbf{x}}_{k})" class="ltx_Math" display="inline" id="A4.Ex86.m2.3"><semantics id="A4.Ex86.m2.3a"><mrow 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\zeta_{k})-\mathbb{E}_{\zeta\sim\mathcal{N}(0,\sigma^{2}\mathbf{I}_{d})}\left(% \nabla\log p_{\sigma}({\mathbf{x}}_{k}+\zeta)\right)\|^{2}|{\mathbf{x}}_{k})</annotation><annotation encoding="application/x-llamapun" id="A4.Ex86.m2.3d">= italic_α start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT blackboard_E ( ∥ ∇ roman_log italic_p start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ( bold_x start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT + italic_ζ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) - blackboard_E start_POSTSUBSCRIPT italic_ζ ∼ caligraphic_N ( 0 , italic_σ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT bold_I start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT ) end_POSTSUBSCRIPT ( ∇ roman_log italic_p start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ( bold_x start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT + italic_ζ ) ) ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT | bold_x start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="A4.Ex87"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_eqn_cell"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle=\alpha^{2}\left(\mathbb{E}(\|\nabla\log p_{\sigma}({\mathbf{x}}_% {k}+\zeta)\|^{2}|{\mathbf{x}}_{k})-\|\mathbb{E}\left(\nabla\log p_{\sigma}({% \mathbf{x}}_{k}+\zeta)\right)\|^{2}|{\mathbf{x}}_{k}\right)" class="ltx_Math" display="inline" id="A4.Ex87.m1.1"><semantics id="A4.Ex87.m1.1a"><mrow id="A4.Ex87.m1.1.1" xref="A4.Ex87.m1.1.1.cmml"><mi id="A4.Ex87.m1.1.1.3" xref="A4.Ex87.m1.1.1.3.cmml"></mi><mo id="A4.Ex87.m1.1.1.2" xref="A4.Ex87.m1.1.1.2.cmml">=</mo><mrow id="A4.Ex87.m1.1.1.1" xref="A4.Ex87.m1.1.1.1.cmml"><msup id="A4.Ex87.m1.1.1.1.3" xref="A4.Ex87.m1.1.1.1.3.cmml"><mi id="A4.Ex87.m1.1.1.1.3.2" xref="A4.Ex87.m1.1.1.1.3.2.cmml">α</mi><mn id="A4.Ex87.m1.1.1.1.3.3" 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id="A4.Ex87.m1.1.1.1.1.1.1.2.1.3.cmml" type="integer" xref="A4.Ex87.m1.1.1.1.1.1.1.2.1.3">2</cn></apply><apply id="A4.Ex87.m1.1.1.1.1.1.1.2.3.cmml" xref="A4.Ex87.m1.1.1.1.1.1.1.2.3"><csymbol cd="ambiguous" id="A4.Ex87.m1.1.1.1.1.1.1.2.3.1.cmml" xref="A4.Ex87.m1.1.1.1.1.1.1.2.3">subscript</csymbol><ci id="A4.Ex87.m1.1.1.1.1.1.1.2.3.2.cmml" xref="A4.Ex87.m1.1.1.1.1.1.1.2.3.2">𝐱</ci><ci id="A4.Ex87.m1.1.1.1.1.1.1.2.3.3.cmml" xref="A4.Ex87.m1.1.1.1.1.1.1.2.3.3">𝑘</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.Ex87.m1.1c">\displaystyle=\alpha^{2}\left(\mathbb{E}(\|\nabla\log p_{\sigma}({\mathbf{x}}_% {k}+\zeta)\|^{2}|{\mathbf{x}}_{k})-\|\mathbb{E}\left(\nabla\log p_{\sigma}({% \mathbf{x}}_{k}+\zeta)\right)\|^{2}|{\mathbf{x}}_{k}\right)</annotation><annotation encoding="application/x-llamapun" id="A4.Ex87.m1.1d">= italic_α start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( blackboard_E ( ∥ ∇ roman_log italic_p start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ( bold_x start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT + italic_ζ ) ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT | bold_x start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) - ∥ blackboard_E ( ∇ roman_log italic_p start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ( bold_x start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT + italic_ζ ) ) ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT | bold_x start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="A4.Ex88"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_eqn_cell"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\leq\alpha^{2}\mathbb{E}(\|\nabla\log p_{\sigma}({\mathbf{x}}_{k}% +\zeta)\|^{2}|{\mathbf{x}}_{k})" class="ltx_Math" display="inline" id="A4.Ex88.m1.1"><semantics id="A4.Ex88.m1.1a"><mrow 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end_POSTSUBSCRIPT ( bold_x start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT + italic_ζ ) ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT | bold_x start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="A4.Ex89"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_eqn_cell"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\leq\alpha^{2}\mathbb{E}(B^{2}\sigma^{2\beta}\left(1+\|{\mathbf{x% }}_{k}+\zeta\|^{r}\right)^{2}|{\mathbf{x}}_{k})" class="ltx_Math" display="inline" id="A4.Ex89.m1.1"><semantics id="A4.Ex89.m1.1a"><mrow id="A4.Ex89.m1.1.1" xref="A4.Ex89.m1.1.1.cmml"><mi id="A4.Ex89.m1.1.1.3" xref="A4.Ex89.m1.1.1.3.cmml"></mi><mo id="A4.Ex89.m1.1.1.2" xref="A4.Ex89.m1.1.1.2.cmml">≤</mo><mrow id="A4.Ex89.m1.1.1.1" xref="A4.Ex89.m1.1.1.1.cmml"><msup id="A4.Ex89.m1.1.1.1.3" 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id="A4.Ex89.m1.1c">\displaystyle\leq\alpha^{2}\mathbb{E}(B^{2}\sigma^{2\beta}\left(1+\|{\mathbf{x% }}_{k}+\zeta\|^{r}\right)^{2}|{\mathbf{x}}_{k})</annotation><annotation encoding="application/x-llamapun" id="A4.Ex89.m1.1d">≤ italic_α start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT blackboard_E ( italic_B start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_σ start_POSTSUPERSCRIPT 2 italic_β end_POSTSUPERSCRIPT ( 1 + ∥ bold_x start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT + italic_ζ ∥ start_POSTSUPERSCRIPT italic_r end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT | bold_x start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="A4.Ex90"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_eqn_cell"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math 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K}}+\|\zeta\|)^{r}\right)^{2}|{\mathbf{x}}_{k}).</annotation><annotation encoding="application/x-llamapun" id="A4.Ex90.m1.2d">≤ italic_α start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_B start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_σ start_POSTSUPERSCRIPT 2 italic_β end_POSTSUPERSCRIPT blackboard_E ( ( 1 + ( italic_R start_POSTSUBSCRIPT bold_K end_POSTSUBSCRIPT + ∥ italic_ζ ∥ ) start_POSTSUPERSCRIPT italic_r end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT | bold_x start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="A4.SS1.SSS3.p2.8">As <math alttext="\mathbb{E}(\left(1+(R_{\mathbf{K}}+\|\zeta\|)^{r}\right)^{2}|{\mathbf{x}}_{k})% =\mathbb{E}(\left(1+(R_{\mathbf{K}}+\|\zeta\|)^{r}\right)^{2})=C_{r,\sigma,d,R% _{\mathbf{K}}}" class="ltx_Math" display="inline" id="A4.SS1.SSS3.p2.2.m1.8"><semantics id="A4.SS1.SSS3.p2.2.m1.8a"><mrow 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_{\mathbf{K}}}</annotation><annotation encoding="application/x-llamapun" id="A4.SS1.SSS3.p2.2.m1.8d">blackboard_E ( ( 1 + ( italic_R start_POSTSUBSCRIPT bold_K end_POSTSUBSCRIPT + ∥ italic_ζ ∥ ) start_POSTSUPERSCRIPT italic_r end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT | bold_x start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) = blackboard_E ( ( 1 + ( italic_R start_POSTSUBSCRIPT bold_K end_POSTSUBSCRIPT + ∥ italic_ζ ∥ ) start_POSTSUPERSCRIPT italic_r end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) = italic_C start_POSTSUBSCRIPT italic_r , italic_σ , italic_d , italic_R start_POSTSUBSCRIPT bold_K end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math> is a constant that only depends on <math alttext="r,\sigma,d" class="ltx_Math" display="inline" id="A4.SS1.SSS3.p2.3.m2.3"><semantics id="A4.SS1.SSS3.p2.3.m2.3a"><mrow id="A4.SS1.SSS3.p2.3.m2.3.4.2" xref="A4.SS1.SSS3.p2.3.m2.3.4.1.cmml"><mi id="A4.SS1.SSS3.p2.3.m2.1.1" xref="A4.SS1.SSS3.p2.3.m2.1.1.cmml">r</mi><mo id="A4.SS1.SSS3.p2.3.m2.3.4.2.1" xref="A4.SS1.SSS3.p2.3.m2.3.4.1.cmml">,</mo><mi id="A4.SS1.SSS3.p2.3.m2.2.2" xref="A4.SS1.SSS3.p2.3.m2.2.2.cmml">σ</mi><mo id="A4.SS1.SSS3.p2.3.m2.3.4.2.2" xref="A4.SS1.SSS3.p2.3.m2.3.4.1.cmml">,</mo><mi id="A4.SS1.SSS3.p2.3.m2.3.3" xref="A4.SS1.SSS3.p2.3.m2.3.3.cmml">d</mi></mrow><annotation-xml encoding="MathML-Content" id="A4.SS1.SSS3.p2.3.m2.3b"><list id="A4.SS1.SSS3.p2.3.m2.3.4.1.cmml" xref="A4.SS1.SSS3.p2.3.m2.3.4.2"><ci id="A4.SS1.SSS3.p2.3.m2.1.1.cmml" xref="A4.SS1.SSS3.p2.3.m2.1.1">𝑟</ci><ci id="A4.SS1.SSS3.p2.3.m2.2.2.cmml" xref="A4.SS1.SSS3.p2.3.m2.2.2">𝜎</ci><ci id="A4.SS1.SSS3.p2.3.m2.3.3.cmml" xref="A4.SS1.SSS3.p2.3.m2.3.3">𝑑</ci></list></annotation-xml><annotation encoding="application/x-tex" id="A4.SS1.SSS3.p2.3.m2.3c">r,\sigma,d</annotation><annotation encoding="application/x-llamapun" id="A4.SS1.SSS3.p2.3.m2.3d">italic_r , italic_σ , italic_d</annotation></semantics></math> and <math alttext="R_{\mathbf{K}}" class="ltx_Math" display="inline" id="A4.SS1.SSS3.p2.4.m3.1"><semantics id="A4.SS1.SSS3.p2.4.m3.1a"><msub id="A4.SS1.SSS3.p2.4.m3.1.1" xref="A4.SS1.SSS3.p2.4.m3.1.1.cmml"><mi id="A4.SS1.SSS3.p2.4.m3.1.1.2" xref="A4.SS1.SSS3.p2.4.m3.1.1.2.cmml">R</mi><mi id="A4.SS1.SSS3.p2.4.m3.1.1.3" xref="A4.SS1.SSS3.p2.4.m3.1.1.3.cmml">𝐊</mi></msub><annotation-xml encoding="MathML-Content" id="A4.SS1.SSS3.p2.4.m3.1b"><apply id="A4.SS1.SSS3.p2.4.m3.1.1.cmml" xref="A4.SS1.SSS3.p2.4.m3.1.1"><csymbol cd="ambiguous" id="A4.SS1.SSS3.p2.4.m3.1.1.1.cmml" xref="A4.SS1.SSS3.p2.4.m3.1.1">subscript</csymbol><ci id="A4.SS1.SSS3.p2.4.m3.1.1.2.cmml" xref="A4.SS1.SSS3.p2.4.m3.1.1.2">𝑅</ci><ci id="A4.SS1.SSS3.p2.4.m3.1.1.3.cmml" xref="A4.SS1.SSS3.p2.4.m3.1.1.3">𝐊</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.SS1.SSS3.p2.4.m3.1c">R_{\mathbf{K}}</annotation><annotation encoding="application/x-llamapun" id="A4.SS1.SSS3.p2.4.m3.1d">italic_R start_POSTSUBSCRIPT bold_K end_POSTSUBSCRIPT</annotation></semantics></math>, we get <math alttext="\mathbb{E}(\|\xi_{k}\|^{2})\leq\tilde{C}" class="ltx_Math" display="inline" id="A4.SS1.SSS3.p2.5.m4.1"><semantics id="A4.SS1.SSS3.p2.5.m4.1a"><mrow id="A4.SS1.SSS3.p2.5.m4.1.1" xref="A4.SS1.SSS3.p2.5.m4.1.1.cmml"><mrow id="A4.SS1.SSS3.p2.5.m4.1.1.1" xref="A4.SS1.SSS3.p2.5.m4.1.1.1.cmml"><mi id="A4.SS1.SSS3.p2.5.m4.1.1.1.3" xref="A4.SS1.SSS3.p2.5.m4.1.1.1.3.cmml">𝔼</mi><mo id="A4.SS1.SSS3.p2.5.m4.1.1.1.2" xref="A4.SS1.SSS3.p2.5.m4.1.1.1.2.cmml">⁢</mo><mrow id="A4.SS1.SSS3.p2.5.m4.1.1.1.1.1" xref="A4.SS1.SSS3.p2.5.m4.1.1.1.1.1.1.cmml"><mo id="A4.SS1.SSS3.p2.5.m4.1.1.1.1.1.2" stretchy="false" xref="A4.SS1.SSS3.p2.5.m4.1.1.1.1.1.1.cmml">(</mo><msup id="A4.SS1.SSS3.p2.5.m4.1.1.1.1.1.1" xref="A4.SS1.SSS3.p2.5.m4.1.1.1.1.1.1.cmml"><mrow id="A4.SS1.SSS3.p2.5.m4.1.1.1.1.1.1.1.1" xref="A4.SS1.SSS3.p2.5.m4.1.1.1.1.1.1.1.2.cmml"><mo id="A4.SS1.SSS3.p2.5.m4.1.1.1.1.1.1.1.1.2" stretchy="false" xref="A4.SS1.SSS3.p2.5.m4.1.1.1.1.1.1.1.2.1.cmml">‖</mo><msub id="A4.SS1.SSS3.p2.5.m4.1.1.1.1.1.1.1.1.1" xref="A4.SS1.SSS3.p2.5.m4.1.1.1.1.1.1.1.1.1.cmml"><mi id="A4.SS1.SSS3.p2.5.m4.1.1.1.1.1.1.1.1.1.2" xref="A4.SS1.SSS3.p2.5.m4.1.1.1.1.1.1.1.1.1.2.cmml">ξ</mi><mi id="A4.SS1.SSS3.p2.5.m4.1.1.1.1.1.1.1.1.1.3" xref="A4.SS1.SSS3.p2.5.m4.1.1.1.1.1.1.1.1.1.3.cmml">k</mi></msub><mo id="A4.SS1.SSS3.p2.5.m4.1.1.1.1.1.1.1.1.3" stretchy="false" xref="A4.SS1.SSS3.p2.5.m4.1.1.1.1.1.1.1.2.1.cmml">‖</mo></mrow><mn id="A4.SS1.SSS3.p2.5.m4.1.1.1.1.1.1.3" xref="A4.SS1.SSS3.p2.5.m4.1.1.1.1.1.1.3.cmml">2</mn></msup><mo id="A4.SS1.SSS3.p2.5.m4.1.1.1.1.1.3" stretchy="false" xref="A4.SS1.SSS3.p2.5.m4.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="A4.SS1.SSS3.p2.5.m4.1.1.2" xref="A4.SS1.SSS3.p2.5.m4.1.1.2.cmml">≤</mo><mover accent="true" id="A4.SS1.SSS3.p2.5.m4.1.1.3" xref="A4.SS1.SSS3.p2.5.m4.1.1.3.cmml"><mi id="A4.SS1.SSS3.p2.5.m4.1.1.3.2" xref="A4.SS1.SSS3.p2.5.m4.1.1.3.2.cmml">C</mi><mo id="A4.SS1.SSS3.p2.5.m4.1.1.3.1" xref="A4.SS1.SSS3.p2.5.m4.1.1.3.1.cmml">~</mo></mover></mrow><annotation-xml encoding="MathML-Content" id="A4.SS1.SSS3.p2.5.m4.1b"><apply id="A4.SS1.SSS3.p2.5.m4.1.1.cmml" xref="A4.SS1.SSS3.p2.5.m4.1.1"><leq id="A4.SS1.SSS3.p2.5.m4.1.1.2.cmml" xref="A4.SS1.SSS3.p2.5.m4.1.1.2"></leq><apply id="A4.SS1.SSS3.p2.5.m4.1.1.1.cmml" xref="A4.SS1.SSS3.p2.5.m4.1.1.1"><times id="A4.SS1.SSS3.p2.5.m4.1.1.1.2.cmml" xref="A4.SS1.SSS3.p2.5.m4.1.1.1.2"></times><ci id="A4.SS1.SSS3.p2.5.m4.1.1.1.3.cmml" xref="A4.SS1.SSS3.p2.5.m4.1.1.1.3">𝔼</ci><apply id="A4.SS1.SSS3.p2.5.m4.1.1.1.1.1.1.cmml" xref="A4.SS1.SSS3.p2.5.m4.1.1.1.1.1"><csymbol cd="ambiguous" id="A4.SS1.SSS3.p2.5.m4.1.1.1.1.1.1.2.cmml" xref="A4.SS1.SSS3.p2.5.m4.1.1.1.1.1">superscript</csymbol><apply id="A4.SS1.SSS3.p2.5.m4.1.1.1.1.1.1.1.2.cmml" xref="A4.SS1.SSS3.p2.5.m4.1.1.1.1.1.1.1.1"><csymbol cd="latexml" id="A4.SS1.SSS3.p2.5.m4.1.1.1.1.1.1.1.2.1.cmml" xref="A4.SS1.SSS3.p2.5.m4.1.1.1.1.1.1.1.1.2">norm</csymbol><apply id="A4.SS1.SSS3.p2.5.m4.1.1.1.1.1.1.1.1.1.cmml" xref="A4.SS1.SSS3.p2.5.m4.1.1.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="A4.SS1.SSS3.p2.5.m4.1.1.1.1.1.1.1.1.1.1.cmml" xref="A4.SS1.SSS3.p2.5.m4.1.1.1.1.1.1.1.1.1">subscript</csymbol><ci id="A4.SS1.SSS3.p2.5.m4.1.1.1.1.1.1.1.1.1.2.cmml" xref="A4.SS1.SSS3.p2.5.m4.1.1.1.1.1.1.1.1.1.2">𝜉</ci><ci id="A4.SS1.SSS3.p2.5.m4.1.1.1.1.1.1.1.1.1.3.cmml" xref="A4.SS1.SSS3.p2.5.m4.1.1.1.1.1.1.1.1.1.3">𝑘</ci></apply></apply><cn id="A4.SS1.SSS3.p2.5.m4.1.1.1.1.1.1.3.cmml" type="integer" xref="A4.SS1.SSS3.p2.5.m4.1.1.1.1.1.1.3">2</cn></apply></apply><apply id="A4.SS1.SSS3.p2.5.m4.1.1.3.cmml" xref="A4.SS1.SSS3.p2.5.m4.1.1.3"><ci id="A4.SS1.SSS3.p2.5.m4.1.1.3.1.cmml" xref="A4.SS1.SSS3.p2.5.m4.1.1.3.1">~</ci><ci id="A4.SS1.SSS3.p2.5.m4.1.1.3.2.cmml" xref="A4.SS1.SSS3.p2.5.m4.1.1.3.2">𝐶</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.SS1.SSS3.p2.5.m4.1c">\mathbb{E}(\|\xi_{k}\|^{2})\leq\tilde{C}</annotation><annotation encoding="application/x-llamapun" id="A4.SS1.SSS3.p2.5.m4.1d">blackboard_E ( ∥ italic_ξ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) ≤ over~ start_ARG italic_C end_ARG</annotation></semantics></math>, with <math alttext="\tilde{C}<+\infty" class="ltx_Math" display="inline" id="A4.SS1.SSS3.p2.6.m5.1"><semantics id="A4.SS1.SSS3.p2.6.m5.1a"><mrow id="A4.SS1.SSS3.p2.6.m5.1.1" xref="A4.SS1.SSS3.p2.6.m5.1.1.cmml"><mover accent="true" id="A4.SS1.SSS3.p2.6.m5.1.1.2" xref="A4.SS1.SSS3.p2.6.m5.1.1.2.cmml"><mi id="A4.SS1.SSS3.p2.6.m5.1.1.2.2" xref="A4.SS1.SSS3.p2.6.m5.1.1.2.2.cmml">C</mi><mo id="A4.SS1.SSS3.p2.6.m5.1.1.2.1" xref="A4.SS1.SSS3.p2.6.m5.1.1.2.1.cmml">~</mo></mover><mo id="A4.SS1.SSS3.p2.6.m5.1.1.1" xref="A4.SS1.SSS3.p2.6.m5.1.1.1.cmml"><</mo><mrow id="A4.SS1.SSS3.p2.6.m5.1.1.3" xref="A4.SS1.SSS3.p2.6.m5.1.1.3.cmml"><mo id="A4.SS1.SSS3.p2.6.m5.1.1.3a" xref="A4.SS1.SSS3.p2.6.m5.1.1.3.cmml">+</mo><mi id="A4.SS1.SSS3.p2.6.m5.1.1.3.2" mathvariant="normal" xref="A4.SS1.SSS3.p2.6.m5.1.1.3.2.cmml">∞</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="A4.SS1.SSS3.p2.6.m5.1b"><apply id="A4.SS1.SSS3.p2.6.m5.1.1.cmml" xref="A4.SS1.SSS3.p2.6.m5.1.1"><lt id="A4.SS1.SSS3.p2.6.m5.1.1.1.cmml" xref="A4.SS1.SSS3.p2.6.m5.1.1.1"></lt><apply id="A4.SS1.SSS3.p2.6.m5.1.1.2.cmml" xref="A4.SS1.SSS3.p2.6.m5.1.1.2"><ci id="A4.SS1.SSS3.p2.6.m5.1.1.2.1.cmml" xref="A4.SS1.SSS3.p2.6.m5.1.1.2.1">~</ci><ci id="A4.SS1.SSS3.p2.6.m5.1.1.2.2.cmml" xref="A4.SS1.SSS3.p2.6.m5.1.1.2.2">𝐶</ci></apply><apply id="A4.SS1.SSS3.p2.6.m5.1.1.3.cmml" xref="A4.SS1.SSS3.p2.6.m5.1.1.3"><plus id="A4.SS1.SSS3.p2.6.m5.1.1.3.1.cmml" xref="A4.SS1.SSS3.p2.6.m5.1.1.3"></plus><infinity id="A4.SS1.SSS3.p2.6.m5.1.1.3.2.cmml" xref="A4.SS1.SSS3.p2.6.m5.1.1.3.2"></infinity></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.SS1.SSS3.p2.6.m5.1c">\tilde{C}<+\infty</annotation><annotation encoding="application/x-llamapun" id="A4.SS1.SSS3.p2.6.m5.1d">over~ start_ARG italic_C end_ARG < + ∞</annotation></semantics></math>, a constant of <math alttext="k" class="ltx_Math" display="inline" id="A4.SS1.SSS3.p2.7.m6.1"><semantics id="A4.SS1.SSS3.p2.7.m6.1a"><mi id="A4.SS1.SSS3.p2.7.m6.1.1" xref="A4.SS1.SSS3.p2.7.m6.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="A4.SS1.SSS3.p2.7.m6.1b"><ci id="A4.SS1.SSS3.p2.7.m6.1.1.cmml" xref="A4.SS1.SSS3.p2.7.m6.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="A4.SS1.SSS3.p2.7.m6.1c">k</annotation><annotation encoding="application/x-llamapun" id="A4.SS1.SSS3.p2.7.m6.1d">italic_k</annotation></semantics></math>. Then <math alttext="\sum_{k\in{\mathbb{N}}}{\delta_{k}^{2}\mathbb{E}(\|\xi_{k}\|^{2})}\leq\tilde{C% }\sum_{k\in{\mathbb{N}}}{\delta_{k}^{2}}<+\infty" class="ltx_Math" display="inline" id="A4.SS1.SSS3.p2.8.m7.1"><semantics id="A4.SS1.SSS3.p2.8.m7.1a"><mrow id="A4.SS1.SSS3.p2.8.m7.1.1" xref="A4.SS1.SSS3.p2.8.m7.1.1.cmml"><mrow id="A4.SS1.SSS3.p2.8.m7.1.1.1" xref="A4.SS1.SSS3.p2.8.m7.1.1.1.cmml"><msub id="A4.SS1.SSS3.p2.8.m7.1.1.1.2" xref="A4.SS1.SSS3.p2.8.m7.1.1.1.2.cmml"><mo id="A4.SS1.SSS3.p2.8.m7.1.1.1.2.2" xref="A4.SS1.SSS3.p2.8.m7.1.1.1.2.2.cmml">∑</mo><mrow id="A4.SS1.SSS3.p2.8.m7.1.1.1.2.3" xref="A4.SS1.SSS3.p2.8.m7.1.1.1.2.3.cmml"><mi id="A4.SS1.SSS3.p2.8.m7.1.1.1.2.3.2" xref="A4.SS1.SSS3.p2.8.m7.1.1.1.2.3.2.cmml">k</mi><mo id="A4.SS1.SSS3.p2.8.m7.1.1.1.2.3.1" xref="A4.SS1.SSS3.p2.8.m7.1.1.1.2.3.1.cmml">∈</mo><mi id="A4.SS1.SSS3.p2.8.m7.1.1.1.2.3.3" xref="A4.SS1.SSS3.p2.8.m7.1.1.1.2.3.3.cmml">ℕ</mi></mrow></msub><mrow id="A4.SS1.SSS3.p2.8.m7.1.1.1.1" xref="A4.SS1.SSS3.p2.8.m7.1.1.1.1.cmml"><msubsup id="A4.SS1.SSS3.p2.8.m7.1.1.1.1.3" xref="A4.SS1.SSS3.p2.8.m7.1.1.1.1.3.cmml"><mi id="A4.SS1.SSS3.p2.8.m7.1.1.1.1.3.2.2" xref="A4.SS1.SSS3.p2.8.m7.1.1.1.1.3.2.2.cmml">δ</mi><mi id="A4.SS1.SSS3.p2.8.m7.1.1.1.1.3.2.3" xref="A4.SS1.SSS3.p2.8.m7.1.1.1.1.3.2.3.cmml">k</mi><mn id="A4.SS1.SSS3.p2.8.m7.1.1.1.1.3.3" xref="A4.SS1.SSS3.p2.8.m7.1.1.1.1.3.3.cmml">2</mn></msubsup><mo id="A4.SS1.SSS3.p2.8.m7.1.1.1.1.2" xref="A4.SS1.SSS3.p2.8.m7.1.1.1.1.2.cmml">⁢</mo><mi id="A4.SS1.SSS3.p2.8.m7.1.1.1.1.4" xref="A4.SS1.SSS3.p2.8.m7.1.1.1.1.4.cmml">𝔼</mi><mo id="A4.SS1.SSS3.p2.8.m7.1.1.1.1.2a" xref="A4.SS1.SSS3.p2.8.m7.1.1.1.1.2.cmml">⁢</mo><mrow id="A4.SS1.SSS3.p2.8.m7.1.1.1.1.1.1" xref="A4.SS1.SSS3.p2.8.m7.1.1.1.1.1.1.1.cmml"><mo id="A4.SS1.SSS3.p2.8.m7.1.1.1.1.1.1.2" stretchy="false" xref="A4.SS1.SSS3.p2.8.m7.1.1.1.1.1.1.1.cmml">(</mo><msup id="A4.SS1.SSS3.p2.8.m7.1.1.1.1.1.1.1" xref="A4.SS1.SSS3.p2.8.m7.1.1.1.1.1.1.1.cmml"><mrow id="A4.SS1.SSS3.p2.8.m7.1.1.1.1.1.1.1.1.1" xref="A4.SS1.SSS3.p2.8.m7.1.1.1.1.1.1.1.1.2.cmml"><mo id="A4.SS1.SSS3.p2.8.m7.1.1.1.1.1.1.1.1.1.2" stretchy="false" xref="A4.SS1.SSS3.p2.8.m7.1.1.1.1.1.1.1.1.2.1.cmml">‖</mo><msub id="A4.SS1.SSS3.p2.8.m7.1.1.1.1.1.1.1.1.1.1" xref="A4.SS1.SSS3.p2.8.m7.1.1.1.1.1.1.1.1.1.1.cmml"><mi id="A4.SS1.SSS3.p2.8.m7.1.1.1.1.1.1.1.1.1.1.2" xref="A4.SS1.SSS3.p2.8.m7.1.1.1.1.1.1.1.1.1.1.2.cmml">ξ</mi><mi id="A4.SS1.SSS3.p2.8.m7.1.1.1.1.1.1.1.1.1.1.3" xref="A4.SS1.SSS3.p2.8.m7.1.1.1.1.1.1.1.1.1.1.3.cmml">k</mi></msub><mo id="A4.SS1.SSS3.p2.8.m7.1.1.1.1.1.1.1.1.1.3" stretchy="false" xref="A4.SS1.SSS3.p2.8.m7.1.1.1.1.1.1.1.1.2.1.cmml">‖</mo></mrow><mn id="A4.SS1.SSS3.p2.8.m7.1.1.1.1.1.1.1.3" xref="A4.SS1.SSS3.p2.8.m7.1.1.1.1.1.1.1.3.cmml">2</mn></msup><mo id="A4.SS1.SSS3.p2.8.m7.1.1.1.1.1.1.3" stretchy="false" xref="A4.SS1.SSS3.p2.8.m7.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><mo id="A4.SS1.SSS3.p2.8.m7.1.1.3" xref="A4.SS1.SSS3.p2.8.m7.1.1.3.cmml">≤</mo><mrow id="A4.SS1.SSS3.p2.8.m7.1.1.4" xref="A4.SS1.SSS3.p2.8.m7.1.1.4.cmml"><mover accent="true" id="A4.SS1.SSS3.p2.8.m7.1.1.4.2" xref="A4.SS1.SSS3.p2.8.m7.1.1.4.2.cmml"><mi id="A4.SS1.SSS3.p2.8.m7.1.1.4.2.2" xref="A4.SS1.SSS3.p2.8.m7.1.1.4.2.2.cmml">C</mi><mo id="A4.SS1.SSS3.p2.8.m7.1.1.4.2.1" xref="A4.SS1.SSS3.p2.8.m7.1.1.4.2.1.cmml">~</mo></mover><mo id="A4.SS1.SSS3.p2.8.m7.1.1.4.1" xref="A4.SS1.SSS3.p2.8.m7.1.1.4.1.cmml">⁢</mo><mrow id="A4.SS1.SSS3.p2.8.m7.1.1.4.3" xref="A4.SS1.SSS3.p2.8.m7.1.1.4.3.cmml"><msub id="A4.SS1.SSS3.p2.8.m7.1.1.4.3.1" xref="A4.SS1.SSS3.p2.8.m7.1.1.4.3.1.cmml"><mo id="A4.SS1.SSS3.p2.8.m7.1.1.4.3.1.2" xref="A4.SS1.SSS3.p2.8.m7.1.1.4.3.1.2.cmml">∑</mo><mrow id="A4.SS1.SSS3.p2.8.m7.1.1.4.3.1.3" xref="A4.SS1.SSS3.p2.8.m7.1.1.4.3.1.3.cmml"><mi id="A4.SS1.SSS3.p2.8.m7.1.1.4.3.1.3.2" xref="A4.SS1.SSS3.p2.8.m7.1.1.4.3.1.3.2.cmml">k</mi><mo id="A4.SS1.SSS3.p2.8.m7.1.1.4.3.1.3.1" xref="A4.SS1.SSS3.p2.8.m7.1.1.4.3.1.3.1.cmml">∈</mo><mi id="A4.SS1.SSS3.p2.8.m7.1.1.4.3.1.3.3" xref="A4.SS1.SSS3.p2.8.m7.1.1.4.3.1.3.3.cmml">ℕ</mi></mrow></msub><msubsup id="A4.SS1.SSS3.p2.8.m7.1.1.4.3.2" xref="A4.SS1.SSS3.p2.8.m7.1.1.4.3.2.cmml"><mi id="A4.SS1.SSS3.p2.8.m7.1.1.4.3.2.2.2" xref="A4.SS1.SSS3.p2.8.m7.1.1.4.3.2.2.2.cmml">δ</mi><mi id="A4.SS1.SSS3.p2.8.m7.1.1.4.3.2.2.3" xref="A4.SS1.SSS3.p2.8.m7.1.1.4.3.2.2.3.cmml">k</mi><mn id="A4.SS1.SSS3.p2.8.m7.1.1.4.3.2.3" xref="A4.SS1.SSS3.p2.8.m7.1.1.4.3.2.3.cmml">2</mn></msubsup></mrow></mrow><mo id="A4.SS1.SSS3.p2.8.m7.1.1.5" xref="A4.SS1.SSS3.p2.8.m7.1.1.5.cmml"><</mo><mrow id="A4.SS1.SSS3.p2.8.m7.1.1.6" xref="A4.SS1.SSS3.p2.8.m7.1.1.6.cmml"><mo id="A4.SS1.SSS3.p2.8.m7.1.1.6a" xref="A4.SS1.SSS3.p2.8.m7.1.1.6.cmml">+</mo><mi id="A4.SS1.SSS3.p2.8.m7.1.1.6.2" mathvariant="normal" xref="A4.SS1.SSS3.p2.8.m7.1.1.6.2.cmml">∞</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="A4.SS1.SSS3.p2.8.m7.1b"><apply id="A4.SS1.SSS3.p2.8.m7.1.1.cmml" xref="A4.SS1.SSS3.p2.8.m7.1.1"><and id="A4.SS1.SSS3.p2.8.m7.1.1a.cmml" xref="A4.SS1.SSS3.p2.8.m7.1.1"></and><apply id="A4.SS1.SSS3.p2.8.m7.1.1b.cmml" xref="A4.SS1.SSS3.p2.8.m7.1.1"><leq id="A4.SS1.SSS3.p2.8.m7.1.1.3.cmml" xref="A4.SS1.SSS3.p2.8.m7.1.1.3"></leq><apply id="A4.SS1.SSS3.p2.8.m7.1.1.1.cmml" xref="A4.SS1.SSS3.p2.8.m7.1.1.1"><apply id="A4.SS1.SSS3.p2.8.m7.1.1.1.2.cmml" xref="A4.SS1.SSS3.p2.8.m7.1.1.1.2"><csymbol cd="ambiguous" id="A4.SS1.SSS3.p2.8.m7.1.1.1.2.1.cmml" xref="A4.SS1.SSS3.p2.8.m7.1.1.1.2">subscript</csymbol><sum id="A4.SS1.SSS3.p2.8.m7.1.1.1.2.2.cmml" xref="A4.SS1.SSS3.p2.8.m7.1.1.1.2.2"></sum><apply id="A4.SS1.SSS3.p2.8.m7.1.1.1.2.3.cmml" xref="A4.SS1.SSS3.p2.8.m7.1.1.1.2.3"><in id="A4.SS1.SSS3.p2.8.m7.1.1.1.2.3.1.cmml" xref="A4.SS1.SSS3.p2.8.m7.1.1.1.2.3.1"></in><ci id="A4.SS1.SSS3.p2.8.m7.1.1.1.2.3.2.cmml" xref="A4.SS1.SSS3.p2.8.m7.1.1.1.2.3.2">𝑘</ci><ci id="A4.SS1.SSS3.p2.8.m7.1.1.1.2.3.3.cmml" xref="A4.SS1.SSS3.p2.8.m7.1.1.1.2.3.3">ℕ</ci></apply></apply><apply id="A4.SS1.SSS3.p2.8.m7.1.1.1.1.cmml" xref="A4.SS1.SSS3.p2.8.m7.1.1.1.1"><times id="A4.SS1.SSS3.p2.8.m7.1.1.1.1.2.cmml" xref="A4.SS1.SSS3.p2.8.m7.1.1.1.1.2"></times><apply id="A4.SS1.SSS3.p2.8.m7.1.1.1.1.3.cmml" xref="A4.SS1.SSS3.p2.8.m7.1.1.1.1.3"><csymbol cd="ambiguous" id="A4.SS1.SSS3.p2.8.m7.1.1.1.1.3.1.cmml" xref="A4.SS1.SSS3.p2.8.m7.1.1.1.1.3">superscript</csymbol><apply id="A4.SS1.SSS3.p2.8.m7.1.1.1.1.3.2.cmml" xref="A4.SS1.SSS3.p2.8.m7.1.1.1.1.3"><csymbol cd="ambiguous" id="A4.SS1.SSS3.p2.8.m7.1.1.1.1.3.2.1.cmml" xref="A4.SS1.SSS3.p2.8.m7.1.1.1.1.3">subscript</csymbol><ci id="A4.SS1.SSS3.p2.8.m7.1.1.1.1.3.2.2.cmml" xref="A4.SS1.SSS3.p2.8.m7.1.1.1.1.3.2.2">𝛿</ci><ci id="A4.SS1.SSS3.p2.8.m7.1.1.1.1.3.2.3.cmml" xref="A4.SS1.SSS3.p2.8.m7.1.1.1.1.3.2.3">𝑘</ci></apply><cn id="A4.SS1.SSS3.p2.8.m7.1.1.1.1.3.3.cmml" type="integer" xref="A4.SS1.SSS3.p2.8.m7.1.1.1.1.3.3">2</cn></apply><ci id="A4.SS1.SSS3.p2.8.m7.1.1.1.1.4.cmml" xref="A4.SS1.SSS3.p2.8.m7.1.1.1.1.4">𝔼</ci><apply id="A4.SS1.SSS3.p2.8.m7.1.1.1.1.1.1.1.cmml" xref="A4.SS1.SSS3.p2.8.m7.1.1.1.1.1.1"><csymbol cd="ambiguous" id="A4.SS1.SSS3.p2.8.m7.1.1.1.1.1.1.1.2.cmml" xref="A4.SS1.SSS3.p2.8.m7.1.1.1.1.1.1">superscript</csymbol><apply id="A4.SS1.SSS3.p2.8.m7.1.1.1.1.1.1.1.1.2.cmml" xref="A4.SS1.SSS3.p2.8.m7.1.1.1.1.1.1.1.1.1"><csymbol cd="latexml" id="A4.SS1.SSS3.p2.8.m7.1.1.1.1.1.1.1.1.2.1.cmml" xref="A4.SS1.SSS3.p2.8.m7.1.1.1.1.1.1.1.1.1.2">norm</csymbol><apply id="A4.SS1.SSS3.p2.8.m7.1.1.1.1.1.1.1.1.1.1.cmml" xref="A4.SS1.SSS3.p2.8.m7.1.1.1.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="A4.SS1.SSS3.p2.8.m7.1.1.1.1.1.1.1.1.1.1.1.cmml" xref="A4.SS1.SSS3.p2.8.m7.1.1.1.1.1.1.1.1.1.1">subscript</csymbol><ci id="A4.SS1.SSS3.p2.8.m7.1.1.1.1.1.1.1.1.1.1.2.cmml" xref="A4.SS1.SSS3.p2.8.m7.1.1.1.1.1.1.1.1.1.1.2">𝜉</ci><ci id="A4.SS1.SSS3.p2.8.m7.1.1.1.1.1.1.1.1.1.1.3.cmml" xref="A4.SS1.SSS3.p2.8.m7.1.1.1.1.1.1.1.1.1.1.3">𝑘</ci></apply></apply><cn id="A4.SS1.SSS3.p2.8.m7.1.1.1.1.1.1.1.3.cmml" type="integer" xref="A4.SS1.SSS3.p2.8.m7.1.1.1.1.1.1.1.3">2</cn></apply></apply></apply><apply id="A4.SS1.SSS3.p2.8.m7.1.1.4.cmml" xref="A4.SS1.SSS3.p2.8.m7.1.1.4"><times id="A4.SS1.SSS3.p2.8.m7.1.1.4.1.cmml" xref="A4.SS1.SSS3.p2.8.m7.1.1.4.1"></times><apply id="A4.SS1.SSS3.p2.8.m7.1.1.4.2.cmml" xref="A4.SS1.SSS3.p2.8.m7.1.1.4.2"><ci id="A4.SS1.SSS3.p2.8.m7.1.1.4.2.1.cmml" xref="A4.SS1.SSS3.p2.8.m7.1.1.4.2.1">~</ci><ci id="A4.SS1.SSS3.p2.8.m7.1.1.4.2.2.cmml" xref="A4.SS1.SSS3.p2.8.m7.1.1.4.2.2">𝐶</ci></apply><apply id="A4.SS1.SSS3.p2.8.m7.1.1.4.3.cmml" xref="A4.SS1.SSS3.p2.8.m7.1.1.4.3"><apply id="A4.SS1.SSS3.p2.8.m7.1.1.4.3.1.cmml" xref="A4.SS1.SSS3.p2.8.m7.1.1.4.3.1"><csymbol cd="ambiguous" id="A4.SS1.SSS3.p2.8.m7.1.1.4.3.1.1.cmml" xref="A4.SS1.SSS3.p2.8.m7.1.1.4.3.1">subscript</csymbol><sum id="A4.SS1.SSS3.p2.8.m7.1.1.4.3.1.2.cmml" xref="A4.SS1.SSS3.p2.8.m7.1.1.4.3.1.2"></sum><apply id="A4.SS1.SSS3.p2.8.m7.1.1.4.3.1.3.cmml" xref="A4.SS1.SSS3.p2.8.m7.1.1.4.3.1.3"><in id="A4.SS1.SSS3.p2.8.m7.1.1.4.3.1.3.1.cmml" xref="A4.SS1.SSS3.p2.8.m7.1.1.4.3.1.3.1"></in><ci id="A4.SS1.SSS3.p2.8.m7.1.1.4.3.1.3.2.cmml" xref="A4.SS1.SSS3.p2.8.m7.1.1.4.3.1.3.2">𝑘</ci><ci id="A4.SS1.SSS3.p2.8.m7.1.1.4.3.1.3.3.cmml" xref="A4.SS1.SSS3.p2.8.m7.1.1.4.3.1.3.3">ℕ</ci></apply></apply><apply id="A4.SS1.SSS3.p2.8.m7.1.1.4.3.2.cmml" xref="A4.SS1.SSS3.p2.8.m7.1.1.4.3.2"><csymbol cd="ambiguous" id="A4.SS1.SSS3.p2.8.m7.1.1.4.3.2.1.cmml" xref="A4.SS1.SSS3.p2.8.m7.1.1.4.3.2">superscript</csymbol><apply id="A4.SS1.SSS3.p2.8.m7.1.1.4.3.2.2.cmml" xref="A4.SS1.SSS3.p2.8.m7.1.1.4.3.2"><csymbol cd="ambiguous" id="A4.SS1.SSS3.p2.8.m7.1.1.4.3.2.2.1.cmml" xref="A4.SS1.SSS3.p2.8.m7.1.1.4.3.2">subscript</csymbol><ci id="A4.SS1.SSS3.p2.8.m7.1.1.4.3.2.2.2.cmml" xref="A4.SS1.SSS3.p2.8.m7.1.1.4.3.2.2.2">𝛿</ci><ci id="A4.SS1.SSS3.p2.8.m7.1.1.4.3.2.2.3.cmml" xref="A4.SS1.SSS3.p2.8.m7.1.1.4.3.2.2.3">𝑘</ci></apply><cn id="A4.SS1.SSS3.p2.8.m7.1.1.4.3.2.3.cmml" type="integer" xref="A4.SS1.SSS3.p2.8.m7.1.1.4.3.2.3">2</cn></apply></apply></apply></apply><apply id="A4.SS1.SSS3.p2.8.m7.1.1c.cmml" xref="A4.SS1.SSS3.p2.8.m7.1.1"><lt id="A4.SS1.SSS3.p2.8.m7.1.1.5.cmml" xref="A4.SS1.SSS3.p2.8.m7.1.1.5"></lt><share href="#A4.SS1.SSS3.p2.8.m7.1.1.4.cmml" id="A4.SS1.SSS3.p2.8.m7.1.1d.cmml" xref="A4.SS1.SSS3.p2.8.m7.1.1"></share><apply id="A4.SS1.SSS3.p2.8.m7.1.1.6.cmml" xref="A4.SS1.SSS3.p2.8.m7.1.1.6"><plus id="A4.SS1.SSS3.p2.8.m7.1.1.6.1.cmml" xref="A4.SS1.SSS3.p2.8.m7.1.1.6"></plus><infinity id="A4.SS1.SSS3.p2.8.m7.1.1.6.2.cmml" xref="A4.SS1.SSS3.p2.8.m7.1.1.6.2"></infinity></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.SS1.SSS3.p2.8.m7.1c">\sum_{k\in{\mathbb{N}}}{\delta_{k}^{2}\mathbb{E}(\|\xi_{k}\|^{2})}\leq\tilde{C% }\sum_{k\in{\mathbb{N}}}{\delta_{k}^{2}}<+\infty</annotation><annotation encoding="application/x-llamapun" id="A4.SS1.SSS3.p2.8.m7.1d">∑ start_POSTSUBSCRIPT italic_k ∈ blackboard_N end_POSTSUBSCRIPT italic_δ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT blackboard_E ( ∥ italic_ξ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) ≤ over~ start_ARG italic_C end_ARG ∑ start_POSTSUBSCRIPT italic_k ∈ blackboard_N end_POSTSUBSCRIPT italic_δ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT < + ∞</annotation></semantics></math>, by Assumption <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#S3.Thmtheorem3" title="Assumption 3.3. ‣ 3.2 Unbiased algorithm analysis ‣ 3 Convergence Analysis ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_tag">3.3</span></a>.</p> </div> <div class="ltx_para" id="A4.SS1.SSS3.p3"> <p class="ltx_p" id="A4.SS1.SSS3.p3.1">By following the classical Robbins-Monro (RM) situation (see for example <cite class="ltx_cite ltx_citemacro_citet">Metivier & Priouret (<a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#bib.bib47" title="">1984</a>, Part II-D)</cite>), we deduce from the Doob inequality (which holds because <math alttext="\sum_{k=n}^{l}\xi_{k}" class="ltx_Math" display="inline" id="A4.SS1.SSS3.p3.1.m1.1"><semantics id="A4.SS1.SSS3.p3.1.m1.1a"><mrow id="A4.SS1.SSS3.p3.1.m1.1.1" xref="A4.SS1.SSS3.p3.1.m1.1.1.cmml"><msubsup id="A4.SS1.SSS3.p3.1.m1.1.1.1" xref="A4.SS1.SSS3.p3.1.m1.1.1.1.cmml"><mo id="A4.SS1.SSS3.p3.1.m1.1.1.1.2.2" xref="A4.SS1.SSS3.p3.1.m1.1.1.1.2.2.cmml">∑</mo><mrow id="A4.SS1.SSS3.p3.1.m1.1.1.1.2.3" xref="A4.SS1.SSS3.p3.1.m1.1.1.1.2.3.cmml"><mi id="A4.SS1.SSS3.p3.1.m1.1.1.1.2.3.2" xref="A4.SS1.SSS3.p3.1.m1.1.1.1.2.3.2.cmml">k</mi><mo id="A4.SS1.SSS3.p3.1.m1.1.1.1.2.3.1" xref="A4.SS1.SSS3.p3.1.m1.1.1.1.2.3.1.cmml">=</mo><mi id="A4.SS1.SSS3.p3.1.m1.1.1.1.2.3.3" xref="A4.SS1.SSS3.p3.1.m1.1.1.1.2.3.3.cmml">n</mi></mrow><mi id="A4.SS1.SSS3.p3.1.m1.1.1.1.3" xref="A4.SS1.SSS3.p3.1.m1.1.1.1.3.cmml">l</mi></msubsup><msub id="A4.SS1.SSS3.p3.1.m1.1.1.2" xref="A4.SS1.SSS3.p3.1.m1.1.1.2.cmml"><mi id="A4.SS1.SSS3.p3.1.m1.1.1.2.2" xref="A4.SS1.SSS3.p3.1.m1.1.1.2.2.cmml">ξ</mi><mi id="A4.SS1.SSS3.p3.1.m1.1.1.2.3" xref="A4.SS1.SSS3.p3.1.m1.1.1.2.3.cmml">k</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="A4.SS1.SSS3.p3.1.m1.1b"><apply id="A4.SS1.SSS3.p3.1.m1.1.1.cmml" xref="A4.SS1.SSS3.p3.1.m1.1.1"><apply id="A4.SS1.SSS3.p3.1.m1.1.1.1.cmml" xref="A4.SS1.SSS3.p3.1.m1.1.1.1"><csymbol cd="ambiguous" id="A4.SS1.SSS3.p3.1.m1.1.1.1.1.cmml" xref="A4.SS1.SSS3.p3.1.m1.1.1.1">superscript</csymbol><apply id="A4.SS1.SSS3.p3.1.m1.1.1.1.2.cmml" xref="A4.SS1.SSS3.p3.1.m1.1.1.1"><csymbol cd="ambiguous" id="A4.SS1.SSS3.p3.1.m1.1.1.1.2.1.cmml" xref="A4.SS1.SSS3.p3.1.m1.1.1.1">subscript</csymbol><sum id="A4.SS1.SSS3.p3.1.m1.1.1.1.2.2.cmml" xref="A4.SS1.SSS3.p3.1.m1.1.1.1.2.2"></sum><apply id="A4.SS1.SSS3.p3.1.m1.1.1.1.2.3.cmml" xref="A4.SS1.SSS3.p3.1.m1.1.1.1.2.3"><eq id="A4.SS1.SSS3.p3.1.m1.1.1.1.2.3.1.cmml" xref="A4.SS1.SSS3.p3.1.m1.1.1.1.2.3.1"></eq><ci id="A4.SS1.SSS3.p3.1.m1.1.1.1.2.3.2.cmml" xref="A4.SS1.SSS3.p3.1.m1.1.1.1.2.3.2">𝑘</ci><ci id="A4.SS1.SSS3.p3.1.m1.1.1.1.2.3.3.cmml" xref="A4.SS1.SSS3.p3.1.m1.1.1.1.2.3.3">𝑛</ci></apply></apply><ci id="A4.SS1.SSS3.p3.1.m1.1.1.1.3.cmml" xref="A4.SS1.SSS3.p3.1.m1.1.1.1.3">𝑙</ci></apply><apply id="A4.SS1.SSS3.p3.1.m1.1.1.2.cmml" xref="A4.SS1.SSS3.p3.1.m1.1.1.2"><csymbol cd="ambiguous" id="A4.SS1.SSS3.p3.1.m1.1.1.2.1.cmml" xref="A4.SS1.SSS3.p3.1.m1.1.1.2">subscript</csymbol><ci id="A4.SS1.SSS3.p3.1.m1.1.1.2.2.cmml" xref="A4.SS1.SSS3.p3.1.m1.1.1.2.2">𝜉</ci><ci id="A4.SS1.SSS3.p3.1.m1.1.1.2.3.cmml" xref="A4.SS1.SSS3.p3.1.m1.1.1.2.3">𝑘</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.SS1.SSS3.p3.1.m1.1c">\sum_{k=n}^{l}\xi_{k}</annotation><annotation encoding="application/x-llamapun" id="A4.SS1.SSS3.p3.1.m1.1d">∑ start_POSTSUBSCRIPT italic_k = italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_l end_POSTSUPERSCRIPT italic_ξ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT</annotation></semantics></math> is a martingale), </p> <table class="ltx_equation ltx_eqn_table" id="A4.Ex92"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\mathbb{E}(\sup_{n\leq l\leq m}{\|\sum_{k=n}^{l}{\delta_{k}\xi_{k}}\|})\leq 4% \sum_{k=n}^{m}{\delta_{k}^{2}\mathbb{E}(\|\xi_{k}\|^{2})}." class="ltx_Math" display="block" id="A4.Ex92.m1.1"><semantics id="A4.Ex92.m1.1a"><mrow id="A4.Ex92.m1.1.1.1" xref="A4.Ex92.m1.1.1.1.1.cmml"><mrow id="A4.Ex92.m1.1.1.1.1" xref="A4.Ex92.m1.1.1.1.1.cmml"><mrow id="A4.Ex92.m1.1.1.1.1.1" xref="A4.Ex92.m1.1.1.1.1.1.cmml"><mi id="A4.Ex92.m1.1.1.1.1.1.3" xref="A4.Ex92.m1.1.1.1.1.1.3.cmml">𝔼</mi><mo id="A4.Ex92.m1.1.1.1.1.1.2" xref="A4.Ex92.m1.1.1.1.1.1.2.cmml">⁢</mo><mrow id="A4.Ex92.m1.1.1.1.1.1.1.1" xref="A4.Ex92.m1.1.1.1.1.1.1.1.1.cmml"><mo id="A4.Ex92.m1.1.1.1.1.1.1.1.2" stretchy="false" xref="A4.Ex92.m1.1.1.1.1.1.1.1.1.cmml">(</mo><mrow id="A4.Ex92.m1.1.1.1.1.1.1.1.1" xref="A4.Ex92.m1.1.1.1.1.1.1.1.1.cmml"><munder id="A4.Ex92.m1.1.1.1.1.1.1.1.1.2" xref="A4.Ex92.m1.1.1.1.1.1.1.1.1.2.cmml"><mo id="A4.Ex92.m1.1.1.1.1.1.1.1.1.2.2" lspace="0em" movablelimits="false" rspace="0em" xref="A4.Ex92.m1.1.1.1.1.1.1.1.1.2.2.cmml">sup</mo><mrow id="A4.Ex92.m1.1.1.1.1.1.1.1.1.2.3" xref="A4.Ex92.m1.1.1.1.1.1.1.1.1.2.3.cmml"><mi id="A4.Ex92.m1.1.1.1.1.1.1.1.1.2.3.2" xref="A4.Ex92.m1.1.1.1.1.1.1.1.1.2.3.2.cmml">n</mi><mo id="A4.Ex92.m1.1.1.1.1.1.1.1.1.2.3.3" xref="A4.Ex92.m1.1.1.1.1.1.1.1.1.2.3.3.cmml">≤</mo><mi id="A4.Ex92.m1.1.1.1.1.1.1.1.1.2.3.4" xref="A4.Ex92.m1.1.1.1.1.1.1.1.1.2.3.4.cmml">l</mi><mo id="A4.Ex92.m1.1.1.1.1.1.1.1.1.2.3.5" xref="A4.Ex92.m1.1.1.1.1.1.1.1.1.2.3.5.cmml">≤</mo><mi id="A4.Ex92.m1.1.1.1.1.1.1.1.1.2.3.6" xref="A4.Ex92.m1.1.1.1.1.1.1.1.1.2.3.6.cmml">m</mi></mrow></munder><mrow id="A4.Ex92.m1.1.1.1.1.1.1.1.1.1.1" xref="A4.Ex92.m1.1.1.1.1.1.1.1.1.1.2.cmml"><mo id="A4.Ex92.m1.1.1.1.1.1.1.1.1.1.1.2" stretchy="false" xref="A4.Ex92.m1.1.1.1.1.1.1.1.1.1.2.1.cmml">‖</mo><mrow id="A4.Ex92.m1.1.1.1.1.1.1.1.1.1.1.1" xref="A4.Ex92.m1.1.1.1.1.1.1.1.1.1.1.1.cmml"><munderover id="A4.Ex92.m1.1.1.1.1.1.1.1.1.1.1.1.1" xref="A4.Ex92.m1.1.1.1.1.1.1.1.1.1.1.1.1.cmml"><mo id="A4.Ex92.m1.1.1.1.1.1.1.1.1.1.1.1.1.2.2" lspace="0em" movablelimits="false" xref="A4.Ex92.m1.1.1.1.1.1.1.1.1.1.1.1.1.2.2.cmml">∑</mo><mrow id="A4.Ex92.m1.1.1.1.1.1.1.1.1.1.1.1.1.2.3" xref="A4.Ex92.m1.1.1.1.1.1.1.1.1.1.1.1.1.2.3.cmml"><mi id="A4.Ex92.m1.1.1.1.1.1.1.1.1.1.1.1.1.2.3.2" xref="A4.Ex92.m1.1.1.1.1.1.1.1.1.1.1.1.1.2.3.2.cmml">k</mi><mo id="A4.Ex92.m1.1.1.1.1.1.1.1.1.1.1.1.1.2.3.1" xref="A4.Ex92.m1.1.1.1.1.1.1.1.1.1.1.1.1.2.3.1.cmml">=</mo><mi id="A4.Ex92.m1.1.1.1.1.1.1.1.1.1.1.1.1.2.3.3" xref="A4.Ex92.m1.1.1.1.1.1.1.1.1.1.1.1.1.2.3.3.cmml">n</mi></mrow><mi id="A4.Ex92.m1.1.1.1.1.1.1.1.1.1.1.1.1.3" xref="A4.Ex92.m1.1.1.1.1.1.1.1.1.1.1.1.1.3.cmml">l</mi></munderover><mrow id="A4.Ex92.m1.1.1.1.1.1.1.1.1.1.1.1.2" xref="A4.Ex92.m1.1.1.1.1.1.1.1.1.1.1.1.2.cmml"><msub id="A4.Ex92.m1.1.1.1.1.1.1.1.1.1.1.1.2.2" xref="A4.Ex92.m1.1.1.1.1.1.1.1.1.1.1.1.2.2.cmml"><mi id="A4.Ex92.m1.1.1.1.1.1.1.1.1.1.1.1.2.2.2" xref="A4.Ex92.m1.1.1.1.1.1.1.1.1.1.1.1.2.2.2.cmml">δ</mi><mi id="A4.Ex92.m1.1.1.1.1.1.1.1.1.1.1.1.2.2.3" xref="A4.Ex92.m1.1.1.1.1.1.1.1.1.1.1.1.2.2.3.cmml">k</mi></msub><mo id="A4.Ex92.m1.1.1.1.1.1.1.1.1.1.1.1.2.1" xref="A4.Ex92.m1.1.1.1.1.1.1.1.1.1.1.1.2.1.cmml">⁢</mo><msub id="A4.Ex92.m1.1.1.1.1.1.1.1.1.1.1.1.2.3" xref="A4.Ex92.m1.1.1.1.1.1.1.1.1.1.1.1.2.3.cmml"><mi id="A4.Ex92.m1.1.1.1.1.1.1.1.1.1.1.1.2.3.2" 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xref="A4.Ex92.m1.1.1.1.1.2.1.1.1.1.1.3">2</cn></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.Ex92.m1.1c">\mathbb{E}(\sup_{n\leq l\leq m}{\|\sum_{k=n}^{l}{\delta_{k}\xi_{k}}\|})\leq 4% \sum_{k=n}^{m}{\delta_{k}^{2}\mathbb{E}(\|\xi_{k}\|^{2})}.</annotation><annotation encoding="application/x-llamapun" id="A4.Ex92.m1.1d">blackboard_E ( roman_sup start_POSTSUBSCRIPT italic_n ≤ italic_l ≤ italic_m end_POSTSUBSCRIPT ∥ ∑ start_POSTSUBSCRIPT italic_k = italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_l end_POSTSUPERSCRIPT italic_δ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT italic_ξ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ∥ ) ≤ 4 ∑ start_POSTSUBSCRIPT italic_k = italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_m end_POSTSUPERSCRIPT italic_δ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT blackboard_E ( ∥ italic_ξ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="A4.SS1.SSS3.p3.4">By the monotone convergence theorem, this implies</p> <table class="ltx_equation ltx_eqn_table" id="A4.Ex93"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\mathbb{E}(\sup_{n\leq l}{\|\sum_{k=n}^{l}{\delta_{k}\xi_{k}}\|})\leq 4\sum_{k% =n}^{\infty}{\delta_{k}^{2}\mathbb{E}(\|\xi_{k}\|^{2})}\leq 4C_{r,\sigma,d,R_{% \mathbf{K}}}\sum_{k=n}^{\infty}\delta_{k}^{2}." class="ltx_Math" display="block" id="A4.Ex93.m1.5"><semantics id="A4.Ex93.m1.5a"><mrow id="A4.Ex93.m1.5.5.1" xref="A4.Ex93.m1.5.5.1.1.cmml"><mrow id="A4.Ex93.m1.5.5.1.1" xref="A4.Ex93.m1.5.5.1.1.cmml"><mrow id="A4.Ex93.m1.5.5.1.1.1" xref="A4.Ex93.m1.5.5.1.1.1.cmml"><mi id="A4.Ex93.m1.5.5.1.1.1.3" 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4C_{r,\sigma,d,R_{% \mathbf{K}}}\sum_{k=n}^{\infty}\delta_{k}^{2}.</annotation><annotation encoding="application/x-llamapun" id="A4.Ex93.m1.5d">blackboard_E ( roman_sup start_POSTSUBSCRIPT italic_n ≤ italic_l end_POSTSUBSCRIPT ∥ ∑ start_POSTSUBSCRIPT italic_k = italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_l end_POSTSUPERSCRIPT italic_δ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT italic_ξ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ∥ ) ≤ 4 ∑ start_POSTSUBSCRIPT italic_k = italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT italic_δ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT blackboard_E ( ∥ italic_ξ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) ≤ 4 italic_C start_POSTSUBSCRIPT italic_r , italic_σ , italic_d , italic_R start_POSTSUBSCRIPT bold_K end_POSTSUBSCRIPT end_POSTSUBSCRIPT ∑ start_POSTSUBSCRIPT italic_k = italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT italic_δ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="A4.SS1.SSS3.p3.3">Thus the sequence <math alttext="(\sup_{n\leq l}{\|\sum_{k=n}^{l}{\delta_{k}\xi_{k}}\|})_{n}" class="ltx_Math" display="inline" id="A4.SS1.SSS3.p3.2.m1.1"><semantics id="A4.SS1.SSS3.p3.2.m1.1a"><msub id="A4.SS1.SSS3.p3.2.m1.1.1" xref="A4.SS1.SSS3.p3.2.m1.1.1.cmml"><mrow id="A4.SS1.SSS3.p3.2.m1.1.1.1.1" xref="A4.SS1.SSS3.p3.2.m1.1.1.1.1.1.cmml"><mo id="A4.SS1.SSS3.p3.2.m1.1.1.1.1.2" stretchy="false" xref="A4.SS1.SSS3.p3.2.m1.1.1.1.1.1.cmml">(</mo><mrow id="A4.SS1.SSS3.p3.2.m1.1.1.1.1.1" xref="A4.SS1.SSS3.p3.2.m1.1.1.1.1.1.cmml"><msub id="A4.SS1.SSS3.p3.2.m1.1.1.1.1.1.2" xref="A4.SS1.SSS3.p3.2.m1.1.1.1.1.1.2.cmml"><mo id="A4.SS1.SSS3.p3.2.m1.1.1.1.1.1.2.2" lspace="0em" rspace="0em" 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xref="A4.SS1.SSS3.p3.2.m1.1.1.1.1.1.1.1.1.2.2.2">𝛿</ci><ci id="A4.SS1.SSS3.p3.2.m1.1.1.1.1.1.1.1.1.2.2.3.cmml" xref="A4.SS1.SSS3.p3.2.m1.1.1.1.1.1.1.1.1.2.2.3">𝑘</ci></apply><apply id="A4.SS1.SSS3.p3.2.m1.1.1.1.1.1.1.1.1.2.3.cmml" xref="A4.SS1.SSS3.p3.2.m1.1.1.1.1.1.1.1.1.2.3"><csymbol cd="ambiguous" id="A4.SS1.SSS3.p3.2.m1.1.1.1.1.1.1.1.1.2.3.1.cmml" xref="A4.SS1.SSS3.p3.2.m1.1.1.1.1.1.1.1.1.2.3">subscript</csymbol><ci id="A4.SS1.SSS3.p3.2.m1.1.1.1.1.1.1.1.1.2.3.2.cmml" xref="A4.SS1.SSS3.p3.2.m1.1.1.1.1.1.1.1.1.2.3.2">𝜉</ci><ci id="A4.SS1.SSS3.p3.2.m1.1.1.1.1.1.1.1.1.2.3.3.cmml" xref="A4.SS1.SSS3.p3.2.m1.1.1.1.1.1.1.1.1.2.3.3">𝑘</ci></apply></apply></apply></apply></apply><ci id="A4.SS1.SSS3.p3.2.m1.1.1.3.cmml" xref="A4.SS1.SSS3.p3.2.m1.1.1.3">𝑛</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.SS1.SSS3.p3.2.m1.1c">(\sup_{n\leq l}{\|\sum_{k=n}^{l}{\delta_{k}\xi_{k}}\|})_{n}</annotation><annotation encoding="application/x-llamapun" id="A4.SS1.SSS3.p3.2.m1.1d">( roman_sup start_POSTSUBSCRIPT italic_n ≤ italic_l end_POSTSUBSCRIPT ∥ ∑ start_POSTSUBSCRIPT italic_k = italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_l end_POSTSUPERSCRIPT italic_δ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT italic_ξ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ∥ ) start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT</annotation></semantics></math> tends to zero in <math alttext="L^{1}" class="ltx_Math" display="inline" id="A4.SS1.SSS3.p3.3.m2.1"><semantics id="A4.SS1.SSS3.p3.3.m2.1a"><msup id="A4.SS1.SSS3.p3.3.m2.1.1" xref="A4.SS1.SSS3.p3.3.m2.1.1.cmml"><mi id="A4.SS1.SSS3.p3.3.m2.1.1.2" xref="A4.SS1.SSS3.p3.3.m2.1.1.2.cmml">L</mi><mn id="A4.SS1.SSS3.p3.3.m2.1.1.3" xref="A4.SS1.SSS3.p3.3.m2.1.1.3.cmml">1</mn></msup><annotation-xml encoding="MathML-Content" id="A4.SS1.SSS3.p3.3.m2.1b"><apply id="A4.SS1.SSS3.p3.3.m2.1.1.cmml" xref="A4.SS1.SSS3.p3.3.m2.1.1"><csymbol cd="ambiguous" id="A4.SS1.SSS3.p3.3.m2.1.1.1.cmml" xref="A4.SS1.SSS3.p3.3.m2.1.1">superscript</csymbol><ci id="A4.SS1.SSS3.p3.3.m2.1.1.2.cmml" xref="A4.SS1.SSS3.p3.3.m2.1.1.2">𝐿</ci><cn id="A4.SS1.SSS3.p3.3.m2.1.1.3.cmml" type="integer" xref="A4.SS1.SSS3.p3.3.m2.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.SS1.SSS3.p3.3.m2.1c">L^{1}</annotation><annotation encoding="application/x-llamapun" id="A4.SS1.SSS3.p3.3.m2.1d">italic_L start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT</annotation></semantics></math> and also almost surely (because it is non-increasing). This allows us to conclude that our process verifies <cite class="ltx_cite ltx_citemacro_citep">(Tadic & Doucet, <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#bib.bib72" title="">2017</a>, Assumption 2.2)</cite> almost surely.</p> </div> <div class="ltx_para" id="A4.SS1.SSS3.p4"> <p class="ltx_p" id="A4.SS1.SSS3.p4.9">Also, <math alttext="p_{\sigma}" class="ltx_Math" display="inline" id="A4.SS1.SSS3.p4.1.m1.1"><semantics id="A4.SS1.SSS3.p4.1.m1.1a"><msub id="A4.SS1.SSS3.p4.1.m1.1.1" xref="A4.SS1.SSS3.p4.1.m1.1.1.cmml"><mi id="A4.SS1.SSS3.p4.1.m1.1.1.2" xref="A4.SS1.SSS3.p4.1.m1.1.1.2.cmml">p</mi><mi id="A4.SS1.SSS3.p4.1.m1.1.1.3" xref="A4.SS1.SSS3.p4.1.m1.1.1.3.cmml">σ</mi></msub><annotation-xml encoding="MathML-Content" id="A4.SS1.SSS3.p4.1.m1.1b"><apply id="A4.SS1.SSS3.p4.1.m1.1.1.cmml" xref="A4.SS1.SSS3.p4.1.m1.1.1"><csymbol cd="ambiguous" id="A4.SS1.SSS3.p4.1.m1.1.1.1.cmml" xref="A4.SS1.SSS3.p4.1.m1.1.1">subscript</csymbol><ci id="A4.SS1.SSS3.p4.1.m1.1.1.2.cmml" xref="A4.SS1.SSS3.p4.1.m1.1.1.2">𝑝</ci><ci id="A4.SS1.SSS3.p4.1.m1.1.1.3.cmml" xref="A4.SS1.SSS3.p4.1.m1.1.1.3">𝜎</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.SS1.SSS3.p4.1.m1.1c">p_{\sigma}</annotation><annotation encoding="application/x-llamapun" id="A4.SS1.SSS3.p4.1.m1.1d">italic_p start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT</annotation></semantics></math> is <math alttext="\mathcal{C}^{\infty}" class="ltx_Math" display="inline" id="A4.SS1.SSS3.p4.2.m2.1"><semantics id="A4.SS1.SSS3.p4.2.m2.1a"><msup id="A4.SS1.SSS3.p4.2.m2.1.1" xref="A4.SS1.SSS3.p4.2.m2.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="A4.SS1.SSS3.p4.2.m2.1.1.2" xref="A4.SS1.SSS3.p4.2.m2.1.1.2.cmml">𝒞</mi><mi id="A4.SS1.SSS3.p4.2.m2.1.1.3" mathvariant="normal" xref="A4.SS1.SSS3.p4.2.m2.1.1.3.cmml">∞</mi></msup><annotation-xml encoding="MathML-Content" id="A4.SS1.SSS3.p4.2.m2.1b"><apply id="A4.SS1.SSS3.p4.2.m2.1.1.cmml" xref="A4.SS1.SSS3.p4.2.m2.1.1"><csymbol cd="ambiguous" id="A4.SS1.SSS3.p4.2.m2.1.1.1.cmml" xref="A4.SS1.SSS3.p4.2.m2.1.1">superscript</csymbol><ci id="A4.SS1.SSS3.p4.2.m2.1.1.2.cmml" xref="A4.SS1.SSS3.p4.2.m2.1.1.2">𝒞</ci><infinity id="A4.SS1.SSS3.p4.2.m2.1.1.3.cmml" xref="A4.SS1.SSS3.p4.2.m2.1.1.3"></infinity></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.SS1.SSS3.p4.2.m2.1c">\mathcal{C}^{\infty}</annotation><annotation encoding="application/x-llamapun" id="A4.SS1.SSS3.p4.2.m2.1d">caligraphic_C start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT</annotation></semantics></math> by convolution with a Gaussian and then <math alttext="\log p_{\sigma}" class="ltx_Math" display="inline" id="A4.SS1.SSS3.p4.3.m3.1"><semantics id="A4.SS1.SSS3.p4.3.m3.1a"><mrow id="A4.SS1.SSS3.p4.3.m3.1.1" xref="A4.SS1.SSS3.p4.3.m3.1.1.cmml"><mi id="A4.SS1.SSS3.p4.3.m3.1.1.1" xref="A4.SS1.SSS3.p4.3.m3.1.1.1.cmml">log</mi><mo id="A4.SS1.SSS3.p4.3.m3.1.1a" lspace="0.167em" xref="A4.SS1.SSS3.p4.3.m3.1.1.cmml">⁡</mo><msub id="A4.SS1.SSS3.p4.3.m3.1.1.2" xref="A4.SS1.SSS3.p4.3.m3.1.1.2.cmml"><mi id="A4.SS1.SSS3.p4.3.m3.1.1.2.2" xref="A4.SS1.SSS3.p4.3.m3.1.1.2.2.cmml">p</mi><mi id="A4.SS1.SSS3.p4.3.m3.1.1.2.3" xref="A4.SS1.SSS3.p4.3.m3.1.1.2.3.cmml">σ</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="A4.SS1.SSS3.p4.3.m3.1b"><apply id="A4.SS1.SSS3.p4.3.m3.1.1.cmml" xref="A4.SS1.SSS3.p4.3.m3.1.1"><log id="A4.SS1.SSS3.p4.3.m3.1.1.1.cmml" xref="A4.SS1.SSS3.p4.3.m3.1.1.1"></log><apply id="A4.SS1.SSS3.p4.3.m3.1.1.2.cmml" xref="A4.SS1.SSS3.p4.3.m3.1.1.2"><csymbol cd="ambiguous" id="A4.SS1.SSS3.p4.3.m3.1.1.2.1.cmml" xref="A4.SS1.SSS3.p4.3.m3.1.1.2">subscript</csymbol><ci id="A4.SS1.SSS3.p4.3.m3.1.1.2.2.cmml" xref="A4.SS1.SSS3.p4.3.m3.1.1.2.2">𝑝</ci><ci id="A4.SS1.SSS3.p4.3.m3.1.1.2.3.cmml" xref="A4.SS1.SSS3.p4.3.m3.1.1.2.3">𝜎</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.SS1.SSS3.p4.3.m3.1c">\log p_{\sigma}</annotation><annotation encoding="application/x-llamapun" id="A4.SS1.SSS3.p4.3.m3.1d">roman_log italic_p start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT</annotation></semantics></math> is also <math alttext="\mathcal{C}^{\infty}" class="ltx_Math" display="inline" id="A4.SS1.SSS3.p4.4.m4.1"><semantics id="A4.SS1.SSS3.p4.4.m4.1a"><msup id="A4.SS1.SSS3.p4.4.m4.1.1" xref="A4.SS1.SSS3.p4.4.m4.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="A4.SS1.SSS3.p4.4.m4.1.1.2" xref="A4.SS1.SSS3.p4.4.m4.1.1.2.cmml">𝒞</mi><mi id="A4.SS1.SSS3.p4.4.m4.1.1.3" mathvariant="normal" xref="A4.SS1.SSS3.p4.4.m4.1.1.3.cmml">∞</mi></msup><annotation-xml encoding="MathML-Content" id="A4.SS1.SSS3.p4.4.m4.1b"><apply id="A4.SS1.SSS3.p4.4.m4.1.1.cmml" xref="A4.SS1.SSS3.p4.4.m4.1.1"><csymbol cd="ambiguous" id="A4.SS1.SSS3.p4.4.m4.1.1.1.cmml" xref="A4.SS1.SSS3.p4.4.m4.1.1">superscript</csymbol><ci id="A4.SS1.SSS3.p4.4.m4.1.1.2.cmml" xref="A4.SS1.SSS3.p4.4.m4.1.1.2">𝒞</ci><infinity id="A4.SS1.SSS3.p4.4.m4.1.1.3.cmml" xref="A4.SS1.SSS3.p4.4.m4.1.1.3"></infinity></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.SS1.SSS3.p4.4.m4.1c">\mathcal{C}^{\infty}</annotation><annotation encoding="application/x-llamapun" id="A4.SS1.SSS3.p4.4.m4.1d">caligraphic_C start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT</annotation></semantics></math>. Again, by convolution, <math alttext="\mathcal{R}_{\sigma}" class="ltx_Math" display="inline" id="A4.SS1.SSS3.p4.5.m5.1"><semantics id="A4.SS1.SSS3.p4.5.m5.1a"><msub id="A4.SS1.SSS3.p4.5.m5.1.1" xref="A4.SS1.SSS3.p4.5.m5.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="A4.SS1.SSS3.p4.5.m5.1.1.2" xref="A4.SS1.SSS3.p4.5.m5.1.1.2.cmml">ℛ</mi><mi id="A4.SS1.SSS3.p4.5.m5.1.1.3" xref="A4.SS1.SSS3.p4.5.m5.1.1.3.cmml">σ</mi></msub><annotation-xml encoding="MathML-Content" id="A4.SS1.SSS3.p4.5.m5.1b"><apply id="A4.SS1.SSS3.p4.5.m5.1.1.cmml" xref="A4.SS1.SSS3.p4.5.m5.1.1"><csymbol cd="ambiguous" id="A4.SS1.SSS3.p4.5.m5.1.1.1.cmml" xref="A4.SS1.SSS3.p4.5.m5.1.1">subscript</csymbol><ci id="A4.SS1.SSS3.p4.5.m5.1.1.2.cmml" xref="A4.SS1.SSS3.p4.5.m5.1.1.2">ℛ</ci><ci id="A4.SS1.SSS3.p4.5.m5.1.1.3.cmml" xref="A4.SS1.SSS3.p4.5.m5.1.1.3">𝜎</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.SS1.SSS3.p4.5.m5.1c">\mathcal{R}_{\sigma}</annotation><annotation encoding="application/x-llamapun" id="A4.SS1.SSS3.p4.5.m5.1d">caligraphic_R start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT</annotation></semantics></math> is clearly <math alttext="\mathcal{C}^{\infty}" class="ltx_Math" display="inline" id="A4.SS1.SSS3.p4.6.m6.1"><semantics id="A4.SS1.SSS3.p4.6.m6.1a"><msup id="A4.SS1.SSS3.p4.6.m6.1.1" xref="A4.SS1.SSS3.p4.6.m6.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="A4.SS1.SSS3.p4.6.m6.1.1.2" xref="A4.SS1.SSS3.p4.6.m6.1.1.2.cmml">𝒞</mi><mi id="A4.SS1.SSS3.p4.6.m6.1.1.3" mathvariant="normal" xref="A4.SS1.SSS3.p4.6.m6.1.1.3.cmml">∞</mi></msup><annotation-xml encoding="MathML-Content" id="A4.SS1.SSS3.p4.6.m6.1b"><apply id="A4.SS1.SSS3.p4.6.m6.1.1.cmml" xref="A4.SS1.SSS3.p4.6.m6.1.1"><csymbol cd="ambiguous" id="A4.SS1.SSS3.p4.6.m6.1.1.1.cmml" xref="A4.SS1.SSS3.p4.6.m6.1.1">superscript</csymbol><ci id="A4.SS1.SSS3.p4.6.m6.1.1.2.cmml" xref="A4.SS1.SSS3.p4.6.m6.1.1.2">𝒞</ci><infinity id="A4.SS1.SSS3.p4.6.m6.1.1.3.cmml" xref="A4.SS1.SSS3.p4.6.m6.1.1.3"></infinity></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.SS1.SSS3.p4.6.m6.1c">\mathcal{C}^{\infty}</annotation><annotation encoding="application/x-llamapun" id="A4.SS1.SSS3.p4.6.m6.1d">caligraphic_C start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT</annotation></semantics></math>. By Assumption <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#S3.Thmtheorem4" title="Assumption 3.4. ‣ 3.2 Unbiased algorithm analysis ‣ 3 Convergence Analysis ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_tag">3.4</span></a>, <math alttext="\mathcal{J}=\mathcal{F}(\cdot,{\mathbf{y}})+\mathcal{R}_{\sigma}" class="ltx_Math" display="inline" id="A4.SS1.SSS3.p4.7.m7.2"><semantics id="A4.SS1.SSS3.p4.7.m7.2a"><mrow id="A4.SS1.SSS3.p4.7.m7.2.3" xref="A4.SS1.SSS3.p4.7.m7.2.3.cmml"><mi class="ltx_font_mathcaligraphic" id="A4.SS1.SSS3.p4.7.m7.2.3.2" xref="A4.SS1.SSS3.p4.7.m7.2.3.2.cmml">𝒥</mi><mo id="A4.SS1.SSS3.p4.7.m7.2.3.1" xref="A4.SS1.SSS3.p4.7.m7.2.3.1.cmml">=</mo><mrow id="A4.SS1.SSS3.p4.7.m7.2.3.3" xref="A4.SS1.SSS3.p4.7.m7.2.3.3.cmml"><mrow id="A4.SS1.SSS3.p4.7.m7.2.3.3.2" xref="A4.SS1.SSS3.p4.7.m7.2.3.3.2.cmml"><mi class="ltx_font_mathcaligraphic" id="A4.SS1.SSS3.p4.7.m7.2.3.3.2.2" xref="A4.SS1.SSS3.p4.7.m7.2.3.3.2.2.cmml">ℱ</mi><mo id="A4.SS1.SSS3.p4.7.m7.2.3.3.2.1" xref="A4.SS1.SSS3.p4.7.m7.2.3.3.2.1.cmml">⁢</mo><mrow id="A4.SS1.SSS3.p4.7.m7.2.3.3.2.3.2" xref="A4.SS1.SSS3.p4.7.m7.2.3.3.2.3.1.cmml"><mo id="A4.SS1.SSS3.p4.7.m7.2.3.3.2.3.2.1" stretchy="false" xref="A4.SS1.SSS3.p4.7.m7.2.3.3.2.3.1.cmml">(</mo><mo id="A4.SS1.SSS3.p4.7.m7.1.1" lspace="0em" rspace="0em" xref="A4.SS1.SSS3.p4.7.m7.1.1.cmml">⋅</mo><mo id="A4.SS1.SSS3.p4.7.m7.2.3.3.2.3.2.2" xref="A4.SS1.SSS3.p4.7.m7.2.3.3.2.3.1.cmml">,</mo><mi id="A4.SS1.SSS3.p4.7.m7.2.2" xref="A4.SS1.SSS3.p4.7.m7.2.2.cmml">𝐲</mi><mo id="A4.SS1.SSS3.p4.7.m7.2.3.3.2.3.2.3" stretchy="false" xref="A4.SS1.SSS3.p4.7.m7.2.3.3.2.3.1.cmml">)</mo></mrow></mrow><mo id="A4.SS1.SSS3.p4.7.m7.2.3.3.1" xref="A4.SS1.SSS3.p4.7.m7.2.3.3.1.cmml">+</mo><msub id="A4.SS1.SSS3.p4.7.m7.2.3.3.3" xref="A4.SS1.SSS3.p4.7.m7.2.3.3.3.cmml"><mi class="ltx_font_mathcaligraphic" id="A4.SS1.SSS3.p4.7.m7.2.3.3.3.2" xref="A4.SS1.SSS3.p4.7.m7.2.3.3.3.2.cmml">ℛ</mi><mi id="A4.SS1.SSS3.p4.7.m7.2.3.3.3.3" xref="A4.SS1.SSS3.p4.7.m7.2.3.3.3.3.cmml">σ</mi></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="A4.SS1.SSS3.p4.7.m7.2b"><apply id="A4.SS1.SSS3.p4.7.m7.2.3.cmml" xref="A4.SS1.SSS3.p4.7.m7.2.3"><eq id="A4.SS1.SSS3.p4.7.m7.2.3.1.cmml" xref="A4.SS1.SSS3.p4.7.m7.2.3.1"></eq><ci id="A4.SS1.SSS3.p4.7.m7.2.3.2.cmml" xref="A4.SS1.SSS3.p4.7.m7.2.3.2">𝒥</ci><apply id="A4.SS1.SSS3.p4.7.m7.2.3.3.cmml" xref="A4.SS1.SSS3.p4.7.m7.2.3.3"><plus id="A4.SS1.SSS3.p4.7.m7.2.3.3.1.cmml" xref="A4.SS1.SSS3.p4.7.m7.2.3.3.1"></plus><apply id="A4.SS1.SSS3.p4.7.m7.2.3.3.2.cmml" xref="A4.SS1.SSS3.p4.7.m7.2.3.3.2"><times id="A4.SS1.SSS3.p4.7.m7.2.3.3.2.1.cmml" xref="A4.SS1.SSS3.p4.7.m7.2.3.3.2.1"></times><ci id="A4.SS1.SSS3.p4.7.m7.2.3.3.2.2.cmml" xref="A4.SS1.SSS3.p4.7.m7.2.3.3.2.2">ℱ</ci><interval closure="open" id="A4.SS1.SSS3.p4.7.m7.2.3.3.2.3.1.cmml" xref="A4.SS1.SSS3.p4.7.m7.2.3.3.2.3.2"><ci id="A4.SS1.SSS3.p4.7.m7.1.1.cmml" xref="A4.SS1.SSS3.p4.7.m7.1.1">⋅</ci><ci id="A4.SS1.SSS3.p4.7.m7.2.2.cmml" xref="A4.SS1.SSS3.p4.7.m7.2.2">𝐲</ci></interval></apply><apply id="A4.SS1.SSS3.p4.7.m7.2.3.3.3.cmml" xref="A4.SS1.SSS3.p4.7.m7.2.3.3.3"><csymbol cd="ambiguous" id="A4.SS1.SSS3.p4.7.m7.2.3.3.3.1.cmml" xref="A4.SS1.SSS3.p4.7.m7.2.3.3.3">subscript</csymbol><ci id="A4.SS1.SSS3.p4.7.m7.2.3.3.3.2.cmml" xref="A4.SS1.SSS3.p4.7.m7.2.3.3.3.2">ℛ</ci><ci id="A4.SS1.SSS3.p4.7.m7.2.3.3.3.3.cmml" xref="A4.SS1.SSS3.p4.7.m7.2.3.3.3.3">𝜎</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.SS1.SSS3.p4.7.m7.2c">\mathcal{J}=\mathcal{F}(\cdot,{\mathbf{y}})+\mathcal{R}_{\sigma}</annotation><annotation encoding="application/x-llamapun" id="A4.SS1.SSS3.p4.7.m7.2d">caligraphic_J = caligraphic_F ( ⋅ , bold_y ) + caligraphic_R start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT</annotation></semantics></math> is <math alttext="\mathcal{C}^{\infty}" class="ltx_Math" display="inline" id="A4.SS1.SSS3.p4.8.m8.1"><semantics id="A4.SS1.SSS3.p4.8.m8.1a"><msup id="A4.SS1.SSS3.p4.8.m8.1.1" xref="A4.SS1.SSS3.p4.8.m8.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="A4.SS1.SSS3.p4.8.m8.1.1.2" xref="A4.SS1.SSS3.p4.8.m8.1.1.2.cmml">𝒞</mi><mi id="A4.SS1.SSS3.p4.8.m8.1.1.3" mathvariant="normal" xref="A4.SS1.SSS3.p4.8.m8.1.1.3.cmml">∞</mi></msup><annotation-xml encoding="MathML-Content" id="A4.SS1.SSS3.p4.8.m8.1b"><apply id="A4.SS1.SSS3.p4.8.m8.1.1.cmml" xref="A4.SS1.SSS3.p4.8.m8.1.1"><csymbol cd="ambiguous" id="A4.SS1.SSS3.p4.8.m8.1.1.1.cmml" xref="A4.SS1.SSS3.p4.8.m8.1.1">superscript</csymbol><ci id="A4.SS1.SSS3.p4.8.m8.1.1.2.cmml" xref="A4.SS1.SSS3.p4.8.m8.1.1.2">𝒞</ci><infinity id="A4.SS1.SSS3.p4.8.m8.1.1.3.cmml" xref="A4.SS1.SSS3.p4.8.m8.1.1.3"></infinity></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.SS1.SSS3.p4.8.m8.1c">\mathcal{C}^{\infty}</annotation><annotation encoding="application/x-llamapun" id="A4.SS1.SSS3.p4.8.m8.1d">caligraphic_C start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT</annotation></semantics></math>. So <cite class="ltx_cite ltx_citemacro_citep">(Tadic & Doucet, <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#bib.bib72" title="">2017</a>, Assumption 2.b)</cite> is verified. We can thus apply <cite class="ltx_cite ltx_citemacro_citep">(Tadic & Doucet, <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#bib.bib72" title="">2017</a>, Theorem 2.1. (ii))</cite> with <math alttext="\eta=0" class="ltx_Math" display="inline" id="A4.SS1.SSS3.p4.9.m9.1"><semantics id="A4.SS1.SSS3.p4.9.m9.1a"><mrow id="A4.SS1.SSS3.p4.9.m9.1.1" xref="A4.SS1.SSS3.p4.9.m9.1.1.cmml"><mi id="A4.SS1.SSS3.p4.9.m9.1.1.2" xref="A4.SS1.SSS3.p4.9.m9.1.1.2.cmml">η</mi><mo id="A4.SS1.SSS3.p4.9.m9.1.1.1" xref="A4.SS1.SSS3.p4.9.m9.1.1.1.cmml">=</mo><mn id="A4.SS1.SSS3.p4.9.m9.1.1.3" xref="A4.SS1.SSS3.p4.9.m9.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="A4.SS1.SSS3.p4.9.m9.1b"><apply id="A4.SS1.SSS3.p4.9.m9.1.1.cmml" xref="A4.SS1.SSS3.p4.9.m9.1.1"><eq id="A4.SS1.SSS3.p4.9.m9.1.1.1.cmml" xref="A4.SS1.SSS3.p4.9.m9.1.1.1"></eq><ci id="A4.SS1.SSS3.p4.9.m9.1.1.2.cmml" xref="A4.SS1.SSS3.p4.9.m9.1.1.2">𝜂</ci><cn id="A4.SS1.SSS3.p4.9.m9.1.1.3.cmml" type="integer" xref="A4.SS1.SSS3.p4.9.m9.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.SS1.SSS3.p4.9.m9.1c">\eta=0</annotation><annotation encoding="application/x-llamapun" id="A4.SS1.SSS3.p4.9.m9.1d">italic_η = 0</annotation></semantics></math> to our process, which concludes the proof.</p> </div> </section> <section class="ltx_subsubsection" id="A4.SS1.SSS4"> <h4 class="ltx_title ltx_title_subsubsection"> <span class="ltx_tag ltx_tag_subsubsection">D.1.4 </span>Proof of Proposition <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#S3.Thmproposition4" title="Proposition 3.4. ‣ 3.2 Unbiased algorithm analysis ‣ 3 Convergence Analysis ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_tag">3.4</span></a> </h4> <div class="ltx_para" id="A4.SS1.SSS4.p1"> <p class="ltx_p" id="A4.SS1.SSS4.p1.6">We only need to notice that Assumption <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#S3.Thmtheorem5" title="Assumption 3.5 (Manifold hypothesis). ‣ 3.2 Unbiased algorithm analysis ‣ 3 Convergence Analysis ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_tag">3.5</span></a> implies Assumption <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#S3.Thmtheorem2" title="Assumption 3.2. ‣ 3.1 Asymptotics of critical points when 𝜎→0 ‣ 3 Convergence Analysis ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_tag">3.2</span></a> and apply Proposition <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#S3.Thmproposition3" title="Proposition 3.3. ‣ 3.2 Unbiased algorithm analysis ‣ 3 Convergence Analysis ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_tag">3.3</span></a>. One can directly apply <cite class="ltx_cite ltx_citemacro_citep">(De Bortoli, <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#bib.bib17" title="">2023</a>, Lemma C.1)</cite> but we give a proof here for the sake of clarity. Observe that</p> <table class="ltx_equation ltx_eqn_table" id="A4.Ex94"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="p_{\sigma}({\mathbf{x}})=(p\star\mathcal{N}_{\sigma})({\mathbf{x}})=\int_{{% \mathbb{R}}^{d}}{p({\mathbf{y}})\mathcal{N}_{\sigma}({\mathbf{x}}-{\mathbf{y}}% )d{\mathbf{y}}}=\int_{\mathcal{B}(0,R)}{p({\mathbf{y}})\mathcal{N}_{\sigma}({% \mathbf{x}}-{\mathbf{y}})d{\mathbf{y}}}." class="ltx_Math" display="block" id="A4.Ex94.m1.7"><semantics id="A4.Ex94.m1.7a"><mrow id="A4.Ex94.m1.7.7.1" xref="A4.Ex94.m1.7.7.1.1.cmml"><mrow id="A4.Ex94.m1.7.7.1.1" xref="A4.Ex94.m1.7.7.1.1.cmml"><mrow id="A4.Ex94.m1.7.7.1.1.5" xref="A4.Ex94.m1.7.7.1.1.5.cmml"><msub id="A4.Ex94.m1.7.7.1.1.5.2" xref="A4.Ex94.m1.7.7.1.1.5.2.cmml"><mi id="A4.Ex94.m1.7.7.1.1.5.2.2" xref="A4.Ex94.m1.7.7.1.1.5.2.2.cmml">p</mi><mi id="A4.Ex94.m1.7.7.1.1.5.2.3" 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id="A4.Ex94.m1.7c">p_{\sigma}({\mathbf{x}})=(p\star\mathcal{N}_{\sigma})({\mathbf{x}})=\int_{{% \mathbb{R}}^{d}}{p({\mathbf{y}})\mathcal{N}_{\sigma}({\mathbf{x}}-{\mathbf{y}}% )d{\mathbf{y}}}=\int_{\mathcal{B}(0,R)}{p({\mathbf{y}})\mathcal{N}_{\sigma}({% \mathbf{x}}-{\mathbf{y}})d{\mathbf{y}}}.</annotation><annotation encoding="application/x-llamapun" id="A4.Ex94.m1.7d">italic_p start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ( bold_x ) = ( italic_p ⋆ caligraphic_N start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ) ( bold_x ) = ∫ start_POSTSUBSCRIPT blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT end_POSTSUBSCRIPT italic_p ( bold_y ) caligraphic_N start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ( bold_x - bold_y ) italic_d bold_y = ∫ start_POSTSUBSCRIPT caligraphic_B ( 0 , italic_R ) end_POSTSUBSCRIPT italic_p ( bold_y ) caligraphic_N start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ( bold_x - bold_y ) italic_d bold_y .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="A4.SS1.SSS4.p1.2">By taking <math alttext="R>0" class="ltx_Math" display="inline" id="A4.SS1.SSS4.p1.1.m1.1"><semantics id="A4.SS1.SSS4.p1.1.m1.1a"><mrow id="A4.SS1.SSS4.p1.1.m1.1.1" xref="A4.SS1.SSS4.p1.1.m1.1.1.cmml"><mi id="A4.SS1.SSS4.p1.1.m1.1.1.2" xref="A4.SS1.SSS4.p1.1.m1.1.1.2.cmml">R</mi><mo id="A4.SS1.SSS4.p1.1.m1.1.1.1" xref="A4.SS1.SSS4.p1.1.m1.1.1.1.cmml">></mo><mn id="A4.SS1.SSS4.p1.1.m1.1.1.3" xref="A4.SS1.SSS4.p1.1.m1.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="A4.SS1.SSS4.p1.1.m1.1b"><apply id="A4.SS1.SSS4.p1.1.m1.1.1.cmml" xref="A4.SS1.SSS4.p1.1.m1.1.1"><gt id="A4.SS1.SSS4.p1.1.m1.1.1.1.cmml" xref="A4.SS1.SSS4.p1.1.m1.1.1.1"></gt><ci id="A4.SS1.SSS4.p1.1.m1.1.1.2.cmml" xref="A4.SS1.SSS4.p1.1.m1.1.1.2">𝑅</ci><cn id="A4.SS1.SSS4.p1.1.m1.1.1.3.cmml" type="integer" xref="A4.SS1.SSS4.p1.1.m1.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.SS1.SSS4.p1.1.m1.1c">R>0</annotation><annotation encoding="application/x-llamapun" id="A4.SS1.SSS4.p1.1.m1.1d">italic_R > 0</annotation></semantics></math> such that <math alttext="\mathcal{M}\subset\mathcal{B}(0,R)" class="ltx_Math" display="inline" id="A4.SS1.SSS4.p1.2.m2.2"><semantics id="A4.SS1.SSS4.p1.2.m2.2a"><mrow id="A4.SS1.SSS4.p1.2.m2.2.3" xref="A4.SS1.SSS4.p1.2.m2.2.3.cmml"><mi class="ltx_font_mathcaligraphic" id="A4.SS1.SSS4.p1.2.m2.2.3.2" xref="A4.SS1.SSS4.p1.2.m2.2.3.2.cmml">ℳ</mi><mo id="A4.SS1.SSS4.p1.2.m2.2.3.1" xref="A4.SS1.SSS4.p1.2.m2.2.3.1.cmml">⊂</mo><mrow id="A4.SS1.SSS4.p1.2.m2.2.3.3" xref="A4.SS1.SSS4.p1.2.m2.2.3.3.cmml"><mi class="ltx_font_mathcaligraphic" id="A4.SS1.SSS4.p1.2.m2.2.3.3.2" xref="A4.SS1.SSS4.p1.2.m2.2.3.3.2.cmml">ℬ</mi><mo id="A4.SS1.SSS4.p1.2.m2.2.3.3.1" xref="A4.SS1.SSS4.p1.2.m2.2.3.3.1.cmml">⁢</mo><mrow id="A4.SS1.SSS4.p1.2.m2.2.3.3.3.2" xref="A4.SS1.SSS4.p1.2.m2.2.3.3.3.1.cmml"><mo id="A4.SS1.SSS4.p1.2.m2.2.3.3.3.2.1" stretchy="false" xref="A4.SS1.SSS4.p1.2.m2.2.3.3.3.1.cmml">(</mo><mn id="A4.SS1.SSS4.p1.2.m2.1.1" xref="A4.SS1.SSS4.p1.2.m2.1.1.cmml">0</mn><mo id="A4.SS1.SSS4.p1.2.m2.2.3.3.3.2.2" xref="A4.SS1.SSS4.p1.2.m2.2.3.3.3.1.cmml">,</mo><mi id="A4.SS1.SSS4.p1.2.m2.2.2" xref="A4.SS1.SSS4.p1.2.m2.2.2.cmml">R</mi><mo id="A4.SS1.SSS4.p1.2.m2.2.3.3.3.2.3" stretchy="false" xref="A4.SS1.SSS4.p1.2.m2.2.3.3.3.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="A4.SS1.SSS4.p1.2.m2.2b"><apply id="A4.SS1.SSS4.p1.2.m2.2.3.cmml" xref="A4.SS1.SSS4.p1.2.m2.2.3"><subset id="A4.SS1.SSS4.p1.2.m2.2.3.1.cmml" xref="A4.SS1.SSS4.p1.2.m2.2.3.1"></subset><ci id="A4.SS1.SSS4.p1.2.m2.2.3.2.cmml" xref="A4.SS1.SSS4.p1.2.m2.2.3.2">ℳ</ci><apply id="A4.SS1.SSS4.p1.2.m2.2.3.3.cmml" xref="A4.SS1.SSS4.p1.2.m2.2.3.3"><times id="A4.SS1.SSS4.p1.2.m2.2.3.3.1.cmml" xref="A4.SS1.SSS4.p1.2.m2.2.3.3.1"></times><ci id="A4.SS1.SSS4.p1.2.m2.2.3.3.2.cmml" xref="A4.SS1.SSS4.p1.2.m2.2.3.3.2">ℬ</ci><interval closure="open" id="A4.SS1.SSS4.p1.2.m2.2.3.3.3.1.cmml" xref="A4.SS1.SSS4.p1.2.m2.2.3.3.3.2"><cn id="A4.SS1.SSS4.p1.2.m2.1.1.cmml" type="integer" xref="A4.SS1.SSS4.p1.2.m2.1.1">0</cn><ci id="A4.SS1.SSS4.p1.2.m2.2.2.cmml" xref="A4.SS1.SSS4.p1.2.m2.2.2">𝑅</ci></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.SS1.SSS4.p1.2.m2.2c">\mathcal{M}\subset\mathcal{B}(0,R)</annotation><annotation encoding="application/x-llamapun" id="A4.SS1.SSS4.p1.2.m2.2d">caligraphic_M ⊂ caligraphic_B ( 0 , italic_R )</annotation></semantics></math>, we get </p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A8.EGx53"> <tbody id="A4.Ex95"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\|\nabla\log p_{\sigma}({\mathbf{x}})\|" class="ltx_Math" display="inline" id="A4.Ex95.m1.2"><semantics id="A4.Ex95.m1.2a"><mrow id="A4.Ex95.m1.2.2.1" xref="A4.Ex95.m1.2.2.2.cmml"><mo id="A4.Ex95.m1.2.2.1.2" stretchy="false" xref="A4.Ex95.m1.2.2.2.1.cmml">‖</mo><mrow id="A4.Ex95.m1.2.2.1.1" xref="A4.Ex95.m1.2.2.1.1.cmml"><mrow id="A4.Ex95.m1.2.2.1.1.2" xref="A4.Ex95.m1.2.2.1.1.2.cmml"><mrow id="A4.Ex95.m1.2.2.1.1.2.1" xref="A4.Ex95.m1.2.2.1.1.2.1.cmml"><mo id="A4.Ex95.m1.2.2.1.1.2.1.1" rspace="0.167em" xref="A4.Ex95.m1.2.2.1.1.2.1.1.cmml">∇</mo><mi id="A4.Ex95.m1.2.2.1.1.2.1.2" xref="A4.Ex95.m1.2.2.1.1.2.1.2.cmml">log</mi></mrow><mo id="A4.Ex95.m1.2.2.1.1.2a" lspace="0.167em" xref="A4.Ex95.m1.2.2.1.1.2.cmml">⁡</mo><msub id="A4.Ex95.m1.2.2.1.1.2.2" xref="A4.Ex95.m1.2.2.1.1.2.2.cmml"><mi id="A4.Ex95.m1.2.2.1.1.2.2.2" xref="A4.Ex95.m1.2.2.1.1.2.2.2.cmml">p</mi><mi id="A4.Ex95.m1.2.2.1.1.2.2.3" xref="A4.Ex95.m1.2.2.1.1.2.2.3.cmml">σ</mi></msub></mrow><mo id="A4.Ex95.m1.2.2.1.1.1" xref="A4.Ex95.m1.2.2.1.1.1.cmml">⁢</mo><mrow id="A4.Ex95.m1.2.2.1.1.3.2" xref="A4.Ex95.m1.2.2.1.1.cmml"><mo id="A4.Ex95.m1.2.2.1.1.3.2.1" stretchy="false" xref="A4.Ex95.m1.2.2.1.1.cmml">(</mo><mi id="A4.Ex95.m1.1.1" xref="A4.Ex95.m1.1.1.cmml">𝐱</mi><mo id="A4.Ex95.m1.2.2.1.1.3.2.2" stretchy="false" xref="A4.Ex95.m1.2.2.1.1.cmml">)</mo></mrow></mrow><mo id="A4.Ex95.m1.2.2.1.3" stretchy="false" xref="A4.Ex95.m1.2.2.2.1.cmml">‖</mo></mrow><annotation-xml encoding="MathML-Content" id="A4.Ex95.m1.2b"><apply id="A4.Ex95.m1.2.2.2.cmml" xref="A4.Ex95.m1.2.2.1"><csymbol cd="latexml" id="A4.Ex95.m1.2.2.2.1.cmml" xref="A4.Ex95.m1.2.2.1.2">norm</csymbol><apply id="A4.Ex95.m1.2.2.1.1.cmml" xref="A4.Ex95.m1.2.2.1.1"><times id="A4.Ex95.m1.2.2.1.1.1.cmml" xref="A4.Ex95.m1.2.2.1.1.1"></times><apply id="A4.Ex95.m1.2.2.1.1.2.cmml" xref="A4.Ex95.m1.2.2.1.1.2"><apply id="A4.Ex95.m1.2.2.1.1.2.1.cmml" xref="A4.Ex95.m1.2.2.1.1.2.1"><ci id="A4.Ex95.m1.2.2.1.1.2.1.1.cmml" xref="A4.Ex95.m1.2.2.1.1.2.1.1">∇</ci><log id="A4.Ex95.m1.2.2.1.1.2.1.2.cmml" xref="A4.Ex95.m1.2.2.1.1.2.1.2"></log></apply><apply id="A4.Ex95.m1.2.2.1.1.2.2.cmml" xref="A4.Ex95.m1.2.2.1.1.2.2"><csymbol cd="ambiguous" id="A4.Ex95.m1.2.2.1.1.2.2.1.cmml" xref="A4.Ex95.m1.2.2.1.1.2.2">subscript</csymbol><ci id="A4.Ex95.m1.2.2.1.1.2.2.2.cmml" xref="A4.Ex95.m1.2.2.1.1.2.2.2">𝑝</ci><ci id="A4.Ex95.m1.2.2.1.1.2.2.3.cmml" xref="A4.Ex95.m1.2.2.1.1.2.2.3">𝜎</ci></apply></apply><ci id="A4.Ex95.m1.1.1.cmml" xref="A4.Ex95.m1.1.1">𝐱</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.Ex95.m1.2c">\displaystyle\|\nabla\log p_{\sigma}({\mathbf{x}})\|</annotation><annotation encoding="application/x-llamapun" id="A4.Ex95.m1.2d">∥ ∇ roman_log italic_p start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ( bold_x ) ∥</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle=\left\|\frac{\nabla p_{\sigma}({\mathbf{x}})}{p_{\sigma}({% 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encoding="application/x-tex" id="A4.Ex95.m2.7c">\displaystyle=\left\|\frac{\nabla p_{\sigma}({\mathbf{x}})}{p_{\sigma}({% \mathbf{x}})}\right\|=\frac{1}{p_{\sigma}({\mathbf{x}})}\left\|\int_{\mathcal{% B}(0,R)}{p({\mathbf{y}})\frac{{\mathbf{x}}-{\mathbf{y}}}{\sigma^{2}}\mathcal{N% }_{\sigma}({\mathbf{x}}-{\mathbf{y}})d{\mathbf{y}}}\right\|</annotation><annotation encoding="application/x-llamapun" id="A4.Ex95.m2.7d">= ∥ divide start_ARG ∇ italic_p start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ( bold_x ) end_ARG start_ARG italic_p start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ( bold_x ) end_ARG ∥ = divide start_ARG 1 end_ARG start_ARG italic_p start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ( bold_x ) end_ARG ∥ ∫ start_POSTSUBSCRIPT caligraphic_B ( 0 , italic_R ) end_POSTSUBSCRIPT italic_p ( bold_y ) divide start_ARG bold_x - bold_y end_ARG start_ARG italic_σ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG caligraphic_N start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ( bold_x - 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id="A4.Ex96.m1.6c">\displaystyle\leq\frac{1}{p_{\sigma}({\mathbf{x}})}\int_{\mathcal{B}(0,R)}{p({% \mathbf{y}})\frac{\|{\mathbf{x}}-{\mathbf{y}}\|}{\sigma^{2}}\mathcal{N}_{% \sigma}({\mathbf{x}}-{\mathbf{y}})d{\mathbf{y}}}</annotation><annotation encoding="application/x-llamapun" id="A4.Ex96.m1.6d">≤ divide start_ARG 1 end_ARG start_ARG italic_p start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ( bold_x ) end_ARG ∫ start_POSTSUBSCRIPT caligraphic_B ( 0 , italic_R ) end_POSTSUBSCRIPT italic_p ( bold_y ) divide start_ARG ∥ bold_x - bold_y ∥ end_ARG start_ARG italic_σ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG caligraphic_N start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ( bold_x - bold_y ) italic_d bold_y</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="A4.Ex97"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_eqn_cell"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\leq\frac{\|{\mathbf{x}}\|+R}{\sigma^{2}}," class="ltx_Math" display="inline" id="A4.Ex97.m1.2"><semantics id="A4.Ex97.m1.2a"><mrow id="A4.Ex97.m1.2.2.1" xref="A4.Ex97.m1.2.2.1.1.cmml"><mrow id="A4.Ex97.m1.2.2.1.1" xref="A4.Ex97.m1.2.2.1.1.cmml"><mi id="A4.Ex97.m1.2.2.1.1.2" xref="A4.Ex97.m1.2.2.1.1.2.cmml"></mi><mo id="A4.Ex97.m1.2.2.1.1.1" xref="A4.Ex97.m1.2.2.1.1.1.cmml">≤</mo><mstyle displaystyle="true" id="A4.Ex97.m1.1.1" xref="A4.Ex97.m1.1.1.cmml"><mfrac id="A4.Ex97.m1.1.1a" xref="A4.Ex97.m1.1.1.cmml"><mrow id="A4.Ex97.m1.1.1.1" xref="A4.Ex97.m1.1.1.1.cmml"><mrow id="A4.Ex97.m1.1.1.1.3.2" xref="A4.Ex97.m1.1.1.1.3.1.cmml"><mo id="A4.Ex97.m1.1.1.1.3.2.1" stretchy="false" xref="A4.Ex97.m1.1.1.1.3.1.1.cmml">‖</mo><mi id="A4.Ex97.m1.1.1.1.1" xref="A4.Ex97.m1.1.1.1.1.cmml">𝐱</mi><mo id="A4.Ex97.m1.1.1.1.3.2.2" stretchy="false" xref="A4.Ex97.m1.1.1.1.3.1.1.cmml">‖</mo></mrow><mo id="A4.Ex97.m1.1.1.1.2" xref="A4.Ex97.m1.1.1.1.2.cmml">+</mo><mi id="A4.Ex97.m1.1.1.1.4" xref="A4.Ex97.m1.1.1.1.4.cmml">R</mi></mrow><msup id="A4.Ex97.m1.1.1.3" xref="A4.Ex97.m1.1.1.3.cmml"><mi id="A4.Ex97.m1.1.1.3.2" xref="A4.Ex97.m1.1.1.3.2.cmml">σ</mi><mn id="A4.Ex97.m1.1.1.3.3" xref="A4.Ex97.m1.1.1.3.3.cmml">2</mn></msup></mfrac></mstyle></mrow><mo id="A4.Ex97.m1.2.2.1.2" xref="A4.Ex97.m1.2.2.1.1.cmml">,</mo></mrow><annotation-xml encoding="MathML-Content" id="A4.Ex97.m1.2b"><apply id="A4.Ex97.m1.2.2.1.1.cmml" xref="A4.Ex97.m1.2.2.1"><leq id="A4.Ex97.m1.2.2.1.1.1.cmml" xref="A4.Ex97.m1.2.2.1.1.1"></leq><csymbol cd="latexml" id="A4.Ex97.m1.2.2.1.1.2.cmml" xref="A4.Ex97.m1.2.2.1.1.2">absent</csymbol><apply id="A4.Ex97.m1.1.1.cmml" xref="A4.Ex97.m1.1.1"><divide id="A4.Ex97.m1.1.1.2.cmml" xref="A4.Ex97.m1.1.1"></divide><apply id="A4.Ex97.m1.1.1.1.cmml" xref="A4.Ex97.m1.1.1.1"><plus id="A4.Ex97.m1.1.1.1.2.cmml" xref="A4.Ex97.m1.1.1.1.2"></plus><apply id="A4.Ex97.m1.1.1.1.3.1.cmml" 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ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="A4.SS1.SSS4.p1.5">so Assumption <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#S3.Thmtheorem2" title="Assumption 3.2. ‣ 3.1 Asymptotics of critical points when 𝜎→0 ‣ 3 Convergence Analysis ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_tag">3.2</span></a> holds with <math alttext="B=\max(1,R)" class="ltx_Math" display="inline" id="A4.SS1.SSS4.p1.3.m1.3"><semantics id="A4.SS1.SSS4.p1.3.m1.3a"><mrow id="A4.SS1.SSS4.p1.3.m1.3.4" xref="A4.SS1.SSS4.p1.3.m1.3.4.cmml"><mi id="A4.SS1.SSS4.p1.3.m1.3.4.2" xref="A4.SS1.SSS4.p1.3.m1.3.4.2.cmml">B</mi><mo id="A4.SS1.SSS4.p1.3.m1.3.4.1" xref="A4.SS1.SSS4.p1.3.m1.3.4.1.cmml">=</mo><mrow id="A4.SS1.SSS4.p1.3.m1.3.4.3.2" xref="A4.SS1.SSS4.p1.3.m1.3.4.3.1.cmml"><mi id="A4.SS1.SSS4.p1.3.m1.1.1" xref="A4.SS1.SSS4.p1.3.m1.1.1.cmml">max</mi><mo id="A4.SS1.SSS4.p1.3.m1.3.4.3.2a" xref="A4.SS1.SSS4.p1.3.m1.3.4.3.1.cmml">⁡</mo><mrow id="A4.SS1.SSS4.p1.3.m1.3.4.3.2.1" xref="A4.SS1.SSS4.p1.3.m1.3.4.3.1.cmml"><mo id="A4.SS1.SSS4.p1.3.m1.3.4.3.2.1.1" stretchy="false" xref="A4.SS1.SSS4.p1.3.m1.3.4.3.1.cmml">(</mo><mn id="A4.SS1.SSS4.p1.3.m1.2.2" xref="A4.SS1.SSS4.p1.3.m1.2.2.cmml">1</mn><mo id="A4.SS1.SSS4.p1.3.m1.3.4.3.2.1.2" xref="A4.SS1.SSS4.p1.3.m1.3.4.3.1.cmml">,</mo><mi id="A4.SS1.SSS4.p1.3.m1.3.3" xref="A4.SS1.SSS4.p1.3.m1.3.3.cmml">R</mi><mo id="A4.SS1.SSS4.p1.3.m1.3.4.3.2.1.3" stretchy="false" xref="A4.SS1.SSS4.p1.3.m1.3.4.3.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="A4.SS1.SSS4.p1.3.m1.3b"><apply id="A4.SS1.SSS4.p1.3.m1.3.4.cmml" xref="A4.SS1.SSS4.p1.3.m1.3.4"><eq id="A4.SS1.SSS4.p1.3.m1.3.4.1.cmml" xref="A4.SS1.SSS4.p1.3.m1.3.4.1"></eq><ci id="A4.SS1.SSS4.p1.3.m1.3.4.2.cmml" xref="A4.SS1.SSS4.p1.3.m1.3.4.2">𝐵</ci><apply id="A4.SS1.SSS4.p1.3.m1.3.4.3.1.cmml" xref="A4.SS1.SSS4.p1.3.m1.3.4.3.2"><max id="A4.SS1.SSS4.p1.3.m1.1.1.cmml" xref="A4.SS1.SSS4.p1.3.m1.1.1"></max><cn id="A4.SS1.SSS4.p1.3.m1.2.2.cmml" type="integer" xref="A4.SS1.SSS4.p1.3.m1.2.2">1</cn><ci id="A4.SS1.SSS4.p1.3.m1.3.3.cmml" xref="A4.SS1.SSS4.p1.3.m1.3.3">𝑅</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.SS1.SSS4.p1.3.m1.3c">B=\max(1,R)</annotation><annotation encoding="application/x-llamapun" id="A4.SS1.SSS4.p1.3.m1.3d">italic_B = roman_max ( 1 , italic_R )</annotation></semantics></math>, <math alttext="\beta=-2" class="ltx_Math" display="inline" id="A4.SS1.SSS4.p1.4.m2.1"><semantics id="A4.SS1.SSS4.p1.4.m2.1a"><mrow id="A4.SS1.SSS4.p1.4.m2.1.1" xref="A4.SS1.SSS4.p1.4.m2.1.1.cmml"><mi id="A4.SS1.SSS4.p1.4.m2.1.1.2" xref="A4.SS1.SSS4.p1.4.m2.1.1.2.cmml">β</mi><mo id="A4.SS1.SSS4.p1.4.m2.1.1.1" xref="A4.SS1.SSS4.p1.4.m2.1.1.1.cmml">=</mo><mrow id="A4.SS1.SSS4.p1.4.m2.1.1.3" xref="A4.SS1.SSS4.p1.4.m2.1.1.3.cmml"><mo id="A4.SS1.SSS4.p1.4.m2.1.1.3a" xref="A4.SS1.SSS4.p1.4.m2.1.1.3.cmml">−</mo><mn id="A4.SS1.SSS4.p1.4.m2.1.1.3.2" xref="A4.SS1.SSS4.p1.4.m2.1.1.3.2.cmml">2</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="A4.SS1.SSS4.p1.4.m2.1b"><apply id="A4.SS1.SSS4.p1.4.m2.1.1.cmml" xref="A4.SS1.SSS4.p1.4.m2.1.1"><eq id="A4.SS1.SSS4.p1.4.m2.1.1.1.cmml" xref="A4.SS1.SSS4.p1.4.m2.1.1.1"></eq><ci id="A4.SS1.SSS4.p1.4.m2.1.1.2.cmml" xref="A4.SS1.SSS4.p1.4.m2.1.1.2">𝛽</ci><apply id="A4.SS1.SSS4.p1.4.m2.1.1.3.cmml" xref="A4.SS1.SSS4.p1.4.m2.1.1.3"><minus id="A4.SS1.SSS4.p1.4.m2.1.1.3.1.cmml" xref="A4.SS1.SSS4.p1.4.m2.1.1.3"></minus><cn id="A4.SS1.SSS4.p1.4.m2.1.1.3.2.cmml" type="integer" xref="A4.SS1.SSS4.p1.4.m2.1.1.3.2">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.SS1.SSS4.p1.4.m2.1c">\beta=-2</annotation><annotation encoding="application/x-llamapun" id="A4.SS1.SSS4.p1.4.m2.1d">italic_β = - 2</annotation></semantics></math> and <math alttext="r=1" class="ltx_Math" display="inline" id="A4.SS1.SSS4.p1.5.m3.1"><semantics id="A4.SS1.SSS4.p1.5.m3.1a"><mrow id="A4.SS1.SSS4.p1.5.m3.1.1" xref="A4.SS1.SSS4.p1.5.m3.1.1.cmml"><mi id="A4.SS1.SSS4.p1.5.m3.1.1.2" xref="A4.SS1.SSS4.p1.5.m3.1.1.2.cmml">r</mi><mo id="A4.SS1.SSS4.p1.5.m3.1.1.1" xref="A4.SS1.SSS4.p1.5.m3.1.1.1.cmml">=</mo><mn id="A4.SS1.SSS4.p1.5.m3.1.1.3" xref="A4.SS1.SSS4.p1.5.m3.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="A4.SS1.SSS4.p1.5.m3.1b"><apply id="A4.SS1.SSS4.p1.5.m3.1.1.cmml" xref="A4.SS1.SSS4.p1.5.m3.1.1"><eq id="A4.SS1.SSS4.p1.5.m3.1.1.1.cmml" xref="A4.SS1.SSS4.p1.5.m3.1.1.1"></eq><ci id="A4.SS1.SSS4.p1.5.m3.1.1.2.cmml" xref="A4.SS1.SSS4.p1.5.m3.1.1.2">𝑟</ci><cn id="A4.SS1.SSS4.p1.5.m3.1.1.3.cmml" type="integer" xref="A4.SS1.SSS4.p1.5.m3.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.SS1.SSS4.p1.5.m3.1c">r=1</annotation><annotation encoding="application/x-llamapun" id="A4.SS1.SSS4.p1.5.m3.1d">italic_r = 1</annotation></semantics></math>.</p> </div> </section> <section class="ltx_subsubsection" id="A4.SS1.SSS5"> <h4 class="ltx_title ltx_title_subsubsection"> <span class="ltx_tag ltx_tag_subsubsection">D.1.5 </span>Proof of Proposition <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#S3.Thmproposition5" title="Proposition 3.5. ‣ 3.3 Biased algorithm analysis ‣ 3 Convergence Analysis ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_tag">3.5</span></a> </h4> <div class="ltx_para" id="A4.SS1.SSS5.p1"> <p class="ltx_p" id="A4.SS1.SSS5.p1.6">We can define <math alttext="\eta_{k}" class="ltx_Math" display="inline" id="A4.SS1.SSS5.p1.1.m1.1"><semantics id="A4.SS1.SSS5.p1.1.m1.1a"><msub id="A4.SS1.SSS5.p1.1.m1.1.1" xref="A4.SS1.SSS5.p1.1.m1.1.1.cmml"><mi id="A4.SS1.SSS5.p1.1.m1.1.1.2" xref="A4.SS1.SSS5.p1.1.m1.1.1.2.cmml">η</mi><mi id="A4.SS1.SSS5.p1.1.m1.1.1.3" xref="A4.SS1.SSS5.p1.1.m1.1.1.3.cmml">k</mi></msub><annotation-xml encoding="MathML-Content" id="A4.SS1.SSS5.p1.1.m1.1b"><apply id="A4.SS1.SSS5.p1.1.m1.1.1.cmml" xref="A4.SS1.SSS5.p1.1.m1.1.1"><csymbol cd="ambiguous" id="A4.SS1.SSS5.p1.1.m1.1.1.1.cmml" xref="A4.SS1.SSS5.p1.1.m1.1.1">subscript</csymbol><ci id="A4.SS1.SSS5.p1.1.m1.1.1.2.cmml" xref="A4.SS1.SSS5.p1.1.m1.1.1.2">𝜂</ci><ci id="A4.SS1.SSS5.p1.1.m1.1.1.3.cmml" xref="A4.SS1.SSS5.p1.1.m1.1.1.3">𝑘</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.SS1.SSS5.p1.1.m1.1c">\eta_{k}</annotation><annotation encoding="application/x-llamapun" id="A4.SS1.SSS5.p1.1.m1.1d">italic_η start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT</annotation></semantics></math> by <math alttext="\eta_{k}=\mathbb{E}(\xi_{k})" class="ltx_Math" display="inline" id="A4.SS1.SSS5.p1.2.m2.1"><semantics id="A4.SS1.SSS5.p1.2.m2.1a"><mrow id="A4.SS1.SSS5.p1.2.m2.1.1" xref="A4.SS1.SSS5.p1.2.m2.1.1.cmml"><msub id="A4.SS1.SSS5.p1.2.m2.1.1.3" xref="A4.SS1.SSS5.p1.2.m2.1.1.3.cmml"><mi id="A4.SS1.SSS5.p1.2.m2.1.1.3.2" xref="A4.SS1.SSS5.p1.2.m2.1.1.3.2.cmml">η</mi><mi id="A4.SS1.SSS5.p1.2.m2.1.1.3.3" xref="A4.SS1.SSS5.p1.2.m2.1.1.3.3.cmml">k</mi></msub><mo id="A4.SS1.SSS5.p1.2.m2.1.1.2" xref="A4.SS1.SSS5.p1.2.m2.1.1.2.cmml">=</mo><mrow id="A4.SS1.SSS5.p1.2.m2.1.1.1" xref="A4.SS1.SSS5.p1.2.m2.1.1.1.cmml"><mi id="A4.SS1.SSS5.p1.2.m2.1.1.1.3" xref="A4.SS1.SSS5.p1.2.m2.1.1.1.3.cmml">𝔼</mi><mo id="A4.SS1.SSS5.p1.2.m2.1.1.1.2" xref="A4.SS1.SSS5.p1.2.m2.1.1.1.2.cmml">⁢</mo><mrow id="A4.SS1.SSS5.p1.2.m2.1.1.1.1.1" xref="A4.SS1.SSS5.p1.2.m2.1.1.1.1.1.1.cmml"><mo id="A4.SS1.SSS5.p1.2.m2.1.1.1.1.1.2" stretchy="false" xref="A4.SS1.SSS5.p1.2.m2.1.1.1.1.1.1.cmml">(</mo><msub id="A4.SS1.SSS5.p1.2.m2.1.1.1.1.1.1" xref="A4.SS1.SSS5.p1.2.m2.1.1.1.1.1.1.cmml"><mi id="A4.SS1.SSS5.p1.2.m2.1.1.1.1.1.1.2" xref="A4.SS1.SSS5.p1.2.m2.1.1.1.1.1.1.2.cmml">ξ</mi><mi id="A4.SS1.SSS5.p1.2.m2.1.1.1.1.1.1.3" xref="A4.SS1.SSS5.p1.2.m2.1.1.1.1.1.1.3.cmml">k</mi></msub><mo id="A4.SS1.SSS5.p1.2.m2.1.1.1.1.1.3" stretchy="false" xref="A4.SS1.SSS5.p1.2.m2.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="A4.SS1.SSS5.p1.2.m2.1b"><apply id="A4.SS1.SSS5.p1.2.m2.1.1.cmml" xref="A4.SS1.SSS5.p1.2.m2.1.1"><eq id="A4.SS1.SSS5.p1.2.m2.1.1.2.cmml" xref="A4.SS1.SSS5.p1.2.m2.1.1.2"></eq><apply id="A4.SS1.SSS5.p1.2.m2.1.1.3.cmml" xref="A4.SS1.SSS5.p1.2.m2.1.1.3"><csymbol cd="ambiguous" id="A4.SS1.SSS5.p1.2.m2.1.1.3.1.cmml" xref="A4.SS1.SSS5.p1.2.m2.1.1.3">subscript</csymbol><ci id="A4.SS1.SSS5.p1.2.m2.1.1.3.2.cmml" xref="A4.SS1.SSS5.p1.2.m2.1.1.3.2">𝜂</ci><ci id="A4.SS1.SSS5.p1.2.m2.1.1.3.3.cmml" xref="A4.SS1.SSS5.p1.2.m2.1.1.3.3">𝑘</ci></apply><apply id="A4.SS1.SSS5.p1.2.m2.1.1.1.cmml" xref="A4.SS1.SSS5.p1.2.m2.1.1.1"><times id="A4.SS1.SSS5.p1.2.m2.1.1.1.2.cmml" xref="A4.SS1.SSS5.p1.2.m2.1.1.1.2"></times><ci id="A4.SS1.SSS5.p1.2.m2.1.1.1.3.cmml" xref="A4.SS1.SSS5.p1.2.m2.1.1.1.3">𝔼</ci><apply id="A4.SS1.SSS5.p1.2.m2.1.1.1.1.1.1.cmml" xref="A4.SS1.SSS5.p1.2.m2.1.1.1.1.1"><csymbol cd="ambiguous" id="A4.SS1.SSS5.p1.2.m2.1.1.1.1.1.1.1.cmml" xref="A4.SS1.SSS5.p1.2.m2.1.1.1.1.1">subscript</csymbol><ci id="A4.SS1.SSS5.p1.2.m2.1.1.1.1.1.1.2.cmml" xref="A4.SS1.SSS5.p1.2.m2.1.1.1.1.1.1.2">𝜉</ci><ci id="A4.SS1.SSS5.p1.2.m2.1.1.1.1.1.1.3.cmml" xref="A4.SS1.SSS5.p1.2.m2.1.1.1.1.1.1.3">𝑘</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.SS1.SSS5.p1.2.m2.1c">\eta_{k}=\mathbb{E}(\xi_{k})</annotation><annotation encoding="application/x-llamapun" id="A4.SS1.SSS5.p1.2.m2.1d">italic_η start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT = blackboard_E ( italic_ξ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT )</annotation></semantics></math> and <math alttext="\gamma_{k}=\xi_{k}-\mathbb{E}(\xi_{k})" class="ltx_Math" display="inline" id="A4.SS1.SSS5.p1.3.m3.1"><semantics id="A4.SS1.SSS5.p1.3.m3.1a"><mrow id="A4.SS1.SSS5.p1.3.m3.1.1" xref="A4.SS1.SSS5.p1.3.m3.1.1.cmml"><msub id="A4.SS1.SSS5.p1.3.m3.1.1.3" xref="A4.SS1.SSS5.p1.3.m3.1.1.3.cmml"><mi id="A4.SS1.SSS5.p1.3.m3.1.1.3.2" xref="A4.SS1.SSS5.p1.3.m3.1.1.3.2.cmml">γ</mi><mi id="A4.SS1.SSS5.p1.3.m3.1.1.3.3" xref="A4.SS1.SSS5.p1.3.m3.1.1.3.3.cmml">k</mi></msub><mo id="A4.SS1.SSS5.p1.3.m3.1.1.2" xref="A4.SS1.SSS5.p1.3.m3.1.1.2.cmml">=</mo><mrow id="A4.SS1.SSS5.p1.3.m3.1.1.1" xref="A4.SS1.SSS5.p1.3.m3.1.1.1.cmml"><msub id="A4.SS1.SSS5.p1.3.m3.1.1.1.3" xref="A4.SS1.SSS5.p1.3.m3.1.1.1.3.cmml"><mi id="A4.SS1.SSS5.p1.3.m3.1.1.1.3.2" xref="A4.SS1.SSS5.p1.3.m3.1.1.1.3.2.cmml">ξ</mi><mi id="A4.SS1.SSS5.p1.3.m3.1.1.1.3.3" xref="A4.SS1.SSS5.p1.3.m3.1.1.1.3.3.cmml">k</mi></msub><mo id="A4.SS1.SSS5.p1.3.m3.1.1.1.2" xref="A4.SS1.SSS5.p1.3.m3.1.1.1.2.cmml">−</mo><mrow id="A4.SS1.SSS5.p1.3.m3.1.1.1.1" xref="A4.SS1.SSS5.p1.3.m3.1.1.1.1.cmml"><mi id="A4.SS1.SSS5.p1.3.m3.1.1.1.1.3" xref="A4.SS1.SSS5.p1.3.m3.1.1.1.1.3.cmml">𝔼</mi><mo id="A4.SS1.SSS5.p1.3.m3.1.1.1.1.2" xref="A4.SS1.SSS5.p1.3.m3.1.1.1.1.2.cmml">⁢</mo><mrow id="A4.SS1.SSS5.p1.3.m3.1.1.1.1.1.1" xref="A4.SS1.SSS5.p1.3.m3.1.1.1.1.1.1.1.cmml"><mo id="A4.SS1.SSS5.p1.3.m3.1.1.1.1.1.1.2" stretchy="false" xref="A4.SS1.SSS5.p1.3.m3.1.1.1.1.1.1.1.cmml">(</mo><msub id="A4.SS1.SSS5.p1.3.m3.1.1.1.1.1.1.1" xref="A4.SS1.SSS5.p1.3.m3.1.1.1.1.1.1.1.cmml"><mi id="A4.SS1.SSS5.p1.3.m3.1.1.1.1.1.1.1.2" xref="A4.SS1.SSS5.p1.3.m3.1.1.1.1.1.1.1.2.cmml">ξ</mi><mi id="A4.SS1.SSS5.p1.3.m3.1.1.1.1.1.1.1.3" xref="A4.SS1.SSS5.p1.3.m3.1.1.1.1.1.1.1.3.cmml">k</mi></msub><mo id="A4.SS1.SSS5.p1.3.m3.1.1.1.1.1.1.3" stretchy="false" xref="A4.SS1.SSS5.p1.3.m3.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="A4.SS1.SSS5.p1.3.m3.1b"><apply id="A4.SS1.SSS5.p1.3.m3.1.1.cmml" xref="A4.SS1.SSS5.p1.3.m3.1.1"><eq id="A4.SS1.SSS5.p1.3.m3.1.1.2.cmml" xref="A4.SS1.SSS5.p1.3.m3.1.1.2"></eq><apply id="A4.SS1.SSS5.p1.3.m3.1.1.3.cmml" xref="A4.SS1.SSS5.p1.3.m3.1.1.3"><csymbol cd="ambiguous" id="A4.SS1.SSS5.p1.3.m3.1.1.3.1.cmml" xref="A4.SS1.SSS5.p1.3.m3.1.1.3">subscript</csymbol><ci id="A4.SS1.SSS5.p1.3.m3.1.1.3.2.cmml" xref="A4.SS1.SSS5.p1.3.m3.1.1.3.2">𝛾</ci><ci id="A4.SS1.SSS5.p1.3.m3.1.1.3.3.cmml" xref="A4.SS1.SSS5.p1.3.m3.1.1.3.3">𝑘</ci></apply><apply id="A4.SS1.SSS5.p1.3.m3.1.1.1.cmml" xref="A4.SS1.SSS5.p1.3.m3.1.1.1"><minus id="A4.SS1.SSS5.p1.3.m3.1.1.1.2.cmml" xref="A4.SS1.SSS5.p1.3.m3.1.1.1.2"></minus><apply id="A4.SS1.SSS5.p1.3.m3.1.1.1.3.cmml" xref="A4.SS1.SSS5.p1.3.m3.1.1.1.3"><csymbol cd="ambiguous" id="A4.SS1.SSS5.p1.3.m3.1.1.1.3.1.cmml" xref="A4.SS1.SSS5.p1.3.m3.1.1.1.3">subscript</csymbol><ci id="A4.SS1.SSS5.p1.3.m3.1.1.1.3.2.cmml" xref="A4.SS1.SSS5.p1.3.m3.1.1.1.3.2">𝜉</ci><ci id="A4.SS1.SSS5.p1.3.m3.1.1.1.3.3.cmml" xref="A4.SS1.SSS5.p1.3.m3.1.1.1.3.3">𝑘</ci></apply><apply id="A4.SS1.SSS5.p1.3.m3.1.1.1.1.cmml" xref="A4.SS1.SSS5.p1.3.m3.1.1.1.1"><times id="A4.SS1.SSS5.p1.3.m3.1.1.1.1.2.cmml" xref="A4.SS1.SSS5.p1.3.m3.1.1.1.1.2"></times><ci id="A4.SS1.SSS5.p1.3.m3.1.1.1.1.3.cmml" xref="A4.SS1.SSS5.p1.3.m3.1.1.1.1.3">𝔼</ci><apply id="A4.SS1.SSS5.p1.3.m3.1.1.1.1.1.1.1.cmml" xref="A4.SS1.SSS5.p1.3.m3.1.1.1.1.1.1"><csymbol cd="ambiguous" id="A4.SS1.SSS5.p1.3.m3.1.1.1.1.1.1.1.1.cmml" xref="A4.SS1.SSS5.p1.3.m3.1.1.1.1.1.1">subscript</csymbol><ci id="A4.SS1.SSS5.p1.3.m3.1.1.1.1.1.1.1.2.cmml" xref="A4.SS1.SSS5.p1.3.m3.1.1.1.1.1.1.1.2">𝜉</ci><ci id="A4.SS1.SSS5.p1.3.m3.1.1.1.1.1.1.1.3.cmml" xref="A4.SS1.SSS5.p1.3.m3.1.1.1.1.1.1.1.3">𝑘</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.SS1.SSS5.p1.3.m3.1c">\gamma_{k}=\xi_{k}-\mathbb{E}(\xi_{k})</annotation><annotation encoding="application/x-llamapun" id="A4.SS1.SSS5.p1.3.m3.1d">italic_γ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT = italic_ξ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT - blackboard_E ( italic_ξ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT )</annotation></semantics></math>. So we have <math alttext="\xi_{k}=\gamma_{k}+\eta_{k}" class="ltx_Math" display="inline" id="A4.SS1.SSS5.p1.4.m4.1"><semantics id="A4.SS1.SSS5.p1.4.m4.1a"><mrow id="A4.SS1.SSS5.p1.4.m4.1.1" xref="A4.SS1.SSS5.p1.4.m4.1.1.cmml"><msub id="A4.SS1.SSS5.p1.4.m4.1.1.2" xref="A4.SS1.SSS5.p1.4.m4.1.1.2.cmml"><mi id="A4.SS1.SSS5.p1.4.m4.1.1.2.2" xref="A4.SS1.SSS5.p1.4.m4.1.1.2.2.cmml">ξ</mi><mi id="A4.SS1.SSS5.p1.4.m4.1.1.2.3" xref="A4.SS1.SSS5.p1.4.m4.1.1.2.3.cmml">k</mi></msub><mo id="A4.SS1.SSS5.p1.4.m4.1.1.1" xref="A4.SS1.SSS5.p1.4.m4.1.1.1.cmml">=</mo><mrow id="A4.SS1.SSS5.p1.4.m4.1.1.3" xref="A4.SS1.SSS5.p1.4.m4.1.1.3.cmml"><msub id="A4.SS1.SSS5.p1.4.m4.1.1.3.2" xref="A4.SS1.SSS5.p1.4.m4.1.1.3.2.cmml"><mi id="A4.SS1.SSS5.p1.4.m4.1.1.3.2.2" xref="A4.SS1.SSS5.p1.4.m4.1.1.3.2.2.cmml">γ</mi><mi id="A4.SS1.SSS5.p1.4.m4.1.1.3.2.3" xref="A4.SS1.SSS5.p1.4.m4.1.1.3.2.3.cmml">k</mi></msub><mo id="A4.SS1.SSS5.p1.4.m4.1.1.3.1" xref="A4.SS1.SSS5.p1.4.m4.1.1.3.1.cmml">+</mo><msub id="A4.SS1.SSS5.p1.4.m4.1.1.3.3" xref="A4.SS1.SSS5.p1.4.m4.1.1.3.3.cmml"><mi id="A4.SS1.SSS5.p1.4.m4.1.1.3.3.2" xref="A4.SS1.SSS5.p1.4.m4.1.1.3.3.2.cmml">η</mi><mi id="A4.SS1.SSS5.p1.4.m4.1.1.3.3.3" xref="A4.SS1.SSS5.p1.4.m4.1.1.3.3.3.cmml">k</mi></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="A4.SS1.SSS5.p1.4.m4.1b"><apply id="A4.SS1.SSS5.p1.4.m4.1.1.cmml" xref="A4.SS1.SSS5.p1.4.m4.1.1"><eq id="A4.SS1.SSS5.p1.4.m4.1.1.1.cmml" xref="A4.SS1.SSS5.p1.4.m4.1.1.1"></eq><apply id="A4.SS1.SSS5.p1.4.m4.1.1.2.cmml" xref="A4.SS1.SSS5.p1.4.m4.1.1.2"><csymbol cd="ambiguous" id="A4.SS1.SSS5.p1.4.m4.1.1.2.1.cmml" xref="A4.SS1.SSS5.p1.4.m4.1.1.2">subscript</csymbol><ci id="A4.SS1.SSS5.p1.4.m4.1.1.2.2.cmml" xref="A4.SS1.SSS5.p1.4.m4.1.1.2.2">𝜉</ci><ci id="A4.SS1.SSS5.p1.4.m4.1.1.2.3.cmml" xref="A4.SS1.SSS5.p1.4.m4.1.1.2.3">𝑘</ci></apply><apply id="A4.SS1.SSS5.p1.4.m4.1.1.3.cmml" xref="A4.SS1.SSS5.p1.4.m4.1.1.3"><plus id="A4.SS1.SSS5.p1.4.m4.1.1.3.1.cmml" xref="A4.SS1.SSS5.p1.4.m4.1.1.3.1"></plus><apply id="A4.SS1.SSS5.p1.4.m4.1.1.3.2.cmml" xref="A4.SS1.SSS5.p1.4.m4.1.1.3.2"><csymbol cd="ambiguous" id="A4.SS1.SSS5.p1.4.m4.1.1.3.2.1.cmml" xref="A4.SS1.SSS5.p1.4.m4.1.1.3.2">subscript</csymbol><ci id="A4.SS1.SSS5.p1.4.m4.1.1.3.2.2.cmml" xref="A4.SS1.SSS5.p1.4.m4.1.1.3.2.2">𝛾</ci><ci id="A4.SS1.SSS5.p1.4.m4.1.1.3.2.3.cmml" xref="A4.SS1.SSS5.p1.4.m4.1.1.3.2.3">𝑘</ci></apply><apply id="A4.SS1.SSS5.p1.4.m4.1.1.3.3.cmml" xref="A4.SS1.SSS5.p1.4.m4.1.1.3.3"><csymbol cd="ambiguous" id="A4.SS1.SSS5.p1.4.m4.1.1.3.3.1.cmml" xref="A4.SS1.SSS5.p1.4.m4.1.1.3.3">subscript</csymbol><ci id="A4.SS1.SSS5.p1.4.m4.1.1.3.3.2.cmml" xref="A4.SS1.SSS5.p1.4.m4.1.1.3.3.2">𝜂</ci><ci id="A4.SS1.SSS5.p1.4.m4.1.1.3.3.3.cmml" xref="A4.SS1.SSS5.p1.4.m4.1.1.3.3.3">𝑘</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.SS1.SSS5.p1.4.m4.1c">\xi_{k}=\gamma_{k}+\eta_{k}</annotation><annotation encoding="application/x-llamapun" id="A4.SS1.SSS5.p1.4.m4.1d">italic_ξ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT = italic_γ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT + italic_η start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="\mathbb{E}(\gamma_{k})=0" class="ltx_Math" display="inline" id="A4.SS1.SSS5.p1.5.m5.1"><semantics id="A4.SS1.SSS5.p1.5.m5.1a"><mrow id="A4.SS1.SSS5.p1.5.m5.1.1" xref="A4.SS1.SSS5.p1.5.m5.1.1.cmml"><mrow id="A4.SS1.SSS5.p1.5.m5.1.1.1" xref="A4.SS1.SSS5.p1.5.m5.1.1.1.cmml"><mi id="A4.SS1.SSS5.p1.5.m5.1.1.1.3" xref="A4.SS1.SSS5.p1.5.m5.1.1.1.3.cmml">𝔼</mi><mo id="A4.SS1.SSS5.p1.5.m5.1.1.1.2" xref="A4.SS1.SSS5.p1.5.m5.1.1.1.2.cmml">⁢</mo><mrow id="A4.SS1.SSS5.p1.5.m5.1.1.1.1.1" xref="A4.SS1.SSS5.p1.5.m5.1.1.1.1.1.1.cmml"><mo id="A4.SS1.SSS5.p1.5.m5.1.1.1.1.1.2" stretchy="false" xref="A4.SS1.SSS5.p1.5.m5.1.1.1.1.1.1.cmml">(</mo><msub id="A4.SS1.SSS5.p1.5.m5.1.1.1.1.1.1" xref="A4.SS1.SSS5.p1.5.m5.1.1.1.1.1.1.cmml"><mi id="A4.SS1.SSS5.p1.5.m5.1.1.1.1.1.1.2" xref="A4.SS1.SSS5.p1.5.m5.1.1.1.1.1.1.2.cmml">γ</mi><mi id="A4.SS1.SSS5.p1.5.m5.1.1.1.1.1.1.3" xref="A4.SS1.SSS5.p1.5.m5.1.1.1.1.1.1.3.cmml">k</mi></msub><mo id="A4.SS1.SSS5.p1.5.m5.1.1.1.1.1.3" stretchy="false" xref="A4.SS1.SSS5.p1.5.m5.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="A4.SS1.SSS5.p1.5.m5.1.1.2" xref="A4.SS1.SSS5.p1.5.m5.1.1.2.cmml">=</mo><mn id="A4.SS1.SSS5.p1.5.m5.1.1.3" xref="A4.SS1.SSS5.p1.5.m5.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="A4.SS1.SSS5.p1.5.m5.1b"><apply id="A4.SS1.SSS5.p1.5.m5.1.1.cmml" xref="A4.SS1.SSS5.p1.5.m5.1.1"><eq id="A4.SS1.SSS5.p1.5.m5.1.1.2.cmml" xref="A4.SS1.SSS5.p1.5.m5.1.1.2"></eq><apply id="A4.SS1.SSS5.p1.5.m5.1.1.1.cmml" xref="A4.SS1.SSS5.p1.5.m5.1.1.1"><times id="A4.SS1.SSS5.p1.5.m5.1.1.1.2.cmml" xref="A4.SS1.SSS5.p1.5.m5.1.1.1.2"></times><ci id="A4.SS1.SSS5.p1.5.m5.1.1.1.3.cmml" xref="A4.SS1.SSS5.p1.5.m5.1.1.1.3">𝔼</ci><apply id="A4.SS1.SSS5.p1.5.m5.1.1.1.1.1.1.cmml" xref="A4.SS1.SSS5.p1.5.m5.1.1.1.1.1"><csymbol cd="ambiguous" id="A4.SS1.SSS5.p1.5.m5.1.1.1.1.1.1.1.cmml" xref="A4.SS1.SSS5.p1.5.m5.1.1.1.1.1">subscript</csymbol><ci id="A4.SS1.SSS5.p1.5.m5.1.1.1.1.1.1.2.cmml" xref="A4.SS1.SSS5.p1.5.m5.1.1.1.1.1.1.2">𝛾</ci><ci id="A4.SS1.SSS5.p1.5.m5.1.1.1.1.1.1.3.cmml" xref="A4.SS1.SSS5.p1.5.m5.1.1.1.1.1.1.3">𝑘</ci></apply></apply><cn id="A4.SS1.SSS5.p1.5.m5.1.1.3.cmml" type="integer" xref="A4.SS1.SSS5.p1.5.m5.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.SS1.SSS5.p1.5.m5.1c">\mathbb{E}(\gamma_{k})=0</annotation><annotation encoding="application/x-llamapun" id="A4.SS1.SSS5.p1.5.m5.1d">blackboard_E ( italic_γ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) = 0</annotation></semantics></math>. We will apply <cite class="ltx_cite ltx_citemacro_citep">(Tadic & Doucet, <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#bib.bib72" title="">2017</a>, Theorem 2.1. (ii))</cite>. Assumption 2.1. and 2.3.b. of this paper are already verified because of Assumption <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#S3.Thmtheorem3" title="Assumption 3.3. ‣ 3.2 Unbiased algorithm analysis ‣ 3 Convergence Analysis ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_tag">3.3</span></a>, Assumption <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#S3.Thmtheorem4" title="Assumption 3.4. ‣ 3.2 Unbiased algorithm analysis ‣ 3 Convergence Analysis ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_tag">3.4</span></a>, and Assumption <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#S3.Thmtheorem6" title="Assumption 3.6. ‣ 3.3 Biased algorithm analysis ‣ 3 Convergence Analysis ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_tag">3.6</span></a>. Then, using the definition of <math alttext="\xi_{k}" class="ltx_Math" display="inline" id="A4.SS1.SSS5.p1.6.m6.1"><semantics id="A4.SS1.SSS5.p1.6.m6.1a"><msub id="A4.SS1.SSS5.p1.6.m6.1.1" xref="A4.SS1.SSS5.p1.6.m6.1.1.cmml"><mi id="A4.SS1.SSS5.p1.6.m6.1.1.2" xref="A4.SS1.SSS5.p1.6.m6.1.1.2.cmml">ξ</mi><mi id="A4.SS1.SSS5.p1.6.m6.1.1.3" xref="A4.SS1.SSS5.p1.6.m6.1.1.3.cmml">k</mi></msub><annotation-xml encoding="MathML-Content" id="A4.SS1.SSS5.p1.6.m6.1b"><apply id="A4.SS1.SSS5.p1.6.m6.1.1.cmml" xref="A4.SS1.SSS5.p1.6.m6.1.1"><csymbol cd="ambiguous" id="A4.SS1.SSS5.p1.6.m6.1.1.1.cmml" xref="A4.SS1.SSS5.p1.6.m6.1.1">subscript</csymbol><ci id="A4.SS1.SSS5.p1.6.m6.1.1.2.cmml" xref="A4.SS1.SSS5.p1.6.m6.1.1.2">𝜉</ci><ci id="A4.SS1.SSS5.p1.6.m6.1.1.3.cmml" xref="A4.SS1.SSS5.p1.6.m6.1.1.3">𝑘</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.SS1.SSS5.p1.6.m6.1c">\xi_{k}</annotation><annotation encoding="application/x-llamapun" id="A4.SS1.SSS5.p1.6.m6.1d">italic_ξ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT</annotation></semantics></math> and Assumption <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#S3.Thmtheorem7" title="Assumption 3.7. ‣ 3.3 Biased algorithm analysis ‣ 3 Convergence Analysis ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_tag">3.7</span></a>, we have</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A8.EGx54"> <tbody id="A4.Ex98"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\mathbb{E}(\|\gamma_{k}\|^{2}|{\mathbf{x}}_{k})" class="ltx_Math" display="inline" id="A4.Ex98.m1.1"><semantics id="A4.Ex98.m1.1a"><mrow id="A4.Ex98.m1.1.1" xref="A4.Ex98.m1.1.1.cmml"><mi id="A4.Ex98.m1.1.1.3" xref="A4.Ex98.m1.1.1.3.cmml">𝔼</mi><mo id="A4.Ex98.m1.1.1.2" xref="A4.Ex98.m1.1.1.2.cmml">⁢</mo><mrow id="A4.Ex98.m1.1.1.1.1" xref="A4.Ex98.m1.1.1.1.1.1.cmml"><mo id="A4.Ex98.m1.1.1.1.1.2" stretchy="false" xref="A4.Ex98.m1.1.1.1.1.1.cmml">(</mo><mrow id="A4.Ex98.m1.1.1.1.1.1" xref="A4.Ex98.m1.1.1.1.1.1.cmml"><msup id="A4.Ex98.m1.1.1.1.1.1.1" xref="A4.Ex98.m1.1.1.1.1.1.1.cmml"><mrow id="A4.Ex98.m1.1.1.1.1.1.1.1.1" xref="A4.Ex98.m1.1.1.1.1.1.1.1.2.cmml"><mo id="A4.Ex98.m1.1.1.1.1.1.1.1.1.2" stretchy="false" xref="A4.Ex98.m1.1.1.1.1.1.1.1.2.1.cmml">‖</mo><msub id="A4.Ex98.m1.1.1.1.1.1.1.1.1.1" xref="A4.Ex98.m1.1.1.1.1.1.1.1.1.1.cmml"><mi id="A4.Ex98.m1.1.1.1.1.1.1.1.1.1.2" xref="A4.Ex98.m1.1.1.1.1.1.1.1.1.1.2.cmml">γ</mi><mi id="A4.Ex98.m1.1.1.1.1.1.1.1.1.1.3" xref="A4.Ex98.m1.1.1.1.1.1.1.1.1.1.3.cmml">k</mi></msub><mo id="A4.Ex98.m1.1.1.1.1.1.1.1.1.3" stretchy="false" xref="A4.Ex98.m1.1.1.1.1.1.1.1.2.1.cmml">‖</mo></mrow><mn id="A4.Ex98.m1.1.1.1.1.1.1.3" xref="A4.Ex98.m1.1.1.1.1.1.1.3.cmml">2</mn></msup><mo fence="false" id="A4.Ex98.m1.1.1.1.1.1.2" xref="A4.Ex98.m1.1.1.1.1.1.2.cmml">|</mo><msub id="A4.Ex98.m1.1.1.1.1.1.3" xref="A4.Ex98.m1.1.1.1.1.1.3.cmml"><mi id="A4.Ex98.m1.1.1.1.1.1.3.2" xref="A4.Ex98.m1.1.1.1.1.1.3.2.cmml">𝐱</mi><mi id="A4.Ex98.m1.1.1.1.1.1.3.3" xref="A4.Ex98.m1.1.1.1.1.1.3.3.cmml">k</mi></msub></mrow><mo id="A4.Ex98.m1.1.1.1.1.3" stretchy="false" xref="A4.Ex98.m1.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A4.Ex98.m1.1b"><apply id="A4.Ex98.m1.1.1.cmml" xref="A4.Ex98.m1.1.1"><times id="A4.Ex98.m1.1.1.2.cmml" xref="A4.Ex98.m1.1.1.2"></times><ci id="A4.Ex98.m1.1.1.3.cmml" xref="A4.Ex98.m1.1.1.3">𝔼</ci><apply id="A4.Ex98.m1.1.1.1.1.1.cmml" xref="A4.Ex98.m1.1.1.1.1"><csymbol cd="latexml" id="A4.Ex98.m1.1.1.1.1.1.2.cmml" xref="A4.Ex98.m1.1.1.1.1.1.2">conditional</csymbol><apply id="A4.Ex98.m1.1.1.1.1.1.1.cmml" xref="A4.Ex98.m1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="A4.Ex98.m1.1.1.1.1.1.1.2.cmml" xref="A4.Ex98.m1.1.1.1.1.1.1">superscript</csymbol><apply id="A4.Ex98.m1.1.1.1.1.1.1.1.2.cmml" xref="A4.Ex98.m1.1.1.1.1.1.1.1.1"><csymbol cd="latexml" id="A4.Ex98.m1.1.1.1.1.1.1.1.2.1.cmml" xref="A4.Ex98.m1.1.1.1.1.1.1.1.1.2">norm</csymbol><apply id="A4.Ex98.m1.1.1.1.1.1.1.1.1.1.cmml" xref="A4.Ex98.m1.1.1.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="A4.Ex98.m1.1.1.1.1.1.1.1.1.1.1.cmml" xref="A4.Ex98.m1.1.1.1.1.1.1.1.1.1">subscript</csymbol><ci id="A4.Ex98.m1.1.1.1.1.1.1.1.1.1.2.cmml" xref="A4.Ex98.m1.1.1.1.1.1.1.1.1.1.2">𝛾</ci><ci id="A4.Ex98.m1.1.1.1.1.1.1.1.1.1.3.cmml" xref="A4.Ex98.m1.1.1.1.1.1.1.1.1.1.3">𝑘</ci></apply></apply><cn id="A4.Ex98.m1.1.1.1.1.1.1.3.cmml" type="integer" xref="A4.Ex98.m1.1.1.1.1.1.1.3">2</cn></apply><apply id="A4.Ex98.m1.1.1.1.1.1.3.cmml" xref="A4.Ex98.m1.1.1.1.1.1.3"><csymbol cd="ambiguous" id="A4.Ex98.m1.1.1.1.1.1.3.1.cmml" xref="A4.Ex98.m1.1.1.1.1.1.3">subscript</csymbol><ci id="A4.Ex98.m1.1.1.1.1.1.3.2.cmml" xref="A4.Ex98.m1.1.1.1.1.1.3.2">𝐱</ci><ci id="A4.Ex98.m1.1.1.1.1.1.3.3.cmml" xref="A4.Ex98.m1.1.1.1.1.1.3.3">𝑘</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.Ex98.m1.1c">\displaystyle\mathbb{E}(\|\gamma_{k}\|^{2}|{\mathbf{x}}_{k})</annotation><annotation encoding="application/x-llamapun" id="A4.Ex98.m1.1d">blackboard_E ( ∥ italic_γ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT | bold_x start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT )</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle=\mathbb{E}(\|\xi_{k}-\mathbb{E}(\xi_{k})\|^{2}|{\mathbf{x}}_{k})" class="ltx_Math" display="inline" id="A4.Ex98.m2.1"><semantics id="A4.Ex98.m2.1a"><mrow id="A4.Ex98.m2.1.1" xref="A4.Ex98.m2.1.1.cmml"><mi id="A4.Ex98.m2.1.1.3" xref="A4.Ex98.m2.1.1.3.cmml"></mi><mo id="A4.Ex98.m2.1.1.2" xref="A4.Ex98.m2.1.1.2.cmml">=</mo><mrow id="A4.Ex98.m2.1.1.1" xref="A4.Ex98.m2.1.1.1.cmml"><mi id="A4.Ex98.m2.1.1.1.3" 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xref="A4.Ex98.m2.1.1.1.1.1.1.1.1.1.1.1.1.1.1.2">𝜉</ci><ci id="A4.Ex98.m2.1.1.1.1.1.1.1.1.1.1.1.1.1.1.3.cmml" xref="A4.Ex98.m2.1.1.1.1.1.1.1.1.1.1.1.1.1.1.3">𝑘</ci></apply></apply></apply></apply><cn id="A4.Ex98.m2.1.1.1.1.1.1.1.3.cmml" type="integer" xref="A4.Ex98.m2.1.1.1.1.1.1.1.3">2</cn></apply><apply id="A4.Ex98.m2.1.1.1.1.1.1.3.cmml" xref="A4.Ex98.m2.1.1.1.1.1.1.3"><csymbol cd="ambiguous" id="A4.Ex98.m2.1.1.1.1.1.1.3.1.cmml" xref="A4.Ex98.m2.1.1.1.1.1.1.3">subscript</csymbol><ci id="A4.Ex98.m2.1.1.1.1.1.1.3.2.cmml" xref="A4.Ex98.m2.1.1.1.1.1.1.3.2">𝐱</ci><ci id="A4.Ex98.m2.1.1.1.1.1.1.3.3.cmml" xref="A4.Ex98.m2.1.1.1.1.1.1.3.3">𝑘</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.Ex98.m2.1c">\displaystyle=\mathbb{E}(\|\xi_{k}-\mathbb{E}(\xi_{k})\|^{2}|{\mathbf{x}}_{k})</annotation><annotation encoding="application/x-llamapun" id="A4.Ex98.m2.1d">= blackboard_E ( ∥ italic_ξ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT - 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\mathbf{x}}_{k}\right)</annotation><annotation encoding="application/x-llamapun" id="A4.Ex99.m1.1d">= divide start_ARG italic_α start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG italic_σ start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT end_ARG blackboard_E ( ∥ italic_D start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ( bold_x start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT + italic_ζ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) - bold_x start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT - italic_ζ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT - blackboard_E start_POSTSUBSCRIPT italic_ζ end_POSTSUBSCRIPT ( italic_D start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ( bold_x start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT + italic_ζ ) - bold_x start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT - italic_ζ ) ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT | bold_x start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="A4.Ex100"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_eqn_cell"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle=\frac{\alpha^{2}}{\sigma^{4}}\mathbb{E}\left(\|D_{\sigma}({% \mathbf{x}}_{k}+\zeta)-\zeta-\mathbb{E}_{\zeta}\left(D_{\sigma}({\mathbf{x}}_{% k}+\zeta)-\zeta\right)\|^{2}|{\mathbf{x}}_{k}\right)" class="ltx_Math" display="inline" id="A4.Ex100.m1.1"><semantics id="A4.Ex100.m1.1a"><mrow id="A4.Ex100.m1.1.1" xref="A4.Ex100.m1.1.1.cmml"><mi id="A4.Ex100.m1.1.1.3" xref="A4.Ex100.m1.1.1.3.cmml"></mi><mo id="A4.Ex100.m1.1.1.2" xref="A4.Ex100.m1.1.1.2.cmml">=</mo><mrow id="A4.Ex100.m1.1.1.1" xref="A4.Ex100.m1.1.1.1.cmml"><mstyle displaystyle="true" id="A4.Ex100.m1.1.1.1.3" xref="A4.Ex100.m1.1.1.1.3.cmml"><mfrac id="A4.Ex100.m1.1.1.1.3a" xref="A4.Ex100.m1.1.1.1.3.cmml"><msup id="A4.Ex100.m1.1.1.1.3.2" 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id="A4.Ex101.m1.1.3.7.7.3">k</mi></msub><mo id="A4.Ex101.m1.1.3.7.8">)</mo></mrow><msup id="A4.Ex101.m1.1.3.8"><mo id="A4.Ex101.m1.1.3.8.2" lspace="0em" rspace="0.167em">∥</mo><mn id="A4.Ex101.m1.1.3.8.3">2</mn></msup><mo id="A4.Ex101.m1.1.3.9">)</mo></mrow></mrow><annotation encoding="application/x-tex" id="A4.Ex101.m1.1c">\displaystyle=\frac{\alpha^{2}}{\sigma^{4}}\left(\mathbb{E}\left(\|D_{\sigma}(% {\mathbf{x}}_{k}+\zeta)-\zeta\|^{2}|{\mathbf{x}}_{k}\right)-\|\mathbb{E}_{% \zeta}\left(D_{\sigma}({\mathbf{x}}_{k}+\zeta)-\zeta|{\mathbf{x}}_{k}\right)\|% ^{2}\right)</annotation><annotation encoding="application/x-llamapun" id="A4.Ex101.m1.1d">= divide start_ARG italic_α start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG italic_σ start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT end_ARG ( blackboard_E ( ∥ italic_D start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ( bold_x start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT + italic_ζ ) - italic_ζ ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT | bold_x start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) - ∥ blackboard_E start_POSTSUBSCRIPT italic_ζ end_POSTSUBSCRIPT ( italic_D start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ( bold_x start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT + italic_ζ ) - italic_ζ | bold_x start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="A4.Ex102"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_eqn_cell"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\leq\frac{\alpha^{2}}{\sigma^{4}}\mathbb{E}\left(\|D_{\sigma}({% \mathbf{x}}_{k}+\zeta)-\zeta\|^{2}|{\mathbf{x}}_{k}\right)" class="ltx_Math" display="inline" id="A4.Ex102.m1.1"><semantics id="A4.Ex102.m1.1a"><mrow id="A4.Ex102.m1.1.1" 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id="A4.Ex102.m1.1.1.1.1.1.1.3.1.cmml" xref="A4.Ex102.m1.1.1.1.1.1.1.3">subscript</csymbol><ci id="A4.Ex102.m1.1.1.1.1.1.1.3.2.cmml" xref="A4.Ex102.m1.1.1.1.1.1.1.3.2">𝐱</ci><ci id="A4.Ex102.m1.1.1.1.1.1.1.3.3.cmml" xref="A4.Ex102.m1.1.1.1.1.1.1.3.3">𝑘</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.Ex102.m1.1c">\displaystyle\leq\frac{\alpha^{2}}{\sigma^{4}}\mathbb{E}\left(\|D_{\sigma}({% \mathbf{x}}_{k}+\zeta)-\zeta\|^{2}|{\mathbf{x}}_{k}\right)</annotation><annotation encoding="application/x-llamapun" id="A4.Ex102.m1.1d">≤ divide start_ARG italic_α start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG italic_σ start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT end_ARG blackboard_E ( ∥ italic_D start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ( bold_x start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT + italic_ζ ) - italic_ζ ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT | bold_x start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="A4.Ex103"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_eqn_cell"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\leq\frac{2\alpha^{2}}{\sigma^{4}}\left(\mathbb{E}\left(\|D_{% \sigma}({\mathbf{x}}_{k}+\zeta)\|^{2}|{\mathbf{x}}_{k}\right)+\mathbb{E}\left(% \|\zeta\|^{2}|{\mathbf{x}}_{k}\right)\right)" class="ltx_Math" display="inline" id="A4.Ex103.m1.2"><semantics id="A4.Ex103.m1.2a"><mrow id="A4.Ex103.m1.2.2" xref="A4.Ex103.m1.2.2.cmml"><mi id="A4.Ex103.m1.2.2.3" xref="A4.Ex103.m1.2.2.3.cmml"></mi><mo id="A4.Ex103.m1.2.2.2" xref="A4.Ex103.m1.2.2.2.cmml">≤</mo><mrow id="A4.Ex103.m1.2.2.1" xref="A4.Ex103.m1.2.2.1.cmml"><mstyle displaystyle="true" id="A4.Ex103.m1.2.2.1.3" xref="A4.Ex103.m1.2.2.1.3.cmml"><mfrac id="A4.Ex103.m1.2.2.1.3a" 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id="A4.Ex103.m1.2d">≤ divide start_ARG 2 italic_α start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG italic_σ start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT end_ARG ( blackboard_E ( ∥ italic_D start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ( bold_x start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT + italic_ζ ) ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT | bold_x start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) + blackboard_E ( ∥ italic_ζ ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT | bold_x start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="A4.Ex104"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_eqn_cell"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\leq\frac{2\alpha^{2}}{\sigma^{4}}\left(\mathbb{E}\left(\left(\|{% 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id="A4.Ex104.m1.1.1.1.1.1.1.3.3.1.cmml" xref="A4.Ex104.m1.1.1.1.1.1.1.3.3">superscript</csymbol><ci id="A4.Ex104.m1.1.1.1.1.1.1.3.3.2.cmml" xref="A4.Ex104.m1.1.1.1.1.1.1.3.3.2">𝜎</ci><cn id="A4.Ex104.m1.1.1.1.1.1.1.3.3.3.cmml" type="integer" xref="A4.Ex104.m1.1.1.1.1.1.1.3.3.3">2</cn></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.Ex104.m1.1c">\displaystyle\leq\frac{2\alpha^{2}}{\sigma^{4}}\left(\mathbb{E}\left(\left(\|{% \mathbf{x}}_{k}+\zeta\|+C\sigma\right)^{2}|{\mathbf{x}}_{k}\right)+d\sigma^{2}\right)</annotation><annotation encoding="application/x-llamapun" id="A4.Ex104.m1.1d">≤ divide start_ARG 2 italic_α start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG italic_σ start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT end_ARG ( blackboard_E ( ( ∥ bold_x start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT + italic_ζ ∥ + italic_C italic_σ ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT | bold_x start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) + italic_d italic_σ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="A4.Ex105"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_eqn_cell"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\leq\frac{2\alpha^{2}}{\sigma^{4}}\left(2\mathbb{E}\left(\left(\|% {\mathbf{x}}_{k}\|+\|\zeta\|\right)^{2}|{\mathbf{x}}_{k}\right)+2C^{2}\sigma^{% 2}+d\sigma^{2}\right)" class="ltx_Math" display="inline" id="A4.Ex105.m1.2"><semantics id="A4.Ex105.m1.2a"><mrow id="A4.Ex105.m1.2.2" xref="A4.Ex105.m1.2.2.cmml"><mi id="A4.Ex105.m1.2.2.3" xref="A4.Ex105.m1.2.2.3.cmml"></mi><mo id="A4.Ex105.m1.2.2.2" xref="A4.Ex105.m1.2.2.2.cmml">≤</mo><mrow id="A4.Ex105.m1.2.2.1" xref="A4.Ex105.m1.2.2.1.cmml"><mstyle displaystyle="true" id="A4.Ex105.m1.2.2.1.3" 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id="A4.Ex105.m1.2.2.1.1.1.1.4.3.2.cmml" xref="A4.Ex105.m1.2.2.1.1.1.1.4.3.2">𝜎</ci><cn id="A4.Ex105.m1.2.2.1.1.1.1.4.3.3.cmml" type="integer" xref="A4.Ex105.m1.2.2.1.1.1.1.4.3.3">2</cn></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.Ex105.m1.2c">\displaystyle\leq\frac{2\alpha^{2}}{\sigma^{4}}\left(2\mathbb{E}\left(\left(\|% {\mathbf{x}}_{k}\|+\|\zeta\|\right)^{2}|{\mathbf{x}}_{k}\right)+2C^{2}\sigma^{% 2}+d\sigma^{2}\right)</annotation><annotation encoding="application/x-llamapun" id="A4.Ex105.m1.2d">≤ divide start_ARG 2 italic_α start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG italic_σ start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT end_ARG ( 2 blackboard_E ( ( ∥ bold_x start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ∥ + ∥ italic_ζ ∥ ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT | bold_x start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) + 2 italic_C start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_σ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + italic_d italic_σ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="A4.Ex106"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_eqn_cell"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\leq\frac{2\alpha^{2}}{\sigma^{4}}\left(4R_{\mathbf{K}}^{2}+4d% \sigma^{2}+2C^{2}\sigma^{2}+d\sigma^{2}\right)<+\infty." class="ltx_Math" display="inline" id="A4.Ex106.m1.1"><semantics id="A4.Ex106.m1.1a"><mrow id="A4.Ex106.m1.1.1.1" xref="A4.Ex106.m1.1.1.1.1.cmml"><mrow id="A4.Ex106.m1.1.1.1.1" xref="A4.Ex106.m1.1.1.1.1.cmml"><mi id="A4.Ex106.m1.1.1.1.1.3" xref="A4.Ex106.m1.1.1.1.1.3.cmml"></mi><mo id="A4.Ex106.m1.1.1.1.1.4" xref="A4.Ex106.m1.1.1.1.1.4.cmml">≤</mo><mrow id="A4.Ex106.m1.1.1.1.1.1" 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href="#A4.Ex106.m1.1.1.1.1.1.cmml" id="A4.Ex106.m1.1.1.1.1d.cmml" xref="A4.Ex106.m1.1.1.1"></share><apply id="A4.Ex106.m1.1.1.1.1.6.cmml" xref="A4.Ex106.m1.1.1.1.1.6"><plus id="A4.Ex106.m1.1.1.1.1.6.1.cmml" xref="A4.Ex106.m1.1.1.1.1.6"></plus><infinity id="A4.Ex106.m1.1.1.1.1.6.2.cmml" xref="A4.Ex106.m1.1.1.1.1.6.2"></infinity></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.Ex106.m1.1c">\displaystyle\leq\frac{2\alpha^{2}}{\sigma^{4}}\left(4R_{\mathbf{K}}^{2}+4d% \sigma^{2}+2C^{2}\sigma^{2}+d\sigma^{2}\right)<+\infty.</annotation><annotation encoding="application/x-llamapun" id="A4.Ex106.m1.1d">≤ divide start_ARG 2 italic_α start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG italic_σ start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT end_ARG ( 4 italic_R start_POSTSUBSCRIPT bold_K end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + 4 italic_d italic_σ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + 2 italic_C start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_σ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + italic_d italic_σ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) < + ∞ .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="A4.SS1.SSS5.p1.7">By using the Doob inequality as in Appendix <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#A4.SS1.SSS3" title="D.1.3 Proof of Proposition 3.3 ‣ D.1 Proof of critical Point Analysis ‣ Appendix D Proofs of section 3 ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_tag">D.1.3</span></a>, we have demonstrated the first part of Assumption 2.2. of <cite class="ltx_cite ltx_citemacro_citep">(Tadic & Doucet, <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#bib.bib72" title="">2017</a>)</cite>.</p> </div> <div class="ltx_para" id="A4.SS1.SSS5.p2"> <p class="ltx_p" id="A4.SS1.SSS5.p2.2">Now we study the asymptotic behavior of <math alttext="\eta_{k}" class="ltx_Math" display="inline" id="A4.SS1.SSS5.p2.1.m1.1"><semantics id="A4.SS1.SSS5.p2.1.m1.1a"><msub id="A4.SS1.SSS5.p2.1.m1.1.1" xref="A4.SS1.SSS5.p2.1.m1.1.1.cmml"><mi id="A4.SS1.SSS5.p2.1.m1.1.1.2" xref="A4.SS1.SSS5.p2.1.m1.1.1.2.cmml">η</mi><mi id="A4.SS1.SSS5.p2.1.m1.1.1.3" xref="A4.SS1.SSS5.p2.1.m1.1.1.3.cmml">k</mi></msub><annotation-xml encoding="MathML-Content" id="A4.SS1.SSS5.p2.1.m1.1b"><apply id="A4.SS1.SSS5.p2.1.m1.1.1.cmml" xref="A4.SS1.SSS5.p2.1.m1.1.1"><csymbol cd="ambiguous" id="A4.SS1.SSS5.p2.1.m1.1.1.1.cmml" xref="A4.SS1.SSS5.p2.1.m1.1.1">subscript</csymbol><ci id="A4.SS1.SSS5.p2.1.m1.1.1.2.cmml" xref="A4.SS1.SSS5.p2.1.m1.1.1.2">𝜂</ci><ci id="A4.SS1.SSS5.p2.1.m1.1.1.3.cmml" xref="A4.SS1.SSS5.p2.1.m1.1.1.3">𝑘</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.SS1.SSS5.p2.1.m1.1c">\eta_{k}</annotation><annotation encoding="application/x-llamapun" id="A4.SS1.SSS5.p2.1.m1.1d">italic_η start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT</annotation></semantics></math>, for <math alttext="R>0" class="ltx_Math" display="inline" id="A4.SS1.SSS5.p2.2.m2.1"><semantics id="A4.SS1.SSS5.p2.2.m2.1a"><mrow id="A4.SS1.SSS5.p2.2.m2.1.1" xref="A4.SS1.SSS5.p2.2.m2.1.1.cmml"><mi id="A4.SS1.SSS5.p2.2.m2.1.1.2" xref="A4.SS1.SSS5.p2.2.m2.1.1.2.cmml">R</mi><mo id="A4.SS1.SSS5.p2.2.m2.1.1.1" xref="A4.SS1.SSS5.p2.2.m2.1.1.1.cmml">></mo><mn id="A4.SS1.SSS5.p2.2.m2.1.1.3" xref="A4.SS1.SSS5.p2.2.m2.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="A4.SS1.SSS5.p2.2.m2.1b"><apply id="A4.SS1.SSS5.p2.2.m2.1.1.cmml" xref="A4.SS1.SSS5.p2.2.m2.1.1"><gt id="A4.SS1.SSS5.p2.2.m2.1.1.1.cmml" xref="A4.SS1.SSS5.p2.2.m2.1.1.1"></gt><ci id="A4.SS1.SSS5.p2.2.m2.1.1.2.cmml" xref="A4.SS1.SSS5.p2.2.m2.1.1.2">𝑅</ci><cn id="A4.SS1.SSS5.p2.2.m2.1.1.3.cmml" type="integer" xref="A4.SS1.SSS5.p2.2.m2.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.SS1.SSS5.p2.2.m2.1c">R>0</annotation><annotation encoding="application/x-llamapun" id="A4.SS1.SSS5.p2.2.m2.1d">italic_R > 0</annotation></semantics></math></p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A8.EGx55"> <tbody id="A4.Ex108"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\|\eta_{k}\|" class="ltx_Math" display="inline" id="A4.Ex108.m1.1"><semantics id="A4.Ex108.m1.1a"><mrow id="A4.Ex108.m1.1.1.1" xref="A4.Ex108.m1.1.1.2.cmml"><mo id="A4.Ex108.m1.1.1.1.2" stretchy="false" xref="A4.Ex108.m1.1.1.2.1.cmml">‖</mo><msub id="A4.Ex108.m1.1.1.1.1" xref="A4.Ex108.m1.1.1.1.1.cmml"><mi id="A4.Ex108.m1.1.1.1.1.2" xref="A4.Ex108.m1.1.1.1.1.2.cmml">η</mi><mi id="A4.Ex108.m1.1.1.1.1.3" xref="A4.Ex108.m1.1.1.1.1.3.cmml">k</mi></msub><mo id="A4.Ex108.m1.1.1.1.3" stretchy="false" xref="A4.Ex108.m1.1.1.2.1.cmml">‖</mo></mrow><annotation-xml encoding="MathML-Content" id="A4.Ex108.m1.1b"><apply id="A4.Ex108.m1.1.1.2.cmml" xref="A4.Ex108.m1.1.1.1"><csymbol cd="latexml" id="A4.Ex108.m1.1.1.2.1.cmml" xref="A4.Ex108.m1.1.1.1.2">norm</csymbol><apply id="A4.Ex108.m1.1.1.1.1.cmml" xref="A4.Ex108.m1.1.1.1.1"><csymbol cd="ambiguous" id="A4.Ex108.m1.1.1.1.1.1.cmml" xref="A4.Ex108.m1.1.1.1.1">subscript</csymbol><ci id="A4.Ex108.m1.1.1.1.1.2.cmml" xref="A4.Ex108.m1.1.1.1.1.2">𝜂</ci><ci id="A4.Ex108.m1.1.1.1.1.3.cmml" xref="A4.Ex108.m1.1.1.1.1.3">𝑘</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.Ex108.m1.1c">\displaystyle\|\eta_{k}\|</annotation><annotation encoding="application/x-llamapun" id="A4.Ex108.m1.1d">∥ italic_η start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ∥</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle=\|\mathbb{E}(\xi_{k})\|=\|\mathbb{E}(\frac{\alpha}{\sigma^{2}}% \left(D_{\sigma}({\mathbf{x}}_{k}+\zeta_{k})-{\mathbf{x}}_{k}-\zeta_{k}\right)% -\frac{\alpha}{\sigma^{2}}\mathbb{E}_{\zeta}\left(D^{\star}_{\sigma}({\mathbf{% x}}_{k}+\zeta)-{\mathbf{x}}_{k}-\zeta\right)\|" class="ltx_math_unparsed" display="inline" id="A4.Ex108.m2.1"><semantics id="A4.Ex108.m2.1a"><mrow id="A4.Ex108.m2.1b"><mo id="A4.Ex108.m2.1.1" rspace="0em">=</mo><mo id="A4.Ex108.m2.1.2" lspace="0em" rspace="0.167em">∥</mo><mi id="A4.Ex108.m2.1.3">𝔼</mi><mrow id="A4.Ex108.m2.1.4"><mo id="A4.Ex108.m2.1.4.1" stretchy="false">(</mo><msub id="A4.Ex108.m2.1.4.2"><mi id="A4.Ex108.m2.1.4.2.2">ξ</mi><mi id="A4.Ex108.m2.1.4.2.3">k</mi></msub><mo id="A4.Ex108.m2.1.4.3" stretchy="false">)</mo></mrow><mo id="A4.Ex108.m2.1.5" lspace="0em" rspace="0.0835em">∥</mo><mo id="A4.Ex108.m2.1.6" lspace="0.0835em" rspace="0em">=</mo><mo id="A4.Ex108.m2.1.7" lspace="0em" rspace="0.167em">∥</mo><mi id="A4.Ex108.m2.1.8">𝔼</mi><mrow id="A4.Ex108.m2.1.9"><mo id="A4.Ex108.m2.1.9.1" stretchy="false">(</mo><mstyle displaystyle="true" id="A4.Ex108.m2.1.9.2"><mfrac id="A4.Ex108.m2.1.9.2a"><mi id="A4.Ex108.m2.1.9.2.2">α</mi><msup id="A4.Ex108.m2.1.9.2.3"><mi id="A4.Ex108.m2.1.9.2.3.2">σ</mi><mn id="A4.Ex108.m2.1.9.2.3.3">2</mn></msup></mfrac></mstyle><mrow id="A4.Ex108.m2.1.9.3"><mo id="A4.Ex108.m2.1.9.3.1">(</mo><msub id="A4.Ex108.m2.1.9.3.2"><mi id="A4.Ex108.m2.1.9.3.2.2">D</mi><mi id="A4.Ex108.m2.1.9.3.2.3">σ</mi></msub><mrow id="A4.Ex108.m2.1.9.3.3"><mo id="A4.Ex108.m2.1.9.3.3.1" stretchy="false">(</mo><msub id="A4.Ex108.m2.1.9.3.3.2"><mi id="A4.Ex108.m2.1.9.3.3.2.2">𝐱</mi><mi id="A4.Ex108.m2.1.9.3.3.2.3">k</mi></msub><mo id="A4.Ex108.m2.1.9.3.3.3">+</mo><msub id="A4.Ex108.m2.1.9.3.3.4"><mi id="A4.Ex108.m2.1.9.3.3.4.2">ζ</mi><mi id="A4.Ex108.m2.1.9.3.3.4.3">k</mi></msub><mo id="A4.Ex108.m2.1.9.3.3.5" stretchy="false">)</mo></mrow><mo id="A4.Ex108.m2.1.9.3.4">−</mo><msub id="A4.Ex108.m2.1.9.3.5"><mi id="A4.Ex108.m2.1.9.3.5.2">𝐱</mi><mi id="A4.Ex108.m2.1.9.3.5.3">k</mi></msub><mo id="A4.Ex108.m2.1.9.3.6">−</mo><msub id="A4.Ex108.m2.1.9.3.7"><mi id="A4.Ex108.m2.1.9.3.7.2">ζ</mi><mi id="A4.Ex108.m2.1.9.3.7.3">k</mi></msub><mo id="A4.Ex108.m2.1.9.3.8">)</mo></mrow><mo id="A4.Ex108.m2.1.9.4">−</mo><mstyle displaystyle="true" id="A4.Ex108.m2.1.9.5"><mfrac id="A4.Ex108.m2.1.9.5a"><mi id="A4.Ex108.m2.1.9.5.2">α</mi><msup id="A4.Ex108.m2.1.9.5.3"><mi id="A4.Ex108.m2.1.9.5.3.2">σ</mi><mn id="A4.Ex108.m2.1.9.5.3.3">2</mn></msup></mfrac></mstyle><msub id="A4.Ex108.m2.1.9.6"><mi id="A4.Ex108.m2.1.9.6.2">𝔼</mi><mi id="A4.Ex108.m2.1.9.6.3">ζ</mi></msub><mrow id="A4.Ex108.m2.1.9.7"><mo id="A4.Ex108.m2.1.9.7.1">(</mo><msubsup id="A4.Ex108.m2.1.9.7.2"><mi id="A4.Ex108.m2.1.9.7.2.2.2">D</mi><mi id="A4.Ex108.m2.1.9.7.2.3">σ</mi><mo id="A4.Ex108.m2.1.9.7.2.2.3">⋆</mo></msubsup><mrow id="A4.Ex108.m2.1.9.7.3"><mo id="A4.Ex108.m2.1.9.7.3.1" stretchy="false">(</mo><msub id="A4.Ex108.m2.1.9.7.3.2"><mi id="A4.Ex108.m2.1.9.7.3.2.2">𝐱</mi><mi id="A4.Ex108.m2.1.9.7.3.2.3">k</mi></msub><mo id="A4.Ex108.m2.1.9.7.3.3">+</mo><mi id="A4.Ex108.m2.1.9.7.3.4">ζ</mi><mo id="A4.Ex108.m2.1.9.7.3.5" stretchy="false">)</mo></mrow><mo id="A4.Ex108.m2.1.9.7.4">−</mo><msub id="A4.Ex108.m2.1.9.7.5"><mi id="A4.Ex108.m2.1.9.7.5.2">𝐱</mi><mi id="A4.Ex108.m2.1.9.7.5.3">k</mi></msub><mo id="A4.Ex108.m2.1.9.7.6">−</mo><mi id="A4.Ex108.m2.1.9.7.7">ζ</mi><mo id="A4.Ex108.m2.1.9.7.8">)</mo></mrow><mo id="A4.Ex108.m2.1.9.8" lspace="0em">∥</mo></mrow></mrow><annotation encoding="application/x-tex" id="A4.Ex108.m2.1c">\displaystyle=\|\mathbb{E}(\xi_{k})\|=\|\mathbb{E}(\frac{\alpha}{\sigma^{2}}% \left(D_{\sigma}({\mathbf{x}}_{k}+\zeta_{k})-{\mathbf{x}}_{k}-\zeta_{k}\right)% -\frac{\alpha}{\sigma^{2}}\mathbb{E}_{\zeta}\left(D^{\star}_{\sigma}({\mathbf{% x}}_{k}+\zeta)-{\mathbf{x}}_{k}-\zeta\right)\|</annotation><annotation encoding="application/x-llamapun" id="A4.Ex108.m2.1d">= ∥ blackboard_E ( italic_ξ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) ∥ = ∥ blackboard_E ( divide start_ARG italic_α end_ARG start_ARG italic_σ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG ( italic_D start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ( bold_x start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT + italic_ζ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) - bold_x start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT - italic_ζ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) - divide start_ARG italic_α end_ARG start_ARG italic_σ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG blackboard_E start_POSTSUBSCRIPT italic_ζ end_POSTSUBSCRIPT ( italic_D start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ( bold_x start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT + italic_ζ ) - bold_x start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT - italic_ζ ) ∥</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="A4.Ex109"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_eqn_cell"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle=\|\frac{\alpha}{\sigma^{2}}\mathbb{E}\left(D_{\sigma}({\mathbf{x% }}_{k}+\zeta)-D^{\star}_{\sigma}({\mathbf{x}}_{k}+\zeta)\right)\|" class="ltx_Math" display="inline" id="A4.Ex109.m1.1"><semantics id="A4.Ex109.m1.1a"><mrow id="A4.Ex109.m1.1.1" xref="A4.Ex109.m1.1.1.cmml"><mi id="A4.Ex109.m1.1.1.3" xref="A4.Ex109.m1.1.1.3.cmml"></mi><mo id="A4.Ex109.m1.1.1.2" xref="A4.Ex109.m1.1.1.2.cmml">=</mo><mrow id="A4.Ex109.m1.1.1.1.1" xref="A4.Ex109.m1.1.1.1.2.cmml"><mo id="A4.Ex109.m1.1.1.1.1.2" stretchy="false" xref="A4.Ex109.m1.1.1.1.2.1.cmml">‖</mo><mrow id="A4.Ex109.m1.1.1.1.1.1" xref="A4.Ex109.m1.1.1.1.1.1.cmml"><mstyle displaystyle="true" id="A4.Ex109.m1.1.1.1.1.1.3" xref="A4.Ex109.m1.1.1.1.1.1.3.cmml"><mfrac id="A4.Ex109.m1.1.1.1.1.1.3a" xref="A4.Ex109.m1.1.1.1.1.1.3.cmml"><mi id="A4.Ex109.m1.1.1.1.1.1.3.2" xref="A4.Ex109.m1.1.1.1.1.1.3.2.cmml">α</mi><msup id="A4.Ex109.m1.1.1.1.1.1.3.3" xref="A4.Ex109.m1.1.1.1.1.1.3.3.cmml"><mi id="A4.Ex109.m1.1.1.1.1.1.3.3.2" xref="A4.Ex109.m1.1.1.1.1.1.3.3.2.cmml">σ</mi><mn id="A4.Ex109.m1.1.1.1.1.1.3.3.3" xref="A4.Ex109.m1.1.1.1.1.1.3.3.3.cmml">2</mn></msup></mfrac></mstyle><mo id="A4.Ex109.m1.1.1.1.1.1.2" xref="A4.Ex109.m1.1.1.1.1.1.2.cmml">⁢</mo><mi id="A4.Ex109.m1.1.1.1.1.1.4" xref="A4.Ex109.m1.1.1.1.1.1.4.cmml">𝔼</mi><mo id="A4.Ex109.m1.1.1.1.1.1.2a" xref="A4.Ex109.m1.1.1.1.1.1.2.cmml">⁢</mo><mrow id="A4.Ex109.m1.1.1.1.1.1.1.1" xref="A4.Ex109.m1.1.1.1.1.1.1.1.1.cmml"><mo id="A4.Ex109.m1.1.1.1.1.1.1.1.2" xref="A4.Ex109.m1.1.1.1.1.1.1.1.1.cmml">(</mo><mrow id="A4.Ex109.m1.1.1.1.1.1.1.1.1" xref="A4.Ex109.m1.1.1.1.1.1.1.1.1.cmml"><mrow id="A4.Ex109.m1.1.1.1.1.1.1.1.1.1" xref="A4.Ex109.m1.1.1.1.1.1.1.1.1.1.cmml"><msub id="A4.Ex109.m1.1.1.1.1.1.1.1.1.1.3" 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id="A4.Ex109.m1.1c">\displaystyle=\|\frac{\alpha}{\sigma^{2}}\mathbb{E}\left(D_{\sigma}({\mathbf{x% }}_{k}+\zeta)-D^{\star}_{\sigma}({\mathbf{x}}_{k}+\zeta)\right)\|</annotation><annotation encoding="application/x-llamapun" id="A4.Ex109.m1.1d">= ∥ divide start_ARG italic_α end_ARG start_ARG italic_σ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG blackboard_E ( italic_D start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ( bold_x start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT + italic_ζ ) - italic_D start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ( bold_x start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT + italic_ζ ) ) ∥</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="A4.Ex110"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_eqn_cell"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle=\frac{\alpha}{\sigma^{2}}\|\int_{\zeta\in{\mathbb{R}}^{d}}(D_{% \sigma}({\mathbf{x}}_{k}+\zeta)-D^{\star}_{\sigma}({\mathbf{x}}_{k}+\zeta))% \mathcal{N}(\zeta;0,\sigma^{2}\mathbf{I}_{d})d\zeta\|" class="ltx_Math" display="inline" id="A4.Ex110.m1.3"><semantics id="A4.Ex110.m1.3a"><mrow id="A4.Ex110.m1.3.3" xref="A4.Ex110.m1.3.3.cmml"><mi id="A4.Ex110.m1.3.3.3" xref="A4.Ex110.m1.3.3.3.cmml"></mi><mo id="A4.Ex110.m1.3.3.2" xref="A4.Ex110.m1.3.3.2.cmml">=</mo><mrow id="A4.Ex110.m1.3.3.1" xref="A4.Ex110.m1.3.3.1.cmml"><mstyle displaystyle="true" id="A4.Ex110.m1.3.3.1.3" xref="A4.Ex110.m1.3.3.1.3.cmml"><mfrac id="A4.Ex110.m1.3.3.1.3a" xref="A4.Ex110.m1.3.3.1.3.cmml"><mi id="A4.Ex110.m1.3.3.1.3.2" xref="A4.Ex110.m1.3.3.1.3.2.cmml">α</mi><msup id="A4.Ex110.m1.3.3.1.3.3" xref="A4.Ex110.m1.3.3.1.3.3.cmml"><mi id="A4.Ex110.m1.3.3.1.3.3.2" xref="A4.Ex110.m1.3.3.1.3.3.2.cmml">σ</mi><mn id="A4.Ex110.m1.3.3.1.3.3.3" 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xref="A4.Ex110.m1.3.3.1.1.1.1.2.5.2">𝜁</ci></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.Ex110.m1.3c">\displaystyle=\frac{\alpha}{\sigma^{2}}\|\int_{\zeta\in{\mathbb{R}}^{d}}(D_{% \sigma}({\mathbf{x}}_{k}+\zeta)-D^{\star}_{\sigma}({\mathbf{x}}_{k}+\zeta))% \mathcal{N}(\zeta;0,\sigma^{2}\mathbf{I}_{d})d\zeta\|</annotation><annotation encoding="application/x-llamapun" id="A4.Ex110.m1.3d">= divide start_ARG italic_α end_ARG start_ARG italic_σ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG ∥ ∫ start_POSTSUBSCRIPT italic_ζ ∈ blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ( italic_D start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ( bold_x start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT + italic_ζ ) - italic_D start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ( bold_x start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT + italic_ζ ) ) caligraphic_N ( italic_ζ ; 0 , italic_σ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT bold_I start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT ) italic_d italic_ζ ∥</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="A4.Ex111"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_eqn_cell"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\leq\frac{\alpha}{\sigma^{2}}\left(\int_{\|{\mathbf{x}}_{k}+\zeta% \|\leq R}\|D_{\sigma}({\mathbf{x}}_{k}+\zeta)-D^{\star}_{\sigma}({\mathbf{x}}_% {k}+\zeta)\|\mathcal{N}(\zeta;0,\sigma^{2}\mathbf{I}_{d})d\zeta+\int_{\|{% \mathbf{x}}_{k}+\zeta\|>R}\|D_{\sigma}({\mathbf{x}}_{k}+\zeta)-D^{\star}_{% \sigma}({\mathbf{x}}_{k}+\zeta)\|\mathcal{N}(\zeta;0,\sigma^{2}\mathbf{I}_{d})% d\zeta\right)" class="ltx_Math" display="inline" id="A4.Ex111.m1.7"><semantics id="A4.Ex111.m1.7a"><mrow id="A4.Ex111.m1.7.7" 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xref="A4.Ex111.m1.1.1.1.1.1.1.3.cmml">ζ</mi></mrow><mo id="A4.Ex111.m1.1.1.1.1.1.3" stretchy="false" xref="A4.Ex111.m1.1.1.1.1.2.1.cmml">‖</mo></mrow><mo id="A4.Ex111.m1.1.1.1.2" xref="A4.Ex111.m1.1.1.1.2.cmml">≤</mo><mi id="A4.Ex111.m1.1.1.1.3" xref="A4.Ex111.m1.1.1.1.3.cmml">R</mi></mrow></msub></mstyle><mrow id="A4.Ex111.m1.7.7.1.1.1.1.2.2" xref="A4.Ex111.m1.7.7.1.1.1.1.2.2.cmml"><mrow id="A4.Ex111.m1.7.7.1.1.1.1.1.1.1.1" xref="A4.Ex111.m1.7.7.1.1.1.1.1.1.1.2.cmml"><mo id="A4.Ex111.m1.7.7.1.1.1.1.1.1.1.1.2" stretchy="false" xref="A4.Ex111.m1.7.7.1.1.1.1.1.1.1.2.1.cmml">‖</mo><mrow id="A4.Ex111.m1.7.7.1.1.1.1.1.1.1.1.1" xref="A4.Ex111.m1.7.7.1.1.1.1.1.1.1.1.1.cmml"><mrow id="A4.Ex111.m1.7.7.1.1.1.1.1.1.1.1.1.1" xref="A4.Ex111.m1.7.7.1.1.1.1.1.1.1.1.1.1.cmml"><msub id="A4.Ex111.m1.7.7.1.1.1.1.1.1.1.1.1.1.3" xref="A4.Ex111.m1.7.7.1.1.1.1.1.1.1.1.1.1.3.cmml"><mi id="A4.Ex111.m1.7.7.1.1.1.1.1.1.1.1.1.1.3.2" xref="A4.Ex111.m1.7.7.1.1.1.1.1.1.1.1.1.1.3.2.cmml">D</mi><mi id="A4.Ex111.m1.7.7.1.1.1.1.1.1.1.1.1.1.3.3" xref="A4.Ex111.m1.7.7.1.1.1.1.1.1.1.1.1.1.3.3.cmml">σ</mi></msub><mo id="A4.Ex111.m1.7.7.1.1.1.1.1.1.1.1.1.1.2" xref="A4.Ex111.m1.7.7.1.1.1.1.1.1.1.1.1.1.2.cmml">⁢</mo><mrow id="A4.Ex111.m1.7.7.1.1.1.1.1.1.1.1.1.1.1.1" xref="A4.Ex111.m1.7.7.1.1.1.1.1.1.1.1.1.1.1.1.1.cmml"><mo id="A4.Ex111.m1.7.7.1.1.1.1.1.1.1.1.1.1.1.1.2" stretchy="false" xref="A4.Ex111.m1.7.7.1.1.1.1.1.1.1.1.1.1.1.1.1.cmml">(</mo><mrow id="A4.Ex111.m1.7.7.1.1.1.1.1.1.1.1.1.1.1.1.1" xref="A4.Ex111.m1.7.7.1.1.1.1.1.1.1.1.1.1.1.1.1.cmml"><msub id="A4.Ex111.m1.7.7.1.1.1.1.1.1.1.1.1.1.1.1.1.2" xref="A4.Ex111.m1.7.7.1.1.1.1.1.1.1.1.1.1.1.1.1.2.cmml"><mi id="A4.Ex111.m1.7.7.1.1.1.1.1.1.1.1.1.1.1.1.1.2.2" xref="A4.Ex111.m1.7.7.1.1.1.1.1.1.1.1.1.1.1.1.1.2.2.cmml">𝐱</mi><mi id="A4.Ex111.m1.7.7.1.1.1.1.1.1.1.1.1.1.1.1.1.2.3" xref="A4.Ex111.m1.7.7.1.1.1.1.1.1.1.1.1.1.1.1.1.2.3.cmml">k</mi></msub><mo id="A4.Ex111.m1.7.7.1.1.1.1.1.1.1.1.1.1.1.1.1.1" xref="A4.Ex111.m1.7.7.1.1.1.1.1.1.1.1.1.1.1.1.1.1.cmml">+</mo><mi id="A4.Ex111.m1.7.7.1.1.1.1.1.1.1.1.1.1.1.1.1.3" xref="A4.Ex111.m1.7.7.1.1.1.1.1.1.1.1.1.1.1.1.1.3.cmml">ζ</mi></mrow><mo id="A4.Ex111.m1.7.7.1.1.1.1.1.1.1.1.1.1.1.1.3" stretchy="false" xref="A4.Ex111.m1.7.7.1.1.1.1.1.1.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="A4.Ex111.m1.7.7.1.1.1.1.1.1.1.1.1.3" xref="A4.Ex111.m1.7.7.1.1.1.1.1.1.1.1.1.3.cmml">−</mo><mrow id="A4.Ex111.m1.7.7.1.1.1.1.1.1.1.1.1.2" xref="A4.Ex111.m1.7.7.1.1.1.1.1.1.1.1.1.2.cmml"><msubsup id="A4.Ex111.m1.7.7.1.1.1.1.1.1.1.1.1.2.3" xref="A4.Ex111.m1.7.7.1.1.1.1.1.1.1.1.1.2.3.cmml"><mi id="A4.Ex111.m1.7.7.1.1.1.1.1.1.1.1.1.2.3.2.2" xref="A4.Ex111.m1.7.7.1.1.1.1.1.1.1.1.1.2.3.2.2.cmml">D</mi><mi id="A4.Ex111.m1.7.7.1.1.1.1.1.1.1.1.1.2.3.3" xref="A4.Ex111.m1.7.7.1.1.1.1.1.1.1.1.1.2.3.3.cmml">σ</mi><mo id="A4.Ex111.m1.7.7.1.1.1.1.1.1.1.1.1.2.3.2.3" xref="A4.Ex111.m1.7.7.1.1.1.1.1.1.1.1.1.2.3.2.3.cmml">⋆</mo></msubsup><mo id="A4.Ex111.m1.7.7.1.1.1.1.1.1.1.1.1.2.2" xref="A4.Ex111.m1.7.7.1.1.1.1.1.1.1.1.1.2.2.cmml">⁢</mo><mrow id="A4.Ex111.m1.7.7.1.1.1.1.1.1.1.1.1.2.1.1" xref="A4.Ex111.m1.7.7.1.1.1.1.1.1.1.1.1.2.1.1.1.cmml"><mo id="A4.Ex111.m1.7.7.1.1.1.1.1.1.1.1.1.2.1.1.2" stretchy="false" xref="A4.Ex111.m1.7.7.1.1.1.1.1.1.1.1.1.2.1.1.1.cmml">(</mo><mrow id="A4.Ex111.m1.7.7.1.1.1.1.1.1.1.1.1.2.1.1.1" xref="A4.Ex111.m1.7.7.1.1.1.1.1.1.1.1.1.2.1.1.1.cmml"><msub id="A4.Ex111.m1.7.7.1.1.1.1.1.1.1.1.1.2.1.1.1.2" xref="A4.Ex111.m1.7.7.1.1.1.1.1.1.1.1.1.2.1.1.1.2.cmml"><mi id="A4.Ex111.m1.7.7.1.1.1.1.1.1.1.1.1.2.1.1.1.2.2" xref="A4.Ex111.m1.7.7.1.1.1.1.1.1.1.1.1.2.1.1.1.2.2.cmml">𝐱</mi><mi id="A4.Ex111.m1.7.7.1.1.1.1.1.1.1.1.1.2.1.1.1.2.3" xref="A4.Ex111.m1.7.7.1.1.1.1.1.1.1.1.1.2.1.1.1.2.3.cmml">k</mi></msub><mo id="A4.Ex111.m1.7.7.1.1.1.1.1.1.1.1.1.2.1.1.1.1" xref="A4.Ex111.m1.7.7.1.1.1.1.1.1.1.1.1.2.1.1.1.1.cmml">+</mo><mi id="A4.Ex111.m1.7.7.1.1.1.1.1.1.1.1.1.2.1.1.1.3" xref="A4.Ex111.m1.7.7.1.1.1.1.1.1.1.1.1.2.1.1.1.3.cmml">ζ</mi></mrow><mo id="A4.Ex111.m1.7.7.1.1.1.1.1.1.1.1.1.2.1.1.3" stretchy="false" xref="A4.Ex111.m1.7.7.1.1.1.1.1.1.1.1.1.2.1.1.1.cmml">)</mo></mrow></mrow></mrow><mo id="A4.Ex111.m1.7.7.1.1.1.1.1.1.1.1.3" stretchy="false" xref="A4.Ex111.m1.7.7.1.1.1.1.1.1.1.2.1.cmml">‖</mo></mrow><mo id="A4.Ex111.m1.7.7.1.1.1.1.2.2.3" xref="A4.Ex111.m1.7.7.1.1.1.1.2.2.3.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="A4.Ex111.m1.7.7.1.1.1.1.2.2.4" xref="A4.Ex111.m1.7.7.1.1.1.1.2.2.4.cmml">𝒩</mi><mo id="A4.Ex111.m1.7.7.1.1.1.1.2.2.3a" xref="A4.Ex111.m1.7.7.1.1.1.1.2.2.3.cmml">⁢</mo><mrow id="A4.Ex111.m1.7.7.1.1.1.1.2.2.2.1" xref="A4.Ex111.m1.7.7.1.1.1.1.2.2.2.2.cmml"><mo id="A4.Ex111.m1.7.7.1.1.1.1.2.2.2.1.2" stretchy="false" xref="A4.Ex111.m1.7.7.1.1.1.1.2.2.2.2.cmml">(</mo><mi id="A4.Ex111.m1.3.3" xref="A4.Ex111.m1.3.3.cmml">ζ</mi><mo id="A4.Ex111.m1.7.7.1.1.1.1.2.2.2.1.3" xref="A4.Ex111.m1.7.7.1.1.1.1.2.2.2.2.cmml">;</mo><mn id="A4.Ex111.m1.4.4" xref="A4.Ex111.m1.4.4.cmml">0</mn><mo id="A4.Ex111.m1.7.7.1.1.1.1.2.2.2.1.4" xref="A4.Ex111.m1.7.7.1.1.1.1.2.2.2.2.cmml">,</mo><mrow id="A4.Ex111.m1.7.7.1.1.1.1.2.2.2.1.1" xref="A4.Ex111.m1.7.7.1.1.1.1.2.2.2.1.1.cmml"><msup id="A4.Ex111.m1.7.7.1.1.1.1.2.2.2.1.1.2" xref="A4.Ex111.m1.7.7.1.1.1.1.2.2.2.1.1.2.cmml"><mi id="A4.Ex111.m1.7.7.1.1.1.1.2.2.2.1.1.2.2" xref="A4.Ex111.m1.7.7.1.1.1.1.2.2.2.1.1.2.2.cmml">σ</mi><mn id="A4.Ex111.m1.7.7.1.1.1.1.2.2.2.1.1.2.3" xref="A4.Ex111.m1.7.7.1.1.1.1.2.2.2.1.1.2.3.cmml">2</mn></msup><mo id="A4.Ex111.m1.7.7.1.1.1.1.2.2.2.1.1.1" xref="A4.Ex111.m1.7.7.1.1.1.1.2.2.2.1.1.1.cmml">⁢</mo><msub id="A4.Ex111.m1.7.7.1.1.1.1.2.2.2.1.1.3" xref="A4.Ex111.m1.7.7.1.1.1.1.2.2.2.1.1.3.cmml"><mi id="A4.Ex111.m1.7.7.1.1.1.1.2.2.2.1.1.3.2" xref="A4.Ex111.m1.7.7.1.1.1.1.2.2.2.1.1.3.2.cmml">𝐈</mi><mi id="A4.Ex111.m1.7.7.1.1.1.1.2.2.2.1.1.3.3" xref="A4.Ex111.m1.7.7.1.1.1.1.2.2.2.1.1.3.3.cmml">d</mi></msub></mrow><mo id="A4.Ex111.m1.7.7.1.1.1.1.2.2.2.1.5" stretchy="false" xref="A4.Ex111.m1.7.7.1.1.1.1.2.2.2.2.cmml">)</mo></mrow><mo id="A4.Ex111.m1.7.7.1.1.1.1.2.2.3b" lspace="0em" xref="A4.Ex111.m1.7.7.1.1.1.1.2.2.3.cmml">⁢</mo><mrow id="A4.Ex111.m1.7.7.1.1.1.1.2.2.5" xref="A4.Ex111.m1.7.7.1.1.1.1.2.2.5.cmml"><mo id="A4.Ex111.m1.7.7.1.1.1.1.2.2.5.1" rspace="0em" xref="A4.Ex111.m1.7.7.1.1.1.1.2.2.5.1.cmml">𝑑</mo><mi id="A4.Ex111.m1.7.7.1.1.1.1.2.2.5.2" xref="A4.Ex111.m1.7.7.1.1.1.1.2.2.5.2.cmml">ζ</mi></mrow></mrow></mrow><mo id="A4.Ex111.m1.7.7.1.1.1.1.5" xref="A4.Ex111.m1.7.7.1.1.1.1.5.cmml">+</mo><mrow id="A4.Ex111.m1.7.7.1.1.1.1.4" xref="A4.Ex111.m1.7.7.1.1.1.1.4.cmml"><mstyle displaystyle="true" id="A4.Ex111.m1.7.7.1.1.1.1.4.3" xref="A4.Ex111.m1.7.7.1.1.1.1.4.3.cmml"><msub id="A4.Ex111.m1.7.7.1.1.1.1.4.3a" xref="A4.Ex111.m1.7.7.1.1.1.1.4.3.cmml"><mo id="A4.Ex111.m1.7.7.1.1.1.1.4.3.2" xref="A4.Ex111.m1.7.7.1.1.1.1.4.3.2.cmml">∫</mo><mrow id="A4.Ex111.m1.2.2.1" xref="A4.Ex111.m1.2.2.1.cmml"><mrow id="A4.Ex111.m1.2.2.1.1.1" xref="A4.Ex111.m1.2.2.1.1.2.cmml"><mo id="A4.Ex111.m1.2.2.1.1.1.2" stretchy="false" xref="A4.Ex111.m1.2.2.1.1.2.1.cmml">‖</mo><mrow id="A4.Ex111.m1.2.2.1.1.1.1" xref="A4.Ex111.m1.2.2.1.1.1.1.cmml"><msub id="A4.Ex111.m1.2.2.1.1.1.1.2" xref="A4.Ex111.m1.2.2.1.1.1.1.2.cmml"><mi id="A4.Ex111.m1.2.2.1.1.1.1.2.2" xref="A4.Ex111.m1.2.2.1.1.1.1.2.2.cmml">𝐱</mi><mi id="A4.Ex111.m1.2.2.1.1.1.1.2.3" xref="A4.Ex111.m1.2.2.1.1.1.1.2.3.cmml">k</mi></msub><mo id="A4.Ex111.m1.2.2.1.1.1.1.1" xref="A4.Ex111.m1.2.2.1.1.1.1.1.cmml">+</mo><mi id="A4.Ex111.m1.2.2.1.1.1.1.3" xref="A4.Ex111.m1.2.2.1.1.1.1.3.cmml">ζ</mi></mrow><mo id="A4.Ex111.m1.2.2.1.1.1.3" stretchy="false" xref="A4.Ex111.m1.2.2.1.1.2.1.cmml">‖</mo></mrow><mo id="A4.Ex111.m1.2.2.1.2" xref="A4.Ex111.m1.2.2.1.2.cmml">></mo><mi id="A4.Ex111.m1.2.2.1.3" xref="A4.Ex111.m1.2.2.1.3.cmml">R</mi></mrow></msub></mstyle><mrow id="A4.Ex111.m1.7.7.1.1.1.1.4.2" xref="A4.Ex111.m1.7.7.1.1.1.1.4.2.cmml"><mrow id="A4.Ex111.m1.7.7.1.1.1.1.3.1.1.1" xref="A4.Ex111.m1.7.7.1.1.1.1.3.1.1.2.cmml"><mo id="A4.Ex111.m1.7.7.1.1.1.1.3.1.1.1.2" stretchy="false" xref="A4.Ex111.m1.7.7.1.1.1.1.3.1.1.2.1.cmml">‖</mo><mrow id="A4.Ex111.m1.7.7.1.1.1.1.3.1.1.1.1" xref="A4.Ex111.m1.7.7.1.1.1.1.3.1.1.1.1.cmml"><mrow id="A4.Ex111.m1.7.7.1.1.1.1.3.1.1.1.1.1" xref="A4.Ex111.m1.7.7.1.1.1.1.3.1.1.1.1.1.cmml"><msub id="A4.Ex111.m1.7.7.1.1.1.1.3.1.1.1.1.1.3" xref="A4.Ex111.m1.7.7.1.1.1.1.3.1.1.1.1.1.3.cmml"><mi id="A4.Ex111.m1.7.7.1.1.1.1.3.1.1.1.1.1.3.2" xref="A4.Ex111.m1.7.7.1.1.1.1.3.1.1.1.1.1.3.2.cmml">D</mi><mi id="A4.Ex111.m1.7.7.1.1.1.1.3.1.1.1.1.1.3.3" xref="A4.Ex111.m1.7.7.1.1.1.1.3.1.1.1.1.1.3.3.cmml">σ</mi></msub><mo id="A4.Ex111.m1.7.7.1.1.1.1.3.1.1.1.1.1.2" xref="A4.Ex111.m1.7.7.1.1.1.1.3.1.1.1.1.1.2.cmml">⁢</mo><mrow id="A4.Ex111.m1.7.7.1.1.1.1.3.1.1.1.1.1.1.1" xref="A4.Ex111.m1.7.7.1.1.1.1.3.1.1.1.1.1.1.1.1.cmml"><mo id="A4.Ex111.m1.7.7.1.1.1.1.3.1.1.1.1.1.1.1.2" stretchy="false" 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id="A4.Ex111.m1.7c">\displaystyle\leq\frac{\alpha}{\sigma^{2}}\left(\int_{\|{\mathbf{x}}_{k}+\zeta% \|\leq R}\|D_{\sigma}({\mathbf{x}}_{k}+\zeta)-D^{\star}_{\sigma}({\mathbf{x}}_% {k}+\zeta)\|\mathcal{N}(\zeta;0,\sigma^{2}\mathbf{I}_{d})d\zeta+\int_{\|{% \mathbf{x}}_{k}+\zeta\|>R}\|D_{\sigma}({\mathbf{x}}_{k}+\zeta)-D^{\star}_{% \sigma}({\mathbf{x}}_{k}+\zeta)\|\mathcal{N}(\zeta;0,\sigma^{2}\mathbf{I}_{d})% d\zeta\right)</annotation><annotation encoding="application/x-llamapun" id="A4.Ex111.m1.7d">≤ divide start_ARG italic_α end_ARG start_ARG italic_σ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG ( ∫ start_POSTSUBSCRIPT ∥ bold_x start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT + italic_ζ ∥ ≤ italic_R end_POSTSUBSCRIPT ∥ italic_D start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ( bold_x start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT + italic_ζ ) - italic_D start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ( bold_x start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT + italic_ζ ) ∥ caligraphic_N ( italic_ζ ; 0 , italic_σ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT bold_I start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT ) italic_d italic_ζ + ∫ start_POSTSUBSCRIPT ∥ bold_x start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT + italic_ζ ∥ > italic_R end_POSTSUBSCRIPT ∥ italic_D start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ( bold_x start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT + italic_ζ ) - italic_D start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ( bold_x start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT + italic_ζ ) ∥ caligraphic_N ( italic_ζ ; 0 , italic_σ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT bold_I start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT ) italic_d italic_ζ )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="A4.Ex112"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_eqn_cell"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\leq\frac{\alpha}{\sigma^{2}}\left(M(R)+\int_{\|{\mathbf{x}}_{k}+% \zeta\|>R}\left(\|D_{\sigma}({\mathbf{x}}_{k}+\zeta)\|+\|D^{\star}_{\sigma}({% \mathbf{x}}_{k}+\zeta)\|\right)\mathcal{N}(\zeta;0,\sigma^{2}\mathbf{I}_{d})d% \zeta\right)" class="ltx_Math" display="inline" id="A4.Ex112.m1.5"><semantics id="A4.Ex112.m1.5a"><mrow id="A4.Ex112.m1.5.5" xref="A4.Ex112.m1.5.5.cmml"><mi id="A4.Ex112.m1.5.5.3" xref="A4.Ex112.m1.5.5.3.cmml"></mi><mo id="A4.Ex112.m1.5.5.2" xref="A4.Ex112.m1.5.5.2.cmml">≤</mo><mrow id="A4.Ex112.m1.5.5.1" xref="A4.Ex112.m1.5.5.1.cmml"><mstyle displaystyle="true" id="A4.Ex112.m1.5.5.1.3" xref="A4.Ex112.m1.5.5.1.3.cmml"><mfrac id="A4.Ex112.m1.5.5.1.3a" xref="A4.Ex112.m1.5.5.1.3.cmml"><mi id="A4.Ex112.m1.5.5.1.3.2" xref="A4.Ex112.m1.5.5.1.3.2.cmml">α</mi><msup id="A4.Ex112.m1.5.5.1.3.3" xref="A4.Ex112.m1.5.5.1.3.3.cmml"><mi 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id="A4.Ex112.m1.5c">\displaystyle\leq\frac{\alpha}{\sigma^{2}}\left(M(R)+\int_{\|{\mathbf{x}}_{k}+% \zeta\|>R}\left(\|D_{\sigma}({\mathbf{x}}_{k}+\zeta)\|+\|D^{\star}_{\sigma}({% \mathbf{x}}_{k}+\zeta)\|\right)\mathcal{N}(\zeta;0,\sigma^{2}\mathbf{I}_{d})d% \zeta\right)</annotation><annotation encoding="application/x-llamapun" id="A4.Ex112.m1.5d">≤ divide start_ARG italic_α end_ARG start_ARG italic_σ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG ( italic_M ( italic_R ) + ∫ start_POSTSUBSCRIPT ∥ bold_x start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT + italic_ζ ∥ > italic_R end_POSTSUBSCRIPT ( ∥ italic_D start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ( bold_x start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT + italic_ζ ) ∥ + ∥ italic_D start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ( bold_x start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT + italic_ζ ) ∥ ) caligraphic_N ( italic_ζ ; 0 , italic_σ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT bold_I start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT ) italic_d italic_ζ )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="A4.Ex113"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_eqn_cell"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\leq\frac{\alpha}{\sigma^{2}}\left(M(R)+\int_{\|{\mathbf{x}}_{k}+% \zeta\|>R}\left(2\|{\mathbf{x}}_{k}+\zeta\|+2C\sigma\right)\mathcal{N}(\zeta;0% ,\sigma^{2}\mathbf{I}_{d})d\zeta\right)" class="ltx_Math" display="inline" id="A4.Ex113.m1.5"><semantics id="A4.Ex113.m1.5a"><mrow id="A4.Ex113.m1.5.5" xref="A4.Ex113.m1.5.5.cmml"><mi id="A4.Ex113.m1.5.5.3" xref="A4.Ex113.m1.5.5.3.cmml"></mi><mo id="A4.Ex113.m1.5.5.2" xref="A4.Ex113.m1.5.5.2.cmml">≤</mo><mrow id="A4.Ex113.m1.5.5.1" xref="A4.Ex113.m1.5.5.1.cmml"><mstyle displaystyle="true" id="A4.Ex113.m1.5.5.1.3" 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,\sigma^{2}\mathbf{I}_{d})d\zeta\right)</annotation><annotation encoding="application/x-llamapun" id="A4.Ex113.m1.5d">≤ divide start_ARG italic_α end_ARG start_ARG italic_σ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG ( italic_M ( italic_R ) + ∫ start_POSTSUBSCRIPT ∥ bold_x start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT + italic_ζ ∥ > italic_R end_POSTSUBSCRIPT ( 2 ∥ bold_x start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT + italic_ζ ∥ + 2 italic_C italic_σ ) caligraphic_N ( italic_ζ ; 0 , italic_σ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT bold_I start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT ) italic_d italic_ζ )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="A4.Ex114"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_eqn_cell"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math 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id="A4.Ex114.m1.6.6.1.1.1.1.1.1.2.2.2.1.1.2.1.cmml" xref="A4.Ex114.m1.6.6.1.1.1.1.1.1.2.2.2.1.1.2">superscript</csymbol><ci id="A4.Ex114.m1.6.6.1.1.1.1.1.1.2.2.2.1.1.2.2.cmml" xref="A4.Ex114.m1.6.6.1.1.1.1.1.1.2.2.2.1.1.2.2">𝜎</ci><cn id="A4.Ex114.m1.6.6.1.1.1.1.1.1.2.2.2.1.1.2.3.cmml" type="integer" xref="A4.Ex114.m1.6.6.1.1.1.1.1.1.2.2.2.1.1.2.3">2</cn></apply><apply id="A4.Ex114.m1.6.6.1.1.1.1.1.1.2.2.2.1.1.3.cmml" xref="A4.Ex114.m1.6.6.1.1.1.1.1.1.2.2.2.1.1.3"><csymbol cd="ambiguous" id="A4.Ex114.m1.6.6.1.1.1.1.1.1.2.2.2.1.1.3.1.cmml" xref="A4.Ex114.m1.6.6.1.1.1.1.1.1.2.2.2.1.1.3">subscript</csymbol><ci id="A4.Ex114.m1.6.6.1.1.1.1.1.1.2.2.2.1.1.3.2.cmml" xref="A4.Ex114.m1.6.6.1.1.1.1.1.1.2.2.2.1.1.3.2">𝐈</ci><ci id="A4.Ex114.m1.6.6.1.1.1.1.1.1.2.2.2.1.1.3.3.cmml" xref="A4.Ex114.m1.6.6.1.1.1.1.1.1.2.2.2.1.1.3.3">𝑑</ci></apply></apply></list><apply id="A4.Ex114.m1.6.6.1.1.1.1.1.1.2.2.5.cmml" xref="A4.Ex114.m1.6.6.1.1.1.1.1.1.2.2.5"><csymbol cd="latexml" id="A4.Ex114.m1.6.6.1.1.1.1.1.1.2.2.5.1.cmml" xref="A4.Ex114.m1.6.6.1.1.1.1.1.1.2.2.5.1">differential-d</csymbol><ci id="A4.Ex114.m1.6.6.1.1.1.1.1.1.2.2.5.2.cmml" xref="A4.Ex114.m1.6.6.1.1.1.1.1.1.2.2.5.2">𝜁</ci></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.Ex114.m1.6c">\displaystyle\leq\frac{\alpha}{\sigma^{2}}\left(M(R)+\int_{\|{\mathbf{x}}_{k}+% \zeta\|>R}\left(2\|{\mathbf{x}}_{k}\|+2\|\zeta\|+2C\sigma\right)\mathcal{N}(% \zeta;0,\sigma^{2}\mathbf{I}_{d})d\zeta\right).</annotation><annotation encoding="application/x-llamapun" id="A4.Ex114.m1.6d">≤ divide start_ARG italic_α end_ARG start_ARG italic_σ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG ( italic_M ( italic_R ) + ∫ start_POSTSUBSCRIPT ∥ bold_x start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT + italic_ζ ∥ > italic_R end_POSTSUBSCRIPT ( 2 ∥ bold_x start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ∥ + 2 ∥ italic_ζ ∥ + 2 italic_C italic_σ ) caligraphic_N ( italic_ζ ; 0 , italic_σ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT bold_I start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT ) italic_d italic_ζ ) .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="A4.SS1.SSS5.p2.6">For <math alttext="R=R_{\mathbf{K}}+1" class="ltx_Math" display="inline" id="A4.SS1.SSS5.p2.3.m1.1"><semantics id="A4.SS1.SSS5.p2.3.m1.1a"><mrow id="A4.SS1.SSS5.p2.3.m1.1.1" xref="A4.SS1.SSS5.p2.3.m1.1.1.cmml"><mi id="A4.SS1.SSS5.p2.3.m1.1.1.2" xref="A4.SS1.SSS5.p2.3.m1.1.1.2.cmml">R</mi><mo id="A4.SS1.SSS5.p2.3.m1.1.1.1" xref="A4.SS1.SSS5.p2.3.m1.1.1.1.cmml">=</mo><mrow id="A4.SS1.SSS5.p2.3.m1.1.1.3" xref="A4.SS1.SSS5.p2.3.m1.1.1.3.cmml"><msub id="A4.SS1.SSS5.p2.3.m1.1.1.3.2" xref="A4.SS1.SSS5.p2.3.m1.1.1.3.2.cmml"><mi id="A4.SS1.SSS5.p2.3.m1.1.1.3.2.2" xref="A4.SS1.SSS5.p2.3.m1.1.1.3.2.2.cmml">R</mi><mi id="A4.SS1.SSS5.p2.3.m1.1.1.3.2.3" xref="A4.SS1.SSS5.p2.3.m1.1.1.3.2.3.cmml">𝐊</mi></msub><mo id="A4.SS1.SSS5.p2.3.m1.1.1.3.1" xref="A4.SS1.SSS5.p2.3.m1.1.1.3.1.cmml">+</mo><mn id="A4.SS1.SSS5.p2.3.m1.1.1.3.3" xref="A4.SS1.SSS5.p2.3.m1.1.1.3.3.cmml">1</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="A4.SS1.SSS5.p2.3.m1.1b"><apply id="A4.SS1.SSS5.p2.3.m1.1.1.cmml" xref="A4.SS1.SSS5.p2.3.m1.1.1"><eq id="A4.SS1.SSS5.p2.3.m1.1.1.1.cmml" xref="A4.SS1.SSS5.p2.3.m1.1.1.1"></eq><ci id="A4.SS1.SSS5.p2.3.m1.1.1.2.cmml" xref="A4.SS1.SSS5.p2.3.m1.1.1.2">𝑅</ci><apply id="A4.SS1.SSS5.p2.3.m1.1.1.3.cmml" xref="A4.SS1.SSS5.p2.3.m1.1.1.3"><plus id="A4.SS1.SSS5.p2.3.m1.1.1.3.1.cmml" xref="A4.SS1.SSS5.p2.3.m1.1.1.3.1"></plus><apply id="A4.SS1.SSS5.p2.3.m1.1.1.3.2.cmml" xref="A4.SS1.SSS5.p2.3.m1.1.1.3.2"><csymbol cd="ambiguous" id="A4.SS1.SSS5.p2.3.m1.1.1.3.2.1.cmml" xref="A4.SS1.SSS5.p2.3.m1.1.1.3.2">subscript</csymbol><ci id="A4.SS1.SSS5.p2.3.m1.1.1.3.2.2.cmml" xref="A4.SS1.SSS5.p2.3.m1.1.1.3.2.2">𝑅</ci><ci id="A4.SS1.SSS5.p2.3.m1.1.1.3.2.3.cmml" xref="A4.SS1.SSS5.p2.3.m1.1.1.3.2.3">𝐊</ci></apply><cn id="A4.SS1.SSS5.p2.3.m1.1.1.3.3.cmml" type="integer" xref="A4.SS1.SSS5.p2.3.m1.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.SS1.SSS5.p2.3.m1.1c">R=R_{\mathbf{K}}+1</annotation><annotation encoding="application/x-llamapun" id="A4.SS1.SSS5.p2.3.m1.1d">italic_R = italic_R start_POSTSUBSCRIPT bold_K end_POSTSUBSCRIPT + 1</annotation></semantics></math>, if <math alttext="\|{\mathbf{x}}_{k}+\zeta\|>R" class="ltx_Math" display="inline" id="A4.SS1.SSS5.p2.4.m2.1"><semantics id="A4.SS1.SSS5.p2.4.m2.1a"><mrow id="A4.SS1.SSS5.p2.4.m2.1.1" xref="A4.SS1.SSS5.p2.4.m2.1.1.cmml"><mrow id="A4.SS1.SSS5.p2.4.m2.1.1.1.1" xref="A4.SS1.SSS5.p2.4.m2.1.1.1.2.cmml"><mo id="A4.SS1.SSS5.p2.4.m2.1.1.1.1.2" stretchy="false" xref="A4.SS1.SSS5.p2.4.m2.1.1.1.2.1.cmml">‖</mo><mrow id="A4.SS1.SSS5.p2.4.m2.1.1.1.1.1" xref="A4.SS1.SSS5.p2.4.m2.1.1.1.1.1.cmml"><msub id="A4.SS1.SSS5.p2.4.m2.1.1.1.1.1.2" xref="A4.SS1.SSS5.p2.4.m2.1.1.1.1.1.2.cmml"><mi id="A4.SS1.SSS5.p2.4.m2.1.1.1.1.1.2.2" xref="A4.SS1.SSS5.p2.4.m2.1.1.1.1.1.2.2.cmml">𝐱</mi><mi id="A4.SS1.SSS5.p2.4.m2.1.1.1.1.1.2.3" xref="A4.SS1.SSS5.p2.4.m2.1.1.1.1.1.2.3.cmml">k</mi></msub><mo id="A4.SS1.SSS5.p2.4.m2.1.1.1.1.1.1" xref="A4.SS1.SSS5.p2.4.m2.1.1.1.1.1.1.cmml">+</mo><mi id="A4.SS1.SSS5.p2.4.m2.1.1.1.1.1.3" xref="A4.SS1.SSS5.p2.4.m2.1.1.1.1.1.3.cmml">ζ</mi></mrow><mo id="A4.SS1.SSS5.p2.4.m2.1.1.1.1.3" stretchy="false" xref="A4.SS1.SSS5.p2.4.m2.1.1.1.2.1.cmml">‖</mo></mrow><mo id="A4.SS1.SSS5.p2.4.m2.1.1.2" xref="A4.SS1.SSS5.p2.4.m2.1.1.2.cmml">></mo><mi id="A4.SS1.SSS5.p2.4.m2.1.1.3" xref="A4.SS1.SSS5.p2.4.m2.1.1.3.cmml">R</mi></mrow><annotation-xml encoding="MathML-Content" id="A4.SS1.SSS5.p2.4.m2.1b"><apply id="A4.SS1.SSS5.p2.4.m2.1.1.cmml" xref="A4.SS1.SSS5.p2.4.m2.1.1"><gt id="A4.SS1.SSS5.p2.4.m2.1.1.2.cmml" xref="A4.SS1.SSS5.p2.4.m2.1.1.2"></gt><apply id="A4.SS1.SSS5.p2.4.m2.1.1.1.2.cmml" xref="A4.SS1.SSS5.p2.4.m2.1.1.1.1"><csymbol cd="latexml" id="A4.SS1.SSS5.p2.4.m2.1.1.1.2.1.cmml" xref="A4.SS1.SSS5.p2.4.m2.1.1.1.1.2">norm</csymbol><apply id="A4.SS1.SSS5.p2.4.m2.1.1.1.1.1.cmml" xref="A4.SS1.SSS5.p2.4.m2.1.1.1.1.1"><plus id="A4.SS1.SSS5.p2.4.m2.1.1.1.1.1.1.cmml" xref="A4.SS1.SSS5.p2.4.m2.1.1.1.1.1.1"></plus><apply id="A4.SS1.SSS5.p2.4.m2.1.1.1.1.1.2.cmml" xref="A4.SS1.SSS5.p2.4.m2.1.1.1.1.1.2"><csymbol cd="ambiguous" id="A4.SS1.SSS5.p2.4.m2.1.1.1.1.1.2.1.cmml" xref="A4.SS1.SSS5.p2.4.m2.1.1.1.1.1.2">subscript</csymbol><ci id="A4.SS1.SSS5.p2.4.m2.1.1.1.1.1.2.2.cmml" xref="A4.SS1.SSS5.p2.4.m2.1.1.1.1.1.2.2">𝐱</ci><ci id="A4.SS1.SSS5.p2.4.m2.1.1.1.1.1.2.3.cmml" xref="A4.SS1.SSS5.p2.4.m2.1.1.1.1.1.2.3">𝑘</ci></apply><ci id="A4.SS1.SSS5.p2.4.m2.1.1.1.1.1.3.cmml" xref="A4.SS1.SSS5.p2.4.m2.1.1.1.1.1.3">𝜁</ci></apply></apply><ci id="A4.SS1.SSS5.p2.4.m2.1.1.3.cmml" xref="A4.SS1.SSS5.p2.4.m2.1.1.3">𝑅</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.SS1.SSS5.p2.4.m2.1c">\|{\mathbf{x}}_{k}+\zeta\|>R</annotation><annotation encoding="application/x-llamapun" id="A4.SS1.SSS5.p2.4.m2.1d">∥ bold_x start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT + italic_ζ ∥ > italic_R</annotation></semantics></math>, then <math alttext="\|\zeta\|>1" class="ltx_Math" display="inline" id="A4.SS1.SSS5.p2.5.m3.1"><semantics id="A4.SS1.SSS5.p2.5.m3.1a"><mrow id="A4.SS1.SSS5.p2.5.m3.1.2" xref="A4.SS1.SSS5.p2.5.m3.1.2.cmml"><mrow id="A4.SS1.SSS5.p2.5.m3.1.2.2.2" xref="A4.SS1.SSS5.p2.5.m3.1.2.2.1.cmml"><mo id="A4.SS1.SSS5.p2.5.m3.1.2.2.2.1" stretchy="false" xref="A4.SS1.SSS5.p2.5.m3.1.2.2.1.1.cmml">‖</mo><mi id="A4.SS1.SSS5.p2.5.m3.1.1" xref="A4.SS1.SSS5.p2.5.m3.1.1.cmml">ζ</mi><mo id="A4.SS1.SSS5.p2.5.m3.1.2.2.2.2" stretchy="false" xref="A4.SS1.SSS5.p2.5.m3.1.2.2.1.1.cmml">‖</mo></mrow><mo id="A4.SS1.SSS5.p2.5.m3.1.2.1" xref="A4.SS1.SSS5.p2.5.m3.1.2.1.cmml">></mo><mn id="A4.SS1.SSS5.p2.5.m3.1.2.3" xref="A4.SS1.SSS5.p2.5.m3.1.2.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="A4.SS1.SSS5.p2.5.m3.1b"><apply id="A4.SS1.SSS5.p2.5.m3.1.2.cmml" xref="A4.SS1.SSS5.p2.5.m3.1.2"><gt id="A4.SS1.SSS5.p2.5.m3.1.2.1.cmml" xref="A4.SS1.SSS5.p2.5.m3.1.2.1"></gt><apply id="A4.SS1.SSS5.p2.5.m3.1.2.2.1.cmml" xref="A4.SS1.SSS5.p2.5.m3.1.2.2.2"><csymbol cd="latexml" id="A4.SS1.SSS5.p2.5.m3.1.2.2.1.1.cmml" xref="A4.SS1.SSS5.p2.5.m3.1.2.2.2.1">norm</csymbol><ci id="A4.SS1.SSS5.p2.5.m3.1.1.cmml" xref="A4.SS1.SSS5.p2.5.m3.1.1">𝜁</ci></apply><cn id="A4.SS1.SSS5.p2.5.m3.1.2.3.cmml" type="integer" xref="A4.SS1.SSS5.p2.5.m3.1.2.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.SS1.SSS5.p2.5.m3.1c">\|\zeta\|>1</annotation><annotation encoding="application/x-llamapun" id="A4.SS1.SSS5.p2.5.m3.1d">∥ italic_ζ ∥ > 1</annotation></semantics></math> because <math alttext="\|{\mathbf{x}}_{k}\|\leq R_{\mathbf{K}}" class="ltx_Math" display="inline" id="A4.SS1.SSS5.p2.6.m4.1"><semantics id="A4.SS1.SSS5.p2.6.m4.1a"><mrow id="A4.SS1.SSS5.p2.6.m4.1.1" xref="A4.SS1.SSS5.p2.6.m4.1.1.cmml"><mrow id="A4.SS1.SSS5.p2.6.m4.1.1.1.1" xref="A4.SS1.SSS5.p2.6.m4.1.1.1.2.cmml"><mo id="A4.SS1.SSS5.p2.6.m4.1.1.1.1.2" stretchy="false" xref="A4.SS1.SSS5.p2.6.m4.1.1.1.2.1.cmml">‖</mo><msub id="A4.SS1.SSS5.p2.6.m4.1.1.1.1.1" xref="A4.SS1.SSS5.p2.6.m4.1.1.1.1.1.cmml"><mi id="A4.SS1.SSS5.p2.6.m4.1.1.1.1.1.2" xref="A4.SS1.SSS5.p2.6.m4.1.1.1.1.1.2.cmml">𝐱</mi><mi id="A4.SS1.SSS5.p2.6.m4.1.1.1.1.1.3" xref="A4.SS1.SSS5.p2.6.m4.1.1.1.1.1.3.cmml">k</mi></msub><mo id="A4.SS1.SSS5.p2.6.m4.1.1.1.1.3" stretchy="false" xref="A4.SS1.SSS5.p2.6.m4.1.1.1.2.1.cmml">‖</mo></mrow><mo id="A4.SS1.SSS5.p2.6.m4.1.1.2" xref="A4.SS1.SSS5.p2.6.m4.1.1.2.cmml">≤</mo><msub id="A4.SS1.SSS5.p2.6.m4.1.1.3" xref="A4.SS1.SSS5.p2.6.m4.1.1.3.cmml"><mi id="A4.SS1.SSS5.p2.6.m4.1.1.3.2" xref="A4.SS1.SSS5.p2.6.m4.1.1.3.2.cmml">R</mi><mi id="A4.SS1.SSS5.p2.6.m4.1.1.3.3" xref="A4.SS1.SSS5.p2.6.m4.1.1.3.3.cmml">𝐊</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="A4.SS1.SSS5.p2.6.m4.1b"><apply id="A4.SS1.SSS5.p2.6.m4.1.1.cmml" xref="A4.SS1.SSS5.p2.6.m4.1.1"><leq id="A4.SS1.SSS5.p2.6.m4.1.1.2.cmml" xref="A4.SS1.SSS5.p2.6.m4.1.1.2"></leq><apply id="A4.SS1.SSS5.p2.6.m4.1.1.1.2.cmml" xref="A4.SS1.SSS5.p2.6.m4.1.1.1.1"><csymbol cd="latexml" id="A4.SS1.SSS5.p2.6.m4.1.1.1.2.1.cmml" xref="A4.SS1.SSS5.p2.6.m4.1.1.1.1.2">norm</csymbol><apply id="A4.SS1.SSS5.p2.6.m4.1.1.1.1.1.cmml" xref="A4.SS1.SSS5.p2.6.m4.1.1.1.1.1"><csymbol cd="ambiguous" id="A4.SS1.SSS5.p2.6.m4.1.1.1.1.1.1.cmml" xref="A4.SS1.SSS5.p2.6.m4.1.1.1.1.1">subscript</csymbol><ci id="A4.SS1.SSS5.p2.6.m4.1.1.1.1.1.2.cmml" xref="A4.SS1.SSS5.p2.6.m4.1.1.1.1.1.2">𝐱</ci><ci id="A4.SS1.SSS5.p2.6.m4.1.1.1.1.1.3.cmml" xref="A4.SS1.SSS5.p2.6.m4.1.1.1.1.1.3">𝑘</ci></apply></apply><apply id="A4.SS1.SSS5.p2.6.m4.1.1.3.cmml" xref="A4.SS1.SSS5.p2.6.m4.1.1.3"><csymbol cd="ambiguous" id="A4.SS1.SSS5.p2.6.m4.1.1.3.1.cmml" xref="A4.SS1.SSS5.p2.6.m4.1.1.3">subscript</csymbol><ci id="A4.SS1.SSS5.p2.6.m4.1.1.3.2.cmml" xref="A4.SS1.SSS5.p2.6.m4.1.1.3.2">𝑅</ci><ci id="A4.SS1.SSS5.p2.6.m4.1.1.3.3.cmml" xref="A4.SS1.SSS5.p2.6.m4.1.1.3.3">𝐊</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.SS1.SSS5.p2.6.m4.1c">\|{\mathbf{x}}_{k}\|\leq R_{\mathbf{K}}</annotation><annotation encoding="application/x-llamapun" id="A4.SS1.SSS5.p2.6.m4.1d">∥ bold_x start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ∥ ≤ italic_R start_POSTSUBSCRIPT bold_K end_POSTSUBSCRIPT</annotation></semantics></math>. This implies that</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A8.EGx56"> <tbody id="A4.Ex116"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\|\eta_{k}\|" class="ltx_Math" display="inline" id="A4.Ex116.m1.1"><semantics id="A4.Ex116.m1.1a"><mrow id="A4.Ex116.m1.1.1.1" xref="A4.Ex116.m1.1.1.2.cmml"><mo id="A4.Ex116.m1.1.1.1.2" stretchy="false" xref="A4.Ex116.m1.1.1.2.1.cmml">‖</mo><msub id="A4.Ex116.m1.1.1.1.1" xref="A4.Ex116.m1.1.1.1.1.cmml"><mi id="A4.Ex116.m1.1.1.1.1.2" xref="A4.Ex116.m1.1.1.1.1.2.cmml">η</mi><mi id="A4.Ex116.m1.1.1.1.1.3" xref="A4.Ex116.m1.1.1.1.1.3.cmml">k</mi></msub><mo id="A4.Ex116.m1.1.1.1.3" stretchy="false" xref="A4.Ex116.m1.1.1.2.1.cmml">‖</mo></mrow><annotation-xml encoding="MathML-Content" id="A4.Ex116.m1.1b"><apply id="A4.Ex116.m1.1.1.2.cmml" xref="A4.Ex116.m1.1.1.1"><csymbol cd="latexml" id="A4.Ex116.m1.1.1.2.1.cmml" xref="A4.Ex116.m1.1.1.1.2">norm</csymbol><apply id="A4.Ex116.m1.1.1.1.1.cmml" xref="A4.Ex116.m1.1.1.1.1"><csymbol cd="ambiguous" id="A4.Ex116.m1.1.1.1.1.1.cmml" xref="A4.Ex116.m1.1.1.1.1">subscript</csymbol><ci id="A4.Ex116.m1.1.1.1.1.2.cmml" xref="A4.Ex116.m1.1.1.1.1.2">𝜂</ci><ci id="A4.Ex116.m1.1.1.1.1.3.cmml" xref="A4.Ex116.m1.1.1.1.1.3">𝑘</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.Ex116.m1.1c">\displaystyle\|\eta_{k}\|</annotation><annotation encoding="application/x-llamapun" id="A4.Ex116.m1.1d">∥ italic_η start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ∥</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\leq\frac{\alpha}{\sigma^{2}}\left(M(R)+\int_{\|\zeta\|>1}\left(2% \|{\mathbf{x}}_{k}\|+2\|\zeta\|+2C\sigma\right)\mathcal{N}(\zeta;0,\sigma^{2}% \mathbf{I}_{d})d\zeta\right)." class="ltx_Math" display="inline" id="A4.Ex116.m2.6"><semantics 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xref="A4.Ex116.m2.6.6.1.1.1.1.1.1.2.2.5.2">𝜁</ci></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.Ex116.m2.6c">\displaystyle\leq\frac{\alpha}{\sigma^{2}}\left(M(R)+\int_{\|\zeta\|>1}\left(2% \|{\mathbf{x}}_{k}\|+2\|\zeta\|+2C\sigma\right)\mathcal{N}(\zeta;0,\sigma^{2}% \mathbf{I}_{d})d\zeta\right).</annotation><annotation encoding="application/x-llamapun" id="A4.Ex116.m2.6d">≤ divide start_ARG italic_α end_ARG start_ARG italic_σ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG ( italic_M ( italic_R ) + ∫ start_POSTSUBSCRIPT ∥ italic_ζ ∥ > 1 end_POSTSUBSCRIPT ( 2 ∥ bold_x start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ∥ + 2 ∥ italic_ζ ∥ + 2 italic_C italic_σ ) caligraphic_N ( italic_ζ ; 0 , italic_σ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT bold_I start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT ) italic_d italic_ζ ) .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="A4.SS1.SSS5.p2.9">Then if <math alttext="\|\zeta\|>1" class="ltx_Math" display="inline" id="A4.SS1.SSS5.p2.7.m1.1"><semantics id="A4.SS1.SSS5.p2.7.m1.1a"><mrow id="A4.SS1.SSS5.p2.7.m1.1.2" xref="A4.SS1.SSS5.p2.7.m1.1.2.cmml"><mrow id="A4.SS1.SSS5.p2.7.m1.1.2.2.2" xref="A4.SS1.SSS5.p2.7.m1.1.2.2.1.cmml"><mo id="A4.SS1.SSS5.p2.7.m1.1.2.2.2.1" stretchy="false" xref="A4.SS1.SSS5.p2.7.m1.1.2.2.1.1.cmml">‖</mo><mi id="A4.SS1.SSS5.p2.7.m1.1.1" xref="A4.SS1.SSS5.p2.7.m1.1.1.cmml">ζ</mi><mo id="A4.SS1.SSS5.p2.7.m1.1.2.2.2.2" stretchy="false" xref="A4.SS1.SSS5.p2.7.m1.1.2.2.1.1.cmml">‖</mo></mrow><mo id="A4.SS1.SSS5.p2.7.m1.1.2.1" xref="A4.SS1.SSS5.p2.7.m1.1.2.1.cmml">></mo><mn id="A4.SS1.SSS5.p2.7.m1.1.2.3" xref="A4.SS1.SSS5.p2.7.m1.1.2.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="A4.SS1.SSS5.p2.7.m1.1b"><apply id="A4.SS1.SSS5.p2.7.m1.1.2.cmml" xref="A4.SS1.SSS5.p2.7.m1.1.2"><gt id="A4.SS1.SSS5.p2.7.m1.1.2.1.cmml" xref="A4.SS1.SSS5.p2.7.m1.1.2.1"></gt><apply id="A4.SS1.SSS5.p2.7.m1.1.2.2.1.cmml" xref="A4.SS1.SSS5.p2.7.m1.1.2.2.2"><csymbol cd="latexml" id="A4.SS1.SSS5.p2.7.m1.1.2.2.1.1.cmml" xref="A4.SS1.SSS5.p2.7.m1.1.2.2.2.1">norm</csymbol><ci id="A4.SS1.SSS5.p2.7.m1.1.1.cmml" xref="A4.SS1.SSS5.p2.7.m1.1.1">𝜁</ci></apply><cn id="A4.SS1.SSS5.p2.7.m1.1.2.3.cmml" type="integer" xref="A4.SS1.SSS5.p2.7.m1.1.2.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.SS1.SSS5.p2.7.m1.1c">\|\zeta\|>1</annotation><annotation encoding="application/x-llamapun" id="A4.SS1.SSS5.p2.7.m1.1d">∥ italic_ζ ∥ > 1</annotation></semantics></math>, <math alttext="\|\zeta\|^{2}\leq\frac{1}{2}+\frac{\|\zeta\|^{2}}{2}" class="ltx_Math" display="inline" id="A4.SS1.SSS5.p2.8.m2.2"><semantics id="A4.SS1.SSS5.p2.8.m2.2a"><mrow id="A4.SS1.SSS5.p2.8.m2.2.3" xref="A4.SS1.SSS5.p2.8.m2.2.3.cmml"><msup id="A4.SS1.SSS5.p2.8.m2.2.3.2" xref="A4.SS1.SSS5.p2.8.m2.2.3.2.cmml"><mrow 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id="A4.SS1.SSS5.p2.8.m2.2c">\|\zeta\|^{2}\leq\frac{1}{2}+\frac{\|\zeta\|^{2}}{2}</annotation><annotation encoding="application/x-llamapun" id="A4.SS1.SSS5.p2.8.m2.2d">∥ italic_ζ ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≤ divide start_ARG 1 end_ARG start_ARG 2 end_ARG + divide start_ARG ∥ italic_ζ ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG 2 end_ARG</annotation></semantics></math>. 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xref="A4.SS1.SSS5.p2.9.m3.8.8.3.2.1.1.1.2.3.2">4</cn><apply id="A4.SS1.SSS5.p2.9.m3.8.8.3.2.1.1.1.2.3.3.cmml" xref="A4.SS1.SSS5.p2.9.m3.8.8.3.2.1.1.1.2.3.3"><csymbol cd="ambiguous" id="A4.SS1.SSS5.p2.9.m3.8.8.3.2.1.1.1.2.3.3.1.cmml" xref="A4.SS1.SSS5.p2.9.m3.8.8.3.2.1.1.1.2.3.3">superscript</csymbol><ci id="A4.SS1.SSS5.p2.9.m3.8.8.3.2.1.1.1.2.3.3.2.cmml" xref="A4.SS1.SSS5.p2.9.m3.8.8.3.2.1.1.1.2.3.3.2">𝜎</ci><cn id="A4.SS1.SSS5.p2.9.m3.8.8.3.2.1.1.1.2.3.3.3.cmml" type="integer" xref="A4.SS1.SSS5.p2.9.m3.8.8.3.2.1.1.1.2.3.3.3">2</cn></apply></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.SS1.SSS5.p2.9.m3.8c">\mathcal{N}(\zeta;0,\sigma^{2}\mathbf{I}_{d})\leq 2^{-\frac{d}{2}}\mathcal{N}(% \zeta;0,2\sigma^{2}\mathbf{I}_{d})\exp\left(-\frac{1}{4\sigma^{2}}\right)</annotation><annotation encoding="application/x-llamapun" id="A4.SS1.SSS5.p2.9.m3.8d">caligraphic_N ( italic_ζ ; 0 , italic_σ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT bold_I start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT ) ≤ 2 start_POSTSUPERSCRIPT - divide start_ARG italic_d end_ARG start_ARG 2 end_ARG end_POSTSUPERSCRIPT caligraphic_N ( italic_ζ ; 0 , 2 italic_σ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT bold_I start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT ) roman_exp ( - divide start_ARG 1 end_ARG start_ARG 4 italic_σ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG )</annotation></semantics></math>. So</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A8.EGx57"> <tbody id="A4.Ex118"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\|\eta_{k}\|" class="ltx_Math" display="inline" id="A4.Ex118.m1.1"><semantics id="A4.Ex118.m1.1a"><mrow id="A4.Ex118.m1.1.1.1" xref="A4.Ex118.m1.1.1.2.cmml"><mo id="A4.Ex118.m1.1.1.1.2" stretchy="false" xref="A4.Ex118.m1.1.1.2.1.cmml">‖</mo><msub id="A4.Ex118.m1.1.1.1.1" xref="A4.Ex118.m1.1.1.1.1.cmml"><mi id="A4.Ex118.m1.1.1.1.1.2" xref="A4.Ex118.m1.1.1.1.1.2.cmml">η</mi><mi id="A4.Ex118.m1.1.1.1.1.3" xref="A4.Ex118.m1.1.1.1.1.3.cmml">k</mi></msub><mo id="A4.Ex118.m1.1.1.1.3" stretchy="false" xref="A4.Ex118.m1.1.1.2.1.cmml">‖</mo></mrow><annotation-xml encoding="MathML-Content" id="A4.Ex118.m1.1b"><apply id="A4.Ex118.m1.1.1.2.cmml" xref="A4.Ex118.m1.1.1.1"><csymbol cd="latexml" id="A4.Ex118.m1.1.1.2.1.cmml" xref="A4.Ex118.m1.1.1.1.2">norm</csymbol><apply id="A4.Ex118.m1.1.1.1.1.cmml" xref="A4.Ex118.m1.1.1.1.1"><csymbol cd="ambiguous" id="A4.Ex118.m1.1.1.1.1.1.cmml" xref="A4.Ex118.m1.1.1.1.1">subscript</csymbol><ci id="A4.Ex118.m1.1.1.1.1.2.cmml" xref="A4.Ex118.m1.1.1.1.1.2">𝜂</ci><ci id="A4.Ex118.m1.1.1.1.1.3.cmml" xref="A4.Ex118.m1.1.1.1.1.3">𝑘</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.Ex118.m1.1c">\displaystyle\|\eta_{k}\|</annotation><annotation encoding="application/x-llamapun" id="A4.Ex118.m1.1d">∥ italic_η start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ∥</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\leq\frac{\alpha}{\sigma^{2}}\left(M(R)+2^{-\frac{d}{2}}\exp\left% (-\frac{1}{4\sigma^{2}}\right)\int_{\|\zeta\|>1}\left(2\|{\mathbf{x}}_{k}\|+2% \|\zeta\|+2C\sigma\right)\mathcal{N}(\zeta;0,2\sigma^{2}\mathbf{I}_{d})d\zeta\right)" class="ltx_Math" display="inline" id="A4.Ex118.m2.7"><semantics id="A4.Ex118.m2.7a"><mrow id="A4.Ex118.m2.7.7" xref="A4.Ex118.m2.7.7.cmml"><mi id="A4.Ex118.m2.7.7.3" xref="A4.Ex118.m2.7.7.3.cmml"></mi><mo id="A4.Ex118.m2.7.7.2" xref="A4.Ex118.m2.7.7.2.cmml">≤</mo><mrow id="A4.Ex118.m2.7.7.1" xref="A4.Ex118.m2.7.7.1.cmml"><mstyle displaystyle="true" id="A4.Ex118.m2.7.7.1.3" xref="A4.Ex118.m2.7.7.1.3.cmml"><mfrac id="A4.Ex118.m2.7.7.1.3a" xref="A4.Ex118.m2.7.7.1.3.cmml"><mi id="A4.Ex118.m2.7.7.1.3.2" xref="A4.Ex118.m2.7.7.1.3.2.cmml">α</mi><msup id="A4.Ex118.m2.7.7.1.3.3" xref="A4.Ex118.m2.7.7.1.3.3.cmml"><mi id="A4.Ex118.m2.7.7.1.3.3.2" xref="A4.Ex118.m2.7.7.1.3.3.2.cmml">σ</mi><mn id="A4.Ex118.m2.7.7.1.3.3.3" xref="A4.Ex118.m2.7.7.1.3.3.3.cmml">2</mn></msup></mfrac></mstyle><mo id="A4.Ex118.m2.7.7.1.2" xref="A4.Ex118.m2.7.7.1.2.cmml">⁢</mo><mrow id="A4.Ex118.m2.7.7.1.1.1" xref="A4.Ex118.m2.7.7.1.1.1.1.cmml"><mo id="A4.Ex118.m2.7.7.1.1.1.2" xref="A4.Ex118.m2.7.7.1.1.1.1.cmml">(</mo><mrow 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id="A4.Ex118.m2.7c">\displaystyle\leq\frac{\alpha}{\sigma^{2}}\left(M(R)+2^{-\frac{d}{2}}\exp\left% (-\frac{1}{4\sigma^{2}}\right)\int_{\|\zeta\|>1}\left(2\|{\mathbf{x}}_{k}\|+2% \|\zeta\|+2C\sigma\right)\mathcal{N}(\zeta;0,2\sigma^{2}\mathbf{I}_{d})d\zeta\right)</annotation><annotation encoding="application/x-llamapun" id="A4.Ex118.m2.7d">≤ divide start_ARG italic_α end_ARG start_ARG italic_σ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG ( italic_M ( italic_R ) + 2 start_POSTSUPERSCRIPT - divide start_ARG italic_d end_ARG start_ARG 2 end_ARG end_POSTSUPERSCRIPT roman_exp ( - divide start_ARG 1 end_ARG start_ARG 4 italic_σ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG ) ∫ start_POSTSUBSCRIPT ∥ italic_ζ ∥ > 1 end_POSTSUBSCRIPT ( 2 ∥ bold_x start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ∥ + 2 ∥ italic_ζ ∥ + 2 italic_C italic_σ ) caligraphic_N ( italic_ζ ; 0 , 2 italic_σ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT bold_I start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT ) italic_d italic_ζ )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="A4.Ex119"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_eqn_cell"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\leq\frac{\alpha}{\sigma^{2}}\left(M(R)+2^{-\frac{d}{2}}\exp\left% (-\frac{1}{4\sigma^{2}}\right)\left(2R_{\mathbf{K}}+2\sqrt{d}\sigma+2C\sigma% \right)\right)." class="ltx_Math" display="inline" id="A4.Ex119.m1.3"><semantics id="A4.Ex119.m1.3a"><mrow id="A4.Ex119.m1.3.3.1" xref="A4.Ex119.m1.3.3.1.1.cmml"><mrow id="A4.Ex119.m1.3.3.1.1" xref="A4.Ex119.m1.3.3.1.1.cmml"><mi id="A4.Ex119.m1.3.3.1.1.3" xref="A4.Ex119.m1.3.3.1.1.3.cmml"></mi><mo id="A4.Ex119.m1.3.3.1.1.2" xref="A4.Ex119.m1.3.3.1.1.2.cmml">≤</mo><mrow id="A4.Ex119.m1.3.3.1.1.1" xref="A4.Ex119.m1.3.3.1.1.1.cmml"><mstyle displaystyle="true" 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type="integer" xref="A4.Ex119.m1.3.3.1.1.1.1.1.1.2.2.1.1.4.2">2</cn><ci id="A4.Ex119.m1.3.3.1.1.1.1.1.1.2.2.1.1.4.3.cmml" xref="A4.Ex119.m1.3.3.1.1.1.1.1.1.2.2.1.1.4.3">𝐶</ci><ci id="A4.Ex119.m1.3.3.1.1.1.1.1.1.2.2.1.1.4.4.cmml" xref="A4.Ex119.m1.3.3.1.1.1.1.1.1.2.2.1.1.4.4">𝜎</ci></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.Ex119.m1.3c">\displaystyle\leq\frac{\alpha}{\sigma^{2}}\left(M(R)+2^{-\frac{d}{2}}\exp\left% (-\frac{1}{4\sigma^{2}}\right)\left(2R_{\mathbf{K}}+2\sqrt{d}\sigma+2C\sigma% \right)\right).</annotation><annotation encoding="application/x-llamapun" id="A4.Ex119.m1.3d">≤ divide start_ARG italic_α end_ARG start_ARG italic_σ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG ( italic_M ( italic_R ) + 2 start_POSTSUPERSCRIPT - divide start_ARG italic_d end_ARG start_ARG 2 end_ARG end_POSTSUPERSCRIPT roman_exp ( - divide start_ARG 1 end_ARG start_ARG 4 italic_σ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG ) ( 2 italic_R start_POSTSUBSCRIPT bold_K end_POSTSUBSCRIPT + 2 square-root start_ARG italic_d end_ARG italic_σ + 2 italic_C italic_σ ) ) .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="A4.SS1.SSS5.p2.10">This shows that</p> <table class="ltx_equation ltx_eqn_table" id="A4.Ex121"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\eta=\limsup_{k\in{\mathbb{N}}}{\|\eta_{k}\|}\underset{\sigma\to 0}{\leq}\frac% {\alpha}{\sigma^{2}}M(R)+o(\sigma).\\ " class="ltx_Math" display="block" id="A4.Ex121.m1.3"><semantics id="A4.Ex121.m1.3a"><mrow id="A4.Ex121.m1.3.3.1" xref="A4.Ex121.m1.3.3.1.1.cmml"><mrow id="A4.Ex121.m1.3.3.1.1" xref="A4.Ex121.m1.3.3.1.1.cmml"><mi id="A4.Ex121.m1.3.3.1.1.3" xref="A4.Ex121.m1.3.3.1.1.3.cmml">η</mi><mo id="A4.Ex121.m1.3.3.1.1.2" rspace="0.1389em" xref="A4.Ex121.m1.3.3.1.1.2.cmml">=</mo><mrow id="A4.Ex121.m1.3.3.1.1.1" xref="A4.Ex121.m1.3.3.1.1.1.cmml"><mrow id="A4.Ex121.m1.3.3.1.1.1.1" xref="A4.Ex121.m1.3.3.1.1.1.1.cmml"><munder id="A4.Ex121.m1.3.3.1.1.1.1.2" xref="A4.Ex121.m1.3.3.1.1.1.1.2.cmml"><mo id="A4.Ex121.m1.3.3.1.1.1.1.2.2" lspace="0.1389em" movablelimits="false" rspace="0em" xref="A4.Ex121.m1.3.3.1.1.1.1.2.2.cmml">lim sup</mo><mrow id="A4.Ex121.m1.3.3.1.1.1.1.2.3" xref="A4.Ex121.m1.3.3.1.1.1.1.2.3.cmml"><mi id="A4.Ex121.m1.3.3.1.1.1.1.2.3.2" xref="A4.Ex121.m1.3.3.1.1.1.1.2.3.2.cmml">k</mi><mo id="A4.Ex121.m1.3.3.1.1.1.1.2.3.1" xref="A4.Ex121.m1.3.3.1.1.1.1.2.3.1.cmml">∈</mo><mi id="A4.Ex121.m1.3.3.1.1.1.1.2.3.3" xref="A4.Ex121.m1.3.3.1.1.1.1.2.3.3.cmml">ℕ</mi></mrow></munder><mrow id="A4.Ex121.m1.3.3.1.1.1.1.1" xref="A4.Ex121.m1.3.3.1.1.1.1.1.cmml"><mrow id="A4.Ex121.m1.3.3.1.1.1.1.1.1.1" xref="A4.Ex121.m1.3.3.1.1.1.1.1.1.2.cmml"><mo id="A4.Ex121.m1.3.3.1.1.1.1.1.1.1.2" stretchy="false" 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xref="A4.Ex121.m1.3.3.1.1.1.1.1.3.1.1.cmml">→</mo><mn id="A4.Ex121.m1.3.3.1.1.1.1.1.3.1.3" xref="A4.Ex121.m1.3.3.1.1.1.1.1.3.1.3.cmml">0</mn></mrow></munder><mo id="A4.Ex121.m1.3.3.1.1.1.1.1.2a" xref="A4.Ex121.m1.3.3.1.1.1.1.1.2.cmml">⁢</mo><mfrac id="A4.Ex121.m1.3.3.1.1.1.1.1.4" xref="A4.Ex121.m1.3.3.1.1.1.1.1.4.cmml"><mi id="A4.Ex121.m1.3.3.1.1.1.1.1.4.2" xref="A4.Ex121.m1.3.3.1.1.1.1.1.4.2.cmml">α</mi><msup id="A4.Ex121.m1.3.3.1.1.1.1.1.4.3" xref="A4.Ex121.m1.3.3.1.1.1.1.1.4.3.cmml"><mi id="A4.Ex121.m1.3.3.1.1.1.1.1.4.3.2" xref="A4.Ex121.m1.3.3.1.1.1.1.1.4.3.2.cmml">σ</mi><mn id="A4.Ex121.m1.3.3.1.1.1.1.1.4.3.3" xref="A4.Ex121.m1.3.3.1.1.1.1.1.4.3.3.cmml">2</mn></msup></mfrac><mo id="A4.Ex121.m1.3.3.1.1.1.1.1.2b" xref="A4.Ex121.m1.3.3.1.1.1.1.1.2.cmml">⁢</mo><mi id="A4.Ex121.m1.3.3.1.1.1.1.1.5" xref="A4.Ex121.m1.3.3.1.1.1.1.1.5.cmml">M</mi><mo id="A4.Ex121.m1.3.3.1.1.1.1.1.2c" xref="A4.Ex121.m1.3.3.1.1.1.1.1.2.cmml">⁢</mo><mrow id="A4.Ex121.m1.3.3.1.1.1.1.1.6.2" 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id="A4.Ex121.m1.3.3.1.1.1.1.1.4.3.1.cmml" xref="A4.Ex121.m1.3.3.1.1.1.1.1.4.3">superscript</csymbol><ci id="A4.Ex121.m1.3.3.1.1.1.1.1.4.3.2.cmml" xref="A4.Ex121.m1.3.3.1.1.1.1.1.4.3.2">𝜎</ci><cn id="A4.Ex121.m1.3.3.1.1.1.1.1.4.3.3.cmml" type="integer" xref="A4.Ex121.m1.3.3.1.1.1.1.1.4.3.3">2</cn></apply></apply><ci id="A4.Ex121.m1.3.3.1.1.1.1.1.5.cmml" xref="A4.Ex121.m1.3.3.1.1.1.1.1.5">𝑀</ci><ci id="A4.Ex121.m1.1.1.cmml" xref="A4.Ex121.m1.1.1">𝑅</ci></apply></apply><apply id="A4.Ex121.m1.3.3.1.1.1.3.cmml" xref="A4.Ex121.m1.3.3.1.1.1.3"><times id="A4.Ex121.m1.3.3.1.1.1.3.1.cmml" xref="A4.Ex121.m1.3.3.1.1.1.3.1"></times><ci id="A4.Ex121.m1.3.3.1.1.1.3.2.cmml" xref="A4.Ex121.m1.3.3.1.1.1.3.2">𝑜</ci><ci id="A4.Ex121.m1.2.2.cmml" xref="A4.Ex121.m1.2.2">𝜎</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.Ex121.m1.3c">\eta=\limsup_{k\in{\mathbb{N}}}{\|\eta_{k}\|}\underset{\sigma\to 0}{\leq}\frac% {\alpha}{\sigma^{2}}M(R)+o(\sigma).\\ </annotation><annotation encoding="application/x-llamapun" id="A4.Ex121.m1.3d">italic_η = lim sup start_POSTSUBSCRIPT italic_k ∈ blackboard_N end_POSTSUBSCRIPT ∥ italic_η start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ∥ start_UNDERACCENT italic_σ → 0 end_UNDERACCENT start_ARG ≤ end_ARG divide start_ARG italic_α end_ARG start_ARG italic_σ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG italic_M ( italic_R ) + italic_o ( italic_σ ) .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> </div> <div class="ltx_para" id="A4.SS1.SSS5.p3"> <p class="ltx_p" id="A4.SS1.SSS5.p3.1">Then Assumption 2.2. of <cite class="ltx_cite ltx_citemacro_citep">(Tadic & Doucet, <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#bib.bib72" title="">2017</a>)</cite> is verified and <cite class="ltx_cite ltx_citemacro_citep">(Tadic & Doucet, <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#bib.bib72" title="">2017</a>, Theorem 2.1. (ii))</cite> applies.</p> </div> <figure class="ltx_figure" id="A4.F5"><img alt="Refer to caption" class="ltx_graphics ltx_centering ltx_img_square" height="577" id="A4.F5.g1" src="extracted/5518834/images/set_of_results_deblurring.png" width="598"/> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure"><span class="ltx_text" id="A4.F5.6.3.1" style="font-size:90%;">Figure 5</span>: </span><span class="ltx_text" id="A4.F5.4.2" style="font-size:90%;">Results of SNORE, RED and DiffPIR algorithms on deblurring tasks with various blur kernels and various images from the dataset CBSD10. The input noise is <math alttext="\sigma_{{\mathbf{y}}}=10/255" class="ltx_Math" display="inline" id="A4.F5.3.1.m1.1"><semantics id="A4.F5.3.1.m1.1b"><mrow id="A4.F5.3.1.m1.1.1" xref="A4.F5.3.1.m1.1.1.cmml"><msub id="A4.F5.3.1.m1.1.1.2" xref="A4.F5.3.1.m1.1.1.2.cmml"><mi id="A4.F5.3.1.m1.1.1.2.2" xref="A4.F5.3.1.m1.1.1.2.2.cmml">σ</mi><mi id="A4.F5.3.1.m1.1.1.2.3" xref="A4.F5.3.1.m1.1.1.2.3.cmml">𝐲</mi></msub><mo id="A4.F5.3.1.m1.1.1.1" xref="A4.F5.3.1.m1.1.1.1.cmml">=</mo><mrow id="A4.F5.3.1.m1.1.1.3" xref="A4.F5.3.1.m1.1.1.3.cmml"><mn id="A4.F5.3.1.m1.1.1.3.2" xref="A4.F5.3.1.m1.1.1.3.2.cmml">10</mn><mo id="A4.F5.3.1.m1.1.1.3.1" xref="A4.F5.3.1.m1.1.1.3.1.cmml">/</mo><mn id="A4.F5.3.1.m1.1.1.3.3" xref="A4.F5.3.1.m1.1.1.3.3.cmml">255</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="A4.F5.3.1.m1.1c"><apply id="A4.F5.3.1.m1.1.1.cmml" xref="A4.F5.3.1.m1.1.1"><eq id="A4.F5.3.1.m1.1.1.1.cmml" xref="A4.F5.3.1.m1.1.1.1"></eq><apply id="A4.F5.3.1.m1.1.1.2.cmml" xref="A4.F5.3.1.m1.1.1.2"><csymbol cd="ambiguous" id="A4.F5.3.1.m1.1.1.2.1.cmml" xref="A4.F5.3.1.m1.1.1.2">subscript</csymbol><ci id="A4.F5.3.1.m1.1.1.2.2.cmml" xref="A4.F5.3.1.m1.1.1.2.2">𝜎</ci><ci id="A4.F5.3.1.m1.1.1.2.3.cmml" xref="A4.F5.3.1.m1.1.1.2.3">𝐲</ci></apply><apply id="A4.F5.3.1.m1.1.1.3.cmml" xref="A4.F5.3.1.m1.1.1.3"><divide id="A4.F5.3.1.m1.1.1.3.1.cmml" xref="A4.F5.3.1.m1.1.1.3.1"></divide><cn id="A4.F5.3.1.m1.1.1.3.2.cmml" type="integer" xref="A4.F5.3.1.m1.1.1.3.2">10</cn><cn id="A4.F5.3.1.m1.1.1.3.3.cmml" type="integer" xref="A4.F5.3.1.m1.1.1.3.3">255</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.F5.3.1.m1.1d">\sigma_{{\mathbf{y}}}=10/255</annotation><annotation encoding="application/x-llamapun" id="A4.F5.3.1.m1.1e">italic_σ start_POSTSUBSCRIPT bold_y end_POSTSUBSCRIPT = 10 / 255</annotation></semantics></math>. Kernels have various sizes and have been resized for clarity of the figure. The last kernel, uniform of size <math alttext="9\times 9" class="ltx_Math" display="inline" id="A4.F5.4.2.m2.1"><semantics id="A4.F5.4.2.m2.1b"><mrow id="A4.F5.4.2.m2.1.1" xref="A4.F5.4.2.m2.1.1.cmml"><mn id="A4.F5.4.2.m2.1.1.2" xref="A4.F5.4.2.m2.1.1.2.cmml">9</mn><mo id="A4.F5.4.2.m2.1.1.1" lspace="0.222em" rspace="0.222em" xref="A4.F5.4.2.m2.1.1.1.cmml">×</mo><mn id="A4.F5.4.2.m2.1.1.3" xref="A4.F5.4.2.m2.1.1.3.cmml">9</mn></mrow><annotation-xml encoding="MathML-Content" id="A4.F5.4.2.m2.1c"><apply id="A4.F5.4.2.m2.1.1.cmml" xref="A4.F5.4.2.m2.1.1"><times id="A4.F5.4.2.m2.1.1.1.cmml" xref="A4.F5.4.2.m2.1.1.1"></times><cn id="A4.F5.4.2.m2.1.1.2.cmml" type="integer" xref="A4.F5.4.2.m2.1.1.2">9</cn><cn id="A4.F5.4.2.m2.1.1.3.cmml" type="integer" xref="A4.F5.4.2.m2.1.1.3">9</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.F5.4.2.m2.1d">9\times 9</annotation><annotation encoding="application/x-llamapun" id="A4.F5.4.2.m2.1e">9 × 9</annotation></semantics></math> is plotted on a black background for visibility.</span></figcaption> </figure> </section> </section> </section> <section class="ltx_appendix" id="A5"> <h2 class="ltx_title ltx_title_appendix"> <span class="ltx_tag ltx_tag_appendix">Appendix E </span>Discussion on Assumptions</h2> <div class="ltx_para" id="A5.p1"> <p class="ltx_p" id="A5.p1.1">In this section, we explicitly list and comment all the assumptions used for results presented in Section <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#S3" title="3 Convergence Analysis ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_tag">3</span></a>. We detail in which case each assumption is verified, especially in the PnP context. A special discussion on the boundedness assumption is postponed in Appendix <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#A8" title="Appendix H Discussion on the boundedness of (𝐱_𝑘)_{𝑘∈ℕ} ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_tag">H</span></a>.</p> </div> <div class="ltx_para" id="A5.p2"> <ul class="ltx_itemize" id="A5.I1"> <li class="ltx_item" id="A5.I1.i1" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">•</span> <div class="ltx_para" id="A5.I1.i1.p1"> <p class="ltx_p" id="A5.I1.i1.p1.6">Assumption <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#S3.Thmtheorem1" title="Assumption 3.1. ‣ 3.1 Asymptotics of critical points when 𝜎→0 ‣ 3 Convergence Analysis ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_tag">3.1</span></a> <span class="ltx_text ltx_font_bold" id="A5.I1.i1.p1.6.1">(a)</span> implies that the score is non-localized with mass <math alttext="p>0" class="ltx_Math" display="inline" id="A5.I1.i1.p1.1.m1.1"><semantics id="A5.I1.i1.p1.1.m1.1a"><mrow id="A5.I1.i1.p1.1.m1.1.1" xref="A5.I1.i1.p1.1.m1.1.1.cmml"><mi id="A5.I1.i1.p1.1.m1.1.1.2" xref="A5.I1.i1.p1.1.m1.1.1.2.cmml">p</mi><mo id="A5.I1.i1.p1.1.m1.1.1.1" xref="A5.I1.i1.p1.1.m1.1.1.1.cmml">></mo><mn id="A5.I1.i1.p1.1.m1.1.1.3" xref="A5.I1.i1.p1.1.m1.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="A5.I1.i1.p1.1.m1.1b"><apply id="A5.I1.i1.p1.1.m1.1.1.cmml" xref="A5.I1.i1.p1.1.m1.1.1"><gt id="A5.I1.i1.p1.1.m1.1.1.1.cmml" xref="A5.I1.i1.p1.1.m1.1.1.1"></gt><ci id="A5.I1.i1.p1.1.m1.1.1.2.cmml" xref="A5.I1.i1.p1.1.m1.1.1.2">𝑝</ci><cn id="A5.I1.i1.p1.1.m1.1.1.3.cmml" type="integer" xref="A5.I1.i1.p1.1.m1.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A5.I1.i1.p1.1.m1.1c">p>0</annotation><annotation encoding="application/x-llamapun" id="A5.I1.i1.p1.1.m1.1d">italic_p > 0</annotation></semantics></math> everywhere, and the density is bounded and Lipschitz. Moreover the density is <math alttext="C^{1}" class="ltx_Math" display="inline" id="A5.I1.i1.p1.2.m2.1"><semantics id="A5.I1.i1.p1.2.m2.1a"><msup id="A5.I1.i1.p1.2.m2.1.1" xref="A5.I1.i1.p1.2.m2.1.1.cmml"><mi id="A5.I1.i1.p1.2.m2.1.1.2" xref="A5.I1.i1.p1.2.m2.1.1.2.cmml">C</mi><mn id="A5.I1.i1.p1.2.m2.1.1.3" xref="A5.I1.i1.p1.2.m2.1.1.3.cmml">1</mn></msup><annotation-xml encoding="MathML-Content" id="A5.I1.i1.p1.2.m2.1b"><apply id="A5.I1.i1.p1.2.m2.1.1.cmml" xref="A5.I1.i1.p1.2.m2.1.1"><csymbol cd="ambiguous" id="A5.I1.i1.p1.2.m2.1.1.1.cmml" xref="A5.I1.i1.p1.2.m2.1.1">superscript</csymbol><ci id="A5.I1.i1.p1.2.m2.1.1.2.cmml" xref="A5.I1.i1.p1.2.m2.1.1.2">𝐶</ci><cn id="A5.I1.i1.p1.2.m2.1.1.3.cmml" type="integer" xref="A5.I1.i1.p1.2.m2.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A5.I1.i1.p1.2.m2.1c">C^{1}</annotation><annotation encoding="application/x-llamapun" id="A5.I1.i1.p1.2.m2.1d">italic_C start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT</annotation></semantics></math>, which is necessary to define the score <math alttext="\nabla\log p" class="ltx_Math" display="inline" id="A5.I1.i1.p1.3.m3.1"><semantics id="A5.I1.i1.p1.3.m3.1a"><mrow id="A5.I1.i1.p1.3.m3.1.1" xref="A5.I1.i1.p1.3.m3.1.1.cmml"><mrow id="A5.I1.i1.p1.3.m3.1.1.1" xref="A5.I1.i1.p1.3.m3.1.1.1.cmml"><mo id="A5.I1.i1.p1.3.m3.1.1.1.1" rspace="0.167em" xref="A5.I1.i1.p1.3.m3.1.1.1.1.cmml">∇</mo><mi id="A5.I1.i1.p1.3.m3.1.1.1.2" xref="A5.I1.i1.p1.3.m3.1.1.1.2.cmml">log</mi></mrow><mo id="A5.I1.i1.p1.3.m3.1.1a" lspace="0.167em" xref="A5.I1.i1.p1.3.m3.1.1.cmml">⁡</mo><mi id="A5.I1.i1.p1.3.m3.1.1.2" xref="A5.I1.i1.p1.3.m3.1.1.2.cmml">p</mi></mrow><annotation-xml encoding="MathML-Content" id="A5.I1.i1.p1.3.m3.1b"><apply id="A5.I1.i1.p1.3.m3.1.1.cmml" xref="A5.I1.i1.p1.3.m3.1.1"><apply id="A5.I1.i1.p1.3.m3.1.1.1.cmml" xref="A5.I1.i1.p1.3.m3.1.1.1"><ci id="A5.I1.i1.p1.3.m3.1.1.1.1.cmml" xref="A5.I1.i1.p1.3.m3.1.1.1.1">∇</ci><log id="A5.I1.i1.p1.3.m3.1.1.1.2.cmml" xref="A5.I1.i1.p1.3.m3.1.1.1.2"></log></apply><ci id="A5.I1.i1.p1.3.m3.1.1.2.cmml" xref="A5.I1.i1.p1.3.m3.1.1.2">𝑝</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A5.I1.i1.p1.3.m3.1c">\nabla\log p</annotation><annotation encoding="application/x-llamapun" id="A5.I1.i1.p1.3.m3.1d">∇ roman_log italic_p</annotation></semantics></math>. Note that image distributions are typically supported on a compact set, so their score <math alttext="\nabla\log p" class="ltx_Math" display="inline" id="A5.I1.i1.p1.4.m4.1"><semantics id="A5.I1.i1.p1.4.m4.1a"><mrow id="A5.I1.i1.p1.4.m4.1.1" xref="A5.I1.i1.p1.4.m4.1.1.cmml"><mrow id="A5.I1.i1.p1.4.m4.1.1.1" xref="A5.I1.i1.p1.4.m4.1.1.1.cmml"><mo id="A5.I1.i1.p1.4.m4.1.1.1.1" rspace="0.167em" xref="A5.I1.i1.p1.4.m4.1.1.1.1.cmml">∇</mo><mi id="A5.I1.i1.p1.4.m4.1.1.1.2" xref="A5.I1.i1.p1.4.m4.1.1.1.2.cmml">log</mi></mrow><mo id="A5.I1.i1.p1.4.m4.1.1a" lspace="0.167em" xref="A5.I1.i1.p1.4.m4.1.1.cmml">⁡</mo><mi id="A5.I1.i1.p1.4.m4.1.1.2" xref="A5.I1.i1.p1.4.m4.1.1.2.cmml">p</mi></mrow><annotation-xml encoding="MathML-Content" id="A5.I1.i1.p1.4.m4.1b"><apply id="A5.I1.i1.p1.4.m4.1.1.cmml" xref="A5.I1.i1.p1.4.m4.1.1"><apply id="A5.I1.i1.p1.4.m4.1.1.1.cmml" xref="A5.I1.i1.p1.4.m4.1.1.1"><ci id="A5.I1.i1.p1.4.m4.1.1.1.1.cmml" xref="A5.I1.i1.p1.4.m4.1.1.1.1">∇</ci><log id="A5.I1.i1.p1.4.m4.1.1.1.2.cmml" xref="A5.I1.i1.p1.4.m4.1.1.1.2"></log></apply><ci id="A5.I1.i1.p1.4.m4.1.1.2.cmml" xref="A5.I1.i1.p1.4.m4.1.1.2">𝑝</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A5.I1.i1.p1.4.m4.1c">\nabla\log p</annotation><annotation encoding="application/x-llamapun" id="A5.I1.i1.p1.4.m4.1d">∇ roman_log italic_p</annotation></semantics></math> is not defined everywhere. Thus, this hypothesis is not realistic in imaging context but it is necessary to realize an analysis of the score <math alttext="\nabla\log p" class="ltx_Math" display="inline" id="A5.I1.i1.p1.5.m5.1"><semantics id="A5.I1.i1.p1.5.m5.1a"><mrow id="A5.I1.i1.p1.5.m5.1.1" xref="A5.I1.i1.p1.5.m5.1.1.cmml"><mrow id="A5.I1.i1.p1.5.m5.1.1.1" xref="A5.I1.i1.p1.5.m5.1.1.1.cmml"><mo id="A5.I1.i1.p1.5.m5.1.1.1.1" rspace="0.167em" xref="A5.I1.i1.p1.5.m5.1.1.1.1.cmml">∇</mo><mi id="A5.I1.i1.p1.5.m5.1.1.1.2" xref="A5.I1.i1.p1.5.m5.1.1.1.2.cmml">log</mi></mrow><mo id="A5.I1.i1.p1.5.m5.1.1a" lspace="0.167em" xref="A5.I1.i1.p1.5.m5.1.1.cmml">⁡</mo><mi id="A5.I1.i1.p1.5.m5.1.1.2" xref="A5.I1.i1.p1.5.m5.1.1.2.cmml">p</mi></mrow><annotation-xml encoding="MathML-Content" id="A5.I1.i1.p1.5.m5.1b"><apply id="A5.I1.i1.p1.5.m5.1.1.cmml" xref="A5.I1.i1.p1.5.m5.1.1"><apply id="A5.I1.i1.p1.5.m5.1.1.1.cmml" xref="A5.I1.i1.p1.5.m5.1.1.1"><ci id="A5.I1.i1.p1.5.m5.1.1.1.1.cmml" xref="A5.I1.i1.p1.5.m5.1.1.1.1">∇</ci><log id="A5.I1.i1.p1.5.m5.1.1.1.2.cmml" xref="A5.I1.i1.p1.5.m5.1.1.1.2"></log></apply><ci id="A5.I1.i1.p1.5.m5.1.1.2.cmml" xref="A5.I1.i1.p1.5.m5.1.1.2">𝑝</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A5.I1.i1.p1.5.m5.1c">\nabla\log p</annotation><annotation encoding="application/x-llamapun" id="A5.I1.i1.p1.5.m5.1d">∇ roman_log italic_p</annotation></semantics></math>. However, if <math alttext="\log p" class="ltx_Math" display="inline" id="A5.I1.i1.p1.6.m6.1"><semantics id="A5.I1.i1.p1.6.m6.1a"><mrow id="A5.I1.i1.p1.6.m6.1.1" xref="A5.I1.i1.p1.6.m6.1.1.cmml"><mi id="A5.I1.i1.p1.6.m6.1.1.1" xref="A5.I1.i1.p1.6.m6.1.1.1.cmml">log</mi><mo id="A5.I1.i1.p1.6.m6.1.1a" lspace="0.167em" xref="A5.I1.i1.p1.6.m6.1.1.cmml">⁡</mo><mi id="A5.I1.i1.p1.6.m6.1.1.2" xref="A5.I1.i1.p1.6.m6.1.1.2.cmml">p</mi></mrow><annotation-xml encoding="MathML-Content" id="A5.I1.i1.p1.6.m6.1b"><apply id="A5.I1.i1.p1.6.m6.1.1.cmml" xref="A5.I1.i1.p1.6.m6.1.1"><log id="A5.I1.i1.p1.6.m6.1.1.1.cmml" xref="A5.I1.i1.p1.6.m6.1.1.1"></log><ci id="A5.I1.i1.p1.6.m6.1.1.2.cmml" xref="A5.I1.i1.p1.6.m6.1.1.2">𝑝</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A5.I1.i1.p1.6.m6.1c">\log p</annotation><annotation encoding="application/x-llamapun" id="A5.I1.i1.p1.6.m6.1d">roman_log italic_p</annotation></semantics></math> is taken among classical image models (e.g. related to Total Variation or Tychonov regularization), it is likely that Assumption <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#S3.Thmtheorem1" title="Assumption 3.1. ‣ 3.1 Asymptotics of critical points when 𝜎→0 ‣ 3 Convergence Analysis ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_tag">3.1</span></a> <span class="ltx_text ltx_font_bold" id="A5.I1.i1.p1.6.2">(a)</span> is verified. In particular, we conjecture that this hypothesis is true for the PnP regularization with explicit potential proposed in <cite class="ltx_cite ltx_citemacro_citep">(Hurault et al., <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#bib.bib30" title="">2022a</a>)</cite>.</p> </div> </li> <li class="ltx_item" id="A5.I1.i2" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">•</span> <div class="ltx_para" id="A5.I1.i2.p1"> <p class="ltx_p" id="A5.I1.i2.p1.1">Assumption <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#S3.Thmtheorem1" title="Assumption 3.1. ‣ 3.1 Asymptotics of critical points when 𝜎→0 ‣ 3 Convergence Analysis ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_tag">3.1</span></a> <span class="ltx_text ltx_font_bold" id="A5.I1.i2.p1.1.1">(b)</span> is verified for Gaussian, Gaussian Mixture or Cauchy distributions. This is a technical assumption to allow a critical point analysis. This hypothesis is non-restrictive because <math alttext="q" class="ltx_Math" display="inline" id="A5.I1.i2.p1.1.m1.1"><semantics id="A5.I1.i2.p1.1.m1.1a"><mi id="A5.I1.i2.p1.1.m1.1.1" xref="A5.I1.i2.p1.1.m1.1.1.cmml">q</mi><annotation-xml encoding="MathML-Content" id="A5.I1.i2.p1.1.m1.1b"><ci id="A5.I1.i2.p1.1.m1.1.1.cmml" xref="A5.I1.i2.p1.1.m1.1.1">𝑞</ci></annotation-xml><annotation encoding="application/x-tex" id="A5.I1.i2.p1.1.m1.1c">q</annotation><annotation encoding="application/x-llamapun" id="A5.I1.i2.p1.1.m1.1d">italic_q</annotation></semantics></math> can be arbitrarily large.</p> </div> </li> <li class="ltx_item" id="A5.I1.i3" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">•</span> <div class="ltx_para" id="A5.I1.i3.p1"> <p class="ltx_p" id="A5.I1.i3.p1.12">Assumption <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#S3.Thmtheorem2" title="Assumption 3.2. ‣ 3.1 Asymptotics of critical points when 𝜎→0 ‣ 3 Convergence Analysis ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_tag">3.2</span></a> is implied by Assumption <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#S3.Thmtheorem5" title="Assumption 3.5 (Manifold hypothesis). ‣ 3.2 Unbiased algorithm analysis ‣ 3 Convergence Analysis ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_tag">3.5</span></a> with <math alttext="r=1" class="ltx_Math" display="inline" id="A5.I1.i3.p1.1.m1.1"><semantics id="A5.I1.i3.p1.1.m1.1a"><mrow id="A5.I1.i3.p1.1.m1.1.1" xref="A5.I1.i3.p1.1.m1.1.1.cmml"><mi id="A5.I1.i3.p1.1.m1.1.1.2" xref="A5.I1.i3.p1.1.m1.1.1.2.cmml">r</mi><mo id="A5.I1.i3.p1.1.m1.1.1.1" xref="A5.I1.i3.p1.1.m1.1.1.1.cmml">=</mo><mn id="A5.I1.i3.p1.1.m1.1.1.3" xref="A5.I1.i3.p1.1.m1.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="A5.I1.i3.p1.1.m1.1b"><apply id="A5.I1.i3.p1.1.m1.1.1.cmml" xref="A5.I1.i3.p1.1.m1.1.1"><eq id="A5.I1.i3.p1.1.m1.1.1.1.cmml" xref="A5.I1.i3.p1.1.m1.1.1.1"></eq><ci id="A5.I1.i3.p1.1.m1.1.1.2.cmml" xref="A5.I1.i3.p1.1.m1.1.1.2">𝑟</ci><cn id="A5.I1.i3.p1.1.m1.1.1.3.cmml" type="integer" xref="A5.I1.i3.p1.1.m1.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A5.I1.i3.p1.1.m1.1c">r=1</annotation><annotation encoding="application/x-llamapun" id="A5.I1.i3.p1.1.m1.1d">italic_r = 1</annotation></semantics></math> and <math alttext="\beta=-2" class="ltx_Math" display="inline" id="A5.I1.i3.p1.2.m2.1"><semantics id="A5.I1.i3.p1.2.m2.1a"><mrow id="A5.I1.i3.p1.2.m2.1.1" xref="A5.I1.i3.p1.2.m2.1.1.cmml"><mi id="A5.I1.i3.p1.2.m2.1.1.2" xref="A5.I1.i3.p1.2.m2.1.1.2.cmml">β</mi><mo id="A5.I1.i3.p1.2.m2.1.1.1" xref="A5.I1.i3.p1.2.m2.1.1.1.cmml">=</mo><mrow id="A5.I1.i3.p1.2.m2.1.1.3" xref="A5.I1.i3.p1.2.m2.1.1.3.cmml"><mo id="A5.I1.i3.p1.2.m2.1.1.3a" xref="A5.I1.i3.p1.2.m2.1.1.3.cmml">−</mo><mn id="A5.I1.i3.p1.2.m2.1.1.3.2" xref="A5.I1.i3.p1.2.m2.1.1.3.2.cmml">2</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="A5.I1.i3.p1.2.m2.1b"><apply id="A5.I1.i3.p1.2.m2.1.1.cmml" xref="A5.I1.i3.p1.2.m2.1.1"><eq id="A5.I1.i3.p1.2.m2.1.1.1.cmml" xref="A5.I1.i3.p1.2.m2.1.1.1"></eq><ci id="A5.I1.i3.p1.2.m2.1.1.2.cmml" xref="A5.I1.i3.p1.2.m2.1.1.2">𝛽</ci><apply id="A5.I1.i3.p1.2.m2.1.1.3.cmml" xref="A5.I1.i3.p1.2.m2.1.1.3"><minus id="A5.I1.i3.p1.2.m2.1.1.3.1.cmml" xref="A5.I1.i3.p1.2.m2.1.1.3"></minus><cn id="A5.I1.i3.p1.2.m2.1.1.3.2.cmml" type="integer" xref="A5.I1.i3.p1.2.m2.1.1.3.2">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A5.I1.i3.p1.2.m2.1c">\beta=-2</annotation><annotation encoding="application/x-llamapun" id="A5.I1.i3.p1.2.m2.1d">italic_β = - 2</annotation></semantics></math> <cite class="ltx_cite ltx_citemacro_citep">(De Bortoli, <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#bib.bib17" title="">2023</a>)</cite>. We provided a simplified proof of this result in Section <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#A4.SS1.SSS4" title="D.1.4 Proof of Proposition 3.4 ‣ D.1 Proof of critical Point Analysis ‣ Appendix D Proofs of section 3 ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_tag">D.1.4</span></a>. <cite class="ltx_cite ltx_citemacro_citet">De Bortoli et al. (<a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#bib.bib18" title="">2023</a>)</cite> have also proved Assumption <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#S3.Thmtheorem2" title="Assumption 3.2. ‣ 3.1 Asymptotics of critical points when 𝜎→0 ‣ 3 Convergence Analysis ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_tag">3.2</span></a> with <math alttext="r=1" class="ltx_Math" display="inline" id="A5.I1.i3.p1.3.m3.1"><semantics id="A5.I1.i3.p1.3.m3.1a"><mrow id="A5.I1.i3.p1.3.m3.1.1" xref="A5.I1.i3.p1.3.m3.1.1.cmml"><mi id="A5.I1.i3.p1.3.m3.1.1.2" xref="A5.I1.i3.p1.3.m3.1.1.2.cmml">r</mi><mo id="A5.I1.i3.p1.3.m3.1.1.1" xref="A5.I1.i3.p1.3.m3.1.1.1.cmml">=</mo><mn id="A5.I1.i3.p1.3.m3.1.1.3" xref="A5.I1.i3.p1.3.m3.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="A5.I1.i3.p1.3.m3.1b"><apply id="A5.I1.i3.p1.3.m3.1.1.cmml" xref="A5.I1.i3.p1.3.m3.1.1"><eq id="A5.I1.i3.p1.3.m3.1.1.1.cmml" xref="A5.I1.i3.p1.3.m3.1.1.1"></eq><ci id="A5.I1.i3.p1.3.m3.1.1.2.cmml" xref="A5.I1.i3.p1.3.m3.1.1.2">𝑟</ci><cn id="A5.I1.i3.p1.3.m3.1.1.3.cmml" type="integer" xref="A5.I1.i3.p1.3.m3.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A5.I1.i3.p1.3.m3.1c">r=1</annotation><annotation encoding="application/x-llamapun" id="A5.I1.i3.p1.3.m3.1d">italic_r = 1</annotation></semantics></math> and <math alttext="\beta=0" class="ltx_Math" display="inline" id="A5.I1.i3.p1.4.m4.1"><semantics id="A5.I1.i3.p1.4.m4.1a"><mrow id="A5.I1.i3.p1.4.m4.1.1" xref="A5.I1.i3.p1.4.m4.1.1.cmml"><mi id="A5.I1.i3.p1.4.m4.1.1.2" xref="A5.I1.i3.p1.4.m4.1.1.2.cmml">β</mi><mo id="A5.I1.i3.p1.4.m4.1.1.1" xref="A5.I1.i3.p1.4.m4.1.1.1.cmml">=</mo><mn id="A5.I1.i3.p1.4.m4.1.1.3" xref="A5.I1.i3.p1.4.m4.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="A5.I1.i3.p1.4.m4.1b"><apply id="A5.I1.i3.p1.4.m4.1.1.cmml" xref="A5.I1.i3.p1.4.m4.1.1"><eq id="A5.I1.i3.p1.4.m4.1.1.1.cmml" xref="A5.I1.i3.p1.4.m4.1.1.1"></eq><ci id="A5.I1.i3.p1.4.m4.1.1.2.cmml" xref="A5.I1.i3.p1.4.m4.1.1.2">𝛽</ci><cn id="A5.I1.i3.p1.4.m4.1.1.3.cmml" type="integer" xref="A5.I1.i3.p1.4.m4.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A5.I1.i3.p1.4.m4.1c">\beta=0</annotation><annotation encoding="application/x-llamapun" id="A5.I1.i3.p1.4.m4.1d">italic_β = 0</annotation></semantics></math> under the Assumption <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#S3.Thmtheorem1" title="Assumption 3.1. ‣ 3.1 Asymptotics of critical points when 𝜎→0 ‣ 3 Convergence Analysis ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_tag">3.1</span></a><span class="ltx_text ltx_font_bold" id="A5.I1.i3.p1.12.1">(b)</span> with <math alttext="q=1" class="ltx_Math" display="inline" id="A5.I1.i3.p1.5.m5.1"><semantics id="A5.I1.i3.p1.5.m5.1a"><mrow id="A5.I1.i3.p1.5.m5.1.1" xref="A5.I1.i3.p1.5.m5.1.1.cmml"><mi id="A5.I1.i3.p1.5.m5.1.1.2" xref="A5.I1.i3.p1.5.m5.1.1.2.cmml">q</mi><mo id="A5.I1.i3.p1.5.m5.1.1.1" xref="A5.I1.i3.p1.5.m5.1.1.1.cmml">=</mo><mn id="A5.I1.i3.p1.5.m5.1.1.3" xref="A5.I1.i3.p1.5.m5.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="A5.I1.i3.p1.5.m5.1b"><apply id="A5.I1.i3.p1.5.m5.1.1.cmml" xref="A5.I1.i3.p1.5.m5.1.1"><eq id="A5.I1.i3.p1.5.m5.1.1.1.cmml" xref="A5.I1.i3.p1.5.m5.1.1.1"></eq><ci id="A5.I1.i3.p1.5.m5.1.1.2.cmml" xref="A5.I1.i3.p1.5.m5.1.1.2">𝑞</ci><cn id="A5.I1.i3.p1.5.m5.1.1.3.cmml" type="integer" xref="A5.I1.i3.p1.5.m5.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A5.I1.i3.p1.5.m5.1c">q=1</annotation><annotation encoding="application/x-llamapun" id="A5.I1.i3.p1.5.m5.1d">italic_q = 1</annotation></semantics></math> and an extra assumption difficult to verify: there exist <math alttext="m_{0}>0" class="ltx_Math" display="inline" id="A5.I1.i3.p1.6.m6.1"><semantics id="A5.I1.i3.p1.6.m6.1a"><mrow id="A5.I1.i3.p1.6.m6.1.1" xref="A5.I1.i3.p1.6.m6.1.1.cmml"><msub id="A5.I1.i3.p1.6.m6.1.1.2" xref="A5.I1.i3.p1.6.m6.1.1.2.cmml"><mi id="A5.I1.i3.p1.6.m6.1.1.2.2" xref="A5.I1.i3.p1.6.m6.1.1.2.2.cmml">m</mi><mn id="A5.I1.i3.p1.6.m6.1.1.2.3" xref="A5.I1.i3.p1.6.m6.1.1.2.3.cmml">0</mn></msub><mo id="A5.I1.i3.p1.6.m6.1.1.1" xref="A5.I1.i3.p1.6.m6.1.1.1.cmml">></mo><mn id="A5.I1.i3.p1.6.m6.1.1.3" xref="A5.I1.i3.p1.6.m6.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="A5.I1.i3.p1.6.m6.1b"><apply id="A5.I1.i3.p1.6.m6.1.1.cmml" xref="A5.I1.i3.p1.6.m6.1.1"><gt id="A5.I1.i3.p1.6.m6.1.1.1.cmml" xref="A5.I1.i3.p1.6.m6.1.1.1"></gt><apply id="A5.I1.i3.p1.6.m6.1.1.2.cmml" xref="A5.I1.i3.p1.6.m6.1.1.2"><csymbol cd="ambiguous" id="A5.I1.i3.p1.6.m6.1.1.2.1.cmml" xref="A5.I1.i3.p1.6.m6.1.1.2">subscript</csymbol><ci id="A5.I1.i3.p1.6.m6.1.1.2.2.cmml" xref="A5.I1.i3.p1.6.m6.1.1.2.2">𝑚</ci><cn id="A5.I1.i3.p1.6.m6.1.1.2.3.cmml" type="integer" xref="A5.I1.i3.p1.6.m6.1.1.2.3">0</cn></apply><cn id="A5.I1.i3.p1.6.m6.1.1.3.cmml" type="integer" xref="A5.I1.i3.p1.6.m6.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A5.I1.i3.p1.6.m6.1c">m_{0}>0</annotation><annotation encoding="application/x-llamapun" id="A5.I1.i3.p1.6.m6.1d">italic_m start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT > 0</annotation></semantics></math> and <math alttext="d_{0}\geq 0" class="ltx_Math" display="inline" id="A5.I1.i3.p1.7.m7.1"><semantics id="A5.I1.i3.p1.7.m7.1a"><mrow id="A5.I1.i3.p1.7.m7.1.1" xref="A5.I1.i3.p1.7.m7.1.1.cmml"><msub id="A5.I1.i3.p1.7.m7.1.1.2" xref="A5.I1.i3.p1.7.m7.1.1.2.cmml"><mi id="A5.I1.i3.p1.7.m7.1.1.2.2" xref="A5.I1.i3.p1.7.m7.1.1.2.2.cmml">d</mi><mn id="A5.I1.i3.p1.7.m7.1.1.2.3" xref="A5.I1.i3.p1.7.m7.1.1.2.3.cmml">0</mn></msub><mo id="A5.I1.i3.p1.7.m7.1.1.1" xref="A5.I1.i3.p1.7.m7.1.1.1.cmml">≥</mo><mn id="A5.I1.i3.p1.7.m7.1.1.3" xref="A5.I1.i3.p1.7.m7.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="A5.I1.i3.p1.7.m7.1b"><apply id="A5.I1.i3.p1.7.m7.1.1.cmml" xref="A5.I1.i3.p1.7.m7.1.1"><geq id="A5.I1.i3.p1.7.m7.1.1.1.cmml" xref="A5.I1.i3.p1.7.m7.1.1.1"></geq><apply id="A5.I1.i3.p1.7.m7.1.1.2.cmml" xref="A5.I1.i3.p1.7.m7.1.1.2"><csymbol cd="ambiguous" id="A5.I1.i3.p1.7.m7.1.1.2.1.cmml" xref="A5.I1.i3.p1.7.m7.1.1.2">subscript</csymbol><ci id="A5.I1.i3.p1.7.m7.1.1.2.2.cmml" xref="A5.I1.i3.p1.7.m7.1.1.2.2">𝑑</ci><cn id="A5.I1.i3.p1.7.m7.1.1.2.3.cmml" type="integer" xref="A5.I1.i3.p1.7.m7.1.1.2.3">0</cn></apply><cn id="A5.I1.i3.p1.7.m7.1.1.3.cmml" type="integer" xref="A5.I1.i3.p1.7.m7.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A5.I1.i3.p1.7.m7.1c">d_{0}\geq 0</annotation><annotation encoding="application/x-llamapun" id="A5.I1.i3.p1.7.m7.1d">italic_d start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ≥ 0</annotation></semantics></math> such that <math alttext="\forall{\mathbf{x}}\in{\mathbb{R}}^{d}" class="ltx_Math" display="inline" id="A5.I1.i3.p1.8.m8.1"><semantics id="A5.I1.i3.p1.8.m8.1a"><mrow id="A5.I1.i3.p1.8.m8.1.1" xref="A5.I1.i3.p1.8.m8.1.1.cmml"><mrow id="A5.I1.i3.p1.8.m8.1.1.2" xref="A5.I1.i3.p1.8.m8.1.1.2.cmml"><mo id="A5.I1.i3.p1.8.m8.1.1.2.1" rspace="0.167em" xref="A5.I1.i3.p1.8.m8.1.1.2.1.cmml">∀</mo><mi id="A5.I1.i3.p1.8.m8.1.1.2.2" xref="A5.I1.i3.p1.8.m8.1.1.2.2.cmml">𝐱</mi></mrow><mo id="A5.I1.i3.p1.8.m8.1.1.1" xref="A5.I1.i3.p1.8.m8.1.1.1.cmml">∈</mo><msup id="A5.I1.i3.p1.8.m8.1.1.3" xref="A5.I1.i3.p1.8.m8.1.1.3.cmml"><mi id="A5.I1.i3.p1.8.m8.1.1.3.2" xref="A5.I1.i3.p1.8.m8.1.1.3.2.cmml">ℝ</mi><mi id="A5.I1.i3.p1.8.m8.1.1.3.3" xref="A5.I1.i3.p1.8.m8.1.1.3.3.cmml">d</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="A5.I1.i3.p1.8.m8.1b"><apply id="A5.I1.i3.p1.8.m8.1.1.cmml" xref="A5.I1.i3.p1.8.m8.1.1"><in id="A5.I1.i3.p1.8.m8.1.1.1.cmml" xref="A5.I1.i3.p1.8.m8.1.1.1"></in><apply 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id="A5.I1.i3.p1.9.m9.5.5.1.1.1.2.2" xref="A5.I1.i3.p1.9.m9.5.5.1.1.1.2.2.cmml">p</mi></mrow><mo id="A5.I1.i3.p1.9.m9.5.5.1.1.1.1" xref="A5.I1.i3.p1.9.m9.5.5.1.1.1.1.cmml">⁢</mo><mrow id="A5.I1.i3.p1.9.m9.5.5.1.1.1.3.2" xref="A5.I1.i3.p1.9.m9.5.5.1.1.1.cmml"><mo id="A5.I1.i3.p1.9.m9.5.5.1.1.1.3.2.1" stretchy="false" xref="A5.I1.i3.p1.9.m9.5.5.1.1.1.cmml">(</mo><mi id="A5.I1.i3.p1.9.m9.1.1" xref="A5.I1.i3.p1.9.m9.1.1.cmml">𝐱</mi><mo id="A5.I1.i3.p1.9.m9.5.5.1.1.1.3.2.2" stretchy="false" xref="A5.I1.i3.p1.9.m9.5.5.1.1.1.cmml">)</mo></mrow></mrow><mo id="A5.I1.i3.p1.9.m9.5.5.1.1.3" xref="A5.I1.i3.p1.9.m9.5.5.1.2.cmml">,</mo><mi id="A5.I1.i3.p1.9.m9.2.2" xref="A5.I1.i3.p1.9.m9.2.2.cmml">𝐱</mi><mo id="A5.I1.i3.p1.9.m9.5.5.1.1.4" stretchy="false" xref="A5.I1.i3.p1.9.m9.5.5.1.2.cmml">⟩</mo></mrow><mo id="A5.I1.i3.p1.9.m9.5.5.2" xref="A5.I1.i3.p1.9.m9.5.5.2.cmml">≤</mo><mrow id="A5.I1.i3.p1.9.m9.5.5.3" xref="A5.I1.i3.p1.9.m9.5.5.3.cmml"><mrow id="A5.I1.i3.p1.9.m9.5.5.3.2" xref="A5.I1.i3.p1.9.m9.5.5.3.2.cmml"><mo id="A5.I1.i3.p1.9.m9.5.5.3.2a" xref="A5.I1.i3.p1.9.m9.5.5.3.2.cmml">−</mo><mrow id="A5.I1.i3.p1.9.m9.5.5.3.2.2" xref="A5.I1.i3.p1.9.m9.5.5.3.2.2.cmml"><msub id="A5.I1.i3.p1.9.m9.5.5.3.2.2.2" xref="A5.I1.i3.p1.9.m9.5.5.3.2.2.2.cmml"><mi id="A5.I1.i3.p1.9.m9.5.5.3.2.2.2.2" xref="A5.I1.i3.p1.9.m9.5.5.3.2.2.2.2.cmml">m</mi><mn id="A5.I1.i3.p1.9.m9.5.5.3.2.2.2.3" xref="A5.I1.i3.p1.9.m9.5.5.3.2.2.2.3.cmml">0</mn></msub><mo id="A5.I1.i3.p1.9.m9.5.5.3.2.2.1" xref="A5.I1.i3.p1.9.m9.5.5.3.2.2.1.cmml">⁢</mo><msup id="A5.I1.i3.p1.9.m9.5.5.3.2.2.3" xref="A5.I1.i3.p1.9.m9.5.5.3.2.2.3.cmml"><mrow id="A5.I1.i3.p1.9.m9.5.5.3.2.2.3.2.2" xref="A5.I1.i3.p1.9.m9.5.5.3.2.2.3.2.1.cmml"><mo id="A5.I1.i3.p1.9.m9.5.5.3.2.2.3.2.2.1" stretchy="false" xref="A5.I1.i3.p1.9.m9.5.5.3.2.2.3.2.1.1.cmml">‖</mo><mi id="A5.I1.i3.p1.9.m9.3.3" xref="A5.I1.i3.p1.9.m9.3.3.cmml">𝐱</mi><mo id="A5.I1.i3.p1.9.m9.5.5.3.2.2.3.2.2.2" stretchy="false" 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xref="A5.I1.i3.p1.9.m9.5.5.3.3.3.1.1.cmml">‖</mo></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="A5.I1.i3.p1.9.m9.5b"><apply id="A5.I1.i3.p1.9.m9.5.5.cmml" xref="A5.I1.i3.p1.9.m9.5.5"><leq id="A5.I1.i3.p1.9.m9.5.5.2.cmml" xref="A5.I1.i3.p1.9.m9.5.5.2"></leq><list id="A5.I1.i3.p1.9.m9.5.5.1.2.cmml" xref="A5.I1.i3.p1.9.m9.5.5.1.1"><apply id="A5.I1.i3.p1.9.m9.5.5.1.1.1.cmml" xref="A5.I1.i3.p1.9.m9.5.5.1.1.1"><times id="A5.I1.i3.p1.9.m9.5.5.1.1.1.1.cmml" xref="A5.I1.i3.p1.9.m9.5.5.1.1.1.1"></times><apply id="A5.I1.i3.p1.9.m9.5.5.1.1.1.2.cmml" xref="A5.I1.i3.p1.9.m9.5.5.1.1.1.2"><apply id="A5.I1.i3.p1.9.m9.5.5.1.1.1.2.1.cmml" xref="A5.I1.i3.p1.9.m9.5.5.1.1.1.2.1"><ci id="A5.I1.i3.p1.9.m9.5.5.1.1.1.2.1.1.cmml" xref="A5.I1.i3.p1.9.m9.5.5.1.1.1.2.1.1">∇</ci><log id="A5.I1.i3.p1.9.m9.5.5.1.1.1.2.1.2.cmml" xref="A5.I1.i3.p1.9.m9.5.5.1.1.1.2.1.2"></log></apply><ci id="A5.I1.i3.p1.9.m9.5.5.1.1.1.2.2.cmml" xref="A5.I1.i3.p1.9.m9.5.5.1.1.1.2.2">𝑝</ci></apply><ci 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xref="A5.I1.i3.p1.9.m9.5.5.3.3.2">subscript</csymbol><ci id="A5.I1.i3.p1.9.m9.5.5.3.3.2.2.cmml" xref="A5.I1.i3.p1.9.m9.5.5.3.3.2.2">𝑑</ci><cn id="A5.I1.i3.p1.9.m9.5.5.3.3.2.3.cmml" type="integer" xref="A5.I1.i3.p1.9.m9.5.5.3.3.2.3">0</cn></apply><apply id="A5.I1.i3.p1.9.m9.5.5.3.3.3.1.cmml" xref="A5.I1.i3.p1.9.m9.5.5.3.3.3.2"><csymbol cd="latexml" id="A5.I1.i3.p1.9.m9.5.5.3.3.3.1.1.cmml" xref="A5.I1.i3.p1.9.m9.5.5.3.3.3.2.1">norm</csymbol><ci id="A5.I1.i3.p1.9.m9.4.4.cmml" xref="A5.I1.i3.p1.9.m9.4.4">𝐱</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A5.I1.i3.p1.9.m9.5c">\langle\nabla\log p({\mathbf{x}}),{\mathbf{x}}\rangle\leq-m_{0}\|{\mathbf{x}}% \|^{2}+d_{0}\|{\mathbf{x}}\|</annotation><annotation encoding="application/x-llamapun" id="A5.I1.i3.p1.9.m9.5d">⟨ ∇ roman_log italic_p ( bold_x ) , bold_x ⟩ ≤ - italic_m start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ∥ bold_x ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + italic_d start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ∥ bold_x ∥</annotation></semantics></math>. However, to our knowledge, there is no general study of the score approximation <math alttext="\nabla\log p_{\sigma}" class="ltx_Math" display="inline" id="A5.I1.i3.p1.10.m10.1"><semantics id="A5.I1.i3.p1.10.m10.1a"><mrow id="A5.I1.i3.p1.10.m10.1.1" xref="A5.I1.i3.p1.10.m10.1.1.cmml"><mrow id="A5.I1.i3.p1.10.m10.1.1.1" xref="A5.I1.i3.p1.10.m10.1.1.1.cmml"><mo id="A5.I1.i3.p1.10.m10.1.1.1.1" rspace="0.167em" xref="A5.I1.i3.p1.10.m10.1.1.1.1.cmml">∇</mo><mi id="A5.I1.i3.p1.10.m10.1.1.1.2" xref="A5.I1.i3.p1.10.m10.1.1.1.2.cmml">log</mi></mrow><mo id="A5.I1.i3.p1.10.m10.1.1a" lspace="0.167em" xref="A5.I1.i3.p1.10.m10.1.1.cmml">⁡</mo><msub id="A5.I1.i3.p1.10.m10.1.1.2" xref="A5.I1.i3.p1.10.m10.1.1.2.cmml"><mi id="A5.I1.i3.p1.10.m10.1.1.2.2" xref="A5.I1.i3.p1.10.m10.1.1.2.2.cmml">p</mi><mi id="A5.I1.i3.p1.10.m10.1.1.2.3" xref="A5.I1.i3.p1.10.m10.1.1.2.3.cmml">σ</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="A5.I1.i3.p1.10.m10.1b"><apply id="A5.I1.i3.p1.10.m10.1.1.cmml" xref="A5.I1.i3.p1.10.m10.1.1"><apply id="A5.I1.i3.p1.10.m10.1.1.1.cmml" xref="A5.I1.i3.p1.10.m10.1.1.1"><ci id="A5.I1.i3.p1.10.m10.1.1.1.1.cmml" xref="A5.I1.i3.p1.10.m10.1.1.1.1">∇</ci><log id="A5.I1.i3.p1.10.m10.1.1.1.2.cmml" xref="A5.I1.i3.p1.10.m10.1.1.1.2"></log></apply><apply id="A5.I1.i3.p1.10.m10.1.1.2.cmml" xref="A5.I1.i3.p1.10.m10.1.1.2"><csymbol cd="ambiguous" id="A5.I1.i3.p1.10.m10.1.1.2.1.cmml" xref="A5.I1.i3.p1.10.m10.1.1.2">subscript</csymbol><ci id="A5.I1.i3.p1.10.m10.1.1.2.2.cmml" xref="A5.I1.i3.p1.10.m10.1.1.2.2">𝑝</ci><ci id="A5.I1.i3.p1.10.m10.1.1.2.3.cmml" xref="A5.I1.i3.p1.10.m10.1.1.2.3">𝜎</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A5.I1.i3.p1.10.m10.1c">\nabla\log p_{\sigma}</annotation><annotation encoding="application/x-llamapun" id="A5.I1.i3.p1.10.m10.1d">∇ roman_log italic_p start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT</annotation></semantics></math> for a general smooth distribution or a non-bounded and non-smooth distribution. Assumption <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#S3.Thmtheorem2" title="Assumption 3.2. ‣ 3.1 Asymptotics of critical points when 𝜎→0 ‣ 3 Convergence Analysis ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_tag">3.2</span></a> seems non-restrictive because <math alttext="r" class="ltx_Math" display="inline" id="A5.I1.i3.p1.11.m11.1"><semantics id="A5.I1.i3.p1.11.m11.1a"><mi id="A5.I1.i3.p1.11.m11.1.1" xref="A5.I1.i3.p1.11.m11.1.1.cmml">r</mi><annotation-xml encoding="MathML-Content" id="A5.I1.i3.p1.11.m11.1b"><ci id="A5.I1.i3.p1.11.m11.1.1.cmml" xref="A5.I1.i3.p1.11.m11.1.1">𝑟</ci></annotation-xml><annotation encoding="application/x-tex" id="A5.I1.i3.p1.11.m11.1c">r</annotation><annotation encoding="application/x-llamapun" id="A5.I1.i3.p1.11.m11.1d">italic_r</annotation></semantics></math> can be arbitrarily large and <math alttext="\beta" class="ltx_Math" display="inline" id="A5.I1.i3.p1.12.m12.1"><semantics id="A5.I1.i3.p1.12.m12.1a"><mi id="A5.I1.i3.p1.12.m12.1.1" xref="A5.I1.i3.p1.12.m12.1.1.cmml">β</mi><annotation-xml encoding="MathML-Content" id="A5.I1.i3.p1.12.m12.1b"><ci id="A5.I1.i3.p1.12.m12.1.1.cmml" xref="A5.I1.i3.p1.12.m12.1.1">𝛽</ci></annotation-xml><annotation encoding="application/x-tex" id="A5.I1.i3.p1.12.m12.1c">\beta</annotation><annotation encoding="application/x-llamapun" id="A5.I1.i3.p1.12.m12.1d">italic_β</annotation></semantics></math> highly negative.</p> </div> </li> <li class="ltx_item" id="A5.I1.i4" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">•</span> <div class="ltx_para" id="A5.I1.i4.p1"> <p class="ltx_p" id="A5.I1.i4.p1.1">Assumption <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#S3.Thmtheorem3" title="Assumption 3.3. ‣ 3.2 Unbiased algorithm analysis ‣ 3 Convergence Analysis ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_tag">3.3</span></a> is standard in stochastic gradient descent <cite class="ltx_cite ltx_citemacro_citep">(Metivier & Priouret, <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#bib.bib47" title="">1984</a>; Benaïm, <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#bib.bib2" title="">1999</a>; Tadic & Doucet, <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#bib.bib72" title="">2017</a>)</cite>. A stochastic process imposes a decreasing step-size to ensure convergence. If the step-size is constant, the noise (with a non-decreasing variance) that is added at each step makes the process explore even after a large number of steps, so there is no convergence to expect. In practice, we take a constant step-size to converge faster. In fact, we run the algorithm for only a small number of iterations (few hundreds).</p> </div> </li> <li class="ltx_item" id="A5.I1.i5" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">•</span> <div class="ltx_para" id="A5.I1.i5.p1"> <p class="ltx_p" id="A5.I1.i5.p1.1">Assumption <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#S3.Thmtheorem4" title="Assumption 3.4. ‣ 3.2 Unbiased algorithm analysis ‣ 3 Convergence Analysis ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_tag">3.4</span></a> makes the data-fidelity <math alttext="\mathcal{F}" class="ltx_Math" display="inline" id="A5.I1.i5.p1.1.m1.1"><semantics id="A5.I1.i5.p1.1.m1.1a"><mi class="ltx_font_mathcaligraphic" id="A5.I1.i5.p1.1.m1.1.1" xref="A5.I1.i5.p1.1.m1.1.1.cmml">ℱ</mi><annotation-xml encoding="MathML-Content" id="A5.I1.i5.p1.1.m1.1b"><ci id="A5.I1.i5.p1.1.m1.1.1.cmml" xref="A5.I1.i5.p1.1.m1.1.1">ℱ</ci></annotation-xml><annotation encoding="application/x-tex" id="A5.I1.i5.p1.1.m1.1c">\mathcal{F}</annotation><annotation encoding="application/x-llamapun" id="A5.I1.i5.p1.1.m1.1d">caligraphic_F</annotation></semantics></math> smooth. It is verified for every linear inverse problem with a Gaussian noise. Assumption <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#S3.Thmtheorem4" title="Assumption 3.4. ‣ 3.2 Unbiased algorithm analysis ‣ 3 Convergence Analysis ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_tag">3.4</span></a> can also be verified in the case of a non-Gaussian noise, for instance with a Fischer-Tippett noise <cite class="ltx_cite ltx_citemacro_citep">(Deledalle et al., <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#bib.bib19" title="">2017</a>)</cite>. However, some specific data-fidelity terms do not verified this assumption, such as the ones related to salt and pepper noise <cite class="ltx_cite ltx_citemacro_citep">(Chan et al., <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#bib.bib8" title="">2005</a>; Nikolova, <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#bib.bib51" title="">2004</a>)</cite> or Laplacian noise <cite class="ltx_cite ltx_citemacro_citep">(Huang et al., <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#bib.bib29" title="">2017</a>)</cite>, or other data-fidelity used for image segmentation <cite class="ltx_cite ltx_citemacro_citep">(Chan et al., <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#bib.bib10" title="">2006</a>)</cite>.</p> </div> </li> <li class="ltx_item" id="A5.I1.i6" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">•</span> <div class="ltx_para" id="A5.I1.i6.p1"> <p class="ltx_p" id="A5.I1.i6.p1.2">Assumption <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#S3.Thmtheorem5" title="Assumption 3.5 (Manifold hypothesis). ‣ 3.2 Unbiased algorithm analysis ‣ 3 Convergence Analysis ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_tag">3.5</span></a> has been used in the same form as presented by <cite class="ltx_cite ltx_citemacro_citet">De Bortoli (<a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#bib.bib17" title="">2023</a>)</cite>. This assumption is validated by real image distributions, typically encoded within a finite range, such as <math alttext="[0,1]" class="ltx_Math" display="inline" id="A5.I1.i6.p1.1.m1.2"><semantics id="A5.I1.i6.p1.1.m1.2a"><mrow id="A5.I1.i6.p1.1.m1.2.3.2" xref="A5.I1.i6.p1.1.m1.2.3.1.cmml"><mo id="A5.I1.i6.p1.1.m1.2.3.2.1" stretchy="false" xref="A5.I1.i6.p1.1.m1.2.3.1.cmml">[</mo><mn id="A5.I1.i6.p1.1.m1.1.1" xref="A5.I1.i6.p1.1.m1.1.1.cmml">0</mn><mo id="A5.I1.i6.p1.1.m1.2.3.2.2" xref="A5.I1.i6.p1.1.m1.2.3.1.cmml">,</mo><mn id="A5.I1.i6.p1.1.m1.2.2" xref="A5.I1.i6.p1.1.m1.2.2.cmml">1</mn><mo id="A5.I1.i6.p1.1.m1.2.3.2.3" stretchy="false" xref="A5.I1.i6.p1.1.m1.2.3.1.cmml">]</mo></mrow><annotation-xml encoding="MathML-Content" id="A5.I1.i6.p1.1.m1.2b"><interval closure="closed" id="A5.I1.i6.p1.1.m1.2.3.1.cmml" xref="A5.I1.i6.p1.1.m1.2.3.2"><cn id="A5.I1.i6.p1.1.m1.1.1.cmml" type="integer" xref="A5.I1.i6.p1.1.m1.1.1">0</cn><cn id="A5.I1.i6.p1.1.m1.2.2.cmml" type="integer" xref="A5.I1.i6.p1.1.m1.2.2">1</cn></interval></annotation-xml><annotation encoding="application/x-tex" id="A5.I1.i6.p1.1.m1.2c">[0,1]</annotation><annotation encoding="application/x-llamapun" id="A5.I1.i6.p1.1.m1.2d">[ 0 , 1 ]</annotation></semantics></math> or <math alttext="[0,255]" class="ltx_Math" display="inline" id="A5.I1.i6.p1.2.m2.2"><semantics id="A5.I1.i6.p1.2.m2.2a"><mrow id="A5.I1.i6.p1.2.m2.2.3.2" xref="A5.I1.i6.p1.2.m2.2.3.1.cmml"><mo id="A5.I1.i6.p1.2.m2.2.3.2.1" stretchy="false" xref="A5.I1.i6.p1.2.m2.2.3.1.cmml">[</mo><mn id="A5.I1.i6.p1.2.m2.1.1" xref="A5.I1.i6.p1.2.m2.1.1.cmml">0</mn><mo id="A5.I1.i6.p1.2.m2.2.3.2.2" xref="A5.I1.i6.p1.2.m2.2.3.1.cmml">,</mo><mn id="A5.I1.i6.p1.2.m2.2.2" xref="A5.I1.i6.p1.2.m2.2.2.cmml">255</mn><mo id="A5.I1.i6.p1.2.m2.2.3.2.3" stretchy="false" xref="A5.I1.i6.p1.2.m2.2.3.1.cmml">]</mo></mrow><annotation-xml encoding="MathML-Content" id="A5.I1.i6.p1.2.m2.2b"><interval closure="closed" id="A5.I1.i6.p1.2.m2.2.3.1.cmml" xref="A5.I1.i6.p1.2.m2.2.3.2"><cn id="A5.I1.i6.p1.2.m2.1.1.cmml" type="integer" xref="A5.I1.i6.p1.2.m2.1.1">0</cn><cn id="A5.I1.i6.p1.2.m2.2.2.cmml" type="integer" xref="A5.I1.i6.p1.2.m2.2.2">255</cn></interval></annotation-xml><annotation encoding="application/x-tex" id="A5.I1.i6.p1.2.m2.2c">[0,255]</annotation><annotation encoding="application/x-llamapun" id="A5.I1.i6.p1.2.m2.2d">[ 0 , 255 ]</annotation></semantics></math>. Traditionally, the manifold hypothesis asserts that the distribution of images is confined to a low-dimensional manifold. <cite class="ltx_cite ltx_citemacro_citet">Fefferman et al. (<a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#bib.bib22" title="">2013</a>)</cite> conducted tests to evaluate this hypothesis, and more recently, <cite class="ltx_cite ltx_citemacro_citet">Brown et al. (<a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#bib.bib6" title="">2023</a>)</cite> have focused on analyzing image distributions. However, within the scope of our study, we refer to the <span class="ltx_text ltx_font_italic" id="A5.I1.i6.p1.2.1">manifold hypothesis</span> as defined in <cite class="ltx_cite ltx_citemacro_citep">(De Bortoli, <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#bib.bib17" title="">2023</a>)</cite>, suggesting that images are supported within a compact set. This diverges from the conventional manifold hypothesis, constituting a relaxed version of the original assumption. </p> </div> </li> <li class="ltx_item" id="A5.I1.i7" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">•</span> <div class="ltx_para" id="A5.I1.i7.p1"> <p class="ltx_p" id="A5.I1.i7.p1.2">Assumption <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#S3.Thmtheorem6" title="Assumption 3.6. ‣ 3.3 Biased algorithm analysis ‣ 3 Convergence Analysis ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_tag">3.6</span></a> quantifies the uniform distance between the exact and the inexact denoiser. If the activation function of the denoiser is <math alttext="\mathcal{C}^{\infty}" class="ltx_Math" display="inline" id="A5.I1.i7.p1.1.m1.1"><semantics id="A5.I1.i7.p1.1.m1.1a"><msup id="A5.I1.i7.p1.1.m1.1.1" xref="A5.I1.i7.p1.1.m1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="A5.I1.i7.p1.1.m1.1.1.2" xref="A5.I1.i7.p1.1.m1.1.1.2.cmml">𝒞</mi><mi id="A5.I1.i7.p1.1.m1.1.1.3" mathvariant="normal" xref="A5.I1.i7.p1.1.m1.1.1.3.cmml">∞</mi></msup><annotation-xml encoding="MathML-Content" id="A5.I1.i7.p1.1.m1.1b"><apply id="A5.I1.i7.p1.1.m1.1.1.cmml" xref="A5.I1.i7.p1.1.m1.1.1"><csymbol cd="ambiguous" id="A5.I1.i7.p1.1.m1.1.1.1.cmml" xref="A5.I1.i7.p1.1.m1.1.1">superscript</csymbol><ci id="A5.I1.i7.p1.1.m1.1.1.2.cmml" xref="A5.I1.i7.p1.1.m1.1.1.2">𝒞</ci><infinity id="A5.I1.i7.p1.1.m1.1.1.3.cmml" xref="A5.I1.i7.p1.1.m1.1.1.3"></infinity></apply></annotation-xml><annotation encoding="application/x-tex" id="A5.I1.i7.p1.1.m1.1c">\mathcal{C}^{\infty}</annotation><annotation encoding="application/x-llamapun" id="A5.I1.i7.p1.1.m1.1d">caligraphic_C start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT</annotation></semantics></math>, then the denoiser is <math alttext="\mathcal{C}^{\infty}" class="ltx_Math" display="inline" id="A5.I1.i7.p1.2.m2.1"><semantics id="A5.I1.i7.p1.2.m2.1a"><msup id="A5.I1.i7.p1.2.m2.1.1" xref="A5.I1.i7.p1.2.m2.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="A5.I1.i7.p1.2.m2.1.1.2" xref="A5.I1.i7.p1.2.m2.1.1.2.cmml">𝒞</mi><mi id="A5.I1.i7.p1.2.m2.1.1.3" mathvariant="normal" xref="A5.I1.i7.p1.2.m2.1.1.3.cmml">∞</mi></msup><annotation-xml encoding="MathML-Content" id="A5.I1.i7.p1.2.m2.1b"><apply id="A5.I1.i7.p1.2.m2.1.1.cmml" xref="A5.I1.i7.p1.2.m2.1.1"><csymbol cd="ambiguous" id="A5.I1.i7.p1.2.m2.1.1.1.cmml" xref="A5.I1.i7.p1.2.m2.1.1">superscript</csymbol><ci id="A5.I1.i7.p1.2.m2.1.1.2.cmml" xref="A5.I1.i7.p1.2.m2.1.1.2">𝒞</ci><infinity id="A5.I1.i7.p1.2.m2.1.1.3.cmml" xref="A5.I1.i7.p1.2.m2.1.1.3"></infinity></apply></annotation-xml><annotation encoding="application/x-tex" id="A5.I1.i7.p1.2.m2.1c">\mathcal{C}^{\infty}</annotation><annotation encoding="application/x-llamapun" id="A5.I1.i7.p1.2.m2.1d">caligraphic_C start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT</annotation></semantics></math>. <cite class="ltx_cite ltx_citemacro_citet">Laumont et al. (<a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#bib.bib37" title="">2022b</a>)</cite> make a similar assumption. In the literature, other types of assumptions have been made to control the error of the inexact MMSE (see for instance <cite class="ltx_cite ltx_citemacro_citep">(Shoushtari et al., <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#bib.bib62" title="">2023</a>, Assumption 5)</cite>). We choose to take this form of assumption because <cite class="ltx_cite ltx_citemacro_citet">Laumont et al. (<a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#bib.bib36" title="">2022a</a>, Proposition 4)</cite> have proved that Assumption <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#S3.Thmtheorem6" title="Assumption 3.6. ‣ 3.3 Biased algorithm analysis ‣ 3 Convergence Analysis ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_tag">3.6</span></a> can be ensured if the denoiser is learned with the Noise2Noise loss <cite class="ltx_cite ltx_citemacro_citep">(Lehtinen et al., <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#bib.bib38" title="">2018</a>)</cite>.</p> </div> </li> <li class="ltx_item" id="A5.I1.i8" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">•</span> <div class="ltx_para" id="A5.I1.i8.p1"> <p class="ltx_p" id="A5.I1.i8.p1.2">Assumption <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#S3.Thmtheorem7" title="Assumption 3.7. ‣ 3.3 Biased algorithm analysis ‣ 3 Convergence Analysis ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_tag">3.7</span></a> is natural for a well trained denoiser. It means that the denoiser <math alttext="D_{\sigma}" class="ltx_Math" display="inline" id="A5.I1.i8.p1.1.m1.1"><semantics id="A5.I1.i8.p1.1.m1.1a"><msub id="A5.I1.i8.p1.1.m1.1.1" xref="A5.I1.i8.p1.1.m1.1.1.cmml"><mi id="A5.I1.i8.p1.1.m1.1.1.2" xref="A5.I1.i8.p1.1.m1.1.1.2.cmml">D</mi><mi id="A5.I1.i8.p1.1.m1.1.1.3" xref="A5.I1.i8.p1.1.m1.1.1.3.cmml">σ</mi></msub><annotation-xml encoding="MathML-Content" id="A5.I1.i8.p1.1.m1.1b"><apply id="A5.I1.i8.p1.1.m1.1.1.cmml" xref="A5.I1.i8.p1.1.m1.1.1"><csymbol cd="ambiguous" id="A5.I1.i8.p1.1.m1.1.1.1.cmml" xref="A5.I1.i8.p1.1.m1.1.1">subscript</csymbol><ci id="A5.I1.i8.p1.1.m1.1.1.2.cmml" xref="A5.I1.i8.p1.1.m1.1.1.2">𝐷</ci><ci id="A5.I1.i8.p1.1.m1.1.1.3.cmml" xref="A5.I1.i8.p1.1.m1.1.1.3">𝜎</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A5.I1.i8.p1.1.m1.1c">D_{\sigma}</annotation><annotation encoding="application/x-llamapun" id="A5.I1.i8.p1.1.m1.1d">italic_D start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT</annotation></semantics></math> can only modify the image by a quantity bounded by the level of noise <math alttext="\sigma" class="ltx_Math" display="inline" id="A5.I1.i8.p1.2.m2.1"><semantics id="A5.I1.i8.p1.2.m2.1a"><mi id="A5.I1.i8.p1.2.m2.1.1" xref="A5.I1.i8.p1.2.m2.1.1.cmml">σ</mi><annotation-xml encoding="MathML-Content" id="A5.I1.i8.p1.2.m2.1b"><ci id="A5.I1.i8.p1.2.m2.1.1.cmml" xref="A5.I1.i8.p1.2.m2.1.1">𝜎</ci></annotation-xml><annotation encoding="application/x-tex" id="A5.I1.i8.p1.2.m2.1c">\sigma</annotation><annotation encoding="application/x-llamapun" id="A5.I1.i8.p1.2.m2.1d">italic_σ</annotation></semantics></math>. As a practical example, a bounded denoiser <cite class="ltx_cite ltx_citemacro_citep">(Chan et al., <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#bib.bib9" title="">2016</a>, Definition 1)</cite> verifies Assumption <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#S3.Thmtheorem7" title="Assumption 3.7. ‣ 3.3 Biased algorithm analysis ‣ 3 Convergence Analysis ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_tag">3.7</span></a>. However, Assumption <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#S3.Thmtheorem7" title="Assumption 3.7. ‣ 3.3 Biased algorithm analysis ‣ 3 Convergence Analysis ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_tag">3.7</span></a> is difficult to verify in practice (especially outside the training domain). To our knowledge there is no theoretical argument stronger than intuition to support this assumption. However, Assumption <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#S3.Thmtheorem7" title="Assumption 3.7. ‣ 3.3 Biased algorithm analysis ‣ 3 Convergence Analysis ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_tag">3.7</span></a> is necessary to analyze the stability of SNORE with an inexact denoiser, presented in Proposition <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#S3.Thmproposition5" title="Proposition 3.5. ‣ 3.3 Biased algorithm analysis ‣ 3 Convergence Analysis ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_tag">3.5</span></a>.</p> </div> </li> </ul> </div> </section> <section class="ltx_appendix" id="A6"> <h2 class="ltx_title ltx_title_appendix"> <span class="ltx_tag ltx_tag_appendix">Appendix F </span>Additional Experiments</h2> <div class="ltx_para" id="A6.p1"> <p class="ltx_p" id="A6.p1.1">In this section, we provide more details about our experiments. First we present general technical details. Then, we present more experiment for image deblurring including parameters influence and uncertainty of SNORE. Moreover, we give all parameter setting for image deblurring and inpainting. Finally, a preliminary experiment on image super-resolution is shown.</p> </div> <section class="ltx_paragraph" id="A6.SS0.SSS0.Px1"> <h5 class="ltx_title ltx_title_paragraph">Metrics</h5> <div class="ltx_para" id="A6.SS0.SSS0.Px1.p1"> <p class="ltx_p" id="A6.SS0.SSS0.Px1.p1.4">We use four metrics to evaluate our results. Structural SIMilarity (SSIM) and Peak Signal to Noise Ratio (PSNR) are two common distortions metrics. BRISQUE is a no-reference metric based on natural scene statistics that was proved to correlate well with human perception <cite class="ltx_cite ltx_citemacro_citep">(Mittal et al., <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#bib.bib49" title="">2012</a>)</cite>. This metric gives a score between <math alttext="0" class="ltx_Math" display="inline" id="A6.SS0.SSS0.Px1.p1.1.m1.1"><semantics id="A6.SS0.SSS0.Px1.p1.1.m1.1a"><mn id="A6.SS0.SSS0.Px1.p1.1.m1.1.1" xref="A6.SS0.SSS0.Px1.p1.1.m1.1.1.cmml">0</mn><annotation-xml encoding="MathML-Content" id="A6.SS0.SSS0.Px1.p1.1.m1.1b"><cn id="A6.SS0.SSS0.Px1.p1.1.m1.1.1.cmml" type="integer" xref="A6.SS0.SSS0.Px1.p1.1.m1.1.1">0</cn></annotation-xml></semantics></math> (best) and <math alttext="100" class="ltx_Math" display="inline" id="A6.SS0.SSS0.Px1.p1.2.m2.1"><semantics id="A6.SS0.SSS0.Px1.p1.2.m2.1a"><mn id="A6.SS0.SSS0.Px1.p1.2.m2.1.1" xref="A6.SS0.SSS0.Px1.p1.2.m2.1.1.cmml">100</mn><annotation-xml encoding="MathML-Content" id="A6.SS0.SSS0.Px1.p1.2.m2.1b"><cn id="A6.SS0.SSS0.Px1.p1.2.m2.1.1.cmml" type="integer" xref="A6.SS0.SSS0.Px1.p1.2.m2.1.1">100</cn></annotation-xml><annotation encoding="application/x-tex" id="A6.SS0.SSS0.Px1.p1.2.m2.1c">100</annotation><annotation encoding="application/x-llamapun" id="A6.SS0.SSS0.Px1.p1.2.m2.1d">100</annotation></semantics></math> (worst). We use the Python library “brisque”, with which we sometimes observe some incoherence with some outlier outputs (smaller than <math alttext="0" class="ltx_Math" display="inline" id="A6.SS0.SSS0.Px1.p1.3.m3.1"><semantics id="A6.SS0.SSS0.Px1.p1.3.m3.1a"><mn id="A6.SS0.SSS0.Px1.p1.3.m3.1.1" xref="A6.SS0.SSS0.Px1.p1.3.m3.1.1.cmml">0</mn><annotation-xml encoding="MathML-Content" id="A6.SS0.SSS0.Px1.p1.3.m3.1b"><cn id="A6.SS0.SSS0.Px1.p1.3.m3.1.1.cmml" type="integer" xref="A6.SS0.SSS0.Px1.p1.3.m3.1.1">0</cn></annotation-xml></semantics></math> or larger than <math alttext="100" class="ltx_Math" display="inline" id="A6.SS0.SSS0.Px1.p1.4.m4.1"><semantics id="A6.SS0.SSS0.Px1.p1.4.m4.1a"><mn id="A6.SS0.SSS0.Px1.p1.4.m4.1.1" xref="A6.SS0.SSS0.Px1.p1.4.m4.1.1.cmml">100</mn><annotation-xml encoding="MathML-Content" id="A6.SS0.SSS0.Px1.p1.4.m4.1b"><cn id="A6.SS0.SSS0.Px1.p1.4.m4.1.1.cmml" type="integer" xref="A6.SS0.SSS0.Px1.p1.4.m4.1.1">100</cn></annotation-xml><annotation encoding="application/x-tex" id="A6.SS0.SSS0.Px1.p1.4.m4.1c">100</annotation><annotation encoding="application/x-llamapun" id="A6.SS0.SSS0.Px1.p1.4.m4.1d">100</annotation></semantics></math>). These outputs where rare so we kept this standard implementation for reproducibility purpose. LPIPS <cite class="ltx_cite ltx_citemacro_citep">(Zhang et al., <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#bib.bib77" title="">2017</a>)</cite> is another perceptual metric that compares the original image and the reconstructed one by measuring their differences in terms of deep features. As suggested in <cite class="ltx_cite ltx_citemacro_citep">(Ren et al., <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#bib.bib56" title="">2023</a>)</cite>, looking at such perceptual metrics is relevant as they are correlated with human perception. We use the Python library ”lpips” to compute this metric.</p> </div> </section> <section class="ltx_paragraph" id="A6.SS0.SSS0.Px2"> <h5 class="ltx_title ltx_title_paragraph">Denoiser</h5> <div class="ltx_para" id="A6.SS0.SSS0.Px2.p1"> <p class="ltx_p" id="A6.SS0.SSS0.Px2.p1.1">We use the denoiser proposed by <cite class="ltx_cite ltx_citemacro_citep">(Hurault et al., <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#bib.bib30" title="">2022a</a>)</cite> based on the DRUNet <cite class="ltx_cite ltx_citemacro_citep">(Zhang et al., <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#bib.bib78" title="">2021</a>)</cite> trained on a dataset of natural images composed of Berkeley segmentation dataset (CBSD) <cite class="ltx_cite ltx_citemacro_citep">(Martin et al., <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#bib.bib46" title="">2001</a>)</cite>, Waterloo Exploration dataset <cite class="ltx_cite ltx_citemacro_citep">(Ma et al., <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#bib.bib43" title="">2017</a>)</cite>, DIV2K dataset <cite class="ltx_cite ltx_citemacro_citep">(Agustsson & Timofte, <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#bib.bib1" title="">2017</a>)</cite> and Flick2K <cite class="ltx_cite ltx_citemacro_citep">(Lim et al., <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#bib.bib40" title="">2017</a>)</cite>. We take the training weights of <cite class="ltx_cite ltx_citemacro_citep">(Hurault et al., <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#bib.bib30" title="">2022a</a>)</cite>. In order to better analyze the advantages and drawbacks of other PnP and RED approaches, this denoiser is used for all our experiments and comparisons.</p> </div> </section> <section class="ltx_paragraph" id="A6.SS0.SSS0.Px3"> <h5 class="ltx_title ltx_title_paragraph">RED Prox</h5> <div class="ltx_para" id="A6.SS0.SSS0.Px3.p1"> <p class="ltx_p" id="A6.SS0.SSS0.Px3.p1.1">We name <span class="ltx_text ltx_font_italic" id="A6.SS0.SSS0.Px3.p1.1.1">RED Prox</span> (Algorithm <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#alg4" title="Algorithm 4 ‣ RED Prox ‣ Appendix F Additional Experiments ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_tag">4</span></a>) the splitting algorithm with a gradient descent step on the regularization and a proximal step on the data-fidelity. Just like RED (Algorithm <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#alg1" title="Algorithm 1 ‣ 2.1 Background ‣ 2 Stochastic deNOising REgularization (SNORE) ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_tag">1</span></a>), RED Prox minimizes Problem (<a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#S1.E1" title="1 ‣ 1 Introduction ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_tag">1</span></a>) with the regularization <math alttext="\mathcal{P}_{\sigma}" class="ltx_Math" display="inline" id="A6.SS0.SSS0.Px3.p1.1.m1.1"><semantics id="A6.SS0.SSS0.Px3.p1.1.m1.1a"><msub id="A6.SS0.SSS0.Px3.p1.1.m1.1.1" xref="A6.SS0.SSS0.Px3.p1.1.m1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="A6.SS0.SSS0.Px3.p1.1.m1.1.1.2" xref="A6.SS0.SSS0.Px3.p1.1.m1.1.1.2.cmml">𝒫</mi><mi id="A6.SS0.SSS0.Px3.p1.1.m1.1.1.3" xref="A6.SS0.SSS0.Px3.p1.1.m1.1.1.3.cmml">σ</mi></msub><annotation-xml encoding="MathML-Content" id="A6.SS0.SSS0.Px3.p1.1.m1.1b"><apply id="A6.SS0.SSS0.Px3.p1.1.m1.1.1.cmml" xref="A6.SS0.SSS0.Px3.p1.1.m1.1.1"><csymbol cd="ambiguous" id="A6.SS0.SSS0.Px3.p1.1.m1.1.1.1.cmml" xref="A6.SS0.SSS0.Px3.p1.1.m1.1.1">subscript</csymbol><ci id="A6.SS0.SSS0.Px3.p1.1.m1.1.1.2.cmml" xref="A6.SS0.SSS0.Px3.p1.1.m1.1.1.2">𝒫</ci><ci id="A6.SS0.SSS0.Px3.p1.1.m1.1.1.3.cmml" xref="A6.SS0.SSS0.Px3.p1.1.m1.1.1.3">𝜎</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A6.SS0.SSS0.Px3.p1.1.m1.1c">\mathcal{P}_{\sigma}</annotation><annotation encoding="application/x-llamapun" id="A6.SS0.SSS0.Px3.p1.1.m1.1d">caligraphic_P start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT</annotation></semantics></math>, defined in Equation (<a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#S2.E2" title="2 ‣ 2.1 Background ‣ 2 Stochastic deNOising REgularization (SNORE) ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_tag">2</span></a>). That kind of splitting algorithm is commonly used in the PnP field <cite class="ltx_cite ltx_citemacro_citep">(Ryu et al., <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#bib.bib60" title="">2019</a>)</cite>. As proposed in <cite class="ltx_cite ltx_citemacro_citep">(Hurault et al., <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#bib.bib30" title="">2022a</a>)</cite>, thanks to the special form of gradient-step denoiser, we implement a backtracking procedure for RED and RED Prox.</p> </div> <figure class="ltx_float ltx_float_algorithm ltx_framed_top" id="alg4"> <figcaption class="ltx_caption"><span class="ltx_tag ltx_tag_float"><span class="ltx_text ltx_font_bold" id="alg4.2.1.1">Algorithm 4</span> </span> RED Prox</figcaption> <div class="ltx_listing ltx_listing" id="alg4.3"> <div class="ltx_listingline" id="alg4.l1"> <span class="ltx_tag ltx_tag_listingline"><span class="ltx_text" id="alg4.l1.1.1.1" style="font-size:80%;">1:</span></span> <span class="ltx_text ltx_font_bold" id="alg4.l1.2">input:</span> <math alttext="{\mathbf{x}}_{0}\in{\mathbb{R}}^{d}" class="ltx_Math" display="inline" id="alg4.l1.m1.1"><semantics id="alg4.l1.m1.1a"><mrow id="alg4.l1.m1.1.1" xref="alg4.l1.m1.1.1.cmml"><msub id="alg4.l1.m1.1.1.2" xref="alg4.l1.m1.1.1.2.cmml"><mi id="alg4.l1.m1.1.1.2.2" xref="alg4.l1.m1.1.1.2.2.cmml">𝐱</mi><mn id="alg4.l1.m1.1.1.2.3" xref="alg4.l1.m1.1.1.2.3.cmml">0</mn></msub><mo id="alg4.l1.m1.1.1.1" xref="alg4.l1.m1.1.1.1.cmml">∈</mo><msup id="alg4.l1.m1.1.1.3" xref="alg4.l1.m1.1.1.3.cmml"><mi id="alg4.l1.m1.1.1.3.2" xref="alg4.l1.m1.1.1.3.2.cmml">ℝ</mi><mi id="alg4.l1.m1.1.1.3.3" xref="alg4.l1.m1.1.1.3.3.cmml">d</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="alg4.l1.m1.1b"><apply id="alg4.l1.m1.1.1.cmml" xref="alg4.l1.m1.1.1"><in id="alg4.l1.m1.1.1.1.cmml" xref="alg4.l1.m1.1.1.1"></in><apply id="alg4.l1.m1.1.1.2.cmml" xref="alg4.l1.m1.1.1.2"><csymbol cd="ambiguous" id="alg4.l1.m1.1.1.2.1.cmml" xref="alg4.l1.m1.1.1.2">subscript</csymbol><ci id="alg4.l1.m1.1.1.2.2.cmml" xref="alg4.l1.m1.1.1.2.2">𝐱</ci><cn id="alg4.l1.m1.1.1.2.3.cmml" type="integer" xref="alg4.l1.m1.1.1.2.3">0</cn></apply><apply id="alg4.l1.m1.1.1.3.cmml" xref="alg4.l1.m1.1.1.3"><csymbol cd="ambiguous" id="alg4.l1.m1.1.1.3.1.cmml" xref="alg4.l1.m1.1.1.3">superscript</csymbol><ci id="alg4.l1.m1.1.1.3.2.cmml" xref="alg4.l1.m1.1.1.3.2">ℝ</ci><ci id="alg4.l1.m1.1.1.3.3.cmml" xref="alg4.l1.m1.1.1.3.3">𝑑</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="alg4.l1.m1.1c">{\mathbf{x}}_{0}\in{\mathbb{R}}^{d}</annotation><annotation encoding="application/x-llamapun" id="alg4.l1.m1.1d">bold_x start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ∈ blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT</annotation></semantics></math>, <math alttext="\sigma>0" class="ltx_Math" display="inline" id="alg4.l1.m2.1"><semantics id="alg4.l1.m2.1a"><mrow id="alg4.l1.m2.1.1" xref="alg4.l1.m2.1.1.cmml"><mi id="alg4.l1.m2.1.1.2" xref="alg4.l1.m2.1.1.2.cmml">σ</mi><mo id="alg4.l1.m2.1.1.1" xref="alg4.l1.m2.1.1.1.cmml">></mo><mn id="alg4.l1.m2.1.1.3" xref="alg4.l1.m2.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="alg4.l1.m2.1b"><apply id="alg4.l1.m2.1.1.cmml" xref="alg4.l1.m2.1.1"><gt id="alg4.l1.m2.1.1.1.cmml" xref="alg4.l1.m2.1.1.1"></gt><ci id="alg4.l1.m2.1.1.2.cmml" xref="alg4.l1.m2.1.1.2">𝜎</ci><cn id="alg4.l1.m2.1.1.3.cmml" type="integer" xref="alg4.l1.m2.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="alg4.l1.m2.1c">\sigma>0</annotation><annotation encoding="application/x-llamapun" id="alg4.l1.m2.1d">italic_σ > 0</annotation></semantics></math>, <math alttext="\alpha>0" class="ltx_Math" display="inline" id="alg4.l1.m3.1"><semantics id="alg4.l1.m3.1a"><mrow id="alg4.l1.m3.1.1" xref="alg4.l1.m3.1.1.cmml"><mi id="alg4.l1.m3.1.1.2" xref="alg4.l1.m3.1.1.2.cmml">α</mi><mo id="alg4.l1.m3.1.1.1" xref="alg4.l1.m3.1.1.1.cmml">></mo><mn id="alg4.l1.m3.1.1.3" xref="alg4.l1.m3.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="alg4.l1.m3.1b"><apply id="alg4.l1.m3.1.1.cmml" xref="alg4.l1.m3.1.1"><gt id="alg4.l1.m3.1.1.1.cmml" xref="alg4.l1.m3.1.1.1"></gt><ci id="alg4.l1.m3.1.1.2.cmml" xref="alg4.l1.m3.1.1.2">𝛼</ci><cn id="alg4.l1.m3.1.1.3.cmml" type="integer" xref="alg4.l1.m3.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="alg4.l1.m3.1c">\alpha>0</annotation><annotation encoding="application/x-llamapun" id="alg4.l1.m3.1d">italic_α > 0</annotation></semantics></math>, <math alttext="\delta>0" class="ltx_Math" display="inline" id="alg4.l1.m4.1"><semantics id="alg4.l1.m4.1a"><mrow id="alg4.l1.m4.1.1" xref="alg4.l1.m4.1.1.cmml"><mi id="alg4.l1.m4.1.1.2" xref="alg4.l1.m4.1.1.2.cmml">δ</mi><mo id="alg4.l1.m4.1.1.1" xref="alg4.l1.m4.1.1.1.cmml">></mo><mn id="alg4.l1.m4.1.1.3" xref="alg4.l1.m4.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="alg4.l1.m4.1b"><apply id="alg4.l1.m4.1.1.cmml" xref="alg4.l1.m4.1.1"><gt id="alg4.l1.m4.1.1.1.cmml" xref="alg4.l1.m4.1.1.1"></gt><ci id="alg4.l1.m4.1.1.2.cmml" xref="alg4.l1.m4.1.1.2">𝛿</ci><cn id="alg4.l1.m4.1.1.3.cmml" type="integer" xref="alg4.l1.m4.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="alg4.l1.m4.1c">\delta>0</annotation><annotation encoding="application/x-llamapun" id="alg4.l1.m4.1d">italic_δ > 0</annotation></semantics></math>, <math alttext="N\in{\mathbb{N}}" class="ltx_Math" display="inline" id="alg4.l1.m5.1"><semantics id="alg4.l1.m5.1a"><mrow id="alg4.l1.m5.1.1" xref="alg4.l1.m5.1.1.cmml"><mi id="alg4.l1.m5.1.1.2" xref="alg4.l1.m5.1.1.2.cmml">N</mi><mo id="alg4.l1.m5.1.1.1" xref="alg4.l1.m5.1.1.1.cmml">∈</mo><mi id="alg4.l1.m5.1.1.3" xref="alg4.l1.m5.1.1.3.cmml">ℕ</mi></mrow><annotation-xml encoding="MathML-Content" id="alg4.l1.m5.1b"><apply id="alg4.l1.m5.1.1.cmml" xref="alg4.l1.m5.1.1"><in id="alg4.l1.m5.1.1.1.cmml" xref="alg4.l1.m5.1.1.1"></in><ci id="alg4.l1.m5.1.1.2.cmml" xref="alg4.l1.m5.1.1.2">𝑁</ci><ci id="alg4.l1.m5.1.1.3.cmml" xref="alg4.l1.m5.1.1.3">ℕ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="alg4.l1.m5.1c">N\in{\mathbb{N}}</annotation><annotation encoding="application/x-llamapun" id="alg4.l1.m5.1d">italic_N ∈ blackboard_N</annotation></semantics></math> </div> <div class="ltx_listingline" id="alg4.l2"> <span class="ltx_tag ltx_tag_listingline"><span class="ltx_text" id="alg4.l2.1.1.1" style="font-size:80%;">2:</span></span> <span class="ltx_text ltx_font_bold" id="alg4.l2.2">for</span> <math alttext="k=0,1,\dots,N-1" class="ltx_Math" display="inline" id="alg4.l2.m1.4"><semantics id="alg4.l2.m1.4a"><mrow id="alg4.l2.m1.4.4" xref="alg4.l2.m1.4.4.cmml"><mi id="alg4.l2.m1.4.4.3" xref="alg4.l2.m1.4.4.3.cmml">k</mi><mo id="alg4.l2.m1.4.4.2" xref="alg4.l2.m1.4.4.2.cmml">=</mo><mrow id="alg4.l2.m1.4.4.1.1" xref="alg4.l2.m1.4.4.1.2.cmml"><mn id="alg4.l2.m1.1.1" xref="alg4.l2.m1.1.1.cmml">0</mn><mo id="alg4.l2.m1.4.4.1.1.2" xref="alg4.l2.m1.4.4.1.2.cmml">,</mo><mn id="alg4.l2.m1.2.2" xref="alg4.l2.m1.2.2.cmml">1</mn><mo id="alg4.l2.m1.4.4.1.1.3" xref="alg4.l2.m1.4.4.1.2.cmml">,</mo><mi id="alg4.l2.m1.3.3" mathvariant="normal" xref="alg4.l2.m1.3.3.cmml">…</mi><mo id="alg4.l2.m1.4.4.1.1.4" xref="alg4.l2.m1.4.4.1.2.cmml">,</mo><mrow id="alg4.l2.m1.4.4.1.1.1" xref="alg4.l2.m1.4.4.1.1.1.cmml"><mi id="alg4.l2.m1.4.4.1.1.1.2" xref="alg4.l2.m1.4.4.1.1.1.2.cmml">N</mi><mo id="alg4.l2.m1.4.4.1.1.1.1" xref="alg4.l2.m1.4.4.1.1.1.1.cmml">−</mo><mn id="alg4.l2.m1.4.4.1.1.1.3" xref="alg4.l2.m1.4.4.1.1.1.3.cmml">1</mn></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="alg4.l2.m1.4b"><apply id="alg4.l2.m1.4.4.cmml" xref="alg4.l2.m1.4.4"><eq id="alg4.l2.m1.4.4.2.cmml" xref="alg4.l2.m1.4.4.2"></eq><ci id="alg4.l2.m1.4.4.3.cmml" xref="alg4.l2.m1.4.4.3">𝑘</ci><list id="alg4.l2.m1.4.4.1.2.cmml" xref="alg4.l2.m1.4.4.1.1"><cn id="alg4.l2.m1.1.1.cmml" type="integer" xref="alg4.l2.m1.1.1">0</cn><cn id="alg4.l2.m1.2.2.cmml" type="integer" xref="alg4.l2.m1.2.2">1</cn><ci id="alg4.l2.m1.3.3.cmml" xref="alg4.l2.m1.3.3">…</ci><apply id="alg4.l2.m1.4.4.1.1.1.cmml" xref="alg4.l2.m1.4.4.1.1.1"><minus id="alg4.l2.m1.4.4.1.1.1.1.cmml" xref="alg4.l2.m1.4.4.1.1.1.1"></minus><ci id="alg4.l2.m1.4.4.1.1.1.2.cmml" xref="alg4.l2.m1.4.4.1.1.1.2">𝑁</ci><cn id="alg4.l2.m1.4.4.1.1.1.3.cmml" type="integer" xref="alg4.l2.m1.4.4.1.1.1.3">1</cn></apply></list></apply></annotation-xml><annotation encoding="application/x-tex" id="alg4.l2.m1.4c">k=0,1,\dots,N-1</annotation><annotation encoding="application/x-llamapun" id="alg4.l2.m1.4d">italic_k = 0 , 1 , … , italic_N - 1</annotation></semantics></math> <span class="ltx_text ltx_font_bold" id="alg4.l2.3">do</span> </div> <div class="ltx_listingline" id="alg4.l3"> <span class="ltx_tag ltx_tag_listingline"><span class="ltx_text" id="alg4.l3.1.1.1" style="font-size:80%;">3:</span></span> <math alttext="{\mathbf{z}}_{k}\leftarrow{\mathbf{x}}_{k}-\frac{\alpha\delta}{\sigma^{2}}% \left({\mathbf{x}}_{k}-D_{\sigma}({\mathbf{x}}_{k})\right)" class="ltx_Math" display="inline" id="alg4.l3.m1.1"><semantics id="alg4.l3.m1.1a"><mrow id="alg4.l3.m1.1.1" xref="alg4.l3.m1.1.1.cmml"><msub id="alg4.l3.m1.1.1.3" xref="alg4.l3.m1.1.1.3.cmml"><mi id="alg4.l3.m1.1.1.3.2" xref="alg4.l3.m1.1.1.3.2.cmml">𝐳</mi><mi id="alg4.l3.m1.1.1.3.3" xref="alg4.l3.m1.1.1.3.3.cmml">k</mi></msub><mo id="alg4.l3.m1.1.1.2" stretchy="false" xref="alg4.l3.m1.1.1.2.cmml">←</mo><mrow id="alg4.l3.m1.1.1.1" xref="alg4.l3.m1.1.1.1.cmml"><msub id="alg4.l3.m1.1.1.1.3" xref="alg4.l3.m1.1.1.1.3.cmml"><mi id="alg4.l3.m1.1.1.1.3.2" xref="alg4.l3.m1.1.1.1.3.2.cmml">𝐱</mi><mi id="alg4.l3.m1.1.1.1.3.3" xref="alg4.l3.m1.1.1.1.3.3.cmml">k</mi></msub><mo id="alg4.l3.m1.1.1.1.2" xref="alg4.l3.m1.1.1.1.2.cmml">−</mo><mrow id="alg4.l3.m1.1.1.1.1" xref="alg4.l3.m1.1.1.1.1.cmml"><mfrac id="alg4.l3.m1.1.1.1.1.3" xref="alg4.l3.m1.1.1.1.1.3.cmml"><mrow id="alg4.l3.m1.1.1.1.1.3.2" xref="alg4.l3.m1.1.1.1.1.3.2.cmml"><mi id="alg4.l3.m1.1.1.1.1.3.2.2" xref="alg4.l3.m1.1.1.1.1.3.2.2.cmml">α</mi><mo id="alg4.l3.m1.1.1.1.1.3.2.1" xref="alg4.l3.m1.1.1.1.1.3.2.1.cmml">⁢</mo><mi id="alg4.l3.m1.1.1.1.1.3.2.3" xref="alg4.l3.m1.1.1.1.1.3.2.3.cmml">δ</mi></mrow><msup id="alg4.l3.m1.1.1.1.1.3.3" xref="alg4.l3.m1.1.1.1.1.3.3.cmml"><mi id="alg4.l3.m1.1.1.1.1.3.3.2" xref="alg4.l3.m1.1.1.1.1.3.3.2.cmml">σ</mi><mn id="alg4.l3.m1.1.1.1.1.3.3.3" xref="alg4.l3.m1.1.1.1.1.3.3.3.cmml">2</mn></msup></mfrac><mo id="alg4.l3.m1.1.1.1.1.2" xref="alg4.l3.m1.1.1.1.1.2.cmml">⁢</mo><mrow id="alg4.l3.m1.1.1.1.1.1.1" xref="alg4.l3.m1.1.1.1.1.1.1.1.cmml"><mo id="alg4.l3.m1.1.1.1.1.1.1.2" xref="alg4.l3.m1.1.1.1.1.1.1.1.cmml">(</mo><mrow id="alg4.l3.m1.1.1.1.1.1.1.1" xref="alg4.l3.m1.1.1.1.1.1.1.1.cmml"><msub id="alg4.l3.m1.1.1.1.1.1.1.1.3" xref="alg4.l3.m1.1.1.1.1.1.1.1.3.cmml"><mi id="alg4.l3.m1.1.1.1.1.1.1.1.3.2" xref="alg4.l3.m1.1.1.1.1.1.1.1.3.2.cmml">𝐱</mi><mi id="alg4.l3.m1.1.1.1.1.1.1.1.3.3" xref="alg4.l3.m1.1.1.1.1.1.1.1.3.3.cmml">k</mi></msub><mo id="alg4.l3.m1.1.1.1.1.1.1.1.2" xref="alg4.l3.m1.1.1.1.1.1.1.1.2.cmml">−</mo><mrow id="alg4.l3.m1.1.1.1.1.1.1.1.1" xref="alg4.l3.m1.1.1.1.1.1.1.1.1.cmml"><msub id="alg4.l3.m1.1.1.1.1.1.1.1.1.3" xref="alg4.l3.m1.1.1.1.1.1.1.1.1.3.cmml"><mi id="alg4.l3.m1.1.1.1.1.1.1.1.1.3.2" xref="alg4.l3.m1.1.1.1.1.1.1.1.1.3.2.cmml">D</mi><mi id="alg4.l3.m1.1.1.1.1.1.1.1.1.3.3" xref="alg4.l3.m1.1.1.1.1.1.1.1.1.3.3.cmml">σ</mi></msub><mo id="alg4.l3.m1.1.1.1.1.1.1.1.1.2" xref="alg4.l3.m1.1.1.1.1.1.1.1.1.2.cmml">⁢</mo><mrow id="alg4.l3.m1.1.1.1.1.1.1.1.1.1.1" xref="alg4.l3.m1.1.1.1.1.1.1.1.1.1.1.1.cmml"><mo id="alg4.l3.m1.1.1.1.1.1.1.1.1.1.1.2" stretchy="false" xref="alg4.l3.m1.1.1.1.1.1.1.1.1.1.1.1.cmml">(</mo><msub id="alg4.l3.m1.1.1.1.1.1.1.1.1.1.1.1" xref="alg4.l3.m1.1.1.1.1.1.1.1.1.1.1.1.cmml"><mi id="alg4.l3.m1.1.1.1.1.1.1.1.1.1.1.1.2" xref="alg4.l3.m1.1.1.1.1.1.1.1.1.1.1.1.2.cmml">𝐱</mi><mi id="alg4.l3.m1.1.1.1.1.1.1.1.1.1.1.1.3" xref="alg4.l3.m1.1.1.1.1.1.1.1.1.1.1.1.3.cmml">k</mi></msub><mo id="alg4.l3.m1.1.1.1.1.1.1.1.1.1.1.3" stretchy="false" xref="alg4.l3.m1.1.1.1.1.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><mo id="alg4.l3.m1.1.1.1.1.1.1.3" xref="alg4.l3.m1.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="alg4.l3.m1.1b"><apply id="alg4.l3.m1.1.1.cmml" xref="alg4.l3.m1.1.1"><ci id="alg4.l3.m1.1.1.2.cmml" xref="alg4.l3.m1.1.1.2">←</ci><apply id="alg4.l3.m1.1.1.3.cmml" xref="alg4.l3.m1.1.1.3"><csymbol cd="ambiguous" id="alg4.l3.m1.1.1.3.1.cmml" xref="alg4.l3.m1.1.1.3">subscript</csymbol><ci id="alg4.l3.m1.1.1.3.2.cmml" xref="alg4.l3.m1.1.1.3.2">𝐳</ci><ci id="alg4.l3.m1.1.1.3.3.cmml" xref="alg4.l3.m1.1.1.3.3">𝑘</ci></apply><apply id="alg4.l3.m1.1.1.1.cmml" xref="alg4.l3.m1.1.1.1"><minus id="alg4.l3.m1.1.1.1.2.cmml" xref="alg4.l3.m1.1.1.1.2"></minus><apply id="alg4.l3.m1.1.1.1.3.cmml" xref="alg4.l3.m1.1.1.1.3"><csymbol cd="ambiguous" id="alg4.l3.m1.1.1.1.3.1.cmml" xref="alg4.l3.m1.1.1.1.3">subscript</csymbol><ci 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id="alg4.l3.m1.1.1.1.1.3.3.3.cmml" type="integer" xref="alg4.l3.m1.1.1.1.1.3.3.3">2</cn></apply></apply><apply id="alg4.l3.m1.1.1.1.1.1.1.1.cmml" xref="alg4.l3.m1.1.1.1.1.1.1"><minus id="alg4.l3.m1.1.1.1.1.1.1.1.2.cmml" xref="alg4.l3.m1.1.1.1.1.1.1.1.2"></minus><apply id="alg4.l3.m1.1.1.1.1.1.1.1.3.cmml" xref="alg4.l3.m1.1.1.1.1.1.1.1.3"><csymbol cd="ambiguous" id="alg4.l3.m1.1.1.1.1.1.1.1.3.1.cmml" xref="alg4.l3.m1.1.1.1.1.1.1.1.3">subscript</csymbol><ci id="alg4.l3.m1.1.1.1.1.1.1.1.3.2.cmml" xref="alg4.l3.m1.1.1.1.1.1.1.1.3.2">𝐱</ci><ci id="alg4.l3.m1.1.1.1.1.1.1.1.3.3.cmml" xref="alg4.l3.m1.1.1.1.1.1.1.1.3.3">𝑘</ci></apply><apply id="alg4.l3.m1.1.1.1.1.1.1.1.1.cmml" xref="alg4.l3.m1.1.1.1.1.1.1.1.1"><times id="alg4.l3.m1.1.1.1.1.1.1.1.1.2.cmml" xref="alg4.l3.m1.1.1.1.1.1.1.1.1.2"></times><apply id="alg4.l3.m1.1.1.1.1.1.1.1.1.3.cmml" xref="alg4.l3.m1.1.1.1.1.1.1.1.1.3"><csymbol cd="ambiguous" id="alg4.l3.m1.1.1.1.1.1.1.1.1.3.1.cmml" xref="alg4.l3.m1.1.1.1.1.1.1.1.1.3">subscript</csymbol><ci id="alg4.l3.m1.1.1.1.1.1.1.1.1.3.2.cmml" xref="alg4.l3.m1.1.1.1.1.1.1.1.1.3.2">𝐷</ci><ci id="alg4.l3.m1.1.1.1.1.1.1.1.1.3.3.cmml" xref="alg4.l3.m1.1.1.1.1.1.1.1.1.3.3">𝜎</ci></apply><apply id="alg4.l3.m1.1.1.1.1.1.1.1.1.1.1.1.cmml" xref="alg4.l3.m1.1.1.1.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="alg4.l3.m1.1.1.1.1.1.1.1.1.1.1.1.1.cmml" xref="alg4.l3.m1.1.1.1.1.1.1.1.1.1.1">subscript</csymbol><ci id="alg4.l3.m1.1.1.1.1.1.1.1.1.1.1.1.2.cmml" xref="alg4.l3.m1.1.1.1.1.1.1.1.1.1.1.1.2">𝐱</ci><ci id="alg4.l3.m1.1.1.1.1.1.1.1.1.1.1.1.3.cmml" xref="alg4.l3.m1.1.1.1.1.1.1.1.1.1.1.1.3">𝑘</ci></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="alg4.l3.m1.1c">{\mathbf{z}}_{k}\leftarrow{\mathbf{x}}_{k}-\frac{\alpha\delta}{\sigma^{2}}% \left({\mathbf{x}}_{k}-D_{\sigma}({\mathbf{x}}_{k})\right)</annotation><annotation encoding="application/x-llamapun" id="alg4.l3.m1.1d">bold_z start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ← bold_x start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT - divide start_ARG italic_α italic_δ end_ARG start_ARG italic_σ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG ( bold_x start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT - italic_D start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ( bold_x start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) )</annotation></semantics></math> </div> <div class="ltx_listingline" id="alg4.l4"> <span class="ltx_tag ltx_tag_listingline"><span class="ltx_text" id="alg4.l4.1.1.1" style="font-size:80%;">4:</span></span> <math alttext="{\mathbf{x}}_{k+1}\leftarrow\text{Prox}_{\delta\mathcal{F}}({\mathbf{z}}_{k})" class="ltx_Math" display="inline" id="alg4.l4.m1.1"><semantics id="alg4.l4.m1.1a"><mrow id="alg4.l4.m1.1.1" xref="alg4.l4.m1.1.1.cmml"><msub id="alg4.l4.m1.1.1.3" xref="alg4.l4.m1.1.1.3.cmml"><mi id="alg4.l4.m1.1.1.3.2" xref="alg4.l4.m1.1.1.3.2.cmml">𝐱</mi><mrow id="alg4.l4.m1.1.1.3.3" xref="alg4.l4.m1.1.1.3.3.cmml"><mi id="alg4.l4.m1.1.1.3.3.2" xref="alg4.l4.m1.1.1.3.3.2.cmml">k</mi><mo id="alg4.l4.m1.1.1.3.3.1" xref="alg4.l4.m1.1.1.3.3.1.cmml">+</mo><mn id="alg4.l4.m1.1.1.3.3.3" xref="alg4.l4.m1.1.1.3.3.3.cmml">1</mn></mrow></msub><mo id="alg4.l4.m1.1.1.2" stretchy="false" xref="alg4.l4.m1.1.1.2.cmml">←</mo><mrow id="alg4.l4.m1.1.1.1" xref="alg4.l4.m1.1.1.1.cmml"><msub id="alg4.l4.m1.1.1.1.3" xref="alg4.l4.m1.1.1.1.3.cmml"><mtext id="alg4.l4.m1.1.1.1.3.2" xref="alg4.l4.m1.1.1.1.3.2a.cmml">Prox</mtext><mrow id="alg4.l4.m1.1.1.1.3.3" xref="alg4.l4.m1.1.1.1.3.3.cmml"><mi id="alg4.l4.m1.1.1.1.3.3.2" xref="alg4.l4.m1.1.1.1.3.3.2.cmml">δ</mi><mo id="alg4.l4.m1.1.1.1.3.3.1" xref="alg4.l4.m1.1.1.1.3.3.1.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="alg4.l4.m1.1.1.1.3.3.3" xref="alg4.l4.m1.1.1.1.3.3.3.cmml">ℱ</mi></mrow></msub><mo id="alg4.l4.m1.1.1.1.2" xref="alg4.l4.m1.1.1.1.2.cmml">⁢</mo><mrow id="alg4.l4.m1.1.1.1.1.1" xref="alg4.l4.m1.1.1.1.1.1.1.cmml"><mo id="alg4.l4.m1.1.1.1.1.1.2" stretchy="false" xref="alg4.l4.m1.1.1.1.1.1.1.cmml">(</mo><msub id="alg4.l4.m1.1.1.1.1.1.1" xref="alg4.l4.m1.1.1.1.1.1.1.cmml"><mi id="alg4.l4.m1.1.1.1.1.1.1.2" xref="alg4.l4.m1.1.1.1.1.1.1.2.cmml">𝐳</mi><mi id="alg4.l4.m1.1.1.1.1.1.1.3" xref="alg4.l4.m1.1.1.1.1.1.1.3.cmml">k</mi></msub><mo id="alg4.l4.m1.1.1.1.1.1.3" stretchy="false" xref="alg4.l4.m1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="alg4.l4.m1.1b"><apply id="alg4.l4.m1.1.1.cmml" xref="alg4.l4.m1.1.1"><ci id="alg4.l4.m1.1.1.2.cmml" xref="alg4.l4.m1.1.1.2">←</ci><apply id="alg4.l4.m1.1.1.3.cmml" xref="alg4.l4.m1.1.1.3"><csymbol cd="ambiguous" id="alg4.l4.m1.1.1.3.1.cmml" xref="alg4.l4.m1.1.1.3">subscript</csymbol><ci id="alg4.l4.m1.1.1.3.2.cmml" xref="alg4.l4.m1.1.1.3.2">𝐱</ci><apply id="alg4.l4.m1.1.1.3.3.cmml" xref="alg4.l4.m1.1.1.3.3"><plus id="alg4.l4.m1.1.1.3.3.1.cmml" xref="alg4.l4.m1.1.1.3.3.1"></plus><ci id="alg4.l4.m1.1.1.3.3.2.cmml" xref="alg4.l4.m1.1.1.3.3.2">𝑘</ci><cn id="alg4.l4.m1.1.1.3.3.3.cmml" type="integer" xref="alg4.l4.m1.1.1.3.3.3">1</cn></apply></apply><apply id="alg4.l4.m1.1.1.1.cmml" xref="alg4.l4.m1.1.1.1"><times id="alg4.l4.m1.1.1.1.2.cmml" xref="alg4.l4.m1.1.1.1.2"></times><apply id="alg4.l4.m1.1.1.1.3.cmml" xref="alg4.l4.m1.1.1.1.3"><csymbol cd="ambiguous" id="alg4.l4.m1.1.1.1.3.1.cmml" xref="alg4.l4.m1.1.1.1.3">subscript</csymbol><ci id="alg4.l4.m1.1.1.1.3.2a.cmml" xref="alg4.l4.m1.1.1.1.3.2"><mtext id="alg4.l4.m1.1.1.1.3.2.cmml" xref="alg4.l4.m1.1.1.1.3.2">Prox</mtext></ci><apply id="alg4.l4.m1.1.1.1.3.3.cmml" xref="alg4.l4.m1.1.1.1.3.3"><times id="alg4.l4.m1.1.1.1.3.3.1.cmml" xref="alg4.l4.m1.1.1.1.3.3.1"></times><ci id="alg4.l4.m1.1.1.1.3.3.2.cmml" xref="alg4.l4.m1.1.1.1.3.3.2">𝛿</ci><ci id="alg4.l4.m1.1.1.1.3.3.3.cmml" xref="alg4.l4.m1.1.1.1.3.3.3">ℱ</ci></apply></apply><apply id="alg4.l4.m1.1.1.1.1.1.1.cmml" xref="alg4.l4.m1.1.1.1.1.1"><csymbol cd="ambiguous" id="alg4.l4.m1.1.1.1.1.1.1.1.cmml" xref="alg4.l4.m1.1.1.1.1.1">subscript</csymbol><ci id="alg4.l4.m1.1.1.1.1.1.1.2.cmml" xref="alg4.l4.m1.1.1.1.1.1.1.2">𝐳</ci><ci id="alg4.l4.m1.1.1.1.1.1.1.3.cmml" xref="alg4.l4.m1.1.1.1.1.1.1.3">𝑘</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="alg4.l4.m1.1c">{\mathbf{x}}_{k+1}\leftarrow\text{Prox}_{\delta\mathcal{F}}({\mathbf{z}}_{k})</annotation><annotation encoding="application/x-llamapun" id="alg4.l4.m1.1d">bold_x start_POSTSUBSCRIPT italic_k + 1 end_POSTSUBSCRIPT ← Prox start_POSTSUBSCRIPT italic_δ caligraphic_F end_POSTSUBSCRIPT ( bold_z start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT )</annotation></semantics></math> </div> <div class="ltx_listingline" id="alg4.l5"> <span class="ltx_tag ltx_tag_listingline"><span class="ltx_text" id="alg4.l5.1.1.1" style="font-size:80%;">5:</span></span> <span class="ltx_text ltx_font_bold" id="alg4.l5.2">end</span> <span class="ltx_text ltx_font_bold" id="alg4.l5.3">for</span> </div> </div> </figure> <figure class="ltx_figure" id="A6.F6"> <div class="ltx_flex_figure"> <div class="ltx_flex_cell ltx_flex_size_2"> <figure class="ltx_figure ltx_flex_size_2 ltx_align_center" id="A6.F6.1"><img alt="Refer to caption" class="ltx_graphics ltx_img_portrait" height="957" id="A6.F6.1.g1" src="extracted/5518834/images/Results_restoration_annexe.png" width="598"/> </figure> </div> <div class="ltx_flex_cell ltx_flex_size_2"> <figure class="ltx_figure ltx_flex_size_2 ltx_align_center" id="A6.F6.2"><img alt="Refer to caption" class="ltx_graphics ltx_img_portrait" height="957" id="A6.F6.2.g1" src="extracted/5518834/images/Results_restoration_annexe_2.png" width="598"/> </figure> </div> </div> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure"><span class="ltx_text" id="A6.F6.30.14.1" style="font-size:90%;">Figure 6</span>: </span><span class="ltx_text" id="A6.F6.28.13" style="font-size:90%;">Image deblurring with an input noise level <math alttext="\sigma_{{\mathbf{y}}}=5/255" class="ltx_Math" display="inline" id="A6.F6.16.1.m1.1"><semantics id="A6.F6.16.1.m1.1b"><mrow id="A6.F6.16.1.m1.1.1" xref="A6.F6.16.1.m1.1.1.cmml"><msub id="A6.F6.16.1.m1.1.1.2" xref="A6.F6.16.1.m1.1.1.2.cmml"><mi id="A6.F6.16.1.m1.1.1.2.2" xref="A6.F6.16.1.m1.1.1.2.2.cmml">σ</mi><mi id="A6.F6.16.1.m1.1.1.2.3" xref="A6.F6.16.1.m1.1.1.2.3.cmml">𝐲</mi></msub><mo id="A6.F6.16.1.m1.1.1.1" xref="A6.F6.16.1.m1.1.1.1.cmml">=</mo><mrow id="A6.F6.16.1.m1.1.1.3" xref="A6.F6.16.1.m1.1.1.3.cmml"><mn id="A6.F6.16.1.m1.1.1.3.2" xref="A6.F6.16.1.m1.1.1.3.2.cmml">5</mn><mo id="A6.F6.16.1.m1.1.1.3.1" xref="A6.F6.16.1.m1.1.1.3.1.cmml">/</mo><mn id="A6.F6.16.1.m1.1.1.3.3" xref="A6.F6.16.1.m1.1.1.3.3.cmml">255</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="A6.F6.16.1.m1.1c"><apply id="A6.F6.16.1.m1.1.1.cmml" xref="A6.F6.16.1.m1.1.1"><eq id="A6.F6.16.1.m1.1.1.1.cmml" xref="A6.F6.16.1.m1.1.1.1"></eq><apply 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display="inline" id="A6.F6.17.2.m2.1"><semantics id="A6.F6.17.2.m2.1b"><msub id="A6.F6.17.2.m2.1.1" xref="A6.F6.17.2.m2.1.1.cmml"><mi id="A6.F6.17.2.m2.1.1.2" xref="A6.F6.17.2.m2.1.1.2.cmml">α</mi><mi id="A6.F6.17.2.m2.1.1.3" xref="A6.F6.17.2.m2.1.1.3.cmml">k</mi></msub><annotation-xml encoding="MathML-Content" id="A6.F6.17.2.m2.1c"><apply id="A6.F6.17.2.m2.1.1.cmml" xref="A6.F6.17.2.m2.1.1"><csymbol cd="ambiguous" id="A6.F6.17.2.m2.1.1.1.cmml" xref="A6.F6.17.2.m2.1.1">subscript</csymbol><ci id="A6.F6.17.2.m2.1.1.2.cmml" xref="A6.F6.17.2.m2.1.1.2">𝛼</ci><ci id="A6.F6.17.2.m2.1.1.3.cmml" xref="A6.F6.17.2.m2.1.1.3">𝑘</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A6.F6.17.2.m2.1d">\alpha_{k}</annotation><annotation encoding="application/x-llamapun" id="A6.F6.17.2.m2.1e">italic_α start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="\sigma_{k}" class="ltx_Math" display="inline" id="A6.F6.18.3.m3.1"><semantics id="A6.F6.18.3.m3.1b"><msub id="A6.F6.18.3.m3.1.1" xref="A6.F6.18.3.m3.1.1.cmml"><mi id="A6.F6.18.3.m3.1.1.2" xref="A6.F6.18.3.m3.1.1.2.cmml">σ</mi><mi id="A6.F6.18.3.m3.1.1.3" xref="A6.F6.18.3.m3.1.1.3.cmml">k</mi></msub><annotation-xml encoding="MathML-Content" id="A6.F6.18.3.m3.1c"><apply id="A6.F6.18.3.m3.1.1.cmml" xref="A6.F6.18.3.m3.1.1"><csymbol cd="ambiguous" id="A6.F6.18.3.m3.1.1.1.cmml" xref="A6.F6.18.3.m3.1.1">subscript</csymbol><ci id="A6.F6.18.3.m3.1.1.2.cmml" xref="A6.F6.18.3.m3.1.1.2">𝜎</ci><ci id="A6.F6.18.3.m3.1.1.3.cmml" xref="A6.F6.18.3.m3.1.1.3">𝑘</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A6.F6.18.3.m3.1d">\sigma_{k}</annotation><annotation encoding="application/x-llamapun" id="A6.F6.18.3.m3.1e">italic_σ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT</annotation></semantics></math> and the potential <math alttext="\mathcal{F}+\alpha_{k}\mathcal{R}_{\sigma_{k}}" class="ltx_Math" display="inline" id="A6.F6.19.4.m4.1"><semantics id="A6.F6.19.4.m4.1b"><mrow id="A6.F6.19.4.m4.1.1" xref="A6.F6.19.4.m4.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="A6.F6.19.4.m4.1.1.2" xref="A6.F6.19.4.m4.1.1.2.cmml">ℱ</mi><mo id="A6.F6.19.4.m4.1.1.1" xref="A6.F6.19.4.m4.1.1.1.cmml">+</mo><mrow id="A6.F6.19.4.m4.1.1.3" xref="A6.F6.19.4.m4.1.1.3.cmml"><msub id="A6.F6.19.4.m4.1.1.3.2" xref="A6.F6.19.4.m4.1.1.3.2.cmml"><mi id="A6.F6.19.4.m4.1.1.3.2.2" xref="A6.F6.19.4.m4.1.1.3.2.2.cmml">α</mi><mi id="A6.F6.19.4.m4.1.1.3.2.3" xref="A6.F6.19.4.m4.1.1.3.2.3.cmml">k</mi></msub><mo id="A6.F6.19.4.m4.1.1.3.1" xref="A6.F6.19.4.m4.1.1.3.1.cmml">⁢</mo><msub id="A6.F6.19.4.m4.1.1.3.3" xref="A6.F6.19.4.m4.1.1.3.3.cmml"><mi class="ltx_font_mathcaligraphic" id="A6.F6.19.4.m4.1.1.3.3.2" xref="A6.F6.19.4.m4.1.1.3.3.2.cmml">ℛ</mi><msub id="A6.F6.19.4.m4.1.1.3.3.3" xref="A6.F6.19.4.m4.1.1.3.3.3.cmml"><mi id="A6.F6.19.4.m4.1.1.3.3.3.2" xref="A6.F6.19.4.m4.1.1.3.3.3.2.cmml">σ</mi><mi id="A6.F6.19.4.m4.1.1.3.3.3.3" xref="A6.F6.19.4.m4.1.1.3.3.3.3.cmml">k</mi></msub></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="A6.F6.19.4.m4.1c"><apply id="A6.F6.19.4.m4.1.1.cmml" xref="A6.F6.19.4.m4.1.1"><plus id="A6.F6.19.4.m4.1.1.1.cmml" xref="A6.F6.19.4.m4.1.1.1"></plus><ci id="A6.F6.19.4.m4.1.1.2.cmml" xref="A6.F6.19.4.m4.1.1.2">ℱ</ci><apply id="A6.F6.19.4.m4.1.1.3.cmml" xref="A6.F6.19.4.m4.1.1.3"><times id="A6.F6.19.4.m4.1.1.3.1.cmml" xref="A6.F6.19.4.m4.1.1.3.1"></times><apply id="A6.F6.19.4.m4.1.1.3.2.cmml" xref="A6.F6.19.4.m4.1.1.3.2"><csymbol cd="ambiguous" id="A6.F6.19.4.m4.1.1.3.2.1.cmml" xref="A6.F6.19.4.m4.1.1.3.2">subscript</csymbol><ci id="A6.F6.19.4.m4.1.1.3.2.2.cmml" xref="A6.F6.19.4.m4.1.1.3.2.2">𝛼</ci><ci id="A6.F6.19.4.m4.1.1.3.2.3.cmml" xref="A6.F6.19.4.m4.1.1.3.2.3">𝑘</ci></apply><apply id="A6.F6.19.4.m4.1.1.3.3.cmml" xref="A6.F6.19.4.m4.1.1.3.3"><csymbol cd="ambiguous" id="A6.F6.19.4.m4.1.1.3.3.1.cmml" xref="A6.F6.19.4.m4.1.1.3.3">subscript</csymbol><ci id="A6.F6.19.4.m4.1.1.3.3.2.cmml" xref="A6.F6.19.4.m4.1.1.3.3.2">ℛ</ci><apply id="A6.F6.19.4.m4.1.1.3.3.3.cmml" xref="A6.F6.19.4.m4.1.1.3.3.3"><csymbol cd="ambiguous" id="A6.F6.19.4.m4.1.1.3.3.3.1.cmml" xref="A6.F6.19.4.m4.1.1.3.3.3">subscript</csymbol><ci id="A6.F6.19.4.m4.1.1.3.3.3.2.cmml" xref="A6.F6.19.4.m4.1.1.3.3.3.2">𝜎</ci><ci id="A6.F6.19.4.m4.1.1.3.3.3.3.cmml" xref="A6.F6.19.4.m4.1.1.3.3.3.3">𝑘</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A6.F6.19.4.m4.1d">\mathcal{F}+\alpha_{k}\mathcal{R}_{\sigma_{k}}</annotation><annotation encoding="application/x-llamapun" id="A6.F6.19.4.m4.1e">caligraphic_F + italic_α start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT caligraphic_R start_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math> which is optimized by Annealed SNORE algorithm (Algorithm <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#alg3" title="Algorithm 3 ‣ Optimization algorithms ‣ 2.2 SNORE regularization ‣ 2 Stochastic deNOising REgularization (SNORE) ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_tag">3</span></a>). There are <math alttext="16" class="ltx_Math" display="inline" id="A6.F6.20.5.m5.1"><semantics id="A6.F6.20.5.m5.1b"><mn id="A6.F6.20.5.m5.1.1" xref="A6.F6.20.5.m5.1.1.cmml">16</mn><annotation-xml encoding="MathML-Content" id="A6.F6.20.5.m5.1c"><cn id="A6.F6.20.5.m5.1.1.cmml" type="integer" xref="A6.F6.20.5.m5.1.1">16</cn></annotation-xml><annotation encoding="application/x-tex" id="A6.F6.20.5.m5.1d">16</annotation><annotation encoding="application/x-llamapun" id="A6.F6.20.5.m5.1e">16</annotation></semantics></math> annealing levels in the first <math alttext="1200" class="ltx_Math" display="inline" id="A6.F6.21.6.m6.1"><semantics id="A6.F6.21.6.m6.1b"><mn id="A6.F6.21.6.m6.1.1" xref="A6.F6.21.6.m6.1.1.cmml">1200</mn><annotation-xml encoding="MathML-Content" id="A6.F6.21.6.m6.1c"><cn id="A6.F6.21.6.m6.1.1.cmml" type="integer" xref="A6.F6.21.6.m6.1.1">1200</cn></annotation-xml><annotation encoding="application/x-tex" id="A6.F6.21.6.m6.1d">1200</annotation><annotation encoding="application/x-llamapun" id="A6.F6.21.6.m6.1e">1200</annotation></semantics></math> iterations and <math alttext="300" class="ltx_Math" display="inline" id="A6.F6.22.7.m7.1"><semantics id="A6.F6.22.7.m7.1b"><mn id="A6.F6.22.7.m7.1.1" xref="A6.F6.22.7.m7.1.1.cmml">300</mn><annotation-xml encoding="MathML-Content" id="A6.F6.22.7.m7.1c"><cn id="A6.F6.22.7.m7.1.1.cmml" type="integer" xref="A6.F6.22.7.m7.1.1">300</cn></annotation-xml><annotation encoding="application/x-tex" id="A6.F6.22.7.m7.1d">300</annotation><annotation encoding="application/x-llamapun" id="A6.F6.22.7.m7.1e">300</annotation></semantics></math> iterations with the final parameters. Note that the potential is minimized at each level for fixed <math alttext="\alpha_{k}" class="ltx_Math" display="inline" id="A6.F6.23.8.m8.1"><semantics id="A6.F6.23.8.m8.1b"><msub id="A6.F6.23.8.m8.1.1" xref="A6.F6.23.8.m8.1.1.cmml"><mi id="A6.F6.23.8.m8.1.1.2" xref="A6.F6.23.8.m8.1.1.2.cmml">α</mi><mi id="A6.F6.23.8.m8.1.1.3" xref="A6.F6.23.8.m8.1.1.3.cmml">k</mi></msub><annotation-xml encoding="MathML-Content" id="A6.F6.23.8.m8.1c"><apply id="A6.F6.23.8.m8.1.1.cmml" xref="A6.F6.23.8.m8.1.1"><csymbol cd="ambiguous" id="A6.F6.23.8.m8.1.1.1.cmml" xref="A6.F6.23.8.m8.1.1">subscript</csymbol><ci id="A6.F6.23.8.m8.1.1.2.cmml" xref="A6.F6.23.8.m8.1.1.2">𝛼</ci><ci id="A6.F6.23.8.m8.1.1.3.cmml" xref="A6.F6.23.8.m8.1.1.3">𝑘</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A6.F6.23.8.m8.1d">\alpha_{k}</annotation><annotation encoding="application/x-llamapun" id="A6.F6.23.8.m8.1e">italic_α start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="\sigma_{k}" class="ltx_Math" display="inline" id="A6.F6.24.9.m9.1"><semantics id="A6.F6.24.9.m9.1b"><msub id="A6.F6.24.9.m9.1.1" xref="A6.F6.24.9.m9.1.1.cmml"><mi id="A6.F6.24.9.m9.1.1.2" xref="A6.F6.24.9.m9.1.1.2.cmml">σ</mi><mi id="A6.F6.24.9.m9.1.1.3" xref="A6.F6.24.9.m9.1.1.3.cmml">k</mi></msub><annotation-xml encoding="MathML-Content" id="A6.F6.24.9.m9.1c"><apply id="A6.F6.24.9.m9.1.1.cmml" xref="A6.F6.24.9.m9.1.1"><csymbol cd="ambiguous" id="A6.F6.24.9.m9.1.1.1.cmml" xref="A6.F6.24.9.m9.1.1">subscript</csymbol><ci id="A6.F6.24.9.m9.1.1.2.cmml" xref="A6.F6.24.9.m9.1.1.2">𝜎</ci><ci id="A6.F6.24.9.m9.1.1.3.cmml" xref="A6.F6.24.9.m9.1.1.3">𝑘</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A6.F6.24.9.m9.1d">\sigma_{k}</annotation><annotation encoding="application/x-llamapun" id="A6.F6.24.9.m9.1e">italic_σ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT</annotation></semantics></math>. The optimization can be understood in two steps. 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Then a refinement is realized with <math alttext="\alpha_{k}" class="ltx_Math" display="inline" id="A6.F6.27.12.m12.1"><semantics id="A6.F6.27.12.m12.1b"><msub id="A6.F6.27.12.m12.1.1" xref="A6.F6.27.12.m12.1.1.cmml"><mi id="A6.F6.27.12.m12.1.1.2" xref="A6.F6.27.12.m12.1.1.2.cmml">α</mi><mi id="A6.F6.27.12.m12.1.1.3" xref="A6.F6.27.12.m12.1.1.3.cmml">k</mi></msub><annotation-xml encoding="MathML-Content" id="A6.F6.27.12.m12.1c"><apply id="A6.F6.27.12.m12.1.1.cmml" xref="A6.F6.27.12.m12.1.1"><csymbol cd="ambiguous" id="A6.F6.27.12.m12.1.1.1.cmml" xref="A6.F6.27.12.m12.1.1">subscript</csymbol><ci id="A6.F6.27.12.m12.1.1.2.cmml" xref="A6.F6.27.12.m12.1.1.2">𝛼</ci><ci id="A6.F6.27.12.m12.1.1.3.cmml" xref="A6.F6.27.12.m12.1.1.3">𝑘</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A6.F6.27.12.m12.1d">\alpha_{k}</annotation><annotation encoding="application/x-llamapun" id="A6.F6.27.12.m12.1e">italic_α start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT</annotation></semantics></math> large and <math alttext="\sigma_{k}" class="ltx_Math" display="inline" id="A6.F6.28.13.m13.1"><semantics id="A6.F6.28.13.m13.1b"><msub id="A6.F6.28.13.m13.1.1" xref="A6.F6.28.13.m13.1.1.cmml"><mi id="A6.F6.28.13.m13.1.1.2" xref="A6.F6.28.13.m13.1.1.2.cmml">σ</mi><mi id="A6.F6.28.13.m13.1.1.3" xref="A6.F6.28.13.m13.1.1.3.cmml">k</mi></msub><annotation-xml encoding="MathML-Content" id="A6.F6.28.13.m13.1c"><apply id="A6.F6.28.13.m13.1.1.cmml" xref="A6.F6.28.13.m13.1.1"><csymbol cd="ambiguous" id="A6.F6.28.13.m13.1.1.1.cmml" xref="A6.F6.28.13.m13.1.1">subscript</csymbol><ci id="A6.F6.28.13.m13.1.1.2.cmml" xref="A6.F6.28.13.m13.1.1.2">𝜎</ci><ci id="A6.F6.28.13.m13.1.1.3.cmml" xref="A6.F6.28.13.m13.1.1.3">𝑘</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A6.F6.28.13.m13.1d">\sigma_{k}</annotation><annotation encoding="application/x-llamapun" id="A6.F6.28.13.m13.1e">italic_σ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT</annotation></semantics></math> small, to generate a realistic image.</span></figcaption> </figure> </section> <section class="ltx_paragraph" id="A6.SS0.SSS0.Px4"> <h5 class="ltx_title ltx_title_paragraph">SNORE Prox</h5> <div class="ltx_para" id="A6.SS0.SSS0.Px4.p1"> <p class="ltx_p" id="A6.SS0.SSS0.Px4.p1.1">We name SNORE Prox the algorithm detailed in Algorithm <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#alg5" title="Algorithm 5 ‣ SNORE Prox ‣ Appendix F Additional Experiments ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_tag">5</span></a>. A Proximal descent step is computed on the data-fidelity instead of a gradient-descent step. Note that our convergence analysis of Section <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#S3" title="3 Convergence Analysis ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_tag">3</span></a> does not apply to this algorithm. We test this algorithm for a comparison with the original SNORE (Algorithm <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#alg3" title="Algorithm 3 ‣ Optimization algorithms ‣ 2.2 SNORE regularization ‣ 2 Stochastic deNOising REgularization (SNORE) ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_tag">3</span></a>). Experimentally, we observe that both algorithms reach similar performance (see Table <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#S4.T1" title="Table 1 ‣ 4.1 Deblurring ‣ 4 Experiments ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_tag">1</span></a> or Table <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#S4.T2" title="Table 2 ‣ 4.2 Inpainting ‣ 4 Experiments ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_tag">2</span></a>).</p> </div> <figure class="ltx_float ltx_float_algorithm ltx_framed_top" id="alg5"> <figcaption class="ltx_caption"><span class="ltx_tag ltx_tag_float"><span class="ltx_text ltx_font_bold" id="alg5.2.1.1">Algorithm 5</span> </span> Annealed SNORE Prox</figcaption> <div class="ltx_listing ltx_listing" id="alg5.3"> <div class="ltx_listingline" id="alg5.l1"> <span class="ltx_tag ltx_tag_listingline"><span class="ltx_text" 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xref="alg5.l1.m4.1.1.4.3">1</cn></apply></apply><apply id="alg5.l1.m4.1.1c.cmml" xref="alg5.l1.m4.1.1"><gt id="alg5.l1.m4.1.1.5.cmml" xref="alg5.l1.m4.1.1.5"></gt><share href="#alg5.l1.m4.1.1.4.cmml" id="alg5.l1.m4.1.1d.cmml" xref="alg5.l1.m4.1.1"></share><ci id="alg5.l1.m4.1.1.6.cmml" xref="alg5.l1.m4.1.1.6">⋯</ci></apply><apply id="alg5.l1.m4.1.1e.cmml" xref="alg5.l1.m4.1.1"><gt id="alg5.l1.m4.1.1.7.cmml" xref="alg5.l1.m4.1.1.7"></gt><share href="#alg5.l1.m4.1.1.6.cmml" id="alg5.l1.m4.1.1f.cmml" xref="alg5.l1.m4.1.1"></share><apply id="alg5.l1.m4.1.1.8.cmml" xref="alg5.l1.m4.1.1.8"><csymbol cd="ambiguous" id="alg5.l1.m4.1.1.8.1.cmml" xref="alg5.l1.m4.1.1.8">subscript</csymbol><ci id="alg5.l1.m4.1.1.8.2.cmml" xref="alg5.l1.m4.1.1.8.2">𝜎</ci><apply id="alg5.l1.m4.1.1.8.3.cmml" xref="alg5.l1.m4.1.1.8.3"><minus id="alg5.l1.m4.1.1.8.3.1.cmml" xref="alg5.l1.m4.1.1.8.3.1"></minus><ci id="alg5.l1.m4.1.1.8.3.2.cmml" xref="alg5.l1.m4.1.1.8.3.2">𝑚</ci><cn id="alg5.l1.m4.1.1.8.3.3.cmml" type="integer" xref="alg5.l1.m4.1.1.8.3.3">1</cn></apply></apply></apply><apply id="alg5.l1.m4.1.1g.cmml" xref="alg5.l1.m4.1.1"><approx id="alg5.l1.m4.1.1.9.cmml" xref="alg5.l1.m4.1.1.9"></approx><share href="#alg5.l1.m4.1.1.8.cmml" id="alg5.l1.m4.1.1h.cmml" xref="alg5.l1.m4.1.1"></share><cn id="alg5.l1.m4.1.1.10.cmml" type="integer" xref="alg5.l1.m4.1.1.10">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="alg5.l1.m4.1c">\sigma_{0}>\sigma_{1}>\dots>\sigma_{m-1}\approx 0</annotation><annotation encoding="application/x-llamapun" id="alg5.l1.m4.1d">italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT > italic_σ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT > ⋯ > italic_σ start_POSTSUBSCRIPT italic_m - 1 end_POSTSUBSCRIPT ≈ 0</annotation></semantics></math>, <math alttext="\alpha_{0},\alpha_{1},\dots,\alpha_{m-1}>0" class="ltx_Math" display="inline" id="alg5.l1.m5.4"><semantics id="alg5.l1.m5.4a"><mrow id="alg5.l1.m5.4.4" xref="alg5.l1.m5.4.4.cmml"><mrow id="alg5.l1.m5.4.4.3.3" xref="alg5.l1.m5.4.4.3.4.cmml"><msub id="alg5.l1.m5.2.2.1.1.1" xref="alg5.l1.m5.2.2.1.1.1.cmml"><mi id="alg5.l1.m5.2.2.1.1.1.2" xref="alg5.l1.m5.2.2.1.1.1.2.cmml">α</mi><mn id="alg5.l1.m5.2.2.1.1.1.3" xref="alg5.l1.m5.2.2.1.1.1.3.cmml">0</mn></msub><mo id="alg5.l1.m5.4.4.3.3.4" xref="alg5.l1.m5.4.4.3.4.cmml">,</mo><msub id="alg5.l1.m5.3.3.2.2.2" xref="alg5.l1.m5.3.3.2.2.2.cmml"><mi id="alg5.l1.m5.3.3.2.2.2.2" xref="alg5.l1.m5.3.3.2.2.2.2.cmml">α</mi><mn id="alg5.l1.m5.3.3.2.2.2.3" xref="alg5.l1.m5.3.3.2.2.2.3.cmml">1</mn></msub><mo id="alg5.l1.m5.4.4.3.3.5" xref="alg5.l1.m5.4.4.3.4.cmml">,</mo><mi id="alg5.l1.m5.1.1" mathvariant="normal" xref="alg5.l1.m5.1.1.cmml">…</mi><mo id="alg5.l1.m5.4.4.3.3.6" xref="alg5.l1.m5.4.4.3.4.cmml">,</mo><msub id="alg5.l1.m5.4.4.3.3.3" xref="alg5.l1.m5.4.4.3.3.3.cmml"><mi id="alg5.l1.m5.4.4.3.3.3.2" xref="alg5.l1.m5.4.4.3.3.3.2.cmml">α</mi><mrow id="alg5.l1.m5.4.4.3.3.3.3" xref="alg5.l1.m5.4.4.3.3.3.3.cmml"><mi id="alg5.l1.m5.4.4.3.3.3.3.2" xref="alg5.l1.m5.4.4.3.3.3.3.2.cmml">m</mi><mo id="alg5.l1.m5.4.4.3.3.3.3.1" xref="alg5.l1.m5.4.4.3.3.3.3.1.cmml">−</mo><mn id="alg5.l1.m5.4.4.3.3.3.3.3" xref="alg5.l1.m5.4.4.3.3.3.3.3.cmml">1</mn></mrow></msub></mrow><mo id="alg5.l1.m5.4.4.4" xref="alg5.l1.m5.4.4.4.cmml">></mo><mn id="alg5.l1.m5.4.4.5" xref="alg5.l1.m5.4.4.5.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="alg5.l1.m5.4b"><apply id="alg5.l1.m5.4.4.cmml" xref="alg5.l1.m5.4.4"><gt id="alg5.l1.m5.4.4.4.cmml" xref="alg5.l1.m5.4.4.4"></gt><list id="alg5.l1.m5.4.4.3.4.cmml" xref="alg5.l1.m5.4.4.3.3"><apply id="alg5.l1.m5.2.2.1.1.1.cmml" xref="alg5.l1.m5.2.2.1.1.1"><csymbol cd="ambiguous" id="alg5.l1.m5.2.2.1.1.1.1.cmml" xref="alg5.l1.m5.2.2.1.1.1">subscript</csymbol><ci id="alg5.l1.m5.2.2.1.1.1.2.cmml" xref="alg5.l1.m5.2.2.1.1.1.2">𝛼</ci><cn id="alg5.l1.m5.2.2.1.1.1.3.cmml" type="integer" xref="alg5.l1.m5.2.2.1.1.1.3">0</cn></apply><apply id="alg5.l1.m5.3.3.2.2.2.cmml" xref="alg5.l1.m5.3.3.2.2.2"><csymbol cd="ambiguous" id="alg5.l1.m5.3.3.2.2.2.1.cmml" xref="alg5.l1.m5.3.3.2.2.2">subscript</csymbol><ci id="alg5.l1.m5.3.3.2.2.2.2.cmml" xref="alg5.l1.m5.3.3.2.2.2.2">𝛼</ci><cn id="alg5.l1.m5.3.3.2.2.2.3.cmml" type="integer" xref="alg5.l1.m5.3.3.2.2.2.3">1</cn></apply><ci id="alg5.l1.m5.1.1.cmml" xref="alg5.l1.m5.1.1">…</ci><apply id="alg5.l1.m5.4.4.3.3.3.cmml" xref="alg5.l1.m5.4.4.3.3.3"><csymbol cd="ambiguous" id="alg5.l1.m5.4.4.3.3.3.1.cmml" xref="alg5.l1.m5.4.4.3.3.3">subscript</csymbol><ci id="alg5.l1.m5.4.4.3.3.3.2.cmml" xref="alg5.l1.m5.4.4.3.3.3.2">𝛼</ci><apply id="alg5.l1.m5.4.4.3.3.3.3.cmml" xref="alg5.l1.m5.4.4.3.3.3.3"><minus id="alg5.l1.m5.4.4.3.3.3.3.1.cmml" xref="alg5.l1.m5.4.4.3.3.3.3.1"></minus><ci id="alg5.l1.m5.4.4.3.3.3.3.2.cmml" xref="alg5.l1.m5.4.4.3.3.3.3.2">𝑚</ci><cn id="alg5.l1.m5.4.4.3.3.3.3.3.cmml" type="integer" xref="alg5.l1.m5.4.4.3.3.3.3.3">1</cn></apply></apply></list><cn id="alg5.l1.m5.4.4.5.cmml" type="integer" 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xref="alg5.l1.m6.1.1">…</ci><apply id="alg5.l1.m6.4.4.3.3.3.cmml" xref="alg5.l1.m6.4.4.3.3.3"><csymbol cd="ambiguous" id="alg5.l1.m6.4.4.3.3.3.1.cmml" xref="alg5.l1.m6.4.4.3.3.3">subscript</csymbol><ci id="alg5.l1.m6.4.4.3.3.3.2.cmml" xref="alg5.l1.m6.4.4.3.3.3.2">𝑁</ci><apply id="alg5.l1.m6.4.4.3.3.3.3.cmml" xref="alg5.l1.m6.4.4.3.3.3.3"><minus id="alg5.l1.m6.4.4.3.3.3.3.1.cmml" xref="alg5.l1.m6.4.4.3.3.3.3.1"></minus><ci id="alg5.l1.m6.4.4.3.3.3.3.2.cmml" xref="alg5.l1.m6.4.4.3.3.3.3.2">𝑚</ci><cn id="alg5.l1.m6.4.4.3.3.3.3.3.cmml" type="integer" xref="alg5.l1.m6.4.4.3.3.3.3.3">1</cn></apply></apply></list><ci id="alg5.l1.m6.4.4.5.cmml" xref="alg5.l1.m6.4.4.5">ℕ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="alg5.l1.m6.4c">N_{0},N_{1},\dots,N_{m-1}\in{\mathbb{N}}</annotation><annotation encoding="application/x-llamapun" id="alg5.l1.m6.4d">italic_N start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_N start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_N start_POSTSUBSCRIPT italic_m - 1 end_POSTSUBSCRIPT ∈ blackboard_N</annotation></semantics></math> </div> <div class="ltx_listingline" id="alg5.l2"> <span class="ltx_tag ltx_tag_listingline"><span class="ltx_text" id="alg5.l2.1.1.1" style="font-size:80%;">2:</span></span> <span class="ltx_text ltx_font_bold" id="alg5.l2.2">for</span> <math alttext="i=0,1,\dots,m-1" class="ltx_Math" display="inline" id="alg5.l2.m1.4"><semantics id="alg5.l2.m1.4a"><mrow id="alg5.l2.m1.4.4" xref="alg5.l2.m1.4.4.cmml"><mi id="alg5.l2.m1.4.4.3" xref="alg5.l2.m1.4.4.3.cmml">i</mi><mo id="alg5.l2.m1.4.4.2" xref="alg5.l2.m1.4.4.2.cmml">=</mo><mrow id="alg5.l2.m1.4.4.1.1" xref="alg5.l2.m1.4.4.1.2.cmml"><mn id="alg5.l2.m1.1.1" xref="alg5.l2.m1.1.1.cmml">0</mn><mo id="alg5.l2.m1.4.4.1.1.2" xref="alg5.l2.m1.4.4.1.2.cmml">,</mo><mn id="alg5.l2.m1.2.2" xref="alg5.l2.m1.2.2.cmml">1</mn><mo id="alg5.l2.m1.4.4.1.1.3" xref="alg5.l2.m1.4.4.1.2.cmml">,</mo><mi id="alg5.l2.m1.3.3" mathvariant="normal" xref="alg5.l2.m1.3.3.cmml">…</mi><mo id="alg5.l2.m1.4.4.1.1.4" xref="alg5.l2.m1.4.4.1.2.cmml">,</mo><mrow id="alg5.l2.m1.4.4.1.1.1" xref="alg5.l2.m1.4.4.1.1.1.cmml"><mi id="alg5.l2.m1.4.4.1.1.1.2" xref="alg5.l2.m1.4.4.1.1.1.2.cmml">m</mi><mo id="alg5.l2.m1.4.4.1.1.1.1" xref="alg5.l2.m1.4.4.1.1.1.1.cmml">−</mo><mn id="alg5.l2.m1.4.4.1.1.1.3" xref="alg5.l2.m1.4.4.1.1.1.3.cmml">1</mn></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="alg5.l2.m1.4b"><apply id="alg5.l2.m1.4.4.cmml" xref="alg5.l2.m1.4.4"><eq id="alg5.l2.m1.4.4.2.cmml" xref="alg5.l2.m1.4.4.2"></eq><ci id="alg5.l2.m1.4.4.3.cmml" xref="alg5.l2.m1.4.4.3">𝑖</ci><list id="alg5.l2.m1.4.4.1.2.cmml" xref="alg5.l2.m1.4.4.1.1"><cn id="alg5.l2.m1.1.1.cmml" type="integer" xref="alg5.l2.m1.1.1">0</cn><cn id="alg5.l2.m1.2.2.cmml" type="integer" xref="alg5.l2.m1.2.2">1</cn><ci id="alg5.l2.m1.3.3.cmml" xref="alg5.l2.m1.3.3">…</ci><apply id="alg5.l2.m1.4.4.1.1.1.cmml" xref="alg5.l2.m1.4.4.1.1.1"><minus id="alg5.l2.m1.4.4.1.1.1.1.cmml" xref="alg5.l2.m1.4.4.1.1.1.1"></minus><ci id="alg5.l2.m1.4.4.1.1.1.2.cmml" xref="alg5.l2.m1.4.4.1.1.1.2">𝑚</ci><cn id="alg5.l2.m1.4.4.1.1.1.3.cmml" type="integer" xref="alg5.l2.m1.4.4.1.1.1.3">1</cn></apply></list></apply></annotation-xml><annotation encoding="application/x-tex" id="alg5.l2.m1.4c">i=0,1,\dots,m-1</annotation><annotation encoding="application/x-llamapun" id="alg5.l2.m1.4d">italic_i = 0 , 1 , … , italic_m - 1</annotation></semantics></math> <span class="ltx_text ltx_font_bold" id="alg5.l2.3">do</span> </div> <div class="ltx_listingline" id="alg5.l3"> <span class="ltx_tag ltx_tag_listingline"><span class="ltx_text" id="alg5.l3.1.1.1" style="font-size:80%;">3:</span></span> <span class="ltx_text ltx_font_bold" id="alg5.l3.2">for</span> <math alttext="k=0,1,\dots,N_{i}-1" class="ltx_Math" display="inline" id="alg5.l3.m1.4"><semantics id="alg5.l3.m1.4a"><mrow id="alg5.l3.m1.4.4" xref="alg5.l3.m1.4.4.cmml"><mi id="alg5.l3.m1.4.4.3" xref="alg5.l3.m1.4.4.3.cmml">k</mi><mo id="alg5.l3.m1.4.4.2" xref="alg5.l3.m1.4.4.2.cmml">=</mo><mrow id="alg5.l3.m1.4.4.1.1" xref="alg5.l3.m1.4.4.1.2.cmml"><mn id="alg5.l3.m1.1.1" xref="alg5.l3.m1.1.1.cmml">0</mn><mo id="alg5.l3.m1.4.4.1.1.2" xref="alg5.l3.m1.4.4.1.2.cmml">,</mo><mn id="alg5.l3.m1.2.2" xref="alg5.l3.m1.2.2.cmml">1</mn><mo id="alg5.l3.m1.4.4.1.1.3" xref="alg5.l3.m1.4.4.1.2.cmml">,</mo><mi id="alg5.l3.m1.3.3" mathvariant="normal" xref="alg5.l3.m1.3.3.cmml">…</mi><mo id="alg5.l3.m1.4.4.1.1.4" xref="alg5.l3.m1.4.4.1.2.cmml">,</mo><mrow id="alg5.l3.m1.4.4.1.1.1" xref="alg5.l3.m1.4.4.1.1.1.cmml"><msub id="alg5.l3.m1.4.4.1.1.1.2" xref="alg5.l3.m1.4.4.1.1.1.2.cmml"><mi id="alg5.l3.m1.4.4.1.1.1.2.2" xref="alg5.l3.m1.4.4.1.1.1.2.2.cmml">N</mi><mi id="alg5.l3.m1.4.4.1.1.1.2.3" xref="alg5.l3.m1.4.4.1.1.1.2.3.cmml">i</mi></msub><mo id="alg5.l3.m1.4.4.1.1.1.1" xref="alg5.l3.m1.4.4.1.1.1.1.cmml">−</mo><mn id="alg5.l3.m1.4.4.1.1.1.3" xref="alg5.l3.m1.4.4.1.1.1.3.cmml">1</mn></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="alg5.l3.m1.4b"><apply id="alg5.l3.m1.4.4.cmml" xref="alg5.l3.m1.4.4"><eq id="alg5.l3.m1.4.4.2.cmml" xref="alg5.l3.m1.4.4.2"></eq><ci id="alg5.l3.m1.4.4.3.cmml" xref="alg5.l3.m1.4.4.3">𝑘</ci><list id="alg5.l3.m1.4.4.1.2.cmml" xref="alg5.l3.m1.4.4.1.1"><cn id="alg5.l3.m1.1.1.cmml" type="integer" xref="alg5.l3.m1.1.1">0</cn><cn id="alg5.l3.m1.2.2.cmml" type="integer" xref="alg5.l3.m1.2.2">1</cn><ci id="alg5.l3.m1.3.3.cmml" xref="alg5.l3.m1.3.3">…</ci><apply id="alg5.l3.m1.4.4.1.1.1.cmml" xref="alg5.l3.m1.4.4.1.1.1"><minus id="alg5.l3.m1.4.4.1.1.1.1.cmml" xref="alg5.l3.m1.4.4.1.1.1.1"></minus><apply id="alg5.l3.m1.4.4.1.1.1.2.cmml" xref="alg5.l3.m1.4.4.1.1.1.2"><csymbol cd="ambiguous" id="alg5.l3.m1.4.4.1.1.1.2.1.cmml" xref="alg5.l3.m1.4.4.1.1.1.2">subscript</csymbol><ci id="alg5.l3.m1.4.4.1.1.1.2.2.cmml" xref="alg5.l3.m1.4.4.1.1.1.2.2">𝑁</ci><ci id="alg5.l3.m1.4.4.1.1.1.2.3.cmml" xref="alg5.l3.m1.4.4.1.1.1.2.3">𝑖</ci></apply><cn id="alg5.l3.m1.4.4.1.1.1.3.cmml" type="integer" xref="alg5.l3.m1.4.4.1.1.1.3">1</cn></apply></list></apply></annotation-xml><annotation encoding="application/x-tex" id="alg5.l3.m1.4c">k=0,1,\dots,N_{i}-1</annotation><annotation encoding="application/x-llamapun" id="alg5.l3.m1.4d">italic_k = 0 , 1 , … , italic_N start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT - 1</annotation></semantics></math> <span class="ltx_text ltx_font_bold" id="alg5.l3.3">do</span> </div> <div class="ltx_listingline" id="alg5.l4"> <span class="ltx_tag ltx_tag_listingline"><span class="ltx_text" id="alg5.l4.1.1.1" style="font-size:80%;">4:</span></span> <math alttext="{\boldsymbol{\epsilon}}\leftarrow\mathcal{N}(0,\mathbf{I}_{d})" class="ltx_Math" display="inline" id="alg5.l4.m1.2"><semantics id="alg5.l4.m1.2a"><mrow id="alg5.l4.m1.2.2" xref="alg5.l4.m1.2.2.cmml"><mi id="alg5.l4.m1.2.2.3" mathvariant="bold-italic" xref="alg5.l4.m1.2.2.3.cmml">ϵ</mi><mo id="alg5.l4.m1.2.2.2" stretchy="false" xref="alg5.l4.m1.2.2.2.cmml">←</mo><mrow id="alg5.l4.m1.2.2.1" xref="alg5.l4.m1.2.2.1.cmml"><mi class="ltx_font_mathcaligraphic" id="alg5.l4.m1.2.2.1.3" xref="alg5.l4.m1.2.2.1.3.cmml">𝒩</mi><mo id="alg5.l4.m1.2.2.1.2" xref="alg5.l4.m1.2.2.1.2.cmml">⁢</mo><mrow id="alg5.l4.m1.2.2.1.1.1" xref="alg5.l4.m1.2.2.1.1.2.cmml"><mo id="alg5.l4.m1.2.2.1.1.1.2" stretchy="false" xref="alg5.l4.m1.2.2.1.1.2.cmml">(</mo><mn id="alg5.l4.m1.1.1" xref="alg5.l4.m1.1.1.cmml">0</mn><mo id="alg5.l4.m1.2.2.1.1.1.3" xref="alg5.l4.m1.2.2.1.1.2.cmml">,</mo><msub id="alg5.l4.m1.2.2.1.1.1.1" xref="alg5.l4.m1.2.2.1.1.1.1.cmml"><mi id="alg5.l4.m1.2.2.1.1.1.1.2" xref="alg5.l4.m1.2.2.1.1.1.1.2.cmml">𝐈</mi><mi id="alg5.l4.m1.2.2.1.1.1.1.3" xref="alg5.l4.m1.2.2.1.1.1.1.3.cmml">d</mi></msub><mo id="alg5.l4.m1.2.2.1.1.1.4" stretchy="false" xref="alg5.l4.m1.2.2.1.1.2.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="alg5.l4.m1.2b"><apply id="alg5.l4.m1.2.2.cmml" xref="alg5.l4.m1.2.2"><ci id="alg5.l4.m1.2.2.2.cmml" xref="alg5.l4.m1.2.2.2">←</ci><ci id="alg5.l4.m1.2.2.3.cmml" xref="alg5.l4.m1.2.2.3">bold-italic-ϵ</ci><apply id="alg5.l4.m1.2.2.1.cmml" xref="alg5.l4.m1.2.2.1"><times id="alg5.l4.m1.2.2.1.2.cmml" xref="alg5.l4.m1.2.2.1.2"></times><ci id="alg5.l4.m1.2.2.1.3.cmml" xref="alg5.l4.m1.2.2.1.3">𝒩</ci><interval closure="open" id="alg5.l4.m1.2.2.1.1.2.cmml" xref="alg5.l4.m1.2.2.1.1.1"><cn id="alg5.l4.m1.1.1.cmml" type="integer" xref="alg5.l4.m1.1.1">0</cn><apply id="alg5.l4.m1.2.2.1.1.1.1.cmml" xref="alg5.l4.m1.2.2.1.1.1.1"><csymbol cd="ambiguous" id="alg5.l4.m1.2.2.1.1.1.1.1.cmml" xref="alg5.l4.m1.2.2.1.1.1.1">subscript</csymbol><ci id="alg5.l4.m1.2.2.1.1.1.1.2.cmml" xref="alg5.l4.m1.2.2.1.1.1.1.2">𝐈</ci><ci id="alg5.l4.m1.2.2.1.1.1.1.3.cmml" xref="alg5.l4.m1.2.2.1.1.1.1.3">𝑑</ci></apply></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="alg5.l4.m1.2c">{\boldsymbol{\epsilon}}\leftarrow\mathcal{N}(0,\mathbf{I}_{d})</annotation><annotation encoding="application/x-llamapun" id="alg5.l4.m1.2d">bold_italic_ϵ ← caligraphic_N ( 0 , bold_I start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT )</annotation></semantics></math> </div> <div class="ltx_listingline" id="alg5.l5"> <span class="ltx_tag ltx_tag_listingline"><span class="ltx_text" id="alg5.l5.1.1.1" style="font-size:80%;">5:</span></span> <math alttext="{\tilde{\mathbf{x}}}_{k}\leftarrow{\mathbf{x}}_{k}+\sigma_{i}{\boldsymbol{% \epsilon}}" class="ltx_Math" display="inline" id="alg5.l5.m1.1"><semantics id="alg5.l5.m1.1a"><mrow id="alg5.l5.m1.1.1" xref="alg5.l5.m1.1.1.cmml"><msub id="alg5.l5.m1.1.1.2" xref="alg5.l5.m1.1.1.2.cmml"><mover accent="true" id="alg5.l5.m1.1.1.2.2" xref="alg5.l5.m1.1.1.2.2.cmml"><mi id="alg5.l5.m1.1.1.2.2.2" xref="alg5.l5.m1.1.1.2.2.2.cmml">𝐱</mi><mo id="alg5.l5.m1.1.1.2.2.1" xref="alg5.l5.m1.1.1.2.2.1.cmml">~</mo></mover><mi id="alg5.l5.m1.1.1.2.3" xref="alg5.l5.m1.1.1.2.3.cmml">k</mi></msub><mo id="alg5.l5.m1.1.1.1" stretchy="false" xref="alg5.l5.m1.1.1.1.cmml">←</mo><mrow id="alg5.l5.m1.1.1.3" xref="alg5.l5.m1.1.1.3.cmml"><msub id="alg5.l5.m1.1.1.3.2" xref="alg5.l5.m1.1.1.3.2.cmml"><mi id="alg5.l5.m1.1.1.3.2.2" xref="alg5.l5.m1.1.1.3.2.2.cmml">𝐱</mi><mi id="alg5.l5.m1.1.1.3.2.3" xref="alg5.l5.m1.1.1.3.2.3.cmml">k</mi></msub><mo id="alg5.l5.m1.1.1.3.1" xref="alg5.l5.m1.1.1.3.1.cmml">+</mo><mrow id="alg5.l5.m1.1.1.3.3" xref="alg5.l5.m1.1.1.3.3.cmml"><msub id="alg5.l5.m1.1.1.3.3.2" xref="alg5.l5.m1.1.1.3.3.2.cmml"><mi id="alg5.l5.m1.1.1.3.3.2.2" xref="alg5.l5.m1.1.1.3.3.2.2.cmml">σ</mi><mi id="alg5.l5.m1.1.1.3.3.2.3" xref="alg5.l5.m1.1.1.3.3.2.3.cmml">i</mi></msub><mo id="alg5.l5.m1.1.1.3.3.1" xref="alg5.l5.m1.1.1.3.3.1.cmml">⁢</mo><mi id="alg5.l5.m1.1.1.3.3.3" mathvariant="bold-italic" xref="alg5.l5.m1.1.1.3.3.3.cmml">ϵ</mi></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="alg5.l5.m1.1b"><apply id="alg5.l5.m1.1.1.cmml" xref="alg5.l5.m1.1.1"><ci id="alg5.l5.m1.1.1.1.cmml" xref="alg5.l5.m1.1.1.1">←</ci><apply id="alg5.l5.m1.1.1.2.cmml" xref="alg5.l5.m1.1.1.2"><csymbol cd="ambiguous" id="alg5.l5.m1.1.1.2.1.cmml" xref="alg5.l5.m1.1.1.2">subscript</csymbol><apply id="alg5.l5.m1.1.1.2.2.cmml" xref="alg5.l5.m1.1.1.2.2"><ci id="alg5.l5.m1.1.1.2.2.1.cmml" xref="alg5.l5.m1.1.1.2.2.1">~</ci><ci id="alg5.l5.m1.1.1.2.2.2.cmml" xref="alg5.l5.m1.1.1.2.2.2">𝐱</ci></apply><ci id="alg5.l5.m1.1.1.2.3.cmml" xref="alg5.l5.m1.1.1.2.3">𝑘</ci></apply><apply id="alg5.l5.m1.1.1.3.cmml" xref="alg5.l5.m1.1.1.3"><plus id="alg5.l5.m1.1.1.3.1.cmml" xref="alg5.l5.m1.1.1.3.1"></plus><apply id="alg5.l5.m1.1.1.3.2.cmml" xref="alg5.l5.m1.1.1.3.2"><csymbol cd="ambiguous" id="alg5.l5.m1.1.1.3.2.1.cmml" xref="alg5.l5.m1.1.1.3.2">subscript</csymbol><ci id="alg5.l5.m1.1.1.3.2.2.cmml" xref="alg5.l5.m1.1.1.3.2.2">𝐱</ci><ci id="alg5.l5.m1.1.1.3.2.3.cmml" xref="alg5.l5.m1.1.1.3.2.3">𝑘</ci></apply><apply id="alg5.l5.m1.1.1.3.3.cmml" xref="alg5.l5.m1.1.1.3.3"><times id="alg5.l5.m1.1.1.3.3.1.cmml" xref="alg5.l5.m1.1.1.3.3.1"></times><apply id="alg5.l5.m1.1.1.3.3.2.cmml" xref="alg5.l5.m1.1.1.3.3.2"><csymbol cd="ambiguous" id="alg5.l5.m1.1.1.3.3.2.1.cmml" xref="alg5.l5.m1.1.1.3.3.2">subscript</csymbol><ci id="alg5.l5.m1.1.1.3.3.2.2.cmml" xref="alg5.l5.m1.1.1.3.3.2.2">𝜎</ci><ci id="alg5.l5.m1.1.1.3.3.2.3.cmml" xref="alg5.l5.m1.1.1.3.3.2.3">𝑖</ci></apply><ci id="alg5.l5.m1.1.1.3.3.3.cmml" xref="alg5.l5.m1.1.1.3.3.3">bold-italic-ϵ</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="alg5.l5.m1.1c">{\tilde{\mathbf{x}}}_{k}\leftarrow{\mathbf{x}}_{k}+\sigma_{i}{\boldsymbol{% \epsilon}}</annotation><annotation encoding="application/x-llamapun" id="alg5.l5.m1.1d">over~ start_ARG bold_x end_ARG start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ← bold_x start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT + italic_σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT bold_italic_ϵ</annotation></semantics></math> </div> <div class="ltx_listingline" id="alg5.l6"> <span class="ltx_tag ltx_tag_listingline"><span class="ltx_text" id="alg5.l6.1.1.1" style="font-size:80%;">6:</span></span> <math alttext="{\mathbf{z}}_{k}\leftarrow{\mathbf{x}}_{k}-\frac{\alpha_{i}\delta}{\sigma_{i}^% {2}}\left({\mathbf{x}}_{k}-D_{\sigma_{i}}({\tilde{\mathbf{x}}}_{k})\right)" class="ltx_Math" display="inline" id="alg5.l6.m1.1"><semantics id="alg5.l6.m1.1a"><mrow id="alg5.l6.m1.1.1" xref="alg5.l6.m1.1.1.cmml"><msub id="alg5.l6.m1.1.1.3" xref="alg5.l6.m1.1.1.3.cmml"><mi id="alg5.l6.m1.1.1.3.2" xref="alg5.l6.m1.1.1.3.2.cmml">𝐳</mi><mi id="alg5.l6.m1.1.1.3.3" xref="alg5.l6.m1.1.1.3.3.cmml">k</mi></msub><mo id="alg5.l6.m1.1.1.2" stretchy="false" xref="alg5.l6.m1.1.1.2.cmml">←</mo><mrow id="alg5.l6.m1.1.1.1" xref="alg5.l6.m1.1.1.1.cmml"><msub id="alg5.l6.m1.1.1.1.3" xref="alg5.l6.m1.1.1.1.3.cmml"><mi id="alg5.l6.m1.1.1.1.3.2" xref="alg5.l6.m1.1.1.1.3.2.cmml">𝐱</mi><mi id="alg5.l6.m1.1.1.1.3.3" xref="alg5.l6.m1.1.1.1.3.3.cmml">k</mi></msub><mo id="alg5.l6.m1.1.1.1.2" xref="alg5.l6.m1.1.1.1.2.cmml">−</mo><mrow id="alg5.l6.m1.1.1.1.1" xref="alg5.l6.m1.1.1.1.1.cmml"><mfrac id="alg5.l6.m1.1.1.1.1.3" xref="alg5.l6.m1.1.1.1.1.3.cmml"><mrow id="alg5.l6.m1.1.1.1.1.3.2" xref="alg5.l6.m1.1.1.1.1.3.2.cmml"><msub id="alg5.l6.m1.1.1.1.1.3.2.2" xref="alg5.l6.m1.1.1.1.1.3.2.2.cmml"><mi id="alg5.l6.m1.1.1.1.1.3.2.2.2" xref="alg5.l6.m1.1.1.1.1.3.2.2.2.cmml">α</mi><mi id="alg5.l6.m1.1.1.1.1.3.2.2.3" xref="alg5.l6.m1.1.1.1.1.3.2.2.3.cmml">i</mi></msub><mo id="alg5.l6.m1.1.1.1.1.3.2.1" xref="alg5.l6.m1.1.1.1.1.3.2.1.cmml">⁢</mo><mi id="alg5.l6.m1.1.1.1.1.3.2.3" xref="alg5.l6.m1.1.1.1.1.3.2.3.cmml">δ</mi></mrow><msubsup id="alg5.l6.m1.1.1.1.1.3.3" xref="alg5.l6.m1.1.1.1.1.3.3.cmml"><mi id="alg5.l6.m1.1.1.1.1.3.3.2.2" xref="alg5.l6.m1.1.1.1.1.3.3.2.2.cmml">σ</mi><mi id="alg5.l6.m1.1.1.1.1.3.3.2.3" xref="alg5.l6.m1.1.1.1.1.3.3.2.3.cmml">i</mi><mn id="alg5.l6.m1.1.1.1.1.3.3.3" xref="alg5.l6.m1.1.1.1.1.3.3.3.cmml">2</mn></msubsup></mfrac><mo id="alg5.l6.m1.1.1.1.1.2" xref="alg5.l6.m1.1.1.1.1.2.cmml">⁢</mo><mrow id="alg5.l6.m1.1.1.1.1.1.1" xref="alg5.l6.m1.1.1.1.1.1.1.1.cmml"><mo id="alg5.l6.m1.1.1.1.1.1.1.2" xref="alg5.l6.m1.1.1.1.1.1.1.1.cmml">(</mo><mrow id="alg5.l6.m1.1.1.1.1.1.1.1" xref="alg5.l6.m1.1.1.1.1.1.1.1.cmml"><msub id="alg5.l6.m1.1.1.1.1.1.1.1.3" xref="alg5.l6.m1.1.1.1.1.1.1.1.3.cmml"><mi id="alg5.l6.m1.1.1.1.1.1.1.1.3.2" xref="alg5.l6.m1.1.1.1.1.1.1.1.3.2.cmml">𝐱</mi><mi id="alg5.l6.m1.1.1.1.1.1.1.1.3.3" xref="alg5.l6.m1.1.1.1.1.1.1.1.3.3.cmml">k</mi></msub><mo id="alg5.l6.m1.1.1.1.1.1.1.1.2" xref="alg5.l6.m1.1.1.1.1.1.1.1.2.cmml">−</mo><mrow id="alg5.l6.m1.1.1.1.1.1.1.1.1" xref="alg5.l6.m1.1.1.1.1.1.1.1.1.cmml"><msub id="alg5.l6.m1.1.1.1.1.1.1.1.1.3" xref="alg5.l6.m1.1.1.1.1.1.1.1.1.3.cmml"><mi id="alg5.l6.m1.1.1.1.1.1.1.1.1.3.2" 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xref="alg5.l6.m1.1.1.1.1.1.1.1.1.1.1.1.2.1.cmml">~</mo></mover><mi id="alg5.l6.m1.1.1.1.1.1.1.1.1.1.1.1.3" xref="alg5.l6.m1.1.1.1.1.1.1.1.1.1.1.1.3.cmml">k</mi></msub><mo id="alg5.l6.m1.1.1.1.1.1.1.1.1.1.1.3" stretchy="false" xref="alg5.l6.m1.1.1.1.1.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><mo id="alg5.l6.m1.1.1.1.1.1.1.3" xref="alg5.l6.m1.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="alg5.l6.m1.1b"><apply id="alg5.l6.m1.1.1.cmml" xref="alg5.l6.m1.1.1"><ci id="alg5.l6.m1.1.1.2.cmml" xref="alg5.l6.m1.1.1.2">←</ci><apply id="alg5.l6.m1.1.1.3.cmml" xref="alg5.l6.m1.1.1.3"><csymbol cd="ambiguous" id="alg5.l6.m1.1.1.3.1.cmml" xref="alg5.l6.m1.1.1.3">subscript</csymbol><ci id="alg5.l6.m1.1.1.3.2.cmml" xref="alg5.l6.m1.1.1.3.2">𝐳</ci><ci id="alg5.l6.m1.1.1.3.3.cmml" xref="alg5.l6.m1.1.1.3.3">𝑘</ci></apply><apply id="alg5.l6.m1.1.1.1.cmml" xref="alg5.l6.m1.1.1.1"><minus id="alg5.l6.m1.1.1.1.2.cmml" xref="alg5.l6.m1.1.1.1.2"></minus><apply id="alg5.l6.m1.1.1.1.3.cmml" xref="alg5.l6.m1.1.1.1.3"><csymbol cd="ambiguous" id="alg5.l6.m1.1.1.1.3.1.cmml" xref="alg5.l6.m1.1.1.1.3">subscript</csymbol><ci id="alg5.l6.m1.1.1.1.3.2.cmml" xref="alg5.l6.m1.1.1.1.3.2">𝐱</ci><ci id="alg5.l6.m1.1.1.1.3.3.cmml" xref="alg5.l6.m1.1.1.1.3.3">𝑘</ci></apply><apply id="alg5.l6.m1.1.1.1.1.cmml" xref="alg5.l6.m1.1.1.1.1"><times id="alg5.l6.m1.1.1.1.1.2.cmml" xref="alg5.l6.m1.1.1.1.1.2"></times><apply id="alg5.l6.m1.1.1.1.1.3.cmml" xref="alg5.l6.m1.1.1.1.1.3"><divide id="alg5.l6.m1.1.1.1.1.3.1.cmml" xref="alg5.l6.m1.1.1.1.1.3"></divide><apply id="alg5.l6.m1.1.1.1.1.3.2.cmml" xref="alg5.l6.m1.1.1.1.1.3.2"><times id="alg5.l6.m1.1.1.1.1.3.2.1.cmml" xref="alg5.l6.m1.1.1.1.1.3.2.1"></times><apply id="alg5.l6.m1.1.1.1.1.3.2.2.cmml" xref="alg5.l6.m1.1.1.1.1.3.2.2"><csymbol cd="ambiguous" id="alg5.l6.m1.1.1.1.1.3.2.2.1.cmml" xref="alg5.l6.m1.1.1.1.1.3.2.2">subscript</csymbol><ci id="alg5.l6.m1.1.1.1.1.3.2.2.2.cmml" xref="alg5.l6.m1.1.1.1.1.3.2.2.2">𝛼</ci><ci id="alg5.l6.m1.1.1.1.1.3.2.2.3.cmml" xref="alg5.l6.m1.1.1.1.1.3.2.2.3">𝑖</ci></apply><ci id="alg5.l6.m1.1.1.1.1.3.2.3.cmml" xref="alg5.l6.m1.1.1.1.1.3.2.3">𝛿</ci></apply><apply id="alg5.l6.m1.1.1.1.1.3.3.cmml" xref="alg5.l6.m1.1.1.1.1.3.3"><csymbol cd="ambiguous" id="alg5.l6.m1.1.1.1.1.3.3.1.cmml" xref="alg5.l6.m1.1.1.1.1.3.3">superscript</csymbol><apply id="alg5.l6.m1.1.1.1.1.3.3.2.cmml" xref="alg5.l6.m1.1.1.1.1.3.3"><csymbol cd="ambiguous" id="alg5.l6.m1.1.1.1.1.3.3.2.1.cmml" xref="alg5.l6.m1.1.1.1.1.3.3">subscript</csymbol><ci id="alg5.l6.m1.1.1.1.1.3.3.2.2.cmml" xref="alg5.l6.m1.1.1.1.1.3.3.2.2">𝜎</ci><ci id="alg5.l6.m1.1.1.1.1.3.3.2.3.cmml" xref="alg5.l6.m1.1.1.1.1.3.3.2.3">𝑖</ci></apply><cn id="alg5.l6.m1.1.1.1.1.3.3.3.cmml" type="integer" xref="alg5.l6.m1.1.1.1.1.3.3.3">2</cn></apply></apply><apply id="alg5.l6.m1.1.1.1.1.1.1.1.cmml" xref="alg5.l6.m1.1.1.1.1.1.1"><minus id="alg5.l6.m1.1.1.1.1.1.1.1.2.cmml" xref="alg5.l6.m1.1.1.1.1.1.1.1.2"></minus><apply id="alg5.l6.m1.1.1.1.1.1.1.1.3.cmml" xref="alg5.l6.m1.1.1.1.1.1.1.1.3"><csymbol cd="ambiguous" id="alg5.l6.m1.1.1.1.1.1.1.1.3.1.cmml" xref="alg5.l6.m1.1.1.1.1.1.1.1.3">subscript</csymbol><ci id="alg5.l6.m1.1.1.1.1.1.1.1.3.2.cmml" xref="alg5.l6.m1.1.1.1.1.1.1.1.3.2">𝐱</ci><ci id="alg5.l6.m1.1.1.1.1.1.1.1.3.3.cmml" xref="alg5.l6.m1.1.1.1.1.1.1.1.3.3">𝑘</ci></apply><apply id="alg5.l6.m1.1.1.1.1.1.1.1.1.cmml" xref="alg5.l6.m1.1.1.1.1.1.1.1.1"><times id="alg5.l6.m1.1.1.1.1.1.1.1.1.2.cmml" xref="alg5.l6.m1.1.1.1.1.1.1.1.1.2"></times><apply id="alg5.l6.m1.1.1.1.1.1.1.1.1.3.cmml" xref="alg5.l6.m1.1.1.1.1.1.1.1.1.3"><csymbol cd="ambiguous" id="alg5.l6.m1.1.1.1.1.1.1.1.1.3.1.cmml" xref="alg5.l6.m1.1.1.1.1.1.1.1.1.3">subscript</csymbol><ci id="alg5.l6.m1.1.1.1.1.1.1.1.1.3.2.cmml" xref="alg5.l6.m1.1.1.1.1.1.1.1.1.3.2">𝐷</ci><apply id="alg5.l6.m1.1.1.1.1.1.1.1.1.3.3.cmml" xref="alg5.l6.m1.1.1.1.1.1.1.1.1.3.3"><csymbol cd="ambiguous" id="alg5.l6.m1.1.1.1.1.1.1.1.1.3.3.1.cmml" xref="alg5.l6.m1.1.1.1.1.1.1.1.1.3.3">subscript</csymbol><ci id="alg5.l6.m1.1.1.1.1.1.1.1.1.3.3.2.cmml" xref="alg5.l6.m1.1.1.1.1.1.1.1.1.3.3.2">𝜎</ci><ci id="alg5.l6.m1.1.1.1.1.1.1.1.1.3.3.3.cmml" xref="alg5.l6.m1.1.1.1.1.1.1.1.1.3.3.3">𝑖</ci></apply></apply><apply id="alg5.l6.m1.1.1.1.1.1.1.1.1.1.1.1.cmml" xref="alg5.l6.m1.1.1.1.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="alg5.l6.m1.1.1.1.1.1.1.1.1.1.1.1.1.cmml" xref="alg5.l6.m1.1.1.1.1.1.1.1.1.1.1">subscript</csymbol><apply id="alg5.l6.m1.1.1.1.1.1.1.1.1.1.1.1.2.cmml" xref="alg5.l6.m1.1.1.1.1.1.1.1.1.1.1.1.2"><ci id="alg5.l6.m1.1.1.1.1.1.1.1.1.1.1.1.2.1.cmml" xref="alg5.l6.m1.1.1.1.1.1.1.1.1.1.1.1.2.1">~</ci><ci id="alg5.l6.m1.1.1.1.1.1.1.1.1.1.1.1.2.2.cmml" xref="alg5.l6.m1.1.1.1.1.1.1.1.1.1.1.1.2.2">𝐱</ci></apply><ci id="alg5.l6.m1.1.1.1.1.1.1.1.1.1.1.1.3.cmml" xref="alg5.l6.m1.1.1.1.1.1.1.1.1.1.1.1.3">𝑘</ci></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="alg5.l6.m1.1c">{\mathbf{z}}_{k}\leftarrow{\mathbf{x}}_{k}-\frac{\alpha_{i}\delta}{\sigma_{i}^% {2}}\left({\mathbf{x}}_{k}-D_{\sigma_{i}}({\tilde{\mathbf{x}}}_{k})\right)</annotation><annotation encoding="application/x-llamapun" id="alg5.l6.m1.1d">bold_z start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ← bold_x start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT - divide start_ARG italic_α start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_δ end_ARG start_ARG italic_σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG ( bold_x start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT - italic_D start_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( over~ start_ARG bold_x end_ARG start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) )</annotation></semantics></math> </div> <div class="ltx_listingline" id="alg5.l7"> <span class="ltx_tag ltx_tag_listingline"><span class="ltx_text" id="alg5.l7.1.1.1" style="font-size:80%;">7:</span></span> <math alttext="{\mathbf{x}}_{k+1}\leftarrow\text{Prox}_{\delta\mathcal{F}}({\mathbf{z}}_{k})" class="ltx_Math" display="inline" id="alg5.l7.m1.1"><semantics id="alg5.l7.m1.1a"><mrow id="alg5.l7.m1.1.1" xref="alg5.l7.m1.1.1.cmml"><msub id="alg5.l7.m1.1.1.3" xref="alg5.l7.m1.1.1.3.cmml"><mi id="alg5.l7.m1.1.1.3.2" xref="alg5.l7.m1.1.1.3.2.cmml">𝐱</mi><mrow id="alg5.l7.m1.1.1.3.3" xref="alg5.l7.m1.1.1.3.3.cmml"><mi id="alg5.l7.m1.1.1.3.3.2" xref="alg5.l7.m1.1.1.3.3.2.cmml">k</mi><mo id="alg5.l7.m1.1.1.3.3.1" xref="alg5.l7.m1.1.1.3.3.1.cmml">+</mo><mn id="alg5.l7.m1.1.1.3.3.3" xref="alg5.l7.m1.1.1.3.3.3.cmml">1</mn></mrow></msub><mo id="alg5.l7.m1.1.1.2" stretchy="false" xref="alg5.l7.m1.1.1.2.cmml">←</mo><mrow id="alg5.l7.m1.1.1.1" xref="alg5.l7.m1.1.1.1.cmml"><msub id="alg5.l7.m1.1.1.1.3" xref="alg5.l7.m1.1.1.1.3.cmml"><mtext id="alg5.l7.m1.1.1.1.3.2" xref="alg5.l7.m1.1.1.1.3.2a.cmml">Prox</mtext><mrow id="alg5.l7.m1.1.1.1.3.3" 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xref="alg5.l7.m1.1.1.1.3.2">Prox</mtext></ci><apply id="alg5.l7.m1.1.1.1.3.3.cmml" xref="alg5.l7.m1.1.1.1.3.3"><times id="alg5.l7.m1.1.1.1.3.3.1.cmml" xref="alg5.l7.m1.1.1.1.3.3.1"></times><ci id="alg5.l7.m1.1.1.1.3.3.2.cmml" xref="alg5.l7.m1.1.1.1.3.3.2">𝛿</ci><ci id="alg5.l7.m1.1.1.1.3.3.3.cmml" xref="alg5.l7.m1.1.1.1.3.3.3">ℱ</ci></apply></apply><apply id="alg5.l7.m1.1.1.1.1.1.1.cmml" xref="alg5.l7.m1.1.1.1.1.1"><csymbol cd="ambiguous" id="alg5.l7.m1.1.1.1.1.1.1.1.cmml" xref="alg5.l7.m1.1.1.1.1.1">subscript</csymbol><ci id="alg5.l7.m1.1.1.1.1.1.1.2.cmml" xref="alg5.l7.m1.1.1.1.1.1.1.2">𝐳</ci><ci id="alg5.l7.m1.1.1.1.1.1.1.3.cmml" xref="alg5.l7.m1.1.1.1.1.1.1.3">𝑘</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="alg5.l7.m1.1c">{\mathbf{x}}_{k+1}\leftarrow\text{Prox}_{\delta\mathcal{F}}({\mathbf{z}}_{k})</annotation><annotation encoding="application/x-llamapun" id="alg5.l7.m1.1d">bold_x start_POSTSUBSCRIPT italic_k + 1 end_POSTSUBSCRIPT ← Prox start_POSTSUBSCRIPT italic_δ caligraphic_F end_POSTSUBSCRIPT ( bold_z start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT )</annotation></semantics></math> </div> <div class="ltx_listingline" id="alg5.l8"> <span class="ltx_tag ltx_tag_listingline"><span class="ltx_text" id="alg5.l8.1.1.1" style="font-size:80%;">8:</span></span> <span class="ltx_text ltx_font_bold" id="alg5.l8.2">end</span> <span class="ltx_text ltx_font_bold" id="alg5.l8.3">for</span> </div> <div class="ltx_listingline" id="alg5.l9"> <span class="ltx_tag ltx_tag_listingline"><span class="ltx_text" id="alg5.l9.1.1.1" style="font-size:80%;">9:</span></span> <span class="ltx_text ltx_font_bold" id="alg5.l9.2">end</span> <span class="ltx_text ltx_font_bold" id="alg5.l9.3">for</span> </div> </div> </figure> </section> <section class="ltx_paragraph" id="A6.SS0.SSS0.Px5"> <h5 class="ltx_title ltx_title_paragraph">DiffPIR</h5> <div class="ltx_para" id="A6.SS0.SSS0.Px5.p1"> <p class="ltx_p" id="A6.SS0.SSS0.Px5.p1.3">We compare our algorithm to state-of-the-art methods, including a Diffusion Model. Among existing methods in the field of Diffusion Models, we choose to test DiffPIR <cite class="ltx_cite ltx_citemacro_citep">(Zhu et al., <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#bib.bib80" title="">2023</a>)</cite>. DiffPIR is a recent algorithm which makes a connection between PnP and Diffusion Models. We use the implementation of the Python library DeepInverse modified to add a time <math alttext="t_{\text{start}}<T" class="ltx_Math" display="inline" id="A6.SS0.SSS0.Px5.p1.1.m1.1"><semantics id="A6.SS0.SSS0.Px5.p1.1.m1.1a"><mrow id="A6.SS0.SSS0.Px5.p1.1.m1.1.1" xref="A6.SS0.SSS0.Px5.p1.1.m1.1.1.cmml"><msub id="A6.SS0.SSS0.Px5.p1.1.m1.1.1.2" xref="A6.SS0.SSS0.Px5.p1.1.m1.1.1.2.cmml"><mi id="A6.SS0.SSS0.Px5.p1.1.m1.1.1.2.2" xref="A6.SS0.SSS0.Px5.p1.1.m1.1.1.2.2.cmml">t</mi><mtext id="A6.SS0.SSS0.Px5.p1.1.m1.1.1.2.3" xref="A6.SS0.SSS0.Px5.p1.1.m1.1.1.2.3a.cmml">start</mtext></msub><mo id="A6.SS0.SSS0.Px5.p1.1.m1.1.1.1" xref="A6.SS0.SSS0.Px5.p1.1.m1.1.1.1.cmml"><</mo><mi id="A6.SS0.SSS0.Px5.p1.1.m1.1.1.3" xref="A6.SS0.SSS0.Px5.p1.1.m1.1.1.3.cmml">T</mi></mrow><annotation-xml encoding="MathML-Content" id="A6.SS0.SSS0.Px5.p1.1.m1.1b"><apply id="A6.SS0.SSS0.Px5.p1.1.m1.1.1.cmml" xref="A6.SS0.SSS0.Px5.p1.1.m1.1.1"><lt id="A6.SS0.SSS0.Px5.p1.1.m1.1.1.1.cmml" xref="A6.SS0.SSS0.Px5.p1.1.m1.1.1.1"></lt><apply id="A6.SS0.SSS0.Px5.p1.1.m1.1.1.2.cmml" xref="A6.SS0.SSS0.Px5.p1.1.m1.1.1.2"><csymbol cd="ambiguous" id="A6.SS0.SSS0.Px5.p1.1.m1.1.1.2.1.cmml" xref="A6.SS0.SSS0.Px5.p1.1.m1.1.1.2">subscript</csymbol><ci id="A6.SS0.SSS0.Px5.p1.1.m1.1.1.2.2.cmml" xref="A6.SS0.SSS0.Px5.p1.1.m1.1.1.2.2">𝑡</ci><ci id="A6.SS0.SSS0.Px5.p1.1.m1.1.1.2.3a.cmml" xref="A6.SS0.SSS0.Px5.p1.1.m1.1.1.2.3"><mtext id="A6.SS0.SSS0.Px5.p1.1.m1.1.1.2.3.cmml" mathsize="70%" xref="A6.SS0.SSS0.Px5.p1.1.m1.1.1.2.3">start</mtext></ci></apply><ci id="A6.SS0.SSS0.Px5.p1.1.m1.1.1.3.cmml" xref="A6.SS0.SSS0.Px5.p1.1.m1.1.1.3">𝑇</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A6.SS0.SSS0.Px5.p1.1.m1.1c">t_{\text{start}}<T</annotation><annotation encoding="application/x-llamapun" id="A6.SS0.SSS0.Px5.p1.1.m1.1d">italic_t start_POSTSUBSCRIPT start end_POSTSUBSCRIPT < italic_T</annotation></semantics></math> such as proposed by the authors <cite class="ltx_cite ltx_citemacro_citep">(Zhu et al., <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#bib.bib80" title="">2023</a>, Section 4.4)</cite>. We need to add this parameter <math alttext="t_{\text{start}}" class="ltx_Math" display="inline" id="A6.SS0.SSS0.Px5.p1.2.m2.1"><semantics id="A6.SS0.SSS0.Px5.p1.2.m2.1a"><msub id="A6.SS0.SSS0.Px5.p1.2.m2.1.1" xref="A6.SS0.SSS0.Px5.p1.2.m2.1.1.cmml"><mi id="A6.SS0.SSS0.Px5.p1.2.m2.1.1.2" xref="A6.SS0.SSS0.Px5.p1.2.m2.1.1.2.cmml">t</mi><mtext id="A6.SS0.SSS0.Px5.p1.2.m2.1.1.3" xref="A6.SS0.SSS0.Px5.p1.2.m2.1.1.3a.cmml">start</mtext></msub><annotation-xml encoding="MathML-Content" id="A6.SS0.SSS0.Px5.p1.2.m2.1b"><apply id="A6.SS0.SSS0.Px5.p1.2.m2.1.1.cmml" xref="A6.SS0.SSS0.Px5.p1.2.m2.1.1"><csymbol cd="ambiguous" id="A6.SS0.SSS0.Px5.p1.2.m2.1.1.1.cmml" xref="A6.SS0.SSS0.Px5.p1.2.m2.1.1">subscript</csymbol><ci id="A6.SS0.SSS0.Px5.p1.2.m2.1.1.2.cmml" xref="A6.SS0.SSS0.Px5.p1.2.m2.1.1.2">𝑡</ci><ci id="A6.SS0.SSS0.Px5.p1.2.m2.1.1.3a.cmml" xref="A6.SS0.SSS0.Px5.p1.2.m2.1.1.3"><mtext id="A6.SS0.SSS0.Px5.p1.2.m2.1.1.3.cmml" mathsize="70%" xref="A6.SS0.SSS0.Px5.p1.2.m2.1.1.3">start</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A6.SS0.SSS0.Px5.p1.2.m2.1c">t_{\text{start}}</annotation><annotation encoding="application/x-llamapun" id="A6.SS0.SSS0.Px5.p1.2.m2.1d">italic_t start_POSTSUBSCRIPT start end_POSTSUBSCRIPT</annotation></semantics></math> as the denoiser is not trained to generate relevant outputs for highly noisy images (compared to neural networks used for diffusion models), since the model has only been trained for noise with standard deviations in the set <math alttext="[0,50]/255" class="ltx_Math" display="inline" id="A6.SS0.SSS0.Px5.p1.3.m3.2"><semantics id="A6.SS0.SSS0.Px5.p1.3.m3.2a"><mrow id="A6.SS0.SSS0.Px5.p1.3.m3.2.3" xref="A6.SS0.SSS0.Px5.p1.3.m3.2.3.cmml"><mrow id="A6.SS0.SSS0.Px5.p1.3.m3.2.3.2.2" xref="A6.SS0.SSS0.Px5.p1.3.m3.2.3.2.1.cmml"><mo id="A6.SS0.SSS0.Px5.p1.3.m3.2.3.2.2.1" stretchy="false" xref="A6.SS0.SSS0.Px5.p1.3.m3.2.3.2.1.cmml">[</mo><mn 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xref="A6.SS0.SSS0.Px5.p1.3.m3.1.1">0</cn><cn id="A6.SS0.SSS0.Px5.p1.3.m3.2.2.cmml" type="integer" xref="A6.SS0.SSS0.Px5.p1.3.m3.2.2">50</cn></interval><cn id="A6.SS0.SSS0.Px5.p1.3.m3.2.3.3.cmml" type="integer" xref="A6.SS0.SSS0.Px5.p1.3.m3.2.3.3">255</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A6.SS0.SSS0.Px5.p1.3.m3.2c">[0,50]/255</annotation><annotation encoding="application/x-llamapun" id="A6.SS0.SSS0.Px5.p1.3.m3.2d">[ 0 , 50 ] / 255</annotation></semantics></math>.</p> </div> <figure class="ltx_float ltx_float_algorithm ltx_framed_top" id="alg6"> <figcaption class="ltx_caption"><span class="ltx_tag ltx_tag_float"><span class="ltx_text ltx_font_bold" id="alg6.2.1.1">Algorithm 6</span> </span> DiffPIR <cite class="ltx_cite ltx_citemacro_citep">(Zhu et al., <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#bib.bib80" title="">2023</a>)</cite></figcaption> <div class="ltx_listing ltx_listing" id="alg6.3"> <div class="ltx_listingline" id="alg6.l1"> <span class="ltx_tag ltx_tag_listingline"><span class="ltx_text" id="alg6.l1.1.1.1" style="font-size:80%;">1:</span></span> <span class="ltx_text ltx_font_bold" id="alg6.l1.2">input:</span> <math alttext="D_{\sigma}" class="ltx_Math" display="inline" id="alg6.l1.m1.1"><semantics id="alg6.l1.m1.1a"><msub id="alg6.l1.m1.1.1" xref="alg6.l1.m1.1.1.cmml"><mi id="alg6.l1.m1.1.1.2" xref="alg6.l1.m1.1.1.2.cmml">D</mi><mi id="alg6.l1.m1.1.1.3" xref="alg6.l1.m1.1.1.3.cmml">σ</mi></msub><annotation-xml encoding="MathML-Content" id="alg6.l1.m1.1b"><apply id="alg6.l1.m1.1.1.cmml" xref="alg6.l1.m1.1.1"><csymbol cd="ambiguous" id="alg6.l1.m1.1.1.1.cmml" xref="alg6.l1.m1.1.1">subscript</csymbol><ci id="alg6.l1.m1.1.1.2.cmml" xref="alg6.l1.m1.1.1.2">𝐷</ci><ci id="alg6.l1.m1.1.1.3.cmml" xref="alg6.l1.m1.1.1.3">𝜎</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="alg6.l1.m1.1c">D_{\sigma}</annotation><annotation encoding="application/x-llamapun" id="alg6.l1.m1.1d">italic_D start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT</annotation></semantics></math>, <math alttext="T>0" class="ltx_Math" display="inline" id="alg6.l1.m2.1"><semantics id="alg6.l1.m2.1a"><mrow id="alg6.l1.m2.1.1" xref="alg6.l1.m2.1.1.cmml"><mi id="alg6.l1.m2.1.1.2" xref="alg6.l1.m2.1.1.2.cmml">T</mi><mo id="alg6.l1.m2.1.1.1" xref="alg6.l1.m2.1.1.1.cmml">></mo><mn id="alg6.l1.m2.1.1.3" xref="alg6.l1.m2.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="alg6.l1.m2.1b"><apply id="alg6.l1.m2.1.1.cmml" xref="alg6.l1.m2.1.1"><gt id="alg6.l1.m2.1.1.1.cmml" xref="alg6.l1.m2.1.1.1"></gt><ci id="alg6.l1.m2.1.1.2.cmml" xref="alg6.l1.m2.1.1.2">𝑇</ci><cn id="alg6.l1.m2.1.1.3.cmml" type="integer" xref="alg6.l1.m2.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="alg6.l1.m2.1c">T>0</annotation><annotation encoding="application/x-llamapun" id="alg6.l1.m2.1d">italic_T > 0</annotation></semantics></math>, <math alttext="{\mathbf{y}}\in{\mathbb{R}}^{q}" class="ltx_Math" display="inline" id="alg6.l1.m3.1"><semantics id="alg6.l1.m3.1a"><mrow id="alg6.l1.m3.1.1" xref="alg6.l1.m3.1.1.cmml"><mi id="alg6.l1.m3.1.1.2" xref="alg6.l1.m3.1.1.2.cmml">𝐲</mi><mo id="alg6.l1.m3.1.1.1" xref="alg6.l1.m3.1.1.1.cmml">∈</mo><msup id="alg6.l1.m3.1.1.3" xref="alg6.l1.m3.1.1.3.cmml"><mi id="alg6.l1.m3.1.1.3.2" xref="alg6.l1.m3.1.1.3.2.cmml">ℝ</mi><mi id="alg6.l1.m3.1.1.3.3" xref="alg6.l1.m3.1.1.3.3.cmml">q</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="alg6.l1.m3.1b"><apply id="alg6.l1.m3.1.1.cmml" xref="alg6.l1.m3.1.1"><in id="alg6.l1.m3.1.1.1.cmml" xref="alg6.l1.m3.1.1.1"></in><ci id="alg6.l1.m3.1.1.2.cmml" xref="alg6.l1.m3.1.1.2">𝐲</ci><apply id="alg6.l1.m3.1.1.3.cmml" xref="alg6.l1.m3.1.1.3"><csymbol cd="ambiguous" id="alg6.l1.m3.1.1.3.1.cmml" xref="alg6.l1.m3.1.1.3">superscript</csymbol><ci id="alg6.l1.m3.1.1.3.2.cmml" xref="alg6.l1.m3.1.1.3.2">ℝ</ci><ci id="alg6.l1.m3.1.1.3.3.cmml" xref="alg6.l1.m3.1.1.3.3">𝑞</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="alg6.l1.m3.1c">{\mathbf{y}}\in{\mathbb{R}}^{q}</annotation><annotation encoding="application/x-llamapun" id="alg6.l1.m3.1d">bold_y ∈ blackboard_R start_POSTSUPERSCRIPT italic_q end_POSTSUPERSCRIPT</annotation></semantics></math>, <math alttext="0<t_{\text{start}<T}" class="ltx_Math" display="inline" id="alg6.l1.m4.1"><semantics id="alg6.l1.m4.1a"><mrow id="alg6.l1.m4.1.1" xref="alg6.l1.m4.1.1.cmml"><mn id="alg6.l1.m4.1.1.2" xref="alg6.l1.m4.1.1.2.cmml">0</mn><mo id="alg6.l1.m4.1.1.1" xref="alg6.l1.m4.1.1.1.cmml"><</mo><msub id="alg6.l1.m4.1.1.3" xref="alg6.l1.m4.1.1.3.cmml"><mi id="alg6.l1.m4.1.1.3.2" xref="alg6.l1.m4.1.1.3.2.cmml">t</mi><mrow id="alg6.l1.m4.1.1.3.3" xref="alg6.l1.m4.1.1.3.3.cmml"><mtext id="alg6.l1.m4.1.1.3.3.2" xref="alg6.l1.m4.1.1.3.3.2a.cmml">start</mtext><mo id="alg6.l1.m4.1.1.3.3.1" xref="alg6.l1.m4.1.1.3.3.1.cmml"><</mo><mi id="alg6.l1.m4.1.1.3.3.3" xref="alg6.l1.m4.1.1.3.3.3.cmml">T</mi></mrow></msub></mrow><annotation-xml encoding="MathML-Content" id="alg6.l1.m4.1b"><apply id="alg6.l1.m4.1.1.cmml" xref="alg6.l1.m4.1.1"><lt id="alg6.l1.m4.1.1.1.cmml" xref="alg6.l1.m4.1.1.1"></lt><cn id="alg6.l1.m4.1.1.2.cmml" type="integer" xref="alg6.l1.m4.1.1.2">0</cn><apply id="alg6.l1.m4.1.1.3.cmml" xref="alg6.l1.m4.1.1.3"><csymbol cd="ambiguous" id="alg6.l1.m4.1.1.3.1.cmml" xref="alg6.l1.m4.1.1.3">subscript</csymbol><ci id="alg6.l1.m4.1.1.3.2.cmml" xref="alg6.l1.m4.1.1.3.2">𝑡</ci><apply id="alg6.l1.m4.1.1.3.3.cmml" xref="alg6.l1.m4.1.1.3.3"><lt id="alg6.l1.m4.1.1.3.3.1.cmml" xref="alg6.l1.m4.1.1.3.3.1"></lt><ci id="alg6.l1.m4.1.1.3.3.2a.cmml" xref="alg6.l1.m4.1.1.3.3.2"><mtext id="alg6.l1.m4.1.1.3.3.2.cmml" mathsize="70%" xref="alg6.l1.m4.1.1.3.3.2">start</mtext></ci><ci id="alg6.l1.m4.1.1.3.3.3.cmml" xref="alg6.l1.m4.1.1.3.3.3">𝑇</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="alg6.l1.m4.1c">0<t_{\text{start}<T}</annotation><annotation encoding="application/x-llamapun" id="alg6.l1.m4.1d">0 < italic_t start_POSTSUBSCRIPT start < italic_T end_POSTSUBSCRIPT</annotation></semantics></math>, <math alttext="\zeta>0" class="ltx_Math" display="inline" id="alg6.l1.m5.1"><semantics id="alg6.l1.m5.1a"><mrow id="alg6.l1.m5.1.1" xref="alg6.l1.m5.1.1.cmml"><mi id="alg6.l1.m5.1.1.2" xref="alg6.l1.m5.1.1.2.cmml">ζ</mi><mo id="alg6.l1.m5.1.1.1" xref="alg6.l1.m5.1.1.1.cmml">></mo><mn id="alg6.l1.m5.1.1.3" xref="alg6.l1.m5.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="alg6.l1.m5.1b"><apply id="alg6.l1.m5.1.1.cmml" xref="alg6.l1.m5.1.1"><gt id="alg6.l1.m5.1.1.1.cmml" xref="alg6.l1.m5.1.1.1"></gt><ci id="alg6.l1.m5.1.1.2.cmml" xref="alg6.l1.m5.1.1.2">𝜁</ci><cn id="alg6.l1.m5.1.1.3.cmml" type="integer" xref="alg6.l1.m5.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="alg6.l1.m5.1c">\zeta>0</annotation><annotation encoding="application/x-llamapun" id="alg6.l1.m5.1d">italic_ζ > 0</annotation></semantics></math>, <math alttext="(\beta_{t})_{0<t<T}" class="ltx_Math" display="inline" id="alg6.l1.m6.1"><semantics id="alg6.l1.m6.1a"><msub id="alg6.l1.m6.1.1" xref="alg6.l1.m6.1.1.cmml"><mrow id="alg6.l1.m6.1.1.1.1" xref="alg6.l1.m6.1.1.1.1.1.cmml"><mo id="alg6.l1.m6.1.1.1.1.2" stretchy="false" xref="alg6.l1.m6.1.1.1.1.1.cmml">(</mo><msub id="alg6.l1.m6.1.1.1.1.1" xref="alg6.l1.m6.1.1.1.1.1.cmml"><mi id="alg6.l1.m6.1.1.1.1.1.2" xref="alg6.l1.m6.1.1.1.1.1.2.cmml">β</mi><mi id="alg6.l1.m6.1.1.1.1.1.3" xref="alg6.l1.m6.1.1.1.1.1.3.cmml">t</mi></msub><mo id="alg6.l1.m6.1.1.1.1.3" stretchy="false" xref="alg6.l1.m6.1.1.1.1.1.cmml">)</mo></mrow><mrow id="alg6.l1.m6.1.1.3" xref="alg6.l1.m6.1.1.3.cmml"><mn id="alg6.l1.m6.1.1.3.2" xref="alg6.l1.m6.1.1.3.2.cmml">0</mn><mo id="alg6.l1.m6.1.1.3.3" xref="alg6.l1.m6.1.1.3.3.cmml"><</mo><mi id="alg6.l1.m6.1.1.3.4" xref="alg6.l1.m6.1.1.3.4.cmml">t</mi><mo id="alg6.l1.m6.1.1.3.5" xref="alg6.l1.m6.1.1.3.5.cmml"><</mo><mi id="alg6.l1.m6.1.1.3.6" xref="alg6.l1.m6.1.1.3.6.cmml">T</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="alg6.l1.m6.1b"><apply id="alg6.l1.m6.1.1.cmml" xref="alg6.l1.m6.1.1"><csymbol cd="ambiguous" id="alg6.l1.m6.1.1.2.cmml" xref="alg6.l1.m6.1.1">subscript</csymbol><apply id="alg6.l1.m6.1.1.1.1.1.cmml" xref="alg6.l1.m6.1.1.1.1"><csymbol cd="ambiguous" id="alg6.l1.m6.1.1.1.1.1.1.cmml" xref="alg6.l1.m6.1.1.1.1">subscript</csymbol><ci id="alg6.l1.m6.1.1.1.1.1.2.cmml" xref="alg6.l1.m6.1.1.1.1.1.2">𝛽</ci><ci id="alg6.l1.m6.1.1.1.1.1.3.cmml" xref="alg6.l1.m6.1.1.1.1.1.3">𝑡</ci></apply><apply id="alg6.l1.m6.1.1.3.cmml" xref="alg6.l1.m6.1.1.3"><and id="alg6.l1.m6.1.1.3a.cmml" xref="alg6.l1.m6.1.1.3"></and><apply id="alg6.l1.m6.1.1.3b.cmml" xref="alg6.l1.m6.1.1.3"><lt id="alg6.l1.m6.1.1.3.3.cmml" xref="alg6.l1.m6.1.1.3.3"></lt><cn id="alg6.l1.m6.1.1.3.2.cmml" type="integer" xref="alg6.l1.m6.1.1.3.2">0</cn><ci id="alg6.l1.m6.1.1.3.4.cmml" xref="alg6.l1.m6.1.1.3.4">𝑡</ci></apply><apply id="alg6.l1.m6.1.1.3c.cmml" xref="alg6.l1.m6.1.1.3"><lt id="alg6.l1.m6.1.1.3.5.cmml" xref="alg6.l1.m6.1.1.3.5"></lt><share href="#alg6.l1.m6.1.1.3.4.cmml" id="alg6.l1.m6.1.1.3d.cmml" xref="alg6.l1.m6.1.1.3"></share><ci id="alg6.l1.m6.1.1.3.6.cmml" xref="alg6.l1.m6.1.1.3.6">𝑇</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="alg6.l1.m6.1c">(\beta_{t})_{0<t<T}</annotation><annotation encoding="application/x-llamapun" id="alg6.l1.m6.1d">( italic_β start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT ) start_POSTSUBSCRIPT 0 < italic_t < italic_T end_POSTSUBSCRIPT</annotation></semantics></math>, <math alttext="\lambda>0" class="ltx_Math" display="inline" id="alg6.l1.m7.1"><semantics id="alg6.l1.m7.1a"><mrow id="alg6.l1.m7.1.1" xref="alg6.l1.m7.1.1.cmml"><mi id="alg6.l1.m7.1.1.2" xref="alg6.l1.m7.1.1.2.cmml">λ</mi><mo id="alg6.l1.m7.1.1.1" xref="alg6.l1.m7.1.1.1.cmml">></mo><mn id="alg6.l1.m7.1.1.3" xref="alg6.l1.m7.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="alg6.l1.m7.1b"><apply id="alg6.l1.m7.1.1.cmml" xref="alg6.l1.m7.1.1"><gt id="alg6.l1.m7.1.1.1.cmml" xref="alg6.l1.m7.1.1.1"></gt><ci id="alg6.l1.m7.1.1.2.cmml" xref="alg6.l1.m7.1.1.2">𝜆</ci><cn id="alg6.l1.m7.1.1.3.cmml" type="integer" xref="alg6.l1.m7.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="alg6.l1.m7.1c">\lambda>0</annotation><annotation encoding="application/x-llamapun" id="alg6.l1.m7.1d">italic_λ > 0</annotation></semantics></math> </div> <div class="ltx_listingline" id="alg6.l2"> <span class="ltx_tag ltx_tag_listingline"><span class="ltx_text" id="alg6.l2.1.1.1" style="font-size:80%;">2:</span></span> <span class="ltx_text" id="alg6.l2.2">Initialize </span><math alttext="{\boldsymbol{\epsilon}}_{t_{\text{start}}}\sim\mathcal{N}(0,\mathbf{I}_{d})" class="ltx_Math" display="inline" id="alg6.l2.m1.2"><semantics id="alg6.l2.m1.2a"><mrow id="alg6.l2.m1.2.2" xref="alg6.l2.m1.2.2.cmml"><msub id="alg6.l2.m1.2.2.3" xref="alg6.l2.m1.2.2.3.cmml"><mi id="alg6.l2.m1.2.2.3.2" mathvariant="bold-italic" xref="alg6.l2.m1.2.2.3.2.cmml">ϵ</mi><msub id="alg6.l2.m1.2.2.3.3" xref="alg6.l2.m1.2.2.3.3.cmml"><mi id="alg6.l2.m1.2.2.3.3.2" xref="alg6.l2.m1.2.2.3.3.2.cmml">t</mi><mtext id="alg6.l2.m1.2.2.3.3.3" xref="alg6.l2.m1.2.2.3.3.3a.cmml">start</mtext></msub></msub><mo id="alg6.l2.m1.2.2.2" xref="alg6.l2.m1.2.2.2.cmml">∼</mo><mrow id="alg6.l2.m1.2.2.1" xref="alg6.l2.m1.2.2.1.cmml"><mi class="ltx_font_mathcaligraphic" id="alg6.l2.m1.2.2.1.3" xref="alg6.l2.m1.2.2.1.3.cmml">𝒩</mi><mo id="alg6.l2.m1.2.2.1.2" xref="alg6.l2.m1.2.2.1.2.cmml">⁢</mo><mrow id="alg6.l2.m1.2.2.1.1.1" xref="alg6.l2.m1.2.2.1.1.2.cmml"><mo id="alg6.l2.m1.2.2.1.1.1.2" stretchy="false" xref="alg6.l2.m1.2.2.1.1.2.cmml">(</mo><mn id="alg6.l2.m1.1.1" xref="alg6.l2.m1.1.1.cmml">0</mn><mo id="alg6.l2.m1.2.2.1.1.1.3" xref="alg6.l2.m1.2.2.1.1.2.cmml">,</mo><msub id="alg6.l2.m1.2.2.1.1.1.1" xref="alg6.l2.m1.2.2.1.1.1.1.cmml"><mi id="alg6.l2.m1.2.2.1.1.1.1.2" xref="alg6.l2.m1.2.2.1.1.1.1.2.cmml">𝐈</mi><mi id="alg6.l2.m1.2.2.1.1.1.1.3" xref="alg6.l2.m1.2.2.1.1.1.1.3.cmml">d</mi></msub><mo id="alg6.l2.m1.2.2.1.1.1.4" stretchy="false" xref="alg6.l2.m1.2.2.1.1.2.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="alg6.l2.m1.2b"><apply id="alg6.l2.m1.2.2.cmml" xref="alg6.l2.m1.2.2"><csymbol cd="latexml" id="alg6.l2.m1.2.2.2.cmml" xref="alg6.l2.m1.2.2.2">similar-to</csymbol><apply id="alg6.l2.m1.2.2.3.cmml" xref="alg6.l2.m1.2.2.3"><csymbol cd="ambiguous" id="alg6.l2.m1.2.2.3.1.cmml" xref="alg6.l2.m1.2.2.3">subscript</csymbol><ci id="alg6.l2.m1.2.2.3.2.cmml" xref="alg6.l2.m1.2.2.3.2">bold-italic-ϵ</ci><apply id="alg6.l2.m1.2.2.3.3.cmml" xref="alg6.l2.m1.2.2.3.3"><csymbol cd="ambiguous" id="alg6.l2.m1.2.2.3.3.1.cmml" xref="alg6.l2.m1.2.2.3.3">subscript</csymbol><ci id="alg6.l2.m1.2.2.3.3.2.cmml" xref="alg6.l2.m1.2.2.3.3.2">𝑡</ci><ci id="alg6.l2.m1.2.2.3.3.3a.cmml" xref="alg6.l2.m1.2.2.3.3.3"><mtext id="alg6.l2.m1.2.2.3.3.3.cmml" mathsize="50%" xref="alg6.l2.m1.2.2.3.3.3">start</mtext></ci></apply></apply><apply id="alg6.l2.m1.2.2.1.cmml" xref="alg6.l2.m1.2.2.1"><times id="alg6.l2.m1.2.2.1.2.cmml" xref="alg6.l2.m1.2.2.1.2"></times><ci id="alg6.l2.m1.2.2.1.3.cmml" xref="alg6.l2.m1.2.2.1.3">𝒩</ci><interval closure="open" id="alg6.l2.m1.2.2.1.1.2.cmml" xref="alg6.l2.m1.2.2.1.1.1"><cn id="alg6.l2.m1.1.1.cmml" type="integer" xref="alg6.l2.m1.1.1">0</cn><apply id="alg6.l2.m1.2.2.1.1.1.1.cmml" xref="alg6.l2.m1.2.2.1.1.1.1"><csymbol cd="ambiguous" id="alg6.l2.m1.2.2.1.1.1.1.1.cmml" xref="alg6.l2.m1.2.2.1.1.1.1">subscript</csymbol><ci id="alg6.l2.m1.2.2.1.1.1.1.2.cmml" xref="alg6.l2.m1.2.2.1.1.1.1.2">𝐈</ci><ci id="alg6.l2.m1.2.2.1.1.1.1.3.cmml" xref="alg6.l2.m1.2.2.1.1.1.1.3">𝑑</ci></apply></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="alg6.l2.m1.2c">{\boldsymbol{\epsilon}}_{t_{\text{start}}}\sim\mathcal{N}(0,\mathbf{I}_{d})</annotation><annotation encoding="application/x-llamapun" id="alg6.l2.m1.2d">bold_italic_ϵ start_POSTSUBSCRIPT italic_t start_POSTSUBSCRIPT start end_POSTSUBSCRIPT end_POSTSUBSCRIPT ∼ caligraphic_N ( 0 , bold_I start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT )</annotation></semantics></math>, <span class="ltx_text" id="alg6.l2.3">pre-calculate </span><math alttext="(\bar{\alpha}_{t})_{0<t<T}" class="ltx_Math" display="inline" id="alg6.l2.m2.1"><semantics id="alg6.l2.m2.1a"><msub id="alg6.l2.m2.1.1" xref="alg6.l2.m2.1.1.cmml"><mrow id="alg6.l2.m2.1.1.1.1" xref="alg6.l2.m2.1.1.1.1.1.cmml"><mo id="alg6.l2.m2.1.1.1.1.2" stretchy="false" xref="alg6.l2.m2.1.1.1.1.1.cmml">(</mo><msub id="alg6.l2.m2.1.1.1.1.1" xref="alg6.l2.m2.1.1.1.1.1.cmml"><mover accent="true" id="alg6.l2.m2.1.1.1.1.1.2" xref="alg6.l2.m2.1.1.1.1.1.2.cmml"><mi id="alg6.l2.m2.1.1.1.1.1.2.2" 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id="alg6.l2.m2.1.1.1.1.1.1.cmml" xref="alg6.l2.m2.1.1.1.1">subscript</csymbol><apply id="alg6.l2.m2.1.1.1.1.1.2.cmml" xref="alg6.l2.m2.1.1.1.1.1.2"><ci id="alg6.l2.m2.1.1.1.1.1.2.1.cmml" xref="alg6.l2.m2.1.1.1.1.1.2.1">¯</ci><ci id="alg6.l2.m2.1.1.1.1.1.2.2.cmml" xref="alg6.l2.m2.1.1.1.1.1.2.2">𝛼</ci></apply><ci id="alg6.l2.m2.1.1.1.1.1.3.cmml" xref="alg6.l2.m2.1.1.1.1.1.3">𝑡</ci></apply><apply id="alg6.l2.m2.1.1.3.cmml" xref="alg6.l2.m2.1.1.3"><and id="alg6.l2.m2.1.1.3a.cmml" xref="alg6.l2.m2.1.1.3"></and><apply id="alg6.l2.m2.1.1.3b.cmml" xref="alg6.l2.m2.1.1.3"><lt id="alg6.l2.m2.1.1.3.3.cmml" xref="alg6.l2.m2.1.1.3.3"></lt><cn id="alg6.l2.m2.1.1.3.2.cmml" type="integer" xref="alg6.l2.m2.1.1.3.2">0</cn><ci id="alg6.l2.m2.1.1.3.4.cmml" xref="alg6.l2.m2.1.1.3.4">𝑡</ci></apply><apply id="alg6.l2.m2.1.1.3c.cmml" xref="alg6.l2.m2.1.1.3"><lt id="alg6.l2.m2.1.1.3.5.cmml" xref="alg6.l2.m2.1.1.3.5"></lt><share href="#alg6.l2.m2.1.1.3.4.cmml" id="alg6.l2.m2.1.1.3d.cmml" xref="alg6.l2.m2.1.1.3"></share><ci id="alg6.l2.m2.1.1.3.6.cmml" xref="alg6.l2.m2.1.1.3.6">𝑇</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="alg6.l2.m2.1c">(\bar{\alpha}_{t})_{0<t<T}</annotation><annotation encoding="application/x-llamapun" id="alg6.l2.m2.1d">( over¯ start_ARG italic_α end_ARG start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT ) start_POSTSUBSCRIPT 0 < italic_t < italic_T end_POSTSUBSCRIPT</annotation></semantics></math>, <math alttext="(\sigma_{t})_{0<t<T}" class="ltx_Math" display="inline" id="alg6.l2.m3.1"><semantics id="alg6.l2.m3.1a"><msub id="alg6.l2.m3.1.1" xref="alg6.l2.m3.1.1.cmml"><mrow id="alg6.l2.m3.1.1.1.1" xref="alg6.l2.m3.1.1.1.1.1.cmml"><mo id="alg6.l2.m3.1.1.1.1.2" stretchy="false" xref="alg6.l2.m3.1.1.1.1.1.cmml">(</mo><msub id="alg6.l2.m3.1.1.1.1.1" xref="alg6.l2.m3.1.1.1.1.1.cmml"><mi id="alg6.l2.m3.1.1.1.1.1.2" xref="alg6.l2.m3.1.1.1.1.1.2.cmml">σ</mi><mi id="alg6.l2.m3.1.1.1.1.1.3" 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id="alg6.l2.m3.1.1.1.1.1.3.cmml" xref="alg6.l2.m3.1.1.1.1.1.3">𝑡</ci></apply><apply id="alg6.l2.m3.1.1.3.cmml" xref="alg6.l2.m3.1.1.3"><and id="alg6.l2.m3.1.1.3a.cmml" xref="alg6.l2.m3.1.1.3"></and><apply id="alg6.l2.m3.1.1.3b.cmml" xref="alg6.l2.m3.1.1.3"><lt id="alg6.l2.m3.1.1.3.3.cmml" xref="alg6.l2.m3.1.1.3.3"></lt><cn id="alg6.l2.m3.1.1.3.2.cmml" type="integer" xref="alg6.l2.m3.1.1.3.2">0</cn><ci id="alg6.l2.m3.1.1.3.4.cmml" xref="alg6.l2.m3.1.1.3.4">𝑡</ci></apply><apply id="alg6.l2.m3.1.1.3c.cmml" xref="alg6.l2.m3.1.1.3"><lt id="alg6.l2.m3.1.1.3.5.cmml" xref="alg6.l2.m3.1.1.3.5"></lt><share href="#alg6.l2.m3.1.1.3.4.cmml" id="alg6.l2.m3.1.1.3d.cmml" xref="alg6.l2.m3.1.1.3"></share><ci id="alg6.l2.m3.1.1.3.6.cmml" xref="alg6.l2.m3.1.1.3.6">𝑇</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="alg6.l2.m3.1c">(\sigma_{t})_{0<t<T}</annotation><annotation encoding="application/x-llamapun" id="alg6.l2.m3.1d">( italic_σ start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT ) start_POSTSUBSCRIPT 0 < italic_t < italic_T end_POSTSUBSCRIPT</annotation></semantics></math> <span class="ltx_text" id="alg6.l2.4"> and </span> <math alttext="(\rho_{t})_{0<t<T}" class="ltx_Math" display="inline" id="alg6.l2.m4.1"><semantics id="alg6.l2.m4.1a"><msub id="alg6.l2.m4.1.1" xref="alg6.l2.m4.1.1.cmml"><mrow id="alg6.l2.m4.1.1.1.1" xref="alg6.l2.m4.1.1.1.1.1.cmml"><mo id="alg6.l2.m4.1.1.1.1.2" stretchy="false" xref="alg6.l2.m4.1.1.1.1.1.cmml">(</mo><msub id="alg6.l2.m4.1.1.1.1.1" xref="alg6.l2.m4.1.1.1.1.1.cmml"><mi id="alg6.l2.m4.1.1.1.1.1.2" xref="alg6.l2.m4.1.1.1.1.1.2.cmml">ρ</mi><mi id="alg6.l2.m4.1.1.1.1.1.3" xref="alg6.l2.m4.1.1.1.1.1.3.cmml">t</mi></msub><mo id="alg6.l2.m4.1.1.1.1.3" stretchy="false" xref="alg6.l2.m4.1.1.1.1.1.cmml">)</mo></mrow><mrow id="alg6.l2.m4.1.1.3" xref="alg6.l2.m4.1.1.3.cmml"><mn id="alg6.l2.m4.1.1.3.2" xref="alg6.l2.m4.1.1.3.2.cmml">0</mn><mo id="alg6.l2.m4.1.1.3.3" xref="alg6.l2.m4.1.1.3.3.cmml"><</mo><mi id="alg6.l2.m4.1.1.3.4" xref="alg6.l2.m4.1.1.3.4.cmml">t</mi><mo id="alg6.l2.m4.1.1.3.5" xref="alg6.l2.m4.1.1.3.5.cmml"><</mo><mi id="alg6.l2.m4.1.1.3.6" xref="alg6.l2.m4.1.1.3.6.cmml">T</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="alg6.l2.m4.1b"><apply id="alg6.l2.m4.1.1.cmml" xref="alg6.l2.m4.1.1"><csymbol cd="ambiguous" id="alg6.l2.m4.1.1.2.cmml" xref="alg6.l2.m4.1.1">subscript</csymbol><apply id="alg6.l2.m4.1.1.1.1.1.cmml" xref="alg6.l2.m4.1.1.1.1"><csymbol cd="ambiguous" id="alg6.l2.m4.1.1.1.1.1.1.cmml" xref="alg6.l2.m4.1.1.1.1">subscript</csymbol><ci id="alg6.l2.m4.1.1.1.1.1.2.cmml" xref="alg6.l2.m4.1.1.1.1.1.2">𝜌</ci><ci id="alg6.l2.m4.1.1.1.1.1.3.cmml" xref="alg6.l2.m4.1.1.1.1.1.3">𝑡</ci></apply><apply id="alg6.l2.m4.1.1.3.cmml" xref="alg6.l2.m4.1.1.3"><and id="alg6.l2.m4.1.1.3a.cmml" xref="alg6.l2.m4.1.1.3"></and><apply id="alg6.l2.m4.1.1.3b.cmml" xref="alg6.l2.m4.1.1.3"><lt id="alg6.l2.m4.1.1.3.3.cmml" xref="alg6.l2.m4.1.1.3.3"></lt><cn id="alg6.l2.m4.1.1.3.2.cmml" type="integer" xref="alg6.l2.m4.1.1.3.2">0</cn><ci id="alg6.l2.m4.1.1.3.4.cmml" xref="alg6.l2.m4.1.1.3.4">𝑡</ci></apply><apply id="alg6.l2.m4.1.1.3c.cmml" xref="alg6.l2.m4.1.1.3"><lt id="alg6.l2.m4.1.1.3.5.cmml" xref="alg6.l2.m4.1.1.3.5"></lt><share href="#alg6.l2.m4.1.1.3.4.cmml" id="alg6.l2.m4.1.1.3d.cmml" xref="alg6.l2.m4.1.1.3"></share><ci id="alg6.l2.m4.1.1.3.6.cmml" xref="alg6.l2.m4.1.1.3.6">𝑇</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="alg6.l2.m4.1c">(\rho_{t})_{0<t<T}</annotation><annotation encoding="application/x-llamapun" id="alg6.l2.m4.1d">( italic_ρ start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT ) start_POSTSUBSCRIPT 0 < italic_t < italic_T end_POSTSUBSCRIPT</annotation></semantics></math> </div> <div class="ltx_listingline" id="alg6.l3"> <span class="ltx_tag ltx_tag_listingline"><span class="ltx_text" id="alg6.l3.1.1.1" style="font-size:80%;">3:</span></span> <math alttext="x_{t_{\text{start}}}=\sqrt{\bar{\alpha}_{\text{start}}}{\mathbf{y}}+\sqrt{1-% \bar{\alpha}_{\text{start}}}{\boldsymbol{\epsilon}}_{t_{\text{start}}}" class="ltx_Math" display="inline" id="alg6.l3.m1.1"><semantics id="alg6.l3.m1.1a"><mrow id="alg6.l3.m1.1.1" xref="alg6.l3.m1.1.1.cmml"><msub id="alg6.l3.m1.1.1.2" xref="alg6.l3.m1.1.1.2.cmml"><mi id="alg6.l3.m1.1.1.2.2" xref="alg6.l3.m1.1.1.2.2.cmml">x</mi><msub id="alg6.l3.m1.1.1.2.3" xref="alg6.l3.m1.1.1.2.3.cmml"><mi id="alg6.l3.m1.1.1.2.3.2" xref="alg6.l3.m1.1.1.2.3.2.cmml">t</mi><mtext id="alg6.l3.m1.1.1.2.3.3" xref="alg6.l3.m1.1.1.2.3.3a.cmml">start</mtext></msub></msub><mo id="alg6.l3.m1.1.1.1" xref="alg6.l3.m1.1.1.1.cmml">=</mo><mrow id="alg6.l3.m1.1.1.3" xref="alg6.l3.m1.1.1.3.cmml"><mrow id="alg6.l3.m1.1.1.3.2" xref="alg6.l3.m1.1.1.3.2.cmml"><msqrt id="alg6.l3.m1.1.1.3.2.2" xref="alg6.l3.m1.1.1.3.2.2.cmml"><msub id="alg6.l3.m1.1.1.3.2.2.2" xref="alg6.l3.m1.1.1.3.2.2.2.cmml"><mover accent="true" id="alg6.l3.m1.1.1.3.2.2.2.2" xref="alg6.l3.m1.1.1.3.2.2.2.2.cmml"><mi id="alg6.l3.m1.1.1.3.2.2.2.2.2" xref="alg6.l3.m1.1.1.3.2.2.2.2.2.cmml">α</mi><mo id="alg6.l3.m1.1.1.3.2.2.2.2.1" xref="alg6.l3.m1.1.1.3.2.2.2.2.1.cmml">¯</mo></mover><mtext id="alg6.l3.m1.1.1.3.2.2.2.3" xref="alg6.l3.m1.1.1.3.2.2.2.3a.cmml">start</mtext></msub></msqrt><mo id="alg6.l3.m1.1.1.3.2.1" xref="alg6.l3.m1.1.1.3.2.1.cmml">⁢</mo><mi id="alg6.l3.m1.1.1.3.2.3" xref="alg6.l3.m1.1.1.3.2.3.cmml">𝐲</mi></mrow><mo id="alg6.l3.m1.1.1.3.1" xref="alg6.l3.m1.1.1.3.1.cmml">+</mo><mrow id="alg6.l3.m1.1.1.3.3" xref="alg6.l3.m1.1.1.3.3.cmml"><msqrt id="alg6.l3.m1.1.1.3.3.2" xref="alg6.l3.m1.1.1.3.3.2.cmml"><mrow id="alg6.l3.m1.1.1.3.3.2.2" xref="alg6.l3.m1.1.1.3.3.2.2.cmml"><mn id="alg6.l3.m1.1.1.3.3.2.2.2" xref="alg6.l3.m1.1.1.3.3.2.2.2.cmml">1</mn><mo id="alg6.l3.m1.1.1.3.3.2.2.1" xref="alg6.l3.m1.1.1.3.3.2.2.1.cmml">−</mo><msub id="alg6.l3.m1.1.1.3.3.2.2.3" xref="alg6.l3.m1.1.1.3.3.2.2.3.cmml"><mover accent="true" 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xref="alg6.l3.m1.1.1.3.3.2.2.3.3">start</mtext></ci></apply></apply></apply><apply id="alg6.l3.m1.1.1.3.3.3.cmml" xref="alg6.l3.m1.1.1.3.3.3"><csymbol cd="ambiguous" id="alg6.l3.m1.1.1.3.3.3.1.cmml" xref="alg6.l3.m1.1.1.3.3.3">subscript</csymbol><ci id="alg6.l3.m1.1.1.3.3.3.2.cmml" xref="alg6.l3.m1.1.1.3.3.3.2">bold-italic-ϵ</ci><apply id="alg6.l3.m1.1.1.3.3.3.3.cmml" xref="alg6.l3.m1.1.1.3.3.3.3"><csymbol cd="ambiguous" id="alg6.l3.m1.1.1.3.3.3.3.1.cmml" xref="alg6.l3.m1.1.1.3.3.3.3">subscript</csymbol><ci id="alg6.l3.m1.1.1.3.3.3.3.2.cmml" xref="alg6.l3.m1.1.1.3.3.3.3.2">𝑡</ci><ci id="alg6.l3.m1.1.1.3.3.3.3.3a.cmml" xref="alg6.l3.m1.1.1.3.3.3.3.3"><mtext id="alg6.l3.m1.1.1.3.3.3.3.3.cmml" mathsize="50%" xref="alg6.l3.m1.1.1.3.3.3.3.3">start</mtext></ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="alg6.l3.m1.1c">x_{t_{\text{start}}}=\sqrt{\bar{\alpha}_{\text{start}}}{\mathbf{y}}+\sqrt{1-% \bar{\alpha}_{\text{start}}}{\boldsymbol{\epsilon}}_{t_{\text{start}}}</annotation><annotation encoding="application/x-llamapun" id="alg6.l3.m1.1d">italic_x start_POSTSUBSCRIPT italic_t start_POSTSUBSCRIPT start end_POSTSUBSCRIPT end_POSTSUBSCRIPT = square-root start_ARG over¯ start_ARG italic_α end_ARG start_POSTSUBSCRIPT start end_POSTSUBSCRIPT end_ARG bold_y + square-root start_ARG 1 - over¯ start_ARG italic_α end_ARG start_POSTSUBSCRIPT start end_POSTSUBSCRIPT end_ARG bold_italic_ϵ start_POSTSUBSCRIPT italic_t start_POSTSUBSCRIPT start end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math> </div> <div class="ltx_listingline" id="alg6.l4"> <span class="ltx_tag ltx_tag_listingline"><span class="ltx_text" id="alg6.l4.1.1.1" style="font-size:80%;">4:</span></span> <span class="ltx_text ltx_font_bold" id="alg6.l4.2">for</span> <math alttext="t=t_{\text{start}},t_{\text{start}}-1,\dots,1" class="ltx_Math" display="inline" id="alg6.l4.m1.4"><semantics id="alg6.l4.m1.4a"><mrow id="alg6.l4.m1.4.4" xref="alg6.l4.m1.4.4.cmml"><mi id="alg6.l4.m1.4.4.4" xref="alg6.l4.m1.4.4.4.cmml">t</mi><mo id="alg6.l4.m1.4.4.3" xref="alg6.l4.m1.4.4.3.cmml">=</mo><mrow id="alg6.l4.m1.4.4.2.2" xref="alg6.l4.m1.4.4.2.3.cmml"><msub id="alg6.l4.m1.3.3.1.1.1" xref="alg6.l4.m1.3.3.1.1.1.cmml"><mi id="alg6.l4.m1.3.3.1.1.1.2" xref="alg6.l4.m1.3.3.1.1.1.2.cmml">t</mi><mtext id="alg6.l4.m1.3.3.1.1.1.3" xref="alg6.l4.m1.3.3.1.1.1.3a.cmml">start</mtext></msub><mo id="alg6.l4.m1.4.4.2.2.3" xref="alg6.l4.m1.4.4.2.3.cmml">,</mo><mrow id="alg6.l4.m1.4.4.2.2.2" xref="alg6.l4.m1.4.4.2.2.2.cmml"><msub id="alg6.l4.m1.4.4.2.2.2.2" xref="alg6.l4.m1.4.4.2.2.2.2.cmml"><mi id="alg6.l4.m1.4.4.2.2.2.2.2" xref="alg6.l4.m1.4.4.2.2.2.2.2.cmml">t</mi><mtext id="alg6.l4.m1.4.4.2.2.2.2.3" xref="alg6.l4.m1.4.4.2.2.2.2.3a.cmml">start</mtext></msub><mo id="alg6.l4.m1.4.4.2.2.2.1" xref="alg6.l4.m1.4.4.2.2.2.1.cmml">−</mo><mn id="alg6.l4.m1.4.4.2.2.2.3" xref="alg6.l4.m1.4.4.2.2.2.3.cmml">1</mn></mrow><mo 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xref="alg6.l5.m1.1.1.1.3"><csymbol cd="ambiguous" id="alg6.l5.m1.1.1.1.3.1.cmml" xref="alg6.l5.m1.1.1.1.3">subscript</csymbol><ci id="alg6.l5.m1.1.1.1.3.2.cmml" xref="alg6.l5.m1.1.1.1.3.2">𝐷</ci><apply id="alg6.l5.m1.1.1.1.3.3.cmml" xref="alg6.l5.m1.1.1.1.3.3"><csymbol cd="ambiguous" id="alg6.l5.m1.1.1.1.3.3.1.cmml" xref="alg6.l5.m1.1.1.1.3.3">subscript</csymbol><ci id="alg6.l5.m1.1.1.1.3.3.2.cmml" xref="alg6.l5.m1.1.1.1.3.3.2">𝜎</ci><ci id="alg6.l5.m1.1.1.1.3.3.3.cmml" xref="alg6.l5.m1.1.1.1.3.3.3">𝑡</ci></apply></apply><apply id="alg6.l5.m1.1.1.1.1.1.1.cmml" xref="alg6.l5.m1.1.1.1.1.1"><csymbol cd="ambiguous" id="alg6.l5.m1.1.1.1.1.1.1.1.cmml" xref="alg6.l5.m1.1.1.1.1.1">subscript</csymbol><ci id="alg6.l5.m1.1.1.1.1.1.1.2.cmml" xref="alg6.l5.m1.1.1.1.1.1.1.2">𝑥</ci><ci id="alg6.l5.m1.1.1.1.1.1.1.3.cmml" xref="alg6.l5.m1.1.1.1.1.1.1.3">𝑡</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="alg6.l5.m1.1c">x_{0}^{t}\leftarrow D_{\sigma_{t}}(x_{t})</annotation><annotation encoding="application/x-llamapun" id="alg6.l5.m1.1d">italic_x start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT ← italic_D start_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_x start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT )</annotation></semantics></math> </div> <div class="ltx_listingline" id="alg6.l6"> <span class="ltx_tag ltx_tag_listingline"><span class="ltx_text" id="alg6.l6.1.1.1" style="font-size:80%;">6:</span></span> <math alttext="\hat{x_{0}}^{t}\leftarrow\text{Prox}_{2\mathcal{F}(\cdot,{\mathbf{y}})/\rho_{t% }}(x_{0}^{t})" class="ltx_Math" display="inline" id="alg6.l6.m1.3"><semantics id="alg6.l6.m1.3a"><mrow id="alg6.l6.m1.3.3" xref="alg6.l6.m1.3.3.cmml"><msup id="alg6.l6.m1.3.3.3" xref="alg6.l6.m1.3.3.3.cmml"><mover accent="true" id="alg6.l6.m1.3.3.3.2" xref="alg6.l6.m1.3.3.3.2.cmml"><msub id="alg6.l6.m1.3.3.3.2.2" xref="alg6.l6.m1.3.3.3.2.2.cmml"><mi id="alg6.l6.m1.3.3.3.2.2.2" xref="alg6.l6.m1.3.3.3.2.2.2.cmml">x</mi><mn id="alg6.l6.m1.3.3.3.2.2.3" xref="alg6.l6.m1.3.3.3.2.2.3.cmml">0</mn></msub><mo id="alg6.l6.m1.3.3.3.2.1" xref="alg6.l6.m1.3.3.3.2.1.cmml">^</mo></mover><mi id="alg6.l6.m1.3.3.3.3" xref="alg6.l6.m1.3.3.3.3.cmml">t</mi></msup><mo id="alg6.l6.m1.3.3.2" stretchy="false" xref="alg6.l6.m1.3.3.2.cmml">←</mo><mrow id="alg6.l6.m1.3.3.1" xref="alg6.l6.m1.3.3.1.cmml"><msub id="alg6.l6.m1.3.3.1.3" xref="alg6.l6.m1.3.3.1.3.cmml"><mtext id="alg6.l6.m1.3.3.1.3.2" xref="alg6.l6.m1.3.3.1.3.2a.cmml">Prox</mtext><mrow id="alg6.l6.m1.2.2.2" xref="alg6.l6.m1.2.2.2.cmml"><mrow id="alg6.l6.m1.2.2.2.4" xref="alg6.l6.m1.2.2.2.4.cmml"><mn id="alg6.l6.m1.2.2.2.4.2" xref="alg6.l6.m1.2.2.2.4.2.cmml">2</mn><mo id="alg6.l6.m1.2.2.2.4.1" xref="alg6.l6.m1.2.2.2.4.1.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="alg6.l6.m1.2.2.2.4.3" xref="alg6.l6.m1.2.2.2.4.3.cmml">ℱ</mi><mo id="alg6.l6.m1.2.2.2.4.1a" xref="alg6.l6.m1.2.2.2.4.1.cmml">⁢</mo><mrow id="alg6.l6.m1.2.2.2.4.4.2" xref="alg6.l6.m1.2.2.2.4.4.1.cmml"><mo id="alg6.l6.m1.2.2.2.4.4.2.1" stretchy="false" xref="alg6.l6.m1.2.2.2.4.4.1.cmml">(</mo><mo id="alg6.l6.m1.1.1.1.1" lspace="0em" rspace="0em" xref="alg6.l6.m1.1.1.1.1.cmml">⋅</mo><mo id="alg6.l6.m1.2.2.2.4.4.2.2" xref="alg6.l6.m1.2.2.2.4.4.1.cmml">,</mo><mi id="alg6.l6.m1.2.2.2.2" xref="alg6.l6.m1.2.2.2.2.cmml">𝐲</mi><mo id="alg6.l6.m1.2.2.2.4.4.2.3" stretchy="false" xref="alg6.l6.m1.2.2.2.4.4.1.cmml">)</mo></mrow></mrow><mo id="alg6.l6.m1.2.2.2.3" xref="alg6.l6.m1.2.2.2.3.cmml">/</mo><msub id="alg6.l6.m1.2.2.2.5" xref="alg6.l6.m1.2.2.2.5.cmml"><mi id="alg6.l6.m1.2.2.2.5.2" xref="alg6.l6.m1.2.2.2.5.2.cmml">ρ</mi><mi id="alg6.l6.m1.2.2.2.5.3" xref="alg6.l6.m1.2.2.2.5.3.cmml">t</mi></msub></mrow></msub><mo id="alg6.l6.m1.3.3.1.2" xref="alg6.l6.m1.3.3.1.2.cmml">⁢</mo><mrow id="alg6.l6.m1.3.3.1.1.1" xref="alg6.l6.m1.3.3.1.1.1.1.cmml"><mo id="alg6.l6.m1.3.3.1.1.1.2" stretchy="false" xref="alg6.l6.m1.3.3.1.1.1.1.cmml">(</mo><msubsup id="alg6.l6.m1.3.3.1.1.1.1" xref="alg6.l6.m1.3.3.1.1.1.1.cmml"><mi id="alg6.l6.m1.3.3.1.1.1.1.2.2" xref="alg6.l6.m1.3.3.1.1.1.1.2.2.cmml">x</mi><mn id="alg6.l6.m1.3.3.1.1.1.1.2.3" xref="alg6.l6.m1.3.3.1.1.1.1.2.3.cmml">0</mn><mi id="alg6.l6.m1.3.3.1.1.1.1.3" xref="alg6.l6.m1.3.3.1.1.1.1.3.cmml">t</mi></msubsup><mo id="alg6.l6.m1.3.3.1.1.1.3" stretchy="false" xref="alg6.l6.m1.3.3.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="alg6.l6.m1.3b"><apply id="alg6.l6.m1.3.3.cmml" xref="alg6.l6.m1.3.3"><ci id="alg6.l6.m1.3.3.2.cmml" xref="alg6.l6.m1.3.3.2">←</ci><apply id="alg6.l6.m1.3.3.3.cmml" xref="alg6.l6.m1.3.3.3"><csymbol cd="ambiguous" id="alg6.l6.m1.3.3.3.1.cmml" xref="alg6.l6.m1.3.3.3">superscript</csymbol><apply id="alg6.l6.m1.3.3.3.2.cmml" xref="alg6.l6.m1.3.3.3.2"><ci id="alg6.l6.m1.3.3.3.2.1.cmml" xref="alg6.l6.m1.3.3.3.2.1">^</ci><apply id="alg6.l6.m1.3.3.3.2.2.cmml" xref="alg6.l6.m1.3.3.3.2.2"><csymbol cd="ambiguous" id="alg6.l6.m1.3.3.3.2.2.1.cmml" xref="alg6.l6.m1.3.3.3.2.2">subscript</csymbol><ci id="alg6.l6.m1.3.3.3.2.2.2.cmml" xref="alg6.l6.m1.3.3.3.2.2.2">𝑥</ci><cn id="alg6.l6.m1.3.3.3.2.2.3.cmml" type="integer" xref="alg6.l6.m1.3.3.3.2.2.3">0</cn></apply></apply><ci id="alg6.l6.m1.3.3.3.3.cmml" xref="alg6.l6.m1.3.3.3.3">𝑡</ci></apply><apply id="alg6.l6.m1.3.3.1.cmml" xref="alg6.l6.m1.3.3.1"><times id="alg6.l6.m1.3.3.1.2.cmml" xref="alg6.l6.m1.3.3.1.2"></times><apply id="alg6.l6.m1.3.3.1.3.cmml" xref="alg6.l6.m1.3.3.1.3"><csymbol cd="ambiguous" id="alg6.l6.m1.3.3.1.3.1.cmml" xref="alg6.l6.m1.3.3.1.3">subscript</csymbol><ci id="alg6.l6.m1.3.3.1.3.2a.cmml" xref="alg6.l6.m1.3.3.1.3.2"><mtext id="alg6.l6.m1.3.3.1.3.2.cmml" xref="alg6.l6.m1.3.3.1.3.2">Prox</mtext></ci><apply id="alg6.l6.m1.2.2.2.cmml" xref="alg6.l6.m1.2.2.2"><divide id="alg6.l6.m1.2.2.2.3.cmml" xref="alg6.l6.m1.2.2.2.3"></divide><apply id="alg6.l6.m1.2.2.2.4.cmml" xref="alg6.l6.m1.2.2.2.4"><times id="alg6.l6.m1.2.2.2.4.1.cmml" xref="alg6.l6.m1.2.2.2.4.1"></times><cn id="alg6.l6.m1.2.2.2.4.2.cmml" type="integer" xref="alg6.l6.m1.2.2.2.4.2">2</cn><ci id="alg6.l6.m1.2.2.2.4.3.cmml" xref="alg6.l6.m1.2.2.2.4.3">ℱ</ci><interval closure="open" id="alg6.l6.m1.2.2.2.4.4.1.cmml" xref="alg6.l6.m1.2.2.2.4.4.2"><ci id="alg6.l6.m1.1.1.1.1.cmml" xref="alg6.l6.m1.1.1.1.1">⋅</ci><ci id="alg6.l6.m1.2.2.2.2.cmml" xref="alg6.l6.m1.2.2.2.2">𝐲</ci></interval></apply><apply id="alg6.l6.m1.2.2.2.5.cmml" xref="alg6.l6.m1.2.2.2.5"><csymbol cd="ambiguous" id="alg6.l6.m1.2.2.2.5.1.cmml" xref="alg6.l6.m1.2.2.2.5">subscript</csymbol><ci id="alg6.l6.m1.2.2.2.5.2.cmml" xref="alg6.l6.m1.2.2.2.5.2">𝜌</ci><ci id="alg6.l6.m1.2.2.2.5.3.cmml" xref="alg6.l6.m1.2.2.2.5.3">𝑡</ci></apply></apply></apply><apply id="alg6.l6.m1.3.3.1.1.1.1.cmml" xref="alg6.l6.m1.3.3.1.1.1"><csymbol cd="ambiguous" id="alg6.l6.m1.3.3.1.1.1.1.1.cmml" xref="alg6.l6.m1.3.3.1.1.1">superscript</csymbol><apply id="alg6.l6.m1.3.3.1.1.1.1.2.cmml" xref="alg6.l6.m1.3.3.1.1.1"><csymbol cd="ambiguous" id="alg6.l6.m1.3.3.1.1.1.1.2.1.cmml" xref="alg6.l6.m1.3.3.1.1.1">subscript</csymbol><ci id="alg6.l6.m1.3.3.1.1.1.1.2.2.cmml" xref="alg6.l6.m1.3.3.1.1.1.1.2.2">𝑥</ci><cn id="alg6.l6.m1.3.3.1.1.1.1.2.3.cmml" type="integer" xref="alg6.l6.m1.3.3.1.1.1.1.2.3">0</cn></apply><ci id="alg6.l6.m1.3.3.1.1.1.1.3.cmml" xref="alg6.l6.m1.3.3.1.1.1.1.3">𝑡</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="alg6.l6.m1.3c">\hat{x_{0}}^{t}\leftarrow\text{Prox}_{2\mathcal{F}(\cdot,{\mathbf{y}})/\rho_{t% }}(x_{0}^{t})</annotation><annotation encoding="application/x-llamapun" id="alg6.l6.m1.3d">over^ start_ARG italic_x start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_ARG start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT ← Prox start_POSTSUBSCRIPT 2 caligraphic_F ( ⋅ , bold_y ) / italic_ρ start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_x start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT )</annotation></semantics></math> </div> <div class="ltx_listingline" id="alg6.l7"> <span class="ltx_tag ltx_tag_listingline"><span class="ltx_text" id="alg6.l7.1.1.1" style="font-size:80%;">7:</span></span> <math alttext="\hat{{\boldsymbol{\epsilon}}}\leftarrow\left(x_{t}-\sqrt{\bar{\alpha}_{t}}\hat% {x_{0}}^{t}\right)/\sqrt{1-\bar{\alpha}_{t}}" class="ltx_Math" display="inline" id="alg6.l7.m1.1"><semantics id="alg6.l7.m1.1a"><mrow id="alg6.l7.m1.1.1" xref="alg6.l7.m1.1.1.cmml"><mover accent="true" id="alg6.l7.m1.1.1.3" xref="alg6.l7.m1.1.1.3.cmml"><mi id="alg6.l7.m1.1.1.3.2" mathvariant="bold-italic" xref="alg6.l7.m1.1.1.3.2.cmml">ϵ</mi><mo id="alg6.l7.m1.1.1.3.1" xref="alg6.l7.m1.1.1.3.1.cmml">^</mo></mover><mo id="alg6.l7.m1.1.1.2" stretchy="false" xref="alg6.l7.m1.1.1.2.cmml">←</mo><mrow id="alg6.l7.m1.1.1.1" xref="alg6.l7.m1.1.1.1.cmml"><mrow id="alg6.l7.m1.1.1.1.1.1" xref="alg6.l7.m1.1.1.1.1.1.1.cmml"><mo 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id="alg6.l7.m1.1.1.1.1.1.1.3.2.2.3" xref="alg6.l7.m1.1.1.1.1.1.1.3.2.2.3.cmml">t</mi></msub></msqrt><mo id="alg6.l7.m1.1.1.1.1.1.1.3.1" xref="alg6.l7.m1.1.1.1.1.1.1.3.1.cmml">⁢</mo><msup id="alg6.l7.m1.1.1.1.1.1.1.3.3" xref="alg6.l7.m1.1.1.1.1.1.1.3.3.cmml"><mover accent="true" id="alg6.l7.m1.1.1.1.1.1.1.3.3.2" xref="alg6.l7.m1.1.1.1.1.1.1.3.3.2.cmml"><msub id="alg6.l7.m1.1.1.1.1.1.1.3.3.2.2" xref="alg6.l7.m1.1.1.1.1.1.1.3.3.2.2.cmml"><mi id="alg6.l7.m1.1.1.1.1.1.1.3.3.2.2.2" xref="alg6.l7.m1.1.1.1.1.1.1.3.3.2.2.2.cmml">x</mi><mn id="alg6.l7.m1.1.1.1.1.1.1.3.3.2.2.3" xref="alg6.l7.m1.1.1.1.1.1.1.3.3.2.2.3.cmml">0</mn></msub><mo id="alg6.l7.m1.1.1.1.1.1.1.3.3.2.1" xref="alg6.l7.m1.1.1.1.1.1.1.3.3.2.1.cmml">^</mo></mover><mi id="alg6.l7.m1.1.1.1.1.1.1.3.3.3" xref="alg6.l7.m1.1.1.1.1.1.1.3.3.3.cmml">t</mi></msup></mrow></mrow><mo id="alg6.l7.m1.1.1.1.1.1.3" xref="alg6.l7.m1.1.1.1.1.1.1.cmml">)</mo></mrow><mo id="alg6.l7.m1.1.1.1.2" xref="alg6.l7.m1.1.1.1.2.cmml">/</mo><msqrt id="alg6.l7.m1.1.1.1.3" xref="alg6.l7.m1.1.1.1.3.cmml"><mrow id="alg6.l7.m1.1.1.1.3.2" xref="alg6.l7.m1.1.1.1.3.2.cmml"><mn id="alg6.l7.m1.1.1.1.3.2.2" xref="alg6.l7.m1.1.1.1.3.2.2.cmml">1</mn><mo id="alg6.l7.m1.1.1.1.3.2.1" xref="alg6.l7.m1.1.1.1.3.2.1.cmml">−</mo><msub id="alg6.l7.m1.1.1.1.3.2.3" xref="alg6.l7.m1.1.1.1.3.2.3.cmml"><mover accent="true" id="alg6.l7.m1.1.1.1.3.2.3.2" xref="alg6.l7.m1.1.1.1.3.2.3.2.cmml"><mi id="alg6.l7.m1.1.1.1.3.2.3.2.2" xref="alg6.l7.m1.1.1.1.3.2.3.2.2.cmml">α</mi><mo id="alg6.l7.m1.1.1.1.3.2.3.2.1" xref="alg6.l7.m1.1.1.1.3.2.3.2.1.cmml">¯</mo></mover><mi id="alg6.l7.m1.1.1.1.3.2.3.3" xref="alg6.l7.m1.1.1.1.3.2.3.3.cmml">t</mi></msub></mrow></msqrt></mrow></mrow><annotation-xml encoding="MathML-Content" id="alg6.l7.m1.1b"><apply id="alg6.l7.m1.1.1.cmml" xref="alg6.l7.m1.1.1"><ci id="alg6.l7.m1.1.1.2.cmml" xref="alg6.l7.m1.1.1.2">←</ci><apply id="alg6.l7.m1.1.1.3.cmml" xref="alg6.l7.m1.1.1.3"><ci id="alg6.l7.m1.1.1.3.1.cmml" 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id="alg6.l7.m1.1.1.1.1.1.1.3.2a.cmml" xref="alg6.l7.m1.1.1.1.1.1.1.3.2"></root><apply id="alg6.l7.m1.1.1.1.1.1.1.3.2.2.cmml" xref="alg6.l7.m1.1.1.1.1.1.1.3.2.2"><csymbol cd="ambiguous" id="alg6.l7.m1.1.1.1.1.1.1.3.2.2.1.cmml" xref="alg6.l7.m1.1.1.1.1.1.1.3.2.2">subscript</csymbol><apply id="alg6.l7.m1.1.1.1.1.1.1.3.2.2.2.cmml" xref="alg6.l7.m1.1.1.1.1.1.1.3.2.2.2"><ci id="alg6.l7.m1.1.1.1.1.1.1.3.2.2.2.1.cmml" xref="alg6.l7.m1.1.1.1.1.1.1.3.2.2.2.1">¯</ci><ci id="alg6.l7.m1.1.1.1.1.1.1.3.2.2.2.2.cmml" xref="alg6.l7.m1.1.1.1.1.1.1.3.2.2.2.2">𝛼</ci></apply><ci id="alg6.l7.m1.1.1.1.1.1.1.3.2.2.3.cmml" xref="alg6.l7.m1.1.1.1.1.1.1.3.2.2.3">𝑡</ci></apply></apply><apply id="alg6.l7.m1.1.1.1.1.1.1.3.3.cmml" xref="alg6.l7.m1.1.1.1.1.1.1.3.3"><csymbol cd="ambiguous" id="alg6.l7.m1.1.1.1.1.1.1.3.3.1.cmml" xref="alg6.l7.m1.1.1.1.1.1.1.3.3">superscript</csymbol><apply id="alg6.l7.m1.1.1.1.1.1.1.3.3.2.cmml" xref="alg6.l7.m1.1.1.1.1.1.1.3.3.2"><ci id="alg6.l7.m1.1.1.1.1.1.1.3.3.2.1.cmml" xref="alg6.l7.m1.1.1.1.1.1.1.3.3.2.1">^</ci><apply id="alg6.l7.m1.1.1.1.1.1.1.3.3.2.2.cmml" xref="alg6.l7.m1.1.1.1.1.1.1.3.3.2.2"><csymbol cd="ambiguous" id="alg6.l7.m1.1.1.1.1.1.1.3.3.2.2.1.cmml" xref="alg6.l7.m1.1.1.1.1.1.1.3.3.2.2">subscript</csymbol><ci id="alg6.l7.m1.1.1.1.1.1.1.3.3.2.2.2.cmml" xref="alg6.l7.m1.1.1.1.1.1.1.3.3.2.2.2">𝑥</ci><cn id="alg6.l7.m1.1.1.1.1.1.1.3.3.2.2.3.cmml" type="integer" xref="alg6.l7.m1.1.1.1.1.1.1.3.3.2.2.3">0</cn></apply></apply><ci id="alg6.l7.m1.1.1.1.1.1.1.3.3.3.cmml" xref="alg6.l7.m1.1.1.1.1.1.1.3.3.3">𝑡</ci></apply></apply></apply><apply id="alg6.l7.m1.1.1.1.3.cmml" xref="alg6.l7.m1.1.1.1.3"><root id="alg6.l7.m1.1.1.1.3a.cmml" xref="alg6.l7.m1.1.1.1.3"></root><apply id="alg6.l7.m1.1.1.1.3.2.cmml" xref="alg6.l7.m1.1.1.1.3.2"><minus id="alg6.l7.m1.1.1.1.3.2.1.cmml" xref="alg6.l7.m1.1.1.1.3.2.1"></minus><cn id="alg6.l7.m1.1.1.1.3.2.2.cmml" type="integer" xref="alg6.l7.m1.1.1.1.3.2.2">1</cn><apply id="alg6.l7.m1.1.1.1.3.2.3.cmml" xref="alg6.l7.m1.1.1.1.3.2.3"><csymbol cd="ambiguous" id="alg6.l7.m1.1.1.1.3.2.3.1.cmml" xref="alg6.l7.m1.1.1.1.3.2.3">subscript</csymbol><apply id="alg6.l7.m1.1.1.1.3.2.3.2.cmml" xref="alg6.l7.m1.1.1.1.3.2.3.2"><ci id="alg6.l7.m1.1.1.1.3.2.3.2.1.cmml" xref="alg6.l7.m1.1.1.1.3.2.3.2.1">¯</ci><ci id="alg6.l7.m1.1.1.1.3.2.3.2.2.cmml" xref="alg6.l7.m1.1.1.1.3.2.3.2.2">𝛼</ci></apply><ci id="alg6.l7.m1.1.1.1.3.2.3.3.cmml" xref="alg6.l7.m1.1.1.1.3.2.3.3">𝑡</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="alg6.l7.m1.1c">\hat{{\boldsymbol{\epsilon}}}\leftarrow\left(x_{t}-\sqrt{\bar{\alpha}_{t}}\hat% {x_{0}}^{t}\right)/\sqrt{1-\bar{\alpha}_{t}}</annotation><annotation encoding="application/x-llamapun" id="alg6.l7.m1.1d">over^ start_ARG bold_italic_ϵ end_ARG ← ( italic_x start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT - square-root start_ARG over¯ start_ARG italic_α end_ARG start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT end_ARG over^ start_ARG italic_x start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_ARG start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT ) / square-root start_ARG 1 - over¯ start_ARG italic_α end_ARG start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT end_ARG</annotation></semantics></math> </div> <div class="ltx_listingline" id="alg6.l8"> <span class="ltx_tag ltx_tag_listingline"><span class="ltx_text" id="alg6.l8.1.1.1" style="font-size:80%;">8:</span></span> <math alttext="{\boldsymbol{\epsilon}}_{t}\leftarrow\mathcal{N}(0,\mathbf{I}_{d})" class="ltx_Math" display="inline" id="alg6.l8.m1.2"><semantics id="alg6.l8.m1.2a"><mrow id="alg6.l8.m1.2.2" xref="alg6.l8.m1.2.2.cmml"><msub id="alg6.l8.m1.2.2.3" xref="alg6.l8.m1.2.2.3.cmml"><mi id="alg6.l8.m1.2.2.3.2" mathvariant="bold-italic" xref="alg6.l8.m1.2.2.3.2.cmml">ϵ</mi><mi id="alg6.l8.m1.2.2.3.3" xref="alg6.l8.m1.2.2.3.3.cmml">t</mi></msub><mo id="alg6.l8.m1.2.2.2" stretchy="false" xref="alg6.l8.m1.2.2.2.cmml">←</mo><mrow id="alg6.l8.m1.2.2.1" xref="alg6.l8.m1.2.2.1.cmml"><mi class="ltx_font_mathcaligraphic" id="alg6.l8.m1.2.2.1.3" xref="alg6.l8.m1.2.2.1.3.cmml">𝒩</mi><mo id="alg6.l8.m1.2.2.1.2" xref="alg6.l8.m1.2.2.1.2.cmml">⁢</mo><mrow id="alg6.l8.m1.2.2.1.1.1" xref="alg6.l8.m1.2.2.1.1.2.cmml"><mo id="alg6.l8.m1.2.2.1.1.1.2" stretchy="false" xref="alg6.l8.m1.2.2.1.1.2.cmml">(</mo><mn id="alg6.l8.m1.1.1" xref="alg6.l8.m1.1.1.cmml">0</mn><mo id="alg6.l8.m1.2.2.1.1.1.3" xref="alg6.l8.m1.2.2.1.1.2.cmml">,</mo><msub id="alg6.l8.m1.2.2.1.1.1.1" xref="alg6.l8.m1.2.2.1.1.1.1.cmml"><mi id="alg6.l8.m1.2.2.1.1.1.1.2" xref="alg6.l8.m1.2.2.1.1.1.1.2.cmml">𝐈</mi><mi id="alg6.l8.m1.2.2.1.1.1.1.3" xref="alg6.l8.m1.2.2.1.1.1.1.3.cmml">d</mi></msub><mo id="alg6.l8.m1.2.2.1.1.1.4" stretchy="false" xref="alg6.l8.m1.2.2.1.1.2.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="alg6.l8.m1.2b"><apply id="alg6.l8.m1.2.2.cmml" xref="alg6.l8.m1.2.2"><ci id="alg6.l8.m1.2.2.2.cmml" xref="alg6.l8.m1.2.2.2">←</ci><apply id="alg6.l8.m1.2.2.3.cmml" xref="alg6.l8.m1.2.2.3"><csymbol cd="ambiguous" id="alg6.l8.m1.2.2.3.1.cmml" xref="alg6.l8.m1.2.2.3">subscript</csymbol><ci id="alg6.l8.m1.2.2.3.2.cmml" xref="alg6.l8.m1.2.2.3.2">bold-italic-ϵ</ci><ci id="alg6.l8.m1.2.2.3.3.cmml" xref="alg6.l8.m1.2.2.3.3">𝑡</ci></apply><apply id="alg6.l8.m1.2.2.1.cmml" xref="alg6.l8.m1.2.2.1"><times id="alg6.l8.m1.2.2.1.2.cmml" xref="alg6.l8.m1.2.2.1.2"></times><ci id="alg6.l8.m1.2.2.1.3.cmml" xref="alg6.l8.m1.2.2.1.3">𝒩</ci><interval closure="open" id="alg6.l8.m1.2.2.1.1.2.cmml" xref="alg6.l8.m1.2.2.1.1.1"><cn id="alg6.l8.m1.1.1.cmml" type="integer" xref="alg6.l8.m1.1.1">0</cn><apply id="alg6.l8.m1.2.2.1.1.1.1.cmml" xref="alg6.l8.m1.2.2.1.1.1.1"><csymbol cd="ambiguous" id="alg6.l8.m1.2.2.1.1.1.1.1.cmml" xref="alg6.l8.m1.2.2.1.1.1.1">subscript</csymbol><ci id="alg6.l8.m1.2.2.1.1.1.1.2.cmml" xref="alg6.l8.m1.2.2.1.1.1.1.2">𝐈</ci><ci id="alg6.l8.m1.2.2.1.1.1.1.3.cmml" xref="alg6.l8.m1.2.2.1.1.1.1.3">𝑑</ci></apply></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="alg6.l8.m1.2c">{\boldsymbol{\epsilon}}_{t}\leftarrow\mathcal{N}(0,\mathbf{I}_{d})</annotation><annotation encoding="application/x-llamapun" id="alg6.l8.m1.2d">bold_italic_ϵ start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT ← caligraphic_N ( 0 , bold_I start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT )</annotation></semantics></math> </div> <div class="ltx_listingline" id="alg6.l9"> <span class="ltx_tag ltx_tag_listingline"><span class="ltx_text" id="alg6.l9.1.1.1" style="font-size:80%;">9:</span></span> <math alttext="x_{t_{1}}\leftarrow\sqrt{\bar{\alpha}_{t}}\hat{x_{0}}^{t}+\sqrt{1-\bar{\alpha}% _{t}}\left(\sqrt{1-\zeta}\hat{{\boldsymbol{\epsilon}}}+\sqrt{\zeta}{% \boldsymbol{\epsilon}}_{t}\right)" class="ltx_Math" display="inline" id="alg6.l9.m1.1"><semantics id="alg6.l9.m1.1a"><mrow id="alg6.l9.m1.1.1" xref="alg6.l9.m1.1.1.cmml"><msub 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xref="alg6.l9.m1.1.1.1.1.1.1.1.3.3"><csymbol cd="ambiguous" id="alg6.l9.m1.1.1.1.1.1.1.1.3.3.1.cmml" xref="alg6.l9.m1.1.1.1.1.1.1.1.3.3">subscript</csymbol><ci id="alg6.l9.m1.1.1.1.1.1.1.1.3.3.2.cmml" xref="alg6.l9.m1.1.1.1.1.1.1.1.3.3.2">bold-italic-ϵ</ci><ci id="alg6.l9.m1.1.1.1.1.1.1.1.3.3.3.cmml" xref="alg6.l9.m1.1.1.1.1.1.1.1.3.3.3">𝑡</ci></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="alg6.l9.m1.1c">x_{t_{1}}\leftarrow\sqrt{\bar{\alpha}_{t}}\hat{x_{0}}^{t}+\sqrt{1-\bar{\alpha}% _{t}}\left(\sqrt{1-\zeta}\hat{{\boldsymbol{\epsilon}}}+\sqrt{\zeta}{% \boldsymbol{\epsilon}}_{t}\right)</annotation><annotation encoding="application/x-llamapun" id="alg6.l9.m1.1d">italic_x start_POSTSUBSCRIPT italic_t start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ← square-root start_ARG over¯ start_ARG italic_α end_ARG start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT end_ARG over^ start_ARG italic_x start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_ARG start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT + square-root start_ARG 1 - over¯ start_ARG italic_α end_ARG start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT end_ARG ( square-root start_ARG 1 - italic_ζ end_ARG over^ start_ARG bold_italic_ϵ end_ARG + square-root start_ARG italic_ζ end_ARG bold_italic_ϵ start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT )</annotation></semantics></math> </div> <div class="ltx_listingline" id="alg6.l10"> <span class="ltx_tag ltx_tag_listingline"><span class="ltx_text" id="alg6.l10.1.1.1" style="font-size:80%;">10:</span></span> <span class="ltx_text ltx_font_bold" id="alg6.l10.2">end</span> <span class="ltx_text ltx_font_bold" id="alg6.l10.3">for</span> </div> </div> </figure> <div class="ltx_para" id="A6.SS0.SSS0.Px5.p2"> <p class="ltx_p" id="A6.SS0.SSS0.Px5.p2.7">The parameter schemes <math alttext="(\bar{\alpha}_{t})_{0<t<T}" class="ltx_Math" display="inline" id="A6.SS0.SSS0.Px5.p2.1.m1.1"><semantics 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id="A6.SS0.SSS0.Px5.p2.3.m3.1.1.3.4.cmml" xref="A6.SS0.SSS0.Px5.p2.3.m3.1.1.3.4">𝑡</ci></apply><apply id="A6.SS0.SSS0.Px5.p2.3.m3.1.1.3c.cmml" xref="A6.SS0.SSS0.Px5.p2.3.m3.1.1.3"><lt id="A6.SS0.SSS0.Px5.p2.3.m3.1.1.3.5.cmml" xref="A6.SS0.SSS0.Px5.p2.3.m3.1.1.3.5"></lt><share href="#A6.SS0.SSS0.Px5.p2.3.m3.1.1.3.4.cmml" id="A6.SS0.SSS0.Px5.p2.3.m3.1.1.3d.cmml" xref="A6.SS0.SSS0.Px5.p2.3.m3.1.1.3"></share><ci id="A6.SS0.SSS0.Px5.p2.3.m3.1.1.3.6.cmml" xref="A6.SS0.SSS0.Px5.p2.3.m3.1.1.3.6">𝑇</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A6.SS0.SSS0.Px5.p2.3.m3.1c">(\rho_{t})_{0<t<T}</annotation><annotation encoding="application/x-llamapun" id="A6.SS0.SSS0.Px5.p2.3.m3.1d">( italic_ρ start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT ) start_POSTSUBSCRIPT 0 < italic_t < italic_T end_POSTSUBSCRIPT</annotation></semantics></math> are described in <cite class="ltx_cite ltx_citemacro_citep">(Zhu et al., <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#bib.bib80" title="">2023</a>)</cite>. For DiffPIR, we choose the following parameters for all inverse problems: <math alttext="\zeta=0.8" class="ltx_Math" display="inline" id="A6.SS0.SSS0.Px5.p2.4.m4.1"><semantics id="A6.SS0.SSS0.Px5.p2.4.m4.1a"><mrow id="A6.SS0.SSS0.Px5.p2.4.m4.1.1" xref="A6.SS0.SSS0.Px5.p2.4.m4.1.1.cmml"><mi id="A6.SS0.SSS0.Px5.p2.4.m4.1.1.2" xref="A6.SS0.SSS0.Px5.p2.4.m4.1.1.2.cmml">ζ</mi><mo id="A6.SS0.SSS0.Px5.p2.4.m4.1.1.1" xref="A6.SS0.SSS0.Px5.p2.4.m4.1.1.1.cmml">=</mo><mn id="A6.SS0.SSS0.Px5.p2.4.m4.1.1.3" xref="A6.SS0.SSS0.Px5.p2.4.m4.1.1.3.cmml">0.8</mn></mrow><annotation-xml encoding="MathML-Content" id="A6.SS0.SSS0.Px5.p2.4.m4.1b"><apply id="A6.SS0.SSS0.Px5.p2.4.m4.1.1.cmml" xref="A6.SS0.SSS0.Px5.p2.4.m4.1.1"><eq id="A6.SS0.SSS0.Px5.p2.4.m4.1.1.1.cmml" xref="A6.SS0.SSS0.Px5.p2.4.m4.1.1.1"></eq><ci id="A6.SS0.SSS0.Px5.p2.4.m4.1.1.2.cmml" xref="A6.SS0.SSS0.Px5.p2.4.m4.1.1.2">𝜁</ci><cn id="A6.SS0.SSS0.Px5.p2.4.m4.1.1.3.cmml" type="float" xref="A6.SS0.SSS0.Px5.p2.4.m4.1.1.3">0.8</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A6.SS0.SSS0.Px5.p2.4.m4.1c">\zeta=0.8</annotation><annotation encoding="application/x-llamapun" id="A6.SS0.SSS0.Px5.p2.4.m4.1d">italic_ζ = 0.8</annotation></semantics></math>, <math alttext="T=1000" class="ltx_Math" display="inline" id="A6.SS0.SSS0.Px5.p2.5.m5.1"><semantics id="A6.SS0.SSS0.Px5.p2.5.m5.1a"><mrow id="A6.SS0.SSS0.Px5.p2.5.m5.1.1" xref="A6.SS0.SSS0.Px5.p2.5.m5.1.1.cmml"><mi id="A6.SS0.SSS0.Px5.p2.5.m5.1.1.2" xref="A6.SS0.SSS0.Px5.p2.5.m5.1.1.2.cmml">T</mi><mo id="A6.SS0.SSS0.Px5.p2.5.m5.1.1.1" xref="A6.SS0.SSS0.Px5.p2.5.m5.1.1.1.cmml">=</mo><mn id="A6.SS0.SSS0.Px5.p2.5.m5.1.1.3" xref="A6.SS0.SSS0.Px5.p2.5.m5.1.1.3.cmml">1000</mn></mrow><annotation-xml encoding="MathML-Content" id="A6.SS0.SSS0.Px5.p2.5.m5.1b"><apply id="A6.SS0.SSS0.Px5.p2.5.m5.1.1.cmml" xref="A6.SS0.SSS0.Px5.p2.5.m5.1.1"><eq id="A6.SS0.SSS0.Px5.p2.5.m5.1.1.1.cmml" xref="A6.SS0.SSS0.Px5.p2.5.m5.1.1.1"></eq><ci id="A6.SS0.SSS0.Px5.p2.5.m5.1.1.2.cmml" xref="A6.SS0.SSS0.Px5.p2.5.m5.1.1.2">𝑇</ci><cn id="A6.SS0.SSS0.Px5.p2.5.m5.1.1.3.cmml" type="integer" xref="A6.SS0.SSS0.Px5.p2.5.m5.1.1.3">1000</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A6.SS0.SSS0.Px5.p2.5.m5.1c">T=1000</annotation><annotation encoding="application/x-llamapun" id="A6.SS0.SSS0.Px5.p2.5.m5.1d">italic_T = 1000</annotation></semantics></math>, <math alttext="t_{\text{start}}=200" class="ltx_Math" display="inline" id="A6.SS0.SSS0.Px5.p2.6.m6.1"><semantics id="A6.SS0.SSS0.Px5.p2.6.m6.1a"><mrow id="A6.SS0.SSS0.Px5.p2.6.m6.1.1" xref="A6.SS0.SSS0.Px5.p2.6.m6.1.1.cmml"><msub id="A6.SS0.SSS0.Px5.p2.6.m6.1.1.2" xref="A6.SS0.SSS0.Px5.p2.6.m6.1.1.2.cmml"><mi id="A6.SS0.SSS0.Px5.p2.6.m6.1.1.2.2" xref="A6.SS0.SSS0.Px5.p2.6.m6.1.1.2.2.cmml">t</mi><mtext id="A6.SS0.SSS0.Px5.p2.6.m6.1.1.2.3" xref="A6.SS0.SSS0.Px5.p2.6.m6.1.1.2.3a.cmml">start</mtext></msub><mo 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xref="A6.SS0.SSS0.Px5.p2.6.m6.1.1.3">200</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A6.SS0.SSS0.Px5.p2.6.m6.1c">t_{\text{start}}=200</annotation><annotation encoding="application/x-llamapun" id="A6.SS0.SSS0.Px5.p2.6.m6.1d">italic_t start_POSTSUBSCRIPT start end_POSTSUBSCRIPT = 200</annotation></semantics></math> and <math alttext="\lambda=0.13" class="ltx_Math" display="inline" id="A6.SS0.SSS0.Px5.p2.7.m7.1"><semantics id="A6.SS0.SSS0.Px5.p2.7.m7.1a"><mrow id="A6.SS0.SSS0.Px5.p2.7.m7.1.1" xref="A6.SS0.SSS0.Px5.p2.7.m7.1.1.cmml"><mi id="A6.SS0.SSS0.Px5.p2.7.m7.1.1.2" xref="A6.SS0.SSS0.Px5.p2.7.m7.1.1.2.cmml">λ</mi><mo id="A6.SS0.SSS0.Px5.p2.7.m7.1.1.1" xref="A6.SS0.SSS0.Px5.p2.7.m7.1.1.1.cmml">=</mo><mn id="A6.SS0.SSS0.Px5.p2.7.m7.1.1.3" xref="A6.SS0.SSS0.Px5.p2.7.m7.1.1.3.cmml">0.13</mn></mrow><annotation-xml encoding="MathML-Content" id="A6.SS0.SSS0.Px5.p2.7.m7.1b"><apply id="A6.SS0.SSS0.Px5.p2.7.m7.1.1.cmml" xref="A6.SS0.SSS0.Px5.p2.7.m7.1.1"><eq id="A6.SS0.SSS0.Px5.p2.7.m7.1.1.1.cmml" xref="A6.SS0.SSS0.Px5.p2.7.m7.1.1.1"></eq><ci id="A6.SS0.SSS0.Px5.p2.7.m7.1.1.2.cmml" xref="A6.SS0.SSS0.Px5.p2.7.m7.1.1.2">𝜆</ci><cn id="A6.SS0.SSS0.Px5.p2.7.m7.1.1.3.cmml" type="float" xref="A6.SS0.SSS0.Px5.p2.7.m7.1.1.3">0.13</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A6.SS0.SSS0.Px5.p2.7.m7.1c">\lambda=0.13</annotation><annotation encoding="application/x-llamapun" id="A6.SS0.SSS0.Px5.p2.7.m7.1d">italic_λ = 0.13</annotation></semantics></math>.</p> </div> </section> <section class="ltx_paragraph" id="A6.SS0.SSS0.Px6"> <h5 class="ltx_title ltx_title_paragraph">PnP SGD</h5> <div class="ltx_para" id="A6.SS0.SSS0.Px6.p1"> <p class="ltx_p" id="A6.SS0.SSS0.Px6.p1.3">We compare our algorithm to an other stochastic PnP method, PnP SGD <cite class="ltx_cite ltx_citemacro_citep">(Laumont et al., <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#bib.bib37" title="">2022b</a>)</cite>, which approximates the maximum of the Posterior Law of Problem <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#S1.E1" title="1 ‣ 1 Introduction ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_tag">1</span></a>. In the implementation of the method, we need to add an other parameter in PnP SGD, <math alttext="\beta>0" class="ltx_Math" display="inline" id="A6.SS0.SSS0.Px6.p1.1.m1.1"><semantics id="A6.SS0.SSS0.Px6.p1.1.m1.1a"><mrow id="A6.SS0.SSS0.Px6.p1.1.m1.1.1" xref="A6.SS0.SSS0.Px6.p1.1.m1.1.1.cmml"><mi id="A6.SS0.SSS0.Px6.p1.1.m1.1.1.2" xref="A6.SS0.SSS0.Px6.p1.1.m1.1.1.2.cmml">β</mi><mo id="A6.SS0.SSS0.Px6.p1.1.m1.1.1.1" xref="A6.SS0.SSS0.Px6.p1.1.m1.1.1.1.cmml">></mo><mn id="A6.SS0.SSS0.Px6.p1.1.m1.1.1.3" xref="A6.SS0.SSS0.Px6.p1.1.m1.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="A6.SS0.SSS0.Px6.p1.1.m1.1b"><apply id="A6.SS0.SSS0.Px6.p1.1.m1.1.1.cmml" xref="A6.SS0.SSS0.Px6.p1.1.m1.1.1"><gt id="A6.SS0.SSS0.Px6.p1.1.m1.1.1.1.cmml" xref="A6.SS0.SSS0.Px6.p1.1.m1.1.1.1"></gt><ci id="A6.SS0.SSS0.Px6.p1.1.m1.1.1.2.cmml" xref="A6.SS0.SSS0.Px6.p1.1.m1.1.1.2">𝛽</ci><cn id="A6.SS0.SSS0.Px6.p1.1.m1.1.1.3.cmml" type="integer" xref="A6.SS0.SSS0.Px6.p1.1.m1.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A6.SS0.SSS0.Px6.p1.1.m1.1c">\beta>0</annotation><annotation encoding="application/x-llamapun" id="A6.SS0.SSS0.Px6.p1.1.m1.1d">italic_β > 0</annotation></semantics></math>, to control the power of the additive noise at each iteration and optimize the performance of the algorithm. 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As noticed by <cite class="ltx_cite ltx_citemacro_citet">Laumont et al. (<a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#bib.bib37" title="">2022b</a>)</cite>, keeping a fixed step-size allows us to obtain the best performance, so we decided to keep the step-size <math alttext="\delta>0" class="ltx_Math" display="inline" id="A6.SS0.SSS0.Px6.p1.3.m3.1"><semantics id="A6.SS0.SSS0.Px6.p1.3.m3.1a"><mrow id="A6.SS0.SSS0.Px6.p1.3.m3.1.1" xref="A6.SS0.SSS0.Px6.p1.3.m3.1.1.cmml"><mi id="A6.SS0.SSS0.Px6.p1.3.m3.1.1.2" xref="A6.SS0.SSS0.Px6.p1.3.m3.1.1.2.cmml">δ</mi><mo id="A6.SS0.SSS0.Px6.p1.3.m3.1.1.1" xref="A6.SS0.SSS0.Px6.p1.3.m3.1.1.1.cmml">></mo><mn id="A6.SS0.SSS0.Px6.p1.3.m3.1.1.3" xref="A6.SS0.SSS0.Px6.p1.3.m3.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="A6.SS0.SSS0.Px6.p1.3.m3.1b"><apply id="A6.SS0.SSS0.Px6.p1.3.m3.1.1.cmml" xref="A6.SS0.SSS0.Px6.p1.3.m3.1.1"><gt id="A6.SS0.SSS0.Px6.p1.3.m3.1.1.1.cmml" xref="A6.SS0.SSS0.Px6.p1.3.m3.1.1.1"></gt><ci id="A6.SS0.SSS0.Px6.p1.3.m3.1.1.2.cmml" xref="A6.SS0.SSS0.Px6.p1.3.m3.1.1.2">𝛿</ci><cn id="A6.SS0.SSS0.Px6.p1.3.m3.1.1.3.cmml" type="integer" xref="A6.SS0.SSS0.Px6.p1.3.m3.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A6.SS0.SSS0.Px6.p1.3.m3.1c">\delta>0</annotation><annotation encoding="application/x-llamapun" id="A6.SS0.SSS0.Px6.p1.3.m3.1d">italic_δ > 0</annotation></semantics></math> fixed. We give the pseudo-code of PnP SGD in Algorithm <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#alg7" title="Algorithm 7 ‣ PnP SGD ‣ Appendix F Additional Experiments ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_tag">7</span></a> and the used parameters for deblurring in Table <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#A6.T4" title="Table 4 ‣ Parameters setting ‣ F.1 Deblurring ‣ Appendix F Additional Experiments ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_tag">4</span></a>. We do not succeed to make the method competitive for image inpainting so we do not include in this paper the output of PnP SGD for inpainting.</p> </div> <figure class="ltx_float ltx_float_algorithm ltx_framed_top" id="alg7"> <figcaption class="ltx_caption"><span class="ltx_tag ltx_tag_float"><span class="ltx_text ltx_font_bold" id="alg7.2.1.1">Algorithm 7</span> </span> PnP SGD</figcaption> <div class="ltx_listing ltx_listing" id="alg7.3"> <div class="ltx_listingline" id="alg7.l1"> <span class="ltx_tag ltx_tag_listingline"><span class="ltx_text" id="alg7.l1.1.1.1" style="font-size:80%;">1:</span></span> <span class="ltx_text ltx_font_bold" id="alg7.l1.2">input:</span> <math alttext="{\mathbf{x}}_{0}\in{\mathbb{R}}^{d}" class="ltx_Math" display="inline" id="alg7.l1.m1.1"><semantics id="alg7.l1.m1.1a"><mrow id="alg7.l1.m1.1.1" xref="alg7.l1.m1.1.1.cmml"><msub id="alg7.l1.m1.1.1.2" xref="alg7.l1.m1.1.1.2.cmml"><mi id="alg7.l1.m1.1.1.2.2" xref="alg7.l1.m1.1.1.2.2.cmml">𝐱</mi><mn 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ltx_tag_listingline"><span class="ltx_text" id="alg7.l2.1.1.1" style="font-size:80%;">2:</span></span> <span class="ltx_text ltx_font_bold" id="alg7.l2.2">for</span> <math alttext="k=0,1,\dots,N-1" class="ltx_Math" display="inline" id="alg7.l2.m1.4"><semantics id="alg7.l2.m1.4a"><mrow id="alg7.l2.m1.4.4" xref="alg7.l2.m1.4.4.cmml"><mi id="alg7.l2.m1.4.4.3" xref="alg7.l2.m1.4.4.3.cmml">k</mi><mo id="alg7.l2.m1.4.4.2" xref="alg7.l2.m1.4.4.2.cmml">=</mo><mrow id="alg7.l2.m1.4.4.1.1" xref="alg7.l2.m1.4.4.1.2.cmml"><mn id="alg7.l2.m1.1.1" xref="alg7.l2.m1.1.1.cmml">0</mn><mo id="alg7.l2.m1.4.4.1.1.2" xref="alg7.l2.m1.4.4.1.2.cmml">,</mo><mn id="alg7.l2.m1.2.2" xref="alg7.l2.m1.2.2.cmml">1</mn><mo id="alg7.l2.m1.4.4.1.1.3" xref="alg7.l2.m1.4.4.1.2.cmml">,</mo><mi id="alg7.l2.m1.3.3" mathvariant="normal" xref="alg7.l2.m1.3.3.cmml">…</mi><mo id="alg7.l2.m1.4.4.1.1.4" xref="alg7.l2.m1.4.4.1.2.cmml">,</mo><mrow id="alg7.l2.m1.4.4.1.1.1" xref="alg7.l2.m1.4.4.1.1.1.cmml"><mi id="alg7.l2.m1.4.4.1.1.1.2" xref="alg7.l2.m1.4.4.1.1.1.2.cmml">N</mi><mo id="alg7.l2.m1.4.4.1.1.1.1" xref="alg7.l2.m1.4.4.1.1.1.1.cmml">−</mo><mn id="alg7.l2.m1.4.4.1.1.1.3" xref="alg7.l2.m1.4.4.1.1.1.3.cmml">1</mn></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="alg7.l2.m1.4b"><apply id="alg7.l2.m1.4.4.cmml" xref="alg7.l2.m1.4.4"><eq id="alg7.l2.m1.4.4.2.cmml" xref="alg7.l2.m1.4.4.2"></eq><ci id="alg7.l2.m1.4.4.3.cmml" xref="alg7.l2.m1.4.4.3">𝑘</ci><list id="alg7.l2.m1.4.4.1.2.cmml" xref="alg7.l2.m1.4.4.1.1"><cn id="alg7.l2.m1.1.1.cmml" type="integer" xref="alg7.l2.m1.1.1">0</cn><cn id="alg7.l2.m1.2.2.cmml" type="integer" xref="alg7.l2.m1.2.2">1</cn><ci id="alg7.l2.m1.3.3.cmml" xref="alg7.l2.m1.3.3">…</ci><apply id="alg7.l2.m1.4.4.1.1.1.cmml" xref="alg7.l2.m1.4.4.1.1.1"><minus id="alg7.l2.m1.4.4.1.1.1.1.cmml" xref="alg7.l2.m1.4.4.1.1.1.1"></minus><ci id="alg7.l2.m1.4.4.1.1.1.2.cmml" xref="alg7.l2.m1.4.4.1.1.1.2">𝑁</ci><cn id="alg7.l2.m1.4.4.1.1.1.3.cmml" type="integer" xref="alg7.l2.m1.4.4.1.1.1.3">1</cn></apply></list></apply></annotation-xml><annotation encoding="application/x-tex" id="alg7.l2.m1.4c">k=0,1,\dots,N-1</annotation><annotation encoding="application/x-llamapun" id="alg7.l2.m1.4d">italic_k = 0 , 1 , … , italic_N - 1</annotation></semantics></math> <span class="ltx_text ltx_font_bold" id="alg7.l2.3">do</span> </div> <div class="ltx_listingline" id="alg7.l3"> <span class="ltx_tag ltx_tag_listingline"><span class="ltx_text" id="alg7.l3.1.1.1" style="font-size:80%;">3:</span></span> <math alttext="{\mathbf{z}}_{k}\sim\mathcal{N}(0,{{\operatorname{Id}}})" class="ltx_Math" display="inline" id="alg7.l3.m1.2"><semantics id="alg7.l3.m1.2a"><mrow id="alg7.l3.m1.2.3" xref="alg7.l3.m1.2.3.cmml"><msub id="alg7.l3.m1.2.3.2" xref="alg7.l3.m1.2.3.2.cmml"><mi id="alg7.l3.m1.2.3.2.2" xref="alg7.l3.m1.2.3.2.2.cmml">𝐳</mi><mi id="alg7.l3.m1.2.3.2.3" xref="alg7.l3.m1.2.3.2.3.cmml">k</mi></msub><mo id="alg7.l3.m1.2.3.1" xref="alg7.l3.m1.2.3.1.cmml">∼</mo><mrow id="alg7.l3.m1.2.3.3" xref="alg7.l3.m1.2.3.3.cmml"><mi class="ltx_font_mathcaligraphic" id="alg7.l3.m1.2.3.3.2" xref="alg7.l3.m1.2.3.3.2.cmml">𝒩</mi><mo id="alg7.l3.m1.2.3.3.1" xref="alg7.l3.m1.2.3.3.1.cmml">⁢</mo><mrow id="alg7.l3.m1.2.3.3.3.2" xref="alg7.l3.m1.2.3.3.3.1.cmml"><mo id="alg7.l3.m1.2.3.3.3.2.1" stretchy="false" xref="alg7.l3.m1.2.3.3.3.1.cmml">(</mo><mn id="alg7.l3.m1.1.1" xref="alg7.l3.m1.1.1.cmml">0</mn><mo id="alg7.l3.m1.2.3.3.3.2.2" xref="alg7.l3.m1.2.3.3.3.1.cmml">,</mo><mi id="alg7.l3.m1.2.2" xref="alg7.l3.m1.2.2.cmml">Id</mi><mo id="alg7.l3.m1.2.3.3.3.2.3" stretchy="false" xref="alg7.l3.m1.2.3.3.3.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="alg7.l3.m1.2b"><apply id="alg7.l3.m1.2.3.cmml" xref="alg7.l3.m1.2.3"><csymbol cd="latexml" id="alg7.l3.m1.2.3.1.cmml" xref="alg7.l3.m1.2.3.1">similar-to</csymbol><apply id="alg7.l3.m1.2.3.2.cmml" xref="alg7.l3.m1.2.3.2"><csymbol cd="ambiguous" id="alg7.l3.m1.2.3.2.1.cmml" xref="alg7.l3.m1.2.3.2">subscript</csymbol><ci id="alg7.l3.m1.2.3.2.2.cmml" xref="alg7.l3.m1.2.3.2.2">𝐳</ci><ci id="alg7.l3.m1.2.3.2.3.cmml" xref="alg7.l3.m1.2.3.2.3">𝑘</ci></apply><apply id="alg7.l3.m1.2.3.3.cmml" xref="alg7.l3.m1.2.3.3"><times id="alg7.l3.m1.2.3.3.1.cmml" xref="alg7.l3.m1.2.3.3.1"></times><ci id="alg7.l3.m1.2.3.3.2.cmml" xref="alg7.l3.m1.2.3.3.2">𝒩</ci><interval closure="open" id="alg7.l3.m1.2.3.3.3.1.cmml" xref="alg7.l3.m1.2.3.3.3.2"><cn id="alg7.l3.m1.1.1.cmml" type="integer" xref="alg7.l3.m1.1.1">0</cn><ci id="alg7.l3.m1.2.2.cmml" xref="alg7.l3.m1.2.2">Id</ci></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="alg7.l3.m1.2c">{\mathbf{z}}_{k}\sim\mathcal{N}(0,{{\operatorname{Id}}})</annotation><annotation encoding="application/x-llamapun" id="alg7.l3.m1.2d">bold_z start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ∼ caligraphic_N ( 0 , roman_Id )</annotation></semantics></math> </div> <div class="ltx_listingline" id="alg7.l4"> <span class="ltx_tag ltx_tag_listingline"><span class="ltx_text" id="alg7.l4.1.1.1" style="font-size:80%;">4:</span></span> <math alttext="{\mathbf{x}}_{k+1}\leftarrow{\mathbf{x}}_{k}-\delta\nabla\mathcal{F}({\mathbf{% x}}_{k})-\frac{\alpha\delta}{\sigma^{2}}\left({\mathbf{x}}_{k}-D_{\sigma}({% \mathbf{x}}_{k})\right)+\beta\delta{\mathbf{z}}_{k}" class="ltx_Math" display="inline" id="alg7.l4.m1.2"><semantics id="alg7.l4.m1.2a"><mrow id="alg7.l4.m1.2.2" xref="alg7.l4.m1.2.2.cmml"><msub id="alg7.l4.m1.2.2.4" xref="alg7.l4.m1.2.2.4.cmml"><mi id="alg7.l4.m1.2.2.4.2" xref="alg7.l4.m1.2.2.4.2.cmml">𝐱</mi><mrow id="alg7.l4.m1.2.2.4.3" xref="alg7.l4.m1.2.2.4.3.cmml"><mi id="alg7.l4.m1.2.2.4.3.2" xref="alg7.l4.m1.2.2.4.3.2.cmml">k</mi><mo id="alg7.l4.m1.2.2.4.3.1" xref="alg7.l4.m1.2.2.4.3.1.cmml">+</mo><mn id="alg7.l4.m1.2.2.4.3.3" xref="alg7.l4.m1.2.2.4.3.3.cmml">1</mn></mrow></msub><mo id="alg7.l4.m1.2.2.3" stretchy="false" xref="alg7.l4.m1.2.2.3.cmml">←</mo><mrow id="alg7.l4.m1.2.2.2" xref="alg7.l4.m1.2.2.2.cmml"><mrow id="alg7.l4.m1.2.2.2.2" xref="alg7.l4.m1.2.2.2.2.cmml"><msub id="alg7.l4.m1.2.2.2.2.4" xref="alg7.l4.m1.2.2.2.2.4.cmml"><mi id="alg7.l4.m1.2.2.2.2.4.2" xref="alg7.l4.m1.2.2.2.2.4.2.cmml">𝐱</mi><mi id="alg7.l4.m1.2.2.2.2.4.3" xref="alg7.l4.m1.2.2.2.2.4.3.cmml">k</mi></msub><mo id="alg7.l4.m1.2.2.2.2.3" xref="alg7.l4.m1.2.2.2.2.3.cmml">−</mo><mrow id="alg7.l4.m1.1.1.1.1.1" xref="alg7.l4.m1.1.1.1.1.1.cmml"><mi id="alg7.l4.m1.1.1.1.1.1.3" xref="alg7.l4.m1.1.1.1.1.1.3.cmml">δ</mi><mo id="alg7.l4.m1.1.1.1.1.1.2" lspace="0.167em" xref="alg7.l4.m1.1.1.1.1.1.2.cmml">⁢</mo><mrow id="alg7.l4.m1.1.1.1.1.1.4" xref="alg7.l4.m1.1.1.1.1.1.4.cmml"><mo id="alg7.l4.m1.1.1.1.1.1.4.1" rspace="0.167em" xref="alg7.l4.m1.1.1.1.1.1.4.1.cmml">∇</mo><mi class="ltx_font_mathcaligraphic" id="alg7.l4.m1.1.1.1.1.1.4.2" xref="alg7.l4.m1.1.1.1.1.1.4.2.cmml">ℱ</mi></mrow><mo id="alg7.l4.m1.1.1.1.1.1.2a" xref="alg7.l4.m1.1.1.1.1.1.2.cmml">⁢</mo><mrow id="alg7.l4.m1.1.1.1.1.1.1.1" xref="alg7.l4.m1.1.1.1.1.1.1.1.1.cmml"><mo id="alg7.l4.m1.1.1.1.1.1.1.1.2" stretchy="false" xref="alg7.l4.m1.1.1.1.1.1.1.1.1.cmml">(</mo><msub id="alg7.l4.m1.1.1.1.1.1.1.1.1" xref="alg7.l4.m1.1.1.1.1.1.1.1.1.cmml"><mi id="alg7.l4.m1.1.1.1.1.1.1.1.1.2" xref="alg7.l4.m1.1.1.1.1.1.1.1.1.2.cmml">𝐱</mi><mi id="alg7.l4.m1.1.1.1.1.1.1.1.1.3" xref="alg7.l4.m1.1.1.1.1.1.1.1.1.3.cmml">k</mi></msub><mo id="alg7.l4.m1.1.1.1.1.1.1.1.3" stretchy="false" xref="alg7.l4.m1.1.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="alg7.l4.m1.2.2.2.2.3a" xref="alg7.l4.m1.2.2.2.2.3.cmml">−</mo><mrow id="alg7.l4.m1.2.2.2.2.2" xref="alg7.l4.m1.2.2.2.2.2.cmml"><mfrac id="alg7.l4.m1.2.2.2.2.2.3" xref="alg7.l4.m1.2.2.2.2.2.3.cmml"><mrow id="alg7.l4.m1.2.2.2.2.2.3.2" xref="alg7.l4.m1.2.2.2.2.2.3.2.cmml"><mi id="alg7.l4.m1.2.2.2.2.2.3.2.2" xref="alg7.l4.m1.2.2.2.2.2.3.2.2.cmml">α</mi><mo id="alg7.l4.m1.2.2.2.2.2.3.2.1" xref="alg7.l4.m1.2.2.2.2.2.3.2.1.cmml">⁢</mo><mi id="alg7.l4.m1.2.2.2.2.2.3.2.3" xref="alg7.l4.m1.2.2.2.2.2.3.2.3.cmml">δ</mi></mrow><msup id="alg7.l4.m1.2.2.2.2.2.3.3" xref="alg7.l4.m1.2.2.2.2.2.3.3.cmml"><mi id="alg7.l4.m1.2.2.2.2.2.3.3.2" xref="alg7.l4.m1.2.2.2.2.2.3.3.2.cmml">σ</mi><mn id="alg7.l4.m1.2.2.2.2.2.3.3.3" xref="alg7.l4.m1.2.2.2.2.2.3.3.3.cmml">2</mn></msup></mfrac><mo id="alg7.l4.m1.2.2.2.2.2.2" xref="alg7.l4.m1.2.2.2.2.2.2.cmml">⁢</mo><mrow id="alg7.l4.m1.2.2.2.2.2.1.1" xref="alg7.l4.m1.2.2.2.2.2.1.1.1.cmml"><mo id="alg7.l4.m1.2.2.2.2.2.1.1.2" xref="alg7.l4.m1.2.2.2.2.2.1.1.1.cmml">(</mo><mrow id="alg7.l4.m1.2.2.2.2.2.1.1.1" xref="alg7.l4.m1.2.2.2.2.2.1.1.1.cmml"><msub id="alg7.l4.m1.2.2.2.2.2.1.1.1.3" xref="alg7.l4.m1.2.2.2.2.2.1.1.1.3.cmml"><mi id="alg7.l4.m1.2.2.2.2.2.1.1.1.3.2" xref="alg7.l4.m1.2.2.2.2.2.1.1.1.3.2.cmml">𝐱</mi><mi id="alg7.l4.m1.2.2.2.2.2.1.1.1.3.3" xref="alg7.l4.m1.2.2.2.2.2.1.1.1.3.3.cmml">k</mi></msub><mo id="alg7.l4.m1.2.2.2.2.2.1.1.1.2" xref="alg7.l4.m1.2.2.2.2.2.1.1.1.2.cmml">−</mo><mrow id="alg7.l4.m1.2.2.2.2.2.1.1.1.1" xref="alg7.l4.m1.2.2.2.2.2.1.1.1.1.cmml"><msub id="alg7.l4.m1.2.2.2.2.2.1.1.1.1.3" xref="alg7.l4.m1.2.2.2.2.2.1.1.1.1.3.cmml"><mi id="alg7.l4.m1.2.2.2.2.2.1.1.1.1.3.2" xref="alg7.l4.m1.2.2.2.2.2.1.1.1.1.3.2.cmml">D</mi><mi id="alg7.l4.m1.2.2.2.2.2.1.1.1.1.3.3" xref="alg7.l4.m1.2.2.2.2.2.1.1.1.1.3.3.cmml">σ</mi></msub><mo id="alg7.l4.m1.2.2.2.2.2.1.1.1.1.2" xref="alg7.l4.m1.2.2.2.2.2.1.1.1.1.2.cmml">⁢</mo><mrow id="alg7.l4.m1.2.2.2.2.2.1.1.1.1.1.1" xref="alg7.l4.m1.2.2.2.2.2.1.1.1.1.1.1.1.cmml"><mo id="alg7.l4.m1.2.2.2.2.2.1.1.1.1.1.1.2" stretchy="false" xref="alg7.l4.m1.2.2.2.2.2.1.1.1.1.1.1.1.cmml">(</mo><msub id="alg7.l4.m1.2.2.2.2.2.1.1.1.1.1.1.1" xref="alg7.l4.m1.2.2.2.2.2.1.1.1.1.1.1.1.cmml"><mi id="alg7.l4.m1.2.2.2.2.2.1.1.1.1.1.1.1.2" xref="alg7.l4.m1.2.2.2.2.2.1.1.1.1.1.1.1.2.cmml">𝐱</mi><mi id="alg7.l4.m1.2.2.2.2.2.1.1.1.1.1.1.1.3" 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end_POSTSUPERSCRIPT end_ARG ( bold_x start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT - italic_D start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ( bold_x start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) ) + italic_β italic_δ bold_z start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT</annotation></semantics></math> </div> <div class="ltx_listingline" id="alg7.l5"> <span class="ltx_tag ltx_tag_listingline"><span class="ltx_text" id="alg7.l5.1.1.1" style="font-size:80%;">5:</span></span> <span class="ltx_text ltx_font_bold" id="alg7.l5.2">end</span> <span class="ltx_text ltx_font_bold" id="alg7.l5.3">for</span> </div> </div> </figure> </section> <section class="ltx_paragraph" id="A6.SS0.SSS0.Px7"> <h5 class="ltx_title ltx_title_paragraph">Computational time</h5> <div class="ltx_para" id="A6.SS0.SSS0.Px7.p1"> <p class="ltx_p" id="A6.SS0.SSS0.Px7.p1.2">On Table <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#A6.T3" title="Table 3 ‣ Computational time ‣ Appendix F Additional Experiments ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_tag">3</span></a>, we compare the computational time of various methods. For deblurring, SNORE and SNORE Prox are slow compare to other methods. This is due to the number of iterations require at each parameters level to converge, leading to a large number of iteration (<math alttext="1500" class="ltx_Math" display="inline" id="A6.SS0.SSS0.Px7.p1.1.m1.1"><semantics id="A6.SS0.SSS0.Px7.p1.1.m1.1a"><mn id="A6.SS0.SSS0.Px7.p1.1.m1.1.1" xref="A6.SS0.SSS0.Px7.p1.1.m1.1.1.cmml">1500</mn><annotation-xml encoding="MathML-Content" id="A6.SS0.SSS0.Px7.p1.1.m1.1b"><cn id="A6.SS0.SSS0.Px7.p1.1.m1.1.1.cmml" type="integer" xref="A6.SS0.SSS0.Px7.p1.1.m1.1.1">1500</cn></annotation-xml><annotation encoding="application/x-tex" id="A6.SS0.SSS0.Px7.p1.1.m1.1c">1500</annotation><annotation encoding="application/x-llamapun" id="A6.SS0.SSS0.Px7.p1.1.m1.1d">1500</annotation></semantics></math>). We observe on Figure <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#A6.F9" title="Figure 9 ‣ On the step-size 𝛿 ‣ F.1 Deblurring ‣ Appendix F Additional Experiments ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_tag">9</span></a> that with less annealing level, metrics performance are similar. However, we observe a qualitative impact of this parameter. For inpainting, out method outperform RED and RED Prox with a fixed number of iteration. Only DiffPIR remains faster. For this inverse problem, we have observed that a smaller number of iteration (<math alttext="500" class="ltx_Math" display="inline" id="A6.SS0.SSS0.Px7.p1.2.m2.1"><semantics id="A6.SS0.SSS0.Px7.p1.2.m2.1a"><mn id="A6.SS0.SSS0.Px7.p1.2.m2.1.1" xref="A6.SS0.SSS0.Px7.p1.2.m2.1.1.cmml">500</mn><annotation-xml encoding="MathML-Content" id="A6.SS0.SSS0.Px7.p1.2.m2.1b"><cn id="A6.SS0.SSS0.Px7.p1.2.m2.1.1.cmml" type="integer" xref="A6.SS0.SSS0.Px7.p1.2.m2.1.1">500</cn></annotation-xml><annotation encoding="application/x-tex" id="A6.SS0.SSS0.Px7.p1.2.m2.1c">500</annotation><annotation encoding="application/x-llamapun" id="A6.SS0.SSS0.Px7.p1.2.m2.1d">500</annotation></semantics></math>) is sufficient.</p> </div> <div class="ltx_para" id="A6.SS0.SSS0.Px7.p2"> <p class="ltx_p" id="A6.SS0.SSS0.Px7.p2.1">Computing all the necessary experiments to generate Table <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#S4.T1" title="Table 1 ‣ 4.1 Deblurring ‣ 4 Experiments ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_tag">1</span></a> requires 9 hours and 40 minutes on a GPU NVIDIA Quadro RTX 8000. Similarly, generating Table <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#S4.T2" title="Table 2 ‣ 4.2 Inpainting ‣ 4 Experiments ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_tag">2</span></a> requires 8 hours of computation on a GPU NVIDIA Quadro RTX 8000. The whole computational resources used for this paper are 17 hours and 40 minutes of computation on a GPU NVIDIA Quadro RTX 8000.</p> </div> <figure class="ltx_table" id="A6.T3"> <div class="ltx_inline-block ltx_align_center ltx_transformed_outer" id="A6.T3.4" style="width:281.9pt;height:55.6pt;vertical-align:-0.0pt;"><span class="ltx_transformed_inner" style="transform:translate(4.2pt,-0.8pt) scale(1.03052957443732,1.03052957443732) ;"> <table class="ltx_tabular ltx_guessed_headers ltx_align_middle" id="A6.T3.4.1"> <thead class="ltx_thead"> <tr class="ltx_tr" id="A6.T3.4.1.1.1"> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_th_row ltx_border_l ltx_border_rr ltx_border_t" id="A6.T3.4.1.1.1.1">Inverse Problem</th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_r ltx_border_t" id="A6.T3.4.1.1.1.2">RED</th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_r ltx_border_t" id="A6.T3.4.1.1.1.3">RED Prox</th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_r ltx_border_t" id="A6.T3.4.1.1.1.4">SNORE</th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_r ltx_border_t" id="A6.T3.4.1.1.1.5">SNORE Prox</th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_r ltx_border_t" id="A6.T3.4.1.1.1.6">DiffPIR</th> </tr> </thead> <tbody class="ltx_tbody"> <tr class="ltx_tr" id="A6.T3.4.1.2.1"> <th class="ltx_td ltx_align_center ltx_th ltx_th_row ltx_border_l ltx_border_rr ltx_border_t" id="A6.T3.4.1.2.1.1">Deblurring</th> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="A6.T3.4.1.2.1.2">5</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="A6.T3.4.1.2.1.3">2</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="A6.T3.4.1.2.1.4">67</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="A6.T3.4.1.2.1.5">41</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="A6.T3.4.1.2.1.6">1</td> </tr> <tr class="ltx_tr" id="A6.T3.4.1.3.2"> <th class="ltx_td ltx_align_center ltx_th ltx_th_row ltx_border_b ltx_border_l ltx_border_rr ltx_border_t" id="A6.T3.4.1.3.2.1">Inpainting</th> <td class="ltx_td ltx_align_center ltx_border_b ltx_border_r ltx_border_t" id="A6.T3.4.1.3.2.2">39</td> <td class="ltx_td ltx_align_center ltx_border_b ltx_border_r ltx_border_t" id="A6.T3.4.1.3.2.3">40</td> <td class="ltx_td ltx_align_center ltx_border_b ltx_border_r ltx_border_t" id="A6.T3.4.1.3.2.4">40</td> <td class="ltx_td ltx_align_center ltx_border_b ltx_border_r ltx_border_t" id="A6.T3.4.1.3.2.5">40</td> <td class="ltx_td ltx_align_center ltx_border_b ltx_border_r ltx_border_t" id="A6.T3.4.1.3.2.6">1</td> </tr> </tbody> </table> </span></div> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_table"><span class="ltx_text" id="A6.T3.5.2.1" style="font-size:90%;">Table 3</span>: </span><span class="ltx_text" id="A6.T3.2.1" style="font-size:90%;">Computational time in second (averaged on <math alttext="4" class="ltx_Math" display="inline" id="A6.T3.2.1.m1.1"><semantics id="A6.T3.2.1.m1.1b"><mn id="A6.T3.2.1.m1.1.1" xref="A6.T3.2.1.m1.1.1.cmml">4</mn><annotation-xml encoding="MathML-Content" id="A6.T3.2.1.m1.1c"><cn id="A6.T3.2.1.m1.1.1.cmml" type="integer" xref="A6.T3.2.1.m1.1.1">4</cn></annotation-xml><annotation encoding="application/x-tex" id="A6.T3.2.1.m1.1d">4</annotation><annotation encoding="application/x-llamapun" id="A6.T3.2.1.m1.1e">4</annotation></semantics></math> images) for various methods and inverse problems on a GPU NVIDIA Quadro RTX 8000.</span></figcaption> </figure> </section> <section class="ltx_subsection" id="A6.SS1"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">F.1 </span>Deblurring</h3> <div class="ltx_para" id="A6.SS1.p1"> <p class="ltx_p" id="A6.SS1.p1.3">In this part, we give more details on our experiments for image deblurring. We also discuss the influence of the parameters <math alttext="m" class="ltx_Math" display="inline" id="A6.SS1.p1.1.m1.1"><semantics id="A6.SS1.p1.1.m1.1a"><mi id="A6.SS1.p1.1.m1.1.1" xref="A6.SS1.p1.1.m1.1.1.cmml">m</mi><annotation-xml encoding="MathML-Content" id="A6.SS1.p1.1.m1.1b"><ci id="A6.SS1.p1.1.m1.1.1.cmml" xref="A6.SS1.p1.1.m1.1.1">𝑚</ci></annotation-xml><annotation encoding="application/x-tex" id="A6.SS1.p1.1.m1.1c">m</annotation><annotation encoding="application/x-llamapun" id="A6.SS1.p1.1.m1.1d">italic_m</annotation></semantics></math>, <math alttext="\alpha_{m-1}" class="ltx_Math" display="inline" id="A6.SS1.p1.2.m2.1"><semantics id="A6.SS1.p1.2.m2.1a"><msub id="A6.SS1.p1.2.m2.1.1" xref="A6.SS1.p1.2.m2.1.1.cmml"><mi id="A6.SS1.p1.2.m2.1.1.2" xref="A6.SS1.p1.2.m2.1.1.2.cmml">α</mi><mrow id="A6.SS1.p1.2.m2.1.1.3" xref="A6.SS1.p1.2.m2.1.1.3.cmml"><mi id="A6.SS1.p1.2.m2.1.1.3.2" xref="A6.SS1.p1.2.m2.1.1.3.2.cmml">m</mi><mo id="A6.SS1.p1.2.m2.1.1.3.1" xref="A6.SS1.p1.2.m2.1.1.3.1.cmml">−</mo><mn id="A6.SS1.p1.2.m2.1.1.3.3" xref="A6.SS1.p1.2.m2.1.1.3.3.cmml">1</mn></mrow></msub><annotation-xml encoding="MathML-Content" id="A6.SS1.p1.2.m2.1b"><apply id="A6.SS1.p1.2.m2.1.1.cmml" xref="A6.SS1.p1.2.m2.1.1"><csymbol cd="ambiguous" id="A6.SS1.p1.2.m2.1.1.1.cmml" xref="A6.SS1.p1.2.m2.1.1">subscript</csymbol><ci id="A6.SS1.p1.2.m2.1.1.2.cmml" xref="A6.SS1.p1.2.m2.1.1.2">𝛼</ci><apply id="A6.SS1.p1.2.m2.1.1.3.cmml" xref="A6.SS1.p1.2.m2.1.1.3"><minus id="A6.SS1.p1.2.m2.1.1.3.1.cmml" xref="A6.SS1.p1.2.m2.1.1.3.1"></minus><ci id="A6.SS1.p1.2.m2.1.1.3.2.cmml" xref="A6.SS1.p1.2.m2.1.1.3.2">𝑚</ci><cn id="A6.SS1.p1.2.m2.1.1.3.3.cmml" type="integer" xref="A6.SS1.p1.2.m2.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A6.SS1.p1.2.m2.1c">\alpha_{m-1}</annotation><annotation encoding="application/x-llamapun" id="A6.SS1.p1.2.m2.1d">italic_α start_POSTSUBSCRIPT italic_m - 1 end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="\sigma_{m-1}" class="ltx_Math" display="inline" id="A6.SS1.p1.3.m3.1"><semantics id="A6.SS1.p1.3.m3.1a"><msub id="A6.SS1.p1.3.m3.1.1" xref="A6.SS1.p1.3.m3.1.1.cmml"><mi id="A6.SS1.p1.3.m3.1.1.2" xref="A6.SS1.p1.3.m3.1.1.2.cmml">σ</mi><mrow id="A6.SS1.p1.3.m3.1.1.3" xref="A6.SS1.p1.3.m3.1.1.3.cmml"><mi id="A6.SS1.p1.3.m3.1.1.3.2" xref="A6.SS1.p1.3.m3.1.1.3.2.cmml">m</mi><mo id="A6.SS1.p1.3.m3.1.1.3.1" xref="A6.SS1.p1.3.m3.1.1.3.1.cmml">−</mo><mn id="A6.SS1.p1.3.m3.1.1.3.3" xref="A6.SS1.p1.3.m3.1.1.3.3.cmml">1</mn></mrow></msub><annotation-xml encoding="MathML-Content" id="A6.SS1.p1.3.m3.1b"><apply id="A6.SS1.p1.3.m3.1.1.cmml" xref="A6.SS1.p1.3.m3.1.1"><csymbol cd="ambiguous" id="A6.SS1.p1.3.m3.1.1.1.cmml" xref="A6.SS1.p1.3.m3.1.1">subscript</csymbol><ci id="A6.SS1.p1.3.m3.1.1.2.cmml" xref="A6.SS1.p1.3.m3.1.1.2">𝜎</ci><apply id="A6.SS1.p1.3.m3.1.1.3.cmml" xref="A6.SS1.p1.3.m3.1.1.3"><minus id="A6.SS1.p1.3.m3.1.1.3.1.cmml" xref="A6.SS1.p1.3.m3.1.1.3.1"></minus><ci id="A6.SS1.p1.3.m3.1.1.3.2.cmml" xref="A6.SS1.p1.3.m3.1.1.3.2">𝑚</ci><cn id="A6.SS1.p1.3.m3.1.1.3.3.cmml" type="integer" xref="A6.SS1.p1.3.m3.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A6.SS1.p1.3.m3.1c">\sigma_{m-1}</annotation><annotation encoding="application/x-llamapun" id="A6.SS1.p1.3.m3.1d">italic_σ start_POSTSUBSCRIPT italic_m - 1 end_POSTSUBSCRIPT</annotation></semantics></math> on Annealing SNORE outputs (Algorithm <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#alg3" title="Algorithm 3 ‣ Optimization algorithms ‣ 2.2 SNORE regularization ‣ 2 Stochastic deNOising REgularization (SNORE) ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_tag">3</span></a>).</p> </div> <section class="ltx_paragraph" id="A6.SS1.SSS0.Px1"> <h5 class="ltx_title ltx_title_paragraph">Parameters setting</h5> <div class="ltx_para" id="A6.SS1.SSS0.Px1.p1"> <p class="ltx_p" id="A6.SS1.SSS0.Px1.p1.1">In Table <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#A6.T4" title="Table 4 ‣ Parameters setting ‣ F.1 Deblurring ‣ Appendix F Additional Experiments ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_tag">4</span></a>, we give the values of the different parameters used in our experiments.</p> </div> <figure class="ltx_table" id="A6.T4"> <div class="ltx_inline-block ltx_align_center ltx_transformed_outer" id="A6.T4.21" style="width:281.9pt;height:209.8pt;vertical-align:-0.0pt;"><span class="ltx_transformed_inner" style="transform:translate(-28.4pt,21.1pt) scale(0.832513830352246,0.832513830352246) ;"> <table class="ltx_tabular ltx_align_middle" id="A6.T4.21.21"> <tbody class="ltx_tbody"> <tr class="ltx_tr" id="A6.T4.21.21.22.1"> <td class="ltx_td ltx_align_center ltx_border_l ltx_border_rr ltx_border_t" id="A6.T4.21.21.22.1.1">Parameters</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="A6.T4.21.21.22.1.2">RED</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="A6.T4.21.21.22.1.3">RED Prox</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="A6.T4.21.21.22.1.4">SNORE</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="A6.T4.21.21.22.1.5">SNORE Prox</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="A6.T4.21.21.22.1.6">PnP SGD</td> </tr> <tr class="ltx_tr" id="A6.T4.2.2.2"> <td class="ltx_td ltx_align_center ltx_border_l ltx_border_rr ltx_border_t" id="A6.T4.2.2.2.2"> <math alttext="\alpha" class="ltx_Math" display="inline" id="A6.T4.1.1.1.1.m1.1"><semantics id="A6.T4.1.1.1.1.m1.1a"><mi id="A6.T4.1.1.1.1.m1.1.1" xref="A6.T4.1.1.1.1.m1.1.1.cmml">α</mi><annotation-xml encoding="MathML-Content" id="A6.T4.1.1.1.1.m1.1b"><ci id="A6.T4.1.1.1.1.m1.1.1.cmml" xref="A6.T4.1.1.1.1.m1.1.1">𝛼</ci></annotation-xml><annotation encoding="application/x-tex" id="A6.T4.1.1.1.1.m1.1c">\alpha</annotation><annotation encoding="application/x-llamapun" id="A6.T4.1.1.1.1.m1.1d">italic_α</annotation></semantics></math> (motion blur, <math alttext="\sigma_{{\mathbf{y}}}<20/255" class="ltx_Math" display="inline" id="A6.T4.2.2.2.2.m2.1"><semantics id="A6.T4.2.2.2.2.m2.1a"><mrow id="A6.T4.2.2.2.2.m2.1.1" xref="A6.T4.2.2.2.2.m2.1.1.cmml"><msub id="A6.T4.2.2.2.2.m2.1.1.2" xref="A6.T4.2.2.2.2.m2.1.1.2.cmml"><mi id="A6.T4.2.2.2.2.m2.1.1.2.2" xref="A6.T4.2.2.2.2.m2.1.1.2.2.cmml">σ</mi><mi id="A6.T4.2.2.2.2.m2.1.1.2.3" xref="A6.T4.2.2.2.2.m2.1.1.2.3.cmml">𝐲</mi></msub><mo id="A6.T4.2.2.2.2.m2.1.1.1" xref="A6.T4.2.2.2.2.m2.1.1.1.cmml"><</mo><mrow id="A6.T4.2.2.2.2.m2.1.1.3" xref="A6.T4.2.2.2.2.m2.1.1.3.cmml"><mn id="A6.T4.2.2.2.2.m2.1.1.3.2" xref="A6.T4.2.2.2.2.m2.1.1.3.2.cmml">20</mn><mo id="A6.T4.2.2.2.2.m2.1.1.3.1" xref="A6.T4.2.2.2.2.m2.1.1.3.1.cmml">/</mo><mn id="A6.T4.2.2.2.2.m2.1.1.3.3" xref="A6.T4.2.2.2.2.m2.1.1.3.3.cmml">255</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="A6.T4.2.2.2.2.m2.1b"><apply id="A6.T4.2.2.2.2.m2.1.1.cmml" xref="A6.T4.2.2.2.2.m2.1.1"><lt id="A6.T4.2.2.2.2.m2.1.1.1.cmml" xref="A6.T4.2.2.2.2.m2.1.1.1"></lt><apply id="A6.T4.2.2.2.2.m2.1.1.2.cmml" xref="A6.T4.2.2.2.2.m2.1.1.2"><csymbol cd="ambiguous" id="A6.T4.2.2.2.2.m2.1.1.2.1.cmml" xref="A6.T4.2.2.2.2.m2.1.1.2">subscript</csymbol><ci id="A6.T4.2.2.2.2.m2.1.1.2.2.cmml" xref="A6.T4.2.2.2.2.m2.1.1.2.2">𝜎</ci><ci id="A6.T4.2.2.2.2.m2.1.1.2.3.cmml" xref="A6.T4.2.2.2.2.m2.1.1.2.3">𝐲</ci></apply><apply id="A6.T4.2.2.2.2.m2.1.1.3.cmml" xref="A6.T4.2.2.2.2.m2.1.1.3"><divide id="A6.T4.2.2.2.2.m2.1.1.3.1.cmml" xref="A6.T4.2.2.2.2.m2.1.1.3.1"></divide><cn id="A6.T4.2.2.2.2.m2.1.1.3.2.cmml" type="integer" xref="A6.T4.2.2.2.2.m2.1.1.3.2">20</cn><cn id="A6.T4.2.2.2.2.m2.1.1.3.3.cmml" type="integer" 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id="A6.T4.3.3.3.1.m1.1a"><mi id="A6.T4.3.3.3.1.m1.1.1" xref="A6.T4.3.3.3.1.m1.1.1.cmml">α</mi><annotation-xml encoding="MathML-Content" id="A6.T4.3.3.3.1.m1.1b"><ci id="A6.T4.3.3.3.1.m1.1.1.cmml" xref="A6.T4.3.3.3.1.m1.1.1">𝛼</ci></annotation-xml><annotation encoding="application/x-tex" id="A6.T4.3.3.3.1.m1.1c">\alpha</annotation><annotation encoding="application/x-llamapun" id="A6.T4.3.3.3.1.m1.1d">italic_α</annotation></semantics></math> (motion blur, <math alttext="\sigma_{{\mathbf{y}}}\geq 20/255" class="ltx_Math" display="inline" id="A6.T4.4.4.4.2.m2.1"><semantics id="A6.T4.4.4.4.2.m2.1a"><mrow id="A6.T4.4.4.4.2.m2.1.1" xref="A6.T4.4.4.4.2.m2.1.1.cmml"><msub id="A6.T4.4.4.4.2.m2.1.1.2" xref="A6.T4.4.4.4.2.m2.1.1.2.cmml"><mi id="A6.T4.4.4.4.2.m2.1.1.2.2" xref="A6.T4.4.4.4.2.m2.1.1.2.2.cmml">σ</mi><mi id="A6.T4.4.4.4.2.m2.1.1.2.3" xref="A6.T4.4.4.4.2.m2.1.1.2.3.cmml">𝐲</mi></msub><mo id="A6.T4.4.4.4.2.m2.1.1.1" xref="A6.T4.4.4.4.2.m2.1.1.1.cmml">≥</mo><mrow id="A6.T4.4.4.4.2.m2.1.1.3" xref="A6.T4.4.4.4.2.m2.1.1.3.cmml"><mn id="A6.T4.4.4.4.2.m2.1.1.3.2" xref="A6.T4.4.4.4.2.m2.1.1.3.2.cmml">20</mn><mo id="A6.T4.4.4.4.2.m2.1.1.3.1" xref="A6.T4.4.4.4.2.m2.1.1.3.1.cmml">/</mo><mn id="A6.T4.4.4.4.2.m2.1.1.3.3" xref="A6.T4.4.4.4.2.m2.1.1.3.3.cmml">255</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="A6.T4.4.4.4.2.m2.1b"><apply id="A6.T4.4.4.4.2.m2.1.1.cmml" xref="A6.T4.4.4.4.2.m2.1.1"><geq id="A6.T4.4.4.4.2.m2.1.1.1.cmml" xref="A6.T4.4.4.4.2.m2.1.1.1"></geq><apply id="A6.T4.4.4.4.2.m2.1.1.2.cmml" xref="A6.T4.4.4.4.2.m2.1.1.2"><csymbol cd="ambiguous" id="A6.T4.4.4.4.2.m2.1.1.2.1.cmml" xref="A6.T4.4.4.4.2.m2.1.1.2">subscript</csymbol><ci id="A6.T4.4.4.4.2.m2.1.1.2.2.cmml" xref="A6.T4.4.4.4.2.m2.1.1.2.2">𝜎</ci><ci id="A6.T4.4.4.4.2.m2.1.1.2.3.cmml" xref="A6.T4.4.4.4.2.m2.1.1.2.3">𝐲</ci></apply><apply id="A6.T4.4.4.4.2.m2.1.1.3.cmml" xref="A6.T4.4.4.4.2.m2.1.1.3"><divide id="A6.T4.4.4.4.2.m2.1.1.3.1.cmml" xref="A6.T4.4.4.4.2.m2.1.1.3.1"></divide><cn id="A6.T4.4.4.4.2.m2.1.1.3.2.cmml" type="integer" xref="A6.T4.4.4.4.2.m2.1.1.3.2">20</cn><cn id="A6.T4.4.4.4.2.m2.1.1.3.3.cmml" type="integer" xref="A6.T4.4.4.4.2.m2.1.1.3.3">255</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A6.T4.4.4.4.2.m2.1c">\sigma_{{\mathbf{y}}}\geq 20/255</annotation><annotation encoding="application/x-llamapun" id="A6.T4.4.4.4.2.m2.1d">italic_σ start_POSTSUBSCRIPT bold_y end_POSTSUBSCRIPT ≥ 20 / 255</annotation></semantics></math>)</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="A6.T4.4.4.4.3">0.1</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="A6.T4.4.4.4.4">0.3</td> <td class="ltx_td ltx_border_r ltx_border_t" id="A6.T4.4.4.4.5"></td> <td class="ltx_td ltx_border_r ltx_border_t" id="A6.T4.4.4.4.6"></td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="A6.T4.4.4.4.7">0.5</td> </tr> <tr class="ltx_tr" id="A6.T4.6.6.6"> <td class="ltx_td ltx_align_center ltx_border_l ltx_border_rr ltx_border_t" id="A6.T4.6.6.6.2"> <math alttext="\alpha" class="ltx_Math" display="inline" id="A6.T4.5.5.5.1.m1.1"><semantics id="A6.T4.5.5.5.1.m1.1a"><mi id="A6.T4.5.5.5.1.m1.1.1" xref="A6.T4.5.5.5.1.m1.1.1.cmml">α</mi><annotation-xml encoding="MathML-Content" id="A6.T4.5.5.5.1.m1.1b"><ci id="A6.T4.5.5.5.1.m1.1.1.cmml" xref="A6.T4.5.5.5.1.m1.1.1">𝛼</ci></annotation-xml><annotation encoding="application/x-tex" id="A6.T4.5.5.5.1.m1.1c">\alpha</annotation><annotation encoding="application/x-llamapun" id="A6.T4.5.5.5.1.m1.1d">italic_α</annotation></semantics></math> (fixed blur, <math alttext="\sigma_{{\mathbf{y}}}<20/255" class="ltx_Math" display="inline" id="A6.T4.6.6.6.2.m2.1"><semantics id="A6.T4.6.6.6.2.m2.1a"><mrow id="A6.T4.6.6.6.2.m2.1.1" xref="A6.T4.6.6.6.2.m2.1.1.cmml"><msub id="A6.T4.6.6.6.2.m2.1.1.2" xref="A6.T4.6.6.6.2.m2.1.1.2.cmml"><mi id="A6.T4.6.6.6.2.m2.1.1.2.2" xref="A6.T4.6.6.6.2.m2.1.1.2.2.cmml">σ</mi><mi id="A6.T4.6.6.6.2.m2.1.1.2.3" xref="A6.T4.6.6.6.2.m2.1.1.2.3.cmml">𝐲</mi></msub><mo id="A6.T4.6.6.6.2.m2.1.1.1" xref="A6.T4.6.6.6.2.m2.1.1.1.cmml"><</mo><mrow id="A6.T4.6.6.6.2.m2.1.1.3" xref="A6.T4.6.6.6.2.m2.1.1.3.cmml"><mn id="A6.T4.6.6.6.2.m2.1.1.3.2" xref="A6.T4.6.6.6.2.m2.1.1.3.2.cmml">20</mn><mo id="A6.T4.6.6.6.2.m2.1.1.3.1" xref="A6.T4.6.6.6.2.m2.1.1.3.1.cmml">/</mo><mn id="A6.T4.6.6.6.2.m2.1.1.3.3" xref="A6.T4.6.6.6.2.m2.1.1.3.3.cmml">255</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="A6.T4.6.6.6.2.m2.1b"><apply id="A6.T4.6.6.6.2.m2.1.1.cmml" xref="A6.T4.6.6.6.2.m2.1.1"><lt id="A6.T4.6.6.6.2.m2.1.1.1.cmml" xref="A6.T4.6.6.6.2.m2.1.1.1"></lt><apply id="A6.T4.6.6.6.2.m2.1.1.2.cmml" xref="A6.T4.6.6.6.2.m2.1.1.2"><csymbol cd="ambiguous" id="A6.T4.6.6.6.2.m2.1.1.2.1.cmml" xref="A6.T4.6.6.6.2.m2.1.1.2">subscript</csymbol><ci id="A6.T4.6.6.6.2.m2.1.1.2.2.cmml" xref="A6.T4.6.6.6.2.m2.1.1.2.2">𝜎</ci><ci id="A6.T4.6.6.6.2.m2.1.1.2.3.cmml" xref="A6.T4.6.6.6.2.m2.1.1.2.3">𝐲</ci></apply><apply id="A6.T4.6.6.6.2.m2.1.1.3.cmml" xref="A6.T4.6.6.6.2.m2.1.1.3"><divide id="A6.T4.6.6.6.2.m2.1.1.3.1.cmml" xref="A6.T4.6.6.6.2.m2.1.1.3.1"></divide><cn id="A6.T4.6.6.6.2.m2.1.1.3.2.cmml" type="integer" xref="A6.T4.6.6.6.2.m2.1.1.3.2">20</cn><cn id="A6.T4.6.6.6.2.m2.1.1.3.3.cmml" type="integer" xref="A6.T4.6.6.6.2.m2.1.1.3.3">255</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A6.T4.6.6.6.2.m2.1c">\sigma_{{\mathbf{y}}}<20/255</annotation><annotation encoding="application/x-llamapun" id="A6.T4.6.6.6.2.m2.1d">italic_σ start_POSTSUBSCRIPT bold_y end_POSTSUBSCRIPT < 20 / 255</annotation></semantics></math>)</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="A6.T4.6.6.6.3">0.075</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="A6.T4.6.6.6.4">0.2</td> <td class="ltx_td ltx_border_r ltx_border_t" id="A6.T4.6.6.6.5"></td> <td class="ltx_td ltx_border_r ltx_border_t" id="A6.T4.6.6.6.6"></td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="A6.T4.6.6.6.7">0.5</td> </tr> <tr class="ltx_tr" id="A6.T4.8.8.8"> <td class="ltx_td ltx_align_center ltx_border_l ltx_border_rr ltx_border_t" id="A6.T4.8.8.8.2"> <math alttext="\alpha" class="ltx_Math" display="inline" id="A6.T4.7.7.7.1.m1.1"><semantics id="A6.T4.7.7.7.1.m1.1a"><mi id="A6.T4.7.7.7.1.m1.1.1" xref="A6.T4.7.7.7.1.m1.1.1.cmml">α</mi><annotation-xml encoding="MathML-Content" id="A6.T4.7.7.7.1.m1.1b"><ci id="A6.T4.7.7.7.1.m1.1.1.cmml" xref="A6.T4.7.7.7.1.m1.1.1">𝛼</ci></annotation-xml><annotation encoding="application/x-tex" id="A6.T4.7.7.7.1.m1.1c">\alpha</annotation><annotation encoding="application/x-llamapun" id="A6.T4.7.7.7.1.m1.1d">italic_α</annotation></semantics></math> (fixed blur, <math alttext="\sigma_{{\mathbf{y}}}\geq 20/255" class="ltx_Math" display="inline" id="A6.T4.8.8.8.2.m2.1"><semantics id="A6.T4.8.8.8.2.m2.1a"><mrow id="A6.T4.8.8.8.2.m2.1.1" xref="A6.T4.8.8.8.2.m2.1.1.cmml"><msub id="A6.T4.8.8.8.2.m2.1.1.2" xref="A6.T4.8.8.8.2.m2.1.1.2.cmml"><mi id="A6.T4.8.8.8.2.m2.1.1.2.2" xref="A6.T4.8.8.8.2.m2.1.1.2.2.cmml">σ</mi><mi id="A6.T4.8.8.8.2.m2.1.1.2.3" xref="A6.T4.8.8.8.2.m2.1.1.2.3.cmml">𝐲</mi></msub><mo id="A6.T4.8.8.8.2.m2.1.1.1" xref="A6.T4.8.8.8.2.m2.1.1.1.cmml">≥</mo><mrow id="A6.T4.8.8.8.2.m2.1.1.3" xref="A6.T4.8.8.8.2.m2.1.1.3.cmml"><mn id="A6.T4.8.8.8.2.m2.1.1.3.2" xref="A6.T4.8.8.8.2.m2.1.1.3.2.cmml">20</mn><mo id="A6.T4.8.8.8.2.m2.1.1.3.1" xref="A6.T4.8.8.8.2.m2.1.1.3.1.cmml">/</mo><mn id="A6.T4.8.8.8.2.m2.1.1.3.3" xref="A6.T4.8.8.8.2.m2.1.1.3.3.cmml">255</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="A6.T4.8.8.8.2.m2.1b"><apply id="A6.T4.8.8.8.2.m2.1.1.cmml" xref="A6.T4.8.8.8.2.m2.1.1"><geq id="A6.T4.8.8.8.2.m2.1.1.1.cmml" xref="A6.T4.8.8.8.2.m2.1.1.1"></geq><apply id="A6.T4.8.8.8.2.m2.1.1.2.cmml" xref="A6.T4.8.8.8.2.m2.1.1.2"><csymbol cd="ambiguous" id="A6.T4.8.8.8.2.m2.1.1.2.1.cmml" xref="A6.T4.8.8.8.2.m2.1.1.2">subscript</csymbol><ci id="A6.T4.8.8.8.2.m2.1.1.2.2.cmml" xref="A6.T4.8.8.8.2.m2.1.1.2.2">𝜎</ci><ci id="A6.T4.8.8.8.2.m2.1.1.2.3.cmml" xref="A6.T4.8.8.8.2.m2.1.1.2.3">𝐲</ci></apply><apply id="A6.T4.8.8.8.2.m2.1.1.3.cmml" xref="A6.T4.8.8.8.2.m2.1.1.3"><divide id="A6.T4.8.8.8.2.m2.1.1.3.1.cmml" xref="A6.T4.8.8.8.2.m2.1.1.3.1"></divide><cn id="A6.T4.8.8.8.2.m2.1.1.3.2.cmml" type="integer" xref="A6.T4.8.8.8.2.m2.1.1.3.2">20</cn><cn id="A6.T4.8.8.8.2.m2.1.1.3.3.cmml" type="integer" xref="A6.T4.8.8.8.2.m2.1.1.3.3">255</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A6.T4.8.8.8.2.m2.1c">\sigma_{{\mathbf{y}}}\geq 20/255</annotation><annotation encoding="application/x-llamapun" id="A6.T4.8.8.8.2.m2.1d">italic_σ start_POSTSUBSCRIPT bold_y end_POSTSUBSCRIPT ≥ 20 / 255</annotation></semantics></math>)</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="A6.T4.8.8.8.3">0.075</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="A6.T4.8.8.8.4">0.3</td> <td class="ltx_td ltx_border_r ltx_border_t" id="A6.T4.8.8.8.5"></td> <td class="ltx_td ltx_border_r ltx_border_t" id="A6.T4.8.8.8.6"></td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="A6.T4.8.8.8.7">0.5</td> </tr> <tr class="ltx_tr" id="A6.T4.9.9.9"> <td class="ltx_td ltx_align_center ltx_border_l ltx_border_rr ltx_border_t" id="A6.T4.9.9.9.1"><math alttext="\alpha_{0}" class="ltx_Math" display="inline" id="A6.T4.9.9.9.1.m1.1"><semantics id="A6.T4.9.9.9.1.m1.1a"><msub id="A6.T4.9.9.9.1.m1.1.1" xref="A6.T4.9.9.9.1.m1.1.1.cmml"><mi id="A6.T4.9.9.9.1.m1.1.1.2" xref="A6.T4.9.9.9.1.m1.1.1.2.cmml">α</mi><mn id="A6.T4.9.9.9.1.m1.1.1.3" xref="A6.T4.9.9.9.1.m1.1.1.3.cmml">0</mn></msub><annotation-xml encoding="MathML-Content" id="A6.T4.9.9.9.1.m1.1b"><apply id="A6.T4.9.9.9.1.m1.1.1.cmml" xref="A6.T4.9.9.9.1.m1.1.1"><csymbol cd="ambiguous" id="A6.T4.9.9.9.1.m1.1.1.1.cmml" xref="A6.T4.9.9.9.1.m1.1.1">subscript</csymbol><ci id="A6.T4.9.9.9.1.m1.1.1.2.cmml" xref="A6.T4.9.9.9.1.m1.1.1.2">𝛼</ci><cn id="A6.T4.9.9.9.1.m1.1.1.3.cmml" type="integer" xref="A6.T4.9.9.9.1.m1.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A6.T4.9.9.9.1.m1.1c">\alpha_{0}</annotation><annotation encoding="application/x-llamapun" id="A6.T4.9.9.9.1.m1.1d">italic_α start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_td ltx_border_r ltx_border_t" id="A6.T4.9.9.9.2"></td> <td class="ltx_td ltx_border_r ltx_border_t" id="A6.T4.9.9.9.3"></td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="A6.T4.9.9.9.4">0.1</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="A6.T4.9.9.9.5">0.1</td> <td class="ltx_td ltx_border_r ltx_border_t" id="A6.T4.9.9.9.6"></td> </tr> <tr class="ltx_tr" id="A6.T4.10.10.10"> <td class="ltx_td ltx_align_center ltx_border_l ltx_border_rr ltx_border_t" id="A6.T4.10.10.10.1"><math alttext="\alpha_{m-1}" class="ltx_Math" display="inline" id="A6.T4.10.10.10.1.m1.1"><semantics id="A6.T4.10.10.10.1.m1.1a"><msub id="A6.T4.10.10.10.1.m1.1.1" xref="A6.T4.10.10.10.1.m1.1.1.cmml"><mi id="A6.T4.10.10.10.1.m1.1.1.2" xref="A6.T4.10.10.10.1.m1.1.1.2.cmml">α</mi><mrow id="A6.T4.10.10.10.1.m1.1.1.3" xref="A6.T4.10.10.10.1.m1.1.1.3.cmml"><mi id="A6.T4.10.10.10.1.m1.1.1.3.2" xref="A6.T4.10.10.10.1.m1.1.1.3.2.cmml">m</mi><mo id="A6.T4.10.10.10.1.m1.1.1.3.1" xref="A6.T4.10.10.10.1.m1.1.1.3.1.cmml">−</mo><mn id="A6.T4.10.10.10.1.m1.1.1.3.3" xref="A6.T4.10.10.10.1.m1.1.1.3.3.cmml">1</mn></mrow></msub><annotation-xml encoding="MathML-Content" id="A6.T4.10.10.10.1.m1.1b"><apply id="A6.T4.10.10.10.1.m1.1.1.cmml" xref="A6.T4.10.10.10.1.m1.1.1"><csymbol cd="ambiguous" id="A6.T4.10.10.10.1.m1.1.1.1.cmml" xref="A6.T4.10.10.10.1.m1.1.1">subscript</csymbol><ci id="A6.T4.10.10.10.1.m1.1.1.2.cmml" xref="A6.T4.10.10.10.1.m1.1.1.2">𝛼</ci><apply id="A6.T4.10.10.10.1.m1.1.1.3.cmml" xref="A6.T4.10.10.10.1.m1.1.1.3"><minus id="A6.T4.10.10.10.1.m1.1.1.3.1.cmml" xref="A6.T4.10.10.10.1.m1.1.1.3.1"></minus><ci id="A6.T4.10.10.10.1.m1.1.1.3.2.cmml" xref="A6.T4.10.10.10.1.m1.1.1.3.2">𝑚</ci><cn id="A6.T4.10.10.10.1.m1.1.1.3.3.cmml" type="integer" xref="A6.T4.10.10.10.1.m1.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A6.T4.10.10.10.1.m1.1c">\alpha_{m-1}</annotation><annotation encoding="application/x-llamapun" id="A6.T4.10.10.10.1.m1.1d">italic_α start_POSTSUBSCRIPT italic_m - 1 end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_td ltx_border_r ltx_border_t" id="A6.T4.10.10.10.2"></td> <td class="ltx_td ltx_border_r ltx_border_t" id="A6.T4.10.10.10.3"></td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="A6.T4.10.10.10.4">1</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="A6.T4.10.10.10.5">1</td> <td class="ltx_td ltx_border_r ltx_border_t" id="A6.T4.10.10.10.6"></td> </tr> <tr class="ltx_tr" id="A6.T4.12.12.12"> <td class="ltx_td ltx_align_center ltx_border_l ltx_border_rr ltx_border_t" id="A6.T4.12.12.12.2"> <math alttext="\sigma/\sigma_{{\mathbf{y}}}" class="ltx_Math" display="inline" id="A6.T4.11.11.11.1.m1.1"><semantics id="A6.T4.11.11.11.1.m1.1a"><mrow id="A6.T4.11.11.11.1.m1.1.1" xref="A6.T4.11.11.11.1.m1.1.1.cmml"><mi id="A6.T4.11.11.11.1.m1.1.1.2" xref="A6.T4.11.11.11.1.m1.1.1.2.cmml">σ</mi><mo id="A6.T4.11.11.11.1.m1.1.1.1" xref="A6.T4.11.11.11.1.m1.1.1.1.cmml">/</mo><msub id="A6.T4.11.11.11.1.m1.1.1.3" xref="A6.T4.11.11.11.1.m1.1.1.3.cmml"><mi id="A6.T4.11.11.11.1.m1.1.1.3.2" xref="A6.T4.11.11.11.1.m1.1.1.3.2.cmml">σ</mi><mi id="A6.T4.11.11.11.1.m1.1.1.3.3" xref="A6.T4.11.11.11.1.m1.1.1.3.3.cmml">𝐲</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="A6.T4.11.11.11.1.m1.1b"><apply id="A6.T4.11.11.11.1.m1.1.1.cmml" xref="A6.T4.11.11.11.1.m1.1.1"><divide id="A6.T4.11.11.11.1.m1.1.1.1.cmml" xref="A6.T4.11.11.11.1.m1.1.1.1"></divide><ci id="A6.T4.11.11.11.1.m1.1.1.2.cmml" xref="A6.T4.11.11.11.1.m1.1.1.2">𝜎</ci><apply id="A6.T4.11.11.11.1.m1.1.1.3.cmml" xref="A6.T4.11.11.11.1.m1.1.1.3"><csymbol cd="ambiguous" id="A6.T4.11.11.11.1.m1.1.1.3.1.cmml" xref="A6.T4.11.11.11.1.m1.1.1.3">subscript</csymbol><ci id="A6.T4.11.11.11.1.m1.1.1.3.2.cmml" xref="A6.T4.11.11.11.1.m1.1.1.3.2">𝜎</ci><ci id="A6.T4.11.11.11.1.m1.1.1.3.3.cmml" xref="A6.T4.11.11.11.1.m1.1.1.3.3">𝐲</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A6.T4.11.11.11.1.m1.1c">\sigma/\sigma_{{\mathbf{y}}}</annotation><annotation encoding="application/x-llamapun" id="A6.T4.11.11.11.1.m1.1d">italic_σ / italic_σ start_POSTSUBSCRIPT bold_y end_POSTSUBSCRIPT</annotation></semantics></math> (<math alttext="\sigma_{{\mathbf{y}}}<20/255" class="ltx_Math" display="inline" id="A6.T4.12.12.12.2.m2.1"><semantics id="A6.T4.12.12.12.2.m2.1a"><mrow id="A6.T4.12.12.12.2.m2.1.1" xref="A6.T4.12.12.12.2.m2.1.1.cmml"><msub id="A6.T4.12.12.12.2.m2.1.1.2" xref="A6.T4.12.12.12.2.m2.1.1.2.cmml"><mi id="A6.T4.12.12.12.2.m2.1.1.2.2" xref="A6.T4.12.12.12.2.m2.1.1.2.2.cmml">σ</mi><mi id="A6.T4.12.12.12.2.m2.1.1.2.3" xref="A6.T4.12.12.12.2.m2.1.1.2.3.cmml">𝐲</mi></msub><mo id="A6.T4.12.12.12.2.m2.1.1.1" xref="A6.T4.12.12.12.2.m2.1.1.1.cmml"><</mo><mrow id="A6.T4.12.12.12.2.m2.1.1.3" xref="A6.T4.12.12.12.2.m2.1.1.3.cmml"><mn id="A6.T4.12.12.12.2.m2.1.1.3.2" xref="A6.T4.12.12.12.2.m2.1.1.3.2.cmml">20</mn><mo id="A6.T4.12.12.12.2.m2.1.1.3.1" xref="A6.T4.12.12.12.2.m2.1.1.3.1.cmml">/</mo><mn id="A6.T4.12.12.12.2.m2.1.1.3.3" xref="A6.T4.12.12.12.2.m2.1.1.3.3.cmml">255</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="A6.T4.12.12.12.2.m2.1b"><apply id="A6.T4.12.12.12.2.m2.1.1.cmml" xref="A6.T4.12.12.12.2.m2.1.1"><lt id="A6.T4.12.12.12.2.m2.1.1.1.cmml" xref="A6.T4.12.12.12.2.m2.1.1.1"></lt><apply id="A6.T4.12.12.12.2.m2.1.1.2.cmml" xref="A6.T4.12.12.12.2.m2.1.1.2"><csymbol cd="ambiguous" id="A6.T4.12.12.12.2.m2.1.1.2.1.cmml" xref="A6.T4.12.12.12.2.m2.1.1.2">subscript</csymbol><ci id="A6.T4.12.12.12.2.m2.1.1.2.2.cmml" xref="A6.T4.12.12.12.2.m2.1.1.2.2">𝜎</ci><ci id="A6.T4.12.12.12.2.m2.1.1.2.3.cmml" xref="A6.T4.12.12.12.2.m2.1.1.2.3">𝐲</ci></apply><apply id="A6.T4.12.12.12.2.m2.1.1.3.cmml" xref="A6.T4.12.12.12.2.m2.1.1.3"><divide id="A6.T4.12.12.12.2.m2.1.1.3.1.cmml" xref="A6.T4.12.12.12.2.m2.1.1.3.1"></divide><cn id="A6.T4.12.12.12.2.m2.1.1.3.2.cmml" type="integer" xref="A6.T4.12.12.12.2.m2.1.1.3.2">20</cn><cn id="A6.T4.12.12.12.2.m2.1.1.3.3.cmml" type="integer" xref="A6.T4.12.12.12.2.m2.1.1.3.3">255</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A6.T4.12.12.12.2.m2.1c">\sigma_{{\mathbf{y}}}<20/255</annotation><annotation encoding="application/x-llamapun" id="A6.T4.12.12.12.2.m2.1d">italic_σ start_POSTSUBSCRIPT bold_y end_POSTSUBSCRIPT < 20 / 255</annotation></semantics></math>)</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="A6.T4.12.12.12.3">1.8</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="A6.T4.12.12.12.4">1.4</td> <td class="ltx_td ltx_border_r ltx_border_t" id="A6.T4.12.12.12.5"></td> <td class="ltx_td ltx_border_r ltx_border_t" id="A6.T4.12.12.12.6"></td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="A6.T4.12.12.12.7">1.</td> </tr> <tr class="ltx_tr" id="A6.T4.14.14.14"> <td class="ltx_td ltx_align_center ltx_border_l ltx_border_rr ltx_border_t" id="A6.T4.14.14.14.2"> <math alttext="\sigma/\sigma_{{\mathbf{y}}}" class="ltx_Math" display="inline" id="A6.T4.13.13.13.1.m1.1"><semantics id="A6.T4.13.13.13.1.m1.1a"><mrow id="A6.T4.13.13.13.1.m1.1.1" xref="A6.T4.13.13.13.1.m1.1.1.cmml"><mi id="A6.T4.13.13.13.1.m1.1.1.2" xref="A6.T4.13.13.13.1.m1.1.1.2.cmml">σ</mi><mo id="A6.T4.13.13.13.1.m1.1.1.1" xref="A6.T4.13.13.13.1.m1.1.1.1.cmml">/</mo><msub id="A6.T4.13.13.13.1.m1.1.1.3" xref="A6.T4.13.13.13.1.m1.1.1.3.cmml"><mi id="A6.T4.13.13.13.1.m1.1.1.3.2" xref="A6.T4.13.13.13.1.m1.1.1.3.2.cmml">σ</mi><mi id="A6.T4.13.13.13.1.m1.1.1.3.3" xref="A6.T4.13.13.13.1.m1.1.1.3.3.cmml">𝐲</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="A6.T4.13.13.13.1.m1.1b"><apply id="A6.T4.13.13.13.1.m1.1.1.cmml" xref="A6.T4.13.13.13.1.m1.1.1"><divide id="A6.T4.13.13.13.1.m1.1.1.1.cmml" xref="A6.T4.13.13.13.1.m1.1.1.1"></divide><ci id="A6.T4.13.13.13.1.m1.1.1.2.cmml" xref="A6.T4.13.13.13.1.m1.1.1.2">𝜎</ci><apply id="A6.T4.13.13.13.1.m1.1.1.3.cmml" xref="A6.T4.13.13.13.1.m1.1.1.3"><csymbol cd="ambiguous" id="A6.T4.13.13.13.1.m1.1.1.3.1.cmml" xref="A6.T4.13.13.13.1.m1.1.1.3">subscript</csymbol><ci id="A6.T4.13.13.13.1.m1.1.1.3.2.cmml" xref="A6.T4.13.13.13.1.m1.1.1.3.2">𝜎</ci><ci id="A6.T4.13.13.13.1.m1.1.1.3.3.cmml" xref="A6.T4.13.13.13.1.m1.1.1.3.3">𝐲</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A6.T4.13.13.13.1.m1.1c">\sigma/\sigma_{{\mathbf{y}}}</annotation><annotation encoding="application/x-llamapun" id="A6.T4.13.13.13.1.m1.1d">italic_σ / italic_σ start_POSTSUBSCRIPT bold_y end_POSTSUBSCRIPT</annotation></semantics></math> (<math alttext="\sigma_{{\mathbf{y}}}\geq 20/255" class="ltx_Math" display="inline" id="A6.T4.14.14.14.2.m2.1"><semantics id="A6.T4.14.14.14.2.m2.1a"><mrow id="A6.T4.14.14.14.2.m2.1.1" xref="A6.T4.14.14.14.2.m2.1.1.cmml"><msub id="A6.T4.14.14.14.2.m2.1.1.2" xref="A6.T4.14.14.14.2.m2.1.1.2.cmml"><mi id="A6.T4.14.14.14.2.m2.1.1.2.2" xref="A6.T4.14.14.14.2.m2.1.1.2.2.cmml">σ</mi><mi id="A6.T4.14.14.14.2.m2.1.1.2.3" xref="A6.T4.14.14.14.2.m2.1.1.2.3.cmml">𝐲</mi></msub><mo id="A6.T4.14.14.14.2.m2.1.1.1" xref="A6.T4.14.14.14.2.m2.1.1.1.cmml">≥</mo><mrow id="A6.T4.14.14.14.2.m2.1.1.3" xref="A6.T4.14.14.14.2.m2.1.1.3.cmml"><mn id="A6.T4.14.14.14.2.m2.1.1.3.2" xref="A6.T4.14.14.14.2.m2.1.1.3.2.cmml">20</mn><mo id="A6.T4.14.14.14.2.m2.1.1.3.1" xref="A6.T4.14.14.14.2.m2.1.1.3.1.cmml">/</mo><mn id="A6.T4.14.14.14.2.m2.1.1.3.3" xref="A6.T4.14.14.14.2.m2.1.1.3.3.cmml">255</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="A6.T4.14.14.14.2.m2.1b"><apply id="A6.T4.14.14.14.2.m2.1.1.cmml" xref="A6.T4.14.14.14.2.m2.1.1"><geq id="A6.T4.14.14.14.2.m2.1.1.1.cmml" xref="A6.T4.14.14.14.2.m2.1.1.1"></geq><apply id="A6.T4.14.14.14.2.m2.1.1.2.cmml" xref="A6.T4.14.14.14.2.m2.1.1.2"><csymbol cd="ambiguous" id="A6.T4.14.14.14.2.m2.1.1.2.1.cmml" xref="A6.T4.14.14.14.2.m2.1.1.2">subscript</csymbol><ci id="A6.T4.14.14.14.2.m2.1.1.2.2.cmml" xref="A6.T4.14.14.14.2.m2.1.1.2.2">𝜎</ci><ci id="A6.T4.14.14.14.2.m2.1.1.2.3.cmml" xref="A6.T4.14.14.14.2.m2.1.1.2.3">𝐲</ci></apply><apply id="A6.T4.14.14.14.2.m2.1.1.3.cmml" xref="A6.T4.14.14.14.2.m2.1.1.3"><divide id="A6.T4.14.14.14.2.m2.1.1.3.1.cmml" xref="A6.T4.14.14.14.2.m2.1.1.3.1"></divide><cn id="A6.T4.14.14.14.2.m2.1.1.3.2.cmml" type="integer" xref="A6.T4.14.14.14.2.m2.1.1.3.2">20</cn><cn id="A6.T4.14.14.14.2.m2.1.1.3.3.cmml" type="integer" xref="A6.T4.14.14.14.2.m2.1.1.3.3">255</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A6.T4.14.14.14.2.m2.1c">\sigma_{{\mathbf{y}}}\geq 20/255</annotation><annotation encoding="application/x-llamapun" id="A6.T4.14.14.14.2.m2.1d">italic_σ start_POSTSUBSCRIPT bold_y end_POSTSUBSCRIPT ≥ 20 / 255</annotation></semantics></math>)</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="A6.T4.14.14.14.3">1.8</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="A6.T4.14.14.14.4">1.8</td> <td class="ltx_td ltx_border_r ltx_border_t" id="A6.T4.14.14.14.5"></td> <td class="ltx_td ltx_border_r ltx_border_t" id="A6.T4.14.14.14.6"></td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="A6.T4.14.14.14.7">1.</td> </tr> <tr class="ltx_tr" id="A6.T4.15.15.15"> <td class="ltx_td ltx_align_center ltx_border_l ltx_border_rr ltx_border_t" id="A6.T4.15.15.15.1"><math alttext="\sigma_{0}/\sigma_{{\mathbf{y}}}" class="ltx_Math" display="inline" id="A6.T4.15.15.15.1.m1.1"><semantics id="A6.T4.15.15.15.1.m1.1a"><mrow id="A6.T4.15.15.15.1.m1.1.1" xref="A6.T4.15.15.15.1.m1.1.1.cmml"><msub id="A6.T4.15.15.15.1.m1.1.1.2" xref="A6.T4.15.15.15.1.m1.1.1.2.cmml"><mi id="A6.T4.15.15.15.1.m1.1.1.2.2" xref="A6.T4.15.15.15.1.m1.1.1.2.2.cmml">σ</mi><mn id="A6.T4.15.15.15.1.m1.1.1.2.3" xref="A6.T4.15.15.15.1.m1.1.1.2.3.cmml">0</mn></msub><mo id="A6.T4.15.15.15.1.m1.1.1.1" xref="A6.T4.15.15.15.1.m1.1.1.1.cmml">/</mo><msub id="A6.T4.15.15.15.1.m1.1.1.3" xref="A6.T4.15.15.15.1.m1.1.1.3.cmml"><mi id="A6.T4.15.15.15.1.m1.1.1.3.2" xref="A6.T4.15.15.15.1.m1.1.1.3.2.cmml">σ</mi><mi id="A6.T4.15.15.15.1.m1.1.1.3.3" xref="A6.T4.15.15.15.1.m1.1.1.3.3.cmml">𝐲</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="A6.T4.15.15.15.1.m1.1b"><apply id="A6.T4.15.15.15.1.m1.1.1.cmml" xref="A6.T4.15.15.15.1.m1.1.1"><divide id="A6.T4.15.15.15.1.m1.1.1.1.cmml" 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id="A6.T4.15.15.15.1.m1.1d">italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT / italic_σ start_POSTSUBSCRIPT bold_y end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_td ltx_border_r ltx_border_t" id="A6.T4.15.15.15.2"></td> <td class="ltx_td ltx_border_r ltx_border_t" id="A6.T4.15.15.15.3"></td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="A6.T4.15.15.15.4">1.8</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="A6.T4.15.15.15.5">1.8</td> <td class="ltx_td ltx_border_r ltx_border_t" id="A6.T4.15.15.15.6"></td> </tr> <tr class="ltx_tr" id="A6.T4.16.16.16"> <td class="ltx_td ltx_align_center ltx_border_l ltx_border_rr ltx_border_t" id="A6.T4.16.16.16.1"><math alttext="\sigma_{m-1}/\sigma_{{\mathbf{y}}}" class="ltx_Math" display="inline" id="A6.T4.16.16.16.1.m1.1"><semantics id="A6.T4.16.16.16.1.m1.1a"><mrow id="A6.T4.16.16.16.1.m1.1.1" xref="A6.T4.16.16.16.1.m1.1.1.cmml"><msub id="A6.T4.16.16.16.1.m1.1.1.2" 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id="A6.T4.17.17.17.1"><math alttext="\text{max}_{\text{itr}}" class="ltx_Math" display="inline" id="A6.T4.17.17.17.1.m1.1"><semantics id="A6.T4.17.17.17.1.m1.1a"><msub id="A6.T4.17.17.17.1.m1.1.1" xref="A6.T4.17.17.17.1.m1.1.1.cmml"><mtext id="A6.T4.17.17.17.1.m1.1.1.2" xref="A6.T4.17.17.17.1.m1.1.1.2a.cmml">max</mtext><mtext id="A6.T4.17.17.17.1.m1.1.1.3" xref="A6.T4.17.17.17.1.m1.1.1.3a.cmml">itr</mtext></msub><annotation-xml encoding="MathML-Content" id="A6.T4.17.17.17.1.m1.1b"><apply id="A6.T4.17.17.17.1.m1.1.1.cmml" xref="A6.T4.17.17.17.1.m1.1.1"><csymbol cd="ambiguous" id="A6.T4.17.17.17.1.m1.1.1.1.cmml" xref="A6.T4.17.17.17.1.m1.1.1">subscript</csymbol><ci id="A6.T4.17.17.17.1.m1.1.1.2a.cmml" xref="A6.T4.17.17.17.1.m1.1.1.2"><mtext id="A6.T4.17.17.17.1.m1.1.1.2.cmml" xref="A6.T4.17.17.17.1.m1.1.1.2">max</mtext></ci><ci id="A6.T4.17.17.17.1.m1.1.1.3a.cmml" xref="A6.T4.17.17.17.1.m1.1.1.3"><mtext id="A6.T4.17.17.17.1.m1.1.1.3.cmml" mathsize="70%" 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display="inline" id="A6.T4.18.18.18.1.m1.1"><semantics id="A6.T4.18.18.18.1.m1.1a"><mi id="A6.T4.18.18.18.1.m1.1.1" xref="A6.T4.18.18.18.1.m1.1.1.cmml">δ</mi><annotation-xml encoding="MathML-Content" id="A6.T4.18.18.18.1.m1.1b"><ci id="A6.T4.18.18.18.1.m1.1.1.cmml" xref="A6.T4.18.18.18.1.m1.1.1">𝛿</ci></annotation-xml><annotation encoding="application/x-tex" id="A6.T4.18.18.18.1.m1.1c">\delta</annotation><annotation encoding="application/x-llamapun" id="A6.T4.18.18.18.1.m1.1d">italic_δ</annotation></semantics></math></td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="A6.T4.19.19.19.2"><math alttext="1/\alpha" class="ltx_Math" display="inline" id="A6.T4.19.19.19.2.m1.1"><semantics id="A6.T4.19.19.19.2.m1.1a"><mrow id="A6.T4.19.19.19.2.m1.1.1" xref="A6.T4.19.19.19.2.m1.1.1.cmml"><mn id="A6.T4.19.19.19.2.m1.1.1.2" xref="A6.T4.19.19.19.2.m1.1.1.2.cmml">1</mn><mo id="A6.T4.19.19.19.2.m1.1.1.1" xref="A6.T4.19.19.19.2.m1.1.1.1.cmml">/</mo><mi id="A6.T4.19.19.19.2.m1.1.1.3" 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id="A6.T4.20.20.20.3.m1.1.1.2" xref="A6.T4.20.20.20.3.m1.1.1.2.cmml">1</mn><mo id="A6.T4.20.20.20.3.m1.1.1.1" xref="A6.T4.20.20.20.3.m1.1.1.1.cmml">/</mo><mi id="A6.T4.20.20.20.3.m1.1.1.3" xref="A6.T4.20.20.20.3.m1.1.1.3.cmml">α</mi></mrow><annotation-xml encoding="MathML-Content" id="A6.T4.20.20.20.3.m1.1b"><apply id="A6.T4.20.20.20.3.m1.1.1.cmml" xref="A6.T4.20.20.20.3.m1.1.1"><divide id="A6.T4.20.20.20.3.m1.1.1.1.cmml" xref="A6.T4.20.20.20.3.m1.1.1.1"></divide><cn id="A6.T4.20.20.20.3.m1.1.1.2.cmml" type="integer" xref="A6.T4.20.20.20.3.m1.1.1.2">1</cn><ci id="A6.T4.20.20.20.3.m1.1.1.3.cmml" xref="A6.T4.20.20.20.3.m1.1.1.3">𝛼</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A6.T4.20.20.20.3.m1.1c">1/\alpha</annotation><annotation encoding="application/x-llamapun" id="A6.T4.20.20.20.3.m1.1d">1 / italic_α</annotation></semantics></math></td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="A6.T4.20.20.20.4">0.1</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="A6.T4.20.20.20.5">0.1</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="A6.T4.20.20.20.6">0.1</td> </tr> <tr class="ltx_tr" id="A6.T4.21.21.21"> <td class="ltx_td ltx_align_center ltx_border_b ltx_border_l ltx_border_rr ltx_border_t" id="A6.T4.21.21.21.1"><math alttext="\beta" class="ltx_Math" display="inline" id="A6.T4.21.21.21.1.m1.1"><semantics id="A6.T4.21.21.21.1.m1.1a"><mi id="A6.T4.21.21.21.1.m1.1.1" xref="A6.T4.21.21.21.1.m1.1.1.cmml">β</mi><annotation-xml encoding="MathML-Content" id="A6.T4.21.21.21.1.m1.1b"><ci id="A6.T4.21.21.21.1.m1.1.1.cmml" xref="A6.T4.21.21.21.1.m1.1.1">𝛽</ci></annotation-xml><annotation encoding="application/x-tex" id="A6.T4.21.21.21.1.m1.1c">\beta</annotation><annotation encoding="application/x-llamapun" id="A6.T4.21.21.21.1.m1.1d">italic_β</annotation></semantics></math></td> <td class="ltx_td ltx_border_b ltx_border_r ltx_border_t" id="A6.T4.21.21.21.2"></td> <td class="ltx_td ltx_border_b ltx_border_r ltx_border_t" id="A6.T4.21.21.21.3"></td> <td class="ltx_td ltx_border_b ltx_border_r ltx_border_t" id="A6.T4.21.21.21.4"></td> <td class="ltx_td ltx_border_b ltx_border_r ltx_border_t" id="A6.T4.21.21.21.5"></td> <td class="ltx_td ltx_align_center ltx_border_b ltx_border_r ltx_border_t" id="A6.T4.21.21.21.6">0.01</td> </tr> </tbody> </table> </span></div> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_table"><span class="ltx_text" id="A6.T4.23.1.1" style="font-size:90%;">Table 4</span>: </span><span class="ltx_text" id="A6.T4.24.2" style="font-size:90%;">Parameters setting for image deblurring for the different implemented methods.</span></figcaption> </figure> </section> <section class="ltx_paragraph" id="A6.SS1.SSS0.Px2"> <h5 class="ltx_title ltx_title_paragraph">More deblurring results</h5> <div class="ltx_para" id="A6.SS1.SSS0.Px2.p1"> <p class="ltx_p" id="A6.SS1.SSS0.Px2.p1.4">For a better qualitative comparison between methods, we present in Figure <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#A4.F5" title="Figure 5 ‣ D.1.5 Proof of Proposition 3.5 ‣ D.1 Proof of critical Point Analysis ‣ Appendix D Proofs of section 3 ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_tag">5</span></a> several image deblurring results obtained with various blur kernels. In Figure <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#A6.F6" title="Figure 6 ‣ RED Prox ‣ Appendix F Additional Experiments ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_tag">6</span></a>, we present the parameters evolution during the optimization process for the annealing SNORE algorithm (Algorithm <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#alg3" title="Algorithm 3 ‣ Optimization algorithms ‣ 2.2 SNORE regularization ‣ 2 Stochastic deNOising REgularization (SNORE) ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_tag">3</span></a>). The annealing levels between <math alttext="\alpha_{0}" class="ltx_Math" display="inline" id="A6.SS1.SSS0.Px2.p1.1.m1.1"><semantics id="A6.SS1.SSS0.Px2.p1.1.m1.1a"><msub id="A6.SS1.SSS0.Px2.p1.1.m1.1.1" xref="A6.SS1.SSS0.Px2.p1.1.m1.1.1.cmml"><mi id="A6.SS1.SSS0.Px2.p1.1.m1.1.1.2" xref="A6.SS1.SSS0.Px2.p1.1.m1.1.1.2.cmml">α</mi><mn id="A6.SS1.SSS0.Px2.p1.1.m1.1.1.3" xref="A6.SS1.SSS0.Px2.p1.1.m1.1.1.3.cmml">0</mn></msub><annotation-xml encoding="MathML-Content" id="A6.SS1.SSS0.Px2.p1.1.m1.1b"><apply id="A6.SS1.SSS0.Px2.p1.1.m1.1.1.cmml" xref="A6.SS1.SSS0.Px2.p1.1.m1.1.1"><csymbol cd="ambiguous" id="A6.SS1.SSS0.Px2.p1.1.m1.1.1.1.cmml" xref="A6.SS1.SSS0.Px2.p1.1.m1.1.1">subscript</csymbol><ci id="A6.SS1.SSS0.Px2.p1.1.m1.1.1.2.cmml" xref="A6.SS1.SSS0.Px2.p1.1.m1.1.1.2">𝛼</ci><cn id="A6.SS1.SSS0.Px2.p1.1.m1.1.1.3.cmml" type="integer" xref="A6.SS1.SSS0.Px2.p1.1.m1.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A6.SS1.SSS0.Px2.p1.1.m1.1c">\alpha_{0}</annotation><annotation encoding="application/x-llamapun" id="A6.SS1.SSS0.Px2.p1.1.m1.1d">italic_α start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math>, <math alttext="\alpha_{m-1}" class="ltx_Math" display="inline" id="A6.SS1.SSS0.Px2.p1.2.m2.1"><semantics id="A6.SS1.SSS0.Px2.p1.2.m2.1a"><msub id="A6.SS1.SSS0.Px2.p1.2.m2.1.1" xref="A6.SS1.SSS0.Px2.p1.2.m2.1.1.cmml"><mi id="A6.SS1.SSS0.Px2.p1.2.m2.1.1.2" xref="A6.SS1.SSS0.Px2.p1.2.m2.1.1.2.cmml">α</mi><mrow id="A6.SS1.SSS0.Px2.p1.2.m2.1.1.3" xref="A6.SS1.SSS0.Px2.p1.2.m2.1.1.3.cmml"><mi id="A6.SS1.SSS0.Px2.p1.2.m2.1.1.3.2" xref="A6.SS1.SSS0.Px2.p1.2.m2.1.1.3.2.cmml">m</mi><mo id="A6.SS1.SSS0.Px2.p1.2.m2.1.1.3.1" xref="A6.SS1.SSS0.Px2.p1.2.m2.1.1.3.1.cmml">−</mo><mn id="A6.SS1.SSS0.Px2.p1.2.m2.1.1.3.3" xref="A6.SS1.SSS0.Px2.p1.2.m2.1.1.3.3.cmml">1</mn></mrow></msub><annotation-xml encoding="MathML-Content" id="A6.SS1.SSS0.Px2.p1.2.m2.1b"><apply id="A6.SS1.SSS0.Px2.p1.2.m2.1.1.cmml" xref="A6.SS1.SSS0.Px2.p1.2.m2.1.1"><csymbol cd="ambiguous" id="A6.SS1.SSS0.Px2.p1.2.m2.1.1.1.cmml" xref="A6.SS1.SSS0.Px2.p1.2.m2.1.1">subscript</csymbol><ci id="A6.SS1.SSS0.Px2.p1.2.m2.1.1.2.cmml" xref="A6.SS1.SSS0.Px2.p1.2.m2.1.1.2">𝛼</ci><apply id="A6.SS1.SSS0.Px2.p1.2.m2.1.1.3.cmml" xref="A6.SS1.SSS0.Px2.p1.2.m2.1.1.3"><minus id="A6.SS1.SSS0.Px2.p1.2.m2.1.1.3.1.cmml" xref="A6.SS1.SSS0.Px2.p1.2.m2.1.1.3.1"></minus><ci id="A6.SS1.SSS0.Px2.p1.2.m2.1.1.3.2.cmml" xref="A6.SS1.SSS0.Px2.p1.2.m2.1.1.3.2">𝑚</ci><cn id="A6.SS1.SSS0.Px2.p1.2.m2.1.1.3.3.cmml" type="integer" xref="A6.SS1.SSS0.Px2.p1.2.m2.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A6.SS1.SSS0.Px2.p1.2.m2.1c">\alpha_{m-1}</annotation><annotation encoding="application/x-llamapun" id="A6.SS1.SSS0.Px2.p1.2.m2.1d">italic_α start_POSTSUBSCRIPT italic_m - 1 end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="\sigma_{0}" class="ltx_Math" display="inline" id="A6.SS1.SSS0.Px2.p1.3.m3.1"><semantics id="A6.SS1.SSS0.Px2.p1.3.m3.1a"><msub id="A6.SS1.SSS0.Px2.p1.3.m3.1.1" xref="A6.SS1.SSS0.Px2.p1.3.m3.1.1.cmml"><mi id="A6.SS1.SSS0.Px2.p1.3.m3.1.1.2" xref="A6.SS1.SSS0.Px2.p1.3.m3.1.1.2.cmml">σ</mi><mn id="A6.SS1.SSS0.Px2.p1.3.m3.1.1.3" xref="A6.SS1.SSS0.Px2.p1.3.m3.1.1.3.cmml">0</mn></msub><annotation-xml encoding="MathML-Content" id="A6.SS1.SSS0.Px2.p1.3.m3.1b"><apply id="A6.SS1.SSS0.Px2.p1.3.m3.1.1.cmml" xref="A6.SS1.SSS0.Px2.p1.3.m3.1.1"><csymbol cd="ambiguous" id="A6.SS1.SSS0.Px2.p1.3.m3.1.1.1.cmml" xref="A6.SS1.SSS0.Px2.p1.3.m3.1.1">subscript</csymbol><ci id="A6.SS1.SSS0.Px2.p1.3.m3.1.1.2.cmml" xref="A6.SS1.SSS0.Px2.p1.3.m3.1.1.2">𝜎</ci><cn id="A6.SS1.SSS0.Px2.p1.3.m3.1.1.3.cmml" type="integer" xref="A6.SS1.SSS0.Px2.p1.3.m3.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A6.SS1.SSS0.Px2.p1.3.m3.1c">\sigma_{0}</annotation><annotation encoding="application/x-llamapun" id="A6.SS1.SSS0.Px2.p1.3.m3.1d">italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math>, <math alttext="\sigma_{m-1}" class="ltx_Math" display="inline" id="A6.SS1.SSS0.Px2.p1.4.m4.1"><semantics id="A6.SS1.SSS0.Px2.p1.4.m4.1a"><msub id="A6.SS1.SSS0.Px2.p1.4.m4.1.1" xref="A6.SS1.SSS0.Px2.p1.4.m4.1.1.cmml"><mi id="A6.SS1.SSS0.Px2.p1.4.m4.1.1.2" xref="A6.SS1.SSS0.Px2.p1.4.m4.1.1.2.cmml">σ</mi><mrow id="A6.SS1.SSS0.Px2.p1.4.m4.1.1.3" xref="A6.SS1.SSS0.Px2.p1.4.m4.1.1.3.cmml"><mi id="A6.SS1.SSS0.Px2.p1.4.m4.1.1.3.2" xref="A6.SS1.SSS0.Px2.p1.4.m4.1.1.3.2.cmml">m</mi><mo id="A6.SS1.SSS0.Px2.p1.4.m4.1.1.3.1" xref="A6.SS1.SSS0.Px2.p1.4.m4.1.1.3.1.cmml">−</mo><mn id="A6.SS1.SSS0.Px2.p1.4.m4.1.1.3.3" xref="A6.SS1.SSS0.Px2.p1.4.m4.1.1.3.3.cmml">1</mn></mrow></msub><annotation-xml encoding="MathML-Content" id="A6.SS1.SSS0.Px2.p1.4.m4.1b"><apply id="A6.SS1.SSS0.Px2.p1.4.m4.1.1.cmml" xref="A6.SS1.SSS0.Px2.p1.4.m4.1.1"><csymbol cd="ambiguous" id="A6.SS1.SSS0.Px2.p1.4.m4.1.1.1.cmml" xref="A6.SS1.SSS0.Px2.p1.4.m4.1.1">subscript</csymbol><ci id="A6.SS1.SSS0.Px2.p1.4.m4.1.1.2.cmml" xref="A6.SS1.SSS0.Px2.p1.4.m4.1.1.2">𝜎</ci><apply id="A6.SS1.SSS0.Px2.p1.4.m4.1.1.3.cmml" xref="A6.SS1.SSS0.Px2.p1.4.m4.1.1.3"><minus id="A6.SS1.SSS0.Px2.p1.4.m4.1.1.3.1.cmml" xref="A6.SS1.SSS0.Px2.p1.4.m4.1.1.3.1"></minus><ci id="A6.SS1.SSS0.Px2.p1.4.m4.1.1.3.2.cmml" xref="A6.SS1.SSS0.Px2.p1.4.m4.1.1.3.2">𝑚</ci><cn id="A6.SS1.SSS0.Px2.p1.4.m4.1.1.3.3.cmml" type="integer" xref="A6.SS1.SSS0.Px2.p1.4.m4.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A6.SS1.SSS0.Px2.p1.4.m4.1c">\sigma_{m-1}</annotation><annotation encoding="application/x-llamapun" id="A6.SS1.SSS0.Px2.p1.4.m4.1d">italic_σ start_POSTSUBSCRIPT italic_m - 1 end_POSTSUBSCRIPT</annotation></semantics></math> are chosen by a linear interpolation.</p> </div> <div class="ltx_para" id="A6.SS1.SSS0.Px2.p2"> <p class="ltx_p" id="A6.SS1.SSS0.Px2.p2.4">Next we focus on the choice of the final parameters <math alttext="(\alpha_{m-1},\sigma_{m-1})" class="ltx_Math" display="inline" id="A6.SS1.SSS0.Px2.p2.1.m1.2"><semantics id="A6.SS1.SSS0.Px2.p2.1.m1.2a"><mrow id="A6.SS1.SSS0.Px2.p2.1.m1.2.2.2" xref="A6.SS1.SSS0.Px2.p2.1.m1.2.2.3.cmml"><mo id="A6.SS1.SSS0.Px2.p2.1.m1.2.2.2.3" stretchy="false" xref="A6.SS1.SSS0.Px2.p2.1.m1.2.2.3.cmml">(</mo><msub id="A6.SS1.SSS0.Px2.p2.1.m1.1.1.1.1" xref="A6.SS1.SSS0.Px2.p2.1.m1.1.1.1.1.cmml"><mi id="A6.SS1.SSS0.Px2.p2.1.m1.1.1.1.1.2" xref="A6.SS1.SSS0.Px2.p2.1.m1.1.1.1.1.2.cmml">α</mi><mrow id="A6.SS1.SSS0.Px2.p2.1.m1.1.1.1.1.3" xref="A6.SS1.SSS0.Px2.p2.1.m1.1.1.1.1.3.cmml"><mi id="A6.SS1.SSS0.Px2.p2.1.m1.1.1.1.1.3.2" xref="A6.SS1.SSS0.Px2.p2.1.m1.1.1.1.1.3.2.cmml">m</mi><mo id="A6.SS1.SSS0.Px2.p2.1.m1.1.1.1.1.3.1" xref="A6.SS1.SSS0.Px2.p2.1.m1.1.1.1.1.3.1.cmml">−</mo><mn id="A6.SS1.SSS0.Px2.p2.1.m1.1.1.1.1.3.3" xref="A6.SS1.SSS0.Px2.p2.1.m1.1.1.1.1.3.3.cmml">1</mn></mrow></msub><mo id="A6.SS1.SSS0.Px2.p2.1.m1.2.2.2.4" xref="A6.SS1.SSS0.Px2.p2.1.m1.2.2.3.cmml">,</mo><msub id="A6.SS1.SSS0.Px2.p2.1.m1.2.2.2.2" xref="A6.SS1.SSS0.Px2.p2.1.m1.2.2.2.2.cmml"><mi id="A6.SS1.SSS0.Px2.p2.1.m1.2.2.2.2.2" xref="A6.SS1.SSS0.Px2.p2.1.m1.2.2.2.2.2.cmml">σ</mi><mrow id="A6.SS1.SSS0.Px2.p2.1.m1.2.2.2.2.3" xref="A6.SS1.SSS0.Px2.p2.1.m1.2.2.2.2.3.cmml"><mi id="A6.SS1.SSS0.Px2.p2.1.m1.2.2.2.2.3.2" xref="A6.SS1.SSS0.Px2.p2.1.m1.2.2.2.2.3.2.cmml">m</mi><mo id="A6.SS1.SSS0.Px2.p2.1.m1.2.2.2.2.3.1" xref="A6.SS1.SSS0.Px2.p2.1.m1.2.2.2.2.3.1.cmml">−</mo><mn id="A6.SS1.SSS0.Px2.p2.1.m1.2.2.2.2.3.3" xref="A6.SS1.SSS0.Px2.p2.1.m1.2.2.2.2.3.3.cmml">1</mn></mrow></msub><mo id="A6.SS1.SSS0.Px2.p2.1.m1.2.2.2.5" stretchy="false" xref="A6.SS1.SSS0.Px2.p2.1.m1.2.2.3.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="A6.SS1.SSS0.Px2.p2.1.m1.2b"><interval closure="open" id="A6.SS1.SSS0.Px2.p2.1.m1.2.2.3.cmml" xref="A6.SS1.SSS0.Px2.p2.1.m1.2.2.2"><apply id="A6.SS1.SSS0.Px2.p2.1.m1.1.1.1.1.cmml" xref="A6.SS1.SSS0.Px2.p2.1.m1.1.1.1.1"><csymbol cd="ambiguous" id="A6.SS1.SSS0.Px2.p2.1.m1.1.1.1.1.1.cmml" xref="A6.SS1.SSS0.Px2.p2.1.m1.1.1.1.1">subscript</csymbol><ci id="A6.SS1.SSS0.Px2.p2.1.m1.1.1.1.1.2.cmml" xref="A6.SS1.SSS0.Px2.p2.1.m1.1.1.1.1.2">𝛼</ci><apply id="A6.SS1.SSS0.Px2.p2.1.m1.1.1.1.1.3.cmml" xref="A6.SS1.SSS0.Px2.p2.1.m1.1.1.1.1.3"><minus id="A6.SS1.SSS0.Px2.p2.1.m1.1.1.1.1.3.1.cmml" xref="A6.SS1.SSS0.Px2.p2.1.m1.1.1.1.1.3.1"></minus><ci id="A6.SS1.SSS0.Px2.p2.1.m1.1.1.1.1.3.2.cmml" xref="A6.SS1.SSS0.Px2.p2.1.m1.1.1.1.1.3.2">𝑚</ci><cn id="A6.SS1.SSS0.Px2.p2.1.m1.1.1.1.1.3.3.cmml" type="integer" xref="A6.SS1.SSS0.Px2.p2.1.m1.1.1.1.1.3.3">1</cn></apply></apply><apply id="A6.SS1.SSS0.Px2.p2.1.m1.2.2.2.2.cmml" xref="A6.SS1.SSS0.Px2.p2.1.m1.2.2.2.2"><csymbol cd="ambiguous" id="A6.SS1.SSS0.Px2.p2.1.m1.2.2.2.2.1.cmml" xref="A6.SS1.SSS0.Px2.p2.1.m1.2.2.2.2">subscript</csymbol><ci id="A6.SS1.SSS0.Px2.p2.1.m1.2.2.2.2.2.cmml" xref="A6.SS1.SSS0.Px2.p2.1.m1.2.2.2.2.2">𝜎</ci><apply id="A6.SS1.SSS0.Px2.p2.1.m1.2.2.2.2.3.cmml" xref="A6.SS1.SSS0.Px2.p2.1.m1.2.2.2.2.3"><minus id="A6.SS1.SSS0.Px2.p2.1.m1.2.2.2.2.3.1.cmml" xref="A6.SS1.SSS0.Px2.p2.1.m1.2.2.2.2.3.1"></minus><ci id="A6.SS1.SSS0.Px2.p2.1.m1.2.2.2.2.3.2.cmml" xref="A6.SS1.SSS0.Px2.p2.1.m1.2.2.2.2.3.2">𝑚</ci><cn id="A6.SS1.SSS0.Px2.p2.1.m1.2.2.2.2.3.3.cmml" type="integer" xref="A6.SS1.SSS0.Px2.p2.1.m1.2.2.2.2.3.3">1</cn></apply></apply></interval></annotation-xml><annotation encoding="application/x-tex" id="A6.SS1.SSS0.Px2.p2.1.m1.2c">(\alpha_{m-1},\sigma_{m-1})</annotation><annotation encoding="application/x-llamapun" id="A6.SS1.SSS0.Px2.p2.1.m1.2d">( italic_α start_POSTSUBSCRIPT italic_m - 1 end_POSTSUBSCRIPT , italic_σ start_POSTSUBSCRIPT italic_m - 1 end_POSTSUBSCRIPT )</annotation></semantics></math>, because these parameters define the optimization problem from which a critical point is finally computed. 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We choose to optimize at different levels <math alttext="(\alpha_{i},\sigma_{i})" class="ltx_Math" display="inline" id="A6.SS1.SSS0.Px2.p2.3.m3.2"><semantics id="A6.SS1.SSS0.Px2.p2.3.m3.2a"><mrow id="A6.SS1.SSS0.Px2.p2.3.m3.2.2.2" xref="A6.SS1.SSS0.Px2.p2.3.m3.2.2.3.cmml"><mo id="A6.SS1.SSS0.Px2.p2.3.m3.2.2.2.3" stretchy="false" xref="A6.SS1.SSS0.Px2.p2.3.m3.2.2.3.cmml">(</mo><msub id="A6.SS1.SSS0.Px2.p2.3.m3.1.1.1.1" xref="A6.SS1.SSS0.Px2.p2.3.m3.1.1.1.1.cmml"><mi id="A6.SS1.SSS0.Px2.p2.3.m3.1.1.1.1.2" xref="A6.SS1.SSS0.Px2.p2.3.m3.1.1.1.1.2.cmml">α</mi><mi id="A6.SS1.SSS0.Px2.p2.3.m3.1.1.1.1.3" xref="A6.SS1.SSS0.Px2.p2.3.m3.1.1.1.1.3.cmml">i</mi></msub><mo id="A6.SS1.SSS0.Px2.p2.3.m3.2.2.2.4" xref="A6.SS1.SSS0.Px2.p2.3.m3.2.2.3.cmml">,</mo><msub id="A6.SS1.SSS0.Px2.p2.3.m3.2.2.2.2" xref="A6.SS1.SSS0.Px2.p2.3.m3.2.2.2.2.cmml"><mi id="A6.SS1.SSS0.Px2.p2.3.m3.2.2.2.2.2" xref="A6.SS1.SSS0.Px2.p2.3.m3.2.2.2.2.2.cmml">σ</mi><mi id="A6.SS1.SSS0.Px2.p2.3.m3.2.2.2.2.3" xref="A6.SS1.SSS0.Px2.p2.3.m3.2.2.2.2.3.cmml">i</mi></msub><mo id="A6.SS1.SSS0.Px2.p2.3.m3.2.2.2.5" stretchy="false" xref="A6.SS1.SSS0.Px2.p2.3.m3.2.2.3.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="A6.SS1.SSS0.Px2.p2.3.m3.2b"><interval closure="open" id="A6.SS1.SSS0.Px2.p2.3.m3.2.2.3.cmml" xref="A6.SS1.SSS0.Px2.p2.3.m3.2.2.2"><apply id="A6.SS1.SSS0.Px2.p2.3.m3.1.1.1.1.cmml" xref="A6.SS1.SSS0.Px2.p2.3.m3.1.1.1.1"><csymbol cd="ambiguous" id="A6.SS1.SSS0.Px2.p2.3.m3.1.1.1.1.1.cmml" xref="A6.SS1.SSS0.Px2.p2.3.m3.1.1.1.1">subscript</csymbol><ci id="A6.SS1.SSS0.Px2.p2.3.m3.1.1.1.1.2.cmml" xref="A6.SS1.SSS0.Px2.p2.3.m3.1.1.1.1.2">𝛼</ci><ci id="A6.SS1.SSS0.Px2.p2.3.m3.1.1.1.1.3.cmml" xref="A6.SS1.SSS0.Px2.p2.3.m3.1.1.1.1.3">𝑖</ci></apply><apply id="A6.SS1.SSS0.Px2.p2.3.m3.2.2.2.2.cmml" xref="A6.SS1.SSS0.Px2.p2.3.m3.2.2.2.2"><csymbol cd="ambiguous" id="A6.SS1.SSS0.Px2.p2.3.m3.2.2.2.2.1.cmml" xref="A6.SS1.SSS0.Px2.p2.3.m3.2.2.2.2">subscript</csymbol><ci 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start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , italic_σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT )</annotation></semantics></math> and to fit with the setting of Proposition <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#S3.Thmproposition2" title="Proposition 3.2. ‣ 3.1 Asymptotics of critical points when 𝜎→0 ‣ 3 Convergence Analysis ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_tag">3.2</span></a>.</p> </div> </section> <section class="ltx_paragraph" id="A6.SS1.SSS0.Px3"> <h5 class="ltx_title ltx_title_paragraph">Influence of <math alttext="\alpha_{m-1}" class="ltx_Math" display="inline" id="A6.SS1.SSS0.Px3.1.m1.1"><semantics id="A6.SS1.SSS0.Px3.1.m1.1b"><msub id="A6.SS1.SSS0.Px3.1.m1.1.1" xref="A6.SS1.SSS0.Px3.1.m1.1.1.cmml"><mi id="A6.SS1.SSS0.Px3.1.m1.1.1.2" xref="A6.SS1.SSS0.Px3.1.m1.1.1.2.cmml">α</mi><mrow id="A6.SS1.SSS0.Px3.1.m1.1.1.3" xref="A6.SS1.SSS0.Px3.1.m1.1.1.3.cmml"><mi id="A6.SS1.SSS0.Px3.1.m1.1.1.3.2" xref="A6.SS1.SSS0.Px3.1.m1.1.1.3.2.cmml">m</mi><mo id="A6.SS1.SSS0.Px3.1.m1.1.1.3.1" xref="A6.SS1.SSS0.Px3.1.m1.1.1.3.1.cmml">−</mo><mn id="A6.SS1.SSS0.Px3.1.m1.1.1.3.3" xref="A6.SS1.SSS0.Px3.1.m1.1.1.3.3.cmml">1</mn></mrow></msub><annotation-xml encoding="MathML-Content" id="A6.SS1.SSS0.Px3.1.m1.1c"><apply id="A6.SS1.SSS0.Px3.1.m1.1.1.cmml" xref="A6.SS1.SSS0.Px3.1.m1.1.1"><csymbol cd="ambiguous" id="A6.SS1.SSS0.Px3.1.m1.1.1.1.cmml" xref="A6.SS1.SSS0.Px3.1.m1.1.1">subscript</csymbol><ci id="A6.SS1.SSS0.Px3.1.m1.1.1.2.cmml" xref="A6.SS1.SSS0.Px3.1.m1.1.1.2">𝛼</ci><apply id="A6.SS1.SSS0.Px3.1.m1.1.1.3.cmml" xref="A6.SS1.SSS0.Px3.1.m1.1.1.3"><minus id="A6.SS1.SSS0.Px3.1.m1.1.1.3.1.cmml" xref="A6.SS1.SSS0.Px3.1.m1.1.1.3.1"></minus><ci id="A6.SS1.SSS0.Px3.1.m1.1.1.3.2.cmml" xref="A6.SS1.SSS0.Px3.1.m1.1.1.3.2">𝑚</ci><cn id="A6.SS1.SSS0.Px3.1.m1.1.1.3.3.cmml" type="integer" xref="A6.SS1.SSS0.Px3.1.m1.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A6.SS1.SSS0.Px3.1.m1.1d">\alpha_{m-1}</annotation><annotation encoding="application/x-llamapun" id="A6.SS1.SSS0.Px3.1.m1.1e">italic_α start_POSTSUBSCRIPT italic_m - 1 end_POSTSUBSCRIPT</annotation></semantics></math> </h5> <div class="ltx_para" id="A6.SS1.SSS0.Px3.p1"> <p class="ltx_p" id="A6.SS1.SSS0.Px3.p1.3">On Figure <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#A6.F7" title="Figure 7 ‣ Influence of 𝛼_{𝑚-1} ‣ F.1 Deblurring ‣ Appendix F Additional Experiments ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_tag">7</span></a>, we study the influence of the final weighting parameter <math alttext="\alpha_{m-1}" class="ltx_Math" display="inline" id="A6.SS1.SSS0.Px3.p1.1.m1.1"><semantics id="A6.SS1.SSS0.Px3.p1.1.m1.1a"><msub id="A6.SS1.SSS0.Px3.p1.1.m1.1.1" xref="A6.SS1.SSS0.Px3.p1.1.m1.1.1.cmml"><mi id="A6.SS1.SSS0.Px3.p1.1.m1.1.1.2" xref="A6.SS1.SSS0.Px3.p1.1.m1.1.1.2.cmml">α</mi><mrow id="A6.SS1.SSS0.Px3.p1.1.m1.1.1.3" xref="A6.SS1.SSS0.Px3.p1.1.m1.1.1.3.cmml"><mi id="A6.SS1.SSS0.Px3.p1.1.m1.1.1.3.2" xref="A6.SS1.SSS0.Px3.p1.1.m1.1.1.3.2.cmml">m</mi><mo id="A6.SS1.SSS0.Px3.p1.1.m1.1.1.3.1" xref="A6.SS1.SSS0.Px3.p1.1.m1.1.1.3.1.cmml">−</mo><mn id="A6.SS1.SSS0.Px3.p1.1.m1.1.1.3.3" xref="A6.SS1.SSS0.Px3.p1.1.m1.1.1.3.3.cmml">1</mn></mrow></msub><annotation-xml encoding="MathML-Content" id="A6.SS1.SSS0.Px3.p1.1.m1.1b"><apply id="A6.SS1.SSS0.Px3.p1.1.m1.1.1.cmml" xref="A6.SS1.SSS0.Px3.p1.1.m1.1.1"><csymbol cd="ambiguous" id="A6.SS1.SSS0.Px3.p1.1.m1.1.1.1.cmml" xref="A6.SS1.SSS0.Px3.p1.1.m1.1.1">subscript</csymbol><ci id="A6.SS1.SSS0.Px3.p1.1.m1.1.1.2.cmml" xref="A6.SS1.SSS0.Px3.p1.1.m1.1.1.2">𝛼</ci><apply id="A6.SS1.SSS0.Px3.p1.1.m1.1.1.3.cmml" xref="A6.SS1.SSS0.Px3.p1.1.m1.1.1.3"><minus id="A6.SS1.SSS0.Px3.p1.1.m1.1.1.3.1.cmml" xref="A6.SS1.SSS0.Px3.p1.1.m1.1.1.3.1"></minus><ci id="A6.SS1.SSS0.Px3.p1.1.m1.1.1.3.2.cmml" xref="A6.SS1.SSS0.Px3.p1.1.m1.1.1.3.2">𝑚</ci><cn id="A6.SS1.SSS0.Px3.p1.1.m1.1.1.3.3.cmml" type="integer" xref="A6.SS1.SSS0.Px3.p1.1.m1.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A6.SS1.SSS0.Px3.p1.1.m1.1c">\alpha_{m-1}</annotation><annotation encoding="application/x-llamapun" id="A6.SS1.SSS0.Px3.p1.1.m1.1d">italic_α start_POSTSUBSCRIPT italic_m - 1 end_POSTSUBSCRIPT</annotation></semantics></math> on the restoration provided by the SNORE algorithm. One can see that if <math alttext="\alpha_{m-1}" class="ltx_Math" display="inline" id="A6.SS1.SSS0.Px3.p1.2.m2.1"><semantics id="A6.SS1.SSS0.Px3.p1.2.m2.1a"><msub id="A6.SS1.SSS0.Px3.p1.2.m2.1.1" xref="A6.SS1.SSS0.Px3.p1.2.m2.1.1.cmml"><mi id="A6.SS1.SSS0.Px3.p1.2.m2.1.1.2" xref="A6.SS1.SSS0.Px3.p1.2.m2.1.1.2.cmml">α</mi><mrow id="A6.SS1.SSS0.Px3.p1.2.m2.1.1.3" xref="A6.SS1.SSS0.Px3.p1.2.m2.1.1.3.cmml"><mi id="A6.SS1.SSS0.Px3.p1.2.m2.1.1.3.2" xref="A6.SS1.SSS0.Px3.p1.2.m2.1.1.3.2.cmml">m</mi><mo id="A6.SS1.SSS0.Px3.p1.2.m2.1.1.3.1" xref="A6.SS1.SSS0.Px3.p1.2.m2.1.1.3.1.cmml">−</mo><mn id="A6.SS1.SSS0.Px3.p1.2.m2.1.1.3.3" xref="A6.SS1.SSS0.Px3.p1.2.m2.1.1.3.3.cmml">1</mn></mrow></msub><annotation-xml encoding="MathML-Content" id="A6.SS1.SSS0.Px3.p1.2.m2.1b"><apply id="A6.SS1.SSS0.Px3.p1.2.m2.1.1.cmml" xref="A6.SS1.SSS0.Px3.p1.2.m2.1.1"><csymbol cd="ambiguous" id="A6.SS1.SSS0.Px3.p1.2.m2.1.1.1.cmml" xref="A6.SS1.SSS0.Px3.p1.2.m2.1.1">subscript</csymbol><ci id="A6.SS1.SSS0.Px3.p1.2.m2.1.1.2.cmml" xref="A6.SS1.SSS0.Px3.p1.2.m2.1.1.2">𝛼</ci><apply id="A6.SS1.SSS0.Px3.p1.2.m2.1.1.3.cmml" xref="A6.SS1.SSS0.Px3.p1.2.m2.1.1.3"><minus id="A6.SS1.SSS0.Px3.p1.2.m2.1.1.3.1.cmml" xref="A6.SS1.SSS0.Px3.p1.2.m2.1.1.3.1"></minus><ci id="A6.SS1.SSS0.Px3.p1.2.m2.1.1.3.2.cmml" xref="A6.SS1.SSS0.Px3.p1.2.m2.1.1.3.2">𝑚</ci><cn id="A6.SS1.SSS0.Px3.p1.2.m2.1.1.3.3.cmml" type="integer" xref="A6.SS1.SSS0.Px3.p1.2.m2.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A6.SS1.SSS0.Px3.p1.2.m2.1c">\alpha_{m-1}</annotation><annotation encoding="application/x-llamapun" id="A6.SS1.SSS0.Px3.p1.2.m2.1d">italic_α start_POSTSUBSCRIPT italic_m - 1 end_POSTSUBSCRIPT</annotation></semantics></math> is too small, the problem is less regularized so there is a residual noise. On the other hand, if <math alttext="\alpha_{m-1}" class="ltx_Math" display="inline" id="A6.SS1.SSS0.Px3.p1.3.m3.1"><semantics id="A6.SS1.SSS0.Px3.p1.3.m3.1a"><msub id="A6.SS1.SSS0.Px3.p1.3.m3.1.1" xref="A6.SS1.SSS0.Px3.p1.3.m3.1.1.cmml"><mi id="A6.SS1.SSS0.Px3.p1.3.m3.1.1.2" xref="A6.SS1.SSS0.Px3.p1.3.m3.1.1.2.cmml">α</mi><mrow id="A6.SS1.SSS0.Px3.p1.3.m3.1.1.3" xref="A6.SS1.SSS0.Px3.p1.3.m3.1.1.3.cmml"><mi id="A6.SS1.SSS0.Px3.p1.3.m3.1.1.3.2" xref="A6.SS1.SSS0.Px3.p1.3.m3.1.1.3.2.cmml">m</mi><mo id="A6.SS1.SSS0.Px3.p1.3.m3.1.1.3.1" xref="A6.SS1.SSS0.Px3.p1.3.m3.1.1.3.1.cmml">−</mo><mn id="A6.SS1.SSS0.Px3.p1.3.m3.1.1.3.3" xref="A6.SS1.SSS0.Px3.p1.3.m3.1.1.3.3.cmml">1</mn></mrow></msub><annotation-xml encoding="MathML-Content" id="A6.SS1.SSS0.Px3.p1.3.m3.1b"><apply id="A6.SS1.SSS0.Px3.p1.3.m3.1.1.cmml" xref="A6.SS1.SSS0.Px3.p1.3.m3.1.1"><csymbol cd="ambiguous" id="A6.SS1.SSS0.Px3.p1.3.m3.1.1.1.cmml" xref="A6.SS1.SSS0.Px3.p1.3.m3.1.1">subscript</csymbol><ci id="A6.SS1.SSS0.Px3.p1.3.m3.1.1.2.cmml" xref="A6.SS1.SSS0.Px3.p1.3.m3.1.1.2">𝛼</ci><apply id="A6.SS1.SSS0.Px3.p1.3.m3.1.1.3.cmml" xref="A6.SS1.SSS0.Px3.p1.3.m3.1.1.3"><minus id="A6.SS1.SSS0.Px3.p1.3.m3.1.1.3.1.cmml" xref="A6.SS1.SSS0.Px3.p1.3.m3.1.1.3.1"></minus><ci id="A6.SS1.SSS0.Px3.p1.3.m3.1.1.3.2.cmml" xref="A6.SS1.SSS0.Px3.p1.3.m3.1.1.3.2">𝑚</ci><cn id="A6.SS1.SSS0.Px3.p1.3.m3.1.1.3.3.cmml" type="integer" xref="A6.SS1.SSS0.Px3.p1.3.m3.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A6.SS1.SSS0.Px3.p1.3.m3.1c">\alpha_{m-1}</annotation><annotation encoding="application/x-llamapun" id="A6.SS1.SSS0.Px3.p1.3.m3.1d">italic_α start_POSTSUBSCRIPT italic_m - 1 end_POSTSUBSCRIPT</annotation></semantics></math> is too large, the restored image is very flat.</p> </div> <figure class="ltx_figure" id="A6.F7"><img alt="Refer to caption" class="ltx_graphics ltx_centering ltx_img_landscape" height="167" id="A6.F7.g1" src="extracted/5518834/images/All_results_lamb_end_Average_PnP.png" width="598"/> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure"><span class="ltx_text" id="A6.F7.6.3.1" style="font-size:90%;">Figure 7</span>: </span><span class="ltx_text" id="A6.F7.4.2" style="font-size:90%;">Influence of the parameter <math alttext="\alpha_{m-1}" class="ltx_Math" display="inline" id="A6.F7.3.1.m1.1"><semantics id="A6.F7.3.1.m1.1b"><msub id="A6.F7.3.1.m1.1.1" xref="A6.F7.3.1.m1.1.1.cmml"><mi id="A6.F7.3.1.m1.1.1.2" xref="A6.F7.3.1.m1.1.1.2.cmml">α</mi><mrow id="A6.F7.3.1.m1.1.1.3" xref="A6.F7.3.1.m1.1.1.3.cmml"><mi id="A6.F7.3.1.m1.1.1.3.2" xref="A6.F7.3.1.m1.1.1.3.2.cmml">m</mi><mo id="A6.F7.3.1.m1.1.1.3.1" xref="A6.F7.3.1.m1.1.1.3.1.cmml">−</mo><mn id="A6.F7.3.1.m1.1.1.3.3" xref="A6.F7.3.1.m1.1.1.3.3.cmml">1</mn></mrow></msub><annotation-xml encoding="MathML-Content" id="A6.F7.3.1.m1.1c"><apply id="A6.F7.3.1.m1.1.1.cmml" xref="A6.F7.3.1.m1.1.1"><csymbol cd="ambiguous" id="A6.F7.3.1.m1.1.1.1.cmml" xref="A6.F7.3.1.m1.1.1">subscript</csymbol><ci id="A6.F7.3.1.m1.1.1.2.cmml" xref="A6.F7.3.1.m1.1.1.2">𝛼</ci><apply id="A6.F7.3.1.m1.1.1.3.cmml" xref="A6.F7.3.1.m1.1.1.3"><minus id="A6.F7.3.1.m1.1.1.3.1.cmml" xref="A6.F7.3.1.m1.1.1.3.1"></minus><ci id="A6.F7.3.1.m1.1.1.3.2.cmml" xref="A6.F7.3.1.m1.1.1.3.2">𝑚</ci><cn id="A6.F7.3.1.m1.1.1.3.3.cmml" type="integer" xref="A6.F7.3.1.m1.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A6.F7.3.1.m1.1d">\alpha_{m-1}</annotation><annotation encoding="application/x-llamapun" id="A6.F7.3.1.m1.1e">italic_α start_POSTSUBSCRIPT italic_m - 1 end_POSTSUBSCRIPT</annotation></semantics></math> in SNORE algorithm on deblurring with an input noise of level <math alttext="\sigma_{{\mathbf{y}}}=10/255" class="ltx_Math" display="inline" id="A6.F7.4.2.m2.1"><semantics id="A6.F7.4.2.m2.1b"><mrow id="A6.F7.4.2.m2.1.1" xref="A6.F7.4.2.m2.1.1.cmml"><msub id="A6.F7.4.2.m2.1.1.2" xref="A6.F7.4.2.m2.1.1.2.cmml"><mi id="A6.F7.4.2.m2.1.1.2.2" xref="A6.F7.4.2.m2.1.1.2.2.cmml">σ</mi><mi id="A6.F7.4.2.m2.1.1.2.3" xref="A6.F7.4.2.m2.1.1.2.3.cmml">𝐲</mi></msub><mo id="A6.F7.4.2.m2.1.1.1" xref="A6.F7.4.2.m2.1.1.1.cmml">=</mo><mrow id="A6.F7.4.2.m2.1.1.3" xref="A6.F7.4.2.m2.1.1.3.cmml"><mn id="A6.F7.4.2.m2.1.1.3.2" xref="A6.F7.4.2.m2.1.1.3.2.cmml">10</mn><mo id="A6.F7.4.2.m2.1.1.3.1" xref="A6.F7.4.2.m2.1.1.3.1.cmml">/</mo><mn id="A6.F7.4.2.m2.1.1.3.3" xref="A6.F7.4.2.m2.1.1.3.3.cmml">255</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="A6.F7.4.2.m2.1c"><apply id="A6.F7.4.2.m2.1.1.cmml" xref="A6.F7.4.2.m2.1.1"><eq id="A6.F7.4.2.m2.1.1.1.cmml" xref="A6.F7.4.2.m2.1.1.1"></eq><apply id="A6.F7.4.2.m2.1.1.2.cmml" xref="A6.F7.4.2.m2.1.1.2"><csymbol cd="ambiguous" id="A6.F7.4.2.m2.1.1.2.1.cmml" xref="A6.F7.4.2.m2.1.1.2">subscript</csymbol><ci id="A6.F7.4.2.m2.1.1.2.2.cmml" xref="A6.F7.4.2.m2.1.1.2.2">𝜎</ci><ci id="A6.F7.4.2.m2.1.1.2.3.cmml" xref="A6.F7.4.2.m2.1.1.2.3">𝐲</ci></apply><apply id="A6.F7.4.2.m2.1.1.3.cmml" xref="A6.F7.4.2.m2.1.1.3"><divide id="A6.F7.4.2.m2.1.1.3.1.cmml" xref="A6.F7.4.2.m2.1.1.3.1"></divide><cn id="A6.F7.4.2.m2.1.1.3.2.cmml" type="integer" xref="A6.F7.4.2.m2.1.1.3.2">10</cn><cn id="A6.F7.4.2.m2.1.1.3.3.cmml" type="integer" xref="A6.F7.4.2.m2.1.1.3.3">255</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A6.F7.4.2.m2.1d">\sigma_{{\mathbf{y}}}=10/255</annotation><annotation encoding="application/x-llamapun" id="A6.F7.4.2.m2.1e">italic_σ start_POSTSUBSCRIPT bold_y end_POSTSUBSCRIPT = 10 / 255</annotation></semantics></math>.</span></figcaption> </figure> </section> <section class="ltx_paragraph" id="A6.SS1.SSS0.Px4"> <h5 class="ltx_title ltx_title_paragraph">Influence of <math alttext="\sigma_{m-1}" class="ltx_Math" display="inline" id="A6.SS1.SSS0.Px4.1.m1.1"><semantics id="A6.SS1.SSS0.Px4.1.m1.1b"><msub id="A6.SS1.SSS0.Px4.1.m1.1.1" xref="A6.SS1.SSS0.Px4.1.m1.1.1.cmml"><mi id="A6.SS1.SSS0.Px4.1.m1.1.1.2" xref="A6.SS1.SSS0.Px4.1.m1.1.1.2.cmml">σ</mi><mrow id="A6.SS1.SSS0.Px4.1.m1.1.1.3" xref="A6.SS1.SSS0.Px4.1.m1.1.1.3.cmml"><mi id="A6.SS1.SSS0.Px4.1.m1.1.1.3.2" xref="A6.SS1.SSS0.Px4.1.m1.1.1.3.2.cmml">m</mi><mo id="A6.SS1.SSS0.Px4.1.m1.1.1.3.1" xref="A6.SS1.SSS0.Px4.1.m1.1.1.3.1.cmml">−</mo><mn id="A6.SS1.SSS0.Px4.1.m1.1.1.3.3" xref="A6.SS1.SSS0.Px4.1.m1.1.1.3.3.cmml">1</mn></mrow></msub><annotation-xml encoding="MathML-Content" id="A6.SS1.SSS0.Px4.1.m1.1c"><apply id="A6.SS1.SSS0.Px4.1.m1.1.1.cmml" xref="A6.SS1.SSS0.Px4.1.m1.1.1"><csymbol cd="ambiguous" id="A6.SS1.SSS0.Px4.1.m1.1.1.1.cmml" xref="A6.SS1.SSS0.Px4.1.m1.1.1">subscript</csymbol><ci id="A6.SS1.SSS0.Px4.1.m1.1.1.2.cmml" xref="A6.SS1.SSS0.Px4.1.m1.1.1.2">𝜎</ci><apply id="A6.SS1.SSS0.Px4.1.m1.1.1.3.cmml" xref="A6.SS1.SSS0.Px4.1.m1.1.1.3"><minus id="A6.SS1.SSS0.Px4.1.m1.1.1.3.1.cmml" xref="A6.SS1.SSS0.Px4.1.m1.1.1.3.1"></minus><ci id="A6.SS1.SSS0.Px4.1.m1.1.1.3.2.cmml" xref="A6.SS1.SSS0.Px4.1.m1.1.1.3.2">𝑚</ci><cn id="A6.SS1.SSS0.Px4.1.m1.1.1.3.3.cmml" type="integer" xref="A6.SS1.SSS0.Px4.1.m1.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A6.SS1.SSS0.Px4.1.m1.1d">\sigma_{m-1}</annotation><annotation encoding="application/x-llamapun" id="A6.SS1.SSS0.Px4.1.m1.1e">italic_σ start_POSTSUBSCRIPT italic_m - 1 end_POSTSUBSCRIPT</annotation></semantics></math> </h5> <div class="ltx_para" id="A6.SS1.SSS0.Px4.p1"> <p class="ltx_p" id="A6.SS1.SSS0.Px4.p1.3">On Figure <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#A6.F8" title="Figure 8 ‣ Influence of 𝜎_{𝑚-1} ‣ F.1 Deblurring ‣ Appendix F Additional Experiments ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_tag">8</span></a>, we illustrate the influence of the final denoiser parameter <math alttext="\sigma_{m-1}" class="ltx_Math" display="inline" id="A6.SS1.SSS0.Px4.p1.1.m1.1"><semantics id="A6.SS1.SSS0.Px4.p1.1.m1.1a"><msub id="A6.SS1.SSS0.Px4.p1.1.m1.1.1" xref="A6.SS1.SSS0.Px4.p1.1.m1.1.1.cmml"><mi id="A6.SS1.SSS0.Px4.p1.1.m1.1.1.2" xref="A6.SS1.SSS0.Px4.p1.1.m1.1.1.2.cmml">σ</mi><mrow id="A6.SS1.SSS0.Px4.p1.1.m1.1.1.3" xref="A6.SS1.SSS0.Px4.p1.1.m1.1.1.3.cmml"><mi id="A6.SS1.SSS0.Px4.p1.1.m1.1.1.3.2" xref="A6.SS1.SSS0.Px4.p1.1.m1.1.1.3.2.cmml">m</mi><mo id="A6.SS1.SSS0.Px4.p1.1.m1.1.1.3.1" xref="A6.SS1.SSS0.Px4.p1.1.m1.1.1.3.1.cmml">−</mo><mn id="A6.SS1.SSS0.Px4.p1.1.m1.1.1.3.3" xref="A6.SS1.SSS0.Px4.p1.1.m1.1.1.3.3.cmml">1</mn></mrow></msub><annotation-xml encoding="MathML-Content" id="A6.SS1.SSS0.Px4.p1.1.m1.1b"><apply id="A6.SS1.SSS0.Px4.p1.1.m1.1.1.cmml" xref="A6.SS1.SSS0.Px4.p1.1.m1.1.1"><csymbol cd="ambiguous" id="A6.SS1.SSS0.Px4.p1.1.m1.1.1.1.cmml" xref="A6.SS1.SSS0.Px4.p1.1.m1.1.1">subscript</csymbol><ci id="A6.SS1.SSS0.Px4.p1.1.m1.1.1.2.cmml" xref="A6.SS1.SSS0.Px4.p1.1.m1.1.1.2">𝜎</ci><apply id="A6.SS1.SSS0.Px4.p1.1.m1.1.1.3.cmml" xref="A6.SS1.SSS0.Px4.p1.1.m1.1.1.3"><minus id="A6.SS1.SSS0.Px4.p1.1.m1.1.1.3.1.cmml" xref="A6.SS1.SSS0.Px4.p1.1.m1.1.1.3.1"></minus><ci id="A6.SS1.SSS0.Px4.p1.1.m1.1.1.3.2.cmml" xref="A6.SS1.SSS0.Px4.p1.1.m1.1.1.3.2">𝑚</ci><cn id="A6.SS1.SSS0.Px4.p1.1.m1.1.1.3.3.cmml" type="integer" xref="A6.SS1.SSS0.Px4.p1.1.m1.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A6.SS1.SSS0.Px4.p1.1.m1.1c">\sigma_{m-1}</annotation><annotation encoding="application/x-llamapun" id="A6.SS1.SSS0.Px4.p1.1.m1.1d">italic_σ start_POSTSUBSCRIPT italic_m - 1 end_POSTSUBSCRIPT</annotation></semantics></math> on the restoration obtained with the SNORE algorithm. One can see that if <math alttext="\sigma_{m-1}" class="ltx_Math" display="inline" id="A6.SS1.SSS0.Px4.p1.2.m2.1"><semantics id="A6.SS1.SSS0.Px4.p1.2.m2.1a"><msub id="A6.SS1.SSS0.Px4.p1.2.m2.1.1" xref="A6.SS1.SSS0.Px4.p1.2.m2.1.1.cmml"><mi id="A6.SS1.SSS0.Px4.p1.2.m2.1.1.2" xref="A6.SS1.SSS0.Px4.p1.2.m2.1.1.2.cmml">σ</mi><mrow id="A6.SS1.SSS0.Px4.p1.2.m2.1.1.3" xref="A6.SS1.SSS0.Px4.p1.2.m2.1.1.3.cmml"><mi id="A6.SS1.SSS0.Px4.p1.2.m2.1.1.3.2" xref="A6.SS1.SSS0.Px4.p1.2.m2.1.1.3.2.cmml">m</mi><mo id="A6.SS1.SSS0.Px4.p1.2.m2.1.1.3.1" xref="A6.SS1.SSS0.Px4.p1.2.m2.1.1.3.1.cmml">−</mo><mn id="A6.SS1.SSS0.Px4.p1.2.m2.1.1.3.3" xref="A6.SS1.SSS0.Px4.p1.2.m2.1.1.3.3.cmml">1</mn></mrow></msub><annotation-xml encoding="MathML-Content" id="A6.SS1.SSS0.Px4.p1.2.m2.1b"><apply id="A6.SS1.SSS0.Px4.p1.2.m2.1.1.cmml" xref="A6.SS1.SSS0.Px4.p1.2.m2.1.1"><csymbol cd="ambiguous" id="A6.SS1.SSS0.Px4.p1.2.m2.1.1.1.cmml" xref="A6.SS1.SSS0.Px4.p1.2.m2.1.1">subscript</csymbol><ci id="A6.SS1.SSS0.Px4.p1.2.m2.1.1.2.cmml" xref="A6.SS1.SSS0.Px4.p1.2.m2.1.1.2">𝜎</ci><apply id="A6.SS1.SSS0.Px4.p1.2.m2.1.1.3.cmml" xref="A6.SS1.SSS0.Px4.p1.2.m2.1.1.3"><minus id="A6.SS1.SSS0.Px4.p1.2.m2.1.1.3.1.cmml" xref="A6.SS1.SSS0.Px4.p1.2.m2.1.1.3.1"></minus><ci id="A6.SS1.SSS0.Px4.p1.2.m2.1.1.3.2.cmml" xref="A6.SS1.SSS0.Px4.p1.2.m2.1.1.3.2">𝑚</ci><cn id="A6.SS1.SSS0.Px4.p1.2.m2.1.1.3.3.cmml" type="integer" xref="A6.SS1.SSS0.Px4.p1.2.m2.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A6.SS1.SSS0.Px4.p1.2.m2.1c">\sigma_{m-1}</annotation><annotation encoding="application/x-llamapun" id="A6.SS1.SSS0.Px4.p1.2.m2.1d">italic_σ start_POSTSUBSCRIPT italic_m - 1 end_POSTSUBSCRIPT</annotation></semantics></math> is too small, the problem is less regularized and a residual noise is present. On the other side, if <math alttext="\sigma_{m-1}" class="ltx_Math" display="inline" id="A6.SS1.SSS0.Px4.p1.3.m3.1"><semantics id="A6.SS1.SSS0.Px4.p1.3.m3.1a"><msub id="A6.SS1.SSS0.Px4.p1.3.m3.1.1" xref="A6.SS1.SSS0.Px4.p1.3.m3.1.1.cmml"><mi id="A6.SS1.SSS0.Px4.p1.3.m3.1.1.2" xref="A6.SS1.SSS0.Px4.p1.3.m3.1.1.2.cmml">σ</mi><mrow id="A6.SS1.SSS0.Px4.p1.3.m3.1.1.3" xref="A6.SS1.SSS0.Px4.p1.3.m3.1.1.3.cmml"><mi id="A6.SS1.SSS0.Px4.p1.3.m3.1.1.3.2" xref="A6.SS1.SSS0.Px4.p1.3.m3.1.1.3.2.cmml">m</mi><mo id="A6.SS1.SSS0.Px4.p1.3.m3.1.1.3.1" xref="A6.SS1.SSS0.Px4.p1.3.m3.1.1.3.1.cmml">−</mo><mn id="A6.SS1.SSS0.Px4.p1.3.m3.1.1.3.3" xref="A6.SS1.SSS0.Px4.p1.3.m3.1.1.3.3.cmml">1</mn></mrow></msub><annotation-xml encoding="MathML-Content" id="A6.SS1.SSS0.Px4.p1.3.m3.1b"><apply id="A6.SS1.SSS0.Px4.p1.3.m3.1.1.cmml" xref="A6.SS1.SSS0.Px4.p1.3.m3.1.1"><csymbol cd="ambiguous" id="A6.SS1.SSS0.Px4.p1.3.m3.1.1.1.cmml" xref="A6.SS1.SSS0.Px4.p1.3.m3.1.1">subscript</csymbol><ci id="A6.SS1.SSS0.Px4.p1.3.m3.1.1.2.cmml" xref="A6.SS1.SSS0.Px4.p1.3.m3.1.1.2">𝜎</ci><apply id="A6.SS1.SSS0.Px4.p1.3.m3.1.1.3.cmml" xref="A6.SS1.SSS0.Px4.p1.3.m3.1.1.3"><minus id="A6.SS1.SSS0.Px4.p1.3.m3.1.1.3.1.cmml" xref="A6.SS1.SSS0.Px4.p1.3.m3.1.1.3.1"></minus><ci id="A6.SS1.SSS0.Px4.p1.3.m3.1.1.3.2.cmml" xref="A6.SS1.SSS0.Px4.p1.3.m3.1.1.3.2">𝑚</ci><cn id="A6.SS1.SSS0.Px4.p1.3.m3.1.1.3.3.cmml" type="integer" xref="A6.SS1.SSS0.Px4.p1.3.m3.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A6.SS1.SSS0.Px4.p1.3.m3.1c">\sigma_{m-1}</annotation><annotation encoding="application/x-llamapun" id="A6.SS1.SSS0.Px4.p1.3.m3.1d">italic_σ start_POSTSUBSCRIPT italic_m - 1 end_POSTSUBSCRIPT</annotation></semantics></math> is too large, the restored image is too flat.</p> </div> <div class="ltx_para" id="A6.SS1.SSS0.Px4.p2"> <p class="ltx_p" id="A6.SS1.SSS0.Px4.p2.2">The influence of parameters <math alttext="\sigma_{m-1}" class="ltx_Math" display="inline" id="A6.SS1.SSS0.Px4.p2.1.m1.1"><semantics id="A6.SS1.SSS0.Px4.p2.1.m1.1a"><msub id="A6.SS1.SSS0.Px4.p2.1.m1.1.1" xref="A6.SS1.SSS0.Px4.p2.1.m1.1.1.cmml"><mi id="A6.SS1.SSS0.Px4.p2.1.m1.1.1.2" xref="A6.SS1.SSS0.Px4.p2.1.m1.1.1.2.cmml">σ</mi><mrow id="A6.SS1.SSS0.Px4.p2.1.m1.1.1.3" xref="A6.SS1.SSS0.Px4.p2.1.m1.1.1.3.cmml"><mi id="A6.SS1.SSS0.Px4.p2.1.m1.1.1.3.2" xref="A6.SS1.SSS0.Px4.p2.1.m1.1.1.3.2.cmml">m</mi><mo id="A6.SS1.SSS0.Px4.p2.1.m1.1.1.3.1" xref="A6.SS1.SSS0.Px4.p2.1.m1.1.1.3.1.cmml">−</mo><mn id="A6.SS1.SSS0.Px4.p2.1.m1.1.1.3.3" xref="A6.SS1.SSS0.Px4.p2.1.m1.1.1.3.3.cmml">1</mn></mrow></msub><annotation-xml encoding="MathML-Content" id="A6.SS1.SSS0.Px4.p2.1.m1.1b"><apply id="A6.SS1.SSS0.Px4.p2.1.m1.1.1.cmml" xref="A6.SS1.SSS0.Px4.p2.1.m1.1.1"><csymbol cd="ambiguous" id="A6.SS1.SSS0.Px4.p2.1.m1.1.1.1.cmml" xref="A6.SS1.SSS0.Px4.p2.1.m1.1.1">subscript</csymbol><ci id="A6.SS1.SSS0.Px4.p2.1.m1.1.1.2.cmml" xref="A6.SS1.SSS0.Px4.p2.1.m1.1.1.2">𝜎</ci><apply id="A6.SS1.SSS0.Px4.p2.1.m1.1.1.3.cmml" xref="A6.SS1.SSS0.Px4.p2.1.m1.1.1.3"><minus id="A6.SS1.SSS0.Px4.p2.1.m1.1.1.3.1.cmml" xref="A6.SS1.SSS0.Px4.p2.1.m1.1.1.3.1"></minus><ci id="A6.SS1.SSS0.Px4.p2.1.m1.1.1.3.2.cmml" xref="A6.SS1.SSS0.Px4.p2.1.m1.1.1.3.2">𝑚</ci><cn id="A6.SS1.SSS0.Px4.p2.1.m1.1.1.3.3.cmml" type="integer" xref="A6.SS1.SSS0.Px4.p2.1.m1.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A6.SS1.SSS0.Px4.p2.1.m1.1c">\sigma_{m-1}</annotation><annotation encoding="application/x-llamapun" id="A6.SS1.SSS0.Px4.p2.1.m1.1d">italic_σ start_POSTSUBSCRIPT italic_m - 1 end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="\alpha_{m-1}" class="ltx_Math" display="inline" id="A6.SS1.SSS0.Px4.p2.2.m2.1"><semantics id="A6.SS1.SSS0.Px4.p2.2.m2.1a"><msub id="A6.SS1.SSS0.Px4.p2.2.m2.1.1" xref="A6.SS1.SSS0.Px4.p2.2.m2.1.1.cmml"><mi id="A6.SS1.SSS0.Px4.p2.2.m2.1.1.2" xref="A6.SS1.SSS0.Px4.p2.2.m2.1.1.2.cmml">α</mi><mrow id="A6.SS1.SSS0.Px4.p2.2.m2.1.1.3" xref="A6.SS1.SSS0.Px4.p2.2.m2.1.1.3.cmml"><mi id="A6.SS1.SSS0.Px4.p2.2.m2.1.1.3.2" xref="A6.SS1.SSS0.Px4.p2.2.m2.1.1.3.2.cmml">m</mi><mo id="A6.SS1.SSS0.Px4.p2.2.m2.1.1.3.1" xref="A6.SS1.SSS0.Px4.p2.2.m2.1.1.3.1.cmml">−</mo><mn id="A6.SS1.SSS0.Px4.p2.2.m2.1.1.3.3" xref="A6.SS1.SSS0.Px4.p2.2.m2.1.1.3.3.cmml">1</mn></mrow></msub><annotation-xml encoding="MathML-Content" id="A6.SS1.SSS0.Px4.p2.2.m2.1b"><apply id="A6.SS1.SSS0.Px4.p2.2.m2.1.1.cmml" xref="A6.SS1.SSS0.Px4.p2.2.m2.1.1"><csymbol cd="ambiguous" id="A6.SS1.SSS0.Px4.p2.2.m2.1.1.1.cmml" xref="A6.SS1.SSS0.Px4.p2.2.m2.1.1">subscript</csymbol><ci id="A6.SS1.SSS0.Px4.p2.2.m2.1.1.2.cmml" xref="A6.SS1.SSS0.Px4.p2.2.m2.1.1.2">𝛼</ci><apply id="A6.SS1.SSS0.Px4.p2.2.m2.1.1.3.cmml" xref="A6.SS1.SSS0.Px4.p2.2.m2.1.1.3"><minus id="A6.SS1.SSS0.Px4.p2.2.m2.1.1.3.1.cmml" xref="A6.SS1.SSS0.Px4.p2.2.m2.1.1.3.1"></minus><ci id="A6.SS1.SSS0.Px4.p2.2.m2.1.1.3.2.cmml" xref="A6.SS1.SSS0.Px4.p2.2.m2.1.1.3.2">𝑚</ci><cn id="A6.SS1.SSS0.Px4.p2.2.m2.1.1.3.3.cmml" type="integer" xref="A6.SS1.SSS0.Px4.p2.2.m2.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A6.SS1.SSS0.Px4.p2.2.m2.1c">\alpha_{m-1}</annotation><annotation encoding="application/x-llamapun" id="A6.SS1.SSS0.Px4.p2.2.m2.1d">italic_α start_POSTSUBSCRIPT italic_m - 1 end_POSTSUBSCRIPT</annotation></semantics></math> is therefore similar. However, we observed experimentally that having these two free parameters allows to obtain better restoration results.</p> </div> <figure class="ltx_figure" id="A6.F8"><img alt="Refer to caption" class="ltx_graphics ltx_centering ltx_img_landscape" height="168" id="A6.F8.g1" src="extracted/5518834/images/All_results_std_end_Average_PnP.png" width="598"/> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure"><span class="ltx_text" id="A6.F8.6.3.1" style="font-size:90%;">Figure 8</span>: </span><span class="ltx_text" id="A6.F8.4.2" style="font-size:90%;">Influence of the parameter <math alttext="\sigma_{m-1}" class="ltx_Math" display="inline" id="A6.F8.3.1.m1.1"><semantics id="A6.F8.3.1.m1.1b"><msub id="A6.F8.3.1.m1.1.1" xref="A6.F8.3.1.m1.1.1.cmml"><mi id="A6.F8.3.1.m1.1.1.2" xref="A6.F8.3.1.m1.1.1.2.cmml">σ</mi><mrow id="A6.F8.3.1.m1.1.1.3" xref="A6.F8.3.1.m1.1.1.3.cmml"><mi id="A6.F8.3.1.m1.1.1.3.2" xref="A6.F8.3.1.m1.1.1.3.2.cmml">m</mi><mo id="A6.F8.3.1.m1.1.1.3.1" xref="A6.F8.3.1.m1.1.1.3.1.cmml">−</mo><mn id="A6.F8.3.1.m1.1.1.3.3" xref="A6.F8.3.1.m1.1.1.3.3.cmml">1</mn></mrow></msub><annotation-xml encoding="MathML-Content" id="A6.F8.3.1.m1.1c"><apply id="A6.F8.3.1.m1.1.1.cmml" xref="A6.F8.3.1.m1.1.1"><csymbol cd="ambiguous" id="A6.F8.3.1.m1.1.1.1.cmml" xref="A6.F8.3.1.m1.1.1">subscript</csymbol><ci id="A6.F8.3.1.m1.1.1.2.cmml" xref="A6.F8.3.1.m1.1.1.2">𝜎</ci><apply id="A6.F8.3.1.m1.1.1.3.cmml" xref="A6.F8.3.1.m1.1.1.3"><minus id="A6.F8.3.1.m1.1.1.3.1.cmml" xref="A6.F8.3.1.m1.1.1.3.1"></minus><ci id="A6.F8.3.1.m1.1.1.3.2.cmml" xref="A6.F8.3.1.m1.1.1.3.2">𝑚</ci><cn id="A6.F8.3.1.m1.1.1.3.3.cmml" type="integer" xref="A6.F8.3.1.m1.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A6.F8.3.1.m1.1d">\sigma_{m-1}</annotation><annotation encoding="application/x-llamapun" id="A6.F8.3.1.m1.1e">italic_σ start_POSTSUBSCRIPT italic_m - 1 end_POSTSUBSCRIPT</annotation></semantics></math> in SNORE algorithm on deblurring with an input noise of level <math alttext="\sigma_{{\mathbf{y}}}=10/255" class="ltx_Math" display="inline" id="A6.F8.4.2.m2.1"><semantics id="A6.F8.4.2.m2.1b"><mrow id="A6.F8.4.2.m2.1.1" xref="A6.F8.4.2.m2.1.1.cmml"><msub id="A6.F8.4.2.m2.1.1.2" xref="A6.F8.4.2.m2.1.1.2.cmml"><mi id="A6.F8.4.2.m2.1.1.2.2" xref="A6.F8.4.2.m2.1.1.2.2.cmml">σ</mi><mi id="A6.F8.4.2.m2.1.1.2.3" xref="A6.F8.4.2.m2.1.1.2.3.cmml">𝐲</mi></msub><mo id="A6.F8.4.2.m2.1.1.1" xref="A6.F8.4.2.m2.1.1.1.cmml">=</mo><mrow id="A6.F8.4.2.m2.1.1.3" xref="A6.F8.4.2.m2.1.1.3.cmml"><mn id="A6.F8.4.2.m2.1.1.3.2" xref="A6.F8.4.2.m2.1.1.3.2.cmml">10</mn><mo id="A6.F8.4.2.m2.1.1.3.1" xref="A6.F8.4.2.m2.1.1.3.1.cmml">/</mo><mn id="A6.F8.4.2.m2.1.1.3.3" xref="A6.F8.4.2.m2.1.1.3.3.cmml">255</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="A6.F8.4.2.m2.1c"><apply id="A6.F8.4.2.m2.1.1.cmml" xref="A6.F8.4.2.m2.1.1"><eq id="A6.F8.4.2.m2.1.1.1.cmml" xref="A6.F8.4.2.m2.1.1.1"></eq><apply id="A6.F8.4.2.m2.1.1.2.cmml" xref="A6.F8.4.2.m2.1.1.2"><csymbol cd="ambiguous" id="A6.F8.4.2.m2.1.1.2.1.cmml" xref="A6.F8.4.2.m2.1.1.2">subscript</csymbol><ci id="A6.F8.4.2.m2.1.1.2.2.cmml" xref="A6.F8.4.2.m2.1.1.2.2">𝜎</ci><ci id="A6.F8.4.2.m2.1.1.2.3.cmml" xref="A6.F8.4.2.m2.1.1.2.3">𝐲</ci></apply><apply id="A6.F8.4.2.m2.1.1.3.cmml" xref="A6.F8.4.2.m2.1.1.3"><divide id="A6.F8.4.2.m2.1.1.3.1.cmml" xref="A6.F8.4.2.m2.1.1.3.1"></divide><cn id="A6.F8.4.2.m2.1.1.3.2.cmml" type="integer" xref="A6.F8.4.2.m2.1.1.3.2">10</cn><cn id="A6.F8.4.2.m2.1.1.3.3.cmml" type="integer" xref="A6.F8.4.2.m2.1.1.3.3">255</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A6.F8.4.2.m2.1d">\sigma_{{\mathbf{y}}}=10/255</annotation><annotation encoding="application/x-llamapun" id="A6.F8.4.2.m2.1e">italic_σ start_POSTSUBSCRIPT bold_y end_POSTSUBSCRIPT = 10 / 255</annotation></semantics></math>.</span></figcaption> </figure> </section> <section class="ltx_paragraph" id="A6.SS1.SSS0.Px5"> <h5 class="ltx_title ltx_title_paragraph">Influence of <math alttext="m" class="ltx_Math" display="inline" id="A6.SS1.SSS0.Px5.1.m1.1"><semantics id="A6.SS1.SSS0.Px5.1.m1.1b"><mi id="A6.SS1.SSS0.Px5.1.m1.1.1" xref="A6.SS1.SSS0.Px5.1.m1.1.1.cmml">m</mi><annotation-xml encoding="MathML-Content" id="A6.SS1.SSS0.Px5.1.m1.1c"><ci id="A6.SS1.SSS0.Px5.1.m1.1.1.cmml" xref="A6.SS1.SSS0.Px5.1.m1.1.1">𝑚</ci></annotation-xml><annotation encoding="application/x-tex" id="A6.SS1.SSS0.Px5.1.m1.1d">m</annotation><annotation encoding="application/x-llamapun" id="A6.SS1.SSS0.Px5.1.m1.1e">italic_m</annotation></semantics></math> </h5> <div class="ltx_para" id="A6.SS1.SSS0.Px5.p1"> <p class="ltx_p" id="A6.SS1.SSS0.Px5.p1.1">On Figure <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#A6.F9" title="Figure 9 ‣ On the step-size 𝛿 ‣ F.1 Deblurring ‣ Appendix F Additional Experiments ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_tag">9</span></a>, we observe the influence of the number of annealing levels <math alttext="m" class="ltx_Math" display="inline" id="A6.SS1.SSS0.Px5.p1.1.m1.1"><semantics id="A6.SS1.SSS0.Px5.p1.1.m1.1a"><mi id="A6.SS1.SSS0.Px5.p1.1.m1.1.1" xref="A6.SS1.SSS0.Px5.p1.1.m1.1.1.cmml">m</mi><annotation-xml encoding="MathML-Content" id="A6.SS1.SSS0.Px5.p1.1.m1.1b"><ci id="A6.SS1.SSS0.Px5.p1.1.m1.1.1.cmml" xref="A6.SS1.SSS0.Px5.p1.1.m1.1.1">𝑚</ci></annotation-xml><annotation encoding="application/x-tex" id="A6.SS1.SSS0.Px5.p1.1.m1.1c">m</annotation><annotation encoding="application/x-llamapun" id="A6.SS1.SSS0.Px5.p1.1.m1.1d">italic_m</annotation></semantics></math> on the quality of the reconstruction. Metrics are not sensitive to this parameter but we observe on images that some artifacts are reduced with additional annealing levels. Our experiments with the SNORE algorithm suggest that with more annealing levels, the algorithm performs better to inverse the degradation and less artifacts are visible. Images of Figure <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#A6.F9" title="Figure 9 ‣ On the step-size 𝛿 ‣ F.1 Deblurring ‣ Appendix F Additional Experiments ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_tag">9</span></a> support this claim. Note that local artifacts do not seem to have a significant influence on the metric values.</p> </div> </section> <section class="ltx_paragraph" id="A6.SS1.SSS0.Px6"> <h5 class="ltx_title ltx_title_paragraph">On the step-size <math alttext="\delta" class="ltx_Math" display="inline" id="A6.SS1.SSS0.Px6.1.m1.1"><semantics id="A6.SS1.SSS0.Px6.1.m1.1b"><mi id="A6.SS1.SSS0.Px6.1.m1.1.1" xref="A6.SS1.SSS0.Px6.1.m1.1.1.cmml">δ</mi><annotation-xml encoding="MathML-Content" id="A6.SS1.SSS0.Px6.1.m1.1c"><ci id="A6.SS1.SSS0.Px6.1.m1.1.1.cmml" xref="A6.SS1.SSS0.Px6.1.m1.1.1">𝛿</ci></annotation-xml><annotation encoding="application/x-tex" id="A6.SS1.SSS0.Px6.1.m1.1d">\delta</annotation><annotation encoding="application/x-llamapun" id="A6.SS1.SSS0.Px6.1.m1.1e">italic_δ</annotation></semantics></math> </h5> <div class="ltx_para" id="A6.SS1.SSS0.Px6.p1"> <p class="ltx_p" id="A6.SS1.SSS0.Px6.p1.5"><cite class="ltx_cite ltx_citemacro_citet">Laumont et al. 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xref="A6.SS1.SSS0.Px6.p1.3.m3.1.1.3.cmml">0.8</mn></mrow><annotation-xml encoding="MathML-Content" id="A6.SS1.SSS0.Px6.p1.3.m3.1b"><apply id="A6.SS1.SSS0.Px6.p1.3.m3.1.1.cmml" xref="A6.SS1.SSS0.Px6.p1.3.m3.1.1"><eq id="A6.SS1.SSS0.Px6.p1.3.m3.1.1.1.cmml" xref="A6.SS1.SSS0.Px6.p1.3.m3.1.1.1"></eq><ci id="A6.SS1.SSS0.Px6.p1.3.m3.1.1.2.cmml" xref="A6.SS1.SSS0.Px6.p1.3.m3.1.1.2">𝛾</ci><cn id="A6.SS1.SSS0.Px6.p1.3.m3.1.1.3.cmml" type="float" xref="A6.SS1.SSS0.Px6.p1.3.m3.1.1.3">0.8</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A6.SS1.SSS0.Px6.p1.3.m3.1c">\gamma=0.8</annotation><annotation encoding="application/x-llamapun" id="A6.SS1.SSS0.Px6.p1.3.m3.1d">italic_γ = 0.8</annotation></semantics></math>). They ensure that Assumption <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#S3.Thmtheorem3" title="Assumption 3.3. ‣ 3.2 Unbiased algorithm analysis ‣ 3 Convergence Analysis ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_tag">3.3</span></a> is verified. However, they observe that the second phase has no impact on the output of the algorithm. We try the same framework (with various <math alttext="\gamma\in]\frac{1}{2},1]" class="ltx_math_unparsed" display="inline" id="A6.SS1.SSS0.Px6.p1.4.m4.1"><semantics id="A6.SS1.SSS0.Px6.p1.4.m4.1a"><mrow id="A6.SS1.SSS0.Px6.p1.4.m4.1b"><mrow id="A6.SS1.SSS0.Px6.p1.4.m4.1.1"><mi id="A6.SS1.SSS0.Px6.p1.4.m4.1.1.1">γ</mi><mo id="A6.SS1.SSS0.Px6.p1.4.m4.1.1.2" rspace="0em">∈</mo><mo id="A6.SS1.SSS0.Px6.p1.4.m4.1.1.3" stretchy="false">]</mo></mrow><mfrac id="A6.SS1.SSS0.Px6.p1.4.m4.1.2"><mn id="A6.SS1.SSS0.Px6.p1.4.m4.1.2.2">1</mn><mn id="A6.SS1.SSS0.Px6.p1.4.m4.1.2.3">2</mn></mfrac><mo id="A6.SS1.SSS0.Px6.p1.4.m4.1.3">,</mo><mn id="A6.SS1.SSS0.Px6.p1.4.m4.1.4">1</mn><mo id="A6.SS1.SSS0.Px6.p1.4.m4.1.5" stretchy="false">]</mo></mrow><annotation encoding="application/x-tex" id="A6.SS1.SSS0.Px6.p1.4.m4.1c">\gamma\in]\frac{1}{2},1]</annotation><annotation encoding="application/x-llamapun" id="A6.SS1.SSS0.Px6.p1.4.m4.1d">italic_γ ∈ ] divide start_ARG 1 end_ARG start_ARG 2 end_ARG , 1 ]</annotation></semantics></math>) and also observe that the second phase is useless. For efficiency, we choose to only compute the first phase with fixed <math alttext="\delta>0" class="ltx_Math" display="inline" id="A6.SS1.SSS0.Px6.p1.5.m5.1"><semantics id="A6.SS1.SSS0.Px6.p1.5.m5.1a"><mrow id="A6.SS1.SSS0.Px6.p1.5.m5.1.1" xref="A6.SS1.SSS0.Px6.p1.5.m5.1.1.cmml"><mi id="A6.SS1.SSS0.Px6.p1.5.m5.1.1.2" xref="A6.SS1.SSS0.Px6.p1.5.m5.1.1.2.cmml">δ</mi><mo id="A6.SS1.SSS0.Px6.p1.5.m5.1.1.1" xref="A6.SS1.SSS0.Px6.p1.5.m5.1.1.1.cmml">></mo><mn id="A6.SS1.SSS0.Px6.p1.5.m5.1.1.3" xref="A6.SS1.SSS0.Px6.p1.5.m5.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="A6.SS1.SSS0.Px6.p1.5.m5.1b"><apply id="A6.SS1.SSS0.Px6.p1.5.m5.1.1.cmml" xref="A6.SS1.SSS0.Px6.p1.5.m5.1.1"><gt id="A6.SS1.SSS0.Px6.p1.5.m5.1.1.1.cmml" xref="A6.SS1.SSS0.Px6.p1.5.m5.1.1.1"></gt><ci id="A6.SS1.SSS0.Px6.p1.5.m5.1.1.2.cmml" xref="A6.SS1.SSS0.Px6.p1.5.m5.1.1.2">𝛿</ci><cn id="A6.SS1.SSS0.Px6.p1.5.m5.1.1.3.cmml" type="integer" xref="A6.SS1.SSS0.Px6.p1.5.m5.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A6.SS1.SSS0.Px6.p1.5.m5.1c">\delta>0</annotation><annotation encoding="application/x-llamapun" id="A6.SS1.SSS0.Px6.p1.5.m5.1d">italic_δ > 0</annotation></semantics></math>.</p> </div> <figure class="ltx_figure" id="A6.F9"> <div class="ltx_flex_figure"> <div class="ltx_flex_cell ltx_flex_size_1"> <figure class="ltx_figure ltx_flex_size_1 ltx_align_center" id="A6.F8.sf1"><img alt="Refer to caption" class="ltx_graphics ltx_img_landscape" height="185" id="A6.F8.sf1.g1" src="extracted/5518834/images/Annealing_influence.png" width="598"/> <figcaption class="ltx_caption"><span class="ltx_tag ltx_tag_figure"><span class="ltx_text" id="A6.F8.sf1.2.1.1" style="font-size:90%;">(a)</span> </span></figcaption> </figure> </div> <div class="ltx_flex_break"></div> <div class="ltx_flex_cell ltx_flex_size_1"> <figure class="ltx_figure ltx_flex_size_1 ltx_align_center" id="A6.F8.sf2"><img alt="Refer to caption" class="ltx_graphics ltx_img_landscape" height="312" id="A6.F8.sf2.g1" src="extracted/5518834/images/Annealing_influence_images.png" width="598"/> <figcaption class="ltx_caption"><span class="ltx_tag ltx_tag_figure"><span class="ltx_text" id="A6.F8.sf2.2.1.1" style="font-size:90%;">(b)</span> </span></figcaption> </figure> </div> </div> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure"><span class="ltx_text" id="A6.F9.18.8.1" style="font-size:90%;">Figure 9</span>: </span><span class="ltx_text" id="A6.F9.14.7" style="font-size:90%;">Influence of the number of annealing levels <math alttext="m" class="ltx_Math" display="inline" id="A6.F9.8.1.m1.1"><semantics id="A6.F9.8.1.m1.1b"><mi id="A6.F9.8.1.m1.1.1" xref="A6.F9.8.1.m1.1.1.cmml">m</mi><annotation-xml encoding="MathML-Content" id="A6.F9.8.1.m1.1c"><ci id="A6.F9.8.1.m1.1.1.cmml" xref="A6.F9.8.1.m1.1.1">𝑚</ci></annotation-xml><annotation encoding="application/x-tex" id="A6.F9.8.1.m1.1d">m</annotation><annotation encoding="application/x-llamapun" id="A6.F9.8.1.m1.1e">italic_m</annotation></semantics></math> on the reconstruction with SNORE algorithm for a motion blur with a noise of standard deviation <math alttext="\sigma_{{\mathbf{y}}}=10/255" class="ltx_Math" display="inline" id="A6.F9.9.2.m2.1"><semantics id="A6.F9.9.2.m2.1b"><mrow id="A6.F9.9.2.m2.1.1" xref="A6.F9.9.2.m2.1.1.cmml"><msub id="A6.F9.9.2.m2.1.1.2" xref="A6.F9.9.2.m2.1.1.2.cmml"><mi id="A6.F9.9.2.m2.1.1.2.2" xref="A6.F9.9.2.m2.1.1.2.2.cmml">σ</mi><mi id="A6.F9.9.2.m2.1.1.2.3" xref="A6.F9.9.2.m2.1.1.2.3.cmml">𝐲</mi></msub><mo id="A6.F9.9.2.m2.1.1.1" xref="A6.F9.9.2.m2.1.1.1.cmml">=</mo><mrow id="A6.F9.9.2.m2.1.1.3" xref="A6.F9.9.2.m2.1.1.3.cmml"><mn id="A6.F9.9.2.m2.1.1.3.2" xref="A6.F9.9.2.m2.1.1.3.2.cmml">10</mn><mo id="A6.F9.9.2.m2.1.1.3.1" xref="A6.F9.9.2.m2.1.1.3.1.cmml">/</mo><mn id="A6.F9.9.2.m2.1.1.3.3" xref="A6.F9.9.2.m2.1.1.3.3.cmml">255</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="A6.F9.9.2.m2.1c"><apply id="A6.F9.9.2.m2.1.1.cmml" xref="A6.F9.9.2.m2.1.1"><eq id="A6.F9.9.2.m2.1.1.1.cmml" xref="A6.F9.9.2.m2.1.1.1"></eq><apply id="A6.F9.9.2.m2.1.1.2.cmml" xref="A6.F9.9.2.m2.1.1.2"><csymbol cd="ambiguous" id="A6.F9.9.2.m2.1.1.2.1.cmml" xref="A6.F9.9.2.m2.1.1.2">subscript</csymbol><ci id="A6.F9.9.2.m2.1.1.2.2.cmml" xref="A6.F9.9.2.m2.1.1.2.2">𝜎</ci><ci id="A6.F9.9.2.m2.1.1.2.3.cmml" xref="A6.F9.9.2.m2.1.1.2.3">𝐲</ci></apply><apply id="A6.F9.9.2.m2.1.1.3.cmml" xref="A6.F9.9.2.m2.1.1.3"><divide id="A6.F9.9.2.m2.1.1.3.1.cmml" xref="A6.F9.9.2.m2.1.1.3.1"></divide><cn id="A6.F9.9.2.m2.1.1.3.2.cmml" type="integer" xref="A6.F9.9.2.m2.1.1.3.2">10</cn><cn id="A6.F9.9.2.m2.1.1.3.3.cmml" type="integer" xref="A6.F9.9.2.m2.1.1.3.3">255</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A6.F9.9.2.m2.1d">\sigma_{{\mathbf{y}}}=10/255</annotation><annotation encoding="application/x-llamapun" id="A6.F9.9.2.m2.1e">italic_σ start_POSTSUBSCRIPT bold_y end_POSTSUBSCRIPT = 10 / 255</annotation></semantics></math> and a fixed number of <math alttext="1500" class="ltx_Math" display="inline" id="A6.F9.10.3.m3.1"><semantics id="A6.F9.10.3.m3.1b"><mn id="A6.F9.10.3.m3.1.1" xref="A6.F9.10.3.m3.1.1.cmml">1500</mn><annotation-xml encoding="MathML-Content" id="A6.F9.10.3.m3.1c"><cn id="A6.F9.10.3.m3.1.1.cmml" type="integer" xref="A6.F9.10.3.m3.1.1">1500</cn></annotation-xml><annotation encoding="application/x-tex" id="A6.F9.10.3.m3.1d">1500</annotation><annotation encoding="application/x-llamapun" id="A6.F9.10.3.m3.1e">1500</annotation></semantics></math> iterations. <span class="ltx_text ltx_font_italic" id="A6.F9.14.7.1">Figure <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#A6.F8.sf1" title="8(a) ‣ Figure 9 ‣ On the step-size 𝛿 ‣ F.1 Deblurring ‣ Appendix F Additional Experiments ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_tag">8(a)</span></a>:</span> Metrics evolution with different <math alttext="m" class="ltx_Math" display="inline" id="A6.F9.11.4.m4.1"><semantics id="A6.F9.11.4.m4.1b"><mi id="A6.F9.11.4.m4.1.1" xref="A6.F9.11.4.m4.1.1.cmml">m</mi><annotation-xml encoding="MathML-Content" id="A6.F9.11.4.m4.1c"><ci id="A6.F9.11.4.m4.1.1.cmml" xref="A6.F9.11.4.m4.1.1">𝑚</ci></annotation-xml><annotation encoding="application/x-tex" id="A6.F9.11.4.m4.1d">m</annotation><annotation encoding="application/x-llamapun" id="A6.F9.11.4.m4.1e">italic_m</annotation></semantics></math>. One can note that the number of annealing levels <math alttext="m" class="ltx_Math" display="inline" id="A6.F9.12.5.m5.1"><semantics id="A6.F9.12.5.m5.1b"><mi id="A6.F9.12.5.m5.1.1" xref="A6.F9.12.5.m5.1.1.cmml">m</mi><annotation-xml encoding="MathML-Content" id="A6.F9.12.5.m5.1c"><ci id="A6.F9.12.5.m5.1.1.cmml" xref="A6.F9.12.5.m5.1.1">𝑚</ci></annotation-xml><annotation encoding="application/x-tex" id="A6.F9.12.5.m5.1d">m</annotation><annotation encoding="application/x-llamapun" id="A6.F9.12.5.m5.1e">italic_m</annotation></semantics></math> has a low influence on metrics values. <span class="ltx_text ltx_font_italic" id="A6.F9.14.7.2">Figure <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#A6.F8.sf2" title="8(b) ‣ Figure 9 ‣ On the step-size 𝛿 ‣ F.1 Deblurring ‣ Appendix F Additional Experiments ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_tag">8(b)</span></a>:</span> Reconstructed images with the SNORE algorithm for different numbers of annealing levels <math alttext="m" class="ltx_Math" display="inline" id="A6.F9.13.6.m6.1"><semantics id="A6.F9.13.6.m6.1b"><mi id="A6.F9.13.6.m6.1.1" xref="A6.F9.13.6.m6.1.1.cmml">m</mi><annotation-xml encoding="MathML-Content" id="A6.F9.13.6.m6.1c"><ci id="A6.F9.13.6.m6.1.1.cmml" xref="A6.F9.13.6.m6.1.1">𝑚</ci></annotation-xml><annotation encoding="application/x-tex" id="A6.F9.13.6.m6.1d">m</annotation><annotation encoding="application/x-llamapun" id="A6.F9.13.6.m6.1e">italic_m</annotation></semantics></math>. One can note that, the larger <math alttext="m" class="ltx_Math" display="inline" id="A6.F9.14.7.m7.1"><semantics id="A6.F9.14.7.m7.1b"><mi id="A6.F9.14.7.m7.1.1" xref="A6.F9.14.7.m7.1.1.cmml">m</mi><annotation-xml encoding="MathML-Content" id="A6.F9.14.7.m7.1c"><ci id="A6.F9.14.7.m7.1.1.cmml" xref="A6.F9.14.7.m7.1.1">𝑚</ci></annotation-xml><annotation encoding="application/x-tex" id="A6.F9.14.7.m7.1d">m</annotation><annotation encoding="application/x-llamapun" id="A6.F9.14.7.m7.1e">italic_m</annotation></semantics></math> is, the less artifacts are visible. However, compute a larger number of annealing levels impose to compute a larger number of iterations, to converge for each annealing parameters.</span></figcaption> </figure> </section> </section> <section class="ltx_subsection" id="A6.SS2"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">F.2 </span>Uncertainty of SNORE</h3> <section class="ltx_paragraph" id="A6.SS2.SSS0.Px1"> <h5 class="ltx_title ltx_title_paragraph">Seed sensitivity</h5> <div class="ltx_para" id="A6.SS2.SSS0.Px1.p1"> <p class="ltx_p" id="A6.SS2.SSS0.Px1.p1.1">On Figure <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#A6.F10" title="Figure 10 ‣ Seed sensitivity ‣ F.2 Uncertainty of SNORE ‣ Appendix F Additional Experiments ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_tag">10</span></a>, we illustrate the robustness of of SNORE to stochasticity, by running the algorithm with different random seed and looking at the standard deviation of the corresponding reconstructions. We observe that our restoration has a low variability and thus a low uncertainty. This is a crucial behavior for the reliability of the algorithm. This experiment suggests that the restoration, and especially the reconstructed structures, is stable and thus reliable.</p> </div> <figure class="ltx_figure" id="A6.F10"><img alt="Refer to caption" class="ltx_graphics ltx_centering ltx_img_landscape" height="200" id="A6.F10.g1" src="extracted/5518834/images/Restoration_deblurring_seed_sensitivity.png" width="598"/> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure"><span class="ltx_text" id="A6.F10.11.5.1" style="font-size:90%;">Figure 10</span>: </span><span class="ltx_text" id="A6.F10.8.4" style="font-size:90%;">Uncertainty of SNORE to the algorithm randomness. <span class="ltx_text ltx_font_italic" id="A6.F10.8.4.1">Rightmost:</span> The algorithm has been run with <math alttext="10" class="ltx_Math" display="inline" id="A6.F10.5.1.m1.1"><semantics id="A6.F10.5.1.m1.1b"><mn id="A6.F10.5.1.m1.1.1" xref="A6.F10.5.1.m1.1.1.cmml">10</mn><annotation-xml encoding="MathML-Content" id="A6.F10.5.1.m1.1c"><cn id="A6.F10.5.1.m1.1.1.cmml" type="integer" xref="A6.F10.5.1.m1.1.1">10</cn></annotation-xml><annotation encoding="application/x-tex" id="A6.F10.5.1.m1.1d">10</annotation><annotation encoding="application/x-llamapun" id="A6.F10.5.1.m1.1e">10</annotation></semantics></math> different seeds and the standard deviation of restored images is shown. The blur kernel is shown on the observed images and the input noise level is <math alttext="\sigma_{{\mathbf{y}}}=5/255" class="ltx_Math" display="inline" id="A6.F10.6.2.m2.1"><semantics id="A6.F10.6.2.m2.1b"><mrow id="A6.F10.6.2.m2.1.1" xref="A6.F10.6.2.m2.1.1.cmml"><msub id="A6.F10.6.2.m2.1.1.2" xref="A6.F10.6.2.m2.1.1.2.cmml"><mi id="A6.F10.6.2.m2.1.1.2.2" xref="A6.F10.6.2.m2.1.1.2.2.cmml">σ</mi><mi id="A6.F10.6.2.m2.1.1.2.3" xref="A6.F10.6.2.m2.1.1.2.3.cmml">𝐲</mi></msub><mo id="A6.F10.6.2.m2.1.1.1" xref="A6.F10.6.2.m2.1.1.1.cmml">=</mo><mrow id="A6.F10.6.2.m2.1.1.3" xref="A6.F10.6.2.m2.1.1.3.cmml"><mn id="A6.F10.6.2.m2.1.1.3.2" xref="A6.F10.6.2.m2.1.1.3.2.cmml">5</mn><mo id="A6.F10.6.2.m2.1.1.3.1" xref="A6.F10.6.2.m2.1.1.3.1.cmml">/</mo><mn id="A6.F10.6.2.m2.1.1.3.3" xref="A6.F10.6.2.m2.1.1.3.3.cmml">255</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="A6.F10.6.2.m2.1c"><apply id="A6.F10.6.2.m2.1.1.cmml" xref="A6.F10.6.2.m2.1.1"><eq id="A6.F10.6.2.m2.1.1.1.cmml" xref="A6.F10.6.2.m2.1.1.1"></eq><apply id="A6.F10.6.2.m2.1.1.2.cmml" xref="A6.F10.6.2.m2.1.1.2"><csymbol cd="ambiguous" id="A6.F10.6.2.m2.1.1.2.1.cmml" xref="A6.F10.6.2.m2.1.1.2">subscript</csymbol><ci id="A6.F10.6.2.m2.1.1.2.2.cmml" xref="A6.F10.6.2.m2.1.1.2.2">𝜎</ci><ci id="A6.F10.6.2.m2.1.1.2.3.cmml" xref="A6.F10.6.2.m2.1.1.2.3">𝐲</ci></apply><apply id="A6.F10.6.2.m2.1.1.3.cmml" xref="A6.F10.6.2.m2.1.1.3"><divide id="A6.F10.6.2.m2.1.1.3.1.cmml" xref="A6.F10.6.2.m2.1.1.3.1"></divide><cn id="A6.F10.6.2.m2.1.1.3.2.cmml" type="integer" xref="A6.F10.6.2.m2.1.1.3.2">5</cn><cn id="A6.F10.6.2.m2.1.1.3.3.cmml" type="integer" xref="A6.F10.6.2.m2.1.1.3.3">255</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A6.F10.6.2.m2.1d">\sigma_{{\mathbf{y}}}=5/255</annotation><annotation encoding="application/x-llamapun" id="A6.F10.6.2.m2.1e">italic_σ start_POSTSUBSCRIPT bold_y end_POSTSUBSCRIPT = 5 / 255</annotation></semantics></math>. Note how the algorithm is stable with a small standard deviation of maximum <math alttext="0.01" class="ltx_Math" display="inline" id="A6.F10.7.3.m3.1"><semantics id="A6.F10.7.3.m3.1b"><mn id="A6.F10.7.3.m3.1.1" xref="A6.F10.7.3.m3.1.1.cmml">0.01</mn><annotation-xml encoding="MathML-Content" id="A6.F10.7.3.m3.1c"><cn id="A6.F10.7.3.m3.1.1.cmml" type="float" xref="A6.F10.7.3.m3.1.1">0.01</cn></annotation-xml><annotation encoding="application/x-tex" id="A6.F10.7.3.m3.1d">0.01</annotation><annotation encoding="application/x-llamapun" id="A6.F10.7.3.m3.1e">0.01</annotation></semantics></math> (pixel-values are in <math alttext="[0,1]" class="ltx_Math" display="inline" id="A6.F10.8.4.m4.2"><semantics id="A6.F10.8.4.m4.2b"><mrow id="A6.F10.8.4.m4.2.3.2" xref="A6.F10.8.4.m4.2.3.1.cmml"><mo id="A6.F10.8.4.m4.2.3.2.1" stretchy="false" xref="A6.F10.8.4.m4.2.3.1.cmml">[</mo><mn id="A6.F10.8.4.m4.1.1" xref="A6.F10.8.4.m4.1.1.cmml">0</mn><mo id="A6.F10.8.4.m4.2.3.2.2" xref="A6.F10.8.4.m4.2.3.1.cmml">,</mo><mn id="A6.F10.8.4.m4.2.2" xref="A6.F10.8.4.m4.2.2.cmml">1</mn><mo id="A6.F10.8.4.m4.2.3.2.3" stretchy="false" xref="A6.F10.8.4.m4.2.3.1.cmml">]</mo></mrow><annotation-xml encoding="MathML-Content" id="A6.F10.8.4.m4.2c"><interval closure="closed" id="A6.F10.8.4.m4.2.3.1.cmml" xref="A6.F10.8.4.m4.2.3.2"><cn id="A6.F10.8.4.m4.1.1.cmml" type="integer" xref="A6.F10.8.4.m4.1.1">0</cn><cn id="A6.F10.8.4.m4.2.2.cmml" type="integer" xref="A6.F10.8.4.m4.2.2">1</cn></interval></annotation-xml><annotation encoding="application/x-tex" id="A6.F10.8.4.m4.2d">[0,1]</annotation><annotation encoding="application/x-llamapun" id="A6.F10.8.4.m4.2e">[ 0 , 1 ]</annotation></semantics></math>). Especially, structures are stable with a particularly low standard deviation.</span></figcaption> </figure> </section> <section class="ltx_paragraph" id="A6.SS2.SSS0.Px2"> <h5 class="ltx_title ltx_title_paragraph">Initialization sensitivity</h5> <div class="ltx_para" id="A6.SS2.SSS0.Px2.p1"> <p class="ltx_p" id="A6.SS2.SSS0.Px2.p1.1">On Figure <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#A6.F11" title="Figure 11 ‣ Initialization sensitivity ‣ F.2 Uncertainty of SNORE ‣ Appendix F Additional Experiments ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_tag">11</span></a>, three different initialization are shown for a deblurring task. We notice that the SNORE algorithm does not diverge, even with a random initialization. This is a remarkable property since we face a non-convex optimization for which the initialization of the algorithm is crucial. With a pure noise initialization, artifacts are nevertheless present on the restored images. We also observe that the gap between the algorithm run with an oracle initialization or the observation is tight. This observation suggests that the observation is a good initialization for the SNORE algorithm.</p> </div> <figure class="ltx_figure" id="A6.F11"><img alt="Refer to caption" class="ltx_graphics ltx_centering ltx_img_landscape" height="316" id="A6.F11.g1" src="extracted/5518834/images/initialisation_sensitivity.png" width="598"/> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure"><span class="ltx_text" id="A6.F11.8.3.1" style="font-size:90%;">Figure 11</span>: </span><span class="ltx_text" id="A6.F11.4.2" style="font-size:90%;">Sensitivity of SNORE to the algorithm initialization. A motion blur kernel and a noise of standard deviation <math alttext="\sigma_{{\mathbf{y}}}=10/255" class="ltx_Math" display="inline" id="A6.F11.3.1.m1.1"><semantics id="A6.F11.3.1.m1.1b"><mrow id="A6.F11.3.1.m1.1.1" xref="A6.F11.3.1.m1.1.1.cmml"><msub id="A6.F11.3.1.m1.1.1.2" xref="A6.F11.3.1.m1.1.1.2.cmml"><mi id="A6.F11.3.1.m1.1.1.2.2" xref="A6.F11.3.1.m1.1.1.2.2.cmml">σ</mi><mi id="A6.F11.3.1.m1.1.1.2.3" xref="A6.F11.3.1.m1.1.1.2.3.cmml">𝐲</mi></msub><mo id="A6.F11.3.1.m1.1.1.1" xref="A6.F11.3.1.m1.1.1.1.cmml">=</mo><mrow id="A6.F11.3.1.m1.1.1.3" xref="A6.F11.3.1.m1.1.1.3.cmml"><mn id="A6.F11.3.1.m1.1.1.3.2" xref="A6.F11.3.1.m1.1.1.3.2.cmml">10</mn><mo id="A6.F11.3.1.m1.1.1.3.1" xref="A6.F11.3.1.m1.1.1.3.1.cmml">/</mo><mn id="A6.F11.3.1.m1.1.1.3.3" xref="A6.F11.3.1.m1.1.1.3.3.cmml">255</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="A6.F11.3.1.m1.1c"><apply id="A6.F11.3.1.m1.1.1.cmml" xref="A6.F11.3.1.m1.1.1"><eq id="A6.F11.3.1.m1.1.1.1.cmml" xref="A6.F11.3.1.m1.1.1.1"></eq><apply id="A6.F11.3.1.m1.1.1.2.cmml" xref="A6.F11.3.1.m1.1.1.2"><csymbol cd="ambiguous" id="A6.F11.3.1.m1.1.1.2.1.cmml" xref="A6.F11.3.1.m1.1.1.2">subscript</csymbol><ci id="A6.F11.3.1.m1.1.1.2.2.cmml" xref="A6.F11.3.1.m1.1.1.2.2">𝜎</ci><ci id="A6.F11.3.1.m1.1.1.2.3.cmml" xref="A6.F11.3.1.m1.1.1.2.3">𝐲</ci></apply><apply id="A6.F11.3.1.m1.1.1.3.cmml" xref="A6.F11.3.1.m1.1.1.3"><divide id="A6.F11.3.1.m1.1.1.3.1.cmml" xref="A6.F11.3.1.m1.1.1.3.1"></divide><cn id="A6.F11.3.1.m1.1.1.3.2.cmml" type="integer" xref="A6.F11.3.1.m1.1.1.3.2">10</cn><cn id="A6.F11.3.1.m1.1.1.3.3.cmml" type="integer" xref="A6.F11.3.1.m1.1.1.3.3">255</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A6.F11.3.1.m1.1d">\sigma_{{\mathbf{y}}}=10/255</annotation><annotation encoding="application/x-llamapun" id="A6.F11.3.1.m1.1e">italic_σ start_POSTSUBSCRIPT bold_y end_POSTSUBSCRIPT = 10 / 255</annotation></semantics></math> are used to degrade the image. <span class="ltx_text ltx_font_italic" id="A6.F11.4.2.1">Top Three Leftmost:</span> Three initializations are used to start the SNORE algorithm: the ground-truth image (Oracle), the observation and a random image where each pixel is sampled uniformly in <math alttext="[0,1]" class="ltx_Math" display="inline" id="A6.F11.4.2.m2.2"><semantics id="A6.F11.4.2.m2.2b"><mrow id="A6.F11.4.2.m2.2.3.2" xref="A6.F11.4.2.m2.2.3.1.cmml"><mo id="A6.F11.4.2.m2.2.3.2.1" stretchy="false" xref="A6.F11.4.2.m2.2.3.1.cmml">[</mo><mn id="A6.F11.4.2.m2.1.1" xref="A6.F11.4.2.m2.1.1.cmml">0</mn><mo id="A6.F11.4.2.m2.2.3.2.2" xref="A6.F11.4.2.m2.2.3.1.cmml">,</mo><mn id="A6.F11.4.2.m2.2.2" xref="A6.F11.4.2.m2.2.2.cmml">1</mn><mo id="A6.F11.4.2.m2.2.3.2.3" stretchy="false" xref="A6.F11.4.2.m2.2.3.1.cmml">]</mo></mrow><annotation-xml encoding="MathML-Content" id="A6.F11.4.2.m2.2c"><interval closure="closed" id="A6.F11.4.2.m2.2.3.1.cmml" xref="A6.F11.4.2.m2.2.3.2"><cn id="A6.F11.4.2.m2.1.1.cmml" type="integer" xref="A6.F11.4.2.m2.1.1">0</cn><cn id="A6.F11.4.2.m2.2.2.cmml" type="integer" xref="A6.F11.4.2.m2.2.2">1</cn></interval></annotation-xml><annotation encoding="application/x-tex" id="A6.F11.4.2.m2.2d">[0,1]</annotation><annotation encoding="application/x-llamapun" id="A6.F11.4.2.m2.2e">[ 0 , 1 ]</annotation></semantics></math>. <span class="ltx_text ltx_font_italic" id="A6.F11.4.2.2">Bottom Three Rightmost:</span> Three corresponding reconstructions. Note how the algorithm succeeds to reconstruct a relevant image even with a random initialization.</span></figcaption> </figure> </section> </section> <section class="ltx_subsection" id="A6.SS3"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">F.3 </span>Inpainting</h3> <div class="ltx_para" id="A6.SS3.p1"> <p class="ltx_p" id="A6.SS3.p1.1">The inpainting mask is created by sampling a Bernouilli law of success probability <math alttext="p=0.5" class="ltx_Math" display="inline" id="A6.SS3.p1.1.m1.1"><semantics id="A6.SS3.p1.1.m1.1a"><mrow id="A6.SS3.p1.1.m1.1.1" xref="A6.SS3.p1.1.m1.1.1.cmml"><mi id="A6.SS3.p1.1.m1.1.1.2" xref="A6.SS3.p1.1.m1.1.1.2.cmml">p</mi><mo id="A6.SS3.p1.1.m1.1.1.1" xref="A6.SS3.p1.1.m1.1.1.1.cmml">=</mo><mn id="A6.SS3.p1.1.m1.1.1.3" xref="A6.SS3.p1.1.m1.1.1.3.cmml">0.5</mn></mrow><annotation-xml encoding="MathML-Content" id="A6.SS3.p1.1.m1.1b"><apply id="A6.SS3.p1.1.m1.1.1.cmml" xref="A6.SS3.p1.1.m1.1.1"><eq id="A6.SS3.p1.1.m1.1.1.1.cmml" xref="A6.SS3.p1.1.m1.1.1.1"></eq><ci id="A6.SS3.p1.1.m1.1.1.2.cmml" xref="A6.SS3.p1.1.m1.1.1.2">𝑝</ci><cn id="A6.SS3.p1.1.m1.1.1.3.cmml" type="float" xref="A6.SS3.p1.1.m1.1.1.3">0.5</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A6.SS3.p1.1.m1.1c">p=0.5</annotation><annotation encoding="application/x-llamapun" id="A6.SS3.p1.1.m1.1d">italic_p = 0.5</annotation></semantics></math> for each pixel of the image.</p> </div> <section class="ltx_paragraph" id="A6.SS3.SSS0.Px1"> <h5 class="ltx_title ltx_title_paragraph">Parameters setting</h5> <div class="ltx_para" id="A6.SS3.SSS0.Px1.p1"> <p class="ltx_p" id="A6.SS3.SSS0.Px1.p1.2">On Table <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#A6.T5" title="Table 5 ‣ Parameters setting ‣ F.3 Inpainting ‣ Appendix F Additional Experiments ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_tag">5</span></a>, we detail the practical choice of parameters we made. As suggested by <cite class="ltx_cite ltx_citemacro_citep">(Hurault et al., <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#bib.bib30" title="">2022a</a>)</cite>, for image inpainting, we start by running the algorithm with a larger value <math alttext="\sigma=50/255" class="ltx_Math" display="inline" id="A6.SS3.SSS0.Px1.p1.1.m1.1"><semantics id="A6.SS3.SSS0.Px1.p1.1.m1.1a"><mrow id="A6.SS3.SSS0.Px1.p1.1.m1.1.1" xref="A6.SS3.SSS0.Px1.p1.1.m1.1.1.cmml"><mi id="A6.SS3.SSS0.Px1.p1.1.m1.1.1.2" xref="A6.SS3.SSS0.Px1.p1.1.m1.1.1.2.cmml">σ</mi><mo id="A6.SS3.SSS0.Px1.p1.1.m1.1.1.1" xref="A6.SS3.SSS0.Px1.p1.1.m1.1.1.1.cmml">=</mo><mrow id="A6.SS3.SSS0.Px1.p1.1.m1.1.1.3" xref="A6.SS3.SSS0.Px1.p1.1.m1.1.1.3.cmml"><mn id="A6.SS3.SSS0.Px1.p1.1.m1.1.1.3.2" xref="A6.SS3.SSS0.Px1.p1.1.m1.1.1.3.2.cmml">50</mn><mo id="A6.SS3.SSS0.Px1.p1.1.m1.1.1.3.1" xref="A6.SS3.SSS0.Px1.p1.1.m1.1.1.3.1.cmml">/</mo><mn id="A6.SS3.SSS0.Px1.p1.1.m1.1.1.3.3" xref="A6.SS3.SSS0.Px1.p1.1.m1.1.1.3.3.cmml">255</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="A6.SS3.SSS0.Px1.p1.1.m1.1b"><apply id="A6.SS3.SSS0.Px1.p1.1.m1.1.1.cmml" xref="A6.SS3.SSS0.Px1.p1.1.m1.1.1"><eq id="A6.SS3.SSS0.Px1.p1.1.m1.1.1.1.cmml" xref="A6.SS3.SSS0.Px1.p1.1.m1.1.1.1"></eq><ci id="A6.SS3.SSS0.Px1.p1.1.m1.1.1.2.cmml" xref="A6.SS3.SSS0.Px1.p1.1.m1.1.1.2">𝜎</ci><apply id="A6.SS3.SSS0.Px1.p1.1.m1.1.1.3.cmml" xref="A6.SS3.SSS0.Px1.p1.1.m1.1.1.3"><divide id="A6.SS3.SSS0.Px1.p1.1.m1.1.1.3.1.cmml" xref="A6.SS3.SSS0.Px1.p1.1.m1.1.1.3.1"></divide><cn id="A6.SS3.SSS0.Px1.p1.1.m1.1.1.3.2.cmml" type="integer" xref="A6.SS3.SSS0.Px1.p1.1.m1.1.1.3.2">50</cn><cn id="A6.SS3.SSS0.Px1.p1.1.m1.1.1.3.3.cmml" type="integer" xref="A6.SS3.SSS0.Px1.p1.1.m1.1.1.3.3">255</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A6.SS3.SSS0.Px1.p1.1.m1.1c">\sigma=50/255</annotation><annotation encoding="application/x-llamapun" id="A6.SS3.SSS0.Px1.p1.1.m1.1d">italic_σ = 50 / 255</annotation></semantics></math> for a number of iterations <math alttext="n_{\text{init}}" class="ltx_Math" display="inline" id="A6.SS3.SSS0.Px1.p1.2.m2.1"><semantics id="A6.SS3.SSS0.Px1.p1.2.m2.1a"><msub id="A6.SS3.SSS0.Px1.p1.2.m2.1.1" xref="A6.SS3.SSS0.Px1.p1.2.m2.1.1.cmml"><mi id="A6.SS3.SSS0.Px1.p1.2.m2.1.1.2" xref="A6.SS3.SSS0.Px1.p1.2.m2.1.1.2.cmml">n</mi><mtext id="A6.SS3.SSS0.Px1.p1.2.m2.1.1.3" xref="A6.SS3.SSS0.Px1.p1.2.m2.1.1.3a.cmml">init</mtext></msub><annotation-xml encoding="MathML-Content" id="A6.SS3.SSS0.Px1.p1.2.m2.1b"><apply id="A6.SS3.SSS0.Px1.p1.2.m2.1.1.cmml" xref="A6.SS3.SSS0.Px1.p1.2.m2.1.1"><csymbol cd="ambiguous" id="A6.SS3.SSS0.Px1.p1.2.m2.1.1.1.cmml" xref="A6.SS3.SSS0.Px1.p1.2.m2.1.1">subscript</csymbol><ci id="A6.SS3.SSS0.Px1.p1.2.m2.1.1.2.cmml" xref="A6.SS3.SSS0.Px1.p1.2.m2.1.1.2">𝑛</ci><ci id="A6.SS3.SSS0.Px1.p1.2.m2.1.1.3a.cmml" xref="A6.SS3.SSS0.Px1.p1.2.m2.1.1.3"><mtext id="A6.SS3.SSS0.Px1.p1.2.m2.1.1.3.cmml" mathsize="70%" xref="A6.SS3.SSS0.Px1.p1.2.m2.1.1.3">init</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A6.SS3.SSS0.Px1.p1.2.m2.1c">n_{\text{init}}</annotation><annotation encoding="application/x-llamapun" id="A6.SS3.SSS0.Px1.p1.2.m2.1d">italic_n start_POSTSUBSCRIPT init end_POSTSUBSCRIPT</annotation></semantics></math>. This allows the algorithm to tackle the ill-posedness of the inpainting task.</p> </div> <figure class="ltx_table" id="A6.T5"> <div class="ltx_inline-block ltx_align_center ltx_transformed_outer" id="A6.T5.10" style="width:281.9pt;height:208.1pt;vertical-align:-0.0pt;"><span class="ltx_transformed_inner" style="transform:translate(19.0pt,-14.0pt) scale(1.15604306801395,1.15604306801395) ;"> <table class="ltx_tabular ltx_guessed_headers ltx_align_middle" id="A6.T5.10.10"> <tbody class="ltx_tbody"> <tr class="ltx_tr" id="A6.T5.10.10.11.1"> <th class="ltx_td ltx_align_center ltx_th ltx_th_row ltx_border_l ltx_border_rr ltx_border_t" id="A6.T5.10.10.11.1.1">Parameters</th> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="A6.T5.10.10.11.1.2">RED</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="A6.T5.10.10.11.1.3">RED Prox</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="A6.T5.10.10.11.1.4">SNORE</td> <td class="ltx_td 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id="A6.T5.3.3.3.1.m1.1.1.2" xref="A6.T5.3.3.3.1.m1.1.1.2.cmml">α</mi><mrow id="A6.T5.3.3.3.1.m1.1.1.3" xref="A6.T5.3.3.3.1.m1.1.1.3.cmml"><mi id="A6.T5.3.3.3.1.m1.1.1.3.2" xref="A6.T5.3.3.3.1.m1.1.1.3.2.cmml">m</mi><mo id="A6.T5.3.3.3.1.m1.1.1.3.1" xref="A6.T5.3.3.3.1.m1.1.1.3.1.cmml">−</mo><mn id="A6.T5.3.3.3.1.m1.1.1.3.3" xref="A6.T5.3.3.3.1.m1.1.1.3.3.cmml">1</mn></mrow></msub><annotation-xml encoding="MathML-Content" id="A6.T5.3.3.3.1.m1.1b"><apply id="A6.T5.3.3.3.1.m1.1.1.cmml" xref="A6.T5.3.3.3.1.m1.1.1"><csymbol cd="ambiguous" id="A6.T5.3.3.3.1.m1.1.1.1.cmml" xref="A6.T5.3.3.3.1.m1.1.1">subscript</csymbol><ci id="A6.T5.3.3.3.1.m1.1.1.2.cmml" xref="A6.T5.3.3.3.1.m1.1.1.2">𝛼</ci><apply id="A6.T5.3.3.3.1.m1.1.1.3.cmml" xref="A6.T5.3.3.3.1.m1.1.1.3"><minus id="A6.T5.3.3.3.1.m1.1.1.3.1.cmml" xref="A6.T5.3.3.3.1.m1.1.1.3.1"></minus><ci id="A6.T5.3.3.3.1.m1.1.1.3.2.cmml" xref="A6.T5.3.3.3.1.m1.1.1.3.2">𝑚</ci><cn id="A6.T5.3.3.3.1.m1.1.1.3.3.cmml" type="integer" xref="A6.T5.3.3.3.1.m1.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A6.T5.3.3.3.1.m1.1c">\alpha_{m-1}</annotation><annotation encoding="application/x-llamapun" id="A6.T5.3.3.3.1.m1.1d">italic_α start_POSTSUBSCRIPT italic_m - 1 end_POSTSUBSCRIPT</annotation></semantics></math></th> <td class="ltx_td ltx_border_r ltx_border_t" id="A6.T5.3.3.3.2"></td> <td class="ltx_td ltx_border_r ltx_border_t" id="A6.T5.3.3.3.3"></td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="A6.T5.3.3.3.4">0.4</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="A6.T5.3.3.3.5">0.15</td> </tr> <tr class="ltx_tr" id="A6.T5.4.4.4"> <th class="ltx_td ltx_align_center ltx_th ltx_th_row ltx_border_l ltx_border_rr ltx_border_t" id="A6.T5.4.4.4.1"><math alttext="\sigma" class="ltx_Math" display="inline" id="A6.T5.4.4.4.1.m1.1"><semantics id="A6.T5.4.4.4.1.m1.1a"><mi id="A6.T5.4.4.4.1.m1.1.1" xref="A6.T5.4.4.4.1.m1.1.1.cmml">σ</mi><annotation-xml encoding="MathML-Content" id="A6.T5.4.4.4.1.m1.1b"><ci id="A6.T5.4.4.4.1.m1.1.1.cmml" xref="A6.T5.4.4.4.1.m1.1.1">𝜎</ci></annotation-xml><annotation encoding="application/x-tex" id="A6.T5.4.4.4.1.m1.1c">\sigma</annotation><annotation encoding="application/x-llamapun" id="A6.T5.4.4.4.1.m1.1d">italic_σ</annotation></semantics></math></th> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="A6.T5.4.4.4.2">10/255</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="A6.T5.4.4.4.3">10/255</td> <td class="ltx_td ltx_border_r ltx_border_t" id="A6.T5.4.4.4.4"></td> <td class="ltx_td ltx_border_r ltx_border_t" id="A6.T5.4.4.4.5"></td> </tr> <tr class="ltx_tr" id="A6.T5.5.5.5"> <th class="ltx_td ltx_align_center ltx_th ltx_th_row ltx_border_l ltx_border_rr ltx_border_t" id="A6.T5.5.5.5.1"><math alttext="\sigma_{0}" class="ltx_Math" display="inline" id="A6.T5.5.5.5.1.m1.1"><semantics id="A6.T5.5.5.5.1.m1.1a"><msub id="A6.T5.5.5.5.1.m1.1.1" xref="A6.T5.5.5.5.1.m1.1.1.cmml"><mi id="A6.T5.5.5.5.1.m1.1.1.2" xref="A6.T5.5.5.5.1.m1.1.1.2.cmml">σ</mi><mn id="A6.T5.5.5.5.1.m1.1.1.3" xref="A6.T5.5.5.5.1.m1.1.1.3.cmml">0</mn></msub><annotation-xml encoding="MathML-Content" id="A6.T5.5.5.5.1.m1.1b"><apply id="A6.T5.5.5.5.1.m1.1.1.cmml" xref="A6.T5.5.5.5.1.m1.1.1"><csymbol cd="ambiguous" id="A6.T5.5.5.5.1.m1.1.1.1.cmml" xref="A6.T5.5.5.5.1.m1.1.1">subscript</csymbol><ci id="A6.T5.5.5.5.1.m1.1.1.2.cmml" xref="A6.T5.5.5.5.1.m1.1.1.2">𝜎</ci><cn id="A6.T5.5.5.5.1.m1.1.1.3.cmml" type="integer" xref="A6.T5.5.5.5.1.m1.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A6.T5.5.5.5.1.m1.1c">\sigma_{0}</annotation><annotation encoding="application/x-llamapun" id="A6.T5.5.5.5.1.m1.1d">italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math></th> <td class="ltx_td ltx_border_r ltx_border_t" id="A6.T5.5.5.5.2"></td> <td class="ltx_td ltx_border_r ltx_border_t" id="A6.T5.5.5.5.3"></td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="A6.T5.5.5.5.4">50/255</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="A6.T5.5.5.5.5">50/255</td> </tr> <tr class="ltx_tr" id="A6.T5.6.6.6"> <th class="ltx_td ltx_align_center ltx_th ltx_th_row ltx_border_l ltx_border_rr ltx_border_t" id="A6.T5.6.6.6.1"><math alttext="\sigma_{m-1}" class="ltx_Math" display="inline" id="A6.T5.6.6.6.1.m1.1"><semantics id="A6.T5.6.6.6.1.m1.1a"><msub id="A6.T5.6.6.6.1.m1.1.1" xref="A6.T5.6.6.6.1.m1.1.1.cmml"><mi id="A6.T5.6.6.6.1.m1.1.1.2" xref="A6.T5.6.6.6.1.m1.1.1.2.cmml">σ</mi><mrow id="A6.T5.6.6.6.1.m1.1.1.3" xref="A6.T5.6.6.6.1.m1.1.1.3.cmml"><mi id="A6.T5.6.6.6.1.m1.1.1.3.2" xref="A6.T5.6.6.6.1.m1.1.1.3.2.cmml">m</mi><mo id="A6.T5.6.6.6.1.m1.1.1.3.1" xref="A6.T5.6.6.6.1.m1.1.1.3.1.cmml">−</mo><mn id="A6.T5.6.6.6.1.m1.1.1.3.3" xref="A6.T5.6.6.6.1.m1.1.1.3.3.cmml">1</mn></mrow></msub><annotation-xml encoding="MathML-Content" id="A6.T5.6.6.6.1.m1.1b"><apply id="A6.T5.6.6.6.1.m1.1.1.cmml" xref="A6.T5.6.6.6.1.m1.1.1"><csymbol cd="ambiguous" id="A6.T5.6.6.6.1.m1.1.1.1.cmml" xref="A6.T5.6.6.6.1.m1.1.1">subscript</csymbol><ci id="A6.T5.6.6.6.1.m1.1.1.2.cmml" xref="A6.T5.6.6.6.1.m1.1.1.2">𝜎</ci><apply id="A6.T5.6.6.6.1.m1.1.1.3.cmml" xref="A6.T5.6.6.6.1.m1.1.1.3"><minus id="A6.T5.6.6.6.1.m1.1.1.3.1.cmml" xref="A6.T5.6.6.6.1.m1.1.1.3.1"></minus><ci id="A6.T5.6.6.6.1.m1.1.1.3.2.cmml" xref="A6.T5.6.6.6.1.m1.1.1.3.2">𝑚</ci><cn id="A6.T5.6.6.6.1.m1.1.1.3.3.cmml" type="integer" xref="A6.T5.6.6.6.1.m1.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A6.T5.6.6.6.1.m1.1c">\sigma_{m-1}</annotation><annotation encoding="application/x-llamapun" id="A6.T5.6.6.6.1.m1.1d">italic_σ start_POSTSUBSCRIPT italic_m - 1 end_POSTSUBSCRIPT</annotation></semantics></math></th> <td class="ltx_td ltx_border_r ltx_border_t" id="A6.T5.6.6.6.2"></td> <td class="ltx_td ltx_border_r ltx_border_t" id="A6.T5.6.6.6.3"></td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="A6.T5.6.6.6.4">5/255</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="A6.T5.6.6.6.5">5/255</td> </tr> <tr class="ltx_tr" id="A6.T5.7.7.7"> <th class="ltx_td ltx_align_center ltx_th ltx_th_row ltx_border_l ltx_border_rr ltx_border_t" id="A6.T5.7.7.7.1"><math alttext="n_{\text{init}}" class="ltx_Math" display="inline" id="A6.T5.7.7.7.1.m1.1"><semantics id="A6.T5.7.7.7.1.m1.1a"><msub id="A6.T5.7.7.7.1.m1.1.1" xref="A6.T5.7.7.7.1.m1.1.1.cmml"><mi id="A6.T5.7.7.7.1.m1.1.1.2" xref="A6.T5.7.7.7.1.m1.1.1.2.cmml">n</mi><mtext id="A6.T5.7.7.7.1.m1.1.1.3" xref="A6.T5.7.7.7.1.m1.1.1.3a.cmml">init</mtext></msub><annotation-xml encoding="MathML-Content" id="A6.T5.7.7.7.1.m1.1b"><apply id="A6.T5.7.7.7.1.m1.1.1.cmml" xref="A6.T5.7.7.7.1.m1.1.1"><csymbol cd="ambiguous" id="A6.T5.7.7.7.1.m1.1.1.1.cmml" xref="A6.T5.7.7.7.1.m1.1.1">subscript</csymbol><ci id="A6.T5.7.7.7.1.m1.1.1.2.cmml" xref="A6.T5.7.7.7.1.m1.1.1.2">𝑛</ci><ci id="A6.T5.7.7.7.1.m1.1.1.3a.cmml" xref="A6.T5.7.7.7.1.m1.1.1.3"><mtext id="A6.T5.7.7.7.1.m1.1.1.3.cmml" mathsize="70%" xref="A6.T5.7.7.7.1.m1.1.1.3">init</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A6.T5.7.7.7.1.m1.1c">n_{\text{init}}</annotation><annotation encoding="application/x-llamapun" id="A6.T5.7.7.7.1.m1.1d">italic_n start_POSTSUBSCRIPT init end_POSTSUBSCRIPT</annotation></semantics></math></th> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="A6.T5.7.7.7.2">10</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="A6.T5.7.7.7.3">100</td> <td class="ltx_td ltx_border_r ltx_border_t" id="A6.T5.7.7.7.4"></td> <td class="ltx_td ltx_border_r ltx_border_t" id="A6.T5.7.7.7.5"></td> </tr> <tr class="ltx_tr" id="A6.T5.8.8.8"> <th class="ltx_td ltx_align_center ltx_th ltx_th_row ltx_border_l ltx_border_rr ltx_border_t" id="A6.T5.8.8.8.1"><math alttext="\text{max}_{\text{itr}}" class="ltx_Math" display="inline" id="A6.T5.8.8.8.1.m1.1"><semantics id="A6.T5.8.8.8.1.m1.1a"><msub id="A6.T5.8.8.8.1.m1.1.1" xref="A6.T5.8.8.8.1.m1.1.1.cmml"><mtext id="A6.T5.8.8.8.1.m1.1.1.2" xref="A6.T5.8.8.8.1.m1.1.1.2a.cmml">max</mtext><mtext id="A6.T5.8.8.8.1.m1.1.1.3" xref="A6.T5.8.8.8.1.m1.1.1.3a.cmml">itr</mtext></msub><annotation-xml encoding="MathML-Content" id="A6.T5.8.8.8.1.m1.1b"><apply id="A6.T5.8.8.8.1.m1.1.1.cmml" xref="A6.T5.8.8.8.1.m1.1.1"><csymbol cd="ambiguous" id="A6.T5.8.8.8.1.m1.1.1.1.cmml" xref="A6.T5.8.8.8.1.m1.1.1">subscript</csymbol><ci id="A6.T5.8.8.8.1.m1.1.1.2a.cmml" xref="A6.T5.8.8.8.1.m1.1.1.2"><mtext id="A6.T5.8.8.8.1.m1.1.1.2.cmml" xref="A6.T5.8.8.8.1.m1.1.1.2">max</mtext></ci><ci id="A6.T5.8.8.8.1.m1.1.1.3a.cmml" xref="A6.T5.8.8.8.1.m1.1.1.3"><mtext id="A6.T5.8.8.8.1.m1.1.1.3.cmml" mathsize="70%" xref="A6.T5.8.8.8.1.m1.1.1.3">itr</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A6.T5.8.8.8.1.m1.1c">\text{max}_{\text{itr}}</annotation><annotation encoding="application/x-llamapun" id="A6.T5.8.8.8.1.m1.1d">max start_POSTSUBSCRIPT itr end_POSTSUBSCRIPT</annotation></semantics></math></th> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="A6.T5.8.8.8.2">500</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="A6.T5.8.8.8.3">500</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="A6.T5.8.8.8.4">500</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="A6.T5.8.8.8.5">500</td> </tr> <tr class="ltx_tr" id="A6.T5.10.10.10"> <th class="ltx_td ltx_align_center ltx_th ltx_th_row ltx_border_b ltx_border_l ltx_border_rr ltx_border_t" id="A6.T5.9.9.9.1"> <math alttext="\delta" class="ltx_Math" display="inline" id="A6.T5.9.9.9.1.m1.1"><semantics id="A6.T5.9.9.9.1.m1.1a"><mi id="A6.T5.9.9.9.1.m1.1.1" xref="A6.T5.9.9.9.1.m1.1.1.cmml">δ</mi><annotation-xml encoding="MathML-Content" id="A6.T5.9.9.9.1.m1.1b"><ci id="A6.T5.9.9.9.1.m1.1.1.cmml" xref="A6.T5.9.9.9.1.m1.1.1">𝛿</ci></annotation-xml><annotation encoding="application/x-tex" id="A6.T5.9.9.9.1.m1.1c">\delta</annotation><annotation encoding="application/x-llamapun" id="A6.T5.9.9.9.1.m1.1d">italic_δ</annotation></semantics></math> (initialization)</th> <td class="ltx_td ltx_align_center ltx_border_b ltx_border_r ltx_border_t" id="A6.T5.10.10.10.2"><math alttext="1/\alpha" class="ltx_Math" display="inline" id="A6.T5.10.10.10.2.m1.1"><semantics id="A6.T5.10.10.10.2.m1.1a"><mrow id="A6.T5.10.10.10.2.m1.1.1" xref="A6.T5.10.10.10.2.m1.1.1.cmml"><mn id="A6.T5.10.10.10.2.m1.1.1.2" xref="A6.T5.10.10.10.2.m1.1.1.2.cmml">1</mn><mo id="A6.T5.10.10.10.2.m1.1.1.1" xref="A6.T5.10.10.10.2.m1.1.1.1.cmml">/</mo><mi id="A6.T5.10.10.10.2.m1.1.1.3" xref="A6.T5.10.10.10.2.m1.1.1.3.cmml">α</mi></mrow><annotation-xml encoding="MathML-Content" id="A6.T5.10.10.10.2.m1.1b"><apply id="A6.T5.10.10.10.2.m1.1.1.cmml" xref="A6.T5.10.10.10.2.m1.1.1"><divide id="A6.T5.10.10.10.2.m1.1.1.1.cmml" xref="A6.T5.10.10.10.2.m1.1.1.1"></divide><cn id="A6.T5.10.10.10.2.m1.1.1.2.cmml" type="integer" xref="A6.T5.10.10.10.2.m1.1.1.2">1</cn><ci id="A6.T5.10.10.10.2.m1.1.1.3.cmml" xref="A6.T5.10.10.10.2.m1.1.1.3">𝛼</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A6.T5.10.10.10.2.m1.1c">1/\alpha</annotation><annotation encoding="application/x-llamapun" id="A6.T5.10.10.10.2.m1.1d">1 / italic_α</annotation></semantics></math></td> <td class="ltx_td ltx_align_center ltx_border_b ltx_border_r ltx_border_t" id="A6.T5.10.10.10.3">0.5</td> <td class="ltx_td ltx_align_center ltx_border_b ltx_border_r ltx_border_t" id="A6.T5.10.10.10.4">0.5</td> <td class="ltx_td ltx_align_center ltx_border_b ltx_border_r ltx_border_t" id="A6.T5.10.10.10.5">1</td> </tr> </tbody> </table> </span></div> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_table"><span class="ltx_text" id="A6.T5.12.1.1" style="font-size:90%;">Table 5</span>: </span><span class="ltx_text" id="A6.T5.13.2" style="font-size:90%;">Parameters setting for image inpainting for the different implemented methods.</span></figcaption> </figure> <figure class="ltx_figure" id="A6.F12"><img alt="Refer to caption" class="ltx_graphics ltx_centering ltx_img_square" height="731" id="A6.F12.g1" src="extracted/5518834/images/set_of_results_inpainting.png" width="598"/> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure"><span class="ltx_text" id="A6.F12.4.2.1" style="font-size:90%;">Figure 12</span>: </span><span class="ltx_text" id="A6.F12.2.1" style="font-size:90%;">Restorations obtained with SNORE, RED and DiffPIR algorithms for various images from the dataset CBSD68 on the inpainting task with a random mask with a proportion <math alttext="0.5" class="ltx_Math" display="inline" id="A6.F12.2.1.m1.1"><semantics id="A6.F12.2.1.m1.1b"><mn id="A6.F12.2.1.m1.1.1" xref="A6.F12.2.1.m1.1.1.cmml">0.5</mn><annotation-xml encoding="MathML-Content" id="A6.F12.2.1.m1.1c"><cn id="A6.F12.2.1.m1.1.1.cmml" type="float" xref="A6.F12.2.1.m1.1.1">0.5</cn></annotation-xml><annotation encoding="application/x-tex" id="A6.F12.2.1.m1.1d">0.5</annotation><annotation encoding="application/x-llamapun" id="A6.F12.2.1.m1.1e">0.5</annotation></semantics></math> of masked pixels.</span></figcaption> </figure> </section> <section class="ltx_paragraph" id="A6.SS3.SSS0.Px2"> <h5 class="ltx_title ltx_title_paragraph">More results</h5> <div class="ltx_para" id="A6.SS3.SSS0.Px2.p1"> <p class="ltx_p" id="A6.SS3.SSS0.Px2.p1.1">On Figure <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#A6.F12" title="Figure 12 ‣ Parameters setting ‣ F.3 Inpainting ‣ Appendix F Additional Experiments ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_tag">12</span></a>, we present various results of image inpainting for a better qualitative comparison between methods.</p> </div> <figure class="ltx_figure" id="A6.F13"><img alt="Refer to caption" class="ltx_graphics ltx_centering ltx_img_landscape" height="125" id="A6.F13.g1" src="extracted/5518834/images/Restoration_SNORE_various_inpainting.png" width="598"/> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure"><span class="ltx_text" id="A6.F13.6.3.1" style="font-size:90%;">Figure 13</span>: </span><span class="ltx_text" id="A6.F13.4.2" style="font-size:90%;">Restorations obtained with SNORE for inpainting, with a random mask of proportion <math alttext="p" class="ltx_Math" display="inline" id="A6.F13.3.1.m1.1"><semantics id="A6.F13.3.1.m1.1b"><mi id="A6.F13.3.1.m1.1.1" xref="A6.F13.3.1.m1.1.1.cmml">p</mi><annotation-xml encoding="MathML-Content" id="A6.F13.3.1.m1.1c"><ci id="A6.F13.3.1.m1.1.1.cmml" xref="A6.F13.3.1.m1.1.1">𝑝</ci></annotation-xml><annotation encoding="application/x-tex" id="A6.F13.3.1.m1.1d">p</annotation><annotation encoding="application/x-llamapun" id="A6.F13.3.1.m1.1e">italic_p</annotation></semantics></math> of masked pixels, on one image from the dataset CBSD68. The last annealing parameters <math alttext="(\alpha_{m-1},\sigma_{m-1})" class="ltx_Math" display="inline" id="A6.F13.4.2.m2.2"><semantics id="A6.F13.4.2.m2.2b"><mrow id="A6.F13.4.2.m2.2.2.2" xref="A6.F13.4.2.m2.2.2.3.cmml"><mo id="A6.F13.4.2.m2.2.2.2.3" stretchy="false" xref="A6.F13.4.2.m2.2.2.3.cmml">(</mo><msub id="A6.F13.4.2.m2.1.1.1.1" xref="A6.F13.4.2.m2.1.1.1.1.cmml"><mi id="A6.F13.4.2.m2.1.1.1.1.2" xref="A6.F13.4.2.m2.1.1.1.1.2.cmml">α</mi><mrow id="A6.F13.4.2.m2.1.1.1.1.3" xref="A6.F13.4.2.m2.1.1.1.1.3.cmml"><mi id="A6.F13.4.2.m2.1.1.1.1.3.2" xref="A6.F13.4.2.m2.1.1.1.1.3.2.cmml">m</mi><mo id="A6.F13.4.2.m2.1.1.1.1.3.1" xref="A6.F13.4.2.m2.1.1.1.1.3.1.cmml">−</mo><mn id="A6.F13.4.2.m2.1.1.1.1.3.3" xref="A6.F13.4.2.m2.1.1.1.1.3.3.cmml">1</mn></mrow></msub><mo id="A6.F13.4.2.m2.2.2.2.4" xref="A6.F13.4.2.m2.2.2.3.cmml">,</mo><msub id="A6.F13.4.2.m2.2.2.2.2" xref="A6.F13.4.2.m2.2.2.2.2.cmml"><mi id="A6.F13.4.2.m2.2.2.2.2.2" xref="A6.F13.4.2.m2.2.2.2.2.2.cmml">σ</mi><mrow id="A6.F13.4.2.m2.2.2.2.2.3" xref="A6.F13.4.2.m2.2.2.2.2.3.cmml"><mi id="A6.F13.4.2.m2.2.2.2.2.3.2" xref="A6.F13.4.2.m2.2.2.2.2.3.2.cmml">m</mi><mo id="A6.F13.4.2.m2.2.2.2.2.3.1" xref="A6.F13.4.2.m2.2.2.2.2.3.1.cmml">−</mo><mn id="A6.F13.4.2.m2.2.2.2.2.3.3" xref="A6.F13.4.2.m2.2.2.2.2.3.3.cmml">1</mn></mrow></msub><mo id="A6.F13.4.2.m2.2.2.2.5" stretchy="false" xref="A6.F13.4.2.m2.2.2.3.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="A6.F13.4.2.m2.2c"><interval closure="open" id="A6.F13.4.2.m2.2.2.3.cmml" xref="A6.F13.4.2.m2.2.2.2"><apply id="A6.F13.4.2.m2.1.1.1.1.cmml" xref="A6.F13.4.2.m2.1.1.1.1"><csymbol cd="ambiguous" id="A6.F13.4.2.m2.1.1.1.1.1.cmml" xref="A6.F13.4.2.m2.1.1.1.1">subscript</csymbol><ci id="A6.F13.4.2.m2.1.1.1.1.2.cmml" xref="A6.F13.4.2.m2.1.1.1.1.2">𝛼</ci><apply id="A6.F13.4.2.m2.1.1.1.1.3.cmml" xref="A6.F13.4.2.m2.1.1.1.1.3"><minus id="A6.F13.4.2.m2.1.1.1.1.3.1.cmml" xref="A6.F13.4.2.m2.1.1.1.1.3.1"></minus><ci id="A6.F13.4.2.m2.1.1.1.1.3.2.cmml" xref="A6.F13.4.2.m2.1.1.1.1.3.2">𝑚</ci><cn id="A6.F13.4.2.m2.1.1.1.1.3.3.cmml" type="integer" xref="A6.F13.4.2.m2.1.1.1.1.3.3">1</cn></apply></apply><apply id="A6.F13.4.2.m2.2.2.2.2.cmml" xref="A6.F13.4.2.m2.2.2.2.2"><csymbol cd="ambiguous" id="A6.F13.4.2.m2.2.2.2.2.1.cmml" xref="A6.F13.4.2.m2.2.2.2.2">subscript</csymbol><ci id="A6.F13.4.2.m2.2.2.2.2.2.cmml" xref="A6.F13.4.2.m2.2.2.2.2.2">𝜎</ci><apply id="A6.F13.4.2.m2.2.2.2.2.3.cmml" xref="A6.F13.4.2.m2.2.2.2.2.3"><minus id="A6.F13.4.2.m2.2.2.2.2.3.1.cmml" xref="A6.F13.4.2.m2.2.2.2.2.3.1"></minus><ci id="A6.F13.4.2.m2.2.2.2.2.3.2.cmml" xref="A6.F13.4.2.m2.2.2.2.2.3.2">𝑚</ci><cn id="A6.F13.4.2.m2.2.2.2.2.3.3.cmml" type="integer" xref="A6.F13.4.2.m2.2.2.2.2.3.3">1</cn></apply></apply></interval></annotation-xml><annotation encoding="application/x-tex" id="A6.F13.4.2.m2.2d">(\alpha_{m-1},\sigma_{m-1})</annotation><annotation encoding="application/x-llamapun" id="A6.F13.4.2.m2.2e">( italic_α start_POSTSUBSCRIPT italic_m - 1 end_POSTSUBSCRIPT , italic_σ start_POSTSUBSCRIPT italic_m - 1 end_POSTSUBSCRIPT )</annotation></semantics></math> are given for each restored image.</span></figcaption> </figure> <div class="ltx_para" id="A6.SS3.SSS0.Px2.p2"> <p class="ltx_p" id="A6.SS3.SSS0.Px2.p2.1">On Figure <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#A6.F13" title="Figure 13 ‣ More results ‣ F.3 Inpainting ‣ Appendix F Additional Experiments ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_tag">13</span></a>, we provide more results of SNORE algorithm on various inpainting problems (with of proportion <math alttext="p\geq 0.5" class="ltx_Math" display="inline" id="A6.SS3.SSS0.Px2.p2.1.m1.1"><semantics id="A6.SS3.SSS0.Px2.p2.1.m1.1a"><mrow id="A6.SS3.SSS0.Px2.p2.1.m1.1.1" xref="A6.SS3.SSS0.Px2.p2.1.m1.1.1.cmml"><mi id="A6.SS3.SSS0.Px2.p2.1.m1.1.1.2" xref="A6.SS3.SSS0.Px2.p2.1.m1.1.1.2.cmml">p</mi><mo id="A6.SS3.SSS0.Px2.p2.1.m1.1.1.1" xref="A6.SS3.SSS0.Px2.p2.1.m1.1.1.1.cmml">≥</mo><mn id="A6.SS3.SSS0.Px2.p2.1.m1.1.1.3" xref="A6.SS3.SSS0.Px2.p2.1.m1.1.1.3.cmml">0.5</mn></mrow><annotation-xml encoding="MathML-Content" id="A6.SS3.SSS0.Px2.p2.1.m1.1b"><apply id="A6.SS3.SSS0.Px2.p2.1.m1.1.1.cmml" xref="A6.SS3.SSS0.Px2.p2.1.m1.1.1"><geq id="A6.SS3.SSS0.Px2.p2.1.m1.1.1.1.cmml" xref="A6.SS3.SSS0.Px2.p2.1.m1.1.1.1"></geq><ci id="A6.SS3.SSS0.Px2.p2.1.m1.1.1.2.cmml" xref="A6.SS3.SSS0.Px2.p2.1.m1.1.1.2">𝑝</ci><cn id="A6.SS3.SSS0.Px2.p2.1.m1.1.1.3.cmml" type="float" xref="A6.SS3.SSS0.Px2.p2.1.m1.1.1.3">0.5</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A6.SS3.SSS0.Px2.p2.1.m1.1c">p\geq 0.5</annotation><annotation encoding="application/x-llamapun" id="A6.SS3.SSS0.Px2.p2.1.m1.1d">italic_p ≥ 0.5</annotation></semantics></math> of masked pixels). As expected, we observe that the quality of the restoration decrease with the proportion of missing pixels.</p> </div> <figure class="ltx_figure" id="A6.F14"><img alt="Refer to caption" class="ltx_graphics ltx_centering ltx_img_square" height="598" id="A6.F14.g1" src="extracted/5518834/images/various_results_despeckle.png" width="598"/> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure"><span class="ltx_text" id="A6.F14.4.2.1" style="font-size:90%;">Figure 14</span>: </span><span class="ltx_text" id="A6.F14.2.1" style="font-size:90%;">Image despeckling (<math alttext="L=1" class="ltx_Math" display="inline" id="A6.F14.2.1.m1.1"><semantics id="A6.F14.2.1.m1.1b"><mrow id="A6.F14.2.1.m1.1.1" xref="A6.F14.2.1.m1.1.1.cmml"><mi id="A6.F14.2.1.m1.1.1.2" xref="A6.F14.2.1.m1.1.1.2.cmml">L</mi><mo id="A6.F14.2.1.m1.1.1.1" xref="A6.F14.2.1.m1.1.1.1.cmml">=</mo><mn id="A6.F14.2.1.m1.1.1.3" xref="A6.F14.2.1.m1.1.1.3.cmml">1</mn></mrow><annotation-xml 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id="A6.SS4.p1.1.m1.1.1.3.2.cmml" xref="A6.SS4.p1.1.m1.1.1.3.2">ℝ</ci><ci id="A6.SS4.p1.1.m1.1.1.3.3.cmml" xref="A6.SS4.p1.1.m1.1.1.3.3">𝑞</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A6.SS4.p1.1.m1.1c">{\mathbf{y}}\in{\mathbb{R}}^{q}</annotation><annotation encoding="application/x-llamapun" id="A6.SS4.p1.1.m1.1d">bold_y ∈ blackboard_R start_POSTSUPERSCRIPT italic_q end_POSTSUPERSCRIPT</annotation></semantics></math> is the low-resolution version of <math alttext="{\mathbf{x}}\in{\mathbb{R}}^{d}" class="ltx_Math" display="inline" id="A6.SS4.p1.2.m2.1"><semantics id="A6.SS4.p1.2.m2.1a"><mrow id="A6.SS4.p1.2.m2.1.1" xref="A6.SS4.p1.2.m2.1.1.cmml"><mi id="A6.SS4.p1.2.m2.1.1.2" xref="A6.SS4.p1.2.m2.1.1.2.cmml">𝐱</mi><mo id="A6.SS4.p1.2.m2.1.1.1" xref="A6.SS4.p1.2.m2.1.1.1.cmml">∈</mo><msup id="A6.SS4.p1.2.m2.1.1.3" xref="A6.SS4.p1.2.m2.1.1.3.cmml"><mi id="A6.SS4.p1.2.m2.1.1.3.2" xref="A6.SS4.p1.2.m2.1.1.3.2.cmml">ℝ</mi><mi id="A6.SS4.p1.2.m2.1.1.3.3" xref="A6.SS4.p1.2.m2.1.1.3.3.cmml">d</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="A6.SS4.p1.2.m2.1b"><apply id="A6.SS4.p1.2.m2.1.1.cmml" xref="A6.SS4.p1.2.m2.1.1"><in id="A6.SS4.p1.2.m2.1.1.1.cmml" xref="A6.SS4.p1.2.m2.1.1.1"></in><ci id="A6.SS4.p1.2.m2.1.1.2.cmml" xref="A6.SS4.p1.2.m2.1.1.2">𝐱</ci><apply id="A6.SS4.p1.2.m2.1.1.3.cmml" xref="A6.SS4.p1.2.m2.1.1.3"><csymbol cd="ambiguous" id="A6.SS4.p1.2.m2.1.1.3.1.cmml" xref="A6.SS4.p1.2.m2.1.1.3">superscript</csymbol><ci id="A6.SS4.p1.2.m2.1.1.3.2.cmml" xref="A6.SS4.p1.2.m2.1.1.3.2">ℝ</ci><ci id="A6.SS4.p1.2.m2.1.1.3.3.cmml" xref="A6.SS4.p1.2.m2.1.1.3.3">𝑑</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A6.SS4.p1.2.m2.1c">{\mathbf{x}}\in{\mathbb{R}}^{d}</annotation><annotation encoding="application/x-llamapun" id="A6.SS4.p1.2.m2.1d">bold_x ∈ blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT</annotation></semantics></math>, obtained by <math alttext="{\mathbf{y}}={\mathbf{S}}{\mathbf{H}}{\mathbf{x}}+{\mathbf{n}}" class="ltx_Math" display="inline" id="A6.SS4.p1.3.m3.1"><semantics id="A6.SS4.p1.3.m3.1a"><mrow id="A6.SS4.p1.3.m3.1.1" xref="A6.SS4.p1.3.m3.1.1.cmml"><mi id="A6.SS4.p1.3.m3.1.1.2" xref="A6.SS4.p1.3.m3.1.1.2.cmml">𝐲</mi><mo id="A6.SS4.p1.3.m3.1.1.1" xref="A6.SS4.p1.3.m3.1.1.1.cmml">=</mo><mrow id="A6.SS4.p1.3.m3.1.1.3" xref="A6.SS4.p1.3.m3.1.1.3.cmml"><mi id="A6.SS4.p1.3.m3.1.1.3.2" xref="A6.SS4.p1.3.m3.1.1.3.2.cmml">𝐒𝐇𝐱</mi><mo id="A6.SS4.p1.3.m3.1.1.3.1" xref="A6.SS4.p1.3.m3.1.1.3.1.cmml">+</mo><mi id="A6.SS4.p1.3.m3.1.1.3.3" xref="A6.SS4.p1.3.m3.1.1.3.3.cmml">𝐧</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="A6.SS4.p1.3.m3.1b"><apply id="A6.SS4.p1.3.m3.1.1.cmml" xref="A6.SS4.p1.3.m3.1.1"><eq id="A6.SS4.p1.3.m3.1.1.1.cmml" xref="A6.SS4.p1.3.m3.1.1.1"></eq><ci id="A6.SS4.p1.3.m3.1.1.2.cmml" xref="A6.SS4.p1.3.m3.1.1.2">𝐲</ci><apply id="A6.SS4.p1.3.m3.1.1.3.cmml" xref="A6.SS4.p1.3.m3.1.1.3"><plus id="A6.SS4.p1.3.m3.1.1.3.1.cmml" xref="A6.SS4.p1.3.m3.1.1.3.1"></plus><ci id="A6.SS4.p1.3.m3.1.1.3.2.cmml" xref="A6.SS4.p1.3.m3.1.1.3.2">𝐒𝐇𝐱</ci><ci id="A6.SS4.p1.3.m3.1.1.3.3.cmml" xref="A6.SS4.p1.3.m3.1.1.3.3">𝐧</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A6.SS4.p1.3.m3.1c">{\mathbf{y}}={\mathbf{S}}{\mathbf{H}}{\mathbf{x}}+{\mathbf{n}}</annotation><annotation encoding="application/x-llamapun" id="A6.SS4.p1.3.m3.1d">bold_y = bold_SHx + bold_n</annotation></semantics></math>, where <math alttext="{\mathbf{H}}" class="ltx_Math" display="inline" id="A6.SS4.p1.4.m4.1"><semantics id="A6.SS4.p1.4.m4.1a"><mi id="A6.SS4.p1.4.m4.1.1" xref="A6.SS4.p1.4.m4.1.1.cmml">𝐇</mi><annotation-xml encoding="MathML-Content" id="A6.SS4.p1.4.m4.1b"><ci id="A6.SS4.p1.4.m4.1.1.cmml" xref="A6.SS4.p1.4.m4.1.1">𝐇</ci></annotation-xml><annotation encoding="application/x-tex" id="A6.SS4.p1.4.m4.1c">{\mathbf{H}}</annotation><annotation encoding="application/x-llamapun" id="A6.SS4.p1.4.m4.1d">bold_H</annotation></semantics></math> is an anti-aliasing blur kernel and <math alttext="{\mathbf{n}}\sim\mathcal{N}(0,\sigma_{{\mathbf{y}}})" class="ltx_Math" display="inline" id="A6.SS4.p1.5.m5.2"><semantics id="A6.SS4.p1.5.m5.2a"><mrow id="A6.SS4.p1.5.m5.2.2" xref="A6.SS4.p1.5.m5.2.2.cmml"><mi id="A6.SS4.p1.5.m5.2.2.3" xref="A6.SS4.p1.5.m5.2.2.3.cmml">𝐧</mi><mo id="A6.SS4.p1.5.m5.2.2.2" xref="A6.SS4.p1.5.m5.2.2.2.cmml">∼</mo><mrow id="A6.SS4.p1.5.m5.2.2.1" xref="A6.SS4.p1.5.m5.2.2.1.cmml"><mi class="ltx_font_mathcaligraphic" id="A6.SS4.p1.5.m5.2.2.1.3" xref="A6.SS4.p1.5.m5.2.2.1.3.cmml">𝒩</mi><mo id="A6.SS4.p1.5.m5.2.2.1.2" xref="A6.SS4.p1.5.m5.2.2.1.2.cmml">⁢</mo><mrow id="A6.SS4.p1.5.m5.2.2.1.1.1" xref="A6.SS4.p1.5.m5.2.2.1.1.2.cmml"><mo id="A6.SS4.p1.5.m5.2.2.1.1.1.2" stretchy="false" xref="A6.SS4.p1.5.m5.2.2.1.1.2.cmml">(</mo><mn id="A6.SS4.p1.5.m5.1.1" xref="A6.SS4.p1.5.m5.1.1.cmml">0</mn><mo id="A6.SS4.p1.5.m5.2.2.1.1.1.3" xref="A6.SS4.p1.5.m5.2.2.1.1.2.cmml">,</mo><msub id="A6.SS4.p1.5.m5.2.2.1.1.1.1" xref="A6.SS4.p1.5.m5.2.2.1.1.1.1.cmml"><mi id="A6.SS4.p1.5.m5.2.2.1.1.1.1.2" xref="A6.SS4.p1.5.m5.2.2.1.1.1.1.2.cmml">σ</mi><mi id="A6.SS4.p1.5.m5.2.2.1.1.1.1.3" xref="A6.SS4.p1.5.m5.2.2.1.1.1.1.3.cmml">𝐲</mi></msub><mo id="A6.SS4.p1.5.m5.2.2.1.1.1.4" stretchy="false" xref="A6.SS4.p1.5.m5.2.2.1.1.2.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="A6.SS4.p1.5.m5.2b"><apply id="A6.SS4.p1.5.m5.2.2.cmml" xref="A6.SS4.p1.5.m5.2.2"><csymbol cd="latexml" id="A6.SS4.p1.5.m5.2.2.2.cmml" xref="A6.SS4.p1.5.m5.2.2.2">similar-to</csymbol><ci id="A6.SS4.p1.5.m5.2.2.3.cmml" xref="A6.SS4.p1.5.m5.2.2.3">𝐧</ci><apply id="A6.SS4.p1.5.m5.2.2.1.cmml" xref="A6.SS4.p1.5.m5.2.2.1"><times id="A6.SS4.p1.5.m5.2.2.1.2.cmml" xref="A6.SS4.p1.5.m5.2.2.1.2"></times><ci id="A6.SS4.p1.5.m5.2.2.1.3.cmml" xref="A6.SS4.p1.5.m5.2.2.1.3">𝒩</ci><interval closure="open" id="A6.SS4.p1.5.m5.2.2.1.1.2.cmml" xref="A6.SS4.p1.5.m5.2.2.1.1.1"><cn id="A6.SS4.p1.5.m5.1.1.cmml" type="integer" xref="A6.SS4.p1.5.m5.1.1">0</cn><apply id="A6.SS4.p1.5.m5.2.2.1.1.1.1.cmml" xref="A6.SS4.p1.5.m5.2.2.1.1.1.1"><csymbol cd="ambiguous" id="A6.SS4.p1.5.m5.2.2.1.1.1.1.1.cmml" xref="A6.SS4.p1.5.m5.2.2.1.1.1.1">subscript</csymbol><ci id="A6.SS4.p1.5.m5.2.2.1.1.1.1.2.cmml" xref="A6.SS4.p1.5.m5.2.2.1.1.1.1.2">𝜎</ci><ci id="A6.SS4.p1.5.m5.2.2.1.1.1.1.3.cmml" xref="A6.SS4.p1.5.m5.2.2.1.1.1.1.3">𝐲</ci></apply></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A6.SS4.p1.5.m5.2c">{\mathbf{n}}\sim\mathcal{N}(0,\sigma_{{\mathbf{y}}})</annotation><annotation encoding="application/x-llamapun" id="A6.SS4.p1.5.m5.2d">bold_n ∼ caligraphic_N ( 0 , italic_σ start_POSTSUBSCRIPT bold_y end_POSTSUBSCRIPT )</annotation></semantics></math>. <math alttext="{\mathbf{S}}" class="ltx_Math" display="inline" id="A6.SS4.p1.6.m6.1"><semantics id="A6.SS4.p1.6.m6.1a"><mi id="A6.SS4.p1.6.m6.1.1" xref="A6.SS4.p1.6.m6.1.1.cmml">𝐒</mi><annotation-xml encoding="MathML-Content" id="A6.SS4.p1.6.m6.1b"><ci id="A6.SS4.p1.6.m6.1.1.cmml" xref="A6.SS4.p1.6.m6.1.1">𝐒</ci></annotation-xml><annotation encoding="application/x-tex" id="A6.SS4.p1.6.m6.1c">{\mathbf{S}}</annotation><annotation encoding="application/x-llamapun" id="A6.SS4.p1.6.m6.1d">bold_S</annotation></semantics></math> is the standard down-sampling matrix with the super-resolution factor <math alttext="s_{f}" class="ltx_Math" display="inline" id="A6.SS4.p1.7.m7.1"><semantics id="A6.SS4.p1.7.m7.1a"><msub id="A6.SS4.p1.7.m7.1.1" xref="A6.SS4.p1.7.m7.1.1.cmml"><mi id="A6.SS4.p1.7.m7.1.1.2" xref="A6.SS4.p1.7.m7.1.1.2.cmml">s</mi><mi id="A6.SS4.p1.7.m7.1.1.3" xref="A6.SS4.p1.7.m7.1.1.3.cmml">f</mi></msub><annotation-xml encoding="MathML-Content" id="A6.SS4.p1.7.m7.1b"><apply id="A6.SS4.p1.7.m7.1.1.cmml" xref="A6.SS4.p1.7.m7.1.1"><csymbol cd="ambiguous" id="A6.SS4.p1.7.m7.1.1.1.cmml" xref="A6.SS4.p1.7.m7.1.1">subscript</csymbol><ci id="A6.SS4.p1.7.m7.1.1.2.cmml" xref="A6.SS4.p1.7.m7.1.1.2">𝑠</ci><ci id="A6.SS4.p1.7.m7.1.1.3.cmml" xref="A6.SS4.p1.7.m7.1.1.3">𝑓</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A6.SS4.p1.7.m7.1c">s_{f}</annotation><annotation encoding="application/x-llamapun" id="A6.SS4.p1.7.m7.1d">italic_s start_POSTSUBSCRIPT italic_f end_POSTSUBSCRIPT</annotation></semantics></math>. The data-fidelity is given by <math alttext="\mathcal{F}({\mathbf{x}},{\mathbf{y}})=\frac{1}{2\sigma_{{\mathbf{y}}}}\|{% \mathbf{S}}{\mathbf{H}}{\mathbf{x}}-{\mathbf{y}}\|^{2}" class="ltx_Math" display="inline" id="A6.SS4.p1.8.m8.3"><semantics id="A6.SS4.p1.8.m8.3a"><mrow id="A6.SS4.p1.8.m8.3.3" xref="A6.SS4.p1.8.m8.3.3.cmml"><mrow id="A6.SS4.p1.8.m8.3.3.3" xref="A6.SS4.p1.8.m8.3.3.3.cmml"><mi class="ltx_font_mathcaligraphic" id="A6.SS4.p1.8.m8.3.3.3.2" xref="A6.SS4.p1.8.m8.3.3.3.2.cmml">ℱ</mi><mo id="A6.SS4.p1.8.m8.3.3.3.1" xref="A6.SS4.p1.8.m8.3.3.3.1.cmml">⁢</mo><mrow id="A6.SS4.p1.8.m8.3.3.3.3.2" xref="A6.SS4.p1.8.m8.3.3.3.3.1.cmml"><mo id="A6.SS4.p1.8.m8.3.3.3.3.2.1" stretchy="false" xref="A6.SS4.p1.8.m8.3.3.3.3.1.cmml">(</mo><mi id="A6.SS4.p1.8.m8.1.1" xref="A6.SS4.p1.8.m8.1.1.cmml">𝐱</mi><mo id="A6.SS4.p1.8.m8.3.3.3.3.2.2" xref="A6.SS4.p1.8.m8.3.3.3.3.1.cmml">,</mo><mi id="A6.SS4.p1.8.m8.2.2" xref="A6.SS4.p1.8.m8.2.2.cmml">𝐲</mi><mo id="A6.SS4.p1.8.m8.3.3.3.3.2.3" stretchy="false" xref="A6.SS4.p1.8.m8.3.3.3.3.1.cmml">)</mo></mrow></mrow><mo id="A6.SS4.p1.8.m8.3.3.2" xref="A6.SS4.p1.8.m8.3.3.2.cmml">=</mo><mrow id="A6.SS4.p1.8.m8.3.3.1" xref="A6.SS4.p1.8.m8.3.3.1.cmml"><mfrac id="A6.SS4.p1.8.m8.3.3.1.3" xref="A6.SS4.p1.8.m8.3.3.1.3.cmml"><mn id="A6.SS4.p1.8.m8.3.3.1.3.2" xref="A6.SS4.p1.8.m8.3.3.1.3.2.cmml">1</mn><mrow id="A6.SS4.p1.8.m8.3.3.1.3.3" xref="A6.SS4.p1.8.m8.3.3.1.3.3.cmml"><mn id="A6.SS4.p1.8.m8.3.3.1.3.3.2" xref="A6.SS4.p1.8.m8.3.3.1.3.3.2.cmml">2</mn><mo id="A6.SS4.p1.8.m8.3.3.1.3.3.1" xref="A6.SS4.p1.8.m8.3.3.1.3.3.1.cmml">⁢</mo><msub id="A6.SS4.p1.8.m8.3.3.1.3.3.3" xref="A6.SS4.p1.8.m8.3.3.1.3.3.3.cmml"><mi id="A6.SS4.p1.8.m8.3.3.1.3.3.3.2" xref="A6.SS4.p1.8.m8.3.3.1.3.3.3.2.cmml">σ</mi><mi id="A6.SS4.p1.8.m8.3.3.1.3.3.3.3" xref="A6.SS4.p1.8.m8.3.3.1.3.3.3.3.cmml">𝐲</mi></msub></mrow></mfrac><mo id="A6.SS4.p1.8.m8.3.3.1.2" xref="A6.SS4.p1.8.m8.3.3.1.2.cmml">⁢</mo><msup id="A6.SS4.p1.8.m8.3.3.1.1" xref="A6.SS4.p1.8.m8.3.3.1.1.cmml"><mrow id="A6.SS4.p1.8.m8.3.3.1.1.1.1" xref="A6.SS4.p1.8.m8.3.3.1.1.1.2.cmml"><mo id="A6.SS4.p1.8.m8.3.3.1.1.1.1.2" stretchy="false" xref="A6.SS4.p1.8.m8.3.3.1.1.1.2.1.cmml">‖</mo><mrow id="A6.SS4.p1.8.m8.3.3.1.1.1.1.1" xref="A6.SS4.p1.8.m8.3.3.1.1.1.1.1.cmml"><mi id="A6.SS4.p1.8.m8.3.3.1.1.1.1.1.2" xref="A6.SS4.p1.8.m8.3.3.1.1.1.1.1.2.cmml">𝐒𝐇𝐱</mi><mo id="A6.SS4.p1.8.m8.3.3.1.1.1.1.1.1" xref="A6.SS4.p1.8.m8.3.3.1.1.1.1.1.1.cmml">−</mo><mi id="A6.SS4.p1.8.m8.3.3.1.1.1.1.1.3" xref="A6.SS4.p1.8.m8.3.3.1.1.1.1.1.3.cmml">𝐲</mi></mrow><mo id="A6.SS4.p1.8.m8.3.3.1.1.1.1.3" stretchy="false" xref="A6.SS4.p1.8.m8.3.3.1.1.1.2.1.cmml">‖</mo></mrow><mn id="A6.SS4.p1.8.m8.3.3.1.1.3" xref="A6.SS4.p1.8.m8.3.3.1.1.3.cmml">2</mn></msup></mrow></mrow><annotation-xml encoding="MathML-Content" id="A6.SS4.p1.8.m8.3b"><apply id="A6.SS4.p1.8.m8.3.3.cmml" xref="A6.SS4.p1.8.m8.3.3"><eq id="A6.SS4.p1.8.m8.3.3.2.cmml" xref="A6.SS4.p1.8.m8.3.3.2"></eq><apply id="A6.SS4.p1.8.m8.3.3.3.cmml" xref="A6.SS4.p1.8.m8.3.3.3"><times id="A6.SS4.p1.8.m8.3.3.3.1.cmml" xref="A6.SS4.p1.8.m8.3.3.3.1"></times><ci id="A6.SS4.p1.8.m8.3.3.3.2.cmml" xref="A6.SS4.p1.8.m8.3.3.3.2">ℱ</ci><interval closure="open" id="A6.SS4.p1.8.m8.3.3.3.3.1.cmml" xref="A6.SS4.p1.8.m8.3.3.3.3.2"><ci id="A6.SS4.p1.8.m8.1.1.cmml" xref="A6.SS4.p1.8.m8.1.1">𝐱</ci><ci id="A6.SS4.p1.8.m8.2.2.cmml" xref="A6.SS4.p1.8.m8.2.2">𝐲</ci></interval></apply><apply id="A6.SS4.p1.8.m8.3.3.1.cmml" xref="A6.SS4.p1.8.m8.3.3.1"><times id="A6.SS4.p1.8.m8.3.3.1.2.cmml" xref="A6.SS4.p1.8.m8.3.3.1.2"></times><apply id="A6.SS4.p1.8.m8.3.3.1.3.cmml" xref="A6.SS4.p1.8.m8.3.3.1.3"><divide id="A6.SS4.p1.8.m8.3.3.1.3.1.cmml" xref="A6.SS4.p1.8.m8.3.3.1.3"></divide><cn id="A6.SS4.p1.8.m8.3.3.1.3.2.cmml" type="integer" xref="A6.SS4.p1.8.m8.3.3.1.3.2">1</cn><apply id="A6.SS4.p1.8.m8.3.3.1.3.3.cmml" xref="A6.SS4.p1.8.m8.3.3.1.3.3"><times id="A6.SS4.p1.8.m8.3.3.1.3.3.1.cmml" xref="A6.SS4.p1.8.m8.3.3.1.3.3.1"></times><cn id="A6.SS4.p1.8.m8.3.3.1.3.3.2.cmml" type="integer" xref="A6.SS4.p1.8.m8.3.3.1.3.3.2">2</cn><apply id="A6.SS4.p1.8.m8.3.3.1.3.3.3.cmml" xref="A6.SS4.p1.8.m8.3.3.1.3.3.3"><csymbol cd="ambiguous" id="A6.SS4.p1.8.m8.3.3.1.3.3.3.1.cmml" xref="A6.SS4.p1.8.m8.3.3.1.3.3.3">subscript</csymbol><ci id="A6.SS4.p1.8.m8.3.3.1.3.3.3.2.cmml" xref="A6.SS4.p1.8.m8.3.3.1.3.3.3.2">𝜎</ci><ci id="A6.SS4.p1.8.m8.3.3.1.3.3.3.3.cmml" xref="A6.SS4.p1.8.m8.3.3.1.3.3.3.3">𝐲</ci></apply></apply></apply><apply id="A6.SS4.p1.8.m8.3.3.1.1.cmml" xref="A6.SS4.p1.8.m8.3.3.1.1"><csymbol cd="ambiguous" id="A6.SS4.p1.8.m8.3.3.1.1.2.cmml" xref="A6.SS4.p1.8.m8.3.3.1.1">superscript</csymbol><apply id="A6.SS4.p1.8.m8.3.3.1.1.1.2.cmml" xref="A6.SS4.p1.8.m8.3.3.1.1.1.1"><csymbol cd="latexml" id="A6.SS4.p1.8.m8.3.3.1.1.1.2.1.cmml" xref="A6.SS4.p1.8.m8.3.3.1.1.1.1.2">norm</csymbol><apply id="A6.SS4.p1.8.m8.3.3.1.1.1.1.1.cmml" xref="A6.SS4.p1.8.m8.3.3.1.1.1.1.1"><minus id="A6.SS4.p1.8.m8.3.3.1.1.1.1.1.1.cmml" xref="A6.SS4.p1.8.m8.3.3.1.1.1.1.1.1"></minus><ci id="A6.SS4.p1.8.m8.3.3.1.1.1.1.1.2.cmml" xref="A6.SS4.p1.8.m8.3.3.1.1.1.1.1.2">𝐒𝐇𝐱</ci><ci id="A6.SS4.p1.8.m8.3.3.1.1.1.1.1.3.cmml" xref="A6.SS4.p1.8.m8.3.3.1.1.1.1.1.3">𝐲</ci></apply></apply><cn id="A6.SS4.p1.8.m8.3.3.1.1.3.cmml" type="integer" xref="A6.SS4.p1.8.m8.3.3.1.1.3">2</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A6.SS4.p1.8.m8.3c">\mathcal{F}({\mathbf{x}},{\mathbf{y}})=\frac{1}{2\sigma_{{\mathbf{y}}}}\|{% \mathbf{S}}{\mathbf{H}}{\mathbf{x}}-{\mathbf{y}}\|^{2}</annotation><annotation encoding="application/x-llamapun" id="A6.SS4.p1.8.m8.3d">caligraphic_F ( bold_x , bold_y ) = divide start_ARG 1 end_ARG start_ARG 2 italic_σ start_POSTSUBSCRIPT bold_y end_POSTSUBSCRIPT end_ARG ∥ bold_SHx - bold_y ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT</annotation></semantics></math> and its proximal operator <cite class="ltx_cite ltx_citemacro_citep">(Zhao et al., <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#bib.bib79" title="">2016</a>)</cite> by <math alttext="\text{Prox}_{\delta\mathcal{F}}({\mathbf{z}})=\hat{{\mathbf{z}}}-\frac{1}{s_{f% }^{2}}{\mathbf{F}}^{\star}\bar{{\mathbf{\Lambda}}}^{\star}\left(\mathbf{I}_{q}% +\frac{\delta}{s_{f}^{2}}\bar{{\mathbf{\Lambda}}}\bar{{\mathbf{\Lambda}}}^{% \star}\right)^{-1}\bar{{\mathbf{\Lambda}}}{\mathbf{F}}\hat{{\mathbf{z}}}" class="ltx_Math" display="inline" id="A6.SS4.p1.9.m9.2"><semantics id="A6.SS4.p1.9.m9.2a"><mrow id="A6.SS4.p1.9.m9.2.2" xref="A6.SS4.p1.9.m9.2.2.cmml"><mrow id="A6.SS4.p1.9.m9.2.2.3" xref="A6.SS4.p1.9.m9.2.2.3.cmml"><msub id="A6.SS4.p1.9.m9.2.2.3.2" xref="A6.SS4.p1.9.m9.2.2.3.2.cmml"><mtext id="A6.SS4.p1.9.m9.2.2.3.2.2" xref="A6.SS4.p1.9.m9.2.2.3.2.2a.cmml">Prox</mtext><mrow id="A6.SS4.p1.9.m9.2.2.3.2.3" xref="A6.SS4.p1.9.m9.2.2.3.2.3.cmml"><mi id="A6.SS4.p1.9.m9.2.2.3.2.3.2" xref="A6.SS4.p1.9.m9.2.2.3.2.3.2.cmml">δ</mi><mo 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encoding="application/x-tex" id="A6.SS4.p1.9.m9.2c">\text{Prox}_{\delta\mathcal{F}}({\mathbf{z}})=\hat{{\mathbf{z}}}-\frac{1}{s_{f% }^{2}}{\mathbf{F}}^{\star}\bar{{\mathbf{\Lambda}}}^{\star}\left(\mathbf{I}_{q}% +\frac{\delta}{s_{f}^{2}}\bar{{\mathbf{\Lambda}}}\bar{{\mathbf{\Lambda}}}^{% \star}\right)^{-1}\bar{{\mathbf{\Lambda}}}{\mathbf{F}}\hat{{\mathbf{z}}}</annotation><annotation encoding="application/x-llamapun" id="A6.SS4.p1.9.m9.2d">Prox start_POSTSUBSCRIPT italic_δ caligraphic_F end_POSTSUBSCRIPT ( bold_z ) = over^ start_ARG bold_z end_ARG - divide start_ARG 1 end_ARG start_ARG italic_s start_POSTSUBSCRIPT italic_f end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG bold_F start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT over¯ start_ARG bold_Λ end_ARG start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT ( bold_I start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT + divide start_ARG italic_δ end_ARG start_ARG italic_s start_POSTSUBSCRIPT italic_f end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG over¯ start_ARG bold_Λ end_ARG over¯ start_ARG bold_Λ end_ARG start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT over¯ start_ARG bold_Λ end_ARG bold_F over^ start_ARG bold_z end_ARG</annotation></semantics></math>, where <math alttext="\hat{{\mathbf{z}}}=\delta{\mathbf{H}}^{T}{\mathbf{S}}^{T}{\mathbf{y}}+{\mathbf% {z}}" class="ltx_Math" display="inline" id="A6.SS4.p1.10.m10.1"><semantics id="A6.SS4.p1.10.m10.1a"><mrow id="A6.SS4.p1.10.m10.1.1" xref="A6.SS4.p1.10.m10.1.1.cmml"><mover accent="true" id="A6.SS4.p1.10.m10.1.1.2" xref="A6.SS4.p1.10.m10.1.1.2.cmml"><mi id="A6.SS4.p1.10.m10.1.1.2.2" xref="A6.SS4.p1.10.m10.1.1.2.2.cmml">𝐳</mi><mo id="A6.SS4.p1.10.m10.1.1.2.1" xref="A6.SS4.p1.10.m10.1.1.2.1.cmml">^</mo></mover><mo id="A6.SS4.p1.10.m10.1.1.1" xref="A6.SS4.p1.10.m10.1.1.1.cmml">=</mo><mrow id="A6.SS4.p1.10.m10.1.1.3" xref="A6.SS4.p1.10.m10.1.1.3.cmml"><mrow id="A6.SS4.p1.10.m10.1.1.3.2" 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id="A6.SS4.p1.10.m10.1c">\hat{{\mathbf{z}}}=\delta{\mathbf{H}}^{T}{\mathbf{S}}^{T}{\mathbf{y}}+{\mathbf% {z}}</annotation><annotation encoding="application/x-llamapun" id="A6.SS4.p1.10.m10.1d">over^ start_ARG bold_z end_ARG = italic_δ bold_H start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT bold_S start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT bold_y + bold_z</annotation></semantics></math> and <math alttext="\bar{{\mathbf{\Lambda}}}=\left({\mathbf{\Lambda}}_{1},\dots\mathbf{\Lambda}_{s% _{f}^{2}}\right)\in{\mathbb{R}}^{q\times d}" class="ltx_Math" display="inline" id="A6.SS4.p1.11.m11.2"><semantics id="A6.SS4.p1.11.m11.2a"><mrow id="A6.SS4.p1.11.m11.2.2" xref="A6.SS4.p1.11.m11.2.2.cmml"><mover accent="true" id="A6.SS4.p1.11.m11.2.2.4" xref="A6.SS4.p1.11.m11.2.2.4.cmml"><mi id="A6.SS4.p1.11.m11.2.2.4.2" xref="A6.SS4.p1.11.m11.2.2.4.2.cmml">𝚲</mi><mo id="A6.SS4.p1.11.m11.2.2.4.1" xref="A6.SS4.p1.11.m11.2.2.4.1.cmml">¯</mo></mover><mo id="A6.SS4.p1.11.m11.2.2.5" xref="A6.SS4.p1.11.m11.2.2.5.cmml">=</mo><mrow id="A6.SS4.p1.11.m11.2.2.2.2" xref="A6.SS4.p1.11.m11.2.2.2.3.cmml"><mo id="A6.SS4.p1.11.m11.2.2.2.2.3" xref="A6.SS4.p1.11.m11.2.2.2.3.cmml">(</mo><msub id="A6.SS4.p1.11.m11.1.1.1.1.1" xref="A6.SS4.p1.11.m11.1.1.1.1.1.cmml"><mi id="A6.SS4.p1.11.m11.1.1.1.1.1.2" xref="A6.SS4.p1.11.m11.1.1.1.1.1.2.cmml">𝚲</mi><mn id="A6.SS4.p1.11.m11.1.1.1.1.1.3" xref="A6.SS4.p1.11.m11.1.1.1.1.1.3.cmml">1</mn></msub><mo id="A6.SS4.p1.11.m11.2.2.2.2.4" xref="A6.SS4.p1.11.m11.2.2.2.3.cmml">,</mo><mrow id="A6.SS4.p1.11.m11.2.2.2.2.2" xref="A6.SS4.p1.11.m11.2.2.2.2.2.cmml"><mi id="A6.SS4.p1.11.m11.2.2.2.2.2.2" mathvariant="normal" xref="A6.SS4.p1.11.m11.2.2.2.2.2.2.cmml">…</mi><mo id="A6.SS4.p1.11.m11.2.2.2.2.2.1" xref="A6.SS4.p1.11.m11.2.2.2.2.2.1.cmml">⁢</mo><msub id="A6.SS4.p1.11.m11.2.2.2.2.2.3" xref="A6.SS4.p1.11.m11.2.2.2.2.2.3.cmml"><mi id="A6.SS4.p1.11.m11.2.2.2.2.2.3.2" xref="A6.SS4.p1.11.m11.2.2.2.2.2.3.2.cmml">𝚲</mi><msubsup 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encoding="application/x-tex" id="A6.SS4.p1.13.m13.1c">s_{f}\times s_{f}</annotation><annotation encoding="application/x-llamapun" id="A6.SS4.p1.13.m13.1d">italic_s start_POSTSUBSCRIPT italic_f end_POSTSUBSCRIPT × italic_s start_POSTSUBSCRIPT italic_f end_POSTSUBSCRIPT</annotation></semantics></math> paving of the Fourier domain.</p> </div> <div class="ltx_para" id="A6.SS4.p2"> <p class="ltx_p" id="A6.SS4.p2.1">Thanks to the previous expression of the Proximal operator, SNORE Prox and RED Prox can be computed for super-resolution. On Figure <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#A6.F15" title="Figure 15 ‣ F.4 Super-resolution ‣ Appendix F Additional Experiments ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_tag">15</span></a>, we give qualitative results for image super-resolutions on a kernel of blur and various images. Note that SNORE Prox produce better perceptual results (LPIPS, BRISQUE) by creating local texture. However, SNORE Prox is worst in distortion metrics (PSNR, SSIM) as for image deblurring. These are preliminaries experiments and we leave for futur work to adapt each methods for image super-resolution and make a quantitative evaluation.</p> </div> <figure class="ltx_figure" id="A6.F15"><img alt="Refer to caption" class="ltx_graphics ltx_centering ltx_img_square" height="605" id="A6.F15.g1" src="extracted/5518834/images/various_results_SR.png" width="598"/> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure"><span class="ltx_text" id="A6.F15.6.3.1" style="font-size:90%;">Figure 15</span>: </span><span class="ltx_text" id="A6.F15.4.2" style="font-size:90%;">Image super-resolution obtained with SNORE Prox and RED Prox, with a super-resolution factor <math alttext="s_{f}=2" class="ltx_Math" display="inline" id="A6.F15.3.1.m1.1"><semantics id="A6.F15.3.1.m1.1b"><mrow id="A6.F15.3.1.m1.1.1" xref="A6.F15.3.1.m1.1.1.cmml"><msub id="A6.F15.3.1.m1.1.1.2" xref="A6.F15.3.1.m1.1.1.2.cmml"><mi id="A6.F15.3.1.m1.1.1.2.2" xref="A6.F15.3.1.m1.1.1.2.2.cmml">s</mi><mi id="A6.F15.3.1.m1.1.1.2.3" xref="A6.F15.3.1.m1.1.1.2.3.cmml">f</mi></msub><mo id="A6.F15.3.1.m1.1.1.1" xref="A6.F15.3.1.m1.1.1.1.cmml">=</mo><mn id="A6.F15.3.1.m1.1.1.3" xref="A6.F15.3.1.m1.1.1.3.cmml">2</mn></mrow><annotation-xml encoding="MathML-Content" id="A6.F15.3.1.m1.1c"><apply id="A6.F15.3.1.m1.1.1.cmml" xref="A6.F15.3.1.m1.1.1"><eq id="A6.F15.3.1.m1.1.1.1.cmml" xref="A6.F15.3.1.m1.1.1.1"></eq><apply id="A6.F15.3.1.m1.1.1.2.cmml" xref="A6.F15.3.1.m1.1.1.2"><csymbol cd="ambiguous" id="A6.F15.3.1.m1.1.1.2.1.cmml" xref="A6.F15.3.1.m1.1.1.2">subscript</csymbol><ci id="A6.F15.3.1.m1.1.1.2.2.cmml" xref="A6.F15.3.1.m1.1.1.2.2">𝑠</ci><ci id="A6.F15.3.1.m1.1.1.2.3.cmml" xref="A6.F15.3.1.m1.1.1.2.3">𝑓</ci></apply><cn id="A6.F15.3.1.m1.1.1.3.cmml" type="integer" xref="A6.F15.3.1.m1.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A6.F15.3.1.m1.1d">s_{f}=2</annotation><annotation encoding="application/x-llamapun" id="A6.F15.3.1.m1.1e">italic_s start_POSTSUBSCRIPT italic_f end_POSTSUBSCRIPT = 2</annotation></semantics></math> and an input noise <math alttext="\sigma_{{\mathbf{y}}}=5/255" class="ltx_Math" display="inline" id="A6.F15.4.2.m2.1"><semantics id="A6.F15.4.2.m2.1b"><mrow id="A6.F15.4.2.m2.1.1" xref="A6.F15.4.2.m2.1.1.cmml"><msub id="A6.F15.4.2.m2.1.1.2" xref="A6.F15.4.2.m2.1.1.2.cmml"><mi id="A6.F15.4.2.m2.1.1.2.2" xref="A6.F15.4.2.m2.1.1.2.2.cmml">σ</mi><mi id="A6.F15.4.2.m2.1.1.2.3" xref="A6.F15.4.2.m2.1.1.2.3.cmml">𝐲</mi></msub><mo id="A6.F15.4.2.m2.1.1.1" xref="A6.F15.4.2.m2.1.1.1.cmml">=</mo><mrow id="A6.F15.4.2.m2.1.1.3" xref="A6.F15.4.2.m2.1.1.3.cmml"><mn id="A6.F15.4.2.m2.1.1.3.2" xref="A6.F15.4.2.m2.1.1.3.2.cmml">5</mn><mo id="A6.F15.4.2.m2.1.1.3.1" xref="A6.F15.4.2.m2.1.1.3.1.cmml">/</mo><mn id="A6.F15.4.2.m2.1.1.3.3" xref="A6.F15.4.2.m2.1.1.3.3.cmml">255</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="A6.F15.4.2.m2.1c"><apply id="A6.F15.4.2.m2.1.1.cmml" xref="A6.F15.4.2.m2.1.1"><eq id="A6.F15.4.2.m2.1.1.1.cmml" xref="A6.F15.4.2.m2.1.1.1"></eq><apply id="A6.F15.4.2.m2.1.1.2.cmml" xref="A6.F15.4.2.m2.1.1.2"><csymbol cd="ambiguous" id="A6.F15.4.2.m2.1.1.2.1.cmml" xref="A6.F15.4.2.m2.1.1.2">subscript</csymbol><ci id="A6.F15.4.2.m2.1.1.2.2.cmml" xref="A6.F15.4.2.m2.1.1.2.2">𝜎</ci><ci id="A6.F15.4.2.m2.1.1.2.3.cmml" xref="A6.F15.4.2.m2.1.1.2.3">𝐲</ci></apply><apply id="A6.F15.4.2.m2.1.1.3.cmml" xref="A6.F15.4.2.m2.1.1.3"><divide id="A6.F15.4.2.m2.1.1.3.1.cmml" xref="A6.F15.4.2.m2.1.1.3.1"></divide><cn id="A6.F15.4.2.m2.1.1.3.2.cmml" type="integer" xref="A6.F15.4.2.m2.1.1.3.2">5</cn><cn id="A6.F15.4.2.m2.1.1.3.3.cmml" type="integer" xref="A6.F15.4.2.m2.1.1.3.3">255</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A6.F15.4.2.m2.1d">\sigma_{{\mathbf{y}}}=5/255</annotation><annotation encoding="application/x-llamapun" id="A6.F15.4.2.m2.1e">italic_σ start_POSTSUBSCRIPT bold_y end_POSTSUBSCRIPT = 5 / 255</annotation></semantics></math>, on various images from the dataset CBSD68.</span></figcaption> </figure> <section class="ltx_paragraph" id="A6.SS4.SSS0.Px1"> <h5 class="ltx_title ltx_title_paragraph">Parameters setting</h5> <div class="ltx_para" id="A6.SS4.SSS0.Px1.p1"> <p class="ltx_p" id="A6.SS4.SSS0.Px1.p1.1">On Table <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#A6.T6" title="Table 6 ‣ Parameters setting ‣ F.4 Super-resolution ‣ Appendix F Additional Experiments ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_tag">6</span></a>, we details the practical choice of parameters we made. Note that the number of iterations of the algorithm are the same for image super-resolution.</p> </div> <figure class="ltx_table" id="A6.T6"> <div class="ltx_inline-block ltx_align_center ltx_transformed_outer" id="A6.T6.9" style="width:195.1pt;height:176.8pt;vertical-align:-0.0pt;"><span class="ltx_transformed_inner" style="transform:translate(8.2pt,-7.4pt) scale(1.09128191342972,1.09128191342972) ;"> <table class="ltx_tabular ltx_guessed_headers ltx_align_middle" id="A6.T6.9.9"> <thead class="ltx_thead"> <tr class="ltx_tr" id="A6.T6.9.9.10.1"> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_th_row ltx_border_l ltx_border_rr ltx_border_t" id="A6.T6.9.9.10.1.1">Parameters</th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_r ltx_border_t" id="A6.T6.9.9.10.1.2">RED Prox</th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_r ltx_border_t" id="A6.T6.9.9.10.1.3">SNORE Prox</th> </tr> </thead> <tbody class="ltx_tbody"> <tr class="ltx_tr" id="A6.T6.1.1.1"> <th class="ltx_td ltx_align_center ltx_th ltx_th_row ltx_border_l ltx_border_rr ltx_border_t" id="A6.T6.1.1.1.1"><math alttext="\alpha" class="ltx_Math" display="inline" id="A6.T6.1.1.1.1.m1.1"><semantics id="A6.T6.1.1.1.1.m1.1a"><mi id="A6.T6.1.1.1.1.m1.1.1" xref="A6.T6.1.1.1.1.m1.1.1.cmml">α</mi><annotation-xml encoding="MathML-Content" id="A6.T6.1.1.1.1.m1.1b"><ci id="A6.T6.1.1.1.1.m1.1.1.cmml" xref="A6.T6.1.1.1.1.m1.1.1">𝛼</ci></annotation-xml><annotation encoding="application/x-tex" id="A6.T6.1.1.1.1.m1.1c">\alpha</annotation><annotation encoding="application/x-llamapun" id="A6.T6.1.1.1.1.m1.1d">italic_α</annotation></semantics></math></th> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="A6.T6.1.1.1.2">0.065</td> <td class="ltx_td ltx_border_r ltx_border_t" id="A6.T6.1.1.1.3"></td> </tr> <tr class="ltx_tr" id="A6.T6.2.2.2"> <th class="ltx_td ltx_align_center ltx_th ltx_th_row ltx_border_l ltx_border_rr ltx_border_t" id="A6.T6.2.2.2.1"><math alttext="\alpha_{0}" class="ltx_Math" display="inline" id="A6.T6.2.2.2.1.m1.1"><semantics id="A6.T6.2.2.2.1.m1.1a"><msub id="A6.T6.2.2.2.1.m1.1.1" xref="A6.T6.2.2.2.1.m1.1.1.cmml"><mi id="A6.T6.2.2.2.1.m1.1.1.2" xref="A6.T6.2.2.2.1.m1.1.1.2.cmml">α</mi><mn id="A6.T6.2.2.2.1.m1.1.1.3" xref="A6.T6.2.2.2.1.m1.1.1.3.cmml">0</mn></msub><annotation-xml encoding="MathML-Content" id="A6.T6.2.2.2.1.m1.1b"><apply id="A6.T6.2.2.2.1.m1.1.1.cmml" xref="A6.T6.2.2.2.1.m1.1.1"><csymbol cd="ambiguous" id="A6.T6.2.2.2.1.m1.1.1.1.cmml" xref="A6.T6.2.2.2.1.m1.1.1">subscript</csymbol><ci id="A6.T6.2.2.2.1.m1.1.1.2.cmml" xref="A6.T6.2.2.2.1.m1.1.1.2">𝛼</ci><cn id="A6.T6.2.2.2.1.m1.1.1.3.cmml" type="integer" xref="A6.T6.2.2.2.1.m1.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A6.T6.2.2.2.1.m1.1c">\alpha_{0}</annotation><annotation encoding="application/x-llamapun" id="A6.T6.2.2.2.1.m1.1d">italic_α start_POSTSUBSCRIPT 0 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id="A6.T6.6.6.6.2"></td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="A6.T6.6.6.6.3">2</td> </tr> <tr class="ltx_tr" id="A6.T6.7.7.7"> <th class="ltx_td ltx_align_center ltx_th ltx_th_row ltx_border_l ltx_border_rr ltx_border_t" id="A6.T6.7.7.7.1"><math alttext="\text{max}_{\text{itr}}" class="ltx_Math" display="inline" id="A6.T6.7.7.7.1.m1.1"><semantics id="A6.T6.7.7.7.1.m1.1a"><msub id="A6.T6.7.7.7.1.m1.1.1" xref="A6.T6.7.7.7.1.m1.1.1.cmml"><mtext id="A6.T6.7.7.7.1.m1.1.1.2" xref="A6.T6.7.7.7.1.m1.1.1.2a.cmml">max</mtext><mtext id="A6.T6.7.7.7.1.m1.1.1.3" xref="A6.T6.7.7.7.1.m1.1.1.3a.cmml">itr</mtext></msub><annotation-xml encoding="MathML-Content" id="A6.T6.7.7.7.1.m1.1b"><apply id="A6.T6.7.7.7.1.m1.1.1.cmml" xref="A6.T6.7.7.7.1.m1.1.1"><csymbol cd="ambiguous" id="A6.T6.7.7.7.1.m1.1.1.1.cmml" xref="A6.T6.7.7.7.1.m1.1.1">subscript</csymbol><ci id="A6.T6.7.7.7.1.m1.1.1.2a.cmml" xref="A6.T6.7.7.7.1.m1.1.1.2"><mtext id="A6.T6.7.7.7.1.m1.1.1.2.cmml" 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xref="A6.T6.8.8.8.1.m1.1.1.cmml">δ</mi><annotation-xml encoding="MathML-Content" id="A6.T6.8.8.8.1.m1.1b"><ci id="A6.T6.8.8.8.1.m1.1.1.cmml" xref="A6.T6.8.8.8.1.m1.1.1">𝛿</ci></annotation-xml><annotation encoding="application/x-tex" id="A6.T6.8.8.8.1.m1.1c">\delta</annotation><annotation encoding="application/x-llamapun" id="A6.T6.8.8.8.1.m1.1d">italic_δ</annotation></semantics></math> (initialization)</th> <td class="ltx_td ltx_align_center ltx_border_b ltx_border_r ltx_border_t" id="A6.T6.9.9.9.2"><math alttext="1/\alpha" class="ltx_Math" display="inline" id="A6.T6.9.9.9.2.m1.1"><semantics id="A6.T6.9.9.9.2.m1.1a"><mrow id="A6.T6.9.9.9.2.m1.1.1" xref="A6.T6.9.9.9.2.m1.1.1.cmml"><mn id="A6.T6.9.9.9.2.m1.1.1.2" xref="A6.T6.9.9.9.2.m1.1.1.2.cmml">1</mn><mo id="A6.T6.9.9.9.2.m1.1.1.1" xref="A6.T6.9.9.9.2.m1.1.1.1.cmml">/</mo><mi id="A6.T6.9.9.9.2.m1.1.1.3" xref="A6.T6.9.9.9.2.m1.1.1.3.cmml">α</mi></mrow><annotation-xml encoding="MathML-Content" id="A6.T6.9.9.9.2.m1.1b"><apply 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methods.</span></figcaption> </figure> </section> </section> <section class="ltx_subsection" id="A6.SS5"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">F.5 </span>SAR despeckling</h3> <div class="ltx_para" id="A6.SS5.p1"> <p class="ltx_p" id="A6.SS5.p1.10">For Synthetic Aperture Radar (SAR) despeckling, we consider the Goodman’s model <cite class="ltx_cite ltx_citemacro_citep">(Goodman, <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#bib.bib27" title="">1976</a>)</cite> in which we aim at recovering the underlying reflexivity <math alttext="{\mathbf{R}}\in{\mathbb{R}}_{+}^{d}" class="ltx_Math" display="inline" id="A6.SS5.p1.1.m1.1"><semantics id="A6.SS5.p1.1.m1.1a"><mrow id="A6.SS5.p1.1.m1.1.1" xref="A6.SS5.p1.1.m1.1.1.cmml"><mi id="A6.SS5.p1.1.m1.1.1.2" xref="A6.SS5.p1.1.m1.1.1.2.cmml">𝐑</mi><mo id="A6.SS5.p1.1.m1.1.1.1" xref="A6.SS5.p1.1.m1.1.1.1.cmml">∈</mo><msubsup id="A6.SS5.p1.1.m1.1.1.3" xref="A6.SS5.p1.1.m1.1.1.3.cmml"><mi id="A6.SS5.p1.1.m1.1.1.3.2.2" xref="A6.SS5.p1.1.m1.1.1.3.2.2.cmml">ℝ</mi><mo id="A6.SS5.p1.1.m1.1.1.3.2.3" xref="A6.SS5.p1.1.m1.1.1.3.2.3.cmml">+</mo><mi id="A6.SS5.p1.1.m1.1.1.3.3" xref="A6.SS5.p1.1.m1.1.1.3.3.cmml">d</mi></msubsup></mrow><annotation-xml encoding="MathML-Content" id="A6.SS5.p1.1.m1.1b"><apply id="A6.SS5.p1.1.m1.1.1.cmml" xref="A6.SS5.p1.1.m1.1.1"><in id="A6.SS5.p1.1.m1.1.1.1.cmml" xref="A6.SS5.p1.1.m1.1.1.1"></in><ci id="A6.SS5.p1.1.m1.1.1.2.cmml" xref="A6.SS5.p1.1.m1.1.1.2">𝐑</ci><apply id="A6.SS5.p1.1.m1.1.1.3.cmml" xref="A6.SS5.p1.1.m1.1.1.3"><csymbol cd="ambiguous" id="A6.SS5.p1.1.m1.1.1.3.1.cmml" xref="A6.SS5.p1.1.m1.1.1.3">superscript</csymbol><apply id="A6.SS5.p1.1.m1.1.1.3.2.cmml" xref="A6.SS5.p1.1.m1.1.1.3"><csymbol cd="ambiguous" id="A6.SS5.p1.1.m1.1.1.3.2.1.cmml" xref="A6.SS5.p1.1.m1.1.1.3">subscript</csymbol><ci id="A6.SS5.p1.1.m1.1.1.3.2.2.cmml" xref="A6.SS5.p1.1.m1.1.1.3.2.2">ℝ</ci><plus id="A6.SS5.p1.1.m1.1.1.3.2.3.cmml" xref="A6.SS5.p1.1.m1.1.1.3.2.3"></plus></apply><ci id="A6.SS5.p1.1.m1.1.1.3.3.cmml" xref="A6.SS5.p1.1.m1.1.1.3.3">𝑑</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A6.SS5.p1.1.m1.1c">{\mathbf{R}}\in{\mathbb{R}}_{+}^{d}</annotation><annotation encoding="application/x-llamapun" id="A6.SS5.p1.1.m1.1d">bold_R ∈ blackboard_R start_POSTSUBSCRIPT + end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT</annotation></semantics></math> from the observed intensity <math alttext="{\mathbf{I}}\in{\mathbb{R}}_{+}^{d}" class="ltx_Math" display="inline" id="A6.SS5.p1.2.m2.1"><semantics id="A6.SS5.p1.2.m2.1a"><mrow id="A6.SS5.p1.2.m2.1.1" xref="A6.SS5.p1.2.m2.1.1.cmml"><mi id="A6.SS5.p1.2.m2.1.1.2" xref="A6.SS5.p1.2.m2.1.1.2.cmml">𝐈</mi><mo id="A6.SS5.p1.2.m2.1.1.1" xref="A6.SS5.p1.2.m2.1.1.1.cmml">∈</mo><msubsup id="A6.SS5.p1.2.m2.1.1.3" xref="A6.SS5.p1.2.m2.1.1.3.cmml"><mi id="A6.SS5.p1.2.m2.1.1.3.2.2" xref="A6.SS5.p1.2.m2.1.1.3.2.2.cmml">ℝ</mi><mo id="A6.SS5.p1.2.m2.1.1.3.2.3" xref="A6.SS5.p1.2.m2.1.1.3.2.3.cmml">+</mo><mi id="A6.SS5.p1.2.m2.1.1.3.3" xref="A6.SS5.p1.2.m2.1.1.3.3.cmml">d</mi></msubsup></mrow><annotation-xml encoding="MathML-Content" id="A6.SS5.p1.2.m2.1b"><apply id="A6.SS5.p1.2.m2.1.1.cmml" xref="A6.SS5.p1.2.m2.1.1"><in id="A6.SS5.p1.2.m2.1.1.1.cmml" xref="A6.SS5.p1.2.m2.1.1.1"></in><ci id="A6.SS5.p1.2.m2.1.1.2.cmml" xref="A6.SS5.p1.2.m2.1.1.2">𝐈</ci><apply id="A6.SS5.p1.2.m2.1.1.3.cmml" xref="A6.SS5.p1.2.m2.1.1.3"><csymbol cd="ambiguous" id="A6.SS5.p1.2.m2.1.1.3.1.cmml" xref="A6.SS5.p1.2.m2.1.1.3">superscript</csymbol><apply id="A6.SS5.p1.2.m2.1.1.3.2.cmml" xref="A6.SS5.p1.2.m2.1.1.3"><csymbol cd="ambiguous" id="A6.SS5.p1.2.m2.1.1.3.2.1.cmml" xref="A6.SS5.p1.2.m2.1.1.3">subscript</csymbol><ci id="A6.SS5.p1.2.m2.1.1.3.2.2.cmml" xref="A6.SS5.p1.2.m2.1.1.3.2.2">ℝ</ci><plus id="A6.SS5.p1.2.m2.1.1.3.2.3.cmml" xref="A6.SS5.p1.2.m2.1.1.3.2.3"></plus></apply><ci id="A6.SS5.p1.2.m2.1.1.3.3.cmml" xref="A6.SS5.p1.2.m2.1.1.3.3">𝑑</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A6.SS5.p1.2.m2.1c">{\mathbf{I}}\in{\mathbb{R}}_{+}^{d}</annotation><annotation encoding="application/x-llamapun" id="A6.SS5.p1.2.m2.1d">bold_I ∈ blackboard_R start_POSTSUBSCRIPT + end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT</annotation></semantics></math>, that is a noisy version of <math alttext="{\mathbf{R}}" class="ltx_Math" display="inline" id="A6.SS5.p1.3.m3.1"><semantics id="A6.SS5.p1.3.m3.1a"><mi id="A6.SS5.p1.3.m3.1.1" xref="A6.SS5.p1.3.m3.1.1.cmml">𝐑</mi><annotation-xml encoding="MathML-Content" id="A6.SS5.p1.3.m3.1b"><ci id="A6.SS5.p1.3.m3.1.1.cmml" xref="A6.SS5.p1.3.m3.1.1">𝐑</ci></annotation-xml><annotation encoding="application/x-tex" id="A6.SS5.p1.3.m3.1c">{\mathbf{R}}</annotation><annotation encoding="application/x-llamapun" id="A6.SS5.p1.3.m3.1d">bold_R</annotation></semantics></math> with a multiplicative gamma noise, <math alttext="{\mathbf{I}}={\mathbf{N}}{\mathbf{R}}" class="ltx_Math" display="inline" id="A6.SS5.p1.4.m4.1"><semantics id="A6.SS5.p1.4.m4.1a"><mrow id="A6.SS5.p1.4.m4.1.1" xref="A6.SS5.p1.4.m4.1.1.cmml"><mi id="A6.SS5.p1.4.m4.1.1.2" xref="A6.SS5.p1.4.m4.1.1.2.cmml">𝐈</mi><mo id="A6.SS5.p1.4.m4.1.1.1" xref="A6.SS5.p1.4.m4.1.1.1.cmml">=</mo><mi id="A6.SS5.p1.4.m4.1.1.3" xref="A6.SS5.p1.4.m4.1.1.3.cmml">𝐍𝐑</mi></mrow><annotation-xml encoding="MathML-Content" id="A6.SS5.p1.4.m4.1b"><apply id="A6.SS5.p1.4.m4.1.1.cmml" xref="A6.SS5.p1.4.m4.1.1"><eq id="A6.SS5.p1.4.m4.1.1.1.cmml" xref="A6.SS5.p1.4.m4.1.1.1"></eq><ci id="A6.SS5.p1.4.m4.1.1.2.cmml" xref="A6.SS5.p1.4.m4.1.1.2">𝐈</ci><ci id="A6.SS5.p1.4.m4.1.1.3.cmml" xref="A6.SS5.p1.4.m4.1.1.3">𝐍𝐑</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A6.SS5.p1.4.m4.1c">{\mathbf{I}}={\mathbf{N}}{\mathbf{R}}</annotation><annotation encoding="application/x-llamapun" id="A6.SS5.p1.4.m4.1d">bold_I = bold_NR</annotation></semantics></math>, where <math alttext="{\mathbf{N}}\sim\Gamma(1,L)" class="ltx_Math" display="inline" id="A6.SS5.p1.5.m5.2"><semantics id="A6.SS5.p1.5.m5.2a"><mrow id="A6.SS5.p1.5.m5.2.3" xref="A6.SS5.p1.5.m5.2.3.cmml"><mi id="A6.SS5.p1.5.m5.2.3.2" xref="A6.SS5.p1.5.m5.2.3.2.cmml">𝐍</mi><mo id="A6.SS5.p1.5.m5.2.3.1" xref="A6.SS5.p1.5.m5.2.3.1.cmml">∼</mo><mrow id="A6.SS5.p1.5.m5.2.3.3" xref="A6.SS5.p1.5.m5.2.3.3.cmml"><mi id="A6.SS5.p1.5.m5.2.3.3.2" mathvariant="normal" xref="A6.SS5.p1.5.m5.2.3.3.2.cmml">Γ</mi><mo id="A6.SS5.p1.5.m5.2.3.3.1" xref="A6.SS5.p1.5.m5.2.3.3.1.cmml">⁢</mo><mrow id="A6.SS5.p1.5.m5.2.3.3.3.2" xref="A6.SS5.p1.5.m5.2.3.3.3.1.cmml"><mo id="A6.SS5.p1.5.m5.2.3.3.3.2.1" stretchy="false" xref="A6.SS5.p1.5.m5.2.3.3.3.1.cmml">(</mo><mn id="A6.SS5.p1.5.m5.1.1" xref="A6.SS5.p1.5.m5.1.1.cmml">1</mn><mo id="A6.SS5.p1.5.m5.2.3.3.3.2.2" xref="A6.SS5.p1.5.m5.2.3.3.3.1.cmml">,</mo><mi id="A6.SS5.p1.5.m5.2.2" xref="A6.SS5.p1.5.m5.2.2.cmml">L</mi><mo id="A6.SS5.p1.5.m5.2.3.3.3.2.3" stretchy="false" xref="A6.SS5.p1.5.m5.2.3.3.3.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="A6.SS5.p1.5.m5.2b"><apply id="A6.SS5.p1.5.m5.2.3.cmml" xref="A6.SS5.p1.5.m5.2.3"><csymbol cd="latexml" id="A6.SS5.p1.5.m5.2.3.1.cmml" xref="A6.SS5.p1.5.m5.2.3.1">similar-to</csymbol><ci id="A6.SS5.p1.5.m5.2.3.2.cmml" xref="A6.SS5.p1.5.m5.2.3.2">𝐍</ci><apply id="A6.SS5.p1.5.m5.2.3.3.cmml" xref="A6.SS5.p1.5.m5.2.3.3"><times id="A6.SS5.p1.5.m5.2.3.3.1.cmml" xref="A6.SS5.p1.5.m5.2.3.3.1"></times><ci id="A6.SS5.p1.5.m5.2.3.3.2.cmml" xref="A6.SS5.p1.5.m5.2.3.3.2">Γ</ci><interval closure="open" id="A6.SS5.p1.5.m5.2.3.3.3.1.cmml" xref="A6.SS5.p1.5.m5.2.3.3.3.2"><cn id="A6.SS5.p1.5.m5.1.1.cmml" type="integer" xref="A6.SS5.p1.5.m5.1.1">1</cn><ci id="A6.SS5.p1.5.m5.2.2.cmml" xref="A6.SS5.p1.5.m5.2.2">𝐿</ci></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A6.SS5.p1.5.m5.2c">{\mathbf{N}}\sim\Gamma(1,L)</annotation><annotation encoding="application/x-llamapun" id="A6.SS5.p1.5.m5.2d">bold_N ∼ roman_Γ ( 1 , italic_L )</annotation></semantics></math> and <math alttext="L>0" class="ltx_Math" display="inline" id="A6.SS5.p1.6.m6.1"><semantics id="A6.SS5.p1.6.m6.1a"><mrow id="A6.SS5.p1.6.m6.1.1" xref="A6.SS5.p1.6.m6.1.1.cmml"><mi id="A6.SS5.p1.6.m6.1.1.2" xref="A6.SS5.p1.6.m6.1.1.2.cmml">L</mi><mo id="A6.SS5.p1.6.m6.1.1.1" xref="A6.SS5.p1.6.m6.1.1.1.cmml">></mo><mn id="A6.SS5.p1.6.m6.1.1.3" xref="A6.SS5.p1.6.m6.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="A6.SS5.p1.6.m6.1b"><apply id="A6.SS5.p1.6.m6.1.1.cmml" xref="A6.SS5.p1.6.m6.1.1"><gt id="A6.SS5.p1.6.m6.1.1.1.cmml" xref="A6.SS5.p1.6.m6.1.1.1"></gt><ci id="A6.SS5.p1.6.m6.1.1.2.cmml" xref="A6.SS5.p1.6.m6.1.1.2">𝐿</ci><cn id="A6.SS5.p1.6.m6.1.1.3.cmml" type="integer" xref="A6.SS5.p1.6.m6.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A6.SS5.p1.6.m6.1c">L>0</annotation><annotation encoding="application/x-llamapun" id="A6.SS5.p1.6.m6.1d">italic_L > 0</annotation></semantics></math> is called the number of looks <cite class="ltx_cite ltx_citemacro_citep">(Goodman, <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#bib.bib27" title="">1976</a>)</cite>. By tacking the log of the previous model and denoting <math alttext="{\mathbf{y}}=\log{{\mathbf{I}}}" class="ltx_Math" display="inline" id="A6.SS5.p1.7.m7.1"><semantics id="A6.SS5.p1.7.m7.1a"><mrow id="A6.SS5.p1.7.m7.1.1" xref="A6.SS5.p1.7.m7.1.1.cmml"><mi id="A6.SS5.p1.7.m7.1.1.2" xref="A6.SS5.p1.7.m7.1.1.2.cmml">𝐲</mi><mo id="A6.SS5.p1.7.m7.1.1.1" xref="A6.SS5.p1.7.m7.1.1.1.cmml">=</mo><mrow id="A6.SS5.p1.7.m7.1.1.3" xref="A6.SS5.p1.7.m7.1.1.3.cmml"><mi id="A6.SS5.p1.7.m7.1.1.3.1" xref="A6.SS5.p1.7.m7.1.1.3.1.cmml">log</mi><mo id="A6.SS5.p1.7.m7.1.1.3a" lspace="0.167em" xref="A6.SS5.p1.7.m7.1.1.3.cmml">⁡</mo><mi id="A6.SS5.p1.7.m7.1.1.3.2" xref="A6.SS5.p1.7.m7.1.1.3.2.cmml">𝐈</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="A6.SS5.p1.7.m7.1b"><apply id="A6.SS5.p1.7.m7.1.1.cmml" xref="A6.SS5.p1.7.m7.1.1"><eq id="A6.SS5.p1.7.m7.1.1.1.cmml" xref="A6.SS5.p1.7.m7.1.1.1"></eq><ci id="A6.SS5.p1.7.m7.1.1.2.cmml" xref="A6.SS5.p1.7.m7.1.1.2">𝐲</ci><apply id="A6.SS5.p1.7.m7.1.1.3.cmml" xref="A6.SS5.p1.7.m7.1.1.3"><log id="A6.SS5.p1.7.m7.1.1.3.1.cmml" xref="A6.SS5.p1.7.m7.1.1.3.1"></log><ci id="A6.SS5.p1.7.m7.1.1.3.2.cmml" xref="A6.SS5.p1.7.m7.1.1.3.2">𝐈</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A6.SS5.p1.7.m7.1c">{\mathbf{y}}=\log{{\mathbf{I}}}</annotation><annotation encoding="application/x-llamapun" id="A6.SS5.p1.7.m7.1d">bold_y = roman_log bold_I</annotation></semantics></math> and <math alttext="{\mathbf{x}}=\log{{\mathbf{R}}}" class="ltx_Math" display="inline" id="A6.SS5.p1.8.m8.1"><semantics id="A6.SS5.p1.8.m8.1a"><mrow id="A6.SS5.p1.8.m8.1.1" xref="A6.SS5.p1.8.m8.1.1.cmml"><mi id="A6.SS5.p1.8.m8.1.1.2" xref="A6.SS5.p1.8.m8.1.1.2.cmml">𝐱</mi><mo id="A6.SS5.p1.8.m8.1.1.1" xref="A6.SS5.p1.8.m8.1.1.1.cmml">=</mo><mrow id="A6.SS5.p1.8.m8.1.1.3" xref="A6.SS5.p1.8.m8.1.1.3.cmml"><mi id="A6.SS5.p1.8.m8.1.1.3.1" xref="A6.SS5.p1.8.m8.1.1.3.1.cmml">log</mi><mo id="A6.SS5.p1.8.m8.1.1.3a" lspace="0.167em" xref="A6.SS5.p1.8.m8.1.1.3.cmml">⁡</mo><mi id="A6.SS5.p1.8.m8.1.1.3.2" xref="A6.SS5.p1.8.m8.1.1.3.2.cmml">𝐑</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="A6.SS5.p1.8.m8.1b"><apply id="A6.SS5.p1.8.m8.1.1.cmml" xref="A6.SS5.p1.8.m8.1.1"><eq id="A6.SS5.p1.8.m8.1.1.1.cmml" xref="A6.SS5.p1.8.m8.1.1.1"></eq><ci id="A6.SS5.p1.8.m8.1.1.2.cmml" xref="A6.SS5.p1.8.m8.1.1.2">𝐱</ci><apply id="A6.SS5.p1.8.m8.1.1.3.cmml" xref="A6.SS5.p1.8.m8.1.1.3"><log id="A6.SS5.p1.8.m8.1.1.3.1.cmml" xref="A6.SS5.p1.8.m8.1.1.3.1"></log><ci id="A6.SS5.p1.8.m8.1.1.3.2.cmml" xref="A6.SS5.p1.8.m8.1.1.3.2">𝐑</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A6.SS5.p1.8.m8.1c">{\mathbf{x}}=\log{{\mathbf{R}}}</annotation><annotation encoding="application/x-llamapun" id="A6.SS5.p1.8.m8.1d">bold_x = roman_log bold_R</annotation></semantics></math>, we turn the multiplicative noise into an additive noise, <math alttext="{\mathbf{y}}={\mathbf{x}}+{\mathbf{n}}" class="ltx_Math" display="inline" id="A6.SS5.p1.9.m9.1"><semantics id="A6.SS5.p1.9.m9.1a"><mrow id="A6.SS5.p1.9.m9.1.1" xref="A6.SS5.p1.9.m9.1.1.cmml"><mi id="A6.SS5.p1.9.m9.1.1.2" xref="A6.SS5.p1.9.m9.1.1.2.cmml">𝐲</mi><mo id="A6.SS5.p1.9.m9.1.1.1" xref="A6.SS5.p1.9.m9.1.1.1.cmml">=</mo><mrow id="A6.SS5.p1.9.m9.1.1.3" xref="A6.SS5.p1.9.m9.1.1.3.cmml"><mi id="A6.SS5.p1.9.m9.1.1.3.2" xref="A6.SS5.p1.9.m9.1.1.3.2.cmml">𝐱</mi><mo id="A6.SS5.p1.9.m9.1.1.3.1" xref="A6.SS5.p1.9.m9.1.1.3.1.cmml">+</mo><mi id="A6.SS5.p1.9.m9.1.1.3.3" xref="A6.SS5.p1.9.m9.1.1.3.3.cmml">𝐧</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="A6.SS5.p1.9.m9.1b"><apply id="A6.SS5.p1.9.m9.1.1.cmml" xref="A6.SS5.p1.9.m9.1.1"><eq id="A6.SS5.p1.9.m9.1.1.1.cmml" xref="A6.SS5.p1.9.m9.1.1.1"></eq><ci id="A6.SS5.p1.9.m9.1.1.2.cmml" xref="A6.SS5.p1.9.m9.1.1.2">𝐲</ci><apply id="A6.SS5.p1.9.m9.1.1.3.cmml" xref="A6.SS5.p1.9.m9.1.1.3"><plus id="A6.SS5.p1.9.m9.1.1.3.1.cmml" xref="A6.SS5.p1.9.m9.1.1.3.1"></plus><ci id="A6.SS5.p1.9.m9.1.1.3.2.cmml" xref="A6.SS5.p1.9.m9.1.1.3.2">𝐱</ci><ci id="A6.SS5.p1.9.m9.1.1.3.3.cmml" xref="A6.SS5.p1.9.m9.1.1.3.3">𝐧</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A6.SS5.p1.9.m9.1c">{\mathbf{y}}={\mathbf{x}}+{\mathbf{n}}</annotation><annotation encoding="application/x-llamapun" id="A6.SS5.p1.9.m9.1d">bold_y = bold_x + bold_n</annotation></semantics></math>, where <math alttext="{\mathbf{n}}" class="ltx_Math" display="inline" id="A6.SS5.p1.10.m10.1"><semantics id="A6.SS5.p1.10.m10.1a"><mi id="A6.SS5.p1.10.m10.1.1" xref="A6.SS5.p1.10.m10.1.1.cmml">𝐧</mi><annotation-xml encoding="MathML-Content" id="A6.SS5.p1.10.m10.1b"><ci id="A6.SS5.p1.10.m10.1.1.cmml" xref="A6.SS5.p1.10.m10.1.1">𝐧</ci></annotation-xml><annotation encoding="application/x-tex" id="A6.SS5.p1.10.m10.1c">{\mathbf{n}}</annotation><annotation encoding="application/x-llamapun" id="A6.SS5.p1.10.m10.1d">bold_n</annotation></semantics></math> is following the Fisher-Tippett distribution.</p> </div> <div class="ltx_para" id="A6.SS5.p2"> <p class="ltx_p" id="A6.SS5.p2.6">To solve this problem, we can solve the following variation problem <cite class="ltx_cite ltx_citemacro_citep">(Deledalle et al., <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#bib.bib19" title="">2017</a>)</cite></p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="A8.EGx58"> <tbody id="A6.Ex122"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\operatorname*{arg\,min}_{{\mathbf{x}}\in{\mathbb{R}}^{d}}{-\log{% p({\mathbf{y}}|{\mathbf{x}})}+\alpha\mathcal{R}({\mathbf{x}})}," class="ltx_Math" display="inline" id="A6.Ex122.m1.2"><semantics id="A6.Ex122.m1.2a"><mrow id="A6.Ex122.m1.2.2.1" xref="A6.Ex122.m1.2.2.1.1.cmml"><mrow id="A6.Ex122.m1.2.2.1.1" xref="A6.Ex122.m1.2.2.1.1.cmml"><mrow id="A6.Ex122.m1.2.2.1.1.1" xref="A6.Ex122.m1.2.2.1.1.1.cmml"><munder id="A6.Ex122.m1.2.2.1.1.1.3" xref="A6.Ex122.m1.2.2.1.1.1.3.cmml"><mrow id="A6.Ex122.m1.2.2.1.1.1.3.2" xref="A6.Ex122.m1.2.2.1.1.1.3.2.cmml"><mi id="A6.Ex122.m1.2.2.1.1.1.3.2.2" xref="A6.Ex122.m1.2.2.1.1.1.3.2.2.cmml">arg</mi><mo id="A6.Ex122.m1.2.2.1.1.1.3.2.1" lspace="0.170em" xref="A6.Ex122.m1.2.2.1.1.1.3.2.1.cmml">⁢</mo><mi id="A6.Ex122.m1.2.2.1.1.1.3.2.3" xref="A6.Ex122.m1.2.2.1.1.1.3.2.3.cmml">min</mi></mrow><mrow id="A6.Ex122.m1.2.2.1.1.1.3.3" xref="A6.Ex122.m1.2.2.1.1.1.3.3.cmml"><mi id="A6.Ex122.m1.2.2.1.1.1.3.3.2" xref="A6.Ex122.m1.2.2.1.1.1.3.3.2.cmml">𝐱</mi><mo id="A6.Ex122.m1.2.2.1.1.1.3.3.1" xref="A6.Ex122.m1.2.2.1.1.1.3.3.1.cmml">∈</mo><msup id="A6.Ex122.m1.2.2.1.1.1.3.3.3" xref="A6.Ex122.m1.2.2.1.1.1.3.3.3.cmml"><mi id="A6.Ex122.m1.2.2.1.1.1.3.3.3.2" xref="A6.Ex122.m1.2.2.1.1.1.3.3.3.2.cmml">ℝ</mi><mi id="A6.Ex122.m1.2.2.1.1.1.3.3.3.3" xref="A6.Ex122.m1.2.2.1.1.1.3.3.3.3.cmml">d</mi></msup></mrow></munder><mo id="A6.Ex122.m1.2.2.1.1.1.2" lspace="0em" xref="A6.Ex122.m1.2.2.1.1.1.2.cmml">−</mo><mrow id="A6.Ex122.m1.2.2.1.1.1.1" xref="A6.Ex122.m1.2.2.1.1.1.1.cmml"><mrow id="A6.Ex122.m1.2.2.1.1.1.1.3" xref="A6.Ex122.m1.2.2.1.1.1.1.3.cmml"><mi id="A6.Ex122.m1.2.2.1.1.1.1.3.1" xref="A6.Ex122.m1.2.2.1.1.1.1.3.1.cmml">log</mi><mo id="A6.Ex122.m1.2.2.1.1.1.1.3a" lspace="0.167em" xref="A6.Ex122.m1.2.2.1.1.1.1.3.cmml">⁡</mo><mi id="A6.Ex122.m1.2.2.1.1.1.1.3.2" xref="A6.Ex122.m1.2.2.1.1.1.1.3.2.cmml">p</mi></mrow><mo id="A6.Ex122.m1.2.2.1.1.1.1.2" xref="A6.Ex122.m1.2.2.1.1.1.1.2.cmml">⁢</mo><mrow id="A6.Ex122.m1.2.2.1.1.1.1.1.1" xref="A6.Ex122.m1.2.2.1.1.1.1.1.1.1.cmml"><mo id="A6.Ex122.m1.2.2.1.1.1.1.1.1.2" stretchy="false" xref="A6.Ex122.m1.2.2.1.1.1.1.1.1.1.cmml">(</mo><mrow id="A6.Ex122.m1.2.2.1.1.1.1.1.1.1" xref="A6.Ex122.m1.2.2.1.1.1.1.1.1.1.cmml"><mi id="A6.Ex122.m1.2.2.1.1.1.1.1.1.1.2" xref="A6.Ex122.m1.2.2.1.1.1.1.1.1.1.2.cmml">𝐲</mi><mo fence="false" id="A6.Ex122.m1.2.2.1.1.1.1.1.1.1.1" xref="A6.Ex122.m1.2.2.1.1.1.1.1.1.1.1.cmml">|</mo><mi id="A6.Ex122.m1.2.2.1.1.1.1.1.1.1.3" xref="A6.Ex122.m1.2.2.1.1.1.1.1.1.1.3.cmml">𝐱</mi></mrow><mo id="A6.Ex122.m1.2.2.1.1.1.1.1.1.3" stretchy="false" xref="A6.Ex122.m1.2.2.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><mo id="A6.Ex122.m1.2.2.1.1.2" xref="A6.Ex122.m1.2.2.1.1.2.cmml">+</mo><mrow id="A6.Ex122.m1.2.2.1.1.3" xref="A6.Ex122.m1.2.2.1.1.3.cmml"><mi id="A6.Ex122.m1.2.2.1.1.3.2" xref="A6.Ex122.m1.2.2.1.1.3.2.cmml">α</mi><mo id="A6.Ex122.m1.2.2.1.1.3.1" xref="A6.Ex122.m1.2.2.1.1.3.1.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="A6.Ex122.m1.2.2.1.1.3.3" xref="A6.Ex122.m1.2.2.1.1.3.3.cmml">ℛ</mi><mo id="A6.Ex122.m1.2.2.1.1.3.1a" xref="A6.Ex122.m1.2.2.1.1.3.1.cmml">⁢</mo><mrow id="A6.Ex122.m1.2.2.1.1.3.4.2" xref="A6.Ex122.m1.2.2.1.1.3.cmml"><mo id="A6.Ex122.m1.2.2.1.1.3.4.2.1" stretchy="false" xref="A6.Ex122.m1.2.2.1.1.3.cmml">(</mo><mi id="A6.Ex122.m1.1.1" xref="A6.Ex122.m1.1.1.cmml">𝐱</mi><mo id="A6.Ex122.m1.2.2.1.1.3.4.2.2" stretchy="false" xref="A6.Ex122.m1.2.2.1.1.3.cmml">)</mo></mrow></mrow></mrow><mo id="A6.Ex122.m1.2.2.1.2" xref="A6.Ex122.m1.2.2.1.1.cmml">,</mo></mrow><annotation-xml encoding="MathML-Content" id="A6.Ex122.m1.2b"><apply id="A6.Ex122.m1.2.2.1.1.cmml" xref="A6.Ex122.m1.2.2.1"><plus id="A6.Ex122.m1.2.2.1.1.2.cmml" xref="A6.Ex122.m1.2.2.1.1.2"></plus><apply id="A6.Ex122.m1.2.2.1.1.1.cmml" xref="A6.Ex122.m1.2.2.1.1.1"><minus id="A6.Ex122.m1.2.2.1.1.1.2.cmml" xref="A6.Ex122.m1.2.2.1.1.1.2"></minus><apply id="A6.Ex122.m1.2.2.1.1.1.3.cmml" xref="A6.Ex122.m1.2.2.1.1.1.3"><csymbol cd="ambiguous" id="A6.Ex122.m1.2.2.1.1.1.3.1.cmml" xref="A6.Ex122.m1.2.2.1.1.1.3">subscript</csymbol><apply id="A6.Ex122.m1.2.2.1.1.1.3.2.cmml" xref="A6.Ex122.m1.2.2.1.1.1.3.2"><times id="A6.Ex122.m1.2.2.1.1.1.3.2.1.cmml" xref="A6.Ex122.m1.2.2.1.1.1.3.2.1"></times><ci id="A6.Ex122.m1.2.2.1.1.1.3.2.2.cmml" xref="A6.Ex122.m1.2.2.1.1.1.3.2.2">arg</ci><ci id="A6.Ex122.m1.2.2.1.1.1.3.2.3.cmml" xref="A6.Ex122.m1.2.2.1.1.1.3.2.3">min</ci></apply><apply id="A6.Ex122.m1.2.2.1.1.1.3.3.cmml" xref="A6.Ex122.m1.2.2.1.1.1.3.3"><in id="A6.Ex122.m1.2.2.1.1.1.3.3.1.cmml" xref="A6.Ex122.m1.2.2.1.1.1.3.3.1"></in><ci id="A6.Ex122.m1.2.2.1.1.1.3.3.2.cmml" xref="A6.Ex122.m1.2.2.1.1.1.3.3.2">𝐱</ci><apply id="A6.Ex122.m1.2.2.1.1.1.3.3.3.cmml" xref="A6.Ex122.m1.2.2.1.1.1.3.3.3"><csymbol cd="ambiguous" id="A6.Ex122.m1.2.2.1.1.1.3.3.3.1.cmml" xref="A6.Ex122.m1.2.2.1.1.1.3.3.3">superscript</csymbol><ci id="A6.Ex122.m1.2.2.1.1.1.3.3.3.2.cmml" xref="A6.Ex122.m1.2.2.1.1.1.3.3.3.2">ℝ</ci><ci id="A6.Ex122.m1.2.2.1.1.1.3.3.3.3.cmml" xref="A6.Ex122.m1.2.2.1.1.1.3.3.3.3">𝑑</ci></apply></apply></apply><apply id="A6.Ex122.m1.2.2.1.1.1.1.cmml" xref="A6.Ex122.m1.2.2.1.1.1.1"><times id="A6.Ex122.m1.2.2.1.1.1.1.2.cmml" xref="A6.Ex122.m1.2.2.1.1.1.1.2"></times><apply id="A6.Ex122.m1.2.2.1.1.1.1.3.cmml" xref="A6.Ex122.m1.2.2.1.1.1.1.3"><log id="A6.Ex122.m1.2.2.1.1.1.1.3.1.cmml" xref="A6.Ex122.m1.2.2.1.1.1.1.3.1"></log><ci id="A6.Ex122.m1.2.2.1.1.1.1.3.2.cmml" xref="A6.Ex122.m1.2.2.1.1.1.1.3.2">𝑝</ci></apply><apply id="A6.Ex122.m1.2.2.1.1.1.1.1.1.1.cmml" xref="A6.Ex122.m1.2.2.1.1.1.1.1.1"><csymbol cd="latexml" id="A6.Ex122.m1.2.2.1.1.1.1.1.1.1.1.cmml" xref="A6.Ex122.m1.2.2.1.1.1.1.1.1.1.1">conditional</csymbol><ci id="A6.Ex122.m1.2.2.1.1.1.1.1.1.1.2.cmml" xref="A6.Ex122.m1.2.2.1.1.1.1.1.1.1.2">𝐲</ci><ci id="A6.Ex122.m1.2.2.1.1.1.1.1.1.1.3.cmml" xref="A6.Ex122.m1.2.2.1.1.1.1.1.1.1.3">𝐱</ci></apply></apply></apply><apply id="A6.Ex122.m1.2.2.1.1.3.cmml" xref="A6.Ex122.m1.2.2.1.1.3"><times id="A6.Ex122.m1.2.2.1.1.3.1.cmml" xref="A6.Ex122.m1.2.2.1.1.3.1"></times><ci id="A6.Ex122.m1.2.2.1.1.3.2.cmml" xref="A6.Ex122.m1.2.2.1.1.3.2">𝛼</ci><ci id="A6.Ex122.m1.2.2.1.1.3.3.cmml" xref="A6.Ex122.m1.2.2.1.1.3.3">ℛ</ci><ci id="A6.Ex122.m1.1.1.cmml" xref="A6.Ex122.m1.1.1">𝐱</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A6.Ex122.m1.2c">\displaystyle\operatorname*{arg\,min}_{{\mathbf{x}}\in{\mathbb{R}}^{d}}{-\log{% p({\mathbf{y}}|{\mathbf{x}})}+\alpha\mathcal{R}({\mathbf{x}})},</annotation><annotation encoding="application/x-llamapun" id="A6.Ex122.m1.2d">start_OPERATOR roman_arg roman_min end_OPERATOR start_POSTSUBSCRIPT bold_x ∈ blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT end_POSTSUBSCRIPT - roman_log italic_p ( bold_y | bold_x ) + italic_α caligraphic_R ( bold_x ) ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="A6.SS5.p2.5">where <math alttext="-\log{p({\mathbf{y}}|{\mathbf{x}})}=L\sum_{k=1}^{d}x_{k}+e^{y_{k}-x_{k}}+\text% {Cst.}" class="ltx_Math" display="inline" id="A6.SS5.p2.1.m1.1"><semantics id="A6.SS5.p2.1.m1.1a"><mrow id="A6.SS5.p2.1.m1.1.1" xref="A6.SS5.p2.1.m1.1.1.cmml"><mrow id="A6.SS5.p2.1.m1.1.1.1" xref="A6.SS5.p2.1.m1.1.1.1.cmml"><mo 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xref="A6.SS5.p2.1.m1.1.1.3.4">Cst.</mtext></ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A6.SS5.p2.1.m1.1c">-\log{p({\mathbf{y}}|{\mathbf{x}})}=L\sum_{k=1}^{d}x_{k}+e^{y_{k}-x_{k}}+\text% {Cst.}</annotation><annotation encoding="application/x-llamapun" id="A6.SS5.p2.1.m1.1d">- roman_log italic_p ( bold_y | bold_x ) = italic_L ∑ start_POSTSUBSCRIPT italic_k = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT italic_x start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT + italic_e start_POSTSUPERSCRIPT italic_y start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT - italic_x start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUPERSCRIPT + Cst.</annotation></semantics></math> (<math alttext="x_{k}" class="ltx_Math" display="inline" id="A6.SS5.p2.2.m2.1"><semantics id="A6.SS5.p2.2.m2.1a"><msub id="A6.SS5.p2.2.m2.1.1" xref="A6.SS5.p2.2.m2.1.1.cmml"><mi id="A6.SS5.p2.2.m2.1.1.2" xref="A6.SS5.p2.2.m2.1.1.2.cmml">x</mi><mi 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xref="A6.SS5.p2.5.m5.1.1.cmml">ℛ</mi><annotation-xml encoding="MathML-Content" id="A6.SS5.p2.5.m5.1b"><ci id="A6.SS5.p2.5.m5.1.1.cmml" xref="A6.SS5.p2.5.m5.1.1">ℛ</ci></annotation-xml><annotation encoding="application/x-tex" id="A6.SS5.p2.5.m5.1c">\mathcal{R}</annotation><annotation encoding="application/x-llamapun" id="A6.SS5.p2.5.m5.1d">caligraphic_R</annotation></semantics></math> the regularization.</p> </div> <div class="ltx_para" id="A6.SS5.p3"> <p class="ltx_p" id="A6.SS5.p3.1">Working in the log-domain allow us to solve an unconstrained optimization with a convex data-fidelity term. However the gradient of this data-fidelity term is not Lipschitz. By solving this problem with the SNORE regularization, our theoretical analysis still holds (the data-fidelity is <math alttext="C^{\infty}" class="ltx_Math" display="inline" id="A6.SS5.p3.1.m1.1"><semantics id="A6.SS5.p3.1.m1.1a"><msup id="A6.SS5.p3.1.m1.1.1" xref="A6.SS5.p3.1.m1.1.1.cmml"><mi id="A6.SS5.p3.1.m1.1.1.2" xref="A6.SS5.p3.1.m1.1.1.2.cmml">C</mi><mi id="A6.SS5.p3.1.m1.1.1.3" mathvariant="normal" xref="A6.SS5.p3.1.m1.1.1.3.cmml">∞</mi></msup><annotation-xml encoding="MathML-Content" id="A6.SS5.p3.1.m1.1b"><apply id="A6.SS5.p3.1.m1.1.1.cmml" xref="A6.SS5.p3.1.m1.1.1"><csymbol cd="ambiguous" id="A6.SS5.p3.1.m1.1.1.1.cmml" xref="A6.SS5.p3.1.m1.1.1">superscript</csymbol><ci id="A6.SS5.p3.1.m1.1.1.2.cmml" xref="A6.SS5.p3.1.m1.1.1.2">𝐶</ci><infinity id="A6.SS5.p3.1.m1.1.1.3.cmml" xref="A6.SS5.p3.1.m1.1.1.3"></infinity></apply></annotation-xml><annotation encoding="application/x-tex" id="A6.SS5.p3.1.m1.1c">C^{\infty}</annotation><annotation encoding="application/x-llamapun" id="A6.SS5.p3.1.m1.1d">italic_C start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT</annotation></semantics></math>). However, the boundedness hypothesis might be harder to verify because the objective function is not necessarily coercive (see Appendix G for more details).</p> </div> <div class="ltx_para" id="A6.SS5.p4"> <p class="ltx_p" id="A6.SS5.p4.1">To solve experimentally this problem, we have trained a GS-denoiser <cite class="ltx_cite ltx_citemacro_citep">(Hurault et al., <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#bib.bib30" title="">2022a</a>)</cite> on the SAR-speckle free dataset developed by <cite class="ltx_cite ltx_citemacro_citet">Dalsasso et al. (<a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#bib.bib15" title="">2020</a>)</cite>. This denoiser is trained to remove additive gaussian noise to SAR images for <math alttext="\sigma\in[0,50]" class="ltx_Math" display="inline" id="A6.SS5.p4.1.m1.2"><semantics id="A6.SS5.p4.1.m1.2a"><mrow id="A6.SS5.p4.1.m1.2.3" xref="A6.SS5.p4.1.m1.2.3.cmml"><mi id="A6.SS5.p4.1.m1.2.3.2" xref="A6.SS5.p4.1.m1.2.3.2.cmml">σ</mi><mo id="A6.SS5.p4.1.m1.2.3.1" xref="A6.SS5.p4.1.m1.2.3.1.cmml">∈</mo><mrow id="A6.SS5.p4.1.m1.2.3.3.2" xref="A6.SS5.p4.1.m1.2.3.3.1.cmml"><mo id="A6.SS5.p4.1.m1.2.3.3.2.1" stretchy="false" xref="A6.SS5.p4.1.m1.2.3.3.1.cmml">[</mo><mn id="A6.SS5.p4.1.m1.1.1" xref="A6.SS5.p4.1.m1.1.1.cmml">0</mn><mo id="A6.SS5.p4.1.m1.2.3.3.2.2" xref="A6.SS5.p4.1.m1.2.3.3.1.cmml">,</mo><mn id="A6.SS5.p4.1.m1.2.2" xref="A6.SS5.p4.1.m1.2.2.cmml">50</mn><mo id="A6.SS5.p4.1.m1.2.3.3.2.3" stretchy="false" xref="A6.SS5.p4.1.m1.2.3.3.1.cmml">]</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A6.SS5.p4.1.m1.2b"><apply id="A6.SS5.p4.1.m1.2.3.cmml" xref="A6.SS5.p4.1.m1.2.3"><in id="A6.SS5.p4.1.m1.2.3.1.cmml" xref="A6.SS5.p4.1.m1.2.3.1"></in><ci id="A6.SS5.p4.1.m1.2.3.2.cmml" xref="A6.SS5.p4.1.m1.2.3.2">𝜎</ci><interval closure="closed" id="A6.SS5.p4.1.m1.2.3.3.1.cmml" xref="A6.SS5.p4.1.m1.2.3.3.2"><cn id="A6.SS5.p4.1.m1.1.1.cmml" type="integer" xref="A6.SS5.p4.1.m1.1.1">0</cn><cn id="A6.SS5.p4.1.m1.2.2.cmml" type="integer" xref="A6.SS5.p4.1.m1.2.2">50</cn></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="A6.SS5.p4.1.m1.2c">\sigma\in[0,50]</annotation><annotation encoding="application/x-llamapun" id="A6.SS5.p4.1.m1.2d">italic_σ ∈ [ 0 , 50 ]</annotation></semantics></math> with the same traning parameters than proposed by <cite class="ltx_cite ltx_citemacro_citet">Hurault et al. (<a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#bib.bib30" title="">2022a</a>)</cite>.</p> </div> <div class="ltx_para" id="A6.SS5.p5"> <p class="ltx_p" id="A6.SS5.p5.1">We use RED and SNORE algorithm to despeckle images with the parameters setting details in Table <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#A6.T7" title="Table 7 ‣ F.5 SAR despeckling ‣ Appendix F Additional Experiments ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_tag">7</span></a>. On Figure <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#A6.F14" title="Figure 14 ‣ More results ‣ F.3 Inpainting ‣ Appendix F Additional Experiments ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_tag">14</span></a>, we show SNORE and RED algorithm qualitative performance on various SAR images. We only compute the PSNR and SSIM metrics because the LPIPS and the BRISQUE are designed for color images. We can notice that SNORE and RED succeed to restore good quality images for the non-standard data-fidelity term of image despeckling.</p> </div> <figure class="ltx_table" id="A6.T7"> <div class="ltx_inline-block ltx_align_center ltx_transformed_outer" id="A6.T7.7" style="width:173.4pt;height:228pt;vertical-align:-0.0pt;"><span class="ltx_transformed_inner" style="transform:translate(31.9pt,-42.0pt) scale(1.58316983166071,1.58316983166071) ;"> <table class="ltx_tabular ltx_guessed_headers ltx_align_middle" id="A6.T7.7.7"> <tbody class="ltx_tbody"> <tr class="ltx_tr" id="A6.T7.7.7.8.1"> <th class="ltx_td ltx_align_center ltx_th ltx_th_row ltx_border_l ltx_border_rr ltx_border_t" id="A6.T7.7.7.8.1.1">Parameters</th> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="A6.T7.7.7.8.1.2">RED</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="A6.T7.7.7.8.1.3">SNORE</td> </tr> <tr class="ltx_tr" id="A6.T7.1.1.1"> <th class="ltx_td ltx_align_center ltx_th ltx_th_row ltx_border_l ltx_border_rr ltx_border_t" id="A6.T7.1.1.1.1"><math alttext="\alpha" class="ltx_Math" display="inline" id="A6.T7.1.1.1.1.m1.1"><semantics id="A6.T7.1.1.1.1.m1.1a"><mi id="A6.T7.1.1.1.1.m1.1.1" xref="A6.T7.1.1.1.1.m1.1.1.cmml">α</mi><annotation-xml encoding="MathML-Content" id="A6.T7.1.1.1.1.m1.1b"><ci id="A6.T7.1.1.1.1.m1.1.1.cmml" xref="A6.T7.1.1.1.1.m1.1.1">𝛼</ci></annotation-xml><annotation encoding="application/x-tex" id="A6.T7.1.1.1.1.m1.1c">\alpha</annotation><annotation encoding="application/x-llamapun" id="A6.T7.1.1.1.1.m1.1d">italic_α</annotation></semantics></math></th> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="A6.T7.1.1.1.2">80</td> <td class="ltx_td ltx_border_r ltx_border_t" id="A6.T7.1.1.1.3"></td> </tr> <tr class="ltx_tr" id="A6.T7.2.2.2"> <th class="ltx_td ltx_align_center ltx_th ltx_th_row ltx_border_l ltx_border_rr ltx_border_t" id="A6.T7.2.2.2.1"><math alttext="\alpha_{0}=\alpha_{m-1}" class="ltx_Math" display="inline" 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id="A6.T7.2.2.2.1.m1.1.1.3.3.3.cmml" type="integer" xref="A6.T7.2.2.2.1.m1.1.1.3.3.3">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A6.T7.2.2.2.1.m1.1c">\alpha_{0}=\alpha_{m-1}</annotation><annotation encoding="application/x-llamapun" id="A6.T7.2.2.2.1.m1.1d">italic_α start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT = italic_α start_POSTSUBSCRIPT italic_m - 1 end_POSTSUBSCRIPT</annotation></semantics></math></th> <td class="ltx_td ltx_border_r ltx_border_t" id="A6.T7.2.2.2.2"></td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="A6.T7.2.2.2.3">80</td> </tr> <tr class="ltx_tr" id="A6.T7.3.3.3"> <th class="ltx_td ltx_align_center ltx_th ltx_th_row ltx_border_l ltx_border_rr ltx_border_t" id="A6.T7.3.3.3.1"><math alttext="\sigma\times 255" class="ltx_Math" display="inline" id="A6.T7.3.3.3.1.m1.1"><semantics id="A6.T7.3.3.3.1.m1.1a"><mrow id="A6.T7.3.3.3.1.m1.1.1" xref="A6.T7.3.3.3.1.m1.1.1.cmml"><mi id="A6.T7.3.3.3.1.m1.1.1.2" 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ltx_th_row ltx_border_b ltx_border_l ltx_border_rr ltx_border_t" id="A6.T7.7.7.7.1"><math alttext="\delta" class="ltx_Math" display="inline" id="A6.T7.7.7.7.1.m1.1"><semantics id="A6.T7.7.7.7.1.m1.1a"><mi id="A6.T7.7.7.7.1.m1.1.1" xref="A6.T7.7.7.7.1.m1.1.1.cmml">δ</mi><annotation-xml encoding="MathML-Content" id="A6.T7.7.7.7.1.m1.1b"><ci id="A6.T7.7.7.7.1.m1.1.1.cmml" xref="A6.T7.7.7.7.1.m1.1.1">𝛿</ci></annotation-xml><annotation encoding="application/x-tex" id="A6.T7.7.7.7.1.m1.1c">\delta</annotation><annotation encoding="application/x-llamapun" id="A6.T7.7.7.7.1.m1.1d">italic_δ</annotation></semantics></math></th> <td class="ltx_td ltx_align_center ltx_border_b ltx_border_r ltx_border_t" id="A6.T7.7.7.7.2">0.01</td> <td class="ltx_td ltx_align_center ltx_border_b ltx_border_r ltx_border_t" id="A6.T7.7.7.7.3">0.01</td> </tr> </tbody> </table> </span></div> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_table"><span class="ltx_text" id="A6.T7.9.1.1" style="font-size:90%;">Table 7</span>: </span><span class="ltx_text" id="A6.T7.10.2" style="font-size:90%;">Parameters setting for image despeckling for the different implemented methods.</span></figcaption> </figure> </section> </section> <section class="ltx_appendix" id="A7"> <h2 class="ltx_title ltx_title_appendix"> <span class="ltx_tag ltx_tag_appendix">Appendix G </span>SNORE applies the denoiser on its training domain</h2> <div class="ltx_para" id="A7.p1"> <p class="ltx_p" id="A7.p1.6">One motivation to use SNORE is to force the denoiser to be applied on its training domain. 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In this noisy version <math alttext="{\tilde{\mathbf{x}}}_{k}" class="ltx_Math" display="inline" id="A7.p1.6.m6.1"><semantics id="A7.p1.6.m6.1a"><msub id="A7.p1.6.m6.1.1" xref="A7.p1.6.m6.1.1.cmml"><mover accent="true" id="A7.p1.6.m6.1.1.2" xref="A7.p1.6.m6.1.1.2.cmml"><mi id="A7.p1.6.m6.1.1.2.2" xref="A7.p1.6.m6.1.1.2.2.cmml">𝐱</mi><mo id="A7.p1.6.m6.1.1.2.1" xref="A7.p1.6.m6.1.1.2.1.cmml">~</mo></mover><mi id="A7.p1.6.m6.1.1.3" xref="A7.p1.6.m6.1.1.3.cmml">k</mi></msub><annotation-xml encoding="MathML-Content" id="A7.p1.6.m6.1b"><apply id="A7.p1.6.m6.1.1.cmml" xref="A7.p1.6.m6.1.1"><csymbol cd="ambiguous" id="A7.p1.6.m6.1.1.1.cmml" xref="A7.p1.6.m6.1.1">subscript</csymbol><apply id="A7.p1.6.m6.1.1.2.cmml" xref="A7.p1.6.m6.1.1.2"><ci id="A7.p1.6.m6.1.1.2.1.cmml" xref="A7.p1.6.m6.1.1.2.1">~</ci><ci id="A7.p1.6.m6.1.1.2.2.cmml" xref="A7.p1.6.m6.1.1.2.2">𝐱</ci></apply><ci id="A7.p1.6.m6.1.1.3.cmml" xref="A7.p1.6.m6.1.1.3">𝑘</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A7.p1.6.m6.1c">{\tilde{\mathbf{x}}}_{k}</annotation><annotation encoding="application/x-llamapun" id="A7.p1.6.m6.1d">over~ start_ARG bold_x end_ARG start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT</annotation></semantics></math>, the input noise level is exactly the noise level of the denoiser.</p> </div> <div class="ltx_para" id="A7.p2"> <p class="ltx_p" id="A7.p2.5">However, in pratice, there might be residual noise in the iteration <math alttext="{\mathbf{x}}_{k}" class="ltx_Math" display="inline" id="A7.p2.1.m1.1"><semantics id="A7.p2.1.m1.1a"><msub id="A7.p2.1.m1.1.1" xref="A7.p2.1.m1.1.1.cmml"><mi id="A7.p2.1.m1.1.1.2" xref="A7.p2.1.m1.1.1.2.cmml">𝐱</mi><mi id="A7.p2.1.m1.1.1.3" xref="A7.p2.1.m1.1.1.3.cmml">k</mi></msub><annotation-xml encoding="MathML-Content" id="A7.p2.1.m1.1b"><apply id="A7.p2.1.m1.1.1.cmml" xref="A7.p2.1.m1.1.1"><csymbol cd="ambiguous" id="A7.p2.1.m1.1.1.1.cmml" xref="A7.p2.1.m1.1.1">subscript</csymbol><ci id="A7.p2.1.m1.1.1.2.cmml" xref="A7.p2.1.m1.1.1.2">𝐱</ci><ci id="A7.p2.1.m1.1.1.3.cmml" xref="A7.p2.1.m1.1.1.3">𝑘</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A7.p2.1.m1.1c">{\mathbf{x}}_{k}</annotation><annotation encoding="application/x-llamapun" id="A7.p2.1.m1.1d">bold_x start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT</annotation></semantics></math>, so the noise level of <math alttext="{\tilde{\mathbf{x}}}_{k}" class="ltx_Math" display="inline" id="A7.p2.2.m2.1"><semantics id="A7.p2.2.m2.1a"><msub id="A7.p2.2.m2.1.1" xref="A7.p2.2.m2.1.1.cmml"><mover accent="true" id="A7.p2.2.m2.1.1.2" xref="A7.p2.2.m2.1.1.2.cmml"><mi id="A7.p2.2.m2.1.1.2.2" xref="A7.p2.2.m2.1.1.2.2.cmml">𝐱</mi><mo id="A7.p2.2.m2.1.1.2.1" xref="A7.p2.2.m2.1.1.2.1.cmml">~</mo></mover><mi id="A7.p2.2.m2.1.1.3" xref="A7.p2.2.m2.1.1.3.cmml">k</mi></msub><annotation-xml encoding="MathML-Content" id="A7.p2.2.m2.1b"><apply id="A7.p2.2.m2.1.1.cmml" xref="A7.p2.2.m2.1.1"><csymbol cd="ambiguous" id="A7.p2.2.m2.1.1.1.cmml" xref="A7.p2.2.m2.1.1">subscript</csymbol><apply id="A7.p2.2.m2.1.1.2.cmml" xref="A7.p2.2.m2.1.1.2"><ci id="A7.p2.2.m2.1.1.2.1.cmml" xref="A7.p2.2.m2.1.1.2.1">~</ci><ci id="A7.p2.2.m2.1.1.2.2.cmml" xref="A7.p2.2.m2.1.1.2.2">𝐱</ci></apply><ci id="A7.p2.2.m2.1.1.3.cmml" xref="A7.p2.2.m2.1.1.3">𝑘</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A7.p2.2.m2.1c">{\tilde{\mathbf{x}}}_{k}</annotation><annotation encoding="application/x-llamapun" id="A7.p2.2.m2.1d">over~ start_ARG bold_x end_ARG start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT</annotation></semantics></math> might be higher than <math alttext="\sigma" class="ltx_Math" display="inline" id="A7.p2.3.m3.1"><semantics id="A7.p2.3.m3.1a"><mi id="A7.p2.3.m3.1.1" xref="A7.p2.3.m3.1.1.cmml">σ</mi><annotation-xml encoding="MathML-Content" id="A7.p2.3.m3.1b"><ci id="A7.p2.3.m3.1.1.cmml" xref="A7.p2.3.m3.1.1">𝜎</ci></annotation-xml><annotation encoding="application/x-tex" id="A7.p2.3.m3.1c">\sigma</annotation><annotation encoding="application/x-llamapun" id="A7.p2.3.m3.1d">italic_σ</annotation></semantics></math>. In order to verify this experimentally, we use a robust wavelet-based noise estimator <cite class="ltx_cite ltx_citemacro_citep">(Donoho & Johnstone, <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#bib.bib20" title="">1994</a>)</cite> implemented in the library library scikit-image (aka skimage) as the function estimate_sigma(). We define <math alttext="\hat{\sigma}({\mathbf{x}})" class="ltx_Math" display="inline" id="A7.p2.4.m4.1"><semantics id="A7.p2.4.m4.1a"><mrow id="A7.p2.4.m4.1.2" xref="A7.p2.4.m4.1.2.cmml"><mover accent="true" id="A7.p2.4.m4.1.2.2" xref="A7.p2.4.m4.1.2.2.cmml"><mi id="A7.p2.4.m4.1.2.2.2" xref="A7.p2.4.m4.1.2.2.2.cmml">σ</mi><mo id="A7.p2.4.m4.1.2.2.1" xref="A7.p2.4.m4.1.2.2.1.cmml">^</mo></mover><mo id="A7.p2.4.m4.1.2.1" xref="A7.p2.4.m4.1.2.1.cmml">⁢</mo><mrow id="A7.p2.4.m4.1.2.3.2" xref="A7.p2.4.m4.1.2.cmml"><mo id="A7.p2.4.m4.1.2.3.2.1" stretchy="false" xref="A7.p2.4.m4.1.2.cmml">(</mo><mi id="A7.p2.4.m4.1.1" xref="A7.p2.4.m4.1.1.cmml">𝐱</mi><mo id="A7.p2.4.m4.1.2.3.2.2" stretchy="false" xref="A7.p2.4.m4.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A7.p2.4.m4.1b"><apply id="A7.p2.4.m4.1.2.cmml" xref="A7.p2.4.m4.1.2"><times id="A7.p2.4.m4.1.2.1.cmml" xref="A7.p2.4.m4.1.2.1"></times><apply id="A7.p2.4.m4.1.2.2.cmml" xref="A7.p2.4.m4.1.2.2"><ci id="A7.p2.4.m4.1.2.2.1.cmml" xref="A7.p2.4.m4.1.2.2.1">^</ci><ci id="A7.p2.4.m4.1.2.2.2.cmml" xref="A7.p2.4.m4.1.2.2.2">𝜎</ci></apply><ci id="A7.p2.4.m4.1.1.cmml" xref="A7.p2.4.m4.1.1">𝐱</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A7.p2.4.m4.1c">\hat{\sigma}({\mathbf{x}})</annotation><annotation encoding="application/x-llamapun" id="A7.p2.4.m4.1d">over^ start_ARG italic_σ end_ARG ( bold_x )</annotation></semantics></math> the noise estimation of the image <math alttext="{\mathbf{x}}" class="ltx_Math" display="inline" id="A7.p2.5.m5.1"><semantics id="A7.p2.5.m5.1a"><mi id="A7.p2.5.m5.1.1" xref="A7.p2.5.m5.1.1.cmml">𝐱</mi><annotation-xml encoding="MathML-Content" id="A7.p2.5.m5.1b"><ci id="A7.p2.5.m5.1.1.cmml" xref="A7.p2.5.m5.1.1">𝐱</ci></annotation-xml><annotation encoding="application/x-tex" id="A7.p2.5.m5.1c">{\mathbf{x}}</annotation><annotation encoding="application/x-llamapun" id="A7.p2.5.m5.1d">bold_x</annotation></semantics></math>.</p> </div> <figure class="ltx_figure" id="A7.F16"><img alt="Refer to caption" class="ltx_graphics ltx_centering ltx_img_square" height="299" id="A7.F16.g1" src="extracted/5518834/images/noise_input_denoiser_inp.png" width="299"/> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure"><span class="ltx_text" id="A7.F16.8.4.1" style="font-size:90%;">Figure 16</span>: </span><span class="ltx_text" id="A7.F16.6.3" style="font-size:90%;">Difference between the estimated noise <math alttext="\hat{\sigma}" class="ltx_Math" display="inline" id="A7.F16.4.1.m1.1"><semantics id="A7.F16.4.1.m1.1b"><mover accent="true" id="A7.F16.4.1.m1.1.1" xref="A7.F16.4.1.m1.1.1.cmml"><mi id="A7.F16.4.1.m1.1.1.2" xref="A7.F16.4.1.m1.1.1.2.cmml">σ</mi><mo id="A7.F16.4.1.m1.1.1.1" xref="A7.F16.4.1.m1.1.1.1.cmml">^</mo></mover><annotation-xml encoding="MathML-Content" id="A7.F16.4.1.m1.1c"><apply id="A7.F16.4.1.m1.1.1.cmml" xref="A7.F16.4.1.m1.1.1"><ci id="A7.F16.4.1.m1.1.1.1.cmml" xref="A7.F16.4.1.m1.1.1.1">^</ci><ci id="A7.F16.4.1.m1.1.1.2.cmml" xref="A7.F16.4.1.m1.1.1.2">𝜎</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A7.F16.4.1.m1.1d">\hat{\sigma}</annotation><annotation encoding="application/x-llamapun" id="A7.F16.4.1.m1.1e">over^ start_ARG italic_σ end_ARG</annotation></semantics></math> and the denoiser parameter <math alttext="\sigma" class="ltx_Math" display="inline" id="A7.F16.5.2.m2.1"><semantics id="A7.F16.5.2.m2.1b"><mi id="A7.F16.5.2.m2.1.1" xref="A7.F16.5.2.m2.1.1.cmml">σ</mi><annotation-xml encoding="MathML-Content" id="A7.F16.5.2.m2.1c"><ci id="A7.F16.5.2.m2.1.1.cmml" xref="A7.F16.5.2.m2.1.1">𝜎</ci></annotation-xml><annotation encoding="application/x-tex" id="A7.F16.5.2.m2.1d">\sigma</annotation><annotation encoding="application/x-llamapun" id="A7.F16.5.2.m2.1e">italic_σ</annotation></semantics></math> for iterations of SNORE and RED algorithms for inpainting (with a proportion <math alttext="p=0.5" class="ltx_Math" display="inline" id="A7.F16.6.3.m3.1"><semantics id="A7.F16.6.3.m3.1b"><mrow id="A7.F16.6.3.m3.1.1" xref="A7.F16.6.3.m3.1.1.cmml"><mi id="A7.F16.6.3.m3.1.1.2" xref="A7.F16.6.3.m3.1.1.2.cmml">p</mi><mo id="A7.F16.6.3.m3.1.1.1" xref="A7.F16.6.3.m3.1.1.1.cmml">=</mo><mn id="A7.F16.6.3.m3.1.1.3" xref="A7.F16.6.3.m3.1.1.3.cmml">0.5</mn></mrow><annotation-xml encoding="MathML-Content" id="A7.F16.6.3.m3.1c"><apply id="A7.F16.6.3.m3.1.1.cmml" xref="A7.F16.6.3.m3.1.1"><eq id="A7.F16.6.3.m3.1.1.1.cmml" xref="A7.F16.6.3.m3.1.1.1"></eq><ci id="A7.F16.6.3.m3.1.1.2.cmml" xref="A7.F16.6.3.m3.1.1.2">𝑝</ci><cn id="A7.F16.6.3.m3.1.1.3.cmml" type="float" xref="A7.F16.6.3.m3.1.1.3">0.5</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A7.F16.6.3.m3.1d">p=0.5</annotation><annotation encoding="application/x-llamapun" id="A7.F16.6.3.m3.1e">italic_p = 0.5</annotation></semantics></math> of masked pixels) on one image of the dataset CBSD68. In orange, the estimated noise on the clean image. </span></figcaption> </figure> <div class="ltx_para" id="A7.p3"> <p class="ltx_p" id="A7.p3.1">On Figure <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#A7.F16" title="Figure 16 ‣ Appendix G SNORE applies the denoiser on its training domain ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_tag">16</span></a>, we see that the annealing level are not visible for SNORE, which suggests that the algorithm is well adapt to the noise level. Moreover, the residual noise is decreasing and converging to the natural noise shift for SNORE. After a first phase of removing the noise of the initialization, SNORE algorithm effectively applies the denoiser to an image with the right noise level. For RED, the shift between the input noise level and the noise level of the denoiser is still large at convergence. This suggests that the denoiser is applied on an image out-of-distribution (without the right level of noise) at convergence.</p> </div> </section> <section class="ltx_appendix" id="A8"> <h2 class="ltx_title ltx_title_appendix"> <span class="ltx_tag ltx_tag_appendix">Appendix H </span>Discussion on the boundedness of <math alttext="({\mathbf{x}}_{k})_{k\in{\mathbb{N}}}" class="ltx_Math" display="inline" id="A8.1.m1.1"><semantics id="A8.1.m1.1b"><msub id="A8.1.m1.1.1" xref="A8.1.m1.1.1.cmml"><mrow id="A8.1.m1.1.1.1.1" xref="A8.1.m1.1.1.1.1.1.cmml"><mo id="A8.1.m1.1.1.1.1.2" stretchy="false" xref="A8.1.m1.1.1.1.1.1.cmml">(</mo><msub id="A8.1.m1.1.1.1.1.1" xref="A8.1.m1.1.1.1.1.1.cmml"><mi id="A8.1.m1.1.1.1.1.1.2" xref="A8.1.m1.1.1.1.1.1.2.cmml">𝐱</mi><mi id="A8.1.m1.1.1.1.1.1.3" xref="A8.1.m1.1.1.1.1.1.3.cmml">k</mi></msub><mo id="A8.1.m1.1.1.1.1.3" stretchy="false" xref="A8.1.m1.1.1.1.1.1.cmml">)</mo></mrow><mrow id="A8.1.m1.1.1.3" xref="A8.1.m1.1.1.3.cmml"><mi id="A8.1.m1.1.1.3.2" xref="A8.1.m1.1.1.3.2.cmml">k</mi><mo id="A8.1.m1.1.1.3.1" xref="A8.1.m1.1.1.3.1.cmml">∈</mo><mi id="A8.1.m1.1.1.3.3" xref="A8.1.m1.1.1.3.3.cmml">ℕ</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="A8.1.m1.1c"><apply id="A8.1.m1.1.1.cmml" xref="A8.1.m1.1.1"><csymbol cd="ambiguous" id="A8.1.m1.1.1.2.cmml" xref="A8.1.m1.1.1">subscript</csymbol><apply id="A8.1.m1.1.1.1.1.1.cmml" xref="A8.1.m1.1.1.1.1"><csymbol cd="ambiguous" id="A8.1.m1.1.1.1.1.1.1.cmml" xref="A8.1.m1.1.1.1.1">subscript</csymbol><ci id="A8.1.m1.1.1.1.1.1.2.cmml" xref="A8.1.m1.1.1.1.1.1.2">𝐱</ci><ci id="A8.1.m1.1.1.1.1.1.3.cmml" xref="A8.1.m1.1.1.1.1.1.3">𝑘</ci></apply><apply id="A8.1.m1.1.1.3.cmml" xref="A8.1.m1.1.1.3"><in id="A8.1.m1.1.1.3.1.cmml" xref="A8.1.m1.1.1.3.1"></in><ci id="A8.1.m1.1.1.3.2.cmml" xref="A8.1.m1.1.1.3.2">𝑘</ci><ci id="A8.1.m1.1.1.3.3.cmml" xref="A8.1.m1.1.1.3.3">ℕ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A8.1.m1.1d">({\mathbf{x}}_{k})_{k\in{\mathbb{N}}}</annotation><annotation encoding="application/x-llamapun" id="A8.1.m1.1e">( bold_x start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) start_POSTSUBSCRIPT italic_k ∈ blackboard_N end_POSTSUBSCRIPT</annotation></semantics></math> </h2> <div class="ltx_para" id="A8.p1"> <p class="ltx_p" id="A8.p1.3">In Proposition <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#S3.Thmproposition3" title="Proposition 3.3. ‣ 3.2 Unbiased algorithm analysis ‣ 3 Convergence Analysis ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_tag">3.3</span></a>-<a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#S3.Thmproposition5" title="Proposition 3.5. ‣ 3.3 Biased algorithm analysis ‣ 3 Convergence Analysis ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_tag">3.5</span></a>, the convergence of Algorithm <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#alg2" title="Algorithm 2 ‣ Optimization algorithms ‣ 2.2 SNORE regularization ‣ 2 Stochastic deNOising REgularization (SNORE) ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_tag">2</span></a> is studied almost surely on <math alttext="\Lambda_{\mathbf{K}}" class="ltx_Math" display="inline" id="A8.p1.1.m1.1"><semantics id="A8.p1.1.m1.1a"><msub id="A8.p1.1.m1.1.1" xref="A8.p1.1.m1.1.1.cmml"><mi id="A8.p1.1.m1.1.1.2" mathvariant="normal" xref="A8.p1.1.m1.1.1.2.cmml">Λ</mi><mi id="A8.p1.1.m1.1.1.3" xref="A8.p1.1.m1.1.1.3.cmml">𝐊</mi></msub><annotation-xml encoding="MathML-Content" id="A8.p1.1.m1.1b"><apply id="A8.p1.1.m1.1.1.cmml" xref="A8.p1.1.m1.1.1"><csymbol cd="ambiguous" id="A8.p1.1.m1.1.1.1.cmml" xref="A8.p1.1.m1.1.1">subscript</csymbol><ci id="A8.p1.1.m1.1.1.2.cmml" xref="A8.p1.1.m1.1.1.2">Λ</ci><ci id="A8.p1.1.m1.1.1.3.cmml" xref="A8.p1.1.m1.1.1.3">𝐊</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A8.p1.1.m1.1c">\Lambda_{\mathbf{K}}</annotation><annotation encoding="application/x-llamapun" id="A8.p1.1.m1.1d">roman_Λ start_POSTSUBSCRIPT bold_K end_POSTSUBSCRIPT</annotation></semantics></math>, the set of realizations where <math alttext="({\mathbf{x}}_{k})_{k\in{\mathbb{N}}}" class="ltx_Math" display="inline" id="A8.p1.2.m2.1"><semantics id="A8.p1.2.m2.1a"><msub id="A8.p1.2.m2.1.1" xref="A8.p1.2.m2.1.1.cmml"><mrow id="A8.p1.2.m2.1.1.1.1" xref="A8.p1.2.m2.1.1.1.1.1.cmml"><mo id="A8.p1.2.m2.1.1.1.1.2" stretchy="false" xref="A8.p1.2.m2.1.1.1.1.1.cmml">(</mo><msub id="A8.p1.2.m2.1.1.1.1.1" xref="A8.p1.2.m2.1.1.1.1.1.cmml"><mi id="A8.p1.2.m2.1.1.1.1.1.2" xref="A8.p1.2.m2.1.1.1.1.1.2.cmml">𝐱</mi><mi id="A8.p1.2.m2.1.1.1.1.1.3" xref="A8.p1.2.m2.1.1.1.1.1.3.cmml">k</mi></msub><mo id="A8.p1.2.m2.1.1.1.1.3" stretchy="false" xref="A8.p1.2.m2.1.1.1.1.1.cmml">)</mo></mrow><mrow id="A8.p1.2.m2.1.1.3" xref="A8.p1.2.m2.1.1.3.cmml"><mi id="A8.p1.2.m2.1.1.3.2" xref="A8.p1.2.m2.1.1.3.2.cmml">k</mi><mo id="A8.p1.2.m2.1.1.3.1" xref="A8.p1.2.m2.1.1.3.1.cmml">∈</mo><mi id="A8.p1.2.m2.1.1.3.3" xref="A8.p1.2.m2.1.1.3.3.cmml">ℕ</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="A8.p1.2.m2.1b"><apply id="A8.p1.2.m2.1.1.cmml" xref="A8.p1.2.m2.1.1"><csymbol cd="ambiguous" id="A8.p1.2.m2.1.1.2.cmml" xref="A8.p1.2.m2.1.1">subscript</csymbol><apply id="A8.p1.2.m2.1.1.1.1.1.cmml" xref="A8.p1.2.m2.1.1.1.1"><csymbol cd="ambiguous" id="A8.p1.2.m2.1.1.1.1.1.1.cmml" xref="A8.p1.2.m2.1.1.1.1">subscript</csymbol><ci id="A8.p1.2.m2.1.1.1.1.1.2.cmml" xref="A8.p1.2.m2.1.1.1.1.1.2">𝐱</ci><ci id="A8.p1.2.m2.1.1.1.1.1.3.cmml" xref="A8.p1.2.m2.1.1.1.1.1.3">𝑘</ci></apply><apply id="A8.p1.2.m2.1.1.3.cmml" xref="A8.p1.2.m2.1.1.3"><in id="A8.p1.2.m2.1.1.3.1.cmml" xref="A8.p1.2.m2.1.1.3.1"></in><ci id="A8.p1.2.m2.1.1.3.2.cmml" xref="A8.p1.2.m2.1.1.3.2">𝑘</ci><ci 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In what follows, we name that the <span class="ltx_text ltx_font_italic" id="A8.p1.3.1">boundedness assumption</span>. That kind of assumption is standard in stochastic gradient descent analysis with non-convex objective functions <cite class="ltx_cite ltx_citemacro_citep">(Benaïm, <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#bib.bib2" title="">1999</a>; Tadic & Doucet, <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#bib.bib72" title="">2017</a>)</cite>. However, one can remark that in similar non-stochastic Plug-and-Play methods <cite class="ltx_cite ltx_citemacro_citep">(Hurault et al., <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#bib.bib30" title="">2022a</a>, Appendix D)</cite> or in posterior sampling algorithms <cite class="ltx_cite ltx_citemacro_citep">(Laumont et al., <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#bib.bib36" title="">2022a</a>; Renaud et al., <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#bib.bib57" title="">2023</a>)</cite>, a projection or a penalty term can be added to guarantee a bounded sequence. Unfortunately, to our knowledge, a simple projected stochastic gradient descent step is not simple to analyze.</p> </div> <div class="ltx_para" id="A8.p2"> <p class="ltx_p" id="A8.p2.1"><cite class="ltx_cite ltx_citemacro_citet">Davis & Drusvyatskiy (<a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#bib.bib16" title="">2018</a>)</cite> prove convergence of a projected stochastic gradient descent algorithm, but the convergence analysis relies on a random choice of the ending step. <cite class="ltx_cite ltx_citemacro_citet">Ghadimi & Lan (<a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#bib.bib26" title="">2013</a>)</cite> develop a similar approach. The random choice of the ending step is not satisfying in our setting as we want to fix the number of iterations for a fair comparison with deterministic methods.</p> </div> <figure class="ltx_float ltx_float_algorithm ltx_framed_top" id="alg8"> <figcaption class="ltx_caption"><span class="ltx_tag ltx_tag_float"><span class="ltx_text ltx_font_bold" id="alg8.2.1.1">Algorithm 8</span> </span> Randomly Projected SNORE</figcaption> <div class="ltx_listing ltx_listing" id="alg8.3"> <div class="ltx_listingline" id="alg8.l1"> <span class="ltx_tag ltx_tag_listingline"><span class="ltx_text" id="alg8.l1.1.1.1" style="font-size:80%;">1:</span></span> <span class="ltx_text ltx_font_bold" id="alg8.l1.2">input:</span> <math alttext="{\mathbf{x}}_{0}\in{\mathbb{R}}^{d}" class="ltx_Math" display="inline" id="alg8.l1.m1.1"><semantics id="alg8.l1.m1.1a"><mrow id="alg8.l1.m1.1.1" xref="alg8.l1.m1.1.1.cmml"><msub id="alg8.l1.m1.1.1.2" xref="alg8.l1.m1.1.1.2.cmml"><mi id="alg8.l1.m1.1.1.2.2" xref="alg8.l1.m1.1.1.2.2.cmml">𝐱</mi><mn id="alg8.l1.m1.1.1.2.3" xref="alg8.l1.m1.1.1.2.3.cmml">0</mn></msub><mo id="alg8.l1.m1.1.1.1" xref="alg8.l1.m1.1.1.1.cmml">∈</mo><msup id="alg8.l1.m1.1.1.3" xref="alg8.l1.m1.1.1.3.cmml"><mi id="alg8.l1.m1.1.1.3.2" xref="alg8.l1.m1.1.1.3.2.cmml">ℝ</mi><mi id="alg8.l1.m1.1.1.3.3" xref="alg8.l1.m1.1.1.3.3.cmml">d</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="alg8.l1.m1.1b"><apply id="alg8.l1.m1.1.1.cmml" xref="alg8.l1.m1.1.1"><in id="alg8.l1.m1.1.1.1.cmml" xref="alg8.l1.m1.1.1.1"></in><apply id="alg8.l1.m1.1.1.2.cmml" xref="alg8.l1.m1.1.1.2"><csymbol cd="ambiguous" id="alg8.l1.m1.1.1.2.1.cmml" xref="alg8.l1.m1.1.1.2">subscript</csymbol><ci id="alg8.l1.m1.1.1.2.2.cmml" xref="alg8.l1.m1.1.1.2.2">𝐱</ci><cn id="alg8.l1.m1.1.1.2.3.cmml" type="integer" xref="alg8.l1.m1.1.1.2.3">0</cn></apply><apply id="alg8.l1.m1.1.1.3.cmml" xref="alg8.l1.m1.1.1.3"><csymbol cd="ambiguous" id="alg8.l1.m1.1.1.3.1.cmml" xref="alg8.l1.m1.1.1.3">superscript</csymbol><ci id="alg8.l1.m1.1.1.3.2.cmml" xref="alg8.l1.m1.1.1.3.2">ℝ</ci><ci id="alg8.l1.m1.1.1.3.3.cmml" xref="alg8.l1.m1.1.1.3.3">𝑑</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="alg8.l1.m1.1c">{\mathbf{x}}_{0}\in{\mathbb{R}}^{d}</annotation><annotation encoding="application/x-llamapun" id="alg8.l1.m1.1d">bold_x start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ∈ blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT</annotation></semantics></math>, <math alttext="m\in{\mathbb{N}}" class="ltx_Math" display="inline" id="alg8.l1.m2.1"><semantics id="alg8.l1.m2.1a"><mrow id="alg8.l1.m2.1.1" xref="alg8.l1.m2.1.1.cmml"><mi id="alg8.l1.m2.1.1.2" xref="alg8.l1.m2.1.1.2.cmml">m</mi><mo id="alg8.l1.m2.1.1.1" xref="alg8.l1.m2.1.1.1.cmml">∈</mo><mi id="alg8.l1.m2.1.1.3" xref="alg8.l1.m2.1.1.3.cmml">ℕ</mi></mrow><annotation-xml encoding="MathML-Content" id="alg8.l1.m2.1b"><apply id="alg8.l1.m2.1.1.cmml" xref="alg8.l1.m2.1.1"><in id="alg8.l1.m2.1.1.1.cmml" xref="alg8.l1.m2.1.1.1"></in><ci id="alg8.l1.m2.1.1.2.cmml" xref="alg8.l1.m2.1.1.2">𝑚</ci><ci id="alg8.l1.m2.1.1.3.cmml" xref="alg8.l1.m2.1.1.3">ℕ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="alg8.l1.m2.1c">m\in{\mathbb{N}}</annotation><annotation encoding="application/x-llamapun" id="alg8.l1.m2.1d">italic_m ∈ blackboard_N</annotation></semantics></math>, <math alttext="\delta>0" class="ltx_Math" display="inline" id="alg8.l1.m3.1"><semantics id="alg8.l1.m3.1a"><mrow id="alg8.l1.m3.1.1" xref="alg8.l1.m3.1.1.cmml"><mi id="alg8.l1.m3.1.1.2" xref="alg8.l1.m3.1.1.2.cmml">δ</mi><mo id="alg8.l1.m3.1.1.1" xref="alg8.l1.m3.1.1.1.cmml">></mo><mn id="alg8.l1.m3.1.1.3" xref="alg8.l1.m3.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="alg8.l1.m3.1b"><apply id="alg8.l1.m3.1.1.cmml" xref="alg8.l1.m3.1.1"><gt id="alg8.l1.m3.1.1.1.cmml" xref="alg8.l1.m3.1.1.1"></gt><ci id="alg8.l1.m3.1.1.2.cmml" xref="alg8.l1.m3.1.1.2">𝛿</ci><cn id="alg8.l1.m3.1.1.3.cmml" type="integer" xref="alg8.l1.m3.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="alg8.l1.m3.1c">\delta>0</annotation><annotation encoding="application/x-llamapun" id="alg8.l1.m3.1d">italic_δ > 0</annotation></semantics></math>, <math alttext="\sigma>0" class="ltx_Math" display="inline" id="alg8.l1.m4.1"><semantics id="alg8.l1.m4.1a"><mrow id="alg8.l1.m4.1.1" xref="alg8.l1.m4.1.1.cmml"><mi id="alg8.l1.m4.1.1.2" xref="alg8.l1.m4.1.1.2.cmml">σ</mi><mo id="alg8.l1.m4.1.1.1" xref="alg8.l1.m4.1.1.1.cmml">></mo><mn id="alg8.l1.m4.1.1.3" xref="alg8.l1.m4.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="alg8.l1.m4.1b"><apply id="alg8.l1.m4.1.1.cmml" xref="alg8.l1.m4.1.1"><gt id="alg8.l1.m4.1.1.1.cmml" xref="alg8.l1.m4.1.1.1"></gt><ci id="alg8.l1.m4.1.1.2.cmml" xref="alg8.l1.m4.1.1.2">𝜎</ci><cn id="alg8.l1.m4.1.1.3.cmml" type="integer" xref="alg8.l1.m4.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="alg8.l1.m4.1c">\sigma>0</annotation><annotation encoding="application/x-llamapun" id="alg8.l1.m4.1d">italic_σ > 0</annotation></semantics></math>, <math alttext="\alpha>0" class="ltx_Math" display="inline" id="alg8.l1.m5.1"><semantics id="alg8.l1.m5.1a"><mrow id="alg8.l1.m5.1.1" xref="alg8.l1.m5.1.1.cmml"><mi id="alg8.l1.m5.1.1.2" xref="alg8.l1.m5.1.1.2.cmml">α</mi><mo id="alg8.l1.m5.1.1.1" xref="alg8.l1.m5.1.1.1.cmml">></mo><mn id="alg8.l1.m5.1.1.3" xref="alg8.l1.m5.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="alg8.l1.m5.1b"><apply id="alg8.l1.m5.1.1.cmml" xref="alg8.l1.m5.1.1"><gt id="alg8.l1.m5.1.1.1.cmml" xref="alg8.l1.m5.1.1.1"></gt><ci id="alg8.l1.m5.1.1.2.cmml" xref="alg8.l1.m5.1.1.2">𝛼</ci><cn id="alg8.l1.m5.1.1.3.cmml" type="integer" xref="alg8.l1.m5.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="alg8.l1.m5.1c">\alpha>0</annotation><annotation encoding="application/x-llamapun" id="alg8.l1.m5.1d">italic_α > 0</annotation></semantics></math>, <math alttext="N\in{\mathbb{N}}" class="ltx_Math" display="inline" id="alg8.l1.m6.1"><semantics id="alg8.l1.m6.1a"><mrow id="alg8.l1.m6.1.1" xref="alg8.l1.m6.1.1.cmml"><mi id="alg8.l1.m6.1.1.2" xref="alg8.l1.m6.1.1.2.cmml">N</mi><mo id="alg8.l1.m6.1.1.1" xref="alg8.l1.m6.1.1.1.cmml">∈</mo><mi id="alg8.l1.m6.1.1.3" xref="alg8.l1.m6.1.1.3.cmml">ℕ</mi></mrow><annotation-xml encoding="MathML-Content" id="alg8.l1.m6.1b"><apply id="alg8.l1.m6.1.1.cmml" xref="alg8.l1.m6.1.1"><in id="alg8.l1.m6.1.1.1.cmml" xref="alg8.l1.m6.1.1.1"></in><ci id="alg8.l1.m6.1.1.2.cmml" xref="alg8.l1.m6.1.1.2">𝑁</ci><ci id="alg8.l1.m6.1.1.3.cmml" xref="alg8.l1.m6.1.1.3">ℕ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="alg8.l1.m6.1c">N\in{\mathbb{N}}</annotation><annotation encoding="application/x-llamapun" id="alg8.l1.m6.1d">italic_N ∈ blackboard_N</annotation></semantics></math>, <math alttext="\beta_{0}>0" class="ltx_Math" display="inline" id="alg8.l1.m7.1"><semantics id="alg8.l1.m7.1a"><mrow id="alg8.l1.m7.1.1" xref="alg8.l1.m7.1.1.cmml"><msub id="alg8.l1.m7.1.1.2" xref="alg8.l1.m7.1.1.2.cmml"><mi id="alg8.l1.m7.1.1.2.2" xref="alg8.l1.m7.1.1.2.2.cmml">β</mi><mn id="alg8.l1.m7.1.1.2.3" xref="alg8.l1.m7.1.1.2.3.cmml">0</mn></msub><mo id="alg8.l1.m7.1.1.1" xref="alg8.l1.m7.1.1.1.cmml">></mo><mn id="alg8.l1.m7.1.1.3" xref="alg8.l1.m7.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="alg8.l1.m7.1b"><apply id="alg8.l1.m7.1.1.cmml" xref="alg8.l1.m7.1.1"><gt id="alg8.l1.m7.1.1.1.cmml" xref="alg8.l1.m7.1.1.1"></gt><apply id="alg8.l1.m7.1.1.2.cmml" xref="alg8.l1.m7.1.1.2"><csymbol cd="ambiguous" id="alg8.l1.m7.1.1.2.1.cmml" xref="alg8.l1.m7.1.1.2">subscript</csymbol><ci id="alg8.l1.m7.1.1.2.2.cmml" xref="alg8.l1.m7.1.1.2.2">𝛽</ci><cn id="alg8.l1.m7.1.1.2.3.cmml" type="integer" xref="alg8.l1.m7.1.1.2.3">0</cn></apply><cn id="alg8.l1.m7.1.1.3.cmml" type="integer" xref="alg8.l1.m7.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="alg8.l1.m7.1c">\beta_{0}>0</annotation><annotation encoding="application/x-llamapun" id="alg8.l1.m7.1d">italic_β start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT > 0</annotation></semantics></math>, <math alttext="\lambda_{0}=0" class="ltx_Math" display="inline" id="alg8.l1.m8.1"><semantics id="alg8.l1.m8.1a"><mrow id="alg8.l1.m8.1.1" xref="alg8.l1.m8.1.1.cmml"><msub id="alg8.l1.m8.1.1.2" xref="alg8.l1.m8.1.1.2.cmml"><mi id="alg8.l1.m8.1.1.2.2" xref="alg8.l1.m8.1.1.2.2.cmml">λ</mi><mn id="alg8.l1.m8.1.1.2.3" xref="alg8.l1.m8.1.1.2.3.cmml">0</mn></msub><mo id="alg8.l1.m8.1.1.1" xref="alg8.l1.m8.1.1.1.cmml">=</mo><mn id="alg8.l1.m8.1.1.3" xref="alg8.l1.m8.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="alg8.l1.m8.1b"><apply id="alg8.l1.m8.1.1.cmml" xref="alg8.l1.m8.1.1"><eq 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display="inline" id="alg8.l2.m1.4"><semantics id="alg8.l2.m1.4a"><mrow id="alg8.l2.m1.4.4" xref="alg8.l2.m1.4.4.cmml"><mi id="alg8.l2.m1.4.4.3" xref="alg8.l2.m1.4.4.3.cmml">k</mi><mo id="alg8.l2.m1.4.4.2" xref="alg8.l2.m1.4.4.2.cmml">=</mo><mrow id="alg8.l2.m1.4.4.1.1" xref="alg8.l2.m1.4.4.1.2.cmml"><mn id="alg8.l2.m1.1.1" xref="alg8.l2.m1.1.1.cmml">0</mn><mo id="alg8.l2.m1.4.4.1.1.2" xref="alg8.l2.m1.4.4.1.2.cmml">,</mo><mn id="alg8.l2.m1.2.2" xref="alg8.l2.m1.2.2.cmml">1</mn><mo id="alg8.l2.m1.4.4.1.1.3" xref="alg8.l2.m1.4.4.1.2.cmml">,</mo><mi id="alg8.l2.m1.3.3" mathvariant="normal" xref="alg8.l2.m1.3.3.cmml">…</mi><mo id="alg8.l2.m1.4.4.1.1.4" xref="alg8.l2.m1.4.4.1.2.cmml">,</mo><mrow id="alg8.l2.m1.4.4.1.1.1" xref="alg8.l2.m1.4.4.1.1.1.cmml"><mi id="alg8.l2.m1.4.4.1.1.1.2" xref="alg8.l2.m1.4.4.1.1.1.2.cmml">N</mi><mo id="alg8.l2.m1.4.4.1.1.1.1" xref="alg8.l2.m1.4.4.1.1.1.1.cmml">−</mo><mn id="alg8.l2.m1.4.4.1.1.1.3" xref="alg8.l2.m1.4.4.1.1.1.3.cmml">1</mn></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="alg8.l2.m1.4b"><apply id="alg8.l2.m1.4.4.cmml" xref="alg8.l2.m1.4.4"><eq id="alg8.l2.m1.4.4.2.cmml" xref="alg8.l2.m1.4.4.2"></eq><ci id="alg8.l2.m1.4.4.3.cmml" xref="alg8.l2.m1.4.4.3">𝑘</ci><list id="alg8.l2.m1.4.4.1.2.cmml" xref="alg8.l2.m1.4.4.1.1"><cn id="alg8.l2.m1.1.1.cmml" type="integer" xref="alg8.l2.m1.1.1">0</cn><cn id="alg8.l2.m1.2.2.cmml" type="integer" xref="alg8.l2.m1.2.2">1</cn><ci id="alg8.l2.m1.3.3.cmml" xref="alg8.l2.m1.3.3">…</ci><apply id="alg8.l2.m1.4.4.1.1.1.cmml" xref="alg8.l2.m1.4.4.1.1.1"><minus id="alg8.l2.m1.4.4.1.1.1.1.cmml" xref="alg8.l2.m1.4.4.1.1.1.1"></minus><ci id="alg8.l2.m1.4.4.1.1.1.2.cmml" xref="alg8.l2.m1.4.4.1.1.1.2">𝑁</ci><cn id="alg8.l2.m1.4.4.1.1.1.3.cmml" type="integer" xref="alg8.l2.m1.4.4.1.1.1.3">1</cn></apply></list></apply></annotation-xml><annotation encoding="application/x-tex" id="alg8.l2.m1.4c">k=0,1,\dots,N-1</annotation><annotation encoding="application/x-llamapun" id="alg8.l2.m1.4d">italic_k = 0 , 1 , … , italic_N - 1</annotation></semantics></math> <span class="ltx_text ltx_font_bold" id="alg8.l2.3">do</span> </div> <div class="ltx_listingline" id="alg8.l3"> <span class="ltx_tag ltx_tag_listingline"><span class="ltx_text" id="alg8.l3.1.1.1" style="font-size:80%;">3:</span></span> <math alttext="\delta_{k}\leftarrow\frac{\delta}{k+1}" class="ltx_Math" display="inline" id="alg8.l3.m1.1"><semantics id="alg8.l3.m1.1a"><mrow id="alg8.l3.m1.1.1" xref="alg8.l3.m1.1.1.cmml"><msub id="alg8.l3.m1.1.1.2" xref="alg8.l3.m1.1.1.2.cmml"><mi id="alg8.l3.m1.1.1.2.2" xref="alg8.l3.m1.1.1.2.2.cmml">δ</mi><mi id="alg8.l3.m1.1.1.2.3" xref="alg8.l3.m1.1.1.2.3.cmml">k</mi></msub><mo id="alg8.l3.m1.1.1.1" stretchy="false" xref="alg8.l3.m1.1.1.1.cmml">←</mo><mfrac id="alg8.l3.m1.1.1.3" xref="alg8.l3.m1.1.1.3.cmml"><mi id="alg8.l3.m1.1.1.3.2" xref="alg8.l3.m1.1.1.3.2.cmml">δ</mi><mrow id="alg8.l3.m1.1.1.3.3" xref="alg8.l3.m1.1.1.3.3.cmml"><mi id="alg8.l3.m1.1.1.3.3.2" xref="alg8.l3.m1.1.1.3.3.2.cmml">k</mi><mo id="alg8.l3.m1.1.1.3.3.1" xref="alg8.l3.m1.1.1.3.3.1.cmml">+</mo><mn id="alg8.l3.m1.1.1.3.3.3" xref="alg8.l3.m1.1.1.3.3.3.cmml">1</mn></mrow></mfrac></mrow><annotation-xml encoding="MathML-Content" id="alg8.l3.m1.1b"><apply id="alg8.l3.m1.1.1.cmml" xref="alg8.l3.m1.1.1"><ci id="alg8.l3.m1.1.1.1.cmml" xref="alg8.l3.m1.1.1.1">←</ci><apply id="alg8.l3.m1.1.1.2.cmml" xref="alg8.l3.m1.1.1.2"><csymbol cd="ambiguous" id="alg8.l3.m1.1.1.2.1.cmml" xref="alg8.l3.m1.1.1.2">subscript</csymbol><ci id="alg8.l3.m1.1.1.2.2.cmml" xref="alg8.l3.m1.1.1.2.2">𝛿</ci><ci id="alg8.l3.m1.1.1.2.3.cmml" xref="alg8.l3.m1.1.1.2.3">𝑘</ci></apply><apply id="alg8.l3.m1.1.1.3.cmml" xref="alg8.l3.m1.1.1.3"><divide id="alg8.l3.m1.1.1.3.1.cmml" xref="alg8.l3.m1.1.1.3"></divide><ci id="alg8.l3.m1.1.1.3.2.cmml" xref="alg8.l3.m1.1.1.3.2">𝛿</ci><apply id="alg8.l3.m1.1.1.3.3.cmml" xref="alg8.l3.m1.1.1.3.3"><plus id="alg8.l3.m1.1.1.3.3.1.cmml" xref="alg8.l3.m1.1.1.3.3.1"></plus><ci id="alg8.l3.m1.1.1.3.3.2.cmml" xref="alg8.l3.m1.1.1.3.3.2">𝑘</ci><cn id="alg8.l3.m1.1.1.3.3.3.cmml" type="integer" xref="alg8.l3.m1.1.1.3.3.3">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="alg8.l3.m1.1c">\delta_{k}\leftarrow\frac{\delta}{k+1}</annotation><annotation encoding="application/x-llamapun" id="alg8.l3.m1.1d">italic_δ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ← divide start_ARG italic_δ end_ARG start_ARG italic_k + 1 end_ARG</annotation></semantics></math> </div> <div class="ltx_listingline" id="alg8.l4"> <span class="ltx_tag ltx_tag_listingline"><span class="ltx_text" id="alg8.l4.1.1.1" style="font-size:80%;">4:</span></span> <math alttext="{\boldsymbol{\epsilon}}\leftarrow\mathcal{N}(0,\mathbf{I}_{d})" class="ltx_Math" display="inline" id="alg8.l4.m1.2"><semantics id="alg8.l4.m1.2a"><mrow id="alg8.l4.m1.2.2" xref="alg8.l4.m1.2.2.cmml"><mi id="alg8.l4.m1.2.2.3" mathvariant="bold-italic" xref="alg8.l4.m1.2.2.3.cmml">ϵ</mi><mo id="alg8.l4.m1.2.2.2" stretchy="false" xref="alg8.l4.m1.2.2.2.cmml">←</mo><mrow id="alg8.l4.m1.2.2.1" xref="alg8.l4.m1.2.2.1.cmml"><mi class="ltx_font_mathcaligraphic" id="alg8.l4.m1.2.2.1.3" xref="alg8.l4.m1.2.2.1.3.cmml">𝒩</mi><mo id="alg8.l4.m1.2.2.1.2" xref="alg8.l4.m1.2.2.1.2.cmml">⁢</mo><mrow id="alg8.l4.m1.2.2.1.1.1" xref="alg8.l4.m1.2.2.1.1.2.cmml"><mo id="alg8.l4.m1.2.2.1.1.1.2" stretchy="false" xref="alg8.l4.m1.2.2.1.1.2.cmml">(</mo><mn id="alg8.l4.m1.1.1" xref="alg8.l4.m1.1.1.cmml">0</mn><mo id="alg8.l4.m1.2.2.1.1.1.3" xref="alg8.l4.m1.2.2.1.1.2.cmml">,</mo><msub id="alg8.l4.m1.2.2.1.1.1.1" xref="alg8.l4.m1.2.2.1.1.1.1.cmml"><mi id="alg8.l4.m1.2.2.1.1.1.1.2" xref="alg8.l4.m1.2.2.1.1.1.1.2.cmml">𝐈</mi><mi id="alg8.l4.m1.2.2.1.1.1.1.3" xref="alg8.l4.m1.2.2.1.1.1.1.3.cmml">d</mi></msub><mo id="alg8.l4.m1.2.2.1.1.1.4" stretchy="false" xref="alg8.l4.m1.2.2.1.1.2.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="alg8.l4.m1.2b"><apply id="alg8.l4.m1.2.2.cmml" xref="alg8.l4.m1.2.2"><ci id="alg8.l4.m1.2.2.2.cmml" xref="alg8.l4.m1.2.2.2">←</ci><ci id="alg8.l4.m1.2.2.3.cmml" xref="alg8.l4.m1.2.2.3">bold-italic-ϵ</ci><apply id="alg8.l4.m1.2.2.1.cmml" xref="alg8.l4.m1.2.2.1"><times id="alg8.l4.m1.2.2.1.2.cmml" xref="alg8.l4.m1.2.2.1.2"></times><ci id="alg8.l4.m1.2.2.1.3.cmml" xref="alg8.l4.m1.2.2.1.3">𝒩</ci><interval closure="open" id="alg8.l4.m1.2.2.1.1.2.cmml" xref="alg8.l4.m1.2.2.1.1.1"><cn id="alg8.l4.m1.1.1.cmml" type="integer" xref="alg8.l4.m1.1.1">0</cn><apply id="alg8.l4.m1.2.2.1.1.1.1.cmml" xref="alg8.l4.m1.2.2.1.1.1.1"><csymbol cd="ambiguous" id="alg8.l4.m1.2.2.1.1.1.1.1.cmml" xref="alg8.l4.m1.2.2.1.1.1.1">subscript</csymbol><ci id="alg8.l4.m1.2.2.1.1.1.1.2.cmml" xref="alg8.l4.m1.2.2.1.1.1.1.2">𝐈</ci><ci id="alg8.l4.m1.2.2.1.1.1.1.3.cmml" xref="alg8.l4.m1.2.2.1.1.1.1.3">𝑑</ci></apply></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="alg8.l4.m1.2c">{\boldsymbol{\epsilon}}\leftarrow\mathcal{N}(0,\mathbf{I}_{d})</annotation><annotation encoding="application/x-llamapun" id="alg8.l4.m1.2d">bold_italic_ϵ ← caligraphic_N ( 0 , bold_I start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT )</annotation></semantics></math> </div> <div class="ltx_listingline" id="alg8.l5"> <span class="ltx_tag ltx_tag_listingline"><span class="ltx_text" id="alg8.l5.1.1.1" style="font-size:80%;">5:</span></span> <math alttext="{\tilde{\mathbf{x}}}_{k}\leftarrow{\mathbf{x}}_{k}+\sigma{\boldsymbol{\epsilon}}" class="ltx_Math" display="inline" id="alg8.l5.m1.1"><semantics id="alg8.l5.m1.1a"><mrow id="alg8.l5.m1.1.1" xref="alg8.l5.m1.1.1.cmml"><msub id="alg8.l5.m1.1.1.2" xref="alg8.l5.m1.1.1.2.cmml"><mover accent="true" id="alg8.l5.m1.1.1.2.2" xref="alg8.l5.m1.1.1.2.2.cmml"><mi id="alg8.l5.m1.1.1.2.2.2" xref="alg8.l5.m1.1.1.2.2.2.cmml">𝐱</mi><mo id="alg8.l5.m1.1.1.2.2.1" xref="alg8.l5.m1.1.1.2.2.1.cmml">~</mo></mover><mi id="alg8.l5.m1.1.1.2.3" xref="alg8.l5.m1.1.1.2.3.cmml">k</mi></msub><mo id="alg8.l5.m1.1.1.1" stretchy="false" xref="alg8.l5.m1.1.1.1.cmml">←</mo><mrow id="alg8.l5.m1.1.1.3" xref="alg8.l5.m1.1.1.3.cmml"><msub id="alg8.l5.m1.1.1.3.2" xref="alg8.l5.m1.1.1.3.2.cmml"><mi id="alg8.l5.m1.1.1.3.2.2" xref="alg8.l5.m1.1.1.3.2.2.cmml">𝐱</mi><mi id="alg8.l5.m1.1.1.3.2.3" xref="alg8.l5.m1.1.1.3.2.3.cmml">k</mi></msub><mo id="alg8.l5.m1.1.1.3.1" xref="alg8.l5.m1.1.1.3.1.cmml">+</mo><mrow id="alg8.l5.m1.1.1.3.3" xref="alg8.l5.m1.1.1.3.3.cmml"><mi id="alg8.l5.m1.1.1.3.3.2" xref="alg8.l5.m1.1.1.3.3.2.cmml">σ</mi><mo id="alg8.l5.m1.1.1.3.3.1" xref="alg8.l5.m1.1.1.3.3.1.cmml">⁢</mo><mi id="alg8.l5.m1.1.1.3.3.3" mathvariant="bold-italic" xref="alg8.l5.m1.1.1.3.3.3.cmml">ϵ</mi></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="alg8.l5.m1.1b"><apply id="alg8.l5.m1.1.1.cmml" xref="alg8.l5.m1.1.1"><ci id="alg8.l5.m1.1.1.1.cmml" xref="alg8.l5.m1.1.1.1">←</ci><apply id="alg8.l5.m1.1.1.2.cmml" xref="alg8.l5.m1.1.1.2"><csymbol cd="ambiguous" id="alg8.l5.m1.1.1.2.1.cmml" xref="alg8.l5.m1.1.1.2">subscript</csymbol><apply id="alg8.l5.m1.1.1.2.2.cmml" xref="alg8.l5.m1.1.1.2.2"><ci id="alg8.l5.m1.1.1.2.2.1.cmml" xref="alg8.l5.m1.1.1.2.2.1">~</ci><ci id="alg8.l5.m1.1.1.2.2.2.cmml" xref="alg8.l5.m1.1.1.2.2.2">𝐱</ci></apply><ci id="alg8.l5.m1.1.1.2.3.cmml" xref="alg8.l5.m1.1.1.2.3">𝑘</ci></apply><apply id="alg8.l5.m1.1.1.3.cmml" xref="alg8.l5.m1.1.1.3"><plus id="alg8.l5.m1.1.1.3.1.cmml" xref="alg8.l5.m1.1.1.3.1"></plus><apply id="alg8.l5.m1.1.1.3.2.cmml" xref="alg8.l5.m1.1.1.3.2"><csymbol cd="ambiguous" id="alg8.l5.m1.1.1.3.2.1.cmml" xref="alg8.l5.m1.1.1.3.2">subscript</csymbol><ci id="alg8.l5.m1.1.1.3.2.2.cmml" xref="alg8.l5.m1.1.1.3.2.2">𝐱</ci><ci id="alg8.l5.m1.1.1.3.2.3.cmml" xref="alg8.l5.m1.1.1.3.2.3">𝑘</ci></apply><apply id="alg8.l5.m1.1.1.3.3.cmml" xref="alg8.l5.m1.1.1.3.3"><times id="alg8.l5.m1.1.1.3.3.1.cmml" xref="alg8.l5.m1.1.1.3.3.1"></times><ci id="alg8.l5.m1.1.1.3.3.2.cmml" xref="alg8.l5.m1.1.1.3.3.2">𝜎</ci><ci id="alg8.l5.m1.1.1.3.3.3.cmml" xref="alg8.l5.m1.1.1.3.3.3">bold-italic-ϵ</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="alg8.l5.m1.1c">{\tilde{\mathbf{x}}}_{k}\leftarrow{\mathbf{x}}_{k}+\sigma{\boldsymbol{\epsilon}}</annotation><annotation encoding="application/x-llamapun" id="alg8.l5.m1.1d">over~ start_ARG bold_x end_ARG start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ← bold_x start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT + italic_σ bold_italic_ϵ</annotation></semantics></math> </div> <div class="ltx_listingline" id="alg8.l6"> <span class="ltx_tag ltx_tag_listingline"><span class="ltx_text" id="alg8.l6.1.1.1" style="font-size:80%;">6:</span></span> <math alttext="{\mathbf{z}}_{k+1}\leftarrow{\mathbf{x}}_{k}-\delta_{k}\nabla\mathcal{F}({% \mathbf{x}}_{k},{\mathbf{y}})-\frac{\alpha\delta_{k}}{\sigma^{2}}\left({% \mathbf{x}}_{k}-D_{\sigma}({\tilde{\mathbf{x}}}_{k})\right)" class="ltx_Math" display="inline" id="alg8.l6.m1.3"><semantics id="alg8.l6.m1.3a"><mrow id="alg8.l6.m1.3.3" xref="alg8.l6.m1.3.3.cmml"><msub id="alg8.l6.m1.3.3.4" xref="alg8.l6.m1.3.3.4.cmml"><mi id="alg8.l6.m1.3.3.4.2" xref="alg8.l6.m1.3.3.4.2.cmml">𝐳</mi><mrow id="alg8.l6.m1.3.3.4.3" xref="alg8.l6.m1.3.3.4.3.cmml"><mi id="alg8.l6.m1.3.3.4.3.2" xref="alg8.l6.m1.3.3.4.3.2.cmml">k</mi><mo id="alg8.l6.m1.3.3.4.3.1" xref="alg8.l6.m1.3.3.4.3.1.cmml">+</mo><mn id="alg8.l6.m1.3.3.4.3.3" xref="alg8.l6.m1.3.3.4.3.3.cmml">1</mn></mrow></msub><mo id="alg8.l6.m1.3.3.3" stretchy="false" xref="alg8.l6.m1.3.3.3.cmml">←</mo><mrow id="alg8.l6.m1.3.3.2" xref="alg8.l6.m1.3.3.2.cmml"><msub id="alg8.l6.m1.3.3.2.4" xref="alg8.l6.m1.3.3.2.4.cmml"><mi id="alg8.l6.m1.3.3.2.4.2" xref="alg8.l6.m1.3.3.2.4.2.cmml">𝐱</mi><mi id="alg8.l6.m1.3.3.2.4.3" xref="alg8.l6.m1.3.3.2.4.3.cmml">k</mi></msub><mo id="alg8.l6.m1.3.3.2.3" xref="alg8.l6.m1.3.3.2.3.cmml">−</mo><mrow id="alg8.l6.m1.2.2.1.1" xref="alg8.l6.m1.2.2.1.1.cmml"><msub id="alg8.l6.m1.2.2.1.1.3" xref="alg8.l6.m1.2.2.1.1.3.cmml"><mi id="alg8.l6.m1.2.2.1.1.3.2" xref="alg8.l6.m1.2.2.1.1.3.2.cmml">δ</mi><mi id="alg8.l6.m1.2.2.1.1.3.3" xref="alg8.l6.m1.2.2.1.1.3.3.cmml">k</mi></msub><mo id="alg8.l6.m1.2.2.1.1.2" lspace="0.167em" xref="alg8.l6.m1.2.2.1.1.2.cmml">⁢</mo><mrow id="alg8.l6.m1.2.2.1.1.4" xref="alg8.l6.m1.2.2.1.1.4.cmml"><mo id="alg8.l6.m1.2.2.1.1.4.1" rspace="0.167em" xref="alg8.l6.m1.2.2.1.1.4.1.cmml">∇</mo><mi class="ltx_font_mathcaligraphic" id="alg8.l6.m1.2.2.1.1.4.2" xref="alg8.l6.m1.2.2.1.1.4.2.cmml">ℱ</mi></mrow><mo id="alg8.l6.m1.2.2.1.1.2a" xref="alg8.l6.m1.2.2.1.1.2.cmml">⁢</mo><mrow id="alg8.l6.m1.2.2.1.1.1.1" xref="alg8.l6.m1.2.2.1.1.1.2.cmml"><mo id="alg8.l6.m1.2.2.1.1.1.1.2" stretchy="false" xref="alg8.l6.m1.2.2.1.1.1.2.cmml">(</mo><msub id="alg8.l6.m1.2.2.1.1.1.1.1" xref="alg8.l6.m1.2.2.1.1.1.1.1.cmml"><mi id="alg8.l6.m1.2.2.1.1.1.1.1.2" xref="alg8.l6.m1.2.2.1.1.1.1.1.2.cmml">𝐱</mi><mi id="alg8.l6.m1.2.2.1.1.1.1.1.3" xref="alg8.l6.m1.2.2.1.1.1.1.1.3.cmml">k</mi></msub><mo id="alg8.l6.m1.2.2.1.1.1.1.3" xref="alg8.l6.m1.2.2.1.1.1.2.cmml">,</mo><mi id="alg8.l6.m1.1.1" xref="alg8.l6.m1.1.1.cmml">𝐲</mi><mo id="alg8.l6.m1.2.2.1.1.1.1.4" stretchy="false" xref="alg8.l6.m1.2.2.1.1.1.2.cmml">)</mo></mrow></mrow><mo id="alg8.l6.m1.3.3.2.3a" xref="alg8.l6.m1.3.3.2.3.cmml">−</mo><mrow id="alg8.l6.m1.3.3.2.2" xref="alg8.l6.m1.3.3.2.2.cmml"><mfrac id="alg8.l6.m1.3.3.2.2.3" xref="alg8.l6.m1.3.3.2.2.3.cmml"><mrow id="alg8.l6.m1.3.3.2.2.3.2" xref="alg8.l6.m1.3.3.2.2.3.2.cmml"><mi id="alg8.l6.m1.3.3.2.2.3.2.2" xref="alg8.l6.m1.3.3.2.2.3.2.2.cmml">α</mi><mo id="alg8.l6.m1.3.3.2.2.3.2.1" xref="alg8.l6.m1.3.3.2.2.3.2.1.cmml">⁢</mo><msub id="alg8.l6.m1.3.3.2.2.3.2.3" xref="alg8.l6.m1.3.3.2.2.3.2.3.cmml"><mi id="alg8.l6.m1.3.3.2.2.3.2.3.2" xref="alg8.l6.m1.3.3.2.2.3.2.3.2.cmml">δ</mi><mi id="alg8.l6.m1.3.3.2.2.3.2.3.3" xref="alg8.l6.m1.3.3.2.2.3.2.3.3.cmml">k</mi></msub></mrow><msup id="alg8.l6.m1.3.3.2.2.3.3" xref="alg8.l6.m1.3.3.2.2.3.3.cmml"><mi id="alg8.l6.m1.3.3.2.2.3.3.2" xref="alg8.l6.m1.3.3.2.2.3.3.2.cmml">σ</mi><mn id="alg8.l6.m1.3.3.2.2.3.3.3" xref="alg8.l6.m1.3.3.2.2.3.3.3.cmml">2</mn></msup></mfrac><mo id="alg8.l6.m1.3.3.2.2.2" xref="alg8.l6.m1.3.3.2.2.2.cmml">⁢</mo><mrow id="alg8.l6.m1.3.3.2.2.1.1" xref="alg8.l6.m1.3.3.2.2.1.1.1.cmml"><mo id="alg8.l6.m1.3.3.2.2.1.1.2" xref="alg8.l6.m1.3.3.2.2.1.1.1.cmml">(</mo><mrow id="alg8.l6.m1.3.3.2.2.1.1.1" xref="alg8.l6.m1.3.3.2.2.1.1.1.cmml"><msub id="alg8.l6.m1.3.3.2.2.1.1.1.3" xref="alg8.l6.m1.3.3.2.2.1.1.1.3.cmml"><mi id="alg8.l6.m1.3.3.2.2.1.1.1.3.2" xref="alg8.l6.m1.3.3.2.2.1.1.1.3.2.cmml">𝐱</mi><mi id="alg8.l6.m1.3.3.2.2.1.1.1.3.3" xref="alg8.l6.m1.3.3.2.2.1.1.1.3.3.cmml">k</mi></msub><mo id="alg8.l6.m1.3.3.2.2.1.1.1.2" 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\mathbf{x}}_{k},{\mathbf{y}})-\frac{\alpha\delta_{k}}{\sigma^{2}}\left({% \mathbf{x}}_{k}-D_{\sigma}({\tilde{\mathbf{x}}}_{k})\right)</annotation><annotation encoding="application/x-llamapun" id="alg8.l6.m1.3d">bold_z start_POSTSUBSCRIPT italic_k + 1 end_POSTSUBSCRIPT ← bold_x start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT - italic_δ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ∇ caligraphic_F ( bold_x start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT , bold_y ) - divide start_ARG italic_α italic_δ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_ARG start_ARG italic_σ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG ( bold_x start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT - italic_D start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ( over~ start_ARG bold_x end_ARG start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) )</annotation></semantics></math> </div> <div class="ltx_listingline" id="alg8.l7"> <span class="ltx_tag ltx_tag_listingline"><span class="ltx_text" id="alg8.l7.1.1.1" style="font-size:80%;">7:</span></span> <math alttext="{\mathbf{x}}_{k+1}\leftarrow{\mathbf{z}}_{k+1}\mathbf{1}_{\|{\mathbf{z}}_{k+1}% \|\leq\beta_{\lambda_{k}}}+{\mathbf{x}}_{0}\mathbf{1}_{\|{\mathbf{z}}_{k+1}\|>% \beta_{\lambda_{k}}}" class="ltx_Math" display="inline" id="alg8.l7.m1.2"><semantics id="alg8.l7.m1.2a"><mrow id="alg8.l7.m1.2.3" xref="alg8.l7.m1.2.3.cmml"><msub id="alg8.l7.m1.2.3.2" xref="alg8.l7.m1.2.3.2.cmml"><mi id="alg8.l7.m1.2.3.2.2" xref="alg8.l7.m1.2.3.2.2.cmml">𝐱</mi><mrow id="alg8.l7.m1.2.3.2.3" xref="alg8.l7.m1.2.3.2.3.cmml"><mi id="alg8.l7.m1.2.3.2.3.2" xref="alg8.l7.m1.2.3.2.3.2.cmml">k</mi><mo id="alg8.l7.m1.2.3.2.3.1" xref="alg8.l7.m1.2.3.2.3.1.cmml">+</mo><mn id="alg8.l7.m1.2.3.2.3.3" xref="alg8.l7.m1.2.3.2.3.3.cmml">1</mn></mrow></msub><mo id="alg8.l7.m1.2.3.1" stretchy="false" xref="alg8.l7.m1.2.3.1.cmml">←</mo><mrow id="alg8.l7.m1.2.3.3" xref="alg8.l7.m1.2.3.3.cmml"><mrow id="alg8.l7.m1.2.3.3.2" xref="alg8.l7.m1.2.3.3.2.cmml"><msub 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xref="alg8.l7.m1.1.1.1.1.1.1.2.cmml">𝐳</mi><mrow id="alg8.l7.m1.1.1.1.1.1.1.3" xref="alg8.l7.m1.1.1.1.1.1.1.3.cmml"><mi id="alg8.l7.m1.1.1.1.1.1.1.3.2" xref="alg8.l7.m1.1.1.1.1.1.1.3.2.cmml">k</mi><mo id="alg8.l7.m1.1.1.1.1.1.1.3.1" xref="alg8.l7.m1.1.1.1.1.1.1.3.1.cmml">+</mo><mn id="alg8.l7.m1.1.1.1.1.1.1.3.3" xref="alg8.l7.m1.1.1.1.1.1.1.3.3.cmml">1</mn></mrow></msub><mo id="alg8.l7.m1.1.1.1.1.1.3" stretchy="false" xref="alg8.l7.m1.1.1.1.1.2.1.cmml">‖</mo></mrow><mo id="alg8.l7.m1.1.1.1.2" xref="alg8.l7.m1.1.1.1.2.cmml">≤</mo><msub id="alg8.l7.m1.1.1.1.3" xref="alg8.l7.m1.1.1.1.3.cmml"><mi id="alg8.l7.m1.1.1.1.3.2" xref="alg8.l7.m1.1.1.1.3.2.cmml">β</mi><msub id="alg8.l7.m1.1.1.1.3.3" xref="alg8.l7.m1.1.1.1.3.3.cmml"><mi id="alg8.l7.m1.1.1.1.3.3.2" xref="alg8.l7.m1.1.1.1.3.3.2.cmml">λ</mi><mi id="alg8.l7.m1.1.1.1.3.3.3" xref="alg8.l7.m1.1.1.1.3.3.3.cmml">k</mi></msub></msub></mrow></msub></mrow><mo id="alg8.l7.m1.2.3.3.1" xref="alg8.l7.m1.2.3.3.1.cmml">+</mo><mrow 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xref="alg8.l7.m1.2.3.3.2.2.3"><plus id="alg8.l7.m1.2.3.3.2.2.3.1.cmml" xref="alg8.l7.m1.2.3.3.2.2.3.1"></plus><ci id="alg8.l7.m1.2.3.3.2.2.3.2.cmml" xref="alg8.l7.m1.2.3.3.2.2.3.2">𝑘</ci><cn id="alg8.l7.m1.2.3.3.2.2.3.3.cmml" type="integer" xref="alg8.l7.m1.2.3.3.2.2.3.3">1</cn></apply></apply><apply id="alg8.l7.m1.2.3.3.2.3.cmml" xref="alg8.l7.m1.2.3.3.2.3"><csymbol cd="ambiguous" id="alg8.l7.m1.2.3.3.2.3.1.cmml" xref="alg8.l7.m1.2.3.3.2.3">subscript</csymbol><cn id="alg8.l7.m1.2.3.3.2.3.2.cmml" type="integer" xref="alg8.l7.m1.2.3.3.2.3.2">1</cn><apply id="alg8.l7.m1.1.1.1.cmml" xref="alg8.l7.m1.1.1.1"><leq id="alg8.l7.m1.1.1.1.2.cmml" xref="alg8.l7.m1.1.1.1.2"></leq><apply id="alg8.l7.m1.1.1.1.1.2.cmml" xref="alg8.l7.m1.1.1.1.1.1"><csymbol cd="latexml" id="alg8.l7.m1.1.1.1.1.2.1.cmml" xref="alg8.l7.m1.1.1.1.1.1.2">norm</csymbol><apply id="alg8.l7.m1.1.1.1.1.1.1.cmml" xref="alg8.l7.m1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="alg8.l7.m1.1.1.1.1.1.1.1.cmml" xref="alg8.l7.m1.1.1.1.1.1.1">subscript</csymbol><ci id="alg8.l7.m1.1.1.1.1.1.1.2.cmml" xref="alg8.l7.m1.1.1.1.1.1.1.2">𝐳</ci><apply id="alg8.l7.m1.1.1.1.1.1.1.3.cmml" xref="alg8.l7.m1.1.1.1.1.1.1.3"><plus id="alg8.l7.m1.1.1.1.1.1.1.3.1.cmml" xref="alg8.l7.m1.1.1.1.1.1.1.3.1"></plus><ci id="alg8.l7.m1.1.1.1.1.1.1.3.2.cmml" xref="alg8.l7.m1.1.1.1.1.1.1.3.2">𝑘</ci><cn id="alg8.l7.m1.1.1.1.1.1.1.3.3.cmml" type="integer" xref="alg8.l7.m1.1.1.1.1.1.1.3.3">1</cn></apply></apply></apply><apply id="alg8.l7.m1.1.1.1.3.cmml" xref="alg8.l7.m1.1.1.1.3"><csymbol cd="ambiguous" id="alg8.l7.m1.1.1.1.3.1.cmml" xref="alg8.l7.m1.1.1.1.3">subscript</csymbol><ci id="alg8.l7.m1.1.1.1.3.2.cmml" xref="alg8.l7.m1.1.1.1.3.2">𝛽</ci><apply id="alg8.l7.m1.1.1.1.3.3.cmml" xref="alg8.l7.m1.1.1.1.3.3"><csymbol cd="ambiguous" id="alg8.l7.m1.1.1.1.3.3.1.cmml" xref="alg8.l7.m1.1.1.1.3.3">subscript</csymbol><ci id="alg8.l7.m1.1.1.1.3.3.2.cmml" xref="alg8.l7.m1.1.1.1.3.3.2">𝜆</ci><ci id="alg8.l7.m1.1.1.1.3.3.3.cmml" xref="alg8.l7.m1.1.1.1.3.3.3">𝑘</ci></apply></apply></apply></apply></apply><apply id="alg8.l7.m1.2.3.3.3.cmml" xref="alg8.l7.m1.2.3.3.3"><times id="alg8.l7.m1.2.3.3.3.1.cmml" xref="alg8.l7.m1.2.3.3.3.1"></times><apply id="alg8.l7.m1.2.3.3.3.2.cmml" xref="alg8.l7.m1.2.3.3.3.2"><csymbol cd="ambiguous" id="alg8.l7.m1.2.3.3.3.2.1.cmml" xref="alg8.l7.m1.2.3.3.3.2">subscript</csymbol><ci id="alg8.l7.m1.2.3.3.3.2.2.cmml" xref="alg8.l7.m1.2.3.3.3.2.2">𝐱</ci><cn id="alg8.l7.m1.2.3.3.3.2.3.cmml" type="integer" xref="alg8.l7.m1.2.3.3.3.2.3">0</cn></apply><apply id="alg8.l7.m1.2.3.3.3.3.cmml" xref="alg8.l7.m1.2.3.3.3.3"><csymbol cd="ambiguous" id="alg8.l7.m1.2.3.3.3.3.1.cmml" xref="alg8.l7.m1.2.3.3.3.3">subscript</csymbol><cn id="alg8.l7.m1.2.3.3.3.3.2.cmml" type="integer" xref="alg8.l7.m1.2.3.3.3.3.2">1</cn><apply id="alg8.l7.m1.2.2.1.cmml" xref="alg8.l7.m1.2.2.1"><gt id="alg8.l7.m1.2.2.1.2.cmml" xref="alg8.l7.m1.2.2.1.2"></gt><apply id="alg8.l7.m1.2.2.1.1.2.cmml" xref="alg8.l7.m1.2.2.1.1.1"><csymbol cd="latexml" id="alg8.l7.m1.2.2.1.1.2.1.cmml" xref="alg8.l7.m1.2.2.1.1.1.2">norm</csymbol><apply id="alg8.l7.m1.2.2.1.1.1.1.cmml" xref="alg8.l7.m1.2.2.1.1.1.1"><csymbol cd="ambiguous" id="alg8.l7.m1.2.2.1.1.1.1.1.cmml" xref="alg8.l7.m1.2.2.1.1.1.1">subscript</csymbol><ci id="alg8.l7.m1.2.2.1.1.1.1.2.cmml" xref="alg8.l7.m1.2.2.1.1.1.1.2">𝐳</ci><apply id="alg8.l7.m1.2.2.1.1.1.1.3.cmml" xref="alg8.l7.m1.2.2.1.1.1.1.3"><plus id="alg8.l7.m1.2.2.1.1.1.1.3.1.cmml" xref="alg8.l7.m1.2.2.1.1.1.1.3.1"></plus><ci id="alg8.l7.m1.2.2.1.1.1.1.3.2.cmml" xref="alg8.l7.m1.2.2.1.1.1.1.3.2">𝑘</ci><cn id="alg8.l7.m1.2.2.1.1.1.1.3.3.cmml" type="integer" xref="alg8.l7.m1.2.2.1.1.1.1.3.3">1</cn></apply></apply></apply><apply id="alg8.l7.m1.2.2.1.3.cmml" xref="alg8.l7.m1.2.2.1.3"><csymbol cd="ambiguous" id="alg8.l7.m1.2.2.1.3.1.cmml" xref="alg8.l7.m1.2.2.1.3">subscript</csymbol><ci id="alg8.l7.m1.2.2.1.3.2.cmml" xref="alg8.l7.m1.2.2.1.3.2">𝛽</ci><apply 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end_POSTSUBSCRIPT end_POSTSUBSCRIPT + bold_x start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT bold_1 start_POSTSUBSCRIPT ∥ bold_z start_POSTSUBSCRIPT italic_k + 1 end_POSTSUBSCRIPT ∥ > italic_β start_POSTSUBSCRIPT italic_λ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math> </div> <div class="ltx_listingline" id="alg8.l8"> <span class="ltx_tag ltx_tag_listingline"><span class="ltx_text" id="alg8.l8.1.1.1" style="font-size:80%;">8:</span></span> <math alttext="\lambda_{k+1}\leftarrow\lambda_{k}+\mathbf{1}_{\|{\mathbf{z}}_{k+1}\|>\beta_{% \lambda_{k}}}" class="ltx_Math" display="inline" id="alg8.l8.m1.1"><semantics id="alg8.l8.m1.1a"><mrow id="alg8.l8.m1.1.2" xref="alg8.l8.m1.1.2.cmml"><msub id="alg8.l8.m1.1.2.2" xref="alg8.l8.m1.1.2.2.cmml"><mi id="alg8.l8.m1.1.2.2.2" xref="alg8.l8.m1.1.2.2.2.cmml">λ</mi><mrow id="alg8.l8.m1.1.2.2.3" xref="alg8.l8.m1.1.2.2.3.cmml"><mi id="alg8.l8.m1.1.2.2.3.2" 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</div> </div> </figure> <div class="ltx_para" id="A8.p3"> <p class="ltx_p" id="A8.p3.2">Another way to ensure convergence is to use the random projected stochastic gradient descent algorithm proposed by <cite class="ltx_cite ltx_citemacro_citep">(Nurminski, <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#bib.bib52" title="">1973</a>)</cite>. As detailed in Algorithm <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#alg8" title="Algorithm 8 ‣ Appendix H Discussion on the boundedness of (𝐱_𝑘)_{𝑘∈ℕ} ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_tag">8</span></a>, at each iteration, this algorithm realizes a projection onto a ball parameterized by an increasing sequence of positive real numbers <math alttext="(\beta_{n})_{n\in\mathbf{N}}" class="ltx_Math" display="inline" id="A8.p3.1.m1.1"><semantics id="A8.p3.1.m1.1a"><msub id="A8.p3.1.m1.1.1" xref="A8.p3.1.m1.1.1.cmml"><mrow id="A8.p3.1.m1.1.1.1.1" xref="A8.p3.1.m1.1.1.1.1.1.cmml"><mo id="A8.p3.1.m1.1.1.1.1.2" stretchy="false" xref="A8.p3.1.m1.1.1.1.1.1.cmml">(</mo><msub id="A8.p3.1.m1.1.1.1.1.1" xref="A8.p3.1.m1.1.1.1.1.1.cmml"><mi id="A8.p3.1.m1.1.1.1.1.1.2" xref="A8.p3.1.m1.1.1.1.1.1.2.cmml">β</mi><mi id="A8.p3.1.m1.1.1.1.1.1.3" xref="A8.p3.1.m1.1.1.1.1.1.3.cmml">n</mi></msub><mo id="A8.p3.1.m1.1.1.1.1.3" stretchy="false" xref="A8.p3.1.m1.1.1.1.1.1.cmml">)</mo></mrow><mrow id="A8.p3.1.m1.1.1.3" xref="A8.p3.1.m1.1.1.3.cmml"><mi id="A8.p3.1.m1.1.1.3.2" xref="A8.p3.1.m1.1.1.3.2.cmml">n</mi><mo id="A8.p3.1.m1.1.1.3.1" xref="A8.p3.1.m1.1.1.3.1.cmml">∈</mo><mi id="A8.p3.1.m1.1.1.3.3" xref="A8.p3.1.m1.1.1.3.3.cmml">𝐍</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="A8.p3.1.m1.1b"><apply id="A8.p3.1.m1.1.1.cmml" xref="A8.p3.1.m1.1.1"><csymbol cd="ambiguous" id="A8.p3.1.m1.1.1.2.cmml" xref="A8.p3.1.m1.1.1">subscript</csymbol><apply id="A8.p3.1.m1.1.1.1.1.1.cmml" xref="A8.p3.1.m1.1.1.1.1"><csymbol cd="ambiguous" id="A8.p3.1.m1.1.1.1.1.1.1.cmml" xref="A8.p3.1.m1.1.1.1.1">subscript</csymbol><ci id="A8.p3.1.m1.1.1.1.1.1.2.cmml" xref="A8.p3.1.m1.1.1.1.1.1.2">𝛽</ci><ci id="A8.p3.1.m1.1.1.1.1.1.3.cmml" xref="A8.p3.1.m1.1.1.1.1.1.3">𝑛</ci></apply><apply id="A8.p3.1.m1.1.1.3.cmml" xref="A8.p3.1.m1.1.1.3"><in id="A8.p3.1.m1.1.1.3.1.cmml" xref="A8.p3.1.m1.1.1.3.1"></in><ci id="A8.p3.1.m1.1.1.3.2.cmml" xref="A8.p3.1.m1.1.1.3.2">𝑛</ci><ci id="A8.p3.1.m1.1.1.3.3.cmml" xref="A8.p3.1.m1.1.1.3.3">𝐍</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A8.p3.1.m1.1c">(\beta_{n})_{n\in\mathbf{N}}</annotation><annotation encoding="application/x-llamapun" id="A8.p3.1.m1.1d">( italic_β start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ) start_POSTSUBSCRIPT italic_n ∈ bold_N end_POSTSUBSCRIPT</annotation></semantics></math>. This algorithm is proved to converge without the boundedness assumption. <cite class="ltx_cite ltx_citemacro_citet">Tadic & Doucet (<a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#bib.bib72" title="">2017</a>, Theorem A1.1.)</cite> explored this perspective and demonstrated that the iterates of this algorithm are bounded. However, in our context, <cite class="ltx_cite ltx_citemacro_citep">(Tadic & Doucet, <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#bib.bib72" title="">2017</a>, Assumption 1.2.)</cite> is difficult to verify. In fact, our demonstrations (Proof of Proposition <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#S3.Thmproposition3" title="Proposition 3.3. ‣ 3.2 Unbiased algorithm analysis ‣ 3 Convergence Analysis ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_tag">3.3</span></a> in Appendix <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#A4.SS1.SSS3" title="D.1.3 Proof of Proposition 3.3 ‣ D.1 Proof of critical Point Analysis ‣ Appendix D Proofs of section 3 ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_tag">D.1.3</span></a>) rely on an upper bound <math alttext="\mathbb{E}\left(\|\xi_{k}\|^{2}\right)" class="ltx_Math" display="inline" id="A8.p3.2.m2.1"><semantics id="A8.p3.2.m2.1a"><mrow id="A8.p3.2.m2.1.1" xref="A8.p3.2.m2.1.1.cmml"><mi id="A8.p3.2.m2.1.1.3" xref="A8.p3.2.m2.1.1.3.cmml">𝔼</mi><mo id="A8.p3.2.m2.1.1.2" xref="A8.p3.2.m2.1.1.2.cmml">⁢</mo><mrow id="A8.p3.2.m2.1.1.1.1" xref="A8.p3.2.m2.1.1.1.1.1.cmml"><mo id="A8.p3.2.m2.1.1.1.1.2" xref="A8.p3.2.m2.1.1.1.1.1.cmml">(</mo><msup id="A8.p3.2.m2.1.1.1.1.1" xref="A8.p3.2.m2.1.1.1.1.1.cmml"><mrow id="A8.p3.2.m2.1.1.1.1.1.1.1" xref="A8.p3.2.m2.1.1.1.1.1.1.2.cmml"><mo id="A8.p3.2.m2.1.1.1.1.1.1.1.2" stretchy="false" xref="A8.p3.2.m2.1.1.1.1.1.1.2.1.cmml">‖</mo><msub id="A8.p3.2.m2.1.1.1.1.1.1.1.1" xref="A8.p3.2.m2.1.1.1.1.1.1.1.1.cmml"><mi id="A8.p3.2.m2.1.1.1.1.1.1.1.1.2" xref="A8.p3.2.m2.1.1.1.1.1.1.1.1.2.cmml">ξ</mi><mi id="A8.p3.2.m2.1.1.1.1.1.1.1.1.3" xref="A8.p3.2.m2.1.1.1.1.1.1.1.1.3.cmml">k</mi></msub><mo id="A8.p3.2.m2.1.1.1.1.1.1.1.3" stretchy="false" xref="A8.p3.2.m2.1.1.1.1.1.1.2.1.cmml">‖</mo></mrow><mn id="A8.p3.2.m2.1.1.1.1.1.3" xref="A8.p3.2.m2.1.1.1.1.1.3.cmml">2</mn></msup><mo id="A8.p3.2.m2.1.1.1.1.3" xref="A8.p3.2.m2.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A8.p3.2.m2.1b"><apply id="A8.p3.2.m2.1.1.cmml" xref="A8.p3.2.m2.1.1"><times id="A8.p3.2.m2.1.1.2.cmml" xref="A8.p3.2.m2.1.1.2"></times><ci id="A8.p3.2.m2.1.1.3.cmml" xref="A8.p3.2.m2.1.1.3">𝔼</ci><apply id="A8.p3.2.m2.1.1.1.1.1.cmml" xref="A8.p3.2.m2.1.1.1.1"><csymbol cd="ambiguous" id="A8.p3.2.m2.1.1.1.1.1.2.cmml" xref="A8.p3.2.m2.1.1.1.1">superscript</csymbol><apply id="A8.p3.2.m2.1.1.1.1.1.1.2.cmml" xref="A8.p3.2.m2.1.1.1.1.1.1.1"><csymbol cd="latexml" id="A8.p3.2.m2.1.1.1.1.1.1.2.1.cmml" xref="A8.p3.2.m2.1.1.1.1.1.1.1.2">norm</csymbol><apply id="A8.p3.2.m2.1.1.1.1.1.1.1.1.cmml" xref="A8.p3.2.m2.1.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="A8.p3.2.m2.1.1.1.1.1.1.1.1.1.cmml" xref="A8.p3.2.m2.1.1.1.1.1.1.1.1">subscript</csymbol><ci id="A8.p3.2.m2.1.1.1.1.1.1.1.1.2.cmml" xref="A8.p3.2.m2.1.1.1.1.1.1.1.1.2">𝜉</ci><ci id="A8.p3.2.m2.1.1.1.1.1.1.1.1.3.cmml" xref="A8.p3.2.m2.1.1.1.1.1.1.1.1.3">𝑘</ci></apply></apply><cn id="A8.p3.2.m2.1.1.1.1.1.3.cmml" type="integer" xref="A8.p3.2.m2.1.1.1.1.1.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A8.p3.2.m2.1c">\mathbb{E}\left(\|\xi_{k}\|^{2}\right)</annotation><annotation encoding="application/x-llamapun" id="A8.p3.2.m2.1d">blackboard_E ( ∥ italic_ξ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT )</annotation></semantics></math> which is obtained thanks to the boundedness assumption. In our context, Algorithm <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#alg8" title="Algorithm 8 ‣ Appendix H Discussion on the boundedness of (𝐱_𝑘)_{𝑘∈ℕ} ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_tag">8</span></a> has not been proved to converge.</p> </div> <div class="ltx_para" id="A8.p4"> <p class="ltx_p" id="A8.p4.1">Finally, we leave for future work the exploration of a strategy to demonstrate the convergence of Algorithm <a class="ltx_ref" href="https://arxiv.org/html/2402.01779v2#alg2" title="Algorithm 2 ‣ Optimization algorithms ‣ 2.2 SNORE regularization ‣ 2 Stochastic deNOising REgularization (SNORE) ‣ Plug-and-Play image restoration with Stochastic deNOising REgularization"><span class="ltx_text ltx_ref_tag">2</span></a> without any boundedness assumption.</p> </div> <div class="ltx_pagination ltx_role_newpage"></div> </section> </article> </div> <footer class="ltx_page_footer"> <div class="ltx_page_logo">Generated on Fri Apr 5 15:01:22 2024 by <a class="ltx_LaTeXML_logo" href="http://dlmf.nist.gov/LaTeXML/"><span style="letter-spacing:-0.2em; margin-right:0.1em;">L<span style="font-size:70%;position:relative; bottom:2.2pt;">A</span>T<span style="position:relative; bottom:-0.4ex;">E</span></span><span class="ltx_font_smallcaps">xml</span><img alt="[LOGO]" src="data:image/png;base64,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"/></a> </div></footer> </div> </body> </html>