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<!DOCTYPE html> <html lang="en"> <head> <meta content="text/html; charset=utf-8" http-equiv="content-type"/> <title>Regularized Parameter Estimation in Mixed Model Trace Regression</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.0.7.9.min.css" rel="stylesheet" type="text/css"/> <link href="/static/browse/0.3.4/css/ar5iv-fonts.0.7.9.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_new.js"></script> <script src="/static/browse/0.3.4/js/feedbackOverlay.js"></script> <base href="/html/2503.13782v1/"/></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/2503.13782v1#S1" title="In Regularized Parameter Estimation in Mixed Model Trace Regression"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">1 </span>Introduction</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/2503.13782v1#S1.SS1" title="In 1 Introduction ‣ Regularized Parameter Estimation in Mixed Model Trace Regression"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">1.1 </span>Prior Literature</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.13782v1#S1.SS2" title="In 1 Introduction ‣ Regularized Parameter Estimation in Mixed Model Trace Regression"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">1.2 </span>Our Contributions</span></a></li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_section"> <a class="ltx_ref" href="https://arxiv.org/html/2503.13782v1#S2" title="In Regularized Parameter Estimation in Mixed Model Trace Regression"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">2 </span>Mixed Model Trace Regression</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/2503.13782v1#S2.SS1" title="In 2 Mixed Model Trace Regression ‣ Regularized Parameter Estimation in Mixed Model Trace Regression"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">2.1 </span>Model</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.13782v1#S2.SS2" title="In 2 Mixed Model Trace Regression ‣ Regularized Parameter Estimation in Mixed Model Trace Regression"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">2.2 </span>Reduction to a Structured Mixed Model</span></a></li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_section"><a class="ltx_ref" href="https://arxiv.org/html/2503.13782v1#S3" title="In Regularized Parameter Estimation in Mixed Model Trace Regression"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">3 </span>Regularized Parameter Estimation</span></a></li> <li class="ltx_tocentry ltx_tocentry_section"> <a class="ltx_ref" href="https://arxiv.org/html/2503.13782v1#S4" title="In Regularized Parameter Estimation in Mixed Model Trace Regression"><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/2503.13782v1#S4.SS1" title="In 4 Experiments ‣ Regularized Parameter Estimation in Mixed Model Trace Regression"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">4.1 </span>Setup</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.13782v1#S4.SS2" title="In 4 Experiments ‣ Regularized Parameter Estimation in Mixed Model Trace Regression"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">4.2 </span>Simulation Using the MMTR Model</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.13782v1#S4.SS3" title="In 4 Experiments ‣ Regularized Parameter Estimation in Mixed Model Trace Regression"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">4.3 </span>Simulations Using the Equicorrelation GEE model</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.13782v1#S4.SS4" title="In 4 Experiments ‣ Regularized Parameter Estimation in Mixed Model Trace Regression"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">4.4 </span>Real Data: Labeled Faces in the Wild</span></a></li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_section"><a class="ltx_ref" href="https://arxiv.org/html/2503.13782v1#S5" title="In Regularized Parameter Estimation in Mixed Model Trace Regression"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">5 </span>Discussion</span></a></li> <li class="ltx_tocentry ltx_tocentry_section"> <a class="ltx_ref" href="https://arxiv.org/html/2503.13782v1#S1a" title="In Regularized Parameter Estimation in Mixed Model Trace Regression"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">1 </span>Proof of the Propositions</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/2503.13782v1#S1.SS1a" title="In 1 Proof of the Propositions ‣ Regularized Parameter Estimation in Mixed Model Trace Regression"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">1.1 </span>Proof of Proposition 2.1</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.13782v1#S1.SS2a" title="In 1 Proof of the Propositions ‣ Regularized Parameter Estimation in Mixed Model Trace Regression"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">1.2 </span>Proof of Proposition 2.2</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.13782v1#S1.SS3" title="In 1 Proof of the Propositions ‣ Regularized Parameter Estimation in Mixed Model Trace Regression"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">1.3 </span>Proof of Proposition 3.1</span></a></li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_section"> <a class="ltx_ref" href="https://arxiv.org/html/2503.13782v1#S2a" title="In Regularized Parameter Estimation in Mixed Model Trace Regression"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">2 </span>Derivation of the AECM Algorithm Updates</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/2503.13782v1#S2.SS1a" title="In 2 Derivation of the AECM Algorithm Updates ‣ Regularized Parameter Estimation in Mixed Model Trace Regression"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">2.1 </span>Derivation of <math alttext="\operatorname{\mathbf{B}}^{(t+1)}" class="ltx_Math" display="inline"><semantics><msup><mi>𝐁</mi><mrow><mo stretchy="false">(</mo><mrow><mi>t</mi><mo>+</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow></msup><annotation-xml encoding="MathML-Content"><apply><csymbol cd="ambiguous">superscript</csymbol><ci>𝐁</ci><apply><plus></plus><ci>𝑡</ci><cn type="integer">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex">\operatorname{\mathbf{B}}^{(t+1)}</annotation><annotation encoding="application/x-llamapun">bold_B start_POSTSUPERSCRIPT ( italic_t + 1 ) end_POSTSUPERSCRIPT</annotation></semantics></math> and <math alttext="\tau^{2(t+1)}" class="ltx_Math" display="inline"><semantics><msup><mi>τ</mi><mrow><mn>2</mn><mo>⁢</mo><mrow><mo stretchy="false">(</mo><mrow><mi>t</mi><mo>+</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow></mrow></msup><annotation-xml encoding="MathML-Content"><apply><csymbol cd="ambiguous">superscript</csymbol><ci>𝜏</ci><apply><times></times><cn type="integer">2</cn><apply><plus></plus><ci>𝑡</ci><cn type="integer">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex">\tau^{2(t+1)}</annotation><annotation encoding="application/x-llamapun">italic_τ start_POSTSUPERSCRIPT 2 ( italic_t + 1 ) end_POSTSUPERSCRIPT</annotation></semantics></math></span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.13782v1#S2.SS2a" title="In 2 Derivation of the AECM Algorithm Updates ‣ Regularized Parameter Estimation in Mixed Model Trace Regression"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">2.2 </span>Derivation of <math alttext="\operatorname{\mathbf{L}}_{1}^{(t+1)}" class="ltx_Math" display="inline"><semantics><msubsup><mi>𝐋</mi><mn>1</mn><mrow><mo stretchy="false">(</mo><mrow><mi>t</mi><mo>+</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow></msubsup><annotation-xml encoding="MathML-Content"><apply><csymbol cd="ambiguous">superscript</csymbol><apply><csymbol cd="ambiguous">subscript</csymbol><ci>𝐋</ci><cn type="integer">1</cn></apply><apply><plus></plus><ci>𝑡</ci><cn type="integer">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex">\operatorname{\mathbf{L}}_{1}^{(t+1)}</annotation><annotation encoding="application/x-llamapun">bold_L start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_t + 1 ) end_POSTSUPERSCRIPT</annotation></semantics></math> and <math alttext="\operatorname{\mathbf{L}}_{2}^{(t+1)}" class="ltx_Math" display="inline"><semantics><msubsup><mi>𝐋</mi><mn>2</mn><mrow><mo stretchy="false">(</mo><mrow><mi>t</mi><mo>+</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow></msubsup><annotation-xml encoding="MathML-Content"><apply><csymbol cd="ambiguous">superscript</csymbol><apply><csymbol cd="ambiguous">subscript</csymbol><ci>𝐋</ci><cn type="integer">2</cn></apply><apply><plus></plus><ci>𝑡</ci><cn type="integer">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex">\operatorname{\mathbf{L}}_{2}^{(t+1)}</annotation><annotation encoding="application/x-llamapun">bold_L start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_t + 1 ) end_POSTSUPERSCRIPT</annotation></semantics></math></span></a></li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_section"><a class="ltx_ref" href="https://arxiv.org/html/2503.13782v1#S3a" title="In Regularized Parameter Estimation in Mixed Model Trace Regression"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">3 </span>MMTR Misspecification Plots</span></a></li> </ol></nav> </nav> <div class="ltx_page_main"> <div class="ltx_page_content"> <article class="ltx_document ltx_authors_1line"> <h1 class="ltx_title ltx_font_bold ltx_title_document">Regularized Parameter Estimation in Mixed Model Trace Regression</h1> <div class="ltx_authors"> <span class="ltx_creator ltx_role_author"> <span class="ltx_personname">Ian Hultman <br class="ltx_break"/>Department of Statistics and Actuarial Science, University of Iowa <br class="ltx_break"/>and <br class="ltx_break"/>Sanvesh Srivastava <br class="ltx_break"/>Department of Statistics and Actuarial Science, University of Iowa </span></span> </div> <div class="ltx_abstract"> <h6 class="ltx_title ltx_title_abstract">Abstract</h6> <p class="ltx_p" id="id1.id1">We introduce mixed model trace regression (MMTR), a mixed model linear regression extension for scalar responses and high-dimensional matrix-valued covariates. MMTR’s fixed effects component is equivalent to trace regression, with an element-wise lasso penalty imposed on the regression coefficients matrix to facilitate the estimation of a sparse mean parameter. MMTR’s key innovation lies in modeling the covariance structure of matrix-variate random effects as a Kronecker product of low-rank row and column covariance matrices, enabling sparse estimation of the covariance parameter through low-rank constraints. We establish identifiability conditions for the estimation of row and column covariance matrices and use them for rank selection by applying group lasso regularization on the columns of their respective Cholesky factors. We develop an Expectation-Maximization (EM) algorithm extension for numerically stable parameter estimation in high-dimensional applications. MMTR achieves estimation accuracy comparable to leading regularized quasi-likelihood competitors across diverse simulation studies and attains the lowest mean square prediction error compared to its competitors on a publicly available image dataset.</p> </div> <div class="ltx_para" id="p1"> <br class="ltx_break"/> </div> <div class="ltx_para ltx_noindent" id="p2"> <p class="ltx_p" id="p2.1"><span class="ltx_text ltx_font_italic" id="p2.1.1">Keywords:</span> High-dimensional regularization; matrix normal distribution; mixed model; separable covariance; trace regression</p> </div> <div class="ltx_pagination ltx_role_newpage"></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.p1.1">We introduce trace regression models that include matrix-valued random effects, referred to as <em class="ltx_emph ltx_font_italic" id="S1.p1.1.1">mixed model trace regression</em> (MMTR). MMTR extends mixed model linear regression to include matrix-valued fixed and random effects covariates, but the response remains a vector of correlated observations. The parameter dimension in these models easily exceeds the number of observations, requiring specific sparsity assumptions on the model parameters. MMTR models the mean parameter by assuming that the regression coefficients matrix is sparse. The random effects’ row and column covariance matrices in MMTR are assumed to have low ranks, leading to sparse covariance parameters. We develop an EM algorithm for regularized parameter estimation in MMTR, which exploits the sparsity patterns in the parameter components for efficiency and stability in the high-dimensional regime.</p> </div> <div class="ltx_para" id="S1.p2"> <p class="ltx_p" id="S1.p2.14">Consider the MMTR setup. Let <math alttext="n" class="ltx_Math" display="inline" id="S1.p2.1.m1.1"><semantics id="S1.p2.1.m1.1a"><mi id="S1.p2.1.m1.1.1" xref="S1.p2.1.m1.1.1.cmml">n</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">𝑛</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p2.1.m1.1c">n</annotation><annotation encoding="application/x-llamapun" id="S1.p2.1.m1.1d">italic_n</annotation></semantics></math> be the number of subjects or clusters, <math alttext="m_{i}" class="ltx_Math" display="inline" id="S1.p2.2.m2.1"><semantics id="S1.p2.2.m2.1a"><msub id="S1.p2.2.m2.1.1" xref="S1.p2.2.m2.1.1.cmml"><mi id="S1.p2.2.m2.1.1.2" xref="S1.p2.2.m2.1.1.2.cmml">m</mi><mi id="S1.p2.2.m2.1.1.3" xref="S1.p2.2.m2.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S1.p2.2.m2.1b"><apply id="S1.p2.2.m2.1.1.cmml" xref="S1.p2.2.m2.1.1"><csymbol cd="ambiguous" id="S1.p2.2.m2.1.1.1.cmml" 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xref="S1.p2.4.m4.1.1">1</cn><ci id="S1.p2.4.m4.2.2.cmml" xref="S1.p2.4.m4.2.2">…</ci><ci id="S1.p2.4.m4.3.3.cmml" xref="S1.p2.4.m4.3.3">𝑛</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p2.4.m4.3c">i=1,\ldots,n</annotation><annotation encoding="application/x-llamapun" id="S1.p2.4.m4.3d">italic_i = 1 , … , italic_n</annotation></semantics></math>), and <math alttext="N=\sum_{i=1}^{n}m_{i}" class="ltx_Math" display="inline" id="S1.p2.5.m5.1"><semantics id="S1.p2.5.m5.1a"><mrow id="S1.p2.5.m5.1.1" xref="S1.p2.5.m5.1.1.cmml"><mi id="S1.p2.5.m5.1.1.2" xref="S1.p2.5.m5.1.1.2.cmml">N</mi><mo id="S1.p2.5.m5.1.1.1" rspace="0.111em" xref="S1.p2.5.m5.1.1.1.cmml">=</mo><mrow id="S1.p2.5.m5.1.1.3" xref="S1.p2.5.m5.1.1.3.cmml"><msubsup id="S1.p2.5.m5.1.1.3.1" xref="S1.p2.5.m5.1.1.3.1.cmml"><mo id="S1.p2.5.m5.1.1.3.1.2.2" xref="S1.p2.5.m5.1.1.3.1.2.2.cmml">∑</mo><mrow id="S1.p2.5.m5.1.1.3.1.2.3" xref="S1.p2.5.m5.1.1.3.1.2.3.cmml"><mi id="S1.p2.5.m5.1.1.3.1.2.3.2" xref="S1.p2.5.m5.1.1.3.1.2.3.2.cmml">i</mi><mo id="S1.p2.5.m5.1.1.3.1.2.3.1" xref="S1.p2.5.m5.1.1.3.1.2.3.1.cmml">=</mo><mn id="S1.p2.5.m5.1.1.3.1.2.3.3" xref="S1.p2.5.m5.1.1.3.1.2.3.3.cmml">1</mn></mrow><mi id="S1.p2.5.m5.1.1.3.1.3" xref="S1.p2.5.m5.1.1.3.1.3.cmml">n</mi></msubsup><msub id="S1.p2.5.m5.1.1.3.2" xref="S1.p2.5.m5.1.1.3.2.cmml"><mi id="S1.p2.5.m5.1.1.3.2.2" xref="S1.p2.5.m5.1.1.3.2.2.cmml">m</mi><mi id="S1.p2.5.m5.1.1.3.2.3" xref="S1.p2.5.m5.1.1.3.2.3.cmml">i</mi></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.p2.5.m5.1b"><apply id="S1.p2.5.m5.1.1.cmml" xref="S1.p2.5.m5.1.1"><eq id="S1.p2.5.m5.1.1.1.cmml" xref="S1.p2.5.m5.1.1.1"></eq><ci id="S1.p2.5.m5.1.1.2.cmml" xref="S1.p2.5.m5.1.1.2">𝑁</ci><apply id="S1.p2.5.m5.1.1.3.cmml" xref="S1.p2.5.m5.1.1.3"><apply id="S1.p2.5.m5.1.1.3.1.cmml" xref="S1.p2.5.m5.1.1.3.1"><csymbol cd="ambiguous" id="S1.p2.5.m5.1.1.3.1.1.cmml" xref="S1.p2.5.m5.1.1.3.1">superscript</csymbol><apply id="S1.p2.5.m5.1.1.3.1.2.cmml" xref="S1.p2.5.m5.1.1.3.1"><csymbol cd="ambiguous" id="S1.p2.5.m5.1.1.3.1.2.1.cmml" xref="S1.p2.5.m5.1.1.3.1">subscript</csymbol><sum id="S1.p2.5.m5.1.1.3.1.2.2.cmml" xref="S1.p2.5.m5.1.1.3.1.2.2"></sum><apply id="S1.p2.5.m5.1.1.3.1.2.3.cmml" xref="S1.p2.5.m5.1.1.3.1.2.3"><eq id="S1.p2.5.m5.1.1.3.1.2.3.1.cmml" xref="S1.p2.5.m5.1.1.3.1.2.3.1"></eq><ci id="S1.p2.5.m5.1.1.3.1.2.3.2.cmml" xref="S1.p2.5.m5.1.1.3.1.2.3.2">𝑖</ci><cn id="S1.p2.5.m5.1.1.3.1.2.3.3.cmml" type="integer" xref="S1.p2.5.m5.1.1.3.1.2.3.3">1</cn></apply></apply><ci id="S1.p2.5.m5.1.1.3.1.3.cmml" xref="S1.p2.5.m5.1.1.3.1.3">𝑛</ci></apply><apply id="S1.p2.5.m5.1.1.3.2.cmml" xref="S1.p2.5.m5.1.1.3.2"><csymbol cd="ambiguous" id="S1.p2.5.m5.1.1.3.2.1.cmml" xref="S1.p2.5.m5.1.1.3.2">subscript</csymbol><ci id="S1.p2.5.m5.1.1.3.2.2.cmml" xref="S1.p2.5.m5.1.1.3.2.2">𝑚</ci><ci id="S1.p2.5.m5.1.1.3.2.3.cmml" xref="S1.p2.5.m5.1.1.3.2.3">𝑖</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p2.5.m5.1c">N=\sum_{i=1}^{n}m_{i}</annotation><annotation encoding="application/x-llamapun" id="S1.p2.5.m5.1d">italic_N = ∑ start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT italic_m start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> be the total number of observations. The response vector for cluster <math alttext="i" class="ltx_Math" display="inline" id="S1.p2.6.m6.1"><semantics id="S1.p2.6.m6.1a"><mi id="S1.p2.6.m6.1.1" xref="S1.p2.6.m6.1.1.cmml">i</mi><annotation-xml encoding="MathML-Content" id="S1.p2.6.m6.1b"><ci id="S1.p2.6.m6.1.1.cmml" xref="S1.p2.6.m6.1.1">𝑖</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p2.6.m6.1c">i</annotation><annotation encoding="application/x-llamapun" id="S1.p2.6.m6.1d">italic_i</annotation></semantics></math> is <math alttext="\operatorname{\mathbf{y}}_{i}\in\operatorname{\mathbb{R}}^{m_{i}}" class="ltx_Math" display="inline" id="S1.p2.7.m7.1"><semantics id="S1.p2.7.m7.1a"><mrow id="S1.p2.7.m7.1.1" xref="S1.p2.7.m7.1.1.cmml"><msub id="S1.p2.7.m7.1.1.2" xref="S1.p2.7.m7.1.1.2.cmml"><mi id="S1.p2.7.m7.1.1.2.2" xref="S1.p2.7.m7.1.1.2.2.cmml">𝐲</mi><mi id="S1.p2.7.m7.1.1.2.3" xref="S1.p2.7.m7.1.1.2.3.cmml">i</mi></msub><mo id="S1.p2.7.m7.1.1.1" xref="S1.p2.7.m7.1.1.1.cmml">∈</mo><msup id="S1.p2.7.m7.1.1.3" xref="S1.p2.7.m7.1.1.3.cmml"><mi id="S1.p2.7.m7.1.1.3.2" xref="S1.p2.7.m7.1.1.3.2.cmml">ℝ</mi><msub id="S1.p2.7.m7.1.1.3.3" xref="S1.p2.7.m7.1.1.3.3.cmml"><mi id="S1.p2.7.m7.1.1.3.3.2" xref="S1.p2.7.m7.1.1.3.3.2.cmml">m</mi><mi id="S1.p2.7.m7.1.1.3.3.3" xref="S1.p2.7.m7.1.1.3.3.3.cmml">i</mi></msub></msup></mrow><annotation-xml encoding="MathML-Content" id="S1.p2.7.m7.1b"><apply id="S1.p2.7.m7.1.1.cmml" xref="S1.p2.7.m7.1.1"><in id="S1.p2.7.m7.1.1.1.cmml" xref="S1.p2.7.m7.1.1.1"></in><apply id="S1.p2.7.m7.1.1.2.cmml" xref="S1.p2.7.m7.1.1.2"><csymbol cd="ambiguous" id="S1.p2.7.m7.1.1.2.1.cmml" xref="S1.p2.7.m7.1.1.2">subscript</csymbol><ci id="S1.p2.7.m7.1.1.2.2.cmml" xref="S1.p2.7.m7.1.1.2.2">𝐲</ci><ci id="S1.p2.7.m7.1.1.2.3.cmml" xref="S1.p2.7.m7.1.1.2.3">𝑖</ci></apply><apply id="S1.p2.7.m7.1.1.3.cmml" xref="S1.p2.7.m7.1.1.3"><csymbol cd="ambiguous" id="S1.p2.7.m7.1.1.3.1.cmml" xref="S1.p2.7.m7.1.1.3">superscript</csymbol><ci id="S1.p2.7.m7.1.1.3.2.cmml" xref="S1.p2.7.m7.1.1.3.2">ℝ</ci><apply id="S1.p2.7.m7.1.1.3.3.cmml" xref="S1.p2.7.m7.1.1.3.3"><csymbol cd="ambiguous" id="S1.p2.7.m7.1.1.3.3.1.cmml" xref="S1.p2.7.m7.1.1.3.3">subscript</csymbol><ci id="S1.p2.7.m7.1.1.3.3.2.cmml" xref="S1.p2.7.m7.1.1.3.3.2">𝑚</ci><ci id="S1.p2.7.m7.1.1.3.3.3.cmml" xref="S1.p2.7.m7.1.1.3.3.3">𝑖</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p2.7.m7.1c">\operatorname{\mathbf{y}}_{i}\in\operatorname{\mathbb{R}}^{m_{i}}</annotation><annotation encoding="application/x-llamapun" id="S1.p2.7.m7.1d">bold_y start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∈ blackboard_R start_POSTSUPERSCRIPT italic_m start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUPERSCRIPT</annotation></semantics></math>, with <math alttext="j" class="ltx_Math" display="inline" id="S1.p2.8.m8.1"><semantics id="S1.p2.8.m8.1a"><mi id="S1.p2.8.m8.1.1" xref="S1.p2.8.m8.1.1.cmml">j</mi><annotation-xml encoding="MathML-Content" id="S1.p2.8.m8.1b"><ci id="S1.p2.8.m8.1.1.cmml" xref="S1.p2.8.m8.1.1">𝑗</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p2.8.m8.1c">j</annotation><annotation encoding="application/x-llamapun" id="S1.p2.8.m8.1d">italic_j</annotation></semantics></math>th element <math alttext="y_{ij}\in\operatorname{\mathbb{R}}" class="ltx_Math" display="inline" id="S1.p2.9.m9.1"><semantics id="S1.p2.9.m9.1a"><mrow id="S1.p2.9.m9.1.1" xref="S1.p2.9.m9.1.1.cmml"><msub id="S1.p2.9.m9.1.1.2" xref="S1.p2.9.m9.1.1.2.cmml"><mi id="S1.p2.9.m9.1.1.2.2" xref="S1.p2.9.m9.1.1.2.2.cmml">y</mi><mrow id="S1.p2.9.m9.1.1.2.3" xref="S1.p2.9.m9.1.1.2.3.cmml"><mi id="S1.p2.9.m9.1.1.2.3.2" xref="S1.p2.9.m9.1.1.2.3.2.cmml">i</mi><mo id="S1.p2.9.m9.1.1.2.3.1" xref="S1.p2.9.m9.1.1.2.3.1.cmml">⁢</mo><mi id="S1.p2.9.m9.1.1.2.3.3" xref="S1.p2.9.m9.1.1.2.3.3.cmml">j</mi></mrow></msub><mo id="S1.p2.9.m9.1.1.1" xref="S1.p2.9.m9.1.1.1.cmml">∈</mo><mi id="S1.p2.9.m9.1.1.3" xref="S1.p2.9.m9.1.1.3.cmml">ℝ</mi></mrow><annotation-xml encoding="MathML-Content" id="S1.p2.9.m9.1b"><apply id="S1.p2.9.m9.1.1.cmml" xref="S1.p2.9.m9.1.1"><in id="S1.p2.9.m9.1.1.1.cmml" xref="S1.p2.9.m9.1.1.1"></in><apply id="S1.p2.9.m9.1.1.2.cmml" xref="S1.p2.9.m9.1.1.2"><csymbol cd="ambiguous" id="S1.p2.9.m9.1.1.2.1.cmml" xref="S1.p2.9.m9.1.1.2">subscript</csymbol><ci id="S1.p2.9.m9.1.1.2.2.cmml" xref="S1.p2.9.m9.1.1.2.2">𝑦</ci><apply id="S1.p2.9.m9.1.1.2.3.cmml" xref="S1.p2.9.m9.1.1.2.3"><times id="S1.p2.9.m9.1.1.2.3.1.cmml" xref="S1.p2.9.m9.1.1.2.3.1"></times><ci id="S1.p2.9.m9.1.1.2.3.2.cmml" xref="S1.p2.9.m9.1.1.2.3.2">𝑖</ci><ci id="S1.p2.9.m9.1.1.2.3.3.cmml" xref="S1.p2.9.m9.1.1.2.3.3">𝑗</ci></apply></apply><ci id="S1.p2.9.m9.1.1.3.cmml" xref="S1.p2.9.m9.1.1.3">ℝ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p2.9.m9.1c">y_{ij}\in\operatorname{\mathbb{R}}</annotation><annotation encoding="application/x-llamapun" id="S1.p2.9.m9.1d">italic_y start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT ∈ blackboard_R</annotation></semantics></math>. The fixed and random effects covariates matrices specific to <math alttext="y_{ij}" class="ltx_Math" display="inline" id="S1.p2.10.m10.1"><semantics id="S1.p2.10.m10.1a"><msub id="S1.p2.10.m10.1.1" xref="S1.p2.10.m10.1.1.cmml"><mi id="S1.p2.10.m10.1.1.2" xref="S1.p2.10.m10.1.1.2.cmml">y</mi><mrow id="S1.p2.10.m10.1.1.3" xref="S1.p2.10.m10.1.1.3.cmml"><mi id="S1.p2.10.m10.1.1.3.2" xref="S1.p2.10.m10.1.1.3.2.cmml">i</mi><mo id="S1.p2.10.m10.1.1.3.1" xref="S1.p2.10.m10.1.1.3.1.cmml">⁢</mo><mi id="S1.p2.10.m10.1.1.3.3" xref="S1.p2.10.m10.1.1.3.3.cmml">j</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="S1.p2.10.m10.1b"><apply id="S1.p2.10.m10.1.1.cmml" xref="S1.p2.10.m10.1.1"><csymbol cd="ambiguous" id="S1.p2.10.m10.1.1.1.cmml" xref="S1.p2.10.m10.1.1">subscript</csymbol><ci id="S1.p2.10.m10.1.1.2.cmml" xref="S1.p2.10.m10.1.1.2">𝑦</ci><apply id="S1.p2.10.m10.1.1.3.cmml" xref="S1.p2.10.m10.1.1.3"><times id="S1.p2.10.m10.1.1.3.1.cmml" xref="S1.p2.10.m10.1.1.3.1"></times><ci id="S1.p2.10.m10.1.1.3.2.cmml" xref="S1.p2.10.m10.1.1.3.2">𝑖</ci><ci id="S1.p2.10.m10.1.1.3.3.cmml" xref="S1.p2.10.m10.1.1.3.3">𝑗</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p2.10.m10.1c">y_{ij}</annotation><annotation encoding="application/x-llamapun" id="S1.p2.10.m10.1d">italic_y start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT</annotation></semantics></math> are <math alttext="\operatorname{\mathbf{X}}_{ij}\in\operatorname{\mathbb{R}}^{P_{1}\times P_{2}}" class="ltx_Math" display="inline" id="S1.p2.11.m11.1"><semantics id="S1.p2.11.m11.1a"><mrow id="S1.p2.11.m11.1.1" xref="S1.p2.11.m11.1.1.cmml"><msub id="S1.p2.11.m11.1.1.2" xref="S1.p2.11.m11.1.1.2.cmml"><mi id="S1.p2.11.m11.1.1.2.2" xref="S1.p2.11.m11.1.1.2.2.cmml">𝐗</mi><mrow id="S1.p2.11.m11.1.1.2.3" xref="S1.p2.11.m11.1.1.2.3.cmml"><mi id="S1.p2.11.m11.1.1.2.3.2" xref="S1.p2.11.m11.1.1.2.3.2.cmml">i</mi><mo id="S1.p2.11.m11.1.1.2.3.1" xref="S1.p2.11.m11.1.1.2.3.1.cmml">⁢</mo><mi id="S1.p2.11.m11.1.1.2.3.3" xref="S1.p2.11.m11.1.1.2.3.3.cmml">j</mi></mrow></msub><mo id="S1.p2.11.m11.1.1.1" xref="S1.p2.11.m11.1.1.1.cmml">∈</mo><msup id="S1.p2.11.m11.1.1.3" xref="S1.p2.11.m11.1.1.3.cmml"><mi id="S1.p2.11.m11.1.1.3.2" xref="S1.p2.11.m11.1.1.3.2.cmml">ℝ</mi><mrow id="S1.p2.11.m11.1.1.3.3" xref="S1.p2.11.m11.1.1.3.3.cmml"><msub id="S1.p2.11.m11.1.1.3.3.2" xref="S1.p2.11.m11.1.1.3.3.2.cmml"><mi id="S1.p2.11.m11.1.1.3.3.2.2" xref="S1.p2.11.m11.1.1.3.3.2.2.cmml">P</mi><mn id="S1.p2.11.m11.1.1.3.3.2.3" xref="S1.p2.11.m11.1.1.3.3.2.3.cmml">1</mn></msub><mo id="S1.p2.11.m11.1.1.3.3.1" lspace="0.222em" rspace="0.222em" xref="S1.p2.11.m11.1.1.3.3.1.cmml">×</mo><msub id="S1.p2.11.m11.1.1.3.3.3" xref="S1.p2.11.m11.1.1.3.3.3.cmml"><mi id="S1.p2.11.m11.1.1.3.3.3.2" xref="S1.p2.11.m11.1.1.3.3.3.2.cmml">P</mi><mn id="S1.p2.11.m11.1.1.3.3.3.3" xref="S1.p2.11.m11.1.1.3.3.3.3.cmml">2</mn></msub></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="S1.p2.11.m11.1b"><apply 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xref="S1.p2.12.m12.1.1.3.3.2"><csymbol cd="ambiguous" id="S1.p2.12.m12.1.1.3.3.2.1.cmml" xref="S1.p2.12.m12.1.1.3.3.2">subscript</csymbol><ci id="S1.p2.12.m12.1.1.3.3.2.2.cmml" xref="S1.p2.12.m12.1.1.3.3.2.2">𝑄</ci><cn id="S1.p2.12.m12.1.1.3.3.2.3.cmml" type="integer" xref="S1.p2.12.m12.1.1.3.3.2.3">1</cn></apply><apply id="S1.p2.12.m12.1.1.3.3.3.cmml" xref="S1.p2.12.m12.1.1.3.3.3"><csymbol cd="ambiguous" id="S1.p2.12.m12.1.1.3.3.3.1.cmml" xref="S1.p2.12.m12.1.1.3.3.3">subscript</csymbol><ci id="S1.p2.12.m12.1.1.3.3.3.2.cmml" xref="S1.p2.12.m12.1.1.3.3.3.2">𝑄</ci><cn id="S1.p2.12.m12.1.1.3.3.3.3.cmml" type="integer" xref="S1.p2.12.m12.1.1.3.3.3.3">2</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p2.12.m12.1c">\operatorname{\mathbf{Z}}_{ij}\in\operatorname{\mathbb{R}}^{Q_{1}\times Q_{2}}</annotation><annotation encoding="application/x-llamapun" id="S1.p2.12.m12.1d">bold_Z start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT ∈ blackboard_R start_POSTSUPERSCRIPT italic_Q start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT × italic_Q start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT</annotation></semantics></math> (<math alttext="i=1,\ldots,n" class="ltx_Math" display="inline" id="S1.p2.13.m13.3"><semantics id="S1.p2.13.m13.3a"><mrow id="S1.p2.13.m13.3.4" xref="S1.p2.13.m13.3.4.cmml"><mi id="S1.p2.13.m13.3.4.2" xref="S1.p2.13.m13.3.4.2.cmml">i</mi><mo id="S1.p2.13.m13.3.4.1" xref="S1.p2.13.m13.3.4.1.cmml">=</mo><mrow id="S1.p2.13.m13.3.4.3.2" xref="S1.p2.13.m13.3.4.3.1.cmml"><mn id="S1.p2.13.m13.1.1" xref="S1.p2.13.m13.1.1.cmml">1</mn><mo id="S1.p2.13.m13.3.4.3.2.1" xref="S1.p2.13.m13.3.4.3.1.cmml">,</mo><mi id="S1.p2.13.m13.2.2" mathvariant="normal" xref="S1.p2.13.m13.2.2.cmml">…</mi><mo id="S1.p2.13.m13.3.4.3.2.2" xref="S1.p2.13.m13.3.4.3.1.cmml">,</mo><mi id="S1.p2.13.m13.3.3" xref="S1.p2.13.m13.3.3.cmml">n</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.p2.13.m13.3b"><apply id="S1.p2.13.m13.3.4.cmml" xref="S1.p2.13.m13.3.4"><eq id="S1.p2.13.m13.3.4.1.cmml" xref="S1.p2.13.m13.3.4.1"></eq><ci id="S1.p2.13.m13.3.4.2.cmml" xref="S1.p2.13.m13.3.4.2">𝑖</ci><list id="S1.p2.13.m13.3.4.3.1.cmml" xref="S1.p2.13.m13.3.4.3.2"><cn id="S1.p2.13.m13.1.1.cmml" type="integer" xref="S1.p2.13.m13.1.1">1</cn><ci id="S1.p2.13.m13.2.2.cmml" xref="S1.p2.13.m13.2.2">…</ci><ci id="S1.p2.13.m13.3.3.cmml" xref="S1.p2.13.m13.3.3">𝑛</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p2.13.m13.3c">i=1,\ldots,n</annotation><annotation encoding="application/x-llamapun" id="S1.p2.13.m13.3d">italic_i = 1 , … , italic_n</annotation></semantics></math>; <math alttext="j=1,\ldots,m_{i}" class="ltx_Math" display="inline" id="S1.p2.14.m14.3"><semantics id="S1.p2.14.m14.3a"><mrow id="S1.p2.14.m14.3.3" xref="S1.p2.14.m14.3.3.cmml"><mi id="S1.p2.14.m14.3.3.3" xref="S1.p2.14.m14.3.3.3.cmml">j</mi><mo id="S1.p2.14.m14.3.3.2" xref="S1.p2.14.m14.3.3.2.cmml">=</mo><mrow id="S1.p2.14.m14.3.3.1.1" xref="S1.p2.14.m14.3.3.1.2.cmml"><mn id="S1.p2.14.m14.1.1" xref="S1.p2.14.m14.1.1.cmml">1</mn><mo id="S1.p2.14.m14.3.3.1.1.2" xref="S1.p2.14.m14.3.3.1.2.cmml">,</mo><mi id="S1.p2.14.m14.2.2" mathvariant="normal" xref="S1.p2.14.m14.2.2.cmml">…</mi><mo id="S1.p2.14.m14.3.3.1.1.3" xref="S1.p2.14.m14.3.3.1.2.cmml">,</mo><msub id="S1.p2.14.m14.3.3.1.1.1" xref="S1.p2.14.m14.3.3.1.1.1.cmml"><mi id="S1.p2.14.m14.3.3.1.1.1.2" xref="S1.p2.14.m14.3.3.1.1.1.2.cmml">m</mi><mi id="S1.p2.14.m14.3.3.1.1.1.3" xref="S1.p2.14.m14.3.3.1.1.1.3.cmml">i</mi></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.p2.14.m14.3b"><apply id="S1.p2.14.m14.3.3.cmml" xref="S1.p2.14.m14.3.3"><eq id="S1.p2.14.m14.3.3.2.cmml" xref="S1.p2.14.m14.3.3.2"></eq><ci id="S1.p2.14.m14.3.3.3.cmml" xref="S1.p2.14.m14.3.3.3">𝑗</ci><list id="S1.p2.14.m14.3.3.1.2.cmml" xref="S1.p2.14.m14.3.3.1.1"><cn id="S1.p2.14.m14.1.1.cmml" type="integer" xref="S1.p2.14.m14.1.1">1</cn><ci id="S1.p2.14.m14.2.2.cmml" xref="S1.p2.14.m14.2.2">…</ci><apply id="S1.p2.14.m14.3.3.1.1.1.cmml" xref="S1.p2.14.m14.3.3.1.1.1"><csymbol cd="ambiguous" id="S1.p2.14.m14.3.3.1.1.1.1.cmml" xref="S1.p2.14.m14.3.3.1.1.1">subscript</csymbol><ci id="S1.p2.14.m14.3.3.1.1.1.2.cmml" xref="S1.p2.14.m14.3.3.1.1.1.2">𝑚</ci><ci id="S1.p2.14.m14.3.3.1.1.1.3.cmml" xref="S1.p2.14.m14.3.3.1.1.1.3">𝑖</ci></apply></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p2.14.m14.3c">j=1,\ldots,m_{i}</annotation><annotation encoding="application/x-llamapun" id="S1.p2.14.m14.3d">italic_j = 1 , … , italic_m start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math>). Then, MMTR sets</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="S3.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 y_{ij}=\operatorname{\mathop{\mathrm{tr}}}(\operatorname{\mathbf% {X}}_{ij}^{\top}\operatorname{\mathbf{B}})+\operatorname{\mathop{\mathrm{tr}}}% (\operatorname{\mathbf{Z}}_{ij}^{\top}\operatorname{\mathbf{A}}_{i})+e_{ij},% \quad\operatorname{\mathbf{A}}_{i}\sim N_{Q_{1},Q_{2}}(\operatorname{\mathbf{0% }},\tau^{2}\operatorname{{\mathbf{\Sigma}}}_{1},\operatorname{{\mathbf{\Sigma}% }}_{2}),\quad\operatorname{\mathbf{e}}_{i}\sim N_{m_{i}}(\operatorname{\mathbf% {0}},\tau^{2}\operatorname{\mathbf{I}}_{m_{i}})," class="ltx_Math" display="inline" id="S1.E1.m1.7"><semantics id="S1.E1.m1.7a"><mrow id="S1.E1.m1.7.7.1"><mrow id="S1.E1.m1.7.7.1.1.2" xref="S1.E1.m1.7.7.1.1.3.cmml"><mrow id="S1.E1.m1.7.7.1.1.1.1" xref="S1.E1.m1.7.7.1.1.1.1.cmml"><msub id="S1.E1.m1.7.7.1.1.1.1.4" xref="S1.E1.m1.7.7.1.1.1.1.4.cmml"><mi id="S1.E1.m1.7.7.1.1.1.1.4.2" xref="S1.E1.m1.7.7.1.1.1.1.4.2.cmml">y</mi><mrow id="S1.E1.m1.7.7.1.1.1.1.4.3" xref="S1.E1.m1.7.7.1.1.1.1.4.3.cmml"><mi id="S1.E1.m1.7.7.1.1.1.1.4.3.2" xref="S1.E1.m1.7.7.1.1.1.1.4.3.2.cmml">i</mi><mo id="S1.E1.m1.7.7.1.1.1.1.4.3.1" xref="S1.E1.m1.7.7.1.1.1.1.4.3.1.cmml">⁢</mo><mi id="S1.E1.m1.7.7.1.1.1.1.4.3.3" xref="S1.E1.m1.7.7.1.1.1.1.4.3.3.cmml">j</mi></mrow></msub><mo id="S1.E1.m1.7.7.1.1.1.1.3" xref="S1.E1.m1.7.7.1.1.1.1.3.cmml">=</mo><mrow id="S1.E1.m1.7.7.1.1.1.1.2" xref="S1.E1.m1.7.7.1.1.1.1.2.cmml"><mrow id="S1.E1.m1.7.7.1.1.1.1.1.1.1" xref="S1.E1.m1.7.7.1.1.1.1.1.1.2.cmml"><mi id="S1.E1.m1.3.3" xref="S1.E1.m1.3.3.cmml">tr</mi><mo id="S1.E1.m1.7.7.1.1.1.1.1.1.1a" xref="S1.E1.m1.7.7.1.1.1.1.1.1.2.cmml">⁡</mo><mrow id="S1.E1.m1.7.7.1.1.1.1.1.1.1.1" xref="S1.E1.m1.7.7.1.1.1.1.1.1.2.cmml"><mo id="S1.E1.m1.7.7.1.1.1.1.1.1.1.1.2" stretchy="false" xref="S1.E1.m1.7.7.1.1.1.1.1.1.2.cmml">(</mo><mrow id="S1.E1.m1.7.7.1.1.1.1.1.1.1.1.1" xref="S1.E1.m1.7.7.1.1.1.1.1.1.1.1.1.cmml"><msubsup id="S1.E1.m1.7.7.1.1.1.1.1.1.1.1.1.1" xref="S1.E1.m1.7.7.1.1.1.1.1.1.1.1.1.1.cmml"><mi id="S1.E1.m1.7.7.1.1.1.1.1.1.1.1.1.1.2.2" xref="S1.E1.m1.7.7.1.1.1.1.1.1.1.1.1.1.2.2.cmml">𝐗</mi><mrow id="S1.E1.m1.7.7.1.1.1.1.1.1.1.1.1.1.2.3" xref="S1.E1.m1.7.7.1.1.1.1.1.1.1.1.1.1.2.3.cmml"><mi id="S1.E1.m1.7.7.1.1.1.1.1.1.1.1.1.1.2.3.2" xref="S1.E1.m1.7.7.1.1.1.1.1.1.1.1.1.1.2.3.2.cmml">i</mi><mo id="S1.E1.m1.7.7.1.1.1.1.1.1.1.1.1.1.2.3.1" xref="S1.E1.m1.7.7.1.1.1.1.1.1.1.1.1.1.2.3.1.cmml">⁢</mo><mi id="S1.E1.m1.7.7.1.1.1.1.1.1.1.1.1.1.2.3.3" xref="S1.E1.m1.7.7.1.1.1.1.1.1.1.1.1.1.2.3.3.cmml">j</mi></mrow><mo id="S1.E1.m1.7.7.1.1.1.1.1.1.1.1.1.1.3" xref="S1.E1.m1.7.7.1.1.1.1.1.1.1.1.1.1.3.cmml">⊤</mo></msubsup><mo id="S1.E1.m1.7.7.1.1.1.1.1.1.1.1.1a" lspace="0.167em" 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xref="S1.E1.m1.7.7.1.1.2.2.2.2.1.1.1.1.3.2">𝐈</ci><apply id="S1.E1.m1.7.7.1.1.2.2.2.2.1.1.1.1.3.3.cmml" xref="S1.E1.m1.7.7.1.1.2.2.2.2.1.1.1.1.3.3"><csymbol cd="ambiguous" id="S1.E1.m1.7.7.1.1.2.2.2.2.1.1.1.1.3.3.1.cmml" xref="S1.E1.m1.7.7.1.1.2.2.2.2.1.1.1.1.3.3">subscript</csymbol><ci id="S1.E1.m1.7.7.1.1.2.2.2.2.1.1.1.1.3.3.2.cmml" xref="S1.E1.m1.7.7.1.1.2.2.2.2.1.1.1.1.3.3.2">𝑚</ci><ci id="S1.E1.m1.7.7.1.1.2.2.2.2.1.1.1.1.3.3.3.cmml" xref="S1.E1.m1.7.7.1.1.2.2.2.2.1.1.1.1.3.3.3">𝑖</ci></apply></apply></apply></interval></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.E1.m1.7c">\displaystyle y_{ij}=\operatorname{\mathop{\mathrm{tr}}}(\operatorname{\mathbf% {X}}_{ij}^{\top}\operatorname{\mathbf{B}})+\operatorname{\mathop{\mathrm{tr}}}% (\operatorname{\mathbf{Z}}_{ij}^{\top}\operatorname{\mathbf{A}}_{i})+e_{ij},% \quad\operatorname{\mathbf{A}}_{i}\sim N_{Q_{1},Q_{2}}(\operatorname{\mathbf{0% }},\tau^{2}\operatorname{{\mathbf{\Sigma}}}_{1},\operatorname{{\mathbf{\Sigma}% }}_{2}),\quad\operatorname{\mathbf{e}}_{i}\sim N_{m_{i}}(\operatorname{\mathbf% {0}},\tau^{2}\operatorname{\mathbf{I}}_{m_{i}}),</annotation><annotation encoding="application/x-llamapun" id="S1.E1.m1.7d">italic_y start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT = roman_tr ( bold_X start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ⊤ end_POSTSUPERSCRIPT bold_B ) + roman_tr ( bold_Z start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ⊤ end_POSTSUPERSCRIPT bold_A start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) + italic_e start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT , bold_A start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∼ italic_N start_POSTSUBSCRIPT italic_Q start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_Q start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( bold_0 , italic_τ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT bold_Σ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , bold_Σ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) , bold_e start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∼ italic_N start_POSTSUBSCRIPT italic_m start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( bold_0 , italic_τ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT bold_I start_POSTSUBSCRIPT italic_m start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT 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">(1)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S1.p2.28">where <math alttext="\operatorname{\mathop{\mathrm{tr}}}" class="ltx_Math" display="inline" id="S1.p2.15.m1.1"><semantics id="S1.p2.15.m1.1a"><mi id="S1.p2.15.m1.1.1" xref="S1.p2.15.m1.1.1.cmml">tr</mi><annotation-xml encoding="MathML-Content" id="S1.p2.15.m1.1b"><ci id="S1.p2.15.m1.1.1.cmml" xref="S1.p2.15.m1.1.1">tr</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p2.15.m1.1c">\operatorname{\mathop{\mathrm{tr}}}</annotation><annotation encoding="application/x-llamapun" id="S1.p2.15.m1.1d">roman_tr</annotation></semantics></math> is the <em class="ltx_emph ltx_font_italic" id="S1.p2.28.1">trace</em> operator, <math alttext="\operatorname{\mathbf{B}}\in\operatorname{\mathbb{R}}^{P_{1}\times P_{2}}" class="ltx_Math" display="inline" id="S1.p2.16.m2.1"><semantics id="S1.p2.16.m2.1a"><mrow id="S1.p2.16.m2.1.1" xref="S1.p2.16.m2.1.1.cmml"><mi id="S1.p2.16.m2.1.1.2" xref="S1.p2.16.m2.1.1.2.cmml">𝐁</mi><mo id="S1.p2.16.m2.1.1.1" xref="S1.p2.16.m2.1.1.1.cmml">∈</mo><msup id="S1.p2.16.m2.1.1.3" xref="S1.p2.16.m2.1.1.3.cmml"><mi id="S1.p2.16.m2.1.1.3.2" xref="S1.p2.16.m2.1.1.3.2.cmml">ℝ</mi><mrow id="S1.p2.16.m2.1.1.3.3" xref="S1.p2.16.m2.1.1.3.3.cmml"><msub id="S1.p2.16.m2.1.1.3.3.2" xref="S1.p2.16.m2.1.1.3.3.2.cmml"><mi 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id="S1.p2.16.m2.1.1.3.3.cmml" xref="S1.p2.16.m2.1.1.3.3"><times id="S1.p2.16.m2.1.1.3.3.1.cmml" xref="S1.p2.16.m2.1.1.3.3.1"></times><apply id="S1.p2.16.m2.1.1.3.3.2.cmml" xref="S1.p2.16.m2.1.1.3.3.2"><csymbol cd="ambiguous" id="S1.p2.16.m2.1.1.3.3.2.1.cmml" xref="S1.p2.16.m2.1.1.3.3.2">subscript</csymbol><ci id="S1.p2.16.m2.1.1.3.3.2.2.cmml" xref="S1.p2.16.m2.1.1.3.3.2.2">𝑃</ci><cn id="S1.p2.16.m2.1.1.3.3.2.3.cmml" type="integer" xref="S1.p2.16.m2.1.1.3.3.2.3">1</cn></apply><apply id="S1.p2.16.m2.1.1.3.3.3.cmml" xref="S1.p2.16.m2.1.1.3.3.3"><csymbol cd="ambiguous" id="S1.p2.16.m2.1.1.3.3.3.1.cmml" xref="S1.p2.16.m2.1.1.3.3.3">subscript</csymbol><ci id="S1.p2.16.m2.1.1.3.3.3.2.cmml" xref="S1.p2.16.m2.1.1.3.3.3.2">𝑃</ci><cn id="S1.p2.16.m2.1.1.3.3.3.3.cmml" type="integer" xref="S1.p2.16.m2.1.1.3.3.3.3">2</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p2.16.m2.1c">\operatorname{\mathbf{B}}\in\operatorname{\mathbb{R}}^{P_{1}\times P_{2}}</annotation><annotation encoding="application/x-llamapun" id="S1.p2.16.m2.1d">bold_B ∈ blackboard_R start_POSTSUPERSCRIPT italic_P start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT × italic_P start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT</annotation></semantics></math> is the fixed effects parameter matrix, <math alttext="\operatorname{\mathbf{A}}_{i}\in\operatorname{\mathbb{R}}^{Q_{1}\times Q_{2}}" class="ltx_Math" display="inline" id="S1.p2.17.m3.1"><semantics id="S1.p2.17.m3.1a"><mrow id="S1.p2.17.m3.1.1" xref="S1.p2.17.m3.1.1.cmml"><msub id="S1.p2.17.m3.1.1.2" xref="S1.p2.17.m3.1.1.2.cmml"><mi id="S1.p2.17.m3.1.1.2.2" xref="S1.p2.17.m3.1.1.2.2.cmml">𝐀</mi><mi id="S1.p2.17.m3.1.1.2.3" xref="S1.p2.17.m3.1.1.2.3.cmml">i</mi></msub><mo id="S1.p2.17.m3.1.1.1" xref="S1.p2.17.m3.1.1.1.cmml">∈</mo><msup id="S1.p2.17.m3.1.1.3" xref="S1.p2.17.m3.1.1.3.cmml"><mi id="S1.p2.17.m3.1.1.3.2" xref="S1.p2.17.m3.1.1.3.2.cmml">ℝ</mi><mrow id="S1.p2.17.m3.1.1.3.3" xref="S1.p2.17.m3.1.1.3.3.cmml"><msub id="S1.p2.17.m3.1.1.3.3.2" xref="S1.p2.17.m3.1.1.3.3.2.cmml"><mi id="S1.p2.17.m3.1.1.3.3.2.2" xref="S1.p2.17.m3.1.1.3.3.2.2.cmml">Q</mi><mn id="S1.p2.17.m3.1.1.3.3.2.3" xref="S1.p2.17.m3.1.1.3.3.2.3.cmml">1</mn></msub><mo id="S1.p2.17.m3.1.1.3.3.1" lspace="0.222em" rspace="0.222em" xref="S1.p2.17.m3.1.1.3.3.1.cmml">×</mo><msub id="S1.p2.17.m3.1.1.3.3.3" xref="S1.p2.17.m3.1.1.3.3.3.cmml"><mi id="S1.p2.17.m3.1.1.3.3.3.2" xref="S1.p2.17.m3.1.1.3.3.3.2.cmml">Q</mi><mn id="S1.p2.17.m3.1.1.3.3.3.3" xref="S1.p2.17.m3.1.1.3.3.3.3.cmml">2</mn></msub></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="S1.p2.17.m3.1b"><apply id="S1.p2.17.m3.1.1.cmml" xref="S1.p2.17.m3.1.1"><in id="S1.p2.17.m3.1.1.1.cmml" xref="S1.p2.17.m3.1.1.1"></in><apply id="S1.p2.17.m3.1.1.2.cmml" xref="S1.p2.17.m3.1.1.2"><csymbol cd="ambiguous" id="S1.p2.17.m3.1.1.2.1.cmml" xref="S1.p2.17.m3.1.1.2">subscript</csymbol><ci id="S1.p2.17.m3.1.1.2.2.cmml" xref="S1.p2.17.m3.1.1.2.2">𝐀</ci><ci id="S1.p2.17.m3.1.1.2.3.cmml" xref="S1.p2.17.m3.1.1.2.3">𝑖</ci></apply><apply id="S1.p2.17.m3.1.1.3.cmml" xref="S1.p2.17.m3.1.1.3"><csymbol cd="ambiguous" id="S1.p2.17.m3.1.1.3.1.cmml" xref="S1.p2.17.m3.1.1.3">superscript</csymbol><ci id="S1.p2.17.m3.1.1.3.2.cmml" xref="S1.p2.17.m3.1.1.3.2">ℝ</ci><apply id="S1.p2.17.m3.1.1.3.3.cmml" xref="S1.p2.17.m3.1.1.3.3"><times id="S1.p2.17.m3.1.1.3.3.1.cmml" xref="S1.p2.17.m3.1.1.3.3.1"></times><apply id="S1.p2.17.m3.1.1.3.3.2.cmml" xref="S1.p2.17.m3.1.1.3.3.2"><csymbol cd="ambiguous" id="S1.p2.17.m3.1.1.3.3.2.1.cmml" xref="S1.p2.17.m3.1.1.3.3.2">subscript</csymbol><ci id="S1.p2.17.m3.1.1.3.3.2.2.cmml" xref="S1.p2.17.m3.1.1.3.3.2.2">𝑄</ci><cn id="S1.p2.17.m3.1.1.3.3.2.3.cmml" type="integer" xref="S1.p2.17.m3.1.1.3.3.2.3">1</cn></apply><apply id="S1.p2.17.m3.1.1.3.3.3.cmml" xref="S1.p2.17.m3.1.1.3.3.3"><csymbol cd="ambiguous" id="S1.p2.17.m3.1.1.3.3.3.1.cmml" xref="S1.p2.17.m3.1.1.3.3.3">subscript</csymbol><ci id="S1.p2.17.m3.1.1.3.3.3.2.cmml" xref="S1.p2.17.m3.1.1.3.3.3.2">𝑄</ci><cn id="S1.p2.17.m3.1.1.3.3.3.3.cmml" type="integer" xref="S1.p2.17.m3.1.1.3.3.3.3">2</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p2.17.m3.1c">\operatorname{\mathbf{A}}_{i}\in\operatorname{\mathbb{R}}^{Q_{1}\times Q_{2}}</annotation><annotation encoding="application/x-llamapun" id="S1.p2.17.m3.1d">bold_A start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∈ blackboard_R start_POSTSUPERSCRIPT italic_Q start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT × italic_Q start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT</annotation></semantics></math> is the random effects matrix, <math alttext="N_{Q_{1},Q_{2}}(\operatorname{\mathbf{0}},\tau^{2}\operatorname{{\mathbf{% \Sigma}}}_{1},\operatorname{{\mathbf{\Sigma}}}_{2})" class="ltx_Math" display="inline" id="S1.p2.18.m4.5"><semantics id="S1.p2.18.m4.5a"><mrow id="S1.p2.18.m4.5.5" xref="S1.p2.18.m4.5.5.cmml"><msub id="S1.p2.18.m4.5.5.4" xref="S1.p2.18.m4.5.5.4.cmml"><mi id="S1.p2.18.m4.5.5.4.2" xref="S1.p2.18.m4.5.5.4.2.cmml">N</mi><mrow id="S1.p2.18.m4.2.2.2.2" xref="S1.p2.18.m4.2.2.2.3.cmml"><msub id="S1.p2.18.m4.1.1.1.1.1" xref="S1.p2.18.m4.1.1.1.1.1.cmml"><mi id="S1.p2.18.m4.1.1.1.1.1.2" xref="S1.p2.18.m4.1.1.1.1.1.2.cmml">Q</mi><mn id="S1.p2.18.m4.1.1.1.1.1.3" xref="S1.p2.18.m4.1.1.1.1.1.3.cmml">1</mn></msub><mo id="S1.p2.18.m4.2.2.2.2.3" xref="S1.p2.18.m4.2.2.2.3.cmml">,</mo><msub id="S1.p2.18.m4.2.2.2.2.2" xref="S1.p2.18.m4.2.2.2.2.2.cmml"><mi id="S1.p2.18.m4.2.2.2.2.2.2" xref="S1.p2.18.m4.2.2.2.2.2.2.cmml">Q</mi><mn id="S1.p2.18.m4.2.2.2.2.2.3" xref="S1.p2.18.m4.2.2.2.2.2.3.cmml">2</mn></msub></mrow></msub><mo id="S1.p2.18.m4.5.5.3" xref="S1.p2.18.m4.5.5.3.cmml">⁢</mo><mrow id="S1.p2.18.m4.5.5.2.2" xref="S1.p2.18.m4.5.5.2.3.cmml"><mo id="S1.p2.18.m4.5.5.2.2.3" stretchy="false" xref="S1.p2.18.m4.5.5.2.3.cmml">(</mo><mi id="S1.p2.18.m4.3.3" xref="S1.p2.18.m4.3.3.cmml">𝟎</mi><mo id="S1.p2.18.m4.5.5.2.2.4" xref="S1.p2.18.m4.5.5.2.3.cmml">,</mo><mrow id="S1.p2.18.m4.4.4.1.1.1" xref="S1.p2.18.m4.4.4.1.1.1.cmml"><msup id="S1.p2.18.m4.4.4.1.1.1.2" xref="S1.p2.18.m4.4.4.1.1.1.2.cmml"><mi id="S1.p2.18.m4.4.4.1.1.1.2.2" xref="S1.p2.18.m4.4.4.1.1.1.2.2.cmml">τ</mi><mn id="S1.p2.18.m4.4.4.1.1.1.2.3" xref="S1.p2.18.m4.4.4.1.1.1.2.3.cmml">2</mn></msup><mo id="S1.p2.18.m4.4.4.1.1.1.1" lspace="0.167em" xref="S1.p2.18.m4.4.4.1.1.1.1.cmml">⁢</mo><msub id="S1.p2.18.m4.4.4.1.1.1.3" xref="S1.p2.18.m4.4.4.1.1.1.3.cmml"><mi id="S1.p2.18.m4.4.4.1.1.1.3.2" xref="S1.p2.18.m4.4.4.1.1.1.3.2.cmml">𝚺</mi><mn id="S1.p2.18.m4.4.4.1.1.1.3.3" xref="S1.p2.18.m4.4.4.1.1.1.3.3.cmml">1</mn></msub></mrow><mo id="S1.p2.18.m4.5.5.2.2.5" xref="S1.p2.18.m4.5.5.2.3.cmml">,</mo><msub id="S1.p2.18.m4.5.5.2.2.2" xref="S1.p2.18.m4.5.5.2.2.2.cmml"><mi id="S1.p2.18.m4.5.5.2.2.2.2" xref="S1.p2.18.m4.5.5.2.2.2.2.cmml">𝚺</mi><mn id="S1.p2.18.m4.5.5.2.2.2.3" xref="S1.p2.18.m4.5.5.2.2.2.3.cmml">2</mn></msub><mo id="S1.p2.18.m4.5.5.2.2.6" stretchy="false" xref="S1.p2.18.m4.5.5.2.3.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.p2.18.m4.5b"><apply id="S1.p2.18.m4.5.5.cmml" xref="S1.p2.18.m4.5.5"><times id="S1.p2.18.m4.5.5.3.cmml" xref="S1.p2.18.m4.5.5.3"></times><apply id="S1.p2.18.m4.5.5.4.cmml" xref="S1.p2.18.m4.5.5.4"><csymbol cd="ambiguous" id="S1.p2.18.m4.5.5.4.1.cmml" xref="S1.p2.18.m4.5.5.4">subscript</csymbol><ci id="S1.p2.18.m4.5.5.4.2.cmml" xref="S1.p2.18.m4.5.5.4.2">𝑁</ci><list id="S1.p2.18.m4.2.2.2.3.cmml" xref="S1.p2.18.m4.2.2.2.2"><apply id="S1.p2.18.m4.1.1.1.1.1.cmml" xref="S1.p2.18.m4.1.1.1.1.1"><csymbol cd="ambiguous" id="S1.p2.18.m4.1.1.1.1.1.1.cmml" xref="S1.p2.18.m4.1.1.1.1.1">subscript</csymbol><ci id="S1.p2.18.m4.1.1.1.1.1.2.cmml" xref="S1.p2.18.m4.1.1.1.1.1.2">𝑄</ci><cn id="S1.p2.18.m4.1.1.1.1.1.3.cmml" type="integer" xref="S1.p2.18.m4.1.1.1.1.1.3">1</cn></apply><apply id="S1.p2.18.m4.2.2.2.2.2.cmml" xref="S1.p2.18.m4.2.2.2.2.2"><csymbol cd="ambiguous" id="S1.p2.18.m4.2.2.2.2.2.1.cmml" xref="S1.p2.18.m4.2.2.2.2.2">subscript</csymbol><ci id="S1.p2.18.m4.2.2.2.2.2.2.cmml" xref="S1.p2.18.m4.2.2.2.2.2.2">𝑄</ci><cn id="S1.p2.18.m4.2.2.2.2.2.3.cmml" type="integer" xref="S1.p2.18.m4.2.2.2.2.2.3">2</cn></apply></list></apply><vector id="S1.p2.18.m4.5.5.2.3.cmml" xref="S1.p2.18.m4.5.5.2.2"><ci id="S1.p2.18.m4.3.3.cmml" xref="S1.p2.18.m4.3.3">𝟎</ci><apply id="S1.p2.18.m4.4.4.1.1.1.cmml" xref="S1.p2.18.m4.4.4.1.1.1"><times id="S1.p2.18.m4.4.4.1.1.1.1.cmml" xref="S1.p2.18.m4.4.4.1.1.1.1"></times><apply id="S1.p2.18.m4.4.4.1.1.1.2.cmml" xref="S1.p2.18.m4.4.4.1.1.1.2"><csymbol cd="ambiguous" id="S1.p2.18.m4.4.4.1.1.1.2.1.cmml" xref="S1.p2.18.m4.4.4.1.1.1.2">superscript</csymbol><ci id="S1.p2.18.m4.4.4.1.1.1.2.2.cmml" xref="S1.p2.18.m4.4.4.1.1.1.2.2">𝜏</ci><cn id="S1.p2.18.m4.4.4.1.1.1.2.3.cmml" type="integer" xref="S1.p2.18.m4.4.4.1.1.1.2.3">2</cn></apply><apply id="S1.p2.18.m4.4.4.1.1.1.3.cmml" xref="S1.p2.18.m4.4.4.1.1.1.3"><csymbol cd="ambiguous" id="S1.p2.18.m4.4.4.1.1.1.3.1.cmml" xref="S1.p2.18.m4.4.4.1.1.1.3">subscript</csymbol><ci id="S1.p2.18.m4.4.4.1.1.1.3.2.cmml" xref="S1.p2.18.m4.4.4.1.1.1.3.2">𝚺</ci><cn id="S1.p2.18.m4.4.4.1.1.1.3.3.cmml" type="integer" xref="S1.p2.18.m4.4.4.1.1.1.3.3">1</cn></apply></apply><apply id="S1.p2.18.m4.5.5.2.2.2.cmml" xref="S1.p2.18.m4.5.5.2.2.2"><csymbol cd="ambiguous" id="S1.p2.18.m4.5.5.2.2.2.1.cmml" xref="S1.p2.18.m4.5.5.2.2.2">subscript</csymbol><ci id="S1.p2.18.m4.5.5.2.2.2.2.cmml" xref="S1.p2.18.m4.5.5.2.2.2.2">𝚺</ci><cn id="S1.p2.18.m4.5.5.2.2.2.3.cmml" type="integer" xref="S1.p2.18.m4.5.5.2.2.2.3">2</cn></apply></vector></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p2.18.m4.5c">N_{Q_{1},Q_{2}}(\operatorname{\mathbf{0}},\tau^{2}\operatorname{{\mathbf{% \Sigma}}}_{1},\operatorname{{\mathbf{\Sigma}}}_{2})</annotation><annotation encoding="application/x-llamapun" id="S1.p2.18.m4.5d">italic_N start_POSTSUBSCRIPT italic_Q start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_Q start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( bold_0 , italic_τ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT bold_Σ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , bold_Σ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT )</annotation></semantics></math> is the matrix variate Gaussian distribution with row and column covariance matrices <math alttext="\tau^{2}\operatorname{{\mathbf{\Sigma}}}_{1}" class="ltx_Math" display="inline" id="S1.p2.19.m5.1"><semantics id="S1.p2.19.m5.1a"><mrow id="S1.p2.19.m5.1.1" xref="S1.p2.19.m5.1.1.cmml"><msup id="S1.p2.19.m5.1.1.2" xref="S1.p2.19.m5.1.1.2.cmml"><mi id="S1.p2.19.m5.1.1.2.2" xref="S1.p2.19.m5.1.1.2.2.cmml">τ</mi><mn id="S1.p2.19.m5.1.1.2.3" xref="S1.p2.19.m5.1.1.2.3.cmml">2</mn></msup><mo id="S1.p2.19.m5.1.1.1" lspace="0.167em" xref="S1.p2.19.m5.1.1.1.cmml">⁢</mo><msub id="S1.p2.19.m5.1.1.3" xref="S1.p2.19.m5.1.1.3.cmml"><mi id="S1.p2.19.m5.1.1.3.2" xref="S1.p2.19.m5.1.1.3.2.cmml">𝚺</mi><mn id="S1.p2.19.m5.1.1.3.3" xref="S1.p2.19.m5.1.1.3.3.cmml">1</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S1.p2.19.m5.1b"><apply id="S1.p2.19.m5.1.1.cmml" xref="S1.p2.19.m5.1.1"><times id="S1.p2.19.m5.1.1.1.cmml" xref="S1.p2.19.m5.1.1.1"></times><apply id="S1.p2.19.m5.1.1.2.cmml" xref="S1.p2.19.m5.1.1.2"><csymbol cd="ambiguous" id="S1.p2.19.m5.1.1.2.1.cmml" xref="S1.p2.19.m5.1.1.2">superscript</csymbol><ci id="S1.p2.19.m5.1.1.2.2.cmml" xref="S1.p2.19.m5.1.1.2.2">𝜏</ci><cn id="S1.p2.19.m5.1.1.2.3.cmml" type="integer" xref="S1.p2.19.m5.1.1.2.3">2</cn></apply><apply id="S1.p2.19.m5.1.1.3.cmml" xref="S1.p2.19.m5.1.1.3"><csymbol cd="ambiguous" id="S1.p2.19.m5.1.1.3.1.cmml" xref="S1.p2.19.m5.1.1.3">subscript</csymbol><ci id="S1.p2.19.m5.1.1.3.2.cmml" xref="S1.p2.19.m5.1.1.3.2">𝚺</ci><cn id="S1.p2.19.m5.1.1.3.3.cmml" type="integer" xref="S1.p2.19.m5.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p2.19.m5.1c">\tau^{2}\operatorname{{\mathbf{\Sigma}}}_{1}</annotation><annotation encoding="application/x-llamapun" id="S1.p2.19.m5.1d">italic_τ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT bold_Σ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="\operatorname{{\mathbf{\Sigma}}}_{2}" class="ltx_Math" display="inline" id="S1.p2.20.m6.1"><semantics id="S1.p2.20.m6.1a"><msub id="S1.p2.20.m6.1.1" xref="S1.p2.20.m6.1.1.cmml"><mi id="S1.p2.20.m6.1.1.2" xref="S1.p2.20.m6.1.1.2.cmml">𝚺</mi><mn id="S1.p2.20.m6.1.1.3" xref="S1.p2.20.m6.1.1.3.cmml">2</mn></msub><annotation-xml encoding="MathML-Content" id="S1.p2.20.m6.1b"><apply id="S1.p2.20.m6.1.1.cmml" xref="S1.p2.20.m6.1.1"><csymbol cd="ambiguous" id="S1.p2.20.m6.1.1.1.cmml" xref="S1.p2.20.m6.1.1">subscript</csymbol><ci id="S1.p2.20.m6.1.1.2.cmml" xref="S1.p2.20.m6.1.1.2">𝚺</ci><cn id="S1.p2.20.m6.1.1.3.cmml" type="integer" xref="S1.p2.20.m6.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p2.20.m6.1c">\operatorname{{\mathbf{\Sigma}}}_{2}</annotation><annotation encoding="application/x-llamapun" id="S1.p2.20.m6.1d">bold_Σ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math>, respectively, <math alttext="\operatorname{\mathbf{e}}_{i}=(e_{i1},\ldots,e_{im_{i}})^{\top}" class="ltx_Math" display="inline" id="S1.p2.21.m7.3"><semantics id="S1.p2.21.m7.3a"><mrow id="S1.p2.21.m7.3.3" xref="S1.p2.21.m7.3.3.cmml"><msub id="S1.p2.21.m7.3.3.4" xref="S1.p2.21.m7.3.3.4.cmml"><mi id="S1.p2.21.m7.3.3.4.2" xref="S1.p2.21.m7.3.3.4.2.cmml">𝐞</mi><mi id="S1.p2.21.m7.3.3.4.3" xref="S1.p2.21.m7.3.3.4.3.cmml">i</mi></msub><mo id="S1.p2.21.m7.3.3.3" xref="S1.p2.21.m7.3.3.3.cmml">=</mo><msup id="S1.p2.21.m7.3.3.2" xref="S1.p2.21.m7.3.3.2.cmml"><mrow id="S1.p2.21.m7.3.3.2.2.2" xref="S1.p2.21.m7.3.3.2.2.3.cmml"><mo id="S1.p2.21.m7.3.3.2.2.2.3" stretchy="false" xref="S1.p2.21.m7.3.3.2.2.3.cmml">(</mo><msub id="S1.p2.21.m7.2.2.1.1.1.1" xref="S1.p2.21.m7.2.2.1.1.1.1.cmml"><mi id="S1.p2.21.m7.2.2.1.1.1.1.2" xref="S1.p2.21.m7.2.2.1.1.1.1.2.cmml">e</mi><mrow id="S1.p2.21.m7.2.2.1.1.1.1.3" xref="S1.p2.21.m7.2.2.1.1.1.1.3.cmml"><mi id="S1.p2.21.m7.2.2.1.1.1.1.3.2" xref="S1.p2.21.m7.2.2.1.1.1.1.3.2.cmml">i</mi><mo id="S1.p2.21.m7.2.2.1.1.1.1.3.1" xref="S1.p2.21.m7.2.2.1.1.1.1.3.1.cmml">⁢</mo><mn id="S1.p2.21.m7.2.2.1.1.1.1.3.3" xref="S1.p2.21.m7.2.2.1.1.1.1.3.3.cmml">1</mn></mrow></msub><mo id="S1.p2.21.m7.3.3.2.2.2.4" xref="S1.p2.21.m7.3.3.2.2.3.cmml">,</mo><mi id="S1.p2.21.m7.1.1" mathvariant="normal" xref="S1.p2.21.m7.1.1.cmml">…</mi><mo id="S1.p2.21.m7.3.3.2.2.2.5" xref="S1.p2.21.m7.3.3.2.2.3.cmml">,</mo><msub id="S1.p2.21.m7.3.3.2.2.2.2" xref="S1.p2.21.m7.3.3.2.2.2.2.cmml"><mi id="S1.p2.21.m7.3.3.2.2.2.2.2" xref="S1.p2.21.m7.3.3.2.2.2.2.2.cmml">e</mi><mrow id="S1.p2.21.m7.3.3.2.2.2.2.3" xref="S1.p2.21.m7.3.3.2.2.2.2.3.cmml"><mi id="S1.p2.21.m7.3.3.2.2.2.2.3.2" xref="S1.p2.21.m7.3.3.2.2.2.2.3.2.cmml">i</mi><mo id="S1.p2.21.m7.3.3.2.2.2.2.3.1" xref="S1.p2.21.m7.3.3.2.2.2.2.3.1.cmml">⁢</mo><msub id="S1.p2.21.m7.3.3.2.2.2.2.3.3" xref="S1.p2.21.m7.3.3.2.2.2.2.3.3.cmml"><mi id="S1.p2.21.m7.3.3.2.2.2.2.3.3.2" xref="S1.p2.21.m7.3.3.2.2.2.2.3.3.2.cmml">m</mi><mi id="S1.p2.21.m7.3.3.2.2.2.2.3.3.3" xref="S1.p2.21.m7.3.3.2.2.2.2.3.3.3.cmml">i</mi></msub></mrow></msub><mo id="S1.p2.21.m7.3.3.2.2.2.6" stretchy="false" xref="S1.p2.21.m7.3.3.2.2.3.cmml">)</mo></mrow><mo id="S1.p2.21.m7.3.3.2.4" xref="S1.p2.21.m7.3.3.2.4.cmml">⊤</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="S1.p2.21.m7.3b"><apply id="S1.p2.21.m7.3.3.cmml" xref="S1.p2.21.m7.3.3"><eq id="S1.p2.21.m7.3.3.3.cmml" xref="S1.p2.21.m7.3.3.3"></eq><apply id="S1.p2.21.m7.3.3.4.cmml" xref="S1.p2.21.m7.3.3.4"><csymbol cd="ambiguous" id="S1.p2.21.m7.3.3.4.1.cmml" xref="S1.p2.21.m7.3.3.4">subscript</csymbol><ci id="S1.p2.21.m7.3.3.4.2.cmml" xref="S1.p2.21.m7.3.3.4.2">𝐞</ci><ci id="S1.p2.21.m7.3.3.4.3.cmml" xref="S1.p2.21.m7.3.3.4.3">𝑖</ci></apply><apply id="S1.p2.21.m7.3.3.2.cmml" xref="S1.p2.21.m7.3.3.2"><csymbol cd="ambiguous" id="S1.p2.21.m7.3.3.2.3.cmml" xref="S1.p2.21.m7.3.3.2">superscript</csymbol><vector id="S1.p2.21.m7.3.3.2.2.3.cmml" xref="S1.p2.21.m7.3.3.2.2.2"><apply id="S1.p2.21.m7.2.2.1.1.1.1.cmml" xref="S1.p2.21.m7.2.2.1.1.1.1"><csymbol cd="ambiguous" id="S1.p2.21.m7.2.2.1.1.1.1.1.cmml" xref="S1.p2.21.m7.2.2.1.1.1.1">subscript</csymbol><ci id="S1.p2.21.m7.2.2.1.1.1.1.2.cmml" xref="S1.p2.21.m7.2.2.1.1.1.1.2">𝑒</ci><apply id="S1.p2.21.m7.2.2.1.1.1.1.3.cmml" xref="S1.p2.21.m7.2.2.1.1.1.1.3"><times id="S1.p2.21.m7.2.2.1.1.1.1.3.1.cmml" xref="S1.p2.21.m7.2.2.1.1.1.1.3.1"></times><ci id="S1.p2.21.m7.2.2.1.1.1.1.3.2.cmml" xref="S1.p2.21.m7.2.2.1.1.1.1.3.2">𝑖</ci><cn id="S1.p2.21.m7.2.2.1.1.1.1.3.3.cmml" type="integer" xref="S1.p2.21.m7.2.2.1.1.1.1.3.3">1</cn></apply></apply><ci id="S1.p2.21.m7.1.1.cmml" xref="S1.p2.21.m7.1.1">…</ci><apply id="S1.p2.21.m7.3.3.2.2.2.2.cmml" xref="S1.p2.21.m7.3.3.2.2.2.2"><csymbol cd="ambiguous" id="S1.p2.21.m7.3.3.2.2.2.2.1.cmml" xref="S1.p2.21.m7.3.3.2.2.2.2">subscript</csymbol><ci id="S1.p2.21.m7.3.3.2.2.2.2.2.cmml" xref="S1.p2.21.m7.3.3.2.2.2.2.2">𝑒</ci><apply id="S1.p2.21.m7.3.3.2.2.2.2.3.cmml" xref="S1.p2.21.m7.3.3.2.2.2.2.3"><times id="S1.p2.21.m7.3.3.2.2.2.2.3.1.cmml" xref="S1.p2.21.m7.3.3.2.2.2.2.3.1"></times><ci id="S1.p2.21.m7.3.3.2.2.2.2.3.2.cmml" xref="S1.p2.21.m7.3.3.2.2.2.2.3.2">𝑖</ci><apply id="S1.p2.21.m7.3.3.2.2.2.2.3.3.cmml" xref="S1.p2.21.m7.3.3.2.2.2.2.3.3"><csymbol cd="ambiguous" id="S1.p2.21.m7.3.3.2.2.2.2.3.3.1.cmml" xref="S1.p2.21.m7.3.3.2.2.2.2.3.3">subscript</csymbol><ci id="S1.p2.21.m7.3.3.2.2.2.2.3.3.2.cmml" xref="S1.p2.21.m7.3.3.2.2.2.2.3.3.2">𝑚</ci><ci id="S1.p2.21.m7.3.3.2.2.2.2.3.3.3.cmml" xref="S1.p2.21.m7.3.3.2.2.2.2.3.3.3">𝑖</ci></apply></apply></apply></vector><csymbol cd="latexml" id="S1.p2.21.m7.3.3.2.4.cmml" xref="S1.p2.21.m7.3.3.2.4">top</csymbol></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p2.21.m7.3c">\operatorname{\mathbf{e}}_{i}=(e_{i1},\ldots,e_{im_{i}})^{\top}</annotation><annotation encoding="application/x-llamapun" id="S1.p2.21.m7.3d">bold_e start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = ( italic_e start_POSTSUBSCRIPT italic_i 1 end_POSTSUBSCRIPT , … , italic_e start_POSTSUBSCRIPT italic_i italic_m start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT ⊤ end_POSTSUPERSCRIPT</annotation></semantics></math> is the <math alttext="i" class="ltx_Math" display="inline" id="S1.p2.22.m8.1"><semantics id="S1.p2.22.m8.1a"><mi id="S1.p2.22.m8.1.1" xref="S1.p2.22.m8.1.1.cmml">i</mi><annotation-xml encoding="MathML-Content" id="S1.p2.22.m8.1b"><ci id="S1.p2.22.m8.1.1.cmml" xref="S1.p2.22.m8.1.1">𝑖</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p2.22.m8.1c">i</annotation><annotation encoding="application/x-llamapun" id="S1.p2.22.m8.1d">italic_i</annotation></semantics></math>th idiosyncratic error vector, <math alttext="\operatorname{\mathbf{I}}_{m_{i}}" class="ltx_Math" display="inline" id="S1.p2.23.m9.1"><semantics id="S1.p2.23.m9.1a"><msub id="S1.p2.23.m9.1.1" xref="S1.p2.23.m9.1.1.cmml"><mi id="S1.p2.23.m9.1.1.2" xref="S1.p2.23.m9.1.1.2.cmml">𝐈</mi><msub id="S1.p2.23.m9.1.1.3" xref="S1.p2.23.m9.1.1.3.cmml"><mi id="S1.p2.23.m9.1.1.3.2" xref="S1.p2.23.m9.1.1.3.2.cmml">m</mi><mi id="S1.p2.23.m9.1.1.3.3" xref="S1.p2.23.m9.1.1.3.3.cmml">i</mi></msub></msub><annotation-xml encoding="MathML-Content" id="S1.p2.23.m9.1b"><apply id="S1.p2.23.m9.1.1.cmml" xref="S1.p2.23.m9.1.1"><csymbol cd="ambiguous" id="S1.p2.23.m9.1.1.1.cmml" xref="S1.p2.23.m9.1.1">subscript</csymbol><ci id="S1.p2.23.m9.1.1.2.cmml" xref="S1.p2.23.m9.1.1.2">𝐈</ci><apply id="S1.p2.23.m9.1.1.3.cmml" xref="S1.p2.23.m9.1.1.3"><csymbol cd="ambiguous" id="S1.p2.23.m9.1.1.3.1.cmml" xref="S1.p2.23.m9.1.1.3">subscript</csymbol><ci id="S1.p2.23.m9.1.1.3.2.cmml" xref="S1.p2.23.m9.1.1.3.2">𝑚</ci><ci id="S1.p2.23.m9.1.1.3.3.cmml" xref="S1.p2.23.m9.1.1.3.3">𝑖</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p2.23.m9.1c">\operatorname{\mathbf{I}}_{m_{i}}</annotation><annotation encoding="application/x-llamapun" id="S1.p2.23.m9.1d">bold_I start_POSTSUBSCRIPT italic_m start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math> is an <math alttext="m_{i}\times m_{i}" class="ltx_Math" display="inline" id="S1.p2.24.m10.1"><semantics id="S1.p2.24.m10.1a"><mrow id="S1.p2.24.m10.1.1" xref="S1.p2.24.m10.1.1.cmml"><msub id="S1.p2.24.m10.1.1.2" xref="S1.p2.24.m10.1.1.2.cmml"><mi id="S1.p2.24.m10.1.1.2.2" xref="S1.p2.24.m10.1.1.2.2.cmml">m</mi><mi id="S1.p2.24.m10.1.1.2.3" xref="S1.p2.24.m10.1.1.2.3.cmml">i</mi></msub><mo id="S1.p2.24.m10.1.1.1" lspace="0.222em" rspace="0.222em" xref="S1.p2.24.m10.1.1.1.cmml">×</mo><msub id="S1.p2.24.m10.1.1.3" xref="S1.p2.24.m10.1.1.3.cmml"><mi id="S1.p2.24.m10.1.1.3.2" xref="S1.p2.24.m10.1.1.3.2.cmml">m</mi><mi id="S1.p2.24.m10.1.1.3.3" xref="S1.p2.24.m10.1.1.3.3.cmml">i</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S1.p2.24.m10.1b"><apply id="S1.p2.24.m10.1.1.cmml" xref="S1.p2.24.m10.1.1"><times id="S1.p2.24.m10.1.1.1.cmml" xref="S1.p2.24.m10.1.1.1"></times><apply id="S1.p2.24.m10.1.1.2.cmml" xref="S1.p2.24.m10.1.1.2"><csymbol cd="ambiguous" id="S1.p2.24.m10.1.1.2.1.cmml" xref="S1.p2.24.m10.1.1.2">subscript</csymbol><ci id="S1.p2.24.m10.1.1.2.2.cmml" xref="S1.p2.24.m10.1.1.2.2">𝑚</ci><ci id="S1.p2.24.m10.1.1.2.3.cmml" xref="S1.p2.24.m10.1.1.2.3">𝑖</ci></apply><apply id="S1.p2.24.m10.1.1.3.cmml" xref="S1.p2.24.m10.1.1.3"><csymbol cd="ambiguous" id="S1.p2.24.m10.1.1.3.1.cmml" xref="S1.p2.24.m10.1.1.3">subscript</csymbol><ci id="S1.p2.24.m10.1.1.3.2.cmml" xref="S1.p2.24.m10.1.1.3.2">𝑚</ci><ci id="S1.p2.24.m10.1.1.3.3.cmml" xref="S1.p2.24.m10.1.1.3.3">𝑖</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p2.24.m10.1c">m_{i}\times m_{i}</annotation><annotation encoding="application/x-llamapun" id="S1.p2.24.m10.1d">italic_m start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT × italic_m start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> identity matrix, and <math alttext="\operatorname{\mathbf{A}}_{i},\operatorname{\mathbf{e}}_{i}" class="ltx_Math" display="inline" id="S1.p2.25.m11.2"><semantics id="S1.p2.25.m11.2a"><mrow id="S1.p2.25.m11.2.2.2" xref="S1.p2.25.m11.2.2.3.cmml"><msub id="S1.p2.25.m11.1.1.1.1" xref="S1.p2.25.m11.1.1.1.1.cmml"><mi id="S1.p2.25.m11.1.1.1.1.2" xref="S1.p2.25.m11.1.1.1.1.2.cmml">𝐀</mi><mi id="S1.p2.25.m11.1.1.1.1.3" xref="S1.p2.25.m11.1.1.1.1.3.cmml">i</mi></msub><mo id="S1.p2.25.m11.2.2.2.3" xref="S1.p2.25.m11.2.2.3.cmml">,</mo><msub id="S1.p2.25.m11.2.2.2.2" xref="S1.p2.25.m11.2.2.2.2.cmml"><mi id="S1.p2.25.m11.2.2.2.2.2" xref="S1.p2.25.m11.2.2.2.2.2.cmml">𝐞</mi><mi id="S1.p2.25.m11.2.2.2.2.3" xref="S1.p2.25.m11.2.2.2.2.3.cmml">i</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S1.p2.25.m11.2b"><list id="S1.p2.25.m11.2.2.3.cmml" xref="S1.p2.25.m11.2.2.2"><apply id="S1.p2.25.m11.1.1.1.1.cmml" xref="S1.p2.25.m11.1.1.1.1"><csymbol cd="ambiguous" id="S1.p2.25.m11.1.1.1.1.1.cmml" xref="S1.p2.25.m11.1.1.1.1">subscript</csymbol><ci id="S1.p2.25.m11.1.1.1.1.2.cmml" xref="S1.p2.25.m11.1.1.1.1.2">𝐀</ci><ci id="S1.p2.25.m11.1.1.1.1.3.cmml" xref="S1.p2.25.m11.1.1.1.1.3">𝑖</ci></apply><apply id="S1.p2.25.m11.2.2.2.2.cmml" xref="S1.p2.25.m11.2.2.2.2"><csymbol cd="ambiguous" id="S1.p2.25.m11.2.2.2.2.1.cmml" xref="S1.p2.25.m11.2.2.2.2">subscript</csymbol><ci id="S1.p2.25.m11.2.2.2.2.2.cmml" xref="S1.p2.25.m11.2.2.2.2.2">𝐞</ci><ci id="S1.p2.25.m11.2.2.2.2.3.cmml" xref="S1.p2.25.m11.2.2.2.2.3">𝑖</ci></apply></list></annotation-xml><annotation encoding="application/x-tex" id="S1.p2.25.m11.2c">\operatorname{\mathbf{A}}_{i},\operatorname{\mathbf{e}}_{i}</annotation><annotation encoding="application/x-llamapun" id="S1.p2.25.m11.2d">bold_A start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , bold_e start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> are mutually independent for every <math alttext="i" class="ltx_Math" display="inline" id="S1.p2.26.m12.1"><semantics id="S1.p2.26.m12.1a"><mi id="S1.p2.26.m12.1.1" xref="S1.p2.26.m12.1.1.cmml">i</mi><annotation-xml encoding="MathML-Content" id="S1.p2.26.m12.1b"><ci id="S1.p2.26.m12.1.1.cmml" xref="S1.p2.26.m12.1.1">𝑖</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p2.26.m12.1c">i</annotation><annotation encoding="application/x-llamapun" id="S1.p2.26.m12.1d">italic_i</annotation></semantics></math>. The model parameters are <math alttext="\operatorname{\mathbf{B}},\operatorname{{\mathbf{\Sigma}}}_{1},\operatorname{{% \mathbf{\Sigma}}}_{2}," class="ltx_Math" display="inline" id="S1.p2.27.m13.2"><semantics id="S1.p2.27.m13.2a"><mrow id="S1.p2.27.m13.2.2.1"><mrow id="S1.p2.27.m13.2.2.1.1.2" xref="S1.p2.27.m13.2.2.1.1.3.cmml"><mi id="S1.p2.27.m13.1.1" xref="S1.p2.27.m13.1.1.cmml">𝐁</mi><mo id="S1.p2.27.m13.2.2.1.1.2.3" xref="S1.p2.27.m13.2.2.1.1.3.cmml">,</mo><msub id="S1.p2.27.m13.2.2.1.1.1.1" xref="S1.p2.27.m13.2.2.1.1.1.1.cmml"><mi id="S1.p2.27.m13.2.2.1.1.1.1.2" xref="S1.p2.27.m13.2.2.1.1.1.1.2.cmml">𝚺</mi><mn id="S1.p2.27.m13.2.2.1.1.1.1.3" xref="S1.p2.27.m13.2.2.1.1.1.1.3.cmml">1</mn></msub><mo id="S1.p2.27.m13.2.2.1.1.2.4" xref="S1.p2.27.m13.2.2.1.1.3.cmml">,</mo><msub id="S1.p2.27.m13.2.2.1.1.2.2" xref="S1.p2.27.m13.2.2.1.1.2.2.cmml"><mi id="S1.p2.27.m13.2.2.1.1.2.2.2" xref="S1.p2.27.m13.2.2.1.1.2.2.2.cmml">𝚺</mi><mn id="S1.p2.27.m13.2.2.1.1.2.2.3" xref="S1.p2.27.m13.2.2.1.1.2.2.3.cmml">2</mn></msub></mrow><mo id="S1.p2.27.m13.2.2.1.2">,</mo></mrow><annotation-xml encoding="MathML-Content" id="S1.p2.27.m13.2b"><list id="S1.p2.27.m13.2.2.1.1.3.cmml" xref="S1.p2.27.m13.2.2.1.1.2"><ci id="S1.p2.27.m13.1.1.cmml" xref="S1.p2.27.m13.1.1">𝐁</ci><apply id="S1.p2.27.m13.2.2.1.1.1.1.cmml" xref="S1.p2.27.m13.2.2.1.1.1.1"><csymbol cd="ambiguous" id="S1.p2.27.m13.2.2.1.1.1.1.1.cmml" xref="S1.p2.27.m13.2.2.1.1.1.1">subscript</csymbol><ci id="S1.p2.27.m13.2.2.1.1.1.1.2.cmml" xref="S1.p2.27.m13.2.2.1.1.1.1.2">𝚺</ci><cn id="S1.p2.27.m13.2.2.1.1.1.1.3.cmml" type="integer" xref="S1.p2.27.m13.2.2.1.1.1.1.3">1</cn></apply><apply id="S1.p2.27.m13.2.2.1.1.2.2.cmml" xref="S1.p2.27.m13.2.2.1.1.2.2"><csymbol cd="ambiguous" id="S1.p2.27.m13.2.2.1.1.2.2.1.cmml" xref="S1.p2.27.m13.2.2.1.1.2.2">subscript</csymbol><ci id="S1.p2.27.m13.2.2.1.1.2.2.2.cmml" xref="S1.p2.27.m13.2.2.1.1.2.2.2">𝚺</ci><cn id="S1.p2.27.m13.2.2.1.1.2.2.3.cmml" type="integer" xref="S1.p2.27.m13.2.2.1.1.2.2.3">2</cn></apply></list></annotation-xml><annotation encoding="application/x-tex" id="S1.p2.27.m13.2c">\operatorname{\mathbf{B}},\operatorname{{\mathbf{\Sigma}}}_{1},\operatorname{{% \mathbf{\Sigma}}}_{2},</annotation><annotation encoding="application/x-llamapun" id="S1.p2.27.m13.2d">bold_B , bold_Σ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , bold_Σ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ,</annotation></semantics></math> and <math alttext="\tau^{2}" class="ltx_Math" display="inline" id="S1.p2.28.m14.1"><semantics id="S1.p2.28.m14.1a"><msup id="S1.p2.28.m14.1.1" xref="S1.p2.28.m14.1.1.cmml"><mi id="S1.p2.28.m14.1.1.2" xref="S1.p2.28.m14.1.1.2.cmml">τ</mi><mn id="S1.p2.28.m14.1.1.3" xref="S1.p2.28.m14.1.1.3.cmml">2</mn></msup><annotation-xml encoding="MathML-Content" id="S1.p2.28.m14.1b"><apply id="S1.p2.28.m14.1.1.cmml" xref="S1.p2.28.m14.1.1"><csymbol cd="ambiguous" id="S1.p2.28.m14.1.1.1.cmml" xref="S1.p2.28.m14.1.1">superscript</csymbol><ci id="S1.p2.28.m14.1.1.2.cmml" xref="S1.p2.28.m14.1.1.2">𝜏</ci><cn id="S1.p2.28.m14.1.1.3.cmml" type="integer" xref="S1.p2.28.m14.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p2.28.m14.1c">\tau^{2}</annotation><annotation encoding="application/x-llamapun" id="S1.p2.28.m14.1d">italic_τ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S1.p3"> <p class="ltx_p" id="S1.p3.4">Mixed model linear and trace regression models are special cases of MMTR in (<a class="ltx_ref" href="https://arxiv.org/html/2503.13782v1#S1.E1" title="In 1 Introduction ‣ Regularized Parameter Estimation in Mixed Model Trace Regression"><span class="ltx_text ltx_ref_tag">1</span></a>). MMTR reduces to mixed model linear regression if <math alttext="P_{2}=Q_{2}=1" class="ltx_Math" display="inline" id="S1.p3.1.m1.1"><semantics id="S1.p3.1.m1.1a"><mrow id="S1.p3.1.m1.1.1" xref="S1.p3.1.m1.1.1.cmml"><msub id="S1.p3.1.m1.1.1.2" xref="S1.p3.1.m1.1.1.2.cmml"><mi id="S1.p3.1.m1.1.1.2.2" xref="S1.p3.1.m1.1.1.2.2.cmml">P</mi><mn id="S1.p3.1.m1.1.1.2.3" xref="S1.p3.1.m1.1.1.2.3.cmml">2</mn></msub><mo id="S1.p3.1.m1.1.1.3" xref="S1.p3.1.m1.1.1.3.cmml">=</mo><msub id="S1.p3.1.m1.1.1.4" xref="S1.p3.1.m1.1.1.4.cmml"><mi id="S1.p3.1.m1.1.1.4.2" xref="S1.p3.1.m1.1.1.4.2.cmml">Q</mi><mn id="S1.p3.1.m1.1.1.4.3" xref="S1.p3.1.m1.1.1.4.3.cmml">2</mn></msub><mo id="S1.p3.1.m1.1.1.5" xref="S1.p3.1.m1.1.1.5.cmml">=</mo><mn id="S1.p3.1.m1.1.1.6" xref="S1.p3.1.m1.1.1.6.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S1.p3.1.m1.1b"><apply id="S1.p3.1.m1.1.1.cmml" xref="S1.p3.1.m1.1.1"><and id="S1.p3.1.m1.1.1a.cmml" xref="S1.p3.1.m1.1.1"></and><apply id="S1.p3.1.m1.1.1b.cmml" xref="S1.p3.1.m1.1.1"><eq id="S1.p3.1.m1.1.1.3.cmml" xref="S1.p3.1.m1.1.1.3"></eq><apply id="S1.p3.1.m1.1.1.2.cmml" xref="S1.p3.1.m1.1.1.2"><csymbol cd="ambiguous" id="S1.p3.1.m1.1.1.2.1.cmml" xref="S1.p3.1.m1.1.1.2">subscript</csymbol><ci id="S1.p3.1.m1.1.1.2.2.cmml" xref="S1.p3.1.m1.1.1.2.2">𝑃</ci><cn id="S1.p3.1.m1.1.1.2.3.cmml" type="integer" xref="S1.p3.1.m1.1.1.2.3">2</cn></apply><apply id="S1.p3.1.m1.1.1.4.cmml" xref="S1.p3.1.m1.1.1.4"><csymbol cd="ambiguous" id="S1.p3.1.m1.1.1.4.1.cmml" xref="S1.p3.1.m1.1.1.4">subscript</csymbol><ci id="S1.p3.1.m1.1.1.4.2.cmml" xref="S1.p3.1.m1.1.1.4.2">𝑄</ci><cn id="S1.p3.1.m1.1.1.4.3.cmml" type="integer" xref="S1.p3.1.m1.1.1.4.3">2</cn></apply></apply><apply id="S1.p3.1.m1.1.1c.cmml" xref="S1.p3.1.m1.1.1"><eq id="S1.p3.1.m1.1.1.5.cmml" xref="S1.p3.1.m1.1.1.5"></eq><share href="https://arxiv.org/html/2503.13782v1#S1.p3.1.m1.1.1.4.cmml" id="S1.p3.1.m1.1.1d.cmml" xref="S1.p3.1.m1.1.1"></share><cn id="S1.p3.1.m1.1.1.6.cmml" type="integer" xref="S1.p3.1.m1.1.1.6">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p3.1.m1.1c">P_{2}=Q_{2}=1</annotation><annotation encoding="application/x-llamapun" id="S1.p3.1.m1.1d">italic_P start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT = italic_Q start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT = 1</annotation></semantics></math>. This model is widely used for analyzing repeated measures and clustered data, but its application is limited to vector-valued covariates <cite class="ltx_cite ltx_citemacro_citep">(Verbeke et al., <a class="ltx_ref" href="https://arxiv.org/html/2503.13782v1#bib.bib33" title="">1997</a>)</cite>. If <math alttext="\operatorname{\mathbf{A}}_{i}=\operatorname{\mathbf{0}}" class="ltx_Math" display="inline" id="S1.p3.2.m2.1"><semantics id="S1.p3.2.m2.1a"><mrow id="S1.p3.2.m2.1.1" xref="S1.p3.2.m2.1.1.cmml"><msub id="S1.p3.2.m2.1.1.2" xref="S1.p3.2.m2.1.1.2.cmml"><mi id="S1.p3.2.m2.1.1.2.2" xref="S1.p3.2.m2.1.1.2.2.cmml">𝐀</mi><mi id="S1.p3.2.m2.1.1.2.3" xref="S1.p3.2.m2.1.1.2.3.cmml">i</mi></msub><mo id="S1.p3.2.m2.1.1.1" xref="S1.p3.2.m2.1.1.1.cmml">=</mo><mi id="S1.p3.2.m2.1.1.3" xref="S1.p3.2.m2.1.1.3.cmml">𝟎</mi></mrow><annotation-xml encoding="MathML-Content" id="S1.p3.2.m2.1b"><apply id="S1.p3.2.m2.1.1.cmml" xref="S1.p3.2.m2.1.1"><eq id="S1.p3.2.m2.1.1.1.cmml" xref="S1.p3.2.m2.1.1.1"></eq><apply id="S1.p3.2.m2.1.1.2.cmml" xref="S1.p3.2.m2.1.1.2"><csymbol cd="ambiguous" id="S1.p3.2.m2.1.1.2.1.cmml" xref="S1.p3.2.m2.1.1.2">subscript</csymbol><ci id="S1.p3.2.m2.1.1.2.2.cmml" xref="S1.p3.2.m2.1.1.2.2">𝐀</ci><ci id="S1.p3.2.m2.1.1.2.3.cmml" xref="S1.p3.2.m2.1.1.2.3">𝑖</ci></apply><ci id="S1.p3.2.m2.1.1.3.cmml" xref="S1.p3.2.m2.1.1.3">𝟎</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p3.2.m2.1c">\operatorname{\mathbf{A}}_{i}=\operatorname{\mathbf{0}}</annotation><annotation encoding="application/x-llamapun" id="S1.p3.2.m2.1d">bold_A start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = bold_0</annotation></semantics></math> and <math alttext="m_{i}=1" class="ltx_Math" display="inline" id="S1.p3.3.m3.1"><semantics id="S1.p3.3.m3.1a"><mrow id="S1.p3.3.m3.1.1" xref="S1.p3.3.m3.1.1.cmml"><msub id="S1.p3.3.m3.1.1.2" xref="S1.p3.3.m3.1.1.2.cmml"><mi id="S1.p3.3.m3.1.1.2.2" xref="S1.p3.3.m3.1.1.2.2.cmml">m</mi><mi id="S1.p3.3.m3.1.1.2.3" xref="S1.p3.3.m3.1.1.2.3.cmml">i</mi></msub><mo id="S1.p3.3.m3.1.1.1" xref="S1.p3.3.m3.1.1.1.cmml">=</mo><mn id="S1.p3.3.m3.1.1.3" xref="S1.p3.3.m3.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S1.p3.3.m3.1b"><apply id="S1.p3.3.m3.1.1.cmml" xref="S1.p3.3.m3.1.1"><eq id="S1.p3.3.m3.1.1.1.cmml" xref="S1.p3.3.m3.1.1.1"></eq><apply id="S1.p3.3.m3.1.1.2.cmml" xref="S1.p3.3.m3.1.1.2"><csymbol cd="ambiguous" id="S1.p3.3.m3.1.1.2.1.cmml" xref="S1.p3.3.m3.1.1.2">subscript</csymbol><ci id="S1.p3.3.m3.1.1.2.2.cmml" xref="S1.p3.3.m3.1.1.2.2">𝑚</ci><ci id="S1.p3.3.m3.1.1.2.3.cmml" xref="S1.p3.3.m3.1.1.2.3">𝑖</ci></apply><cn id="S1.p3.3.m3.1.1.3.cmml" type="integer" xref="S1.p3.3.m3.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p3.3.m3.1c">m_{i}=1</annotation><annotation encoding="application/x-llamapun" id="S1.p3.3.m3.1d">italic_m start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = 1</annotation></semantics></math> for every <math alttext="i" class="ltx_Math" display="inline" id="S1.p3.4.m4.1"><semantics id="S1.p3.4.m4.1a"><mi id="S1.p3.4.m4.1.1" xref="S1.p3.4.m4.1.1.cmml">i</mi><annotation-xml encoding="MathML-Content" id="S1.p3.4.m4.1b"><ci id="S1.p3.4.m4.1.1.cmml" xref="S1.p3.4.m4.1.1">𝑖</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p3.4.m4.1c">i</annotation><annotation encoding="application/x-llamapun" id="S1.p3.4.m4.1d">italic_i</annotation></semantics></math> in (<a class="ltx_ref" href="https://arxiv.org/html/2503.13782v1#S1.E1" title="In 1 Introduction ‣ Regularized Parameter Estimation in Mixed Model Trace Regression"><span class="ltx_text ltx_ref_tag">1</span></a>), then MMTR is equivalent to the trace regression model <cite class="ltx_cite ltx_citemacro_citep">(Fan et al., <a class="ltx_ref" href="https://arxiv.org/html/2503.13782v1#bib.bib7" title="">2019</a>)</cite>. This model accommodates matrix-valued covariates but fails to model the correlation in the responses. MMTR integrates the advantages of these two frameworks and enables realistic models for repeated measures and clustered data with matrix covariates and scalar responses.</p> </div> <section class="ltx_subsection" id="S1.SS1"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">1.1 </span>Prior Literature</h3> <div class="ltx_para" id="S1.SS1.p1"> <p class="ltx_p" id="S1.SS1.p1.15">The vectorization of covariates reduces MMTR to a “structured” mixed model linear regression. If <math alttext="P=P_{1}P_{2}" class="ltx_Math" display="inline" id="S1.SS1.p1.1.m1.1"><semantics id="S1.SS1.p1.1.m1.1a"><mrow id="S1.SS1.p1.1.m1.1.1" xref="S1.SS1.p1.1.m1.1.1.cmml"><mi id="S1.SS1.p1.1.m1.1.1.2" xref="S1.SS1.p1.1.m1.1.1.2.cmml">P</mi><mo id="S1.SS1.p1.1.m1.1.1.1" xref="S1.SS1.p1.1.m1.1.1.1.cmml">=</mo><mrow id="S1.SS1.p1.1.m1.1.1.3" xref="S1.SS1.p1.1.m1.1.1.3.cmml"><msub id="S1.SS1.p1.1.m1.1.1.3.2" xref="S1.SS1.p1.1.m1.1.1.3.2.cmml"><mi id="S1.SS1.p1.1.m1.1.1.3.2.2" xref="S1.SS1.p1.1.m1.1.1.3.2.2.cmml">P</mi><mn id="S1.SS1.p1.1.m1.1.1.3.2.3" xref="S1.SS1.p1.1.m1.1.1.3.2.3.cmml">1</mn></msub><mo id="S1.SS1.p1.1.m1.1.1.3.1" xref="S1.SS1.p1.1.m1.1.1.3.1.cmml">⁢</mo><msub id="S1.SS1.p1.1.m1.1.1.3.3" xref="S1.SS1.p1.1.m1.1.1.3.3.cmml"><mi id="S1.SS1.p1.1.m1.1.1.3.3.2" xref="S1.SS1.p1.1.m1.1.1.3.3.2.cmml">P</mi><mn id="S1.SS1.p1.1.m1.1.1.3.3.3" xref="S1.SS1.p1.1.m1.1.1.3.3.3.cmml">2</mn></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.SS1.p1.1.m1.1b"><apply id="S1.SS1.p1.1.m1.1.1.cmml" xref="S1.SS1.p1.1.m1.1.1"><eq id="S1.SS1.p1.1.m1.1.1.1.cmml" xref="S1.SS1.p1.1.m1.1.1.1"></eq><ci id="S1.SS1.p1.1.m1.1.1.2.cmml" xref="S1.SS1.p1.1.m1.1.1.2">𝑃</ci><apply id="S1.SS1.p1.1.m1.1.1.3.cmml" xref="S1.SS1.p1.1.m1.1.1.3"><times id="S1.SS1.p1.1.m1.1.1.3.1.cmml" xref="S1.SS1.p1.1.m1.1.1.3.1"></times><apply id="S1.SS1.p1.1.m1.1.1.3.2.cmml" xref="S1.SS1.p1.1.m1.1.1.3.2"><csymbol cd="ambiguous" id="S1.SS1.p1.1.m1.1.1.3.2.1.cmml" xref="S1.SS1.p1.1.m1.1.1.3.2">subscript</csymbol><ci id="S1.SS1.p1.1.m1.1.1.3.2.2.cmml" xref="S1.SS1.p1.1.m1.1.1.3.2.2">𝑃</ci><cn id="S1.SS1.p1.1.m1.1.1.3.2.3.cmml" type="integer" xref="S1.SS1.p1.1.m1.1.1.3.2.3">1</cn></apply><apply id="S1.SS1.p1.1.m1.1.1.3.3.cmml" xref="S1.SS1.p1.1.m1.1.1.3.3"><csymbol cd="ambiguous" id="S1.SS1.p1.1.m1.1.1.3.3.1.cmml" xref="S1.SS1.p1.1.m1.1.1.3.3">subscript</csymbol><ci id="S1.SS1.p1.1.m1.1.1.3.3.2.cmml" xref="S1.SS1.p1.1.m1.1.1.3.3.2">𝑃</ci><cn id="S1.SS1.p1.1.m1.1.1.3.3.3.cmml" type="integer" xref="S1.SS1.p1.1.m1.1.1.3.3.3">2</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.p1.1.m1.1c">P=P_{1}P_{2}</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.p1.1.m1.1d">italic_P = italic_P start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_P start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="Q=Q_{1}Q_{2}" class="ltx_Math" display="inline" id="S1.SS1.p1.2.m2.1"><semantics id="S1.SS1.p1.2.m2.1a"><mrow id="S1.SS1.p1.2.m2.1.1" xref="S1.SS1.p1.2.m2.1.1.cmml"><mi id="S1.SS1.p1.2.m2.1.1.2" xref="S1.SS1.p1.2.m2.1.1.2.cmml">Q</mi><mo id="S1.SS1.p1.2.m2.1.1.1" xref="S1.SS1.p1.2.m2.1.1.1.cmml">=</mo><mrow id="S1.SS1.p1.2.m2.1.1.3" xref="S1.SS1.p1.2.m2.1.1.3.cmml"><msub id="S1.SS1.p1.2.m2.1.1.3.2" xref="S1.SS1.p1.2.m2.1.1.3.2.cmml"><mi id="S1.SS1.p1.2.m2.1.1.3.2.2" xref="S1.SS1.p1.2.m2.1.1.3.2.2.cmml">Q</mi><mn id="S1.SS1.p1.2.m2.1.1.3.2.3" xref="S1.SS1.p1.2.m2.1.1.3.2.3.cmml">1</mn></msub><mo id="S1.SS1.p1.2.m2.1.1.3.1" xref="S1.SS1.p1.2.m2.1.1.3.1.cmml">⁢</mo><msub id="S1.SS1.p1.2.m2.1.1.3.3" xref="S1.SS1.p1.2.m2.1.1.3.3.cmml"><mi id="S1.SS1.p1.2.m2.1.1.3.3.2" xref="S1.SS1.p1.2.m2.1.1.3.3.2.cmml">Q</mi><mn id="S1.SS1.p1.2.m2.1.1.3.3.3" xref="S1.SS1.p1.2.m2.1.1.3.3.3.cmml">2</mn></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.SS1.p1.2.m2.1b"><apply id="S1.SS1.p1.2.m2.1.1.cmml" xref="S1.SS1.p1.2.m2.1.1"><eq id="S1.SS1.p1.2.m2.1.1.1.cmml" xref="S1.SS1.p1.2.m2.1.1.1"></eq><ci id="S1.SS1.p1.2.m2.1.1.2.cmml" xref="S1.SS1.p1.2.m2.1.1.2">𝑄</ci><apply id="S1.SS1.p1.2.m2.1.1.3.cmml" xref="S1.SS1.p1.2.m2.1.1.3"><times id="S1.SS1.p1.2.m2.1.1.3.1.cmml" xref="S1.SS1.p1.2.m2.1.1.3.1"></times><apply id="S1.SS1.p1.2.m2.1.1.3.2.cmml" xref="S1.SS1.p1.2.m2.1.1.3.2"><csymbol cd="ambiguous" id="S1.SS1.p1.2.m2.1.1.3.2.1.cmml" xref="S1.SS1.p1.2.m2.1.1.3.2">subscript</csymbol><ci id="S1.SS1.p1.2.m2.1.1.3.2.2.cmml" xref="S1.SS1.p1.2.m2.1.1.3.2.2">𝑄</ci><cn id="S1.SS1.p1.2.m2.1.1.3.2.3.cmml" type="integer" xref="S1.SS1.p1.2.m2.1.1.3.2.3">1</cn></apply><apply id="S1.SS1.p1.2.m2.1.1.3.3.cmml" xref="S1.SS1.p1.2.m2.1.1.3.3"><csymbol cd="ambiguous" id="S1.SS1.p1.2.m2.1.1.3.3.1.cmml" xref="S1.SS1.p1.2.m2.1.1.3.3">subscript</csymbol><ci id="S1.SS1.p1.2.m2.1.1.3.3.2.cmml" xref="S1.SS1.p1.2.m2.1.1.3.3.2">𝑄</ci><cn id="S1.SS1.p1.2.m2.1.1.3.3.3.cmml" type="integer" xref="S1.SS1.p1.2.m2.1.1.3.3.3">2</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.p1.2.m2.1c">Q=Q_{1}Q_{2}</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.p1.2.m2.1d">italic_Q = italic_Q start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_Q start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math>, then (<a class="ltx_ref" href="https://arxiv.org/html/2503.13782v1#S1.E1" title="In 1 Introduction ‣ Regularized Parameter Estimation in Mixed Model Trace Regression"><span class="ltx_text ltx_ref_tag">1</span></a>) is equivalent to a mixed model with fixed and random effects covariates <math alttext="\text{vec}(\operatorname{\mathbf{X}}_{{ij}})\in\operatorname{\mathbb{R}}^{P}" class="ltx_Math" display="inline" id="S1.SS1.p1.3.m3.1"><semantics id="S1.SS1.p1.3.m3.1a"><mrow id="S1.SS1.p1.3.m3.1.1" xref="S1.SS1.p1.3.m3.1.1.cmml"><mrow id="S1.SS1.p1.3.m3.1.1.1" xref="S1.SS1.p1.3.m3.1.1.1.cmml"><mtext id="S1.SS1.p1.3.m3.1.1.1.3" xref="S1.SS1.p1.3.m3.1.1.1.3a.cmml">vec</mtext><mo id="S1.SS1.p1.3.m3.1.1.1.2" xref="S1.SS1.p1.3.m3.1.1.1.2.cmml">⁢</mo><mrow id="S1.SS1.p1.3.m3.1.1.1.1.1" xref="S1.SS1.p1.3.m3.1.1.1.1.1.1.cmml"><mo id="S1.SS1.p1.3.m3.1.1.1.1.1.2" stretchy="false" xref="S1.SS1.p1.3.m3.1.1.1.1.1.1.cmml">(</mo><msub id="S1.SS1.p1.3.m3.1.1.1.1.1.1" xref="S1.SS1.p1.3.m3.1.1.1.1.1.1.cmml"><mi id="S1.SS1.p1.3.m3.1.1.1.1.1.1.2" xref="S1.SS1.p1.3.m3.1.1.1.1.1.1.2.cmml">𝐗</mi><mrow id="S1.SS1.p1.3.m3.1.1.1.1.1.1.3" xref="S1.SS1.p1.3.m3.1.1.1.1.1.1.3.cmml"><mi 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xref="S1.SS1.p1.3.m3.1.1.3">superscript</csymbol><ci id="S1.SS1.p1.3.m3.1.1.3.2.cmml" xref="S1.SS1.p1.3.m3.1.1.3.2">ℝ</ci><ci id="S1.SS1.p1.3.m3.1.1.3.3.cmml" xref="S1.SS1.p1.3.m3.1.1.3.3">𝑃</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.p1.3.m3.1c">\text{vec}(\operatorname{\mathbf{X}}_{{ij}})\in\operatorname{\mathbb{R}}^{P}</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.p1.3.m3.1d">vec ( bold_X start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT ) ∈ blackboard_R start_POSTSUPERSCRIPT italic_P end_POSTSUPERSCRIPT</annotation></semantics></math> and <math alttext="\text{vec}(\operatorname{\mathbf{Z}}_{ij})\in\operatorname{\mathbb{R}}^{Q}" class="ltx_Math" display="inline" id="S1.SS1.p1.4.m4.1"><semantics id="S1.SS1.p1.4.m4.1a"><mrow id="S1.SS1.p1.4.m4.1.1" xref="S1.SS1.p1.4.m4.1.1.cmml"><mrow id="S1.SS1.p1.4.m4.1.1.1" xref="S1.SS1.p1.4.m4.1.1.1.cmml"><mtext id="S1.SS1.p1.4.m4.1.1.1.3" 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xref="S1.SS1.p1.4.m4.1.1.1.1.1.1.2">𝐙</ci><apply id="S1.SS1.p1.4.m4.1.1.1.1.1.1.3.cmml" xref="S1.SS1.p1.4.m4.1.1.1.1.1.1.3"><times id="S1.SS1.p1.4.m4.1.1.1.1.1.1.3.1.cmml" xref="S1.SS1.p1.4.m4.1.1.1.1.1.1.3.1"></times><ci id="S1.SS1.p1.4.m4.1.1.1.1.1.1.3.2.cmml" xref="S1.SS1.p1.4.m4.1.1.1.1.1.1.3.2">𝑖</ci><ci id="S1.SS1.p1.4.m4.1.1.1.1.1.1.3.3.cmml" xref="S1.SS1.p1.4.m4.1.1.1.1.1.1.3.3">𝑗</ci></apply></apply></apply><apply id="S1.SS1.p1.4.m4.1.1.3.cmml" xref="S1.SS1.p1.4.m4.1.1.3"><csymbol cd="ambiguous" id="S1.SS1.p1.4.m4.1.1.3.1.cmml" xref="S1.SS1.p1.4.m4.1.1.3">superscript</csymbol><ci id="S1.SS1.p1.4.m4.1.1.3.2.cmml" xref="S1.SS1.p1.4.m4.1.1.3.2">ℝ</ci><ci id="S1.SS1.p1.4.m4.1.1.3.3.cmml" xref="S1.SS1.p1.4.m4.1.1.3.3">𝑄</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.p1.4.m4.1c">\text{vec}(\operatorname{\mathbf{Z}}_{ij})\in\operatorname{\mathbb{R}}^{Q}</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.p1.4.m4.1d">vec ( bold_Z start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT ) ∈ blackboard_R start_POSTSUPERSCRIPT italic_Q end_POSTSUPERSCRIPT</annotation></semantics></math>, <math alttext="P" class="ltx_Math" display="inline" id="S1.SS1.p1.5.m5.1"><semantics id="S1.SS1.p1.5.m5.1a"><mi id="S1.SS1.p1.5.m5.1.1" xref="S1.SS1.p1.5.m5.1.1.cmml">P</mi><annotation-xml encoding="MathML-Content" id="S1.SS1.p1.5.m5.1b"><ci id="S1.SS1.p1.5.m5.1.1.cmml" xref="S1.SS1.p1.5.m5.1.1">𝑃</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.p1.5.m5.1c">P</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.p1.5.m5.1d">italic_P</annotation></semantics></math>-dimensional regression coefficient <math alttext="\text{vec}(\operatorname{\mathbf{B}})" class="ltx_Math" display="inline" id="S1.SS1.p1.6.m6.1"><semantics id="S1.SS1.p1.6.m6.1a"><mrow id="S1.SS1.p1.6.m6.1.2" xref="S1.SS1.p1.6.m6.1.2.cmml"><mtext id="S1.SS1.p1.6.m6.1.2.2" xref="S1.SS1.p1.6.m6.1.2.2a.cmml">vec</mtext><mo 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id="S1.SS1.p1.6.m6.1d">vec ( bold_B )</annotation></semantics></math>, and <math alttext="Q" class="ltx_Math" display="inline" id="S1.SS1.p1.7.m7.1"><semantics id="S1.SS1.p1.7.m7.1a"><mi id="S1.SS1.p1.7.m7.1.1" xref="S1.SS1.p1.7.m7.1.1.cmml">Q</mi><annotation-xml encoding="MathML-Content" id="S1.SS1.p1.7.m7.1b"><ci id="S1.SS1.p1.7.m7.1.1.cmml" xref="S1.SS1.p1.7.m7.1.1">𝑄</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.p1.7.m7.1c">Q</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.p1.7.m7.1d">italic_Q</annotation></semantics></math>-dimensional random effects <math alttext="\text{vec}(\operatorname{\mathbf{A}}_{i})" class="ltx_Math" display="inline" id="S1.SS1.p1.8.m8.1"><semantics id="S1.SS1.p1.8.m8.1a"><mrow id="S1.SS1.p1.8.m8.1.1" xref="S1.SS1.p1.8.m8.1.1.cmml"><mtext id="S1.SS1.p1.8.m8.1.1.3" xref="S1.SS1.p1.8.m8.1.1.3a.cmml">vec</mtext><mo id="S1.SS1.p1.8.m8.1.1.2" xref="S1.SS1.p1.8.m8.1.1.2.cmml">⁢</mo><mrow id="S1.SS1.p1.8.m8.1.1.1.1" 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xref="S1.SS1.p1.10.m10.1.1.2.2.2.cmml">τ</mi><mn id="S1.SS1.p1.10.m10.1.1.2.2.3" xref="S1.SS1.p1.10.m10.1.1.2.2.3.cmml">2</mn></msup><mo id="S1.SS1.p1.10.m10.1.1.2.1" lspace="0.167em" xref="S1.SS1.p1.10.m10.1.1.2.1.cmml">⁢</mo><msub id="S1.SS1.p1.10.m10.1.1.2.3" xref="S1.SS1.p1.10.m10.1.1.2.3.cmml"><mi id="S1.SS1.p1.10.m10.1.1.2.3.2" xref="S1.SS1.p1.10.m10.1.1.2.3.2.cmml">𝚺</mi><mn id="S1.SS1.p1.10.m10.1.1.2.3.3" xref="S1.SS1.p1.10.m10.1.1.2.3.3.cmml">2</mn></msub></mrow><mo id="S1.SS1.p1.10.m10.1.1.1" lspace="0em" rspace="0.222em" xref="S1.SS1.p1.10.m10.1.1.1.cmml">⊗</mo><msub id="S1.SS1.p1.10.m10.1.1.3" xref="S1.SS1.p1.10.m10.1.1.3.cmml"><mi id="S1.SS1.p1.10.m10.1.1.3.2" xref="S1.SS1.p1.10.m10.1.1.3.2.cmml">𝚺</mi><mn id="S1.SS1.p1.10.m10.1.1.3.3" xref="S1.SS1.p1.10.m10.1.1.3.3.cmml">1</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S1.SS1.p1.10.m10.1b"><apply id="S1.SS1.p1.10.m10.1.1.cmml" xref="S1.SS1.p1.10.m10.1.1"><csymbol cd="latexml" id="S1.SS1.p1.10.m10.1.1.1.cmml" xref="S1.SS1.p1.10.m10.1.1.1">tensor-product</csymbol><apply id="S1.SS1.p1.10.m10.1.1.2.cmml" xref="S1.SS1.p1.10.m10.1.1.2"><times id="S1.SS1.p1.10.m10.1.1.2.1.cmml" xref="S1.SS1.p1.10.m10.1.1.2.1"></times><apply id="S1.SS1.p1.10.m10.1.1.2.2.cmml" xref="S1.SS1.p1.10.m10.1.1.2.2"><csymbol cd="ambiguous" id="S1.SS1.p1.10.m10.1.1.2.2.1.cmml" xref="S1.SS1.p1.10.m10.1.1.2.2">superscript</csymbol><ci id="S1.SS1.p1.10.m10.1.1.2.2.2.cmml" xref="S1.SS1.p1.10.m10.1.1.2.2.2">𝜏</ci><cn id="S1.SS1.p1.10.m10.1.1.2.2.3.cmml" type="integer" xref="S1.SS1.p1.10.m10.1.1.2.2.3">2</cn></apply><apply id="S1.SS1.p1.10.m10.1.1.2.3.cmml" xref="S1.SS1.p1.10.m10.1.1.2.3"><csymbol cd="ambiguous" id="S1.SS1.p1.10.m10.1.1.2.3.1.cmml" xref="S1.SS1.p1.10.m10.1.1.2.3">subscript</csymbol><ci id="S1.SS1.p1.10.m10.1.1.2.3.2.cmml" xref="S1.SS1.p1.10.m10.1.1.2.3.2">𝚺</ci><cn id="S1.SS1.p1.10.m10.1.1.2.3.3.cmml" type="integer" xref="S1.SS1.p1.10.m10.1.1.2.3.3">2</cn></apply></apply><apply 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id="S1.SS1.p1.12.m12.1"><semantics id="S1.SS1.p1.12.m12.1a"><mo id="S1.SS1.p1.12.m12.1.1" xref="S1.SS1.p1.12.m12.1.1.cmml">⊗</mo><annotation-xml encoding="MathML-Content" id="S1.SS1.p1.12.m12.1b"><csymbol cd="latexml" id="S1.SS1.p1.12.m12.1.1.cmml" xref="S1.SS1.p1.12.m12.1.1">tensor-product</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.p1.12.m12.1c">\otimes</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.p1.12.m12.1d">⊗</annotation></semantics></math> is the Kronecker product. Fitting these structured mixed models using classical algorithms, such as those in <span class="ltx_text ltx_font_typewriter" id="S1.SS1.p1.15.2">lme4</span> <cite class="ltx_cite ltx_citemacro_citep">(Bates et al., <a class="ltx_ref" href="https://arxiv.org/html/2503.13782v1#bib.bib2" title="">2015</a>)</cite>, has poor empirical performance for two main reasons. First, the estimation algorithms do not exploit the sparse and the separable matrix-variate structures of <math alttext="\operatorname{\mathbf{B}}" class="ltx_Math" display="inline" id="S1.SS1.p1.13.m13.1"><semantics id="S1.SS1.p1.13.m13.1a"><mi id="S1.SS1.p1.13.m13.1.1" xref="S1.SS1.p1.13.m13.1.1.cmml">𝐁</mi><annotation-xml encoding="MathML-Content" id="S1.SS1.p1.13.m13.1b"><ci id="S1.SS1.p1.13.m13.1.1.cmml" xref="S1.SS1.p1.13.m13.1.1">𝐁</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.p1.13.m13.1c">\operatorname{\mathbf{B}}</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.p1.13.m13.1d">bold_B</annotation></semantics></math> and <math alttext="\tau^{2}\operatorname{{\mathbf{\Sigma}}}_{2}\otimes\operatorname{{\mathbf{% \Sigma}}}_{1}" class="ltx_Math" display="inline" id="S1.SS1.p1.14.m14.1"><semantics id="S1.SS1.p1.14.m14.1a"><mrow id="S1.SS1.p1.14.m14.1.1" xref="S1.SS1.p1.14.m14.1.1.cmml"><mrow id="S1.SS1.p1.14.m14.1.1.2" xref="S1.SS1.p1.14.m14.1.1.2.cmml"><msup id="S1.SS1.p1.14.m14.1.1.2.2" xref="S1.SS1.p1.14.m14.1.1.2.2.cmml"><mi id="S1.SS1.p1.14.m14.1.1.2.2.2" xref="S1.SS1.p1.14.m14.1.1.2.2.2.cmml">τ</mi><mn id="S1.SS1.p1.14.m14.1.1.2.2.3" xref="S1.SS1.p1.14.m14.1.1.2.2.3.cmml">2</mn></msup><mo id="S1.SS1.p1.14.m14.1.1.2.1" lspace="0.167em" xref="S1.SS1.p1.14.m14.1.1.2.1.cmml">⁢</mo><msub id="S1.SS1.p1.14.m14.1.1.2.3" xref="S1.SS1.p1.14.m14.1.1.2.3.cmml"><mi id="S1.SS1.p1.14.m14.1.1.2.3.2" xref="S1.SS1.p1.14.m14.1.1.2.3.2.cmml">𝚺</mi><mn id="S1.SS1.p1.14.m14.1.1.2.3.3" xref="S1.SS1.p1.14.m14.1.1.2.3.3.cmml">2</mn></msub></mrow><mo id="S1.SS1.p1.14.m14.1.1.1" lspace="0em" rspace="0.222em" xref="S1.SS1.p1.14.m14.1.1.1.cmml">⊗</mo><msub id="S1.SS1.p1.14.m14.1.1.3" xref="S1.SS1.p1.14.m14.1.1.3.cmml"><mi id="S1.SS1.p1.14.m14.1.1.3.2" xref="S1.SS1.p1.14.m14.1.1.3.2.cmml">𝚺</mi><mn id="S1.SS1.p1.14.m14.1.1.3.3" xref="S1.SS1.p1.14.m14.1.1.3.3.cmml">1</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S1.SS1.p1.14.m14.1b"><apply id="S1.SS1.p1.14.m14.1.1.cmml" xref="S1.SS1.p1.14.m14.1.1"><csymbol cd="latexml" id="S1.SS1.p1.14.m14.1.1.1.cmml" xref="S1.SS1.p1.14.m14.1.1.1">tensor-product</csymbol><apply id="S1.SS1.p1.14.m14.1.1.2.cmml" xref="S1.SS1.p1.14.m14.1.1.2"><times id="S1.SS1.p1.14.m14.1.1.2.1.cmml" xref="S1.SS1.p1.14.m14.1.1.2.1"></times><apply id="S1.SS1.p1.14.m14.1.1.2.2.cmml" xref="S1.SS1.p1.14.m14.1.1.2.2"><csymbol cd="ambiguous" id="S1.SS1.p1.14.m14.1.1.2.2.1.cmml" xref="S1.SS1.p1.14.m14.1.1.2.2">superscript</csymbol><ci id="S1.SS1.p1.14.m14.1.1.2.2.2.cmml" xref="S1.SS1.p1.14.m14.1.1.2.2.2">𝜏</ci><cn id="S1.SS1.p1.14.m14.1.1.2.2.3.cmml" type="integer" xref="S1.SS1.p1.14.m14.1.1.2.2.3">2</cn></apply><apply id="S1.SS1.p1.14.m14.1.1.2.3.cmml" xref="S1.SS1.p1.14.m14.1.1.2.3"><csymbol cd="ambiguous" id="S1.SS1.p1.14.m14.1.1.2.3.1.cmml" xref="S1.SS1.p1.14.m14.1.1.2.3">subscript</csymbol><ci id="S1.SS1.p1.14.m14.1.1.2.3.2.cmml" xref="S1.SS1.p1.14.m14.1.1.2.3.2">𝚺</ci><cn id="S1.SS1.p1.14.m14.1.1.2.3.3.cmml" type="integer" xref="S1.SS1.p1.14.m14.1.1.2.3.3">2</cn></apply></apply><apply id="S1.SS1.p1.14.m14.1.1.3.cmml" xref="S1.SS1.p1.14.m14.1.1.3"><csymbol cd="ambiguous" id="S1.SS1.p1.14.m14.1.1.3.1.cmml" xref="S1.SS1.p1.14.m14.1.1.3">subscript</csymbol><ci id="S1.SS1.p1.14.m14.1.1.3.2.cmml" xref="S1.SS1.p1.14.m14.1.1.3.2">𝚺</ci><cn id="S1.SS1.p1.14.m14.1.1.3.3.cmml" type="integer" xref="S1.SS1.p1.14.m14.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.p1.14.m14.1c">\tau^{2}\operatorname{{\mathbf{\Sigma}}}_{2}\otimes\operatorname{{\mathbf{% \Sigma}}}_{1}</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.p1.14.m14.1d">italic_τ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT bold_Σ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ⊗ bold_Σ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math>, respectively. Second, they are prohibitively slow because the parameter dimension grows rapidly as <math alttext="O(P+Q^{2})" class="ltx_Math" display="inline" id="S1.SS1.p1.15.m15.1"><semantics id="S1.SS1.p1.15.m15.1a"><mrow id="S1.SS1.p1.15.m15.1.1" xref="S1.SS1.p1.15.m15.1.1.cmml"><mi id="S1.SS1.p1.15.m15.1.1.3" xref="S1.SS1.p1.15.m15.1.1.3.cmml">O</mi><mo id="S1.SS1.p1.15.m15.1.1.2" xref="S1.SS1.p1.15.m15.1.1.2.cmml">⁢</mo><mrow id="S1.SS1.p1.15.m15.1.1.1.1" xref="S1.SS1.p1.15.m15.1.1.1.1.1.cmml"><mo id="S1.SS1.p1.15.m15.1.1.1.1.2" stretchy="false" xref="S1.SS1.p1.15.m15.1.1.1.1.1.cmml">(</mo><mrow id="S1.SS1.p1.15.m15.1.1.1.1.1" xref="S1.SS1.p1.15.m15.1.1.1.1.1.cmml"><mi id="S1.SS1.p1.15.m15.1.1.1.1.1.2" xref="S1.SS1.p1.15.m15.1.1.1.1.1.2.cmml">P</mi><mo id="S1.SS1.p1.15.m15.1.1.1.1.1.1" xref="S1.SS1.p1.15.m15.1.1.1.1.1.1.cmml">+</mo><msup id="S1.SS1.p1.15.m15.1.1.1.1.1.3" xref="S1.SS1.p1.15.m15.1.1.1.1.1.3.cmml"><mi id="S1.SS1.p1.15.m15.1.1.1.1.1.3.2" xref="S1.SS1.p1.15.m15.1.1.1.1.1.3.2.cmml">Q</mi><mn id="S1.SS1.p1.15.m15.1.1.1.1.1.3.3" xref="S1.SS1.p1.15.m15.1.1.1.1.1.3.3.cmml">2</mn></msup></mrow><mo id="S1.SS1.p1.15.m15.1.1.1.1.3" stretchy="false" xref="S1.SS1.p1.15.m15.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.SS1.p1.15.m15.1b"><apply id="S1.SS1.p1.15.m15.1.1.cmml" xref="S1.SS1.p1.15.m15.1.1"><times id="S1.SS1.p1.15.m15.1.1.2.cmml" xref="S1.SS1.p1.15.m15.1.1.2"></times><ci id="S1.SS1.p1.15.m15.1.1.3.cmml" xref="S1.SS1.p1.15.m15.1.1.3">𝑂</ci><apply id="S1.SS1.p1.15.m15.1.1.1.1.1.cmml" xref="S1.SS1.p1.15.m15.1.1.1.1"><plus id="S1.SS1.p1.15.m15.1.1.1.1.1.1.cmml" xref="S1.SS1.p1.15.m15.1.1.1.1.1.1"></plus><ci id="S1.SS1.p1.15.m15.1.1.1.1.1.2.cmml" xref="S1.SS1.p1.15.m15.1.1.1.1.1.2">𝑃</ci><apply id="S1.SS1.p1.15.m15.1.1.1.1.1.3.cmml" xref="S1.SS1.p1.15.m15.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S1.SS1.p1.15.m15.1.1.1.1.1.3.1.cmml" xref="S1.SS1.p1.15.m15.1.1.1.1.1.3">superscript</csymbol><ci id="S1.SS1.p1.15.m15.1.1.1.1.1.3.2.cmml" xref="S1.SS1.p1.15.m15.1.1.1.1.1.3.2">𝑄</ci><cn id="S1.SS1.p1.15.m15.1.1.1.1.1.3.3.cmml" type="integer" xref="S1.SS1.p1.15.m15.1.1.1.1.1.3.3">2</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.p1.15.m15.1c">O(P+Q^{2})</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.p1.15.m15.1d">italic_O ( italic_P + italic_Q start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT )</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S1.SS1.p2"> <p class="ltx_p" id="S1.SS1.p2.2">High-dimensional mixed model extensions address these issues through penalization but fail to exploit the separable covariance structures. The main idea of these methods is to define a quasi-likelihood by replacing the covariance matrix with a proxy matrix <cite class="ltx_cite ltx_citemacro_citep">(Fan and Li, <a class="ltx_ref" href="https://arxiv.org/html/2503.13782v1#bib.bib8" title="">2012</a>; Hui et al., <a class="ltx_ref" href="https://arxiv.org/html/2503.13782v1#bib.bib17" title="">2017</a>; Bradic et al., <a class="ltx_ref" href="https://arxiv.org/html/2503.13782v1#bib.bib4" title="">2020</a>; Li et al., <a class="ltx_ref" href="https://arxiv.org/html/2503.13782v1#bib.bib21" title="">2022</a>)</cite>. A sparse estimate of <math alttext="\operatorname{\mathbf{B}}" class="ltx_Math" display="inline" id="S1.SS1.p2.1.m1.1"><semantics id="S1.SS1.p2.1.m1.1a"><mi id="S1.SS1.p2.1.m1.1.1" xref="S1.SS1.p2.1.m1.1.1.cmml">𝐁</mi><annotation-xml encoding="MathML-Content" id="S1.SS1.p2.1.m1.1b"><ci id="S1.SS1.p2.1.m1.1.1.cmml" xref="S1.SS1.p2.1.m1.1.1">𝐁</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.p2.1.m1.1c">\operatorname{\mathbf{B}}</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.p2.1.m1.1d">bold_B</annotation></semantics></math> is obtained using lasso-type penalties, which are optimal under various high-dimensional asymptotic regimes. The estimate of the random effects covariance matrix is only available when <math alttext="Q<\min_{i}m_{i}" class="ltx_Math" display="inline" id="S1.SS1.p2.2.m2.1"><semantics id="S1.SS1.p2.2.m2.1a"><mrow id="S1.SS1.p2.2.m2.1.1" xref="S1.SS1.p2.2.m2.1.1.cmml"><mi id="S1.SS1.p2.2.m2.1.1.2" xref="S1.SS1.p2.2.m2.1.1.2.cmml">Q</mi><mo id="S1.SS1.p2.2.m2.1.1.1" xref="S1.SS1.p2.2.m2.1.1.1.cmml"><</mo><mrow id="S1.SS1.p2.2.m2.1.1.3" xref="S1.SS1.p2.2.m2.1.1.3.cmml"><msub id="S1.SS1.p2.2.m2.1.1.3.1" xref="S1.SS1.p2.2.m2.1.1.3.1.cmml"><mi id="S1.SS1.p2.2.m2.1.1.3.1.2" xref="S1.SS1.p2.2.m2.1.1.3.1.2.cmml">min</mi><mi id="S1.SS1.p2.2.m2.1.1.3.1.3" xref="S1.SS1.p2.2.m2.1.1.3.1.3.cmml">i</mi></msub><mo id="S1.SS1.p2.2.m2.1.1.3a" lspace="0.167em" xref="S1.SS1.p2.2.m2.1.1.3.cmml">⁡</mo><msub id="S1.SS1.p2.2.m2.1.1.3.2" xref="S1.SS1.p2.2.m2.1.1.3.2.cmml"><mi id="S1.SS1.p2.2.m2.1.1.3.2.2" xref="S1.SS1.p2.2.m2.1.1.3.2.2.cmml">m</mi><mi id="S1.SS1.p2.2.m2.1.1.3.2.3" xref="S1.SS1.p2.2.m2.1.1.3.2.3.cmml">i</mi></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.SS1.p2.2.m2.1b"><apply id="S1.SS1.p2.2.m2.1.1.cmml" xref="S1.SS1.p2.2.m2.1.1"><lt id="S1.SS1.p2.2.m2.1.1.1.cmml" xref="S1.SS1.p2.2.m2.1.1.1"></lt><ci id="S1.SS1.p2.2.m2.1.1.2.cmml" xref="S1.SS1.p2.2.m2.1.1.2">𝑄</ci><apply id="S1.SS1.p2.2.m2.1.1.3.cmml" xref="S1.SS1.p2.2.m2.1.1.3"><apply id="S1.SS1.p2.2.m2.1.1.3.1.cmml" xref="S1.SS1.p2.2.m2.1.1.3.1"><csymbol cd="ambiguous" id="S1.SS1.p2.2.m2.1.1.3.1.1.cmml" xref="S1.SS1.p2.2.m2.1.1.3.1">subscript</csymbol><min id="S1.SS1.p2.2.m2.1.1.3.1.2.cmml" xref="S1.SS1.p2.2.m2.1.1.3.1.2"></min><ci id="S1.SS1.p2.2.m2.1.1.3.1.3.cmml" xref="S1.SS1.p2.2.m2.1.1.3.1.3">𝑖</ci></apply><apply id="S1.SS1.p2.2.m2.1.1.3.2.cmml" xref="S1.SS1.p2.2.m2.1.1.3.2"><csymbol cd="ambiguous" id="S1.SS1.p2.2.m2.1.1.3.2.1.cmml" xref="S1.SS1.p2.2.m2.1.1.3.2">subscript</csymbol><ci id="S1.SS1.p2.2.m2.1.1.3.2.2.cmml" xref="S1.SS1.p2.2.m2.1.1.3.2.2">𝑚</ci><ci id="S1.SS1.p2.2.m2.1.1.3.2.3.cmml" xref="S1.SS1.p2.2.m2.1.1.3.2.3">𝑖</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.p2.2.m2.1c">Q<\min_{i}m_{i}</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.p2.2.m2.1d">italic_Q < roman_min start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_m start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math>, a condition often violated in practice <cite class="ltx_cite ltx_citemacro_citep">(Li et al., <a class="ltx_ref" href="https://arxiv.org/html/2503.13782v1#bib.bib21" title="">2022</a>)</cite>. Furthermore, even when the estimate exists, it does not have the separable structure of the true covariance matrix.</p> </div> <div class="ltx_para" id="S1.SS1.p3"> <p class="ltx_p" id="S1.SS1.p3.15">The literature on separable covariance estimation addresses such problems. Consider a modification of (<a class="ltx_ref" href="https://arxiv.org/html/2503.13782v1#S1.E1" title="In 1 Introduction ‣ Regularized Parameter Estimation in Mixed Model Trace Regression"><span class="ltx_text ltx_ref_tag">1</span></a>) with <math alttext="\operatorname{\mathbf{B}}=\operatorname{\mathbf{0}}" class="ltx_Math" display="inline" id="S1.SS1.p3.1.m1.1"><semantics id="S1.SS1.p3.1.m1.1a"><mrow id="S1.SS1.p3.1.m1.1.1" xref="S1.SS1.p3.1.m1.1.1.cmml"><mi id="S1.SS1.p3.1.m1.1.1.2" xref="S1.SS1.p3.1.m1.1.1.2.cmml">𝐁</mi><mo id="S1.SS1.p3.1.m1.1.1.1" xref="S1.SS1.p3.1.m1.1.1.1.cmml">=</mo><mi id="S1.SS1.p3.1.m1.1.1.3" xref="S1.SS1.p3.1.m1.1.1.3.cmml">𝟎</mi></mrow><annotation-xml encoding="MathML-Content" id="S1.SS1.p3.1.m1.1b"><apply id="S1.SS1.p3.1.m1.1.1.cmml" xref="S1.SS1.p3.1.m1.1.1"><eq id="S1.SS1.p3.1.m1.1.1.1.cmml" xref="S1.SS1.p3.1.m1.1.1.1"></eq><ci id="S1.SS1.p3.1.m1.1.1.2.cmml" xref="S1.SS1.p3.1.m1.1.1.2">𝐁</ci><ci id="S1.SS1.p3.1.m1.1.1.3.cmml" xref="S1.SS1.p3.1.m1.1.1.3">𝟎</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.p3.1.m1.1c">\operatorname{\mathbf{B}}=\operatorname{\mathbf{0}}</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.p3.1.m1.1d">bold_B = bold_0</annotation></semantics></math> and <math alttext="\tau^{2}=1" class="ltx_Math" display="inline" id="S1.SS1.p3.2.m2.1"><semantics id="S1.SS1.p3.2.m2.1a"><mrow id="S1.SS1.p3.2.m2.1.1" xref="S1.SS1.p3.2.m2.1.1.cmml"><msup id="S1.SS1.p3.2.m2.1.1.2" xref="S1.SS1.p3.2.m2.1.1.2.cmml"><mi id="S1.SS1.p3.2.m2.1.1.2.2" xref="S1.SS1.p3.2.m2.1.1.2.2.cmml">τ</mi><mn id="S1.SS1.p3.2.m2.1.1.2.3" xref="S1.SS1.p3.2.m2.1.1.2.3.cmml">2</mn></msup><mo id="S1.SS1.p3.2.m2.1.1.1" xref="S1.SS1.p3.2.m2.1.1.1.cmml">=</mo><mn id="S1.SS1.p3.2.m2.1.1.3" xref="S1.SS1.p3.2.m2.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S1.SS1.p3.2.m2.1b"><apply id="S1.SS1.p3.2.m2.1.1.cmml" xref="S1.SS1.p3.2.m2.1.1"><eq id="S1.SS1.p3.2.m2.1.1.1.cmml" xref="S1.SS1.p3.2.m2.1.1.1"></eq><apply id="S1.SS1.p3.2.m2.1.1.2.cmml" xref="S1.SS1.p3.2.m2.1.1.2"><csymbol cd="ambiguous" id="S1.SS1.p3.2.m2.1.1.2.1.cmml" xref="S1.SS1.p3.2.m2.1.1.2">superscript</csymbol><ci id="S1.SS1.p3.2.m2.1.1.2.2.cmml" xref="S1.SS1.p3.2.m2.1.1.2.2">𝜏</ci><cn id="S1.SS1.p3.2.m2.1.1.2.3.cmml" type="integer" xref="S1.SS1.p3.2.m2.1.1.2.3">2</cn></apply><cn id="S1.SS1.p3.2.m2.1.1.3.cmml" type="integer" xref="S1.SS1.p3.2.m2.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.p3.2.m2.1c">\tau^{2}=1</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.p3.2.m2.1d">italic_τ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT = 1</annotation></semantics></math> that sets <math alttext="\operatorname{\mathbf{Y}}_{i}=\operatorname{\mathbf{A}}_{i}\in\operatorname{% \mathbb{R}}^{Q_{1}\times Q_{2}}" class="ltx_Math" display="inline" id="S1.SS1.p3.3.m3.1"><semantics id="S1.SS1.p3.3.m3.1a"><mrow id="S1.SS1.p3.3.m3.1.1" xref="S1.SS1.p3.3.m3.1.1.cmml"><msub id="S1.SS1.p3.3.m3.1.1.2" xref="S1.SS1.p3.3.m3.1.1.2.cmml"><mi id="S1.SS1.p3.3.m3.1.1.2.2" xref="S1.SS1.p3.3.m3.1.1.2.2.cmml">𝐘</mi><mi id="S1.SS1.p3.3.m3.1.1.2.3" xref="S1.SS1.p3.3.m3.1.1.2.3.cmml">i</mi></msub><mo id="S1.SS1.p3.3.m3.1.1.3" xref="S1.SS1.p3.3.m3.1.1.3.cmml">=</mo><msub id="S1.SS1.p3.3.m3.1.1.4" xref="S1.SS1.p3.3.m3.1.1.4.cmml"><mi id="S1.SS1.p3.3.m3.1.1.4.2" xref="S1.SS1.p3.3.m3.1.1.4.2.cmml">𝐀</mi><mi id="S1.SS1.p3.3.m3.1.1.4.3" xref="S1.SS1.p3.3.m3.1.1.4.3.cmml">i</mi></msub><mo id="S1.SS1.p3.3.m3.1.1.5" xref="S1.SS1.p3.3.m3.1.1.5.cmml">∈</mo><msup id="S1.SS1.p3.3.m3.1.1.6" xref="S1.SS1.p3.3.m3.1.1.6.cmml"><mi id="S1.SS1.p3.3.m3.1.1.6.2" xref="S1.SS1.p3.3.m3.1.1.6.2.cmml">ℝ</mi><mrow id="S1.SS1.p3.3.m3.1.1.6.3" xref="S1.SS1.p3.3.m3.1.1.6.3.cmml"><msub id="S1.SS1.p3.3.m3.1.1.6.3.2" xref="S1.SS1.p3.3.m3.1.1.6.3.2.cmml"><mi id="S1.SS1.p3.3.m3.1.1.6.3.2.2" xref="S1.SS1.p3.3.m3.1.1.6.3.2.2.cmml">Q</mi><mn id="S1.SS1.p3.3.m3.1.1.6.3.2.3" xref="S1.SS1.p3.3.m3.1.1.6.3.2.3.cmml">1</mn></msub><mo id="S1.SS1.p3.3.m3.1.1.6.3.1" lspace="0.222em" rspace="0.222em" xref="S1.SS1.p3.3.m3.1.1.6.3.1.cmml">×</mo><msub id="S1.SS1.p3.3.m3.1.1.6.3.3" xref="S1.SS1.p3.3.m3.1.1.6.3.3.cmml"><mi id="S1.SS1.p3.3.m3.1.1.6.3.3.2" xref="S1.SS1.p3.3.m3.1.1.6.3.3.2.cmml">Q</mi><mn id="S1.SS1.p3.3.m3.1.1.6.3.3.3" xref="S1.SS1.p3.3.m3.1.1.6.3.3.3.cmml">2</mn></msub></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="S1.SS1.p3.3.m3.1b"><apply id="S1.SS1.p3.3.m3.1.1.cmml" xref="S1.SS1.p3.3.m3.1.1"><and id="S1.SS1.p3.3.m3.1.1a.cmml" xref="S1.SS1.p3.3.m3.1.1"></and><apply id="S1.SS1.p3.3.m3.1.1b.cmml" xref="S1.SS1.p3.3.m3.1.1"><eq id="S1.SS1.p3.3.m3.1.1.3.cmml" xref="S1.SS1.p3.3.m3.1.1.3"></eq><apply id="S1.SS1.p3.3.m3.1.1.2.cmml" xref="S1.SS1.p3.3.m3.1.1.2"><csymbol cd="ambiguous" id="S1.SS1.p3.3.m3.1.1.2.1.cmml" xref="S1.SS1.p3.3.m3.1.1.2">subscript</csymbol><ci id="S1.SS1.p3.3.m3.1.1.2.2.cmml" xref="S1.SS1.p3.3.m3.1.1.2.2">𝐘</ci><ci id="S1.SS1.p3.3.m3.1.1.2.3.cmml" xref="S1.SS1.p3.3.m3.1.1.2.3">𝑖</ci></apply><apply id="S1.SS1.p3.3.m3.1.1.4.cmml" xref="S1.SS1.p3.3.m3.1.1.4"><csymbol cd="ambiguous" id="S1.SS1.p3.3.m3.1.1.4.1.cmml" xref="S1.SS1.p3.3.m3.1.1.4">subscript</csymbol><ci id="S1.SS1.p3.3.m3.1.1.4.2.cmml" xref="S1.SS1.p3.3.m3.1.1.4.2">𝐀</ci><ci id="S1.SS1.p3.3.m3.1.1.4.3.cmml" xref="S1.SS1.p3.3.m3.1.1.4.3">𝑖</ci></apply></apply><apply id="S1.SS1.p3.3.m3.1.1c.cmml" xref="S1.SS1.p3.3.m3.1.1"><in id="S1.SS1.p3.3.m3.1.1.5.cmml" xref="S1.SS1.p3.3.m3.1.1.5"></in><share href="https://arxiv.org/html/2503.13782v1#S1.SS1.p3.3.m3.1.1.4.cmml" id="S1.SS1.p3.3.m3.1.1d.cmml" xref="S1.SS1.p3.3.m3.1.1"></share><apply id="S1.SS1.p3.3.m3.1.1.6.cmml" xref="S1.SS1.p3.3.m3.1.1.6"><csymbol cd="ambiguous" id="S1.SS1.p3.3.m3.1.1.6.1.cmml" xref="S1.SS1.p3.3.m3.1.1.6">superscript</csymbol><ci id="S1.SS1.p3.3.m3.1.1.6.2.cmml" xref="S1.SS1.p3.3.m3.1.1.6.2">ℝ</ci><apply id="S1.SS1.p3.3.m3.1.1.6.3.cmml" xref="S1.SS1.p3.3.m3.1.1.6.3"><times id="S1.SS1.p3.3.m3.1.1.6.3.1.cmml" xref="S1.SS1.p3.3.m3.1.1.6.3.1"></times><apply id="S1.SS1.p3.3.m3.1.1.6.3.2.cmml" xref="S1.SS1.p3.3.m3.1.1.6.3.2"><csymbol cd="ambiguous" id="S1.SS1.p3.3.m3.1.1.6.3.2.1.cmml" xref="S1.SS1.p3.3.m3.1.1.6.3.2">subscript</csymbol><ci id="S1.SS1.p3.3.m3.1.1.6.3.2.2.cmml" xref="S1.SS1.p3.3.m3.1.1.6.3.2.2">𝑄</ci><cn id="S1.SS1.p3.3.m3.1.1.6.3.2.3.cmml" type="integer" xref="S1.SS1.p3.3.m3.1.1.6.3.2.3">1</cn></apply><apply id="S1.SS1.p3.3.m3.1.1.6.3.3.cmml" xref="S1.SS1.p3.3.m3.1.1.6.3.3"><csymbol cd="ambiguous" id="S1.SS1.p3.3.m3.1.1.6.3.3.1.cmml" xref="S1.SS1.p3.3.m3.1.1.6.3.3">subscript</csymbol><ci id="S1.SS1.p3.3.m3.1.1.6.3.3.2.cmml" xref="S1.SS1.p3.3.m3.1.1.6.3.3.2">𝑄</ci><cn id="S1.SS1.p3.3.m3.1.1.6.3.3.3.cmml" type="integer" xref="S1.SS1.p3.3.m3.1.1.6.3.3.3">2</cn></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.p3.3.m3.1c">\operatorname{\mathbf{Y}}_{i}=\operatorname{\mathbf{A}}_{i}\in\operatorname{% \mathbb{R}}^{Q_{1}\times Q_{2}}</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.p3.3.m3.1d">bold_Y start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = bold_A start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∈ blackboard_R start_POSTSUPERSCRIPT italic_Q start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT × italic_Q start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT</annotation></semantics></math> for <math alttext="i=1,\ldots,n" class="ltx_Math" display="inline" id="S1.SS1.p3.4.m4.3"><semantics id="S1.SS1.p3.4.m4.3a"><mrow id="S1.SS1.p3.4.m4.3.4" xref="S1.SS1.p3.4.m4.3.4.cmml"><mi id="S1.SS1.p3.4.m4.3.4.2" xref="S1.SS1.p3.4.m4.3.4.2.cmml">i</mi><mo id="S1.SS1.p3.4.m4.3.4.1" xref="S1.SS1.p3.4.m4.3.4.1.cmml">=</mo><mrow id="S1.SS1.p3.4.m4.3.4.3.2" xref="S1.SS1.p3.4.m4.3.4.3.1.cmml"><mn id="S1.SS1.p3.4.m4.1.1" xref="S1.SS1.p3.4.m4.1.1.cmml">1</mn><mo id="S1.SS1.p3.4.m4.3.4.3.2.1" xref="S1.SS1.p3.4.m4.3.4.3.1.cmml">,</mo><mi id="S1.SS1.p3.4.m4.2.2" mathvariant="normal" xref="S1.SS1.p3.4.m4.2.2.cmml">…</mi><mo id="S1.SS1.p3.4.m4.3.4.3.2.2" xref="S1.SS1.p3.4.m4.3.4.3.1.cmml">,</mo><mi id="S1.SS1.p3.4.m4.3.3" xref="S1.SS1.p3.4.m4.3.3.cmml">n</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.SS1.p3.4.m4.3b"><apply id="S1.SS1.p3.4.m4.3.4.cmml" xref="S1.SS1.p3.4.m4.3.4"><eq id="S1.SS1.p3.4.m4.3.4.1.cmml" xref="S1.SS1.p3.4.m4.3.4.1"></eq><ci id="S1.SS1.p3.4.m4.3.4.2.cmml" xref="S1.SS1.p3.4.m4.3.4.2">𝑖</ci><list id="S1.SS1.p3.4.m4.3.4.3.1.cmml" xref="S1.SS1.p3.4.m4.3.4.3.2"><cn id="S1.SS1.p3.4.m4.1.1.cmml" type="integer" xref="S1.SS1.p3.4.m4.1.1">1</cn><ci id="S1.SS1.p3.4.m4.2.2.cmml" xref="S1.SS1.p3.4.m4.2.2">…</ci><ci id="S1.SS1.p3.4.m4.3.3.cmml" xref="S1.SS1.p3.4.m4.3.3">𝑛</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.p3.4.m4.3c">i=1,\ldots,n</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.p3.4.m4.3d">italic_i = 1 , … , italic_n</annotation></semantics></math>. In this model, <math alttext="\text{Cov}\{\text{vec}(\operatorname{\mathbf{A}}_{i})\}" class="ltx_Math" display="inline" id="S1.SS1.p3.5.m5.1"><semantics id="S1.SS1.p3.5.m5.1a"><mrow id="S1.SS1.p3.5.m5.1.1" xref="S1.SS1.p3.5.m5.1.1.cmml"><mtext id="S1.SS1.p3.5.m5.1.1.3" xref="S1.SS1.p3.5.m5.1.1.3a.cmml">Cov</mtext><mo id="S1.SS1.p3.5.m5.1.1.2" xref="S1.SS1.p3.5.m5.1.1.2.cmml">⁢</mo><mrow id="S1.SS1.p3.5.m5.1.1.1.1" xref="S1.SS1.p3.5.m5.1.1.1.2.cmml"><mo id="S1.SS1.p3.5.m5.1.1.1.1.2" stretchy="false" xref="S1.SS1.p3.5.m5.1.1.1.2.cmml">{</mo><mrow id="S1.SS1.p3.5.m5.1.1.1.1.1" xref="S1.SS1.p3.5.m5.1.1.1.1.1.cmml"><mtext id="S1.SS1.p3.5.m5.1.1.1.1.1.3" xref="S1.SS1.p3.5.m5.1.1.1.1.1.3a.cmml">vec</mtext><mo id="S1.SS1.p3.5.m5.1.1.1.1.1.2" xref="S1.SS1.p3.5.m5.1.1.1.1.1.2.cmml">⁢</mo><mrow id="S1.SS1.p3.5.m5.1.1.1.1.1.1.1" xref="S1.SS1.p3.5.m5.1.1.1.1.1.1.1.1.cmml"><mo id="S1.SS1.p3.5.m5.1.1.1.1.1.1.1.2" stretchy="false" xref="S1.SS1.p3.5.m5.1.1.1.1.1.1.1.1.cmml">(</mo><msub id="S1.SS1.p3.5.m5.1.1.1.1.1.1.1.1" xref="S1.SS1.p3.5.m5.1.1.1.1.1.1.1.1.cmml"><mi id="S1.SS1.p3.5.m5.1.1.1.1.1.1.1.1.2" xref="S1.SS1.p3.5.m5.1.1.1.1.1.1.1.1.2.cmml">𝐀</mi><mi id="S1.SS1.p3.5.m5.1.1.1.1.1.1.1.1.3" xref="S1.SS1.p3.5.m5.1.1.1.1.1.1.1.1.3.cmml">i</mi></msub><mo id="S1.SS1.p3.5.m5.1.1.1.1.1.1.1.3" stretchy="false" xref="S1.SS1.p3.5.m5.1.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S1.SS1.p3.5.m5.1.1.1.1.3" stretchy="false" xref="S1.SS1.p3.5.m5.1.1.1.2.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.SS1.p3.5.m5.1b"><apply id="S1.SS1.p3.5.m5.1.1.cmml" xref="S1.SS1.p3.5.m5.1.1"><times id="S1.SS1.p3.5.m5.1.1.2.cmml" xref="S1.SS1.p3.5.m5.1.1.2"></times><ci id="S1.SS1.p3.5.m5.1.1.3a.cmml" xref="S1.SS1.p3.5.m5.1.1.3"><mtext id="S1.SS1.p3.5.m5.1.1.3.cmml" xref="S1.SS1.p3.5.m5.1.1.3">Cov</mtext></ci><set id="S1.SS1.p3.5.m5.1.1.1.2.cmml" xref="S1.SS1.p3.5.m5.1.1.1.1"><apply id="S1.SS1.p3.5.m5.1.1.1.1.1.cmml" xref="S1.SS1.p3.5.m5.1.1.1.1.1"><times id="S1.SS1.p3.5.m5.1.1.1.1.1.2.cmml" xref="S1.SS1.p3.5.m5.1.1.1.1.1.2"></times><ci id="S1.SS1.p3.5.m5.1.1.1.1.1.3a.cmml" xref="S1.SS1.p3.5.m5.1.1.1.1.1.3"><mtext id="S1.SS1.p3.5.m5.1.1.1.1.1.3.cmml" xref="S1.SS1.p3.5.m5.1.1.1.1.1.3">vec</mtext></ci><apply id="S1.SS1.p3.5.m5.1.1.1.1.1.1.1.1.cmml" xref="S1.SS1.p3.5.m5.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S1.SS1.p3.5.m5.1.1.1.1.1.1.1.1.1.cmml" xref="S1.SS1.p3.5.m5.1.1.1.1.1.1.1">subscript</csymbol><ci id="S1.SS1.p3.5.m5.1.1.1.1.1.1.1.1.2.cmml" xref="S1.SS1.p3.5.m5.1.1.1.1.1.1.1.1.2">𝐀</ci><ci id="S1.SS1.p3.5.m5.1.1.1.1.1.1.1.1.3.cmml" xref="S1.SS1.p3.5.m5.1.1.1.1.1.1.1.1.3">𝑖</ci></apply></apply></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.p3.5.m5.1c">\text{Cov}\{\text{vec}(\operatorname{\mathbf{A}}_{i})\}</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.p3.5.m5.1d">Cov { vec ( bold_A start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) }</annotation></semantics></math> has the separable form <math alttext="\operatorname{{\mathbf{\Sigma}}}_{2}\otimes\operatorname{{\mathbf{\Sigma}}}_{1}" class="ltx_Math" display="inline" id="S1.SS1.p3.6.m6.1"><semantics id="S1.SS1.p3.6.m6.1a"><mrow id="S1.SS1.p3.6.m6.1.1" xref="S1.SS1.p3.6.m6.1.1.cmml"><msub id="S1.SS1.p3.6.m6.1.1.2" xref="S1.SS1.p3.6.m6.1.1.2.cmml"><mi id="S1.SS1.p3.6.m6.1.1.2.2" xref="S1.SS1.p3.6.m6.1.1.2.2.cmml">𝚺</mi><mn id="S1.SS1.p3.6.m6.1.1.2.3" xref="S1.SS1.p3.6.m6.1.1.2.3.cmml">2</mn></msub><mo id="S1.SS1.p3.6.m6.1.1.1" lspace="0em" rspace="0.222em" xref="S1.SS1.p3.6.m6.1.1.1.cmml">⊗</mo><msub id="S1.SS1.p3.6.m6.1.1.3" xref="S1.SS1.p3.6.m6.1.1.3.cmml"><mi id="S1.SS1.p3.6.m6.1.1.3.2" xref="S1.SS1.p3.6.m6.1.1.3.2.cmml">𝚺</mi><mn id="S1.SS1.p3.6.m6.1.1.3.3" xref="S1.SS1.p3.6.m6.1.1.3.3.cmml">1</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S1.SS1.p3.6.m6.1b"><apply id="S1.SS1.p3.6.m6.1.1.cmml" xref="S1.SS1.p3.6.m6.1.1"><csymbol cd="latexml" id="S1.SS1.p3.6.m6.1.1.1.cmml" xref="S1.SS1.p3.6.m6.1.1.1">tensor-product</csymbol><apply id="S1.SS1.p3.6.m6.1.1.2.cmml" xref="S1.SS1.p3.6.m6.1.1.2"><csymbol cd="ambiguous" id="S1.SS1.p3.6.m6.1.1.2.1.cmml" xref="S1.SS1.p3.6.m6.1.1.2">subscript</csymbol><ci id="S1.SS1.p3.6.m6.1.1.2.2.cmml" xref="S1.SS1.p3.6.m6.1.1.2.2">𝚺</ci><cn id="S1.SS1.p3.6.m6.1.1.2.3.cmml" type="integer" xref="S1.SS1.p3.6.m6.1.1.2.3">2</cn></apply><apply id="S1.SS1.p3.6.m6.1.1.3.cmml" xref="S1.SS1.p3.6.m6.1.1.3"><csymbol cd="ambiguous" id="S1.SS1.p3.6.m6.1.1.3.1.cmml" xref="S1.SS1.p3.6.m6.1.1.3">subscript</csymbol><ci id="S1.SS1.p3.6.m6.1.1.3.2.cmml" xref="S1.SS1.p3.6.m6.1.1.3.2">𝚺</ci><cn id="S1.SS1.p3.6.m6.1.1.3.3.cmml" type="integer" xref="S1.SS1.p3.6.m6.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.p3.6.m6.1c">\operatorname{{\mathbf{\Sigma}}}_{2}\otimes\operatorname{{\mathbf{\Sigma}}}_{1}</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.p3.6.m6.1d">bold_Σ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ⊗ bold_Σ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math>, where <math alttext="\operatorname{{\mathbf{\Sigma}}}_{1}" class="ltx_Math" display="inline" id="S1.SS1.p3.7.m7.1"><semantics id="S1.SS1.p3.7.m7.1a"><msub id="S1.SS1.p3.7.m7.1.1" xref="S1.SS1.p3.7.m7.1.1.cmml"><mi id="S1.SS1.p3.7.m7.1.1.2" xref="S1.SS1.p3.7.m7.1.1.2.cmml">𝚺</mi><mn id="S1.SS1.p3.7.m7.1.1.3" xref="S1.SS1.p3.7.m7.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S1.SS1.p3.7.m7.1b"><apply id="S1.SS1.p3.7.m7.1.1.cmml" xref="S1.SS1.p3.7.m7.1.1"><csymbol cd="ambiguous" id="S1.SS1.p3.7.m7.1.1.1.cmml" xref="S1.SS1.p3.7.m7.1.1">subscript</csymbol><ci id="S1.SS1.p3.7.m7.1.1.2.cmml" xref="S1.SS1.p3.7.m7.1.1.2">𝚺</ci><cn id="S1.SS1.p3.7.m7.1.1.3.cmml" type="integer" xref="S1.SS1.p3.7.m7.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.p3.7.m7.1c">\operatorname{{\mathbf{\Sigma}}}_{1}</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.p3.7.m7.1d">bold_Σ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="\operatorname{{\mathbf{\Sigma}}}_{2}" class="ltx_Math" display="inline" id="S1.SS1.p3.8.m8.1"><semantics id="S1.SS1.p3.8.m8.1a"><msub id="S1.SS1.p3.8.m8.1.1" xref="S1.SS1.p3.8.m8.1.1.cmml"><mi id="S1.SS1.p3.8.m8.1.1.2" xref="S1.SS1.p3.8.m8.1.1.2.cmml">𝚺</mi><mn id="S1.SS1.p3.8.m8.1.1.3" xref="S1.SS1.p3.8.m8.1.1.3.cmml">2</mn></msub><annotation-xml encoding="MathML-Content" id="S1.SS1.p3.8.m8.1b"><apply id="S1.SS1.p3.8.m8.1.1.cmml" xref="S1.SS1.p3.8.m8.1.1"><csymbol cd="ambiguous" id="S1.SS1.p3.8.m8.1.1.1.cmml" xref="S1.SS1.p3.8.m8.1.1">subscript</csymbol><ci id="S1.SS1.p3.8.m8.1.1.2.cmml" xref="S1.SS1.p3.8.m8.1.1.2">𝚺</ci><cn id="S1.SS1.p3.8.m8.1.1.3.cmml" type="integer" xref="S1.SS1.p3.8.m8.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.p3.8.m8.1c">\operatorname{{\mathbf{\Sigma}}}_{2}</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.p3.8.m8.1d">bold_Σ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math> are defined in (<a class="ltx_ref" href="https://arxiv.org/html/2503.13782v1#S1.E1" title="In 1 Introduction ‣ Regularized Parameter Estimation in Mixed Model Trace Regression"><span class="ltx_text ltx_ref_tag">1</span></a>). In the low dimensional setting, the two parameters are estimated using a “flip-flop” algorithm that estimates <math alttext="\operatorname{{\mathbf{\Sigma}}}_{2}" class="ltx_Math" display="inline" id="S1.SS1.p3.9.m9.1"><semantics id="S1.SS1.p3.9.m9.1a"><msub id="S1.SS1.p3.9.m9.1.1" xref="S1.SS1.p3.9.m9.1.1.cmml"><mi id="S1.SS1.p3.9.m9.1.1.2" xref="S1.SS1.p3.9.m9.1.1.2.cmml">𝚺</mi><mn id="S1.SS1.p3.9.m9.1.1.3" xref="S1.SS1.p3.9.m9.1.1.3.cmml">2</mn></msub><annotation-xml encoding="MathML-Content" id="S1.SS1.p3.9.m9.1b"><apply id="S1.SS1.p3.9.m9.1.1.cmml" xref="S1.SS1.p3.9.m9.1.1"><csymbol cd="ambiguous" id="S1.SS1.p3.9.m9.1.1.1.cmml" xref="S1.SS1.p3.9.m9.1.1">subscript</csymbol><ci id="S1.SS1.p3.9.m9.1.1.2.cmml" xref="S1.SS1.p3.9.m9.1.1.2">𝚺</ci><cn id="S1.SS1.p3.9.m9.1.1.3.cmml" type="integer" xref="S1.SS1.p3.9.m9.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.p3.9.m9.1c">\operatorname{{\mathbf{\Sigma}}}_{2}</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.p3.9.m9.1d">bold_Σ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math> given <math alttext="\operatorname{{\mathbf{\Sigma}}}_{1}" class="ltx_Math" display="inline" id="S1.SS1.p3.10.m10.1"><semantics id="S1.SS1.p3.10.m10.1a"><msub id="S1.SS1.p3.10.m10.1.1" xref="S1.SS1.p3.10.m10.1.1.cmml"><mi id="S1.SS1.p3.10.m10.1.1.2" xref="S1.SS1.p3.10.m10.1.1.2.cmml">𝚺</mi><mn id="S1.SS1.p3.10.m10.1.1.3" xref="S1.SS1.p3.10.m10.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S1.SS1.p3.10.m10.1b"><apply id="S1.SS1.p3.10.m10.1.1.cmml" xref="S1.SS1.p3.10.m10.1.1"><csymbol cd="ambiguous" id="S1.SS1.p3.10.m10.1.1.1.cmml" xref="S1.SS1.p3.10.m10.1.1">subscript</csymbol><ci id="S1.SS1.p3.10.m10.1.1.2.cmml" xref="S1.SS1.p3.10.m10.1.1.2">𝚺</ci><cn id="S1.SS1.p3.10.m10.1.1.3.cmml" type="integer" xref="S1.SS1.p3.10.m10.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.p3.10.m10.1c">\operatorname{{\mathbf{\Sigma}}}_{1}</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.p3.10.m10.1d">bold_Σ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> and vice versa <cite class="ltx_cite ltx_citemacro_citep">(Dutilleul, <a class="ltx_ref" href="https://arxiv.org/html/2503.13782v1#bib.bib6" title="">1999</a>; Lu and Zimmerman, <a class="ltx_ref" href="https://arxiv.org/html/2503.13782v1#bib.bib23" title="">2005</a>; Srivastava et al., <a class="ltx_ref" href="https://arxiv.org/html/2503.13782v1#bib.bib27" title="">2008</a>)</cite>. <cite class="ltx_cite ltx_citemacro_citet">Hoff (<a class="ltx_ref" href="https://arxiv.org/html/2503.13782v1#bib.bib14" title="">2011</a>)</cite> develops tensor-variate extensions of these algorithms; however, all these approaches assume that <math alttext="Q\ll n" class="ltx_Math" display="inline" id="S1.SS1.p3.11.m11.1"><semantics id="S1.SS1.p3.11.m11.1a"><mrow id="S1.SS1.p3.11.m11.1.1" xref="S1.SS1.p3.11.m11.1.1.cmml"><mi id="S1.SS1.p3.11.m11.1.1.2" xref="S1.SS1.p3.11.m11.1.1.2.cmml">Q</mi><mo id="S1.SS1.p3.11.m11.1.1.1" xref="S1.SS1.p3.11.m11.1.1.1.cmml">≪</mo><mi id="S1.SS1.p3.11.m11.1.1.3" xref="S1.SS1.p3.11.m11.1.1.3.cmml">n</mi></mrow><annotation-xml encoding="MathML-Content" id="S1.SS1.p3.11.m11.1b"><apply id="S1.SS1.p3.11.m11.1.1.cmml" xref="S1.SS1.p3.11.m11.1.1"><csymbol cd="latexml" id="S1.SS1.p3.11.m11.1.1.1.cmml" xref="S1.SS1.p3.11.m11.1.1.1">much-less-than</csymbol><ci id="S1.SS1.p3.11.m11.1.1.2.cmml" xref="S1.SS1.p3.11.m11.1.1.2">𝑄</ci><ci id="S1.SS1.p3.11.m11.1.1.3.cmml" xref="S1.SS1.p3.11.m11.1.1.3">𝑛</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.p3.11.m11.1c">Q\ll n</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.p3.11.m11.1d">italic_Q ≪ italic_n</annotation></semantics></math>. Extending methods from the literature on low-rank covariance estimation, <cite class="ltx_cite ltx_citemacro_citet">Zhang et al. (<a class="ltx_ref" href="https://arxiv.org/html/2503.13782v1#bib.bib37" title="">2023</a>)</cite> estimate <math alttext="\text{Cov}\{\text{vec}(\operatorname{\mathbf{A}}_{i})\}" class="ltx_Math" display="inline" id="S1.SS1.p3.12.m12.1"><semantics id="S1.SS1.p3.12.m12.1a"><mrow id="S1.SS1.p3.12.m12.1.1" xref="S1.SS1.p3.12.m12.1.1.cmml"><mtext id="S1.SS1.p3.12.m12.1.1.3" xref="S1.SS1.p3.12.m12.1.1.3a.cmml">Cov</mtext><mo id="S1.SS1.p3.12.m12.1.1.2" xref="S1.SS1.p3.12.m12.1.1.2.cmml">⁢</mo><mrow id="S1.SS1.p3.12.m12.1.1.1.1" xref="S1.SS1.p3.12.m12.1.1.1.2.cmml"><mo id="S1.SS1.p3.12.m12.1.1.1.1.2" stretchy="false" xref="S1.SS1.p3.12.m12.1.1.1.2.cmml">{</mo><mrow id="S1.SS1.p3.12.m12.1.1.1.1.1" xref="S1.SS1.p3.12.m12.1.1.1.1.1.cmml"><mtext id="S1.SS1.p3.12.m12.1.1.1.1.1.3" xref="S1.SS1.p3.12.m12.1.1.1.1.1.3a.cmml">vec</mtext><mo id="S1.SS1.p3.12.m12.1.1.1.1.1.2" xref="S1.SS1.p3.12.m12.1.1.1.1.1.2.cmml">⁢</mo><mrow id="S1.SS1.p3.12.m12.1.1.1.1.1.1.1" xref="S1.SS1.p3.12.m12.1.1.1.1.1.1.1.1.cmml"><mo id="S1.SS1.p3.12.m12.1.1.1.1.1.1.1.2" stretchy="false" xref="S1.SS1.p3.12.m12.1.1.1.1.1.1.1.1.cmml">(</mo><msub id="S1.SS1.p3.12.m12.1.1.1.1.1.1.1.1" xref="S1.SS1.p3.12.m12.1.1.1.1.1.1.1.1.cmml"><mi id="S1.SS1.p3.12.m12.1.1.1.1.1.1.1.1.2" xref="S1.SS1.p3.12.m12.1.1.1.1.1.1.1.1.2.cmml">𝐀</mi><mi id="S1.SS1.p3.12.m12.1.1.1.1.1.1.1.1.3" xref="S1.SS1.p3.12.m12.1.1.1.1.1.1.1.1.3.cmml">i</mi></msub><mo id="S1.SS1.p3.12.m12.1.1.1.1.1.1.1.3" stretchy="false" xref="S1.SS1.p3.12.m12.1.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S1.SS1.p3.12.m12.1.1.1.1.3" stretchy="false" xref="S1.SS1.p3.12.m12.1.1.1.2.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.SS1.p3.12.m12.1b"><apply id="S1.SS1.p3.12.m12.1.1.cmml" xref="S1.SS1.p3.12.m12.1.1"><times id="S1.SS1.p3.12.m12.1.1.2.cmml" xref="S1.SS1.p3.12.m12.1.1.2"></times><ci id="S1.SS1.p3.12.m12.1.1.3a.cmml" xref="S1.SS1.p3.12.m12.1.1.3"><mtext id="S1.SS1.p3.12.m12.1.1.3.cmml" xref="S1.SS1.p3.12.m12.1.1.3">Cov</mtext></ci><set id="S1.SS1.p3.12.m12.1.1.1.2.cmml" xref="S1.SS1.p3.12.m12.1.1.1.1"><apply id="S1.SS1.p3.12.m12.1.1.1.1.1.cmml" xref="S1.SS1.p3.12.m12.1.1.1.1.1"><times id="S1.SS1.p3.12.m12.1.1.1.1.1.2.cmml" xref="S1.SS1.p3.12.m12.1.1.1.1.1.2"></times><ci id="S1.SS1.p3.12.m12.1.1.1.1.1.3a.cmml" xref="S1.SS1.p3.12.m12.1.1.1.1.1.3"><mtext id="S1.SS1.p3.12.m12.1.1.1.1.1.3.cmml" xref="S1.SS1.p3.12.m12.1.1.1.1.1.3">vec</mtext></ci><apply id="S1.SS1.p3.12.m12.1.1.1.1.1.1.1.1.cmml" xref="S1.SS1.p3.12.m12.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S1.SS1.p3.12.m12.1.1.1.1.1.1.1.1.1.cmml" xref="S1.SS1.p3.12.m12.1.1.1.1.1.1.1">subscript</csymbol><ci id="S1.SS1.p3.12.m12.1.1.1.1.1.1.1.1.2.cmml" xref="S1.SS1.p3.12.m12.1.1.1.1.1.1.1.1.2">𝐀</ci><ci id="S1.SS1.p3.12.m12.1.1.1.1.1.1.1.1.3.cmml" xref="S1.SS1.p3.12.m12.1.1.1.1.1.1.1.1.3">𝑖</ci></apply></apply></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.p3.12.m12.1c">\text{Cov}\{\text{vec}(\operatorname{\mathbf{A}}_{i})\}</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.p3.12.m12.1d">Cov { vec ( bold_A start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) }</annotation></semantics></math> via regularized banded estimates of <math alttext="\operatorname{{\mathbf{\Sigma}}}_{1}" class="ltx_Math" display="inline" id="S1.SS1.p3.13.m13.1"><semantics id="S1.SS1.p3.13.m13.1a"><msub id="S1.SS1.p3.13.m13.1.1" xref="S1.SS1.p3.13.m13.1.1.cmml"><mi id="S1.SS1.p3.13.m13.1.1.2" xref="S1.SS1.p3.13.m13.1.1.2.cmml">𝚺</mi><mn id="S1.SS1.p3.13.m13.1.1.3" xref="S1.SS1.p3.13.m13.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S1.SS1.p3.13.m13.1b"><apply id="S1.SS1.p3.13.m13.1.1.cmml" xref="S1.SS1.p3.13.m13.1.1"><csymbol cd="ambiguous" id="S1.SS1.p3.13.m13.1.1.1.cmml" xref="S1.SS1.p3.13.m13.1.1">subscript</csymbol><ci id="S1.SS1.p3.13.m13.1.1.2.cmml" xref="S1.SS1.p3.13.m13.1.1.2">𝚺</ci><cn id="S1.SS1.p3.13.m13.1.1.3.cmml" type="integer" xref="S1.SS1.p3.13.m13.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.p3.13.m13.1c">\operatorname{{\mathbf{\Sigma}}}_{1}</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.p3.13.m13.1d">bold_Σ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="\operatorname{{\mathbf{\Sigma}}}_{2}" class="ltx_Math" display="inline" id="S1.SS1.p3.14.m14.1"><semantics id="S1.SS1.p3.14.m14.1a"><msub id="S1.SS1.p3.14.m14.1.1" xref="S1.SS1.p3.14.m14.1.1.cmml"><mi id="S1.SS1.p3.14.m14.1.1.2" xref="S1.SS1.p3.14.m14.1.1.2.cmml">𝚺</mi><mn id="S1.SS1.p3.14.m14.1.1.3" xref="S1.SS1.p3.14.m14.1.1.3.cmml">2</mn></msub><annotation-xml encoding="MathML-Content" id="S1.SS1.p3.14.m14.1b"><apply id="S1.SS1.p3.14.m14.1.1.cmml" xref="S1.SS1.p3.14.m14.1.1"><csymbol cd="ambiguous" id="S1.SS1.p3.14.m14.1.1.1.cmml" xref="S1.SS1.p3.14.m14.1.1">subscript</csymbol><ci id="S1.SS1.p3.14.m14.1.1.2.cmml" xref="S1.SS1.p3.14.m14.1.1.2">𝚺</ci><cn id="S1.SS1.p3.14.m14.1.1.3.cmml" type="integer" xref="S1.SS1.p3.14.m14.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.p3.14.m14.1c">\operatorname{{\mathbf{\Sigma}}}_{2}</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.p3.14.m14.1d">bold_Σ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math>. Unlike the setup in these models, the random effects <math alttext="\operatorname{\mathbf{A}}_{i}" class="ltx_Math" display="inline" id="S1.SS1.p3.15.m15.1"><semantics id="S1.SS1.p3.15.m15.1a"><msub id="S1.SS1.p3.15.m15.1.1" xref="S1.SS1.p3.15.m15.1.1.cmml"><mi id="S1.SS1.p3.15.m15.1.1.2" xref="S1.SS1.p3.15.m15.1.1.2.cmml">𝐀</mi><mi id="S1.SS1.p3.15.m15.1.1.3" xref="S1.SS1.p3.15.m15.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S1.SS1.p3.15.m15.1b"><apply id="S1.SS1.p3.15.m15.1.1.cmml" xref="S1.SS1.p3.15.m15.1.1"><csymbol cd="ambiguous" id="S1.SS1.p3.15.m15.1.1.1.cmml" xref="S1.SS1.p3.15.m15.1.1">subscript</csymbol><ci id="S1.SS1.p3.15.m15.1.1.2.cmml" xref="S1.SS1.p3.15.m15.1.1.2">𝐀</ci><ci id="S1.SS1.p3.15.m15.1.1.3.cmml" xref="S1.SS1.p3.15.m15.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.p3.15.m15.1c">\operatorname{\mathbf{A}}_{i}</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.p3.15.m15.1d">bold_A start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math>’s are unobserved in (<a class="ltx_ref" href="https://arxiv.org/html/2503.13782v1#S1.E1" title="In 1 Introduction ‣ Regularized Parameter Estimation in Mixed Model Trace Regression"><span class="ltx_text ltx_ref_tag">1</span></a>), making it impossible to apply these algorithms directly for parameter estimation in MMTR.</p> </div> <div class="ltx_para" id="S1.SS1.p4"> <p class="ltx_p" id="S1.SS1.p4.8">Tensor regression directly handles matrix-variate covariates but assumes <math alttext="\operatorname{\mathbf{A}}_{i}=\operatorname{\mathbf{0}}" class="ltx_Math" display="inline" id="S1.SS1.p4.1.m1.1"><semantics id="S1.SS1.p4.1.m1.1a"><mrow id="S1.SS1.p4.1.m1.1.1" xref="S1.SS1.p4.1.m1.1.1.cmml"><msub id="S1.SS1.p4.1.m1.1.1.2" xref="S1.SS1.p4.1.m1.1.1.2.cmml"><mi id="S1.SS1.p4.1.m1.1.1.2.2" xref="S1.SS1.p4.1.m1.1.1.2.2.cmml">𝐀</mi><mi id="S1.SS1.p4.1.m1.1.1.2.3" xref="S1.SS1.p4.1.m1.1.1.2.3.cmml">i</mi></msub><mo id="S1.SS1.p4.1.m1.1.1.1" xref="S1.SS1.p4.1.m1.1.1.1.cmml">=</mo><mi id="S1.SS1.p4.1.m1.1.1.3" xref="S1.SS1.p4.1.m1.1.1.3.cmml">𝟎</mi></mrow><annotation-xml encoding="MathML-Content" id="S1.SS1.p4.1.m1.1b"><apply id="S1.SS1.p4.1.m1.1.1.cmml" xref="S1.SS1.p4.1.m1.1.1"><eq id="S1.SS1.p4.1.m1.1.1.1.cmml" xref="S1.SS1.p4.1.m1.1.1.1"></eq><apply id="S1.SS1.p4.1.m1.1.1.2.cmml" xref="S1.SS1.p4.1.m1.1.1.2"><csymbol cd="ambiguous" id="S1.SS1.p4.1.m1.1.1.2.1.cmml" xref="S1.SS1.p4.1.m1.1.1.2">subscript</csymbol><ci id="S1.SS1.p4.1.m1.1.1.2.2.cmml" xref="S1.SS1.p4.1.m1.1.1.2.2">𝐀</ci><ci id="S1.SS1.p4.1.m1.1.1.2.3.cmml" xref="S1.SS1.p4.1.m1.1.1.2.3">𝑖</ci></apply><ci id="S1.SS1.p4.1.m1.1.1.3.cmml" xref="S1.SS1.p4.1.m1.1.1.3">𝟎</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.p4.1.m1.1c">\operatorname{\mathbf{A}}_{i}=\operatorname{\mathbf{0}}</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.p4.1.m1.1d">bold_A start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = bold_0</annotation></semantics></math> and <math alttext="m_{i}=1" class="ltx_Math" display="inline" id="S1.SS1.p4.2.m2.1"><semantics id="S1.SS1.p4.2.m2.1a"><mrow id="S1.SS1.p4.2.m2.1.1" xref="S1.SS1.p4.2.m2.1.1.cmml"><msub id="S1.SS1.p4.2.m2.1.1.2" xref="S1.SS1.p4.2.m2.1.1.2.cmml"><mi id="S1.SS1.p4.2.m2.1.1.2.2" xref="S1.SS1.p4.2.m2.1.1.2.2.cmml">m</mi><mi id="S1.SS1.p4.2.m2.1.1.2.3" xref="S1.SS1.p4.2.m2.1.1.2.3.cmml">i</mi></msub><mo id="S1.SS1.p4.2.m2.1.1.1" xref="S1.SS1.p4.2.m2.1.1.1.cmml">=</mo><mn id="S1.SS1.p4.2.m2.1.1.3" xref="S1.SS1.p4.2.m2.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S1.SS1.p4.2.m2.1b"><apply id="S1.SS1.p4.2.m2.1.1.cmml" xref="S1.SS1.p4.2.m2.1.1"><eq id="S1.SS1.p4.2.m2.1.1.1.cmml" xref="S1.SS1.p4.2.m2.1.1.1"></eq><apply id="S1.SS1.p4.2.m2.1.1.2.cmml" xref="S1.SS1.p4.2.m2.1.1.2"><csymbol cd="ambiguous" id="S1.SS1.p4.2.m2.1.1.2.1.cmml" xref="S1.SS1.p4.2.m2.1.1.2">subscript</csymbol><ci id="S1.SS1.p4.2.m2.1.1.2.2.cmml" xref="S1.SS1.p4.2.m2.1.1.2.2">𝑚</ci><ci id="S1.SS1.p4.2.m2.1.1.2.3.cmml" xref="S1.SS1.p4.2.m2.1.1.2.3">𝑖</ci></apply><cn id="S1.SS1.p4.2.m2.1.1.3.cmml" type="integer" xref="S1.SS1.p4.2.m2.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.p4.2.m2.1c">m_{i}=1</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.p4.2.m2.1d">italic_m start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = 1</annotation></semantics></math> for every <math alttext="i" class="ltx_Math" display="inline" id="S1.SS1.p4.3.m3.1"><semantics id="S1.SS1.p4.3.m3.1a"><mi id="S1.SS1.p4.3.m3.1.1" xref="S1.SS1.p4.3.m3.1.1.cmml">i</mi><annotation-xml encoding="MathML-Content" id="S1.SS1.p4.3.m3.1b"><ci id="S1.SS1.p4.3.m3.1.1.cmml" xref="S1.SS1.p4.3.m3.1.1">𝑖</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.p4.3.m3.1c">i</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.p4.3.m3.1d">italic_i</annotation></semantics></math> in (<a class="ltx_ref" href="https://arxiv.org/html/2503.13782v1#S1.E1" title="In 1 Introduction ‣ Regularized Parameter Estimation in Mixed Model Trace Regression"><span class="ltx_text ltx_ref_tag">1</span></a>). The main focus is on estimating <math alttext="\operatorname{\mathbf{B}}" class="ltx_Math" display="inline" id="S1.SS1.p4.4.m4.1"><semantics id="S1.SS1.p4.4.m4.1a"><mi id="S1.SS1.p4.4.m4.1.1" xref="S1.SS1.p4.4.m4.1.1.cmml">𝐁</mi><annotation-xml encoding="MathML-Content" id="S1.SS1.p4.4.m4.1b"><ci id="S1.SS1.p4.4.m4.1.1.cmml" xref="S1.SS1.p4.4.m4.1.1">𝐁</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.p4.4.m4.1c">\operatorname{\mathbf{B}}</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.p4.4.m4.1d">bold_B</annotation></semantics></math> under sparsity inducing penalties. If the correlation in <math alttext="\operatorname{\mathbf{y}}_{i}" class="ltx_Math" display="inline" id="S1.SS1.p4.5.m5.1"><semantics id="S1.SS1.p4.5.m5.1a"><msub id="S1.SS1.p4.5.m5.1.1" xref="S1.SS1.p4.5.m5.1.1.cmml"><mi id="S1.SS1.p4.5.m5.1.1.2" xref="S1.SS1.p4.5.m5.1.1.2.cmml">𝐲</mi><mi id="S1.SS1.p4.5.m5.1.1.3" xref="S1.SS1.p4.5.m5.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S1.SS1.p4.5.m5.1b"><apply id="S1.SS1.p4.5.m5.1.1.cmml" xref="S1.SS1.p4.5.m5.1.1"><csymbol cd="ambiguous" id="S1.SS1.p4.5.m5.1.1.1.cmml" xref="S1.SS1.p4.5.m5.1.1">subscript</csymbol><ci id="S1.SS1.p4.5.m5.1.1.2.cmml" xref="S1.SS1.p4.5.m5.1.1.2">𝐲</ci><ci id="S1.SS1.p4.5.m5.1.1.3.cmml" xref="S1.SS1.p4.5.m5.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.p4.5.m5.1c">\operatorname{\mathbf{y}}_{i}</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.p4.5.m5.1d">bold_y start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> induced by <math alttext="\operatorname{\mathbf{A}}_{i}" class="ltx_Math" display="inline" id="S1.SS1.p4.6.m6.1"><semantics id="S1.SS1.p4.6.m6.1a"><msub id="S1.SS1.p4.6.m6.1.1" xref="S1.SS1.p4.6.m6.1.1.cmml"><mi id="S1.SS1.p4.6.m6.1.1.2" xref="S1.SS1.p4.6.m6.1.1.2.cmml">𝐀</mi><mi id="S1.SS1.p4.6.m6.1.1.3" xref="S1.SS1.p4.6.m6.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S1.SS1.p4.6.m6.1b"><apply id="S1.SS1.p4.6.m6.1.1.cmml" xref="S1.SS1.p4.6.m6.1.1"><csymbol cd="ambiguous" id="S1.SS1.p4.6.m6.1.1.1.cmml" xref="S1.SS1.p4.6.m6.1.1">subscript</csymbol><ci id="S1.SS1.p4.6.m6.1.1.2.cmml" xref="S1.SS1.p4.6.m6.1.1.2">𝐀</ci><ci id="S1.SS1.p4.6.m6.1.1.3.cmml" xref="S1.SS1.p4.6.m6.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.p4.6.m6.1c">\operatorname{\mathbf{A}}_{i}</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.p4.6.m6.1d">bold_A start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> in (<a class="ltx_ref" href="https://arxiv.org/html/2503.13782v1#S1.E1" title="In 1 Introduction ‣ Regularized Parameter Estimation in Mixed Model Trace Regression"><span class="ltx_text ltx_ref_tag">1</span></a>) is ignored, then existing trace regression algorithms estimate low-rank, row-sparse, or column-sparse estimates of <math alttext="\operatorname{\mathbf{B}}" class="ltx_Math" display="inline" id="S1.SS1.p4.7.m7.1"><semantics id="S1.SS1.p4.7.m7.1a"><mi id="S1.SS1.p4.7.m7.1.1" xref="S1.SS1.p4.7.m7.1.1.cmml">𝐁</mi><annotation-xml encoding="MathML-Content" id="S1.SS1.p4.7.m7.1b"><ci id="S1.SS1.p4.7.m7.1.1.cmml" xref="S1.SS1.p4.7.m7.1.1">𝐁</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.p4.7.m7.1c">\operatorname{\mathbf{B}}</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.p4.7.m7.1d">bold_B</annotation></semantics></math> in (<a class="ltx_ref" href="https://arxiv.org/html/2503.13782v1#S1.E1" title="In 1 Introduction ‣ Regularized Parameter Estimation in Mixed Model Trace Regression"><span class="ltx_text ltx_ref_tag">1</span></a>) using nuclear norm and group lasso penalties <cite class="ltx_cite ltx_citemacro_citep">(Zhao et al., <a class="ltx_ref" href="https://arxiv.org/html/2503.13782v1#bib.bib38" title="">2017</a>; Slawski et al., <a class="ltx_ref" href="https://arxiv.org/html/2503.13782v1#bib.bib26" title="">2015</a>; Fan et al., <a class="ltx_ref" href="https://arxiv.org/html/2503.13782v1#bib.bib7" title="">2019</a>)</cite>. While such estimates remain under-studied, similar estimates, which assume the independence of <math alttext="\operatorname{\mathbf{y}}_{i}" class="ltx_Math" display="inline" id="S1.SS1.p4.8.m8.1"><semantics id="S1.SS1.p4.8.m8.1a"><msub id="S1.SS1.p4.8.m8.1.1" xref="S1.SS1.p4.8.m8.1.1.cmml"><mi id="S1.SS1.p4.8.m8.1.1.2" xref="S1.SS1.p4.8.m8.1.1.2.cmml">𝐲</mi><mi id="S1.SS1.p4.8.m8.1.1.3" xref="S1.SS1.p4.8.m8.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S1.SS1.p4.8.m8.1b"><apply id="S1.SS1.p4.8.m8.1.1.cmml" xref="S1.SS1.p4.8.m8.1.1"><csymbol cd="ambiguous" id="S1.SS1.p4.8.m8.1.1.1.cmml" xref="S1.SS1.p4.8.m8.1.1">subscript</csymbol><ci id="S1.SS1.p4.8.m8.1.1.2.cmml" xref="S1.SS1.p4.8.m8.1.1.2">𝐲</ci><ci id="S1.SS1.p4.8.m8.1.1.3.cmml" xref="S1.SS1.p4.8.m8.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.p4.8.m8.1c">\operatorname{\mathbf{y}}_{i}</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.p4.8.m8.1d">bold_y start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> entries, in mixed model linear regression have poor inferential and predictive performance <cite class="ltx_cite ltx_citemacro_citep">(Hui et al., <a class="ltx_ref" href="https://arxiv.org/html/2503.13782v1#bib.bib18" title="">2021</a>; Li et al., <a class="ltx_ref" href="https://arxiv.org/html/2503.13782v1#bib.bib21" title="">2022</a>)</cite>. We conjecture that such results extend to the MMTR due to the equivalence between the two model classes.</p> </div> <div class="ltx_para" id="S1.SS1.p5"> <p class="ltx_p" id="S1.SS1.p5.4">MMTR belongs to the class of tensor mixed models, where the responses and covariates are structured as tensors. The simplest models in this class have no random effects and estimate the regression coefficients under low rank and sparse constraints <cite class="ltx_cite ltx_citemacro_citep">(Zhou et al., <a class="ltx_ref" href="https://arxiv.org/html/2503.13782v1#bib.bib42" title="">2013</a>; Zhou and Li, <a class="ltx_ref" href="https://arxiv.org/html/2503.13782v1#bib.bib41" title="">2014</a>)</cite>. There is limited literature on tensor mixed models that jointly estimate mean and covariance parameters. <cite class="ltx_cite ltx_citemacro_citet">Yue et al. (<a class="ltx_ref" href="https://arxiv.org/html/2503.13782v1#bib.bib35" title="">2020</a>)</cite> develop one such model, but it excludes random effects covariates. The random effects tensor has a separable structure in this model and the estimation algorithm is a variant of the flip-flop algorithm; however, their approach is only applicable when <math alttext="n\ll Q" class="ltx_Math" display="inline" id="S1.SS1.p5.1.m1.1"><semantics id="S1.SS1.p5.1.m1.1a"><mrow id="S1.SS1.p5.1.m1.1.1" xref="S1.SS1.p5.1.m1.1.1.cmml"><mi id="S1.SS1.p5.1.m1.1.1.2" xref="S1.SS1.p5.1.m1.1.1.2.cmml">n</mi><mo id="S1.SS1.p5.1.m1.1.1.1" xref="S1.SS1.p5.1.m1.1.1.1.cmml">≪</mo><mi id="S1.SS1.p5.1.m1.1.1.3" xref="S1.SS1.p5.1.m1.1.1.3.cmml">Q</mi></mrow><annotation-xml encoding="MathML-Content" id="S1.SS1.p5.1.m1.1b"><apply id="S1.SS1.p5.1.m1.1.1.cmml" xref="S1.SS1.p5.1.m1.1.1"><csymbol cd="latexml" id="S1.SS1.p5.1.m1.1.1.1.cmml" xref="S1.SS1.p5.1.m1.1.1.1">much-less-than</csymbol><ci id="S1.SS1.p5.1.m1.1.1.2.cmml" xref="S1.SS1.p5.1.m1.1.1.2">𝑛</ci><ci id="S1.SS1.p5.1.m1.1.1.3.cmml" xref="S1.SS1.p5.1.m1.1.1.3">𝑄</ci></apply></annotation-xml><annotation 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italic_i italic_j end_POSTSUBSCRIPT = bold_I start_POSTSUBSCRIPT italic_Q start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math> for every <math alttext="i" class="ltx_Math" display="inline" id="S1.SS1.p5.3.m3.1"><semantics id="S1.SS1.p5.3.m3.1a"><mi id="S1.SS1.p5.3.m3.1.1" xref="S1.SS1.p5.3.m3.1.1.cmml">i</mi><annotation-xml encoding="MathML-Content" id="S1.SS1.p5.3.m3.1b"><ci id="S1.SS1.p5.3.m3.1.1.cmml" xref="S1.SS1.p5.3.m3.1.1">𝑖</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.p5.3.m3.1c">i</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.p5.3.m3.1d">italic_i</annotation></semantics></math> and <math alttext="j" class="ltx_Math" display="inline" id="S1.SS1.p5.4.m4.1"><semantics id="S1.SS1.p5.4.m4.1a"><mi id="S1.SS1.p5.4.m4.1.1" xref="S1.SS1.p5.4.m4.1.1.cmml">j</mi><annotation-xml encoding="MathML-Content" id="S1.SS1.p5.4.m4.1b"><ci id="S1.SS1.p5.4.m4.1.1.cmml" xref="S1.SS1.p5.4.m4.1.1">𝑗</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.p5.4.m4.1c">j</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.p5.4.m4.1d">italic_j</annotation></semantics></math>, implying that this method cannot be used for parameter estimation in MMTR.</p> </div> <div class="ltx_para" id="S1.SS1.p6"> <p class="ltx_p" id="S1.SS1.p6.3">MMTR is closely related to tensor mixed models based on generalized estimating equations (GEE), which are tuned for modeling longitudinal imaging data <cite class="ltx_cite ltx_citemacro_citep">(Zhang et al., <a class="ltx_ref" href="https://arxiv.org/html/2503.13782v1#bib.bib36" title="">2019</a>)</cite>. The GEE-based model has scalar responses and tensor covariates. For two dimensional tensors, the GEE-based models and MMTR have the same mean parametrization; however, the GEE-based model assumes that the regression coefficients have a low tensor rank, whereas MMTR assumes they are sparse. The random effects terms are absent in the GEE-based model, which replaces them with sample-specific marginal “working” covariance matrices. The GEE-based model requires selection of the rank, working covariance matrix form, and penalty tuning parameters for parameter estimation using a minorization maximization (MM) algorithm. In contrast, MMTR’s parameter estimation algorithm only requires two tuning parameters that determine the sparsity of <math alttext="\operatorname{\mathbf{B}}" class="ltx_Math" display="inline" id="S1.SS1.p6.1.m1.1"><semantics id="S1.SS1.p6.1.m1.1a"><mi id="S1.SS1.p6.1.m1.1.1" xref="S1.SS1.p6.1.m1.1.1.cmml">𝐁</mi><annotation-xml encoding="MathML-Content" id="S1.SS1.p6.1.m1.1b"><ci id="S1.SS1.p6.1.m1.1.1.cmml" xref="S1.SS1.p6.1.m1.1.1">𝐁</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.p6.1.m1.1c">\operatorname{\mathbf{B}}</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.p6.1.m1.1d">bold_B</annotation></semantics></math> and ranks of <math alttext="\operatorname{{\mathbf{\Sigma}}}_{1}" class="ltx_Math" display="inline" id="S1.SS1.p6.2.m2.1"><semantics id="S1.SS1.p6.2.m2.1a"><msub id="S1.SS1.p6.2.m2.1.1" xref="S1.SS1.p6.2.m2.1.1.cmml"><mi id="S1.SS1.p6.2.m2.1.1.2" xref="S1.SS1.p6.2.m2.1.1.2.cmml">𝚺</mi><mn id="S1.SS1.p6.2.m2.1.1.3" xref="S1.SS1.p6.2.m2.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S1.SS1.p6.2.m2.1b"><apply id="S1.SS1.p6.2.m2.1.1.cmml" xref="S1.SS1.p6.2.m2.1.1"><csymbol cd="ambiguous" id="S1.SS1.p6.2.m2.1.1.1.cmml" xref="S1.SS1.p6.2.m2.1.1">subscript</csymbol><ci id="S1.SS1.p6.2.m2.1.1.2.cmml" xref="S1.SS1.p6.2.m2.1.1.2">𝚺</ci><cn id="S1.SS1.p6.2.m2.1.1.3.cmml" type="integer" xref="S1.SS1.p6.2.m2.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.p6.2.m2.1c">\operatorname{{\mathbf{\Sigma}}}_{1}</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.p6.2.m2.1d">bold_Σ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="\operatorname{{\mathbf{\Sigma}}}_{2}" class="ltx_Math" display="inline" id="S1.SS1.p6.3.m3.1"><semantics id="S1.SS1.p6.3.m3.1a"><msub id="S1.SS1.p6.3.m3.1.1" xref="S1.SS1.p6.3.m3.1.1.cmml"><mi id="S1.SS1.p6.3.m3.1.1.2" xref="S1.SS1.p6.3.m3.1.1.2.cmml">𝚺</mi><mn id="S1.SS1.p6.3.m3.1.1.3" xref="S1.SS1.p6.3.m3.1.1.3.cmml">2</mn></msub><annotation-xml encoding="MathML-Content" id="S1.SS1.p6.3.m3.1b"><apply id="S1.SS1.p6.3.m3.1.1.cmml" xref="S1.SS1.p6.3.m3.1.1"><csymbol cd="ambiguous" id="S1.SS1.p6.3.m3.1.1.1.cmml" xref="S1.SS1.p6.3.m3.1.1">subscript</csymbol><ci id="S1.SS1.p6.3.m3.1.1.2.cmml" xref="S1.SS1.p6.3.m3.1.1.2">𝚺</ci><cn id="S1.SS1.p6.3.m3.1.1.3.cmml" type="integer" xref="S1.SS1.p6.3.m3.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1.p6.3.m3.1c">\operatorname{{\mathbf{\Sigma}}}_{2}</annotation><annotation encoding="application/x-llamapun" id="S1.SS1.p6.3.m3.1d">bold_Σ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math>.</p> </div> </section> <section class="ltx_subsection" id="S1.SS2"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">1.2 </span>Our Contributions</h3> <div class="ltx_para" id="S1.SS2.p1"> <p class="ltx_p" id="S1.SS2.p1.7">MMTR employs a regularized EM for estimating the parameters in (<a class="ltx_ref" href="https://arxiv.org/html/2503.13782v1#S1.E1" title="In 1 Introduction ‣ Regularized Parameter Estimation in Mixed Model Trace Regression"><span class="ltx_text ltx_ref_tag">1</span></a>). Let <math alttext="\operatorname{\mathbf{L}}_{1}" class="ltx_Math" display="inline" id="S1.SS2.p1.1.m1.1"><semantics id="S1.SS2.p1.1.m1.1a"><msub id="S1.SS2.p1.1.m1.1.1" xref="S1.SS2.p1.1.m1.1.1.cmml"><mi id="S1.SS2.p1.1.m1.1.1.2" xref="S1.SS2.p1.1.m1.1.1.2.cmml">𝐋</mi><mn id="S1.SS2.p1.1.m1.1.1.3" xref="S1.SS2.p1.1.m1.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S1.SS2.p1.1.m1.1b"><apply id="S1.SS2.p1.1.m1.1.1.cmml" xref="S1.SS2.p1.1.m1.1.1"><csymbol cd="ambiguous" id="S1.SS2.p1.1.m1.1.1.1.cmml" xref="S1.SS2.p1.1.m1.1.1">subscript</csymbol><ci id="S1.SS2.p1.1.m1.1.1.2.cmml" xref="S1.SS2.p1.1.m1.1.1.2">𝐋</ci><cn id="S1.SS2.p1.1.m1.1.1.3.cmml" type="integer" xref="S1.SS2.p1.1.m1.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.p1.1.m1.1c">\operatorname{\mathbf{L}}_{1}</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.p1.1.m1.1d">bold_L start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="\operatorname{\mathbf{L}}_{2}" class="ltx_Math" display="inline" id="S1.SS2.p1.2.m2.1"><semantics id="S1.SS2.p1.2.m2.1a"><msub id="S1.SS2.p1.2.m2.1.1" xref="S1.SS2.p1.2.m2.1.1.cmml"><mi id="S1.SS2.p1.2.m2.1.1.2" xref="S1.SS2.p1.2.m2.1.1.2.cmml">𝐋</mi><mn id="S1.SS2.p1.2.m2.1.1.3" xref="S1.SS2.p1.2.m2.1.1.3.cmml">2</mn></msub><annotation-xml encoding="MathML-Content" id="S1.SS2.p1.2.m2.1b"><apply id="S1.SS2.p1.2.m2.1.1.cmml" xref="S1.SS2.p1.2.m2.1.1"><csymbol cd="ambiguous" id="S1.SS2.p1.2.m2.1.1.1.cmml" xref="S1.SS2.p1.2.m2.1.1">subscript</csymbol><ci id="S1.SS2.p1.2.m2.1.1.2.cmml" xref="S1.SS2.p1.2.m2.1.1.2">𝐋</ci><cn id="S1.SS2.p1.2.m2.1.1.3.cmml" type="integer" xref="S1.SS2.p1.2.m2.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.p1.2.m2.1c">\operatorname{\mathbf{L}}_{2}</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.p1.2.m2.1d">bold_L start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math> be two matrices such that <math alttext="\operatorname{{\mathbf{\Sigma}}}_{1}=\operatorname{\mathbf{L}}_{1}% \operatorname{\mathbf{L}}_{1}^{\top}" class="ltx_Math" display="inline" id="S1.SS2.p1.3.m3.1"><semantics id="S1.SS2.p1.3.m3.1a"><mrow id="S1.SS2.p1.3.m3.1.1" xref="S1.SS2.p1.3.m3.1.1.cmml"><msub id="S1.SS2.p1.3.m3.1.1.2" xref="S1.SS2.p1.3.m3.1.1.2.cmml"><mi id="S1.SS2.p1.3.m3.1.1.2.2" xref="S1.SS2.p1.3.m3.1.1.2.2.cmml">𝚺</mi><mn id="S1.SS2.p1.3.m3.1.1.2.3" xref="S1.SS2.p1.3.m3.1.1.2.3.cmml">1</mn></msub><mo id="S1.SS2.p1.3.m3.1.1.1" xref="S1.SS2.p1.3.m3.1.1.1.cmml">=</mo><mrow id="S1.SS2.p1.3.m3.1.1.3" xref="S1.SS2.p1.3.m3.1.1.3.cmml"><msub id="S1.SS2.p1.3.m3.1.1.3.1" xref="S1.SS2.p1.3.m3.1.1.3.1.cmml"><mi id="S1.SS2.p1.3.m3.1.1.3.1.2" xref="S1.SS2.p1.3.m3.1.1.3.1.2.cmml">𝐋</mi><mn id="S1.SS2.p1.3.m3.1.1.3.1.3" xref="S1.SS2.p1.3.m3.1.1.3.1.3.cmml">1</mn></msub><mo id="S1.SS2.p1.3.m3.1.1.3a" lspace="0.167em" xref="S1.SS2.p1.3.m3.1.1.3.cmml">⁡</mo><msubsup id="S1.SS2.p1.3.m3.1.1.3.2" xref="S1.SS2.p1.3.m3.1.1.3.2.cmml"><mi id="S1.SS2.p1.3.m3.1.1.3.2.2.2" xref="S1.SS2.p1.3.m3.1.1.3.2.2.2.cmml">𝐋</mi><mn id="S1.SS2.p1.3.m3.1.1.3.2.2.3" xref="S1.SS2.p1.3.m3.1.1.3.2.2.3.cmml">1</mn><mo id="S1.SS2.p1.3.m3.1.1.3.2.3" xref="S1.SS2.p1.3.m3.1.1.3.2.3.cmml">⊤</mo></msubsup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.SS2.p1.3.m3.1b"><apply id="S1.SS2.p1.3.m3.1.1.cmml" xref="S1.SS2.p1.3.m3.1.1"><eq id="S1.SS2.p1.3.m3.1.1.1.cmml" xref="S1.SS2.p1.3.m3.1.1.1"></eq><apply id="S1.SS2.p1.3.m3.1.1.2.cmml" xref="S1.SS2.p1.3.m3.1.1.2"><csymbol cd="ambiguous" id="S1.SS2.p1.3.m3.1.1.2.1.cmml" xref="S1.SS2.p1.3.m3.1.1.2">subscript</csymbol><ci id="S1.SS2.p1.3.m3.1.1.2.2.cmml" xref="S1.SS2.p1.3.m3.1.1.2.2">𝚺</ci><cn id="S1.SS2.p1.3.m3.1.1.2.3.cmml" type="integer" xref="S1.SS2.p1.3.m3.1.1.2.3">1</cn></apply><apply id="S1.SS2.p1.3.m3.1.1.3.cmml" xref="S1.SS2.p1.3.m3.1.1.3"><apply id="S1.SS2.p1.3.m3.1.1.3.1.cmml" xref="S1.SS2.p1.3.m3.1.1.3.1"><csymbol cd="ambiguous" id="S1.SS2.p1.3.m3.1.1.3.1.1.cmml" xref="S1.SS2.p1.3.m3.1.1.3.1">subscript</csymbol><ci id="S1.SS2.p1.3.m3.1.1.3.1.2.cmml" xref="S1.SS2.p1.3.m3.1.1.3.1.2">𝐋</ci><cn id="S1.SS2.p1.3.m3.1.1.3.1.3.cmml" type="integer" xref="S1.SS2.p1.3.m3.1.1.3.1.3">1</cn></apply><apply id="S1.SS2.p1.3.m3.1.1.3.2.cmml" xref="S1.SS2.p1.3.m3.1.1.3.2"><csymbol cd="ambiguous" id="S1.SS2.p1.3.m3.1.1.3.2.1.cmml" xref="S1.SS2.p1.3.m3.1.1.3.2">superscript</csymbol><apply id="S1.SS2.p1.3.m3.1.1.3.2.2.cmml" xref="S1.SS2.p1.3.m3.1.1.3.2"><csymbol cd="ambiguous" id="S1.SS2.p1.3.m3.1.1.3.2.2.1.cmml" xref="S1.SS2.p1.3.m3.1.1.3.2">subscript</csymbol><ci id="S1.SS2.p1.3.m3.1.1.3.2.2.2.cmml" xref="S1.SS2.p1.3.m3.1.1.3.2.2.2">𝐋</ci><cn id="S1.SS2.p1.3.m3.1.1.3.2.2.3.cmml" type="integer" xref="S1.SS2.p1.3.m3.1.1.3.2.2.3">1</cn></apply><csymbol cd="latexml" id="S1.SS2.p1.3.m3.1.1.3.2.3.cmml" xref="S1.SS2.p1.3.m3.1.1.3.2.3">top</csymbol></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.p1.3.m3.1c">\operatorname{{\mathbf{\Sigma}}}_{1}=\operatorname{\mathbf{L}}_{1}% \operatorname{\mathbf{L}}_{1}^{\top}</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.p1.3.m3.1d">bold_Σ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = bold_L start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT bold_L start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ⊤ end_POSTSUPERSCRIPT</annotation></semantics></math> and <math alttext="\operatorname{{\mathbf{\Sigma}}}_{2}=\operatorname{\mathbf{L}}_{2}% \operatorname{\mathbf{L}}_{2}^{\top}" class="ltx_Math" display="inline" id="S1.SS2.p1.4.m4.1"><semantics id="S1.SS2.p1.4.m4.1a"><mrow id="S1.SS2.p1.4.m4.1.1" xref="S1.SS2.p1.4.m4.1.1.cmml"><msub id="S1.SS2.p1.4.m4.1.1.2" xref="S1.SS2.p1.4.m4.1.1.2.cmml"><mi id="S1.SS2.p1.4.m4.1.1.2.2" xref="S1.SS2.p1.4.m4.1.1.2.2.cmml">𝚺</mi><mn id="S1.SS2.p1.4.m4.1.1.2.3" xref="S1.SS2.p1.4.m4.1.1.2.3.cmml">2</mn></msub><mo id="S1.SS2.p1.4.m4.1.1.1" xref="S1.SS2.p1.4.m4.1.1.1.cmml">=</mo><mrow id="S1.SS2.p1.4.m4.1.1.3" xref="S1.SS2.p1.4.m4.1.1.3.cmml"><msub id="S1.SS2.p1.4.m4.1.1.3.1" xref="S1.SS2.p1.4.m4.1.1.3.1.cmml"><mi id="S1.SS2.p1.4.m4.1.1.3.1.2" xref="S1.SS2.p1.4.m4.1.1.3.1.2.cmml">𝐋</mi><mn id="S1.SS2.p1.4.m4.1.1.3.1.3" xref="S1.SS2.p1.4.m4.1.1.3.1.3.cmml">2</mn></msub><mo id="S1.SS2.p1.4.m4.1.1.3a" lspace="0.167em" xref="S1.SS2.p1.4.m4.1.1.3.cmml">⁡</mo><msubsup id="S1.SS2.p1.4.m4.1.1.3.2" xref="S1.SS2.p1.4.m4.1.1.3.2.cmml"><mi id="S1.SS2.p1.4.m4.1.1.3.2.2.2" xref="S1.SS2.p1.4.m4.1.1.3.2.2.2.cmml">𝐋</mi><mn id="S1.SS2.p1.4.m4.1.1.3.2.2.3" xref="S1.SS2.p1.4.m4.1.1.3.2.2.3.cmml">2</mn><mo id="S1.SS2.p1.4.m4.1.1.3.2.3" xref="S1.SS2.p1.4.m4.1.1.3.2.3.cmml">⊤</mo></msubsup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.SS2.p1.4.m4.1b"><apply id="S1.SS2.p1.4.m4.1.1.cmml" xref="S1.SS2.p1.4.m4.1.1"><eq id="S1.SS2.p1.4.m4.1.1.1.cmml" xref="S1.SS2.p1.4.m4.1.1.1"></eq><apply id="S1.SS2.p1.4.m4.1.1.2.cmml" xref="S1.SS2.p1.4.m4.1.1.2"><csymbol cd="ambiguous" id="S1.SS2.p1.4.m4.1.1.2.1.cmml" xref="S1.SS2.p1.4.m4.1.1.2">subscript</csymbol><ci id="S1.SS2.p1.4.m4.1.1.2.2.cmml" xref="S1.SS2.p1.4.m4.1.1.2.2">𝚺</ci><cn id="S1.SS2.p1.4.m4.1.1.2.3.cmml" type="integer" xref="S1.SS2.p1.4.m4.1.1.2.3">2</cn></apply><apply id="S1.SS2.p1.4.m4.1.1.3.cmml" xref="S1.SS2.p1.4.m4.1.1.3"><apply id="S1.SS2.p1.4.m4.1.1.3.1.cmml" xref="S1.SS2.p1.4.m4.1.1.3.1"><csymbol cd="ambiguous" id="S1.SS2.p1.4.m4.1.1.3.1.1.cmml" xref="S1.SS2.p1.4.m4.1.1.3.1">subscript</csymbol><ci id="S1.SS2.p1.4.m4.1.1.3.1.2.cmml" xref="S1.SS2.p1.4.m4.1.1.3.1.2">𝐋</ci><cn id="S1.SS2.p1.4.m4.1.1.3.1.3.cmml" type="integer" xref="S1.SS2.p1.4.m4.1.1.3.1.3">2</cn></apply><apply id="S1.SS2.p1.4.m4.1.1.3.2.cmml" xref="S1.SS2.p1.4.m4.1.1.3.2"><csymbol cd="ambiguous" id="S1.SS2.p1.4.m4.1.1.3.2.1.cmml" xref="S1.SS2.p1.4.m4.1.1.3.2">superscript</csymbol><apply id="S1.SS2.p1.4.m4.1.1.3.2.2.cmml" xref="S1.SS2.p1.4.m4.1.1.3.2"><csymbol cd="ambiguous" id="S1.SS2.p1.4.m4.1.1.3.2.2.1.cmml" xref="S1.SS2.p1.4.m4.1.1.3.2">subscript</csymbol><ci id="S1.SS2.p1.4.m4.1.1.3.2.2.2.cmml" xref="S1.SS2.p1.4.m4.1.1.3.2.2.2">𝐋</ci><cn id="S1.SS2.p1.4.m4.1.1.3.2.2.3.cmml" type="integer" xref="S1.SS2.p1.4.m4.1.1.3.2.2.3">2</cn></apply><csymbol cd="latexml" id="S1.SS2.p1.4.m4.1.1.3.2.3.cmml" xref="S1.SS2.p1.4.m4.1.1.3.2.3">top</csymbol></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.p1.4.m4.1c">\operatorname{{\mathbf{\Sigma}}}_{2}=\operatorname{\mathbf{L}}_{2}% \operatorname{\mathbf{L}}_{2}^{\top}</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.p1.4.m4.1d">bold_Σ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT = bold_L start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT bold_L start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ⊤ end_POSTSUPERSCRIPT</annotation></semantics></math>. Then, the parameters in (<a class="ltx_ref" href="https://arxiv.org/html/2503.13782v1#S1.E1" title="In 1 Introduction ‣ Regularized Parameter Estimation in Mixed Model Trace Regression"><span class="ltx_text ltx_ref_tag">1</span></a>) are <math alttext="\operatorname{{\bm{\theta}}}=\{\operatorname{\mathbf{B}},\operatorname{\mathbf% {L}}_{1},\operatorname{\mathbf{L}}_{2},\tau^{2}\}" class="ltx_Math" display="inline" id="S1.SS2.p1.5.m5.4"><semantics id="S1.SS2.p1.5.m5.4a"><mrow id="S1.SS2.p1.5.m5.4.4" xref="S1.SS2.p1.5.m5.4.4.cmml"><mi id="S1.SS2.p1.5.m5.4.4.5" xref="S1.SS2.p1.5.m5.4.4.5.cmml">𝜽</mi><mo id="S1.SS2.p1.5.m5.4.4.4" xref="S1.SS2.p1.5.m5.4.4.4.cmml">=</mo><mrow id="S1.SS2.p1.5.m5.4.4.3.3" xref="S1.SS2.p1.5.m5.4.4.3.4.cmml"><mo id="S1.SS2.p1.5.m5.4.4.3.3.4" stretchy="false" xref="S1.SS2.p1.5.m5.4.4.3.4.cmml">{</mo><mi id="S1.SS2.p1.5.m5.1.1" xref="S1.SS2.p1.5.m5.1.1.cmml">𝐁</mi><mo id="S1.SS2.p1.5.m5.4.4.3.3.5" xref="S1.SS2.p1.5.m5.4.4.3.4.cmml">,</mo><msub id="S1.SS2.p1.5.m5.2.2.1.1.1" xref="S1.SS2.p1.5.m5.2.2.1.1.1.cmml"><mi id="S1.SS2.p1.5.m5.2.2.1.1.1.2" xref="S1.SS2.p1.5.m5.2.2.1.1.1.2.cmml">𝐋</mi><mn id="S1.SS2.p1.5.m5.2.2.1.1.1.3" xref="S1.SS2.p1.5.m5.2.2.1.1.1.3.cmml">1</mn></msub><mo id="S1.SS2.p1.5.m5.4.4.3.3.6" xref="S1.SS2.p1.5.m5.4.4.3.4.cmml">,</mo><msub id="S1.SS2.p1.5.m5.3.3.2.2.2" xref="S1.SS2.p1.5.m5.3.3.2.2.2.cmml"><mi id="S1.SS2.p1.5.m5.3.3.2.2.2.2" xref="S1.SS2.p1.5.m5.3.3.2.2.2.2.cmml">𝐋</mi><mn id="S1.SS2.p1.5.m5.3.3.2.2.2.3" xref="S1.SS2.p1.5.m5.3.3.2.2.2.3.cmml">2</mn></msub><mo id="S1.SS2.p1.5.m5.4.4.3.3.7" xref="S1.SS2.p1.5.m5.4.4.3.4.cmml">,</mo><msup id="S1.SS2.p1.5.m5.4.4.3.3.3" xref="S1.SS2.p1.5.m5.4.4.3.3.3.cmml"><mi id="S1.SS2.p1.5.m5.4.4.3.3.3.2" xref="S1.SS2.p1.5.m5.4.4.3.3.3.2.cmml">τ</mi><mn id="S1.SS2.p1.5.m5.4.4.3.3.3.3" xref="S1.SS2.p1.5.m5.4.4.3.3.3.3.cmml">2</mn></msup><mo id="S1.SS2.p1.5.m5.4.4.3.3.8" stretchy="false" xref="S1.SS2.p1.5.m5.4.4.3.4.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.SS2.p1.5.m5.4b"><apply id="S1.SS2.p1.5.m5.4.4.cmml" xref="S1.SS2.p1.5.m5.4.4"><eq id="S1.SS2.p1.5.m5.4.4.4.cmml" xref="S1.SS2.p1.5.m5.4.4.4"></eq><ci id="S1.SS2.p1.5.m5.4.4.5.cmml" xref="S1.SS2.p1.5.m5.4.4.5">𝜽</ci><set id="S1.SS2.p1.5.m5.4.4.3.4.cmml" xref="S1.SS2.p1.5.m5.4.4.3.3"><ci id="S1.SS2.p1.5.m5.1.1.cmml" xref="S1.SS2.p1.5.m5.1.1">𝐁</ci><apply id="S1.SS2.p1.5.m5.2.2.1.1.1.cmml" xref="S1.SS2.p1.5.m5.2.2.1.1.1"><csymbol cd="ambiguous" id="S1.SS2.p1.5.m5.2.2.1.1.1.1.cmml" xref="S1.SS2.p1.5.m5.2.2.1.1.1">subscript</csymbol><ci id="S1.SS2.p1.5.m5.2.2.1.1.1.2.cmml" xref="S1.SS2.p1.5.m5.2.2.1.1.1.2">𝐋</ci><cn id="S1.SS2.p1.5.m5.2.2.1.1.1.3.cmml" type="integer" xref="S1.SS2.p1.5.m5.2.2.1.1.1.3">1</cn></apply><apply id="S1.SS2.p1.5.m5.3.3.2.2.2.cmml" xref="S1.SS2.p1.5.m5.3.3.2.2.2"><csymbol cd="ambiguous" id="S1.SS2.p1.5.m5.3.3.2.2.2.1.cmml" xref="S1.SS2.p1.5.m5.3.3.2.2.2">subscript</csymbol><ci id="S1.SS2.p1.5.m5.3.3.2.2.2.2.cmml" xref="S1.SS2.p1.5.m5.3.3.2.2.2.2">𝐋</ci><cn id="S1.SS2.p1.5.m5.3.3.2.2.2.3.cmml" type="integer" xref="S1.SS2.p1.5.m5.3.3.2.2.2.3">2</cn></apply><apply id="S1.SS2.p1.5.m5.4.4.3.3.3.cmml" xref="S1.SS2.p1.5.m5.4.4.3.3.3"><csymbol cd="ambiguous" id="S1.SS2.p1.5.m5.4.4.3.3.3.1.cmml" xref="S1.SS2.p1.5.m5.4.4.3.3.3">superscript</csymbol><ci id="S1.SS2.p1.5.m5.4.4.3.3.3.2.cmml" xref="S1.SS2.p1.5.m5.4.4.3.3.3.2">𝜏</ci><cn id="S1.SS2.p1.5.m5.4.4.3.3.3.3.cmml" type="integer" xref="S1.SS2.p1.5.m5.4.4.3.3.3.3">2</cn></apply></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.p1.5.m5.4c">\operatorname{{\bm{\theta}}}=\{\operatorname{\mathbf{B}},\operatorname{\mathbf% {L}}_{1},\operatorname{\mathbf{L}}_{2},\tau^{2}\}</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.p1.5.m5.4d">bold_italic_θ = { bold_B , bold_L start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , bold_L start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , italic_τ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT }</annotation></semantics></math>. The E step treats the random effects <math alttext="\operatorname{\mathbf{A}}_{1},\ldots,\operatorname{\mathbf{A}}_{n}" class="ltx_Math" display="inline" id="S1.SS2.p1.6.m6.3"><semantics id="S1.SS2.p1.6.m6.3a"><mrow id="S1.SS2.p1.6.m6.3.3.2" xref="S1.SS2.p1.6.m6.3.3.3.cmml"><msub id="S1.SS2.p1.6.m6.2.2.1.1" xref="S1.SS2.p1.6.m6.2.2.1.1.cmml"><mi id="S1.SS2.p1.6.m6.2.2.1.1.2" xref="S1.SS2.p1.6.m6.2.2.1.1.2.cmml">𝐀</mi><mn id="S1.SS2.p1.6.m6.2.2.1.1.3" xref="S1.SS2.p1.6.m6.2.2.1.1.3.cmml">1</mn></msub><mo id="S1.SS2.p1.6.m6.3.3.2.3" xref="S1.SS2.p1.6.m6.3.3.3.cmml">,</mo><mi id="S1.SS2.p1.6.m6.1.1" mathvariant="normal" xref="S1.SS2.p1.6.m6.1.1.cmml">…</mi><mo id="S1.SS2.p1.6.m6.3.3.2.4" xref="S1.SS2.p1.6.m6.3.3.3.cmml">,</mo><msub id="S1.SS2.p1.6.m6.3.3.2.2" xref="S1.SS2.p1.6.m6.3.3.2.2.cmml"><mi id="S1.SS2.p1.6.m6.3.3.2.2.2" xref="S1.SS2.p1.6.m6.3.3.2.2.2.cmml">𝐀</mi><mi id="S1.SS2.p1.6.m6.3.3.2.2.3" xref="S1.SS2.p1.6.m6.3.3.2.2.3.cmml">n</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S1.SS2.p1.6.m6.3b"><list id="S1.SS2.p1.6.m6.3.3.3.cmml" xref="S1.SS2.p1.6.m6.3.3.2"><apply id="S1.SS2.p1.6.m6.2.2.1.1.cmml" xref="S1.SS2.p1.6.m6.2.2.1.1"><csymbol cd="ambiguous" id="S1.SS2.p1.6.m6.2.2.1.1.1.cmml" xref="S1.SS2.p1.6.m6.2.2.1.1">subscript</csymbol><ci id="S1.SS2.p1.6.m6.2.2.1.1.2.cmml" xref="S1.SS2.p1.6.m6.2.2.1.1.2">𝐀</ci><cn id="S1.SS2.p1.6.m6.2.2.1.1.3.cmml" type="integer" xref="S1.SS2.p1.6.m6.2.2.1.1.3">1</cn></apply><ci id="S1.SS2.p1.6.m6.1.1.cmml" xref="S1.SS2.p1.6.m6.1.1">…</ci><apply id="S1.SS2.p1.6.m6.3.3.2.2.cmml" xref="S1.SS2.p1.6.m6.3.3.2.2"><csymbol cd="ambiguous" id="S1.SS2.p1.6.m6.3.3.2.2.1.cmml" xref="S1.SS2.p1.6.m6.3.3.2.2">subscript</csymbol><ci id="S1.SS2.p1.6.m6.3.3.2.2.2.cmml" xref="S1.SS2.p1.6.m6.3.3.2.2.2">𝐀</ci><ci id="S1.SS2.p1.6.m6.3.3.2.2.3.cmml" xref="S1.SS2.p1.6.m6.3.3.2.2.3">𝑛</ci></apply></list></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.p1.6.m6.3c">\operatorname{\mathbf{A}}_{1},\ldots,\operatorname{\mathbf{A}}_{n}</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.p1.6.m6.3d">bold_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , bold_A start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT</annotation></semantics></math> as missing data and replaces the log-likelihood with a minorizer, which is analytically tractable due to the set up in (<a class="ltx_ref" href="https://arxiv.org/html/2503.13782v1#S1.E1" title="In 1 Introduction ‣ Regularized Parameter Estimation in Mixed Model Trace Regression"><span class="ltx_text ltx_ref_tag">1</span></a>). The M step maximizes the E step minorizer through a series of closed-form conditional maximizations; however, this approach requires <math alttext="P+Q_{1}^{2}+Q_{2}^{2}\ll n" class="ltx_Math" display="inline" id="S1.SS2.p1.7.m7.1"><semantics id="S1.SS2.p1.7.m7.1a"><mrow id="S1.SS2.p1.7.m7.1.1" xref="S1.SS2.p1.7.m7.1.1.cmml"><mrow id="S1.SS2.p1.7.m7.1.1.2" xref="S1.SS2.p1.7.m7.1.1.2.cmml"><mi id="S1.SS2.p1.7.m7.1.1.2.2" xref="S1.SS2.p1.7.m7.1.1.2.2.cmml">P</mi><mo id="S1.SS2.p1.7.m7.1.1.2.1" xref="S1.SS2.p1.7.m7.1.1.2.1.cmml">+</mo><msubsup id="S1.SS2.p1.7.m7.1.1.2.3" xref="S1.SS2.p1.7.m7.1.1.2.3.cmml"><mi id="S1.SS2.p1.7.m7.1.1.2.3.2.2" xref="S1.SS2.p1.7.m7.1.1.2.3.2.2.cmml">Q</mi><mn id="S1.SS2.p1.7.m7.1.1.2.3.2.3" xref="S1.SS2.p1.7.m7.1.1.2.3.2.3.cmml">1</mn><mn id="S1.SS2.p1.7.m7.1.1.2.3.3" xref="S1.SS2.p1.7.m7.1.1.2.3.3.cmml">2</mn></msubsup><mo id="S1.SS2.p1.7.m7.1.1.2.1a" xref="S1.SS2.p1.7.m7.1.1.2.1.cmml">+</mo><msubsup id="S1.SS2.p1.7.m7.1.1.2.4" xref="S1.SS2.p1.7.m7.1.1.2.4.cmml"><mi id="S1.SS2.p1.7.m7.1.1.2.4.2.2" xref="S1.SS2.p1.7.m7.1.1.2.4.2.2.cmml">Q</mi><mn id="S1.SS2.p1.7.m7.1.1.2.4.2.3" xref="S1.SS2.p1.7.m7.1.1.2.4.2.3.cmml">2</mn><mn id="S1.SS2.p1.7.m7.1.1.2.4.3" xref="S1.SS2.p1.7.m7.1.1.2.4.3.cmml">2</mn></msubsup></mrow><mo id="S1.SS2.p1.7.m7.1.1.1" xref="S1.SS2.p1.7.m7.1.1.1.cmml">≪</mo><mi id="S1.SS2.p1.7.m7.1.1.3" xref="S1.SS2.p1.7.m7.1.1.3.cmml">n</mi></mrow><annotation-xml encoding="MathML-Content" id="S1.SS2.p1.7.m7.1b"><apply id="S1.SS2.p1.7.m7.1.1.cmml" xref="S1.SS2.p1.7.m7.1.1"><csymbol cd="latexml" id="S1.SS2.p1.7.m7.1.1.1.cmml" xref="S1.SS2.p1.7.m7.1.1.1">much-less-than</csymbol><apply id="S1.SS2.p1.7.m7.1.1.2.cmml" xref="S1.SS2.p1.7.m7.1.1.2"><plus id="S1.SS2.p1.7.m7.1.1.2.1.cmml" xref="S1.SS2.p1.7.m7.1.1.2.1"></plus><ci id="S1.SS2.p1.7.m7.1.1.2.2.cmml" xref="S1.SS2.p1.7.m7.1.1.2.2">𝑃</ci><apply id="S1.SS2.p1.7.m7.1.1.2.3.cmml" xref="S1.SS2.p1.7.m7.1.1.2.3"><csymbol cd="ambiguous" id="S1.SS2.p1.7.m7.1.1.2.3.1.cmml" xref="S1.SS2.p1.7.m7.1.1.2.3">superscript</csymbol><apply id="S1.SS2.p1.7.m7.1.1.2.3.2.cmml" xref="S1.SS2.p1.7.m7.1.1.2.3"><csymbol cd="ambiguous" id="S1.SS2.p1.7.m7.1.1.2.3.2.1.cmml" xref="S1.SS2.p1.7.m7.1.1.2.3">subscript</csymbol><ci id="S1.SS2.p1.7.m7.1.1.2.3.2.2.cmml" xref="S1.SS2.p1.7.m7.1.1.2.3.2.2">𝑄</ci><cn id="S1.SS2.p1.7.m7.1.1.2.3.2.3.cmml" type="integer" xref="S1.SS2.p1.7.m7.1.1.2.3.2.3">1</cn></apply><cn id="S1.SS2.p1.7.m7.1.1.2.3.3.cmml" type="integer" xref="S1.SS2.p1.7.m7.1.1.2.3.3">2</cn></apply><apply id="S1.SS2.p1.7.m7.1.1.2.4.cmml" xref="S1.SS2.p1.7.m7.1.1.2.4"><csymbol cd="ambiguous" id="S1.SS2.p1.7.m7.1.1.2.4.1.cmml" xref="S1.SS2.p1.7.m7.1.1.2.4">superscript</csymbol><apply id="S1.SS2.p1.7.m7.1.1.2.4.2.cmml" xref="S1.SS2.p1.7.m7.1.1.2.4"><csymbol cd="ambiguous" id="S1.SS2.p1.7.m7.1.1.2.4.2.1.cmml" xref="S1.SS2.p1.7.m7.1.1.2.4">subscript</csymbol><ci id="S1.SS2.p1.7.m7.1.1.2.4.2.2.cmml" xref="S1.SS2.p1.7.m7.1.1.2.4.2.2">𝑄</ci><cn id="S1.SS2.p1.7.m7.1.1.2.4.2.3.cmml" type="integer" xref="S1.SS2.p1.7.m7.1.1.2.4.2.3">2</cn></apply><cn id="S1.SS2.p1.7.m7.1.1.2.4.3.cmml" type="integer" xref="S1.SS2.p1.7.m7.1.1.2.4.3">2</cn></apply></apply><ci id="S1.SS2.p1.7.m7.1.1.3.cmml" xref="S1.SS2.p1.7.m7.1.1.3">𝑛</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.p1.7.m7.1c">P+Q_{1}^{2}+Q_{2}^{2}\ll n</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.p1.7.m7.1d">italic_P + italic_Q start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + italic_Q start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≪ italic_n</annotation></semantics></math> for numerical stability. For vector-valued covariates, this algorithm reduces to the EM algorithm in <cite class="ltx_cite ltx_citemacro_citet">van Dyk (<a class="ltx_ref" href="https://arxiv.org/html/2503.13782v1#bib.bib31" title="">2000</a>)</cite> for parameter estimation in mixed model linear regression.</p> </div> <div class="ltx_para" id="S1.SS2.p2"> <p class="ltx_p" id="S1.SS2.p2.19">In the high-dimensional settings where <math alttext="n\ll P+Q_{1}^{2}+Q_{2}^{2}" class="ltx_Math" display="inline" id="S1.SS2.p2.1.m1.1"><semantics id="S1.SS2.p2.1.m1.1a"><mrow id="S1.SS2.p2.1.m1.1.1" xref="S1.SS2.p2.1.m1.1.1.cmml"><mi id="S1.SS2.p2.1.m1.1.1.2" xref="S1.SS2.p2.1.m1.1.1.2.cmml">n</mi><mo id="S1.SS2.p2.1.m1.1.1.1" xref="S1.SS2.p2.1.m1.1.1.1.cmml">≪</mo><mrow id="S1.SS2.p2.1.m1.1.1.3" xref="S1.SS2.p2.1.m1.1.1.3.cmml"><mi id="S1.SS2.p2.1.m1.1.1.3.2" xref="S1.SS2.p2.1.m1.1.1.3.2.cmml">P</mi><mo id="S1.SS2.p2.1.m1.1.1.3.1" xref="S1.SS2.p2.1.m1.1.1.3.1.cmml">+</mo><msubsup id="S1.SS2.p2.1.m1.1.1.3.3" xref="S1.SS2.p2.1.m1.1.1.3.3.cmml"><mi id="S1.SS2.p2.1.m1.1.1.3.3.2.2" xref="S1.SS2.p2.1.m1.1.1.3.3.2.2.cmml">Q</mi><mn id="S1.SS2.p2.1.m1.1.1.3.3.2.3" xref="S1.SS2.p2.1.m1.1.1.3.3.2.3.cmml">1</mn><mn id="S1.SS2.p2.1.m1.1.1.3.3.3" xref="S1.SS2.p2.1.m1.1.1.3.3.3.cmml">2</mn></msubsup><mo id="S1.SS2.p2.1.m1.1.1.3.1a" xref="S1.SS2.p2.1.m1.1.1.3.1.cmml">+</mo><msubsup id="S1.SS2.p2.1.m1.1.1.3.4" xref="S1.SS2.p2.1.m1.1.1.3.4.cmml"><mi id="S1.SS2.p2.1.m1.1.1.3.4.2.2" xref="S1.SS2.p2.1.m1.1.1.3.4.2.2.cmml">Q</mi><mn id="S1.SS2.p2.1.m1.1.1.3.4.2.3" xref="S1.SS2.p2.1.m1.1.1.3.4.2.3.cmml">2</mn><mn id="S1.SS2.p2.1.m1.1.1.3.4.3" xref="S1.SS2.p2.1.m1.1.1.3.4.3.cmml">2</mn></msubsup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.SS2.p2.1.m1.1b"><apply id="S1.SS2.p2.1.m1.1.1.cmml" xref="S1.SS2.p2.1.m1.1.1"><csymbol cd="latexml" id="S1.SS2.p2.1.m1.1.1.1.cmml" xref="S1.SS2.p2.1.m1.1.1.1">much-less-than</csymbol><ci id="S1.SS2.p2.1.m1.1.1.2.cmml" xref="S1.SS2.p2.1.m1.1.1.2">𝑛</ci><apply id="S1.SS2.p2.1.m1.1.1.3.cmml" xref="S1.SS2.p2.1.m1.1.1.3"><plus id="S1.SS2.p2.1.m1.1.1.3.1.cmml" xref="S1.SS2.p2.1.m1.1.1.3.1"></plus><ci id="S1.SS2.p2.1.m1.1.1.3.2.cmml" xref="S1.SS2.p2.1.m1.1.1.3.2">𝑃</ci><apply id="S1.SS2.p2.1.m1.1.1.3.3.cmml" xref="S1.SS2.p2.1.m1.1.1.3.3"><csymbol cd="ambiguous" id="S1.SS2.p2.1.m1.1.1.3.3.1.cmml" xref="S1.SS2.p2.1.m1.1.1.3.3">superscript</csymbol><apply id="S1.SS2.p2.1.m1.1.1.3.3.2.cmml" xref="S1.SS2.p2.1.m1.1.1.3.3"><csymbol cd="ambiguous" id="S1.SS2.p2.1.m1.1.1.3.3.2.1.cmml" xref="S1.SS2.p2.1.m1.1.1.3.3">subscript</csymbol><ci id="S1.SS2.p2.1.m1.1.1.3.3.2.2.cmml" xref="S1.SS2.p2.1.m1.1.1.3.3.2.2">𝑄</ci><cn id="S1.SS2.p2.1.m1.1.1.3.3.2.3.cmml" type="integer" xref="S1.SS2.p2.1.m1.1.1.3.3.2.3">1</cn></apply><cn id="S1.SS2.p2.1.m1.1.1.3.3.3.cmml" type="integer" xref="S1.SS2.p2.1.m1.1.1.3.3.3">2</cn></apply><apply id="S1.SS2.p2.1.m1.1.1.3.4.cmml" xref="S1.SS2.p2.1.m1.1.1.3.4"><csymbol cd="ambiguous" id="S1.SS2.p2.1.m1.1.1.3.4.1.cmml" xref="S1.SS2.p2.1.m1.1.1.3.4">superscript</csymbol><apply id="S1.SS2.p2.1.m1.1.1.3.4.2.cmml" xref="S1.SS2.p2.1.m1.1.1.3.4"><csymbol cd="ambiguous" id="S1.SS2.p2.1.m1.1.1.3.4.2.1.cmml" xref="S1.SS2.p2.1.m1.1.1.3.4">subscript</csymbol><ci id="S1.SS2.p2.1.m1.1.1.3.4.2.2.cmml" xref="S1.SS2.p2.1.m1.1.1.3.4.2.2">𝑄</ci><cn id="S1.SS2.p2.1.m1.1.1.3.4.2.3.cmml" type="integer" xref="S1.SS2.p2.1.m1.1.1.3.4.2.3">2</cn></apply><cn id="S1.SS2.p2.1.m1.1.1.3.4.3.cmml" type="integer" xref="S1.SS2.p2.1.m1.1.1.3.4.3">2</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.p2.1.m1.1c">n\ll P+Q_{1}^{2}+Q_{2}^{2}</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.p2.1.m1.1d">italic_n ≪ italic_P + italic_Q start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + italic_Q start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT</annotation></semantics></math>, the previous EM algorithm requires regularization. We assume that <math alttext="\operatorname{\mathbf{B}}" class="ltx_Math" display="inline" id="S1.SS2.p2.2.m2.1"><semantics id="S1.SS2.p2.2.m2.1a"><mi id="S1.SS2.p2.2.m2.1.1" xref="S1.SS2.p2.2.m2.1.1.cmml">𝐁</mi><annotation-xml encoding="MathML-Content" id="S1.SS2.p2.2.m2.1b"><ci id="S1.SS2.p2.2.m2.1.1.cmml" xref="S1.SS2.p2.2.m2.1.1">𝐁</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.p2.2.m2.1c">\operatorname{\mathbf{B}}</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.p2.2.m2.1d">bold_B</annotation></semantics></math> is sparse and that <math alttext="\operatorname{{\mathbf{\Sigma}}}_{1}" class="ltx_Math" display="inline" id="S1.SS2.p2.3.m3.1"><semantics id="S1.SS2.p2.3.m3.1a"><msub id="S1.SS2.p2.3.m3.1.1" xref="S1.SS2.p2.3.m3.1.1.cmml"><mi id="S1.SS2.p2.3.m3.1.1.2" xref="S1.SS2.p2.3.m3.1.1.2.cmml">𝚺</mi><mn id="S1.SS2.p2.3.m3.1.1.3" xref="S1.SS2.p2.3.m3.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S1.SS2.p2.3.m3.1b"><apply id="S1.SS2.p2.3.m3.1.1.cmml" xref="S1.SS2.p2.3.m3.1.1"><csymbol cd="ambiguous" id="S1.SS2.p2.3.m3.1.1.1.cmml" xref="S1.SS2.p2.3.m3.1.1">subscript</csymbol><ci id="S1.SS2.p2.3.m3.1.1.2.cmml" xref="S1.SS2.p2.3.m3.1.1.2">𝚺</ci><cn id="S1.SS2.p2.3.m3.1.1.3.cmml" type="integer" xref="S1.SS2.p2.3.m3.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.p2.3.m3.1c">\operatorname{{\mathbf{\Sigma}}}_{1}</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.p2.3.m3.1d">bold_Σ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="\operatorname{{\mathbf{\Sigma}}}_{2}" class="ltx_Math" display="inline" id="S1.SS2.p2.4.m4.1"><semantics id="S1.SS2.p2.4.m4.1a"><msub id="S1.SS2.p2.4.m4.1.1" xref="S1.SS2.p2.4.m4.1.1.cmml"><mi id="S1.SS2.p2.4.m4.1.1.2" xref="S1.SS2.p2.4.m4.1.1.2.cmml">𝚺</mi><mn id="S1.SS2.p2.4.m4.1.1.3" xref="S1.SS2.p2.4.m4.1.1.3.cmml">2</mn></msub><annotation-xml encoding="MathML-Content" id="S1.SS2.p2.4.m4.1b"><apply id="S1.SS2.p2.4.m4.1.1.cmml" xref="S1.SS2.p2.4.m4.1.1"><csymbol cd="ambiguous" id="S1.SS2.p2.4.m4.1.1.1.cmml" xref="S1.SS2.p2.4.m4.1.1">subscript</csymbol><ci id="S1.SS2.p2.4.m4.1.1.2.cmml" xref="S1.SS2.p2.4.m4.1.1.2">𝚺</ci><cn id="S1.SS2.p2.4.m4.1.1.3.cmml" type="integer" xref="S1.SS2.p2.4.m4.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.p2.4.m4.1c">\operatorname{{\mathbf{\Sigma}}}_{2}</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.p2.4.m4.1d">bold_Σ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math> are approximately low rank; that is, <math alttext="\operatorname{{\mathbf{\Sigma}}}_{1}\approx\operatorname{\mathbf{L}}_{1}% \operatorname{\mathbf{L}}_{1}^{\top}" class="ltx_Math" display="inline" id="S1.SS2.p2.5.m5.1"><semantics id="S1.SS2.p2.5.m5.1a"><mrow id="S1.SS2.p2.5.m5.1.1" xref="S1.SS2.p2.5.m5.1.1.cmml"><msub id="S1.SS2.p2.5.m5.1.1.2" xref="S1.SS2.p2.5.m5.1.1.2.cmml"><mi id="S1.SS2.p2.5.m5.1.1.2.2" xref="S1.SS2.p2.5.m5.1.1.2.2.cmml">𝚺</mi><mn id="S1.SS2.p2.5.m5.1.1.2.3" xref="S1.SS2.p2.5.m5.1.1.2.3.cmml">1</mn></msub><mo id="S1.SS2.p2.5.m5.1.1.1" xref="S1.SS2.p2.5.m5.1.1.1.cmml">≈</mo><mrow id="S1.SS2.p2.5.m5.1.1.3" xref="S1.SS2.p2.5.m5.1.1.3.cmml"><msub id="S1.SS2.p2.5.m5.1.1.3.1" xref="S1.SS2.p2.5.m5.1.1.3.1.cmml"><mi id="S1.SS2.p2.5.m5.1.1.3.1.2" xref="S1.SS2.p2.5.m5.1.1.3.1.2.cmml">𝐋</mi><mn id="S1.SS2.p2.5.m5.1.1.3.1.3" xref="S1.SS2.p2.5.m5.1.1.3.1.3.cmml">1</mn></msub><mo id="S1.SS2.p2.5.m5.1.1.3a" lspace="0.167em" xref="S1.SS2.p2.5.m5.1.1.3.cmml">⁡</mo><msubsup id="S1.SS2.p2.5.m5.1.1.3.2" xref="S1.SS2.p2.5.m5.1.1.3.2.cmml"><mi id="S1.SS2.p2.5.m5.1.1.3.2.2.2" xref="S1.SS2.p2.5.m5.1.1.3.2.2.2.cmml">𝐋</mi><mn id="S1.SS2.p2.5.m5.1.1.3.2.2.3" xref="S1.SS2.p2.5.m5.1.1.3.2.2.3.cmml">1</mn><mo id="S1.SS2.p2.5.m5.1.1.3.2.3" xref="S1.SS2.p2.5.m5.1.1.3.2.3.cmml">⊤</mo></msubsup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.SS2.p2.5.m5.1b"><apply id="S1.SS2.p2.5.m5.1.1.cmml" xref="S1.SS2.p2.5.m5.1.1"><approx id="S1.SS2.p2.5.m5.1.1.1.cmml" xref="S1.SS2.p2.5.m5.1.1.1"></approx><apply id="S1.SS2.p2.5.m5.1.1.2.cmml" xref="S1.SS2.p2.5.m5.1.1.2"><csymbol cd="ambiguous" id="S1.SS2.p2.5.m5.1.1.2.1.cmml" xref="S1.SS2.p2.5.m5.1.1.2">subscript</csymbol><ci id="S1.SS2.p2.5.m5.1.1.2.2.cmml" xref="S1.SS2.p2.5.m5.1.1.2.2">𝚺</ci><cn id="S1.SS2.p2.5.m5.1.1.2.3.cmml" type="integer" xref="S1.SS2.p2.5.m5.1.1.2.3">1</cn></apply><apply id="S1.SS2.p2.5.m5.1.1.3.cmml" xref="S1.SS2.p2.5.m5.1.1.3"><apply id="S1.SS2.p2.5.m5.1.1.3.1.cmml" xref="S1.SS2.p2.5.m5.1.1.3.1"><csymbol cd="ambiguous" id="S1.SS2.p2.5.m5.1.1.3.1.1.cmml" xref="S1.SS2.p2.5.m5.1.1.3.1">subscript</csymbol><ci id="S1.SS2.p2.5.m5.1.1.3.1.2.cmml" xref="S1.SS2.p2.5.m5.1.1.3.1.2">𝐋</ci><cn id="S1.SS2.p2.5.m5.1.1.3.1.3.cmml" type="integer" xref="S1.SS2.p2.5.m5.1.1.3.1.3">1</cn></apply><apply id="S1.SS2.p2.5.m5.1.1.3.2.cmml" xref="S1.SS2.p2.5.m5.1.1.3.2"><csymbol cd="ambiguous" id="S1.SS2.p2.5.m5.1.1.3.2.1.cmml" xref="S1.SS2.p2.5.m5.1.1.3.2">superscript</csymbol><apply id="S1.SS2.p2.5.m5.1.1.3.2.2.cmml" xref="S1.SS2.p2.5.m5.1.1.3.2"><csymbol cd="ambiguous" id="S1.SS2.p2.5.m5.1.1.3.2.2.1.cmml" xref="S1.SS2.p2.5.m5.1.1.3.2">subscript</csymbol><ci id="S1.SS2.p2.5.m5.1.1.3.2.2.2.cmml" xref="S1.SS2.p2.5.m5.1.1.3.2.2.2">𝐋</ci><cn id="S1.SS2.p2.5.m5.1.1.3.2.2.3.cmml" type="integer" xref="S1.SS2.p2.5.m5.1.1.3.2.2.3">1</cn></apply><csymbol cd="latexml" id="S1.SS2.p2.5.m5.1.1.3.2.3.cmml" xref="S1.SS2.p2.5.m5.1.1.3.2.3">top</csymbol></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.p2.5.m5.1c">\operatorname{{\mathbf{\Sigma}}}_{1}\approx\operatorname{\mathbf{L}}_{1}% \operatorname{\mathbf{L}}_{1}^{\top}</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.p2.5.m5.1d">bold_Σ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ≈ bold_L start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT bold_L start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ⊤ end_POSTSUPERSCRIPT</annotation></semantics></math> and <math alttext="\operatorname{{\mathbf{\Sigma}}}_{2}\approx\operatorname{\mathbf{L}}_{2}% \operatorname{\mathbf{L}}_{2}^{\top}" class="ltx_Math" display="inline" id="S1.SS2.p2.6.m6.1"><semantics id="S1.SS2.p2.6.m6.1a"><mrow id="S1.SS2.p2.6.m6.1.1" xref="S1.SS2.p2.6.m6.1.1.cmml"><msub id="S1.SS2.p2.6.m6.1.1.2" xref="S1.SS2.p2.6.m6.1.1.2.cmml"><mi id="S1.SS2.p2.6.m6.1.1.2.2" xref="S1.SS2.p2.6.m6.1.1.2.2.cmml">𝚺</mi><mn id="S1.SS2.p2.6.m6.1.1.2.3" xref="S1.SS2.p2.6.m6.1.1.2.3.cmml">2</mn></msub><mo id="S1.SS2.p2.6.m6.1.1.1" xref="S1.SS2.p2.6.m6.1.1.1.cmml">≈</mo><mrow id="S1.SS2.p2.6.m6.1.1.3" xref="S1.SS2.p2.6.m6.1.1.3.cmml"><msub id="S1.SS2.p2.6.m6.1.1.3.1" xref="S1.SS2.p2.6.m6.1.1.3.1.cmml"><mi id="S1.SS2.p2.6.m6.1.1.3.1.2" xref="S1.SS2.p2.6.m6.1.1.3.1.2.cmml">𝐋</mi><mn id="S1.SS2.p2.6.m6.1.1.3.1.3" xref="S1.SS2.p2.6.m6.1.1.3.1.3.cmml">2</mn></msub><mo id="S1.SS2.p2.6.m6.1.1.3a" lspace="0.167em" xref="S1.SS2.p2.6.m6.1.1.3.cmml">⁡</mo><msubsup id="S1.SS2.p2.6.m6.1.1.3.2" xref="S1.SS2.p2.6.m6.1.1.3.2.cmml"><mi id="S1.SS2.p2.6.m6.1.1.3.2.2.2" xref="S1.SS2.p2.6.m6.1.1.3.2.2.2.cmml">𝐋</mi><mn id="S1.SS2.p2.6.m6.1.1.3.2.2.3" xref="S1.SS2.p2.6.m6.1.1.3.2.2.3.cmml">2</mn><mo id="S1.SS2.p2.6.m6.1.1.3.2.3" xref="S1.SS2.p2.6.m6.1.1.3.2.3.cmml">⊤</mo></msubsup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.SS2.p2.6.m6.1b"><apply id="S1.SS2.p2.6.m6.1.1.cmml" xref="S1.SS2.p2.6.m6.1.1"><approx id="S1.SS2.p2.6.m6.1.1.1.cmml" xref="S1.SS2.p2.6.m6.1.1.1"></approx><apply id="S1.SS2.p2.6.m6.1.1.2.cmml" xref="S1.SS2.p2.6.m6.1.1.2"><csymbol cd="ambiguous" id="S1.SS2.p2.6.m6.1.1.2.1.cmml" xref="S1.SS2.p2.6.m6.1.1.2">subscript</csymbol><ci id="S1.SS2.p2.6.m6.1.1.2.2.cmml" xref="S1.SS2.p2.6.m6.1.1.2.2">𝚺</ci><cn id="S1.SS2.p2.6.m6.1.1.2.3.cmml" type="integer" xref="S1.SS2.p2.6.m6.1.1.2.3">2</cn></apply><apply id="S1.SS2.p2.6.m6.1.1.3.cmml" xref="S1.SS2.p2.6.m6.1.1.3"><apply id="S1.SS2.p2.6.m6.1.1.3.1.cmml" xref="S1.SS2.p2.6.m6.1.1.3.1"><csymbol cd="ambiguous" id="S1.SS2.p2.6.m6.1.1.3.1.1.cmml" xref="S1.SS2.p2.6.m6.1.1.3.1">subscript</csymbol><ci id="S1.SS2.p2.6.m6.1.1.3.1.2.cmml" xref="S1.SS2.p2.6.m6.1.1.3.1.2">𝐋</ci><cn id="S1.SS2.p2.6.m6.1.1.3.1.3.cmml" type="integer" xref="S1.SS2.p2.6.m6.1.1.3.1.3">2</cn></apply><apply id="S1.SS2.p2.6.m6.1.1.3.2.cmml" xref="S1.SS2.p2.6.m6.1.1.3.2"><csymbol cd="ambiguous" id="S1.SS2.p2.6.m6.1.1.3.2.1.cmml" xref="S1.SS2.p2.6.m6.1.1.3.2">superscript</csymbol><apply id="S1.SS2.p2.6.m6.1.1.3.2.2.cmml" xref="S1.SS2.p2.6.m6.1.1.3.2"><csymbol cd="ambiguous" id="S1.SS2.p2.6.m6.1.1.3.2.2.1.cmml" xref="S1.SS2.p2.6.m6.1.1.3.2">subscript</csymbol><ci id="S1.SS2.p2.6.m6.1.1.3.2.2.2.cmml" xref="S1.SS2.p2.6.m6.1.1.3.2.2.2">𝐋</ci><cn id="S1.SS2.p2.6.m6.1.1.3.2.2.3.cmml" type="integer" xref="S1.SS2.p2.6.m6.1.1.3.2.2.3">2</cn></apply><csymbol cd="latexml" id="S1.SS2.p2.6.m6.1.1.3.2.3.cmml" xref="S1.SS2.p2.6.m6.1.1.3.2.3">top</csymbol></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.p2.6.m6.1c">\operatorname{{\mathbf{\Sigma}}}_{2}\approx\operatorname{\mathbf{L}}_{2}% \operatorname{\mathbf{L}}_{2}^{\top}</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.p2.6.m6.1d">bold_Σ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ≈ bold_L start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT bold_L start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ⊤ end_POSTSUPERSCRIPT</annotation></semantics></math>, where <math alttext="\operatorname{\mathbf{L}}_{k}\in\operatorname{\mathbb{R}}^{Q_{k}\times S_{k}}" class="ltx_Math" display="inline" id="S1.SS2.p2.7.m7.1"><semantics id="S1.SS2.p2.7.m7.1a"><mrow id="S1.SS2.p2.7.m7.1.1" xref="S1.SS2.p2.7.m7.1.1.cmml"><msub id="S1.SS2.p2.7.m7.1.1.2" xref="S1.SS2.p2.7.m7.1.1.2.cmml"><mi id="S1.SS2.p2.7.m7.1.1.2.2" xref="S1.SS2.p2.7.m7.1.1.2.2.cmml">𝐋</mi><mi id="S1.SS2.p2.7.m7.1.1.2.3" xref="S1.SS2.p2.7.m7.1.1.2.3.cmml">k</mi></msub><mo id="S1.SS2.p2.7.m7.1.1.1" xref="S1.SS2.p2.7.m7.1.1.1.cmml">∈</mo><msup id="S1.SS2.p2.7.m7.1.1.3" xref="S1.SS2.p2.7.m7.1.1.3.cmml"><mi id="S1.SS2.p2.7.m7.1.1.3.2" xref="S1.SS2.p2.7.m7.1.1.3.2.cmml">ℝ</mi><mrow id="S1.SS2.p2.7.m7.1.1.3.3" xref="S1.SS2.p2.7.m7.1.1.3.3.cmml"><msub id="S1.SS2.p2.7.m7.1.1.3.3.2" xref="S1.SS2.p2.7.m7.1.1.3.3.2.cmml"><mi id="S1.SS2.p2.7.m7.1.1.3.3.2.2" xref="S1.SS2.p2.7.m7.1.1.3.3.2.2.cmml">Q</mi><mi id="S1.SS2.p2.7.m7.1.1.3.3.2.3" xref="S1.SS2.p2.7.m7.1.1.3.3.2.3.cmml">k</mi></msub><mo id="S1.SS2.p2.7.m7.1.1.3.3.1" lspace="0.222em" rspace="0.222em" xref="S1.SS2.p2.7.m7.1.1.3.3.1.cmml">×</mo><msub id="S1.SS2.p2.7.m7.1.1.3.3.3" xref="S1.SS2.p2.7.m7.1.1.3.3.3.cmml"><mi id="S1.SS2.p2.7.m7.1.1.3.3.3.2" xref="S1.SS2.p2.7.m7.1.1.3.3.3.2.cmml">S</mi><mi id="S1.SS2.p2.7.m7.1.1.3.3.3.3" xref="S1.SS2.p2.7.m7.1.1.3.3.3.3.cmml">k</mi></msub></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="S1.SS2.p2.7.m7.1b"><apply id="S1.SS2.p2.7.m7.1.1.cmml" xref="S1.SS2.p2.7.m7.1.1"><in id="S1.SS2.p2.7.m7.1.1.1.cmml" xref="S1.SS2.p2.7.m7.1.1.1"></in><apply id="S1.SS2.p2.7.m7.1.1.2.cmml" xref="S1.SS2.p2.7.m7.1.1.2"><csymbol cd="ambiguous" id="S1.SS2.p2.7.m7.1.1.2.1.cmml" xref="S1.SS2.p2.7.m7.1.1.2">subscript</csymbol><ci id="S1.SS2.p2.7.m7.1.1.2.2.cmml" xref="S1.SS2.p2.7.m7.1.1.2.2">𝐋</ci><ci id="S1.SS2.p2.7.m7.1.1.2.3.cmml" xref="S1.SS2.p2.7.m7.1.1.2.3">𝑘</ci></apply><apply id="S1.SS2.p2.7.m7.1.1.3.cmml" xref="S1.SS2.p2.7.m7.1.1.3"><csymbol cd="ambiguous" id="S1.SS2.p2.7.m7.1.1.3.1.cmml" xref="S1.SS2.p2.7.m7.1.1.3">superscript</csymbol><ci id="S1.SS2.p2.7.m7.1.1.3.2.cmml" xref="S1.SS2.p2.7.m7.1.1.3.2">ℝ</ci><apply id="S1.SS2.p2.7.m7.1.1.3.3.cmml" xref="S1.SS2.p2.7.m7.1.1.3.3"><times id="S1.SS2.p2.7.m7.1.1.3.3.1.cmml" xref="S1.SS2.p2.7.m7.1.1.3.3.1"></times><apply id="S1.SS2.p2.7.m7.1.1.3.3.2.cmml" xref="S1.SS2.p2.7.m7.1.1.3.3.2"><csymbol cd="ambiguous" id="S1.SS2.p2.7.m7.1.1.3.3.2.1.cmml" xref="S1.SS2.p2.7.m7.1.1.3.3.2">subscript</csymbol><ci id="S1.SS2.p2.7.m7.1.1.3.3.2.2.cmml" xref="S1.SS2.p2.7.m7.1.1.3.3.2.2">𝑄</ci><ci id="S1.SS2.p2.7.m7.1.1.3.3.2.3.cmml" xref="S1.SS2.p2.7.m7.1.1.3.3.2.3">𝑘</ci></apply><apply id="S1.SS2.p2.7.m7.1.1.3.3.3.cmml" xref="S1.SS2.p2.7.m7.1.1.3.3.3"><csymbol cd="ambiguous" id="S1.SS2.p2.7.m7.1.1.3.3.3.1.cmml" xref="S1.SS2.p2.7.m7.1.1.3.3.3">subscript</csymbol><ci id="S1.SS2.p2.7.m7.1.1.3.3.3.2.cmml" xref="S1.SS2.p2.7.m7.1.1.3.3.3.2">𝑆</ci><ci id="S1.SS2.p2.7.m7.1.1.3.3.3.3.cmml" xref="S1.SS2.p2.7.m7.1.1.3.3.3.3">𝑘</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.p2.7.m7.1c">\operatorname{\mathbf{L}}_{k}\in\operatorname{\mathbb{R}}^{Q_{k}\times S_{k}}</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.p2.7.m7.1d">bold_L start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ∈ blackboard_R start_POSTSUPERSCRIPT italic_Q start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT × italic_S start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUPERSCRIPT</annotation></semantics></math>, and <math alttext="S_{k}\ll Q_{k}" class="ltx_Math" display="inline" id="S1.SS2.p2.8.m8.1"><semantics id="S1.SS2.p2.8.m8.1a"><mrow id="S1.SS2.p2.8.m8.1.1" xref="S1.SS2.p2.8.m8.1.1.cmml"><msub id="S1.SS2.p2.8.m8.1.1.2" xref="S1.SS2.p2.8.m8.1.1.2.cmml"><mi id="S1.SS2.p2.8.m8.1.1.2.2" xref="S1.SS2.p2.8.m8.1.1.2.2.cmml">S</mi><mi id="S1.SS2.p2.8.m8.1.1.2.3" xref="S1.SS2.p2.8.m8.1.1.2.3.cmml">k</mi></msub><mo id="S1.SS2.p2.8.m8.1.1.1" xref="S1.SS2.p2.8.m8.1.1.1.cmml">≪</mo><msub id="S1.SS2.p2.8.m8.1.1.3" xref="S1.SS2.p2.8.m8.1.1.3.cmml"><mi id="S1.SS2.p2.8.m8.1.1.3.2" xref="S1.SS2.p2.8.m8.1.1.3.2.cmml">Q</mi><mi id="S1.SS2.p2.8.m8.1.1.3.3" xref="S1.SS2.p2.8.m8.1.1.3.3.cmml">k</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S1.SS2.p2.8.m8.1b"><apply id="S1.SS2.p2.8.m8.1.1.cmml" xref="S1.SS2.p2.8.m8.1.1"><csymbol cd="latexml" id="S1.SS2.p2.8.m8.1.1.1.cmml" xref="S1.SS2.p2.8.m8.1.1.1">much-less-than</csymbol><apply id="S1.SS2.p2.8.m8.1.1.2.cmml" xref="S1.SS2.p2.8.m8.1.1.2"><csymbol cd="ambiguous" id="S1.SS2.p2.8.m8.1.1.2.1.cmml" xref="S1.SS2.p2.8.m8.1.1.2">subscript</csymbol><ci id="S1.SS2.p2.8.m8.1.1.2.2.cmml" xref="S1.SS2.p2.8.m8.1.1.2.2">𝑆</ci><ci id="S1.SS2.p2.8.m8.1.1.2.3.cmml" xref="S1.SS2.p2.8.m8.1.1.2.3">𝑘</ci></apply><apply id="S1.SS2.p2.8.m8.1.1.3.cmml" xref="S1.SS2.p2.8.m8.1.1.3"><csymbol cd="ambiguous" id="S1.SS2.p2.8.m8.1.1.3.1.cmml" xref="S1.SS2.p2.8.m8.1.1.3">subscript</csymbol><ci id="S1.SS2.p2.8.m8.1.1.3.2.cmml" xref="S1.SS2.p2.8.m8.1.1.3.2">𝑄</ci><ci id="S1.SS2.p2.8.m8.1.1.3.3.cmml" xref="S1.SS2.p2.8.m8.1.1.3.3">𝑘</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.p2.8.m8.1c">S_{k}\ll Q_{k}</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.p2.8.m8.1d">italic_S start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ≪ italic_Q start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT</annotation></semantics></math> for <math alttext="k=1,2" class="ltx_Math" display="inline" id="S1.SS2.p2.9.m9.2"><semantics id="S1.SS2.p2.9.m9.2a"><mrow id="S1.SS2.p2.9.m9.2.3" xref="S1.SS2.p2.9.m9.2.3.cmml"><mi id="S1.SS2.p2.9.m9.2.3.2" xref="S1.SS2.p2.9.m9.2.3.2.cmml">k</mi><mo id="S1.SS2.p2.9.m9.2.3.1" xref="S1.SS2.p2.9.m9.2.3.1.cmml">=</mo><mrow id="S1.SS2.p2.9.m9.2.3.3.2" xref="S1.SS2.p2.9.m9.2.3.3.1.cmml"><mn id="S1.SS2.p2.9.m9.1.1" xref="S1.SS2.p2.9.m9.1.1.cmml">1</mn><mo id="S1.SS2.p2.9.m9.2.3.3.2.1" xref="S1.SS2.p2.9.m9.2.3.3.1.cmml">,</mo><mn id="S1.SS2.p2.9.m9.2.2" xref="S1.SS2.p2.9.m9.2.2.cmml">2</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.SS2.p2.9.m9.2b"><apply id="S1.SS2.p2.9.m9.2.3.cmml" xref="S1.SS2.p2.9.m9.2.3"><eq id="S1.SS2.p2.9.m9.2.3.1.cmml" xref="S1.SS2.p2.9.m9.2.3.1"></eq><ci id="S1.SS2.p2.9.m9.2.3.2.cmml" xref="S1.SS2.p2.9.m9.2.3.2">𝑘</ci><list id="S1.SS2.p2.9.m9.2.3.3.1.cmml" xref="S1.SS2.p2.9.m9.2.3.3.2"><cn id="S1.SS2.p2.9.m9.1.1.cmml" type="integer" xref="S1.SS2.p2.9.m9.1.1">1</cn><cn id="S1.SS2.p2.9.m9.2.2.cmml" type="integer" xref="S1.SS2.p2.9.m9.2.2">2</cn></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.p2.9.m9.2c">k=1,2</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.p2.9.m9.2d">italic_k = 1 , 2</annotation></semantics></math>, reducing the parameter dimension from <math alttext="O(P_{1}P_{2}+Q_{1}^{2}+Q_{2}^{2})" class="ltx_Math" display="inline" id="S1.SS2.p2.10.m10.1"><semantics id="S1.SS2.p2.10.m10.1a"><mrow id="S1.SS2.p2.10.m10.1.1" xref="S1.SS2.p2.10.m10.1.1.cmml"><mi id="S1.SS2.p2.10.m10.1.1.3" xref="S1.SS2.p2.10.m10.1.1.3.cmml">O</mi><mo id="S1.SS2.p2.10.m10.1.1.2" xref="S1.SS2.p2.10.m10.1.1.2.cmml">⁢</mo><mrow id="S1.SS2.p2.10.m10.1.1.1.1" xref="S1.SS2.p2.10.m10.1.1.1.1.1.cmml"><mo id="S1.SS2.p2.10.m10.1.1.1.1.2" stretchy="false" xref="S1.SS2.p2.10.m10.1.1.1.1.1.cmml">(</mo><mrow id="S1.SS2.p2.10.m10.1.1.1.1.1" xref="S1.SS2.p2.10.m10.1.1.1.1.1.cmml"><mrow id="S1.SS2.p2.10.m10.1.1.1.1.1.2" xref="S1.SS2.p2.10.m10.1.1.1.1.1.2.cmml"><msub id="S1.SS2.p2.10.m10.1.1.1.1.1.2.2" xref="S1.SS2.p2.10.m10.1.1.1.1.1.2.2.cmml"><mi id="S1.SS2.p2.10.m10.1.1.1.1.1.2.2.2" xref="S1.SS2.p2.10.m10.1.1.1.1.1.2.2.2.cmml">P</mi><mn id="S1.SS2.p2.10.m10.1.1.1.1.1.2.2.3" xref="S1.SS2.p2.10.m10.1.1.1.1.1.2.2.3.cmml">1</mn></msub><mo id="S1.SS2.p2.10.m10.1.1.1.1.1.2.1" xref="S1.SS2.p2.10.m10.1.1.1.1.1.2.1.cmml">⁢</mo><msub id="S1.SS2.p2.10.m10.1.1.1.1.1.2.3" xref="S1.SS2.p2.10.m10.1.1.1.1.1.2.3.cmml"><mi id="S1.SS2.p2.10.m10.1.1.1.1.1.2.3.2" xref="S1.SS2.p2.10.m10.1.1.1.1.1.2.3.2.cmml">P</mi><mn id="S1.SS2.p2.10.m10.1.1.1.1.1.2.3.3" xref="S1.SS2.p2.10.m10.1.1.1.1.1.2.3.3.cmml">2</mn></msub></mrow><mo id="S1.SS2.p2.10.m10.1.1.1.1.1.1" xref="S1.SS2.p2.10.m10.1.1.1.1.1.1.cmml">+</mo><msubsup id="S1.SS2.p2.10.m10.1.1.1.1.1.3" xref="S1.SS2.p2.10.m10.1.1.1.1.1.3.cmml"><mi id="S1.SS2.p2.10.m10.1.1.1.1.1.3.2.2" xref="S1.SS2.p2.10.m10.1.1.1.1.1.3.2.2.cmml">Q</mi><mn id="S1.SS2.p2.10.m10.1.1.1.1.1.3.2.3" xref="S1.SS2.p2.10.m10.1.1.1.1.1.3.2.3.cmml">1</mn><mn id="S1.SS2.p2.10.m10.1.1.1.1.1.3.3" xref="S1.SS2.p2.10.m10.1.1.1.1.1.3.3.cmml">2</mn></msubsup><mo id="S1.SS2.p2.10.m10.1.1.1.1.1.1a" xref="S1.SS2.p2.10.m10.1.1.1.1.1.1.cmml">+</mo><msubsup id="S1.SS2.p2.10.m10.1.1.1.1.1.4" xref="S1.SS2.p2.10.m10.1.1.1.1.1.4.cmml"><mi id="S1.SS2.p2.10.m10.1.1.1.1.1.4.2.2" xref="S1.SS2.p2.10.m10.1.1.1.1.1.4.2.2.cmml">Q</mi><mn id="S1.SS2.p2.10.m10.1.1.1.1.1.4.2.3" xref="S1.SS2.p2.10.m10.1.1.1.1.1.4.2.3.cmml">2</mn><mn id="S1.SS2.p2.10.m10.1.1.1.1.1.4.3" xref="S1.SS2.p2.10.m10.1.1.1.1.1.4.3.cmml">2</mn></msubsup></mrow><mo id="S1.SS2.p2.10.m10.1.1.1.1.3" stretchy="false" xref="S1.SS2.p2.10.m10.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.SS2.p2.10.m10.1b"><apply id="S1.SS2.p2.10.m10.1.1.cmml" xref="S1.SS2.p2.10.m10.1.1"><times id="S1.SS2.p2.10.m10.1.1.2.cmml" xref="S1.SS2.p2.10.m10.1.1.2"></times><ci id="S1.SS2.p2.10.m10.1.1.3.cmml" xref="S1.SS2.p2.10.m10.1.1.3">𝑂</ci><apply id="S1.SS2.p2.10.m10.1.1.1.1.1.cmml" xref="S1.SS2.p2.10.m10.1.1.1.1"><plus id="S1.SS2.p2.10.m10.1.1.1.1.1.1.cmml" xref="S1.SS2.p2.10.m10.1.1.1.1.1.1"></plus><apply id="S1.SS2.p2.10.m10.1.1.1.1.1.2.cmml" xref="S1.SS2.p2.10.m10.1.1.1.1.1.2"><times id="S1.SS2.p2.10.m10.1.1.1.1.1.2.1.cmml" xref="S1.SS2.p2.10.m10.1.1.1.1.1.2.1"></times><apply id="S1.SS2.p2.10.m10.1.1.1.1.1.2.2.cmml" xref="S1.SS2.p2.10.m10.1.1.1.1.1.2.2"><csymbol cd="ambiguous" id="S1.SS2.p2.10.m10.1.1.1.1.1.2.2.1.cmml" xref="S1.SS2.p2.10.m10.1.1.1.1.1.2.2">subscript</csymbol><ci id="S1.SS2.p2.10.m10.1.1.1.1.1.2.2.2.cmml" xref="S1.SS2.p2.10.m10.1.1.1.1.1.2.2.2">𝑃</ci><cn id="S1.SS2.p2.10.m10.1.1.1.1.1.2.2.3.cmml" type="integer" xref="S1.SS2.p2.10.m10.1.1.1.1.1.2.2.3">1</cn></apply><apply id="S1.SS2.p2.10.m10.1.1.1.1.1.2.3.cmml" xref="S1.SS2.p2.10.m10.1.1.1.1.1.2.3"><csymbol cd="ambiguous" id="S1.SS2.p2.10.m10.1.1.1.1.1.2.3.1.cmml" xref="S1.SS2.p2.10.m10.1.1.1.1.1.2.3">subscript</csymbol><ci id="S1.SS2.p2.10.m10.1.1.1.1.1.2.3.2.cmml" xref="S1.SS2.p2.10.m10.1.1.1.1.1.2.3.2">𝑃</ci><cn id="S1.SS2.p2.10.m10.1.1.1.1.1.2.3.3.cmml" type="integer" xref="S1.SS2.p2.10.m10.1.1.1.1.1.2.3.3">2</cn></apply></apply><apply id="S1.SS2.p2.10.m10.1.1.1.1.1.3.cmml" xref="S1.SS2.p2.10.m10.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S1.SS2.p2.10.m10.1.1.1.1.1.3.1.cmml" xref="S1.SS2.p2.10.m10.1.1.1.1.1.3">superscript</csymbol><apply id="S1.SS2.p2.10.m10.1.1.1.1.1.3.2.cmml" xref="S1.SS2.p2.10.m10.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S1.SS2.p2.10.m10.1.1.1.1.1.3.2.1.cmml" xref="S1.SS2.p2.10.m10.1.1.1.1.1.3">subscript</csymbol><ci id="S1.SS2.p2.10.m10.1.1.1.1.1.3.2.2.cmml" xref="S1.SS2.p2.10.m10.1.1.1.1.1.3.2.2">𝑄</ci><cn id="S1.SS2.p2.10.m10.1.1.1.1.1.3.2.3.cmml" type="integer" xref="S1.SS2.p2.10.m10.1.1.1.1.1.3.2.3">1</cn></apply><cn id="S1.SS2.p2.10.m10.1.1.1.1.1.3.3.cmml" type="integer" xref="S1.SS2.p2.10.m10.1.1.1.1.1.3.3">2</cn></apply><apply id="S1.SS2.p2.10.m10.1.1.1.1.1.4.cmml" xref="S1.SS2.p2.10.m10.1.1.1.1.1.4"><csymbol cd="ambiguous" id="S1.SS2.p2.10.m10.1.1.1.1.1.4.1.cmml" xref="S1.SS2.p2.10.m10.1.1.1.1.1.4">superscript</csymbol><apply id="S1.SS2.p2.10.m10.1.1.1.1.1.4.2.cmml" xref="S1.SS2.p2.10.m10.1.1.1.1.1.4"><csymbol cd="ambiguous" id="S1.SS2.p2.10.m10.1.1.1.1.1.4.2.1.cmml" xref="S1.SS2.p2.10.m10.1.1.1.1.1.4">subscript</csymbol><ci id="S1.SS2.p2.10.m10.1.1.1.1.1.4.2.2.cmml" xref="S1.SS2.p2.10.m10.1.1.1.1.1.4.2.2">𝑄</ci><cn id="S1.SS2.p2.10.m10.1.1.1.1.1.4.2.3.cmml" type="integer" xref="S1.SS2.p2.10.m10.1.1.1.1.1.4.2.3">2</cn></apply><cn id="S1.SS2.p2.10.m10.1.1.1.1.1.4.3.cmml" type="integer" xref="S1.SS2.p2.10.m10.1.1.1.1.1.4.3">2</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.p2.10.m10.1c">O(P_{1}P_{2}+Q_{1}^{2}+Q_{2}^{2})</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.p2.10.m10.1d">italic_O ( italic_P start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_P start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT + italic_Q start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + italic_Q start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT )</annotation></semantics></math> to <math alttext="O(P_{1}P_{2}+Q_{1}S_{1}+Q_{2}S_{2})" class="ltx_Math" display="inline" id="S1.SS2.p2.11.m11.1"><semantics id="S1.SS2.p2.11.m11.1a"><mrow id="S1.SS2.p2.11.m11.1.1" xref="S1.SS2.p2.11.m11.1.1.cmml"><mi id="S1.SS2.p2.11.m11.1.1.3" xref="S1.SS2.p2.11.m11.1.1.3.cmml">O</mi><mo id="S1.SS2.p2.11.m11.1.1.2" xref="S1.SS2.p2.11.m11.1.1.2.cmml">⁢</mo><mrow id="S1.SS2.p2.11.m11.1.1.1.1" xref="S1.SS2.p2.11.m11.1.1.1.1.1.cmml"><mo id="S1.SS2.p2.11.m11.1.1.1.1.2" stretchy="false" xref="S1.SS2.p2.11.m11.1.1.1.1.1.cmml">(</mo><mrow id="S1.SS2.p2.11.m11.1.1.1.1.1" xref="S1.SS2.p2.11.m11.1.1.1.1.1.cmml"><mrow id="S1.SS2.p2.11.m11.1.1.1.1.1.2" xref="S1.SS2.p2.11.m11.1.1.1.1.1.2.cmml"><msub id="S1.SS2.p2.11.m11.1.1.1.1.1.2.2" xref="S1.SS2.p2.11.m11.1.1.1.1.1.2.2.cmml"><mi id="S1.SS2.p2.11.m11.1.1.1.1.1.2.2.2" xref="S1.SS2.p2.11.m11.1.1.1.1.1.2.2.2.cmml">P</mi><mn id="S1.SS2.p2.11.m11.1.1.1.1.1.2.2.3" xref="S1.SS2.p2.11.m11.1.1.1.1.1.2.2.3.cmml">1</mn></msub><mo id="S1.SS2.p2.11.m11.1.1.1.1.1.2.1" xref="S1.SS2.p2.11.m11.1.1.1.1.1.2.1.cmml">⁢</mo><msub id="S1.SS2.p2.11.m11.1.1.1.1.1.2.3" xref="S1.SS2.p2.11.m11.1.1.1.1.1.2.3.cmml"><mi id="S1.SS2.p2.11.m11.1.1.1.1.1.2.3.2" xref="S1.SS2.p2.11.m11.1.1.1.1.1.2.3.2.cmml">P</mi><mn id="S1.SS2.p2.11.m11.1.1.1.1.1.2.3.3" xref="S1.SS2.p2.11.m11.1.1.1.1.1.2.3.3.cmml">2</mn></msub></mrow><mo id="S1.SS2.p2.11.m11.1.1.1.1.1.1" xref="S1.SS2.p2.11.m11.1.1.1.1.1.1.cmml">+</mo><mrow id="S1.SS2.p2.11.m11.1.1.1.1.1.3" xref="S1.SS2.p2.11.m11.1.1.1.1.1.3.cmml"><msub id="S1.SS2.p2.11.m11.1.1.1.1.1.3.2" xref="S1.SS2.p2.11.m11.1.1.1.1.1.3.2.cmml"><mi id="S1.SS2.p2.11.m11.1.1.1.1.1.3.2.2" xref="S1.SS2.p2.11.m11.1.1.1.1.1.3.2.2.cmml">Q</mi><mn id="S1.SS2.p2.11.m11.1.1.1.1.1.3.2.3" xref="S1.SS2.p2.11.m11.1.1.1.1.1.3.2.3.cmml">1</mn></msub><mo id="S1.SS2.p2.11.m11.1.1.1.1.1.3.1" xref="S1.SS2.p2.11.m11.1.1.1.1.1.3.1.cmml">⁢</mo><msub id="S1.SS2.p2.11.m11.1.1.1.1.1.3.3" xref="S1.SS2.p2.11.m11.1.1.1.1.1.3.3.cmml"><mi id="S1.SS2.p2.11.m11.1.1.1.1.1.3.3.2" xref="S1.SS2.p2.11.m11.1.1.1.1.1.3.3.2.cmml">S</mi><mn id="S1.SS2.p2.11.m11.1.1.1.1.1.3.3.3" xref="S1.SS2.p2.11.m11.1.1.1.1.1.3.3.3.cmml">1</mn></msub></mrow><mo id="S1.SS2.p2.11.m11.1.1.1.1.1.1a" xref="S1.SS2.p2.11.m11.1.1.1.1.1.1.cmml">+</mo><mrow id="S1.SS2.p2.11.m11.1.1.1.1.1.4" xref="S1.SS2.p2.11.m11.1.1.1.1.1.4.cmml"><msub id="S1.SS2.p2.11.m11.1.1.1.1.1.4.2" xref="S1.SS2.p2.11.m11.1.1.1.1.1.4.2.cmml"><mi id="S1.SS2.p2.11.m11.1.1.1.1.1.4.2.2" xref="S1.SS2.p2.11.m11.1.1.1.1.1.4.2.2.cmml">Q</mi><mn id="S1.SS2.p2.11.m11.1.1.1.1.1.4.2.3" xref="S1.SS2.p2.11.m11.1.1.1.1.1.4.2.3.cmml">2</mn></msub><mo id="S1.SS2.p2.11.m11.1.1.1.1.1.4.1" xref="S1.SS2.p2.11.m11.1.1.1.1.1.4.1.cmml">⁢</mo><msub id="S1.SS2.p2.11.m11.1.1.1.1.1.4.3" xref="S1.SS2.p2.11.m11.1.1.1.1.1.4.3.cmml"><mi id="S1.SS2.p2.11.m11.1.1.1.1.1.4.3.2" xref="S1.SS2.p2.11.m11.1.1.1.1.1.4.3.2.cmml">S</mi><mn id="S1.SS2.p2.11.m11.1.1.1.1.1.4.3.3" xref="S1.SS2.p2.11.m11.1.1.1.1.1.4.3.3.cmml">2</mn></msub></mrow></mrow><mo id="S1.SS2.p2.11.m11.1.1.1.1.3" stretchy="false" xref="S1.SS2.p2.11.m11.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.SS2.p2.11.m11.1b"><apply id="S1.SS2.p2.11.m11.1.1.cmml" xref="S1.SS2.p2.11.m11.1.1"><times id="S1.SS2.p2.11.m11.1.1.2.cmml" xref="S1.SS2.p2.11.m11.1.1.2"></times><ci id="S1.SS2.p2.11.m11.1.1.3.cmml" xref="S1.SS2.p2.11.m11.1.1.3">𝑂</ci><apply id="S1.SS2.p2.11.m11.1.1.1.1.1.cmml" xref="S1.SS2.p2.11.m11.1.1.1.1"><plus id="S1.SS2.p2.11.m11.1.1.1.1.1.1.cmml" xref="S1.SS2.p2.11.m11.1.1.1.1.1.1"></plus><apply id="S1.SS2.p2.11.m11.1.1.1.1.1.2.cmml" xref="S1.SS2.p2.11.m11.1.1.1.1.1.2"><times id="S1.SS2.p2.11.m11.1.1.1.1.1.2.1.cmml" xref="S1.SS2.p2.11.m11.1.1.1.1.1.2.1"></times><apply id="S1.SS2.p2.11.m11.1.1.1.1.1.2.2.cmml" xref="S1.SS2.p2.11.m11.1.1.1.1.1.2.2"><csymbol cd="ambiguous" id="S1.SS2.p2.11.m11.1.1.1.1.1.2.2.1.cmml" xref="S1.SS2.p2.11.m11.1.1.1.1.1.2.2">subscript</csymbol><ci id="S1.SS2.p2.11.m11.1.1.1.1.1.2.2.2.cmml" xref="S1.SS2.p2.11.m11.1.1.1.1.1.2.2.2">𝑃</ci><cn id="S1.SS2.p2.11.m11.1.1.1.1.1.2.2.3.cmml" type="integer" xref="S1.SS2.p2.11.m11.1.1.1.1.1.2.2.3">1</cn></apply><apply id="S1.SS2.p2.11.m11.1.1.1.1.1.2.3.cmml" xref="S1.SS2.p2.11.m11.1.1.1.1.1.2.3"><csymbol cd="ambiguous" id="S1.SS2.p2.11.m11.1.1.1.1.1.2.3.1.cmml" xref="S1.SS2.p2.11.m11.1.1.1.1.1.2.3">subscript</csymbol><ci id="S1.SS2.p2.11.m11.1.1.1.1.1.2.3.2.cmml" xref="S1.SS2.p2.11.m11.1.1.1.1.1.2.3.2">𝑃</ci><cn id="S1.SS2.p2.11.m11.1.1.1.1.1.2.3.3.cmml" type="integer" xref="S1.SS2.p2.11.m11.1.1.1.1.1.2.3.3">2</cn></apply></apply><apply id="S1.SS2.p2.11.m11.1.1.1.1.1.3.cmml" xref="S1.SS2.p2.11.m11.1.1.1.1.1.3"><times id="S1.SS2.p2.11.m11.1.1.1.1.1.3.1.cmml" xref="S1.SS2.p2.11.m11.1.1.1.1.1.3.1"></times><apply id="S1.SS2.p2.11.m11.1.1.1.1.1.3.2.cmml" xref="S1.SS2.p2.11.m11.1.1.1.1.1.3.2"><csymbol cd="ambiguous" id="S1.SS2.p2.11.m11.1.1.1.1.1.3.2.1.cmml" xref="S1.SS2.p2.11.m11.1.1.1.1.1.3.2">subscript</csymbol><ci id="S1.SS2.p2.11.m11.1.1.1.1.1.3.2.2.cmml" xref="S1.SS2.p2.11.m11.1.1.1.1.1.3.2.2">𝑄</ci><cn id="S1.SS2.p2.11.m11.1.1.1.1.1.3.2.3.cmml" type="integer" xref="S1.SS2.p2.11.m11.1.1.1.1.1.3.2.3">1</cn></apply><apply id="S1.SS2.p2.11.m11.1.1.1.1.1.3.3.cmml" xref="S1.SS2.p2.11.m11.1.1.1.1.1.3.3"><csymbol cd="ambiguous" id="S1.SS2.p2.11.m11.1.1.1.1.1.3.3.1.cmml" xref="S1.SS2.p2.11.m11.1.1.1.1.1.3.3">subscript</csymbol><ci id="S1.SS2.p2.11.m11.1.1.1.1.1.3.3.2.cmml" xref="S1.SS2.p2.11.m11.1.1.1.1.1.3.3.2">𝑆</ci><cn id="S1.SS2.p2.11.m11.1.1.1.1.1.3.3.3.cmml" type="integer" xref="S1.SS2.p2.11.m11.1.1.1.1.1.3.3.3">1</cn></apply></apply><apply id="S1.SS2.p2.11.m11.1.1.1.1.1.4.cmml" xref="S1.SS2.p2.11.m11.1.1.1.1.1.4"><times id="S1.SS2.p2.11.m11.1.1.1.1.1.4.1.cmml" xref="S1.SS2.p2.11.m11.1.1.1.1.1.4.1"></times><apply id="S1.SS2.p2.11.m11.1.1.1.1.1.4.2.cmml" xref="S1.SS2.p2.11.m11.1.1.1.1.1.4.2"><csymbol cd="ambiguous" id="S1.SS2.p2.11.m11.1.1.1.1.1.4.2.1.cmml" xref="S1.SS2.p2.11.m11.1.1.1.1.1.4.2">subscript</csymbol><ci id="S1.SS2.p2.11.m11.1.1.1.1.1.4.2.2.cmml" xref="S1.SS2.p2.11.m11.1.1.1.1.1.4.2.2">𝑄</ci><cn id="S1.SS2.p2.11.m11.1.1.1.1.1.4.2.3.cmml" type="integer" xref="S1.SS2.p2.11.m11.1.1.1.1.1.4.2.3">2</cn></apply><apply id="S1.SS2.p2.11.m11.1.1.1.1.1.4.3.cmml" xref="S1.SS2.p2.11.m11.1.1.1.1.1.4.3"><csymbol cd="ambiguous" id="S1.SS2.p2.11.m11.1.1.1.1.1.4.3.1.cmml" xref="S1.SS2.p2.11.m11.1.1.1.1.1.4.3">subscript</csymbol><ci id="S1.SS2.p2.11.m11.1.1.1.1.1.4.3.2.cmml" xref="S1.SS2.p2.11.m11.1.1.1.1.1.4.3.2">𝑆</ci><cn id="S1.SS2.p2.11.m11.1.1.1.1.1.4.3.3.cmml" type="integer" xref="S1.SS2.p2.11.m11.1.1.1.1.1.4.3.3">2</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.p2.11.m11.1c">O(P_{1}P_{2}+Q_{1}S_{1}+Q_{2}S_{2})</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.p2.11.m11.1d">italic_O ( italic_P start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_P start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT + italic_Q start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_S start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + italic_Q start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT italic_S start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT )</annotation></semantics></math>. The <math alttext="\operatorname{{\mathbf{\Sigma}}}_{k}" class="ltx_Math" display="inline" id="S1.SS2.p2.12.m12.1"><semantics id="S1.SS2.p2.12.m12.1a"><msub id="S1.SS2.p2.12.m12.1.1" xref="S1.SS2.p2.12.m12.1.1.cmml"><mi id="S1.SS2.p2.12.m12.1.1.2" xref="S1.SS2.p2.12.m12.1.1.2.cmml">𝚺</mi><mi id="S1.SS2.p2.12.m12.1.1.3" xref="S1.SS2.p2.12.m12.1.1.3.cmml">k</mi></msub><annotation-xml encoding="MathML-Content" id="S1.SS2.p2.12.m12.1b"><apply id="S1.SS2.p2.12.m12.1.1.cmml" xref="S1.SS2.p2.12.m12.1.1"><csymbol cd="ambiguous" id="S1.SS2.p2.12.m12.1.1.1.cmml" xref="S1.SS2.p2.12.m12.1.1">subscript</csymbol><ci id="S1.SS2.p2.12.m12.1.1.2.cmml" xref="S1.SS2.p2.12.m12.1.1.2">𝚺</ci><ci id="S1.SS2.p2.12.m12.1.1.3.cmml" xref="S1.SS2.p2.12.m12.1.1.3">𝑘</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.p2.12.m12.1c">\operatorname{{\mathbf{\Sigma}}}_{k}</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.p2.12.m12.1d">bold_Σ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT</annotation></semantics></math> estimates depend on <math alttext="S_{k}" class="ltx_Math" display="inline" id="S1.SS2.p2.13.m13.1"><semantics id="S1.SS2.p2.13.m13.1a"><msub id="S1.SS2.p2.13.m13.1.1" xref="S1.SS2.p2.13.m13.1.1.cmml"><mi id="S1.SS2.p2.13.m13.1.1.2" xref="S1.SS2.p2.13.m13.1.1.2.cmml">S</mi><mi id="S1.SS2.p2.13.m13.1.1.3" xref="S1.SS2.p2.13.m13.1.1.3.cmml">k</mi></msub><annotation-xml encoding="MathML-Content" id="S1.SS2.p2.13.m13.1b"><apply id="S1.SS2.p2.13.m13.1.1.cmml" xref="S1.SS2.p2.13.m13.1.1"><csymbol cd="ambiguous" id="S1.SS2.p2.13.m13.1.1.1.cmml" xref="S1.SS2.p2.13.m13.1.1">subscript</csymbol><ci id="S1.SS2.p2.13.m13.1.1.2.cmml" xref="S1.SS2.p2.13.m13.1.1.2">𝑆</ci><ci id="S1.SS2.p2.13.m13.1.1.3.cmml" xref="S1.SS2.p2.13.m13.1.1.3">𝑘</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.p2.13.m13.1c">S_{k}</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.p2.13.m13.1d">italic_S start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT</annotation></semantics></math> choice, so we use group lasso penalties on the columns of <math alttext="\operatorname{\mathbf{L}}_{k}" class="ltx_Math" display="inline" id="S1.SS2.p2.14.m14.1"><semantics id="S1.SS2.p2.14.m14.1a"><msub id="S1.SS2.p2.14.m14.1.1" xref="S1.SS2.p2.14.m14.1.1.cmml"><mi id="S1.SS2.p2.14.m14.1.1.2" xref="S1.SS2.p2.14.m14.1.1.2.cmml">𝐋</mi><mi id="S1.SS2.p2.14.m14.1.1.3" xref="S1.SS2.p2.14.m14.1.1.3.cmml">k</mi></msub><annotation-xml encoding="MathML-Content" id="S1.SS2.p2.14.m14.1b"><apply id="S1.SS2.p2.14.m14.1.1.cmml" xref="S1.SS2.p2.14.m14.1.1"><csymbol cd="ambiguous" id="S1.SS2.p2.14.m14.1.1.1.cmml" xref="S1.SS2.p2.14.m14.1.1">subscript</csymbol><ci id="S1.SS2.p2.14.m14.1.1.2.cmml" xref="S1.SS2.p2.14.m14.1.1.2">𝐋</ci><ci id="S1.SS2.p2.14.m14.1.1.3.cmml" xref="S1.SS2.p2.14.m14.1.1.3">𝑘</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.p2.14.m14.1c">\operatorname{\mathbf{L}}_{k}</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.p2.14.m14.1d">bold_L start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT</annotation></semantics></math> for a data-driven choice of <math alttext="S_{k}" class="ltx_Math" display="inline" id="S1.SS2.p2.15.m15.1"><semantics id="S1.SS2.p2.15.m15.1a"><msub id="S1.SS2.p2.15.m15.1.1" xref="S1.SS2.p2.15.m15.1.1.cmml"><mi id="S1.SS2.p2.15.m15.1.1.2" xref="S1.SS2.p2.15.m15.1.1.2.cmml">S</mi><mi id="S1.SS2.p2.15.m15.1.1.3" xref="S1.SS2.p2.15.m15.1.1.3.cmml">k</mi></msub><annotation-xml encoding="MathML-Content" id="S1.SS2.p2.15.m15.1b"><apply id="S1.SS2.p2.15.m15.1.1.cmml" xref="S1.SS2.p2.15.m15.1.1"><csymbol cd="ambiguous" id="S1.SS2.p2.15.m15.1.1.1.cmml" xref="S1.SS2.p2.15.m15.1.1">subscript</csymbol><ci id="S1.SS2.p2.15.m15.1.1.2.cmml" xref="S1.SS2.p2.15.m15.1.1.2">𝑆</ci><ci id="S1.SS2.p2.15.m15.1.1.3.cmml" xref="S1.SS2.p2.15.m15.1.1.3">𝑘</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.p2.15.m15.1c">S_{k}</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.p2.15.m15.1d">italic_S start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT</annotation></semantics></math>. The low-rank factorization of <math alttext="\operatorname{{\mathbf{\Sigma}}}_{k}" class="ltx_Math" display="inline" id="S1.SS2.p2.16.m16.1"><semantics id="S1.SS2.p2.16.m16.1a"><msub id="S1.SS2.p2.16.m16.1.1" xref="S1.SS2.p2.16.m16.1.1.cmml"><mi id="S1.SS2.p2.16.m16.1.1.2" xref="S1.SS2.p2.16.m16.1.1.2.cmml">𝚺</mi><mi id="S1.SS2.p2.16.m16.1.1.3" xref="S1.SS2.p2.16.m16.1.1.3.cmml">k</mi></msub><annotation-xml encoding="MathML-Content" id="S1.SS2.p2.16.m16.1b"><apply id="S1.SS2.p2.16.m16.1.1.cmml" xref="S1.SS2.p2.16.m16.1.1"><csymbol cd="ambiguous" id="S1.SS2.p2.16.m16.1.1.1.cmml" xref="S1.SS2.p2.16.m16.1.1">subscript</csymbol><ci id="S1.SS2.p2.16.m16.1.1.2.cmml" xref="S1.SS2.p2.16.m16.1.1.2">𝚺</ci><ci id="S1.SS2.p2.16.m16.1.1.3.cmml" xref="S1.SS2.p2.16.m16.1.1.3">𝑘</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.p2.16.m16.1c">\operatorname{{\mathbf{\Sigma}}}_{k}</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.p2.16.m16.1d">bold_Σ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT</annotation></semantics></math> resembles a factor-analytic structure, so we set <math alttext="S_{k}=O(\log Q_{k})" class="ltx_Math" display="inline" id="S1.SS2.p2.17.m17.1"><semantics id="S1.SS2.p2.17.m17.1a"><mrow id="S1.SS2.p2.17.m17.1.1" xref="S1.SS2.p2.17.m17.1.1.cmml"><msub id="S1.SS2.p2.17.m17.1.1.3" xref="S1.SS2.p2.17.m17.1.1.3.cmml"><mi id="S1.SS2.p2.17.m17.1.1.3.2" xref="S1.SS2.p2.17.m17.1.1.3.2.cmml">S</mi><mi id="S1.SS2.p2.17.m17.1.1.3.3" xref="S1.SS2.p2.17.m17.1.1.3.3.cmml">k</mi></msub><mo id="S1.SS2.p2.17.m17.1.1.2" xref="S1.SS2.p2.17.m17.1.1.2.cmml">=</mo><mrow id="S1.SS2.p2.17.m17.1.1.1" xref="S1.SS2.p2.17.m17.1.1.1.cmml"><mi id="S1.SS2.p2.17.m17.1.1.1.3" xref="S1.SS2.p2.17.m17.1.1.1.3.cmml">O</mi><mo id="S1.SS2.p2.17.m17.1.1.1.2" xref="S1.SS2.p2.17.m17.1.1.1.2.cmml">⁢</mo><mrow id="S1.SS2.p2.17.m17.1.1.1.1.1" xref="S1.SS2.p2.17.m17.1.1.1.1.1.1.cmml"><mo id="S1.SS2.p2.17.m17.1.1.1.1.1.2" stretchy="false" xref="S1.SS2.p2.17.m17.1.1.1.1.1.1.cmml">(</mo><mrow id="S1.SS2.p2.17.m17.1.1.1.1.1.1" xref="S1.SS2.p2.17.m17.1.1.1.1.1.1.cmml"><mi id="S1.SS2.p2.17.m17.1.1.1.1.1.1.1" xref="S1.SS2.p2.17.m17.1.1.1.1.1.1.1.cmml">log</mi><mo id="S1.SS2.p2.17.m17.1.1.1.1.1.1a" lspace="0.167em" xref="S1.SS2.p2.17.m17.1.1.1.1.1.1.cmml">⁡</mo><msub id="S1.SS2.p2.17.m17.1.1.1.1.1.1.2" xref="S1.SS2.p2.17.m17.1.1.1.1.1.1.2.cmml"><mi id="S1.SS2.p2.17.m17.1.1.1.1.1.1.2.2" xref="S1.SS2.p2.17.m17.1.1.1.1.1.1.2.2.cmml">Q</mi><mi id="S1.SS2.p2.17.m17.1.1.1.1.1.1.2.3" xref="S1.SS2.p2.17.m17.1.1.1.1.1.1.2.3.cmml">k</mi></msub></mrow><mo id="S1.SS2.p2.17.m17.1.1.1.1.1.3" stretchy="false" xref="S1.SS2.p2.17.m17.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.SS2.p2.17.m17.1b"><apply id="S1.SS2.p2.17.m17.1.1.cmml" xref="S1.SS2.p2.17.m17.1.1"><eq id="S1.SS2.p2.17.m17.1.1.2.cmml" xref="S1.SS2.p2.17.m17.1.1.2"></eq><apply id="S1.SS2.p2.17.m17.1.1.3.cmml" xref="S1.SS2.p2.17.m17.1.1.3"><csymbol cd="ambiguous" id="S1.SS2.p2.17.m17.1.1.3.1.cmml" xref="S1.SS2.p2.17.m17.1.1.3">subscript</csymbol><ci id="S1.SS2.p2.17.m17.1.1.3.2.cmml" xref="S1.SS2.p2.17.m17.1.1.3.2">𝑆</ci><ci id="S1.SS2.p2.17.m17.1.1.3.3.cmml" xref="S1.SS2.p2.17.m17.1.1.3.3">𝑘</ci></apply><apply id="S1.SS2.p2.17.m17.1.1.1.cmml" xref="S1.SS2.p2.17.m17.1.1.1"><times id="S1.SS2.p2.17.m17.1.1.1.2.cmml" xref="S1.SS2.p2.17.m17.1.1.1.2"></times><ci id="S1.SS2.p2.17.m17.1.1.1.3.cmml" xref="S1.SS2.p2.17.m17.1.1.1.3">𝑂</ci><apply id="S1.SS2.p2.17.m17.1.1.1.1.1.1.cmml" xref="S1.SS2.p2.17.m17.1.1.1.1.1"><log id="S1.SS2.p2.17.m17.1.1.1.1.1.1.1.cmml" xref="S1.SS2.p2.17.m17.1.1.1.1.1.1.1"></log><apply id="S1.SS2.p2.17.m17.1.1.1.1.1.1.2.cmml" xref="S1.SS2.p2.17.m17.1.1.1.1.1.1.2"><csymbol cd="ambiguous" id="S1.SS2.p2.17.m17.1.1.1.1.1.1.2.1.cmml" xref="S1.SS2.p2.17.m17.1.1.1.1.1.1.2">subscript</csymbol><ci id="S1.SS2.p2.17.m17.1.1.1.1.1.1.2.2.cmml" xref="S1.SS2.p2.17.m17.1.1.1.1.1.1.2.2">𝑄</ci><ci id="S1.SS2.p2.17.m17.1.1.1.1.1.1.2.3.cmml" xref="S1.SS2.p2.17.m17.1.1.1.1.1.1.2.3">𝑘</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.p2.17.m17.1c">S_{k}=O(\log Q_{k})</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.p2.17.m17.1d">italic_S start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT = italic_O ( roman_log italic_Q start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT )</annotation></semantics></math> initially <cite class="ltx_cite ltx_citemacro_citep">(Ročková and George, <a class="ltx_ref" href="https://arxiv.org/html/2503.13782v1#bib.bib24" title="">2016</a>; Srivastava et al., <a class="ltx_ref" href="https://arxiv.org/html/2503.13782v1#bib.bib28" title="">2017</a>)</cite>. This idea extends the low-rank random effects in mixed model linear regression to low-rank matrix-variate random effects in MMTR <cite class="ltx_cite ltx_citemacro_citep">(James et al., <a class="ltx_ref" href="https://arxiv.org/html/2503.13782v1#bib.bib19" title="">2000</a>; Heiling et al., <a class="ltx_ref" href="https://arxiv.org/html/2503.13782v1#bib.bib13" title="">2024</a>)</cite>. This setup implies that the maximization of the E step objective reduces to a series of regularized least squares problems, enabling estimation of <math alttext="\operatorname{\mathbf{B}},\tau^{2},\operatorname{\mathbf{L}}_{1}" class="ltx_Math" display="inline" id="S1.SS2.p2.18.m18.3"><semantics id="S1.SS2.p2.18.m18.3a"><mrow id="S1.SS2.p2.18.m18.3.3.2" xref="S1.SS2.p2.18.m18.3.3.3.cmml"><mi id="S1.SS2.p2.18.m18.1.1" xref="S1.SS2.p2.18.m18.1.1.cmml">𝐁</mi><mo id="S1.SS2.p2.18.m18.3.3.2.3" xref="S1.SS2.p2.18.m18.3.3.3.cmml">,</mo><msup id="S1.SS2.p2.18.m18.2.2.1.1" xref="S1.SS2.p2.18.m18.2.2.1.1.cmml"><mi id="S1.SS2.p2.18.m18.2.2.1.1.2" xref="S1.SS2.p2.18.m18.2.2.1.1.2.cmml">τ</mi><mn id="S1.SS2.p2.18.m18.2.2.1.1.3" xref="S1.SS2.p2.18.m18.2.2.1.1.3.cmml">2</mn></msup><mo id="S1.SS2.p2.18.m18.3.3.2.4" xref="S1.SS2.p2.18.m18.3.3.3.cmml">,</mo><msub id="S1.SS2.p2.18.m18.3.3.2.2" xref="S1.SS2.p2.18.m18.3.3.2.2.cmml"><mi id="S1.SS2.p2.18.m18.3.3.2.2.2" xref="S1.SS2.p2.18.m18.3.3.2.2.2.cmml">𝐋</mi><mn id="S1.SS2.p2.18.m18.3.3.2.2.3" xref="S1.SS2.p2.18.m18.3.3.2.2.3.cmml">1</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S1.SS2.p2.18.m18.3b"><list id="S1.SS2.p2.18.m18.3.3.3.cmml" xref="S1.SS2.p2.18.m18.3.3.2"><ci id="S1.SS2.p2.18.m18.1.1.cmml" xref="S1.SS2.p2.18.m18.1.1">𝐁</ci><apply id="S1.SS2.p2.18.m18.2.2.1.1.cmml" xref="S1.SS2.p2.18.m18.2.2.1.1"><csymbol cd="ambiguous" id="S1.SS2.p2.18.m18.2.2.1.1.1.cmml" xref="S1.SS2.p2.18.m18.2.2.1.1">superscript</csymbol><ci id="S1.SS2.p2.18.m18.2.2.1.1.2.cmml" xref="S1.SS2.p2.18.m18.2.2.1.1.2">𝜏</ci><cn id="S1.SS2.p2.18.m18.2.2.1.1.3.cmml" type="integer" xref="S1.SS2.p2.18.m18.2.2.1.1.3">2</cn></apply><apply id="S1.SS2.p2.18.m18.3.3.2.2.cmml" xref="S1.SS2.p2.18.m18.3.3.2.2"><csymbol cd="ambiguous" id="S1.SS2.p2.18.m18.3.3.2.2.1.cmml" xref="S1.SS2.p2.18.m18.3.3.2.2">subscript</csymbol><ci id="S1.SS2.p2.18.m18.3.3.2.2.2.cmml" xref="S1.SS2.p2.18.m18.3.3.2.2.2">𝐋</ci><cn id="S1.SS2.p2.18.m18.3.3.2.2.3.cmml" type="integer" xref="S1.SS2.p2.18.m18.3.3.2.2.3">1</cn></apply></list></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.p2.18.m18.3c">\operatorname{\mathbf{B}},\tau^{2},\operatorname{\mathbf{L}}_{1}</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.p2.18.m18.3d">bold_B , italic_τ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT , bold_L start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="\operatorname{\mathbf{L}}_{2}" class="ltx_Math" display="inline" id="S1.SS2.p2.19.m19.1"><semantics id="S1.SS2.p2.19.m19.1a"><msub id="S1.SS2.p2.19.m19.1.1" xref="S1.SS2.p2.19.m19.1.1.cmml"><mi id="S1.SS2.p2.19.m19.1.1.2" xref="S1.SS2.p2.19.m19.1.1.2.cmml">𝐋</mi><mn id="S1.SS2.p2.19.m19.1.1.3" xref="S1.SS2.p2.19.m19.1.1.3.cmml">2</mn></msub><annotation-xml encoding="MathML-Content" id="S1.SS2.p2.19.m19.1b"><apply id="S1.SS2.p2.19.m19.1.1.cmml" xref="S1.SS2.p2.19.m19.1.1"><csymbol cd="ambiguous" id="S1.SS2.p2.19.m19.1.1.1.cmml" xref="S1.SS2.p2.19.m19.1.1">subscript</csymbol><ci id="S1.SS2.p2.19.m19.1.1.2.cmml" xref="S1.SS2.p2.19.m19.1.1.2">𝐋</ci><cn id="S1.SS2.p2.19.m19.1.1.3.cmml" type="integer" xref="S1.SS2.p2.19.m19.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.p2.19.m19.1c">\operatorname{\mathbf{L}}_{2}</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.p2.19.m19.1d">bold_L start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math> via <span class="ltx_text ltx_font_typewriter" id="S1.SS2.p2.19.1">scalreg</span> <cite class="ltx_cite ltx_citemacro_citep">(Sun, <a class="ltx_ref" href="https://arxiv.org/html/2503.13782v1#bib.bib29" title="">2019</a>)</cite> and <span class="ltx_text ltx_font_typewriter" id="S1.SS2.p2.19.2">gglasso</span> <cite class="ltx_cite ltx_citemacro_citep">(Yang et al., <a class="ltx_ref" href="https://arxiv.org/html/2503.13782v1#bib.bib34" title="">2024</a>)</cite> <em class="ltx_emph ltx_font_italic" id="S1.SS2.p2.19.3">R</em> packages; see Section <a class="ltx_ref" href="https://arxiv.org/html/2503.13782v1#S3" title="3 Regularized Parameter Estimation ‣ Regularized Parameter Estimation in Mixed Model Trace Regression"><span class="ltx_text ltx_ref_tag">3</span></a> for the details.</p> </div> <div class="ltx_para" id="S1.SS2.p3"> <p class="ltx_p" id="S1.SS2.p3.9">Our main contributions are threefold. First, MMTR is a novel extension of mixed models for matrix-valued covariates. We establish conditions for the separate estimation of <math alttext="\operatorname{{\mathbf{\Sigma}}}_{1}" class="ltx_Math" display="inline" id="S1.SS2.p3.1.m1.1"><semantics id="S1.SS2.p3.1.m1.1a"><msub id="S1.SS2.p3.1.m1.1.1" xref="S1.SS2.p3.1.m1.1.1.cmml"><mi id="S1.SS2.p3.1.m1.1.1.2" xref="S1.SS2.p3.1.m1.1.1.2.cmml">𝚺</mi><mn id="S1.SS2.p3.1.m1.1.1.3" xref="S1.SS2.p3.1.m1.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S1.SS2.p3.1.m1.1b"><apply id="S1.SS2.p3.1.m1.1.1.cmml" xref="S1.SS2.p3.1.m1.1.1"><csymbol cd="ambiguous" id="S1.SS2.p3.1.m1.1.1.1.cmml" xref="S1.SS2.p3.1.m1.1.1">subscript</csymbol><ci id="S1.SS2.p3.1.m1.1.1.2.cmml" xref="S1.SS2.p3.1.m1.1.1.2">𝚺</ci><cn id="S1.SS2.p3.1.m1.1.1.3.cmml" type="integer" xref="S1.SS2.p3.1.m1.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.p3.1.m1.1c">\operatorname{{\mathbf{\Sigma}}}_{1}</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.p3.1.m1.1d">bold_Σ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="\operatorname{{\mathbf{\Sigma}}}_{2}" class="ltx_Math" display="inline" id="S1.SS2.p3.2.m2.1"><semantics id="S1.SS2.p3.2.m2.1a"><msub id="S1.SS2.p3.2.m2.1.1" xref="S1.SS2.p3.2.m2.1.1.cmml"><mi id="S1.SS2.p3.2.m2.1.1.2" xref="S1.SS2.p3.2.m2.1.1.2.cmml">𝚺</mi><mn id="S1.SS2.p3.2.m2.1.1.3" xref="S1.SS2.p3.2.m2.1.1.3.cmml">2</mn></msub><annotation-xml encoding="MathML-Content" id="S1.SS2.p3.2.m2.1b"><apply id="S1.SS2.p3.2.m2.1.1.cmml" xref="S1.SS2.p3.2.m2.1.1"><csymbol cd="ambiguous" id="S1.SS2.p3.2.m2.1.1.1.cmml" xref="S1.SS2.p3.2.m2.1.1">subscript</csymbol><ci id="S1.SS2.p3.2.m2.1.1.2.cmml" xref="S1.SS2.p3.2.m2.1.1.2">𝚺</ci><cn id="S1.SS2.p3.2.m2.1.1.3.cmml" type="integer" xref="S1.SS2.p3.2.m2.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.p3.2.m2.1c">\operatorname{{\mathbf{\Sigma}}}_{2}</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.p3.2.m2.1d">bold_Σ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math>, given that only <math alttext="\operatorname{{\mathbf{\Sigma}}}_{2}\otimes\operatorname{{\mathbf{\Sigma}}}_{1}" class="ltx_Math" display="inline" id="S1.SS2.p3.3.m3.1"><semantics id="S1.SS2.p3.3.m3.1a"><mrow id="S1.SS2.p3.3.m3.1.1" xref="S1.SS2.p3.3.m3.1.1.cmml"><msub id="S1.SS2.p3.3.m3.1.1.2" xref="S1.SS2.p3.3.m3.1.1.2.cmml"><mi id="S1.SS2.p3.3.m3.1.1.2.2" xref="S1.SS2.p3.3.m3.1.1.2.2.cmml">𝚺</mi><mn id="S1.SS2.p3.3.m3.1.1.2.3" xref="S1.SS2.p3.3.m3.1.1.2.3.cmml">2</mn></msub><mo id="S1.SS2.p3.3.m3.1.1.1" lspace="0em" rspace="0.222em" xref="S1.SS2.p3.3.m3.1.1.1.cmml">⊗</mo><msub id="S1.SS2.p3.3.m3.1.1.3" xref="S1.SS2.p3.3.m3.1.1.3.cmml"><mi id="S1.SS2.p3.3.m3.1.1.3.2" xref="S1.SS2.p3.3.m3.1.1.3.2.cmml">𝚺</mi><mn id="S1.SS2.p3.3.m3.1.1.3.3" xref="S1.SS2.p3.3.m3.1.1.3.3.cmml">1</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S1.SS2.p3.3.m3.1b"><apply id="S1.SS2.p3.3.m3.1.1.cmml" xref="S1.SS2.p3.3.m3.1.1"><csymbol cd="latexml" id="S1.SS2.p3.3.m3.1.1.1.cmml" xref="S1.SS2.p3.3.m3.1.1.1">tensor-product</csymbol><apply id="S1.SS2.p3.3.m3.1.1.2.cmml" xref="S1.SS2.p3.3.m3.1.1.2"><csymbol cd="ambiguous" id="S1.SS2.p3.3.m3.1.1.2.1.cmml" xref="S1.SS2.p3.3.m3.1.1.2">subscript</csymbol><ci id="S1.SS2.p3.3.m3.1.1.2.2.cmml" xref="S1.SS2.p3.3.m3.1.1.2.2">𝚺</ci><cn id="S1.SS2.p3.3.m3.1.1.2.3.cmml" type="integer" xref="S1.SS2.p3.3.m3.1.1.2.3">2</cn></apply><apply id="S1.SS2.p3.3.m3.1.1.3.cmml" xref="S1.SS2.p3.3.m3.1.1.3"><csymbol cd="ambiguous" id="S1.SS2.p3.3.m3.1.1.3.1.cmml" xref="S1.SS2.p3.3.m3.1.1.3">subscript</csymbol><ci id="S1.SS2.p3.3.m3.1.1.3.2.cmml" xref="S1.SS2.p3.3.m3.1.1.3.2">𝚺</ci><cn id="S1.SS2.p3.3.m3.1.1.3.3.cmml" type="integer" xref="S1.SS2.p3.3.m3.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.p3.3.m3.1c">\operatorname{{\mathbf{\Sigma}}}_{2}\otimes\operatorname{{\mathbf{\Sigma}}}_{1}</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.p3.3.m3.1d">bold_Σ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ⊗ bold_Σ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> is identified in (<a class="ltx_ref" href="https://arxiv.org/html/2503.13782v1#S1.E1" title="In 1 Introduction ‣ Regularized Parameter Estimation in Mixed Model Trace Regression"><span class="ltx_text ltx_ref_tag">1</span></a>); see Section <a class="ltx_ref" href="https://arxiv.org/html/2503.13782v1#S2.SS1" title="2.1 Model ‣ 2 Mixed Model Trace Regression ‣ Regularized Parameter Estimation in Mixed Model Trace Regression"><span class="ltx_text ltx_ref_tag">2.1</span></a>. Second, random effects <math alttext="\operatorname{\mathbf{A}}_{i}" class="ltx_Math" display="inline" id="S1.SS2.p3.4.m4.1"><semantics id="S1.SS2.p3.4.m4.1a"><msub id="S1.SS2.p3.4.m4.1.1" xref="S1.SS2.p3.4.m4.1.1.cmml"><mi id="S1.SS2.p3.4.m4.1.1.2" xref="S1.SS2.p3.4.m4.1.1.2.cmml">𝐀</mi><mi id="S1.SS2.p3.4.m4.1.1.3" xref="S1.SS2.p3.4.m4.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S1.SS2.p3.4.m4.1b"><apply id="S1.SS2.p3.4.m4.1.1.cmml" xref="S1.SS2.p3.4.m4.1.1"><csymbol cd="ambiguous" id="S1.SS2.p3.4.m4.1.1.1.cmml" xref="S1.SS2.p3.4.m4.1.1">subscript</csymbol><ci id="S1.SS2.p3.4.m4.1.1.2.cmml" xref="S1.SS2.p3.4.m4.1.1.2">𝐀</ci><ci id="S1.SS2.p3.4.m4.1.1.3.cmml" xref="S1.SS2.p3.4.m4.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.p3.4.m4.1c">\operatorname{\mathbf{A}}_{i}</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.p3.4.m4.1d">bold_A start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math>’s in (<a class="ltx_ref" href="https://arxiv.org/html/2503.13782v1#S1.E1" title="In 1 Introduction ‣ Regularized Parameter Estimation in Mixed Model Trace Regression"><span class="ltx_text ltx_ref_tag">1</span></a>) have a low-rank separable covariance structure, which is a natural extension of separable covariance arrays to high-dimensional settings <cite class="ltx_cite ltx_citemacro_citep">(Hoff, <a class="ltx_ref" href="https://arxiv.org/html/2503.13782v1#bib.bib14" title="">2011</a>)</cite>. MMTR is also related to separable factor analysis (SFA) in that SFA adds diagonal matrices with positive entries to <math alttext="\operatorname{{\mathbf{\Sigma}}}_{1}" class="ltx_Math" display="inline" id="S1.SS2.p3.5.m5.1"><semantics id="S1.SS2.p3.5.m5.1a"><msub id="S1.SS2.p3.5.m5.1.1" xref="S1.SS2.p3.5.m5.1.1.cmml"><mi id="S1.SS2.p3.5.m5.1.1.2" xref="S1.SS2.p3.5.m5.1.1.2.cmml">𝚺</mi><mn id="S1.SS2.p3.5.m5.1.1.3" xref="S1.SS2.p3.5.m5.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S1.SS2.p3.5.m5.1b"><apply id="S1.SS2.p3.5.m5.1.1.cmml" xref="S1.SS2.p3.5.m5.1.1"><csymbol cd="ambiguous" id="S1.SS2.p3.5.m5.1.1.1.cmml" xref="S1.SS2.p3.5.m5.1.1">subscript</csymbol><ci id="S1.SS2.p3.5.m5.1.1.2.cmml" xref="S1.SS2.p3.5.m5.1.1.2">𝚺</ci><cn id="S1.SS2.p3.5.m5.1.1.3.cmml" type="integer" xref="S1.SS2.p3.5.m5.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.p3.5.m5.1c">\operatorname{{\mathbf{\Sigma}}}_{1}</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.p3.5.m5.1d">bold_Σ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="\operatorname{{\mathbf{\Sigma}}}_{2}" class="ltx_Math" display="inline" id="S1.SS2.p3.6.m6.1"><semantics id="S1.SS2.p3.6.m6.1a"><msub id="S1.SS2.p3.6.m6.1.1" xref="S1.SS2.p3.6.m6.1.1.cmml"><mi id="S1.SS2.p3.6.m6.1.1.2" xref="S1.SS2.p3.6.m6.1.1.2.cmml">𝚺</mi><mn id="S1.SS2.p3.6.m6.1.1.3" xref="S1.SS2.p3.6.m6.1.1.3.cmml">2</mn></msub><annotation-xml encoding="MathML-Content" id="S1.SS2.p3.6.m6.1b"><apply id="S1.SS2.p3.6.m6.1.1.cmml" xref="S1.SS2.p3.6.m6.1.1"><csymbol cd="ambiguous" id="S1.SS2.p3.6.m6.1.1.1.cmml" xref="S1.SS2.p3.6.m6.1.1">subscript</csymbol><ci id="S1.SS2.p3.6.m6.1.1.2.cmml" xref="S1.SS2.p3.6.m6.1.1.2">𝚺</ci><cn id="S1.SS2.p3.6.m6.1.1.3.cmml" type="integer" xref="S1.SS2.p3.6.m6.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.p3.6.m6.1c">\operatorname{{\mathbf{\Sigma}}}_{2}</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.p3.6.m6.1d">bold_Σ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math>, implying that MMTR’s covariance structure is more parsimonious than SFA when <math alttext="\min(P_{1},P_{2})" class="ltx_Math" display="inline" id="S1.SS2.p3.7.m7.3"><semantics id="S1.SS2.p3.7.m7.3a"><mrow id="S1.SS2.p3.7.m7.3.3.2" xref="S1.SS2.p3.7.m7.3.3.3.cmml"><mi id="S1.SS2.p3.7.m7.1.1" xref="S1.SS2.p3.7.m7.1.1.cmml">min</mi><mo id="S1.SS2.p3.7.m7.3.3.2a" xref="S1.SS2.p3.7.m7.3.3.3.cmml">⁡</mo><mrow id="S1.SS2.p3.7.m7.3.3.2.2" xref="S1.SS2.p3.7.m7.3.3.3.cmml"><mo id="S1.SS2.p3.7.m7.3.3.2.2.3" stretchy="false" xref="S1.SS2.p3.7.m7.3.3.3.cmml">(</mo><msub id="S1.SS2.p3.7.m7.2.2.1.1.1" xref="S1.SS2.p3.7.m7.2.2.1.1.1.cmml"><mi id="S1.SS2.p3.7.m7.2.2.1.1.1.2" xref="S1.SS2.p3.7.m7.2.2.1.1.1.2.cmml">P</mi><mn id="S1.SS2.p3.7.m7.2.2.1.1.1.3" xref="S1.SS2.p3.7.m7.2.2.1.1.1.3.cmml">1</mn></msub><mo id="S1.SS2.p3.7.m7.3.3.2.2.4" xref="S1.SS2.p3.7.m7.3.3.3.cmml">,</mo><msub id="S1.SS2.p3.7.m7.3.3.2.2.2" xref="S1.SS2.p3.7.m7.3.3.2.2.2.cmml"><mi id="S1.SS2.p3.7.m7.3.3.2.2.2.2" xref="S1.SS2.p3.7.m7.3.3.2.2.2.2.cmml">P</mi><mn id="S1.SS2.p3.7.m7.3.3.2.2.2.3" xref="S1.SS2.p3.7.m7.3.3.2.2.2.3.cmml">2</mn></msub><mo id="S1.SS2.p3.7.m7.3.3.2.2.5" stretchy="false" xref="S1.SS2.p3.7.m7.3.3.3.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.SS2.p3.7.m7.3b"><apply id="S1.SS2.p3.7.m7.3.3.3.cmml" xref="S1.SS2.p3.7.m7.3.3.2"><min id="S1.SS2.p3.7.m7.1.1.cmml" xref="S1.SS2.p3.7.m7.1.1"></min><apply id="S1.SS2.p3.7.m7.2.2.1.1.1.cmml" xref="S1.SS2.p3.7.m7.2.2.1.1.1"><csymbol cd="ambiguous" id="S1.SS2.p3.7.m7.2.2.1.1.1.1.cmml" xref="S1.SS2.p3.7.m7.2.2.1.1.1">subscript</csymbol><ci id="S1.SS2.p3.7.m7.2.2.1.1.1.2.cmml" xref="S1.SS2.p3.7.m7.2.2.1.1.1.2">𝑃</ci><cn id="S1.SS2.p3.7.m7.2.2.1.1.1.3.cmml" type="integer" xref="S1.SS2.p3.7.m7.2.2.1.1.1.3">1</cn></apply><apply id="S1.SS2.p3.7.m7.3.3.2.2.2.cmml" xref="S1.SS2.p3.7.m7.3.3.2.2.2"><csymbol cd="ambiguous" id="S1.SS2.p3.7.m7.3.3.2.2.2.1.cmml" xref="S1.SS2.p3.7.m7.3.3.2.2.2">subscript</csymbol><ci id="S1.SS2.p3.7.m7.3.3.2.2.2.2.cmml" xref="S1.SS2.p3.7.m7.3.3.2.2.2.2">𝑃</ci><cn id="S1.SS2.p3.7.m7.3.3.2.2.2.3.cmml" type="integer" xref="S1.SS2.p3.7.m7.3.3.2.2.2.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.p3.7.m7.3c">\min(P_{1},P_{2})</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.p3.7.m7.3d">roman_min ( italic_P start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_P start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT )</annotation></semantics></math> is large <cite class="ltx_cite ltx_citemacro_citep">(Fosdick and Hoff, <a class="ltx_ref" href="https://arxiv.org/html/2503.13782v1#bib.bib9" title="">2014</a>)</cite>; see Section <a class="ltx_ref" href="https://arxiv.org/html/2503.13782v1#S2.SS2" title="2.2 Reduction to a Structured Mixed Model ‣ 2 Mixed Model Trace Regression ‣ Regularized Parameter Estimation in Mixed Model Trace Regression"><span class="ltx_text ltx_ref_tag">2.2</span></a>. Finally, we develop an EM algorithm for efficient parameter regularized estimation and data-driven choice of the ranks of <math alttext="\operatorname{{\mathbf{\Sigma}}}_{1}" class="ltx_Math" display="inline" id="S1.SS2.p3.8.m8.1"><semantics id="S1.SS2.p3.8.m8.1a"><msub id="S1.SS2.p3.8.m8.1.1" xref="S1.SS2.p3.8.m8.1.1.cmml"><mi id="S1.SS2.p3.8.m8.1.1.2" xref="S1.SS2.p3.8.m8.1.1.2.cmml">𝚺</mi><mn id="S1.SS2.p3.8.m8.1.1.3" xref="S1.SS2.p3.8.m8.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S1.SS2.p3.8.m8.1b"><apply id="S1.SS2.p3.8.m8.1.1.cmml" xref="S1.SS2.p3.8.m8.1.1"><csymbol cd="ambiguous" id="S1.SS2.p3.8.m8.1.1.1.cmml" xref="S1.SS2.p3.8.m8.1.1">subscript</csymbol><ci id="S1.SS2.p3.8.m8.1.1.2.cmml" xref="S1.SS2.p3.8.m8.1.1.2">𝚺</ci><cn id="S1.SS2.p3.8.m8.1.1.3.cmml" type="integer" xref="S1.SS2.p3.8.m8.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.p3.8.m8.1c">\operatorname{{\mathbf{\Sigma}}}_{1}</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.p3.8.m8.1d">bold_Σ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="\operatorname{{\mathbf{\Sigma}}}_{2}" class="ltx_Math" display="inline" id="S1.SS2.p3.9.m9.1"><semantics id="S1.SS2.p3.9.m9.1a"><msub id="S1.SS2.p3.9.m9.1.1" xref="S1.SS2.p3.9.m9.1.1.cmml"><mi id="S1.SS2.p3.9.m9.1.1.2" xref="S1.SS2.p3.9.m9.1.1.2.cmml">𝚺</mi><mn id="S1.SS2.p3.9.m9.1.1.3" xref="S1.SS2.p3.9.m9.1.1.3.cmml">2</mn></msub><annotation-xml encoding="MathML-Content" id="S1.SS2.p3.9.m9.1b"><apply id="S1.SS2.p3.9.m9.1.1.cmml" xref="S1.SS2.p3.9.m9.1.1"><csymbol cd="ambiguous" id="S1.SS2.p3.9.m9.1.1.1.cmml" xref="S1.SS2.p3.9.m9.1.1">subscript</csymbol><ci id="S1.SS2.p3.9.m9.1.1.2.cmml" xref="S1.SS2.p3.9.m9.1.1.2">𝚺</ci><cn id="S1.SS2.p3.9.m9.1.1.3.cmml" type="integer" xref="S1.SS2.p3.9.m9.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2.p3.9.m9.1c">\operatorname{{\mathbf{\Sigma}}}_{2}</annotation><annotation encoding="application/x-llamapun" id="S1.SS2.p3.9.m9.1d">bold_Σ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math>; see Section <a class="ltx_ref" href="https://arxiv.org/html/2503.13782v1#S3" title="3 Regularized Parameter Estimation ‣ Regularized Parameter Estimation in Mixed Model Trace Regression"><span class="ltx_text ltx_ref_tag">3</span></a>. MMTR achieves lower estimation and prediction errors than its high-dimensional mixed model competitors across diverse simulated and real data analyses.</p> </div> </section> </section> <section class="ltx_section" id="S2"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">2 </span>Mixed Model Trace Regression</h2> <section class="ltx_subsection" id="S2.SS1"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">2.1 </span>Model</h3> <div class="ltx_para" id="S2.SS1.p1"> <p class="ltx_p" id="S2.SS1.p1.15">Consider the parameter expanded form of (<a class="ltx_ref" href="https://arxiv.org/html/2503.13782v1#S1.E1" title="In 1 Introduction ‣ Regularized Parameter Estimation in Mixed Model Trace Regression"><span class="ltx_text ltx_ref_tag">1</span></a>). Let <math alttext="\operatorname{\mathbf{L}}_{k}\in\operatorname{\mathbb{R}}^{Q_{k}\times S_{k}}" 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"><msub id="S2.SS1.p1.1.m1.1.1.2" xref="S2.SS1.p1.1.m1.1.1.2.cmml"><mi id="S2.SS1.p1.1.m1.1.1.2.2" xref="S2.SS1.p1.1.m1.1.1.2.2.cmml">𝐋</mi><mi id="S2.SS1.p1.1.m1.1.1.2.3" xref="S2.SS1.p1.1.m1.1.1.2.3.cmml">k</mi></msub><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><mrow id="S2.SS1.p1.1.m1.1.1.3.3" xref="S2.SS1.p1.1.m1.1.1.3.3.cmml"><msub id="S2.SS1.p1.1.m1.1.1.3.3.2" xref="S2.SS1.p1.1.m1.1.1.3.3.2.cmml"><mi id="S2.SS1.p1.1.m1.1.1.3.3.2.2" xref="S2.SS1.p1.1.m1.1.1.3.3.2.2.cmml">Q</mi><mi id="S2.SS1.p1.1.m1.1.1.3.3.2.3" xref="S2.SS1.p1.1.m1.1.1.3.3.2.3.cmml">k</mi></msub><mo id="S2.SS1.p1.1.m1.1.1.3.3.1" lspace="0.222em" rspace="0.222em" xref="S2.SS1.p1.1.m1.1.1.3.3.1.cmml">×</mo><msub id="S2.SS1.p1.1.m1.1.1.3.3.3" xref="S2.SS1.p1.1.m1.1.1.3.3.3.cmml"><mi id="S2.SS1.p1.1.m1.1.1.3.3.3.2" xref="S2.SS1.p1.1.m1.1.1.3.3.3.2.cmml">S</mi><mi id="S2.SS1.p1.1.m1.1.1.3.3.3.3" xref="S2.SS1.p1.1.m1.1.1.3.3.3.3.cmml">k</mi></msub></mrow></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><apply id="S2.SS1.p1.1.m1.1.1.2.cmml" xref="S2.SS1.p1.1.m1.1.1.2"><csymbol cd="ambiguous" id="S2.SS1.p1.1.m1.1.1.2.1.cmml" xref="S2.SS1.p1.1.m1.1.1.2">subscript</csymbol><ci id="S2.SS1.p1.1.m1.1.1.2.2.cmml" xref="S2.SS1.p1.1.m1.1.1.2.2">𝐋</ci><ci id="S2.SS1.p1.1.m1.1.1.2.3.cmml" xref="S2.SS1.p1.1.m1.1.1.2.3">𝑘</ci></apply><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><apply id="S2.SS1.p1.1.m1.1.1.3.3.cmml" xref="S2.SS1.p1.1.m1.1.1.3.3"><times id="S2.SS1.p1.1.m1.1.1.3.3.1.cmml" xref="S2.SS1.p1.1.m1.1.1.3.3.1"></times><apply id="S2.SS1.p1.1.m1.1.1.3.3.2.cmml" xref="S2.SS1.p1.1.m1.1.1.3.3.2"><csymbol cd="ambiguous" id="S2.SS1.p1.1.m1.1.1.3.3.2.1.cmml" xref="S2.SS1.p1.1.m1.1.1.3.3.2">subscript</csymbol><ci id="S2.SS1.p1.1.m1.1.1.3.3.2.2.cmml" xref="S2.SS1.p1.1.m1.1.1.3.3.2.2">𝑄</ci><ci id="S2.SS1.p1.1.m1.1.1.3.3.2.3.cmml" xref="S2.SS1.p1.1.m1.1.1.3.3.2.3">𝑘</ci></apply><apply id="S2.SS1.p1.1.m1.1.1.3.3.3.cmml" xref="S2.SS1.p1.1.m1.1.1.3.3.3"><csymbol cd="ambiguous" id="S2.SS1.p1.1.m1.1.1.3.3.3.1.cmml" xref="S2.SS1.p1.1.m1.1.1.3.3.3">subscript</csymbol><ci id="S2.SS1.p1.1.m1.1.1.3.3.3.2.cmml" xref="S2.SS1.p1.1.m1.1.1.3.3.3.2">𝑆</ci><ci id="S2.SS1.p1.1.m1.1.1.3.3.3.3.cmml" xref="S2.SS1.p1.1.m1.1.1.3.3.3.3">𝑘</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p1.1.m1.1c">\operatorname{\mathbf{L}}_{k}\in\operatorname{\mathbb{R}}^{Q_{k}\times S_{k}}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p1.1.m1.1d">bold_L start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ∈ blackboard_R start_POSTSUPERSCRIPT italic_Q start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT × italic_S start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUPERSCRIPT</annotation></semantics></math> be the square root matrix factor of <math alttext="\operatorname{{\mathbf{\Sigma}}}_{k}" class="ltx_Math" display="inline" id="S2.SS1.p1.2.m2.1"><semantics id="S2.SS1.p1.2.m2.1a"><msub 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">𝚺</mi><mi id="S2.SS1.p1.2.m2.1.1.3" xref="S2.SS1.p1.2.m2.1.1.3.cmml">k</mi></msub><annotation-xml encoding="MathML-Content" id="S2.SS1.p1.2.m2.1b"><apply id="S2.SS1.p1.2.m2.1.1.cmml" xref="S2.SS1.p1.2.m2.1.1"><csymbol cd="ambiguous" id="S2.SS1.p1.2.m2.1.1.1.cmml" xref="S2.SS1.p1.2.m2.1.1">subscript</csymbol><ci id="S2.SS1.p1.2.m2.1.1.2.cmml" xref="S2.SS1.p1.2.m2.1.1.2">𝚺</ci><ci id="S2.SS1.p1.2.m2.1.1.3.cmml" xref="S2.SS1.p1.2.m2.1.1.3">𝑘</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p1.2.m2.1c">\operatorname{{\mathbf{\Sigma}}}_{k}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p1.2.m2.1d">bold_Σ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT</annotation></semantics></math> for <math alttext="k=1,2" class="ltx_Math" display="inline" id="S2.SS1.p1.3.m3.2"><semantics id="S2.SS1.p1.3.m3.2a"><mrow id="S2.SS1.p1.3.m3.2.3" xref="S2.SS1.p1.3.m3.2.3.cmml"><mi id="S2.SS1.p1.3.m3.2.3.2" xref="S2.SS1.p1.3.m3.2.3.2.cmml">k</mi><mo id="S2.SS1.p1.3.m3.2.3.1" xref="S2.SS1.p1.3.m3.2.3.1.cmml">=</mo><mrow id="S2.SS1.p1.3.m3.2.3.3.2" xref="S2.SS1.p1.3.m3.2.3.3.1.cmml"><mn id="S2.SS1.p1.3.m3.1.1" xref="S2.SS1.p1.3.m3.1.1.cmml">1</mn><mo id="S2.SS1.p1.3.m3.2.3.3.2.1" xref="S2.SS1.p1.3.m3.2.3.3.1.cmml">,</mo><mn id="S2.SS1.p1.3.m3.2.2" xref="S2.SS1.p1.3.m3.2.2.cmml">2</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p1.3.m3.2b"><apply id="S2.SS1.p1.3.m3.2.3.cmml" xref="S2.SS1.p1.3.m3.2.3"><eq id="S2.SS1.p1.3.m3.2.3.1.cmml" xref="S2.SS1.p1.3.m3.2.3.1"></eq><ci id="S2.SS1.p1.3.m3.2.3.2.cmml" xref="S2.SS1.p1.3.m3.2.3.2">𝑘</ci><list id="S2.SS1.p1.3.m3.2.3.3.1.cmml" xref="S2.SS1.p1.3.m3.2.3.3.2"><cn id="S2.SS1.p1.3.m3.1.1.cmml" type="integer" xref="S2.SS1.p1.3.m3.1.1">1</cn><cn id="S2.SS1.p1.3.m3.2.2.cmml" type="integer" xref="S2.SS1.p1.3.m3.2.2">2</cn></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p1.3.m3.2c">k=1,2</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p1.3.m3.2d">italic_k = 1 , 2</annotation></semantics></math>. If <math alttext="\operatorname{\mathbf{L}}_{k}" class="ltx_Math" display="inline" id="S2.SS1.p1.4.m4.1"><semantics id="S2.SS1.p1.4.m4.1a"><msub id="S2.SS1.p1.4.m4.1.1" xref="S2.SS1.p1.4.m4.1.1.cmml"><mi id="S2.SS1.p1.4.m4.1.1.2" xref="S2.SS1.p1.4.m4.1.1.2.cmml">𝐋</mi><mi id="S2.SS1.p1.4.m4.1.1.3" xref="S2.SS1.p1.4.m4.1.1.3.cmml">k</mi></msub><annotation-xml encoding="MathML-Content" id="S2.SS1.p1.4.m4.1b"><apply id="S2.SS1.p1.4.m4.1.1.cmml" xref="S2.SS1.p1.4.m4.1.1"><csymbol cd="ambiguous" id="S2.SS1.p1.4.m4.1.1.1.cmml" xref="S2.SS1.p1.4.m4.1.1">subscript</csymbol><ci id="S2.SS1.p1.4.m4.1.1.2.cmml" xref="S2.SS1.p1.4.m4.1.1.2">𝐋</ci><ci id="S2.SS1.p1.4.m4.1.1.3.cmml" xref="S2.SS1.p1.4.m4.1.1.3">𝑘</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p1.4.m4.1c">\operatorname{\mathbf{L}}_{k}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p1.4.m4.1d">bold_L start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT</annotation></semantics></math> has full rank for <math alttext="k=1,2" class="ltx_Math" display="inline" id="S2.SS1.p1.5.m5.2"><semantics id="S2.SS1.p1.5.m5.2a"><mrow id="S2.SS1.p1.5.m5.2.3" xref="S2.SS1.p1.5.m5.2.3.cmml"><mi id="S2.SS1.p1.5.m5.2.3.2" xref="S2.SS1.p1.5.m5.2.3.2.cmml">k</mi><mo id="S2.SS1.p1.5.m5.2.3.1" xref="S2.SS1.p1.5.m5.2.3.1.cmml">=</mo><mrow id="S2.SS1.p1.5.m5.2.3.3.2" xref="S2.SS1.p1.5.m5.2.3.3.1.cmml"><mn id="S2.SS1.p1.5.m5.1.1" xref="S2.SS1.p1.5.m5.1.1.cmml">1</mn><mo id="S2.SS1.p1.5.m5.2.3.3.2.1" xref="S2.SS1.p1.5.m5.2.3.3.1.cmml">,</mo><mn id="S2.SS1.p1.5.m5.2.2" xref="S2.SS1.p1.5.m5.2.2.cmml">2</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p1.5.m5.2b"><apply id="S2.SS1.p1.5.m5.2.3.cmml" xref="S2.SS1.p1.5.m5.2.3"><eq id="S2.SS1.p1.5.m5.2.3.1.cmml" xref="S2.SS1.p1.5.m5.2.3.1"></eq><ci id="S2.SS1.p1.5.m5.2.3.2.cmml" xref="S2.SS1.p1.5.m5.2.3.2">𝑘</ci><list id="S2.SS1.p1.5.m5.2.3.3.1.cmml" xref="S2.SS1.p1.5.m5.2.3.3.2"><cn id="S2.SS1.p1.5.m5.1.1.cmml" type="integer" xref="S2.SS1.p1.5.m5.1.1">1</cn><cn id="S2.SS1.p1.5.m5.2.2.cmml" type="integer" xref="S2.SS1.p1.5.m5.2.2">2</cn></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p1.5.m5.2c">k=1,2</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p1.5.m5.2d">italic_k = 1 , 2</annotation></semantics></math>, then <math alttext="Q_{k}=S_{k}" class="ltx_Math" display="inline" id="S2.SS1.p1.6.m6.1"><semantics id="S2.SS1.p1.6.m6.1a"><mrow id="S2.SS1.p1.6.m6.1.1" xref="S2.SS1.p1.6.m6.1.1.cmml"><msub id="S2.SS1.p1.6.m6.1.1.2" xref="S2.SS1.p1.6.m6.1.1.2.cmml"><mi id="S2.SS1.p1.6.m6.1.1.2.2" xref="S2.SS1.p1.6.m6.1.1.2.2.cmml">Q</mi><mi id="S2.SS1.p1.6.m6.1.1.2.3" xref="S2.SS1.p1.6.m6.1.1.2.3.cmml">k</mi></msub><mo id="S2.SS1.p1.6.m6.1.1.1" xref="S2.SS1.p1.6.m6.1.1.1.cmml">=</mo><msub id="S2.SS1.p1.6.m6.1.1.3" xref="S2.SS1.p1.6.m6.1.1.3.cmml"><mi id="S2.SS1.p1.6.m6.1.1.3.2" xref="S2.SS1.p1.6.m6.1.1.3.2.cmml">S</mi><mi id="S2.SS1.p1.6.m6.1.1.3.3" xref="S2.SS1.p1.6.m6.1.1.3.3.cmml">k</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p1.6.m6.1b"><apply id="S2.SS1.p1.6.m6.1.1.cmml" xref="S2.SS1.p1.6.m6.1.1"><eq id="S2.SS1.p1.6.m6.1.1.1.cmml" xref="S2.SS1.p1.6.m6.1.1.1"></eq><apply id="S2.SS1.p1.6.m6.1.1.2.cmml" xref="S2.SS1.p1.6.m6.1.1.2"><csymbol cd="ambiguous" id="S2.SS1.p1.6.m6.1.1.2.1.cmml" xref="S2.SS1.p1.6.m6.1.1.2">subscript</csymbol><ci id="S2.SS1.p1.6.m6.1.1.2.2.cmml" xref="S2.SS1.p1.6.m6.1.1.2.2">𝑄</ci><ci id="S2.SS1.p1.6.m6.1.1.2.3.cmml" xref="S2.SS1.p1.6.m6.1.1.2.3">𝑘</ci></apply><apply id="S2.SS1.p1.6.m6.1.1.3.cmml" xref="S2.SS1.p1.6.m6.1.1.3"><csymbol cd="ambiguous" id="S2.SS1.p1.6.m6.1.1.3.1.cmml" xref="S2.SS1.p1.6.m6.1.1.3">subscript</csymbol><ci id="S2.SS1.p1.6.m6.1.1.3.2.cmml" xref="S2.SS1.p1.6.m6.1.1.3.2">𝑆</ci><ci id="S2.SS1.p1.6.m6.1.1.3.3.cmml" xref="S2.SS1.p1.6.m6.1.1.3.3">𝑘</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p1.6.m6.1c">Q_{k}=S_{k}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p1.6.m6.1d">italic_Q start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT = italic_S start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="\operatorname{{\mathbf{\Sigma}}}_{k}=\operatorname{\mathbf{L}}_{k}% \operatorname{\mathbf{L}}_{k}^{\top}" class="ltx_Math" display="inline" id="S2.SS1.p1.7.m7.1"><semantics id="S2.SS1.p1.7.m7.1a"><mrow id="S2.SS1.p1.7.m7.1.1" xref="S2.SS1.p1.7.m7.1.1.cmml"><msub id="S2.SS1.p1.7.m7.1.1.2" xref="S2.SS1.p1.7.m7.1.1.2.cmml"><mi id="S2.SS1.p1.7.m7.1.1.2.2" xref="S2.SS1.p1.7.m7.1.1.2.2.cmml">𝚺</mi><mi id="S2.SS1.p1.7.m7.1.1.2.3" xref="S2.SS1.p1.7.m7.1.1.2.3.cmml">k</mi></msub><mo id="S2.SS1.p1.7.m7.1.1.1" xref="S2.SS1.p1.7.m7.1.1.1.cmml">=</mo><mrow id="S2.SS1.p1.7.m7.1.1.3" xref="S2.SS1.p1.7.m7.1.1.3.cmml"><msub id="S2.SS1.p1.7.m7.1.1.3.1" xref="S2.SS1.p1.7.m7.1.1.3.1.cmml"><mi id="S2.SS1.p1.7.m7.1.1.3.1.2" xref="S2.SS1.p1.7.m7.1.1.3.1.2.cmml">𝐋</mi><mi id="S2.SS1.p1.7.m7.1.1.3.1.3" xref="S2.SS1.p1.7.m7.1.1.3.1.3.cmml">k</mi></msub><mo id="S2.SS1.p1.7.m7.1.1.3a" lspace="0.167em" xref="S2.SS1.p1.7.m7.1.1.3.cmml">⁡</mo><msubsup id="S2.SS1.p1.7.m7.1.1.3.2" xref="S2.SS1.p1.7.m7.1.1.3.2.cmml"><mi id="S2.SS1.p1.7.m7.1.1.3.2.2.2" xref="S2.SS1.p1.7.m7.1.1.3.2.2.2.cmml">𝐋</mi><mi id="S2.SS1.p1.7.m7.1.1.3.2.2.3" xref="S2.SS1.p1.7.m7.1.1.3.2.2.3.cmml">k</mi><mo id="S2.SS1.p1.7.m7.1.1.3.2.3" xref="S2.SS1.p1.7.m7.1.1.3.2.3.cmml">⊤</mo></msubsup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p1.7.m7.1b"><apply id="S2.SS1.p1.7.m7.1.1.cmml" xref="S2.SS1.p1.7.m7.1.1"><eq id="S2.SS1.p1.7.m7.1.1.1.cmml" xref="S2.SS1.p1.7.m7.1.1.1"></eq><apply id="S2.SS1.p1.7.m7.1.1.2.cmml" xref="S2.SS1.p1.7.m7.1.1.2"><csymbol cd="ambiguous" id="S2.SS1.p1.7.m7.1.1.2.1.cmml" xref="S2.SS1.p1.7.m7.1.1.2">subscript</csymbol><ci id="S2.SS1.p1.7.m7.1.1.2.2.cmml" xref="S2.SS1.p1.7.m7.1.1.2.2">𝚺</ci><ci id="S2.SS1.p1.7.m7.1.1.2.3.cmml" xref="S2.SS1.p1.7.m7.1.1.2.3">𝑘</ci></apply><apply id="S2.SS1.p1.7.m7.1.1.3.cmml" xref="S2.SS1.p1.7.m7.1.1.3"><apply id="S2.SS1.p1.7.m7.1.1.3.1.cmml" xref="S2.SS1.p1.7.m7.1.1.3.1"><csymbol cd="ambiguous" id="S2.SS1.p1.7.m7.1.1.3.1.1.cmml" xref="S2.SS1.p1.7.m7.1.1.3.1">subscript</csymbol><ci id="S2.SS1.p1.7.m7.1.1.3.1.2.cmml" xref="S2.SS1.p1.7.m7.1.1.3.1.2">𝐋</ci><ci id="S2.SS1.p1.7.m7.1.1.3.1.3.cmml" xref="S2.SS1.p1.7.m7.1.1.3.1.3">𝑘</ci></apply><apply id="S2.SS1.p1.7.m7.1.1.3.2.cmml" xref="S2.SS1.p1.7.m7.1.1.3.2"><csymbol cd="ambiguous" id="S2.SS1.p1.7.m7.1.1.3.2.1.cmml" xref="S2.SS1.p1.7.m7.1.1.3.2">superscript</csymbol><apply id="S2.SS1.p1.7.m7.1.1.3.2.2.cmml" xref="S2.SS1.p1.7.m7.1.1.3.2"><csymbol cd="ambiguous" id="S2.SS1.p1.7.m7.1.1.3.2.2.1.cmml" xref="S2.SS1.p1.7.m7.1.1.3.2">subscript</csymbol><ci id="S2.SS1.p1.7.m7.1.1.3.2.2.2.cmml" xref="S2.SS1.p1.7.m7.1.1.3.2.2.2">𝐋</ci><ci id="S2.SS1.p1.7.m7.1.1.3.2.2.3.cmml" xref="S2.SS1.p1.7.m7.1.1.3.2.2.3">𝑘</ci></apply><csymbol cd="latexml" id="S2.SS1.p1.7.m7.1.1.3.2.3.cmml" xref="S2.SS1.p1.7.m7.1.1.3.2.3">top</csymbol></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p1.7.m7.1c">\operatorname{{\mathbf{\Sigma}}}_{k}=\operatorname{\mathbf{L}}_{k}% \operatorname{\mathbf{L}}_{k}^{\top}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p1.7.m7.1d">bold_Σ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT = bold_L start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT bold_L start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ⊤ end_POSTSUPERSCRIPT</annotation></semantics></math>. MMTR imposes a low-rank structure on <math alttext="\operatorname{{\mathbf{\Sigma}}}_{1}" class="ltx_Math" display="inline" id="S2.SS1.p1.8.m8.1"><semantics id="S2.SS1.p1.8.m8.1a"><msub id="S2.SS1.p1.8.m8.1.1" xref="S2.SS1.p1.8.m8.1.1.cmml"><mi id="S2.SS1.p1.8.m8.1.1.2" xref="S2.SS1.p1.8.m8.1.1.2.cmml">𝚺</mi><mn id="S2.SS1.p1.8.m8.1.1.3" xref="S2.SS1.p1.8.m8.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S2.SS1.p1.8.m8.1b"><apply id="S2.SS1.p1.8.m8.1.1.cmml" xref="S2.SS1.p1.8.m8.1.1"><csymbol cd="ambiguous" id="S2.SS1.p1.8.m8.1.1.1.cmml" xref="S2.SS1.p1.8.m8.1.1">subscript</csymbol><ci id="S2.SS1.p1.8.m8.1.1.2.cmml" xref="S2.SS1.p1.8.m8.1.1.2">𝚺</ci><cn id="S2.SS1.p1.8.m8.1.1.3.cmml" type="integer" xref="S2.SS1.p1.8.m8.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p1.8.m8.1c">\operatorname{{\mathbf{\Sigma}}}_{1}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p1.8.m8.1d">bold_Σ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="\operatorname{{\mathbf{\Sigma}}}_{2}" class="ltx_Math" display="inline" id="S2.SS1.p1.9.m9.1"><semantics id="S2.SS1.p1.9.m9.1a"><msub id="S2.SS1.p1.9.m9.1.1" xref="S2.SS1.p1.9.m9.1.1.cmml"><mi id="S2.SS1.p1.9.m9.1.1.2" xref="S2.SS1.p1.9.m9.1.1.2.cmml">𝚺</mi><mn id="S2.SS1.p1.9.m9.1.1.3" xref="S2.SS1.p1.9.m9.1.1.3.cmml">2</mn></msub><annotation-xml encoding="MathML-Content" id="S2.SS1.p1.9.m9.1b"><apply id="S2.SS1.p1.9.m9.1.1.cmml" xref="S2.SS1.p1.9.m9.1.1"><csymbol cd="ambiguous" id="S2.SS1.p1.9.m9.1.1.1.cmml" xref="S2.SS1.p1.9.m9.1.1">subscript</csymbol><ci id="S2.SS1.p1.9.m9.1.1.2.cmml" xref="S2.SS1.p1.9.m9.1.1.2">𝚺</ci><cn id="S2.SS1.p1.9.m9.1.1.3.cmml" type="integer" xref="S2.SS1.p1.9.m9.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p1.9.m9.1c">\operatorname{{\mathbf{\Sigma}}}_{2}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p1.9.m9.1d">bold_Σ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math> by assuming that <math alttext="S_{1}\ll Q_{1}" class="ltx_Math" display="inline" id="S2.SS1.p1.10.m10.1"><semantics id="S2.SS1.p1.10.m10.1a"><mrow id="S2.SS1.p1.10.m10.1.1" xref="S2.SS1.p1.10.m10.1.1.cmml"><msub id="S2.SS1.p1.10.m10.1.1.2" xref="S2.SS1.p1.10.m10.1.1.2.cmml"><mi id="S2.SS1.p1.10.m10.1.1.2.2" xref="S2.SS1.p1.10.m10.1.1.2.2.cmml">S</mi><mn id="S2.SS1.p1.10.m10.1.1.2.3" xref="S2.SS1.p1.10.m10.1.1.2.3.cmml">1</mn></msub><mo id="S2.SS1.p1.10.m10.1.1.1" xref="S2.SS1.p1.10.m10.1.1.1.cmml">≪</mo><msub id="S2.SS1.p1.10.m10.1.1.3" xref="S2.SS1.p1.10.m10.1.1.3.cmml"><mi id="S2.SS1.p1.10.m10.1.1.3.2" xref="S2.SS1.p1.10.m10.1.1.3.2.cmml">Q</mi><mn id="S2.SS1.p1.10.m10.1.1.3.3" xref="S2.SS1.p1.10.m10.1.1.3.3.cmml">1</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p1.10.m10.1b"><apply id="S2.SS1.p1.10.m10.1.1.cmml" xref="S2.SS1.p1.10.m10.1.1"><csymbol cd="latexml" id="S2.SS1.p1.10.m10.1.1.1.cmml" xref="S2.SS1.p1.10.m10.1.1.1">much-less-than</csymbol><apply id="S2.SS1.p1.10.m10.1.1.2.cmml" xref="S2.SS1.p1.10.m10.1.1.2"><csymbol cd="ambiguous" id="S2.SS1.p1.10.m10.1.1.2.1.cmml" xref="S2.SS1.p1.10.m10.1.1.2">subscript</csymbol><ci id="S2.SS1.p1.10.m10.1.1.2.2.cmml" xref="S2.SS1.p1.10.m10.1.1.2.2">𝑆</ci><cn id="S2.SS1.p1.10.m10.1.1.2.3.cmml" type="integer" xref="S2.SS1.p1.10.m10.1.1.2.3">1</cn></apply><apply id="S2.SS1.p1.10.m10.1.1.3.cmml" xref="S2.SS1.p1.10.m10.1.1.3"><csymbol cd="ambiguous" id="S2.SS1.p1.10.m10.1.1.3.1.cmml" xref="S2.SS1.p1.10.m10.1.1.3">subscript</csymbol><ci id="S2.SS1.p1.10.m10.1.1.3.2.cmml" xref="S2.SS1.p1.10.m10.1.1.3.2">𝑄</ci><cn id="S2.SS1.p1.10.m10.1.1.3.3.cmml" type="integer" xref="S2.SS1.p1.10.m10.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p1.10.m10.1c">S_{1}\ll Q_{1}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p1.10.m10.1d">italic_S start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ≪ italic_Q start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="S_{2}\ll Q_{2}" class="ltx_Math" display="inline" id="S2.SS1.p1.11.m11.1"><semantics id="S2.SS1.p1.11.m11.1a"><mrow id="S2.SS1.p1.11.m11.1.1" xref="S2.SS1.p1.11.m11.1.1.cmml"><msub id="S2.SS1.p1.11.m11.1.1.2" xref="S2.SS1.p1.11.m11.1.1.2.cmml"><mi id="S2.SS1.p1.11.m11.1.1.2.2" xref="S2.SS1.p1.11.m11.1.1.2.2.cmml">S</mi><mn id="S2.SS1.p1.11.m11.1.1.2.3" xref="S2.SS1.p1.11.m11.1.1.2.3.cmml">2</mn></msub><mo id="S2.SS1.p1.11.m11.1.1.1" xref="S2.SS1.p1.11.m11.1.1.1.cmml">≪</mo><msub id="S2.SS1.p1.11.m11.1.1.3" xref="S2.SS1.p1.11.m11.1.1.3.cmml"><mi id="S2.SS1.p1.11.m11.1.1.3.2" xref="S2.SS1.p1.11.m11.1.1.3.2.cmml">Q</mi><mn id="S2.SS1.p1.11.m11.1.1.3.3" xref="S2.SS1.p1.11.m11.1.1.3.3.cmml">2</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p1.11.m11.1b"><apply id="S2.SS1.p1.11.m11.1.1.cmml" xref="S2.SS1.p1.11.m11.1.1"><csymbol cd="latexml" id="S2.SS1.p1.11.m11.1.1.1.cmml" xref="S2.SS1.p1.11.m11.1.1.1">much-less-than</csymbol><apply id="S2.SS1.p1.11.m11.1.1.2.cmml" xref="S2.SS1.p1.11.m11.1.1.2"><csymbol cd="ambiguous" id="S2.SS1.p1.11.m11.1.1.2.1.cmml" xref="S2.SS1.p1.11.m11.1.1.2">subscript</csymbol><ci id="S2.SS1.p1.11.m11.1.1.2.2.cmml" xref="S2.SS1.p1.11.m11.1.1.2.2">𝑆</ci><cn id="S2.SS1.p1.11.m11.1.1.2.3.cmml" type="integer" xref="S2.SS1.p1.11.m11.1.1.2.3">2</cn></apply><apply id="S2.SS1.p1.11.m11.1.1.3.cmml" xref="S2.SS1.p1.11.m11.1.1.3"><csymbol cd="ambiguous" id="S2.SS1.p1.11.m11.1.1.3.1.cmml" xref="S2.SS1.p1.11.m11.1.1.3">subscript</csymbol><ci id="S2.SS1.p1.11.m11.1.1.3.2.cmml" xref="S2.SS1.p1.11.m11.1.1.3.2">𝑄</ci><cn id="S2.SS1.p1.11.m11.1.1.3.3.cmml" type="integer" xref="S2.SS1.p1.11.m11.1.1.3.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p1.11.m11.1c">S_{2}\ll Q_{2}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p1.11.m11.1d">italic_S start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ≪ italic_Q start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math>, so that <math alttext="\operatorname{\mathbf{L}}_{1}\operatorname{\mathbf{L}}_{1}^{\top}\approx% \operatorname{{\mathbf{\Sigma}}}_{1}" class="ltx_Math" display="inline" id="S2.SS1.p1.12.m12.1"><semantics id="S2.SS1.p1.12.m12.1a"><mrow id="S2.SS1.p1.12.m12.1.1" xref="S2.SS1.p1.12.m12.1.1.cmml"><mrow id="S2.SS1.p1.12.m12.1.1.2" xref="S2.SS1.p1.12.m12.1.1.2.cmml"><msub id="S2.SS1.p1.12.m12.1.1.2.1" xref="S2.SS1.p1.12.m12.1.1.2.1.cmml"><mi id="S2.SS1.p1.12.m12.1.1.2.1.2" xref="S2.SS1.p1.12.m12.1.1.2.1.2.cmml">𝐋</mi><mn id="S2.SS1.p1.12.m12.1.1.2.1.3" xref="S2.SS1.p1.12.m12.1.1.2.1.3.cmml">1</mn></msub><mo id="S2.SS1.p1.12.m12.1.1.2a" lspace="0.167em" xref="S2.SS1.p1.12.m12.1.1.2.cmml">⁡</mo><msubsup id="S2.SS1.p1.12.m12.1.1.2.2" xref="S2.SS1.p1.12.m12.1.1.2.2.cmml"><mi id="S2.SS1.p1.12.m12.1.1.2.2.2.2" xref="S2.SS1.p1.12.m12.1.1.2.2.2.2.cmml">𝐋</mi><mn id="S2.SS1.p1.12.m12.1.1.2.2.2.3" xref="S2.SS1.p1.12.m12.1.1.2.2.2.3.cmml">1</mn><mo id="S2.SS1.p1.12.m12.1.1.2.2.3" xref="S2.SS1.p1.12.m12.1.1.2.2.3.cmml">⊤</mo></msubsup></mrow><mo id="S2.SS1.p1.12.m12.1.1.1" xref="S2.SS1.p1.12.m12.1.1.1.cmml">≈</mo><msub id="S2.SS1.p1.12.m12.1.1.3" xref="S2.SS1.p1.12.m12.1.1.3.cmml"><mi id="S2.SS1.p1.12.m12.1.1.3.2" xref="S2.SS1.p1.12.m12.1.1.3.2.cmml">𝚺</mi><mn id="S2.SS1.p1.12.m12.1.1.3.3" xref="S2.SS1.p1.12.m12.1.1.3.3.cmml">1</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p1.12.m12.1b"><apply id="S2.SS1.p1.12.m12.1.1.cmml" xref="S2.SS1.p1.12.m12.1.1"><approx id="S2.SS1.p1.12.m12.1.1.1.cmml" xref="S2.SS1.p1.12.m12.1.1.1"></approx><apply id="S2.SS1.p1.12.m12.1.1.2.cmml" xref="S2.SS1.p1.12.m12.1.1.2"><apply id="S2.SS1.p1.12.m12.1.1.2.1.cmml" xref="S2.SS1.p1.12.m12.1.1.2.1"><csymbol cd="ambiguous" id="S2.SS1.p1.12.m12.1.1.2.1.1.cmml" xref="S2.SS1.p1.12.m12.1.1.2.1">subscript</csymbol><ci id="S2.SS1.p1.12.m12.1.1.2.1.2.cmml" xref="S2.SS1.p1.12.m12.1.1.2.1.2">𝐋</ci><cn id="S2.SS1.p1.12.m12.1.1.2.1.3.cmml" type="integer" xref="S2.SS1.p1.12.m12.1.1.2.1.3">1</cn></apply><apply id="S2.SS1.p1.12.m12.1.1.2.2.cmml" xref="S2.SS1.p1.12.m12.1.1.2.2"><csymbol cd="ambiguous" id="S2.SS1.p1.12.m12.1.1.2.2.1.cmml" xref="S2.SS1.p1.12.m12.1.1.2.2">superscript</csymbol><apply id="S2.SS1.p1.12.m12.1.1.2.2.2.cmml" xref="S2.SS1.p1.12.m12.1.1.2.2"><csymbol cd="ambiguous" id="S2.SS1.p1.12.m12.1.1.2.2.2.1.cmml" xref="S2.SS1.p1.12.m12.1.1.2.2">subscript</csymbol><ci id="S2.SS1.p1.12.m12.1.1.2.2.2.2.cmml" xref="S2.SS1.p1.12.m12.1.1.2.2.2.2">𝐋</ci><cn id="S2.SS1.p1.12.m12.1.1.2.2.2.3.cmml" type="integer" xref="S2.SS1.p1.12.m12.1.1.2.2.2.3">1</cn></apply><csymbol cd="latexml" id="S2.SS1.p1.12.m12.1.1.2.2.3.cmml" xref="S2.SS1.p1.12.m12.1.1.2.2.3">top</csymbol></apply></apply><apply id="S2.SS1.p1.12.m12.1.1.3.cmml" xref="S2.SS1.p1.12.m12.1.1.3"><csymbol cd="ambiguous" id="S2.SS1.p1.12.m12.1.1.3.1.cmml" xref="S2.SS1.p1.12.m12.1.1.3">subscript</csymbol><ci id="S2.SS1.p1.12.m12.1.1.3.2.cmml" xref="S2.SS1.p1.12.m12.1.1.3.2">𝚺</ci><cn id="S2.SS1.p1.12.m12.1.1.3.3.cmml" type="integer" xref="S2.SS1.p1.12.m12.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p1.12.m12.1c">\operatorname{\mathbf{L}}_{1}\operatorname{\mathbf{L}}_{1}^{\top}\approx% \operatorname{{\mathbf{\Sigma}}}_{1}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p1.12.m12.1d">bold_L start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT bold_L start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ⊤ end_POSTSUPERSCRIPT ≈ bold_Σ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="\operatorname{\mathbf{L}}_{2}\operatorname{\mathbf{L}}_{2}^{\top}\approx% \operatorname{{\mathbf{\Sigma}}}_{2}" class="ltx_Math" display="inline" id="S2.SS1.p1.13.m13.1"><semantics id="S2.SS1.p1.13.m13.1a"><mrow id="S2.SS1.p1.13.m13.1.1" xref="S2.SS1.p1.13.m13.1.1.cmml"><mrow id="S2.SS1.p1.13.m13.1.1.2" xref="S2.SS1.p1.13.m13.1.1.2.cmml"><msub id="S2.SS1.p1.13.m13.1.1.2.1" xref="S2.SS1.p1.13.m13.1.1.2.1.cmml"><mi id="S2.SS1.p1.13.m13.1.1.2.1.2" xref="S2.SS1.p1.13.m13.1.1.2.1.2.cmml">𝐋</mi><mn id="S2.SS1.p1.13.m13.1.1.2.1.3" xref="S2.SS1.p1.13.m13.1.1.2.1.3.cmml">2</mn></msub><mo id="S2.SS1.p1.13.m13.1.1.2a" lspace="0.167em" xref="S2.SS1.p1.13.m13.1.1.2.cmml">⁡</mo><msubsup id="S2.SS1.p1.13.m13.1.1.2.2" xref="S2.SS1.p1.13.m13.1.1.2.2.cmml"><mi id="S2.SS1.p1.13.m13.1.1.2.2.2.2" xref="S2.SS1.p1.13.m13.1.1.2.2.2.2.cmml">𝐋</mi><mn id="S2.SS1.p1.13.m13.1.1.2.2.2.3" xref="S2.SS1.p1.13.m13.1.1.2.2.2.3.cmml">2</mn><mo id="S2.SS1.p1.13.m13.1.1.2.2.3" xref="S2.SS1.p1.13.m13.1.1.2.2.3.cmml">⊤</mo></msubsup></mrow><mo id="S2.SS1.p1.13.m13.1.1.1" xref="S2.SS1.p1.13.m13.1.1.1.cmml">≈</mo><msub id="S2.SS1.p1.13.m13.1.1.3" xref="S2.SS1.p1.13.m13.1.1.3.cmml"><mi id="S2.SS1.p1.13.m13.1.1.3.2" xref="S2.SS1.p1.13.m13.1.1.3.2.cmml">𝚺</mi><mn id="S2.SS1.p1.13.m13.1.1.3.3" xref="S2.SS1.p1.13.m13.1.1.3.3.cmml">2</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p1.13.m13.1b"><apply id="S2.SS1.p1.13.m13.1.1.cmml" xref="S2.SS1.p1.13.m13.1.1"><approx id="S2.SS1.p1.13.m13.1.1.1.cmml" xref="S2.SS1.p1.13.m13.1.1.1"></approx><apply id="S2.SS1.p1.13.m13.1.1.2.cmml" xref="S2.SS1.p1.13.m13.1.1.2"><apply id="S2.SS1.p1.13.m13.1.1.2.1.cmml" xref="S2.SS1.p1.13.m13.1.1.2.1"><csymbol cd="ambiguous" id="S2.SS1.p1.13.m13.1.1.2.1.1.cmml" xref="S2.SS1.p1.13.m13.1.1.2.1">subscript</csymbol><ci id="S2.SS1.p1.13.m13.1.1.2.1.2.cmml" xref="S2.SS1.p1.13.m13.1.1.2.1.2">𝐋</ci><cn id="S2.SS1.p1.13.m13.1.1.2.1.3.cmml" type="integer" xref="S2.SS1.p1.13.m13.1.1.2.1.3">2</cn></apply><apply id="S2.SS1.p1.13.m13.1.1.2.2.cmml" xref="S2.SS1.p1.13.m13.1.1.2.2"><csymbol cd="ambiguous" id="S2.SS1.p1.13.m13.1.1.2.2.1.cmml" xref="S2.SS1.p1.13.m13.1.1.2.2">superscript</csymbol><apply id="S2.SS1.p1.13.m13.1.1.2.2.2.cmml" xref="S2.SS1.p1.13.m13.1.1.2.2"><csymbol cd="ambiguous" id="S2.SS1.p1.13.m13.1.1.2.2.2.1.cmml" xref="S2.SS1.p1.13.m13.1.1.2.2">subscript</csymbol><ci id="S2.SS1.p1.13.m13.1.1.2.2.2.2.cmml" xref="S2.SS1.p1.13.m13.1.1.2.2.2.2">𝐋</ci><cn id="S2.SS1.p1.13.m13.1.1.2.2.2.3.cmml" type="integer" xref="S2.SS1.p1.13.m13.1.1.2.2.2.3">2</cn></apply><csymbol cd="latexml" id="S2.SS1.p1.13.m13.1.1.2.2.3.cmml" xref="S2.SS1.p1.13.m13.1.1.2.2.3">top</csymbol></apply></apply><apply id="S2.SS1.p1.13.m13.1.1.3.cmml" xref="S2.SS1.p1.13.m13.1.1.3"><csymbol cd="ambiguous" id="S2.SS1.p1.13.m13.1.1.3.1.cmml" xref="S2.SS1.p1.13.m13.1.1.3">subscript</csymbol><ci id="S2.SS1.p1.13.m13.1.1.3.2.cmml" xref="S2.SS1.p1.13.m13.1.1.3.2">𝚺</ci><cn id="S2.SS1.p1.13.m13.1.1.3.3.cmml" type="integer" xref="S2.SS1.p1.13.m13.1.1.3.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p1.13.m13.1c">\operatorname{\mathbf{L}}_{2}\operatorname{\mathbf{L}}_{2}^{\top}\approx% \operatorname{{\mathbf{\Sigma}}}_{2}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p1.13.m13.1d">bold_L start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT bold_L start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ⊤ end_POSTSUPERSCRIPT ≈ bold_Σ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math>. The random effects are expressed using <math alttext="\operatorname{\mathbf{L}}_{1}" class="ltx_Math" display="inline" id="S2.SS1.p1.14.m14.1"><semantics id="S2.SS1.p1.14.m14.1a"><msub id="S2.SS1.p1.14.m14.1.1" xref="S2.SS1.p1.14.m14.1.1.cmml"><mi id="S2.SS1.p1.14.m14.1.1.2" xref="S2.SS1.p1.14.m14.1.1.2.cmml">𝐋</mi><mn id="S2.SS1.p1.14.m14.1.1.3" xref="S2.SS1.p1.14.m14.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S2.SS1.p1.14.m14.1b"><apply id="S2.SS1.p1.14.m14.1.1.cmml" xref="S2.SS1.p1.14.m14.1.1"><csymbol cd="ambiguous" id="S2.SS1.p1.14.m14.1.1.1.cmml" xref="S2.SS1.p1.14.m14.1.1">subscript</csymbol><ci id="S2.SS1.p1.14.m14.1.1.2.cmml" xref="S2.SS1.p1.14.m14.1.1.2">𝐋</ci><cn id="S2.SS1.p1.14.m14.1.1.3.cmml" type="integer" xref="S2.SS1.p1.14.m14.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p1.14.m14.1c">\operatorname{\mathbf{L}}_{1}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p1.14.m14.1d">bold_L start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="\operatorname{\mathbf{L}}_{2}" class="ltx_Math" display="inline" id="S2.SS1.p1.15.m15.1"><semantics id="S2.SS1.p1.15.m15.1a"><msub id="S2.SS1.p1.15.m15.1.1" xref="S2.SS1.p1.15.m15.1.1.cmml"><mi id="S2.SS1.p1.15.m15.1.1.2" xref="S2.SS1.p1.15.m15.1.1.2.cmml">𝐋</mi><mn id="S2.SS1.p1.15.m15.1.1.3" xref="S2.SS1.p1.15.m15.1.1.3.cmml">2</mn></msub><annotation-xml encoding="MathML-Content" id="S2.SS1.p1.15.m15.1b"><apply id="S2.SS1.p1.15.m15.1.1.cmml" xref="S2.SS1.p1.15.m15.1.1"><csymbol cd="ambiguous" id="S2.SS1.p1.15.m15.1.1.1.cmml" xref="S2.SS1.p1.15.m15.1.1">subscript</csymbol><ci id="S2.SS1.p1.15.m15.1.1.2.cmml" xref="S2.SS1.p1.15.m15.1.1.2">𝐋</ci><cn id="S2.SS1.p1.15.m15.1.1.3.cmml" type="integer" xref="S2.SS1.p1.15.m15.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p1.15.m15.1c">\operatorname{\mathbf{L}}_{2}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p1.15.m15.1d">bold_L start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math> as</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="S3.EGx2"> <tbody id="S2.E2"><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{\mathbf{A}}_{i}=\operatorname{\mathbf{L}}_{1}% \operatorname{\mathbf{C}}_{i}\operatorname{\mathbf{L}}_{2}^{\top},\quad% \operatorname{\mathbf{C}}_{i}\sim N_{S_{1},S_{2}}(\operatorname{\mathbf{0}},% \tau^{2}\operatorname{\mathbf{I}}_{S_{1}},\operatorname{\mathbf{I}}_{S_{2}})," class="ltx_Math" display="inline" id="S2.E2.m1.4"><semantics id="S2.E2.m1.4a"><mrow id="S2.E2.m1.4.4.1"><mrow id="S2.E2.m1.4.4.1.1.2" xref="S2.E2.m1.4.4.1.1.3.cmml"><mrow id="S2.E2.m1.4.4.1.1.1.1" 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xref="S2.E2.m1.4.4.1.1.2.2.2.2.2.2.3">subscript</csymbol><ci id="S2.E2.m1.4.4.1.1.2.2.2.2.2.2.3.2.cmml" xref="S2.E2.m1.4.4.1.1.2.2.2.2.2.2.3.2">𝑆</ci><cn id="S2.E2.m1.4.4.1.1.2.2.2.2.2.2.3.3.cmml" type="integer" xref="S2.E2.m1.4.4.1.1.2.2.2.2.2.2.3.3">2</cn></apply></apply></vector></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.E2.m1.4c">\displaystyle\operatorname{\mathbf{A}}_{i}=\operatorname{\mathbf{L}}_{1}% \operatorname{\mathbf{C}}_{i}\operatorname{\mathbf{L}}_{2}^{\top},\quad% \operatorname{\mathbf{C}}_{i}\sim N_{S_{1},S_{2}}(\operatorname{\mathbf{0}},% \tau^{2}\operatorname{\mathbf{I}}_{S_{1}},\operatorname{\mathbf{I}}_{S_{2}}),</annotation><annotation encoding="application/x-llamapun" id="S2.E2.m1.4d">bold_A start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = bold_L start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT bold_C start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT bold_L start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ⊤ end_POSTSUPERSCRIPT , bold_C start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∼ italic_N start_POSTSUBSCRIPT italic_S start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_S start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( bold_0 , italic_τ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT bold_I start_POSTSUBSCRIPT italic_S start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT , bold_I start_POSTSUBSCRIPT italic_S start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT 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">(2)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S2.SS1.p1.18">where <math alttext="\operatorname{\mathbf{C}}_{i}" class="ltx_Math" display="inline" id="S2.SS1.p1.16.m1.1"><semantics id="S2.SS1.p1.16.m1.1a"><msub id="S2.SS1.p1.16.m1.1.1" xref="S2.SS1.p1.16.m1.1.1.cmml"><mi id="S2.SS1.p1.16.m1.1.1.2" xref="S2.SS1.p1.16.m1.1.1.2.cmml">𝐂</mi><mi id="S2.SS1.p1.16.m1.1.1.3" xref="S2.SS1.p1.16.m1.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S2.SS1.p1.16.m1.1b"><apply id="S2.SS1.p1.16.m1.1.1.cmml" xref="S2.SS1.p1.16.m1.1.1"><csymbol cd="ambiguous" id="S2.SS1.p1.16.m1.1.1.1.cmml" xref="S2.SS1.p1.16.m1.1.1">subscript</csymbol><ci id="S2.SS1.p1.16.m1.1.1.2.cmml" xref="S2.SS1.p1.16.m1.1.1.2">𝐂</ci><ci id="S2.SS1.p1.16.m1.1.1.3.cmml" xref="S2.SS1.p1.16.m1.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p1.16.m1.1c">\operatorname{\mathbf{C}}_{i}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p1.16.m1.1d">bold_C start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> is a <math alttext="S_{1}\times S_{2}" class="ltx_Math" display="inline" id="S2.SS1.p1.17.m2.1"><semantics id="S2.SS1.p1.17.m2.1a"><mrow id="S2.SS1.p1.17.m2.1.1" xref="S2.SS1.p1.17.m2.1.1.cmml"><msub id="S2.SS1.p1.17.m2.1.1.2" xref="S2.SS1.p1.17.m2.1.1.2.cmml"><mi id="S2.SS1.p1.17.m2.1.1.2.2" xref="S2.SS1.p1.17.m2.1.1.2.2.cmml">S</mi><mn id="S2.SS1.p1.17.m2.1.1.2.3" xref="S2.SS1.p1.17.m2.1.1.2.3.cmml">1</mn></msub><mo id="S2.SS1.p1.17.m2.1.1.1" lspace="0.222em" rspace="0.222em" xref="S2.SS1.p1.17.m2.1.1.1.cmml">×</mo><msub id="S2.SS1.p1.17.m2.1.1.3" xref="S2.SS1.p1.17.m2.1.1.3.cmml"><mi id="S2.SS1.p1.17.m2.1.1.3.2" xref="S2.SS1.p1.17.m2.1.1.3.2.cmml">S</mi><mn id="S2.SS1.p1.17.m2.1.1.3.3" xref="S2.SS1.p1.17.m2.1.1.3.3.cmml">2</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p1.17.m2.1b"><apply id="S2.SS1.p1.17.m2.1.1.cmml" xref="S2.SS1.p1.17.m2.1.1"><times id="S2.SS1.p1.17.m2.1.1.1.cmml" xref="S2.SS1.p1.17.m2.1.1.1"></times><apply id="S2.SS1.p1.17.m2.1.1.2.cmml" xref="S2.SS1.p1.17.m2.1.1.2"><csymbol cd="ambiguous" id="S2.SS1.p1.17.m2.1.1.2.1.cmml" xref="S2.SS1.p1.17.m2.1.1.2">subscript</csymbol><ci id="S2.SS1.p1.17.m2.1.1.2.2.cmml" xref="S2.SS1.p1.17.m2.1.1.2.2">𝑆</ci><cn id="S2.SS1.p1.17.m2.1.1.2.3.cmml" type="integer" xref="S2.SS1.p1.17.m2.1.1.2.3">1</cn></apply><apply id="S2.SS1.p1.17.m2.1.1.3.cmml" xref="S2.SS1.p1.17.m2.1.1.3"><csymbol cd="ambiguous" id="S2.SS1.p1.17.m2.1.1.3.1.cmml" xref="S2.SS1.p1.17.m2.1.1.3">subscript</csymbol><ci id="S2.SS1.p1.17.m2.1.1.3.2.cmml" xref="S2.SS1.p1.17.m2.1.1.3.2">𝑆</ci><cn id="S2.SS1.p1.17.m2.1.1.3.3.cmml" type="integer" xref="S2.SS1.p1.17.m2.1.1.3.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p1.17.m2.1c">S_{1}\times S_{2}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p1.17.m2.1d">italic_S start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT × italic_S start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math> matrix of independent normal random variables with mean 0 and variance <math alttext="\tau^{2}" class="ltx_Math" display="inline" id="S2.SS1.p1.18.m3.1"><semantics id="S2.SS1.p1.18.m3.1a"><msup id="S2.SS1.p1.18.m3.1.1" xref="S2.SS1.p1.18.m3.1.1.cmml"><mi id="S2.SS1.p1.18.m3.1.1.2" xref="S2.SS1.p1.18.m3.1.1.2.cmml">τ</mi><mn id="S2.SS1.p1.18.m3.1.1.3" xref="S2.SS1.p1.18.m3.1.1.3.cmml">2</mn></msup><annotation-xml encoding="MathML-Content" id="S2.SS1.p1.18.m3.1b"><apply id="S2.SS1.p1.18.m3.1.1.cmml" xref="S2.SS1.p1.18.m3.1.1"><csymbol cd="ambiguous" id="S2.SS1.p1.18.m3.1.1.1.cmml" xref="S2.SS1.p1.18.m3.1.1">superscript</csymbol><ci id="S2.SS1.p1.18.m3.1.1.2.cmml" xref="S2.SS1.p1.18.m3.1.1.2">𝜏</ci><cn id="S2.SS1.p1.18.m3.1.1.3.cmml" type="integer" xref="S2.SS1.p1.18.m3.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p1.18.m3.1c">\tau^{2}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p1.18.m3.1d">italic_τ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT</annotation></semantics></math>. Based on these assumptions, the parameter expanded form of MMTR in (<a class="ltx_ref" href="https://arxiv.org/html/2503.13782v1#S1.E1" title="In 1 Introduction ‣ Regularized Parameter Estimation in Mixed Model Trace Regression"><span class="ltx_text ltx_ref_tag">1</span></a>) is</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="S3.EGx3"> <tbody id="S2.E3"><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 y_{ij}=\operatorname{\mathop{\mathrm{tr}}}(\operatorname{\mathbf% {X}}_{ij}^{\top}\operatorname{\mathbf{B}})+\operatorname{\mathop{\mathrm{tr}}}% (\operatorname{\mathbf{Z}}_{ij}^{\top}\operatorname{\mathbf{L}}_{1}% \operatorname{\mathbf{C}}_{i}\operatorname{\mathbf{L}}_{2}^{\top})+e_{ij},% \quad e_{ij}\sim N(0,\tau^{2})." class="ltx_Math" display="inline" id="S2.E3.m1.4"><semantics id="S2.E3.m1.4a"><mrow id="S2.E3.m1.4.4.1"><mrow 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encoding="application/x-tex" id="S2.E3.m1.4c">\displaystyle y_{ij}=\operatorname{\mathop{\mathrm{tr}}}(\operatorname{\mathbf% {X}}_{ij}^{\top}\operatorname{\mathbf{B}})+\operatorname{\mathop{\mathrm{tr}}}% (\operatorname{\mathbf{Z}}_{ij}^{\top}\operatorname{\mathbf{L}}_{1}% \operatorname{\mathbf{C}}_{i}\operatorname{\mathbf{L}}_{2}^{\top})+e_{ij},% \quad e_{ij}\sim N(0,\tau^{2}).</annotation><annotation encoding="application/x-llamapun" id="S2.E3.m1.4d">italic_y start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT = roman_tr ( bold_X start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ⊤ end_POSTSUPERSCRIPT bold_B ) + roman_tr ( bold_Z start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ⊤ end_POSTSUPERSCRIPT bold_L start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT bold_C start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT bold_L start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ⊤ end_POSTSUPERSCRIPT ) + italic_e start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT , italic_e start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT ∼ italic_N ( 0 , italic_τ 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">(3)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S2.SS1.p1.29">The model parameters are <math alttext="\operatorname{{\bm{\theta}}}=(\operatorname{\mathbf{B}},\operatorname{\mathbf{% L}}_{1},\operatorname{\mathbf{L}}_{2},\tau^{2})" class="ltx_Math" display="inline" id="S2.SS1.p1.19.m1.4"><semantics id="S2.SS1.p1.19.m1.4a"><mrow id="S2.SS1.p1.19.m1.4.4" xref="S2.SS1.p1.19.m1.4.4.cmml"><mi id="S2.SS1.p1.19.m1.4.4.5" xref="S2.SS1.p1.19.m1.4.4.5.cmml">𝜽</mi><mo id="S2.SS1.p1.19.m1.4.4.4" xref="S2.SS1.p1.19.m1.4.4.4.cmml">=</mo><mrow id="S2.SS1.p1.19.m1.4.4.3.3" 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id="S2.SS1.p1.19.m1.4c">\operatorname{{\bm{\theta}}}=(\operatorname{\mathbf{B}},\operatorname{\mathbf{% L}}_{1},\operatorname{\mathbf{L}}_{2},\tau^{2})</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p1.19.m1.4d">bold_italic_θ = ( bold_B , bold_L start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , bold_L start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , italic_τ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT )</annotation></semantics></math>, and <math alttext="\operatorname{\mathbf{L}}_{k}" class="ltx_Math" display="inline" id="S2.SS1.p1.20.m2.1"><semantics id="S2.SS1.p1.20.m2.1a"><msub id="S2.SS1.p1.20.m2.1.1" xref="S2.SS1.p1.20.m2.1.1.cmml"><mi id="S2.SS1.p1.20.m2.1.1.2" xref="S2.SS1.p1.20.m2.1.1.2.cmml">𝐋</mi><mi id="S2.SS1.p1.20.m2.1.1.3" xref="S2.SS1.p1.20.m2.1.1.3.cmml">k</mi></msub><annotation-xml encoding="MathML-Content" id="S2.SS1.p1.20.m2.1b"><apply id="S2.SS1.p1.20.m2.1.1.cmml" xref="S2.SS1.p1.20.m2.1.1"><csymbol cd="ambiguous" id="S2.SS1.p1.20.m2.1.1.1.cmml" xref="S2.SS1.p1.20.m2.1.1">subscript</csymbol><ci id="S2.SS1.p1.20.m2.1.1.2.cmml" xref="S2.SS1.p1.20.m2.1.1.2">𝐋</ci><ci id="S2.SS1.p1.20.m2.1.1.3.cmml" xref="S2.SS1.p1.20.m2.1.1.3">𝑘</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p1.20.m2.1c">\operatorname{\mathbf{L}}_{k}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p1.20.m2.1d">bold_L start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT</annotation></semantics></math> for <math alttext="k=1,2" class="ltx_Math" display="inline" id="S2.SS1.p1.21.m3.2"><semantics id="S2.SS1.p1.21.m3.2a"><mrow id="S2.SS1.p1.21.m3.2.3" xref="S2.SS1.p1.21.m3.2.3.cmml"><mi id="S2.SS1.p1.21.m3.2.3.2" xref="S2.SS1.p1.21.m3.2.3.2.cmml">k</mi><mo id="S2.SS1.p1.21.m3.2.3.1" xref="S2.SS1.p1.21.m3.2.3.1.cmml">=</mo><mrow id="S2.SS1.p1.21.m3.2.3.3.2" xref="S2.SS1.p1.21.m3.2.3.3.1.cmml"><mn id="S2.SS1.p1.21.m3.1.1" xref="S2.SS1.p1.21.m3.1.1.cmml">1</mn><mo id="S2.SS1.p1.21.m3.2.3.3.2.1" xref="S2.SS1.p1.21.m3.2.3.3.1.cmml">,</mo><mn id="S2.SS1.p1.21.m3.2.2" xref="S2.SS1.p1.21.m3.2.2.cmml">2</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p1.21.m3.2b"><apply id="S2.SS1.p1.21.m3.2.3.cmml" xref="S2.SS1.p1.21.m3.2.3"><eq id="S2.SS1.p1.21.m3.2.3.1.cmml" xref="S2.SS1.p1.21.m3.2.3.1"></eq><ci id="S2.SS1.p1.21.m3.2.3.2.cmml" xref="S2.SS1.p1.21.m3.2.3.2">𝑘</ci><list id="S2.SS1.p1.21.m3.2.3.3.1.cmml" xref="S2.SS1.p1.21.m3.2.3.3.2"><cn id="S2.SS1.p1.21.m3.1.1.cmml" type="integer" xref="S2.SS1.p1.21.m3.1.1">1</cn><cn id="S2.SS1.p1.21.m3.2.2.cmml" type="integer" xref="S2.SS1.p1.21.m3.2.2">2</cn></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p1.21.m3.2c">k=1,2</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p1.21.m3.2d">italic_k = 1 , 2</annotation></semantics></math> is non-identified because replacing <math alttext="\operatorname{\mathbf{L}}_{k}" class="ltx_Math" display="inline" id="S2.SS1.p1.22.m4.1"><semantics id="S2.SS1.p1.22.m4.1a"><msub id="S2.SS1.p1.22.m4.1.1" xref="S2.SS1.p1.22.m4.1.1.cmml"><mi id="S2.SS1.p1.22.m4.1.1.2" xref="S2.SS1.p1.22.m4.1.1.2.cmml">𝐋</mi><mi id="S2.SS1.p1.22.m4.1.1.3" xref="S2.SS1.p1.22.m4.1.1.3.cmml">k</mi></msub><annotation-xml encoding="MathML-Content" id="S2.SS1.p1.22.m4.1b"><apply id="S2.SS1.p1.22.m4.1.1.cmml" xref="S2.SS1.p1.22.m4.1.1"><csymbol cd="ambiguous" id="S2.SS1.p1.22.m4.1.1.1.cmml" xref="S2.SS1.p1.22.m4.1.1">subscript</csymbol><ci id="S2.SS1.p1.22.m4.1.1.2.cmml" xref="S2.SS1.p1.22.m4.1.1.2">𝐋</ci><ci id="S2.SS1.p1.22.m4.1.1.3.cmml" xref="S2.SS1.p1.22.m4.1.1.3">𝑘</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p1.22.m4.1c">\operatorname{\mathbf{L}}_{k}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p1.22.m4.1d">bold_L start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT</annotation></semantics></math> with <math alttext="\operatorname{\mathbf{L}}_{k}\operatorname{\mathbf{O}}_{k}" class="ltx_Math" display="inline" id="S2.SS1.p1.23.m5.1"><semantics id="S2.SS1.p1.23.m5.1a"><mrow id="S2.SS1.p1.23.m5.1.1" xref="S2.SS1.p1.23.m5.1.1.cmml"><msub id="S2.SS1.p1.23.m5.1.1.1" xref="S2.SS1.p1.23.m5.1.1.1.cmml"><mi id="S2.SS1.p1.23.m5.1.1.1.2" xref="S2.SS1.p1.23.m5.1.1.1.2.cmml">𝐋</mi><mi id="S2.SS1.p1.23.m5.1.1.1.3" xref="S2.SS1.p1.23.m5.1.1.1.3.cmml">k</mi></msub><mo id="S2.SS1.p1.23.m5.1.1a" lspace="0.167em" xref="S2.SS1.p1.23.m5.1.1.cmml">⁡</mo><msub id="S2.SS1.p1.23.m5.1.1.2" xref="S2.SS1.p1.23.m5.1.1.2.cmml"><mi id="S2.SS1.p1.23.m5.1.1.2.2" xref="S2.SS1.p1.23.m5.1.1.2.2.cmml">𝐎</mi><mi id="S2.SS1.p1.23.m5.1.1.2.3" xref="S2.SS1.p1.23.m5.1.1.2.3.cmml">k</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p1.23.m5.1b"><apply id="S2.SS1.p1.23.m5.1.1.cmml" xref="S2.SS1.p1.23.m5.1.1"><apply id="S2.SS1.p1.23.m5.1.1.1.cmml" xref="S2.SS1.p1.23.m5.1.1.1"><csymbol cd="ambiguous" id="S2.SS1.p1.23.m5.1.1.1.1.cmml" xref="S2.SS1.p1.23.m5.1.1.1">subscript</csymbol><ci id="S2.SS1.p1.23.m5.1.1.1.2.cmml" xref="S2.SS1.p1.23.m5.1.1.1.2">𝐋</ci><ci id="S2.SS1.p1.23.m5.1.1.1.3.cmml" xref="S2.SS1.p1.23.m5.1.1.1.3">𝑘</ci></apply><apply id="S2.SS1.p1.23.m5.1.1.2.cmml" xref="S2.SS1.p1.23.m5.1.1.2"><csymbol cd="ambiguous" id="S2.SS1.p1.23.m5.1.1.2.1.cmml" xref="S2.SS1.p1.23.m5.1.1.2">subscript</csymbol><ci id="S2.SS1.p1.23.m5.1.1.2.2.cmml" xref="S2.SS1.p1.23.m5.1.1.2.2">𝐎</ci><ci id="S2.SS1.p1.23.m5.1.1.2.3.cmml" xref="S2.SS1.p1.23.m5.1.1.2.3">𝑘</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p1.23.m5.1c">\operatorname{\mathbf{L}}_{k}\operatorname{\mathbf{O}}_{k}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p1.23.m5.1d">bold_L start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT bold_O start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT</annotation></semantics></math> for any orthonormal matrix <math alttext="\operatorname{\mathbf{O}}_{k}" class="ltx_Math" display="inline" id="S2.SS1.p1.24.m6.1"><semantics id="S2.SS1.p1.24.m6.1a"><msub id="S2.SS1.p1.24.m6.1.1" xref="S2.SS1.p1.24.m6.1.1.cmml"><mi id="S2.SS1.p1.24.m6.1.1.2" xref="S2.SS1.p1.24.m6.1.1.2.cmml">𝐎</mi><mi id="S2.SS1.p1.24.m6.1.1.3" xref="S2.SS1.p1.24.m6.1.1.3.cmml">k</mi></msub><annotation-xml encoding="MathML-Content" id="S2.SS1.p1.24.m6.1b"><apply id="S2.SS1.p1.24.m6.1.1.cmml" xref="S2.SS1.p1.24.m6.1.1"><csymbol cd="ambiguous" id="S2.SS1.p1.24.m6.1.1.1.cmml" xref="S2.SS1.p1.24.m6.1.1">subscript</csymbol><ci id="S2.SS1.p1.24.m6.1.1.2.cmml" xref="S2.SS1.p1.24.m6.1.1.2">𝐎</ci><ci id="S2.SS1.p1.24.m6.1.1.3.cmml" xref="S2.SS1.p1.24.m6.1.1.3">𝑘</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p1.24.m6.1c">\operatorname{\mathbf{O}}_{k}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p1.24.m6.1d">bold_O start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT</annotation></semantics></math> leaves the decomposition of <math alttext="\operatorname{{\mathbf{\Sigma}}}_{k}" class="ltx_Math" display="inline" id="S2.SS1.p1.25.m7.1"><semantics id="S2.SS1.p1.25.m7.1a"><msub id="S2.SS1.p1.25.m7.1.1" xref="S2.SS1.p1.25.m7.1.1.cmml"><mi id="S2.SS1.p1.25.m7.1.1.2" xref="S2.SS1.p1.25.m7.1.1.2.cmml">𝚺</mi><mi id="S2.SS1.p1.25.m7.1.1.3" xref="S2.SS1.p1.25.m7.1.1.3.cmml">k</mi></msub><annotation-xml encoding="MathML-Content" id="S2.SS1.p1.25.m7.1b"><apply id="S2.SS1.p1.25.m7.1.1.cmml" xref="S2.SS1.p1.25.m7.1.1"><csymbol cd="ambiguous" id="S2.SS1.p1.25.m7.1.1.1.cmml" xref="S2.SS1.p1.25.m7.1.1">subscript</csymbol><ci id="S2.SS1.p1.25.m7.1.1.2.cmml" xref="S2.SS1.p1.25.m7.1.1.2">𝚺</ci><ci id="S2.SS1.p1.25.m7.1.1.3.cmml" xref="S2.SS1.p1.25.m7.1.1.3">𝑘</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p1.25.m7.1c">\operatorname{{\mathbf{\Sigma}}}_{k}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p1.25.m7.1d">bold_Σ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT</annotation></semantics></math> in (<a class="ltx_ref" href="https://arxiv.org/html/2503.13782v1#S2.E2" title="In 2.1 Model ‣ 2 Mixed Model Trace Regression ‣ Regularized Parameter Estimation in Mixed Model Trace Regression"><span class="ltx_text ltx_ref_tag">2</span></a>) unchanged. We do not impose identifiability conditions on <math alttext="\operatorname{\mathbf{L}}_{1}" class="ltx_Math" display="inline" id="S2.SS1.p1.26.m8.1"><semantics id="S2.SS1.p1.26.m8.1a"><msub id="S2.SS1.p1.26.m8.1.1" xref="S2.SS1.p1.26.m8.1.1.cmml"><mi id="S2.SS1.p1.26.m8.1.1.2" xref="S2.SS1.p1.26.m8.1.1.2.cmml">𝐋</mi><mn id="S2.SS1.p1.26.m8.1.1.3" xref="S2.SS1.p1.26.m8.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S2.SS1.p1.26.m8.1b"><apply id="S2.SS1.p1.26.m8.1.1.cmml" xref="S2.SS1.p1.26.m8.1.1"><csymbol cd="ambiguous" id="S2.SS1.p1.26.m8.1.1.1.cmml" xref="S2.SS1.p1.26.m8.1.1">subscript</csymbol><ci id="S2.SS1.p1.26.m8.1.1.2.cmml" xref="S2.SS1.p1.26.m8.1.1.2">𝐋</ci><cn id="S2.SS1.p1.26.m8.1.1.3.cmml" type="integer" xref="S2.SS1.p1.26.m8.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p1.26.m8.1c">\operatorname{\mathbf{L}}_{1}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p1.26.m8.1d">bold_L start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="\operatorname{\mathbf{L}}_{2}" class="ltx_Math" display="inline" id="S2.SS1.p1.27.m9.1"><semantics id="S2.SS1.p1.27.m9.1a"><msub id="S2.SS1.p1.27.m9.1.1" xref="S2.SS1.p1.27.m9.1.1.cmml"><mi id="S2.SS1.p1.27.m9.1.1.2" xref="S2.SS1.p1.27.m9.1.1.2.cmml">𝐋</mi><mn id="S2.SS1.p1.27.m9.1.1.3" xref="S2.SS1.p1.27.m9.1.1.3.cmml">2</mn></msub><annotation-xml encoding="MathML-Content" id="S2.SS1.p1.27.m9.1b"><apply id="S2.SS1.p1.27.m9.1.1.cmml" xref="S2.SS1.p1.27.m9.1.1"><csymbol cd="ambiguous" id="S2.SS1.p1.27.m9.1.1.1.cmml" xref="S2.SS1.p1.27.m9.1.1">subscript</csymbol><ci id="S2.SS1.p1.27.m9.1.1.2.cmml" xref="S2.SS1.p1.27.m9.1.1.2">𝐋</ci><cn id="S2.SS1.p1.27.m9.1.1.3.cmml" type="integer" xref="S2.SS1.p1.27.m9.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p1.27.m9.1c">\operatorname{\mathbf{L}}_{2}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p1.27.m9.1d">bold_L start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math> because they do not impact the estimates of <math alttext="\operatorname{{\mathbf{\Sigma}}}_{1}" class="ltx_Math" display="inline" id="S2.SS1.p1.28.m10.1"><semantics id="S2.SS1.p1.28.m10.1a"><msub id="S2.SS1.p1.28.m10.1.1" xref="S2.SS1.p1.28.m10.1.1.cmml"><mi id="S2.SS1.p1.28.m10.1.1.2" xref="S2.SS1.p1.28.m10.1.1.2.cmml">𝚺</mi><mn id="S2.SS1.p1.28.m10.1.1.3" xref="S2.SS1.p1.28.m10.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S2.SS1.p1.28.m10.1b"><apply id="S2.SS1.p1.28.m10.1.1.cmml" xref="S2.SS1.p1.28.m10.1.1"><csymbol cd="ambiguous" id="S2.SS1.p1.28.m10.1.1.1.cmml" xref="S2.SS1.p1.28.m10.1.1">subscript</csymbol><ci id="S2.SS1.p1.28.m10.1.1.2.cmml" xref="S2.SS1.p1.28.m10.1.1.2">𝚺</ci><cn id="S2.SS1.p1.28.m10.1.1.3.cmml" type="integer" xref="S2.SS1.p1.28.m10.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p1.28.m10.1c">\operatorname{{\mathbf{\Sigma}}}_{1}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p1.28.m10.1d">bold_Σ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="\operatorname{{\mathbf{\Sigma}}}_{2}" class="ltx_Math" display="inline" id="S2.SS1.p1.29.m11.1"><semantics id="S2.SS1.p1.29.m11.1a"><msub id="S2.SS1.p1.29.m11.1.1" xref="S2.SS1.p1.29.m11.1.1.cmml"><mi id="S2.SS1.p1.29.m11.1.1.2" xref="S2.SS1.p1.29.m11.1.1.2.cmml">𝚺</mi><mn id="S2.SS1.p1.29.m11.1.1.3" xref="S2.SS1.p1.29.m11.1.1.3.cmml">2</mn></msub><annotation-xml encoding="MathML-Content" id="S2.SS1.p1.29.m11.1b"><apply id="S2.SS1.p1.29.m11.1.1.cmml" xref="S2.SS1.p1.29.m11.1.1"><csymbol cd="ambiguous" id="S2.SS1.p1.29.m11.1.1.1.cmml" xref="S2.SS1.p1.29.m11.1.1">subscript</csymbol><ci id="S2.SS1.p1.29.m11.1.1.2.cmml" xref="S2.SS1.p1.29.m11.1.1.2">𝚺</ci><cn id="S2.SS1.p1.29.m11.1.1.3.cmml" type="integer" xref="S2.SS1.p1.29.m11.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p1.29.m11.1c">\operatorname{{\mathbf{\Sigma}}}_{2}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p1.29.m11.1d">bold_Σ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S2.SS1.p2"> <p class="ltx_p" id="S2.SS1.p2.20">MMTR in (<a class="ltx_ref" href="https://arxiv.org/html/2503.13782v1#S2.E3" title="In 2.1 Model ‣ 2 Mixed Model Trace Regression ‣ Regularized Parameter Estimation in Mixed Model Trace Regression"><span class="ltx_text ltx_ref_tag">3</span></a>) reduces to the trace regression model when <math alttext="m_{i}=1" 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"><msub id="S2.SS1.p2.1.m1.1.1.2" xref="S2.SS1.p2.1.m1.1.1.2.cmml"><mi id="S2.SS1.p2.1.m1.1.1.2.2" xref="S2.SS1.p2.1.m1.1.1.2.2.cmml">m</mi><mi id="S2.SS1.p2.1.m1.1.1.2.3" xref="S2.SS1.p2.1.m1.1.1.2.3.cmml">i</mi></msub><mo id="S2.SS1.p2.1.m1.1.1.1" xref="S2.SS1.p2.1.m1.1.1.1.cmml">=</mo><mn id="S2.SS1.p2.1.m1.1.1.3" xref="S2.SS1.p2.1.m1.1.1.3.cmml">1</mn></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"><eq id="S2.SS1.p2.1.m1.1.1.1.cmml" xref="S2.SS1.p2.1.m1.1.1.1"></eq><apply id="S2.SS1.p2.1.m1.1.1.2.cmml" xref="S2.SS1.p2.1.m1.1.1.2"><csymbol cd="ambiguous" id="S2.SS1.p2.1.m1.1.1.2.1.cmml" xref="S2.SS1.p2.1.m1.1.1.2">subscript</csymbol><ci id="S2.SS1.p2.1.m1.1.1.2.2.cmml" xref="S2.SS1.p2.1.m1.1.1.2.2">𝑚</ci><ci id="S2.SS1.p2.1.m1.1.1.2.3.cmml" xref="S2.SS1.p2.1.m1.1.1.2.3">𝑖</ci></apply><cn id="S2.SS1.p2.1.m1.1.1.3.cmml" type="integer" xref="S2.SS1.p2.1.m1.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p2.1.m1.1c">m_{i}=1</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p2.1.m1.1d">italic_m start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = 1</annotation></semantics></math> and <math alttext="\operatorname{\mathbf{A}}_{i}=\operatorname{\mathbf{0}}" class="ltx_Math" display="inline" id="S2.SS1.p2.2.m2.1"><semantics id="S2.SS1.p2.2.m2.1a"><mrow id="S2.SS1.p2.2.m2.1.1" xref="S2.SS1.p2.2.m2.1.1.cmml"><msub id="S2.SS1.p2.2.m2.1.1.2" xref="S2.SS1.p2.2.m2.1.1.2.cmml"><mi id="S2.SS1.p2.2.m2.1.1.2.2" xref="S2.SS1.p2.2.m2.1.1.2.2.cmml">𝐀</mi><mi id="S2.SS1.p2.2.m2.1.1.2.3" xref="S2.SS1.p2.2.m2.1.1.2.3.cmml">i</mi></msub><mo id="S2.SS1.p2.2.m2.1.1.1" xref="S2.SS1.p2.2.m2.1.1.1.cmml">=</mo><mi id="S2.SS1.p2.2.m2.1.1.3" xref="S2.SS1.p2.2.m2.1.1.3.cmml">𝟎</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p2.2.m2.1b"><apply id="S2.SS1.p2.2.m2.1.1.cmml" xref="S2.SS1.p2.2.m2.1.1"><eq id="S2.SS1.p2.2.m2.1.1.1.cmml" xref="S2.SS1.p2.2.m2.1.1.1"></eq><apply id="S2.SS1.p2.2.m2.1.1.2.cmml" xref="S2.SS1.p2.2.m2.1.1.2"><csymbol cd="ambiguous" id="S2.SS1.p2.2.m2.1.1.2.1.cmml" xref="S2.SS1.p2.2.m2.1.1.2">subscript</csymbol><ci id="S2.SS1.p2.2.m2.1.1.2.2.cmml" xref="S2.SS1.p2.2.m2.1.1.2.2">𝐀</ci><ci id="S2.SS1.p2.2.m2.1.1.2.3.cmml" xref="S2.SS1.p2.2.m2.1.1.2.3">𝑖</ci></apply><ci id="S2.SS1.p2.2.m2.1.1.3.cmml" xref="S2.SS1.p2.2.m2.1.1.3">𝟎</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p2.2.m2.1c">\operatorname{\mathbf{A}}_{i}=\operatorname{\mathbf{0}}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p2.2.m2.1d">bold_A start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = bold_0</annotation></semantics></math> for every <math alttext="i=1,\ldots,n" class="ltx_Math" display="inline" id="S2.SS1.p2.3.m3.3"><semantics id="S2.SS1.p2.3.m3.3a"><mrow id="S2.SS1.p2.3.m3.3.4" xref="S2.SS1.p2.3.m3.3.4.cmml"><mi id="S2.SS1.p2.3.m3.3.4.2" xref="S2.SS1.p2.3.m3.3.4.2.cmml">i</mi><mo id="S2.SS1.p2.3.m3.3.4.1" xref="S2.SS1.p2.3.m3.3.4.1.cmml">=</mo><mrow id="S2.SS1.p2.3.m3.3.4.3.2" xref="S2.SS1.p2.3.m3.3.4.3.1.cmml"><mn id="S2.SS1.p2.3.m3.1.1" xref="S2.SS1.p2.3.m3.1.1.cmml">1</mn><mo id="S2.SS1.p2.3.m3.3.4.3.2.1" xref="S2.SS1.p2.3.m3.3.4.3.1.cmml">,</mo><mi id="S2.SS1.p2.3.m3.2.2" mathvariant="normal" xref="S2.SS1.p2.3.m3.2.2.cmml">…</mi><mo id="S2.SS1.p2.3.m3.3.4.3.2.2" xref="S2.SS1.p2.3.m3.3.4.3.1.cmml">,</mo><mi id="S2.SS1.p2.3.m3.3.3" xref="S2.SS1.p2.3.m3.3.3.cmml">n</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p2.3.m3.3b"><apply id="S2.SS1.p2.3.m3.3.4.cmml" xref="S2.SS1.p2.3.m3.3.4"><eq id="S2.SS1.p2.3.m3.3.4.1.cmml" xref="S2.SS1.p2.3.m3.3.4.1"></eq><ci id="S2.SS1.p2.3.m3.3.4.2.cmml" xref="S2.SS1.p2.3.m3.3.4.2">𝑖</ci><list id="S2.SS1.p2.3.m3.3.4.3.1.cmml" xref="S2.SS1.p2.3.m3.3.4.3.2"><cn id="S2.SS1.p2.3.m3.1.1.cmml" type="integer" xref="S2.SS1.p2.3.m3.1.1">1</cn><ci id="S2.SS1.p2.3.m3.2.2.cmml" xref="S2.SS1.p2.3.m3.2.2">…</ci><ci id="S2.SS1.p2.3.m3.3.3.cmml" xref="S2.SS1.p2.3.m3.3.3">𝑛</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p2.3.m3.3c">i=1,\ldots,n</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p2.3.m3.3d">italic_i = 1 , … , italic_n</annotation></semantics></math> <cite class="ltx_cite ltx_citemacro_citep">(Slawski et al., <a class="ltx_ref" href="https://arxiv.org/html/2503.13782v1#bib.bib26" title="">2015</a>; Zhao et al., <a class="ltx_ref" href="https://arxiv.org/html/2503.13782v1#bib.bib38" title="">2017</a>; Fan et al., <a class="ltx_ref" href="https://arxiv.org/html/2503.13782v1#bib.bib7" title="">2019</a>)</cite>. These methods estimate <math alttext="\operatorname{\mathbf{B}}" class="ltx_Math" display="inline" id="S2.SS1.p2.4.m4.1"><semantics id="S2.SS1.p2.4.m4.1a"><mi id="S2.SS1.p2.4.m4.1.1" xref="S2.SS1.p2.4.m4.1.1.cmml">𝐁</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.p2.4.m4.1b"><ci id="S2.SS1.p2.4.m4.1.1.cmml" xref="S2.SS1.p2.4.m4.1.1">𝐁</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p2.4.m4.1c">\operatorname{\mathbf{B}}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p2.4.m4.1d">bold_B</annotation></semantics></math> under different regularization schemes using the lasso-type and nuclear norm penalties. For example, <math alttext="\|\operatorname{\mathbf{B}}\|_{1}=\sum_{ij}|b_{ij}|" class="ltx_Math" display="inline" id="S2.SS1.p2.5.m5.2"><semantics id="S2.SS1.p2.5.m5.2a"><mrow id="S2.SS1.p2.5.m5.2.2" xref="S2.SS1.p2.5.m5.2.2.cmml"><msub id="S2.SS1.p2.5.m5.2.2.3" xref="S2.SS1.p2.5.m5.2.2.3.cmml"><mrow id="S2.SS1.p2.5.m5.2.2.3.2.2" xref="S2.SS1.p2.5.m5.2.2.3.2.1.cmml"><mo id="S2.SS1.p2.5.m5.2.2.3.2.2.1" stretchy="false" xref="S2.SS1.p2.5.m5.2.2.3.2.1.1.cmml">‖</mo><mi id="S2.SS1.p2.5.m5.1.1" xref="S2.SS1.p2.5.m5.1.1.cmml">𝐁</mi><mo id="S2.SS1.p2.5.m5.2.2.3.2.2.2" stretchy="false" xref="S2.SS1.p2.5.m5.2.2.3.2.1.1.cmml">‖</mo></mrow><mn id="S2.SS1.p2.5.m5.2.2.3.3" xref="S2.SS1.p2.5.m5.2.2.3.3.cmml">1</mn></msub><mo id="S2.SS1.p2.5.m5.2.2.2" rspace="0.111em" xref="S2.SS1.p2.5.m5.2.2.2.cmml">=</mo><mrow id="S2.SS1.p2.5.m5.2.2.1" xref="S2.SS1.p2.5.m5.2.2.1.cmml"><msub id="S2.SS1.p2.5.m5.2.2.1.2" xref="S2.SS1.p2.5.m5.2.2.1.2.cmml"><mo id="S2.SS1.p2.5.m5.2.2.1.2.2" rspace="0em" 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xref="S2.SS1.p2.6.m6.4.4.1.2.3.3">1</cn></apply></apply><apply id="S2.SS1.p2.6.m6.4.4.1.1.cmml" xref="S2.SS1.p2.6.m6.4.4.1.1"><csymbol cd="ambiguous" id="S2.SS1.p2.6.m6.4.4.1.1.2.cmml" xref="S2.SS1.p2.6.m6.4.4.1.1">subscript</csymbol><apply id="S2.SS1.p2.6.m6.4.4.1.1.1.2.cmml" xref="S2.SS1.p2.6.m6.4.4.1.1.1.1"><csymbol cd="latexml" id="S2.SS1.p2.6.m6.4.4.1.1.1.2.1.cmml" xref="S2.SS1.p2.6.m6.4.4.1.1.1.1.2">norm</csymbol><apply id="S2.SS1.p2.6.m6.4.4.1.1.1.1.1.cmml" xref="S2.SS1.p2.6.m6.4.4.1.1.1.1.1"><csymbol cd="ambiguous" id="S2.SS1.p2.6.m6.4.4.1.1.1.1.1.1.cmml" xref="S2.SS1.p2.6.m6.4.4.1.1.1.1.1">subscript</csymbol><ci id="S2.SS1.p2.6.m6.4.4.1.1.1.1.1.2.cmml" xref="S2.SS1.p2.6.m6.4.4.1.1.1.1.1.2">𝐛</ci><apply id="S2.SS1.p2.6.m6.4.4.1.1.1.1.1.3.cmml" xref="S2.SS1.p2.6.m6.4.4.1.1.1.1.1.3"><ci id="S2.SS1.p2.6.m6.4.4.1.1.1.1.1.3.1.cmml" xref="S2.SS1.p2.6.m6.4.4.1.1.1.1.1.3.1">:</ci><ci id="S2.SS1.p2.6.m6.4.4.1.1.1.1.1.3.2.cmml" xref="S2.SS1.p2.6.m6.4.4.1.1.1.1.1.3.2">𝑖</ci><csymbol cd="latexml" id="S2.SS1.p2.6.m6.4.4.1.1.1.1.1.3.3.cmml" xref="S2.SS1.p2.6.m6.4.4.1.1.1.1.1.3.3">absent</csymbol></apply></apply></apply><cn id="S2.SS1.p2.6.m6.4.4.1.1.3.cmml" type="integer" xref="S2.SS1.p2.6.m6.4.4.1.1.3">2</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p2.6.m6.4c">\|\operatorname{\mathbf{B}}\|_{1,2}=\sum_{i=1}^{P_{1}}\|\operatorname{\mathbf{% b}}_{i:}\|_{2}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p2.6.m6.4d">∥ bold_B ∥ start_POSTSUBSCRIPT 1 , 2 end_POSTSUBSCRIPT = ∑ start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_P start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT ∥ bold_b start_POSTSUBSCRIPT italic_i : end_POSTSUBSCRIPT ∥ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math>, and <math alttext="\|\operatorname{\mathbf{B}}\|_{2,1}=\sum_{j=1}^{P_{2}}\|\operatorname{\mathbf{% b}}_{:j}\|_{2}" class="ltx_Math" display="inline" id="S2.SS1.p2.7.m7.4"><semantics id="S2.SS1.p2.7.m7.4a"><mrow id="S2.SS1.p2.7.m7.4.4" xref="S2.SS1.p2.7.m7.4.4.cmml"><msub id="S2.SS1.p2.7.m7.4.4.3" xref="S2.SS1.p2.7.m7.4.4.3.cmml"><mrow id="S2.SS1.p2.7.m7.4.4.3.2.2" xref="S2.SS1.p2.7.m7.4.4.3.2.1.cmml"><mo id="S2.SS1.p2.7.m7.4.4.3.2.2.1" stretchy="false" xref="S2.SS1.p2.7.m7.4.4.3.2.1.1.cmml">‖</mo><mi id="S2.SS1.p2.7.m7.3.3" xref="S2.SS1.p2.7.m7.3.3.cmml">𝐁</mi><mo id="S2.SS1.p2.7.m7.4.4.3.2.2.2" stretchy="false" xref="S2.SS1.p2.7.m7.4.4.3.2.1.1.cmml">‖</mo></mrow><mrow id="S2.SS1.p2.7.m7.2.2.2.4" xref="S2.SS1.p2.7.m7.2.2.2.3.cmml"><mn id="S2.SS1.p2.7.m7.1.1.1.1" xref="S2.SS1.p2.7.m7.1.1.1.1.cmml">2</mn><mo id="S2.SS1.p2.7.m7.2.2.2.4.1" xref="S2.SS1.p2.7.m7.2.2.2.3.cmml">,</mo><mn id="S2.SS1.p2.7.m7.2.2.2.2" xref="S2.SS1.p2.7.m7.2.2.2.2.cmml">1</mn></mrow></msub><mo id="S2.SS1.p2.7.m7.4.4.2" rspace="0.111em" xref="S2.SS1.p2.7.m7.4.4.2.cmml">=</mo><mrow id="S2.SS1.p2.7.m7.4.4.1" xref="S2.SS1.p2.7.m7.4.4.1.cmml"><msubsup id="S2.SS1.p2.7.m7.4.4.1.2" xref="S2.SS1.p2.7.m7.4.4.1.2.cmml"><mo id="S2.SS1.p2.7.m7.4.4.1.2.2.2" rspace="0em" xref="S2.SS1.p2.7.m7.4.4.1.2.2.2.cmml">∑</mo><mrow id="S2.SS1.p2.7.m7.4.4.1.2.2.3" xref="S2.SS1.p2.7.m7.4.4.1.2.2.3.cmml"><mi id="S2.SS1.p2.7.m7.4.4.1.2.2.3.2" xref="S2.SS1.p2.7.m7.4.4.1.2.2.3.2.cmml">j</mi><mo id="S2.SS1.p2.7.m7.4.4.1.2.2.3.1" xref="S2.SS1.p2.7.m7.4.4.1.2.2.3.1.cmml">=</mo><mn id="S2.SS1.p2.7.m7.4.4.1.2.2.3.3" xref="S2.SS1.p2.7.m7.4.4.1.2.2.3.3.cmml">1</mn></mrow><msub id="S2.SS1.p2.7.m7.4.4.1.2.3" xref="S2.SS1.p2.7.m7.4.4.1.2.3.cmml"><mi id="S2.SS1.p2.7.m7.4.4.1.2.3.2" xref="S2.SS1.p2.7.m7.4.4.1.2.3.2.cmml">P</mi><mn id="S2.SS1.p2.7.m7.4.4.1.2.3.3" xref="S2.SS1.p2.7.m7.4.4.1.2.3.3.cmml">2</mn></msub></msubsup><msub id="S2.SS1.p2.7.m7.4.4.1.1" xref="S2.SS1.p2.7.m7.4.4.1.1.cmml"><mrow id="S2.SS1.p2.7.m7.4.4.1.1.1.1" xref="S2.SS1.p2.7.m7.4.4.1.1.1.2.cmml"><mo id="S2.SS1.p2.7.m7.4.4.1.1.1.1.2" stretchy="false" xref="S2.SS1.p2.7.m7.4.4.1.1.1.2.1.cmml">‖</mo><msub id="S2.SS1.p2.7.m7.4.4.1.1.1.1.1" xref="S2.SS1.p2.7.m7.4.4.1.1.1.1.1.cmml"><mi id="S2.SS1.p2.7.m7.4.4.1.1.1.1.1.2" xref="S2.SS1.p2.7.m7.4.4.1.1.1.1.1.2.cmml">𝐛</mi><mrow id="S2.SS1.p2.7.m7.4.4.1.1.1.1.1.3" xref="S2.SS1.p2.7.m7.4.4.1.1.1.1.1.3.cmml"><mi id="S2.SS1.p2.7.m7.4.4.1.1.1.1.1.3.2" xref="S2.SS1.p2.7.m7.4.4.1.1.1.1.1.3.2.cmml"></mi><mo id="S2.SS1.p2.7.m7.4.4.1.1.1.1.1.3.1" lspace="0.278em" rspace="0.278em" xref="S2.SS1.p2.7.m7.4.4.1.1.1.1.1.3.1.cmml">:</mo><mi id="S2.SS1.p2.7.m7.4.4.1.1.1.1.1.3.3" xref="S2.SS1.p2.7.m7.4.4.1.1.1.1.1.3.3.cmml">j</mi></mrow></msub><mo id="S2.SS1.p2.7.m7.4.4.1.1.1.1.3" stretchy="false" xref="S2.SS1.p2.7.m7.4.4.1.1.1.2.1.cmml">‖</mo></mrow><mn id="S2.SS1.p2.7.m7.4.4.1.1.3" xref="S2.SS1.p2.7.m7.4.4.1.1.3.cmml">2</mn></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p2.7.m7.4b"><apply id="S2.SS1.p2.7.m7.4.4.cmml" xref="S2.SS1.p2.7.m7.4.4"><eq id="S2.SS1.p2.7.m7.4.4.2.cmml" xref="S2.SS1.p2.7.m7.4.4.2"></eq><apply id="S2.SS1.p2.7.m7.4.4.3.cmml" xref="S2.SS1.p2.7.m7.4.4.3"><csymbol cd="ambiguous" id="S2.SS1.p2.7.m7.4.4.3.1.cmml" xref="S2.SS1.p2.7.m7.4.4.3">subscript</csymbol><apply id="S2.SS1.p2.7.m7.4.4.3.2.1.cmml" xref="S2.SS1.p2.7.m7.4.4.3.2.2"><csymbol cd="latexml" id="S2.SS1.p2.7.m7.4.4.3.2.1.1.cmml" xref="S2.SS1.p2.7.m7.4.4.3.2.2.1">norm</csymbol><ci id="S2.SS1.p2.7.m7.3.3.cmml" xref="S2.SS1.p2.7.m7.3.3">𝐁</ci></apply><list id="S2.SS1.p2.7.m7.2.2.2.3.cmml" xref="S2.SS1.p2.7.m7.2.2.2.4"><cn id="S2.SS1.p2.7.m7.1.1.1.1.cmml" type="integer" xref="S2.SS1.p2.7.m7.1.1.1.1">2</cn><cn id="S2.SS1.p2.7.m7.2.2.2.2.cmml" type="integer" xref="S2.SS1.p2.7.m7.2.2.2.2">1</cn></list></apply><apply id="S2.SS1.p2.7.m7.4.4.1.cmml" xref="S2.SS1.p2.7.m7.4.4.1"><apply id="S2.SS1.p2.7.m7.4.4.1.2.cmml" xref="S2.SS1.p2.7.m7.4.4.1.2"><csymbol cd="ambiguous" id="S2.SS1.p2.7.m7.4.4.1.2.1.cmml" xref="S2.SS1.p2.7.m7.4.4.1.2">superscript</csymbol><apply id="S2.SS1.p2.7.m7.4.4.1.2.2.cmml" xref="S2.SS1.p2.7.m7.4.4.1.2"><csymbol cd="ambiguous" id="S2.SS1.p2.7.m7.4.4.1.2.2.1.cmml" xref="S2.SS1.p2.7.m7.4.4.1.2">subscript</csymbol><sum id="S2.SS1.p2.7.m7.4.4.1.2.2.2.cmml" xref="S2.SS1.p2.7.m7.4.4.1.2.2.2"></sum><apply id="S2.SS1.p2.7.m7.4.4.1.2.2.3.cmml" xref="S2.SS1.p2.7.m7.4.4.1.2.2.3"><eq id="S2.SS1.p2.7.m7.4.4.1.2.2.3.1.cmml" xref="S2.SS1.p2.7.m7.4.4.1.2.2.3.1"></eq><ci id="S2.SS1.p2.7.m7.4.4.1.2.2.3.2.cmml" xref="S2.SS1.p2.7.m7.4.4.1.2.2.3.2">𝑗</ci><cn id="S2.SS1.p2.7.m7.4.4.1.2.2.3.3.cmml" type="integer" xref="S2.SS1.p2.7.m7.4.4.1.2.2.3.3">1</cn></apply></apply><apply id="S2.SS1.p2.7.m7.4.4.1.2.3.cmml" xref="S2.SS1.p2.7.m7.4.4.1.2.3"><csymbol cd="ambiguous" id="S2.SS1.p2.7.m7.4.4.1.2.3.1.cmml" xref="S2.SS1.p2.7.m7.4.4.1.2.3">subscript</csymbol><ci id="S2.SS1.p2.7.m7.4.4.1.2.3.2.cmml" xref="S2.SS1.p2.7.m7.4.4.1.2.3.2">𝑃</ci><cn id="S2.SS1.p2.7.m7.4.4.1.2.3.3.cmml" type="integer" xref="S2.SS1.p2.7.m7.4.4.1.2.3.3">2</cn></apply></apply><apply id="S2.SS1.p2.7.m7.4.4.1.1.cmml" xref="S2.SS1.p2.7.m7.4.4.1.1"><csymbol cd="ambiguous" id="S2.SS1.p2.7.m7.4.4.1.1.2.cmml" xref="S2.SS1.p2.7.m7.4.4.1.1">subscript</csymbol><apply id="S2.SS1.p2.7.m7.4.4.1.1.1.2.cmml" xref="S2.SS1.p2.7.m7.4.4.1.1.1.1"><csymbol cd="latexml" id="S2.SS1.p2.7.m7.4.4.1.1.1.2.1.cmml" xref="S2.SS1.p2.7.m7.4.4.1.1.1.1.2">norm</csymbol><apply id="S2.SS1.p2.7.m7.4.4.1.1.1.1.1.cmml" xref="S2.SS1.p2.7.m7.4.4.1.1.1.1.1"><csymbol cd="ambiguous" id="S2.SS1.p2.7.m7.4.4.1.1.1.1.1.1.cmml" xref="S2.SS1.p2.7.m7.4.4.1.1.1.1.1">subscript</csymbol><ci id="S2.SS1.p2.7.m7.4.4.1.1.1.1.1.2.cmml" xref="S2.SS1.p2.7.m7.4.4.1.1.1.1.1.2">𝐛</ci><apply id="S2.SS1.p2.7.m7.4.4.1.1.1.1.1.3.cmml" xref="S2.SS1.p2.7.m7.4.4.1.1.1.1.1.3"><ci id="S2.SS1.p2.7.m7.4.4.1.1.1.1.1.3.1.cmml" xref="S2.SS1.p2.7.m7.4.4.1.1.1.1.1.3.1">:</ci><csymbol cd="latexml" id="S2.SS1.p2.7.m7.4.4.1.1.1.1.1.3.2.cmml" xref="S2.SS1.p2.7.m7.4.4.1.1.1.1.1.3.2">absent</csymbol><ci id="S2.SS1.p2.7.m7.4.4.1.1.1.1.1.3.3.cmml" xref="S2.SS1.p2.7.m7.4.4.1.1.1.1.1.3.3">𝑗</ci></apply></apply></apply><cn id="S2.SS1.p2.7.m7.4.4.1.1.3.cmml" type="integer" xref="S2.SS1.p2.7.m7.4.4.1.1.3">2</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p2.7.m7.4c">\|\operatorname{\mathbf{B}}\|_{2,1}=\sum_{j=1}^{P_{2}}\|\operatorname{\mathbf{% b}}_{:j}\|_{2}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p2.7.m7.4d">∥ bold_B ∥ start_POSTSUBSCRIPT 2 , 1 end_POSTSUBSCRIPT = ∑ start_POSTSUBSCRIPT italic_j = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_P start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT ∥ bold_b start_POSTSUBSCRIPT : italic_j end_POSTSUBSCRIPT ∥ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math>, where <math alttext="b_{ij}" class="ltx_Math" display="inline" id="S2.SS1.p2.8.m8.1"><semantics id="S2.SS1.p2.8.m8.1a"><msub id="S2.SS1.p2.8.m8.1.1" xref="S2.SS1.p2.8.m8.1.1.cmml"><mi id="S2.SS1.p2.8.m8.1.1.2" xref="S2.SS1.p2.8.m8.1.1.2.cmml">b</mi><mrow id="S2.SS1.p2.8.m8.1.1.3" xref="S2.SS1.p2.8.m8.1.1.3.cmml"><mi id="S2.SS1.p2.8.m8.1.1.3.2" xref="S2.SS1.p2.8.m8.1.1.3.2.cmml">i</mi><mo id="S2.SS1.p2.8.m8.1.1.3.1" xref="S2.SS1.p2.8.m8.1.1.3.1.cmml">⁢</mo><mi id="S2.SS1.p2.8.m8.1.1.3.3" xref="S2.SS1.p2.8.m8.1.1.3.3.cmml">j</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="S2.SS1.p2.8.m8.1b"><apply id="S2.SS1.p2.8.m8.1.1.cmml" xref="S2.SS1.p2.8.m8.1.1"><csymbol cd="ambiguous" id="S2.SS1.p2.8.m8.1.1.1.cmml" xref="S2.SS1.p2.8.m8.1.1">subscript</csymbol><ci id="S2.SS1.p2.8.m8.1.1.2.cmml" xref="S2.SS1.p2.8.m8.1.1.2">𝑏</ci><apply id="S2.SS1.p2.8.m8.1.1.3.cmml" xref="S2.SS1.p2.8.m8.1.1.3"><times id="S2.SS1.p2.8.m8.1.1.3.1.cmml" xref="S2.SS1.p2.8.m8.1.1.3.1"></times><ci id="S2.SS1.p2.8.m8.1.1.3.2.cmml" xref="S2.SS1.p2.8.m8.1.1.3.2">𝑖</ci><ci id="S2.SS1.p2.8.m8.1.1.3.3.cmml" xref="S2.SS1.p2.8.m8.1.1.3.3">𝑗</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p2.8.m8.1c">b_{ij}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p2.8.m8.1d">italic_b start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT</annotation></semantics></math>, <math alttext="\operatorname{\mathbf{b}}_{i:}" class="ltx_Math" display="inline" id="S2.SS1.p2.9.m9.1"><semantics id="S2.SS1.p2.9.m9.1a"><msub id="S2.SS1.p2.9.m9.1.1" xref="S2.SS1.p2.9.m9.1.1.cmml"><mi id="S2.SS1.p2.9.m9.1.1.2" xref="S2.SS1.p2.9.m9.1.1.2.cmml">𝐛</mi><mrow id="S2.SS1.p2.9.m9.1.1.3" xref="S2.SS1.p2.9.m9.1.1.3.cmml"><mi id="S2.SS1.p2.9.m9.1.1.3.2" xref="S2.SS1.p2.9.m9.1.1.3.2.cmml">i</mi><mo id="S2.SS1.p2.9.m9.1.1.3.1" lspace="0.278em" rspace="0.278em" xref="S2.SS1.p2.9.m9.1.1.3.1.cmml">:</mo><mi id="S2.SS1.p2.9.m9.1.1.3.3" xref="S2.SS1.p2.9.m9.1.1.3.3.cmml"></mi></mrow></msub><annotation-xml encoding="MathML-Content" id="S2.SS1.p2.9.m9.1b"><apply id="S2.SS1.p2.9.m9.1.1.cmml" xref="S2.SS1.p2.9.m9.1.1"><csymbol cd="ambiguous" id="S2.SS1.p2.9.m9.1.1.1.cmml" xref="S2.SS1.p2.9.m9.1.1">subscript</csymbol><ci id="S2.SS1.p2.9.m9.1.1.2.cmml" xref="S2.SS1.p2.9.m9.1.1.2">𝐛</ci><apply id="S2.SS1.p2.9.m9.1.1.3.cmml" xref="S2.SS1.p2.9.m9.1.1.3"><ci id="S2.SS1.p2.9.m9.1.1.3.1.cmml" xref="S2.SS1.p2.9.m9.1.1.3.1">:</ci><ci id="S2.SS1.p2.9.m9.1.1.3.2.cmml" xref="S2.SS1.p2.9.m9.1.1.3.2">𝑖</ci><csymbol cd="latexml" id="S2.SS1.p2.9.m9.1.1.3.3.cmml" xref="S2.SS1.p2.9.m9.1.1.3.3">absent</csymbol></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p2.9.m9.1c">\operatorname{\mathbf{b}}_{i:}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p2.9.m9.1d">bold_b start_POSTSUBSCRIPT italic_i : end_POSTSUBSCRIPT</annotation></semantics></math>, and <math alttext="\operatorname{\mathbf{b}}_{:j}" class="ltx_Math" display="inline" id="S2.SS1.p2.10.m10.1"><semantics id="S2.SS1.p2.10.m10.1a"><msub id="S2.SS1.p2.10.m10.1.1" xref="S2.SS1.p2.10.m10.1.1.cmml"><mi id="S2.SS1.p2.10.m10.1.1.2" xref="S2.SS1.p2.10.m10.1.1.2.cmml">𝐛</mi><mrow id="S2.SS1.p2.10.m10.1.1.3" xref="S2.SS1.p2.10.m10.1.1.3.cmml"><mi id="S2.SS1.p2.10.m10.1.1.3.2" xref="S2.SS1.p2.10.m10.1.1.3.2.cmml"></mi><mo id="S2.SS1.p2.10.m10.1.1.3.1" lspace="0.278em" rspace="0.278em" xref="S2.SS1.p2.10.m10.1.1.3.1.cmml">:</mo><mi id="S2.SS1.p2.10.m10.1.1.3.3" xref="S2.SS1.p2.10.m10.1.1.3.3.cmml">j</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="S2.SS1.p2.10.m10.1b"><apply id="S2.SS1.p2.10.m10.1.1.cmml" xref="S2.SS1.p2.10.m10.1.1"><csymbol cd="ambiguous" id="S2.SS1.p2.10.m10.1.1.1.cmml" xref="S2.SS1.p2.10.m10.1.1">subscript</csymbol><ci id="S2.SS1.p2.10.m10.1.1.2.cmml" xref="S2.SS1.p2.10.m10.1.1.2">𝐛</ci><apply id="S2.SS1.p2.10.m10.1.1.3.cmml" xref="S2.SS1.p2.10.m10.1.1.3"><ci id="S2.SS1.p2.10.m10.1.1.3.1.cmml" xref="S2.SS1.p2.10.m10.1.1.3.1">:</ci><csymbol cd="latexml" id="S2.SS1.p2.10.m10.1.1.3.2.cmml" xref="S2.SS1.p2.10.m10.1.1.3.2">absent</csymbol><ci id="S2.SS1.p2.10.m10.1.1.3.3.cmml" xref="S2.SS1.p2.10.m10.1.1.3.3">𝑗</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p2.10.m10.1c">\operatorname{\mathbf{b}}_{:j}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p2.10.m10.1d">bold_b start_POSTSUBSCRIPT : italic_j end_POSTSUBSCRIPT</annotation></semantics></math> are the <math alttext="(i,j)" class="ltx_Math" display="inline" id="S2.SS1.p2.11.m11.2"><semantics id="S2.SS1.p2.11.m11.2a"><mrow id="S2.SS1.p2.11.m11.2.3.2" xref="S2.SS1.p2.11.m11.2.3.1.cmml"><mo id="S2.SS1.p2.11.m11.2.3.2.1" stretchy="false" xref="S2.SS1.p2.11.m11.2.3.1.cmml">(</mo><mi id="S2.SS1.p2.11.m11.1.1" xref="S2.SS1.p2.11.m11.1.1.cmml">i</mi><mo id="S2.SS1.p2.11.m11.2.3.2.2" xref="S2.SS1.p2.11.m11.2.3.1.cmml">,</mo><mi id="S2.SS1.p2.11.m11.2.2" xref="S2.SS1.p2.11.m11.2.2.cmml">j</mi><mo id="S2.SS1.p2.11.m11.2.3.2.3" stretchy="false" xref="S2.SS1.p2.11.m11.2.3.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p2.11.m11.2b"><interval closure="open" id="S2.SS1.p2.11.m11.2.3.1.cmml" xref="S2.SS1.p2.11.m11.2.3.2"><ci id="S2.SS1.p2.11.m11.1.1.cmml" xref="S2.SS1.p2.11.m11.1.1">𝑖</ci><ci id="S2.SS1.p2.11.m11.2.2.cmml" xref="S2.SS1.p2.11.m11.2.2">𝑗</ci></interval></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p2.11.m11.2c">(i,j)</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p2.11.m11.2d">( italic_i , italic_j )</annotation></semantics></math>th entry, <math alttext="i" class="ltx_Math" display="inline" id="S2.SS1.p2.12.m12.1"><semantics id="S2.SS1.p2.12.m12.1a"><mi id="S2.SS1.p2.12.m12.1.1" xref="S2.SS1.p2.12.m12.1.1.cmml">i</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.p2.12.m12.1b"><ci id="S2.SS1.p2.12.m12.1.1.cmml" xref="S2.SS1.p2.12.m12.1.1">𝑖</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p2.12.m12.1c">i</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p2.12.m12.1d">italic_i</annotation></semantics></math>th row, and <math alttext="j" class="ltx_Math" display="inline" id="S2.SS1.p2.13.m13.1"><semantics id="S2.SS1.p2.13.m13.1a"><mi id="S2.SS1.p2.13.m13.1.1" xref="S2.SS1.p2.13.m13.1.1.cmml">j</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.p2.13.m13.1b"><ci id="S2.SS1.p2.13.m13.1.1.cmml" xref="S2.SS1.p2.13.m13.1.1">𝑗</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p2.13.m13.1c">j</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p2.13.m13.1d">italic_j</annotation></semantics></math>th column of <math alttext="\operatorname{\mathbf{B}}" class="ltx_Math" display="inline" id="S2.SS1.p2.14.m14.1"><semantics id="S2.SS1.p2.14.m14.1a"><mi id="S2.SS1.p2.14.m14.1.1" xref="S2.SS1.p2.14.m14.1.1.cmml">𝐁</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.p2.14.m14.1b"><ci id="S2.SS1.p2.14.m14.1.1.cmml" xref="S2.SS1.p2.14.m14.1.1">𝐁</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p2.14.m14.1c">\operatorname{\mathbf{B}}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p2.14.m14.1d">bold_B</annotation></semantics></math>. The application of these three penalties results in sparse, row sparse, and column sparse <math alttext="\operatorname{\mathbf{B}}" class="ltx_Math" display="inline" id="S2.SS1.p2.15.m15.1"><semantics id="S2.SS1.p2.15.m15.1a"><mi id="S2.SS1.p2.15.m15.1.1" xref="S2.SS1.p2.15.m15.1.1.cmml">𝐁</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.p2.15.m15.1b"><ci id="S2.SS1.p2.15.m15.1.1.cmml" xref="S2.SS1.p2.15.m15.1.1">𝐁</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p2.15.m15.1c">\operatorname{\mathbf{B}}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p2.15.m15.1d">bold_B</annotation></semantics></math> estimates, which are efficiently estimated using the alternating direction method of multipliers or projected gradient methods. If <math alttext="\operatorname{\mathbf{L}}_{1},\operatorname{\mathbf{L}}_{2},\tau^{2}" class="ltx_Math" display="inline" id="S2.SS1.p2.16.m16.3"><semantics id="S2.SS1.p2.16.m16.3a"><mrow id="S2.SS1.p2.16.m16.3.3.3" xref="S2.SS1.p2.16.m16.3.3.4.cmml"><msub id="S2.SS1.p2.16.m16.1.1.1.1" xref="S2.SS1.p2.16.m16.1.1.1.1.cmml"><mi id="S2.SS1.p2.16.m16.1.1.1.1.2" xref="S2.SS1.p2.16.m16.1.1.1.1.2.cmml">𝐋</mi><mn id="S2.SS1.p2.16.m16.1.1.1.1.3" xref="S2.SS1.p2.16.m16.1.1.1.1.3.cmml">1</mn></msub><mo id="S2.SS1.p2.16.m16.3.3.3.4" xref="S2.SS1.p2.16.m16.3.3.4.cmml">,</mo><msub id="S2.SS1.p2.16.m16.2.2.2.2" xref="S2.SS1.p2.16.m16.2.2.2.2.cmml"><mi id="S2.SS1.p2.16.m16.2.2.2.2.2" xref="S2.SS1.p2.16.m16.2.2.2.2.2.cmml">𝐋</mi><mn id="S2.SS1.p2.16.m16.2.2.2.2.3" xref="S2.SS1.p2.16.m16.2.2.2.2.3.cmml">2</mn></msub><mo id="S2.SS1.p2.16.m16.3.3.3.5" xref="S2.SS1.p2.16.m16.3.3.4.cmml">,</mo><msup id="S2.SS1.p2.16.m16.3.3.3.3" xref="S2.SS1.p2.16.m16.3.3.3.3.cmml"><mi id="S2.SS1.p2.16.m16.3.3.3.3.2" xref="S2.SS1.p2.16.m16.3.3.3.3.2.cmml">τ</mi><mn id="S2.SS1.p2.16.m16.3.3.3.3.3" xref="S2.SS1.p2.16.m16.3.3.3.3.3.cmml">2</mn></msup></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p2.16.m16.3b"><list id="S2.SS1.p2.16.m16.3.3.4.cmml" xref="S2.SS1.p2.16.m16.3.3.3"><apply id="S2.SS1.p2.16.m16.1.1.1.1.cmml" xref="S2.SS1.p2.16.m16.1.1.1.1"><csymbol cd="ambiguous" id="S2.SS1.p2.16.m16.1.1.1.1.1.cmml" xref="S2.SS1.p2.16.m16.1.1.1.1">subscript</csymbol><ci id="S2.SS1.p2.16.m16.1.1.1.1.2.cmml" xref="S2.SS1.p2.16.m16.1.1.1.1.2">𝐋</ci><cn id="S2.SS1.p2.16.m16.1.1.1.1.3.cmml" type="integer" xref="S2.SS1.p2.16.m16.1.1.1.1.3">1</cn></apply><apply id="S2.SS1.p2.16.m16.2.2.2.2.cmml" xref="S2.SS1.p2.16.m16.2.2.2.2"><csymbol cd="ambiguous" id="S2.SS1.p2.16.m16.2.2.2.2.1.cmml" xref="S2.SS1.p2.16.m16.2.2.2.2">subscript</csymbol><ci id="S2.SS1.p2.16.m16.2.2.2.2.2.cmml" xref="S2.SS1.p2.16.m16.2.2.2.2.2">𝐋</ci><cn id="S2.SS1.p2.16.m16.2.2.2.2.3.cmml" type="integer" xref="S2.SS1.p2.16.m16.2.2.2.2.3">2</cn></apply><apply id="S2.SS1.p2.16.m16.3.3.3.3.cmml" xref="S2.SS1.p2.16.m16.3.3.3.3"><csymbol cd="ambiguous" id="S2.SS1.p2.16.m16.3.3.3.3.1.cmml" xref="S2.SS1.p2.16.m16.3.3.3.3">superscript</csymbol><ci id="S2.SS1.p2.16.m16.3.3.3.3.2.cmml" xref="S2.SS1.p2.16.m16.3.3.3.3.2">𝜏</ci><cn id="S2.SS1.p2.16.m16.3.3.3.3.3.cmml" type="integer" xref="S2.SS1.p2.16.m16.3.3.3.3.3">2</cn></apply></list></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p2.16.m16.3c">\operatorname{\mathbf{L}}_{1},\operatorname{\mathbf{L}}_{2},\tau^{2}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p2.16.m16.3d">bold_L start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , bold_L start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , italic_τ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT</annotation></semantics></math> are known, then we can use these algorithms for sparse estimation of <math alttext="\operatorname{\mathbf{B}}" class="ltx_Math" display="inline" id="S2.SS1.p2.17.m17.1"><semantics id="S2.SS1.p2.17.m17.1a"><mi id="S2.SS1.p2.17.m17.1.1" xref="S2.SS1.p2.17.m17.1.1.cmml">𝐁</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.p2.17.m17.1b"><ci id="S2.SS1.p2.17.m17.1.1.cmml" xref="S2.SS1.p2.17.m17.1.1">𝐁</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p2.17.m17.1c">\operatorname{\mathbf{B}}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p2.17.m17.1d">bold_B</annotation></semantics></math>. MMTR extensions with <math alttext="\|\operatorname{\mathbf{B}}\|_{1,2}" class="ltx_Math" display="inline" id="S2.SS1.p2.18.m18.3"><semantics id="S2.SS1.p2.18.m18.3a"><msub id="S2.SS1.p2.18.m18.3.4" xref="S2.SS1.p2.18.m18.3.4.cmml"><mrow id="S2.SS1.p2.18.m18.3.4.2.2" xref="S2.SS1.p2.18.m18.3.4.2.1.cmml"><mo id="S2.SS1.p2.18.m18.3.4.2.2.1" stretchy="false" xref="S2.SS1.p2.18.m18.3.4.2.1.1.cmml">‖</mo><mi id="S2.SS1.p2.18.m18.3.3" xref="S2.SS1.p2.18.m18.3.3.cmml">𝐁</mi><mo id="S2.SS1.p2.18.m18.3.4.2.2.2" stretchy="false" xref="S2.SS1.p2.18.m18.3.4.2.1.1.cmml">‖</mo></mrow><mrow id="S2.SS1.p2.18.m18.2.2.2.4" xref="S2.SS1.p2.18.m18.2.2.2.3.cmml"><mn id="S2.SS1.p2.18.m18.1.1.1.1" xref="S2.SS1.p2.18.m18.1.1.1.1.cmml">1</mn><mo id="S2.SS1.p2.18.m18.2.2.2.4.1" xref="S2.SS1.p2.18.m18.2.2.2.3.cmml">,</mo><mn id="S2.SS1.p2.18.m18.2.2.2.2" xref="S2.SS1.p2.18.m18.2.2.2.2.cmml">2</mn></mrow></msub><annotation-xml encoding="MathML-Content" id="S2.SS1.p2.18.m18.3b"><apply id="S2.SS1.p2.18.m18.3.4.cmml" xref="S2.SS1.p2.18.m18.3.4"><csymbol cd="ambiguous" id="S2.SS1.p2.18.m18.3.4.1.cmml" xref="S2.SS1.p2.18.m18.3.4">subscript</csymbol><apply id="S2.SS1.p2.18.m18.3.4.2.1.cmml" xref="S2.SS1.p2.18.m18.3.4.2.2"><csymbol cd="latexml" id="S2.SS1.p2.18.m18.3.4.2.1.1.cmml" xref="S2.SS1.p2.18.m18.3.4.2.2.1">norm</csymbol><ci id="S2.SS1.p2.18.m18.3.3.cmml" xref="S2.SS1.p2.18.m18.3.3">𝐁</ci></apply><list id="S2.SS1.p2.18.m18.2.2.2.3.cmml" xref="S2.SS1.p2.18.m18.2.2.2.4"><cn id="S2.SS1.p2.18.m18.1.1.1.1.cmml" type="integer" xref="S2.SS1.p2.18.m18.1.1.1.1">1</cn><cn id="S2.SS1.p2.18.m18.2.2.2.2.cmml" type="integer" xref="S2.SS1.p2.18.m18.2.2.2.2">2</cn></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p2.18.m18.3c">\|\operatorname{\mathbf{B}}\|_{1,2}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p2.18.m18.3d">∥ bold_B ∥ start_POSTSUBSCRIPT 1 , 2 end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="\|\operatorname{\mathbf{B}}\|_{2,1}" class="ltx_Math" display="inline" id="S2.SS1.p2.19.m19.3"><semantics id="S2.SS1.p2.19.m19.3a"><msub id="S2.SS1.p2.19.m19.3.4" xref="S2.SS1.p2.19.m19.3.4.cmml"><mrow id="S2.SS1.p2.19.m19.3.4.2.2" xref="S2.SS1.p2.19.m19.3.4.2.1.cmml"><mo id="S2.SS1.p2.19.m19.3.4.2.2.1" stretchy="false" xref="S2.SS1.p2.19.m19.3.4.2.1.1.cmml">‖</mo><mi id="S2.SS1.p2.19.m19.3.3" xref="S2.SS1.p2.19.m19.3.3.cmml">𝐁</mi><mo id="S2.SS1.p2.19.m19.3.4.2.2.2" stretchy="false" xref="S2.SS1.p2.19.m19.3.4.2.1.1.cmml">‖</mo></mrow><mrow id="S2.SS1.p2.19.m19.2.2.2.4" xref="S2.SS1.p2.19.m19.2.2.2.3.cmml"><mn id="S2.SS1.p2.19.m19.1.1.1.1" xref="S2.SS1.p2.19.m19.1.1.1.1.cmml">2</mn><mo id="S2.SS1.p2.19.m19.2.2.2.4.1" xref="S2.SS1.p2.19.m19.2.2.2.3.cmml">,</mo><mn id="S2.SS1.p2.19.m19.2.2.2.2" xref="S2.SS1.p2.19.m19.2.2.2.2.cmml">1</mn></mrow></msub><annotation-xml encoding="MathML-Content" id="S2.SS1.p2.19.m19.3b"><apply id="S2.SS1.p2.19.m19.3.4.cmml" xref="S2.SS1.p2.19.m19.3.4"><csymbol cd="ambiguous" id="S2.SS1.p2.19.m19.3.4.1.cmml" xref="S2.SS1.p2.19.m19.3.4">subscript</csymbol><apply id="S2.SS1.p2.19.m19.3.4.2.1.cmml" xref="S2.SS1.p2.19.m19.3.4.2.2"><csymbol cd="latexml" id="S2.SS1.p2.19.m19.3.4.2.1.1.cmml" xref="S2.SS1.p2.19.m19.3.4.2.2.1">norm</csymbol><ci id="S2.SS1.p2.19.m19.3.3.cmml" xref="S2.SS1.p2.19.m19.3.3">𝐁</ci></apply><list id="S2.SS1.p2.19.m19.2.2.2.3.cmml" xref="S2.SS1.p2.19.m19.2.2.2.4"><cn id="S2.SS1.p2.19.m19.1.1.1.1.cmml" type="integer" xref="S2.SS1.p2.19.m19.1.1.1.1">2</cn><cn id="S2.SS1.p2.19.m19.2.2.2.2.cmml" type="integer" xref="S2.SS1.p2.19.m19.2.2.2.2">1</cn></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p2.19.m19.3c">\|\operatorname{\mathbf{B}}\|_{2,1}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p2.19.m19.3d">∥ bold_B ∥ start_POSTSUBSCRIPT 2 , 1 end_POSTSUBSCRIPT</annotation></semantics></math> penalties can leverage these estimation algorithms; however, we employ only the <math alttext="\|\operatorname{\mathbf{B}}\|_{1}" class="ltx_Math" display="inline" id="S2.SS1.p2.20.m20.1"><semantics id="S2.SS1.p2.20.m20.1a"><msub id="S2.SS1.p2.20.m20.1.2" xref="S2.SS1.p2.20.m20.1.2.cmml"><mrow id="S2.SS1.p2.20.m20.1.2.2.2" xref="S2.SS1.p2.20.m20.1.2.2.1.cmml"><mo id="S2.SS1.p2.20.m20.1.2.2.2.1" stretchy="false" xref="S2.SS1.p2.20.m20.1.2.2.1.1.cmml">‖</mo><mi id="S2.SS1.p2.20.m20.1.1" xref="S2.SS1.p2.20.m20.1.1.cmml">𝐁</mi><mo id="S2.SS1.p2.20.m20.1.2.2.2.2" stretchy="false" xref="S2.SS1.p2.20.m20.1.2.2.1.1.cmml">‖</mo></mrow><mn id="S2.SS1.p2.20.m20.1.2.3" xref="S2.SS1.p2.20.m20.1.2.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S2.SS1.p2.20.m20.1b"><apply id="S2.SS1.p2.20.m20.1.2.cmml" xref="S2.SS1.p2.20.m20.1.2"><csymbol cd="ambiguous" id="S2.SS1.p2.20.m20.1.2.1.cmml" xref="S2.SS1.p2.20.m20.1.2">subscript</csymbol><apply id="S2.SS1.p2.20.m20.1.2.2.1.cmml" xref="S2.SS1.p2.20.m20.1.2.2.2"><csymbol cd="latexml" id="S2.SS1.p2.20.m20.1.2.2.1.1.cmml" xref="S2.SS1.p2.20.m20.1.2.2.2.1">norm</csymbol><ci id="S2.SS1.p2.20.m20.1.1.cmml" xref="S2.SS1.p2.20.m20.1.1">𝐁</ci></apply><cn id="S2.SS1.p2.20.m20.1.2.3.cmml" type="integer" xref="S2.SS1.p2.20.m20.1.2.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p2.20.m20.1c">\|\operatorname{\mathbf{B}}\|_{1}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p2.20.m20.1d">∥ bold_B ∥ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> penalty for simplicity, enabling the direct application of existing algorithms designed for scaled sparse regression <cite class="ltx_cite ltx_citemacro_citep">(Sun and Zhang, <a class="ltx_ref" href="https://arxiv.org/html/2503.13782v1#bib.bib30" title="">2012</a>)</cite>.</p> </div> <div class="ltx_para" id="S2.SS1.p3"> <p class="ltx_p" id="S2.SS1.p3.17">The covariance matrices <math alttext="\operatorname{{\mathbf{\Sigma}}}_{1}" class="ltx_Math" display="inline" id="S2.SS1.p3.1.m1.1"><semantics id="S2.SS1.p3.1.m1.1a"><msub id="S2.SS1.p3.1.m1.1.1" xref="S2.SS1.p3.1.m1.1.1.cmml"><mi id="S2.SS1.p3.1.m1.1.1.2" xref="S2.SS1.p3.1.m1.1.1.2.cmml">𝚺</mi><mn id="S2.SS1.p3.1.m1.1.1.3" xref="S2.SS1.p3.1.m1.1.1.3.cmml">1</mn></msub><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"><csymbol cd="ambiguous" id="S2.SS1.p3.1.m1.1.1.1.cmml" xref="S2.SS1.p3.1.m1.1.1">subscript</csymbol><ci id="S2.SS1.p3.1.m1.1.1.2.cmml" xref="S2.SS1.p3.1.m1.1.1.2">𝚺</ci><cn id="S2.SS1.p3.1.m1.1.1.3.cmml" type="integer" xref="S2.SS1.p3.1.m1.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p3.1.m1.1c">\operatorname{{\mathbf{\Sigma}}}_{1}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p3.1.m1.1d">bold_Σ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="\operatorname{{\mathbf{\Sigma}}}_{2}" class="ltx_Math" display="inline" id="S2.SS1.p3.2.m2.1"><semantics id="S2.SS1.p3.2.m2.1a"><msub id="S2.SS1.p3.2.m2.1.1" xref="S2.SS1.p3.2.m2.1.1.cmml"><mi id="S2.SS1.p3.2.m2.1.1.2" xref="S2.SS1.p3.2.m2.1.1.2.cmml">𝚺</mi><mn id="S2.SS1.p3.2.m2.1.1.3" xref="S2.SS1.p3.2.m2.1.1.3.cmml">2</mn></msub><annotation-xml encoding="MathML-Content" id="S2.SS1.p3.2.m2.1b"><apply id="S2.SS1.p3.2.m2.1.1.cmml" xref="S2.SS1.p3.2.m2.1.1"><csymbol cd="ambiguous" id="S2.SS1.p3.2.m2.1.1.1.cmml" xref="S2.SS1.p3.2.m2.1.1">subscript</csymbol><ci id="S2.SS1.p3.2.m2.1.1.2.cmml" xref="S2.SS1.p3.2.m2.1.1.2">𝚺</ci><cn id="S2.SS1.p3.2.m2.1.1.3.cmml" type="integer" xref="S2.SS1.p3.2.m2.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p3.2.m2.1c">\operatorname{{\mathbf{\Sigma}}}_{2}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p3.2.m2.1d">bold_Σ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math> in (<a class="ltx_ref" href="https://arxiv.org/html/2503.13782v1#S1.E1" title="In 1 Introduction ‣ Regularized Parameter Estimation in Mixed Model Trace Regression"><span class="ltx_text ltx_ref_tag">1</span></a>) are identified up to a constant. For <math alttext="i=1,\ldots,n" class="ltx_Math" display="inline" id="S2.SS1.p3.3.m3.3"><semantics id="S2.SS1.p3.3.m3.3a"><mrow id="S2.SS1.p3.3.m3.3.4" xref="S2.SS1.p3.3.m3.3.4.cmml"><mi id="S2.SS1.p3.3.m3.3.4.2" xref="S2.SS1.p3.3.m3.3.4.2.cmml">i</mi><mo id="S2.SS1.p3.3.m3.3.4.1" xref="S2.SS1.p3.3.m3.3.4.1.cmml">=</mo><mrow id="S2.SS1.p3.3.m3.3.4.3.2" xref="S2.SS1.p3.3.m3.3.4.3.1.cmml"><mn id="S2.SS1.p3.3.m3.1.1" xref="S2.SS1.p3.3.m3.1.1.cmml">1</mn><mo id="S2.SS1.p3.3.m3.3.4.3.2.1" xref="S2.SS1.p3.3.m3.3.4.3.1.cmml">,</mo><mi id="S2.SS1.p3.3.m3.2.2" mathvariant="normal" xref="S2.SS1.p3.3.m3.2.2.cmml">…</mi><mo id="S2.SS1.p3.3.m3.3.4.3.2.2" xref="S2.SS1.p3.3.m3.3.4.3.1.cmml">,</mo><mi id="S2.SS1.p3.3.m3.3.3" xref="S2.SS1.p3.3.m3.3.3.cmml">n</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p3.3.m3.3b"><apply id="S2.SS1.p3.3.m3.3.4.cmml" xref="S2.SS1.p3.3.m3.3.4"><eq id="S2.SS1.p3.3.m3.3.4.1.cmml" xref="S2.SS1.p3.3.m3.3.4.1"></eq><ci id="S2.SS1.p3.3.m3.3.4.2.cmml" 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start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ⊗ bold_Σ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> <cite class="ltx_cite ltx_citemacro_citep">(Dutilleul, <a class="ltx_ref" href="https://arxiv.org/html/2503.13782v1#bib.bib6" title="">1999</a>; Lu and Zimmerman, <a class="ltx_ref" href="https://arxiv.org/html/2503.13782v1#bib.bib23" title="">2005</a>; Hoff, <a class="ltx_ref" href="https://arxiv.org/html/2503.13782v1#bib.bib14" title="">2011</a>)</cite>. The covariance array remains unchanged if <math alttext="\operatorname{{\mathbf{\Sigma}}}_{1}" class="ltx_Math" display="inline" id="S2.SS1.p3.6.m6.1"><semantics id="S2.SS1.p3.6.m6.1a"><msub id="S2.SS1.p3.6.m6.1.1" xref="S2.SS1.p3.6.m6.1.1.cmml"><mi id="S2.SS1.p3.6.m6.1.1.2" xref="S2.SS1.p3.6.m6.1.1.2.cmml">𝚺</mi><mn id="S2.SS1.p3.6.m6.1.1.3" xref="S2.SS1.p3.6.m6.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S2.SS1.p3.6.m6.1b"><apply id="S2.SS1.p3.6.m6.1.1.cmml" xref="S2.SS1.p3.6.m6.1.1"><csymbol cd="ambiguous" id="S2.SS1.p3.6.m6.1.1.1.cmml" xref="S2.SS1.p3.6.m6.1.1">subscript</csymbol><ci id="S2.SS1.p3.6.m6.1.1.2.cmml" xref="S2.SS1.p3.6.m6.1.1.2">𝚺</ci><cn id="S2.SS1.p3.6.m6.1.1.3.cmml" type="integer" xref="S2.SS1.p3.6.m6.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p3.6.m6.1c">\operatorname{{\mathbf{\Sigma}}}_{1}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p3.6.m6.1d">bold_Σ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="\operatorname{{\mathbf{\Sigma}}}_{2}" class="ltx_Math" display="inline" id="S2.SS1.p3.7.m7.1"><semantics id="S2.SS1.p3.7.m7.1a"><msub id="S2.SS1.p3.7.m7.1.1" xref="S2.SS1.p3.7.m7.1.1.cmml"><mi id="S2.SS1.p3.7.m7.1.1.2" xref="S2.SS1.p3.7.m7.1.1.2.cmml">𝚺</mi><mn id="S2.SS1.p3.7.m7.1.1.3" xref="S2.SS1.p3.7.m7.1.1.3.cmml">2</mn></msub><annotation-xml encoding="MathML-Content" id="S2.SS1.p3.7.m7.1b"><apply id="S2.SS1.p3.7.m7.1.1.cmml" xref="S2.SS1.p3.7.m7.1.1"><csymbol cd="ambiguous" id="S2.SS1.p3.7.m7.1.1.1.cmml" xref="S2.SS1.p3.7.m7.1.1">subscript</csymbol><ci id="S2.SS1.p3.7.m7.1.1.2.cmml" xref="S2.SS1.p3.7.m7.1.1.2">𝚺</ci><cn id="S2.SS1.p3.7.m7.1.1.3.cmml" type="integer" xref="S2.SS1.p3.7.m7.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p3.7.m7.1c">\operatorname{{\mathbf{\Sigma}}}_{2}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p3.7.m7.1d">bold_Σ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math> are modified to <math alttext="c^{\prime}\operatorname{{\mathbf{\Sigma}}}_{1}" class="ltx_Math" display="inline" id="S2.SS1.p3.8.m8.1"><semantics id="S2.SS1.p3.8.m8.1a"><mrow id="S2.SS1.p3.8.m8.1.1" xref="S2.SS1.p3.8.m8.1.1.cmml"><msup id="S2.SS1.p3.8.m8.1.1.2" xref="S2.SS1.p3.8.m8.1.1.2.cmml"><mi id="S2.SS1.p3.8.m8.1.1.2.2" xref="S2.SS1.p3.8.m8.1.1.2.2.cmml">c</mi><mo id="S2.SS1.p3.8.m8.1.1.2.3" xref="S2.SS1.p3.8.m8.1.1.2.3.cmml">′</mo></msup><mo id="S2.SS1.p3.8.m8.1.1.1" lspace="0.167em" xref="S2.SS1.p3.8.m8.1.1.1.cmml">⁢</mo><msub id="S2.SS1.p3.8.m8.1.1.3" xref="S2.SS1.p3.8.m8.1.1.3.cmml"><mi id="S2.SS1.p3.8.m8.1.1.3.2" xref="S2.SS1.p3.8.m8.1.1.3.2.cmml">𝚺</mi><mn id="S2.SS1.p3.8.m8.1.1.3.3" xref="S2.SS1.p3.8.m8.1.1.3.3.cmml">1</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p3.8.m8.1b"><apply id="S2.SS1.p3.8.m8.1.1.cmml" xref="S2.SS1.p3.8.m8.1.1"><times id="S2.SS1.p3.8.m8.1.1.1.cmml" xref="S2.SS1.p3.8.m8.1.1.1"></times><apply id="S2.SS1.p3.8.m8.1.1.2.cmml" xref="S2.SS1.p3.8.m8.1.1.2"><csymbol cd="ambiguous" id="S2.SS1.p3.8.m8.1.1.2.1.cmml" xref="S2.SS1.p3.8.m8.1.1.2">superscript</csymbol><ci id="S2.SS1.p3.8.m8.1.1.2.2.cmml" xref="S2.SS1.p3.8.m8.1.1.2.2">𝑐</ci><ci id="S2.SS1.p3.8.m8.1.1.2.3.cmml" xref="S2.SS1.p3.8.m8.1.1.2.3">′</ci></apply><apply id="S2.SS1.p3.8.m8.1.1.3.cmml" xref="S2.SS1.p3.8.m8.1.1.3"><csymbol cd="ambiguous" id="S2.SS1.p3.8.m8.1.1.3.1.cmml" xref="S2.SS1.p3.8.m8.1.1.3">subscript</csymbol><ci id="S2.SS1.p3.8.m8.1.1.3.2.cmml" xref="S2.SS1.p3.8.m8.1.1.3.2">𝚺</ci><cn id="S2.SS1.p3.8.m8.1.1.3.3.cmml" type="integer" xref="S2.SS1.p3.8.m8.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p3.8.m8.1c">c^{\prime}\operatorname{{\mathbf{\Sigma}}}_{1}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p3.8.m8.1d">italic_c start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT bold_Σ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="\operatorname{{\mathbf{\Sigma}}}_{2}/c^{\prime}" class="ltx_Math" display="inline" id="S2.SS1.p3.9.m9.1"><semantics id="S2.SS1.p3.9.m9.1a"><mrow id="S2.SS1.p3.9.m9.1.1" xref="S2.SS1.p3.9.m9.1.1.cmml"><msub id="S2.SS1.p3.9.m9.1.1.2" xref="S2.SS1.p3.9.m9.1.1.2.cmml"><mi id="S2.SS1.p3.9.m9.1.1.2.2" xref="S2.SS1.p3.9.m9.1.1.2.2.cmml">𝚺</mi><mn id="S2.SS1.p3.9.m9.1.1.2.3" xref="S2.SS1.p3.9.m9.1.1.2.3.cmml">2</mn></msub><mo id="S2.SS1.p3.9.m9.1.1.1" lspace="0em" xref="S2.SS1.p3.9.m9.1.1.1.cmml">/</mo><msup id="S2.SS1.p3.9.m9.1.1.3" xref="S2.SS1.p3.9.m9.1.1.3.cmml"><mi id="S2.SS1.p3.9.m9.1.1.3.2" xref="S2.SS1.p3.9.m9.1.1.3.2.cmml">c</mi><mo id="S2.SS1.p3.9.m9.1.1.3.3" xref="S2.SS1.p3.9.m9.1.1.3.3.cmml">′</mo></msup></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p3.9.m9.1b"><apply id="S2.SS1.p3.9.m9.1.1.cmml" xref="S2.SS1.p3.9.m9.1.1"><divide id="S2.SS1.p3.9.m9.1.1.1.cmml" xref="S2.SS1.p3.9.m9.1.1.1"></divide><apply id="S2.SS1.p3.9.m9.1.1.2.cmml" xref="S2.SS1.p3.9.m9.1.1.2"><csymbol cd="ambiguous" id="S2.SS1.p3.9.m9.1.1.2.1.cmml" xref="S2.SS1.p3.9.m9.1.1.2">subscript</csymbol><ci id="S2.SS1.p3.9.m9.1.1.2.2.cmml" xref="S2.SS1.p3.9.m9.1.1.2.2">𝚺</ci><cn id="S2.SS1.p3.9.m9.1.1.2.3.cmml" type="integer" xref="S2.SS1.p3.9.m9.1.1.2.3">2</cn></apply><apply id="S2.SS1.p3.9.m9.1.1.3.cmml" xref="S2.SS1.p3.9.m9.1.1.3"><csymbol cd="ambiguous" id="S2.SS1.p3.9.m9.1.1.3.1.cmml" xref="S2.SS1.p3.9.m9.1.1.3">superscript</csymbol><ci id="S2.SS1.p3.9.m9.1.1.3.2.cmml" xref="S2.SS1.p3.9.m9.1.1.3.2">𝑐</ci><ci id="S2.SS1.p3.9.m9.1.1.3.3.cmml" xref="S2.SS1.p3.9.m9.1.1.3.3">′</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p3.9.m9.1c">\operatorname{{\mathbf{\Sigma}}}_{2}/c^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p3.9.m9.1d">bold_Σ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT / italic_c start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> for any <math alttext="c^{\prime}>0" class="ltx_Math" display="inline" id="S2.SS1.p3.10.m10.1"><semantics id="S2.SS1.p3.10.m10.1a"><mrow id="S2.SS1.p3.10.m10.1.1" xref="S2.SS1.p3.10.m10.1.1.cmml"><msup id="S2.SS1.p3.10.m10.1.1.2" xref="S2.SS1.p3.10.m10.1.1.2.cmml"><mi id="S2.SS1.p3.10.m10.1.1.2.2" xref="S2.SS1.p3.10.m10.1.1.2.2.cmml">c</mi><mo id="S2.SS1.p3.10.m10.1.1.2.3" xref="S2.SS1.p3.10.m10.1.1.2.3.cmml">′</mo></msup><mo id="S2.SS1.p3.10.m10.1.1.1" xref="S2.SS1.p3.10.m10.1.1.1.cmml">></mo><mn id="S2.SS1.p3.10.m10.1.1.3" xref="S2.SS1.p3.10.m10.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p3.10.m10.1b"><apply id="S2.SS1.p3.10.m10.1.1.cmml" xref="S2.SS1.p3.10.m10.1.1"><gt id="S2.SS1.p3.10.m10.1.1.1.cmml" xref="S2.SS1.p3.10.m10.1.1.1"></gt><apply id="S2.SS1.p3.10.m10.1.1.2.cmml" xref="S2.SS1.p3.10.m10.1.1.2"><csymbol cd="ambiguous" id="S2.SS1.p3.10.m10.1.1.2.1.cmml" xref="S2.SS1.p3.10.m10.1.1.2">superscript</csymbol><ci id="S2.SS1.p3.10.m10.1.1.2.2.cmml" xref="S2.SS1.p3.10.m10.1.1.2.2">𝑐</ci><ci id="S2.SS1.p3.10.m10.1.1.2.3.cmml" xref="S2.SS1.p3.10.m10.1.1.2.3">′</ci></apply><cn id="S2.SS1.p3.10.m10.1.1.3.cmml" type="integer" xref="S2.SS1.p3.10.m10.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p3.10.m10.1c">c^{\prime}>0</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p3.10.m10.1d">italic_c start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT > 0</annotation></semantics></math>. If <math alttext="\sigma^{2}" class="ltx_Math" display="inline" id="S2.SS1.p3.11.m11.1"><semantics id="S2.SS1.p3.11.m11.1a"><msup id="S2.SS1.p3.11.m11.1.1" xref="S2.SS1.p3.11.m11.1.1.cmml"><mi id="S2.SS1.p3.11.m11.1.1.2" xref="S2.SS1.p3.11.m11.1.1.2.cmml">σ</mi><mn id="S2.SS1.p3.11.m11.1.1.3" xref="S2.SS1.p3.11.m11.1.1.3.cmml">2</mn></msup><annotation-xml encoding="MathML-Content" id="S2.SS1.p3.11.m11.1b"><apply id="S2.SS1.p3.11.m11.1.1.cmml" xref="S2.SS1.p3.11.m11.1.1"><csymbol cd="ambiguous" id="S2.SS1.p3.11.m11.1.1.1.cmml" xref="S2.SS1.p3.11.m11.1.1">superscript</csymbol><ci id="S2.SS1.p3.11.m11.1.1.2.cmml" xref="S2.SS1.p3.11.m11.1.1.2">𝜎</ci><cn id="S2.SS1.p3.11.m11.1.1.3.cmml" type="integer" xref="S2.SS1.p3.11.m11.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p3.11.m11.1c">\sigma^{2}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p3.11.m11.1d">italic_σ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT</annotation></semantics></math> is the positive “overall variance” parameter, then <math alttext="\operatorname{{\mathbf{\Sigma}}}_{1}" class="ltx_Math" display="inline" id="S2.SS1.p3.12.m12.1"><semantics id="S2.SS1.p3.12.m12.1a"><msub id="S2.SS1.p3.12.m12.1.1" xref="S2.SS1.p3.12.m12.1.1.cmml"><mi id="S2.SS1.p3.12.m12.1.1.2" xref="S2.SS1.p3.12.m12.1.1.2.cmml">𝚺</mi><mn id="S2.SS1.p3.12.m12.1.1.3" xref="S2.SS1.p3.12.m12.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S2.SS1.p3.12.m12.1b"><apply id="S2.SS1.p3.12.m12.1.1.cmml" xref="S2.SS1.p3.12.m12.1.1"><csymbol cd="ambiguous" id="S2.SS1.p3.12.m12.1.1.1.cmml" xref="S2.SS1.p3.12.m12.1.1">subscript</csymbol><ci id="S2.SS1.p3.12.m12.1.1.2.cmml" xref="S2.SS1.p3.12.m12.1.1.2">𝚺</ci><cn id="S2.SS1.p3.12.m12.1.1.3.cmml" type="integer" xref="S2.SS1.p3.12.m12.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p3.12.m12.1c">\operatorname{{\mathbf{\Sigma}}}_{1}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p3.12.m12.1d">bold_Σ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="\operatorname{{\mathbf{\Sigma}}}_{2}" class="ltx_Math" display="inline" id="S2.SS1.p3.13.m13.1"><semantics id="S2.SS1.p3.13.m13.1a"><msub id="S2.SS1.p3.13.m13.1.1" xref="S2.SS1.p3.13.m13.1.1.cmml"><mi id="S2.SS1.p3.13.m13.1.1.2" xref="S2.SS1.p3.13.m13.1.1.2.cmml">𝚺</mi><mn id="S2.SS1.p3.13.m13.1.1.3" xref="S2.SS1.p3.13.m13.1.1.3.cmml">2</mn></msub><annotation-xml encoding="MathML-Content" id="S2.SS1.p3.13.m13.1b"><apply id="S2.SS1.p3.13.m13.1.1.cmml" xref="S2.SS1.p3.13.m13.1.1"><csymbol cd="ambiguous" id="S2.SS1.p3.13.m13.1.1.1.cmml" xref="S2.SS1.p3.13.m13.1.1">subscript</csymbol><ci id="S2.SS1.p3.13.m13.1.1.2.cmml" xref="S2.SS1.p3.13.m13.1.1.2">𝚺</ci><cn id="S2.SS1.p3.13.m13.1.1.3.cmml" type="integer" xref="S2.SS1.p3.13.m13.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p3.13.m13.1c">\operatorname{{\mathbf{\Sigma}}}_{2}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p3.13.m13.1d">bold_Σ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math> are identified by assuming that <math alttext="\det(\operatorname{{\mathbf{\Sigma}}}_{1})=\det(\operatorname{{\mathbf{\Sigma}% }}_{2})=1" class="ltx_Math" display="inline" id="S2.SS1.p3.14.m14.2"><semantics id="S2.SS1.p3.14.m14.2a"><mrow id="S2.SS1.p3.14.m14.2.2" xref="S2.SS1.p3.14.m14.2.2.cmml"><mrow id="S2.SS1.p3.14.m14.1.1.1" xref="S2.SS1.p3.14.m14.1.1.1.cmml"><mo id="S2.SS1.p3.14.m14.1.1.1.2" rspace="0em" xref="S2.SS1.p3.14.m14.1.1.1.2.cmml">det</mo><mrow id="S2.SS1.p3.14.m14.1.1.1.1.1" xref="S2.SS1.p3.14.m14.1.1.1.1.1.1.cmml"><mo id="S2.SS1.p3.14.m14.1.1.1.1.1.2" stretchy="false" xref="S2.SS1.p3.14.m14.1.1.1.1.1.1.cmml">(</mo><msub id="S2.SS1.p3.14.m14.1.1.1.1.1.1" xref="S2.SS1.p3.14.m14.1.1.1.1.1.1.cmml"><mi id="S2.SS1.p3.14.m14.1.1.1.1.1.1.2" xref="S2.SS1.p3.14.m14.1.1.1.1.1.1.2.cmml">𝚺</mi><mn id="S2.SS1.p3.14.m14.1.1.1.1.1.1.3" xref="S2.SS1.p3.14.m14.1.1.1.1.1.1.3.cmml">1</mn></msub><mo id="S2.SS1.p3.14.m14.1.1.1.1.1.3" stretchy="false" xref="S2.SS1.p3.14.m14.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S2.SS1.p3.14.m14.2.2.4" rspace="0.1389em" xref="S2.SS1.p3.14.m14.2.2.4.cmml">=</mo><mrow id="S2.SS1.p3.14.m14.2.2.2" xref="S2.SS1.p3.14.m14.2.2.2.cmml"><mo id="S2.SS1.p3.14.m14.2.2.2.2" lspace="0.1389em" rspace="0em" xref="S2.SS1.p3.14.m14.2.2.2.2.cmml">det</mo><mrow id="S2.SS1.p3.14.m14.2.2.2.1.1" xref="S2.SS1.p3.14.m14.2.2.2.1.1.1.cmml"><mo id="S2.SS1.p3.14.m14.2.2.2.1.1.2" stretchy="false" xref="S2.SS1.p3.14.m14.2.2.2.1.1.1.cmml">(</mo><msub id="S2.SS1.p3.14.m14.2.2.2.1.1.1" xref="S2.SS1.p3.14.m14.2.2.2.1.1.1.cmml"><mi id="S2.SS1.p3.14.m14.2.2.2.1.1.1.2" xref="S2.SS1.p3.14.m14.2.2.2.1.1.1.2.cmml">𝚺</mi><mn id="S2.SS1.p3.14.m14.2.2.2.1.1.1.3" xref="S2.SS1.p3.14.m14.2.2.2.1.1.1.3.cmml">2</mn></msub><mo id="S2.SS1.p3.14.m14.2.2.2.1.1.3" stretchy="false" xref="S2.SS1.p3.14.m14.2.2.2.1.1.1.cmml">)</mo></mrow></mrow><mo id="S2.SS1.p3.14.m14.2.2.5" xref="S2.SS1.p3.14.m14.2.2.5.cmml">=</mo><mn id="S2.SS1.p3.14.m14.2.2.6" xref="S2.SS1.p3.14.m14.2.2.6.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p3.14.m14.2b"><apply id="S2.SS1.p3.14.m14.2.2.cmml" xref="S2.SS1.p3.14.m14.2.2"><and id="S2.SS1.p3.14.m14.2.2a.cmml" xref="S2.SS1.p3.14.m14.2.2"></and><apply id="S2.SS1.p3.14.m14.2.2b.cmml" xref="S2.SS1.p3.14.m14.2.2"><eq id="S2.SS1.p3.14.m14.2.2.4.cmml" xref="S2.SS1.p3.14.m14.2.2.4"></eq><apply id="S2.SS1.p3.14.m14.1.1.1.cmml" xref="S2.SS1.p3.14.m14.1.1.1"><determinant id="S2.SS1.p3.14.m14.1.1.1.2.cmml" xref="S2.SS1.p3.14.m14.1.1.1.2"></determinant><apply id="S2.SS1.p3.14.m14.1.1.1.1.1.1.cmml" xref="S2.SS1.p3.14.m14.1.1.1.1.1"><csymbol cd="ambiguous" id="S2.SS1.p3.14.m14.1.1.1.1.1.1.1.cmml" xref="S2.SS1.p3.14.m14.1.1.1.1.1">subscript</csymbol><ci id="S2.SS1.p3.14.m14.1.1.1.1.1.1.2.cmml" xref="S2.SS1.p3.14.m14.1.1.1.1.1.1.2">𝚺</ci><cn id="S2.SS1.p3.14.m14.1.1.1.1.1.1.3.cmml" type="integer" xref="S2.SS1.p3.14.m14.1.1.1.1.1.1.3">1</cn></apply></apply><apply id="S2.SS1.p3.14.m14.2.2.2.cmml" xref="S2.SS1.p3.14.m14.2.2.2"><determinant id="S2.SS1.p3.14.m14.2.2.2.2.cmml" xref="S2.SS1.p3.14.m14.2.2.2.2"></determinant><apply id="S2.SS1.p3.14.m14.2.2.2.1.1.1.cmml" xref="S2.SS1.p3.14.m14.2.2.2.1.1"><csymbol cd="ambiguous" id="S2.SS1.p3.14.m14.2.2.2.1.1.1.1.cmml" xref="S2.SS1.p3.14.m14.2.2.2.1.1">subscript</csymbol><ci id="S2.SS1.p3.14.m14.2.2.2.1.1.1.2.cmml" xref="S2.SS1.p3.14.m14.2.2.2.1.1.1.2">𝚺</ci><cn id="S2.SS1.p3.14.m14.2.2.2.1.1.1.3.cmml" type="integer" xref="S2.SS1.p3.14.m14.2.2.2.1.1.1.3">2</cn></apply></apply></apply><apply id="S2.SS1.p3.14.m14.2.2c.cmml" xref="S2.SS1.p3.14.m14.2.2"><eq id="S2.SS1.p3.14.m14.2.2.5.cmml" xref="S2.SS1.p3.14.m14.2.2.5"></eq><share href="https://arxiv.org/html/2503.13782v1#S2.SS1.p3.14.m14.2.2.2.cmml" id="S2.SS1.p3.14.m14.2.2d.cmml" xref="S2.SS1.p3.14.m14.2.2"></share><cn id="S2.SS1.p3.14.m14.2.2.6.cmml" type="integer" xref="S2.SS1.p3.14.m14.2.2.6">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p3.14.m14.2c">\det(\operatorname{{\mathbf{\Sigma}}}_{1})=\det(\operatorname{{\mathbf{\Sigma}% }}_{2})=1</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p3.14.m14.2d">roman_det ( bold_Σ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) = roman_det ( bold_Σ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) = 1</annotation></semantics></math> and <math alttext="\text{Cov}\{\text{vec}(\operatorname{\mathbf{A}}_{i})\}=\tau^{2}\sigma^{2}% \operatorname{{\mathbf{\Sigma}}}_{2}\otimes\operatorname{{\mathbf{\Sigma}}}_{1}" class="ltx_Math" display="inline" id="S2.SS1.p3.15.m15.1"><semantics id="S2.SS1.p3.15.m15.1a"><mrow id="S2.SS1.p3.15.m15.1.1" xref="S2.SS1.p3.15.m15.1.1.cmml"><mrow id="S2.SS1.p3.15.m15.1.1.1" xref="S2.SS1.p3.15.m15.1.1.1.cmml"><mtext id="S2.SS1.p3.15.m15.1.1.1.3" xref="S2.SS1.p3.15.m15.1.1.1.3a.cmml">Cov</mtext><mo id="S2.SS1.p3.15.m15.1.1.1.2" xref="S2.SS1.p3.15.m15.1.1.1.2.cmml">⁢</mo><mrow id="S2.SS1.p3.15.m15.1.1.1.1.1" xref="S2.SS1.p3.15.m15.1.1.1.1.2.cmml"><mo id="S2.SS1.p3.15.m15.1.1.1.1.1.2" stretchy="false" xref="S2.SS1.p3.15.m15.1.1.1.1.2.cmml">{</mo><mrow id="S2.SS1.p3.15.m15.1.1.1.1.1.1" xref="S2.SS1.p3.15.m15.1.1.1.1.1.1.cmml"><mtext id="S2.SS1.p3.15.m15.1.1.1.1.1.1.3" xref="S2.SS1.p3.15.m15.1.1.1.1.1.1.3a.cmml">vec</mtext><mo id="S2.SS1.p3.15.m15.1.1.1.1.1.1.2" xref="S2.SS1.p3.15.m15.1.1.1.1.1.1.2.cmml">⁢</mo><mrow id="S2.SS1.p3.15.m15.1.1.1.1.1.1.1.1" xref="S2.SS1.p3.15.m15.1.1.1.1.1.1.1.1.1.cmml"><mo id="S2.SS1.p3.15.m15.1.1.1.1.1.1.1.1.2" stretchy="false" xref="S2.SS1.p3.15.m15.1.1.1.1.1.1.1.1.1.cmml">(</mo><msub id="S2.SS1.p3.15.m15.1.1.1.1.1.1.1.1.1" xref="S2.SS1.p3.15.m15.1.1.1.1.1.1.1.1.1.cmml"><mi id="S2.SS1.p3.15.m15.1.1.1.1.1.1.1.1.1.2" xref="S2.SS1.p3.15.m15.1.1.1.1.1.1.1.1.1.2.cmml">𝐀</mi><mi id="S2.SS1.p3.15.m15.1.1.1.1.1.1.1.1.1.3" xref="S2.SS1.p3.15.m15.1.1.1.1.1.1.1.1.1.3.cmml">i</mi></msub><mo id="S2.SS1.p3.15.m15.1.1.1.1.1.1.1.1.3" stretchy="false" xref="S2.SS1.p3.15.m15.1.1.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S2.SS1.p3.15.m15.1.1.1.1.1.3" stretchy="false" xref="S2.SS1.p3.15.m15.1.1.1.1.2.cmml">}</mo></mrow></mrow><mo id="S2.SS1.p3.15.m15.1.1.2" xref="S2.SS1.p3.15.m15.1.1.2.cmml">=</mo><mrow id="S2.SS1.p3.15.m15.1.1.3" xref="S2.SS1.p3.15.m15.1.1.3.cmml"><mrow id="S2.SS1.p3.15.m15.1.1.3.2" xref="S2.SS1.p3.15.m15.1.1.3.2.cmml"><msup id="S2.SS1.p3.15.m15.1.1.3.2.2" xref="S2.SS1.p3.15.m15.1.1.3.2.2.cmml"><mi id="S2.SS1.p3.15.m15.1.1.3.2.2.2" xref="S2.SS1.p3.15.m15.1.1.3.2.2.2.cmml">τ</mi><mn id="S2.SS1.p3.15.m15.1.1.3.2.2.3" xref="S2.SS1.p3.15.m15.1.1.3.2.2.3.cmml">2</mn></msup><mo id="S2.SS1.p3.15.m15.1.1.3.2.1" xref="S2.SS1.p3.15.m15.1.1.3.2.1.cmml">⁢</mo><msup id="S2.SS1.p3.15.m15.1.1.3.2.3" xref="S2.SS1.p3.15.m15.1.1.3.2.3.cmml"><mi id="S2.SS1.p3.15.m15.1.1.3.2.3.2" xref="S2.SS1.p3.15.m15.1.1.3.2.3.2.cmml">σ</mi><mn id="S2.SS1.p3.15.m15.1.1.3.2.3.3" xref="S2.SS1.p3.15.m15.1.1.3.2.3.3.cmml">2</mn></msup><mo id="S2.SS1.p3.15.m15.1.1.3.2.1a" lspace="0.167em" xref="S2.SS1.p3.15.m15.1.1.3.2.1.cmml">⁢</mo><msub id="S2.SS1.p3.15.m15.1.1.3.2.4" xref="S2.SS1.p3.15.m15.1.1.3.2.4.cmml"><mi id="S2.SS1.p3.15.m15.1.1.3.2.4.2" xref="S2.SS1.p3.15.m15.1.1.3.2.4.2.cmml">𝚺</mi><mn id="S2.SS1.p3.15.m15.1.1.3.2.4.3" xref="S2.SS1.p3.15.m15.1.1.3.2.4.3.cmml">2</mn></msub></mrow><mo id="S2.SS1.p3.15.m15.1.1.3.1" lspace="0em" rspace="0.222em" xref="S2.SS1.p3.15.m15.1.1.3.1.cmml">⊗</mo><msub id="S2.SS1.p3.15.m15.1.1.3.3" xref="S2.SS1.p3.15.m15.1.1.3.3.cmml"><mi id="S2.SS1.p3.15.m15.1.1.3.3.2" xref="S2.SS1.p3.15.m15.1.1.3.3.2.cmml">𝚺</mi><mn id="S2.SS1.p3.15.m15.1.1.3.3.3" xref="S2.SS1.p3.15.m15.1.1.3.3.3.cmml">1</mn></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p3.15.m15.1b"><apply id="S2.SS1.p3.15.m15.1.1.cmml" xref="S2.SS1.p3.15.m15.1.1"><eq id="S2.SS1.p3.15.m15.1.1.2.cmml" xref="S2.SS1.p3.15.m15.1.1.2"></eq><apply id="S2.SS1.p3.15.m15.1.1.1.cmml" xref="S2.SS1.p3.15.m15.1.1.1"><times id="S2.SS1.p3.15.m15.1.1.1.2.cmml" xref="S2.SS1.p3.15.m15.1.1.1.2"></times><ci id="S2.SS1.p3.15.m15.1.1.1.3a.cmml" xref="S2.SS1.p3.15.m15.1.1.1.3"><mtext id="S2.SS1.p3.15.m15.1.1.1.3.cmml" xref="S2.SS1.p3.15.m15.1.1.1.3">Cov</mtext></ci><set id="S2.SS1.p3.15.m15.1.1.1.1.2.cmml" xref="S2.SS1.p3.15.m15.1.1.1.1.1"><apply id="S2.SS1.p3.15.m15.1.1.1.1.1.1.cmml" xref="S2.SS1.p3.15.m15.1.1.1.1.1.1"><times id="S2.SS1.p3.15.m15.1.1.1.1.1.1.2.cmml" xref="S2.SS1.p3.15.m15.1.1.1.1.1.1.2"></times><ci id="S2.SS1.p3.15.m15.1.1.1.1.1.1.3a.cmml" xref="S2.SS1.p3.15.m15.1.1.1.1.1.1.3"><mtext id="S2.SS1.p3.15.m15.1.1.1.1.1.1.3.cmml" xref="S2.SS1.p3.15.m15.1.1.1.1.1.1.3">vec</mtext></ci><apply id="S2.SS1.p3.15.m15.1.1.1.1.1.1.1.1.1.cmml" xref="S2.SS1.p3.15.m15.1.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S2.SS1.p3.15.m15.1.1.1.1.1.1.1.1.1.1.cmml" xref="S2.SS1.p3.15.m15.1.1.1.1.1.1.1.1">subscript</csymbol><ci id="S2.SS1.p3.15.m15.1.1.1.1.1.1.1.1.1.2.cmml" xref="S2.SS1.p3.15.m15.1.1.1.1.1.1.1.1.1.2">𝐀</ci><ci id="S2.SS1.p3.15.m15.1.1.1.1.1.1.1.1.1.3.cmml" xref="S2.SS1.p3.15.m15.1.1.1.1.1.1.1.1.1.3">𝑖</ci></apply></apply></set></apply><apply id="S2.SS1.p3.15.m15.1.1.3.cmml" xref="S2.SS1.p3.15.m15.1.1.3"><csymbol cd="latexml" id="S2.SS1.p3.15.m15.1.1.3.1.cmml" xref="S2.SS1.p3.15.m15.1.1.3.1">tensor-product</csymbol><apply id="S2.SS1.p3.15.m15.1.1.3.2.cmml" xref="S2.SS1.p3.15.m15.1.1.3.2"><times id="S2.SS1.p3.15.m15.1.1.3.2.1.cmml" xref="S2.SS1.p3.15.m15.1.1.3.2.1"></times><apply id="S2.SS1.p3.15.m15.1.1.3.2.2.cmml" xref="S2.SS1.p3.15.m15.1.1.3.2.2"><csymbol cd="ambiguous" id="S2.SS1.p3.15.m15.1.1.3.2.2.1.cmml" xref="S2.SS1.p3.15.m15.1.1.3.2.2">superscript</csymbol><ci id="S2.SS1.p3.15.m15.1.1.3.2.2.2.cmml" xref="S2.SS1.p3.15.m15.1.1.3.2.2.2">𝜏</ci><cn id="S2.SS1.p3.15.m15.1.1.3.2.2.3.cmml" type="integer" xref="S2.SS1.p3.15.m15.1.1.3.2.2.3">2</cn></apply><apply id="S2.SS1.p3.15.m15.1.1.3.2.3.cmml" xref="S2.SS1.p3.15.m15.1.1.3.2.3"><csymbol cd="ambiguous" id="S2.SS1.p3.15.m15.1.1.3.2.3.1.cmml" xref="S2.SS1.p3.15.m15.1.1.3.2.3">superscript</csymbol><ci id="S2.SS1.p3.15.m15.1.1.3.2.3.2.cmml" xref="S2.SS1.p3.15.m15.1.1.3.2.3.2">𝜎</ci><cn id="S2.SS1.p3.15.m15.1.1.3.2.3.3.cmml" type="integer" xref="S2.SS1.p3.15.m15.1.1.3.2.3.3">2</cn></apply><apply id="S2.SS1.p3.15.m15.1.1.3.2.4.cmml" xref="S2.SS1.p3.15.m15.1.1.3.2.4"><csymbol cd="ambiguous" id="S2.SS1.p3.15.m15.1.1.3.2.4.1.cmml" xref="S2.SS1.p3.15.m15.1.1.3.2.4">subscript</csymbol><ci id="S2.SS1.p3.15.m15.1.1.3.2.4.2.cmml" xref="S2.SS1.p3.15.m15.1.1.3.2.4.2">𝚺</ci><cn id="S2.SS1.p3.15.m15.1.1.3.2.4.3.cmml" type="integer" xref="S2.SS1.p3.15.m15.1.1.3.2.4.3">2</cn></apply></apply><apply id="S2.SS1.p3.15.m15.1.1.3.3.cmml" xref="S2.SS1.p3.15.m15.1.1.3.3"><csymbol cd="ambiguous" id="S2.SS1.p3.15.m15.1.1.3.3.1.cmml" xref="S2.SS1.p3.15.m15.1.1.3.3">subscript</csymbol><ci id="S2.SS1.p3.15.m15.1.1.3.3.2.cmml" xref="S2.SS1.p3.15.m15.1.1.3.3.2">𝚺</ci><cn id="S2.SS1.p3.15.m15.1.1.3.3.3.cmml" type="integer" xref="S2.SS1.p3.15.m15.1.1.3.3.3">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p3.15.m15.1c">\text{Cov}\{\text{vec}(\operatorname{\mathbf{A}}_{i})\}=\tau^{2}\sigma^{2}% \operatorname{{\mathbf{\Sigma}}}_{2}\otimes\operatorname{{\mathbf{\Sigma}}}_{1}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p3.15.m15.1d">Cov { vec ( bold_A start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) } = italic_τ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_σ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT bold_Σ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ⊗ bold_Σ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> <cite class="ltx_cite ltx_citemacro_citep">(Gerard and Hoff, <a class="ltx_ref" href="https://arxiv.org/html/2503.13782v1#bib.bib11" title="">2015</a>)</cite>. This condition is inapplicable for identifying <math alttext="\operatorname{{\mathbf{\Sigma}}}_{1}" class="ltx_Math" display="inline" id="S2.SS1.p3.16.m16.1"><semantics id="S2.SS1.p3.16.m16.1a"><msub id="S2.SS1.p3.16.m16.1.1" xref="S2.SS1.p3.16.m16.1.1.cmml"><mi id="S2.SS1.p3.16.m16.1.1.2" xref="S2.SS1.p3.16.m16.1.1.2.cmml">𝚺</mi><mn id="S2.SS1.p3.16.m16.1.1.3" xref="S2.SS1.p3.16.m16.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S2.SS1.p3.16.m16.1b"><apply id="S2.SS1.p3.16.m16.1.1.cmml" xref="S2.SS1.p3.16.m16.1.1"><csymbol cd="ambiguous" id="S2.SS1.p3.16.m16.1.1.1.cmml" xref="S2.SS1.p3.16.m16.1.1">subscript</csymbol><ci id="S2.SS1.p3.16.m16.1.1.2.cmml" xref="S2.SS1.p3.16.m16.1.1.2">𝚺</ci><cn id="S2.SS1.p3.16.m16.1.1.3.cmml" type="integer" xref="S2.SS1.p3.16.m16.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p3.16.m16.1c">\operatorname{{\mathbf{\Sigma}}}_{1}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p3.16.m16.1d">bold_Σ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="\operatorname{{\mathbf{\Sigma}}}_{2}" class="ltx_Math" display="inline" id="S2.SS1.p3.17.m17.1"><semantics id="S2.SS1.p3.17.m17.1a"><msub id="S2.SS1.p3.17.m17.1.1" xref="S2.SS1.p3.17.m17.1.1.cmml"><mi id="S2.SS1.p3.17.m17.1.1.2" xref="S2.SS1.p3.17.m17.1.1.2.cmml">𝚺</mi><mn id="S2.SS1.p3.17.m17.1.1.3" xref="S2.SS1.p3.17.m17.1.1.3.cmml">2</mn></msub><annotation-xml encoding="MathML-Content" id="S2.SS1.p3.17.m17.1b"><apply id="S2.SS1.p3.17.m17.1.1.cmml" xref="S2.SS1.p3.17.m17.1.1"><csymbol cd="ambiguous" id="S2.SS1.p3.17.m17.1.1.1.cmml" xref="S2.SS1.p3.17.m17.1.1">subscript</csymbol><ci id="S2.SS1.p3.17.m17.1.1.2.cmml" xref="S2.SS1.p3.17.m17.1.1.2">𝚺</ci><cn id="S2.SS1.p3.17.m17.1.1.3.cmml" type="integer" xref="S2.SS1.p3.17.m17.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p3.17.m17.1c">\operatorname{{\mathbf{\Sigma}}}_{2}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p3.17.m17.1d">bold_Σ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math> in (<a class="ltx_ref" href="https://arxiv.org/html/2503.13782v1#S1.E1" title="In 1 Introduction ‣ Regularized Parameter Estimation in Mixed Model Trace Regression"><span class="ltx_text ltx_ref_tag">1</span></a>) because they are low-rank matrices.</p> </div> <div class="ltx_para" id="S2.SS1.p4"> <p class="ltx_p" id="S2.SS1.p4.22">We enforce the identifiability of <math alttext="\operatorname{{\mathbf{\Sigma}}}_{1}" class="ltx_Math" display="inline" id="S2.SS1.p4.1.m1.1"><semantics id="S2.SS1.p4.1.m1.1a"><msub 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" xref="S2.SS1.p4.1.m1.1.1.2.cmml">𝚺</mi><mn id="S2.SS1.p4.1.m1.1.1.3" xref="S2.SS1.p4.1.m1.1.1.3.cmml">1</mn></msub><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">subscript</csymbol><ci id="S2.SS1.p4.1.m1.1.1.2.cmml" xref="S2.SS1.p4.1.m1.1.1.2">𝚺</ci><cn id="S2.SS1.p4.1.m1.1.1.3.cmml" type="integer" xref="S2.SS1.p4.1.m1.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p4.1.m1.1c">\operatorname{{\mathbf{\Sigma}}}_{1}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p4.1.m1.1d">bold_Σ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="\operatorname{{\mathbf{\Sigma}}}_{2}" 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">𝚺</mi><mn id="S2.SS1.p4.2.m2.1.1.3" xref="S2.SS1.p4.2.m2.1.1.3.cmml">2</mn></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">𝚺</ci><cn id="S2.SS1.p4.2.m2.1.1.3.cmml" type="integer" xref="S2.SS1.p4.2.m2.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p4.2.m2.1c">\operatorname{{\mathbf{\Sigma}}}_{2}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p4.2.m2.1d">bold_Σ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math> based on <math alttext="\operatorname{\mathbf{L}}_{2}" class="ltx_Math" display="inline" id="S2.SS1.p4.3.m3.1"><semantics id="S2.SS1.p4.3.m3.1a"><msub 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" xref="S2.SS1.p4.3.m3.1.1.2.cmml">𝐋</mi><mn id="S2.SS1.p4.3.m3.1.1.3" xref="S2.SS1.p4.3.m3.1.1.3.cmml">2</mn></msub><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">subscript</csymbol><ci id="S2.SS1.p4.3.m3.1.1.2.cmml" xref="S2.SS1.p4.3.m3.1.1.2">𝐋</ci><cn id="S2.SS1.p4.3.m3.1.1.3.cmml" type="integer" xref="S2.SS1.p4.3.m3.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p4.3.m3.1c">\operatorname{\mathbf{L}}_{2}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p4.3.m3.1d">bold_L start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math>. Let <math alttext="(\operatorname{\mathbf{M}})_{ij}" class="ltx_Math" display="inline" id="S2.SS1.p4.4.m4.1"><semantics id="S2.SS1.p4.4.m4.1a"><msub id="S2.SS1.p4.4.m4.1.2" xref="S2.SS1.p4.4.m4.1.2.cmml"><mrow id="S2.SS1.p4.4.m4.1.2.2.2" xref="S2.SS1.p4.4.m4.1.2.cmml"><mo id="S2.SS1.p4.4.m4.1.2.2.2.1" stretchy="false" xref="S2.SS1.p4.4.m4.1.2.cmml">(</mo><mi id="S2.SS1.p4.4.m4.1.1" xref="S2.SS1.p4.4.m4.1.1.cmml">𝐌</mi><mo id="S2.SS1.p4.4.m4.1.2.2.2.2" stretchy="false" xref="S2.SS1.p4.4.m4.1.2.cmml">)</mo></mrow><mrow id="S2.SS1.p4.4.m4.1.2.3" xref="S2.SS1.p4.4.m4.1.2.3.cmml"><mi id="S2.SS1.p4.4.m4.1.2.3.2" xref="S2.SS1.p4.4.m4.1.2.3.2.cmml">i</mi><mo id="S2.SS1.p4.4.m4.1.2.3.1" xref="S2.SS1.p4.4.m4.1.2.3.1.cmml">⁢</mo><mi id="S2.SS1.p4.4.m4.1.2.3.3" xref="S2.SS1.p4.4.m4.1.2.3.3.cmml">j</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="S2.SS1.p4.4.m4.1b"><apply id="S2.SS1.p4.4.m4.1.2.cmml" xref="S2.SS1.p4.4.m4.1.2"><csymbol cd="ambiguous" id="S2.SS1.p4.4.m4.1.2.1.cmml" xref="S2.SS1.p4.4.m4.1.2">subscript</csymbol><ci id="S2.SS1.p4.4.m4.1.1.cmml" xref="S2.SS1.p4.4.m4.1.1">𝐌</ci><apply id="S2.SS1.p4.4.m4.1.2.3.cmml" xref="S2.SS1.p4.4.m4.1.2.3"><times id="S2.SS1.p4.4.m4.1.2.3.1.cmml" xref="S2.SS1.p4.4.m4.1.2.3.1"></times><ci id="S2.SS1.p4.4.m4.1.2.3.2.cmml" xref="S2.SS1.p4.4.m4.1.2.3.2">𝑖</ci><ci id="S2.SS1.p4.4.m4.1.2.3.3.cmml" xref="S2.SS1.p4.4.m4.1.2.3.3">𝑗</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p4.4.m4.1c">(\operatorname{\mathbf{M}})_{ij}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p4.4.m4.1d">( bold_M ) start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT</annotation></semantics></math> denote the <math alttext="(i,j)" class="ltx_Math" display="inline" id="S2.SS1.p4.5.m5.2"><semantics id="S2.SS1.p4.5.m5.2a"><mrow id="S2.SS1.p4.5.m5.2.3.2" xref="S2.SS1.p4.5.m5.2.3.1.cmml"><mo id="S2.SS1.p4.5.m5.2.3.2.1" stretchy="false" xref="S2.SS1.p4.5.m5.2.3.1.cmml">(</mo><mi id="S2.SS1.p4.5.m5.1.1" xref="S2.SS1.p4.5.m5.1.1.cmml">i</mi><mo id="S2.SS1.p4.5.m5.2.3.2.2" xref="S2.SS1.p4.5.m5.2.3.1.cmml">,</mo><mi id="S2.SS1.p4.5.m5.2.2" xref="S2.SS1.p4.5.m5.2.2.cmml">j</mi><mo id="S2.SS1.p4.5.m5.2.3.2.3" stretchy="false" xref="S2.SS1.p4.5.m5.2.3.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p4.5.m5.2b"><interval closure="open" id="S2.SS1.p4.5.m5.2.3.1.cmml" xref="S2.SS1.p4.5.m5.2.3.2"><ci id="S2.SS1.p4.5.m5.1.1.cmml" xref="S2.SS1.p4.5.m5.1.1">𝑖</ci><ci id="S2.SS1.p4.5.m5.2.2.cmml" xref="S2.SS1.p4.5.m5.2.2">𝑗</ci></interval></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p4.5.m5.2c">(i,j)</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p4.5.m5.2d">( italic_i , italic_j )</annotation></semantics></math>th element of some matrix <math alttext="\operatorname{\mathbf{M}}" class="ltx_Math" display="inline" id="S2.SS1.p4.6.m6.1"><semantics id="S2.SS1.p4.6.m6.1a"><mi id="S2.SS1.p4.6.m6.1.1" xref="S2.SS1.p4.6.m6.1.1.cmml">𝐌</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.p4.6.m6.1b"><ci id="S2.SS1.p4.6.m6.1.1.cmml" xref="S2.SS1.p4.6.m6.1.1">𝐌</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p4.6.m6.1c">\operatorname{\mathbf{M}}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p4.6.m6.1d">bold_M</annotation></semantics></math>. A popular identifiability condition assumes that <math alttext="(\operatorname{{\mathbf{\Sigma}}}_{2})_{11}=1" class="ltx_Math" display="inline" id="S2.SS1.p4.7.m7.1"><semantics id="S2.SS1.p4.7.m7.1a"><mrow id="S2.SS1.p4.7.m7.1.1" xref="S2.SS1.p4.7.m7.1.1.cmml"><msub id="S2.SS1.p4.7.m7.1.1.1" xref="S2.SS1.p4.7.m7.1.1.1.cmml"><mrow id="S2.SS1.p4.7.m7.1.1.1.1.1" xref="S2.SS1.p4.7.m7.1.1.1.1.1.1.cmml"><mo id="S2.SS1.p4.7.m7.1.1.1.1.1.2" stretchy="false" xref="S2.SS1.p4.7.m7.1.1.1.1.1.1.cmml">(</mo><msub id="S2.SS1.p4.7.m7.1.1.1.1.1.1" xref="S2.SS1.p4.7.m7.1.1.1.1.1.1.cmml"><mi id="S2.SS1.p4.7.m7.1.1.1.1.1.1.2" xref="S2.SS1.p4.7.m7.1.1.1.1.1.1.2.cmml">𝚺</mi><mn id="S2.SS1.p4.7.m7.1.1.1.1.1.1.3" xref="S2.SS1.p4.7.m7.1.1.1.1.1.1.3.cmml">2</mn></msub><mo id="S2.SS1.p4.7.m7.1.1.1.1.1.3" stretchy="false" xref="S2.SS1.p4.7.m7.1.1.1.1.1.1.cmml">)</mo></mrow><mn id="S2.SS1.p4.7.m7.1.1.1.3" xref="S2.SS1.p4.7.m7.1.1.1.3.cmml">11</mn></msub><mo id="S2.SS1.p4.7.m7.1.1.2" xref="S2.SS1.p4.7.m7.1.1.2.cmml">=</mo><mn id="S2.SS1.p4.7.m7.1.1.3" xref="S2.SS1.p4.7.m7.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p4.7.m7.1b"><apply id="S2.SS1.p4.7.m7.1.1.cmml" xref="S2.SS1.p4.7.m7.1.1"><eq id="S2.SS1.p4.7.m7.1.1.2.cmml" xref="S2.SS1.p4.7.m7.1.1.2"></eq><apply id="S2.SS1.p4.7.m7.1.1.1.cmml" xref="S2.SS1.p4.7.m7.1.1.1"><csymbol cd="ambiguous" id="S2.SS1.p4.7.m7.1.1.1.2.cmml" xref="S2.SS1.p4.7.m7.1.1.1">subscript</csymbol><apply id="S2.SS1.p4.7.m7.1.1.1.1.1.1.cmml" xref="S2.SS1.p4.7.m7.1.1.1.1.1"><csymbol cd="ambiguous" id="S2.SS1.p4.7.m7.1.1.1.1.1.1.1.cmml" xref="S2.SS1.p4.7.m7.1.1.1.1.1">subscript</csymbol><ci id="S2.SS1.p4.7.m7.1.1.1.1.1.1.2.cmml" xref="S2.SS1.p4.7.m7.1.1.1.1.1.1.2">𝚺</ci><cn id="S2.SS1.p4.7.m7.1.1.1.1.1.1.3.cmml" type="integer" xref="S2.SS1.p4.7.m7.1.1.1.1.1.1.3">2</cn></apply><cn id="S2.SS1.p4.7.m7.1.1.1.3.cmml" type="integer" xref="S2.SS1.p4.7.m7.1.1.1.3">11</cn></apply><cn id="S2.SS1.p4.7.m7.1.1.3.cmml" type="integer" xref="S2.SS1.p4.7.m7.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p4.7.m7.1c">(\operatorname{{\mathbf{\Sigma}}}_{2})_{11}=1</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p4.7.m7.1d">( bold_Σ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) start_POSTSUBSCRIPT 11 end_POSTSUBSCRIPT = 1</annotation></semantics></math>, implying that the first <math alttext="Q_{1}\times Q_{1}" class="ltx_Math" display="inline" id="S2.SS1.p4.8.m8.1"><semantics id="S2.SS1.p4.8.m8.1a"><mrow id="S2.SS1.p4.8.m8.1.1" xref="S2.SS1.p4.8.m8.1.1.cmml"><msub id="S2.SS1.p4.8.m8.1.1.2" xref="S2.SS1.p4.8.m8.1.1.2.cmml"><mi id="S2.SS1.p4.8.m8.1.1.2.2" xref="S2.SS1.p4.8.m8.1.1.2.2.cmml">Q</mi><mn id="S2.SS1.p4.8.m8.1.1.2.3" xref="S2.SS1.p4.8.m8.1.1.2.3.cmml">1</mn></msub><mo id="S2.SS1.p4.8.m8.1.1.1" lspace="0.222em" rspace="0.222em" xref="S2.SS1.p4.8.m8.1.1.1.cmml">×</mo><msub id="S2.SS1.p4.8.m8.1.1.3" xref="S2.SS1.p4.8.m8.1.1.3.cmml"><mi id="S2.SS1.p4.8.m8.1.1.3.2" xref="S2.SS1.p4.8.m8.1.1.3.2.cmml">Q</mi><mn id="S2.SS1.p4.8.m8.1.1.3.3" xref="S2.SS1.p4.8.m8.1.1.3.3.cmml">1</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p4.8.m8.1b"><apply id="S2.SS1.p4.8.m8.1.1.cmml" xref="S2.SS1.p4.8.m8.1.1"><times id="S2.SS1.p4.8.m8.1.1.1.cmml" xref="S2.SS1.p4.8.m8.1.1.1"></times><apply id="S2.SS1.p4.8.m8.1.1.2.cmml" xref="S2.SS1.p4.8.m8.1.1.2"><csymbol cd="ambiguous" id="S2.SS1.p4.8.m8.1.1.2.1.cmml" xref="S2.SS1.p4.8.m8.1.1.2">subscript</csymbol><ci id="S2.SS1.p4.8.m8.1.1.2.2.cmml" xref="S2.SS1.p4.8.m8.1.1.2.2">𝑄</ci><cn id="S2.SS1.p4.8.m8.1.1.2.3.cmml" type="integer" xref="S2.SS1.p4.8.m8.1.1.2.3">1</cn></apply><apply id="S2.SS1.p4.8.m8.1.1.3.cmml" xref="S2.SS1.p4.8.m8.1.1.3"><csymbol cd="ambiguous" id="S2.SS1.p4.8.m8.1.1.3.1.cmml" xref="S2.SS1.p4.8.m8.1.1.3">subscript</csymbol><ci id="S2.SS1.p4.8.m8.1.1.3.2.cmml" xref="S2.SS1.p4.8.m8.1.1.3.2">𝑄</ci><cn id="S2.SS1.p4.8.m8.1.1.3.3.cmml" type="integer" xref="S2.SS1.p4.8.m8.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p4.8.m8.1c">Q_{1}\times Q_{1}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p4.8.m8.1d">italic_Q start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT × italic_Q start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> block of <math alttext="\operatorname{{\mathbf{\Sigma}}}_{2}\otimes\operatorname{{\mathbf{\Sigma}}}_{1}" class="ltx_Math" display="inline" id="S2.SS1.p4.9.m9.1"><semantics id="S2.SS1.p4.9.m9.1a"><mrow id="S2.SS1.p4.9.m9.1.1" xref="S2.SS1.p4.9.m9.1.1.cmml"><msub id="S2.SS1.p4.9.m9.1.1.2" xref="S2.SS1.p4.9.m9.1.1.2.cmml"><mi id="S2.SS1.p4.9.m9.1.1.2.2" xref="S2.SS1.p4.9.m9.1.1.2.2.cmml">𝚺</mi><mn id="S2.SS1.p4.9.m9.1.1.2.3" xref="S2.SS1.p4.9.m9.1.1.2.3.cmml">2</mn></msub><mo id="S2.SS1.p4.9.m9.1.1.1" lspace="0em" rspace="0.222em" xref="S2.SS1.p4.9.m9.1.1.1.cmml">⊗</mo><msub id="S2.SS1.p4.9.m9.1.1.3" xref="S2.SS1.p4.9.m9.1.1.3.cmml"><mi id="S2.SS1.p4.9.m9.1.1.3.2" xref="S2.SS1.p4.9.m9.1.1.3.2.cmml">𝚺</mi><mn id="S2.SS1.p4.9.m9.1.1.3.3" xref="S2.SS1.p4.9.m9.1.1.3.3.cmml">1</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p4.9.m9.1b"><apply id="S2.SS1.p4.9.m9.1.1.cmml" xref="S2.SS1.p4.9.m9.1.1"><csymbol cd="latexml" id="S2.SS1.p4.9.m9.1.1.1.cmml" xref="S2.SS1.p4.9.m9.1.1.1">tensor-product</csymbol><apply id="S2.SS1.p4.9.m9.1.1.2.cmml" xref="S2.SS1.p4.9.m9.1.1.2"><csymbol cd="ambiguous" id="S2.SS1.p4.9.m9.1.1.2.1.cmml" xref="S2.SS1.p4.9.m9.1.1.2">subscript</csymbol><ci id="S2.SS1.p4.9.m9.1.1.2.2.cmml" xref="S2.SS1.p4.9.m9.1.1.2.2">𝚺</ci><cn id="S2.SS1.p4.9.m9.1.1.2.3.cmml" type="integer" xref="S2.SS1.p4.9.m9.1.1.2.3">2</cn></apply><apply id="S2.SS1.p4.9.m9.1.1.3.cmml" xref="S2.SS1.p4.9.m9.1.1.3"><csymbol cd="ambiguous" id="S2.SS1.p4.9.m9.1.1.3.1.cmml" xref="S2.SS1.p4.9.m9.1.1.3">subscript</csymbol><ci id="S2.SS1.p4.9.m9.1.1.3.2.cmml" xref="S2.SS1.p4.9.m9.1.1.3.2">𝚺</ci><cn id="S2.SS1.p4.9.m9.1.1.3.3.cmml" type="integer" xref="S2.SS1.p4.9.m9.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p4.9.m9.1c">\operatorname{{\mathbf{\Sigma}}}_{2}\otimes\operatorname{{\mathbf{\Sigma}}}_{1}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p4.9.m9.1d">bold_Σ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ⊗ bold_Σ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> equals <math alttext="\operatorname{{\mathbf{\Sigma}}}_{1}" class="ltx_Math" display="inline" id="S2.SS1.p4.10.m10.1"><semantics id="S2.SS1.p4.10.m10.1a"><msub id="S2.SS1.p4.10.m10.1.1" xref="S2.SS1.p4.10.m10.1.1.cmml"><mi id="S2.SS1.p4.10.m10.1.1.2" xref="S2.SS1.p4.10.m10.1.1.2.cmml">𝚺</mi><mn id="S2.SS1.p4.10.m10.1.1.3" xref="S2.SS1.p4.10.m10.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S2.SS1.p4.10.m10.1b"><apply id="S2.SS1.p4.10.m10.1.1.cmml" xref="S2.SS1.p4.10.m10.1.1"><csymbol cd="ambiguous" id="S2.SS1.p4.10.m10.1.1.1.cmml" xref="S2.SS1.p4.10.m10.1.1">subscript</csymbol><ci id="S2.SS1.p4.10.m10.1.1.2.cmml" xref="S2.SS1.p4.10.m10.1.1.2">𝚺</ci><cn id="S2.SS1.p4.10.m10.1.1.3.cmml" type="integer" xref="S2.SS1.p4.10.m10.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p4.10.m10.1c">\operatorname{{\mathbf{\Sigma}}}_{1}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p4.10.m10.1d">bold_Σ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> <cite class="ltx_cite ltx_citemacro_citep">(Dutilleul, <a class="ltx_ref" href="https://arxiv.org/html/2503.13782v1#bib.bib6" title="">1999</a>)</cite>. We can extend this condition to the case when <math alttext="(\operatorname{{\mathbf{\Sigma}}}_{2})_{jj}=1" class="ltx_Math" display="inline" id="S2.SS1.p4.11.m11.1"><semantics id="S2.SS1.p4.11.m11.1a"><mrow id="S2.SS1.p4.11.m11.1.1" xref="S2.SS1.p4.11.m11.1.1.cmml"><msub id="S2.SS1.p4.11.m11.1.1.1" xref="S2.SS1.p4.11.m11.1.1.1.cmml"><mrow id="S2.SS1.p4.11.m11.1.1.1.1.1" xref="S2.SS1.p4.11.m11.1.1.1.1.1.1.cmml"><mo id="S2.SS1.p4.11.m11.1.1.1.1.1.2" stretchy="false" xref="S2.SS1.p4.11.m11.1.1.1.1.1.1.cmml">(</mo><msub id="S2.SS1.p4.11.m11.1.1.1.1.1.1" xref="S2.SS1.p4.11.m11.1.1.1.1.1.1.cmml"><mi id="S2.SS1.p4.11.m11.1.1.1.1.1.1.2" xref="S2.SS1.p4.11.m11.1.1.1.1.1.1.2.cmml">𝚺</mi><mn id="S2.SS1.p4.11.m11.1.1.1.1.1.1.3" xref="S2.SS1.p4.11.m11.1.1.1.1.1.1.3.cmml">2</mn></msub><mo id="S2.SS1.p4.11.m11.1.1.1.1.1.3" stretchy="false" xref="S2.SS1.p4.11.m11.1.1.1.1.1.1.cmml">)</mo></mrow><mrow id="S2.SS1.p4.11.m11.1.1.1.3" xref="S2.SS1.p4.11.m11.1.1.1.3.cmml"><mi id="S2.SS1.p4.11.m11.1.1.1.3.2" xref="S2.SS1.p4.11.m11.1.1.1.3.2.cmml">j</mi><mo id="S2.SS1.p4.11.m11.1.1.1.3.1" xref="S2.SS1.p4.11.m11.1.1.1.3.1.cmml">⁢</mo><mi id="S2.SS1.p4.11.m11.1.1.1.3.3" xref="S2.SS1.p4.11.m11.1.1.1.3.3.cmml">j</mi></mrow></msub><mo id="S2.SS1.p4.11.m11.1.1.2" xref="S2.SS1.p4.11.m11.1.1.2.cmml">=</mo><mn id="S2.SS1.p4.11.m11.1.1.3" xref="S2.SS1.p4.11.m11.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p4.11.m11.1b"><apply id="S2.SS1.p4.11.m11.1.1.cmml" xref="S2.SS1.p4.11.m11.1.1"><eq id="S2.SS1.p4.11.m11.1.1.2.cmml" xref="S2.SS1.p4.11.m11.1.1.2"></eq><apply id="S2.SS1.p4.11.m11.1.1.1.cmml" xref="S2.SS1.p4.11.m11.1.1.1"><csymbol cd="ambiguous" id="S2.SS1.p4.11.m11.1.1.1.2.cmml" xref="S2.SS1.p4.11.m11.1.1.1">subscript</csymbol><apply id="S2.SS1.p4.11.m11.1.1.1.1.1.1.cmml" xref="S2.SS1.p4.11.m11.1.1.1.1.1"><csymbol cd="ambiguous" id="S2.SS1.p4.11.m11.1.1.1.1.1.1.1.cmml" xref="S2.SS1.p4.11.m11.1.1.1.1.1">subscript</csymbol><ci id="S2.SS1.p4.11.m11.1.1.1.1.1.1.2.cmml" xref="S2.SS1.p4.11.m11.1.1.1.1.1.1.2">𝚺</ci><cn id="S2.SS1.p4.11.m11.1.1.1.1.1.1.3.cmml" type="integer" xref="S2.SS1.p4.11.m11.1.1.1.1.1.1.3">2</cn></apply><apply id="S2.SS1.p4.11.m11.1.1.1.3.cmml" xref="S2.SS1.p4.11.m11.1.1.1.3"><times id="S2.SS1.p4.11.m11.1.1.1.3.1.cmml" xref="S2.SS1.p4.11.m11.1.1.1.3.1"></times><ci id="S2.SS1.p4.11.m11.1.1.1.3.2.cmml" xref="S2.SS1.p4.11.m11.1.1.1.3.2">𝑗</ci><ci id="S2.SS1.p4.11.m11.1.1.1.3.3.cmml" xref="S2.SS1.p4.11.m11.1.1.1.3.3">𝑗</ci></apply></apply><cn id="S2.SS1.p4.11.m11.1.1.3.cmml" type="integer" xref="S2.SS1.p4.11.m11.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p4.11.m11.1c">(\operatorname{{\mathbf{\Sigma}}}_{2})_{jj}=1</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p4.11.m11.1d">( bold_Σ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) start_POSTSUBSCRIPT italic_j italic_j end_POSTSUBSCRIPT = 1</annotation></semantics></math>, which results in the <math alttext="j" class="ltx_Math" display="inline" id="S2.SS1.p4.12.m12.1"><semantics id="S2.SS1.p4.12.m12.1a"><mi id="S2.SS1.p4.12.m12.1.1" xref="S2.SS1.p4.12.m12.1.1.cmml">j</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.p4.12.m12.1b"><ci id="S2.SS1.p4.12.m12.1.1.cmml" xref="S2.SS1.p4.12.m12.1.1">𝑗</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p4.12.m12.1c">j</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p4.12.m12.1d">italic_j</annotation></semantics></math>th diagonal <math alttext="Q_{1}\times Q_{1}" class="ltx_Math" display="inline" id="S2.SS1.p4.13.m13.1"><semantics id="S2.SS1.p4.13.m13.1a"><mrow id="S2.SS1.p4.13.m13.1.1" xref="S2.SS1.p4.13.m13.1.1.cmml"><msub id="S2.SS1.p4.13.m13.1.1.2" xref="S2.SS1.p4.13.m13.1.1.2.cmml"><mi id="S2.SS1.p4.13.m13.1.1.2.2" xref="S2.SS1.p4.13.m13.1.1.2.2.cmml">Q</mi><mn id="S2.SS1.p4.13.m13.1.1.2.3" xref="S2.SS1.p4.13.m13.1.1.2.3.cmml">1</mn></msub><mo id="S2.SS1.p4.13.m13.1.1.1" lspace="0.222em" rspace="0.222em" xref="S2.SS1.p4.13.m13.1.1.1.cmml">×</mo><msub id="S2.SS1.p4.13.m13.1.1.3" xref="S2.SS1.p4.13.m13.1.1.3.cmml"><mi id="S2.SS1.p4.13.m13.1.1.3.2" xref="S2.SS1.p4.13.m13.1.1.3.2.cmml">Q</mi><mn id="S2.SS1.p4.13.m13.1.1.3.3" xref="S2.SS1.p4.13.m13.1.1.3.3.cmml">1</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p4.13.m13.1b"><apply id="S2.SS1.p4.13.m13.1.1.cmml" xref="S2.SS1.p4.13.m13.1.1"><times id="S2.SS1.p4.13.m13.1.1.1.cmml" xref="S2.SS1.p4.13.m13.1.1.1"></times><apply id="S2.SS1.p4.13.m13.1.1.2.cmml" xref="S2.SS1.p4.13.m13.1.1.2"><csymbol cd="ambiguous" id="S2.SS1.p4.13.m13.1.1.2.1.cmml" xref="S2.SS1.p4.13.m13.1.1.2">subscript</csymbol><ci id="S2.SS1.p4.13.m13.1.1.2.2.cmml" xref="S2.SS1.p4.13.m13.1.1.2.2">𝑄</ci><cn id="S2.SS1.p4.13.m13.1.1.2.3.cmml" type="integer" xref="S2.SS1.p4.13.m13.1.1.2.3">1</cn></apply><apply id="S2.SS1.p4.13.m13.1.1.3.cmml" xref="S2.SS1.p4.13.m13.1.1.3"><csymbol cd="ambiguous" id="S2.SS1.p4.13.m13.1.1.3.1.cmml" xref="S2.SS1.p4.13.m13.1.1.3">subscript</csymbol><ci id="S2.SS1.p4.13.m13.1.1.3.2.cmml" xref="S2.SS1.p4.13.m13.1.1.3.2">𝑄</ci><cn id="S2.SS1.p4.13.m13.1.1.3.3.cmml" type="integer" xref="S2.SS1.p4.13.m13.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p4.13.m13.1c">Q_{1}\times Q_{1}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p4.13.m13.1d">italic_Q start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT × italic_Q start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> block of <math alttext="\operatorname{{\mathbf{\Sigma}}}_{2}\otimes\operatorname{{\mathbf{\Sigma}}}_{1}" class="ltx_Math" display="inline" id="S2.SS1.p4.14.m14.1"><semantics id="S2.SS1.p4.14.m14.1a"><mrow id="S2.SS1.p4.14.m14.1.1" xref="S2.SS1.p4.14.m14.1.1.cmml"><msub id="S2.SS1.p4.14.m14.1.1.2" xref="S2.SS1.p4.14.m14.1.1.2.cmml"><mi id="S2.SS1.p4.14.m14.1.1.2.2" xref="S2.SS1.p4.14.m14.1.1.2.2.cmml">𝚺</mi><mn id="S2.SS1.p4.14.m14.1.1.2.3" xref="S2.SS1.p4.14.m14.1.1.2.3.cmml">2</mn></msub><mo id="S2.SS1.p4.14.m14.1.1.1" lspace="0em" rspace="0.222em" xref="S2.SS1.p4.14.m14.1.1.1.cmml">⊗</mo><msub id="S2.SS1.p4.14.m14.1.1.3" xref="S2.SS1.p4.14.m14.1.1.3.cmml"><mi id="S2.SS1.p4.14.m14.1.1.3.2" xref="S2.SS1.p4.14.m14.1.1.3.2.cmml">𝚺</mi><mn id="S2.SS1.p4.14.m14.1.1.3.3" xref="S2.SS1.p4.14.m14.1.1.3.3.cmml">1</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p4.14.m14.1b"><apply id="S2.SS1.p4.14.m14.1.1.cmml" xref="S2.SS1.p4.14.m14.1.1"><csymbol cd="latexml" id="S2.SS1.p4.14.m14.1.1.1.cmml" xref="S2.SS1.p4.14.m14.1.1.1">tensor-product</csymbol><apply id="S2.SS1.p4.14.m14.1.1.2.cmml" xref="S2.SS1.p4.14.m14.1.1.2"><csymbol cd="ambiguous" id="S2.SS1.p4.14.m14.1.1.2.1.cmml" xref="S2.SS1.p4.14.m14.1.1.2">subscript</csymbol><ci id="S2.SS1.p4.14.m14.1.1.2.2.cmml" xref="S2.SS1.p4.14.m14.1.1.2.2">𝚺</ci><cn id="S2.SS1.p4.14.m14.1.1.2.3.cmml" type="integer" xref="S2.SS1.p4.14.m14.1.1.2.3">2</cn></apply><apply id="S2.SS1.p4.14.m14.1.1.3.cmml" xref="S2.SS1.p4.14.m14.1.1.3"><csymbol cd="ambiguous" id="S2.SS1.p4.14.m14.1.1.3.1.cmml" xref="S2.SS1.p4.14.m14.1.1.3">subscript</csymbol><ci id="S2.SS1.p4.14.m14.1.1.3.2.cmml" xref="S2.SS1.p4.14.m14.1.1.3.2">𝚺</ci><cn id="S2.SS1.p4.14.m14.1.1.3.3.cmml" type="integer" xref="S2.SS1.p4.14.m14.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p4.14.m14.1c">\operatorname{{\mathbf{\Sigma}}}_{2}\otimes\operatorname{{\mathbf{\Sigma}}}_{1}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p4.14.m14.1d">bold_Σ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ⊗ bold_Σ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> equaling <math alttext="\operatorname{{\mathbf{\Sigma}}}_{1}" class="ltx_Math" display="inline" id="S2.SS1.p4.15.m15.1"><semantics id="S2.SS1.p4.15.m15.1a"><msub id="S2.SS1.p4.15.m15.1.1" xref="S2.SS1.p4.15.m15.1.1.cmml"><mi id="S2.SS1.p4.15.m15.1.1.2" xref="S2.SS1.p4.15.m15.1.1.2.cmml">𝚺</mi><mn id="S2.SS1.p4.15.m15.1.1.3" xref="S2.SS1.p4.15.m15.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S2.SS1.p4.15.m15.1b"><apply id="S2.SS1.p4.15.m15.1.1.cmml" xref="S2.SS1.p4.15.m15.1.1"><csymbol cd="ambiguous" id="S2.SS1.p4.15.m15.1.1.1.cmml" xref="S2.SS1.p4.15.m15.1.1">subscript</csymbol><ci id="S2.SS1.p4.15.m15.1.1.2.cmml" xref="S2.SS1.p4.15.m15.1.1.2">𝚺</ci><cn id="S2.SS1.p4.15.m15.1.1.3.cmml" type="integer" xref="S2.SS1.p4.15.m15.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p4.15.m15.1c">\operatorname{{\mathbf{\Sigma}}}_{1}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p4.15.m15.1d">bold_Σ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math>. This condition is useful when <math alttext="\operatorname{{\mathbf{\Sigma}}}_{1}" class="ltx_Math" display="inline" id="S2.SS1.p4.16.m16.1"><semantics id="S2.SS1.p4.16.m16.1a"><msub id="S2.SS1.p4.16.m16.1.1" xref="S2.SS1.p4.16.m16.1.1.cmml"><mi id="S2.SS1.p4.16.m16.1.1.2" xref="S2.SS1.p4.16.m16.1.1.2.cmml">𝚺</mi><mn id="S2.SS1.p4.16.m16.1.1.3" xref="S2.SS1.p4.16.m16.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S2.SS1.p4.16.m16.1b"><apply id="S2.SS1.p4.16.m16.1.1.cmml" xref="S2.SS1.p4.16.m16.1.1"><csymbol cd="ambiguous" id="S2.SS1.p4.16.m16.1.1.1.cmml" xref="S2.SS1.p4.16.m16.1.1">subscript</csymbol><ci id="S2.SS1.p4.16.m16.1.1.2.cmml" xref="S2.SS1.p4.16.m16.1.1.2">𝚺</ci><cn id="S2.SS1.p4.16.m16.1.1.3.cmml" type="integer" xref="S2.SS1.p4.16.m16.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p4.16.m16.1c">\operatorname{{\mathbf{\Sigma}}}_{1}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p4.16.m16.1d">bold_Σ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="\operatorname{{\mathbf{\Sigma}}}_{2}" class="ltx_Math" display="inline" id="S2.SS1.p4.17.m17.1"><semantics id="S2.SS1.p4.17.m17.1a"><msub id="S2.SS1.p4.17.m17.1.1" xref="S2.SS1.p4.17.m17.1.1.cmml"><mi id="S2.SS1.p4.17.m17.1.1.2" xref="S2.SS1.p4.17.m17.1.1.2.cmml">𝚺</mi><mn id="S2.SS1.p4.17.m17.1.1.3" xref="S2.SS1.p4.17.m17.1.1.3.cmml">2</mn></msub><annotation-xml encoding="MathML-Content" id="S2.SS1.p4.17.m17.1b"><apply id="S2.SS1.p4.17.m17.1.1.cmml" xref="S2.SS1.p4.17.m17.1.1"><csymbol cd="ambiguous" id="S2.SS1.p4.17.m17.1.1.1.cmml" xref="S2.SS1.p4.17.m17.1.1">subscript</csymbol><ci id="S2.SS1.p4.17.m17.1.1.2.cmml" xref="S2.SS1.p4.17.m17.1.1.2">𝚺</ci><cn id="S2.SS1.p4.17.m17.1.1.3.cmml" type="integer" xref="S2.SS1.p4.17.m17.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p4.17.m17.1c">\operatorname{{\mathbf{\Sigma}}}_{2}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p4.17.m17.1d">bold_Σ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math> have full ranks and are estimated directly; however, MMTR estimates <math alttext="\operatorname{{\mathbf{\Sigma}}}_{1}" class="ltx_Math" display="inline" id="S2.SS1.p4.18.m18.1"><semantics id="S2.SS1.p4.18.m18.1a"><msub id="S2.SS1.p4.18.m18.1.1" xref="S2.SS1.p4.18.m18.1.1.cmml"><mi id="S2.SS1.p4.18.m18.1.1.2" xref="S2.SS1.p4.18.m18.1.1.2.cmml">𝚺</mi><mn id="S2.SS1.p4.18.m18.1.1.3" xref="S2.SS1.p4.18.m18.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S2.SS1.p4.18.m18.1b"><apply id="S2.SS1.p4.18.m18.1.1.cmml" xref="S2.SS1.p4.18.m18.1.1"><csymbol cd="ambiguous" id="S2.SS1.p4.18.m18.1.1.1.cmml" xref="S2.SS1.p4.18.m18.1.1">subscript</csymbol><ci id="S2.SS1.p4.18.m18.1.1.2.cmml" xref="S2.SS1.p4.18.m18.1.1.2">𝚺</ci><cn id="S2.SS1.p4.18.m18.1.1.3.cmml" type="integer" xref="S2.SS1.p4.18.m18.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p4.18.m18.1c">\operatorname{{\mathbf{\Sigma}}}_{1}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p4.18.m18.1d">bold_Σ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="\operatorname{{\mathbf{\Sigma}}}_{2}" class="ltx_Math" display="inline" id="S2.SS1.p4.19.m19.1"><semantics id="S2.SS1.p4.19.m19.1a"><msub id="S2.SS1.p4.19.m19.1.1" xref="S2.SS1.p4.19.m19.1.1.cmml"><mi id="S2.SS1.p4.19.m19.1.1.2" xref="S2.SS1.p4.19.m19.1.1.2.cmml">𝚺</mi><mn id="S2.SS1.p4.19.m19.1.1.3" xref="S2.SS1.p4.19.m19.1.1.3.cmml">2</mn></msub><annotation-xml encoding="MathML-Content" id="S2.SS1.p4.19.m19.1b"><apply id="S2.SS1.p4.19.m19.1.1.cmml" xref="S2.SS1.p4.19.m19.1.1"><csymbol cd="ambiguous" id="S2.SS1.p4.19.m19.1.1.1.cmml" xref="S2.SS1.p4.19.m19.1.1">subscript</csymbol><ci id="S2.SS1.p4.19.m19.1.1.2.cmml" xref="S2.SS1.p4.19.m19.1.1.2">𝚺</ci><cn id="S2.SS1.p4.19.m19.1.1.3.cmml" type="integer" xref="S2.SS1.p4.19.m19.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p4.19.m19.1c">\operatorname{{\mathbf{\Sigma}}}_{2}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p4.19.m19.1d">bold_Σ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math> via <math alttext="\operatorname{\mathbf{L}}_{1}" class="ltx_Math" display="inline" id="S2.SS1.p4.20.m20.1"><semantics id="S2.SS1.p4.20.m20.1a"><msub id="S2.SS1.p4.20.m20.1.1" xref="S2.SS1.p4.20.m20.1.1.cmml"><mi id="S2.SS1.p4.20.m20.1.1.2" xref="S2.SS1.p4.20.m20.1.1.2.cmml">𝐋</mi><mn id="S2.SS1.p4.20.m20.1.1.3" xref="S2.SS1.p4.20.m20.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S2.SS1.p4.20.m20.1b"><apply id="S2.SS1.p4.20.m20.1.1.cmml" xref="S2.SS1.p4.20.m20.1.1"><csymbol cd="ambiguous" id="S2.SS1.p4.20.m20.1.1.1.cmml" xref="S2.SS1.p4.20.m20.1.1">subscript</csymbol><ci id="S2.SS1.p4.20.m20.1.1.2.cmml" xref="S2.SS1.p4.20.m20.1.1.2">𝐋</ci><cn id="S2.SS1.p4.20.m20.1.1.3.cmml" type="integer" xref="S2.SS1.p4.20.m20.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p4.20.m20.1c">\operatorname{\mathbf{L}}_{1}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p4.20.m20.1d">bold_L start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="\operatorname{\mathbf{L}}_{2}" class="ltx_Math" display="inline" id="S2.SS1.p4.21.m21.1"><semantics id="S2.SS1.p4.21.m21.1a"><msub id="S2.SS1.p4.21.m21.1.1" xref="S2.SS1.p4.21.m21.1.1.cmml"><mi id="S2.SS1.p4.21.m21.1.1.2" xref="S2.SS1.p4.21.m21.1.1.2.cmml">𝐋</mi><mn id="S2.SS1.p4.21.m21.1.1.3" xref="S2.SS1.p4.21.m21.1.1.3.cmml">2</mn></msub><annotation-xml encoding="MathML-Content" id="S2.SS1.p4.21.m21.1b"><apply id="S2.SS1.p4.21.m21.1.1.cmml" xref="S2.SS1.p4.21.m21.1.1"><csymbol cd="ambiguous" id="S2.SS1.p4.21.m21.1.1.1.cmml" xref="S2.SS1.p4.21.m21.1.1">subscript</csymbol><ci id="S2.SS1.p4.21.m21.1.1.2.cmml" xref="S2.SS1.p4.21.m21.1.1.2">𝐋</ci><cn id="S2.SS1.p4.21.m21.1.1.3.cmml" type="integer" xref="S2.SS1.p4.21.m21.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p4.21.m21.1c">\operatorname{\mathbf{L}}_{2}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p4.21.m21.1d">bold_L start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math>, so this condition requires modification. The following proposition restates this identifiability condition using <math alttext="\operatorname{\mathbf{L}}_{2}" class="ltx_Math" display="inline" id="S2.SS1.p4.22.m22.1"><semantics id="S2.SS1.p4.22.m22.1a"><msub id="S2.SS1.p4.22.m22.1.1" xref="S2.SS1.p4.22.m22.1.1.cmml"><mi id="S2.SS1.p4.22.m22.1.1.2" xref="S2.SS1.p4.22.m22.1.1.2.cmml">𝐋</mi><mn id="S2.SS1.p4.22.m22.1.1.3" xref="S2.SS1.p4.22.m22.1.1.3.cmml">2</mn></msub><annotation-xml encoding="MathML-Content" id="S2.SS1.p4.22.m22.1b"><apply id="S2.SS1.p4.22.m22.1.1.cmml" xref="S2.SS1.p4.22.m22.1.1"><csymbol cd="ambiguous" id="S2.SS1.p4.22.m22.1.1.1.cmml" xref="S2.SS1.p4.22.m22.1.1">subscript</csymbol><ci id="S2.SS1.p4.22.m22.1.1.2.cmml" xref="S2.SS1.p4.22.m22.1.1.2">𝐋</ci><cn id="S2.SS1.p4.22.m22.1.1.3.cmml" type="integer" xref="S2.SS1.p4.22.m22.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p4.22.m22.1c">\operatorname{\mathbf{L}}_{2}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p4.22.m22.1d">bold_L start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math>, which is suited for high-dimensional MMTR applications.</p> </div> <div class="ltx_theorem ltx_theorem_proposition" id="S2.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="S2.Thmproposition1.1.1.1">Proposition 2.1</span></span></h6> <div class="ltx_para" id="S2.Thmproposition1.p1"> <p class="ltx_p" id="S2.Thmproposition1.p1.7"><span class="ltx_text ltx_font_italic" id="S2.Thmproposition1.p1.7.7">Assume that <math alttext="(\operatorname{\mathbf{L}}_{2})_{j1}=1" class="ltx_Math" display="inline" id="S2.Thmproposition1.p1.1.1.m1.1"><semantics id="S2.Thmproposition1.p1.1.1.m1.1a"><mrow id="S2.Thmproposition1.p1.1.1.m1.1.1" xref="S2.Thmproposition1.p1.1.1.m1.1.1.cmml"><msub id="S2.Thmproposition1.p1.1.1.m1.1.1.1" xref="S2.Thmproposition1.p1.1.1.m1.1.1.1.cmml"><mrow id="S2.Thmproposition1.p1.1.1.m1.1.1.1.1.1" xref="S2.Thmproposition1.p1.1.1.m1.1.1.1.1.1.1.cmml"><mo id="S2.Thmproposition1.p1.1.1.m1.1.1.1.1.1.2" stretchy="false" xref="S2.Thmproposition1.p1.1.1.m1.1.1.1.1.1.1.cmml">(</mo><msub id="S2.Thmproposition1.p1.1.1.m1.1.1.1.1.1.1" xref="S2.Thmproposition1.p1.1.1.m1.1.1.1.1.1.1.cmml"><mi id="S2.Thmproposition1.p1.1.1.m1.1.1.1.1.1.1.2" xref="S2.Thmproposition1.p1.1.1.m1.1.1.1.1.1.1.2.cmml">𝐋</mi><mn id="S2.Thmproposition1.p1.1.1.m1.1.1.1.1.1.1.3" xref="S2.Thmproposition1.p1.1.1.m1.1.1.1.1.1.1.3.cmml">2</mn></msub><mo id="S2.Thmproposition1.p1.1.1.m1.1.1.1.1.1.3" stretchy="false" xref="S2.Thmproposition1.p1.1.1.m1.1.1.1.1.1.1.cmml">)</mo></mrow><mrow id="S2.Thmproposition1.p1.1.1.m1.1.1.1.3" xref="S2.Thmproposition1.p1.1.1.m1.1.1.1.3.cmml"><mi id="S2.Thmproposition1.p1.1.1.m1.1.1.1.3.2" xref="S2.Thmproposition1.p1.1.1.m1.1.1.1.3.2.cmml">j</mi><mo id="S2.Thmproposition1.p1.1.1.m1.1.1.1.3.1" xref="S2.Thmproposition1.p1.1.1.m1.1.1.1.3.1.cmml">⁢</mo><mn id="S2.Thmproposition1.p1.1.1.m1.1.1.1.3.3" xref="S2.Thmproposition1.p1.1.1.m1.1.1.1.3.3.cmml">1</mn></mrow></msub><mo id="S2.Thmproposition1.p1.1.1.m1.1.1.2" xref="S2.Thmproposition1.p1.1.1.m1.1.1.2.cmml">=</mo><mn id="S2.Thmproposition1.p1.1.1.m1.1.1.3" xref="S2.Thmproposition1.p1.1.1.m1.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmproposition1.p1.1.1.m1.1b"><apply id="S2.Thmproposition1.p1.1.1.m1.1.1.cmml" xref="S2.Thmproposition1.p1.1.1.m1.1.1"><eq id="S2.Thmproposition1.p1.1.1.m1.1.1.2.cmml" xref="S2.Thmproposition1.p1.1.1.m1.1.1.2"></eq><apply id="S2.Thmproposition1.p1.1.1.m1.1.1.1.cmml" xref="S2.Thmproposition1.p1.1.1.m1.1.1.1"><csymbol cd="ambiguous" id="S2.Thmproposition1.p1.1.1.m1.1.1.1.2.cmml" xref="S2.Thmproposition1.p1.1.1.m1.1.1.1">subscript</csymbol><apply id="S2.Thmproposition1.p1.1.1.m1.1.1.1.1.1.1.cmml" xref="S2.Thmproposition1.p1.1.1.m1.1.1.1.1.1"><csymbol cd="ambiguous" id="S2.Thmproposition1.p1.1.1.m1.1.1.1.1.1.1.1.cmml" xref="S2.Thmproposition1.p1.1.1.m1.1.1.1.1.1">subscript</csymbol><ci id="S2.Thmproposition1.p1.1.1.m1.1.1.1.1.1.1.2.cmml" xref="S2.Thmproposition1.p1.1.1.m1.1.1.1.1.1.1.2">𝐋</ci><cn id="S2.Thmproposition1.p1.1.1.m1.1.1.1.1.1.1.3.cmml" type="integer" xref="S2.Thmproposition1.p1.1.1.m1.1.1.1.1.1.1.3">2</cn></apply><apply id="S2.Thmproposition1.p1.1.1.m1.1.1.1.3.cmml" xref="S2.Thmproposition1.p1.1.1.m1.1.1.1.3"><times id="S2.Thmproposition1.p1.1.1.m1.1.1.1.3.1.cmml" xref="S2.Thmproposition1.p1.1.1.m1.1.1.1.3.1"></times><ci id="S2.Thmproposition1.p1.1.1.m1.1.1.1.3.2.cmml" xref="S2.Thmproposition1.p1.1.1.m1.1.1.1.3.2">𝑗</ci><cn id="S2.Thmproposition1.p1.1.1.m1.1.1.1.3.3.cmml" type="integer" xref="S2.Thmproposition1.p1.1.1.m1.1.1.1.3.3">1</cn></apply></apply><cn id="S2.Thmproposition1.p1.1.1.m1.1.1.3.cmml" type="integer" xref="S2.Thmproposition1.p1.1.1.m1.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmproposition1.p1.1.1.m1.1c">(\operatorname{\mathbf{L}}_{2})_{j1}=1</annotation><annotation encoding="application/x-llamapun" id="S2.Thmproposition1.p1.1.1.m1.1d">( bold_L start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) start_POSTSUBSCRIPT italic_j 1 end_POSTSUBSCRIPT = 1</annotation></semantics></math> and <math alttext="(\operatorname{\mathbf{L}}_{2})_{j2}=\cdots=(\operatorname{\mathbf{L}}_{2})_{% jS_{2}}=0" class="ltx_Math" display="inline" id="S2.Thmproposition1.p1.2.2.m2.2"><semantics id="S2.Thmproposition1.p1.2.2.m2.2a"><mrow id="S2.Thmproposition1.p1.2.2.m2.2.2" xref="S2.Thmproposition1.p1.2.2.m2.2.2.cmml"><msub id="S2.Thmproposition1.p1.2.2.m2.1.1.1" xref="S2.Thmproposition1.p1.2.2.m2.1.1.1.cmml"><mrow id="S2.Thmproposition1.p1.2.2.m2.1.1.1.1.1" xref="S2.Thmproposition1.p1.2.2.m2.1.1.1.1.1.1.cmml"><mo id="S2.Thmproposition1.p1.2.2.m2.1.1.1.1.1.2" stretchy="false" xref="S2.Thmproposition1.p1.2.2.m2.1.1.1.1.1.1.cmml">(</mo><msub id="S2.Thmproposition1.p1.2.2.m2.1.1.1.1.1.1" xref="S2.Thmproposition1.p1.2.2.m2.1.1.1.1.1.1.cmml"><mi id="S2.Thmproposition1.p1.2.2.m2.1.1.1.1.1.1.2" xref="S2.Thmproposition1.p1.2.2.m2.1.1.1.1.1.1.2.cmml">𝐋</mi><mn id="S2.Thmproposition1.p1.2.2.m2.1.1.1.1.1.1.3" xref="S2.Thmproposition1.p1.2.2.m2.1.1.1.1.1.1.3.cmml">2</mn></msub><mo id="S2.Thmproposition1.p1.2.2.m2.1.1.1.1.1.3" stretchy="false" xref="S2.Thmproposition1.p1.2.2.m2.1.1.1.1.1.1.cmml">)</mo></mrow><mrow id="S2.Thmproposition1.p1.2.2.m2.1.1.1.3" xref="S2.Thmproposition1.p1.2.2.m2.1.1.1.3.cmml"><mi id="S2.Thmproposition1.p1.2.2.m2.1.1.1.3.2" xref="S2.Thmproposition1.p1.2.2.m2.1.1.1.3.2.cmml">j</mi><mo id="S2.Thmproposition1.p1.2.2.m2.1.1.1.3.1" xref="S2.Thmproposition1.p1.2.2.m2.1.1.1.3.1.cmml">⁢</mo><mn id="S2.Thmproposition1.p1.2.2.m2.1.1.1.3.3" xref="S2.Thmproposition1.p1.2.2.m2.1.1.1.3.3.cmml">2</mn></mrow></msub><mo id="S2.Thmproposition1.p1.2.2.m2.2.2.4" xref="S2.Thmproposition1.p1.2.2.m2.2.2.4.cmml">=</mo><mi id="S2.Thmproposition1.p1.2.2.m2.2.2.5" mathvariant="normal" xref="S2.Thmproposition1.p1.2.2.m2.2.2.5.cmml">⋯</mi><mo id="S2.Thmproposition1.p1.2.2.m2.2.2.6" xref="S2.Thmproposition1.p1.2.2.m2.2.2.6.cmml">=</mo><msub id="S2.Thmproposition1.p1.2.2.m2.2.2.2" xref="S2.Thmproposition1.p1.2.2.m2.2.2.2.cmml"><mrow id="S2.Thmproposition1.p1.2.2.m2.2.2.2.1.1" xref="S2.Thmproposition1.p1.2.2.m2.2.2.2.1.1.1.cmml"><mo id="S2.Thmproposition1.p1.2.2.m2.2.2.2.1.1.2" stretchy="false" xref="S2.Thmproposition1.p1.2.2.m2.2.2.2.1.1.1.cmml">(</mo><msub id="S2.Thmproposition1.p1.2.2.m2.2.2.2.1.1.1" xref="S2.Thmproposition1.p1.2.2.m2.2.2.2.1.1.1.cmml"><mi id="S2.Thmproposition1.p1.2.2.m2.2.2.2.1.1.1.2" xref="S2.Thmproposition1.p1.2.2.m2.2.2.2.1.1.1.2.cmml">𝐋</mi><mn id="S2.Thmproposition1.p1.2.2.m2.2.2.2.1.1.1.3" xref="S2.Thmproposition1.p1.2.2.m2.2.2.2.1.1.1.3.cmml">2</mn></msub><mo id="S2.Thmproposition1.p1.2.2.m2.2.2.2.1.1.3" stretchy="false" xref="S2.Thmproposition1.p1.2.2.m2.2.2.2.1.1.1.cmml">)</mo></mrow><mrow id="S2.Thmproposition1.p1.2.2.m2.2.2.2.3" xref="S2.Thmproposition1.p1.2.2.m2.2.2.2.3.cmml"><mi id="S2.Thmproposition1.p1.2.2.m2.2.2.2.3.2" xref="S2.Thmproposition1.p1.2.2.m2.2.2.2.3.2.cmml">j</mi><mo id="S2.Thmproposition1.p1.2.2.m2.2.2.2.3.1" xref="S2.Thmproposition1.p1.2.2.m2.2.2.2.3.1.cmml">⁢</mo><msub id="S2.Thmproposition1.p1.2.2.m2.2.2.2.3.3" xref="S2.Thmproposition1.p1.2.2.m2.2.2.2.3.3.cmml"><mi id="S2.Thmproposition1.p1.2.2.m2.2.2.2.3.3.2" xref="S2.Thmproposition1.p1.2.2.m2.2.2.2.3.3.2.cmml">S</mi><mn id="S2.Thmproposition1.p1.2.2.m2.2.2.2.3.3.3" xref="S2.Thmproposition1.p1.2.2.m2.2.2.2.3.3.3.cmml">2</mn></msub></mrow></msub><mo id="S2.Thmproposition1.p1.2.2.m2.2.2.7" xref="S2.Thmproposition1.p1.2.2.m2.2.2.7.cmml">=</mo><mn id="S2.Thmproposition1.p1.2.2.m2.2.2.8" xref="S2.Thmproposition1.p1.2.2.m2.2.2.8.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmproposition1.p1.2.2.m2.2b"><apply id="S2.Thmproposition1.p1.2.2.m2.2.2.cmml" xref="S2.Thmproposition1.p1.2.2.m2.2.2"><and id="S2.Thmproposition1.p1.2.2.m2.2.2a.cmml" xref="S2.Thmproposition1.p1.2.2.m2.2.2"></and><apply id="S2.Thmproposition1.p1.2.2.m2.2.2b.cmml" xref="S2.Thmproposition1.p1.2.2.m2.2.2"><eq id="S2.Thmproposition1.p1.2.2.m2.2.2.4.cmml" xref="S2.Thmproposition1.p1.2.2.m2.2.2.4"></eq><apply id="S2.Thmproposition1.p1.2.2.m2.1.1.1.cmml" xref="S2.Thmproposition1.p1.2.2.m2.1.1.1"><csymbol cd="ambiguous" id="S2.Thmproposition1.p1.2.2.m2.1.1.1.2.cmml" xref="S2.Thmproposition1.p1.2.2.m2.1.1.1">subscript</csymbol><apply id="S2.Thmproposition1.p1.2.2.m2.1.1.1.1.1.1.cmml" xref="S2.Thmproposition1.p1.2.2.m2.1.1.1.1.1"><csymbol cd="ambiguous" id="S2.Thmproposition1.p1.2.2.m2.1.1.1.1.1.1.1.cmml" xref="S2.Thmproposition1.p1.2.2.m2.1.1.1.1.1">subscript</csymbol><ci id="S2.Thmproposition1.p1.2.2.m2.1.1.1.1.1.1.2.cmml" xref="S2.Thmproposition1.p1.2.2.m2.1.1.1.1.1.1.2">𝐋</ci><cn id="S2.Thmproposition1.p1.2.2.m2.1.1.1.1.1.1.3.cmml" type="integer" xref="S2.Thmproposition1.p1.2.2.m2.1.1.1.1.1.1.3">2</cn></apply><apply id="S2.Thmproposition1.p1.2.2.m2.1.1.1.3.cmml" xref="S2.Thmproposition1.p1.2.2.m2.1.1.1.3"><times id="S2.Thmproposition1.p1.2.2.m2.1.1.1.3.1.cmml" xref="S2.Thmproposition1.p1.2.2.m2.1.1.1.3.1"></times><ci id="S2.Thmproposition1.p1.2.2.m2.1.1.1.3.2.cmml" xref="S2.Thmproposition1.p1.2.2.m2.1.1.1.3.2">𝑗</ci><cn id="S2.Thmproposition1.p1.2.2.m2.1.1.1.3.3.cmml" type="integer" xref="S2.Thmproposition1.p1.2.2.m2.1.1.1.3.3">2</cn></apply></apply><ci id="S2.Thmproposition1.p1.2.2.m2.2.2.5.cmml" xref="S2.Thmproposition1.p1.2.2.m2.2.2.5">⋯</ci></apply><apply id="S2.Thmproposition1.p1.2.2.m2.2.2c.cmml" xref="S2.Thmproposition1.p1.2.2.m2.2.2"><eq id="S2.Thmproposition1.p1.2.2.m2.2.2.6.cmml" xref="S2.Thmproposition1.p1.2.2.m2.2.2.6"></eq><share href="https://arxiv.org/html/2503.13782v1#S2.Thmproposition1.p1.2.2.m2.2.2.5.cmml" id="S2.Thmproposition1.p1.2.2.m2.2.2d.cmml" xref="S2.Thmproposition1.p1.2.2.m2.2.2"></share><apply id="S2.Thmproposition1.p1.2.2.m2.2.2.2.cmml" xref="S2.Thmproposition1.p1.2.2.m2.2.2.2"><csymbol cd="ambiguous" id="S2.Thmproposition1.p1.2.2.m2.2.2.2.2.cmml" xref="S2.Thmproposition1.p1.2.2.m2.2.2.2">subscript</csymbol><apply id="S2.Thmproposition1.p1.2.2.m2.2.2.2.1.1.1.cmml" xref="S2.Thmproposition1.p1.2.2.m2.2.2.2.1.1"><csymbol cd="ambiguous" id="S2.Thmproposition1.p1.2.2.m2.2.2.2.1.1.1.1.cmml" xref="S2.Thmproposition1.p1.2.2.m2.2.2.2.1.1">subscript</csymbol><ci id="S2.Thmproposition1.p1.2.2.m2.2.2.2.1.1.1.2.cmml" xref="S2.Thmproposition1.p1.2.2.m2.2.2.2.1.1.1.2">𝐋</ci><cn id="S2.Thmproposition1.p1.2.2.m2.2.2.2.1.1.1.3.cmml" type="integer" xref="S2.Thmproposition1.p1.2.2.m2.2.2.2.1.1.1.3">2</cn></apply><apply id="S2.Thmproposition1.p1.2.2.m2.2.2.2.3.cmml" xref="S2.Thmproposition1.p1.2.2.m2.2.2.2.3"><times id="S2.Thmproposition1.p1.2.2.m2.2.2.2.3.1.cmml" xref="S2.Thmproposition1.p1.2.2.m2.2.2.2.3.1"></times><ci id="S2.Thmproposition1.p1.2.2.m2.2.2.2.3.2.cmml" xref="S2.Thmproposition1.p1.2.2.m2.2.2.2.3.2">𝑗</ci><apply id="S2.Thmproposition1.p1.2.2.m2.2.2.2.3.3.cmml" xref="S2.Thmproposition1.p1.2.2.m2.2.2.2.3.3"><csymbol cd="ambiguous" id="S2.Thmproposition1.p1.2.2.m2.2.2.2.3.3.1.cmml" xref="S2.Thmproposition1.p1.2.2.m2.2.2.2.3.3">subscript</csymbol><ci id="S2.Thmproposition1.p1.2.2.m2.2.2.2.3.3.2.cmml" xref="S2.Thmproposition1.p1.2.2.m2.2.2.2.3.3.2">𝑆</ci><cn id="S2.Thmproposition1.p1.2.2.m2.2.2.2.3.3.3.cmml" type="integer" xref="S2.Thmproposition1.p1.2.2.m2.2.2.2.3.3.3">2</cn></apply></apply></apply></apply><apply id="S2.Thmproposition1.p1.2.2.m2.2.2e.cmml" xref="S2.Thmproposition1.p1.2.2.m2.2.2"><eq id="S2.Thmproposition1.p1.2.2.m2.2.2.7.cmml" xref="S2.Thmproposition1.p1.2.2.m2.2.2.7"></eq><share href="https://arxiv.org/html/2503.13782v1#S2.Thmproposition1.p1.2.2.m2.2.2.2.cmml" id="S2.Thmproposition1.p1.2.2.m2.2.2f.cmml" xref="S2.Thmproposition1.p1.2.2.m2.2.2"></share><cn id="S2.Thmproposition1.p1.2.2.m2.2.2.8.cmml" type="integer" xref="S2.Thmproposition1.p1.2.2.m2.2.2.8">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmproposition1.p1.2.2.m2.2c">(\operatorname{\mathbf{L}}_{2})_{j2}=\cdots=(\operatorname{\mathbf{L}}_{2})_{% jS_{2}}=0</annotation><annotation encoding="application/x-llamapun" id="S2.Thmproposition1.p1.2.2.m2.2d">( bold_L start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) start_POSTSUBSCRIPT italic_j 2 end_POSTSUBSCRIPT = ⋯ = ( bold_L start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) start_POSTSUBSCRIPT italic_j italic_S start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT = 0</annotation></semantics></math>. Then, <math alttext="(\operatorname{{\mathbf{\Sigma}}}_{2})_{jj}=1" class="ltx_Math" display="inline" id="S2.Thmproposition1.p1.3.3.m3.1"><semantics id="S2.Thmproposition1.p1.3.3.m3.1a"><mrow id="S2.Thmproposition1.p1.3.3.m3.1.1" xref="S2.Thmproposition1.p1.3.3.m3.1.1.cmml"><msub id="S2.Thmproposition1.p1.3.3.m3.1.1.1" xref="S2.Thmproposition1.p1.3.3.m3.1.1.1.cmml"><mrow id="S2.Thmproposition1.p1.3.3.m3.1.1.1.1.1" xref="S2.Thmproposition1.p1.3.3.m3.1.1.1.1.1.1.cmml"><mo id="S2.Thmproposition1.p1.3.3.m3.1.1.1.1.1.2" stretchy="false" xref="S2.Thmproposition1.p1.3.3.m3.1.1.1.1.1.1.cmml">(</mo><msub id="S2.Thmproposition1.p1.3.3.m3.1.1.1.1.1.1" xref="S2.Thmproposition1.p1.3.3.m3.1.1.1.1.1.1.cmml"><mi id="S2.Thmproposition1.p1.3.3.m3.1.1.1.1.1.1.2" xref="S2.Thmproposition1.p1.3.3.m3.1.1.1.1.1.1.2.cmml">𝚺</mi><mn id="S2.Thmproposition1.p1.3.3.m3.1.1.1.1.1.1.3" xref="S2.Thmproposition1.p1.3.3.m3.1.1.1.1.1.1.3.cmml">2</mn></msub><mo id="S2.Thmproposition1.p1.3.3.m3.1.1.1.1.1.3" stretchy="false" xref="S2.Thmproposition1.p1.3.3.m3.1.1.1.1.1.1.cmml">)</mo></mrow><mrow id="S2.Thmproposition1.p1.3.3.m3.1.1.1.3" xref="S2.Thmproposition1.p1.3.3.m3.1.1.1.3.cmml"><mi id="S2.Thmproposition1.p1.3.3.m3.1.1.1.3.2" xref="S2.Thmproposition1.p1.3.3.m3.1.1.1.3.2.cmml">j</mi><mo id="S2.Thmproposition1.p1.3.3.m3.1.1.1.3.1" xref="S2.Thmproposition1.p1.3.3.m3.1.1.1.3.1.cmml">⁢</mo><mi id="S2.Thmproposition1.p1.3.3.m3.1.1.1.3.3" xref="S2.Thmproposition1.p1.3.3.m3.1.1.1.3.3.cmml">j</mi></mrow></msub><mo id="S2.Thmproposition1.p1.3.3.m3.1.1.2" xref="S2.Thmproposition1.p1.3.3.m3.1.1.2.cmml">=</mo><mn id="S2.Thmproposition1.p1.3.3.m3.1.1.3" xref="S2.Thmproposition1.p1.3.3.m3.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmproposition1.p1.3.3.m3.1b"><apply id="S2.Thmproposition1.p1.3.3.m3.1.1.cmml" xref="S2.Thmproposition1.p1.3.3.m3.1.1"><eq id="S2.Thmproposition1.p1.3.3.m3.1.1.2.cmml" xref="S2.Thmproposition1.p1.3.3.m3.1.1.2"></eq><apply id="S2.Thmproposition1.p1.3.3.m3.1.1.1.cmml" xref="S2.Thmproposition1.p1.3.3.m3.1.1.1"><csymbol cd="ambiguous" id="S2.Thmproposition1.p1.3.3.m3.1.1.1.2.cmml" xref="S2.Thmproposition1.p1.3.3.m3.1.1.1">subscript</csymbol><apply id="S2.Thmproposition1.p1.3.3.m3.1.1.1.1.1.1.cmml" xref="S2.Thmproposition1.p1.3.3.m3.1.1.1.1.1"><csymbol cd="ambiguous" id="S2.Thmproposition1.p1.3.3.m3.1.1.1.1.1.1.1.cmml" xref="S2.Thmproposition1.p1.3.3.m3.1.1.1.1.1">subscript</csymbol><ci id="S2.Thmproposition1.p1.3.3.m3.1.1.1.1.1.1.2.cmml" xref="S2.Thmproposition1.p1.3.3.m3.1.1.1.1.1.1.2">𝚺</ci><cn id="S2.Thmproposition1.p1.3.3.m3.1.1.1.1.1.1.3.cmml" type="integer" xref="S2.Thmproposition1.p1.3.3.m3.1.1.1.1.1.1.3">2</cn></apply><apply id="S2.Thmproposition1.p1.3.3.m3.1.1.1.3.cmml" xref="S2.Thmproposition1.p1.3.3.m3.1.1.1.3"><times id="S2.Thmproposition1.p1.3.3.m3.1.1.1.3.1.cmml" xref="S2.Thmproposition1.p1.3.3.m3.1.1.1.3.1"></times><ci id="S2.Thmproposition1.p1.3.3.m3.1.1.1.3.2.cmml" xref="S2.Thmproposition1.p1.3.3.m3.1.1.1.3.2">𝑗</ci><ci id="S2.Thmproposition1.p1.3.3.m3.1.1.1.3.3.cmml" xref="S2.Thmproposition1.p1.3.3.m3.1.1.1.3.3">𝑗</ci></apply></apply><cn id="S2.Thmproposition1.p1.3.3.m3.1.1.3.cmml" type="integer" xref="S2.Thmproposition1.p1.3.3.m3.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmproposition1.p1.3.3.m3.1c">(\operatorname{{\mathbf{\Sigma}}}_{2})_{jj}=1</annotation><annotation encoding="application/x-llamapun" id="S2.Thmproposition1.p1.3.3.m3.1d">( bold_Σ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) start_POSTSUBSCRIPT italic_j italic_j end_POSTSUBSCRIPT = 1</annotation></semantics></math> and the <math alttext="j" class="ltx_Math" display="inline" id="S2.Thmproposition1.p1.4.4.m4.1"><semantics id="S2.Thmproposition1.p1.4.4.m4.1a"><mi id="S2.Thmproposition1.p1.4.4.m4.1.1" xref="S2.Thmproposition1.p1.4.4.m4.1.1.cmml">j</mi><annotation-xml encoding="MathML-Content" id="S2.Thmproposition1.p1.4.4.m4.1b"><ci id="S2.Thmproposition1.p1.4.4.m4.1.1.cmml" xref="S2.Thmproposition1.p1.4.4.m4.1.1">𝑗</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmproposition1.p1.4.4.m4.1c">j</annotation><annotation encoding="application/x-llamapun" id="S2.Thmproposition1.p1.4.4.m4.1d">italic_j</annotation></semantics></math>th diagonal <math alttext="Q_{1}\times Q_{1}" class="ltx_Math" display="inline" id="S2.Thmproposition1.p1.5.5.m5.1"><semantics id="S2.Thmproposition1.p1.5.5.m5.1a"><mrow id="S2.Thmproposition1.p1.5.5.m5.1.1" xref="S2.Thmproposition1.p1.5.5.m5.1.1.cmml"><msub id="S2.Thmproposition1.p1.5.5.m5.1.1.2" xref="S2.Thmproposition1.p1.5.5.m5.1.1.2.cmml"><mi id="S2.Thmproposition1.p1.5.5.m5.1.1.2.2" xref="S2.Thmproposition1.p1.5.5.m5.1.1.2.2.cmml">Q</mi><mn id="S2.Thmproposition1.p1.5.5.m5.1.1.2.3" xref="S2.Thmproposition1.p1.5.5.m5.1.1.2.3.cmml">1</mn></msub><mo id="S2.Thmproposition1.p1.5.5.m5.1.1.1" lspace="0.222em" rspace="0.222em" xref="S2.Thmproposition1.p1.5.5.m5.1.1.1.cmml">×</mo><msub id="S2.Thmproposition1.p1.5.5.m5.1.1.3" xref="S2.Thmproposition1.p1.5.5.m5.1.1.3.cmml"><mi id="S2.Thmproposition1.p1.5.5.m5.1.1.3.2" xref="S2.Thmproposition1.p1.5.5.m5.1.1.3.2.cmml">Q</mi><mn id="S2.Thmproposition1.p1.5.5.m5.1.1.3.3" xref="S2.Thmproposition1.p1.5.5.m5.1.1.3.3.cmml">1</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmproposition1.p1.5.5.m5.1b"><apply id="S2.Thmproposition1.p1.5.5.m5.1.1.cmml" xref="S2.Thmproposition1.p1.5.5.m5.1.1"><times id="S2.Thmproposition1.p1.5.5.m5.1.1.1.cmml" xref="S2.Thmproposition1.p1.5.5.m5.1.1.1"></times><apply id="S2.Thmproposition1.p1.5.5.m5.1.1.2.cmml" xref="S2.Thmproposition1.p1.5.5.m5.1.1.2"><csymbol cd="ambiguous" id="S2.Thmproposition1.p1.5.5.m5.1.1.2.1.cmml" xref="S2.Thmproposition1.p1.5.5.m5.1.1.2">subscript</csymbol><ci id="S2.Thmproposition1.p1.5.5.m5.1.1.2.2.cmml" xref="S2.Thmproposition1.p1.5.5.m5.1.1.2.2">𝑄</ci><cn id="S2.Thmproposition1.p1.5.5.m5.1.1.2.3.cmml" type="integer" xref="S2.Thmproposition1.p1.5.5.m5.1.1.2.3">1</cn></apply><apply id="S2.Thmproposition1.p1.5.5.m5.1.1.3.cmml" xref="S2.Thmproposition1.p1.5.5.m5.1.1.3"><csymbol cd="ambiguous" id="S2.Thmproposition1.p1.5.5.m5.1.1.3.1.cmml" xref="S2.Thmproposition1.p1.5.5.m5.1.1.3">subscript</csymbol><ci id="S2.Thmproposition1.p1.5.5.m5.1.1.3.2.cmml" xref="S2.Thmproposition1.p1.5.5.m5.1.1.3.2">𝑄</ci><cn id="S2.Thmproposition1.p1.5.5.m5.1.1.3.3.cmml" type="integer" xref="S2.Thmproposition1.p1.5.5.m5.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmproposition1.p1.5.5.m5.1c">Q_{1}\times Q_{1}</annotation><annotation encoding="application/x-llamapun" id="S2.Thmproposition1.p1.5.5.m5.1d">italic_Q start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT × italic_Q start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> block of <math 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xref="S2.Thmproposition1.p1.6.6.m6.1.1.3.3.cmml">1</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmproposition1.p1.6.6.m6.1b"><apply id="S2.Thmproposition1.p1.6.6.m6.1.1.cmml" xref="S2.Thmproposition1.p1.6.6.m6.1.1"><csymbol cd="latexml" id="S2.Thmproposition1.p1.6.6.m6.1.1.1.cmml" xref="S2.Thmproposition1.p1.6.6.m6.1.1.1">tensor-product</csymbol><apply id="S2.Thmproposition1.p1.6.6.m6.1.1.2.cmml" xref="S2.Thmproposition1.p1.6.6.m6.1.1.2"><csymbol cd="ambiguous" id="S2.Thmproposition1.p1.6.6.m6.1.1.2.1.cmml" xref="S2.Thmproposition1.p1.6.6.m6.1.1.2">subscript</csymbol><ci id="S2.Thmproposition1.p1.6.6.m6.1.1.2.2.cmml" xref="S2.Thmproposition1.p1.6.6.m6.1.1.2.2">𝚺</ci><cn id="S2.Thmproposition1.p1.6.6.m6.1.1.2.3.cmml" type="integer" xref="S2.Thmproposition1.p1.6.6.m6.1.1.2.3">2</cn></apply><apply id="S2.Thmproposition1.p1.6.6.m6.1.1.3.cmml" xref="S2.Thmproposition1.p1.6.6.m6.1.1.3"><csymbol cd="ambiguous" id="S2.Thmproposition1.p1.6.6.m6.1.1.3.1.cmml" xref="S2.Thmproposition1.p1.6.6.m6.1.1.3">subscript</csymbol><ci id="S2.Thmproposition1.p1.6.6.m6.1.1.3.2.cmml" xref="S2.Thmproposition1.p1.6.6.m6.1.1.3.2">𝚺</ci><cn id="S2.Thmproposition1.p1.6.6.m6.1.1.3.3.cmml" type="integer" xref="S2.Thmproposition1.p1.6.6.m6.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmproposition1.p1.6.6.m6.1c">\operatorname{{\mathbf{\Sigma}}}_{2}\otimes\operatorname{{\mathbf{\Sigma}}}_{1}</annotation><annotation encoding="application/x-llamapun" id="S2.Thmproposition1.p1.6.6.m6.1d">bold_Σ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ⊗ bold_Σ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> is <math alttext="\operatorname{{\mathbf{\Sigma}}}_{1}" class="ltx_Math" display="inline" id="S2.Thmproposition1.p1.7.7.m7.1"><semantics id="S2.Thmproposition1.p1.7.7.m7.1a"><msub id="S2.Thmproposition1.p1.7.7.m7.1.1" xref="S2.Thmproposition1.p1.7.7.m7.1.1.cmml"><mi id="S2.Thmproposition1.p1.7.7.m7.1.1.2" xref="S2.Thmproposition1.p1.7.7.m7.1.1.2.cmml">𝚺</mi><mn id="S2.Thmproposition1.p1.7.7.m7.1.1.3" xref="S2.Thmproposition1.p1.7.7.m7.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S2.Thmproposition1.p1.7.7.m7.1b"><apply id="S2.Thmproposition1.p1.7.7.m7.1.1.cmml" xref="S2.Thmproposition1.p1.7.7.m7.1.1"><csymbol cd="ambiguous" id="S2.Thmproposition1.p1.7.7.m7.1.1.1.cmml" xref="S2.Thmproposition1.p1.7.7.m7.1.1">subscript</csymbol><ci id="S2.Thmproposition1.p1.7.7.m7.1.1.2.cmml" xref="S2.Thmproposition1.p1.7.7.m7.1.1.2">𝚺</ci><cn id="S2.Thmproposition1.p1.7.7.m7.1.1.3.cmml" type="integer" xref="S2.Thmproposition1.p1.7.7.m7.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmproposition1.p1.7.7.m7.1c">\operatorname{{\mathbf{\Sigma}}}_{1}</annotation><annotation encoding="application/x-llamapun" id="S2.Thmproposition1.p1.7.7.m7.1d">bold_Σ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math>.</span></p> </div> </div> <div class="ltx_para" id="S2.SS1.p5"> <p class="ltx_p" id="S2.SS1.p5.5">The proof of this proposition is given in the supplementary material along with other proofs. Proposition <a class="ltx_ref" href="https://arxiv.org/html/2503.13782v1#S1.Thmproposition1" title="Proposition 1.1 ‣ 1.1 Proof of Proposition 2.1 ‣ 1 Proof of the Propositions ‣ Regularized Parameter Estimation in Mixed Model Trace Regression"><span class="ltx_text ltx_ref_tag">1.1</span></a> does not impose any assumption on the ranks of <math alttext="\operatorname{{\mathbf{\Sigma}}}_{1}" class="ltx_Math" display="inline" id="S2.SS1.p5.1.m1.1"><semantics id="S2.SS1.p5.1.m1.1a"><msub id="S2.SS1.p5.1.m1.1.1" xref="S2.SS1.p5.1.m1.1.1.cmml"><mi id="S2.SS1.p5.1.m1.1.1.2" xref="S2.SS1.p5.1.m1.1.1.2.cmml">𝚺</mi><mn id="S2.SS1.p5.1.m1.1.1.3" xref="S2.SS1.p5.1.m1.1.1.3.cmml">1</mn></msub><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"><csymbol cd="ambiguous" id="S2.SS1.p5.1.m1.1.1.1.cmml" xref="S2.SS1.p5.1.m1.1.1">subscript</csymbol><ci id="S2.SS1.p5.1.m1.1.1.2.cmml" xref="S2.SS1.p5.1.m1.1.1.2">𝚺</ci><cn id="S2.SS1.p5.1.m1.1.1.3.cmml" type="integer" xref="S2.SS1.p5.1.m1.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p5.1.m1.1c">\operatorname{{\mathbf{\Sigma}}}_{1}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p5.1.m1.1d">bold_Σ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="\operatorname{{\mathbf{\Sigma}}}_{2}" 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">𝚺</mi><mn id="S2.SS1.p5.2.m2.1.1.3" xref="S2.SS1.p5.2.m2.1.1.3.cmml">2</mn></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">𝚺</ci><cn id="S2.SS1.p5.2.m2.1.1.3.cmml" type="integer" xref="S2.SS1.p5.2.m2.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p5.2.m2.1c">\operatorname{{\mathbf{\Sigma}}}_{2}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p5.2.m2.1d">bold_Σ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math>, so it is applicable when <math alttext="\operatorname{{\mathbf{\Sigma}}}_{1}" 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">𝚺</mi><mn id="S2.SS1.p5.3.m3.1.1.3" xref="S2.SS1.p5.3.m3.1.1.3.cmml">1</mn></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">𝚺</ci><cn id="S2.SS1.p5.3.m3.1.1.3.cmml" type="integer" xref="S2.SS1.p5.3.m3.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p5.3.m3.1c">\operatorname{{\mathbf{\Sigma}}}_{1}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p5.3.m3.1d">bold_Σ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> or <math alttext="\operatorname{{\mathbf{\Sigma}}}_{2}" class="ltx_Math" display="inline" id="S2.SS1.p5.4.m4.1"><semantics id="S2.SS1.p5.4.m4.1a"><msub id="S2.SS1.p5.4.m4.1.1" xref="S2.SS1.p5.4.m4.1.1.cmml"><mi id="S2.SS1.p5.4.m4.1.1.2" xref="S2.SS1.p5.4.m4.1.1.2.cmml">𝚺</mi><mn id="S2.SS1.p5.4.m4.1.1.3" xref="S2.SS1.p5.4.m4.1.1.3.cmml">2</mn></msub><annotation-xml encoding="MathML-Content" id="S2.SS1.p5.4.m4.1b"><apply id="S2.SS1.p5.4.m4.1.1.cmml" xref="S2.SS1.p5.4.m4.1.1"><csymbol cd="ambiguous" id="S2.SS1.p5.4.m4.1.1.1.cmml" xref="S2.SS1.p5.4.m4.1.1">subscript</csymbol><ci id="S2.SS1.p5.4.m4.1.1.2.cmml" xref="S2.SS1.p5.4.m4.1.1.2">𝚺</ci><cn id="S2.SS1.p5.4.m4.1.1.3.cmml" type="integer" xref="S2.SS1.p5.4.m4.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p5.4.m4.1c">\operatorname{{\mathbf{\Sigma}}}_{2}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p5.4.m4.1d">bold_Σ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math> are rank-deficient. Furthermore, the following proposition shows that we can convert any square-root covariance matrix <math alttext="\operatorname{\mathbf{L}}_{2}" class="ltx_Math" display="inline" id="S2.SS1.p5.5.m5.1"><semantics id="S2.SS1.p5.5.m5.1a"><msub id="S2.SS1.p5.5.m5.1.1" xref="S2.SS1.p5.5.m5.1.1.cmml"><mi id="S2.SS1.p5.5.m5.1.1.2" xref="S2.SS1.p5.5.m5.1.1.2.cmml">𝐋</mi><mn id="S2.SS1.p5.5.m5.1.1.3" xref="S2.SS1.p5.5.m5.1.1.3.cmml">2</mn></msub><annotation-xml encoding="MathML-Content" id="S2.SS1.p5.5.m5.1b"><apply id="S2.SS1.p5.5.m5.1.1.cmml" xref="S2.SS1.p5.5.m5.1.1"><csymbol cd="ambiguous" id="S2.SS1.p5.5.m5.1.1.1.cmml" xref="S2.SS1.p5.5.m5.1.1">subscript</csymbol><ci id="S2.SS1.p5.5.m5.1.1.2.cmml" xref="S2.SS1.p5.5.m5.1.1.2">𝐋</ci><cn id="S2.SS1.p5.5.m5.1.1.3.cmml" type="integer" xref="S2.SS1.p5.5.m5.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p5.5.m5.1c">\operatorname{\mathbf{L}}_{2}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p5.5.m5.1d">bold_L start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math> into the form described in Proposition <a class="ltx_ref" href="https://arxiv.org/html/2503.13782v1#S1.Thmproposition1" title="Proposition 1.1 ‣ 1.1 Proof of Proposition 2.1 ‣ 1 Proof of the Propositions ‣ Regularized Parameter Estimation in Mixed Model Trace Regression"><span class="ltx_text ltx_ref_tag">1.1</span></a></p> </div> <div class="ltx_theorem ltx_theorem_proposition" id="S2.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="S2.Thmproposition2.1.1.1">Proposition 2.2</span></span></h6> <div class="ltx_para" id="S2.Thmproposition2.p1"> <p class="ltx_p" id="S2.Thmproposition2.p1.6"><span class="ltx_text ltx_font_italic" id="S2.Thmproposition2.p1.6.6">For any matrix <math alttext="\operatorname{\mathbf{L}}\in\operatorname{\mathbb{R}}^{Q\times S}" class="ltx_Math" display="inline" id="S2.Thmproposition2.p1.1.1.m1.1"><semantics id="S2.Thmproposition2.p1.1.1.m1.1a"><mrow id="S2.Thmproposition2.p1.1.1.m1.1.1" xref="S2.Thmproposition2.p1.1.1.m1.1.1.cmml"><mi id="S2.Thmproposition2.p1.1.1.m1.1.1.2" xref="S2.Thmproposition2.p1.1.1.m1.1.1.2.cmml">𝐋</mi><mo id="S2.Thmproposition2.p1.1.1.m1.1.1.1" xref="S2.Thmproposition2.p1.1.1.m1.1.1.1.cmml">∈</mo><msup id="S2.Thmproposition2.p1.1.1.m1.1.1.3" xref="S2.Thmproposition2.p1.1.1.m1.1.1.3.cmml"><mi id="S2.Thmproposition2.p1.1.1.m1.1.1.3.2" xref="S2.Thmproposition2.p1.1.1.m1.1.1.3.2.cmml">ℝ</mi><mrow id="S2.Thmproposition2.p1.1.1.m1.1.1.3.3" xref="S2.Thmproposition2.p1.1.1.m1.1.1.3.3.cmml"><mi id="S2.Thmproposition2.p1.1.1.m1.1.1.3.3.2" xref="S2.Thmproposition2.p1.1.1.m1.1.1.3.3.2.cmml">Q</mi><mo id="S2.Thmproposition2.p1.1.1.m1.1.1.3.3.1" lspace="0.222em" rspace="0.222em" xref="S2.Thmproposition2.p1.1.1.m1.1.1.3.3.1.cmml">×</mo><mi id="S2.Thmproposition2.p1.1.1.m1.1.1.3.3.3" xref="S2.Thmproposition2.p1.1.1.m1.1.1.3.3.3.cmml">S</mi></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmproposition2.p1.1.1.m1.1b"><apply id="S2.Thmproposition2.p1.1.1.m1.1.1.cmml" xref="S2.Thmproposition2.p1.1.1.m1.1.1"><in id="S2.Thmproposition2.p1.1.1.m1.1.1.1.cmml" xref="S2.Thmproposition2.p1.1.1.m1.1.1.1"></in><ci id="S2.Thmproposition2.p1.1.1.m1.1.1.2.cmml" xref="S2.Thmproposition2.p1.1.1.m1.1.1.2">𝐋</ci><apply id="S2.Thmproposition2.p1.1.1.m1.1.1.3.cmml" xref="S2.Thmproposition2.p1.1.1.m1.1.1.3"><csymbol cd="ambiguous" id="S2.Thmproposition2.p1.1.1.m1.1.1.3.1.cmml" xref="S2.Thmproposition2.p1.1.1.m1.1.1.3">superscript</csymbol><ci id="S2.Thmproposition2.p1.1.1.m1.1.1.3.2.cmml" xref="S2.Thmproposition2.p1.1.1.m1.1.1.3.2">ℝ</ci><apply id="S2.Thmproposition2.p1.1.1.m1.1.1.3.3.cmml" xref="S2.Thmproposition2.p1.1.1.m1.1.1.3.3"><times id="S2.Thmproposition2.p1.1.1.m1.1.1.3.3.1.cmml" xref="S2.Thmproposition2.p1.1.1.m1.1.1.3.3.1"></times><ci id="S2.Thmproposition2.p1.1.1.m1.1.1.3.3.2.cmml" xref="S2.Thmproposition2.p1.1.1.m1.1.1.3.3.2">𝑄</ci><ci id="S2.Thmproposition2.p1.1.1.m1.1.1.3.3.3.cmml" xref="S2.Thmproposition2.p1.1.1.m1.1.1.3.3.3">𝑆</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmproposition2.p1.1.1.m1.1c">\operatorname{\mathbf{L}}\in\operatorname{\mathbb{R}}^{Q\times S}</annotation><annotation encoding="application/x-llamapun" id="S2.Thmproposition2.p1.1.1.m1.1d">bold_L ∈ blackboard_R start_POSTSUPERSCRIPT italic_Q × italic_S end_POSTSUPERSCRIPT</annotation></semantics></math>, there exists an orthonormal matrix <math alttext="\operatorname{\mathbf{Q}}_{j}\in\operatorname{\mathbb{R}}^{S\times S}" class="ltx_Math" display="inline" id="S2.Thmproposition2.p1.2.2.m2.1"><semantics id="S2.Thmproposition2.p1.2.2.m2.1a"><mrow id="S2.Thmproposition2.p1.2.2.m2.1.1" xref="S2.Thmproposition2.p1.2.2.m2.1.1.cmml"><msub id="S2.Thmproposition2.p1.2.2.m2.1.1.2" xref="S2.Thmproposition2.p1.2.2.m2.1.1.2.cmml"><mi id="S2.Thmproposition2.p1.2.2.m2.1.1.2.2" xref="S2.Thmproposition2.p1.2.2.m2.1.1.2.2.cmml">𝐐</mi><mi id="S2.Thmproposition2.p1.2.2.m2.1.1.2.3" xref="S2.Thmproposition2.p1.2.2.m2.1.1.2.3.cmml">j</mi></msub><mo id="S2.Thmproposition2.p1.2.2.m2.1.1.1" xref="S2.Thmproposition2.p1.2.2.m2.1.1.1.cmml">∈</mo><msup id="S2.Thmproposition2.p1.2.2.m2.1.1.3" xref="S2.Thmproposition2.p1.2.2.m2.1.1.3.cmml"><mi id="S2.Thmproposition2.p1.2.2.m2.1.1.3.2" xref="S2.Thmproposition2.p1.2.2.m2.1.1.3.2.cmml">ℝ</mi><mrow id="S2.Thmproposition2.p1.2.2.m2.1.1.3.3" xref="S2.Thmproposition2.p1.2.2.m2.1.1.3.3.cmml"><mi id="S2.Thmproposition2.p1.2.2.m2.1.1.3.3.2" xref="S2.Thmproposition2.p1.2.2.m2.1.1.3.3.2.cmml">S</mi><mo id="S2.Thmproposition2.p1.2.2.m2.1.1.3.3.1" lspace="0.222em" rspace="0.222em" xref="S2.Thmproposition2.p1.2.2.m2.1.1.3.3.1.cmml">×</mo><mi id="S2.Thmproposition2.p1.2.2.m2.1.1.3.3.3" xref="S2.Thmproposition2.p1.2.2.m2.1.1.3.3.3.cmml">S</mi></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmproposition2.p1.2.2.m2.1b"><apply id="S2.Thmproposition2.p1.2.2.m2.1.1.cmml" xref="S2.Thmproposition2.p1.2.2.m2.1.1"><in id="S2.Thmproposition2.p1.2.2.m2.1.1.1.cmml" xref="S2.Thmproposition2.p1.2.2.m2.1.1.1"></in><apply id="S2.Thmproposition2.p1.2.2.m2.1.1.2.cmml" xref="S2.Thmproposition2.p1.2.2.m2.1.1.2"><csymbol cd="ambiguous" id="S2.Thmproposition2.p1.2.2.m2.1.1.2.1.cmml" xref="S2.Thmproposition2.p1.2.2.m2.1.1.2">subscript</csymbol><ci id="S2.Thmproposition2.p1.2.2.m2.1.1.2.2.cmml" xref="S2.Thmproposition2.p1.2.2.m2.1.1.2.2">𝐐</ci><ci id="S2.Thmproposition2.p1.2.2.m2.1.1.2.3.cmml" xref="S2.Thmproposition2.p1.2.2.m2.1.1.2.3">𝑗</ci></apply><apply id="S2.Thmproposition2.p1.2.2.m2.1.1.3.cmml" xref="S2.Thmproposition2.p1.2.2.m2.1.1.3"><csymbol cd="ambiguous" id="S2.Thmproposition2.p1.2.2.m2.1.1.3.1.cmml" xref="S2.Thmproposition2.p1.2.2.m2.1.1.3">superscript</csymbol><ci id="S2.Thmproposition2.p1.2.2.m2.1.1.3.2.cmml" xref="S2.Thmproposition2.p1.2.2.m2.1.1.3.2">ℝ</ci><apply id="S2.Thmproposition2.p1.2.2.m2.1.1.3.3.cmml" xref="S2.Thmproposition2.p1.2.2.m2.1.1.3.3"><times id="S2.Thmproposition2.p1.2.2.m2.1.1.3.3.1.cmml" xref="S2.Thmproposition2.p1.2.2.m2.1.1.3.3.1"></times><ci id="S2.Thmproposition2.p1.2.2.m2.1.1.3.3.2.cmml" xref="S2.Thmproposition2.p1.2.2.m2.1.1.3.3.2">𝑆</ci><ci id="S2.Thmproposition2.p1.2.2.m2.1.1.3.3.3.cmml" xref="S2.Thmproposition2.p1.2.2.m2.1.1.3.3.3">𝑆</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmproposition2.p1.2.2.m2.1c">\operatorname{\mathbf{Q}}_{j}\in\operatorname{\mathbb{R}}^{S\times S}</annotation><annotation encoding="application/x-llamapun" id="S2.Thmproposition2.p1.2.2.m2.1d">bold_Q start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ∈ blackboard_R start_POSTSUPERSCRIPT italic_S × italic_S end_POSTSUPERSCRIPT</annotation></semantics></math> and constant <math alttext="c" class="ltx_Math" display="inline" id="S2.Thmproposition2.p1.3.3.m3.1"><semantics id="S2.Thmproposition2.p1.3.3.m3.1a"><mi id="S2.Thmproposition2.p1.3.3.m3.1.1" xref="S2.Thmproposition2.p1.3.3.m3.1.1.cmml">c</mi><annotation-xml encoding="MathML-Content" id="S2.Thmproposition2.p1.3.3.m3.1b"><ci id="S2.Thmproposition2.p1.3.3.m3.1.1.cmml" xref="S2.Thmproposition2.p1.3.3.m3.1.1">𝑐</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmproposition2.p1.3.3.m3.1c">c</annotation><annotation encoding="application/x-llamapun" id="S2.Thmproposition2.p1.3.3.m3.1d">italic_c</annotation></semantics></math> such that <math alttext="c\operatorname{\mathbf{L}}\operatorname{\mathbf{Q}}_{j}^{\top}" class="ltx_Math" display="inline" id="S2.Thmproposition2.p1.4.4.m4.1"><semantics id="S2.Thmproposition2.p1.4.4.m4.1a"><mrow id="S2.Thmproposition2.p1.4.4.m4.1.1" xref="S2.Thmproposition2.p1.4.4.m4.1.1.cmml"><mi id="S2.Thmproposition2.p1.4.4.m4.1.1.2" xref="S2.Thmproposition2.p1.4.4.m4.1.1.2.cmml">c</mi><mo id="S2.Thmproposition2.p1.4.4.m4.1.1.1" lspace="0.167em" xref="S2.Thmproposition2.p1.4.4.m4.1.1.1.cmml">⁢</mo><mrow id="S2.Thmproposition2.p1.4.4.m4.1.1.3" xref="S2.Thmproposition2.p1.4.4.m4.1.1.3.cmml"><mi id="S2.Thmproposition2.p1.4.4.m4.1.1.3.1" xref="S2.Thmproposition2.p1.4.4.m4.1.1.3.1.cmml">𝐋</mi><mo id="S2.Thmproposition2.p1.4.4.m4.1.1.3a" lspace="0.167em" xref="S2.Thmproposition2.p1.4.4.m4.1.1.3.cmml">⁡</mo><msubsup id="S2.Thmproposition2.p1.4.4.m4.1.1.3.2" xref="S2.Thmproposition2.p1.4.4.m4.1.1.3.2.cmml"><mi id="S2.Thmproposition2.p1.4.4.m4.1.1.3.2.2.2" xref="S2.Thmproposition2.p1.4.4.m4.1.1.3.2.2.2.cmml">𝐐</mi><mi id="S2.Thmproposition2.p1.4.4.m4.1.1.3.2.2.3" xref="S2.Thmproposition2.p1.4.4.m4.1.1.3.2.2.3.cmml">j</mi><mo id="S2.Thmproposition2.p1.4.4.m4.1.1.3.2.3" xref="S2.Thmproposition2.p1.4.4.m4.1.1.3.2.3.cmml">⊤</mo></msubsup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmproposition2.p1.4.4.m4.1b"><apply id="S2.Thmproposition2.p1.4.4.m4.1.1.cmml" xref="S2.Thmproposition2.p1.4.4.m4.1.1"><times id="S2.Thmproposition2.p1.4.4.m4.1.1.1.cmml" xref="S2.Thmproposition2.p1.4.4.m4.1.1.1"></times><ci id="S2.Thmproposition2.p1.4.4.m4.1.1.2.cmml" xref="S2.Thmproposition2.p1.4.4.m4.1.1.2">𝑐</ci><apply id="S2.Thmproposition2.p1.4.4.m4.1.1.3.cmml" xref="S2.Thmproposition2.p1.4.4.m4.1.1.3"><ci id="S2.Thmproposition2.p1.4.4.m4.1.1.3.1.cmml" xref="S2.Thmproposition2.p1.4.4.m4.1.1.3.1">𝐋</ci><apply id="S2.Thmproposition2.p1.4.4.m4.1.1.3.2.cmml" xref="S2.Thmproposition2.p1.4.4.m4.1.1.3.2"><csymbol cd="ambiguous" id="S2.Thmproposition2.p1.4.4.m4.1.1.3.2.1.cmml" xref="S2.Thmproposition2.p1.4.4.m4.1.1.3.2">superscript</csymbol><apply id="S2.Thmproposition2.p1.4.4.m4.1.1.3.2.2.cmml" xref="S2.Thmproposition2.p1.4.4.m4.1.1.3.2"><csymbol cd="ambiguous" id="S2.Thmproposition2.p1.4.4.m4.1.1.3.2.2.1.cmml" xref="S2.Thmproposition2.p1.4.4.m4.1.1.3.2">subscript</csymbol><ci id="S2.Thmproposition2.p1.4.4.m4.1.1.3.2.2.2.cmml" xref="S2.Thmproposition2.p1.4.4.m4.1.1.3.2.2.2">𝐐</ci><ci id="S2.Thmproposition2.p1.4.4.m4.1.1.3.2.2.3.cmml" xref="S2.Thmproposition2.p1.4.4.m4.1.1.3.2.2.3">𝑗</ci></apply><csymbol cd="latexml" id="S2.Thmproposition2.p1.4.4.m4.1.1.3.2.3.cmml" xref="S2.Thmproposition2.p1.4.4.m4.1.1.3.2.3">top</csymbol></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmproposition2.p1.4.4.m4.1c">c\operatorname{\mathbf{L}}\operatorname{\mathbf{Q}}_{j}^{\top}</annotation><annotation encoding="application/x-llamapun" id="S2.Thmproposition2.p1.4.4.m4.1d">italic_c bold_L bold_Q start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ⊤ end_POSTSUPERSCRIPT</annotation></semantics></math> is a matrix whose <math alttext="j" class="ltx_Math" display="inline" id="S2.Thmproposition2.p1.5.5.m5.1"><semantics id="S2.Thmproposition2.p1.5.5.m5.1a"><mi id="S2.Thmproposition2.p1.5.5.m5.1.1" xref="S2.Thmproposition2.p1.5.5.m5.1.1.cmml">j</mi><annotation-xml encoding="MathML-Content" id="S2.Thmproposition2.p1.5.5.m5.1b"><ci id="S2.Thmproposition2.p1.5.5.m5.1.1.cmml" xref="S2.Thmproposition2.p1.5.5.m5.1.1">𝑗</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmproposition2.p1.5.5.m5.1c">j</annotation><annotation encoding="application/x-llamapun" id="S2.Thmproposition2.p1.5.5.m5.1d">italic_j</annotation></semantics></math>th row is the first standard basis vector in <math alttext="\operatorname{\mathbb{R}}^{S}" class="ltx_Math" display="inline" id="S2.Thmproposition2.p1.6.6.m6.1"><semantics id="S2.Thmproposition2.p1.6.6.m6.1a"><msup id="S2.Thmproposition2.p1.6.6.m6.1.1" xref="S2.Thmproposition2.p1.6.6.m6.1.1.cmml"><mi id="S2.Thmproposition2.p1.6.6.m6.1.1.2" xref="S2.Thmproposition2.p1.6.6.m6.1.1.2.cmml">ℝ</mi><mi id="S2.Thmproposition2.p1.6.6.m6.1.1.3" xref="S2.Thmproposition2.p1.6.6.m6.1.1.3.cmml">S</mi></msup><annotation-xml encoding="MathML-Content" id="S2.Thmproposition2.p1.6.6.m6.1b"><apply id="S2.Thmproposition2.p1.6.6.m6.1.1.cmml" xref="S2.Thmproposition2.p1.6.6.m6.1.1"><csymbol cd="ambiguous" id="S2.Thmproposition2.p1.6.6.m6.1.1.1.cmml" xref="S2.Thmproposition2.p1.6.6.m6.1.1">superscript</csymbol><ci id="S2.Thmproposition2.p1.6.6.m6.1.1.2.cmml" xref="S2.Thmproposition2.p1.6.6.m6.1.1.2">ℝ</ci><ci id="S2.Thmproposition2.p1.6.6.m6.1.1.3.cmml" xref="S2.Thmproposition2.p1.6.6.m6.1.1.3">𝑆</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmproposition2.p1.6.6.m6.1c">\operatorname{\mathbb{R}}^{S}</annotation><annotation encoding="application/x-llamapun" id="S2.Thmproposition2.p1.6.6.m6.1d">blackboard_R start_POSTSUPERSCRIPT italic_S end_POSTSUPERSCRIPT</annotation></semantics></math>.</span></p> </div> </div> <div class="ltx_para" id="S2.SS1.p6"> <p class="ltx_p" id="S2.SS1.p6.13">Proposition <a class="ltx_ref" href="https://arxiv.org/html/2503.13782v1#S1.Thmproposition2" title="Proposition 1.2 ‣ 1.2 Proof of Proposition 2.2 ‣ 1 Proof of the Propositions ‣ Regularized Parameter Estimation in Mixed Model Trace Regression"><span class="ltx_text ltx_ref_tag">1.2</span></a> states that <math alttext="\operatorname{{\mathbf{\Sigma}}}_{1}" class="ltx_Math" display="inline" id="S2.SS1.p6.1.m1.1"><semantics id="S2.SS1.p6.1.m1.1a"><msub id="S2.SS1.p6.1.m1.1.1" xref="S2.SS1.p6.1.m1.1.1.cmml"><mi id="S2.SS1.p6.1.m1.1.1.2" xref="S2.SS1.p6.1.m1.1.1.2.cmml">𝚺</mi><mn id="S2.SS1.p6.1.m1.1.1.3" xref="S2.SS1.p6.1.m1.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S2.SS1.p6.1.m1.1b"><apply id="S2.SS1.p6.1.m1.1.1.cmml" xref="S2.SS1.p6.1.m1.1.1"><csymbol cd="ambiguous" id="S2.SS1.p6.1.m1.1.1.1.cmml" xref="S2.SS1.p6.1.m1.1.1">subscript</csymbol><ci id="S2.SS1.p6.1.m1.1.1.2.cmml" xref="S2.SS1.p6.1.m1.1.1.2">𝚺</ci><cn id="S2.SS1.p6.1.m1.1.1.3.cmml" type="integer" xref="S2.SS1.p6.1.m1.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p6.1.m1.1c">\operatorname{{\mathbf{\Sigma}}}_{1}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p6.1.m1.1d">bold_Σ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="\operatorname{{\mathbf{\Sigma}}}_{2}" class="ltx_Math" display="inline" id="S2.SS1.p6.2.m2.1"><semantics id="S2.SS1.p6.2.m2.1a"><msub id="S2.SS1.p6.2.m2.1.1" xref="S2.SS1.p6.2.m2.1.1.cmml"><mi id="S2.SS1.p6.2.m2.1.1.2" xref="S2.SS1.p6.2.m2.1.1.2.cmml">𝚺</mi><mn id="S2.SS1.p6.2.m2.1.1.3" xref="S2.SS1.p6.2.m2.1.1.3.cmml">2</mn></msub><annotation-xml encoding="MathML-Content" id="S2.SS1.p6.2.m2.1b"><apply id="S2.SS1.p6.2.m2.1.1.cmml" xref="S2.SS1.p6.2.m2.1.1"><csymbol cd="ambiguous" id="S2.SS1.p6.2.m2.1.1.1.cmml" xref="S2.SS1.p6.2.m2.1.1">subscript</csymbol><ci id="S2.SS1.p6.2.m2.1.1.2.cmml" xref="S2.SS1.p6.2.m2.1.1.2">𝚺</ci><cn id="S2.SS1.p6.2.m2.1.1.3.cmml" type="integer" xref="S2.SS1.p6.2.m2.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p6.2.m2.1c">\operatorname{{\mathbf{\Sigma}}}_{2}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p6.2.m2.1d">bold_Σ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math> are identified if <math alttext="\operatorname{{\mathbf{\Sigma}}}_{2}" class="ltx_Math" display="inline" id="S2.SS1.p6.3.m3.1"><semantics id="S2.SS1.p6.3.m3.1a"><msub id="S2.SS1.p6.3.m3.1.1" xref="S2.SS1.p6.3.m3.1.1.cmml"><mi id="S2.SS1.p6.3.m3.1.1.2" xref="S2.SS1.p6.3.m3.1.1.2.cmml">𝚺</mi><mn id="S2.SS1.p6.3.m3.1.1.3" xref="S2.SS1.p6.3.m3.1.1.3.cmml">2</mn></msub><annotation-xml encoding="MathML-Content" id="S2.SS1.p6.3.m3.1b"><apply id="S2.SS1.p6.3.m3.1.1.cmml" xref="S2.SS1.p6.3.m3.1.1"><csymbol cd="ambiguous" id="S2.SS1.p6.3.m3.1.1.1.cmml" xref="S2.SS1.p6.3.m3.1.1">subscript</csymbol><ci id="S2.SS1.p6.3.m3.1.1.2.cmml" xref="S2.SS1.p6.3.m3.1.1.2">𝚺</ci><cn id="S2.SS1.p6.3.m3.1.1.3.cmml" type="integer" xref="S2.SS1.p6.3.m3.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p6.3.m3.1c">\operatorname{{\mathbf{\Sigma}}}_{2}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p6.3.m3.1d">bold_Σ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math> is scaled so that any one of its diagonal values equals 1. We use Proposition <a class="ltx_ref" href="https://arxiv.org/html/2503.13782v1#S1.Thmproposition2" title="Proposition 1.2 ‣ 1.2 Proof of Proposition 2.2 ‣ 1 Proof of the Propositions ‣ Regularized Parameter Estimation in Mixed Model Trace Regression"><span class="ltx_text ltx_ref_tag">1.2</span></a> to evaluate the estimation accuracy of our estimation algorithm. We compare our estimates <math alttext="\hat{\operatorname{{\mathbf{\Sigma}}}}_{1}" class="ltx_Math" display="inline" id="S2.SS1.p6.4.m4.1"><semantics id="S2.SS1.p6.4.m4.1a"><msub id="S2.SS1.p6.4.m4.1.1" xref="S2.SS1.p6.4.m4.1.1.cmml"><mover accent="true" id="S2.SS1.p6.4.m4.1.1.2" xref="S2.SS1.p6.4.m4.1.1.2.cmml"><mi id="S2.SS1.p6.4.m4.1.1.2.2" xref="S2.SS1.p6.4.m4.1.1.2.2.cmml">𝚺</mi><mo id="S2.SS1.p6.4.m4.1.1.2.1" xref="S2.SS1.p6.4.m4.1.1.2.1.cmml">^</mo></mover><mn id="S2.SS1.p6.4.m4.1.1.3" xref="S2.SS1.p6.4.m4.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S2.SS1.p6.4.m4.1b"><apply id="S2.SS1.p6.4.m4.1.1.cmml" xref="S2.SS1.p6.4.m4.1.1"><csymbol cd="ambiguous" id="S2.SS1.p6.4.m4.1.1.1.cmml" xref="S2.SS1.p6.4.m4.1.1">subscript</csymbol><apply id="S2.SS1.p6.4.m4.1.1.2.cmml" xref="S2.SS1.p6.4.m4.1.1.2"><ci id="S2.SS1.p6.4.m4.1.1.2.1.cmml" xref="S2.SS1.p6.4.m4.1.1.2.1">^</ci><ci id="S2.SS1.p6.4.m4.1.1.2.2.cmml" xref="S2.SS1.p6.4.m4.1.1.2.2">𝚺</ci></apply><cn id="S2.SS1.p6.4.m4.1.1.3.cmml" type="integer" xref="S2.SS1.p6.4.m4.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p6.4.m4.1c">\hat{\operatorname{{\mathbf{\Sigma}}}}_{1}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p6.4.m4.1d">over^ start_ARG bold_Σ end_ARG start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="\hat{\operatorname{{\mathbf{\Sigma}}}}_{2}" class="ltx_Math" display="inline" id="S2.SS1.p6.5.m5.1"><semantics id="S2.SS1.p6.5.m5.1a"><msub id="S2.SS1.p6.5.m5.1.1" xref="S2.SS1.p6.5.m5.1.1.cmml"><mover accent="true" id="S2.SS1.p6.5.m5.1.1.2" xref="S2.SS1.p6.5.m5.1.1.2.cmml"><mi id="S2.SS1.p6.5.m5.1.1.2.2" xref="S2.SS1.p6.5.m5.1.1.2.2.cmml">𝚺</mi><mo id="S2.SS1.p6.5.m5.1.1.2.1" xref="S2.SS1.p6.5.m5.1.1.2.1.cmml">^</mo></mover><mn id="S2.SS1.p6.5.m5.1.1.3" xref="S2.SS1.p6.5.m5.1.1.3.cmml">2</mn></msub><annotation-xml encoding="MathML-Content" id="S2.SS1.p6.5.m5.1b"><apply id="S2.SS1.p6.5.m5.1.1.cmml" xref="S2.SS1.p6.5.m5.1.1"><csymbol cd="ambiguous" id="S2.SS1.p6.5.m5.1.1.1.cmml" xref="S2.SS1.p6.5.m5.1.1">subscript</csymbol><apply id="S2.SS1.p6.5.m5.1.1.2.cmml" xref="S2.SS1.p6.5.m5.1.1.2"><ci id="S2.SS1.p6.5.m5.1.1.2.1.cmml" xref="S2.SS1.p6.5.m5.1.1.2.1">^</ci><ci id="S2.SS1.p6.5.m5.1.1.2.2.cmml" xref="S2.SS1.p6.5.m5.1.1.2.2">𝚺</ci></apply><cn id="S2.SS1.p6.5.m5.1.1.3.cmml" type="integer" xref="S2.SS1.p6.5.m5.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p6.5.m5.1c">\hat{\operatorname{{\mathbf{\Sigma}}}}_{2}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p6.5.m5.1d">over^ start_ARG bold_Σ end_ARG start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math> of the true parameters <math alttext="\operatorname{{\mathbf{\Sigma}}}_{1}" class="ltx_Math" display="inline" id="S2.SS1.p6.6.m6.1"><semantics id="S2.SS1.p6.6.m6.1a"><msub id="S2.SS1.p6.6.m6.1.1" xref="S2.SS1.p6.6.m6.1.1.cmml"><mi id="S2.SS1.p6.6.m6.1.1.2" xref="S2.SS1.p6.6.m6.1.1.2.cmml">𝚺</mi><mn id="S2.SS1.p6.6.m6.1.1.3" xref="S2.SS1.p6.6.m6.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S2.SS1.p6.6.m6.1b"><apply id="S2.SS1.p6.6.m6.1.1.cmml" xref="S2.SS1.p6.6.m6.1.1"><csymbol cd="ambiguous" id="S2.SS1.p6.6.m6.1.1.1.cmml" xref="S2.SS1.p6.6.m6.1.1">subscript</csymbol><ci id="S2.SS1.p6.6.m6.1.1.2.cmml" xref="S2.SS1.p6.6.m6.1.1.2">𝚺</ci><cn id="S2.SS1.p6.6.m6.1.1.3.cmml" type="integer" xref="S2.SS1.p6.6.m6.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p6.6.m6.1c">\operatorname{{\mathbf{\Sigma}}}_{1}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p6.6.m6.1d">bold_Σ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="\operatorname{{\mathbf{\Sigma}}}_{2}" class="ltx_Math" display="inline" id="S2.SS1.p6.7.m7.1"><semantics id="S2.SS1.p6.7.m7.1a"><msub id="S2.SS1.p6.7.m7.1.1" xref="S2.SS1.p6.7.m7.1.1.cmml"><mi id="S2.SS1.p6.7.m7.1.1.2" xref="S2.SS1.p6.7.m7.1.1.2.cmml">𝚺</mi><mn id="S2.SS1.p6.7.m7.1.1.3" xref="S2.SS1.p6.7.m7.1.1.3.cmml">2</mn></msub><annotation-xml encoding="MathML-Content" id="S2.SS1.p6.7.m7.1b"><apply id="S2.SS1.p6.7.m7.1.1.cmml" xref="S2.SS1.p6.7.m7.1.1"><csymbol cd="ambiguous" id="S2.SS1.p6.7.m7.1.1.1.cmml" xref="S2.SS1.p6.7.m7.1.1">subscript</csymbol><ci id="S2.SS1.p6.7.m7.1.1.2.cmml" xref="S2.SS1.p6.7.m7.1.1.2">𝚺</ci><cn id="S2.SS1.p6.7.m7.1.1.3.cmml" type="integer" xref="S2.SS1.p6.7.m7.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p6.7.m7.1c">\operatorname{{\mathbf{\Sigma}}}_{2}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p6.7.m7.1d">bold_Σ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math> in the simulations by scaling <math alttext="\hat{\operatorname{{\mathbf{\Sigma}}}}_{1}" class="ltx_Math" display="inline" id="S2.SS1.p6.8.m8.1"><semantics id="S2.SS1.p6.8.m8.1a"><msub id="S2.SS1.p6.8.m8.1.1" xref="S2.SS1.p6.8.m8.1.1.cmml"><mover accent="true" id="S2.SS1.p6.8.m8.1.1.2" xref="S2.SS1.p6.8.m8.1.1.2.cmml"><mi id="S2.SS1.p6.8.m8.1.1.2.2" xref="S2.SS1.p6.8.m8.1.1.2.2.cmml">𝚺</mi><mo id="S2.SS1.p6.8.m8.1.1.2.1" xref="S2.SS1.p6.8.m8.1.1.2.1.cmml">^</mo></mover><mn id="S2.SS1.p6.8.m8.1.1.3" xref="S2.SS1.p6.8.m8.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S2.SS1.p6.8.m8.1b"><apply id="S2.SS1.p6.8.m8.1.1.cmml" xref="S2.SS1.p6.8.m8.1.1"><csymbol cd="ambiguous" id="S2.SS1.p6.8.m8.1.1.1.cmml" xref="S2.SS1.p6.8.m8.1.1">subscript</csymbol><apply id="S2.SS1.p6.8.m8.1.1.2.cmml" xref="S2.SS1.p6.8.m8.1.1.2"><ci id="S2.SS1.p6.8.m8.1.1.2.1.cmml" xref="S2.SS1.p6.8.m8.1.1.2.1">^</ci><ci id="S2.SS1.p6.8.m8.1.1.2.2.cmml" xref="S2.SS1.p6.8.m8.1.1.2.2">𝚺</ci></apply><cn id="S2.SS1.p6.8.m8.1.1.3.cmml" type="integer" xref="S2.SS1.p6.8.m8.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p6.8.m8.1c">\hat{\operatorname{{\mathbf{\Sigma}}}}_{1}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p6.8.m8.1d">over^ start_ARG bold_Σ end_ARG start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math>, <math alttext="\hat{\operatorname{{\mathbf{\Sigma}}}}_{2}" class="ltx_Math" display="inline" id="S2.SS1.p6.9.m9.1"><semantics id="S2.SS1.p6.9.m9.1a"><msub id="S2.SS1.p6.9.m9.1.1" xref="S2.SS1.p6.9.m9.1.1.cmml"><mover accent="true" id="S2.SS1.p6.9.m9.1.1.2" xref="S2.SS1.p6.9.m9.1.1.2.cmml"><mi id="S2.SS1.p6.9.m9.1.1.2.2" xref="S2.SS1.p6.9.m9.1.1.2.2.cmml">𝚺</mi><mo id="S2.SS1.p6.9.m9.1.1.2.1" xref="S2.SS1.p6.9.m9.1.1.2.1.cmml">^</mo></mover><mn id="S2.SS1.p6.9.m9.1.1.3" xref="S2.SS1.p6.9.m9.1.1.3.cmml">2</mn></msub><annotation-xml encoding="MathML-Content" id="S2.SS1.p6.9.m9.1b"><apply id="S2.SS1.p6.9.m9.1.1.cmml" xref="S2.SS1.p6.9.m9.1.1"><csymbol cd="ambiguous" id="S2.SS1.p6.9.m9.1.1.1.cmml" xref="S2.SS1.p6.9.m9.1.1">subscript</csymbol><apply id="S2.SS1.p6.9.m9.1.1.2.cmml" xref="S2.SS1.p6.9.m9.1.1.2"><ci id="S2.SS1.p6.9.m9.1.1.2.1.cmml" xref="S2.SS1.p6.9.m9.1.1.2.1">^</ci><ci id="S2.SS1.p6.9.m9.1.1.2.2.cmml" xref="S2.SS1.p6.9.m9.1.1.2.2">𝚺</ci></apply><cn id="S2.SS1.p6.9.m9.1.1.3.cmml" type="integer" xref="S2.SS1.p6.9.m9.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p6.9.m9.1c">\hat{\operatorname{{\mathbf{\Sigma}}}}_{2}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p6.9.m9.1d">over^ start_ARG bold_Σ end_ARG start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math>, <math alttext="\operatorname{{\mathbf{\Sigma}}}_{1}" class="ltx_Math" display="inline" id="S2.SS1.p6.10.m10.1"><semantics id="S2.SS1.p6.10.m10.1a"><msub id="S2.SS1.p6.10.m10.1.1" xref="S2.SS1.p6.10.m10.1.1.cmml"><mi id="S2.SS1.p6.10.m10.1.1.2" xref="S2.SS1.p6.10.m10.1.1.2.cmml">𝚺</mi><mn id="S2.SS1.p6.10.m10.1.1.3" xref="S2.SS1.p6.10.m10.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S2.SS1.p6.10.m10.1b"><apply id="S2.SS1.p6.10.m10.1.1.cmml" xref="S2.SS1.p6.10.m10.1.1"><csymbol cd="ambiguous" id="S2.SS1.p6.10.m10.1.1.1.cmml" xref="S2.SS1.p6.10.m10.1.1">subscript</csymbol><ci id="S2.SS1.p6.10.m10.1.1.2.cmml" xref="S2.SS1.p6.10.m10.1.1.2">𝚺</ci><cn id="S2.SS1.p6.10.m10.1.1.3.cmml" type="integer" xref="S2.SS1.p6.10.m10.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p6.10.m10.1c">\operatorname{{\mathbf{\Sigma}}}_{1}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p6.10.m10.1d">bold_Σ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math>, and <math alttext="\operatorname{{\mathbf{\Sigma}}}_{2}" class="ltx_Math" display="inline" id="S2.SS1.p6.11.m11.1"><semantics id="S2.SS1.p6.11.m11.1a"><msub id="S2.SS1.p6.11.m11.1.1" xref="S2.SS1.p6.11.m11.1.1.cmml"><mi id="S2.SS1.p6.11.m11.1.1.2" xref="S2.SS1.p6.11.m11.1.1.2.cmml">𝚺</mi><mn id="S2.SS1.p6.11.m11.1.1.3" xref="S2.SS1.p6.11.m11.1.1.3.cmml">2</mn></msub><annotation-xml encoding="MathML-Content" id="S2.SS1.p6.11.m11.1b"><apply id="S2.SS1.p6.11.m11.1.1.cmml" xref="S2.SS1.p6.11.m11.1.1"><csymbol cd="ambiguous" id="S2.SS1.p6.11.m11.1.1.1.cmml" xref="S2.SS1.p6.11.m11.1.1">subscript</csymbol><ci id="S2.SS1.p6.11.m11.1.1.2.cmml" xref="S2.SS1.p6.11.m11.1.1.2">𝚺</ci><cn id="S2.SS1.p6.11.m11.1.1.3.cmml" type="integer" xref="S2.SS1.p6.11.m11.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p6.11.m11.1c">\operatorname{{\mathbf{\Sigma}}}_{2}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p6.11.m11.1d">bold_Σ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math> so that their respective largest diagonal entries are 1. We also use Proposition <a class="ltx_ref" href="https://arxiv.org/html/2503.13782v1#S1.Thmproposition2" title="Proposition 1.2 ‣ 1.2 Proof of Proposition 2.2 ‣ 1 Proof of the Propositions ‣ Regularized Parameter Estimation in Mixed Model Trace Regression"><span class="ltx_text ltx_ref_tag">1.2</span></a> to scale the <math alttext="{\operatorname{{\mathbf{\Sigma}}}}_{1}" class="ltx_Math" display="inline" id="S2.SS1.p6.12.m12.1"><semantics id="S2.SS1.p6.12.m12.1a"><msub id="S2.SS1.p6.12.m12.1.1" xref="S2.SS1.p6.12.m12.1.1.cmml"><mi id="S2.SS1.p6.12.m12.1.1.2" xref="S2.SS1.p6.12.m12.1.1.2.cmml">𝚺</mi><mn id="S2.SS1.p6.12.m12.1.1.3" xref="S2.SS1.p6.12.m12.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S2.SS1.p6.12.m12.1b"><apply id="S2.SS1.p6.12.m12.1.1.cmml" xref="S2.SS1.p6.12.m12.1.1"><csymbol cd="ambiguous" id="S2.SS1.p6.12.m12.1.1.1.cmml" xref="S2.SS1.p6.12.m12.1.1">subscript</csymbol><ci id="S2.SS1.p6.12.m12.1.1.2.cmml" xref="S2.SS1.p6.12.m12.1.1.2">𝚺</ci><cn id="S2.SS1.p6.12.m12.1.1.3.cmml" type="integer" xref="S2.SS1.p6.12.m12.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p6.12.m12.1c">{\operatorname{{\mathbf{\Sigma}}}}_{1}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p6.12.m12.1d">bold_Σ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="{\operatorname{{\mathbf{\Sigma}}}}_{2}" class="ltx_Math" display="inline" id="S2.SS1.p6.13.m13.1"><semantics id="S2.SS1.p6.13.m13.1a"><msub id="S2.SS1.p6.13.m13.1.1" xref="S2.SS1.p6.13.m13.1.1.cmml"><mi id="S2.SS1.p6.13.m13.1.1.2" xref="S2.SS1.p6.13.m13.1.1.2.cmml">𝚺</mi><mn id="S2.SS1.p6.13.m13.1.1.3" xref="S2.SS1.p6.13.m13.1.1.3.cmml">2</mn></msub><annotation-xml encoding="MathML-Content" id="S2.SS1.p6.13.m13.1b"><apply id="S2.SS1.p6.13.m13.1.1.cmml" xref="S2.SS1.p6.13.m13.1.1"><csymbol cd="ambiguous" id="S2.SS1.p6.13.m13.1.1.1.cmml" xref="S2.SS1.p6.13.m13.1.1">subscript</csymbol><ci id="S2.SS1.p6.13.m13.1.1.2.cmml" xref="S2.SS1.p6.13.m13.1.1.2">𝚺</ci><cn id="S2.SS1.p6.13.m13.1.1.3.cmml" type="integer" xref="S2.SS1.p6.13.m13.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p6.13.m13.1c">{\operatorname{{\mathbf{\Sigma}}}}_{2}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p6.13.m13.1d">bold_Σ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math> estimates at the end of each iteration to further improve the stability of the estimation algorithm.</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>Reduction to a Structured Mixed Model</h3> <div class="ltx_para" id="S2.SS2.p1"> <p class="ltx_p" id="S2.SS2.p1.25">The reduction of MMTR to a structured mixed model linear regression requires the following definitions. 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xref="S2.SS2.p1.10.m10.2.2.1.1.1.1.cmml">(</mo><msubsup id="S2.SS2.p1.10.m10.2.2.1.1.1.1" xref="S2.SS2.p1.10.m10.2.2.1.1.1.1.cmml"><mi id="S2.SS2.p1.10.m10.2.2.1.1.1.1.2.2" xref="S2.SS2.p1.10.m10.2.2.1.1.1.1.2.2.cmml">𝐂</mi><mi id="S2.SS2.p1.10.m10.2.2.1.1.1.1.2.3" xref="S2.SS2.p1.10.m10.2.2.1.1.1.1.2.3.cmml">i</mi><mo id="S2.SS2.p1.10.m10.2.2.1.1.1.1.3" xref="S2.SS2.p1.10.m10.2.2.1.1.1.1.3.cmml">⊤</mo></msubsup><mo id="S2.SS2.p1.10.m10.2.2.1.1.1.3" stretchy="false" xref="S2.SS2.p1.10.m10.2.2.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p1.10.m10.2b"><apply id="S2.SS2.p1.10.m10.2.2.cmml" xref="S2.SS2.p1.10.m10.2.2"><eq id="S2.SS2.p1.10.m10.2.2.2.cmml" xref="S2.SS2.p1.10.m10.2.2.2"></eq><apply id="S2.SS2.p1.10.m10.2.2.3.cmml" xref="S2.SS2.p1.10.m10.2.2.3"><csymbol cd="ambiguous" id="S2.SS2.p1.10.m10.2.2.3.1.cmml" xref="S2.SS2.p1.10.m10.2.2.3">subscript</csymbol><ci id="S2.SS2.p1.10.m10.2.2.3.2.cmml" xref="S2.SS2.p1.10.m10.2.2.3.2">𝐜</ci><apply id="S2.SS2.p1.10.m10.1.1.1.cmml" xref="S2.SS2.p1.10.m10.1.1.1"><times id="S2.SS2.p1.10.m10.1.1.1.2.cmml" xref="S2.SS2.p1.10.m10.1.1.1.2"></times><ci id="S2.SS2.p1.10.m10.1.1.1.3.cmml" xref="S2.SS2.p1.10.m10.1.1.1.3">𝑖</ci><cn id="S2.SS2.p1.10.m10.1.1.1.1.cmml" type="integer" xref="S2.SS2.p1.10.m10.1.1.1.1">2</cn></apply></apply><apply id="S2.SS2.p1.10.m10.2.2.1.cmml" xref="S2.SS2.p1.10.m10.2.2.1"><times id="S2.SS2.p1.10.m10.2.2.1.2.cmml" xref="S2.SS2.p1.10.m10.2.2.1.2"></times><ci id="S2.SS2.p1.10.m10.2.2.1.3a.cmml" xref="S2.SS2.p1.10.m10.2.2.1.3"><mtext id="S2.SS2.p1.10.m10.2.2.1.3.cmml" xref="S2.SS2.p1.10.m10.2.2.1.3">vec</mtext></ci><apply id="S2.SS2.p1.10.m10.2.2.1.1.1.1.cmml" xref="S2.SS2.p1.10.m10.2.2.1.1.1"><csymbol cd="ambiguous" id="S2.SS2.p1.10.m10.2.2.1.1.1.1.1.cmml" xref="S2.SS2.p1.10.m10.2.2.1.1.1">superscript</csymbol><apply id="S2.SS2.p1.10.m10.2.2.1.1.1.1.2.cmml" xref="S2.SS2.p1.10.m10.2.2.1.1.1"><csymbol cd="ambiguous" id="S2.SS2.p1.10.m10.2.2.1.1.1.1.2.1.cmml" xref="S2.SS2.p1.10.m10.2.2.1.1.1">subscript</csymbol><ci id="S2.SS2.p1.10.m10.2.2.1.1.1.1.2.2.cmml" xref="S2.SS2.p1.10.m10.2.2.1.1.1.1.2.2">𝐂</ci><ci id="S2.SS2.p1.10.m10.2.2.1.1.1.1.2.3.cmml" xref="S2.SS2.p1.10.m10.2.2.1.1.1.1.2.3">𝑖</ci></apply><csymbol cd="latexml" id="S2.SS2.p1.10.m10.2.2.1.1.1.1.3.cmml" xref="S2.SS2.p1.10.m10.2.2.1.1.1.1.3">top</csymbol></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p1.10.m10.2c">\operatorname{\mathbf{c}}_{i(2)}=\text{vec}(\operatorname{\mathbf{C}}_{i}^{% \top})</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p1.10.m10.2d">bold_c start_POSTSUBSCRIPT italic_i ( 2 ) end_POSTSUBSCRIPT = vec ( bold_C start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ⊤ end_POSTSUPERSCRIPT )</annotation></semantics></math>, <math alttext="\operatorname{\mathbf{l}}_{1}=\text{vec}(\operatorname{\mathbf{L}}_{1})" class="ltx_Math" display="inline" id="S2.SS2.p1.11.m11.1"><semantics id="S2.SS2.p1.11.m11.1a"><mrow id="S2.SS2.p1.11.m11.1.1" xref="S2.SS2.p1.11.m11.1.1.cmml"><msub id="S2.SS2.p1.11.m11.1.1.3" xref="S2.SS2.p1.11.m11.1.1.3.cmml"><mi id="S2.SS2.p1.11.m11.1.1.3.2" xref="S2.SS2.p1.11.m11.1.1.3.2.cmml">𝐥</mi><mn id="S2.SS2.p1.11.m11.1.1.3.3" xref="S2.SS2.p1.11.m11.1.1.3.3.cmml">1</mn></msub><mo id="S2.SS2.p1.11.m11.1.1.2" xref="S2.SS2.p1.11.m11.1.1.2.cmml">=</mo><mrow id="S2.SS2.p1.11.m11.1.1.1" xref="S2.SS2.p1.11.m11.1.1.1.cmml"><mtext id="S2.SS2.p1.11.m11.1.1.1.3" xref="S2.SS2.p1.11.m11.1.1.1.3a.cmml">vec</mtext><mo id="S2.SS2.p1.11.m11.1.1.1.2" xref="S2.SS2.p1.11.m11.1.1.1.2.cmml">⁢</mo><mrow id="S2.SS2.p1.11.m11.1.1.1.1.1" xref="S2.SS2.p1.11.m11.1.1.1.1.1.1.cmml"><mo id="S2.SS2.p1.11.m11.1.1.1.1.1.2" stretchy="false" xref="S2.SS2.p1.11.m11.1.1.1.1.1.1.cmml">(</mo><msub id="S2.SS2.p1.11.m11.1.1.1.1.1.1" xref="S2.SS2.p1.11.m11.1.1.1.1.1.1.cmml"><mi id="S2.SS2.p1.11.m11.1.1.1.1.1.1.2" xref="S2.SS2.p1.11.m11.1.1.1.1.1.1.2.cmml">𝐋</mi><mn id="S2.SS2.p1.11.m11.1.1.1.1.1.1.3" xref="S2.SS2.p1.11.m11.1.1.1.1.1.1.3.cmml">1</mn></msub><mo id="S2.SS2.p1.11.m11.1.1.1.1.1.3" stretchy="false" xref="S2.SS2.p1.11.m11.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p1.11.m11.1b"><apply id="S2.SS2.p1.11.m11.1.1.cmml" xref="S2.SS2.p1.11.m11.1.1"><eq id="S2.SS2.p1.11.m11.1.1.2.cmml" xref="S2.SS2.p1.11.m11.1.1.2"></eq><apply id="S2.SS2.p1.11.m11.1.1.3.cmml" xref="S2.SS2.p1.11.m11.1.1.3"><csymbol cd="ambiguous" id="S2.SS2.p1.11.m11.1.1.3.1.cmml" xref="S2.SS2.p1.11.m11.1.1.3">subscript</csymbol><ci id="S2.SS2.p1.11.m11.1.1.3.2.cmml" xref="S2.SS2.p1.11.m11.1.1.3.2">𝐥</ci><cn id="S2.SS2.p1.11.m11.1.1.3.3.cmml" type="integer" xref="S2.SS2.p1.11.m11.1.1.3.3">1</cn></apply><apply id="S2.SS2.p1.11.m11.1.1.1.cmml" xref="S2.SS2.p1.11.m11.1.1.1"><times id="S2.SS2.p1.11.m11.1.1.1.2.cmml" xref="S2.SS2.p1.11.m11.1.1.1.2"></times><ci id="S2.SS2.p1.11.m11.1.1.1.3a.cmml" xref="S2.SS2.p1.11.m11.1.1.1.3"><mtext id="S2.SS2.p1.11.m11.1.1.1.3.cmml" xref="S2.SS2.p1.11.m11.1.1.1.3">vec</mtext></ci><apply id="S2.SS2.p1.11.m11.1.1.1.1.1.1.cmml" xref="S2.SS2.p1.11.m11.1.1.1.1.1"><csymbol cd="ambiguous" id="S2.SS2.p1.11.m11.1.1.1.1.1.1.1.cmml" xref="S2.SS2.p1.11.m11.1.1.1.1.1">subscript</csymbol><ci id="S2.SS2.p1.11.m11.1.1.1.1.1.1.2.cmml" xref="S2.SS2.p1.11.m11.1.1.1.1.1.1.2">𝐋</ci><cn id="S2.SS2.p1.11.m11.1.1.1.1.1.1.3.cmml" type="integer" xref="S2.SS2.p1.11.m11.1.1.1.1.1.1.3">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p1.11.m11.1c">\operatorname{\mathbf{l}}_{1}=\text{vec}(\operatorname{\mathbf{L}}_{1})</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p1.11.m11.1d">bold_l start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = vec ( bold_L start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT )</annotation></semantics></math>, <math alttext="\operatorname{\mathbf{l}}_{2}=\text{vec}(\operatorname{\mathbf{L}}_{2})" class="ltx_Math" display="inline" id="S2.SS2.p1.12.m12.1"><semantics id="S2.SS2.p1.12.m12.1a"><mrow id="S2.SS2.p1.12.m12.1.1" xref="S2.SS2.p1.12.m12.1.1.cmml"><msub id="S2.SS2.p1.12.m12.1.1.3" xref="S2.SS2.p1.12.m12.1.1.3.cmml"><mi id="S2.SS2.p1.12.m12.1.1.3.2" xref="S2.SS2.p1.12.m12.1.1.3.2.cmml">𝐥</mi><mn id="S2.SS2.p1.12.m12.1.1.3.3" xref="S2.SS2.p1.12.m12.1.1.3.3.cmml">2</mn></msub><mo id="S2.SS2.p1.12.m12.1.1.2" xref="S2.SS2.p1.12.m12.1.1.2.cmml">=</mo><mrow id="S2.SS2.p1.12.m12.1.1.1" xref="S2.SS2.p1.12.m12.1.1.1.cmml"><mtext id="S2.SS2.p1.12.m12.1.1.1.3" xref="S2.SS2.p1.12.m12.1.1.1.3a.cmml">vec</mtext><mo id="S2.SS2.p1.12.m12.1.1.1.2" xref="S2.SS2.p1.12.m12.1.1.1.2.cmml">⁢</mo><mrow id="S2.SS2.p1.12.m12.1.1.1.1.1" xref="S2.SS2.p1.12.m12.1.1.1.1.1.1.cmml"><mo id="S2.SS2.p1.12.m12.1.1.1.1.1.2" stretchy="false" xref="S2.SS2.p1.12.m12.1.1.1.1.1.1.cmml">(</mo><msub id="S2.SS2.p1.12.m12.1.1.1.1.1.1" xref="S2.SS2.p1.12.m12.1.1.1.1.1.1.cmml"><mi id="S2.SS2.p1.12.m12.1.1.1.1.1.1.2" xref="S2.SS2.p1.12.m12.1.1.1.1.1.1.2.cmml">𝐋</mi><mn id="S2.SS2.p1.12.m12.1.1.1.1.1.1.3" xref="S2.SS2.p1.12.m12.1.1.1.1.1.1.3.cmml">2</mn></msub><mo id="S2.SS2.p1.12.m12.1.1.1.1.1.3" stretchy="false" xref="S2.SS2.p1.12.m12.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p1.12.m12.1b"><apply id="S2.SS2.p1.12.m12.1.1.cmml" xref="S2.SS2.p1.12.m12.1.1"><eq id="S2.SS2.p1.12.m12.1.1.2.cmml" xref="S2.SS2.p1.12.m12.1.1.2"></eq><apply id="S2.SS2.p1.12.m12.1.1.3.cmml" xref="S2.SS2.p1.12.m12.1.1.3"><csymbol cd="ambiguous" id="S2.SS2.p1.12.m12.1.1.3.1.cmml" xref="S2.SS2.p1.12.m12.1.1.3">subscript</csymbol><ci id="S2.SS2.p1.12.m12.1.1.3.2.cmml" xref="S2.SS2.p1.12.m12.1.1.3.2">𝐥</ci><cn id="S2.SS2.p1.12.m12.1.1.3.3.cmml" type="integer" xref="S2.SS2.p1.12.m12.1.1.3.3">2</cn></apply><apply id="S2.SS2.p1.12.m12.1.1.1.cmml" xref="S2.SS2.p1.12.m12.1.1.1"><times id="S2.SS2.p1.12.m12.1.1.1.2.cmml" xref="S2.SS2.p1.12.m12.1.1.1.2"></times><ci id="S2.SS2.p1.12.m12.1.1.1.3a.cmml" xref="S2.SS2.p1.12.m12.1.1.1.3"><mtext id="S2.SS2.p1.12.m12.1.1.1.3.cmml" xref="S2.SS2.p1.12.m12.1.1.1.3">vec</mtext></ci><apply id="S2.SS2.p1.12.m12.1.1.1.1.1.1.cmml" xref="S2.SS2.p1.12.m12.1.1.1.1.1"><csymbol cd="ambiguous" id="S2.SS2.p1.12.m12.1.1.1.1.1.1.1.cmml" xref="S2.SS2.p1.12.m12.1.1.1.1.1">subscript</csymbol><ci id="S2.SS2.p1.12.m12.1.1.1.1.1.1.2.cmml" xref="S2.SS2.p1.12.m12.1.1.1.1.1.1.2">𝐋</ci><cn id="S2.SS2.p1.12.m12.1.1.1.1.1.1.3.cmml" type="integer" xref="S2.SS2.p1.12.m12.1.1.1.1.1.1.3">2</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p1.12.m12.1c">\operatorname{\mathbf{l}}_{2}=\text{vec}(\operatorname{\mathbf{L}}_{2})</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p1.12.m12.1d">bold_l start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT = vec ( bold_L start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT )</annotation></semantics></math>. Then, identities relating <math alttext="\operatorname{\mathop{\mathrm{tr}}}" class="ltx_Math" display="inline" id="S2.SS2.p1.13.m13.1"><semantics id="S2.SS2.p1.13.m13.1a"><mi id="S2.SS2.p1.13.m13.1.1" xref="S2.SS2.p1.13.m13.1.1.cmml">tr</mi><annotation-xml encoding="MathML-Content" id="S2.SS2.p1.13.m13.1b"><ci id="S2.SS2.p1.13.m13.1.1.cmml" xref="S2.SS2.p1.13.m13.1.1">tr</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p1.13.m13.1c">\operatorname{\mathop{\mathrm{tr}}}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p1.13.m13.1d">roman_tr</annotation></semantics></math>, vec, and <math alttext="\otimes" class="ltx_Math" display="inline" id="S2.SS2.p1.14.m14.1"><semantics id="S2.SS2.p1.14.m14.1a"><mo id="S2.SS2.p1.14.m14.1.1" xref="S2.SS2.p1.14.m14.1.1.cmml">⊗</mo><annotation-xml encoding="MathML-Content" id="S2.SS2.p1.14.m14.1b"><csymbol cd="latexml" id="S2.SS2.p1.14.m14.1.1.cmml" xref="S2.SS2.p1.14.m14.1.1">tensor-product</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p1.14.m14.1c">\otimes</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p1.14.m14.1d">⊗</annotation></semantics></math> operators imply that <math alttext="\operatorname{\mathbf{A}}_{i}=\operatorname{\mathbf{L}}_{1}\operatorname{% \mathbf{C}}_{i}\operatorname{\mathbf{L}}_{2}^{\top}" class="ltx_Math" display="inline" id="S2.SS2.p1.15.m15.1"><semantics id="S2.SS2.p1.15.m15.1a"><mrow id="S2.SS2.p1.15.m15.1.1" xref="S2.SS2.p1.15.m15.1.1.cmml"><msub id="S2.SS2.p1.15.m15.1.1.2" xref="S2.SS2.p1.15.m15.1.1.2.cmml"><mi id="S2.SS2.p1.15.m15.1.1.2.2" xref="S2.SS2.p1.15.m15.1.1.2.2.cmml">𝐀</mi><mi id="S2.SS2.p1.15.m15.1.1.2.3" xref="S2.SS2.p1.15.m15.1.1.2.3.cmml">i</mi></msub><mo id="S2.SS2.p1.15.m15.1.1.1" xref="S2.SS2.p1.15.m15.1.1.1.cmml">=</mo><mrow id="S2.SS2.p1.15.m15.1.1.3" xref="S2.SS2.p1.15.m15.1.1.3.cmml"><msub id="S2.SS2.p1.15.m15.1.1.3.1" xref="S2.SS2.p1.15.m15.1.1.3.1.cmml"><mi id="S2.SS2.p1.15.m15.1.1.3.1.2" xref="S2.SS2.p1.15.m15.1.1.3.1.2.cmml">𝐋</mi><mn id="S2.SS2.p1.15.m15.1.1.3.1.3" xref="S2.SS2.p1.15.m15.1.1.3.1.3.cmml">1</mn></msub><mo id="S2.SS2.p1.15.m15.1.1.3a" lspace="0.167em" xref="S2.SS2.p1.15.m15.1.1.3.cmml">⁡</mo><mrow id="S2.SS2.p1.15.m15.1.1.3.2" xref="S2.SS2.p1.15.m15.1.1.3.2.cmml"><msub id="S2.SS2.p1.15.m15.1.1.3.2.1" xref="S2.SS2.p1.15.m15.1.1.3.2.1.cmml"><mi id="S2.SS2.p1.15.m15.1.1.3.2.1.2" xref="S2.SS2.p1.15.m15.1.1.3.2.1.2.cmml">𝐂</mi><mi id="S2.SS2.p1.15.m15.1.1.3.2.1.3" xref="S2.SS2.p1.15.m15.1.1.3.2.1.3.cmml">i</mi></msub><mo id="S2.SS2.p1.15.m15.1.1.3.2a" lspace="0.167em" xref="S2.SS2.p1.15.m15.1.1.3.2.cmml">⁡</mo><msubsup id="S2.SS2.p1.15.m15.1.1.3.2.2" xref="S2.SS2.p1.15.m15.1.1.3.2.2.cmml"><mi id="S2.SS2.p1.15.m15.1.1.3.2.2.2.2" xref="S2.SS2.p1.15.m15.1.1.3.2.2.2.2.cmml">𝐋</mi><mn id="S2.SS2.p1.15.m15.1.1.3.2.2.2.3" xref="S2.SS2.p1.15.m15.1.1.3.2.2.2.3.cmml">2</mn><mo id="S2.SS2.p1.15.m15.1.1.3.2.2.3" xref="S2.SS2.p1.15.m15.1.1.3.2.2.3.cmml">⊤</mo></msubsup></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p1.15.m15.1b"><apply id="S2.SS2.p1.15.m15.1.1.cmml" xref="S2.SS2.p1.15.m15.1.1"><eq id="S2.SS2.p1.15.m15.1.1.1.cmml" xref="S2.SS2.p1.15.m15.1.1.1"></eq><apply id="S2.SS2.p1.15.m15.1.1.2.cmml" xref="S2.SS2.p1.15.m15.1.1.2"><csymbol cd="ambiguous" id="S2.SS2.p1.15.m15.1.1.2.1.cmml" xref="S2.SS2.p1.15.m15.1.1.2">subscript</csymbol><ci id="S2.SS2.p1.15.m15.1.1.2.2.cmml" xref="S2.SS2.p1.15.m15.1.1.2.2">𝐀</ci><ci id="S2.SS2.p1.15.m15.1.1.2.3.cmml" xref="S2.SS2.p1.15.m15.1.1.2.3">𝑖</ci></apply><apply id="S2.SS2.p1.15.m15.1.1.3.cmml" xref="S2.SS2.p1.15.m15.1.1.3"><apply id="S2.SS2.p1.15.m15.1.1.3.1.cmml" xref="S2.SS2.p1.15.m15.1.1.3.1"><csymbol cd="ambiguous" id="S2.SS2.p1.15.m15.1.1.3.1.1.cmml" xref="S2.SS2.p1.15.m15.1.1.3.1">subscript</csymbol><ci id="S2.SS2.p1.15.m15.1.1.3.1.2.cmml" xref="S2.SS2.p1.15.m15.1.1.3.1.2">𝐋</ci><cn id="S2.SS2.p1.15.m15.1.1.3.1.3.cmml" type="integer" xref="S2.SS2.p1.15.m15.1.1.3.1.3">1</cn></apply><apply id="S2.SS2.p1.15.m15.1.1.3.2.cmml" xref="S2.SS2.p1.15.m15.1.1.3.2"><apply id="S2.SS2.p1.15.m15.1.1.3.2.1.cmml" xref="S2.SS2.p1.15.m15.1.1.3.2.1"><csymbol cd="ambiguous" id="S2.SS2.p1.15.m15.1.1.3.2.1.1.cmml" xref="S2.SS2.p1.15.m15.1.1.3.2.1">subscript</csymbol><ci id="S2.SS2.p1.15.m15.1.1.3.2.1.2.cmml" xref="S2.SS2.p1.15.m15.1.1.3.2.1.2">𝐂</ci><ci id="S2.SS2.p1.15.m15.1.1.3.2.1.3.cmml" xref="S2.SS2.p1.15.m15.1.1.3.2.1.3">𝑖</ci></apply><apply id="S2.SS2.p1.15.m15.1.1.3.2.2.cmml" xref="S2.SS2.p1.15.m15.1.1.3.2.2"><csymbol cd="ambiguous" id="S2.SS2.p1.15.m15.1.1.3.2.2.1.cmml" xref="S2.SS2.p1.15.m15.1.1.3.2.2">superscript</csymbol><apply id="S2.SS2.p1.15.m15.1.1.3.2.2.2.cmml" xref="S2.SS2.p1.15.m15.1.1.3.2.2"><csymbol cd="ambiguous" id="S2.SS2.p1.15.m15.1.1.3.2.2.2.1.cmml" xref="S2.SS2.p1.15.m15.1.1.3.2.2">subscript</csymbol><ci id="S2.SS2.p1.15.m15.1.1.3.2.2.2.2.cmml" xref="S2.SS2.p1.15.m15.1.1.3.2.2.2.2">𝐋</ci><cn id="S2.SS2.p1.15.m15.1.1.3.2.2.2.3.cmml" type="integer" xref="S2.SS2.p1.15.m15.1.1.3.2.2.2.3">2</cn></apply><csymbol cd="latexml" id="S2.SS2.p1.15.m15.1.1.3.2.2.3.cmml" xref="S2.SS2.p1.15.m15.1.1.3.2.2.3">top</csymbol></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p1.15.m15.1c">\operatorname{\mathbf{A}}_{i}=\operatorname{\mathbf{L}}_{1}\operatorname{% \mathbf{C}}_{i}\operatorname{\mathbf{L}}_{2}^{\top}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p1.15.m15.1d">bold_A start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = bold_L start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT bold_C start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT bold_L start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ⊤ end_POSTSUPERSCRIPT</annotation></semantics></math> is equivalent to <math alttext="\operatorname{\mathbf{a}}_{i(1)}=\operatorname{\mathbf{L}}_{(1)}\operatorname{% \mathbf{c}}_{(1)}" class="ltx_Math" display="inline" id="S2.SS2.p1.16.m16.3"><semantics id="S2.SS2.p1.16.m16.3a"><mrow id="S2.SS2.p1.16.m16.3.4" xref="S2.SS2.p1.16.m16.3.4.cmml"><msub id="S2.SS2.p1.16.m16.3.4.2" xref="S2.SS2.p1.16.m16.3.4.2.cmml"><mi id="S2.SS2.p1.16.m16.3.4.2.2" xref="S2.SS2.p1.16.m16.3.4.2.2.cmml">𝐚</mi><mrow id="S2.SS2.p1.16.m16.1.1.1" xref="S2.SS2.p1.16.m16.1.1.1.cmml"><mi id="S2.SS2.p1.16.m16.1.1.1.3" xref="S2.SS2.p1.16.m16.1.1.1.3.cmml">i</mi><mo id="S2.SS2.p1.16.m16.1.1.1.2" xref="S2.SS2.p1.16.m16.1.1.1.2.cmml">⁢</mo><mrow id="S2.SS2.p1.16.m16.1.1.1.4.2" xref="S2.SS2.p1.16.m16.1.1.1.cmml"><mo id="S2.SS2.p1.16.m16.1.1.1.4.2.1" stretchy="false" xref="S2.SS2.p1.16.m16.1.1.1.cmml">(</mo><mn id="S2.SS2.p1.16.m16.1.1.1.1" xref="S2.SS2.p1.16.m16.1.1.1.1.cmml">1</mn><mo id="S2.SS2.p1.16.m16.1.1.1.4.2.2" stretchy="false" xref="S2.SS2.p1.16.m16.1.1.1.cmml">)</mo></mrow></mrow></msub><mo id="S2.SS2.p1.16.m16.3.4.1" xref="S2.SS2.p1.16.m16.3.4.1.cmml">=</mo><mrow id="S2.SS2.p1.16.m16.3.4.3" xref="S2.SS2.p1.16.m16.3.4.3.cmml"><msub id="S2.SS2.p1.16.m16.3.4.3.1" xref="S2.SS2.p1.16.m16.3.4.3.1.cmml"><mi id="S2.SS2.p1.16.m16.3.4.3.1.2" xref="S2.SS2.p1.16.m16.3.4.3.1.2.cmml">𝐋</mi><mrow id="S2.SS2.p1.16.m16.2.2.1.3" xref="S2.SS2.p1.16.m16.3.4.3.1.cmml"><mo id="S2.SS2.p1.16.m16.2.2.1.3.1" stretchy="false" xref="S2.SS2.p1.16.m16.3.4.3.1.cmml">(</mo><mn id="S2.SS2.p1.16.m16.2.2.1.1" xref="S2.SS2.p1.16.m16.2.2.1.1.cmml">1</mn><mo id="S2.SS2.p1.16.m16.2.2.1.3.2" stretchy="false" xref="S2.SS2.p1.16.m16.3.4.3.1.cmml">)</mo></mrow></msub><mo id="S2.SS2.p1.16.m16.3.4.3a" lspace="0.167em" xref="S2.SS2.p1.16.m16.3.4.3.cmml">⁡</mo><msub id="S2.SS2.p1.16.m16.3.4.3.2" xref="S2.SS2.p1.16.m16.3.4.3.2.cmml"><mi id="S2.SS2.p1.16.m16.3.4.3.2.2" xref="S2.SS2.p1.16.m16.3.4.3.2.2.cmml">𝐜</mi><mrow id="S2.SS2.p1.16.m16.3.3.1.3" xref="S2.SS2.p1.16.m16.3.4.3.2.cmml"><mo id="S2.SS2.p1.16.m16.3.3.1.3.1" stretchy="false" xref="S2.SS2.p1.16.m16.3.4.3.2.cmml">(</mo><mn id="S2.SS2.p1.16.m16.3.3.1.1" xref="S2.SS2.p1.16.m16.3.3.1.1.cmml">1</mn><mo id="S2.SS2.p1.16.m16.3.3.1.3.2" stretchy="false" xref="S2.SS2.p1.16.m16.3.4.3.2.cmml">)</mo></mrow></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p1.16.m16.3b"><apply id="S2.SS2.p1.16.m16.3.4.cmml" xref="S2.SS2.p1.16.m16.3.4"><eq id="S2.SS2.p1.16.m16.3.4.1.cmml" xref="S2.SS2.p1.16.m16.3.4.1"></eq><apply id="S2.SS2.p1.16.m16.3.4.2.cmml" xref="S2.SS2.p1.16.m16.3.4.2"><csymbol cd="ambiguous" id="S2.SS2.p1.16.m16.3.4.2.1.cmml" xref="S2.SS2.p1.16.m16.3.4.2">subscript</csymbol><ci id="S2.SS2.p1.16.m16.3.4.2.2.cmml" xref="S2.SS2.p1.16.m16.3.4.2.2">𝐚</ci><apply id="S2.SS2.p1.16.m16.1.1.1.cmml" xref="S2.SS2.p1.16.m16.1.1.1"><times id="S2.SS2.p1.16.m16.1.1.1.2.cmml" xref="S2.SS2.p1.16.m16.1.1.1.2"></times><ci id="S2.SS2.p1.16.m16.1.1.1.3.cmml" xref="S2.SS2.p1.16.m16.1.1.1.3">𝑖</ci><cn id="S2.SS2.p1.16.m16.1.1.1.1.cmml" type="integer" xref="S2.SS2.p1.16.m16.1.1.1.1">1</cn></apply></apply><apply id="S2.SS2.p1.16.m16.3.4.3.cmml" xref="S2.SS2.p1.16.m16.3.4.3"><apply id="S2.SS2.p1.16.m16.3.4.3.1.cmml" xref="S2.SS2.p1.16.m16.3.4.3.1"><csymbol cd="ambiguous" id="S2.SS2.p1.16.m16.3.4.3.1.1.cmml" xref="S2.SS2.p1.16.m16.3.4.3.1">subscript</csymbol><ci id="S2.SS2.p1.16.m16.3.4.3.1.2.cmml" xref="S2.SS2.p1.16.m16.3.4.3.1.2">𝐋</ci><cn id="S2.SS2.p1.16.m16.2.2.1.1.cmml" type="integer" xref="S2.SS2.p1.16.m16.2.2.1.1">1</cn></apply><apply id="S2.SS2.p1.16.m16.3.4.3.2.cmml" xref="S2.SS2.p1.16.m16.3.4.3.2"><csymbol cd="ambiguous" id="S2.SS2.p1.16.m16.3.4.3.2.1.cmml" xref="S2.SS2.p1.16.m16.3.4.3.2">subscript</csymbol><ci id="S2.SS2.p1.16.m16.3.4.3.2.2.cmml" xref="S2.SS2.p1.16.m16.3.4.3.2.2">𝐜</ci><cn id="S2.SS2.p1.16.m16.3.3.1.1.cmml" type="integer" xref="S2.SS2.p1.16.m16.3.3.1.1">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p1.16.m16.3c">\operatorname{\mathbf{a}}_{i(1)}=\operatorname{\mathbf{L}}_{(1)}\operatorname{% \mathbf{c}}_{(1)}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p1.16.m16.3d">bold_a start_POSTSUBSCRIPT italic_i ( 1 ) end_POSTSUBSCRIPT = bold_L start_POSTSUBSCRIPT ( 1 ) end_POSTSUBSCRIPT bold_c start_POSTSUBSCRIPT ( 1 ) end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="\operatorname{\mathbf{a}}_{i(2)}=\operatorname{\mathbf{L}}_{(2)}\operatorname{% \mathbf{c}}_{i(2)}" class="ltx_Math" display="inline" id="S2.SS2.p1.17.m17.3"><semantics id="S2.SS2.p1.17.m17.3a"><mrow id="S2.SS2.p1.17.m17.3.4" xref="S2.SS2.p1.17.m17.3.4.cmml"><msub id="S2.SS2.p1.17.m17.3.4.2" xref="S2.SS2.p1.17.m17.3.4.2.cmml"><mi id="S2.SS2.p1.17.m17.3.4.2.2" xref="S2.SS2.p1.17.m17.3.4.2.2.cmml">𝐚</mi><mrow id="S2.SS2.p1.17.m17.1.1.1" xref="S2.SS2.p1.17.m17.1.1.1.cmml"><mi id="S2.SS2.p1.17.m17.1.1.1.3" xref="S2.SS2.p1.17.m17.1.1.1.3.cmml">i</mi><mo id="S2.SS2.p1.17.m17.1.1.1.2" xref="S2.SS2.p1.17.m17.1.1.1.2.cmml">⁢</mo><mrow id="S2.SS2.p1.17.m17.1.1.1.4.2" xref="S2.SS2.p1.17.m17.1.1.1.cmml"><mo id="S2.SS2.p1.17.m17.1.1.1.4.2.1" stretchy="false" xref="S2.SS2.p1.17.m17.1.1.1.cmml">(</mo><mn id="S2.SS2.p1.17.m17.1.1.1.1" xref="S2.SS2.p1.17.m17.1.1.1.1.cmml">2</mn><mo id="S2.SS2.p1.17.m17.1.1.1.4.2.2" stretchy="false" xref="S2.SS2.p1.17.m17.1.1.1.cmml">)</mo></mrow></mrow></msub><mo id="S2.SS2.p1.17.m17.3.4.1" xref="S2.SS2.p1.17.m17.3.4.1.cmml">=</mo><mrow id="S2.SS2.p1.17.m17.3.4.3" xref="S2.SS2.p1.17.m17.3.4.3.cmml"><msub id="S2.SS2.p1.17.m17.3.4.3.1" xref="S2.SS2.p1.17.m17.3.4.3.1.cmml"><mi id="S2.SS2.p1.17.m17.3.4.3.1.2" xref="S2.SS2.p1.17.m17.3.4.3.1.2.cmml">𝐋</mi><mrow id="S2.SS2.p1.17.m17.2.2.1.3" xref="S2.SS2.p1.17.m17.3.4.3.1.cmml"><mo id="S2.SS2.p1.17.m17.2.2.1.3.1" stretchy="false" xref="S2.SS2.p1.17.m17.3.4.3.1.cmml">(</mo><mn id="S2.SS2.p1.17.m17.2.2.1.1" xref="S2.SS2.p1.17.m17.2.2.1.1.cmml">2</mn><mo id="S2.SS2.p1.17.m17.2.2.1.3.2" stretchy="false" xref="S2.SS2.p1.17.m17.3.4.3.1.cmml">)</mo></mrow></msub><mo id="S2.SS2.p1.17.m17.3.4.3a" lspace="0.167em" xref="S2.SS2.p1.17.m17.3.4.3.cmml">⁡</mo><msub id="S2.SS2.p1.17.m17.3.4.3.2" xref="S2.SS2.p1.17.m17.3.4.3.2.cmml"><mi id="S2.SS2.p1.17.m17.3.4.3.2.2" xref="S2.SS2.p1.17.m17.3.4.3.2.2.cmml">𝐜</mi><mrow id="S2.SS2.p1.17.m17.3.3.1" xref="S2.SS2.p1.17.m17.3.3.1.cmml"><mi id="S2.SS2.p1.17.m17.3.3.1.3" xref="S2.SS2.p1.17.m17.3.3.1.3.cmml">i</mi><mo id="S2.SS2.p1.17.m17.3.3.1.2" xref="S2.SS2.p1.17.m17.3.3.1.2.cmml">⁢</mo><mrow id="S2.SS2.p1.17.m17.3.3.1.4.2" xref="S2.SS2.p1.17.m17.3.3.1.cmml"><mo id="S2.SS2.p1.17.m17.3.3.1.4.2.1" stretchy="false" xref="S2.SS2.p1.17.m17.3.3.1.cmml">(</mo><mn id="S2.SS2.p1.17.m17.3.3.1.1" xref="S2.SS2.p1.17.m17.3.3.1.1.cmml">2</mn><mo id="S2.SS2.p1.17.m17.3.3.1.4.2.2" stretchy="false" xref="S2.SS2.p1.17.m17.3.3.1.cmml">)</mo></mrow></mrow></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p1.17.m17.3b"><apply id="S2.SS2.p1.17.m17.3.4.cmml" xref="S2.SS2.p1.17.m17.3.4"><eq id="S2.SS2.p1.17.m17.3.4.1.cmml" xref="S2.SS2.p1.17.m17.3.4.1"></eq><apply id="S2.SS2.p1.17.m17.3.4.2.cmml" xref="S2.SS2.p1.17.m17.3.4.2"><csymbol cd="ambiguous" id="S2.SS2.p1.17.m17.3.4.2.1.cmml" xref="S2.SS2.p1.17.m17.3.4.2">subscript</csymbol><ci id="S2.SS2.p1.17.m17.3.4.2.2.cmml" xref="S2.SS2.p1.17.m17.3.4.2.2">𝐚</ci><apply id="S2.SS2.p1.17.m17.1.1.1.cmml" xref="S2.SS2.p1.17.m17.1.1.1"><times id="S2.SS2.p1.17.m17.1.1.1.2.cmml" xref="S2.SS2.p1.17.m17.1.1.1.2"></times><ci id="S2.SS2.p1.17.m17.1.1.1.3.cmml" xref="S2.SS2.p1.17.m17.1.1.1.3">𝑖</ci><cn id="S2.SS2.p1.17.m17.1.1.1.1.cmml" type="integer" xref="S2.SS2.p1.17.m17.1.1.1.1">2</cn></apply></apply><apply id="S2.SS2.p1.17.m17.3.4.3.cmml" xref="S2.SS2.p1.17.m17.3.4.3"><apply id="S2.SS2.p1.17.m17.3.4.3.1.cmml" xref="S2.SS2.p1.17.m17.3.4.3.1"><csymbol cd="ambiguous" id="S2.SS2.p1.17.m17.3.4.3.1.1.cmml" xref="S2.SS2.p1.17.m17.3.4.3.1">subscript</csymbol><ci id="S2.SS2.p1.17.m17.3.4.3.1.2.cmml" xref="S2.SS2.p1.17.m17.3.4.3.1.2">𝐋</ci><cn id="S2.SS2.p1.17.m17.2.2.1.1.cmml" type="integer" xref="S2.SS2.p1.17.m17.2.2.1.1">2</cn></apply><apply id="S2.SS2.p1.17.m17.3.4.3.2.cmml" xref="S2.SS2.p1.17.m17.3.4.3.2"><csymbol cd="ambiguous" id="S2.SS2.p1.17.m17.3.4.3.2.1.cmml" xref="S2.SS2.p1.17.m17.3.4.3.2">subscript</csymbol><ci id="S2.SS2.p1.17.m17.3.4.3.2.2.cmml" xref="S2.SS2.p1.17.m17.3.4.3.2.2">𝐜</ci><apply id="S2.SS2.p1.17.m17.3.3.1.cmml" xref="S2.SS2.p1.17.m17.3.3.1"><times id="S2.SS2.p1.17.m17.3.3.1.2.cmml" xref="S2.SS2.p1.17.m17.3.3.1.2"></times><ci id="S2.SS2.p1.17.m17.3.3.1.3.cmml" xref="S2.SS2.p1.17.m17.3.3.1.3">𝑖</ci><cn id="S2.SS2.p1.17.m17.3.3.1.1.cmml" type="integer" xref="S2.SS2.p1.17.m17.3.3.1.1">2</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p1.17.m17.3c">\operatorname{\mathbf{a}}_{i(2)}=\operatorname{\mathbf{L}}_{(2)}\operatorname{% \mathbf{c}}_{i(2)}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p1.17.m17.3d">bold_a start_POSTSUBSCRIPT italic_i ( 2 ) end_POSTSUBSCRIPT = bold_L start_POSTSUBSCRIPT ( 2 ) end_POSTSUBSCRIPT bold_c start_POSTSUBSCRIPT italic_i ( 2 ) end_POSTSUBSCRIPT</annotation></semantics></math>, where <math alttext="\operatorname{\mathbf{L}}_{(1)}=\operatorname{\mathbf{L}}_{2}\otimes% \operatorname{\mathbf{L}}_{1}" class="ltx_Math" display="inline" id="S2.SS2.p1.18.m18.1"><semantics id="S2.SS2.p1.18.m18.1a"><mrow id="S2.SS2.p1.18.m18.1.2" xref="S2.SS2.p1.18.m18.1.2.cmml"><msub id="S2.SS2.p1.18.m18.1.2.2" xref="S2.SS2.p1.18.m18.1.2.2.cmml"><mi id="S2.SS2.p1.18.m18.1.2.2.2" xref="S2.SS2.p1.18.m18.1.2.2.2.cmml">𝐋</mi><mrow id="S2.SS2.p1.18.m18.1.1.1.3" xref="S2.SS2.p1.18.m18.1.2.2.cmml"><mo id="S2.SS2.p1.18.m18.1.1.1.3.1" stretchy="false" xref="S2.SS2.p1.18.m18.1.2.2.cmml">(</mo><mn id="S2.SS2.p1.18.m18.1.1.1.1" xref="S2.SS2.p1.18.m18.1.1.1.1.cmml">1</mn><mo id="S2.SS2.p1.18.m18.1.1.1.3.2" stretchy="false" xref="S2.SS2.p1.18.m18.1.2.2.cmml">)</mo></mrow></msub><mo id="S2.SS2.p1.18.m18.1.2.1" xref="S2.SS2.p1.18.m18.1.2.1.cmml">=</mo><mrow id="S2.SS2.p1.18.m18.1.2.3" xref="S2.SS2.p1.18.m18.1.2.3.cmml"><msub id="S2.SS2.p1.18.m18.1.2.3.2" xref="S2.SS2.p1.18.m18.1.2.3.2.cmml"><mi id="S2.SS2.p1.18.m18.1.2.3.2.2" xref="S2.SS2.p1.18.m18.1.2.3.2.2.cmml">𝐋</mi><mn id="S2.SS2.p1.18.m18.1.2.3.2.3" xref="S2.SS2.p1.18.m18.1.2.3.2.3.cmml">2</mn></msub><mo id="S2.SS2.p1.18.m18.1.2.3.1" lspace="0em" rspace="0.222em" xref="S2.SS2.p1.18.m18.1.2.3.1.cmml">⊗</mo><msub id="S2.SS2.p1.18.m18.1.2.3.3" xref="S2.SS2.p1.18.m18.1.2.3.3.cmml"><mi id="S2.SS2.p1.18.m18.1.2.3.3.2" xref="S2.SS2.p1.18.m18.1.2.3.3.2.cmml">𝐋</mi><mn id="S2.SS2.p1.18.m18.1.2.3.3.3" xref="S2.SS2.p1.18.m18.1.2.3.3.3.cmml">1</mn></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p1.18.m18.1b"><apply id="S2.SS2.p1.18.m18.1.2.cmml" xref="S2.SS2.p1.18.m18.1.2"><eq id="S2.SS2.p1.18.m18.1.2.1.cmml" xref="S2.SS2.p1.18.m18.1.2.1"></eq><apply id="S2.SS2.p1.18.m18.1.2.2.cmml" xref="S2.SS2.p1.18.m18.1.2.2"><csymbol cd="ambiguous" id="S2.SS2.p1.18.m18.1.2.2.1.cmml" xref="S2.SS2.p1.18.m18.1.2.2">subscript</csymbol><ci id="S2.SS2.p1.18.m18.1.2.2.2.cmml" xref="S2.SS2.p1.18.m18.1.2.2.2">𝐋</ci><cn id="S2.SS2.p1.18.m18.1.1.1.1.cmml" type="integer" xref="S2.SS2.p1.18.m18.1.1.1.1">1</cn></apply><apply id="S2.SS2.p1.18.m18.1.2.3.cmml" xref="S2.SS2.p1.18.m18.1.2.3"><csymbol cd="latexml" id="S2.SS2.p1.18.m18.1.2.3.1.cmml" xref="S2.SS2.p1.18.m18.1.2.3.1">tensor-product</csymbol><apply id="S2.SS2.p1.18.m18.1.2.3.2.cmml" xref="S2.SS2.p1.18.m18.1.2.3.2"><csymbol cd="ambiguous" id="S2.SS2.p1.18.m18.1.2.3.2.1.cmml" xref="S2.SS2.p1.18.m18.1.2.3.2">subscript</csymbol><ci id="S2.SS2.p1.18.m18.1.2.3.2.2.cmml" xref="S2.SS2.p1.18.m18.1.2.3.2.2">𝐋</ci><cn id="S2.SS2.p1.18.m18.1.2.3.2.3.cmml" type="integer" xref="S2.SS2.p1.18.m18.1.2.3.2.3">2</cn></apply><apply id="S2.SS2.p1.18.m18.1.2.3.3.cmml" xref="S2.SS2.p1.18.m18.1.2.3.3"><csymbol cd="ambiguous" id="S2.SS2.p1.18.m18.1.2.3.3.1.cmml" xref="S2.SS2.p1.18.m18.1.2.3.3">subscript</csymbol><ci id="S2.SS2.p1.18.m18.1.2.3.3.2.cmml" xref="S2.SS2.p1.18.m18.1.2.3.3.2">𝐋</ci><cn id="S2.SS2.p1.18.m18.1.2.3.3.3.cmml" type="integer" xref="S2.SS2.p1.18.m18.1.2.3.3.3">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p1.18.m18.1c">\operatorname{\mathbf{L}}_{(1)}=\operatorname{\mathbf{L}}_{2}\otimes% \operatorname{\mathbf{L}}_{1}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p1.18.m18.1d">bold_L start_POSTSUBSCRIPT ( 1 ) end_POSTSUBSCRIPT = bold_L start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ⊗ bold_L start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="\operatorname{\mathbf{L}}_{(2)}=\operatorname{\mathbf{L}}_{1}\otimes% \operatorname{\mathbf{L}}_{2}" class="ltx_Math" display="inline" id="S2.SS2.p1.19.m19.1"><semantics id="S2.SS2.p1.19.m19.1a"><mrow id="S2.SS2.p1.19.m19.1.2" xref="S2.SS2.p1.19.m19.1.2.cmml"><msub id="S2.SS2.p1.19.m19.1.2.2" xref="S2.SS2.p1.19.m19.1.2.2.cmml"><mi id="S2.SS2.p1.19.m19.1.2.2.2" xref="S2.SS2.p1.19.m19.1.2.2.2.cmml">𝐋</mi><mrow id="S2.SS2.p1.19.m19.1.1.1.3" xref="S2.SS2.p1.19.m19.1.2.2.cmml"><mo id="S2.SS2.p1.19.m19.1.1.1.3.1" stretchy="false" xref="S2.SS2.p1.19.m19.1.2.2.cmml">(</mo><mn id="S2.SS2.p1.19.m19.1.1.1.1" xref="S2.SS2.p1.19.m19.1.1.1.1.cmml">2</mn><mo id="S2.SS2.p1.19.m19.1.1.1.3.2" stretchy="false" xref="S2.SS2.p1.19.m19.1.2.2.cmml">)</mo></mrow></msub><mo id="S2.SS2.p1.19.m19.1.2.1" xref="S2.SS2.p1.19.m19.1.2.1.cmml">=</mo><mrow id="S2.SS2.p1.19.m19.1.2.3" xref="S2.SS2.p1.19.m19.1.2.3.cmml"><msub id="S2.SS2.p1.19.m19.1.2.3.2" xref="S2.SS2.p1.19.m19.1.2.3.2.cmml"><mi id="S2.SS2.p1.19.m19.1.2.3.2.2" xref="S2.SS2.p1.19.m19.1.2.3.2.2.cmml">𝐋</mi><mn id="S2.SS2.p1.19.m19.1.2.3.2.3" xref="S2.SS2.p1.19.m19.1.2.3.2.3.cmml">1</mn></msub><mo id="S2.SS2.p1.19.m19.1.2.3.1" lspace="0em" rspace="0.222em" xref="S2.SS2.p1.19.m19.1.2.3.1.cmml">⊗</mo><msub id="S2.SS2.p1.19.m19.1.2.3.3" xref="S2.SS2.p1.19.m19.1.2.3.3.cmml"><mi id="S2.SS2.p1.19.m19.1.2.3.3.2" xref="S2.SS2.p1.19.m19.1.2.3.3.2.cmml">𝐋</mi><mn id="S2.SS2.p1.19.m19.1.2.3.3.3" xref="S2.SS2.p1.19.m19.1.2.3.3.3.cmml">2</mn></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p1.19.m19.1b"><apply id="S2.SS2.p1.19.m19.1.2.cmml" xref="S2.SS2.p1.19.m19.1.2"><eq id="S2.SS2.p1.19.m19.1.2.1.cmml" xref="S2.SS2.p1.19.m19.1.2.1"></eq><apply id="S2.SS2.p1.19.m19.1.2.2.cmml" xref="S2.SS2.p1.19.m19.1.2.2"><csymbol cd="ambiguous" id="S2.SS2.p1.19.m19.1.2.2.1.cmml" xref="S2.SS2.p1.19.m19.1.2.2">subscript</csymbol><ci id="S2.SS2.p1.19.m19.1.2.2.2.cmml" xref="S2.SS2.p1.19.m19.1.2.2.2">𝐋</ci><cn id="S2.SS2.p1.19.m19.1.1.1.1.cmml" type="integer" xref="S2.SS2.p1.19.m19.1.1.1.1">2</cn></apply><apply id="S2.SS2.p1.19.m19.1.2.3.cmml" xref="S2.SS2.p1.19.m19.1.2.3"><csymbol cd="latexml" id="S2.SS2.p1.19.m19.1.2.3.1.cmml" xref="S2.SS2.p1.19.m19.1.2.3.1">tensor-product</csymbol><apply id="S2.SS2.p1.19.m19.1.2.3.2.cmml" xref="S2.SS2.p1.19.m19.1.2.3.2"><csymbol cd="ambiguous" id="S2.SS2.p1.19.m19.1.2.3.2.1.cmml" xref="S2.SS2.p1.19.m19.1.2.3.2">subscript</csymbol><ci id="S2.SS2.p1.19.m19.1.2.3.2.2.cmml" xref="S2.SS2.p1.19.m19.1.2.3.2.2">𝐋</ci><cn id="S2.SS2.p1.19.m19.1.2.3.2.3.cmml" type="integer" xref="S2.SS2.p1.19.m19.1.2.3.2.3">1</cn></apply><apply id="S2.SS2.p1.19.m19.1.2.3.3.cmml" xref="S2.SS2.p1.19.m19.1.2.3.3"><csymbol cd="ambiguous" id="S2.SS2.p1.19.m19.1.2.3.3.1.cmml" xref="S2.SS2.p1.19.m19.1.2.3.3">subscript</csymbol><ci id="S2.SS2.p1.19.m19.1.2.3.3.2.cmml" xref="S2.SS2.p1.19.m19.1.2.3.3.2">𝐋</ci><cn id="S2.SS2.p1.19.m19.1.2.3.3.3.cmml" type="integer" xref="S2.SS2.p1.19.m19.1.2.3.3.3">2</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p1.19.m19.1c">\operatorname{\mathbf{L}}_{(2)}=\operatorname{\mathbf{L}}_{1}\otimes% \operatorname{\mathbf{L}}_{2}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p1.19.m19.1d">bold_L start_POSTSUBSCRIPT ( 2 ) end_POSTSUBSCRIPT = bold_L start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ⊗ bold_L start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math>; see <cite class="ltx_cite ltx_citemacro_citet">Seber (<a class="ltx_ref" href="https://arxiv.org/html/2503.13782v1#bib.bib25" title="">2007</a>)</cite>. The vectors <math alttext="\operatorname{\mathbf{a}}_{i(1)}" class="ltx_Math" display="inline" id="S2.SS2.p1.20.m20.1"><semantics id="S2.SS2.p1.20.m20.1a"><msub id="S2.SS2.p1.20.m20.1.2" xref="S2.SS2.p1.20.m20.1.2.cmml"><mi id="S2.SS2.p1.20.m20.1.2.2" xref="S2.SS2.p1.20.m20.1.2.2.cmml">𝐚</mi><mrow id="S2.SS2.p1.20.m20.1.1.1" xref="S2.SS2.p1.20.m20.1.1.1.cmml"><mi id="S2.SS2.p1.20.m20.1.1.1.3" xref="S2.SS2.p1.20.m20.1.1.1.3.cmml">i</mi><mo id="S2.SS2.p1.20.m20.1.1.1.2" xref="S2.SS2.p1.20.m20.1.1.1.2.cmml">⁢</mo><mrow id="S2.SS2.p1.20.m20.1.1.1.4.2" xref="S2.SS2.p1.20.m20.1.1.1.cmml"><mo id="S2.SS2.p1.20.m20.1.1.1.4.2.1" stretchy="false" xref="S2.SS2.p1.20.m20.1.1.1.cmml">(</mo><mn id="S2.SS2.p1.20.m20.1.1.1.1" xref="S2.SS2.p1.20.m20.1.1.1.1.cmml">1</mn><mo id="S2.SS2.p1.20.m20.1.1.1.4.2.2" stretchy="false" xref="S2.SS2.p1.20.m20.1.1.1.cmml">)</mo></mrow></mrow></msub><annotation-xml encoding="MathML-Content" id="S2.SS2.p1.20.m20.1b"><apply id="S2.SS2.p1.20.m20.1.2.cmml" xref="S2.SS2.p1.20.m20.1.2"><csymbol cd="ambiguous" id="S2.SS2.p1.20.m20.1.2.1.cmml" xref="S2.SS2.p1.20.m20.1.2">subscript</csymbol><ci id="S2.SS2.p1.20.m20.1.2.2.cmml" xref="S2.SS2.p1.20.m20.1.2.2">𝐚</ci><apply id="S2.SS2.p1.20.m20.1.1.1.cmml" xref="S2.SS2.p1.20.m20.1.1.1"><times id="S2.SS2.p1.20.m20.1.1.1.2.cmml" xref="S2.SS2.p1.20.m20.1.1.1.2"></times><ci id="S2.SS2.p1.20.m20.1.1.1.3.cmml" xref="S2.SS2.p1.20.m20.1.1.1.3">𝑖</ci><cn id="S2.SS2.p1.20.m20.1.1.1.1.cmml" type="integer" xref="S2.SS2.p1.20.m20.1.1.1.1">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p1.20.m20.1c">\operatorname{\mathbf{a}}_{i(1)}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p1.20.m20.1d">bold_a start_POSTSUBSCRIPT italic_i ( 1 ) end_POSTSUBSCRIPT</annotation></semantics></math>’s and <math alttext="\operatorname{\mathbf{a}}_{i(2)}" class="ltx_Math" display="inline" id="S2.SS2.p1.21.m21.1"><semantics id="S2.SS2.p1.21.m21.1a"><msub id="S2.SS2.p1.21.m21.1.2" xref="S2.SS2.p1.21.m21.1.2.cmml"><mi id="S2.SS2.p1.21.m21.1.2.2" xref="S2.SS2.p1.21.m21.1.2.2.cmml">𝐚</mi><mrow id="S2.SS2.p1.21.m21.1.1.1" xref="S2.SS2.p1.21.m21.1.1.1.cmml"><mi id="S2.SS2.p1.21.m21.1.1.1.3" xref="S2.SS2.p1.21.m21.1.1.1.3.cmml">i</mi><mo id="S2.SS2.p1.21.m21.1.1.1.2" xref="S2.SS2.p1.21.m21.1.1.1.2.cmml">⁢</mo><mrow id="S2.SS2.p1.21.m21.1.1.1.4.2" xref="S2.SS2.p1.21.m21.1.1.1.cmml"><mo id="S2.SS2.p1.21.m21.1.1.1.4.2.1" stretchy="false" xref="S2.SS2.p1.21.m21.1.1.1.cmml">(</mo><mn id="S2.SS2.p1.21.m21.1.1.1.1" xref="S2.SS2.p1.21.m21.1.1.1.1.cmml">2</mn><mo id="S2.SS2.p1.21.m21.1.1.1.4.2.2" stretchy="false" xref="S2.SS2.p1.21.m21.1.1.1.cmml">)</mo></mrow></mrow></msub><annotation-xml encoding="MathML-Content" id="S2.SS2.p1.21.m21.1b"><apply id="S2.SS2.p1.21.m21.1.2.cmml" xref="S2.SS2.p1.21.m21.1.2"><csymbol cd="ambiguous" id="S2.SS2.p1.21.m21.1.2.1.cmml" xref="S2.SS2.p1.21.m21.1.2">subscript</csymbol><ci id="S2.SS2.p1.21.m21.1.2.2.cmml" xref="S2.SS2.p1.21.m21.1.2.2">𝐚</ci><apply id="S2.SS2.p1.21.m21.1.1.1.cmml" xref="S2.SS2.p1.21.m21.1.1.1"><times id="S2.SS2.p1.21.m21.1.1.1.2.cmml" xref="S2.SS2.p1.21.m21.1.1.1.2"></times><ci id="S2.SS2.p1.21.m21.1.1.1.3.cmml" xref="S2.SS2.p1.21.m21.1.1.1.3">𝑖</ci><cn id="S2.SS2.p1.21.m21.1.1.1.1.cmml" type="integer" xref="S2.SS2.p1.21.m21.1.1.1.1">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p1.21.m21.1c">\operatorname{\mathbf{a}}_{i(2)}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p1.21.m21.1d">bold_a start_POSTSUBSCRIPT italic_i ( 2 ) end_POSTSUBSCRIPT</annotation></semantics></math>’s are used in estimating <math alttext="\operatorname{\mathbf{L}}_{1}" class="ltx_Math" display="inline" id="S2.SS2.p1.22.m22.1"><semantics id="S2.SS2.p1.22.m22.1a"><msub id="S2.SS2.p1.22.m22.1.1" xref="S2.SS2.p1.22.m22.1.1.cmml"><mi id="S2.SS2.p1.22.m22.1.1.2" xref="S2.SS2.p1.22.m22.1.1.2.cmml">𝐋</mi><mn id="S2.SS2.p1.22.m22.1.1.3" xref="S2.SS2.p1.22.m22.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S2.SS2.p1.22.m22.1b"><apply id="S2.SS2.p1.22.m22.1.1.cmml" xref="S2.SS2.p1.22.m22.1.1"><csymbol cd="ambiguous" id="S2.SS2.p1.22.m22.1.1.1.cmml" xref="S2.SS2.p1.22.m22.1.1">subscript</csymbol><ci id="S2.SS2.p1.22.m22.1.1.2.cmml" xref="S2.SS2.p1.22.m22.1.1.2">𝐋</ci><cn id="S2.SS2.p1.22.m22.1.1.3.cmml" type="integer" xref="S2.SS2.p1.22.m22.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p1.22.m22.1c">\operatorname{\mathbf{L}}_{1}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p1.22.m22.1d">bold_L start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> given <math alttext="\operatorname{\mathbf{L}}_{2}" class="ltx_Math" display="inline" id="S2.SS2.p1.23.m23.1"><semantics id="S2.SS2.p1.23.m23.1a"><msub id="S2.SS2.p1.23.m23.1.1" xref="S2.SS2.p1.23.m23.1.1.cmml"><mi id="S2.SS2.p1.23.m23.1.1.2" xref="S2.SS2.p1.23.m23.1.1.2.cmml">𝐋</mi><mn id="S2.SS2.p1.23.m23.1.1.3" xref="S2.SS2.p1.23.m23.1.1.3.cmml">2</mn></msub><annotation-xml encoding="MathML-Content" id="S2.SS2.p1.23.m23.1b"><apply id="S2.SS2.p1.23.m23.1.1.cmml" xref="S2.SS2.p1.23.m23.1.1"><csymbol cd="ambiguous" id="S2.SS2.p1.23.m23.1.1.1.cmml" xref="S2.SS2.p1.23.m23.1.1">subscript</csymbol><ci id="S2.SS2.p1.23.m23.1.1.2.cmml" xref="S2.SS2.p1.23.m23.1.1.2">𝐋</ci><cn id="S2.SS2.p1.23.m23.1.1.3.cmml" type="integer" xref="S2.SS2.p1.23.m23.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p1.23.m23.1c">\operatorname{\mathbf{L}}_{2}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p1.23.m23.1d">bold_L start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="\operatorname{\mathbf{L}}_{2}" class="ltx_Math" display="inline" id="S2.SS2.p1.24.m24.1"><semantics id="S2.SS2.p1.24.m24.1a"><msub id="S2.SS2.p1.24.m24.1.1" xref="S2.SS2.p1.24.m24.1.1.cmml"><mi id="S2.SS2.p1.24.m24.1.1.2" xref="S2.SS2.p1.24.m24.1.1.2.cmml">𝐋</mi><mn id="S2.SS2.p1.24.m24.1.1.3" xref="S2.SS2.p1.24.m24.1.1.3.cmml">2</mn></msub><annotation-xml encoding="MathML-Content" id="S2.SS2.p1.24.m24.1b"><apply id="S2.SS2.p1.24.m24.1.1.cmml" xref="S2.SS2.p1.24.m24.1.1"><csymbol cd="ambiguous" id="S2.SS2.p1.24.m24.1.1.1.cmml" xref="S2.SS2.p1.24.m24.1.1">subscript</csymbol><ci id="S2.SS2.p1.24.m24.1.1.2.cmml" xref="S2.SS2.p1.24.m24.1.1.2">𝐋</ci><cn id="S2.SS2.p1.24.m24.1.1.3.cmml" type="integer" xref="S2.SS2.p1.24.m24.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p1.24.m24.1c">\operatorname{\mathbf{L}}_{2}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p1.24.m24.1d">bold_L start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math> given <math alttext="\operatorname{\mathbf{L}}_{1}" class="ltx_Math" display="inline" id="S2.SS2.p1.25.m25.1"><semantics id="S2.SS2.p1.25.m25.1a"><msub id="S2.SS2.p1.25.m25.1.1" xref="S2.SS2.p1.25.m25.1.1.cmml"><mi id="S2.SS2.p1.25.m25.1.1.2" xref="S2.SS2.p1.25.m25.1.1.2.cmml">𝐋</mi><mn id="S2.SS2.p1.25.m25.1.1.3" xref="S2.SS2.p1.25.m25.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S2.SS2.p1.25.m25.1b"><apply id="S2.SS2.p1.25.m25.1.1.cmml" xref="S2.SS2.p1.25.m25.1.1"><csymbol cd="ambiguous" id="S2.SS2.p1.25.m25.1.1.1.cmml" xref="S2.SS2.p1.25.m25.1.1">subscript</csymbol><ci id="S2.SS2.p1.25.m25.1.1.2.cmml" xref="S2.SS2.p1.25.m25.1.1.2">𝐋</ci><cn id="S2.SS2.p1.25.m25.1.1.3.cmml" type="integer" xref="S2.SS2.p1.25.m25.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p1.25.m25.1c">\operatorname{\mathbf{L}}_{1}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p1.25.m25.1d">bold_L start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math>, respectively.</p> </div> <div class="ltx_para" id="S2.SS2.p2"> <p class="ltx_p" id="S2.SS2.p2.2">MMTR in (<a class="ltx_ref" href="https://arxiv.org/html/2503.13782v1#S2.E3" title="In 2.1 Model ‣ 2 Mixed Model Trace Regression ‣ Regularized Parameter Estimation in Mixed Model Trace Regression"><span class="ltx_text ltx_ref_tag">3</span></a>) is equivalent to a mixed model linear regression with the mean parameter <math alttext="\operatorname{\mathbf{b}}" class="ltx_Math" display="inline" id="S2.SS2.p2.1.m1.1"><semantics id="S2.SS2.p2.1.m1.1a"><mi id="S2.SS2.p2.1.m1.1.1" xref="S2.SS2.p2.1.m1.1.1.cmml">𝐛</mi><annotation-xml encoding="MathML-Content" id="S2.SS2.p2.1.m1.1b"><ci id="S2.SS2.p2.1.m1.1.1.cmml" xref="S2.SS2.p2.1.m1.1.1">𝐛</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p2.1.m1.1c">\operatorname{\mathbf{b}}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p2.1.m1.1d">bold_b</annotation></semantics></math> and random effects covariance <math alttext="\tau^{2}\operatorname{\mathbf{L}}_{2}\operatorname{\mathbf{L}}_{2}^{\top}% \otimes\operatorname{\mathbf{L}}_{1}\operatorname{\mathbf{L}}_{1}^{\top}" class="ltx_Math" display="inline" id="S2.SS2.p2.2.m2.1"><semantics id="S2.SS2.p2.2.m2.1a"><mrow id="S2.SS2.p2.2.m2.1.1" xref="S2.SS2.p2.2.m2.1.1.cmml"><msup id="S2.SS2.p2.2.m2.1.1.2" xref="S2.SS2.p2.2.m2.1.1.2.cmml"><mi id="S2.SS2.p2.2.m2.1.1.2.2" xref="S2.SS2.p2.2.m2.1.1.2.2.cmml">τ</mi><mn id="S2.SS2.p2.2.m2.1.1.2.3" xref="S2.SS2.p2.2.m2.1.1.2.3.cmml">2</mn></msup><mo id="S2.SS2.p2.2.m2.1.1.1" lspace="0.167em" xref="S2.SS2.p2.2.m2.1.1.1.cmml">⁢</mo><mrow id="S2.SS2.p2.2.m2.1.1.3" xref="S2.SS2.p2.2.m2.1.1.3.cmml"><msub id="S2.SS2.p2.2.m2.1.1.3.1" xref="S2.SS2.p2.2.m2.1.1.3.1.cmml"><mi id="S2.SS2.p2.2.m2.1.1.3.1.2" xref="S2.SS2.p2.2.m2.1.1.3.1.2.cmml">𝐋</mi><mn id="S2.SS2.p2.2.m2.1.1.3.1.3" xref="S2.SS2.p2.2.m2.1.1.3.1.3.cmml">2</mn></msub><mo id="S2.SS2.p2.2.m2.1.1.3a" lspace="0.167em" xref="S2.SS2.p2.2.m2.1.1.3.cmml">⁡</mo><mrow id="S2.SS2.p2.2.m2.1.1.3.2" xref="S2.SS2.p2.2.m2.1.1.3.2.cmml"><msubsup id="S2.SS2.p2.2.m2.1.1.3.2.2" xref="S2.SS2.p2.2.m2.1.1.3.2.2.cmml"><mi id="S2.SS2.p2.2.m2.1.1.3.2.2.2.2" xref="S2.SS2.p2.2.m2.1.1.3.2.2.2.2.cmml">𝐋</mi><mn id="S2.SS2.p2.2.m2.1.1.3.2.2.2.3" xref="S2.SS2.p2.2.m2.1.1.3.2.2.2.3.cmml">2</mn><mo id="S2.SS2.p2.2.m2.1.1.3.2.2.3" xref="S2.SS2.p2.2.m2.1.1.3.2.2.3.cmml">⊤</mo></msubsup><mo id="S2.SS2.p2.2.m2.1.1.3.2.1" lspace="0em" rspace="0.222em" xref="S2.SS2.p2.2.m2.1.1.3.2.1.cmml">⊗</mo><mrow id="S2.SS2.p2.2.m2.1.1.3.2.3" xref="S2.SS2.p2.2.m2.1.1.3.2.3.cmml"><msub id="S2.SS2.p2.2.m2.1.1.3.2.3.1" xref="S2.SS2.p2.2.m2.1.1.3.2.3.1.cmml"><mi id="S2.SS2.p2.2.m2.1.1.3.2.3.1.2" xref="S2.SS2.p2.2.m2.1.1.3.2.3.1.2.cmml">𝐋</mi><mn id="S2.SS2.p2.2.m2.1.1.3.2.3.1.3" xref="S2.SS2.p2.2.m2.1.1.3.2.3.1.3.cmml">1</mn></msub><mo id="S2.SS2.p2.2.m2.1.1.3.2.3a" lspace="0.167em" xref="S2.SS2.p2.2.m2.1.1.3.2.3.cmml">⁡</mo><msubsup id="S2.SS2.p2.2.m2.1.1.3.2.3.2" 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id="S2.SS2.p2.2.m2.1.1.3.2.3.2.2.1.cmml" xref="S2.SS2.p2.2.m2.1.1.3.2.3.2">subscript</csymbol><ci id="S2.SS2.p2.2.m2.1.1.3.2.3.2.2.2.cmml" xref="S2.SS2.p2.2.m2.1.1.3.2.3.2.2.2">𝐋</ci><cn id="S2.SS2.p2.2.m2.1.1.3.2.3.2.2.3.cmml" type="integer" xref="S2.SS2.p2.2.m2.1.1.3.2.3.2.2.3">1</cn></apply><csymbol cd="latexml" id="S2.SS2.p2.2.m2.1.1.3.2.3.2.3.cmml" xref="S2.SS2.p2.2.m2.1.1.3.2.3.2.3">top</csymbol></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p2.2.m2.1c">\tau^{2}\operatorname{\mathbf{L}}_{2}\operatorname{\mathbf{L}}_{2}^{\top}% \otimes\operatorname{\mathbf{L}}_{1}\operatorname{\mathbf{L}}_{1}^{\top}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p2.2.m2.1d">italic_τ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT bold_L start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT bold_L start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ⊤ end_POSTSUPERSCRIPT ⊗ bold_L start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT bold_L start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ⊤ end_POSTSUPERSCRIPT</annotation></semantics></math>. Using (<a class="ltx_ref" href="https://arxiv.org/html/2503.13782v1#S2.E3" title="In 2.1 Model ‣ 2 Mixed Model Trace Regression ‣ Regularized Parameter Estimation in Mixed Model Trace Regression"><span class="ltx_text ltx_ref_tag">3</span></a>) and previous identities,</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="S3.EGx4"> <tbody id="S2.E4"><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 y_{ij}" class="ltx_Math" display="inline" id="S2.E4.m1.1"><semantics id="S2.E4.m1.1a"><msub id="S2.E4.m1.1.1" xref="S2.E4.m1.1.1.cmml"><mi id="S2.E4.m1.1.1.2" xref="S2.E4.m1.1.1.2.cmml">y</mi><mrow id="S2.E4.m1.1.1.3" xref="S2.E4.m1.1.1.3.cmml"><mi id="S2.E4.m1.1.1.3.2" xref="S2.E4.m1.1.1.3.2.cmml">i</mi><mo id="S2.E4.m1.1.1.3.1" xref="S2.E4.m1.1.1.3.1.cmml">⁢</mo><mi id="S2.E4.m1.1.1.3.3" xref="S2.E4.m1.1.1.3.3.cmml">j</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="S2.E4.m1.1b"><apply id="S2.E4.m1.1.1.cmml" xref="S2.E4.m1.1.1"><csymbol cd="ambiguous" id="S2.E4.m1.1.1.1.cmml" xref="S2.E4.m1.1.1">subscript</csymbol><ci id="S2.E4.m1.1.1.2.cmml" xref="S2.E4.m1.1.1.2">𝑦</ci><apply id="S2.E4.m1.1.1.3.cmml" xref="S2.E4.m1.1.1.3"><times id="S2.E4.m1.1.1.3.1.cmml" xref="S2.E4.m1.1.1.3.1"></times><ci id="S2.E4.m1.1.1.3.2.cmml" xref="S2.E4.m1.1.1.3.2">𝑖</ci><ci id="S2.E4.m1.1.1.3.3.cmml" xref="S2.E4.m1.1.1.3.3">𝑗</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.E4.m1.1c">\displaystyle y_{ij}</annotation><annotation encoding="application/x-llamapun" id="S2.E4.m1.1d">italic_y start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle=\operatorname{\mathbf{x}}^{\top}_{ij}\operatorname{\mathbf{b}}+% 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\otimes\operatorname{\mathbf{L}}_{1})\operatorname{\mathbf{c}}_{i(1)}+% \operatorname{\mathbf{e}}_{i},</annotation><annotation encoding="application/x-llamapun" id="S2.E4.m2.5d">= bold_x start_POSTSUPERSCRIPT ⊤ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT bold_b + bold_z start_POSTSUBSCRIPT italic_i italic_j ( 1 ) end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ⊤ end_POSTSUPERSCRIPT ( bold_L start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ⊗ bold_L start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) bold_c start_POSTSUBSCRIPT italic_i ( 1 ) end_POSTSUBSCRIPT + italic_e start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT , bold_y start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = bold_X start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT bold_b + bold_Z start_POSTSUBSCRIPT italic_i ( 1 ) end_POSTSUBSCRIPT ( bold_L start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ⊗ bold_L start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) bold_c start_POSTSUBSCRIPT italic_i ( 1 ) end_POSTSUBSCRIPT + bold_e start_POSTSUBSCRIPT italic_i 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">(4)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S2.SS2.p2.28">for <math alttext="i=1,\ldots,n" class="ltx_Math" display="inline" id="S2.SS2.p2.3.m1.3"><semantics id="S2.SS2.p2.3.m1.3a"><mrow id="S2.SS2.p2.3.m1.3.4" xref="S2.SS2.p2.3.m1.3.4.cmml"><mi id="S2.SS2.p2.3.m1.3.4.2" xref="S2.SS2.p2.3.m1.3.4.2.cmml">i</mi><mo id="S2.SS2.p2.3.m1.3.4.1" xref="S2.SS2.p2.3.m1.3.4.1.cmml">=</mo><mrow id="S2.SS2.p2.3.m1.3.4.3.2" xref="S2.SS2.p2.3.m1.3.4.3.1.cmml"><mn id="S2.SS2.p2.3.m1.1.1" xref="S2.SS2.p2.3.m1.1.1.cmml">1</mn><mo id="S2.SS2.p2.3.m1.3.4.3.2.1" xref="S2.SS2.p2.3.m1.3.4.3.1.cmml">,</mo><mi id="S2.SS2.p2.3.m1.2.2" mathvariant="normal" xref="S2.SS2.p2.3.m1.2.2.cmml">…</mi><mo id="S2.SS2.p2.3.m1.3.4.3.2.2" xref="S2.SS2.p2.3.m1.3.4.3.1.cmml">,</mo><mi id="S2.SS2.p2.3.m1.3.3" xref="S2.SS2.p2.3.m1.3.3.cmml">n</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p2.3.m1.3b"><apply id="S2.SS2.p2.3.m1.3.4.cmml" xref="S2.SS2.p2.3.m1.3.4"><eq id="S2.SS2.p2.3.m1.3.4.1.cmml" xref="S2.SS2.p2.3.m1.3.4.1"></eq><ci id="S2.SS2.p2.3.m1.3.4.2.cmml" xref="S2.SS2.p2.3.m1.3.4.2">𝑖</ci><list id="S2.SS2.p2.3.m1.3.4.3.1.cmml" xref="S2.SS2.p2.3.m1.3.4.3.2"><cn id="S2.SS2.p2.3.m1.1.1.cmml" type="integer" xref="S2.SS2.p2.3.m1.1.1">1</cn><ci id="S2.SS2.p2.3.m1.2.2.cmml" xref="S2.SS2.p2.3.m1.2.2">…</ci><ci id="S2.SS2.p2.3.m1.3.3.cmml" xref="S2.SS2.p2.3.m1.3.3">𝑛</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p2.3.m1.3c">i=1,\ldots,n</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p2.3.m1.3d">italic_i = 1 , … , italic_n</annotation></semantics></math> and <math alttext="j=1,\ldots,m_{i}" class="ltx_Math" display="inline" id="S2.SS2.p2.4.m2.3"><semantics id="S2.SS2.p2.4.m2.3a"><mrow id="S2.SS2.p2.4.m2.3.3" xref="S2.SS2.p2.4.m2.3.3.cmml"><mi id="S2.SS2.p2.4.m2.3.3.3" xref="S2.SS2.p2.4.m2.3.3.3.cmml">j</mi><mo id="S2.SS2.p2.4.m2.3.3.2" xref="S2.SS2.p2.4.m2.3.3.2.cmml">=</mo><mrow id="S2.SS2.p2.4.m2.3.3.1.1" xref="S2.SS2.p2.4.m2.3.3.1.2.cmml"><mn id="S2.SS2.p2.4.m2.1.1" xref="S2.SS2.p2.4.m2.1.1.cmml">1</mn><mo id="S2.SS2.p2.4.m2.3.3.1.1.2" xref="S2.SS2.p2.4.m2.3.3.1.2.cmml">,</mo><mi id="S2.SS2.p2.4.m2.2.2" mathvariant="normal" xref="S2.SS2.p2.4.m2.2.2.cmml">…</mi><mo id="S2.SS2.p2.4.m2.3.3.1.1.3" xref="S2.SS2.p2.4.m2.3.3.1.2.cmml">,</mo><msub id="S2.SS2.p2.4.m2.3.3.1.1.1" xref="S2.SS2.p2.4.m2.3.3.1.1.1.cmml"><mi id="S2.SS2.p2.4.m2.3.3.1.1.1.2" xref="S2.SS2.p2.4.m2.3.3.1.1.1.2.cmml">m</mi><mi id="S2.SS2.p2.4.m2.3.3.1.1.1.3" xref="S2.SS2.p2.4.m2.3.3.1.1.1.3.cmml">i</mi></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p2.4.m2.3b"><apply id="S2.SS2.p2.4.m2.3.3.cmml" xref="S2.SS2.p2.4.m2.3.3"><eq id="S2.SS2.p2.4.m2.3.3.2.cmml" xref="S2.SS2.p2.4.m2.3.3.2"></eq><ci id="S2.SS2.p2.4.m2.3.3.3.cmml" xref="S2.SS2.p2.4.m2.3.3.3">𝑗</ci><list id="S2.SS2.p2.4.m2.3.3.1.2.cmml" xref="S2.SS2.p2.4.m2.3.3.1.1"><cn id="S2.SS2.p2.4.m2.1.1.cmml" type="integer" xref="S2.SS2.p2.4.m2.1.1">1</cn><ci id="S2.SS2.p2.4.m2.2.2.cmml" xref="S2.SS2.p2.4.m2.2.2">…</ci><apply id="S2.SS2.p2.4.m2.3.3.1.1.1.cmml" xref="S2.SS2.p2.4.m2.3.3.1.1.1"><csymbol cd="ambiguous" id="S2.SS2.p2.4.m2.3.3.1.1.1.1.cmml" xref="S2.SS2.p2.4.m2.3.3.1.1.1">subscript</csymbol><ci id="S2.SS2.p2.4.m2.3.3.1.1.1.2.cmml" xref="S2.SS2.p2.4.m2.3.3.1.1.1.2">𝑚</ci><ci id="S2.SS2.p2.4.m2.3.3.1.1.1.3.cmml" xref="S2.SS2.p2.4.m2.3.3.1.1.1.3">𝑖</ci></apply></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p2.4.m2.3c">j=1,\ldots,m_{i}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p2.4.m2.3d">italic_j = 1 , … , italic_m start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math>, where <math alttext="\operatorname{\mathbf{X}}_{i}" class="ltx_Math" display="inline" id="S2.SS2.p2.5.m3.1"><semantics id="S2.SS2.p2.5.m3.1a"><msub id="S2.SS2.p2.5.m3.1.1" xref="S2.SS2.p2.5.m3.1.1.cmml"><mi id="S2.SS2.p2.5.m3.1.1.2" xref="S2.SS2.p2.5.m3.1.1.2.cmml">𝐗</mi><mi id="S2.SS2.p2.5.m3.1.1.3" xref="S2.SS2.p2.5.m3.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S2.SS2.p2.5.m3.1b"><apply id="S2.SS2.p2.5.m3.1.1.cmml" xref="S2.SS2.p2.5.m3.1.1"><csymbol cd="ambiguous" id="S2.SS2.p2.5.m3.1.1.1.cmml" xref="S2.SS2.p2.5.m3.1.1">subscript</csymbol><ci id="S2.SS2.p2.5.m3.1.1.2.cmml" xref="S2.SS2.p2.5.m3.1.1.2">𝐗</ci><ci id="S2.SS2.p2.5.m3.1.1.3.cmml" xref="S2.SS2.p2.5.m3.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p2.5.m3.1c">\operatorname{\mathbf{X}}_{i}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p2.5.m3.1d">bold_X start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> is the matrix whose <math alttext="j" class="ltx_Math" display="inline" id="S2.SS2.p2.6.m4.1"><semantics id="S2.SS2.p2.6.m4.1a"><mi id="S2.SS2.p2.6.m4.1.1" xref="S2.SS2.p2.6.m4.1.1.cmml">j</mi><annotation-xml encoding="MathML-Content" id="S2.SS2.p2.6.m4.1b"><ci id="S2.SS2.p2.6.m4.1.1.cmml" xref="S2.SS2.p2.6.m4.1.1">𝑗</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p2.6.m4.1c">j</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p2.6.m4.1d">italic_j</annotation></semantics></math>th row is <math alttext="\operatorname{\mathbf{x}}_{ij}^{\top}" class="ltx_Math" display="inline" id="S2.SS2.p2.7.m5.1"><semantics id="S2.SS2.p2.7.m5.1a"><msubsup id="S2.SS2.p2.7.m5.1.1" xref="S2.SS2.p2.7.m5.1.1.cmml"><mi id="S2.SS2.p2.7.m5.1.1.2.2" xref="S2.SS2.p2.7.m5.1.1.2.2.cmml">𝐱</mi><mrow id="S2.SS2.p2.7.m5.1.1.2.3" xref="S2.SS2.p2.7.m5.1.1.2.3.cmml"><mi id="S2.SS2.p2.7.m5.1.1.2.3.2" xref="S2.SS2.p2.7.m5.1.1.2.3.2.cmml">i</mi><mo id="S2.SS2.p2.7.m5.1.1.2.3.1" xref="S2.SS2.p2.7.m5.1.1.2.3.1.cmml">⁢</mo><mi id="S2.SS2.p2.7.m5.1.1.2.3.3" xref="S2.SS2.p2.7.m5.1.1.2.3.3.cmml">j</mi></mrow><mo id="S2.SS2.p2.7.m5.1.1.3" xref="S2.SS2.p2.7.m5.1.1.3.cmml">⊤</mo></msubsup><annotation-xml encoding="MathML-Content" id="S2.SS2.p2.7.m5.1b"><apply id="S2.SS2.p2.7.m5.1.1.cmml" xref="S2.SS2.p2.7.m5.1.1"><csymbol cd="ambiguous" id="S2.SS2.p2.7.m5.1.1.1.cmml" xref="S2.SS2.p2.7.m5.1.1">superscript</csymbol><apply id="S2.SS2.p2.7.m5.1.1.2.cmml" xref="S2.SS2.p2.7.m5.1.1"><csymbol cd="ambiguous" id="S2.SS2.p2.7.m5.1.1.2.1.cmml" xref="S2.SS2.p2.7.m5.1.1">subscript</csymbol><ci id="S2.SS2.p2.7.m5.1.1.2.2.cmml" xref="S2.SS2.p2.7.m5.1.1.2.2">𝐱</ci><apply id="S2.SS2.p2.7.m5.1.1.2.3.cmml" xref="S2.SS2.p2.7.m5.1.1.2.3"><times id="S2.SS2.p2.7.m5.1.1.2.3.1.cmml" xref="S2.SS2.p2.7.m5.1.1.2.3.1"></times><ci id="S2.SS2.p2.7.m5.1.1.2.3.2.cmml" xref="S2.SS2.p2.7.m5.1.1.2.3.2">𝑖</ci><ci id="S2.SS2.p2.7.m5.1.1.2.3.3.cmml" xref="S2.SS2.p2.7.m5.1.1.2.3.3">𝑗</ci></apply></apply><csymbol cd="latexml" id="S2.SS2.p2.7.m5.1.1.3.cmml" xref="S2.SS2.p2.7.m5.1.1.3">top</csymbol></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p2.7.m5.1c">\operatorname{\mathbf{x}}_{ij}^{\top}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p2.7.m5.1d">bold_x start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ⊤ end_POSTSUPERSCRIPT</annotation></semantics></math> and <math alttext="\operatorname{\mathbf{Z}}_{i(1)}" class="ltx_Math" display="inline" id="S2.SS2.p2.8.m6.1"><semantics id="S2.SS2.p2.8.m6.1a"><msub id="S2.SS2.p2.8.m6.1.2" xref="S2.SS2.p2.8.m6.1.2.cmml"><mi id="S2.SS2.p2.8.m6.1.2.2" xref="S2.SS2.p2.8.m6.1.2.2.cmml">𝐙</mi><mrow id="S2.SS2.p2.8.m6.1.1.1" xref="S2.SS2.p2.8.m6.1.1.1.cmml"><mi id="S2.SS2.p2.8.m6.1.1.1.3" xref="S2.SS2.p2.8.m6.1.1.1.3.cmml">i</mi><mo id="S2.SS2.p2.8.m6.1.1.1.2" xref="S2.SS2.p2.8.m6.1.1.1.2.cmml">⁢</mo><mrow id="S2.SS2.p2.8.m6.1.1.1.4.2" xref="S2.SS2.p2.8.m6.1.1.1.cmml"><mo id="S2.SS2.p2.8.m6.1.1.1.4.2.1" stretchy="false" xref="S2.SS2.p2.8.m6.1.1.1.cmml">(</mo><mn id="S2.SS2.p2.8.m6.1.1.1.1" xref="S2.SS2.p2.8.m6.1.1.1.1.cmml">1</mn><mo id="S2.SS2.p2.8.m6.1.1.1.4.2.2" stretchy="false" xref="S2.SS2.p2.8.m6.1.1.1.cmml">)</mo></mrow></mrow></msub><annotation-xml encoding="MathML-Content" id="S2.SS2.p2.8.m6.1b"><apply id="S2.SS2.p2.8.m6.1.2.cmml" xref="S2.SS2.p2.8.m6.1.2"><csymbol cd="ambiguous" id="S2.SS2.p2.8.m6.1.2.1.cmml" xref="S2.SS2.p2.8.m6.1.2">subscript</csymbol><ci id="S2.SS2.p2.8.m6.1.2.2.cmml" xref="S2.SS2.p2.8.m6.1.2.2">𝐙</ci><apply id="S2.SS2.p2.8.m6.1.1.1.cmml" xref="S2.SS2.p2.8.m6.1.1.1"><times id="S2.SS2.p2.8.m6.1.1.1.2.cmml" xref="S2.SS2.p2.8.m6.1.1.1.2"></times><ci id="S2.SS2.p2.8.m6.1.1.1.3.cmml" xref="S2.SS2.p2.8.m6.1.1.1.3">𝑖</ci><cn id="S2.SS2.p2.8.m6.1.1.1.1.cmml" type="integer" xref="S2.SS2.p2.8.m6.1.1.1.1">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p2.8.m6.1c">\operatorname{\mathbf{Z}}_{i(1)}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p2.8.m6.1d">bold_Z start_POSTSUBSCRIPT italic_i ( 1 ) end_POSTSUBSCRIPT</annotation></semantics></math> is the matrix whose <math alttext="j" class="ltx_Math" display="inline" id="S2.SS2.p2.9.m7.1"><semantics id="S2.SS2.p2.9.m7.1a"><mi id="S2.SS2.p2.9.m7.1.1" xref="S2.SS2.p2.9.m7.1.1.cmml">j</mi><annotation-xml encoding="MathML-Content" id="S2.SS2.p2.9.m7.1b"><ci id="S2.SS2.p2.9.m7.1.1.cmml" xref="S2.SS2.p2.9.m7.1.1">𝑗</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p2.9.m7.1c">j</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p2.9.m7.1d">italic_j</annotation></semantics></math>th row is <math alttext="\operatorname{\mathbf{z}}_{ij(1)}^{\top}" class="ltx_Math" display="inline" id="S2.SS2.p2.10.m8.1"><semantics id="S2.SS2.p2.10.m8.1a"><msubsup id="S2.SS2.p2.10.m8.1.2" xref="S2.SS2.p2.10.m8.1.2.cmml"><mi id="S2.SS2.p2.10.m8.1.2.2.2" xref="S2.SS2.p2.10.m8.1.2.2.2.cmml">𝐳</mi><mrow id="S2.SS2.p2.10.m8.1.1.1" xref="S2.SS2.p2.10.m8.1.1.1.cmml"><mi id="S2.SS2.p2.10.m8.1.1.1.3" xref="S2.SS2.p2.10.m8.1.1.1.3.cmml">i</mi><mo id="S2.SS2.p2.10.m8.1.1.1.2" xref="S2.SS2.p2.10.m8.1.1.1.2.cmml">⁢</mo><mi id="S2.SS2.p2.10.m8.1.1.1.4" xref="S2.SS2.p2.10.m8.1.1.1.4.cmml">j</mi><mo id="S2.SS2.p2.10.m8.1.1.1.2a" xref="S2.SS2.p2.10.m8.1.1.1.2.cmml">⁢</mo><mrow id="S2.SS2.p2.10.m8.1.1.1.5.2" xref="S2.SS2.p2.10.m8.1.1.1.cmml"><mo id="S2.SS2.p2.10.m8.1.1.1.5.2.1" stretchy="false" xref="S2.SS2.p2.10.m8.1.1.1.cmml">(</mo><mn id="S2.SS2.p2.10.m8.1.1.1.1" xref="S2.SS2.p2.10.m8.1.1.1.1.cmml">1</mn><mo id="S2.SS2.p2.10.m8.1.1.1.5.2.2" stretchy="false" xref="S2.SS2.p2.10.m8.1.1.1.cmml">)</mo></mrow></mrow><mo id="S2.SS2.p2.10.m8.1.2.3" xref="S2.SS2.p2.10.m8.1.2.3.cmml">⊤</mo></msubsup><annotation-xml encoding="MathML-Content" id="S2.SS2.p2.10.m8.1b"><apply id="S2.SS2.p2.10.m8.1.2.cmml" xref="S2.SS2.p2.10.m8.1.2"><csymbol cd="ambiguous" id="S2.SS2.p2.10.m8.1.2.1.cmml" xref="S2.SS2.p2.10.m8.1.2">superscript</csymbol><apply id="S2.SS2.p2.10.m8.1.2.2.cmml" xref="S2.SS2.p2.10.m8.1.2"><csymbol cd="ambiguous" id="S2.SS2.p2.10.m8.1.2.2.1.cmml" xref="S2.SS2.p2.10.m8.1.2">subscript</csymbol><ci id="S2.SS2.p2.10.m8.1.2.2.2.cmml" xref="S2.SS2.p2.10.m8.1.2.2.2">𝐳</ci><apply id="S2.SS2.p2.10.m8.1.1.1.cmml" xref="S2.SS2.p2.10.m8.1.1.1"><times id="S2.SS2.p2.10.m8.1.1.1.2.cmml" xref="S2.SS2.p2.10.m8.1.1.1.2"></times><ci id="S2.SS2.p2.10.m8.1.1.1.3.cmml" xref="S2.SS2.p2.10.m8.1.1.1.3">𝑖</ci><ci id="S2.SS2.p2.10.m8.1.1.1.4.cmml" xref="S2.SS2.p2.10.m8.1.1.1.4">𝑗</ci><cn id="S2.SS2.p2.10.m8.1.1.1.1.cmml" type="integer" xref="S2.SS2.p2.10.m8.1.1.1.1">1</cn></apply></apply><csymbol cd="latexml" id="S2.SS2.p2.10.m8.1.2.3.cmml" xref="S2.SS2.p2.10.m8.1.2.3">top</csymbol></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p2.10.m8.1c">\operatorname{\mathbf{z}}_{ij(1)}^{\top}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p2.10.m8.1d">bold_z start_POSTSUBSCRIPT italic_i italic_j ( 1 ) end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ⊤ end_POSTSUPERSCRIPT</annotation></semantics></math>. The first equation uses <math alttext="\operatorname{\mathop{\mathrm{tr}}}(\operatorname{\mathbf{Z}}_{ij}^{\top}% \operatorname{\mathbf{L}}_{1}\operatorname{\mathbf{C}}_{i}\operatorname{% \mathbf{L}}_{2}^{\top})=\operatorname{\mathbf{z}}_{ij(1)}^{\top}\text{vec}(% \operatorname{\mathbf{L}}_{1}\operatorname{\mathbf{C}}_{i}\operatorname{% \mathbf{L}}_{2}^{\top})=\operatorname{\mathbf{z}}_{ij(1)}^{\top}(\operatorname% {\mathbf{L}}_{2}\otimes\operatorname{\mathbf{L}}_{1})\operatorname{\mathbf{c}}% _{i(1)}" class="ltx_Math" display="inline" id="S2.SS2.p2.11.m9.8"><semantics id="S2.SS2.p2.11.m9.8a"><mrow id="S2.SS2.p2.11.m9.8.8" xref="S2.SS2.p2.11.m9.8.8.cmml"><mrow id="S2.SS2.p2.11.m9.5.5.1.1" xref="S2.SS2.p2.11.m9.5.5.1.2.cmml"><mi id="S2.SS2.p2.11.m9.4.4" xref="S2.SS2.p2.11.m9.4.4.cmml">tr</mi><mo id="S2.SS2.p2.11.m9.5.5.1.1a" xref="S2.SS2.p2.11.m9.5.5.1.2.cmml">⁡</mo><mrow id="S2.SS2.p2.11.m9.5.5.1.1.1" xref="S2.SS2.p2.11.m9.5.5.1.2.cmml"><mo id="S2.SS2.p2.11.m9.5.5.1.1.1.2" 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encoding="application/x-tex" id="S2.SS2.p2.11.m9.8c">\operatorname{\mathop{\mathrm{tr}}}(\operatorname{\mathbf{Z}}_{ij}^{\top}% \operatorname{\mathbf{L}}_{1}\operatorname{\mathbf{C}}_{i}\operatorname{% \mathbf{L}}_{2}^{\top})=\operatorname{\mathbf{z}}_{ij(1)}^{\top}\text{vec}(% \operatorname{\mathbf{L}}_{1}\operatorname{\mathbf{C}}_{i}\operatorname{% \mathbf{L}}_{2}^{\top})=\operatorname{\mathbf{z}}_{ij(1)}^{\top}(\operatorname% {\mathbf{L}}_{2}\otimes\operatorname{\mathbf{L}}_{1})\operatorname{\mathbf{c}}% _{i(1)}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p2.11.m9.8d">roman_tr ( bold_Z start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ⊤ end_POSTSUPERSCRIPT bold_L start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT bold_C start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT bold_L start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ⊤ end_POSTSUPERSCRIPT ) = bold_z start_POSTSUBSCRIPT italic_i italic_j ( 1 ) end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ⊤ end_POSTSUPERSCRIPT vec ( bold_L start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT bold_C start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT bold_L start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ⊤ end_POSTSUPERSCRIPT ) = bold_z start_POSTSUBSCRIPT italic_i italic_j ( 1 ) end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ⊤ end_POSTSUPERSCRIPT ( bold_L start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ⊗ bold_L start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) bold_c start_POSTSUBSCRIPT italic_i ( 1 ) end_POSTSUBSCRIPT</annotation></semantics></math>, which implies that the covariance matrix of random effects is <math alttext="\text{Cov}\{(\operatorname{\mathbf{L}}_{2}\otimes\operatorname{\mathbf{L}}_{1}% )\operatorname{\mathbf{c}}_{i(1)}\}=\tau^{2}\operatorname{\mathbf{L}}_{2}% \operatorname{\mathbf{L}}_{2}^{\top}\otimes\operatorname{\mathbf{L}}_{1}% \operatorname{\mathbf{L}}_{1}^{\top}" class="ltx_Math" display="inline" id="S2.SS2.p2.12.m10.2"><semantics id="S2.SS2.p2.12.m10.2a"><mrow 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id="S2.SS2.p2.12.m10.2.2.3.3.2.1.cmml" xref="S2.SS2.p2.12.m10.2.2.3.3.2.1">tensor-product</csymbol><apply id="S2.SS2.p2.12.m10.2.2.3.3.2.2.cmml" xref="S2.SS2.p2.12.m10.2.2.3.3.2.2"><csymbol cd="ambiguous" id="S2.SS2.p2.12.m10.2.2.3.3.2.2.1.cmml" xref="S2.SS2.p2.12.m10.2.2.3.3.2.2">superscript</csymbol><apply id="S2.SS2.p2.12.m10.2.2.3.3.2.2.2.cmml" xref="S2.SS2.p2.12.m10.2.2.3.3.2.2"><csymbol cd="ambiguous" id="S2.SS2.p2.12.m10.2.2.3.3.2.2.2.1.cmml" xref="S2.SS2.p2.12.m10.2.2.3.3.2.2">subscript</csymbol><ci id="S2.SS2.p2.12.m10.2.2.3.3.2.2.2.2.cmml" xref="S2.SS2.p2.12.m10.2.2.3.3.2.2.2.2">𝐋</ci><cn id="S2.SS2.p2.12.m10.2.2.3.3.2.2.2.3.cmml" type="integer" xref="S2.SS2.p2.12.m10.2.2.3.3.2.2.2.3">2</cn></apply><csymbol cd="latexml" id="S2.SS2.p2.12.m10.2.2.3.3.2.2.3.cmml" xref="S2.SS2.p2.12.m10.2.2.3.3.2.2.3">top</csymbol></apply><apply id="S2.SS2.p2.12.m10.2.2.3.3.2.3.cmml" xref="S2.SS2.p2.12.m10.2.2.3.3.2.3"><apply id="S2.SS2.p2.12.m10.2.2.3.3.2.3.1.cmml" xref="S2.SS2.p2.12.m10.2.2.3.3.2.3.1"><csymbol cd="ambiguous" id="S2.SS2.p2.12.m10.2.2.3.3.2.3.1.1.cmml" xref="S2.SS2.p2.12.m10.2.2.3.3.2.3.1">subscript</csymbol><ci id="S2.SS2.p2.12.m10.2.2.3.3.2.3.1.2.cmml" xref="S2.SS2.p2.12.m10.2.2.3.3.2.3.1.2">𝐋</ci><cn id="S2.SS2.p2.12.m10.2.2.3.3.2.3.1.3.cmml" type="integer" xref="S2.SS2.p2.12.m10.2.2.3.3.2.3.1.3">1</cn></apply><apply id="S2.SS2.p2.12.m10.2.2.3.3.2.3.2.cmml" xref="S2.SS2.p2.12.m10.2.2.3.3.2.3.2"><csymbol cd="ambiguous" id="S2.SS2.p2.12.m10.2.2.3.3.2.3.2.1.cmml" xref="S2.SS2.p2.12.m10.2.2.3.3.2.3.2">superscript</csymbol><apply id="S2.SS2.p2.12.m10.2.2.3.3.2.3.2.2.cmml" xref="S2.SS2.p2.12.m10.2.2.3.3.2.3.2"><csymbol cd="ambiguous" id="S2.SS2.p2.12.m10.2.2.3.3.2.3.2.2.1.cmml" xref="S2.SS2.p2.12.m10.2.2.3.3.2.3.2">subscript</csymbol><ci id="S2.SS2.p2.12.m10.2.2.3.3.2.3.2.2.2.cmml" xref="S2.SS2.p2.12.m10.2.2.3.3.2.3.2.2.2">𝐋</ci><cn id="S2.SS2.p2.12.m10.2.2.3.3.2.3.2.2.3.cmml" type="integer" xref="S2.SS2.p2.12.m10.2.2.3.3.2.3.2.2.3">1</cn></apply><csymbol cd="latexml" id="S2.SS2.p2.12.m10.2.2.3.3.2.3.2.3.cmml" xref="S2.SS2.p2.12.m10.2.2.3.3.2.3.2.3">top</csymbol></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p2.12.m10.2c">\text{Cov}\{(\operatorname{\mathbf{L}}_{2}\otimes\operatorname{\mathbf{L}}_{1}% )\operatorname{\mathbf{c}}_{i(1)}\}=\tau^{2}\operatorname{\mathbf{L}}_{2}% \operatorname{\mathbf{L}}_{2}^{\top}\otimes\operatorname{\mathbf{L}}_{1}% \operatorname{\mathbf{L}}_{1}^{\top}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p2.12.m10.2d">Cov { ( bold_L start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ⊗ bold_L start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) bold_c start_POSTSUBSCRIPT italic_i ( 1 ) end_POSTSUBSCRIPT } = italic_τ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT bold_L start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT bold_L start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ⊤ end_POSTSUPERSCRIPT ⊗ bold_L start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT bold_L start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ⊤ end_POSTSUPERSCRIPT</annotation></semantics></math>. If <math alttext="\operatorname{{\mathbf{\Sigma}}}_{1}" class="ltx_Math" display="inline" id="S2.SS2.p2.13.m11.1"><semantics id="S2.SS2.p2.13.m11.1a"><msub id="S2.SS2.p2.13.m11.1.1" xref="S2.SS2.p2.13.m11.1.1.cmml"><mi id="S2.SS2.p2.13.m11.1.1.2" xref="S2.SS2.p2.13.m11.1.1.2.cmml">𝚺</mi><mn id="S2.SS2.p2.13.m11.1.1.3" xref="S2.SS2.p2.13.m11.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S2.SS2.p2.13.m11.1b"><apply id="S2.SS2.p2.13.m11.1.1.cmml" xref="S2.SS2.p2.13.m11.1.1"><csymbol cd="ambiguous" id="S2.SS2.p2.13.m11.1.1.1.cmml" xref="S2.SS2.p2.13.m11.1.1">subscript</csymbol><ci id="S2.SS2.p2.13.m11.1.1.2.cmml" xref="S2.SS2.p2.13.m11.1.1.2">𝚺</ci><cn id="S2.SS2.p2.13.m11.1.1.3.cmml" type="integer" xref="S2.SS2.p2.13.m11.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p2.13.m11.1c">\operatorname{{\mathbf{\Sigma}}}_{1}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p2.13.m11.1d">bold_Σ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="\operatorname{{\mathbf{\Sigma}}}_{2}" class="ltx_Math" display="inline" id="S2.SS2.p2.14.m12.1"><semantics id="S2.SS2.p2.14.m12.1a"><msub id="S2.SS2.p2.14.m12.1.1" xref="S2.SS2.p2.14.m12.1.1.cmml"><mi id="S2.SS2.p2.14.m12.1.1.2" xref="S2.SS2.p2.14.m12.1.1.2.cmml">𝚺</mi><mn id="S2.SS2.p2.14.m12.1.1.3" xref="S2.SS2.p2.14.m12.1.1.3.cmml">2</mn></msub><annotation-xml encoding="MathML-Content" id="S2.SS2.p2.14.m12.1b"><apply id="S2.SS2.p2.14.m12.1.1.cmml" xref="S2.SS2.p2.14.m12.1.1"><csymbol cd="ambiguous" id="S2.SS2.p2.14.m12.1.1.1.cmml" xref="S2.SS2.p2.14.m12.1.1">subscript</csymbol><ci id="S2.SS2.p2.14.m12.1.1.2.cmml" xref="S2.SS2.p2.14.m12.1.1.2">𝚺</ci><cn id="S2.SS2.p2.14.m12.1.1.3.cmml" type="integer" xref="S2.SS2.p2.14.m12.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p2.14.m12.1c">\operatorname{{\mathbf{\Sigma}}}_{2}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p2.14.m12.1d">bold_Σ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math> are full-rank and <math alttext="n" class="ltx_Math" display="inline" id="S2.SS2.p2.15.m13.1"><semantics id="S2.SS2.p2.15.m13.1a"><mi id="S2.SS2.p2.15.m13.1.1" xref="S2.SS2.p2.15.m13.1.1.cmml">n</mi><annotation-xml encoding="MathML-Content" id="S2.SS2.p2.15.m13.1b"><ci id="S2.SS2.p2.15.m13.1.1.cmml" xref="S2.SS2.p2.15.m13.1.1">𝑛</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p2.15.m13.1c">n</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p2.15.m13.1d">italic_n</annotation></semantics></math> and <math alttext="m_{1},\ldots,m_{n}" class="ltx_Math" display="inline" id="S2.SS2.p2.16.m14.3"><semantics id="S2.SS2.p2.16.m14.3a"><mrow id="S2.SS2.p2.16.m14.3.3.2" xref="S2.SS2.p2.16.m14.3.3.3.cmml"><msub id="S2.SS2.p2.16.m14.2.2.1.1" xref="S2.SS2.p2.16.m14.2.2.1.1.cmml"><mi id="S2.SS2.p2.16.m14.2.2.1.1.2" xref="S2.SS2.p2.16.m14.2.2.1.1.2.cmml">m</mi><mn id="S2.SS2.p2.16.m14.2.2.1.1.3" xref="S2.SS2.p2.16.m14.2.2.1.1.3.cmml">1</mn></msub><mo id="S2.SS2.p2.16.m14.3.3.2.3" xref="S2.SS2.p2.16.m14.3.3.3.cmml">,</mo><mi id="S2.SS2.p2.16.m14.1.1" mathvariant="normal" xref="S2.SS2.p2.16.m14.1.1.cmml">…</mi><mo id="S2.SS2.p2.16.m14.3.3.2.4" xref="S2.SS2.p2.16.m14.3.3.3.cmml">,</mo><msub id="S2.SS2.p2.16.m14.3.3.2.2" xref="S2.SS2.p2.16.m14.3.3.2.2.cmml"><mi id="S2.SS2.p2.16.m14.3.3.2.2.2" xref="S2.SS2.p2.16.m14.3.3.2.2.2.cmml">m</mi><mi id="S2.SS2.p2.16.m14.3.3.2.2.3" xref="S2.SS2.p2.16.m14.3.3.2.2.3.cmml">n</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p2.16.m14.3b"><list id="S2.SS2.p2.16.m14.3.3.3.cmml" xref="S2.SS2.p2.16.m14.3.3.2"><apply id="S2.SS2.p2.16.m14.2.2.1.1.cmml" xref="S2.SS2.p2.16.m14.2.2.1.1"><csymbol cd="ambiguous" id="S2.SS2.p2.16.m14.2.2.1.1.1.cmml" xref="S2.SS2.p2.16.m14.2.2.1.1">subscript</csymbol><ci id="S2.SS2.p2.16.m14.2.2.1.1.2.cmml" xref="S2.SS2.p2.16.m14.2.2.1.1.2">𝑚</ci><cn id="S2.SS2.p2.16.m14.2.2.1.1.3.cmml" type="integer" xref="S2.SS2.p2.16.m14.2.2.1.1.3">1</cn></apply><ci id="S2.SS2.p2.16.m14.1.1.cmml" xref="S2.SS2.p2.16.m14.1.1">…</ci><apply id="S2.SS2.p2.16.m14.3.3.2.2.cmml" xref="S2.SS2.p2.16.m14.3.3.2.2"><csymbol cd="ambiguous" id="S2.SS2.p2.16.m14.3.3.2.2.1.cmml" xref="S2.SS2.p2.16.m14.3.3.2.2">subscript</csymbol><ci id="S2.SS2.p2.16.m14.3.3.2.2.2.cmml" xref="S2.SS2.p2.16.m14.3.3.2.2.2">𝑚</ci><ci id="S2.SS2.p2.16.m14.3.3.2.2.3.cmml" xref="S2.SS2.p2.16.m14.3.3.2.2.3">𝑛</ci></apply></list></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p2.16.m14.3c">m_{1},\ldots,m_{n}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p2.16.m14.3d">italic_m start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_m start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT</annotation></semantics></math> are sufficiently large, then we estimate <math alttext="\operatorname{\mathbf{B}}=\text{vec}^{-1}(\operatorname{\mathbf{b}})" class="ltx_Math" display="inline" id="S2.SS2.p2.17.m15.1"><semantics id="S2.SS2.p2.17.m15.1a"><mrow id="S2.SS2.p2.17.m15.1.2" xref="S2.SS2.p2.17.m15.1.2.cmml"><mi id="S2.SS2.p2.17.m15.1.2.2" xref="S2.SS2.p2.17.m15.1.2.2.cmml">𝐁</mi><mo id="S2.SS2.p2.17.m15.1.2.1" xref="S2.SS2.p2.17.m15.1.2.1.cmml">=</mo><mrow id="S2.SS2.p2.17.m15.1.2.3" xref="S2.SS2.p2.17.m15.1.2.3.cmml"><msup id="S2.SS2.p2.17.m15.1.2.3.2" xref="S2.SS2.p2.17.m15.1.2.3.2.cmml"><mtext id="S2.SS2.p2.17.m15.1.2.3.2.2" xref="S2.SS2.p2.17.m15.1.2.3.2.2a.cmml">vec</mtext><mrow id="S2.SS2.p2.17.m15.1.2.3.2.3" xref="S2.SS2.p2.17.m15.1.2.3.2.3.cmml"><mo id="S2.SS2.p2.17.m15.1.2.3.2.3a" xref="S2.SS2.p2.17.m15.1.2.3.2.3.cmml">−</mo><mn id="S2.SS2.p2.17.m15.1.2.3.2.3.2" xref="S2.SS2.p2.17.m15.1.2.3.2.3.2.cmml">1</mn></mrow></msup><mo id="S2.SS2.p2.17.m15.1.2.3.1" xref="S2.SS2.p2.17.m15.1.2.3.1.cmml">⁢</mo><mrow id="S2.SS2.p2.17.m15.1.2.3.3.2" xref="S2.SS2.p2.17.m15.1.2.3.cmml"><mo id="S2.SS2.p2.17.m15.1.2.3.3.2.1" stretchy="false" xref="S2.SS2.p2.17.m15.1.2.3.cmml">(</mo><mi id="S2.SS2.p2.17.m15.1.1" xref="S2.SS2.p2.17.m15.1.1.cmml">𝐛</mi><mo id="S2.SS2.p2.17.m15.1.2.3.3.2.2" stretchy="false" xref="S2.SS2.p2.17.m15.1.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p2.17.m15.1b"><apply id="S2.SS2.p2.17.m15.1.2.cmml" xref="S2.SS2.p2.17.m15.1.2"><eq id="S2.SS2.p2.17.m15.1.2.1.cmml" xref="S2.SS2.p2.17.m15.1.2.1"></eq><ci id="S2.SS2.p2.17.m15.1.2.2.cmml" xref="S2.SS2.p2.17.m15.1.2.2">𝐁</ci><apply id="S2.SS2.p2.17.m15.1.2.3.cmml" xref="S2.SS2.p2.17.m15.1.2.3"><times id="S2.SS2.p2.17.m15.1.2.3.1.cmml" xref="S2.SS2.p2.17.m15.1.2.3.1"></times><apply id="S2.SS2.p2.17.m15.1.2.3.2.cmml" xref="S2.SS2.p2.17.m15.1.2.3.2"><csymbol cd="ambiguous" id="S2.SS2.p2.17.m15.1.2.3.2.1.cmml" xref="S2.SS2.p2.17.m15.1.2.3.2">superscript</csymbol><ci id="S2.SS2.p2.17.m15.1.2.3.2.2a.cmml" xref="S2.SS2.p2.17.m15.1.2.3.2.2"><mtext id="S2.SS2.p2.17.m15.1.2.3.2.2.cmml" xref="S2.SS2.p2.17.m15.1.2.3.2.2">vec</mtext></ci><apply id="S2.SS2.p2.17.m15.1.2.3.2.3.cmml" xref="S2.SS2.p2.17.m15.1.2.3.2.3"><minus id="S2.SS2.p2.17.m15.1.2.3.2.3.1.cmml" xref="S2.SS2.p2.17.m15.1.2.3.2.3"></minus><cn id="S2.SS2.p2.17.m15.1.2.3.2.3.2.cmml" type="integer" xref="S2.SS2.p2.17.m15.1.2.3.2.3.2">1</cn></apply></apply><ci id="S2.SS2.p2.17.m15.1.1.cmml" xref="S2.SS2.p2.17.m15.1.1">𝐛</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p2.17.m15.1c">\operatorname{\mathbf{B}}=\text{vec}^{-1}(\operatorname{\mathbf{b}})</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p2.17.m15.1d">bold_B = vec start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ( bold_b )</annotation></semantics></math> and <math alttext="\operatorname{{\mathbf{\Sigma}}}=\operatorname{{\mathbf{\Sigma}}}_{2}\otimes% \operatorname{{\mathbf{\Sigma}}}_{1}" class="ltx_Math" display="inline" id="S2.SS2.p2.18.m16.1"><semantics id="S2.SS2.p2.18.m16.1a"><mrow id="S2.SS2.p2.18.m16.1.1" xref="S2.SS2.p2.18.m16.1.1.cmml"><mi id="S2.SS2.p2.18.m16.1.1.2" xref="S2.SS2.p2.18.m16.1.1.2.cmml">𝚺</mi><mo id="S2.SS2.p2.18.m16.1.1.1" xref="S2.SS2.p2.18.m16.1.1.1.cmml">=</mo><mrow id="S2.SS2.p2.18.m16.1.1.3" xref="S2.SS2.p2.18.m16.1.1.3.cmml"><msub id="S2.SS2.p2.18.m16.1.1.3.2" xref="S2.SS2.p2.18.m16.1.1.3.2.cmml"><mi id="S2.SS2.p2.18.m16.1.1.3.2.2" xref="S2.SS2.p2.18.m16.1.1.3.2.2.cmml">𝚺</mi><mn id="S2.SS2.p2.18.m16.1.1.3.2.3" xref="S2.SS2.p2.18.m16.1.1.3.2.3.cmml">2</mn></msub><mo id="S2.SS2.p2.18.m16.1.1.3.1" lspace="0em" rspace="0.222em" xref="S2.SS2.p2.18.m16.1.1.3.1.cmml">⊗</mo><msub id="S2.SS2.p2.18.m16.1.1.3.3" xref="S2.SS2.p2.18.m16.1.1.3.3.cmml"><mi id="S2.SS2.p2.18.m16.1.1.3.3.2" xref="S2.SS2.p2.18.m16.1.1.3.3.2.cmml">𝚺</mi><mn id="S2.SS2.p2.18.m16.1.1.3.3.3" xref="S2.SS2.p2.18.m16.1.1.3.3.3.cmml">1</mn></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p2.18.m16.1b"><apply id="S2.SS2.p2.18.m16.1.1.cmml" xref="S2.SS2.p2.18.m16.1.1"><eq id="S2.SS2.p2.18.m16.1.1.1.cmml" xref="S2.SS2.p2.18.m16.1.1.1"></eq><ci id="S2.SS2.p2.18.m16.1.1.2.cmml" xref="S2.SS2.p2.18.m16.1.1.2">𝚺</ci><apply id="S2.SS2.p2.18.m16.1.1.3.cmml" xref="S2.SS2.p2.18.m16.1.1.3"><csymbol cd="latexml" id="S2.SS2.p2.18.m16.1.1.3.1.cmml" xref="S2.SS2.p2.18.m16.1.1.3.1">tensor-product</csymbol><apply id="S2.SS2.p2.18.m16.1.1.3.2.cmml" xref="S2.SS2.p2.18.m16.1.1.3.2"><csymbol cd="ambiguous" id="S2.SS2.p2.18.m16.1.1.3.2.1.cmml" xref="S2.SS2.p2.18.m16.1.1.3.2">subscript</csymbol><ci id="S2.SS2.p2.18.m16.1.1.3.2.2.cmml" xref="S2.SS2.p2.18.m16.1.1.3.2.2">𝚺</ci><cn id="S2.SS2.p2.18.m16.1.1.3.2.3.cmml" type="integer" xref="S2.SS2.p2.18.m16.1.1.3.2.3">2</cn></apply><apply id="S2.SS2.p2.18.m16.1.1.3.3.cmml" xref="S2.SS2.p2.18.m16.1.1.3.3"><csymbol cd="ambiguous" id="S2.SS2.p2.18.m16.1.1.3.3.1.cmml" xref="S2.SS2.p2.18.m16.1.1.3.3">subscript</csymbol><ci id="S2.SS2.p2.18.m16.1.1.3.3.2.cmml" xref="S2.SS2.p2.18.m16.1.1.3.3.2">𝚺</ci><cn id="S2.SS2.p2.18.m16.1.1.3.3.3.cmml" type="integer" xref="S2.SS2.p2.18.m16.1.1.3.3.3">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p2.18.m16.1c">\operatorname{{\mathbf{\Sigma}}}=\operatorname{{\mathbf{\Sigma}}}_{2}\otimes% \operatorname{{\mathbf{\Sigma}}}_{1}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p2.18.m16.1d">bold_Σ = bold_Σ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ⊗ bold_Σ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> using existing software such as <span class="ltx_text ltx_font_typewriter" id="S2.SS2.p2.28.1">lme4</span>. If <math alttext="\|\cdot\|_{F}" class="ltx_math_unparsed" display="inline" id="S2.SS2.p2.19.m17.1"><semantics id="S2.SS2.p2.19.m17.1a"><mrow id="S2.SS2.p2.19.m17.1b"><mo id="S2.SS2.p2.19.m17.1.1" rspace="0em">∥</mo><mo id="S2.SS2.p2.19.m17.1.2" lspace="0em" rspace="0em">⋅</mo><msub id="S2.SS2.p2.19.m17.1.3"><mo id="S2.SS2.p2.19.m17.1.3.2" lspace="0em">∥</mo><mi id="S2.SS2.p2.19.m17.1.3.3">F</mi></msub></mrow><annotation encoding="application/x-tex" id="S2.SS2.p2.19.m17.1c">\|\cdot\|_{F}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p2.19.m17.1d">∥ ⋅ ∥ start_POSTSUBSCRIPT italic_F end_POSTSUBSCRIPT</annotation></semantics></math> is the Frobenius norm, then the <math alttext="\operatorname{{\mathbf{\Sigma}}}_{1}" class="ltx_Math" display="inline" id="S2.SS2.p2.20.m18.1"><semantics id="S2.SS2.p2.20.m18.1a"><msub id="S2.SS2.p2.20.m18.1.1" xref="S2.SS2.p2.20.m18.1.1.cmml"><mi id="S2.SS2.p2.20.m18.1.1.2" xref="S2.SS2.p2.20.m18.1.1.2.cmml">𝚺</mi><mn id="S2.SS2.p2.20.m18.1.1.3" xref="S2.SS2.p2.20.m18.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S2.SS2.p2.20.m18.1b"><apply id="S2.SS2.p2.20.m18.1.1.cmml" xref="S2.SS2.p2.20.m18.1.1"><csymbol cd="ambiguous" id="S2.SS2.p2.20.m18.1.1.1.cmml" xref="S2.SS2.p2.20.m18.1.1">subscript</csymbol><ci id="S2.SS2.p2.20.m18.1.1.2.cmml" xref="S2.SS2.p2.20.m18.1.1.2">𝚺</ci><cn id="S2.SS2.p2.20.m18.1.1.3.cmml" type="integer" xref="S2.SS2.p2.20.m18.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p2.20.m18.1c">\operatorname{{\mathbf{\Sigma}}}_{1}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p2.20.m18.1d">bold_Σ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="\operatorname{{\mathbf{\Sigma}}}_{2}" class="ltx_Math" display="inline" id="S2.SS2.p2.21.m19.1"><semantics id="S2.SS2.p2.21.m19.1a"><msub id="S2.SS2.p2.21.m19.1.1" xref="S2.SS2.p2.21.m19.1.1.cmml"><mi id="S2.SS2.p2.21.m19.1.1.2" xref="S2.SS2.p2.21.m19.1.1.2.cmml">𝚺</mi><mn id="S2.SS2.p2.21.m19.1.1.3" xref="S2.SS2.p2.21.m19.1.1.3.cmml">2</mn></msub><annotation-xml encoding="MathML-Content" id="S2.SS2.p2.21.m19.1b"><apply id="S2.SS2.p2.21.m19.1.1.cmml" xref="S2.SS2.p2.21.m19.1.1"><csymbol cd="ambiguous" id="S2.SS2.p2.21.m19.1.1.1.cmml" xref="S2.SS2.p2.21.m19.1.1">subscript</csymbol><ci id="S2.SS2.p2.21.m19.1.1.2.cmml" xref="S2.SS2.p2.21.m19.1.1.2">𝚺</ci><cn id="S2.SS2.p2.21.m19.1.1.3.cmml" type="integer" xref="S2.SS2.p2.21.m19.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p2.21.m19.1c">\operatorname{{\mathbf{\Sigma}}}_{2}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p2.21.m19.1d">bold_Σ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math> estimates minimize <math alttext="\|\hat{\operatorname{{\mathbf{\Sigma}}}}-\operatorname{{\mathbf{\Sigma}}}_{2}% \otimes\operatorname{{\mathbf{\Sigma}}}_{1}\|_{F}" class="ltx_Math" display="inline" id="S2.SS2.p2.22.m20.1"><semantics id="S2.SS2.p2.22.m20.1a"><msub id="S2.SS2.p2.22.m20.1.1" xref="S2.SS2.p2.22.m20.1.1.cmml"><mrow id="S2.SS2.p2.22.m20.1.1.1.1" xref="S2.SS2.p2.22.m20.1.1.1.2.cmml"><mo id="S2.SS2.p2.22.m20.1.1.1.1.2" stretchy="false" xref="S2.SS2.p2.22.m20.1.1.1.2.1.cmml">‖</mo><mrow id="S2.SS2.p2.22.m20.1.1.1.1.1" xref="S2.SS2.p2.22.m20.1.1.1.1.1.cmml"><mover accent="true" id="S2.SS2.p2.22.m20.1.1.1.1.1.2" xref="S2.SS2.p2.22.m20.1.1.1.1.1.2.cmml"><mi id="S2.SS2.p2.22.m20.1.1.1.1.1.2.2" xref="S2.SS2.p2.22.m20.1.1.1.1.1.2.2.cmml">𝚺</mi><mo id="S2.SS2.p2.22.m20.1.1.1.1.1.2.1" xref="S2.SS2.p2.22.m20.1.1.1.1.1.2.1.cmml">^</mo></mover><mo id="S2.SS2.p2.22.m20.1.1.1.1.1.1" lspace="0em" xref="S2.SS2.p2.22.m20.1.1.1.1.1.1.cmml">−</mo><mrow id="S2.SS2.p2.22.m20.1.1.1.1.1.3" xref="S2.SS2.p2.22.m20.1.1.1.1.1.3.cmml"><msub id="S2.SS2.p2.22.m20.1.1.1.1.1.3.2" xref="S2.SS2.p2.22.m20.1.1.1.1.1.3.2.cmml"><mi id="S2.SS2.p2.22.m20.1.1.1.1.1.3.2.2" xref="S2.SS2.p2.22.m20.1.1.1.1.1.3.2.2.cmml">𝚺</mi><mn id="S2.SS2.p2.22.m20.1.1.1.1.1.3.2.3" xref="S2.SS2.p2.22.m20.1.1.1.1.1.3.2.3.cmml">2</mn></msub><mo id="S2.SS2.p2.22.m20.1.1.1.1.1.3.1" lspace="0em" rspace="0.222em" xref="S2.SS2.p2.22.m20.1.1.1.1.1.3.1.cmml">⊗</mo><msub id="S2.SS2.p2.22.m20.1.1.1.1.1.3.3" xref="S2.SS2.p2.22.m20.1.1.1.1.1.3.3.cmml"><mi id="S2.SS2.p2.22.m20.1.1.1.1.1.3.3.2" xref="S2.SS2.p2.22.m20.1.1.1.1.1.3.3.2.cmml">𝚺</mi><mn id="S2.SS2.p2.22.m20.1.1.1.1.1.3.3.3" xref="S2.SS2.p2.22.m20.1.1.1.1.1.3.3.3.cmml">1</mn></msub></mrow></mrow><mo id="S2.SS2.p2.22.m20.1.1.1.1.3" stretchy="false" xref="S2.SS2.p2.22.m20.1.1.1.2.1.cmml">‖</mo></mrow><mi id="S2.SS2.p2.22.m20.1.1.3" xref="S2.SS2.p2.22.m20.1.1.3.cmml">F</mi></msub><annotation-xml encoding="MathML-Content" id="S2.SS2.p2.22.m20.1b"><apply id="S2.SS2.p2.22.m20.1.1.cmml" xref="S2.SS2.p2.22.m20.1.1"><csymbol cd="ambiguous" id="S2.SS2.p2.22.m20.1.1.2.cmml" xref="S2.SS2.p2.22.m20.1.1">subscript</csymbol><apply id="S2.SS2.p2.22.m20.1.1.1.2.cmml" xref="S2.SS2.p2.22.m20.1.1.1.1"><csymbol cd="latexml" id="S2.SS2.p2.22.m20.1.1.1.2.1.cmml" xref="S2.SS2.p2.22.m20.1.1.1.1.2">norm</csymbol><apply id="S2.SS2.p2.22.m20.1.1.1.1.1.cmml" xref="S2.SS2.p2.22.m20.1.1.1.1.1"><minus id="S2.SS2.p2.22.m20.1.1.1.1.1.1.cmml" xref="S2.SS2.p2.22.m20.1.1.1.1.1.1"></minus><apply id="S2.SS2.p2.22.m20.1.1.1.1.1.2.cmml" xref="S2.SS2.p2.22.m20.1.1.1.1.1.2"><ci id="S2.SS2.p2.22.m20.1.1.1.1.1.2.1.cmml" xref="S2.SS2.p2.22.m20.1.1.1.1.1.2.1">^</ci><ci id="S2.SS2.p2.22.m20.1.1.1.1.1.2.2.cmml" xref="S2.SS2.p2.22.m20.1.1.1.1.1.2.2">𝚺</ci></apply><apply id="S2.SS2.p2.22.m20.1.1.1.1.1.3.cmml" xref="S2.SS2.p2.22.m20.1.1.1.1.1.3"><csymbol cd="latexml" id="S2.SS2.p2.22.m20.1.1.1.1.1.3.1.cmml" xref="S2.SS2.p2.22.m20.1.1.1.1.1.3.1">tensor-product</csymbol><apply id="S2.SS2.p2.22.m20.1.1.1.1.1.3.2.cmml" xref="S2.SS2.p2.22.m20.1.1.1.1.1.3.2"><csymbol cd="ambiguous" id="S2.SS2.p2.22.m20.1.1.1.1.1.3.2.1.cmml" xref="S2.SS2.p2.22.m20.1.1.1.1.1.3.2">subscript</csymbol><ci id="S2.SS2.p2.22.m20.1.1.1.1.1.3.2.2.cmml" xref="S2.SS2.p2.22.m20.1.1.1.1.1.3.2.2">𝚺</ci><cn id="S2.SS2.p2.22.m20.1.1.1.1.1.3.2.3.cmml" type="integer" xref="S2.SS2.p2.22.m20.1.1.1.1.1.3.2.3">2</cn></apply><apply id="S2.SS2.p2.22.m20.1.1.1.1.1.3.3.cmml" xref="S2.SS2.p2.22.m20.1.1.1.1.1.3.3"><csymbol cd="ambiguous" id="S2.SS2.p2.22.m20.1.1.1.1.1.3.3.1.cmml" xref="S2.SS2.p2.22.m20.1.1.1.1.1.3.3">subscript</csymbol><ci id="S2.SS2.p2.22.m20.1.1.1.1.1.3.3.2.cmml" xref="S2.SS2.p2.22.m20.1.1.1.1.1.3.3.2">𝚺</ci><cn id="S2.SS2.p2.22.m20.1.1.1.1.1.3.3.3.cmml" type="integer" xref="S2.SS2.p2.22.m20.1.1.1.1.1.3.3.3">1</cn></apply></apply></apply></apply><ci id="S2.SS2.p2.22.m20.1.1.3.cmml" xref="S2.SS2.p2.22.m20.1.1.3">𝐹</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p2.22.m20.1c">\|\hat{\operatorname{{\mathbf{\Sigma}}}}-\operatorname{{\mathbf{\Sigma}}}_{2}% \otimes\operatorname{{\mathbf{\Sigma}}}_{1}\|_{F}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p2.22.m20.1d">∥ over^ start_ARG bold_Σ end_ARG - bold_Σ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ⊗ bold_Σ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ∥ start_POSTSUBSCRIPT italic_F end_POSTSUBSCRIPT</annotation></semantics></math> and are obtained using the singular value decomposition (SVD) of a matrix formed by reordering <math alttext="\hat{\operatorname{{\mathbf{\Sigma}}}}" class="ltx_Math" display="inline" id="S2.SS2.p2.23.m21.1"><semantics id="S2.SS2.p2.23.m21.1a"><mover accent="true" id="S2.SS2.p2.23.m21.1.1" xref="S2.SS2.p2.23.m21.1.1.cmml"><mi id="S2.SS2.p2.23.m21.1.1.2" xref="S2.SS2.p2.23.m21.1.1.2.cmml">𝚺</mi><mo id="S2.SS2.p2.23.m21.1.1.1" xref="S2.SS2.p2.23.m21.1.1.1.cmml">^</mo></mover><annotation-xml encoding="MathML-Content" id="S2.SS2.p2.23.m21.1b"><apply id="S2.SS2.p2.23.m21.1.1.cmml" xref="S2.SS2.p2.23.m21.1.1"><ci id="S2.SS2.p2.23.m21.1.1.1.cmml" xref="S2.SS2.p2.23.m21.1.1.1">^</ci><ci id="S2.SS2.p2.23.m21.1.1.2.cmml" xref="S2.SS2.p2.23.m21.1.1.2">𝚺</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p2.23.m21.1c">\hat{\operatorname{{\mathbf{\Sigma}}}}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p2.23.m21.1d">over^ start_ARG bold_Σ end_ARG</annotation></semantics></math> entries <cite class="ltx_cite ltx_citemacro_citep">(Van Loan, <a class="ltx_ref" href="https://arxiv.org/html/2503.13782v1#bib.bib32" title="">2000</a>)</cite>. The complexity of computing <math alttext="\hat{\operatorname{{\mathbf{\Sigma}}}}_{1}" class="ltx_Math" display="inline" id="S2.SS2.p2.24.m22.1"><semantics id="S2.SS2.p2.24.m22.1a"><msub id="S2.SS2.p2.24.m22.1.1" xref="S2.SS2.p2.24.m22.1.1.cmml"><mover accent="true" id="S2.SS2.p2.24.m22.1.1.2" xref="S2.SS2.p2.24.m22.1.1.2.cmml"><mi id="S2.SS2.p2.24.m22.1.1.2.2" xref="S2.SS2.p2.24.m22.1.1.2.2.cmml">𝚺</mi><mo id="S2.SS2.p2.24.m22.1.1.2.1" xref="S2.SS2.p2.24.m22.1.1.2.1.cmml">^</mo></mover><mn id="S2.SS2.p2.24.m22.1.1.3" xref="S2.SS2.p2.24.m22.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S2.SS2.p2.24.m22.1b"><apply id="S2.SS2.p2.24.m22.1.1.cmml" xref="S2.SS2.p2.24.m22.1.1"><csymbol cd="ambiguous" id="S2.SS2.p2.24.m22.1.1.1.cmml" xref="S2.SS2.p2.24.m22.1.1">subscript</csymbol><apply id="S2.SS2.p2.24.m22.1.1.2.cmml" xref="S2.SS2.p2.24.m22.1.1.2"><ci id="S2.SS2.p2.24.m22.1.1.2.1.cmml" xref="S2.SS2.p2.24.m22.1.1.2.1">^</ci><ci id="S2.SS2.p2.24.m22.1.1.2.2.cmml" xref="S2.SS2.p2.24.m22.1.1.2.2">𝚺</ci></apply><cn id="S2.SS2.p2.24.m22.1.1.3.cmml" type="integer" xref="S2.SS2.p2.24.m22.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p2.24.m22.1c">\hat{\operatorname{{\mathbf{\Sigma}}}}_{1}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p2.24.m22.1d">over^ start_ARG bold_Σ end_ARG start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="\hat{\operatorname{{\mathbf{\Sigma}}}}_{2}" class="ltx_Math" display="inline" id="S2.SS2.p2.25.m23.1"><semantics id="S2.SS2.p2.25.m23.1a"><msub id="S2.SS2.p2.25.m23.1.1" xref="S2.SS2.p2.25.m23.1.1.cmml"><mover accent="true" id="S2.SS2.p2.25.m23.1.1.2" xref="S2.SS2.p2.25.m23.1.1.2.cmml"><mi id="S2.SS2.p2.25.m23.1.1.2.2" xref="S2.SS2.p2.25.m23.1.1.2.2.cmml">𝚺</mi><mo id="S2.SS2.p2.25.m23.1.1.2.1" xref="S2.SS2.p2.25.m23.1.1.2.1.cmml">^</mo></mover><mn id="S2.SS2.p2.25.m23.1.1.3" xref="S2.SS2.p2.25.m23.1.1.3.cmml">2</mn></msub><annotation-xml encoding="MathML-Content" id="S2.SS2.p2.25.m23.1b"><apply id="S2.SS2.p2.25.m23.1.1.cmml" xref="S2.SS2.p2.25.m23.1.1"><csymbol cd="ambiguous" id="S2.SS2.p2.25.m23.1.1.1.cmml" xref="S2.SS2.p2.25.m23.1.1">subscript</csymbol><apply id="S2.SS2.p2.25.m23.1.1.2.cmml" xref="S2.SS2.p2.25.m23.1.1.2"><ci id="S2.SS2.p2.25.m23.1.1.2.1.cmml" xref="S2.SS2.p2.25.m23.1.1.2.1">^</ci><ci id="S2.SS2.p2.25.m23.1.1.2.2.cmml" xref="S2.SS2.p2.25.m23.1.1.2.2">𝚺</ci></apply><cn id="S2.SS2.p2.25.m23.1.1.3.cmml" type="integer" xref="S2.SS2.p2.25.m23.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p2.25.m23.1c">\hat{\operatorname{{\mathbf{\Sigma}}}}_{2}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p2.25.m23.1d">over^ start_ARG bold_Σ end_ARG start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math> using this method scales as <math alttext="O(Q_{1}^{3}Q_{2}^{3})" class="ltx_Math" display="inline" id="S2.SS2.p2.26.m24.1"><semantics id="S2.SS2.p2.26.m24.1a"><mrow id="S2.SS2.p2.26.m24.1.1" xref="S2.SS2.p2.26.m24.1.1.cmml"><mi id="S2.SS2.p2.26.m24.1.1.3" xref="S2.SS2.p2.26.m24.1.1.3.cmml">O</mi><mo id="S2.SS2.p2.26.m24.1.1.2" xref="S2.SS2.p2.26.m24.1.1.2.cmml">⁢</mo><mrow id="S2.SS2.p2.26.m24.1.1.1.1" xref="S2.SS2.p2.26.m24.1.1.1.1.1.cmml"><mo id="S2.SS2.p2.26.m24.1.1.1.1.2" stretchy="false" xref="S2.SS2.p2.26.m24.1.1.1.1.1.cmml">(</mo><mrow id="S2.SS2.p2.26.m24.1.1.1.1.1" xref="S2.SS2.p2.26.m24.1.1.1.1.1.cmml"><msubsup id="S2.SS2.p2.26.m24.1.1.1.1.1.2" xref="S2.SS2.p2.26.m24.1.1.1.1.1.2.cmml"><mi id="S2.SS2.p2.26.m24.1.1.1.1.1.2.2.2" xref="S2.SS2.p2.26.m24.1.1.1.1.1.2.2.2.cmml">Q</mi><mn id="S2.SS2.p2.26.m24.1.1.1.1.1.2.2.3" xref="S2.SS2.p2.26.m24.1.1.1.1.1.2.2.3.cmml">1</mn><mn id="S2.SS2.p2.26.m24.1.1.1.1.1.2.3" xref="S2.SS2.p2.26.m24.1.1.1.1.1.2.3.cmml">3</mn></msubsup><mo id="S2.SS2.p2.26.m24.1.1.1.1.1.1" 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xref="S2.SS2.p2.26.m24.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S2.SS2.p2.26.m24.1.1.1.1.1.3.2.1.cmml" xref="S2.SS2.p2.26.m24.1.1.1.1.1.3">subscript</csymbol><ci id="S2.SS2.p2.26.m24.1.1.1.1.1.3.2.2.cmml" xref="S2.SS2.p2.26.m24.1.1.1.1.1.3.2.2">𝑄</ci><cn id="S2.SS2.p2.26.m24.1.1.1.1.1.3.2.3.cmml" type="integer" xref="S2.SS2.p2.26.m24.1.1.1.1.1.3.2.3">2</cn></apply><cn id="S2.SS2.p2.26.m24.1.1.1.1.1.3.3.cmml" type="integer" xref="S2.SS2.p2.26.m24.1.1.1.1.1.3.3">3</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p2.26.m24.1c">O(Q_{1}^{3}Q_{2}^{3})</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p2.26.m24.1d">italic_O ( italic_Q start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT italic_Q start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT )</annotation></semantics></math>, which leads to inefficiency in practice even for small <math alttext="Q_{1}" class="ltx_Math" display="inline" id="S2.SS2.p2.27.m25.1"><semantics id="S2.SS2.p2.27.m25.1a"><msub id="S2.SS2.p2.27.m25.1.1" xref="S2.SS2.p2.27.m25.1.1.cmml"><mi id="S2.SS2.p2.27.m25.1.1.2" xref="S2.SS2.p2.27.m25.1.1.2.cmml">Q</mi><mn id="S2.SS2.p2.27.m25.1.1.3" xref="S2.SS2.p2.27.m25.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S2.SS2.p2.27.m25.1b"><apply id="S2.SS2.p2.27.m25.1.1.cmml" xref="S2.SS2.p2.27.m25.1.1"><csymbol cd="ambiguous" id="S2.SS2.p2.27.m25.1.1.1.cmml" xref="S2.SS2.p2.27.m25.1.1">subscript</csymbol><ci id="S2.SS2.p2.27.m25.1.1.2.cmml" xref="S2.SS2.p2.27.m25.1.1.2">𝑄</ci><cn id="S2.SS2.p2.27.m25.1.1.3.cmml" type="integer" xref="S2.SS2.p2.27.m25.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p2.27.m25.1c">Q_{1}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p2.27.m25.1d">italic_Q start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="Q_{2}" class="ltx_Math" display="inline" id="S2.SS2.p2.28.m26.1"><semantics id="S2.SS2.p2.28.m26.1a"><msub id="S2.SS2.p2.28.m26.1.1" xref="S2.SS2.p2.28.m26.1.1.cmml"><mi id="S2.SS2.p2.28.m26.1.1.2" xref="S2.SS2.p2.28.m26.1.1.2.cmml">Q</mi><mn id="S2.SS2.p2.28.m26.1.1.3" xref="S2.SS2.p2.28.m26.1.1.3.cmml">2</mn></msub><annotation-xml encoding="MathML-Content" id="S2.SS2.p2.28.m26.1b"><apply id="S2.SS2.p2.28.m26.1.1.cmml" xref="S2.SS2.p2.28.m26.1.1"><csymbol cd="ambiguous" id="S2.SS2.p2.28.m26.1.1.1.cmml" xref="S2.SS2.p2.28.m26.1.1">subscript</csymbol><ci id="S2.SS2.p2.28.m26.1.1.2.cmml" xref="S2.SS2.p2.28.m26.1.1.2">𝑄</ci><cn id="S2.SS2.p2.28.m26.1.1.3.cmml" type="integer" xref="S2.SS2.p2.28.m26.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p2.28.m26.1c">Q_{2}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p2.28.m26.1d">italic_Q start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math>. Furthermore, high-dimensional extensions of mixed models face similar issues.</p> </div> <div class="ltx_para" id="S2.SS2.p3"> <p class="ltx_p" id="S2.SS2.p3.7">The quasi-likelihood approaches bypass these problems by replacing <math alttext="\operatorname{{\mathbf{\Sigma}}}_{1}" class="ltx_Math" display="inline" id="S2.SS2.p3.1.m1.1"><semantics id="S2.SS2.p3.1.m1.1a"><msub id="S2.SS2.p3.1.m1.1.1" xref="S2.SS2.p3.1.m1.1.1.cmml"><mi id="S2.SS2.p3.1.m1.1.1.2" xref="S2.SS2.p3.1.m1.1.1.2.cmml">𝚺</mi><mn id="S2.SS2.p3.1.m1.1.1.3" xref="S2.SS2.p3.1.m1.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S2.SS2.p3.1.m1.1b"><apply id="S2.SS2.p3.1.m1.1.1.cmml" xref="S2.SS2.p3.1.m1.1.1"><csymbol cd="ambiguous" id="S2.SS2.p3.1.m1.1.1.1.cmml" xref="S2.SS2.p3.1.m1.1.1">subscript</csymbol><ci id="S2.SS2.p3.1.m1.1.1.2.cmml" xref="S2.SS2.p3.1.m1.1.1.2">𝚺</ci><cn id="S2.SS2.p3.1.m1.1.1.3.cmml" type="integer" xref="S2.SS2.p3.1.m1.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p3.1.m1.1c">\operatorname{{\mathbf{\Sigma}}}_{1}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p3.1.m1.1d">bold_Σ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="\operatorname{{\mathbf{\Sigma}}}_{2}" class="ltx_Math" display="inline" id="S2.SS2.p3.2.m2.1"><semantics id="S2.SS2.p3.2.m2.1a"><msub id="S2.SS2.p3.2.m2.1.1" xref="S2.SS2.p3.2.m2.1.1.cmml"><mi id="S2.SS2.p3.2.m2.1.1.2" xref="S2.SS2.p3.2.m2.1.1.2.cmml">𝚺</mi><mn id="S2.SS2.p3.2.m2.1.1.3" xref="S2.SS2.p3.2.m2.1.1.3.cmml">2</mn></msub><annotation-xml encoding="MathML-Content" id="S2.SS2.p3.2.m2.1b"><apply id="S2.SS2.p3.2.m2.1.1.cmml" xref="S2.SS2.p3.2.m2.1.1"><csymbol cd="ambiguous" id="S2.SS2.p3.2.m2.1.1.1.cmml" xref="S2.SS2.p3.2.m2.1.1">subscript</csymbol><ci id="S2.SS2.p3.2.m2.1.1.2.cmml" xref="S2.SS2.p3.2.m2.1.1.2">𝚺</ci><cn id="S2.SS2.p3.2.m2.1.1.3.cmml" type="integer" xref="S2.SS2.p3.2.m2.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p3.2.m2.1c">\operatorname{{\mathbf{\Sigma}}}_{2}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p3.2.m2.1d">bold_Σ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math> with proxy matrices that have certain asymptotic properties. The resulting loss for the estimation of <math alttext="\operatorname{\mathbf{b}}" class="ltx_Math" display="inline" id="S2.SS2.p3.3.m3.1"><semantics id="S2.SS2.p3.3.m3.1a"><mi id="S2.SS2.p3.3.m3.1.1" xref="S2.SS2.p3.3.m3.1.1.cmml">𝐛</mi><annotation-xml encoding="MathML-Content" id="S2.SS2.p3.3.m3.1b"><ci id="S2.SS2.p3.3.m3.1.1.cmml" xref="S2.SS2.p3.3.m3.1.1">𝐛</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p3.3.m3.1c">\operatorname{\mathbf{b}}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p3.3.m3.1d">bold_b</annotation></semantics></math> is equivalent to that of weighted least squares regression. The penalized <math alttext="\operatorname{\mathbf{b}}" class="ltx_Math" display="inline" id="S2.SS2.p3.4.m4.1"><semantics id="S2.SS2.p3.4.m4.1a"><mi id="S2.SS2.p3.4.m4.1.1" xref="S2.SS2.p3.4.m4.1.1.cmml">𝐛</mi><annotation-xml encoding="MathML-Content" id="S2.SS2.p3.4.m4.1b"><ci id="S2.SS2.p3.4.m4.1.1.cmml" xref="S2.SS2.p3.4.m4.1.1">𝐛</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p3.4.m4.1c">\operatorname{\mathbf{b}}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p3.4.m4.1d">bold_b</annotation></semantics></math> estimate is obtained using a lasso-type penalty on <math alttext="\operatorname{\mathbf{b}}" class="ltx_Math" display="inline" id="S2.SS2.p3.5.m5.1"><semantics id="S2.SS2.p3.5.m5.1a"><mi id="S2.SS2.p3.5.m5.1.1" xref="S2.SS2.p3.5.m5.1.1.cmml">𝐛</mi><annotation-xml encoding="MathML-Content" id="S2.SS2.p3.5.m5.1b"><ci id="S2.SS2.p3.5.m5.1.1.cmml" xref="S2.SS2.p3.5.m5.1.1">𝐛</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p3.5.m5.1c">\operatorname{\mathbf{b}}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p3.5.m5.1d">bold_b</annotation></semantics></math> that has optimal theoretical properties depending on the choice of proxy matrices and tuning parameter <cite class="ltx_cite ltx_citemacro_citep">(Fan and Li, <a class="ltx_ref" href="https://arxiv.org/html/2503.13782v1#bib.bib8" title="">2012</a>; Li et al., <a class="ltx_ref" href="https://arxiv.org/html/2503.13782v1#bib.bib21" title="">2022</a>)</cite>. The estimation of <math alttext="\operatorname{{\mathbf{\Sigma}}}" class="ltx_Math" display="inline" id="S2.SS2.p3.6.m6.1"><semantics id="S2.SS2.p3.6.m6.1a"><mi id="S2.SS2.p3.6.m6.1.1" xref="S2.SS2.p3.6.m6.1.1.cmml">𝚺</mi><annotation-xml encoding="MathML-Content" id="S2.SS2.p3.6.m6.1b"><ci id="S2.SS2.p3.6.m6.1.1.cmml" xref="S2.SS2.p3.6.m6.1.1">𝚺</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p3.6.m6.1c">\operatorname{{\mathbf{\Sigma}}}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p3.6.m6.1d">bold_Σ</annotation></semantics></math> is still challenging in that it requires that <math alttext="Q_{1}Q_{2}<\min_{i}m_{i}" class="ltx_Math" display="inline" id="S2.SS2.p3.7.m7.1"><semantics id="S2.SS2.p3.7.m7.1a"><mrow id="S2.SS2.p3.7.m7.1.1" xref="S2.SS2.p3.7.m7.1.1.cmml"><mrow id="S2.SS2.p3.7.m7.1.1.2" xref="S2.SS2.p3.7.m7.1.1.2.cmml"><msub id="S2.SS2.p3.7.m7.1.1.2.2" xref="S2.SS2.p3.7.m7.1.1.2.2.cmml"><mi id="S2.SS2.p3.7.m7.1.1.2.2.2" xref="S2.SS2.p3.7.m7.1.1.2.2.2.cmml">Q</mi><mn id="S2.SS2.p3.7.m7.1.1.2.2.3" xref="S2.SS2.p3.7.m7.1.1.2.2.3.cmml">1</mn></msub><mo id="S2.SS2.p3.7.m7.1.1.2.1" xref="S2.SS2.p3.7.m7.1.1.2.1.cmml">⁢</mo><msub id="S2.SS2.p3.7.m7.1.1.2.3" xref="S2.SS2.p3.7.m7.1.1.2.3.cmml"><mi id="S2.SS2.p3.7.m7.1.1.2.3.2" xref="S2.SS2.p3.7.m7.1.1.2.3.2.cmml">Q</mi><mn id="S2.SS2.p3.7.m7.1.1.2.3.3" xref="S2.SS2.p3.7.m7.1.1.2.3.3.cmml">2</mn></msub></mrow><mo id="S2.SS2.p3.7.m7.1.1.1" xref="S2.SS2.p3.7.m7.1.1.1.cmml"><</mo><mrow id="S2.SS2.p3.7.m7.1.1.3" xref="S2.SS2.p3.7.m7.1.1.3.cmml"><msub id="S2.SS2.p3.7.m7.1.1.3.1" xref="S2.SS2.p3.7.m7.1.1.3.1.cmml"><mi id="S2.SS2.p3.7.m7.1.1.3.1.2" xref="S2.SS2.p3.7.m7.1.1.3.1.2.cmml">min</mi><mi id="S2.SS2.p3.7.m7.1.1.3.1.3" xref="S2.SS2.p3.7.m7.1.1.3.1.3.cmml">i</mi></msub><mo id="S2.SS2.p3.7.m7.1.1.3a" lspace="0.167em" xref="S2.SS2.p3.7.m7.1.1.3.cmml">⁡</mo><msub id="S2.SS2.p3.7.m7.1.1.3.2" xref="S2.SS2.p3.7.m7.1.1.3.2.cmml"><mi id="S2.SS2.p3.7.m7.1.1.3.2.2" xref="S2.SS2.p3.7.m7.1.1.3.2.2.cmml">m</mi><mi id="S2.SS2.p3.7.m7.1.1.3.2.3" xref="S2.SS2.p3.7.m7.1.1.3.2.3.cmml">i</mi></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p3.7.m7.1b"><apply id="S2.SS2.p3.7.m7.1.1.cmml" xref="S2.SS2.p3.7.m7.1.1"><lt id="S2.SS2.p3.7.m7.1.1.1.cmml" xref="S2.SS2.p3.7.m7.1.1.1"></lt><apply id="S2.SS2.p3.7.m7.1.1.2.cmml" xref="S2.SS2.p3.7.m7.1.1.2"><times id="S2.SS2.p3.7.m7.1.1.2.1.cmml" xref="S2.SS2.p3.7.m7.1.1.2.1"></times><apply id="S2.SS2.p3.7.m7.1.1.2.2.cmml" xref="S2.SS2.p3.7.m7.1.1.2.2"><csymbol cd="ambiguous" id="S2.SS2.p3.7.m7.1.1.2.2.1.cmml" xref="S2.SS2.p3.7.m7.1.1.2.2">subscript</csymbol><ci id="S2.SS2.p3.7.m7.1.1.2.2.2.cmml" xref="S2.SS2.p3.7.m7.1.1.2.2.2">𝑄</ci><cn id="S2.SS2.p3.7.m7.1.1.2.2.3.cmml" type="integer" xref="S2.SS2.p3.7.m7.1.1.2.2.3">1</cn></apply><apply id="S2.SS2.p3.7.m7.1.1.2.3.cmml" xref="S2.SS2.p3.7.m7.1.1.2.3"><csymbol cd="ambiguous" id="S2.SS2.p3.7.m7.1.1.2.3.1.cmml" xref="S2.SS2.p3.7.m7.1.1.2.3">subscript</csymbol><ci id="S2.SS2.p3.7.m7.1.1.2.3.2.cmml" xref="S2.SS2.p3.7.m7.1.1.2.3.2">𝑄</ci><cn id="S2.SS2.p3.7.m7.1.1.2.3.3.cmml" type="integer" xref="S2.SS2.p3.7.m7.1.1.2.3.3">2</cn></apply></apply><apply id="S2.SS2.p3.7.m7.1.1.3.cmml" xref="S2.SS2.p3.7.m7.1.1.3"><apply id="S2.SS2.p3.7.m7.1.1.3.1.cmml" xref="S2.SS2.p3.7.m7.1.1.3.1"><csymbol cd="ambiguous" id="S2.SS2.p3.7.m7.1.1.3.1.1.cmml" xref="S2.SS2.p3.7.m7.1.1.3.1">subscript</csymbol><min id="S2.SS2.p3.7.m7.1.1.3.1.2.cmml" xref="S2.SS2.p3.7.m7.1.1.3.1.2"></min><ci id="S2.SS2.p3.7.m7.1.1.3.1.3.cmml" xref="S2.SS2.p3.7.m7.1.1.3.1.3">𝑖</ci></apply><apply id="S2.SS2.p3.7.m7.1.1.3.2.cmml" xref="S2.SS2.p3.7.m7.1.1.3.2"><csymbol cd="ambiguous" id="S2.SS2.p3.7.m7.1.1.3.2.1.cmml" xref="S2.SS2.p3.7.m7.1.1.3.2">subscript</csymbol><ci id="S2.SS2.p3.7.m7.1.1.3.2.2.cmml" xref="S2.SS2.p3.7.m7.1.1.3.2.2">𝑚</ci><ci id="S2.SS2.p3.7.m7.1.1.3.2.3.cmml" xref="S2.SS2.p3.7.m7.1.1.3.2.3">𝑖</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p3.7.m7.1c">Q_{1}Q_{2}<\min_{i}m_{i}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p3.7.m7.1d">italic_Q start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_Q start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT < roman_min start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_m start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math>, limiting their practical applications <cite class="ltx_cite ltx_citemacro_citep">(Li et al., <a class="ltx_ref" href="https://arxiv.org/html/2503.13782v1#bib.bib21" title="">2022</a>)</cite>.</p> </div> <div class="ltx_para" id="S2.SS2.p4"> <p class="ltx_p" id="S2.SS2.p4.9">Due to these reasons, the structured mixed model for MMTR has two forms that enable estimation of <math alttext="\operatorname{\mathbf{L}}_{1}" 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 id="S2.SS2.p4.1.m1.1.1.2" xref="S2.SS2.p4.1.m1.1.1.2.cmml">𝐋</mi><mn id="S2.SS2.p4.1.m1.1.1.3" xref="S2.SS2.p4.1.m1.1.1.3.cmml">1</mn></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">𝐋</ci><cn id="S2.SS2.p4.1.m1.1.1.3.cmml" type="integer" xref="S2.SS2.p4.1.m1.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p4.1.m1.1c">\operatorname{\mathbf{L}}_{1}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p4.1.m1.1d">bold_L start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> given <math alttext="\operatorname{\mathbf{L}}_{2}" 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 id="S2.SS2.p4.2.m2.1.1.2" xref="S2.SS2.p4.2.m2.1.1.2.cmml">𝐋</mi><mn id="S2.SS2.p4.2.m2.1.1.3" xref="S2.SS2.p4.2.m2.1.1.3.cmml">2</mn></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">𝐋</ci><cn id="S2.SS2.p4.2.m2.1.1.3.cmml" type="integer" xref="S2.SS2.p4.2.m2.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p4.2.m2.1c">\operatorname{\mathbf{L}}_{2}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p4.2.m2.1d">bold_L start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math> and vice versa. These two models facilitate the estimation of <math alttext="\operatorname{{\mathbf{\Sigma}}}_{1}" class="ltx_Math" display="inline" id="S2.SS2.p4.3.m3.1"><semantics id="S2.SS2.p4.3.m3.1a"><msub id="S2.SS2.p4.3.m3.1.1" xref="S2.SS2.p4.3.m3.1.1.cmml"><mi id="S2.SS2.p4.3.m3.1.1.2" xref="S2.SS2.p4.3.m3.1.1.2.cmml">𝚺</mi><mn id="S2.SS2.p4.3.m3.1.1.3" xref="S2.SS2.p4.3.m3.1.1.3.cmml">1</mn></msub><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"><csymbol cd="ambiguous" id="S2.SS2.p4.3.m3.1.1.1.cmml" xref="S2.SS2.p4.3.m3.1.1">subscript</csymbol><ci id="S2.SS2.p4.3.m3.1.1.2.cmml" xref="S2.SS2.p4.3.m3.1.1.2">𝚺</ci><cn id="S2.SS2.p4.3.m3.1.1.3.cmml" type="integer" xref="S2.SS2.p4.3.m3.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p4.3.m3.1c">\operatorname{{\mathbf{\Sigma}}}_{1}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p4.3.m3.1d">bold_Σ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="\operatorname{{\mathbf{\Sigma}}}_{2}" class="ltx_Math" display="inline" id="S2.SS2.p4.4.m4.1"><semantics id="S2.SS2.p4.4.m4.1a"><msub id="S2.SS2.p4.4.m4.1.1" xref="S2.SS2.p4.4.m4.1.1.cmml"><mi id="S2.SS2.p4.4.m4.1.1.2" xref="S2.SS2.p4.4.m4.1.1.2.cmml">𝚺</mi><mn id="S2.SS2.p4.4.m4.1.1.3" xref="S2.SS2.p4.4.m4.1.1.3.cmml">2</mn></msub><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"><csymbol cd="ambiguous" id="S2.SS2.p4.4.m4.1.1.1.cmml" xref="S2.SS2.p4.4.m4.1.1">subscript</csymbol><ci id="S2.SS2.p4.4.m4.1.1.2.cmml" xref="S2.SS2.p4.4.m4.1.1.2">𝚺</ci><cn id="S2.SS2.p4.4.m4.1.1.3.cmml" type="integer" xref="S2.SS2.p4.4.m4.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p4.4.m4.1c">\operatorname{{\mathbf{\Sigma}}}_{2}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p4.4.m4.1d">bold_Σ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math> without any previous restrictions. For <math alttext="i=1,\ldots,n" class="ltx_Math" display="inline" id="S2.SS2.p4.5.m5.3"><semantics id="S2.SS2.p4.5.m5.3a"><mrow id="S2.SS2.p4.5.m5.3.4" xref="S2.SS2.p4.5.m5.3.4.cmml"><mi id="S2.SS2.p4.5.m5.3.4.2" xref="S2.SS2.p4.5.m5.3.4.2.cmml">i</mi><mo id="S2.SS2.p4.5.m5.3.4.1" xref="S2.SS2.p4.5.m5.3.4.1.cmml">=</mo><mrow id="S2.SS2.p4.5.m5.3.4.3.2" xref="S2.SS2.p4.5.m5.3.4.3.1.cmml"><mn id="S2.SS2.p4.5.m5.1.1" xref="S2.SS2.p4.5.m5.1.1.cmml">1</mn><mo id="S2.SS2.p4.5.m5.3.4.3.2.1" xref="S2.SS2.p4.5.m5.3.4.3.1.cmml">,</mo><mi id="S2.SS2.p4.5.m5.2.2" mathvariant="normal" xref="S2.SS2.p4.5.m5.2.2.cmml">…</mi><mo id="S2.SS2.p4.5.m5.3.4.3.2.2" xref="S2.SS2.p4.5.m5.3.4.3.1.cmml">,</mo><mi id="S2.SS2.p4.5.m5.3.3" xref="S2.SS2.p4.5.m5.3.3.cmml">n</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p4.5.m5.3b"><apply id="S2.SS2.p4.5.m5.3.4.cmml" xref="S2.SS2.p4.5.m5.3.4"><eq id="S2.SS2.p4.5.m5.3.4.1.cmml" xref="S2.SS2.p4.5.m5.3.4.1"></eq><ci id="S2.SS2.p4.5.m5.3.4.2.cmml" xref="S2.SS2.p4.5.m5.3.4.2">𝑖</ci><list id="S2.SS2.p4.5.m5.3.4.3.1.cmml" xref="S2.SS2.p4.5.m5.3.4.3.2"><cn id="S2.SS2.p4.5.m5.1.1.cmml" type="integer" xref="S2.SS2.p4.5.m5.1.1">1</cn><ci id="S2.SS2.p4.5.m5.2.2.cmml" xref="S2.SS2.p4.5.m5.2.2">…</ci><ci id="S2.SS2.p4.5.m5.3.3.cmml" xref="S2.SS2.p4.5.m5.3.3">𝑛</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p4.5.m5.3c">i=1,\ldots,n</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p4.5.m5.3d">italic_i = 1 , … , italic_n</annotation></semantics></math> and <math alttext="j=1,\ldots,m_{i}" class="ltx_Math" display="inline" id="S2.SS2.p4.6.m6.3"><semantics id="S2.SS2.p4.6.m6.3a"><mrow id="S2.SS2.p4.6.m6.3.3" xref="S2.SS2.p4.6.m6.3.3.cmml"><mi id="S2.SS2.p4.6.m6.3.3.3" xref="S2.SS2.p4.6.m6.3.3.3.cmml">j</mi><mo id="S2.SS2.p4.6.m6.3.3.2" xref="S2.SS2.p4.6.m6.3.3.2.cmml">=</mo><mrow id="S2.SS2.p4.6.m6.3.3.1.1" xref="S2.SS2.p4.6.m6.3.3.1.2.cmml"><mn id="S2.SS2.p4.6.m6.1.1" xref="S2.SS2.p4.6.m6.1.1.cmml">1</mn><mo id="S2.SS2.p4.6.m6.3.3.1.1.2" xref="S2.SS2.p4.6.m6.3.3.1.2.cmml">,</mo><mi id="S2.SS2.p4.6.m6.2.2" mathvariant="normal" xref="S2.SS2.p4.6.m6.2.2.cmml">…</mi><mo id="S2.SS2.p4.6.m6.3.3.1.1.3" xref="S2.SS2.p4.6.m6.3.3.1.2.cmml">,</mo><msub id="S2.SS2.p4.6.m6.3.3.1.1.1" xref="S2.SS2.p4.6.m6.3.3.1.1.1.cmml"><mi id="S2.SS2.p4.6.m6.3.3.1.1.1.2" xref="S2.SS2.p4.6.m6.3.3.1.1.1.2.cmml">m</mi><mi id="S2.SS2.p4.6.m6.3.3.1.1.1.3" xref="S2.SS2.p4.6.m6.3.3.1.1.1.3.cmml">i</mi></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p4.6.m6.3b"><apply id="S2.SS2.p4.6.m6.3.3.cmml" xref="S2.SS2.p4.6.m6.3.3"><eq id="S2.SS2.p4.6.m6.3.3.2.cmml" xref="S2.SS2.p4.6.m6.3.3.2"></eq><ci id="S2.SS2.p4.6.m6.3.3.3.cmml" xref="S2.SS2.p4.6.m6.3.3.3">𝑗</ci><list id="S2.SS2.p4.6.m6.3.3.1.2.cmml" xref="S2.SS2.p4.6.m6.3.3.1.1"><cn id="S2.SS2.p4.6.m6.1.1.cmml" type="integer" xref="S2.SS2.p4.6.m6.1.1">1</cn><ci id="S2.SS2.p4.6.m6.2.2.cmml" xref="S2.SS2.p4.6.m6.2.2">…</ci><apply id="S2.SS2.p4.6.m6.3.3.1.1.1.cmml" xref="S2.SS2.p4.6.m6.3.3.1.1.1"><csymbol cd="ambiguous" id="S2.SS2.p4.6.m6.3.3.1.1.1.1.cmml" xref="S2.SS2.p4.6.m6.3.3.1.1.1">subscript</csymbol><ci id="S2.SS2.p4.6.m6.3.3.1.1.1.2.cmml" xref="S2.SS2.p4.6.m6.3.3.1.1.1.2">𝑚</ci><ci id="S2.SS2.p4.6.m6.3.3.1.1.1.3.cmml" xref="S2.SS2.p4.6.m6.3.3.1.1.1.3">𝑖</ci></apply></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p4.6.m6.3c">j=1,\ldots,m_{i}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p4.6.m6.3d">italic_j = 1 , … , italic_m start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math>, <math alttext="\operatorname{\mathop{\mathrm{tr}}}(\operatorname{\mathbf{X}}_{ij}^{\top}% \operatorname{\mathbf{B}})=\operatorname{\mathbf{x}}^{\top}_{ij}\operatorname{% \mathbf{b}}" class="ltx_Math" display="inline" id="S2.SS2.p4.7.m7.2"><semantics id="S2.SS2.p4.7.m7.2a"><mrow id="S2.SS2.p4.7.m7.2.2" xref="S2.SS2.p4.7.m7.2.2.cmml"><mrow id="S2.SS2.p4.7.m7.2.2.1.1" xref="S2.SS2.p4.7.m7.2.2.1.2.cmml"><mi id="S2.SS2.p4.7.m7.1.1" xref="S2.SS2.p4.7.m7.1.1.cmml">tr</mi><mo id="S2.SS2.p4.7.m7.2.2.1.1a" xref="S2.SS2.p4.7.m7.2.2.1.2.cmml">⁡</mo><mrow id="S2.SS2.p4.7.m7.2.2.1.1.1" xref="S2.SS2.p4.7.m7.2.2.1.2.cmml"><mo id="S2.SS2.p4.7.m7.2.2.1.1.1.2" stretchy="false" xref="S2.SS2.p4.7.m7.2.2.1.2.cmml">(</mo><mrow id="S2.SS2.p4.7.m7.2.2.1.1.1.1" xref="S2.SS2.p4.7.m7.2.2.1.1.1.1.cmml"><msubsup id="S2.SS2.p4.7.m7.2.2.1.1.1.1.1" xref="S2.SS2.p4.7.m7.2.2.1.1.1.1.1.cmml"><mi id="S2.SS2.p4.7.m7.2.2.1.1.1.1.1.2.2" xref="S2.SS2.p4.7.m7.2.2.1.1.1.1.1.2.2.cmml">𝐗</mi><mrow id="S2.SS2.p4.7.m7.2.2.1.1.1.1.1.2.3" xref="S2.SS2.p4.7.m7.2.2.1.1.1.1.1.2.3.cmml"><mi id="S2.SS2.p4.7.m7.2.2.1.1.1.1.1.2.3.2" xref="S2.SS2.p4.7.m7.2.2.1.1.1.1.1.2.3.2.cmml">i</mi><mo id="S2.SS2.p4.7.m7.2.2.1.1.1.1.1.2.3.1" xref="S2.SS2.p4.7.m7.2.2.1.1.1.1.1.2.3.1.cmml">⁢</mo><mi id="S2.SS2.p4.7.m7.2.2.1.1.1.1.1.2.3.3" xref="S2.SS2.p4.7.m7.2.2.1.1.1.1.1.2.3.3.cmml">j</mi></mrow><mo id="S2.SS2.p4.7.m7.2.2.1.1.1.1.1.3" xref="S2.SS2.p4.7.m7.2.2.1.1.1.1.1.3.cmml">⊤</mo></msubsup><mo id="S2.SS2.p4.7.m7.2.2.1.1.1.1a" lspace="0.167em" xref="S2.SS2.p4.7.m7.2.2.1.1.1.1.cmml">⁡</mo><mi id="S2.SS2.p4.7.m7.2.2.1.1.1.1.2" xref="S2.SS2.p4.7.m7.2.2.1.1.1.1.2.cmml">𝐁</mi></mrow><mo id="S2.SS2.p4.7.m7.2.2.1.1.1.3" stretchy="false" xref="S2.SS2.p4.7.m7.2.2.1.2.cmml">)</mo></mrow></mrow><mo id="S2.SS2.p4.7.m7.2.2.2" xref="S2.SS2.p4.7.m7.2.2.2.cmml">=</mo><mrow id="S2.SS2.p4.7.m7.2.2.3" xref="S2.SS2.p4.7.m7.2.2.3.cmml"><msubsup id="S2.SS2.p4.7.m7.2.2.3.1" xref="S2.SS2.p4.7.m7.2.2.3.1.cmml"><mi id="S2.SS2.p4.7.m7.2.2.3.1.2.2" xref="S2.SS2.p4.7.m7.2.2.3.1.2.2.cmml">𝐱</mi><mrow id="S2.SS2.p4.7.m7.2.2.3.1.3" xref="S2.SS2.p4.7.m7.2.2.3.1.3.cmml"><mi id="S2.SS2.p4.7.m7.2.2.3.1.3.2" xref="S2.SS2.p4.7.m7.2.2.3.1.3.2.cmml">i</mi><mo id="S2.SS2.p4.7.m7.2.2.3.1.3.1" xref="S2.SS2.p4.7.m7.2.2.3.1.3.1.cmml">⁢</mo><mi id="S2.SS2.p4.7.m7.2.2.3.1.3.3" xref="S2.SS2.p4.7.m7.2.2.3.1.3.3.cmml">j</mi></mrow><mo id="S2.SS2.p4.7.m7.2.2.3.1.2.3" xref="S2.SS2.p4.7.m7.2.2.3.1.2.3.cmml">⊤</mo></msubsup><mo id="S2.SS2.p4.7.m7.2.2.3a" lspace="0.167em" xref="S2.SS2.p4.7.m7.2.2.3.cmml">⁡</mo><mi id="S2.SS2.p4.7.m7.2.2.3.2" xref="S2.SS2.p4.7.m7.2.2.3.2.cmml">𝐛</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p4.7.m7.2b"><apply id="S2.SS2.p4.7.m7.2.2.cmml" xref="S2.SS2.p4.7.m7.2.2"><eq id="S2.SS2.p4.7.m7.2.2.2.cmml" xref="S2.SS2.p4.7.m7.2.2.2"></eq><apply id="S2.SS2.p4.7.m7.2.2.1.2.cmml" xref="S2.SS2.p4.7.m7.2.2.1.1"><ci id="S2.SS2.p4.7.m7.1.1.cmml" xref="S2.SS2.p4.7.m7.1.1">tr</ci><apply id="S2.SS2.p4.7.m7.2.2.1.1.1.1.cmml" xref="S2.SS2.p4.7.m7.2.2.1.1.1.1"><apply id="S2.SS2.p4.7.m7.2.2.1.1.1.1.1.cmml" xref="S2.SS2.p4.7.m7.2.2.1.1.1.1.1"><csymbol cd="ambiguous" id="S2.SS2.p4.7.m7.2.2.1.1.1.1.1.1.cmml" xref="S2.SS2.p4.7.m7.2.2.1.1.1.1.1">superscript</csymbol><apply 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xref="S2.SS2.p4.7.m7.2.2.3.2">𝐛</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p4.7.m7.2c">\operatorname{\mathop{\mathrm{tr}}}(\operatorname{\mathbf{X}}_{ij}^{\top}% \operatorname{\mathbf{B}})=\operatorname{\mathbf{x}}^{\top}_{ij}\operatorname{% \mathbf{b}}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p4.7.m7.2d">roman_tr ( bold_X start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ⊤ end_POSTSUPERSCRIPT bold_B ) = bold_x start_POSTSUPERSCRIPT ⊤ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT bold_b</annotation></semantics></math>, <math alttext="\operatorname{\mathop{\mathrm{tr}}}(\operatorname{\mathbf{Z}}_{ij}^{\top}% \operatorname{\mathbf{A}}_{i})=\operatorname{\mathbf{z}}_{ij(1)}^{\top}% \operatorname{\mathbf{a}}_{i(1)}=\operatorname{\mathbf{z}}_{ij(1)}^{\top}% \operatorname{\mathbf{L}}_{(1)}\operatorname{\mathbf{c}}_{(1)}" class="ltx_Math" display="inline" 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id="S2.SS2.p4.8.m8.7c">\operatorname{\mathop{\mathrm{tr}}}(\operatorname{\mathbf{Z}}_{ij}^{\top}% \operatorname{\mathbf{A}}_{i})=\operatorname{\mathbf{z}}_{ij(1)}^{\top}% \operatorname{\mathbf{a}}_{i(1)}=\operatorname{\mathbf{z}}_{ij(1)}^{\top}% \operatorname{\mathbf{L}}_{(1)}\operatorname{\mathbf{c}}_{(1)}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p4.8.m8.7d">roman_tr ( bold_Z start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ⊤ end_POSTSUPERSCRIPT bold_A start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) = bold_z start_POSTSUBSCRIPT italic_i italic_j ( 1 ) end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ⊤ end_POSTSUPERSCRIPT bold_a start_POSTSUBSCRIPT italic_i ( 1 ) end_POSTSUBSCRIPT = bold_z start_POSTSUBSCRIPT italic_i italic_j ( 1 ) end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ⊤ end_POSTSUPERSCRIPT bold_L start_POSTSUBSCRIPT ( 1 ) end_POSTSUBSCRIPT bold_c start_POSTSUBSCRIPT ( 1 ) end_POSTSUBSCRIPT</annotation></semantics></math>, and <math 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type="integer" xref="S2.SS2.p4.9.m9.5.5.1.1">2</cn></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p4.9.m9.7c">\operatorname{\mathop{\mathrm{tr}}}(\operatorname{\mathbf{Z}}_{ij}% \operatorname{\mathbf{A}}_{i}^{\top})=\operatorname{\mathbf{z}}_{ij(2)}^{\top}% \operatorname{\mathbf{a}}_{i(2)}=\operatorname{\mathbf{z}}_{ij(2)}^{\top}% \operatorname{\mathbf{L}}_{(2)}\operatorname{\mathbf{c}}_{(2)}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p4.9.m9.7d">roman_tr ( bold_Z start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT bold_A start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ⊤ end_POSTSUPERSCRIPT ) = bold_z start_POSTSUBSCRIPT italic_i italic_j ( 2 ) end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ⊤ end_POSTSUPERSCRIPT bold_a start_POSTSUBSCRIPT italic_i ( 2 ) end_POSTSUBSCRIPT = bold_z start_POSTSUBSCRIPT italic_i italic_j ( 2 ) end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ⊤ end_POSTSUPERSCRIPT bold_L start_POSTSUBSCRIPT ( 2 ) end_POSTSUBSCRIPT bold_c start_POSTSUBSCRIPT ( 2 ) end_POSTSUBSCRIPT</annotation></semantics></math>. Substituting these identities in (<a class="ltx_ref" href="https://arxiv.org/html/2503.13782v1#S2.E3" title="In 2.1 Model ‣ 2 Mixed Model Trace Regression ‣ Regularized Parameter Estimation in Mixed Model Trace Regression"><span class="ltx_text ltx_ref_tag">3</span></a>) gives</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="S3.EGx5"> <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 y_{ij}" class="ltx_Math" display="inline" id="S2.E5.m1.1"><semantics id="S2.E5.m1.1a"><msub id="S2.E5.m1.1.1" xref="S2.E5.m1.1.1.cmml"><mi id="S2.E5.m1.1.1.2" xref="S2.E5.m1.1.1.2.cmml">y</mi><mrow id="S2.E5.m1.1.1.3" xref="S2.E5.m1.1.1.3.cmml"><mi id="S2.E5.m1.1.1.3.2" xref="S2.E5.m1.1.1.3.2.cmml">i</mi><mo id="S2.E5.m1.1.1.3.1" xref="S2.E5.m1.1.1.3.1.cmml">⁢</mo><mi id="S2.E5.m1.1.1.3.3" xref="S2.E5.m1.1.1.3.3.cmml">j</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="S2.E5.m1.1b"><apply id="S2.E5.m1.1.1.cmml" xref="S2.E5.m1.1.1"><csymbol cd="ambiguous" id="S2.E5.m1.1.1.1.cmml" xref="S2.E5.m1.1.1">subscript</csymbol><ci id="S2.E5.m1.1.1.2.cmml" xref="S2.E5.m1.1.1.2">𝑦</ci><apply id="S2.E5.m1.1.1.3.cmml" xref="S2.E5.m1.1.1.3"><times id="S2.E5.m1.1.1.3.1.cmml" xref="S2.E5.m1.1.1.3.1"></times><ci id="S2.E5.m1.1.1.3.2.cmml" xref="S2.E5.m1.1.1.3.2">𝑖</ci><ci id="S2.E5.m1.1.1.3.3.cmml" xref="S2.E5.m1.1.1.3.3">𝑗</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.E5.m1.1c">\displaystyle y_{ij}</annotation><annotation encoding="application/x-llamapun" id="S2.E5.m1.1d">italic_y start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle=\operatorname{\mathbf{x}}^{\top}_{ij}\operatorname{\mathbf{b}}+% \operatorname{\mathbf{z}}_{ij(1)}^{\top}\operatorname{\mathbf{L}}_{(1)}% \operatorname{\mathbf{c}}_{i(1)}+e_{ij}=\operatorname{\mathbf{x}}^{\top}_{ij}% \operatorname{\mathbf{b}}+\operatorname{\mathbf{z}}_{ij(2)}^{\top}% \operatorname{\mathbf{L}}_{(2)}\operatorname{\mathbf{c}}_{i(2)}+e_{ij},\quad e% _{ij}\sim N(0,\tau^{2})," class="ltx_Math" display="inline" id="S2.E5.m2.8"><semantics id="S2.E5.m2.8a"><mrow id="S2.E5.m2.8.8.1"><mrow id="S2.E5.m2.8.8.1.1.2" xref="S2.E5.m2.8.8.1.1.3.cmml"><mrow id="S2.E5.m2.8.8.1.1.1.1" xref="S2.E5.m2.8.8.1.1.1.1.cmml"><mi id="S2.E5.m2.8.8.1.1.1.1.2" xref="S2.E5.m2.8.8.1.1.1.1.2.cmml"></mi><mo id="S2.E5.m2.8.8.1.1.1.1.3" xref="S2.E5.m2.8.8.1.1.1.1.3.cmml">=</mo><mrow id="S2.E5.m2.8.8.1.1.1.1.4" xref="S2.E5.m2.8.8.1.1.1.1.4.cmml"><mrow id="S2.E5.m2.8.8.1.1.1.1.4.2" xref="S2.E5.m2.8.8.1.1.1.1.4.2.cmml"><msubsup id="S2.E5.m2.8.8.1.1.1.1.4.2.1" xref="S2.E5.m2.8.8.1.1.1.1.4.2.1.cmml"><mi id="S2.E5.m2.8.8.1.1.1.1.4.2.1.2.2" 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id="S2.E5.m2.8.8.1.1.2.2.3.3.cmml" xref="S2.E5.m2.8.8.1.1.2.2.3.3"><times id="S2.E5.m2.8.8.1.1.2.2.3.3.1.cmml" xref="S2.E5.m2.8.8.1.1.2.2.3.3.1"></times><ci id="S2.E5.m2.8.8.1.1.2.2.3.3.2.cmml" xref="S2.E5.m2.8.8.1.1.2.2.3.3.2">𝑖</ci><ci id="S2.E5.m2.8.8.1.1.2.2.3.3.3.cmml" xref="S2.E5.m2.8.8.1.1.2.2.3.3.3">𝑗</ci></apply></apply><apply id="S2.E5.m2.8.8.1.1.2.2.1.cmml" xref="S2.E5.m2.8.8.1.1.2.2.1"><times id="S2.E5.m2.8.8.1.1.2.2.1.2.cmml" xref="S2.E5.m2.8.8.1.1.2.2.1.2"></times><ci id="S2.E5.m2.8.8.1.1.2.2.1.3.cmml" xref="S2.E5.m2.8.8.1.1.2.2.1.3">𝑁</ci><interval closure="open" id="S2.E5.m2.8.8.1.1.2.2.1.1.2.cmml" xref="S2.E5.m2.8.8.1.1.2.2.1.1.1"><cn id="S2.E5.m2.7.7.cmml" type="integer" xref="S2.E5.m2.7.7">0</cn><apply id="S2.E5.m2.8.8.1.1.2.2.1.1.1.1.cmml" xref="S2.E5.m2.8.8.1.1.2.2.1.1.1.1"><csymbol cd="ambiguous" id="S2.E5.m2.8.8.1.1.2.2.1.1.1.1.1.cmml" xref="S2.E5.m2.8.8.1.1.2.2.1.1.1.1">superscript</csymbol><ci id="S2.E5.m2.8.8.1.1.2.2.1.1.1.1.2.cmml" xref="S2.E5.m2.8.8.1.1.2.2.1.1.1.1.2">𝜏</ci><cn id="S2.E5.m2.8.8.1.1.2.2.1.1.1.1.3.cmml" type="integer" xref="S2.E5.m2.8.8.1.1.2.2.1.1.1.1.3">2</cn></apply></interval></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.E5.m2.8c">\displaystyle=\operatorname{\mathbf{x}}^{\top}_{ij}\operatorname{\mathbf{b}}+% \operatorname{\mathbf{z}}_{ij(1)}^{\top}\operatorname{\mathbf{L}}_{(1)}% \operatorname{\mathbf{c}}_{i(1)}+e_{ij}=\operatorname{\mathbf{x}}^{\top}_{ij}% \operatorname{\mathbf{b}}+\operatorname{\mathbf{z}}_{ij(2)}^{\top}% \operatorname{\mathbf{L}}_{(2)}\operatorname{\mathbf{c}}_{i(2)}+e_{ij},\quad e% _{ij}\sim N(0,\tau^{2}),</annotation><annotation encoding="application/x-llamapun" id="S2.E5.m2.8d">= bold_x start_POSTSUPERSCRIPT ⊤ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT bold_b + bold_z start_POSTSUBSCRIPT italic_i italic_j ( 1 ) end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ⊤ end_POSTSUPERSCRIPT bold_L start_POSTSUBSCRIPT ( 1 ) end_POSTSUBSCRIPT bold_c start_POSTSUBSCRIPT italic_i ( 1 ) end_POSTSUBSCRIPT + italic_e start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT = bold_x start_POSTSUPERSCRIPT ⊤ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT bold_b + bold_z start_POSTSUBSCRIPT italic_i italic_j ( 2 ) end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ⊤ end_POSTSUPERSCRIPT bold_L start_POSTSUBSCRIPT ( 2 ) end_POSTSUBSCRIPT bold_c start_POSTSUBSCRIPT italic_i ( 2 ) end_POSTSUBSCRIPT + italic_e start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT , italic_e start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT ∼ italic_N ( 0 , italic_τ 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">(5)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S2.SS2.p4.13">where <math alttext="\operatorname{\mathbf{c}}_{i(1)}" class="ltx_Math" display="inline" id="S2.SS2.p4.10.m1.1"><semantics id="S2.SS2.p4.10.m1.1a"><msub id="S2.SS2.p4.10.m1.1.2" xref="S2.SS2.p4.10.m1.1.2.cmml"><mi id="S2.SS2.p4.10.m1.1.2.2" xref="S2.SS2.p4.10.m1.1.2.2.cmml">𝐜</mi><mrow id="S2.SS2.p4.10.m1.1.1.1" xref="S2.SS2.p4.10.m1.1.1.1.cmml"><mi id="S2.SS2.p4.10.m1.1.1.1.3" xref="S2.SS2.p4.10.m1.1.1.1.3.cmml">i</mi><mo id="S2.SS2.p4.10.m1.1.1.1.2" xref="S2.SS2.p4.10.m1.1.1.1.2.cmml">⁢</mo><mrow id="S2.SS2.p4.10.m1.1.1.1.4.2" xref="S2.SS2.p4.10.m1.1.1.1.cmml"><mo id="S2.SS2.p4.10.m1.1.1.1.4.2.1" stretchy="false" xref="S2.SS2.p4.10.m1.1.1.1.cmml">(</mo><mn id="S2.SS2.p4.10.m1.1.1.1.1" xref="S2.SS2.p4.10.m1.1.1.1.1.cmml">1</mn><mo id="S2.SS2.p4.10.m1.1.1.1.4.2.2" stretchy="false" xref="S2.SS2.p4.10.m1.1.1.1.cmml">)</mo></mrow></mrow></msub><annotation-xml encoding="MathML-Content" id="S2.SS2.p4.10.m1.1b"><apply id="S2.SS2.p4.10.m1.1.2.cmml" xref="S2.SS2.p4.10.m1.1.2"><csymbol cd="ambiguous" id="S2.SS2.p4.10.m1.1.2.1.cmml" xref="S2.SS2.p4.10.m1.1.2">subscript</csymbol><ci id="S2.SS2.p4.10.m1.1.2.2.cmml" xref="S2.SS2.p4.10.m1.1.2.2">𝐜</ci><apply id="S2.SS2.p4.10.m1.1.1.1.cmml" xref="S2.SS2.p4.10.m1.1.1.1"><times id="S2.SS2.p4.10.m1.1.1.1.2.cmml" xref="S2.SS2.p4.10.m1.1.1.1.2"></times><ci id="S2.SS2.p4.10.m1.1.1.1.3.cmml" xref="S2.SS2.p4.10.m1.1.1.1.3">𝑖</ci><cn id="S2.SS2.p4.10.m1.1.1.1.1.cmml" type="integer" xref="S2.SS2.p4.10.m1.1.1.1.1">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p4.10.m1.1c">\operatorname{\mathbf{c}}_{i(1)}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p4.10.m1.1d">bold_c start_POSTSUBSCRIPT italic_i ( 1 ) end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="\operatorname{\mathbf{c}}_{i(2)}" class="ltx_Math" display="inline" id="S2.SS2.p4.11.m2.1"><semantics id="S2.SS2.p4.11.m2.1a"><msub id="S2.SS2.p4.11.m2.1.2" xref="S2.SS2.p4.11.m2.1.2.cmml"><mi id="S2.SS2.p4.11.m2.1.2.2" xref="S2.SS2.p4.11.m2.1.2.2.cmml">𝐜</mi><mrow id="S2.SS2.p4.11.m2.1.1.1" xref="S2.SS2.p4.11.m2.1.1.1.cmml"><mi id="S2.SS2.p4.11.m2.1.1.1.3" xref="S2.SS2.p4.11.m2.1.1.1.3.cmml">i</mi><mo id="S2.SS2.p4.11.m2.1.1.1.2" xref="S2.SS2.p4.11.m2.1.1.1.2.cmml">⁢</mo><mrow id="S2.SS2.p4.11.m2.1.1.1.4.2" xref="S2.SS2.p4.11.m2.1.1.1.cmml"><mo id="S2.SS2.p4.11.m2.1.1.1.4.2.1" stretchy="false" xref="S2.SS2.p4.11.m2.1.1.1.cmml">(</mo><mn id="S2.SS2.p4.11.m2.1.1.1.1" xref="S2.SS2.p4.11.m2.1.1.1.1.cmml">2</mn><mo id="S2.SS2.p4.11.m2.1.1.1.4.2.2" stretchy="false" xref="S2.SS2.p4.11.m2.1.1.1.cmml">)</mo></mrow></mrow></msub><annotation-xml encoding="MathML-Content" id="S2.SS2.p4.11.m2.1b"><apply id="S2.SS2.p4.11.m2.1.2.cmml" xref="S2.SS2.p4.11.m2.1.2"><csymbol cd="ambiguous" id="S2.SS2.p4.11.m2.1.2.1.cmml" xref="S2.SS2.p4.11.m2.1.2">subscript</csymbol><ci id="S2.SS2.p4.11.m2.1.2.2.cmml" xref="S2.SS2.p4.11.m2.1.2.2">𝐜</ci><apply id="S2.SS2.p4.11.m2.1.1.1.cmml" xref="S2.SS2.p4.11.m2.1.1.1"><times id="S2.SS2.p4.11.m2.1.1.1.2.cmml" xref="S2.SS2.p4.11.m2.1.1.1.2"></times><ci id="S2.SS2.p4.11.m2.1.1.1.3.cmml" xref="S2.SS2.p4.11.m2.1.1.1.3">𝑖</ci><cn id="S2.SS2.p4.11.m2.1.1.1.1.cmml" type="integer" xref="S2.SS2.p4.11.m2.1.1.1.1">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p4.11.m2.1c">\operatorname{\mathbf{c}}_{i(2)}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p4.11.m2.1d">bold_c start_POSTSUBSCRIPT italic_i ( 2 ) end_POSTSUBSCRIPT</annotation></semantics></math> are distributed as <math alttext="N(0,\tau^{2}\operatorname{\mathbf{I}}_{S_{1}S_{2}})" class="ltx_Math" display="inline" id="S2.SS2.p4.12.m3.2"><semantics id="S2.SS2.p4.12.m3.2a"><mrow id="S2.SS2.p4.12.m3.2.2" xref="S2.SS2.p4.12.m3.2.2.cmml"><mi id="S2.SS2.p4.12.m3.2.2.3" xref="S2.SS2.p4.12.m3.2.2.3.cmml">N</mi><mo id="S2.SS2.p4.12.m3.2.2.2" xref="S2.SS2.p4.12.m3.2.2.2.cmml">⁢</mo><mrow id="S2.SS2.p4.12.m3.2.2.1.1" xref="S2.SS2.p4.12.m3.2.2.1.2.cmml"><mo id="S2.SS2.p4.12.m3.2.2.1.1.2" stretchy="false" xref="S2.SS2.p4.12.m3.2.2.1.2.cmml">(</mo><mn id="S2.SS2.p4.12.m3.1.1" xref="S2.SS2.p4.12.m3.1.1.cmml">0</mn><mo id="S2.SS2.p4.12.m3.2.2.1.1.3" xref="S2.SS2.p4.12.m3.2.2.1.2.cmml">,</mo><mrow id="S2.SS2.p4.12.m3.2.2.1.1.1" xref="S2.SS2.p4.12.m3.2.2.1.1.1.cmml"><msup id="S2.SS2.p4.12.m3.2.2.1.1.1.2" xref="S2.SS2.p4.12.m3.2.2.1.1.1.2.cmml"><mi id="S2.SS2.p4.12.m3.2.2.1.1.1.2.2" xref="S2.SS2.p4.12.m3.2.2.1.1.1.2.2.cmml">τ</mi><mn id="S2.SS2.p4.12.m3.2.2.1.1.1.2.3" xref="S2.SS2.p4.12.m3.2.2.1.1.1.2.3.cmml">2</mn></msup><mo id="S2.SS2.p4.12.m3.2.2.1.1.1.1" lspace="0.167em" xref="S2.SS2.p4.12.m3.2.2.1.1.1.1.cmml">⁢</mo><msub id="S2.SS2.p4.12.m3.2.2.1.1.1.3" xref="S2.SS2.p4.12.m3.2.2.1.1.1.3.cmml"><mi id="S2.SS2.p4.12.m3.2.2.1.1.1.3.2" xref="S2.SS2.p4.12.m3.2.2.1.1.1.3.2.cmml">𝐈</mi><mrow id="S2.SS2.p4.12.m3.2.2.1.1.1.3.3" xref="S2.SS2.p4.12.m3.2.2.1.1.1.3.3.cmml"><msub 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id="S2.SS2.p4.12.m3.2.2.3.cmml" xref="S2.SS2.p4.12.m3.2.2.3">𝑁</ci><interval closure="open" id="S2.SS2.p4.12.m3.2.2.1.2.cmml" xref="S2.SS2.p4.12.m3.2.2.1.1"><cn id="S2.SS2.p4.12.m3.1.1.cmml" type="integer" xref="S2.SS2.p4.12.m3.1.1">0</cn><apply id="S2.SS2.p4.12.m3.2.2.1.1.1.cmml" xref="S2.SS2.p4.12.m3.2.2.1.1.1"><times id="S2.SS2.p4.12.m3.2.2.1.1.1.1.cmml" xref="S2.SS2.p4.12.m3.2.2.1.1.1.1"></times><apply id="S2.SS2.p4.12.m3.2.2.1.1.1.2.cmml" xref="S2.SS2.p4.12.m3.2.2.1.1.1.2"><csymbol cd="ambiguous" id="S2.SS2.p4.12.m3.2.2.1.1.1.2.1.cmml" xref="S2.SS2.p4.12.m3.2.2.1.1.1.2">superscript</csymbol><ci id="S2.SS2.p4.12.m3.2.2.1.1.1.2.2.cmml" xref="S2.SS2.p4.12.m3.2.2.1.1.1.2.2">𝜏</ci><cn id="S2.SS2.p4.12.m3.2.2.1.1.1.2.3.cmml" type="integer" xref="S2.SS2.p4.12.m3.2.2.1.1.1.2.3">2</cn></apply><apply id="S2.SS2.p4.12.m3.2.2.1.1.1.3.cmml" xref="S2.SS2.p4.12.m3.2.2.1.1.1.3"><csymbol cd="ambiguous" id="S2.SS2.p4.12.m3.2.2.1.1.1.3.1.cmml" xref="S2.SS2.p4.12.m3.2.2.1.1.1.3">subscript</csymbol><ci id="S2.SS2.p4.12.m3.2.2.1.1.1.3.2.cmml" xref="S2.SS2.p4.12.m3.2.2.1.1.1.3.2">𝐈</ci><apply id="S2.SS2.p4.12.m3.2.2.1.1.1.3.3.cmml" xref="S2.SS2.p4.12.m3.2.2.1.1.1.3.3"><times id="S2.SS2.p4.12.m3.2.2.1.1.1.3.3.1.cmml" xref="S2.SS2.p4.12.m3.2.2.1.1.1.3.3.1"></times><apply id="S2.SS2.p4.12.m3.2.2.1.1.1.3.3.2.cmml" xref="S2.SS2.p4.12.m3.2.2.1.1.1.3.3.2"><csymbol cd="ambiguous" id="S2.SS2.p4.12.m3.2.2.1.1.1.3.3.2.1.cmml" xref="S2.SS2.p4.12.m3.2.2.1.1.1.3.3.2">subscript</csymbol><ci id="S2.SS2.p4.12.m3.2.2.1.1.1.3.3.2.2.cmml" xref="S2.SS2.p4.12.m3.2.2.1.1.1.3.3.2.2">𝑆</ci><cn id="S2.SS2.p4.12.m3.2.2.1.1.1.3.3.2.3.cmml" type="integer" xref="S2.SS2.p4.12.m3.2.2.1.1.1.3.3.2.3">1</cn></apply><apply id="S2.SS2.p4.12.m3.2.2.1.1.1.3.3.3.cmml" xref="S2.SS2.p4.12.m3.2.2.1.1.1.3.3.3"><csymbol cd="ambiguous" id="S2.SS2.p4.12.m3.2.2.1.1.1.3.3.3.1.cmml" xref="S2.SS2.p4.12.m3.2.2.1.1.1.3.3.3">subscript</csymbol><ci id="S2.SS2.p4.12.m3.2.2.1.1.1.3.3.3.2.cmml" xref="S2.SS2.p4.12.m3.2.2.1.1.1.3.3.3.2">𝑆</ci><cn id="S2.SS2.p4.12.m3.2.2.1.1.1.3.3.3.3.cmml" type="integer" xref="S2.SS2.p4.12.m3.2.2.1.1.1.3.3.3.3">2</cn></apply></apply></apply></apply></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p4.12.m3.2c">N(0,\tau^{2}\operatorname{\mathbf{I}}_{S_{1}S_{2}})</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p4.12.m3.2d">italic_N ( 0 , italic_τ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT bold_I start_POSTSUBSCRIPT italic_S start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_S start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT )</annotation></semantics></math>. Combining (<a class="ltx_ref" href="https://arxiv.org/html/2503.13782v1#S2.E5" title="In 2.2 Reduction to a Structured Mixed Model ‣ 2 Mixed Model Trace Regression ‣ Regularized Parameter Estimation in Mixed Model Trace Regression"><span class="ltx_text ltx_ref_tag">5</span></a>) for <math alttext="j=1,\ldots,m_{i}" class="ltx_Math" display="inline" id="S2.SS2.p4.13.m4.3"><semantics id="S2.SS2.p4.13.m4.3a"><mrow id="S2.SS2.p4.13.m4.3.3" xref="S2.SS2.p4.13.m4.3.3.cmml"><mi id="S2.SS2.p4.13.m4.3.3.3" xref="S2.SS2.p4.13.m4.3.3.3.cmml">j</mi><mo id="S2.SS2.p4.13.m4.3.3.2" xref="S2.SS2.p4.13.m4.3.3.2.cmml">=</mo><mrow id="S2.SS2.p4.13.m4.3.3.1.1" xref="S2.SS2.p4.13.m4.3.3.1.2.cmml"><mn id="S2.SS2.p4.13.m4.1.1" xref="S2.SS2.p4.13.m4.1.1.cmml">1</mn><mo id="S2.SS2.p4.13.m4.3.3.1.1.2" xref="S2.SS2.p4.13.m4.3.3.1.2.cmml">,</mo><mi id="S2.SS2.p4.13.m4.2.2" mathvariant="normal" xref="S2.SS2.p4.13.m4.2.2.cmml">…</mi><mo id="S2.SS2.p4.13.m4.3.3.1.1.3" xref="S2.SS2.p4.13.m4.3.3.1.2.cmml">,</mo><msub id="S2.SS2.p4.13.m4.3.3.1.1.1" xref="S2.SS2.p4.13.m4.3.3.1.1.1.cmml"><mi id="S2.SS2.p4.13.m4.3.3.1.1.1.2" xref="S2.SS2.p4.13.m4.3.3.1.1.1.2.cmml">m</mi><mi id="S2.SS2.p4.13.m4.3.3.1.1.1.3" xref="S2.SS2.p4.13.m4.3.3.1.1.1.3.cmml">i</mi></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p4.13.m4.3b"><apply id="S2.SS2.p4.13.m4.3.3.cmml" xref="S2.SS2.p4.13.m4.3.3"><eq id="S2.SS2.p4.13.m4.3.3.2.cmml" xref="S2.SS2.p4.13.m4.3.3.2"></eq><ci id="S2.SS2.p4.13.m4.3.3.3.cmml" xref="S2.SS2.p4.13.m4.3.3.3">𝑗</ci><list id="S2.SS2.p4.13.m4.3.3.1.2.cmml" xref="S2.SS2.p4.13.m4.3.3.1.1"><cn id="S2.SS2.p4.13.m4.1.1.cmml" type="integer" xref="S2.SS2.p4.13.m4.1.1">1</cn><ci id="S2.SS2.p4.13.m4.2.2.cmml" xref="S2.SS2.p4.13.m4.2.2">…</ci><apply id="S2.SS2.p4.13.m4.3.3.1.1.1.cmml" xref="S2.SS2.p4.13.m4.3.3.1.1.1"><csymbol cd="ambiguous" id="S2.SS2.p4.13.m4.3.3.1.1.1.1.cmml" xref="S2.SS2.p4.13.m4.3.3.1.1.1">subscript</csymbol><ci id="S2.SS2.p4.13.m4.3.3.1.1.1.2.cmml" xref="S2.SS2.p4.13.m4.3.3.1.1.1.2">𝑚</ci><ci id="S2.SS2.p4.13.m4.3.3.1.1.1.3.cmml" xref="S2.SS2.p4.13.m4.3.3.1.1.1.3">𝑖</ci></apply></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p4.13.m4.3c">j=1,\ldots,m_{i}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p4.13.m4.3d">italic_j = 1 , … , italic_m start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> gives</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="S3.EGx6"> <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\operatorname{\mathbf{y}}_{i}" class="ltx_Math" display="inline" id="S2.E6.m1.1"><semantics id="S2.E6.m1.1a"><msub id="S2.E6.m1.1.1" xref="S2.E6.m1.1.1.cmml"><mi id="S2.E6.m1.1.1.2" xref="S2.E6.m1.1.1.2.cmml">𝐲</mi><mi id="S2.E6.m1.1.1.3" xref="S2.E6.m1.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S2.E6.m1.1b"><apply id="S2.E6.m1.1.1.cmml" xref="S2.E6.m1.1.1"><csymbol cd="ambiguous" id="S2.E6.m1.1.1.1.cmml" xref="S2.E6.m1.1.1">subscript</csymbol><ci id="S2.E6.m1.1.1.2.cmml" xref="S2.E6.m1.1.1.2">𝐲</ci><ci id="S2.E6.m1.1.1.3.cmml" xref="S2.E6.m1.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.E6.m1.1c">\displaystyle\operatorname{\mathbf{y}}_{i}</annotation><annotation encoding="application/x-llamapun" id="S2.E6.m1.1d">bold_y start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle=\operatorname{\mathbf{X}}_{i}\operatorname{\mathbf{b}}+% \operatorname{\mathbf{Z}}_{i(1)}\operatorname{\mathbf{L}}_{(1)}\operatorname{% \mathbf{c}}_{i(1)}+\operatorname{\mathbf{e}}_{i},\quad\operatorname{\mathbf{y}% }_{i}=\operatorname{\mathbf{X}}_{i}\operatorname{\mathbf{b}}+\operatorname{% \mathbf{Z}}_{i(2)}\operatorname{\mathbf{L}}_{(2)}\operatorname{\mathbf{c}}_{i(% 2)}+\operatorname{\mathbf{e}}_{i},\quad\operatorname{\mathbf{e}}_{i}\sim N(0,% \tau^{2}\operatorname{\mathbf{I}}_{m_{i}})," class="ltx_Math" display="inline" id="S2.E6.m2.8"><semantics id="S2.E6.m2.8a"><mrow id="S2.E6.m2.8.8.1"><mrow id="S2.E6.m2.8.8.1.1.2" xref="S2.E6.m2.8.8.1.1.3.cmml"><mrow id="S2.E6.m2.8.8.1.1.1.1" xref="S2.E6.m2.8.8.1.1.1.1.cmml"><mi id="S2.E6.m2.8.8.1.1.1.1.2" xref="S2.E6.m2.8.8.1.1.1.1.2.cmml"></mi><mo id="S2.E6.m2.8.8.1.1.1.1.1" xref="S2.E6.m2.8.8.1.1.1.1.1.cmml">=</mo><mrow id="S2.E6.m2.8.8.1.1.1.1.3" xref="S2.E6.m2.8.8.1.1.1.1.3.cmml"><mrow id="S2.E6.m2.8.8.1.1.1.1.3.2" xref="S2.E6.m2.8.8.1.1.1.1.3.2.cmml"><msub id="S2.E6.m2.8.8.1.1.1.1.3.2.1" xref="S2.E6.m2.8.8.1.1.1.1.3.2.1.cmml"><mi id="S2.E6.m2.8.8.1.1.1.1.3.2.1.2" xref="S2.E6.m2.8.8.1.1.1.1.3.2.1.2.cmml">𝐗</mi><mi id="S2.E6.m2.8.8.1.1.1.1.3.2.1.3" xref="S2.E6.m2.8.8.1.1.1.1.3.2.1.3.cmml">i</mi></msub><mo id="S2.E6.m2.8.8.1.1.1.1.3.2a" lspace="0.167em" xref="S2.E6.m2.8.8.1.1.1.1.3.2.cmml">⁡</mo><mi id="S2.E6.m2.8.8.1.1.1.1.3.2.2" xref="S2.E6.m2.8.8.1.1.1.1.3.2.2.cmml">𝐛</mi></mrow><mo id="S2.E6.m2.8.8.1.1.1.1.3.1" lspace="0em" xref="S2.E6.m2.8.8.1.1.1.1.3.1.cmml">+</mo><mrow id="S2.E6.m2.8.8.1.1.1.1.3.3" xref="S2.E6.m2.8.8.1.1.1.1.3.3.cmml"><msub id="S2.E6.m2.8.8.1.1.1.1.3.3.1" xref="S2.E6.m2.8.8.1.1.1.1.3.3.1.cmml"><mi id="S2.E6.m2.8.8.1.1.1.1.3.3.1.2" xref="S2.E6.m2.8.8.1.1.1.1.3.3.1.2.cmml">𝐙</mi><mrow id="S2.E6.m2.1.1.1" xref="S2.E6.m2.1.1.1.cmml"><mi id="S2.E6.m2.1.1.1.3" xref="S2.E6.m2.1.1.1.3.cmml">i</mi><mo id="S2.E6.m2.1.1.1.2" xref="S2.E6.m2.1.1.1.2.cmml">⁢</mo><mrow id="S2.E6.m2.1.1.1.4.2" xref="S2.E6.m2.1.1.1.cmml"><mo id="S2.E6.m2.1.1.1.4.2.1" stretchy="false" xref="S2.E6.m2.1.1.1.cmml">(</mo><mn id="S2.E6.m2.1.1.1.1" xref="S2.E6.m2.1.1.1.1.cmml">1</mn><mo id="S2.E6.m2.1.1.1.4.2.2" stretchy="false" xref="S2.E6.m2.1.1.1.cmml">)</mo></mrow></mrow></msub><mo id="S2.E6.m2.8.8.1.1.1.1.3.3a" lspace="0.167em" xref="S2.E6.m2.8.8.1.1.1.1.3.3.cmml">⁡</mo><mrow id="S2.E6.m2.8.8.1.1.1.1.3.3.2" xref="S2.E6.m2.8.8.1.1.1.1.3.3.2.cmml"><msub id="S2.E6.m2.8.8.1.1.1.1.3.3.2.1" xref="S2.E6.m2.8.8.1.1.1.1.3.3.2.1.cmml"><mi id="S2.E6.m2.8.8.1.1.1.1.3.3.2.1.2" xref="S2.E6.m2.8.8.1.1.1.1.3.3.2.1.2.cmml">𝐋</mi><mrow id="S2.E6.m2.2.2.1.3" xref="S2.E6.m2.8.8.1.1.1.1.3.3.2.1.cmml"><mo id="S2.E6.m2.2.2.1.3.1" stretchy="false" xref="S2.E6.m2.8.8.1.1.1.1.3.3.2.1.cmml">(</mo><mn id="S2.E6.m2.2.2.1.1" xref="S2.E6.m2.2.2.1.1.cmml">1</mn><mo id="S2.E6.m2.2.2.1.3.2" stretchy="false" xref="S2.E6.m2.8.8.1.1.1.1.3.3.2.1.cmml">)</mo></mrow></msub><mo id="S2.E6.m2.8.8.1.1.1.1.3.3.2a" lspace="0.167em" xref="S2.E6.m2.8.8.1.1.1.1.3.3.2.cmml">⁡</mo><msub id="S2.E6.m2.8.8.1.1.1.1.3.3.2.2" xref="S2.E6.m2.8.8.1.1.1.1.3.3.2.2.cmml"><mi id="S2.E6.m2.8.8.1.1.1.1.3.3.2.2.2" xref="S2.E6.m2.8.8.1.1.1.1.3.3.2.2.2.cmml">𝐜</mi><mrow id="S2.E6.m2.3.3.1" xref="S2.E6.m2.3.3.1.cmml"><mi id="S2.E6.m2.3.3.1.3" xref="S2.E6.m2.3.3.1.3.cmml">i</mi><mo id="S2.E6.m2.3.3.1.2" xref="S2.E6.m2.3.3.1.2.cmml">⁢</mo><mrow id="S2.E6.m2.3.3.1.4.2" xref="S2.E6.m2.3.3.1.cmml"><mo id="S2.E6.m2.3.3.1.4.2.1" stretchy="false" xref="S2.E6.m2.3.3.1.cmml">(</mo><mn id="S2.E6.m2.3.3.1.1" xref="S2.E6.m2.3.3.1.1.cmml">1</mn><mo id="S2.E6.m2.3.3.1.4.2.2" stretchy="false" xref="S2.E6.m2.3.3.1.cmml">)</mo></mrow></mrow></msub></mrow></mrow><mo id="S2.E6.m2.8.8.1.1.1.1.3.1a" lspace="0em" xref="S2.E6.m2.8.8.1.1.1.1.3.1.cmml">+</mo><msub id="S2.E6.m2.8.8.1.1.1.1.3.4" xref="S2.E6.m2.8.8.1.1.1.1.3.4.cmml"><mi id="S2.E6.m2.8.8.1.1.1.1.3.4.2" xref="S2.E6.m2.8.8.1.1.1.1.3.4.2.cmml">𝐞</mi><mi id="S2.E6.m2.8.8.1.1.1.1.3.4.3" xref="S2.E6.m2.8.8.1.1.1.1.3.4.3.cmml">i</mi></msub></mrow></mrow><mo id="S2.E6.m2.8.8.1.1.2.3" rspace="1.167em" xref="S2.E6.m2.8.8.1.1.3a.cmml">,</mo><mrow id="S2.E6.m2.8.8.1.1.2.2.2" xref="S2.E6.m2.8.8.1.1.2.2.3.cmml"><mrow id="S2.E6.m2.8.8.1.1.2.2.1.1" xref="S2.E6.m2.8.8.1.1.2.2.1.1.cmml"><msub id="S2.E6.m2.8.8.1.1.2.2.1.1.2" xref="S2.E6.m2.8.8.1.1.2.2.1.1.2.cmml"><mi id="S2.E6.m2.8.8.1.1.2.2.1.1.2.2" xref="S2.E6.m2.8.8.1.1.2.2.1.1.2.2.cmml">𝐲</mi><mi id="S2.E6.m2.8.8.1.1.2.2.1.1.2.3" xref="S2.E6.m2.8.8.1.1.2.2.1.1.2.3.cmml">i</mi></msub><mo id="S2.E6.m2.8.8.1.1.2.2.1.1.1" xref="S2.E6.m2.8.8.1.1.2.2.1.1.1.cmml">=</mo><mrow id="S2.E6.m2.8.8.1.1.2.2.1.1.3" xref="S2.E6.m2.8.8.1.1.2.2.1.1.3.cmml"><mrow id="S2.E6.m2.8.8.1.1.2.2.1.1.3.2" xref="S2.E6.m2.8.8.1.1.2.2.1.1.3.2.cmml"><msub id="S2.E6.m2.8.8.1.1.2.2.1.1.3.2.1" xref="S2.E6.m2.8.8.1.1.2.2.1.1.3.2.1.cmml"><mi id="S2.E6.m2.8.8.1.1.2.2.1.1.3.2.1.2" xref="S2.E6.m2.8.8.1.1.2.2.1.1.3.2.1.2.cmml">𝐗</mi><mi id="S2.E6.m2.8.8.1.1.2.2.1.1.3.2.1.3" xref="S2.E6.m2.8.8.1.1.2.2.1.1.3.2.1.3.cmml">i</mi></msub><mo id="S2.E6.m2.8.8.1.1.2.2.1.1.3.2a" lspace="0.167em" xref="S2.E6.m2.8.8.1.1.2.2.1.1.3.2.cmml">⁡</mo><mi id="S2.E6.m2.8.8.1.1.2.2.1.1.3.2.2" xref="S2.E6.m2.8.8.1.1.2.2.1.1.3.2.2.cmml">𝐛</mi></mrow><mo id="S2.E6.m2.8.8.1.1.2.2.1.1.3.1" lspace="0em" xref="S2.E6.m2.8.8.1.1.2.2.1.1.3.1.cmml">+</mo><mrow id="S2.E6.m2.8.8.1.1.2.2.1.1.3.3" xref="S2.E6.m2.8.8.1.1.2.2.1.1.3.3.cmml"><msub id="S2.E6.m2.8.8.1.1.2.2.1.1.3.3.1" xref="S2.E6.m2.8.8.1.1.2.2.1.1.3.3.1.cmml"><mi id="S2.E6.m2.8.8.1.1.2.2.1.1.3.3.1.2" xref="S2.E6.m2.8.8.1.1.2.2.1.1.3.3.1.2.cmml">𝐙</mi><mrow id="S2.E6.m2.4.4.1" xref="S2.E6.m2.4.4.1.cmml"><mi id="S2.E6.m2.4.4.1.3" xref="S2.E6.m2.4.4.1.3.cmml">i</mi><mo id="S2.E6.m2.4.4.1.2" xref="S2.E6.m2.4.4.1.2.cmml">⁢</mo><mrow id="S2.E6.m2.4.4.1.4.2" xref="S2.E6.m2.4.4.1.cmml"><mo id="S2.E6.m2.4.4.1.4.2.1" stretchy="false" xref="S2.E6.m2.4.4.1.cmml">(</mo><mn id="S2.E6.m2.4.4.1.1" xref="S2.E6.m2.4.4.1.1.cmml">2</mn><mo id="S2.E6.m2.4.4.1.4.2.2" stretchy="false" xref="S2.E6.m2.4.4.1.cmml">)</mo></mrow></mrow></msub><mo id="S2.E6.m2.8.8.1.1.2.2.1.1.3.3a" lspace="0.167em" xref="S2.E6.m2.8.8.1.1.2.2.1.1.3.3.cmml">⁡</mo><mrow id="S2.E6.m2.8.8.1.1.2.2.1.1.3.3.2" xref="S2.E6.m2.8.8.1.1.2.2.1.1.3.3.2.cmml"><msub id="S2.E6.m2.8.8.1.1.2.2.1.1.3.3.2.1" xref="S2.E6.m2.8.8.1.1.2.2.1.1.3.3.2.1.cmml"><mi id="S2.E6.m2.8.8.1.1.2.2.1.1.3.3.2.1.2" xref="S2.E6.m2.8.8.1.1.2.2.1.1.3.3.2.1.2.cmml">𝐋</mi><mrow id="S2.E6.m2.5.5.1.3" xref="S2.E6.m2.8.8.1.1.2.2.1.1.3.3.2.1.cmml"><mo id="S2.E6.m2.5.5.1.3.1" stretchy="false" xref="S2.E6.m2.8.8.1.1.2.2.1.1.3.3.2.1.cmml">(</mo><mn id="S2.E6.m2.5.5.1.1" xref="S2.E6.m2.5.5.1.1.cmml">2</mn><mo id="S2.E6.m2.5.5.1.3.2" stretchy="false" xref="S2.E6.m2.8.8.1.1.2.2.1.1.3.3.2.1.cmml">)</mo></mrow></msub><mo id="S2.E6.m2.8.8.1.1.2.2.1.1.3.3.2a" lspace="0.167em" xref="S2.E6.m2.8.8.1.1.2.2.1.1.3.3.2.cmml">⁡</mo><msub id="S2.E6.m2.8.8.1.1.2.2.1.1.3.3.2.2" xref="S2.E6.m2.8.8.1.1.2.2.1.1.3.3.2.2.cmml"><mi id="S2.E6.m2.8.8.1.1.2.2.1.1.3.3.2.2.2" xref="S2.E6.m2.8.8.1.1.2.2.1.1.3.3.2.2.2.cmml">𝐜</mi><mrow id="S2.E6.m2.6.6.1" xref="S2.E6.m2.6.6.1.cmml"><mi id="S2.E6.m2.6.6.1.3" xref="S2.E6.m2.6.6.1.3.cmml">i</mi><mo id="S2.E6.m2.6.6.1.2" xref="S2.E6.m2.6.6.1.2.cmml">⁢</mo><mrow id="S2.E6.m2.6.6.1.4.2" xref="S2.E6.m2.6.6.1.cmml"><mo id="S2.E6.m2.6.6.1.4.2.1" stretchy="false" xref="S2.E6.m2.6.6.1.cmml">(</mo><mn id="S2.E6.m2.6.6.1.1" xref="S2.E6.m2.6.6.1.1.cmml">2</mn><mo id="S2.E6.m2.6.6.1.4.2.2" stretchy="false" xref="S2.E6.m2.6.6.1.cmml">)</mo></mrow></mrow></msub></mrow></mrow><mo id="S2.E6.m2.8.8.1.1.2.2.1.1.3.1a" lspace="0em" xref="S2.E6.m2.8.8.1.1.2.2.1.1.3.1.cmml">+</mo><msub id="S2.E6.m2.8.8.1.1.2.2.1.1.3.4" xref="S2.E6.m2.8.8.1.1.2.2.1.1.3.4.cmml"><mi id="S2.E6.m2.8.8.1.1.2.2.1.1.3.4.2" xref="S2.E6.m2.8.8.1.1.2.2.1.1.3.4.2.cmml">𝐞</mi><mi id="S2.E6.m2.8.8.1.1.2.2.1.1.3.4.3" xref="S2.E6.m2.8.8.1.1.2.2.1.1.3.4.3.cmml">i</mi></msub></mrow></mrow><mo id="S2.E6.m2.8.8.1.1.2.2.2.3" rspace="1.167em" xref="S2.E6.m2.8.8.1.1.2.2.3a.cmml">,</mo><mrow id="S2.E6.m2.8.8.1.1.2.2.2.2" xref="S2.E6.m2.8.8.1.1.2.2.2.2.cmml"><msub id="S2.E6.m2.8.8.1.1.2.2.2.2.3" 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id="S2.E6.m2.7.7.cmml" type="integer" xref="S2.E6.m2.7.7">0</cn><apply id="S2.E6.m2.8.8.1.1.2.2.2.2.1.1.1.1.cmml" xref="S2.E6.m2.8.8.1.1.2.2.2.2.1.1.1.1"><times id="S2.E6.m2.8.8.1.1.2.2.2.2.1.1.1.1.1.cmml" xref="S2.E6.m2.8.8.1.1.2.2.2.2.1.1.1.1.1"></times><apply id="S2.E6.m2.8.8.1.1.2.2.2.2.1.1.1.1.2.cmml" xref="S2.E6.m2.8.8.1.1.2.2.2.2.1.1.1.1.2"><csymbol cd="ambiguous" id="S2.E6.m2.8.8.1.1.2.2.2.2.1.1.1.1.2.1.cmml" xref="S2.E6.m2.8.8.1.1.2.2.2.2.1.1.1.1.2">superscript</csymbol><ci id="S2.E6.m2.8.8.1.1.2.2.2.2.1.1.1.1.2.2.cmml" xref="S2.E6.m2.8.8.1.1.2.2.2.2.1.1.1.1.2.2">𝜏</ci><cn id="S2.E6.m2.8.8.1.1.2.2.2.2.1.1.1.1.2.3.cmml" type="integer" xref="S2.E6.m2.8.8.1.1.2.2.2.2.1.1.1.1.2.3">2</cn></apply><apply id="S2.E6.m2.8.8.1.1.2.2.2.2.1.1.1.1.3.cmml" xref="S2.E6.m2.8.8.1.1.2.2.2.2.1.1.1.1.3"><csymbol cd="ambiguous" id="S2.E6.m2.8.8.1.1.2.2.2.2.1.1.1.1.3.1.cmml" xref="S2.E6.m2.8.8.1.1.2.2.2.2.1.1.1.1.3">subscript</csymbol><ci id="S2.E6.m2.8.8.1.1.2.2.2.2.1.1.1.1.3.2.cmml" xref="S2.E6.m2.8.8.1.1.2.2.2.2.1.1.1.1.3.2">𝐈</ci><apply id="S2.E6.m2.8.8.1.1.2.2.2.2.1.1.1.1.3.3.cmml" xref="S2.E6.m2.8.8.1.1.2.2.2.2.1.1.1.1.3.3"><csymbol cd="ambiguous" id="S2.E6.m2.8.8.1.1.2.2.2.2.1.1.1.1.3.3.1.cmml" xref="S2.E6.m2.8.8.1.1.2.2.2.2.1.1.1.1.3.3">subscript</csymbol><ci id="S2.E6.m2.8.8.1.1.2.2.2.2.1.1.1.1.3.3.2.cmml" xref="S2.E6.m2.8.8.1.1.2.2.2.2.1.1.1.1.3.3.2">𝑚</ci><ci id="S2.E6.m2.8.8.1.1.2.2.2.2.1.1.1.1.3.3.3.cmml" xref="S2.E6.m2.8.8.1.1.2.2.2.2.1.1.1.1.3.3.3">𝑖</ci></apply></apply></apply></interval></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.E6.m2.8c">\displaystyle=\operatorname{\mathbf{X}}_{i}\operatorname{\mathbf{b}}+% \operatorname{\mathbf{Z}}_{i(1)}\operatorname{\mathbf{L}}_{(1)}\operatorname{% \mathbf{c}}_{i(1)}+\operatorname{\mathbf{e}}_{i},\quad\operatorname{\mathbf{y}% }_{i}=\operatorname{\mathbf{X}}_{i}\operatorname{\mathbf{b}}+\operatorname{% \mathbf{Z}}_{i(2)}\operatorname{\mathbf{L}}_{(2)}\operatorname{\mathbf{c}}_{i(% 2)}+\operatorname{\mathbf{e}}_{i},\quad\operatorname{\mathbf{e}}_{i}\sim N(0,% \tau^{2}\operatorname{\mathbf{I}}_{m_{i}}),</annotation><annotation encoding="application/x-llamapun" id="S2.E6.m2.8d">= bold_X start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT bold_b + bold_Z start_POSTSUBSCRIPT italic_i ( 1 ) end_POSTSUBSCRIPT bold_L start_POSTSUBSCRIPT ( 1 ) end_POSTSUBSCRIPT bold_c start_POSTSUBSCRIPT italic_i ( 1 ) end_POSTSUBSCRIPT + bold_e start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , bold_y start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = bold_X start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT bold_b + bold_Z start_POSTSUBSCRIPT italic_i ( 2 ) end_POSTSUBSCRIPT bold_L start_POSTSUBSCRIPT ( 2 ) end_POSTSUBSCRIPT bold_c start_POSTSUBSCRIPT italic_i ( 2 ) end_POSTSUBSCRIPT + bold_e start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , bold_e start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∼ italic_N ( 0 , italic_τ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT bold_I start_POSTSUBSCRIPT italic_m start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT 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">(6)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S2.SS2.p4.25">for <math alttext="i=1,\ldots,n," class="ltx_Math" display="inline" id="S2.SS2.p4.14.m1.4"><semantics id="S2.SS2.p4.14.m1.4a"><mrow id="S2.SS2.p4.14.m1.4.4.1" xref="S2.SS2.p4.14.m1.4.4.1.1.cmml"><mrow id="S2.SS2.p4.14.m1.4.4.1.1" xref="S2.SS2.p4.14.m1.4.4.1.1.cmml"><mi id="S2.SS2.p4.14.m1.4.4.1.1.2" xref="S2.SS2.p4.14.m1.4.4.1.1.2.cmml">i</mi><mo id="S2.SS2.p4.14.m1.4.4.1.1.1" xref="S2.SS2.p4.14.m1.4.4.1.1.1.cmml">=</mo><mrow id="S2.SS2.p4.14.m1.4.4.1.1.3.2" xref="S2.SS2.p4.14.m1.4.4.1.1.3.1.cmml"><mn id="S2.SS2.p4.14.m1.1.1" xref="S2.SS2.p4.14.m1.1.1.cmml">1</mn><mo id="S2.SS2.p4.14.m1.4.4.1.1.3.2.1" xref="S2.SS2.p4.14.m1.4.4.1.1.3.1.cmml">,</mo><mi id="S2.SS2.p4.14.m1.2.2" mathvariant="normal" xref="S2.SS2.p4.14.m1.2.2.cmml">…</mi><mo id="S2.SS2.p4.14.m1.4.4.1.1.3.2.2" xref="S2.SS2.p4.14.m1.4.4.1.1.3.1.cmml">,</mo><mi id="S2.SS2.p4.14.m1.3.3" xref="S2.SS2.p4.14.m1.3.3.cmml">n</mi></mrow></mrow><mo id="S2.SS2.p4.14.m1.4.4.1.2" xref="S2.SS2.p4.14.m1.4.4.1.1.cmml">,</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p4.14.m1.4b"><apply id="S2.SS2.p4.14.m1.4.4.1.1.cmml" xref="S2.SS2.p4.14.m1.4.4.1"><eq id="S2.SS2.p4.14.m1.4.4.1.1.1.cmml" xref="S2.SS2.p4.14.m1.4.4.1.1.1"></eq><ci id="S2.SS2.p4.14.m1.4.4.1.1.2.cmml" xref="S2.SS2.p4.14.m1.4.4.1.1.2">𝑖</ci><list id="S2.SS2.p4.14.m1.4.4.1.1.3.1.cmml" xref="S2.SS2.p4.14.m1.4.4.1.1.3.2"><cn id="S2.SS2.p4.14.m1.1.1.cmml" type="integer" xref="S2.SS2.p4.14.m1.1.1">1</cn><ci id="S2.SS2.p4.14.m1.2.2.cmml" xref="S2.SS2.p4.14.m1.2.2">…</ci><ci id="S2.SS2.p4.14.m1.3.3.cmml" xref="S2.SS2.p4.14.m1.3.3">𝑛</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p4.14.m1.4c">i=1,\ldots,n,</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p4.14.m1.4d">italic_i = 1 , … , italic_n ,</annotation></semantics></math>, where the <math alttext="j" class="ltx_Math" display="inline" id="S2.SS2.p4.15.m2.1"><semantics id="S2.SS2.p4.15.m2.1a"><mi id="S2.SS2.p4.15.m2.1.1" xref="S2.SS2.p4.15.m2.1.1.cmml">j</mi><annotation-xml encoding="MathML-Content" id="S2.SS2.p4.15.m2.1b"><ci id="S2.SS2.p4.15.m2.1.1.cmml" xref="S2.SS2.p4.15.m2.1.1">𝑗</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p4.15.m2.1c">j</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p4.15.m2.1d">italic_j</annotation></semantics></math>th row of <math alttext="\operatorname{\mathbf{X}}_{i}" class="ltx_Math" display="inline" id="S2.SS2.p4.16.m3.1"><semantics id="S2.SS2.p4.16.m3.1a"><msub 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id="S2.SS2.p4.17.m4.1.1.2.2.cmml" xref="S2.SS2.p4.17.m4.1.1.2.2">𝐱</ci><apply id="S2.SS2.p4.17.m4.1.1.2.3.cmml" xref="S2.SS2.p4.17.m4.1.1.2.3"><times id="S2.SS2.p4.17.m4.1.1.2.3.1.cmml" xref="S2.SS2.p4.17.m4.1.1.2.3.1"></times><ci id="S2.SS2.p4.17.m4.1.1.2.3.2.cmml" xref="S2.SS2.p4.17.m4.1.1.2.3.2">𝑖</ci><ci id="S2.SS2.p4.17.m4.1.1.2.3.3.cmml" xref="S2.SS2.p4.17.m4.1.1.2.3.3">𝑗</ci></apply></apply><csymbol cd="latexml" id="S2.SS2.p4.17.m4.1.1.3.cmml" xref="S2.SS2.p4.17.m4.1.1.3">top</csymbol></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p4.17.m4.1c">\operatorname{\mathbf{x}}_{ij}^{\top}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p4.17.m4.1d">bold_x start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ⊤ end_POSTSUPERSCRIPT</annotation></semantics></math>, the <math alttext="j" class="ltx_Math" display="inline" id="S2.SS2.p4.18.m5.1"><semantics id="S2.SS2.p4.18.m5.1a"><mi id="S2.SS2.p4.18.m5.1.1" xref="S2.SS2.p4.18.m5.1.1.cmml">j</mi><annotation-xml encoding="MathML-Content" id="S2.SS2.p4.18.m5.1b"><ci id="S2.SS2.p4.18.m5.1.1.cmml" xref="S2.SS2.p4.18.m5.1.1">𝑗</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p4.18.m5.1c">j</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p4.18.m5.1d">italic_j</annotation></semantics></math>th row of <math alttext="\operatorname{\mathbf{Z}}_{i(1)}" class="ltx_Math" display="inline" id="S2.SS2.p4.19.m6.1"><semantics id="S2.SS2.p4.19.m6.1a"><msub id="S2.SS2.p4.19.m6.1.2" xref="S2.SS2.p4.19.m6.1.2.cmml"><mi id="S2.SS2.p4.19.m6.1.2.2" xref="S2.SS2.p4.19.m6.1.2.2.cmml">𝐙</mi><mrow id="S2.SS2.p4.19.m6.1.1.1" xref="S2.SS2.p4.19.m6.1.1.1.cmml"><mi id="S2.SS2.p4.19.m6.1.1.1.3" xref="S2.SS2.p4.19.m6.1.1.1.3.cmml">i</mi><mo id="S2.SS2.p4.19.m6.1.1.1.2" xref="S2.SS2.p4.19.m6.1.1.1.2.cmml">⁢</mo><mrow id="S2.SS2.p4.19.m6.1.1.1.4.2" xref="S2.SS2.p4.19.m6.1.1.1.cmml"><mo id="S2.SS2.p4.19.m6.1.1.1.4.2.1" stretchy="false" xref="S2.SS2.p4.19.m6.1.1.1.cmml">(</mo><mn id="S2.SS2.p4.19.m6.1.1.1.1" xref="S2.SS2.p4.19.m6.1.1.1.1.cmml">1</mn><mo id="S2.SS2.p4.19.m6.1.1.1.4.2.2" stretchy="false" xref="S2.SS2.p4.19.m6.1.1.1.cmml">)</mo></mrow></mrow></msub><annotation-xml encoding="MathML-Content" id="S2.SS2.p4.19.m6.1b"><apply id="S2.SS2.p4.19.m6.1.2.cmml" xref="S2.SS2.p4.19.m6.1.2"><csymbol cd="ambiguous" id="S2.SS2.p4.19.m6.1.2.1.cmml" xref="S2.SS2.p4.19.m6.1.2">subscript</csymbol><ci id="S2.SS2.p4.19.m6.1.2.2.cmml" xref="S2.SS2.p4.19.m6.1.2.2">𝐙</ci><apply id="S2.SS2.p4.19.m6.1.1.1.cmml" xref="S2.SS2.p4.19.m6.1.1.1"><times id="S2.SS2.p4.19.m6.1.1.1.2.cmml" xref="S2.SS2.p4.19.m6.1.1.1.2"></times><ci id="S2.SS2.p4.19.m6.1.1.1.3.cmml" xref="S2.SS2.p4.19.m6.1.1.1.3">𝑖</ci><cn id="S2.SS2.p4.19.m6.1.1.1.1.cmml" type="integer" xref="S2.SS2.p4.19.m6.1.1.1.1">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p4.19.m6.1c">\operatorname{\mathbf{Z}}_{i(1)}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p4.19.m6.1d">bold_Z start_POSTSUBSCRIPT italic_i ( 1 ) end_POSTSUBSCRIPT</annotation></semantics></math> is <math alttext="\operatorname{\mathbf{z}}_{ij(1)}^{\top}" class="ltx_Math" display="inline" id="S2.SS2.p4.20.m7.1"><semantics id="S2.SS2.p4.20.m7.1a"><msubsup id="S2.SS2.p4.20.m7.1.2" xref="S2.SS2.p4.20.m7.1.2.cmml"><mi id="S2.SS2.p4.20.m7.1.2.2.2" xref="S2.SS2.p4.20.m7.1.2.2.2.cmml">𝐳</mi><mrow id="S2.SS2.p4.20.m7.1.1.1" xref="S2.SS2.p4.20.m7.1.1.1.cmml"><mi id="S2.SS2.p4.20.m7.1.1.1.3" xref="S2.SS2.p4.20.m7.1.1.1.3.cmml">i</mi><mo id="S2.SS2.p4.20.m7.1.1.1.2" xref="S2.SS2.p4.20.m7.1.1.1.2.cmml">⁢</mo><mi id="S2.SS2.p4.20.m7.1.1.1.4" xref="S2.SS2.p4.20.m7.1.1.1.4.cmml">j</mi><mo id="S2.SS2.p4.20.m7.1.1.1.2a" xref="S2.SS2.p4.20.m7.1.1.1.2.cmml">⁢</mo><mrow id="S2.SS2.p4.20.m7.1.1.1.5.2" xref="S2.SS2.p4.20.m7.1.1.1.cmml"><mo id="S2.SS2.p4.20.m7.1.1.1.5.2.1" stretchy="false" xref="S2.SS2.p4.20.m7.1.1.1.cmml">(</mo><mn id="S2.SS2.p4.20.m7.1.1.1.1" xref="S2.SS2.p4.20.m7.1.1.1.1.cmml">1</mn><mo id="S2.SS2.p4.20.m7.1.1.1.5.2.2" stretchy="false" xref="S2.SS2.p4.20.m7.1.1.1.cmml">)</mo></mrow></mrow><mo id="S2.SS2.p4.20.m7.1.2.3" xref="S2.SS2.p4.20.m7.1.2.3.cmml">⊤</mo></msubsup><annotation-xml encoding="MathML-Content" id="S2.SS2.p4.20.m7.1b"><apply id="S2.SS2.p4.20.m7.1.2.cmml" xref="S2.SS2.p4.20.m7.1.2"><csymbol cd="ambiguous" id="S2.SS2.p4.20.m7.1.2.1.cmml" xref="S2.SS2.p4.20.m7.1.2">superscript</csymbol><apply id="S2.SS2.p4.20.m7.1.2.2.cmml" xref="S2.SS2.p4.20.m7.1.2"><csymbol cd="ambiguous" id="S2.SS2.p4.20.m7.1.2.2.1.cmml" xref="S2.SS2.p4.20.m7.1.2">subscript</csymbol><ci id="S2.SS2.p4.20.m7.1.2.2.2.cmml" xref="S2.SS2.p4.20.m7.1.2.2.2">𝐳</ci><apply id="S2.SS2.p4.20.m7.1.1.1.cmml" xref="S2.SS2.p4.20.m7.1.1.1"><times id="S2.SS2.p4.20.m7.1.1.1.2.cmml" xref="S2.SS2.p4.20.m7.1.1.1.2"></times><ci id="S2.SS2.p4.20.m7.1.1.1.3.cmml" xref="S2.SS2.p4.20.m7.1.1.1.3">𝑖</ci><ci id="S2.SS2.p4.20.m7.1.1.1.4.cmml" xref="S2.SS2.p4.20.m7.1.1.1.4">𝑗</ci><cn id="S2.SS2.p4.20.m7.1.1.1.1.cmml" type="integer" xref="S2.SS2.p4.20.m7.1.1.1.1">1</cn></apply></apply><csymbol cd="latexml" id="S2.SS2.p4.20.m7.1.2.3.cmml" xref="S2.SS2.p4.20.m7.1.2.3">top</csymbol></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p4.20.m7.1c">\operatorname{\mathbf{z}}_{ij(1)}^{\top}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p4.20.m7.1d">bold_z start_POSTSUBSCRIPT italic_i italic_j ( 1 ) end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ⊤ end_POSTSUPERSCRIPT</annotation></semantics></math>, and the <math alttext="j" class="ltx_Math" display="inline" id="S2.SS2.p4.21.m8.1"><semantics id="S2.SS2.p4.21.m8.1a"><mi id="S2.SS2.p4.21.m8.1.1" xref="S2.SS2.p4.21.m8.1.1.cmml">j</mi><annotation-xml encoding="MathML-Content" id="S2.SS2.p4.21.m8.1b"><ci id="S2.SS2.p4.21.m8.1.1.cmml" xref="S2.SS2.p4.21.m8.1.1">𝑗</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p4.21.m8.1c">j</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p4.21.m8.1d">italic_j</annotation></semantics></math>th row of <math alttext="\operatorname{\mathbf{Z}}_{i(2)}" class="ltx_Math" display="inline" id="S2.SS2.p4.22.m9.1"><semantics id="S2.SS2.p4.22.m9.1a"><msub id="S2.SS2.p4.22.m9.1.2" xref="S2.SS2.p4.22.m9.1.2.cmml"><mi id="S2.SS2.p4.22.m9.1.2.2" xref="S2.SS2.p4.22.m9.1.2.2.cmml">𝐙</mi><mrow id="S2.SS2.p4.22.m9.1.1.1" xref="S2.SS2.p4.22.m9.1.1.1.cmml"><mi id="S2.SS2.p4.22.m9.1.1.1.3" xref="S2.SS2.p4.22.m9.1.1.1.3.cmml">i</mi><mo id="S2.SS2.p4.22.m9.1.1.1.2" xref="S2.SS2.p4.22.m9.1.1.1.2.cmml">⁢</mo><mrow id="S2.SS2.p4.22.m9.1.1.1.4.2" xref="S2.SS2.p4.22.m9.1.1.1.cmml"><mo id="S2.SS2.p4.22.m9.1.1.1.4.2.1" stretchy="false" xref="S2.SS2.p4.22.m9.1.1.1.cmml">(</mo><mn id="S2.SS2.p4.22.m9.1.1.1.1" xref="S2.SS2.p4.22.m9.1.1.1.1.cmml">2</mn><mo id="S2.SS2.p4.22.m9.1.1.1.4.2.2" stretchy="false" xref="S2.SS2.p4.22.m9.1.1.1.cmml">)</mo></mrow></mrow></msub><annotation-xml encoding="MathML-Content" id="S2.SS2.p4.22.m9.1b"><apply id="S2.SS2.p4.22.m9.1.2.cmml" xref="S2.SS2.p4.22.m9.1.2"><csymbol cd="ambiguous" id="S2.SS2.p4.22.m9.1.2.1.cmml" xref="S2.SS2.p4.22.m9.1.2">subscript</csymbol><ci id="S2.SS2.p4.22.m9.1.2.2.cmml" xref="S2.SS2.p4.22.m9.1.2.2">𝐙</ci><apply id="S2.SS2.p4.22.m9.1.1.1.cmml" xref="S2.SS2.p4.22.m9.1.1.1"><times id="S2.SS2.p4.22.m9.1.1.1.2.cmml" xref="S2.SS2.p4.22.m9.1.1.1.2"></times><ci id="S2.SS2.p4.22.m9.1.1.1.3.cmml" xref="S2.SS2.p4.22.m9.1.1.1.3">𝑖</ci><cn id="S2.SS2.p4.22.m9.1.1.1.1.cmml" type="integer" xref="S2.SS2.p4.22.m9.1.1.1.1">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p4.22.m9.1c">\operatorname{\mathbf{Z}}_{i(2)}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p4.22.m9.1d">bold_Z start_POSTSUBSCRIPT italic_i ( 2 ) end_POSTSUBSCRIPT</annotation></semantics></math> is <math alttext="\operatorname{\mathbf{z}}_{ij(2)}^{\top}" class="ltx_Math" display="inline" id="S2.SS2.p4.23.m10.1"><semantics id="S2.SS2.p4.23.m10.1a"><msubsup id="S2.SS2.p4.23.m10.1.2" xref="S2.SS2.p4.23.m10.1.2.cmml"><mi id="S2.SS2.p4.23.m10.1.2.2.2" xref="S2.SS2.p4.23.m10.1.2.2.2.cmml">𝐳</mi><mrow id="S2.SS2.p4.23.m10.1.1.1" xref="S2.SS2.p4.23.m10.1.1.1.cmml"><mi id="S2.SS2.p4.23.m10.1.1.1.3" xref="S2.SS2.p4.23.m10.1.1.1.3.cmml">i</mi><mo id="S2.SS2.p4.23.m10.1.1.1.2" xref="S2.SS2.p4.23.m10.1.1.1.2.cmml">⁢</mo><mi id="S2.SS2.p4.23.m10.1.1.1.4" xref="S2.SS2.p4.23.m10.1.1.1.4.cmml">j</mi><mo id="S2.SS2.p4.23.m10.1.1.1.2a" xref="S2.SS2.p4.23.m10.1.1.1.2.cmml">⁢</mo><mrow id="S2.SS2.p4.23.m10.1.1.1.5.2" xref="S2.SS2.p4.23.m10.1.1.1.cmml"><mo id="S2.SS2.p4.23.m10.1.1.1.5.2.1" stretchy="false" xref="S2.SS2.p4.23.m10.1.1.1.cmml">(</mo><mn id="S2.SS2.p4.23.m10.1.1.1.1" xref="S2.SS2.p4.23.m10.1.1.1.1.cmml">2</mn><mo id="S2.SS2.p4.23.m10.1.1.1.5.2.2" stretchy="false" xref="S2.SS2.p4.23.m10.1.1.1.cmml">)</mo></mrow></mrow><mo id="S2.SS2.p4.23.m10.1.2.3" xref="S2.SS2.p4.23.m10.1.2.3.cmml">⊤</mo></msubsup><annotation-xml encoding="MathML-Content" id="S2.SS2.p4.23.m10.1b"><apply id="S2.SS2.p4.23.m10.1.2.cmml" xref="S2.SS2.p4.23.m10.1.2"><csymbol cd="ambiguous" id="S2.SS2.p4.23.m10.1.2.1.cmml" xref="S2.SS2.p4.23.m10.1.2">superscript</csymbol><apply id="S2.SS2.p4.23.m10.1.2.2.cmml" xref="S2.SS2.p4.23.m10.1.2"><csymbol cd="ambiguous" id="S2.SS2.p4.23.m10.1.2.2.1.cmml" xref="S2.SS2.p4.23.m10.1.2">subscript</csymbol><ci id="S2.SS2.p4.23.m10.1.2.2.2.cmml" xref="S2.SS2.p4.23.m10.1.2.2.2">𝐳</ci><apply id="S2.SS2.p4.23.m10.1.1.1.cmml" xref="S2.SS2.p4.23.m10.1.1.1"><times id="S2.SS2.p4.23.m10.1.1.1.2.cmml" xref="S2.SS2.p4.23.m10.1.1.1.2"></times><ci id="S2.SS2.p4.23.m10.1.1.1.3.cmml" xref="S2.SS2.p4.23.m10.1.1.1.3">𝑖</ci><ci id="S2.SS2.p4.23.m10.1.1.1.4.cmml" xref="S2.SS2.p4.23.m10.1.1.1.4">𝑗</ci><cn id="S2.SS2.p4.23.m10.1.1.1.1.cmml" type="integer" xref="S2.SS2.p4.23.m10.1.1.1.1">2</cn></apply></apply><csymbol cd="latexml" id="S2.SS2.p4.23.m10.1.2.3.cmml" xref="S2.SS2.p4.23.m10.1.2.3">top</csymbol></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p4.23.m10.1c">\operatorname{\mathbf{z}}_{ij(2)}^{\top}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p4.23.m10.1d">bold_z start_POSTSUBSCRIPT italic_i italic_j ( 2 ) end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ⊤ end_POSTSUPERSCRIPT</annotation></semantics></math>. The two models in (<a class="ltx_ref" href="https://arxiv.org/html/2503.13782v1#S2.E6" title="In 2.2 Reduction to a Structured Mixed Model ‣ 2 Mixed Model Trace Regression ‣ Regularized Parameter Estimation in Mixed Model Trace Regression"><span class="ltx_text ltx_ref_tag">6</span></a>) are used for estimating <math alttext="\operatorname{\mathbf{L}}_{1}" class="ltx_Math" display="inline" id="S2.SS2.p4.24.m11.1"><semantics id="S2.SS2.p4.24.m11.1a"><msub id="S2.SS2.p4.24.m11.1.1" xref="S2.SS2.p4.24.m11.1.1.cmml"><mi id="S2.SS2.p4.24.m11.1.1.2" xref="S2.SS2.p4.24.m11.1.1.2.cmml">𝐋</mi><mn id="S2.SS2.p4.24.m11.1.1.3" xref="S2.SS2.p4.24.m11.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S2.SS2.p4.24.m11.1b"><apply id="S2.SS2.p4.24.m11.1.1.cmml" xref="S2.SS2.p4.24.m11.1.1"><csymbol cd="ambiguous" id="S2.SS2.p4.24.m11.1.1.1.cmml" xref="S2.SS2.p4.24.m11.1.1">subscript</csymbol><ci id="S2.SS2.p4.24.m11.1.1.2.cmml" xref="S2.SS2.p4.24.m11.1.1.2">𝐋</ci><cn id="S2.SS2.p4.24.m11.1.1.3.cmml" type="integer" xref="S2.SS2.p4.24.m11.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p4.24.m11.1c">\operatorname{\mathbf{L}}_{1}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p4.24.m11.1d">bold_L start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> given <math alttext="\operatorname{\mathbf{L}}_{2}" class="ltx_Math" display="inline" id="S2.SS2.p4.25.m12.1"><semantics id="S2.SS2.p4.25.m12.1a"><msub id="S2.SS2.p4.25.m12.1.1" xref="S2.SS2.p4.25.m12.1.1.cmml"><mi id="S2.SS2.p4.25.m12.1.1.2" xref="S2.SS2.p4.25.m12.1.1.2.cmml">𝐋</mi><mn id="S2.SS2.p4.25.m12.1.1.3" xref="S2.SS2.p4.25.m12.1.1.3.cmml">2</mn></msub><annotation-xml encoding="MathML-Content" id="S2.SS2.p4.25.m12.1b"><apply id="S2.SS2.p4.25.m12.1.1.cmml" xref="S2.SS2.p4.25.m12.1.1"><csymbol cd="ambiguous" id="S2.SS2.p4.25.m12.1.1.1.cmml" xref="S2.SS2.p4.25.m12.1.1">subscript</csymbol><ci id="S2.SS2.p4.25.m12.1.1.2.cmml" xref="S2.SS2.p4.25.m12.1.1.2">𝐋</ci><cn id="S2.SS2.p4.25.m12.1.1.3.cmml" type="integer" xref="S2.SS2.p4.25.m12.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p4.25.m12.1c">\operatorname{\mathbf{L}}_{2}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p4.25.m12.1d">bold_L start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math> and vice versa.</p> </div> <div class="ltx_para" id="S2.SS2.p5"> <p class="ltx_p" id="S2.SS2.p5.18">We regularize <math alttext="\operatorname{\mathbf{B}},\operatorname{\mathbf{L}}_{1},\operatorname{\mathbf{% L}}_{2}" class="ltx_Math" display="inline" id="S2.SS2.p5.1.m1.3"><semantics id="S2.SS2.p5.1.m1.3a"><mrow id="S2.SS2.p5.1.m1.3.3.2" xref="S2.SS2.p5.1.m1.3.3.3.cmml"><mi id="S2.SS2.p5.1.m1.1.1" xref="S2.SS2.p5.1.m1.1.1.cmml">𝐁</mi><mo id="S2.SS2.p5.1.m1.3.3.2.3" xref="S2.SS2.p5.1.m1.3.3.3.cmml">,</mo><msub id="S2.SS2.p5.1.m1.2.2.1.1" xref="S2.SS2.p5.1.m1.2.2.1.1.cmml"><mi id="S2.SS2.p5.1.m1.2.2.1.1.2" xref="S2.SS2.p5.1.m1.2.2.1.1.2.cmml">𝐋</mi><mn id="S2.SS2.p5.1.m1.2.2.1.1.3" xref="S2.SS2.p5.1.m1.2.2.1.1.3.cmml">1</mn></msub><mo id="S2.SS2.p5.1.m1.3.3.2.4" xref="S2.SS2.p5.1.m1.3.3.3.cmml">,</mo><msub id="S2.SS2.p5.1.m1.3.3.2.2" xref="S2.SS2.p5.1.m1.3.3.2.2.cmml"><mi id="S2.SS2.p5.1.m1.3.3.2.2.2" xref="S2.SS2.p5.1.m1.3.3.2.2.2.cmml">𝐋</mi><mn id="S2.SS2.p5.1.m1.3.3.2.2.3" xref="S2.SS2.p5.1.m1.3.3.2.2.3.cmml">2</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p5.1.m1.3b"><list id="S2.SS2.p5.1.m1.3.3.3.cmml" xref="S2.SS2.p5.1.m1.3.3.2"><ci id="S2.SS2.p5.1.m1.1.1.cmml" xref="S2.SS2.p5.1.m1.1.1">𝐁</ci><apply id="S2.SS2.p5.1.m1.2.2.1.1.cmml" xref="S2.SS2.p5.1.m1.2.2.1.1"><csymbol cd="ambiguous" id="S2.SS2.p5.1.m1.2.2.1.1.1.cmml" xref="S2.SS2.p5.1.m1.2.2.1.1">subscript</csymbol><ci id="S2.SS2.p5.1.m1.2.2.1.1.2.cmml" xref="S2.SS2.p5.1.m1.2.2.1.1.2">𝐋</ci><cn id="S2.SS2.p5.1.m1.2.2.1.1.3.cmml" type="integer" xref="S2.SS2.p5.1.m1.2.2.1.1.3">1</cn></apply><apply id="S2.SS2.p5.1.m1.3.3.2.2.cmml" xref="S2.SS2.p5.1.m1.3.3.2.2"><csymbol cd="ambiguous" id="S2.SS2.p5.1.m1.3.3.2.2.1.cmml" xref="S2.SS2.p5.1.m1.3.3.2.2">subscript</csymbol><ci id="S2.SS2.p5.1.m1.3.3.2.2.2.cmml" xref="S2.SS2.p5.1.m1.3.3.2.2.2">𝐋</ci><cn id="S2.SS2.p5.1.m1.3.3.2.2.3.cmml" type="integer" xref="S2.SS2.p5.1.m1.3.3.2.2.3">2</cn></apply></list></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p5.1.m1.3c">\operatorname{\mathbf{B}},\operatorname{\mathbf{L}}_{1},\operatorname{\mathbf{% L}}_{2}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p5.1.m1.3d">bold_B , bold_L start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , bold_L start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math> to ensure numerically stable parameter estimation in the high-dimensional settings. As noted earlier, the trace regression literature offers a range of penalty options for <math alttext="\operatorname{\mathbf{B}}" class="ltx_Math" display="inline" id="S2.SS2.p5.2.m2.1"><semantics id="S2.SS2.p5.2.m2.1a"><mi id="S2.SS2.p5.2.m2.1.1" xref="S2.SS2.p5.2.m2.1.1.cmml">𝐁</mi><annotation-xml encoding="MathML-Content" id="S2.SS2.p5.2.m2.1b"><ci id="S2.SS2.p5.2.m2.1.1.cmml" xref="S2.SS2.p5.2.m2.1.1">𝐁</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p5.2.m2.1c">\operatorname{\mathbf{B}}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p5.2.m2.1d">bold_B</annotation></semantics></math>, but we use the lasso penalty because it facilitates efficient estimation via the <span class="ltx_text ltx_font_typewriter" id="S2.SS2.p5.18.1">scalreg</span> package. An additional advantage of using <span class="ltx_text ltx_font_typewriter" id="S2.SS2.p5.18.2">scalreg</span> is that its <math alttext="\tau^{2}" class="ltx_Math" display="inline" id="S2.SS2.p5.3.m3.1"><semantics id="S2.SS2.p5.3.m3.1a"><msup id="S2.SS2.p5.3.m3.1.1" xref="S2.SS2.p5.3.m3.1.1.cmml"><mi id="S2.SS2.p5.3.m3.1.1.2" xref="S2.SS2.p5.3.m3.1.1.2.cmml">τ</mi><mn id="S2.SS2.p5.3.m3.1.1.3" xref="S2.SS2.p5.3.m3.1.1.3.cmml">2</mn></msup><annotation-xml encoding="MathML-Content" id="S2.SS2.p5.3.m3.1b"><apply id="S2.SS2.p5.3.m3.1.1.cmml" xref="S2.SS2.p5.3.m3.1.1"><csymbol cd="ambiguous" id="S2.SS2.p5.3.m3.1.1.1.cmml" xref="S2.SS2.p5.3.m3.1.1">superscript</csymbol><ci id="S2.SS2.p5.3.m3.1.1.2.cmml" xref="S2.SS2.p5.3.m3.1.1.2">𝜏</ci><cn id="S2.SS2.p5.3.m3.1.1.3.cmml" type="integer" xref="S2.SS2.p5.3.m3.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p5.3.m3.1c">\tau^{2}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p5.3.m3.1d">italic_τ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT</annotation></semantics></math> estimate has a smaller bias compared to other alternatives, including <span class="ltx_text ltx_font_typewriter" id="S2.SS2.p5.18.3">glmnet</span> <cite class="ltx_cite ltx_citemacro_citep">(Friedman et al., <a class="ltx_ref" href="https://arxiv.org/html/2503.13782v1#bib.bib10" title="">2010</a>)</cite>. The penalties on <math alttext="\operatorname{\mathbf{L}}_{1}" class="ltx_Math" display="inline" id="S2.SS2.p5.4.m4.1"><semantics id="S2.SS2.p5.4.m4.1a"><msub id="S2.SS2.p5.4.m4.1.1" xref="S2.SS2.p5.4.m4.1.1.cmml"><mi id="S2.SS2.p5.4.m4.1.1.2" xref="S2.SS2.p5.4.m4.1.1.2.cmml">𝐋</mi><mn id="S2.SS2.p5.4.m4.1.1.3" xref="S2.SS2.p5.4.m4.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S2.SS2.p5.4.m4.1b"><apply id="S2.SS2.p5.4.m4.1.1.cmml" xref="S2.SS2.p5.4.m4.1.1"><csymbol cd="ambiguous" id="S2.SS2.p5.4.m4.1.1.1.cmml" xref="S2.SS2.p5.4.m4.1.1">subscript</csymbol><ci id="S2.SS2.p5.4.m4.1.1.2.cmml" xref="S2.SS2.p5.4.m4.1.1.2">𝐋</ci><cn id="S2.SS2.p5.4.m4.1.1.3.cmml" type="integer" xref="S2.SS2.p5.4.m4.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p5.4.m4.1c">\operatorname{\mathbf{L}}_{1}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p5.4.m4.1d">bold_L start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="\operatorname{\mathbf{L}}_{2}" class="ltx_Math" display="inline" id="S2.SS2.p5.5.m5.1"><semantics id="S2.SS2.p5.5.m5.1a"><msub id="S2.SS2.p5.5.m5.1.1" xref="S2.SS2.p5.5.m5.1.1.cmml"><mi id="S2.SS2.p5.5.m5.1.1.2" xref="S2.SS2.p5.5.m5.1.1.2.cmml">𝐋</mi><mn id="S2.SS2.p5.5.m5.1.1.3" xref="S2.SS2.p5.5.m5.1.1.3.cmml">2</mn></msub><annotation-xml encoding="MathML-Content" id="S2.SS2.p5.5.m5.1b"><apply id="S2.SS2.p5.5.m5.1.1.cmml" xref="S2.SS2.p5.5.m5.1.1"><csymbol cd="ambiguous" id="S2.SS2.p5.5.m5.1.1.1.cmml" xref="S2.SS2.p5.5.m5.1.1">subscript</csymbol><ci id="S2.SS2.p5.5.m5.1.1.2.cmml" xref="S2.SS2.p5.5.m5.1.1.2">𝐋</ci><cn id="S2.SS2.p5.5.m5.1.1.3.cmml" type="integer" xref="S2.SS2.p5.5.m5.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p5.5.m5.1c">\operatorname{\mathbf{L}}_{2}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p5.5.m5.1d">bold_L start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math> are chosen to avoid the need for the exact specification of <math alttext="S_{1}" class="ltx_Math" display="inline" id="S2.SS2.p5.6.m6.1"><semantics id="S2.SS2.p5.6.m6.1a"><msub id="S2.SS2.p5.6.m6.1.1" xref="S2.SS2.p5.6.m6.1.1.cmml"><mi id="S2.SS2.p5.6.m6.1.1.2" xref="S2.SS2.p5.6.m6.1.1.2.cmml">S</mi><mn id="S2.SS2.p5.6.m6.1.1.3" xref="S2.SS2.p5.6.m6.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S2.SS2.p5.6.m6.1b"><apply id="S2.SS2.p5.6.m6.1.1.cmml" xref="S2.SS2.p5.6.m6.1.1"><csymbol cd="ambiguous" id="S2.SS2.p5.6.m6.1.1.1.cmml" xref="S2.SS2.p5.6.m6.1.1">subscript</csymbol><ci id="S2.SS2.p5.6.m6.1.1.2.cmml" xref="S2.SS2.p5.6.m6.1.1.2">𝑆</ci><cn id="S2.SS2.p5.6.m6.1.1.3.cmml" type="integer" xref="S2.SS2.p5.6.m6.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p5.6.m6.1c">S_{1}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p5.6.m6.1d">italic_S start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="S_{2}" class="ltx_Math" display="inline" id="S2.SS2.p5.7.m7.1"><semantics id="S2.SS2.p5.7.m7.1a"><msub id="S2.SS2.p5.7.m7.1.1" xref="S2.SS2.p5.7.m7.1.1.cmml"><mi id="S2.SS2.p5.7.m7.1.1.2" xref="S2.SS2.p5.7.m7.1.1.2.cmml">S</mi><mn id="S2.SS2.p5.7.m7.1.1.3" xref="S2.SS2.p5.7.m7.1.1.3.cmml">2</mn></msub><annotation-xml encoding="MathML-Content" id="S2.SS2.p5.7.m7.1b"><apply id="S2.SS2.p5.7.m7.1.1.cmml" xref="S2.SS2.p5.7.m7.1.1"><csymbol cd="ambiguous" id="S2.SS2.p5.7.m7.1.1.1.cmml" xref="S2.SS2.p5.7.m7.1.1">subscript</csymbol><ci id="S2.SS2.p5.7.m7.1.1.2.cmml" xref="S2.SS2.p5.7.m7.1.1.2">𝑆</ci><cn id="S2.SS2.p5.7.m7.1.1.3.cmml" type="integer" xref="S2.SS2.p5.7.m7.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p5.7.m7.1c">S_{2}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p5.7.m7.1d">italic_S start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math>, which are typically unknown. The <math alttext="\operatorname{\mathbf{L}}_{1}" class="ltx_Math" display="inline" id="S2.SS2.p5.8.m8.1"><semantics id="S2.SS2.p5.8.m8.1a"><msub id="S2.SS2.p5.8.m8.1.1" xref="S2.SS2.p5.8.m8.1.1.cmml"><mi id="S2.SS2.p5.8.m8.1.1.2" xref="S2.SS2.p5.8.m8.1.1.2.cmml">𝐋</mi><mn id="S2.SS2.p5.8.m8.1.1.3" xref="S2.SS2.p5.8.m8.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S2.SS2.p5.8.m8.1b"><apply id="S2.SS2.p5.8.m8.1.1.cmml" xref="S2.SS2.p5.8.m8.1.1"><csymbol cd="ambiguous" id="S2.SS2.p5.8.m8.1.1.1.cmml" xref="S2.SS2.p5.8.m8.1.1">subscript</csymbol><ci id="S2.SS2.p5.8.m8.1.1.2.cmml" xref="S2.SS2.p5.8.m8.1.1.2">𝐋</ci><cn id="S2.SS2.p5.8.m8.1.1.3.cmml" type="integer" xref="S2.SS2.p5.8.m8.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p5.8.m8.1c">\operatorname{\mathbf{L}}_{1}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p5.8.m8.1d">bold_L start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="\operatorname{\mathbf{L}}_{2}" class="ltx_Math" display="inline" id="S2.SS2.p5.9.m9.1"><semantics id="S2.SS2.p5.9.m9.1a"><msub id="S2.SS2.p5.9.m9.1.1" xref="S2.SS2.p5.9.m9.1.1.cmml"><mi id="S2.SS2.p5.9.m9.1.1.2" xref="S2.SS2.p5.9.m9.1.1.2.cmml">𝐋</mi><mn id="S2.SS2.p5.9.m9.1.1.3" xref="S2.SS2.p5.9.m9.1.1.3.cmml">2</mn></msub><annotation-xml encoding="MathML-Content" id="S2.SS2.p5.9.m9.1b"><apply id="S2.SS2.p5.9.m9.1.1.cmml" xref="S2.SS2.p5.9.m9.1.1"><csymbol cd="ambiguous" id="S2.SS2.p5.9.m9.1.1.1.cmml" xref="S2.SS2.p5.9.m9.1.1">subscript</csymbol><ci id="S2.SS2.p5.9.m9.1.1.2.cmml" xref="S2.SS2.p5.9.m9.1.1.2">𝐋</ci><cn id="S2.SS2.p5.9.m9.1.1.3.cmml" type="integer" xref="S2.SS2.p5.9.m9.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p5.9.m9.1c">\operatorname{\mathbf{L}}_{2}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p5.9.m9.1d">bold_L start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math> matrices define low-rank row and column covariance matrices in that <math alttext="\operatorname{{\mathbf{\Sigma}}}_{k}\approx\operatorname{\mathbf{L}}_{k}% \operatorname{\mathbf{L}}_{k}^{\top}" class="ltx_Math" display="inline" id="S2.SS2.p5.10.m10.1"><semantics id="S2.SS2.p5.10.m10.1a"><mrow id="S2.SS2.p5.10.m10.1.1" xref="S2.SS2.p5.10.m10.1.1.cmml"><msub id="S2.SS2.p5.10.m10.1.1.2" xref="S2.SS2.p5.10.m10.1.1.2.cmml"><mi id="S2.SS2.p5.10.m10.1.1.2.2" xref="S2.SS2.p5.10.m10.1.1.2.2.cmml">𝚺</mi><mi id="S2.SS2.p5.10.m10.1.1.2.3" xref="S2.SS2.p5.10.m10.1.1.2.3.cmml">k</mi></msub><mo id="S2.SS2.p5.10.m10.1.1.1" xref="S2.SS2.p5.10.m10.1.1.1.cmml">≈</mo><mrow id="S2.SS2.p5.10.m10.1.1.3" xref="S2.SS2.p5.10.m10.1.1.3.cmml"><msub id="S2.SS2.p5.10.m10.1.1.3.1" xref="S2.SS2.p5.10.m10.1.1.3.1.cmml"><mi id="S2.SS2.p5.10.m10.1.1.3.1.2" xref="S2.SS2.p5.10.m10.1.1.3.1.2.cmml">𝐋</mi><mi id="S2.SS2.p5.10.m10.1.1.3.1.3" xref="S2.SS2.p5.10.m10.1.1.3.1.3.cmml">k</mi></msub><mo id="S2.SS2.p5.10.m10.1.1.3a" lspace="0.167em" xref="S2.SS2.p5.10.m10.1.1.3.cmml">⁡</mo><msubsup id="S2.SS2.p5.10.m10.1.1.3.2" xref="S2.SS2.p5.10.m10.1.1.3.2.cmml"><mi id="S2.SS2.p5.10.m10.1.1.3.2.2.2" xref="S2.SS2.p5.10.m10.1.1.3.2.2.2.cmml">𝐋</mi><mi id="S2.SS2.p5.10.m10.1.1.3.2.2.3" xref="S2.SS2.p5.10.m10.1.1.3.2.2.3.cmml">k</mi><mo id="S2.SS2.p5.10.m10.1.1.3.2.3" xref="S2.SS2.p5.10.m10.1.1.3.2.3.cmml">⊤</mo></msubsup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p5.10.m10.1b"><apply id="S2.SS2.p5.10.m10.1.1.cmml" xref="S2.SS2.p5.10.m10.1.1"><approx id="S2.SS2.p5.10.m10.1.1.1.cmml" xref="S2.SS2.p5.10.m10.1.1.1"></approx><apply id="S2.SS2.p5.10.m10.1.1.2.cmml" xref="S2.SS2.p5.10.m10.1.1.2"><csymbol cd="ambiguous" id="S2.SS2.p5.10.m10.1.1.2.1.cmml" xref="S2.SS2.p5.10.m10.1.1.2">subscript</csymbol><ci id="S2.SS2.p5.10.m10.1.1.2.2.cmml" xref="S2.SS2.p5.10.m10.1.1.2.2">𝚺</ci><ci id="S2.SS2.p5.10.m10.1.1.2.3.cmml" xref="S2.SS2.p5.10.m10.1.1.2.3">𝑘</ci></apply><apply id="S2.SS2.p5.10.m10.1.1.3.cmml" xref="S2.SS2.p5.10.m10.1.1.3"><apply id="S2.SS2.p5.10.m10.1.1.3.1.cmml" xref="S2.SS2.p5.10.m10.1.1.3.1"><csymbol cd="ambiguous" id="S2.SS2.p5.10.m10.1.1.3.1.1.cmml" xref="S2.SS2.p5.10.m10.1.1.3.1">subscript</csymbol><ci id="S2.SS2.p5.10.m10.1.1.3.1.2.cmml" xref="S2.SS2.p5.10.m10.1.1.3.1.2">𝐋</ci><ci id="S2.SS2.p5.10.m10.1.1.3.1.3.cmml" xref="S2.SS2.p5.10.m10.1.1.3.1.3">𝑘</ci></apply><apply id="S2.SS2.p5.10.m10.1.1.3.2.cmml" xref="S2.SS2.p5.10.m10.1.1.3.2"><csymbol cd="ambiguous" id="S2.SS2.p5.10.m10.1.1.3.2.1.cmml" xref="S2.SS2.p5.10.m10.1.1.3.2">superscript</csymbol><apply id="S2.SS2.p5.10.m10.1.1.3.2.2.cmml" xref="S2.SS2.p5.10.m10.1.1.3.2"><csymbol cd="ambiguous" id="S2.SS2.p5.10.m10.1.1.3.2.2.1.cmml" xref="S2.SS2.p5.10.m10.1.1.3.2">subscript</csymbol><ci id="S2.SS2.p5.10.m10.1.1.3.2.2.2.cmml" xref="S2.SS2.p5.10.m10.1.1.3.2.2.2">𝐋</ci><ci id="S2.SS2.p5.10.m10.1.1.3.2.2.3.cmml" xref="S2.SS2.p5.10.m10.1.1.3.2.2.3">𝑘</ci></apply><csymbol cd="latexml" id="S2.SS2.p5.10.m10.1.1.3.2.3.cmml" xref="S2.SS2.p5.10.m10.1.1.3.2.3">top</csymbol></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p5.10.m10.1c">\operatorname{{\mathbf{\Sigma}}}_{k}\approx\operatorname{\mathbf{L}}_{k}% \operatorname{\mathbf{L}}_{k}^{\top}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p5.10.m10.1d">bold_Σ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ≈ bold_L start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT bold_L start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ⊤ end_POSTSUPERSCRIPT</annotation></semantics></math> and <math alttext="S_{k}\ll Q_{k}" class="ltx_Math" display="inline" id="S2.SS2.p5.11.m11.1"><semantics id="S2.SS2.p5.11.m11.1a"><mrow id="S2.SS2.p5.11.m11.1.1" xref="S2.SS2.p5.11.m11.1.1.cmml"><msub id="S2.SS2.p5.11.m11.1.1.2" xref="S2.SS2.p5.11.m11.1.1.2.cmml"><mi id="S2.SS2.p5.11.m11.1.1.2.2" xref="S2.SS2.p5.11.m11.1.1.2.2.cmml">S</mi><mi id="S2.SS2.p5.11.m11.1.1.2.3" xref="S2.SS2.p5.11.m11.1.1.2.3.cmml">k</mi></msub><mo id="S2.SS2.p5.11.m11.1.1.1" xref="S2.SS2.p5.11.m11.1.1.1.cmml">≪</mo><msub id="S2.SS2.p5.11.m11.1.1.3" xref="S2.SS2.p5.11.m11.1.1.3.cmml"><mi id="S2.SS2.p5.11.m11.1.1.3.2" xref="S2.SS2.p5.11.m11.1.1.3.2.cmml">Q</mi><mi id="S2.SS2.p5.11.m11.1.1.3.3" xref="S2.SS2.p5.11.m11.1.1.3.3.cmml">k</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p5.11.m11.1b"><apply id="S2.SS2.p5.11.m11.1.1.cmml" xref="S2.SS2.p5.11.m11.1.1"><csymbol cd="latexml" id="S2.SS2.p5.11.m11.1.1.1.cmml" xref="S2.SS2.p5.11.m11.1.1.1">much-less-than</csymbol><apply id="S2.SS2.p5.11.m11.1.1.2.cmml" xref="S2.SS2.p5.11.m11.1.1.2"><csymbol cd="ambiguous" id="S2.SS2.p5.11.m11.1.1.2.1.cmml" xref="S2.SS2.p5.11.m11.1.1.2">subscript</csymbol><ci id="S2.SS2.p5.11.m11.1.1.2.2.cmml" xref="S2.SS2.p5.11.m11.1.1.2.2">𝑆</ci><ci id="S2.SS2.p5.11.m11.1.1.2.3.cmml" xref="S2.SS2.p5.11.m11.1.1.2.3">𝑘</ci></apply><apply id="S2.SS2.p5.11.m11.1.1.3.cmml" xref="S2.SS2.p5.11.m11.1.1.3"><csymbol cd="ambiguous" id="S2.SS2.p5.11.m11.1.1.3.1.cmml" xref="S2.SS2.p5.11.m11.1.1.3">subscript</csymbol><ci id="S2.SS2.p5.11.m11.1.1.3.2.cmml" xref="S2.SS2.p5.11.m11.1.1.3.2">𝑄</ci><ci id="S2.SS2.p5.11.m11.1.1.3.3.cmml" xref="S2.SS2.p5.11.m11.1.1.3.3">𝑘</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p5.11.m11.1c">S_{k}\ll Q_{k}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p5.11.m11.1d">italic_S start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ≪ italic_Q start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT</annotation></semantics></math> for <math alttext="k=1,2" class="ltx_Math" display="inline" id="S2.SS2.p5.12.m12.2"><semantics id="S2.SS2.p5.12.m12.2a"><mrow id="S2.SS2.p5.12.m12.2.3" xref="S2.SS2.p5.12.m12.2.3.cmml"><mi id="S2.SS2.p5.12.m12.2.3.2" xref="S2.SS2.p5.12.m12.2.3.2.cmml">k</mi><mo id="S2.SS2.p5.12.m12.2.3.1" xref="S2.SS2.p5.12.m12.2.3.1.cmml">=</mo><mrow id="S2.SS2.p5.12.m12.2.3.3.2" xref="S2.SS2.p5.12.m12.2.3.3.1.cmml"><mn id="S2.SS2.p5.12.m12.1.1" xref="S2.SS2.p5.12.m12.1.1.cmml">1</mn><mo id="S2.SS2.p5.12.m12.2.3.3.2.1" xref="S2.SS2.p5.12.m12.2.3.3.1.cmml">,</mo><mn id="S2.SS2.p5.12.m12.2.2" xref="S2.SS2.p5.12.m12.2.2.cmml">2</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p5.12.m12.2b"><apply id="S2.SS2.p5.12.m12.2.3.cmml" xref="S2.SS2.p5.12.m12.2.3"><eq id="S2.SS2.p5.12.m12.2.3.1.cmml" xref="S2.SS2.p5.12.m12.2.3.1"></eq><ci id="S2.SS2.p5.12.m12.2.3.2.cmml" xref="S2.SS2.p5.12.m12.2.3.2">𝑘</ci><list id="S2.SS2.p5.12.m12.2.3.3.1.cmml" xref="S2.SS2.p5.12.m12.2.3.3.2"><cn id="S2.SS2.p5.12.m12.1.1.cmml" type="integer" xref="S2.SS2.p5.12.m12.1.1">1</cn><cn id="S2.SS2.p5.12.m12.2.2.cmml" type="integer" xref="S2.SS2.p5.12.m12.2.2">2</cn></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p5.12.m12.2c">k=1,2</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p5.12.m12.2d">italic_k = 1 , 2</annotation></semantics></math>. Leveraging similar results in factor analysis that bypass the number of latent factors specification, we set <math alttext="S_{1}=O(\log Q_{1})" class="ltx_Math" display="inline" id="S2.SS2.p5.13.m13.1"><semantics id="S2.SS2.p5.13.m13.1a"><mrow id="S2.SS2.p5.13.m13.1.1" xref="S2.SS2.p5.13.m13.1.1.cmml"><msub id="S2.SS2.p5.13.m13.1.1.3" xref="S2.SS2.p5.13.m13.1.1.3.cmml"><mi id="S2.SS2.p5.13.m13.1.1.3.2" xref="S2.SS2.p5.13.m13.1.1.3.2.cmml">S</mi><mn id="S2.SS2.p5.13.m13.1.1.3.3" xref="S2.SS2.p5.13.m13.1.1.3.3.cmml">1</mn></msub><mo id="S2.SS2.p5.13.m13.1.1.2" xref="S2.SS2.p5.13.m13.1.1.2.cmml">=</mo><mrow id="S2.SS2.p5.13.m13.1.1.1" xref="S2.SS2.p5.13.m13.1.1.1.cmml"><mi id="S2.SS2.p5.13.m13.1.1.1.3" xref="S2.SS2.p5.13.m13.1.1.1.3.cmml">O</mi><mo id="S2.SS2.p5.13.m13.1.1.1.2" xref="S2.SS2.p5.13.m13.1.1.1.2.cmml">⁢</mo><mrow id="S2.SS2.p5.13.m13.1.1.1.1.1" xref="S2.SS2.p5.13.m13.1.1.1.1.1.1.cmml"><mo id="S2.SS2.p5.13.m13.1.1.1.1.1.2" stretchy="false" xref="S2.SS2.p5.13.m13.1.1.1.1.1.1.cmml">(</mo><mrow id="S2.SS2.p5.13.m13.1.1.1.1.1.1" xref="S2.SS2.p5.13.m13.1.1.1.1.1.1.cmml"><mi id="S2.SS2.p5.13.m13.1.1.1.1.1.1.1" xref="S2.SS2.p5.13.m13.1.1.1.1.1.1.1.cmml">log</mi><mo id="S2.SS2.p5.13.m13.1.1.1.1.1.1a" lspace="0.167em" xref="S2.SS2.p5.13.m13.1.1.1.1.1.1.cmml">⁡</mo><msub id="S2.SS2.p5.13.m13.1.1.1.1.1.1.2" xref="S2.SS2.p5.13.m13.1.1.1.1.1.1.2.cmml"><mi id="S2.SS2.p5.13.m13.1.1.1.1.1.1.2.2" xref="S2.SS2.p5.13.m13.1.1.1.1.1.1.2.2.cmml">Q</mi><mn id="S2.SS2.p5.13.m13.1.1.1.1.1.1.2.3" xref="S2.SS2.p5.13.m13.1.1.1.1.1.1.2.3.cmml">1</mn></msub></mrow><mo id="S2.SS2.p5.13.m13.1.1.1.1.1.3" stretchy="false" xref="S2.SS2.p5.13.m13.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p5.13.m13.1b"><apply id="S2.SS2.p5.13.m13.1.1.cmml" xref="S2.SS2.p5.13.m13.1.1"><eq id="S2.SS2.p5.13.m13.1.1.2.cmml" xref="S2.SS2.p5.13.m13.1.1.2"></eq><apply id="S2.SS2.p5.13.m13.1.1.3.cmml" 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<math alttext="\operatorname{\mathbf{L}}_{1}" class="ltx_Math" display="inline" id="S2.SS2.p5.15.m15.1"><semantics id="S2.SS2.p5.15.m15.1a"><msub id="S2.SS2.p5.15.m15.1.1" xref="S2.SS2.p5.15.m15.1.1.cmml"><mi id="S2.SS2.p5.15.m15.1.1.2" xref="S2.SS2.p5.15.m15.1.1.2.cmml">𝐋</mi><mn id="S2.SS2.p5.15.m15.1.1.3" xref="S2.SS2.p5.15.m15.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S2.SS2.p5.15.m15.1b"><apply id="S2.SS2.p5.15.m15.1.1.cmml" xref="S2.SS2.p5.15.m15.1.1"><csymbol cd="ambiguous" id="S2.SS2.p5.15.m15.1.1.1.cmml" xref="S2.SS2.p5.15.m15.1.1">subscript</csymbol><ci id="S2.SS2.p5.15.m15.1.1.2.cmml" xref="S2.SS2.p5.15.m15.1.1.2">𝐋</ci><cn id="S2.SS2.p5.15.m15.1.1.3.cmml" type="integer" xref="S2.SS2.p5.15.m15.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p5.15.m15.1c">\operatorname{\mathbf{L}}_{1}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p5.15.m15.1d">bold_L start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="\operatorname{\mathbf{L}}_{2}" class="ltx_Math" display="inline" id="S2.SS2.p5.16.m16.1"><semantics id="S2.SS2.p5.16.m16.1a"><msub id="S2.SS2.p5.16.m16.1.1" xref="S2.SS2.p5.16.m16.1.1.cmml"><mi id="S2.SS2.p5.16.m16.1.1.2" xref="S2.SS2.p5.16.m16.1.1.2.cmml">𝐋</mi><mn id="S2.SS2.p5.16.m16.1.1.3" xref="S2.SS2.p5.16.m16.1.1.3.cmml">2</mn></msub><annotation-xml encoding="MathML-Content" id="S2.SS2.p5.16.m16.1b"><apply id="S2.SS2.p5.16.m16.1.1.cmml" xref="S2.SS2.p5.16.m16.1.1"><csymbol cd="ambiguous" id="S2.SS2.p5.16.m16.1.1.1.cmml" xref="S2.SS2.p5.16.m16.1.1">subscript</csymbol><ci id="S2.SS2.p5.16.m16.1.1.2.cmml" xref="S2.SS2.p5.16.m16.1.1.2">𝐋</ci><cn id="S2.SS2.p5.16.m16.1.1.3.cmml" type="integer" xref="S2.SS2.p5.16.m16.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p5.16.m16.1c">\operatorname{\mathbf{L}}_{2}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p5.16.m16.1d">bold_L start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math> <cite class="ltx_cite ltx_citemacro_citep">(Ročková and George, <a class="ltx_ref" href="https://arxiv.org/html/2503.13782v1#bib.bib24" title="">2016</a>; Srivastava et al., <a class="ltx_ref" href="https://arxiv.org/html/2503.13782v1#bib.bib28" title="">2017</a>)</cite>. This penalty offers a data-driven approach for estimating <math alttext="S_{1}" class="ltx_Math" display="inline" id="S2.SS2.p5.17.m17.1"><semantics id="S2.SS2.p5.17.m17.1a"><msub id="S2.SS2.p5.17.m17.1.1" xref="S2.SS2.p5.17.m17.1.1.cmml"><mi id="S2.SS2.p5.17.m17.1.1.2" xref="S2.SS2.p5.17.m17.1.1.2.cmml">S</mi><mn id="S2.SS2.p5.17.m17.1.1.3" xref="S2.SS2.p5.17.m17.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S2.SS2.p5.17.m17.1b"><apply id="S2.SS2.p5.17.m17.1.1.cmml" xref="S2.SS2.p5.17.m17.1.1"><csymbol cd="ambiguous" id="S2.SS2.p5.17.m17.1.1.1.cmml" xref="S2.SS2.p5.17.m17.1.1">subscript</csymbol><ci id="S2.SS2.p5.17.m17.1.1.2.cmml" xref="S2.SS2.p5.17.m17.1.1.2">𝑆</ci><cn id="S2.SS2.p5.17.m17.1.1.3.cmml" type="integer" xref="S2.SS2.p5.17.m17.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p5.17.m17.1c">S_{1}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p5.17.m17.1d">italic_S start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="S_{2}" class="ltx_Math" display="inline" id="S2.SS2.p5.18.m18.1"><semantics id="S2.SS2.p5.18.m18.1a"><msub id="S2.SS2.p5.18.m18.1.1" xref="S2.SS2.p5.18.m18.1.1.cmml"><mi id="S2.SS2.p5.18.m18.1.1.2" xref="S2.SS2.p5.18.m18.1.1.2.cmml">S</mi><mn id="S2.SS2.p5.18.m18.1.1.3" xref="S2.SS2.p5.18.m18.1.1.3.cmml">2</mn></msub><annotation-xml encoding="MathML-Content" id="S2.SS2.p5.18.m18.1b"><apply id="S2.SS2.p5.18.m18.1.1.cmml" xref="S2.SS2.p5.18.m18.1.1"><csymbol cd="ambiguous" id="S2.SS2.p5.18.m18.1.1.1.cmml" xref="S2.SS2.p5.18.m18.1.1">subscript</csymbol><ci id="S2.SS2.p5.18.m18.1.1.2.cmml" xref="S2.SS2.p5.18.m18.1.1.2">𝑆</ci><cn id="S2.SS2.p5.18.m18.1.1.3.cmml" type="integer" xref="S2.SS2.p5.18.m18.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p5.18.m18.1c">S_{2}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p5.18.m18.1d">italic_S start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math>, where their estimates correspond to the highest column index with nonzero entries.</p> </div> </section> </section> <section class="ltx_section" id="S3"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">3 </span>Regularized Parameter Estimation</h2> <div class="ltx_para" id="S3.p1"> <p class="ltx_p" id="S3.p1.8">We employ a regularized alternating expected-conditional maximization (AECM) algorithm with three cycles for parameter estimation. This algorithm is an EM extension that estimates <math alttext="(\operatorname{\mathbf{b}},\tau^{2})" class="ltx_Math" display="inline" id="S3.p1.1.m1.2"><semantics id="S3.p1.1.m1.2a"><mrow id="S3.p1.1.m1.2.2.1" xref="S3.p1.1.m1.2.2.2.cmml"><mo id="S3.p1.1.m1.2.2.1.2" stretchy="false" xref="S3.p1.1.m1.2.2.2.cmml">(</mo><mi id="S3.p1.1.m1.1.1" xref="S3.p1.1.m1.1.1.cmml">𝐛</mi><mo id="S3.p1.1.m1.2.2.1.3" xref="S3.p1.1.m1.2.2.2.cmml">,</mo><msup id="S3.p1.1.m1.2.2.1.1" xref="S3.p1.1.m1.2.2.1.1.cmml"><mi id="S3.p1.1.m1.2.2.1.1.2" xref="S3.p1.1.m1.2.2.1.1.2.cmml">τ</mi><mn id="S3.p1.1.m1.2.2.1.1.3" xref="S3.p1.1.m1.2.2.1.1.3.cmml">2</mn></msup><mo id="S3.p1.1.m1.2.2.1.4" stretchy="false" xref="S3.p1.1.m1.2.2.2.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.p1.1.m1.2b"><interval closure="open" id="S3.p1.1.m1.2.2.2.cmml" xref="S3.p1.1.m1.2.2.1"><ci id="S3.p1.1.m1.1.1.cmml" xref="S3.p1.1.m1.1.1">𝐛</ci><apply id="S3.p1.1.m1.2.2.1.1.cmml" xref="S3.p1.1.m1.2.2.1.1"><csymbol cd="ambiguous" id="S3.p1.1.m1.2.2.1.1.1.cmml" xref="S3.p1.1.m1.2.2.1.1">superscript</csymbol><ci id="S3.p1.1.m1.2.2.1.1.2.cmml" xref="S3.p1.1.m1.2.2.1.1.2">𝜏</ci><cn id="S3.p1.1.m1.2.2.1.1.3.cmml" type="integer" xref="S3.p1.1.m1.2.2.1.1.3">2</cn></apply></interval></annotation-xml><annotation encoding="application/x-tex" id="S3.p1.1.m1.2c">(\operatorname{\mathbf{b}},\tau^{2})</annotation><annotation encoding="application/x-llamapun" id="S3.p1.1.m1.2d">( bold_b , italic_τ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT )</annotation></semantics></math>, <math alttext="\operatorname{\mathbf{L}}_{1}" class="ltx_Math" display="inline" id="S3.p1.2.m2.1"><semantics id="S3.p1.2.m2.1a"><msub id="S3.p1.2.m2.1.1" xref="S3.p1.2.m2.1.1.cmml"><mi id="S3.p1.2.m2.1.1.2" xref="S3.p1.2.m2.1.1.2.cmml">𝐋</mi><mn id="S3.p1.2.m2.1.1.3" xref="S3.p1.2.m2.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S3.p1.2.m2.1b"><apply id="S3.p1.2.m2.1.1.cmml" xref="S3.p1.2.m2.1.1"><csymbol cd="ambiguous" id="S3.p1.2.m2.1.1.1.cmml" xref="S3.p1.2.m2.1.1">subscript</csymbol><ci id="S3.p1.2.m2.1.1.2.cmml" xref="S3.p1.2.m2.1.1.2">𝐋</ci><cn id="S3.p1.2.m2.1.1.3.cmml" type="integer" xref="S3.p1.2.m2.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p1.2.m2.1c">\operatorname{\mathbf{L}}_{1}</annotation><annotation encoding="application/x-llamapun" id="S3.p1.2.m2.1d">bold_L start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math>, and <math alttext="\operatorname{\mathbf{L}}_{2}" class="ltx_Math" display="inline" id="S3.p1.3.m3.1"><semantics id="S3.p1.3.m3.1a"><msub id="S3.p1.3.m3.1.1" xref="S3.p1.3.m3.1.1.cmml"><mi id="S3.p1.3.m3.1.1.2" xref="S3.p1.3.m3.1.1.2.cmml">𝐋</mi><mn id="S3.p1.3.m3.1.1.3" xref="S3.p1.3.m3.1.1.3.cmml">2</mn></msub><annotation-xml encoding="MathML-Content" id="S3.p1.3.m3.1b"><apply id="S3.p1.3.m3.1.1.cmml" xref="S3.p1.3.m3.1.1"><csymbol cd="ambiguous" id="S3.p1.3.m3.1.1.1.cmml" xref="S3.p1.3.m3.1.1">subscript</csymbol><ci id="S3.p1.3.m3.1.1.2.cmml" xref="S3.p1.3.m3.1.1.2">𝐋</ci><cn id="S3.p1.3.m3.1.1.3.cmml" type="integer" xref="S3.p1.3.m3.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p1.3.m3.1c">\operatorname{\mathbf{L}}_{2}</annotation><annotation encoding="application/x-llamapun" id="S3.p1.3.m3.1d">bold_L start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math> across its three cycles, with each cycle conditioning on the remaining parameters. The first cycle estimates <math alttext="(\operatorname{\mathbf{b}},\tau^{2})" class="ltx_Math" display="inline" id="S3.p1.4.m4.2"><semantics id="S3.p1.4.m4.2a"><mrow id="S3.p1.4.m4.2.2.1" xref="S3.p1.4.m4.2.2.2.cmml"><mo id="S3.p1.4.m4.2.2.1.2" stretchy="false" xref="S3.p1.4.m4.2.2.2.cmml">(</mo><mi id="S3.p1.4.m4.1.1" xref="S3.p1.4.m4.1.1.cmml">𝐛</mi><mo id="S3.p1.4.m4.2.2.1.3" xref="S3.p1.4.m4.2.2.2.cmml">,</mo><msup id="S3.p1.4.m4.2.2.1.1" xref="S3.p1.4.m4.2.2.1.1.cmml"><mi id="S3.p1.4.m4.2.2.1.1.2" xref="S3.p1.4.m4.2.2.1.1.2.cmml">τ</mi><mn id="S3.p1.4.m4.2.2.1.1.3" xref="S3.p1.4.m4.2.2.1.1.3.cmml">2</mn></msup><mo id="S3.p1.4.m4.2.2.1.4" stretchy="false" xref="S3.p1.4.m4.2.2.2.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.p1.4.m4.2b"><interval closure="open" id="S3.p1.4.m4.2.2.2.cmml" xref="S3.p1.4.m4.2.2.1"><ci id="S3.p1.4.m4.1.1.cmml" xref="S3.p1.4.m4.1.1">𝐛</ci><apply id="S3.p1.4.m4.2.2.1.1.cmml" xref="S3.p1.4.m4.2.2.1.1"><csymbol cd="ambiguous" id="S3.p1.4.m4.2.2.1.1.1.cmml" xref="S3.p1.4.m4.2.2.1.1">superscript</csymbol><ci id="S3.p1.4.m4.2.2.1.1.2.cmml" xref="S3.p1.4.m4.2.2.1.1.2">𝜏</ci><cn id="S3.p1.4.m4.2.2.1.1.3.cmml" type="integer" xref="S3.p1.4.m4.2.2.1.1.3">2</cn></apply></interval></annotation-xml><annotation encoding="application/x-tex" id="S3.p1.4.m4.2c">(\operatorname{\mathbf{b}},\tau^{2})</annotation><annotation encoding="application/x-llamapun" id="S3.p1.4.m4.2d">( bold_b , italic_τ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT )</annotation></semantics></math> using regularized weighted least squares regression. The second and third cycles treat <math alttext="\operatorname{\mathbf{c}}_{i(2)}" class="ltx_Math" display="inline" id="S3.p1.5.m5.1"><semantics id="S3.p1.5.m5.1a"><msub id="S3.p1.5.m5.1.2" xref="S3.p1.5.m5.1.2.cmml"><mi id="S3.p1.5.m5.1.2.2" xref="S3.p1.5.m5.1.2.2.cmml">𝐜</mi><mrow id="S3.p1.5.m5.1.1.1" xref="S3.p1.5.m5.1.1.1.cmml"><mi id="S3.p1.5.m5.1.1.1.3" xref="S3.p1.5.m5.1.1.1.3.cmml">i</mi><mo id="S3.p1.5.m5.1.1.1.2" xref="S3.p1.5.m5.1.1.1.2.cmml">⁢</mo><mrow id="S3.p1.5.m5.1.1.1.4.2" xref="S3.p1.5.m5.1.1.1.cmml"><mo id="S3.p1.5.m5.1.1.1.4.2.1" stretchy="false" xref="S3.p1.5.m5.1.1.1.cmml">(</mo><mn id="S3.p1.5.m5.1.1.1.1" xref="S3.p1.5.m5.1.1.1.1.cmml">2</mn><mo id="S3.p1.5.m5.1.1.1.4.2.2" stretchy="false" xref="S3.p1.5.m5.1.1.1.cmml">)</mo></mrow></mrow></msub><annotation-xml encoding="MathML-Content" id="S3.p1.5.m5.1b"><apply id="S3.p1.5.m5.1.2.cmml" xref="S3.p1.5.m5.1.2"><csymbol cd="ambiguous" id="S3.p1.5.m5.1.2.1.cmml" xref="S3.p1.5.m5.1.2">subscript</csymbol><ci id="S3.p1.5.m5.1.2.2.cmml" xref="S3.p1.5.m5.1.2.2">𝐜</ci><apply id="S3.p1.5.m5.1.1.1.cmml" xref="S3.p1.5.m5.1.1.1"><times id="S3.p1.5.m5.1.1.1.2.cmml" xref="S3.p1.5.m5.1.1.1.2"></times><ci id="S3.p1.5.m5.1.1.1.3.cmml" xref="S3.p1.5.m5.1.1.1.3">𝑖</ci><cn id="S3.p1.5.m5.1.1.1.1.cmml" type="integer" xref="S3.p1.5.m5.1.1.1.1">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p1.5.m5.1c">\operatorname{\mathbf{c}}_{i(2)}</annotation><annotation encoding="application/x-llamapun" id="S3.p1.5.m5.1d">bold_c start_POSTSUBSCRIPT italic_i ( 2 ) end_POSTSUBSCRIPT</annotation></semantics></math>’s and <math alttext="\operatorname{\mathbf{c}}_{i(1)}" class="ltx_Math" display="inline" id="S3.p1.6.m6.1"><semantics id="S3.p1.6.m6.1a"><msub id="S3.p1.6.m6.1.2" xref="S3.p1.6.m6.1.2.cmml"><mi id="S3.p1.6.m6.1.2.2" xref="S3.p1.6.m6.1.2.2.cmml">𝐜</mi><mrow id="S3.p1.6.m6.1.1.1" xref="S3.p1.6.m6.1.1.1.cmml"><mi id="S3.p1.6.m6.1.1.1.3" xref="S3.p1.6.m6.1.1.1.3.cmml">i</mi><mo id="S3.p1.6.m6.1.1.1.2" xref="S3.p1.6.m6.1.1.1.2.cmml">⁢</mo><mrow id="S3.p1.6.m6.1.1.1.4.2" xref="S3.p1.6.m6.1.1.1.cmml"><mo id="S3.p1.6.m6.1.1.1.4.2.1" stretchy="false" xref="S3.p1.6.m6.1.1.1.cmml">(</mo><mn id="S3.p1.6.m6.1.1.1.1" xref="S3.p1.6.m6.1.1.1.1.cmml">1</mn><mo id="S3.p1.6.m6.1.1.1.4.2.2" stretchy="false" xref="S3.p1.6.m6.1.1.1.cmml">)</mo></mrow></mrow></msub><annotation-xml encoding="MathML-Content" id="S3.p1.6.m6.1b"><apply id="S3.p1.6.m6.1.2.cmml" xref="S3.p1.6.m6.1.2"><csymbol cd="ambiguous" id="S3.p1.6.m6.1.2.1.cmml" xref="S3.p1.6.m6.1.2">subscript</csymbol><ci id="S3.p1.6.m6.1.2.2.cmml" xref="S3.p1.6.m6.1.2.2">𝐜</ci><apply id="S3.p1.6.m6.1.1.1.cmml" xref="S3.p1.6.m6.1.1.1"><times id="S3.p1.6.m6.1.1.1.2.cmml" xref="S3.p1.6.m6.1.1.1.2"></times><ci id="S3.p1.6.m6.1.1.1.3.cmml" xref="S3.p1.6.m6.1.1.1.3">𝑖</ci><cn id="S3.p1.6.m6.1.1.1.1.cmml" type="integer" xref="S3.p1.6.m6.1.1.1.1">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p1.6.m6.1c">\operatorname{\mathbf{c}}_{i(1)}</annotation><annotation encoding="application/x-llamapun" id="S3.p1.6.m6.1d">bold_c start_POSTSUBSCRIPT italic_i ( 1 ) end_POSTSUBSCRIPT</annotation></semantics></math>’s in (<a class="ltx_ref" href="https://arxiv.org/html/2503.13782v1#S2.E6" title="In 2.2 Reduction to a Structured Mixed Model ‣ 2 Mixed Model Trace Regression ‣ Regularized Parameter Estimation in Mixed Model Trace Regression"><span class="ltx_text ltx_ref_tag">6</span></a>) as missing data and estimate <math alttext="\operatorname{\mathbf{L}}_{1}" class="ltx_Math" display="inline" id="S3.p1.7.m7.1"><semantics id="S3.p1.7.m7.1a"><msub id="S3.p1.7.m7.1.1" xref="S3.p1.7.m7.1.1.cmml"><mi id="S3.p1.7.m7.1.1.2" xref="S3.p1.7.m7.1.1.2.cmml">𝐋</mi><mn id="S3.p1.7.m7.1.1.3" xref="S3.p1.7.m7.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S3.p1.7.m7.1b"><apply id="S3.p1.7.m7.1.1.cmml" xref="S3.p1.7.m7.1.1"><csymbol cd="ambiguous" id="S3.p1.7.m7.1.1.1.cmml" xref="S3.p1.7.m7.1.1">subscript</csymbol><ci id="S3.p1.7.m7.1.1.2.cmml" xref="S3.p1.7.m7.1.1.2">𝐋</ci><cn id="S3.p1.7.m7.1.1.3.cmml" type="integer" xref="S3.p1.7.m7.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p1.7.m7.1c">\operatorname{\mathbf{L}}_{1}</annotation><annotation encoding="application/x-llamapun" id="S3.p1.7.m7.1d">bold_L start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="\operatorname{\mathbf{L}}_{2}" class="ltx_Math" display="inline" id="S3.p1.8.m8.1"><semantics id="S3.p1.8.m8.1a"><msub id="S3.p1.8.m8.1.1" xref="S3.p1.8.m8.1.1.cmml"><mi id="S3.p1.8.m8.1.1.2" xref="S3.p1.8.m8.1.1.2.cmml">𝐋</mi><mn id="S3.p1.8.m8.1.1.3" xref="S3.p1.8.m8.1.1.3.cmml">2</mn></msub><annotation-xml encoding="MathML-Content" id="S3.p1.8.m8.1b"><apply id="S3.p1.8.m8.1.1.cmml" 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The derivation of the parameter updates is in the supplementary material.</p> </div> <div class="ltx_para" id="S3.p2"> <p class="ltx_p" id="S3.p2.12">Consider the <math alttext="(t+1)" class="ltx_Math" display="inline" id="S3.p2.1.m1.1"><semantics id="S3.p2.1.m1.1a"><mrow id="S3.p2.1.m1.1.1.1" xref="S3.p2.1.m1.1.1.1.1.cmml"><mo id="S3.p2.1.m1.1.1.1.2" stretchy="false" xref="S3.p2.1.m1.1.1.1.1.cmml">(</mo><mrow id="S3.p2.1.m1.1.1.1.1" xref="S3.p2.1.m1.1.1.1.1.cmml"><mi id="S3.p2.1.m1.1.1.1.1.2" xref="S3.p2.1.m1.1.1.1.1.2.cmml">t</mi><mo id="S3.p2.1.m1.1.1.1.1.1" xref="S3.p2.1.m1.1.1.1.1.1.cmml">+</mo><mn id="S3.p2.1.m1.1.1.1.1.3" xref="S3.p2.1.m1.1.1.1.1.3.cmml">1</mn></mrow><mo id="S3.p2.1.m1.1.1.1.3" stretchy="false" xref="S3.p2.1.m1.1.1.1.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.p2.1.m1.1b"><apply id="S3.p2.1.m1.1.1.1.1.cmml" xref="S3.p2.1.m1.1.1.1"><plus id="S3.p2.1.m1.1.1.1.1.1.cmml" xref="S3.p2.1.m1.1.1.1.1.1"></plus><ci id="S3.p2.1.m1.1.1.1.1.2.cmml" xref="S3.p2.1.m1.1.1.1.1.2">𝑡</ci><cn id="S3.p2.1.m1.1.1.1.1.3.cmml" type="integer" xref="S3.p2.1.m1.1.1.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p2.1.m1.1c">(t+1)</annotation><annotation encoding="application/x-llamapun" id="S3.p2.1.m1.1d">( italic_t + 1 )</annotation></semantics></math>th AECM iteration. Let <math alttext="\operatorname{{\bm{\theta}}}^{(t)}=(\operatorname{\mathbf{b}}^{(t)},% \operatorname{\mathbf{L}}_{1}^{(t)},\operatorname{\mathbf{L}}_{2}^{(t)},\tau^{% 2(t)})" class="ltx_Math" display="inline" id="S3.p2.2.m2.9"><semantics id="S3.p2.2.m2.9a"><mrow id="S3.p2.2.m2.9.9" xref="S3.p2.2.m2.9.9.cmml"><msup id="S3.p2.2.m2.9.9.6" xref="S3.p2.2.m2.9.9.6.cmml"><mi id="S3.p2.2.m2.9.9.6.2" xref="S3.p2.2.m2.9.9.6.2.cmml">𝜽</mi><mrow id="S3.p2.2.m2.1.1.1.3" xref="S3.p2.2.m2.9.9.6.cmml"><mo id="S3.p2.2.m2.1.1.1.3.1" stretchy="false" xref="S3.p2.2.m2.9.9.6.cmml">(</mo><mi id="S3.p2.2.m2.1.1.1.1" xref="S3.p2.2.m2.1.1.1.1.cmml">t</mi><mo id="S3.p2.2.m2.1.1.1.3.2" stretchy="false" xref="S3.p2.2.m2.9.9.6.cmml">)</mo></mrow></msup><mo id="S3.p2.2.m2.9.9.5" xref="S3.p2.2.m2.9.9.5.cmml">=</mo><mrow id="S3.p2.2.m2.9.9.4.4" xref="S3.p2.2.m2.9.9.4.5.cmml"><mo id="S3.p2.2.m2.9.9.4.4.5" stretchy="false" xref="S3.p2.2.m2.9.9.4.5.cmml">(</mo><msup id="S3.p2.2.m2.6.6.1.1.1" xref="S3.p2.2.m2.6.6.1.1.1.cmml"><mi id="S3.p2.2.m2.6.6.1.1.1.2" xref="S3.p2.2.m2.6.6.1.1.1.2.cmml">𝐛</mi><mrow id="S3.p2.2.m2.2.2.1.3" xref="S3.p2.2.m2.6.6.1.1.1.cmml"><mo id="S3.p2.2.m2.2.2.1.3.1" stretchy="false" xref="S3.p2.2.m2.6.6.1.1.1.cmml">(</mo><mi id="S3.p2.2.m2.2.2.1.1" xref="S3.p2.2.m2.2.2.1.1.cmml">t</mi><mo id="S3.p2.2.m2.2.2.1.3.2" stretchy="false" xref="S3.p2.2.m2.6.6.1.1.1.cmml">)</mo></mrow></msup><mo id="S3.p2.2.m2.9.9.4.4.6" xref="S3.p2.2.m2.9.9.4.5.cmml">,</mo><msubsup id="S3.p2.2.m2.7.7.2.2.2" xref="S3.p2.2.m2.7.7.2.2.2.cmml"><mi id="S3.p2.2.m2.7.7.2.2.2.2.2" xref="S3.p2.2.m2.7.7.2.2.2.2.2.cmml">𝐋</mi><mn id="S3.p2.2.m2.7.7.2.2.2.2.3" xref="S3.p2.2.m2.7.7.2.2.2.2.3.cmml">1</mn><mrow id="S3.p2.2.m2.3.3.1.3" xref="S3.p2.2.m2.7.7.2.2.2.cmml"><mo id="S3.p2.2.m2.3.3.1.3.1" stretchy="false" xref="S3.p2.2.m2.7.7.2.2.2.cmml">(</mo><mi id="S3.p2.2.m2.3.3.1.1" xref="S3.p2.2.m2.3.3.1.1.cmml">t</mi><mo id="S3.p2.2.m2.3.3.1.3.2" stretchy="false" xref="S3.p2.2.m2.7.7.2.2.2.cmml">)</mo></mrow></msubsup><mo id="S3.p2.2.m2.9.9.4.4.7" xref="S3.p2.2.m2.9.9.4.5.cmml">,</mo><msubsup id="S3.p2.2.m2.8.8.3.3.3" xref="S3.p2.2.m2.8.8.3.3.3.cmml"><mi id="S3.p2.2.m2.8.8.3.3.3.2.2" xref="S3.p2.2.m2.8.8.3.3.3.2.2.cmml">𝐋</mi><mn id="S3.p2.2.m2.8.8.3.3.3.2.3" xref="S3.p2.2.m2.8.8.3.3.3.2.3.cmml">2</mn><mrow id="S3.p2.2.m2.4.4.1.3" xref="S3.p2.2.m2.8.8.3.3.3.cmml"><mo id="S3.p2.2.m2.4.4.1.3.1" stretchy="false" xref="S3.p2.2.m2.8.8.3.3.3.cmml">(</mo><mi id="S3.p2.2.m2.4.4.1.1" xref="S3.p2.2.m2.4.4.1.1.cmml">t</mi><mo id="S3.p2.2.m2.4.4.1.3.2" stretchy="false" xref="S3.p2.2.m2.8.8.3.3.3.cmml">)</mo></mrow></msubsup><mo id="S3.p2.2.m2.9.9.4.4.8" xref="S3.p2.2.m2.9.9.4.5.cmml">,</mo><msup id="S3.p2.2.m2.9.9.4.4.4" xref="S3.p2.2.m2.9.9.4.4.4.cmml"><mi id="S3.p2.2.m2.9.9.4.4.4.2" xref="S3.p2.2.m2.9.9.4.4.4.2.cmml">τ</mi><mrow id="S3.p2.2.m2.5.5.1" xref="S3.p2.2.m2.5.5.1.cmml"><mn id="S3.p2.2.m2.5.5.1.3" xref="S3.p2.2.m2.5.5.1.3.cmml">2</mn><mo id="S3.p2.2.m2.5.5.1.2" xref="S3.p2.2.m2.5.5.1.2.cmml">⁢</mo><mrow id="S3.p2.2.m2.5.5.1.4.2" xref="S3.p2.2.m2.5.5.1.cmml"><mo id="S3.p2.2.m2.5.5.1.4.2.1" stretchy="false" xref="S3.p2.2.m2.5.5.1.cmml">(</mo><mi id="S3.p2.2.m2.5.5.1.1" xref="S3.p2.2.m2.5.5.1.1.cmml">t</mi><mo id="S3.p2.2.m2.5.5.1.4.2.2" stretchy="false" xref="S3.p2.2.m2.5.5.1.cmml">)</mo></mrow></mrow></msup><mo id="S3.p2.2.m2.9.9.4.4.9" stretchy="false" xref="S3.p2.2.m2.9.9.4.5.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.p2.2.m2.9b"><apply id="S3.p2.2.m2.9.9.cmml" xref="S3.p2.2.m2.9.9"><eq id="S3.p2.2.m2.9.9.5.cmml" xref="S3.p2.2.m2.9.9.5"></eq><apply id="S3.p2.2.m2.9.9.6.cmml" xref="S3.p2.2.m2.9.9.6"><csymbol cd="ambiguous" id="S3.p2.2.m2.9.9.6.1.cmml" xref="S3.p2.2.m2.9.9.6">superscript</csymbol><ci id="S3.p2.2.m2.9.9.6.2.cmml" xref="S3.p2.2.m2.9.9.6.2">𝜽</ci><ci id="S3.p2.2.m2.1.1.1.1.cmml" xref="S3.p2.2.m2.1.1.1.1">𝑡</ci></apply><vector id="S3.p2.2.m2.9.9.4.5.cmml" 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xref="S3.p2.2.m2.5.5.1.3">2</cn><ci id="S3.p2.2.m2.5.5.1.1.cmml" xref="S3.p2.2.m2.5.5.1.1">𝑡</ci></apply></apply></vector></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p2.2.m2.9c">\operatorname{{\bm{\theta}}}^{(t)}=(\operatorname{\mathbf{b}}^{(t)},% \operatorname{\mathbf{L}}_{1}^{(t)},\operatorname{\mathbf{L}}_{2}^{(t)},\tau^{% 2(t)})</annotation><annotation encoding="application/x-llamapun" id="S3.p2.2.m2.9d">bold_italic_θ start_POSTSUPERSCRIPT ( italic_t ) end_POSTSUPERSCRIPT = ( bold_b start_POSTSUPERSCRIPT ( italic_t ) end_POSTSUPERSCRIPT , bold_L start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_t ) end_POSTSUPERSCRIPT , bold_L start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_t ) end_POSTSUPERSCRIPT , italic_τ start_POSTSUPERSCRIPT 2 ( italic_t ) end_POSTSUPERSCRIPT )</annotation></semantics></math> be the parameter estimate at the end of the <math alttext="t" 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">t</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">𝑡</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.p2.3.m3.1c">t</annotation><annotation encoding="application/x-llamapun" id="S3.p2.3.m3.1d">italic_t</annotation></semantics></math>th AECM iteration. Define <math alttext="\operatorname{{\mathbf{\Lambda}}}_{i}^{(t)}=\operatorname{\mathbf{Z}}_{i(1)}% \operatorname{\mathbf{L}}_{(1)}^{(t)}\operatorname{\mathbf{L}}_{(1)}^{(t)\top}% \operatorname{\mathbf{Z}}_{i(1)}^{\top}+\operatorname{\mathbf{I}}_{m_{i}}" class="ltx_Math" display="inline" id="S3.p2.4.m4.7"><semantics id="S3.p2.4.m4.7a"><mrow id="S3.p2.4.m4.7.8" xref="S3.p2.4.m4.7.8.cmml"><msubsup id="S3.p2.4.m4.7.8.2" xref="S3.p2.4.m4.7.8.2.cmml"><mi id="S3.p2.4.m4.7.8.2.2.2" xref="S3.p2.4.m4.7.8.2.2.2.cmml">𝚲</mi><mi id="S3.p2.4.m4.7.8.2.2.3" xref="S3.p2.4.m4.7.8.2.2.3.cmml">i</mi><mrow id="S3.p2.4.m4.1.1.1.3" xref="S3.p2.4.m4.7.8.2.cmml"><mo id="S3.p2.4.m4.1.1.1.3.1" stretchy="false" xref="S3.p2.4.m4.7.8.2.cmml">(</mo><mi id="S3.p2.4.m4.1.1.1.1" xref="S3.p2.4.m4.1.1.1.1.cmml">t</mi><mo id="S3.p2.4.m4.1.1.1.3.2" stretchy="false" xref="S3.p2.4.m4.7.8.2.cmml">)</mo></mrow></msubsup><mo id="S3.p2.4.m4.7.8.1" xref="S3.p2.4.m4.7.8.1.cmml">=</mo><mrow id="S3.p2.4.m4.7.8.3" 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\operatorname{\mathbf{Z}}_{i(1)}^{\top}+\operatorname{\mathbf{I}}_{m_{i}}</annotation><annotation encoding="application/x-llamapun" id="S3.p2.4.m4.7d">bold_Λ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_t ) end_POSTSUPERSCRIPT = bold_Z start_POSTSUBSCRIPT italic_i ( 1 ) end_POSTSUBSCRIPT bold_L start_POSTSUBSCRIPT ( 1 ) end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_t ) end_POSTSUPERSCRIPT bold_L start_POSTSUBSCRIPT ( 1 ) end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_t ) ⊤ end_POSTSUPERSCRIPT bold_Z start_POSTSUBSCRIPT italic_i ( 1 ) end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ⊤ end_POSTSUPERSCRIPT + bold_I start_POSTSUBSCRIPT italic_m start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math> for <math alttext="i=1,\ldots,n" class="ltx_Math" display="inline" id="S3.p2.5.m5.3"><semantics id="S3.p2.5.m5.3a"><mrow id="S3.p2.5.m5.3.4" xref="S3.p2.5.m5.3.4.cmml"><mi id="S3.p2.5.m5.3.4.2" 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xref="S3.p2.5.m5.3.3">𝑛</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p2.5.m5.3c">i=1,\ldots,n</annotation><annotation encoding="application/x-llamapun" id="S3.p2.5.m5.3d">italic_i = 1 , … , italic_n</annotation></semantics></math>, where, as defined in Section <a class="ltx_ref" href="https://arxiv.org/html/2503.13782v1#S2.SS2" title="2.2 Reduction to a Structured Mixed Model ‣ 2 Mixed Model Trace Regression ‣ Regularized Parameter Estimation in Mixed Model Trace Regression"><span class="ltx_text ltx_ref_tag">2.2</span></a>, <math alttext="\operatorname{\mathbf{L}}_{(1)}^{(t)}=\operatorname{\mathbf{L}}_{2}^{(t)}% \otimes\operatorname{\mathbf{L}}_{1}^{(t)}" class="ltx_Math" display="inline" id="S3.p2.6.m6.4"><semantics id="S3.p2.6.m6.4a"><mrow id="S3.p2.6.m6.4.5" xref="S3.p2.6.m6.4.5.cmml"><msubsup id="S3.p2.6.m6.4.5.2" xref="S3.p2.6.m6.4.5.2.cmml"><mi id="S3.p2.6.m6.4.5.2.2.2" xref="S3.p2.6.m6.4.5.2.2.2.cmml">𝐋</mi><mrow id="S3.p2.6.m6.1.1.1.3" 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id="S3.p2.6.m6.4c">\operatorname{\mathbf{L}}_{(1)}^{(t)}=\operatorname{\mathbf{L}}_{2}^{(t)}% \otimes\operatorname{\mathbf{L}}_{1}^{(t)}</annotation><annotation encoding="application/x-llamapun" id="S3.p2.6.m6.4d">bold_L start_POSTSUBSCRIPT ( 1 ) end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_t ) end_POSTSUPERSCRIPT = bold_L start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_t ) end_POSTSUPERSCRIPT ⊗ bold_L start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_t ) end_POSTSUPERSCRIPT</annotation></semantics></math> and <math alttext="\operatorname{\mathbf{Z}}_{i(1)}" class="ltx_Math" display="inline" id="S3.p2.7.m7.1"><semantics id="S3.p2.7.m7.1a"><msub id="S3.p2.7.m7.1.2" xref="S3.p2.7.m7.1.2.cmml"><mi id="S3.p2.7.m7.1.2.2" xref="S3.p2.7.m7.1.2.2.cmml">𝐙</mi><mrow id="S3.p2.7.m7.1.1.1" xref="S3.p2.7.m7.1.1.1.cmml"><mi id="S3.p2.7.m7.1.1.1.3" xref="S3.p2.7.m7.1.1.1.3.cmml">i</mi><mo id="S3.p2.7.m7.1.1.1.2" xref="S3.p2.7.m7.1.1.1.2.cmml">⁢</mo><mrow id="S3.p2.7.m7.1.1.1.4.2" xref="S3.p2.7.m7.1.1.1.cmml"><mo id="S3.p2.7.m7.1.1.1.4.2.1" stretchy="false" xref="S3.p2.7.m7.1.1.1.cmml">(</mo><mn id="S3.p2.7.m7.1.1.1.1" xref="S3.p2.7.m7.1.1.1.1.cmml">1</mn><mo id="S3.p2.7.m7.1.1.1.4.2.2" stretchy="false" xref="S3.p2.7.m7.1.1.1.cmml">)</mo></mrow></mrow></msub><annotation-xml encoding="MathML-Content" id="S3.p2.7.m7.1b"><apply id="S3.p2.7.m7.1.2.cmml" xref="S3.p2.7.m7.1.2"><csymbol cd="ambiguous" id="S3.p2.7.m7.1.2.1.cmml" xref="S3.p2.7.m7.1.2">subscript</csymbol><ci id="S3.p2.7.m7.1.2.2.cmml" xref="S3.p2.7.m7.1.2.2">𝐙</ci><apply id="S3.p2.7.m7.1.1.1.cmml" xref="S3.p2.7.m7.1.1.1"><times id="S3.p2.7.m7.1.1.1.2.cmml" xref="S3.p2.7.m7.1.1.1.2"></times><ci id="S3.p2.7.m7.1.1.1.3.cmml" xref="S3.p2.7.m7.1.1.1.3">𝑖</ci><cn id="S3.p2.7.m7.1.1.1.1.cmml" type="integer" xref="S3.p2.7.m7.1.1.1.1">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p2.7.m7.1c">\operatorname{\mathbf{Z}}_{i(1)}</annotation><annotation encoding="application/x-llamapun" id="S3.p2.7.m7.1d">bold_Z start_POSTSUBSCRIPT italic_i ( 1 ) end_POSTSUBSCRIPT</annotation></semantics></math> is the matrix whose <math alttext="j" class="ltx_Math" display="inline" id="S3.p2.8.m8.1"><semantics id="S3.p2.8.m8.1a"><mi id="S3.p2.8.m8.1.1" xref="S3.p2.8.m8.1.1.cmml">j</mi><annotation-xml encoding="MathML-Content" id="S3.p2.8.m8.1b"><ci id="S3.p2.8.m8.1.1.cmml" xref="S3.p2.8.m8.1.1">𝑗</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.p2.8.m8.1c">j</annotation><annotation encoding="application/x-llamapun" id="S3.p2.8.m8.1d">italic_j</annotation></semantics></math>th row is <math alttext="\operatorname{\mathbf{z}}_{ij(1)}^{\top}" class="ltx_Math" display="inline" id="S3.p2.9.m9.1"><semantics id="S3.p2.9.m9.1a"><msubsup id="S3.p2.9.m9.1.2" xref="S3.p2.9.m9.1.2.cmml"><mi id="S3.p2.9.m9.1.2.2.2" xref="S3.p2.9.m9.1.2.2.2.cmml">𝐳</mi><mrow id="S3.p2.9.m9.1.1.1" xref="S3.p2.9.m9.1.1.1.cmml"><mi id="S3.p2.9.m9.1.1.1.3" xref="S3.p2.9.m9.1.1.1.3.cmml">i</mi><mo id="S3.p2.9.m9.1.1.1.2" xref="S3.p2.9.m9.1.1.1.2.cmml">⁢</mo><mi id="S3.p2.9.m9.1.1.1.4" xref="S3.p2.9.m9.1.1.1.4.cmml">j</mi><mo id="S3.p2.9.m9.1.1.1.2a" xref="S3.p2.9.m9.1.1.1.2.cmml">⁢</mo><mrow id="S3.p2.9.m9.1.1.1.5.2" xref="S3.p2.9.m9.1.1.1.cmml"><mo id="S3.p2.9.m9.1.1.1.5.2.1" stretchy="false" xref="S3.p2.9.m9.1.1.1.cmml">(</mo><mn id="S3.p2.9.m9.1.1.1.1" xref="S3.p2.9.m9.1.1.1.1.cmml">1</mn><mo id="S3.p2.9.m9.1.1.1.5.2.2" stretchy="false" xref="S3.p2.9.m9.1.1.1.cmml">)</mo></mrow></mrow><mo id="S3.p2.9.m9.1.2.3" xref="S3.p2.9.m9.1.2.3.cmml">⊤</mo></msubsup><annotation-xml encoding="MathML-Content" id="S3.p2.9.m9.1b"><apply id="S3.p2.9.m9.1.2.cmml" xref="S3.p2.9.m9.1.2"><csymbol cd="ambiguous" id="S3.p2.9.m9.1.2.1.cmml" xref="S3.p2.9.m9.1.2">superscript</csymbol><apply id="S3.p2.9.m9.1.2.2.cmml" xref="S3.p2.9.m9.1.2"><csymbol cd="ambiguous" id="S3.p2.9.m9.1.2.2.1.cmml" xref="S3.p2.9.m9.1.2">subscript</csymbol><ci id="S3.p2.9.m9.1.2.2.2.cmml" xref="S3.p2.9.m9.1.2.2.2">𝐳</ci><apply id="S3.p2.9.m9.1.1.1.cmml" xref="S3.p2.9.m9.1.1.1"><times id="S3.p2.9.m9.1.1.1.2.cmml" xref="S3.p2.9.m9.1.1.1.2"></times><ci id="S3.p2.9.m9.1.1.1.3.cmml" xref="S3.p2.9.m9.1.1.1.3">𝑖</ci><ci id="S3.p2.9.m9.1.1.1.4.cmml" xref="S3.p2.9.m9.1.1.1.4">𝑗</ci><cn id="S3.p2.9.m9.1.1.1.1.cmml" type="integer" xref="S3.p2.9.m9.1.1.1.1">1</cn></apply></apply><csymbol cd="latexml" id="S3.p2.9.m9.1.2.3.cmml" xref="S3.p2.9.m9.1.2.3">top</csymbol></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p2.9.m9.1c">\operatorname{\mathbf{z}}_{ij(1)}^{\top}</annotation><annotation encoding="application/x-llamapun" id="S3.p2.9.m9.1d">bold_z start_POSTSUBSCRIPT italic_i italic_j ( 1 ) end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ⊤ end_POSTSUPERSCRIPT</annotation></semantics></math>. The first AECM cycle updates <math alttext="(\operatorname{\mathbf{b}},\tau^{2})" class="ltx_Math" display="inline" id="S3.p2.10.m10.2"><semantics id="S3.p2.10.m10.2a"><mrow id="S3.p2.10.m10.2.2.1" xref="S3.p2.10.m10.2.2.2.cmml"><mo id="S3.p2.10.m10.2.2.1.2" stretchy="false" xref="S3.p2.10.m10.2.2.2.cmml">(</mo><mi id="S3.p2.10.m10.1.1" xref="S3.p2.10.m10.1.1.cmml">𝐛</mi><mo id="S3.p2.10.m10.2.2.1.3" xref="S3.p2.10.m10.2.2.2.cmml">,</mo><msup id="S3.p2.10.m10.2.2.1.1" xref="S3.p2.10.m10.2.2.1.1.cmml"><mi id="S3.p2.10.m10.2.2.1.1.2" xref="S3.p2.10.m10.2.2.1.1.2.cmml">τ</mi><mn id="S3.p2.10.m10.2.2.1.1.3" xref="S3.p2.10.m10.2.2.1.1.3.cmml">2</mn></msup><mo id="S3.p2.10.m10.2.2.1.4" stretchy="false" xref="S3.p2.10.m10.2.2.2.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.p2.10.m10.2b"><interval closure="open" id="S3.p2.10.m10.2.2.2.cmml" xref="S3.p2.10.m10.2.2.1"><ci id="S3.p2.10.m10.1.1.cmml" xref="S3.p2.10.m10.1.1">𝐛</ci><apply id="S3.p2.10.m10.2.2.1.1.cmml" xref="S3.p2.10.m10.2.2.1.1"><csymbol cd="ambiguous" id="S3.p2.10.m10.2.2.1.1.1.cmml" xref="S3.p2.10.m10.2.2.1.1">superscript</csymbol><ci id="S3.p2.10.m10.2.2.1.1.2.cmml" xref="S3.p2.10.m10.2.2.1.1.2">𝜏</ci><cn id="S3.p2.10.m10.2.2.1.1.3.cmml" type="integer" xref="S3.p2.10.m10.2.2.1.1.3">2</cn></apply></interval></annotation-xml><annotation encoding="application/x-tex" id="S3.p2.10.m10.2c">(\operatorname{\mathbf{b}},\tau^{2})</annotation><annotation encoding="application/x-llamapun" id="S3.p2.10.m10.2d">( bold_b , italic_τ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT )</annotation></semantics></math> given <math alttext="(\operatorname{\mathbf{L}}_{1}^{(t)},\operatorname{\mathbf{L}}_{2}^{(t)})" class="ltx_Math" display="inline" id="S3.p2.11.m11.4"><semantics id="S3.p2.11.m11.4a"><mrow id="S3.p2.11.m11.4.4.2" xref="S3.p2.11.m11.4.4.3.cmml"><mo id="S3.p2.11.m11.4.4.2.3" stretchy="false" xref="S3.p2.11.m11.4.4.3.cmml">(</mo><msubsup id="S3.p2.11.m11.3.3.1.1" xref="S3.p2.11.m11.3.3.1.1.cmml"><mi id="S3.p2.11.m11.3.3.1.1.2.2" xref="S3.p2.11.m11.3.3.1.1.2.2.cmml">𝐋</mi><mn id="S3.p2.11.m11.3.3.1.1.2.3" xref="S3.p2.11.m11.3.3.1.1.2.3.cmml">1</mn><mrow id="S3.p2.11.m11.1.1.1.3" xref="S3.p2.11.m11.3.3.1.1.cmml"><mo id="S3.p2.11.m11.1.1.1.3.1" stretchy="false" xref="S3.p2.11.m11.3.3.1.1.cmml">(</mo><mi id="S3.p2.11.m11.1.1.1.1" xref="S3.p2.11.m11.1.1.1.1.cmml">t</mi><mo id="S3.p2.11.m11.1.1.1.3.2" stretchy="false" xref="S3.p2.11.m11.3.3.1.1.cmml">)</mo></mrow></msubsup><mo id="S3.p2.11.m11.4.4.2.4" xref="S3.p2.11.m11.4.4.3.cmml">,</mo><msubsup id="S3.p2.11.m11.4.4.2.2" xref="S3.p2.11.m11.4.4.2.2.cmml"><mi id="S3.p2.11.m11.4.4.2.2.2.2" xref="S3.p2.11.m11.4.4.2.2.2.2.cmml">𝐋</mi><mn id="S3.p2.11.m11.4.4.2.2.2.3" xref="S3.p2.11.m11.4.4.2.2.2.3.cmml">2</mn><mrow id="S3.p2.11.m11.2.2.1.3" xref="S3.p2.11.m11.4.4.2.2.cmml"><mo id="S3.p2.11.m11.2.2.1.3.1" stretchy="false" xref="S3.p2.11.m11.4.4.2.2.cmml">(</mo><mi id="S3.p2.11.m11.2.2.1.1" xref="S3.p2.11.m11.2.2.1.1.cmml">t</mi><mo id="S3.p2.11.m11.2.2.1.3.2" stretchy="false" xref="S3.p2.11.m11.4.4.2.2.cmml">)</mo></mrow></msubsup><mo id="S3.p2.11.m11.4.4.2.5" stretchy="false" xref="S3.p2.11.m11.4.4.3.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.p2.11.m11.4b"><interval closure="open" id="S3.p2.11.m11.4.4.3.cmml" xref="S3.p2.11.m11.4.4.2"><apply id="S3.p2.11.m11.3.3.1.1.cmml" xref="S3.p2.11.m11.3.3.1.1"><csymbol cd="ambiguous" id="S3.p2.11.m11.3.3.1.1.1.cmml" xref="S3.p2.11.m11.3.3.1.1">superscript</csymbol><apply id="S3.p2.11.m11.3.3.1.1.2.cmml" xref="S3.p2.11.m11.3.3.1.1"><csymbol cd="ambiguous" id="S3.p2.11.m11.3.3.1.1.2.1.cmml" xref="S3.p2.11.m11.3.3.1.1">subscript</csymbol><ci id="S3.p2.11.m11.3.3.1.1.2.2.cmml" xref="S3.p2.11.m11.3.3.1.1.2.2">𝐋</ci><cn id="S3.p2.11.m11.3.3.1.1.2.3.cmml" type="integer" xref="S3.p2.11.m11.3.3.1.1.2.3">1</cn></apply><ci id="S3.p2.11.m11.1.1.1.1.cmml" xref="S3.p2.11.m11.1.1.1.1">𝑡</ci></apply><apply id="S3.p2.11.m11.4.4.2.2.cmml" xref="S3.p2.11.m11.4.4.2.2"><csymbol cd="ambiguous" id="S3.p2.11.m11.4.4.2.2.1.cmml" xref="S3.p2.11.m11.4.4.2.2">superscript</csymbol><apply id="S3.p2.11.m11.4.4.2.2.2.cmml" xref="S3.p2.11.m11.4.4.2.2"><csymbol cd="ambiguous" id="S3.p2.11.m11.4.4.2.2.2.1.cmml" xref="S3.p2.11.m11.4.4.2.2">subscript</csymbol><ci id="S3.p2.11.m11.4.4.2.2.2.2.cmml" xref="S3.p2.11.m11.4.4.2.2.2.2">𝐋</ci><cn id="S3.p2.11.m11.4.4.2.2.2.3.cmml" type="integer" xref="S3.p2.11.m11.4.4.2.2.2.3">2</cn></apply><ci id="S3.p2.11.m11.2.2.1.1.cmml" xref="S3.p2.11.m11.2.2.1.1">𝑡</ci></apply></interval></annotation-xml><annotation encoding="application/x-tex" id="S3.p2.11.m11.4c">(\operatorname{\mathbf{L}}_{1}^{(t)},\operatorname{\mathbf{L}}_{2}^{(t)})</annotation><annotation encoding="application/x-llamapun" id="S3.p2.11.m11.4d">( bold_L start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_t ) end_POSTSUPERSCRIPT , bold_L start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_t ) end_POSTSUPERSCRIPT )</annotation></semantics></math>. Marginalizing over <math alttext="\operatorname{\mathbf{c}}_{i(1)}" class="ltx_Math" display="inline" id="S3.p2.12.m12.1"><semantics id="S3.p2.12.m12.1a"><msub id="S3.p2.12.m12.1.2" xref="S3.p2.12.m12.1.2.cmml"><mi id="S3.p2.12.m12.1.2.2" xref="S3.p2.12.m12.1.2.2.cmml">𝐜</mi><mrow id="S3.p2.12.m12.1.1.1" xref="S3.p2.12.m12.1.1.1.cmml"><mi id="S3.p2.12.m12.1.1.1.3" xref="S3.p2.12.m12.1.1.1.3.cmml">i</mi><mo id="S3.p2.12.m12.1.1.1.2" xref="S3.p2.12.m12.1.1.1.2.cmml">⁢</mo><mrow id="S3.p2.12.m12.1.1.1.4.2" xref="S3.p2.12.m12.1.1.1.cmml"><mo id="S3.p2.12.m12.1.1.1.4.2.1" stretchy="false" xref="S3.p2.12.m12.1.1.1.cmml">(</mo><mn id="S3.p2.12.m12.1.1.1.1" xref="S3.p2.12.m12.1.1.1.1.cmml">1</mn><mo id="S3.p2.12.m12.1.1.1.4.2.2" stretchy="false" xref="S3.p2.12.m12.1.1.1.cmml">)</mo></mrow></mrow></msub><annotation-xml encoding="MathML-Content" id="S3.p2.12.m12.1b"><apply id="S3.p2.12.m12.1.2.cmml" xref="S3.p2.12.m12.1.2"><csymbol cd="ambiguous" id="S3.p2.12.m12.1.2.1.cmml" xref="S3.p2.12.m12.1.2">subscript</csymbol><ci id="S3.p2.12.m12.1.2.2.cmml" xref="S3.p2.12.m12.1.2.2">𝐜</ci><apply id="S3.p2.12.m12.1.1.1.cmml" xref="S3.p2.12.m12.1.1.1"><times id="S3.p2.12.m12.1.1.1.2.cmml" xref="S3.p2.12.m12.1.1.1.2"></times><ci id="S3.p2.12.m12.1.1.1.3.cmml" xref="S3.p2.12.m12.1.1.1.3">𝑖</ci><cn id="S3.p2.12.m12.1.1.1.1.cmml" type="integer" xref="S3.p2.12.m12.1.1.1.1">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p2.12.m12.1c">\operatorname{\mathbf{c}}_{i(1)}</annotation><annotation encoding="application/x-llamapun" id="S3.p2.12.m12.1d">bold_c start_POSTSUBSCRIPT italic_i ( 1 ) end_POSTSUBSCRIPT</annotation></semantics></math> in (<a class="ltx_ref" href="https://arxiv.org/html/2503.13782v1#S2.E4" title="In 2.2 Reduction to a Structured Mixed Model ‣ 2 Mixed Model Trace Regression ‣ Regularized Parameter Estimation in Mixed Model Trace Regression"><span class="ltx_text ltx_ref_tag">4</span></a>) implies that</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="S3.EGx7"> <tbody id="S3.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\operatorname{\mathbf{y}}_{i}=\operatorname{\mathbf{X}}_{i}% \operatorname{\mathbf{b}}+\bar{\operatorname{\mathbf{e}}}_{i},\quad% \operatorname{\mathbb{E}}(\overline{\operatorname{\mathbf{e}}}_{i})=% \operatorname{\mathbf{0}},\quad\text{Cov}(\overline{\operatorname{\mathbf{e}}}% _{i})=\tau^{2}\operatorname{{\mathbf{\Lambda}}}_{i}^{(t)},\quad i=1,\ldots,n." class="ltx_Math" display="inline" id="S3.E7.m1.6"><semantics id="S3.E7.m1.6a"><mrow id="S3.E7.m1.6.6.1"><mrow id="S3.E7.m1.6.6.1.1.2" xref="S3.E7.m1.6.6.1.1.3.cmml"><mrow id="S3.E7.m1.6.6.1.1.1.1" xref="S3.E7.m1.6.6.1.1.1.1.cmml"><msub id="S3.E7.m1.6.6.1.1.1.1.2" xref="S3.E7.m1.6.6.1.1.1.1.2.cmml"><mi id="S3.E7.m1.6.6.1.1.1.1.2.2" xref="S3.E7.m1.6.6.1.1.1.1.2.2.cmml">𝐲</mi><mi id="S3.E7.m1.6.6.1.1.1.1.2.3" xref="S3.E7.m1.6.6.1.1.1.1.2.3.cmml">i</mi></msub><mo id="S3.E7.m1.6.6.1.1.1.1.1" xref="S3.E7.m1.6.6.1.1.1.1.1.cmml">=</mo><mrow id="S3.E7.m1.6.6.1.1.1.1.3" xref="S3.E7.m1.6.6.1.1.1.1.3.cmml"><mrow id="S3.E7.m1.6.6.1.1.1.1.3.2" xref="S3.E7.m1.6.6.1.1.1.1.3.2.cmml"><msub id="S3.E7.m1.6.6.1.1.1.1.3.2.1" xref="S3.E7.m1.6.6.1.1.1.1.3.2.1.cmml"><mi id="S3.E7.m1.6.6.1.1.1.1.3.2.1.2" xref="S3.E7.m1.6.6.1.1.1.1.3.2.1.2.cmml">𝐗</mi><mi id="S3.E7.m1.6.6.1.1.1.1.3.2.1.3" xref="S3.E7.m1.6.6.1.1.1.1.3.2.1.3.cmml">i</mi></msub><mo id="S3.E7.m1.6.6.1.1.1.1.3.2a" lspace="0.167em" xref="S3.E7.m1.6.6.1.1.1.1.3.2.cmml">⁡</mo><mi id="S3.E7.m1.6.6.1.1.1.1.3.2.2" xref="S3.E7.m1.6.6.1.1.1.1.3.2.2.cmml">𝐛</mi></mrow><mo id="S3.E7.m1.6.6.1.1.1.1.3.1" lspace="0em" xref="S3.E7.m1.6.6.1.1.1.1.3.1.cmml">+</mo><msub id="S3.E7.m1.6.6.1.1.1.1.3.3" xref="S3.E7.m1.6.6.1.1.1.1.3.3.cmml"><mover accent="true" id="S3.E7.m1.6.6.1.1.1.1.3.3.2" xref="S3.E7.m1.6.6.1.1.1.1.3.3.2.cmml"><mi id="S3.E7.m1.6.6.1.1.1.1.3.3.2.2" xref="S3.E7.m1.6.6.1.1.1.1.3.3.2.2.cmml">𝐞</mi><mo id="S3.E7.m1.6.6.1.1.1.1.3.3.2.1" xref="S3.E7.m1.6.6.1.1.1.1.3.3.2.1.cmml">¯</mo></mover><mi id="S3.E7.m1.6.6.1.1.1.1.3.3.3" xref="S3.E7.m1.6.6.1.1.1.1.3.3.3.cmml">i</mi></msub></mrow></mrow><mo id="S3.E7.m1.6.6.1.1.2.3" rspace="1.167em" xref="S3.E7.m1.6.6.1.1.3a.cmml">,</mo><mrow id="S3.E7.m1.6.6.1.1.2.2.2" xref="S3.E7.m1.6.6.1.1.2.2.3.cmml"><mrow id="S3.E7.m1.6.6.1.1.2.2.1.1" xref="S3.E7.m1.6.6.1.1.2.2.1.1.cmml"><mrow id="S3.E7.m1.6.6.1.1.2.2.1.1.1.1" xref="S3.E7.m1.6.6.1.1.2.2.1.1.1.2.cmml"><mi id="S3.E7.m1.2.2" xref="S3.E7.m1.2.2.cmml">𝔼</mi><mo id="S3.E7.m1.6.6.1.1.2.2.1.1.1.1a" xref="S3.E7.m1.6.6.1.1.2.2.1.1.1.2.cmml">⁡</mo><mrow id="S3.E7.m1.6.6.1.1.2.2.1.1.1.1.1" xref="S3.E7.m1.6.6.1.1.2.2.1.1.1.2.cmml"><mo id="S3.E7.m1.6.6.1.1.2.2.1.1.1.1.1.2" stretchy="false" xref="S3.E7.m1.6.6.1.1.2.2.1.1.1.2.cmml">(</mo><msub id="S3.E7.m1.6.6.1.1.2.2.1.1.1.1.1.1" 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id="S3.E7.m1.6.6.1.1.2.2.2.2.1.1.3.3.2.cmml" xref="S3.E7.m1.6.6.1.1.2.2.2.2.1.1.3.3"><csymbol cd="ambiguous" id="S3.E7.m1.6.6.1.1.2.2.2.2.1.1.3.3.2.1.cmml" xref="S3.E7.m1.6.6.1.1.2.2.2.2.1.1.3.3">subscript</csymbol><ci id="S3.E7.m1.6.6.1.1.2.2.2.2.1.1.3.3.2.2.cmml" xref="S3.E7.m1.6.6.1.1.2.2.2.2.1.1.3.3.2.2">𝚲</ci><ci id="S3.E7.m1.6.6.1.1.2.2.2.2.1.1.3.3.2.3.cmml" xref="S3.E7.m1.6.6.1.1.2.2.2.2.1.1.3.3.2.3">𝑖</ci></apply><ci id="S3.E7.m1.1.1.1.1.cmml" xref="S3.E7.m1.1.1.1.1">𝑡</ci></apply></apply></apply><apply id="S3.E7.m1.6.6.1.1.2.2.2.2.2.2.cmml" xref="S3.E7.m1.6.6.1.1.2.2.2.2.2.2"><eq id="S3.E7.m1.6.6.1.1.2.2.2.2.2.2.1.cmml" xref="S3.E7.m1.6.6.1.1.2.2.2.2.2.2.1"></eq><ci id="S3.E7.m1.6.6.1.1.2.2.2.2.2.2.2.cmml" xref="S3.E7.m1.6.6.1.1.2.2.2.2.2.2.2">𝑖</ci><list id="S3.E7.m1.6.6.1.1.2.2.2.2.2.2.3.1.cmml" xref="S3.E7.m1.6.6.1.1.2.2.2.2.2.2.3.2"><cn id="S3.E7.m1.3.3.cmml" type="integer" xref="S3.E7.m1.3.3">1</cn><ci id="S3.E7.m1.4.4.cmml" xref="S3.E7.m1.4.4">…</ci><ci id="S3.E7.m1.5.5.cmml" xref="S3.E7.m1.5.5">𝑛</ci></list></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.E7.m1.6c">\displaystyle\operatorname{\mathbf{y}}_{i}=\operatorname{\mathbf{X}}_{i}% \operatorname{\mathbf{b}}+\bar{\operatorname{\mathbf{e}}}_{i},\quad% \operatorname{\mathbb{E}}(\overline{\operatorname{\mathbf{e}}}_{i})=% \operatorname{\mathbf{0}},\quad\text{Cov}(\overline{\operatorname{\mathbf{e}}}% _{i})=\tau^{2}\operatorname{{\mathbf{\Lambda}}}_{i}^{(t)},\quad i=1,\ldots,n.</annotation><annotation encoding="application/x-llamapun" id="S3.E7.m1.6d">bold_y start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = bold_X start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT bold_b + over¯ start_ARG bold_e end_ARG start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , blackboard_E ( over¯ start_ARG bold_e end_ARG start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) = bold_0 , Cov ( over¯ start_ARG bold_e end_ARG start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) = italic_τ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT bold_Λ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_t ) end_POSTSUPERSCRIPT , italic_i = 1 , … , italic_n .</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> <p class="ltx_p" id="S3.p2.22">Let <math alttext="\operatorname{{\mathbf{\Lambda}}}_{i}^{{1}/{2}(t)}" class="ltx_Math" display="inline" id="S3.p2.13.m1.1"><semantics id="S3.p2.13.m1.1a"><msubsup id="S3.p2.13.m1.1.2" xref="S3.p2.13.m1.1.2.cmml"><mi id="S3.p2.13.m1.1.2.2.2" xref="S3.p2.13.m1.1.2.2.2.cmml">𝚲</mi><mi id="S3.p2.13.m1.1.2.2.3" xref="S3.p2.13.m1.1.2.2.3.cmml">i</mi><mrow id="S3.p2.13.m1.1.1.1" xref="S3.p2.13.m1.1.1.1.cmml"><mrow id="S3.p2.13.m1.1.1.1.3" xref="S3.p2.13.m1.1.1.1.3.cmml"><mn id="S3.p2.13.m1.1.1.1.3.2" xref="S3.p2.13.m1.1.1.1.3.2.cmml">1</mn><mo id="S3.p2.13.m1.1.1.1.3.1" xref="S3.p2.13.m1.1.1.1.3.1.cmml">/</mo><mn id="S3.p2.13.m1.1.1.1.3.3" xref="S3.p2.13.m1.1.1.1.3.3.cmml">2</mn></mrow><mo id="S3.p2.13.m1.1.1.1.2" xref="S3.p2.13.m1.1.1.1.2.cmml">⁢</mo><mrow id="S3.p2.13.m1.1.1.1.4.2" xref="S3.p2.13.m1.1.1.1.cmml"><mo id="S3.p2.13.m1.1.1.1.4.2.1" stretchy="false" xref="S3.p2.13.m1.1.1.1.cmml">(</mo><mi id="S3.p2.13.m1.1.1.1.1" xref="S3.p2.13.m1.1.1.1.1.cmml">t</mi><mo id="S3.p2.13.m1.1.1.1.4.2.2" stretchy="false" xref="S3.p2.13.m1.1.1.1.cmml">)</mo></mrow></mrow></msubsup><annotation-xml encoding="MathML-Content" id="S3.p2.13.m1.1b"><apply id="S3.p2.13.m1.1.2.cmml" xref="S3.p2.13.m1.1.2"><csymbol cd="ambiguous" id="S3.p2.13.m1.1.2.1.cmml" xref="S3.p2.13.m1.1.2">superscript</csymbol><apply id="S3.p2.13.m1.1.2.2.cmml" xref="S3.p2.13.m1.1.2"><csymbol cd="ambiguous" id="S3.p2.13.m1.1.2.2.1.cmml" xref="S3.p2.13.m1.1.2">subscript</csymbol><ci id="S3.p2.13.m1.1.2.2.2.cmml" xref="S3.p2.13.m1.1.2.2.2">𝚲</ci><ci id="S3.p2.13.m1.1.2.2.3.cmml" xref="S3.p2.13.m1.1.2.2.3">𝑖</ci></apply><apply id="S3.p2.13.m1.1.1.1.cmml" xref="S3.p2.13.m1.1.1.1"><times id="S3.p2.13.m1.1.1.1.2.cmml" xref="S3.p2.13.m1.1.1.1.2"></times><apply id="S3.p2.13.m1.1.1.1.3.cmml" xref="S3.p2.13.m1.1.1.1.3"><divide id="S3.p2.13.m1.1.1.1.3.1.cmml" xref="S3.p2.13.m1.1.1.1.3.1"></divide><cn id="S3.p2.13.m1.1.1.1.3.2.cmml" type="integer" xref="S3.p2.13.m1.1.1.1.3.2">1</cn><cn id="S3.p2.13.m1.1.1.1.3.3.cmml" type="integer" xref="S3.p2.13.m1.1.1.1.3.3">2</cn></apply><ci id="S3.p2.13.m1.1.1.1.1.cmml" xref="S3.p2.13.m1.1.1.1.1">𝑡</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p2.13.m1.1c">\operatorname{{\mathbf{\Lambda}}}_{i}^{{1}/{2}(t)}</annotation><annotation encoding="application/x-llamapun" id="S3.p2.13.m1.1d">bold_Λ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 1 / 2 ( italic_t ) end_POSTSUPERSCRIPT</annotation></semantics></math> be any matrix such that <math alttext="\operatorname{{\mathbf{\Lambda}}}_{i}^{{1}/{2}(t)}\operatorname{{\mathbf{% \Lambda}}}_{i}^{{1}/{2}(t)\top}=\operatorname{{\mathbf{\Lambda}}}_{i}^{(t)}" class="ltx_Math" display="inline" id="S3.p2.14.m2.3"><semantics id="S3.p2.14.m2.3a"><mrow id="S3.p2.14.m2.3.4" xref="S3.p2.14.m2.3.4.cmml"><mrow id="S3.p2.14.m2.3.4.2" xref="S3.p2.14.m2.3.4.2.cmml"><msubsup id="S3.p2.14.m2.3.4.2.1" xref="S3.p2.14.m2.3.4.2.1.cmml"><mi id="S3.p2.14.m2.3.4.2.1.2.2" xref="S3.p2.14.m2.3.4.2.1.2.2.cmml">𝚲</mi><mi id="S3.p2.14.m2.3.4.2.1.2.3" xref="S3.p2.14.m2.3.4.2.1.2.3.cmml">i</mi><mrow id="S3.p2.14.m2.1.1.1" xref="S3.p2.14.m2.1.1.1.cmml"><mrow id="S3.p2.14.m2.1.1.1.3" xref="S3.p2.14.m2.1.1.1.3.cmml"><mn id="S3.p2.14.m2.1.1.1.3.2" xref="S3.p2.14.m2.1.1.1.3.2.cmml">1</mn><mo id="S3.p2.14.m2.1.1.1.3.1" xref="S3.p2.14.m2.1.1.1.3.1.cmml">/</mo><mn id="S3.p2.14.m2.1.1.1.3.3" xref="S3.p2.14.m2.1.1.1.3.3.cmml">2</mn></mrow><mo id="S3.p2.14.m2.1.1.1.2" 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xref="S3.p2.14.m2.3.4.2.2.2.3">𝑖</ci></apply><apply id="S3.p2.14.m2.2.2.1.cmml" xref="S3.p2.14.m2.2.2.1"><csymbol cd="latexml" id="S3.p2.14.m2.2.2.1.2.cmml" xref="S3.p2.14.m2.2.2.1">limit-from</csymbol><apply id="S3.p2.14.m2.2.2.1.3.cmml" xref="S3.p2.14.m2.2.2.1.3"><times id="S3.p2.14.m2.2.2.1.3.1.cmml" xref="S3.p2.14.m2.2.2.1.3.1"></times><apply id="S3.p2.14.m2.2.2.1.3.2.cmml" xref="S3.p2.14.m2.2.2.1.3.2"><divide id="S3.p2.14.m2.2.2.1.3.2.1.cmml" xref="S3.p2.14.m2.2.2.1.3.2.1"></divide><cn id="S3.p2.14.m2.2.2.1.3.2.2.cmml" type="integer" xref="S3.p2.14.m2.2.2.1.3.2.2">1</cn><cn id="S3.p2.14.m2.2.2.1.3.2.3.cmml" type="integer" xref="S3.p2.14.m2.2.2.1.3.2.3">2</cn></apply><ci id="S3.p2.14.m2.2.2.1.1.cmml" xref="S3.p2.14.m2.2.2.1.1">𝑡</ci></apply><csymbol cd="latexml" id="S3.p2.14.m2.2.2.1.4.cmml" xref="S3.p2.14.m2.2.2.1.4">top</csymbol></apply></apply></apply><apply id="S3.p2.14.m2.3.4.3.cmml" xref="S3.p2.14.m2.3.4.3"><csymbol cd="ambiguous" id="S3.p2.14.m2.3.4.3.1.cmml" xref="S3.p2.14.m2.3.4.3">superscript</csymbol><apply id="S3.p2.14.m2.3.4.3.2.cmml" xref="S3.p2.14.m2.3.4.3"><csymbol cd="ambiguous" id="S3.p2.14.m2.3.4.3.2.1.cmml" xref="S3.p2.14.m2.3.4.3">subscript</csymbol><ci id="S3.p2.14.m2.3.4.3.2.2.cmml" xref="S3.p2.14.m2.3.4.3.2.2">𝚲</ci><ci id="S3.p2.14.m2.3.4.3.2.3.cmml" xref="S3.p2.14.m2.3.4.3.2.3">𝑖</ci></apply><ci id="S3.p2.14.m2.3.3.1.1.cmml" xref="S3.p2.14.m2.3.3.1.1">𝑡</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p2.14.m2.3c">\operatorname{{\mathbf{\Lambda}}}_{i}^{{1}/{2}(t)}\operatorname{{\mathbf{% \Lambda}}}_{i}^{{1}/{2}(t)\top}=\operatorname{{\mathbf{\Lambda}}}_{i}^{(t)}</annotation><annotation encoding="application/x-llamapun" id="S3.p2.14.m2.3d">bold_Λ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 1 / 2 ( italic_t ) end_POSTSUPERSCRIPT bold_Λ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 1 / 2 ( italic_t ) ⊤ end_POSTSUPERSCRIPT = bold_Λ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_t ) end_POSTSUPERSCRIPT</annotation></semantics></math>, and <math alttext="\breve{\operatorname{\mathbf{y}}}^{(t)}\in\operatorname{\mathbb{R}}^{N}" class="ltx_Math" display="inline" id="S3.p2.15.m3.1"><semantics id="S3.p2.15.m3.1a"><mrow id="S3.p2.15.m3.1.2" xref="S3.p2.15.m3.1.2.cmml"><msup id="S3.p2.15.m3.1.2.2" xref="S3.p2.15.m3.1.2.2.cmml"><mover accent="true" id="S3.p2.15.m3.1.2.2.2" xref="S3.p2.15.m3.1.2.2.2.cmml"><mi id="S3.p2.15.m3.1.2.2.2.2" xref="S3.p2.15.m3.1.2.2.2.2.cmml">𝐲</mi><mo id="S3.p2.15.m3.1.2.2.2.1" xref="S3.p2.15.m3.1.2.2.2.1.cmml">˘</mo></mover><mrow id="S3.p2.15.m3.1.1.1.3" xref="S3.p2.15.m3.1.2.2.cmml"><mo id="S3.p2.15.m3.1.1.1.3.1" stretchy="false" xref="S3.p2.15.m3.1.2.2.cmml">(</mo><mi id="S3.p2.15.m3.1.1.1.1" xref="S3.p2.15.m3.1.1.1.1.cmml">t</mi><mo id="S3.p2.15.m3.1.1.1.3.2" stretchy="false" xref="S3.p2.15.m3.1.2.2.cmml">)</mo></mrow></msup><mo id="S3.p2.15.m3.1.2.1" xref="S3.p2.15.m3.1.2.1.cmml">∈</mo><msup id="S3.p2.15.m3.1.2.3" xref="S3.p2.15.m3.1.2.3.cmml"><mi id="S3.p2.15.m3.1.2.3.2" xref="S3.p2.15.m3.1.2.3.2.cmml">ℝ</mi><mi id="S3.p2.15.m3.1.2.3.3" xref="S3.p2.15.m3.1.2.3.3.cmml">N</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S3.p2.15.m3.1b"><apply id="S3.p2.15.m3.1.2.cmml" xref="S3.p2.15.m3.1.2"><in id="S3.p2.15.m3.1.2.1.cmml" xref="S3.p2.15.m3.1.2.1"></in><apply id="S3.p2.15.m3.1.2.2.cmml" xref="S3.p2.15.m3.1.2.2"><csymbol cd="ambiguous" id="S3.p2.15.m3.1.2.2.1.cmml" xref="S3.p2.15.m3.1.2.2">superscript</csymbol><apply id="S3.p2.15.m3.1.2.2.2.cmml" xref="S3.p2.15.m3.1.2.2.2"><ci id="S3.p2.15.m3.1.2.2.2.1.cmml" xref="S3.p2.15.m3.1.2.2.2.1">˘</ci><ci id="S3.p2.15.m3.1.2.2.2.2.cmml" xref="S3.p2.15.m3.1.2.2.2.2">𝐲</ci></apply><ci id="S3.p2.15.m3.1.1.1.1.cmml" xref="S3.p2.15.m3.1.1.1.1">𝑡</ci></apply><apply id="S3.p2.15.m3.1.2.3.cmml" xref="S3.p2.15.m3.1.2.3"><csymbol cd="ambiguous" id="S3.p2.15.m3.1.2.3.1.cmml" xref="S3.p2.15.m3.1.2.3">superscript</csymbol><ci id="S3.p2.15.m3.1.2.3.2.cmml" xref="S3.p2.15.m3.1.2.3.2">ℝ</ci><ci id="S3.p2.15.m3.1.2.3.3.cmml" xref="S3.p2.15.m3.1.2.3.3">𝑁</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p2.15.m3.1c">\breve{\operatorname{\mathbf{y}}}^{(t)}\in\operatorname{\mathbb{R}}^{N}</annotation><annotation encoding="application/x-llamapun" id="S3.p2.15.m3.1d">over˘ start_ARG bold_y end_ARG start_POSTSUPERSCRIPT ( italic_t ) end_POSTSUPERSCRIPT ∈ blackboard_R start_POSTSUPERSCRIPT italic_N end_POSTSUPERSCRIPT</annotation></semantics></math> and <math alttext="\breve{\operatorname{\mathbf{X}}}^{{(t)}}\in\operatorname{\mathbb{R}}^{N\times P% _{1}P_{2}}" class="ltx_Math" display="inline" id="S3.p2.16.m4.1"><semantics id="S3.p2.16.m4.1a"><mrow id="S3.p2.16.m4.1.2" xref="S3.p2.16.m4.1.2.cmml"><msup id="S3.p2.16.m4.1.2.2" xref="S3.p2.16.m4.1.2.2.cmml"><mover accent="true" id="S3.p2.16.m4.1.2.2.2" xref="S3.p2.16.m4.1.2.2.2.cmml"><mi id="S3.p2.16.m4.1.2.2.2.2" xref="S3.p2.16.m4.1.2.2.2.2.cmml">𝐗</mi><mo id="S3.p2.16.m4.1.2.2.2.1" xref="S3.p2.16.m4.1.2.2.2.1.cmml">˘</mo></mover><mrow id="S3.p2.16.m4.1.1.1.3" xref="S3.p2.16.m4.1.2.2.cmml"><mo id="S3.p2.16.m4.1.1.1.3.1" stretchy="false" xref="S3.p2.16.m4.1.2.2.cmml">(</mo><mi id="S3.p2.16.m4.1.1.1.1" xref="S3.p2.16.m4.1.1.1.1.cmml">t</mi><mo id="S3.p2.16.m4.1.1.1.3.2" stretchy="false" xref="S3.p2.16.m4.1.2.2.cmml">)</mo></mrow></msup><mo id="S3.p2.16.m4.1.2.1" xref="S3.p2.16.m4.1.2.1.cmml">∈</mo><msup id="S3.p2.16.m4.1.2.3" xref="S3.p2.16.m4.1.2.3.cmml"><mi id="S3.p2.16.m4.1.2.3.2" xref="S3.p2.16.m4.1.2.3.2.cmml">ℝ</mi><mrow id="S3.p2.16.m4.1.2.3.3" xref="S3.p2.16.m4.1.2.3.3.cmml"><mrow id="S3.p2.16.m4.1.2.3.3.2" xref="S3.p2.16.m4.1.2.3.3.2.cmml"><mi id="S3.p2.16.m4.1.2.3.3.2.2" xref="S3.p2.16.m4.1.2.3.3.2.2.cmml">N</mi><mo id="S3.p2.16.m4.1.2.3.3.2.1" lspace="0.222em" rspace="0.222em" xref="S3.p2.16.m4.1.2.3.3.2.1.cmml">×</mo><msub id="S3.p2.16.m4.1.2.3.3.2.3" xref="S3.p2.16.m4.1.2.3.3.2.3.cmml"><mi id="S3.p2.16.m4.1.2.3.3.2.3.2" xref="S3.p2.16.m4.1.2.3.3.2.3.2.cmml">P</mi><mn id="S3.p2.16.m4.1.2.3.3.2.3.3" xref="S3.p2.16.m4.1.2.3.3.2.3.3.cmml">1</mn></msub></mrow><mo id="S3.p2.16.m4.1.2.3.3.1" xref="S3.p2.16.m4.1.2.3.3.1.cmml">⁢</mo><msub id="S3.p2.16.m4.1.2.3.3.3" xref="S3.p2.16.m4.1.2.3.3.3.cmml"><mi id="S3.p2.16.m4.1.2.3.3.3.2" xref="S3.p2.16.m4.1.2.3.3.3.2.cmml">P</mi><mn id="S3.p2.16.m4.1.2.3.3.3.3" xref="S3.p2.16.m4.1.2.3.3.3.3.cmml">2</mn></msub></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="S3.p2.16.m4.1b"><apply id="S3.p2.16.m4.1.2.cmml" xref="S3.p2.16.m4.1.2"><in id="S3.p2.16.m4.1.2.1.cmml" xref="S3.p2.16.m4.1.2.1"></in><apply id="S3.p2.16.m4.1.2.2.cmml" xref="S3.p2.16.m4.1.2.2"><csymbol cd="ambiguous" id="S3.p2.16.m4.1.2.2.1.cmml" xref="S3.p2.16.m4.1.2.2">superscript</csymbol><apply id="S3.p2.16.m4.1.2.2.2.cmml" xref="S3.p2.16.m4.1.2.2.2"><ci id="S3.p2.16.m4.1.2.2.2.1.cmml" xref="S3.p2.16.m4.1.2.2.2.1">˘</ci><ci id="S3.p2.16.m4.1.2.2.2.2.cmml" xref="S3.p2.16.m4.1.2.2.2.2">𝐗</ci></apply><ci id="S3.p2.16.m4.1.1.1.1.cmml" xref="S3.p2.16.m4.1.1.1.1">𝑡</ci></apply><apply id="S3.p2.16.m4.1.2.3.cmml" xref="S3.p2.16.m4.1.2.3"><csymbol cd="ambiguous" id="S3.p2.16.m4.1.2.3.1.cmml" xref="S3.p2.16.m4.1.2.3">superscript</csymbol><ci id="S3.p2.16.m4.1.2.3.2.cmml" xref="S3.p2.16.m4.1.2.3.2">ℝ</ci><apply id="S3.p2.16.m4.1.2.3.3.cmml" xref="S3.p2.16.m4.1.2.3.3"><times id="S3.p2.16.m4.1.2.3.3.1.cmml" xref="S3.p2.16.m4.1.2.3.3.1"></times><apply id="S3.p2.16.m4.1.2.3.3.2.cmml" xref="S3.p2.16.m4.1.2.3.3.2"><times id="S3.p2.16.m4.1.2.3.3.2.1.cmml" xref="S3.p2.16.m4.1.2.3.3.2.1"></times><ci id="S3.p2.16.m4.1.2.3.3.2.2.cmml" xref="S3.p2.16.m4.1.2.3.3.2.2">𝑁</ci><apply id="S3.p2.16.m4.1.2.3.3.2.3.cmml" xref="S3.p2.16.m4.1.2.3.3.2.3"><csymbol cd="ambiguous" id="S3.p2.16.m4.1.2.3.3.2.3.1.cmml" xref="S3.p2.16.m4.1.2.3.3.2.3">subscript</csymbol><ci id="S3.p2.16.m4.1.2.3.3.2.3.2.cmml" xref="S3.p2.16.m4.1.2.3.3.2.3.2">𝑃</ci><cn id="S3.p2.16.m4.1.2.3.3.2.3.3.cmml" type="integer" xref="S3.p2.16.m4.1.2.3.3.2.3.3">1</cn></apply></apply><apply id="S3.p2.16.m4.1.2.3.3.3.cmml" xref="S3.p2.16.m4.1.2.3.3.3"><csymbol cd="ambiguous" id="S3.p2.16.m4.1.2.3.3.3.1.cmml" xref="S3.p2.16.m4.1.2.3.3.3">subscript</csymbol><ci id="S3.p2.16.m4.1.2.3.3.3.2.cmml" xref="S3.p2.16.m4.1.2.3.3.3.2">𝑃</ci><cn id="S3.p2.16.m4.1.2.3.3.3.3.cmml" type="integer" xref="S3.p2.16.m4.1.2.3.3.3.3">2</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p2.16.m4.1c">\breve{\operatorname{\mathbf{X}}}^{{(t)}}\in\operatorname{\mathbb{R}}^{N\times P% _{1}P_{2}}</annotation><annotation encoding="application/x-llamapun" id="S3.p2.16.m4.1d">over˘ start_ARG bold_X end_ARG start_POSTSUPERSCRIPT ( italic_t ) end_POSTSUPERSCRIPT ∈ blackboard_R start_POSTSUPERSCRIPT italic_N × italic_P start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_P start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT</annotation></semantics></math> be the scaled response and design matrix with <math alttext="\operatorname{{\mathbf{\Lambda}}}_{i}^{-{1}/{2}(t)}\operatorname{\mathbf{y}}_{% i}\in\operatorname{\mathbb{R}}^{m_{i}}" class="ltx_Math" display="inline" id="S3.p2.17.m5.1"><semantics id="S3.p2.17.m5.1a"><mrow id="S3.p2.17.m5.1.2" xref="S3.p2.17.m5.1.2.cmml"><mrow id="S3.p2.17.m5.1.2.2" xref="S3.p2.17.m5.1.2.2.cmml"><msubsup id="S3.p2.17.m5.1.2.2.1" xref="S3.p2.17.m5.1.2.2.1.cmml"><mi id="S3.p2.17.m5.1.2.2.1.2.2" xref="S3.p2.17.m5.1.2.2.1.2.2.cmml">𝚲</mi><mi id="S3.p2.17.m5.1.2.2.1.2.3" xref="S3.p2.17.m5.1.2.2.1.2.3.cmml">i</mi><mrow id="S3.p2.17.m5.1.1.1" xref="S3.p2.17.m5.1.1.1.cmml"><mo id="S3.p2.17.m5.1.1.1a" xref="S3.p2.17.m5.1.1.1.cmml">−</mo><mrow id="S3.p2.17.m5.1.1.1.3" xref="S3.p2.17.m5.1.1.1.3.cmml"><mrow id="S3.p2.17.m5.1.1.1.3.2" xref="S3.p2.17.m5.1.1.1.3.2.cmml"><mn id="S3.p2.17.m5.1.1.1.3.2.2" xref="S3.p2.17.m5.1.1.1.3.2.2.cmml">1</mn><mo id="S3.p2.17.m5.1.1.1.3.2.1" xref="S3.p2.17.m5.1.1.1.3.2.1.cmml">/</mo><mn id="S3.p2.17.m5.1.1.1.3.2.3" xref="S3.p2.17.m5.1.1.1.3.2.3.cmml">2</mn></mrow><mo id="S3.p2.17.m5.1.1.1.3.1" xref="S3.p2.17.m5.1.1.1.3.1.cmml">⁢</mo><mrow id="S3.p2.17.m5.1.1.1.3.3.2" xref="S3.p2.17.m5.1.1.1.3.cmml"><mo id="S3.p2.17.m5.1.1.1.3.3.2.1" stretchy="false" xref="S3.p2.17.m5.1.1.1.3.cmml">(</mo><mi id="S3.p2.17.m5.1.1.1.1" xref="S3.p2.17.m5.1.1.1.1.cmml">t</mi><mo id="S3.p2.17.m5.1.1.1.3.3.2.2" stretchy="false" xref="S3.p2.17.m5.1.1.1.3.cmml">)</mo></mrow></mrow></mrow></msubsup><mo id="S3.p2.17.m5.1.2.2a" lspace="0.167em" xref="S3.p2.17.m5.1.2.2.cmml">⁡</mo><msub id="S3.p2.17.m5.1.2.2.2" xref="S3.p2.17.m5.1.2.2.2.cmml"><mi id="S3.p2.17.m5.1.2.2.2.2" xref="S3.p2.17.m5.1.2.2.2.2.cmml">𝐲</mi><mi id="S3.p2.17.m5.1.2.2.2.3" xref="S3.p2.17.m5.1.2.2.2.3.cmml">i</mi></msub></mrow><mo id="S3.p2.17.m5.1.2.1" xref="S3.p2.17.m5.1.2.1.cmml">∈</mo><msup id="S3.p2.17.m5.1.2.3" xref="S3.p2.17.m5.1.2.3.cmml"><mi id="S3.p2.17.m5.1.2.3.2" xref="S3.p2.17.m5.1.2.3.2.cmml">ℝ</mi><msub id="S3.p2.17.m5.1.2.3.3" xref="S3.p2.17.m5.1.2.3.3.cmml"><mi id="S3.p2.17.m5.1.2.3.3.2" xref="S3.p2.17.m5.1.2.3.3.2.cmml">m</mi><mi id="S3.p2.17.m5.1.2.3.3.3" xref="S3.p2.17.m5.1.2.3.3.3.cmml">i</mi></msub></msup></mrow><annotation-xml encoding="MathML-Content" id="S3.p2.17.m5.1b"><apply id="S3.p2.17.m5.1.2.cmml" xref="S3.p2.17.m5.1.2"><in id="S3.p2.17.m5.1.2.1.cmml" xref="S3.p2.17.m5.1.2.1"></in><apply id="S3.p2.17.m5.1.2.2.cmml" xref="S3.p2.17.m5.1.2.2"><apply id="S3.p2.17.m5.1.2.2.1.cmml" xref="S3.p2.17.m5.1.2.2.1"><csymbol cd="ambiguous" id="S3.p2.17.m5.1.2.2.1.1.cmml" xref="S3.p2.17.m5.1.2.2.1">superscript</csymbol><apply id="S3.p2.17.m5.1.2.2.1.2.cmml" xref="S3.p2.17.m5.1.2.2.1"><csymbol cd="ambiguous" id="S3.p2.17.m5.1.2.2.1.2.1.cmml" xref="S3.p2.17.m5.1.2.2.1">subscript</csymbol><ci id="S3.p2.17.m5.1.2.2.1.2.2.cmml" xref="S3.p2.17.m5.1.2.2.1.2.2">𝚲</ci><ci id="S3.p2.17.m5.1.2.2.1.2.3.cmml" xref="S3.p2.17.m5.1.2.2.1.2.3">𝑖</ci></apply><apply id="S3.p2.17.m5.1.1.1.cmml" xref="S3.p2.17.m5.1.1.1"><minus id="S3.p2.17.m5.1.1.1.2.cmml" xref="S3.p2.17.m5.1.1.1"></minus><apply id="S3.p2.17.m5.1.1.1.3.cmml" xref="S3.p2.17.m5.1.1.1.3"><times id="S3.p2.17.m5.1.1.1.3.1.cmml" xref="S3.p2.17.m5.1.1.1.3.1"></times><apply id="S3.p2.17.m5.1.1.1.3.2.cmml" xref="S3.p2.17.m5.1.1.1.3.2"><divide id="S3.p2.17.m5.1.1.1.3.2.1.cmml" xref="S3.p2.17.m5.1.1.1.3.2.1"></divide><cn id="S3.p2.17.m5.1.1.1.3.2.2.cmml" type="integer" xref="S3.p2.17.m5.1.1.1.3.2.2">1</cn><cn id="S3.p2.17.m5.1.1.1.3.2.3.cmml" type="integer" xref="S3.p2.17.m5.1.1.1.3.2.3">2</cn></apply><ci id="S3.p2.17.m5.1.1.1.1.cmml" xref="S3.p2.17.m5.1.1.1.1">𝑡</ci></apply></apply></apply><apply id="S3.p2.17.m5.1.2.2.2.cmml" xref="S3.p2.17.m5.1.2.2.2"><csymbol cd="ambiguous" id="S3.p2.17.m5.1.2.2.2.1.cmml" xref="S3.p2.17.m5.1.2.2.2">subscript</csymbol><ci id="S3.p2.17.m5.1.2.2.2.2.cmml" xref="S3.p2.17.m5.1.2.2.2.2">𝐲</ci><ci id="S3.p2.17.m5.1.2.2.2.3.cmml" xref="S3.p2.17.m5.1.2.2.2.3">𝑖</ci></apply></apply><apply id="S3.p2.17.m5.1.2.3.cmml" xref="S3.p2.17.m5.1.2.3"><csymbol cd="ambiguous" id="S3.p2.17.m5.1.2.3.1.cmml" xref="S3.p2.17.m5.1.2.3">superscript</csymbol><ci id="S3.p2.17.m5.1.2.3.2.cmml" xref="S3.p2.17.m5.1.2.3.2">ℝ</ci><apply id="S3.p2.17.m5.1.2.3.3.cmml" xref="S3.p2.17.m5.1.2.3.3"><csymbol cd="ambiguous" id="S3.p2.17.m5.1.2.3.3.1.cmml" xref="S3.p2.17.m5.1.2.3.3">subscript</csymbol><ci id="S3.p2.17.m5.1.2.3.3.2.cmml" xref="S3.p2.17.m5.1.2.3.3.2">𝑚</ci><ci id="S3.p2.17.m5.1.2.3.3.3.cmml" xref="S3.p2.17.m5.1.2.3.3.3">𝑖</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p2.17.m5.1c">\operatorname{{\mathbf{\Lambda}}}_{i}^{-{1}/{2}(t)}\operatorname{\mathbf{y}}_{% i}\in\operatorname{\mathbb{R}}^{m_{i}}</annotation><annotation encoding="application/x-llamapun" id="S3.p2.17.m5.1d">bold_Λ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT - 1 / 2 ( italic_t ) end_POSTSUPERSCRIPT bold_y start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∈ blackboard_R start_POSTSUPERSCRIPT italic_m start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUPERSCRIPT</annotation></semantics></math> and <math alttext="\operatorname{{\mathbf{\Lambda}}}_{i}^{-{1}/{2}(t)}\operatorname{\mathbf{X}}_{i}" class="ltx_Math" display="inline" id="S3.p2.18.m6.1"><semantics id="S3.p2.18.m6.1a"><mrow id="S3.p2.18.m6.1.2" xref="S3.p2.18.m6.1.2.cmml"><msubsup id="S3.p2.18.m6.1.2.1" xref="S3.p2.18.m6.1.2.1.cmml"><mi id="S3.p2.18.m6.1.2.1.2.2" xref="S3.p2.18.m6.1.2.1.2.2.cmml">𝚲</mi><mi id="S3.p2.18.m6.1.2.1.2.3" xref="S3.p2.18.m6.1.2.1.2.3.cmml">i</mi><mrow id="S3.p2.18.m6.1.1.1" xref="S3.p2.18.m6.1.1.1.cmml"><mo id="S3.p2.18.m6.1.1.1a" xref="S3.p2.18.m6.1.1.1.cmml">−</mo><mrow id="S3.p2.18.m6.1.1.1.3" xref="S3.p2.18.m6.1.1.1.3.cmml"><mrow id="S3.p2.18.m6.1.1.1.3.2" xref="S3.p2.18.m6.1.1.1.3.2.cmml"><mn id="S3.p2.18.m6.1.1.1.3.2.2" xref="S3.p2.18.m6.1.1.1.3.2.2.cmml">1</mn><mo id="S3.p2.18.m6.1.1.1.3.2.1" xref="S3.p2.18.m6.1.1.1.3.2.1.cmml">/</mo><mn id="S3.p2.18.m6.1.1.1.3.2.3" xref="S3.p2.18.m6.1.1.1.3.2.3.cmml">2</mn></mrow><mo id="S3.p2.18.m6.1.1.1.3.1" xref="S3.p2.18.m6.1.1.1.3.1.cmml">⁢</mo><mrow id="S3.p2.18.m6.1.1.1.3.3.2" xref="S3.p2.18.m6.1.1.1.3.cmml"><mo id="S3.p2.18.m6.1.1.1.3.3.2.1" stretchy="false" xref="S3.p2.18.m6.1.1.1.3.cmml">(</mo><mi id="S3.p2.18.m6.1.1.1.1" xref="S3.p2.18.m6.1.1.1.1.cmml">t</mi><mo id="S3.p2.18.m6.1.1.1.3.3.2.2" stretchy="false" xref="S3.p2.18.m6.1.1.1.3.cmml">)</mo></mrow></mrow></mrow></msubsup><mo id="S3.p2.18.m6.1.2a" lspace="0.167em" xref="S3.p2.18.m6.1.2.cmml">⁡</mo><msub id="S3.p2.18.m6.1.2.2" xref="S3.p2.18.m6.1.2.2.cmml"><mi id="S3.p2.18.m6.1.2.2.2" xref="S3.p2.18.m6.1.2.2.2.cmml">𝐗</mi><mi id="S3.p2.18.m6.1.2.2.3" xref="S3.p2.18.m6.1.2.2.3.cmml">i</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.p2.18.m6.1b"><apply id="S3.p2.18.m6.1.2.cmml" xref="S3.p2.18.m6.1.2"><apply id="S3.p2.18.m6.1.2.1.cmml" xref="S3.p2.18.m6.1.2.1"><csymbol cd="ambiguous" id="S3.p2.18.m6.1.2.1.1.cmml" xref="S3.p2.18.m6.1.2.1">superscript</csymbol><apply id="S3.p2.18.m6.1.2.1.2.cmml" xref="S3.p2.18.m6.1.2.1"><csymbol cd="ambiguous" id="S3.p2.18.m6.1.2.1.2.1.cmml" xref="S3.p2.18.m6.1.2.1">subscript</csymbol><ci id="S3.p2.18.m6.1.2.1.2.2.cmml" xref="S3.p2.18.m6.1.2.1.2.2">𝚲</ci><ci id="S3.p2.18.m6.1.2.1.2.3.cmml" xref="S3.p2.18.m6.1.2.1.2.3">𝑖</ci></apply><apply id="S3.p2.18.m6.1.1.1.cmml" xref="S3.p2.18.m6.1.1.1"><minus id="S3.p2.18.m6.1.1.1.2.cmml" xref="S3.p2.18.m6.1.1.1"></minus><apply id="S3.p2.18.m6.1.1.1.3.cmml" xref="S3.p2.18.m6.1.1.1.3"><times id="S3.p2.18.m6.1.1.1.3.1.cmml" xref="S3.p2.18.m6.1.1.1.3.1"></times><apply id="S3.p2.18.m6.1.1.1.3.2.cmml" xref="S3.p2.18.m6.1.1.1.3.2"><divide id="S3.p2.18.m6.1.1.1.3.2.1.cmml" xref="S3.p2.18.m6.1.1.1.3.2.1"></divide><cn id="S3.p2.18.m6.1.1.1.3.2.2.cmml" type="integer" xref="S3.p2.18.m6.1.1.1.3.2.2">1</cn><cn id="S3.p2.18.m6.1.1.1.3.2.3.cmml" type="integer" xref="S3.p2.18.m6.1.1.1.3.2.3">2</cn></apply><ci id="S3.p2.18.m6.1.1.1.1.cmml" xref="S3.p2.18.m6.1.1.1.1">𝑡</ci></apply></apply></apply><apply id="S3.p2.18.m6.1.2.2.cmml" xref="S3.p2.18.m6.1.2.2"><csymbol cd="ambiguous" id="S3.p2.18.m6.1.2.2.1.cmml" xref="S3.p2.18.m6.1.2.2">subscript</csymbol><ci id="S3.p2.18.m6.1.2.2.2.cmml" xref="S3.p2.18.m6.1.2.2.2">𝐗</ci><ci id="S3.p2.18.m6.1.2.2.3.cmml" xref="S3.p2.18.m6.1.2.2.3">𝑖</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p2.18.m6.1c">\operatorname{{\mathbf{\Lambda}}}_{i}^{-{1}/{2}(t)}\operatorname{\mathbf{X}}_{i}</annotation><annotation encoding="application/x-llamapun" id="S3.p2.18.m6.1d">bold_Λ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT - 1 / 2 ( italic_t ) end_POSTSUPERSCRIPT bold_X start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> as their <math alttext="i" class="ltx_Math" display="inline" id="S3.p2.19.m7.1"><semantics id="S3.p2.19.m7.1a"><mi id="S3.p2.19.m7.1.1" xref="S3.p2.19.m7.1.1.cmml">i</mi><annotation-xml encoding="MathML-Content" id="S3.p2.19.m7.1b"><ci id="S3.p2.19.m7.1.1.cmml" xref="S3.p2.19.m7.1.1">𝑖</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.p2.19.m7.1c">i</annotation><annotation encoding="application/x-llamapun" id="S3.p2.19.m7.1d">italic_i</annotation></semantics></math>th row blocks. Then, (<a class="ltx_ref" href="https://arxiv.org/html/2503.13782v1#S3.E7" title="In 3 Regularized Parameter Estimation ‣ Regularized Parameter Estimation in Mixed Model Trace Regression"><span class="ltx_text ltx_ref_tag">7</span></a>) implies that the loss for estimating <math alttext="\operatorname{\mathbf{b}}" class="ltx_Math" display="inline" id="S3.p2.20.m8.1"><semantics id="S3.p2.20.m8.1a"><mi id="S3.p2.20.m8.1.1" xref="S3.p2.20.m8.1.1.cmml">𝐛</mi><annotation-xml encoding="MathML-Content" id="S3.p2.20.m8.1b"><ci id="S3.p2.20.m8.1.1.cmml" xref="S3.p2.20.m8.1.1">𝐛</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.p2.20.m8.1c">\operatorname{\mathbf{b}}</annotation><annotation encoding="application/x-llamapun" id="S3.p2.20.m8.1d">bold_b</annotation></semantics></math> is <math alttext="\|\breve{\operatorname{\mathbf{y}}}^{{(t)}}-\breve{\operatorname{\mathbf{X}}}^% {{(t)}}\operatorname{\mathbf{b}}\|^{2}" class="ltx_Math" display="inline" id="S3.p2.21.m9.3"><semantics id="S3.p2.21.m9.3a"><msup id="S3.p2.21.m9.3.3" xref="S3.p2.21.m9.3.3.cmml"><mrow id="S3.p2.21.m9.3.3.1.1" xref="S3.p2.21.m9.3.3.1.2.cmml"><mo id="S3.p2.21.m9.3.3.1.1.2" stretchy="false" xref="S3.p2.21.m9.3.3.1.2.1.cmml">‖</mo><mrow id="S3.p2.21.m9.3.3.1.1.1" xref="S3.p2.21.m9.3.3.1.1.1.cmml"><msup id="S3.p2.21.m9.3.3.1.1.1.2" xref="S3.p2.21.m9.3.3.1.1.1.2.cmml"><mover accent="true" id="S3.p2.21.m9.3.3.1.1.1.2.2" xref="S3.p2.21.m9.3.3.1.1.1.2.2.cmml"><mi id="S3.p2.21.m9.3.3.1.1.1.2.2.2" xref="S3.p2.21.m9.3.3.1.1.1.2.2.2.cmml">𝐲</mi><mo id="S3.p2.21.m9.3.3.1.1.1.2.2.1" xref="S3.p2.21.m9.3.3.1.1.1.2.2.1.cmml">˘</mo></mover><mrow id="S3.p2.21.m9.1.1.1.3" xref="S3.p2.21.m9.3.3.1.1.1.2.cmml"><mo id="S3.p2.21.m9.1.1.1.3.1" stretchy="false" xref="S3.p2.21.m9.3.3.1.1.1.2.cmml">(</mo><mi id="S3.p2.21.m9.1.1.1.1" xref="S3.p2.21.m9.1.1.1.1.cmml">t</mi><mo id="S3.p2.21.m9.1.1.1.3.2" stretchy="false" xref="S3.p2.21.m9.3.3.1.1.1.2.cmml">)</mo></mrow></msup><mo id="S3.p2.21.m9.3.3.1.1.1.1" lspace="0em" xref="S3.p2.21.m9.3.3.1.1.1.1.cmml">−</mo><mrow id="S3.p2.21.m9.3.3.1.1.1.3" xref="S3.p2.21.m9.3.3.1.1.1.3.cmml"><msup id="S3.p2.21.m9.3.3.1.1.1.3.2" xref="S3.p2.21.m9.3.3.1.1.1.3.2.cmml"><mover accent="true" id="S3.p2.21.m9.3.3.1.1.1.3.2.2" xref="S3.p2.21.m9.3.3.1.1.1.3.2.2.cmml"><mi id="S3.p2.21.m9.3.3.1.1.1.3.2.2.2" xref="S3.p2.21.m9.3.3.1.1.1.3.2.2.2.cmml">𝐗</mi><mo id="S3.p2.21.m9.3.3.1.1.1.3.2.2.1" xref="S3.p2.21.m9.3.3.1.1.1.3.2.2.1.cmml">˘</mo></mover><mrow id="S3.p2.21.m9.2.2.1.3" xref="S3.p2.21.m9.3.3.1.1.1.3.2.cmml"><mo id="S3.p2.21.m9.2.2.1.3.1" stretchy="false" xref="S3.p2.21.m9.3.3.1.1.1.3.2.cmml">(</mo><mi id="S3.p2.21.m9.2.2.1.1" xref="S3.p2.21.m9.2.2.1.1.cmml">t</mi><mo id="S3.p2.21.m9.2.2.1.3.2" stretchy="false" xref="S3.p2.21.m9.3.3.1.1.1.3.2.cmml">)</mo></mrow></msup><mo id="S3.p2.21.m9.3.3.1.1.1.3.1" lspace="0.167em" xref="S3.p2.21.m9.3.3.1.1.1.3.1.cmml">⁢</mo><mi id="S3.p2.21.m9.3.3.1.1.1.3.3" xref="S3.p2.21.m9.3.3.1.1.1.3.3.cmml">𝐛</mi></mrow></mrow><mo id="S3.p2.21.m9.3.3.1.1.3" stretchy="false" xref="S3.p2.21.m9.3.3.1.2.1.cmml">‖</mo></mrow><mn id="S3.p2.21.m9.3.3.3" xref="S3.p2.21.m9.3.3.3.cmml">2</mn></msup><annotation-xml encoding="MathML-Content" id="S3.p2.21.m9.3b"><apply id="S3.p2.21.m9.3.3.cmml" xref="S3.p2.21.m9.3.3"><csymbol cd="ambiguous" id="S3.p2.21.m9.3.3.2.cmml" xref="S3.p2.21.m9.3.3">superscript</csymbol><apply id="S3.p2.21.m9.3.3.1.2.cmml" xref="S3.p2.21.m9.3.3.1.1"><csymbol cd="latexml" id="S3.p2.21.m9.3.3.1.2.1.cmml" xref="S3.p2.21.m9.3.3.1.1.2">norm</csymbol><apply id="S3.p2.21.m9.3.3.1.1.1.cmml" xref="S3.p2.21.m9.3.3.1.1.1"><minus id="S3.p2.21.m9.3.3.1.1.1.1.cmml" xref="S3.p2.21.m9.3.3.1.1.1.1"></minus><apply id="S3.p2.21.m9.3.3.1.1.1.2.cmml" xref="S3.p2.21.m9.3.3.1.1.1.2"><csymbol cd="ambiguous" id="S3.p2.21.m9.3.3.1.1.1.2.1.cmml" xref="S3.p2.21.m9.3.3.1.1.1.2">superscript</csymbol><apply id="S3.p2.21.m9.3.3.1.1.1.2.2.cmml" xref="S3.p2.21.m9.3.3.1.1.1.2.2"><ci id="S3.p2.21.m9.3.3.1.1.1.2.2.1.cmml" xref="S3.p2.21.m9.3.3.1.1.1.2.2.1">˘</ci><ci id="S3.p2.21.m9.3.3.1.1.1.2.2.2.cmml" xref="S3.p2.21.m9.3.3.1.1.1.2.2.2">𝐲</ci></apply><ci id="S3.p2.21.m9.1.1.1.1.cmml" xref="S3.p2.21.m9.1.1.1.1">𝑡</ci></apply><apply id="S3.p2.21.m9.3.3.1.1.1.3.cmml" xref="S3.p2.21.m9.3.3.1.1.1.3"><times id="S3.p2.21.m9.3.3.1.1.1.3.1.cmml" xref="S3.p2.21.m9.3.3.1.1.1.3.1"></times><apply id="S3.p2.21.m9.3.3.1.1.1.3.2.cmml" xref="S3.p2.21.m9.3.3.1.1.1.3.2"><csymbol cd="ambiguous" id="S3.p2.21.m9.3.3.1.1.1.3.2.1.cmml" xref="S3.p2.21.m9.3.3.1.1.1.3.2">superscript</csymbol><apply id="S3.p2.21.m9.3.3.1.1.1.3.2.2.cmml" xref="S3.p2.21.m9.3.3.1.1.1.3.2.2"><ci id="S3.p2.21.m9.3.3.1.1.1.3.2.2.1.cmml" xref="S3.p2.21.m9.3.3.1.1.1.3.2.2.1">˘</ci><ci id="S3.p2.21.m9.3.3.1.1.1.3.2.2.2.cmml" xref="S3.p2.21.m9.3.3.1.1.1.3.2.2.2">𝐗</ci></apply><ci id="S3.p2.21.m9.2.2.1.1.cmml" xref="S3.p2.21.m9.2.2.1.1">𝑡</ci></apply><ci id="S3.p2.21.m9.3.3.1.1.1.3.3.cmml" xref="S3.p2.21.m9.3.3.1.1.1.3.3">𝐛</ci></apply></apply></apply><cn id="S3.p2.21.m9.3.3.3.cmml" type="integer" xref="S3.p2.21.m9.3.3.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p2.21.m9.3c">\|\breve{\operatorname{\mathbf{y}}}^{{(t)}}-\breve{\operatorname{\mathbf{X}}}^% {{(t)}}\operatorname{\mathbf{b}}\|^{2}</annotation><annotation encoding="application/x-llamapun" id="S3.p2.21.m9.3d">∥ over˘ start_ARG bold_y end_ARG start_POSTSUPERSCRIPT ( italic_t ) end_POSTSUPERSCRIPT - over˘ start_ARG bold_X end_ARG start_POSTSUPERSCRIPT ( italic_t ) end_POSTSUPERSCRIPT bold_b ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT</annotation></semantics></math>. We jointly estimate (<math alttext="\operatorname{\mathbf{b}}^{(t+1)},\tau^{2(t+1)}" class="ltx_Math" display="inline" id="S3.p2.22.m10.4"><semantics id="S3.p2.22.m10.4a"><mrow id="S3.p2.22.m10.4.4.2" xref="S3.p2.22.m10.4.4.3.cmml"><msup id="S3.p2.22.m10.3.3.1.1" xref="S3.p2.22.m10.3.3.1.1.cmml"><mi id="S3.p2.22.m10.3.3.1.1.2" xref="S3.p2.22.m10.3.3.1.1.2.cmml">𝐛</mi><mrow id="S3.p2.22.m10.1.1.1.1" xref="S3.p2.22.m10.1.1.1.1.1.cmml"><mo id="S3.p2.22.m10.1.1.1.1.2" stretchy="false" xref="S3.p2.22.m10.1.1.1.1.1.cmml">(</mo><mrow id="S3.p2.22.m10.1.1.1.1.1" xref="S3.p2.22.m10.1.1.1.1.1.cmml"><mi id="S3.p2.22.m10.1.1.1.1.1.2" xref="S3.p2.22.m10.1.1.1.1.1.2.cmml">t</mi><mo id="S3.p2.22.m10.1.1.1.1.1.1" xref="S3.p2.22.m10.1.1.1.1.1.1.cmml">+</mo><mn id="S3.p2.22.m10.1.1.1.1.1.3" xref="S3.p2.22.m10.1.1.1.1.1.3.cmml">1</mn></mrow><mo id="S3.p2.22.m10.1.1.1.1.3" stretchy="false" xref="S3.p2.22.m10.1.1.1.1.1.cmml">)</mo></mrow></msup><mo id="S3.p2.22.m10.4.4.2.3" xref="S3.p2.22.m10.4.4.3.cmml">,</mo><msup id="S3.p2.22.m10.4.4.2.2" xref="S3.p2.22.m10.4.4.2.2.cmml"><mi id="S3.p2.22.m10.4.4.2.2.2" xref="S3.p2.22.m10.4.4.2.2.2.cmml">τ</mi><mrow id="S3.p2.22.m10.2.2.1" xref="S3.p2.22.m10.2.2.1.cmml"><mn id="S3.p2.22.m10.2.2.1.3" xref="S3.p2.22.m10.2.2.1.3.cmml">2</mn><mo id="S3.p2.22.m10.2.2.1.2" xref="S3.p2.22.m10.2.2.1.2.cmml">⁢</mo><mrow id="S3.p2.22.m10.2.2.1.1.1" xref="S3.p2.22.m10.2.2.1.1.1.1.cmml"><mo id="S3.p2.22.m10.2.2.1.1.1.2" stretchy="false" xref="S3.p2.22.m10.2.2.1.1.1.1.cmml">(</mo><mrow id="S3.p2.22.m10.2.2.1.1.1.1" xref="S3.p2.22.m10.2.2.1.1.1.1.cmml"><mi id="S3.p2.22.m10.2.2.1.1.1.1.2" xref="S3.p2.22.m10.2.2.1.1.1.1.2.cmml">t</mi><mo id="S3.p2.22.m10.2.2.1.1.1.1.1" xref="S3.p2.22.m10.2.2.1.1.1.1.1.cmml">+</mo><mn id="S3.p2.22.m10.2.2.1.1.1.1.3" xref="S3.p2.22.m10.2.2.1.1.1.1.3.cmml">1</mn></mrow><mo id="S3.p2.22.m10.2.2.1.1.1.3" stretchy="false" xref="S3.p2.22.m10.2.2.1.1.1.1.cmml">)</mo></mrow></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="S3.p2.22.m10.4b"><list id="S3.p2.22.m10.4.4.3.cmml" xref="S3.p2.22.m10.4.4.2"><apply id="S3.p2.22.m10.3.3.1.1.cmml" xref="S3.p2.22.m10.3.3.1.1"><csymbol cd="ambiguous" id="S3.p2.22.m10.3.3.1.1.1.cmml" xref="S3.p2.22.m10.3.3.1.1">superscript</csymbol><ci id="S3.p2.22.m10.3.3.1.1.2.cmml" xref="S3.p2.22.m10.3.3.1.1.2">𝐛</ci><apply id="S3.p2.22.m10.1.1.1.1.1.cmml" xref="S3.p2.22.m10.1.1.1.1"><plus id="S3.p2.22.m10.1.1.1.1.1.1.cmml" xref="S3.p2.22.m10.1.1.1.1.1.1"></plus><ci id="S3.p2.22.m10.1.1.1.1.1.2.cmml" xref="S3.p2.22.m10.1.1.1.1.1.2">𝑡</ci><cn id="S3.p2.22.m10.1.1.1.1.1.3.cmml" type="integer" xref="S3.p2.22.m10.1.1.1.1.1.3">1</cn></apply></apply><apply id="S3.p2.22.m10.4.4.2.2.cmml" xref="S3.p2.22.m10.4.4.2.2"><csymbol cd="ambiguous" id="S3.p2.22.m10.4.4.2.2.1.cmml" xref="S3.p2.22.m10.4.4.2.2">superscript</csymbol><ci id="S3.p2.22.m10.4.4.2.2.2.cmml" xref="S3.p2.22.m10.4.4.2.2.2">𝜏</ci><apply id="S3.p2.22.m10.2.2.1.cmml" xref="S3.p2.22.m10.2.2.1"><times id="S3.p2.22.m10.2.2.1.2.cmml" xref="S3.p2.22.m10.2.2.1.2"></times><cn id="S3.p2.22.m10.2.2.1.3.cmml" type="integer" xref="S3.p2.22.m10.2.2.1.3">2</cn><apply id="S3.p2.22.m10.2.2.1.1.1.1.cmml" xref="S3.p2.22.m10.2.2.1.1.1"><plus id="S3.p2.22.m10.2.2.1.1.1.1.1.cmml" xref="S3.p2.22.m10.2.2.1.1.1.1.1"></plus><ci id="S3.p2.22.m10.2.2.1.1.1.1.2.cmml" xref="S3.p2.22.m10.2.2.1.1.1.1.2">𝑡</ci><cn id="S3.p2.22.m10.2.2.1.1.1.1.3.cmml" type="integer" xref="S3.p2.22.m10.2.2.1.1.1.1.3">1</cn></apply></apply></apply></list></annotation-xml><annotation encoding="application/x-tex" id="S3.p2.22.m10.4c">\operatorname{\mathbf{b}}^{(t+1)},\tau^{2(t+1)}</annotation><annotation encoding="application/x-llamapun" id="S3.p2.22.m10.4d">bold_b start_POSTSUPERSCRIPT ( italic_t + 1 ) end_POSTSUPERSCRIPT , italic_τ start_POSTSUPERSCRIPT 2 ( italic_t + 1 ) end_POSTSUPERSCRIPT</annotation></semantics></math>) by solving the following scaled lasso optimization problem <cite class="ltx_cite ltx_citemacro_citep">(Sun and Zhang, <a class="ltx_ref" href="https://arxiv.org/html/2503.13782v1#bib.bib30" title="">2012</a>)</cite>:</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="S3.EGx8"> <tbody id="S3.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_right ltx_eqn_cell"><math alttext="\displaystyle(\operatorname{\mathbf{b}}^{{(t+1)}},\tau^{(t+1)})=\underset{% \operatorname{\mathbf{b}}\in\operatorname{\mathbb{R}}^{P_{1}P_{2}},\tau\in% \operatorname{\mathbb{R}}}{\operatorname*{\mathop{\mathrm{argmin}}}}\frac{% \quad\|\breve{\operatorname{\mathbf{y}}}^{{(t)}}-\breve{\operatorname{\mathbf{% X}}}^{{(t)}}\operatorname{\mathbf{b}}\|^{2}}{2N\tau}+\frac{\tau}{2}+\lambda_{% \operatorname{\mathbf{B}}}\|\operatorname{\mathbf{b}}\|_{1}," class="ltx_Math" display="inline" id="S3.E8.m1.9"><semantics id="S3.E8.m1.9a"><mrow id="S3.E8.m1.9.9.1" xref="S3.E8.m1.9.9.1.1.cmml"><mrow id="S3.E8.m1.9.9.1.1" 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xref="S3.E8.m1.9.9.1.1.4.4.3.3">1</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.E8.m1.9c">\displaystyle(\operatorname{\mathbf{b}}^{{(t+1)}},\tau^{(t+1)})=\underset{% \operatorname{\mathbf{b}}\in\operatorname{\mathbb{R}}^{P_{1}P_{2}},\tau\in% \operatorname{\mathbb{R}}}{\operatorname*{\mathop{\mathrm{argmin}}}}\frac{% \quad\|\breve{\operatorname{\mathbf{y}}}^{{(t)}}-\breve{\operatorname{\mathbf{% X}}}^{{(t)}}\operatorname{\mathbf{b}}\|^{2}}{2N\tau}+\frac{\tau}{2}+\lambda_{% \operatorname{\mathbf{B}}}\|\operatorname{\mathbf{b}}\|_{1},</annotation><annotation encoding="application/x-llamapun" id="S3.E8.m1.9d">( bold_b start_POSTSUPERSCRIPT ( italic_t + 1 ) end_POSTSUPERSCRIPT , italic_τ start_POSTSUPERSCRIPT ( italic_t + 1 ) end_POSTSUPERSCRIPT ) = start_UNDERACCENT bold_b ∈ blackboard_R start_POSTSUPERSCRIPT italic_P start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_P start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT , italic_τ ∈ blackboard_R end_UNDERACCENT start_ARG roman_argmin end_ARG divide start_ARG ∥ over˘ start_ARG bold_y end_ARG start_POSTSUPERSCRIPT ( italic_t ) end_POSTSUPERSCRIPT - over˘ start_ARG bold_X end_ARG start_POSTSUPERSCRIPT ( italic_t ) end_POSTSUPERSCRIPT bold_b ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG 2 italic_N italic_τ end_ARG + divide start_ARG italic_τ end_ARG start_ARG 2 end_ARG + italic_λ start_POSTSUBSCRIPT bold_B end_POSTSUBSCRIPT ∥ bold_b ∥ start_POSTSUBSCRIPT 1 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">(8)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S3.p2.29">where <math alttext="\lambda_{\operatorname{\mathbf{B}}}\geq 0" class="ltx_Math" display="inline" id="S3.p2.23.m1.1"><semantics id="S3.p2.23.m1.1a"><mrow id="S3.p2.23.m1.1.1" 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xref="S3.p2.24.m2.2.2.2.cmml">=</mo><mrow id="S3.p2.24.m2.2.2.1" xref="S3.p2.24.m2.2.2.1.cmml"><msubsup id="S3.p2.24.m2.2.2.1.2" xref="S3.p2.24.m2.2.2.1.2.cmml"><mo id="S3.p2.24.m2.2.2.1.2.2.2" rspace="0em" xref="S3.p2.24.m2.2.2.1.2.2.2.cmml">∑</mo><mrow id="S3.p2.24.m2.2.2.1.2.2.3" xref="S3.p2.24.m2.2.2.1.2.2.3.cmml"><mi id="S3.p2.24.m2.2.2.1.2.2.3.2" xref="S3.p2.24.m2.2.2.1.2.2.3.2.cmml">i</mi><mo id="S3.p2.24.m2.2.2.1.2.2.3.1" xref="S3.p2.24.m2.2.2.1.2.2.3.1.cmml">=</mo><mn id="S3.p2.24.m2.2.2.1.2.2.3.3" xref="S3.p2.24.m2.2.2.1.2.2.3.3.cmml">1</mn></mrow><msub id="S3.p2.24.m2.2.2.1.2.3" xref="S3.p2.24.m2.2.2.1.2.3.cmml"><mi id="S3.p2.24.m2.2.2.1.2.3.2" xref="S3.p2.24.m2.2.2.1.2.3.2.cmml">P</mi><mn id="S3.p2.24.m2.2.2.1.2.3.3" xref="S3.p2.24.m2.2.2.1.2.3.3.cmml">1</mn></msub></msubsup><mrow id="S3.p2.24.m2.2.2.1.1" xref="S3.p2.24.m2.2.2.1.1.cmml"><msubsup id="S3.p2.24.m2.2.2.1.1.2" xref="S3.p2.24.m2.2.2.1.1.2.cmml"><mo id="S3.p2.24.m2.2.2.1.1.2.2.2" rspace="0em" 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xref="S3.p2.24.m2.2.2.1.1.1.1.1.3.cmml"><mi id="S3.p2.24.m2.2.2.1.1.1.1.1.3.2" xref="S3.p2.24.m2.2.2.1.1.1.1.1.3.2.cmml">i</mi><mo id="S3.p2.24.m2.2.2.1.1.1.1.1.3.1" xref="S3.p2.24.m2.2.2.1.1.1.1.1.3.1.cmml">⁢</mo><mi id="S3.p2.24.m2.2.2.1.1.1.1.1.3.3" xref="S3.p2.24.m2.2.2.1.1.1.1.1.3.3.cmml">j</mi></mrow></msub><mo id="S3.p2.24.m2.2.2.1.1.1.1.3" stretchy="false" xref="S3.p2.24.m2.2.2.1.1.1.2.1.cmml">|</mo></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.p2.24.m2.2b"><apply id="S3.p2.24.m2.2.2.cmml" xref="S3.p2.24.m2.2.2"><eq id="S3.p2.24.m2.2.2.2.cmml" xref="S3.p2.24.m2.2.2.2"></eq><apply id="S3.p2.24.m2.2.2.3.cmml" xref="S3.p2.24.m2.2.2.3"><csymbol cd="ambiguous" id="S3.p2.24.m2.2.2.3.1.cmml" xref="S3.p2.24.m2.2.2.3">subscript</csymbol><apply id="S3.p2.24.m2.2.2.3.2.1.cmml" xref="S3.p2.24.m2.2.2.3.2.2"><csymbol cd="latexml" id="S3.p2.24.m2.2.2.3.2.1.1.cmml" xref="S3.p2.24.m2.2.2.3.2.2.1">norm</csymbol><ci id="S3.p2.24.m2.1.1.cmml" xref="S3.p2.24.m2.1.1">𝐛</ci></apply><cn id="S3.p2.24.m2.2.2.3.3.cmml" type="integer" xref="S3.p2.24.m2.2.2.3.3">1</cn></apply><apply id="S3.p2.24.m2.2.2.1.cmml" xref="S3.p2.24.m2.2.2.1"><apply id="S3.p2.24.m2.2.2.1.2.cmml" xref="S3.p2.24.m2.2.2.1.2"><csymbol cd="ambiguous" id="S3.p2.24.m2.2.2.1.2.1.cmml" xref="S3.p2.24.m2.2.2.1.2">superscript</csymbol><apply id="S3.p2.24.m2.2.2.1.2.2.cmml" xref="S3.p2.24.m2.2.2.1.2"><csymbol cd="ambiguous" id="S3.p2.24.m2.2.2.1.2.2.1.cmml" xref="S3.p2.24.m2.2.2.1.2">subscript</csymbol><sum id="S3.p2.24.m2.2.2.1.2.2.2.cmml" xref="S3.p2.24.m2.2.2.1.2.2.2"></sum><apply id="S3.p2.24.m2.2.2.1.2.2.3.cmml" xref="S3.p2.24.m2.2.2.1.2.2.3"><eq id="S3.p2.24.m2.2.2.1.2.2.3.1.cmml" xref="S3.p2.24.m2.2.2.1.2.2.3.1"></eq><ci id="S3.p2.24.m2.2.2.1.2.2.3.2.cmml" xref="S3.p2.24.m2.2.2.1.2.2.3.2">𝑖</ci><cn id="S3.p2.24.m2.2.2.1.2.2.3.3.cmml" type="integer" xref="S3.p2.24.m2.2.2.1.2.2.3.3">1</cn></apply></apply><apply id="S3.p2.24.m2.2.2.1.2.3.cmml" xref="S3.p2.24.m2.2.2.1.2.3"><csymbol cd="ambiguous" id="S3.p2.24.m2.2.2.1.2.3.1.cmml" xref="S3.p2.24.m2.2.2.1.2.3">subscript</csymbol><ci id="S3.p2.24.m2.2.2.1.2.3.2.cmml" xref="S3.p2.24.m2.2.2.1.2.3.2">𝑃</ci><cn id="S3.p2.24.m2.2.2.1.2.3.3.cmml" type="integer" xref="S3.p2.24.m2.2.2.1.2.3.3">1</cn></apply></apply><apply id="S3.p2.24.m2.2.2.1.1.cmml" xref="S3.p2.24.m2.2.2.1.1"><apply id="S3.p2.24.m2.2.2.1.1.2.cmml" xref="S3.p2.24.m2.2.2.1.1.2"><csymbol cd="ambiguous" id="S3.p2.24.m2.2.2.1.1.2.1.cmml" xref="S3.p2.24.m2.2.2.1.1.2">superscript</csymbol><apply id="S3.p2.24.m2.2.2.1.1.2.2.cmml" xref="S3.p2.24.m2.2.2.1.1.2"><csymbol cd="ambiguous" id="S3.p2.24.m2.2.2.1.1.2.2.1.cmml" xref="S3.p2.24.m2.2.2.1.1.2">subscript</csymbol><sum id="S3.p2.24.m2.2.2.1.1.2.2.2.cmml" xref="S3.p2.24.m2.2.2.1.1.2.2.2"></sum><apply id="S3.p2.24.m2.2.2.1.1.2.2.3.cmml" xref="S3.p2.24.m2.2.2.1.1.2.2.3"><eq id="S3.p2.24.m2.2.2.1.1.2.2.3.1.cmml" xref="S3.p2.24.m2.2.2.1.1.2.2.3.1"></eq><ci id="S3.p2.24.m2.2.2.1.1.2.2.3.2.cmml" xref="S3.p2.24.m2.2.2.1.1.2.2.3.2">𝑗</ci><cn id="S3.p2.24.m2.2.2.1.1.2.2.3.3.cmml" type="integer" xref="S3.p2.24.m2.2.2.1.1.2.2.3.3">1</cn></apply></apply><apply id="S3.p2.24.m2.2.2.1.1.2.3.cmml" xref="S3.p2.24.m2.2.2.1.1.2.3"><csymbol cd="ambiguous" id="S3.p2.24.m2.2.2.1.1.2.3.1.cmml" xref="S3.p2.24.m2.2.2.1.1.2.3">subscript</csymbol><ci id="S3.p2.24.m2.2.2.1.1.2.3.2.cmml" xref="S3.p2.24.m2.2.2.1.1.2.3.2">𝑃</ci><cn id="S3.p2.24.m2.2.2.1.1.2.3.3.cmml" type="integer" xref="S3.p2.24.m2.2.2.1.1.2.3.3">2</cn></apply></apply><apply id="S3.p2.24.m2.2.2.1.1.1.2.cmml" xref="S3.p2.24.m2.2.2.1.1.1.1"><abs id="S3.p2.24.m2.2.2.1.1.1.2.1.cmml" xref="S3.p2.24.m2.2.2.1.1.1.1.2"></abs><apply id="S3.p2.24.m2.2.2.1.1.1.1.1.cmml" xref="S3.p2.24.m2.2.2.1.1.1.1.1"><csymbol cd="ambiguous" id="S3.p2.24.m2.2.2.1.1.1.1.1.1.cmml" xref="S3.p2.24.m2.2.2.1.1.1.1.1">subscript</csymbol><ci id="S3.p2.24.m2.2.2.1.1.1.1.1.2.cmml" xref="S3.p2.24.m2.2.2.1.1.1.1.1.2">𝑏</ci><apply id="S3.p2.24.m2.2.2.1.1.1.1.1.3.cmml" xref="S3.p2.24.m2.2.2.1.1.1.1.1.3"><times id="S3.p2.24.m2.2.2.1.1.1.1.1.3.1.cmml" xref="S3.p2.24.m2.2.2.1.1.1.1.1.3.1"></times><ci id="S3.p2.24.m2.2.2.1.1.1.1.1.3.2.cmml" xref="S3.p2.24.m2.2.2.1.1.1.1.1.3.2">𝑖</ci><ci id="S3.p2.24.m2.2.2.1.1.1.1.1.3.3.cmml" xref="S3.p2.24.m2.2.2.1.1.1.1.1.3.3">𝑗</ci></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p2.24.m2.2c">\|\operatorname{\mathbf{b}}\|_{1}=\sum_{i=1}^{P_{1}}\sum_{j=1}^{P_{2}}|b_{ij}|</annotation><annotation encoding="application/x-llamapun" id="S3.p2.24.m2.2d">∥ bold_b ∥ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = ∑ start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_P start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT ∑ start_POSTSUBSCRIPT italic_j = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_P start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT | italic_b start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT |</annotation></semantics></math> is the lasso penalty. Given <math alttext="\lambda_{\operatorname{\mathbf{B}}}" class="ltx_Math" display="inline" id="S3.p2.25.m3.1"><semantics id="S3.p2.25.m3.1a"><msub id="S3.p2.25.m3.1.1" xref="S3.p2.25.m3.1.1.cmml"><mi id="S3.p2.25.m3.1.1.2" xref="S3.p2.25.m3.1.1.2.cmml">λ</mi><mi id="S3.p2.25.m3.1.1.3" xref="S3.p2.25.m3.1.1.3.cmml">𝐁</mi></msub><annotation-xml encoding="MathML-Content" id="S3.p2.25.m3.1b"><apply id="S3.p2.25.m3.1.1.cmml" xref="S3.p2.25.m3.1.1"><csymbol cd="ambiguous" id="S3.p2.25.m3.1.1.1.cmml" xref="S3.p2.25.m3.1.1">subscript</csymbol><ci id="S3.p2.25.m3.1.1.2.cmml" xref="S3.p2.25.m3.1.1.2">𝜆</ci><ci id="S3.p2.25.m3.1.1.3.cmml" xref="S3.p2.25.m3.1.1.3">𝐁</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p2.25.m3.1c">\lambda_{\operatorname{\mathbf{B}}}</annotation><annotation encoding="application/x-llamapun" id="S3.p2.25.m3.1d">italic_λ start_POSTSUBSCRIPT bold_B end_POSTSUBSCRIPT</annotation></semantics></math>, we estimate <math alttext="\operatorname{\mathbf{b}}^{{(t+1)}}" class="ltx_Math" display="inline" id="S3.p2.26.m4.1"><semantics id="S3.p2.26.m4.1a"><msup id="S3.p2.26.m4.1.2" xref="S3.p2.26.m4.1.2.cmml"><mi id="S3.p2.26.m4.1.2.2" xref="S3.p2.26.m4.1.2.2.cmml">𝐛</mi><mrow id="S3.p2.26.m4.1.1.1.1" xref="S3.p2.26.m4.1.1.1.1.1.cmml"><mo id="S3.p2.26.m4.1.1.1.1.2" stretchy="false" xref="S3.p2.26.m4.1.1.1.1.1.cmml">(</mo><mrow id="S3.p2.26.m4.1.1.1.1.1" xref="S3.p2.26.m4.1.1.1.1.1.cmml"><mi id="S3.p2.26.m4.1.1.1.1.1.2" xref="S3.p2.26.m4.1.1.1.1.1.2.cmml">t</mi><mo id="S3.p2.26.m4.1.1.1.1.1.1" xref="S3.p2.26.m4.1.1.1.1.1.1.cmml">+</mo><mn id="S3.p2.26.m4.1.1.1.1.1.3" xref="S3.p2.26.m4.1.1.1.1.1.3.cmml">1</mn></mrow><mo id="S3.p2.26.m4.1.1.1.1.3" stretchy="false" xref="S3.p2.26.m4.1.1.1.1.1.cmml">)</mo></mrow></msup><annotation-xml encoding="MathML-Content" id="S3.p2.26.m4.1b"><apply id="S3.p2.26.m4.1.2.cmml" xref="S3.p2.26.m4.1.2"><csymbol cd="ambiguous" id="S3.p2.26.m4.1.2.1.cmml" xref="S3.p2.26.m4.1.2">superscript</csymbol><ci id="S3.p2.26.m4.1.2.2.cmml" xref="S3.p2.26.m4.1.2.2">𝐛</ci><apply id="S3.p2.26.m4.1.1.1.1.1.cmml" xref="S3.p2.26.m4.1.1.1.1"><plus id="S3.p2.26.m4.1.1.1.1.1.1.cmml" xref="S3.p2.26.m4.1.1.1.1.1.1"></plus><ci id="S3.p2.26.m4.1.1.1.1.1.2.cmml" xref="S3.p2.26.m4.1.1.1.1.1.2">𝑡</ci><cn id="S3.p2.26.m4.1.1.1.1.1.3.cmml" type="integer" xref="S3.p2.26.m4.1.1.1.1.1.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p2.26.m4.1c">\operatorname{\mathbf{b}}^{{(t+1)}}</annotation><annotation encoding="application/x-llamapun" id="S3.p2.26.m4.1d">bold_b start_POSTSUPERSCRIPT ( italic_t + 1 ) end_POSTSUPERSCRIPT</annotation></semantics></math> and <math alttext="\tau^{2(t+1)}" class="ltx_Math" display="inline" id="S3.p2.27.m5.1"><semantics id="S3.p2.27.m5.1a"><msup id="S3.p2.27.m5.1.2" xref="S3.p2.27.m5.1.2.cmml"><mi id="S3.p2.27.m5.1.2.2" xref="S3.p2.27.m5.1.2.2.cmml">τ</mi><mrow id="S3.p2.27.m5.1.1.1" xref="S3.p2.27.m5.1.1.1.cmml"><mn id="S3.p2.27.m5.1.1.1.3" xref="S3.p2.27.m5.1.1.1.3.cmml">2</mn><mo id="S3.p2.27.m5.1.1.1.2" xref="S3.p2.27.m5.1.1.1.2.cmml">⁢</mo><mrow id="S3.p2.27.m5.1.1.1.1.1" xref="S3.p2.27.m5.1.1.1.1.1.1.cmml"><mo id="S3.p2.27.m5.1.1.1.1.1.2" stretchy="false" xref="S3.p2.27.m5.1.1.1.1.1.1.cmml">(</mo><mrow id="S3.p2.27.m5.1.1.1.1.1.1" xref="S3.p2.27.m5.1.1.1.1.1.1.cmml"><mi id="S3.p2.27.m5.1.1.1.1.1.1.2" xref="S3.p2.27.m5.1.1.1.1.1.1.2.cmml">t</mi><mo id="S3.p2.27.m5.1.1.1.1.1.1.1" xref="S3.p2.27.m5.1.1.1.1.1.1.1.cmml">+</mo><mn id="S3.p2.27.m5.1.1.1.1.1.1.3" xref="S3.p2.27.m5.1.1.1.1.1.1.3.cmml">1</mn></mrow><mo id="S3.p2.27.m5.1.1.1.1.1.3" stretchy="false" xref="S3.p2.27.m5.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></msup><annotation-xml encoding="MathML-Content" id="S3.p2.27.m5.1b"><apply id="S3.p2.27.m5.1.2.cmml" xref="S3.p2.27.m5.1.2"><csymbol cd="ambiguous" id="S3.p2.27.m5.1.2.1.cmml" xref="S3.p2.27.m5.1.2">superscript</csymbol><ci id="S3.p2.27.m5.1.2.2.cmml" xref="S3.p2.27.m5.1.2.2">𝜏</ci><apply id="S3.p2.27.m5.1.1.1.cmml" xref="S3.p2.27.m5.1.1.1"><times id="S3.p2.27.m5.1.1.1.2.cmml" xref="S3.p2.27.m5.1.1.1.2"></times><cn id="S3.p2.27.m5.1.1.1.3.cmml" type="integer" xref="S3.p2.27.m5.1.1.1.3">2</cn><apply id="S3.p2.27.m5.1.1.1.1.1.1.cmml" xref="S3.p2.27.m5.1.1.1.1.1"><plus id="S3.p2.27.m5.1.1.1.1.1.1.1.cmml" xref="S3.p2.27.m5.1.1.1.1.1.1.1"></plus><ci id="S3.p2.27.m5.1.1.1.1.1.1.2.cmml" xref="S3.p2.27.m5.1.1.1.1.1.1.2">𝑡</ci><cn id="S3.p2.27.m5.1.1.1.1.1.1.3.cmml" type="integer" xref="S3.p2.27.m5.1.1.1.1.1.1.3">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p2.27.m5.1c">\tau^{2(t+1)}</annotation><annotation encoding="application/x-llamapun" id="S3.p2.27.m5.1d">italic_τ start_POSTSUPERSCRIPT 2 ( italic_t + 1 ) end_POSTSUPERSCRIPT</annotation></semantics></math> using the <span class="ltx_text ltx_font_typewriter" id="S3.p2.29.1">scalreg</span> package and update <math alttext="\operatorname{{\bm{\theta}}}^{(t)}" class="ltx_Math" display="inline" id="S3.p2.28.m6.1"><semantics id="S3.p2.28.m6.1a"><msup id="S3.p2.28.m6.1.2" xref="S3.p2.28.m6.1.2.cmml"><mi id="S3.p2.28.m6.1.2.2" xref="S3.p2.28.m6.1.2.2.cmml">𝜽</mi><mrow id="S3.p2.28.m6.1.1.1.3" xref="S3.p2.28.m6.1.2.cmml"><mo id="S3.p2.28.m6.1.1.1.3.1" stretchy="false" xref="S3.p2.28.m6.1.2.cmml">(</mo><mi id="S3.p2.28.m6.1.1.1.1" xref="S3.p2.28.m6.1.1.1.1.cmml">t</mi><mo id="S3.p2.28.m6.1.1.1.3.2" stretchy="false" xref="S3.p2.28.m6.1.2.cmml">)</mo></mrow></msup><annotation-xml encoding="MathML-Content" id="S3.p2.28.m6.1b"><apply id="S3.p2.28.m6.1.2.cmml" xref="S3.p2.28.m6.1.2"><csymbol cd="ambiguous" id="S3.p2.28.m6.1.2.1.cmml" xref="S3.p2.28.m6.1.2">superscript</csymbol><ci id="S3.p2.28.m6.1.2.2.cmml" xref="S3.p2.28.m6.1.2.2">𝜽</ci><ci id="S3.p2.28.m6.1.1.1.1.cmml" xref="S3.p2.28.m6.1.1.1.1">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p2.28.m6.1c">\operatorname{{\bm{\theta}}}^{(t)}</annotation><annotation encoding="application/x-llamapun" id="S3.p2.28.m6.1d">bold_italic_θ start_POSTSUPERSCRIPT ( italic_t ) end_POSTSUPERSCRIPT</annotation></semantics></math> to <math alttext="\operatorname{{\bm{\theta}}}^{(t+1/3)}=(\operatorname{\mathbf{b}}^{(t+1)},% \operatorname{\mathbf{L}}_{1}^{(t)},\operatorname{\mathbf{L}}_{2}^{(t)},\tau^{% 2(t+1)})" class="ltx_Math" display="inline" id="S3.p2.29.m7.9"><semantics id="S3.p2.29.m7.9a"><mrow id="S3.p2.29.m7.9.9" xref="S3.p2.29.m7.9.9.cmml"><msup id="S3.p2.29.m7.9.9.6" xref="S3.p2.29.m7.9.9.6.cmml"><mi id="S3.p2.29.m7.9.9.6.2" xref="S3.p2.29.m7.9.9.6.2.cmml">𝜽</mi><mrow id="S3.p2.29.m7.1.1.1.1" xref="S3.p2.29.m7.1.1.1.1.1.cmml"><mo id="S3.p2.29.m7.1.1.1.1.2" stretchy="false" xref="S3.p2.29.m7.1.1.1.1.1.cmml">(</mo><mrow id="S3.p2.29.m7.1.1.1.1.1" xref="S3.p2.29.m7.1.1.1.1.1.cmml"><mi id="S3.p2.29.m7.1.1.1.1.1.2" xref="S3.p2.29.m7.1.1.1.1.1.2.cmml">t</mi><mo id="S3.p2.29.m7.1.1.1.1.1.1" xref="S3.p2.29.m7.1.1.1.1.1.1.cmml">+</mo><mrow id="S3.p2.29.m7.1.1.1.1.1.3" xref="S3.p2.29.m7.1.1.1.1.1.3.cmml"><mn id="S3.p2.29.m7.1.1.1.1.1.3.2" xref="S3.p2.29.m7.1.1.1.1.1.3.2.cmml">1</mn><mo id="S3.p2.29.m7.1.1.1.1.1.3.1" xref="S3.p2.29.m7.1.1.1.1.1.3.1.cmml">/</mo><mn id="S3.p2.29.m7.1.1.1.1.1.3.3" xref="S3.p2.29.m7.1.1.1.1.1.3.3.cmml">3</mn></mrow></mrow><mo id="S3.p2.29.m7.1.1.1.1.3" stretchy="false" xref="S3.p2.29.m7.1.1.1.1.1.cmml">)</mo></mrow></msup><mo id="S3.p2.29.m7.9.9.5" xref="S3.p2.29.m7.9.9.5.cmml">=</mo><mrow id="S3.p2.29.m7.9.9.4.4" xref="S3.p2.29.m7.9.9.4.5.cmml"><mo id="S3.p2.29.m7.9.9.4.4.5" stretchy="false" xref="S3.p2.29.m7.9.9.4.5.cmml">(</mo><msup id="S3.p2.29.m7.6.6.1.1.1" xref="S3.p2.29.m7.6.6.1.1.1.cmml"><mi id="S3.p2.29.m7.6.6.1.1.1.2" xref="S3.p2.29.m7.6.6.1.1.1.2.cmml">𝐛</mi><mrow id="S3.p2.29.m7.2.2.1.1" xref="S3.p2.29.m7.2.2.1.1.1.cmml"><mo id="S3.p2.29.m7.2.2.1.1.2" stretchy="false" xref="S3.p2.29.m7.2.2.1.1.1.cmml">(</mo><mrow id="S3.p2.29.m7.2.2.1.1.1" xref="S3.p2.29.m7.2.2.1.1.1.cmml"><mi id="S3.p2.29.m7.2.2.1.1.1.2" xref="S3.p2.29.m7.2.2.1.1.1.2.cmml">t</mi><mo id="S3.p2.29.m7.2.2.1.1.1.1" xref="S3.p2.29.m7.2.2.1.1.1.1.cmml">+</mo><mn id="S3.p2.29.m7.2.2.1.1.1.3" xref="S3.p2.29.m7.2.2.1.1.1.3.cmml">1</mn></mrow><mo id="S3.p2.29.m7.2.2.1.1.3" stretchy="false" xref="S3.p2.29.m7.2.2.1.1.1.cmml">)</mo></mrow></msup><mo id="S3.p2.29.m7.9.9.4.4.6" xref="S3.p2.29.m7.9.9.4.5.cmml">,</mo><msubsup id="S3.p2.29.m7.7.7.2.2.2" xref="S3.p2.29.m7.7.7.2.2.2.cmml"><mi id="S3.p2.29.m7.7.7.2.2.2.2.2" xref="S3.p2.29.m7.7.7.2.2.2.2.2.cmml">𝐋</mi><mn id="S3.p2.29.m7.7.7.2.2.2.2.3" xref="S3.p2.29.m7.7.7.2.2.2.2.3.cmml">1</mn><mrow id="S3.p2.29.m7.3.3.1.3" xref="S3.p2.29.m7.7.7.2.2.2.cmml"><mo id="S3.p2.29.m7.3.3.1.3.1" stretchy="false" xref="S3.p2.29.m7.7.7.2.2.2.cmml">(</mo><mi id="S3.p2.29.m7.3.3.1.1" xref="S3.p2.29.m7.3.3.1.1.cmml">t</mi><mo id="S3.p2.29.m7.3.3.1.3.2" stretchy="false" xref="S3.p2.29.m7.7.7.2.2.2.cmml">)</mo></mrow></msubsup><mo id="S3.p2.29.m7.9.9.4.4.7" xref="S3.p2.29.m7.9.9.4.5.cmml">,</mo><msubsup id="S3.p2.29.m7.8.8.3.3.3" xref="S3.p2.29.m7.8.8.3.3.3.cmml"><mi id="S3.p2.29.m7.8.8.3.3.3.2.2" xref="S3.p2.29.m7.8.8.3.3.3.2.2.cmml">𝐋</mi><mn id="S3.p2.29.m7.8.8.3.3.3.2.3" xref="S3.p2.29.m7.8.8.3.3.3.2.3.cmml">2</mn><mrow id="S3.p2.29.m7.4.4.1.3" xref="S3.p2.29.m7.8.8.3.3.3.cmml"><mo id="S3.p2.29.m7.4.4.1.3.1" stretchy="false" xref="S3.p2.29.m7.8.8.3.3.3.cmml">(</mo><mi id="S3.p2.29.m7.4.4.1.1" xref="S3.p2.29.m7.4.4.1.1.cmml">t</mi><mo id="S3.p2.29.m7.4.4.1.3.2" stretchy="false" xref="S3.p2.29.m7.8.8.3.3.3.cmml">)</mo></mrow></msubsup><mo id="S3.p2.29.m7.9.9.4.4.8" xref="S3.p2.29.m7.9.9.4.5.cmml">,</mo><msup id="S3.p2.29.m7.9.9.4.4.4" xref="S3.p2.29.m7.9.9.4.4.4.cmml"><mi id="S3.p2.29.m7.9.9.4.4.4.2" xref="S3.p2.29.m7.9.9.4.4.4.2.cmml">τ</mi><mrow id="S3.p2.29.m7.5.5.1" xref="S3.p2.29.m7.5.5.1.cmml"><mn id="S3.p2.29.m7.5.5.1.3" xref="S3.p2.29.m7.5.5.1.3.cmml">2</mn><mo id="S3.p2.29.m7.5.5.1.2" xref="S3.p2.29.m7.5.5.1.2.cmml">⁢</mo><mrow id="S3.p2.29.m7.5.5.1.1.1" xref="S3.p2.29.m7.5.5.1.1.1.1.cmml"><mo id="S3.p2.29.m7.5.5.1.1.1.2" stretchy="false" xref="S3.p2.29.m7.5.5.1.1.1.1.cmml">(</mo><mrow id="S3.p2.29.m7.5.5.1.1.1.1" xref="S3.p2.29.m7.5.5.1.1.1.1.cmml"><mi id="S3.p2.29.m7.5.5.1.1.1.1.2" xref="S3.p2.29.m7.5.5.1.1.1.1.2.cmml">t</mi><mo id="S3.p2.29.m7.5.5.1.1.1.1.1" xref="S3.p2.29.m7.5.5.1.1.1.1.1.cmml">+</mo><mn id="S3.p2.29.m7.5.5.1.1.1.1.3" xref="S3.p2.29.m7.5.5.1.1.1.1.3.cmml">1</mn></mrow><mo id="S3.p2.29.m7.5.5.1.1.1.3" stretchy="false" xref="S3.p2.29.m7.5.5.1.1.1.1.cmml">)</mo></mrow></mrow></msup><mo id="S3.p2.29.m7.9.9.4.4.9" stretchy="false" xref="S3.p2.29.m7.9.9.4.5.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.p2.29.m7.9b"><apply id="S3.p2.29.m7.9.9.cmml" xref="S3.p2.29.m7.9.9"><eq id="S3.p2.29.m7.9.9.5.cmml" xref="S3.p2.29.m7.9.9.5"></eq><apply id="S3.p2.29.m7.9.9.6.cmml" xref="S3.p2.29.m7.9.9.6"><csymbol cd="ambiguous" id="S3.p2.29.m7.9.9.6.1.cmml" xref="S3.p2.29.m7.9.9.6">superscript</csymbol><ci id="S3.p2.29.m7.9.9.6.2.cmml" xref="S3.p2.29.m7.9.9.6.2">𝜽</ci><apply id="S3.p2.29.m7.1.1.1.1.1.cmml" xref="S3.p2.29.m7.1.1.1.1"><plus id="S3.p2.29.m7.1.1.1.1.1.1.cmml" xref="S3.p2.29.m7.1.1.1.1.1.1"></plus><ci id="S3.p2.29.m7.1.1.1.1.1.2.cmml" xref="S3.p2.29.m7.1.1.1.1.1.2">𝑡</ci><apply id="S3.p2.29.m7.1.1.1.1.1.3.cmml" xref="S3.p2.29.m7.1.1.1.1.1.3"><divide id="S3.p2.29.m7.1.1.1.1.1.3.1.cmml" xref="S3.p2.29.m7.1.1.1.1.1.3.1"></divide><cn id="S3.p2.29.m7.1.1.1.1.1.3.2.cmml" type="integer" xref="S3.p2.29.m7.1.1.1.1.1.3.2">1</cn><cn id="S3.p2.29.m7.1.1.1.1.1.3.3.cmml" type="integer" xref="S3.p2.29.m7.1.1.1.1.1.3.3">3</cn></apply></apply></apply><vector id="S3.p2.29.m7.9.9.4.5.cmml" xref="S3.p2.29.m7.9.9.4.4"><apply id="S3.p2.29.m7.6.6.1.1.1.cmml" xref="S3.p2.29.m7.6.6.1.1.1"><csymbol cd="ambiguous" id="S3.p2.29.m7.6.6.1.1.1.1.cmml" xref="S3.p2.29.m7.6.6.1.1.1">superscript</csymbol><ci id="S3.p2.29.m7.6.6.1.1.1.2.cmml" xref="S3.p2.29.m7.6.6.1.1.1.2">𝐛</ci><apply id="S3.p2.29.m7.2.2.1.1.1.cmml" xref="S3.p2.29.m7.2.2.1.1"><plus id="S3.p2.29.m7.2.2.1.1.1.1.cmml" xref="S3.p2.29.m7.2.2.1.1.1.1"></plus><ci id="S3.p2.29.m7.2.2.1.1.1.2.cmml" xref="S3.p2.29.m7.2.2.1.1.1.2">𝑡</ci><cn id="S3.p2.29.m7.2.2.1.1.1.3.cmml" type="integer" xref="S3.p2.29.m7.2.2.1.1.1.3">1</cn></apply></apply><apply id="S3.p2.29.m7.7.7.2.2.2.cmml" xref="S3.p2.29.m7.7.7.2.2.2"><csymbol cd="ambiguous" id="S3.p2.29.m7.7.7.2.2.2.1.cmml" xref="S3.p2.29.m7.7.7.2.2.2">superscript</csymbol><apply id="S3.p2.29.m7.7.7.2.2.2.2.cmml" xref="S3.p2.29.m7.7.7.2.2.2"><csymbol cd="ambiguous" id="S3.p2.29.m7.7.7.2.2.2.2.1.cmml" xref="S3.p2.29.m7.7.7.2.2.2">subscript</csymbol><ci id="S3.p2.29.m7.7.7.2.2.2.2.2.cmml" xref="S3.p2.29.m7.7.7.2.2.2.2.2">𝐋</ci><cn id="S3.p2.29.m7.7.7.2.2.2.2.3.cmml" type="integer" xref="S3.p2.29.m7.7.7.2.2.2.2.3">1</cn></apply><ci 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id="S3.p2.29.m7.5.5.1.2.cmml" xref="S3.p2.29.m7.5.5.1.2"></times><cn id="S3.p2.29.m7.5.5.1.3.cmml" type="integer" xref="S3.p2.29.m7.5.5.1.3">2</cn><apply id="S3.p2.29.m7.5.5.1.1.1.1.cmml" xref="S3.p2.29.m7.5.5.1.1.1"><plus id="S3.p2.29.m7.5.5.1.1.1.1.1.cmml" xref="S3.p2.29.m7.5.5.1.1.1.1.1"></plus><ci id="S3.p2.29.m7.5.5.1.1.1.1.2.cmml" xref="S3.p2.29.m7.5.5.1.1.1.1.2">𝑡</ci><cn id="S3.p2.29.m7.5.5.1.1.1.1.3.cmml" type="integer" xref="S3.p2.29.m7.5.5.1.1.1.1.3">1</cn></apply></apply></apply></vector></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p2.29.m7.9c">\operatorname{{\bm{\theta}}}^{(t+1/3)}=(\operatorname{\mathbf{b}}^{(t+1)},% \operatorname{\mathbf{L}}_{1}^{(t)},\operatorname{\mathbf{L}}_{2}^{(t)},\tau^{% 2(t+1)})</annotation><annotation encoding="application/x-llamapun" id="S3.p2.29.m7.9d">bold_italic_θ start_POSTSUPERSCRIPT ( italic_t + 1 / 3 ) end_POSTSUPERSCRIPT = ( bold_b start_POSTSUPERSCRIPT ( italic_t + 1 ) end_POSTSUPERSCRIPT , bold_L start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_t ) end_POSTSUPERSCRIPT , bold_L start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_t ) end_POSTSUPERSCRIPT , italic_τ start_POSTSUPERSCRIPT 2 ( italic_t + 1 ) end_POSTSUPERSCRIPT )</annotation></semantics></math>, ending the first AECM cycle.</p> </div> <div class="ltx_para" id="S3.p3"> <p class="ltx_p" id="S3.p3.9">The second cycle treats <math alttext="\operatorname{\mathbf{c}}_{1(2)},\ldots,\operatorname{\mathbf{c}}_{n(2)}" class="ltx_Math" display="inline" id="S3.p3.1.m1.5"><semantics id="S3.p3.1.m1.5a"><mrow id="S3.p3.1.m1.5.5.2" xref="S3.p3.1.m1.5.5.3.cmml"><msub id="S3.p3.1.m1.4.4.1.1" xref="S3.p3.1.m1.4.4.1.1.cmml"><mi id="S3.p3.1.m1.4.4.1.1.2" xref="S3.p3.1.m1.4.4.1.1.2.cmml">𝐜</mi><mrow id="S3.p3.1.m1.1.1.1" xref="S3.p3.1.m1.1.1.1.cmml"><mn id="S3.p3.1.m1.1.1.1.3" xref="S3.p3.1.m1.1.1.1.3.cmml">1</mn><mo id="S3.p3.1.m1.1.1.1.2" xref="S3.p3.1.m1.1.1.1.2.cmml">⁢</mo><mrow 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id="S3.p3.1.m1.2.2.1.1" xref="S3.p3.1.m1.2.2.1.1.cmml">2</mn><mo id="S3.p3.1.m1.2.2.1.4.2.2" stretchy="false" xref="S3.p3.1.m1.2.2.1.cmml">)</mo></mrow></mrow></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.p3.1.m1.5b"><list id="S3.p3.1.m1.5.5.3.cmml" xref="S3.p3.1.m1.5.5.2"><apply id="S3.p3.1.m1.4.4.1.1.cmml" xref="S3.p3.1.m1.4.4.1.1"><csymbol cd="ambiguous" id="S3.p3.1.m1.4.4.1.1.1.cmml" xref="S3.p3.1.m1.4.4.1.1">subscript</csymbol><ci id="S3.p3.1.m1.4.4.1.1.2.cmml" xref="S3.p3.1.m1.4.4.1.1.2">𝐜</ci><apply id="S3.p3.1.m1.1.1.1.cmml" xref="S3.p3.1.m1.1.1.1"><times id="S3.p3.1.m1.1.1.1.2.cmml" xref="S3.p3.1.m1.1.1.1.2"></times><cn id="S3.p3.1.m1.1.1.1.3.cmml" type="integer" xref="S3.p3.1.m1.1.1.1.3">1</cn><cn id="S3.p3.1.m1.1.1.1.1.cmml" type="integer" xref="S3.p3.1.m1.1.1.1.1">2</cn></apply></apply><ci id="S3.p3.1.m1.3.3.cmml" xref="S3.p3.1.m1.3.3">…</ci><apply id="S3.p3.1.m1.5.5.2.2.cmml" xref="S3.p3.1.m1.5.5.2.2"><csymbol cd="ambiguous" id="S3.p3.1.m1.5.5.2.2.1.cmml" xref="S3.p3.1.m1.5.5.2.2">subscript</csymbol><ci id="S3.p3.1.m1.5.5.2.2.2.cmml" xref="S3.p3.1.m1.5.5.2.2.2">𝐜</ci><apply id="S3.p3.1.m1.2.2.1.cmml" xref="S3.p3.1.m1.2.2.1"><times id="S3.p3.1.m1.2.2.1.2.cmml" xref="S3.p3.1.m1.2.2.1.2"></times><ci id="S3.p3.1.m1.2.2.1.3.cmml" xref="S3.p3.1.m1.2.2.1.3">𝑛</ci><cn id="S3.p3.1.m1.2.2.1.1.cmml" type="integer" xref="S3.p3.1.m1.2.2.1.1">2</cn></apply></apply></list></annotation-xml><annotation encoding="application/x-tex" id="S3.p3.1.m1.5c">\operatorname{\mathbf{c}}_{1(2)},\ldots,\operatorname{\mathbf{c}}_{n(2)}</annotation><annotation encoding="application/x-llamapun" id="S3.p3.1.m1.5d">bold_c start_POSTSUBSCRIPT 1 ( 2 ) end_POSTSUBSCRIPT , … , bold_c start_POSTSUBSCRIPT italic_n ( 2 ) end_POSTSUBSCRIPT</annotation></semantics></math> in (<a class="ltx_ref" href="https://arxiv.org/html/2503.13782v1#S2.E6" title="In 2.2 Reduction to a Structured Mixed Model ‣ 2 Mixed Model Trace Regression ‣ Regularized Parameter Estimation in Mixed Model Trace Regression"><span class="ltx_text ltx_ref_tag">6</span></a>) as missing data and updates <math alttext="\operatorname{\mathbf{L}}_{1}^{(t)}" class="ltx_Math" display="inline" id="S3.p3.2.m2.1"><semantics id="S3.p3.2.m2.1a"><msubsup id="S3.p3.2.m2.1.2" xref="S3.p3.2.m2.1.2.cmml"><mi id="S3.p3.2.m2.1.2.2.2" xref="S3.p3.2.m2.1.2.2.2.cmml">𝐋</mi><mn id="S3.p3.2.m2.1.2.2.3" xref="S3.p3.2.m2.1.2.2.3.cmml">1</mn><mrow id="S3.p3.2.m2.1.1.1.3" xref="S3.p3.2.m2.1.2.cmml"><mo id="S3.p3.2.m2.1.1.1.3.1" stretchy="false" xref="S3.p3.2.m2.1.2.cmml">(</mo><mi id="S3.p3.2.m2.1.1.1.1" xref="S3.p3.2.m2.1.1.1.1.cmml">t</mi><mo id="S3.p3.2.m2.1.1.1.3.2" stretchy="false" xref="S3.p3.2.m2.1.2.cmml">)</mo></mrow></msubsup><annotation-xml encoding="MathML-Content" id="S3.p3.2.m2.1b"><apply id="S3.p3.2.m2.1.2.cmml" xref="S3.p3.2.m2.1.2"><csymbol cd="ambiguous" id="S3.p3.2.m2.1.2.1.cmml" xref="S3.p3.2.m2.1.2">superscript</csymbol><apply id="S3.p3.2.m2.1.2.2.cmml" xref="S3.p3.2.m2.1.2"><csymbol cd="ambiguous" id="S3.p3.2.m2.1.2.2.1.cmml" xref="S3.p3.2.m2.1.2">subscript</csymbol><ci id="S3.p3.2.m2.1.2.2.2.cmml" xref="S3.p3.2.m2.1.2.2.2">𝐋</ci><cn id="S3.p3.2.m2.1.2.2.3.cmml" type="integer" xref="S3.p3.2.m2.1.2.2.3">1</cn></apply><ci id="S3.p3.2.m2.1.1.1.1.cmml" xref="S3.p3.2.m2.1.1.1.1">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p3.2.m2.1c">\operatorname{\mathbf{L}}_{1}^{(t)}</annotation><annotation encoding="application/x-llamapun" id="S3.p3.2.m2.1d">bold_L start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_t ) end_POSTSUPERSCRIPT</annotation></semantics></math> to <math alttext="\operatorname{\mathbf{L}}_{1}^{(t+1)}" class="ltx_Math" display="inline" id="S3.p3.3.m3.1"><semantics id="S3.p3.3.m3.1a"><msubsup id="S3.p3.3.m3.1.2" xref="S3.p3.3.m3.1.2.cmml"><mi id="S3.p3.3.m3.1.2.2.2" xref="S3.p3.3.m3.1.2.2.2.cmml">𝐋</mi><mn id="S3.p3.3.m3.1.2.2.3" xref="S3.p3.3.m3.1.2.2.3.cmml">1</mn><mrow id="S3.p3.3.m3.1.1.1.1" xref="S3.p3.3.m3.1.1.1.1.1.cmml"><mo id="S3.p3.3.m3.1.1.1.1.2" stretchy="false" xref="S3.p3.3.m3.1.1.1.1.1.cmml">(</mo><mrow id="S3.p3.3.m3.1.1.1.1.1" xref="S3.p3.3.m3.1.1.1.1.1.cmml"><mi id="S3.p3.3.m3.1.1.1.1.1.2" xref="S3.p3.3.m3.1.1.1.1.1.2.cmml">t</mi><mo id="S3.p3.3.m3.1.1.1.1.1.1" xref="S3.p3.3.m3.1.1.1.1.1.1.cmml">+</mo><mn id="S3.p3.3.m3.1.1.1.1.1.3" xref="S3.p3.3.m3.1.1.1.1.1.3.cmml">1</mn></mrow><mo id="S3.p3.3.m3.1.1.1.1.3" stretchy="false" xref="S3.p3.3.m3.1.1.1.1.1.cmml">)</mo></mrow></msubsup><annotation-xml encoding="MathML-Content" id="S3.p3.3.m3.1b"><apply id="S3.p3.3.m3.1.2.cmml" xref="S3.p3.3.m3.1.2"><csymbol cd="ambiguous" id="S3.p3.3.m3.1.2.1.cmml" xref="S3.p3.3.m3.1.2">superscript</csymbol><apply id="S3.p3.3.m3.1.2.2.cmml" xref="S3.p3.3.m3.1.2"><csymbol cd="ambiguous" id="S3.p3.3.m3.1.2.2.1.cmml" xref="S3.p3.3.m3.1.2">subscript</csymbol><ci id="S3.p3.3.m3.1.2.2.2.cmml" xref="S3.p3.3.m3.1.2.2.2">𝐋</ci><cn id="S3.p3.3.m3.1.2.2.3.cmml" type="integer" xref="S3.p3.3.m3.1.2.2.3">1</cn></apply><apply id="S3.p3.3.m3.1.1.1.1.1.cmml" xref="S3.p3.3.m3.1.1.1.1"><plus id="S3.p3.3.m3.1.1.1.1.1.1.cmml" xref="S3.p3.3.m3.1.1.1.1.1.1"></plus><ci id="S3.p3.3.m3.1.1.1.1.1.2.cmml" xref="S3.p3.3.m3.1.1.1.1.1.2">𝑡</ci><cn id="S3.p3.3.m3.1.1.1.1.1.3.cmml" type="integer" xref="S3.p3.3.m3.1.1.1.1.1.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p3.3.m3.1c">\operatorname{\mathbf{L}}_{1}^{(t+1)}</annotation><annotation encoding="application/x-llamapun" id="S3.p3.3.m3.1d">bold_L start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_t + 1 ) end_POSTSUPERSCRIPT</annotation></semantics></math> given <math alttext="(\operatorname{\mathbf{b}}^{(t+1)},\operatorname{\mathbf{L}}_{2}^{(t)},\tau^{2% (t+1)})" class="ltx_Math" display="inline" id="S3.p3.4.m4.6"><semantics id="S3.p3.4.m4.6a"><mrow id="S3.p3.4.m4.6.6.3" xref="S3.p3.4.m4.6.6.4.cmml"><mo id="S3.p3.4.m4.6.6.3.4" stretchy="false" 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id="S3.p3.4.m4.3.3.1.3.cmml" type="integer" xref="S3.p3.4.m4.3.3.1.3">2</cn><apply id="S3.p3.4.m4.3.3.1.1.1.1.cmml" xref="S3.p3.4.m4.3.3.1.1.1"><plus id="S3.p3.4.m4.3.3.1.1.1.1.1.cmml" xref="S3.p3.4.m4.3.3.1.1.1.1.1"></plus><ci id="S3.p3.4.m4.3.3.1.1.1.1.2.cmml" xref="S3.p3.4.m4.3.3.1.1.1.1.2">𝑡</ci><cn id="S3.p3.4.m4.3.3.1.1.1.1.3.cmml" type="integer" xref="S3.p3.4.m4.3.3.1.1.1.1.3">1</cn></apply></apply></apply></vector></annotation-xml><annotation encoding="application/x-tex" id="S3.p3.4.m4.6c">(\operatorname{\mathbf{b}}^{(t+1)},\operatorname{\mathbf{L}}_{2}^{(t)},\tau^{2% (t+1)})</annotation><annotation encoding="application/x-llamapun" id="S3.p3.4.m4.6d">( bold_b start_POSTSUPERSCRIPT ( italic_t + 1 ) end_POSTSUPERSCRIPT , bold_L start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_t ) end_POSTSUPERSCRIPT , italic_τ start_POSTSUPERSCRIPT 2 ( italic_t + 1 ) end_POSTSUPERSCRIPT )</annotation></semantics></math>. For <math alttext="i=1,\ldots,n" class="ltx_Math" display="inline" id="S3.p3.5.m5.3"><semantics id="S3.p3.5.m5.3a"><mrow id="S3.p3.5.m5.3.4" xref="S3.p3.5.m5.3.4.cmml"><mi id="S3.p3.5.m5.3.4.2" xref="S3.p3.5.m5.3.4.2.cmml">i</mi><mo id="S3.p3.5.m5.3.4.1" xref="S3.p3.5.m5.3.4.1.cmml">=</mo><mrow id="S3.p3.5.m5.3.4.3.2" xref="S3.p3.5.m5.3.4.3.1.cmml"><mn id="S3.p3.5.m5.1.1" xref="S3.p3.5.m5.1.1.cmml">1</mn><mo id="S3.p3.5.m5.3.4.3.2.1" xref="S3.p3.5.m5.3.4.3.1.cmml">,</mo><mi id="S3.p3.5.m5.2.2" mathvariant="normal" xref="S3.p3.5.m5.2.2.cmml">…</mi><mo id="S3.p3.5.m5.3.4.3.2.2" xref="S3.p3.5.m5.3.4.3.1.cmml">,</mo><mi id="S3.p3.5.m5.3.3" xref="S3.p3.5.m5.3.3.cmml">n</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.p3.5.m5.3b"><apply id="S3.p3.5.m5.3.4.cmml" xref="S3.p3.5.m5.3.4"><eq id="S3.p3.5.m5.3.4.1.cmml" xref="S3.p3.5.m5.3.4.1"></eq><ci id="S3.p3.5.m5.3.4.2.cmml" xref="S3.p3.5.m5.3.4.2">𝑖</ci><list id="S3.p3.5.m5.3.4.3.1.cmml" xref="S3.p3.5.m5.3.4.3.2"><cn id="S3.p3.5.m5.1.1.cmml" type="integer" xref="S3.p3.5.m5.1.1">1</cn><ci id="S3.p3.5.m5.2.2.cmml" xref="S3.p3.5.m5.2.2">…</ci><ci id="S3.p3.5.m5.3.3.cmml" xref="S3.p3.5.m5.3.3">𝑛</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p3.5.m5.3c">i=1,\ldots,n</annotation><annotation encoding="application/x-llamapun" id="S3.p3.5.m5.3d">italic_i = 1 , … , italic_n</annotation></semantics></math>, (<a class="ltx_ref" href="https://arxiv.org/html/2503.13782v1#S2.E6" title="In 2.2 Reduction to a Structured Mixed Model ‣ 2 Mixed Model Trace Regression ‣ Regularized Parameter Estimation in Mixed Model Trace Regression"><span class="ltx_text ltx_ref_tag">6</span></a>) implies that the complete data are <math alttext="(\operatorname{\mathbf{y}}_{i},\operatorname{\mathbf{c}}_{i(2)})" class="ltx_Math" display="inline" id="S3.p3.6.m6.3"><semantics id="S3.p3.6.m6.3a"><mrow id="S3.p3.6.m6.3.3.2" xref="S3.p3.6.m6.3.3.3.cmml"><mo id="S3.p3.6.m6.3.3.2.3" stretchy="false" xref="S3.p3.6.m6.3.3.3.cmml">(</mo><msub id="S3.p3.6.m6.2.2.1.1" xref="S3.p3.6.m6.2.2.1.1.cmml"><mi id="S3.p3.6.m6.2.2.1.1.2" xref="S3.p3.6.m6.2.2.1.1.2.cmml">𝐲</mi><mi id="S3.p3.6.m6.2.2.1.1.3" xref="S3.p3.6.m6.2.2.1.1.3.cmml">i</mi></msub><mo id="S3.p3.6.m6.3.3.2.4" xref="S3.p3.6.m6.3.3.3.cmml">,</mo><msub id="S3.p3.6.m6.3.3.2.2" xref="S3.p3.6.m6.3.3.2.2.cmml"><mi id="S3.p3.6.m6.3.3.2.2.2" xref="S3.p3.6.m6.3.3.2.2.2.cmml">𝐜</mi><mrow id="S3.p3.6.m6.1.1.1" xref="S3.p3.6.m6.1.1.1.cmml"><mi id="S3.p3.6.m6.1.1.1.3" xref="S3.p3.6.m6.1.1.1.3.cmml">i</mi><mo id="S3.p3.6.m6.1.1.1.2" xref="S3.p3.6.m6.1.1.1.2.cmml">⁢</mo><mrow id="S3.p3.6.m6.1.1.1.4.2" xref="S3.p3.6.m6.1.1.1.cmml"><mo id="S3.p3.6.m6.1.1.1.4.2.1" stretchy="false" xref="S3.p3.6.m6.1.1.1.cmml">(</mo><mn id="S3.p3.6.m6.1.1.1.1" xref="S3.p3.6.m6.1.1.1.1.cmml">2</mn><mo id="S3.p3.6.m6.1.1.1.4.2.2" stretchy="false" xref="S3.p3.6.m6.1.1.1.cmml">)</mo></mrow></mrow></msub><mo id="S3.p3.6.m6.3.3.2.5" stretchy="false" xref="S3.p3.6.m6.3.3.3.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.p3.6.m6.3b"><interval closure="open" id="S3.p3.6.m6.3.3.3.cmml" xref="S3.p3.6.m6.3.3.2"><apply id="S3.p3.6.m6.2.2.1.1.cmml" xref="S3.p3.6.m6.2.2.1.1"><csymbol cd="ambiguous" id="S3.p3.6.m6.2.2.1.1.1.cmml" xref="S3.p3.6.m6.2.2.1.1">subscript</csymbol><ci id="S3.p3.6.m6.2.2.1.1.2.cmml" xref="S3.p3.6.m6.2.2.1.1.2">𝐲</ci><ci id="S3.p3.6.m6.2.2.1.1.3.cmml" xref="S3.p3.6.m6.2.2.1.1.3">𝑖</ci></apply><apply id="S3.p3.6.m6.3.3.2.2.cmml" xref="S3.p3.6.m6.3.3.2.2"><csymbol cd="ambiguous" id="S3.p3.6.m6.3.3.2.2.1.cmml" xref="S3.p3.6.m6.3.3.2.2">subscript</csymbol><ci id="S3.p3.6.m6.3.3.2.2.2.cmml" xref="S3.p3.6.m6.3.3.2.2.2">𝐜</ci><apply id="S3.p3.6.m6.1.1.1.cmml" xref="S3.p3.6.m6.1.1.1"><times id="S3.p3.6.m6.1.1.1.2.cmml" xref="S3.p3.6.m6.1.1.1.2"></times><ci id="S3.p3.6.m6.1.1.1.3.cmml" xref="S3.p3.6.m6.1.1.1.3">𝑖</ci><cn id="S3.p3.6.m6.1.1.1.1.cmml" type="integer" 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start_POSTSUPERSCRIPT ( italic_t ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i ( 2 ) end_POSTSUBSCRIPT = blackboard_E ( bold_c start_POSTSUBSCRIPT italic_i ( 2 ) end_POSTSUBSCRIPT ∣ bold_y start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , bold_italic_θ start_POSTSUPERSCRIPT ( italic_t + 1 / 3 ) end_POSTSUPERSCRIPT )</annotation></semantics></math> and <math alttext="\operatorname{{\mathbf{\Gamma}}}^{(t)}_{i(2)}=\operatorname{\mathbb{E}}(% \operatorname{\mathbf{c}}_{i(2)}\operatorname{\mathbf{c}}_{i(2)}^{\top}|% \operatorname{\mathbf{y}}_{i},\operatorname{{\bm{\theta}}}^{{(t+1/3)}})" class="ltx_Math" display="inline" id="S3.p3.8.m8.7"><semantics id="S3.p3.8.m8.7a"><mrow id="S3.p3.8.m8.7.7" xref="S3.p3.8.m8.7.7.cmml"><msubsup id="S3.p3.8.m8.7.7.3" xref="S3.p3.8.m8.7.7.3.cmml"><mi id="S3.p3.8.m8.7.7.3.2.2" xref="S3.p3.8.m8.7.7.3.2.2.cmml">𝚪</mi><mrow id="S3.p3.8.m8.2.2.1" xref="S3.p3.8.m8.2.2.1.cmml"><mi id="S3.p3.8.m8.2.2.1.3" xref="S3.p3.8.m8.2.2.1.3.cmml">i</mi><mo 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xref="S3.p3.8.m8.5.5.1.1"><plus id="S3.p3.8.m8.5.5.1.1.1.1.cmml" xref="S3.p3.8.m8.5.5.1.1.1.1"></plus><ci id="S3.p3.8.m8.5.5.1.1.1.2.cmml" xref="S3.p3.8.m8.5.5.1.1.1.2">𝑡</ci><apply id="S3.p3.8.m8.5.5.1.1.1.3.cmml" xref="S3.p3.8.m8.5.5.1.1.1.3"><divide id="S3.p3.8.m8.5.5.1.1.1.3.1.cmml" xref="S3.p3.8.m8.5.5.1.1.1.3.1"></divide><cn id="S3.p3.8.m8.5.5.1.1.1.3.2.cmml" type="integer" xref="S3.p3.8.m8.5.5.1.1.1.3.2">1</cn><cn id="S3.p3.8.m8.5.5.1.1.1.3.3.cmml" type="integer" xref="S3.p3.8.m8.5.5.1.1.1.3.3">3</cn></apply></apply></apply></list></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p3.8.m8.7c">\operatorname{{\mathbf{\Gamma}}}^{(t)}_{i(2)}=\operatorname{\mathbb{E}}(% \operatorname{\mathbf{c}}_{i(2)}\operatorname{\mathbf{c}}_{i(2)}^{\top}|% \operatorname{\mathbf{y}}_{i},\operatorname{{\bm{\theta}}}^{{(t+1/3)}})</annotation><annotation encoding="application/x-llamapun" id="S3.p3.8.m8.7d">bold_Γ start_POSTSUPERSCRIPT ( italic_t ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i ( 2 ) end_POSTSUBSCRIPT = blackboard_E ( bold_c start_POSTSUBSCRIPT italic_i ( 2 ) end_POSTSUBSCRIPT bold_c start_POSTSUBSCRIPT italic_i ( 2 ) end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ⊤ end_POSTSUPERSCRIPT | bold_y start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , bold_italic_θ start_POSTSUPERSCRIPT ( italic_t + 1 / 3 ) end_POSTSUPERSCRIPT )</annotation></semantics></math> are analytically tractable. If <math alttext="\operatorname{\mathbf{l}}_{1}=\text{vec}(\operatorname{\mathbf{L}}_{1})" class="ltx_Math" display="inline" id="S3.p3.9.m9.1"><semantics id="S3.p3.9.m9.1a"><mrow id="S3.p3.9.m9.1.1" xref="S3.p3.9.m9.1.1.cmml"><msub id="S3.p3.9.m9.1.1.3" xref="S3.p3.9.m9.1.1.3.cmml"><mi id="S3.p3.9.m9.1.1.3.2" xref="S3.p3.9.m9.1.1.3.2.cmml">𝐥</mi><mn id="S3.p3.9.m9.1.1.3.3" xref="S3.p3.9.m9.1.1.3.3.cmml">1</mn></msub><mo id="S3.p3.9.m9.1.1.2" xref="S3.p3.9.m9.1.1.2.cmml">=</mo><mrow id="S3.p3.9.m9.1.1.1" xref="S3.p3.9.m9.1.1.1.cmml"><mtext id="S3.p3.9.m9.1.1.1.3" xref="S3.p3.9.m9.1.1.1.3a.cmml">vec</mtext><mo id="S3.p3.9.m9.1.1.1.2" xref="S3.p3.9.m9.1.1.1.2.cmml">⁢</mo><mrow id="S3.p3.9.m9.1.1.1.1.1" xref="S3.p3.9.m9.1.1.1.1.1.1.cmml"><mo id="S3.p3.9.m9.1.1.1.1.1.2" stretchy="false" xref="S3.p3.9.m9.1.1.1.1.1.1.cmml">(</mo><msub id="S3.p3.9.m9.1.1.1.1.1.1" xref="S3.p3.9.m9.1.1.1.1.1.1.cmml"><mi id="S3.p3.9.m9.1.1.1.1.1.1.2" xref="S3.p3.9.m9.1.1.1.1.1.1.2.cmml">𝐋</mi><mn id="S3.p3.9.m9.1.1.1.1.1.1.3" xref="S3.p3.9.m9.1.1.1.1.1.1.3.cmml">1</mn></msub><mo id="S3.p3.9.m9.1.1.1.1.1.3" stretchy="false" xref="S3.p3.9.m9.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.p3.9.m9.1b"><apply id="S3.p3.9.m9.1.1.cmml" xref="S3.p3.9.m9.1.1"><eq id="S3.p3.9.m9.1.1.2.cmml" xref="S3.p3.9.m9.1.1.2"></eq><apply id="S3.p3.9.m9.1.1.3.cmml" xref="S3.p3.9.m9.1.1.3"><csymbol cd="ambiguous" id="S3.p3.9.m9.1.1.3.1.cmml" xref="S3.p3.9.m9.1.1.3">subscript</csymbol><ci id="S3.p3.9.m9.1.1.3.2.cmml" xref="S3.p3.9.m9.1.1.3.2">𝐥</ci><cn id="S3.p3.9.m9.1.1.3.3.cmml" type="integer" xref="S3.p3.9.m9.1.1.3.3">1</cn></apply><apply id="S3.p3.9.m9.1.1.1.cmml" xref="S3.p3.9.m9.1.1.1"><times id="S3.p3.9.m9.1.1.1.2.cmml" xref="S3.p3.9.m9.1.1.1.2"></times><ci id="S3.p3.9.m9.1.1.1.3a.cmml" xref="S3.p3.9.m9.1.1.1.3"><mtext id="S3.p3.9.m9.1.1.1.3.cmml" xref="S3.p3.9.m9.1.1.1.3">vec</mtext></ci><apply id="S3.p3.9.m9.1.1.1.1.1.1.cmml" xref="S3.p3.9.m9.1.1.1.1.1"><csymbol cd="ambiguous" id="S3.p3.9.m9.1.1.1.1.1.1.1.cmml" xref="S3.p3.9.m9.1.1.1.1.1">subscript</csymbol><ci id="S3.p3.9.m9.1.1.1.1.1.1.2.cmml" xref="S3.p3.9.m9.1.1.1.1.1.1.2">𝐋</ci><cn id="S3.p3.9.m9.1.1.1.1.1.1.3.cmml" type="integer" xref="S3.p3.9.m9.1.1.1.1.1.1.3">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p3.9.m9.1c">\operatorname{\mathbf{l}}_{1}=\text{vec}(\operatorname{\mathbf{L}}_{1})</annotation><annotation encoding="application/x-llamapun" id="S3.p3.9.m9.1d">bold_l start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = vec ( bold_L start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT )</annotation></semantics></math>, then the second cycle E-step’s objective is</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="S3.EGx9"> <tbody id="S3.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{\mathcal{Q}}_{(1)}(\operatorname{\mathbf{l}}_{1})" class="ltx_Math" display="inline" id="S3.E9.m1.3"><semantics id="S3.E9.m1.3a"><mrow id="S3.E9.m1.3.3.2" xref="S3.E9.m1.3.3.3.cmml"><msub id="S3.E9.m1.2.2.1.1" xref="S3.E9.m1.2.2.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.E9.m1.2.2.1.1.2" xref="S3.E9.m1.2.2.1.1.2.cmml">𝒬</mi><mrow id="S3.E9.m1.1.1.1.3" xref="S3.E9.m1.2.2.1.1.cmml"><mo id="S3.E9.m1.1.1.1.3.1" stretchy="false" xref="S3.E9.m1.2.2.1.1.cmml">(</mo><mn id="S3.E9.m1.1.1.1.1" xref="S3.E9.m1.1.1.1.1.cmml">1</mn><mo id="S3.E9.m1.1.1.1.3.2" stretchy="false" xref="S3.E9.m1.2.2.1.1.cmml">)</mo></mrow></msub><mo id="S3.E9.m1.3.3.2a" xref="S3.E9.m1.3.3.3.cmml">⁡</mo><mrow id="S3.E9.m1.3.3.2.2" xref="S3.E9.m1.3.3.3.cmml"><mo id="S3.E9.m1.3.3.2.2.2" stretchy="false" xref="S3.E9.m1.3.3.3.cmml">(</mo><msub id="S3.E9.m1.3.3.2.2.1" xref="S3.E9.m1.3.3.2.2.1.cmml"><mi id="S3.E9.m1.3.3.2.2.1.2" xref="S3.E9.m1.3.3.2.2.1.2.cmml">𝐥</mi><mn id="S3.E9.m1.3.3.2.2.1.3" xref="S3.E9.m1.3.3.2.2.1.3.cmml">1</mn></msub><mo id="S3.E9.m1.3.3.2.2.3" stretchy="false" xref="S3.E9.m1.3.3.3.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.E9.m1.3b"><apply id="S3.E9.m1.3.3.3.cmml" xref="S3.E9.m1.3.3.2"><apply id="S3.E9.m1.2.2.1.1.cmml" xref="S3.E9.m1.2.2.1.1"><csymbol cd="ambiguous" id="S3.E9.m1.2.2.1.1.1.cmml" xref="S3.E9.m1.2.2.1.1">subscript</csymbol><ci id="S3.E9.m1.2.2.1.1.2.cmml" xref="S3.E9.m1.2.2.1.1.2">𝒬</ci><cn id="S3.E9.m1.1.1.1.1.cmml" type="integer" xref="S3.E9.m1.1.1.1.1">1</cn></apply><apply id="S3.E9.m1.3.3.2.2.1.cmml" xref="S3.E9.m1.3.3.2.2.1"><csymbol cd="ambiguous" id="S3.E9.m1.3.3.2.2.1.1.cmml" xref="S3.E9.m1.3.3.2.2.1">subscript</csymbol><ci id="S3.E9.m1.3.3.2.2.1.2.cmml" xref="S3.E9.m1.3.3.2.2.1.2">𝐥</ci><cn id="S3.E9.m1.3.3.2.2.1.3.cmml" type="integer" xref="S3.E9.m1.3.3.2.2.1.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.E9.m1.3c">\displaystyle\operatorname{\mathcal{Q}}_{(1)}(\operatorname{\mathbf{l}}_{1})</annotation><annotation encoding="application/x-llamapun" id="S3.E9.m1.3d">caligraphic_Q start_POSTSUBSCRIPT ( 1 ) end_POSTSUBSCRIPT ( bold_l start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT )</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\propto-(\operatorname{\mathbf{H}}_{(1)}^{-{1}/{2}(t)}% \operatorname{\mathbf{g}}^{(t)}_{(1)}-\operatorname{\mathbf{H}}_{(1)}^{{1}/{2}% (t)\top}\operatorname{\mathbf{l}}_{1})^{\top}(\operatorname{\mathbf{H}}_{(1)}^% {-{1}/{2}(t)}\operatorname{\mathbf{g}}^{(t)}_{(1)}-\operatorname{\mathbf{H}}_{% (1)}^{{1}/{2}(t)\top}\operatorname{\mathbf{l}}_{1})," class="ltx_Math" display="inline" id="S3.E9.m2.13"><semantics id="S3.E9.m2.13a"><mrow id="S3.E9.m2.13.13.1" xref="S3.E9.m2.13.13.1.1.cmml"><mrow id="S3.E9.m2.13.13.1.1" xref="S3.E9.m2.13.13.1.1.cmml"><mi id="S3.E9.m2.13.13.1.1.4" 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xref="S3.E9.m2.13.13.1.1.1.1.1.1.1.1.2.1.cmml"><mo id="S3.E9.m2.1.1.1.3.1" stretchy="false" xref="S3.E9.m2.13.13.1.1.1.1.1.1.1.1.2.1.cmml">(</mo><mn id="S3.E9.m2.1.1.1.1" xref="S3.E9.m2.1.1.1.1.cmml">1</mn><mo id="S3.E9.m2.1.1.1.3.2" stretchy="false" xref="S3.E9.m2.13.13.1.1.1.1.1.1.1.1.2.1.cmml">)</mo></mrow><mrow id="S3.E9.m2.2.2.1" xref="S3.E9.m2.2.2.1.cmml"><mo id="S3.E9.m2.2.2.1a" xref="S3.E9.m2.2.2.1.cmml">−</mo><mrow id="S3.E9.m2.2.2.1.3" xref="S3.E9.m2.2.2.1.3.cmml"><mrow id="S3.E9.m2.2.2.1.3.2" xref="S3.E9.m2.2.2.1.3.2.cmml"><mn id="S3.E9.m2.2.2.1.3.2.2" xref="S3.E9.m2.2.2.1.3.2.2.cmml">1</mn><mo id="S3.E9.m2.2.2.1.3.2.1" xref="S3.E9.m2.2.2.1.3.2.1.cmml">/</mo><mn id="S3.E9.m2.2.2.1.3.2.3" xref="S3.E9.m2.2.2.1.3.2.3.cmml">2</mn></mrow><mo id="S3.E9.m2.2.2.1.3.1" xref="S3.E9.m2.2.2.1.3.1.cmml">⁢</mo><mrow id="S3.E9.m2.2.2.1.3.3.2" xref="S3.E9.m2.2.2.1.3.cmml"><mo id="S3.E9.m2.2.2.1.3.3.2.1" stretchy="false" xref="S3.E9.m2.2.2.1.3.cmml">(</mo><mi id="S3.E9.m2.2.2.1.1" 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xref="S3.E9.m2.3.3.1.1.cmml">t</mi><mo id="S3.E9.m2.3.3.1.3.2" stretchy="false" xref="S3.E9.m2.13.13.1.1.1.1.1.1.1.1.2.2.cmml">)</mo></mrow></msubsup></mrow><mo id="S3.E9.m2.13.13.1.1.1.1.1.1.1.1.1" lspace="0em" xref="S3.E9.m2.13.13.1.1.1.1.1.1.1.1.1.cmml">−</mo><mrow id="S3.E9.m2.13.13.1.1.1.1.1.1.1.1.3" xref="S3.E9.m2.13.13.1.1.1.1.1.1.1.1.3.cmml"><msubsup id="S3.E9.m2.13.13.1.1.1.1.1.1.1.1.3.1" xref="S3.E9.m2.13.13.1.1.1.1.1.1.1.1.3.1.cmml"><mi id="S3.E9.m2.13.13.1.1.1.1.1.1.1.1.3.1.2.2" xref="S3.E9.m2.13.13.1.1.1.1.1.1.1.1.3.1.2.2.cmml">𝐇</mi><mrow id="S3.E9.m2.5.5.1.3" xref="S3.E9.m2.13.13.1.1.1.1.1.1.1.1.3.1.cmml"><mo id="S3.E9.m2.5.5.1.3.1" stretchy="false" xref="S3.E9.m2.13.13.1.1.1.1.1.1.1.1.3.1.cmml">(</mo><mn id="S3.E9.m2.5.5.1.1" xref="S3.E9.m2.5.5.1.1.cmml">1</mn><mo id="S3.E9.m2.5.5.1.3.2" stretchy="false" xref="S3.E9.m2.13.13.1.1.1.1.1.1.1.1.3.1.cmml">)</mo></mrow><mrow id="S3.E9.m2.6.6.1" xref="S3.E9.m2.6.6.1.cmml"><mrow id="S3.E9.m2.6.6.1.3" 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xref="S3.E9.m2.13.13.1.1.2.2.2.1.1.3.2.3">1</cn></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.E9.m2.13c">\displaystyle\propto-(\operatorname{\mathbf{H}}_{(1)}^{-{1}/{2}(t)}% \operatorname{\mathbf{g}}^{(t)}_{(1)}-\operatorname{\mathbf{H}}_{(1)}^{{1}/{2}% (t)\top}\operatorname{\mathbf{l}}_{1})^{\top}(\operatorname{\mathbf{H}}_{(1)}^% {-{1}/{2}(t)}\operatorname{\mathbf{g}}^{(t)}_{(1)}-\operatorname{\mathbf{H}}_{% (1)}^{{1}/{2}(t)\top}\operatorname{\mathbf{l}}_{1}),</annotation><annotation encoding="application/x-llamapun" id="S3.E9.m2.13d">∝ - ( bold_H start_POSTSUBSCRIPT ( 1 ) end_POSTSUBSCRIPT start_POSTSUPERSCRIPT - 1 / 2 ( italic_t ) end_POSTSUPERSCRIPT bold_g start_POSTSUPERSCRIPT ( italic_t ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT ( 1 ) end_POSTSUBSCRIPT - bold_H start_POSTSUBSCRIPT ( 1 ) end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 1 / 2 ( italic_t ) ⊤ end_POSTSUPERSCRIPT bold_l start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT ⊤ end_POSTSUPERSCRIPT ( bold_H start_POSTSUBSCRIPT ( 1 ) end_POSTSUBSCRIPT start_POSTSUPERSCRIPT - 1 / 2 ( italic_t ) end_POSTSUPERSCRIPT bold_g start_POSTSUPERSCRIPT ( italic_t ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT ( 1 ) end_POSTSUBSCRIPT - bold_H start_POSTSUBSCRIPT ( 1 ) end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 1 / 2 ( italic_t ) ⊤ end_POSTSUPERSCRIPT bold_l start_POSTSUBSCRIPT 1 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">(9)</span></td> </tr></tbody> <tbody id="S3.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\operatorname{\mathbf{H}}_{(1)}^{{(t)}}" class="ltx_Math" display="inline" id="S3.Ex1.m1.2"><semantics id="S3.Ex1.m1.2a"><msubsup id="S3.Ex1.m1.2.3" xref="S3.Ex1.m1.2.3.cmml"><mi id="S3.Ex1.m1.2.3.2.2" xref="S3.Ex1.m1.2.3.2.2.cmml">𝐇</mi><mrow id="S3.Ex1.m1.1.1.1.3" xref="S3.Ex1.m1.2.3.cmml"><mo id="S3.Ex1.m1.1.1.1.3.1" stretchy="false" xref="S3.Ex1.m1.2.3.cmml">(</mo><mn id="S3.Ex1.m1.1.1.1.1" xref="S3.Ex1.m1.1.1.1.1.cmml">1</mn><mo id="S3.Ex1.m1.1.1.1.3.2" stretchy="false" xref="S3.Ex1.m1.2.3.cmml">)</mo></mrow><mrow id="S3.Ex1.m1.2.2.1.3" xref="S3.Ex1.m1.2.3.cmml"><mo id="S3.Ex1.m1.2.2.1.3.1" stretchy="false" xref="S3.Ex1.m1.2.3.cmml">(</mo><mi id="S3.Ex1.m1.2.2.1.1" xref="S3.Ex1.m1.2.2.1.1.cmml">t</mi><mo id="S3.Ex1.m1.2.2.1.3.2" stretchy="false" xref="S3.Ex1.m1.2.3.cmml">)</mo></mrow></msubsup><annotation-xml encoding="MathML-Content" id="S3.Ex1.m1.2b"><apply id="S3.Ex1.m1.2.3.cmml" xref="S3.Ex1.m1.2.3"><csymbol cd="ambiguous" id="S3.Ex1.m1.2.3.1.cmml" xref="S3.Ex1.m1.2.3">superscript</csymbol><apply id="S3.Ex1.m1.2.3.2.cmml" xref="S3.Ex1.m1.2.3"><csymbol cd="ambiguous" id="S3.Ex1.m1.2.3.2.1.cmml" xref="S3.Ex1.m1.2.3">subscript</csymbol><ci id="S3.Ex1.m1.2.3.2.2.cmml" xref="S3.Ex1.m1.2.3.2.2">𝐇</ci><cn id="S3.Ex1.m1.1.1.1.1.cmml" type="integer" xref="S3.Ex1.m1.1.1.1.1">1</cn></apply><ci id="S3.Ex1.m1.2.2.1.1.cmml" xref="S3.Ex1.m1.2.2.1.1">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Ex1.m1.2c">\displaystyle\operatorname{\mathbf{H}}_{(1)}^{{(t)}}</annotation><annotation encoding="application/x-llamapun" id="S3.Ex1.m1.2d">bold_H start_POSTSUBSCRIPT ( 1 ) end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_t ) end_POSTSUPERSCRIPT</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle=\sum_{i=1}^{n}\sum_{j=1}^{m_{i}}(\operatorname{\mathbf{I}}_{S_{1% }}\otimes\operatorname{\mathbf{Z}}_{ij(1)}\operatorname{\mathbf{L}}^{(t)}_{2})% \operatorname{{\mathbf{\Gamma}}}_{i(2)}^{(t)}(\operatorname{\mathbf{I}}_{S_{1}% 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}}\otimes\operatorname{\mathbf{Z}}_{ij(1)}\operatorname{\mathbf{L}}^{(t)}_{2})% \operatorname{{\mathbf{\Gamma}}}_{i(2)}^{(t)}(\operatorname{\mathbf{I}}_{S_{1}% }\otimes\operatorname{\mathbf{L}}^{(t)\top}_{2}\operatorname{\mathbf{Z}}_{ij(1% )}^{\top}),</annotation><annotation encoding="application/x-llamapun" id="S3.Ex1.m2.7d">= ∑ start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT ∑ start_POSTSUBSCRIPT italic_j = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_m start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUPERSCRIPT ( bold_I start_POSTSUBSCRIPT italic_S start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ⊗ bold_Z start_POSTSUBSCRIPT italic_i italic_j ( 1 ) end_POSTSUBSCRIPT bold_L start_POSTSUPERSCRIPT ( italic_t ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) bold_Γ start_POSTSUBSCRIPT italic_i ( 2 ) end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_t ) end_POSTSUPERSCRIPT ( bold_I start_POSTSUBSCRIPT italic_S start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ⊗ bold_L start_POSTSUPERSCRIPT ( italic_t ) ⊤ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT bold_Z start_POSTSUBSCRIPT italic_i italic_j ( 1 ) end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ⊤ end_POSTSUPERSCRIPT ) ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="S3.Ex2"><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{\mathbf{g}}_{(1)}^{(t)}" class="ltx_Math" display="inline" id="S3.Ex2.m1.2"><semantics id="S3.Ex2.m1.2a"><msubsup id="S3.Ex2.m1.2.3" xref="S3.Ex2.m1.2.3.cmml"><mi id="S3.Ex2.m1.2.3.2.2" xref="S3.Ex2.m1.2.3.2.2.cmml">𝐠</mi><mrow id="S3.Ex2.m1.1.1.1.3" xref="S3.Ex2.m1.2.3.cmml"><mo id="S3.Ex2.m1.1.1.1.3.1" stretchy="false" xref="S3.Ex2.m1.2.3.cmml">(</mo><mn id="S3.Ex2.m1.1.1.1.1" xref="S3.Ex2.m1.1.1.1.1.cmml">1</mn><mo id="S3.Ex2.m1.1.1.1.3.2" stretchy="false" xref="S3.Ex2.m1.2.3.cmml">)</mo></mrow><mrow id="S3.Ex2.m1.2.2.1.3" xref="S3.Ex2.m1.2.3.cmml"><mo id="S3.Ex2.m1.2.2.1.3.1" stretchy="false" xref="S3.Ex2.m1.2.3.cmml">(</mo><mi id="S3.Ex2.m1.2.2.1.1" xref="S3.Ex2.m1.2.2.1.1.cmml">t</mi><mo id="S3.Ex2.m1.2.2.1.3.2" stretchy="false" xref="S3.Ex2.m1.2.3.cmml">)</mo></mrow></msubsup><annotation-xml encoding="MathML-Content" id="S3.Ex2.m1.2b"><apply id="S3.Ex2.m1.2.3.cmml" xref="S3.Ex2.m1.2.3"><csymbol cd="ambiguous" id="S3.Ex2.m1.2.3.1.cmml" xref="S3.Ex2.m1.2.3">superscript</csymbol><apply id="S3.Ex2.m1.2.3.2.cmml" xref="S3.Ex2.m1.2.3"><csymbol cd="ambiguous" id="S3.Ex2.m1.2.3.2.1.cmml" xref="S3.Ex2.m1.2.3">subscript</csymbol><ci id="S3.Ex2.m1.2.3.2.2.cmml" xref="S3.Ex2.m1.2.3.2.2">𝐠</ci><cn id="S3.Ex2.m1.1.1.1.1.cmml" type="integer" xref="S3.Ex2.m1.1.1.1.1">1</cn></apply><ci id="S3.Ex2.m1.2.2.1.1.cmml" xref="S3.Ex2.m1.2.2.1.1">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Ex2.m1.2c">\displaystyle\operatorname{\mathbf{g}}_{(1)}^{(t)}</annotation><annotation encoding="application/x-llamapun" id="S3.Ex2.m1.2d">bold_g start_POSTSUBSCRIPT ( 1 ) end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_t ) end_POSTSUPERSCRIPT</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle=\sum_{i=1}^{n}\left[\operatorname{\mathbf{I}}_{S_{1}}\otimes% \left\{\sum_{j=1}^{m_{i}}(y_{ij}-\operatorname{\mathbf{x}}_{ij}^{\top}% \operatorname{\mathbf{b}}^{{(t+1)}})\operatorname{\mathbf{Z}}_{ij(1)}\right\}% \operatorname{\mathbf{L}}_{2}^{(t)}\right]\operatorname{{\bm{\mu}}}_{i(2)}^{(t% )}," class="ltx_Math" display="inline" id="S3.Ex2.m2.6"><semantics id="S3.Ex2.m2.6a"><mrow id="S3.Ex2.m2.6.6.1" xref="S3.Ex2.m2.6.6.1.1.cmml"><mrow id="S3.Ex2.m2.6.6.1.1" xref="S3.Ex2.m2.6.6.1.1.cmml"><mi id="S3.Ex2.m2.6.6.1.1.3" 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id="S3.Ex2.m2.6d">= ∑ start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT [ bold_I start_POSTSUBSCRIPT italic_S start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ⊗ { ∑ start_POSTSUBSCRIPT italic_j = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_m start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUPERSCRIPT ( italic_y start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT - bold_x start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ⊤ end_POSTSUPERSCRIPT bold_b start_POSTSUPERSCRIPT ( italic_t + 1 ) end_POSTSUPERSCRIPT ) bold_Z start_POSTSUBSCRIPT italic_i italic_j ( 1 ) end_POSTSUBSCRIPT } bold_L start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_t ) end_POSTSUPERSCRIPT ] bold_italic_μ start_POSTSUBSCRIPT italic_i ( 2 ) end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_t ) end_POSTSUPERSCRIPT ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S3.p3.15">where <math alttext="\operatorname{\mathbf{H}}_{(1)}^{{1}/{2}(t)}" class="ltx_Math" display="inline" id="S3.p3.10.m1.2"><semantics id="S3.p3.10.m1.2a"><msubsup id="S3.p3.10.m1.2.3" xref="S3.p3.10.m1.2.3.cmml"><mi id="S3.p3.10.m1.2.3.2.2" xref="S3.p3.10.m1.2.3.2.2.cmml">𝐇</mi><mrow id="S3.p3.10.m1.1.1.1.3" xref="S3.p3.10.m1.2.3.cmml"><mo id="S3.p3.10.m1.1.1.1.3.1" stretchy="false" xref="S3.p3.10.m1.2.3.cmml">(</mo><mn id="S3.p3.10.m1.1.1.1.1" xref="S3.p3.10.m1.1.1.1.1.cmml">1</mn><mo id="S3.p3.10.m1.1.1.1.3.2" stretchy="false" xref="S3.p3.10.m1.2.3.cmml">)</mo></mrow><mrow id="S3.p3.10.m1.2.2.1" xref="S3.p3.10.m1.2.2.1.cmml"><mrow id="S3.p3.10.m1.2.2.1.3" xref="S3.p3.10.m1.2.2.1.3.cmml"><mn id="S3.p3.10.m1.2.2.1.3.2" xref="S3.p3.10.m1.2.2.1.3.2.cmml">1</mn><mo id="S3.p3.10.m1.2.2.1.3.1" xref="S3.p3.10.m1.2.2.1.3.1.cmml">/</mo><mn id="S3.p3.10.m1.2.2.1.3.3" xref="S3.p3.10.m1.2.2.1.3.3.cmml">2</mn></mrow><mo id="S3.p3.10.m1.2.2.1.2" xref="S3.p3.10.m1.2.2.1.2.cmml">⁢</mo><mrow id="S3.p3.10.m1.2.2.1.4.2" xref="S3.p3.10.m1.2.2.1.cmml"><mo id="S3.p3.10.m1.2.2.1.4.2.1" stretchy="false" xref="S3.p3.10.m1.2.2.1.cmml">(</mo><mi id="S3.p3.10.m1.2.2.1.1" xref="S3.p3.10.m1.2.2.1.1.cmml">t</mi><mo id="S3.p3.10.m1.2.2.1.4.2.2" stretchy="false" xref="S3.p3.10.m1.2.2.1.cmml">)</mo></mrow></mrow></msubsup><annotation-xml encoding="MathML-Content" id="S3.p3.10.m1.2b"><apply id="S3.p3.10.m1.2.3.cmml" xref="S3.p3.10.m1.2.3"><csymbol cd="ambiguous" id="S3.p3.10.m1.2.3.1.cmml" xref="S3.p3.10.m1.2.3">superscript</csymbol><apply id="S3.p3.10.m1.2.3.2.cmml" xref="S3.p3.10.m1.2.3"><csymbol cd="ambiguous" id="S3.p3.10.m1.2.3.2.1.cmml" xref="S3.p3.10.m1.2.3">subscript</csymbol><ci id="S3.p3.10.m1.2.3.2.2.cmml" xref="S3.p3.10.m1.2.3.2.2">𝐇</ci><cn id="S3.p3.10.m1.1.1.1.1.cmml" type="integer" xref="S3.p3.10.m1.1.1.1.1">1</cn></apply><apply id="S3.p3.10.m1.2.2.1.cmml" xref="S3.p3.10.m1.2.2.1"><times id="S3.p3.10.m1.2.2.1.2.cmml" xref="S3.p3.10.m1.2.2.1.2"></times><apply id="S3.p3.10.m1.2.2.1.3.cmml" xref="S3.p3.10.m1.2.2.1.3"><divide id="S3.p3.10.m1.2.2.1.3.1.cmml" xref="S3.p3.10.m1.2.2.1.3.1"></divide><cn id="S3.p3.10.m1.2.2.1.3.2.cmml" type="integer" xref="S3.p3.10.m1.2.2.1.3.2">1</cn><cn id="S3.p3.10.m1.2.2.1.3.3.cmml" type="integer" xref="S3.p3.10.m1.2.2.1.3.3">2</cn></apply><ci id="S3.p3.10.m1.2.2.1.1.cmml" xref="S3.p3.10.m1.2.2.1.1">𝑡</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p3.10.m1.2c">\operatorname{\mathbf{H}}_{(1)}^{{1}/{2}(t)}</annotation><annotation encoding="application/x-llamapun" id="S3.p3.10.m1.2d">bold_H start_POSTSUBSCRIPT ( 1 ) end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 1 / 2 ( italic_t ) end_POSTSUPERSCRIPT</annotation></semantics></math> is any matrix such that <math alttext="\operatorname{\mathbf{H}}_{(1)}^{{1}/{2}(t)}\operatorname{\mathbf{H}}_{(1)}^{{% 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xref="S3.p3.11.m2.2.2.1.3.3.cmml">2</mn></mrow><mo id="S3.p3.11.m2.2.2.1.2" xref="S3.p3.11.m2.2.2.1.2.cmml">⁢</mo><mrow id="S3.p3.11.m2.2.2.1.4.2" xref="S3.p3.11.m2.2.2.1.cmml"><mo id="S3.p3.11.m2.2.2.1.4.2.1" stretchy="false" xref="S3.p3.11.m2.2.2.1.cmml">(</mo><mi id="S3.p3.11.m2.2.2.1.1" xref="S3.p3.11.m2.2.2.1.1.cmml">t</mi><mo id="S3.p3.11.m2.2.2.1.4.2.2" stretchy="false" xref="S3.p3.11.m2.2.2.1.cmml">)</mo></mrow></mrow></msubsup><mo id="S3.p3.11.m2.6.7.2a" lspace="0.167em" xref="S3.p3.11.m2.6.7.2.cmml">⁡</mo><msubsup id="S3.p3.11.m2.6.7.2.2" xref="S3.p3.11.m2.6.7.2.2.cmml"><mi id="S3.p3.11.m2.6.7.2.2.2.2" xref="S3.p3.11.m2.6.7.2.2.2.2.cmml">𝐇</mi><mrow id="S3.p3.11.m2.3.3.1.3" xref="S3.p3.11.m2.6.7.2.2.cmml"><mo id="S3.p3.11.m2.3.3.1.3.1" stretchy="false" xref="S3.p3.11.m2.6.7.2.2.cmml">(</mo><mn id="S3.p3.11.m2.3.3.1.1" xref="S3.p3.11.m2.3.3.1.1.cmml">1</mn><mo id="S3.p3.11.m2.3.3.1.3.2" stretchy="false" xref="S3.p3.11.m2.6.7.2.2.cmml">)</mo></mrow><mrow id="S3.p3.11.m2.4.4.1" xref="S3.p3.11.m2.4.4.1.cmml"><mrow id="S3.p3.11.m2.4.4.1.3" xref="S3.p3.11.m2.4.4.1.3.cmml"><mrow id="S3.p3.11.m2.4.4.1.3.2" xref="S3.p3.11.m2.4.4.1.3.2.cmml"><mn id="S3.p3.11.m2.4.4.1.3.2.2" xref="S3.p3.11.m2.4.4.1.3.2.2.cmml">1</mn><mo id="S3.p3.11.m2.4.4.1.3.2.1" xref="S3.p3.11.m2.4.4.1.3.2.1.cmml">/</mo><mn id="S3.p3.11.m2.4.4.1.3.2.3" xref="S3.p3.11.m2.4.4.1.3.2.3.cmml">2</mn></mrow><mo id="S3.p3.11.m2.4.4.1.3.1" xref="S3.p3.11.m2.4.4.1.3.1.cmml">⁢</mo><mrow id="S3.p3.11.m2.4.4.1.3.3.2" xref="S3.p3.11.m2.4.4.1.3.cmml"><mo id="S3.p3.11.m2.4.4.1.3.3.2.1" stretchy="false" xref="S3.p3.11.m2.4.4.1.3.cmml">(</mo><mi id="S3.p3.11.m2.4.4.1.1" xref="S3.p3.11.m2.4.4.1.1.cmml">t</mi><mo id="S3.p3.11.m2.4.4.1.3.3.2.2" stretchy="false" xref="S3.p3.11.m2.4.4.1.3.cmml">)</mo></mrow></mrow><mo id="S3.p3.11.m2.4.4.1.4" lspace="0.222em" xref="S3.p3.11.m2.4.4.1.4.cmml">⊤</mo></mrow></msubsup></mrow><mo id="S3.p3.11.m2.6.7.1" xref="S3.p3.11.m2.6.7.1.cmml">=</mo><msubsup id="S3.p3.11.m2.6.7.3" xref="S3.p3.11.m2.6.7.3.cmml"><mi id="S3.p3.11.m2.6.7.3.2.2" xref="S3.p3.11.m2.6.7.3.2.2.cmml">𝐇</mi><mrow id="S3.p3.11.m2.5.5.1.3" xref="S3.p3.11.m2.6.7.3.cmml"><mo id="S3.p3.11.m2.5.5.1.3.1" stretchy="false" xref="S3.p3.11.m2.6.7.3.cmml">(</mo><mn id="S3.p3.11.m2.5.5.1.1" xref="S3.p3.11.m2.5.5.1.1.cmml">1</mn><mo id="S3.p3.11.m2.5.5.1.3.2" stretchy="false" xref="S3.p3.11.m2.6.7.3.cmml">)</mo></mrow><mrow id="S3.p3.11.m2.6.6.1.3" xref="S3.p3.11.m2.6.7.3.cmml"><mo id="S3.p3.11.m2.6.6.1.3.1" stretchy="false" xref="S3.p3.11.m2.6.7.3.cmml">(</mo><mi id="S3.p3.11.m2.6.6.1.1" xref="S3.p3.11.m2.6.6.1.1.cmml">t</mi><mo id="S3.p3.11.m2.6.6.1.3.2" stretchy="false" xref="S3.p3.11.m2.6.7.3.cmml">)</mo></mrow></msubsup></mrow><annotation-xml encoding="MathML-Content" id="S3.p3.11.m2.6b"><apply id="S3.p3.11.m2.6.7.cmml" xref="S3.p3.11.m2.6.7"><eq id="S3.p3.11.m2.6.7.1.cmml" xref="S3.p3.11.m2.6.7.1"></eq><apply id="S3.p3.11.m2.6.7.2.cmml" xref="S3.p3.11.m2.6.7.2"><apply id="S3.p3.11.m2.6.7.2.1.cmml" xref="S3.p3.11.m2.6.7.2.1"><csymbol cd="ambiguous" id="S3.p3.11.m2.6.7.2.1.1.cmml" xref="S3.p3.11.m2.6.7.2.1">superscript</csymbol><apply id="S3.p3.11.m2.6.7.2.1.2.cmml" xref="S3.p3.11.m2.6.7.2.1"><csymbol cd="ambiguous" id="S3.p3.11.m2.6.7.2.1.2.1.cmml" xref="S3.p3.11.m2.6.7.2.1">subscript</csymbol><ci id="S3.p3.11.m2.6.7.2.1.2.2.cmml" xref="S3.p3.11.m2.6.7.2.1.2.2">𝐇</ci><cn id="S3.p3.11.m2.1.1.1.1.cmml" type="integer" xref="S3.p3.11.m2.1.1.1.1">1</cn></apply><apply id="S3.p3.11.m2.2.2.1.cmml" xref="S3.p3.11.m2.2.2.1"><times id="S3.p3.11.m2.2.2.1.2.cmml" xref="S3.p3.11.m2.2.2.1.2"></times><apply id="S3.p3.11.m2.2.2.1.3.cmml" xref="S3.p3.11.m2.2.2.1.3"><divide id="S3.p3.11.m2.2.2.1.3.1.cmml" xref="S3.p3.11.m2.2.2.1.3.1"></divide><cn id="S3.p3.11.m2.2.2.1.3.2.cmml" type="integer" xref="S3.p3.11.m2.2.2.1.3.2">1</cn><cn id="S3.p3.11.m2.2.2.1.3.3.cmml" type="integer" xref="S3.p3.11.m2.2.2.1.3.3">2</cn></apply><ci id="S3.p3.11.m2.2.2.1.1.cmml" xref="S3.p3.11.m2.2.2.1.1">𝑡</ci></apply></apply><apply id="S3.p3.11.m2.6.7.2.2.cmml" xref="S3.p3.11.m2.6.7.2.2"><csymbol cd="ambiguous" id="S3.p3.11.m2.6.7.2.2.1.cmml" xref="S3.p3.11.m2.6.7.2.2">superscript</csymbol><apply id="S3.p3.11.m2.6.7.2.2.2.cmml" xref="S3.p3.11.m2.6.7.2.2"><csymbol cd="ambiguous" id="S3.p3.11.m2.6.7.2.2.2.1.cmml" xref="S3.p3.11.m2.6.7.2.2">subscript</csymbol><ci id="S3.p3.11.m2.6.7.2.2.2.2.cmml" xref="S3.p3.11.m2.6.7.2.2.2.2">𝐇</ci><cn id="S3.p3.11.m2.3.3.1.1.cmml" type="integer" xref="S3.p3.11.m2.3.3.1.1">1</cn></apply><apply id="S3.p3.11.m2.4.4.1.cmml" xref="S3.p3.11.m2.4.4.1"><csymbol cd="latexml" id="S3.p3.11.m2.4.4.1.2.cmml" xref="S3.p3.11.m2.4.4.1">limit-from</csymbol><apply id="S3.p3.11.m2.4.4.1.3.cmml" xref="S3.p3.11.m2.4.4.1.3"><times id="S3.p3.11.m2.4.4.1.3.1.cmml" xref="S3.p3.11.m2.4.4.1.3.1"></times><apply id="S3.p3.11.m2.4.4.1.3.2.cmml" xref="S3.p3.11.m2.4.4.1.3.2"><divide id="S3.p3.11.m2.4.4.1.3.2.1.cmml" xref="S3.p3.11.m2.4.4.1.3.2.1"></divide><cn id="S3.p3.11.m2.4.4.1.3.2.2.cmml" type="integer" xref="S3.p3.11.m2.4.4.1.3.2.2">1</cn><cn id="S3.p3.11.m2.4.4.1.3.2.3.cmml" type="integer" xref="S3.p3.11.m2.4.4.1.3.2.3">2</cn></apply><ci id="S3.p3.11.m2.4.4.1.1.cmml" xref="S3.p3.11.m2.4.4.1.1">𝑡</ci></apply><csymbol cd="latexml" id="S3.p3.11.m2.4.4.1.4.cmml" xref="S3.p3.11.m2.4.4.1.4">top</csymbol></apply></apply></apply><apply id="S3.p3.11.m2.6.7.3.cmml" xref="S3.p3.11.m2.6.7.3"><csymbol cd="ambiguous" id="S3.p3.11.m2.6.7.3.1.cmml" xref="S3.p3.11.m2.6.7.3">superscript</csymbol><apply id="S3.p3.11.m2.6.7.3.2.cmml" xref="S3.p3.11.m2.6.7.3"><csymbol cd="ambiguous" id="S3.p3.11.m2.6.7.3.2.1.cmml" xref="S3.p3.11.m2.6.7.3">subscript</csymbol><ci id="S3.p3.11.m2.6.7.3.2.2.cmml" xref="S3.p3.11.m2.6.7.3.2.2">𝐇</ci><cn id="S3.p3.11.m2.5.5.1.1.cmml" type="integer" xref="S3.p3.11.m2.5.5.1.1">1</cn></apply><ci id="S3.p3.11.m2.6.6.1.1.cmml" xref="S3.p3.11.m2.6.6.1.1">𝑡</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p3.11.m2.6c">\operatorname{\mathbf{H}}_{(1)}^{{1}/{2}(t)}\operatorname{\mathbf{H}}_{(1)}^{{% 1}/{2}(t)\top}=\operatorname{\mathbf{H}}_{(1)}^{(t)}</annotation><annotation encoding="application/x-llamapun" id="S3.p3.11.m2.6d">bold_H start_POSTSUBSCRIPT ( 1 ) end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 1 / 2 ( italic_t ) end_POSTSUPERSCRIPT bold_H start_POSTSUBSCRIPT ( 1 ) end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 1 / 2 ( italic_t ) ⊤ end_POSTSUPERSCRIPT = bold_H start_POSTSUBSCRIPT ( 1 ) end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_t ) end_POSTSUPERSCRIPT</annotation></semantics></math> and <math alttext="\operatorname{\mathbf{H}}_{(1)}^{-{1}/{2}(t)}" class="ltx_Math" display="inline" id="S3.p3.12.m3.2"><semantics id="S3.p3.12.m3.2a"><msubsup id="S3.p3.12.m3.2.3" xref="S3.p3.12.m3.2.3.cmml"><mi id="S3.p3.12.m3.2.3.2.2" xref="S3.p3.12.m3.2.3.2.2.cmml">𝐇</mi><mrow id="S3.p3.12.m3.1.1.1.3" xref="S3.p3.12.m3.2.3.cmml"><mo id="S3.p3.12.m3.1.1.1.3.1" stretchy="false" xref="S3.p3.12.m3.2.3.cmml">(</mo><mn id="S3.p3.12.m3.1.1.1.1" xref="S3.p3.12.m3.1.1.1.1.cmml">1</mn><mo id="S3.p3.12.m3.1.1.1.3.2" stretchy="false" xref="S3.p3.12.m3.2.3.cmml">)</mo></mrow><mrow id="S3.p3.12.m3.2.2.1" xref="S3.p3.12.m3.2.2.1.cmml"><mo id="S3.p3.12.m3.2.2.1a" xref="S3.p3.12.m3.2.2.1.cmml">−</mo><mrow id="S3.p3.12.m3.2.2.1.3" xref="S3.p3.12.m3.2.2.1.3.cmml"><mrow id="S3.p3.12.m3.2.2.1.3.2" xref="S3.p3.12.m3.2.2.1.3.2.cmml"><mn id="S3.p3.12.m3.2.2.1.3.2.2" xref="S3.p3.12.m3.2.2.1.3.2.2.cmml">1</mn><mo id="S3.p3.12.m3.2.2.1.3.2.1" xref="S3.p3.12.m3.2.2.1.3.2.1.cmml">/</mo><mn id="S3.p3.12.m3.2.2.1.3.2.3" xref="S3.p3.12.m3.2.2.1.3.2.3.cmml">2</mn></mrow><mo id="S3.p3.12.m3.2.2.1.3.1" xref="S3.p3.12.m3.2.2.1.3.1.cmml">⁢</mo><mrow id="S3.p3.12.m3.2.2.1.3.3.2" xref="S3.p3.12.m3.2.2.1.3.cmml"><mo id="S3.p3.12.m3.2.2.1.3.3.2.1" stretchy="false" xref="S3.p3.12.m3.2.2.1.3.cmml">(</mo><mi id="S3.p3.12.m3.2.2.1.1" xref="S3.p3.12.m3.2.2.1.1.cmml">t</mi><mo id="S3.p3.12.m3.2.2.1.3.3.2.2" stretchy="false" xref="S3.p3.12.m3.2.2.1.3.cmml">)</mo></mrow></mrow></mrow></msubsup><annotation-xml encoding="MathML-Content" id="S3.p3.12.m3.2b"><apply id="S3.p3.12.m3.2.3.cmml" xref="S3.p3.12.m3.2.3"><csymbol cd="ambiguous" id="S3.p3.12.m3.2.3.1.cmml" xref="S3.p3.12.m3.2.3">superscript</csymbol><apply id="S3.p3.12.m3.2.3.2.cmml" xref="S3.p3.12.m3.2.3"><csymbol cd="ambiguous" id="S3.p3.12.m3.2.3.2.1.cmml" xref="S3.p3.12.m3.2.3">subscript</csymbol><ci id="S3.p3.12.m3.2.3.2.2.cmml" xref="S3.p3.12.m3.2.3.2.2">𝐇</ci><cn id="S3.p3.12.m3.1.1.1.1.cmml" type="integer" xref="S3.p3.12.m3.1.1.1.1">1</cn></apply><apply id="S3.p3.12.m3.2.2.1.cmml" xref="S3.p3.12.m3.2.2.1"><minus id="S3.p3.12.m3.2.2.1.2.cmml" xref="S3.p3.12.m3.2.2.1"></minus><apply id="S3.p3.12.m3.2.2.1.3.cmml" xref="S3.p3.12.m3.2.2.1.3"><times id="S3.p3.12.m3.2.2.1.3.1.cmml" xref="S3.p3.12.m3.2.2.1.3.1"></times><apply id="S3.p3.12.m3.2.2.1.3.2.cmml" xref="S3.p3.12.m3.2.2.1.3.2"><divide id="S3.p3.12.m3.2.2.1.3.2.1.cmml" xref="S3.p3.12.m3.2.2.1.3.2.1"></divide><cn id="S3.p3.12.m3.2.2.1.3.2.2.cmml" type="integer" xref="S3.p3.12.m3.2.2.1.3.2.2">1</cn><cn id="S3.p3.12.m3.2.2.1.3.2.3.cmml" type="integer" xref="S3.p3.12.m3.2.2.1.3.2.3">2</cn></apply><ci id="S3.p3.12.m3.2.2.1.1.cmml" xref="S3.p3.12.m3.2.2.1.1">𝑡</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p3.12.m3.2c">\operatorname{\mathbf{H}}_{(1)}^{-{1}/{2}(t)}</annotation><annotation encoding="application/x-llamapun" id="S3.p3.12.m3.2d">bold_H start_POSTSUBSCRIPT ( 1 ) end_POSTSUBSCRIPT start_POSTSUPERSCRIPT - 1 / 2 ( italic_t ) end_POSTSUPERSCRIPT</annotation></semantics></math> is any generalized inverse of <math alttext="\operatorname{\mathbf{H}}_{(1)}^{{1}/{2}(t)}" class="ltx_Math" display="inline" id="S3.p3.13.m4.2"><semantics id="S3.p3.13.m4.2a"><msubsup id="S3.p3.13.m4.2.3" xref="S3.p3.13.m4.2.3.cmml"><mi id="S3.p3.13.m4.2.3.2.2" xref="S3.p3.13.m4.2.3.2.2.cmml">𝐇</mi><mrow id="S3.p3.13.m4.1.1.1.3" xref="S3.p3.13.m4.2.3.cmml"><mo id="S3.p3.13.m4.1.1.1.3.1" stretchy="false" xref="S3.p3.13.m4.2.3.cmml">(</mo><mn id="S3.p3.13.m4.1.1.1.1" xref="S3.p3.13.m4.1.1.1.1.cmml">1</mn><mo id="S3.p3.13.m4.1.1.1.3.2" stretchy="false" xref="S3.p3.13.m4.2.3.cmml">)</mo></mrow><mrow id="S3.p3.13.m4.2.2.1" xref="S3.p3.13.m4.2.2.1.cmml"><mrow id="S3.p3.13.m4.2.2.1.3" xref="S3.p3.13.m4.2.2.1.3.cmml"><mn id="S3.p3.13.m4.2.2.1.3.2" xref="S3.p3.13.m4.2.2.1.3.2.cmml">1</mn><mo id="S3.p3.13.m4.2.2.1.3.1" xref="S3.p3.13.m4.2.2.1.3.1.cmml">/</mo><mn id="S3.p3.13.m4.2.2.1.3.3" xref="S3.p3.13.m4.2.2.1.3.3.cmml">2</mn></mrow><mo id="S3.p3.13.m4.2.2.1.2" xref="S3.p3.13.m4.2.2.1.2.cmml">⁢</mo><mrow id="S3.p3.13.m4.2.2.1.4.2" xref="S3.p3.13.m4.2.2.1.cmml"><mo id="S3.p3.13.m4.2.2.1.4.2.1" stretchy="false" xref="S3.p3.13.m4.2.2.1.cmml">(</mo><mi id="S3.p3.13.m4.2.2.1.1" xref="S3.p3.13.m4.2.2.1.1.cmml">t</mi><mo id="S3.p3.13.m4.2.2.1.4.2.2" stretchy="false" xref="S3.p3.13.m4.2.2.1.cmml">)</mo></mrow></mrow></msubsup><annotation-xml encoding="MathML-Content" id="S3.p3.13.m4.2b"><apply id="S3.p3.13.m4.2.3.cmml" xref="S3.p3.13.m4.2.3"><csymbol cd="ambiguous" id="S3.p3.13.m4.2.3.1.cmml" xref="S3.p3.13.m4.2.3">superscript</csymbol><apply id="S3.p3.13.m4.2.3.2.cmml" xref="S3.p3.13.m4.2.3"><csymbol cd="ambiguous" id="S3.p3.13.m4.2.3.2.1.cmml" xref="S3.p3.13.m4.2.3">subscript</csymbol><ci id="S3.p3.13.m4.2.3.2.2.cmml" xref="S3.p3.13.m4.2.3.2.2">𝐇</ci><cn id="S3.p3.13.m4.1.1.1.1.cmml" type="integer" xref="S3.p3.13.m4.1.1.1.1">1</cn></apply><apply id="S3.p3.13.m4.2.2.1.cmml" xref="S3.p3.13.m4.2.2.1"><times id="S3.p3.13.m4.2.2.1.2.cmml" xref="S3.p3.13.m4.2.2.1.2"></times><apply id="S3.p3.13.m4.2.2.1.3.cmml" xref="S3.p3.13.m4.2.2.1.3"><divide id="S3.p3.13.m4.2.2.1.3.1.cmml" xref="S3.p3.13.m4.2.2.1.3.1"></divide><cn id="S3.p3.13.m4.2.2.1.3.2.cmml" type="integer" xref="S3.p3.13.m4.2.2.1.3.2">1</cn><cn id="S3.p3.13.m4.2.2.1.3.3.cmml" type="integer" xref="S3.p3.13.m4.2.2.1.3.3">2</cn></apply><ci id="S3.p3.13.m4.2.2.1.1.cmml" xref="S3.p3.13.m4.2.2.1.1">𝑡</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p3.13.m4.2c">\operatorname{\mathbf{H}}_{(1)}^{{1}/{2}(t)}</annotation><annotation encoding="application/x-llamapun" id="S3.p3.13.m4.2d">bold_H start_POSTSUBSCRIPT ( 1 ) end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 1 / 2 ( italic_t ) end_POSTSUPERSCRIPT</annotation></semantics></math>. Using (<a class="ltx_ref" href="https://arxiv.org/html/2503.13782v1#S3.E9" title="In 3 Regularized Parameter Estimation ‣ Regularized Parameter Estimation in Mixed Model Trace Regression"><span class="ltx_text ltx_ref_tag">9</span></a>), <math alttext="-\operatorname{\mathcal{Q}}_{(1)}(\operatorname{\mathbf{l}}_{1})" class="ltx_Math" display="inline" id="S3.p3.14.m5.3"><semantics id="S3.p3.14.m5.3a"><mrow id="S3.p3.14.m5.3.3" xref="S3.p3.14.m5.3.3.cmml"><mo id="S3.p3.14.m5.3.3a" rspace="0.167em" xref="S3.p3.14.m5.3.3.cmml">−</mo><mrow id="S3.p3.14.m5.3.3.2.2" xref="S3.p3.14.m5.3.3.2.3.cmml"><msub id="S3.p3.14.m5.2.2.1.1.1" xref="S3.p3.14.m5.2.2.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.p3.14.m5.2.2.1.1.1.2" xref="S3.p3.14.m5.2.2.1.1.1.2.cmml">𝒬</mi><mrow id="S3.p3.14.m5.1.1.1.3" xref="S3.p3.14.m5.2.2.1.1.1.cmml"><mo id="S3.p3.14.m5.1.1.1.3.1" stretchy="false" xref="S3.p3.14.m5.2.2.1.1.1.cmml">(</mo><mn id="S3.p3.14.m5.1.1.1.1" xref="S3.p3.14.m5.1.1.1.1.cmml">1</mn><mo id="S3.p3.14.m5.1.1.1.3.2" stretchy="false" xref="S3.p3.14.m5.2.2.1.1.1.cmml">)</mo></mrow></msub><mo id="S3.p3.14.m5.3.3.2.2a" xref="S3.p3.14.m5.3.3.2.3.cmml">⁡</mo><mrow id="S3.p3.14.m5.3.3.2.2.2" xref="S3.p3.14.m5.3.3.2.3.cmml"><mo id="S3.p3.14.m5.3.3.2.2.2.2" stretchy="false" xref="S3.p3.14.m5.3.3.2.3.cmml">(</mo><msub id="S3.p3.14.m5.3.3.2.2.2.1" xref="S3.p3.14.m5.3.3.2.2.2.1.cmml"><mi id="S3.p3.14.m5.3.3.2.2.2.1.2" xref="S3.p3.14.m5.3.3.2.2.2.1.2.cmml">𝐥</mi><mn id="S3.p3.14.m5.3.3.2.2.2.1.3" xref="S3.p3.14.m5.3.3.2.2.2.1.3.cmml">1</mn></msub><mo id="S3.p3.14.m5.3.3.2.2.2.3" stretchy="false" xref="S3.p3.14.m5.3.3.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.p3.14.m5.3b"><apply id="S3.p3.14.m5.3.3.cmml" xref="S3.p3.14.m5.3.3"><minus id="S3.p3.14.m5.3.3.3.cmml" xref="S3.p3.14.m5.3.3"></minus><apply id="S3.p3.14.m5.3.3.2.3.cmml" xref="S3.p3.14.m5.3.3.2.2"><apply id="S3.p3.14.m5.2.2.1.1.1.cmml" xref="S3.p3.14.m5.2.2.1.1.1"><csymbol cd="ambiguous" id="S3.p3.14.m5.2.2.1.1.1.1.cmml" xref="S3.p3.14.m5.2.2.1.1.1">subscript</csymbol><ci id="S3.p3.14.m5.2.2.1.1.1.2.cmml" xref="S3.p3.14.m5.2.2.1.1.1.2">𝒬</ci><cn id="S3.p3.14.m5.1.1.1.1.cmml" type="integer" xref="S3.p3.14.m5.1.1.1.1">1</cn></apply><apply id="S3.p3.14.m5.3.3.2.2.2.1.cmml" xref="S3.p3.14.m5.3.3.2.2.2.1"><csymbol cd="ambiguous" id="S3.p3.14.m5.3.3.2.2.2.1.1.cmml" xref="S3.p3.14.m5.3.3.2.2.2.1">subscript</csymbol><ci id="S3.p3.14.m5.3.3.2.2.2.1.2.cmml" xref="S3.p3.14.m5.3.3.2.2.2.1.2">𝐥</ci><cn id="S3.p3.14.m5.3.3.2.2.2.1.3.cmml" type="integer" xref="S3.p3.14.m5.3.3.2.2.2.1.3">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p3.14.m5.3c">-\operatorname{\mathcal{Q}}_{(1)}(\operatorname{\mathbf{l}}_{1})</annotation><annotation encoding="application/x-llamapun" id="S3.p3.14.m5.3d">- caligraphic_Q start_POSTSUBSCRIPT ( 1 ) end_POSTSUBSCRIPT ( bold_l start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT )</annotation></semantics></math> is equivalent to a squared loss for estimating <math alttext="\operatorname{\mathbf{l}}_{1}" class="ltx_Math" display="inline" id="S3.p3.15.m6.1"><semantics id="S3.p3.15.m6.1a"><msub id="S3.p3.15.m6.1.1" xref="S3.p3.15.m6.1.1.cmml"><mi id="S3.p3.15.m6.1.1.2" xref="S3.p3.15.m6.1.1.2.cmml">𝐥</mi><mn id="S3.p3.15.m6.1.1.3" xref="S3.p3.15.m6.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S3.p3.15.m6.1b"><apply id="S3.p3.15.m6.1.1.cmml" xref="S3.p3.15.m6.1.1"><csymbol cd="ambiguous" id="S3.p3.15.m6.1.1.1.cmml" xref="S3.p3.15.m6.1.1">subscript</csymbol><ci id="S3.p3.15.m6.1.1.2.cmml" xref="S3.p3.15.m6.1.1.2">𝐥</ci><cn id="S3.p3.15.m6.1.1.3.cmml" type="integer" xref="S3.p3.15.m6.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p3.15.m6.1c">\operatorname{\mathbf{l}}_{1}</annotation><annotation encoding="application/x-llamapun" id="S3.p3.15.m6.1d">bold_l start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S3.p4"> <p class="ltx_p" id="S3.p4.9">The conditional maximization (CM) step in the second cycle minimizes <math alttext="-\operatorname{\mathcal{Q}}_{(1)}(\operatorname{\mathbf{l}}_{1})" class="ltx_Math" display="inline" id="S3.p4.1.m1.3"><semantics id="S3.p4.1.m1.3a"><mrow id="S3.p4.1.m1.3.3" xref="S3.p4.1.m1.3.3.cmml"><mo id="S3.p4.1.m1.3.3a" rspace="0.167em" xref="S3.p4.1.m1.3.3.cmml">−</mo><mrow id="S3.p4.1.m1.3.3.2.2" xref="S3.p4.1.m1.3.3.2.3.cmml"><msub id="S3.p4.1.m1.2.2.1.1.1" xref="S3.p4.1.m1.2.2.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.p4.1.m1.2.2.1.1.1.2" xref="S3.p4.1.m1.2.2.1.1.1.2.cmml">𝒬</mi><mrow id="S3.p4.1.m1.1.1.1.3" xref="S3.p4.1.m1.2.2.1.1.1.cmml"><mo id="S3.p4.1.m1.1.1.1.3.1" stretchy="false" xref="S3.p4.1.m1.2.2.1.1.1.cmml">(</mo><mn id="S3.p4.1.m1.1.1.1.1" xref="S3.p4.1.m1.1.1.1.1.cmml">1</mn><mo id="S3.p4.1.m1.1.1.1.3.2" stretchy="false" xref="S3.p4.1.m1.2.2.1.1.1.cmml">)</mo></mrow></msub><mo id="S3.p4.1.m1.3.3.2.2a" xref="S3.p4.1.m1.3.3.2.3.cmml">⁡</mo><mrow id="S3.p4.1.m1.3.3.2.2.2" xref="S3.p4.1.m1.3.3.2.3.cmml"><mo id="S3.p4.1.m1.3.3.2.2.2.2" stretchy="false" xref="S3.p4.1.m1.3.3.2.3.cmml">(</mo><msub id="S3.p4.1.m1.3.3.2.2.2.1" xref="S3.p4.1.m1.3.3.2.2.2.1.cmml"><mi id="S3.p4.1.m1.3.3.2.2.2.1.2" xref="S3.p4.1.m1.3.3.2.2.2.1.2.cmml">𝐥</mi><mn id="S3.p4.1.m1.3.3.2.2.2.1.3" xref="S3.p4.1.m1.3.3.2.2.2.1.3.cmml">1</mn></msub><mo id="S3.p4.1.m1.3.3.2.2.2.3" stretchy="false" xref="S3.p4.1.m1.3.3.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.p4.1.m1.3b"><apply id="S3.p4.1.m1.3.3.cmml" xref="S3.p4.1.m1.3.3"><minus id="S3.p4.1.m1.3.3.3.cmml" xref="S3.p4.1.m1.3.3"></minus><apply id="S3.p4.1.m1.3.3.2.3.cmml" xref="S3.p4.1.m1.3.3.2.2"><apply id="S3.p4.1.m1.2.2.1.1.1.cmml" xref="S3.p4.1.m1.2.2.1.1.1"><csymbol cd="ambiguous" id="S3.p4.1.m1.2.2.1.1.1.1.cmml" xref="S3.p4.1.m1.2.2.1.1.1">subscript</csymbol><ci id="S3.p4.1.m1.2.2.1.1.1.2.cmml" xref="S3.p4.1.m1.2.2.1.1.1.2">𝒬</ci><cn id="S3.p4.1.m1.1.1.1.1.cmml" type="integer" xref="S3.p4.1.m1.1.1.1.1">1</cn></apply><apply id="S3.p4.1.m1.3.3.2.2.2.1.cmml" xref="S3.p4.1.m1.3.3.2.2.2.1"><csymbol cd="ambiguous" id="S3.p4.1.m1.3.3.2.2.2.1.1.cmml" xref="S3.p4.1.m1.3.3.2.2.2.1">subscript</csymbol><ci id="S3.p4.1.m1.3.3.2.2.2.1.2.cmml" xref="S3.p4.1.m1.3.3.2.2.2.1.2">𝐥</ci><cn id="S3.p4.1.m1.3.3.2.2.2.1.3.cmml" type="integer" xref="S3.p4.1.m1.3.3.2.2.2.1.3">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p4.1.m1.3c">-\operatorname{\mathcal{Q}}_{(1)}(\operatorname{\mathbf{l}}_{1})</annotation><annotation encoding="application/x-llamapun" id="S3.p4.1.m1.3d">- caligraphic_Q start_POSTSUBSCRIPT ( 1 ) end_POSTSUBSCRIPT ( bold_l start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT )</annotation></semantics></math> using the group lasso penalty on the columns of <math alttext="\operatorname{\mathbf{L}}_{1}" class="ltx_Math" display="inline" id="S3.p4.2.m2.1"><semantics id="S3.p4.2.m2.1a"><msub id="S3.p4.2.m2.1.1" xref="S3.p4.2.m2.1.1.cmml"><mi id="S3.p4.2.m2.1.1.2" xref="S3.p4.2.m2.1.1.2.cmml">𝐋</mi><mn id="S3.p4.2.m2.1.1.3" xref="S3.p4.2.m2.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S3.p4.2.m2.1b"><apply id="S3.p4.2.m2.1.1.cmml" xref="S3.p4.2.m2.1.1"><csymbol cd="ambiguous" id="S3.p4.2.m2.1.1.1.cmml" xref="S3.p4.2.m2.1.1">subscript</csymbol><ci id="S3.p4.2.m2.1.1.2.cmml" xref="S3.p4.2.m2.1.1.2">𝐋</ci><cn id="S3.p4.2.m2.1.1.3.cmml" type="integer" xref="S3.p4.2.m2.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p4.2.m2.1c">\operatorname{\mathbf{L}}_{1}</annotation><annotation encoding="application/x-llamapun" id="S3.p4.2.m2.1d">bold_L start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math>. Let <math alttext="\operatorname{\mathbf{l}}_{(1):i}" class="ltx_Math" display="inline" id="S3.p4.3.m3.1"><semantics id="S3.p4.3.m3.1a"><msub id="S3.p4.3.m3.1.2" xref="S3.p4.3.m3.1.2.cmml"><mi id="S3.p4.3.m3.1.2.2" xref="S3.p4.3.m3.1.2.2.cmml">𝐥</mi><mrow id="S3.p4.3.m3.1.1.1" xref="S3.p4.3.m3.1.1.1.cmml"><mrow id="S3.p4.3.m3.1.1.1.3.2" xref="S3.p4.3.m3.1.1.1.cmml"><mo id="S3.p4.3.m3.1.1.1.3.2.1" stretchy="false" xref="S3.p4.3.m3.1.1.1.cmml">(</mo><mn id="S3.p4.3.m3.1.1.1.1" xref="S3.p4.3.m3.1.1.1.1.cmml">1</mn><mo id="S3.p4.3.m3.1.1.1.3.2.2" rspace="0.278em" stretchy="false" xref="S3.p4.3.m3.1.1.1.cmml">)</mo></mrow><mo id="S3.p4.3.m3.1.1.1.2" rspace="0.278em" xref="S3.p4.3.m3.1.1.1.2.cmml">:</mo><mi id="S3.p4.3.m3.1.1.1.4" xref="S3.p4.3.m3.1.1.1.4.cmml">i</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="S3.p4.3.m3.1b"><apply id="S3.p4.3.m3.1.2.cmml" xref="S3.p4.3.m3.1.2"><csymbol cd="ambiguous" id="S3.p4.3.m3.1.2.1.cmml" xref="S3.p4.3.m3.1.2">subscript</csymbol><ci id="S3.p4.3.m3.1.2.2.cmml" xref="S3.p4.3.m3.1.2.2">𝐥</ci><apply id="S3.p4.3.m3.1.1.1.cmml" xref="S3.p4.3.m3.1.1.1"><ci id="S3.p4.3.m3.1.1.1.2.cmml" xref="S3.p4.3.m3.1.1.1.2">:</ci><cn id="S3.p4.3.m3.1.1.1.1.cmml" type="integer" xref="S3.p4.3.m3.1.1.1.1">1</cn><ci id="S3.p4.3.m3.1.1.1.4.cmml" xref="S3.p4.3.m3.1.1.1.4">𝑖</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p4.3.m3.1c">\operatorname{\mathbf{l}}_{(1):i}</annotation><annotation encoding="application/x-llamapun" id="S3.p4.3.m3.1d">bold_l start_POSTSUBSCRIPT ( 1 ) : italic_i end_POSTSUBSCRIPT</annotation></semantics></math> is the <math alttext="i" class="ltx_Math" display="inline" id="S3.p4.4.m4.1"><semantics id="S3.p4.4.m4.1a"><mi id="S3.p4.4.m4.1.1" xref="S3.p4.4.m4.1.1.cmml">i</mi><annotation-xml encoding="MathML-Content" id="S3.p4.4.m4.1b"><ci id="S3.p4.4.m4.1.1.cmml" xref="S3.p4.4.m4.1.1">𝑖</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.p4.4.m4.1c">i</annotation><annotation encoding="application/x-llamapun" id="S3.p4.4.m4.1d">italic_i</annotation></semantics></math>th <math alttext="Q_{1}" class="ltx_Math" display="inline" id="S3.p4.5.m5.1"><semantics id="S3.p4.5.m5.1a"><msub id="S3.p4.5.m5.1.1" xref="S3.p4.5.m5.1.1.cmml"><mi id="S3.p4.5.m5.1.1.2" xref="S3.p4.5.m5.1.1.2.cmml">Q</mi><mn id="S3.p4.5.m5.1.1.3" xref="S3.p4.5.m5.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S3.p4.5.m5.1b"><apply id="S3.p4.5.m5.1.1.cmml" xref="S3.p4.5.m5.1.1"><csymbol cd="ambiguous" id="S3.p4.5.m5.1.1.1.cmml" xref="S3.p4.5.m5.1.1">subscript</csymbol><ci id="S3.p4.5.m5.1.1.2.cmml" xref="S3.p4.5.m5.1.1.2">𝑄</ci><cn id="S3.p4.5.m5.1.1.3.cmml" type="integer" xref="S3.p4.5.m5.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p4.5.m5.1c">Q_{1}</annotation><annotation encoding="application/x-llamapun" id="S3.p4.5.m5.1d">italic_Q start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math>-dimensional block of <math alttext="\operatorname{\mathbf{l}}_{1}" class="ltx_Math" display="inline" id="S3.p4.6.m6.1"><semantics id="S3.p4.6.m6.1a"><msub id="S3.p4.6.m6.1.1" xref="S3.p4.6.m6.1.1.cmml"><mi id="S3.p4.6.m6.1.1.2" xref="S3.p4.6.m6.1.1.2.cmml">𝐥</mi><mn id="S3.p4.6.m6.1.1.3" xref="S3.p4.6.m6.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S3.p4.6.m6.1b"><apply id="S3.p4.6.m6.1.1.cmml" xref="S3.p4.6.m6.1.1"><csymbol cd="ambiguous" id="S3.p4.6.m6.1.1.1.cmml" xref="S3.p4.6.m6.1.1">subscript</csymbol><ci id="S3.p4.6.m6.1.1.2.cmml" xref="S3.p4.6.m6.1.1.2">𝐥</ci><cn id="S3.p4.6.m6.1.1.3.cmml" type="integer" xref="S3.p4.6.m6.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p4.6.m6.1c">\operatorname{\mathbf{l}}_{1}</annotation><annotation encoding="application/x-llamapun" id="S3.p4.6.m6.1d">bold_l start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math>, which corresponds to the <math alttext="i" class="ltx_Math" display="inline" id="S3.p4.7.m7.1"><semantics id="S3.p4.7.m7.1a"><mi id="S3.p4.7.m7.1.1" xref="S3.p4.7.m7.1.1.cmml">i</mi><annotation-xml encoding="MathML-Content" id="S3.p4.7.m7.1b"><ci id="S3.p4.7.m7.1.1.cmml" xref="S3.p4.7.m7.1.1">𝑖</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.p4.7.m7.1c">i</annotation><annotation encoding="application/x-llamapun" id="S3.p4.7.m7.1d">italic_i</annotation></semantics></math>th column of <math alttext="\operatorname{\mathbf{L}}_{1}" class="ltx_Math" display="inline" id="S3.p4.8.m8.1"><semantics id="S3.p4.8.m8.1a"><msub id="S3.p4.8.m8.1.1" xref="S3.p4.8.m8.1.1.cmml"><mi id="S3.p4.8.m8.1.1.2" xref="S3.p4.8.m8.1.1.2.cmml">𝐋</mi><mn id="S3.p4.8.m8.1.1.3" xref="S3.p4.8.m8.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S3.p4.8.m8.1b"><apply id="S3.p4.8.m8.1.1.cmml" xref="S3.p4.8.m8.1.1"><csymbol cd="ambiguous" id="S3.p4.8.m8.1.1.1.cmml" xref="S3.p4.8.m8.1.1">subscript</csymbol><ci id="S3.p4.8.m8.1.1.2.cmml" xref="S3.p4.8.m8.1.1.2">𝐋</ci><cn id="S3.p4.8.m8.1.1.3.cmml" type="integer" xref="S3.p4.8.m8.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p4.8.m8.1c">\operatorname{\mathbf{L}}_{1}</annotation><annotation encoding="application/x-llamapun" id="S3.p4.8.m8.1d">bold_L start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math>. Then, <math alttext="\operatorname{\mathbf{l}}_{1}" class="ltx_Math" display="inline" id="S3.p4.9.m9.1"><semantics id="S3.p4.9.m9.1a"><msub id="S3.p4.9.m9.1.1" xref="S3.p4.9.m9.1.1.cmml"><mi id="S3.p4.9.m9.1.1.2" xref="S3.p4.9.m9.1.1.2.cmml">𝐥</mi><mn id="S3.p4.9.m9.1.1.3" xref="S3.p4.9.m9.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S3.p4.9.m9.1b"><apply id="S3.p4.9.m9.1.1.cmml" xref="S3.p4.9.m9.1.1"><csymbol cd="ambiguous" id="S3.p4.9.m9.1.1.1.cmml" xref="S3.p4.9.m9.1.1">subscript</csymbol><ci id="S3.p4.9.m9.1.1.2.cmml" xref="S3.p4.9.m9.1.1.2">𝐥</ci><cn id="S3.p4.9.m9.1.1.3.cmml" type="integer" xref="S3.p4.9.m9.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p4.9.m9.1c">\operatorname{\mathbf{l}}_{1}</annotation><annotation encoding="application/x-llamapun" id="S3.p4.9.m9.1d">bold_l start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> is estimated using the <span class="ltx_text ltx_font_typewriter" id="S3.p4.9.1">gglasso</span> package as</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="S3.EGx10"> <tbody id="S3.E10"><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{\mathbf{l}}_{1}^{(t+1)}=\operatorname*{\mathop{% \mathrm{argmin}}}_{\operatorname{\mathbf{l}}_{1}\in\operatorname{\mathbb{R}}^{% Q_{1}S_{1}}}\;\big{\lVert}\operatorname{\mathbf{H}}_{(1)}^{-{1}/{2}(t)}% \operatorname{\mathbf{g}}^{(t)}_{(1)}-\operatorname{\mathbf{H}}_{(1)}^{{1}/{2}% (t)\top}\operatorname{\mathbf{l}}_{1}\big{\rVert}_{2}^{2}+\lambda_{% \operatorname{\mathbf{L}}}\sum_{i=1}^{S_{1}}\lVert\operatorname{\mathbf{l}}_{(% 1):i}\rVert_{2}," class="ltx_math_unparsed" display="inline" id="S3.E10.m1.8"><semantics id="S3.E10.m1.8a"><mrow id="S3.E10.m1.8b"><msubsup id="S3.E10.m1.8.9"><mi id="S3.E10.m1.8.9.2.2">𝐥</mi><mn id="S3.E10.m1.8.9.2.3">1</mn><mrow id="S3.E10.m1.1.1.1.1"><mo id="S3.E10.m1.1.1.1.1.2" stretchy="false">(</mo><mrow id="S3.E10.m1.1.1.1.1.1"><mi id="S3.E10.m1.1.1.1.1.1.2">t</mi><mo id="S3.E10.m1.1.1.1.1.1.1">+</mo><mn id="S3.E10.m1.1.1.1.1.1.3">1</mn></mrow><mo id="S3.E10.m1.1.1.1.1.3" stretchy="false">)</mo></mrow></msubsup><mo id="S3.E10.m1.8.10" rspace="0.1389em">=</mo><munder id="S3.E10.m1.8.11"><mo id="S3.E10.m1.8.11.2" lspace="0.1389em" rspace="0em">argmin</mo><mrow id="S3.E10.m1.8.11.3"><msub id="S3.E10.m1.8.11.3.2"><mi id="S3.E10.m1.8.11.3.2.2">𝐥</mi><mn id="S3.E10.m1.8.11.3.2.3">1</mn></msub><mo id="S3.E10.m1.8.11.3.1">∈</mo><msup id="S3.E10.m1.8.11.3.3"><mi id="S3.E10.m1.8.11.3.3.2">ℝ</mi><mrow id="S3.E10.m1.8.11.3.3.3"><msub id="S3.E10.m1.8.11.3.3.3.2"><mi id="S3.E10.m1.8.11.3.3.3.2.2">Q</mi><mn id="S3.E10.m1.8.11.3.3.3.2.3">1</mn></msub><mo id="S3.E10.m1.8.11.3.3.3.1">⁢</mo><msub id="S3.E10.m1.8.11.3.3.3.3"><mi id="S3.E10.m1.8.11.3.3.3.3.2">S</mi><mn id="S3.E10.m1.8.11.3.3.3.3.3">1</mn></msub></mrow></msup></mrow></munder><msubsup id="S3.E10.m1.8.12"><mrow id="S3.E10.m1.8.12.2.2"><mo fence="true" id="S3.E10.m1.8.12.2.2.1" lspace="0em" mathsize="120%" rspace="0em">∥</mo><msubsup id="S3.E10.m1.8.12.2.2.2"><mi id="S3.E10.m1.8.12.2.2.2.2.2">𝐇</mi><mrow id="S3.E10.m1.2.2.1.3"><mo id="S3.E10.m1.2.2.1.3.1" stretchy="false">(</mo><mn id="S3.E10.m1.2.2.1.1">1</mn><mo id="S3.E10.m1.2.2.1.3.2" stretchy="false">)</mo></mrow><mrow id="S3.E10.m1.3.3.1"><mo id="S3.E10.m1.3.3.1a">−</mo><mrow id="S3.E10.m1.3.3.1.3"><mrow id="S3.E10.m1.3.3.1.3.2"><mn id="S3.E10.m1.3.3.1.3.2.2">1</mn><mo id="S3.E10.m1.3.3.1.3.2.1">/</mo><mn id="S3.E10.m1.3.3.1.3.2.3">2</mn></mrow><mo id="S3.E10.m1.3.3.1.3.1">⁢</mo><mrow id="S3.E10.m1.3.3.1.3.3.2"><mo id="S3.E10.m1.3.3.1.3.3.2.1" stretchy="false">(</mo><mi id="S3.E10.m1.3.3.1.1">t</mi><mo id="S3.E10.m1.3.3.1.3.3.2.2" stretchy="false">)</mo></mrow></mrow></mrow></msubsup><msubsup id="S3.E10.m1.8.12.2.2.3"><mi id="S3.E10.m1.8.12.2.2.3.2.2">𝐠</mi><mrow id="S3.E10.m1.5.5.1.3"><mo id="S3.E10.m1.5.5.1.3.1" stretchy="false">(</mo><mn id="S3.E10.m1.5.5.1.1">1</mn><mo id="S3.E10.m1.5.5.1.3.2" stretchy="false">)</mo></mrow><mrow id="S3.E10.m1.4.4.1.3"><mo id="S3.E10.m1.4.4.1.3.1" stretchy="false">(</mo><mi id="S3.E10.m1.4.4.1.1">t</mi><mo id="S3.E10.m1.4.4.1.3.2" stretchy="false">)</mo></mrow></msubsup><mo id="S3.E10.m1.8.12.2.2.4" lspace="0em">−</mo><msubsup id="S3.E10.m1.8.12.2.2.5"><mi id="S3.E10.m1.8.12.2.2.5.2.2">𝐇</mi><mrow id="S3.E10.m1.6.6.1.3"><mo id="S3.E10.m1.6.6.1.3.1" stretchy="false">(</mo><mn id="S3.E10.m1.6.6.1.1">1</mn><mo id="S3.E10.m1.6.6.1.3.2" stretchy="false">)</mo></mrow><mrow id="S3.E10.m1.7.7.1"><mrow id="S3.E10.m1.7.7.1.3"><mrow id="S3.E10.m1.7.7.1.3.2"><mn id="S3.E10.m1.7.7.1.3.2.2">1</mn><mo id="S3.E10.m1.7.7.1.3.2.1">/</mo><mn id="S3.E10.m1.7.7.1.3.2.3">2</mn></mrow><mo id="S3.E10.m1.7.7.1.3.1">⁢</mo><mrow id="S3.E10.m1.7.7.1.3.3.2"><mo id="S3.E10.m1.7.7.1.3.3.2.1" stretchy="false">(</mo><mi id="S3.E10.m1.7.7.1.1">t</mi><mo id="S3.E10.m1.7.7.1.3.3.2.2" stretchy="false">)</mo></mrow></mrow><mo id="S3.E10.m1.7.7.1.4" lspace="0.222em">⊤</mo></mrow></msubsup><msub id="S3.E10.m1.8.12.2.2.6"><mi id="S3.E10.m1.8.12.2.2.6.2">𝐥</mi><mn id="S3.E10.m1.8.12.2.2.6.3">1</mn></msub><mo fence="true" id="S3.E10.m1.8.12.2.2.7" lspace="0em" mathsize="120%" rspace="0em">∥</mo></mrow><mn id="S3.E10.m1.8.12.2.3">2</mn><mn id="S3.E10.m1.8.12.3">2</mn></msubsup><mo id="S3.E10.m1.8.13">+</mo><msub id="S3.E10.m1.8.14"><mi id="S3.E10.m1.8.14.2">λ</mi><mi id="S3.E10.m1.8.14.3">𝐋</mi></msub><mstyle displaystyle="true" id="S3.E10.m1.8.15"><munderover id="S3.E10.m1.8.15a"><mo id="S3.E10.m1.8.15.2.2" movablelimits="false">∑</mo><mrow id="S3.E10.m1.8.15.2.3"><mi id="S3.E10.m1.8.15.2.3.2">i</mi><mo id="S3.E10.m1.8.15.2.3.1">=</mo><mn id="S3.E10.m1.8.15.2.3.3">1</mn></mrow><msub id="S3.E10.m1.8.15.3"><mi id="S3.E10.m1.8.15.3.2">S</mi><mn id="S3.E10.m1.8.15.3.3">1</mn></msub></munderover></mstyle><msub id="S3.E10.m1.8.16"><mrow id="S3.E10.m1.8.16.2"><mo fence="true" id="S3.E10.m1.8.16.2.1" lspace="0em" rspace="0em">∥</mo><msub id="S3.E10.m1.8.16.2.2"><mi id="S3.E10.m1.8.16.2.2.2">𝐥</mi><mrow id="S3.E10.m1.8.8.1"><mrow id="S3.E10.m1.8.8.1.3.2"><mo id="S3.E10.m1.8.8.1.3.2.1" stretchy="false">(</mo><mn id="S3.E10.m1.8.8.1.1">1</mn><mo id="S3.E10.m1.8.8.1.3.2.2" rspace="0.278em" stretchy="false">)</mo></mrow><mo id="S3.E10.m1.8.8.1.2" rspace="0.278em">:</mo><mi id="S3.E10.m1.8.8.1.4">i</mi></mrow></msub><mo fence="true" id="S3.E10.m1.8.16.2.3" lspace="0em" rspace="0em">∥</mo></mrow><mn id="S3.E10.m1.8.16.3">2</mn></msub><mo id="S3.E10.m1.8.17">,</mo></mrow><annotation encoding="application/x-tex" id="S3.E10.m1.8c">\displaystyle\operatorname{\mathbf{l}}_{1}^{(t+1)}=\operatorname*{\mathop{% \mathrm{argmin}}}_{\operatorname{\mathbf{l}}_{1}\in\operatorname{\mathbb{R}}^{% Q_{1}S_{1}}}\;\big{\lVert}\operatorname{\mathbf{H}}_{(1)}^{-{1}/{2}(t)}% \operatorname{\mathbf{g}}^{(t)}_{(1)}-\operatorname{\mathbf{H}}_{(1)}^{{1}/{2}% (t)\top}\operatorname{\mathbf{l}}_{1}\big{\rVert}_{2}^{2}+\lambda_{% \operatorname{\mathbf{L}}}\sum_{i=1}^{S_{1}}\lVert\operatorname{\mathbf{l}}_{(% 1):i}\rVert_{2},</annotation><annotation encoding="application/x-llamapun" id="S3.E10.m1.8d">bold_l start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_t + 1 ) end_POSTSUPERSCRIPT = roman_argmin start_POSTSUBSCRIPT bold_l start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ∈ blackboard_R start_POSTSUPERSCRIPT italic_Q start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_S start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ∥ bold_H start_POSTSUBSCRIPT ( 1 ) end_POSTSUBSCRIPT start_POSTSUPERSCRIPT - 1 / 2 ( italic_t ) end_POSTSUPERSCRIPT bold_g start_POSTSUPERSCRIPT ( italic_t ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT ( 1 ) end_POSTSUBSCRIPT - bold_H start_POSTSUBSCRIPT ( 1 ) end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 1 / 2 ( italic_t ) ⊤ end_POSTSUPERSCRIPT bold_l start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ∥ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + italic_λ start_POSTSUBSCRIPT bold_L end_POSTSUBSCRIPT ∑ start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_S start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT ∥ bold_l start_POSTSUBSCRIPT ( 1 ) : italic_i end_POSTSUBSCRIPT ∥ start_POSTSUBSCRIPT 2 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="S3.p4.14">where <math alttext="\operatorname{\mathbf{L}}^{(t+1)}_{1}=\text{vec}^{{-1}}(\operatorname{\mathbf{% l}}_{1}^{(t+1)})" class="ltx_Math" 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xref="S3.p4.10.m1.3.3.1.3.2">vec</mtext></ci><apply id="S3.p4.10.m1.3.3.1.3.3.cmml" xref="S3.p4.10.m1.3.3.1.3.3"><minus id="S3.p4.10.m1.3.3.1.3.3.1.cmml" xref="S3.p4.10.m1.3.3.1.3.3"></minus><cn id="S3.p4.10.m1.3.3.1.3.3.2.cmml" type="integer" xref="S3.p4.10.m1.3.3.1.3.3.2">1</cn></apply></apply><apply id="S3.p4.10.m1.3.3.1.1.1.1.cmml" xref="S3.p4.10.m1.3.3.1.1.1"><csymbol cd="ambiguous" id="S3.p4.10.m1.3.3.1.1.1.1.1.cmml" xref="S3.p4.10.m1.3.3.1.1.1">superscript</csymbol><apply id="S3.p4.10.m1.3.3.1.1.1.1.2.cmml" xref="S3.p4.10.m1.3.3.1.1.1"><csymbol cd="ambiguous" id="S3.p4.10.m1.3.3.1.1.1.1.2.1.cmml" xref="S3.p4.10.m1.3.3.1.1.1">subscript</csymbol><ci id="S3.p4.10.m1.3.3.1.1.1.1.2.2.cmml" xref="S3.p4.10.m1.3.3.1.1.1.1.2.2">𝐥</ci><cn id="S3.p4.10.m1.3.3.1.1.1.1.2.3.cmml" type="integer" xref="S3.p4.10.m1.3.3.1.1.1.1.2.3">1</cn></apply><apply id="S3.p4.10.m1.2.2.1.1.1.cmml" xref="S3.p4.10.m1.2.2.1.1"><plus id="S3.p4.10.m1.2.2.1.1.1.1.cmml" xref="S3.p4.10.m1.2.2.1.1.1.1"></plus><ci id="S3.p4.10.m1.2.2.1.1.1.2.cmml" xref="S3.p4.10.m1.2.2.1.1.1.2">𝑡</ci><cn id="S3.p4.10.m1.2.2.1.1.1.3.cmml" type="integer" xref="S3.p4.10.m1.2.2.1.1.1.3">1</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p4.10.m1.3c">\operatorname{\mathbf{L}}^{(t+1)}_{1}=\text{vec}^{{-1}}(\operatorname{\mathbf{% l}}_{1}^{(t+1)})</annotation><annotation encoding="application/x-llamapun" id="S3.p4.10.m1.3d">bold_L start_POSTSUPERSCRIPT ( italic_t + 1 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = vec start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ( bold_l start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_t + 1 ) end_POSTSUPERSCRIPT )</annotation></semantics></math>, <math alttext="\lambda_{\operatorname{\mathbf{L}}}\geq 0" class="ltx_Math" display="inline" id="S3.p4.11.m2.1"><semantics id="S3.p4.11.m2.1a"><mrow id="S3.p4.11.m2.1.1" xref="S3.p4.11.m2.1.1.cmml"><msub id="S3.p4.11.m2.1.1.2" xref="S3.p4.11.m2.1.1.2.cmml"><mi id="S3.p4.11.m2.1.1.2.2" xref="S3.p4.11.m2.1.1.2.2.cmml">λ</mi><mi id="S3.p4.11.m2.1.1.2.3" xref="S3.p4.11.m2.1.1.2.3.cmml">𝐋</mi></msub><mo id="S3.p4.11.m2.1.1.1" xref="S3.p4.11.m2.1.1.1.cmml">≥</mo><mn id="S3.p4.11.m2.1.1.3" xref="S3.p4.11.m2.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.p4.11.m2.1b"><apply id="S3.p4.11.m2.1.1.cmml" xref="S3.p4.11.m2.1.1"><geq id="S3.p4.11.m2.1.1.1.cmml" xref="S3.p4.11.m2.1.1.1"></geq><apply id="S3.p4.11.m2.1.1.2.cmml" xref="S3.p4.11.m2.1.1.2"><csymbol cd="ambiguous" id="S3.p4.11.m2.1.1.2.1.cmml" xref="S3.p4.11.m2.1.1.2">subscript</csymbol><ci id="S3.p4.11.m2.1.1.2.2.cmml" xref="S3.p4.11.m2.1.1.2.2">𝜆</ci><ci id="S3.p4.11.m2.1.1.2.3.cmml" xref="S3.p4.11.m2.1.1.2.3">𝐋</ci></apply><cn id="S3.p4.11.m2.1.1.3.cmml" type="integer" xref="S3.p4.11.m2.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p4.11.m2.1c">\lambda_{\operatorname{\mathbf{L}}}\geq 0</annotation><annotation encoding="application/x-llamapun" id="S3.p4.11.m2.1d">italic_λ start_POSTSUBSCRIPT bold_L end_POSTSUBSCRIPT ≥ 0</annotation></semantics></math> is a tuning parameter, and <math alttext="\lVert\cdot\rVert_{2}" class="ltx_Math" display="inline" id="S3.p4.12.m3.1"><semantics id="S3.p4.12.m3.1a"><msub id="S3.p4.12.m3.1.2" xref="S3.p4.12.m3.1.2.cmml"><mrow id="S3.p4.12.m3.1.2.2.2" xref="S3.p4.12.m3.1.2.2.1.cmml"><mo fence="true" id="S3.p4.12.m3.1.2.2.2.1" rspace="0em" xref="S3.p4.12.m3.1.2.2.1.1.cmml">∥</mo><mo id="S3.p4.12.m3.1.1" lspace="0em" rspace="0em" xref="S3.p4.12.m3.1.1.cmml">⋅</mo><mo fence="true" id="S3.p4.12.m3.1.2.2.2.2" lspace="0em" xref="S3.p4.12.m3.1.2.2.1.1.cmml">∥</mo></mrow><mn id="S3.p4.12.m3.1.2.3" xref="S3.p4.12.m3.1.2.3.cmml">2</mn></msub><annotation-xml encoding="MathML-Content" id="S3.p4.12.m3.1b"><apply id="S3.p4.12.m3.1.2.cmml" xref="S3.p4.12.m3.1.2"><csymbol cd="ambiguous" id="S3.p4.12.m3.1.2.1.cmml" xref="S3.p4.12.m3.1.2">subscript</csymbol><apply id="S3.p4.12.m3.1.2.2.1.cmml" xref="S3.p4.12.m3.1.2.2.2"><csymbol cd="latexml" id="S3.p4.12.m3.1.2.2.1.1.cmml" xref="S3.p4.12.m3.1.2.2.2.1">delimited-∥∥</csymbol><ci id="S3.p4.12.m3.1.1.cmml" xref="S3.p4.12.m3.1.1">⋅</ci></apply><cn id="S3.p4.12.m3.1.2.3.cmml" type="integer" xref="S3.p4.12.m3.1.2.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p4.12.m3.1c">\lVert\cdot\rVert_{2}</annotation><annotation encoding="application/x-llamapun" id="S3.p4.12.m3.1d">∥ ⋅ ∥ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math> is the Euclidean norm. The second cycle ends by updating <math alttext="\operatorname{{\bm{\theta}}}^{(t+1/3)}" class="ltx_Math" display="inline" id="S3.p4.13.m4.1"><semantics id="S3.p4.13.m4.1a"><msup id="S3.p4.13.m4.1.2" xref="S3.p4.13.m4.1.2.cmml"><mi id="S3.p4.13.m4.1.2.2" xref="S3.p4.13.m4.1.2.2.cmml">𝜽</mi><mrow id="S3.p4.13.m4.1.1.1.1" xref="S3.p4.13.m4.1.1.1.1.1.cmml"><mo id="S3.p4.13.m4.1.1.1.1.2" stretchy="false" xref="S3.p4.13.m4.1.1.1.1.1.cmml">(</mo><mrow id="S3.p4.13.m4.1.1.1.1.1" xref="S3.p4.13.m4.1.1.1.1.1.cmml"><mi id="S3.p4.13.m4.1.1.1.1.1.2" xref="S3.p4.13.m4.1.1.1.1.1.2.cmml">t</mi><mo id="S3.p4.13.m4.1.1.1.1.1.1" xref="S3.p4.13.m4.1.1.1.1.1.1.cmml">+</mo><mrow id="S3.p4.13.m4.1.1.1.1.1.3" xref="S3.p4.13.m4.1.1.1.1.1.3.cmml"><mn id="S3.p4.13.m4.1.1.1.1.1.3.2" xref="S3.p4.13.m4.1.1.1.1.1.3.2.cmml">1</mn><mo id="S3.p4.13.m4.1.1.1.1.1.3.1" xref="S3.p4.13.m4.1.1.1.1.1.3.1.cmml">/</mo><mn id="S3.p4.13.m4.1.1.1.1.1.3.3" xref="S3.p4.13.m4.1.1.1.1.1.3.3.cmml">3</mn></mrow></mrow><mo 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xref="S3.p4.14.m5.1.1.1.1.1.2.cmml">t</mi><mo id="S3.p4.14.m5.1.1.1.1.1.1" xref="S3.p4.14.m5.1.1.1.1.1.1.cmml">+</mo><mrow id="S3.p4.14.m5.1.1.1.1.1.3" xref="S3.p4.14.m5.1.1.1.1.1.3.cmml"><mn id="S3.p4.14.m5.1.1.1.1.1.3.2" xref="S3.p4.14.m5.1.1.1.1.1.3.2.cmml">2</mn><mo id="S3.p4.14.m5.1.1.1.1.1.3.1" xref="S3.p4.14.m5.1.1.1.1.1.3.1.cmml">/</mo><mn id="S3.p4.14.m5.1.1.1.1.1.3.3" xref="S3.p4.14.m5.1.1.1.1.1.3.3.cmml">3</mn></mrow></mrow><mo id="S3.p4.14.m5.1.1.1.1.3" stretchy="false" xref="S3.p4.14.m5.1.1.1.1.1.cmml">)</mo></mrow></msup><mo id="S3.p4.14.m5.9.9.5" xref="S3.p4.14.m5.9.9.5.cmml">=</mo><mrow id="S3.p4.14.m5.9.9.4.4" xref="S3.p4.14.m5.9.9.4.5.cmml"><mo id="S3.p4.14.m5.9.9.4.4.5" stretchy="false" xref="S3.p4.14.m5.9.9.4.5.cmml">(</mo><msup id="S3.p4.14.m5.6.6.1.1.1" xref="S3.p4.14.m5.6.6.1.1.1.cmml"><mi id="S3.p4.14.m5.6.6.1.1.1.2" xref="S3.p4.14.m5.6.6.1.1.1.2.cmml">𝐛</mi><mrow id="S3.p4.14.m5.2.2.1.1" xref="S3.p4.14.m5.2.2.1.1.1.cmml"><mo id="S3.p4.14.m5.2.2.1.1.2" 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xref="S3.p4.14.m5.5.5.1.1.1.1.3">1</cn></apply></apply></apply></vector></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p4.14.m5.9c">\operatorname{{\bm{\theta}}}^{(t+2/3)}=(\operatorname{\mathbf{b}}^{(t+1)},% \operatorname{\mathbf{L}}_{1}^{(t+1)},\operatorname{\mathbf{L}}_{2}^{(t)},\tau% ^{2(t+1)})</annotation><annotation encoding="application/x-llamapun" id="S3.p4.14.m5.9d">bold_italic_θ start_POSTSUPERSCRIPT ( italic_t + 2 / 3 ) end_POSTSUPERSCRIPT = ( bold_b start_POSTSUPERSCRIPT ( italic_t + 1 ) end_POSTSUPERSCRIPT , bold_L start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_t + 1 ) end_POSTSUPERSCRIPT , bold_L start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_t ) end_POSTSUPERSCRIPT , italic_τ start_POSTSUPERSCRIPT 2 ( italic_t + 1 ) end_POSTSUPERSCRIPT )</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S3.p5"> <p class="ltx_p" id="S3.p5.15">Finally, the third AECM cycle updates <math alttext="\operatorname{{\bm{\theta}}}^{(t+2/3)}" class="ltx_Math" display="inline" id="S3.p5.1.m1.1"><semantics id="S3.p5.1.m1.1a"><msup id="S3.p5.1.m1.1.2" xref="S3.p5.1.m1.1.2.cmml"><mi id="S3.p5.1.m1.1.2.2" xref="S3.p5.1.m1.1.2.2.cmml">𝜽</mi><mrow id="S3.p5.1.m1.1.1.1.1" xref="S3.p5.1.m1.1.1.1.1.1.cmml"><mo id="S3.p5.1.m1.1.1.1.1.2" stretchy="false" xref="S3.p5.1.m1.1.1.1.1.1.cmml">(</mo><mrow id="S3.p5.1.m1.1.1.1.1.1" xref="S3.p5.1.m1.1.1.1.1.1.cmml"><mi id="S3.p5.1.m1.1.1.1.1.1.2" xref="S3.p5.1.m1.1.1.1.1.1.2.cmml">t</mi><mo id="S3.p5.1.m1.1.1.1.1.1.1" xref="S3.p5.1.m1.1.1.1.1.1.1.cmml">+</mo><mrow id="S3.p5.1.m1.1.1.1.1.1.3" xref="S3.p5.1.m1.1.1.1.1.1.3.cmml"><mn id="S3.p5.1.m1.1.1.1.1.1.3.2" xref="S3.p5.1.m1.1.1.1.1.1.3.2.cmml">2</mn><mo id="S3.p5.1.m1.1.1.1.1.1.3.1" xref="S3.p5.1.m1.1.1.1.1.1.3.1.cmml">/</mo><mn id="S3.p5.1.m1.1.1.1.1.1.3.3" xref="S3.p5.1.m1.1.1.1.1.1.3.3.cmml">3</mn></mrow></mrow><mo id="S3.p5.1.m1.1.1.1.1.3" stretchy="false" xref="S3.p5.1.m1.1.1.1.1.1.cmml">)</mo></mrow></msup><annotation-xml encoding="MathML-Content" id="S3.p5.1.m1.1b"><apply id="S3.p5.1.m1.1.2.cmml" xref="S3.p5.1.m1.1.2"><csymbol cd="ambiguous" id="S3.p5.1.m1.1.2.1.cmml" xref="S3.p5.1.m1.1.2">superscript</csymbol><ci id="S3.p5.1.m1.1.2.2.cmml" xref="S3.p5.1.m1.1.2.2">𝜽</ci><apply id="S3.p5.1.m1.1.1.1.1.1.cmml" xref="S3.p5.1.m1.1.1.1.1"><plus id="S3.p5.1.m1.1.1.1.1.1.1.cmml" xref="S3.p5.1.m1.1.1.1.1.1.1"></plus><ci id="S3.p5.1.m1.1.1.1.1.1.2.cmml" xref="S3.p5.1.m1.1.1.1.1.1.2">𝑡</ci><apply id="S3.p5.1.m1.1.1.1.1.1.3.cmml" xref="S3.p5.1.m1.1.1.1.1.1.3"><divide id="S3.p5.1.m1.1.1.1.1.1.3.1.cmml" xref="S3.p5.1.m1.1.1.1.1.1.3.1"></divide><cn id="S3.p5.1.m1.1.1.1.1.1.3.2.cmml" type="integer" xref="S3.p5.1.m1.1.1.1.1.1.3.2">2</cn><cn id="S3.p5.1.m1.1.1.1.1.1.3.3.cmml" type="integer" xref="S3.p5.1.m1.1.1.1.1.1.3.3">3</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" 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xref="S3.p5.4.m4.3.3.1.3.cmml">2</mn><mo id="S3.p5.4.m4.3.3.1.2" xref="S3.p5.4.m4.3.3.1.2.cmml">⁢</mo><mrow id="S3.p5.4.m4.3.3.1.1.1" xref="S3.p5.4.m4.3.3.1.1.1.1.cmml"><mo id="S3.p5.4.m4.3.3.1.1.1.2" stretchy="false" xref="S3.p5.4.m4.3.3.1.1.1.1.cmml">(</mo><mrow id="S3.p5.4.m4.3.3.1.1.1.1" xref="S3.p5.4.m4.3.3.1.1.1.1.cmml"><mi id="S3.p5.4.m4.3.3.1.1.1.1.2" xref="S3.p5.4.m4.3.3.1.1.1.1.2.cmml">t</mi><mo id="S3.p5.4.m4.3.3.1.1.1.1.1" xref="S3.p5.4.m4.3.3.1.1.1.1.1.cmml">+</mo><mn id="S3.p5.4.m4.3.3.1.1.1.1.3" xref="S3.p5.4.m4.3.3.1.1.1.1.3.cmml">1</mn></mrow><mo id="S3.p5.4.m4.3.3.1.1.1.3" stretchy="false" xref="S3.p5.4.m4.3.3.1.1.1.1.cmml">)</mo></mrow></mrow></msup><mo id="S3.p5.4.m4.6.6.3.7" stretchy="false" xref="S3.p5.4.m4.6.6.4.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.p5.4.m4.6b"><vector id="S3.p5.4.m4.6.6.4.cmml" xref="S3.p5.4.m4.6.6.3"><apply id="S3.p5.4.m4.4.4.1.1.cmml" xref="S3.p5.4.m4.4.4.1.1"><csymbol cd="ambiguous" id="S3.p5.4.m4.4.4.1.1.1.cmml" xref="S3.p5.4.m4.4.4.1.1">superscript</csymbol><ci id="S3.p5.4.m4.4.4.1.1.2.cmml" xref="S3.p5.4.m4.4.4.1.1.2">𝐛</ci><apply id="S3.p5.4.m4.1.1.1.1.1.cmml" xref="S3.p5.4.m4.1.1.1.1"><plus id="S3.p5.4.m4.1.1.1.1.1.1.cmml" xref="S3.p5.4.m4.1.1.1.1.1.1"></plus><ci id="S3.p5.4.m4.1.1.1.1.1.2.cmml" xref="S3.p5.4.m4.1.1.1.1.1.2">𝑡</ci><cn id="S3.p5.4.m4.1.1.1.1.1.3.cmml" type="integer" xref="S3.p5.4.m4.1.1.1.1.1.3">1</cn></apply></apply><apply id="S3.p5.4.m4.5.5.2.2.cmml" xref="S3.p5.4.m4.5.5.2.2"><csymbol cd="ambiguous" id="S3.p5.4.m4.5.5.2.2.1.cmml" xref="S3.p5.4.m4.5.5.2.2">superscript</csymbol><apply id="S3.p5.4.m4.5.5.2.2.2.cmml" xref="S3.p5.4.m4.5.5.2.2"><csymbol cd="ambiguous" id="S3.p5.4.m4.5.5.2.2.2.1.cmml" xref="S3.p5.4.m4.5.5.2.2">subscript</csymbol><ci id="S3.p5.4.m4.5.5.2.2.2.2.cmml" xref="S3.p5.4.m4.5.5.2.2.2.2">𝐋</ci><cn id="S3.p5.4.m4.5.5.2.2.2.3.cmml" type="integer" xref="S3.p5.4.m4.5.5.2.2.2.3">1</cn></apply><apply id="S3.p5.4.m4.2.2.1.1.1.cmml" xref="S3.p5.4.m4.2.2.1.1"><plus id="S3.p5.4.m4.2.2.1.1.1.1.cmml" xref="S3.p5.4.m4.2.2.1.1.1.1"></plus><ci id="S3.p5.4.m4.2.2.1.1.1.2.cmml" xref="S3.p5.4.m4.2.2.1.1.1.2">𝑡</ci><cn id="S3.p5.4.m4.2.2.1.1.1.3.cmml" type="integer" xref="S3.p5.4.m4.2.2.1.1.1.3">1</cn></apply></apply><apply id="S3.p5.4.m4.6.6.3.3.cmml" xref="S3.p5.4.m4.6.6.3.3"><csymbol cd="ambiguous" id="S3.p5.4.m4.6.6.3.3.1.cmml" xref="S3.p5.4.m4.6.6.3.3">superscript</csymbol><ci id="S3.p5.4.m4.6.6.3.3.2.cmml" xref="S3.p5.4.m4.6.6.3.3.2">𝜏</ci><apply id="S3.p5.4.m4.3.3.1.cmml" xref="S3.p5.4.m4.3.3.1"><times id="S3.p5.4.m4.3.3.1.2.cmml" xref="S3.p5.4.m4.3.3.1.2"></times><cn id="S3.p5.4.m4.3.3.1.3.cmml" type="integer" xref="S3.p5.4.m4.3.3.1.3">2</cn><apply id="S3.p5.4.m4.3.3.1.1.1.1.cmml" xref="S3.p5.4.m4.3.3.1.1.1"><plus id="S3.p5.4.m4.3.3.1.1.1.1.1.cmml" xref="S3.p5.4.m4.3.3.1.1.1.1.1"></plus><ci id="S3.p5.4.m4.3.3.1.1.1.1.2.cmml" xref="S3.p5.4.m4.3.3.1.1.1.1.2">𝑡</ci><cn id="S3.p5.4.m4.3.3.1.1.1.1.3.cmml" type="integer" xref="S3.p5.4.m4.3.3.1.1.1.1.3">1</cn></apply></apply></apply></vector></annotation-xml><annotation encoding="application/x-tex" id="S3.p5.4.m4.6c">(\operatorname{\mathbf{b}}^{(t+1)},\operatorname{\mathbf{L}}_{1}^{(t+1)},\tau^% {2(t+1)})</annotation><annotation encoding="application/x-llamapun" id="S3.p5.4.m4.6d">( bold_b start_POSTSUPERSCRIPT ( italic_t + 1 ) end_POSTSUPERSCRIPT , bold_L start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_t + 1 ) end_POSTSUPERSCRIPT , italic_τ start_POSTSUPERSCRIPT 2 ( italic_t + 1 ) end_POSTSUPERSCRIPT )</annotation></semantics></math>. The E and CM steps in this cycle change the indices 1 to 2 and vice versa in the second cycle E and CM steps. Specifically, for <math alttext="i=1,\ldots,n" class="ltx_Math" display="inline" id="S3.p5.5.m5.3"><semantics id="S3.p5.5.m5.3a"><mrow id="S3.p5.5.m5.3.4" xref="S3.p5.5.m5.3.4.cmml"><mi id="S3.p5.5.m5.3.4.2" xref="S3.p5.5.m5.3.4.2.cmml">i</mi><mo id="S3.p5.5.m5.3.4.1" xref="S3.p5.5.m5.3.4.1.cmml">=</mo><mrow id="S3.p5.5.m5.3.4.3.2" xref="S3.p5.5.m5.3.4.3.1.cmml"><mn id="S3.p5.5.m5.1.1" xref="S3.p5.5.m5.1.1.cmml">1</mn><mo id="S3.p5.5.m5.3.4.3.2.1" xref="S3.p5.5.m5.3.4.3.1.cmml">,</mo><mi id="S3.p5.5.m5.2.2" mathvariant="normal" xref="S3.p5.5.m5.2.2.cmml">…</mi><mo id="S3.p5.5.m5.3.4.3.2.2" xref="S3.p5.5.m5.3.4.3.1.cmml">,</mo><mi id="S3.p5.5.m5.3.3" xref="S3.p5.5.m5.3.3.cmml">n</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.p5.5.m5.3b"><apply id="S3.p5.5.m5.3.4.cmml" xref="S3.p5.5.m5.3.4"><eq id="S3.p5.5.m5.3.4.1.cmml" xref="S3.p5.5.m5.3.4.1"></eq><ci id="S3.p5.5.m5.3.4.2.cmml" xref="S3.p5.5.m5.3.4.2">𝑖</ci><list id="S3.p5.5.m5.3.4.3.1.cmml" xref="S3.p5.5.m5.3.4.3.2"><cn id="S3.p5.5.m5.1.1.cmml" type="integer" xref="S3.p5.5.m5.1.1">1</cn><ci id="S3.p5.5.m5.2.2.cmml" xref="S3.p5.5.m5.2.2">…</ci><ci id="S3.p5.5.m5.3.3.cmml" xref="S3.p5.5.m5.3.3">𝑛</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p5.5.m5.3c">i=1,\ldots,n</annotation><annotation encoding="application/x-llamapun" id="S3.p5.5.m5.3d">italic_i = 1 , … , italic_n</annotation></semantics></math>, the complete data are <math alttext="(\operatorname{\mathbf{y}}_{i},\operatorname{\mathbf{c}}_{i(1)})" class="ltx_Math" display="inline" id="S3.p5.6.m6.3"><semantics id="S3.p5.6.m6.3a"><mrow id="S3.p5.6.m6.3.3.2" xref="S3.p5.6.m6.3.3.3.cmml"><mo id="S3.p5.6.m6.3.3.2.3" stretchy="false" xref="S3.p5.6.m6.3.3.3.cmml">(</mo><msub id="S3.p5.6.m6.2.2.1.1" xref="S3.p5.6.m6.2.2.1.1.cmml"><mi id="S3.p5.6.m6.2.2.1.1.2" xref="S3.p5.6.m6.2.2.1.1.2.cmml">𝐲</mi><mi id="S3.p5.6.m6.2.2.1.1.3" xref="S3.p5.6.m6.2.2.1.1.3.cmml">i</mi></msub><mo id="S3.p5.6.m6.3.3.2.4" xref="S3.p5.6.m6.3.3.3.cmml">,</mo><msub id="S3.p5.6.m6.3.3.2.2" xref="S3.p5.6.m6.3.3.2.2.cmml"><mi id="S3.p5.6.m6.3.3.2.2.2" xref="S3.p5.6.m6.3.3.2.2.2.cmml">𝐜</mi><mrow id="S3.p5.6.m6.1.1.1" xref="S3.p5.6.m6.1.1.1.cmml"><mi id="S3.p5.6.m6.1.1.1.3" xref="S3.p5.6.m6.1.1.1.3.cmml">i</mi><mo id="S3.p5.6.m6.1.1.1.2" xref="S3.p5.6.m6.1.1.1.2.cmml">⁢</mo><mrow id="S3.p5.6.m6.1.1.1.4.2" xref="S3.p5.6.m6.1.1.1.cmml"><mo id="S3.p5.6.m6.1.1.1.4.2.1" stretchy="false" xref="S3.p5.6.m6.1.1.1.cmml">(</mo><mn id="S3.p5.6.m6.1.1.1.1" xref="S3.p5.6.m6.1.1.1.1.cmml">1</mn><mo id="S3.p5.6.m6.1.1.1.4.2.2" stretchy="false" xref="S3.p5.6.m6.1.1.1.cmml">)</mo></mrow></mrow></msub><mo id="S3.p5.6.m6.3.3.2.5" stretchy="false" xref="S3.p5.6.m6.3.3.3.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.p5.6.m6.3b"><interval closure="open" id="S3.p5.6.m6.3.3.3.cmml" xref="S3.p5.6.m6.3.3.2"><apply id="S3.p5.6.m6.2.2.1.1.cmml" xref="S3.p5.6.m6.2.2.1.1"><csymbol cd="ambiguous" 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italic_i end_POSTSUBSCRIPT , bold_c start_POSTSUBSCRIPT italic_i ( 1 ) end_POSTSUBSCRIPT )</annotation></semantics></math> and <math alttext="\operatorname{{\bm{\mu}}}^{(t)}_{i(1)}=\operatorname{\mathbb{E}}(\operatorname% {\mathbf{c}}_{i(1)}\mid\operatorname{\mathbf{y}}_{i},\operatorname{{\bm{\theta% }}}^{{(t+2/3)}})" class="ltx_Math" display="inline" id="S3.p5.7.m7.6"><semantics id="S3.p5.7.m7.6a"><mrow id="S3.p5.7.m7.6.6" xref="S3.p5.7.m7.6.6.cmml"><msubsup id="S3.p5.7.m7.6.6.3" xref="S3.p5.7.m7.6.6.3.cmml"><mi id="S3.p5.7.m7.6.6.3.2.2" xref="S3.p5.7.m7.6.6.3.2.2.cmml">𝝁</mi><mrow id="S3.p5.7.m7.2.2.1" xref="S3.p5.7.m7.2.2.1.cmml"><mi id="S3.p5.7.m7.2.2.1.3" xref="S3.p5.7.m7.2.2.1.3.cmml">i</mi><mo id="S3.p5.7.m7.2.2.1.2" xref="S3.p5.7.m7.2.2.1.2.cmml">⁢</mo><mrow id="S3.p5.7.m7.2.2.1.4.2" xref="S3.p5.7.m7.2.2.1.cmml"><mo id="S3.p5.7.m7.2.2.1.4.2.1" stretchy="false" xref="S3.p5.7.m7.2.2.1.cmml">(</mo><mn id="S3.p5.7.m7.2.2.1.1" xref="S3.p5.7.m7.2.2.1.1.cmml">1</mn><mo 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id="S3.p5.7.m7.4.4.1.1.1.3.cmml" xref="S3.p5.7.m7.4.4.1.1.1.3"><divide id="S3.p5.7.m7.4.4.1.1.1.3.1.cmml" xref="S3.p5.7.m7.4.4.1.1.1.3.1"></divide><cn id="S3.p5.7.m7.4.4.1.1.1.3.2.cmml" type="integer" xref="S3.p5.7.m7.4.4.1.1.1.3.2">2</cn><cn id="S3.p5.7.m7.4.4.1.1.1.3.3.cmml" type="integer" xref="S3.p5.7.m7.4.4.1.1.1.3.3">3</cn></apply></apply></apply></list></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p5.7.m7.6c">\operatorname{{\bm{\mu}}}^{(t)}_{i(1)}=\operatorname{\mathbb{E}}(\operatorname% {\mathbf{c}}_{i(1)}\mid\operatorname{\mathbf{y}}_{i},\operatorname{{\bm{\theta% }}}^{{(t+2/3)}})</annotation><annotation encoding="application/x-llamapun" id="S3.p5.7.m7.6d">bold_italic_μ start_POSTSUPERSCRIPT ( italic_t ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i ( 1 ) end_POSTSUBSCRIPT = blackboard_E ( bold_c start_POSTSUBSCRIPT italic_i ( 1 ) end_POSTSUBSCRIPT ∣ bold_y start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , bold_italic_θ 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end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ⊤ end_POSTSUPERSCRIPT | bold_y start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , bold_italic_θ start_POSTSUPERSCRIPT ( italic_t + 2 / 3 ) end_POSTSUPERSCRIPT )</annotation></semantics></math> are analytically tractable. The E step objective for the third cycle, <math alttext="\operatorname{\mathcal{Q}}_{(2)}(\operatorname{\mathbf{l}}_{2})" class="ltx_Math" display="inline" id="S3.p5.9.m9.3"><semantics id="S3.p5.9.m9.3a"><mrow id="S3.p5.9.m9.3.3.2" xref="S3.p5.9.m9.3.3.3.cmml"><msub id="S3.p5.9.m9.2.2.1.1" xref="S3.p5.9.m9.2.2.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.p5.9.m9.2.2.1.1.2" xref="S3.p5.9.m9.2.2.1.1.2.cmml">𝒬</mi><mrow id="S3.p5.9.m9.1.1.1.3" xref="S3.p5.9.m9.2.2.1.1.cmml"><mo id="S3.p5.9.m9.1.1.1.3.1" stretchy="false" xref="S3.p5.9.m9.2.2.1.1.cmml">(</mo><mn id="S3.p5.9.m9.1.1.1.1" xref="S3.p5.9.m9.1.1.1.1.cmml">2</mn><mo id="S3.p5.9.m9.1.1.1.3.2" stretchy="false" xref="S3.p5.9.m9.2.2.1.1.cmml">)</mo></mrow></msub><mo id="S3.p5.9.m9.3.3.2a" xref="S3.p5.9.m9.3.3.3.cmml">⁡</mo><mrow id="S3.p5.9.m9.3.3.2.2" xref="S3.p5.9.m9.3.3.3.cmml"><mo id="S3.p5.9.m9.3.3.2.2.2" stretchy="false" xref="S3.p5.9.m9.3.3.3.cmml">(</mo><msub id="S3.p5.9.m9.3.3.2.2.1" xref="S3.p5.9.m9.3.3.2.2.1.cmml"><mi id="S3.p5.9.m9.3.3.2.2.1.2" xref="S3.p5.9.m9.3.3.2.2.1.2.cmml">𝐥</mi><mn id="S3.p5.9.m9.3.3.2.2.1.3" xref="S3.p5.9.m9.3.3.2.2.1.3.cmml">2</mn></msub><mo id="S3.p5.9.m9.3.3.2.2.3" stretchy="false" xref="S3.p5.9.m9.3.3.3.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.p5.9.m9.3b"><apply id="S3.p5.9.m9.3.3.3.cmml" xref="S3.p5.9.m9.3.3.2"><apply id="S3.p5.9.m9.2.2.1.1.cmml" xref="S3.p5.9.m9.2.2.1.1"><csymbol cd="ambiguous" id="S3.p5.9.m9.2.2.1.1.1.cmml" xref="S3.p5.9.m9.2.2.1.1">subscript</csymbol><ci id="S3.p5.9.m9.2.2.1.1.2.cmml" xref="S3.p5.9.m9.2.2.1.1.2">𝒬</ci><cn id="S3.p5.9.m9.1.1.1.1.cmml" type="integer" xref="S3.p5.9.m9.1.1.1.1">2</cn></apply><apply id="S3.p5.9.m9.3.3.2.2.1.cmml" xref="S3.p5.9.m9.3.3.2.2.1"><csymbol cd="ambiguous" id="S3.p5.9.m9.3.3.2.2.1.1.cmml" xref="S3.p5.9.m9.3.3.2.2.1">subscript</csymbol><ci id="S3.p5.9.m9.3.3.2.2.1.2.cmml" xref="S3.p5.9.m9.3.3.2.2.1.2">𝐥</ci><cn id="S3.p5.9.m9.3.3.2.2.1.3.cmml" type="integer" xref="S3.p5.9.m9.3.3.2.2.1.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p5.9.m9.3c">\operatorname{\mathcal{Q}}_{(2)}(\operatorname{\mathbf{l}}_{2})</annotation><annotation encoding="application/x-llamapun" id="S3.p5.9.m9.3d">caligraphic_Q start_POSTSUBSCRIPT ( 2 ) end_POSTSUBSCRIPT ( bold_l start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT )</annotation></semantics></math>, swaps the indices 1 and 2 in <math alttext="\operatorname{\mathcal{Q}}_{(1)}(\operatorname{\mathbf{l}}_{1})" class="ltx_Math" display="inline" id="S3.p5.10.m10.3"><semantics id="S3.p5.10.m10.3a"><mrow id="S3.p5.10.m10.3.3.2" xref="S3.p5.10.m10.3.3.3.cmml"><msub id="S3.p5.10.m10.2.2.1.1" xref="S3.p5.10.m10.2.2.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.p5.10.m10.2.2.1.1.2" xref="S3.p5.10.m10.2.2.1.1.2.cmml">𝒬</mi><mrow id="S3.p5.10.m10.1.1.1.3" xref="S3.p5.10.m10.2.2.1.1.cmml"><mo id="S3.p5.10.m10.1.1.1.3.1" stretchy="false" xref="S3.p5.10.m10.2.2.1.1.cmml">(</mo><mn id="S3.p5.10.m10.1.1.1.1" xref="S3.p5.10.m10.1.1.1.1.cmml">1</mn><mo id="S3.p5.10.m10.1.1.1.3.2" stretchy="false" xref="S3.p5.10.m10.2.2.1.1.cmml">)</mo></mrow></msub><mo id="S3.p5.10.m10.3.3.2a" xref="S3.p5.10.m10.3.3.3.cmml">⁡</mo><mrow id="S3.p5.10.m10.3.3.2.2" xref="S3.p5.10.m10.3.3.3.cmml"><mo id="S3.p5.10.m10.3.3.2.2.2" stretchy="false" xref="S3.p5.10.m10.3.3.3.cmml">(</mo><msub id="S3.p5.10.m10.3.3.2.2.1" xref="S3.p5.10.m10.3.3.2.2.1.cmml"><mi id="S3.p5.10.m10.3.3.2.2.1.2" xref="S3.p5.10.m10.3.3.2.2.1.2.cmml">𝐥</mi><mn id="S3.p5.10.m10.3.3.2.2.1.3" xref="S3.p5.10.m10.3.3.2.2.1.3.cmml">1</mn></msub><mo id="S3.p5.10.m10.3.3.2.2.3" stretchy="false" xref="S3.p5.10.m10.3.3.3.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.p5.10.m10.3b"><apply id="S3.p5.10.m10.3.3.3.cmml" xref="S3.p5.10.m10.3.3.2"><apply id="S3.p5.10.m10.2.2.1.1.cmml" xref="S3.p5.10.m10.2.2.1.1"><csymbol cd="ambiguous" id="S3.p5.10.m10.2.2.1.1.1.cmml" xref="S3.p5.10.m10.2.2.1.1">subscript</csymbol><ci id="S3.p5.10.m10.2.2.1.1.2.cmml" xref="S3.p5.10.m10.2.2.1.1.2">𝒬</ci><cn id="S3.p5.10.m10.1.1.1.1.cmml" type="integer" xref="S3.p5.10.m10.1.1.1.1">1</cn></apply><apply id="S3.p5.10.m10.3.3.2.2.1.cmml" xref="S3.p5.10.m10.3.3.2.2.1"><csymbol cd="ambiguous" id="S3.p5.10.m10.3.3.2.2.1.1.cmml" xref="S3.p5.10.m10.3.3.2.2.1">subscript</csymbol><ci id="S3.p5.10.m10.3.3.2.2.1.2.cmml" xref="S3.p5.10.m10.3.3.2.2.1.2">𝐥</ci><cn id="S3.p5.10.m10.3.3.2.2.1.3.cmml" type="integer" xref="S3.p5.10.m10.3.3.2.2.1.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p5.10.m10.3c">\operatorname{\mathcal{Q}}_{(1)}(\operatorname{\mathbf{l}}_{1})</annotation><annotation encoding="application/x-llamapun" id="S3.p5.10.m10.3d">caligraphic_Q start_POSTSUBSCRIPT ( 1 ) end_POSTSUBSCRIPT ( bold_l start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT )</annotation></semantics></math> as defined in (<a class="ltx_ref" href="https://arxiv.org/html/2503.13782v1#S3.E9" title="In 3 Regularized Parameter Estimation ‣ Regularized Parameter Estimation in Mixed Model Trace Regression"><span class="ltx_text ltx_ref_tag">9</span></a>), where <math alttext="\operatorname{\mathbf{l}}_{2}=\text{vec}(\operatorname{\mathbf{L}}_{2})" class="ltx_Math" display="inline" id="S3.p5.11.m11.1"><semantics id="S3.p5.11.m11.1a"><mrow id="S3.p5.11.m11.1.1" xref="S3.p5.11.m11.1.1.cmml"><msub id="S3.p5.11.m11.1.1.3" xref="S3.p5.11.m11.1.1.3.cmml"><mi id="S3.p5.11.m11.1.1.3.2" xref="S3.p5.11.m11.1.1.3.2.cmml">𝐥</mi><mn id="S3.p5.11.m11.1.1.3.3" xref="S3.p5.11.m11.1.1.3.3.cmml">2</mn></msub><mo id="S3.p5.11.m11.1.1.2" xref="S3.p5.11.m11.1.1.2.cmml">=</mo><mrow id="S3.p5.11.m11.1.1.1" xref="S3.p5.11.m11.1.1.1.cmml"><mtext id="S3.p5.11.m11.1.1.1.3" xref="S3.p5.11.m11.1.1.1.3a.cmml">vec</mtext><mo id="S3.p5.11.m11.1.1.1.2" xref="S3.p5.11.m11.1.1.1.2.cmml">⁢</mo><mrow id="S3.p5.11.m11.1.1.1.1.1" xref="S3.p5.11.m11.1.1.1.1.1.1.cmml"><mo id="S3.p5.11.m11.1.1.1.1.1.2" stretchy="false" xref="S3.p5.11.m11.1.1.1.1.1.1.cmml">(</mo><msub id="S3.p5.11.m11.1.1.1.1.1.1" xref="S3.p5.11.m11.1.1.1.1.1.1.cmml"><mi id="S3.p5.11.m11.1.1.1.1.1.1.2" xref="S3.p5.11.m11.1.1.1.1.1.1.2.cmml">𝐋</mi><mn id="S3.p5.11.m11.1.1.1.1.1.1.3" xref="S3.p5.11.m11.1.1.1.1.1.1.3.cmml">2</mn></msub><mo id="S3.p5.11.m11.1.1.1.1.1.3" stretchy="false" xref="S3.p5.11.m11.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.p5.11.m11.1b"><apply id="S3.p5.11.m11.1.1.cmml" xref="S3.p5.11.m11.1.1"><eq id="S3.p5.11.m11.1.1.2.cmml" xref="S3.p5.11.m11.1.1.2"></eq><apply id="S3.p5.11.m11.1.1.3.cmml" xref="S3.p5.11.m11.1.1.3"><csymbol cd="ambiguous" id="S3.p5.11.m11.1.1.3.1.cmml" xref="S3.p5.11.m11.1.1.3">subscript</csymbol><ci id="S3.p5.11.m11.1.1.3.2.cmml" xref="S3.p5.11.m11.1.1.3.2">𝐥</ci><cn id="S3.p5.11.m11.1.1.3.3.cmml" type="integer" xref="S3.p5.11.m11.1.1.3.3">2</cn></apply><apply id="S3.p5.11.m11.1.1.1.cmml" xref="S3.p5.11.m11.1.1.1"><times id="S3.p5.11.m11.1.1.1.2.cmml" xref="S3.p5.11.m11.1.1.1.2"></times><ci id="S3.p5.11.m11.1.1.1.3a.cmml" xref="S3.p5.11.m11.1.1.1.3"><mtext id="S3.p5.11.m11.1.1.1.3.cmml" xref="S3.p5.11.m11.1.1.1.3">vec</mtext></ci><apply id="S3.p5.11.m11.1.1.1.1.1.1.cmml" xref="S3.p5.11.m11.1.1.1.1.1"><csymbol cd="ambiguous" id="S3.p5.11.m11.1.1.1.1.1.1.1.cmml" xref="S3.p5.11.m11.1.1.1.1.1">subscript</csymbol><ci id="S3.p5.11.m11.1.1.1.1.1.1.2.cmml" xref="S3.p5.11.m11.1.1.1.1.1.1.2">𝐋</ci><cn id="S3.p5.11.m11.1.1.1.1.1.1.3.cmml" type="integer" xref="S3.p5.11.m11.1.1.1.1.1.1.3">2</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p5.11.m11.1c">\operatorname{\mathbf{l}}_{2}=\text{vec}(\operatorname{\mathbf{L}}_{2})</annotation><annotation encoding="application/x-llamapun" id="S3.p5.11.m11.1d">bold_l start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT = vec ( bold_L start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT )</annotation></semantics></math> and <math alttext="\operatorname{\mathbf{H}}_{(2)}^{{(t)}}" class="ltx_Math" display="inline" id="S3.p5.12.m12.2"><semantics id="S3.p5.12.m12.2a"><msubsup id="S3.p5.12.m12.2.3" xref="S3.p5.12.m12.2.3.cmml"><mi id="S3.p5.12.m12.2.3.2.2" xref="S3.p5.12.m12.2.3.2.2.cmml">𝐇</mi><mrow id="S3.p5.12.m12.1.1.1.3" xref="S3.p5.12.m12.2.3.cmml"><mo id="S3.p5.12.m12.1.1.1.3.1" stretchy="false" xref="S3.p5.12.m12.2.3.cmml">(</mo><mn id="S3.p5.12.m12.1.1.1.1" xref="S3.p5.12.m12.1.1.1.1.cmml">2</mn><mo id="S3.p5.12.m12.1.1.1.3.2" stretchy="false" xref="S3.p5.12.m12.2.3.cmml">)</mo></mrow><mrow id="S3.p5.12.m12.2.2.1.3" xref="S3.p5.12.m12.2.3.cmml"><mo id="S3.p5.12.m12.2.2.1.3.1" stretchy="false" xref="S3.p5.12.m12.2.3.cmml">(</mo><mi id="S3.p5.12.m12.2.2.1.1" xref="S3.p5.12.m12.2.2.1.1.cmml">t</mi><mo id="S3.p5.12.m12.2.2.1.3.2" stretchy="false" xref="S3.p5.12.m12.2.3.cmml">)</mo></mrow></msubsup><annotation-xml encoding="MathML-Content" id="S3.p5.12.m12.2b"><apply id="S3.p5.12.m12.2.3.cmml" xref="S3.p5.12.m12.2.3"><csymbol cd="ambiguous" id="S3.p5.12.m12.2.3.1.cmml" xref="S3.p5.12.m12.2.3">superscript</csymbol><apply id="S3.p5.12.m12.2.3.2.cmml" xref="S3.p5.12.m12.2.3"><csymbol cd="ambiguous" id="S3.p5.12.m12.2.3.2.1.cmml" xref="S3.p5.12.m12.2.3">subscript</csymbol><ci id="S3.p5.12.m12.2.3.2.2.cmml" xref="S3.p5.12.m12.2.3.2.2">𝐇</ci><cn id="S3.p5.12.m12.1.1.1.1.cmml" type="integer" xref="S3.p5.12.m12.1.1.1.1">2</cn></apply><ci id="S3.p5.12.m12.2.2.1.1.cmml" xref="S3.p5.12.m12.2.2.1.1">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p5.12.m12.2c">\operatorname{\mathbf{H}}_{(2)}^{{(t)}}</annotation><annotation encoding="application/x-llamapun" id="S3.p5.12.m12.2d">bold_H start_POSTSUBSCRIPT ( 2 ) end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_t ) end_POSTSUPERSCRIPT</annotation></semantics></math> and <math alttext="\operatorname{\mathbf{g}}_{(2)}^{(t)}" class="ltx_Math" display="inline" id="S3.p5.13.m13.2"><semantics id="S3.p5.13.m13.2a"><msubsup id="S3.p5.13.m13.2.3" xref="S3.p5.13.m13.2.3.cmml"><mi id="S3.p5.13.m13.2.3.2.2" xref="S3.p5.13.m13.2.3.2.2.cmml">𝐠</mi><mrow id="S3.p5.13.m13.1.1.1.3" xref="S3.p5.13.m13.2.3.cmml"><mo id="S3.p5.13.m13.1.1.1.3.1" stretchy="false" xref="S3.p5.13.m13.2.3.cmml">(</mo><mn id="S3.p5.13.m13.1.1.1.1" xref="S3.p5.13.m13.1.1.1.1.cmml">2</mn><mo id="S3.p5.13.m13.1.1.1.3.2" stretchy="false" xref="S3.p5.13.m13.2.3.cmml">)</mo></mrow><mrow id="S3.p5.13.m13.2.2.1.3" xref="S3.p5.13.m13.2.3.cmml"><mo id="S3.p5.13.m13.2.2.1.3.1" stretchy="false" xref="S3.p5.13.m13.2.3.cmml">(</mo><mi id="S3.p5.13.m13.2.2.1.1" xref="S3.p5.13.m13.2.2.1.1.cmml">t</mi><mo id="S3.p5.13.m13.2.2.1.3.2" stretchy="false" xref="S3.p5.13.m13.2.3.cmml">)</mo></mrow></msubsup><annotation-xml encoding="MathML-Content" id="S3.p5.13.m13.2b"><apply id="S3.p5.13.m13.2.3.cmml" xref="S3.p5.13.m13.2.3"><csymbol cd="ambiguous" id="S3.p5.13.m13.2.3.1.cmml" xref="S3.p5.13.m13.2.3">superscript</csymbol><apply id="S3.p5.13.m13.2.3.2.cmml" xref="S3.p5.13.m13.2.3"><csymbol cd="ambiguous" id="S3.p5.13.m13.2.3.2.1.cmml" xref="S3.p5.13.m13.2.3">subscript</csymbol><ci id="S3.p5.13.m13.2.3.2.2.cmml" xref="S3.p5.13.m13.2.3.2.2">𝐠</ci><cn id="S3.p5.13.m13.1.1.1.1.cmml" type="integer" xref="S3.p5.13.m13.1.1.1.1">2</cn></apply><ci id="S3.p5.13.m13.2.2.1.1.cmml" xref="S3.p5.13.m13.2.2.1.1">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p5.13.m13.2c">\operatorname{\mathbf{g}}_{(2)}^{(t)}</annotation><annotation encoding="application/x-llamapun" id="S3.p5.13.m13.2d">bold_g start_POSTSUBSCRIPT ( 2 ) end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_t ) end_POSTSUPERSCRIPT</annotation></semantics></math> are obtained by swapping indices 1 and 2 in <math alttext="\operatorname{\mathbf{H}}_{(1)}^{(t)}" class="ltx_Math" display="inline" id="S3.p5.14.m14.2"><semantics id="S3.p5.14.m14.2a"><msubsup id="S3.p5.14.m14.2.3" xref="S3.p5.14.m14.2.3.cmml"><mi id="S3.p5.14.m14.2.3.2.2" xref="S3.p5.14.m14.2.3.2.2.cmml">𝐇</mi><mrow id="S3.p5.14.m14.1.1.1.3" xref="S3.p5.14.m14.2.3.cmml"><mo id="S3.p5.14.m14.1.1.1.3.1" stretchy="false" xref="S3.p5.14.m14.2.3.cmml">(</mo><mn id="S3.p5.14.m14.1.1.1.1" xref="S3.p5.14.m14.1.1.1.1.cmml">1</mn><mo id="S3.p5.14.m14.1.1.1.3.2" stretchy="false" xref="S3.p5.14.m14.2.3.cmml">)</mo></mrow><mrow id="S3.p5.14.m14.2.2.1.3" xref="S3.p5.14.m14.2.3.cmml"><mo id="S3.p5.14.m14.2.2.1.3.1" stretchy="false" xref="S3.p5.14.m14.2.3.cmml">(</mo><mi id="S3.p5.14.m14.2.2.1.1" xref="S3.p5.14.m14.2.2.1.1.cmml">t</mi><mo id="S3.p5.14.m14.2.2.1.3.2" stretchy="false" xref="S3.p5.14.m14.2.3.cmml">)</mo></mrow></msubsup><annotation-xml encoding="MathML-Content" id="S3.p5.14.m14.2b"><apply id="S3.p5.14.m14.2.3.cmml" xref="S3.p5.14.m14.2.3"><csymbol cd="ambiguous" id="S3.p5.14.m14.2.3.1.cmml" xref="S3.p5.14.m14.2.3">superscript</csymbol><apply id="S3.p5.14.m14.2.3.2.cmml" xref="S3.p5.14.m14.2.3"><csymbol cd="ambiguous" id="S3.p5.14.m14.2.3.2.1.cmml" xref="S3.p5.14.m14.2.3">subscript</csymbol><ci id="S3.p5.14.m14.2.3.2.2.cmml" xref="S3.p5.14.m14.2.3.2.2">𝐇</ci><cn id="S3.p5.14.m14.1.1.1.1.cmml" type="integer" xref="S3.p5.14.m14.1.1.1.1">1</cn></apply><ci id="S3.p5.14.m14.2.2.1.1.cmml" xref="S3.p5.14.m14.2.2.1.1">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p5.14.m14.2c">\operatorname{\mathbf{H}}_{(1)}^{(t)}</annotation><annotation encoding="application/x-llamapun" id="S3.p5.14.m14.2d">bold_H start_POSTSUBSCRIPT ( 1 ) end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_t ) end_POSTSUPERSCRIPT</annotation></semantics></math> and <math alttext="\operatorname{\mathbf{g}}_{(1)}^{(t)}" class="ltx_Math" display="inline" id="S3.p5.15.m15.2"><semantics id="S3.p5.15.m15.2a"><msubsup id="S3.p5.15.m15.2.3" xref="S3.p5.15.m15.2.3.cmml"><mi id="S3.p5.15.m15.2.3.2.2" xref="S3.p5.15.m15.2.3.2.2.cmml">𝐠</mi><mrow id="S3.p5.15.m15.1.1.1.3" xref="S3.p5.15.m15.2.3.cmml"><mo id="S3.p5.15.m15.1.1.1.3.1" stretchy="false" xref="S3.p5.15.m15.2.3.cmml">(</mo><mn id="S3.p5.15.m15.1.1.1.1" xref="S3.p5.15.m15.1.1.1.1.cmml">1</mn><mo id="S3.p5.15.m15.1.1.1.3.2" stretchy="false" xref="S3.p5.15.m15.2.3.cmml">)</mo></mrow><mrow id="S3.p5.15.m15.2.2.1.3" xref="S3.p5.15.m15.2.3.cmml"><mo id="S3.p5.15.m15.2.2.1.3.1" stretchy="false" xref="S3.p5.15.m15.2.3.cmml">(</mo><mi id="S3.p5.15.m15.2.2.1.1" xref="S3.p5.15.m15.2.2.1.1.cmml">t</mi><mo id="S3.p5.15.m15.2.2.1.3.2" stretchy="false" xref="S3.p5.15.m15.2.3.cmml">)</mo></mrow></msubsup><annotation-xml encoding="MathML-Content" id="S3.p5.15.m15.2b"><apply id="S3.p5.15.m15.2.3.cmml" xref="S3.p5.15.m15.2.3"><csymbol cd="ambiguous" id="S3.p5.15.m15.2.3.1.cmml" xref="S3.p5.15.m15.2.3">superscript</csymbol><apply id="S3.p5.15.m15.2.3.2.cmml" xref="S3.p5.15.m15.2.3"><csymbol cd="ambiguous" id="S3.p5.15.m15.2.3.2.1.cmml" xref="S3.p5.15.m15.2.3">subscript</csymbol><ci id="S3.p5.15.m15.2.3.2.2.cmml" xref="S3.p5.15.m15.2.3.2.2">𝐠</ci><cn id="S3.p5.15.m15.1.1.1.1.cmml" type="integer" xref="S3.p5.15.m15.1.1.1.1">1</cn></apply><ci id="S3.p5.15.m15.2.2.1.1.cmml" xref="S3.p5.15.m15.2.2.1.1">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p5.15.m15.2c">\operatorname{\mathbf{g}}_{(1)}^{(t)}</annotation><annotation encoding="application/x-llamapun" id="S3.p5.15.m15.2d">bold_g start_POSTSUBSCRIPT ( 1 ) end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_t ) end_POSTSUPERSCRIPT</annotation></semantics></math>. Following (<a class="ltx_ref" href="https://arxiv.org/html/2503.13782v1#S3.E10" title="In 3 Regularized Parameter Estimation ‣ Regularized Parameter Estimation in Mixed Model Trace Regression"><span class="ltx_text ltx_ref_tag">10</span></a>), the CM step in the third cycle is</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="S3.EGx11"> <tbody id="S3.E11"><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{\mathbf{l}}_{2}^{(t+1)}=\operatorname*{\mathop{% \mathrm{argmin}}}_{\operatorname{\mathbf{l}}_{2}\in\operatorname{\mathbb{R}}^{% Q_{2}S_{2}}}\;\big{\lVert}\operatorname{\mathbf{H}}_{(2)}^{-{1}/{2}(t)}% \operatorname{\mathbf{g}}^{(t)}_{(2)}-\operatorname{\mathbf{H}}_{(2)}^{{1}/{2}% (t)\top}\operatorname{\mathbf{l}}_{2}\big{\rVert}_{2}^{2}+\lambda_{% \operatorname{\mathbf{L}}}\sum_{i=1}^{S_{2}}\lVert\operatorname{\mathbf{l}}_{(% 2):i}\rVert_{2}," class="ltx_math_unparsed" display="inline" id="S3.E11.m1.8"><semantics id="S3.E11.m1.8a"><mrow id="S3.E11.m1.8b"><msubsup id="S3.E11.m1.8.9"><mi id="S3.E11.m1.8.9.2.2">𝐥</mi><mn id="S3.E11.m1.8.9.2.3">2</mn><mrow id="S3.E11.m1.1.1.1.1"><mo id="S3.E11.m1.1.1.1.1.2" stretchy="false">(</mo><mrow id="S3.E11.m1.1.1.1.1.1"><mi id="S3.E11.m1.1.1.1.1.1.2">t</mi><mo id="S3.E11.m1.1.1.1.1.1.1">+</mo><mn id="S3.E11.m1.1.1.1.1.1.3">1</mn></mrow><mo id="S3.E11.m1.1.1.1.1.3" stretchy="false">)</mo></mrow></msubsup><mo id="S3.E11.m1.8.10" rspace="0.1389em">=</mo><munder id="S3.E11.m1.8.11"><mo id="S3.E11.m1.8.11.2" lspace="0.1389em" rspace="0em">argmin</mo><mrow id="S3.E11.m1.8.11.3"><msub id="S3.E11.m1.8.11.3.2"><mi id="S3.E11.m1.8.11.3.2.2">𝐥</mi><mn id="S3.E11.m1.8.11.3.2.3">2</mn></msub><mo id="S3.E11.m1.8.11.3.1">∈</mo><msup id="S3.E11.m1.8.11.3.3"><mi id="S3.E11.m1.8.11.3.3.2">ℝ</mi><mrow id="S3.E11.m1.8.11.3.3.3"><msub id="S3.E11.m1.8.11.3.3.3.2"><mi id="S3.E11.m1.8.11.3.3.3.2.2">Q</mi><mn id="S3.E11.m1.8.11.3.3.3.2.3">2</mn></msub><mo id="S3.E11.m1.8.11.3.3.3.1">⁢</mo><msub id="S3.E11.m1.8.11.3.3.3.3"><mi id="S3.E11.m1.8.11.3.3.3.3.2">S</mi><mn id="S3.E11.m1.8.11.3.3.3.3.3">2</mn></msub></mrow></msup></mrow></munder><msubsup id="S3.E11.m1.8.12"><mrow id="S3.E11.m1.8.12.2.2"><mo fence="true" id="S3.E11.m1.8.12.2.2.1" lspace="0em" mathsize="120%" rspace="0em">∥</mo><msubsup id="S3.E11.m1.8.12.2.2.2"><mi id="S3.E11.m1.8.12.2.2.2.2.2">𝐇</mi><mrow id="S3.E11.m1.2.2.1.3"><mo id="S3.E11.m1.2.2.1.3.1" stretchy="false">(</mo><mn id="S3.E11.m1.2.2.1.1">2</mn><mo id="S3.E11.m1.2.2.1.3.2" stretchy="false">)</mo></mrow><mrow id="S3.E11.m1.3.3.1"><mo id="S3.E11.m1.3.3.1a">−</mo><mrow id="S3.E11.m1.3.3.1.3"><mrow id="S3.E11.m1.3.3.1.3.2"><mn id="S3.E11.m1.3.3.1.3.2.2">1</mn><mo id="S3.E11.m1.3.3.1.3.2.1">/</mo><mn id="S3.E11.m1.3.3.1.3.2.3">2</mn></mrow><mo id="S3.E11.m1.3.3.1.3.1">⁢</mo><mrow id="S3.E11.m1.3.3.1.3.3.2"><mo id="S3.E11.m1.3.3.1.3.3.2.1" stretchy="false">(</mo><mi id="S3.E11.m1.3.3.1.1">t</mi><mo id="S3.E11.m1.3.3.1.3.3.2.2" stretchy="false">)</mo></mrow></mrow></mrow></msubsup><msubsup id="S3.E11.m1.8.12.2.2.3"><mi id="S3.E11.m1.8.12.2.2.3.2.2">𝐠</mi><mrow id="S3.E11.m1.5.5.1.3"><mo id="S3.E11.m1.5.5.1.3.1" stretchy="false">(</mo><mn id="S3.E11.m1.5.5.1.1">2</mn><mo id="S3.E11.m1.5.5.1.3.2" stretchy="false">)</mo></mrow><mrow id="S3.E11.m1.4.4.1.3"><mo id="S3.E11.m1.4.4.1.3.1" stretchy="false">(</mo><mi id="S3.E11.m1.4.4.1.1">t</mi><mo id="S3.E11.m1.4.4.1.3.2" stretchy="false">)</mo></mrow></msubsup><mo id="S3.E11.m1.8.12.2.2.4" lspace="0em">−</mo><msubsup id="S3.E11.m1.8.12.2.2.5"><mi id="S3.E11.m1.8.12.2.2.5.2.2">𝐇</mi><mrow id="S3.E11.m1.6.6.1.3"><mo id="S3.E11.m1.6.6.1.3.1" stretchy="false">(</mo><mn id="S3.E11.m1.6.6.1.1">2</mn><mo id="S3.E11.m1.6.6.1.3.2" stretchy="false">)</mo></mrow><mrow id="S3.E11.m1.7.7.1"><mrow id="S3.E11.m1.7.7.1.3"><mrow id="S3.E11.m1.7.7.1.3.2"><mn id="S3.E11.m1.7.7.1.3.2.2">1</mn><mo id="S3.E11.m1.7.7.1.3.2.1">/</mo><mn id="S3.E11.m1.7.7.1.3.2.3">2</mn></mrow><mo id="S3.E11.m1.7.7.1.3.1">⁢</mo><mrow id="S3.E11.m1.7.7.1.3.3.2"><mo id="S3.E11.m1.7.7.1.3.3.2.1" stretchy="false">(</mo><mi id="S3.E11.m1.7.7.1.1">t</mi><mo id="S3.E11.m1.7.7.1.3.3.2.2" stretchy="false">)</mo></mrow></mrow><mo id="S3.E11.m1.7.7.1.4" lspace="0.222em">⊤</mo></mrow></msubsup><msub id="S3.E11.m1.8.12.2.2.6"><mi id="S3.E11.m1.8.12.2.2.6.2">𝐥</mi><mn id="S3.E11.m1.8.12.2.2.6.3">2</mn></msub><mo fence="true" id="S3.E11.m1.8.12.2.2.7" lspace="0em" mathsize="120%" rspace="0em">∥</mo></mrow><mn id="S3.E11.m1.8.12.2.3">2</mn><mn id="S3.E11.m1.8.12.3">2</mn></msubsup><mo id="S3.E11.m1.8.13">+</mo><msub id="S3.E11.m1.8.14"><mi id="S3.E11.m1.8.14.2">λ</mi><mi id="S3.E11.m1.8.14.3">𝐋</mi></msub><mstyle displaystyle="true" id="S3.E11.m1.8.15"><munderover id="S3.E11.m1.8.15a"><mo id="S3.E11.m1.8.15.2.2" movablelimits="false">∑</mo><mrow id="S3.E11.m1.8.15.2.3"><mi id="S3.E11.m1.8.15.2.3.2">i</mi><mo id="S3.E11.m1.8.15.2.3.1">=</mo><mn id="S3.E11.m1.8.15.2.3.3">1</mn></mrow><msub id="S3.E11.m1.8.15.3"><mi id="S3.E11.m1.8.15.3.2">S</mi><mn id="S3.E11.m1.8.15.3.3">2</mn></msub></munderover></mstyle><msub id="S3.E11.m1.8.16"><mrow id="S3.E11.m1.8.16.2"><mo fence="true" id="S3.E11.m1.8.16.2.1" lspace="0em" rspace="0em">∥</mo><msub id="S3.E11.m1.8.16.2.2"><mi id="S3.E11.m1.8.16.2.2.2">𝐥</mi><mrow id="S3.E11.m1.8.8.1"><mrow id="S3.E11.m1.8.8.1.3.2"><mo id="S3.E11.m1.8.8.1.3.2.1" stretchy="false">(</mo><mn id="S3.E11.m1.8.8.1.1">2</mn><mo id="S3.E11.m1.8.8.1.3.2.2" rspace="0.278em" stretchy="false">)</mo></mrow><mo id="S3.E11.m1.8.8.1.2" rspace="0.278em">:</mo><mi id="S3.E11.m1.8.8.1.4">i</mi></mrow></msub><mo fence="true" id="S3.E11.m1.8.16.2.3" lspace="0em" rspace="0em">∥</mo></mrow><mn id="S3.E11.m1.8.16.3">2</mn></msub><mo id="S3.E11.m1.8.17">,</mo></mrow><annotation encoding="application/x-tex" id="S3.E11.m1.8c">\displaystyle\operatorname{\mathbf{l}}_{2}^{(t+1)}=\operatorname*{\mathop{% \mathrm{argmin}}}_{\operatorname{\mathbf{l}}_{2}\in\operatorname{\mathbb{R}}^{% Q_{2}S_{2}}}\;\big{\lVert}\operatorname{\mathbf{H}}_{(2)}^{-{1}/{2}(t)}% \operatorname{\mathbf{g}}^{(t)}_{(2)}-\operatorname{\mathbf{H}}_{(2)}^{{1}/{2}% (t)\top}\operatorname{\mathbf{l}}_{2}\big{\rVert}_{2}^{2}+\lambda_{% \operatorname{\mathbf{L}}}\sum_{i=1}^{S_{2}}\lVert\operatorname{\mathbf{l}}_{(% 2):i}\rVert_{2},</annotation><annotation encoding="application/x-llamapun" id="S3.E11.m1.8d">bold_l start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_t + 1 ) end_POSTSUPERSCRIPT = roman_argmin start_POSTSUBSCRIPT bold_l start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ∈ blackboard_R start_POSTSUPERSCRIPT italic_Q start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT italic_S start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ∥ bold_H start_POSTSUBSCRIPT ( 2 ) end_POSTSUBSCRIPT start_POSTSUPERSCRIPT - 1 / 2 ( italic_t ) end_POSTSUPERSCRIPT bold_g start_POSTSUPERSCRIPT ( italic_t ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT ( 2 ) end_POSTSUBSCRIPT - bold_H start_POSTSUBSCRIPT ( 2 ) end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 1 / 2 ( italic_t ) ⊤ end_POSTSUPERSCRIPT bold_l start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ∥ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + italic_λ start_POSTSUBSCRIPT bold_L end_POSTSUBSCRIPT ∑ start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_S start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT ∥ bold_l start_POSTSUBSCRIPT ( 2 ) : italic_i end_POSTSUBSCRIPT ∥ start_POSTSUBSCRIPT 2 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">(11)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S3.p5.26">where <math alttext="\operatorname{\mathbf{L}}^{(t+1)}_{2}=\text{vec}^{{-1}}(\operatorname{\mathbf{% l}}_{2}^{(t+1)})" class="ltx_Math" display="inline" id="S3.p5.16.m1.3"><semantics id="S3.p5.16.m1.3a"><mrow id="S3.p5.16.m1.3.3" xref="S3.p5.16.m1.3.3.cmml"><msubsup id="S3.p5.16.m1.3.3.3" xref="S3.p5.16.m1.3.3.3.cmml"><mi id="S3.p5.16.m1.3.3.3.2.2" xref="S3.p5.16.m1.3.3.3.2.2.cmml">𝐋</mi><mn id="S3.p5.16.m1.3.3.3.3" xref="S3.p5.16.m1.3.3.3.3.cmml">2</mn><mrow id="S3.p5.16.m1.1.1.1.1" xref="S3.p5.16.m1.1.1.1.1.1.cmml"><mo id="S3.p5.16.m1.1.1.1.1.2" stretchy="false" xref="S3.p5.16.m1.1.1.1.1.1.cmml">(</mo><mrow id="S3.p5.16.m1.1.1.1.1.1" xref="S3.p5.16.m1.1.1.1.1.1.cmml"><mi id="S3.p5.16.m1.1.1.1.1.1.2" xref="S3.p5.16.m1.1.1.1.1.1.2.cmml">t</mi><mo id="S3.p5.16.m1.1.1.1.1.1.1" xref="S3.p5.16.m1.1.1.1.1.1.1.cmml">+</mo><mn id="S3.p5.16.m1.1.1.1.1.1.3" xref="S3.p5.16.m1.1.1.1.1.1.3.cmml">1</mn></mrow><mo id="S3.p5.16.m1.1.1.1.1.3" stretchy="false" xref="S3.p5.16.m1.1.1.1.1.1.cmml">)</mo></mrow></msubsup><mo id="S3.p5.16.m1.3.3.2" xref="S3.p5.16.m1.3.3.2.cmml">=</mo><mrow id="S3.p5.16.m1.3.3.1" xref="S3.p5.16.m1.3.3.1.cmml"><msup id="S3.p5.16.m1.3.3.1.3" xref="S3.p5.16.m1.3.3.1.3.cmml"><mtext id="S3.p5.16.m1.3.3.1.3.2" xref="S3.p5.16.m1.3.3.1.3.2a.cmml">vec</mtext><mrow id="S3.p5.16.m1.3.3.1.3.3" xref="S3.p5.16.m1.3.3.1.3.3.cmml"><mo id="S3.p5.16.m1.3.3.1.3.3a" xref="S3.p5.16.m1.3.3.1.3.3.cmml">−</mo><mn id="S3.p5.16.m1.3.3.1.3.3.2" xref="S3.p5.16.m1.3.3.1.3.3.2.cmml">1</mn></mrow></msup><mo id="S3.p5.16.m1.3.3.1.2" xref="S3.p5.16.m1.3.3.1.2.cmml">⁢</mo><mrow id="S3.p5.16.m1.3.3.1.1.1" xref="S3.p5.16.m1.3.3.1.1.1.1.cmml"><mo id="S3.p5.16.m1.3.3.1.1.1.2" stretchy="false" xref="S3.p5.16.m1.3.3.1.1.1.1.cmml">(</mo><msubsup id="S3.p5.16.m1.3.3.1.1.1.1" xref="S3.p5.16.m1.3.3.1.1.1.1.cmml"><mi id="S3.p5.16.m1.3.3.1.1.1.1.2.2" xref="S3.p5.16.m1.3.3.1.1.1.1.2.2.cmml">𝐥</mi><mn id="S3.p5.16.m1.3.3.1.1.1.1.2.3" xref="S3.p5.16.m1.3.3.1.1.1.1.2.3.cmml">2</mn><mrow id="S3.p5.16.m1.2.2.1.1" xref="S3.p5.16.m1.2.2.1.1.1.cmml"><mo id="S3.p5.16.m1.2.2.1.1.2" stretchy="false" xref="S3.p5.16.m1.2.2.1.1.1.cmml">(</mo><mrow id="S3.p5.16.m1.2.2.1.1.1" xref="S3.p5.16.m1.2.2.1.1.1.cmml"><mi id="S3.p5.16.m1.2.2.1.1.1.2" xref="S3.p5.16.m1.2.2.1.1.1.2.cmml">t</mi><mo id="S3.p5.16.m1.2.2.1.1.1.1" xref="S3.p5.16.m1.2.2.1.1.1.1.cmml">+</mo><mn id="S3.p5.16.m1.2.2.1.1.1.3" xref="S3.p5.16.m1.2.2.1.1.1.3.cmml">1</mn></mrow><mo id="S3.p5.16.m1.2.2.1.1.3" stretchy="false" xref="S3.p5.16.m1.2.2.1.1.1.cmml">)</mo></mrow></msubsup><mo id="S3.p5.16.m1.3.3.1.1.1.3" stretchy="false" xref="S3.p5.16.m1.3.3.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.p5.16.m1.3b"><apply id="S3.p5.16.m1.3.3.cmml" xref="S3.p5.16.m1.3.3"><eq id="S3.p5.16.m1.3.3.2.cmml" xref="S3.p5.16.m1.3.3.2"></eq><apply id="S3.p5.16.m1.3.3.3.cmml" xref="S3.p5.16.m1.3.3.3"><csymbol cd="ambiguous" id="S3.p5.16.m1.3.3.3.1.cmml" xref="S3.p5.16.m1.3.3.3">subscript</csymbol><apply id="S3.p5.16.m1.3.3.3.2.cmml" xref="S3.p5.16.m1.3.3.3"><csymbol cd="ambiguous" id="S3.p5.16.m1.3.3.3.2.1.cmml" xref="S3.p5.16.m1.3.3.3">superscript</csymbol><ci id="S3.p5.16.m1.3.3.3.2.2.cmml" xref="S3.p5.16.m1.3.3.3.2.2">𝐋</ci><apply id="S3.p5.16.m1.1.1.1.1.1.cmml" xref="S3.p5.16.m1.1.1.1.1"><plus id="S3.p5.16.m1.1.1.1.1.1.1.cmml" xref="S3.p5.16.m1.1.1.1.1.1.1"></plus><ci id="S3.p5.16.m1.1.1.1.1.1.2.cmml" xref="S3.p5.16.m1.1.1.1.1.1.2">𝑡</ci><cn id="S3.p5.16.m1.1.1.1.1.1.3.cmml" type="integer" xref="S3.p5.16.m1.1.1.1.1.1.3">1</cn></apply></apply><cn id="S3.p5.16.m1.3.3.3.3.cmml" type="integer" xref="S3.p5.16.m1.3.3.3.3">2</cn></apply><apply id="S3.p5.16.m1.3.3.1.cmml" xref="S3.p5.16.m1.3.3.1"><times id="S3.p5.16.m1.3.3.1.2.cmml" xref="S3.p5.16.m1.3.3.1.2"></times><apply id="S3.p5.16.m1.3.3.1.3.cmml" xref="S3.p5.16.m1.3.3.1.3"><csymbol cd="ambiguous" id="S3.p5.16.m1.3.3.1.3.1.cmml" xref="S3.p5.16.m1.3.3.1.3">superscript</csymbol><ci id="S3.p5.16.m1.3.3.1.3.2a.cmml" xref="S3.p5.16.m1.3.3.1.3.2"><mtext id="S3.p5.16.m1.3.3.1.3.2.cmml" xref="S3.p5.16.m1.3.3.1.3.2">vec</mtext></ci><apply id="S3.p5.16.m1.3.3.1.3.3.cmml" xref="S3.p5.16.m1.3.3.1.3.3"><minus id="S3.p5.16.m1.3.3.1.3.3.1.cmml" xref="S3.p5.16.m1.3.3.1.3.3"></minus><cn id="S3.p5.16.m1.3.3.1.3.3.2.cmml" type="integer" xref="S3.p5.16.m1.3.3.1.3.3.2">1</cn></apply></apply><apply id="S3.p5.16.m1.3.3.1.1.1.1.cmml" xref="S3.p5.16.m1.3.3.1.1.1"><csymbol cd="ambiguous" id="S3.p5.16.m1.3.3.1.1.1.1.1.cmml" xref="S3.p5.16.m1.3.3.1.1.1">superscript</csymbol><apply id="S3.p5.16.m1.3.3.1.1.1.1.2.cmml" xref="S3.p5.16.m1.3.3.1.1.1"><csymbol cd="ambiguous" id="S3.p5.16.m1.3.3.1.1.1.1.2.1.cmml" xref="S3.p5.16.m1.3.3.1.1.1">subscript</csymbol><ci id="S3.p5.16.m1.3.3.1.1.1.1.2.2.cmml" xref="S3.p5.16.m1.3.3.1.1.1.1.2.2">𝐥</ci><cn id="S3.p5.16.m1.3.3.1.1.1.1.2.3.cmml" type="integer" xref="S3.p5.16.m1.3.3.1.1.1.1.2.3">2</cn></apply><apply id="S3.p5.16.m1.2.2.1.1.1.cmml" xref="S3.p5.16.m1.2.2.1.1"><plus id="S3.p5.16.m1.2.2.1.1.1.1.cmml" xref="S3.p5.16.m1.2.2.1.1.1.1"></plus><ci id="S3.p5.16.m1.2.2.1.1.1.2.cmml" xref="S3.p5.16.m1.2.2.1.1.1.2">𝑡</ci><cn id="S3.p5.16.m1.2.2.1.1.1.3.cmml" type="integer" xref="S3.p5.16.m1.2.2.1.1.1.3">1</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p5.16.m1.3c">\operatorname{\mathbf{L}}^{(t+1)}_{2}=\text{vec}^{{-1}}(\operatorname{\mathbf{% l}}_{2}^{(t+1)})</annotation><annotation encoding="application/x-llamapun" id="S3.p5.16.m1.3d">bold_L start_POSTSUPERSCRIPT ( italic_t + 1 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT = vec start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ( bold_l start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_t + 1 ) end_POSTSUPERSCRIPT )</annotation></semantics></math>, <math alttext="\operatorname{\mathbf{l}}_{(2):i}" class="ltx_Math" display="inline" id="S3.p5.17.m2.1"><semantics id="S3.p5.17.m2.1a"><msub id="S3.p5.17.m2.1.2" xref="S3.p5.17.m2.1.2.cmml"><mi id="S3.p5.17.m2.1.2.2" xref="S3.p5.17.m2.1.2.2.cmml">𝐥</mi><mrow id="S3.p5.17.m2.1.1.1" xref="S3.p5.17.m2.1.1.1.cmml"><mrow id="S3.p5.17.m2.1.1.1.3.2" xref="S3.p5.17.m2.1.1.1.cmml"><mo id="S3.p5.17.m2.1.1.1.3.2.1" stretchy="false" xref="S3.p5.17.m2.1.1.1.cmml">(</mo><mn id="S3.p5.17.m2.1.1.1.1" xref="S3.p5.17.m2.1.1.1.1.cmml">2</mn><mo id="S3.p5.17.m2.1.1.1.3.2.2" rspace="0.278em" stretchy="false" xref="S3.p5.17.m2.1.1.1.cmml">)</mo></mrow><mo id="S3.p5.17.m2.1.1.1.2" rspace="0.278em" xref="S3.p5.17.m2.1.1.1.2.cmml">:</mo><mi id="S3.p5.17.m2.1.1.1.4" xref="S3.p5.17.m2.1.1.1.4.cmml">i</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="S3.p5.17.m2.1b"><apply id="S3.p5.17.m2.1.2.cmml" xref="S3.p5.17.m2.1.2"><csymbol cd="ambiguous" id="S3.p5.17.m2.1.2.1.cmml" xref="S3.p5.17.m2.1.2">subscript</csymbol><ci id="S3.p5.17.m2.1.2.2.cmml" xref="S3.p5.17.m2.1.2.2">𝐥</ci><apply id="S3.p5.17.m2.1.1.1.cmml" xref="S3.p5.17.m2.1.1.1"><ci id="S3.p5.17.m2.1.1.1.2.cmml" xref="S3.p5.17.m2.1.1.1.2">:</ci><cn id="S3.p5.17.m2.1.1.1.1.cmml" type="integer" xref="S3.p5.17.m2.1.1.1.1">2</cn><ci id="S3.p5.17.m2.1.1.1.4.cmml" xref="S3.p5.17.m2.1.1.1.4">𝑖</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p5.17.m2.1c">\operatorname{\mathbf{l}}_{(2):i}</annotation><annotation encoding="application/x-llamapun" id="S3.p5.17.m2.1d">bold_l start_POSTSUBSCRIPT ( 2 ) : italic_i end_POSTSUBSCRIPT</annotation></semantics></math> is the <math alttext="i" class="ltx_Math" display="inline" id="S3.p5.18.m3.1"><semantics id="S3.p5.18.m3.1a"><mi id="S3.p5.18.m3.1.1" xref="S3.p5.18.m3.1.1.cmml">i</mi><annotation-xml encoding="MathML-Content" id="S3.p5.18.m3.1b"><ci id="S3.p5.18.m3.1.1.cmml" xref="S3.p5.18.m3.1.1">𝑖</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.p5.18.m3.1c">i</annotation><annotation encoding="application/x-llamapun" id="S3.p5.18.m3.1d">italic_i</annotation></semantics></math>th <math alttext="Q_{2}" class="ltx_Math" display="inline" id="S3.p5.19.m4.1"><semantics id="S3.p5.19.m4.1a"><msub id="S3.p5.19.m4.1.1" xref="S3.p5.19.m4.1.1.cmml"><mi id="S3.p5.19.m4.1.1.2" xref="S3.p5.19.m4.1.1.2.cmml">Q</mi><mn id="S3.p5.19.m4.1.1.3" xref="S3.p5.19.m4.1.1.3.cmml">2</mn></msub><annotation-xml encoding="MathML-Content" id="S3.p5.19.m4.1b"><apply id="S3.p5.19.m4.1.1.cmml" xref="S3.p5.19.m4.1.1"><csymbol cd="ambiguous" id="S3.p5.19.m4.1.1.1.cmml" xref="S3.p5.19.m4.1.1">subscript</csymbol><ci id="S3.p5.19.m4.1.1.2.cmml" xref="S3.p5.19.m4.1.1.2">𝑄</ci><cn id="S3.p5.19.m4.1.1.3.cmml" type="integer" xref="S3.p5.19.m4.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p5.19.m4.1c">Q_{2}</annotation><annotation encoding="application/x-llamapun" id="S3.p5.19.m4.1d">italic_Q start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math>-dimensional block of <math alttext="\operatorname{\mathbf{l}}_{2}" class="ltx_Math" display="inline" id="S3.p5.20.m5.1"><semantics id="S3.p5.20.m5.1a"><msub id="S3.p5.20.m5.1.1" xref="S3.p5.20.m5.1.1.cmml"><mi id="S3.p5.20.m5.1.1.2" xref="S3.p5.20.m5.1.1.2.cmml">𝐥</mi><mn id="S3.p5.20.m5.1.1.3" xref="S3.p5.20.m5.1.1.3.cmml">2</mn></msub><annotation-xml encoding="MathML-Content" id="S3.p5.20.m5.1b"><apply id="S3.p5.20.m5.1.1.cmml" xref="S3.p5.20.m5.1.1"><csymbol cd="ambiguous" id="S3.p5.20.m5.1.1.1.cmml" xref="S3.p5.20.m5.1.1">subscript</csymbol><ci id="S3.p5.20.m5.1.1.2.cmml" xref="S3.p5.20.m5.1.1.2">𝐥</ci><cn id="S3.p5.20.m5.1.1.3.cmml" type="integer" xref="S3.p5.20.m5.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p5.20.m5.1c">\operatorname{\mathbf{l}}_{2}</annotation><annotation encoding="application/x-llamapun" id="S3.p5.20.m5.1d">bold_l start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math>, which corresponds to the <math alttext="i" class="ltx_Math" display="inline" id="S3.p5.21.m6.1"><semantics id="S3.p5.21.m6.1a"><mi id="S3.p5.21.m6.1.1" xref="S3.p5.21.m6.1.1.cmml">i</mi><annotation-xml encoding="MathML-Content" id="S3.p5.21.m6.1b"><ci id="S3.p5.21.m6.1.1.cmml" xref="S3.p5.21.m6.1.1">𝑖</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.p5.21.m6.1c">i</annotation><annotation encoding="application/x-llamapun" id="S3.p5.21.m6.1d">italic_i</annotation></semantics></math>th column of <math alttext="\operatorname{\mathbf{L}}_{2}" class="ltx_Math" display="inline" id="S3.p5.22.m7.1"><semantics id="S3.p5.22.m7.1a"><msub id="S3.p5.22.m7.1.1" xref="S3.p5.22.m7.1.1.cmml"><mi id="S3.p5.22.m7.1.1.2" xref="S3.p5.22.m7.1.1.2.cmml">𝐋</mi><mn id="S3.p5.22.m7.1.1.3" xref="S3.p5.22.m7.1.1.3.cmml">2</mn></msub><annotation-xml encoding="MathML-Content" id="S3.p5.22.m7.1b"><apply id="S3.p5.22.m7.1.1.cmml" xref="S3.p5.22.m7.1.1"><csymbol cd="ambiguous" id="S3.p5.22.m7.1.1.1.cmml" xref="S3.p5.22.m7.1.1">subscript</csymbol><ci id="S3.p5.22.m7.1.1.2.cmml" xref="S3.p5.22.m7.1.1.2">𝐋</ci><cn id="S3.p5.22.m7.1.1.3.cmml" type="integer" xref="S3.p5.22.m7.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p5.22.m7.1c">\operatorname{\mathbf{L}}_{2}</annotation><annotation encoding="application/x-llamapun" id="S3.p5.22.m7.1d">bold_L start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math>, and <math alttext="\operatorname{\mathbf{l}}_{2}^{(t+1)}" class="ltx_Math" display="inline" id="S3.p5.23.m8.1"><semantics id="S3.p5.23.m8.1a"><msubsup id="S3.p5.23.m8.1.2" xref="S3.p5.23.m8.1.2.cmml"><mi id="S3.p5.23.m8.1.2.2.2" xref="S3.p5.23.m8.1.2.2.2.cmml">𝐥</mi><mn id="S3.p5.23.m8.1.2.2.3" xref="S3.p5.23.m8.1.2.2.3.cmml">2</mn><mrow id="S3.p5.23.m8.1.1.1.1" xref="S3.p5.23.m8.1.1.1.1.1.cmml"><mo id="S3.p5.23.m8.1.1.1.1.2" stretchy="false" xref="S3.p5.23.m8.1.1.1.1.1.cmml">(</mo><mrow id="S3.p5.23.m8.1.1.1.1.1" xref="S3.p5.23.m8.1.1.1.1.1.cmml"><mi id="S3.p5.23.m8.1.1.1.1.1.2" xref="S3.p5.23.m8.1.1.1.1.1.2.cmml">t</mi><mo id="S3.p5.23.m8.1.1.1.1.1.1" xref="S3.p5.23.m8.1.1.1.1.1.1.cmml">+</mo><mn id="S3.p5.23.m8.1.1.1.1.1.3" xref="S3.p5.23.m8.1.1.1.1.1.3.cmml">1</mn></mrow><mo id="S3.p5.23.m8.1.1.1.1.3" stretchy="false" xref="S3.p5.23.m8.1.1.1.1.1.cmml">)</mo></mrow></msubsup><annotation-xml encoding="MathML-Content" id="S3.p5.23.m8.1b"><apply id="S3.p5.23.m8.1.2.cmml" xref="S3.p5.23.m8.1.2"><csymbol cd="ambiguous" id="S3.p5.23.m8.1.2.1.cmml" xref="S3.p5.23.m8.1.2">superscript</csymbol><apply id="S3.p5.23.m8.1.2.2.cmml" xref="S3.p5.23.m8.1.2"><csymbol cd="ambiguous" id="S3.p5.23.m8.1.2.2.1.cmml" xref="S3.p5.23.m8.1.2">subscript</csymbol><ci id="S3.p5.23.m8.1.2.2.2.cmml" xref="S3.p5.23.m8.1.2.2.2">𝐥</ci><cn id="S3.p5.23.m8.1.2.2.3.cmml" type="integer" xref="S3.p5.23.m8.1.2.2.3">2</cn></apply><apply id="S3.p5.23.m8.1.1.1.1.1.cmml" xref="S3.p5.23.m8.1.1.1.1"><plus id="S3.p5.23.m8.1.1.1.1.1.1.cmml" xref="S3.p5.23.m8.1.1.1.1.1.1"></plus><ci id="S3.p5.23.m8.1.1.1.1.1.2.cmml" xref="S3.p5.23.m8.1.1.1.1.1.2">𝑡</ci><cn id="S3.p5.23.m8.1.1.1.1.1.3.cmml" type="integer" xref="S3.p5.23.m8.1.1.1.1.1.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p5.23.m8.1c">\operatorname{\mathbf{l}}_{2}^{(t+1)}</annotation><annotation encoding="application/x-llamapun" id="S3.p5.23.m8.1d">bold_l start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_t + 1 ) end_POSTSUPERSCRIPT</annotation></semantics></math> is estimated using the <span class="ltx_text ltx_font_typewriter" id="S3.p5.26.1">gglasso</span> package. This cycle finishes the <math alttext="(t+1)" class="ltx_Math" display="inline" id="S3.p5.24.m9.1"><semantics id="S3.p5.24.m9.1a"><mrow id="S3.p5.24.m9.1.1.1" xref="S3.p5.24.m9.1.1.1.1.cmml"><mo id="S3.p5.24.m9.1.1.1.2" stretchy="false" xref="S3.p5.24.m9.1.1.1.1.cmml">(</mo><mrow id="S3.p5.24.m9.1.1.1.1" xref="S3.p5.24.m9.1.1.1.1.cmml"><mi id="S3.p5.24.m9.1.1.1.1.2" xref="S3.p5.24.m9.1.1.1.1.2.cmml">t</mi><mo id="S3.p5.24.m9.1.1.1.1.1" xref="S3.p5.24.m9.1.1.1.1.1.cmml">+</mo><mn id="S3.p5.24.m9.1.1.1.1.3" xref="S3.p5.24.m9.1.1.1.1.3.cmml">1</mn></mrow><mo id="S3.p5.24.m9.1.1.1.3" stretchy="false" xref="S3.p5.24.m9.1.1.1.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.p5.24.m9.1b"><apply id="S3.p5.24.m9.1.1.1.1.cmml" xref="S3.p5.24.m9.1.1.1"><plus id="S3.p5.24.m9.1.1.1.1.1.cmml" xref="S3.p5.24.m9.1.1.1.1.1"></plus><ci id="S3.p5.24.m9.1.1.1.1.2.cmml" xref="S3.p5.24.m9.1.1.1.1.2">𝑡</ci><cn id="S3.p5.24.m9.1.1.1.1.3.cmml" type="integer" xref="S3.p5.24.m9.1.1.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p5.24.m9.1c">(t+1)</annotation><annotation encoding="application/x-llamapun" id="S3.p5.24.m9.1d">( italic_t + 1 )</annotation></semantics></math>th iteration of the AECM algorithm, updating the <math alttext="\operatorname{{\bm{\theta}}}^{{(t)}}" class="ltx_Math" display="inline" id="S3.p5.25.m10.1"><semantics id="S3.p5.25.m10.1a"><msup id="S3.p5.25.m10.1.2" xref="S3.p5.25.m10.1.2.cmml"><mi id="S3.p5.25.m10.1.2.2" xref="S3.p5.25.m10.1.2.2.cmml">𝜽</mi><mrow id="S3.p5.25.m10.1.1.1.3" xref="S3.p5.25.m10.1.2.cmml"><mo id="S3.p5.25.m10.1.1.1.3.1" stretchy="false" xref="S3.p5.25.m10.1.2.cmml">(</mo><mi id="S3.p5.25.m10.1.1.1.1" xref="S3.p5.25.m10.1.1.1.1.cmml">t</mi><mo id="S3.p5.25.m10.1.1.1.3.2" stretchy="false" xref="S3.p5.25.m10.1.2.cmml">)</mo></mrow></msup><annotation-xml encoding="MathML-Content" id="S3.p5.25.m10.1b"><apply id="S3.p5.25.m10.1.2.cmml" 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Algorithm <a class="ltx_ref" href="https://arxiv.org/html/2503.13782v1#alg1" title="In 3 Regularized Parameter Estimation ‣ Regularized Parameter Estimation in Mixed Model Trace Regression"><span class="ltx_text ltx_ref_tag">1</span></a> summarizes the AECM algorithm.</p> </div> <figure class="ltx_float ltx_algorithm" id="alg1"> <div class="ltx_listing ltx_lst_numbers_left ltx_listing" id="alg1.2"> <div class="ltx_listingline" id="alg1.2.1"> <span class="ltx_text ltx_font_bold" id="alg1.2.1.1">First cycle:</span> <ul class="ltx_itemize" id="S3.I1"> <li class="ltx_item" id="S3.I1.i1" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">•</span> <div class="ltx_para" id="S3.I1.i1.p1"> <p class="ltx_p" id="S3.I1.i1.p1.6"><span class="ltx_text ltx_font_bold" id="S3.I1.i1.p1.6.1">Regularized weighted least squares:</span> Given <math alttext="{\operatorname{\mathbf{L}}}_{1}^{(t)}" class="ltx_Math" display="inline" id="S3.I1.i1.p1.1.m1.1"><semantics 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xref="S3.I1.i1.p1.3.m3.1.1.1.1.1.2.cmml">t</mi><mo id="S3.I1.i1.p1.3.m3.1.1.1.1.1.1" xref="S3.I1.i1.p1.3.m3.1.1.1.1.1.1.cmml">+</mo><mn id="S3.I1.i1.p1.3.m3.1.1.1.1.1.3" xref="S3.I1.i1.p1.3.m3.1.1.1.1.1.3.cmml">1</mn></mrow><mo id="S3.I1.i1.p1.3.m3.1.1.1.1.3" stretchy="false" xref="S3.I1.i1.p1.3.m3.1.1.1.1.1.cmml">)</mo></mrow></msup><annotation-xml encoding="MathML-Content" id="S3.I1.i1.p1.3.m3.1b"><apply id="S3.I1.i1.p1.3.m3.1.2.cmml" xref="S3.I1.i1.p1.3.m3.1.2"><csymbol cd="ambiguous" id="S3.I1.i1.p1.3.m3.1.2.1.cmml" xref="S3.I1.i1.p1.3.m3.1.2">superscript</csymbol><ci id="S3.I1.i1.p1.3.m3.1.2.2.cmml" xref="S3.I1.i1.p1.3.m3.1.2.2">𝐁</ci><apply id="S3.I1.i1.p1.3.m3.1.1.1.1.1.cmml" xref="S3.I1.i1.p1.3.m3.1.1.1.1"><plus id="S3.I1.i1.p1.3.m3.1.1.1.1.1.1.cmml" xref="S3.I1.i1.p1.3.m3.1.1.1.1.1.1"></plus><ci id="S3.I1.i1.p1.3.m3.1.1.1.1.1.2.cmml" xref="S3.I1.i1.p1.3.m3.1.1.1.1.1.2">𝑡</ci><cn id="S3.I1.i1.p1.3.m3.1.1.1.1.1.3.cmml" type="integer" xref="S3.I1.i1.p1.3.m3.1.1.1.1.1.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I1.i1.p1.3.m3.1c">\operatorname{\mathbf{B}}^{{(t+1)}}</annotation><annotation encoding="application/x-llamapun" id="S3.I1.i1.p1.3.m3.1d">bold_B start_POSTSUPERSCRIPT ( italic_t + 1 ) end_POSTSUPERSCRIPT</annotation></semantics></math> and <math alttext="{\tau}^{2(t+1)}" class="ltx_Math" display="inline" id="S3.I1.i1.p1.4.m4.1"><semantics id="S3.I1.i1.p1.4.m4.1a"><msup id="S3.I1.i1.p1.4.m4.1.2" xref="S3.I1.i1.p1.4.m4.1.2.cmml"><mi id="S3.I1.i1.p1.4.m4.1.2.2" xref="S3.I1.i1.p1.4.m4.1.2.2.cmml">τ</mi><mrow id="S3.I1.i1.p1.4.m4.1.1.1" xref="S3.I1.i1.p1.4.m4.1.1.1.cmml"><mn id="S3.I1.i1.p1.4.m4.1.1.1.3" xref="S3.I1.i1.p1.4.m4.1.1.1.3.cmml">2</mn><mo id="S3.I1.i1.p1.4.m4.1.1.1.2" xref="S3.I1.i1.p1.4.m4.1.1.1.2.cmml">⁢</mo><mrow id="S3.I1.i1.p1.4.m4.1.1.1.1.1" xref="S3.I1.i1.p1.4.m4.1.1.1.1.1.1.cmml"><mo id="S3.I1.i1.p1.4.m4.1.1.1.1.1.2" stretchy="false" xref="S3.I1.i1.p1.4.m4.1.1.1.1.1.1.cmml">(</mo><mrow id="S3.I1.i1.p1.4.m4.1.1.1.1.1.1" xref="S3.I1.i1.p1.4.m4.1.1.1.1.1.1.cmml"><mi id="S3.I1.i1.p1.4.m4.1.1.1.1.1.1.2" xref="S3.I1.i1.p1.4.m4.1.1.1.1.1.1.2.cmml">t</mi><mo id="S3.I1.i1.p1.4.m4.1.1.1.1.1.1.1" xref="S3.I1.i1.p1.4.m4.1.1.1.1.1.1.1.cmml">+</mo><mn id="S3.I1.i1.p1.4.m4.1.1.1.1.1.1.3" xref="S3.I1.i1.p1.4.m4.1.1.1.1.1.1.3.cmml">1</mn></mrow><mo id="S3.I1.i1.p1.4.m4.1.1.1.1.1.3" stretchy="false" xref="S3.I1.i1.p1.4.m4.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></msup><annotation-xml encoding="MathML-Content" id="S3.I1.i1.p1.4.m4.1b"><apply id="S3.I1.i1.p1.4.m4.1.2.cmml" xref="S3.I1.i1.p1.4.m4.1.2"><csymbol cd="ambiguous" id="S3.I1.i1.p1.4.m4.1.2.1.cmml" xref="S3.I1.i1.p1.4.m4.1.2">superscript</csymbol><ci id="S3.I1.i1.p1.4.m4.1.2.2.cmml" xref="S3.I1.i1.p1.4.m4.1.2.2">𝜏</ci><apply id="S3.I1.i1.p1.4.m4.1.1.1.cmml" xref="S3.I1.i1.p1.4.m4.1.1.1"><times id="S3.I1.i1.p1.4.m4.1.1.1.2.cmml" xref="S3.I1.i1.p1.4.m4.1.1.1.2"></times><cn id="S3.I1.i1.p1.4.m4.1.1.1.3.cmml" type="integer" xref="S3.I1.i1.p1.4.m4.1.1.1.3">2</cn><apply id="S3.I1.i1.p1.4.m4.1.1.1.1.1.1.cmml" xref="S3.I1.i1.p1.4.m4.1.1.1.1.1"><plus id="S3.I1.i1.p1.4.m4.1.1.1.1.1.1.1.cmml" xref="S3.I1.i1.p1.4.m4.1.1.1.1.1.1.1"></plus><ci id="S3.I1.i1.p1.4.m4.1.1.1.1.1.1.2.cmml" xref="S3.I1.i1.p1.4.m4.1.1.1.1.1.1.2">𝑡</ci><cn id="S3.I1.i1.p1.4.m4.1.1.1.1.1.1.3.cmml" type="integer" xref="S3.I1.i1.p1.4.m4.1.1.1.1.1.1.3">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I1.i1.p1.4.m4.1c">{\tau}^{2(t+1)}</annotation><annotation encoding="application/x-llamapun" id="S3.I1.i1.p1.4.m4.1d">italic_τ start_POSTSUPERSCRIPT 2 ( italic_t + 1 ) end_POSTSUPERSCRIPT</annotation></semantics></math> by solving (<a class="ltx_ref" href="https://arxiv.org/html/2503.13782v1#S3.E8" title="In 3 Regularized Parameter Estimation ‣ Regularized Parameter Estimation in Mixed Model Trace Regression"><span class="ltx_text ltx_ref_tag">8</span></a>). Update <math alttext="\operatorname{{\bm{\theta}}}^{{(t)}}" class="ltx_Math" display="inline" id="S3.I1.i1.p1.5.m5.1"><semantics id="S3.I1.i1.p1.5.m5.1a"><msup id="S3.I1.i1.p1.5.m5.1.2" xref="S3.I1.i1.p1.5.m5.1.2.cmml"><mi id="S3.I1.i1.p1.5.m5.1.2.2" xref="S3.I1.i1.p1.5.m5.1.2.2.cmml">𝜽</mi><mrow id="S3.I1.i1.p1.5.m5.1.1.1.3" xref="S3.I1.i1.p1.5.m5.1.2.cmml"><mo id="S3.I1.i1.p1.5.m5.1.1.1.3.1" stretchy="false" xref="S3.I1.i1.p1.5.m5.1.2.cmml">(</mo><mi id="S3.I1.i1.p1.5.m5.1.1.1.1" xref="S3.I1.i1.p1.5.m5.1.1.1.1.cmml">t</mi><mo id="S3.I1.i1.p1.5.m5.1.1.1.3.2" stretchy="false" xref="S3.I1.i1.p1.5.m5.1.2.cmml">)</mo></mrow></msup><annotation-xml encoding="MathML-Content" id="S3.I1.i1.p1.5.m5.1b"><apply id="S3.I1.i1.p1.5.m5.1.2.cmml" xref="S3.I1.i1.p1.5.m5.1.2"><csymbol cd="ambiguous" id="S3.I1.i1.p1.5.m5.1.2.1.cmml" xref="S3.I1.i1.p1.5.m5.1.2">superscript</csymbol><ci id="S3.I1.i1.p1.5.m5.1.2.2.cmml" xref="S3.I1.i1.p1.5.m5.1.2.2">𝜽</ci><ci id="S3.I1.i1.p1.5.m5.1.1.1.1.cmml" xref="S3.I1.i1.p1.5.m5.1.1.1.1">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I1.i1.p1.5.m5.1c">\operatorname{{\bm{\theta}}}^{{(t)}}</annotation><annotation encoding="application/x-llamapun" id="S3.I1.i1.p1.5.m5.1d">bold_italic_θ start_POSTSUPERSCRIPT ( italic_t ) end_POSTSUPERSCRIPT</annotation></semantics></math> to <math alttext="\operatorname{{\bm{\theta}}}^{(t+1/3)}=(\operatorname{\mathbf{b}}^{(t+1)},% \operatorname{\mathbf{L}}_{1}^{(t)},\operatorname{\mathbf{L}}_{2}^{(t)},\tau^{% 2(t+1)})" class="ltx_Math" display="inline" id="S3.I1.i1.p1.6.m6.9"><semantics id="S3.I1.i1.p1.6.m6.9a"><mrow id="S3.I1.i1.p1.6.m6.9.9" xref="S3.I1.i1.p1.6.m6.9.9.cmml"><msup id="S3.I1.i1.p1.6.m6.9.9.6" xref="S3.I1.i1.p1.6.m6.9.9.6.cmml"><mi id="S3.I1.i1.p1.6.m6.9.9.6.2" xref="S3.I1.i1.p1.6.m6.9.9.6.2.cmml">𝜽</mi><mrow id="S3.I1.i1.p1.6.m6.1.1.1.1" xref="S3.I1.i1.p1.6.m6.1.1.1.1.1.cmml"><mo id="S3.I1.i1.p1.6.m6.1.1.1.1.2" stretchy="false" xref="S3.I1.i1.p1.6.m6.1.1.1.1.1.cmml">(</mo><mrow id="S3.I1.i1.p1.6.m6.1.1.1.1.1" xref="S3.I1.i1.p1.6.m6.1.1.1.1.1.cmml"><mi id="S3.I1.i1.p1.6.m6.1.1.1.1.1.2" xref="S3.I1.i1.p1.6.m6.1.1.1.1.1.2.cmml">t</mi><mo id="S3.I1.i1.p1.6.m6.1.1.1.1.1.1" xref="S3.I1.i1.p1.6.m6.1.1.1.1.1.1.cmml">+</mo><mrow id="S3.I1.i1.p1.6.m6.1.1.1.1.1.3" xref="S3.I1.i1.p1.6.m6.1.1.1.1.1.3.cmml"><mn id="S3.I1.i1.p1.6.m6.1.1.1.1.1.3.2" xref="S3.I1.i1.p1.6.m6.1.1.1.1.1.3.2.cmml">1</mn><mo id="S3.I1.i1.p1.6.m6.1.1.1.1.1.3.1" xref="S3.I1.i1.p1.6.m6.1.1.1.1.1.3.1.cmml">/</mo><mn id="S3.I1.i1.p1.6.m6.1.1.1.1.1.3.3" xref="S3.I1.i1.p1.6.m6.1.1.1.1.1.3.3.cmml">3</mn></mrow></mrow><mo id="S3.I1.i1.p1.6.m6.1.1.1.1.3" stretchy="false" xref="S3.I1.i1.p1.6.m6.1.1.1.1.1.cmml">)</mo></mrow></msup><mo id="S3.I1.i1.p1.6.m6.9.9.5" xref="S3.I1.i1.p1.6.m6.9.9.5.cmml">=</mo><mrow id="S3.I1.i1.p1.6.m6.9.9.4.4" xref="S3.I1.i1.p1.6.m6.9.9.4.5.cmml"><mo id="S3.I1.i1.p1.6.m6.9.9.4.4.5" stretchy="false" 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id="S3.I2.i1.p1.2"><span class="ltx_text ltx_font_bold" id="S3.I2.i1.p1.2.1">E step:</span> Compute <math alttext="\operatorname{\mathbf{H}}_{(1)}^{{(t)}}" class="ltx_Math" display="inline" id="S3.I2.i1.p1.1.m1.2"><semantics id="S3.I2.i1.p1.1.m1.2a"><msubsup id="S3.I2.i1.p1.1.m1.2.3" xref="S3.I2.i1.p1.1.m1.2.3.cmml"><mi id="S3.I2.i1.p1.1.m1.2.3.2.2" xref="S3.I2.i1.p1.1.m1.2.3.2.2.cmml">𝐇</mi><mrow id="S3.I2.i1.p1.1.m1.1.1.1.3" xref="S3.I2.i1.p1.1.m1.2.3.cmml"><mo id="S3.I2.i1.p1.1.m1.1.1.1.3.1" stretchy="false" xref="S3.I2.i1.p1.1.m1.2.3.cmml">(</mo><mn id="S3.I2.i1.p1.1.m1.1.1.1.1" xref="S3.I2.i1.p1.1.m1.1.1.1.1.cmml">1</mn><mo id="S3.I2.i1.p1.1.m1.1.1.1.3.2" stretchy="false" xref="S3.I2.i1.p1.1.m1.2.3.cmml">)</mo></mrow><mrow id="S3.I2.i1.p1.1.m1.2.2.1.3" xref="S3.I2.i1.p1.1.m1.2.3.cmml"><mo id="S3.I2.i1.p1.1.m1.2.2.1.3.1" stretchy="false" xref="S3.I2.i1.p1.1.m1.2.3.cmml">(</mo><mi id="S3.I2.i1.p1.1.m1.2.2.1.1" xref="S3.I2.i1.p1.1.m1.2.2.1.1.cmml">t</mi><mo id="S3.I2.i1.p1.1.m1.2.2.1.3.2" stretchy="false" xref="S3.I2.i1.p1.1.m1.2.3.cmml">)</mo></mrow></msubsup><annotation-xml encoding="MathML-Content" id="S3.I2.i1.p1.1.m1.2b"><apply id="S3.I2.i1.p1.1.m1.2.3.cmml" xref="S3.I2.i1.p1.1.m1.2.3"><csymbol cd="ambiguous" id="S3.I2.i1.p1.1.m1.2.3.1.cmml" xref="S3.I2.i1.p1.1.m1.2.3">superscript</csymbol><apply id="S3.I2.i1.p1.1.m1.2.3.2.cmml" xref="S3.I2.i1.p1.1.m1.2.3"><csymbol cd="ambiguous" id="S3.I2.i1.p1.1.m1.2.3.2.1.cmml" xref="S3.I2.i1.p1.1.m1.2.3">subscript</csymbol><ci id="S3.I2.i1.p1.1.m1.2.3.2.2.cmml" xref="S3.I2.i1.p1.1.m1.2.3.2.2">𝐇</ci><cn id="S3.I2.i1.p1.1.m1.1.1.1.1.cmml" type="integer" xref="S3.I2.i1.p1.1.m1.1.1.1.1">1</cn></apply><ci id="S3.I2.i1.p1.1.m1.2.2.1.1.cmml" xref="S3.I2.i1.p1.1.m1.2.2.1.1">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I2.i1.p1.1.m1.2c">\operatorname{\mathbf{H}}_{(1)}^{{(t)}}</annotation><annotation encoding="application/x-llamapun" id="S3.I2.i1.p1.1.m1.2d">bold_H start_POSTSUBSCRIPT ( 1 ) end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_t ) end_POSTSUPERSCRIPT</annotation></semantics></math> and <math alttext="\operatorname{\mathbf{g}}_{(1)}^{(t)}" class="ltx_Math" display="inline" id="S3.I2.i1.p1.2.m2.2"><semantics id="S3.I2.i1.p1.2.m2.2a"><msubsup id="S3.I2.i1.p1.2.m2.2.3" xref="S3.I2.i1.p1.2.m2.2.3.cmml"><mi id="S3.I2.i1.p1.2.m2.2.3.2.2" xref="S3.I2.i1.p1.2.m2.2.3.2.2.cmml">𝐠</mi><mrow id="S3.I2.i1.p1.2.m2.1.1.1.3" xref="S3.I2.i1.p1.2.m2.2.3.cmml"><mo id="S3.I2.i1.p1.2.m2.1.1.1.3.1" stretchy="false" xref="S3.I2.i1.p1.2.m2.2.3.cmml">(</mo><mn id="S3.I2.i1.p1.2.m2.1.1.1.1" xref="S3.I2.i1.p1.2.m2.1.1.1.1.cmml">1</mn><mo id="S3.I2.i1.p1.2.m2.1.1.1.3.2" stretchy="false" xref="S3.I2.i1.p1.2.m2.2.3.cmml">)</mo></mrow><mrow id="S3.I2.i1.p1.2.m2.2.2.1.3" xref="S3.I2.i1.p1.2.m2.2.3.cmml"><mo id="S3.I2.i1.p1.2.m2.2.2.1.3.1" stretchy="false" xref="S3.I2.i1.p1.2.m2.2.3.cmml">(</mo><mi id="S3.I2.i1.p1.2.m2.2.2.1.1" xref="S3.I2.i1.p1.2.m2.2.2.1.1.cmml">t</mi><mo id="S3.I2.i1.p1.2.m2.2.2.1.3.2" stretchy="false" xref="S3.I2.i1.p1.2.m2.2.3.cmml">)</mo></mrow></msubsup><annotation-xml encoding="MathML-Content" id="S3.I2.i1.p1.2.m2.2b"><apply id="S3.I2.i1.p1.2.m2.2.3.cmml" xref="S3.I2.i1.p1.2.m2.2.3"><csymbol cd="ambiguous" id="S3.I2.i1.p1.2.m2.2.3.1.cmml" xref="S3.I2.i1.p1.2.m2.2.3">superscript</csymbol><apply id="S3.I2.i1.p1.2.m2.2.3.2.cmml" xref="S3.I2.i1.p1.2.m2.2.3"><csymbol cd="ambiguous" id="S3.I2.i1.p1.2.m2.2.3.2.1.cmml" xref="S3.I2.i1.p1.2.m2.2.3">subscript</csymbol><ci id="S3.I2.i1.p1.2.m2.2.3.2.2.cmml" xref="S3.I2.i1.p1.2.m2.2.3.2.2">𝐠</ci><cn id="S3.I2.i1.p1.2.m2.1.1.1.1.cmml" type="integer" xref="S3.I2.i1.p1.2.m2.1.1.1.1">1</cn></apply><ci id="S3.I2.i1.p1.2.m2.2.2.1.1.cmml" xref="S3.I2.i1.p1.2.m2.2.2.1.1">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I2.i1.p1.2.m2.2c">\operatorname{\mathbf{g}}_{(1)}^{(t)}</annotation><annotation encoding="application/x-llamapun" id="S3.I2.i1.p1.2.m2.2d">bold_g start_POSTSUBSCRIPT ( 1 ) end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_t ) end_POSTSUPERSCRIPT</annotation></semantics></math> in (<a class="ltx_ref" href="https://arxiv.org/html/2503.13782v1#S3.E9" title="In 3 Regularized Parameter Estimation ‣ Regularized Parameter Estimation in Mixed Model Trace Regression"><span class="ltx_text ltx_ref_tag">9</span></a>).</p> </div> </li> <li class="ltx_item" id="S3.I2.i2" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">•</span> <div class="ltx_para" id="S3.I2.i2.p1"> <p class="ltx_p" id="S3.I2.i2.p1.4"><span class="ltx_text ltx_font_bold" id="S3.I2.i2.p1.4.1">CM step:</span> Given <math alttext="{\operatorname{{\bm{\theta}}}}^{(t+1/3)}" class="ltx_Math" display="inline" id="S3.I2.i2.p1.1.m1.1"><semantics id="S3.I2.i2.p1.1.m1.1a"><msup id="S3.I2.i2.p1.1.m1.1.2" xref="S3.I2.i2.p1.1.m1.1.2.cmml"><mi id="S3.I2.i2.p1.1.m1.1.2.2" xref="S3.I2.i2.p1.1.m1.1.2.2.cmml">𝜽</mi><mrow id="S3.I2.i2.p1.1.m1.1.1.1.1" xref="S3.I2.i2.p1.1.m1.1.1.1.1.1.cmml"><mo id="S3.I2.i2.p1.1.m1.1.1.1.1.2" stretchy="false" xref="S3.I2.i2.p1.1.m1.1.1.1.1.1.cmml">(</mo><mrow id="S3.I2.i2.p1.1.m1.1.1.1.1.1" xref="S3.I2.i2.p1.1.m1.1.1.1.1.1.cmml"><mi id="S3.I2.i2.p1.1.m1.1.1.1.1.1.2" xref="S3.I2.i2.p1.1.m1.1.1.1.1.1.2.cmml">t</mi><mo id="S3.I2.i2.p1.1.m1.1.1.1.1.1.1" xref="S3.I2.i2.p1.1.m1.1.1.1.1.1.1.cmml">+</mo><mrow id="S3.I2.i2.p1.1.m1.1.1.1.1.1.3" xref="S3.I2.i2.p1.1.m1.1.1.1.1.1.3.cmml"><mn id="S3.I2.i2.p1.1.m1.1.1.1.1.1.3.2" xref="S3.I2.i2.p1.1.m1.1.1.1.1.1.3.2.cmml">1</mn><mo id="S3.I2.i2.p1.1.m1.1.1.1.1.1.3.1" xref="S3.I2.i2.p1.1.m1.1.1.1.1.1.3.1.cmml">/</mo><mn id="S3.I2.i2.p1.1.m1.1.1.1.1.1.3.3" xref="S3.I2.i2.p1.1.m1.1.1.1.1.1.3.3.cmml">3</mn></mrow></mrow><mo id="S3.I2.i2.p1.1.m1.1.1.1.1.3" stretchy="false" xref="S3.I2.i2.p1.1.m1.1.1.1.1.1.cmml">)</mo></mrow></msup><annotation-xml encoding="MathML-Content" id="S3.I2.i2.p1.1.m1.1b"><apply id="S3.I2.i2.p1.1.m1.1.2.cmml" xref="S3.I2.i2.p1.1.m1.1.2"><csymbol cd="ambiguous" id="S3.I2.i2.p1.1.m1.1.2.1.cmml" xref="S3.I2.i2.p1.1.m1.1.2">superscript</csymbol><ci id="S3.I2.i2.p1.1.m1.1.2.2.cmml" xref="S3.I2.i2.p1.1.m1.1.2.2">𝜽</ci><apply id="S3.I2.i2.p1.1.m1.1.1.1.1.1.cmml" xref="S3.I2.i2.p1.1.m1.1.1.1.1"><plus id="S3.I2.i2.p1.1.m1.1.1.1.1.1.1.cmml" xref="S3.I2.i2.p1.1.m1.1.1.1.1.1.1"></plus><ci id="S3.I2.i2.p1.1.m1.1.1.1.1.1.2.cmml" xref="S3.I2.i2.p1.1.m1.1.1.1.1.1.2">𝑡</ci><apply id="S3.I2.i2.p1.1.m1.1.1.1.1.1.3.cmml" xref="S3.I2.i2.p1.1.m1.1.1.1.1.1.3"><divide id="S3.I2.i2.p1.1.m1.1.1.1.1.1.3.1.cmml" xref="S3.I2.i2.p1.1.m1.1.1.1.1.1.3.1"></divide><cn id="S3.I2.i2.p1.1.m1.1.1.1.1.1.3.2.cmml" type="integer" xref="S3.I2.i2.p1.1.m1.1.1.1.1.1.3.2">1</cn><cn id="S3.I2.i2.p1.1.m1.1.1.1.1.1.3.3.cmml" type="integer" xref="S3.I2.i2.p1.1.m1.1.1.1.1.1.3.3">3</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I2.i2.p1.1.m1.1c">{\operatorname{{\bm{\theta}}}}^{(t+1/3)}</annotation><annotation encoding="application/x-llamapun" id="S3.I2.i2.p1.1.m1.1d">bold_italic_θ start_POSTSUPERSCRIPT ( italic_t + 1 / 3 ) end_POSTSUPERSCRIPT</annotation></semantics></math>, compute <math alttext="\operatorname{\mathbf{L}}_{1}^{{(t+1)}}" class="ltx_Math" display="inline" id="S3.I2.i2.p1.2.m2.1"><semantics id="S3.I2.i2.p1.2.m2.1a"><msubsup id="S3.I2.i2.p1.2.m2.1.2" xref="S3.I2.i2.p1.2.m2.1.2.cmml"><mi id="S3.I2.i2.p1.2.m2.1.2.2.2" xref="S3.I2.i2.p1.2.m2.1.2.2.2.cmml">𝐋</mi><mn id="S3.I2.i2.p1.2.m2.1.2.2.3" xref="S3.I2.i2.p1.2.m2.1.2.2.3.cmml">1</mn><mrow id="S3.I2.i2.p1.2.m2.1.1.1.1" xref="S3.I2.i2.p1.2.m2.1.1.1.1.1.cmml"><mo id="S3.I2.i2.p1.2.m2.1.1.1.1.2" stretchy="false" xref="S3.I2.i2.p1.2.m2.1.1.1.1.1.cmml">(</mo><mrow id="S3.I2.i2.p1.2.m2.1.1.1.1.1" xref="S3.I2.i2.p1.2.m2.1.1.1.1.1.cmml"><mi id="S3.I2.i2.p1.2.m2.1.1.1.1.1.2" xref="S3.I2.i2.p1.2.m2.1.1.1.1.1.2.cmml">t</mi><mo id="S3.I2.i2.p1.2.m2.1.1.1.1.1.1" xref="S3.I2.i2.p1.2.m2.1.1.1.1.1.1.cmml">+</mo><mn id="S3.I2.i2.p1.2.m2.1.1.1.1.1.3" xref="S3.I2.i2.p1.2.m2.1.1.1.1.1.3.cmml">1</mn></mrow><mo id="S3.I2.i2.p1.2.m2.1.1.1.1.3" stretchy="false" xref="S3.I2.i2.p1.2.m2.1.1.1.1.1.cmml">)</mo></mrow></msubsup><annotation-xml encoding="MathML-Content" id="S3.I2.i2.p1.2.m2.1b"><apply id="S3.I2.i2.p1.2.m2.1.2.cmml" xref="S3.I2.i2.p1.2.m2.1.2"><csymbol cd="ambiguous" id="S3.I2.i2.p1.2.m2.1.2.1.cmml" xref="S3.I2.i2.p1.2.m2.1.2">superscript</csymbol><apply id="S3.I2.i2.p1.2.m2.1.2.2.cmml" xref="S3.I2.i2.p1.2.m2.1.2"><csymbol cd="ambiguous" id="S3.I2.i2.p1.2.m2.1.2.2.1.cmml" xref="S3.I2.i2.p1.2.m2.1.2">subscript</csymbol><ci id="S3.I2.i2.p1.2.m2.1.2.2.2.cmml" xref="S3.I2.i2.p1.2.m2.1.2.2.2">𝐋</ci><cn id="S3.I2.i2.p1.2.m2.1.2.2.3.cmml" type="integer" xref="S3.I2.i2.p1.2.m2.1.2.2.3">1</cn></apply><apply id="S3.I2.i2.p1.2.m2.1.1.1.1.1.cmml" xref="S3.I2.i2.p1.2.m2.1.1.1.1"><plus id="S3.I2.i2.p1.2.m2.1.1.1.1.1.1.cmml" xref="S3.I2.i2.p1.2.m2.1.1.1.1.1.1"></plus><ci id="S3.I2.i2.p1.2.m2.1.1.1.1.1.2.cmml" xref="S3.I2.i2.p1.2.m2.1.1.1.1.1.2">𝑡</ci><cn id="S3.I2.i2.p1.2.m2.1.1.1.1.1.3.cmml" type="integer" xref="S3.I2.i2.p1.2.m2.1.1.1.1.1.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I2.i2.p1.2.m2.1c">\operatorname{\mathbf{L}}_{1}^{{(t+1)}}</annotation><annotation encoding="application/x-llamapun" id="S3.I2.i2.p1.2.m2.1d">bold_L start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_t + 1 ) end_POSTSUPERSCRIPT</annotation></semantics></math> by solving (<a class="ltx_ref" href="https://arxiv.org/html/2503.13782v1#S3.E10" title="In 3 Regularized Parameter Estimation ‣ Regularized Parameter Estimation in Mixed Model Trace Regression"><span class="ltx_text ltx_ref_tag">10</span></a>). Update <math alttext="\operatorname{{\bm{\theta}}}^{(t+1/3)}" class="ltx_Math" display="inline" id="S3.I2.i2.p1.3.m3.1"><semantics id="S3.I2.i2.p1.3.m3.1a"><msup id="S3.I2.i2.p1.3.m3.1.2" xref="S3.I2.i2.p1.3.m3.1.2.cmml"><mi id="S3.I2.i2.p1.3.m3.1.2.2" xref="S3.I2.i2.p1.3.m3.1.2.2.cmml">𝜽</mi><mrow id="S3.I2.i2.p1.3.m3.1.1.1.1" xref="S3.I2.i2.p1.3.m3.1.1.1.1.1.cmml"><mo id="S3.I2.i2.p1.3.m3.1.1.1.1.2" stretchy="false" xref="S3.I2.i2.p1.3.m3.1.1.1.1.1.cmml">(</mo><mrow id="S3.I2.i2.p1.3.m3.1.1.1.1.1" xref="S3.I2.i2.p1.3.m3.1.1.1.1.1.cmml"><mi id="S3.I2.i2.p1.3.m3.1.1.1.1.1.2" xref="S3.I2.i2.p1.3.m3.1.1.1.1.1.2.cmml">t</mi><mo id="S3.I2.i2.p1.3.m3.1.1.1.1.1.1" xref="S3.I2.i2.p1.3.m3.1.1.1.1.1.1.cmml">+</mo><mrow id="S3.I2.i2.p1.3.m3.1.1.1.1.1.3" xref="S3.I2.i2.p1.3.m3.1.1.1.1.1.3.cmml"><mn id="S3.I2.i2.p1.3.m3.1.1.1.1.1.3.2" xref="S3.I2.i2.p1.3.m3.1.1.1.1.1.3.2.cmml">1</mn><mo id="S3.I2.i2.p1.3.m3.1.1.1.1.1.3.1" xref="S3.I2.i2.p1.3.m3.1.1.1.1.1.3.1.cmml">/</mo><mn id="S3.I2.i2.p1.3.m3.1.1.1.1.1.3.3" xref="S3.I2.i2.p1.3.m3.1.1.1.1.1.3.3.cmml">3</mn></mrow></mrow><mo id="S3.I2.i2.p1.3.m3.1.1.1.1.3" stretchy="false" xref="S3.I2.i2.p1.3.m3.1.1.1.1.1.cmml">)</mo></mrow></msup><annotation-xml encoding="MathML-Content" id="S3.I2.i2.p1.3.m3.1b"><apply id="S3.I2.i2.p1.3.m3.1.2.cmml" xref="S3.I2.i2.p1.3.m3.1.2"><csymbol cd="ambiguous" id="S3.I2.i2.p1.3.m3.1.2.1.cmml" xref="S3.I2.i2.p1.3.m3.1.2">superscript</csymbol><ci id="S3.I2.i2.p1.3.m3.1.2.2.cmml" xref="S3.I2.i2.p1.3.m3.1.2.2">𝜽</ci><apply id="S3.I2.i2.p1.3.m3.1.1.1.1.1.cmml" xref="S3.I2.i2.p1.3.m3.1.1.1.1"><plus id="S3.I2.i2.p1.3.m3.1.1.1.1.1.1.cmml" xref="S3.I2.i2.p1.3.m3.1.1.1.1.1.1"></plus><ci id="S3.I2.i2.p1.3.m3.1.1.1.1.1.2.cmml" xref="S3.I2.i2.p1.3.m3.1.1.1.1.1.2">𝑡</ci><apply id="S3.I2.i2.p1.3.m3.1.1.1.1.1.3.cmml" xref="S3.I2.i2.p1.3.m3.1.1.1.1.1.3"><divide id="S3.I2.i2.p1.3.m3.1.1.1.1.1.3.1.cmml" xref="S3.I2.i2.p1.3.m3.1.1.1.1.1.3.1"></divide><cn id="S3.I2.i2.p1.3.m3.1.1.1.1.1.3.2.cmml" type="integer" xref="S3.I2.i2.p1.3.m3.1.1.1.1.1.3.2">1</cn><cn id="S3.I2.i2.p1.3.m3.1.1.1.1.1.3.3.cmml" type="integer" xref="S3.I2.i2.p1.3.m3.1.1.1.1.1.3.3">3</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I2.i2.p1.3.m3.1c">\operatorname{{\bm{\theta}}}^{(t+1/3)}</annotation><annotation encoding="application/x-llamapun" id="S3.I2.i2.p1.3.m3.1d">bold_italic_θ start_POSTSUPERSCRIPT ( italic_t + 1 / 3 ) end_POSTSUPERSCRIPT</annotation></semantics></math> to <math alttext="\operatorname{{\bm{\theta}}}^{(t+2/3)}=(\operatorname{\mathbf{b}}^{(t+1)},% \operatorname{\mathbf{L}}_{1}^{(t+1)},\operatorname{\mathbf{L}}_{2}^{(t)},\tau% ^{2(t+1)})" class="ltx_Math" display="inline" id="S3.I2.i2.p1.4.m4.9"><semantics id="S3.I2.i2.p1.4.m4.9a"><mrow id="S3.I2.i2.p1.4.m4.9.9" xref="S3.I2.i2.p1.4.m4.9.9.cmml"><msup id="S3.I2.i2.p1.4.m4.9.9.6" xref="S3.I2.i2.p1.4.m4.9.9.6.cmml"><mi id="S3.I2.i2.p1.4.m4.9.9.6.2" xref="S3.I2.i2.p1.4.m4.9.9.6.2.cmml">𝜽</mi><mrow 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xref="S3.I2.i2.p1.4.m4.8.8.3.3.3"><csymbol cd="ambiguous" id="S3.I2.i2.p1.4.m4.8.8.3.3.3.2.1.cmml" xref="S3.I2.i2.p1.4.m4.8.8.3.3.3">subscript</csymbol><ci id="S3.I2.i2.p1.4.m4.8.8.3.3.3.2.2.cmml" xref="S3.I2.i2.p1.4.m4.8.8.3.3.3.2.2">𝐋</ci><cn id="S3.I2.i2.p1.4.m4.8.8.3.3.3.2.3.cmml" type="integer" xref="S3.I2.i2.p1.4.m4.8.8.3.3.3.2.3">2</cn></apply><ci id="S3.I2.i2.p1.4.m4.4.4.1.1.cmml" xref="S3.I2.i2.p1.4.m4.4.4.1.1">𝑡</ci></apply><apply id="S3.I2.i2.p1.4.m4.9.9.4.4.4.cmml" xref="S3.I2.i2.p1.4.m4.9.9.4.4.4"><csymbol cd="ambiguous" id="S3.I2.i2.p1.4.m4.9.9.4.4.4.1.cmml" xref="S3.I2.i2.p1.4.m4.9.9.4.4.4">superscript</csymbol><ci id="S3.I2.i2.p1.4.m4.9.9.4.4.4.2.cmml" xref="S3.I2.i2.p1.4.m4.9.9.4.4.4.2">𝜏</ci><apply id="S3.I2.i2.p1.4.m4.5.5.1.cmml" xref="S3.I2.i2.p1.4.m4.5.5.1"><times id="S3.I2.i2.p1.4.m4.5.5.1.2.cmml" xref="S3.I2.i2.p1.4.m4.5.5.1.2"></times><cn id="S3.I2.i2.p1.4.m4.5.5.1.3.cmml" type="integer" xref="S3.I2.i2.p1.4.m4.5.5.1.3">2</cn><apply id="S3.I2.i2.p1.4.m4.5.5.1.1.1.1.cmml" xref="S3.I2.i2.p1.4.m4.5.5.1.1.1"><plus id="S3.I2.i2.p1.4.m4.5.5.1.1.1.1.1.cmml" xref="S3.I2.i2.p1.4.m4.5.5.1.1.1.1.1"></plus><ci id="S3.I2.i2.p1.4.m4.5.5.1.1.1.1.2.cmml" xref="S3.I2.i2.p1.4.m4.5.5.1.1.1.1.2">𝑡</ci><cn id="S3.I2.i2.p1.4.m4.5.5.1.1.1.1.3.cmml" type="integer" xref="S3.I2.i2.p1.4.m4.5.5.1.1.1.1.3">1</cn></apply></apply></apply></vector></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I2.i2.p1.4.m4.9c">\operatorname{{\bm{\theta}}}^{(t+2/3)}=(\operatorname{\mathbf{b}}^{(t+1)},% \operatorname{\mathbf{L}}_{1}^{(t+1)},\operatorname{\mathbf{L}}_{2}^{(t)},\tau% ^{2(t+1)})</annotation><annotation encoding="application/x-llamapun" id="S3.I2.i2.p1.4.m4.9d">bold_italic_θ start_POSTSUPERSCRIPT ( italic_t + 2 / 3 ) end_POSTSUPERSCRIPT = ( bold_b start_POSTSUPERSCRIPT ( italic_t + 1 ) end_POSTSUPERSCRIPT , bold_L start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_t + 1 ) end_POSTSUPERSCRIPT , bold_L start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_t ) end_POSTSUPERSCRIPT , italic_τ start_POSTSUPERSCRIPT 2 ( italic_t + 1 ) end_POSTSUPERSCRIPT )</annotation></semantics></math>.</p> </div> </li> </ul> <span class="ltx_text ltx_font_bold" id="alg1.2.2.2">Third cycle:</span> <ul class="ltx_itemize" id="S3.I3"> <li class="ltx_item" id="S3.I3.i1" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">•</span> <div class="ltx_para" id="S3.I3.i1.p1"> <p class="ltx_p" id="S3.I3.i1.p1.2"><span class="ltx_text ltx_font_bold" id="S3.I3.i1.p1.2.1">E step:</span> Compute <math alttext="\operatorname{\mathbf{H}}_{(2)}^{{(t)}}" class="ltx_Math" display="inline" id="S3.I3.i1.p1.1.m1.2"><semantics id="S3.I3.i1.p1.1.m1.2a"><msubsup id="S3.I3.i1.p1.1.m1.2.3" xref="S3.I3.i1.p1.1.m1.2.3.cmml"><mi id="S3.I3.i1.p1.1.m1.2.3.2.2" xref="S3.I3.i1.p1.1.m1.2.3.2.2.cmml">𝐇</mi><mrow id="S3.I3.i1.p1.1.m1.1.1.1.3" xref="S3.I3.i1.p1.1.m1.2.3.cmml"><mo id="S3.I3.i1.p1.1.m1.1.1.1.3.1" stretchy="false" xref="S3.I3.i1.p1.1.m1.2.3.cmml">(</mo><mn id="S3.I3.i1.p1.1.m1.1.1.1.1" xref="S3.I3.i1.p1.1.m1.1.1.1.1.cmml">2</mn><mo id="S3.I3.i1.p1.1.m1.1.1.1.3.2" stretchy="false" xref="S3.I3.i1.p1.1.m1.2.3.cmml">)</mo></mrow><mrow id="S3.I3.i1.p1.1.m1.2.2.1.3" xref="S3.I3.i1.p1.1.m1.2.3.cmml"><mo id="S3.I3.i1.p1.1.m1.2.2.1.3.1" stretchy="false" xref="S3.I3.i1.p1.1.m1.2.3.cmml">(</mo><mi id="S3.I3.i1.p1.1.m1.2.2.1.1" xref="S3.I3.i1.p1.1.m1.2.2.1.1.cmml">t</mi><mo id="S3.I3.i1.p1.1.m1.2.2.1.3.2" stretchy="false" xref="S3.I3.i1.p1.1.m1.2.3.cmml">)</mo></mrow></msubsup><annotation-xml encoding="MathML-Content" id="S3.I3.i1.p1.1.m1.2b"><apply id="S3.I3.i1.p1.1.m1.2.3.cmml" xref="S3.I3.i1.p1.1.m1.2.3"><csymbol cd="ambiguous" id="S3.I3.i1.p1.1.m1.2.3.1.cmml" xref="S3.I3.i1.p1.1.m1.2.3">superscript</csymbol><apply id="S3.I3.i1.p1.1.m1.2.3.2.cmml" xref="S3.I3.i1.p1.1.m1.2.3"><csymbol cd="ambiguous" id="S3.I3.i1.p1.1.m1.2.3.2.1.cmml" xref="S3.I3.i1.p1.1.m1.2.3">subscript</csymbol><ci id="S3.I3.i1.p1.1.m1.2.3.2.2.cmml" xref="S3.I3.i1.p1.1.m1.2.3.2.2">𝐇</ci><cn id="S3.I3.i1.p1.1.m1.1.1.1.1.cmml" type="integer" xref="S3.I3.i1.p1.1.m1.1.1.1.1">2</cn></apply><ci id="S3.I3.i1.p1.1.m1.2.2.1.1.cmml" xref="S3.I3.i1.p1.1.m1.2.2.1.1">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I3.i1.p1.1.m1.2c">\operatorname{\mathbf{H}}_{(2)}^{{(t)}}</annotation><annotation encoding="application/x-llamapun" id="S3.I3.i1.p1.1.m1.2d">bold_H start_POSTSUBSCRIPT ( 2 ) end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_t ) end_POSTSUPERSCRIPT</annotation></semantics></math> and <math alttext="\operatorname{\mathbf{g}}_{(2)}^{(t)}" class="ltx_Math" display="inline" id="S3.I3.i1.p1.2.m2.2"><semantics id="S3.I3.i1.p1.2.m2.2a"><msubsup id="S3.I3.i1.p1.2.m2.2.3" xref="S3.I3.i1.p1.2.m2.2.3.cmml"><mi id="S3.I3.i1.p1.2.m2.2.3.2.2" xref="S3.I3.i1.p1.2.m2.2.3.2.2.cmml">𝐠</mi><mrow id="S3.I3.i1.p1.2.m2.1.1.1.3" xref="S3.I3.i1.p1.2.m2.2.3.cmml"><mo id="S3.I3.i1.p1.2.m2.1.1.1.3.1" stretchy="false" xref="S3.I3.i1.p1.2.m2.2.3.cmml">(</mo><mn id="S3.I3.i1.p1.2.m2.1.1.1.1" xref="S3.I3.i1.p1.2.m2.1.1.1.1.cmml">2</mn><mo id="S3.I3.i1.p1.2.m2.1.1.1.3.2" stretchy="false" xref="S3.I3.i1.p1.2.m2.2.3.cmml">)</mo></mrow><mrow id="S3.I3.i1.p1.2.m2.2.2.1.3" xref="S3.I3.i1.p1.2.m2.2.3.cmml"><mo id="S3.I3.i1.p1.2.m2.2.2.1.3.1" stretchy="false" xref="S3.I3.i1.p1.2.m2.2.3.cmml">(</mo><mi id="S3.I3.i1.p1.2.m2.2.2.1.1" xref="S3.I3.i1.p1.2.m2.2.2.1.1.cmml">t</mi><mo id="S3.I3.i1.p1.2.m2.2.2.1.3.2" stretchy="false" xref="S3.I3.i1.p1.2.m2.2.3.cmml">)</mo></mrow></msubsup><annotation-xml encoding="MathML-Content" id="S3.I3.i1.p1.2.m2.2b"><apply id="S3.I3.i1.p1.2.m2.2.3.cmml" xref="S3.I3.i1.p1.2.m2.2.3"><csymbol cd="ambiguous" id="S3.I3.i1.p1.2.m2.2.3.1.cmml" xref="S3.I3.i1.p1.2.m2.2.3">superscript</csymbol><apply id="S3.I3.i1.p1.2.m2.2.3.2.cmml" xref="S3.I3.i1.p1.2.m2.2.3"><csymbol cd="ambiguous" id="S3.I3.i1.p1.2.m2.2.3.2.1.cmml" xref="S3.I3.i1.p1.2.m2.2.3">subscript</csymbol><ci id="S3.I3.i1.p1.2.m2.2.3.2.2.cmml" xref="S3.I3.i1.p1.2.m2.2.3.2.2">𝐠</ci><cn id="S3.I3.i1.p1.2.m2.1.1.1.1.cmml" type="integer" xref="S3.I3.i1.p1.2.m2.1.1.1.1">2</cn></apply><ci id="S3.I3.i1.p1.2.m2.2.2.1.1.cmml" xref="S3.I3.i1.p1.2.m2.2.2.1.1">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I3.i1.p1.2.m2.2c">\operatorname{\mathbf{g}}_{(2)}^{(t)}</annotation><annotation encoding="application/x-llamapun" id="S3.I3.i1.p1.2.m2.2d">bold_g start_POSTSUBSCRIPT ( 2 ) end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_t ) end_POSTSUPERSCRIPT</annotation></semantics></math> by swapping 1 and 2 in (<a class="ltx_ref" href="https://arxiv.org/html/2503.13782v1#S3.E9" title="In 3 Regularized Parameter Estimation ‣ Regularized Parameter Estimation in Mixed Model Trace Regression"><span class="ltx_text ltx_ref_tag">9</span></a>).</p> </div> </li> <li class="ltx_item" id="S3.I3.i2" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">•</span> <div class="ltx_para" id="S3.I3.i2.p1"> <p class="ltx_p" id="S3.I3.i2.p1.4"><span class="ltx_text ltx_font_bold" id="S3.I3.i2.p1.4.1">CM step:</span> Given <math alttext="{\operatorname{{\bm{\theta}}}}^{(t+2/3)}" class="ltx_Math" display="inline" id="S3.I3.i2.p1.1.m1.1"><semantics id="S3.I3.i2.p1.1.m1.1a"><msup id="S3.I3.i2.p1.1.m1.1.2" xref="S3.I3.i2.p1.1.m1.1.2.cmml"><mi id="S3.I3.i2.p1.1.m1.1.2.2" xref="S3.I3.i2.p1.1.m1.1.2.2.cmml">𝜽</mi><mrow id="S3.I3.i2.p1.1.m1.1.1.1.1" xref="S3.I3.i2.p1.1.m1.1.1.1.1.1.cmml"><mo id="S3.I3.i2.p1.1.m1.1.1.1.1.2" stretchy="false" xref="S3.I3.i2.p1.1.m1.1.1.1.1.1.cmml">(</mo><mrow id="S3.I3.i2.p1.1.m1.1.1.1.1.1" xref="S3.I3.i2.p1.1.m1.1.1.1.1.1.cmml"><mi id="S3.I3.i2.p1.1.m1.1.1.1.1.1.2" xref="S3.I3.i2.p1.1.m1.1.1.1.1.1.2.cmml">t</mi><mo id="S3.I3.i2.p1.1.m1.1.1.1.1.1.1" xref="S3.I3.i2.p1.1.m1.1.1.1.1.1.1.cmml">+</mo><mrow id="S3.I3.i2.p1.1.m1.1.1.1.1.1.3" xref="S3.I3.i2.p1.1.m1.1.1.1.1.1.3.cmml"><mn id="S3.I3.i2.p1.1.m1.1.1.1.1.1.3.2" xref="S3.I3.i2.p1.1.m1.1.1.1.1.1.3.2.cmml">2</mn><mo id="S3.I3.i2.p1.1.m1.1.1.1.1.1.3.1" xref="S3.I3.i2.p1.1.m1.1.1.1.1.1.3.1.cmml">/</mo><mn id="S3.I3.i2.p1.1.m1.1.1.1.1.1.3.3" xref="S3.I3.i2.p1.1.m1.1.1.1.1.1.3.3.cmml">3</mn></mrow></mrow><mo id="S3.I3.i2.p1.1.m1.1.1.1.1.3" stretchy="false" xref="S3.I3.i2.p1.1.m1.1.1.1.1.1.cmml">)</mo></mrow></msup><annotation-xml encoding="MathML-Content" id="S3.I3.i2.p1.1.m1.1b"><apply id="S3.I3.i2.p1.1.m1.1.2.cmml" xref="S3.I3.i2.p1.1.m1.1.2"><csymbol cd="ambiguous" id="S3.I3.i2.p1.1.m1.1.2.1.cmml" xref="S3.I3.i2.p1.1.m1.1.2">superscript</csymbol><ci id="S3.I3.i2.p1.1.m1.1.2.2.cmml" xref="S3.I3.i2.p1.1.m1.1.2.2">𝜽</ci><apply id="S3.I3.i2.p1.1.m1.1.1.1.1.1.cmml" xref="S3.I3.i2.p1.1.m1.1.1.1.1"><plus id="S3.I3.i2.p1.1.m1.1.1.1.1.1.1.cmml" xref="S3.I3.i2.p1.1.m1.1.1.1.1.1.1"></plus><ci id="S3.I3.i2.p1.1.m1.1.1.1.1.1.2.cmml" xref="S3.I3.i2.p1.1.m1.1.1.1.1.1.2">𝑡</ci><apply id="S3.I3.i2.p1.1.m1.1.1.1.1.1.3.cmml" xref="S3.I3.i2.p1.1.m1.1.1.1.1.1.3"><divide id="S3.I3.i2.p1.1.m1.1.1.1.1.1.3.1.cmml" xref="S3.I3.i2.p1.1.m1.1.1.1.1.1.3.1"></divide><cn id="S3.I3.i2.p1.1.m1.1.1.1.1.1.3.2.cmml" type="integer" xref="S3.I3.i2.p1.1.m1.1.1.1.1.1.3.2">2</cn><cn id="S3.I3.i2.p1.1.m1.1.1.1.1.1.3.3.cmml" type="integer" xref="S3.I3.i2.p1.1.m1.1.1.1.1.1.3.3">3</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I3.i2.p1.1.m1.1c">{\operatorname{{\bm{\theta}}}}^{(t+2/3)}</annotation><annotation encoding="application/x-llamapun" id="S3.I3.i2.p1.1.m1.1d">bold_italic_θ start_POSTSUPERSCRIPT ( italic_t + 2 / 3 ) end_POSTSUPERSCRIPT</annotation></semantics></math>, compute <math alttext="\operatorname{\mathbf{L}}_{2}^{{(t+1)}}" class="ltx_Math" display="inline" id="S3.I3.i2.p1.2.m2.1"><semantics id="S3.I3.i2.p1.2.m2.1a"><msubsup id="S3.I3.i2.p1.2.m2.1.2" xref="S3.I3.i2.p1.2.m2.1.2.cmml"><mi id="S3.I3.i2.p1.2.m2.1.2.2.2" xref="S3.I3.i2.p1.2.m2.1.2.2.2.cmml">𝐋</mi><mn id="S3.I3.i2.p1.2.m2.1.2.2.3" xref="S3.I3.i2.p1.2.m2.1.2.2.3.cmml">2</mn><mrow id="S3.I3.i2.p1.2.m2.1.1.1.1" xref="S3.I3.i2.p1.2.m2.1.1.1.1.1.cmml"><mo id="S3.I3.i2.p1.2.m2.1.1.1.1.2" stretchy="false" xref="S3.I3.i2.p1.2.m2.1.1.1.1.1.cmml">(</mo><mrow id="S3.I3.i2.p1.2.m2.1.1.1.1.1" xref="S3.I3.i2.p1.2.m2.1.1.1.1.1.cmml"><mi id="S3.I3.i2.p1.2.m2.1.1.1.1.1.2" xref="S3.I3.i2.p1.2.m2.1.1.1.1.1.2.cmml">t</mi><mo id="S3.I3.i2.p1.2.m2.1.1.1.1.1.1" xref="S3.I3.i2.p1.2.m2.1.1.1.1.1.1.cmml">+</mo><mn id="S3.I3.i2.p1.2.m2.1.1.1.1.1.3" xref="S3.I3.i2.p1.2.m2.1.1.1.1.1.3.cmml">1</mn></mrow><mo id="S3.I3.i2.p1.2.m2.1.1.1.1.3" stretchy="false" xref="S3.I3.i2.p1.2.m2.1.1.1.1.1.cmml">)</mo></mrow></msubsup><annotation-xml encoding="MathML-Content" id="S3.I3.i2.p1.2.m2.1b"><apply id="S3.I3.i2.p1.2.m2.1.2.cmml" xref="S3.I3.i2.p1.2.m2.1.2"><csymbol cd="ambiguous" id="S3.I3.i2.p1.2.m2.1.2.1.cmml" xref="S3.I3.i2.p1.2.m2.1.2">superscript</csymbol><apply id="S3.I3.i2.p1.2.m2.1.2.2.cmml" xref="S3.I3.i2.p1.2.m2.1.2"><csymbol cd="ambiguous" id="S3.I3.i2.p1.2.m2.1.2.2.1.cmml" xref="S3.I3.i2.p1.2.m2.1.2">subscript</csymbol><ci id="S3.I3.i2.p1.2.m2.1.2.2.2.cmml" xref="S3.I3.i2.p1.2.m2.1.2.2.2">𝐋</ci><cn id="S3.I3.i2.p1.2.m2.1.2.2.3.cmml" type="integer" xref="S3.I3.i2.p1.2.m2.1.2.2.3">2</cn></apply><apply id="S3.I3.i2.p1.2.m2.1.1.1.1.1.cmml" xref="S3.I3.i2.p1.2.m2.1.1.1.1"><plus id="S3.I3.i2.p1.2.m2.1.1.1.1.1.1.cmml" xref="S3.I3.i2.p1.2.m2.1.1.1.1.1.1"></plus><ci id="S3.I3.i2.p1.2.m2.1.1.1.1.1.2.cmml" xref="S3.I3.i2.p1.2.m2.1.1.1.1.1.2">𝑡</ci><cn id="S3.I3.i2.p1.2.m2.1.1.1.1.1.3.cmml" type="integer" xref="S3.I3.i2.p1.2.m2.1.1.1.1.1.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I3.i2.p1.2.m2.1c">\operatorname{\mathbf{L}}_{2}^{{(t+1)}}</annotation><annotation encoding="application/x-llamapun" id="S3.I3.i2.p1.2.m2.1d">bold_L start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_t + 1 ) end_POSTSUPERSCRIPT</annotation></semantics></math> by solving (<a class="ltx_ref" href="https://arxiv.org/html/2503.13782v1#S3.E11" title="In 3 Regularized Parameter Estimation ‣ Regularized Parameter Estimation in Mixed Model Trace Regression"><span class="ltx_text ltx_ref_tag">11</span></a>). Update <math alttext="\operatorname{{\bm{\theta}}}^{(t+2/3)}" class="ltx_Math" display="inline" id="S3.I3.i2.p1.3.m3.1"><semantics id="S3.I3.i2.p1.3.m3.1a"><msup id="S3.I3.i2.p1.3.m3.1.2" xref="S3.I3.i2.p1.3.m3.1.2.cmml"><mi id="S3.I3.i2.p1.3.m3.1.2.2" xref="S3.I3.i2.p1.3.m3.1.2.2.cmml">𝜽</mi><mrow id="S3.I3.i2.p1.3.m3.1.1.1.1" xref="S3.I3.i2.p1.3.m3.1.1.1.1.1.cmml"><mo id="S3.I3.i2.p1.3.m3.1.1.1.1.2" stretchy="false" xref="S3.I3.i2.p1.3.m3.1.1.1.1.1.cmml">(</mo><mrow id="S3.I3.i2.p1.3.m3.1.1.1.1.1" xref="S3.I3.i2.p1.3.m3.1.1.1.1.1.cmml"><mi id="S3.I3.i2.p1.3.m3.1.1.1.1.1.2" xref="S3.I3.i2.p1.3.m3.1.1.1.1.1.2.cmml">t</mi><mo id="S3.I3.i2.p1.3.m3.1.1.1.1.1.1" xref="S3.I3.i2.p1.3.m3.1.1.1.1.1.1.cmml">+</mo><mrow id="S3.I3.i2.p1.3.m3.1.1.1.1.1.3" xref="S3.I3.i2.p1.3.m3.1.1.1.1.1.3.cmml"><mn id="S3.I3.i2.p1.3.m3.1.1.1.1.1.3.2" xref="S3.I3.i2.p1.3.m3.1.1.1.1.1.3.2.cmml">2</mn><mo id="S3.I3.i2.p1.3.m3.1.1.1.1.1.3.1" xref="S3.I3.i2.p1.3.m3.1.1.1.1.1.3.1.cmml">/</mo><mn id="S3.I3.i2.p1.3.m3.1.1.1.1.1.3.3" xref="S3.I3.i2.p1.3.m3.1.1.1.1.1.3.3.cmml">3</mn></mrow></mrow><mo id="S3.I3.i2.p1.3.m3.1.1.1.1.3" stretchy="false" xref="S3.I3.i2.p1.3.m3.1.1.1.1.1.cmml">)</mo></mrow></msup><annotation-xml encoding="MathML-Content" id="S3.I3.i2.p1.3.m3.1b"><apply id="S3.I3.i2.p1.3.m3.1.2.cmml" xref="S3.I3.i2.p1.3.m3.1.2"><csymbol cd="ambiguous" id="S3.I3.i2.p1.3.m3.1.2.1.cmml" xref="S3.I3.i2.p1.3.m3.1.2">superscript</csymbol><ci id="S3.I3.i2.p1.3.m3.1.2.2.cmml" xref="S3.I3.i2.p1.3.m3.1.2.2">𝜽</ci><apply id="S3.I3.i2.p1.3.m3.1.1.1.1.1.cmml" xref="S3.I3.i2.p1.3.m3.1.1.1.1"><plus id="S3.I3.i2.p1.3.m3.1.1.1.1.1.1.cmml" xref="S3.I3.i2.p1.3.m3.1.1.1.1.1.1"></plus><ci id="S3.I3.i2.p1.3.m3.1.1.1.1.1.2.cmml" xref="S3.I3.i2.p1.3.m3.1.1.1.1.1.2">𝑡</ci><apply id="S3.I3.i2.p1.3.m3.1.1.1.1.1.3.cmml" xref="S3.I3.i2.p1.3.m3.1.1.1.1.1.3"><divide id="S3.I3.i2.p1.3.m3.1.1.1.1.1.3.1.cmml" xref="S3.I3.i2.p1.3.m3.1.1.1.1.1.3.1"></divide><cn id="S3.I3.i2.p1.3.m3.1.1.1.1.1.3.2.cmml" type="integer" xref="S3.I3.i2.p1.3.m3.1.1.1.1.1.3.2">2</cn><cn id="S3.I3.i2.p1.3.m3.1.1.1.1.1.3.3.cmml" type="integer" xref="S3.I3.i2.p1.3.m3.1.1.1.1.1.3.3">3</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I3.i2.p1.3.m3.1c">\operatorname{{\bm{\theta}}}^{(t+2/3)}</annotation><annotation encoding="application/x-llamapun" id="S3.I3.i2.p1.3.m3.1d">bold_italic_θ start_POSTSUPERSCRIPT ( italic_t + 2 / 3 ) end_POSTSUPERSCRIPT</annotation></semantics></math> to <math alttext="\operatorname{{\bm{\theta}}}^{(t+1)}=(\operatorname{\mathbf{b}}^{(t+1)},% \operatorname{\mathbf{L}}_{1}^{(t+1)},\operatorname{\mathbf{L}}_{2}^{(t+1)},% \tau^{2(t+1)})" class="ltx_Math" display="inline" id="S3.I3.i2.p1.4.m4.9"><semantics id="S3.I3.i2.p1.4.m4.9a"><mrow id="S3.I3.i2.p1.4.m4.9.9" xref="S3.I3.i2.p1.4.m4.9.9.cmml"><msup id="S3.I3.i2.p1.4.m4.9.9.6" xref="S3.I3.i2.p1.4.m4.9.9.6.cmml"><mi id="S3.I3.i2.p1.4.m4.9.9.6.2" xref="S3.I3.i2.p1.4.m4.9.9.6.2.cmml">𝜽</mi><mrow id="S3.I3.i2.p1.4.m4.1.1.1.1" xref="S3.I3.i2.p1.4.m4.1.1.1.1.1.cmml"><mo id="S3.I3.i2.p1.4.m4.1.1.1.1.2" stretchy="false" xref="S3.I3.i2.p1.4.m4.1.1.1.1.1.cmml">(</mo><mrow id="S3.I3.i2.p1.4.m4.1.1.1.1.1" xref="S3.I3.i2.p1.4.m4.1.1.1.1.1.cmml"><mi id="S3.I3.i2.p1.4.m4.1.1.1.1.1.2" xref="S3.I3.i2.p1.4.m4.1.1.1.1.1.2.cmml">t</mi><mo id="S3.I3.i2.p1.4.m4.1.1.1.1.1.1" xref="S3.I3.i2.p1.4.m4.1.1.1.1.1.1.cmml">+</mo><mn id="S3.I3.i2.p1.4.m4.1.1.1.1.1.3" xref="S3.I3.i2.p1.4.m4.1.1.1.1.1.3.cmml">1</mn></mrow><mo id="S3.I3.i2.p1.4.m4.1.1.1.1.3" stretchy="false" xref="S3.I3.i2.p1.4.m4.1.1.1.1.1.cmml">)</mo></mrow></msup><mo id="S3.I3.i2.p1.4.m4.9.9.5" xref="S3.I3.i2.p1.4.m4.9.9.5.cmml">=</mo><mrow id="S3.I3.i2.p1.4.m4.9.9.4.4" xref="S3.I3.i2.p1.4.m4.9.9.4.5.cmml"><mo id="S3.I3.i2.p1.4.m4.9.9.4.4.5" stretchy="false" xref="S3.I3.i2.p1.4.m4.9.9.4.5.cmml">(</mo><msup id="S3.I3.i2.p1.4.m4.6.6.1.1.1" xref="S3.I3.i2.p1.4.m4.6.6.1.1.1.cmml"><mi id="S3.I3.i2.p1.4.m4.6.6.1.1.1.2" 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start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_t + 1 ) end_POSTSUPERSCRIPT , italic_τ start_POSTSUPERSCRIPT 2 ( italic_t + 1 ) end_POSTSUPERSCRIPT )</annotation></semantics></math>.</p> </div> </li> </ul> </div> </div> <figcaption class="ltx_caption"><span class="ltx_tag ltx_tag_float"><span class="ltx_text ltx_font_bold" id="alg1.3.1.1">Algorithm 1</span> </span>MMTR AECM Algorithm</figcaption> </figure> <div class="ltx_para" id="S3.p6"> <p class="ltx_p" id="S3.p6.17">We post-process the <math alttext="\operatorname{\mathbf{L}}_{1}" class="ltx_Math" display="inline" id="S3.p6.1.m1.1"><semantics id="S3.p6.1.m1.1a"><msub id="S3.p6.1.m1.1.1" xref="S3.p6.1.m1.1.1.cmml"><mi id="S3.p6.1.m1.1.1.2" xref="S3.p6.1.m1.1.1.2.cmml">𝐋</mi><mn id="S3.p6.1.m1.1.1.3" xref="S3.p6.1.m1.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S3.p6.1.m1.1b"><apply id="S3.p6.1.m1.1.1.cmml" xref="S3.p6.1.m1.1.1"><csymbol cd="ambiguous" id="S3.p6.1.m1.1.1.1.cmml" 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xref="S3.p6.2.m2.1.1.2">𝐋</ci><cn id="S3.p6.2.m2.1.1.3.cmml" type="integer" xref="S3.p6.2.m2.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p6.2.m2.1c">\operatorname{\mathbf{L}}_{2}</annotation><annotation encoding="application/x-llamapun" id="S3.p6.2.m2.1d">bold_L start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math> estimates at the end of every iteration to improve the efficiency and stability of Algorithm <a class="ltx_ref" href="https://arxiv.org/html/2503.13782v1#alg1" title="In 3 Regularized Parameter Estimation ‣ Regularized Parameter Estimation in Mixed Model Trace Regression"><span class="ltx_text ltx_ref_tag">1</span></a>. For <math alttext="t=1,2,\ldots,\infty" class="ltx_Math" display="inline" id="S3.p6.3.m3.4"><semantics id="S3.p6.3.m3.4a"><mrow id="S3.p6.3.m3.4.5" xref="S3.p6.3.m3.4.5.cmml"><mi id="S3.p6.3.m3.4.5.2" xref="S3.p6.3.m3.4.5.2.cmml">t</mi><mo id="S3.p6.3.m3.4.5.1" xref="S3.p6.3.m3.4.5.1.cmml">=</mo><mrow id="S3.p6.3.m3.4.5.3.2" xref="S3.p6.3.m3.4.5.3.1.cmml"><mn id="S3.p6.3.m3.1.1" xref="S3.p6.3.m3.1.1.cmml">1</mn><mo id="S3.p6.3.m3.4.5.3.2.1" xref="S3.p6.3.m3.4.5.3.1.cmml">,</mo><mn id="S3.p6.3.m3.2.2" xref="S3.p6.3.m3.2.2.cmml">2</mn><mo id="S3.p6.3.m3.4.5.3.2.2" xref="S3.p6.3.m3.4.5.3.1.cmml">,</mo><mi id="S3.p6.3.m3.3.3" mathvariant="normal" xref="S3.p6.3.m3.3.3.cmml">…</mi><mo id="S3.p6.3.m3.4.5.3.2.3" xref="S3.p6.3.m3.4.5.3.1.cmml">,</mo><mi id="S3.p6.3.m3.4.4" mathvariant="normal" xref="S3.p6.3.m3.4.4.cmml">∞</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.p6.3.m3.4b"><apply id="S3.p6.3.m3.4.5.cmml" xref="S3.p6.3.m3.4.5"><eq id="S3.p6.3.m3.4.5.1.cmml" xref="S3.p6.3.m3.4.5.1"></eq><ci id="S3.p6.3.m3.4.5.2.cmml" xref="S3.p6.3.m3.4.5.2">𝑡</ci><list id="S3.p6.3.m3.4.5.3.1.cmml" xref="S3.p6.3.m3.4.5.3.2"><cn id="S3.p6.3.m3.1.1.cmml" type="integer" xref="S3.p6.3.m3.1.1">1</cn><cn id="S3.p6.3.m3.2.2.cmml" type="integer" xref="S3.p6.3.m3.2.2">2</cn><ci id="S3.p6.3.m3.3.3.cmml" xref="S3.p6.3.m3.3.3">…</ci><infinity id="S3.p6.3.m3.4.4.cmml" xref="S3.p6.3.m3.4.4"></infinity></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p6.3.m3.4c">t=1,2,\ldots,\infty</annotation><annotation encoding="application/x-llamapun" id="S3.p6.3.m3.4d">italic_t = 1 , 2 , … , ∞</annotation></semantics></math>, any zero columns in <math alttext="\operatorname{\mathbf{L}}_{1}^{(t)}" class="ltx_Math" display="inline" id="S3.p6.4.m4.1"><semantics id="S3.p6.4.m4.1a"><msubsup id="S3.p6.4.m4.1.2" xref="S3.p6.4.m4.1.2.cmml"><mi id="S3.p6.4.m4.1.2.2.2" xref="S3.p6.4.m4.1.2.2.2.cmml">𝐋</mi><mn id="S3.p6.4.m4.1.2.2.3" xref="S3.p6.4.m4.1.2.2.3.cmml">1</mn><mrow id="S3.p6.4.m4.1.1.1.3" xref="S3.p6.4.m4.1.2.cmml"><mo id="S3.p6.4.m4.1.1.1.3.1" stretchy="false" xref="S3.p6.4.m4.1.2.cmml">(</mo><mi id="S3.p6.4.m4.1.1.1.1" xref="S3.p6.4.m4.1.1.1.1.cmml">t</mi><mo id="S3.p6.4.m4.1.1.1.3.2" stretchy="false" xref="S3.p6.4.m4.1.2.cmml">)</mo></mrow></msubsup><annotation-xml encoding="MathML-Content" id="S3.p6.4.m4.1b"><apply id="S3.p6.4.m4.1.2.cmml" xref="S3.p6.4.m4.1.2"><csymbol cd="ambiguous" id="S3.p6.4.m4.1.2.1.cmml" xref="S3.p6.4.m4.1.2">superscript</csymbol><apply id="S3.p6.4.m4.1.2.2.cmml" xref="S3.p6.4.m4.1.2"><csymbol cd="ambiguous" id="S3.p6.4.m4.1.2.2.1.cmml" xref="S3.p6.4.m4.1.2">subscript</csymbol><ci id="S3.p6.4.m4.1.2.2.2.cmml" xref="S3.p6.4.m4.1.2.2.2">𝐋</ci><cn id="S3.p6.4.m4.1.2.2.3.cmml" type="integer" xref="S3.p6.4.m4.1.2.2.3">1</cn></apply><ci id="S3.p6.4.m4.1.1.1.1.cmml" xref="S3.p6.4.m4.1.1.1.1">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p6.4.m4.1c">\operatorname{\mathbf{L}}_{1}^{(t)}</annotation><annotation encoding="application/x-llamapun" id="S3.p6.4.m4.1d">bold_L start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_t ) end_POSTSUPERSCRIPT</annotation></semantics></math> and <math alttext="\operatorname{\mathbf{L}}_{2}^{(t)}" class="ltx_Math" display="inline" id="S3.p6.5.m5.1"><semantics id="S3.p6.5.m5.1a"><msubsup id="S3.p6.5.m5.1.2" xref="S3.p6.5.m5.1.2.cmml"><mi id="S3.p6.5.m5.1.2.2.2" xref="S3.p6.5.m5.1.2.2.2.cmml">𝐋</mi><mn id="S3.p6.5.m5.1.2.2.3" xref="S3.p6.5.m5.1.2.2.3.cmml">2</mn><mrow id="S3.p6.5.m5.1.1.1.3" xref="S3.p6.5.m5.1.2.cmml"><mo id="S3.p6.5.m5.1.1.1.3.1" stretchy="false" xref="S3.p6.5.m5.1.2.cmml">(</mo><mi id="S3.p6.5.m5.1.1.1.1" xref="S3.p6.5.m5.1.1.1.1.cmml">t</mi><mo id="S3.p6.5.m5.1.1.1.3.2" stretchy="false" xref="S3.p6.5.m5.1.2.cmml">)</mo></mrow></msubsup><annotation-xml encoding="MathML-Content" id="S3.p6.5.m5.1b"><apply id="S3.p6.5.m5.1.2.cmml" xref="S3.p6.5.m5.1.2"><csymbol cd="ambiguous" id="S3.p6.5.m5.1.2.1.cmml" xref="S3.p6.5.m5.1.2">superscript</csymbol><apply id="S3.p6.5.m5.1.2.2.cmml" xref="S3.p6.5.m5.1.2"><csymbol cd="ambiguous" id="S3.p6.5.m5.1.2.2.1.cmml" xref="S3.p6.5.m5.1.2">subscript</csymbol><ci id="S3.p6.5.m5.1.2.2.2.cmml" xref="S3.p6.5.m5.1.2.2.2">𝐋</ci><cn id="S3.p6.5.m5.1.2.2.3.cmml" type="integer" xref="S3.p6.5.m5.1.2.2.3">2</cn></apply><ci id="S3.p6.5.m5.1.1.1.1.cmml" xref="S3.p6.5.m5.1.1.1.1">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p6.5.m5.1c">\operatorname{\mathbf{L}}_{2}^{(t)}</annotation><annotation encoding="application/x-llamapun" id="S3.p6.5.m5.1d">bold_L start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_t ) end_POSTSUPERSCRIPT</annotation></semantics></math> are removed. Then, <math alttext="\operatorname{\mathbf{L}}_{1}^{(t)}" class="ltx_Math" display="inline" id="S3.p6.6.m6.1"><semantics id="S3.p6.6.m6.1a"><msubsup id="S3.p6.6.m6.1.2" xref="S3.p6.6.m6.1.2.cmml"><mi id="S3.p6.6.m6.1.2.2.2" xref="S3.p6.6.m6.1.2.2.2.cmml">𝐋</mi><mn id="S3.p6.6.m6.1.2.2.3" xref="S3.p6.6.m6.1.2.2.3.cmml">1</mn><mrow id="S3.p6.6.m6.1.1.1.3" xref="S3.p6.6.m6.1.2.cmml"><mo id="S3.p6.6.m6.1.1.1.3.1" stretchy="false" xref="S3.p6.6.m6.1.2.cmml">(</mo><mi id="S3.p6.6.m6.1.1.1.1" xref="S3.p6.6.m6.1.1.1.1.cmml">t</mi><mo id="S3.p6.6.m6.1.1.1.3.2" stretchy="false" xref="S3.p6.6.m6.1.2.cmml">)</mo></mrow></msubsup><annotation-xml encoding="MathML-Content" id="S3.p6.6.m6.1b"><apply id="S3.p6.6.m6.1.2.cmml" xref="S3.p6.6.m6.1.2"><csymbol cd="ambiguous" id="S3.p6.6.m6.1.2.1.cmml" xref="S3.p6.6.m6.1.2">superscript</csymbol><apply id="S3.p6.6.m6.1.2.2.cmml" xref="S3.p6.6.m6.1.2"><csymbol cd="ambiguous" id="S3.p6.6.m6.1.2.2.1.cmml" xref="S3.p6.6.m6.1.2">subscript</csymbol><ci id="S3.p6.6.m6.1.2.2.2.cmml" xref="S3.p6.6.m6.1.2.2.2">𝐋</ci><cn id="S3.p6.6.m6.1.2.2.3.cmml" type="integer" xref="S3.p6.6.m6.1.2.2.3">1</cn></apply><ci id="S3.p6.6.m6.1.1.1.1.cmml" xref="S3.p6.6.m6.1.1.1.1">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p6.6.m6.1c">\operatorname{\mathbf{L}}_{1}^{(t)}</annotation><annotation encoding="application/x-llamapun" id="S3.p6.6.m6.1d">bold_L start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_t ) end_POSTSUPERSCRIPT</annotation></semantics></math> and <math alttext="\operatorname{\mathbf{L}}_{2}^{(t)}" class="ltx_Math" display="inline" id="S3.p6.7.m7.1"><semantics id="S3.p6.7.m7.1a"><msubsup id="S3.p6.7.m7.1.2" xref="S3.p6.7.m7.1.2.cmml"><mi id="S3.p6.7.m7.1.2.2.2" xref="S3.p6.7.m7.1.2.2.2.cmml">𝐋</mi><mn id="S3.p6.7.m7.1.2.2.3" xref="S3.p6.7.m7.1.2.2.3.cmml">2</mn><mrow id="S3.p6.7.m7.1.1.1.3" xref="S3.p6.7.m7.1.2.cmml"><mo id="S3.p6.7.m7.1.1.1.3.1" stretchy="false" xref="S3.p6.7.m7.1.2.cmml">(</mo><mi id="S3.p6.7.m7.1.1.1.1" xref="S3.p6.7.m7.1.1.1.1.cmml">t</mi><mo id="S3.p6.7.m7.1.1.1.3.2" stretchy="false" xref="S3.p6.7.m7.1.2.cmml">)</mo></mrow></msubsup><annotation-xml encoding="MathML-Content" id="S3.p6.7.m7.1b"><apply id="S3.p6.7.m7.1.2.cmml" xref="S3.p6.7.m7.1.2"><csymbol cd="ambiguous" id="S3.p6.7.m7.1.2.1.cmml" xref="S3.p6.7.m7.1.2">superscript</csymbol><apply id="S3.p6.7.m7.1.2.2.cmml" xref="S3.p6.7.m7.1.2"><csymbol cd="ambiguous" id="S3.p6.7.m7.1.2.2.1.cmml" xref="S3.p6.7.m7.1.2">subscript</csymbol><ci id="S3.p6.7.m7.1.2.2.2.cmml" xref="S3.p6.7.m7.1.2.2.2">𝐋</ci><cn id="S3.p6.7.m7.1.2.2.3.cmml" type="integer" xref="S3.p6.7.m7.1.2.2.3">2</cn></apply><ci id="S3.p6.7.m7.1.1.1.1.cmml" xref="S3.p6.7.m7.1.1.1.1">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p6.7.m7.1c">\operatorname{\mathbf{L}}_{2}^{(t)}</annotation><annotation encoding="application/x-llamapun" id="S3.p6.7.m7.1d">bold_L start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_t ) end_POSTSUPERSCRIPT</annotation></semantics></math> are updated following Proposition <a class="ltx_ref" href="https://arxiv.org/html/2503.13782v1#S1.Thmproposition2" title="Proposition 1.2 ‣ 1.2 Proof of Proposition 2.2 ‣ 1 Proof of the Propositions ‣ Regularized Parameter Estimation in Mixed Model Trace Regression"><span class="ltx_text ltx_ref_tag">1.2</span></a> such that the maximum diagonal values of <math alttext="\operatorname{{\mathbf{\Sigma}}}_{1}^{(t)}" class="ltx_Math" display="inline" id="S3.p6.8.m8.1"><semantics id="S3.p6.8.m8.1a"><msubsup id="S3.p6.8.m8.1.2" xref="S3.p6.8.m8.1.2.cmml"><mi id="S3.p6.8.m8.1.2.2.2" xref="S3.p6.8.m8.1.2.2.2.cmml">𝚺</mi><mn id="S3.p6.8.m8.1.2.2.3" xref="S3.p6.8.m8.1.2.2.3.cmml">1</mn><mrow id="S3.p6.8.m8.1.1.1.3" xref="S3.p6.8.m8.1.2.cmml"><mo id="S3.p6.8.m8.1.1.1.3.1" stretchy="false" xref="S3.p6.8.m8.1.2.cmml">(</mo><mi id="S3.p6.8.m8.1.1.1.1" xref="S3.p6.8.m8.1.1.1.1.cmml">t</mi><mo id="S3.p6.8.m8.1.1.1.3.2" stretchy="false" xref="S3.p6.8.m8.1.2.cmml">)</mo></mrow></msubsup><annotation-xml encoding="MathML-Content" id="S3.p6.8.m8.1b"><apply id="S3.p6.8.m8.1.2.cmml" xref="S3.p6.8.m8.1.2"><csymbol cd="ambiguous" id="S3.p6.8.m8.1.2.1.cmml" xref="S3.p6.8.m8.1.2">superscript</csymbol><apply id="S3.p6.8.m8.1.2.2.cmml" xref="S3.p6.8.m8.1.2"><csymbol cd="ambiguous" id="S3.p6.8.m8.1.2.2.1.cmml" xref="S3.p6.8.m8.1.2">subscript</csymbol><ci id="S3.p6.8.m8.1.2.2.2.cmml" xref="S3.p6.8.m8.1.2.2.2">𝚺</ci><cn id="S3.p6.8.m8.1.2.2.3.cmml" type="integer" xref="S3.p6.8.m8.1.2.2.3">1</cn></apply><ci id="S3.p6.8.m8.1.1.1.1.cmml" xref="S3.p6.8.m8.1.1.1.1">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p6.8.m8.1c">\operatorname{{\mathbf{\Sigma}}}_{1}^{(t)}</annotation><annotation encoding="application/x-llamapun" id="S3.p6.8.m8.1d">bold_Σ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_t ) end_POSTSUPERSCRIPT</annotation></semantics></math> and <math alttext="\operatorname{{\mathbf{\Sigma}}}_{2}^{(t)}" class="ltx_Math" display="inline" id="S3.p6.9.m9.1"><semantics id="S3.p6.9.m9.1a"><msubsup id="S3.p6.9.m9.1.2" xref="S3.p6.9.m9.1.2.cmml"><mi id="S3.p6.9.m9.1.2.2.2" xref="S3.p6.9.m9.1.2.2.2.cmml">𝚺</mi><mn id="S3.p6.9.m9.1.2.2.3" xref="S3.p6.9.m9.1.2.2.3.cmml">2</mn><mrow id="S3.p6.9.m9.1.1.1.3" xref="S3.p6.9.m9.1.2.cmml"><mo id="S3.p6.9.m9.1.1.1.3.1" stretchy="false" xref="S3.p6.9.m9.1.2.cmml">(</mo><mi id="S3.p6.9.m9.1.1.1.1" xref="S3.p6.9.m9.1.1.1.1.cmml">t</mi><mo id="S3.p6.9.m9.1.1.1.3.2" stretchy="false" xref="S3.p6.9.m9.1.2.cmml">)</mo></mrow></msubsup><annotation-xml encoding="MathML-Content" id="S3.p6.9.m9.1b"><apply id="S3.p6.9.m9.1.2.cmml" xref="S3.p6.9.m9.1.2"><csymbol cd="ambiguous" id="S3.p6.9.m9.1.2.1.cmml" xref="S3.p6.9.m9.1.2">superscript</csymbol><apply id="S3.p6.9.m9.1.2.2.cmml" xref="S3.p6.9.m9.1.2"><csymbol cd="ambiguous" id="S3.p6.9.m9.1.2.2.1.cmml" xref="S3.p6.9.m9.1.2">subscript</csymbol><ci id="S3.p6.9.m9.1.2.2.2.cmml" xref="S3.p6.9.m9.1.2.2.2">𝚺</ci><cn id="S3.p6.9.m9.1.2.2.3.cmml" type="integer" xref="S3.p6.9.m9.1.2.2.3">2</cn></apply><ci id="S3.p6.9.m9.1.1.1.1.cmml" xref="S3.p6.9.m9.1.1.1.1">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p6.9.m9.1c">\operatorname{{\mathbf{\Sigma}}}_{2}^{(t)}</annotation><annotation encoding="application/x-llamapun" id="S3.p6.9.m9.1d">bold_Σ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_t ) end_POSTSUPERSCRIPT</annotation></semantics></math> equal 1 and the “overall variance” <math alttext="\sigma^{2(t)}" class="ltx_Math" display="inline" id="S3.p6.10.m10.1"><semantics id="S3.p6.10.m10.1a"><msup id="S3.p6.10.m10.1.2" xref="S3.p6.10.m10.1.2.cmml"><mi id="S3.p6.10.m10.1.2.2" xref="S3.p6.10.m10.1.2.2.cmml">σ</mi><mrow id="S3.p6.10.m10.1.1.1" xref="S3.p6.10.m10.1.1.1.cmml"><mn id="S3.p6.10.m10.1.1.1.3" xref="S3.p6.10.m10.1.1.1.3.cmml">2</mn><mo id="S3.p6.10.m10.1.1.1.2" xref="S3.p6.10.m10.1.1.1.2.cmml">⁢</mo><mrow id="S3.p6.10.m10.1.1.1.4.2" xref="S3.p6.10.m10.1.1.1.cmml"><mo id="S3.p6.10.m10.1.1.1.4.2.1" stretchy="false" xref="S3.p6.10.m10.1.1.1.cmml">(</mo><mi id="S3.p6.10.m10.1.1.1.1" xref="S3.p6.10.m10.1.1.1.1.cmml">t</mi><mo id="S3.p6.10.m10.1.1.1.4.2.2" stretchy="false" xref="S3.p6.10.m10.1.1.1.cmml">)</mo></mrow></mrow></msup><annotation-xml encoding="MathML-Content" id="S3.p6.10.m10.1b"><apply id="S3.p6.10.m10.1.2.cmml" xref="S3.p6.10.m10.1.2"><csymbol cd="ambiguous" id="S3.p6.10.m10.1.2.1.cmml" xref="S3.p6.10.m10.1.2">superscript</csymbol><ci id="S3.p6.10.m10.1.2.2.cmml" xref="S3.p6.10.m10.1.2.2">𝜎</ci><apply id="S3.p6.10.m10.1.1.1.cmml" xref="S3.p6.10.m10.1.1.1"><times id="S3.p6.10.m10.1.1.1.2.cmml" xref="S3.p6.10.m10.1.1.1.2"></times><cn id="S3.p6.10.m10.1.1.1.3.cmml" type="integer" xref="S3.p6.10.m10.1.1.1.3">2</cn><ci id="S3.p6.10.m10.1.1.1.1.cmml" xref="S3.p6.10.m10.1.1.1.1">𝑡</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p6.10.m10.1c">\sigma^{2(t)}</annotation><annotation encoding="application/x-llamapun" id="S3.p6.10.m10.1d">italic_σ start_POSTSUPERSCRIPT 2 ( italic_t ) end_POSTSUPERSCRIPT</annotation></semantics></math> is split equally between <math alttext="\operatorname{{\mathbf{\Sigma}}}_{1}^{(t)}" class="ltx_Math" display="inline" id="S3.p6.11.m11.1"><semantics id="S3.p6.11.m11.1a"><msubsup id="S3.p6.11.m11.1.2" xref="S3.p6.11.m11.1.2.cmml"><mi id="S3.p6.11.m11.1.2.2.2" xref="S3.p6.11.m11.1.2.2.2.cmml">𝚺</mi><mn id="S3.p6.11.m11.1.2.2.3" xref="S3.p6.11.m11.1.2.2.3.cmml">1</mn><mrow id="S3.p6.11.m11.1.1.1.3" xref="S3.p6.11.m11.1.2.cmml"><mo id="S3.p6.11.m11.1.1.1.3.1" stretchy="false" xref="S3.p6.11.m11.1.2.cmml">(</mo><mi id="S3.p6.11.m11.1.1.1.1" xref="S3.p6.11.m11.1.1.1.1.cmml">t</mi><mo id="S3.p6.11.m11.1.1.1.3.2" stretchy="false" 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end_POSTSUPERSCRIPT</annotation></semantics></math> and <math alttext="\operatorname{{\mathbf{\Sigma}}}_{2}^{(t)}" class="ltx_Math" display="inline" id="S3.p6.12.m12.1"><semantics id="S3.p6.12.m12.1a"><msubsup id="S3.p6.12.m12.1.2" xref="S3.p6.12.m12.1.2.cmml"><mi id="S3.p6.12.m12.1.2.2.2" xref="S3.p6.12.m12.1.2.2.2.cmml">𝚺</mi><mn id="S3.p6.12.m12.1.2.2.3" xref="S3.p6.12.m12.1.2.2.3.cmml">2</mn><mrow id="S3.p6.12.m12.1.1.1.3" xref="S3.p6.12.m12.1.2.cmml"><mo id="S3.p6.12.m12.1.1.1.3.1" stretchy="false" xref="S3.p6.12.m12.1.2.cmml">(</mo><mi id="S3.p6.12.m12.1.1.1.1" xref="S3.p6.12.m12.1.1.1.1.cmml">t</mi><mo id="S3.p6.12.m12.1.1.1.3.2" stretchy="false" xref="S3.p6.12.m12.1.2.cmml">)</mo></mrow></msubsup><annotation-xml encoding="MathML-Content" id="S3.p6.12.m12.1b"><apply id="S3.p6.12.m12.1.2.cmml" xref="S3.p6.12.m12.1.2"><csymbol cd="ambiguous" id="S3.p6.12.m12.1.2.1.cmml" xref="S3.p6.12.m12.1.2">superscript</csymbol><apply id="S3.p6.12.m12.1.2.2.cmml" 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This update maintains similar ranks for <math alttext="\operatorname{\mathbf{L}}_{1}^{(t)}" class="ltx_Math" display="inline" id="S3.p6.13.m13.1"><semantics id="S3.p6.13.m13.1a"><msubsup id="S3.p6.13.m13.1.2" xref="S3.p6.13.m13.1.2.cmml"><mi id="S3.p6.13.m13.1.2.2.2" xref="S3.p6.13.m13.1.2.2.2.cmml">𝐋</mi><mn id="S3.p6.13.m13.1.2.2.3" xref="S3.p6.13.m13.1.2.2.3.cmml">1</mn><mrow id="S3.p6.13.m13.1.1.1.3" xref="S3.p6.13.m13.1.2.cmml"><mo id="S3.p6.13.m13.1.1.1.3.1" stretchy="false" xref="S3.p6.13.m13.1.2.cmml">(</mo><mi id="S3.p6.13.m13.1.1.1.1" xref="S3.p6.13.m13.1.1.1.1.cmml">t</mi><mo id="S3.p6.13.m13.1.1.1.3.2" stretchy="false" xref="S3.p6.13.m13.1.2.cmml">)</mo></mrow></msubsup><annotation-xml encoding="MathML-Content" id="S3.p6.13.m13.1b"><apply id="S3.p6.13.m13.1.2.cmml" xref="S3.p6.13.m13.1.2"><csymbol cd="ambiguous" id="S3.p6.13.m13.1.2.1.cmml" xref="S3.p6.13.m13.1.2">superscript</csymbol><apply id="S3.p6.13.m13.1.2.2.cmml" xref="S3.p6.13.m13.1.2"><csymbol cd="ambiguous" id="S3.p6.13.m13.1.2.2.1.cmml" xref="S3.p6.13.m13.1.2">subscript</csymbol><ci id="S3.p6.13.m13.1.2.2.2.cmml" xref="S3.p6.13.m13.1.2.2.2">𝐋</ci><cn id="S3.p6.13.m13.1.2.2.3.cmml" type="integer" xref="S3.p6.13.m13.1.2.2.3">1</cn></apply><ci id="S3.p6.13.m13.1.1.1.1.cmml" xref="S3.p6.13.m13.1.1.1.1">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p6.13.m13.1c">\operatorname{\mathbf{L}}_{1}^{(t)}</annotation><annotation encoding="application/x-llamapun" id="S3.p6.13.m13.1d">bold_L start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_t ) end_POSTSUPERSCRIPT</annotation></semantics></math> and <math alttext="\operatorname{\mathbf{L}}_{2}^{(t)}" class="ltx_Math" display="inline" id="S3.p6.14.m14.1"><semantics id="S3.p6.14.m14.1a"><msubsup id="S3.p6.14.m14.1.2" xref="S3.p6.14.m14.1.2.cmml"><mi id="S3.p6.14.m14.1.2.2.2" xref="S3.p6.14.m14.1.2.2.2.cmml">𝐋</mi><mn id="S3.p6.14.m14.1.2.2.3" xref="S3.p6.14.m14.1.2.2.3.cmml">2</mn><mrow id="S3.p6.14.m14.1.1.1.3" xref="S3.p6.14.m14.1.2.cmml"><mo id="S3.p6.14.m14.1.1.1.3.1" stretchy="false" xref="S3.p6.14.m14.1.2.cmml">(</mo><mi id="S3.p6.14.m14.1.1.1.1" xref="S3.p6.14.m14.1.1.1.1.cmml">t</mi><mo id="S3.p6.14.m14.1.1.1.3.2" stretchy="false" xref="S3.p6.14.m14.1.2.cmml">)</mo></mrow></msubsup><annotation-xml encoding="MathML-Content" id="S3.p6.14.m14.1b"><apply id="S3.p6.14.m14.1.2.cmml" xref="S3.p6.14.m14.1.2"><csymbol cd="ambiguous" id="S3.p6.14.m14.1.2.1.cmml" xref="S3.p6.14.m14.1.2">superscript</csymbol><apply id="S3.p6.14.m14.1.2.2.cmml" xref="S3.p6.14.m14.1.2"><csymbol cd="ambiguous" id="S3.p6.14.m14.1.2.2.1.cmml" xref="S3.p6.14.m14.1.2">subscript</csymbol><ci id="S3.p6.14.m14.1.2.2.2.cmml" xref="S3.p6.14.m14.1.2.2.2">𝐋</ci><cn id="S3.p6.14.m14.1.2.2.3.cmml" type="integer" xref="S3.p6.14.m14.1.2.2.3">2</cn></apply><ci id="S3.p6.14.m14.1.1.1.1.cmml" xref="S3.p6.14.m14.1.1.1.1">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p6.14.m14.1c">\operatorname{\mathbf{L}}_{2}^{(t)}</annotation><annotation encoding="application/x-llamapun" id="S3.p6.14.m14.1d">bold_L start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_t ) end_POSTSUPERSCRIPT</annotation></semantics></math> across different values of tuning parameter <math alttext="\lambda_{\operatorname{\mathbf{L}}}" class="ltx_Math" display="inline" id="S3.p6.15.m15.1"><semantics id="S3.p6.15.m15.1a"><msub id="S3.p6.15.m15.1.1" xref="S3.p6.15.m15.1.1.cmml"><mi id="S3.p6.15.m15.1.1.2" xref="S3.p6.15.m15.1.1.2.cmml">λ</mi><mi id="S3.p6.15.m15.1.1.3" xref="S3.p6.15.m15.1.1.3.cmml">𝐋</mi></msub><annotation-xml encoding="MathML-Content" id="S3.p6.15.m15.1b"><apply id="S3.p6.15.m15.1.1.cmml" xref="S3.p6.15.m15.1.1"><csymbol cd="ambiguous" id="S3.p6.15.m15.1.1.1.cmml" xref="S3.p6.15.m15.1.1">subscript</csymbol><ci id="S3.p6.15.m15.1.1.2.cmml" xref="S3.p6.15.m15.1.1.2">𝜆</ci><ci id="S3.p6.15.m15.1.1.3.cmml" xref="S3.p6.15.m15.1.1.3">𝐋</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p6.15.m15.1c">\lambda_{\operatorname{\mathbf{L}}}</annotation><annotation encoding="application/x-llamapun" id="S3.p6.15.m15.1d">italic_λ start_POSTSUBSCRIPT bold_L end_POSTSUBSCRIPT</annotation></semantics></math> and random initializations <math alttext="\operatorname{\mathbf{L}}_{1}^{(0)}" class="ltx_Math" display="inline" id="S3.p6.16.m16.1"><semantics id="S3.p6.16.m16.1a"><msubsup id="S3.p6.16.m16.1.2" xref="S3.p6.16.m16.1.2.cmml"><mi id="S3.p6.16.m16.1.2.2.2" xref="S3.p6.16.m16.1.2.2.2.cmml">𝐋</mi><mn id="S3.p6.16.m16.1.2.2.3" xref="S3.p6.16.m16.1.2.2.3.cmml">1</mn><mrow id="S3.p6.16.m16.1.1.1.3" xref="S3.p6.16.m16.1.2.cmml"><mo id="S3.p6.16.m16.1.1.1.3.1" stretchy="false" xref="S3.p6.16.m16.1.2.cmml">(</mo><mn id="S3.p6.16.m16.1.1.1.1" xref="S3.p6.16.m16.1.1.1.1.cmml">0</mn><mo id="S3.p6.16.m16.1.1.1.3.2" stretchy="false" xref="S3.p6.16.m16.1.2.cmml">)</mo></mrow></msubsup><annotation-xml encoding="MathML-Content" id="S3.p6.16.m16.1b"><apply id="S3.p6.16.m16.1.2.cmml" xref="S3.p6.16.m16.1.2"><csymbol cd="ambiguous" id="S3.p6.16.m16.1.2.1.cmml" xref="S3.p6.16.m16.1.2">superscript</csymbol><apply id="S3.p6.16.m16.1.2.2.cmml" xref="S3.p6.16.m16.1.2"><csymbol cd="ambiguous" id="S3.p6.16.m16.1.2.2.1.cmml" xref="S3.p6.16.m16.1.2">subscript</csymbol><ci id="S3.p6.16.m16.1.2.2.2.cmml" xref="S3.p6.16.m16.1.2.2.2">𝐋</ci><cn id="S3.p6.16.m16.1.2.2.3.cmml" type="integer" xref="S3.p6.16.m16.1.2.2.3">1</cn></apply><cn id="S3.p6.16.m16.1.1.1.1.cmml" type="integer" xref="S3.p6.16.m16.1.1.1.1">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p6.16.m16.1c">\operatorname{\mathbf{L}}_{1}^{(0)}</annotation><annotation encoding="application/x-llamapun" id="S3.p6.16.m16.1d">bold_L start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( 0 ) end_POSTSUPERSCRIPT</annotation></semantics></math> and <math alttext="\operatorname{\mathbf{L}}_{2}^{(0)}" class="ltx_Math" display="inline" id="S3.p6.17.m17.1"><semantics id="S3.p6.17.m17.1a"><msubsup id="S3.p6.17.m17.1.2" xref="S3.p6.17.m17.1.2.cmml"><mi id="S3.p6.17.m17.1.2.2.2" xref="S3.p6.17.m17.1.2.2.2.cmml">𝐋</mi><mn id="S3.p6.17.m17.1.2.2.3" xref="S3.p6.17.m17.1.2.2.3.cmml">2</mn><mrow id="S3.p6.17.m17.1.1.1.3" xref="S3.p6.17.m17.1.2.cmml"><mo id="S3.p6.17.m17.1.1.1.3.1" stretchy="false" xref="S3.p6.17.m17.1.2.cmml">(</mo><mn id="S3.p6.17.m17.1.1.1.1" xref="S3.p6.17.m17.1.1.1.1.cmml">0</mn><mo id="S3.p6.17.m17.1.1.1.3.2" stretchy="false" xref="S3.p6.17.m17.1.2.cmml">)</mo></mrow></msubsup><annotation-xml encoding="MathML-Content" id="S3.p6.17.m17.1b"><apply id="S3.p6.17.m17.1.2.cmml" xref="S3.p6.17.m17.1.2"><csymbol cd="ambiguous" id="S3.p6.17.m17.1.2.1.cmml" xref="S3.p6.17.m17.1.2">superscript</csymbol><apply id="S3.p6.17.m17.1.2.2.cmml" xref="S3.p6.17.m17.1.2"><csymbol cd="ambiguous" id="S3.p6.17.m17.1.2.2.1.cmml" xref="S3.p6.17.m17.1.2">subscript</csymbol><ci id="S3.p6.17.m17.1.2.2.2.cmml" xref="S3.p6.17.m17.1.2.2.2">𝐋</ci><cn id="S3.p6.17.m17.1.2.2.3.cmml" type="integer" xref="S3.p6.17.m17.1.2.2.3">2</cn></apply><cn id="S3.p6.17.m17.1.1.1.1.cmml" type="integer" xref="S3.p6.17.m17.1.1.1.1">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p6.17.m17.1c">\operatorname{\mathbf{L}}_{2}^{(0)}</annotation><annotation encoding="application/x-llamapun" id="S3.p6.17.m17.1d">bold_L start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( 0 ) end_POSTSUPERSCRIPT</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S3.p7"> <p class="ltx_p" id="S3.p7.14">The MMTR’s parameter estimate <math alttext="\operatorname{{\bm{\theta}}}^{{(\infty)}}" class="ltx_Math" display="inline" id="S3.p7.1.m1.1"><semantics id="S3.p7.1.m1.1a"><msup id="S3.p7.1.m1.1.2" xref="S3.p7.1.m1.1.2.cmml"><mi id="S3.p7.1.m1.1.2.2" xref="S3.p7.1.m1.1.2.2.cmml">𝜽</mi><mrow id="S3.p7.1.m1.1.1.1.3" xref="S3.p7.1.m1.1.2.cmml"><mo id="S3.p7.1.m1.1.1.1.3.1" stretchy="false" xref="S3.p7.1.m1.1.2.cmml">(</mo><mi id="S3.p7.1.m1.1.1.1.1" mathvariant="normal" xref="S3.p7.1.m1.1.1.1.1.cmml">∞</mi><mo id="S3.p7.1.m1.1.1.1.3.2" stretchy="false" xref="S3.p7.1.m1.1.2.cmml">)</mo></mrow></msup><annotation-xml encoding="MathML-Content" id="S3.p7.1.m1.1b"><apply id="S3.p7.1.m1.1.2.cmml" xref="S3.p7.1.m1.1.2"><csymbol cd="ambiguous" id="S3.p7.1.m1.1.2.1.cmml" xref="S3.p7.1.m1.1.2">superscript</csymbol><ci id="S3.p7.1.m1.1.2.2.cmml" xref="S3.p7.1.m1.1.2.2">𝜽</ci><infinity id="S3.p7.1.m1.1.1.1.1.cmml" xref="S3.p7.1.m1.1.1.1.1"></infinity></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p7.1.m1.1c">\operatorname{{\bm{\theta}}}^{{(\infty)}}</annotation><annotation encoding="application/x-llamapun" id="S3.p7.1.m1.1d">bold_italic_θ start_POSTSUPERSCRIPT ( ∞ ) end_POSTSUPERSCRIPT</annotation></semantics></math> is a regularized maximum likehood estimate of <math alttext="\operatorname{{\bm{\theta}}}" class="ltx_Math" display="inline" id="S3.p7.2.m2.1"><semantics id="S3.p7.2.m2.1a"><mi id="S3.p7.2.m2.1.1" xref="S3.p7.2.m2.1.1.cmml">𝜽</mi><annotation-xml encoding="MathML-Content" id="S3.p7.2.m2.1b"><ci id="S3.p7.2.m2.1.1.cmml" xref="S3.p7.2.m2.1.1">𝜽</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.p7.2.m2.1c">\operatorname{{\bm{\theta}}}</annotation><annotation encoding="application/x-llamapun" id="S3.p7.2.m2.1d">bold_italic_θ</annotation></semantics></math>. Let <math alttext="\ell(\operatorname{{\bm{\theta}}})" class="ltx_Math" display="inline" id="S3.p7.3.m3.1"><semantics id="S3.p7.3.m3.1a"><mrow id="S3.p7.3.m3.1.2" xref="S3.p7.3.m3.1.2.cmml"><mi id="S3.p7.3.m3.1.2.2" mathvariant="normal" xref="S3.p7.3.m3.1.2.2.cmml">ℓ</mi><mo id="S3.p7.3.m3.1.2.1" xref="S3.p7.3.m3.1.2.1.cmml">⁢</mo><mrow id="S3.p7.3.m3.1.2.3.2" xref="S3.p7.3.m3.1.2.cmml"><mo id="S3.p7.3.m3.1.2.3.2.1" stretchy="false" xref="S3.p7.3.m3.1.2.cmml">(</mo><mi id="S3.p7.3.m3.1.1" xref="S3.p7.3.m3.1.1.cmml">𝜽</mi><mo id="S3.p7.3.m3.1.2.3.2.2" stretchy="false" xref="S3.p7.3.m3.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.p7.3.m3.1b"><apply id="S3.p7.3.m3.1.2.cmml" xref="S3.p7.3.m3.1.2"><times id="S3.p7.3.m3.1.2.1.cmml" xref="S3.p7.3.m3.1.2.1"></times><ci id="S3.p7.3.m3.1.2.2.cmml" xref="S3.p7.3.m3.1.2.2">ℓ</ci><ci id="S3.p7.3.m3.1.1.cmml" xref="S3.p7.3.m3.1.1">𝜽</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p7.3.m3.1c">\ell(\operatorname{{\bm{\theta}}})</annotation><annotation encoding="application/x-llamapun" id="S3.p7.3.m3.1d">roman_ℓ ( bold_italic_θ )</annotation></semantics></math> be the negative log likehood function implied by (<a class="ltx_ref" href="https://arxiv.org/html/2503.13782v1#S2.E3" title="In 2.1 Model ‣ 2 Mixed Model Trace Regression ‣ Regularized Parameter Estimation in Mixed Model Trace Regression"><span class="ltx_text ltx_ref_tag">3</span></a>), <math alttext="\operatorname{{\bm{\theta}}}_{1}=(\operatorname{\mathbf{B}},\tau^{2})" class="ltx_Math" display="inline" id="S3.p7.4.m4.2"><semantics id="S3.p7.4.m4.2a"><mrow id="S3.p7.4.m4.2.2" xref="S3.p7.4.m4.2.2.cmml"><msub id="S3.p7.4.m4.2.2.3" xref="S3.p7.4.m4.2.2.3.cmml"><mi id="S3.p7.4.m4.2.2.3.2" xref="S3.p7.4.m4.2.2.3.2.cmml">𝜽</mi><mn id="S3.p7.4.m4.2.2.3.3" xref="S3.p7.4.m4.2.2.3.3.cmml">1</mn></msub><mo id="S3.p7.4.m4.2.2.2" xref="S3.p7.4.m4.2.2.2.cmml">=</mo><mrow id="S3.p7.4.m4.2.2.1.1" 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xref="S3.p7.4.m4.2.2.3.3">1</cn></apply><interval closure="open" id="S3.p7.4.m4.2.2.1.2.cmml" xref="S3.p7.4.m4.2.2.1.1"><ci id="S3.p7.4.m4.1.1.cmml" xref="S3.p7.4.m4.1.1">𝐁</ci><apply id="S3.p7.4.m4.2.2.1.1.1.cmml" xref="S3.p7.4.m4.2.2.1.1.1"><csymbol cd="ambiguous" id="S3.p7.4.m4.2.2.1.1.1.1.cmml" xref="S3.p7.4.m4.2.2.1.1.1">superscript</csymbol><ci id="S3.p7.4.m4.2.2.1.1.1.2.cmml" xref="S3.p7.4.m4.2.2.1.1.1.2">𝜏</ci><cn id="S3.p7.4.m4.2.2.1.1.1.3.cmml" type="integer" xref="S3.p7.4.m4.2.2.1.1.1.3">2</cn></apply></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p7.4.m4.2c">\operatorname{{\bm{\theta}}}_{1}=(\operatorname{\mathbf{B}},\tau^{2})</annotation><annotation encoding="application/x-llamapun" id="S3.p7.4.m4.2d">bold_italic_θ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = ( bold_B , italic_τ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT )</annotation></semantics></math>, <math alttext="\operatorname{{\bm{\theta}}}_{2}=\operatorname{\mathbf{L}}_{1}" 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xref="S3.p7.10.m10.1.1.1.1.2">𝒫</ci><apply id="S3.p7.10.m10.1.1.1.1.3.cmml" xref="S3.p7.10.m10.1.1.1.1.3"><csymbol cd="ambiguous" id="S3.p7.10.m10.1.1.1.1.3.1.cmml" xref="S3.p7.10.m10.1.1.1.1.3">subscript</csymbol><ci id="S3.p7.10.m10.1.1.1.1.3.2.cmml" xref="S3.p7.10.m10.1.1.1.1.3.2">𝜆</ci><ci id="S3.p7.10.m10.1.1.1.1.3.3.cmml" xref="S3.p7.10.m10.1.1.1.1.3.3">𝐋</ci></apply></apply><apply id="S3.p7.10.m10.2.2.2.2.1.cmml" xref="S3.p7.10.m10.2.2.2.2.1"><csymbol cd="ambiguous" id="S3.p7.10.m10.2.2.2.2.1.1.cmml" xref="S3.p7.10.m10.2.2.2.2.1">subscript</csymbol><ci id="S3.p7.10.m10.2.2.2.2.1.2.cmml" xref="S3.p7.10.m10.2.2.2.2.1.2">𝜽</ci><cn id="S3.p7.10.m10.2.2.2.2.1.3.cmml" type="integer" xref="S3.p7.10.m10.2.2.2.2.1.3">3</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p7.10.m10.2c">\operatorname{\mathcal{P}}_{\lambda_{\operatorname{\mathbf{L}}}}(\operatorname% {{\bm{\theta}}}_{3})</annotation><annotation encoding="application/x-llamapun" id="S3.p7.10.m10.2d">caligraphic_P start_POSTSUBSCRIPT italic_λ start_POSTSUBSCRIPT bold_L end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( bold_italic_θ start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT )</annotation></semantics></math> are the penalties on <math alttext="\operatorname{\mathbf{B}}" class="ltx_Math" display="inline" id="S3.p7.11.m11.1"><semantics id="S3.p7.11.m11.1a"><mi id="S3.p7.11.m11.1.1" xref="S3.p7.11.m11.1.1.cmml">𝐁</mi><annotation-xml encoding="MathML-Content" id="S3.p7.11.m11.1b"><ci id="S3.p7.11.m11.1.1.cmml" xref="S3.p7.11.m11.1.1">𝐁</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.p7.11.m11.1c">\operatorname{\mathbf{B}}</annotation><annotation encoding="application/x-llamapun" id="S3.p7.11.m11.1d">bold_B</annotation></semantics></math>, <math alttext="\operatorname{\mathbf{L}}_{1}" class="ltx_Math" display="inline" id="S3.p7.12.m12.1"><semantics id="S3.p7.12.m12.1a"><msub id="S3.p7.12.m12.1.1" xref="S3.p7.12.m12.1.1.cmml"><mi id="S3.p7.12.m12.1.1.2" xref="S3.p7.12.m12.1.1.2.cmml">𝐋</mi><mn id="S3.p7.12.m12.1.1.3" xref="S3.p7.12.m12.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S3.p7.12.m12.1b"><apply id="S3.p7.12.m12.1.1.cmml" xref="S3.p7.12.m12.1.1"><csymbol cd="ambiguous" id="S3.p7.12.m12.1.1.1.cmml" xref="S3.p7.12.m12.1.1">subscript</csymbol><ci id="S3.p7.12.m12.1.1.2.cmml" xref="S3.p7.12.m12.1.1.2">𝐋</ci><cn id="S3.p7.12.m12.1.1.3.cmml" type="integer" xref="S3.p7.12.m12.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p7.12.m12.1c">\operatorname{\mathbf{L}}_{1}</annotation><annotation encoding="application/x-llamapun" id="S3.p7.12.m12.1d">bold_L start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math>, and <math alttext="\operatorname{\mathbf{L}}_{2}" class="ltx_Math" display="inline" id="S3.p7.13.m13.1"><semantics id="S3.p7.13.m13.1a"><msub id="S3.p7.13.m13.1.1" xref="S3.p7.13.m13.1.1.cmml"><mi id="S3.p7.13.m13.1.1.2" xref="S3.p7.13.m13.1.1.2.cmml">𝐋</mi><mn id="S3.p7.13.m13.1.1.3" xref="S3.p7.13.m13.1.1.3.cmml">2</mn></msub><annotation-xml encoding="MathML-Content" id="S3.p7.13.m13.1b"><apply id="S3.p7.13.m13.1.1.cmml" xref="S3.p7.13.m13.1.1"><csymbol cd="ambiguous" id="S3.p7.13.m13.1.1.1.cmml" xref="S3.p7.13.m13.1.1">subscript</csymbol><ci id="S3.p7.13.m13.1.1.2.cmml" xref="S3.p7.13.m13.1.1.2">𝐋</ci><cn id="S3.p7.13.m13.1.1.3.cmml" type="integer" xref="S3.p7.13.m13.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p7.13.m13.1c">\operatorname{\mathbf{L}}_{2}</annotation><annotation encoding="application/x-llamapun" id="S3.p7.13.m13.1d">bold_L start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math> in (<a class="ltx_ref" href="https://arxiv.org/html/2503.13782v1#S3.E8" title="In 3 Regularized Parameter Estimation ‣ Regularized Parameter Estimation in Mixed Model Trace Regression"><span class="ltx_text ltx_ref_tag">8</span></a>), (<a class="ltx_ref" href="https://arxiv.org/html/2503.13782v1#S3.E10" title="In 3 Regularized Parameter Estimation ‣ Regularized Parameter Estimation in Mixed Model Trace Regression"><span class="ltx_text ltx_ref_tag">10</span></a>), and (<a class="ltx_ref" href="https://arxiv.org/html/2503.13782v1#S3.E11" title="In 3 Regularized Parameter Estimation ‣ Regularized Parameter Estimation in Mixed Model Trace Regression"><span class="ltx_text ltx_ref_tag">11</span></a>), respectively. Then, the following proposition shows that <math alttext="\{\operatorname{{\bm{\theta}}}^{(t)}\}_{t=1}^{\infty}" class="ltx_Math" display="inline" id="S3.p7.14.m14.2"><semantics id="S3.p7.14.m14.2a"><msubsup id="S3.p7.14.m14.2.2" xref="S3.p7.14.m14.2.2.cmml"><mrow id="S3.p7.14.m14.2.2.1.1.1" xref="S3.p7.14.m14.2.2.1.1.2.cmml"><mo id="S3.p7.14.m14.2.2.1.1.1.2" stretchy="false" xref="S3.p7.14.m14.2.2.1.1.2.cmml">{</mo><msup id="S3.p7.14.m14.2.2.1.1.1.1" xref="S3.p7.14.m14.2.2.1.1.1.1.cmml"><mi id="S3.p7.14.m14.2.2.1.1.1.1.2" xref="S3.p7.14.m14.2.2.1.1.1.1.2.cmml">𝜽</mi><mrow id="S3.p7.14.m14.1.1.1.3" xref="S3.p7.14.m14.2.2.1.1.1.1.cmml"><mo id="S3.p7.14.m14.1.1.1.3.1" stretchy="false" xref="S3.p7.14.m14.2.2.1.1.1.1.cmml">(</mo><mi id="S3.p7.14.m14.1.1.1.1" xref="S3.p7.14.m14.1.1.1.1.cmml">t</mi><mo id="S3.p7.14.m14.1.1.1.3.2" stretchy="false" xref="S3.p7.14.m14.2.2.1.1.1.1.cmml">)</mo></mrow></msup><mo id="S3.p7.14.m14.2.2.1.1.1.3" stretchy="false" xref="S3.p7.14.m14.2.2.1.1.2.cmml">}</mo></mrow><mrow id="S3.p7.14.m14.2.2.1.3" xref="S3.p7.14.m14.2.2.1.3.cmml"><mi id="S3.p7.14.m14.2.2.1.3.2" xref="S3.p7.14.m14.2.2.1.3.2.cmml">t</mi><mo id="S3.p7.14.m14.2.2.1.3.1" xref="S3.p7.14.m14.2.2.1.3.1.cmml">=</mo><mn id="S3.p7.14.m14.2.2.1.3.3" xref="S3.p7.14.m14.2.2.1.3.3.cmml">1</mn></mrow><mi id="S3.p7.14.m14.2.2.3" mathvariant="normal" xref="S3.p7.14.m14.2.2.3.cmml">∞</mi></msubsup><annotation-xml encoding="MathML-Content" id="S3.p7.14.m14.2b"><apply id="S3.p7.14.m14.2.2.cmml" xref="S3.p7.14.m14.2.2"><csymbol cd="ambiguous" id="S3.p7.14.m14.2.2.2.cmml" xref="S3.p7.14.m14.2.2">superscript</csymbol><apply id="S3.p7.14.m14.2.2.1.cmml" xref="S3.p7.14.m14.2.2"><csymbol cd="ambiguous" id="S3.p7.14.m14.2.2.1.2.cmml" xref="S3.p7.14.m14.2.2">subscript</csymbol><set id="S3.p7.14.m14.2.2.1.1.2.cmml" xref="S3.p7.14.m14.2.2.1.1.1"><apply id="S3.p7.14.m14.2.2.1.1.1.1.cmml" xref="S3.p7.14.m14.2.2.1.1.1.1"><csymbol cd="ambiguous" id="S3.p7.14.m14.2.2.1.1.1.1.1.cmml" xref="S3.p7.14.m14.2.2.1.1.1.1">superscript</csymbol><ci id="S3.p7.14.m14.2.2.1.1.1.1.2.cmml" xref="S3.p7.14.m14.2.2.1.1.1.1.2">𝜽</ci><ci id="S3.p7.14.m14.1.1.1.1.cmml" xref="S3.p7.14.m14.1.1.1.1">𝑡</ci></apply></set><apply id="S3.p7.14.m14.2.2.1.3.cmml" xref="S3.p7.14.m14.2.2.1.3"><eq id="S3.p7.14.m14.2.2.1.3.1.cmml" xref="S3.p7.14.m14.2.2.1.3.1"></eq><ci id="S3.p7.14.m14.2.2.1.3.2.cmml" xref="S3.p7.14.m14.2.2.1.3.2">𝑡</ci><cn id="S3.p7.14.m14.2.2.1.3.3.cmml" type="integer" xref="S3.p7.14.m14.2.2.1.3.3">1</cn></apply></apply><infinity id="S3.p7.14.m14.2.2.3.cmml" xref="S3.p7.14.m14.2.2.3"></infinity></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p7.14.m14.2c">\{\operatorname{{\bm{\theta}}}^{(t)}\}_{t=1}^{\infty}</annotation><annotation encoding="application/x-llamapun" id="S3.p7.14.m14.2d">{ bold_italic_θ start_POSTSUPERSCRIPT ( italic_t ) end_POSTSUPERSCRIPT } start_POSTSUBSCRIPT italic_t = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT</annotation></semantics></math> sequence produced by Algorithm <a class="ltx_ref" href="https://arxiv.org/html/2503.13782v1#alg1" title="In 3 Regularized Parameter Estimation ‣ Regularized Parameter Estimation in Mixed Model Trace Regression"><span class="ltx_text ltx_ref_tag">1</span></a> converges under weak assumptions.</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></h6> <div class="ltx_para" id="S3.Thmproposition1.p1"> <p class="ltx_p" id="S3.Thmproposition1.p1.9"><span class="ltx_text ltx_font_italic" id="S3.Thmproposition1.p1.9.9">The MMTR objective is <math alttext="f(\operatorname{{\bm{\theta}}})=\ell(\operatorname{{\bm{\theta}}})+% \operatorname{\mathcal{P}}_{\lambda_{\operatorname{\mathbf{B}}}}(\operatorname% {{\bm{\theta}}}_{1})+\operatorname{\mathcal{P}}_{\lambda_{\operatorname{% \mathbf{L}}}}(\operatorname{{\bm{\theta}}}_{2})+\operatorname{\mathcal{P}}_{% \lambda_{\operatorname{\mathbf{L}}}}(\operatorname{{\bm{\theta}}}_{3})" class="ltx_Math" display="inline" id="S3.Thmproposition1.p1.1.1.m1.8"><semantics id="S3.Thmproposition1.p1.1.1.m1.8a"><mrow id="S3.Thmproposition1.p1.1.1.m1.8.8" xref="S3.Thmproposition1.p1.1.1.m1.8.8.cmml"><mrow id="S3.Thmproposition1.p1.1.1.m1.8.8.8" xref="S3.Thmproposition1.p1.1.1.m1.8.8.8.cmml"><mi id="S3.Thmproposition1.p1.1.1.m1.8.8.8.2" xref="S3.Thmproposition1.p1.1.1.m1.8.8.8.2.cmml">f</mi><mo id="S3.Thmproposition1.p1.1.1.m1.8.8.8.1" xref="S3.Thmproposition1.p1.1.1.m1.8.8.8.1.cmml">⁢</mo><mrow id="S3.Thmproposition1.p1.1.1.m1.8.8.8.3.2" xref="S3.Thmproposition1.p1.1.1.m1.8.8.8.cmml"><mo id="S3.Thmproposition1.p1.1.1.m1.8.8.8.3.2.1" stretchy="false" xref="S3.Thmproposition1.p1.1.1.m1.8.8.8.cmml">(</mo><mi id="S3.Thmproposition1.p1.1.1.m1.1.1" xref="S3.Thmproposition1.p1.1.1.m1.1.1.cmml">𝛉</mi><mo id="S3.Thmproposition1.p1.1.1.m1.8.8.8.3.2.2" stretchy="false" xref="S3.Thmproposition1.p1.1.1.m1.8.8.8.cmml">)</mo></mrow></mrow><mo id="S3.Thmproposition1.p1.1.1.m1.8.8.7" xref="S3.Thmproposition1.p1.1.1.m1.8.8.7.cmml">=</mo><mrow id="S3.Thmproposition1.p1.1.1.m1.8.8.6" xref="S3.Thmproposition1.p1.1.1.m1.8.8.6.cmml"><mrow id="S3.Thmproposition1.p1.1.1.m1.8.8.6.8" xref="S3.Thmproposition1.p1.1.1.m1.8.8.6.8.cmml"><mi id="S3.Thmproposition1.p1.1.1.m1.8.8.6.8.2" mathvariant="normal" xref="S3.Thmproposition1.p1.1.1.m1.8.8.6.8.2.cmml">ℓ</mi><mo id="S3.Thmproposition1.p1.1.1.m1.8.8.6.8.1" xref="S3.Thmproposition1.p1.1.1.m1.8.8.6.8.1.cmml">⁢</mo><mrow id="S3.Thmproposition1.p1.1.1.m1.8.8.6.8.3.2" xref="S3.Thmproposition1.p1.1.1.m1.8.8.6.8.cmml"><mo id="S3.Thmproposition1.p1.1.1.m1.8.8.6.8.3.2.1" stretchy="false" xref="S3.Thmproposition1.p1.1.1.m1.8.8.6.8.cmml">(</mo><mi id="S3.Thmproposition1.p1.1.1.m1.2.2" xref="S3.Thmproposition1.p1.1.1.m1.2.2.cmml">𝛉</mi><mo id="S3.Thmproposition1.p1.1.1.m1.8.8.6.8.3.2.2" stretchy="false" xref="S3.Thmproposition1.p1.1.1.m1.8.8.6.8.cmml">)</mo></mrow></mrow><mo id="S3.Thmproposition1.p1.1.1.m1.8.8.6.7" xref="S3.Thmproposition1.p1.1.1.m1.8.8.6.7.cmml">+</mo><mrow id="S3.Thmproposition1.p1.1.1.m1.4.4.2.2.2" xref="S3.Thmproposition1.p1.1.1.m1.4.4.2.2.3.cmml"><msub id="S3.Thmproposition1.p1.1.1.m1.3.3.1.1.1.1" xref="S3.Thmproposition1.p1.1.1.m1.3.3.1.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.Thmproposition1.p1.1.1.m1.3.3.1.1.1.1.2" xref="S3.Thmproposition1.p1.1.1.m1.3.3.1.1.1.1.2.cmml">𝒫</mi><msub id="S3.Thmproposition1.p1.1.1.m1.3.3.1.1.1.1.3" xref="S3.Thmproposition1.p1.1.1.m1.3.3.1.1.1.1.3.cmml"><mi id="S3.Thmproposition1.p1.1.1.m1.3.3.1.1.1.1.3.2" xref="S3.Thmproposition1.p1.1.1.m1.3.3.1.1.1.1.3.2.cmml">λ</mi><mi id="S3.Thmproposition1.p1.1.1.m1.3.3.1.1.1.1.3.3" xref="S3.Thmproposition1.p1.1.1.m1.3.3.1.1.1.1.3.3.cmml">𝐁</mi></msub></msub><mo id="S3.Thmproposition1.p1.1.1.m1.4.4.2.2.2a" xref="S3.Thmproposition1.p1.1.1.m1.4.4.2.2.3.cmml">⁡</mo><mrow id="S3.Thmproposition1.p1.1.1.m1.4.4.2.2.2.2" xref="S3.Thmproposition1.p1.1.1.m1.4.4.2.2.3.cmml"><mo id="S3.Thmproposition1.p1.1.1.m1.4.4.2.2.2.2.2" stretchy="false" xref="S3.Thmproposition1.p1.1.1.m1.4.4.2.2.3.cmml">(</mo><msub id="S3.Thmproposition1.p1.1.1.m1.4.4.2.2.2.2.1" xref="S3.Thmproposition1.p1.1.1.m1.4.4.2.2.2.2.1.cmml"><mi id="S3.Thmproposition1.p1.1.1.m1.4.4.2.2.2.2.1.2" xref="S3.Thmproposition1.p1.1.1.m1.4.4.2.2.2.2.1.2.cmml">𝛉</mi><mn id="S3.Thmproposition1.p1.1.1.m1.4.4.2.2.2.2.1.3" xref="S3.Thmproposition1.p1.1.1.m1.4.4.2.2.2.2.1.3.cmml">1</mn></msub><mo id="S3.Thmproposition1.p1.1.1.m1.4.4.2.2.2.2.3" stretchy="false" xref="S3.Thmproposition1.p1.1.1.m1.4.4.2.2.3.cmml">)</mo></mrow></mrow><mo id="S3.Thmproposition1.p1.1.1.m1.8.8.6.7a" xref="S3.Thmproposition1.p1.1.1.m1.8.8.6.7.cmml">+</mo><mrow id="S3.Thmproposition1.p1.1.1.m1.6.6.4.4.2" xref="S3.Thmproposition1.p1.1.1.m1.6.6.4.4.3.cmml"><msub id="S3.Thmproposition1.p1.1.1.m1.5.5.3.3.1.1" xref="S3.Thmproposition1.p1.1.1.m1.5.5.3.3.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.Thmproposition1.p1.1.1.m1.5.5.3.3.1.1.2" xref="S3.Thmproposition1.p1.1.1.m1.5.5.3.3.1.1.2.cmml">𝒫</mi><msub id="S3.Thmproposition1.p1.1.1.m1.5.5.3.3.1.1.3" xref="S3.Thmproposition1.p1.1.1.m1.5.5.3.3.1.1.3.cmml"><mi id="S3.Thmproposition1.p1.1.1.m1.5.5.3.3.1.1.3.2" xref="S3.Thmproposition1.p1.1.1.m1.5.5.3.3.1.1.3.2.cmml">λ</mi><mi id="S3.Thmproposition1.p1.1.1.m1.5.5.3.3.1.1.3.3" xref="S3.Thmproposition1.p1.1.1.m1.5.5.3.3.1.1.3.3.cmml">𝐋</mi></msub></msub><mo id="S3.Thmproposition1.p1.1.1.m1.6.6.4.4.2a" xref="S3.Thmproposition1.p1.1.1.m1.6.6.4.4.3.cmml">⁡</mo><mrow id="S3.Thmproposition1.p1.1.1.m1.6.6.4.4.2.2" xref="S3.Thmproposition1.p1.1.1.m1.6.6.4.4.3.cmml"><mo 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xref="S3.Thmproposition1.p1.1.1.m1.8.8.6.6.2.2.1.2.cmml">𝛉</mi><mn id="S3.Thmproposition1.p1.1.1.m1.8.8.6.6.2.2.1.3" xref="S3.Thmproposition1.p1.1.1.m1.8.8.6.6.2.2.1.3.cmml">3</mn></msub><mo id="S3.Thmproposition1.p1.1.1.m1.8.8.6.6.2.2.3" stretchy="false" xref="S3.Thmproposition1.p1.1.1.m1.8.8.6.6.3.cmml">)</mo></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmproposition1.p1.1.1.m1.8b"><apply id="S3.Thmproposition1.p1.1.1.m1.8.8.cmml" xref="S3.Thmproposition1.p1.1.1.m1.8.8"><eq id="S3.Thmproposition1.p1.1.1.m1.8.8.7.cmml" xref="S3.Thmproposition1.p1.1.1.m1.8.8.7"></eq><apply id="S3.Thmproposition1.p1.1.1.m1.8.8.8.cmml" xref="S3.Thmproposition1.p1.1.1.m1.8.8.8"><times id="S3.Thmproposition1.p1.1.1.m1.8.8.8.1.cmml" xref="S3.Thmproposition1.p1.1.1.m1.8.8.8.1"></times><ci id="S3.Thmproposition1.p1.1.1.m1.8.8.8.2.cmml" xref="S3.Thmproposition1.p1.1.1.m1.8.8.8.2">𝑓</ci><ci id="S3.Thmproposition1.p1.1.1.m1.1.1.cmml" 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xref="S3.Thmproposition1.p1.1.1.m1.3.3.1.1.1.1">subscript</csymbol><ci id="S3.Thmproposition1.p1.1.1.m1.3.3.1.1.1.1.2.cmml" xref="S3.Thmproposition1.p1.1.1.m1.3.3.1.1.1.1.2">𝒫</ci><apply id="S3.Thmproposition1.p1.1.1.m1.3.3.1.1.1.1.3.cmml" xref="S3.Thmproposition1.p1.1.1.m1.3.3.1.1.1.1.3"><csymbol cd="ambiguous" id="S3.Thmproposition1.p1.1.1.m1.3.3.1.1.1.1.3.1.cmml" xref="S3.Thmproposition1.p1.1.1.m1.3.3.1.1.1.1.3">subscript</csymbol><ci id="S3.Thmproposition1.p1.1.1.m1.3.3.1.1.1.1.3.2.cmml" xref="S3.Thmproposition1.p1.1.1.m1.3.3.1.1.1.1.3.2">𝜆</ci><ci id="S3.Thmproposition1.p1.1.1.m1.3.3.1.1.1.1.3.3.cmml" xref="S3.Thmproposition1.p1.1.1.m1.3.3.1.1.1.1.3.3">𝐁</ci></apply></apply><apply id="S3.Thmproposition1.p1.1.1.m1.4.4.2.2.2.2.1.cmml" xref="S3.Thmproposition1.p1.1.1.m1.4.4.2.2.2.2.1"><csymbol cd="ambiguous" id="S3.Thmproposition1.p1.1.1.m1.4.4.2.2.2.2.1.1.cmml" xref="S3.Thmproposition1.p1.1.1.m1.4.4.2.2.2.2.1">subscript</csymbol><ci 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id="S3.Thmproposition1.p1.1.1.m1.7.7.5.5.1.1.1.cmml" xref="S3.Thmproposition1.p1.1.1.m1.7.7.5.5.1.1">subscript</csymbol><ci id="S3.Thmproposition1.p1.1.1.m1.7.7.5.5.1.1.2.cmml" xref="S3.Thmproposition1.p1.1.1.m1.7.7.5.5.1.1.2">𝒫</ci><apply id="S3.Thmproposition1.p1.1.1.m1.7.7.5.5.1.1.3.cmml" xref="S3.Thmproposition1.p1.1.1.m1.7.7.5.5.1.1.3"><csymbol cd="ambiguous" id="S3.Thmproposition1.p1.1.1.m1.7.7.5.5.1.1.3.1.cmml" xref="S3.Thmproposition1.p1.1.1.m1.7.7.5.5.1.1.3">subscript</csymbol><ci id="S3.Thmproposition1.p1.1.1.m1.7.7.5.5.1.1.3.2.cmml" xref="S3.Thmproposition1.p1.1.1.m1.7.7.5.5.1.1.3.2">𝜆</ci><ci id="S3.Thmproposition1.p1.1.1.m1.7.7.5.5.1.1.3.3.cmml" xref="S3.Thmproposition1.p1.1.1.m1.7.7.5.5.1.1.3.3">𝐋</ci></apply></apply><apply id="S3.Thmproposition1.p1.1.1.m1.8.8.6.6.2.2.1.cmml" xref="S3.Thmproposition1.p1.1.1.m1.8.8.6.6.2.2.1"><csymbol cd="ambiguous" id="S3.Thmproposition1.p1.1.1.m1.8.8.6.6.2.2.1.1.cmml" xref="S3.Thmproposition1.p1.1.1.m1.8.8.6.6.2.2.1">subscript</csymbol><ci id="S3.Thmproposition1.p1.1.1.m1.8.8.6.6.2.2.1.2.cmml" xref="S3.Thmproposition1.p1.1.1.m1.8.8.6.6.2.2.1.2">𝛉</ci><cn id="S3.Thmproposition1.p1.1.1.m1.8.8.6.6.2.2.1.3.cmml" type="integer" xref="S3.Thmproposition1.p1.1.1.m1.8.8.6.6.2.2.1.3">3</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmproposition1.p1.1.1.m1.8c">f(\operatorname{{\bm{\theta}}})=\ell(\operatorname{{\bm{\theta}}})+% \operatorname{\mathcal{P}}_{\lambda_{\operatorname{\mathbf{B}}}}(\operatorname% {{\bm{\theta}}}_{1})+\operatorname{\mathcal{P}}_{\lambda_{\operatorname{% \mathbf{L}}}}(\operatorname{{\bm{\theta}}}_{2})+\operatorname{\mathcal{P}}_{% \lambda_{\operatorname{\mathbf{L}}}}(\operatorname{{\bm{\theta}}}_{3})</annotation><annotation encoding="application/x-llamapun" id="S3.Thmproposition1.p1.1.1.m1.8d">italic_f ( bold_italic_θ ) = roman_ℓ ( bold_italic_θ ) + caligraphic_P start_POSTSUBSCRIPT italic_λ start_POSTSUBSCRIPT bold_B end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( bold_italic_θ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) + caligraphic_P start_POSTSUBSCRIPT italic_λ start_POSTSUBSCRIPT bold_L end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( bold_italic_θ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) + caligraphic_P start_POSTSUBSCRIPT italic_λ start_POSTSUBSCRIPT bold_L end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( bold_italic_θ start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT )</annotation></semantics></math>. Let <math alttext="\operatorname{\mathcal{M}}(\cdot)" class="ltx_Math" display="inline" id="S3.Thmproposition1.p1.2.2.m2.2"><semantics id="S3.Thmproposition1.p1.2.2.m2.2a"><mrow id="S3.Thmproposition1.p1.2.2.m2.2.3.2" xref="S3.Thmproposition1.p1.2.2.m2.2.3.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.Thmproposition1.p1.2.2.m2.1.1" xref="S3.Thmproposition1.p1.2.2.m2.1.1.cmml">ℳ</mi><mo id="S3.Thmproposition1.p1.2.2.m2.2.3.2a" xref="S3.Thmproposition1.p1.2.2.m2.2.3.1.cmml">⁡</mo><mrow id="S3.Thmproposition1.p1.2.2.m2.2.3.2.1" xref="S3.Thmproposition1.p1.2.2.m2.2.3.1.cmml"><mo id="S3.Thmproposition1.p1.2.2.m2.2.3.2.1.1" stretchy="false" xref="S3.Thmproposition1.p1.2.2.m2.2.3.1.cmml">(</mo><mo id="S3.Thmproposition1.p1.2.2.m2.2.2" lspace="0em" rspace="0em" xref="S3.Thmproposition1.p1.2.2.m2.2.2.cmml">⋅</mo><mo id="S3.Thmproposition1.p1.2.2.m2.2.3.2.1.2" stretchy="false" xref="S3.Thmproposition1.p1.2.2.m2.2.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmproposition1.p1.2.2.m2.2b"><apply id="S3.Thmproposition1.p1.2.2.m2.2.3.1.cmml" xref="S3.Thmproposition1.p1.2.2.m2.2.3.2"><ci id="S3.Thmproposition1.p1.2.2.m2.1.1.cmml" xref="S3.Thmproposition1.p1.2.2.m2.1.1">ℳ</ci><ci id="S3.Thmproposition1.p1.2.2.m2.2.2.cmml" xref="S3.Thmproposition1.p1.2.2.m2.2.2">⋅</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmproposition1.p1.2.2.m2.2c">\operatorname{\mathcal{M}}(\cdot)</annotation><annotation encoding="application/x-llamapun" id="S3.Thmproposition1.p1.2.2.m2.2d">caligraphic_M ( ⋅ )</annotation></semantics></math> be the function that maps <math alttext="\operatorname{{\bm{\theta}}}^{(t)}" class="ltx_Math" display="inline" id="S3.Thmproposition1.p1.3.3.m3.1"><semantics id="S3.Thmproposition1.p1.3.3.m3.1a"><msup id="S3.Thmproposition1.p1.3.3.m3.1.2" xref="S3.Thmproposition1.p1.3.3.m3.1.2.cmml"><mi id="S3.Thmproposition1.p1.3.3.m3.1.2.2" xref="S3.Thmproposition1.p1.3.3.m3.1.2.2.cmml">𝛉</mi><mrow id="S3.Thmproposition1.p1.3.3.m3.1.1.1.3" xref="S3.Thmproposition1.p1.3.3.m3.1.2.cmml"><mo id="S3.Thmproposition1.p1.3.3.m3.1.1.1.3.1" stretchy="false" xref="S3.Thmproposition1.p1.3.3.m3.1.2.cmml">(</mo><mi id="S3.Thmproposition1.p1.3.3.m3.1.1.1.1" xref="S3.Thmproposition1.p1.3.3.m3.1.1.1.1.cmml">t</mi><mo id="S3.Thmproposition1.p1.3.3.m3.1.1.1.3.2" stretchy="false" xref="S3.Thmproposition1.p1.3.3.m3.1.2.cmml">)</mo></mrow></msup><annotation-xml encoding="MathML-Content" id="S3.Thmproposition1.p1.3.3.m3.1b"><apply id="S3.Thmproposition1.p1.3.3.m3.1.2.cmml" xref="S3.Thmproposition1.p1.3.3.m3.1.2"><csymbol cd="ambiguous" id="S3.Thmproposition1.p1.3.3.m3.1.2.1.cmml" xref="S3.Thmproposition1.p1.3.3.m3.1.2">superscript</csymbol><ci id="S3.Thmproposition1.p1.3.3.m3.1.2.2.cmml" xref="S3.Thmproposition1.p1.3.3.m3.1.2.2">𝛉</ci><ci id="S3.Thmproposition1.p1.3.3.m3.1.1.1.1.cmml" xref="S3.Thmproposition1.p1.3.3.m3.1.1.1.1">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmproposition1.p1.3.3.m3.1c">\operatorname{{\bm{\theta}}}^{(t)}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmproposition1.p1.3.3.m3.1d">bold_italic_θ start_POSTSUPERSCRIPT ( italic_t ) end_POSTSUPERSCRIPT</annotation></semantics></math> to <math alttext="\operatorname{{\bm{\theta}}}^{(t+1)}" class="ltx_Math" display="inline" id="S3.Thmproposition1.p1.4.4.m4.1"><semantics id="S3.Thmproposition1.p1.4.4.m4.1a"><msup id="S3.Thmproposition1.p1.4.4.m4.1.2" xref="S3.Thmproposition1.p1.4.4.m4.1.2.cmml"><mi id="S3.Thmproposition1.p1.4.4.m4.1.2.2" xref="S3.Thmproposition1.p1.4.4.m4.1.2.2.cmml">𝛉</mi><mrow id="S3.Thmproposition1.p1.4.4.m4.1.1.1.1" xref="S3.Thmproposition1.p1.4.4.m4.1.1.1.1.1.cmml"><mo id="S3.Thmproposition1.p1.4.4.m4.1.1.1.1.2" stretchy="false" xref="S3.Thmproposition1.p1.4.4.m4.1.1.1.1.1.cmml">(</mo><mrow id="S3.Thmproposition1.p1.4.4.m4.1.1.1.1.1" xref="S3.Thmproposition1.p1.4.4.m4.1.1.1.1.1.cmml"><mi id="S3.Thmproposition1.p1.4.4.m4.1.1.1.1.1.2" xref="S3.Thmproposition1.p1.4.4.m4.1.1.1.1.1.2.cmml">t</mi><mo id="S3.Thmproposition1.p1.4.4.m4.1.1.1.1.1.1" xref="S3.Thmproposition1.p1.4.4.m4.1.1.1.1.1.1.cmml">+</mo><mn id="S3.Thmproposition1.p1.4.4.m4.1.1.1.1.1.3" xref="S3.Thmproposition1.p1.4.4.m4.1.1.1.1.1.3.cmml">1</mn></mrow><mo id="S3.Thmproposition1.p1.4.4.m4.1.1.1.1.3" stretchy="false" xref="S3.Thmproposition1.p1.4.4.m4.1.1.1.1.1.cmml">)</mo></mrow></msup><annotation-xml encoding="MathML-Content" id="S3.Thmproposition1.p1.4.4.m4.1b"><apply id="S3.Thmproposition1.p1.4.4.m4.1.2.cmml" xref="S3.Thmproposition1.p1.4.4.m4.1.2"><csymbol cd="ambiguous" id="S3.Thmproposition1.p1.4.4.m4.1.2.1.cmml" xref="S3.Thmproposition1.p1.4.4.m4.1.2">superscript</csymbol><ci id="S3.Thmproposition1.p1.4.4.m4.1.2.2.cmml" xref="S3.Thmproposition1.p1.4.4.m4.1.2.2">𝛉</ci><apply id="S3.Thmproposition1.p1.4.4.m4.1.1.1.1.1.cmml" xref="S3.Thmproposition1.p1.4.4.m4.1.1.1.1"><plus id="S3.Thmproposition1.p1.4.4.m4.1.1.1.1.1.1.cmml" xref="S3.Thmproposition1.p1.4.4.m4.1.1.1.1.1.1"></plus><ci id="S3.Thmproposition1.p1.4.4.m4.1.1.1.1.1.2.cmml" xref="S3.Thmproposition1.p1.4.4.m4.1.1.1.1.1.2">𝑡</ci><cn id="S3.Thmproposition1.p1.4.4.m4.1.1.1.1.1.3.cmml" type="integer" xref="S3.Thmproposition1.p1.4.4.m4.1.1.1.1.1.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmproposition1.p1.4.4.m4.1c">\operatorname{{\bm{\theta}}}^{(t+1)}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmproposition1.p1.4.4.m4.1d">bold_italic_θ start_POSTSUPERSCRIPT ( italic_t + 1 ) end_POSTSUPERSCRIPT</annotation></semantics></math> using Algorithm <a class="ltx_ref" href="https://arxiv.org/html/2503.13782v1#alg1" title="In 3 Regularized Parameter Estimation ‣ Regularized Parameter Estimation in Mixed Model Trace Regression"><span class="ltx_text ltx_ref_tag">1</span></a>. Then, each iteration of Algorithm <a class="ltx_ref" href="https://arxiv.org/html/2503.13782v1#alg1" title="In 3 Regularized Parameter Estimation ‣ Regularized Parameter Estimation in Mixed Model Trace Regression"><span class="ltx_text ltx_ref_tag">1</span></a> does not increase <math alttext="f(\operatorname{{\bm{\theta}}})" class="ltx_Math" display="inline" id="S3.Thmproposition1.p1.5.5.m5.1"><semantics id="S3.Thmproposition1.p1.5.5.m5.1a"><mrow id="S3.Thmproposition1.p1.5.5.m5.1.2" xref="S3.Thmproposition1.p1.5.5.m5.1.2.cmml"><mi id="S3.Thmproposition1.p1.5.5.m5.1.2.2" xref="S3.Thmproposition1.p1.5.5.m5.1.2.2.cmml">f</mi><mo id="S3.Thmproposition1.p1.5.5.m5.1.2.1" xref="S3.Thmproposition1.p1.5.5.m5.1.2.1.cmml">⁢</mo><mrow id="S3.Thmproposition1.p1.5.5.m5.1.2.3.2" xref="S3.Thmproposition1.p1.5.5.m5.1.2.cmml"><mo id="S3.Thmproposition1.p1.5.5.m5.1.2.3.2.1" stretchy="false" xref="S3.Thmproposition1.p1.5.5.m5.1.2.cmml">(</mo><mi id="S3.Thmproposition1.p1.5.5.m5.1.1" xref="S3.Thmproposition1.p1.5.5.m5.1.1.cmml">𝛉</mi><mo id="S3.Thmproposition1.p1.5.5.m5.1.2.3.2.2" stretchy="false" xref="S3.Thmproposition1.p1.5.5.m5.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmproposition1.p1.5.5.m5.1b"><apply id="S3.Thmproposition1.p1.5.5.m5.1.2.cmml" xref="S3.Thmproposition1.p1.5.5.m5.1.2"><times id="S3.Thmproposition1.p1.5.5.m5.1.2.1.cmml" xref="S3.Thmproposition1.p1.5.5.m5.1.2.1"></times><ci id="S3.Thmproposition1.p1.5.5.m5.1.2.2.cmml" xref="S3.Thmproposition1.p1.5.5.m5.1.2.2">𝑓</ci><ci id="S3.Thmproposition1.p1.5.5.m5.1.1.cmml" xref="S3.Thmproposition1.p1.5.5.m5.1.1">𝛉</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmproposition1.p1.5.5.m5.1c">f(\operatorname{{\bm{\theta}}})</annotation><annotation encoding="application/x-llamapun" id="S3.Thmproposition1.p1.5.5.m5.1d">italic_f ( bold_italic_θ )</annotation></semantics></math>. Furthermore, assume that the parameter space <math alttext="\operatorname{{\mathbf{\Theta}}}" class="ltx_Math" display="inline" id="S3.Thmproposition1.p1.6.6.m6.1"><semantics id="S3.Thmproposition1.p1.6.6.m6.1a"><mi id="S3.Thmproposition1.p1.6.6.m6.1.1" xref="S3.Thmproposition1.p1.6.6.m6.1.1.cmml">𝚯</mi><annotation-xml encoding="MathML-Content" id="S3.Thmproposition1.p1.6.6.m6.1b"><ci id="S3.Thmproposition1.p1.6.6.m6.1.1.cmml" xref="S3.Thmproposition1.p1.6.6.m6.1.1">𝚯</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmproposition1.p1.6.6.m6.1c">\operatorname{{\mathbf{\Theta}}}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmproposition1.p1.6.6.m6.1d">bold_Θ</annotation></semantics></math> is compact and <math alttext="f(\operatorname{{\bm{\theta}}})=f\{\operatorname{\mathcal{M}}(\operatorname{{% \bm{\theta}}})\}" class="ltx_Math" display="inline" id="S3.Thmproposition1.p1.7.7.m7.4"><semantics id="S3.Thmproposition1.p1.7.7.m7.4a"><mrow id="S3.Thmproposition1.p1.7.7.m7.4.4" xref="S3.Thmproposition1.p1.7.7.m7.4.4.cmml"><mrow id="S3.Thmproposition1.p1.7.7.m7.4.4.3" xref="S3.Thmproposition1.p1.7.7.m7.4.4.3.cmml"><mi id="S3.Thmproposition1.p1.7.7.m7.4.4.3.2" xref="S3.Thmproposition1.p1.7.7.m7.4.4.3.2.cmml">f</mi><mo id="S3.Thmproposition1.p1.7.7.m7.4.4.3.1" xref="S3.Thmproposition1.p1.7.7.m7.4.4.3.1.cmml">⁢</mo><mrow id="S3.Thmproposition1.p1.7.7.m7.4.4.3.3.2" xref="S3.Thmproposition1.p1.7.7.m7.4.4.3.cmml"><mo id="S3.Thmproposition1.p1.7.7.m7.4.4.3.3.2.1" stretchy="false" xref="S3.Thmproposition1.p1.7.7.m7.4.4.3.cmml">(</mo><mi id="S3.Thmproposition1.p1.7.7.m7.1.1" xref="S3.Thmproposition1.p1.7.7.m7.1.1.cmml">𝛉</mi><mo id="S3.Thmproposition1.p1.7.7.m7.4.4.3.3.2.2" stretchy="false" xref="S3.Thmproposition1.p1.7.7.m7.4.4.3.cmml">)</mo></mrow></mrow><mo id="S3.Thmproposition1.p1.7.7.m7.4.4.2" xref="S3.Thmproposition1.p1.7.7.m7.4.4.2.cmml">=</mo><mrow id="S3.Thmproposition1.p1.7.7.m7.4.4.1" xref="S3.Thmproposition1.p1.7.7.m7.4.4.1.cmml"><mi id="S3.Thmproposition1.p1.7.7.m7.4.4.1.3" xref="S3.Thmproposition1.p1.7.7.m7.4.4.1.3.cmml">f</mi><mo id="S3.Thmproposition1.p1.7.7.m7.4.4.1.2" xref="S3.Thmproposition1.p1.7.7.m7.4.4.1.2.cmml">⁢</mo><mrow id="S3.Thmproposition1.p1.7.7.m7.4.4.1.1.1" xref="S3.Thmproposition1.p1.7.7.m7.4.4.1.1.2.cmml"><mo id="S3.Thmproposition1.p1.7.7.m7.4.4.1.1.1.2" stretchy="false" xref="S3.Thmproposition1.p1.7.7.m7.4.4.1.1.2.cmml">{</mo><mrow id="S3.Thmproposition1.p1.7.7.m7.4.4.1.1.1.1.2" xref="S3.Thmproposition1.p1.7.7.m7.4.4.1.1.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.Thmproposition1.p1.7.7.m7.2.2" xref="S3.Thmproposition1.p1.7.7.m7.2.2.cmml">ℳ</mi><mo id="S3.Thmproposition1.p1.7.7.m7.4.4.1.1.1.1.2a" xref="S3.Thmproposition1.p1.7.7.m7.4.4.1.1.1.1.1.cmml">⁡</mo><mrow id="S3.Thmproposition1.p1.7.7.m7.4.4.1.1.1.1.2.1" xref="S3.Thmproposition1.p1.7.7.m7.4.4.1.1.1.1.1.cmml"><mo id="S3.Thmproposition1.p1.7.7.m7.4.4.1.1.1.1.2.1.1" stretchy="false" xref="S3.Thmproposition1.p1.7.7.m7.4.4.1.1.1.1.1.cmml">(</mo><mi id="S3.Thmproposition1.p1.7.7.m7.3.3" xref="S3.Thmproposition1.p1.7.7.m7.3.3.cmml">𝛉</mi><mo id="S3.Thmproposition1.p1.7.7.m7.4.4.1.1.1.1.2.1.2" stretchy="false" xref="S3.Thmproposition1.p1.7.7.m7.4.4.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S3.Thmproposition1.p1.7.7.m7.4.4.1.1.1.3" stretchy="false" xref="S3.Thmproposition1.p1.7.7.m7.4.4.1.1.2.cmml">}</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmproposition1.p1.7.7.m7.4b"><apply id="S3.Thmproposition1.p1.7.7.m7.4.4.cmml" xref="S3.Thmproposition1.p1.7.7.m7.4.4"><eq id="S3.Thmproposition1.p1.7.7.m7.4.4.2.cmml" xref="S3.Thmproposition1.p1.7.7.m7.4.4.2"></eq><apply id="S3.Thmproposition1.p1.7.7.m7.4.4.3.cmml" xref="S3.Thmproposition1.p1.7.7.m7.4.4.3"><times id="S3.Thmproposition1.p1.7.7.m7.4.4.3.1.cmml" xref="S3.Thmproposition1.p1.7.7.m7.4.4.3.1"></times><ci id="S3.Thmproposition1.p1.7.7.m7.4.4.3.2.cmml" xref="S3.Thmproposition1.p1.7.7.m7.4.4.3.2">𝑓</ci><ci id="S3.Thmproposition1.p1.7.7.m7.1.1.cmml" xref="S3.Thmproposition1.p1.7.7.m7.1.1">𝛉</ci></apply><apply id="S3.Thmproposition1.p1.7.7.m7.4.4.1.cmml" xref="S3.Thmproposition1.p1.7.7.m7.4.4.1"><times id="S3.Thmproposition1.p1.7.7.m7.4.4.1.2.cmml" xref="S3.Thmproposition1.p1.7.7.m7.4.4.1.2"></times><ci id="S3.Thmproposition1.p1.7.7.m7.4.4.1.3.cmml" xref="S3.Thmproposition1.p1.7.7.m7.4.4.1.3">𝑓</ci><set id="S3.Thmproposition1.p1.7.7.m7.4.4.1.1.2.cmml" xref="S3.Thmproposition1.p1.7.7.m7.4.4.1.1.1"><apply id="S3.Thmproposition1.p1.7.7.m7.4.4.1.1.1.1.1.cmml" xref="S3.Thmproposition1.p1.7.7.m7.4.4.1.1.1.1.2"><ci id="S3.Thmproposition1.p1.7.7.m7.2.2.cmml" xref="S3.Thmproposition1.p1.7.7.m7.2.2">ℳ</ci><ci id="S3.Thmproposition1.p1.7.7.m7.3.3.cmml" xref="S3.Thmproposition1.p1.7.7.m7.3.3">𝛉</ci></apply></set></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmproposition1.p1.7.7.m7.4c">f(\operatorname{{\bm{\theta}}})=f\{\operatorname{\mathcal{M}}(\operatorname{{% \bm{\theta}}})\}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmproposition1.p1.7.7.m7.4d">italic_f ( bold_italic_θ ) = italic_f { caligraphic_M ( bold_italic_θ ) }</annotation></semantics></math> only for the stationary points of <math alttext="f(\operatorname{{\bm{\theta}}})" class="ltx_Math" display="inline" id="S3.Thmproposition1.p1.8.8.m8.1"><semantics id="S3.Thmproposition1.p1.8.8.m8.1a"><mrow id="S3.Thmproposition1.p1.8.8.m8.1.2" xref="S3.Thmproposition1.p1.8.8.m8.1.2.cmml"><mi id="S3.Thmproposition1.p1.8.8.m8.1.2.2" xref="S3.Thmproposition1.p1.8.8.m8.1.2.2.cmml">f</mi><mo id="S3.Thmproposition1.p1.8.8.m8.1.2.1" xref="S3.Thmproposition1.p1.8.8.m8.1.2.1.cmml">⁢</mo><mrow id="S3.Thmproposition1.p1.8.8.m8.1.2.3.2" xref="S3.Thmproposition1.p1.8.8.m8.1.2.cmml"><mo id="S3.Thmproposition1.p1.8.8.m8.1.2.3.2.1" stretchy="false" xref="S3.Thmproposition1.p1.8.8.m8.1.2.cmml">(</mo><mi id="S3.Thmproposition1.p1.8.8.m8.1.1" xref="S3.Thmproposition1.p1.8.8.m8.1.1.cmml">𝛉</mi><mo id="S3.Thmproposition1.p1.8.8.m8.1.2.3.2.2" stretchy="false" xref="S3.Thmproposition1.p1.8.8.m8.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmproposition1.p1.8.8.m8.1b"><apply id="S3.Thmproposition1.p1.8.8.m8.1.2.cmml" xref="S3.Thmproposition1.p1.8.8.m8.1.2"><times id="S3.Thmproposition1.p1.8.8.m8.1.2.1.cmml" xref="S3.Thmproposition1.p1.8.8.m8.1.2.1"></times><ci id="S3.Thmproposition1.p1.8.8.m8.1.2.2.cmml" xref="S3.Thmproposition1.p1.8.8.m8.1.2.2">𝑓</ci><ci id="S3.Thmproposition1.p1.8.8.m8.1.1.cmml" xref="S3.Thmproposition1.p1.8.8.m8.1.1">𝛉</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmproposition1.p1.8.8.m8.1c">f(\operatorname{{\bm{\theta}}})</annotation><annotation encoding="application/x-llamapun" id="S3.Thmproposition1.p1.8.8.m8.1d">italic_f ( bold_italic_θ )</annotation></semantics></math>. Then, the <math alttext="\{\operatorname{{\bm{\theta}}}^{(t)}\}_{t=1}^{\infty}" class="ltx_Math" display="inline" id="S3.Thmproposition1.p1.9.9.m9.2"><semantics id="S3.Thmproposition1.p1.9.9.m9.2a"><msubsup id="S3.Thmproposition1.p1.9.9.m9.2.2" xref="S3.Thmproposition1.p1.9.9.m9.2.2.cmml"><mrow id="S3.Thmproposition1.p1.9.9.m9.2.2.1.1.1" xref="S3.Thmproposition1.p1.9.9.m9.2.2.1.1.2.cmml"><mo id="S3.Thmproposition1.p1.9.9.m9.2.2.1.1.1.2" stretchy="false" xref="S3.Thmproposition1.p1.9.9.m9.2.2.1.1.2.cmml">{</mo><msup id="S3.Thmproposition1.p1.9.9.m9.2.2.1.1.1.1" xref="S3.Thmproposition1.p1.9.9.m9.2.2.1.1.1.1.cmml"><mi id="S3.Thmproposition1.p1.9.9.m9.2.2.1.1.1.1.2" xref="S3.Thmproposition1.p1.9.9.m9.2.2.1.1.1.1.2.cmml">𝛉</mi><mrow id="S3.Thmproposition1.p1.9.9.m9.1.1.1.3" xref="S3.Thmproposition1.p1.9.9.m9.2.2.1.1.1.1.cmml"><mo id="S3.Thmproposition1.p1.9.9.m9.1.1.1.3.1" stretchy="false" xref="S3.Thmproposition1.p1.9.9.m9.2.2.1.1.1.1.cmml">(</mo><mi id="S3.Thmproposition1.p1.9.9.m9.1.1.1.1" xref="S3.Thmproposition1.p1.9.9.m9.1.1.1.1.cmml">t</mi><mo id="S3.Thmproposition1.p1.9.9.m9.1.1.1.3.2" stretchy="false" xref="S3.Thmproposition1.p1.9.9.m9.2.2.1.1.1.1.cmml">)</mo></mrow></msup><mo id="S3.Thmproposition1.p1.9.9.m9.2.2.1.1.1.3" stretchy="false" xref="S3.Thmproposition1.p1.9.9.m9.2.2.1.1.2.cmml">}</mo></mrow><mrow id="S3.Thmproposition1.p1.9.9.m9.2.2.1.3" xref="S3.Thmproposition1.p1.9.9.m9.2.2.1.3.cmml"><mi id="S3.Thmproposition1.p1.9.9.m9.2.2.1.3.2" xref="S3.Thmproposition1.p1.9.9.m9.2.2.1.3.2.cmml">t</mi><mo id="S3.Thmproposition1.p1.9.9.m9.2.2.1.3.1" xref="S3.Thmproposition1.p1.9.9.m9.2.2.1.3.1.cmml">=</mo><mn id="S3.Thmproposition1.p1.9.9.m9.2.2.1.3.3" xref="S3.Thmproposition1.p1.9.9.m9.2.2.1.3.3.cmml">1</mn></mrow><mi id="S3.Thmproposition1.p1.9.9.m9.2.2.3" mathvariant="normal" xref="S3.Thmproposition1.p1.9.9.m9.2.2.3.cmml">∞</mi></msubsup><annotation-xml encoding="MathML-Content" id="S3.Thmproposition1.p1.9.9.m9.2b"><apply id="S3.Thmproposition1.p1.9.9.m9.2.2.cmml" xref="S3.Thmproposition1.p1.9.9.m9.2.2"><csymbol cd="ambiguous" id="S3.Thmproposition1.p1.9.9.m9.2.2.2.cmml" xref="S3.Thmproposition1.p1.9.9.m9.2.2">superscript</csymbol><apply id="S3.Thmproposition1.p1.9.9.m9.2.2.1.cmml" xref="S3.Thmproposition1.p1.9.9.m9.2.2"><csymbol cd="ambiguous" id="S3.Thmproposition1.p1.9.9.m9.2.2.1.2.cmml" xref="S3.Thmproposition1.p1.9.9.m9.2.2">subscript</csymbol><set id="S3.Thmproposition1.p1.9.9.m9.2.2.1.1.2.cmml" xref="S3.Thmproposition1.p1.9.9.m9.2.2.1.1.1"><apply id="S3.Thmproposition1.p1.9.9.m9.2.2.1.1.1.1.cmml" xref="S3.Thmproposition1.p1.9.9.m9.2.2.1.1.1.1"><csymbol cd="ambiguous" id="S3.Thmproposition1.p1.9.9.m9.2.2.1.1.1.1.1.cmml" xref="S3.Thmproposition1.p1.9.9.m9.2.2.1.1.1.1">superscript</csymbol><ci id="S3.Thmproposition1.p1.9.9.m9.2.2.1.1.1.1.2.cmml" xref="S3.Thmproposition1.p1.9.9.m9.2.2.1.1.1.1.2">𝛉</ci><ci id="S3.Thmproposition1.p1.9.9.m9.1.1.1.1.cmml" xref="S3.Thmproposition1.p1.9.9.m9.1.1.1.1">𝑡</ci></apply></set><apply id="S3.Thmproposition1.p1.9.9.m9.2.2.1.3.cmml" xref="S3.Thmproposition1.p1.9.9.m9.2.2.1.3"><eq id="S3.Thmproposition1.p1.9.9.m9.2.2.1.3.1.cmml" xref="S3.Thmproposition1.p1.9.9.m9.2.2.1.3.1"></eq><ci id="S3.Thmproposition1.p1.9.9.m9.2.2.1.3.2.cmml" xref="S3.Thmproposition1.p1.9.9.m9.2.2.1.3.2">𝑡</ci><cn id="S3.Thmproposition1.p1.9.9.m9.2.2.1.3.3.cmml" type="integer" xref="S3.Thmproposition1.p1.9.9.m9.2.2.1.3.3">1</cn></apply></apply><infinity id="S3.Thmproposition1.p1.9.9.m9.2.2.3.cmml" xref="S3.Thmproposition1.p1.9.9.m9.2.2.3"></infinity></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmproposition1.p1.9.9.m9.2c">\{\operatorname{{\bm{\theta}}}^{(t)}\}_{t=1}^{\infty}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmproposition1.p1.9.9.m9.2d">{ bold_italic_θ start_POSTSUPERSCRIPT ( italic_t ) end_POSTSUPERSCRIPT } start_POSTSUBSCRIPT italic_t = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT</annotation></semantics></math> sequence converges to a stationary point.</span></p> </div> </div> </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> <section class="ltx_subsection" id="S4.SS1"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">4.1 </span>Setup</h3> <div class="ltx_para" id="S4.SS1.p1"> <p class="ltx_p" id="S4.SS1.p1.1">MMTR’s performance was evaluated using simulated and real data. The high-dimensional penalized competitors were based on quasi-likelihood <cite class="ltx_cite ltx_citemacro_citep">(Li et al., <a class="ltx_ref" href="https://arxiv.org/html/2503.13782v1#bib.bib21" title="">2022</a>)</cite>, trace regression <cite class="ltx_cite ltx_citemacro_citep">(Zhao et al., <a class="ltx_ref" href="https://arxiv.org/html/2503.13782v1#bib.bib38" title="">2017</a>)</cite>, posterior mode estimation <cite class="ltx_cite ltx_citemacro_citep">(Heiling et al., <a class="ltx_ref" href="https://arxiv.org/html/2503.13782v1#bib.bib12" title="">2023</a>)</cite>, and generalized estimating equations (GEE) <cite class="ltx_cite ltx_citemacro_citep">(Zhang et al., <a class="ltx_ref" href="https://arxiv.org/html/2503.13782v1#bib.bib36" title="">2019</a>)</cite>. The posterior mode estimation algorithm implemented in <span class="ltx_text ltx_font_typewriter" id="S4.SS1.p1.1.1">glmmPen</span> R package failed with an error in the pre-screening step, so we do not present comparisons with this method. In low-dimensional settings, we used <span class="ltx_text ltx_font_typewriter" id="S4.SS1.p1.1.2">lme4</span> <cite class="ltx_cite ltx_citemacro_citep">(Bates et al., <a class="ltx_ref" href="https://arxiv.org/html/2503.13782v1#bib.bib3" title="">2013</a>)</cite> for maximum likelihood estimation, resulting in total four competing methods. The penalized quasi-likelihood (PQL) method, implemented using the <span class="ltx_text ltx_font_typewriter" id="S4.SS1.p1.1.3">scalreg</span> R package, and <span class="ltx_text ltx_font_typewriter" id="S4.SS1.p1.1.4">lme4</span> used MMTR’s vectorized form in (<a class="ltx_ref" href="https://arxiv.org/html/2503.13782v1#S2.E5" title="In 2.2 Reduction to a Structured Mixed Model ‣ 2 Mixed Model Trace Regression ‣ Regularized Parameter Estimation in Mixed Model Trace Regression"><span class="ltx_text ltx_ref_tag">5</span></a>) for parameter estimation. In contrast, trace regression and GEE did not have random effects, so we employed <span class="ltx_text ltx_font_typewriter" id="S4.SS1.p1.1.5">TensorReg</span> <cite class="ltx_cite ltx_citemacro_citep">(Zhou, <a class="ltx_ref" href="https://arxiv.org/html/2503.13782v1#bib.bib39" title="">2017</a>)</cite> and <span class="ltx_text ltx_font_typewriter" id="S4.SS1.p1.1.6">SparseReg</span> <cite class="ltx_cite ltx_citemacro_citep">(Zhou and Gaines, <a class="ltx_ref" href="https://arxiv.org/html/2503.13782v1#bib.bib40" title="">2017</a>)</cite> MATLAB packages to fit trace regression and GEE models. MMTR used Algorithm 1 for parameter estimation.</p> </div> <div class="ltx_para" id="S4.SS1.p2"> <p class="ltx_p" id="S4.SS1.p2.6">The marginal model in (<a class="ltx_ref" href="https://arxiv.org/html/2503.13782v1#S3.E7" title="In 3 Regularized Parameter Estimation ‣ Regularized Parameter Estimation in Mixed Model Trace Regression"><span class="ltx_text ltx_ref_tag">7</span></a>) was used for simulations. Specifically,</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="S3.EGx12"> <tbody id="S4.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\operatorname{\mathbf{y}}=\operatorname{\mathbf{X}}\operatorname{% \mathbf{b}}+\operatorname{\mathbf{e}},\quad\operatorname{\mathbf{e}}\sim N(% \operatorname{\mathbf{0}},\operatorname{{\mathbf{\Lambda}}}),\quad% \operatorname{\mathbf{y}}\in\operatorname{\mathbb{R}}^{N},\;\operatorname{% \mathbf{X}}\in\operatorname{\mathbb{R}}^{{N\times P}},\;\operatorname{\mathbf{% b}}\in\operatorname{\mathbb{R}}^{P},\;\operatorname{\mathbf{e}}\in% \operatorname{\mathbb{R}}^{N}," class="ltx_Math" display="inline" id="S4.E12.m1.3"><semantics id="S4.E12.m1.3a"><mrow id="S4.E12.m1.3.3.1"><mrow id="S4.E12.m1.3.3.1.1.2" xref="S4.E12.m1.3.3.1.1.3.cmml"><mrow id="S4.E12.m1.3.3.1.1.1.1" 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xref="S4.E12.m1.3.3.1.1.2.2.2.2.2.2.2.2.2.2"><in id="S4.E12.m1.3.3.1.1.2.2.2.2.2.2.2.2.2.2.1.cmml" xref="S4.E12.m1.3.3.1.1.2.2.2.2.2.2.2.2.2.2.1"></in><ci id="S4.E12.m1.3.3.1.1.2.2.2.2.2.2.2.2.2.2.2.cmml" xref="S4.E12.m1.3.3.1.1.2.2.2.2.2.2.2.2.2.2.2">𝐞</ci><apply id="S4.E12.m1.3.3.1.1.2.2.2.2.2.2.2.2.2.2.3.cmml" xref="S4.E12.m1.3.3.1.1.2.2.2.2.2.2.2.2.2.2.3"><csymbol cd="ambiguous" id="S4.E12.m1.3.3.1.1.2.2.2.2.2.2.2.2.2.2.3.1.cmml" xref="S4.E12.m1.3.3.1.1.2.2.2.2.2.2.2.2.2.2.3">superscript</csymbol><ci id="S4.E12.m1.3.3.1.1.2.2.2.2.2.2.2.2.2.2.3.2.cmml" xref="S4.E12.m1.3.3.1.1.2.2.2.2.2.2.2.2.2.2.3.2">ℝ</ci><ci id="S4.E12.m1.3.3.1.1.2.2.2.2.2.2.2.2.2.2.3.3.cmml" xref="S4.E12.m1.3.3.1.1.2.2.2.2.2.2.2.2.2.2.3.3">𝑁</ci></apply></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.E12.m1.3c">\displaystyle\operatorname{\mathbf{y}}=\operatorname{\mathbf{X}}\operatorname{% \mathbf{b}}+\operatorname{\mathbf{e}},\quad\operatorname{\mathbf{e}}\sim N(% \operatorname{\mathbf{0}},\operatorname{{\mathbf{\Lambda}}}),\quad% \operatorname{\mathbf{y}}\in\operatorname{\mathbb{R}}^{N},\;\operatorname{% \mathbf{X}}\in\operatorname{\mathbb{R}}^{{N\times P}},\;\operatorname{\mathbf{% b}}\in\operatorname{\mathbb{R}}^{P},\;\operatorname{\mathbf{e}}\in% \operatorname{\mathbb{R}}^{N},</annotation><annotation encoding="application/x-llamapun" id="S4.E12.m1.3d">bold_y = bold_X bold_b + bold_e , bold_e ∼ italic_N ( bold_0 , bold_Λ ) , bold_y ∈ blackboard_R start_POSTSUPERSCRIPT italic_N end_POSTSUPERSCRIPT , bold_X ∈ blackboard_R start_POSTSUPERSCRIPT italic_N × italic_P end_POSTSUPERSCRIPT , bold_b ∈ blackboard_R start_POSTSUPERSCRIPT italic_P end_POSTSUPERSCRIPT , bold_e ∈ blackboard_R start_POSTSUPERSCRIPT italic_N 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">(12)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S4.SS1.p2.5">where <math alttext="\operatorname{{\mathbf{\Lambda}}}" 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">𝚲</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">𝚲</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p2.1.m1.1c">\operatorname{{\mathbf{\Lambda}}}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p2.1.m1.1d">bold_Λ</annotation></semantics></math> was either a diagonal or block diagonal covariance matrix. We used MMTR and GEE as the true models for simulating the data. The GEE model specified <math alttext="\operatorname{{\mathbf{\Lambda}}}" 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">𝚲</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">𝚲</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p2.2.m2.1c">\operatorname{{\mathbf{\Lambda}}}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p2.2.m2.1d">bold_Λ</annotation></semantics></math> directly, whereas MMTR specified <math alttext="\operatorname{{\mathbf{\Lambda}}}" class="ltx_Math" display="inline" id="S4.SS1.p2.3.m3.1"><semantics id="S4.SS1.p2.3.m3.1a"><mi id="S4.SS1.p2.3.m3.1.1" xref="S4.SS1.p2.3.m3.1.1.cmml">𝚲</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.p2.3.m3.1b"><ci id="S4.SS1.p2.3.m3.1.1.cmml" xref="S4.SS1.p2.3.m3.1.1">𝚲</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p2.3.m3.1c">\operatorname{{\mathbf{\Lambda}}}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p2.3.m3.1d">bold_Λ</annotation></semantics></math> through random effects design matrix, <math alttext="\operatorname{{\mathbf{\Sigma}}}_{1},\operatorname{{\mathbf{\Sigma}}}_{2}" class="ltx_Math" display="inline" id="S4.SS1.p2.4.m4.2"><semantics id="S4.SS1.p2.4.m4.2a"><mrow id="S4.SS1.p2.4.m4.2.2.2" xref="S4.SS1.p2.4.m4.2.2.3.cmml"><msub id="S4.SS1.p2.4.m4.1.1.1.1" xref="S4.SS1.p2.4.m4.1.1.1.1.cmml"><mi id="S4.SS1.p2.4.m4.1.1.1.1.2" xref="S4.SS1.p2.4.m4.1.1.1.1.2.cmml">𝚺</mi><mn id="S4.SS1.p2.4.m4.1.1.1.1.3" xref="S4.SS1.p2.4.m4.1.1.1.1.3.cmml">1</mn></msub><mo id="S4.SS1.p2.4.m4.2.2.2.3" xref="S4.SS1.p2.4.m4.2.2.3.cmml">,</mo><msub id="S4.SS1.p2.4.m4.2.2.2.2" xref="S4.SS1.p2.4.m4.2.2.2.2.cmml"><mi id="S4.SS1.p2.4.m4.2.2.2.2.2" xref="S4.SS1.p2.4.m4.2.2.2.2.2.cmml">𝚺</mi><mn id="S4.SS1.p2.4.m4.2.2.2.2.3" xref="S4.SS1.p2.4.m4.2.2.2.2.3.cmml">2</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p2.4.m4.2b"><list id="S4.SS1.p2.4.m4.2.2.3.cmml" xref="S4.SS1.p2.4.m4.2.2.2"><apply id="S4.SS1.p2.4.m4.1.1.1.1.cmml" xref="S4.SS1.p2.4.m4.1.1.1.1"><csymbol cd="ambiguous" id="S4.SS1.p2.4.m4.1.1.1.1.1.cmml" xref="S4.SS1.p2.4.m4.1.1.1.1">subscript</csymbol><ci id="S4.SS1.p2.4.m4.1.1.1.1.2.cmml" xref="S4.SS1.p2.4.m4.1.1.1.1.2">𝚺</ci><cn id="S4.SS1.p2.4.m4.1.1.1.1.3.cmml" type="integer" xref="S4.SS1.p2.4.m4.1.1.1.1.3">1</cn></apply><apply id="S4.SS1.p2.4.m4.2.2.2.2.cmml" xref="S4.SS1.p2.4.m4.2.2.2.2"><csymbol cd="ambiguous" id="S4.SS1.p2.4.m4.2.2.2.2.1.cmml" xref="S4.SS1.p2.4.m4.2.2.2.2">subscript</csymbol><ci id="S4.SS1.p2.4.m4.2.2.2.2.2.cmml" xref="S4.SS1.p2.4.m4.2.2.2.2.2">𝚺</ci><cn id="S4.SS1.p2.4.m4.2.2.2.2.3.cmml" type="integer" xref="S4.SS1.p2.4.m4.2.2.2.2.3">2</cn></apply></list></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p2.4.m4.2c">\operatorname{{\mathbf{\Sigma}}}_{1},\operatorname{{\mathbf{\Sigma}}}_{2}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p2.4.m4.2d">bold_Σ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , bold_Σ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math>, and <math alttext="\tau^{2}" class="ltx_Math" display="inline" id="S4.SS1.p2.5.m5.1"><semantics id="S4.SS1.p2.5.m5.1a"><msup id="S4.SS1.p2.5.m5.1.1" xref="S4.SS1.p2.5.m5.1.1.cmml"><mi id="S4.SS1.p2.5.m5.1.1.2" xref="S4.SS1.p2.5.m5.1.1.2.cmml">τ</mi><mn id="S4.SS1.p2.5.m5.1.1.3" xref="S4.SS1.p2.5.m5.1.1.3.cmml">2</mn></msup><annotation-xml encoding="MathML-Content" id="S4.SS1.p2.5.m5.1b"><apply id="S4.SS1.p2.5.m5.1.1.cmml" xref="S4.SS1.p2.5.m5.1.1"><csymbol cd="ambiguous" id="S4.SS1.p2.5.m5.1.1.1.cmml" xref="S4.SS1.p2.5.m5.1.1">superscript</csymbol><ci id="S4.SS1.p2.5.m5.1.1.2.cmml" xref="S4.SS1.p2.5.m5.1.1.2">𝜏</ci><cn id="S4.SS1.p2.5.m5.1.1.3.cmml" type="integer" xref="S4.SS1.p2.5.m5.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p2.5.m5.1c">\tau^{2}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p2.5.m5.1d">italic_τ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S4.SS1.p3"> <p class="ltx_p" id="S4.SS1.p3.22">Except for <span class="ltx_text ltx_font_typewriter" id="S4.SS1.p3.22.1">lme4</span>, the performance of MMTR and its competitors depended on model-specific choices. The PQL method replaced <math alttext="\operatorname{{\mathbf{\Lambda}}}" class="ltx_Math" display="inline" id="S4.SS1.p3.1.m1.1"><semantics id="S4.SS1.p3.1.m1.1a"><mi id="S4.SS1.p3.1.m1.1.1" xref="S4.SS1.p3.1.m1.1.1.cmml">𝚲</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.p3.1.m1.1b"><ci id="S4.SS1.p3.1.m1.1.1.cmml" xref="S4.SS1.p3.1.m1.1.1">𝚲</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p3.1.m1.1c">\operatorname{{\mathbf{\Lambda}}}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p3.1.m1.1d">bold_Λ</annotation></semantics></math> with a proxy matrix <math alttext="c^{2}\operatorname{\mathbf{I}}" class="ltx_Math" display="inline" id="S4.SS1.p3.2.m2.1"><semantics id="S4.SS1.p3.2.m2.1a"><mrow id="S4.SS1.p3.2.m2.1.1" xref="S4.SS1.p3.2.m2.1.1.cmml"><msup id="S4.SS1.p3.2.m2.1.1.2" xref="S4.SS1.p3.2.m2.1.1.2.cmml"><mi id="S4.SS1.p3.2.m2.1.1.2.2" xref="S4.SS1.p3.2.m2.1.1.2.2.cmml">c</mi><mn id="S4.SS1.p3.2.m2.1.1.2.3" xref="S4.SS1.p3.2.m2.1.1.2.3.cmml">2</mn></msup><mo id="S4.SS1.p3.2.m2.1.1.1" lspace="0.167em" xref="S4.SS1.p3.2.m2.1.1.1.cmml">⁢</mo><mi id="S4.SS1.p3.2.m2.1.1.3" xref="S4.SS1.p3.2.m2.1.1.3.cmml">𝐈</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p3.2.m2.1b"><apply id="S4.SS1.p3.2.m2.1.1.cmml" xref="S4.SS1.p3.2.m2.1.1"><times id="S4.SS1.p3.2.m2.1.1.1.cmml" xref="S4.SS1.p3.2.m2.1.1.1"></times><apply id="S4.SS1.p3.2.m2.1.1.2.cmml" xref="S4.SS1.p3.2.m2.1.1.2"><csymbol cd="ambiguous" id="S4.SS1.p3.2.m2.1.1.2.1.cmml" xref="S4.SS1.p3.2.m2.1.1.2">superscript</csymbol><ci id="S4.SS1.p3.2.m2.1.1.2.2.cmml" xref="S4.SS1.p3.2.m2.1.1.2.2">𝑐</ci><cn id="S4.SS1.p3.2.m2.1.1.2.3.cmml" type="integer" xref="S4.SS1.p3.2.m2.1.1.2.3">2</cn></apply><ci id="S4.SS1.p3.2.m2.1.1.3.cmml" xref="S4.SS1.p3.2.m2.1.1.3">𝐈</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p3.2.m2.1c">c^{2}\operatorname{\mathbf{I}}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p3.2.m2.1d">italic_c start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT bold_I</annotation></semantics></math> and used a lasso penalty on <math alttext="\operatorname{\mathbf{b}}" class="ltx_Math" display="inline" id="S4.SS1.p3.3.m3.1"><semantics id="S4.SS1.p3.3.m3.1a"><mi id="S4.SS1.p3.3.m3.1.1" xref="S4.SS1.p3.3.m3.1.1.cmml">𝐛</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.p3.3.m3.1b"><ci id="S4.SS1.p3.3.m3.1.1.cmml" xref="S4.SS1.p3.3.m3.1.1">𝐛</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p3.3.m3.1c">\operatorname{\mathbf{b}}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p3.3.m3.1d">bold_b</annotation></semantics></math>. Sparse trace regression set <math alttext="\operatorname{{\mathbf{\Lambda}}}=\tau^{2}\operatorname{\mathbf{I}}" 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">𝚲</mi><mo id="S4.SS1.p3.4.m4.1.1.1" xref="S4.SS1.p3.4.m4.1.1.1.cmml">=</mo><mrow id="S4.SS1.p3.4.m4.1.1.3" xref="S4.SS1.p3.4.m4.1.1.3.cmml"><msup id="S4.SS1.p3.4.m4.1.1.3.2" xref="S4.SS1.p3.4.m4.1.1.3.2.cmml"><mi id="S4.SS1.p3.4.m4.1.1.3.2.2" xref="S4.SS1.p3.4.m4.1.1.3.2.2.cmml">τ</mi><mn id="S4.SS1.p3.4.m4.1.1.3.2.3" xref="S4.SS1.p3.4.m4.1.1.3.2.3.cmml">2</mn></msup><mo id="S4.SS1.p3.4.m4.1.1.3.1" lspace="0.167em" xref="S4.SS1.p3.4.m4.1.1.3.1.cmml">⁢</mo><mi id="S4.SS1.p3.4.m4.1.1.3.3" xref="S4.SS1.p3.4.m4.1.1.3.3.cmml">𝐈</mi></mrow></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"><eq id="S4.SS1.p3.4.m4.1.1.1.cmml" xref="S4.SS1.p3.4.m4.1.1.1"></eq><ci id="S4.SS1.p3.4.m4.1.1.2.cmml" xref="S4.SS1.p3.4.m4.1.1.2">𝚲</ci><apply id="S4.SS1.p3.4.m4.1.1.3.cmml" xref="S4.SS1.p3.4.m4.1.1.3"><times id="S4.SS1.p3.4.m4.1.1.3.1.cmml" xref="S4.SS1.p3.4.m4.1.1.3.1"></times><apply id="S4.SS1.p3.4.m4.1.1.3.2.cmml" xref="S4.SS1.p3.4.m4.1.1.3.2"><csymbol cd="ambiguous" id="S4.SS1.p3.4.m4.1.1.3.2.1.cmml" xref="S4.SS1.p3.4.m4.1.1.3.2">superscript</csymbol><ci id="S4.SS1.p3.4.m4.1.1.3.2.2.cmml" xref="S4.SS1.p3.4.m4.1.1.3.2.2">𝜏</ci><cn id="S4.SS1.p3.4.m4.1.1.3.2.3.cmml" type="integer" xref="S4.SS1.p3.4.m4.1.1.3.2.3">2</cn></apply><ci id="S4.SS1.p3.4.m4.1.1.3.3.cmml" xref="S4.SS1.p3.4.m4.1.1.3.3">𝐈</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p3.4.m4.1c">\operatorname{{\mathbf{\Lambda}}}=\tau^{2}\operatorname{\mathbf{I}}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p3.4.m4.1d">bold_Λ = italic_τ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT bold_I</annotation></semantics></math> and <math alttext="\operatorname{\mathbf{B}}=\text{vec}^{-1}(\operatorname{\mathbf{b}})" class="ltx_Math" display="inline" id="S4.SS1.p3.5.m5.1"><semantics id="S4.SS1.p3.5.m5.1a"><mrow id="S4.SS1.p3.5.m5.1.2" xref="S4.SS1.p3.5.m5.1.2.cmml"><mi id="S4.SS1.p3.5.m5.1.2.2" xref="S4.SS1.p3.5.m5.1.2.2.cmml">𝐁</mi><mo id="S4.SS1.p3.5.m5.1.2.1" xref="S4.SS1.p3.5.m5.1.2.1.cmml">=</mo><mrow id="S4.SS1.p3.5.m5.1.2.3" xref="S4.SS1.p3.5.m5.1.2.3.cmml"><msup id="S4.SS1.p3.5.m5.1.2.3.2" xref="S4.SS1.p3.5.m5.1.2.3.2.cmml"><mtext id="S4.SS1.p3.5.m5.1.2.3.2.2" xref="S4.SS1.p3.5.m5.1.2.3.2.2a.cmml">vec</mtext><mrow id="S4.SS1.p3.5.m5.1.2.3.2.3" xref="S4.SS1.p3.5.m5.1.2.3.2.3.cmml"><mo id="S4.SS1.p3.5.m5.1.2.3.2.3a" xref="S4.SS1.p3.5.m5.1.2.3.2.3.cmml">−</mo><mn id="S4.SS1.p3.5.m5.1.2.3.2.3.2" xref="S4.SS1.p3.5.m5.1.2.3.2.3.2.cmml">1</mn></mrow></msup><mo id="S4.SS1.p3.5.m5.1.2.3.1" xref="S4.SS1.p3.5.m5.1.2.3.1.cmml">⁢</mo><mrow id="S4.SS1.p3.5.m5.1.2.3.3.2" xref="S4.SS1.p3.5.m5.1.2.3.cmml"><mo id="S4.SS1.p3.5.m5.1.2.3.3.2.1" stretchy="false" xref="S4.SS1.p3.5.m5.1.2.3.cmml">(</mo><mi id="S4.SS1.p3.5.m5.1.1" xref="S4.SS1.p3.5.m5.1.1.cmml">𝐛</mi><mo id="S4.SS1.p3.5.m5.1.2.3.3.2.2" stretchy="false" xref="S4.SS1.p3.5.m5.1.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p3.5.m5.1b"><apply id="S4.SS1.p3.5.m5.1.2.cmml" xref="S4.SS1.p3.5.m5.1.2"><eq id="S4.SS1.p3.5.m5.1.2.1.cmml" xref="S4.SS1.p3.5.m5.1.2.1"></eq><ci id="S4.SS1.p3.5.m5.1.2.2.cmml" xref="S4.SS1.p3.5.m5.1.2.2">𝐁</ci><apply id="S4.SS1.p3.5.m5.1.2.3.cmml" xref="S4.SS1.p3.5.m5.1.2.3"><times id="S4.SS1.p3.5.m5.1.2.3.1.cmml" xref="S4.SS1.p3.5.m5.1.2.3.1"></times><apply id="S4.SS1.p3.5.m5.1.2.3.2.cmml" xref="S4.SS1.p3.5.m5.1.2.3.2"><csymbol cd="ambiguous" id="S4.SS1.p3.5.m5.1.2.3.2.1.cmml" xref="S4.SS1.p3.5.m5.1.2.3.2">superscript</csymbol><ci id="S4.SS1.p3.5.m5.1.2.3.2.2a.cmml" xref="S4.SS1.p3.5.m5.1.2.3.2.2"><mtext id="S4.SS1.p3.5.m5.1.2.3.2.2.cmml" xref="S4.SS1.p3.5.m5.1.2.3.2.2">vec</mtext></ci><apply id="S4.SS1.p3.5.m5.1.2.3.2.3.cmml" xref="S4.SS1.p3.5.m5.1.2.3.2.3"><minus id="S4.SS1.p3.5.m5.1.2.3.2.3.1.cmml" xref="S4.SS1.p3.5.m5.1.2.3.2.3"></minus><cn id="S4.SS1.p3.5.m5.1.2.3.2.3.2.cmml" type="integer" xref="S4.SS1.p3.5.m5.1.2.3.2.3.2">1</cn></apply></apply><ci id="S4.SS1.p3.5.m5.1.1.cmml" xref="S4.SS1.p3.5.m5.1.1">𝐛</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p3.5.m5.1c">\operatorname{\mathbf{B}}=\text{vec}^{-1}(\operatorname{\mathbf{b}})</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p3.5.m5.1d">bold_B = vec start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ( bold_b )</annotation></semantics></math> to be a low-rank and sparse matrix. We used two sparse GEE models: one with an unstructured covariance and the other with an equicorrelation covariance structure. Both these models employed a lasso penalty on <math alttext="\operatorname{\mathbf{b}}" class="ltx_Math" display="inline" id="S4.SS1.p3.6.m6.1"><semantics id="S4.SS1.p3.6.m6.1a"><mi id="S4.SS1.p3.6.m6.1.1" xref="S4.SS1.p3.6.m6.1.1.cmml">𝐛</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.p3.6.m6.1b"><ci id="S4.SS1.p3.6.m6.1.1.cmml" xref="S4.SS1.p3.6.m6.1.1">𝐛</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p3.6.m6.1c">\operatorname{\mathbf{b}}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p3.6.m6.1d">bold_b</annotation></semantics></math>. The unstructured covariance model assumed that <math alttext="\operatorname{\mathbf{e}}_{i}" class="ltx_Math" display="inline" id="S4.SS1.p3.7.m7.1"><semantics id="S4.SS1.p3.7.m7.1a"><msub id="S4.SS1.p3.7.m7.1.1" xref="S4.SS1.p3.7.m7.1.1.cmml"><mi id="S4.SS1.p3.7.m7.1.1.2" xref="S4.SS1.p3.7.m7.1.1.2.cmml">𝐞</mi><mi id="S4.SS1.p3.7.m7.1.1.3" xref="S4.SS1.p3.7.m7.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS1.p3.7.m7.1b"><apply id="S4.SS1.p3.7.m7.1.1.cmml" xref="S4.SS1.p3.7.m7.1.1"><csymbol cd="ambiguous" id="S4.SS1.p3.7.m7.1.1.1.cmml" xref="S4.SS1.p3.7.m7.1.1">subscript</csymbol><ci id="S4.SS1.p3.7.m7.1.1.2.cmml" xref="S4.SS1.p3.7.m7.1.1.2">𝐞</ci><ci id="S4.SS1.p3.7.m7.1.1.3.cmml" xref="S4.SS1.p3.7.m7.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p3.7.m7.1c">\operatorname{\mathbf{e}}_{i}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p3.7.m7.1d">bold_e start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> was distributed as <math alttext="N(\operatorname{\mathbf{0}},\tilde{\operatorname{{\mathbf{\Lambda}}}})" class="ltx_Math" display="inline" id="S4.SS1.p3.8.m8.2"><semantics id="S4.SS1.p3.8.m8.2a"><mrow id="S4.SS1.p3.8.m8.2.3" xref="S4.SS1.p3.8.m8.2.3.cmml"><mi id="S4.SS1.p3.8.m8.2.3.2" xref="S4.SS1.p3.8.m8.2.3.2.cmml">N</mi><mo id="S4.SS1.p3.8.m8.2.3.1" xref="S4.SS1.p3.8.m8.2.3.1.cmml">⁢</mo><mrow id="S4.SS1.p3.8.m8.2.3.3.2" xref="S4.SS1.p3.8.m8.2.3.3.1.cmml"><mo id="S4.SS1.p3.8.m8.2.3.3.2.1" stretchy="false" xref="S4.SS1.p3.8.m8.2.3.3.1.cmml">(</mo><mi id="S4.SS1.p3.8.m8.1.1" xref="S4.SS1.p3.8.m8.1.1.cmml">𝟎</mi><mo id="S4.SS1.p3.8.m8.2.3.3.2.2" xref="S4.SS1.p3.8.m8.2.3.3.1.cmml">,</mo><mover accent="true" id="S4.SS1.p3.8.m8.2.2" xref="S4.SS1.p3.8.m8.2.2.cmml"><mi id="S4.SS1.p3.8.m8.2.2.2" xref="S4.SS1.p3.8.m8.2.2.2.cmml">𝚲</mi><mo id="S4.SS1.p3.8.m8.2.2.1" xref="S4.SS1.p3.8.m8.2.2.1.cmml">~</mo></mover><mo id="S4.SS1.p3.8.m8.2.3.3.2.3" stretchy="false" xref="S4.SS1.p3.8.m8.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p3.8.m8.2b"><apply id="S4.SS1.p3.8.m8.2.3.cmml" xref="S4.SS1.p3.8.m8.2.3"><times id="S4.SS1.p3.8.m8.2.3.1.cmml" xref="S4.SS1.p3.8.m8.2.3.1"></times><ci id="S4.SS1.p3.8.m8.2.3.2.cmml" xref="S4.SS1.p3.8.m8.2.3.2">𝑁</ci><interval closure="open" id="S4.SS1.p3.8.m8.2.3.3.1.cmml" xref="S4.SS1.p3.8.m8.2.3.3.2"><ci id="S4.SS1.p3.8.m8.1.1.cmml" xref="S4.SS1.p3.8.m8.1.1">𝟎</ci><apply id="S4.SS1.p3.8.m8.2.2.cmml" xref="S4.SS1.p3.8.m8.2.2"><ci id="S4.SS1.p3.8.m8.2.2.1.cmml" xref="S4.SS1.p3.8.m8.2.2.1">~</ci><ci id="S4.SS1.p3.8.m8.2.2.2.cmml" xref="S4.SS1.p3.8.m8.2.2.2">𝚲</ci></apply></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p3.8.m8.2c">N(\operatorname{\mathbf{0}},\tilde{\operatorname{{\mathbf{\Lambda}}}})</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p3.8.m8.2d">italic_N ( bold_0 , over~ start_ARG bold_Λ end_ARG )</annotation></semantics></math> for every <math alttext="i" class="ltx_Math" display="inline" id="S4.SS1.p3.9.m9.1"><semantics id="S4.SS1.p3.9.m9.1a"><mi id="S4.SS1.p3.9.m9.1.1" xref="S4.SS1.p3.9.m9.1.1.cmml">i</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.p3.9.m9.1b"><ci id="S4.SS1.p3.9.m9.1.1.cmml" xref="S4.SS1.p3.9.m9.1.1">𝑖</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p3.9.m9.1c">i</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p3.9.m9.1d">italic_i</annotation></semantics></math>, implying that <math alttext="m_{i}=m" class="ltx_Math" display="inline" id="S4.SS1.p3.10.m10.1"><semantics id="S4.SS1.p3.10.m10.1a"><mrow id="S4.SS1.p3.10.m10.1.1" xref="S4.SS1.p3.10.m10.1.1.cmml"><msub id="S4.SS1.p3.10.m10.1.1.2" xref="S4.SS1.p3.10.m10.1.1.2.cmml"><mi id="S4.SS1.p3.10.m10.1.1.2.2" xref="S4.SS1.p3.10.m10.1.1.2.2.cmml">m</mi><mi id="S4.SS1.p3.10.m10.1.1.2.3" xref="S4.SS1.p3.10.m10.1.1.2.3.cmml">i</mi></msub><mo id="S4.SS1.p3.10.m10.1.1.1" xref="S4.SS1.p3.10.m10.1.1.1.cmml">=</mo><mi id="S4.SS1.p3.10.m10.1.1.3" xref="S4.SS1.p3.10.m10.1.1.3.cmml">m</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p3.10.m10.1b"><apply id="S4.SS1.p3.10.m10.1.1.cmml" xref="S4.SS1.p3.10.m10.1.1"><eq id="S4.SS1.p3.10.m10.1.1.1.cmml" xref="S4.SS1.p3.10.m10.1.1.1"></eq><apply id="S4.SS1.p3.10.m10.1.1.2.cmml" xref="S4.SS1.p3.10.m10.1.1.2"><csymbol cd="ambiguous" id="S4.SS1.p3.10.m10.1.1.2.1.cmml" xref="S4.SS1.p3.10.m10.1.1.2">subscript</csymbol><ci id="S4.SS1.p3.10.m10.1.1.2.2.cmml" xref="S4.SS1.p3.10.m10.1.1.2.2">𝑚</ci><ci id="S4.SS1.p3.10.m10.1.1.2.3.cmml" xref="S4.SS1.p3.10.m10.1.1.2.3">𝑖</ci></apply><ci id="S4.SS1.p3.10.m10.1.1.3.cmml" xref="S4.SS1.p3.10.m10.1.1.3">𝑚</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p3.10.m10.1c">m_{i}=m</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p3.10.m10.1d">italic_m start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = italic_m</annotation></semantics></math> and <math alttext="\operatorname{{\mathbf{\Lambda}}}=\operatorname{\mathbf{I}}_{n}\otimes\tilde{% \operatorname{{\mathbf{\Lambda}}}}" class="ltx_Math" display="inline" id="S4.SS1.p3.11.m11.1"><semantics id="S4.SS1.p3.11.m11.1a"><mrow id="S4.SS1.p3.11.m11.1.1" xref="S4.SS1.p3.11.m11.1.1.cmml"><mi id="S4.SS1.p3.11.m11.1.1.2" xref="S4.SS1.p3.11.m11.1.1.2.cmml">𝚲</mi><mo id="S4.SS1.p3.11.m11.1.1.1" xref="S4.SS1.p3.11.m11.1.1.1.cmml">=</mo><mrow id="S4.SS1.p3.11.m11.1.1.3" xref="S4.SS1.p3.11.m11.1.1.3.cmml"><msub id="S4.SS1.p3.11.m11.1.1.3.2" xref="S4.SS1.p3.11.m11.1.1.3.2.cmml"><mi id="S4.SS1.p3.11.m11.1.1.3.2.2" xref="S4.SS1.p3.11.m11.1.1.3.2.2.cmml">𝐈</mi><mi id="S4.SS1.p3.11.m11.1.1.3.2.3" xref="S4.SS1.p3.11.m11.1.1.3.2.3.cmml">n</mi></msub><mo id="S4.SS1.p3.11.m11.1.1.3.1" lspace="0em" rspace="0.222em" xref="S4.SS1.p3.11.m11.1.1.3.1.cmml">⊗</mo><mover accent="true" id="S4.SS1.p3.11.m11.1.1.3.3" xref="S4.SS1.p3.11.m11.1.1.3.3.cmml"><mi id="S4.SS1.p3.11.m11.1.1.3.3.2" xref="S4.SS1.p3.11.m11.1.1.3.3.2.cmml">𝚲</mi><mo id="S4.SS1.p3.11.m11.1.1.3.3.1" xref="S4.SS1.p3.11.m11.1.1.3.3.1.cmml">~</mo></mover></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p3.11.m11.1b"><apply id="S4.SS1.p3.11.m11.1.1.cmml" xref="S4.SS1.p3.11.m11.1.1"><eq id="S4.SS1.p3.11.m11.1.1.1.cmml" xref="S4.SS1.p3.11.m11.1.1.1"></eq><ci id="S4.SS1.p3.11.m11.1.1.2.cmml" xref="S4.SS1.p3.11.m11.1.1.2">𝚲</ci><apply id="S4.SS1.p3.11.m11.1.1.3.cmml" xref="S4.SS1.p3.11.m11.1.1.3"><csymbol cd="latexml" id="S4.SS1.p3.11.m11.1.1.3.1.cmml" xref="S4.SS1.p3.11.m11.1.1.3.1">tensor-product</csymbol><apply id="S4.SS1.p3.11.m11.1.1.3.2.cmml" xref="S4.SS1.p3.11.m11.1.1.3.2"><csymbol cd="ambiguous" id="S4.SS1.p3.11.m11.1.1.3.2.1.cmml" xref="S4.SS1.p3.11.m11.1.1.3.2">subscript</csymbol><ci id="S4.SS1.p3.11.m11.1.1.3.2.2.cmml" xref="S4.SS1.p3.11.m11.1.1.3.2.2">𝐈</ci><ci id="S4.SS1.p3.11.m11.1.1.3.2.3.cmml" xref="S4.SS1.p3.11.m11.1.1.3.2.3">𝑛</ci></apply><apply id="S4.SS1.p3.11.m11.1.1.3.3.cmml" xref="S4.SS1.p3.11.m11.1.1.3.3"><ci id="S4.SS1.p3.11.m11.1.1.3.3.1.cmml" xref="S4.SS1.p3.11.m11.1.1.3.3.1">~</ci><ci id="S4.SS1.p3.11.m11.1.1.3.3.2.cmml" xref="S4.SS1.p3.11.m11.1.1.3.3.2">𝚲</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p3.11.m11.1c">\operatorname{{\mathbf{\Lambda}}}=\operatorname{\mathbf{I}}_{n}\otimes\tilde{% \operatorname{{\mathbf{\Lambda}}}}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p3.11.m11.1d">bold_Λ = bold_I start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ⊗ over~ start_ARG bold_Λ end_ARG</annotation></semantics></math>. Due to this constraint, we used this model only in simulations. The sparse GEE equicorrelation model assumed that <math alttext="\operatorname{{\mathbf{\Lambda}}}" class="ltx_Math" display="inline" id="S4.SS1.p3.12.m12.1"><semantics id="S4.SS1.p3.12.m12.1a"><mi id="S4.SS1.p3.12.m12.1.1" xref="S4.SS1.p3.12.m12.1.1.cmml">𝚲</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.p3.12.m12.1b"><ci id="S4.SS1.p3.12.m12.1.1.cmml" xref="S4.SS1.p3.12.m12.1.1">𝚲</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p3.12.m12.1c">\operatorname{{\mathbf{\Lambda}}}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p3.12.m12.1d">bold_Λ</annotation></semantics></math> was a correlation matrix with off-diagonal elements equal to <math alttext="\alpha\in(0,1)" class="ltx_Math" display="inline" id="S4.SS1.p3.13.m13.2"><semantics id="S4.SS1.p3.13.m13.2a"><mrow id="S4.SS1.p3.13.m13.2.3" xref="S4.SS1.p3.13.m13.2.3.cmml"><mi id="S4.SS1.p3.13.m13.2.3.2" xref="S4.SS1.p3.13.m13.2.3.2.cmml">α</mi><mo id="S4.SS1.p3.13.m13.2.3.1" xref="S4.SS1.p3.13.m13.2.3.1.cmml">∈</mo><mrow id="S4.SS1.p3.13.m13.2.3.3.2" xref="S4.SS1.p3.13.m13.2.3.3.1.cmml"><mo id="S4.SS1.p3.13.m13.2.3.3.2.1" stretchy="false" xref="S4.SS1.p3.13.m13.2.3.3.1.cmml">(</mo><mn id="S4.SS1.p3.13.m13.1.1" xref="S4.SS1.p3.13.m13.1.1.cmml">0</mn><mo id="S4.SS1.p3.13.m13.2.3.3.2.2" xref="S4.SS1.p3.13.m13.2.3.3.1.cmml">,</mo><mn id="S4.SS1.p3.13.m13.2.2" xref="S4.SS1.p3.13.m13.2.2.cmml">1</mn><mo id="S4.SS1.p3.13.m13.2.3.3.2.3" stretchy="false" xref="S4.SS1.p3.13.m13.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p3.13.m13.2b"><apply id="S4.SS1.p3.13.m13.2.3.cmml" xref="S4.SS1.p3.13.m13.2.3"><in id="S4.SS1.p3.13.m13.2.3.1.cmml" xref="S4.SS1.p3.13.m13.2.3.1"></in><ci id="S4.SS1.p3.13.m13.2.3.2.cmml" xref="S4.SS1.p3.13.m13.2.3.2">𝛼</ci><interval closure="open" id="S4.SS1.p3.13.m13.2.3.3.1.cmml" xref="S4.SS1.p3.13.m13.2.3.3.2"><cn id="S4.SS1.p3.13.m13.1.1.cmml" type="integer" xref="S4.SS1.p3.13.m13.1.1">0</cn><cn id="S4.SS1.p3.13.m13.2.2.cmml" type="integer" xref="S4.SS1.p3.13.m13.2.2">1</cn></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p3.13.m13.2c">\alpha\in(0,1)</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p3.13.m13.2d">italic_α ∈ ( 0 , 1 )</annotation></semantics></math>. Unlike the previous GEE model, the <math alttext="m_{i}" class="ltx_Math" display="inline" id="S4.SS1.p3.14.m14.1"><semantics id="S4.SS1.p3.14.m14.1a"><msub id="S4.SS1.p3.14.m14.1.1" xref="S4.SS1.p3.14.m14.1.1.cmml"><mi id="S4.SS1.p3.14.m14.1.1.2" xref="S4.SS1.p3.14.m14.1.1.2.cmml">m</mi><mi id="S4.SS1.p3.14.m14.1.1.3" xref="S4.SS1.p3.14.m14.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS1.p3.14.m14.1b"><apply id="S4.SS1.p3.14.m14.1.1.cmml" xref="S4.SS1.p3.14.m14.1.1"><csymbol cd="ambiguous" id="S4.SS1.p3.14.m14.1.1.1.cmml" xref="S4.SS1.p3.14.m14.1.1">subscript</csymbol><ci id="S4.SS1.p3.14.m14.1.1.2.cmml" xref="S4.SS1.p3.14.m14.1.1.2">𝑚</ci><ci id="S4.SS1.p3.14.m14.1.1.3.cmml" xref="S4.SS1.p3.14.m14.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p3.14.m14.1c">m_{i}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p3.14.m14.1d">italic_m start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math>s in this model were allowed to be unequal, enabling its use for simulated and real data analyses. The PQL, trace regression, and GEE tuning parameters were selected using the recommended approach or cross-validation. MMTR had two tuning parameters: <math alttext="\lambda_{\operatorname{\mathbf{B}}}" class="ltx_Math" display="inline" id="S4.SS1.p3.15.m15.1"><semantics id="S4.SS1.p3.15.m15.1a"><msub id="S4.SS1.p3.15.m15.1.1" xref="S4.SS1.p3.15.m15.1.1.cmml"><mi id="S4.SS1.p3.15.m15.1.1.2" xref="S4.SS1.p3.15.m15.1.1.2.cmml">λ</mi><mi id="S4.SS1.p3.15.m15.1.1.3" xref="S4.SS1.p3.15.m15.1.1.3.cmml">𝐁</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS1.p3.15.m15.1b"><apply id="S4.SS1.p3.15.m15.1.1.cmml" xref="S4.SS1.p3.15.m15.1.1"><csymbol cd="ambiguous" id="S4.SS1.p3.15.m15.1.1.1.cmml" xref="S4.SS1.p3.15.m15.1.1">subscript</csymbol><ci id="S4.SS1.p3.15.m15.1.1.2.cmml" xref="S4.SS1.p3.15.m15.1.1.2">𝜆</ci><ci id="S4.SS1.p3.15.m15.1.1.3.cmml" xref="S4.SS1.p3.15.m15.1.1.3">𝐁</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p3.15.m15.1c">\lambda_{\operatorname{\mathbf{B}}}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p3.15.m15.1d">italic_λ start_POSTSUBSCRIPT bold_B end_POSTSUBSCRIPT</annotation></semantics></math> in (<a class="ltx_ref" href="https://arxiv.org/html/2503.13782v1#S3.E8" title="In 3 Regularized Parameter Estimation ‣ Regularized Parameter Estimation in Mixed Model Trace Regression"><span class="ltx_text ltx_ref_tag">8</span></a>), which controlled the sparsity of <math alttext="\operatorname{\mathbf{B}}" class="ltx_Math" display="inline" id="S4.SS1.p3.16.m16.1"><semantics id="S4.SS1.p3.16.m16.1a"><mi id="S4.SS1.p3.16.m16.1.1" xref="S4.SS1.p3.16.m16.1.1.cmml">𝐁</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.p3.16.m16.1b"><ci id="S4.SS1.p3.16.m16.1.1.cmml" xref="S4.SS1.p3.16.m16.1.1">𝐁</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p3.16.m16.1c">\operatorname{\mathbf{B}}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p3.16.m16.1d">bold_B</annotation></semantics></math>, and <math alttext="\lambda_{\operatorname{\mathbf{L}}}" class="ltx_Math" display="inline" id="S4.SS1.p3.17.m17.1"><semantics id="S4.SS1.p3.17.m17.1a"><msub id="S4.SS1.p3.17.m17.1.1" xref="S4.SS1.p3.17.m17.1.1.cmml"><mi id="S4.SS1.p3.17.m17.1.1.2" xref="S4.SS1.p3.17.m17.1.1.2.cmml">λ</mi><mi id="S4.SS1.p3.17.m17.1.1.3" xref="S4.SS1.p3.17.m17.1.1.3.cmml">𝐋</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS1.p3.17.m17.1b"><apply id="S4.SS1.p3.17.m17.1.1.cmml" xref="S4.SS1.p3.17.m17.1.1"><csymbol cd="ambiguous" id="S4.SS1.p3.17.m17.1.1.1.cmml" xref="S4.SS1.p3.17.m17.1.1">subscript</csymbol><ci id="S4.SS1.p3.17.m17.1.1.2.cmml" xref="S4.SS1.p3.17.m17.1.1.2">𝜆</ci><ci id="S4.SS1.p3.17.m17.1.1.3.cmml" xref="S4.SS1.p3.17.m17.1.1.3">𝐋</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p3.17.m17.1c">\lambda_{\operatorname{\mathbf{L}}}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p3.17.m17.1d">italic_λ start_POSTSUBSCRIPT bold_L end_POSTSUBSCRIPT</annotation></semantics></math> in (<a class="ltx_ref" href="https://arxiv.org/html/2503.13782v1#S3.E10" title="In 3 Regularized Parameter Estimation ‣ Regularized Parameter Estimation in Mixed Model Trace Regression"><span class="ltx_text ltx_ref_tag">10</span></a>) and (<a class="ltx_ref" href="https://arxiv.org/html/2503.13782v1#S3.E11" title="In 3 Regularized Parameter Estimation ‣ Regularized Parameter Estimation in Mixed Model Trace Regression"><span class="ltx_text ltx_ref_tag">11</span></a>), which determined the ranks of <math alttext="\operatorname{{\mathbf{\Sigma}}}_{1}" class="ltx_Math" display="inline" id="S4.SS1.p3.18.m18.1"><semantics id="S4.SS1.p3.18.m18.1a"><msub id="S4.SS1.p3.18.m18.1.1" xref="S4.SS1.p3.18.m18.1.1.cmml"><mi id="S4.SS1.p3.18.m18.1.1.2" xref="S4.SS1.p3.18.m18.1.1.2.cmml">𝚺</mi><mn id="S4.SS1.p3.18.m18.1.1.3" xref="S4.SS1.p3.18.m18.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S4.SS1.p3.18.m18.1b"><apply id="S4.SS1.p3.18.m18.1.1.cmml" xref="S4.SS1.p3.18.m18.1.1"><csymbol cd="ambiguous" id="S4.SS1.p3.18.m18.1.1.1.cmml" xref="S4.SS1.p3.18.m18.1.1">subscript</csymbol><ci id="S4.SS1.p3.18.m18.1.1.2.cmml" xref="S4.SS1.p3.18.m18.1.1.2">𝚺</ci><cn id="S4.SS1.p3.18.m18.1.1.3.cmml" type="integer" xref="S4.SS1.p3.18.m18.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p3.18.m18.1c">\operatorname{{\mathbf{\Sigma}}}_{1}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p3.18.m18.1d">bold_Σ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="\operatorname{{\mathbf{\Sigma}}}_{2}" class="ltx_Math" display="inline" id="S4.SS1.p3.19.m19.1"><semantics id="S4.SS1.p3.19.m19.1a"><msub id="S4.SS1.p3.19.m19.1.1" xref="S4.SS1.p3.19.m19.1.1.cmml"><mi id="S4.SS1.p3.19.m19.1.1.2" xref="S4.SS1.p3.19.m19.1.1.2.cmml">𝚺</mi><mn id="S4.SS1.p3.19.m19.1.1.3" xref="S4.SS1.p3.19.m19.1.1.3.cmml">2</mn></msub><annotation-xml encoding="MathML-Content" id="S4.SS1.p3.19.m19.1b"><apply id="S4.SS1.p3.19.m19.1.1.cmml" xref="S4.SS1.p3.19.m19.1.1"><csymbol cd="ambiguous" id="S4.SS1.p3.19.m19.1.1.1.cmml" xref="S4.SS1.p3.19.m19.1.1">subscript</csymbol><ci id="S4.SS1.p3.19.m19.1.1.2.cmml" xref="S4.SS1.p3.19.m19.1.1.2">𝚺</ci><cn id="S4.SS1.p3.19.m19.1.1.3.cmml" type="integer" xref="S4.SS1.p3.19.m19.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p3.19.m19.1c">\operatorname{{\mathbf{\Sigma}}}_{2}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p3.19.m19.1d">bold_Σ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math>. We varied <math alttext="\log\lambda_{\operatorname{\mathbf{B}}}" class="ltx_Math" display="inline" id="S4.SS1.p3.20.m20.1"><semantics id="S4.SS1.p3.20.m20.1a"><mrow id="S4.SS1.p3.20.m20.1.1" xref="S4.SS1.p3.20.m20.1.1.cmml"><mi id="S4.SS1.p3.20.m20.1.1.1" xref="S4.SS1.p3.20.m20.1.1.1.cmml">log</mi><mo id="S4.SS1.p3.20.m20.1.1a" lspace="0.167em" xref="S4.SS1.p3.20.m20.1.1.cmml">⁡</mo><msub id="S4.SS1.p3.20.m20.1.1.2" xref="S4.SS1.p3.20.m20.1.1.2.cmml"><mi id="S4.SS1.p3.20.m20.1.1.2.2" xref="S4.SS1.p3.20.m20.1.1.2.2.cmml">λ</mi><mi id="S4.SS1.p3.20.m20.1.1.2.3" xref="S4.SS1.p3.20.m20.1.1.2.3.cmml">𝐁</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p3.20.m20.1b"><apply id="S4.SS1.p3.20.m20.1.1.cmml" xref="S4.SS1.p3.20.m20.1.1"><log id="S4.SS1.p3.20.m20.1.1.1.cmml" xref="S4.SS1.p3.20.m20.1.1.1"></log><apply id="S4.SS1.p3.20.m20.1.1.2.cmml" xref="S4.SS1.p3.20.m20.1.1.2"><csymbol cd="ambiguous" id="S4.SS1.p3.20.m20.1.1.2.1.cmml" xref="S4.SS1.p3.20.m20.1.1.2">subscript</csymbol><ci id="S4.SS1.p3.20.m20.1.1.2.2.cmml" xref="S4.SS1.p3.20.m20.1.1.2.2">𝜆</ci><ci id="S4.SS1.p3.20.m20.1.1.2.3.cmml" xref="S4.SS1.p3.20.m20.1.1.2.3">𝐁</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p3.20.m20.1c">\log\lambda_{\operatorname{\mathbf{B}}}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p3.20.m20.1d">roman_log italic_λ start_POSTSUBSCRIPT bold_B end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="\log\lambda_{\operatorname{\mathbf{L}}}" class="ltx_Math" display="inline" id="S4.SS1.p3.21.m21.1"><semantics id="S4.SS1.p3.21.m21.1a"><mrow id="S4.SS1.p3.21.m21.1.1" xref="S4.SS1.p3.21.m21.1.1.cmml"><mi id="S4.SS1.p3.21.m21.1.1.1" xref="S4.SS1.p3.21.m21.1.1.1.cmml">log</mi><mo id="S4.SS1.p3.21.m21.1.1a" lspace="0.167em" xref="S4.SS1.p3.21.m21.1.1.cmml">⁡</mo><msub id="S4.SS1.p3.21.m21.1.1.2" xref="S4.SS1.p3.21.m21.1.1.2.cmml"><mi id="S4.SS1.p3.21.m21.1.1.2.2" xref="S4.SS1.p3.21.m21.1.1.2.2.cmml">λ</mi><mi id="S4.SS1.p3.21.m21.1.1.2.3" xref="S4.SS1.p3.21.m21.1.1.2.3.cmml">𝐋</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p3.21.m21.1b"><apply id="S4.SS1.p3.21.m21.1.1.cmml" xref="S4.SS1.p3.21.m21.1.1"><log id="S4.SS1.p3.21.m21.1.1.1.cmml" xref="S4.SS1.p3.21.m21.1.1.1"></log><apply id="S4.SS1.p3.21.m21.1.1.2.cmml" xref="S4.SS1.p3.21.m21.1.1.2"><csymbol cd="ambiguous" id="S4.SS1.p3.21.m21.1.1.2.1.cmml" xref="S4.SS1.p3.21.m21.1.1.2">subscript</csymbol><ci id="S4.SS1.p3.21.m21.1.1.2.2.cmml" xref="S4.SS1.p3.21.m21.1.1.2.2">𝜆</ci><ci id="S4.SS1.p3.21.m21.1.1.2.3.cmml" xref="S4.SS1.p3.21.m21.1.1.2.3">𝐋</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p3.21.m21.1c">\log\lambda_{\operatorname{\mathbf{L}}}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p3.21.m21.1d">roman_log italic_λ start_POSTSUBSCRIPT bold_L end_POSTSUBSCRIPT</annotation></semantics></math> on a grid and chose the best <math alttext="(\lambda_{\operatorname{\mathbf{B}}},\lambda_{\operatorname{\mathbf{L}}})" class="ltx_Math" display="inline" id="S4.SS1.p3.22.m22.2"><semantics id="S4.SS1.p3.22.m22.2a"><mrow id="S4.SS1.p3.22.m22.2.2.2" xref="S4.SS1.p3.22.m22.2.2.3.cmml"><mo id="S4.SS1.p3.22.m22.2.2.2.3" stretchy="false" xref="S4.SS1.p3.22.m22.2.2.3.cmml">(</mo><msub id="S4.SS1.p3.22.m22.1.1.1.1" xref="S4.SS1.p3.22.m22.1.1.1.1.cmml"><mi id="S4.SS1.p3.22.m22.1.1.1.1.2" xref="S4.SS1.p3.22.m22.1.1.1.1.2.cmml">λ</mi><mi id="S4.SS1.p3.22.m22.1.1.1.1.3" xref="S4.SS1.p3.22.m22.1.1.1.1.3.cmml">𝐁</mi></msub><mo id="S4.SS1.p3.22.m22.2.2.2.4" xref="S4.SS1.p3.22.m22.2.2.3.cmml">,</mo><msub id="S4.SS1.p3.22.m22.2.2.2.2" xref="S4.SS1.p3.22.m22.2.2.2.2.cmml"><mi id="S4.SS1.p3.22.m22.2.2.2.2.2" xref="S4.SS1.p3.22.m22.2.2.2.2.2.cmml">λ</mi><mi id="S4.SS1.p3.22.m22.2.2.2.2.3" xref="S4.SS1.p3.22.m22.2.2.2.2.3.cmml">𝐋</mi></msub><mo id="S4.SS1.p3.22.m22.2.2.2.5" stretchy="false" xref="S4.SS1.p3.22.m22.2.2.3.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p3.22.m22.2b"><interval closure="open" id="S4.SS1.p3.22.m22.2.2.3.cmml" xref="S4.SS1.p3.22.m22.2.2.2"><apply id="S4.SS1.p3.22.m22.1.1.1.1.cmml" xref="S4.SS1.p3.22.m22.1.1.1.1"><csymbol cd="ambiguous" id="S4.SS1.p3.22.m22.1.1.1.1.1.cmml" xref="S4.SS1.p3.22.m22.1.1.1.1">subscript</csymbol><ci id="S4.SS1.p3.22.m22.1.1.1.1.2.cmml" xref="S4.SS1.p3.22.m22.1.1.1.1.2">𝜆</ci><ci id="S4.SS1.p3.22.m22.1.1.1.1.3.cmml" xref="S4.SS1.p3.22.m22.1.1.1.1.3">𝐁</ci></apply><apply id="S4.SS1.p3.22.m22.2.2.2.2.cmml" xref="S4.SS1.p3.22.m22.2.2.2.2"><csymbol cd="ambiguous" id="S4.SS1.p3.22.m22.2.2.2.2.1.cmml" xref="S4.SS1.p3.22.m22.2.2.2.2">subscript</csymbol><ci id="S4.SS1.p3.22.m22.2.2.2.2.2.cmml" xref="S4.SS1.p3.22.m22.2.2.2.2.2">𝜆</ci><ci id="S4.SS1.p3.22.m22.2.2.2.2.3.cmml" xref="S4.SS1.p3.22.m22.2.2.2.2.3">𝐋</ci></apply></interval></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p3.22.m22.2c">(\lambda_{\operatorname{\mathbf{B}}},\lambda_{\operatorname{\mathbf{L}}})</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p3.22.m22.2d">( italic_λ start_POSTSUBSCRIPT bold_B end_POSTSUBSCRIPT , italic_λ start_POSTSUBSCRIPT bold_L end_POSTSUBSCRIPT )</annotation></semantics></math> pair with the minimum extended Bayesian information criterion <cite class="ltx_cite ltx_citemacro_citep">(Chen and Chen, <a class="ltx_ref" href="https://arxiv.org/html/2503.13782v1#bib.bib5" title="">2008</a>)</cite>.</p> </div> <div class="ltx_para" id="S4.SS1.p4"> <p class="ltx_p" id="S4.SS1.p4.5">The empirical performance in estimation and prediction was quantified using relative estimation error and mean squared prediction error. Let <math alttext="\hat{\operatorname{{\bm{\xi}}}}" class="ltx_Math" display="inline" id="S4.SS1.p4.1.m1.1"><semantics id="S4.SS1.p4.1.m1.1a"><mover accent="true" id="S4.SS1.p4.1.m1.1.1" xref="S4.SS1.p4.1.m1.1.1.cmml"><mi id="S4.SS1.p4.1.m1.1.1.2" xref="S4.SS1.p4.1.m1.1.1.2.cmml">𝝃</mi><mo id="S4.SS1.p4.1.m1.1.1.1" xref="S4.SS1.p4.1.m1.1.1.1.cmml">^</mo></mover><annotation-xml encoding="MathML-Content" id="S4.SS1.p4.1.m1.1b"><apply id="S4.SS1.p4.1.m1.1.1.cmml" xref="S4.SS1.p4.1.m1.1.1"><ci id="S4.SS1.p4.1.m1.1.1.1.cmml" xref="S4.SS1.p4.1.m1.1.1.1">^</ci><ci id="S4.SS1.p4.1.m1.1.1.2.cmml" xref="S4.SS1.p4.1.m1.1.1.2">𝝃</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p4.1.m1.1c">\hat{\operatorname{{\bm{\xi}}}}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p4.1.m1.1d">over^ start_ARG bold_italic_ξ end_ARG</annotation></semantics></math> be the estimate of the parameter <math alttext="\operatorname{{\bm{\xi}}}" class="ltx_Math" display="inline" id="S4.SS1.p4.2.m2.1"><semantics id="S4.SS1.p4.2.m2.1a"><mi id="S4.SS1.p4.2.m2.1.1" xref="S4.SS1.p4.2.m2.1.1.cmml">𝝃</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.p4.2.m2.1b"><ci id="S4.SS1.p4.2.m2.1.1.cmml" xref="S4.SS1.p4.2.m2.1.1">𝝃</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p4.2.m2.1c">\operatorname{{\bm{\xi}}}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p4.2.m2.1d">bold_italic_ξ</annotation></semantics></math>, where <math alttext="\operatorname{{\bm{\xi}}}\in\{\operatorname{\mathbf{B}},\operatorname{{\mathbf% {\Lambda}}},\operatorname{{\mathbf{\Sigma}}}_{1}.\operatorname{{\mathbf{\Sigma% }}}_{2}\}" class="ltx_math_unparsed" display="inline" id="S4.SS1.p4.3.m3.2"><semantics id="S4.SS1.p4.3.m3.2a"><mrow id="S4.SS1.p4.3.m3.2b"><mi id="S4.SS1.p4.3.m3.2.3">𝝃</mi><mo id="S4.SS1.p4.3.m3.2.4">∈</mo><mrow id="S4.SS1.p4.3.m3.2.5"><mo id="S4.SS1.p4.3.m3.2.5.1" stretchy="false">{</mo><mi id="S4.SS1.p4.3.m3.1.1">𝐁</mi><mo id="S4.SS1.p4.3.m3.2.5.2">,</mo><mi id="S4.SS1.p4.3.m3.2.2">𝚲</mi><mo id="S4.SS1.p4.3.m3.2.5.3">,</mo><msub id="S4.SS1.p4.3.m3.2.5.4"><mi id="S4.SS1.p4.3.m3.2.5.4.2">𝚺</mi><mn id="S4.SS1.p4.3.m3.2.5.4.3">1</mn></msub><mo id="S4.SS1.p4.3.m3.2.5.5" lspace="0em" rspace="0.167em">.</mo><msub id="S4.SS1.p4.3.m3.2.5.6"><mi id="S4.SS1.p4.3.m3.2.5.6.2">𝚺</mi><mn id="S4.SS1.p4.3.m3.2.5.6.3">2</mn></msub><mo id="S4.SS1.p4.3.m3.2.5.7" stretchy="false">}</mo></mrow></mrow><annotation encoding="application/x-tex" id="S4.SS1.p4.3.m3.2c">\operatorname{{\bm{\xi}}}\in\{\operatorname{\mathbf{B}},\operatorname{{\mathbf% {\Lambda}}},\operatorname{{\mathbf{\Sigma}}}_{1}.\operatorname{{\mathbf{\Sigma% }}}_{2}\}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p4.3.m3.2d">bold_italic_ξ ∈ { bold_B , bold_Λ , bold_Σ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT . bold_Σ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT }</annotation></semantics></math>, and <math alttext="\hat{\operatorname{\mathbf{y}}}_{i}" class="ltx_Math" display="inline" id="S4.SS1.p4.4.m4.1"><semantics id="S4.SS1.p4.4.m4.1a"><msub id="S4.SS1.p4.4.m4.1.1" xref="S4.SS1.p4.4.m4.1.1.cmml"><mover accent="true" id="S4.SS1.p4.4.m4.1.1.2" xref="S4.SS1.p4.4.m4.1.1.2.cmml"><mi id="S4.SS1.p4.4.m4.1.1.2.2" xref="S4.SS1.p4.4.m4.1.1.2.2.cmml">𝐲</mi><mo id="S4.SS1.p4.4.m4.1.1.2.1" xref="S4.SS1.p4.4.m4.1.1.2.1.cmml">^</mo></mover><mi id="S4.SS1.p4.4.m4.1.1.3" xref="S4.SS1.p4.4.m4.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS1.p4.4.m4.1b"><apply id="S4.SS1.p4.4.m4.1.1.cmml" xref="S4.SS1.p4.4.m4.1.1"><csymbol cd="ambiguous" id="S4.SS1.p4.4.m4.1.1.1.cmml" xref="S4.SS1.p4.4.m4.1.1">subscript</csymbol><apply id="S4.SS1.p4.4.m4.1.1.2.cmml" xref="S4.SS1.p4.4.m4.1.1.2"><ci id="S4.SS1.p4.4.m4.1.1.2.1.cmml" xref="S4.SS1.p4.4.m4.1.1.2.1">^</ci><ci id="S4.SS1.p4.4.m4.1.1.2.2.cmml" xref="S4.SS1.p4.4.m4.1.1.2.2">𝐲</ci></apply><ci id="S4.SS1.p4.4.m4.1.1.3.cmml" xref="S4.SS1.p4.4.m4.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p4.4.m4.1c">\hat{\operatorname{\mathbf{y}}}_{i}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p4.4.m4.1d">over^ start_ARG bold_y end_ARG start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> be the predicted value of <math alttext="\operatorname{\mathbf{y}}_{i}" class="ltx_Math" display="inline" id="S4.SS1.p4.5.m5.1"><semantics id="S4.SS1.p4.5.m5.1a"><msub id="S4.SS1.p4.5.m5.1.1" xref="S4.SS1.p4.5.m5.1.1.cmml"><mi id="S4.SS1.p4.5.m5.1.1.2" xref="S4.SS1.p4.5.m5.1.1.2.cmml">𝐲</mi><mi id="S4.SS1.p4.5.m5.1.1.3" xref="S4.SS1.p4.5.m5.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS1.p4.5.m5.1b"><apply id="S4.SS1.p4.5.m5.1.1.cmml" xref="S4.SS1.p4.5.m5.1.1"><csymbol cd="ambiguous" id="S4.SS1.p4.5.m5.1.1.1.cmml" xref="S4.SS1.p4.5.m5.1.1">subscript</csymbol><ci id="S4.SS1.p4.5.m5.1.1.2.cmml" xref="S4.SS1.p4.5.m5.1.1.2">𝐲</ci><ci id="S4.SS1.p4.5.m5.1.1.3.cmml" xref="S4.SS1.p4.5.m5.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p4.5.m5.1c">\operatorname{\mathbf{y}}_{i}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p4.5.m5.1d">bold_y start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> in the test data. Then, the relative estimation error and mean square prediction errors are</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="S3.EGx13"> <tbody id="S4.E13"><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\text{err}_{\operatorname{{\bm{\xi}}}}={\|\hat{\operatorname{{\bm% {\xi}}}}-\operatorname{{\bm{\xi}}}\|_{F}}/{\|\operatorname{{\bm{\xi}}}\|_{F}},% \quad\operatorname{{\bm{\xi}}}\in\{\operatorname{\mathbf{B}},\operatorname{{% \mathbf{\Lambda}}},\operatorname{{\mathbf{\Sigma}}}_{1},\operatorname{{\mathbf% {\Sigma}}}_{2}\},\quad\text{MSPE}=\sum_{i=1}^{n^{*}}\|\operatorname{\mathbf{y}% }_{i}-\hat{\operatorname{\mathbf{y}}}_{i}\|_{2}^{2}/n^{*}," class="ltx_Math" display="inline" id="S4.E13.m1.4"><semantics id="S4.E13.m1.4a"><mrow id="S4.E13.m1.4.4.1"><mrow id="S4.E13.m1.4.4.1.1.2" xref="S4.E13.m1.4.4.1.1.3.cmml"><mrow 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xref="S4.E13.m1.3.3.cmml">𝚲</mi><mo id="S4.E13.m1.4.4.1.1.2.2.1.1.2.2.5" xref="S4.E13.m1.4.4.1.1.2.2.1.1.2.3.cmml">,</mo><msub id="S4.E13.m1.4.4.1.1.2.2.1.1.1.1.1" xref="S4.E13.m1.4.4.1.1.2.2.1.1.1.1.1.cmml"><mi id="S4.E13.m1.4.4.1.1.2.2.1.1.1.1.1.2" xref="S4.E13.m1.4.4.1.1.2.2.1.1.1.1.1.2.cmml">𝚺</mi><mn id="S4.E13.m1.4.4.1.1.2.2.1.1.1.1.1.3" xref="S4.E13.m1.4.4.1.1.2.2.1.1.1.1.1.3.cmml">1</mn></msub><mo id="S4.E13.m1.4.4.1.1.2.2.1.1.2.2.6" xref="S4.E13.m1.4.4.1.1.2.2.1.1.2.3.cmml">,</mo><msub id="S4.E13.m1.4.4.1.1.2.2.1.1.2.2.2" xref="S4.E13.m1.4.4.1.1.2.2.1.1.2.2.2.cmml"><mi id="S4.E13.m1.4.4.1.1.2.2.1.1.2.2.2.2" xref="S4.E13.m1.4.4.1.1.2.2.1.1.2.2.2.2.cmml">𝚺</mi><mn id="S4.E13.m1.4.4.1.1.2.2.1.1.2.2.2.3" xref="S4.E13.m1.4.4.1.1.2.2.1.1.2.2.2.3.cmml">2</mn></msub><mo id="S4.E13.m1.4.4.1.1.2.2.1.1.2.2.7" stretchy="false" xref="S4.E13.m1.4.4.1.1.2.2.1.1.2.3.cmml">}</mo></mrow></mrow><mo id="S4.E13.m1.4.4.1.1.2.2.2.3" rspace="1.167em" xref="S4.E13.m1.4.4.1.1.2.2.3a.cmml">,</mo><mrow 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{\xi}}}}-\operatorname{{\bm{\xi}}}\|_{F}}/{\|\operatorname{{\bm{\xi}}}\|_{F}},% \quad\operatorname{{\bm{\xi}}}\in\{\operatorname{\mathbf{B}},\operatorname{{% \mathbf{\Lambda}}},\operatorname{{\mathbf{\Sigma}}}_{1},\operatorname{{\mathbf% {\Sigma}}}_{2}\},\quad\text{MSPE}=\sum_{i=1}^{n^{*}}\|\operatorname{\mathbf{y}% }_{i}-\hat{\operatorname{\mathbf{y}}}_{i}\|_{2}^{2}/n^{*},</annotation><annotation encoding="application/x-llamapun" id="S4.E13.m1.4d">err start_POSTSUBSCRIPT bold_italic_ξ end_POSTSUBSCRIPT = ∥ over^ start_ARG bold_italic_ξ end_ARG - bold_italic_ξ ∥ start_POSTSUBSCRIPT italic_F end_POSTSUBSCRIPT / ∥ bold_italic_ξ ∥ start_POSTSUBSCRIPT italic_F end_POSTSUBSCRIPT , bold_italic_ξ ∈ { bold_B , bold_Λ , bold_Σ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , bold_Σ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT } , MSPE = ∑ start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT end_POSTSUPERSCRIPT ∥ bold_y start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT - over^ start_ARG bold_y end_ARG start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∥ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / italic_n start_POSTSUPERSCRIPT ∗ 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">(13)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S4.SS1.p4.8">where <math alttext="n^{*}" class="ltx_Math" display="inline" id="S4.SS1.p4.6.m1.1"><semantics id="S4.SS1.p4.6.m1.1a"><msup id="S4.SS1.p4.6.m1.1.1" xref="S4.SS1.p4.6.m1.1.1.cmml"><mi id="S4.SS1.p4.6.m1.1.1.2" xref="S4.SS1.p4.6.m1.1.1.2.cmml">n</mi><mo id="S4.SS1.p4.6.m1.1.1.3" xref="S4.SS1.p4.6.m1.1.1.3.cmml">∗</mo></msup><annotation-xml encoding="MathML-Content" id="S4.SS1.p4.6.m1.1b"><apply id="S4.SS1.p4.6.m1.1.1.cmml" xref="S4.SS1.p4.6.m1.1.1"><csymbol cd="ambiguous" id="S4.SS1.p4.6.m1.1.1.1.cmml" xref="S4.SS1.p4.6.m1.1.1">superscript</csymbol><ci id="S4.SS1.p4.6.m1.1.1.2.cmml" xref="S4.SS1.p4.6.m1.1.1.2">𝑛</ci><times id="S4.SS1.p4.6.m1.1.1.3.cmml" xref="S4.SS1.p4.6.m1.1.1.3"></times></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p4.6.m1.1c">n^{*}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p4.6.m1.1d">italic_n start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math> is test data sample size and <math alttext="\|\cdot\|_{F}" class="ltx_math_unparsed" display="inline" id="S4.SS1.p4.7.m2.1"><semantics id="S4.SS1.p4.7.m2.1a"><mrow id="S4.SS1.p4.7.m2.1b"><mo id="S4.SS1.p4.7.m2.1.1" rspace="0em">∥</mo><mo id="S4.SS1.p4.7.m2.1.2" lspace="0em" rspace="0em">⋅</mo><msub id="S4.SS1.p4.7.m2.1.3"><mo id="S4.SS1.p4.7.m2.1.3.2" lspace="0em">∥</mo><mi id="S4.SS1.p4.7.m2.1.3.3">F</mi></msub></mrow><annotation encoding="application/x-tex" id="S4.SS1.p4.7.m2.1c">\|\cdot\|_{F}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p4.7.m2.1d">∥ ⋅ ∥ start_POSTSUBSCRIPT italic_F end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="\|\cdot\|_{2}" class="ltx_math_unparsed" display="inline" id="S4.SS1.p4.8.m3.1"><semantics id="S4.SS1.p4.8.m3.1a"><mrow id="S4.SS1.p4.8.m3.1b"><mo id="S4.SS1.p4.8.m3.1.1" rspace="0em">∥</mo><mo id="S4.SS1.p4.8.m3.1.2" lspace="0em" rspace="0em">⋅</mo><msub id="S4.SS1.p4.8.m3.1.3"><mo id="S4.SS1.p4.8.m3.1.3.2" lspace="0em">∥</mo><mn id="S4.SS1.p4.8.m3.1.3.3">2</mn></msub></mrow><annotation encoding="application/x-tex" id="S4.SS1.p4.8.m3.1c">\|\cdot\|_{2}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p4.8.m3.1d">∥ ⋅ ∥ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math> are the Frobenius and Euclidean norms.</p> </div> </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>Simulation Using the MMTR Model</h3> <div class="ltx_para" id="S4.SS2.p1"> <p class="ltx_p" id="S4.SS2.p1.18">We simulated the data from MMTR in (<a class="ltx_ref" href="https://arxiv.org/html/2503.13782v1#S2.E3" title="In 2.1 Model ‣ 2 Mixed Model Trace Regression ‣ Regularized Parameter Estimation in Mixed Model Trace Regression"><span class="ltx_text ltx_ref_tag">3</span></a>), resulting in a misspecified model for the MMTR’s competitors. 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xref="S4.SS2.p1.2.m2.1.1.3.3.1.cmml">×</mo><mn id="S4.SS2.p1.2.m2.1.1.3.3.3" xref="S4.SS2.p1.2.m2.1.1.3.3.3.cmml">2</mn></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p1.2.m2.1b"><apply id="S4.SS2.p1.2.m2.1.1.cmml" xref="S4.SS2.p1.2.m2.1.1"><in id="S4.SS2.p1.2.m2.1.1.1.cmml" xref="S4.SS2.p1.2.m2.1.1.1"></in><apply id="S4.SS2.p1.2.m2.1.1.2.cmml" xref="S4.SS2.p1.2.m2.1.1.2"><csymbol cd="ambiguous" id="S4.SS2.p1.2.m2.1.1.2.1.cmml" xref="S4.SS2.p1.2.m2.1.1.2">subscript</csymbol><ci id="S4.SS2.p1.2.m2.1.1.2.2.cmml" xref="S4.SS2.p1.2.m2.1.1.2.2">𝐋</ci><cn id="S4.SS2.p1.2.m2.1.1.2.3.cmml" type="integer" xref="S4.SS2.p1.2.m2.1.1.2.3">1</cn></apply><apply id="S4.SS2.p1.2.m2.1.1.3.cmml" xref="S4.SS2.p1.2.m2.1.1.3"><csymbol cd="ambiguous" id="S4.SS2.p1.2.m2.1.1.3.1.cmml" xref="S4.SS2.p1.2.m2.1.1.3">superscript</csymbol><ci id="S4.SS2.p1.2.m2.1.1.3.2.cmml" xref="S4.SS2.p1.2.m2.1.1.3.2">ℝ</ci><apply id="S4.SS2.p1.2.m2.1.1.3.3.cmml" xref="S4.SS2.p1.2.m2.1.1.3.3"><times id="S4.SS2.p1.2.m2.1.1.3.3.1.cmml" xref="S4.SS2.p1.2.m2.1.1.3.3.1"></times><cn id="S4.SS2.p1.2.m2.1.1.3.3.2.cmml" type="integer" xref="S4.SS2.p1.2.m2.1.1.3.3.2">5</cn><cn id="S4.SS2.p1.2.m2.1.1.3.3.3.cmml" type="integer" xref="S4.SS2.p1.2.m2.1.1.3.3.3">2</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p1.2.m2.1c">\operatorname{\mathbf{L}}_{1}\in\mathbb{R}^{5\times 2}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p1.2.m2.1d">bold_L start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ∈ blackboard_R start_POSTSUPERSCRIPT 5 × 2 end_POSTSUPERSCRIPT</annotation></semantics></math>, <math alttext="\operatorname{\mathbf{L}}_{2}\in\mathbb{R}^{5\times 2}" 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"><msub id="S4.SS2.p1.3.m3.1.1.2" xref="S4.SS2.p1.3.m3.1.1.2.cmml"><mi id="S4.SS2.p1.3.m3.1.1.2.2" xref="S4.SS2.p1.3.m3.1.1.2.2.cmml">𝐋</mi><mn id="S4.SS2.p1.3.m3.1.1.2.3" xref="S4.SS2.p1.3.m3.1.1.2.3.cmml">2</mn></msub><mo id="S4.SS2.p1.3.m3.1.1.1" xref="S4.SS2.p1.3.m3.1.1.1.cmml">∈</mo><msup id="S4.SS2.p1.3.m3.1.1.3" xref="S4.SS2.p1.3.m3.1.1.3.cmml"><mi id="S4.SS2.p1.3.m3.1.1.3.2" xref="S4.SS2.p1.3.m3.1.1.3.2.cmml">ℝ</mi><mrow id="S4.SS2.p1.3.m3.1.1.3.3" xref="S4.SS2.p1.3.m3.1.1.3.3.cmml"><mn id="S4.SS2.p1.3.m3.1.1.3.3.2" xref="S4.SS2.p1.3.m3.1.1.3.3.2.cmml">5</mn><mo id="S4.SS2.p1.3.m3.1.1.3.3.1" lspace="0.222em" rspace="0.222em" xref="S4.SS2.p1.3.m3.1.1.3.3.1.cmml">×</mo><mn id="S4.SS2.p1.3.m3.1.1.3.3.3" xref="S4.SS2.p1.3.m3.1.1.3.3.3.cmml">2</mn></mrow></msup></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"><in id="S4.SS2.p1.3.m3.1.1.1.cmml" xref="S4.SS2.p1.3.m3.1.1.1"></in><apply id="S4.SS2.p1.3.m3.1.1.2.cmml" xref="S4.SS2.p1.3.m3.1.1.2"><csymbol cd="ambiguous" id="S4.SS2.p1.3.m3.1.1.2.1.cmml" xref="S4.SS2.p1.3.m3.1.1.2">subscript</csymbol><ci id="S4.SS2.p1.3.m3.1.1.2.2.cmml" xref="S4.SS2.p1.3.m3.1.1.2.2">𝐋</ci><cn id="S4.SS2.p1.3.m3.1.1.2.3.cmml" type="integer" xref="S4.SS2.p1.3.m3.1.1.2.3">2</cn></apply><apply id="S4.SS2.p1.3.m3.1.1.3.cmml" xref="S4.SS2.p1.3.m3.1.1.3"><csymbol cd="ambiguous" id="S4.SS2.p1.3.m3.1.1.3.1.cmml" xref="S4.SS2.p1.3.m3.1.1.3">superscript</csymbol><ci id="S4.SS2.p1.3.m3.1.1.3.2.cmml" xref="S4.SS2.p1.3.m3.1.1.3.2">ℝ</ci><apply id="S4.SS2.p1.3.m3.1.1.3.3.cmml" xref="S4.SS2.p1.3.m3.1.1.3.3"><times id="S4.SS2.p1.3.m3.1.1.3.3.1.cmml" xref="S4.SS2.p1.3.m3.1.1.3.3.1"></times><cn id="S4.SS2.p1.3.m3.1.1.3.3.2.cmml" type="integer" xref="S4.SS2.p1.3.m3.1.1.3.3.2">5</cn><cn id="S4.SS2.p1.3.m3.1.1.3.3.3.cmml" type="integer" xref="S4.SS2.p1.3.m3.1.1.3.3.3">2</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p1.3.m3.1c">\operatorname{\mathbf{L}}_{2}\in\mathbb{R}^{5\times 2}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p1.3.m3.1d">bold_L start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ∈ blackboard_R start_POSTSUPERSCRIPT 5 × 2 end_POSTSUPERSCRIPT</annotation></semantics></math> (<span class="ltx_text ltx_font_bold" id="S4.SS2.p1.18.1">Case 1</span>), and (2) <math alttext="\operatorname{\mathbf{B}}\in\mathbb{R}^{10\times 10}" class="ltx_Math" display="inline" id="S4.SS2.p1.4.m4.1"><semantics id="S4.SS2.p1.4.m4.1a"><mrow id="S4.SS2.p1.4.m4.1.1" xref="S4.SS2.p1.4.m4.1.1.cmml"><mi id="S4.SS2.p1.4.m4.1.1.2" xref="S4.SS2.p1.4.m4.1.1.2.cmml">𝐁</mi><mo id="S4.SS2.p1.4.m4.1.1.1" xref="S4.SS2.p1.4.m4.1.1.1.cmml">∈</mo><msup id="S4.SS2.p1.4.m4.1.1.3" xref="S4.SS2.p1.4.m4.1.1.3.cmml"><mi id="S4.SS2.p1.4.m4.1.1.3.2" xref="S4.SS2.p1.4.m4.1.1.3.2.cmml">ℝ</mi><mrow id="S4.SS2.p1.4.m4.1.1.3.3" xref="S4.SS2.p1.4.m4.1.1.3.3.cmml"><mn id="S4.SS2.p1.4.m4.1.1.3.3.2" xref="S4.SS2.p1.4.m4.1.1.3.3.2.cmml">10</mn><mo id="S4.SS2.p1.4.m4.1.1.3.3.1" lspace="0.222em" rspace="0.222em" xref="S4.SS2.p1.4.m4.1.1.3.3.1.cmml">×</mo><mn id="S4.SS2.p1.4.m4.1.1.3.3.3" xref="S4.SS2.p1.4.m4.1.1.3.3.3.cmml">10</mn></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p1.4.m4.1b"><apply id="S4.SS2.p1.4.m4.1.1.cmml" xref="S4.SS2.p1.4.m4.1.1"><in id="S4.SS2.p1.4.m4.1.1.1.cmml" xref="S4.SS2.p1.4.m4.1.1.1"></in><ci id="S4.SS2.p1.4.m4.1.1.2.cmml" xref="S4.SS2.p1.4.m4.1.1.2">𝐁</ci><apply id="S4.SS2.p1.4.m4.1.1.3.cmml" xref="S4.SS2.p1.4.m4.1.1.3"><csymbol cd="ambiguous" id="S4.SS2.p1.4.m4.1.1.3.1.cmml" xref="S4.SS2.p1.4.m4.1.1.3">superscript</csymbol><ci id="S4.SS2.p1.4.m4.1.1.3.2.cmml" xref="S4.SS2.p1.4.m4.1.1.3.2">ℝ</ci><apply id="S4.SS2.p1.4.m4.1.1.3.3.cmml" xref="S4.SS2.p1.4.m4.1.1.3.3"><times id="S4.SS2.p1.4.m4.1.1.3.3.1.cmml" xref="S4.SS2.p1.4.m4.1.1.3.3.1"></times><cn id="S4.SS2.p1.4.m4.1.1.3.3.2.cmml" type="integer" xref="S4.SS2.p1.4.m4.1.1.3.3.2">10</cn><cn id="S4.SS2.p1.4.m4.1.1.3.3.3.cmml" type="integer" 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xref="S4.SS2.p1.5.m5.1.1.3"><csymbol cd="ambiguous" id="S4.SS2.p1.5.m5.1.1.3.1.cmml" xref="S4.SS2.p1.5.m5.1.1.3">superscript</csymbol><ci id="S4.SS2.p1.5.m5.1.1.3.2.cmml" xref="S4.SS2.p1.5.m5.1.1.3.2">ℝ</ci><apply id="S4.SS2.p1.5.m5.1.1.3.3.cmml" xref="S4.SS2.p1.5.m5.1.1.3.3"><times id="S4.SS2.p1.5.m5.1.1.3.3.1.cmml" xref="S4.SS2.p1.5.m5.1.1.3.3.1"></times><cn id="S4.SS2.p1.5.m5.1.1.3.3.2.cmml" type="integer" xref="S4.SS2.p1.5.m5.1.1.3.3.2">10</cn><cn id="S4.SS2.p1.5.m5.1.1.3.3.3.cmml" type="integer" xref="S4.SS2.p1.5.m5.1.1.3.3.3">3</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p1.5.m5.1c">\operatorname{\mathbf{L}}_{1}\in\mathbb{R}^{10\times 3}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p1.5.m5.1d">bold_L start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ∈ blackboard_R start_POSTSUPERSCRIPT 10 × 3 end_POSTSUPERSCRIPT</annotation></semantics></math>, <math alttext="\operatorname{\mathbf{L}}_{2}\in\mathbb{R}^{10\times 3}" 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"><msub id="S4.SS2.p1.6.m6.1.1.2" xref="S4.SS2.p1.6.m6.1.1.2.cmml"><mi id="S4.SS2.p1.6.m6.1.1.2.2" xref="S4.SS2.p1.6.m6.1.1.2.2.cmml">𝐋</mi><mn id="S4.SS2.p1.6.m6.1.1.2.3" xref="S4.SS2.p1.6.m6.1.1.2.3.cmml">2</mn></msub><mo id="S4.SS2.p1.6.m6.1.1.1" xref="S4.SS2.p1.6.m6.1.1.1.cmml">∈</mo><msup id="S4.SS2.p1.6.m6.1.1.3" xref="S4.SS2.p1.6.m6.1.1.3.cmml"><mi id="S4.SS2.p1.6.m6.1.1.3.2" xref="S4.SS2.p1.6.m6.1.1.3.2.cmml">ℝ</mi><mrow id="S4.SS2.p1.6.m6.1.1.3.3" xref="S4.SS2.p1.6.m6.1.1.3.3.cmml"><mn id="S4.SS2.p1.6.m6.1.1.3.3.2" xref="S4.SS2.p1.6.m6.1.1.3.3.2.cmml">10</mn><mo id="S4.SS2.p1.6.m6.1.1.3.3.1" lspace="0.222em" rspace="0.222em" xref="S4.SS2.p1.6.m6.1.1.3.3.1.cmml">×</mo><mn id="S4.SS2.p1.6.m6.1.1.3.3.3" xref="S4.SS2.p1.6.m6.1.1.3.3.3.cmml">3</mn></mrow></msup></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"><in id="S4.SS2.p1.6.m6.1.1.1.cmml" xref="S4.SS2.p1.6.m6.1.1.1"></in><apply id="S4.SS2.p1.6.m6.1.1.2.cmml" xref="S4.SS2.p1.6.m6.1.1.2"><csymbol cd="ambiguous" id="S4.SS2.p1.6.m6.1.1.2.1.cmml" xref="S4.SS2.p1.6.m6.1.1.2">subscript</csymbol><ci id="S4.SS2.p1.6.m6.1.1.2.2.cmml" xref="S4.SS2.p1.6.m6.1.1.2.2">𝐋</ci><cn id="S4.SS2.p1.6.m6.1.1.2.3.cmml" type="integer" xref="S4.SS2.p1.6.m6.1.1.2.3">2</cn></apply><apply id="S4.SS2.p1.6.m6.1.1.3.cmml" xref="S4.SS2.p1.6.m6.1.1.3"><csymbol cd="ambiguous" id="S4.SS2.p1.6.m6.1.1.3.1.cmml" xref="S4.SS2.p1.6.m6.1.1.3">superscript</csymbol><ci id="S4.SS2.p1.6.m6.1.1.3.2.cmml" xref="S4.SS2.p1.6.m6.1.1.3.2">ℝ</ci><apply id="S4.SS2.p1.6.m6.1.1.3.3.cmml" xref="S4.SS2.p1.6.m6.1.1.3.3"><times id="S4.SS2.p1.6.m6.1.1.3.3.1.cmml" xref="S4.SS2.p1.6.m6.1.1.3.3.1"></times><cn id="S4.SS2.p1.6.m6.1.1.3.3.2.cmml" type="integer" xref="S4.SS2.p1.6.m6.1.1.3.3.2">10</cn><cn id="S4.SS2.p1.6.m6.1.1.3.3.3.cmml" type="integer" xref="S4.SS2.p1.6.m6.1.1.3.3.3">3</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p1.6.m6.1c">\operatorname{\mathbf{L}}_{2}\in\mathbb{R}^{10\times 3}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p1.6.m6.1d">bold_L start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ∈ blackboard_R start_POSTSUPERSCRIPT 10 × 3 end_POSTSUPERSCRIPT</annotation></semantics></math> (<span class="ltx_text ltx_font_bold" id="S4.SS2.p1.18.2">Case 2</span>). For the two cases, <math alttext="\lfloor{0.4P_{1}P_{2}}\rfloor" class="ltx_Math" display="inline" id="S4.SS2.p1.7.m7.1"><semantics id="S4.SS2.p1.7.m7.1a"><mrow id="S4.SS2.p1.7.m7.1.1.1" xref="S4.SS2.p1.7.m7.1.1.2.cmml"><mo id="S4.SS2.p1.7.m7.1.1.1.2" stretchy="false" xref="S4.SS2.p1.7.m7.1.1.2.1.cmml">⌊</mo><mrow id="S4.SS2.p1.7.m7.1.1.1.1" xref="S4.SS2.p1.7.m7.1.1.1.1.cmml"><mn id="S4.SS2.p1.7.m7.1.1.1.1.2" xref="S4.SS2.p1.7.m7.1.1.1.1.2.cmml">0.4</mn><mo id="S4.SS2.p1.7.m7.1.1.1.1.1" xref="S4.SS2.p1.7.m7.1.1.1.1.1.cmml">⁢</mo><msub id="S4.SS2.p1.7.m7.1.1.1.1.3" xref="S4.SS2.p1.7.m7.1.1.1.1.3.cmml"><mi id="S4.SS2.p1.7.m7.1.1.1.1.3.2" xref="S4.SS2.p1.7.m7.1.1.1.1.3.2.cmml">P</mi><mn id="S4.SS2.p1.7.m7.1.1.1.1.3.3" xref="S4.SS2.p1.7.m7.1.1.1.1.3.3.cmml">1</mn></msub><mo id="S4.SS2.p1.7.m7.1.1.1.1.1a" xref="S4.SS2.p1.7.m7.1.1.1.1.1.cmml">⁢</mo><msub id="S4.SS2.p1.7.m7.1.1.1.1.4" xref="S4.SS2.p1.7.m7.1.1.1.1.4.cmml"><mi id="S4.SS2.p1.7.m7.1.1.1.1.4.2" xref="S4.SS2.p1.7.m7.1.1.1.1.4.2.cmml">P</mi><mn id="S4.SS2.p1.7.m7.1.1.1.1.4.3" xref="S4.SS2.p1.7.m7.1.1.1.1.4.3.cmml">2</mn></msub></mrow><mo id="S4.SS2.p1.7.m7.1.1.1.3" stretchy="false" xref="S4.SS2.p1.7.m7.1.1.2.1.cmml">⌋</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p1.7.m7.1b"><apply id="S4.SS2.p1.7.m7.1.1.2.cmml" xref="S4.SS2.p1.7.m7.1.1.1"><floor id="S4.SS2.p1.7.m7.1.1.2.1.cmml" xref="S4.SS2.p1.7.m7.1.1.1.2"></floor><apply id="S4.SS2.p1.7.m7.1.1.1.1.cmml" xref="S4.SS2.p1.7.m7.1.1.1.1"><times id="S4.SS2.p1.7.m7.1.1.1.1.1.cmml" xref="S4.SS2.p1.7.m7.1.1.1.1.1"></times><cn id="S4.SS2.p1.7.m7.1.1.1.1.2.cmml" type="float" xref="S4.SS2.p1.7.m7.1.1.1.1.2">0.4</cn><apply id="S4.SS2.p1.7.m7.1.1.1.1.3.cmml" xref="S4.SS2.p1.7.m7.1.1.1.1.3"><csymbol cd="ambiguous" id="S4.SS2.p1.7.m7.1.1.1.1.3.1.cmml" xref="S4.SS2.p1.7.m7.1.1.1.1.3">subscript</csymbol><ci id="S4.SS2.p1.7.m7.1.1.1.1.3.2.cmml" xref="S4.SS2.p1.7.m7.1.1.1.1.3.2">𝑃</ci><cn id="S4.SS2.p1.7.m7.1.1.1.1.3.3.cmml" type="integer" xref="S4.SS2.p1.7.m7.1.1.1.1.3.3">1</cn></apply><apply id="S4.SS2.p1.7.m7.1.1.1.1.4.cmml" xref="S4.SS2.p1.7.m7.1.1.1.1.4"><csymbol cd="ambiguous" id="S4.SS2.p1.7.m7.1.1.1.1.4.1.cmml" xref="S4.SS2.p1.7.m7.1.1.1.1.4">subscript</csymbol><ci id="S4.SS2.p1.7.m7.1.1.1.1.4.2.cmml" xref="S4.SS2.p1.7.m7.1.1.1.1.4.2">𝑃</ci><cn id="S4.SS2.p1.7.m7.1.1.1.1.4.3.cmml" type="integer" xref="S4.SS2.p1.7.m7.1.1.1.1.4.3">2</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p1.7.m7.1c">\lfloor{0.4P_{1}P_{2}}\rfloor</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p1.7.m7.1d">⌊ 0.4 italic_P start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_P start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ⌋</annotation></semantics></math> entries in <math alttext="\operatorname{\mathbf{B}}" class="ltx_Math" display="inline" id="S4.SS2.p1.8.m8.1"><semantics id="S4.SS2.p1.8.m8.1a"><mi id="S4.SS2.p1.8.m8.1.1" xref="S4.SS2.p1.8.m8.1.1.cmml">𝐁</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.p1.8.m8.1b"><ci id="S4.SS2.p1.8.m8.1.1.cmml" xref="S4.SS2.p1.8.m8.1.1">𝐁</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p1.8.m8.1c">\operatorname{\mathbf{B}}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p1.8.m8.1d">bold_B</annotation></semantics></math> were randomly set to zero, and the remaining <math alttext="\operatorname{\mathbf{B}}" class="ltx_Math" display="inline" id="S4.SS2.p1.9.m9.1"><semantics id="S4.SS2.p1.9.m9.1a"><mi id="S4.SS2.p1.9.m9.1.1" xref="S4.SS2.p1.9.m9.1.1.cmml">𝐁</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.p1.9.m9.1b"><ci id="S4.SS2.p1.9.m9.1.1.cmml" xref="S4.SS2.p1.9.m9.1.1">𝐁</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p1.9.m9.1c">\operatorname{\mathbf{B}}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p1.9.m9.1d">bold_B</annotation></semantics></math> entries were sampled with replacement from a uniform grid in <math alttext="[-10,10]" class="ltx_Math" display="inline" id="S4.SS2.p1.10.m10.2"><semantics id="S4.SS2.p1.10.m10.2a"><mrow id="S4.SS2.p1.10.m10.2.2.1" xref="S4.SS2.p1.10.m10.2.2.2.cmml"><mo id="S4.SS2.p1.10.m10.2.2.1.2" stretchy="false" xref="S4.SS2.p1.10.m10.2.2.2.cmml">[</mo><mrow id="S4.SS2.p1.10.m10.2.2.1.1" xref="S4.SS2.p1.10.m10.2.2.1.1.cmml"><mo id="S4.SS2.p1.10.m10.2.2.1.1a" xref="S4.SS2.p1.10.m10.2.2.1.1.cmml">−</mo><mn id="S4.SS2.p1.10.m10.2.2.1.1.2" xref="S4.SS2.p1.10.m10.2.2.1.1.2.cmml">10</mn></mrow><mo id="S4.SS2.p1.10.m10.2.2.1.3" xref="S4.SS2.p1.10.m10.2.2.2.cmml">,</mo><mn id="S4.SS2.p1.10.m10.1.1" xref="S4.SS2.p1.10.m10.1.1.cmml">10</mn><mo id="S4.SS2.p1.10.m10.2.2.1.4" stretchy="false" xref="S4.SS2.p1.10.m10.2.2.2.cmml">]</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p1.10.m10.2b"><interval closure="closed" id="S4.SS2.p1.10.m10.2.2.2.cmml" xref="S4.SS2.p1.10.m10.2.2.1"><apply id="S4.SS2.p1.10.m10.2.2.1.1.cmml" xref="S4.SS2.p1.10.m10.2.2.1.1"><minus id="S4.SS2.p1.10.m10.2.2.1.1.1.cmml" xref="S4.SS2.p1.10.m10.2.2.1.1"></minus><cn id="S4.SS2.p1.10.m10.2.2.1.1.2.cmml" type="integer" xref="S4.SS2.p1.10.m10.2.2.1.1.2">10</cn></apply><cn id="S4.SS2.p1.10.m10.1.1.cmml" type="integer" xref="S4.SS2.p1.10.m10.1.1">10</cn></interval></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p1.10.m10.2c">[-10,10]</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p1.10.m10.2d">[ - 10 , 10 ]</annotation></semantics></math> with a step size of 0.5. The entries in <math alttext="\operatorname{\mathbf{L}}_{1}" class="ltx_Math" display="inline" id="S4.SS2.p1.11.m11.1"><semantics id="S4.SS2.p1.11.m11.1a"><msub id="S4.SS2.p1.11.m11.1.1" xref="S4.SS2.p1.11.m11.1.1.cmml"><mi id="S4.SS2.p1.11.m11.1.1.2" xref="S4.SS2.p1.11.m11.1.1.2.cmml">𝐋</mi><mn id="S4.SS2.p1.11.m11.1.1.3" xref="S4.SS2.p1.11.m11.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S4.SS2.p1.11.m11.1b"><apply id="S4.SS2.p1.11.m11.1.1.cmml" xref="S4.SS2.p1.11.m11.1.1"><csymbol cd="ambiguous" id="S4.SS2.p1.11.m11.1.1.1.cmml" xref="S4.SS2.p1.11.m11.1.1">subscript</csymbol><ci id="S4.SS2.p1.11.m11.1.1.2.cmml" xref="S4.SS2.p1.11.m11.1.1.2">𝐋</ci><cn id="S4.SS2.p1.11.m11.1.1.3.cmml" type="integer" xref="S4.SS2.p1.11.m11.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p1.11.m11.1c">\operatorname{\mathbf{L}}_{1}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p1.11.m11.1d">bold_L start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="\operatorname{\mathbf{L}}_{2}" class="ltx_Math" display="inline" id="S4.SS2.p1.12.m12.1"><semantics id="S4.SS2.p1.12.m12.1a"><msub id="S4.SS2.p1.12.m12.1.1" xref="S4.SS2.p1.12.m12.1.1.cmml"><mi id="S4.SS2.p1.12.m12.1.1.2" xref="S4.SS2.p1.12.m12.1.1.2.cmml">𝐋</mi><mn id="S4.SS2.p1.12.m12.1.1.3" xref="S4.SS2.p1.12.m12.1.1.3.cmml">2</mn></msub><annotation-xml encoding="MathML-Content" id="S4.SS2.p1.12.m12.1b"><apply id="S4.SS2.p1.12.m12.1.1.cmml" xref="S4.SS2.p1.12.m12.1.1"><csymbol cd="ambiguous" id="S4.SS2.p1.12.m12.1.1.1.cmml" xref="S4.SS2.p1.12.m12.1.1">subscript</csymbol><ci id="S4.SS2.p1.12.m12.1.1.2.cmml" xref="S4.SS2.p1.12.m12.1.1.2">𝐋</ci><cn id="S4.SS2.p1.12.m12.1.1.3.cmml" type="integer" xref="S4.SS2.p1.12.m12.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p1.12.m12.1c">\operatorname{\mathbf{L}}_{2}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p1.12.m12.1d">bold_L start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math> were sampled from a continuous uniform distribution on <math alttext="[-1,1]" class="ltx_Math" display="inline" id="S4.SS2.p1.13.m13.2"><semantics id="S4.SS2.p1.13.m13.2a"><mrow id="S4.SS2.p1.13.m13.2.2.1" xref="S4.SS2.p1.13.m13.2.2.2.cmml"><mo id="S4.SS2.p1.13.m13.2.2.1.2" stretchy="false" xref="S4.SS2.p1.13.m13.2.2.2.cmml">[</mo><mrow id="S4.SS2.p1.13.m13.2.2.1.1" xref="S4.SS2.p1.13.m13.2.2.1.1.cmml"><mo id="S4.SS2.p1.13.m13.2.2.1.1a" xref="S4.SS2.p1.13.m13.2.2.1.1.cmml">−</mo><mn id="S4.SS2.p1.13.m13.2.2.1.1.2" xref="S4.SS2.p1.13.m13.2.2.1.1.2.cmml">1</mn></mrow><mo id="S4.SS2.p1.13.m13.2.2.1.3" xref="S4.SS2.p1.13.m13.2.2.2.cmml">,</mo><mn id="S4.SS2.p1.13.m13.1.1" xref="S4.SS2.p1.13.m13.1.1.cmml">1</mn><mo id="S4.SS2.p1.13.m13.2.2.1.4" stretchy="false" xref="S4.SS2.p1.13.m13.2.2.2.cmml">]</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p1.13.m13.2b"><interval closure="closed" id="S4.SS2.p1.13.m13.2.2.2.cmml" xref="S4.SS2.p1.13.m13.2.2.1"><apply id="S4.SS2.p1.13.m13.2.2.1.1.cmml" xref="S4.SS2.p1.13.m13.2.2.1.1"><minus id="S4.SS2.p1.13.m13.2.2.1.1.1.cmml" xref="S4.SS2.p1.13.m13.2.2.1.1"></minus><cn id="S4.SS2.p1.13.m13.2.2.1.1.2.cmml" type="integer" xref="S4.SS2.p1.13.m13.2.2.1.1.2">1</cn></apply><cn id="S4.SS2.p1.13.m13.1.1.cmml" type="integer" xref="S4.SS2.p1.13.m13.1.1">1</cn></interval></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p1.13.m13.2c">[-1,1]</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p1.13.m13.2d">[ - 1 , 1 ]</annotation></semantics></math>, implying that the row and column covariance matrices were <math alttext="\operatorname{{\mathbf{\Sigma}}}_{1}=\operatorname{\mathbf{L}}_{1}% \operatorname{\mathbf{L}}_{1}^{\top}" class="ltx_Math" display="inline" id="S4.SS2.p1.14.m14.1"><semantics id="S4.SS2.p1.14.m14.1a"><mrow id="S4.SS2.p1.14.m14.1.1" xref="S4.SS2.p1.14.m14.1.1.cmml"><msub id="S4.SS2.p1.14.m14.1.1.2" xref="S4.SS2.p1.14.m14.1.1.2.cmml"><mi id="S4.SS2.p1.14.m14.1.1.2.2" xref="S4.SS2.p1.14.m14.1.1.2.2.cmml">𝚺</mi><mn id="S4.SS2.p1.14.m14.1.1.2.3" xref="S4.SS2.p1.14.m14.1.1.2.3.cmml">1</mn></msub><mo id="S4.SS2.p1.14.m14.1.1.1" xref="S4.SS2.p1.14.m14.1.1.1.cmml">=</mo><mrow id="S4.SS2.p1.14.m14.1.1.3" xref="S4.SS2.p1.14.m14.1.1.3.cmml"><msub id="S4.SS2.p1.14.m14.1.1.3.1" xref="S4.SS2.p1.14.m14.1.1.3.1.cmml"><mi id="S4.SS2.p1.14.m14.1.1.3.1.2" xref="S4.SS2.p1.14.m14.1.1.3.1.2.cmml">𝐋</mi><mn id="S4.SS2.p1.14.m14.1.1.3.1.3" xref="S4.SS2.p1.14.m14.1.1.3.1.3.cmml">1</mn></msub><mo id="S4.SS2.p1.14.m14.1.1.3a" lspace="0.167em" xref="S4.SS2.p1.14.m14.1.1.3.cmml">⁡</mo><msubsup id="S4.SS2.p1.14.m14.1.1.3.2" xref="S4.SS2.p1.14.m14.1.1.3.2.cmml"><mi id="S4.SS2.p1.14.m14.1.1.3.2.2.2" xref="S4.SS2.p1.14.m14.1.1.3.2.2.2.cmml">𝐋</mi><mn id="S4.SS2.p1.14.m14.1.1.3.2.2.3" xref="S4.SS2.p1.14.m14.1.1.3.2.2.3.cmml">1</mn><mo id="S4.SS2.p1.14.m14.1.1.3.2.3" xref="S4.SS2.p1.14.m14.1.1.3.2.3.cmml">⊤</mo></msubsup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p1.14.m14.1b"><apply id="S4.SS2.p1.14.m14.1.1.cmml" xref="S4.SS2.p1.14.m14.1.1"><eq id="S4.SS2.p1.14.m14.1.1.1.cmml" xref="S4.SS2.p1.14.m14.1.1.1"></eq><apply id="S4.SS2.p1.14.m14.1.1.2.cmml" xref="S4.SS2.p1.14.m14.1.1.2"><csymbol cd="ambiguous" id="S4.SS2.p1.14.m14.1.1.2.1.cmml" xref="S4.SS2.p1.14.m14.1.1.2">subscript</csymbol><ci id="S4.SS2.p1.14.m14.1.1.2.2.cmml" xref="S4.SS2.p1.14.m14.1.1.2.2">𝚺</ci><cn id="S4.SS2.p1.14.m14.1.1.2.3.cmml" type="integer" xref="S4.SS2.p1.14.m14.1.1.2.3">1</cn></apply><apply id="S4.SS2.p1.14.m14.1.1.3.cmml" xref="S4.SS2.p1.14.m14.1.1.3"><apply id="S4.SS2.p1.14.m14.1.1.3.1.cmml" xref="S4.SS2.p1.14.m14.1.1.3.1"><csymbol cd="ambiguous" id="S4.SS2.p1.14.m14.1.1.3.1.1.cmml" xref="S4.SS2.p1.14.m14.1.1.3.1">subscript</csymbol><ci id="S4.SS2.p1.14.m14.1.1.3.1.2.cmml" xref="S4.SS2.p1.14.m14.1.1.3.1.2">𝐋</ci><cn id="S4.SS2.p1.14.m14.1.1.3.1.3.cmml" type="integer" xref="S4.SS2.p1.14.m14.1.1.3.1.3">1</cn></apply><apply id="S4.SS2.p1.14.m14.1.1.3.2.cmml" xref="S4.SS2.p1.14.m14.1.1.3.2"><csymbol cd="ambiguous" id="S4.SS2.p1.14.m14.1.1.3.2.1.cmml" xref="S4.SS2.p1.14.m14.1.1.3.2">superscript</csymbol><apply id="S4.SS2.p1.14.m14.1.1.3.2.2.cmml" xref="S4.SS2.p1.14.m14.1.1.3.2"><csymbol cd="ambiguous" id="S4.SS2.p1.14.m14.1.1.3.2.2.1.cmml" xref="S4.SS2.p1.14.m14.1.1.3.2">subscript</csymbol><ci id="S4.SS2.p1.14.m14.1.1.3.2.2.2.cmml" xref="S4.SS2.p1.14.m14.1.1.3.2.2.2">𝐋</ci><cn id="S4.SS2.p1.14.m14.1.1.3.2.2.3.cmml" type="integer" xref="S4.SS2.p1.14.m14.1.1.3.2.2.3">1</cn></apply><csymbol cd="latexml" id="S4.SS2.p1.14.m14.1.1.3.2.3.cmml" xref="S4.SS2.p1.14.m14.1.1.3.2.3">top</csymbol></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p1.14.m14.1c">\operatorname{{\mathbf{\Sigma}}}_{1}=\operatorname{\mathbf{L}}_{1}% \operatorname{\mathbf{L}}_{1}^{\top}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p1.14.m14.1d">bold_Σ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = bold_L start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT bold_L start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ⊤ end_POSTSUPERSCRIPT</annotation></semantics></math> and <math alttext="\operatorname{{\mathbf{\Sigma}}}_{2}=\operatorname{\mathbf{L}}_{2}% \operatorname{\mathbf{L}}_{2}^{\top}" class="ltx_Math" display="inline" id="S4.SS2.p1.15.m15.1"><semantics id="S4.SS2.p1.15.m15.1a"><mrow id="S4.SS2.p1.15.m15.1.1" xref="S4.SS2.p1.15.m15.1.1.cmml"><msub id="S4.SS2.p1.15.m15.1.1.2" xref="S4.SS2.p1.15.m15.1.1.2.cmml"><mi id="S4.SS2.p1.15.m15.1.1.2.2" xref="S4.SS2.p1.15.m15.1.1.2.2.cmml">𝚺</mi><mn id="S4.SS2.p1.15.m15.1.1.2.3" xref="S4.SS2.p1.15.m15.1.1.2.3.cmml">2</mn></msub><mo id="S4.SS2.p1.15.m15.1.1.1" xref="S4.SS2.p1.15.m15.1.1.1.cmml">=</mo><mrow id="S4.SS2.p1.15.m15.1.1.3" xref="S4.SS2.p1.15.m15.1.1.3.cmml"><msub id="S4.SS2.p1.15.m15.1.1.3.1" xref="S4.SS2.p1.15.m15.1.1.3.1.cmml"><mi id="S4.SS2.p1.15.m15.1.1.3.1.2" xref="S4.SS2.p1.15.m15.1.1.3.1.2.cmml">𝐋</mi><mn id="S4.SS2.p1.15.m15.1.1.3.1.3" xref="S4.SS2.p1.15.m15.1.1.3.1.3.cmml">2</mn></msub><mo id="S4.SS2.p1.15.m15.1.1.3a" lspace="0.167em" xref="S4.SS2.p1.15.m15.1.1.3.cmml">⁡</mo><msubsup id="S4.SS2.p1.15.m15.1.1.3.2" xref="S4.SS2.p1.15.m15.1.1.3.2.cmml"><mi id="S4.SS2.p1.15.m15.1.1.3.2.2.2" xref="S4.SS2.p1.15.m15.1.1.3.2.2.2.cmml">𝐋</mi><mn id="S4.SS2.p1.15.m15.1.1.3.2.2.3" xref="S4.SS2.p1.15.m15.1.1.3.2.2.3.cmml">2</mn><mo id="S4.SS2.p1.15.m15.1.1.3.2.3" xref="S4.SS2.p1.15.m15.1.1.3.2.3.cmml">⊤</mo></msubsup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p1.15.m15.1b"><apply id="S4.SS2.p1.15.m15.1.1.cmml" xref="S4.SS2.p1.15.m15.1.1"><eq id="S4.SS2.p1.15.m15.1.1.1.cmml" xref="S4.SS2.p1.15.m15.1.1.1"></eq><apply id="S4.SS2.p1.15.m15.1.1.2.cmml" xref="S4.SS2.p1.15.m15.1.1.2"><csymbol cd="ambiguous" id="S4.SS2.p1.15.m15.1.1.2.1.cmml" xref="S4.SS2.p1.15.m15.1.1.2">subscript</csymbol><ci id="S4.SS2.p1.15.m15.1.1.2.2.cmml" xref="S4.SS2.p1.15.m15.1.1.2.2">𝚺</ci><cn id="S4.SS2.p1.15.m15.1.1.2.3.cmml" type="integer" xref="S4.SS2.p1.15.m15.1.1.2.3">2</cn></apply><apply id="S4.SS2.p1.15.m15.1.1.3.cmml" xref="S4.SS2.p1.15.m15.1.1.3"><apply id="S4.SS2.p1.15.m15.1.1.3.1.cmml" xref="S4.SS2.p1.15.m15.1.1.3.1"><csymbol cd="ambiguous" id="S4.SS2.p1.15.m15.1.1.3.1.1.cmml" xref="S4.SS2.p1.15.m15.1.1.3.1">subscript</csymbol><ci id="S4.SS2.p1.15.m15.1.1.3.1.2.cmml" xref="S4.SS2.p1.15.m15.1.1.3.1.2">𝐋</ci><cn id="S4.SS2.p1.15.m15.1.1.3.1.3.cmml" type="integer" xref="S4.SS2.p1.15.m15.1.1.3.1.3">2</cn></apply><apply id="S4.SS2.p1.15.m15.1.1.3.2.cmml" xref="S4.SS2.p1.15.m15.1.1.3.2"><csymbol cd="ambiguous" id="S4.SS2.p1.15.m15.1.1.3.2.1.cmml" xref="S4.SS2.p1.15.m15.1.1.3.2">superscript</csymbol><apply id="S4.SS2.p1.15.m15.1.1.3.2.2.cmml" xref="S4.SS2.p1.15.m15.1.1.3.2"><csymbol cd="ambiguous" id="S4.SS2.p1.15.m15.1.1.3.2.2.1.cmml" xref="S4.SS2.p1.15.m15.1.1.3.2">subscript</csymbol><ci id="S4.SS2.p1.15.m15.1.1.3.2.2.2.cmml" xref="S4.SS2.p1.15.m15.1.1.3.2.2.2">𝐋</ci><cn id="S4.SS2.p1.15.m15.1.1.3.2.2.3.cmml" type="integer" xref="S4.SS2.p1.15.m15.1.1.3.2.2.3">2</cn></apply><csymbol cd="latexml" id="S4.SS2.p1.15.m15.1.1.3.2.3.cmml" xref="S4.SS2.p1.15.m15.1.1.3.2.3">top</csymbol></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p1.15.m15.1c">\operatorname{{\mathbf{\Sigma}}}_{2}=\operatorname{\mathbf{L}}_{2}% \operatorname{\mathbf{L}}_{2}^{\top}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p1.15.m15.1d">bold_Σ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT = bold_L start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT bold_L start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ⊤ end_POSTSUPERSCRIPT</annotation></semantics></math>. The covariance matrices <math alttext="\operatorname{{\mathbf{\Sigma}}}_{1}" class="ltx_Math" display="inline" id="S4.SS2.p1.16.m16.1"><semantics id="S4.SS2.p1.16.m16.1a"><msub id="S4.SS2.p1.16.m16.1.1" xref="S4.SS2.p1.16.m16.1.1.cmml"><mi id="S4.SS2.p1.16.m16.1.1.2" xref="S4.SS2.p1.16.m16.1.1.2.cmml">𝚺</mi><mn id="S4.SS2.p1.16.m16.1.1.3" xref="S4.SS2.p1.16.m16.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S4.SS2.p1.16.m16.1b"><apply id="S4.SS2.p1.16.m16.1.1.cmml" xref="S4.SS2.p1.16.m16.1.1"><csymbol cd="ambiguous" id="S4.SS2.p1.16.m16.1.1.1.cmml" xref="S4.SS2.p1.16.m16.1.1">subscript</csymbol><ci id="S4.SS2.p1.16.m16.1.1.2.cmml" xref="S4.SS2.p1.16.m16.1.1.2">𝚺</ci><cn id="S4.SS2.p1.16.m16.1.1.3.cmml" type="integer" xref="S4.SS2.p1.16.m16.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p1.16.m16.1c">\operatorname{{\mathbf{\Sigma}}}_{1}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p1.16.m16.1d">bold_Σ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="\operatorname{{\mathbf{\Sigma}}}_{2}" class="ltx_Math" display="inline" id="S4.SS2.p1.17.m17.1"><semantics id="S4.SS2.p1.17.m17.1a"><msub id="S4.SS2.p1.17.m17.1.1" xref="S4.SS2.p1.17.m17.1.1.cmml"><mi id="S4.SS2.p1.17.m17.1.1.2" xref="S4.SS2.p1.17.m17.1.1.2.cmml">𝚺</mi><mn id="S4.SS2.p1.17.m17.1.1.3" xref="S4.SS2.p1.17.m17.1.1.3.cmml">2</mn></msub><annotation-xml encoding="MathML-Content" id="S4.SS2.p1.17.m17.1b"><apply id="S4.SS2.p1.17.m17.1.1.cmml" xref="S4.SS2.p1.17.m17.1.1"><csymbol cd="ambiguous" id="S4.SS2.p1.17.m17.1.1.1.cmml" xref="S4.SS2.p1.17.m17.1.1">subscript</csymbol><ci id="S4.SS2.p1.17.m17.1.1.2.cmml" xref="S4.SS2.p1.17.m17.1.1.2">𝚺</ci><cn id="S4.SS2.p1.17.m17.1.1.3.cmml" type="integer" xref="S4.SS2.p1.17.m17.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p1.17.m17.1c">\operatorname{{\mathbf{\Sigma}}}_{2}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p1.17.m17.1d">bold_Σ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math> were identified by scaling their nonzero singular values such that their product equaled one. The <math alttext="\tau^{2}" class="ltx_Math" display="inline" id="S4.SS2.p1.18.m18.1"><semantics id="S4.SS2.p1.18.m18.1a"><msup id="S4.SS2.p1.18.m18.1.1" xref="S4.SS2.p1.18.m18.1.1.cmml"><mi id="S4.SS2.p1.18.m18.1.1.2" xref="S4.SS2.p1.18.m18.1.1.2.cmml">τ</mi><mn id="S4.SS2.p1.18.m18.1.1.3" xref="S4.SS2.p1.18.m18.1.1.3.cmml">2</mn></msup><annotation-xml encoding="MathML-Content" id="S4.SS2.p1.18.m18.1b"><apply id="S4.SS2.p1.18.m18.1.1.cmml" xref="S4.SS2.p1.18.m18.1.1"><csymbol cd="ambiguous" id="S4.SS2.p1.18.m18.1.1.1.cmml" xref="S4.SS2.p1.18.m18.1.1">superscript</csymbol><ci id="S4.SS2.p1.18.m18.1.1.2.cmml" xref="S4.SS2.p1.18.m18.1.1.2">𝜏</ci><cn id="S4.SS2.p1.18.m18.1.1.3.cmml" type="integer" xref="S4.SS2.p1.18.m18.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p1.18.m18.1c">\tau^{2}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p1.18.m18.1d">italic_τ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT</annotation></semantics></math> parameter was fixed at 0.5 for all simulations.</p> </div> <div class="ltx_para" id="S4.SS2.p2"> <p class="ltx_p" id="S4.SS2.p2.9">We varied <math alttext="m_{i}" class="ltx_Math" display="inline" id="S4.SS2.p2.1.m1.1"><semantics id="S4.SS2.p2.1.m1.1a"><msub id="S4.SS2.p2.1.m1.1.1" xref="S4.SS2.p2.1.m1.1.1.cmml"><mi id="S4.SS2.p2.1.m1.1.1.2" xref="S4.SS2.p2.1.m1.1.1.2.cmml">m</mi><mi id="S4.SS2.p2.1.m1.1.1.3" xref="S4.SS2.p2.1.m1.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS2.p2.1.m1.1b"><apply id="S4.SS2.p2.1.m1.1.1.cmml" xref="S4.SS2.p2.1.m1.1.1"><csymbol cd="ambiguous" id="S4.SS2.p2.1.m1.1.1.1.cmml" xref="S4.SS2.p2.1.m1.1.1">subscript</csymbol><ci id="S4.SS2.p2.1.m1.1.1.2.cmml" xref="S4.SS2.p2.1.m1.1.1.2">𝑚</ci><ci id="S4.SS2.p2.1.m1.1.1.3.cmml" xref="S4.SS2.p2.1.m1.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p2.1.m1.1c">m_{i}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p2.1.m1.1d">italic_m start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math>, <math alttext="n" class="ltx_Math" display="inline" id="S4.SS2.p2.2.m2.1"><semantics id="S4.SS2.p2.2.m2.1a"><mi id="S4.SS2.p2.2.m2.1.1" xref="S4.SS2.p2.2.m2.1.1.cmml">n</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.p2.2.m2.1b"><ci id="S4.SS2.p2.2.m2.1.1.cmml" xref="S4.SS2.p2.2.m2.1.1">𝑛</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p2.2.m2.1c">n</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p2.2.m2.1d">italic_n</annotation></semantics></math>, and <math alttext="N" class="ltx_Math" display="inline" id="S4.SS2.p2.3.m3.1"><semantics id="S4.SS2.p2.3.m3.1a"><mi id="S4.SS2.p2.3.m3.1.1" xref="S4.SS2.p2.3.m3.1.1.cmml">N</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.p2.3.m3.1b"><ci id="S4.SS2.p2.3.m3.1.1.cmml" xref="S4.SS2.p2.3.m3.1.1">𝑁</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p2.3.m3.1c">N</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p2.3.m3.1d">italic_N</annotation></semantics></math> to evaluate their impact on MMTR’s performance. For Cases 1 and 2, we simulated data with <math alttext="n\in\{18,27,54,81\}" class="ltx_Math" display="inline" id="S4.SS2.p2.4.m4.4"><semantics id="S4.SS2.p2.4.m4.4a"><mrow id="S4.SS2.p2.4.m4.4.5" xref="S4.SS2.p2.4.m4.4.5.cmml"><mi id="S4.SS2.p2.4.m4.4.5.2" xref="S4.SS2.p2.4.m4.4.5.2.cmml">n</mi><mo id="S4.SS2.p2.4.m4.4.5.1" xref="S4.SS2.p2.4.m4.4.5.1.cmml">∈</mo><mrow id="S4.SS2.p2.4.m4.4.5.3.2" xref="S4.SS2.p2.4.m4.4.5.3.1.cmml"><mo id="S4.SS2.p2.4.m4.4.5.3.2.1" stretchy="false" xref="S4.SS2.p2.4.m4.4.5.3.1.cmml">{</mo><mn id="S4.SS2.p2.4.m4.1.1" xref="S4.SS2.p2.4.m4.1.1.cmml">18</mn><mo id="S4.SS2.p2.4.m4.4.5.3.2.2" xref="S4.SS2.p2.4.m4.4.5.3.1.cmml">,</mo><mn id="S4.SS2.p2.4.m4.2.2" xref="S4.SS2.p2.4.m4.2.2.cmml">27</mn><mo id="S4.SS2.p2.4.m4.4.5.3.2.3" xref="S4.SS2.p2.4.m4.4.5.3.1.cmml">,</mo><mn id="S4.SS2.p2.4.m4.3.3" xref="S4.SS2.p2.4.m4.3.3.cmml">54</mn><mo id="S4.SS2.p2.4.m4.4.5.3.2.4" xref="S4.SS2.p2.4.m4.4.5.3.1.cmml">,</mo><mn id="S4.SS2.p2.4.m4.4.4" xref="S4.SS2.p2.4.m4.4.4.cmml">81</mn><mo id="S4.SS2.p2.4.m4.4.5.3.2.5" stretchy="false" xref="S4.SS2.p2.4.m4.4.5.3.1.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p2.4.m4.4b"><apply id="S4.SS2.p2.4.m4.4.5.cmml" xref="S4.SS2.p2.4.m4.4.5"><in id="S4.SS2.p2.4.m4.4.5.1.cmml" xref="S4.SS2.p2.4.m4.4.5.1"></in><ci id="S4.SS2.p2.4.m4.4.5.2.cmml" xref="S4.SS2.p2.4.m4.4.5.2">𝑛</ci><set id="S4.SS2.p2.4.m4.4.5.3.1.cmml" xref="S4.SS2.p2.4.m4.4.5.3.2"><cn id="S4.SS2.p2.4.m4.1.1.cmml" type="integer" xref="S4.SS2.p2.4.m4.1.1">18</cn><cn id="S4.SS2.p2.4.m4.2.2.cmml" type="integer" xref="S4.SS2.p2.4.m4.2.2">27</cn><cn id="S4.SS2.p2.4.m4.3.3.cmml" type="integer" xref="S4.SS2.p2.4.m4.3.3">54</cn><cn id="S4.SS2.p2.4.m4.4.4.cmml" type="integer" xref="S4.SS2.p2.4.m4.4.4">81</cn></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p2.4.m4.4c">n\in\{18,27,54,81\}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p2.4.m4.4d">italic_n ∈ { 18 , 27 , 54 , 81 }</annotation></semantics></math> and <math alttext="n\in\{64,96,192,288\}" class="ltx_Math" display="inline" id="S4.SS2.p2.5.m5.4"><semantics id="S4.SS2.p2.5.m5.4a"><mrow id="S4.SS2.p2.5.m5.4.5" xref="S4.SS2.p2.5.m5.4.5.cmml"><mi id="S4.SS2.p2.5.m5.4.5.2" xref="S4.SS2.p2.5.m5.4.5.2.cmml">n</mi><mo id="S4.SS2.p2.5.m5.4.5.1" xref="S4.SS2.p2.5.m5.4.5.1.cmml">∈</mo><mrow id="S4.SS2.p2.5.m5.4.5.3.2" xref="S4.SS2.p2.5.m5.4.5.3.1.cmml"><mo id="S4.SS2.p2.5.m5.4.5.3.2.1" stretchy="false" xref="S4.SS2.p2.5.m5.4.5.3.1.cmml">{</mo><mn id="S4.SS2.p2.5.m5.1.1" xref="S4.SS2.p2.5.m5.1.1.cmml">64</mn><mo id="S4.SS2.p2.5.m5.4.5.3.2.2" xref="S4.SS2.p2.5.m5.4.5.3.1.cmml">,</mo><mn id="S4.SS2.p2.5.m5.2.2" xref="S4.SS2.p2.5.m5.2.2.cmml">96</mn><mo id="S4.SS2.p2.5.m5.4.5.3.2.3" xref="S4.SS2.p2.5.m5.4.5.3.1.cmml">,</mo><mn id="S4.SS2.p2.5.m5.3.3" xref="S4.SS2.p2.5.m5.3.3.cmml">192</mn><mo id="S4.SS2.p2.5.m5.4.5.3.2.4" xref="S4.SS2.p2.5.m5.4.5.3.1.cmml">,</mo><mn id="S4.SS2.p2.5.m5.4.4" xref="S4.SS2.p2.5.m5.4.4.cmml">288</mn><mo id="S4.SS2.p2.5.m5.4.5.3.2.5" stretchy="false" xref="S4.SS2.p2.5.m5.4.5.3.1.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p2.5.m5.4b"><apply id="S4.SS2.p2.5.m5.4.5.cmml" xref="S4.SS2.p2.5.m5.4.5"><in id="S4.SS2.p2.5.m5.4.5.1.cmml" xref="S4.SS2.p2.5.m5.4.5.1"></in><ci id="S4.SS2.p2.5.m5.4.5.2.cmml" xref="S4.SS2.p2.5.m5.4.5.2">𝑛</ci><set id="S4.SS2.p2.5.m5.4.5.3.1.cmml" xref="S4.SS2.p2.5.m5.4.5.3.2"><cn id="S4.SS2.p2.5.m5.1.1.cmml" type="integer" xref="S4.SS2.p2.5.m5.1.1">64</cn><cn id="S4.SS2.p2.5.m5.2.2.cmml" type="integer" xref="S4.SS2.p2.5.m5.2.2">96</cn><cn id="S4.SS2.p2.5.m5.3.3.cmml" type="integer" xref="S4.SS2.p2.5.m5.3.3">192</cn><cn id="S4.SS2.p2.5.m5.4.4.cmml" type="integer" xref="S4.SS2.p2.5.m5.4.4">288</cn></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p2.5.m5.4c">n\in\{64,96,192,288\}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p2.5.m5.4d">italic_n ∈ { 64 , 96 , 192 , 288 }</annotation></semantics></math>, respectively, ensuring that the ratio of <math alttext="n" class="ltx_Math" display="inline" id="S4.SS2.p2.6.m6.1"><semantics id="S4.SS2.p2.6.m6.1a"><mi id="S4.SS2.p2.6.m6.1.1" xref="S4.SS2.p2.6.m6.1.1.cmml">n</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.p2.6.m6.1b"><ci id="S4.SS2.p2.6.m6.1.1.cmml" xref="S4.SS2.p2.6.m6.1.1">𝑛</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p2.6.m6.1c">n</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p2.6.m6.1d">italic_n</annotation></semantics></math> to the parameter dimension was 0.4, 0.6, 1.2, and 1.8. For every case and <math alttext="n" class="ltx_Math" display="inline" id="S4.SS2.p2.7.m7.1"><semantics id="S4.SS2.p2.7.m7.1a"><mi id="S4.SS2.p2.7.m7.1.1" xref="S4.SS2.p2.7.m7.1.1.cmml">n</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.p2.7.m7.1b"><ci id="S4.SS2.p2.7.m7.1.1.cmml" xref="S4.SS2.p2.7.m7.1.1">𝑛</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p2.7.m7.1c">n</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p2.7.m7.1d">italic_n</annotation></semantics></math> combination, the number of observations per group was set to <math alttext="m_{i}\in\{2,6\}" class="ltx_Math" display="inline" id="S4.SS2.p2.8.m8.2"><semantics id="S4.SS2.p2.8.m8.2a"><mrow id="S4.SS2.p2.8.m8.2.3" xref="S4.SS2.p2.8.m8.2.3.cmml"><msub id="S4.SS2.p2.8.m8.2.3.2" xref="S4.SS2.p2.8.m8.2.3.2.cmml"><mi id="S4.SS2.p2.8.m8.2.3.2.2" xref="S4.SS2.p2.8.m8.2.3.2.2.cmml">m</mi><mi id="S4.SS2.p2.8.m8.2.3.2.3" xref="S4.SS2.p2.8.m8.2.3.2.3.cmml">i</mi></msub><mo id="S4.SS2.p2.8.m8.2.3.1" xref="S4.SS2.p2.8.m8.2.3.1.cmml">∈</mo><mrow id="S4.SS2.p2.8.m8.2.3.3.2" xref="S4.SS2.p2.8.m8.2.3.3.1.cmml"><mo id="S4.SS2.p2.8.m8.2.3.3.2.1" stretchy="false" xref="S4.SS2.p2.8.m8.2.3.3.1.cmml">{</mo><mn id="S4.SS2.p2.8.m8.1.1" xref="S4.SS2.p2.8.m8.1.1.cmml">2</mn><mo id="S4.SS2.p2.8.m8.2.3.3.2.2" xref="S4.SS2.p2.8.m8.2.3.3.1.cmml">,</mo><mn id="S4.SS2.p2.8.m8.2.2" xref="S4.SS2.p2.8.m8.2.2.cmml">6</mn><mo id="S4.SS2.p2.8.m8.2.3.3.2.3" stretchy="false" xref="S4.SS2.p2.8.m8.2.3.3.1.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p2.8.m8.2b"><apply id="S4.SS2.p2.8.m8.2.3.cmml" xref="S4.SS2.p2.8.m8.2.3"><in id="S4.SS2.p2.8.m8.2.3.1.cmml" xref="S4.SS2.p2.8.m8.2.3.1"></in><apply id="S4.SS2.p2.8.m8.2.3.2.cmml" xref="S4.SS2.p2.8.m8.2.3.2"><csymbol cd="ambiguous" id="S4.SS2.p2.8.m8.2.3.2.1.cmml" xref="S4.SS2.p2.8.m8.2.3.2">subscript</csymbol><ci id="S4.SS2.p2.8.m8.2.3.2.2.cmml" xref="S4.SS2.p2.8.m8.2.3.2.2">𝑚</ci><ci id="S4.SS2.p2.8.m8.2.3.2.3.cmml" xref="S4.SS2.p2.8.m8.2.3.2.3">𝑖</ci></apply><set id="S4.SS2.p2.8.m8.2.3.3.1.cmml" xref="S4.SS2.p2.8.m8.2.3.3.2"><cn id="S4.SS2.p2.8.m8.1.1.cmml" type="integer" xref="S4.SS2.p2.8.m8.1.1">2</cn><cn id="S4.SS2.p2.8.m8.2.2.cmml" type="integer" xref="S4.SS2.p2.8.m8.2.2">6</cn></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p2.8.m8.2c">m_{i}\in\{2,6\}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p2.8.m8.2d">italic_m start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∈ { 2 , 6 }</annotation></semantics></math>, resulting in 16 different simulation scenarios. The <span class="ltx_text ltx_font_typewriter" id="S4.SS2.p2.9.1">lme4</span> package required <math alttext="N\geq nQ_{1}Q_{2}" class="ltx_Math" display="inline" id="S4.SS2.p2.9.m9.1"><semantics id="S4.SS2.p2.9.m9.1a"><mrow id="S4.SS2.p2.9.m9.1.1" xref="S4.SS2.p2.9.m9.1.1.cmml"><mi id="S4.SS2.p2.9.m9.1.1.2" xref="S4.SS2.p2.9.m9.1.1.2.cmml">N</mi><mo id="S4.SS2.p2.9.m9.1.1.1" xref="S4.SS2.p2.9.m9.1.1.1.cmml">≥</mo><mrow id="S4.SS2.p2.9.m9.1.1.3" xref="S4.SS2.p2.9.m9.1.1.3.cmml"><mi id="S4.SS2.p2.9.m9.1.1.3.2" xref="S4.SS2.p2.9.m9.1.1.3.2.cmml">n</mi><mo id="S4.SS2.p2.9.m9.1.1.3.1" xref="S4.SS2.p2.9.m9.1.1.3.1.cmml">⁢</mo><msub id="S4.SS2.p2.9.m9.1.1.3.3" xref="S4.SS2.p2.9.m9.1.1.3.3.cmml"><mi id="S4.SS2.p2.9.m9.1.1.3.3.2" xref="S4.SS2.p2.9.m9.1.1.3.3.2.cmml">Q</mi><mn id="S4.SS2.p2.9.m9.1.1.3.3.3" xref="S4.SS2.p2.9.m9.1.1.3.3.3.cmml">1</mn></msub><mo id="S4.SS2.p2.9.m9.1.1.3.1a" xref="S4.SS2.p2.9.m9.1.1.3.1.cmml">⁢</mo><msub id="S4.SS2.p2.9.m9.1.1.3.4" xref="S4.SS2.p2.9.m9.1.1.3.4.cmml"><mi id="S4.SS2.p2.9.m9.1.1.3.4.2" xref="S4.SS2.p2.9.m9.1.1.3.4.2.cmml">Q</mi><mn id="S4.SS2.p2.9.m9.1.1.3.4.3" xref="S4.SS2.p2.9.m9.1.1.3.4.3.cmml">2</mn></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p2.9.m9.1b"><apply id="S4.SS2.p2.9.m9.1.1.cmml" xref="S4.SS2.p2.9.m9.1.1"><geq id="S4.SS2.p2.9.m9.1.1.1.cmml" xref="S4.SS2.p2.9.m9.1.1.1"></geq><ci id="S4.SS2.p2.9.m9.1.1.2.cmml" xref="S4.SS2.p2.9.m9.1.1.2">𝑁</ci><apply id="S4.SS2.p2.9.m9.1.1.3.cmml" xref="S4.SS2.p2.9.m9.1.1.3"><times id="S4.SS2.p2.9.m9.1.1.3.1.cmml" xref="S4.SS2.p2.9.m9.1.1.3.1"></times><ci id="S4.SS2.p2.9.m9.1.1.3.2.cmml" xref="S4.SS2.p2.9.m9.1.1.3.2">𝑛</ci><apply id="S4.SS2.p2.9.m9.1.1.3.3.cmml" xref="S4.SS2.p2.9.m9.1.1.3.3"><csymbol cd="ambiguous" id="S4.SS2.p2.9.m9.1.1.3.3.1.cmml" xref="S4.SS2.p2.9.m9.1.1.3.3">subscript</csymbol><ci id="S4.SS2.p2.9.m9.1.1.3.3.2.cmml" xref="S4.SS2.p2.9.m9.1.1.3.3.2">𝑄</ci><cn id="S4.SS2.p2.9.m9.1.1.3.3.3.cmml" type="integer" xref="S4.SS2.p2.9.m9.1.1.3.3.3">1</cn></apply><apply id="S4.SS2.p2.9.m9.1.1.3.4.cmml" xref="S4.SS2.p2.9.m9.1.1.3.4"><csymbol cd="ambiguous" id="S4.SS2.p2.9.m9.1.1.3.4.1.cmml" xref="S4.SS2.p2.9.m9.1.1.3.4">subscript</csymbol><ci id="S4.SS2.p2.9.m9.1.1.3.4.2.cmml" xref="S4.SS2.p2.9.m9.1.1.3.4.2">𝑄</ci><cn id="S4.SS2.p2.9.m9.1.1.3.4.3.cmml" type="integer" xref="S4.SS2.p2.9.m9.1.1.3.4.3">2</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p2.9.m9.1c">N\geq nQ_{1}Q_{2}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p2.9.m9.1d">italic_N ≥ italic_n italic_Q start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_Q start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math> for estimating covariance parameters, so it was excluded as a competitor. We performed 100 replications for each scenario and evaluated MMTR’s performance using three competitors.</p> </div> <div class="ltx_para" id="S4.SS2.p3"> <p class="ltx_p" id="S4.SS2.p3.18">We estimated <math alttext="\operatorname{\mathbf{B}}" class="ltx_Math" display="inline" id="S4.SS2.p3.1.m1.1"><semantics id="S4.SS2.p3.1.m1.1a"><mi id="S4.SS2.p3.1.m1.1.1" xref="S4.SS2.p3.1.m1.1.1.cmml">𝐁</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.p3.1.m1.1b"><ci id="S4.SS2.p3.1.m1.1.1.cmml" xref="S4.SS2.p3.1.m1.1.1">𝐁</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p3.1.m1.1c">\operatorname{\mathbf{B}}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p3.1.m1.1d">bold_B</annotation></semantics></math> and <math alttext="\operatorname{{\mathbf{\Lambda}}}" class="ltx_Math" display="inline" id="S4.SS2.p3.2.m2.1"><semantics id="S4.SS2.p3.2.m2.1a"><mi id="S4.SS2.p3.2.m2.1.1" xref="S4.SS2.p3.2.m2.1.1.cmml">𝚲</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.p3.2.m2.1b"><ci id="S4.SS2.p3.2.m2.1.1.cmml" xref="S4.SS2.p3.2.m2.1.1">𝚲</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p3.2.m2.1c">\operatorname{{\mathbf{\Lambda}}}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p3.2.m2.1d">bold_Λ</annotation></semantics></math> in (<a class="ltx_ref" href="https://arxiv.org/html/2503.13782v1#S4.E12" title="In 4.1 Setup ‣ 4 Experiments ‣ Regularized Parameter Estimation in Mixed Model Trace Regression"><span class="ltx_text ltx_ref_tag">12</span></a>) by selecting tuning parameters via ten-fold cross-validation. MMTR’s tuning parameters, <math alttext="\lambda_{\operatorname{\mathbf{B}}}" class="ltx_Math" display="inline" id="S4.SS2.p3.3.m3.1"><semantics id="S4.SS2.p3.3.m3.1a"><msub id="S4.SS2.p3.3.m3.1.1" xref="S4.SS2.p3.3.m3.1.1.cmml"><mi id="S4.SS2.p3.3.m3.1.1.2" xref="S4.SS2.p3.3.m3.1.1.2.cmml">λ</mi><mi id="S4.SS2.p3.3.m3.1.1.3" xref="S4.SS2.p3.3.m3.1.1.3.cmml">𝐁</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS2.p3.3.m3.1b"><apply id="S4.SS2.p3.3.m3.1.1.cmml" xref="S4.SS2.p3.3.m3.1.1"><csymbol cd="ambiguous" id="S4.SS2.p3.3.m3.1.1.1.cmml" xref="S4.SS2.p3.3.m3.1.1">subscript</csymbol><ci id="S4.SS2.p3.3.m3.1.1.2.cmml" xref="S4.SS2.p3.3.m3.1.1.2">𝜆</ci><ci id="S4.SS2.p3.3.m3.1.1.3.cmml" xref="S4.SS2.p3.3.m3.1.1.3">𝐁</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p3.3.m3.1c">\lambda_{\operatorname{\mathbf{B}}}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p3.3.m3.1d">italic_λ start_POSTSUBSCRIPT bold_B end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="\lambda_{\operatorname{\mathbf{L}}}" class="ltx_Math" display="inline" id="S4.SS2.p3.4.m4.1"><semantics id="S4.SS2.p3.4.m4.1a"><msub id="S4.SS2.p3.4.m4.1.1" xref="S4.SS2.p3.4.m4.1.1.cmml"><mi id="S4.SS2.p3.4.m4.1.1.2" xref="S4.SS2.p3.4.m4.1.1.2.cmml">λ</mi><mi id="S4.SS2.p3.4.m4.1.1.3" xref="S4.SS2.p3.4.m4.1.1.3.cmml">𝐋</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS2.p3.4.m4.1b"><apply id="S4.SS2.p3.4.m4.1.1.cmml" xref="S4.SS2.p3.4.m4.1.1"><csymbol cd="ambiguous" id="S4.SS2.p3.4.m4.1.1.1.cmml" xref="S4.SS2.p3.4.m4.1.1">subscript</csymbol><ci id="S4.SS2.p3.4.m4.1.1.2.cmml" xref="S4.SS2.p3.4.m4.1.1.2">𝜆</ci><ci id="S4.SS2.p3.4.m4.1.1.3.cmml" xref="S4.SS2.p3.4.m4.1.1.3">𝐋</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p3.4.m4.1c">\lambda_{\operatorname{\mathbf{L}}}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p3.4.m4.1d">italic_λ start_POSTSUBSCRIPT bold_L end_POSTSUBSCRIPT</annotation></semantics></math>, ranged from <math alttext="10^{-4}" class="ltx_Math" display="inline" id="S4.SS2.p3.5.m5.1"><semantics id="S4.SS2.p3.5.m5.1a"><msup id="S4.SS2.p3.5.m5.1.1" xref="S4.SS2.p3.5.m5.1.1.cmml"><mn id="S4.SS2.p3.5.m5.1.1.2" xref="S4.SS2.p3.5.m5.1.1.2.cmml">10</mn><mrow id="S4.SS2.p3.5.m5.1.1.3" xref="S4.SS2.p3.5.m5.1.1.3.cmml"><mo id="S4.SS2.p3.5.m5.1.1.3a" xref="S4.SS2.p3.5.m5.1.1.3.cmml">−</mo><mn id="S4.SS2.p3.5.m5.1.1.3.2" xref="S4.SS2.p3.5.m5.1.1.3.2.cmml">4</mn></mrow></msup><annotation-xml encoding="MathML-Content" id="S4.SS2.p3.5.m5.1b"><apply id="S4.SS2.p3.5.m5.1.1.cmml" xref="S4.SS2.p3.5.m5.1.1"><csymbol cd="ambiguous" id="S4.SS2.p3.5.m5.1.1.1.cmml" xref="S4.SS2.p3.5.m5.1.1">superscript</csymbol><cn id="S4.SS2.p3.5.m5.1.1.2.cmml" type="integer" xref="S4.SS2.p3.5.m5.1.1.2">10</cn><apply id="S4.SS2.p3.5.m5.1.1.3.cmml" xref="S4.SS2.p3.5.m5.1.1.3"><minus id="S4.SS2.p3.5.m5.1.1.3.1.cmml" xref="S4.SS2.p3.5.m5.1.1.3"></minus><cn id="S4.SS2.p3.5.m5.1.1.3.2.cmml" type="integer" xref="S4.SS2.p3.5.m5.1.1.3.2">4</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p3.5.m5.1c">10^{-4}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p3.5.m5.1d">10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT</annotation></semantics></math> and <math alttext="0.04" class="ltx_Math" display="inline" id="S4.SS2.p3.6.m6.1"><semantics id="S4.SS2.p3.6.m6.1a"><mn id="S4.SS2.p3.6.m6.1.1" xref="S4.SS2.p3.6.m6.1.1.cmml">0.04</mn><annotation-xml encoding="MathML-Content" id="S4.SS2.p3.6.m6.1b"><cn id="S4.SS2.p3.6.m6.1.1.cmml" type="float" xref="S4.SS2.p3.6.m6.1.1">0.04</cn></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p3.6.m6.1c">0.04</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p3.6.m6.1d">0.04</annotation></semantics></math> and <math alttext="10^{{-4}}" class="ltx_Math" display="inline" id="S4.SS2.p3.7.m7.1"><semantics id="S4.SS2.p3.7.m7.1a"><msup id="S4.SS2.p3.7.m7.1.1" xref="S4.SS2.p3.7.m7.1.1.cmml"><mn id="S4.SS2.p3.7.m7.1.1.2" xref="S4.SS2.p3.7.m7.1.1.2.cmml">10</mn><mrow id="S4.SS2.p3.7.m7.1.1.3" xref="S4.SS2.p3.7.m7.1.1.3.cmml"><mo id="S4.SS2.p3.7.m7.1.1.3a" xref="S4.SS2.p3.7.m7.1.1.3.cmml">−</mo><mn id="S4.SS2.p3.7.m7.1.1.3.2" xref="S4.SS2.p3.7.m7.1.1.3.2.cmml">4</mn></mrow></msup><annotation-xml encoding="MathML-Content" id="S4.SS2.p3.7.m7.1b"><apply id="S4.SS2.p3.7.m7.1.1.cmml" xref="S4.SS2.p3.7.m7.1.1"><csymbol cd="ambiguous" id="S4.SS2.p3.7.m7.1.1.1.cmml" xref="S4.SS2.p3.7.m7.1.1">superscript</csymbol><cn id="S4.SS2.p3.7.m7.1.1.2.cmml" type="integer" xref="S4.SS2.p3.7.m7.1.1.2">10</cn><apply id="S4.SS2.p3.7.m7.1.1.3.cmml" xref="S4.SS2.p3.7.m7.1.1.3"><minus id="S4.SS2.p3.7.m7.1.1.3.1.cmml" xref="S4.SS2.p3.7.m7.1.1.3"></minus><cn id="S4.SS2.p3.7.m7.1.1.3.2.cmml" type="integer" xref="S4.SS2.p3.7.m7.1.1.3.2">4</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p3.7.m7.1c">10^{{-4}}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p3.7.m7.1d">10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT</annotation></semantics></math> and 1, respectively, on an equally spaced 10-by-10 log-scale grid. Sparse Kruskal regression estimated only <math alttext="\operatorname{\mathbf{B}}" class="ltx_Math" display="inline" id="S4.SS2.p3.8.m8.1"><semantics id="S4.SS2.p3.8.m8.1a"><mi id="S4.SS2.p3.8.m8.1.1" xref="S4.SS2.p3.8.m8.1.1.cmml">𝐁</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.p3.8.m8.1b"><ci id="S4.SS2.p3.8.m8.1.1.cmml" xref="S4.SS2.p3.8.m8.1.1">𝐁</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p3.8.m8.1c">\operatorname{\mathbf{B}}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p3.8.m8.1d">bold_B</annotation></semantics></math> by varying its tuning parameter from <math alttext="10^{{-2}}" class="ltx_Math" display="inline" id="S4.SS2.p3.9.m9.1"><semantics id="S4.SS2.p3.9.m9.1a"><msup id="S4.SS2.p3.9.m9.1.1" xref="S4.SS2.p3.9.m9.1.1.cmml"><mn id="S4.SS2.p3.9.m9.1.1.2" xref="S4.SS2.p3.9.m9.1.1.2.cmml">10</mn><mrow id="S4.SS2.p3.9.m9.1.1.3" xref="S4.SS2.p3.9.m9.1.1.3.cmml"><mo id="S4.SS2.p3.9.m9.1.1.3a" xref="S4.SS2.p3.9.m9.1.1.3.cmml">−</mo><mn id="S4.SS2.p3.9.m9.1.1.3.2" xref="S4.SS2.p3.9.m9.1.1.3.2.cmml">2</mn></mrow></msup><annotation-xml encoding="MathML-Content" id="S4.SS2.p3.9.m9.1b"><apply id="S4.SS2.p3.9.m9.1.1.cmml" xref="S4.SS2.p3.9.m9.1.1"><csymbol cd="ambiguous" id="S4.SS2.p3.9.m9.1.1.1.cmml" xref="S4.SS2.p3.9.m9.1.1">superscript</csymbol><cn id="S4.SS2.p3.9.m9.1.1.2.cmml" type="integer" xref="S4.SS2.p3.9.m9.1.1.2">10</cn><apply id="S4.SS2.p3.9.m9.1.1.3.cmml" xref="S4.SS2.p3.9.m9.1.1.3"><minus id="S4.SS2.p3.9.m9.1.1.3.1.cmml" xref="S4.SS2.p3.9.m9.1.1.3"></minus><cn id="S4.SS2.p3.9.m9.1.1.3.2.cmml" type="integer" xref="S4.SS2.p3.9.m9.1.1.3.2">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p3.9.m9.1c">10^{{-2}}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p3.9.m9.1d">10 start_POSTSUPERSCRIPT - 2 end_POSTSUPERSCRIPT</annotation></semantics></math> to <math alttext="10" class="ltx_Math" display="inline" id="S4.SS2.p3.10.m10.1"><semantics id="S4.SS2.p3.10.m10.1a"><mn id="S4.SS2.p3.10.m10.1.1" xref="S4.SS2.p3.10.m10.1.1.cmml">10</mn><annotation-xml encoding="MathML-Content" id="S4.SS2.p3.10.m10.1b"><cn id="S4.SS2.p3.10.m10.1.1.cmml" type="integer" xref="S4.SS2.p3.10.m10.1.1">10</cn></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p3.10.m10.1c">10</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p3.10.m10.1d">10</annotation></semantics></math> on an equally spaced log-scale grid. The quasi-likelihood method also required <math alttext="N\geq nQ_{1}Q_{2}" class="ltx_Math" display="inline" id="S4.SS2.p3.11.m11.1"><semantics id="S4.SS2.p3.11.m11.1a"><mrow id="S4.SS2.p3.11.m11.1.1" xref="S4.SS2.p3.11.m11.1.1.cmml"><mi id="S4.SS2.p3.11.m11.1.1.2" xref="S4.SS2.p3.11.m11.1.1.2.cmml">N</mi><mo id="S4.SS2.p3.11.m11.1.1.1" xref="S4.SS2.p3.11.m11.1.1.1.cmml">≥</mo><mrow id="S4.SS2.p3.11.m11.1.1.3" xref="S4.SS2.p3.11.m11.1.1.3.cmml"><mi id="S4.SS2.p3.11.m11.1.1.3.2" xref="S4.SS2.p3.11.m11.1.1.3.2.cmml">n</mi><mo id="S4.SS2.p3.11.m11.1.1.3.1" xref="S4.SS2.p3.11.m11.1.1.3.1.cmml">⁢</mo><msub id="S4.SS2.p3.11.m11.1.1.3.3" xref="S4.SS2.p3.11.m11.1.1.3.3.cmml"><mi id="S4.SS2.p3.11.m11.1.1.3.3.2" xref="S4.SS2.p3.11.m11.1.1.3.3.2.cmml">Q</mi><mn id="S4.SS2.p3.11.m11.1.1.3.3.3" xref="S4.SS2.p3.11.m11.1.1.3.3.3.cmml">1</mn></msub><mo id="S4.SS2.p3.11.m11.1.1.3.1a" xref="S4.SS2.p3.11.m11.1.1.3.1.cmml">⁢</mo><msub id="S4.SS2.p3.11.m11.1.1.3.4" xref="S4.SS2.p3.11.m11.1.1.3.4.cmml"><mi id="S4.SS2.p3.11.m11.1.1.3.4.2" xref="S4.SS2.p3.11.m11.1.1.3.4.2.cmml">Q</mi><mn id="S4.SS2.p3.11.m11.1.1.3.4.3" xref="S4.SS2.p3.11.m11.1.1.3.4.3.cmml">2</mn></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p3.11.m11.1b"><apply id="S4.SS2.p3.11.m11.1.1.cmml" xref="S4.SS2.p3.11.m11.1.1"><geq id="S4.SS2.p3.11.m11.1.1.1.cmml" xref="S4.SS2.p3.11.m11.1.1.1"></geq><ci id="S4.SS2.p3.11.m11.1.1.2.cmml" xref="S4.SS2.p3.11.m11.1.1.2">𝑁</ci><apply id="S4.SS2.p3.11.m11.1.1.3.cmml" xref="S4.SS2.p3.11.m11.1.1.3"><times id="S4.SS2.p3.11.m11.1.1.3.1.cmml" xref="S4.SS2.p3.11.m11.1.1.3.1"></times><ci id="S4.SS2.p3.11.m11.1.1.3.2.cmml" xref="S4.SS2.p3.11.m11.1.1.3.2">𝑛</ci><apply id="S4.SS2.p3.11.m11.1.1.3.3.cmml" xref="S4.SS2.p3.11.m11.1.1.3.3"><csymbol cd="ambiguous" id="S4.SS2.p3.11.m11.1.1.3.3.1.cmml" xref="S4.SS2.p3.11.m11.1.1.3.3">subscript</csymbol><ci id="S4.SS2.p3.11.m11.1.1.3.3.2.cmml" xref="S4.SS2.p3.11.m11.1.1.3.3.2">𝑄</ci><cn id="S4.SS2.p3.11.m11.1.1.3.3.3.cmml" type="integer" xref="S4.SS2.p3.11.m11.1.1.3.3.3">1</cn></apply><apply id="S4.SS2.p3.11.m11.1.1.3.4.cmml" xref="S4.SS2.p3.11.m11.1.1.3.4"><csymbol cd="ambiguous" id="S4.SS2.p3.11.m11.1.1.3.4.1.cmml" xref="S4.SS2.p3.11.m11.1.1.3.4">subscript</csymbol><ci id="S4.SS2.p3.11.m11.1.1.3.4.2.cmml" xref="S4.SS2.p3.11.m11.1.1.3.4.2">𝑄</ci><cn id="S4.SS2.p3.11.m11.1.1.3.4.3.cmml" type="integer" xref="S4.SS2.p3.11.m11.1.1.3.4.3">2</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p3.11.m11.1c">N\geq nQ_{1}Q_{2}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p3.11.m11.1d">italic_N ≥ italic_n italic_Q start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_Q start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math> for estimating <math alttext="\operatorname{{\mathbf{\Lambda}}}" class="ltx_Math" display="inline" id="S4.SS2.p3.12.m12.1"><semantics id="S4.SS2.p3.12.m12.1a"><mi id="S4.SS2.p3.12.m12.1.1" xref="S4.SS2.p3.12.m12.1.1.cmml">𝚲</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.p3.12.m12.1b"><ci id="S4.SS2.p3.12.m12.1.1.cmml" xref="S4.SS2.p3.12.m12.1.1">𝚲</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p3.12.m12.1c">\operatorname{{\mathbf{\Lambda}}}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p3.12.m12.1d">bold_Λ</annotation></semantics></math>, so it estimated only <math alttext="\operatorname{\mathbf{B}}" class="ltx_Math" display="inline" id="S4.SS2.p3.13.m13.1"><semantics id="S4.SS2.p3.13.m13.1a"><mi id="S4.SS2.p3.13.m13.1.1" xref="S4.SS2.p3.13.m13.1.1.cmml">𝐁</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.p3.13.m13.1b"><ci id="S4.SS2.p3.13.m13.1.1.cmml" xref="S4.SS2.p3.13.m13.1.1">𝐁</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p3.13.m13.1c">\operatorname{\mathbf{B}}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p3.13.m13.1d">bold_B</annotation></semantics></math>. For the two GEE models and the quasi-likelihood method, the tuning parameters for estimating <math alttext="\operatorname{\mathbf{B}}" class="ltx_Math" display="inline" id="S4.SS2.p3.14.m14.1"><semantics id="S4.SS2.p3.14.m14.1a"><mi id="S4.SS2.p3.14.m14.1.1" xref="S4.SS2.p3.14.m14.1.1.cmml">𝐁</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.p3.14.m14.1b"><ci id="S4.SS2.p3.14.m14.1.1.cmml" xref="S4.SS2.p3.14.m14.1.1">𝐁</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p3.14.m14.1c">\operatorname{\mathbf{B}}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p3.14.m14.1d">bold_B</annotation></semantics></math> varied evenly over 10 values from <math alttext="10^{-4}" class="ltx_Math" display="inline" id="S4.SS2.p3.15.m15.1"><semantics id="S4.SS2.p3.15.m15.1a"><msup id="S4.SS2.p3.15.m15.1.1" xref="S4.SS2.p3.15.m15.1.1.cmml"><mn id="S4.SS2.p3.15.m15.1.1.2" xref="S4.SS2.p3.15.m15.1.1.2.cmml">10</mn><mrow id="S4.SS2.p3.15.m15.1.1.3" xref="S4.SS2.p3.15.m15.1.1.3.cmml"><mo id="S4.SS2.p3.15.m15.1.1.3a" xref="S4.SS2.p3.15.m15.1.1.3.cmml">−</mo><mn id="S4.SS2.p3.15.m15.1.1.3.2" xref="S4.SS2.p3.15.m15.1.1.3.2.cmml">4</mn></mrow></msup><annotation-xml encoding="MathML-Content" id="S4.SS2.p3.15.m15.1b"><apply id="S4.SS2.p3.15.m15.1.1.cmml" xref="S4.SS2.p3.15.m15.1.1"><csymbol cd="ambiguous" id="S4.SS2.p3.15.m15.1.1.1.cmml" xref="S4.SS2.p3.15.m15.1.1">superscript</csymbol><cn id="S4.SS2.p3.15.m15.1.1.2.cmml" type="integer" xref="S4.SS2.p3.15.m15.1.1.2">10</cn><apply id="S4.SS2.p3.15.m15.1.1.3.cmml" xref="S4.SS2.p3.15.m15.1.1.3"><minus id="S4.SS2.p3.15.m15.1.1.3.1.cmml" xref="S4.SS2.p3.15.m15.1.1.3"></minus><cn id="S4.SS2.p3.15.m15.1.1.3.2.cmml" type="integer" xref="S4.SS2.p3.15.m15.1.1.3.2">4</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p3.15.m15.1c">10^{-4}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p3.15.m15.1d">10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT</annotation></semantics></math> and <math alttext="10" class="ltx_Math" display="inline" id="S4.SS2.p3.16.m16.1"><semantics id="S4.SS2.p3.16.m16.1a"><mn id="S4.SS2.p3.16.m16.1.1" xref="S4.SS2.p3.16.m16.1.1.cmml">10</mn><annotation-xml encoding="MathML-Content" id="S4.SS2.p3.16.m16.1b"><cn id="S4.SS2.p3.16.m16.1.1.cmml" type="integer" xref="S4.SS2.p3.16.m16.1.1">10</cn></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p3.16.m16.1c">10</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p3.16.m16.1d">10</annotation></semantics></math> on a log-scale grid. This resulted in <math alttext="\operatorname{\mathbf{B}}" class="ltx_Math" display="inline" id="S4.SS2.p3.17.m17.1"><semantics id="S4.SS2.p3.17.m17.1a"><mi id="S4.SS2.p3.17.m17.1.1" xref="S4.SS2.p3.17.m17.1.1.cmml">𝐁</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.p3.17.m17.1b"><ci id="S4.SS2.p3.17.m17.1.1.cmml" xref="S4.SS2.p3.17.m17.1.1">𝐁</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p3.17.m17.1c">\operatorname{\mathbf{B}}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p3.17.m17.1d">bold_B</annotation></semantics></math> estimates for four methods and <math alttext="\operatorname{{\mathbf{\Lambda}}}" class="ltx_Math" display="inline" id="S4.SS2.p3.18.m18.1"><semantics id="S4.SS2.p3.18.m18.1a"><mi id="S4.SS2.p3.18.m18.1.1" xref="S4.SS2.p3.18.m18.1.1.cmml">𝚲</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.p3.18.m18.1b"><ci id="S4.SS2.p3.18.m18.1.1.cmml" xref="S4.SS2.p3.18.m18.1.1">𝚲</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p3.18.m18.1c">\operatorname{{\mathbf{\Lambda}}}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p3.18.m18.1d">bold_Λ</annotation></semantics></math> estimates for only two.</p> </div> <div class="ltx_para" id="S4.SS2.p4"> <p class="ltx_p" id="S4.SS2.p4.9">We evaluated the performance of MMTR and its competitors using the metrics in (<a class="ltx_ref" href="https://arxiv.org/html/2503.13782v1#S4.E13" title="In 4.1 Setup ‣ 4 Experiments ‣ Regularized Parameter Estimation in Mixed Model Trace Regression"><span class="ltx_text ltx_ref_tag">13</span></a>) (Tables <a class="ltx_ref" href="https://arxiv.org/html/2503.13782v1#S4.T1" title="Table 1 ‣ 4.2 Simulation Using the MMTR Model ‣ 4 Experiments ‣ Regularized Parameter Estimation in Mixed Model Trace Regression"><span class="ltx_text ltx_ref_tag">1</span></a> and <a class="ltx_ref" href="https://arxiv.org/html/2503.13782v1#S4.T2" title="Table 2 ‣ 4.2 Simulation Using the MMTR Model ‣ 4 Experiments ‣ Regularized Parameter Estimation in Mixed Model Trace Regression"><span class="ltx_text ltx_ref_tag">2</span></a>). For <math alttext="\operatorname{\mathbf{B}}" class="ltx_Math" display="inline" id="S4.SS2.p4.1.m1.1"><semantics id="S4.SS2.p4.1.m1.1a"><mi id="S4.SS2.p4.1.m1.1.1" xref="S4.SS2.p4.1.m1.1.1.cmml">𝐁</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.p4.1.m1.1b"><ci id="S4.SS2.p4.1.m1.1.1.cmml" xref="S4.SS2.p4.1.m1.1.1">𝐁</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p4.1.m1.1c">\operatorname{\mathbf{B}}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p4.1.m1.1d">bold_B</annotation></semantics></math> estimation, all four methods showed similar estimation accuracy across all simulation scenarios, demonstrating that mean parameter estimation was robust to model misspecification. In contrast, accurate <math alttext="\operatorname{{\mathbf{\Lambda}}}" class="ltx_Math" display="inline" id="S4.SS2.p4.2.m2.1"><semantics id="S4.SS2.p4.2.m2.1a"><mi id="S4.SS2.p4.2.m2.1.1" xref="S4.SS2.p4.2.m2.1.1.cmml">𝚲</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.p4.2.m2.1b"><ci id="S4.SS2.p4.2.m2.1.1.cmml" xref="S4.SS2.p4.2.m2.1.1">𝚲</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p4.2.m2.1c">\operatorname{{\mathbf{\Lambda}}}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p4.2.m2.1d">bold_Λ</annotation></semantics></math> estimation was more challenging and required <math alttext="m_{i}" class="ltx_Math" display="inline" id="S4.SS2.p4.3.m3.1"><semantics id="S4.SS2.p4.3.m3.1a"><msub id="S4.SS2.p4.3.m3.1.1" xref="S4.SS2.p4.3.m3.1.1.cmml"><mi id="S4.SS2.p4.3.m3.1.1.2" xref="S4.SS2.p4.3.m3.1.1.2.cmml">m</mi><mi id="S4.SS2.p4.3.m3.1.1.3" xref="S4.SS2.p4.3.m3.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS2.p4.3.m3.1b"><apply id="S4.SS2.p4.3.m3.1.1.cmml" xref="S4.SS2.p4.3.m3.1.1"><csymbol cd="ambiguous" id="S4.SS2.p4.3.m3.1.1.1.cmml" xref="S4.SS2.p4.3.m3.1.1">subscript</csymbol><ci id="S4.SS2.p4.3.m3.1.1.2.cmml" xref="S4.SS2.p4.3.m3.1.1.2">𝑚</ci><ci id="S4.SS2.p4.3.m3.1.1.3.cmml" xref="S4.SS2.p4.3.m3.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p4.3.m3.1c">m_{i}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p4.3.m3.1d">italic_m start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="n" class="ltx_Math" display="inline" id="S4.SS2.p4.4.m4.1"><semantics id="S4.SS2.p4.4.m4.1a"><mi id="S4.SS2.p4.4.m4.1.1" xref="S4.SS2.p4.4.m4.1.1.cmml">n</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.p4.4.m4.1b"><ci id="S4.SS2.p4.4.m4.1.1.cmml" xref="S4.SS2.p4.4.m4.1.1">𝑛</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p4.4.m4.1c">n</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p4.4.m4.1d">italic_n</annotation></semantics></math> to be sufficiently large. Furthermore, MMTR outperformed both GEE models across all scenarios in <math alttext="\operatorname{{\mathbf{\Lambda}}}" class="ltx_Math" display="inline" id="S4.SS2.p4.5.m5.1"><semantics id="S4.SS2.p4.5.m5.1a"><mi id="S4.SS2.p4.5.m5.1.1" xref="S4.SS2.p4.5.m5.1.1.cmml">𝚲</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.p4.5.m5.1b"><ci id="S4.SS2.p4.5.m5.1.1.cmml" xref="S4.SS2.p4.5.m5.1.1">𝚲</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p4.5.m5.1c">\operatorname{{\mathbf{\Lambda}}}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p4.5.m5.1d">bold_Λ</annotation></semantics></math> estimation, indicating that covariance estimation was sensitive to model misspecification. MMTR’s relative accuracy increased with <math alttext="n" class="ltx_Math" display="inline" id="S4.SS2.p4.6.m6.1"><semantics id="S4.SS2.p4.6.m6.1a"><mi id="S4.SS2.p4.6.m6.1.1" xref="S4.SS2.p4.6.m6.1.1.cmml">n</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.p4.6.m6.1b"><ci id="S4.SS2.p4.6.m6.1.1.cmml" xref="S4.SS2.p4.6.m6.1.1">𝑛</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p4.6.m6.1c">n</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p4.6.m6.1d">italic_n</annotation></semantics></math> and <math alttext="m_{i}" class="ltx_Math" display="inline" id="S4.SS2.p4.7.m7.1"><semantics id="S4.SS2.p4.7.m7.1a"><msub id="S4.SS2.p4.7.m7.1.1" xref="S4.SS2.p4.7.m7.1.1.cmml"><mi id="S4.SS2.p4.7.m7.1.1.2" xref="S4.SS2.p4.7.m7.1.1.2.cmml">m</mi><mi id="S4.SS2.p4.7.m7.1.1.3" xref="S4.SS2.p4.7.m7.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS2.p4.7.m7.1b"><apply id="S4.SS2.p4.7.m7.1.1.cmml" xref="S4.SS2.p4.7.m7.1.1"><csymbol cd="ambiguous" id="S4.SS2.p4.7.m7.1.1.1.cmml" xref="S4.SS2.p4.7.m7.1.1">subscript</csymbol><ci id="S4.SS2.p4.7.m7.1.1.2.cmml" xref="S4.SS2.p4.7.m7.1.1.2">𝑚</ci><ci id="S4.SS2.p4.7.m7.1.1.3.cmml" xref="S4.SS2.p4.7.m7.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p4.7.m7.1c">m_{i}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p4.7.m7.1d">italic_m start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math>, whereas both GEE models’ performance remained unaffected by increasing <math alttext="n" class="ltx_Math" display="inline" id="S4.SS2.p4.8.m8.1"><semantics id="S4.SS2.p4.8.m8.1a"><mi id="S4.SS2.p4.8.m8.1.1" xref="S4.SS2.p4.8.m8.1.1.cmml">n</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.p4.8.m8.1b"><ci id="S4.SS2.p4.8.m8.1.1.cmml" xref="S4.SS2.p4.8.m8.1.1">𝑛</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p4.8.m8.1c">n</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p4.8.m8.1d">italic_n</annotation></semantics></math> and <math alttext="m_{i}" class="ltx_Math" display="inline" id="S4.SS2.p4.9.m9.1"><semantics id="S4.SS2.p4.9.m9.1a"><msub id="S4.SS2.p4.9.m9.1.1" xref="S4.SS2.p4.9.m9.1.1.cmml"><mi id="S4.SS2.p4.9.m9.1.1.2" xref="S4.SS2.p4.9.m9.1.1.2.cmml">m</mi><mi id="S4.SS2.p4.9.m9.1.1.3" xref="S4.SS2.p4.9.m9.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS2.p4.9.m9.1b"><apply id="S4.SS2.p4.9.m9.1.1.cmml" xref="S4.SS2.p4.9.m9.1.1"><csymbol cd="ambiguous" id="S4.SS2.p4.9.m9.1.1.1.cmml" xref="S4.SS2.p4.9.m9.1.1">subscript</csymbol><ci id="S4.SS2.p4.9.m9.1.1.2.cmml" xref="S4.SS2.p4.9.m9.1.1.2">𝑚</ci><ci id="S4.SS2.p4.9.m9.1.1.3.cmml" xref="S4.SS2.p4.9.m9.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p4.9.m9.1c">m_{i}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p4.9.m9.1d">italic_m start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S4.SS2.p5"> <p class="ltx_p" id="S4.SS2.p5.14">Unlike its competitors, MMTR estimated <math alttext="\operatorname{{\mathbf{\Sigma}}}_{1}" class="ltx_Math" display="inline" id="S4.SS2.p5.1.m1.1"><semantics id="S4.SS2.p5.1.m1.1a"><msub id="S4.SS2.p5.1.m1.1.1" xref="S4.SS2.p5.1.m1.1.1.cmml"><mi id="S4.SS2.p5.1.m1.1.1.2" xref="S4.SS2.p5.1.m1.1.1.2.cmml">𝚺</mi><mn id="S4.SS2.p5.1.m1.1.1.3" xref="S4.SS2.p5.1.m1.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S4.SS2.p5.1.m1.1b"><apply id="S4.SS2.p5.1.m1.1.1.cmml" xref="S4.SS2.p5.1.m1.1.1"><csymbol cd="ambiguous" id="S4.SS2.p5.1.m1.1.1.1.cmml" xref="S4.SS2.p5.1.m1.1.1">subscript</csymbol><ci id="S4.SS2.p5.1.m1.1.1.2.cmml" xref="S4.SS2.p5.1.m1.1.1.2">𝚺</ci><cn id="S4.SS2.p5.1.m1.1.1.3.cmml" type="integer" xref="S4.SS2.p5.1.m1.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p5.1.m1.1c">\operatorname{{\mathbf{\Sigma}}}_{1}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p5.1.m1.1d">bold_Σ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="\operatorname{{\mathbf{\Sigma}}}_{2}" class="ltx_Math" display="inline" id="S4.SS2.p5.2.m2.1"><semantics id="S4.SS2.p5.2.m2.1a"><msub id="S4.SS2.p5.2.m2.1.1" xref="S4.SS2.p5.2.m2.1.1.cmml"><mi id="S4.SS2.p5.2.m2.1.1.2" xref="S4.SS2.p5.2.m2.1.1.2.cmml">𝚺</mi><mn id="S4.SS2.p5.2.m2.1.1.3" xref="S4.SS2.p5.2.m2.1.1.3.cmml">2</mn></msub><annotation-xml encoding="MathML-Content" id="S4.SS2.p5.2.m2.1b"><apply id="S4.SS2.p5.2.m2.1.1.cmml" xref="S4.SS2.p5.2.m2.1.1"><csymbol cd="ambiguous" id="S4.SS2.p5.2.m2.1.1.1.cmml" xref="S4.SS2.p5.2.m2.1.1">subscript</csymbol><ci id="S4.SS2.p5.2.m2.1.1.2.cmml" xref="S4.SS2.p5.2.m2.1.1.2">𝚺</ci><cn id="S4.SS2.p5.2.m2.1.1.3.cmml" type="integer" xref="S4.SS2.p5.2.m2.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p5.2.m2.1c">\operatorname{{\mathbf{\Sigma}}}_{2}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p5.2.m2.1d">bold_Σ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math> using Proposition <a class="ltx_ref" href="https://arxiv.org/html/2503.13782v1#S1.Thmproposition2" title="Proposition 1.2 ‣ 1.2 Proof of Proposition 2.2 ‣ 1 Proof of the Propositions ‣ Regularized Parameter Estimation in Mixed Model Trace Regression"><span class="ltx_text ltx_ref_tag">1.2</span></a> (Tables <a class="ltx_ref" href="https://arxiv.org/html/2503.13782v1#S4.T4" title="Table 4 ‣ 4.2 Simulation Using the MMTR Model ‣ 4 Experiments ‣ Regularized Parameter Estimation in Mixed Model Trace Regression"><span class="ltx_text ltx_ref_tag">4</span></a> and <a class="ltx_ref" href="https://arxiv.org/html/2503.13782v1#S4.T4" title="Table 4 ‣ 4.2 Simulation Using the MMTR Model ‣ 4 Experiments ‣ Regularized Parameter Estimation in Mixed Model Trace Regression"><span class="ltx_text ltx_ref_tag">4</span></a>). The results showed that <math alttext="m_{i}" class="ltx_Math" display="inline" id="S4.SS2.p5.3.m3.1"><semantics id="S4.SS2.p5.3.m3.1a"><msub id="S4.SS2.p5.3.m3.1.1" xref="S4.SS2.p5.3.m3.1.1.cmml"><mi id="S4.SS2.p5.3.m3.1.1.2" xref="S4.SS2.p5.3.m3.1.1.2.cmml">m</mi><mi id="S4.SS2.p5.3.m3.1.1.3" xref="S4.SS2.p5.3.m3.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS2.p5.3.m3.1b"><apply id="S4.SS2.p5.3.m3.1.1.cmml" xref="S4.SS2.p5.3.m3.1.1"><csymbol cd="ambiguous" id="S4.SS2.p5.3.m3.1.1.1.cmml" xref="S4.SS2.p5.3.m3.1.1">subscript</csymbol><ci id="S4.SS2.p5.3.m3.1.1.2.cmml" xref="S4.SS2.p5.3.m3.1.1.2">𝑚</ci><ci id="S4.SS2.p5.3.m3.1.1.3.cmml" xref="S4.SS2.p5.3.m3.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p5.3.m3.1c">m_{i}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p5.3.m3.1d">italic_m start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> had a greater impact on estimation accuracy of <math alttext="\operatorname{{\mathbf{\Sigma}}}_{1}" class="ltx_Math" display="inline" id="S4.SS2.p5.4.m4.1"><semantics id="S4.SS2.p5.4.m4.1a"><msub id="S4.SS2.p5.4.m4.1.1" xref="S4.SS2.p5.4.m4.1.1.cmml"><mi id="S4.SS2.p5.4.m4.1.1.2" xref="S4.SS2.p5.4.m4.1.1.2.cmml">𝚺</mi><mn id="S4.SS2.p5.4.m4.1.1.3" xref="S4.SS2.p5.4.m4.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S4.SS2.p5.4.m4.1b"><apply id="S4.SS2.p5.4.m4.1.1.cmml" xref="S4.SS2.p5.4.m4.1.1"><csymbol cd="ambiguous" id="S4.SS2.p5.4.m4.1.1.1.cmml" xref="S4.SS2.p5.4.m4.1.1">subscript</csymbol><ci id="S4.SS2.p5.4.m4.1.1.2.cmml" xref="S4.SS2.p5.4.m4.1.1.2">𝚺</ci><cn id="S4.SS2.p5.4.m4.1.1.3.cmml" type="integer" xref="S4.SS2.p5.4.m4.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p5.4.m4.1c">\operatorname{{\mathbf{\Sigma}}}_{1}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p5.4.m4.1d">bold_Σ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="\operatorname{{\mathbf{\Sigma}}}_{2}" class="ltx_Math" display="inline" id="S4.SS2.p5.5.m5.1"><semantics id="S4.SS2.p5.5.m5.1a"><msub id="S4.SS2.p5.5.m5.1.1" xref="S4.SS2.p5.5.m5.1.1.cmml"><mi id="S4.SS2.p5.5.m5.1.1.2" xref="S4.SS2.p5.5.m5.1.1.2.cmml">𝚺</mi><mn id="S4.SS2.p5.5.m5.1.1.3" xref="S4.SS2.p5.5.m5.1.1.3.cmml">2</mn></msub><annotation-xml encoding="MathML-Content" id="S4.SS2.p5.5.m5.1b"><apply id="S4.SS2.p5.5.m5.1.1.cmml" xref="S4.SS2.p5.5.m5.1.1"><csymbol cd="ambiguous" id="S4.SS2.p5.5.m5.1.1.1.cmml" xref="S4.SS2.p5.5.m5.1.1">subscript</csymbol><ci id="S4.SS2.p5.5.m5.1.1.2.cmml" xref="S4.SS2.p5.5.m5.1.1.2">𝚺</ci><cn id="S4.SS2.p5.5.m5.1.1.3.cmml" type="integer" xref="S4.SS2.p5.5.m5.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p5.5.m5.1c">\operatorname{{\mathbf{\Sigma}}}_{2}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p5.5.m5.1d">bold_Σ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math> than <math alttext="N" class="ltx_Math" display="inline" id="S4.SS2.p5.6.m6.1"><semantics id="S4.SS2.p5.6.m6.1a"><mi id="S4.SS2.p5.6.m6.1.1" xref="S4.SS2.p5.6.m6.1.1.cmml">N</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.p5.6.m6.1b"><ci id="S4.SS2.p5.6.m6.1.1.cmml" xref="S4.SS2.p5.6.m6.1.1">𝑁</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p5.6.m6.1c">N</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p5.6.m6.1d">italic_N</annotation></semantics></math>. Specifically, for a fixed <math alttext="N" class="ltx_Math" display="inline" id="S4.SS2.p5.7.m7.1"><semantics id="S4.SS2.p5.7.m7.1a"><mi id="S4.SS2.p5.7.m7.1.1" xref="S4.SS2.p5.7.m7.1.1.cmml">N</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.p5.7.m7.1b"><ci id="S4.SS2.p5.7.m7.1.1.cmml" xref="S4.SS2.p5.7.m7.1.1">𝑁</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p5.7.m7.1c">N</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p5.7.m7.1d">italic_N</annotation></semantics></math>, MMTR achieved higher accuracy when <math alttext="m_{i}=6" class="ltx_Math" display="inline" id="S4.SS2.p5.8.m8.1"><semantics id="S4.SS2.p5.8.m8.1a"><mrow id="S4.SS2.p5.8.m8.1.1" xref="S4.SS2.p5.8.m8.1.1.cmml"><msub id="S4.SS2.p5.8.m8.1.1.2" xref="S4.SS2.p5.8.m8.1.1.2.cmml"><mi id="S4.SS2.p5.8.m8.1.1.2.2" xref="S4.SS2.p5.8.m8.1.1.2.2.cmml">m</mi><mi id="S4.SS2.p5.8.m8.1.1.2.3" xref="S4.SS2.p5.8.m8.1.1.2.3.cmml">i</mi></msub><mo id="S4.SS2.p5.8.m8.1.1.1" xref="S4.SS2.p5.8.m8.1.1.1.cmml">=</mo><mn id="S4.SS2.p5.8.m8.1.1.3" xref="S4.SS2.p5.8.m8.1.1.3.cmml">6</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p5.8.m8.1b"><apply id="S4.SS2.p5.8.m8.1.1.cmml" xref="S4.SS2.p5.8.m8.1.1"><eq id="S4.SS2.p5.8.m8.1.1.1.cmml" xref="S4.SS2.p5.8.m8.1.1.1"></eq><apply id="S4.SS2.p5.8.m8.1.1.2.cmml" xref="S4.SS2.p5.8.m8.1.1.2"><csymbol cd="ambiguous" id="S4.SS2.p5.8.m8.1.1.2.1.cmml" xref="S4.SS2.p5.8.m8.1.1.2">subscript</csymbol><ci id="S4.SS2.p5.8.m8.1.1.2.2.cmml" xref="S4.SS2.p5.8.m8.1.1.2.2">𝑚</ci><ci id="S4.SS2.p5.8.m8.1.1.2.3.cmml" xref="S4.SS2.p5.8.m8.1.1.2.3">𝑖</ci></apply><cn id="S4.SS2.p5.8.m8.1.1.3.cmml" type="integer" xref="S4.SS2.p5.8.m8.1.1.3">6</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p5.8.m8.1c">m_{i}=6</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p5.8.m8.1d">italic_m start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = 6</annotation></semantics></math> than when <math alttext="m_{i}=2" class="ltx_Math" display="inline" id="S4.SS2.p5.9.m9.1"><semantics id="S4.SS2.p5.9.m9.1a"><mrow id="S4.SS2.p5.9.m9.1.1" xref="S4.SS2.p5.9.m9.1.1.cmml"><msub id="S4.SS2.p5.9.m9.1.1.2" xref="S4.SS2.p5.9.m9.1.1.2.cmml"><mi id="S4.SS2.p5.9.m9.1.1.2.2" xref="S4.SS2.p5.9.m9.1.1.2.2.cmml">m</mi><mi id="S4.SS2.p5.9.m9.1.1.2.3" xref="S4.SS2.p5.9.m9.1.1.2.3.cmml">i</mi></msub><mo id="S4.SS2.p5.9.m9.1.1.1" xref="S4.SS2.p5.9.m9.1.1.1.cmml">=</mo><mn id="S4.SS2.p5.9.m9.1.1.3" xref="S4.SS2.p5.9.m9.1.1.3.cmml">2</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p5.9.m9.1b"><apply id="S4.SS2.p5.9.m9.1.1.cmml" xref="S4.SS2.p5.9.m9.1.1"><eq id="S4.SS2.p5.9.m9.1.1.1.cmml" xref="S4.SS2.p5.9.m9.1.1.1"></eq><apply id="S4.SS2.p5.9.m9.1.1.2.cmml" xref="S4.SS2.p5.9.m9.1.1.2"><csymbol cd="ambiguous" id="S4.SS2.p5.9.m9.1.1.2.1.cmml" xref="S4.SS2.p5.9.m9.1.1.2">subscript</csymbol><ci id="S4.SS2.p5.9.m9.1.1.2.2.cmml" xref="S4.SS2.p5.9.m9.1.1.2.2">𝑚</ci><ci id="S4.SS2.p5.9.m9.1.1.2.3.cmml" xref="S4.SS2.p5.9.m9.1.1.2.3">𝑖</ci></apply><cn id="S4.SS2.p5.9.m9.1.1.3.cmml" type="integer" xref="S4.SS2.p5.9.m9.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p5.9.m9.1c">m_{i}=2</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p5.9.m9.1d">italic_m start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = 2</annotation></semantics></math>; see the blue and red highlighted rows in Tables <a class="ltx_ref" href="https://arxiv.org/html/2503.13782v1#S4.T4" title="Table 4 ‣ 4.2 Simulation Using the MMTR Model ‣ 4 Experiments ‣ Regularized Parameter Estimation in Mixed Model Trace Regression"><span class="ltx_text ltx_ref_tag">4</span></a> and <a class="ltx_ref" href="https://arxiv.org/html/2503.13782v1#S4.T4" title="Table 4 ‣ 4.2 Simulation Using the MMTR Model ‣ 4 Experiments ‣ Regularized Parameter Estimation in Mixed Model Trace Regression"><span class="ltx_text ltx_ref_tag">4</span></a>. In summary, while all the methods had similar accuracy in estimating <math alttext="\operatorname{\mathbf{B}}" class="ltx_Math" display="inline" id="S4.SS2.p5.10.m10.1"><semantics id="S4.SS2.p5.10.m10.1a"><mi id="S4.SS2.p5.10.m10.1.1" xref="S4.SS2.p5.10.m10.1.1.cmml">𝐁</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.p5.10.m10.1b"><ci id="S4.SS2.p5.10.m10.1.1.cmml" xref="S4.SS2.p5.10.m10.1.1">𝐁</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p5.10.m10.1c">\operatorname{\mathbf{B}}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p5.10.m10.1d">bold_B</annotation></semantics></math>, MMTR outperformed its competitors in accurately estimating <math alttext="\operatorname{{\mathbf{\Sigma}}}_{1},\operatorname{{\mathbf{\Sigma}}}_{2}" class="ltx_Math" display="inline" id="S4.SS2.p5.11.m11.2"><semantics id="S4.SS2.p5.11.m11.2a"><mrow id="S4.SS2.p5.11.m11.2.2.2" xref="S4.SS2.p5.11.m11.2.2.3.cmml"><msub id="S4.SS2.p5.11.m11.1.1.1.1" xref="S4.SS2.p5.11.m11.1.1.1.1.cmml"><mi id="S4.SS2.p5.11.m11.1.1.1.1.2" xref="S4.SS2.p5.11.m11.1.1.1.1.2.cmml">𝚺</mi><mn id="S4.SS2.p5.11.m11.1.1.1.1.3" xref="S4.SS2.p5.11.m11.1.1.1.1.3.cmml">1</mn></msub><mo id="S4.SS2.p5.11.m11.2.2.2.3" xref="S4.SS2.p5.11.m11.2.2.3.cmml">,</mo><msub id="S4.SS2.p5.11.m11.2.2.2.2" xref="S4.SS2.p5.11.m11.2.2.2.2.cmml"><mi id="S4.SS2.p5.11.m11.2.2.2.2.2" xref="S4.SS2.p5.11.m11.2.2.2.2.2.cmml">𝚺</mi><mn id="S4.SS2.p5.11.m11.2.2.2.2.3" xref="S4.SS2.p5.11.m11.2.2.2.2.3.cmml">2</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p5.11.m11.2b"><list id="S4.SS2.p5.11.m11.2.2.3.cmml" xref="S4.SS2.p5.11.m11.2.2.2"><apply id="S4.SS2.p5.11.m11.1.1.1.1.cmml" xref="S4.SS2.p5.11.m11.1.1.1.1"><csymbol cd="ambiguous" id="S4.SS2.p5.11.m11.1.1.1.1.1.cmml" xref="S4.SS2.p5.11.m11.1.1.1.1">subscript</csymbol><ci id="S4.SS2.p5.11.m11.1.1.1.1.2.cmml" xref="S4.SS2.p5.11.m11.1.1.1.1.2">𝚺</ci><cn id="S4.SS2.p5.11.m11.1.1.1.1.3.cmml" type="integer" xref="S4.SS2.p5.11.m11.1.1.1.1.3">1</cn></apply><apply id="S4.SS2.p5.11.m11.2.2.2.2.cmml" xref="S4.SS2.p5.11.m11.2.2.2.2"><csymbol cd="ambiguous" id="S4.SS2.p5.11.m11.2.2.2.2.1.cmml" xref="S4.SS2.p5.11.m11.2.2.2.2">subscript</csymbol><ci id="S4.SS2.p5.11.m11.2.2.2.2.2.cmml" xref="S4.SS2.p5.11.m11.2.2.2.2.2">𝚺</ci><cn id="S4.SS2.p5.11.m11.2.2.2.2.3.cmml" type="integer" xref="S4.SS2.p5.11.m11.2.2.2.2.3">2</cn></apply></list></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p5.11.m11.2c">\operatorname{{\mathbf{\Sigma}}}_{1},\operatorname{{\mathbf{\Sigma}}}_{2}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p5.11.m11.2d">bold_Σ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , bold_Σ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math>, and <math alttext="\operatorname{{\mathbf{\Lambda}}}" class="ltx_Math" display="inline" id="S4.SS2.p5.12.m12.1"><semantics id="S4.SS2.p5.12.m12.1a"><mi id="S4.SS2.p5.12.m12.1.1" xref="S4.SS2.p5.12.m12.1.1.cmml">𝚲</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.p5.12.m12.1b"><ci id="S4.SS2.p5.12.m12.1.1.cmml" xref="S4.SS2.p5.12.m12.1.1">𝚲</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p5.12.m12.1c">\operatorname{{\mathbf{\Lambda}}}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p5.12.m12.1d">bold_Λ</annotation></semantics></math>, particularly for large <math alttext="n" class="ltx_Math" display="inline" id="S4.SS2.p5.13.m13.1"><semantics id="S4.SS2.p5.13.m13.1a"><mi id="S4.SS2.p5.13.m13.1.1" xref="S4.SS2.p5.13.m13.1.1.cmml">n</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.p5.13.m13.1b"><ci id="S4.SS2.p5.13.m13.1.1.cmml" xref="S4.SS2.p5.13.m13.1.1">𝑛</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p5.13.m13.1c">n</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p5.13.m13.1d">italic_n</annotation></semantics></math> and <math alttext="m_{i}" class="ltx_Math" display="inline" id="S4.SS2.p5.14.m14.1"><semantics id="S4.SS2.p5.14.m14.1a"><msub id="S4.SS2.p5.14.m14.1.1" xref="S4.SS2.p5.14.m14.1.1.cmml"><mi id="S4.SS2.p5.14.m14.1.1.2" xref="S4.SS2.p5.14.m14.1.1.2.cmml">m</mi><mi id="S4.SS2.p5.14.m14.1.1.3" xref="S4.SS2.p5.14.m14.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS2.p5.14.m14.1b"><apply id="S4.SS2.p5.14.m14.1.1.cmml" xref="S4.SS2.p5.14.m14.1.1"><csymbol cd="ambiguous" id="S4.SS2.p5.14.m14.1.1.1.cmml" xref="S4.SS2.p5.14.m14.1.1">subscript</csymbol><ci id="S4.SS2.p5.14.m14.1.1.2.cmml" xref="S4.SS2.p5.14.m14.1.1.2">𝑚</ci><ci id="S4.SS2.p5.14.m14.1.1.3.cmml" xref="S4.SS2.p5.14.m14.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p5.14.m14.1c">m_{i}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p5.14.m14.1d">italic_m start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math>.</p> </div> <figure class="ltx_table" id="S4.T1"> <figcaption class="ltx_caption"><span class="ltx_tag ltx_tag_table">Table 1: </span> Relative estimation error in Case 1. The simulation follows MMTR in (<a class="ltx_ref" href="https://arxiv.org/html/2503.13782v1#S2.E3" title="In 2.1 Model ‣ 2 Mixed Model Trace Regression ‣ Regularized Parameter Estimation in Mixed Model Trace Regression"><span class="ltx_text ltx_ref_tag">3</span></a>). Every estimation error is averaged over 100 simulation replications, with the Monte Carlo errors in parenthesis. A small relative error indicates high estimation accuracy. GEE un. and GEE eq. denote the GEE models with unstructured and equicorrelated covariance matrix. Kruskal and Quasi denote the sparse Kruskal and quasi-likelihood methods.</figcaption> <div class="ltx_inline-block ltx_align_center ltx_transformed_outer" id="S4.T1.10" style="width:505.9pt;height:207.2pt;vertical-align:-0.0pt;"><span class="ltx_transformed_inner" style="transform:translate(-54.7pt,22.4pt) scale(0.822175034223395,0.822175034223395) ;"> <table class="ltx_tabular ltx_guessed_headers ltx_align_top" id="S4.T1.10.10"> <tbody class="ltx_tbody"> <tr class="ltx_tr" id="S4.T1.2.2.2"> <th class="ltx_td ltx_align_right ltx_th ltx_th_row ltx_border_l ltx_border_r ltx_border_t" id="S4.T1.2.2.2.3">group size</th> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" colspan="5" id="S4.T1.1.1.1.1">err<math alttext="{}_{\operatorname{\mathbf{B}}}" class="ltx_Math" display="inline" id="S4.T1.1.1.1.1.m1.1"><semantics id="S4.T1.1.1.1.1.m1.1a"><msub id="S4.T1.1.1.1.1.m1.1.1" xref="S4.T1.1.1.1.1.m1.1.1.cmml"><mi id="S4.T1.1.1.1.1.m1.1.1a" xref="S4.T1.1.1.1.1.m1.1.1.cmml"></mi><mi id="S4.T1.1.1.1.1.m1.1.1.1" xref="S4.T1.1.1.1.1.m1.1.1.1.cmml">𝐁</mi></msub><annotation-xml encoding="MathML-Content" id="S4.T1.1.1.1.1.m1.1b"><apply id="S4.T1.1.1.1.1.m1.1.1.cmml" xref="S4.T1.1.1.1.1.m1.1.1"><ci id="S4.T1.1.1.1.1.m1.1.1.1.cmml" xref="S4.T1.1.1.1.1.m1.1.1.1">𝐁</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.T1.1.1.1.1.m1.1c">{}_{\operatorname{\mathbf{B}}}</annotation><annotation encoding="application/x-llamapun" id="S4.T1.1.1.1.1.m1.1d">start_FLOATSUBSCRIPT bold_B end_FLOATSUBSCRIPT</annotation></semantics></math> </td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" colspan="3" id="S4.T1.2.2.2.2">err<math alttext="{}_{\operatorname{{\mathbf{\Lambda}}}}" class="ltx_Math" display="inline" id="S4.T1.2.2.2.2.m1.1"><semantics id="S4.T1.2.2.2.2.m1.1a"><msub id="S4.T1.2.2.2.2.m1.1.1" xref="S4.T1.2.2.2.2.m1.1.1.cmml"><mi id="S4.T1.2.2.2.2.m1.1.1a" xref="S4.T1.2.2.2.2.m1.1.1.cmml"></mi><mi id="S4.T1.2.2.2.2.m1.1.1.1" xref="S4.T1.2.2.2.2.m1.1.1.1.cmml">𝚲</mi></msub><annotation-xml encoding="MathML-Content" id="S4.T1.2.2.2.2.m1.1b"><apply id="S4.T1.2.2.2.2.m1.1.1.cmml" xref="S4.T1.2.2.2.2.m1.1.1"><ci id="S4.T1.2.2.2.2.m1.1.1.1.cmml" xref="S4.T1.2.2.2.2.m1.1.1.1">𝚲</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.T1.2.2.2.2.m1.1c">{}_{\operatorname{{\mathbf{\Lambda}}}}</annotation><annotation encoding="application/x-llamapun" id="S4.T1.2.2.2.2.m1.1d">start_FLOATSUBSCRIPT bold_Λ end_FLOATSUBSCRIPT</annotation></semantics></math> </td> </tr> <tr class="ltx_tr" id="S4.T1.10.10.11.1"> <th class="ltx_td ltx_th ltx_th_row ltx_border_l ltx_border_r ltx_border_t" id="S4.T1.10.10.11.1.1"></th> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T1.10.10.11.1.2">MMTR</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T1.10.10.11.1.3">Kruskal</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T1.10.10.11.1.4">GEE un.</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T1.10.10.11.1.5">GEE eq.</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T1.10.10.11.1.6">Quasi</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T1.10.10.11.1.7">MMTR</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T1.10.10.11.1.8">GEE un.</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T1.10.10.11.1.9">GEE eq.</td> </tr> <tr class="ltx_tr" id="S4.T1.10.10.12.2"> <th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l ltx_border_r ltx_border_t" colspan="9" id="S4.T1.10.10.12.2.1"><span class="ltx_text ltx_font_bold" id="S4.T1.10.10.12.2.1.1">n = 18</span></th> </tr> <tr class="ltx_tr" id="S4.T1.3.3.3"> <th class="ltx_td ltx_align_right ltx_th ltx_th_row ltx_border_l ltx_border_r ltx_border_t" id="S4.T1.3.3.3.1"> <math alttext="m_{i}=2" class="ltx_Math" display="inline" id="S4.T1.3.3.3.1.m1.1"><semantics id="S4.T1.3.3.3.1.m1.1a"><mrow id="S4.T1.3.3.3.1.m1.1.1" xref="S4.T1.3.3.3.1.m1.1.1.cmml"><msub id="S4.T1.3.3.3.1.m1.1.1.2" xref="S4.T1.3.3.3.1.m1.1.1.2.cmml"><mi id="S4.T1.3.3.3.1.m1.1.1.2.2" xref="S4.T1.3.3.3.1.m1.1.1.2.2.cmml">m</mi><mi id="S4.T1.3.3.3.1.m1.1.1.2.3" xref="S4.T1.3.3.3.1.m1.1.1.2.3.cmml">i</mi></msub><mo id="S4.T1.3.3.3.1.m1.1.1.1" xref="S4.T1.3.3.3.1.m1.1.1.1.cmml">=</mo><mn id="S4.T1.3.3.3.1.m1.1.1.3" xref="S4.T1.3.3.3.1.m1.1.1.3.cmml">2</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.T1.3.3.3.1.m1.1b"><apply id="S4.T1.3.3.3.1.m1.1.1.cmml" xref="S4.T1.3.3.3.1.m1.1.1"><eq id="S4.T1.3.3.3.1.m1.1.1.1.cmml" xref="S4.T1.3.3.3.1.m1.1.1.1"></eq><apply id="S4.T1.3.3.3.1.m1.1.1.2.cmml" xref="S4.T1.3.3.3.1.m1.1.1.2"><csymbol cd="ambiguous" id="S4.T1.3.3.3.1.m1.1.1.2.1.cmml" xref="S4.T1.3.3.3.1.m1.1.1.2">subscript</csymbol><ci id="S4.T1.3.3.3.1.m1.1.1.2.2.cmml" xref="S4.T1.3.3.3.1.m1.1.1.2.2">𝑚</ci><ci id="S4.T1.3.3.3.1.m1.1.1.2.3.cmml" xref="S4.T1.3.3.3.1.m1.1.1.2.3">𝑖</ci></apply><cn id="S4.T1.3.3.3.1.m1.1.1.3.cmml" type="integer" xref="S4.T1.3.3.3.1.m1.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.T1.3.3.3.1.m1.1c">m_{i}=2</annotation><annotation encoding="application/x-llamapun" id="S4.T1.3.3.3.1.m1.1d">italic_m start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = 2</annotation></semantics></math> </th> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T1.3.3.3.2">0.130 (0.0440)</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T1.3.3.3.3">0.130 (0.0500)</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T1.3.3.3.4">0.170 (0.0650)</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T1.3.3.3.5">0.110 (0.0360)</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T1.3.3.3.6">0.620 (0.2000)</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T1.3.3.3.7">0.91 (0.052)</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T1.3.3.3.8">0.95 (0.094)</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T1.3.3.3.9">0.86 (0.052)</td> </tr> <tr class="ltx_tr" id="S4.T1.4.4.4"> <th class="ltx_td ltx_align_right ltx_th ltx_th_row ltx_border_l ltx_border_r ltx_border_t" id="S4.T1.4.4.4.1"> <math alttext="m_{i}=6" class="ltx_Math" display="inline" id="S4.T1.4.4.4.1.m1.1"><semantics id="S4.T1.4.4.4.1.m1.1a"><mrow id="S4.T1.4.4.4.1.m1.1.1" xref="S4.T1.4.4.4.1.m1.1.1.cmml"><msub id="S4.T1.4.4.4.1.m1.1.1.2" xref="S4.T1.4.4.4.1.m1.1.1.2.cmml"><mi id="S4.T1.4.4.4.1.m1.1.1.2.2" xref="S4.T1.4.4.4.1.m1.1.1.2.2.cmml">m</mi><mi id="S4.T1.4.4.4.1.m1.1.1.2.3" xref="S4.T1.4.4.4.1.m1.1.1.2.3.cmml">i</mi></msub><mo id="S4.T1.4.4.4.1.m1.1.1.1" xref="S4.T1.4.4.4.1.m1.1.1.1.cmml">=</mo><mn id="S4.T1.4.4.4.1.m1.1.1.3" xref="S4.T1.4.4.4.1.m1.1.1.3.cmml">6</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.T1.4.4.4.1.m1.1b"><apply id="S4.T1.4.4.4.1.m1.1.1.cmml" xref="S4.T1.4.4.4.1.m1.1.1"><eq id="S4.T1.4.4.4.1.m1.1.1.1.cmml" xref="S4.T1.4.4.4.1.m1.1.1.1"></eq><apply id="S4.T1.4.4.4.1.m1.1.1.2.cmml" xref="S4.T1.4.4.4.1.m1.1.1.2"><csymbol cd="ambiguous" id="S4.T1.4.4.4.1.m1.1.1.2.1.cmml" xref="S4.T1.4.4.4.1.m1.1.1.2">subscript</csymbol><ci id="S4.T1.4.4.4.1.m1.1.1.2.2.cmml" xref="S4.T1.4.4.4.1.m1.1.1.2.2">𝑚</ci><ci id="S4.T1.4.4.4.1.m1.1.1.2.3.cmml" xref="S4.T1.4.4.4.1.m1.1.1.2.3">𝑖</ci></apply><cn id="S4.T1.4.4.4.1.m1.1.1.3.cmml" type="integer" xref="S4.T1.4.4.4.1.m1.1.1.3">6</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.T1.4.4.4.1.m1.1c">m_{i}=6</annotation><annotation encoding="application/x-llamapun" id="S4.T1.4.4.4.1.m1.1d">italic_m start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = 6</annotation></semantics></math> </th> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T1.4.4.4.2">0.044 (0.0100)</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T1.4.4.4.3">0.051 (0.0120)</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T1.4.4.4.4">0.082 (0.0340)</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T1.4.4.4.5">0.047 (0.0120)</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T1.4.4.4.6">0.110 (0.0310)</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T1.4.4.4.7">0.74 (0.330)</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T1.4.4.4.8">0.93 (0.027)</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T1.4.4.4.9">0.91 (0.037)</td> </tr> <tr class="ltx_tr" id="S4.T1.10.10.13.3"> <th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l ltx_border_r ltx_border_t" colspan="9" id="S4.T1.10.10.13.3.1"><span class="ltx_text ltx_font_bold" id="S4.T1.10.10.13.3.1.1">n = 27</span></th> </tr> <tr class="ltx_tr" id="S4.T1.5.5.5"> <th class="ltx_td ltx_align_right ltx_th ltx_th_row ltx_border_l ltx_border_r ltx_border_t" id="S4.T1.5.5.5.1"> <math alttext="m_{i}=2" class="ltx_Math" display="inline" id="S4.T1.5.5.5.1.m1.1"><semantics id="S4.T1.5.5.5.1.m1.1a"><mrow id="S4.T1.5.5.5.1.m1.1.1" xref="S4.T1.5.5.5.1.m1.1.1.cmml"><msub id="S4.T1.5.5.5.1.m1.1.1.2" xref="S4.T1.5.5.5.1.m1.1.1.2.cmml"><mi id="S4.T1.5.5.5.1.m1.1.1.2.2" xref="S4.T1.5.5.5.1.m1.1.1.2.2.cmml">m</mi><mi id="S4.T1.5.5.5.1.m1.1.1.2.3" xref="S4.T1.5.5.5.1.m1.1.1.2.3.cmml">i</mi></msub><mo id="S4.T1.5.5.5.1.m1.1.1.1" xref="S4.T1.5.5.5.1.m1.1.1.1.cmml">=</mo><mn id="S4.T1.5.5.5.1.m1.1.1.3" xref="S4.T1.5.5.5.1.m1.1.1.3.cmml">2</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.T1.5.5.5.1.m1.1b"><apply id="S4.T1.5.5.5.1.m1.1.1.cmml" xref="S4.T1.5.5.5.1.m1.1.1"><eq id="S4.T1.5.5.5.1.m1.1.1.1.cmml" xref="S4.T1.5.5.5.1.m1.1.1.1"></eq><apply id="S4.T1.5.5.5.1.m1.1.1.2.cmml" xref="S4.T1.5.5.5.1.m1.1.1.2"><csymbol cd="ambiguous" id="S4.T1.5.5.5.1.m1.1.1.2.1.cmml" xref="S4.T1.5.5.5.1.m1.1.1.2">subscript</csymbol><ci id="S4.T1.5.5.5.1.m1.1.1.2.2.cmml" xref="S4.T1.5.5.5.1.m1.1.1.2.2">𝑚</ci><ci id="S4.T1.5.5.5.1.m1.1.1.2.3.cmml" xref="S4.T1.5.5.5.1.m1.1.1.2.3">𝑖</ci></apply><cn id="S4.T1.5.5.5.1.m1.1.1.3.cmml" type="integer" xref="S4.T1.5.5.5.1.m1.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.T1.5.5.5.1.m1.1c">m_{i}=2</annotation><annotation encoding="application/x-llamapun" id="S4.T1.5.5.5.1.m1.1d">italic_m start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = 2</annotation></semantics></math> </th> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T1.5.5.5.2">0.099 (0.0290)</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T1.5.5.5.3">0.077 (0.0250)</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T1.5.5.5.4">0.087 (0.0430)</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T1.5.5.5.5">0.073 (0.0230)</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T1.5.5.5.6">0.310 (0.1500)</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T1.5.5.5.7">0.99 (0.250)</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T1.5.5.5.8">0.81 (0.160)</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T1.5.5.5.9">0.86 (0.055)</td> </tr> <tr class="ltx_tr" id="S4.T1.6.6.6"> <th class="ltx_td ltx_align_right ltx_th ltx_th_row ltx_border_l ltx_border_r ltx_border_t" id="S4.T1.6.6.6.1"> <math alttext="m_{i}=6" class="ltx_Math" display="inline" id="S4.T1.6.6.6.1.m1.1"><semantics id="S4.T1.6.6.6.1.m1.1a"><mrow id="S4.T1.6.6.6.1.m1.1.1" xref="S4.T1.6.6.6.1.m1.1.1.cmml"><msub id="S4.T1.6.6.6.1.m1.1.1.2" xref="S4.T1.6.6.6.1.m1.1.1.2.cmml"><mi id="S4.T1.6.6.6.1.m1.1.1.2.2" xref="S4.T1.6.6.6.1.m1.1.1.2.2.cmml">m</mi><mi id="S4.T1.6.6.6.1.m1.1.1.2.3" xref="S4.T1.6.6.6.1.m1.1.1.2.3.cmml">i</mi></msub><mo id="S4.T1.6.6.6.1.m1.1.1.1" xref="S4.T1.6.6.6.1.m1.1.1.1.cmml">=</mo><mn id="S4.T1.6.6.6.1.m1.1.1.3" xref="S4.T1.6.6.6.1.m1.1.1.3.cmml">6</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.T1.6.6.6.1.m1.1b"><apply id="S4.T1.6.6.6.1.m1.1.1.cmml" xref="S4.T1.6.6.6.1.m1.1.1"><eq id="S4.T1.6.6.6.1.m1.1.1.1.cmml" xref="S4.T1.6.6.6.1.m1.1.1.1"></eq><apply id="S4.T1.6.6.6.1.m1.1.1.2.cmml" xref="S4.T1.6.6.6.1.m1.1.1.2"><csymbol cd="ambiguous" id="S4.T1.6.6.6.1.m1.1.1.2.1.cmml" xref="S4.T1.6.6.6.1.m1.1.1.2">subscript</csymbol><ci id="S4.T1.6.6.6.1.m1.1.1.2.2.cmml" xref="S4.T1.6.6.6.1.m1.1.1.2.2">𝑚</ci><ci id="S4.T1.6.6.6.1.m1.1.1.2.3.cmml" xref="S4.T1.6.6.6.1.m1.1.1.2.3">𝑖</ci></apply><cn id="S4.T1.6.6.6.1.m1.1.1.3.cmml" type="integer" xref="S4.T1.6.6.6.1.m1.1.1.3">6</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.T1.6.6.6.1.m1.1c">m_{i}=6</annotation><annotation encoding="application/x-llamapun" id="S4.T1.6.6.6.1.m1.1d">italic_m start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = 6</annotation></semantics></math> </th> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T1.6.6.6.2">0.028 (0.0076)</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T1.6.6.6.3">0.037 (0.0100)</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T1.6.6.6.4">0.043 (0.0120)</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T1.6.6.6.5">0.035 (0.0097)</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T1.6.6.6.6">0.083 (0.0220)</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T1.6.6.6.7">0.49 (0.150)</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T1.6.6.6.8">1.20 (0.470)</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T1.6.6.6.9">0.91 (0.036)</td> </tr> <tr class="ltx_tr" id="S4.T1.10.10.14.4"> <th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l ltx_border_r ltx_border_t" colspan="9" id="S4.T1.10.10.14.4.1"><span class="ltx_text ltx_font_bold" id="S4.T1.10.10.14.4.1.1">n = 54</span></th> </tr> <tr class="ltx_tr" id="S4.T1.7.7.7"> <th class="ltx_td ltx_align_right ltx_th ltx_th_row ltx_border_l ltx_border_r ltx_border_t" id="S4.T1.7.7.7.1"> <math alttext="m_{i}=2" class="ltx_Math" display="inline" id="S4.T1.7.7.7.1.m1.1"><semantics id="S4.T1.7.7.7.1.m1.1a"><mrow id="S4.T1.7.7.7.1.m1.1.1" xref="S4.T1.7.7.7.1.m1.1.1.cmml"><msub id="S4.T1.7.7.7.1.m1.1.1.2" xref="S4.T1.7.7.7.1.m1.1.1.2.cmml"><mi id="S4.T1.7.7.7.1.m1.1.1.2.2" xref="S4.T1.7.7.7.1.m1.1.1.2.2.cmml">m</mi><mi id="S4.T1.7.7.7.1.m1.1.1.2.3" xref="S4.T1.7.7.7.1.m1.1.1.2.3.cmml">i</mi></msub><mo id="S4.T1.7.7.7.1.m1.1.1.1" xref="S4.T1.7.7.7.1.m1.1.1.1.cmml">=</mo><mn id="S4.T1.7.7.7.1.m1.1.1.3" xref="S4.T1.7.7.7.1.m1.1.1.3.cmml">2</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.T1.7.7.7.1.m1.1b"><apply id="S4.T1.7.7.7.1.m1.1.1.cmml" xref="S4.T1.7.7.7.1.m1.1.1"><eq id="S4.T1.7.7.7.1.m1.1.1.1.cmml" xref="S4.T1.7.7.7.1.m1.1.1.1"></eq><apply id="S4.T1.7.7.7.1.m1.1.1.2.cmml" xref="S4.T1.7.7.7.1.m1.1.1.2"><csymbol cd="ambiguous" id="S4.T1.7.7.7.1.m1.1.1.2.1.cmml" xref="S4.T1.7.7.7.1.m1.1.1.2">subscript</csymbol><ci id="S4.T1.7.7.7.1.m1.1.1.2.2.cmml" xref="S4.T1.7.7.7.1.m1.1.1.2.2">𝑚</ci><ci id="S4.T1.7.7.7.1.m1.1.1.2.3.cmml" xref="S4.T1.7.7.7.1.m1.1.1.2.3">𝑖</ci></apply><cn id="S4.T1.7.7.7.1.m1.1.1.3.cmml" type="integer" xref="S4.T1.7.7.7.1.m1.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.T1.7.7.7.1.m1.1c">m_{i}=2</annotation><annotation encoding="application/x-llamapun" id="S4.T1.7.7.7.1.m1.1d">italic_m start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = 2</annotation></semantics></math> </th> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T1.7.7.7.2">0.051 (0.0110)</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T1.7.7.7.3">0.048 (0.0110)</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T1.7.7.7.4">0.046 (0.0120)</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T1.7.7.7.5">0.045 (0.0120)</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T1.7.7.7.6">0.110 (0.0280)</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T1.7.7.7.7">0.92 (0.290)</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T1.7.7.7.8">0.81 (0.071)</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T1.7.7.7.9">0.85 (0.056)</td> </tr> <tr class="ltx_tr" id="S4.T1.8.8.8"> <th class="ltx_td ltx_align_right ltx_th ltx_th_row ltx_border_l ltx_border_r ltx_border_t" id="S4.T1.8.8.8.1"> <math alttext="m_{i}=6" class="ltx_Math" display="inline" id="S4.T1.8.8.8.1.m1.1"><semantics id="S4.T1.8.8.8.1.m1.1a"><mrow id="S4.T1.8.8.8.1.m1.1.1" xref="S4.T1.8.8.8.1.m1.1.1.cmml"><msub id="S4.T1.8.8.8.1.m1.1.1.2" xref="S4.T1.8.8.8.1.m1.1.1.2.cmml"><mi id="S4.T1.8.8.8.1.m1.1.1.2.2" xref="S4.T1.8.8.8.1.m1.1.1.2.2.cmml">m</mi><mi id="S4.T1.8.8.8.1.m1.1.1.2.3" xref="S4.T1.8.8.8.1.m1.1.1.2.3.cmml">i</mi></msub><mo id="S4.T1.8.8.8.1.m1.1.1.1" xref="S4.T1.8.8.8.1.m1.1.1.1.cmml">=</mo><mn id="S4.T1.8.8.8.1.m1.1.1.3" xref="S4.T1.8.8.8.1.m1.1.1.3.cmml">6</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.T1.8.8.8.1.m1.1b"><apply id="S4.T1.8.8.8.1.m1.1.1.cmml" xref="S4.T1.8.8.8.1.m1.1.1"><eq id="S4.T1.8.8.8.1.m1.1.1.1.cmml" xref="S4.T1.8.8.8.1.m1.1.1.1"></eq><apply id="S4.T1.8.8.8.1.m1.1.1.2.cmml" xref="S4.T1.8.8.8.1.m1.1.1.2"><csymbol cd="ambiguous" id="S4.T1.8.8.8.1.m1.1.1.2.1.cmml" xref="S4.T1.8.8.8.1.m1.1.1.2">subscript</csymbol><ci id="S4.T1.8.8.8.1.m1.1.1.2.2.cmml" xref="S4.T1.8.8.8.1.m1.1.1.2.2">𝑚</ci><ci id="S4.T1.8.8.8.1.m1.1.1.2.3.cmml" xref="S4.T1.8.8.8.1.m1.1.1.2.3">𝑖</ci></apply><cn id="S4.T1.8.8.8.1.m1.1.1.3.cmml" type="integer" xref="S4.T1.8.8.8.1.m1.1.1.3">6</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.T1.8.8.8.1.m1.1c">m_{i}=6</annotation><annotation encoding="application/x-llamapun" id="S4.T1.8.8.8.1.m1.1d">italic_m start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = 6</annotation></semantics></math> </th> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T1.8.8.8.2">0.016 (0.0040)</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T1.8.8.8.3">0.026 (0.0060)</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T1.8.8.8.4">0.026 (0.0066)</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T1.8.8.8.5">0.024 (0.0058)</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T1.8.8.8.6">0.052 (0.0110)</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T1.8.8.8.7">0.29 (0.090)</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T1.8.8.8.8">0.88 (0.046)</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T1.8.8.8.9">0.91 (0.037)</td> </tr> <tr class="ltx_tr" id="S4.T1.10.10.15.5"> <th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l ltx_border_r ltx_border_t" colspan="9" id="S4.T1.10.10.15.5.1"><span class="ltx_text ltx_font_bold" id="S4.T1.10.10.15.5.1.1">n = 81</span></th> </tr> <tr class="ltx_tr" id="S4.T1.9.9.9"> <th class="ltx_td ltx_align_right ltx_th ltx_th_row ltx_border_l ltx_border_r ltx_border_t" id="S4.T1.9.9.9.1"> <math alttext="m_{i}=2" class="ltx_Math" display="inline" id="S4.T1.9.9.9.1.m1.1"><semantics id="S4.T1.9.9.9.1.m1.1a"><mrow id="S4.T1.9.9.9.1.m1.1.1" xref="S4.T1.9.9.9.1.m1.1.1.cmml"><msub id="S4.T1.9.9.9.1.m1.1.1.2" xref="S4.T1.9.9.9.1.m1.1.1.2.cmml"><mi id="S4.T1.9.9.9.1.m1.1.1.2.2" xref="S4.T1.9.9.9.1.m1.1.1.2.2.cmml">m</mi><mi id="S4.T1.9.9.9.1.m1.1.1.2.3" xref="S4.T1.9.9.9.1.m1.1.1.2.3.cmml">i</mi></msub><mo id="S4.T1.9.9.9.1.m1.1.1.1" xref="S4.T1.9.9.9.1.m1.1.1.1.cmml">=</mo><mn id="S4.T1.9.9.9.1.m1.1.1.3" xref="S4.T1.9.9.9.1.m1.1.1.3.cmml">2</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.T1.9.9.9.1.m1.1b"><apply id="S4.T1.9.9.9.1.m1.1.1.cmml" xref="S4.T1.9.9.9.1.m1.1.1"><eq id="S4.T1.9.9.9.1.m1.1.1.1.cmml" xref="S4.T1.9.9.9.1.m1.1.1.1"></eq><apply id="S4.T1.9.9.9.1.m1.1.1.2.cmml" xref="S4.T1.9.9.9.1.m1.1.1.2"><csymbol cd="ambiguous" id="S4.T1.9.9.9.1.m1.1.1.2.1.cmml" xref="S4.T1.9.9.9.1.m1.1.1.2">subscript</csymbol><ci id="S4.T1.9.9.9.1.m1.1.1.2.2.cmml" xref="S4.T1.9.9.9.1.m1.1.1.2.2">𝑚</ci><ci id="S4.T1.9.9.9.1.m1.1.1.2.3.cmml" xref="S4.T1.9.9.9.1.m1.1.1.2.3">𝑖</ci></apply><cn id="S4.T1.9.9.9.1.m1.1.1.3.cmml" type="integer" xref="S4.T1.9.9.9.1.m1.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.T1.9.9.9.1.m1.1c">m_{i}=2</annotation><annotation encoding="application/x-llamapun" id="S4.T1.9.9.9.1.m1.1d">italic_m start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = 2</annotation></semantics></math> </th> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T1.9.9.9.2">0.034 (0.0066)</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T1.9.9.9.3">0.037 (0.0096)</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T1.9.9.9.4">0.035 (0.0092)</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T1.9.9.9.5">0.035 (0.0091)</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T1.9.9.9.6">0.082 (0.0210)</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T1.9.9.9.7">0.59 (0.220)</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T1.9.9.9.8">0.83 (0.062)</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T1.9.9.9.9">0.85 (0.054)</td> </tr> <tr class="ltx_tr" id="S4.T1.10.10.10"> <th class="ltx_td ltx_align_right ltx_th ltx_th_row ltx_border_b ltx_border_l ltx_border_r ltx_border_t" id="S4.T1.10.10.10.1"> <math alttext="m_{i}=6" class="ltx_Math" display="inline" id="S4.T1.10.10.10.1.m1.1"><semantics id="S4.T1.10.10.10.1.m1.1a"><mrow id="S4.T1.10.10.10.1.m1.1.1" xref="S4.T1.10.10.10.1.m1.1.1.cmml"><msub id="S4.T1.10.10.10.1.m1.1.1.2" xref="S4.T1.10.10.10.1.m1.1.1.2.cmml"><mi id="S4.T1.10.10.10.1.m1.1.1.2.2" xref="S4.T1.10.10.10.1.m1.1.1.2.2.cmml">m</mi><mi id="S4.T1.10.10.10.1.m1.1.1.2.3" xref="S4.T1.10.10.10.1.m1.1.1.2.3.cmml">i</mi></msub><mo id="S4.T1.10.10.10.1.m1.1.1.1" xref="S4.T1.10.10.10.1.m1.1.1.1.cmml">=</mo><mn id="S4.T1.10.10.10.1.m1.1.1.3" xref="S4.T1.10.10.10.1.m1.1.1.3.cmml">6</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.T1.10.10.10.1.m1.1b"><apply id="S4.T1.10.10.10.1.m1.1.1.cmml" xref="S4.T1.10.10.10.1.m1.1.1"><eq id="S4.T1.10.10.10.1.m1.1.1.1.cmml" xref="S4.T1.10.10.10.1.m1.1.1.1"></eq><apply id="S4.T1.10.10.10.1.m1.1.1.2.cmml" xref="S4.T1.10.10.10.1.m1.1.1.2"><csymbol cd="ambiguous" id="S4.T1.10.10.10.1.m1.1.1.2.1.cmml" xref="S4.T1.10.10.10.1.m1.1.1.2">subscript</csymbol><ci id="S4.T1.10.10.10.1.m1.1.1.2.2.cmml" xref="S4.T1.10.10.10.1.m1.1.1.2.2">𝑚</ci><ci id="S4.T1.10.10.10.1.m1.1.1.2.3.cmml" xref="S4.T1.10.10.10.1.m1.1.1.2.3">𝑖</ci></apply><cn id="S4.T1.10.10.10.1.m1.1.1.3.cmml" type="integer" xref="S4.T1.10.10.10.1.m1.1.1.3">6</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.T1.10.10.10.1.m1.1c">m_{i}=6</annotation><annotation encoding="application/x-llamapun" id="S4.T1.10.10.10.1.m1.1d">italic_m start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = 6</annotation></semantics></math> </th> <td class="ltx_td ltx_align_center ltx_border_b ltx_border_r ltx_border_t" id="S4.T1.10.10.10.2">0.013 (0.0034)</td> <td class="ltx_td ltx_align_center ltx_border_b ltx_border_r ltx_border_t" id="S4.T1.10.10.10.3">0.021 (0.0057)</td> <td class="ltx_td ltx_align_center ltx_border_b ltx_border_r ltx_border_t" id="S4.T1.10.10.10.4">0.021 (0.0056)</td> <td class="ltx_td ltx_align_center ltx_border_b ltx_border_r ltx_border_t" id="S4.T1.10.10.10.5">0.020 (0.0056)</td> <td class="ltx_td ltx_align_center ltx_border_b ltx_border_r ltx_border_t" id="S4.T1.10.10.10.6">0.042 (0.0094)</td> <td class="ltx_td ltx_align_center ltx_border_b ltx_border_r ltx_border_t" id="S4.T1.10.10.10.7">0.24 (0.074)</td> <td class="ltx_td ltx_align_center ltx_border_b ltx_border_r ltx_border_t" id="S4.T1.10.10.10.8">0.89 (0.043)</td> <td class="ltx_td ltx_align_center ltx_border_b ltx_border_r ltx_border_t" id="S4.T1.10.10.10.9">0.91 (0.036)</td> </tr> </tbody> </table> </span></div> </figure> <figure class="ltx_table" id="S4.T2"> <figcaption class="ltx_caption"><span class="ltx_tag ltx_tag_table">Table 2: </span> Relative estimation error in Case 2. The simulation follows MMTR in (<a class="ltx_ref" href="https://arxiv.org/html/2503.13782v1#S2.E3" title="In 2.1 Model ‣ 2 Mixed Model Trace Regression ‣ Regularized Parameter Estimation in Mixed Model Trace Regression"><span class="ltx_text ltx_ref_tag">3</span></a>). Every estimation error is averaged over 100 simulation replications, with the Monte Carlo errors in parenthesis. A small relative error indicates high estimation accuracy. GEE un. and GEE eq. denote the GEE models with unstructured and equicorrelated covariance matrix. Kruskal and Quasi denote the sparse Kruskal and quasi-likelihood methods.</figcaption> <div class="ltx_inline-block ltx_align_center ltx_transformed_outer" id="S4.T2.10" style="width:505.9pt;height:208.9pt;vertical-align:-0.0pt;"><span class="ltx_transformed_inner" style="transform:translate(-52.2pt,21.6pt) scale(0.828910805548196,0.828910805548196) ;"> <table class="ltx_tabular ltx_guessed_headers ltx_align_top" id="S4.T2.10.10"> <tbody class="ltx_tbody"> <tr class="ltx_tr" id="S4.T2.2.2.2"> <th class="ltx_td ltx_align_right ltx_th ltx_th_row ltx_border_l ltx_border_r ltx_border_t" id="S4.T2.2.2.2.3">group size</th> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" colspan="5" id="S4.T2.1.1.1.1">err<math alttext="{}_{\operatorname{\mathbf{B}}}" class="ltx_Math" display="inline" id="S4.T2.1.1.1.1.m1.1"><semantics id="S4.T2.1.1.1.1.m1.1a"><msub id="S4.T2.1.1.1.1.m1.1.1" xref="S4.T2.1.1.1.1.m1.1.1.cmml"><mi id="S4.T2.1.1.1.1.m1.1.1a" xref="S4.T2.1.1.1.1.m1.1.1.cmml"></mi><mi id="S4.T2.1.1.1.1.m1.1.1.1" xref="S4.T2.1.1.1.1.m1.1.1.1.cmml">𝐁</mi></msub><annotation-xml encoding="MathML-Content" id="S4.T2.1.1.1.1.m1.1b"><apply id="S4.T2.1.1.1.1.m1.1.1.cmml" xref="S4.T2.1.1.1.1.m1.1.1"><ci id="S4.T2.1.1.1.1.m1.1.1.1.cmml" xref="S4.T2.1.1.1.1.m1.1.1.1">𝐁</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.T2.1.1.1.1.m1.1c">{}_{\operatorname{\mathbf{B}}}</annotation><annotation encoding="application/x-llamapun" id="S4.T2.1.1.1.1.m1.1d">start_FLOATSUBSCRIPT bold_B end_FLOATSUBSCRIPT</annotation></semantics></math> </td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" colspan="3" id="S4.T2.2.2.2.2">err<math alttext="{}_{\operatorname{{\mathbf{\Lambda}}}}" class="ltx_Math" display="inline" id="S4.T2.2.2.2.2.m1.1"><semantics id="S4.T2.2.2.2.2.m1.1a"><msub id="S4.T2.2.2.2.2.m1.1.1" xref="S4.T2.2.2.2.2.m1.1.1.cmml"><mi id="S4.T2.2.2.2.2.m1.1.1a" xref="S4.T2.2.2.2.2.m1.1.1.cmml"></mi><mi id="S4.T2.2.2.2.2.m1.1.1.1" xref="S4.T2.2.2.2.2.m1.1.1.1.cmml">𝚲</mi></msub><annotation-xml encoding="MathML-Content" id="S4.T2.2.2.2.2.m1.1b"><apply id="S4.T2.2.2.2.2.m1.1.1.cmml" xref="S4.T2.2.2.2.2.m1.1.1"><ci id="S4.T2.2.2.2.2.m1.1.1.1.cmml" xref="S4.T2.2.2.2.2.m1.1.1.1">𝚲</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.T2.2.2.2.2.m1.1c">{}_{\operatorname{{\mathbf{\Lambda}}}}</annotation><annotation encoding="application/x-llamapun" id="S4.T2.2.2.2.2.m1.1d">start_FLOATSUBSCRIPT bold_Λ end_FLOATSUBSCRIPT</annotation></semantics></math> </td> </tr> <tr class="ltx_tr" id="S4.T2.10.10.11.1"> <th class="ltx_td ltx_th ltx_th_row ltx_border_l ltx_border_r ltx_border_t" id="S4.T2.10.10.11.1.1"></th> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T2.10.10.11.1.2">MMTR</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T2.10.10.11.1.3">Kruskal</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T2.10.10.11.1.4">GEE un.</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T2.10.10.11.1.5">GEE eq.</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T2.10.10.11.1.6">Quasi</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T2.10.10.11.1.7">MMTR</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T2.10.10.11.1.8">GEE un.</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T2.10.10.11.1.9">GEE eq.</td> </tr> <tr class="ltx_tr" id="S4.T2.10.10.12.2"> <th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l ltx_border_r ltx_border_t" colspan="9" id="S4.T2.10.10.12.2.1"><span class="ltx_text ltx_font_bold" id="S4.T2.10.10.12.2.1.1">n = 64</span></th> </tr> <tr class="ltx_tr" id="S4.T2.3.3.3"> <th class="ltx_td ltx_align_right ltx_th ltx_th_row ltx_border_l ltx_border_r ltx_border_t" id="S4.T2.3.3.3.1"> <math alttext="m_{i}=2" class="ltx_Math" display="inline" id="S4.T2.3.3.3.1.m1.1"><semantics id="S4.T2.3.3.3.1.m1.1a"><mrow id="S4.T2.3.3.3.1.m1.1.1" xref="S4.T2.3.3.3.1.m1.1.1.cmml"><msub id="S4.T2.3.3.3.1.m1.1.1.2" xref="S4.T2.3.3.3.1.m1.1.1.2.cmml"><mi id="S4.T2.3.3.3.1.m1.1.1.2.2" xref="S4.T2.3.3.3.1.m1.1.1.2.2.cmml">m</mi><mi id="S4.T2.3.3.3.1.m1.1.1.2.3" xref="S4.T2.3.3.3.1.m1.1.1.2.3.cmml">i</mi></msub><mo id="S4.T2.3.3.3.1.m1.1.1.1" xref="S4.T2.3.3.3.1.m1.1.1.1.cmml">=</mo><mn id="S4.T2.3.3.3.1.m1.1.1.3" xref="S4.T2.3.3.3.1.m1.1.1.3.cmml">2</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.T2.3.3.3.1.m1.1b"><apply id="S4.T2.3.3.3.1.m1.1.1.cmml" xref="S4.T2.3.3.3.1.m1.1.1"><eq id="S4.T2.3.3.3.1.m1.1.1.1.cmml" xref="S4.T2.3.3.3.1.m1.1.1.1"></eq><apply id="S4.T2.3.3.3.1.m1.1.1.2.cmml" xref="S4.T2.3.3.3.1.m1.1.1.2"><csymbol cd="ambiguous" id="S4.T2.3.3.3.1.m1.1.1.2.1.cmml" xref="S4.T2.3.3.3.1.m1.1.1.2">subscript</csymbol><ci id="S4.T2.3.3.3.1.m1.1.1.2.2.cmml" xref="S4.T2.3.3.3.1.m1.1.1.2.2">𝑚</ci><ci id="S4.T2.3.3.3.1.m1.1.1.2.3.cmml" xref="S4.T2.3.3.3.1.m1.1.1.2.3">𝑖</ci></apply><cn id="S4.T2.3.3.3.1.m1.1.1.3.cmml" type="integer" xref="S4.T2.3.3.3.1.m1.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.T2.3.3.3.1.m1.1c">m_{i}=2</annotation><annotation encoding="application/x-llamapun" id="S4.T2.3.3.3.1.m1.1d">italic_m start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = 2</annotation></semantics></math> </th> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T2.3.3.3.2">0.120 (0.0200)</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T2.3.3.3.3">0.55 (0.097)</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T2.3.3.3.4">0.120 (0.0230)</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T2.3.3.3.5">0.096 (0.0160)</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T2.3.3.3.6">0.890 (0.0440)</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T2.3.3.3.7">0.92 (0.019)</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T2.3.3.3.8">0.98 (0.0045)</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T2.3.3.3.9">0.90 (0.019)</td> </tr> <tr class="ltx_tr" id="S4.T2.4.4.4"> <th class="ltx_td ltx_align_right ltx_th ltx_th_row ltx_border_l ltx_border_r ltx_border_t" id="S4.T2.4.4.4.1"> <math alttext="m_{i}=6" class="ltx_Math" display="inline" id="S4.T2.4.4.4.1.m1.1"><semantics id="S4.T2.4.4.4.1.m1.1a"><mrow id="S4.T2.4.4.4.1.m1.1.1" xref="S4.T2.4.4.4.1.m1.1.1.cmml"><msub id="S4.T2.4.4.4.1.m1.1.1.2" xref="S4.T2.4.4.4.1.m1.1.1.2.cmml"><mi id="S4.T2.4.4.4.1.m1.1.1.2.2" xref="S4.T2.4.4.4.1.m1.1.1.2.2.cmml">m</mi><mi id="S4.T2.4.4.4.1.m1.1.1.2.3" xref="S4.T2.4.4.4.1.m1.1.1.2.3.cmml">i</mi></msub><mo id="S4.T2.4.4.4.1.m1.1.1.1" xref="S4.T2.4.4.4.1.m1.1.1.1.cmml">=</mo><mn id="S4.T2.4.4.4.1.m1.1.1.3" xref="S4.T2.4.4.4.1.m1.1.1.3.cmml">6</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.T2.4.4.4.1.m1.1b"><apply id="S4.T2.4.4.4.1.m1.1.1.cmml" xref="S4.T2.4.4.4.1.m1.1.1"><eq id="S4.T2.4.4.4.1.m1.1.1.1.cmml" xref="S4.T2.4.4.4.1.m1.1.1.1"></eq><apply id="S4.T2.4.4.4.1.m1.1.1.2.cmml" xref="S4.T2.4.4.4.1.m1.1.1.2"><csymbol cd="ambiguous" id="S4.T2.4.4.4.1.m1.1.1.2.1.cmml" xref="S4.T2.4.4.4.1.m1.1.1.2">subscript</csymbol><ci id="S4.T2.4.4.4.1.m1.1.1.2.2.cmml" xref="S4.T2.4.4.4.1.m1.1.1.2.2">𝑚</ci><ci id="S4.T2.4.4.4.1.m1.1.1.2.3.cmml" xref="S4.T2.4.4.4.1.m1.1.1.2.3">𝑖</ci></apply><cn id="S4.T2.4.4.4.1.m1.1.1.3.cmml" type="integer" xref="S4.T2.4.4.4.1.m1.1.1.3">6</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.T2.4.4.4.1.m1.1c">m_{i}=6</annotation><annotation encoding="application/x-llamapun" id="S4.T2.4.4.4.1.m1.1d">italic_m start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = 6</annotation></semantics></math> </th> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T2.4.4.4.2">0.040 (0.0066)</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T2.4.4.4.3">0.36 (0.050)</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T2.4.4.4.4">0.039 (0.0091)</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T2.4.4.4.5">0.033 (0.0050)</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T2.4.4.4.6">0.150 (0.0350)</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T2.4.4.4.7">0.87 (0.240)</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T2.4.4.4.8">0.95 (0.0110)</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T2.4.4.4.9">0.93 (0.015)</td> </tr> <tr class="ltx_tr" id="S4.T2.10.10.13.3"> <th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l ltx_border_r ltx_border_t" colspan="9" id="S4.T2.10.10.13.3.1"><span class="ltx_text ltx_font_bold" id="S4.T2.10.10.13.3.1.1">n = 96</span></th> </tr> <tr class="ltx_tr" id="S4.T2.5.5.5"> <th class="ltx_td ltx_align_right ltx_th ltx_th_row ltx_border_l ltx_border_r ltx_border_t" id="S4.T2.5.5.5.1"> <math alttext="m_{i}=2" class="ltx_Math" display="inline" id="S4.T2.5.5.5.1.m1.1"><semantics id="S4.T2.5.5.5.1.m1.1a"><mrow id="S4.T2.5.5.5.1.m1.1.1" xref="S4.T2.5.5.5.1.m1.1.1.cmml"><msub id="S4.T2.5.5.5.1.m1.1.1.2" xref="S4.T2.5.5.5.1.m1.1.1.2.cmml"><mi id="S4.T2.5.5.5.1.m1.1.1.2.2" xref="S4.T2.5.5.5.1.m1.1.1.2.2.cmml">m</mi><mi id="S4.T2.5.5.5.1.m1.1.1.2.3" xref="S4.T2.5.5.5.1.m1.1.1.2.3.cmml">i</mi></msub><mo id="S4.T2.5.5.5.1.m1.1.1.1" xref="S4.T2.5.5.5.1.m1.1.1.1.cmml">=</mo><mn id="S4.T2.5.5.5.1.m1.1.1.3" xref="S4.T2.5.5.5.1.m1.1.1.3.cmml">2</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.T2.5.5.5.1.m1.1b"><apply id="S4.T2.5.5.5.1.m1.1.1.cmml" xref="S4.T2.5.5.5.1.m1.1.1"><eq id="S4.T2.5.5.5.1.m1.1.1.1.cmml" xref="S4.T2.5.5.5.1.m1.1.1.1"></eq><apply id="S4.T2.5.5.5.1.m1.1.1.2.cmml" xref="S4.T2.5.5.5.1.m1.1.1.2"><csymbol cd="ambiguous" id="S4.T2.5.5.5.1.m1.1.1.2.1.cmml" xref="S4.T2.5.5.5.1.m1.1.1.2">subscript</csymbol><ci id="S4.T2.5.5.5.1.m1.1.1.2.2.cmml" xref="S4.T2.5.5.5.1.m1.1.1.2.2">𝑚</ci><ci id="S4.T2.5.5.5.1.m1.1.1.2.3.cmml" xref="S4.T2.5.5.5.1.m1.1.1.2.3">𝑖</ci></apply><cn id="S4.T2.5.5.5.1.m1.1.1.3.cmml" type="integer" xref="S4.T2.5.5.5.1.m1.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.T2.5.5.5.1.m1.1c">m_{i}=2</annotation><annotation encoding="application/x-llamapun" id="S4.T2.5.5.5.1.m1.1d">italic_m start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = 2</annotation></semantics></math> </th> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T2.5.5.5.2">0.085 (0.0130)</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T2.5.5.5.3">0.42 (0.062)</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T2.5.5.5.4">0.071 (0.0240)</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T2.5.5.5.5">0.057 (0.0090)</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T2.5.5.5.6">0.740 (0.0980)</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T2.5.5.5.7">1.10 (0.140)</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T2.5.5.5.8">0.76 (0.0700)</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T2.5.5.5.9">0.90 (0.018)</td> </tr> <tr class="ltx_tr" id="S4.T2.6.6.6"> <th class="ltx_td ltx_align_right ltx_th ltx_th_row ltx_border_l ltx_border_r ltx_border_t" id="S4.T2.6.6.6.1"> <math alttext="m_{i}=6" class="ltx_Math" display="inline" id="S4.T2.6.6.6.1.m1.1"><semantics id="S4.T2.6.6.6.1.m1.1a"><mrow id="S4.T2.6.6.6.1.m1.1.1" xref="S4.T2.6.6.6.1.m1.1.1.cmml"><msub id="S4.T2.6.6.6.1.m1.1.1.2" xref="S4.T2.6.6.6.1.m1.1.1.2.cmml"><mi id="S4.T2.6.6.6.1.m1.1.1.2.2" xref="S4.T2.6.6.6.1.m1.1.1.2.2.cmml">m</mi><mi id="S4.T2.6.6.6.1.m1.1.1.2.3" xref="S4.T2.6.6.6.1.m1.1.1.2.3.cmml">i</mi></msub><mo id="S4.T2.6.6.6.1.m1.1.1.1" xref="S4.T2.6.6.6.1.m1.1.1.1.cmml">=</mo><mn id="S4.T2.6.6.6.1.m1.1.1.3" xref="S4.T2.6.6.6.1.m1.1.1.3.cmml">6</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.T2.6.6.6.1.m1.1b"><apply id="S4.T2.6.6.6.1.m1.1.1.cmml" xref="S4.T2.6.6.6.1.m1.1.1"><eq id="S4.T2.6.6.6.1.m1.1.1.1.cmml" xref="S4.T2.6.6.6.1.m1.1.1.1"></eq><apply id="S4.T2.6.6.6.1.m1.1.1.2.cmml" xref="S4.T2.6.6.6.1.m1.1.1.2"><csymbol cd="ambiguous" id="S4.T2.6.6.6.1.m1.1.1.2.1.cmml" xref="S4.T2.6.6.6.1.m1.1.1.2">subscript</csymbol><ci id="S4.T2.6.6.6.1.m1.1.1.2.2.cmml" xref="S4.T2.6.6.6.1.m1.1.1.2.2">𝑚</ci><ci id="S4.T2.6.6.6.1.m1.1.1.2.3.cmml" xref="S4.T2.6.6.6.1.m1.1.1.2.3">𝑖</ci></apply><cn id="S4.T2.6.6.6.1.m1.1.1.3.cmml" type="integer" xref="S4.T2.6.6.6.1.m1.1.1.3">6</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.T2.6.6.6.1.m1.1c">m_{i}=6</annotation><annotation encoding="application/x-llamapun" id="S4.T2.6.6.6.1.m1.1d">italic_m start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = 6</annotation></semantics></math> </th> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T2.6.6.6.2">0.022 (0.0028)</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T2.6.6.6.3">0.34 (0.044)</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T2.6.6.6.4">0.028 (0.0081)</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T2.6.6.6.5">0.025 (0.0037)</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T2.6.6.6.6">0.084 (0.0130)</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T2.6.6.6.7">0.46 (0.098)</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T2.6.6.6.8">0.84 (0.1800)</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T2.6.6.6.9">0.93 (0.015)</td> </tr> <tr class="ltx_tr" id="S4.T2.10.10.14.4"> <th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l ltx_border_r ltx_border_t" colspan="9" id="S4.T2.10.10.14.4.1"><span class="ltx_text ltx_font_bold" id="S4.T2.10.10.14.4.1.1">n = 192</span></th> </tr> <tr class="ltx_tr" id="S4.T2.7.7.7"> <th class="ltx_td ltx_align_right ltx_th ltx_th_row ltx_border_l ltx_border_r ltx_border_t" id="S4.T2.7.7.7.1"> <math alttext="m_{i}=2" class="ltx_Math" display="inline" id="S4.T2.7.7.7.1.m1.1"><semantics id="S4.T2.7.7.7.1.m1.1a"><mrow id="S4.T2.7.7.7.1.m1.1.1" xref="S4.T2.7.7.7.1.m1.1.1.cmml"><msub id="S4.T2.7.7.7.1.m1.1.1.2" xref="S4.T2.7.7.7.1.m1.1.1.2.cmml"><mi id="S4.T2.7.7.7.1.m1.1.1.2.2" xref="S4.T2.7.7.7.1.m1.1.1.2.2.cmml">m</mi><mi id="S4.T2.7.7.7.1.m1.1.1.2.3" xref="S4.T2.7.7.7.1.m1.1.1.2.3.cmml">i</mi></msub><mo id="S4.T2.7.7.7.1.m1.1.1.1" xref="S4.T2.7.7.7.1.m1.1.1.1.cmml">=</mo><mn id="S4.T2.7.7.7.1.m1.1.1.3" xref="S4.T2.7.7.7.1.m1.1.1.3.cmml">2</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.T2.7.7.7.1.m1.1b"><apply id="S4.T2.7.7.7.1.m1.1.1.cmml" xref="S4.T2.7.7.7.1.m1.1.1"><eq id="S4.T2.7.7.7.1.m1.1.1.1.cmml" xref="S4.T2.7.7.7.1.m1.1.1.1"></eq><apply id="S4.T2.7.7.7.1.m1.1.1.2.cmml" xref="S4.T2.7.7.7.1.m1.1.1.2"><csymbol cd="ambiguous" id="S4.T2.7.7.7.1.m1.1.1.2.1.cmml" xref="S4.T2.7.7.7.1.m1.1.1.2">subscript</csymbol><ci id="S4.T2.7.7.7.1.m1.1.1.2.2.cmml" xref="S4.T2.7.7.7.1.m1.1.1.2.2">𝑚</ci><ci id="S4.T2.7.7.7.1.m1.1.1.2.3.cmml" xref="S4.T2.7.7.7.1.m1.1.1.2.3">𝑖</ci></apply><cn id="S4.T2.7.7.7.1.m1.1.1.3.cmml" type="integer" xref="S4.T2.7.7.7.1.m1.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.T2.7.7.7.1.m1.1c">m_{i}=2</annotation><annotation encoding="application/x-llamapun" id="S4.T2.7.7.7.1.m1.1d">italic_m start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = 2</annotation></semantics></math> </th> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T2.7.7.7.2">0.043 (0.0075)</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T2.7.7.7.3">0.36 (0.049)</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T2.7.7.7.4">0.033 (0.0057)</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T2.7.7.7.5">0.033 (0.0059)</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T2.7.7.7.6">0.150 (0.0280)</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T2.7.7.7.7">1.10 (0.200)</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T2.7.7.7.8">0.86 (0.0260)</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T2.7.7.7.9">0.90 (0.019)</td> </tr> <tr class="ltx_tr" id="S4.T2.8.8.8"> <th class="ltx_td ltx_align_right ltx_th ltx_th_row ltx_border_l ltx_border_r ltx_border_t" id="S4.T2.8.8.8.1"> <math alttext="m_{i}=6" class="ltx_Math" display="inline" id="S4.T2.8.8.8.1.m1.1"><semantics id="S4.T2.8.8.8.1.m1.1a"><mrow id="S4.T2.8.8.8.1.m1.1.1" xref="S4.T2.8.8.8.1.m1.1.1.cmml"><msub id="S4.T2.8.8.8.1.m1.1.1.2" xref="S4.T2.8.8.8.1.m1.1.1.2.cmml"><mi id="S4.T2.8.8.8.1.m1.1.1.2.2" xref="S4.T2.8.8.8.1.m1.1.1.2.2.cmml">m</mi><mi id="S4.T2.8.8.8.1.m1.1.1.2.3" xref="S4.T2.8.8.8.1.m1.1.1.2.3.cmml">i</mi></msub><mo id="S4.T2.8.8.8.1.m1.1.1.1" xref="S4.T2.8.8.8.1.m1.1.1.1.cmml">=</mo><mn id="S4.T2.8.8.8.1.m1.1.1.3" xref="S4.T2.8.8.8.1.m1.1.1.3.cmml">6</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.T2.8.8.8.1.m1.1b"><apply id="S4.T2.8.8.8.1.m1.1.1.cmml" xref="S4.T2.8.8.8.1.m1.1.1"><eq id="S4.T2.8.8.8.1.m1.1.1.1.cmml" xref="S4.T2.8.8.8.1.m1.1.1.1"></eq><apply id="S4.T2.8.8.8.1.m1.1.1.2.cmml" xref="S4.T2.8.8.8.1.m1.1.1.2"><csymbol cd="ambiguous" id="S4.T2.8.8.8.1.m1.1.1.2.1.cmml" xref="S4.T2.8.8.8.1.m1.1.1.2">subscript</csymbol><ci id="S4.T2.8.8.8.1.m1.1.1.2.2.cmml" xref="S4.T2.8.8.8.1.m1.1.1.2.2">𝑚</ci><ci id="S4.T2.8.8.8.1.m1.1.1.2.3.cmml" xref="S4.T2.8.8.8.1.m1.1.1.2.3">𝑖</ci></apply><cn id="S4.T2.8.8.8.1.m1.1.1.3.cmml" type="integer" xref="S4.T2.8.8.8.1.m1.1.1.3">6</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.T2.8.8.8.1.m1.1c">m_{i}=6</annotation><annotation encoding="application/x-llamapun" id="S4.T2.8.8.8.1.m1.1d">italic_m start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = 6</annotation></semantics></math> </th> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T2.8.8.8.2">0.012 (0.0018)</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T2.8.8.8.3">0.33 (0.043)</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T2.8.8.8.4">0.018 (0.0026)</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T2.8.8.8.5">0.018 (0.0026)</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T2.8.8.8.6">0.049 (0.0063)</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T2.8.8.8.7">0.24 (0.041)</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T2.8.8.8.8">0.88 (0.0240)</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T2.8.8.8.9">0.93 (0.015)</td> </tr> <tr class="ltx_tr" id="S4.T2.10.10.15.5"> <th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l ltx_border_r ltx_border_t" colspan="9" id="S4.T2.10.10.15.5.1"><span class="ltx_text ltx_font_bold" id="S4.T2.10.10.15.5.1.1">n = 288</span></th> </tr> <tr class="ltx_tr" id="S4.T2.9.9.9"> <th class="ltx_td ltx_align_right ltx_th ltx_th_row ltx_border_l ltx_border_r ltx_border_t" id="S4.T2.9.9.9.1"> <math alttext="m_{i}=2" class="ltx_Math" display="inline" id="S4.T2.9.9.9.1.m1.1"><semantics id="S4.T2.9.9.9.1.m1.1a"><mrow id="S4.T2.9.9.9.1.m1.1.1" xref="S4.T2.9.9.9.1.m1.1.1.cmml"><msub id="S4.T2.9.9.9.1.m1.1.1.2" xref="S4.T2.9.9.9.1.m1.1.1.2.cmml"><mi id="S4.T2.9.9.9.1.m1.1.1.2.2" xref="S4.T2.9.9.9.1.m1.1.1.2.2.cmml">m</mi><mi id="S4.T2.9.9.9.1.m1.1.1.2.3" xref="S4.T2.9.9.9.1.m1.1.1.2.3.cmml">i</mi></msub><mo id="S4.T2.9.9.9.1.m1.1.1.1" xref="S4.T2.9.9.9.1.m1.1.1.1.cmml">=</mo><mn id="S4.T2.9.9.9.1.m1.1.1.3" xref="S4.T2.9.9.9.1.m1.1.1.3.cmml">2</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.T2.9.9.9.1.m1.1b"><apply id="S4.T2.9.9.9.1.m1.1.1.cmml" xref="S4.T2.9.9.9.1.m1.1.1"><eq id="S4.T2.9.9.9.1.m1.1.1.1.cmml" xref="S4.T2.9.9.9.1.m1.1.1.1"></eq><apply id="S4.T2.9.9.9.1.m1.1.1.2.cmml" xref="S4.T2.9.9.9.1.m1.1.1.2"><csymbol cd="ambiguous" id="S4.T2.9.9.9.1.m1.1.1.2.1.cmml" xref="S4.T2.9.9.9.1.m1.1.1.2">subscript</csymbol><ci id="S4.T2.9.9.9.1.m1.1.1.2.2.cmml" xref="S4.T2.9.9.9.1.m1.1.1.2.2">𝑚</ci><ci id="S4.T2.9.9.9.1.m1.1.1.2.3.cmml" xref="S4.T2.9.9.9.1.m1.1.1.2.3">𝑖</ci></apply><cn id="S4.T2.9.9.9.1.m1.1.1.3.cmml" type="integer" xref="S4.T2.9.9.9.1.m1.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.T2.9.9.9.1.m1.1c">m_{i}=2</annotation><annotation encoding="application/x-llamapun" id="S4.T2.9.9.9.1.m1.1d">italic_m start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = 2</annotation></semantics></math> </th> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T2.9.9.9.2">0.028 (0.0045)</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T2.9.9.9.3">0.34 (0.045)</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T2.9.9.9.4">0.025 (0.0036)</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T2.9.9.9.5">0.025 (0.0036)</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T2.9.9.9.6">0.086 (0.0130)</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T2.9.9.9.7">0.63 (0.130)</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T2.9.9.9.8">0.88 (0.0230)</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T2.9.9.9.9">0.90 (0.019)</td> </tr> <tr class="ltx_tr" id="S4.T2.10.10.10"> <th class="ltx_td ltx_align_right ltx_th ltx_th_row ltx_border_b ltx_border_l ltx_border_r ltx_border_t" id="S4.T2.10.10.10.1"> <math alttext="m_{i}=6" class="ltx_Math" display="inline" id="S4.T2.10.10.10.1.m1.1"><semantics id="S4.T2.10.10.10.1.m1.1a"><mrow id="S4.T2.10.10.10.1.m1.1.1" xref="S4.T2.10.10.10.1.m1.1.1.cmml"><msub id="S4.T2.10.10.10.1.m1.1.1.2" xref="S4.T2.10.10.10.1.m1.1.1.2.cmml"><mi id="S4.T2.10.10.10.1.m1.1.1.2.2" xref="S4.T2.10.10.10.1.m1.1.1.2.2.cmml">m</mi><mi id="S4.T2.10.10.10.1.m1.1.1.2.3" xref="S4.T2.10.10.10.1.m1.1.1.2.3.cmml">i</mi></msub><mo id="S4.T2.10.10.10.1.m1.1.1.1" xref="S4.T2.10.10.10.1.m1.1.1.1.cmml">=</mo><mn id="S4.T2.10.10.10.1.m1.1.1.3" xref="S4.T2.10.10.10.1.m1.1.1.3.cmml">6</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.T2.10.10.10.1.m1.1b"><apply id="S4.T2.10.10.10.1.m1.1.1.cmml" xref="S4.T2.10.10.10.1.m1.1.1"><eq id="S4.T2.10.10.10.1.m1.1.1.1.cmml" xref="S4.T2.10.10.10.1.m1.1.1.1"></eq><apply id="S4.T2.10.10.10.1.m1.1.1.2.cmml" xref="S4.T2.10.10.10.1.m1.1.1.2"><csymbol cd="ambiguous" id="S4.T2.10.10.10.1.m1.1.1.2.1.cmml" xref="S4.T2.10.10.10.1.m1.1.1.2">subscript</csymbol><ci id="S4.T2.10.10.10.1.m1.1.1.2.2.cmml" xref="S4.T2.10.10.10.1.m1.1.1.2.2">𝑚</ci><ci id="S4.T2.10.10.10.1.m1.1.1.2.3.cmml" xref="S4.T2.10.10.10.1.m1.1.1.2.3">𝑖</ci></apply><cn id="S4.T2.10.10.10.1.m1.1.1.3.cmml" type="integer" xref="S4.T2.10.10.10.1.m1.1.1.3">6</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.T2.10.10.10.1.m1.1c">m_{i}=6</annotation><annotation encoding="application/x-llamapun" id="S4.T2.10.10.10.1.m1.1d">italic_m start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = 6</annotation></semantics></math> </th> <td class="ltx_td ltx_align_center ltx_border_b ltx_border_r ltx_border_t" id="S4.T2.10.10.10.2">0.010 (0.0015)</td> <td class="ltx_td ltx_align_center ltx_border_b ltx_border_r ltx_border_t" id="S4.T2.10.10.10.3">0.32 (0.042)</td> <td class="ltx_td ltx_align_center ltx_border_b ltx_border_r ltx_border_t" id="S4.T2.10.10.10.4">0.014 (0.0021)</td> <td class="ltx_td ltx_align_center ltx_border_b ltx_border_r ltx_border_t" id="S4.T2.10.10.10.5">0.014 (0.0021)</td> <td class="ltx_td ltx_align_center ltx_border_b ltx_border_r ltx_border_t" id="S4.T2.10.10.10.6">0.038 (0.0049)</td> <td class="ltx_td ltx_align_center ltx_border_b ltx_border_r ltx_border_t" id="S4.T2.10.10.10.7">0.18 (0.032)</td> <td class="ltx_td ltx_align_center ltx_border_b ltx_border_r ltx_border_t" id="S4.T2.10.10.10.8">0.90 (0.0200)</td> <td class="ltx_td ltx_align_center ltx_border_b ltx_border_r ltx_border_t" id="S4.T2.10.10.10.9">0.93 (0.015)</td> </tr> </tbody> </table> </span></div> </figure> <figure class="ltx_table" id="S4.T4"> <div class="ltx_flex_figure ltx_flex_table"> <div class="ltx_flex_cell ltx_flex_size_2"> <figure class="ltx_figure ltx_figure_panel ltx_minipage ltx_align_middle" id="S4.T4.16" style="width:195.1pt;"> <figcaption class="ltx_caption"><span class="ltx_tag ltx_tag_figure">Table 3: </span>MMTR’s relative estimation error for <math alttext="\operatorname{{\mathbf{\Sigma}}}_{1}" class="ltx_Math" display="inline" id="S4.T4.4.4.m1.1"><semantics id="S4.T4.4.4.m1.1b"><msub id="S4.T4.4.4.m1.1.1" xref="S4.T4.4.4.m1.1.1.cmml"><mi id="S4.T4.4.4.m1.1.1.2" xref="S4.T4.4.4.m1.1.1.2.cmml">𝚺</mi><mn id="S4.T4.4.4.m1.1.1.3" xref="S4.T4.4.4.m1.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S4.T4.4.4.m1.1c"><apply id="S4.T4.4.4.m1.1.1.cmml" xref="S4.T4.4.4.m1.1.1"><csymbol cd="ambiguous" id="S4.T4.4.4.m1.1.1.1.cmml" xref="S4.T4.4.4.m1.1.1">subscript</csymbol><ci id="S4.T4.4.4.m1.1.1.2.cmml" xref="S4.T4.4.4.m1.1.1.2">𝚺</ci><cn id="S4.T4.4.4.m1.1.1.3.cmml" type="integer" xref="S4.T4.4.4.m1.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.T4.4.4.m1.1d">\operatorname{{\mathbf{\Sigma}}}_{1}</annotation><annotation encoding="application/x-llamapun" id="S4.T4.4.4.m1.1e">bold_Σ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="\operatorname{{\mathbf{\Sigma}}}_{2}" class="ltx_Math" display="inline" id="S4.T4.5.5.m2.1"><semantics id="S4.T4.5.5.m2.1b"><msub id="S4.T4.5.5.m2.1.1" xref="S4.T4.5.5.m2.1.1.cmml"><mi id="S4.T4.5.5.m2.1.1.2" xref="S4.T4.5.5.m2.1.1.2.cmml">𝚺</mi><mn id="S4.T4.5.5.m2.1.1.3" xref="S4.T4.5.5.m2.1.1.3.cmml">2</mn></msub><annotation-xml encoding="MathML-Content" id="S4.T4.5.5.m2.1c"><apply id="S4.T4.5.5.m2.1.1.cmml" xref="S4.T4.5.5.m2.1.1"><csymbol cd="ambiguous" id="S4.T4.5.5.m2.1.1.1.cmml" xref="S4.T4.5.5.m2.1.1">subscript</csymbol><ci id="S4.T4.5.5.m2.1.1.2.cmml" xref="S4.T4.5.5.m2.1.1.2">𝚺</ci><cn id="S4.T4.5.5.m2.1.1.3.cmml" type="integer" xref="S4.T4.5.5.m2.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.T4.5.5.m2.1d">\operatorname{{\mathbf{\Sigma}}}_{2}</annotation><annotation encoding="application/x-llamapun" id="S4.T4.5.5.m2.1e">bold_Σ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math> in Case 1. Every estimation error is averaged over 100 simulation replications, with the Monte Carlo errors in parenthesis. The red and blue lines represent simulations with equal <math alttext="N" class="ltx_Math" display="inline" id="S4.T4.6.6.m3.1"><semantics id="S4.T4.6.6.m3.1b"><mi id="S4.T4.6.6.m3.1.1" xref="S4.T4.6.6.m3.1.1.cmml">N</mi><annotation-xml encoding="MathML-Content" id="S4.T4.6.6.m3.1c"><ci id="S4.T4.6.6.m3.1.1.cmml" xref="S4.T4.6.6.m3.1.1">𝑁</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.T4.6.6.m3.1d">N</annotation><annotation encoding="application/x-llamapun" id="S4.T4.6.6.m3.1e">italic_N</annotation></semantics></math> values.</figcaption> <div class="ltx_inline-block ltx_align_center ltx_transformed_outer" id="S4.T4.16.16" style="width:303.5pt;height:387.5pt;vertical-align:-0.0pt;"><span class="ltx_transformed_inner" style="transform:translate(60.1pt,-76.7pt) scale(1.65590441996162,1.65590441996162) ;"> <table class="ltx_tabular ltx_guessed_headers ltx_align_top" id="S4.T4.16.16.10"> <tbody class="ltx_tbody"> <tr class="ltx_tr" id="S4.T4.8.8.2.2"> <th class="ltx_td ltx_align_right ltx_th ltx_th_row ltx_border_l ltx_border_r ltx_border_t" id="S4.T4.8.8.2.2.3">group size</th> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T4.7.7.1.1.1">err<math alttext="{}_{\operatorname{{\mathbf{\Sigma}}}_{1}}" class="ltx_Math" display="inline" id="S4.T4.7.7.1.1.1.m1.1"><semantics id="S4.T4.7.7.1.1.1.m1.1a"><msub id="S4.T4.7.7.1.1.1.m1.1.1" xref="S4.T4.7.7.1.1.1.m1.1.1.cmml"><mi id="S4.T4.7.7.1.1.1.m1.1.1a" xref="S4.T4.7.7.1.1.1.m1.1.1.cmml"></mi><msub id="S4.T4.7.7.1.1.1.m1.1.1.1" xref="S4.T4.7.7.1.1.1.m1.1.1.1.cmml"><mi id="S4.T4.7.7.1.1.1.m1.1.1.1.2" xref="S4.T4.7.7.1.1.1.m1.1.1.1.2.cmml">𝚺</mi><mn id="S4.T4.7.7.1.1.1.m1.1.1.1.3" xref="S4.T4.7.7.1.1.1.m1.1.1.1.3.cmml">1</mn></msub></msub><annotation-xml encoding="MathML-Content" id="S4.T4.7.7.1.1.1.m1.1b"><apply id="S4.T4.7.7.1.1.1.m1.1.1.cmml" xref="S4.T4.7.7.1.1.1.m1.1.1"><apply id="S4.T4.7.7.1.1.1.m1.1.1.1.cmml" xref="S4.T4.7.7.1.1.1.m1.1.1.1"><csymbol cd="ambiguous" id="S4.T4.7.7.1.1.1.m1.1.1.1.1.cmml" xref="S4.T4.7.7.1.1.1.m1.1.1.1">subscript</csymbol><ci id="S4.T4.7.7.1.1.1.m1.1.1.1.2.cmml" xref="S4.T4.7.7.1.1.1.m1.1.1.1.2">𝚺</ci><cn id="S4.T4.7.7.1.1.1.m1.1.1.1.3.cmml" type="integer" xref="S4.T4.7.7.1.1.1.m1.1.1.1.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.T4.7.7.1.1.1.m1.1c">{}_{\operatorname{{\mathbf{\Sigma}}}_{1}}</annotation><annotation encoding="application/x-llamapun" id="S4.T4.7.7.1.1.1.m1.1d">start_FLOATSUBSCRIPT bold_Σ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_FLOATSUBSCRIPT</annotation></semantics></math> </td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T4.8.8.2.2.2">err<math alttext="{}_{\operatorname{{\mathbf{\Sigma}}}_{2}}" class="ltx_Math" display="inline" id="S4.T4.8.8.2.2.2.m1.1"><semantics id="S4.T4.8.8.2.2.2.m1.1a"><msub id="S4.T4.8.8.2.2.2.m1.1.1" xref="S4.T4.8.8.2.2.2.m1.1.1.cmml"><mi id="S4.T4.8.8.2.2.2.m1.1.1a" xref="S4.T4.8.8.2.2.2.m1.1.1.cmml"></mi><msub id="S4.T4.8.8.2.2.2.m1.1.1.1" xref="S4.T4.8.8.2.2.2.m1.1.1.1.cmml"><mi id="S4.T4.8.8.2.2.2.m1.1.1.1.2" xref="S4.T4.8.8.2.2.2.m1.1.1.1.2.cmml">𝚺</mi><mn id="S4.T4.8.8.2.2.2.m1.1.1.1.3" xref="S4.T4.8.8.2.2.2.m1.1.1.1.3.cmml">2</mn></msub></msub><annotation-xml encoding="MathML-Content" id="S4.T4.8.8.2.2.2.m1.1b"><apply id="S4.T4.8.8.2.2.2.m1.1.1.cmml" xref="S4.T4.8.8.2.2.2.m1.1.1"><apply id="S4.T4.8.8.2.2.2.m1.1.1.1.cmml" xref="S4.T4.8.8.2.2.2.m1.1.1.1"><csymbol cd="ambiguous" id="S4.T4.8.8.2.2.2.m1.1.1.1.1.cmml" xref="S4.T4.8.8.2.2.2.m1.1.1.1">subscript</csymbol><ci id="S4.T4.8.8.2.2.2.m1.1.1.1.2.cmml" xref="S4.T4.8.8.2.2.2.m1.1.1.1.2">𝚺</ci><cn id="S4.T4.8.8.2.2.2.m1.1.1.1.3.cmml" type="integer" xref="S4.T4.8.8.2.2.2.m1.1.1.1.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.T4.8.8.2.2.2.m1.1c">{}_{\operatorname{{\mathbf{\Sigma}}}_{2}}</annotation><annotation encoding="application/x-llamapun" id="S4.T4.8.8.2.2.2.m1.1d">start_FLOATSUBSCRIPT bold_Σ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_FLOATSUBSCRIPT</annotation></semantics></math> </td> </tr> <tr class="ltx_tr" id="S4.T4.16.16.10.11.1"> <th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l ltx_border_r ltx_border_t" colspan="3" id="S4.T4.16.16.10.11.1.1"><span class="ltx_text ltx_font_bold" id="S4.T4.16.16.10.11.1.1.1">n = 18</span></th> </tr> <tr class="ltx_tr" id="S4.T4.9.9.3.3"> <th class="ltx_td ltx_align_right ltx_th ltx_th_row ltx_border_l ltx_border_r ltx_border_t" id="S4.T4.9.9.3.3.1"> <math alttext="m_{i}=2" class="ltx_Math" display="inline" id="S4.T4.9.9.3.3.1.m1.1"><semantics id="S4.T4.9.9.3.3.1.m1.1a"><mrow id="S4.T4.9.9.3.3.1.m1.1.1" xref="S4.T4.9.9.3.3.1.m1.1.1.cmml"><msub id="S4.T4.9.9.3.3.1.m1.1.1.2" xref="S4.T4.9.9.3.3.1.m1.1.1.2.cmml"><mi id="S4.T4.9.9.3.3.1.m1.1.1.2.2" xref="S4.T4.9.9.3.3.1.m1.1.1.2.2.cmml">m</mi><mi id="S4.T4.9.9.3.3.1.m1.1.1.2.3" xref="S4.T4.9.9.3.3.1.m1.1.1.2.3.cmml">i</mi></msub><mo id="S4.T4.9.9.3.3.1.m1.1.1.1" xref="S4.T4.9.9.3.3.1.m1.1.1.1.cmml">=</mo><mn id="S4.T4.9.9.3.3.1.m1.1.1.3" xref="S4.T4.9.9.3.3.1.m1.1.1.3.cmml">2</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.T4.9.9.3.3.1.m1.1b"><apply id="S4.T4.9.9.3.3.1.m1.1.1.cmml" xref="S4.T4.9.9.3.3.1.m1.1.1"><eq id="S4.T4.9.9.3.3.1.m1.1.1.1.cmml" xref="S4.T4.9.9.3.3.1.m1.1.1.1"></eq><apply id="S4.T4.9.9.3.3.1.m1.1.1.2.cmml" xref="S4.T4.9.9.3.3.1.m1.1.1.2"><csymbol cd="ambiguous" id="S4.T4.9.9.3.3.1.m1.1.1.2.1.cmml" xref="S4.T4.9.9.3.3.1.m1.1.1.2">subscript</csymbol><ci id="S4.T4.9.9.3.3.1.m1.1.1.2.2.cmml" xref="S4.T4.9.9.3.3.1.m1.1.1.2.2">𝑚</ci><ci id="S4.T4.9.9.3.3.1.m1.1.1.2.3.cmml" xref="S4.T4.9.9.3.3.1.m1.1.1.2.3">𝑖</ci></apply><cn id="S4.T4.9.9.3.3.1.m1.1.1.3.cmml" type="integer" xref="S4.T4.9.9.3.3.1.m1.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.T4.9.9.3.3.1.m1.1c">m_{i}=2</annotation><annotation encoding="application/x-llamapun" id="S4.T4.9.9.3.3.1.m1.1d">italic_m start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = 2</annotation></semantics></math> </th> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T4.9.9.3.3.2">1.10 (0.230)</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T4.9.9.3.3.3">1.10 (0.270)</td> </tr> <tr class="ltx_tr" id="S4.T4.10.10.4.4" style="background-color:#B2DFEE;"> <th class="ltx_td ltx_align_right ltx_th ltx_th_row ltx_border_l ltx_border_r ltx_border_t" id="S4.T4.10.10.4.4.1"><span class="ltx_text" id="S4.T4.10.10.4.4.1.1" style="background-color:#B2DFEE;"> <math alttext="m_{i}=6" class="ltx_Math" display="inline" id="S4.T4.10.10.4.4.1.1.m1.1"><semantics id="S4.T4.10.10.4.4.1.1.m1.1a"><mrow id="S4.T4.10.10.4.4.1.1.m1.1.1" xref="S4.T4.10.10.4.4.1.1.m1.1.1.cmml"><msub id="S4.T4.10.10.4.4.1.1.m1.1.1.2" xref="S4.T4.10.10.4.4.1.1.m1.1.1.2.cmml"><mi id="S4.T4.10.10.4.4.1.1.m1.1.1.2.2" mathbackground="#B2DFEE" xref="S4.T4.10.10.4.4.1.1.m1.1.1.2.2.cmml">m</mi><mi id="S4.T4.10.10.4.4.1.1.m1.1.1.2.3" mathbackground="#B2DFEE" xref="S4.T4.10.10.4.4.1.1.m1.1.1.2.3.cmml">i</mi></msub><mo id="S4.T4.10.10.4.4.1.1.m1.1.1.1" mathbackground="#B2DFEE" xref="S4.T4.10.10.4.4.1.1.m1.1.1.1.cmml">=</mo><mn id="S4.T4.10.10.4.4.1.1.m1.1.1.3" mathbackground="#B2DFEE" xref="S4.T4.10.10.4.4.1.1.m1.1.1.3.cmml">6</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.T4.10.10.4.4.1.1.m1.1b"><apply id="S4.T4.10.10.4.4.1.1.m1.1.1.cmml" xref="S4.T4.10.10.4.4.1.1.m1.1.1"><eq id="S4.T4.10.10.4.4.1.1.m1.1.1.1.cmml" xref="S4.T4.10.10.4.4.1.1.m1.1.1.1"></eq><apply id="S4.T4.10.10.4.4.1.1.m1.1.1.2.cmml" xref="S4.T4.10.10.4.4.1.1.m1.1.1.2"><csymbol cd="ambiguous" id="S4.T4.10.10.4.4.1.1.m1.1.1.2.1.cmml" xref="S4.T4.10.10.4.4.1.1.m1.1.1.2">subscript</csymbol><ci id="S4.T4.10.10.4.4.1.1.m1.1.1.2.2.cmml" xref="S4.T4.10.10.4.4.1.1.m1.1.1.2.2">𝑚</ci><ci id="S4.T4.10.10.4.4.1.1.m1.1.1.2.3.cmml" xref="S4.T4.10.10.4.4.1.1.m1.1.1.2.3">𝑖</ci></apply><cn id="S4.T4.10.10.4.4.1.1.m1.1.1.3.cmml" type="integer" xref="S4.T4.10.10.4.4.1.1.m1.1.1.3">6</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.T4.10.10.4.4.1.1.m1.1c">m_{i}=6</annotation><annotation encoding="application/x-llamapun" id="S4.T4.10.10.4.4.1.1.m1.1d">italic_m start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = 6</annotation></semantics></math></span></th> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T4.10.10.4.4.2"><span class="ltx_text" id="S4.T4.10.10.4.4.2.1" style="background-color:#B2DFEE;">0.57 (0.260)</span></td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T4.10.10.4.4.3"><span class="ltx_text" id="S4.T4.10.10.4.4.3.1" style="background-color:#B2DFEE;">0.54 (0.250)</span></td> </tr> <tr class="ltx_tr" id="S4.T4.16.16.10.12.2"> <th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l ltx_border_r ltx_border_t" colspan="3" id="S4.T4.16.16.10.12.2.1"><span class="ltx_text ltx_font_bold" id="S4.T4.16.16.10.12.2.1.1">n = 27</span></th> </tr> <tr class="ltx_tr" id="S4.T4.11.11.5.5"> <th class="ltx_td ltx_align_right ltx_th ltx_th_row ltx_border_l ltx_border_r ltx_border_t" id="S4.T4.11.11.5.5.1"> <math alttext="m_{i}=2" class="ltx_Math" display="inline" id="S4.T4.11.11.5.5.1.m1.1"><semantics id="S4.T4.11.11.5.5.1.m1.1a"><mrow id="S4.T4.11.11.5.5.1.m1.1.1" xref="S4.T4.11.11.5.5.1.m1.1.1.cmml"><msub id="S4.T4.11.11.5.5.1.m1.1.1.2" xref="S4.T4.11.11.5.5.1.m1.1.1.2.cmml"><mi id="S4.T4.11.11.5.5.1.m1.1.1.2.2" xref="S4.T4.11.11.5.5.1.m1.1.1.2.2.cmml">m</mi><mi id="S4.T4.11.11.5.5.1.m1.1.1.2.3" xref="S4.T4.11.11.5.5.1.m1.1.1.2.3.cmml">i</mi></msub><mo id="S4.T4.11.11.5.5.1.m1.1.1.1" xref="S4.T4.11.11.5.5.1.m1.1.1.1.cmml">=</mo><mn id="S4.T4.11.11.5.5.1.m1.1.1.3" xref="S4.T4.11.11.5.5.1.m1.1.1.3.cmml">2</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.T4.11.11.5.5.1.m1.1b"><apply id="S4.T4.11.11.5.5.1.m1.1.1.cmml" xref="S4.T4.11.11.5.5.1.m1.1.1"><eq id="S4.T4.11.11.5.5.1.m1.1.1.1.cmml" xref="S4.T4.11.11.5.5.1.m1.1.1.1"></eq><apply id="S4.T4.11.11.5.5.1.m1.1.1.2.cmml" xref="S4.T4.11.11.5.5.1.m1.1.1.2"><csymbol cd="ambiguous" id="S4.T4.11.11.5.5.1.m1.1.1.2.1.cmml" xref="S4.T4.11.11.5.5.1.m1.1.1.2">subscript</csymbol><ci id="S4.T4.11.11.5.5.1.m1.1.1.2.2.cmml" xref="S4.T4.11.11.5.5.1.m1.1.1.2.2">𝑚</ci><ci id="S4.T4.11.11.5.5.1.m1.1.1.2.3.cmml" xref="S4.T4.11.11.5.5.1.m1.1.1.2.3">𝑖</ci></apply><cn id="S4.T4.11.11.5.5.1.m1.1.1.3.cmml" type="integer" xref="S4.T4.11.11.5.5.1.m1.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.T4.11.11.5.5.1.m1.1c">m_{i}=2</annotation><annotation encoding="application/x-llamapun" id="S4.T4.11.11.5.5.1.m1.1d">italic_m start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = 2</annotation></semantics></math> </th> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T4.11.11.5.5.2">1.00 (0.290)</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T4.11.11.5.5.3">1.10 (0.300)</td> </tr> <tr class="ltx_tr" id="S4.T4.12.12.6.6" style="background-color:#EEA2AD;"> <th class="ltx_td ltx_align_right ltx_th ltx_th_row ltx_border_l ltx_border_r ltx_border_t" id="S4.T4.12.12.6.6.1"><span class="ltx_text" id="S4.T4.12.12.6.6.1.1" style="background-color:#EEA2AD;"> <math alttext="m_{i}=6" class="ltx_Math" display="inline" id="S4.T4.12.12.6.6.1.1.m1.1"><semantics id="S4.T4.12.12.6.6.1.1.m1.1a"><mrow id="S4.T4.12.12.6.6.1.1.m1.1.1" xref="S4.T4.12.12.6.6.1.1.m1.1.1.cmml"><msub id="S4.T4.12.12.6.6.1.1.m1.1.1.2" xref="S4.T4.12.12.6.6.1.1.m1.1.1.2.cmml"><mi id="S4.T4.12.12.6.6.1.1.m1.1.1.2.2" mathbackground="#EEA2AD" xref="S4.T4.12.12.6.6.1.1.m1.1.1.2.2.cmml">m</mi><mi id="S4.T4.12.12.6.6.1.1.m1.1.1.2.3" mathbackground="#EEA2AD" xref="S4.T4.12.12.6.6.1.1.m1.1.1.2.3.cmml">i</mi></msub><mo id="S4.T4.12.12.6.6.1.1.m1.1.1.1" mathbackground="#EEA2AD" xref="S4.T4.12.12.6.6.1.1.m1.1.1.1.cmml">=</mo><mn id="S4.T4.12.12.6.6.1.1.m1.1.1.3" mathbackground="#EEA2AD" xref="S4.T4.12.12.6.6.1.1.m1.1.1.3.cmml">6</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.T4.12.12.6.6.1.1.m1.1b"><apply id="S4.T4.12.12.6.6.1.1.m1.1.1.cmml" xref="S4.T4.12.12.6.6.1.1.m1.1.1"><eq id="S4.T4.12.12.6.6.1.1.m1.1.1.1.cmml" xref="S4.T4.12.12.6.6.1.1.m1.1.1.1"></eq><apply id="S4.T4.12.12.6.6.1.1.m1.1.1.2.cmml" xref="S4.T4.12.12.6.6.1.1.m1.1.1.2"><csymbol cd="ambiguous" id="S4.T4.12.12.6.6.1.1.m1.1.1.2.1.cmml" xref="S4.T4.12.12.6.6.1.1.m1.1.1.2">subscript</csymbol><ci id="S4.T4.12.12.6.6.1.1.m1.1.1.2.2.cmml" xref="S4.T4.12.12.6.6.1.1.m1.1.1.2.2">𝑚</ci><ci id="S4.T4.12.12.6.6.1.1.m1.1.1.2.3.cmml" xref="S4.T4.12.12.6.6.1.1.m1.1.1.2.3">𝑖</ci></apply><cn id="S4.T4.12.12.6.6.1.1.m1.1.1.3.cmml" type="integer" xref="S4.T4.12.12.6.6.1.1.m1.1.1.3">6</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.T4.12.12.6.6.1.1.m1.1c">m_{i}=6</annotation><annotation encoding="application/x-llamapun" id="S4.T4.12.12.6.6.1.1.m1.1d">italic_m start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = 6</annotation></semantics></math></span></th> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T4.12.12.6.6.2"><span class="ltx_text" id="S4.T4.12.12.6.6.2.1" style="background-color:#EEA2AD;">0.40 (0.170)</span></td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T4.12.12.6.6.3"><span class="ltx_text" id="S4.T4.12.12.6.6.3.1" style="background-color:#EEA2AD;">0.39 (0.150)</span></td> </tr> <tr class="ltx_tr" id="S4.T4.16.16.10.13.3"> <th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l ltx_border_r ltx_border_t" colspan="3" id="S4.T4.16.16.10.13.3.1"><span class="ltx_text ltx_font_bold" id="S4.T4.16.16.10.13.3.1.1">n = 54</span></th> </tr> <tr class="ltx_tr" id="S4.T4.13.13.7.7" style="background-color:#B2DFEE;"> <th class="ltx_td ltx_align_right ltx_th ltx_th_row ltx_border_l ltx_border_r ltx_border_t" id="S4.T4.13.13.7.7.1"><span class="ltx_text" id="S4.T4.13.13.7.7.1.1" style="background-color:#B2DFEE;"> <math alttext="m_{i}=2" class="ltx_Math" display="inline" id="S4.T4.13.13.7.7.1.1.m1.1"><semantics id="S4.T4.13.13.7.7.1.1.m1.1a"><mrow id="S4.T4.13.13.7.7.1.1.m1.1.1" xref="S4.T4.13.13.7.7.1.1.m1.1.1.cmml"><msub id="S4.T4.13.13.7.7.1.1.m1.1.1.2" xref="S4.T4.13.13.7.7.1.1.m1.1.1.2.cmml"><mi id="S4.T4.13.13.7.7.1.1.m1.1.1.2.2" mathbackground="#B2DFEE" xref="S4.T4.13.13.7.7.1.1.m1.1.1.2.2.cmml">m</mi><mi id="S4.T4.13.13.7.7.1.1.m1.1.1.2.3" mathbackground="#B2DFEE" xref="S4.T4.13.13.7.7.1.1.m1.1.1.2.3.cmml">i</mi></msub><mo id="S4.T4.13.13.7.7.1.1.m1.1.1.1" mathbackground="#B2DFEE" xref="S4.T4.13.13.7.7.1.1.m1.1.1.1.cmml">=</mo><mn id="S4.T4.13.13.7.7.1.1.m1.1.1.3" mathbackground="#B2DFEE" xref="S4.T4.13.13.7.7.1.1.m1.1.1.3.cmml">2</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.T4.13.13.7.7.1.1.m1.1b"><apply id="S4.T4.13.13.7.7.1.1.m1.1.1.cmml" xref="S4.T4.13.13.7.7.1.1.m1.1.1"><eq id="S4.T4.13.13.7.7.1.1.m1.1.1.1.cmml" xref="S4.T4.13.13.7.7.1.1.m1.1.1.1"></eq><apply id="S4.T4.13.13.7.7.1.1.m1.1.1.2.cmml" xref="S4.T4.13.13.7.7.1.1.m1.1.1.2"><csymbol cd="ambiguous" id="S4.T4.13.13.7.7.1.1.m1.1.1.2.1.cmml" xref="S4.T4.13.13.7.7.1.1.m1.1.1.2">subscript</csymbol><ci id="S4.T4.13.13.7.7.1.1.m1.1.1.2.2.cmml" xref="S4.T4.13.13.7.7.1.1.m1.1.1.2.2">𝑚</ci><ci id="S4.T4.13.13.7.7.1.1.m1.1.1.2.3.cmml" xref="S4.T4.13.13.7.7.1.1.m1.1.1.2.3">𝑖</ci></apply><cn id="S4.T4.13.13.7.7.1.1.m1.1.1.3.cmml" type="integer" xref="S4.T4.13.13.7.7.1.1.m1.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.T4.13.13.7.7.1.1.m1.1c">m_{i}=2</annotation><annotation encoding="application/x-llamapun" id="S4.T4.13.13.7.7.1.1.m1.1d">italic_m start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = 2</annotation></semantics></math></span></th> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T4.13.13.7.7.2"><span class="ltx_text" id="S4.T4.13.13.7.7.2.1" style="background-color:#B2DFEE;">0.84 (0.290)</span></td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T4.13.13.7.7.3"><span class="ltx_text" id="S4.T4.13.13.7.7.3.1" style="background-color:#B2DFEE;">0.83 (0.340)</span></td> </tr> <tr class="ltx_tr" id="S4.T4.14.14.8.8"> <th class="ltx_td ltx_align_right ltx_th ltx_th_row ltx_border_l ltx_border_r ltx_border_t" id="S4.T4.14.14.8.8.1"> <math alttext="m_{i}=6" class="ltx_Math" display="inline" id="S4.T4.14.14.8.8.1.m1.1"><semantics id="S4.T4.14.14.8.8.1.m1.1a"><mrow id="S4.T4.14.14.8.8.1.m1.1.1" xref="S4.T4.14.14.8.8.1.m1.1.1.cmml"><msub id="S4.T4.14.14.8.8.1.m1.1.1.2" xref="S4.T4.14.14.8.8.1.m1.1.1.2.cmml"><mi id="S4.T4.14.14.8.8.1.m1.1.1.2.2" xref="S4.T4.14.14.8.8.1.m1.1.1.2.2.cmml">m</mi><mi id="S4.T4.14.14.8.8.1.m1.1.1.2.3" xref="S4.T4.14.14.8.8.1.m1.1.1.2.3.cmml">i</mi></msub><mo id="S4.T4.14.14.8.8.1.m1.1.1.1" xref="S4.T4.14.14.8.8.1.m1.1.1.1.cmml">=</mo><mn id="S4.T4.14.14.8.8.1.m1.1.1.3" xref="S4.T4.14.14.8.8.1.m1.1.1.3.cmml">6</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.T4.14.14.8.8.1.m1.1b"><apply id="S4.T4.14.14.8.8.1.m1.1.1.cmml" xref="S4.T4.14.14.8.8.1.m1.1.1"><eq id="S4.T4.14.14.8.8.1.m1.1.1.1.cmml" xref="S4.T4.14.14.8.8.1.m1.1.1.1"></eq><apply id="S4.T4.14.14.8.8.1.m1.1.1.2.cmml" xref="S4.T4.14.14.8.8.1.m1.1.1.2"><csymbol cd="ambiguous" id="S4.T4.14.14.8.8.1.m1.1.1.2.1.cmml" xref="S4.T4.14.14.8.8.1.m1.1.1.2">subscript</csymbol><ci id="S4.T4.14.14.8.8.1.m1.1.1.2.2.cmml" xref="S4.T4.14.14.8.8.1.m1.1.1.2.2">𝑚</ci><ci id="S4.T4.14.14.8.8.1.m1.1.1.2.3.cmml" xref="S4.T4.14.14.8.8.1.m1.1.1.2.3">𝑖</ci></apply><cn id="S4.T4.14.14.8.8.1.m1.1.1.3.cmml" type="integer" xref="S4.T4.14.14.8.8.1.m1.1.1.3">6</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.T4.14.14.8.8.1.m1.1c">m_{i}=6</annotation><annotation encoding="application/x-llamapun" id="S4.T4.14.14.8.8.1.m1.1d">italic_m start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = 6</annotation></semantics></math> </th> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T4.14.14.8.8.2">0.21 (0.100)</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T4.14.14.8.8.3">0.21 (0.085)</td> </tr> <tr class="ltx_tr" id="S4.T4.16.16.10.14.4"> <th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l ltx_border_r ltx_border_t" colspan="3" id="S4.T4.16.16.10.14.4.1"><span class="ltx_text ltx_font_bold" id="S4.T4.16.16.10.14.4.1.1">n = 81</span></th> </tr> <tr class="ltx_tr" id="S4.T4.15.15.9.9" style="background-color:#EEA2AD;"> <th class="ltx_td ltx_align_right ltx_th ltx_th_row ltx_border_l ltx_border_r ltx_border_t" id="S4.T4.15.15.9.9.1"><span class="ltx_text" id="S4.T4.15.15.9.9.1.1" style="background-color:#EEA2AD;"> <math alttext="m_{i}=2" class="ltx_Math" display="inline" id="S4.T4.15.15.9.9.1.1.m1.1"><semantics id="S4.T4.15.15.9.9.1.1.m1.1a"><mrow id="S4.T4.15.15.9.9.1.1.m1.1.1" xref="S4.T4.15.15.9.9.1.1.m1.1.1.cmml"><msub id="S4.T4.15.15.9.9.1.1.m1.1.1.2" xref="S4.T4.15.15.9.9.1.1.m1.1.1.2.cmml"><mi id="S4.T4.15.15.9.9.1.1.m1.1.1.2.2" mathbackground="#EEA2AD" xref="S4.T4.15.15.9.9.1.1.m1.1.1.2.2.cmml">m</mi><mi id="S4.T4.15.15.9.9.1.1.m1.1.1.2.3" mathbackground="#EEA2AD" xref="S4.T4.15.15.9.9.1.1.m1.1.1.2.3.cmml">i</mi></msub><mo id="S4.T4.15.15.9.9.1.1.m1.1.1.1" mathbackground="#EEA2AD" xref="S4.T4.15.15.9.9.1.1.m1.1.1.1.cmml">=</mo><mn id="S4.T4.15.15.9.9.1.1.m1.1.1.3" mathbackground="#EEA2AD" xref="S4.T4.15.15.9.9.1.1.m1.1.1.3.cmml">2</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.T4.15.15.9.9.1.1.m1.1b"><apply id="S4.T4.15.15.9.9.1.1.m1.1.1.cmml" xref="S4.T4.15.15.9.9.1.1.m1.1.1"><eq id="S4.T4.15.15.9.9.1.1.m1.1.1.1.cmml" xref="S4.T4.15.15.9.9.1.1.m1.1.1.1"></eq><apply id="S4.T4.15.15.9.9.1.1.m1.1.1.2.cmml" xref="S4.T4.15.15.9.9.1.1.m1.1.1.2"><csymbol cd="ambiguous" id="S4.T4.15.15.9.9.1.1.m1.1.1.2.1.cmml" xref="S4.T4.15.15.9.9.1.1.m1.1.1.2">subscript</csymbol><ci id="S4.T4.15.15.9.9.1.1.m1.1.1.2.2.cmml" xref="S4.T4.15.15.9.9.1.1.m1.1.1.2.2">𝑚</ci><ci id="S4.T4.15.15.9.9.1.1.m1.1.1.2.3.cmml" xref="S4.T4.15.15.9.9.1.1.m1.1.1.2.3">𝑖</ci></apply><cn id="S4.T4.15.15.9.9.1.1.m1.1.1.3.cmml" type="integer" xref="S4.T4.15.15.9.9.1.1.m1.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.T4.15.15.9.9.1.1.m1.1c">m_{i}=2</annotation><annotation encoding="application/x-llamapun" id="S4.T4.15.15.9.9.1.1.m1.1d">italic_m start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = 2</annotation></semantics></math></span></th> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T4.15.15.9.9.2"><span class="ltx_text" id="S4.T4.15.15.9.9.2.1" style="background-color:#EEA2AD;">0.64 (0.320)</span></td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T4.15.15.9.9.3"><span class="ltx_text" id="S4.T4.15.15.9.9.3.1" style="background-color:#EEA2AD;">0.63 (0.280)</span></td> </tr> <tr class="ltx_tr" id="S4.T4.16.16.10.10"> <th class="ltx_td ltx_align_right ltx_th ltx_th_row ltx_border_b ltx_border_l ltx_border_r ltx_border_t" id="S4.T4.16.16.10.10.1"> <math alttext="m_{i}=6" class="ltx_Math" display="inline" id="S4.T4.16.16.10.10.1.m1.1"><semantics id="S4.T4.16.16.10.10.1.m1.1a"><mrow id="S4.T4.16.16.10.10.1.m1.1.1" xref="S4.T4.16.16.10.10.1.m1.1.1.cmml"><msub id="S4.T4.16.16.10.10.1.m1.1.1.2" xref="S4.T4.16.16.10.10.1.m1.1.1.2.cmml"><mi id="S4.T4.16.16.10.10.1.m1.1.1.2.2" xref="S4.T4.16.16.10.10.1.m1.1.1.2.2.cmml">m</mi><mi id="S4.T4.16.16.10.10.1.m1.1.1.2.3" xref="S4.T4.16.16.10.10.1.m1.1.1.2.3.cmml">i</mi></msub><mo id="S4.T4.16.16.10.10.1.m1.1.1.1" xref="S4.T4.16.16.10.10.1.m1.1.1.1.cmml">=</mo><mn id="S4.T4.16.16.10.10.1.m1.1.1.3" xref="S4.T4.16.16.10.10.1.m1.1.1.3.cmml">6</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.T4.16.16.10.10.1.m1.1b"><apply id="S4.T4.16.16.10.10.1.m1.1.1.cmml" xref="S4.T4.16.16.10.10.1.m1.1.1"><eq id="S4.T4.16.16.10.10.1.m1.1.1.1.cmml" xref="S4.T4.16.16.10.10.1.m1.1.1.1"></eq><apply id="S4.T4.16.16.10.10.1.m1.1.1.2.cmml" xref="S4.T4.16.16.10.10.1.m1.1.1.2"><csymbol cd="ambiguous" id="S4.T4.16.16.10.10.1.m1.1.1.2.1.cmml" xref="S4.T4.16.16.10.10.1.m1.1.1.2">subscript</csymbol><ci id="S4.T4.16.16.10.10.1.m1.1.1.2.2.cmml" xref="S4.T4.16.16.10.10.1.m1.1.1.2.2">𝑚</ci><ci id="S4.T4.16.16.10.10.1.m1.1.1.2.3.cmml" xref="S4.T4.16.16.10.10.1.m1.1.1.2.3">𝑖</ci></apply><cn id="S4.T4.16.16.10.10.1.m1.1.1.3.cmml" type="integer" xref="S4.T4.16.16.10.10.1.m1.1.1.3">6</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.T4.16.16.10.10.1.m1.1c">m_{i}=6</annotation><annotation encoding="application/x-llamapun" id="S4.T4.16.16.10.10.1.m1.1d">italic_m start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = 6</annotation></semantics></math> </th> <td class="ltx_td ltx_align_center ltx_border_b ltx_border_r ltx_border_t" id="S4.T4.16.16.10.10.2">0.17 (0.069)</td> <td class="ltx_td ltx_align_center ltx_border_b ltx_border_r ltx_border_t" id="S4.T4.16.16.10.10.3">0.16 (0.066)</td> </tr> </tbody> </table> </span></div> </figure> </div> <div class="ltx_flex_cell ltx_flex_size_2"> <figure class="ltx_figure ltx_figure_panel ltx_minipage ltx_align_middle" id="S4.T4.32" style="width:195.1pt;"> <figcaption class="ltx_caption"><span class="ltx_tag ltx_tag_figure">Table 4: </span>MMTR’s relative estimation error for <math alttext="\operatorname{{\mathbf{\Sigma}}}_{1}" class="ltx_Math" display="inline" id="S4.T4.20.4.m1.1"><semantics id="S4.T4.20.4.m1.1b"><msub id="S4.T4.20.4.m1.1.1" xref="S4.T4.20.4.m1.1.1.cmml"><mi id="S4.T4.20.4.m1.1.1.2" xref="S4.T4.20.4.m1.1.1.2.cmml">𝚺</mi><mn id="S4.T4.20.4.m1.1.1.3" xref="S4.T4.20.4.m1.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S4.T4.20.4.m1.1c"><apply id="S4.T4.20.4.m1.1.1.cmml" xref="S4.T4.20.4.m1.1.1"><csymbol cd="ambiguous" id="S4.T4.20.4.m1.1.1.1.cmml" xref="S4.T4.20.4.m1.1.1">subscript</csymbol><ci id="S4.T4.20.4.m1.1.1.2.cmml" xref="S4.T4.20.4.m1.1.1.2">𝚺</ci><cn id="S4.T4.20.4.m1.1.1.3.cmml" type="integer" xref="S4.T4.20.4.m1.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.T4.20.4.m1.1d">\operatorname{{\mathbf{\Sigma}}}_{1}</annotation><annotation encoding="application/x-llamapun" id="S4.T4.20.4.m1.1e">bold_Σ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="\operatorname{{\mathbf{\Sigma}}}_{2}" class="ltx_Math" display="inline" id="S4.T4.21.5.m2.1"><semantics id="S4.T4.21.5.m2.1b"><msub id="S4.T4.21.5.m2.1.1" xref="S4.T4.21.5.m2.1.1.cmml"><mi id="S4.T4.21.5.m2.1.1.2" xref="S4.T4.21.5.m2.1.1.2.cmml">𝚺</mi><mn id="S4.T4.21.5.m2.1.1.3" xref="S4.T4.21.5.m2.1.1.3.cmml">2</mn></msub><annotation-xml encoding="MathML-Content" id="S4.T4.21.5.m2.1c"><apply id="S4.T4.21.5.m2.1.1.cmml" xref="S4.T4.21.5.m2.1.1"><csymbol cd="ambiguous" id="S4.T4.21.5.m2.1.1.1.cmml" xref="S4.T4.21.5.m2.1.1">subscript</csymbol><ci id="S4.T4.21.5.m2.1.1.2.cmml" xref="S4.T4.21.5.m2.1.1.2">𝚺</ci><cn id="S4.T4.21.5.m2.1.1.3.cmml" type="integer" xref="S4.T4.21.5.m2.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.T4.21.5.m2.1d">\operatorname{{\mathbf{\Sigma}}}_{2}</annotation><annotation encoding="application/x-llamapun" id="S4.T4.21.5.m2.1e">bold_Σ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math> in Case 2. Every estimation error is averaged over 100 simulation replications, with the Monte Carlo errors in parenthesis. The red and blue lines represent simulations with equal <math alttext="N" class="ltx_Math" display="inline" id="S4.T4.22.6.m3.1"><semantics id="S4.T4.22.6.m3.1b"><mi id="S4.T4.22.6.m3.1.1" xref="S4.T4.22.6.m3.1.1.cmml">N</mi><annotation-xml encoding="MathML-Content" id="S4.T4.22.6.m3.1c"><ci id="S4.T4.22.6.m3.1.1.cmml" xref="S4.T4.22.6.m3.1.1">𝑁</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.T4.22.6.m3.1d">N</annotation><annotation encoding="application/x-llamapun" id="S4.T4.22.6.m3.1e">italic_N</annotation></semantics></math> values.</figcaption> <div class="ltx_inline-block ltx_align_center ltx_transformed_outer" id="S4.T4.32.16" style="width:303.5pt;height:387.5pt;vertical-align:-0.0pt;"><span class="ltx_transformed_inner" style="transform:translate(60.1pt,-76.7pt) scale(1.65590441996162,1.65590441996162) ;"> <table class="ltx_tabular ltx_guessed_headers ltx_align_top" id="S4.T4.32.16.10"> <tbody class="ltx_tbody"> <tr class="ltx_tr" id="S4.T4.24.8.2.2"> <th class="ltx_td ltx_align_right ltx_th ltx_th_row ltx_border_l ltx_border_r ltx_border_t" id="S4.T4.24.8.2.2.3">group size</th> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T4.23.7.1.1.1">err<math alttext="{}_{\operatorname{{\mathbf{\Sigma}}}_{1}}" class="ltx_Math" display="inline" id="S4.T4.23.7.1.1.1.m1.1"><semantics id="S4.T4.23.7.1.1.1.m1.1a"><msub id="S4.T4.23.7.1.1.1.m1.1.1" xref="S4.T4.23.7.1.1.1.m1.1.1.cmml"><mi id="S4.T4.23.7.1.1.1.m1.1.1a" xref="S4.T4.23.7.1.1.1.m1.1.1.cmml"></mi><msub id="S4.T4.23.7.1.1.1.m1.1.1.1" xref="S4.T4.23.7.1.1.1.m1.1.1.1.cmml"><mi id="S4.T4.23.7.1.1.1.m1.1.1.1.2" xref="S4.T4.23.7.1.1.1.m1.1.1.1.2.cmml">𝚺</mi><mn id="S4.T4.23.7.1.1.1.m1.1.1.1.3" xref="S4.T4.23.7.1.1.1.m1.1.1.1.3.cmml">1</mn></msub></msub><annotation-xml encoding="MathML-Content" id="S4.T4.23.7.1.1.1.m1.1b"><apply id="S4.T4.23.7.1.1.1.m1.1.1.cmml" xref="S4.T4.23.7.1.1.1.m1.1.1"><apply id="S4.T4.23.7.1.1.1.m1.1.1.1.cmml" xref="S4.T4.23.7.1.1.1.m1.1.1.1"><csymbol cd="ambiguous" id="S4.T4.23.7.1.1.1.m1.1.1.1.1.cmml" xref="S4.T4.23.7.1.1.1.m1.1.1.1">subscript</csymbol><ci id="S4.T4.23.7.1.1.1.m1.1.1.1.2.cmml" xref="S4.T4.23.7.1.1.1.m1.1.1.1.2">𝚺</ci><cn id="S4.T4.23.7.1.1.1.m1.1.1.1.3.cmml" type="integer" xref="S4.T4.23.7.1.1.1.m1.1.1.1.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.T4.23.7.1.1.1.m1.1c">{}_{\operatorname{{\mathbf{\Sigma}}}_{1}}</annotation><annotation encoding="application/x-llamapun" id="S4.T4.23.7.1.1.1.m1.1d">start_FLOATSUBSCRIPT bold_Σ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_FLOATSUBSCRIPT</annotation></semantics></math> </td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T4.24.8.2.2.2">err<math alttext="{}_{\operatorname{{\mathbf{\Sigma}}}_{2}}" class="ltx_Math" display="inline" id="S4.T4.24.8.2.2.2.m1.1"><semantics id="S4.T4.24.8.2.2.2.m1.1a"><msub id="S4.T4.24.8.2.2.2.m1.1.1" xref="S4.T4.24.8.2.2.2.m1.1.1.cmml"><mi id="S4.T4.24.8.2.2.2.m1.1.1a" xref="S4.T4.24.8.2.2.2.m1.1.1.cmml"></mi><msub id="S4.T4.24.8.2.2.2.m1.1.1.1" xref="S4.T4.24.8.2.2.2.m1.1.1.1.cmml"><mi id="S4.T4.24.8.2.2.2.m1.1.1.1.2" xref="S4.T4.24.8.2.2.2.m1.1.1.1.2.cmml">𝚺</mi><mn id="S4.T4.24.8.2.2.2.m1.1.1.1.3" xref="S4.T4.24.8.2.2.2.m1.1.1.1.3.cmml">2</mn></msub></msub><annotation-xml encoding="MathML-Content" id="S4.T4.24.8.2.2.2.m1.1b"><apply id="S4.T4.24.8.2.2.2.m1.1.1.cmml" xref="S4.T4.24.8.2.2.2.m1.1.1"><apply id="S4.T4.24.8.2.2.2.m1.1.1.1.cmml" xref="S4.T4.24.8.2.2.2.m1.1.1.1"><csymbol cd="ambiguous" id="S4.T4.24.8.2.2.2.m1.1.1.1.1.cmml" xref="S4.T4.24.8.2.2.2.m1.1.1.1">subscript</csymbol><ci id="S4.T4.24.8.2.2.2.m1.1.1.1.2.cmml" xref="S4.T4.24.8.2.2.2.m1.1.1.1.2">𝚺</ci><cn id="S4.T4.24.8.2.2.2.m1.1.1.1.3.cmml" type="integer" xref="S4.T4.24.8.2.2.2.m1.1.1.1.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.T4.24.8.2.2.2.m1.1c">{}_{\operatorname{{\mathbf{\Sigma}}}_{2}}</annotation><annotation encoding="application/x-llamapun" id="S4.T4.24.8.2.2.2.m1.1d">start_FLOATSUBSCRIPT bold_Σ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_FLOATSUBSCRIPT</annotation></semantics></math> </td> </tr> <tr class="ltx_tr" id="S4.T4.32.16.10.11.1"> <th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l ltx_border_r ltx_border_t" colspan="3" id="S4.T4.32.16.10.11.1.1"><span class="ltx_text ltx_font_bold" id="S4.T4.32.16.10.11.1.1.1">n = 64</span></th> </tr> <tr class="ltx_tr" id="S4.T4.25.9.3.3"> <th class="ltx_td ltx_align_right ltx_th ltx_th_row ltx_border_l ltx_border_r ltx_border_t" id="S4.T4.25.9.3.3.1"> <math alttext="m_{i}=2" class="ltx_Math" display="inline" id="S4.T4.25.9.3.3.1.m1.1"><semantics id="S4.T4.25.9.3.3.1.m1.1a"><mrow id="S4.T4.25.9.3.3.1.m1.1.1" xref="S4.T4.25.9.3.3.1.m1.1.1.cmml"><msub id="S4.T4.25.9.3.3.1.m1.1.1.2" xref="S4.T4.25.9.3.3.1.m1.1.1.2.cmml"><mi id="S4.T4.25.9.3.3.1.m1.1.1.2.2" xref="S4.T4.25.9.3.3.1.m1.1.1.2.2.cmml">m</mi><mi id="S4.T4.25.9.3.3.1.m1.1.1.2.3" xref="S4.T4.25.9.3.3.1.m1.1.1.2.3.cmml">i</mi></msub><mo id="S4.T4.25.9.3.3.1.m1.1.1.1" xref="S4.T4.25.9.3.3.1.m1.1.1.1.cmml">=</mo><mn id="S4.T4.25.9.3.3.1.m1.1.1.3" xref="S4.T4.25.9.3.3.1.m1.1.1.3.cmml">2</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.T4.25.9.3.3.1.m1.1b"><apply id="S4.T4.25.9.3.3.1.m1.1.1.cmml" xref="S4.T4.25.9.3.3.1.m1.1.1"><eq id="S4.T4.25.9.3.3.1.m1.1.1.1.cmml" xref="S4.T4.25.9.3.3.1.m1.1.1.1"></eq><apply id="S4.T4.25.9.3.3.1.m1.1.1.2.cmml" xref="S4.T4.25.9.3.3.1.m1.1.1.2"><csymbol cd="ambiguous" id="S4.T4.25.9.3.3.1.m1.1.1.2.1.cmml" xref="S4.T4.25.9.3.3.1.m1.1.1.2">subscript</csymbol><ci id="S4.T4.25.9.3.3.1.m1.1.1.2.2.cmml" xref="S4.T4.25.9.3.3.1.m1.1.1.2.2">𝑚</ci><ci id="S4.T4.25.9.3.3.1.m1.1.1.2.3.cmml" xref="S4.T4.25.9.3.3.1.m1.1.1.2.3">𝑖</ci></apply><cn id="S4.T4.25.9.3.3.1.m1.1.1.3.cmml" type="integer" xref="S4.T4.25.9.3.3.1.m1.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.T4.25.9.3.3.1.m1.1c">m_{i}=2</annotation><annotation encoding="application/x-llamapun" id="S4.T4.25.9.3.3.1.m1.1d">italic_m start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = 2</annotation></semantics></math> </th> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T4.25.9.3.3.2">1.20 (0.200)</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T4.25.9.3.3.3">1.20 (0.170)</td> </tr> <tr class="ltx_tr" id="S4.T4.26.10.4.4" style="background-color:#B2DFEE;"> <th class="ltx_td ltx_align_right ltx_th ltx_th_row ltx_border_l ltx_border_r ltx_border_t" id="S4.T4.26.10.4.4.1"><span class="ltx_text" id="S4.T4.26.10.4.4.1.1" style="background-color:#B2DFEE;"> <math alttext="m_{i}=6" class="ltx_Math" display="inline" id="S4.T4.26.10.4.4.1.1.m1.1"><semantics id="S4.T4.26.10.4.4.1.1.m1.1a"><mrow id="S4.T4.26.10.4.4.1.1.m1.1.1" xref="S4.T4.26.10.4.4.1.1.m1.1.1.cmml"><msub id="S4.T4.26.10.4.4.1.1.m1.1.1.2" xref="S4.T4.26.10.4.4.1.1.m1.1.1.2.cmml"><mi id="S4.T4.26.10.4.4.1.1.m1.1.1.2.2" mathbackground="#B2DFEE" xref="S4.T4.26.10.4.4.1.1.m1.1.1.2.2.cmml">m</mi><mi id="S4.T4.26.10.4.4.1.1.m1.1.1.2.3" mathbackground="#B2DFEE" xref="S4.T4.26.10.4.4.1.1.m1.1.1.2.3.cmml">i</mi></msub><mo id="S4.T4.26.10.4.4.1.1.m1.1.1.1" mathbackground="#B2DFEE" xref="S4.T4.26.10.4.4.1.1.m1.1.1.1.cmml">=</mo><mn id="S4.T4.26.10.4.4.1.1.m1.1.1.3" mathbackground="#B2DFEE" xref="S4.T4.26.10.4.4.1.1.m1.1.1.3.cmml">6</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.T4.26.10.4.4.1.1.m1.1b"><apply id="S4.T4.26.10.4.4.1.1.m1.1.1.cmml" xref="S4.T4.26.10.4.4.1.1.m1.1.1"><eq id="S4.T4.26.10.4.4.1.1.m1.1.1.1.cmml" xref="S4.T4.26.10.4.4.1.1.m1.1.1.1"></eq><apply id="S4.T4.26.10.4.4.1.1.m1.1.1.2.cmml" xref="S4.T4.26.10.4.4.1.1.m1.1.1.2"><csymbol cd="ambiguous" id="S4.T4.26.10.4.4.1.1.m1.1.1.2.1.cmml" xref="S4.T4.26.10.4.4.1.1.m1.1.1.2">subscript</csymbol><ci id="S4.T4.26.10.4.4.1.1.m1.1.1.2.2.cmml" xref="S4.T4.26.10.4.4.1.1.m1.1.1.2.2">𝑚</ci><ci id="S4.T4.26.10.4.4.1.1.m1.1.1.2.3.cmml" xref="S4.T4.26.10.4.4.1.1.m1.1.1.2.3">𝑖</ci></apply><cn id="S4.T4.26.10.4.4.1.1.m1.1.1.3.cmml" type="integer" xref="S4.T4.26.10.4.4.1.1.m1.1.1.3">6</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.T4.26.10.4.4.1.1.m1.1c">m_{i}=6</annotation><annotation encoding="application/x-llamapun" id="S4.T4.26.10.4.4.1.1.m1.1d">italic_m start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = 6</annotation></semantics></math></span></th> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T4.26.10.4.4.2"><span class="ltx_text" id="S4.T4.26.10.4.4.2.1" style="background-color:#B2DFEE;">0.80 (0.230)</span></td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T4.26.10.4.4.3"><span class="ltx_text" id="S4.T4.26.10.4.4.3.1" style="background-color:#B2DFEE;">0.80 (0.210)</span></td> </tr> <tr class="ltx_tr" id="S4.T4.32.16.10.12.2"> <th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l ltx_border_r ltx_border_t" colspan="3" id="S4.T4.32.16.10.12.2.1"><span class="ltx_text ltx_font_bold" id="S4.T4.32.16.10.12.2.1.1">n = 96</span></th> </tr> <tr class="ltx_tr" id="S4.T4.27.11.5.5"> <th class="ltx_td ltx_align_right ltx_th ltx_th_row ltx_border_l ltx_border_r ltx_border_t" id="S4.T4.27.11.5.5.1"> <math alttext="m_{i}=2" class="ltx_Math" display="inline" id="S4.T4.27.11.5.5.1.m1.1"><semantics id="S4.T4.27.11.5.5.1.m1.1a"><mrow id="S4.T4.27.11.5.5.1.m1.1.1" xref="S4.T4.27.11.5.5.1.m1.1.1.cmml"><msub id="S4.T4.27.11.5.5.1.m1.1.1.2" xref="S4.T4.27.11.5.5.1.m1.1.1.2.cmml"><mi id="S4.T4.27.11.5.5.1.m1.1.1.2.2" xref="S4.T4.27.11.5.5.1.m1.1.1.2.2.cmml">m</mi><mi id="S4.T4.27.11.5.5.1.m1.1.1.2.3" xref="S4.T4.27.11.5.5.1.m1.1.1.2.3.cmml">i</mi></msub><mo id="S4.T4.27.11.5.5.1.m1.1.1.1" xref="S4.T4.27.11.5.5.1.m1.1.1.1.cmml">=</mo><mn id="S4.T4.27.11.5.5.1.m1.1.1.3" xref="S4.T4.27.11.5.5.1.m1.1.1.3.cmml">2</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.T4.27.11.5.5.1.m1.1b"><apply id="S4.T4.27.11.5.5.1.m1.1.1.cmml" xref="S4.T4.27.11.5.5.1.m1.1.1"><eq id="S4.T4.27.11.5.5.1.m1.1.1.1.cmml" xref="S4.T4.27.11.5.5.1.m1.1.1.1"></eq><apply id="S4.T4.27.11.5.5.1.m1.1.1.2.cmml" xref="S4.T4.27.11.5.5.1.m1.1.1.2"><csymbol cd="ambiguous" id="S4.T4.27.11.5.5.1.m1.1.1.2.1.cmml" xref="S4.T4.27.11.5.5.1.m1.1.1.2">subscript</csymbol><ci id="S4.T4.27.11.5.5.1.m1.1.1.2.2.cmml" xref="S4.T4.27.11.5.5.1.m1.1.1.2.2">𝑚</ci><ci id="S4.T4.27.11.5.5.1.m1.1.1.2.3.cmml" xref="S4.T4.27.11.5.5.1.m1.1.1.2.3">𝑖</ci></apply><cn id="S4.T4.27.11.5.5.1.m1.1.1.3.cmml" type="integer" xref="S4.T4.27.11.5.5.1.m1.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.T4.27.11.5.5.1.m1.1c">m_{i}=2</annotation><annotation encoding="application/x-llamapun" id="S4.T4.27.11.5.5.1.m1.1d">italic_m start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = 2</annotation></semantics></math> </th> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T4.27.11.5.5.2">1.20 (0.190)</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T4.27.11.5.5.3">1.20 (0.190)</td> </tr> <tr class="ltx_tr" id="S4.T4.28.12.6.6" style="background-color:#EEA2AD;"> <th class="ltx_td ltx_align_right ltx_th ltx_th_row ltx_border_l ltx_border_r ltx_border_t" id="S4.T4.28.12.6.6.1"><span class="ltx_text" id="S4.T4.28.12.6.6.1.1" style="background-color:#EEA2AD;"> <math alttext="m_{i}=6" class="ltx_Math" display="inline" id="S4.T4.28.12.6.6.1.1.m1.1"><semantics id="S4.T4.28.12.6.6.1.1.m1.1a"><mrow id="S4.T4.28.12.6.6.1.1.m1.1.1" xref="S4.T4.28.12.6.6.1.1.m1.1.1.cmml"><msub id="S4.T4.28.12.6.6.1.1.m1.1.1.2" xref="S4.T4.28.12.6.6.1.1.m1.1.1.2.cmml"><mi id="S4.T4.28.12.6.6.1.1.m1.1.1.2.2" mathbackground="#EEA2AD" xref="S4.T4.28.12.6.6.1.1.m1.1.1.2.2.cmml">m</mi><mi id="S4.T4.28.12.6.6.1.1.m1.1.1.2.3" mathbackground="#EEA2AD" xref="S4.T4.28.12.6.6.1.1.m1.1.1.2.3.cmml">i</mi></msub><mo id="S4.T4.28.12.6.6.1.1.m1.1.1.1" mathbackground="#EEA2AD" xref="S4.T4.28.12.6.6.1.1.m1.1.1.1.cmml">=</mo><mn id="S4.T4.28.12.6.6.1.1.m1.1.1.3" mathbackground="#EEA2AD" xref="S4.T4.28.12.6.6.1.1.m1.1.1.3.cmml">6</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.T4.28.12.6.6.1.1.m1.1b"><apply id="S4.T4.28.12.6.6.1.1.m1.1.1.cmml" xref="S4.T4.28.12.6.6.1.1.m1.1.1"><eq id="S4.T4.28.12.6.6.1.1.m1.1.1.1.cmml" xref="S4.T4.28.12.6.6.1.1.m1.1.1.1"></eq><apply id="S4.T4.28.12.6.6.1.1.m1.1.1.2.cmml" xref="S4.T4.28.12.6.6.1.1.m1.1.1.2"><csymbol cd="ambiguous" id="S4.T4.28.12.6.6.1.1.m1.1.1.2.1.cmml" xref="S4.T4.28.12.6.6.1.1.m1.1.1.2">subscript</csymbol><ci id="S4.T4.28.12.6.6.1.1.m1.1.1.2.2.cmml" xref="S4.T4.28.12.6.6.1.1.m1.1.1.2.2">𝑚</ci><ci id="S4.T4.28.12.6.6.1.1.m1.1.1.2.3.cmml" xref="S4.T4.28.12.6.6.1.1.m1.1.1.2.3">𝑖</ci></apply><cn id="S4.T4.28.12.6.6.1.1.m1.1.1.3.cmml" type="integer" xref="S4.T4.28.12.6.6.1.1.m1.1.1.3">6</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.T4.28.12.6.6.1.1.m1.1c">m_{i}=6</annotation><annotation encoding="application/x-llamapun" id="S4.T4.28.12.6.6.1.1.m1.1d">italic_m start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = 6</annotation></semantics></math></span></th> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T4.28.12.6.6.2"><span class="ltx_text" id="S4.T4.28.12.6.6.2.1" style="background-color:#EEA2AD;">0.47 (0.160)</span></td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T4.28.12.6.6.3"><span class="ltx_text" id="S4.T4.28.12.6.6.3.1" style="background-color:#EEA2AD;">0.49 (0.160)</span></td> </tr> <tr class="ltx_tr" id="S4.T4.32.16.10.13.3"> <th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l ltx_border_r ltx_border_t" colspan="3" id="S4.T4.32.16.10.13.3.1"><span class="ltx_text ltx_font_bold" id="S4.T4.32.16.10.13.3.1.1">n = 192</span></th> </tr> <tr class="ltx_tr" id="S4.T4.29.13.7.7" style="background-color:#B2DFEE;"> <th class="ltx_td ltx_align_right ltx_th ltx_th_row ltx_border_l ltx_border_r ltx_border_t" id="S4.T4.29.13.7.7.1"><span class="ltx_text" id="S4.T4.29.13.7.7.1.1" style="background-color:#B2DFEE;"> <math alttext="m_{i}=2" class="ltx_Math" display="inline" id="S4.T4.29.13.7.7.1.1.m1.1"><semantics id="S4.T4.29.13.7.7.1.1.m1.1a"><mrow id="S4.T4.29.13.7.7.1.1.m1.1.1" xref="S4.T4.29.13.7.7.1.1.m1.1.1.cmml"><msub id="S4.T4.29.13.7.7.1.1.m1.1.1.2" xref="S4.T4.29.13.7.7.1.1.m1.1.1.2.cmml"><mi id="S4.T4.29.13.7.7.1.1.m1.1.1.2.2" mathbackground="#B2DFEE" xref="S4.T4.29.13.7.7.1.1.m1.1.1.2.2.cmml">m</mi><mi id="S4.T4.29.13.7.7.1.1.m1.1.1.2.3" mathbackground="#B2DFEE" xref="S4.T4.29.13.7.7.1.1.m1.1.1.2.3.cmml">i</mi></msub><mo id="S4.T4.29.13.7.7.1.1.m1.1.1.1" mathbackground="#B2DFEE" xref="S4.T4.29.13.7.7.1.1.m1.1.1.1.cmml">=</mo><mn id="S4.T4.29.13.7.7.1.1.m1.1.1.3" mathbackground="#B2DFEE" xref="S4.T4.29.13.7.7.1.1.m1.1.1.3.cmml">2</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.T4.29.13.7.7.1.1.m1.1b"><apply id="S4.T4.29.13.7.7.1.1.m1.1.1.cmml" xref="S4.T4.29.13.7.7.1.1.m1.1.1"><eq id="S4.T4.29.13.7.7.1.1.m1.1.1.1.cmml" xref="S4.T4.29.13.7.7.1.1.m1.1.1.1"></eq><apply id="S4.T4.29.13.7.7.1.1.m1.1.1.2.cmml" xref="S4.T4.29.13.7.7.1.1.m1.1.1.2"><csymbol cd="ambiguous" id="S4.T4.29.13.7.7.1.1.m1.1.1.2.1.cmml" xref="S4.T4.29.13.7.7.1.1.m1.1.1.2">subscript</csymbol><ci id="S4.T4.29.13.7.7.1.1.m1.1.1.2.2.cmml" xref="S4.T4.29.13.7.7.1.1.m1.1.1.2.2">𝑚</ci><ci id="S4.T4.29.13.7.7.1.1.m1.1.1.2.3.cmml" xref="S4.T4.29.13.7.7.1.1.m1.1.1.2.3">𝑖</ci></apply><cn id="S4.T4.29.13.7.7.1.1.m1.1.1.3.cmml" type="integer" xref="S4.T4.29.13.7.7.1.1.m1.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.T4.29.13.7.7.1.1.m1.1c">m_{i}=2</annotation><annotation encoding="application/x-llamapun" id="S4.T4.29.13.7.7.1.1.m1.1d">italic_m start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = 2</annotation></semantics></math></span></th> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T4.29.13.7.7.2"><span class="ltx_text" id="S4.T4.29.13.7.7.2.1" style="background-color:#B2DFEE;">1.10 (0.200)</span></td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T4.29.13.7.7.3"><span class="ltx_text" id="S4.T4.29.13.7.7.3.1" style="background-color:#B2DFEE;">1.10 (0.210)</span></td> </tr> <tr class="ltx_tr" id="S4.T4.30.14.8.8"> <th class="ltx_td ltx_align_right ltx_th ltx_th_row ltx_border_l ltx_border_r ltx_border_t" id="S4.T4.30.14.8.8.1"> <math alttext="m_{i}=6" class="ltx_Math" display="inline" id="S4.T4.30.14.8.8.1.m1.1"><semantics id="S4.T4.30.14.8.8.1.m1.1a"><mrow id="S4.T4.30.14.8.8.1.m1.1.1" xref="S4.T4.30.14.8.8.1.m1.1.1.cmml"><msub id="S4.T4.30.14.8.8.1.m1.1.1.2" xref="S4.T4.30.14.8.8.1.m1.1.1.2.cmml"><mi id="S4.T4.30.14.8.8.1.m1.1.1.2.2" xref="S4.T4.30.14.8.8.1.m1.1.1.2.2.cmml">m</mi><mi id="S4.T4.30.14.8.8.1.m1.1.1.2.3" xref="S4.T4.30.14.8.8.1.m1.1.1.2.3.cmml">i</mi></msub><mo id="S4.T4.30.14.8.8.1.m1.1.1.1" xref="S4.T4.30.14.8.8.1.m1.1.1.1.cmml">=</mo><mn id="S4.T4.30.14.8.8.1.m1.1.1.3" xref="S4.T4.30.14.8.8.1.m1.1.1.3.cmml">6</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.T4.30.14.8.8.1.m1.1b"><apply id="S4.T4.30.14.8.8.1.m1.1.1.cmml" xref="S4.T4.30.14.8.8.1.m1.1.1"><eq id="S4.T4.30.14.8.8.1.m1.1.1.1.cmml" xref="S4.T4.30.14.8.8.1.m1.1.1.1"></eq><apply id="S4.T4.30.14.8.8.1.m1.1.1.2.cmml" xref="S4.T4.30.14.8.8.1.m1.1.1.2"><csymbol cd="ambiguous" id="S4.T4.30.14.8.8.1.m1.1.1.2.1.cmml" xref="S4.T4.30.14.8.8.1.m1.1.1.2">subscript</csymbol><ci id="S4.T4.30.14.8.8.1.m1.1.1.2.2.cmml" xref="S4.T4.30.14.8.8.1.m1.1.1.2.2">𝑚</ci><ci id="S4.T4.30.14.8.8.1.m1.1.1.2.3.cmml" xref="S4.T4.30.14.8.8.1.m1.1.1.2.3">𝑖</ci></apply><cn id="S4.T4.30.14.8.8.1.m1.1.1.3.cmml" type="integer" xref="S4.T4.30.14.8.8.1.m1.1.1.3">6</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.T4.30.14.8.8.1.m1.1c">m_{i}=6</annotation><annotation encoding="application/x-llamapun" id="S4.T4.30.14.8.8.1.m1.1d">italic_m start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = 6</annotation></semantics></math> </th> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T4.30.14.8.8.2">0.24 (0.056)</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T4.30.14.8.8.3">0.23 (0.048)</td> </tr> <tr class="ltx_tr" id="S4.T4.32.16.10.14.4"> <th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l ltx_border_r ltx_border_t" colspan="3" id="S4.T4.32.16.10.14.4.1"><span class="ltx_text ltx_font_bold" id="S4.T4.32.16.10.14.4.1.1">n = 288</span></th> </tr> <tr class="ltx_tr" id="S4.T4.31.15.9.9" style="background-color:#EEA2AD;"> <th class="ltx_td ltx_align_right ltx_th ltx_th_row ltx_border_l ltx_border_r ltx_border_t" id="S4.T4.31.15.9.9.1"><span class="ltx_text" id="S4.T4.31.15.9.9.1.1" style="background-color:#EEA2AD;"> <math alttext="m_{i}=2" class="ltx_Math" display="inline" id="S4.T4.31.15.9.9.1.1.m1.1"><semantics id="S4.T4.31.15.9.9.1.1.m1.1a"><mrow id="S4.T4.31.15.9.9.1.1.m1.1.1" xref="S4.T4.31.15.9.9.1.1.m1.1.1.cmml"><msub id="S4.T4.31.15.9.9.1.1.m1.1.1.2" xref="S4.T4.31.15.9.9.1.1.m1.1.1.2.cmml"><mi id="S4.T4.31.15.9.9.1.1.m1.1.1.2.2" mathbackground="#EEA2AD" xref="S4.T4.31.15.9.9.1.1.m1.1.1.2.2.cmml">m</mi><mi id="S4.T4.31.15.9.9.1.1.m1.1.1.2.3" mathbackground="#EEA2AD" xref="S4.T4.31.15.9.9.1.1.m1.1.1.2.3.cmml">i</mi></msub><mo id="S4.T4.31.15.9.9.1.1.m1.1.1.1" mathbackground="#EEA2AD" xref="S4.T4.31.15.9.9.1.1.m1.1.1.1.cmml">=</mo><mn id="S4.T4.31.15.9.9.1.1.m1.1.1.3" mathbackground="#EEA2AD" xref="S4.T4.31.15.9.9.1.1.m1.1.1.3.cmml">2</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.T4.31.15.9.9.1.1.m1.1b"><apply id="S4.T4.31.15.9.9.1.1.m1.1.1.cmml" xref="S4.T4.31.15.9.9.1.1.m1.1.1"><eq id="S4.T4.31.15.9.9.1.1.m1.1.1.1.cmml" xref="S4.T4.31.15.9.9.1.1.m1.1.1.1"></eq><apply id="S4.T4.31.15.9.9.1.1.m1.1.1.2.cmml" xref="S4.T4.31.15.9.9.1.1.m1.1.1.2"><csymbol cd="ambiguous" id="S4.T4.31.15.9.9.1.1.m1.1.1.2.1.cmml" xref="S4.T4.31.15.9.9.1.1.m1.1.1.2">subscript</csymbol><ci id="S4.T4.31.15.9.9.1.1.m1.1.1.2.2.cmml" xref="S4.T4.31.15.9.9.1.1.m1.1.1.2.2">𝑚</ci><ci id="S4.T4.31.15.9.9.1.1.m1.1.1.2.3.cmml" xref="S4.T4.31.15.9.9.1.1.m1.1.1.2.3">𝑖</ci></apply><cn id="S4.T4.31.15.9.9.1.1.m1.1.1.3.cmml" type="integer" xref="S4.T4.31.15.9.9.1.1.m1.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.T4.31.15.9.9.1.1.m1.1c">m_{i}=2</annotation><annotation encoding="application/x-llamapun" id="S4.T4.31.15.9.9.1.1.m1.1d">italic_m start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = 2</annotation></semantics></math></span></th> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T4.31.15.9.9.2"><span class="ltx_text" id="S4.T4.31.15.9.9.2.1" style="background-color:#EEA2AD;">0.86 (0.280)</span></td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T4.31.15.9.9.3"><span class="ltx_text" id="S4.T4.31.15.9.9.3.1" style="background-color:#EEA2AD;">0.87 (0.260)</span></td> </tr> <tr class="ltx_tr" id="S4.T4.32.16.10.10"> <th class="ltx_td ltx_align_right ltx_th ltx_th_row ltx_border_b ltx_border_l ltx_border_r ltx_border_t" id="S4.T4.32.16.10.10.1"> <math alttext="m_{i}=6" class="ltx_Math" display="inline" id="S4.T4.32.16.10.10.1.m1.1"><semantics id="S4.T4.32.16.10.10.1.m1.1a"><mrow id="S4.T4.32.16.10.10.1.m1.1.1" xref="S4.T4.32.16.10.10.1.m1.1.1.cmml"><msub id="S4.T4.32.16.10.10.1.m1.1.1.2" xref="S4.T4.32.16.10.10.1.m1.1.1.2.cmml"><mi id="S4.T4.32.16.10.10.1.m1.1.1.2.2" xref="S4.T4.32.16.10.10.1.m1.1.1.2.2.cmml">m</mi><mi id="S4.T4.32.16.10.10.1.m1.1.1.2.3" xref="S4.T4.32.16.10.10.1.m1.1.1.2.3.cmml">i</mi></msub><mo id="S4.T4.32.16.10.10.1.m1.1.1.1" xref="S4.T4.32.16.10.10.1.m1.1.1.1.cmml">=</mo><mn id="S4.T4.32.16.10.10.1.m1.1.1.3" xref="S4.T4.32.16.10.10.1.m1.1.1.3.cmml">6</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.T4.32.16.10.10.1.m1.1b"><apply id="S4.T4.32.16.10.10.1.m1.1.1.cmml" xref="S4.T4.32.16.10.10.1.m1.1.1"><eq id="S4.T4.32.16.10.10.1.m1.1.1.1.cmml" xref="S4.T4.32.16.10.10.1.m1.1.1.1"></eq><apply id="S4.T4.32.16.10.10.1.m1.1.1.2.cmml" xref="S4.T4.32.16.10.10.1.m1.1.1.2"><csymbol cd="ambiguous" id="S4.T4.32.16.10.10.1.m1.1.1.2.1.cmml" xref="S4.T4.32.16.10.10.1.m1.1.1.2">subscript</csymbol><ci id="S4.T4.32.16.10.10.1.m1.1.1.2.2.cmml" xref="S4.T4.32.16.10.10.1.m1.1.1.2.2">𝑚</ci><ci id="S4.T4.32.16.10.10.1.m1.1.1.2.3.cmml" xref="S4.T4.32.16.10.10.1.m1.1.1.2.3">𝑖</ci></apply><cn id="S4.T4.32.16.10.10.1.m1.1.1.3.cmml" type="integer" xref="S4.T4.32.16.10.10.1.m1.1.1.3">6</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.T4.32.16.10.10.1.m1.1c">m_{i}=6</annotation><annotation encoding="application/x-llamapun" id="S4.T4.32.16.10.10.1.m1.1d">italic_m start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = 6</annotation></semantics></math> </th> <td class="ltx_td ltx_align_center ltx_border_b ltx_border_r ltx_border_t" id="S4.T4.32.16.10.10.2">0.17 (0.041)</td> <td class="ltx_td ltx_align_center ltx_border_b ltx_border_r ltx_border_t" id="S4.T4.32.16.10.10.3">0.18 (0.048)</td> </tr> </tbody> </table> </span></div> </figure> </div> </div> </figure> </section> <section class="ltx_subsection" id="S4.SS3"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">4.3 </span>Simulations Using the Equicorrelation GEE model</h3> <div class="ltx_para" id="S4.SS3.p1"> <p class="ltx_p" id="S4.SS3.p1.9">We simulated the data using the equicorrelation GEE model based on (<a class="ltx_ref" href="https://arxiv.org/html/2503.13782v1#S4.E12" title="In 4.1 Setup ‣ 4 Experiments ‣ Regularized Parameter Estimation in Mixed Model Trace Regression"><span class="ltx_text ltx_ref_tag">12</span></a>) to evaluate the impact of misspecification on MMTR’s performance. The mean parameter in (<a class="ltx_ref" href="https://arxiv.org/html/2503.13782v1#S4.E12" title="In 4.1 Setup ‣ 4 Experiments ‣ Regularized Parameter Estimation in Mixed Model Trace Regression"><span class="ltx_text ltx_ref_tag">12</span></a>) was generated as in the previous simulation, resulting in two cases: <math alttext="\operatorname{\mathbf{B}}\in\mathbb{R}^{5\times 5}" class="ltx_Math" display="inline" id="S4.SS3.p1.1.m1.1"><semantics id="S4.SS3.p1.1.m1.1a"><mrow id="S4.SS3.p1.1.m1.1.1" xref="S4.SS3.p1.1.m1.1.1.cmml"><mi id="S4.SS3.p1.1.m1.1.1.2" xref="S4.SS3.p1.1.m1.1.1.2.cmml">𝐁</mi><mo id="S4.SS3.p1.1.m1.1.1.1" xref="S4.SS3.p1.1.m1.1.1.1.cmml">∈</mo><msup id="S4.SS3.p1.1.m1.1.1.3" xref="S4.SS3.p1.1.m1.1.1.3.cmml"><mi id="S4.SS3.p1.1.m1.1.1.3.2" xref="S4.SS3.p1.1.m1.1.1.3.2.cmml">ℝ</mi><mrow id="S4.SS3.p1.1.m1.1.1.3.3" xref="S4.SS3.p1.1.m1.1.1.3.3.cmml"><mn id="S4.SS3.p1.1.m1.1.1.3.3.2" xref="S4.SS3.p1.1.m1.1.1.3.3.2.cmml">5</mn><mo id="S4.SS3.p1.1.m1.1.1.3.3.1" lspace="0.222em" rspace="0.222em" xref="S4.SS3.p1.1.m1.1.1.3.3.1.cmml">×</mo><mn id="S4.SS3.p1.1.m1.1.1.3.3.3" xref="S4.SS3.p1.1.m1.1.1.3.3.3.cmml">5</mn></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.p1.1.m1.1b"><apply id="S4.SS3.p1.1.m1.1.1.cmml" xref="S4.SS3.p1.1.m1.1.1"><in id="S4.SS3.p1.1.m1.1.1.1.cmml" xref="S4.SS3.p1.1.m1.1.1.1"></in><ci id="S4.SS3.p1.1.m1.1.1.2.cmml" xref="S4.SS3.p1.1.m1.1.1.2">𝐁</ci><apply id="S4.SS3.p1.1.m1.1.1.3.cmml" xref="S4.SS3.p1.1.m1.1.1.3"><csymbol cd="ambiguous" id="S4.SS3.p1.1.m1.1.1.3.1.cmml" xref="S4.SS3.p1.1.m1.1.1.3">superscript</csymbol><ci id="S4.SS3.p1.1.m1.1.1.3.2.cmml" xref="S4.SS3.p1.1.m1.1.1.3.2">ℝ</ci><apply id="S4.SS3.p1.1.m1.1.1.3.3.cmml" xref="S4.SS3.p1.1.m1.1.1.3.3"><times id="S4.SS3.p1.1.m1.1.1.3.3.1.cmml" xref="S4.SS3.p1.1.m1.1.1.3.3.1"></times><cn id="S4.SS3.p1.1.m1.1.1.3.3.2.cmml" type="integer" xref="S4.SS3.p1.1.m1.1.1.3.3.2">5</cn><cn id="S4.SS3.p1.1.m1.1.1.3.3.3.cmml" type="integer" xref="S4.SS3.p1.1.m1.1.1.3.3.3">5</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p1.1.m1.1c">\operatorname{\mathbf{B}}\in\mathbb{R}^{5\times 5}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p1.1.m1.1d">bold_B ∈ blackboard_R start_POSTSUPERSCRIPT 5 × 5 end_POSTSUPERSCRIPT</annotation></semantics></math> (<span class="ltx_text ltx_font_bold" id="S4.SS3.p1.9.1">Case 1</span>) and <math alttext="\operatorname{\mathbf{B}}\in\mathbb{R}^{10\times 10}(\textbf{Case 2})" class="ltx_Math" display="inline" id="S4.SS3.p1.2.m2.1"><semantics id="S4.SS3.p1.2.m2.1a"><mrow id="S4.SS3.p1.2.m2.1.2" xref="S4.SS3.p1.2.m2.1.2.cmml"><mi id="S4.SS3.p1.2.m2.1.2.2" xref="S4.SS3.p1.2.m2.1.2.2.cmml">𝐁</mi><mo id="S4.SS3.p1.2.m2.1.2.1" xref="S4.SS3.p1.2.m2.1.2.1.cmml">∈</mo><mrow id="S4.SS3.p1.2.m2.1.2.3" xref="S4.SS3.p1.2.m2.1.2.3.cmml"><msup id="S4.SS3.p1.2.m2.1.2.3.2" xref="S4.SS3.p1.2.m2.1.2.3.2.cmml"><mi id="S4.SS3.p1.2.m2.1.2.3.2.2" xref="S4.SS3.p1.2.m2.1.2.3.2.2.cmml">ℝ</mi><mrow id="S4.SS3.p1.2.m2.1.2.3.2.3" xref="S4.SS3.p1.2.m2.1.2.3.2.3.cmml"><mn id="S4.SS3.p1.2.m2.1.2.3.2.3.2" xref="S4.SS3.p1.2.m2.1.2.3.2.3.2.cmml">10</mn><mo id="S4.SS3.p1.2.m2.1.2.3.2.3.1" lspace="0.222em" rspace="0.222em" xref="S4.SS3.p1.2.m2.1.2.3.2.3.1.cmml">×</mo><mn id="S4.SS3.p1.2.m2.1.2.3.2.3.3" xref="S4.SS3.p1.2.m2.1.2.3.2.3.3.cmml">10</mn></mrow></msup><mo id="S4.SS3.p1.2.m2.1.2.3.1" xref="S4.SS3.p1.2.m2.1.2.3.1.cmml">⁢</mo><mrow id="S4.SS3.p1.2.m2.1.2.3.3.2" xref="S4.SS3.p1.2.m2.1.1a.cmml"><mo id="S4.SS3.p1.2.m2.1.2.3.3.2.1" stretchy="false" xref="S4.SS3.p1.2.m2.1.1a.cmml">(</mo><mtext class="ltx_mathvariant_bold" id="S4.SS3.p1.2.m2.1.1" xref="S4.SS3.p1.2.m2.1.1.cmml">Case 2</mtext><mo id="S4.SS3.p1.2.m2.1.2.3.3.2.2" stretchy="false" xref="S4.SS3.p1.2.m2.1.1a.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.p1.2.m2.1b"><apply id="S4.SS3.p1.2.m2.1.2.cmml" xref="S4.SS3.p1.2.m2.1.2"><in id="S4.SS3.p1.2.m2.1.2.1.cmml" xref="S4.SS3.p1.2.m2.1.2.1"></in><ci id="S4.SS3.p1.2.m2.1.2.2.cmml" xref="S4.SS3.p1.2.m2.1.2.2">𝐁</ci><apply id="S4.SS3.p1.2.m2.1.2.3.cmml" xref="S4.SS3.p1.2.m2.1.2.3"><times id="S4.SS3.p1.2.m2.1.2.3.1.cmml" xref="S4.SS3.p1.2.m2.1.2.3.1"></times><apply id="S4.SS3.p1.2.m2.1.2.3.2.cmml" xref="S4.SS3.p1.2.m2.1.2.3.2"><csymbol cd="ambiguous" id="S4.SS3.p1.2.m2.1.2.3.2.1.cmml" xref="S4.SS3.p1.2.m2.1.2.3.2">superscript</csymbol><ci id="S4.SS3.p1.2.m2.1.2.3.2.2.cmml" xref="S4.SS3.p1.2.m2.1.2.3.2.2">ℝ</ci><apply id="S4.SS3.p1.2.m2.1.2.3.2.3.cmml" xref="S4.SS3.p1.2.m2.1.2.3.2.3"><times id="S4.SS3.p1.2.m2.1.2.3.2.3.1.cmml" xref="S4.SS3.p1.2.m2.1.2.3.2.3.1"></times><cn id="S4.SS3.p1.2.m2.1.2.3.2.3.2.cmml" type="integer" xref="S4.SS3.p1.2.m2.1.2.3.2.3.2">10</cn><cn id="S4.SS3.p1.2.m2.1.2.3.2.3.3.cmml" type="integer" xref="S4.SS3.p1.2.m2.1.2.3.2.3.3">10</cn></apply></apply><ci id="S4.SS3.p1.2.m2.1.1a.cmml" xref="S4.SS3.p1.2.m2.1.2.3.3.2"><mtext class="ltx_mathvariant_bold" id="S4.SS3.p1.2.m2.1.1.cmml" xref="S4.SS3.p1.2.m2.1.1">Case 2</mtext></ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p1.2.m2.1c">\operatorname{\mathbf{B}}\in\mathbb{R}^{10\times 10}(\textbf{Case 2})</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p1.2.m2.1d">bold_B ∈ blackboard_R start_POSTSUPERSCRIPT 10 × 10 end_POSTSUPERSCRIPT ( Case 2 )</annotation></semantics></math>. The <math alttext="n" class="ltx_Math" display="inline" id="S4.SS3.p1.3.m3.1"><semantics id="S4.SS3.p1.3.m3.1a"><mi id="S4.SS3.p1.3.m3.1.1" xref="S4.SS3.p1.3.m3.1.1.cmml">n</mi><annotation-xml encoding="MathML-Content" id="S4.SS3.p1.3.m3.1b"><ci id="S4.SS3.p1.3.m3.1.1.cmml" xref="S4.SS3.p1.3.m3.1.1">𝑛</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p1.3.m3.1c">n</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p1.3.m3.1d">italic_n</annotation></semantics></math> choices for these two cases were the same as in the previous simulation to facilitate comparisons. Based on the previous results, we set <math alttext="m_{i}=6" class="ltx_Math" display="inline" id="S4.SS3.p1.4.m4.1"><semantics id="S4.SS3.p1.4.m4.1a"><mrow id="S4.SS3.p1.4.m4.1.1" xref="S4.SS3.p1.4.m4.1.1.cmml"><msub id="S4.SS3.p1.4.m4.1.1.2" xref="S4.SS3.p1.4.m4.1.1.2.cmml"><mi id="S4.SS3.p1.4.m4.1.1.2.2" xref="S4.SS3.p1.4.m4.1.1.2.2.cmml">m</mi><mi id="S4.SS3.p1.4.m4.1.1.2.3" xref="S4.SS3.p1.4.m4.1.1.2.3.cmml">i</mi></msub><mo id="S4.SS3.p1.4.m4.1.1.1" xref="S4.SS3.p1.4.m4.1.1.1.cmml">=</mo><mn id="S4.SS3.p1.4.m4.1.1.3" xref="S4.SS3.p1.4.m4.1.1.3.cmml">6</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.p1.4.m4.1b"><apply id="S4.SS3.p1.4.m4.1.1.cmml" xref="S4.SS3.p1.4.m4.1.1"><eq id="S4.SS3.p1.4.m4.1.1.1.cmml" xref="S4.SS3.p1.4.m4.1.1.1"></eq><apply id="S4.SS3.p1.4.m4.1.1.2.cmml" xref="S4.SS3.p1.4.m4.1.1.2"><csymbol cd="ambiguous" id="S4.SS3.p1.4.m4.1.1.2.1.cmml" xref="S4.SS3.p1.4.m4.1.1.2">subscript</csymbol><ci id="S4.SS3.p1.4.m4.1.1.2.2.cmml" xref="S4.SS3.p1.4.m4.1.1.2.2">𝑚</ci><ci id="S4.SS3.p1.4.m4.1.1.2.3.cmml" xref="S4.SS3.p1.4.m4.1.1.2.3">𝑖</ci></apply><cn id="S4.SS3.p1.4.m4.1.1.3.cmml" type="integer" xref="S4.SS3.p1.4.m4.1.1.3">6</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p1.4.m4.1c">m_{i}=6</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p1.4.m4.1d">italic_m start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = 6</annotation></semantics></math> in both cases because MMTR demonstrated satisfactory performance only with sufficiently large <math alttext="m_{i}" class="ltx_Math" display="inline" id="S4.SS3.p1.5.m5.1"><semantics id="S4.SS3.p1.5.m5.1a"><msub id="S4.SS3.p1.5.m5.1.1" xref="S4.SS3.p1.5.m5.1.1.cmml"><mi id="S4.SS3.p1.5.m5.1.1.2" xref="S4.SS3.p1.5.m5.1.1.2.cmml">m</mi><mi id="S4.SS3.p1.5.m5.1.1.3" xref="S4.SS3.p1.5.m5.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS3.p1.5.m5.1b"><apply id="S4.SS3.p1.5.m5.1.1.cmml" xref="S4.SS3.p1.5.m5.1.1"><csymbol cd="ambiguous" id="S4.SS3.p1.5.m5.1.1.1.cmml" xref="S4.SS3.p1.5.m5.1.1">subscript</csymbol><ci id="S4.SS3.p1.5.m5.1.1.2.cmml" xref="S4.SS3.p1.5.m5.1.1.2">𝑚</ci><ci id="S4.SS3.p1.5.m5.1.1.3.cmml" xref="S4.SS3.p1.5.m5.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p1.5.m5.1c">m_{i}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p1.5.m5.1d">italic_m start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math>. This setup resulted in eight different simulation scenarios. The covariance parameter <math alttext="\operatorname{{\mathbf{\Lambda}}}" class="ltx_Math" display="inline" id="S4.SS3.p1.6.m6.1"><semantics id="S4.SS3.p1.6.m6.1a"><mi id="S4.SS3.p1.6.m6.1.1" xref="S4.SS3.p1.6.m6.1.1.cmml">𝚲</mi><annotation-xml encoding="MathML-Content" id="S4.SS3.p1.6.m6.1b"><ci id="S4.SS3.p1.6.m6.1.1.cmml" xref="S4.SS3.p1.6.m6.1.1">𝚲</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p1.6.m6.1c">\operatorname{{\mathbf{\Lambda}}}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p1.6.m6.1d">bold_Λ</annotation></semantics></math> in this simulation was specified by one correlation parameter <math alttext="\alpha" class="ltx_Math" display="inline" id="S4.SS3.p1.7.m7.1"><semantics id="S4.SS3.p1.7.m7.1a"><mi id="S4.SS3.p1.7.m7.1.1" xref="S4.SS3.p1.7.m7.1.1.cmml">α</mi><annotation-xml encoding="MathML-Content" id="S4.SS3.p1.7.m7.1b"><ci id="S4.SS3.p1.7.m7.1.1.cmml" xref="S4.SS3.p1.7.m7.1.1">𝛼</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p1.7.m7.1c">\alpha</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p1.7.m7.1d">italic_α</annotation></semantics></math>. For each of the 100 simulation replications across the eight scenarios, <math alttext="\alpha" class="ltx_Math" display="inline" id="S4.SS3.p1.8.m8.1"><semantics id="S4.SS3.p1.8.m8.1a"><mi id="S4.SS3.p1.8.m8.1.1" xref="S4.SS3.p1.8.m8.1.1.cmml">α</mi><annotation-xml encoding="MathML-Content" id="S4.SS3.p1.8.m8.1b"><ci id="S4.SS3.p1.8.m8.1.1.cmml" xref="S4.SS3.p1.8.m8.1.1">𝛼</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p1.8.m8.1c">\alpha</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p1.8.m8.1d">italic_α</annotation></semantics></math> was drawn from a continuous uniform distribution on <math alttext="[0.2,0.8]" class="ltx_Math" display="inline" id="S4.SS3.p1.9.m9.2"><semantics id="S4.SS3.p1.9.m9.2a"><mrow id="S4.SS3.p1.9.m9.2.3.2" xref="S4.SS3.p1.9.m9.2.3.1.cmml"><mo id="S4.SS3.p1.9.m9.2.3.2.1" stretchy="false" xref="S4.SS3.p1.9.m9.2.3.1.cmml">[</mo><mn id="S4.SS3.p1.9.m9.1.1" xref="S4.SS3.p1.9.m9.1.1.cmml">0.2</mn><mo id="S4.SS3.p1.9.m9.2.3.2.2" xref="S4.SS3.p1.9.m9.2.3.1.cmml">,</mo><mn id="S4.SS3.p1.9.m9.2.2" xref="S4.SS3.p1.9.m9.2.2.cmml">0.8</mn><mo id="S4.SS3.p1.9.m9.2.3.2.3" stretchy="false" xref="S4.SS3.p1.9.m9.2.3.1.cmml">]</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.p1.9.m9.2b"><interval closure="closed" id="S4.SS3.p1.9.m9.2.3.1.cmml" xref="S4.SS3.p1.9.m9.2.3.2"><cn id="S4.SS3.p1.9.m9.1.1.cmml" type="float" xref="S4.SS3.p1.9.m9.1.1">0.2</cn><cn id="S4.SS3.p1.9.m9.2.2.cmml" type="float" xref="S4.SS3.p1.9.m9.2.2">0.8</cn></interval></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p1.9.m9.2c">[0.2,0.8]</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p1.9.m9.2d">[ 0.2 , 0.8 ]</annotation></semantics></math>. The three MMTR competitors (GEE, Kruskal, and quasi-likelihood), tuning parameter selection methods, and error metrics were identical to those used in the previous simulation.</p> </div> <div class="ltx_para" id="S4.SS3.p2"> <p class="ltx_p" id="S4.SS3.p2.11">All methods demonstrated similar accuracy in <math alttext="\operatorname{\mathbf{B}}" class="ltx_Math" display="inline" id="S4.SS3.p2.1.m1.1"><semantics id="S4.SS3.p2.1.m1.1a"><mi id="S4.SS3.p2.1.m1.1.1" xref="S4.SS3.p2.1.m1.1.1.cmml">𝐁</mi><annotation-xml encoding="MathML-Content" id="S4.SS3.p2.1.m1.1b"><ci id="S4.SS3.p2.1.m1.1.1.cmml" xref="S4.SS3.p2.1.m1.1.1">𝐁</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p2.1.m1.1c">\operatorname{\mathbf{B}}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p2.1.m1.1d">bold_B</annotation></semantics></math> estimation, but the equicorrelation GEE model outperformed the others in <math alttext="\operatorname{{\mathbf{\Lambda}}}" class="ltx_Math" display="inline" id="S4.SS3.p2.2.m2.1"><semantics id="S4.SS3.p2.2.m2.1a"><mi id="S4.SS3.p2.2.m2.1.1" xref="S4.SS3.p2.2.m2.1.1.cmml">𝚲</mi><annotation-xml encoding="MathML-Content" id="S4.SS3.p2.2.m2.1b"><ci id="S4.SS3.p2.2.m2.1.1.cmml" xref="S4.SS3.p2.2.m2.1.1">𝚲</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p2.2.m2.1c">\operatorname{{\mathbf{\Lambda}}}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p2.2.m2.1d">bold_Λ</annotation></semantics></math> estimation (Tables <a class="ltx_ref" href="https://arxiv.org/html/2503.13782v1#S4.T5" title="Table 5 ‣ 4.3 Simulations Using the Equicorrelation GEE model ‣ 4 Experiments ‣ Regularized Parameter Estimation in Mixed Model Trace Regression"><span class="ltx_text ltx_ref_tag">5</span></a> and <a class="ltx_ref" href="https://arxiv.org/html/2503.13782v1#S4.T6" title="Table 6 ‣ 4.3 Simulations Using the Equicorrelation GEE model ‣ 4 Experiments ‣ Regularized Parameter Estimation in Mixed Model Trace Regression"><span class="ltx_text ltx_ref_tag">6</span></a>). For both cases, <math alttext="\operatorname{\mathbf{B}}" class="ltx_Math" display="inline" id="S4.SS3.p2.3.m3.1"><semantics id="S4.SS3.p2.3.m3.1a"><mi id="S4.SS3.p2.3.m3.1.1" xref="S4.SS3.p2.3.m3.1.1.cmml">𝐁</mi><annotation-xml encoding="MathML-Content" id="S4.SS3.p2.3.m3.1b"><ci id="S4.SS3.p2.3.m3.1.1.cmml" xref="S4.SS3.p2.3.m3.1.1">𝐁</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p2.3.m3.1c">\operatorname{\mathbf{B}}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p2.3.m3.1d">bold_B</annotation></semantics></math> estimation accuracy improved as <math alttext="n" class="ltx_Math" display="inline" id="S4.SS3.p2.4.m4.1"><semantics id="S4.SS3.p2.4.m4.1a"><mi id="S4.SS3.p2.4.m4.1.1" xref="S4.SS3.p2.4.m4.1.1.cmml">n</mi><annotation-xml encoding="MathML-Content" id="S4.SS3.p2.4.m4.1b"><ci id="S4.SS3.p2.4.m4.1.1.cmml" xref="S4.SS3.p2.4.m4.1.1">𝑛</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p2.4.m4.1c">n</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p2.4.m4.1d">italic_n</annotation></semantics></math> increased, providing further evidence that the mean parameter estimation was robust to model misspecification. Although MMTR’s covariance parameter was misspecified relative to the simulation model, its accuracy in estimating <math alttext="\operatorname{{\mathbf{\Lambda}}}" class="ltx_Math" display="inline" id="S4.SS3.p2.5.m5.1"><semantics id="S4.SS3.p2.5.m5.1a"><mi id="S4.SS3.p2.5.m5.1.1" xref="S4.SS3.p2.5.m5.1.1.cmml">𝚲</mi><annotation-xml encoding="MathML-Content" id="S4.SS3.p2.5.m5.1b"><ci id="S4.SS3.p2.5.m5.1.1.cmml" xref="S4.SS3.p2.5.m5.1.1">𝚲</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p2.5.m5.1c">\operatorname{{\mathbf{\Lambda}}}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p2.5.m5.1d">bold_Λ</annotation></semantics></math> improved up to a point with increasing <math alttext="n" class="ltx_Math" display="inline" id="S4.SS3.p2.6.m6.1"><semantics id="S4.SS3.p2.6.m6.1a"><mi id="S4.SS3.p2.6.m6.1.1" xref="S4.SS3.p2.6.m6.1.1.cmml">n</mi><annotation-xml encoding="MathML-Content" id="S4.SS3.p2.6.m6.1b"><ci id="S4.SS3.p2.6.m6.1.1.cmml" xref="S4.SS3.p2.6.m6.1.1">𝑛</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p2.6.m6.1c">n</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p2.6.m6.1d">italic_n</annotation></semantics></math>. Furthermore, MMTR’s covariance estimation errors were significantly smaller for lower <math alttext="\alpha" class="ltx_Math" display="inline" id="S4.SS3.p2.7.m7.1"><semantics id="S4.SS3.p2.7.m7.1a"><mi id="S4.SS3.p2.7.m7.1.1" xref="S4.SS3.p2.7.m7.1.1.cmml">α</mi><annotation-xml encoding="MathML-Content" id="S4.SS3.p2.7.m7.1b"><ci id="S4.SS3.p2.7.m7.1.1.cmml" xref="S4.SS3.p2.7.m7.1.1">𝛼</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p2.7.m7.1c">\alpha</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p2.7.m7.1d">italic_α</annotation></semantics></math> values than higher values. A larger <math alttext="\alpha" class="ltx_Math" display="inline" id="S4.SS3.p2.8.m8.1"><semantics id="S4.SS3.p2.8.m8.1a"><mi id="S4.SS3.p2.8.m8.1.1" xref="S4.SS3.p2.8.m8.1.1.cmml">α</mi><annotation-xml encoding="MathML-Content" id="S4.SS3.p2.8.m8.1b"><ci id="S4.SS3.p2.8.m8.1.1.cmml" xref="S4.SS3.p2.8.m8.1.1">𝛼</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p2.8.m8.1c">\alpha</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p2.8.m8.1d">italic_α</annotation></semantics></math> corresponds to a heavily misspecified covariance model, making it more challenging to approximate using a Kronecker product structure; see supplementary materials for additional simulation results. In summary, while model misspecification implied that a larger <math alttext="n" class="ltx_Math" display="inline" id="S4.SS3.p2.9.m9.1"><semantics id="S4.SS3.p2.9.m9.1a"><mi id="S4.SS3.p2.9.m9.1.1" xref="S4.SS3.p2.9.m9.1.1.cmml">n</mi><annotation-xml encoding="MathML-Content" id="S4.SS3.p2.9.m9.1b"><ci id="S4.SS3.p2.9.m9.1.1.cmml" xref="S4.SS3.p2.9.m9.1.1">𝑛</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p2.9.m9.1c">n</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p2.9.m9.1d">italic_n</annotation></semantics></math> and smaller <math alttext="\alpha" class="ltx_Math" display="inline" id="S4.SS3.p2.10.m10.1"><semantics id="S4.SS3.p2.10.m10.1a"><mi id="S4.SS3.p2.10.m10.1.1" xref="S4.SS3.p2.10.m10.1.1.cmml">α</mi><annotation-xml encoding="MathML-Content" id="S4.SS3.p2.10.m10.1b"><ci id="S4.SS3.p2.10.m10.1.1.cmml" xref="S4.SS3.p2.10.m10.1.1">𝛼</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p2.10.m10.1c">\alpha</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p2.10.m10.1d">italic_α</annotation></semantics></math> were required for MMTR’s accurate covariance estimation, MMTR showed excellent performance in estimating the mean parameter across all <math alttext="n" class="ltx_Math" display="inline" id="S4.SS3.p2.11.m11.1"><semantics id="S4.SS3.p2.11.m11.1a"><mi id="S4.SS3.p2.11.m11.1.1" xref="S4.SS3.p2.11.m11.1.1.cmml">n</mi><annotation-xml encoding="MathML-Content" id="S4.SS3.p2.11.m11.1b"><ci id="S4.SS3.p2.11.m11.1.1.cmml" xref="S4.SS3.p2.11.m11.1.1">𝑛</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p2.11.m11.1c">n</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p2.11.m11.1d">italic_n</annotation></semantics></math> choices.</p> </div> <figure class="ltx_table" id="S4.T5"> <figcaption class="ltx_caption"><span class="ltx_tag ltx_tag_table">Table 5: </span>Relative estimation error in Case 1 with a misspecified equicorrelated covariance structure in (<a class="ltx_ref" href="https://arxiv.org/html/2503.13782v1#S4.E12" title="In 4.1 Setup ‣ 4 Experiments ‣ Regularized Parameter Estimation in Mixed Model Trace Regression"><span class="ltx_text ltx_ref_tag">12</span></a>). Every estimation error is averaged over 100 simulation replications, with the Monte Carlo errors in parenthesis. A small relative error indicates high estimation accuracy. GEE un. and GEE eq. denote the GEE models with unstructured and equicorrelated covariance matrix. Kruskal and Quasi denote the sparse Kruskal and quasi-likelihood methods.</figcaption> <div class="ltx_inline-block ltx_align_center ltx_transformed_outer" id="S4.T5.6" style="width:505.9pt;height:83.5pt;vertical-align:-0.0pt;"><span class="ltx_transformed_inner" style="transform:translate(-74.4pt,12.3pt) scale(0.772689745709282,0.772689745709282) ;"> <table class="ltx_tabular ltx_guessed_headers ltx_align_top" id="S4.T5.6.6"> <thead class="ltx_thead"> <tr class="ltx_tr" id="S4.T5.2.2.2"> <th class="ltx_td ltx_align_right ltx_th ltx_th_column ltx_th_row ltx_border_l ltx_border_r ltx_border_t" id="S4.T5.2.2.2.3">group size</th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_r ltx_border_t" colspan="5" id="S4.T5.1.1.1.1">err<math alttext="{}_{\operatorname{\mathbf{B}}}" class="ltx_Math" display="inline" id="S4.T5.1.1.1.1.m1.1"><semantics id="S4.T5.1.1.1.1.m1.1a"><msub id="S4.T5.1.1.1.1.m1.1.1" xref="S4.T5.1.1.1.1.m1.1.1.cmml"><mi id="S4.T5.1.1.1.1.m1.1.1a" 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xref="S4.T5.2.2.2.2.m1.1.1.cmml"></mi><mi id="S4.T5.2.2.2.2.m1.1.1.1" xref="S4.T5.2.2.2.2.m1.1.1.1.cmml">𝚲</mi></msub><annotation-xml encoding="MathML-Content" id="S4.T5.2.2.2.2.m1.1b"><apply id="S4.T5.2.2.2.2.m1.1.1.cmml" xref="S4.T5.2.2.2.2.m1.1.1"><ci id="S4.T5.2.2.2.2.m1.1.1.1.cmml" xref="S4.T5.2.2.2.2.m1.1.1.1">𝚲</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.T5.2.2.2.2.m1.1c">{}_{\operatorname{{\mathbf{\Lambda}}}}</annotation><annotation encoding="application/x-llamapun" id="S4.T5.2.2.2.2.m1.1d">start_FLOATSUBSCRIPT bold_Λ end_FLOATSUBSCRIPT</annotation></semantics></math> </th> </tr> </thead> <tbody class="ltx_tbody"> <tr class="ltx_tr" id="S4.T5.6.6.7.1"> <th class="ltx_td ltx_th ltx_th_row ltx_border_l ltx_border_r ltx_border_t" id="S4.T5.6.6.7.1.1"></th> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T5.6.6.7.1.2">MMTR</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T5.6.6.7.1.3">Kruskal</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T5.6.6.7.1.4">GEE un.</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T5.6.6.7.1.5">GEE eq.</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T5.6.6.7.1.6">Quasi</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T5.6.6.7.1.7">MMTR</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T5.6.6.7.1.8">GEE un.</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T5.6.6.7.1.9">GEE eq.</td> </tr> <tr class="ltx_tr" id="S4.T5.3.3.3"> <th class="ltx_td ltx_align_right ltx_th ltx_th_row ltx_border_l ltx_border_r ltx_border_t" id="S4.T5.3.3.3.1"> <math alttext="n=18,m_{i}=6" class="ltx_Math" display="inline" id="S4.T5.3.3.3.1.m1.2"><semantics id="S4.T5.3.3.3.1.m1.2a"><mrow id="S4.T5.3.3.3.1.m1.2.2.2" xref="S4.T5.3.3.3.1.m1.2.2.3.cmml"><mrow id="S4.T5.3.3.3.1.m1.1.1.1.1" xref="S4.T5.3.3.3.1.m1.1.1.1.1.cmml"><mi 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type="integer" xref="S4.T5.3.3.3.1.m1.2.2.2.2.3">6</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.T5.3.3.3.1.m1.2c">n=18,m_{i}=6</annotation><annotation encoding="application/x-llamapun" id="S4.T5.3.3.3.1.m1.2d">italic_n = 18 , italic_m start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = 6</annotation></semantics></math> </th> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T5.3.3.3.2">0.038 (0.0110)</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T5.3.3.3.3">0.025 (0.0071)</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T5.3.3.3.4">0.0480 (0.0270)</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T5.3.3.3.5">0.0200 (0.0053)</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T5.3.3.3.6">0.062 (0.0160)</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T5.3.3.3.7">1.90 (0.75)</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T5.3.3.3.8">1.20 (2.200)</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T5.3.3.3.9">0.150 (0.110)</td> </tr> <tr class="ltx_tr" id="S4.T5.4.4.4"> <th class="ltx_td ltx_align_right ltx_th ltx_th_row ltx_border_l ltx_border_r ltx_border_t" id="S4.T5.4.4.4.1"> <math alttext="n=27,m_{i}=6" class="ltx_Math" display="inline" id="S4.T5.4.4.4.1.m1.2"><semantics id="S4.T5.4.4.4.1.m1.2a"><mrow id="S4.T5.4.4.4.1.m1.2.2.2" xref="S4.T5.4.4.4.1.m1.2.2.3.cmml"><mrow id="S4.T5.4.4.4.1.m1.1.1.1.1" xref="S4.T5.4.4.4.1.m1.1.1.1.1.cmml"><mi id="S4.T5.4.4.4.1.m1.1.1.1.1.2" xref="S4.T5.4.4.4.1.m1.1.1.1.1.2.cmml">n</mi><mo id="S4.T5.4.4.4.1.m1.1.1.1.1.1" xref="S4.T5.4.4.4.1.m1.1.1.1.1.1.cmml">=</mo><mn id="S4.T5.4.4.4.1.m1.1.1.1.1.3" xref="S4.T5.4.4.4.1.m1.1.1.1.1.3.cmml">27</mn></mrow><mo id="S4.T5.4.4.4.1.m1.2.2.2.3" xref="S4.T5.4.4.4.1.m1.2.2.3a.cmml">,</mo><mrow id="S4.T5.4.4.4.1.m1.2.2.2.2" xref="S4.T5.4.4.4.1.m1.2.2.2.2.cmml"><msub id="S4.T5.4.4.4.1.m1.2.2.2.2.2" xref="S4.T5.4.4.4.1.m1.2.2.2.2.2.cmml"><mi id="S4.T5.4.4.4.1.m1.2.2.2.2.2.2" xref="S4.T5.4.4.4.1.m1.2.2.2.2.2.2.cmml">m</mi><mi id="S4.T5.4.4.4.1.m1.2.2.2.2.2.3" xref="S4.T5.4.4.4.1.m1.2.2.2.2.2.3.cmml">i</mi></msub><mo id="S4.T5.4.4.4.1.m1.2.2.2.2.1" xref="S4.T5.4.4.4.1.m1.2.2.2.2.1.cmml">=</mo><mn id="S4.T5.4.4.4.1.m1.2.2.2.2.3" xref="S4.T5.4.4.4.1.m1.2.2.2.2.3.cmml">6</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.T5.4.4.4.1.m1.2b"><apply id="S4.T5.4.4.4.1.m1.2.2.3.cmml" xref="S4.T5.4.4.4.1.m1.2.2.2"><csymbol cd="ambiguous" id="S4.T5.4.4.4.1.m1.2.2.3a.cmml" xref="S4.T5.4.4.4.1.m1.2.2.2.3">formulae-sequence</csymbol><apply id="S4.T5.4.4.4.1.m1.1.1.1.1.cmml" xref="S4.T5.4.4.4.1.m1.1.1.1.1"><eq id="S4.T5.4.4.4.1.m1.1.1.1.1.1.cmml" xref="S4.T5.4.4.4.1.m1.1.1.1.1.1"></eq><ci id="S4.T5.4.4.4.1.m1.1.1.1.1.2.cmml" xref="S4.T5.4.4.4.1.m1.1.1.1.1.2">𝑛</ci><cn id="S4.T5.4.4.4.1.m1.1.1.1.1.3.cmml" type="integer" xref="S4.T5.4.4.4.1.m1.1.1.1.1.3">27</cn></apply><apply id="S4.T5.4.4.4.1.m1.2.2.2.2.cmml" xref="S4.T5.4.4.4.1.m1.2.2.2.2"><eq id="S4.T5.4.4.4.1.m1.2.2.2.2.1.cmml" xref="S4.T5.4.4.4.1.m1.2.2.2.2.1"></eq><apply id="S4.T5.4.4.4.1.m1.2.2.2.2.2.cmml" xref="S4.T5.4.4.4.1.m1.2.2.2.2.2"><csymbol cd="ambiguous" id="S4.T5.4.4.4.1.m1.2.2.2.2.2.1.cmml" xref="S4.T5.4.4.4.1.m1.2.2.2.2.2">subscript</csymbol><ci id="S4.T5.4.4.4.1.m1.2.2.2.2.2.2.cmml" xref="S4.T5.4.4.4.1.m1.2.2.2.2.2.2">𝑚</ci><ci id="S4.T5.4.4.4.1.m1.2.2.2.2.2.3.cmml" xref="S4.T5.4.4.4.1.m1.2.2.2.2.2.3">𝑖</ci></apply><cn id="S4.T5.4.4.4.1.m1.2.2.2.2.3.cmml" type="integer" xref="S4.T5.4.4.4.1.m1.2.2.2.2.3">6</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.T5.4.4.4.1.m1.2c">n=27,m_{i}=6</annotation><annotation encoding="application/x-llamapun" id="S4.T5.4.4.4.1.m1.2d">italic_n = 27 , italic_m start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = 6</annotation></semantics></math> </th> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T5.4.4.4.2">0.022 (0.0065)</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T5.4.4.4.3">0.020 (0.0046)</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T5.4.4.4.4">0.0220 (0.0061)</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T5.4.4.4.5">0.0160 (0.0038)</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T5.4.4.4.6">0.046 (0.0100)</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T5.4.4.4.7">1.10 (0.39)</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T5.4.4.4.8">5.20 (3.000)</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T5.4.4.4.9">0.110 (0.074)</td> </tr> <tr class="ltx_tr" id="S4.T5.5.5.5"> <th class="ltx_td ltx_align_right ltx_th ltx_th_row ltx_border_l ltx_border_r ltx_border_t" id="S4.T5.5.5.5.1"> <math alttext="n=54,m_{i}=6" class="ltx_Math" display="inline" id="S4.T5.5.5.5.1.m1.2"><semantics id="S4.T5.5.5.5.1.m1.2a"><mrow id="S4.T5.5.5.5.1.m1.2.2.2" xref="S4.T5.5.5.5.1.m1.2.2.3.cmml"><mrow id="S4.T5.5.5.5.1.m1.1.1.1.1" xref="S4.T5.5.5.5.1.m1.1.1.1.1.cmml"><mi id="S4.T5.5.5.5.1.m1.1.1.1.1.2" xref="S4.T5.5.5.5.1.m1.1.1.1.1.2.cmml">n</mi><mo id="S4.T5.5.5.5.1.m1.1.1.1.1.1" xref="S4.T5.5.5.5.1.m1.1.1.1.1.1.cmml">=</mo><mn id="S4.T5.5.5.5.1.m1.1.1.1.1.3" xref="S4.T5.5.5.5.1.m1.1.1.1.1.3.cmml">54</mn></mrow><mo id="S4.T5.5.5.5.1.m1.2.2.2.3" xref="S4.T5.5.5.5.1.m1.2.2.3a.cmml">,</mo><mrow id="S4.T5.5.5.5.1.m1.2.2.2.2" xref="S4.T5.5.5.5.1.m1.2.2.2.2.cmml"><msub id="S4.T5.5.5.5.1.m1.2.2.2.2.2" xref="S4.T5.5.5.5.1.m1.2.2.2.2.2.cmml"><mi id="S4.T5.5.5.5.1.m1.2.2.2.2.2.2" xref="S4.T5.5.5.5.1.m1.2.2.2.2.2.2.cmml">m</mi><mi id="S4.T5.5.5.5.1.m1.2.2.2.2.2.3" xref="S4.T5.5.5.5.1.m1.2.2.2.2.2.3.cmml">i</mi></msub><mo id="S4.T5.5.5.5.1.m1.2.2.2.2.1" xref="S4.T5.5.5.5.1.m1.2.2.2.2.1.cmml">=</mo><mn id="S4.T5.5.5.5.1.m1.2.2.2.2.3" xref="S4.T5.5.5.5.1.m1.2.2.2.2.3.cmml">6</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.T5.5.5.5.1.m1.2b"><apply id="S4.T5.5.5.5.1.m1.2.2.3.cmml" xref="S4.T5.5.5.5.1.m1.2.2.2"><csymbol cd="ambiguous" id="S4.T5.5.5.5.1.m1.2.2.3a.cmml" xref="S4.T5.5.5.5.1.m1.2.2.2.3">formulae-sequence</csymbol><apply id="S4.T5.5.5.5.1.m1.1.1.1.1.cmml" xref="S4.T5.5.5.5.1.m1.1.1.1.1"><eq id="S4.T5.5.5.5.1.m1.1.1.1.1.1.cmml" xref="S4.T5.5.5.5.1.m1.1.1.1.1.1"></eq><ci id="S4.T5.5.5.5.1.m1.1.1.1.1.2.cmml" xref="S4.T5.5.5.5.1.m1.1.1.1.1.2">𝑛</ci><cn id="S4.T5.5.5.5.1.m1.1.1.1.1.3.cmml" type="integer" xref="S4.T5.5.5.5.1.m1.1.1.1.1.3">54</cn></apply><apply id="S4.T5.5.5.5.1.m1.2.2.2.2.cmml" xref="S4.T5.5.5.5.1.m1.2.2.2.2"><eq id="S4.T5.5.5.5.1.m1.2.2.2.2.1.cmml" xref="S4.T5.5.5.5.1.m1.2.2.2.2.1"></eq><apply id="S4.T5.5.5.5.1.m1.2.2.2.2.2.cmml" xref="S4.T5.5.5.5.1.m1.2.2.2.2.2"><csymbol cd="ambiguous" id="S4.T5.5.5.5.1.m1.2.2.2.2.2.1.cmml" xref="S4.T5.5.5.5.1.m1.2.2.2.2.2">subscript</csymbol><ci id="S4.T5.5.5.5.1.m1.2.2.2.2.2.2.cmml" xref="S4.T5.5.5.5.1.m1.2.2.2.2.2.2">𝑚</ci><ci id="S4.T5.5.5.5.1.m1.2.2.2.2.2.3.cmml" xref="S4.T5.5.5.5.1.m1.2.2.2.2.2.3">𝑖</ci></apply><cn id="S4.T5.5.5.5.1.m1.2.2.2.2.3.cmml" type="integer" xref="S4.T5.5.5.5.1.m1.2.2.2.2.3">6</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.T5.5.5.5.1.m1.2c">n=54,m_{i}=6</annotation><annotation encoding="application/x-llamapun" id="S4.T5.5.5.5.1.m1.2d">italic_n = 54 , italic_m start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = 6</annotation></semantics></math> </th> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T5.5.5.5.2">0.013 (0.0031)</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T5.5.5.5.3">0.013 (0.0031)</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T5.5.5.5.4">0.0120 (0.0030)</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T5.5.5.5.5">0.0110 (0.0026)</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T5.5.5.5.6">0.029 (0.0057)</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T5.5.5.5.7">0.76 (0.12)</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T5.5.5.5.8">0.68 (0.130)</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T5.5.5.5.9">0.088 (0.066)</td> </tr> <tr class="ltx_tr" id="S4.T5.6.6.6"> <th class="ltx_td ltx_align_right ltx_th ltx_th_row ltx_border_b ltx_border_l ltx_border_r ltx_border_t" id="S4.T5.6.6.6.1"> <math alttext="n=81,m_{i}=6" class="ltx_Math" display="inline" id="S4.T5.6.6.6.1.m1.2"><semantics id="S4.T5.6.6.6.1.m1.2a"><mrow id="S4.T5.6.6.6.1.m1.2.2.2" xref="S4.T5.6.6.6.1.m1.2.2.3.cmml"><mrow id="S4.T5.6.6.6.1.m1.1.1.1.1" xref="S4.T5.6.6.6.1.m1.1.1.1.1.cmml"><mi id="S4.T5.6.6.6.1.m1.1.1.1.1.2" xref="S4.T5.6.6.6.1.m1.1.1.1.1.2.cmml">n</mi><mo id="S4.T5.6.6.6.1.m1.1.1.1.1.1" xref="S4.T5.6.6.6.1.m1.1.1.1.1.1.cmml">=</mo><mn id="S4.T5.6.6.6.1.m1.1.1.1.1.3" xref="S4.T5.6.6.6.1.m1.1.1.1.1.3.cmml">81</mn></mrow><mo id="S4.T5.6.6.6.1.m1.2.2.2.3" xref="S4.T5.6.6.6.1.m1.2.2.3a.cmml">,</mo><mrow id="S4.T5.6.6.6.1.m1.2.2.2.2" xref="S4.T5.6.6.6.1.m1.2.2.2.2.cmml"><msub id="S4.T5.6.6.6.1.m1.2.2.2.2.2" xref="S4.T5.6.6.6.1.m1.2.2.2.2.2.cmml"><mi id="S4.T5.6.6.6.1.m1.2.2.2.2.2.2" xref="S4.T5.6.6.6.1.m1.2.2.2.2.2.2.cmml">m</mi><mi id="S4.T5.6.6.6.1.m1.2.2.2.2.2.3" xref="S4.T5.6.6.6.1.m1.2.2.2.2.2.3.cmml">i</mi></msub><mo id="S4.T5.6.6.6.1.m1.2.2.2.2.1" xref="S4.T5.6.6.6.1.m1.2.2.2.2.1.cmml">=</mo><mn id="S4.T5.6.6.6.1.m1.2.2.2.2.3" xref="S4.T5.6.6.6.1.m1.2.2.2.2.3.cmml">6</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.T5.6.6.6.1.m1.2b"><apply id="S4.T5.6.6.6.1.m1.2.2.3.cmml" xref="S4.T5.6.6.6.1.m1.2.2.2"><csymbol cd="ambiguous" id="S4.T5.6.6.6.1.m1.2.2.3a.cmml" xref="S4.T5.6.6.6.1.m1.2.2.2.3">formulae-sequence</csymbol><apply id="S4.T5.6.6.6.1.m1.1.1.1.1.cmml" xref="S4.T5.6.6.6.1.m1.1.1.1.1"><eq id="S4.T5.6.6.6.1.m1.1.1.1.1.1.cmml" xref="S4.T5.6.6.6.1.m1.1.1.1.1.1"></eq><ci id="S4.T5.6.6.6.1.m1.1.1.1.1.2.cmml" xref="S4.T5.6.6.6.1.m1.1.1.1.1.2">𝑛</ci><cn id="S4.T5.6.6.6.1.m1.1.1.1.1.3.cmml" type="integer" xref="S4.T5.6.6.6.1.m1.1.1.1.1.3">81</cn></apply><apply id="S4.T5.6.6.6.1.m1.2.2.2.2.cmml" xref="S4.T5.6.6.6.1.m1.2.2.2.2"><eq id="S4.T5.6.6.6.1.m1.2.2.2.2.1.cmml" xref="S4.T5.6.6.6.1.m1.2.2.2.2.1"></eq><apply id="S4.T5.6.6.6.1.m1.2.2.2.2.2.cmml" xref="S4.T5.6.6.6.1.m1.2.2.2.2.2"><csymbol cd="ambiguous" id="S4.T5.6.6.6.1.m1.2.2.2.2.2.1.cmml" xref="S4.T5.6.6.6.1.m1.2.2.2.2.2">subscript</csymbol><ci id="S4.T5.6.6.6.1.m1.2.2.2.2.2.2.cmml" xref="S4.T5.6.6.6.1.m1.2.2.2.2.2.2">𝑚</ci><ci id="S4.T5.6.6.6.1.m1.2.2.2.2.2.3.cmml" xref="S4.T5.6.6.6.1.m1.2.2.2.2.2.3">𝑖</ci></apply><cn id="S4.T5.6.6.6.1.m1.2.2.2.2.3.cmml" type="integer" xref="S4.T5.6.6.6.1.m1.2.2.2.2.3">6</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.T5.6.6.6.1.m1.2c">n=81,m_{i}=6</annotation><annotation encoding="application/x-llamapun" id="S4.T5.6.6.6.1.m1.2d">italic_n = 81 , italic_m start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = 6</annotation></semantics></math> </th> <td class="ltx_td ltx_align_center ltx_border_b ltx_border_r ltx_border_t" id="S4.T5.6.6.6.2">0.011 (0.0029)</td> <td class="ltx_td ltx_align_center ltx_border_b ltx_border_r ltx_border_t" id="S4.T5.6.6.6.3">0.011 (0.0024)</td> <td class="ltx_td ltx_align_center ltx_border_b ltx_border_r ltx_border_t" id="S4.T5.6.6.6.4">0.0095 (0.0022)</td> <td class="ltx_td ltx_align_center ltx_border_b ltx_border_r ltx_border_t" id="S4.T5.6.6.6.5">0.0088 (0.0020)</td> <td class="ltx_td ltx_align_center ltx_border_b ltx_border_r ltx_border_t" id="S4.T5.6.6.6.6">0.023 (0.0048)</td> <td class="ltx_td ltx_align_center ltx_border_b ltx_border_r ltx_border_t" id="S4.T5.6.6.6.7">0.73 (0.11)</td> <td class="ltx_td ltx_align_center ltx_border_b ltx_border_r ltx_border_t" id="S4.T5.6.6.6.8">0.40 (0.085)</td> <td class="ltx_td ltx_align_center ltx_border_b ltx_border_r ltx_border_t" id="S4.T5.6.6.6.9">0.071 (0.056)</td> </tr> </tbody> </table> </span></div> </figure> <figure class="ltx_table" id="S4.T6"> <figcaption class="ltx_caption"><span class="ltx_tag ltx_tag_table">Table 6: </span>Relative estimation error in Case 2 with a misspecified equicorrelated covariance structure in (<a class="ltx_ref" href="https://arxiv.org/html/2503.13782v1#S4.E12" title="In 4.1 Setup ‣ 4 Experiments ‣ Regularized Parameter Estimation in Mixed Model Trace Regression"><span class="ltx_text ltx_ref_tag">12</span></a>). Every estimation error is averaged over 100 simulation replications, with the Monte Carlo errors in parenthesis. A small relative error indicates high estimation accuracy. GEE un. and GEE eq. denote the GEE models with unstructured and equicorrelated covariance matrix. Kruskal and Quasi denote the sparse Kruskal and quasi-likelihood methods.</figcaption> <div class="ltx_inline-block ltx_align_center ltx_transformed_outer" id="S4.T6.6" style="width:505.9pt;height:82.2pt;vertical-align:-0.0pt;"><span class="ltx_transformed_inner" style="transform:translate(-79.4pt,12.9pt) scale(0.76106529839911,0.76106529839911) ;"> <table class="ltx_tabular ltx_guessed_headers ltx_align_top" id="S4.T6.6.6"> <thead class="ltx_thead"> <tr class="ltx_tr" id="S4.T6.2.2.2"> <th class="ltx_td ltx_align_right ltx_th ltx_th_column ltx_th_row ltx_border_l ltx_border_r ltx_border_t" id="S4.T6.2.2.2.3">group size</th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_r ltx_border_t" colspan="5" id="S4.T6.1.1.1.1">err<math alttext="{}_{\operatorname{\mathbf{B}}}" class="ltx_Math" display="inline" id="S4.T6.1.1.1.1.m1.1"><semantics id="S4.T6.1.1.1.1.m1.1a"><msub id="S4.T6.1.1.1.1.m1.1.1" xref="S4.T6.1.1.1.1.m1.1.1.cmml"><mi id="S4.T6.1.1.1.1.m1.1.1a" xref="S4.T6.1.1.1.1.m1.1.1.cmml"></mi><mi id="S4.T6.1.1.1.1.m1.1.1.1" xref="S4.T6.1.1.1.1.m1.1.1.1.cmml">𝐁</mi></msub><annotation-xml encoding="MathML-Content" id="S4.T6.1.1.1.1.m1.1b"><apply id="S4.T6.1.1.1.1.m1.1.1.cmml" xref="S4.T6.1.1.1.1.m1.1.1"><ci id="S4.T6.1.1.1.1.m1.1.1.1.cmml" xref="S4.T6.1.1.1.1.m1.1.1.1">𝐁</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.T6.1.1.1.1.m1.1c">{}_{\operatorname{\mathbf{B}}}</annotation><annotation encoding="application/x-llamapun" id="S4.T6.1.1.1.1.m1.1d">start_FLOATSUBSCRIPT bold_B end_FLOATSUBSCRIPT</annotation></semantics></math> </th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_r ltx_border_t" colspan="3" id="S4.T6.2.2.2.2">err<math alttext="{}_{\operatorname{{\mathbf{\Lambda}}}}" class="ltx_Math" display="inline" id="S4.T6.2.2.2.2.m1.1"><semantics id="S4.T6.2.2.2.2.m1.1a"><msub id="S4.T6.2.2.2.2.m1.1.1" xref="S4.T6.2.2.2.2.m1.1.1.cmml"><mi id="S4.T6.2.2.2.2.m1.1.1a" xref="S4.T6.2.2.2.2.m1.1.1.cmml"></mi><mi id="S4.T6.2.2.2.2.m1.1.1.1" xref="S4.T6.2.2.2.2.m1.1.1.1.cmml">𝚲</mi></msub><annotation-xml encoding="MathML-Content" id="S4.T6.2.2.2.2.m1.1b"><apply id="S4.T6.2.2.2.2.m1.1.1.cmml" xref="S4.T6.2.2.2.2.m1.1.1"><ci id="S4.T6.2.2.2.2.m1.1.1.1.cmml" xref="S4.T6.2.2.2.2.m1.1.1.1">𝚲</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.T6.2.2.2.2.m1.1c">{}_{\operatorname{{\mathbf{\Lambda}}}}</annotation><annotation encoding="application/x-llamapun" id="S4.T6.2.2.2.2.m1.1d">start_FLOATSUBSCRIPT bold_Λ end_FLOATSUBSCRIPT</annotation></semantics></math> </th> </tr> </thead> <tbody class="ltx_tbody"> <tr class="ltx_tr" id="S4.T6.6.6.7.1"> <th class="ltx_td ltx_th ltx_th_row ltx_border_l ltx_border_r ltx_border_t" id="S4.T6.6.6.7.1.1"></th> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T6.6.6.7.1.2">MMTR</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T6.6.6.7.1.3">Kruskal</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T6.6.6.7.1.4">GEE un.</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T6.6.6.7.1.5">GEE eq.</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T6.6.6.7.1.6">Quasi</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T6.6.6.7.1.7">MMTR</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T6.6.6.7.1.8">GEE un.</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T6.6.6.7.1.9">GEE eq.</td> </tr> <tr class="ltx_tr" id="S4.T6.3.3.3"> <th class="ltx_td ltx_align_right ltx_th ltx_th_row ltx_border_l ltx_border_r ltx_border_t" id="S4.T6.3.3.3.1"> <math alttext="n=64,m_{i}=6" class="ltx_Math" display="inline" id="S4.T6.3.3.3.1.m1.2"><semantics id="S4.T6.3.3.3.1.m1.2a"><mrow id="S4.T6.3.3.3.1.m1.2.2.2" xref="S4.T6.3.3.3.1.m1.2.2.3.cmml"><mrow id="S4.T6.3.3.3.1.m1.1.1.1.1" xref="S4.T6.3.3.3.1.m1.1.1.1.1.cmml"><mi id="S4.T6.3.3.3.1.m1.1.1.1.1.2" xref="S4.T6.3.3.3.1.m1.1.1.1.1.2.cmml">n</mi><mo id="S4.T6.3.3.3.1.m1.1.1.1.1.1" xref="S4.T6.3.3.3.1.m1.1.1.1.1.1.cmml">=</mo><mn id="S4.T6.3.3.3.1.m1.1.1.1.1.3" xref="S4.T6.3.3.3.1.m1.1.1.1.1.3.cmml">64</mn></mrow><mo id="S4.T6.3.3.3.1.m1.2.2.2.3" xref="S4.T6.3.3.3.1.m1.2.2.3a.cmml">,</mo><mrow id="S4.T6.3.3.3.1.m1.2.2.2.2" xref="S4.T6.3.3.3.1.m1.2.2.2.2.cmml"><msub id="S4.T6.3.3.3.1.m1.2.2.2.2.2" xref="S4.T6.3.3.3.1.m1.2.2.2.2.2.cmml"><mi id="S4.T6.3.3.3.1.m1.2.2.2.2.2.2" xref="S4.T6.3.3.3.1.m1.2.2.2.2.2.2.cmml">m</mi><mi id="S4.T6.3.3.3.1.m1.2.2.2.2.2.3" xref="S4.T6.3.3.3.1.m1.2.2.2.2.2.3.cmml">i</mi></msub><mo id="S4.T6.3.3.3.1.m1.2.2.2.2.1" xref="S4.T6.3.3.3.1.m1.2.2.2.2.1.cmml">=</mo><mn id="S4.T6.3.3.3.1.m1.2.2.2.2.3" xref="S4.T6.3.3.3.1.m1.2.2.2.2.3.cmml">6</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.T6.3.3.3.1.m1.2b"><apply id="S4.T6.3.3.3.1.m1.2.2.3.cmml" xref="S4.T6.3.3.3.1.m1.2.2.2"><csymbol cd="ambiguous" 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type="integer" xref="S4.T6.3.3.3.1.m1.2.2.2.2.3">6</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.T6.3.3.3.1.m1.2c">n=64,m_{i}=6</annotation><annotation encoding="application/x-llamapun" id="S4.T6.3.3.3.1.m1.2d">italic_n = 64 , italic_m start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = 6</annotation></semantics></math> </th> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T6.3.3.3.2">0.0180 (0.0023)</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T6.3.3.3.3">0.36 (0.042)</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T6.3.3.3.4">0.0140 (0.00490)</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T6.3.3.3.5">0.0100 (0.00130)</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T6.3.3.3.6">0.060 (0.0170)</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T6.3.3.3.7">1.20 (0.30)</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T6.3.3.3.8">0.42 (0.090)</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T6.3.3.3.9">0.130 (0.077)</td> </tr> <tr class="ltx_tr" id="S4.T6.4.4.4"> <th class="ltx_td ltx_align_right ltx_th ltx_th_row ltx_border_l ltx_border_r ltx_border_t" id="S4.T6.4.4.4.1"> <math alttext="n=96,m_{i}=6" class="ltx_Math" display="inline" id="S4.T6.4.4.4.1.m1.2"><semantics id="S4.T6.4.4.4.1.m1.2a"><mrow id="S4.T6.4.4.4.1.m1.2.2.2" xref="S4.T6.4.4.4.1.m1.2.2.3.cmml"><mrow id="S4.T6.4.4.4.1.m1.1.1.1.1" xref="S4.T6.4.4.4.1.m1.1.1.1.1.cmml"><mi id="S4.T6.4.4.4.1.m1.1.1.1.1.2" xref="S4.T6.4.4.4.1.m1.1.1.1.1.2.cmml">n</mi><mo id="S4.T6.4.4.4.1.m1.1.1.1.1.1" xref="S4.T6.4.4.4.1.m1.1.1.1.1.1.cmml">=</mo><mn id="S4.T6.4.4.4.1.m1.1.1.1.1.3" xref="S4.T6.4.4.4.1.m1.1.1.1.1.3.cmml">96</mn></mrow><mo id="S4.T6.4.4.4.1.m1.2.2.2.3" xref="S4.T6.4.4.4.1.m1.2.2.3a.cmml">,</mo><mrow id="S4.T6.4.4.4.1.m1.2.2.2.2" 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type="integer" xref="S4.T6.4.4.4.1.m1.1.1.1.1.3">96</cn></apply><apply id="S4.T6.4.4.4.1.m1.2.2.2.2.cmml" xref="S4.T6.4.4.4.1.m1.2.2.2.2"><eq id="S4.T6.4.4.4.1.m1.2.2.2.2.1.cmml" xref="S4.T6.4.4.4.1.m1.2.2.2.2.1"></eq><apply id="S4.T6.4.4.4.1.m1.2.2.2.2.2.cmml" xref="S4.T6.4.4.4.1.m1.2.2.2.2.2"><csymbol cd="ambiguous" id="S4.T6.4.4.4.1.m1.2.2.2.2.2.1.cmml" xref="S4.T6.4.4.4.1.m1.2.2.2.2.2">subscript</csymbol><ci id="S4.T6.4.4.4.1.m1.2.2.2.2.2.2.cmml" xref="S4.T6.4.4.4.1.m1.2.2.2.2.2.2">𝑚</ci><ci id="S4.T6.4.4.4.1.m1.2.2.2.2.2.3.cmml" xref="S4.T6.4.4.4.1.m1.2.2.2.2.2.3">𝑖</ci></apply><cn id="S4.T6.4.4.4.1.m1.2.2.2.2.3.cmml" type="integer" xref="S4.T6.4.4.4.1.m1.2.2.2.2.3">6</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.T6.4.4.4.1.m1.2c">n=96,m_{i}=6</annotation><annotation encoding="application/x-llamapun" id="S4.T6.4.4.4.1.m1.2d">italic_n = 96 , italic_m start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = 6</annotation></semantics></math> </th> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T6.4.4.4.2">0.0110 (0.0015)</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T6.4.4.4.3">0.34 (0.039)</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T6.4.4.4.4">0.0094 (0.00130)</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T6.4.4.4.5">0.0081 (0.00089)</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T6.4.4.4.6">0.033 (0.0039)</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T6.4.4.4.7">0.90 (0.13)</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T6.4.4.4.8">2.50 (2.800)</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T6.4.4.4.9">0.098 (0.054)</td> </tr> <tr class="ltx_tr" id="S4.T6.5.5.5"> <th class="ltx_td ltx_align_right ltx_th ltx_th_row ltx_border_l ltx_border_r ltx_border_t" id="S4.T6.5.5.5.1"> <math alttext="n=192,m_{i}=6" 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id="S4.T6.5.5.5.1.m1.2.2.2.2.3" xref="S4.T6.5.5.5.1.m1.2.2.2.2.3.cmml">6</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.T6.5.5.5.1.m1.2b"><apply id="S4.T6.5.5.5.1.m1.2.2.3.cmml" xref="S4.T6.5.5.5.1.m1.2.2.2"><csymbol cd="ambiguous" id="S4.T6.5.5.5.1.m1.2.2.3a.cmml" xref="S4.T6.5.5.5.1.m1.2.2.2.3">formulae-sequence</csymbol><apply id="S4.T6.5.5.5.1.m1.1.1.1.1.cmml" xref="S4.T6.5.5.5.1.m1.1.1.1.1"><eq id="S4.T6.5.5.5.1.m1.1.1.1.1.1.cmml" xref="S4.T6.5.5.5.1.m1.1.1.1.1.1"></eq><ci id="S4.T6.5.5.5.1.m1.1.1.1.1.2.cmml" xref="S4.T6.5.5.5.1.m1.1.1.1.1.2">𝑛</ci><cn id="S4.T6.5.5.5.1.m1.1.1.1.1.3.cmml" type="integer" xref="S4.T6.5.5.5.1.m1.1.1.1.1.3">192</cn></apply><apply id="S4.T6.5.5.5.1.m1.2.2.2.2.cmml" xref="S4.T6.5.5.5.1.m1.2.2.2.2"><eq id="S4.T6.5.5.5.1.m1.2.2.2.2.1.cmml" xref="S4.T6.5.5.5.1.m1.2.2.2.2.1"></eq><apply id="S4.T6.5.5.5.1.m1.2.2.2.2.2.cmml" xref="S4.T6.5.5.5.1.m1.2.2.2.2.2"><csymbol cd="ambiguous" id="S4.T6.5.5.5.1.m1.2.2.2.2.2.1.cmml" xref="S4.T6.5.5.5.1.m1.2.2.2.2.2">subscript</csymbol><ci id="S4.T6.5.5.5.1.m1.2.2.2.2.2.2.cmml" xref="S4.T6.5.5.5.1.m1.2.2.2.2.2.2">𝑚</ci><ci id="S4.T6.5.5.5.1.m1.2.2.2.2.2.3.cmml" xref="S4.T6.5.5.5.1.m1.2.2.2.2.2.3">𝑖</ci></apply><cn id="S4.T6.5.5.5.1.m1.2.2.2.2.3.cmml" type="integer" xref="S4.T6.5.5.5.1.m1.2.2.2.2.3">6</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.T6.5.5.5.1.m1.2c">n=192,m_{i}=6</annotation><annotation encoding="application/x-llamapun" id="S4.T6.5.5.5.1.m1.2d">italic_n = 192 , italic_m start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = 6</annotation></semantics></math> </th> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T6.5.5.5.2">0.0079 (0.0016)</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T6.5.5.5.3">0.33 (0.037)</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T6.5.5.5.4">0.0059 (0.00070)</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T6.5.5.5.5">0.0057 (0.00066)</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T6.5.5.5.6">0.019 (0.0016)</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T6.5.5.5.7">0.72 (0.12)</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T6.5.5.5.8">0.77 (0.082)</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T6.5.5.5.9">0.054 (0.033)</td> </tr> <tr class="ltx_tr" id="S4.T6.6.6.6"> <th class="ltx_td ltx_align_right ltx_th ltx_th_row ltx_border_b ltx_border_l ltx_border_r ltx_border_t" id="S4.T6.6.6.6.1"> <math alttext="n=288,m_{i}=6" class="ltx_Math" display="inline" id="S4.T6.6.6.6.1.m1.2"><semantics id="S4.T6.6.6.6.1.m1.2a"><mrow id="S4.T6.6.6.6.1.m1.2.2.2" xref="S4.T6.6.6.6.1.m1.2.2.3.cmml"><mrow id="S4.T6.6.6.6.1.m1.1.1.1.1" xref="S4.T6.6.6.6.1.m1.1.1.1.1.cmml"><mi id="S4.T6.6.6.6.1.m1.1.1.1.1.2" xref="S4.T6.6.6.6.1.m1.1.1.1.1.2.cmml">n</mi><mo id="S4.T6.6.6.6.1.m1.1.1.1.1.1" xref="S4.T6.6.6.6.1.m1.1.1.1.1.1.cmml">=</mo><mn id="S4.T6.6.6.6.1.m1.1.1.1.1.3" xref="S4.T6.6.6.6.1.m1.1.1.1.1.3.cmml">288</mn></mrow><mo id="S4.T6.6.6.6.1.m1.2.2.2.3" xref="S4.T6.6.6.6.1.m1.2.2.3a.cmml">,</mo><mrow id="S4.T6.6.6.6.1.m1.2.2.2.2" xref="S4.T6.6.6.6.1.m1.2.2.2.2.cmml"><msub id="S4.T6.6.6.6.1.m1.2.2.2.2.2" xref="S4.T6.6.6.6.1.m1.2.2.2.2.2.cmml"><mi id="S4.T6.6.6.6.1.m1.2.2.2.2.2.2" xref="S4.T6.6.6.6.1.m1.2.2.2.2.2.2.cmml">m</mi><mi id="S4.T6.6.6.6.1.m1.2.2.2.2.2.3" xref="S4.T6.6.6.6.1.m1.2.2.2.2.2.3.cmml">i</mi></msub><mo id="S4.T6.6.6.6.1.m1.2.2.2.2.1" xref="S4.T6.6.6.6.1.m1.2.2.2.2.1.cmml">=</mo><mn id="S4.T6.6.6.6.1.m1.2.2.2.2.3" xref="S4.T6.6.6.6.1.m1.2.2.2.2.3.cmml">6</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.T6.6.6.6.1.m1.2b"><apply id="S4.T6.6.6.6.1.m1.2.2.3.cmml" xref="S4.T6.6.6.6.1.m1.2.2.2"><csymbol cd="ambiguous" id="S4.T6.6.6.6.1.m1.2.2.3a.cmml" xref="S4.T6.6.6.6.1.m1.2.2.2.3">formulae-sequence</csymbol><apply id="S4.T6.6.6.6.1.m1.1.1.1.1.cmml" xref="S4.T6.6.6.6.1.m1.1.1.1.1"><eq id="S4.T6.6.6.6.1.m1.1.1.1.1.1.cmml" xref="S4.T6.6.6.6.1.m1.1.1.1.1.1"></eq><ci id="S4.T6.6.6.6.1.m1.1.1.1.1.2.cmml" xref="S4.T6.6.6.6.1.m1.1.1.1.1.2">𝑛</ci><cn id="S4.T6.6.6.6.1.m1.1.1.1.1.3.cmml" type="integer" xref="S4.T6.6.6.6.1.m1.1.1.1.1.3">288</cn></apply><apply id="S4.T6.6.6.6.1.m1.2.2.2.2.cmml" xref="S4.T6.6.6.6.1.m1.2.2.2.2"><eq id="S4.T6.6.6.6.1.m1.2.2.2.2.1.cmml" xref="S4.T6.6.6.6.1.m1.2.2.2.2.1"></eq><apply id="S4.T6.6.6.6.1.m1.2.2.2.2.2.cmml" xref="S4.T6.6.6.6.1.m1.2.2.2.2.2"><csymbol cd="ambiguous" id="S4.T6.6.6.6.1.m1.2.2.2.2.2.1.cmml" xref="S4.T6.6.6.6.1.m1.2.2.2.2.2">subscript</csymbol><ci id="S4.T6.6.6.6.1.m1.2.2.2.2.2.2.cmml" xref="S4.T6.6.6.6.1.m1.2.2.2.2.2.2">𝑚</ci><ci id="S4.T6.6.6.6.1.m1.2.2.2.2.2.3.cmml" xref="S4.T6.6.6.6.1.m1.2.2.2.2.2.3">𝑖</ci></apply><cn id="S4.T6.6.6.6.1.m1.2.2.2.2.3.cmml" type="integer" xref="S4.T6.6.6.6.1.m1.2.2.2.2.3">6</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.T6.6.6.6.1.m1.2c">n=288,m_{i}=6</annotation><annotation encoding="application/x-llamapun" id="S4.T6.6.6.6.1.m1.2d">italic_n = 288 , italic_m start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = 6</annotation></semantics></math> </th> <td class="ltx_td ltx_align_center ltx_border_b ltx_border_r ltx_border_t" id="S4.T6.6.6.6.2">0.0066 (0.0015)</td> <td class="ltx_td ltx_align_center ltx_border_b ltx_border_r ltx_border_t" id="S4.T6.6.6.6.3">0.32 (0.036)</td> <td class="ltx_td ltx_align_center ltx_border_b ltx_border_r ltx_border_t" id="S4.T6.6.6.6.4">0.0047 (0.00053)</td> <td class="ltx_td ltx_align_center ltx_border_b ltx_border_r ltx_border_t" id="S4.T6.6.6.6.5">0.0046 (0.00046)</td> <td class="ltx_td ltx_align_center ltx_border_b ltx_border_r ltx_border_t" id="S4.T6.6.6.6.6">0.014 (0.0013)</td> <td class="ltx_td ltx_align_center ltx_border_b ltx_border_r ltx_border_t" id="S4.T6.6.6.6.7">0.70 (0.13)</td> <td class="ltx_td ltx_align_center ltx_border_b ltx_border_r ltx_border_t" id="S4.T6.6.6.6.8">0.42 (0.045)</td> <td class="ltx_td ltx_align_center ltx_border_b ltx_border_r ltx_border_t" id="S4.T6.6.6.6.9">0.043 (0.031)</td> </tr> </tbody> </table> </span></div> </figure> </section> <section class="ltx_subsection" id="S4.SS4"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">4.4 </span>Real Data: Labeled Faces in the Wild</h3> <div class="ltx_para" id="S4.SS4.p1"> <p class="ltx_p" id="S4.SS4.p1.1">We used the Labeled Faces in the Wild (LFW) database for our real data analysis, a benchmark dataset widely used to evaluate the empirical performance of array-variate models <cite class="ltx_cite ltx_citemacro_citep">(Huang et al., <a class="ltx_ref" href="https://arxiv.org/html/2503.13782v1#bib.bib16" title="">2007</a>; Lock, <a class="ltx_ref" href="https://arxiv.org/html/2503.13782v1#bib.bib22" title="">2018</a>)</cite>. The LFW data contains multiple images of famous individuals and 73 real-valued image attributes for every image <cite class="ltx_cite ltx_citemacro_citep">(Kumar et al., <a class="ltx_ref" href="https://arxiv.org/html/2503.13782v1#bib.bib20" title="">2009</a>)</cite>. For this analysis, we focused on a subset of 5749 individuals whose images were aligned using deep funneling <cite class="ltx_cite ltx_citemacro_citep">(Huang et al., <a class="ltx_ref" href="https://arxiv.org/html/2503.13782v1#bib.bib15" title="">2012</a>)</cite>. To ensure sufficiently large <math alttext="m_{i}" class="ltx_Math" display="inline" id="S4.SS4.p1.1.m1.1"><semantics id="S4.SS4.p1.1.m1.1a"><msub id="S4.SS4.p1.1.m1.1.1" xref="S4.SS4.p1.1.m1.1.1.cmml"><mi id="S4.SS4.p1.1.m1.1.1.2" xref="S4.SS4.p1.1.m1.1.1.2.cmml">m</mi><mi id="S4.SS4.p1.1.m1.1.1.3" xref="S4.SS4.p1.1.m1.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS4.p1.1.m1.1b"><apply id="S4.SS4.p1.1.m1.1.1.cmml" xref="S4.SS4.p1.1.m1.1.1"><csymbol cd="ambiguous" id="S4.SS4.p1.1.m1.1.1.1.cmml" xref="S4.SS4.p1.1.m1.1.1">subscript</csymbol><ci id="S4.SS4.p1.1.m1.1.1.2.cmml" xref="S4.SS4.p1.1.m1.1.1.2">𝑚</ci><ci id="S4.SS4.p1.1.m1.1.1.3.cmml" xref="S4.SS4.p1.1.m1.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS4.p1.1.m1.1c">m_{i}</annotation><annotation encoding="application/x-llamapun" id="S4.SS4.p1.1.m1.1d">italic_m start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> values for MMTR, we selected individuals with 4 to 50 images, resulting in a final dataset of 594 individuals and a total of 5124 images.</p> </div> <div class="ltx_para" id="S4.SS4.p2"> <p class="ltx_p" id="S4.SS4.p2.2">The data were pre-processed to define the covariates. Each image was converted to grayscale and resized to <math alttext="32\times 32" class="ltx_Math" display="inline" id="S4.SS4.p2.1.m1.1"><semantics id="S4.SS4.p2.1.m1.1a"><mrow id="S4.SS4.p2.1.m1.1.1" xref="S4.SS4.p2.1.m1.1.1.cmml"><mn id="S4.SS4.p2.1.m1.1.1.2" xref="S4.SS4.p2.1.m1.1.1.2.cmml">32</mn><mo id="S4.SS4.p2.1.m1.1.1.1" lspace="0.222em" rspace="0.222em" xref="S4.SS4.p2.1.m1.1.1.1.cmml">×</mo><mn id="S4.SS4.p2.1.m1.1.1.3" xref="S4.SS4.p2.1.m1.1.1.3.cmml">32</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS4.p2.1.m1.1b"><apply id="S4.SS4.p2.1.m1.1.1.cmml" xref="S4.SS4.p2.1.m1.1.1"><times id="S4.SS4.p2.1.m1.1.1.1.cmml" xref="S4.SS4.p2.1.m1.1.1.1"></times><cn id="S4.SS4.p2.1.m1.1.1.2.cmml" type="integer" xref="S4.SS4.p2.1.m1.1.1.2">32</cn><cn id="S4.SS4.p2.1.m1.1.1.3.cmml" type="integer" xref="S4.SS4.p2.1.m1.1.1.3">32</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS4.p2.1.m1.1c">32\times 32</annotation><annotation encoding="application/x-llamapun" id="S4.SS4.p2.1.m1.1d">32 × 32</annotation></semantics></math> dimensions using <span class="ltx_text ltx_font_italic" id="S4.SS4.p2.2.1">grayscale</span> and <span class="ltx_text ltx_font_italic" id="S4.SS4.p2.2.2">resize</span> functions in the <span class="ltx_text ltx_font_typewriter" id="S4.SS4.p2.2.3">imager</span> library <cite class="ltx_cite ltx_citemacro_citep">(Barthelme, <a class="ltx_ref" href="https://arxiv.org/html/2503.13782v1#bib.bib1" title="">2024</a>)</cite>. This process generated <math alttext="32\times 32" class="ltx_Math" display="inline" id="S4.SS4.p2.2.m2.1"><semantics id="S4.SS4.p2.2.m2.1a"><mrow id="S4.SS4.p2.2.m2.1.1" xref="S4.SS4.p2.2.m2.1.1.cmml"><mn id="S4.SS4.p2.2.m2.1.1.2" xref="S4.SS4.p2.2.m2.1.1.2.cmml">32</mn><mo id="S4.SS4.p2.2.m2.1.1.1" lspace="0.222em" rspace="0.222em" xref="S4.SS4.p2.2.m2.1.1.1.cmml">×</mo><mn id="S4.SS4.p2.2.m2.1.1.3" xref="S4.SS4.p2.2.m2.1.1.3.cmml">32</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS4.p2.2.m2.1b"><apply id="S4.SS4.p2.2.m2.1.1.cmml" xref="S4.SS4.p2.2.m2.1.1"><times id="S4.SS4.p2.2.m2.1.1.1.cmml" xref="S4.SS4.p2.2.m2.1.1.1"></times><cn id="S4.SS4.p2.2.m2.1.1.2.cmml" type="integer" xref="S4.SS4.p2.2.m2.1.1.2">32</cn><cn id="S4.SS4.p2.2.m2.1.1.3.cmml" type="integer" xref="S4.SS4.p2.2.m2.1.1.3">32</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS4.p2.2.m2.1c">32\times 32</annotation><annotation encoding="application/x-llamapun" id="S4.SS4.p2.2.m2.1d">32 × 32</annotation></semantics></math> fixed and random effects covariate matrices for all the 5124 images. The response variable was defined as the first principal component of the 73 image attributes for each image.</p> </div> <div class="ltx_para" id="S4.SS4.p3"> <p class="ltx_p" id="S4.SS4.p3.2">We randomly split the images into ten training and testing sets. Each test set consisted of four randomly selected images from each of the 115 individuals with at least twelve images, resulting in ten training sets of 4664 images and ten testing sets of 460 images. We fit MMTR and its four competitors to each training set. Tuning parameter selection methods remain unchanged from the simulations. Unlike the simulations, <span class="ltx_text ltx_font_typewriter" id="S4.SS4.p3.2.1">lme4</span> and quasi-likelihood methods included random intercepts to account for dependencies within images of the same individual. The GEE model with an unstructured covariance parameter was inapplicable because the <math alttext="m_{i}" class="ltx_Math" display="inline" id="S4.SS4.p3.1.m1.1"><semantics id="S4.SS4.p3.1.m1.1a"><msub id="S4.SS4.p3.1.m1.1.1" xref="S4.SS4.p3.1.m1.1.1.cmml"><mi id="S4.SS4.p3.1.m1.1.1.2" xref="S4.SS4.p3.1.m1.1.1.2.cmml">m</mi><mi id="S4.SS4.p3.1.m1.1.1.3" xref="S4.SS4.p3.1.m1.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS4.p3.1.m1.1b"><apply id="S4.SS4.p3.1.m1.1.1.cmml" xref="S4.SS4.p3.1.m1.1.1"><csymbol cd="ambiguous" id="S4.SS4.p3.1.m1.1.1.1.cmml" xref="S4.SS4.p3.1.m1.1.1">subscript</csymbol><ci id="S4.SS4.p3.1.m1.1.1.2.cmml" xref="S4.SS4.p3.1.m1.1.1.2">𝑚</ci><ci id="S4.SS4.p3.1.m1.1.1.3.cmml" xref="S4.SS4.p3.1.m1.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS4.p3.1.m1.1c">m_{i}</annotation><annotation encoding="application/x-llamapun" id="S4.SS4.p3.1.m1.1d">italic_m start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> values varied across the selected individuals. Their performance was evaluated on the test sets using MSPE in (<a class="ltx_ref" href="https://arxiv.org/html/2503.13782v1#S4.E13" title="In 4.1 Setup ‣ 4 Experiments ‣ Regularized Parameter Estimation in Mixed Model Trace Regression"><span class="ltx_text ltx_ref_tag">13</span></a>) and prediction <math alttext="R^{2}" class="ltx_Math" display="inline" id="S4.SS4.p3.2.m2.1"><semantics id="S4.SS4.p3.2.m2.1a"><msup id="S4.SS4.p3.2.m2.1.1" xref="S4.SS4.p3.2.m2.1.1.cmml"><mi id="S4.SS4.p3.2.m2.1.1.2" xref="S4.SS4.p3.2.m2.1.1.2.cmml">R</mi><mn id="S4.SS4.p3.2.m2.1.1.3" xref="S4.SS4.p3.2.m2.1.1.3.cmml">2</mn></msup><annotation-xml encoding="MathML-Content" id="S4.SS4.p3.2.m2.1b"><apply id="S4.SS4.p3.2.m2.1.1.cmml" xref="S4.SS4.p3.2.m2.1.1"><csymbol cd="ambiguous" id="S4.SS4.p3.2.m2.1.1.1.cmml" xref="S4.SS4.p3.2.m2.1.1">superscript</csymbol><ci id="S4.SS4.p3.2.m2.1.1.2.cmml" xref="S4.SS4.p3.2.m2.1.1.2">𝑅</ci><cn id="S4.SS4.p3.2.m2.1.1.3.cmml" type="integer" xref="S4.SS4.p3.2.m2.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS4.p3.2.m2.1c">R^{2}</annotation><annotation encoding="application/x-llamapun" id="S4.SS4.p3.2.m2.1d">italic_R start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S4.SS4.p4"> <p class="ltx_p" id="S4.SS4.p4.3">MMTR achieved the lowest MSPE and highest prediction <math alttext="R^{2}" class="ltx_Math" display="inline" id="S4.SS4.p4.1.m1.1"><semantics id="S4.SS4.p4.1.m1.1a"><msup id="S4.SS4.p4.1.m1.1.1" xref="S4.SS4.p4.1.m1.1.1.cmml"><mi id="S4.SS4.p4.1.m1.1.1.2" xref="S4.SS4.p4.1.m1.1.1.2.cmml">R</mi><mn id="S4.SS4.p4.1.m1.1.1.3" xref="S4.SS4.p4.1.m1.1.1.3.cmml">2</mn></msup><annotation-xml encoding="MathML-Content" id="S4.SS4.p4.1.m1.1b"><apply id="S4.SS4.p4.1.m1.1.1.cmml" xref="S4.SS4.p4.1.m1.1.1"><csymbol cd="ambiguous" id="S4.SS4.p4.1.m1.1.1.1.cmml" xref="S4.SS4.p4.1.m1.1.1">superscript</csymbol><ci id="S4.SS4.p4.1.m1.1.1.2.cmml" xref="S4.SS4.p4.1.m1.1.1.2">𝑅</ci><cn id="S4.SS4.p4.1.m1.1.1.3.cmml" type="integer" xref="S4.SS4.p4.1.m1.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS4.p4.1.m1.1c">R^{2}</annotation><annotation encoding="application/x-llamapun" id="S4.SS4.p4.1.m1.1d">italic_R start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT</annotation></semantics></math> among all the methods (Table <a class="ltx_ref" href="https://arxiv.org/html/2503.13782v1#S4.T7" title="Table 7 ‣ 4.4 Real Data: Labeled Faces in the Wild ‣ 4 Experiments ‣ Regularized Parameter Estimation in Mixed Model Trace Regression"><span class="ltx_text ltx_ref_tag">7</span></a>). The averaged estimate of <math alttext="\operatorname{\mathbf{B}}" class="ltx_Math" display="inline" id="S4.SS4.p4.2.m2.1"><semantics id="S4.SS4.p4.2.m2.1a"><mi id="S4.SS4.p4.2.m2.1.1" xref="S4.SS4.p4.2.m2.1.1.cmml">𝐁</mi><annotation-xml encoding="MathML-Content" id="S4.SS4.p4.2.m2.1b"><ci id="S4.SS4.p4.2.m2.1.1.cmml" xref="S4.SS4.p4.2.m2.1.1">𝐁</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS4.p4.2.m2.1c">\operatorname{\mathbf{B}}</annotation><annotation encoding="application/x-llamapun" id="S4.SS4.p4.2.m2.1d">bold_B</annotation></semantics></math> across the ten training sets revealed key facial features in the images (Figure <a class="ltx_ref" href="https://arxiv.org/html/2503.13782v1#S4.F1" title="Figure 1 ‣ 4.4 Real Data: Labeled Faces in the Wild ‣ 4 Experiments ‣ Regularized Parameter Estimation in Mixed Model Trace Regression"><span class="ltx_text ltx_ref_tag">1</span></a>). Based on the simulation results, MSPE, and prediction <math alttext="R^{2}" class="ltx_Math" display="inline" id="S4.SS4.p4.3.m3.1"><semantics id="S4.SS4.p4.3.m3.1a"><msup id="S4.SS4.p4.3.m3.1.1" xref="S4.SS4.p4.3.m3.1.1.cmml"><mi id="S4.SS4.p4.3.m3.1.1.2" xref="S4.SS4.p4.3.m3.1.1.2.cmml">R</mi><mn id="S4.SS4.p4.3.m3.1.1.3" xref="S4.SS4.p4.3.m3.1.1.3.cmml">2</mn></msup><annotation-xml encoding="MathML-Content" id="S4.SS4.p4.3.m3.1b"><apply id="S4.SS4.p4.3.m3.1.1.cmml" xref="S4.SS4.p4.3.m3.1.1"><csymbol cd="ambiguous" id="S4.SS4.p4.3.m3.1.1.1.cmml" xref="S4.SS4.p4.3.m3.1.1">superscript</csymbol><ci id="S4.SS4.p4.3.m3.1.1.2.cmml" xref="S4.SS4.p4.3.m3.1.1.2">𝑅</ci><cn id="S4.SS4.p4.3.m3.1.1.3.cmml" type="integer" xref="S4.SS4.p4.3.m3.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS4.p4.3.m3.1c">R^{2}</annotation><annotation encoding="application/x-llamapun" id="S4.SS4.p4.3.m3.1d">italic_R start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT</annotation></semantics></math>, we concluded that MMTR outperforms its competitors in accurately predicting resized LFW images.</p> </div> <figure class="ltx_table" id="S4.T7"> <figcaption class="ltx_caption"><span class="ltx_tag ltx_tag_table">Table 7: </span>MSPE and prediction <math alttext="R^{2}" class="ltx_Math" display="inline" id="S4.T7.3.m1.1"><semantics id="S4.T7.3.m1.1b"><msup id="S4.T7.3.m1.1.1" xref="S4.T7.3.m1.1.1.cmml"><mi id="S4.T7.3.m1.1.1.2" xref="S4.T7.3.m1.1.1.2.cmml">R</mi><mn id="S4.T7.3.m1.1.1.3" xref="S4.T7.3.m1.1.1.3.cmml">2</mn></msup><annotation-xml encoding="MathML-Content" id="S4.T7.3.m1.1c"><apply id="S4.T7.3.m1.1.1.cmml" xref="S4.T7.3.m1.1.1"><csymbol cd="ambiguous" id="S4.T7.3.m1.1.1.1.cmml" xref="S4.T7.3.m1.1.1">superscript</csymbol><ci id="S4.T7.3.m1.1.1.2.cmml" xref="S4.T7.3.m1.1.1.2">𝑅</ci><cn id="S4.T7.3.m1.1.1.3.cmml" type="integer" xref="S4.T7.3.m1.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.T7.3.m1.1d">R^{2}</annotation><annotation encoding="application/x-llamapun" id="S4.T7.3.m1.1e">italic_R start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT</annotation></semantics></math>. The MSPE and prediction <math alttext="R^{2}" class="ltx_Math" display="inline" id="S4.T7.4.m2.1"><semantics id="S4.T7.4.m2.1b"><msup id="S4.T7.4.m2.1.1" xref="S4.T7.4.m2.1.1.cmml"><mi id="S4.T7.4.m2.1.1.2" xref="S4.T7.4.m2.1.1.2.cmml">R</mi><mn id="S4.T7.4.m2.1.1.3" xref="S4.T7.4.m2.1.1.3.cmml">2</mn></msup><annotation-xml encoding="MathML-Content" id="S4.T7.4.m2.1c"><apply id="S4.T7.4.m2.1.1.cmml" xref="S4.T7.4.m2.1.1"><csymbol cd="ambiguous" id="S4.T7.4.m2.1.1.1.cmml" xref="S4.T7.4.m2.1.1">superscript</csymbol><ci id="S4.T7.4.m2.1.1.2.cmml" xref="S4.T7.4.m2.1.1.2">𝑅</ci><cn id="S4.T7.4.m2.1.1.3.cmml" type="integer" xref="S4.T7.4.m2.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.T7.4.m2.1d">R^{2}</annotation><annotation encoding="application/x-llamapun" id="S4.T7.4.m2.1e">italic_R start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT</annotation></semantics></math> values are averaged across ten testing sets of the LFW data, with the Monte Carlo errors in parenthesis. GEE equicorr. Kruskal, and Quasi denote the GEE with equicorrelated covariance matrix, sparse Kruskal, and quasi-likelihood methods.</figcaption> <table class="ltx_tabular ltx_centering ltx_guessed_headers ltx_align_top" id="S4.T7.5"> <tbody class="ltx_tbody"> <tr class="ltx_tr" id="S4.T7.5.2.1"> <th class="ltx_td ltx_th ltx_th_row ltx_border_l ltx_border_r ltx_border_t" id="S4.T7.5.2.1.1"></th> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T7.5.2.1.2">MMTR</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T7.5.2.1.3">Quasi</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T7.5.2.1.4">LME4</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T7.5.2.1.5">GEE equicorr.</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T7.5.2.1.6">Kruskal</td> </tr> <tr class="ltx_tr" id="S4.T7.5.3.2"> <th class="ltx_td ltx_align_right ltx_th ltx_th_row ltx_border_l ltx_border_r ltx_border_t" id="S4.T7.5.3.2.1">MSPE</th> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T7.5.3.2.2">2.19 (0.13)</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T7.5.3.2.3">2.32 (0.14)</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T7.5.3.2.4">2.81 (0.20)</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T7.5.3.2.5">3.18 (0.24)</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S4.T7.5.3.2.6">5.47 (0.42)</td> </tr> <tr class="ltx_tr" id="S4.T7.5.1"> <th class="ltx_td ltx_align_right ltx_th ltx_th_row ltx_border_b ltx_border_l ltx_border_r ltx_border_t" id="S4.T7.5.1.1"><math alttext="R^{2}" class="ltx_Math" display="inline" id="S4.T7.5.1.1.m1.1"><semantics id="S4.T7.5.1.1.m1.1a"><msup id="S4.T7.5.1.1.m1.1.1" xref="S4.T7.5.1.1.m1.1.1.cmml"><mi id="S4.T7.5.1.1.m1.1.1.2" xref="S4.T7.5.1.1.m1.1.1.2.cmml">R</mi><mn id="S4.T7.5.1.1.m1.1.1.3" xref="S4.T7.5.1.1.m1.1.1.3.cmml">2</mn></msup><annotation-xml encoding="MathML-Content" id="S4.T7.5.1.1.m1.1b"><apply id="S4.T7.5.1.1.m1.1.1.cmml" xref="S4.T7.5.1.1.m1.1.1"><csymbol cd="ambiguous" id="S4.T7.5.1.1.m1.1.1.1.cmml" xref="S4.T7.5.1.1.m1.1.1">superscript</csymbol><ci id="S4.T7.5.1.1.m1.1.1.2.cmml" xref="S4.T7.5.1.1.m1.1.1.2">𝑅</ci><cn id="S4.T7.5.1.1.m1.1.1.3.cmml" type="integer" xref="S4.T7.5.1.1.m1.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.T7.5.1.1.m1.1c">R^{2}</annotation><annotation encoding="application/x-llamapun" id="S4.T7.5.1.1.m1.1d">italic_R start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT</annotation></semantics></math></th> <td class="ltx_td ltx_align_center ltx_border_b ltx_border_r ltx_border_t" id="S4.T7.5.1.2">0.84 (0.012)</td> <td class="ltx_td ltx_align_center ltx_border_b ltx_border_r ltx_border_t" id="S4.T7.5.1.3">0.83 (0.013)</td> <td class="ltx_td ltx_align_center ltx_border_b ltx_border_r ltx_border_t" id="S4.T7.5.1.4">0.79 (0.009)</td> <td class="ltx_td ltx_align_center ltx_border_b ltx_border_r ltx_border_t" id="S4.T7.5.1.5">0.76 (0.012)</td> <td class="ltx_td ltx_align_center ltx_border_b ltx_border_r ltx_border_t" id="S4.T7.5.1.6">0.60 (0.018)</td> </tr> </tbody> </table> </figure> <figure class="ltx_figure" id="S4.F1"><img alt="Refer to caption" class="ltx_graphics ltx_img_landscape" height="490" id="S4.F1.g1" src="extracted/6288390/mean_B_hat.png" width="698"/> <figcaption class="ltx_caption"><span class="ltx_tag ltx_tag_figure">Figure 1: </span>Mean parameter <math alttext="{\operatorname{\mathbf{B}}}" class="ltx_Math" display="inline" id="S4.F1.3.m1.1"><semantics id="S4.F1.3.m1.1b"><mi id="S4.F1.3.m1.1.1" xref="S4.F1.3.m1.1.1.cmml">𝐁</mi><annotation-xml encoding="MathML-Content" id="S4.F1.3.m1.1c"><ci id="S4.F1.3.m1.1.1.cmml" xref="S4.F1.3.m1.1.1">𝐁</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.F1.3.m1.1d">{\operatorname{\mathbf{B}}}</annotation><annotation encoding="application/x-llamapun" id="S4.F1.3.m1.1e">bold_B</annotation></semantics></math> estimated by MMTR, quasi-likelihood, lme4, GEE with equicorrelation covariance matrix, and sparse Kruskal methods. The <math alttext="\operatorname{\mathbf{B}}" class="ltx_Math" display="inline" id="S4.F1.4.m2.1"><semantics id="S4.F1.4.m2.1b"><mi id="S4.F1.4.m2.1.1" xref="S4.F1.4.m2.1.1.cmml">𝐁</mi><annotation-xml encoding="MathML-Content" id="S4.F1.4.m2.1c"><ci id="S4.F1.4.m2.1.1.cmml" xref="S4.F1.4.m2.1.1">𝐁</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.F1.4.m2.1d">\operatorname{\mathbf{B}}</annotation><annotation encoding="application/x-llamapun" id="S4.F1.4.m2.1e">bold_B</annotation></semantics></math> estimates are averaged across ten training sets of the LFW data.</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>Discussion</h2> <div class="ltx_para" id="S5.p1"> <p class="ltx_p" id="S5.p1.21">We have focused on a correlated scalar responses and matrix-variate covariates, but MMTR generalizes to array-variate covariates. If the random effect <math alttext="\operatorname{\mathbf{A}}_{i}" class="ltx_Math" display="inline" id="S5.p1.1.m1.1"><semantics id="S5.p1.1.m1.1a"><msub id="S5.p1.1.m1.1.1" xref="S5.p1.1.m1.1.1.cmml"><mi id="S5.p1.1.m1.1.1.2" xref="S5.p1.1.m1.1.1.2.cmml">𝐀</mi><mi id="S5.p1.1.m1.1.1.3" xref="S5.p1.1.m1.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S5.p1.1.m1.1b"><apply id="S5.p1.1.m1.1.1.cmml" xref="S5.p1.1.m1.1.1"><csymbol cd="ambiguous" id="S5.p1.1.m1.1.1.1.cmml" xref="S5.p1.1.m1.1.1">subscript</csymbol><ci id="S5.p1.1.m1.1.1.2.cmml" xref="S5.p1.1.m1.1.1.2">𝐀</ci><ci id="S5.p1.1.m1.1.1.3.cmml" xref="S5.p1.1.m1.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.p1.1.m1.1c">\operatorname{\mathbf{A}}_{i}</annotation><annotation encoding="application/x-llamapun" id="S5.p1.1.m1.1d">bold_A start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> in (<a class="ltx_ref" href="https://arxiv.org/html/2503.13782v1#S1.E1" title="In 1 Introduction ‣ Regularized Parameter Estimation in Mixed Model Trace Regression"><span class="ltx_text ltx_ref_tag">1</span></a>) is a <math alttext="Q_{1}\times\cdots\times Q_{d}" class="ltx_Math" display="inline" id="S5.p1.2.m2.1"><semantics id="S5.p1.2.m2.1a"><mrow id="S5.p1.2.m2.1.1" xref="S5.p1.2.m2.1.1.cmml"><msub id="S5.p1.2.m2.1.1.2" xref="S5.p1.2.m2.1.1.2.cmml"><mi id="S5.p1.2.m2.1.1.2.2" xref="S5.p1.2.m2.1.1.2.2.cmml">Q</mi><mn id="S5.p1.2.m2.1.1.2.3" xref="S5.p1.2.m2.1.1.2.3.cmml">1</mn></msub><mo id="S5.p1.2.m2.1.1.1" lspace="0.222em" rspace="0.222em" xref="S5.p1.2.m2.1.1.1.cmml">×</mo><mi id="S5.p1.2.m2.1.1.3" mathvariant="normal" xref="S5.p1.2.m2.1.1.3.cmml">⋯</mi><mo id="S5.p1.2.m2.1.1.1a" lspace="0.222em" rspace="0.222em" xref="S5.p1.2.m2.1.1.1.cmml">×</mo><msub id="S5.p1.2.m2.1.1.4" xref="S5.p1.2.m2.1.1.4.cmml"><mi id="S5.p1.2.m2.1.1.4.2" xref="S5.p1.2.m2.1.1.4.2.cmml">Q</mi><mi id="S5.p1.2.m2.1.1.4.3" 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id="S5.p1.4.m4.1.1.3.2.2.3.cmml" type="integer" xref="S5.p1.4.m4.1.1.3.2.2.3">2</cn></apply><apply id="S5.p1.4.m4.1.1.3.2.3.cmml" xref="S5.p1.4.m4.1.1.3.2.3"><csymbol cd="ambiguous" id="S5.p1.4.m4.1.1.3.2.3.1.cmml" xref="S5.p1.4.m4.1.1.3.2.3">subscript</csymbol><ci id="S5.p1.4.m4.1.1.3.2.3.2.cmml" xref="S5.p1.4.m4.1.1.3.2.3.2">𝚺</ci><ci id="S5.p1.4.m4.1.1.3.2.3.3.cmml" xref="S5.p1.4.m4.1.1.3.2.3.3">𝑑</ci></apply></apply><ci id="S5.p1.4.m4.1.1.3.3.cmml" xref="S5.p1.4.m4.1.1.3.3">⋯</ci><apply id="S5.p1.4.m4.1.1.3.4.cmml" xref="S5.p1.4.m4.1.1.3.4"><csymbol cd="ambiguous" id="S5.p1.4.m4.1.1.3.4.1.cmml" xref="S5.p1.4.m4.1.1.3.4">subscript</csymbol><ci id="S5.p1.4.m4.1.1.3.4.2.cmml" xref="S5.p1.4.m4.1.1.3.4.2">𝚺</ci><cn id="S5.p1.4.m4.1.1.3.4.3.cmml" type="integer" xref="S5.p1.4.m4.1.1.3.4.3">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.p1.4.m4.1c">\text{Cov}\{\text{vec}(\operatorname{\mathbf{A}}_{i})\}=\tau^{2}\operatorname{% {\mathbf{\Sigma}}}_{d}\otimes\cdots\otimes\operatorname{{\mathbf{\Sigma}}}_{1}</annotation><annotation encoding="application/x-llamapun" id="S5.p1.4.m4.1d">Cov { vec ( bold_A start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) } = italic_τ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT bold_Σ start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT ⊗ ⋯ ⊗ bold_Σ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math>, where <math alttext="\operatorname{\mathbf{A}}_{i}" class="ltx_Math" display="inline" id="S5.p1.5.m5.1"><semantics id="S5.p1.5.m5.1a"><msub id="S5.p1.5.m5.1.1" xref="S5.p1.5.m5.1.1.cmml"><mi id="S5.p1.5.m5.1.1.2" xref="S5.p1.5.m5.1.1.2.cmml">𝐀</mi><mi id="S5.p1.5.m5.1.1.3" xref="S5.p1.5.m5.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S5.p1.5.m5.1b"><apply id="S5.p1.5.m5.1.1.cmml" xref="S5.p1.5.m5.1.1"><csymbol cd="ambiguous" id="S5.p1.5.m5.1.1.1.cmml" xref="S5.p1.5.m5.1.1">subscript</csymbol><ci id="S5.p1.5.m5.1.1.2.cmml" xref="S5.p1.5.m5.1.1.2">𝐀</ci><ci id="S5.p1.5.m5.1.1.3.cmml" xref="S5.p1.5.m5.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.p1.5.m5.1c">\operatorname{\mathbf{A}}_{i}</annotation><annotation encoding="application/x-llamapun" id="S5.p1.5.m5.1d">bold_A start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> is an array-variate normal distribution with zero mean array and dimension <math alttext="j" class="ltx_Math" display="inline" id="S5.p1.6.m6.1"><semantics id="S5.p1.6.m6.1a"><mi id="S5.p1.6.m6.1.1" xref="S5.p1.6.m6.1.1.cmml">j</mi><annotation-xml encoding="MathML-Content" id="S5.p1.6.m6.1b"><ci id="S5.p1.6.m6.1.1.cmml" xref="S5.p1.6.m6.1.1">𝑗</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.p1.6.m6.1c">j</annotation><annotation encoding="application/x-llamapun" id="S5.p1.6.m6.1d">italic_j</annotation></semantics></math>-specific covariance matrix <math alttext="\operatorname{{\mathbf{\Sigma}}}_{j}" class="ltx_Math" display="inline" id="S5.p1.7.m7.1"><semantics id="S5.p1.7.m7.1a"><msub id="S5.p1.7.m7.1.1" xref="S5.p1.7.m7.1.1.cmml"><mi id="S5.p1.7.m7.1.1.2" xref="S5.p1.7.m7.1.1.2.cmml">𝚺</mi><mi id="S5.p1.7.m7.1.1.3" xref="S5.p1.7.m7.1.1.3.cmml">j</mi></msub><annotation-xml encoding="MathML-Content" id="S5.p1.7.m7.1b"><apply id="S5.p1.7.m7.1.1.cmml" xref="S5.p1.7.m7.1.1"><csymbol cd="ambiguous" id="S5.p1.7.m7.1.1.1.cmml" xref="S5.p1.7.m7.1.1">subscript</csymbol><ci id="S5.p1.7.m7.1.1.2.cmml" xref="S5.p1.7.m7.1.1.2">𝚺</ci><ci id="S5.p1.7.m7.1.1.3.cmml" xref="S5.p1.7.m7.1.1.3">𝑗</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.p1.7.m7.1c">\operatorname{{\mathbf{\Sigma}}}_{j}</annotation><annotation encoding="application/x-llamapun" id="S5.p1.7.m7.1d">bold_Σ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT</annotation></semantics></math> (<math alttext="j=1,\ldots,d" class="ltx_Math" display="inline" id="S5.p1.8.m8.3"><semantics id="S5.p1.8.m8.3a"><mrow id="S5.p1.8.m8.3.4" xref="S5.p1.8.m8.3.4.cmml"><mi id="S5.p1.8.m8.3.4.2" xref="S5.p1.8.m8.3.4.2.cmml">j</mi><mo id="S5.p1.8.m8.3.4.1" xref="S5.p1.8.m8.3.4.1.cmml">=</mo><mrow id="S5.p1.8.m8.3.4.3.2" xref="S5.p1.8.m8.3.4.3.1.cmml"><mn id="S5.p1.8.m8.1.1" xref="S5.p1.8.m8.1.1.cmml">1</mn><mo id="S5.p1.8.m8.3.4.3.2.1" xref="S5.p1.8.m8.3.4.3.1.cmml">,</mo><mi id="S5.p1.8.m8.2.2" mathvariant="normal" xref="S5.p1.8.m8.2.2.cmml">…</mi><mo id="S5.p1.8.m8.3.4.3.2.2" xref="S5.p1.8.m8.3.4.3.1.cmml">,</mo><mi id="S5.p1.8.m8.3.3" xref="S5.p1.8.m8.3.3.cmml">d</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.p1.8.m8.3b"><apply id="S5.p1.8.m8.3.4.cmml" xref="S5.p1.8.m8.3.4"><eq id="S5.p1.8.m8.3.4.1.cmml" xref="S5.p1.8.m8.3.4.1"></eq><ci id="S5.p1.8.m8.3.4.2.cmml" xref="S5.p1.8.m8.3.4.2">𝑗</ci><list id="S5.p1.8.m8.3.4.3.1.cmml" xref="S5.p1.8.m8.3.4.3.2"><cn id="S5.p1.8.m8.1.1.cmml" type="integer" xref="S5.p1.8.m8.1.1">1</cn><ci id="S5.p1.8.m8.2.2.cmml" xref="S5.p1.8.m8.2.2">…</ci><ci id="S5.p1.8.m8.3.3.cmml" xref="S5.p1.8.m8.3.3">𝑑</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.p1.8.m8.3c">j=1,\ldots,d</annotation><annotation encoding="application/x-llamapun" id="S5.p1.8.m8.3d">italic_j = 1 , … , italic_d</annotation></semantics></math>) <cite class="ltx_cite ltx_citemacro_citep">(Hoff, <a class="ltx_ref" href="https://arxiv.org/html/2503.13782v1#bib.bib14" title="">2011</a>)</cite>. For <math alttext="j=1,\ldots,d" class="ltx_Math" display="inline" id="S5.p1.9.m9.3"><semantics id="S5.p1.9.m9.3a"><mrow id="S5.p1.9.m9.3.4" xref="S5.p1.9.m9.3.4.cmml"><mi id="S5.p1.9.m9.3.4.2" xref="S5.p1.9.m9.3.4.2.cmml">j</mi><mo id="S5.p1.9.m9.3.4.1" xref="S5.p1.9.m9.3.4.1.cmml">=</mo><mrow id="S5.p1.9.m9.3.4.3.2" xref="S5.p1.9.m9.3.4.3.1.cmml"><mn id="S5.p1.9.m9.1.1" xref="S5.p1.9.m9.1.1.cmml">1</mn><mo id="S5.p1.9.m9.3.4.3.2.1" xref="S5.p1.9.m9.3.4.3.1.cmml">,</mo><mi id="S5.p1.9.m9.2.2" mathvariant="normal" xref="S5.p1.9.m9.2.2.cmml">…</mi><mo id="S5.p1.9.m9.3.4.3.2.2" xref="S5.p1.9.m9.3.4.3.1.cmml">,</mo><mi id="S5.p1.9.m9.3.3" xref="S5.p1.9.m9.3.3.cmml">d</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.p1.9.m9.3b"><apply id="S5.p1.9.m9.3.4.cmml" xref="S5.p1.9.m9.3.4"><eq id="S5.p1.9.m9.3.4.1.cmml" xref="S5.p1.9.m9.3.4.1"></eq><ci id="S5.p1.9.m9.3.4.2.cmml" xref="S5.p1.9.m9.3.4.2">𝑗</ci><list id="S5.p1.9.m9.3.4.3.1.cmml" xref="S5.p1.9.m9.3.4.3.2"><cn id="S5.p1.9.m9.1.1.cmml" type="integer" xref="S5.p1.9.m9.1.1">1</cn><ci id="S5.p1.9.m9.2.2.cmml" xref="S5.p1.9.m9.2.2">…</ci><ci id="S5.p1.9.m9.3.3.cmml" xref="S5.p1.9.m9.3.3">𝑑</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.p1.9.m9.3c">j=1,\ldots,d</annotation><annotation encoding="application/x-llamapun" id="S5.p1.9.m9.3d">italic_j = 1 , … , italic_d</annotation></semantics></math> and <math alttext="S_{j}\ll Q_{j}" class="ltx_Math" display="inline" id="S5.p1.10.m10.1"><semantics id="S5.p1.10.m10.1a"><mrow id="S5.p1.10.m10.1.1" xref="S5.p1.10.m10.1.1.cmml"><msub id="S5.p1.10.m10.1.1.2" xref="S5.p1.10.m10.1.1.2.cmml"><mi id="S5.p1.10.m10.1.1.2.2" xref="S5.p1.10.m10.1.1.2.2.cmml">S</mi><mi id="S5.p1.10.m10.1.1.2.3" xref="S5.p1.10.m10.1.1.2.3.cmml">j</mi></msub><mo id="S5.p1.10.m10.1.1.1" xref="S5.p1.10.m10.1.1.1.cmml">≪</mo><msub id="S5.p1.10.m10.1.1.3" xref="S5.p1.10.m10.1.1.3.cmml"><mi id="S5.p1.10.m10.1.1.3.2" xref="S5.p1.10.m10.1.1.3.2.cmml">Q</mi><mi id="S5.p1.10.m10.1.1.3.3" xref="S5.p1.10.m10.1.1.3.3.cmml">j</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S5.p1.10.m10.1b"><apply id="S5.p1.10.m10.1.1.cmml" xref="S5.p1.10.m10.1.1"><csymbol cd="latexml" id="S5.p1.10.m10.1.1.1.cmml" xref="S5.p1.10.m10.1.1.1">much-less-than</csymbol><apply id="S5.p1.10.m10.1.1.2.cmml" xref="S5.p1.10.m10.1.1.2"><csymbol cd="ambiguous" id="S5.p1.10.m10.1.1.2.1.cmml" xref="S5.p1.10.m10.1.1.2">subscript</csymbol><ci id="S5.p1.10.m10.1.1.2.2.cmml" xref="S5.p1.10.m10.1.1.2.2">𝑆</ci><ci id="S5.p1.10.m10.1.1.2.3.cmml" xref="S5.p1.10.m10.1.1.2.3">𝑗</ci></apply><apply id="S5.p1.10.m10.1.1.3.cmml" xref="S5.p1.10.m10.1.1.3"><csymbol cd="ambiguous" id="S5.p1.10.m10.1.1.3.1.cmml" xref="S5.p1.10.m10.1.1.3">subscript</csymbol><ci id="S5.p1.10.m10.1.1.3.2.cmml" xref="S5.p1.10.m10.1.1.3.2">𝑄</ci><ci id="S5.p1.10.m10.1.1.3.3.cmml" xref="S5.p1.10.m10.1.1.3.3">𝑗</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.p1.10.m10.1c">S_{j}\ll Q_{j}</annotation><annotation encoding="application/x-llamapun" id="S5.p1.10.m10.1d">italic_S start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ≪ italic_Q start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT</annotation></semantics></math>, we assume that there exists <math alttext="\operatorname{\mathbf{L}}_{j}\in\operatorname{\mathbb{R}}^{Q_{j}\times S_{j}}" class="ltx_Math" display="inline" id="S5.p1.11.m11.1"><semantics id="S5.p1.11.m11.1a"><mrow id="S5.p1.11.m11.1.1" xref="S5.p1.11.m11.1.1.cmml"><msub id="S5.p1.11.m11.1.1.2" xref="S5.p1.11.m11.1.1.2.cmml"><mi id="S5.p1.11.m11.1.1.2.2" xref="S5.p1.11.m11.1.1.2.2.cmml">𝐋</mi><mi id="S5.p1.11.m11.1.1.2.3" xref="S5.p1.11.m11.1.1.2.3.cmml">j</mi></msub><mo id="S5.p1.11.m11.1.1.1" xref="S5.p1.11.m11.1.1.1.cmml">∈</mo><msup id="S5.p1.11.m11.1.1.3" xref="S5.p1.11.m11.1.1.3.cmml"><mi id="S5.p1.11.m11.1.1.3.2" xref="S5.p1.11.m11.1.1.3.2.cmml">ℝ</mi><mrow id="S5.p1.11.m11.1.1.3.3" xref="S5.p1.11.m11.1.1.3.3.cmml"><msub id="S5.p1.11.m11.1.1.3.3.2" xref="S5.p1.11.m11.1.1.3.3.2.cmml"><mi id="S5.p1.11.m11.1.1.3.3.2.2" xref="S5.p1.11.m11.1.1.3.3.2.2.cmml">Q</mi><mi id="S5.p1.11.m11.1.1.3.3.2.3" xref="S5.p1.11.m11.1.1.3.3.2.3.cmml">j</mi></msub><mo id="S5.p1.11.m11.1.1.3.3.1" lspace="0.222em" rspace="0.222em" xref="S5.p1.11.m11.1.1.3.3.1.cmml">×</mo><msub id="S5.p1.11.m11.1.1.3.3.3" xref="S5.p1.11.m11.1.1.3.3.3.cmml"><mi id="S5.p1.11.m11.1.1.3.3.3.2" xref="S5.p1.11.m11.1.1.3.3.3.2.cmml">S</mi><mi id="S5.p1.11.m11.1.1.3.3.3.3" xref="S5.p1.11.m11.1.1.3.3.3.3.cmml">j</mi></msub></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="S5.p1.11.m11.1b"><apply id="S5.p1.11.m11.1.1.cmml" xref="S5.p1.11.m11.1.1"><in id="S5.p1.11.m11.1.1.1.cmml" xref="S5.p1.11.m11.1.1.1"></in><apply id="S5.p1.11.m11.1.1.2.cmml" xref="S5.p1.11.m11.1.1.2"><csymbol cd="ambiguous" id="S5.p1.11.m11.1.1.2.1.cmml" xref="S5.p1.11.m11.1.1.2">subscript</csymbol><ci id="S5.p1.11.m11.1.1.2.2.cmml" xref="S5.p1.11.m11.1.1.2.2">𝐋</ci><ci id="S5.p1.11.m11.1.1.2.3.cmml" xref="S5.p1.11.m11.1.1.2.3">𝑗</ci></apply><apply id="S5.p1.11.m11.1.1.3.cmml" xref="S5.p1.11.m11.1.1.3"><csymbol cd="ambiguous" id="S5.p1.11.m11.1.1.3.1.cmml" xref="S5.p1.11.m11.1.1.3">superscript</csymbol><ci id="S5.p1.11.m11.1.1.3.2.cmml" xref="S5.p1.11.m11.1.1.3.2">ℝ</ci><apply id="S5.p1.11.m11.1.1.3.3.cmml" xref="S5.p1.11.m11.1.1.3.3"><times id="S5.p1.11.m11.1.1.3.3.1.cmml" xref="S5.p1.11.m11.1.1.3.3.1"></times><apply id="S5.p1.11.m11.1.1.3.3.2.cmml" xref="S5.p1.11.m11.1.1.3.3.2"><csymbol cd="ambiguous" id="S5.p1.11.m11.1.1.3.3.2.1.cmml" xref="S5.p1.11.m11.1.1.3.3.2">subscript</csymbol><ci id="S5.p1.11.m11.1.1.3.3.2.2.cmml" xref="S5.p1.11.m11.1.1.3.3.2.2">𝑄</ci><ci id="S5.p1.11.m11.1.1.3.3.2.3.cmml" xref="S5.p1.11.m11.1.1.3.3.2.3">𝑗</ci></apply><apply id="S5.p1.11.m11.1.1.3.3.3.cmml" xref="S5.p1.11.m11.1.1.3.3.3"><csymbol cd="ambiguous" id="S5.p1.11.m11.1.1.3.3.3.1.cmml" xref="S5.p1.11.m11.1.1.3.3.3">subscript</csymbol><ci id="S5.p1.11.m11.1.1.3.3.3.2.cmml" xref="S5.p1.11.m11.1.1.3.3.3.2">𝑆</ci><ci id="S5.p1.11.m11.1.1.3.3.3.3.cmml" xref="S5.p1.11.m11.1.1.3.3.3.3">𝑗</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.p1.11.m11.1c">\operatorname{\mathbf{L}}_{j}\in\operatorname{\mathbb{R}}^{Q_{j}\times S_{j}}</annotation><annotation encoding="application/x-llamapun" id="S5.p1.11.m11.1d">bold_L start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ∈ blackboard_R start_POSTSUPERSCRIPT italic_Q start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT × italic_S start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT end_POSTSUPERSCRIPT</annotation></semantics></math> such that <math alttext="\operatorname{{\mathbf{\Sigma}}}_{j}\approx\operatorname{\mathbf{L}}_{j}% \operatorname{\mathbf{L}}_{j}^{\top}" class="ltx_Math" display="inline" id="S5.p1.12.m12.1"><semantics id="S5.p1.12.m12.1a"><mrow id="S5.p1.12.m12.1.1" xref="S5.p1.12.m12.1.1.cmml"><msub id="S5.p1.12.m12.1.1.2" xref="S5.p1.12.m12.1.1.2.cmml"><mi id="S5.p1.12.m12.1.1.2.2" xref="S5.p1.12.m12.1.1.2.2.cmml">𝚺</mi><mi id="S5.p1.12.m12.1.1.2.3" xref="S5.p1.12.m12.1.1.2.3.cmml">j</mi></msub><mo id="S5.p1.12.m12.1.1.1" xref="S5.p1.12.m12.1.1.1.cmml">≈</mo><mrow id="S5.p1.12.m12.1.1.3" xref="S5.p1.12.m12.1.1.3.cmml"><msub id="S5.p1.12.m12.1.1.3.1" xref="S5.p1.12.m12.1.1.3.1.cmml"><mi id="S5.p1.12.m12.1.1.3.1.2" xref="S5.p1.12.m12.1.1.3.1.2.cmml">𝐋</mi><mi id="S5.p1.12.m12.1.1.3.1.3" xref="S5.p1.12.m12.1.1.3.1.3.cmml">j</mi></msub><mo id="S5.p1.12.m12.1.1.3a" lspace="0.167em" xref="S5.p1.12.m12.1.1.3.cmml">⁡</mo><msubsup id="S5.p1.12.m12.1.1.3.2" xref="S5.p1.12.m12.1.1.3.2.cmml"><mi id="S5.p1.12.m12.1.1.3.2.2.2" xref="S5.p1.12.m12.1.1.3.2.2.2.cmml">𝐋</mi><mi id="S5.p1.12.m12.1.1.3.2.2.3" xref="S5.p1.12.m12.1.1.3.2.2.3.cmml">j</mi><mo id="S5.p1.12.m12.1.1.3.2.3" xref="S5.p1.12.m12.1.1.3.2.3.cmml">⊤</mo></msubsup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.p1.12.m12.1b"><apply id="S5.p1.12.m12.1.1.cmml" xref="S5.p1.12.m12.1.1"><approx id="S5.p1.12.m12.1.1.1.cmml" xref="S5.p1.12.m12.1.1.1"></approx><apply id="S5.p1.12.m12.1.1.2.cmml" xref="S5.p1.12.m12.1.1.2"><csymbol cd="ambiguous" id="S5.p1.12.m12.1.1.2.1.cmml" xref="S5.p1.12.m12.1.1.2">subscript</csymbol><ci id="S5.p1.12.m12.1.1.2.2.cmml" xref="S5.p1.12.m12.1.1.2.2">𝚺</ci><ci id="S5.p1.12.m12.1.1.2.3.cmml" xref="S5.p1.12.m12.1.1.2.3">𝑗</ci></apply><apply id="S5.p1.12.m12.1.1.3.cmml" xref="S5.p1.12.m12.1.1.3"><apply id="S5.p1.12.m12.1.1.3.1.cmml" xref="S5.p1.12.m12.1.1.3.1"><csymbol cd="ambiguous" id="S5.p1.12.m12.1.1.3.1.1.cmml" xref="S5.p1.12.m12.1.1.3.1">subscript</csymbol><ci id="S5.p1.12.m12.1.1.3.1.2.cmml" xref="S5.p1.12.m12.1.1.3.1.2">𝐋</ci><ci id="S5.p1.12.m12.1.1.3.1.3.cmml" xref="S5.p1.12.m12.1.1.3.1.3">𝑗</ci></apply><apply id="S5.p1.12.m12.1.1.3.2.cmml" xref="S5.p1.12.m12.1.1.3.2"><csymbol cd="ambiguous" id="S5.p1.12.m12.1.1.3.2.1.cmml" xref="S5.p1.12.m12.1.1.3.2">superscript</csymbol><apply id="S5.p1.12.m12.1.1.3.2.2.cmml" xref="S5.p1.12.m12.1.1.3.2"><csymbol cd="ambiguous" id="S5.p1.12.m12.1.1.3.2.2.1.cmml" xref="S5.p1.12.m12.1.1.3.2">subscript</csymbol><ci id="S5.p1.12.m12.1.1.3.2.2.2.cmml" xref="S5.p1.12.m12.1.1.3.2.2.2">𝐋</ci><ci id="S5.p1.12.m12.1.1.3.2.2.3.cmml" xref="S5.p1.12.m12.1.1.3.2.2.3">𝑗</ci></apply><csymbol cd="latexml" id="S5.p1.12.m12.1.1.3.2.3.cmml" xref="S5.p1.12.m12.1.1.3.2.3">top</csymbol></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.p1.12.m12.1c">\operatorname{{\mathbf{\Sigma}}}_{j}\approx\operatorname{\mathbf{L}}_{j}% \operatorname{\mathbf{L}}_{j}^{\top}</annotation><annotation encoding="application/x-llamapun" id="S5.p1.12.m12.1d">bold_Σ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ≈ bold_L start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT bold_L start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ⊤ end_POSTSUPERSCRIPT</annotation></semantics></math>, <math alttext="\operatorname{\mathbf{C}}_{i}" class="ltx_Math" display="inline" id="S5.p1.13.m13.1"><semantics id="S5.p1.13.m13.1a"><msub id="S5.p1.13.m13.1.1" xref="S5.p1.13.m13.1.1.cmml"><mi id="S5.p1.13.m13.1.1.2" xref="S5.p1.13.m13.1.1.2.cmml">𝐂</mi><mi id="S5.p1.13.m13.1.1.3" xref="S5.p1.13.m13.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S5.p1.13.m13.1b"><apply id="S5.p1.13.m13.1.1.cmml" xref="S5.p1.13.m13.1.1"><csymbol cd="ambiguous" id="S5.p1.13.m13.1.1.1.cmml" xref="S5.p1.13.m13.1.1">subscript</csymbol><ci id="S5.p1.13.m13.1.1.2.cmml" xref="S5.p1.13.m13.1.1.2">𝐂</ci><ci id="S5.p1.13.m13.1.1.3.cmml" xref="S5.p1.13.m13.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.p1.13.m13.1c">\operatorname{\mathbf{C}}_{i}</annotation><annotation encoding="application/x-llamapun" id="S5.p1.13.m13.1d">bold_C start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> in (<a class="ltx_ref" href="https://arxiv.org/html/2503.13782v1#S2.E3" title="In 2.1 Model ‣ 2 Mixed Model Trace Regression ‣ Regularized Parameter Estimation in Mixed Model Trace Regression"><span class="ltx_text ltx_ref_tag">3</span></a>) is an <math alttext="S_{1}\times\cdots\times S_{d}" class="ltx_Math" display="inline" id="S5.p1.14.m14.1"><semantics id="S5.p1.14.m14.1a"><mrow id="S5.p1.14.m14.1.1" xref="S5.p1.14.m14.1.1.cmml"><msub id="S5.p1.14.m14.1.1.2" xref="S5.p1.14.m14.1.1.2.cmml"><mi id="S5.p1.14.m14.1.1.2.2" xref="S5.p1.14.m14.1.1.2.2.cmml">S</mi><mn id="S5.p1.14.m14.1.1.2.3" xref="S5.p1.14.m14.1.1.2.3.cmml">1</mn></msub><mo id="S5.p1.14.m14.1.1.1" lspace="0.222em" rspace="0.222em" xref="S5.p1.14.m14.1.1.1.cmml">×</mo><mi id="S5.p1.14.m14.1.1.3" mathvariant="normal" xref="S5.p1.14.m14.1.1.3.cmml">⋯</mi><mo id="S5.p1.14.m14.1.1.1a" lspace="0.222em" rspace="0.222em" xref="S5.p1.14.m14.1.1.1.cmml">×</mo><msub id="S5.p1.14.m14.1.1.4" xref="S5.p1.14.m14.1.1.4.cmml"><mi id="S5.p1.14.m14.1.1.4.2" xref="S5.p1.14.m14.1.1.4.2.cmml">S</mi><mi id="S5.p1.14.m14.1.1.4.3" xref="S5.p1.14.m14.1.1.4.3.cmml">d</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S5.p1.14.m14.1b"><apply id="S5.p1.14.m14.1.1.cmml" xref="S5.p1.14.m14.1.1"><times id="S5.p1.14.m14.1.1.1.cmml" xref="S5.p1.14.m14.1.1.1"></times><apply id="S5.p1.14.m14.1.1.2.cmml" xref="S5.p1.14.m14.1.1.2"><csymbol cd="ambiguous" id="S5.p1.14.m14.1.1.2.1.cmml" xref="S5.p1.14.m14.1.1.2">subscript</csymbol><ci id="S5.p1.14.m14.1.1.2.2.cmml" xref="S5.p1.14.m14.1.1.2.2">𝑆</ci><cn id="S5.p1.14.m14.1.1.2.3.cmml" type="integer" xref="S5.p1.14.m14.1.1.2.3">1</cn></apply><ci id="S5.p1.14.m14.1.1.3.cmml" xref="S5.p1.14.m14.1.1.3">⋯</ci><apply id="S5.p1.14.m14.1.1.4.cmml" xref="S5.p1.14.m14.1.1.4"><csymbol cd="ambiguous" id="S5.p1.14.m14.1.1.4.1.cmml" xref="S5.p1.14.m14.1.1.4">subscript</csymbol><ci id="S5.p1.14.m14.1.1.4.2.cmml" xref="S5.p1.14.m14.1.1.4.2">𝑆</ci><ci id="S5.p1.14.m14.1.1.4.3.cmml" xref="S5.p1.14.m14.1.1.4.3">𝑑</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.p1.14.m14.1c">S_{1}\times\cdots\times S_{d}</annotation><annotation encoding="application/x-llamapun" id="S5.p1.14.m14.1d">italic_S start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT × ⋯ × italic_S start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT</annotation></semantics></math> array, and <math alttext="\operatorname{\mathbf{L}}_{j}" class="ltx_Math" display="inline" id="S5.p1.15.m15.1"><semantics id="S5.p1.15.m15.1a"><msub id="S5.p1.15.m15.1.1" xref="S5.p1.15.m15.1.1.cmml"><mi id="S5.p1.15.m15.1.1.2" xref="S5.p1.15.m15.1.1.2.cmml">𝐋</mi><mi id="S5.p1.15.m15.1.1.3" xref="S5.p1.15.m15.1.1.3.cmml">j</mi></msub><annotation-xml encoding="MathML-Content" id="S5.p1.15.m15.1b"><apply id="S5.p1.15.m15.1.1.cmml" xref="S5.p1.15.m15.1.1"><csymbol cd="ambiguous" id="S5.p1.15.m15.1.1.1.cmml" xref="S5.p1.15.m15.1.1">subscript</csymbol><ci id="S5.p1.15.m15.1.1.2.cmml" xref="S5.p1.15.m15.1.1.2">𝐋</ci><ci id="S5.p1.15.m15.1.1.3.cmml" xref="S5.p1.15.m15.1.1.3">𝑗</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.p1.15.m15.1c">\operatorname{\mathbf{L}}_{j}</annotation><annotation encoding="application/x-llamapun" id="S5.p1.15.m15.1d">bold_L start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT</annotation></semantics></math> is regularized by imposing group lasso penalty on its columns to enable sparse estimation. Similarly, Propositions <a class="ltx_ref" href="https://arxiv.org/html/2503.13782v1#S1.Thmproposition1" title="Proposition 1.1 ‣ 1.1 Proof of Proposition 2.1 ‣ 1 Proof of the Propositions ‣ Regularized Parameter Estimation in Mixed Model Trace Regression"><span class="ltx_text ltx_ref_tag">1.1</span></a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.13782v1#S1.Thmproposition2" title="Proposition 1.2 ‣ 1.2 Proof of Proposition 2.2 ‣ 1 Proof of the Propositions ‣ Regularized Parameter Estimation in Mixed Model Trace Regression"><span class="ltx_text ltx_ref_tag">1.2</span></a>, and <a class="ltx_ref" href="https://arxiv.org/html/2503.13782v1#S1.Thmproposition3" title="Proposition 1.3 ‣ 1.3 Proof of Proposition 3.1 ‣ 1 Proof of the Propositions ‣ Regularized Parameter Estimation in Mixed Model Trace Regression"><span class="ltx_text ltx_ref_tag">1.3</span></a> naturally generalize to the array-variate MMTR extension. Finally, the array-variate extension of Algorithm <a class="ltx_ref" href="https://arxiv.org/html/2503.13782v1#alg1" title="In 3 Regularized Parameter Estimation ‣ Regularized Parameter Estimation in Mixed Model Trace Regression"><span class="ltx_text ltx_ref_tag">1</span></a> has <math alttext="d+1" class="ltx_Math" display="inline" id="S5.p1.16.m16.1"><semantics id="S5.p1.16.m16.1a"><mrow id="S5.p1.16.m16.1.1" xref="S5.p1.16.m16.1.1.cmml"><mi id="S5.p1.16.m16.1.1.2" xref="S5.p1.16.m16.1.1.2.cmml">d</mi><mo id="S5.p1.16.m16.1.1.1" xref="S5.p1.16.m16.1.1.1.cmml">+</mo><mn id="S5.p1.16.m16.1.1.3" xref="S5.p1.16.m16.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S5.p1.16.m16.1b"><apply id="S5.p1.16.m16.1.1.cmml" xref="S5.p1.16.m16.1.1"><plus id="S5.p1.16.m16.1.1.1.cmml" xref="S5.p1.16.m16.1.1.1"></plus><ci id="S5.p1.16.m16.1.1.2.cmml" xref="S5.p1.16.m16.1.1.2">𝑑</ci><cn id="S5.p1.16.m16.1.1.3.cmml" type="integer" xref="S5.p1.16.m16.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.p1.16.m16.1c">d+1</annotation><annotation encoding="application/x-llamapun" id="S5.p1.16.m16.1d">italic_d + 1</annotation></semantics></math> cycles, where the first cycle estimates <math alttext="\operatorname{\mathbf{B}}" class="ltx_Math" display="inline" id="S5.p1.17.m17.1"><semantics id="S5.p1.17.m17.1a"><mi id="S5.p1.17.m17.1.1" xref="S5.p1.17.m17.1.1.cmml">𝐁</mi><annotation-xml encoding="MathML-Content" id="S5.p1.17.m17.1b"><ci id="S5.p1.17.m17.1.1.cmml" xref="S5.p1.17.m17.1.1">𝐁</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.p1.17.m17.1c">\operatorname{\mathbf{B}}</annotation><annotation encoding="application/x-llamapun" id="S5.p1.17.m17.1d">bold_B</annotation></semantics></math> and <math alttext="\tau^{2}" class="ltx_Math" display="inline" id="S5.p1.18.m18.1"><semantics id="S5.p1.18.m18.1a"><msup id="S5.p1.18.m18.1.1" xref="S5.p1.18.m18.1.1.cmml"><mi id="S5.p1.18.m18.1.1.2" xref="S5.p1.18.m18.1.1.2.cmml">τ</mi><mn id="S5.p1.18.m18.1.1.3" xref="S5.p1.18.m18.1.1.3.cmml">2</mn></msup><annotation-xml encoding="MathML-Content" id="S5.p1.18.m18.1b"><apply id="S5.p1.18.m18.1.1.cmml" xref="S5.p1.18.m18.1.1"><csymbol cd="ambiguous" id="S5.p1.18.m18.1.1.1.cmml" xref="S5.p1.18.m18.1.1">superscript</csymbol><ci id="S5.p1.18.m18.1.1.2.cmml" xref="S5.p1.18.m18.1.1.2">𝜏</ci><cn id="S5.p1.18.m18.1.1.3.cmml" type="integer" xref="S5.p1.18.m18.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.p1.18.m18.1c">\tau^{2}</annotation><annotation encoding="application/x-llamapun" id="S5.p1.18.m18.1d">italic_τ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT</annotation></semantics></math> using regularized array-variate regression <cite class="ltx_cite ltx_citemacro_citep">(Zhou et al., <a class="ltx_ref" href="https://arxiv.org/html/2503.13782v1#bib.bib42" title="">2013</a>)</cite> and the next <math alttext="d" class="ltx_Math" display="inline" id="S5.p1.19.m19.1"><semantics id="S5.p1.19.m19.1a"><mi id="S5.p1.19.m19.1.1" xref="S5.p1.19.m19.1.1.cmml">d</mi><annotation-xml encoding="MathML-Content" id="S5.p1.19.m19.1b"><ci id="S5.p1.19.m19.1.1.cmml" xref="S5.p1.19.m19.1.1">𝑑</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.p1.19.m19.1c">d</annotation><annotation encoding="application/x-llamapun" id="S5.p1.19.m19.1d">italic_d</annotation></semantics></math> cycles estimate <math alttext="\operatorname{\mathbf{L}}_{j}" class="ltx_Math" display="inline" id="S5.p1.20.m20.1"><semantics id="S5.p1.20.m20.1a"><msub id="S5.p1.20.m20.1.1" xref="S5.p1.20.m20.1.1.cmml"><mi id="S5.p1.20.m20.1.1.2" xref="S5.p1.20.m20.1.1.2.cmml">𝐋</mi><mi id="S5.p1.20.m20.1.1.3" xref="S5.p1.20.m20.1.1.3.cmml">j</mi></msub><annotation-xml encoding="MathML-Content" id="S5.p1.20.m20.1b"><apply id="S5.p1.20.m20.1.1.cmml" xref="S5.p1.20.m20.1.1"><csymbol cd="ambiguous" id="S5.p1.20.m20.1.1.1.cmml" xref="S5.p1.20.m20.1.1">subscript</csymbol><ci id="S5.p1.20.m20.1.1.2.cmml" xref="S5.p1.20.m20.1.1.2">𝐋</ci><ci id="S5.p1.20.m20.1.1.3.cmml" xref="S5.p1.20.m20.1.1.3">𝑗</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.p1.20.m20.1c">\operatorname{\mathbf{L}}_{j}</annotation><annotation encoding="application/x-llamapun" id="S5.p1.20.m20.1d">bold_L start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT</annotation></semantics></math> given the remaining parameters using a version of (<a class="ltx_ref" href="https://arxiv.org/html/2503.13782v1#S3.E10" title="In 3 Regularized Parameter Estimation ‣ Regularized Parameter Estimation in Mixed Model Trace Regression"><span class="ltx_text ltx_ref_tag">10</span></a>) for <math alttext="j=1,\ldots,d" class="ltx_Math" display="inline" id="S5.p1.21.m21.3"><semantics id="S5.p1.21.m21.3a"><mrow id="S5.p1.21.m21.3.4" xref="S5.p1.21.m21.3.4.cmml"><mi id="S5.p1.21.m21.3.4.2" xref="S5.p1.21.m21.3.4.2.cmml">j</mi><mo id="S5.p1.21.m21.3.4.1" xref="S5.p1.21.m21.3.4.1.cmml">=</mo><mrow id="S5.p1.21.m21.3.4.3.2" xref="S5.p1.21.m21.3.4.3.1.cmml"><mn id="S5.p1.21.m21.1.1" xref="S5.p1.21.m21.1.1.cmml">1</mn><mo id="S5.p1.21.m21.3.4.3.2.1" xref="S5.p1.21.m21.3.4.3.1.cmml">,</mo><mi id="S5.p1.21.m21.2.2" mathvariant="normal" xref="S5.p1.21.m21.2.2.cmml">…</mi><mo id="S5.p1.21.m21.3.4.3.2.2" xref="S5.p1.21.m21.3.4.3.1.cmml">,</mo><mi id="S5.p1.21.m21.3.3" xref="S5.p1.21.m21.3.3.cmml">d</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.p1.21.m21.3b"><apply id="S5.p1.21.m21.3.4.cmml" xref="S5.p1.21.m21.3.4"><eq id="S5.p1.21.m21.3.4.1.cmml" xref="S5.p1.21.m21.3.4.1"></eq><ci id="S5.p1.21.m21.3.4.2.cmml" xref="S5.p1.21.m21.3.4.2">𝑗</ci><list id="S5.p1.21.m21.3.4.3.1.cmml" xref="S5.p1.21.m21.3.4.3.2"><cn id="S5.p1.21.m21.1.1.cmml" type="integer" xref="S5.p1.21.m21.1.1">1</cn><ci id="S5.p1.21.m21.2.2.cmml" xref="S5.p1.21.m21.2.2">…</ci><ci id="S5.p1.21.m21.3.3.cmml" xref="S5.p1.21.m21.3.3">𝑑</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.p1.21.m21.3c">j=1,\ldots,d</annotation><annotation encoding="application/x-llamapun" id="S5.p1.21.m21.3d">italic_j = 1 , … , italic_d</annotation></semantics></math>. We are developing an MMTR extension for correlated array-variate responses.</p> </div> <div class="ltx_para" id="S5.p2"> <p class="ltx_p" id="S5.p2.6">The objectives for estimating <math alttext="\operatorname{\mathbf{L}}_{1}" class="ltx_Math" display="inline" id="S5.p2.1.m1.1"><semantics id="S5.p2.1.m1.1a"><msub id="S5.p2.1.m1.1.1" xref="S5.p2.1.m1.1.1.cmml"><mi id="S5.p2.1.m1.1.1.2" xref="S5.p2.1.m1.1.1.2.cmml">𝐋</mi><mn id="S5.p2.1.m1.1.1.3" xref="S5.p2.1.m1.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S5.p2.1.m1.1b"><apply id="S5.p2.1.m1.1.1.cmml" xref="S5.p2.1.m1.1.1"><csymbol cd="ambiguous" id="S5.p2.1.m1.1.1.1.cmml" xref="S5.p2.1.m1.1.1">subscript</csymbol><ci id="S5.p2.1.m1.1.1.2.cmml" xref="S5.p2.1.m1.1.1.2">𝐋</ci><cn id="S5.p2.1.m1.1.1.3.cmml" type="integer" xref="S5.p2.1.m1.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.p2.1.m1.1c">\operatorname{\mathbf{L}}_{1}</annotation><annotation encoding="application/x-llamapun" id="S5.p2.1.m1.1d">bold_L start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="\operatorname{\mathbf{L}}_{2}" class="ltx_Math" display="inline" id="S5.p2.2.m2.1"><semantics id="S5.p2.2.m2.1a"><msub id="S5.p2.2.m2.1.1" xref="S5.p2.2.m2.1.1.cmml"><mi id="S5.p2.2.m2.1.1.2" xref="S5.p2.2.m2.1.1.2.cmml">𝐋</mi><mn id="S5.p2.2.m2.1.1.3" xref="S5.p2.2.m2.1.1.3.cmml">2</mn></msub><annotation-xml encoding="MathML-Content" id="S5.p2.2.m2.1b"><apply id="S5.p2.2.m2.1.1.cmml" xref="S5.p2.2.m2.1.1"><csymbol cd="ambiguous" id="S5.p2.2.m2.1.1.1.cmml" xref="S5.p2.2.m2.1.1">subscript</csymbol><ci id="S5.p2.2.m2.1.1.2.cmml" xref="S5.p2.2.m2.1.1.2">𝐋</ci><cn id="S5.p2.2.m2.1.1.3.cmml" type="integer" xref="S5.p2.2.m2.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.p2.2.m2.1c">\operatorname{\mathbf{L}}_{2}</annotation><annotation encoding="application/x-llamapun" id="S5.p2.2.m2.1d">bold_L start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math> in Algorithm <a class="ltx_ref" href="https://arxiv.org/html/2503.13782v1#alg1" title="In 3 Regularized Parameter Estimation ‣ Regularized Parameter Estimation in Mixed Model Trace Regression"><span class="ltx_text ltx_ref_tag">1</span></a> rely on the assumption that the random effects and error terms are Gaussian, but our parameter estimation algorithm generalizes to the case where we only assume the first and second moments of the random effects exist. In this case, Algorithm <a class="ltx_ref" href="https://arxiv.org/html/2503.13782v1#alg1" title="In 3 Regularized Parameter Estimation ‣ Regularized Parameter Estimation in Mixed Model Trace Regression"><span class="ltx_text ltx_ref_tag">1</span></a> iteratively replaces the first two moments of the random effects, <math alttext="\operatorname{\mathbb{E}}(\operatorname{\mathbf{A}}_{i}\mid\operatorname{% \mathbf{y}}_{i},\operatorname{{\bm{\theta}}}^{(t)})" class="ltx_Math" display="inline" id="S5.p2.3.m3.3"><semantics id="S5.p2.3.m3.3a"><mrow id="S5.p2.3.m3.3.3.1" xref="S5.p2.3.m3.3.3.2.cmml"><mi id="S5.p2.3.m3.2.2" xref="S5.p2.3.m3.2.2.cmml">𝔼</mi><mo id="S5.p2.3.m3.3.3.1a" xref="S5.p2.3.m3.3.3.2.cmml">⁡</mo><mrow id="S5.p2.3.m3.3.3.1.1" xref="S5.p2.3.m3.3.3.2.cmml"><mo id="S5.p2.3.m3.3.3.1.1.2" stretchy="false" xref="S5.p2.3.m3.3.3.2.cmml">(</mo><mrow id="S5.p2.3.m3.3.3.1.1.1" xref="S5.p2.3.m3.3.3.1.1.1.cmml"><msub id="S5.p2.3.m3.3.3.1.1.1.4" xref="S5.p2.3.m3.3.3.1.1.1.4.cmml"><mi 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id="S5.p2.6.m6.1a"><msub id="S5.p2.6.m6.1.1" xref="S5.p2.6.m6.1.1.cmml"><mi id="S5.p2.6.m6.1.1.2" xref="S5.p2.6.m6.1.1.2.cmml">𝐀</mi><mi id="S5.p2.6.m6.1.1.3" xref="S5.p2.6.m6.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S5.p2.6.m6.1b"><apply id="S5.p2.6.m6.1.1.cmml" xref="S5.p2.6.m6.1.1"><csymbol cd="ambiguous" id="S5.p2.6.m6.1.1.1.cmml" xref="S5.p2.6.m6.1.1">subscript</csymbol><ci id="S5.p2.6.m6.1.1.2.cmml" xref="S5.p2.6.m6.1.1.2">𝐀</ci><ci id="S5.p2.6.m6.1.1.3.cmml" xref="S5.p2.6.m6.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.p2.6.m6.1c">\operatorname{\mathbf{A}}_{i}</annotation><annotation encoding="application/x-llamapun" id="S5.p2.6.m6.1d">bold_A start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> follows a matrix normal distribution, resulting in the objectives (<a class="ltx_ref" href="https://arxiv.org/html/2503.13782v1#S3.E10" title="In 3 Regularized Parameter Estimation ‣ Regularized Parameter Estimation in Mixed Model Trace Regression"><span class="ltx_text ltx_ref_tag">10</span></a>) and (<a class="ltx_ref" href="https://arxiv.org/html/2503.13782v1#S3.E11" title="In 3 Regularized Parameter Estimation ‣ Regularized Parameter Estimation in Mixed Model Trace Regression"><span class="ltx_text ltx_ref_tag">11</span></a>).</p> </div> </section> <section class="ltx_section" id="Sx1"> <h2 class="ltx_title ltx_title_section">Acknowledgments</h2> <div class="ltx_para" id="Sx1.p1"> <p class="ltx_p" id="Sx1.p1.1">Ian Hultman and Sanvesh Srivastava were partially supported by grants from the National Institutes of Health (1DP2MH126377-01) and the National Science Foundation (DMS-1854667). The authors thank Kshitij Khare, Joe Lang, Boxiang Wang, and Dale Zimmerman for their valuable feedback on an earlier version of this manuscript. The code used in the experiments is available at <a class="ltx_ref ltx_url ltx_font_typewriter" href="https://github.com/IHultman/MMTR" title="">https://github.com/IHultman/MMTR</a>.</p> </div> </section> <section class="ltx_bibliography" id="bib"> <h2 class="ltx_title ltx_title_bibliography">References</h2> <ul class="ltx_biblist"> <li class="ltx_bibitem" id="bib.bib1"> <span class="ltx_tag ltx_role_refnum ltx_tag_bibitem">Barthelme (2024)</span> <span class="ltx_bibblock"> Barthelme, S. (2024). </span> <span class="ltx_bibblock"><span class="ltx_text ltx_font_italic" id="bib.bib1.1.1">imager: Image Processing Library Based on ’CImg’</span>. </span> <span class="ltx_bibblock">R package version 1.0.2. </span> </li> <li class="ltx_bibitem" id="bib.bib2"> <span class="ltx_tag ltx_role_refnum ltx_tag_bibitem">Bates et al. 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Zhu (2013). </span> <span class="ltx_bibblock">Tensor regression with applications in neuroimaging data analysis. </span> <span class="ltx_bibblock"><span class="ltx_text ltx_font_italic" id="bib.bib42.1.1">Journal of the American Statistical Association</span> <span class="ltx_text ltx_font_italic" id="bib.bib42.2.2">108</span>(502), 540–552. </span> </li> </ul> </section> <div class="ltx_pagination ltx_role_newpage"></div> <div class="ltx_para" id="p3"> <p class="ltx_p ltx_align_center" id="p3.1"><span class="ltx_text ltx_font_bold" id="p3.1.1" style="font-size:173%;">Supplementary Material: Regularized Parameter Estimation in Mixed Model Trace Regression</span></p> </div> <section class="ltx_section" id="S1a"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">1 </span>Proof of the Propositions</h2> <section class="ltx_subsection" id="S1.SS1a"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">1.1 </span>Proof of Proposition 2.1</h3> <div class="ltx_para" id="S1.SS1a.p1"> <p class="ltx_p" id="S1.SS1a.p1.1">Recall that Proposition 2.1 from the main manuscript.</p> </div> <div class="ltx_theorem ltx_theorem_proposition" id="S1.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="S1.Thmproposition1.1.1.1">Proposition 1.1</span></span></h6> <div class="ltx_para" id="S1.Thmproposition1.p1"> <p class="ltx_p" id="S1.Thmproposition1.p1.7"><span class="ltx_text ltx_font_italic" id="S1.Thmproposition1.p1.7.7">Assume that <math alttext="(\operatorname{\mathbf{L}}_{2})_{j1}=1" class="ltx_Math" display="inline" id="S1.Thmproposition1.p1.1.1.m1.1"><semantics id="S1.Thmproposition1.p1.1.1.m1.1a"><mrow id="S1.Thmproposition1.p1.1.1.m1.1.1" xref="S1.Thmproposition1.p1.1.1.m1.1.1.cmml"><msub id="S1.Thmproposition1.p1.1.1.m1.1.1.1" xref="S1.Thmproposition1.p1.1.1.m1.1.1.1.cmml"><mrow 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xref="S1.Thmproposition1.p1.2.2.m2.2.2.8">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmproposition1.p1.2.2.m2.2c">(\operatorname{\mathbf{L}}_{2})_{j2}=\cdots=(\operatorname{\mathbf{L}}_{2})_{% jS_{2}}=0</annotation><annotation encoding="application/x-llamapun" id="S1.Thmproposition1.p1.2.2.m2.2d">( bold_L start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) start_POSTSUBSCRIPT italic_j 2 end_POSTSUBSCRIPT = ⋯ = ( bold_L start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) start_POSTSUBSCRIPT italic_j italic_S start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT = 0</annotation></semantics></math>. Then, <math alttext="(\operatorname{{\mathbf{\Sigma}}}_{2})_{jj}=1" class="ltx_Math" display="inline" id="S1.Thmproposition1.p1.3.3.m3.1"><semantics id="S1.Thmproposition1.p1.3.3.m3.1a"><mrow id="S1.Thmproposition1.p1.3.3.m3.1.1" xref="S1.Thmproposition1.p1.3.3.m3.1.1.cmml"><msub id="S1.Thmproposition1.p1.3.3.m3.1.1.1" xref="S1.Thmproposition1.p1.3.3.m3.1.1.1.cmml"><mrow id="S1.Thmproposition1.p1.3.3.m3.1.1.1.1.1" xref="S1.Thmproposition1.p1.3.3.m3.1.1.1.1.1.1.cmml"><mo id="S1.Thmproposition1.p1.3.3.m3.1.1.1.1.1.2" stretchy="false" xref="S1.Thmproposition1.p1.3.3.m3.1.1.1.1.1.1.cmml">(</mo><msub id="S1.Thmproposition1.p1.3.3.m3.1.1.1.1.1.1" xref="S1.Thmproposition1.p1.3.3.m3.1.1.1.1.1.1.cmml"><mi id="S1.Thmproposition1.p1.3.3.m3.1.1.1.1.1.1.2" xref="S1.Thmproposition1.p1.3.3.m3.1.1.1.1.1.1.2.cmml">𝚺</mi><mn id="S1.Thmproposition1.p1.3.3.m3.1.1.1.1.1.1.3" xref="S1.Thmproposition1.p1.3.3.m3.1.1.1.1.1.1.3.cmml">2</mn></msub><mo id="S1.Thmproposition1.p1.3.3.m3.1.1.1.1.1.3" stretchy="false" xref="S1.Thmproposition1.p1.3.3.m3.1.1.1.1.1.1.cmml">)</mo></mrow><mrow id="S1.Thmproposition1.p1.3.3.m3.1.1.1.3" xref="S1.Thmproposition1.p1.3.3.m3.1.1.1.3.cmml"><mi id="S1.Thmproposition1.p1.3.3.m3.1.1.1.3.2" xref="S1.Thmproposition1.p1.3.3.m3.1.1.1.3.2.cmml">j</mi><mo id="S1.Thmproposition1.p1.3.3.m3.1.1.1.3.1" xref="S1.Thmproposition1.p1.3.3.m3.1.1.1.3.1.cmml">⁢</mo><mi id="S1.Thmproposition1.p1.3.3.m3.1.1.1.3.3" xref="S1.Thmproposition1.p1.3.3.m3.1.1.1.3.3.cmml">j</mi></mrow></msub><mo id="S1.Thmproposition1.p1.3.3.m3.1.1.2" xref="S1.Thmproposition1.p1.3.3.m3.1.1.2.cmml">=</mo><mn id="S1.Thmproposition1.p1.3.3.m3.1.1.3" xref="S1.Thmproposition1.p1.3.3.m3.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S1.Thmproposition1.p1.3.3.m3.1b"><apply id="S1.Thmproposition1.p1.3.3.m3.1.1.cmml" xref="S1.Thmproposition1.p1.3.3.m3.1.1"><eq id="S1.Thmproposition1.p1.3.3.m3.1.1.2.cmml" xref="S1.Thmproposition1.p1.3.3.m3.1.1.2"></eq><apply id="S1.Thmproposition1.p1.3.3.m3.1.1.1.cmml" xref="S1.Thmproposition1.p1.3.3.m3.1.1.1"><csymbol cd="ambiguous" id="S1.Thmproposition1.p1.3.3.m3.1.1.1.2.cmml" xref="S1.Thmproposition1.p1.3.3.m3.1.1.1">subscript</csymbol><apply id="S1.Thmproposition1.p1.3.3.m3.1.1.1.1.1.1.cmml" xref="S1.Thmproposition1.p1.3.3.m3.1.1.1.1.1"><csymbol cd="ambiguous" id="S1.Thmproposition1.p1.3.3.m3.1.1.1.1.1.1.1.cmml" xref="S1.Thmproposition1.p1.3.3.m3.1.1.1.1.1">subscript</csymbol><ci id="S1.Thmproposition1.p1.3.3.m3.1.1.1.1.1.1.2.cmml" xref="S1.Thmproposition1.p1.3.3.m3.1.1.1.1.1.1.2">𝚺</ci><cn id="S1.Thmproposition1.p1.3.3.m3.1.1.1.1.1.1.3.cmml" type="integer" xref="S1.Thmproposition1.p1.3.3.m3.1.1.1.1.1.1.3">2</cn></apply><apply id="S1.Thmproposition1.p1.3.3.m3.1.1.1.3.cmml" xref="S1.Thmproposition1.p1.3.3.m3.1.1.1.3"><times id="S1.Thmproposition1.p1.3.3.m3.1.1.1.3.1.cmml" xref="S1.Thmproposition1.p1.3.3.m3.1.1.1.3.1"></times><ci id="S1.Thmproposition1.p1.3.3.m3.1.1.1.3.2.cmml" xref="S1.Thmproposition1.p1.3.3.m3.1.1.1.3.2">𝑗</ci><ci id="S1.Thmproposition1.p1.3.3.m3.1.1.1.3.3.cmml" xref="S1.Thmproposition1.p1.3.3.m3.1.1.1.3.3">𝑗</ci></apply></apply><cn id="S1.Thmproposition1.p1.3.3.m3.1.1.3.cmml" type="integer" xref="S1.Thmproposition1.p1.3.3.m3.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmproposition1.p1.3.3.m3.1c">(\operatorname{{\mathbf{\Sigma}}}_{2})_{jj}=1</annotation><annotation encoding="application/x-llamapun" id="S1.Thmproposition1.p1.3.3.m3.1d">( bold_Σ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) start_POSTSUBSCRIPT italic_j italic_j end_POSTSUBSCRIPT = 1</annotation></semantics></math> and the <math alttext="j" class="ltx_Math" display="inline" id="S1.Thmproposition1.p1.4.4.m4.1"><semantics id="S1.Thmproposition1.p1.4.4.m4.1a"><mi id="S1.Thmproposition1.p1.4.4.m4.1.1" xref="S1.Thmproposition1.p1.4.4.m4.1.1.cmml">j</mi><annotation-xml encoding="MathML-Content" id="S1.Thmproposition1.p1.4.4.m4.1b"><ci id="S1.Thmproposition1.p1.4.4.m4.1.1.cmml" xref="S1.Thmproposition1.p1.4.4.m4.1.1">𝑗</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmproposition1.p1.4.4.m4.1c">j</annotation><annotation encoding="application/x-llamapun" id="S1.Thmproposition1.p1.4.4.m4.1d">italic_j</annotation></semantics></math>th diagonal <math alttext="Q_{1}\times Q_{1}" class="ltx_Math" display="inline" id="S1.Thmproposition1.p1.5.5.m5.1"><semantics id="S1.Thmproposition1.p1.5.5.m5.1a"><mrow id="S1.Thmproposition1.p1.5.5.m5.1.1" xref="S1.Thmproposition1.p1.5.5.m5.1.1.cmml"><msub id="S1.Thmproposition1.p1.5.5.m5.1.1.2" xref="S1.Thmproposition1.p1.5.5.m5.1.1.2.cmml"><mi id="S1.Thmproposition1.p1.5.5.m5.1.1.2.2" xref="S1.Thmproposition1.p1.5.5.m5.1.1.2.2.cmml">Q</mi><mn id="S1.Thmproposition1.p1.5.5.m5.1.1.2.3" xref="S1.Thmproposition1.p1.5.5.m5.1.1.2.3.cmml">1</mn></msub><mo id="S1.Thmproposition1.p1.5.5.m5.1.1.1" lspace="0.222em" rspace="0.222em" xref="S1.Thmproposition1.p1.5.5.m5.1.1.1.cmml">×</mo><msub id="S1.Thmproposition1.p1.5.5.m5.1.1.3" xref="S1.Thmproposition1.p1.5.5.m5.1.1.3.cmml"><mi id="S1.Thmproposition1.p1.5.5.m5.1.1.3.2" xref="S1.Thmproposition1.p1.5.5.m5.1.1.3.2.cmml">Q</mi><mn id="S1.Thmproposition1.p1.5.5.m5.1.1.3.3" xref="S1.Thmproposition1.p1.5.5.m5.1.1.3.3.cmml">1</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S1.Thmproposition1.p1.5.5.m5.1b"><apply id="S1.Thmproposition1.p1.5.5.m5.1.1.cmml" xref="S1.Thmproposition1.p1.5.5.m5.1.1"><times id="S1.Thmproposition1.p1.5.5.m5.1.1.1.cmml" xref="S1.Thmproposition1.p1.5.5.m5.1.1.1"></times><apply id="S1.Thmproposition1.p1.5.5.m5.1.1.2.cmml" xref="S1.Thmproposition1.p1.5.5.m5.1.1.2"><csymbol cd="ambiguous" id="S1.Thmproposition1.p1.5.5.m5.1.1.2.1.cmml" xref="S1.Thmproposition1.p1.5.5.m5.1.1.2">subscript</csymbol><ci id="S1.Thmproposition1.p1.5.5.m5.1.1.2.2.cmml" xref="S1.Thmproposition1.p1.5.5.m5.1.1.2.2">𝑄</ci><cn id="S1.Thmproposition1.p1.5.5.m5.1.1.2.3.cmml" type="integer" xref="S1.Thmproposition1.p1.5.5.m5.1.1.2.3">1</cn></apply><apply id="S1.Thmproposition1.p1.5.5.m5.1.1.3.cmml" xref="S1.Thmproposition1.p1.5.5.m5.1.1.3"><csymbol cd="ambiguous" id="S1.Thmproposition1.p1.5.5.m5.1.1.3.1.cmml" xref="S1.Thmproposition1.p1.5.5.m5.1.1.3">subscript</csymbol><ci id="S1.Thmproposition1.p1.5.5.m5.1.1.3.2.cmml" xref="S1.Thmproposition1.p1.5.5.m5.1.1.3.2">𝑄</ci><cn id="S1.Thmproposition1.p1.5.5.m5.1.1.3.3.cmml" type="integer" xref="S1.Thmproposition1.p1.5.5.m5.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmproposition1.p1.5.5.m5.1c">Q_{1}\times Q_{1}</annotation><annotation encoding="application/x-llamapun" id="S1.Thmproposition1.p1.5.5.m5.1d">italic_Q start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT × italic_Q start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> block of <math alttext="\operatorname{{\mathbf{\Sigma}}}_{2}\otimes\operatorname{{\mathbf{\Sigma}}}_{1}" class="ltx_Math" display="inline" id="S1.Thmproposition1.p1.6.6.m6.1"><semantics id="S1.Thmproposition1.p1.6.6.m6.1a"><mrow id="S1.Thmproposition1.p1.6.6.m6.1.1" xref="S1.Thmproposition1.p1.6.6.m6.1.1.cmml"><msub id="S1.Thmproposition1.p1.6.6.m6.1.1.2" xref="S1.Thmproposition1.p1.6.6.m6.1.1.2.cmml"><mi id="S1.Thmproposition1.p1.6.6.m6.1.1.2.2" xref="S1.Thmproposition1.p1.6.6.m6.1.1.2.2.cmml">𝚺</mi><mn id="S1.Thmproposition1.p1.6.6.m6.1.1.2.3" xref="S1.Thmproposition1.p1.6.6.m6.1.1.2.3.cmml">2</mn></msub><mo id="S1.Thmproposition1.p1.6.6.m6.1.1.1" lspace="0em" rspace="0.222em" xref="S1.Thmproposition1.p1.6.6.m6.1.1.1.cmml">⊗</mo><msub id="S1.Thmproposition1.p1.6.6.m6.1.1.3" xref="S1.Thmproposition1.p1.6.6.m6.1.1.3.cmml"><mi id="S1.Thmproposition1.p1.6.6.m6.1.1.3.2" xref="S1.Thmproposition1.p1.6.6.m6.1.1.3.2.cmml">𝚺</mi><mn id="S1.Thmproposition1.p1.6.6.m6.1.1.3.3" xref="S1.Thmproposition1.p1.6.6.m6.1.1.3.3.cmml">1</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S1.Thmproposition1.p1.6.6.m6.1b"><apply id="S1.Thmproposition1.p1.6.6.m6.1.1.cmml" xref="S1.Thmproposition1.p1.6.6.m6.1.1"><csymbol cd="latexml" id="S1.Thmproposition1.p1.6.6.m6.1.1.1.cmml" xref="S1.Thmproposition1.p1.6.6.m6.1.1.1">tensor-product</csymbol><apply id="S1.Thmproposition1.p1.6.6.m6.1.1.2.cmml" xref="S1.Thmproposition1.p1.6.6.m6.1.1.2"><csymbol cd="ambiguous" id="S1.Thmproposition1.p1.6.6.m6.1.1.2.1.cmml" xref="S1.Thmproposition1.p1.6.6.m6.1.1.2">subscript</csymbol><ci id="S1.Thmproposition1.p1.6.6.m6.1.1.2.2.cmml" xref="S1.Thmproposition1.p1.6.6.m6.1.1.2.2">𝚺</ci><cn id="S1.Thmproposition1.p1.6.6.m6.1.1.2.3.cmml" type="integer" xref="S1.Thmproposition1.p1.6.6.m6.1.1.2.3">2</cn></apply><apply id="S1.Thmproposition1.p1.6.6.m6.1.1.3.cmml" xref="S1.Thmproposition1.p1.6.6.m6.1.1.3"><csymbol cd="ambiguous" id="S1.Thmproposition1.p1.6.6.m6.1.1.3.1.cmml" xref="S1.Thmproposition1.p1.6.6.m6.1.1.3">subscript</csymbol><ci id="S1.Thmproposition1.p1.6.6.m6.1.1.3.2.cmml" xref="S1.Thmproposition1.p1.6.6.m6.1.1.3.2">𝚺</ci><cn id="S1.Thmproposition1.p1.6.6.m6.1.1.3.3.cmml" type="integer" xref="S1.Thmproposition1.p1.6.6.m6.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmproposition1.p1.6.6.m6.1c">\operatorname{{\mathbf{\Sigma}}}_{2}\otimes\operatorname{{\mathbf{\Sigma}}}_{1}</annotation><annotation encoding="application/x-llamapun" id="S1.Thmproposition1.p1.6.6.m6.1d">bold_Σ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ⊗ bold_Σ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> is <math alttext="\operatorname{{\mathbf{\Sigma}}}_{1}" class="ltx_Math" display="inline" id="S1.Thmproposition1.p1.7.7.m7.1"><semantics id="S1.Thmproposition1.p1.7.7.m7.1a"><msub id="S1.Thmproposition1.p1.7.7.m7.1.1" xref="S1.Thmproposition1.p1.7.7.m7.1.1.cmml"><mi id="S1.Thmproposition1.p1.7.7.m7.1.1.2" xref="S1.Thmproposition1.p1.7.7.m7.1.1.2.cmml">𝚺</mi><mn id="S1.Thmproposition1.p1.7.7.m7.1.1.3" xref="S1.Thmproposition1.p1.7.7.m7.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S1.Thmproposition1.p1.7.7.m7.1b"><apply id="S1.Thmproposition1.p1.7.7.m7.1.1.cmml" xref="S1.Thmproposition1.p1.7.7.m7.1.1"><csymbol cd="ambiguous" id="S1.Thmproposition1.p1.7.7.m7.1.1.1.cmml" xref="S1.Thmproposition1.p1.7.7.m7.1.1">subscript</csymbol><ci id="S1.Thmproposition1.p1.7.7.m7.1.1.2.cmml" xref="S1.Thmproposition1.p1.7.7.m7.1.1.2">𝚺</ci><cn id="S1.Thmproposition1.p1.7.7.m7.1.1.3.cmml" type="integer" xref="S1.Thmproposition1.p1.7.7.m7.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmproposition1.p1.7.7.m7.1c">\operatorname{{\mathbf{\Sigma}}}_{1}</annotation><annotation encoding="application/x-llamapun" id="S1.Thmproposition1.p1.7.7.m7.1d">bold_Σ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math>.</span></p> </div> </div> <div class="ltx_para ltx_noindent" id="S1.SS1a.p2"> <p class="ltx_p" id="S1.SS1a.p2.10"><span class="ltx_text ltx_font_bold" id="S1.SS1a.p2.10.1">Proof </span>[Proof of Proposition <a class="ltx_ref" href="https://arxiv.org/html/2503.13782v1#S1.Thmproposition1" title="Proposition 1.1 ‣ 1.1 Proof of Proposition 2.1 ‣ 1 Proof of the Propositions ‣ Regularized Parameter Estimation in Mixed Model Trace Regression"><span class="ltx_text ltx_ref_tag">1.1</span></a>] Let <math alttext="\operatorname{\mathbf{L}}_{2}\in\mathbb{R}^{Q_{2}\times S_{2}}" class="ltx_Math" display="inline" id="S1.SS1a.p2.1.m1.1"><semantics id="S1.SS1a.p2.1.m1.1a"><mrow id="S1.SS1a.p2.1.m1.1.1" xref="S1.SS1a.p2.1.m1.1.1.cmml"><msub id="S1.SS1a.p2.1.m1.1.1.2" xref="S1.SS1a.p2.1.m1.1.1.2.cmml"><mi id="S1.SS1a.p2.1.m1.1.1.2.2" xref="S1.SS1a.p2.1.m1.1.1.2.2.cmml">𝐋</mi><mn id="S1.SS1a.p2.1.m1.1.1.2.3" xref="S1.SS1a.p2.1.m1.1.1.2.3.cmml">2</mn></msub><mo id="S1.SS1a.p2.1.m1.1.1.1" xref="S1.SS1a.p2.1.m1.1.1.1.cmml">∈</mo><msup id="S1.SS1a.p2.1.m1.1.1.3" xref="S1.SS1a.p2.1.m1.1.1.3.cmml"><mi id="S1.SS1a.p2.1.m1.1.1.3.2" xref="S1.SS1a.p2.1.m1.1.1.3.2.cmml">ℝ</mi><mrow id="S1.SS1a.p2.1.m1.1.1.3.3" xref="S1.SS1a.p2.1.m1.1.1.3.3.cmml"><msub id="S1.SS1a.p2.1.m1.1.1.3.3.2" xref="S1.SS1a.p2.1.m1.1.1.3.3.2.cmml"><mi id="S1.SS1a.p2.1.m1.1.1.3.3.2.2" xref="S1.SS1a.p2.1.m1.1.1.3.3.2.2.cmml">Q</mi><mn id="S1.SS1a.p2.1.m1.1.1.3.3.2.3" xref="S1.SS1a.p2.1.m1.1.1.3.3.2.3.cmml">2</mn></msub><mo id="S1.SS1a.p2.1.m1.1.1.3.3.1" lspace="0.222em" rspace="0.222em" xref="S1.SS1a.p2.1.m1.1.1.3.3.1.cmml">×</mo><msub id="S1.SS1a.p2.1.m1.1.1.3.3.3" xref="S1.SS1a.p2.1.m1.1.1.3.3.3.cmml"><mi id="S1.SS1a.p2.1.m1.1.1.3.3.3.2" xref="S1.SS1a.p2.1.m1.1.1.3.3.3.2.cmml">S</mi><mn id="S1.SS1a.p2.1.m1.1.1.3.3.3.3" xref="S1.SS1a.p2.1.m1.1.1.3.3.3.3.cmml">2</mn></msub></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="S1.SS1a.p2.1.m1.1b"><apply id="S1.SS1a.p2.1.m1.1.1.cmml" xref="S1.SS1a.p2.1.m1.1.1"><in id="S1.SS1a.p2.1.m1.1.1.1.cmml" xref="S1.SS1a.p2.1.m1.1.1.1"></in><apply id="S1.SS1a.p2.1.m1.1.1.2.cmml" xref="S1.SS1a.p2.1.m1.1.1.2"><csymbol cd="ambiguous" id="S1.SS1a.p2.1.m1.1.1.2.1.cmml" xref="S1.SS1a.p2.1.m1.1.1.2">subscript</csymbol><ci id="S1.SS1a.p2.1.m1.1.1.2.2.cmml" xref="S1.SS1a.p2.1.m1.1.1.2.2">𝐋</ci><cn id="S1.SS1a.p2.1.m1.1.1.2.3.cmml" type="integer" xref="S1.SS1a.p2.1.m1.1.1.2.3">2</cn></apply><apply id="S1.SS1a.p2.1.m1.1.1.3.cmml" xref="S1.SS1a.p2.1.m1.1.1.3"><csymbol cd="ambiguous" id="S1.SS1a.p2.1.m1.1.1.3.1.cmml" xref="S1.SS1a.p2.1.m1.1.1.3">superscript</csymbol><ci id="S1.SS1a.p2.1.m1.1.1.3.2.cmml" xref="S1.SS1a.p2.1.m1.1.1.3.2">ℝ</ci><apply id="S1.SS1a.p2.1.m1.1.1.3.3.cmml" xref="S1.SS1a.p2.1.m1.1.1.3.3"><times id="S1.SS1a.p2.1.m1.1.1.3.3.1.cmml" xref="S1.SS1a.p2.1.m1.1.1.3.3.1"></times><apply id="S1.SS1a.p2.1.m1.1.1.3.3.2.cmml" xref="S1.SS1a.p2.1.m1.1.1.3.3.2"><csymbol cd="ambiguous" id="S1.SS1a.p2.1.m1.1.1.3.3.2.1.cmml" xref="S1.SS1a.p2.1.m1.1.1.3.3.2">subscript</csymbol><ci id="S1.SS1a.p2.1.m1.1.1.3.3.2.2.cmml" xref="S1.SS1a.p2.1.m1.1.1.3.3.2.2">𝑄</ci><cn id="S1.SS1a.p2.1.m1.1.1.3.3.2.3.cmml" type="integer" xref="S1.SS1a.p2.1.m1.1.1.3.3.2.3">2</cn></apply><apply id="S1.SS1a.p2.1.m1.1.1.3.3.3.cmml" xref="S1.SS1a.p2.1.m1.1.1.3.3.3"><csymbol cd="ambiguous" id="S1.SS1a.p2.1.m1.1.1.3.3.3.1.cmml" xref="S1.SS1a.p2.1.m1.1.1.3.3.3">subscript</csymbol><ci id="S1.SS1a.p2.1.m1.1.1.3.3.3.2.cmml" xref="S1.SS1a.p2.1.m1.1.1.3.3.3.2">𝑆</ci><cn id="S1.SS1a.p2.1.m1.1.1.3.3.3.3.cmml" type="integer" xref="S1.SS1a.p2.1.m1.1.1.3.3.3.3">2</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1a.p2.1.m1.1c">\operatorname{\mathbf{L}}_{2}\in\mathbb{R}^{Q_{2}\times S_{2}}</annotation><annotation encoding="application/x-llamapun" id="S1.SS1a.p2.1.m1.1d">bold_L start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ∈ blackboard_R start_POSTSUPERSCRIPT italic_Q start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT × italic_S start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT</annotation></semantics></math> denote the matrix such that <math alttext="\operatorname{\mathbf{L}}_{2}\operatorname{\mathbf{L}}_{2}^{\top}=% \operatorname{{\mathbf{\Sigma}}}_{2}" class="ltx_Math" display="inline" id="S1.SS1a.p2.2.m2.1"><semantics id="S1.SS1a.p2.2.m2.1a"><mrow id="S1.SS1a.p2.2.m2.1.1" xref="S1.SS1a.p2.2.m2.1.1.cmml"><mrow id="S1.SS1a.p2.2.m2.1.1.2" xref="S1.SS1a.p2.2.m2.1.1.2.cmml"><msub id="S1.SS1a.p2.2.m2.1.1.2.1" xref="S1.SS1a.p2.2.m2.1.1.2.1.cmml"><mi id="S1.SS1a.p2.2.m2.1.1.2.1.2" xref="S1.SS1a.p2.2.m2.1.1.2.1.2.cmml">𝐋</mi><mn id="S1.SS1a.p2.2.m2.1.1.2.1.3" xref="S1.SS1a.p2.2.m2.1.1.2.1.3.cmml">2</mn></msub><mo id="S1.SS1a.p2.2.m2.1.1.2a" lspace="0.167em" xref="S1.SS1a.p2.2.m2.1.1.2.cmml">⁡</mo><msubsup id="S1.SS1a.p2.2.m2.1.1.2.2" xref="S1.SS1a.p2.2.m2.1.1.2.2.cmml"><mi id="S1.SS1a.p2.2.m2.1.1.2.2.2.2" xref="S1.SS1a.p2.2.m2.1.1.2.2.2.2.cmml">𝐋</mi><mn id="S1.SS1a.p2.2.m2.1.1.2.2.2.3" xref="S1.SS1a.p2.2.m2.1.1.2.2.2.3.cmml">2</mn><mo id="S1.SS1a.p2.2.m2.1.1.2.2.3" xref="S1.SS1a.p2.2.m2.1.1.2.2.3.cmml">⊤</mo></msubsup></mrow><mo id="S1.SS1a.p2.2.m2.1.1.1" xref="S1.SS1a.p2.2.m2.1.1.1.cmml">=</mo><msub id="S1.SS1a.p2.2.m2.1.1.3" xref="S1.SS1a.p2.2.m2.1.1.3.cmml"><mi id="S1.SS1a.p2.2.m2.1.1.3.2" xref="S1.SS1a.p2.2.m2.1.1.3.2.cmml">𝚺</mi><mn id="S1.SS1a.p2.2.m2.1.1.3.3" xref="S1.SS1a.p2.2.m2.1.1.3.3.cmml">2</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S1.SS1a.p2.2.m2.1b"><apply id="S1.SS1a.p2.2.m2.1.1.cmml" xref="S1.SS1a.p2.2.m2.1.1"><eq id="S1.SS1a.p2.2.m2.1.1.1.cmml" xref="S1.SS1a.p2.2.m2.1.1.1"></eq><apply id="S1.SS1a.p2.2.m2.1.1.2.cmml" xref="S1.SS1a.p2.2.m2.1.1.2"><apply id="S1.SS1a.p2.2.m2.1.1.2.1.cmml" xref="S1.SS1a.p2.2.m2.1.1.2.1"><csymbol cd="ambiguous" id="S1.SS1a.p2.2.m2.1.1.2.1.1.cmml" xref="S1.SS1a.p2.2.m2.1.1.2.1">subscript</csymbol><ci id="S1.SS1a.p2.2.m2.1.1.2.1.2.cmml" xref="S1.SS1a.p2.2.m2.1.1.2.1.2">𝐋</ci><cn id="S1.SS1a.p2.2.m2.1.1.2.1.3.cmml" type="integer" xref="S1.SS1a.p2.2.m2.1.1.2.1.3">2</cn></apply><apply id="S1.SS1a.p2.2.m2.1.1.2.2.cmml" xref="S1.SS1a.p2.2.m2.1.1.2.2"><csymbol cd="ambiguous" id="S1.SS1a.p2.2.m2.1.1.2.2.1.cmml" xref="S1.SS1a.p2.2.m2.1.1.2.2">superscript</csymbol><apply id="S1.SS1a.p2.2.m2.1.1.2.2.2.cmml" xref="S1.SS1a.p2.2.m2.1.1.2.2"><csymbol cd="ambiguous" id="S1.SS1a.p2.2.m2.1.1.2.2.2.1.cmml" xref="S1.SS1a.p2.2.m2.1.1.2.2">subscript</csymbol><ci id="S1.SS1a.p2.2.m2.1.1.2.2.2.2.cmml" xref="S1.SS1a.p2.2.m2.1.1.2.2.2.2">𝐋</ci><cn id="S1.SS1a.p2.2.m2.1.1.2.2.2.3.cmml" type="integer" xref="S1.SS1a.p2.2.m2.1.1.2.2.2.3">2</cn></apply><csymbol cd="latexml" id="S1.SS1a.p2.2.m2.1.1.2.2.3.cmml" xref="S1.SS1a.p2.2.m2.1.1.2.2.3">top</csymbol></apply></apply><apply id="S1.SS1a.p2.2.m2.1.1.3.cmml" xref="S1.SS1a.p2.2.m2.1.1.3"><csymbol cd="ambiguous" id="S1.SS1a.p2.2.m2.1.1.3.1.cmml" xref="S1.SS1a.p2.2.m2.1.1.3">subscript</csymbol><ci id="S1.SS1a.p2.2.m2.1.1.3.2.cmml" xref="S1.SS1a.p2.2.m2.1.1.3.2">𝚺</ci><cn id="S1.SS1a.p2.2.m2.1.1.3.3.cmml" type="integer" xref="S1.SS1a.p2.2.m2.1.1.3.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1a.p2.2.m2.1c">\operatorname{\mathbf{L}}_{2}\operatorname{\mathbf{L}}_{2}^{\top}=% \operatorname{{\mathbf{\Sigma}}}_{2}</annotation><annotation encoding="application/x-llamapun" id="S1.SS1a.p2.2.m2.1d">bold_L start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT bold_L start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ⊤ end_POSTSUPERSCRIPT = bold_Σ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math>, let <math alttext="\operatorname{\mathbf{l}}_{i:}" class="ltx_Math" display="inline" id="S1.SS1a.p2.3.m3.1"><semantics id="S1.SS1a.p2.3.m3.1a"><msub id="S1.SS1a.p2.3.m3.1.1" xref="S1.SS1a.p2.3.m3.1.1.cmml"><mi id="S1.SS1a.p2.3.m3.1.1.2" xref="S1.SS1a.p2.3.m3.1.1.2.cmml">𝐥</mi><mrow id="S1.SS1a.p2.3.m3.1.1.3" xref="S1.SS1a.p2.3.m3.1.1.3.cmml"><mi id="S1.SS1a.p2.3.m3.1.1.3.2" xref="S1.SS1a.p2.3.m3.1.1.3.2.cmml">i</mi><mo id="S1.SS1a.p2.3.m3.1.1.3.1" lspace="0.278em" rspace="0.278em" xref="S1.SS1a.p2.3.m3.1.1.3.1.cmml">:</mo><mi id="S1.SS1a.p2.3.m3.1.1.3.3" xref="S1.SS1a.p2.3.m3.1.1.3.3.cmml"></mi></mrow></msub><annotation-xml encoding="MathML-Content" id="S1.SS1a.p2.3.m3.1b"><apply id="S1.SS1a.p2.3.m3.1.1.cmml" xref="S1.SS1a.p2.3.m3.1.1"><csymbol cd="ambiguous" id="S1.SS1a.p2.3.m3.1.1.1.cmml" xref="S1.SS1a.p2.3.m3.1.1">subscript</csymbol><ci id="S1.SS1a.p2.3.m3.1.1.2.cmml" xref="S1.SS1a.p2.3.m3.1.1.2">𝐥</ci><apply id="S1.SS1a.p2.3.m3.1.1.3.cmml" xref="S1.SS1a.p2.3.m3.1.1.3"><ci id="S1.SS1a.p2.3.m3.1.1.3.1.cmml" xref="S1.SS1a.p2.3.m3.1.1.3.1">:</ci><ci id="S1.SS1a.p2.3.m3.1.1.3.2.cmml" xref="S1.SS1a.p2.3.m3.1.1.3.2">𝑖</ci><csymbol cd="latexml" id="S1.SS1a.p2.3.m3.1.1.3.3.cmml" xref="S1.SS1a.p2.3.m3.1.1.3.3">absent</csymbol></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1a.p2.3.m3.1c">\operatorname{\mathbf{l}}_{i:}</annotation><annotation encoding="application/x-llamapun" id="S1.SS1a.p2.3.m3.1d">bold_l start_POSTSUBSCRIPT italic_i : end_POSTSUBSCRIPT</annotation></semantics></math> denote the <math alttext="i" class="ltx_Math" display="inline" id="S1.SS1a.p2.4.m4.1"><semantics id="S1.SS1a.p2.4.m4.1a"><mi id="S1.SS1a.p2.4.m4.1.1" xref="S1.SS1a.p2.4.m4.1.1.cmml">i</mi><annotation-xml encoding="MathML-Content" id="S1.SS1a.p2.4.m4.1b"><ci id="S1.SS1a.p2.4.m4.1.1.cmml" xref="S1.SS1a.p2.4.m4.1.1">𝑖</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1a.p2.4.m4.1c">i</annotation><annotation encoding="application/x-llamapun" id="S1.SS1a.p2.4.m4.1d">italic_i</annotation></semantics></math>th row vector of <math alttext="\operatorname{\mathbf{L}}_{2}" class="ltx_Math" display="inline" id="S1.SS1a.p2.5.m5.1"><semantics id="S1.SS1a.p2.5.m5.1a"><msub id="S1.SS1a.p2.5.m5.1.1" xref="S1.SS1a.p2.5.m5.1.1.cmml"><mi id="S1.SS1a.p2.5.m5.1.1.2" xref="S1.SS1a.p2.5.m5.1.1.2.cmml">𝐋</mi><mn id="S1.SS1a.p2.5.m5.1.1.3" xref="S1.SS1a.p2.5.m5.1.1.3.cmml">2</mn></msub><annotation-xml encoding="MathML-Content" id="S1.SS1a.p2.5.m5.1b"><apply id="S1.SS1a.p2.5.m5.1.1.cmml" xref="S1.SS1a.p2.5.m5.1.1"><csymbol cd="ambiguous" id="S1.SS1a.p2.5.m5.1.1.1.cmml" xref="S1.SS1a.p2.5.m5.1.1">subscript</csymbol><ci id="S1.SS1a.p2.5.m5.1.1.2.cmml" xref="S1.SS1a.p2.5.m5.1.1.2">𝐋</ci><cn id="S1.SS1a.p2.5.m5.1.1.3.cmml" type="integer" xref="S1.SS1a.p2.5.m5.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1a.p2.5.m5.1c">\operatorname{\mathbf{L}}_{2}</annotation><annotation encoding="application/x-llamapun" id="S1.SS1a.p2.5.m5.1d">bold_L start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math> and let <math alttext="\operatorname{\mathbf{e}}_{1}\in\mathbb{R}^{S_{2}}" class="ltx_Math" display="inline" id="S1.SS1a.p2.6.m6.1"><semantics id="S1.SS1a.p2.6.m6.1a"><mrow id="S1.SS1a.p2.6.m6.1.1" xref="S1.SS1a.p2.6.m6.1.1.cmml"><msub id="S1.SS1a.p2.6.m6.1.1.2" xref="S1.SS1a.p2.6.m6.1.1.2.cmml"><mi id="S1.SS1a.p2.6.m6.1.1.2.2" xref="S1.SS1a.p2.6.m6.1.1.2.2.cmml">𝐞</mi><mn id="S1.SS1a.p2.6.m6.1.1.2.3" xref="S1.SS1a.p2.6.m6.1.1.2.3.cmml">1</mn></msub><mo id="S1.SS1a.p2.6.m6.1.1.1" xref="S1.SS1a.p2.6.m6.1.1.1.cmml">∈</mo><msup id="S1.SS1a.p2.6.m6.1.1.3" xref="S1.SS1a.p2.6.m6.1.1.3.cmml"><mi id="S1.SS1a.p2.6.m6.1.1.3.2" xref="S1.SS1a.p2.6.m6.1.1.3.2.cmml">ℝ</mi><msub id="S1.SS1a.p2.6.m6.1.1.3.3" xref="S1.SS1a.p2.6.m6.1.1.3.3.cmml"><mi id="S1.SS1a.p2.6.m6.1.1.3.3.2" xref="S1.SS1a.p2.6.m6.1.1.3.3.2.cmml">S</mi><mn id="S1.SS1a.p2.6.m6.1.1.3.3.3" xref="S1.SS1a.p2.6.m6.1.1.3.3.3.cmml">2</mn></msub></msup></mrow><annotation-xml encoding="MathML-Content" id="S1.SS1a.p2.6.m6.1b"><apply id="S1.SS1a.p2.6.m6.1.1.cmml" xref="S1.SS1a.p2.6.m6.1.1"><in id="S1.SS1a.p2.6.m6.1.1.1.cmml" xref="S1.SS1a.p2.6.m6.1.1.1"></in><apply id="S1.SS1a.p2.6.m6.1.1.2.cmml" xref="S1.SS1a.p2.6.m6.1.1.2"><csymbol cd="ambiguous" id="S1.SS1a.p2.6.m6.1.1.2.1.cmml" xref="S1.SS1a.p2.6.m6.1.1.2">subscript</csymbol><ci id="S1.SS1a.p2.6.m6.1.1.2.2.cmml" xref="S1.SS1a.p2.6.m6.1.1.2.2">𝐞</ci><cn id="S1.SS1a.p2.6.m6.1.1.2.3.cmml" type="integer" xref="S1.SS1a.p2.6.m6.1.1.2.3">1</cn></apply><apply id="S1.SS1a.p2.6.m6.1.1.3.cmml" xref="S1.SS1a.p2.6.m6.1.1.3"><csymbol cd="ambiguous" id="S1.SS1a.p2.6.m6.1.1.3.1.cmml" xref="S1.SS1a.p2.6.m6.1.1.3">superscript</csymbol><ci id="S1.SS1a.p2.6.m6.1.1.3.2.cmml" xref="S1.SS1a.p2.6.m6.1.1.3.2">ℝ</ci><apply id="S1.SS1a.p2.6.m6.1.1.3.3.cmml" xref="S1.SS1a.p2.6.m6.1.1.3.3"><csymbol cd="ambiguous" id="S1.SS1a.p2.6.m6.1.1.3.3.1.cmml" xref="S1.SS1a.p2.6.m6.1.1.3.3">subscript</csymbol><ci id="S1.SS1a.p2.6.m6.1.1.3.3.2.cmml" xref="S1.SS1a.p2.6.m6.1.1.3.3.2">𝑆</ci><cn id="S1.SS1a.p2.6.m6.1.1.3.3.3.cmml" type="integer" xref="S1.SS1a.p2.6.m6.1.1.3.3.3">2</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1a.p2.6.m6.1c">\operatorname{\mathbf{e}}_{1}\in\mathbb{R}^{S_{2}}</annotation><annotation encoding="application/x-llamapun" id="S1.SS1a.p2.6.m6.1d">bold_e start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ∈ blackboard_R start_POSTSUPERSCRIPT italic_S start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT</annotation></semantics></math> denote the first standard basis vector in <math alttext="\mathbb{R}^{S_{2}}" class="ltx_Math" display="inline" id="S1.SS1a.p2.7.m7.1"><semantics id="S1.SS1a.p2.7.m7.1a"><msup id="S1.SS1a.p2.7.m7.1.1" xref="S1.SS1a.p2.7.m7.1.1.cmml"><mi id="S1.SS1a.p2.7.m7.1.1.2" xref="S1.SS1a.p2.7.m7.1.1.2.cmml">ℝ</mi><msub id="S1.SS1a.p2.7.m7.1.1.3" xref="S1.SS1a.p2.7.m7.1.1.3.cmml"><mi id="S1.SS1a.p2.7.m7.1.1.3.2" xref="S1.SS1a.p2.7.m7.1.1.3.2.cmml">S</mi><mn id="S1.SS1a.p2.7.m7.1.1.3.3" xref="S1.SS1a.p2.7.m7.1.1.3.3.cmml">2</mn></msub></msup><annotation-xml encoding="MathML-Content" id="S1.SS1a.p2.7.m7.1b"><apply id="S1.SS1a.p2.7.m7.1.1.cmml" xref="S1.SS1a.p2.7.m7.1.1"><csymbol cd="ambiguous" id="S1.SS1a.p2.7.m7.1.1.1.cmml" xref="S1.SS1a.p2.7.m7.1.1">superscript</csymbol><ci id="S1.SS1a.p2.7.m7.1.1.2.cmml" xref="S1.SS1a.p2.7.m7.1.1.2">ℝ</ci><apply id="S1.SS1a.p2.7.m7.1.1.3.cmml" xref="S1.SS1a.p2.7.m7.1.1.3"><csymbol cd="ambiguous" id="S1.SS1a.p2.7.m7.1.1.3.1.cmml" xref="S1.SS1a.p2.7.m7.1.1.3">subscript</csymbol><ci id="S1.SS1a.p2.7.m7.1.1.3.2.cmml" xref="S1.SS1a.p2.7.m7.1.1.3.2">𝑆</ci><cn id="S1.SS1a.p2.7.m7.1.1.3.3.cmml" type="integer" xref="S1.SS1a.p2.7.m7.1.1.3.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1a.p2.7.m7.1c">\mathbb{R}^{S_{2}}</annotation><annotation encoding="application/x-llamapun" id="S1.SS1a.p2.7.m7.1d">blackboard_R start_POSTSUPERSCRIPT italic_S start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT</annotation></semantics></math>. If the <math alttext="j" class="ltx_Math" display="inline" id="S1.SS1a.p2.8.m8.1"><semantics id="S1.SS1a.p2.8.m8.1a"><mi id="S1.SS1a.p2.8.m8.1.1" xref="S1.SS1a.p2.8.m8.1.1.cmml">j</mi><annotation-xml encoding="MathML-Content" id="S1.SS1a.p2.8.m8.1b"><ci id="S1.SS1a.p2.8.m8.1.1.cmml" xref="S1.SS1a.p2.8.m8.1.1">𝑗</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1a.p2.8.m8.1c">j</annotation><annotation encoding="application/x-llamapun" id="S1.SS1a.p2.8.m8.1d">italic_j</annotation></semantics></math>th row of <math alttext="\operatorname{\mathbf{L}}_{2}" class="ltx_Math" display="inline" id="S1.SS1a.p2.9.m9.1"><semantics id="S1.SS1a.p2.9.m9.1a"><msub id="S1.SS1a.p2.9.m9.1.1" xref="S1.SS1a.p2.9.m9.1.1.cmml"><mi id="S1.SS1a.p2.9.m9.1.1.2" xref="S1.SS1a.p2.9.m9.1.1.2.cmml">𝐋</mi><mn id="S1.SS1a.p2.9.m9.1.1.3" xref="S1.SS1a.p2.9.m9.1.1.3.cmml">2</mn></msub><annotation-xml encoding="MathML-Content" id="S1.SS1a.p2.9.m9.1b"><apply id="S1.SS1a.p2.9.m9.1.1.cmml" xref="S1.SS1a.p2.9.m9.1.1"><csymbol cd="ambiguous" id="S1.SS1a.p2.9.m9.1.1.1.cmml" xref="S1.SS1a.p2.9.m9.1.1">subscript</csymbol><ci id="S1.SS1a.p2.9.m9.1.1.2.cmml" xref="S1.SS1a.p2.9.m9.1.1.2">𝐋</ci><cn id="S1.SS1a.p2.9.m9.1.1.3.cmml" type="integer" xref="S1.SS1a.p2.9.m9.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1a.p2.9.m9.1c">\operatorname{\mathbf{L}}_{2}</annotation><annotation encoding="application/x-llamapun" id="S1.SS1a.p2.9.m9.1d">bold_L start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math> is <math alttext="\operatorname{\mathbf{e}}_{1}" class="ltx_Math" display="inline" id="S1.SS1a.p2.10.m10.1"><semantics id="S1.SS1a.p2.10.m10.1a"><msub id="S1.SS1a.p2.10.m10.1.1" xref="S1.SS1a.p2.10.m10.1.1.cmml"><mi id="S1.SS1a.p2.10.m10.1.1.2" xref="S1.SS1a.p2.10.m10.1.1.2.cmml">𝐞</mi><mn id="S1.SS1a.p2.10.m10.1.1.3" xref="S1.SS1a.p2.10.m10.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S1.SS1a.p2.10.m10.1b"><apply id="S1.SS1a.p2.10.m10.1.1.cmml" xref="S1.SS1a.p2.10.m10.1.1"><csymbol cd="ambiguous" id="S1.SS1a.p2.10.m10.1.1.1.cmml" xref="S1.SS1a.p2.10.m10.1.1">subscript</csymbol><ci id="S1.SS1a.p2.10.m10.1.1.2.cmml" xref="S1.SS1a.p2.10.m10.1.1.2">𝐞</ci><cn id="S1.SS1a.p2.10.m10.1.1.3.cmml" type="integer" xref="S1.SS1a.p2.10.m10.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1a.p2.10.m10.1c">\operatorname{\mathbf{e}}_{1}</annotation><annotation encoding="application/x-llamapun" id="S1.SS1a.p2.10.m10.1d">bold_e start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math>, then:</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="S3.EGx14"> <tbody id="S1.Ex3"><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{\mathbf{L}}_{2}=\begin{bmatrix}\operatorname{% \mathbf{l}}_{1:}^{\top}\\ \vdots\\ \operatorname{\mathbf{e}}_{1}^{\top}\\ \vdots\\ \operatorname{\mathbf{l}}_{Q_{2}:}^{\top}\end{bmatrix}\implies\operatorname{% \mathbf{L}}_{2}\operatorname{\mathbf{L}}_{2}^{\top}=\begin{bmatrix}% \operatorname{\mathbf{l}}_{1:}^{\top}\operatorname{\mathbf{l}}_{1:}&\ldots&(% \operatorname{\mathbf{L}}_{2})_{11}&\ldots&\operatorname{\mathbf{l}}_{1:}^{% \top}\operatorname{\mathbf{l}}_{Q_{2}:}\\ \vdots&\ddots&\vdots&&\vdots\\ (\operatorname{\mathbf{L}}_{2})_{11}&\ldots&1&\ldots&(\operatorname{\mathbf{L}% }_{2})_{Q_{2}1}\\ \vdots&&\vdots&\ddots&\vdots\\ \operatorname{\mathbf{l}}_{Q_{2}:}^{\top}\operatorname{\mathbf{l}}_{1:}&\ldots% &(\operatorname{\mathbf{L}}_{2})_{Q_{2}1}&\ldots&\operatorname{\mathbf{l}}_{Q_% {2}:}^{\top}\operatorname{\mathbf{l}}_{Q_{2}:}\end{bmatrix}," class="ltx_Math" display="inline" id="S1.Ex3.m1.3"><semantics id="S1.Ex3.m1.3a"><mrow id="S1.Ex3.m1.3.3.1" xref="S1.Ex3.m1.3.3.1.1.cmml"><mrow id="S1.Ex3.m1.3.3.1.1" xref="S1.Ex3.m1.3.3.1.1.cmml"><msub id="S1.Ex3.m1.3.3.1.1.2" xref="S1.Ex3.m1.3.3.1.1.2.cmml"><mi id="S1.Ex3.m1.3.3.1.1.2.2" xref="S1.Ex3.m1.3.3.1.1.2.2.cmml">𝐋</mi><mn id="S1.Ex3.m1.3.3.1.1.2.3" xref="S1.Ex3.m1.3.3.1.1.2.3.cmml">2</mn></msub><mo id="S1.Ex3.m1.3.3.1.1.3" xref="S1.Ex3.m1.3.3.1.1.3.cmml">=</mo><mrow id="S1.Ex3.m1.1.1a.3" xref="S1.Ex3.m1.1.1a.2.cmml"><mo id="S1.Ex3.m1.1.1a.3.1" xref="S1.Ex3.m1.1.1a.2.1.cmml">[</mo><mtable id="S1.Ex3.m1.1.1.1.1" rowspacing="0pt" xref="S1.Ex3.m1.1.1.1.1.cmml"><mtr id="S1.Ex3.m1.1.1.1.1a" xref="S1.Ex3.m1.1.1.1.1.cmml"><mtd id="S1.Ex3.m1.1.1.1.1b" xref="S1.Ex3.m1.1.1.1.1.cmml"><msubsup id="S1.Ex3.m1.1.1.1.1.1.1.1" xref="S1.Ex3.m1.1.1.1.1.1.1.1.cmml"><mi id="S1.Ex3.m1.1.1.1.1.1.1.1.2.2" xref="S1.Ex3.m1.1.1.1.1.1.1.1.2.2.cmml">𝐥</mi><mrow id="S1.Ex3.m1.1.1.1.1.1.1.1.2.3" xref="S1.Ex3.m1.1.1.1.1.1.1.1.2.3.cmml"><mn id="S1.Ex3.m1.1.1.1.1.1.1.1.2.3.2" xref="S1.Ex3.m1.1.1.1.1.1.1.1.2.3.2.cmml">1</mn><mo id="S1.Ex3.m1.1.1.1.1.1.1.1.2.3.1" lspace="0.278em" rspace="0.278em" xref="S1.Ex3.m1.1.1.1.1.1.1.1.2.3.1.cmml">:</mo><mi id="S1.Ex3.m1.1.1.1.1.1.1.1.2.3.3" xref="S1.Ex3.m1.1.1.1.1.1.1.1.2.3.3.cmml"></mi></mrow><mo id="S1.Ex3.m1.1.1.1.1.1.1.1.3" xref="S1.Ex3.m1.1.1.1.1.1.1.1.3.cmml">⊤</mo></msubsup></mtd></mtr><mtr id="S1.Ex3.m1.1.1.1.1c" xref="S1.Ex3.m1.1.1.1.1.cmml"><mtd id="S1.Ex3.m1.1.1.1.1d" xref="S1.Ex3.m1.1.1.1.1.cmml"><mi id="S1.Ex3.m1.1.1.1.1.2.1.1" mathvariant="normal" xref="S1.Ex3.m1.1.1.1.1.2.1.1.cmml">⋮</mi></mtd></mtr><mtr id="S1.Ex3.m1.1.1.1.1e" xref="S1.Ex3.m1.1.1.1.1.cmml"><mtd id="S1.Ex3.m1.1.1.1.1f" xref="S1.Ex3.m1.1.1.1.1.cmml"><msubsup id="S1.Ex3.m1.1.1.1.1.3.1.1" xref="S1.Ex3.m1.1.1.1.1.3.1.1.cmml"><mi id="S1.Ex3.m1.1.1.1.1.3.1.1.2.2" xref="S1.Ex3.m1.1.1.1.1.3.1.1.2.2.cmml">𝐞</mi><mn id="S1.Ex3.m1.1.1.1.1.3.1.1.2.3" xref="S1.Ex3.m1.1.1.1.1.3.1.1.2.3.cmml">1</mn><mo id="S1.Ex3.m1.1.1.1.1.3.1.1.3" xref="S1.Ex3.m1.1.1.1.1.3.1.1.3.cmml">⊤</mo></msubsup></mtd></mtr><mtr id="S1.Ex3.m1.1.1.1.1g" xref="S1.Ex3.m1.1.1.1.1.cmml"><mtd id="S1.Ex3.m1.1.1.1.1h" xref="S1.Ex3.m1.1.1.1.1.cmml"><mi id="S1.Ex3.m1.1.1.1.1.4.1.1" mathvariant="normal" xref="S1.Ex3.m1.1.1.1.1.4.1.1.cmml">⋮</mi></mtd></mtr><mtr id="S1.Ex3.m1.1.1.1.1i" xref="S1.Ex3.m1.1.1.1.1.cmml"><mtd id="S1.Ex3.m1.1.1.1.1j" xref="S1.Ex3.m1.1.1.1.1.cmml"><msubsup id="S1.Ex3.m1.1.1.1.1.5.1.1" xref="S1.Ex3.m1.1.1.1.1.5.1.1.cmml"><mi id="S1.Ex3.m1.1.1.1.1.5.1.1.2.2" xref="S1.Ex3.m1.1.1.1.1.5.1.1.2.2.cmml">𝐥</mi><mrow id="S1.Ex3.m1.1.1.1.1.5.1.1.2.3" xref="S1.Ex3.m1.1.1.1.1.5.1.1.2.3.cmml"><msub id="S1.Ex3.m1.1.1.1.1.5.1.1.2.3.2" xref="S1.Ex3.m1.1.1.1.1.5.1.1.2.3.2.cmml"><mi id="S1.Ex3.m1.1.1.1.1.5.1.1.2.3.2.2" xref="S1.Ex3.m1.1.1.1.1.5.1.1.2.3.2.2.cmml">Q</mi><mn id="S1.Ex3.m1.1.1.1.1.5.1.1.2.3.2.3" xref="S1.Ex3.m1.1.1.1.1.5.1.1.2.3.2.3.cmml">2</mn></msub><mo id="S1.Ex3.m1.1.1.1.1.5.1.1.2.3.1" lspace="0.278em" rspace="0.278em" xref="S1.Ex3.m1.1.1.1.1.5.1.1.2.3.1.cmml">:</mo><mi id="S1.Ex3.m1.1.1.1.1.5.1.1.2.3.3" xref="S1.Ex3.m1.1.1.1.1.5.1.1.2.3.3.cmml"></mi></mrow><mo id="S1.Ex3.m1.1.1.1.1.5.1.1.3" xref="S1.Ex3.m1.1.1.1.1.5.1.1.3.cmml">⊤</mo></msubsup></mtd></mtr></mtable><mo id="S1.Ex3.m1.1.1a.3.2" xref="S1.Ex3.m1.1.1a.2.1.cmml">]</mo></mrow><mo id="S1.Ex3.m1.3.3.1.1.4" stretchy="false" xref="S1.Ex3.m1.3.3.1.1.4.cmml">⟹</mo><mrow id="S1.Ex3.m1.3.3.1.1.5" xref="S1.Ex3.m1.3.3.1.1.5.cmml"><msub id="S1.Ex3.m1.3.3.1.1.5.1" xref="S1.Ex3.m1.3.3.1.1.5.1.cmml"><mi id="S1.Ex3.m1.3.3.1.1.5.1.2" xref="S1.Ex3.m1.3.3.1.1.5.1.2.cmml">𝐋</mi><mn id="S1.Ex3.m1.3.3.1.1.5.1.3" xref="S1.Ex3.m1.3.3.1.1.5.1.3.cmml">2</mn></msub><mo id="S1.Ex3.m1.3.3.1.1.5a" lspace="0.167em" xref="S1.Ex3.m1.3.3.1.1.5.cmml">⁡</mo><msubsup id="S1.Ex3.m1.3.3.1.1.5.2" xref="S1.Ex3.m1.3.3.1.1.5.2.cmml"><mi id="S1.Ex3.m1.3.3.1.1.5.2.2.2" xref="S1.Ex3.m1.3.3.1.1.5.2.2.2.cmml">𝐋</mi><mn id="S1.Ex3.m1.3.3.1.1.5.2.2.3" 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encoding="application/x-tex" id="S1.Ex3.m1.3c">\displaystyle\operatorname{\mathbf{L}}_{2}=\begin{bmatrix}\operatorname{% \mathbf{l}}_{1:}^{\top}\\ \vdots\\ \operatorname{\mathbf{e}}_{1}^{\top}\\ \vdots\\ \operatorname{\mathbf{l}}_{Q_{2}:}^{\top}\end{bmatrix}\implies\operatorname{% \mathbf{L}}_{2}\operatorname{\mathbf{L}}_{2}^{\top}=\begin{bmatrix}% \operatorname{\mathbf{l}}_{1:}^{\top}\operatorname{\mathbf{l}}_{1:}&\ldots&(% \operatorname{\mathbf{L}}_{2})_{11}&\ldots&\operatorname{\mathbf{l}}_{1:}^{% \top}\operatorname{\mathbf{l}}_{Q_{2}:}\\ \vdots&\ddots&\vdots&&\vdots\\ (\operatorname{\mathbf{L}}_{2})_{11}&\ldots&1&\ldots&(\operatorname{\mathbf{L}% }_{2})_{Q_{2}1}\\ \vdots&&\vdots&\ddots&\vdots\\ \operatorname{\mathbf{l}}_{Q_{2}:}^{\top}\operatorname{\mathbf{l}}_{1:}&\ldots% &(\operatorname{\mathbf{L}}_{2})_{Q_{2}1}&\ldots&\operatorname{\mathbf{l}}_{Q_% {2}:}^{\top}\operatorname{\mathbf{l}}_{Q_{2}:}\end{bmatrix},</annotation><annotation encoding="application/x-llamapun" id="S1.Ex3.m1.3d">bold_L start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT = [ start_ARG start_ROW start_CELL bold_l start_POSTSUBSCRIPT 1 : end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ⊤ end_POSTSUPERSCRIPT end_CELL end_ROW start_ROW start_CELL ⋮ end_CELL end_ROW start_ROW start_CELL bold_e start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ⊤ end_POSTSUPERSCRIPT end_CELL end_ROW start_ROW start_CELL ⋮ end_CELL end_ROW start_ROW start_CELL bold_l start_POSTSUBSCRIPT italic_Q start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT : end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ⊤ end_POSTSUPERSCRIPT end_CELL end_ROW end_ARG ] ⟹ bold_L start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT bold_L start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ⊤ end_POSTSUPERSCRIPT = [ start_ARG start_ROW start_CELL bold_l start_POSTSUBSCRIPT 1 : end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ⊤ end_POSTSUPERSCRIPT bold_l start_POSTSUBSCRIPT 1 : end_POSTSUBSCRIPT end_CELL start_CELL … end_CELL start_CELL ( bold_L start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) start_POSTSUBSCRIPT 11 end_POSTSUBSCRIPT end_CELL start_CELL … end_CELL start_CELL bold_l start_POSTSUBSCRIPT 1 : end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ⊤ end_POSTSUPERSCRIPT bold_l start_POSTSUBSCRIPT italic_Q start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT : end_POSTSUBSCRIPT end_CELL end_ROW start_ROW start_CELL ⋮ end_CELL start_CELL ⋱ end_CELL start_CELL ⋮ end_CELL start_CELL end_CELL start_CELL ⋮ end_CELL end_ROW start_ROW start_CELL ( bold_L start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) start_POSTSUBSCRIPT 11 end_POSTSUBSCRIPT end_CELL start_CELL … end_CELL start_CELL 1 end_CELL start_CELL … end_CELL start_CELL ( bold_L start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) start_POSTSUBSCRIPT italic_Q start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_CELL end_ROW start_ROW start_CELL ⋮ end_CELL start_CELL end_CELL start_CELL ⋮ end_CELL start_CELL ⋱ end_CELL start_CELL ⋮ end_CELL end_ROW start_ROW start_CELL bold_l start_POSTSUBSCRIPT italic_Q start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT : end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ⊤ end_POSTSUPERSCRIPT bold_l start_POSTSUBSCRIPT 1 : end_POSTSUBSCRIPT end_CELL start_CELL … end_CELL start_CELL ( bold_L start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) start_POSTSUBSCRIPT italic_Q start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_CELL start_CELL … end_CELL start_CELL bold_l start_POSTSUBSCRIPT italic_Q start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT : end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ⊤ end_POSTSUPERSCRIPT bold_l start_POSTSUBSCRIPT italic_Q start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT : end_POSTSUBSCRIPT end_CELL end_ROW end_ARG ] ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S1.SS1a.p2.15">demonstrating that <math alttext="(\operatorname{{\mathbf{\Sigma}}}_{2})_{jj}=1" class="ltx_Math" display="inline" id="S1.SS1a.p2.11.m1.1"><semantics id="S1.SS1a.p2.11.m1.1a"><mrow id="S1.SS1a.p2.11.m1.1.1" xref="S1.SS1a.p2.11.m1.1.1.cmml"><msub id="S1.SS1a.p2.11.m1.1.1.1" xref="S1.SS1a.p2.11.m1.1.1.1.cmml"><mrow id="S1.SS1a.p2.11.m1.1.1.1.1.1" xref="S1.SS1a.p2.11.m1.1.1.1.1.1.1.cmml"><mo id="S1.SS1a.p2.11.m1.1.1.1.1.1.2" stretchy="false" xref="S1.SS1a.p2.11.m1.1.1.1.1.1.1.cmml">(</mo><msub id="S1.SS1a.p2.11.m1.1.1.1.1.1.1" xref="S1.SS1a.p2.11.m1.1.1.1.1.1.1.cmml"><mi id="S1.SS1a.p2.11.m1.1.1.1.1.1.1.2" xref="S1.SS1a.p2.11.m1.1.1.1.1.1.1.2.cmml">𝚺</mi><mn id="S1.SS1a.p2.11.m1.1.1.1.1.1.1.3" xref="S1.SS1a.p2.11.m1.1.1.1.1.1.1.3.cmml">2</mn></msub><mo id="S1.SS1a.p2.11.m1.1.1.1.1.1.3" stretchy="false" xref="S1.SS1a.p2.11.m1.1.1.1.1.1.1.cmml">)</mo></mrow><mrow id="S1.SS1a.p2.11.m1.1.1.1.3" xref="S1.SS1a.p2.11.m1.1.1.1.3.cmml"><mi id="S1.SS1a.p2.11.m1.1.1.1.3.2" xref="S1.SS1a.p2.11.m1.1.1.1.3.2.cmml">j</mi><mo id="S1.SS1a.p2.11.m1.1.1.1.3.1" xref="S1.SS1a.p2.11.m1.1.1.1.3.1.cmml">⁢</mo><mi id="S1.SS1a.p2.11.m1.1.1.1.3.3" xref="S1.SS1a.p2.11.m1.1.1.1.3.3.cmml">j</mi></mrow></msub><mo id="S1.SS1a.p2.11.m1.1.1.2" xref="S1.SS1a.p2.11.m1.1.1.2.cmml">=</mo><mn id="S1.SS1a.p2.11.m1.1.1.3" xref="S1.SS1a.p2.11.m1.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S1.SS1a.p2.11.m1.1b"><apply id="S1.SS1a.p2.11.m1.1.1.cmml" xref="S1.SS1a.p2.11.m1.1.1"><eq id="S1.SS1a.p2.11.m1.1.1.2.cmml" xref="S1.SS1a.p2.11.m1.1.1.2"></eq><apply id="S1.SS1a.p2.11.m1.1.1.1.cmml" xref="S1.SS1a.p2.11.m1.1.1.1"><csymbol cd="ambiguous" id="S1.SS1a.p2.11.m1.1.1.1.2.cmml" xref="S1.SS1a.p2.11.m1.1.1.1">subscript</csymbol><apply id="S1.SS1a.p2.11.m1.1.1.1.1.1.1.cmml" xref="S1.SS1a.p2.11.m1.1.1.1.1.1"><csymbol cd="ambiguous" id="S1.SS1a.p2.11.m1.1.1.1.1.1.1.1.cmml" xref="S1.SS1a.p2.11.m1.1.1.1.1.1">subscript</csymbol><ci id="S1.SS1a.p2.11.m1.1.1.1.1.1.1.2.cmml" xref="S1.SS1a.p2.11.m1.1.1.1.1.1.1.2">𝚺</ci><cn id="S1.SS1a.p2.11.m1.1.1.1.1.1.1.3.cmml" type="integer" xref="S1.SS1a.p2.11.m1.1.1.1.1.1.1.3">2</cn></apply><apply id="S1.SS1a.p2.11.m1.1.1.1.3.cmml" xref="S1.SS1a.p2.11.m1.1.1.1.3"><times id="S1.SS1a.p2.11.m1.1.1.1.3.1.cmml" xref="S1.SS1a.p2.11.m1.1.1.1.3.1"></times><ci id="S1.SS1a.p2.11.m1.1.1.1.3.2.cmml" xref="S1.SS1a.p2.11.m1.1.1.1.3.2">𝑗</ci><ci id="S1.SS1a.p2.11.m1.1.1.1.3.3.cmml" xref="S1.SS1a.p2.11.m1.1.1.1.3.3">𝑗</ci></apply></apply><cn id="S1.SS1a.p2.11.m1.1.1.3.cmml" type="integer" xref="S1.SS1a.p2.11.m1.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1a.p2.11.m1.1c">(\operatorname{{\mathbf{\Sigma}}}_{2})_{jj}=1</annotation><annotation encoding="application/x-llamapun" id="S1.SS1a.p2.11.m1.1d">( bold_Σ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) start_POSTSUBSCRIPT italic_j italic_j end_POSTSUBSCRIPT = 1</annotation></semantics></math> which implies, by definition of the Kronecker product, that the <math alttext="j" class="ltx_Math" display="inline" id="S1.SS1a.p2.12.m2.1"><semantics id="S1.SS1a.p2.12.m2.1a"><mi id="S1.SS1a.p2.12.m2.1.1" xref="S1.SS1a.p2.12.m2.1.1.cmml">j</mi><annotation-xml encoding="MathML-Content" id="S1.SS1a.p2.12.m2.1b"><ci id="S1.SS1a.p2.12.m2.1.1.cmml" xref="S1.SS1a.p2.12.m2.1.1">𝑗</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1a.p2.12.m2.1c">j</annotation><annotation encoding="application/x-llamapun" id="S1.SS1a.p2.12.m2.1d">italic_j</annotation></semantics></math>th diagonal <math alttext="Q_{1}\times Q_{1}" class="ltx_Math" display="inline" id="S1.SS1a.p2.13.m3.1"><semantics id="S1.SS1a.p2.13.m3.1a"><mrow id="S1.SS1a.p2.13.m3.1.1" xref="S1.SS1a.p2.13.m3.1.1.cmml"><msub id="S1.SS1a.p2.13.m3.1.1.2" xref="S1.SS1a.p2.13.m3.1.1.2.cmml"><mi id="S1.SS1a.p2.13.m3.1.1.2.2" xref="S1.SS1a.p2.13.m3.1.1.2.2.cmml">Q</mi><mn id="S1.SS1a.p2.13.m3.1.1.2.3" xref="S1.SS1a.p2.13.m3.1.1.2.3.cmml">1</mn></msub><mo id="S1.SS1a.p2.13.m3.1.1.1" lspace="0.222em" rspace="0.222em" xref="S1.SS1a.p2.13.m3.1.1.1.cmml">×</mo><msub id="S1.SS1a.p2.13.m3.1.1.3" xref="S1.SS1a.p2.13.m3.1.1.3.cmml"><mi id="S1.SS1a.p2.13.m3.1.1.3.2" xref="S1.SS1a.p2.13.m3.1.1.3.2.cmml">Q</mi><mn id="S1.SS1a.p2.13.m3.1.1.3.3" xref="S1.SS1a.p2.13.m3.1.1.3.3.cmml">1</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S1.SS1a.p2.13.m3.1b"><apply id="S1.SS1a.p2.13.m3.1.1.cmml" xref="S1.SS1a.p2.13.m3.1.1"><times id="S1.SS1a.p2.13.m3.1.1.1.cmml" xref="S1.SS1a.p2.13.m3.1.1.1"></times><apply id="S1.SS1a.p2.13.m3.1.1.2.cmml" xref="S1.SS1a.p2.13.m3.1.1.2"><csymbol cd="ambiguous" id="S1.SS1a.p2.13.m3.1.1.2.1.cmml" xref="S1.SS1a.p2.13.m3.1.1.2">subscript</csymbol><ci id="S1.SS1a.p2.13.m3.1.1.2.2.cmml" xref="S1.SS1a.p2.13.m3.1.1.2.2">𝑄</ci><cn id="S1.SS1a.p2.13.m3.1.1.2.3.cmml" type="integer" xref="S1.SS1a.p2.13.m3.1.1.2.3">1</cn></apply><apply id="S1.SS1a.p2.13.m3.1.1.3.cmml" xref="S1.SS1a.p2.13.m3.1.1.3"><csymbol cd="ambiguous" id="S1.SS1a.p2.13.m3.1.1.3.1.cmml" xref="S1.SS1a.p2.13.m3.1.1.3">subscript</csymbol><ci id="S1.SS1a.p2.13.m3.1.1.3.2.cmml" xref="S1.SS1a.p2.13.m3.1.1.3.2">𝑄</ci><cn id="S1.SS1a.p2.13.m3.1.1.3.3.cmml" type="integer" xref="S1.SS1a.p2.13.m3.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1a.p2.13.m3.1c">Q_{1}\times Q_{1}</annotation><annotation encoding="application/x-llamapun" id="S1.SS1a.p2.13.m3.1d">italic_Q start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT × italic_Q start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> block of <math alttext="\operatorname{{\mathbf{\Sigma}}}_{2}\otimes\operatorname{{\mathbf{\Sigma}}}_{1}" class="ltx_Math" display="inline" id="S1.SS1a.p2.14.m4.1"><semantics id="S1.SS1a.p2.14.m4.1a"><mrow id="S1.SS1a.p2.14.m4.1.1" xref="S1.SS1a.p2.14.m4.1.1.cmml"><msub id="S1.SS1a.p2.14.m4.1.1.2" xref="S1.SS1a.p2.14.m4.1.1.2.cmml"><mi id="S1.SS1a.p2.14.m4.1.1.2.2" xref="S1.SS1a.p2.14.m4.1.1.2.2.cmml">𝚺</mi><mn id="S1.SS1a.p2.14.m4.1.1.2.3" xref="S1.SS1a.p2.14.m4.1.1.2.3.cmml">2</mn></msub><mo id="S1.SS1a.p2.14.m4.1.1.1" lspace="0em" rspace="0.222em" xref="S1.SS1a.p2.14.m4.1.1.1.cmml">⊗</mo><msub id="S1.SS1a.p2.14.m4.1.1.3" xref="S1.SS1a.p2.14.m4.1.1.3.cmml"><mi id="S1.SS1a.p2.14.m4.1.1.3.2" xref="S1.SS1a.p2.14.m4.1.1.3.2.cmml">𝚺</mi><mn id="S1.SS1a.p2.14.m4.1.1.3.3" xref="S1.SS1a.p2.14.m4.1.1.3.3.cmml">1</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S1.SS1a.p2.14.m4.1b"><apply id="S1.SS1a.p2.14.m4.1.1.cmml" xref="S1.SS1a.p2.14.m4.1.1"><csymbol cd="latexml" id="S1.SS1a.p2.14.m4.1.1.1.cmml" xref="S1.SS1a.p2.14.m4.1.1.1">tensor-product</csymbol><apply id="S1.SS1a.p2.14.m4.1.1.2.cmml" xref="S1.SS1a.p2.14.m4.1.1.2"><csymbol cd="ambiguous" id="S1.SS1a.p2.14.m4.1.1.2.1.cmml" xref="S1.SS1a.p2.14.m4.1.1.2">subscript</csymbol><ci id="S1.SS1a.p2.14.m4.1.1.2.2.cmml" xref="S1.SS1a.p2.14.m4.1.1.2.2">𝚺</ci><cn id="S1.SS1a.p2.14.m4.1.1.2.3.cmml" type="integer" xref="S1.SS1a.p2.14.m4.1.1.2.3">2</cn></apply><apply id="S1.SS1a.p2.14.m4.1.1.3.cmml" xref="S1.SS1a.p2.14.m4.1.1.3"><csymbol cd="ambiguous" id="S1.SS1a.p2.14.m4.1.1.3.1.cmml" xref="S1.SS1a.p2.14.m4.1.1.3">subscript</csymbol><ci id="S1.SS1a.p2.14.m4.1.1.3.2.cmml" xref="S1.SS1a.p2.14.m4.1.1.3.2">𝚺</ci><cn id="S1.SS1a.p2.14.m4.1.1.3.3.cmml" type="integer" xref="S1.SS1a.p2.14.m4.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1a.p2.14.m4.1c">\operatorname{{\mathbf{\Sigma}}}_{2}\otimes\operatorname{{\mathbf{\Sigma}}}_{1}</annotation><annotation encoding="application/x-llamapun" id="S1.SS1a.p2.14.m4.1d">bold_Σ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ⊗ bold_Σ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> is <math alttext="\operatorname{{\mathbf{\Sigma}}}_{1}" class="ltx_Math" display="inline" id="S1.SS1a.p2.15.m5.1"><semantics id="S1.SS1a.p2.15.m5.1a"><msub id="S1.SS1a.p2.15.m5.1.1" xref="S1.SS1a.p2.15.m5.1.1.cmml"><mi id="S1.SS1a.p2.15.m5.1.1.2" xref="S1.SS1a.p2.15.m5.1.1.2.cmml">𝚺</mi><mn id="S1.SS1a.p2.15.m5.1.1.3" xref="S1.SS1a.p2.15.m5.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S1.SS1a.p2.15.m5.1b"><apply id="S1.SS1a.p2.15.m5.1.1.cmml" xref="S1.SS1a.p2.15.m5.1.1"><csymbol cd="ambiguous" id="S1.SS1a.p2.15.m5.1.1.1.cmml" xref="S1.SS1a.p2.15.m5.1.1">subscript</csymbol><ci id="S1.SS1a.p2.15.m5.1.1.2.cmml" xref="S1.SS1a.p2.15.m5.1.1.2">𝚺</ci><cn id="S1.SS1a.p2.15.m5.1.1.3.cmml" type="integer" xref="S1.SS1a.p2.15.m5.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS1a.p2.15.m5.1c">\operatorname{{\mathbf{\Sigma}}}_{1}</annotation><annotation encoding="application/x-llamapun" id="S1.SS1a.p2.15.m5.1d">bold_Σ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math>. <span class="ltx_rule" style="width:6.5pt;height:6.5pt;background:black;display:inline-block;"> </span> <br class="ltx_break"/></p> </div> </section> <section class="ltx_subsection" id="S1.SS2a"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">1.2 </span>Proof of Proposition 2.2</h3> <div class="ltx_para" id="S1.SS2a.p1"> <p class="ltx_p" id="S1.SS2a.p1.1">Recall that Proposition 2.2 from the main manuscript.</p> </div> <div class="ltx_theorem ltx_theorem_proposition" id="S1.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="S1.Thmproposition2.1.1.1">Proposition 1.2</span></span></h6> <div class="ltx_para" id="S1.Thmproposition2.p1"> <p class="ltx_p" id="S1.Thmproposition2.p1.6"><span class="ltx_text ltx_font_italic" id="S1.Thmproposition2.p1.6.6">For any matrix <math alttext="\operatorname{\mathbf{L}}\in\operatorname{\mathbb{R}}^{Q\times S}" class="ltx_Math" display="inline" id="S1.Thmproposition2.p1.1.1.m1.1"><semantics id="S1.Thmproposition2.p1.1.1.m1.1a"><mrow id="S1.Thmproposition2.p1.1.1.m1.1.1" xref="S1.Thmproposition2.p1.1.1.m1.1.1.cmml"><mi id="S1.Thmproposition2.p1.1.1.m1.1.1.2" xref="S1.Thmproposition2.p1.1.1.m1.1.1.2.cmml">𝐋</mi><mo id="S1.Thmproposition2.p1.1.1.m1.1.1.1" xref="S1.Thmproposition2.p1.1.1.m1.1.1.1.cmml">∈</mo><msup id="S1.Thmproposition2.p1.1.1.m1.1.1.3" xref="S1.Thmproposition2.p1.1.1.m1.1.1.3.cmml"><mi id="S1.Thmproposition2.p1.1.1.m1.1.1.3.2" xref="S1.Thmproposition2.p1.1.1.m1.1.1.3.2.cmml">ℝ</mi><mrow id="S1.Thmproposition2.p1.1.1.m1.1.1.3.3" xref="S1.Thmproposition2.p1.1.1.m1.1.1.3.3.cmml"><mi id="S1.Thmproposition2.p1.1.1.m1.1.1.3.3.2" xref="S1.Thmproposition2.p1.1.1.m1.1.1.3.3.2.cmml">Q</mi><mo id="S1.Thmproposition2.p1.1.1.m1.1.1.3.3.1" lspace="0.222em" rspace="0.222em" xref="S1.Thmproposition2.p1.1.1.m1.1.1.3.3.1.cmml">×</mo><mi id="S1.Thmproposition2.p1.1.1.m1.1.1.3.3.3" xref="S1.Thmproposition2.p1.1.1.m1.1.1.3.3.3.cmml">S</mi></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="S1.Thmproposition2.p1.1.1.m1.1b"><apply id="S1.Thmproposition2.p1.1.1.m1.1.1.cmml" xref="S1.Thmproposition2.p1.1.1.m1.1.1"><in id="S1.Thmproposition2.p1.1.1.m1.1.1.1.cmml" xref="S1.Thmproposition2.p1.1.1.m1.1.1.1"></in><ci id="S1.Thmproposition2.p1.1.1.m1.1.1.2.cmml" xref="S1.Thmproposition2.p1.1.1.m1.1.1.2">𝐋</ci><apply id="S1.Thmproposition2.p1.1.1.m1.1.1.3.cmml" xref="S1.Thmproposition2.p1.1.1.m1.1.1.3"><csymbol cd="ambiguous" id="S1.Thmproposition2.p1.1.1.m1.1.1.3.1.cmml" xref="S1.Thmproposition2.p1.1.1.m1.1.1.3">superscript</csymbol><ci id="S1.Thmproposition2.p1.1.1.m1.1.1.3.2.cmml" xref="S1.Thmproposition2.p1.1.1.m1.1.1.3.2">ℝ</ci><apply id="S1.Thmproposition2.p1.1.1.m1.1.1.3.3.cmml" xref="S1.Thmproposition2.p1.1.1.m1.1.1.3.3"><times id="S1.Thmproposition2.p1.1.1.m1.1.1.3.3.1.cmml" xref="S1.Thmproposition2.p1.1.1.m1.1.1.3.3.1"></times><ci id="S1.Thmproposition2.p1.1.1.m1.1.1.3.3.2.cmml" xref="S1.Thmproposition2.p1.1.1.m1.1.1.3.3.2">𝑄</ci><ci id="S1.Thmproposition2.p1.1.1.m1.1.1.3.3.3.cmml" xref="S1.Thmproposition2.p1.1.1.m1.1.1.3.3.3">𝑆</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmproposition2.p1.1.1.m1.1c">\operatorname{\mathbf{L}}\in\operatorname{\mathbb{R}}^{Q\times S}</annotation><annotation encoding="application/x-llamapun" id="S1.Thmproposition2.p1.1.1.m1.1d">bold_L ∈ blackboard_R start_POSTSUPERSCRIPT italic_Q × italic_S end_POSTSUPERSCRIPT</annotation></semantics></math>, there exists an orthonormal matrix <math alttext="\operatorname{\mathbf{Q}}_{j}\in\operatorname{\mathbb{R}}^{S\times S}" class="ltx_Math" display="inline" id="S1.Thmproposition2.p1.2.2.m2.1"><semantics id="S1.Thmproposition2.p1.2.2.m2.1a"><mrow id="S1.Thmproposition2.p1.2.2.m2.1.1" xref="S1.Thmproposition2.p1.2.2.m2.1.1.cmml"><msub id="S1.Thmproposition2.p1.2.2.m2.1.1.2" xref="S1.Thmproposition2.p1.2.2.m2.1.1.2.cmml"><mi id="S1.Thmproposition2.p1.2.2.m2.1.1.2.2" xref="S1.Thmproposition2.p1.2.2.m2.1.1.2.2.cmml">𝐐</mi><mi id="S1.Thmproposition2.p1.2.2.m2.1.1.2.3" xref="S1.Thmproposition2.p1.2.2.m2.1.1.2.3.cmml">j</mi></msub><mo id="S1.Thmproposition2.p1.2.2.m2.1.1.1" xref="S1.Thmproposition2.p1.2.2.m2.1.1.1.cmml">∈</mo><msup id="S1.Thmproposition2.p1.2.2.m2.1.1.3" xref="S1.Thmproposition2.p1.2.2.m2.1.1.3.cmml"><mi id="S1.Thmproposition2.p1.2.2.m2.1.1.3.2" xref="S1.Thmproposition2.p1.2.2.m2.1.1.3.2.cmml">ℝ</mi><mrow id="S1.Thmproposition2.p1.2.2.m2.1.1.3.3" xref="S1.Thmproposition2.p1.2.2.m2.1.1.3.3.cmml"><mi id="S1.Thmproposition2.p1.2.2.m2.1.1.3.3.2" xref="S1.Thmproposition2.p1.2.2.m2.1.1.3.3.2.cmml">S</mi><mo id="S1.Thmproposition2.p1.2.2.m2.1.1.3.3.1" lspace="0.222em" rspace="0.222em" xref="S1.Thmproposition2.p1.2.2.m2.1.1.3.3.1.cmml">×</mo><mi id="S1.Thmproposition2.p1.2.2.m2.1.1.3.3.3" xref="S1.Thmproposition2.p1.2.2.m2.1.1.3.3.3.cmml">S</mi></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="S1.Thmproposition2.p1.2.2.m2.1b"><apply id="S1.Thmproposition2.p1.2.2.m2.1.1.cmml" xref="S1.Thmproposition2.p1.2.2.m2.1.1"><in id="S1.Thmproposition2.p1.2.2.m2.1.1.1.cmml" xref="S1.Thmproposition2.p1.2.2.m2.1.1.1"></in><apply id="S1.Thmproposition2.p1.2.2.m2.1.1.2.cmml" xref="S1.Thmproposition2.p1.2.2.m2.1.1.2"><csymbol cd="ambiguous" id="S1.Thmproposition2.p1.2.2.m2.1.1.2.1.cmml" xref="S1.Thmproposition2.p1.2.2.m2.1.1.2">subscript</csymbol><ci id="S1.Thmproposition2.p1.2.2.m2.1.1.2.2.cmml" xref="S1.Thmproposition2.p1.2.2.m2.1.1.2.2">𝐐</ci><ci id="S1.Thmproposition2.p1.2.2.m2.1.1.2.3.cmml" xref="S1.Thmproposition2.p1.2.2.m2.1.1.2.3">𝑗</ci></apply><apply id="S1.Thmproposition2.p1.2.2.m2.1.1.3.cmml" xref="S1.Thmproposition2.p1.2.2.m2.1.1.3"><csymbol cd="ambiguous" id="S1.Thmproposition2.p1.2.2.m2.1.1.3.1.cmml" xref="S1.Thmproposition2.p1.2.2.m2.1.1.3">superscript</csymbol><ci id="S1.Thmproposition2.p1.2.2.m2.1.1.3.2.cmml" xref="S1.Thmproposition2.p1.2.2.m2.1.1.3.2">ℝ</ci><apply id="S1.Thmproposition2.p1.2.2.m2.1.1.3.3.cmml" xref="S1.Thmproposition2.p1.2.2.m2.1.1.3.3"><times id="S1.Thmproposition2.p1.2.2.m2.1.1.3.3.1.cmml" xref="S1.Thmproposition2.p1.2.2.m2.1.1.3.3.1"></times><ci id="S1.Thmproposition2.p1.2.2.m2.1.1.3.3.2.cmml" xref="S1.Thmproposition2.p1.2.2.m2.1.1.3.3.2">𝑆</ci><ci id="S1.Thmproposition2.p1.2.2.m2.1.1.3.3.3.cmml" xref="S1.Thmproposition2.p1.2.2.m2.1.1.3.3.3">𝑆</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmproposition2.p1.2.2.m2.1c">\operatorname{\mathbf{Q}}_{j}\in\operatorname{\mathbb{R}}^{S\times S}</annotation><annotation encoding="application/x-llamapun" id="S1.Thmproposition2.p1.2.2.m2.1d">bold_Q start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ∈ blackboard_R start_POSTSUPERSCRIPT italic_S × italic_S end_POSTSUPERSCRIPT</annotation></semantics></math> and constant <math alttext="c" class="ltx_Math" display="inline" id="S1.Thmproposition2.p1.3.3.m3.1"><semantics id="S1.Thmproposition2.p1.3.3.m3.1a"><mi id="S1.Thmproposition2.p1.3.3.m3.1.1" xref="S1.Thmproposition2.p1.3.3.m3.1.1.cmml">c</mi><annotation-xml encoding="MathML-Content" id="S1.Thmproposition2.p1.3.3.m3.1b"><ci id="S1.Thmproposition2.p1.3.3.m3.1.1.cmml" xref="S1.Thmproposition2.p1.3.3.m3.1.1">𝑐</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmproposition2.p1.3.3.m3.1c">c</annotation><annotation encoding="application/x-llamapun" id="S1.Thmproposition2.p1.3.3.m3.1d">italic_c</annotation></semantics></math> such that <math alttext="c\operatorname{\mathbf{L}}\operatorname{\mathbf{Q}}_{j}^{\top}" class="ltx_Math" display="inline" id="S1.Thmproposition2.p1.4.4.m4.1"><semantics id="S1.Thmproposition2.p1.4.4.m4.1a"><mrow id="S1.Thmproposition2.p1.4.4.m4.1.1" xref="S1.Thmproposition2.p1.4.4.m4.1.1.cmml"><mi id="S1.Thmproposition2.p1.4.4.m4.1.1.2" xref="S1.Thmproposition2.p1.4.4.m4.1.1.2.cmml">c</mi><mo id="S1.Thmproposition2.p1.4.4.m4.1.1.1" lspace="0.167em" xref="S1.Thmproposition2.p1.4.4.m4.1.1.1.cmml">⁢</mo><mrow id="S1.Thmproposition2.p1.4.4.m4.1.1.3" xref="S1.Thmproposition2.p1.4.4.m4.1.1.3.cmml"><mi id="S1.Thmproposition2.p1.4.4.m4.1.1.3.1" xref="S1.Thmproposition2.p1.4.4.m4.1.1.3.1.cmml">𝐋</mi><mo id="S1.Thmproposition2.p1.4.4.m4.1.1.3a" lspace="0.167em" xref="S1.Thmproposition2.p1.4.4.m4.1.1.3.cmml">⁡</mo><msubsup id="S1.Thmproposition2.p1.4.4.m4.1.1.3.2" xref="S1.Thmproposition2.p1.4.4.m4.1.1.3.2.cmml"><mi id="S1.Thmproposition2.p1.4.4.m4.1.1.3.2.2.2" xref="S1.Thmproposition2.p1.4.4.m4.1.1.3.2.2.2.cmml">𝐐</mi><mi id="S1.Thmproposition2.p1.4.4.m4.1.1.3.2.2.3" xref="S1.Thmproposition2.p1.4.4.m4.1.1.3.2.2.3.cmml">j</mi><mo id="S1.Thmproposition2.p1.4.4.m4.1.1.3.2.3" xref="S1.Thmproposition2.p1.4.4.m4.1.1.3.2.3.cmml">⊤</mo></msubsup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.Thmproposition2.p1.4.4.m4.1b"><apply id="S1.Thmproposition2.p1.4.4.m4.1.1.cmml" xref="S1.Thmproposition2.p1.4.4.m4.1.1"><times id="S1.Thmproposition2.p1.4.4.m4.1.1.1.cmml" xref="S1.Thmproposition2.p1.4.4.m4.1.1.1"></times><ci id="S1.Thmproposition2.p1.4.4.m4.1.1.2.cmml" xref="S1.Thmproposition2.p1.4.4.m4.1.1.2">𝑐</ci><apply id="S1.Thmproposition2.p1.4.4.m4.1.1.3.cmml" xref="S1.Thmproposition2.p1.4.4.m4.1.1.3"><ci id="S1.Thmproposition2.p1.4.4.m4.1.1.3.1.cmml" xref="S1.Thmproposition2.p1.4.4.m4.1.1.3.1">𝐋</ci><apply id="S1.Thmproposition2.p1.4.4.m4.1.1.3.2.cmml" xref="S1.Thmproposition2.p1.4.4.m4.1.1.3.2"><csymbol cd="ambiguous" id="S1.Thmproposition2.p1.4.4.m4.1.1.3.2.1.cmml" xref="S1.Thmproposition2.p1.4.4.m4.1.1.3.2">superscript</csymbol><apply id="S1.Thmproposition2.p1.4.4.m4.1.1.3.2.2.cmml" xref="S1.Thmproposition2.p1.4.4.m4.1.1.3.2"><csymbol cd="ambiguous" id="S1.Thmproposition2.p1.4.4.m4.1.1.3.2.2.1.cmml" xref="S1.Thmproposition2.p1.4.4.m4.1.1.3.2">subscript</csymbol><ci id="S1.Thmproposition2.p1.4.4.m4.1.1.3.2.2.2.cmml" xref="S1.Thmproposition2.p1.4.4.m4.1.1.3.2.2.2">𝐐</ci><ci id="S1.Thmproposition2.p1.4.4.m4.1.1.3.2.2.3.cmml" xref="S1.Thmproposition2.p1.4.4.m4.1.1.3.2.2.3">𝑗</ci></apply><csymbol cd="latexml" id="S1.Thmproposition2.p1.4.4.m4.1.1.3.2.3.cmml" xref="S1.Thmproposition2.p1.4.4.m4.1.1.3.2.3">top</csymbol></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmproposition2.p1.4.4.m4.1c">c\operatorname{\mathbf{L}}\operatorname{\mathbf{Q}}_{j}^{\top}</annotation><annotation encoding="application/x-llamapun" id="S1.Thmproposition2.p1.4.4.m4.1d">italic_c bold_L bold_Q start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ⊤ end_POSTSUPERSCRIPT</annotation></semantics></math> is a matrix whose <math alttext="j" class="ltx_Math" display="inline" id="S1.Thmproposition2.p1.5.5.m5.1"><semantics id="S1.Thmproposition2.p1.5.5.m5.1a"><mi id="S1.Thmproposition2.p1.5.5.m5.1.1" xref="S1.Thmproposition2.p1.5.5.m5.1.1.cmml">j</mi><annotation-xml encoding="MathML-Content" id="S1.Thmproposition2.p1.5.5.m5.1b"><ci id="S1.Thmproposition2.p1.5.5.m5.1.1.cmml" xref="S1.Thmproposition2.p1.5.5.m5.1.1">𝑗</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmproposition2.p1.5.5.m5.1c">j</annotation><annotation encoding="application/x-llamapun" id="S1.Thmproposition2.p1.5.5.m5.1d">italic_j</annotation></semantics></math>th row is the first standard basis vector in <math alttext="\operatorname{\mathbb{R}}^{S}" class="ltx_Math" display="inline" id="S1.Thmproposition2.p1.6.6.m6.1"><semantics id="S1.Thmproposition2.p1.6.6.m6.1a"><msup id="S1.Thmproposition2.p1.6.6.m6.1.1" xref="S1.Thmproposition2.p1.6.6.m6.1.1.cmml"><mi id="S1.Thmproposition2.p1.6.6.m6.1.1.2" xref="S1.Thmproposition2.p1.6.6.m6.1.1.2.cmml">ℝ</mi><mi id="S1.Thmproposition2.p1.6.6.m6.1.1.3" xref="S1.Thmproposition2.p1.6.6.m6.1.1.3.cmml">S</mi></msup><annotation-xml encoding="MathML-Content" id="S1.Thmproposition2.p1.6.6.m6.1b"><apply id="S1.Thmproposition2.p1.6.6.m6.1.1.cmml" xref="S1.Thmproposition2.p1.6.6.m6.1.1"><csymbol cd="ambiguous" id="S1.Thmproposition2.p1.6.6.m6.1.1.1.cmml" xref="S1.Thmproposition2.p1.6.6.m6.1.1">superscript</csymbol><ci id="S1.Thmproposition2.p1.6.6.m6.1.1.2.cmml" xref="S1.Thmproposition2.p1.6.6.m6.1.1.2">ℝ</ci><ci id="S1.Thmproposition2.p1.6.6.m6.1.1.3.cmml" xref="S1.Thmproposition2.p1.6.6.m6.1.1.3">𝑆</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmproposition2.p1.6.6.m6.1c">\operatorname{\mathbb{R}}^{S}</annotation><annotation encoding="application/x-llamapun" id="S1.Thmproposition2.p1.6.6.m6.1d">blackboard_R start_POSTSUPERSCRIPT italic_S end_POSTSUPERSCRIPT</annotation></semantics></math>.</span></p> </div> </div> <div class="ltx_para ltx_noindent" id="S1.SS2a.p2"> <p class="ltx_p" id="S1.SS2a.p2.11"><span class="ltx_text ltx_font_bold" id="S1.SS2a.p2.11.1">Proof </span>[Proof of Proposition <a class="ltx_ref" href="https://arxiv.org/html/2503.13782v1#S1.Thmproposition2" title="Proposition 1.2 ‣ 1.2 Proof of Proposition 2.2 ‣ 1 Proof of the Propositions ‣ Regularized Parameter Estimation in Mixed Model Trace Regression"><span class="ltx_text ltx_ref_tag">1.2</span></a>] Let <math alttext="\operatorname{\mathbf{L}}" class="ltx_Math" display="inline" id="S1.SS2a.p2.1.m1.1"><semantics id="S1.SS2a.p2.1.m1.1a"><mi id="S1.SS2a.p2.1.m1.1.1" xref="S1.SS2a.p2.1.m1.1.1.cmml">𝐋</mi><annotation-xml encoding="MathML-Content" id="S1.SS2a.p2.1.m1.1b"><ci id="S1.SS2a.p2.1.m1.1.1.cmml" xref="S1.SS2a.p2.1.m1.1.1">𝐋</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2a.p2.1.m1.1c">\operatorname{\mathbf{L}}</annotation><annotation encoding="application/x-llamapun" id="S1.SS2a.p2.1.m1.1d">bold_L</annotation></semantics></math> be any matrix in <math alttext="\mathbb{R}^{Q\times S}" class="ltx_Math" display="inline" id="S1.SS2a.p2.2.m2.1"><semantics id="S1.SS2a.p2.2.m2.1a"><msup id="S1.SS2a.p2.2.m2.1.1" xref="S1.SS2a.p2.2.m2.1.1.cmml"><mi id="S1.SS2a.p2.2.m2.1.1.2" xref="S1.SS2a.p2.2.m2.1.1.2.cmml">ℝ</mi><mrow id="S1.SS2a.p2.2.m2.1.1.3" xref="S1.SS2a.p2.2.m2.1.1.3.cmml"><mi id="S1.SS2a.p2.2.m2.1.1.3.2" xref="S1.SS2a.p2.2.m2.1.1.3.2.cmml">Q</mi><mo id="S1.SS2a.p2.2.m2.1.1.3.1" lspace="0.222em" rspace="0.222em" xref="S1.SS2a.p2.2.m2.1.1.3.1.cmml">×</mo><mi id="S1.SS2a.p2.2.m2.1.1.3.3" xref="S1.SS2a.p2.2.m2.1.1.3.3.cmml">S</mi></mrow></msup><annotation-xml encoding="MathML-Content" id="S1.SS2a.p2.2.m2.1b"><apply id="S1.SS2a.p2.2.m2.1.1.cmml" xref="S1.SS2a.p2.2.m2.1.1"><csymbol cd="ambiguous" id="S1.SS2a.p2.2.m2.1.1.1.cmml" xref="S1.SS2a.p2.2.m2.1.1">superscript</csymbol><ci id="S1.SS2a.p2.2.m2.1.1.2.cmml" xref="S1.SS2a.p2.2.m2.1.1.2">ℝ</ci><apply id="S1.SS2a.p2.2.m2.1.1.3.cmml" xref="S1.SS2a.p2.2.m2.1.1.3"><times id="S1.SS2a.p2.2.m2.1.1.3.1.cmml" xref="S1.SS2a.p2.2.m2.1.1.3.1"></times><ci id="S1.SS2a.p2.2.m2.1.1.3.2.cmml" xref="S1.SS2a.p2.2.m2.1.1.3.2">𝑄</ci><ci id="S1.SS2a.p2.2.m2.1.1.3.3.cmml" xref="S1.SS2a.p2.2.m2.1.1.3.3">𝑆</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2a.p2.2.m2.1c">\mathbb{R}^{Q\times S}</annotation><annotation encoding="application/x-llamapun" id="S1.SS2a.p2.2.m2.1d">blackboard_R start_POSTSUPERSCRIPT italic_Q × italic_S end_POSTSUPERSCRIPT</annotation></semantics></math> such that <math alttext="\operatorname{\mathbf{L}}\operatorname{\mathbf{L}}^{\top}=\operatorname{{% \mathbf{\Sigma}}}" class="ltx_Math" display="inline" id="S1.SS2a.p2.3.m3.1"><semantics id="S1.SS2a.p2.3.m3.1a"><mrow id="S1.SS2a.p2.3.m3.1.1" xref="S1.SS2a.p2.3.m3.1.1.cmml"><mrow id="S1.SS2a.p2.3.m3.1.1.2" xref="S1.SS2a.p2.3.m3.1.1.2.cmml"><mi id="S1.SS2a.p2.3.m3.1.1.2.1" xref="S1.SS2a.p2.3.m3.1.1.2.1.cmml">𝐋</mi><mo id="S1.SS2a.p2.3.m3.1.1.2a" lspace="0.167em" xref="S1.SS2a.p2.3.m3.1.1.2.cmml">⁡</mo><msup id="S1.SS2a.p2.3.m3.1.1.2.2" xref="S1.SS2a.p2.3.m3.1.1.2.2.cmml"><mi id="S1.SS2a.p2.3.m3.1.1.2.2.2" xref="S1.SS2a.p2.3.m3.1.1.2.2.2.cmml">𝐋</mi><mo id="S1.SS2a.p2.3.m3.1.1.2.2.3" xref="S1.SS2a.p2.3.m3.1.1.2.2.3.cmml">⊤</mo></msup></mrow><mo id="S1.SS2a.p2.3.m3.1.1.1" xref="S1.SS2a.p2.3.m3.1.1.1.cmml">=</mo><mi id="S1.SS2a.p2.3.m3.1.1.3" xref="S1.SS2a.p2.3.m3.1.1.3.cmml">𝚺</mi></mrow><annotation-xml encoding="MathML-Content" id="S1.SS2a.p2.3.m3.1b"><apply id="S1.SS2a.p2.3.m3.1.1.cmml" xref="S1.SS2a.p2.3.m3.1.1"><eq id="S1.SS2a.p2.3.m3.1.1.1.cmml" xref="S1.SS2a.p2.3.m3.1.1.1"></eq><apply id="S1.SS2a.p2.3.m3.1.1.2.cmml" xref="S1.SS2a.p2.3.m3.1.1.2"><ci id="S1.SS2a.p2.3.m3.1.1.2.1.cmml" xref="S1.SS2a.p2.3.m3.1.1.2.1">𝐋</ci><apply id="S1.SS2a.p2.3.m3.1.1.2.2.cmml" xref="S1.SS2a.p2.3.m3.1.1.2.2"><csymbol cd="ambiguous" id="S1.SS2a.p2.3.m3.1.1.2.2.1.cmml" xref="S1.SS2a.p2.3.m3.1.1.2.2">superscript</csymbol><ci id="S1.SS2a.p2.3.m3.1.1.2.2.2.cmml" xref="S1.SS2a.p2.3.m3.1.1.2.2.2">𝐋</ci><csymbol cd="latexml" id="S1.SS2a.p2.3.m3.1.1.2.2.3.cmml" xref="S1.SS2a.p2.3.m3.1.1.2.2.3">top</csymbol></apply></apply><ci id="S1.SS2a.p2.3.m3.1.1.3.cmml" xref="S1.SS2a.p2.3.m3.1.1.3">𝚺</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2a.p2.3.m3.1c">\operatorname{\mathbf{L}}\operatorname{\mathbf{L}}^{\top}=\operatorname{{% \mathbf{\Sigma}}}</annotation><annotation encoding="application/x-llamapun" id="S1.SS2a.p2.3.m3.1d">bold_L bold_L start_POSTSUPERSCRIPT ⊤ end_POSTSUPERSCRIPT = bold_Σ</annotation></semantics></math>, and let <math alttext="\operatorname{\mathbf{Q}}_{j}\in\operatorname{\mathbb{R}}^{S\times S}" class="ltx_Math" display="inline" id="S1.SS2a.p2.4.m4.1"><semantics id="S1.SS2a.p2.4.m4.1a"><mrow id="S1.SS2a.p2.4.m4.1.1" xref="S1.SS2a.p2.4.m4.1.1.cmml"><msub id="S1.SS2a.p2.4.m4.1.1.2" xref="S1.SS2a.p2.4.m4.1.1.2.cmml"><mi id="S1.SS2a.p2.4.m4.1.1.2.2" xref="S1.SS2a.p2.4.m4.1.1.2.2.cmml">𝐐</mi><mi id="S1.SS2a.p2.4.m4.1.1.2.3" xref="S1.SS2a.p2.4.m4.1.1.2.3.cmml">j</mi></msub><mo id="S1.SS2a.p2.4.m4.1.1.1" xref="S1.SS2a.p2.4.m4.1.1.1.cmml">∈</mo><msup id="S1.SS2a.p2.4.m4.1.1.3" xref="S1.SS2a.p2.4.m4.1.1.3.cmml"><mi id="S1.SS2a.p2.4.m4.1.1.3.2" xref="S1.SS2a.p2.4.m4.1.1.3.2.cmml">ℝ</mi><mrow id="S1.SS2a.p2.4.m4.1.1.3.3" xref="S1.SS2a.p2.4.m4.1.1.3.3.cmml"><mi id="S1.SS2a.p2.4.m4.1.1.3.3.2" xref="S1.SS2a.p2.4.m4.1.1.3.3.2.cmml">S</mi><mo id="S1.SS2a.p2.4.m4.1.1.3.3.1" lspace="0.222em" rspace="0.222em" xref="S1.SS2a.p2.4.m4.1.1.3.3.1.cmml">×</mo><mi id="S1.SS2a.p2.4.m4.1.1.3.3.3" xref="S1.SS2a.p2.4.m4.1.1.3.3.3.cmml">S</mi></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="S1.SS2a.p2.4.m4.1b"><apply id="S1.SS2a.p2.4.m4.1.1.cmml" xref="S1.SS2a.p2.4.m4.1.1"><in id="S1.SS2a.p2.4.m4.1.1.1.cmml" xref="S1.SS2a.p2.4.m4.1.1.1"></in><apply id="S1.SS2a.p2.4.m4.1.1.2.cmml" xref="S1.SS2a.p2.4.m4.1.1.2"><csymbol cd="ambiguous" id="S1.SS2a.p2.4.m4.1.1.2.1.cmml" xref="S1.SS2a.p2.4.m4.1.1.2">subscript</csymbol><ci id="S1.SS2a.p2.4.m4.1.1.2.2.cmml" xref="S1.SS2a.p2.4.m4.1.1.2.2">𝐐</ci><ci id="S1.SS2a.p2.4.m4.1.1.2.3.cmml" xref="S1.SS2a.p2.4.m4.1.1.2.3">𝑗</ci></apply><apply id="S1.SS2a.p2.4.m4.1.1.3.cmml" xref="S1.SS2a.p2.4.m4.1.1.3"><csymbol cd="ambiguous" id="S1.SS2a.p2.4.m4.1.1.3.1.cmml" xref="S1.SS2a.p2.4.m4.1.1.3">superscript</csymbol><ci id="S1.SS2a.p2.4.m4.1.1.3.2.cmml" xref="S1.SS2a.p2.4.m4.1.1.3.2">ℝ</ci><apply id="S1.SS2a.p2.4.m4.1.1.3.3.cmml" xref="S1.SS2a.p2.4.m4.1.1.3.3"><times id="S1.SS2a.p2.4.m4.1.1.3.3.1.cmml" xref="S1.SS2a.p2.4.m4.1.1.3.3.1"></times><ci id="S1.SS2a.p2.4.m4.1.1.3.3.2.cmml" xref="S1.SS2a.p2.4.m4.1.1.3.3.2">𝑆</ci><ci id="S1.SS2a.p2.4.m4.1.1.3.3.3.cmml" xref="S1.SS2a.p2.4.m4.1.1.3.3.3">𝑆</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2a.p2.4.m4.1c">\operatorname{\mathbf{Q}}_{j}\in\operatorname{\mathbb{R}}^{S\times S}</annotation><annotation encoding="application/x-llamapun" id="S1.SS2a.p2.4.m4.1d">bold_Q start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ∈ blackboard_R start_POSTSUPERSCRIPT italic_S × italic_S end_POSTSUPERSCRIPT</annotation></semantics></math> be the Householder matrix that reflects the <math alttext="j" class="ltx_Math" display="inline" id="S1.SS2a.p2.5.m5.1"><semantics id="S1.SS2a.p2.5.m5.1a"><mi id="S1.SS2a.p2.5.m5.1.1" xref="S1.SS2a.p2.5.m5.1.1.cmml">j</mi><annotation-xml encoding="MathML-Content" id="S1.SS2a.p2.5.m5.1b"><ci id="S1.SS2a.p2.5.m5.1.1.cmml" xref="S1.SS2a.p2.5.m5.1.1">𝑗</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2a.p2.5.m5.1c">j</annotation><annotation encoding="application/x-llamapun" id="S1.SS2a.p2.5.m5.1d">italic_j</annotation></semantics></math>th row vector of <math alttext="\operatorname{\mathbf{L}}" class="ltx_Math" display="inline" id="S1.SS2a.p2.6.m6.1"><semantics id="S1.SS2a.p2.6.m6.1a"><mi id="S1.SS2a.p2.6.m6.1.1" xref="S1.SS2a.p2.6.m6.1.1.cmml">𝐋</mi><annotation-xml encoding="MathML-Content" id="S1.SS2a.p2.6.m6.1b"><ci id="S1.SS2a.p2.6.m6.1.1.cmml" xref="S1.SS2a.p2.6.m6.1.1">𝐋</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2a.p2.6.m6.1c">\operatorname{\mathbf{L}}</annotation><annotation encoding="application/x-llamapun" id="S1.SS2a.p2.6.m6.1d">bold_L</annotation></semantics></math> onto the first standard basis vector in <math alttext="\mathbb{R}^{S}" class="ltx_Math" display="inline" id="S1.SS2a.p2.7.m7.1"><semantics id="S1.SS2a.p2.7.m7.1a"><msup id="S1.SS2a.p2.7.m7.1.1" xref="S1.SS2a.p2.7.m7.1.1.cmml"><mi id="S1.SS2a.p2.7.m7.1.1.2" xref="S1.SS2a.p2.7.m7.1.1.2.cmml">ℝ</mi><mi id="S1.SS2a.p2.7.m7.1.1.3" xref="S1.SS2a.p2.7.m7.1.1.3.cmml">S</mi></msup><annotation-xml encoding="MathML-Content" id="S1.SS2a.p2.7.m7.1b"><apply id="S1.SS2a.p2.7.m7.1.1.cmml" xref="S1.SS2a.p2.7.m7.1.1"><csymbol cd="ambiguous" id="S1.SS2a.p2.7.m7.1.1.1.cmml" xref="S1.SS2a.p2.7.m7.1.1">superscript</csymbol><ci id="S1.SS2a.p2.7.m7.1.1.2.cmml" xref="S1.SS2a.p2.7.m7.1.1.2">ℝ</ci><ci id="S1.SS2a.p2.7.m7.1.1.3.cmml" xref="S1.SS2a.p2.7.m7.1.1.3">𝑆</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2a.p2.7.m7.1c">\mathbb{R}^{S}</annotation><annotation encoding="application/x-llamapun" id="S1.SS2a.p2.7.m7.1d">blackboard_R start_POSTSUPERSCRIPT italic_S end_POSTSUPERSCRIPT</annotation></semantics></math>, denoted as <math alttext="\operatorname{\mathbf{e}}_{1}" class="ltx_Math" display="inline" id="S1.SS2a.p2.8.m8.1"><semantics id="S1.SS2a.p2.8.m8.1a"><msub id="S1.SS2a.p2.8.m8.1.1" xref="S1.SS2a.p2.8.m8.1.1.cmml"><mi id="S1.SS2a.p2.8.m8.1.1.2" xref="S1.SS2a.p2.8.m8.1.1.2.cmml">𝐞</mi><mn id="S1.SS2a.p2.8.m8.1.1.3" xref="S1.SS2a.p2.8.m8.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S1.SS2a.p2.8.m8.1b"><apply id="S1.SS2a.p2.8.m8.1.1.cmml" xref="S1.SS2a.p2.8.m8.1.1"><csymbol cd="ambiguous" id="S1.SS2a.p2.8.m8.1.1.1.cmml" xref="S1.SS2a.p2.8.m8.1.1">subscript</csymbol><ci id="S1.SS2a.p2.8.m8.1.1.2.cmml" xref="S1.SS2a.p2.8.m8.1.1.2">𝐞</ci><cn id="S1.SS2a.p2.8.m8.1.1.3.cmml" type="integer" xref="S1.SS2a.p2.8.m8.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2a.p2.8.m8.1c">\operatorname{\mathbf{e}}_{1}</annotation><annotation encoding="application/x-llamapun" id="S1.SS2a.p2.8.m8.1d">bold_e start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math>. Let <math alttext="\operatorname{\mathbf{l}}_{i:}" class="ltx_Math" display="inline" id="S1.SS2a.p2.9.m9.1"><semantics id="S1.SS2a.p2.9.m9.1a"><msub id="S1.SS2a.p2.9.m9.1.1" xref="S1.SS2a.p2.9.m9.1.1.cmml"><mi id="S1.SS2a.p2.9.m9.1.1.2" xref="S1.SS2a.p2.9.m9.1.1.2.cmml">𝐥</mi><mrow id="S1.SS2a.p2.9.m9.1.1.3" xref="S1.SS2a.p2.9.m9.1.1.3.cmml"><mi id="S1.SS2a.p2.9.m9.1.1.3.2" xref="S1.SS2a.p2.9.m9.1.1.3.2.cmml">i</mi><mo id="S1.SS2a.p2.9.m9.1.1.3.1" lspace="0.278em" rspace="0.278em" xref="S1.SS2a.p2.9.m9.1.1.3.1.cmml">:</mo><mi id="S1.SS2a.p2.9.m9.1.1.3.3" xref="S1.SS2a.p2.9.m9.1.1.3.3.cmml"></mi></mrow></msub><annotation-xml encoding="MathML-Content" id="S1.SS2a.p2.9.m9.1b"><apply id="S1.SS2a.p2.9.m9.1.1.cmml" xref="S1.SS2a.p2.9.m9.1.1"><csymbol cd="ambiguous" id="S1.SS2a.p2.9.m9.1.1.1.cmml" xref="S1.SS2a.p2.9.m9.1.1">subscript</csymbol><ci id="S1.SS2a.p2.9.m9.1.1.2.cmml" xref="S1.SS2a.p2.9.m9.1.1.2">𝐥</ci><apply id="S1.SS2a.p2.9.m9.1.1.3.cmml" xref="S1.SS2a.p2.9.m9.1.1.3"><ci id="S1.SS2a.p2.9.m9.1.1.3.1.cmml" xref="S1.SS2a.p2.9.m9.1.1.3.1">:</ci><ci id="S1.SS2a.p2.9.m9.1.1.3.2.cmml" xref="S1.SS2a.p2.9.m9.1.1.3.2">𝑖</ci><csymbol cd="latexml" id="S1.SS2a.p2.9.m9.1.1.3.3.cmml" xref="S1.SS2a.p2.9.m9.1.1.3.3">absent</csymbol></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2a.p2.9.m9.1c">\operatorname{\mathbf{l}}_{i:}</annotation><annotation encoding="application/x-llamapun" id="S1.SS2a.p2.9.m9.1d">bold_l start_POSTSUBSCRIPT italic_i : end_POSTSUBSCRIPT</annotation></semantics></math> denote the <math alttext="i" class="ltx_Math" display="inline" id="S1.SS2a.p2.10.m10.1"><semantics id="S1.SS2a.p2.10.m10.1a"><mi id="S1.SS2a.p2.10.m10.1.1" xref="S1.SS2a.p2.10.m10.1.1.cmml">i</mi><annotation-xml encoding="MathML-Content" id="S1.SS2a.p2.10.m10.1b"><ci id="S1.SS2a.p2.10.m10.1.1.cmml" xref="S1.SS2a.p2.10.m10.1.1">𝑖</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2a.p2.10.m10.1c">i</annotation><annotation encoding="application/x-llamapun" id="S1.SS2a.p2.10.m10.1d">italic_i</annotation></semantics></math>th row vector of <math alttext="\operatorname{\mathbf{L}}" class="ltx_Math" display="inline" id="S1.SS2a.p2.11.m11.1"><semantics id="S1.SS2a.p2.11.m11.1a"><mi id="S1.SS2a.p2.11.m11.1.1" xref="S1.SS2a.p2.11.m11.1.1.cmml">𝐋</mi><annotation-xml encoding="MathML-Content" id="S1.SS2a.p2.11.m11.1b"><ci id="S1.SS2a.p2.11.m11.1.1.cmml" xref="S1.SS2a.p2.11.m11.1.1">𝐋</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2a.p2.11.m11.1c">\operatorname{\mathbf{L}}</annotation><annotation encoding="application/x-llamapun" id="S1.SS2a.p2.11.m11.1d">bold_L</annotation></semantics></math>, then</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="S3.EGx15"> <tbody id="S1.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_right ltx_eqn_cell"><math alttext="\displaystyle\operatorname{\mathbf{L}}=\begin{bmatrix}\operatorname{\mathbf{l}% }_{1:}^{\top}\\ \vdots\\ \operatorname{\mathbf{l}}_{j:}^{\top}\\ \vdots\\ \operatorname{\mathbf{l}}_{Q:}^{\top}\end{bmatrix}\implies\operatorname{% \mathbf{L}}\operatorname{\mathbf{Q}}_{j}^{\top}=\begin{bmatrix}\operatorname{% \mathbf{l}}_{1:}^{\top}\operatorname{\mathbf{Q}}_{j}^{\top}\\ \vdots\\ \left\lVert\operatorname{\mathbf{l}}_{j:}\right\rVert_{2}\operatorname{\mathbf% {e}}_{1}^{\top}\\ \vdots\\ \operatorname{\mathbf{l}}_{Q:}^{\top}\operatorname{\mathbf{Q}}_{j}^{\top}\end{% bmatrix}\implies\frac{1}{\left\lVert\operatorname{\mathbf{l}}_{j:}\right\rVert% _{2}}\operatorname{\mathbf{L}}\operatorname{\mathbf{Q}}_{j}^{\top}=\begin{% bmatrix}\frac{1}{\left\lVert\operatorname{\mathbf{l}}_{j:}\right\rVert_{2}}% \operatorname{\mathbf{l}}_{1:}^{\top}\operatorname{\mathbf{Q}}_{j}^{\top}\\ \vdots\\ \operatorname{\mathbf{e}}_{1}^{\top}\\ \vdots\\ \frac{1}{\left\lVert\operatorname{\mathbf{l}}_{j:}\right\rVert_{2}}% \operatorname{\mathbf{l}}_{Q:}^{\top}\operatorname{\mathbf{Q}}_{j}^{\top}\end{% bmatrix}," class="ltx_Math" display="inline" id="S1.Ex4.m1.5"><semantics id="S1.Ex4.m1.5a"><mrow id="S1.Ex4.m1.5.5.1" xref="S1.Ex4.m1.5.5.1.1.cmml"><mrow id="S1.Ex4.m1.5.5.1.1" xref="S1.Ex4.m1.5.5.1.1.cmml"><mi id="S1.Ex4.m1.5.5.1.1.2" xref="S1.Ex4.m1.5.5.1.1.2.cmml">𝐋</mi><mo id="S1.Ex4.m1.5.5.1.1.3" xref="S1.Ex4.m1.5.5.1.1.3.cmml">=</mo><mrow id="S1.Ex4.m1.1.1a.3" xref="S1.Ex4.m1.1.1a.2.cmml"><mo id="S1.Ex4.m1.1.1a.3.1" xref="S1.Ex4.m1.1.1a.2.1.cmml">[</mo><mtable id="S1.Ex4.m1.1.1.1.1" rowspacing="0pt" xref="S1.Ex4.m1.1.1.1.1.cmml"><mtr id="S1.Ex4.m1.1.1.1.1a" xref="S1.Ex4.m1.1.1.1.1.cmml"><mtd id="S1.Ex4.m1.1.1.1.1b" xref="S1.Ex4.m1.1.1.1.1.cmml"><msubsup id="S1.Ex4.m1.1.1.1.1.1.1.1" xref="S1.Ex4.m1.1.1.1.1.1.1.1.cmml"><mi id="S1.Ex4.m1.1.1.1.1.1.1.1.2.2" xref="S1.Ex4.m1.1.1.1.1.1.1.1.2.2.cmml">𝐥</mi><mrow id="S1.Ex4.m1.1.1.1.1.1.1.1.2.3" xref="S1.Ex4.m1.1.1.1.1.1.1.1.2.3.cmml"><mn id="S1.Ex4.m1.1.1.1.1.1.1.1.2.3.2" xref="S1.Ex4.m1.1.1.1.1.1.1.1.2.3.2.cmml">1</mn><mo id="S1.Ex4.m1.1.1.1.1.1.1.1.2.3.1" lspace="0.278em" rspace="0.278em" xref="S1.Ex4.m1.1.1.1.1.1.1.1.2.3.1.cmml">:</mo><mi id="S1.Ex4.m1.1.1.1.1.1.1.1.2.3.3" xref="S1.Ex4.m1.1.1.1.1.1.1.1.2.3.3.cmml"></mi></mrow><mo id="S1.Ex4.m1.1.1.1.1.1.1.1.3" xref="S1.Ex4.m1.1.1.1.1.1.1.1.3.cmml">⊤</mo></msubsup></mtd></mtr><mtr id="S1.Ex4.m1.1.1.1.1c" xref="S1.Ex4.m1.1.1.1.1.cmml"><mtd id="S1.Ex4.m1.1.1.1.1d" xref="S1.Ex4.m1.1.1.1.1.cmml"><mi id="S1.Ex4.m1.1.1.1.1.2.1.1" mathvariant="normal" xref="S1.Ex4.m1.1.1.1.1.2.1.1.cmml">⋮</mi></mtd></mtr><mtr id="S1.Ex4.m1.1.1.1.1e" xref="S1.Ex4.m1.1.1.1.1.cmml"><mtd id="S1.Ex4.m1.1.1.1.1f" xref="S1.Ex4.m1.1.1.1.1.cmml"><msubsup id="S1.Ex4.m1.1.1.1.1.3.1.1" xref="S1.Ex4.m1.1.1.1.1.3.1.1.cmml"><mi id="S1.Ex4.m1.1.1.1.1.3.1.1.2.2" xref="S1.Ex4.m1.1.1.1.1.3.1.1.2.2.cmml">𝐥</mi><mrow id="S1.Ex4.m1.1.1.1.1.3.1.1.2.3" xref="S1.Ex4.m1.1.1.1.1.3.1.1.2.3.cmml"><mi id="S1.Ex4.m1.1.1.1.1.3.1.1.2.3.2" xref="S1.Ex4.m1.1.1.1.1.3.1.1.2.3.2.cmml">j</mi><mo id="S1.Ex4.m1.1.1.1.1.3.1.1.2.3.1" lspace="0.278em" rspace="0.278em" xref="S1.Ex4.m1.1.1.1.1.3.1.1.2.3.1.cmml">:</mo><mi id="S1.Ex4.m1.1.1.1.1.3.1.1.2.3.3" xref="S1.Ex4.m1.1.1.1.1.3.1.1.2.3.3.cmml"></mi></mrow><mo id="S1.Ex4.m1.1.1.1.1.3.1.1.3" xref="S1.Ex4.m1.1.1.1.1.3.1.1.3.cmml">⊤</mo></msubsup></mtd></mtr><mtr id="S1.Ex4.m1.1.1.1.1g" xref="S1.Ex4.m1.1.1.1.1.cmml"><mtd id="S1.Ex4.m1.1.1.1.1h" xref="S1.Ex4.m1.1.1.1.1.cmml"><mi id="S1.Ex4.m1.1.1.1.1.4.1.1" mathvariant="normal" xref="S1.Ex4.m1.1.1.1.1.4.1.1.cmml">⋮</mi></mtd></mtr><mtr id="S1.Ex4.m1.1.1.1.1i" xref="S1.Ex4.m1.1.1.1.1.cmml"><mtd id="S1.Ex4.m1.1.1.1.1j" xref="S1.Ex4.m1.1.1.1.1.cmml"><msubsup id="S1.Ex4.m1.1.1.1.1.5.1.1" xref="S1.Ex4.m1.1.1.1.1.5.1.1.cmml"><mi id="S1.Ex4.m1.1.1.1.1.5.1.1.2.2" xref="S1.Ex4.m1.1.1.1.1.5.1.1.2.2.cmml">𝐥</mi><mrow id="S1.Ex4.m1.1.1.1.1.5.1.1.2.3" xref="S1.Ex4.m1.1.1.1.1.5.1.1.2.3.cmml"><mi id="S1.Ex4.m1.1.1.1.1.5.1.1.2.3.2" xref="S1.Ex4.m1.1.1.1.1.5.1.1.2.3.2.cmml">Q</mi><mo id="S1.Ex4.m1.1.1.1.1.5.1.1.2.3.1" lspace="0.278em" rspace="0.278em" xref="S1.Ex4.m1.1.1.1.1.5.1.1.2.3.1.cmml">:</mo><mi id="S1.Ex4.m1.1.1.1.1.5.1.1.2.3.3" xref="S1.Ex4.m1.1.1.1.1.5.1.1.2.3.3.cmml"></mi></mrow><mo id="S1.Ex4.m1.1.1.1.1.5.1.1.3" xref="S1.Ex4.m1.1.1.1.1.5.1.1.3.cmml">⊤</mo></msubsup></mtd></mtr></mtable><mo id="S1.Ex4.m1.1.1a.3.2" xref="S1.Ex4.m1.1.1a.2.1.cmml">]</mo></mrow><mo id="S1.Ex4.m1.5.5.1.1.4" stretchy="false" xref="S1.Ex4.m1.5.5.1.1.4.cmml">⟹</mo><mrow id="S1.Ex4.m1.5.5.1.1.5" xref="S1.Ex4.m1.5.5.1.1.5.cmml"><mi id="S1.Ex4.m1.5.5.1.1.5.1" xref="S1.Ex4.m1.5.5.1.1.5.1.cmml">𝐋</mi><mo id="S1.Ex4.m1.5.5.1.1.5a" lspace="0.167em" xref="S1.Ex4.m1.5.5.1.1.5.cmml">⁡</mo><msubsup id="S1.Ex4.m1.5.5.1.1.5.2" xref="S1.Ex4.m1.5.5.1.1.5.2.cmml"><mi id="S1.Ex4.m1.5.5.1.1.5.2.2.2" xref="S1.Ex4.m1.5.5.1.1.5.2.2.2.cmml">𝐐</mi><mi id="S1.Ex4.m1.5.5.1.1.5.2.2.3" xref="S1.Ex4.m1.5.5.1.1.5.2.2.3.cmml">j</mi><mo id="S1.Ex4.m1.5.5.1.1.5.2.3" xref="S1.Ex4.m1.5.5.1.1.5.2.3.cmml">⊤</mo></msubsup></mrow><mo id="S1.Ex4.m1.5.5.1.1.6" xref="S1.Ex4.m1.5.5.1.1.6.cmml">=</mo><mrow id="S1.Ex4.m1.2.2a.3" xref="S1.Ex4.m1.2.2a.2.cmml"><mo id="S1.Ex4.m1.2.2a.3.1" xref="S1.Ex4.m1.2.2a.2.1.cmml">[</mo><mtable id="S1.Ex4.m1.2.2.1.1" rowspacing="0pt" xref="S1.Ex4.m1.2.2.1.1.cmml"><mtr id="S1.Ex4.m1.2.2.1.1a" xref="S1.Ex4.m1.2.2.1.1.cmml"><mtd id="S1.Ex4.m1.2.2.1.1b" xref="S1.Ex4.m1.2.2.1.1.cmml"><mrow id="S1.Ex4.m1.2.2.1.1.2.1.1" xref="S1.Ex4.m1.2.2.1.1.2.1.1.cmml"><msubsup id="S1.Ex4.m1.2.2.1.1.2.1.1.1" xref="S1.Ex4.m1.2.2.1.1.2.1.1.1.cmml"><mi id="S1.Ex4.m1.2.2.1.1.2.1.1.1.2.2" xref="S1.Ex4.m1.2.2.1.1.2.1.1.1.2.2.cmml">𝐥</mi><mrow id="S1.Ex4.m1.2.2.1.1.2.1.1.1.2.3" xref="S1.Ex4.m1.2.2.1.1.2.1.1.1.2.3.cmml"><mn id="S1.Ex4.m1.2.2.1.1.2.1.1.1.2.3.2" xref="S1.Ex4.m1.2.2.1.1.2.1.1.1.2.3.2.cmml">1</mn><mo id="S1.Ex4.m1.2.2.1.1.2.1.1.1.2.3.1" lspace="0.278em" rspace="0.278em" xref="S1.Ex4.m1.2.2.1.1.2.1.1.1.2.3.1.cmml">:</mo><mi id="S1.Ex4.m1.2.2.1.1.2.1.1.1.2.3.3" xref="S1.Ex4.m1.2.2.1.1.2.1.1.1.2.3.3.cmml"></mi></mrow><mo id="S1.Ex4.m1.2.2.1.1.2.1.1.1.3" xref="S1.Ex4.m1.2.2.1.1.2.1.1.1.3.cmml">⊤</mo></msubsup><mo id="S1.Ex4.m1.2.2.1.1.2.1.1a" lspace="0.167em" xref="S1.Ex4.m1.2.2.1.1.2.1.1.cmml">⁡</mo><msubsup id="S1.Ex4.m1.2.2.1.1.2.1.1.2" xref="S1.Ex4.m1.2.2.1.1.2.1.1.2.cmml"><mi id="S1.Ex4.m1.2.2.1.1.2.1.1.2.2.2" xref="S1.Ex4.m1.2.2.1.1.2.1.1.2.2.2.cmml">𝐐</mi><mi id="S1.Ex4.m1.2.2.1.1.2.1.1.2.2.3" xref="S1.Ex4.m1.2.2.1.1.2.1.1.2.2.3.cmml">j</mi><mo id="S1.Ex4.m1.2.2.1.1.2.1.1.2.3" xref="S1.Ex4.m1.2.2.1.1.2.1.1.2.3.cmml">⊤</mo></msubsup></mrow></mtd></mtr><mtr id="S1.Ex4.m1.2.2.1.1c" xref="S1.Ex4.m1.2.2.1.1.cmml"><mtd id="S1.Ex4.m1.2.2.1.1d" xref="S1.Ex4.m1.2.2.1.1.cmml"><mi id="S1.Ex4.m1.2.2.1.1.3.1.1" mathvariant="normal" xref="S1.Ex4.m1.2.2.1.1.3.1.1.cmml">⋮</mi></mtd></mtr><mtr id="S1.Ex4.m1.2.2.1.1e" xref="S1.Ex4.m1.2.2.1.1.cmml"><mtd id="S1.Ex4.m1.2.2.1.1f" xref="S1.Ex4.m1.2.2.1.1.cmml"><mrow id="S1.Ex4.m1.2.2.1.1.1.1.1.1" xref="S1.Ex4.m1.2.2.1.1.1.1.1.1.cmml"><msub id="S1.Ex4.m1.2.2.1.1.1.1.1.1.1" xref="S1.Ex4.m1.2.2.1.1.1.1.1.1.1.cmml"><mrow id="S1.Ex4.m1.2.2.1.1.1.1.1.1.1.1.1" xref="S1.Ex4.m1.2.2.1.1.1.1.1.1.1.1.2.cmml"><mo fence="true" id="S1.Ex4.m1.2.2.1.1.1.1.1.1.1.1.1.2" rspace="0em" stretchy="true" xref="S1.Ex4.m1.2.2.1.1.1.1.1.1.1.1.2.1.cmml">∥</mo><msub id="S1.Ex4.m1.2.2.1.1.1.1.1.1.1.1.1.1" xref="S1.Ex4.m1.2.2.1.1.1.1.1.1.1.1.1.1.cmml"><mi id="S1.Ex4.m1.2.2.1.1.1.1.1.1.1.1.1.1.2" xref="S1.Ex4.m1.2.2.1.1.1.1.1.1.1.1.1.1.2.cmml">𝐥</mi><mrow id="S1.Ex4.m1.2.2.1.1.1.1.1.1.1.1.1.1.3" xref="S1.Ex4.m1.2.2.1.1.1.1.1.1.1.1.1.1.3.cmml"><mi id="S1.Ex4.m1.2.2.1.1.1.1.1.1.1.1.1.1.3.2" xref="S1.Ex4.m1.2.2.1.1.1.1.1.1.1.1.1.1.3.2.cmml">j</mi><mo id="S1.Ex4.m1.2.2.1.1.1.1.1.1.1.1.1.1.3.1" lspace="0.278em" rspace="0.278em" xref="S1.Ex4.m1.2.2.1.1.1.1.1.1.1.1.1.1.3.1.cmml">:</mo><mi id="S1.Ex4.m1.2.2.1.1.1.1.1.1.1.1.1.1.3.3" xref="S1.Ex4.m1.2.2.1.1.1.1.1.1.1.1.1.1.3.3.cmml"></mi></mrow></msub><mo fence="true" id="S1.Ex4.m1.2.2.1.1.1.1.1.1.1.1.1.3" lspace="0em" stretchy="true" xref="S1.Ex4.m1.2.2.1.1.1.1.1.1.1.1.2.1.cmml">∥</mo></mrow><mn id="S1.Ex4.m1.2.2.1.1.1.1.1.1.1.3" xref="S1.Ex4.m1.2.2.1.1.1.1.1.1.1.3.cmml">2</mn></msub><mo id="S1.Ex4.m1.2.2.1.1.1.1.1.1.2" lspace="0.167em" xref="S1.Ex4.m1.2.2.1.1.1.1.1.1.2.cmml">⁢</mo><msubsup id="S1.Ex4.m1.2.2.1.1.1.1.1.1.3" xref="S1.Ex4.m1.2.2.1.1.1.1.1.1.3.cmml"><mi id="S1.Ex4.m1.2.2.1.1.1.1.1.1.3.2.2" xref="S1.Ex4.m1.2.2.1.1.1.1.1.1.3.2.2.cmml">𝐞</mi><mn id="S1.Ex4.m1.2.2.1.1.1.1.1.1.3.2.3" xref="S1.Ex4.m1.2.2.1.1.1.1.1.1.3.2.3.cmml">1</mn><mo id="S1.Ex4.m1.2.2.1.1.1.1.1.1.3.3" xref="S1.Ex4.m1.2.2.1.1.1.1.1.1.3.3.cmml">⊤</mo></msubsup></mrow></mtd></mtr><mtr id="S1.Ex4.m1.2.2.1.1g" xref="S1.Ex4.m1.2.2.1.1.cmml"><mtd id="S1.Ex4.m1.2.2.1.1h" xref="S1.Ex4.m1.2.2.1.1.cmml"><mi id="S1.Ex4.m1.2.2.1.1.4.1.1" mathvariant="normal" xref="S1.Ex4.m1.2.2.1.1.4.1.1.cmml">⋮</mi></mtd></mtr><mtr id="S1.Ex4.m1.2.2.1.1i" xref="S1.Ex4.m1.2.2.1.1.cmml"><mtd id="S1.Ex4.m1.2.2.1.1j" xref="S1.Ex4.m1.2.2.1.1.cmml"><mrow id="S1.Ex4.m1.2.2.1.1.5.1.1" xref="S1.Ex4.m1.2.2.1.1.5.1.1.cmml"><msubsup id="S1.Ex4.m1.2.2.1.1.5.1.1.1" xref="S1.Ex4.m1.2.2.1.1.5.1.1.1.cmml"><mi id="S1.Ex4.m1.2.2.1.1.5.1.1.1.2.2" xref="S1.Ex4.m1.2.2.1.1.5.1.1.1.2.2.cmml">𝐥</mi><mrow id="S1.Ex4.m1.2.2.1.1.5.1.1.1.2.3" xref="S1.Ex4.m1.2.2.1.1.5.1.1.1.2.3.cmml"><mi id="S1.Ex4.m1.2.2.1.1.5.1.1.1.2.3.2" xref="S1.Ex4.m1.2.2.1.1.5.1.1.1.2.3.2.cmml">Q</mi><mo id="S1.Ex4.m1.2.2.1.1.5.1.1.1.2.3.1" lspace="0.278em" rspace="0.278em" xref="S1.Ex4.m1.2.2.1.1.5.1.1.1.2.3.1.cmml">:</mo><mi id="S1.Ex4.m1.2.2.1.1.5.1.1.1.2.3.3" xref="S1.Ex4.m1.2.2.1.1.5.1.1.1.2.3.3.cmml"></mi></mrow><mo id="S1.Ex4.m1.2.2.1.1.5.1.1.1.3" xref="S1.Ex4.m1.2.2.1.1.5.1.1.1.3.cmml">⊤</mo></msubsup><mo id="S1.Ex4.m1.2.2.1.1.5.1.1a" lspace="0.167em" xref="S1.Ex4.m1.2.2.1.1.5.1.1.cmml">⁡</mo><msubsup id="S1.Ex4.m1.2.2.1.1.5.1.1.2" xref="S1.Ex4.m1.2.2.1.1.5.1.1.2.cmml"><mi id="S1.Ex4.m1.2.2.1.1.5.1.1.2.2.2" xref="S1.Ex4.m1.2.2.1.1.5.1.1.2.2.2.cmml">𝐐</mi><mi id="S1.Ex4.m1.2.2.1.1.5.1.1.2.2.3" xref="S1.Ex4.m1.2.2.1.1.5.1.1.2.2.3.cmml">j</mi><mo id="S1.Ex4.m1.2.2.1.1.5.1.1.2.3" xref="S1.Ex4.m1.2.2.1.1.5.1.1.2.3.cmml">⊤</mo></msubsup></mrow></mtd></mtr></mtable><mo id="S1.Ex4.m1.2.2a.3.2" xref="S1.Ex4.m1.2.2a.2.1.cmml">]</mo></mrow><mo id="S1.Ex4.m1.5.5.1.1.7" stretchy="false" xref="S1.Ex4.m1.5.5.1.1.7.cmml">⟹</mo><mrow id="S1.Ex4.m1.5.5.1.1.8" xref="S1.Ex4.m1.5.5.1.1.8.cmml"><mstyle displaystyle="true" id="S1.Ex4.m1.4.4" xref="S1.Ex4.m1.4.4.cmml"><mfrac id="S1.Ex4.m1.4.4a" xref="S1.Ex4.m1.4.4.cmml"><mn id="S1.Ex4.m1.4.4.3" xref="S1.Ex4.m1.4.4.3.cmml">1</mn><msub id="S1.Ex4.m1.4.4.1" xref="S1.Ex4.m1.4.4.1.cmml"><mrow id="S1.Ex4.m1.4.4.1.1.1" xref="S1.Ex4.m1.4.4.1.1.2.cmml"><mo fence="true" id="S1.Ex4.m1.4.4.1.1.1.2" rspace="0em" stretchy="true" xref="S1.Ex4.m1.4.4.1.1.2.1.cmml">∥</mo><msub id="S1.Ex4.m1.4.4.1.1.1.1" xref="S1.Ex4.m1.4.4.1.1.1.1.cmml"><mi id="S1.Ex4.m1.4.4.1.1.1.1.2" xref="S1.Ex4.m1.4.4.1.1.1.1.2.cmml">𝐥</mi><mrow id="S1.Ex4.m1.4.4.1.1.1.1.3" xref="S1.Ex4.m1.4.4.1.1.1.1.3.cmml"><mi id="S1.Ex4.m1.4.4.1.1.1.1.3.2" xref="S1.Ex4.m1.4.4.1.1.1.1.3.2.cmml">j</mi><mo id="S1.Ex4.m1.4.4.1.1.1.1.3.1" lspace="0.278em" rspace="0.278em" xref="S1.Ex4.m1.4.4.1.1.1.1.3.1.cmml">:</mo><mi id="S1.Ex4.m1.4.4.1.1.1.1.3.3" xref="S1.Ex4.m1.4.4.1.1.1.1.3.3.cmml"></mi></mrow></msub><mo fence="true" id="S1.Ex4.m1.4.4.1.1.1.3" lspace="0em" stretchy="true" xref="S1.Ex4.m1.4.4.1.1.2.1.cmml">∥</mo></mrow><mn id="S1.Ex4.m1.4.4.1.3" xref="S1.Ex4.m1.4.4.1.3.cmml">2</mn></msub></mfrac></mstyle><mo id="S1.Ex4.m1.5.5.1.1.8.1" lspace="0.167em" xref="S1.Ex4.m1.5.5.1.1.8.1.cmml">⁢</mo><mrow id="S1.Ex4.m1.5.5.1.1.8.2" xref="S1.Ex4.m1.5.5.1.1.8.2.cmml"><mi id="S1.Ex4.m1.5.5.1.1.8.2.1" xref="S1.Ex4.m1.5.5.1.1.8.2.1.cmml">𝐋</mi><mo id="S1.Ex4.m1.5.5.1.1.8.2a" lspace="0.167em" xref="S1.Ex4.m1.5.5.1.1.8.2.cmml">⁡</mo><msubsup id="S1.Ex4.m1.5.5.1.1.8.2.2" xref="S1.Ex4.m1.5.5.1.1.8.2.2.cmml"><mi id="S1.Ex4.m1.5.5.1.1.8.2.2.2.2" xref="S1.Ex4.m1.5.5.1.1.8.2.2.2.2.cmml">𝐐</mi><mi id="S1.Ex4.m1.5.5.1.1.8.2.2.2.3" xref="S1.Ex4.m1.5.5.1.1.8.2.2.2.3.cmml">j</mi><mo id="S1.Ex4.m1.5.5.1.1.8.2.2.3" xref="S1.Ex4.m1.5.5.1.1.8.2.2.3.cmml">⊤</mo></msubsup></mrow></mrow><mo id="S1.Ex4.m1.5.5.1.1.9" xref="S1.Ex4.m1.5.5.1.1.9.cmml">=</mo><mrow id="S1.Ex4.m1.3.3a.3" xref="S1.Ex4.m1.3.3a.2.cmml"><mo id="S1.Ex4.m1.3.3a.3.1" xref="S1.Ex4.m1.3.3a.2.1.cmml">[</mo><mtable id="S1.Ex4.m1.3.3.1.1" rowspacing="0pt" xref="S1.Ex4.m1.3.3.1.1.cmml"><mtr id="S1.Ex4.m1.3.3.1.1a" xref="S1.Ex4.m1.3.3.1.1.cmml"><mtd id="S1.Ex4.m1.3.3.1.1b" xref="S1.Ex4.m1.3.3.1.1.cmml"><mrow id="S1.Ex4.m1.3.3.1.1.1.1.1.1" 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id="S1.Ex4.m1.5c">\displaystyle\operatorname{\mathbf{L}}=\begin{bmatrix}\operatorname{\mathbf{l}% }_{1:}^{\top}\\ \vdots\\ \operatorname{\mathbf{l}}_{j:}^{\top}\\ \vdots\\ \operatorname{\mathbf{l}}_{Q:}^{\top}\end{bmatrix}\implies\operatorname{% \mathbf{L}}\operatorname{\mathbf{Q}}_{j}^{\top}=\begin{bmatrix}\operatorname{% \mathbf{l}}_{1:}^{\top}\operatorname{\mathbf{Q}}_{j}^{\top}\\ \vdots\\ \left\lVert\operatorname{\mathbf{l}}_{j:}\right\rVert_{2}\operatorname{\mathbf% {e}}_{1}^{\top}\\ \vdots\\ \operatorname{\mathbf{l}}_{Q:}^{\top}\operatorname{\mathbf{Q}}_{j}^{\top}\end{% bmatrix}\implies\frac{1}{\left\lVert\operatorname{\mathbf{l}}_{j:}\right\rVert% _{2}}\operatorname{\mathbf{L}}\operatorname{\mathbf{Q}}_{j}^{\top}=\begin{% bmatrix}\frac{1}{\left\lVert\operatorname{\mathbf{l}}_{j:}\right\rVert_{2}}% \operatorname{\mathbf{l}}_{1:}^{\top}\operatorname{\mathbf{Q}}_{j}^{\top}\\ \vdots\\ \operatorname{\mathbf{e}}_{1}^{\top}\\ \vdots\\ \frac{1}{\left\lVert\operatorname{\mathbf{l}}_{j:}\right\rVert_{2}}% \operatorname{\mathbf{l}}_{Q:}^{\top}\operatorname{\mathbf{Q}}_{j}^{\top}\end{% bmatrix},</annotation><annotation encoding="application/x-llamapun" id="S1.Ex4.m1.5d">bold_L = [ start_ARG start_ROW start_CELL bold_l start_POSTSUBSCRIPT 1 : end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ⊤ end_POSTSUPERSCRIPT end_CELL end_ROW start_ROW start_CELL ⋮ end_CELL end_ROW start_ROW start_CELL bold_l start_POSTSUBSCRIPT italic_j : end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ⊤ end_POSTSUPERSCRIPT end_CELL end_ROW start_ROW start_CELL ⋮ end_CELL end_ROW start_ROW start_CELL bold_l start_POSTSUBSCRIPT italic_Q : end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ⊤ end_POSTSUPERSCRIPT end_CELL end_ROW end_ARG ] ⟹ bold_L bold_Q start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ⊤ end_POSTSUPERSCRIPT = [ start_ARG start_ROW start_CELL bold_l start_POSTSUBSCRIPT 1 : end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ⊤ end_POSTSUPERSCRIPT bold_Q start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ⊤ end_POSTSUPERSCRIPT end_CELL end_ROW start_ROW start_CELL ⋮ end_CELL end_ROW start_ROW start_CELL ∥ bold_l start_POSTSUBSCRIPT italic_j : end_POSTSUBSCRIPT ∥ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT bold_e start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ⊤ end_POSTSUPERSCRIPT end_CELL end_ROW start_ROW start_CELL ⋮ end_CELL end_ROW start_ROW start_CELL bold_l start_POSTSUBSCRIPT italic_Q : end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ⊤ end_POSTSUPERSCRIPT bold_Q start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ⊤ end_POSTSUPERSCRIPT end_CELL end_ROW end_ARG ] ⟹ divide start_ARG 1 end_ARG start_ARG ∥ bold_l start_POSTSUBSCRIPT italic_j : end_POSTSUBSCRIPT ∥ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_ARG bold_L bold_Q start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ⊤ end_POSTSUPERSCRIPT = [ start_ARG start_ROW start_CELL divide start_ARG 1 end_ARG start_ARG ∥ bold_l start_POSTSUBSCRIPT italic_j : end_POSTSUBSCRIPT ∥ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_ARG bold_l start_POSTSUBSCRIPT 1 : end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ⊤ end_POSTSUPERSCRIPT bold_Q start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ⊤ end_POSTSUPERSCRIPT end_CELL end_ROW start_ROW start_CELL ⋮ end_CELL end_ROW start_ROW start_CELL bold_e start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ⊤ end_POSTSUPERSCRIPT end_CELL end_ROW start_ROW start_CELL ⋮ end_CELL end_ROW start_ROW start_CELL divide start_ARG 1 end_ARG start_ARG ∥ bold_l start_POSTSUBSCRIPT italic_j : end_POSTSUBSCRIPT ∥ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_ARG bold_l start_POSTSUBSCRIPT italic_Q : end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ⊤ end_POSTSUPERSCRIPT bold_Q start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ⊤ end_POSTSUPERSCRIPT end_CELL end_ROW end_ARG ] ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S1.SS2a.p2.14">which has the form described in Proposition <a class="ltx_ref" href="https://arxiv.org/html/2503.13782v1#S1.Thmproposition1" title="Proposition 1.1 ‣ 1.1 Proof of Proposition 2.1 ‣ 1 Proof of the Propositions ‣ Regularized Parameter Estimation in Mixed Model Trace Regression"><span class="ltx_text ltx_ref_tag">1.1</span></a>. Because Householder transformations are orthonormal,</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="S3.EGx16"> <tbody id="S1.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_right ltx_eqn_cell"><math alttext="\displaystyle\frac{1}{\left\lVert\operatorname{\mathbf{l}}_{j:}\right\rVert_{2% }^{2}}\operatorname{\mathbf{L}}\operatorname{\mathbf{Q}}_{j}^{\top}% \operatorname{\mathbf{Q}}_{j}\operatorname{\mathbf{L}}^{\top}=\frac{1}{\left% \lVert\operatorname{\mathbf{l}}_{j:}\right\rVert_{2}^{2}}\operatorname{{% \mathbf{\Sigma}}}," class="ltx_Math" display="inline" id="S1.Ex5.m1.3"><semantics id="S1.Ex5.m1.3a"><mrow id="S1.Ex5.m1.3.3.1" xref="S1.Ex5.m1.3.3.1.1.cmml"><mrow id="S1.Ex5.m1.3.3.1.1" xref="S1.Ex5.m1.3.3.1.1.cmml"><mrow id="S1.Ex5.m1.3.3.1.1.2" xref="S1.Ex5.m1.3.3.1.1.2.cmml"><mstyle displaystyle="true" id="S1.Ex5.m1.1.1" xref="S1.Ex5.m1.1.1.cmml"><mfrac 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}^{2}}\operatorname{\mathbf{L}}\operatorname{\mathbf{Q}}_{j}^{\top}% \operatorname{\mathbf{Q}}_{j}\operatorname{\mathbf{L}}^{\top}=\frac{1}{\left% \lVert\operatorname{\mathbf{l}}_{j:}\right\rVert_{2}^{2}}\operatorname{{% \mathbf{\Sigma}}},</annotation><annotation encoding="application/x-llamapun" id="S1.Ex5.m1.3d">divide start_ARG 1 end_ARG start_ARG ∥ bold_l start_POSTSUBSCRIPT italic_j : end_POSTSUBSCRIPT ∥ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG bold_L bold_Q start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ⊤ end_POSTSUPERSCRIPT bold_Q start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT bold_L start_POSTSUPERSCRIPT ⊤ end_POSTSUPERSCRIPT = divide start_ARG 1 end_ARG start_ARG ∥ bold_l start_POSTSUBSCRIPT italic_j : end_POSTSUBSCRIPT ∥ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG bold_Σ ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S1.SS2a.p2.13">which is just the initial covariance matrix <math alttext="\operatorname{{\mathbf{\Sigma}}}" class="ltx_Math" display="inline" id="S1.SS2a.p2.12.m1.1"><semantics id="S1.SS2a.p2.12.m1.1a"><mi id="S1.SS2a.p2.12.m1.1.1" xref="S1.SS2a.p2.12.m1.1.1.cmml">𝚺</mi><annotation-xml encoding="MathML-Content" id="S1.SS2a.p2.12.m1.1b"><ci id="S1.SS2a.p2.12.m1.1.1.cmml" xref="S1.SS2a.p2.12.m1.1.1">𝚺</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2a.p2.12.m1.1c">\operatorname{{\mathbf{\Sigma}}}</annotation><annotation encoding="application/x-llamapun" id="S1.SS2a.p2.12.m1.1d">bold_Σ</annotation></semantics></math> scaled such that its <math alttext="j" class="ltx_Math" display="inline" id="S1.SS2a.p2.13.m2.1"><semantics id="S1.SS2a.p2.13.m2.1a"><mi id="S1.SS2a.p2.13.m2.1.1" xref="S1.SS2a.p2.13.m2.1.1.cmml">j</mi><annotation-xml encoding="MathML-Content" id="S1.SS2a.p2.13.m2.1b"><ci id="S1.SS2a.p2.13.m2.1.1.cmml" xref="S1.SS2a.p2.13.m2.1.1">𝑗</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS2a.p2.13.m2.1c">j</annotation><annotation encoding="application/x-llamapun" id="S1.SS2a.p2.13.m2.1d">italic_j</annotation></semantics></math>th diagonal entry is equal to one. <span class="ltx_rule" style="width:6.5pt;height:6.5pt;background:black;display:inline-block;"> </span> <br class="ltx_break"/></p> </div> </section> <section class="ltx_subsection" id="S1.SS3"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">1.3 </span>Proof of Proposition 3.1</h3> <div class="ltx_para" id="S1.SS3.p1"> <p class="ltx_p" id="S1.SS3.p1.11">Recall Proposition 3.1 from the main manuscript. Let <math alttext="\ell(\operatorname{{\bm{\theta}}})" class="ltx_Math" display="inline" id="S1.SS3.p1.1.m1.1"><semantics id="S1.SS3.p1.1.m1.1a"><mrow id="S1.SS3.p1.1.m1.1.2" xref="S1.SS3.p1.1.m1.1.2.cmml"><mi id="S1.SS3.p1.1.m1.1.2.2" mathvariant="normal" xref="S1.SS3.p1.1.m1.1.2.2.cmml">ℓ</mi><mo id="S1.SS3.p1.1.m1.1.2.1" xref="S1.SS3.p1.1.m1.1.2.1.cmml">⁢</mo><mrow id="S1.SS3.p1.1.m1.1.2.3.2" xref="S1.SS3.p1.1.m1.1.2.cmml"><mo id="S1.SS3.p1.1.m1.1.2.3.2.1" stretchy="false" xref="S1.SS3.p1.1.m1.1.2.cmml">(</mo><mi id="S1.SS3.p1.1.m1.1.1" xref="S1.SS3.p1.1.m1.1.1.cmml">𝜽</mi><mo id="S1.SS3.p1.1.m1.1.2.3.2.2" stretchy="false" xref="S1.SS3.p1.1.m1.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.SS3.p1.1.m1.1b"><apply id="S1.SS3.p1.1.m1.1.2.cmml" xref="S1.SS3.p1.1.m1.1.2"><times id="S1.SS3.p1.1.m1.1.2.1.cmml" xref="S1.SS3.p1.1.m1.1.2.1"></times><ci id="S1.SS3.p1.1.m1.1.2.2.cmml" xref="S1.SS3.p1.1.m1.1.2.2">ℓ</ci><ci id="S1.SS3.p1.1.m1.1.1.cmml" xref="S1.SS3.p1.1.m1.1.1">𝜽</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS3.p1.1.m1.1c">\ell(\operatorname{{\bm{\theta}}})</annotation><annotation encoding="application/x-llamapun" id="S1.SS3.p1.1.m1.1d">roman_ℓ ( bold_italic_θ )</annotation></semantics></math> be the negative log likehood function implied by the MMTR, <math alttext="\operatorname{{\bm{\theta}}}_{1}=(\operatorname{\mathbf{B}},\tau^{2})" class="ltx_Math" display="inline" id="S1.SS3.p1.2.m2.2"><semantics id="S1.SS3.p1.2.m2.2a"><mrow id="S1.SS3.p1.2.m2.2.2" xref="S1.SS3.p1.2.m2.2.2.cmml"><msub id="S1.SS3.p1.2.m2.2.2.3" xref="S1.SS3.p1.2.m2.2.2.3.cmml"><mi id="S1.SS3.p1.2.m2.2.2.3.2" xref="S1.SS3.p1.2.m2.2.2.3.2.cmml">𝜽</mi><mn id="S1.SS3.p1.2.m2.2.2.3.3" xref="S1.SS3.p1.2.m2.2.2.3.3.cmml">1</mn></msub><mo id="S1.SS3.p1.2.m2.2.2.2" xref="S1.SS3.p1.2.m2.2.2.2.cmml">=</mo><mrow id="S1.SS3.p1.2.m2.2.2.1.1" xref="S1.SS3.p1.2.m2.2.2.1.2.cmml"><mo id="S1.SS3.p1.2.m2.2.2.1.1.2" stretchy="false" xref="S1.SS3.p1.2.m2.2.2.1.2.cmml">(</mo><mi id="S1.SS3.p1.2.m2.1.1" xref="S1.SS3.p1.2.m2.1.1.cmml">𝐁</mi><mo id="S1.SS3.p1.2.m2.2.2.1.1.3" xref="S1.SS3.p1.2.m2.2.2.1.2.cmml">,</mo><msup id="S1.SS3.p1.2.m2.2.2.1.1.1" xref="S1.SS3.p1.2.m2.2.2.1.1.1.cmml"><mi id="S1.SS3.p1.2.m2.2.2.1.1.1.2" xref="S1.SS3.p1.2.m2.2.2.1.1.1.2.cmml">τ</mi><mn id="S1.SS3.p1.2.m2.2.2.1.1.1.3" xref="S1.SS3.p1.2.m2.2.2.1.1.1.3.cmml">2</mn></msup><mo id="S1.SS3.p1.2.m2.2.2.1.1.4" stretchy="false" xref="S1.SS3.p1.2.m2.2.2.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.SS3.p1.2.m2.2b"><apply id="S1.SS3.p1.2.m2.2.2.cmml" xref="S1.SS3.p1.2.m2.2.2"><eq id="S1.SS3.p1.2.m2.2.2.2.cmml" xref="S1.SS3.p1.2.m2.2.2.2"></eq><apply id="S1.SS3.p1.2.m2.2.2.3.cmml" xref="S1.SS3.p1.2.m2.2.2.3"><csymbol cd="ambiguous" id="S1.SS3.p1.2.m2.2.2.3.1.cmml" xref="S1.SS3.p1.2.m2.2.2.3">subscript</csymbol><ci id="S1.SS3.p1.2.m2.2.2.3.2.cmml" xref="S1.SS3.p1.2.m2.2.2.3.2">𝜽</ci><cn id="S1.SS3.p1.2.m2.2.2.3.3.cmml" type="integer" xref="S1.SS3.p1.2.m2.2.2.3.3">1</cn></apply><interval closure="open" id="S1.SS3.p1.2.m2.2.2.1.2.cmml" xref="S1.SS3.p1.2.m2.2.2.1.1"><ci id="S1.SS3.p1.2.m2.1.1.cmml" xref="S1.SS3.p1.2.m2.1.1">𝐁</ci><apply id="S1.SS3.p1.2.m2.2.2.1.1.1.cmml" xref="S1.SS3.p1.2.m2.2.2.1.1.1"><csymbol cd="ambiguous" id="S1.SS3.p1.2.m2.2.2.1.1.1.1.cmml" xref="S1.SS3.p1.2.m2.2.2.1.1.1">superscript</csymbol><ci id="S1.SS3.p1.2.m2.2.2.1.1.1.2.cmml" xref="S1.SS3.p1.2.m2.2.2.1.1.1.2">𝜏</ci><cn id="S1.SS3.p1.2.m2.2.2.1.1.1.3.cmml" type="integer" xref="S1.SS3.p1.2.m2.2.2.1.1.1.3">2</cn></apply></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS3.p1.2.m2.2c">\operatorname{{\bm{\theta}}}_{1}=(\operatorname{\mathbf{B}},\tau^{2})</annotation><annotation encoding="application/x-llamapun" id="S1.SS3.p1.2.m2.2d">bold_italic_θ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = ( bold_B , italic_τ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT )</annotation></semantics></math>, <math alttext="\operatorname{{\bm{\theta}}}_{2}=\operatorname{\mathbf{L}}_{1}" class="ltx_Math" display="inline" id="S1.SS3.p1.3.m3.1"><semantics id="S1.SS3.p1.3.m3.1a"><mrow id="S1.SS3.p1.3.m3.1.1" xref="S1.SS3.p1.3.m3.1.1.cmml"><msub id="S1.SS3.p1.3.m3.1.1.2" xref="S1.SS3.p1.3.m3.1.1.2.cmml"><mi id="S1.SS3.p1.3.m3.1.1.2.2" xref="S1.SS3.p1.3.m3.1.1.2.2.cmml">𝜽</mi><mn id="S1.SS3.p1.3.m3.1.1.2.3" xref="S1.SS3.p1.3.m3.1.1.2.3.cmml">2</mn></msub><mo id="S1.SS3.p1.3.m3.1.1.1" xref="S1.SS3.p1.3.m3.1.1.1.cmml">=</mo><msub id="S1.SS3.p1.3.m3.1.1.3" xref="S1.SS3.p1.3.m3.1.1.3.cmml"><mi id="S1.SS3.p1.3.m3.1.1.3.2" xref="S1.SS3.p1.3.m3.1.1.3.2.cmml">𝐋</mi><mn id="S1.SS3.p1.3.m3.1.1.3.3" xref="S1.SS3.p1.3.m3.1.1.3.3.cmml">1</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S1.SS3.p1.3.m3.1b"><apply id="S1.SS3.p1.3.m3.1.1.cmml" xref="S1.SS3.p1.3.m3.1.1"><eq id="S1.SS3.p1.3.m3.1.1.1.cmml" xref="S1.SS3.p1.3.m3.1.1.1"></eq><apply id="S1.SS3.p1.3.m3.1.1.2.cmml" xref="S1.SS3.p1.3.m3.1.1.2"><csymbol cd="ambiguous" id="S1.SS3.p1.3.m3.1.1.2.1.cmml" xref="S1.SS3.p1.3.m3.1.1.2">subscript</csymbol><ci id="S1.SS3.p1.3.m3.1.1.2.2.cmml" xref="S1.SS3.p1.3.m3.1.1.2.2">𝜽</ci><cn id="S1.SS3.p1.3.m3.1.1.2.3.cmml" type="integer" xref="S1.SS3.p1.3.m3.1.1.2.3">2</cn></apply><apply id="S1.SS3.p1.3.m3.1.1.3.cmml" xref="S1.SS3.p1.3.m3.1.1.3"><csymbol cd="ambiguous" id="S1.SS3.p1.3.m3.1.1.3.1.cmml" xref="S1.SS3.p1.3.m3.1.1.3">subscript</csymbol><ci id="S1.SS3.p1.3.m3.1.1.3.2.cmml" xref="S1.SS3.p1.3.m3.1.1.3.2">𝐋</ci><cn id="S1.SS3.p1.3.m3.1.1.3.3.cmml" type="integer" xref="S1.SS3.p1.3.m3.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS3.p1.3.m3.1c">\operatorname{{\bm{\theta}}}_{2}=\operatorname{\mathbf{L}}_{1}</annotation><annotation encoding="application/x-llamapun" id="S1.SS3.p1.3.m3.1d">bold_italic_θ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT = bold_L start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math>, <math alttext="\operatorname{{\bm{\theta}}}_{3}=\operatorname{\mathbf{L}}_{2}" class="ltx_Math" display="inline" id="S1.SS3.p1.4.m4.1"><semantics id="S1.SS3.p1.4.m4.1a"><mrow id="S1.SS3.p1.4.m4.1.1" xref="S1.SS3.p1.4.m4.1.1.cmml"><msub id="S1.SS3.p1.4.m4.1.1.2" xref="S1.SS3.p1.4.m4.1.1.2.cmml"><mi id="S1.SS3.p1.4.m4.1.1.2.2" xref="S1.SS3.p1.4.m4.1.1.2.2.cmml">𝜽</mi><mn id="S1.SS3.p1.4.m4.1.1.2.3" xref="S1.SS3.p1.4.m4.1.1.2.3.cmml">3</mn></msub><mo id="S1.SS3.p1.4.m4.1.1.1" xref="S1.SS3.p1.4.m4.1.1.1.cmml">=</mo><msub id="S1.SS3.p1.4.m4.1.1.3" xref="S1.SS3.p1.4.m4.1.1.3.cmml"><mi id="S1.SS3.p1.4.m4.1.1.3.2" xref="S1.SS3.p1.4.m4.1.1.3.2.cmml">𝐋</mi><mn id="S1.SS3.p1.4.m4.1.1.3.3" xref="S1.SS3.p1.4.m4.1.1.3.3.cmml">2</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S1.SS3.p1.4.m4.1b"><apply id="S1.SS3.p1.4.m4.1.1.cmml" xref="S1.SS3.p1.4.m4.1.1"><eq id="S1.SS3.p1.4.m4.1.1.1.cmml" xref="S1.SS3.p1.4.m4.1.1.1"></eq><apply id="S1.SS3.p1.4.m4.1.1.2.cmml" xref="S1.SS3.p1.4.m4.1.1.2"><csymbol cd="ambiguous" id="S1.SS3.p1.4.m4.1.1.2.1.cmml" xref="S1.SS3.p1.4.m4.1.1.2">subscript</csymbol><ci id="S1.SS3.p1.4.m4.1.1.2.2.cmml" xref="S1.SS3.p1.4.m4.1.1.2.2">𝜽</ci><cn id="S1.SS3.p1.4.m4.1.1.2.3.cmml" type="integer" xref="S1.SS3.p1.4.m4.1.1.2.3">3</cn></apply><apply id="S1.SS3.p1.4.m4.1.1.3.cmml" xref="S1.SS3.p1.4.m4.1.1.3"><csymbol cd="ambiguous" id="S1.SS3.p1.4.m4.1.1.3.1.cmml" xref="S1.SS3.p1.4.m4.1.1.3">subscript</csymbol><ci id="S1.SS3.p1.4.m4.1.1.3.2.cmml" xref="S1.SS3.p1.4.m4.1.1.3.2">𝐋</ci><cn id="S1.SS3.p1.4.m4.1.1.3.3.cmml" type="integer" xref="S1.SS3.p1.4.m4.1.1.3.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS3.p1.4.m4.1c">\operatorname{{\bm{\theta}}}_{3}=\operatorname{\mathbf{L}}_{2}</annotation><annotation encoding="application/x-llamapun" id="S1.SS3.p1.4.m4.1d">bold_italic_θ start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT = bold_L start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math>, <math alttext="\operatorname{{\bm{\theta}}}=(\operatorname{{\bm{\theta}}}_{1},\operatorname{{% \bm{\theta}}}_{2},\operatorname{{\bm{\theta}}}_{3})" class="ltx_Math" display="inline" id="S1.SS3.p1.5.m5.3"><semantics id="S1.SS3.p1.5.m5.3a"><mrow id="S1.SS3.p1.5.m5.3.3" xref="S1.SS3.p1.5.m5.3.3.cmml"><mi id="S1.SS3.p1.5.m5.3.3.5" xref="S1.SS3.p1.5.m5.3.3.5.cmml">𝜽</mi><mo id="S1.SS3.p1.5.m5.3.3.4" xref="S1.SS3.p1.5.m5.3.3.4.cmml">=</mo><mrow id="S1.SS3.p1.5.m5.3.3.3.3" xref="S1.SS3.p1.5.m5.3.3.3.4.cmml"><mo id="S1.SS3.p1.5.m5.3.3.3.3.4" stretchy="false" xref="S1.SS3.p1.5.m5.3.3.3.4.cmml">(</mo><msub id="S1.SS3.p1.5.m5.1.1.1.1.1" xref="S1.SS3.p1.5.m5.1.1.1.1.1.cmml"><mi id="S1.SS3.p1.5.m5.1.1.1.1.1.2" xref="S1.SS3.p1.5.m5.1.1.1.1.1.2.cmml">𝜽</mi><mn id="S1.SS3.p1.5.m5.1.1.1.1.1.3" xref="S1.SS3.p1.5.m5.1.1.1.1.1.3.cmml">1</mn></msub><mo id="S1.SS3.p1.5.m5.3.3.3.3.5" xref="S1.SS3.p1.5.m5.3.3.3.4.cmml">,</mo><msub id="S1.SS3.p1.5.m5.2.2.2.2.2" xref="S1.SS3.p1.5.m5.2.2.2.2.2.cmml"><mi id="S1.SS3.p1.5.m5.2.2.2.2.2.2" xref="S1.SS3.p1.5.m5.2.2.2.2.2.2.cmml">𝜽</mi><mn id="S1.SS3.p1.5.m5.2.2.2.2.2.3" xref="S1.SS3.p1.5.m5.2.2.2.2.2.3.cmml">2</mn></msub><mo id="S1.SS3.p1.5.m5.3.3.3.3.6" xref="S1.SS3.p1.5.m5.3.3.3.4.cmml">,</mo><msub id="S1.SS3.p1.5.m5.3.3.3.3.3" xref="S1.SS3.p1.5.m5.3.3.3.3.3.cmml"><mi id="S1.SS3.p1.5.m5.3.3.3.3.3.2" xref="S1.SS3.p1.5.m5.3.3.3.3.3.2.cmml">𝜽</mi><mn id="S1.SS3.p1.5.m5.3.3.3.3.3.3" xref="S1.SS3.p1.5.m5.3.3.3.3.3.3.cmml">3</mn></msub><mo id="S1.SS3.p1.5.m5.3.3.3.3.7" stretchy="false" xref="S1.SS3.p1.5.m5.3.3.3.4.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.SS3.p1.5.m5.3b"><apply id="S1.SS3.p1.5.m5.3.3.cmml" xref="S1.SS3.p1.5.m5.3.3"><eq id="S1.SS3.p1.5.m5.3.3.4.cmml" xref="S1.SS3.p1.5.m5.3.3.4"></eq><ci id="S1.SS3.p1.5.m5.3.3.5.cmml" xref="S1.SS3.p1.5.m5.3.3.5">𝜽</ci><vector id="S1.SS3.p1.5.m5.3.3.3.4.cmml" xref="S1.SS3.p1.5.m5.3.3.3.3"><apply id="S1.SS3.p1.5.m5.1.1.1.1.1.cmml" xref="S1.SS3.p1.5.m5.1.1.1.1.1"><csymbol cd="ambiguous" id="S1.SS3.p1.5.m5.1.1.1.1.1.1.cmml" xref="S1.SS3.p1.5.m5.1.1.1.1.1">subscript</csymbol><ci id="S1.SS3.p1.5.m5.1.1.1.1.1.2.cmml" xref="S1.SS3.p1.5.m5.1.1.1.1.1.2">𝜽</ci><cn id="S1.SS3.p1.5.m5.1.1.1.1.1.3.cmml" type="integer" xref="S1.SS3.p1.5.m5.1.1.1.1.1.3">1</cn></apply><apply id="S1.SS3.p1.5.m5.2.2.2.2.2.cmml" xref="S1.SS3.p1.5.m5.2.2.2.2.2"><csymbol cd="ambiguous" id="S1.SS3.p1.5.m5.2.2.2.2.2.1.cmml" xref="S1.SS3.p1.5.m5.2.2.2.2.2">subscript</csymbol><ci id="S1.SS3.p1.5.m5.2.2.2.2.2.2.cmml" xref="S1.SS3.p1.5.m5.2.2.2.2.2.2">𝜽</ci><cn id="S1.SS3.p1.5.m5.2.2.2.2.2.3.cmml" type="integer" xref="S1.SS3.p1.5.m5.2.2.2.2.2.3">2</cn></apply><apply id="S1.SS3.p1.5.m5.3.3.3.3.3.cmml" xref="S1.SS3.p1.5.m5.3.3.3.3.3"><csymbol cd="ambiguous" id="S1.SS3.p1.5.m5.3.3.3.3.3.1.cmml" xref="S1.SS3.p1.5.m5.3.3.3.3.3">subscript</csymbol><ci id="S1.SS3.p1.5.m5.3.3.3.3.3.2.cmml" xref="S1.SS3.p1.5.m5.3.3.3.3.3.2">𝜽</ci><cn id="S1.SS3.p1.5.m5.3.3.3.3.3.3.cmml" type="integer" xref="S1.SS3.p1.5.m5.3.3.3.3.3.3">3</cn></apply></vector></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS3.p1.5.m5.3c">\operatorname{{\bm{\theta}}}=(\operatorname{{\bm{\theta}}}_{1},\operatorname{{% \bm{\theta}}}_{2},\operatorname{{\bm{\theta}}}_{3})</annotation><annotation encoding="application/x-llamapun" id="S1.SS3.p1.5.m5.3d">bold_italic_θ = ( bold_italic_θ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , bold_italic_θ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , bold_italic_θ start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT )</annotation></semantics></math>, and <math alttext="\operatorname{\mathcal{P}}_{\lambda_{\operatorname{\mathbf{B}}}}(\operatorname% {{\bm{\theta}}}_{1})" class="ltx_Math" display="inline" id="S1.SS3.p1.6.m6.2"><semantics id="S1.SS3.p1.6.m6.2a"><mrow id="S1.SS3.p1.6.m6.2.2.2" xref="S1.SS3.p1.6.m6.2.2.3.cmml"><msub id="S1.SS3.p1.6.m6.1.1.1.1" 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encoding="MathML-Content" id="S1.SS3.p1.6.m6.2b"><apply id="S1.SS3.p1.6.m6.2.2.3.cmml" xref="S1.SS3.p1.6.m6.2.2.2"><apply id="S1.SS3.p1.6.m6.1.1.1.1.cmml" xref="S1.SS3.p1.6.m6.1.1.1.1"><csymbol cd="ambiguous" id="S1.SS3.p1.6.m6.1.1.1.1.1.cmml" xref="S1.SS3.p1.6.m6.1.1.1.1">subscript</csymbol><ci id="S1.SS3.p1.6.m6.1.1.1.1.2.cmml" xref="S1.SS3.p1.6.m6.1.1.1.1.2">𝒫</ci><apply id="S1.SS3.p1.6.m6.1.1.1.1.3.cmml" xref="S1.SS3.p1.6.m6.1.1.1.1.3"><csymbol cd="ambiguous" id="S1.SS3.p1.6.m6.1.1.1.1.3.1.cmml" xref="S1.SS3.p1.6.m6.1.1.1.1.3">subscript</csymbol><ci id="S1.SS3.p1.6.m6.1.1.1.1.3.2.cmml" xref="S1.SS3.p1.6.m6.1.1.1.1.3.2">𝜆</ci><ci id="S1.SS3.p1.6.m6.1.1.1.1.3.3.cmml" xref="S1.SS3.p1.6.m6.1.1.1.1.3.3">𝐁</ci></apply></apply><apply id="S1.SS3.p1.6.m6.2.2.2.2.1.cmml" xref="S1.SS3.p1.6.m6.2.2.2.2.1"><csymbol cd="ambiguous" id="S1.SS3.p1.6.m6.2.2.2.2.1.1.cmml" xref="S1.SS3.p1.6.m6.2.2.2.2.1">subscript</csymbol><ci id="S1.SS3.p1.6.m6.2.2.2.2.1.2.cmml" xref="S1.SS3.p1.6.m6.2.2.2.2.1.2">𝜽</ci><cn id="S1.SS3.p1.6.m6.2.2.2.2.1.3.cmml" type="integer" xref="S1.SS3.p1.6.m6.2.2.2.2.1.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS3.p1.6.m6.2c">\operatorname{\mathcal{P}}_{\lambda_{\operatorname{\mathbf{B}}}}(\operatorname% {{\bm{\theta}}}_{1})</annotation><annotation encoding="application/x-llamapun" id="S1.SS3.p1.6.m6.2d">caligraphic_P start_POSTSUBSCRIPT italic_λ start_POSTSUBSCRIPT bold_B end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( bold_italic_θ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT )</annotation></semantics></math>, <math alttext="\operatorname{\mathcal{P}}_{\lambda_{\operatorname{\mathbf{L}}}}(\operatorname% {{\bm{\theta}}}_{2})" class="ltx_Math" display="inline" id="S1.SS3.p1.7.m7.2"><semantics id="S1.SS3.p1.7.m7.2a"><mrow id="S1.SS3.p1.7.m7.2.2.2" xref="S1.SS3.p1.7.m7.2.2.3.cmml"><msub id="S1.SS3.p1.7.m7.1.1.1.1" xref="S1.SS3.p1.7.m7.1.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S1.SS3.p1.7.m7.1.1.1.1.2" xref="S1.SS3.p1.7.m7.1.1.1.1.2.cmml">𝒫</mi><msub id="S1.SS3.p1.7.m7.1.1.1.1.3" xref="S1.SS3.p1.7.m7.1.1.1.1.3.cmml"><mi id="S1.SS3.p1.7.m7.1.1.1.1.3.2" xref="S1.SS3.p1.7.m7.1.1.1.1.3.2.cmml">λ</mi><mi id="S1.SS3.p1.7.m7.1.1.1.1.3.3" xref="S1.SS3.p1.7.m7.1.1.1.1.3.3.cmml">𝐋</mi></msub></msub><mo id="S1.SS3.p1.7.m7.2.2.2a" xref="S1.SS3.p1.7.m7.2.2.3.cmml">⁡</mo><mrow id="S1.SS3.p1.7.m7.2.2.2.2" xref="S1.SS3.p1.7.m7.2.2.3.cmml"><mo id="S1.SS3.p1.7.m7.2.2.2.2.2" stretchy="false" xref="S1.SS3.p1.7.m7.2.2.3.cmml">(</mo><msub id="S1.SS3.p1.7.m7.2.2.2.2.1" xref="S1.SS3.p1.7.m7.2.2.2.2.1.cmml"><mi id="S1.SS3.p1.7.m7.2.2.2.2.1.2" xref="S1.SS3.p1.7.m7.2.2.2.2.1.2.cmml">𝜽</mi><mn id="S1.SS3.p1.7.m7.2.2.2.2.1.3" xref="S1.SS3.p1.7.m7.2.2.2.2.1.3.cmml">2</mn></msub><mo id="S1.SS3.p1.7.m7.2.2.2.2.3" stretchy="false" xref="S1.SS3.p1.7.m7.2.2.3.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.SS3.p1.7.m7.2b"><apply id="S1.SS3.p1.7.m7.2.2.3.cmml" xref="S1.SS3.p1.7.m7.2.2.2"><apply id="S1.SS3.p1.7.m7.1.1.1.1.cmml" xref="S1.SS3.p1.7.m7.1.1.1.1"><csymbol cd="ambiguous" id="S1.SS3.p1.7.m7.1.1.1.1.1.cmml" xref="S1.SS3.p1.7.m7.1.1.1.1">subscript</csymbol><ci id="S1.SS3.p1.7.m7.1.1.1.1.2.cmml" xref="S1.SS3.p1.7.m7.1.1.1.1.2">𝒫</ci><apply id="S1.SS3.p1.7.m7.1.1.1.1.3.cmml" xref="S1.SS3.p1.7.m7.1.1.1.1.3"><csymbol cd="ambiguous" id="S1.SS3.p1.7.m7.1.1.1.1.3.1.cmml" xref="S1.SS3.p1.7.m7.1.1.1.1.3">subscript</csymbol><ci id="S1.SS3.p1.7.m7.1.1.1.1.3.2.cmml" xref="S1.SS3.p1.7.m7.1.1.1.1.3.2">𝜆</ci><ci id="S1.SS3.p1.7.m7.1.1.1.1.3.3.cmml" xref="S1.SS3.p1.7.m7.1.1.1.1.3.3">𝐋</ci></apply></apply><apply id="S1.SS3.p1.7.m7.2.2.2.2.1.cmml" xref="S1.SS3.p1.7.m7.2.2.2.2.1"><csymbol cd="ambiguous" id="S1.SS3.p1.7.m7.2.2.2.2.1.1.cmml" xref="S1.SS3.p1.7.m7.2.2.2.2.1">subscript</csymbol><ci id="S1.SS3.p1.7.m7.2.2.2.2.1.2.cmml" xref="S1.SS3.p1.7.m7.2.2.2.2.1.2">𝜽</ci><cn id="S1.SS3.p1.7.m7.2.2.2.2.1.3.cmml" type="integer" xref="S1.SS3.p1.7.m7.2.2.2.2.1.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS3.p1.7.m7.2c">\operatorname{\mathcal{P}}_{\lambda_{\operatorname{\mathbf{L}}}}(\operatorname% {{\bm{\theta}}}_{2})</annotation><annotation encoding="application/x-llamapun" id="S1.SS3.p1.7.m7.2d">caligraphic_P start_POSTSUBSCRIPT italic_λ start_POSTSUBSCRIPT bold_L end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( bold_italic_θ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT )</annotation></semantics></math>, and <math alttext="\operatorname{\mathcal{P}}_{\lambda_{\operatorname{\mathbf{L}}}}(\operatorname% {{\bm{\theta}}}_{3})" class="ltx_Math" display="inline" id="S1.SS3.p1.8.m8.2"><semantics id="S1.SS3.p1.8.m8.2a"><mrow id="S1.SS3.p1.8.m8.2.2.2" xref="S1.SS3.p1.8.m8.2.2.3.cmml"><msub id="S1.SS3.p1.8.m8.1.1.1.1" xref="S1.SS3.p1.8.m8.1.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S1.SS3.p1.8.m8.1.1.1.1.2" xref="S1.SS3.p1.8.m8.1.1.1.1.2.cmml">𝒫</mi><msub id="S1.SS3.p1.8.m8.1.1.1.1.3" xref="S1.SS3.p1.8.m8.1.1.1.1.3.cmml"><mi id="S1.SS3.p1.8.m8.1.1.1.1.3.2" xref="S1.SS3.p1.8.m8.1.1.1.1.3.2.cmml">λ</mi><mi id="S1.SS3.p1.8.m8.1.1.1.1.3.3" xref="S1.SS3.p1.8.m8.1.1.1.1.3.3.cmml">𝐋</mi></msub></msub><mo id="S1.SS3.p1.8.m8.2.2.2a" xref="S1.SS3.p1.8.m8.2.2.3.cmml">⁡</mo><mrow id="S1.SS3.p1.8.m8.2.2.2.2" xref="S1.SS3.p1.8.m8.2.2.3.cmml"><mo id="S1.SS3.p1.8.m8.2.2.2.2.2" stretchy="false" xref="S1.SS3.p1.8.m8.2.2.3.cmml">(</mo><msub id="S1.SS3.p1.8.m8.2.2.2.2.1" xref="S1.SS3.p1.8.m8.2.2.2.2.1.cmml"><mi id="S1.SS3.p1.8.m8.2.2.2.2.1.2" xref="S1.SS3.p1.8.m8.2.2.2.2.1.2.cmml">𝜽</mi><mn id="S1.SS3.p1.8.m8.2.2.2.2.1.3" xref="S1.SS3.p1.8.m8.2.2.2.2.1.3.cmml">3</mn></msub><mo id="S1.SS3.p1.8.m8.2.2.2.2.3" stretchy="false" xref="S1.SS3.p1.8.m8.2.2.3.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.SS3.p1.8.m8.2b"><apply id="S1.SS3.p1.8.m8.2.2.3.cmml" xref="S1.SS3.p1.8.m8.2.2.2"><apply id="S1.SS3.p1.8.m8.1.1.1.1.cmml" xref="S1.SS3.p1.8.m8.1.1.1.1"><csymbol cd="ambiguous" id="S1.SS3.p1.8.m8.1.1.1.1.1.cmml" xref="S1.SS3.p1.8.m8.1.1.1.1">subscript</csymbol><ci id="S1.SS3.p1.8.m8.1.1.1.1.2.cmml" xref="S1.SS3.p1.8.m8.1.1.1.1.2">𝒫</ci><apply id="S1.SS3.p1.8.m8.1.1.1.1.3.cmml" xref="S1.SS3.p1.8.m8.1.1.1.1.3"><csymbol cd="ambiguous" id="S1.SS3.p1.8.m8.1.1.1.1.3.1.cmml" xref="S1.SS3.p1.8.m8.1.1.1.1.3">subscript</csymbol><ci id="S1.SS3.p1.8.m8.1.1.1.1.3.2.cmml" xref="S1.SS3.p1.8.m8.1.1.1.1.3.2">𝜆</ci><ci id="S1.SS3.p1.8.m8.1.1.1.1.3.3.cmml" xref="S1.SS3.p1.8.m8.1.1.1.1.3.3">𝐋</ci></apply></apply><apply id="S1.SS3.p1.8.m8.2.2.2.2.1.cmml" xref="S1.SS3.p1.8.m8.2.2.2.2.1"><csymbol cd="ambiguous" id="S1.SS3.p1.8.m8.2.2.2.2.1.1.cmml" xref="S1.SS3.p1.8.m8.2.2.2.2.1">subscript</csymbol><ci id="S1.SS3.p1.8.m8.2.2.2.2.1.2.cmml" xref="S1.SS3.p1.8.m8.2.2.2.2.1.2">𝜽</ci><cn id="S1.SS3.p1.8.m8.2.2.2.2.1.3.cmml" type="integer" xref="S1.SS3.p1.8.m8.2.2.2.2.1.3">3</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS3.p1.8.m8.2c">\operatorname{\mathcal{P}}_{\lambda_{\operatorname{\mathbf{L}}}}(\operatorname% {{\bm{\theta}}}_{3})</annotation><annotation encoding="application/x-llamapun" id="S1.SS3.p1.8.m8.2d">caligraphic_P start_POSTSUBSCRIPT italic_λ start_POSTSUBSCRIPT bold_L end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( bold_italic_θ start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT )</annotation></semantics></math> are the penalties on <math alttext="\operatorname{\mathbf{B}}" class="ltx_Math" display="inline" id="S1.SS3.p1.9.m9.1"><semantics id="S1.SS3.p1.9.m9.1a"><mi id="S1.SS3.p1.9.m9.1.1" xref="S1.SS3.p1.9.m9.1.1.cmml">𝐁</mi><annotation-xml encoding="MathML-Content" id="S1.SS3.p1.9.m9.1b"><ci id="S1.SS3.p1.9.m9.1.1.cmml" xref="S1.SS3.p1.9.m9.1.1">𝐁</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS3.p1.9.m9.1c">\operatorname{\mathbf{B}}</annotation><annotation encoding="application/x-llamapun" id="S1.SS3.p1.9.m9.1d">bold_B</annotation></semantics></math>, <math alttext="\operatorname{\mathbf{L}}_{1}" class="ltx_Math" display="inline" id="S1.SS3.p1.10.m10.1"><semantics id="S1.SS3.p1.10.m10.1a"><msub id="S1.SS3.p1.10.m10.1.1" xref="S1.SS3.p1.10.m10.1.1.cmml"><mi id="S1.SS3.p1.10.m10.1.1.2" xref="S1.SS3.p1.10.m10.1.1.2.cmml">𝐋</mi><mn id="S1.SS3.p1.10.m10.1.1.3" xref="S1.SS3.p1.10.m10.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S1.SS3.p1.10.m10.1b"><apply id="S1.SS3.p1.10.m10.1.1.cmml" xref="S1.SS3.p1.10.m10.1.1"><csymbol cd="ambiguous" id="S1.SS3.p1.10.m10.1.1.1.cmml" xref="S1.SS3.p1.10.m10.1.1">subscript</csymbol><ci id="S1.SS3.p1.10.m10.1.1.2.cmml" xref="S1.SS3.p1.10.m10.1.1.2">𝐋</ci><cn id="S1.SS3.p1.10.m10.1.1.3.cmml" type="integer" xref="S1.SS3.p1.10.m10.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS3.p1.10.m10.1c">\operatorname{\mathbf{L}}_{1}</annotation><annotation encoding="application/x-llamapun" id="S1.SS3.p1.10.m10.1d">bold_L start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math>, and <math alttext="\operatorname{\mathbf{L}}_{2}" class="ltx_Math" display="inline" id="S1.SS3.p1.11.m11.1"><semantics id="S1.SS3.p1.11.m11.1a"><msub id="S1.SS3.p1.11.m11.1.1" xref="S1.SS3.p1.11.m11.1.1.cmml"><mi id="S1.SS3.p1.11.m11.1.1.2" xref="S1.SS3.p1.11.m11.1.1.2.cmml">𝐋</mi><mn id="S1.SS3.p1.11.m11.1.1.3" xref="S1.SS3.p1.11.m11.1.1.3.cmml">2</mn></msub><annotation-xml encoding="MathML-Content" id="S1.SS3.p1.11.m11.1b"><apply id="S1.SS3.p1.11.m11.1.1.cmml" xref="S1.SS3.p1.11.m11.1.1"><csymbol cd="ambiguous" id="S1.SS3.p1.11.m11.1.1.1.cmml" xref="S1.SS3.p1.11.m11.1.1">subscript</csymbol><ci id="S1.SS3.p1.11.m11.1.1.2.cmml" xref="S1.SS3.p1.11.m11.1.1.2">𝐋</ci><cn id="S1.SS3.p1.11.m11.1.1.3.cmml" type="integer" xref="S1.SS3.p1.11.m11.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS3.p1.11.m11.1c">\operatorname{\mathbf{L}}_{2}</annotation><annotation encoding="application/x-llamapun" id="S1.SS3.p1.11.m11.1d">bold_L start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math>, respectively. Then, the Proposition 3.1 from the main manuscript is as follows.</p> </div> <div class="ltx_theorem ltx_theorem_proposition" id="S1.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="S1.Thmproposition3.1.1.1">Proposition 1.3</span></span></h6> <div class="ltx_para" id="S1.Thmproposition3.p1"> <p class="ltx_p" id="S1.Thmproposition3.p1.9"><span class="ltx_text ltx_font_italic" id="S1.Thmproposition3.p1.9.9">The MMTR objective is <math alttext="f(\operatorname{{\bm{\theta}}})=\ell(\operatorname{{\bm{\theta}}})+% \operatorname{\mathcal{P}}_{\lambda_{\operatorname{\mathbf{B}}}}(\operatorname% {{\bm{\theta}}}_{1})+\operatorname{\mathcal{P}}_{\lambda_{\operatorname{% \mathbf{L}}}}(\operatorname{{\bm{\theta}}}_{2})+\operatorname{\mathcal{P}}_{% \lambda_{\operatorname{\mathbf{L}}}}(\operatorname{{\bm{\theta}}}_{3})" class="ltx_Math" display="inline" id="S1.Thmproposition3.p1.1.1.m1.8"><semantics 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xref="S1.Thmproposition3.p1.1.1.m1.6.6.4.4.2.2.1">subscript</csymbol><ci id="S1.Thmproposition3.p1.1.1.m1.6.6.4.4.2.2.1.2.cmml" xref="S1.Thmproposition3.p1.1.1.m1.6.6.4.4.2.2.1.2">𝛉</ci><cn id="S1.Thmproposition3.p1.1.1.m1.6.6.4.4.2.2.1.3.cmml" type="integer" xref="S1.Thmproposition3.p1.1.1.m1.6.6.4.4.2.2.1.3">2</cn></apply></apply><apply id="S1.Thmproposition3.p1.1.1.m1.8.8.6.6.3.cmml" xref="S1.Thmproposition3.p1.1.1.m1.8.8.6.6.2"><apply id="S1.Thmproposition3.p1.1.1.m1.7.7.5.5.1.1.cmml" xref="S1.Thmproposition3.p1.1.1.m1.7.7.5.5.1.1"><csymbol cd="ambiguous" id="S1.Thmproposition3.p1.1.1.m1.7.7.5.5.1.1.1.cmml" xref="S1.Thmproposition3.p1.1.1.m1.7.7.5.5.1.1">subscript</csymbol><ci id="S1.Thmproposition3.p1.1.1.m1.7.7.5.5.1.1.2.cmml" xref="S1.Thmproposition3.p1.1.1.m1.7.7.5.5.1.1.2">𝒫</ci><apply id="S1.Thmproposition3.p1.1.1.m1.7.7.5.5.1.1.3.cmml" xref="S1.Thmproposition3.p1.1.1.m1.7.7.5.5.1.1.3"><csymbol cd="ambiguous" id="S1.Thmproposition3.p1.1.1.m1.7.7.5.5.1.1.3.1.cmml" xref="S1.Thmproposition3.p1.1.1.m1.7.7.5.5.1.1.3">subscript</csymbol><ci id="S1.Thmproposition3.p1.1.1.m1.7.7.5.5.1.1.3.2.cmml" xref="S1.Thmproposition3.p1.1.1.m1.7.7.5.5.1.1.3.2">𝜆</ci><ci id="S1.Thmproposition3.p1.1.1.m1.7.7.5.5.1.1.3.3.cmml" xref="S1.Thmproposition3.p1.1.1.m1.7.7.5.5.1.1.3.3">𝐋</ci></apply></apply><apply id="S1.Thmproposition3.p1.1.1.m1.8.8.6.6.2.2.1.cmml" xref="S1.Thmproposition3.p1.1.1.m1.8.8.6.6.2.2.1"><csymbol cd="ambiguous" id="S1.Thmproposition3.p1.1.1.m1.8.8.6.6.2.2.1.1.cmml" xref="S1.Thmproposition3.p1.1.1.m1.8.8.6.6.2.2.1">subscript</csymbol><ci id="S1.Thmproposition3.p1.1.1.m1.8.8.6.6.2.2.1.2.cmml" xref="S1.Thmproposition3.p1.1.1.m1.8.8.6.6.2.2.1.2">𝛉</ci><cn id="S1.Thmproposition3.p1.1.1.m1.8.8.6.6.2.2.1.3.cmml" type="integer" xref="S1.Thmproposition3.p1.1.1.m1.8.8.6.6.2.2.1.3">3</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmproposition3.p1.1.1.m1.8c">f(\operatorname{{\bm{\theta}}})=\ell(\operatorname{{\bm{\theta}}})+% \operatorname{\mathcal{P}}_{\lambda_{\operatorname{\mathbf{B}}}}(\operatorname% {{\bm{\theta}}}_{1})+\operatorname{\mathcal{P}}_{\lambda_{\operatorname{% \mathbf{L}}}}(\operatorname{{\bm{\theta}}}_{2})+\operatorname{\mathcal{P}}_{% \lambda_{\operatorname{\mathbf{L}}}}(\operatorname{{\bm{\theta}}}_{3})</annotation><annotation encoding="application/x-llamapun" id="S1.Thmproposition3.p1.1.1.m1.8d">italic_f ( bold_italic_θ ) = roman_ℓ ( bold_italic_θ ) + caligraphic_P start_POSTSUBSCRIPT italic_λ start_POSTSUBSCRIPT bold_B end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( bold_italic_θ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) + caligraphic_P start_POSTSUBSCRIPT italic_λ start_POSTSUBSCRIPT bold_L end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( bold_italic_θ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) + caligraphic_P start_POSTSUBSCRIPT italic_λ start_POSTSUBSCRIPT bold_L end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( bold_italic_θ start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT )</annotation></semantics></math>. Let <math alttext="\operatorname{\mathcal{M}}(\cdot)" class="ltx_Math" display="inline" id="S1.Thmproposition3.p1.2.2.m2.2"><semantics id="S1.Thmproposition3.p1.2.2.m2.2a"><mrow id="S1.Thmproposition3.p1.2.2.m2.2.3.2" xref="S1.Thmproposition3.p1.2.2.m2.2.3.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S1.Thmproposition3.p1.2.2.m2.1.1" xref="S1.Thmproposition3.p1.2.2.m2.1.1.cmml">ℳ</mi><mo id="S1.Thmproposition3.p1.2.2.m2.2.3.2a" xref="S1.Thmproposition3.p1.2.2.m2.2.3.1.cmml">⁡</mo><mrow id="S1.Thmproposition3.p1.2.2.m2.2.3.2.1" xref="S1.Thmproposition3.p1.2.2.m2.2.3.1.cmml"><mo id="S1.Thmproposition3.p1.2.2.m2.2.3.2.1.1" stretchy="false" xref="S1.Thmproposition3.p1.2.2.m2.2.3.1.cmml">(</mo><mo id="S1.Thmproposition3.p1.2.2.m2.2.2" lspace="0em" rspace="0em" xref="S1.Thmproposition3.p1.2.2.m2.2.2.cmml">⋅</mo><mo id="S1.Thmproposition3.p1.2.2.m2.2.3.2.1.2" stretchy="false" xref="S1.Thmproposition3.p1.2.2.m2.2.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.Thmproposition3.p1.2.2.m2.2b"><apply id="S1.Thmproposition3.p1.2.2.m2.2.3.1.cmml" xref="S1.Thmproposition3.p1.2.2.m2.2.3.2"><ci id="S1.Thmproposition3.p1.2.2.m2.1.1.cmml" xref="S1.Thmproposition3.p1.2.2.m2.1.1">ℳ</ci><ci id="S1.Thmproposition3.p1.2.2.m2.2.2.cmml" xref="S1.Thmproposition3.p1.2.2.m2.2.2">⋅</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmproposition3.p1.2.2.m2.2c">\operatorname{\mathcal{M}}(\cdot)</annotation><annotation encoding="application/x-llamapun" id="S1.Thmproposition3.p1.2.2.m2.2d">caligraphic_M ( ⋅ )</annotation></semantics></math> be the function that maps <math alttext="\operatorname{{\bm{\theta}}}^{(t)}" class="ltx_Math" display="inline" id="S1.Thmproposition3.p1.3.3.m3.1"><semantics id="S1.Thmproposition3.p1.3.3.m3.1a"><msup id="S1.Thmproposition3.p1.3.3.m3.1.2" xref="S1.Thmproposition3.p1.3.3.m3.1.2.cmml"><mi id="S1.Thmproposition3.p1.3.3.m3.1.2.2" xref="S1.Thmproposition3.p1.3.3.m3.1.2.2.cmml">𝛉</mi><mrow id="S1.Thmproposition3.p1.3.3.m3.1.1.1.3" xref="S1.Thmproposition3.p1.3.3.m3.1.2.cmml"><mo id="S1.Thmproposition3.p1.3.3.m3.1.1.1.3.1" stretchy="false" xref="S1.Thmproposition3.p1.3.3.m3.1.2.cmml">(</mo><mi id="S1.Thmproposition3.p1.3.3.m3.1.1.1.1" xref="S1.Thmproposition3.p1.3.3.m3.1.1.1.1.cmml">t</mi><mo id="S1.Thmproposition3.p1.3.3.m3.1.1.1.3.2" stretchy="false" xref="S1.Thmproposition3.p1.3.3.m3.1.2.cmml">)</mo></mrow></msup><annotation-xml encoding="MathML-Content" id="S1.Thmproposition3.p1.3.3.m3.1b"><apply id="S1.Thmproposition3.p1.3.3.m3.1.2.cmml" xref="S1.Thmproposition3.p1.3.3.m3.1.2"><csymbol cd="ambiguous" id="S1.Thmproposition3.p1.3.3.m3.1.2.1.cmml" xref="S1.Thmproposition3.p1.3.3.m3.1.2">superscript</csymbol><ci id="S1.Thmproposition3.p1.3.3.m3.1.2.2.cmml" xref="S1.Thmproposition3.p1.3.3.m3.1.2.2">𝛉</ci><ci id="S1.Thmproposition3.p1.3.3.m3.1.1.1.1.cmml" xref="S1.Thmproposition3.p1.3.3.m3.1.1.1.1">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmproposition3.p1.3.3.m3.1c">\operatorname{{\bm{\theta}}}^{(t)}</annotation><annotation encoding="application/x-llamapun" id="S1.Thmproposition3.p1.3.3.m3.1d">bold_italic_θ start_POSTSUPERSCRIPT ( italic_t ) end_POSTSUPERSCRIPT</annotation></semantics></math> to <math alttext="\operatorname{{\bm{\theta}}}^{(t+1)}" class="ltx_Math" display="inline" id="S1.Thmproposition3.p1.4.4.m4.1"><semantics id="S1.Thmproposition3.p1.4.4.m4.1a"><msup id="S1.Thmproposition3.p1.4.4.m4.1.2" xref="S1.Thmproposition3.p1.4.4.m4.1.2.cmml"><mi id="S1.Thmproposition3.p1.4.4.m4.1.2.2" xref="S1.Thmproposition3.p1.4.4.m4.1.2.2.cmml">𝛉</mi><mrow id="S1.Thmproposition3.p1.4.4.m4.1.1.1.1" xref="S1.Thmproposition3.p1.4.4.m4.1.1.1.1.1.cmml"><mo id="S1.Thmproposition3.p1.4.4.m4.1.1.1.1.2" stretchy="false" xref="S1.Thmproposition3.p1.4.4.m4.1.1.1.1.1.cmml">(</mo><mrow id="S1.Thmproposition3.p1.4.4.m4.1.1.1.1.1" xref="S1.Thmproposition3.p1.4.4.m4.1.1.1.1.1.cmml"><mi id="S1.Thmproposition3.p1.4.4.m4.1.1.1.1.1.2" xref="S1.Thmproposition3.p1.4.4.m4.1.1.1.1.1.2.cmml">t</mi><mo id="S1.Thmproposition3.p1.4.4.m4.1.1.1.1.1.1" xref="S1.Thmproposition3.p1.4.4.m4.1.1.1.1.1.1.cmml">+</mo><mn id="S1.Thmproposition3.p1.4.4.m4.1.1.1.1.1.3" xref="S1.Thmproposition3.p1.4.4.m4.1.1.1.1.1.3.cmml">1</mn></mrow><mo id="S1.Thmproposition3.p1.4.4.m4.1.1.1.1.3" stretchy="false" xref="S1.Thmproposition3.p1.4.4.m4.1.1.1.1.1.cmml">)</mo></mrow></msup><annotation-xml encoding="MathML-Content" id="S1.Thmproposition3.p1.4.4.m4.1b"><apply id="S1.Thmproposition3.p1.4.4.m4.1.2.cmml" xref="S1.Thmproposition3.p1.4.4.m4.1.2"><csymbol cd="ambiguous" id="S1.Thmproposition3.p1.4.4.m4.1.2.1.cmml" xref="S1.Thmproposition3.p1.4.4.m4.1.2">superscript</csymbol><ci id="S1.Thmproposition3.p1.4.4.m4.1.2.2.cmml" xref="S1.Thmproposition3.p1.4.4.m4.1.2.2">𝛉</ci><apply id="S1.Thmproposition3.p1.4.4.m4.1.1.1.1.1.cmml" xref="S1.Thmproposition3.p1.4.4.m4.1.1.1.1"><plus id="S1.Thmproposition3.p1.4.4.m4.1.1.1.1.1.1.cmml" xref="S1.Thmproposition3.p1.4.4.m4.1.1.1.1.1.1"></plus><ci id="S1.Thmproposition3.p1.4.4.m4.1.1.1.1.1.2.cmml" xref="S1.Thmproposition3.p1.4.4.m4.1.1.1.1.1.2">𝑡</ci><cn id="S1.Thmproposition3.p1.4.4.m4.1.1.1.1.1.3.cmml" type="integer" xref="S1.Thmproposition3.p1.4.4.m4.1.1.1.1.1.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmproposition3.p1.4.4.m4.1c">\operatorname{{\bm{\theta}}}^{(t+1)}</annotation><annotation encoding="application/x-llamapun" id="S1.Thmproposition3.p1.4.4.m4.1d">bold_italic_θ start_POSTSUPERSCRIPT ( italic_t + 1 ) end_POSTSUPERSCRIPT</annotation></semantics></math> using Algorithm 1 in the main manuscript. Then, each iteration of Algorithm 1 does not increase <math alttext="f(\operatorname{{\bm{\theta}}})" class="ltx_Math" display="inline" id="S1.Thmproposition3.p1.5.5.m5.1"><semantics id="S1.Thmproposition3.p1.5.5.m5.1a"><mrow id="S1.Thmproposition3.p1.5.5.m5.1.2" xref="S1.Thmproposition3.p1.5.5.m5.1.2.cmml"><mi id="S1.Thmproposition3.p1.5.5.m5.1.2.2" xref="S1.Thmproposition3.p1.5.5.m5.1.2.2.cmml">f</mi><mo id="S1.Thmproposition3.p1.5.5.m5.1.2.1" xref="S1.Thmproposition3.p1.5.5.m5.1.2.1.cmml">⁢</mo><mrow id="S1.Thmproposition3.p1.5.5.m5.1.2.3.2" xref="S1.Thmproposition3.p1.5.5.m5.1.2.cmml"><mo id="S1.Thmproposition3.p1.5.5.m5.1.2.3.2.1" stretchy="false" xref="S1.Thmproposition3.p1.5.5.m5.1.2.cmml">(</mo><mi id="S1.Thmproposition3.p1.5.5.m5.1.1" xref="S1.Thmproposition3.p1.5.5.m5.1.1.cmml">𝛉</mi><mo id="S1.Thmproposition3.p1.5.5.m5.1.2.3.2.2" stretchy="false" xref="S1.Thmproposition3.p1.5.5.m5.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.Thmproposition3.p1.5.5.m5.1b"><apply id="S1.Thmproposition3.p1.5.5.m5.1.2.cmml" xref="S1.Thmproposition3.p1.5.5.m5.1.2"><times id="S1.Thmproposition3.p1.5.5.m5.1.2.1.cmml" xref="S1.Thmproposition3.p1.5.5.m5.1.2.1"></times><ci id="S1.Thmproposition3.p1.5.5.m5.1.2.2.cmml" xref="S1.Thmproposition3.p1.5.5.m5.1.2.2">𝑓</ci><ci id="S1.Thmproposition3.p1.5.5.m5.1.1.cmml" xref="S1.Thmproposition3.p1.5.5.m5.1.1">𝛉</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmproposition3.p1.5.5.m5.1c">f(\operatorname{{\bm{\theta}}})</annotation><annotation encoding="application/x-llamapun" id="S1.Thmproposition3.p1.5.5.m5.1d">italic_f ( bold_italic_θ )</annotation></semantics></math>. Furthermore, assume that the parameter space <math alttext="\operatorname{{\mathbf{\Theta}}}" class="ltx_Math" display="inline" id="S1.Thmproposition3.p1.6.6.m6.1"><semantics id="S1.Thmproposition3.p1.6.6.m6.1a"><mi id="S1.Thmproposition3.p1.6.6.m6.1.1" xref="S1.Thmproposition3.p1.6.6.m6.1.1.cmml">𝚯</mi><annotation-xml encoding="MathML-Content" id="S1.Thmproposition3.p1.6.6.m6.1b"><ci id="S1.Thmproposition3.p1.6.6.m6.1.1.cmml" xref="S1.Thmproposition3.p1.6.6.m6.1.1">𝚯</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmproposition3.p1.6.6.m6.1c">\operatorname{{\mathbf{\Theta}}}</annotation><annotation encoding="application/x-llamapun" id="S1.Thmproposition3.p1.6.6.m6.1d">bold_Θ</annotation></semantics></math> is compact and <math alttext="f(\operatorname{{\bm{\theta}}})=f\{\operatorname{\mathcal{M}}(\operatorname{{% \bm{\theta}}})\}" class="ltx_Math" display="inline" id="S1.Thmproposition3.p1.7.7.m7.4"><semantics id="S1.Thmproposition3.p1.7.7.m7.4a"><mrow id="S1.Thmproposition3.p1.7.7.m7.4.4" xref="S1.Thmproposition3.p1.7.7.m7.4.4.cmml"><mrow id="S1.Thmproposition3.p1.7.7.m7.4.4.3" xref="S1.Thmproposition3.p1.7.7.m7.4.4.3.cmml"><mi id="S1.Thmproposition3.p1.7.7.m7.4.4.3.2" xref="S1.Thmproposition3.p1.7.7.m7.4.4.3.2.cmml">f</mi><mo id="S1.Thmproposition3.p1.7.7.m7.4.4.3.1" xref="S1.Thmproposition3.p1.7.7.m7.4.4.3.1.cmml">⁢</mo><mrow id="S1.Thmproposition3.p1.7.7.m7.4.4.3.3.2" xref="S1.Thmproposition3.p1.7.7.m7.4.4.3.cmml"><mo id="S1.Thmproposition3.p1.7.7.m7.4.4.3.3.2.1" stretchy="false" xref="S1.Thmproposition3.p1.7.7.m7.4.4.3.cmml">(</mo><mi id="S1.Thmproposition3.p1.7.7.m7.1.1" xref="S1.Thmproposition3.p1.7.7.m7.1.1.cmml">𝛉</mi><mo id="S1.Thmproposition3.p1.7.7.m7.4.4.3.3.2.2" stretchy="false" xref="S1.Thmproposition3.p1.7.7.m7.4.4.3.cmml">)</mo></mrow></mrow><mo id="S1.Thmproposition3.p1.7.7.m7.4.4.2" xref="S1.Thmproposition3.p1.7.7.m7.4.4.2.cmml">=</mo><mrow id="S1.Thmproposition3.p1.7.7.m7.4.4.1" xref="S1.Thmproposition3.p1.7.7.m7.4.4.1.cmml"><mi id="S1.Thmproposition3.p1.7.7.m7.4.4.1.3" xref="S1.Thmproposition3.p1.7.7.m7.4.4.1.3.cmml">f</mi><mo id="S1.Thmproposition3.p1.7.7.m7.4.4.1.2" xref="S1.Thmproposition3.p1.7.7.m7.4.4.1.2.cmml">⁢</mo><mrow id="S1.Thmproposition3.p1.7.7.m7.4.4.1.1.1" xref="S1.Thmproposition3.p1.7.7.m7.4.4.1.1.2.cmml"><mo id="S1.Thmproposition3.p1.7.7.m7.4.4.1.1.1.2" stretchy="false" xref="S1.Thmproposition3.p1.7.7.m7.4.4.1.1.2.cmml">{</mo><mrow id="S1.Thmproposition3.p1.7.7.m7.4.4.1.1.1.1.2" xref="S1.Thmproposition3.p1.7.7.m7.4.4.1.1.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S1.Thmproposition3.p1.7.7.m7.2.2" xref="S1.Thmproposition3.p1.7.7.m7.2.2.cmml">ℳ</mi><mo id="S1.Thmproposition3.p1.7.7.m7.4.4.1.1.1.1.2a" xref="S1.Thmproposition3.p1.7.7.m7.4.4.1.1.1.1.1.cmml">⁡</mo><mrow id="S1.Thmproposition3.p1.7.7.m7.4.4.1.1.1.1.2.1" xref="S1.Thmproposition3.p1.7.7.m7.4.4.1.1.1.1.1.cmml"><mo id="S1.Thmproposition3.p1.7.7.m7.4.4.1.1.1.1.2.1.1" stretchy="false" xref="S1.Thmproposition3.p1.7.7.m7.4.4.1.1.1.1.1.cmml">(</mo><mi id="S1.Thmproposition3.p1.7.7.m7.3.3" xref="S1.Thmproposition3.p1.7.7.m7.3.3.cmml">𝛉</mi><mo id="S1.Thmproposition3.p1.7.7.m7.4.4.1.1.1.1.2.1.2" stretchy="false" xref="S1.Thmproposition3.p1.7.7.m7.4.4.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S1.Thmproposition3.p1.7.7.m7.4.4.1.1.1.3" stretchy="false" xref="S1.Thmproposition3.p1.7.7.m7.4.4.1.1.2.cmml">}</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.Thmproposition3.p1.7.7.m7.4b"><apply id="S1.Thmproposition3.p1.7.7.m7.4.4.cmml" xref="S1.Thmproposition3.p1.7.7.m7.4.4"><eq id="S1.Thmproposition3.p1.7.7.m7.4.4.2.cmml" xref="S1.Thmproposition3.p1.7.7.m7.4.4.2"></eq><apply id="S1.Thmproposition3.p1.7.7.m7.4.4.3.cmml" xref="S1.Thmproposition3.p1.7.7.m7.4.4.3"><times id="S1.Thmproposition3.p1.7.7.m7.4.4.3.1.cmml" xref="S1.Thmproposition3.p1.7.7.m7.4.4.3.1"></times><ci id="S1.Thmproposition3.p1.7.7.m7.4.4.3.2.cmml" xref="S1.Thmproposition3.p1.7.7.m7.4.4.3.2">𝑓</ci><ci id="S1.Thmproposition3.p1.7.7.m7.1.1.cmml" xref="S1.Thmproposition3.p1.7.7.m7.1.1">𝛉</ci></apply><apply id="S1.Thmproposition3.p1.7.7.m7.4.4.1.cmml" xref="S1.Thmproposition3.p1.7.7.m7.4.4.1"><times id="S1.Thmproposition3.p1.7.7.m7.4.4.1.2.cmml" xref="S1.Thmproposition3.p1.7.7.m7.4.4.1.2"></times><ci id="S1.Thmproposition3.p1.7.7.m7.4.4.1.3.cmml" xref="S1.Thmproposition3.p1.7.7.m7.4.4.1.3">𝑓</ci><set id="S1.Thmproposition3.p1.7.7.m7.4.4.1.1.2.cmml" xref="S1.Thmproposition3.p1.7.7.m7.4.4.1.1.1"><apply id="S1.Thmproposition3.p1.7.7.m7.4.4.1.1.1.1.1.cmml" xref="S1.Thmproposition3.p1.7.7.m7.4.4.1.1.1.1.2"><ci id="S1.Thmproposition3.p1.7.7.m7.2.2.cmml" xref="S1.Thmproposition3.p1.7.7.m7.2.2">ℳ</ci><ci id="S1.Thmproposition3.p1.7.7.m7.3.3.cmml" xref="S1.Thmproposition3.p1.7.7.m7.3.3">𝛉</ci></apply></set></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmproposition3.p1.7.7.m7.4c">f(\operatorname{{\bm{\theta}}})=f\{\operatorname{\mathcal{M}}(\operatorname{{% \bm{\theta}}})\}</annotation><annotation encoding="application/x-llamapun" id="S1.Thmproposition3.p1.7.7.m7.4d">italic_f ( bold_italic_θ ) = italic_f { caligraphic_M ( bold_italic_θ ) }</annotation></semantics></math> only for the stationary points of <math alttext="f(\operatorname{{\bm{\theta}}})" class="ltx_Math" display="inline" id="S1.Thmproposition3.p1.8.8.m8.1"><semantics id="S1.Thmproposition3.p1.8.8.m8.1a"><mrow id="S1.Thmproposition3.p1.8.8.m8.1.2" xref="S1.Thmproposition3.p1.8.8.m8.1.2.cmml"><mi id="S1.Thmproposition3.p1.8.8.m8.1.2.2" xref="S1.Thmproposition3.p1.8.8.m8.1.2.2.cmml">f</mi><mo id="S1.Thmproposition3.p1.8.8.m8.1.2.1" xref="S1.Thmproposition3.p1.8.8.m8.1.2.1.cmml">⁢</mo><mrow id="S1.Thmproposition3.p1.8.8.m8.1.2.3.2" xref="S1.Thmproposition3.p1.8.8.m8.1.2.cmml"><mo id="S1.Thmproposition3.p1.8.8.m8.1.2.3.2.1" stretchy="false" xref="S1.Thmproposition3.p1.8.8.m8.1.2.cmml">(</mo><mi id="S1.Thmproposition3.p1.8.8.m8.1.1" xref="S1.Thmproposition3.p1.8.8.m8.1.1.cmml">𝛉</mi><mo id="S1.Thmproposition3.p1.8.8.m8.1.2.3.2.2" stretchy="false" xref="S1.Thmproposition3.p1.8.8.m8.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.Thmproposition3.p1.8.8.m8.1b"><apply id="S1.Thmproposition3.p1.8.8.m8.1.2.cmml" xref="S1.Thmproposition3.p1.8.8.m8.1.2"><times id="S1.Thmproposition3.p1.8.8.m8.1.2.1.cmml" xref="S1.Thmproposition3.p1.8.8.m8.1.2.1"></times><ci id="S1.Thmproposition3.p1.8.8.m8.1.2.2.cmml" xref="S1.Thmproposition3.p1.8.8.m8.1.2.2">𝑓</ci><ci id="S1.Thmproposition3.p1.8.8.m8.1.1.cmml" xref="S1.Thmproposition3.p1.8.8.m8.1.1">𝛉</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmproposition3.p1.8.8.m8.1c">f(\operatorname{{\bm{\theta}}})</annotation><annotation encoding="application/x-llamapun" id="S1.Thmproposition3.p1.8.8.m8.1d">italic_f ( bold_italic_θ )</annotation></semantics></math>. Then, the <math alttext="\{\operatorname{{\bm{\theta}}}^{(t)}\}_{t=1}^{\infty}" class="ltx_Math" display="inline" id="S1.Thmproposition3.p1.9.9.m9.2"><semantics id="S1.Thmproposition3.p1.9.9.m9.2a"><msubsup id="S1.Thmproposition3.p1.9.9.m9.2.2" xref="S1.Thmproposition3.p1.9.9.m9.2.2.cmml"><mrow id="S1.Thmproposition3.p1.9.9.m9.2.2.1.1.1" xref="S1.Thmproposition3.p1.9.9.m9.2.2.1.1.2.cmml"><mo id="S1.Thmproposition3.p1.9.9.m9.2.2.1.1.1.2" stretchy="false" xref="S1.Thmproposition3.p1.9.9.m9.2.2.1.1.2.cmml">{</mo><msup id="S1.Thmproposition3.p1.9.9.m9.2.2.1.1.1.1" xref="S1.Thmproposition3.p1.9.9.m9.2.2.1.1.1.1.cmml"><mi id="S1.Thmproposition3.p1.9.9.m9.2.2.1.1.1.1.2" xref="S1.Thmproposition3.p1.9.9.m9.2.2.1.1.1.1.2.cmml">𝛉</mi><mrow id="S1.Thmproposition3.p1.9.9.m9.1.1.1.3" xref="S1.Thmproposition3.p1.9.9.m9.2.2.1.1.1.1.cmml"><mo id="S1.Thmproposition3.p1.9.9.m9.1.1.1.3.1" stretchy="false" xref="S1.Thmproposition3.p1.9.9.m9.2.2.1.1.1.1.cmml">(</mo><mi id="S1.Thmproposition3.p1.9.9.m9.1.1.1.1" xref="S1.Thmproposition3.p1.9.9.m9.1.1.1.1.cmml">t</mi><mo id="S1.Thmproposition3.p1.9.9.m9.1.1.1.3.2" stretchy="false" xref="S1.Thmproposition3.p1.9.9.m9.2.2.1.1.1.1.cmml">)</mo></mrow></msup><mo id="S1.Thmproposition3.p1.9.9.m9.2.2.1.1.1.3" stretchy="false" xref="S1.Thmproposition3.p1.9.9.m9.2.2.1.1.2.cmml">}</mo></mrow><mrow id="S1.Thmproposition3.p1.9.9.m9.2.2.1.3" xref="S1.Thmproposition3.p1.9.9.m9.2.2.1.3.cmml"><mi id="S1.Thmproposition3.p1.9.9.m9.2.2.1.3.2" xref="S1.Thmproposition3.p1.9.9.m9.2.2.1.3.2.cmml">t</mi><mo id="S1.Thmproposition3.p1.9.9.m9.2.2.1.3.1" xref="S1.Thmproposition3.p1.9.9.m9.2.2.1.3.1.cmml">=</mo><mn id="S1.Thmproposition3.p1.9.9.m9.2.2.1.3.3" xref="S1.Thmproposition3.p1.9.9.m9.2.2.1.3.3.cmml">1</mn></mrow><mi id="S1.Thmproposition3.p1.9.9.m9.2.2.3" mathvariant="normal" xref="S1.Thmproposition3.p1.9.9.m9.2.2.3.cmml">∞</mi></msubsup><annotation-xml encoding="MathML-Content" id="S1.Thmproposition3.p1.9.9.m9.2b"><apply id="S1.Thmproposition3.p1.9.9.m9.2.2.cmml" xref="S1.Thmproposition3.p1.9.9.m9.2.2"><csymbol cd="ambiguous" id="S1.Thmproposition3.p1.9.9.m9.2.2.2.cmml" xref="S1.Thmproposition3.p1.9.9.m9.2.2">superscript</csymbol><apply id="S1.Thmproposition3.p1.9.9.m9.2.2.1.cmml" xref="S1.Thmproposition3.p1.9.9.m9.2.2"><csymbol cd="ambiguous" id="S1.Thmproposition3.p1.9.9.m9.2.2.1.2.cmml" xref="S1.Thmproposition3.p1.9.9.m9.2.2">subscript</csymbol><set id="S1.Thmproposition3.p1.9.9.m9.2.2.1.1.2.cmml" xref="S1.Thmproposition3.p1.9.9.m9.2.2.1.1.1"><apply id="S1.Thmproposition3.p1.9.9.m9.2.2.1.1.1.1.cmml" xref="S1.Thmproposition3.p1.9.9.m9.2.2.1.1.1.1"><csymbol cd="ambiguous" id="S1.Thmproposition3.p1.9.9.m9.2.2.1.1.1.1.1.cmml" xref="S1.Thmproposition3.p1.9.9.m9.2.2.1.1.1.1">superscript</csymbol><ci id="S1.Thmproposition3.p1.9.9.m9.2.2.1.1.1.1.2.cmml" xref="S1.Thmproposition3.p1.9.9.m9.2.2.1.1.1.1.2">𝛉</ci><ci id="S1.Thmproposition3.p1.9.9.m9.1.1.1.1.cmml" xref="S1.Thmproposition3.p1.9.9.m9.1.1.1.1">𝑡</ci></apply></set><apply id="S1.Thmproposition3.p1.9.9.m9.2.2.1.3.cmml" xref="S1.Thmproposition3.p1.9.9.m9.2.2.1.3"><eq id="S1.Thmproposition3.p1.9.9.m9.2.2.1.3.1.cmml" xref="S1.Thmproposition3.p1.9.9.m9.2.2.1.3.1"></eq><ci id="S1.Thmproposition3.p1.9.9.m9.2.2.1.3.2.cmml" xref="S1.Thmproposition3.p1.9.9.m9.2.2.1.3.2">𝑡</ci><cn id="S1.Thmproposition3.p1.9.9.m9.2.2.1.3.3.cmml" type="integer" xref="S1.Thmproposition3.p1.9.9.m9.2.2.1.3.3">1</cn></apply></apply><infinity id="S1.Thmproposition3.p1.9.9.m9.2.2.3.cmml" xref="S1.Thmproposition3.p1.9.9.m9.2.2.3"></infinity></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmproposition3.p1.9.9.m9.2c">\{\operatorname{{\bm{\theta}}}^{(t)}\}_{t=1}^{\infty}</annotation><annotation encoding="application/x-llamapun" id="S1.Thmproposition3.p1.9.9.m9.2d">{ bold_italic_θ start_POSTSUPERSCRIPT ( italic_t ) end_POSTSUPERSCRIPT } start_POSTSUBSCRIPT italic_t = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT</annotation></semantics></math> sequence converges to a stationary point.</span></p> </div> </div> <div class="ltx_para ltx_noindent" id="S1.SS3.p2"> <p class="ltx_p" id="S1.SS3.p2.4"><span class="ltx_text ltx_font_bold" id="S1.SS3.p2.4.1">Proof </span>[Proof of Proposition <a class="ltx_ref" href="https://arxiv.org/html/2503.13782v1#S1.Thmproposition3" title="Proposition 1.3 ‣ 1.3 Proof of Proposition 3.1 ‣ 1 Proof of the Propositions ‣ Regularized Parameter Estimation in Mixed Model Trace Regression"><span class="ltx_text ltx_ref_tag">1.3</span></a>] Let <math alttext="\operatorname{{\bm{\theta}}}^{(t)}=(\operatorname{{\bm{\theta}}}_{1}^{(t)},% \operatorname{{\bm{\theta}}}_{2}^{(t)},\operatorname{{\bm{\theta}}}_{3}^{(t)})" class="ltx_Math" display="inline" id="S1.SS3.p2.1.m1.7"><semantics id="S1.SS3.p2.1.m1.7a"><mrow id="S1.SS3.p2.1.m1.7.7" xref="S1.SS3.p2.1.m1.7.7.cmml"><msup id="S1.SS3.p2.1.m1.7.7.5" xref="S1.SS3.p2.1.m1.7.7.5.cmml"><mi id="S1.SS3.p2.1.m1.7.7.5.2" xref="S1.SS3.p2.1.m1.7.7.5.2.cmml">𝜽</mi><mrow id="S1.SS3.p2.1.m1.1.1.1.3" xref="S1.SS3.p2.1.m1.7.7.5.cmml"><mo id="S1.SS3.p2.1.m1.1.1.1.3.1" stretchy="false" xref="S1.SS3.p2.1.m1.7.7.5.cmml">(</mo><mi id="S1.SS3.p2.1.m1.1.1.1.1" xref="S1.SS3.p2.1.m1.1.1.1.1.cmml">t</mi><mo id="S1.SS3.p2.1.m1.1.1.1.3.2" stretchy="false" xref="S1.SS3.p2.1.m1.7.7.5.cmml">)</mo></mrow></msup><mo id="S1.SS3.p2.1.m1.7.7.4" xref="S1.SS3.p2.1.m1.7.7.4.cmml">=</mo><mrow id="S1.SS3.p2.1.m1.7.7.3.3" xref="S1.SS3.p2.1.m1.7.7.3.4.cmml"><mo id="S1.SS3.p2.1.m1.7.7.3.3.4" stretchy="false" xref="S1.SS3.p2.1.m1.7.7.3.4.cmml">(</mo><msubsup id="S1.SS3.p2.1.m1.5.5.1.1.1" xref="S1.SS3.p2.1.m1.5.5.1.1.1.cmml"><mi id="S1.SS3.p2.1.m1.5.5.1.1.1.2.2" xref="S1.SS3.p2.1.m1.5.5.1.1.1.2.2.cmml">𝜽</mi><mn id="S1.SS3.p2.1.m1.5.5.1.1.1.2.3" xref="S1.SS3.p2.1.m1.5.5.1.1.1.2.3.cmml">1</mn><mrow id="S1.SS3.p2.1.m1.2.2.1.3" xref="S1.SS3.p2.1.m1.5.5.1.1.1.cmml"><mo id="S1.SS3.p2.1.m1.2.2.1.3.1" stretchy="false" xref="S1.SS3.p2.1.m1.5.5.1.1.1.cmml">(</mo><mi id="S1.SS3.p2.1.m1.2.2.1.1" xref="S1.SS3.p2.1.m1.2.2.1.1.cmml">t</mi><mo id="S1.SS3.p2.1.m1.2.2.1.3.2" stretchy="false" xref="S1.SS3.p2.1.m1.5.5.1.1.1.cmml">)</mo></mrow></msubsup><mo id="S1.SS3.p2.1.m1.7.7.3.3.5" xref="S1.SS3.p2.1.m1.7.7.3.4.cmml">,</mo><msubsup id="S1.SS3.p2.1.m1.6.6.2.2.2" xref="S1.SS3.p2.1.m1.6.6.2.2.2.cmml"><mi id="S1.SS3.p2.1.m1.6.6.2.2.2.2.2" xref="S1.SS3.p2.1.m1.6.6.2.2.2.2.2.cmml">𝜽</mi><mn id="S1.SS3.p2.1.m1.6.6.2.2.2.2.3" xref="S1.SS3.p2.1.m1.6.6.2.2.2.2.3.cmml">2</mn><mrow id="S1.SS3.p2.1.m1.3.3.1.3" xref="S1.SS3.p2.1.m1.6.6.2.2.2.cmml"><mo id="S1.SS3.p2.1.m1.3.3.1.3.1" stretchy="false" xref="S1.SS3.p2.1.m1.6.6.2.2.2.cmml">(</mo><mi id="S1.SS3.p2.1.m1.3.3.1.1" xref="S1.SS3.p2.1.m1.3.3.1.1.cmml">t</mi><mo id="S1.SS3.p2.1.m1.3.3.1.3.2" stretchy="false" xref="S1.SS3.p2.1.m1.6.6.2.2.2.cmml">)</mo></mrow></msubsup><mo id="S1.SS3.p2.1.m1.7.7.3.3.6" xref="S1.SS3.p2.1.m1.7.7.3.4.cmml">,</mo><msubsup id="S1.SS3.p2.1.m1.7.7.3.3.3" xref="S1.SS3.p2.1.m1.7.7.3.3.3.cmml"><mi id="S1.SS3.p2.1.m1.7.7.3.3.3.2.2" xref="S1.SS3.p2.1.m1.7.7.3.3.3.2.2.cmml">𝜽</mi><mn id="S1.SS3.p2.1.m1.7.7.3.3.3.2.3" xref="S1.SS3.p2.1.m1.7.7.3.3.3.2.3.cmml">3</mn><mrow id="S1.SS3.p2.1.m1.4.4.1.3" xref="S1.SS3.p2.1.m1.7.7.3.3.3.cmml"><mo id="S1.SS3.p2.1.m1.4.4.1.3.1" stretchy="false" xref="S1.SS3.p2.1.m1.7.7.3.3.3.cmml">(</mo><mi id="S1.SS3.p2.1.m1.4.4.1.1" xref="S1.SS3.p2.1.m1.4.4.1.1.cmml">t</mi><mo id="S1.SS3.p2.1.m1.4.4.1.3.2" stretchy="false" xref="S1.SS3.p2.1.m1.7.7.3.3.3.cmml">)</mo></mrow></msubsup><mo id="S1.SS3.p2.1.m1.7.7.3.3.7" stretchy="false" xref="S1.SS3.p2.1.m1.7.7.3.4.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.SS3.p2.1.m1.7b"><apply id="S1.SS3.p2.1.m1.7.7.cmml" xref="S1.SS3.p2.1.m1.7.7"><eq id="S1.SS3.p2.1.m1.7.7.4.cmml" xref="S1.SS3.p2.1.m1.7.7.4"></eq><apply id="S1.SS3.p2.1.m1.7.7.5.cmml" xref="S1.SS3.p2.1.m1.7.7.5"><csymbol cd="ambiguous" id="S1.SS3.p2.1.m1.7.7.5.1.cmml" xref="S1.SS3.p2.1.m1.7.7.5">superscript</csymbol><ci id="S1.SS3.p2.1.m1.7.7.5.2.cmml" xref="S1.SS3.p2.1.m1.7.7.5.2">𝜽</ci><ci id="S1.SS3.p2.1.m1.1.1.1.1.cmml" xref="S1.SS3.p2.1.m1.1.1.1.1">𝑡</ci></apply><vector id="S1.SS3.p2.1.m1.7.7.3.4.cmml" xref="S1.SS3.p2.1.m1.7.7.3.3"><apply id="S1.SS3.p2.1.m1.5.5.1.1.1.cmml" xref="S1.SS3.p2.1.m1.5.5.1.1.1"><csymbol cd="ambiguous" id="S1.SS3.p2.1.m1.5.5.1.1.1.1.cmml" xref="S1.SS3.p2.1.m1.5.5.1.1.1">superscript</csymbol><apply id="S1.SS3.p2.1.m1.5.5.1.1.1.2.cmml" xref="S1.SS3.p2.1.m1.5.5.1.1.1"><csymbol cd="ambiguous" id="S1.SS3.p2.1.m1.5.5.1.1.1.2.1.cmml" xref="S1.SS3.p2.1.m1.5.5.1.1.1">subscript</csymbol><ci id="S1.SS3.p2.1.m1.5.5.1.1.1.2.2.cmml" xref="S1.SS3.p2.1.m1.5.5.1.1.1.2.2">𝜽</ci><cn id="S1.SS3.p2.1.m1.5.5.1.1.1.2.3.cmml" type="integer" xref="S1.SS3.p2.1.m1.5.5.1.1.1.2.3">1</cn></apply><ci id="S1.SS3.p2.1.m1.2.2.1.1.cmml" xref="S1.SS3.p2.1.m1.2.2.1.1">𝑡</ci></apply><apply id="S1.SS3.p2.1.m1.6.6.2.2.2.cmml" xref="S1.SS3.p2.1.m1.6.6.2.2.2"><csymbol cd="ambiguous" id="S1.SS3.p2.1.m1.6.6.2.2.2.1.cmml" xref="S1.SS3.p2.1.m1.6.6.2.2.2">superscript</csymbol><apply id="S1.SS3.p2.1.m1.6.6.2.2.2.2.cmml" xref="S1.SS3.p2.1.m1.6.6.2.2.2"><csymbol cd="ambiguous" id="S1.SS3.p2.1.m1.6.6.2.2.2.2.1.cmml" xref="S1.SS3.p2.1.m1.6.6.2.2.2">subscript</csymbol><ci id="S1.SS3.p2.1.m1.6.6.2.2.2.2.2.cmml" xref="S1.SS3.p2.1.m1.6.6.2.2.2.2.2">𝜽</ci><cn id="S1.SS3.p2.1.m1.6.6.2.2.2.2.3.cmml" type="integer" xref="S1.SS3.p2.1.m1.6.6.2.2.2.2.3">2</cn></apply><ci id="S1.SS3.p2.1.m1.3.3.1.1.cmml" xref="S1.SS3.p2.1.m1.3.3.1.1">𝑡</ci></apply><apply id="S1.SS3.p2.1.m1.7.7.3.3.3.cmml" xref="S1.SS3.p2.1.m1.7.7.3.3.3"><csymbol cd="ambiguous" id="S1.SS3.p2.1.m1.7.7.3.3.3.1.cmml" xref="S1.SS3.p2.1.m1.7.7.3.3.3">superscript</csymbol><apply id="S1.SS3.p2.1.m1.7.7.3.3.3.2.cmml" xref="S1.SS3.p2.1.m1.7.7.3.3.3"><csymbol cd="ambiguous" id="S1.SS3.p2.1.m1.7.7.3.3.3.2.1.cmml" xref="S1.SS3.p2.1.m1.7.7.3.3.3">subscript</csymbol><ci id="S1.SS3.p2.1.m1.7.7.3.3.3.2.2.cmml" xref="S1.SS3.p2.1.m1.7.7.3.3.3.2.2">𝜽</ci><cn id="S1.SS3.p2.1.m1.7.7.3.3.3.2.3.cmml" type="integer" xref="S1.SS3.p2.1.m1.7.7.3.3.3.2.3">3</cn></apply><ci id="S1.SS3.p2.1.m1.4.4.1.1.cmml" xref="S1.SS3.p2.1.m1.4.4.1.1">𝑡</ci></apply></vector></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS3.p2.1.m1.7c">\operatorname{{\bm{\theta}}}^{(t)}=(\operatorname{{\bm{\theta}}}_{1}^{(t)},% \operatorname{{\bm{\theta}}}_{2}^{(t)},\operatorname{{\bm{\theta}}}_{3}^{(t)})</annotation><annotation encoding="application/x-llamapun" id="S1.SS3.p2.1.m1.7d">bold_italic_θ start_POSTSUPERSCRIPT ( italic_t ) end_POSTSUPERSCRIPT = ( bold_italic_θ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_t ) end_POSTSUPERSCRIPT , bold_italic_θ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_t ) end_POSTSUPERSCRIPT , bold_italic_θ start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_t ) end_POSTSUPERSCRIPT )</annotation></semantics></math> for any <math alttext="t=0,1,2,\ldots,\infty" class="ltx_Math" display="inline" id="S1.SS3.p2.2.m2.5"><semantics id="S1.SS3.p2.2.m2.5a"><mrow id="S1.SS3.p2.2.m2.5.6" xref="S1.SS3.p2.2.m2.5.6.cmml"><mi id="S1.SS3.p2.2.m2.5.6.2" xref="S1.SS3.p2.2.m2.5.6.2.cmml">t</mi><mo id="S1.SS3.p2.2.m2.5.6.1" xref="S1.SS3.p2.2.m2.5.6.1.cmml">=</mo><mrow id="S1.SS3.p2.2.m2.5.6.3.2" xref="S1.SS3.p2.2.m2.5.6.3.1.cmml"><mn id="S1.SS3.p2.2.m2.1.1" xref="S1.SS3.p2.2.m2.1.1.cmml">0</mn><mo id="S1.SS3.p2.2.m2.5.6.3.2.1" xref="S1.SS3.p2.2.m2.5.6.3.1.cmml">,</mo><mn id="S1.SS3.p2.2.m2.2.2" xref="S1.SS3.p2.2.m2.2.2.cmml">1</mn><mo id="S1.SS3.p2.2.m2.5.6.3.2.2" xref="S1.SS3.p2.2.m2.5.6.3.1.cmml">,</mo><mn id="S1.SS3.p2.2.m2.3.3" xref="S1.SS3.p2.2.m2.3.3.cmml">2</mn><mo id="S1.SS3.p2.2.m2.5.6.3.2.3" xref="S1.SS3.p2.2.m2.5.6.3.1.cmml">,</mo><mi id="S1.SS3.p2.2.m2.4.4" mathvariant="normal" xref="S1.SS3.p2.2.m2.4.4.cmml">…</mi><mo id="S1.SS3.p2.2.m2.5.6.3.2.4" xref="S1.SS3.p2.2.m2.5.6.3.1.cmml">,</mo><mi id="S1.SS3.p2.2.m2.5.5" mathvariant="normal" xref="S1.SS3.p2.2.m2.5.5.cmml">∞</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.SS3.p2.2.m2.5b"><apply id="S1.SS3.p2.2.m2.5.6.cmml" xref="S1.SS3.p2.2.m2.5.6"><eq id="S1.SS3.p2.2.m2.5.6.1.cmml" xref="S1.SS3.p2.2.m2.5.6.1"></eq><ci id="S1.SS3.p2.2.m2.5.6.2.cmml" xref="S1.SS3.p2.2.m2.5.6.2">𝑡</ci><list id="S1.SS3.p2.2.m2.5.6.3.1.cmml" xref="S1.SS3.p2.2.m2.5.6.3.2"><cn id="S1.SS3.p2.2.m2.1.1.cmml" type="integer" xref="S1.SS3.p2.2.m2.1.1">0</cn><cn id="S1.SS3.p2.2.m2.2.2.cmml" type="integer" xref="S1.SS3.p2.2.m2.2.2">1</cn><cn id="S1.SS3.p2.2.m2.3.3.cmml" type="integer" xref="S1.SS3.p2.2.m2.3.3">2</cn><ci id="S1.SS3.p2.2.m2.4.4.cmml" xref="S1.SS3.p2.2.m2.4.4">…</ci><infinity id="S1.SS3.p2.2.m2.5.5.cmml" xref="S1.SS3.p2.2.m2.5.5"></infinity></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS3.p2.2.m2.5c">t=0,1,2,\ldots,\infty</annotation><annotation encoding="application/x-llamapun" id="S1.SS3.p2.2.m2.5d">italic_t = 0 , 1 , 2 , … , ∞</annotation></semantics></math>. Then, for any non-negative <math alttext="\lambda_{\operatorname{\mathbf{B}}}" class="ltx_Math" display="inline" id="S1.SS3.p2.3.m3.1"><semantics id="S1.SS3.p2.3.m3.1a"><msub id="S1.SS3.p2.3.m3.1.1" xref="S1.SS3.p2.3.m3.1.1.cmml"><mi id="S1.SS3.p2.3.m3.1.1.2" xref="S1.SS3.p2.3.m3.1.1.2.cmml">λ</mi><mi id="S1.SS3.p2.3.m3.1.1.3" xref="S1.SS3.p2.3.m3.1.1.3.cmml">𝐁</mi></msub><annotation-xml encoding="MathML-Content" id="S1.SS3.p2.3.m3.1b"><apply id="S1.SS3.p2.3.m3.1.1.cmml" xref="S1.SS3.p2.3.m3.1.1"><csymbol cd="ambiguous" id="S1.SS3.p2.3.m3.1.1.1.cmml" xref="S1.SS3.p2.3.m3.1.1">subscript</csymbol><ci id="S1.SS3.p2.3.m3.1.1.2.cmml" xref="S1.SS3.p2.3.m3.1.1.2">𝜆</ci><ci id="S1.SS3.p2.3.m3.1.1.3.cmml" xref="S1.SS3.p2.3.m3.1.1.3">𝐁</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS3.p2.3.m3.1c">\lambda_{\operatorname{\mathbf{B}}}</annotation><annotation encoding="application/x-llamapun" id="S1.SS3.p2.3.m3.1d">italic_λ start_POSTSUBSCRIPT bold_B end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="\lambda_{\operatorname{\mathbf{L}}}" class="ltx_Math" display="inline" id="S1.SS3.p2.4.m4.1"><semantics id="S1.SS3.p2.4.m4.1a"><msub id="S1.SS3.p2.4.m4.1.1" xref="S1.SS3.p2.4.m4.1.1.cmml"><mi id="S1.SS3.p2.4.m4.1.1.2" xref="S1.SS3.p2.4.m4.1.1.2.cmml">λ</mi><mi id="S1.SS3.p2.4.m4.1.1.3" xref="S1.SS3.p2.4.m4.1.1.3.cmml">𝐋</mi></msub><annotation-xml encoding="MathML-Content" id="S1.SS3.p2.4.m4.1b"><apply id="S1.SS3.p2.4.m4.1.1.cmml" xref="S1.SS3.p2.4.m4.1.1"><csymbol cd="ambiguous" id="S1.SS3.p2.4.m4.1.1.1.cmml" xref="S1.SS3.p2.4.m4.1.1">subscript</csymbol><ci id="S1.SS3.p2.4.m4.1.1.2.cmml" xref="S1.SS3.p2.4.m4.1.1.2">𝜆</ci><ci id="S1.SS3.p2.4.m4.1.1.3.cmml" xref="S1.SS3.p2.4.m4.1.1.3">𝐋</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS3.p2.4.m4.1c">\lambda_{\operatorname{\mathbf{L}}}</annotation><annotation encoding="application/x-llamapun" id="S1.SS3.p2.4.m4.1d">italic_λ start_POSTSUBSCRIPT bold_L end_POSTSUBSCRIPT</annotation></semantics></math>,</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="S3.EGx17"> <tbody id="S1.E1a"><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(\operatorname{{\bm{\theta}}}_{1}^{(t+1)},\operatorname{{\bm{% \theta}}}_{2}^{(t)},\operatorname{{\bm{\theta}}}_{3}^{(t)})\leq f(% \operatorname{{\bm{\theta}}}_{1}^{(t)},\operatorname{{\bm{\theta}}}_{2}^{(t)},% \operatorname{{\bm{\theta}}}_{3}^{(t)})" class="ltx_Math" display="inline" id="S1.E1a.m1.12"><semantics id="S1.E1a.m1.12a"><mrow id="S1.E1a.m1.12.12" xref="S1.E1a.m1.12.12.cmml"><mrow id="S1.E1a.m1.9.9.3" xref="S1.E1a.m1.9.9.3.cmml"><mi id="S1.E1a.m1.9.9.3.5" xref="S1.E1a.m1.9.9.3.5.cmml">f</mi><mo id="S1.E1a.m1.9.9.3.4" xref="S1.E1a.m1.9.9.3.4.cmml">⁢</mo><mrow id="S1.E1a.m1.9.9.3.3.3" 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start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_t ) end_POSTSUPERSCRIPT , bold_italic_θ start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_t ) end_POSTSUPERSCRIPT ) ≤ italic_f ( bold_italic_θ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_t ) end_POSTSUPERSCRIPT , bold_italic_θ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_t ) end_POSTSUPERSCRIPT , bold_italic_θ start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_t ) 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">(1)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S1.SS3.p2.11">because the first cycle implies that <math alttext="\operatorname{{\bm{\theta}}}_{1}^{(t+1)}=\underset{\operatorname{{\bm{\theta}}% 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_{1},\operatorname{{\bm{\theta}}}_{2}^{(t)},\operatorname{{\bm{\theta}}}_{3}^{% (t)})</annotation><annotation encoding="application/x-llamapun" id="S1.SS3.p2.5.m1.6d">bold_italic_θ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_t + 1 ) end_POSTSUPERSCRIPT = start_UNDERACCENT bold_italic_θ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_UNDERACCENT start_ARG roman_argmin end_ARG italic_f ( bold_italic_θ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , bold_italic_θ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_t ) end_POSTSUPERSCRIPT , bold_italic_θ start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_t ) end_POSTSUPERSCRIPT )</annotation></semantics></math>; see (8) in the main manuscript. We use two properties of the AECM algorithm’s second cycle. 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id="S1.SS3.p2.6.m2.5.5.4.2.2.1.cmml" xref="S1.SS3.p2.6.m2.5.5.4.2.2.1"><csymbol cd="ambiguous" id="S1.SS3.p2.6.m2.5.5.4.2.2.1.1.cmml" xref="S1.SS3.p2.6.m2.5.5.4.2.2.1">subscript</csymbol><ci id="S1.SS3.p2.6.m2.5.5.4.2.2.1.2.cmml" xref="S1.SS3.p2.6.m2.5.5.4.2.2.1.2">𝜽</ci><cn id="S1.SS3.p2.6.m2.5.5.4.2.2.1.3.cmml" type="integer" xref="S1.SS3.p2.6.m2.5.5.4.2.2.1.3">2</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS3.p2.6.m2.5c">-\operatorname{\mathcal{Q}}_{(1)}(\operatorname{{\bm{\theta}}}_{2})+% \operatorname{\mathcal{P}}_{\lambda_{\operatorname{\mathbf{L}}}}(\operatorname% {{\bm{\theta}}}_{2})</annotation><annotation encoding="application/x-llamapun" id="S1.SS3.p2.6.m2.5d">- caligraphic_Q start_POSTSUBSCRIPT ( 1 ) end_POSTSUBSCRIPT ( bold_italic_θ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) + caligraphic_P start_POSTSUBSCRIPT italic_λ start_POSTSUBSCRIPT bold_L end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( bold_italic_θ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT )</annotation></semantics></math> majorizes <math alttext="f(\operatorname{{\bm{\theta}}}_{1}^{(t+1)},\operatorname{{\bm{\theta}}}_{2},% \operatorname{{\bm{\theta}}}_{3}^{(t)})" class="ltx_Math" display="inline" id="S1.SS3.p2.7.m3.5"><semantics id="S1.SS3.p2.7.m3.5a"><mrow id="S1.SS3.p2.7.m3.5.5" xref="S1.SS3.p2.7.m3.5.5.cmml"><mi id="S1.SS3.p2.7.m3.5.5.5" xref="S1.SS3.p2.7.m3.5.5.5.cmml">f</mi><mo id="S1.SS3.p2.7.m3.5.5.4" xref="S1.SS3.p2.7.m3.5.5.4.cmml">⁢</mo><mrow id="S1.SS3.p2.7.m3.5.5.3.3" xref="S1.SS3.p2.7.m3.5.5.3.4.cmml"><mo id="S1.SS3.p2.7.m3.5.5.3.3.4" stretchy="false" xref="S1.SS3.p2.7.m3.5.5.3.4.cmml">(</mo><msubsup id="S1.SS3.p2.7.m3.3.3.1.1.1" xref="S1.SS3.p2.7.m3.3.3.1.1.1.cmml"><mi id="S1.SS3.p2.7.m3.3.3.1.1.1.2.2" xref="S1.SS3.p2.7.m3.3.3.1.1.1.2.2.cmml">𝜽</mi><mn id="S1.SS3.p2.7.m3.3.3.1.1.1.2.3" xref="S1.SS3.p2.7.m3.3.3.1.1.1.2.3.cmml">1</mn><mrow id="S1.SS3.p2.7.m3.1.1.1.1" xref="S1.SS3.p2.7.m3.1.1.1.1.1.cmml"><mo id="S1.SS3.p2.7.m3.1.1.1.1.2" stretchy="false" xref="S1.SS3.p2.7.m3.1.1.1.1.1.cmml">(</mo><mrow id="S1.SS3.p2.7.m3.1.1.1.1.1" xref="S1.SS3.p2.7.m3.1.1.1.1.1.cmml"><mi id="S1.SS3.p2.7.m3.1.1.1.1.1.2" xref="S1.SS3.p2.7.m3.1.1.1.1.1.2.cmml">t</mi><mo id="S1.SS3.p2.7.m3.1.1.1.1.1.1" xref="S1.SS3.p2.7.m3.1.1.1.1.1.1.cmml">+</mo><mn id="S1.SS3.p2.7.m3.1.1.1.1.1.3" xref="S1.SS3.p2.7.m3.1.1.1.1.1.3.cmml">1</mn></mrow><mo id="S1.SS3.p2.7.m3.1.1.1.1.3" stretchy="false" xref="S1.SS3.p2.7.m3.1.1.1.1.1.cmml">)</mo></mrow></msubsup><mo id="S1.SS3.p2.7.m3.5.5.3.3.5" xref="S1.SS3.p2.7.m3.5.5.3.4.cmml">,</mo><msub id="S1.SS3.p2.7.m3.4.4.2.2.2" xref="S1.SS3.p2.7.m3.4.4.2.2.2.cmml"><mi id="S1.SS3.p2.7.m3.4.4.2.2.2.2" xref="S1.SS3.p2.7.m3.4.4.2.2.2.2.cmml">𝜽</mi><mn id="S1.SS3.p2.7.m3.4.4.2.2.2.3" xref="S1.SS3.p2.7.m3.4.4.2.2.2.3.cmml">2</mn></msub><mo id="S1.SS3.p2.7.m3.5.5.3.3.6" xref="S1.SS3.p2.7.m3.5.5.3.4.cmml">,</mo><msubsup id="S1.SS3.p2.7.m3.5.5.3.3.3" xref="S1.SS3.p2.7.m3.5.5.3.3.3.cmml"><mi id="S1.SS3.p2.7.m3.5.5.3.3.3.2.2" 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id="S1.SS3.p2.7.m3.4.4.2.2.2.1.cmml" xref="S1.SS3.p2.7.m3.4.4.2.2.2">subscript</csymbol><ci id="S1.SS3.p2.7.m3.4.4.2.2.2.2.cmml" xref="S1.SS3.p2.7.m3.4.4.2.2.2.2">𝜽</ci><cn id="S1.SS3.p2.7.m3.4.4.2.2.2.3.cmml" type="integer" xref="S1.SS3.p2.7.m3.4.4.2.2.2.3">2</cn></apply><apply id="S1.SS3.p2.7.m3.5.5.3.3.3.cmml" xref="S1.SS3.p2.7.m3.5.5.3.3.3"><csymbol cd="ambiguous" id="S1.SS3.p2.7.m3.5.5.3.3.3.1.cmml" xref="S1.SS3.p2.7.m3.5.5.3.3.3">superscript</csymbol><apply id="S1.SS3.p2.7.m3.5.5.3.3.3.2.cmml" xref="S1.SS3.p2.7.m3.5.5.3.3.3"><csymbol cd="ambiguous" id="S1.SS3.p2.7.m3.5.5.3.3.3.2.1.cmml" xref="S1.SS3.p2.7.m3.5.5.3.3.3">subscript</csymbol><ci id="S1.SS3.p2.7.m3.5.5.3.3.3.2.2.cmml" xref="S1.SS3.p2.7.m3.5.5.3.3.3.2.2">𝜽</ci><cn id="S1.SS3.p2.7.m3.5.5.3.3.3.2.3.cmml" type="integer" xref="S1.SS3.p2.7.m3.5.5.3.3.3.2.3">3</cn></apply><ci id="S1.SS3.p2.7.m3.2.2.1.1.cmml" xref="S1.SS3.p2.7.m3.2.2.1.1">𝑡</ci></apply></vector></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS3.p2.7.m3.5c">f(\operatorname{{\bm{\theta}}}_{1}^{(t+1)},\operatorname{{\bm{\theta}}}_{2},% \operatorname{{\bm{\theta}}}_{3}^{(t)})</annotation><annotation encoding="application/x-llamapun" id="S1.SS3.p2.7.m3.5d">italic_f ( bold_italic_θ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_t + 1 ) end_POSTSUPERSCRIPT , bold_italic_θ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , bold_italic_θ start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_t ) end_POSTSUPERSCRIPT )</annotation></semantics></math> for every <math alttext="\operatorname{{\bm{\theta}}}_{2}" class="ltx_Math" display="inline" id="S1.SS3.p2.8.m4.1"><semantics id="S1.SS3.p2.8.m4.1a"><msub id="S1.SS3.p2.8.m4.1.1" xref="S1.SS3.p2.8.m4.1.1.cmml"><mi id="S1.SS3.p2.8.m4.1.1.2" xref="S1.SS3.p2.8.m4.1.1.2.cmml">𝜽</mi><mn id="S1.SS3.p2.8.m4.1.1.3" xref="S1.SS3.p2.8.m4.1.1.3.cmml">2</mn></msub><annotation-xml encoding="MathML-Content" id="S1.SS3.p2.8.m4.1b"><apply id="S1.SS3.p2.8.m4.1.1.cmml" xref="S1.SS3.p2.8.m4.1.1"><csymbol cd="ambiguous" id="S1.SS3.p2.8.m4.1.1.1.cmml" xref="S1.SS3.p2.8.m4.1.1">subscript</csymbol><ci id="S1.SS3.p2.8.m4.1.1.2.cmml" xref="S1.SS3.p2.8.m4.1.1.2">𝜽</ci><cn id="S1.SS3.p2.8.m4.1.1.3.cmml" type="integer" xref="S1.SS3.p2.8.m4.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS3.p2.8.m4.1c">\operatorname{{\bm{\theta}}}_{2}</annotation><annotation encoding="application/x-llamapun" id="S1.SS3.p2.8.m4.1d">bold_italic_θ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math>, where <math alttext="\operatorname{\mathcal{Q}}_{(1)}" class="ltx_Math" display="inline" id="S1.SS3.p2.9.m5.1"><semantics id="S1.SS3.p2.9.m5.1a"><msub id="S1.SS3.p2.9.m5.1.2" xref="S1.SS3.p2.9.m5.1.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S1.SS3.p2.9.m5.1.2.2" xref="S1.SS3.p2.9.m5.1.2.2.cmml">𝒬</mi><mrow id="S1.SS3.p2.9.m5.1.1.1.3" xref="S1.SS3.p2.9.m5.1.2.cmml"><mo id="S1.SS3.p2.9.m5.1.1.1.3.1" stretchy="false" xref="S1.SS3.p2.9.m5.1.2.cmml">(</mo><mn id="S1.SS3.p2.9.m5.1.1.1.1" xref="S1.SS3.p2.9.m5.1.1.1.1.cmml">1</mn><mo id="S1.SS3.p2.9.m5.1.1.1.3.2" stretchy="false" xref="S1.SS3.p2.9.m5.1.2.cmml">)</mo></mrow></msub><annotation-xml encoding="MathML-Content" id="S1.SS3.p2.9.m5.1b"><apply id="S1.SS3.p2.9.m5.1.2.cmml" xref="S1.SS3.p2.9.m5.1.2"><csymbol cd="ambiguous" id="S1.SS3.p2.9.m5.1.2.1.cmml" xref="S1.SS3.p2.9.m5.1.2">subscript</csymbol><ci id="S1.SS3.p2.9.m5.1.2.2.cmml" xref="S1.SS3.p2.9.m5.1.2.2">𝒬</ci><cn id="S1.SS3.p2.9.m5.1.1.1.1.cmml" type="integer" xref="S1.SS3.p2.9.m5.1.1.1.1">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS3.p2.9.m5.1c">\operatorname{\mathcal{Q}}_{(1)}</annotation><annotation encoding="application/x-llamapun" id="S1.SS3.p2.9.m5.1d">caligraphic_Q start_POSTSUBSCRIPT ( 1 ) end_POSTSUBSCRIPT</annotation></semantics></math> is defined in (9) in the main manuscript; therefore, <math 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id="S1.SS3.p2.10.m6.10.10.7.3.3.3.2.3.cmml" type="integer" xref="S1.SS3.p2.10.m6.10.10.7.3.3.3.2.3">3</cn></apply><ci id="S1.SS3.p2.10.m6.3.3.1.1.cmml" xref="S1.SS3.p2.10.m6.3.3.1.1">𝑡</ci></apply></vector></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS3.p2.10.m6.10c">-\operatorname{\mathcal{Q}}_{(1)}(\operatorname{{\bm{\theta}}}_{2})+% \operatorname{\mathcal{P}}_{\lambda_{\operatorname{\mathbf{L}}}}(\operatorname% {{\bm{\theta}}}_{2})\leq f(\operatorname{{\bm{\theta}}}_{1}^{(t+1)},% \operatorname{{\bm{\theta}}}_{2},\operatorname{{\bm{\theta}}}_{3}^{(t)})</annotation><annotation encoding="application/x-llamapun" id="S1.SS3.p2.10.m6.10d">- caligraphic_Q start_POSTSUBSCRIPT ( 1 ) end_POSTSUBSCRIPT ( bold_italic_θ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) + caligraphic_P start_POSTSUBSCRIPT italic_λ start_POSTSUBSCRIPT bold_L end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( bold_italic_θ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) ≤ italic_f ( bold_italic_θ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_t + 1 ) end_POSTSUPERSCRIPT , bold_italic_θ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , bold_italic_θ start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_t ) end_POSTSUPERSCRIPT )</annotation></semantics></math>. 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end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_t ) end_POSTSUPERSCRIPT ) + caligraphic_P start_POSTSUBSCRIPT italic_λ start_POSTSUBSCRIPT bold_L end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( bold_italic_θ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_t ) end_POSTSUPERSCRIPT ) = italic_f ( bold_italic_θ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_t + 1 ) end_POSTSUPERSCRIPT , bold_italic_θ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_t ) end_POSTSUPERSCRIPT , bold_italic_θ start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_t ) end_POSTSUPERSCRIPT )</annotation></semantics></math>; see (9) and (10) in the main manuscript for details. We use these properties to obtain that</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="S3.EGx18"> <tbody id="S1.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 f(\operatorname{{\bm{\theta}}}_{1}^{(t+1)},\operatorname{{\bm{% \theta}}}_{2}^{(t+1)},\operatorname{{\bm{\theta}}}_{3}^{(t)})" class="ltx_Math" display="inline" id="S1.Ex6.m1.6"><semantics id="S1.Ex6.m1.6a"><mrow id="S1.Ex6.m1.6.6" xref="S1.Ex6.m1.6.6.cmml"><mi id="S1.Ex6.m1.6.6.5" xref="S1.Ex6.m1.6.6.5.cmml">f</mi><mo id="S1.Ex6.m1.6.6.4" xref="S1.Ex6.m1.6.6.4.cmml">⁢</mo><mrow id="S1.Ex6.m1.6.6.3.3" xref="S1.Ex6.m1.6.6.3.4.cmml"><mo id="S1.Ex6.m1.6.6.3.3.4" stretchy="false" xref="S1.Ex6.m1.6.6.3.4.cmml">(</mo><msubsup id="S1.Ex6.m1.4.4.1.1.1" xref="S1.Ex6.m1.4.4.1.1.1.cmml"><mi id="S1.Ex6.m1.4.4.1.1.1.2.2" xref="S1.Ex6.m1.4.4.1.1.1.2.2.cmml">𝜽</mi><mn id="S1.Ex6.m1.4.4.1.1.1.2.3" xref="S1.Ex6.m1.4.4.1.1.1.2.3.cmml">1</mn><mrow id="S1.Ex6.m1.1.1.1.1" xref="S1.Ex6.m1.1.1.1.1.1.cmml"><mo id="S1.Ex6.m1.1.1.1.1.2" stretchy="false" xref="S1.Ex6.m1.1.1.1.1.1.cmml">(</mo><mrow id="S1.Ex6.m1.1.1.1.1.1" xref="S1.Ex6.m1.1.1.1.1.1.cmml"><mi id="S1.Ex6.m1.1.1.1.1.1.2" xref="S1.Ex6.m1.1.1.1.1.1.2.cmml">t</mi><mo id="S1.Ex6.m1.1.1.1.1.1.1" xref="S1.Ex6.m1.1.1.1.1.1.1.cmml">+</mo><mn id="S1.Ex6.m1.1.1.1.1.1.3" xref="S1.Ex6.m1.1.1.1.1.1.3.cmml">1</mn></mrow><mo id="S1.Ex6.m1.1.1.1.1.3" stretchy="false" xref="S1.Ex6.m1.1.1.1.1.1.cmml">)</mo></mrow></msubsup><mo id="S1.Ex6.m1.6.6.3.3.5" xref="S1.Ex6.m1.6.6.3.4.cmml">,</mo><msubsup id="S1.Ex6.m1.5.5.2.2.2" xref="S1.Ex6.m1.5.5.2.2.2.cmml"><mi id="S1.Ex6.m1.5.5.2.2.2.2.2" xref="S1.Ex6.m1.5.5.2.2.2.2.2.cmml">𝜽</mi><mn id="S1.Ex6.m1.5.5.2.2.2.2.3" xref="S1.Ex6.m1.5.5.2.2.2.2.3.cmml">2</mn><mrow id="S1.Ex6.m1.2.2.1.1" xref="S1.Ex6.m1.2.2.1.1.1.cmml"><mo id="S1.Ex6.m1.2.2.1.1.2" stretchy="false" xref="S1.Ex6.m1.2.2.1.1.1.cmml">(</mo><mrow id="S1.Ex6.m1.2.2.1.1.1" xref="S1.Ex6.m1.2.2.1.1.1.cmml"><mi id="S1.Ex6.m1.2.2.1.1.1.2" xref="S1.Ex6.m1.2.2.1.1.1.2.cmml">t</mi><mo id="S1.Ex6.m1.2.2.1.1.1.1" xref="S1.Ex6.m1.2.2.1.1.1.1.cmml">+</mo><mn id="S1.Ex6.m1.2.2.1.1.1.3" xref="S1.Ex6.m1.2.2.1.1.1.3.cmml">1</mn></mrow><mo id="S1.Ex6.m1.2.2.1.1.3" stretchy="false" xref="S1.Ex6.m1.2.2.1.1.1.cmml">)</mo></mrow></msubsup><mo id="S1.Ex6.m1.6.6.3.3.6" xref="S1.Ex6.m1.6.6.3.4.cmml">,</mo><msubsup id="S1.Ex6.m1.6.6.3.3.3" xref="S1.Ex6.m1.6.6.3.3.3.cmml"><mi id="S1.Ex6.m1.6.6.3.3.3.2.2" xref="S1.Ex6.m1.6.6.3.3.3.2.2.cmml">𝜽</mi><mn id="S1.Ex6.m1.6.6.3.3.3.2.3" xref="S1.Ex6.m1.6.6.3.3.3.2.3.cmml">3</mn><mrow id="S1.Ex6.m1.3.3.1.3" xref="S1.Ex6.m1.6.6.3.3.3.cmml"><mo id="S1.Ex6.m1.3.3.1.3.1" stretchy="false" xref="S1.Ex6.m1.6.6.3.3.3.cmml">(</mo><mi id="S1.Ex6.m1.3.3.1.1" xref="S1.Ex6.m1.3.3.1.1.cmml">t</mi><mo id="S1.Ex6.m1.3.3.1.3.2" stretchy="false" xref="S1.Ex6.m1.6.6.3.3.3.cmml">)</mo></mrow></msubsup><mo id="S1.Ex6.m1.6.6.3.3.7" stretchy="false" xref="S1.Ex6.m1.6.6.3.4.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.Ex6.m1.6b"><apply id="S1.Ex6.m1.6.6.cmml" xref="S1.Ex6.m1.6.6"><times id="S1.Ex6.m1.6.6.4.cmml" xref="S1.Ex6.m1.6.6.4"></times><ci id="S1.Ex6.m1.6.6.5.cmml" xref="S1.Ex6.m1.6.6.5">𝑓</ci><vector id="S1.Ex6.m1.6.6.3.4.cmml" xref="S1.Ex6.m1.6.6.3.3"><apply id="S1.Ex6.m1.4.4.1.1.1.cmml" xref="S1.Ex6.m1.4.4.1.1.1"><csymbol cd="ambiguous" id="S1.Ex6.m1.4.4.1.1.1.1.cmml" xref="S1.Ex6.m1.4.4.1.1.1">superscript</csymbol><apply id="S1.Ex6.m1.4.4.1.1.1.2.cmml" xref="S1.Ex6.m1.4.4.1.1.1"><csymbol cd="ambiguous" id="S1.Ex6.m1.4.4.1.1.1.2.1.cmml" xref="S1.Ex6.m1.4.4.1.1.1">subscript</csymbol><ci id="S1.Ex6.m1.4.4.1.1.1.2.2.cmml" xref="S1.Ex6.m1.4.4.1.1.1.2.2">𝜽</ci><cn id="S1.Ex6.m1.4.4.1.1.1.2.3.cmml" type="integer" xref="S1.Ex6.m1.4.4.1.1.1.2.3">1</cn></apply><apply id="S1.Ex6.m1.1.1.1.1.1.cmml" 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xref="S1.Ex7.m1.10.10.4.1.1.4.2.2.1.2.3">2</cn></apply><apply id="S1.Ex7.m1.6.6.1.1.1.cmml" xref="S1.Ex7.m1.6.6.1.1"><plus id="S1.Ex7.m1.6.6.1.1.1.1.cmml" xref="S1.Ex7.m1.6.6.1.1.1.1"></plus><ci id="S1.Ex7.m1.6.6.1.1.1.2.cmml" xref="S1.Ex7.m1.6.6.1.1.1.2">𝑡</ci><cn id="S1.Ex7.m1.6.6.1.1.1.3.cmml" type="integer" xref="S1.Ex7.m1.6.6.1.1.1.3">1</cn></apply></apply></apply></apply></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.Ex7.m1.10c">\displaystyle\quad\;f(\operatorname{{\bm{\theta}}}_{1}^{(t+1)},\operatorname{{% \bm{\theta}}}_{2}^{(t+1)},\operatorname{{\bm{\theta}}}_{3}^{(t)})-\{-% \operatorname{\mathcal{Q}}_{(1)}(\operatorname{{\bm{\theta}}}_{2}^{(t+1)})+% \operatorname{\mathcal{P}}_{\lambda_{\operatorname{\mathbf{L}}}}(\operatorname% {{\bm{\theta}}}_{2}^{(t+1)})\}</annotation><annotation encoding="application/x-llamapun" id="S1.Ex7.m1.10d">italic_f ( bold_italic_θ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_t + 1 ) end_POSTSUPERSCRIPT , bold_italic_θ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_t + 1 ) end_POSTSUPERSCRIPT , bold_italic_θ start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_t ) end_POSTSUPERSCRIPT ) - { - caligraphic_Q start_POSTSUBSCRIPT ( 1 ) end_POSTSUBSCRIPT ( bold_italic_θ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_t + 1 ) end_POSTSUPERSCRIPT ) + caligraphic_P start_POSTSUBSCRIPT italic_λ start_POSTSUBSCRIPT bold_L end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( bold_italic_θ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_t + 1 ) end_POSTSUPERSCRIPT ) }</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="S1.Ex8"><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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xref="S1.Ex8.m1.8.8.4.2.2.1.2.2">𝜽</ci><cn id="S1.Ex8.m1.8.8.4.2.2.1.2.3.cmml" type="integer" xref="S1.Ex8.m1.8.8.4.2.2.1.2.3">2</cn></apply><apply id="S1.Ex8.m1.4.4.1.1.1.cmml" xref="S1.Ex8.m1.4.4.1.1"><plus id="S1.Ex8.m1.4.4.1.1.1.1.cmml" xref="S1.Ex8.m1.4.4.1.1.1.1"></plus><ci id="S1.Ex8.m1.4.4.1.1.1.2.cmml" xref="S1.Ex8.m1.4.4.1.1.1.2">𝑡</ci><cn id="S1.Ex8.m1.4.4.1.1.1.3.cmml" type="integer" xref="S1.Ex8.m1.4.4.1.1.1.3">1</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.Ex8.m1.8c">\displaystyle\overset{(i)}{\leq}-\operatorname{\mathcal{Q}}_{(1)}(% \operatorname{{\bm{\theta}}}_{2}^{(t+1)})+\operatorname{\mathcal{P}}_{\lambda_% {\operatorname{\mathbf{L}}}}(\operatorname{{\bm{\theta}}}_{2}^{(t+1)})</annotation><annotation encoding="application/x-llamapun" id="S1.Ex8.m1.8d">start_OVERACCENT ( italic_i ) end_OVERACCENT start_ARG ≤ end_ARG - caligraphic_Q start_POSTSUBSCRIPT ( 1 ) end_POSTSUBSCRIPT ( bold_italic_θ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_t + 1 ) end_POSTSUPERSCRIPT ) + caligraphic_P start_POSTSUBSCRIPT italic_λ start_POSTSUBSCRIPT bold_L end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( bold_italic_θ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_t + 1 ) end_POSTSUPERSCRIPT )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="S1.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_eqn_cell"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\overset{(ii)}{\leq}-\operatorname{\mathcal{Q}}_{(1)}(% \operatorname{{\bm{\theta}}}_{2}^{(t)})+\operatorname{\mathcal{P}}_{\lambda_{% \operatorname{\mathbf{L}}}}(\operatorname{{\bm{\theta}}}_{2}^{(t)})" class="ltx_Math" display="inline" id="S1.Ex9.m1.8"><semantics id="S1.Ex9.m1.8a"><mrow id="S1.Ex9.m1.8.8" xref="S1.Ex9.m1.8.8.cmml"><mrow 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start_ARG = end_ARG italic_f ( bold_italic_θ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_t + 1 ) end_POSTSUPERSCRIPT , bold_italic_θ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_t ) end_POSTSUPERSCRIPT , bold_italic_θ start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_t ) 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">(2)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S1.SS3.p2.17">where <math alttext="(i)" class="ltx_Math" display="inline" id="S1.SS3.p2.12.m1.1"><semantics id="S1.SS3.p2.12.m1.1a"><mrow id="S1.SS3.p2.12.m1.1.2.2"><mo id="S1.SS3.p2.12.m1.1.2.2.1" stretchy="false">(</mo><mi id="S1.SS3.p2.12.m1.1.1" xref="S1.SS3.p2.12.m1.1.1.cmml">i</mi><mo id="S1.SS3.p2.12.m1.1.2.2.2" stretchy="false">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S1.SS3.p2.12.m1.1b"><ci id="S1.SS3.p2.12.m1.1.1.cmml" xref="S1.SS3.p2.12.m1.1.1">𝑖</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS3.p2.12.m1.1c">(i)</annotation><annotation encoding="application/x-llamapun" id="S1.SS3.p2.12.m1.1d">( italic_i )</annotation></semantics></math> follows because <math alttext="f(\operatorname{{\bm{\theta}}}_{1}^{(t+1)},\operatorname{{\bm{\theta}}}_{2}^{(% t+1)},\operatorname{{\bm{\theta}}}_{3}^{(t)})-\{-\operatorname{\mathcal{Q}}_{(% 1)}(\operatorname{{\bm{\theta}}}_{2}^{(t+1)})+\operatorname{\mathcal{P}}_{% \lambda_{\operatorname{\mathbf{L}}}}(\operatorname{{\bm{\theta}}}_{2}^{(t+1)})\geq 0" class="ltx_math_unparsed" display="inline" id="S1.SS3.p2.13.m2.6"><semantics id="S1.SS3.p2.13.m2.6a"><mrow id="S1.SS3.p2.13.m2.6b"><mi id="S1.SS3.p2.13.m2.6.7">f</mi><mrow id="S1.SS3.p2.13.m2.6.8"><mo id="S1.SS3.p2.13.m2.6.8.1" stretchy="false">(</mo><msubsup id="S1.SS3.p2.13.m2.6.8.2"><mi id="S1.SS3.p2.13.m2.6.8.2.2.2">𝜽</mi><mn id="S1.SS3.p2.13.m2.6.8.2.2.3">1</mn><mrow id="S1.SS3.p2.13.m2.1.1.1.1"><mo id="S1.SS3.p2.13.m2.1.1.1.1.2" stretchy="false">(</mo><mrow id="S1.SS3.p2.13.m2.1.1.1.1.1"><mi id="S1.SS3.p2.13.m2.1.1.1.1.1.2">t</mi><mo id="S1.SS3.p2.13.m2.1.1.1.1.1.1">+</mo><mn id="S1.SS3.p2.13.m2.1.1.1.1.1.3">1</mn></mrow><mo id="S1.SS3.p2.13.m2.1.1.1.1.3" stretchy="false">)</mo></mrow></msubsup><mo id="S1.SS3.p2.13.m2.6.8.3">,</mo><msubsup id="S1.SS3.p2.13.m2.6.8.4"><mi id="S1.SS3.p2.13.m2.6.8.4.2.2">𝜽</mi><mn id="S1.SS3.p2.13.m2.6.8.4.2.3">2</mn><mrow id="S1.SS3.p2.13.m2.2.2.1.1"><mo id="S1.SS3.p2.13.m2.2.2.1.1.2" stretchy="false">(</mo><mrow id="S1.SS3.p2.13.m2.2.2.1.1.1"><mi id="S1.SS3.p2.13.m2.2.2.1.1.1.2">t</mi><mo id="S1.SS3.p2.13.m2.2.2.1.1.1.1">+</mo><mn id="S1.SS3.p2.13.m2.2.2.1.1.1.3">1</mn></mrow><mo id="S1.SS3.p2.13.m2.2.2.1.1.3" stretchy="false">)</mo></mrow></msubsup><mo id="S1.SS3.p2.13.m2.6.8.5">,</mo><msubsup id="S1.SS3.p2.13.m2.6.8.6"><mi id="S1.SS3.p2.13.m2.6.8.6.2.2">𝜽</mi><mn id="S1.SS3.p2.13.m2.6.8.6.2.3">3</mn><mrow id="S1.SS3.p2.13.m2.3.3.1.3"><mo id="S1.SS3.p2.13.m2.3.3.1.3.1" stretchy="false">(</mo><mi id="S1.SS3.p2.13.m2.3.3.1.1">t</mi><mo id="S1.SS3.p2.13.m2.3.3.1.3.2" stretchy="false">)</mo></mrow></msubsup><mo id="S1.SS3.p2.13.m2.6.8.7" stretchy="false">)</mo></mrow><mo id="S1.SS3.p2.13.m2.6.9">−</mo><mrow id="S1.SS3.p2.13.m2.6.10"><mo id="S1.SS3.p2.13.m2.6.10.1" stretchy="false">{</mo><mo id="S1.SS3.p2.13.m2.6.10.2" lspace="0em">−</mo><msub id="S1.SS3.p2.13.m2.6.10.3"><mi class="ltx_font_mathcaligraphic" id="S1.SS3.p2.13.m2.6.10.3.2">𝒬</mi><mrow id="S1.SS3.p2.13.m2.4.4.1.3"><mo id="S1.SS3.p2.13.m2.4.4.1.3.1" stretchy="false">(</mo><mn id="S1.SS3.p2.13.m2.4.4.1.1">1</mn><mo id="S1.SS3.p2.13.m2.4.4.1.3.2" stretchy="false">)</mo></mrow></msub><mrow id="S1.SS3.p2.13.m2.6.10.4"><mo id="S1.SS3.p2.13.m2.6.10.4.1" stretchy="false">(</mo><msubsup id="S1.SS3.p2.13.m2.6.10.4.2"><mi id="S1.SS3.p2.13.m2.6.10.4.2.2.2">𝜽</mi><mn id="S1.SS3.p2.13.m2.6.10.4.2.2.3">2</mn><mrow id="S1.SS3.p2.13.m2.5.5.1.1"><mo id="S1.SS3.p2.13.m2.5.5.1.1.2" stretchy="false">(</mo><mrow id="S1.SS3.p2.13.m2.5.5.1.1.1"><mi id="S1.SS3.p2.13.m2.5.5.1.1.1.2">t</mi><mo id="S1.SS3.p2.13.m2.5.5.1.1.1.1">+</mo><mn id="S1.SS3.p2.13.m2.5.5.1.1.1.3">1</mn></mrow><mo id="S1.SS3.p2.13.m2.5.5.1.1.3" stretchy="false">)</mo></mrow></msubsup><mo id="S1.SS3.p2.13.m2.6.10.4.3" stretchy="false">)</mo></mrow><mo id="S1.SS3.p2.13.m2.6.10.5">+</mo><msub id="S1.SS3.p2.13.m2.6.10.6"><mi class="ltx_font_mathcaligraphic" id="S1.SS3.p2.13.m2.6.10.6.2">𝒫</mi><msub id="S1.SS3.p2.13.m2.6.10.6.3"><mi id="S1.SS3.p2.13.m2.6.10.6.3.2">λ</mi><mi id="S1.SS3.p2.13.m2.6.10.6.3.3">𝐋</mi></msub></msub><mrow id="S1.SS3.p2.13.m2.6.10.7"><mo id="S1.SS3.p2.13.m2.6.10.7.1" stretchy="false">(</mo><msubsup id="S1.SS3.p2.13.m2.6.10.7.2"><mi id="S1.SS3.p2.13.m2.6.10.7.2.2.2">𝜽</mi><mn id="S1.SS3.p2.13.m2.6.10.7.2.2.3">2</mn><mrow id="S1.SS3.p2.13.m2.6.6.1.1"><mo id="S1.SS3.p2.13.m2.6.6.1.1.2" stretchy="false">(</mo><mrow id="S1.SS3.p2.13.m2.6.6.1.1.1"><mi id="S1.SS3.p2.13.m2.6.6.1.1.1.2">t</mi><mo id="S1.SS3.p2.13.m2.6.6.1.1.1.1">+</mo><mn id="S1.SS3.p2.13.m2.6.6.1.1.1.3">1</mn></mrow><mo id="S1.SS3.p2.13.m2.6.6.1.1.3" stretchy="false">)</mo></mrow></msubsup><mo id="S1.SS3.p2.13.m2.6.10.7.3" stretchy="false">)</mo></mrow><mo id="S1.SS3.p2.13.m2.6.10.8">≥</mo><mn id="S1.SS3.p2.13.m2.6.10.9">0</mn></mrow></mrow><annotation encoding="application/x-tex" id="S1.SS3.p2.13.m2.6c">f(\operatorname{{\bm{\theta}}}_{1}^{(t+1)},\operatorname{{\bm{\theta}}}_{2}^{(% t+1)},\operatorname{{\bm{\theta}}}_{3}^{(t)})-\{-\operatorname{\mathcal{Q}}_{(% 1)}(\operatorname{{\bm{\theta}}}_{2}^{(t+1)})+\operatorname{\mathcal{P}}_{% \lambda_{\operatorname{\mathbf{L}}}}(\operatorname{{\bm{\theta}}}_{2}^{(t+1)})\geq 0</annotation><annotation encoding="application/x-llamapun" id="S1.SS3.p2.13.m2.6d">italic_f ( bold_italic_θ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_t + 1 ) end_POSTSUPERSCRIPT , bold_italic_θ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_t + 1 ) end_POSTSUPERSCRIPT , bold_italic_θ start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_t ) end_POSTSUPERSCRIPT ) - { - caligraphic_Q start_POSTSUBSCRIPT ( 1 ) end_POSTSUBSCRIPT ( bold_italic_θ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_t + 1 ) end_POSTSUPERSCRIPT ) + caligraphic_P start_POSTSUBSCRIPT italic_λ start_POSTSUBSCRIPT bold_L end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( bold_italic_θ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_t + 1 ) end_POSTSUPERSCRIPT ) ≥ 0</annotation></semantics></math> using the first property of AECM algorithm’s second cycle, <math alttext="(ii)" class="ltx_Math" display="inline" id="S1.SS3.p2.14.m3.1"><semantics id="S1.SS3.p2.14.m3.1a"><mrow id="S1.SS3.p2.14.m3.1.1.1" 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id="S1.SS3.p2.14.m3.1c">(ii)</annotation><annotation encoding="application/x-llamapun" id="S1.SS3.p2.14.m3.1d">( italic_i italic_i )</annotation></semantics></math> follows because <math alttext="\operatorname{{\bm{\theta}}}_{2}^{(t+1)}=\underset{\operatorname{{\bm{\theta}}% }_{2}}{\operatorname*{\mathop{\mathrm{argmin}}}}-\operatorname{\mathcal{Q}}_{(% 1)}(\operatorname{{\bm{\theta}}}_{2})+\operatorname{\mathcal{P}}_{\lambda_{% \operatorname{\mathbf{L}}}}(\operatorname{{\bm{\theta}}}_{2})" class="ltx_Math" display="inline" id="S1.SS3.p2.15.m4.6"><semantics id="S1.SS3.p2.15.m4.6a"><mrow id="S1.SS3.p2.15.m4.6.6" xref="S1.SS3.p2.15.m4.6.6.cmml"><msubsup id="S1.SS3.p2.15.m4.6.6.6" xref="S1.SS3.p2.15.m4.6.6.6.cmml"><mi id="S1.SS3.p2.15.m4.6.6.6.2.2" xref="S1.SS3.p2.15.m4.6.6.6.2.2.cmml">𝜽</mi><mn id="S1.SS3.p2.15.m4.6.6.6.2.3" xref="S1.SS3.p2.15.m4.6.6.6.2.3.cmml">2</mn><mrow id="S1.SS3.p2.15.m4.1.1.1.1" xref="S1.SS3.p2.15.m4.1.1.1.1.1.cmml"><mo id="S1.SS3.p2.15.m4.1.1.1.1.2" 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id="S1.SS3.p2.15.m4.6.6.4.4.2.2.1.3.cmml" type="integer" xref="S1.SS3.p2.15.m4.6.6.4.4.2.2.1.3">2</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS3.p2.15.m4.6c">\operatorname{{\bm{\theta}}}_{2}^{(t+1)}=\underset{\operatorname{{\bm{\theta}}% }_{2}}{\operatorname*{\mathop{\mathrm{argmin}}}}-\operatorname{\mathcal{Q}}_{(% 1)}(\operatorname{{\bm{\theta}}}_{2})+\operatorname{\mathcal{P}}_{\lambda_{% \operatorname{\mathbf{L}}}}(\operatorname{{\bm{\theta}}}_{2})</annotation><annotation encoding="application/x-llamapun" id="S1.SS3.p2.15.m4.6d">bold_italic_θ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_t + 1 ) end_POSTSUPERSCRIPT = start_UNDERACCENT bold_italic_θ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_UNDERACCENT start_ARG roman_argmin end_ARG - caligraphic_Q start_POSTSUBSCRIPT ( 1 ) end_POSTSUBSCRIPT ( bold_italic_θ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) + caligraphic_P start_POSTSUBSCRIPT italic_λ start_POSTSUBSCRIPT bold_L end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( bold_italic_θ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT )</annotation></semantics></math>, and <math alttext="(iii)" class="ltx_Math" display="inline" id="S1.SS3.p2.16.m5.1"><semantics id="S1.SS3.p2.16.m5.1a"><mrow id="S1.SS3.p2.16.m5.1.1.1" xref="S1.SS3.p2.16.m5.1.1.1.1.cmml"><mo id="S1.SS3.p2.16.m5.1.1.1.2" stretchy="false" xref="S1.SS3.p2.16.m5.1.1.1.1.cmml">(</mo><mrow id="S1.SS3.p2.16.m5.1.1.1.1" xref="S1.SS3.p2.16.m5.1.1.1.1.cmml"><mi id="S1.SS3.p2.16.m5.1.1.1.1.2" xref="S1.SS3.p2.16.m5.1.1.1.1.2.cmml">i</mi><mo id="S1.SS3.p2.16.m5.1.1.1.1.1" xref="S1.SS3.p2.16.m5.1.1.1.1.1.cmml">⁢</mo><mi id="S1.SS3.p2.16.m5.1.1.1.1.3" xref="S1.SS3.p2.16.m5.1.1.1.1.3.cmml">i</mi><mo id="S1.SS3.p2.16.m5.1.1.1.1.1a" xref="S1.SS3.p2.16.m5.1.1.1.1.1.cmml">⁢</mo><mi id="S1.SS3.p2.16.m5.1.1.1.1.4" xref="S1.SS3.p2.16.m5.1.1.1.1.4.cmml">i</mi></mrow><mo id="S1.SS3.p2.16.m5.1.1.1.3" stretchy="false" xref="S1.SS3.p2.16.m5.1.1.1.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S1.SS3.p2.16.m5.1b"><apply id="S1.SS3.p2.16.m5.1.1.1.1.cmml" xref="S1.SS3.p2.16.m5.1.1.1"><times id="S1.SS3.p2.16.m5.1.1.1.1.1.cmml" xref="S1.SS3.p2.16.m5.1.1.1.1.1"></times><ci id="S1.SS3.p2.16.m5.1.1.1.1.2.cmml" xref="S1.SS3.p2.16.m5.1.1.1.1.2">𝑖</ci><ci id="S1.SS3.p2.16.m5.1.1.1.1.3.cmml" xref="S1.SS3.p2.16.m5.1.1.1.1.3">𝑖</ci><ci id="S1.SS3.p2.16.m5.1.1.1.1.4.cmml" xref="S1.SS3.p2.16.m5.1.1.1.1.4">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS3.p2.16.m5.1c">(iii)</annotation><annotation encoding="application/x-llamapun" id="S1.SS3.p2.16.m5.1d">( italic_i italic_i italic_i )</annotation></semantics></math> follows from the second property of AECM algorithm’s second cycle, which implies that <math alttext="\operatorname{\mathcal{Q}}_{(1)}(\operatorname{{\bm{\theta}}}_{2}^{(t)})+% 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id="S1.SS3.p2.17.m6.13c">\operatorname{\mathcal{Q}}_{(1)}(\operatorname{{\bm{\theta}}}_{2}^{(t)})+% \operatorname{\mathcal{P}}_{\lambda_{\operatorname{\mathbf{L}}}}(\operatorname% {{\bm{\theta}}}_{2}^{(t)})=f(\operatorname{{\bm{\theta}}}_{1}^{(t+1)},% \operatorname{{\bm{\theta}}}_{2}^{(t)},\operatorname{{\bm{\theta}}}_{3}^{(t)})</annotation><annotation encoding="application/x-llamapun" id="S1.SS3.p2.17.m6.13d">caligraphic_Q start_POSTSUBSCRIPT ( 1 ) end_POSTSUBSCRIPT ( bold_italic_θ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_t ) end_POSTSUPERSCRIPT ) + caligraphic_P start_POSTSUBSCRIPT italic_λ start_POSTSUBSCRIPT bold_L end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( bold_italic_θ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_t ) end_POSTSUPERSCRIPT ) = italic_f ( bold_italic_θ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_t + 1 ) end_POSTSUPERSCRIPT , bold_italic_θ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_t ) end_POSTSUPERSCRIPT , bold_italic_θ start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_t ) end_POSTSUPERSCRIPT )</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S1.SS3.p3"> <p class="ltx_p" id="S1.SS3.p3.3">We follow a similar sequence of arguments after swapping the subscripts <math alttext="1" class="ltx_Math" display="inline" id="S1.SS3.p3.1.m1.1"><semantics id="S1.SS3.p3.1.m1.1a"><mn id="S1.SS3.p3.1.m1.1.1" xref="S1.SS3.p3.1.m1.1.1.cmml">1</mn><annotation-xml encoding="MathML-Content" id="S1.SS3.p3.1.m1.1b"><cn id="S1.SS3.p3.1.m1.1.1.cmml" type="integer" xref="S1.SS3.p3.1.m1.1.1">1</cn></annotation-xml><annotation encoding="application/x-tex" id="S1.SS3.p3.1.m1.1c">1</annotation><annotation encoding="application/x-llamapun" id="S1.SS3.p3.1.m1.1d">1</annotation></semantics></math> and <math alttext="2" class="ltx_Math" display="inline" id="S1.SS3.p3.2.m2.1"><semantics id="S1.SS3.p3.2.m2.1a"><mn id="S1.SS3.p3.2.m2.1.1" xref="S1.SS3.p3.2.m2.1.1.cmml">2</mn><annotation-xml encoding="MathML-Content" id="S1.SS3.p3.2.m2.1b"><cn id="S1.SS3.p3.2.m2.1.1.cmml" type="integer" xref="S1.SS3.p3.2.m2.1.1">2</cn></annotation-xml><annotation encoding="application/x-tex" id="S1.SS3.p3.2.m2.1c">2</annotation><annotation encoding="application/x-llamapun" id="S1.SS3.p3.2.m2.1d">2</annotation></semantics></math> to obtain lower bound for <math alttext="f(\operatorname{{\bm{\theta}}}_{1}^{(t+1)},\operatorname{{\bm{\theta}}}_{2}^{(% t+1)},\operatorname{{\bm{\theta}}}_{3}^{(t)})" class="ltx_Math" display="inline" id="S1.SS3.p3.3.m3.6"><semantics id="S1.SS3.p3.3.m3.6a"><mrow id="S1.SS3.p3.3.m3.6.6" xref="S1.SS3.p3.3.m3.6.6.cmml"><mi id="S1.SS3.p3.3.m3.6.6.5" xref="S1.SS3.p3.3.m3.6.6.5.cmml">f</mi><mo id="S1.SS3.p3.3.m3.6.6.4" xref="S1.SS3.p3.3.m3.6.6.4.cmml">⁢</mo><mrow id="S1.SS3.p3.3.m3.6.6.3.3" xref="S1.SS3.p3.3.m3.6.6.3.4.cmml"><mo id="S1.SS3.p3.3.m3.6.6.3.3.4" stretchy="false" 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end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_t + 1 ) end_POSTSUPERSCRIPT , bold_italic_θ start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_t ) end_POSTSUPERSCRIPT )</annotation></semantics></math> in (<a class="ltx_ref" href="https://arxiv.org/html/2503.13782v1#S1.Ex6" title="1.3 Proof of Proposition 3.1 ‣ 1 Proof of the Propositions ‣ Regularized Parameter Estimation in Mixed Model Trace Regression"><span class="ltx_text ltx_ref_tag">1.3</span></a>) at the end for the AECM algorithm’s third cycle. 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xref="S1.Ex10.m1.6.6.3.3.3.2.3">3</cn></apply><apply id="S1.Ex10.m1.3.3.1.1.1.cmml" xref="S1.Ex10.m1.3.3.1.1"><plus id="S1.Ex10.m1.3.3.1.1.1.1.cmml" xref="S1.Ex10.m1.3.3.1.1.1.1"></plus><ci id="S1.Ex10.m1.3.3.1.1.1.2.cmml" xref="S1.Ex10.m1.3.3.1.1.1.2">𝑡</ci><cn id="S1.Ex10.m1.3.3.1.1.1.3.cmml" type="integer" xref="S1.Ex10.m1.3.3.1.1.1.3">1</cn></apply></apply></vector></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.Ex10.m1.6c">\displaystyle f(\operatorname{{\bm{\theta}}}_{1}^{(t+1)},\operatorname{{\bm{% \theta}}}_{2}^{(t+1)},\operatorname{{\bm{\theta}}}_{3}^{(t+1)})</annotation><annotation encoding="application/x-llamapun" id="S1.Ex10.m1.6d">italic_f ( bold_italic_θ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_t + 1 ) end_POSTSUPERSCRIPT , bold_italic_θ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_t + 1 ) end_POSTSUPERSCRIPT , bold_italic_θ start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( 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xref="S1.Ex10.m2.3.3.1.1.1.1"></plus><ci id="S1.Ex10.m2.3.3.1.1.1.2.cmml" xref="S1.Ex10.m2.3.3.1.1.1.2">𝑡</ci><cn id="S1.Ex10.m2.3.3.1.1.1.3.cmml" type="integer" xref="S1.Ex10.m2.3.3.1.1.1.3">1</cn></apply></apply></apply><plus id="S1.Ex10.m2.7.7.4.4.4.cmml" xref="S1.Ex10.m2.7.7.4.4.4"></plus></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.Ex10.m2.7c">\displaystyle=-\operatorname{\mathcal{Q}}_{(2)}(\operatorname{{\bm{\theta}}}_{% 3}^{(t+1)})+\operatorname{\mathcal{P}}_{\lambda_{\operatorname{\mathbf{L}}}}(% \operatorname{{\bm{\theta}}}_{3}^{(t+1)})+</annotation><annotation encoding="application/x-llamapun" id="S1.Ex10.m2.7d">= - caligraphic_Q start_POSTSUBSCRIPT ( 2 ) end_POSTSUBSCRIPT ( bold_italic_θ start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_t + 1 ) end_POSTSUPERSCRIPT ) + caligraphic_P start_POSTSUBSCRIPT italic_λ start_POSTSUBSCRIPT bold_L end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( bold_italic_θ start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_t + 1 ) end_POSTSUPERSCRIPT ) +</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="S1.Ex11"><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\quad\;f(\operatorname{{\bm{\theta}}}_{1}^{(t+1)},\operatorname{{% \bm{\theta}}}_{2}^{(t+1)},\operatorname{{\bm{\theta}}}_{3}^{(t+1)})-\{-% \operatorname{\mathcal{Q}}_{(2)}(\operatorname{{\bm{\theta}}}_{3}^{(t+1)})+% \operatorname{\mathcal{P}}_{\lambda_{\operatorname{\mathbf{L}}}}(\operatorname% {{\bm{\theta}}}_{3}^{(t+1)})\}" class="ltx_Math" display="inline" id="S1.Ex11.m1.10"><semantics id="S1.Ex11.m1.10a"><mrow id="S1.Ex11.m1.10.10" xref="S1.Ex11.m1.10.10.cmml"><mrow id="S1.Ex11.m1.9.9.3" xref="S1.Ex11.m1.9.9.3.cmml"><mi id="S1.Ex11.m1.9.9.3.5" 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xref="S1.Ex11.m1.6.6.1.1.1.3">1</cn></apply></apply></apply></apply></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.Ex11.m1.10c">\displaystyle\quad\;f(\operatorname{{\bm{\theta}}}_{1}^{(t+1)},\operatorname{{% \bm{\theta}}}_{2}^{(t+1)},\operatorname{{\bm{\theta}}}_{3}^{(t+1)})-\{-% \operatorname{\mathcal{Q}}_{(2)}(\operatorname{{\bm{\theta}}}_{3}^{(t+1)})+% \operatorname{\mathcal{P}}_{\lambda_{\operatorname{\mathbf{L}}}}(\operatorname% {{\bm{\theta}}}_{3}^{(t+1)})\}</annotation><annotation encoding="application/x-llamapun" id="S1.Ex11.m1.10d">italic_f ( bold_italic_θ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_t + 1 ) end_POSTSUPERSCRIPT , bold_italic_θ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_t + 1 ) end_POSTSUPERSCRIPT , bold_italic_θ start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_t + 1 ) end_POSTSUPERSCRIPT ) - { - caligraphic_Q start_POSTSUBSCRIPT ( 2 ) end_POSTSUBSCRIPT ( bold_italic_θ start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_t + 1 ) end_POSTSUPERSCRIPT ) + caligraphic_P start_POSTSUBSCRIPT italic_λ start_POSTSUBSCRIPT bold_L end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( bold_italic_θ start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_t + 1 ) end_POSTSUPERSCRIPT ) }</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="S1.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_eqn_cell"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\overset{(i)}{\leq}-\operatorname{\mathcal{Q}}_{(2)}(% \operatorname{{\bm{\theta}}}_{3}^{(t+1)})+\operatorname{\mathcal{P}}_{\lambda_% {\operatorname{\mathbf{L}}}}(\operatorname{{\bm{\theta}}}_{3}^{(t+1)})" class="ltx_Math" display="inline" id="S1.Ex12.m1.8"><semantics id="S1.Ex12.m1.8a"><mrow 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xref="S1.Ex12.m1.7.7.3.1.1">subscript</csymbol><ci id="S1.Ex12.m1.7.7.3.1.1.2.cmml" xref="S1.Ex12.m1.7.7.3.1.1.2">𝒫</ci><apply id="S1.Ex12.m1.7.7.3.1.1.3.cmml" xref="S1.Ex12.m1.7.7.3.1.1.3"><csymbol cd="ambiguous" id="S1.Ex12.m1.7.7.3.1.1.3.1.cmml" xref="S1.Ex12.m1.7.7.3.1.1.3">subscript</csymbol><ci id="S1.Ex12.m1.7.7.3.1.1.3.2.cmml" xref="S1.Ex12.m1.7.7.3.1.1.3.2">𝜆</ci><ci id="S1.Ex12.m1.7.7.3.1.1.3.3.cmml" xref="S1.Ex12.m1.7.7.3.1.1.3.3">𝐋</ci></apply></apply><apply id="S1.Ex12.m1.8.8.4.2.2.1.cmml" xref="S1.Ex12.m1.8.8.4.2.2.1"><csymbol cd="ambiguous" id="S1.Ex12.m1.8.8.4.2.2.1.1.cmml" xref="S1.Ex12.m1.8.8.4.2.2.1">superscript</csymbol><apply id="S1.Ex12.m1.8.8.4.2.2.1.2.cmml" xref="S1.Ex12.m1.8.8.4.2.2.1"><csymbol cd="ambiguous" id="S1.Ex12.m1.8.8.4.2.2.1.2.1.cmml" xref="S1.Ex12.m1.8.8.4.2.2.1">subscript</csymbol><ci id="S1.Ex12.m1.8.8.4.2.2.1.2.2.cmml" xref="S1.Ex12.m1.8.8.4.2.2.1.2.2">𝜽</ci><cn id="S1.Ex12.m1.8.8.4.2.2.1.2.3.cmml" type="integer" xref="S1.Ex12.m1.8.8.4.2.2.1.2.3">3</cn></apply><apply id="S1.Ex12.m1.4.4.1.1.1.cmml" xref="S1.Ex12.m1.4.4.1.1"><plus id="S1.Ex12.m1.4.4.1.1.1.1.cmml" xref="S1.Ex12.m1.4.4.1.1.1.1"></plus><ci id="S1.Ex12.m1.4.4.1.1.1.2.cmml" xref="S1.Ex12.m1.4.4.1.1.1.2">𝑡</ci><cn id="S1.Ex12.m1.4.4.1.1.1.3.cmml" type="integer" xref="S1.Ex12.m1.4.4.1.1.1.3">1</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.Ex12.m1.8c">\displaystyle\overset{(i)}{\leq}-\operatorname{\mathcal{Q}}_{(2)}(% \operatorname{{\bm{\theta}}}_{3}^{(t+1)})+\operatorname{\mathcal{P}}_{\lambda_% {\operatorname{\mathbf{L}}}}(\operatorname{{\bm{\theta}}}_{3}^{(t+1)})</annotation><annotation encoding="application/x-llamapun" id="S1.Ex12.m1.8d">start_OVERACCENT ( italic_i ) end_OVERACCENT start_ARG ≤ end_ARG - caligraphic_Q start_POSTSUBSCRIPT ( 2 ) end_POSTSUBSCRIPT ( bold_italic_θ start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_t + 1 ) end_POSTSUPERSCRIPT ) + caligraphic_P start_POSTSUBSCRIPT italic_λ start_POSTSUBSCRIPT bold_L end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( bold_italic_θ start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_t + 1 ) end_POSTSUPERSCRIPT )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="S1.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_eqn_cell"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\overset{(ii)}{\leq}-\operatorname{\mathcal{Q}}_{(2)}(% \operatorname{{\bm{\theta}}}_{3}^{(t)})+\operatorname{\mathcal{P}}_{\lambda_{% \operatorname{\mathbf{L}}}}(\operatorname{{\bm{\theta}}}_{3}^{(t)})" class="ltx_Math" display="inline" id="S1.Ex13.m1.8"><semantics id="S1.Ex13.m1.8a"><mrow id="S1.Ex13.m1.8.8" xref="S1.Ex13.m1.8.8.cmml"><mrow id="S1.Ex13.m1.6.6.2" xref="S1.Ex13.m1.6.6.2.cmml"><mover accent="true" 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xref="S1.E3.m1.4.4.1.1">𝑡</ci></apply></vector></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.E3.m1.5c">\displaystyle\overset{(iii)}{=}f(\operatorname{{\bm{\theta}}}_{1}^{(t+1)},% \operatorname{{\bm{\theta}}}_{2}^{(t+1)},\operatorname{{\bm{\theta}}}_{3}^{(t)% }),</annotation><annotation encoding="application/x-llamapun" id="S1.E3.m1.5d">start_OVERACCENT ( italic_i italic_i italic_i ) end_OVERACCENT start_ARG = end_ARG italic_f ( bold_italic_θ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_t + 1 ) end_POSTSUPERSCRIPT , bold_italic_θ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_t + 1 ) end_POSTSUPERSCRIPT , bold_italic_θ start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_t ) 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">(3)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S1.SS3.p3.7">where <math alttext="(i),(ii)," class="ltx_Math" display="inline" id="S1.SS3.p3.4.m1.2"><semantics id="S1.SS3.p3.4.m1.2a"><mrow id="S1.SS3.p3.4.m1.2.2.1"><mrow id="S1.SS3.p3.4.m1.2.2.1.1.2" xref="S1.SS3.p3.4.m1.2.2.1.1.3.cmml"><mrow id="S1.SS3.p3.4.m1.2.2.1.1.1.1.2" xref="S1.SS3.p3.4.m1.2.2.1.1.3.cmml"><mo id="S1.SS3.p3.4.m1.2.2.1.1.1.1.2.1" stretchy="false" xref="S1.SS3.p3.4.m1.2.2.1.1.3.cmml">(</mo><mi id="S1.SS3.p3.4.m1.1.1" xref="S1.SS3.p3.4.m1.1.1.cmml">i</mi><mo id="S1.SS3.p3.4.m1.2.2.1.1.1.1.2.2" stretchy="false" xref="S1.SS3.p3.4.m1.2.2.1.1.3.cmml">)</mo></mrow><mo id="S1.SS3.p3.4.m1.2.2.1.1.2.3" xref="S1.SS3.p3.4.m1.2.2.1.1.3.cmml">,</mo><mrow id="S1.SS3.p3.4.m1.2.2.1.1.2.2.1" xref="S1.SS3.p3.4.m1.2.2.1.1.2.2.1.1.cmml"><mo id="S1.SS3.p3.4.m1.2.2.1.1.2.2.1.2" stretchy="false" xref="S1.SS3.p3.4.m1.2.2.1.1.2.2.1.1.cmml">(</mo><mrow id="S1.SS3.p3.4.m1.2.2.1.1.2.2.1.1" xref="S1.SS3.p3.4.m1.2.2.1.1.2.2.1.1.cmml"><mi id="S1.SS3.p3.4.m1.2.2.1.1.2.2.1.1.2" xref="S1.SS3.p3.4.m1.2.2.1.1.2.2.1.1.2.cmml">i</mi><mo id="S1.SS3.p3.4.m1.2.2.1.1.2.2.1.1.1" xref="S1.SS3.p3.4.m1.2.2.1.1.2.2.1.1.1.cmml">⁢</mo><mi id="S1.SS3.p3.4.m1.2.2.1.1.2.2.1.1.3" xref="S1.SS3.p3.4.m1.2.2.1.1.2.2.1.1.3.cmml">i</mi></mrow><mo id="S1.SS3.p3.4.m1.2.2.1.1.2.2.1.3" stretchy="false" xref="S1.SS3.p3.4.m1.2.2.1.1.2.2.1.1.cmml">)</mo></mrow></mrow><mo id="S1.SS3.p3.4.m1.2.2.1.2">,</mo></mrow><annotation-xml encoding="MathML-Content" id="S1.SS3.p3.4.m1.2b"><list id="S1.SS3.p3.4.m1.2.2.1.1.3.cmml" xref="S1.SS3.p3.4.m1.2.2.1.1.2"><ci id="S1.SS3.p3.4.m1.1.1.cmml" xref="S1.SS3.p3.4.m1.1.1">𝑖</ci><apply id="S1.SS3.p3.4.m1.2.2.1.1.2.2.1.1.cmml" xref="S1.SS3.p3.4.m1.2.2.1.1.2.2.1"><times id="S1.SS3.p3.4.m1.2.2.1.1.2.2.1.1.1.cmml" xref="S1.SS3.p3.4.m1.2.2.1.1.2.2.1.1.1"></times><ci id="S1.SS3.p3.4.m1.2.2.1.1.2.2.1.1.2.cmml" xref="S1.SS3.p3.4.m1.2.2.1.1.2.2.1.1.2">𝑖</ci><ci id="S1.SS3.p3.4.m1.2.2.1.1.2.2.1.1.3.cmml" xref="S1.SS3.p3.4.m1.2.2.1.1.2.2.1.1.3">𝑖</ci></apply></list></annotation-xml><annotation encoding="application/x-tex" id="S1.SS3.p3.4.m1.2c">(i),(ii),</annotation><annotation encoding="application/x-llamapun" id="S1.SS3.p3.4.m1.2d">( italic_i ) , ( italic_i italic_i ) ,</annotation></semantics></math> and <math alttext="(iii)" class="ltx_Math" display="inline" id="S1.SS3.p3.5.m2.1"><semantics id="S1.SS3.p3.5.m2.1a"><mrow id="S1.SS3.p3.5.m2.1.1.1" xref="S1.SS3.p3.5.m2.1.1.1.1.cmml"><mo id="S1.SS3.p3.5.m2.1.1.1.2" stretchy="false" xref="S1.SS3.p3.5.m2.1.1.1.1.cmml">(</mo><mrow id="S1.SS3.p3.5.m2.1.1.1.1" xref="S1.SS3.p3.5.m2.1.1.1.1.cmml"><mi id="S1.SS3.p3.5.m2.1.1.1.1.2" xref="S1.SS3.p3.5.m2.1.1.1.1.2.cmml">i</mi><mo id="S1.SS3.p3.5.m2.1.1.1.1.1" xref="S1.SS3.p3.5.m2.1.1.1.1.1.cmml">⁢</mo><mi id="S1.SS3.p3.5.m2.1.1.1.1.3" xref="S1.SS3.p3.5.m2.1.1.1.1.3.cmml">i</mi><mo id="S1.SS3.p3.5.m2.1.1.1.1.1a" xref="S1.SS3.p3.5.m2.1.1.1.1.1.cmml">⁢</mo><mi id="S1.SS3.p3.5.m2.1.1.1.1.4" xref="S1.SS3.p3.5.m2.1.1.1.1.4.cmml">i</mi></mrow><mo id="S1.SS3.p3.5.m2.1.1.1.3" stretchy="false" xref="S1.SS3.p3.5.m2.1.1.1.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S1.SS3.p3.5.m2.1b"><apply id="S1.SS3.p3.5.m2.1.1.1.1.cmml" xref="S1.SS3.p3.5.m2.1.1.1"><times id="S1.SS3.p3.5.m2.1.1.1.1.1.cmml" xref="S1.SS3.p3.5.m2.1.1.1.1.1"></times><ci id="S1.SS3.p3.5.m2.1.1.1.1.2.cmml" xref="S1.SS3.p3.5.m2.1.1.1.1.2">𝑖</ci><ci id="S1.SS3.p3.5.m2.1.1.1.1.3.cmml" xref="S1.SS3.p3.5.m2.1.1.1.1.3">𝑖</ci><ci id="S1.SS3.p3.5.m2.1.1.1.1.4.cmml" xref="S1.SS3.p3.5.m2.1.1.1.1.4">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS3.p3.5.m2.1c">(iii)</annotation><annotation encoding="application/x-llamapun" id="S1.SS3.p3.5.m2.1d">( italic_i italic_i italic_i )</annotation></semantics></math> follow from the same arguments used in (<a class="ltx_ref" href="https://arxiv.org/html/2503.13782v1#S1.Ex6" title="1.3 Proof of Proposition 3.1 ‣ 1 Proof of the Propositions ‣ Regularized Parameter Estimation in Mixed Model Trace Regression"><span class="ltx_text ltx_ref_tag">1.3</span></a>), except we swap <math alttext="1" class="ltx_Math" display="inline" id="S1.SS3.p3.6.m3.1"><semantics id="S1.SS3.p3.6.m3.1a"><mn id="S1.SS3.p3.6.m3.1.1" xref="S1.SS3.p3.6.m3.1.1.cmml">1</mn><annotation-xml encoding="MathML-Content" id="S1.SS3.p3.6.m3.1b"><cn id="S1.SS3.p3.6.m3.1.1.cmml" type="integer" xref="S1.SS3.p3.6.m3.1.1">1</cn></annotation-xml><annotation encoding="application/x-tex" id="S1.SS3.p3.6.m3.1c">1</annotation><annotation encoding="application/x-llamapun" id="S1.SS3.p3.6.m3.1d">1</annotation></semantics></math> and <math alttext="2" class="ltx_Math" display="inline" id="S1.SS3.p3.7.m4.1"><semantics id="S1.SS3.p3.7.m4.1a"><mn id="S1.SS3.p3.7.m4.1.1" xref="S1.SS3.p3.7.m4.1.1.cmml">2</mn><annotation-xml encoding="MathML-Content" id="S1.SS3.p3.7.m4.1b"><cn id="S1.SS3.p3.7.m4.1.1.cmml" type="integer" xref="S1.SS3.p3.7.m4.1.1">2</cn></annotation-xml><annotation encoding="application/x-tex" id="S1.SS3.p3.7.m4.1c">2</annotation><annotation encoding="application/x-llamapun" id="S1.SS3.p3.7.m4.1d">2</annotation></semantics></math>. Finally, (<a class="ltx_ref" href="https://arxiv.org/html/2503.13782v1#S1.E1a" title="In 1.3 Proof of Proposition 3.1 ‣ 1 Proof of the Propositions ‣ Regularized Parameter Estimation in Mixed Model Trace Regression"><span class="ltx_text ltx_ref_tag">1</span></a>), (<a class="ltx_ref" href="https://arxiv.org/html/2503.13782v1#S1.Ex6" title="1.3 Proof of Proposition 3.1 ‣ 1 Proof of the Propositions ‣ Regularized Parameter Estimation in Mixed Model Trace Regression"><span class="ltx_text ltx_ref_tag">1.3</span></a>), and (<a class="ltx_ref" href="https://arxiv.org/html/2503.13782v1#S1.Ex10" title="1.3 Proof of Proposition 3.1 ‣ 1 Proof of the Propositions ‣ Regularized Parameter Estimation in Mixed Model Trace Regression"><span class="ltx_text ltx_ref_tag">1.3</span></a>) imply that</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="S3.EGx20"> <tbody id="S1.E4"><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(\operatorname{{\bm{\theta}}}_{1}^{(t+1)},\operatorname{{\bm{% \theta}}}_{2}^{(t+1)},\operatorname{{\bm{\theta}}}_{3}^{(t+1)})\leq f(% \operatorname{{\bm{\theta}}}_{1}^{(t+1)},\operatorname{{\bm{\theta}}}_{2}^{(t+% 1)},\operatorname{{\bm{\theta}}}_{3}^{(t)})\leq f(\operatorname{{\bm{\theta}}}% _{1}^{(t+1)},\operatorname{{\bm{\theta}}}_{2}^{(t)},\operatorname{{\bm{\theta}% }}_{3}^{(t)})\leq f(\operatorname{{\bm{\theta}}}_{1}^{(t)},\operatorname{{\bm{% \theta}}}_{2}^{(t)},\operatorname{{\bm{\theta}}}_{3}^{(t)})," class="ltx_Math" display="inline" id="S1.E4.m1.13"><semantics id="S1.E4.m1.13a"><mrow id="S1.E4.m1.13.13.1" xref="S1.E4.m1.13.13.1.1.cmml"><mrow id="S1.E4.m1.13.13.1.1" xref="S1.E4.m1.13.13.1.1.cmml"><mrow id="S1.E4.m1.13.13.1.1.3" xref="S1.E4.m1.13.13.1.1.3.cmml"><mi id="S1.E4.m1.13.13.1.1.3.5" xref="S1.E4.m1.13.13.1.1.3.5.cmml">f</mi><mo id="S1.E4.m1.13.13.1.1.3.4" xref="S1.E4.m1.13.13.1.1.3.4.cmml">⁢</mo><mrow id="S1.E4.m1.13.13.1.1.3.3.3" xref="S1.E4.m1.13.13.1.1.3.3.4.cmml"><mo id="S1.E4.m1.13.13.1.1.3.3.3.4" stretchy="false" xref="S1.E4.m1.13.13.1.1.3.3.4.cmml">(</mo><msubsup id="S1.E4.m1.13.13.1.1.1.1.1.1" xref="S1.E4.m1.13.13.1.1.1.1.1.1.cmml"><mi id="S1.E4.m1.13.13.1.1.1.1.1.1.2.2" xref="S1.E4.m1.13.13.1.1.1.1.1.1.2.2.cmml">𝜽</mi><mn id="S1.E4.m1.13.13.1.1.1.1.1.1.2.3" xref="S1.E4.m1.13.13.1.1.1.1.1.1.2.3.cmml">1</mn><mrow id="S1.E4.m1.1.1.1.1" xref="S1.E4.m1.1.1.1.1.1.cmml"><mo id="S1.E4.m1.1.1.1.1.2" stretchy="false" xref="S1.E4.m1.1.1.1.1.1.cmml">(</mo><mrow id="S1.E4.m1.1.1.1.1.1" xref="S1.E4.m1.1.1.1.1.1.cmml"><mi id="S1.E4.m1.1.1.1.1.1.2" xref="S1.E4.m1.1.1.1.1.1.2.cmml">t</mi><mo id="S1.E4.m1.1.1.1.1.1.1" xref="S1.E4.m1.1.1.1.1.1.1.cmml">+</mo><mn id="S1.E4.m1.1.1.1.1.1.3" xref="S1.E4.m1.1.1.1.1.1.3.cmml">1</mn></mrow><mo id="S1.E4.m1.1.1.1.1.3" stretchy="false" xref="S1.E4.m1.1.1.1.1.1.cmml">)</mo></mrow></msubsup><mo id="S1.E4.m1.13.13.1.1.3.3.3.5" xref="S1.E4.m1.13.13.1.1.3.3.4.cmml">,</mo><msubsup id="S1.E4.m1.13.13.1.1.2.2.2.2" xref="S1.E4.m1.13.13.1.1.2.2.2.2.cmml"><mi id="S1.E4.m1.13.13.1.1.2.2.2.2.2.2" xref="S1.E4.m1.13.13.1.1.2.2.2.2.2.2.cmml">𝜽</mi><mn id="S1.E4.m1.13.13.1.1.2.2.2.2.2.3" xref="S1.E4.m1.13.13.1.1.2.2.2.2.2.3.cmml">2</mn><mrow id="S1.E4.m1.2.2.1.1" xref="S1.E4.m1.2.2.1.1.1.cmml"><mo id="S1.E4.m1.2.2.1.1.2" stretchy="false" xref="S1.E4.m1.2.2.1.1.1.cmml">(</mo><mrow id="S1.E4.m1.2.2.1.1.1" xref="S1.E4.m1.2.2.1.1.1.cmml"><mi id="S1.E4.m1.2.2.1.1.1.2" xref="S1.E4.m1.2.2.1.1.1.2.cmml">t</mi><mo id="S1.E4.m1.2.2.1.1.1.1" xref="S1.E4.m1.2.2.1.1.1.1.cmml">+</mo><mn id="S1.E4.m1.2.2.1.1.1.3" xref="S1.E4.m1.2.2.1.1.1.3.cmml">1</mn></mrow><mo id="S1.E4.m1.2.2.1.1.3" stretchy="false" xref="S1.E4.m1.2.2.1.1.1.cmml">)</mo></mrow></msubsup><mo id="S1.E4.m1.13.13.1.1.3.3.3.6" xref="S1.E4.m1.13.13.1.1.3.3.4.cmml">,</mo><msubsup id="S1.E4.m1.13.13.1.1.3.3.3.3" xref="S1.E4.m1.13.13.1.1.3.3.3.3.cmml"><mi id="S1.E4.m1.13.13.1.1.3.3.3.3.2.2" xref="S1.E4.m1.13.13.1.1.3.3.3.3.2.2.cmml">𝜽</mi><mn id="S1.E4.m1.13.13.1.1.3.3.3.3.2.3" xref="S1.E4.m1.13.13.1.1.3.3.3.3.2.3.cmml">3</mn><mrow id="S1.E4.m1.3.3.1.1" xref="S1.E4.m1.3.3.1.1.1.cmml"><mo id="S1.E4.m1.3.3.1.1.2" stretchy="false" xref="S1.E4.m1.3.3.1.1.1.cmml">(</mo><mrow id="S1.E4.m1.3.3.1.1.1" xref="S1.E4.m1.3.3.1.1.1.cmml"><mi id="S1.E4.m1.3.3.1.1.1.2" xref="S1.E4.m1.3.3.1.1.1.2.cmml">t</mi><mo id="S1.E4.m1.3.3.1.1.1.1" xref="S1.E4.m1.3.3.1.1.1.1.cmml">+</mo><mn id="S1.E4.m1.3.3.1.1.1.3" xref="S1.E4.m1.3.3.1.1.1.3.cmml">1</mn></mrow><mo id="S1.E4.m1.3.3.1.1.3" stretchy="false" xref="S1.E4.m1.3.3.1.1.1.cmml">)</mo></mrow></msubsup><mo id="S1.E4.m1.13.13.1.1.3.3.3.7" stretchy="false" xref="S1.E4.m1.13.13.1.1.3.3.4.cmml">)</mo></mrow></mrow><mo id="S1.E4.m1.13.13.1.1.14" xref="S1.E4.m1.13.13.1.1.14.cmml">≤</mo><mrow id="S1.E4.m1.13.13.1.1.6" xref="S1.E4.m1.13.13.1.1.6.cmml"><mi id="S1.E4.m1.13.13.1.1.6.5" xref="S1.E4.m1.13.13.1.1.6.5.cmml">f</mi><mo id="S1.E4.m1.13.13.1.1.6.4" xref="S1.E4.m1.13.13.1.1.6.4.cmml">⁢</mo><mrow id="S1.E4.m1.13.13.1.1.6.3.3" xref="S1.E4.m1.13.13.1.1.6.3.4.cmml"><mo id="S1.E4.m1.13.13.1.1.6.3.3.4" stretchy="false" xref="S1.E4.m1.13.13.1.1.6.3.4.cmml">(</mo><msubsup id="S1.E4.m1.13.13.1.1.4.1.1.1" xref="S1.E4.m1.13.13.1.1.4.1.1.1.cmml"><mi id="S1.E4.m1.13.13.1.1.4.1.1.1.2.2" xref="S1.E4.m1.13.13.1.1.4.1.1.1.2.2.cmml">𝜽</mi><mn id="S1.E4.m1.13.13.1.1.4.1.1.1.2.3" xref="S1.E4.m1.13.13.1.1.4.1.1.1.2.3.cmml">1</mn><mrow id="S1.E4.m1.4.4.1.1" xref="S1.E4.m1.4.4.1.1.1.cmml"><mo id="S1.E4.m1.4.4.1.1.2" stretchy="false" xref="S1.E4.m1.4.4.1.1.1.cmml">(</mo><mrow id="S1.E4.m1.4.4.1.1.1" xref="S1.E4.m1.4.4.1.1.1.cmml"><mi id="S1.E4.m1.4.4.1.1.1.2" xref="S1.E4.m1.4.4.1.1.1.2.cmml">t</mi><mo id="S1.E4.m1.4.4.1.1.1.1" xref="S1.E4.m1.4.4.1.1.1.1.cmml">+</mo><mn id="S1.E4.m1.4.4.1.1.1.3" xref="S1.E4.m1.4.4.1.1.1.3.cmml">1</mn></mrow><mo id="S1.E4.m1.4.4.1.1.3" stretchy="false" xref="S1.E4.m1.4.4.1.1.1.cmml">)</mo></mrow></msubsup><mo id="S1.E4.m1.13.13.1.1.6.3.3.5" xref="S1.E4.m1.13.13.1.1.6.3.4.cmml">,</mo><msubsup id="S1.E4.m1.13.13.1.1.5.2.2.2" xref="S1.E4.m1.13.13.1.1.5.2.2.2.cmml"><mi id="S1.E4.m1.13.13.1.1.5.2.2.2.2.2" xref="S1.E4.m1.13.13.1.1.5.2.2.2.2.2.cmml">𝜽</mi><mn id="S1.E4.m1.13.13.1.1.5.2.2.2.2.3" xref="S1.E4.m1.13.13.1.1.5.2.2.2.2.3.cmml">2</mn><mrow id="S1.E4.m1.5.5.1.1" xref="S1.E4.m1.5.5.1.1.1.cmml"><mo id="S1.E4.m1.5.5.1.1.2" stretchy="false" xref="S1.E4.m1.5.5.1.1.1.cmml">(</mo><mrow id="S1.E4.m1.5.5.1.1.1" xref="S1.E4.m1.5.5.1.1.1.cmml"><mi id="S1.E4.m1.5.5.1.1.1.2" xref="S1.E4.m1.5.5.1.1.1.2.cmml">t</mi><mo id="S1.E4.m1.5.5.1.1.1.1" xref="S1.E4.m1.5.5.1.1.1.1.cmml">+</mo><mn id="S1.E4.m1.5.5.1.1.1.3" xref="S1.E4.m1.5.5.1.1.1.3.cmml">1</mn></mrow><mo id="S1.E4.m1.5.5.1.1.3" stretchy="false" xref="S1.E4.m1.5.5.1.1.1.cmml">)</mo></mrow></msubsup><mo id="S1.E4.m1.13.13.1.1.6.3.3.6" xref="S1.E4.m1.13.13.1.1.6.3.4.cmml">,</mo><msubsup id="S1.E4.m1.13.13.1.1.6.3.3.3" xref="S1.E4.m1.13.13.1.1.6.3.3.3.cmml"><mi id="S1.E4.m1.13.13.1.1.6.3.3.3.2.2" xref="S1.E4.m1.13.13.1.1.6.3.3.3.2.2.cmml">𝜽</mi><mn id="S1.E4.m1.13.13.1.1.6.3.3.3.2.3" xref="S1.E4.m1.13.13.1.1.6.3.3.3.2.3.cmml">3</mn><mrow id="S1.E4.m1.6.6.1.3" xref="S1.E4.m1.13.13.1.1.6.3.3.3.cmml"><mo id="S1.E4.m1.6.6.1.3.1" stretchy="false" xref="S1.E4.m1.13.13.1.1.6.3.3.3.cmml">(</mo><mi id="S1.E4.m1.6.6.1.1" xref="S1.E4.m1.6.6.1.1.cmml">t</mi><mo id="S1.E4.m1.6.6.1.3.2" stretchy="false" xref="S1.E4.m1.13.13.1.1.6.3.3.3.cmml">)</mo></mrow></msubsup><mo id="S1.E4.m1.13.13.1.1.6.3.3.7" stretchy="false" xref="S1.E4.m1.13.13.1.1.6.3.4.cmml">)</mo></mrow></mrow><mo id="S1.E4.m1.13.13.1.1.15" xref="S1.E4.m1.13.13.1.1.15.cmml">≤</mo><mrow id="S1.E4.m1.13.13.1.1.9" xref="S1.E4.m1.13.13.1.1.9.cmml"><mi id="S1.E4.m1.13.13.1.1.9.5" xref="S1.E4.m1.13.13.1.1.9.5.cmml">f</mi><mo id="S1.E4.m1.13.13.1.1.9.4" xref="S1.E4.m1.13.13.1.1.9.4.cmml">⁢</mo><mrow id="S1.E4.m1.13.13.1.1.9.3.3" xref="S1.E4.m1.13.13.1.1.9.3.4.cmml"><mo id="S1.E4.m1.13.13.1.1.9.3.3.4" stretchy="false" xref="S1.E4.m1.13.13.1.1.9.3.4.cmml">(</mo><msubsup id="S1.E4.m1.13.13.1.1.7.1.1.1" xref="S1.E4.m1.13.13.1.1.7.1.1.1.cmml"><mi id="S1.E4.m1.13.13.1.1.7.1.1.1.2.2" xref="S1.E4.m1.13.13.1.1.7.1.1.1.2.2.cmml">𝜽</mi><mn id="S1.E4.m1.13.13.1.1.7.1.1.1.2.3" xref="S1.E4.m1.13.13.1.1.7.1.1.1.2.3.cmml">1</mn><mrow id="S1.E4.m1.7.7.1.1" xref="S1.E4.m1.7.7.1.1.1.cmml"><mo id="S1.E4.m1.7.7.1.1.2" stretchy="false" xref="S1.E4.m1.7.7.1.1.1.cmml">(</mo><mrow id="S1.E4.m1.7.7.1.1.1" xref="S1.E4.m1.7.7.1.1.1.cmml"><mi id="S1.E4.m1.7.7.1.1.1.2" xref="S1.E4.m1.7.7.1.1.1.2.cmml">t</mi><mo id="S1.E4.m1.7.7.1.1.1.1" xref="S1.E4.m1.7.7.1.1.1.1.cmml">+</mo><mn id="S1.E4.m1.7.7.1.1.1.3" xref="S1.E4.m1.7.7.1.1.1.3.cmml">1</mn></mrow><mo id="S1.E4.m1.7.7.1.1.3" stretchy="false" xref="S1.E4.m1.7.7.1.1.1.cmml">)</mo></mrow></msubsup><mo id="S1.E4.m1.13.13.1.1.9.3.3.5" xref="S1.E4.m1.13.13.1.1.9.3.4.cmml">,</mo><msubsup id="S1.E4.m1.13.13.1.1.8.2.2.2" xref="S1.E4.m1.13.13.1.1.8.2.2.2.cmml"><mi id="S1.E4.m1.13.13.1.1.8.2.2.2.2.2" xref="S1.E4.m1.13.13.1.1.8.2.2.2.2.2.cmml">𝜽</mi><mn id="S1.E4.m1.13.13.1.1.8.2.2.2.2.3" xref="S1.E4.m1.13.13.1.1.8.2.2.2.2.3.cmml">2</mn><mrow id="S1.E4.m1.8.8.1.3" xref="S1.E4.m1.13.13.1.1.8.2.2.2.cmml"><mo id="S1.E4.m1.8.8.1.3.1" stretchy="false" xref="S1.E4.m1.13.13.1.1.8.2.2.2.cmml">(</mo><mi id="S1.E4.m1.8.8.1.1" xref="S1.E4.m1.8.8.1.1.cmml">t</mi><mo id="S1.E4.m1.8.8.1.3.2" stretchy="false" xref="S1.E4.m1.13.13.1.1.8.2.2.2.cmml">)</mo></mrow></msubsup><mo id="S1.E4.m1.13.13.1.1.9.3.3.6" xref="S1.E4.m1.13.13.1.1.9.3.4.cmml">,</mo><msubsup id="S1.E4.m1.13.13.1.1.9.3.3.3" xref="S1.E4.m1.13.13.1.1.9.3.3.3.cmml"><mi id="S1.E4.m1.13.13.1.1.9.3.3.3.2.2" xref="S1.E4.m1.13.13.1.1.9.3.3.3.2.2.cmml">𝜽</mi><mn id="S1.E4.m1.13.13.1.1.9.3.3.3.2.3" xref="S1.E4.m1.13.13.1.1.9.3.3.3.2.3.cmml">3</mn><mrow id="S1.E4.m1.9.9.1.3" xref="S1.E4.m1.13.13.1.1.9.3.3.3.cmml"><mo id="S1.E4.m1.9.9.1.3.1" stretchy="false" xref="S1.E4.m1.13.13.1.1.9.3.3.3.cmml">(</mo><mi id="S1.E4.m1.9.9.1.1" xref="S1.E4.m1.9.9.1.1.cmml">t</mi><mo id="S1.E4.m1.9.9.1.3.2" stretchy="false" xref="S1.E4.m1.13.13.1.1.9.3.3.3.cmml">)</mo></mrow></msubsup><mo id="S1.E4.m1.13.13.1.1.9.3.3.7" stretchy="false" xref="S1.E4.m1.13.13.1.1.9.3.4.cmml">)</mo></mrow></mrow><mo id="S1.E4.m1.13.13.1.1.16" xref="S1.E4.m1.13.13.1.1.16.cmml">≤</mo><mrow id="S1.E4.m1.13.13.1.1.12" xref="S1.E4.m1.13.13.1.1.12.cmml"><mi id="S1.E4.m1.13.13.1.1.12.5" xref="S1.E4.m1.13.13.1.1.12.5.cmml">f</mi><mo id="S1.E4.m1.13.13.1.1.12.4" xref="S1.E4.m1.13.13.1.1.12.4.cmml">⁢</mo><mrow id="S1.E4.m1.13.13.1.1.12.3.3" xref="S1.E4.m1.13.13.1.1.12.3.4.cmml"><mo id="S1.E4.m1.13.13.1.1.12.3.3.4" stretchy="false" xref="S1.E4.m1.13.13.1.1.12.3.4.cmml">(</mo><msubsup id="S1.E4.m1.13.13.1.1.10.1.1.1" xref="S1.E4.m1.13.13.1.1.10.1.1.1.cmml"><mi id="S1.E4.m1.13.13.1.1.10.1.1.1.2.2" xref="S1.E4.m1.13.13.1.1.10.1.1.1.2.2.cmml">𝜽</mi><mn id="S1.E4.m1.13.13.1.1.10.1.1.1.2.3" xref="S1.E4.m1.13.13.1.1.10.1.1.1.2.3.cmml">1</mn><mrow id="S1.E4.m1.10.10.1.3" xref="S1.E4.m1.13.13.1.1.10.1.1.1.cmml"><mo id="S1.E4.m1.10.10.1.3.1" stretchy="false" xref="S1.E4.m1.13.13.1.1.10.1.1.1.cmml">(</mo><mi id="S1.E4.m1.10.10.1.1" xref="S1.E4.m1.10.10.1.1.cmml">t</mi><mo id="S1.E4.m1.10.10.1.3.2" stretchy="false" xref="S1.E4.m1.13.13.1.1.10.1.1.1.cmml">)</mo></mrow></msubsup><mo id="S1.E4.m1.13.13.1.1.12.3.3.5" xref="S1.E4.m1.13.13.1.1.12.3.4.cmml">,</mo><msubsup id="S1.E4.m1.13.13.1.1.11.2.2.2" xref="S1.E4.m1.13.13.1.1.11.2.2.2.cmml"><mi id="S1.E4.m1.13.13.1.1.11.2.2.2.2.2" xref="S1.E4.m1.13.13.1.1.11.2.2.2.2.2.cmml">𝜽</mi><mn id="S1.E4.m1.13.13.1.1.11.2.2.2.2.3" xref="S1.E4.m1.13.13.1.1.11.2.2.2.2.3.cmml">2</mn><mrow id="S1.E4.m1.11.11.1.3" xref="S1.E4.m1.13.13.1.1.11.2.2.2.cmml"><mo id="S1.E4.m1.11.11.1.3.1" stretchy="false" 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xref="S1.E4.m1.13.13.1.1.11.2.2.2.2.3">2</cn></apply><ci id="S1.E4.m1.11.11.1.1.cmml" xref="S1.E4.m1.11.11.1.1">𝑡</ci></apply><apply id="S1.E4.m1.13.13.1.1.12.3.3.3.cmml" xref="S1.E4.m1.13.13.1.1.12.3.3.3"><csymbol cd="ambiguous" id="S1.E4.m1.13.13.1.1.12.3.3.3.1.cmml" xref="S1.E4.m1.13.13.1.1.12.3.3.3">superscript</csymbol><apply id="S1.E4.m1.13.13.1.1.12.3.3.3.2.cmml" xref="S1.E4.m1.13.13.1.1.12.3.3.3"><csymbol cd="ambiguous" id="S1.E4.m1.13.13.1.1.12.3.3.3.2.1.cmml" xref="S1.E4.m1.13.13.1.1.12.3.3.3">subscript</csymbol><ci id="S1.E4.m1.13.13.1.1.12.3.3.3.2.2.cmml" xref="S1.E4.m1.13.13.1.1.12.3.3.3.2.2">𝜽</ci><cn id="S1.E4.m1.13.13.1.1.12.3.3.3.2.3.cmml" type="integer" xref="S1.E4.m1.13.13.1.1.12.3.3.3.2.3">3</cn></apply><ci id="S1.E4.m1.12.12.1.1.cmml" xref="S1.E4.m1.12.12.1.1">𝑡</ci></apply></vector></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.E4.m1.13c">\displaystyle f(\operatorname{{\bm{\theta}}}_{1}^{(t+1)},\operatorname{{\bm{% \theta}}}_{2}^{(t+1)},\operatorname{{\bm{\theta}}}_{3}^{(t+1)})\leq f(% \operatorname{{\bm{\theta}}}_{1}^{(t+1)},\operatorname{{\bm{\theta}}}_{2}^{(t+% 1)},\operatorname{{\bm{\theta}}}_{3}^{(t)})\leq f(\operatorname{{\bm{\theta}}}% _{1}^{(t+1)},\operatorname{{\bm{\theta}}}_{2}^{(t)},\operatorname{{\bm{\theta}% }}_{3}^{(t)})\leq f(\operatorname{{\bm{\theta}}}_{1}^{(t)},\operatorname{{\bm{% \theta}}}_{2}^{(t)},\operatorname{{\bm{\theta}}}_{3}^{(t)}),</annotation><annotation encoding="application/x-llamapun" id="S1.E4.m1.13d">italic_f ( bold_italic_θ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_t + 1 ) end_POSTSUPERSCRIPT , bold_italic_θ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_t + 1 ) end_POSTSUPERSCRIPT , bold_italic_θ start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_t + 1 ) end_POSTSUPERSCRIPT ) ≤ italic_f ( bold_italic_θ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_t + 1 ) end_POSTSUPERSCRIPT , bold_italic_θ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_t + 1 ) end_POSTSUPERSCRIPT , bold_italic_θ start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_t ) end_POSTSUPERSCRIPT ) ≤ italic_f ( bold_italic_θ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_t + 1 ) end_POSTSUPERSCRIPT , bold_italic_θ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_t ) end_POSTSUPERSCRIPT , bold_italic_θ start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_t ) end_POSTSUPERSCRIPT ) ≤ italic_f ( bold_italic_θ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_t ) end_POSTSUPERSCRIPT , bold_italic_θ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_t ) end_POSTSUPERSCRIPT , bold_italic_θ start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_t ) 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">(4)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S1.SS3.p3.15">showing that the AECM algorithm does not increase the objective in every iteration. The objective function <math alttext="f(\operatorname{{\bm{\theta}}})" class="ltx_Math" display="inline" id="S1.SS3.p3.8.m1.1"><semantics id="S1.SS3.p3.8.m1.1a"><mrow id="S1.SS3.p3.8.m1.1.2" xref="S1.SS3.p3.8.m1.1.2.cmml"><mi id="S1.SS3.p3.8.m1.1.2.2" xref="S1.SS3.p3.8.m1.1.2.2.cmml">f</mi><mo id="S1.SS3.p3.8.m1.1.2.1" xref="S1.SS3.p3.8.m1.1.2.1.cmml">⁢</mo><mrow id="S1.SS3.p3.8.m1.1.2.3.2" xref="S1.SS3.p3.8.m1.1.2.cmml"><mo id="S1.SS3.p3.8.m1.1.2.3.2.1" stretchy="false" xref="S1.SS3.p3.8.m1.1.2.cmml">(</mo><mi id="S1.SS3.p3.8.m1.1.1" xref="S1.SS3.p3.8.m1.1.1.cmml">𝜽</mi><mo id="S1.SS3.p3.8.m1.1.2.3.2.2" stretchy="false" xref="S1.SS3.p3.8.m1.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.SS3.p3.8.m1.1b"><apply id="S1.SS3.p3.8.m1.1.2.cmml" xref="S1.SS3.p3.8.m1.1.2"><times id="S1.SS3.p3.8.m1.1.2.1.cmml" xref="S1.SS3.p3.8.m1.1.2.1"></times><ci id="S1.SS3.p3.8.m1.1.2.2.cmml" xref="S1.SS3.p3.8.m1.1.2.2">𝑓</ci><ci id="S1.SS3.p3.8.m1.1.1.cmml" xref="S1.SS3.p3.8.m1.1.1">𝜽</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS3.p3.8.m1.1c">f(\operatorname{{\bm{\theta}}})</annotation><annotation encoding="application/x-llamapun" id="S1.SS3.p3.8.m1.1d">italic_f ( bold_italic_θ )</annotation></semantics></math> is bounded for every <math alttext="\operatorname{{\bm{\theta}}}\in\operatorname{{\mathbf{\Theta}}}" class="ltx_Math" display="inline" id="S1.SS3.p3.9.m2.1"><semantics id="S1.SS3.p3.9.m2.1a"><mrow id="S1.SS3.p3.9.m2.1.1" xref="S1.SS3.p3.9.m2.1.1.cmml"><mi id="S1.SS3.p3.9.m2.1.1.2" xref="S1.SS3.p3.9.m2.1.1.2.cmml">𝜽</mi><mo id="S1.SS3.p3.9.m2.1.1.1" xref="S1.SS3.p3.9.m2.1.1.1.cmml">∈</mo><mi id="S1.SS3.p3.9.m2.1.1.3" xref="S1.SS3.p3.9.m2.1.1.3.cmml">𝚯</mi></mrow><annotation-xml encoding="MathML-Content" id="S1.SS3.p3.9.m2.1b"><apply id="S1.SS3.p3.9.m2.1.1.cmml" xref="S1.SS3.p3.9.m2.1.1"><in id="S1.SS3.p3.9.m2.1.1.1.cmml" xref="S1.SS3.p3.9.m2.1.1.1"></in><ci id="S1.SS3.p3.9.m2.1.1.2.cmml" xref="S1.SS3.p3.9.m2.1.1.2">𝜽</ci><ci id="S1.SS3.p3.9.m2.1.1.3.cmml" xref="S1.SS3.p3.9.m2.1.1.3">𝚯</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS3.p3.9.m2.1c">\operatorname{{\bm{\theta}}}\in\operatorname{{\mathbf{\Theta}}}</annotation><annotation encoding="application/x-llamapun" id="S1.SS3.p3.9.m2.1d">bold_italic_θ ∈ bold_Θ</annotation></semantics></math> because <math alttext="\operatorname{{\mathbf{\Theta}}}" class="ltx_Math" display="inline" id="S1.SS3.p3.10.m3.1"><semantics id="S1.SS3.p3.10.m3.1a"><mi id="S1.SS3.p3.10.m3.1.1" xref="S1.SS3.p3.10.m3.1.1.cmml">𝚯</mi><annotation-xml encoding="MathML-Content" id="S1.SS3.p3.10.m3.1b"><ci id="S1.SS3.p3.10.m3.1.1.cmml" xref="S1.SS3.p3.10.m3.1.1">𝚯</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.SS3.p3.10.m3.1c">\operatorname{{\mathbf{\Theta}}}</annotation><annotation encoding="application/x-llamapun" id="S1.SS3.p3.10.m3.1d">bold_Θ</annotation></semantics></math> is compact; therefore, 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id="S1.SS3.p3.11.m4.2d">italic_f ( bold_italic_θ start_POSTSUPERSCRIPT ( italic_t ) end_POSTSUPERSCRIPT )</annotation></semantics></math> is a bounded non-increasing sequence, so <math alttext="f(\operatorname{{\bm{\theta}}}^{(t)})" class="ltx_Math" display="inline" id="S1.SS3.p3.12.m5.2"><semantics id="S1.SS3.p3.12.m5.2a"><mrow id="S1.SS3.p3.12.m5.2.2" xref="S1.SS3.p3.12.m5.2.2.cmml"><mi id="S1.SS3.p3.12.m5.2.2.3" xref="S1.SS3.p3.12.m5.2.2.3.cmml">f</mi><mo id="S1.SS3.p3.12.m5.2.2.2" xref="S1.SS3.p3.12.m5.2.2.2.cmml">⁢</mo><mrow id="S1.SS3.p3.12.m5.2.2.1.1" xref="S1.SS3.p3.12.m5.2.2.1.1.1.cmml"><mo id="S1.SS3.p3.12.m5.2.2.1.1.2" stretchy="false" xref="S1.SS3.p3.12.m5.2.2.1.1.1.cmml">(</mo><msup id="S1.SS3.p3.12.m5.2.2.1.1.1" xref="S1.SS3.p3.12.m5.2.2.1.1.1.cmml"><mi id="S1.SS3.p3.12.m5.2.2.1.1.1.2" xref="S1.SS3.p3.12.m5.2.2.1.1.1.2.cmml">𝜽</mi><mrow id="S1.SS3.p3.12.m5.1.1.1.3" xref="S1.SS3.p3.12.m5.2.2.1.1.1.cmml"><mo id="S1.SS3.p3.12.m5.1.1.1.3.1" stretchy="false" 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Our assumption implies that convergence happens only for stationary points, so <math alttext="\operatorname{{\bm{\theta}}}^{(\infty)}" class="ltx_Math" display="inline" id="S1.SS3.p3.14.m7.1"><semantics id="S1.SS3.p3.14.m7.1a"><msup id="S1.SS3.p3.14.m7.1.2" xref="S1.SS3.p3.14.m7.1.2.cmml"><mi id="S1.SS3.p3.14.m7.1.2.2" xref="S1.SS3.p3.14.m7.1.2.2.cmml">𝜽</mi><mrow id="S1.SS3.p3.14.m7.1.1.1.3" xref="S1.SS3.p3.14.m7.1.2.cmml"><mo id="S1.SS3.p3.14.m7.1.1.1.3.1" stretchy="false" xref="S1.SS3.p3.14.m7.1.2.cmml">(</mo><mi id="S1.SS3.p3.14.m7.1.1.1.1" mathvariant="normal" xref="S1.SS3.p3.14.m7.1.1.1.1.cmml">∞</mi><mo id="S1.SS3.p3.14.m7.1.1.1.3.2" stretchy="false" xref="S1.SS3.p3.14.m7.1.2.cmml">)</mo></mrow></msup><annotation-xml encoding="MathML-Content" id="S1.SS3.p3.14.m7.1b"><apply id="S1.SS3.p3.14.m7.1.2.cmml" xref="S1.SS3.p3.14.m7.1.2"><csymbol cd="ambiguous" id="S1.SS3.p3.14.m7.1.2.1.cmml" xref="S1.SS3.p3.14.m7.1.2">superscript</csymbol><ci id="S1.SS3.p3.14.m7.1.2.2.cmml" xref="S1.SS3.p3.14.m7.1.2.2">𝜽</ci><infinity id="S1.SS3.p3.14.m7.1.1.1.1.cmml" xref="S1.SS3.p3.14.m7.1.1.1.1"></infinity></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS3.p3.14.m7.1c">\operatorname{{\bm{\theta}}}^{(\infty)}</annotation><annotation encoding="application/x-llamapun" id="S1.SS3.p3.14.m7.1d">bold_italic_θ start_POSTSUPERSCRIPT ( ∞ ) end_POSTSUPERSCRIPT</annotation></semantics></math> is a stationary point of <math alttext="\operatorname{{\bm{\theta}}}^{(t)}" class="ltx_Math" display="inline" id="S1.SS3.p3.15.m8.1"><semantics id="S1.SS3.p3.15.m8.1a"><msup id="S1.SS3.p3.15.m8.1.2" xref="S1.SS3.p3.15.m8.1.2.cmml"><mi id="S1.SS3.p3.15.m8.1.2.2" xref="S1.SS3.p3.15.m8.1.2.2.cmml">𝜽</mi><mrow id="S1.SS3.p3.15.m8.1.1.1.3" xref="S1.SS3.p3.15.m8.1.2.cmml"><mo id="S1.SS3.p3.15.m8.1.1.1.3.1" stretchy="false" xref="S1.SS3.p3.15.m8.1.2.cmml">(</mo><mi id="S1.SS3.p3.15.m8.1.1.1.1" xref="S1.SS3.p3.15.m8.1.1.1.1.cmml">t</mi><mo id="S1.SS3.p3.15.m8.1.1.1.3.2" stretchy="false" xref="S1.SS3.p3.15.m8.1.2.cmml">)</mo></mrow></msup><annotation-xml encoding="MathML-Content" id="S1.SS3.p3.15.m8.1b"><apply id="S1.SS3.p3.15.m8.1.2.cmml" xref="S1.SS3.p3.15.m8.1.2"><csymbol cd="ambiguous" id="S1.SS3.p3.15.m8.1.2.1.cmml" xref="S1.SS3.p3.15.m8.1.2">superscript</csymbol><ci id="S1.SS3.p3.15.m8.1.2.2.cmml" xref="S1.SS3.p3.15.m8.1.2.2">𝜽</ci><ci id="S1.SS3.p3.15.m8.1.1.1.1.cmml" xref="S1.SS3.p3.15.m8.1.1.1.1">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.SS3.p3.15.m8.1c">\operatorname{{\bm{\theta}}}^{(t)}</annotation><annotation encoding="application/x-llamapun" id="S1.SS3.p3.15.m8.1d">bold_italic_θ start_POSTSUPERSCRIPT ( italic_t ) end_POSTSUPERSCRIPT</annotation></semantics></math> sequence. The proposition is proved. <span class="ltx_rule" style="width:6.5pt;height:6.5pt;background:black;display:inline-block;"> </span> <br class="ltx_break"/></p> </div> </section> </section> <section class="ltx_section" id="S2a"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">2 </span>Derivation of the AECM Algorithm Updates</h2> <section class="ltx_subsection" id="S2.SS1a"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">2.1 </span>Derivation of <math alttext="\operatorname{\mathbf{B}}^{(t+1)}" class="ltx_Math" display="inline" id="S2.SS1a.1.m1.1"><semantics id="S2.SS1a.1.m1.1b"><msup id="S2.SS1a.1.m1.1.2" xref="S2.SS1a.1.m1.1.2.cmml"><mi id="S2.SS1a.1.m1.1.2.2" xref="S2.SS1a.1.m1.1.2.2.cmml">𝐁</mi><mrow id="S2.SS1a.1.m1.1.1.1.1" xref="S2.SS1a.1.m1.1.1.1.1.1.cmml"><mo id="S2.SS1a.1.m1.1.1.1.1.2" stretchy="false" xref="S2.SS1a.1.m1.1.1.1.1.1.cmml">(</mo><mrow id="S2.SS1a.1.m1.1.1.1.1.1" 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xref="S2.SS1a.1.m1.1.1.1.1.1.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1a.1.m1.1d">\operatorname{\mathbf{B}}^{(t+1)}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1a.1.m1.1e">bold_B start_POSTSUPERSCRIPT ( italic_t + 1 ) end_POSTSUPERSCRIPT</annotation></semantics></math> and <math alttext="\tau^{2(t+1)}" class="ltx_Math" display="inline" id="S2.SS1a.2.m2.1"><semantics id="S2.SS1a.2.m2.1b"><msup id="S2.SS1a.2.m2.1.2" xref="S2.SS1a.2.m2.1.2.cmml"><mi id="S2.SS1a.2.m2.1.2.2" xref="S2.SS1a.2.m2.1.2.2.cmml">τ</mi><mrow id="S2.SS1a.2.m2.1.1.1" xref="S2.SS1a.2.m2.1.1.1.cmml"><mn id="S2.SS1a.2.m2.1.1.1.3" xref="S2.SS1a.2.m2.1.1.1.3.cmml">2</mn><mo id="S2.SS1a.2.m2.1.1.1.2" xref="S2.SS1a.2.m2.1.1.1.2.cmml">⁢</mo><mrow id="S2.SS1a.2.m2.1.1.1.1.1" xref="S2.SS1a.2.m2.1.1.1.1.1.1.cmml"><mo id="S2.SS1a.2.m2.1.1.1.1.1.2" stretchy="false" xref="S2.SS1a.2.m2.1.1.1.1.1.1.cmml">(</mo><mrow id="S2.SS1a.2.m2.1.1.1.1.1.1" xref="S2.SS1a.2.m2.1.1.1.1.1.1.cmml"><mi id="S2.SS1a.2.m2.1.1.1.1.1.1.2" xref="S2.SS1a.2.m2.1.1.1.1.1.1.2.cmml">t</mi><mo id="S2.SS1a.2.m2.1.1.1.1.1.1.1" xref="S2.SS1a.2.m2.1.1.1.1.1.1.1.cmml">+</mo><mn id="S2.SS1a.2.m2.1.1.1.1.1.1.3" xref="S2.SS1a.2.m2.1.1.1.1.1.1.3.cmml">1</mn></mrow><mo id="S2.SS1a.2.m2.1.1.1.1.1.3" stretchy="false" xref="S2.SS1a.2.m2.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></msup><annotation-xml encoding="MathML-Content" id="S2.SS1a.2.m2.1c"><apply id="S2.SS1a.2.m2.1.2.cmml" xref="S2.SS1a.2.m2.1.2"><csymbol cd="ambiguous" id="S2.SS1a.2.m2.1.2.1.cmml" xref="S2.SS1a.2.m2.1.2">superscript</csymbol><ci id="S2.SS1a.2.m2.1.2.2.cmml" xref="S2.SS1a.2.m2.1.2.2">𝜏</ci><apply id="S2.SS1a.2.m2.1.1.1.cmml" xref="S2.SS1a.2.m2.1.1.1"><times id="S2.SS1a.2.m2.1.1.1.2.cmml" xref="S2.SS1a.2.m2.1.1.1.2"></times><cn id="S2.SS1a.2.m2.1.1.1.3.cmml" type="integer" xref="S2.SS1a.2.m2.1.1.1.3">2</cn><apply id="S2.SS1a.2.m2.1.1.1.1.1.1.cmml" xref="S2.SS1a.2.m2.1.1.1.1.1"><plus id="S2.SS1a.2.m2.1.1.1.1.1.1.1.cmml" xref="S2.SS1a.2.m2.1.1.1.1.1.1.1"></plus><ci id="S2.SS1a.2.m2.1.1.1.1.1.1.2.cmml" xref="S2.SS1a.2.m2.1.1.1.1.1.1.2">𝑡</ci><cn id="S2.SS1a.2.m2.1.1.1.1.1.1.3.cmml" type="integer" xref="S2.SS1a.2.m2.1.1.1.1.1.1.3">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1a.2.m2.1d">\tau^{2(t+1)}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1a.2.m2.1e">italic_τ start_POSTSUPERSCRIPT 2 ( italic_t + 1 ) end_POSTSUPERSCRIPT</annotation></semantics></math> </h3> <div class="ltx_para" id="S2.SS1a.p1"> <p class="ltx_p" id="S2.SS1a.p1.8">To estimate <math alttext="\operatorname{\mathbf{B}}^{(t+1)}" class="ltx_Math" display="inline" id="S2.SS1a.p1.1.m1.1"><semantics id="S2.SS1a.p1.1.m1.1a"><msup id="S2.SS1a.p1.1.m1.1.2" xref="S2.SS1a.p1.1.m1.1.2.cmml"><mi id="S2.SS1a.p1.1.m1.1.2.2" xref="S2.SS1a.p1.1.m1.1.2.2.cmml">𝐁</mi><mrow id="S2.SS1a.p1.1.m1.1.1.1.1" xref="S2.SS1a.p1.1.m1.1.1.1.1.1.cmml"><mo id="S2.SS1a.p1.1.m1.1.1.1.1.2" stretchy="false" xref="S2.SS1a.p1.1.m1.1.1.1.1.1.cmml">(</mo><mrow id="S2.SS1a.p1.1.m1.1.1.1.1.1" xref="S2.SS1a.p1.1.m1.1.1.1.1.1.cmml"><mi id="S2.SS1a.p1.1.m1.1.1.1.1.1.2" xref="S2.SS1a.p1.1.m1.1.1.1.1.1.2.cmml">t</mi><mo id="S2.SS1a.p1.1.m1.1.1.1.1.1.1" xref="S2.SS1a.p1.1.m1.1.1.1.1.1.1.cmml">+</mo><mn id="S2.SS1a.p1.1.m1.1.1.1.1.1.3" xref="S2.SS1a.p1.1.m1.1.1.1.1.1.3.cmml">1</mn></mrow><mo id="S2.SS1a.p1.1.m1.1.1.1.1.3" stretchy="false" xref="S2.SS1a.p1.1.m1.1.1.1.1.1.cmml">)</mo></mrow></msup><annotation-xml encoding="MathML-Content" id="S2.SS1a.p1.1.m1.1b"><apply id="S2.SS1a.p1.1.m1.1.2.cmml" xref="S2.SS1a.p1.1.m1.1.2"><csymbol cd="ambiguous" id="S2.SS1a.p1.1.m1.1.2.1.cmml" xref="S2.SS1a.p1.1.m1.1.2">superscript</csymbol><ci id="S2.SS1a.p1.1.m1.1.2.2.cmml" xref="S2.SS1a.p1.1.m1.1.2.2">𝐁</ci><apply id="S2.SS1a.p1.1.m1.1.1.1.1.1.cmml" xref="S2.SS1a.p1.1.m1.1.1.1.1"><plus id="S2.SS1a.p1.1.m1.1.1.1.1.1.1.cmml" xref="S2.SS1a.p1.1.m1.1.1.1.1.1.1"></plus><ci id="S2.SS1a.p1.1.m1.1.1.1.1.1.2.cmml" xref="S2.SS1a.p1.1.m1.1.1.1.1.1.2">𝑡</ci><cn id="S2.SS1a.p1.1.m1.1.1.1.1.1.3.cmml" type="integer" xref="S2.SS1a.p1.1.m1.1.1.1.1.1.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1a.p1.1.m1.1c">\operatorname{\mathbf{B}}^{(t+1)}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1a.p1.1.m1.1d">bold_B start_POSTSUPERSCRIPT ( italic_t + 1 ) end_POSTSUPERSCRIPT</annotation></semantics></math> and <math alttext="\tau^{2(t+1)}" class="ltx_Math" display="inline" id="S2.SS1a.p1.2.m2.1"><semantics id="S2.SS1a.p1.2.m2.1a"><msup id="S2.SS1a.p1.2.m2.1.2" xref="S2.SS1a.p1.2.m2.1.2.cmml"><mi id="S2.SS1a.p1.2.m2.1.2.2" xref="S2.SS1a.p1.2.m2.1.2.2.cmml">τ</mi><mrow id="S2.SS1a.p1.2.m2.1.1.1" xref="S2.SS1a.p1.2.m2.1.1.1.cmml"><mn id="S2.SS1a.p1.2.m2.1.1.1.3" xref="S2.SS1a.p1.2.m2.1.1.1.3.cmml">2</mn><mo id="S2.SS1a.p1.2.m2.1.1.1.2" xref="S2.SS1a.p1.2.m2.1.1.1.2.cmml">⁢</mo><mrow id="S2.SS1a.p1.2.m2.1.1.1.1.1" xref="S2.SS1a.p1.2.m2.1.1.1.1.1.1.cmml"><mo id="S2.SS1a.p1.2.m2.1.1.1.1.1.2" stretchy="false" xref="S2.SS1a.p1.2.m2.1.1.1.1.1.1.cmml">(</mo><mrow id="S2.SS1a.p1.2.m2.1.1.1.1.1.1" xref="S2.SS1a.p1.2.m2.1.1.1.1.1.1.cmml"><mi id="S2.SS1a.p1.2.m2.1.1.1.1.1.1.2" xref="S2.SS1a.p1.2.m2.1.1.1.1.1.1.2.cmml">t</mi><mo id="S2.SS1a.p1.2.m2.1.1.1.1.1.1.1" xref="S2.SS1a.p1.2.m2.1.1.1.1.1.1.1.cmml">+</mo><mn id="S2.SS1a.p1.2.m2.1.1.1.1.1.1.3" xref="S2.SS1a.p1.2.m2.1.1.1.1.1.1.3.cmml">1</mn></mrow><mo id="S2.SS1a.p1.2.m2.1.1.1.1.1.3" stretchy="false" xref="S2.SS1a.p1.2.m2.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></msup><annotation-xml encoding="MathML-Content" id="S2.SS1a.p1.2.m2.1b"><apply id="S2.SS1a.p1.2.m2.1.2.cmml" xref="S2.SS1a.p1.2.m2.1.2"><csymbol cd="ambiguous" id="S2.SS1a.p1.2.m2.1.2.1.cmml" xref="S2.SS1a.p1.2.m2.1.2">superscript</csymbol><ci id="S2.SS1a.p1.2.m2.1.2.2.cmml" xref="S2.SS1a.p1.2.m2.1.2.2">𝜏</ci><apply id="S2.SS1a.p1.2.m2.1.1.1.cmml" xref="S2.SS1a.p1.2.m2.1.1.1"><times id="S2.SS1a.p1.2.m2.1.1.1.2.cmml" xref="S2.SS1a.p1.2.m2.1.1.1.2"></times><cn id="S2.SS1a.p1.2.m2.1.1.1.3.cmml" type="integer" xref="S2.SS1a.p1.2.m2.1.1.1.3">2</cn><apply id="S2.SS1a.p1.2.m2.1.1.1.1.1.1.cmml" xref="S2.SS1a.p1.2.m2.1.1.1.1.1"><plus id="S2.SS1a.p1.2.m2.1.1.1.1.1.1.1.cmml" xref="S2.SS1a.p1.2.m2.1.1.1.1.1.1.1"></plus><ci id="S2.SS1a.p1.2.m2.1.1.1.1.1.1.2.cmml" xref="S2.SS1a.p1.2.m2.1.1.1.1.1.1.2">𝑡</ci><cn id="S2.SS1a.p1.2.m2.1.1.1.1.1.1.3.cmml" type="integer" xref="S2.SS1a.p1.2.m2.1.1.1.1.1.1.3">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1a.p1.2.m2.1c">\tau^{2(t+1)}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1a.p1.2.m2.1d">italic_τ start_POSTSUPERSCRIPT 2 ( italic_t + 1 ) end_POSTSUPERSCRIPT</annotation></semantics></math>, we directly use the observed data log likelihood, <math alttext="\log f_{\operatorname{\mathbf{y}}|\operatorname{{\bm{\theta}}}}(\operatorname{% \mathbf{y}})" class="ltx_Math" display="inline" id="S2.SS1a.p1.3.m3.1"><semantics id="S2.SS1a.p1.3.m3.1a"><mrow id="S2.SS1a.p1.3.m3.1.2" xref="S2.SS1a.p1.3.m3.1.2.cmml"><mrow id="S2.SS1a.p1.3.m3.1.2.2" xref="S2.SS1a.p1.3.m3.1.2.2.cmml"><mi id="S2.SS1a.p1.3.m3.1.2.2.1" xref="S2.SS1a.p1.3.m3.1.2.2.1.cmml">log</mi><mo id="S2.SS1a.p1.3.m3.1.2.2a" lspace="0.167em" xref="S2.SS1a.p1.3.m3.1.2.2.cmml">⁡</mo><msub id="S2.SS1a.p1.3.m3.1.2.2.2" xref="S2.SS1a.p1.3.m3.1.2.2.2.cmml"><mi id="S2.SS1a.p1.3.m3.1.2.2.2.2" xref="S2.SS1a.p1.3.m3.1.2.2.2.2.cmml">f</mi><mrow id="S2.SS1a.p1.3.m3.1.2.2.2.3" xref="S2.SS1a.p1.3.m3.1.2.2.2.3.cmml"><mi id="S2.SS1a.p1.3.m3.1.2.2.2.3.2" xref="S2.SS1a.p1.3.m3.1.2.2.2.3.2.cmml">𝐲</mi><mo fence="false" id="S2.SS1a.p1.3.m3.1.2.2.2.3.1" xref="S2.SS1a.p1.3.m3.1.2.2.2.3.1.cmml">|</mo><mi id="S2.SS1a.p1.3.m3.1.2.2.2.3.3" xref="S2.SS1a.p1.3.m3.1.2.2.2.3.3.cmml">𝜽</mi></mrow></msub></mrow><mo id="S2.SS1a.p1.3.m3.1.2.1" xref="S2.SS1a.p1.3.m3.1.2.1.cmml">⁢</mo><mrow id="S2.SS1a.p1.3.m3.1.2.3.2" xref="S2.SS1a.p1.3.m3.1.2.cmml"><mo id="S2.SS1a.p1.3.m3.1.2.3.2.1" stretchy="false" xref="S2.SS1a.p1.3.m3.1.2.cmml">(</mo><mi id="S2.SS1a.p1.3.m3.1.1" xref="S2.SS1a.p1.3.m3.1.1.cmml">𝐲</mi><mo id="S2.SS1a.p1.3.m3.1.2.3.2.2" stretchy="false" xref="S2.SS1a.p1.3.m3.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1a.p1.3.m3.1b"><apply id="S2.SS1a.p1.3.m3.1.2.cmml" xref="S2.SS1a.p1.3.m3.1.2"><times id="S2.SS1a.p1.3.m3.1.2.1.cmml" xref="S2.SS1a.p1.3.m3.1.2.1"></times><apply id="S2.SS1a.p1.3.m3.1.2.2.cmml" xref="S2.SS1a.p1.3.m3.1.2.2"><log id="S2.SS1a.p1.3.m3.1.2.2.1.cmml" xref="S2.SS1a.p1.3.m3.1.2.2.1"></log><apply id="S2.SS1a.p1.3.m3.1.2.2.2.cmml" xref="S2.SS1a.p1.3.m3.1.2.2.2"><csymbol cd="ambiguous" id="S2.SS1a.p1.3.m3.1.2.2.2.1.cmml" xref="S2.SS1a.p1.3.m3.1.2.2.2">subscript</csymbol><ci id="S2.SS1a.p1.3.m3.1.2.2.2.2.cmml" xref="S2.SS1a.p1.3.m3.1.2.2.2.2">𝑓</ci><apply id="S2.SS1a.p1.3.m3.1.2.2.2.3.cmml" xref="S2.SS1a.p1.3.m3.1.2.2.2.3"><csymbol cd="latexml" id="S2.SS1a.p1.3.m3.1.2.2.2.3.1.cmml" xref="S2.SS1a.p1.3.m3.1.2.2.2.3.1">conditional</csymbol><ci id="S2.SS1a.p1.3.m3.1.2.2.2.3.2.cmml" xref="S2.SS1a.p1.3.m3.1.2.2.2.3.2">𝐲</ci><ci id="S2.SS1a.p1.3.m3.1.2.2.2.3.3.cmml" xref="S2.SS1a.p1.3.m3.1.2.2.2.3.3">𝜽</ci></apply></apply></apply><ci id="S2.SS1a.p1.3.m3.1.1.cmml" xref="S2.SS1a.p1.3.m3.1.1">𝐲</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1a.p1.3.m3.1c">\log f_{\operatorname{\mathbf{y}}|\operatorname{{\bm{\theta}}}}(\operatorname{% \mathbf{y}})</annotation><annotation encoding="application/x-llamapun" id="S2.SS1a.p1.3.m3.1d">roman_log italic_f start_POSTSUBSCRIPT bold_y | bold_italic_θ end_POSTSUBSCRIPT ( bold_y )</annotation></semantics></math>, while conditioning on <math alttext="\operatorname{\mathbf{L}}_{1}^{(t)}" class="ltx_Math" display="inline" id="S2.SS1a.p1.4.m4.1"><semantics id="S2.SS1a.p1.4.m4.1a"><msubsup id="S2.SS1a.p1.4.m4.1.2" xref="S2.SS1a.p1.4.m4.1.2.cmml"><mi id="S2.SS1a.p1.4.m4.1.2.2.2" xref="S2.SS1a.p1.4.m4.1.2.2.2.cmml">𝐋</mi><mn id="S2.SS1a.p1.4.m4.1.2.2.3" xref="S2.SS1a.p1.4.m4.1.2.2.3.cmml">1</mn><mrow id="S2.SS1a.p1.4.m4.1.1.1.3" xref="S2.SS1a.p1.4.m4.1.2.cmml"><mo id="S2.SS1a.p1.4.m4.1.1.1.3.1" stretchy="false" xref="S2.SS1a.p1.4.m4.1.2.cmml">(</mo><mi id="S2.SS1a.p1.4.m4.1.1.1.1" xref="S2.SS1a.p1.4.m4.1.1.1.1.cmml">t</mi><mo id="S2.SS1a.p1.4.m4.1.1.1.3.2" stretchy="false" xref="S2.SS1a.p1.4.m4.1.2.cmml">)</mo></mrow></msubsup><annotation-xml encoding="MathML-Content" id="S2.SS1a.p1.4.m4.1b"><apply id="S2.SS1a.p1.4.m4.1.2.cmml" xref="S2.SS1a.p1.4.m4.1.2"><csymbol cd="ambiguous" id="S2.SS1a.p1.4.m4.1.2.1.cmml" xref="S2.SS1a.p1.4.m4.1.2">superscript</csymbol><apply id="S2.SS1a.p1.4.m4.1.2.2.cmml" xref="S2.SS1a.p1.4.m4.1.2"><csymbol cd="ambiguous" id="S2.SS1a.p1.4.m4.1.2.2.1.cmml" xref="S2.SS1a.p1.4.m4.1.2">subscript</csymbol><ci id="S2.SS1a.p1.4.m4.1.2.2.2.cmml" xref="S2.SS1a.p1.4.m4.1.2.2.2">𝐋</ci><cn id="S2.SS1a.p1.4.m4.1.2.2.3.cmml" type="integer" xref="S2.SS1a.p1.4.m4.1.2.2.3">1</cn></apply><ci id="S2.SS1a.p1.4.m4.1.1.1.1.cmml" xref="S2.SS1a.p1.4.m4.1.1.1.1">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1a.p1.4.m4.1c">\operatorname{\mathbf{L}}_{1}^{(t)}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1a.p1.4.m4.1d">bold_L start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_t ) end_POSTSUPERSCRIPT</annotation></semantics></math> and <math alttext="\operatorname{\mathbf{L}}_{2}^{(t)}" class="ltx_Math" display="inline" id="S2.SS1a.p1.5.m5.1"><semantics id="S2.SS1a.p1.5.m5.1a"><msubsup id="S2.SS1a.p1.5.m5.1.2" xref="S2.SS1a.p1.5.m5.1.2.cmml"><mi id="S2.SS1a.p1.5.m5.1.2.2.2" xref="S2.SS1a.p1.5.m5.1.2.2.2.cmml">𝐋</mi><mn id="S2.SS1a.p1.5.m5.1.2.2.3" xref="S2.SS1a.p1.5.m5.1.2.2.3.cmml">2</mn><mrow id="S2.SS1a.p1.5.m5.1.1.1.3" xref="S2.SS1a.p1.5.m5.1.2.cmml"><mo id="S2.SS1a.p1.5.m5.1.1.1.3.1" stretchy="false" xref="S2.SS1a.p1.5.m5.1.2.cmml">(</mo><mi id="S2.SS1a.p1.5.m5.1.1.1.1" xref="S2.SS1a.p1.5.m5.1.1.1.1.cmml">t</mi><mo id="S2.SS1a.p1.5.m5.1.1.1.3.2" stretchy="false" xref="S2.SS1a.p1.5.m5.1.2.cmml">)</mo></mrow></msubsup><annotation-xml encoding="MathML-Content" id="S2.SS1a.p1.5.m5.1b"><apply id="S2.SS1a.p1.5.m5.1.2.cmml" xref="S2.SS1a.p1.5.m5.1.2"><csymbol cd="ambiguous" id="S2.SS1a.p1.5.m5.1.2.1.cmml" xref="S2.SS1a.p1.5.m5.1.2">superscript</csymbol><apply id="S2.SS1a.p1.5.m5.1.2.2.cmml" xref="S2.SS1a.p1.5.m5.1.2"><csymbol cd="ambiguous" id="S2.SS1a.p1.5.m5.1.2.2.1.cmml" xref="S2.SS1a.p1.5.m5.1.2">subscript</csymbol><ci id="S2.SS1a.p1.5.m5.1.2.2.2.cmml" xref="S2.SS1a.p1.5.m5.1.2.2.2">𝐋</ci><cn id="S2.SS1a.p1.5.m5.1.2.2.3.cmml" type="integer" xref="S2.SS1a.p1.5.m5.1.2.2.3">2</cn></apply><ci id="S2.SS1a.p1.5.m5.1.1.1.1.cmml" 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Given <math alttext="\operatorname{\mathbf{L}}_{1}^{(t)}" class="ltx_Math" display="inline" id="S2.SS1a.p1.6.m6.1"><semantics id="S2.SS1a.p1.6.m6.1a"><msubsup id="S2.SS1a.p1.6.m6.1.2" xref="S2.SS1a.p1.6.m6.1.2.cmml"><mi id="S2.SS1a.p1.6.m6.1.2.2.2" xref="S2.SS1a.p1.6.m6.1.2.2.2.cmml">𝐋</mi><mn id="S2.SS1a.p1.6.m6.1.2.2.3" xref="S2.SS1a.p1.6.m6.1.2.2.3.cmml">1</mn><mrow id="S2.SS1a.p1.6.m6.1.1.1.3" xref="S2.SS1a.p1.6.m6.1.2.cmml"><mo id="S2.SS1a.p1.6.m6.1.1.1.3.1" stretchy="false" xref="S2.SS1a.p1.6.m6.1.2.cmml">(</mo><mi id="S2.SS1a.p1.6.m6.1.1.1.1" xref="S2.SS1a.p1.6.m6.1.1.1.1.cmml">t</mi><mo id="S2.SS1a.p1.6.m6.1.1.1.3.2" stretchy="false" xref="S2.SS1a.p1.6.m6.1.2.cmml">)</mo></mrow></msubsup><annotation-xml encoding="MathML-Content" id="S2.SS1a.p1.6.m6.1b"><apply id="S2.SS1a.p1.6.m6.1.2.cmml" xref="S2.SS1a.p1.6.m6.1.2"><csymbol cd="ambiguous" id="S2.SS1a.p1.6.m6.1.2.1.cmml" xref="S2.SS1a.p1.6.m6.1.2">superscript</csymbol><apply id="S2.SS1a.p1.6.m6.1.2.2.cmml" xref="S2.SS1a.p1.6.m6.1.2"><csymbol cd="ambiguous" id="S2.SS1a.p1.6.m6.1.2.2.1.cmml" xref="S2.SS1a.p1.6.m6.1.2">subscript</csymbol><ci id="S2.SS1a.p1.6.m6.1.2.2.2.cmml" xref="S2.SS1a.p1.6.m6.1.2.2.2">𝐋</ci><cn id="S2.SS1a.p1.6.m6.1.2.2.3.cmml" type="integer" xref="S2.SS1a.p1.6.m6.1.2.2.3">1</cn></apply><ci id="S2.SS1a.p1.6.m6.1.1.1.1.cmml" xref="S2.SS1a.p1.6.m6.1.1.1.1">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1a.p1.6.m6.1c">\operatorname{\mathbf{L}}_{1}^{(t)}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1a.p1.6.m6.1d">bold_L start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_t ) end_POSTSUPERSCRIPT</annotation></semantics></math> and <math alttext="\operatorname{\mathbf{L}}_{2}^{(t)}" class="ltx_Math" display="inline" id="S2.SS1a.p1.7.m7.1"><semantics id="S2.SS1a.p1.7.m7.1a"><msubsup id="S2.SS1a.p1.7.m7.1.2" xref="S2.SS1a.p1.7.m7.1.2.cmml"><mi id="S2.SS1a.p1.7.m7.1.2.2.2" xref="S2.SS1a.p1.7.m7.1.2.2.2.cmml">𝐋</mi><mn id="S2.SS1a.p1.7.m7.1.2.2.3" xref="S2.SS1a.p1.7.m7.1.2.2.3.cmml">2</mn><mrow id="S2.SS1a.p1.7.m7.1.1.1.3" xref="S2.SS1a.p1.7.m7.1.2.cmml"><mo id="S2.SS1a.p1.7.m7.1.1.1.3.1" stretchy="false" xref="S2.SS1a.p1.7.m7.1.2.cmml">(</mo><mi id="S2.SS1a.p1.7.m7.1.1.1.1" xref="S2.SS1a.p1.7.m7.1.1.1.1.cmml">t</mi><mo id="S2.SS1a.p1.7.m7.1.1.1.3.2" stretchy="false" xref="S2.SS1a.p1.7.m7.1.2.cmml">)</mo></mrow></msubsup><annotation-xml encoding="MathML-Content" id="S2.SS1a.p1.7.m7.1b"><apply id="S2.SS1a.p1.7.m7.1.2.cmml" xref="S2.SS1a.p1.7.m7.1.2"><csymbol cd="ambiguous" id="S2.SS1a.p1.7.m7.1.2.1.cmml" xref="S2.SS1a.p1.7.m7.1.2">superscript</csymbol><apply id="S2.SS1a.p1.7.m7.1.2.2.cmml" xref="S2.SS1a.p1.7.m7.1.2"><csymbol cd="ambiguous" id="S2.SS1a.p1.7.m7.1.2.2.1.cmml" xref="S2.SS1a.p1.7.m7.1.2">subscript</csymbol><ci id="S2.SS1a.p1.7.m7.1.2.2.2.cmml" xref="S2.SS1a.p1.7.m7.1.2.2.2">𝐋</ci><cn id="S2.SS1a.p1.7.m7.1.2.2.3.cmml" type="integer" 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italic_f start_POSTSUBSCRIPT bold_y | bold_italic_θ end_POSTSUBSCRIPT ( bold_y )</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle=\sum_{i=1}^{n}\log f_{\operatorname{\mathbf{y}}_{i}|% \operatorname{{\bm{\theta}}}}(\operatorname{\mathbf{y}}_{i})" class="ltx_Math" display="inline" id="S2.Ex14.m2.1"><semantics id="S2.Ex14.m2.1a"><mrow id="S2.Ex14.m2.1.1" xref="S2.Ex14.m2.1.1.cmml"><mi id="S2.Ex14.m2.1.1.3" xref="S2.Ex14.m2.1.1.3.cmml"></mi><mo id="S2.Ex14.m2.1.1.2" xref="S2.Ex14.m2.1.1.2.cmml">=</mo><mrow id="S2.Ex14.m2.1.1.1" xref="S2.Ex14.m2.1.1.1.cmml"><mstyle displaystyle="true" id="S2.Ex14.m2.1.1.1.2" xref="S2.Ex14.m2.1.1.1.2.cmml"><munderover id="S2.Ex14.m2.1.1.1.2a" xref="S2.Ex14.m2.1.1.1.2.cmml"><mo id="S2.Ex14.m2.1.1.1.2.2.2" movablelimits="false" xref="S2.Ex14.m2.1.1.1.2.2.2.cmml">∑</mo><mrow id="S2.Ex14.m2.1.1.1.2.2.3" xref="S2.Ex14.m2.1.1.1.2.2.3.cmml"><mi id="S2.Ex14.m2.1.1.1.2.2.3.2" 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end_POSTSUPERSCRIPT roman_log italic_f start_POSTSUBSCRIPT bold_y start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT | bold_italic_θ end_POSTSUBSCRIPT ( bold_y start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="S2.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_eqn_cell"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle=-\frac{N}{2}\log 2\pi-\frac{1}{2}\sum_{i=1}^{n}\text{logdet}(% \operatorname{{\mathbf{\Sigma}}}_{\operatorname{\mathbf{y}}_{i}}^{(t)})-\frac{% 1}{2}\sum_{i=1}^{n}(\operatorname{\mathbf{y}}_{i}-\operatorname{\mathbf{X}}_{i% }\operatorname{\mathbf{b}})^{\top}\operatorname{{\mathbf{\Sigma}}}_{% \operatorname{\mathbf{y}}_{i}}^{-1(t)}(\operatorname{\mathbf{y}}_{i}-% \operatorname{\mathbf{X}}_{i}\operatorname{\mathbf{b}})," class="ltx_Math" 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xref="S2.Ex15.m1.3.3.1.1.4.4.3.3.3.2.2.1.3.1.3">𝑖</ci></apply><ci id="S2.Ex15.m1.3.3.1.1.4.4.3.3.3.2.2.1.3.2.cmml" xref="S2.Ex15.m1.3.3.1.1.4.4.3.3.3.2.2.1.3.2">𝐛</ci></apply></apply></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Ex15.m1.3c">\displaystyle=-\frac{N}{2}\log 2\pi-\frac{1}{2}\sum_{i=1}^{n}\text{logdet}(% \operatorname{{\mathbf{\Sigma}}}_{\operatorname{\mathbf{y}}_{i}}^{(t)})-\frac{% 1}{2}\sum_{i=1}^{n}(\operatorname{\mathbf{y}}_{i}-\operatorname{\mathbf{X}}_{i% }\operatorname{\mathbf{b}})^{\top}\operatorname{{\mathbf{\Sigma}}}_{% \operatorname{\mathbf{y}}_{i}}^{-1(t)}(\operatorname{\mathbf{y}}_{i}-% \operatorname{\mathbf{X}}_{i}\operatorname{\mathbf{b}}),</annotation><annotation encoding="application/x-llamapun" id="S2.Ex15.m1.3d">= - divide start_ARG italic_N end_ARG start_ARG 2 end_ARG roman_log 2 italic_π - divide start_ARG 1 end_ARG start_ARG 2 end_ARG ∑ start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT logdet ( bold_Σ start_POSTSUBSCRIPT bold_y start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_t ) end_POSTSUPERSCRIPT ) - divide start_ARG 1 end_ARG start_ARG 2 end_ARG ∑ start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT ( bold_y start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT - bold_X start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT bold_b ) start_POSTSUPERSCRIPT ⊤ end_POSTSUPERSCRIPT bold_Σ start_POSTSUBSCRIPT bold_y start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT - 1 ( italic_t ) end_POSTSUPERSCRIPT ( bold_y start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT - bold_X start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT bold_b ) ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S2.SS1a.p1.19">where <math alttext="\log f_{\operatorname{\mathbf{y}}_{i}|\operatorname{{\bm{\theta}}}}(% \operatorname{\mathbf{y}}_{i})" class="ltx_Math" display="inline" id="S2.SS1a.p1.9.m1.1"><semantics id="S2.SS1a.p1.9.m1.1a"><mrow id="S2.SS1a.p1.9.m1.1.1" xref="S2.SS1a.p1.9.m1.1.1.cmml"><mrow id="S2.SS1a.p1.9.m1.1.1.3" xref="S2.SS1a.p1.9.m1.1.1.3.cmml"><mi id="S2.SS1a.p1.9.m1.1.1.3.1" xref="S2.SS1a.p1.9.m1.1.1.3.1.cmml">log</mi><mo id="S2.SS1a.p1.9.m1.1.1.3a" lspace="0.167em" xref="S2.SS1a.p1.9.m1.1.1.3.cmml">⁡</mo><msub id="S2.SS1a.p1.9.m1.1.1.3.2" xref="S2.SS1a.p1.9.m1.1.1.3.2.cmml"><mi id="S2.SS1a.p1.9.m1.1.1.3.2.2" xref="S2.SS1a.p1.9.m1.1.1.3.2.2.cmml">f</mi><mrow id="S2.SS1a.p1.9.m1.1.1.3.2.3" xref="S2.SS1a.p1.9.m1.1.1.3.2.3.cmml"><msub id="S2.SS1a.p1.9.m1.1.1.3.2.3.2" xref="S2.SS1a.p1.9.m1.1.1.3.2.3.2.cmml"><mi id="S2.SS1a.p1.9.m1.1.1.3.2.3.2.2" xref="S2.SS1a.p1.9.m1.1.1.3.2.3.2.2.cmml">𝐲</mi><mi id="S2.SS1a.p1.9.m1.1.1.3.2.3.2.3" xref="S2.SS1a.p1.9.m1.1.1.3.2.3.2.3.cmml">i</mi></msub><mo fence="false" id="S2.SS1a.p1.9.m1.1.1.3.2.3.1" xref="S2.SS1a.p1.9.m1.1.1.3.2.3.1.cmml">|</mo><mi id="S2.SS1a.p1.9.m1.1.1.3.2.3.3" xref="S2.SS1a.p1.9.m1.1.1.3.2.3.3.cmml">𝜽</mi></mrow></msub></mrow><mo id="S2.SS1a.p1.9.m1.1.1.2" xref="S2.SS1a.p1.9.m1.1.1.2.cmml">⁢</mo><mrow id="S2.SS1a.p1.9.m1.1.1.1.1" xref="S2.SS1a.p1.9.m1.1.1.1.1.1.cmml"><mo id="S2.SS1a.p1.9.m1.1.1.1.1.2" stretchy="false" xref="S2.SS1a.p1.9.m1.1.1.1.1.1.cmml">(</mo><msub id="S2.SS1a.p1.9.m1.1.1.1.1.1" xref="S2.SS1a.p1.9.m1.1.1.1.1.1.cmml"><mi id="S2.SS1a.p1.9.m1.1.1.1.1.1.2" xref="S2.SS1a.p1.9.m1.1.1.1.1.1.2.cmml">𝐲</mi><mi id="S2.SS1a.p1.9.m1.1.1.1.1.1.3" xref="S2.SS1a.p1.9.m1.1.1.1.1.1.3.cmml">i</mi></msub><mo id="S2.SS1a.p1.9.m1.1.1.1.1.3" stretchy="false" xref="S2.SS1a.p1.9.m1.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1a.p1.9.m1.1b"><apply id="S2.SS1a.p1.9.m1.1.1.cmml" xref="S2.SS1a.p1.9.m1.1.1"><times id="S2.SS1a.p1.9.m1.1.1.2.cmml" xref="S2.SS1a.p1.9.m1.1.1.2"></times><apply id="S2.SS1a.p1.9.m1.1.1.3.cmml" xref="S2.SS1a.p1.9.m1.1.1.3"><log id="S2.SS1a.p1.9.m1.1.1.3.1.cmml" xref="S2.SS1a.p1.9.m1.1.1.3.1"></log><apply id="S2.SS1a.p1.9.m1.1.1.3.2.cmml" xref="S2.SS1a.p1.9.m1.1.1.3.2"><csymbol cd="ambiguous" id="S2.SS1a.p1.9.m1.1.1.3.2.1.cmml" xref="S2.SS1a.p1.9.m1.1.1.3.2">subscript</csymbol><ci id="S2.SS1a.p1.9.m1.1.1.3.2.2.cmml" xref="S2.SS1a.p1.9.m1.1.1.3.2.2">𝑓</ci><apply id="S2.SS1a.p1.9.m1.1.1.3.2.3.cmml" xref="S2.SS1a.p1.9.m1.1.1.3.2.3"><csymbol cd="latexml" id="S2.SS1a.p1.9.m1.1.1.3.2.3.1.cmml" xref="S2.SS1a.p1.9.m1.1.1.3.2.3.1">conditional</csymbol><apply id="S2.SS1a.p1.9.m1.1.1.3.2.3.2.cmml" xref="S2.SS1a.p1.9.m1.1.1.3.2.3.2"><csymbol cd="ambiguous" id="S2.SS1a.p1.9.m1.1.1.3.2.3.2.1.cmml" xref="S2.SS1a.p1.9.m1.1.1.3.2.3.2">subscript</csymbol><ci id="S2.SS1a.p1.9.m1.1.1.3.2.3.2.2.cmml" xref="S2.SS1a.p1.9.m1.1.1.3.2.3.2.2">𝐲</ci><ci id="S2.SS1a.p1.9.m1.1.1.3.2.3.2.3.cmml" xref="S2.SS1a.p1.9.m1.1.1.3.2.3.2.3">𝑖</ci></apply><ci id="S2.SS1a.p1.9.m1.1.1.3.2.3.3.cmml" xref="S2.SS1a.p1.9.m1.1.1.3.2.3.3">𝜽</ci></apply></apply></apply><apply id="S2.SS1a.p1.9.m1.1.1.1.1.1.cmml" xref="S2.SS1a.p1.9.m1.1.1.1.1"><csymbol cd="ambiguous" id="S2.SS1a.p1.9.m1.1.1.1.1.1.1.cmml" xref="S2.SS1a.p1.9.m1.1.1.1.1">subscript</csymbol><ci id="S2.SS1a.p1.9.m1.1.1.1.1.1.2.cmml" xref="S2.SS1a.p1.9.m1.1.1.1.1.1.2">𝐲</ci><ci id="S2.SS1a.p1.9.m1.1.1.1.1.1.3.cmml" xref="S2.SS1a.p1.9.m1.1.1.1.1.1.3">𝑖</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1a.p1.9.m1.1c">\log f_{\operatorname{\mathbf{y}}_{i}|\operatorname{{\bm{\theta}}}}(% \operatorname{\mathbf{y}}_{i})</annotation><annotation encoding="application/x-llamapun" id="S2.SS1a.p1.9.m1.1d">roman_log italic_f start_POSTSUBSCRIPT bold_y start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT | bold_italic_θ end_POSTSUBSCRIPT ( bold_y start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT )</annotation></semantics></math> is the log likelihood sample <math alttext="i" class="ltx_Math" display="inline" id="S2.SS1a.p1.10.m2.1"><semantics id="S2.SS1a.p1.10.m2.1a"><mi id="S2.SS1a.p1.10.m2.1.1" xref="S2.SS1a.p1.10.m2.1.1.cmml">i</mi><annotation-xml encoding="MathML-Content" id="S2.SS1a.p1.10.m2.1b"><ci id="S2.SS1a.p1.10.m2.1.1.cmml" xref="S2.SS1a.p1.10.m2.1.1">𝑖</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1a.p1.10.m2.1c">i</annotation><annotation encoding="application/x-llamapun" id="S2.SS1a.p1.10.m2.1d">italic_i</annotation></semantics></math>, <math alttext="\operatorname{{\mathbf{\Sigma}}}_{\operatorname{\mathbf{y}}_{i}}^{(t)}=\tau^{2% }(\operatorname{\mathbf{Z}}_{i(1)}\operatorname{\mathbf{L}}_{(1)}^{(t)}% \operatorname{\mathbf{L}}_{(1)}^{(t)\top}\operatorname{\mathbf{Z}}_{i(1)}^{% \top}+\operatorname{\mathbf{I}}_{m_{i}})" class="ltx_Math" display="inline" id="S2.SS1a.p1.11.m3.8"><semantics id="S2.SS1a.p1.11.m3.8a"><mrow id="S2.SS1a.p1.11.m3.8.8" xref="S2.SS1a.p1.11.m3.8.8.cmml"><msubsup id="S2.SS1a.p1.11.m3.8.8.3" 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xref="S2.SS1a.p1.11.m3.8.8.1.3.2.cmml">τ</mi><mn id="S2.SS1a.p1.11.m3.8.8.1.3.3" xref="S2.SS1a.p1.11.m3.8.8.1.3.3.cmml">2</mn></msup><mo id="S2.SS1a.p1.11.m3.8.8.1.2" xref="S2.SS1a.p1.11.m3.8.8.1.2.cmml">⁢</mo><mrow id="S2.SS1a.p1.11.m3.8.8.1.1.1" xref="S2.SS1a.p1.11.m3.8.8.1.1.1.1.cmml"><mo id="S2.SS1a.p1.11.m3.8.8.1.1.1.2" stretchy="false" xref="S2.SS1a.p1.11.m3.8.8.1.1.1.1.cmml">(</mo><mrow id="S2.SS1a.p1.11.m3.8.8.1.1.1.1" xref="S2.SS1a.p1.11.m3.8.8.1.1.1.1.cmml"><mrow id="S2.SS1a.p1.11.m3.8.8.1.1.1.1.2" xref="S2.SS1a.p1.11.m3.8.8.1.1.1.1.2.cmml"><msub id="S2.SS1a.p1.11.m3.8.8.1.1.1.1.2.1" xref="S2.SS1a.p1.11.m3.8.8.1.1.1.1.2.1.cmml"><mi id="S2.SS1a.p1.11.m3.8.8.1.1.1.1.2.1.2" xref="S2.SS1a.p1.11.m3.8.8.1.1.1.1.2.1.2.cmml">𝐙</mi><mrow id="S2.SS1a.p1.11.m3.2.2.1" xref="S2.SS1a.p1.11.m3.2.2.1.cmml"><mi id="S2.SS1a.p1.11.m3.2.2.1.3" xref="S2.SS1a.p1.11.m3.2.2.1.3.cmml">i</mi><mo id="S2.SS1a.p1.11.m3.2.2.1.2" xref="S2.SS1a.p1.11.m3.2.2.1.2.cmml">⁢</mo><mrow 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bold_L start_POSTSUBSCRIPT ( 1 ) end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_t ) ⊤ end_POSTSUPERSCRIPT bold_Z start_POSTSUBSCRIPT italic_i ( 1 ) end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ⊤ end_POSTSUPERSCRIPT + bold_I start_POSTSUBSCRIPT italic_m start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUBSCRIPT )</annotation></semantics></math>, <math alttext="\operatorname{\mathbf{Z}}_{i(1)}" class="ltx_Math" display="inline" id="S2.SS1a.p1.12.m4.1"><semantics id="S2.SS1a.p1.12.m4.1a"><msub id="S2.SS1a.p1.12.m4.1.2" xref="S2.SS1a.p1.12.m4.1.2.cmml"><mi id="S2.SS1a.p1.12.m4.1.2.2" xref="S2.SS1a.p1.12.m4.1.2.2.cmml">𝐙</mi><mrow id="S2.SS1a.p1.12.m4.1.1.1" xref="S2.SS1a.p1.12.m4.1.1.1.cmml"><mi id="S2.SS1a.p1.12.m4.1.1.1.3" xref="S2.SS1a.p1.12.m4.1.1.1.3.cmml">i</mi><mo id="S2.SS1a.p1.12.m4.1.1.1.2" xref="S2.SS1a.p1.12.m4.1.1.1.2.cmml">⁢</mo><mrow id="S2.SS1a.p1.12.m4.1.1.1.4.2" xref="S2.SS1a.p1.12.m4.1.1.1.cmml"><mo id="S2.SS1a.p1.12.m4.1.1.1.4.2.1" stretchy="false" xref="S2.SS1a.p1.12.m4.1.1.1.cmml">(</mo><mn id="S2.SS1a.p1.12.m4.1.1.1.1" xref="S2.SS1a.p1.12.m4.1.1.1.1.cmml">1</mn><mo id="S2.SS1a.p1.12.m4.1.1.1.4.2.2" stretchy="false" xref="S2.SS1a.p1.12.m4.1.1.1.cmml">)</mo></mrow></mrow></msub><annotation-xml encoding="MathML-Content" id="S2.SS1a.p1.12.m4.1b"><apply id="S2.SS1a.p1.12.m4.1.2.cmml" xref="S2.SS1a.p1.12.m4.1.2"><csymbol cd="ambiguous" id="S2.SS1a.p1.12.m4.1.2.1.cmml" xref="S2.SS1a.p1.12.m4.1.2">subscript</csymbol><ci id="S2.SS1a.p1.12.m4.1.2.2.cmml" xref="S2.SS1a.p1.12.m4.1.2.2">𝐙</ci><apply id="S2.SS1a.p1.12.m4.1.1.1.cmml" xref="S2.SS1a.p1.12.m4.1.1.1"><times id="S2.SS1a.p1.12.m4.1.1.1.2.cmml" xref="S2.SS1a.p1.12.m4.1.1.1.2"></times><ci id="S2.SS1a.p1.12.m4.1.1.1.3.cmml" xref="S2.SS1a.p1.12.m4.1.1.1.3">𝑖</ci><cn id="S2.SS1a.p1.12.m4.1.1.1.1.cmml" type="integer" xref="S2.SS1a.p1.12.m4.1.1.1.1">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1a.p1.12.m4.1c">\operatorname{\mathbf{Z}}_{i(1)}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1a.p1.12.m4.1d">bold_Z start_POSTSUBSCRIPT italic_i ( 1 ) end_POSTSUBSCRIPT</annotation></semantics></math> is the matrix whose <math alttext="j" class="ltx_Math" display="inline" id="S2.SS1a.p1.13.m5.1"><semantics id="S2.SS1a.p1.13.m5.1a"><mi id="S2.SS1a.p1.13.m5.1.1" xref="S2.SS1a.p1.13.m5.1.1.cmml">j</mi><annotation-xml encoding="MathML-Content" id="S2.SS1a.p1.13.m5.1b"><ci id="S2.SS1a.p1.13.m5.1.1.cmml" xref="S2.SS1a.p1.13.m5.1.1">𝑗</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1a.p1.13.m5.1c">j</annotation><annotation encoding="application/x-llamapun" id="S2.SS1a.p1.13.m5.1d">italic_j</annotation></semantics></math>th row is <math alttext="\text{vec}(\operatorname{\mathbf{Z}}_{ij})^{\top}" class="ltx_Math" display="inline" id="S2.SS1a.p1.14.m6.1"><semantics id="S2.SS1a.p1.14.m6.1a"><mrow id="S2.SS1a.p1.14.m6.1.1" 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xref="S2.SS1a.p1.14.m6.1.1.1.1.1.1.2">𝐙</ci><apply id="S2.SS1a.p1.14.m6.1.1.1.1.1.1.3.cmml" xref="S2.SS1a.p1.14.m6.1.1.1.1.1.1.3"><times id="S2.SS1a.p1.14.m6.1.1.1.1.1.1.3.1.cmml" xref="S2.SS1a.p1.14.m6.1.1.1.1.1.1.3.1"></times><ci id="S2.SS1a.p1.14.m6.1.1.1.1.1.1.3.2.cmml" xref="S2.SS1a.p1.14.m6.1.1.1.1.1.1.3.2">𝑖</ci><ci id="S2.SS1a.p1.14.m6.1.1.1.1.1.1.3.3.cmml" xref="S2.SS1a.p1.14.m6.1.1.1.1.1.1.3.3">𝑗</ci></apply></apply><csymbol cd="latexml" id="S2.SS1a.p1.14.m6.1.1.1.3.cmml" xref="S2.SS1a.p1.14.m6.1.1.1.3">top</csymbol></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1a.p1.14.m6.1c">\text{vec}(\operatorname{\mathbf{Z}}_{ij})^{\top}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1a.p1.14.m6.1d">vec ( bold_Z start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT ⊤ end_POSTSUPERSCRIPT</annotation></semantics></math>, <math alttext="\operatorname{\mathbf{b}}=\text{vec}(\operatorname{\mathbf{B}})" class="ltx_Math" display="inline" id="S2.SS1a.p1.15.m7.1"><semantics id="S2.SS1a.p1.15.m7.1a"><mrow id="S2.SS1a.p1.15.m7.1.2" xref="S2.SS1a.p1.15.m7.1.2.cmml"><mi id="S2.SS1a.p1.15.m7.1.2.2" xref="S2.SS1a.p1.15.m7.1.2.2.cmml">𝐛</mi><mo id="S2.SS1a.p1.15.m7.1.2.1" xref="S2.SS1a.p1.15.m7.1.2.1.cmml">=</mo><mrow id="S2.SS1a.p1.15.m7.1.2.3" xref="S2.SS1a.p1.15.m7.1.2.3.cmml"><mtext id="S2.SS1a.p1.15.m7.1.2.3.2" xref="S2.SS1a.p1.15.m7.1.2.3.2a.cmml">vec</mtext><mo id="S2.SS1a.p1.15.m7.1.2.3.1" xref="S2.SS1a.p1.15.m7.1.2.3.1.cmml">⁢</mo><mrow id="S2.SS1a.p1.15.m7.1.2.3.3.2" xref="S2.SS1a.p1.15.m7.1.2.3.cmml"><mo id="S2.SS1a.p1.15.m7.1.2.3.3.2.1" stretchy="false" xref="S2.SS1a.p1.15.m7.1.2.3.cmml">(</mo><mi id="S2.SS1a.p1.15.m7.1.1" xref="S2.SS1a.p1.15.m7.1.1.cmml">𝐁</mi><mo id="S2.SS1a.p1.15.m7.1.2.3.3.2.2" stretchy="false" xref="S2.SS1a.p1.15.m7.1.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1a.p1.15.m7.1b"><apply id="S2.SS1a.p1.15.m7.1.2.cmml" xref="S2.SS1a.p1.15.m7.1.2"><eq id="S2.SS1a.p1.15.m7.1.2.1.cmml" xref="S2.SS1a.p1.15.m7.1.2.1"></eq><ci id="S2.SS1a.p1.15.m7.1.2.2.cmml" xref="S2.SS1a.p1.15.m7.1.2.2">𝐛</ci><apply id="S2.SS1a.p1.15.m7.1.2.3.cmml" xref="S2.SS1a.p1.15.m7.1.2.3"><times id="S2.SS1a.p1.15.m7.1.2.3.1.cmml" xref="S2.SS1a.p1.15.m7.1.2.3.1"></times><ci id="S2.SS1a.p1.15.m7.1.2.3.2a.cmml" xref="S2.SS1a.p1.15.m7.1.2.3.2"><mtext id="S2.SS1a.p1.15.m7.1.2.3.2.cmml" xref="S2.SS1a.p1.15.m7.1.2.3.2">vec</mtext></ci><ci id="S2.SS1a.p1.15.m7.1.1.cmml" xref="S2.SS1a.p1.15.m7.1.1">𝐁</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1a.p1.15.m7.1c">\operatorname{\mathbf{b}}=\text{vec}(\operatorname{\mathbf{B}})</annotation><annotation encoding="application/x-llamapun" id="S2.SS1a.p1.15.m7.1d">bold_b = vec ( bold_B )</annotation></semantics></math>, and <math alttext="\operatorname{\mathbf{L}}_{(1)}^{(t)}=\operatorname{\mathbf{L}}_{2}^{(t)}% \otimes\operatorname{\mathbf{L}}_{1}^{(t)}" 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xref="S2.SS1a.p1.16.m8.4.5.2.cmml">)</mo></mrow></msubsup><mo id="S2.SS1a.p1.16.m8.4.5.1" xref="S2.SS1a.p1.16.m8.4.5.1.cmml">=</mo><mrow id="S2.SS1a.p1.16.m8.4.5.3" xref="S2.SS1a.p1.16.m8.4.5.3.cmml"><msubsup id="S2.SS1a.p1.16.m8.4.5.3.2" xref="S2.SS1a.p1.16.m8.4.5.3.2.cmml"><mi id="S2.SS1a.p1.16.m8.4.5.3.2.2.2" xref="S2.SS1a.p1.16.m8.4.5.3.2.2.2.cmml">𝐋</mi><mn id="S2.SS1a.p1.16.m8.4.5.3.2.2.3" xref="S2.SS1a.p1.16.m8.4.5.3.2.2.3.cmml">2</mn><mrow id="S2.SS1a.p1.16.m8.3.3.1.3" xref="S2.SS1a.p1.16.m8.4.5.3.2.cmml"><mo id="S2.SS1a.p1.16.m8.3.3.1.3.1" stretchy="false" xref="S2.SS1a.p1.16.m8.4.5.3.2.cmml">(</mo><mi id="S2.SS1a.p1.16.m8.3.3.1.1" xref="S2.SS1a.p1.16.m8.3.3.1.1.cmml">t</mi><mo id="S2.SS1a.p1.16.m8.3.3.1.3.2" stretchy="false" xref="S2.SS1a.p1.16.m8.4.5.3.2.cmml">)</mo></mrow></msubsup><mo id="S2.SS1a.p1.16.m8.4.5.3.1" lspace="0em" rspace="0.222em" xref="S2.SS1a.p1.16.m8.4.5.3.1.cmml">⊗</mo><msubsup id="S2.SS1a.p1.16.m8.4.5.3.3" xref="S2.SS1a.p1.16.m8.4.5.3.3.cmml"><mi id="S2.SS1a.p1.16.m8.4.5.3.3.2.2" xref="S2.SS1a.p1.16.m8.4.5.3.3.2.2.cmml">𝐋</mi><mn id="S2.SS1a.p1.16.m8.4.5.3.3.2.3" xref="S2.SS1a.p1.16.m8.4.5.3.3.2.3.cmml">1</mn><mrow id="S2.SS1a.p1.16.m8.4.4.1.3" xref="S2.SS1a.p1.16.m8.4.5.3.3.cmml"><mo id="S2.SS1a.p1.16.m8.4.4.1.3.1" stretchy="false" xref="S2.SS1a.p1.16.m8.4.5.3.3.cmml">(</mo><mi id="S2.SS1a.p1.16.m8.4.4.1.1" xref="S2.SS1a.p1.16.m8.4.4.1.1.cmml">t</mi><mo id="S2.SS1a.p1.16.m8.4.4.1.3.2" stretchy="false" xref="S2.SS1a.p1.16.m8.4.5.3.3.cmml">)</mo></mrow></msubsup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1a.p1.16.m8.4b"><apply id="S2.SS1a.p1.16.m8.4.5.cmml" xref="S2.SS1a.p1.16.m8.4.5"><eq id="S2.SS1a.p1.16.m8.4.5.1.cmml" xref="S2.SS1a.p1.16.m8.4.5.1"></eq><apply id="S2.SS1a.p1.16.m8.4.5.2.cmml" xref="S2.SS1a.p1.16.m8.4.5.2"><csymbol cd="ambiguous" id="S2.SS1a.p1.16.m8.4.5.2.1.cmml" xref="S2.SS1a.p1.16.m8.4.5.2">superscript</csymbol><apply id="S2.SS1a.p1.16.m8.4.5.2.2.cmml" xref="S2.SS1a.p1.16.m8.4.5.2"><csymbol cd="ambiguous" id="S2.SS1a.p1.16.m8.4.5.2.2.1.cmml" xref="S2.SS1a.p1.16.m8.4.5.2">subscript</csymbol><ci id="S2.SS1a.p1.16.m8.4.5.2.2.2.cmml" xref="S2.SS1a.p1.16.m8.4.5.2.2.2">𝐋</ci><cn id="S2.SS1a.p1.16.m8.1.1.1.1.cmml" type="integer" xref="S2.SS1a.p1.16.m8.1.1.1.1">1</cn></apply><ci id="S2.SS1a.p1.16.m8.2.2.1.1.cmml" xref="S2.SS1a.p1.16.m8.2.2.1.1">𝑡</ci></apply><apply id="S2.SS1a.p1.16.m8.4.5.3.cmml" xref="S2.SS1a.p1.16.m8.4.5.3"><csymbol cd="latexml" id="S2.SS1a.p1.16.m8.4.5.3.1.cmml" xref="S2.SS1a.p1.16.m8.4.5.3.1">tensor-product</csymbol><apply id="S2.SS1a.p1.16.m8.4.5.3.2.cmml" xref="S2.SS1a.p1.16.m8.4.5.3.2"><csymbol cd="ambiguous" id="S2.SS1a.p1.16.m8.4.5.3.2.1.cmml" xref="S2.SS1a.p1.16.m8.4.5.3.2">superscript</csymbol><apply id="S2.SS1a.p1.16.m8.4.5.3.2.2.cmml" xref="S2.SS1a.p1.16.m8.4.5.3.2"><csymbol cd="ambiguous" id="S2.SS1a.p1.16.m8.4.5.3.2.2.1.cmml" xref="S2.SS1a.p1.16.m8.4.5.3.2">subscript</csymbol><ci id="S2.SS1a.p1.16.m8.4.5.3.2.2.2.cmml" xref="S2.SS1a.p1.16.m8.4.5.3.2.2.2">𝐋</ci><cn id="S2.SS1a.p1.16.m8.4.5.3.2.2.3.cmml" type="integer" xref="S2.SS1a.p1.16.m8.4.5.3.2.2.3">2</cn></apply><ci id="S2.SS1a.p1.16.m8.3.3.1.1.cmml" xref="S2.SS1a.p1.16.m8.3.3.1.1">𝑡</ci></apply><apply id="S2.SS1a.p1.16.m8.4.5.3.3.cmml" xref="S2.SS1a.p1.16.m8.4.5.3.3"><csymbol cd="ambiguous" id="S2.SS1a.p1.16.m8.4.5.3.3.1.cmml" xref="S2.SS1a.p1.16.m8.4.5.3.3">superscript</csymbol><apply id="S2.SS1a.p1.16.m8.4.5.3.3.2.cmml" xref="S2.SS1a.p1.16.m8.4.5.3.3"><csymbol cd="ambiguous" id="S2.SS1a.p1.16.m8.4.5.3.3.2.1.cmml" xref="S2.SS1a.p1.16.m8.4.5.3.3">subscript</csymbol><ci id="S2.SS1a.p1.16.m8.4.5.3.3.2.2.cmml" xref="S2.SS1a.p1.16.m8.4.5.3.3.2.2">𝐋</ci><cn id="S2.SS1a.p1.16.m8.4.5.3.3.2.3.cmml" type="integer" xref="S2.SS1a.p1.16.m8.4.5.3.3.2.3">1</cn></apply><ci id="S2.SS1a.p1.16.m8.4.4.1.1.cmml" xref="S2.SS1a.p1.16.m8.4.4.1.1">𝑡</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1a.p1.16.m8.4c">\operatorname{\mathbf{L}}_{(1)}^{(t)}=\operatorname{\mathbf{L}}_{2}^{(t)}% \otimes\operatorname{\mathbf{L}}_{1}^{(t)}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1a.p1.16.m8.4d">bold_L start_POSTSUBSCRIPT ( 1 ) end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_t ) end_POSTSUPERSCRIPT = bold_L start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_t ) end_POSTSUPERSCRIPT ⊗ bold_L start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_t ) end_POSTSUPERSCRIPT</annotation></semantics></math>; see Section 2.1 in the main manuscript about the notation. Let <math alttext="\operatorname{{\mathbf{\Lambda}}}_{i}^{(t)}=\operatorname{\mathbf{Z}}_{i(1)}% \operatorname{\mathbf{L}}_{(1)}^{(t)}\operatorname{\mathbf{L}}_{(1)}^{(t)\top}% \operatorname{\mathbf{Z}}_{i(1)}^{\top}+\operatorname{\mathbf{I}}_{m_{i}}" class="ltx_Math" display="inline" id="S2.SS1a.p1.17.m9.7"><semantics id="S2.SS1a.p1.17.m9.7a"><mrow id="S2.SS1a.p1.17.m9.7.8" xref="S2.SS1a.p1.17.m9.7.8.cmml"><msubsup id="S2.SS1a.p1.17.m9.7.8.2" xref="S2.SS1a.p1.17.m9.7.8.2.cmml"><mi id="S2.SS1a.p1.17.m9.7.8.2.2.2" xref="S2.SS1a.p1.17.m9.7.8.2.2.2.cmml">𝚲</mi><mi id="S2.SS1a.p1.17.m9.7.8.2.2.3" xref="S2.SS1a.p1.17.m9.7.8.2.2.3.cmml">i</mi><mrow id="S2.SS1a.p1.17.m9.1.1.1.3" xref="S2.SS1a.p1.17.m9.7.8.2.cmml"><mo id="S2.SS1a.p1.17.m9.1.1.1.3.1" stretchy="false" xref="S2.SS1a.p1.17.m9.7.8.2.cmml">(</mo><mi id="S2.SS1a.p1.17.m9.1.1.1.1" xref="S2.SS1a.p1.17.m9.1.1.1.1.cmml">t</mi><mo id="S2.SS1a.p1.17.m9.1.1.1.3.2" stretchy="false" xref="S2.SS1a.p1.17.m9.7.8.2.cmml">)</mo></mrow></msubsup><mo id="S2.SS1a.p1.17.m9.7.8.1" xref="S2.SS1a.p1.17.m9.7.8.1.cmml">=</mo><mrow id="S2.SS1a.p1.17.m9.7.8.3" xref="S2.SS1a.p1.17.m9.7.8.3.cmml"><mrow id="S2.SS1a.p1.17.m9.7.8.3.2" xref="S2.SS1a.p1.17.m9.7.8.3.2.cmml"><msub id="S2.SS1a.p1.17.m9.7.8.3.2.1" xref="S2.SS1a.p1.17.m9.7.8.3.2.1.cmml"><mi id="S2.SS1a.p1.17.m9.7.8.3.2.1.2" xref="S2.SS1a.p1.17.m9.7.8.3.2.1.2.cmml">𝐙</mi><mrow id="S2.SS1a.p1.17.m9.2.2.1" xref="S2.SS1a.p1.17.m9.2.2.1.cmml"><mi id="S2.SS1a.p1.17.m9.2.2.1.3" xref="S2.SS1a.p1.17.m9.2.2.1.3.cmml">i</mi><mo id="S2.SS1a.p1.17.m9.2.2.1.2" xref="S2.SS1a.p1.17.m9.2.2.1.2.cmml">⁢</mo><mrow id="S2.SS1a.p1.17.m9.2.2.1.4.2" xref="S2.SS1a.p1.17.m9.2.2.1.cmml"><mo id="S2.SS1a.p1.17.m9.2.2.1.4.2.1" stretchy="false" xref="S2.SS1a.p1.17.m9.2.2.1.cmml">(</mo><mn id="S2.SS1a.p1.17.m9.2.2.1.1" xref="S2.SS1a.p1.17.m9.2.2.1.1.cmml">1</mn><mo id="S2.SS1a.p1.17.m9.2.2.1.4.2.2" stretchy="false" xref="S2.SS1a.p1.17.m9.2.2.1.cmml">)</mo></mrow></mrow></msub><mo id="S2.SS1a.p1.17.m9.7.8.3.2a" lspace="0.167em" xref="S2.SS1a.p1.17.m9.7.8.3.2.cmml">⁡</mo><mrow id="S2.SS1a.p1.17.m9.7.8.3.2.2" xref="S2.SS1a.p1.17.m9.7.8.3.2.2.cmml"><msubsup id="S2.SS1a.p1.17.m9.7.8.3.2.2.1" xref="S2.SS1a.p1.17.m9.7.8.3.2.2.1.cmml"><mi id="S2.SS1a.p1.17.m9.7.8.3.2.2.1.2.2" xref="S2.SS1a.p1.17.m9.7.8.3.2.2.1.2.2.cmml">𝐋</mi><mrow id="S2.SS1a.p1.17.m9.3.3.1.3" xref="S2.SS1a.p1.17.m9.7.8.3.2.2.1.cmml"><mo id="S2.SS1a.p1.17.m9.3.3.1.3.1" stretchy="false" xref="S2.SS1a.p1.17.m9.7.8.3.2.2.1.cmml">(</mo><mn id="S2.SS1a.p1.17.m9.3.3.1.1" xref="S2.SS1a.p1.17.m9.3.3.1.1.cmml">1</mn><mo id="S2.SS1a.p1.17.m9.3.3.1.3.2" stretchy="false" xref="S2.SS1a.p1.17.m9.7.8.3.2.2.1.cmml">)</mo></mrow><mrow id="S2.SS1a.p1.17.m9.4.4.1.3" xref="S2.SS1a.p1.17.m9.7.8.3.2.2.1.cmml"><mo id="S2.SS1a.p1.17.m9.4.4.1.3.1" stretchy="false" xref="S2.SS1a.p1.17.m9.7.8.3.2.2.1.cmml">(</mo><mi id="S2.SS1a.p1.17.m9.4.4.1.1" xref="S2.SS1a.p1.17.m9.4.4.1.1.cmml">t</mi><mo id="S2.SS1a.p1.17.m9.4.4.1.3.2" stretchy="false" xref="S2.SS1a.p1.17.m9.7.8.3.2.2.1.cmml">)</mo></mrow></msubsup><mo id="S2.SS1a.p1.17.m9.7.8.3.2.2a" lspace="0.167em" xref="S2.SS1a.p1.17.m9.7.8.3.2.2.cmml">⁡</mo><mrow id="S2.SS1a.p1.17.m9.7.8.3.2.2.2" xref="S2.SS1a.p1.17.m9.7.8.3.2.2.2.cmml"><msubsup id="S2.SS1a.p1.17.m9.7.8.3.2.2.2.1" xref="S2.SS1a.p1.17.m9.7.8.3.2.2.2.1.cmml"><mi id="S2.SS1a.p1.17.m9.7.8.3.2.2.2.1.2.2" xref="S2.SS1a.p1.17.m9.7.8.3.2.2.2.1.2.2.cmml">𝐋</mi><mrow id="S2.SS1a.p1.17.m9.5.5.1.3" xref="S2.SS1a.p1.17.m9.7.8.3.2.2.2.1.cmml"><mo id="S2.SS1a.p1.17.m9.5.5.1.3.1" stretchy="false" xref="S2.SS1a.p1.17.m9.7.8.3.2.2.2.1.cmml">(</mo><mn id="S2.SS1a.p1.17.m9.5.5.1.1" xref="S2.SS1a.p1.17.m9.5.5.1.1.cmml">1</mn><mo id="S2.SS1a.p1.17.m9.5.5.1.3.2" stretchy="false" xref="S2.SS1a.p1.17.m9.7.8.3.2.2.2.1.cmml">)</mo></mrow><mrow id="S2.SS1a.p1.17.m9.6.6.1" xref="S2.SS1a.p1.17.m9.6.6.1.cmml"><mrow id="S2.SS1a.p1.17.m9.6.6.1.3.2" xref="S2.SS1a.p1.17.m9.6.6.1.cmml"><mo id="S2.SS1a.p1.17.m9.6.6.1.3.2.1" stretchy="false" xref="S2.SS1a.p1.17.m9.6.6.1.cmml">(</mo><mi id="S2.SS1a.p1.17.m9.6.6.1.1" xref="S2.SS1a.p1.17.m9.6.6.1.1.cmml">t</mi><mo id="S2.SS1a.p1.17.m9.6.6.1.3.2.2" stretchy="false" xref="S2.SS1a.p1.17.m9.6.6.1.cmml">)</mo></mrow><mo id="S2.SS1a.p1.17.m9.6.6.1.4" lspace="0.222em" xref="S2.SS1a.p1.17.m9.6.6.1.4.cmml">⊤</mo></mrow></msubsup><mo id="S2.SS1a.p1.17.m9.7.8.3.2.2.2a" lspace="0.167em" xref="S2.SS1a.p1.17.m9.7.8.3.2.2.2.cmml">⁡</mo><msubsup id="S2.SS1a.p1.17.m9.7.8.3.2.2.2.2" xref="S2.SS1a.p1.17.m9.7.8.3.2.2.2.2.cmml"><mi id="S2.SS1a.p1.17.m9.7.8.3.2.2.2.2.2.2" xref="S2.SS1a.p1.17.m9.7.8.3.2.2.2.2.2.2.cmml">𝐙</mi><mrow id="S2.SS1a.p1.17.m9.7.7.1" xref="S2.SS1a.p1.17.m9.7.7.1.cmml"><mi id="S2.SS1a.p1.17.m9.7.7.1.3" xref="S2.SS1a.p1.17.m9.7.7.1.3.cmml">i</mi><mo id="S2.SS1a.p1.17.m9.7.7.1.2" xref="S2.SS1a.p1.17.m9.7.7.1.2.cmml">⁢</mo><mrow id="S2.SS1a.p1.17.m9.7.7.1.4.2" 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id="S2.SS1a.p1.17.m9.7.8.3.3.3.2.cmml" xref="S2.SS1a.p1.17.m9.7.8.3.3.3.2">𝑚</ci><ci id="S2.SS1a.p1.17.m9.7.8.3.3.3.3.cmml" xref="S2.SS1a.p1.17.m9.7.8.3.3.3.3">𝑖</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1a.p1.17.m9.7c">\operatorname{{\mathbf{\Lambda}}}_{i}^{(t)}=\operatorname{\mathbf{Z}}_{i(1)}% \operatorname{\mathbf{L}}_{(1)}^{(t)}\operatorname{\mathbf{L}}_{(1)}^{(t)\top}% \operatorname{\mathbf{Z}}_{i(1)}^{\top}+\operatorname{\mathbf{I}}_{m_{i}}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1a.p1.17.m9.7d">bold_Λ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_t ) end_POSTSUPERSCRIPT = bold_Z start_POSTSUBSCRIPT italic_i ( 1 ) end_POSTSUBSCRIPT bold_L start_POSTSUBSCRIPT ( 1 ) end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_t ) end_POSTSUPERSCRIPT bold_L start_POSTSUBSCRIPT ( 1 ) end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_t ) ⊤ end_POSTSUPERSCRIPT bold_Z start_POSTSUBSCRIPT italic_i ( 1 ) end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ⊤ end_POSTSUPERSCRIPT + bold_I start_POSTSUBSCRIPT italic_m start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math>, <math alttext="\operatorname{{\mathbf{\Lambda}}}_{i}^{1/2(t)}" class="ltx_Math" display="inline" id="S2.SS1a.p1.18.m10.1"><semantics id="S2.SS1a.p1.18.m10.1a"><msubsup id="S2.SS1a.p1.18.m10.1.2" xref="S2.SS1a.p1.18.m10.1.2.cmml"><mi id="S2.SS1a.p1.18.m10.1.2.2.2" xref="S2.SS1a.p1.18.m10.1.2.2.2.cmml">𝚲</mi><mi id="S2.SS1a.p1.18.m10.1.2.2.3" xref="S2.SS1a.p1.18.m10.1.2.2.3.cmml">i</mi><mrow id="S2.SS1a.p1.18.m10.1.1.1" xref="S2.SS1a.p1.18.m10.1.1.1.cmml"><mrow id="S2.SS1a.p1.18.m10.1.1.1.3" xref="S2.SS1a.p1.18.m10.1.1.1.3.cmml"><mn id="S2.SS1a.p1.18.m10.1.1.1.3.2" xref="S2.SS1a.p1.18.m10.1.1.1.3.2.cmml">1</mn><mo id="S2.SS1a.p1.18.m10.1.1.1.3.1" xref="S2.SS1a.p1.18.m10.1.1.1.3.1.cmml">/</mo><mn id="S2.SS1a.p1.18.m10.1.1.1.3.3" 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start_POSTSUPERSCRIPT 1 / 2 ( italic_t ) ⊤ end_POSTSUPERSCRIPT = bold_Λ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_t ) end_POSTSUPERSCRIPT</annotation></semantics></math>, and the scaled response and predictor matrices be</p> <table class="ltx_equation ltx_eqn_table" id="S2.Ex16"> <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="\breve{\operatorname{\mathbf{y}}}=\begin{bmatrix}\operatorname{{\mathbf{% \Lambda}}}_{1}^{-1/2(t)}\operatorname{\mathbf{y}}_{1}\vspace{0.4mm}\\ \operatorname{{\mathbf{\Lambda}}}_{2}^{-1/2(t)}\operatorname{\mathbf{y}}_{2}% \vspace{0.4mm}\\ \vdots\vspace{0.4mm}\\ \operatorname{{\mathbf{\Lambda}}}_{n}^{-1/2(t)}\operatorname{\mathbf{y}}_{n}% \end{bmatrix},\quad\breve{\operatorname{\mathbf{X}}}=\begin{bmatrix}% \operatorname{{\mathbf{\Lambda}}}_{1}^{-1/2(t)}\operatorname{\mathbf{X}}_{1}% \vspace{0.4mm}\\ \operatorname{{\mathbf{\Lambda}}}_{2}^{-1/2(t)}\operatorname{\mathbf{X}}_{2}% \vspace{0.4mm}\\ \vdots\vspace{0.4mm}\\ \operatorname{{\mathbf{\Lambda}}}_{n}^{-1/2(t)}\operatorname{\mathbf{X}}_{n}% \end{bmatrix}." class="ltx_Math" display="block" id="S2.Ex16.m1.3"><semantics id="S2.Ex16.m1.3a"><mrow id="S2.Ex16.m1.3.3.1"><mrow id="S2.Ex16.m1.3.3.1.1.2" xref="S2.Ex16.m1.3.3.1.1.3.cmml"><mrow id="S2.Ex16.m1.3.3.1.1.1.1" xref="S2.Ex16.m1.3.3.1.1.1.1.cmml"><mover accent="true" id="S2.Ex16.m1.3.3.1.1.1.1.2" xref="S2.Ex16.m1.3.3.1.1.1.1.2.cmml"><mi id="S2.Ex16.m1.3.3.1.1.1.1.2.2" xref="S2.Ex16.m1.3.3.1.1.1.1.2.2.cmml">𝐲</mi><mo id="S2.Ex16.m1.3.3.1.1.1.1.2.1" xref="S2.Ex16.m1.3.3.1.1.1.1.2.1.cmml">˘</mo></mover><mo id="S2.Ex16.m1.3.3.1.1.1.1.1" xref="S2.Ex16.m1.3.3.1.1.1.1.1.cmml">=</mo><mrow id="S2.Ex16.m1.1.1.3" xref="S2.Ex16.m1.1.1.2.cmml"><mo id="S2.Ex16.m1.1.1.3.1" xref="S2.Ex16.m1.1.1.2.1.cmml">[</mo><mtable displaystyle="true" id="S2.Ex16.m1.1.1.1.1" rowspacing="0pt" 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xref="S2.Ex16.m1.1.1.1.1.1.1.1.1.3.2.cmml">𝐲</mi><mn id="S2.Ex16.m1.1.1.1.1.1.1.1.1.3.3" xref="S2.Ex16.m1.1.1.1.1.1.1.1.1.3.3.cmml">1</mn></msub></mrow></mtd></mtr><mtr id="S2.Ex16.m1.1.1.1.1c" xref="S2.Ex16.m1.1.1.1.1.cmml"><mtd id="S2.Ex16.m1.1.1.1.1d" xref="S2.Ex16.m1.1.1.1.1.cmml"><mrow id="S2.Ex16.m1.1.1.1.1.2.2.1.1" xref="S2.Ex16.m1.1.1.1.1.2.2.1.1.cmml"><msubsup id="S2.Ex16.m1.1.1.1.1.2.2.1.1.2" xref="S2.Ex16.m1.1.1.1.1.2.2.1.1.2.cmml"><mi id="S2.Ex16.m1.1.1.1.1.2.2.1.1.2.2.2" xref="S2.Ex16.m1.1.1.1.1.2.2.1.1.2.2.2.cmml">𝚲</mi><mn id="S2.Ex16.m1.1.1.1.1.2.2.1.1.2.2.3" xref="S2.Ex16.m1.1.1.1.1.2.2.1.1.2.2.3.cmml">2</mn><mrow id="S2.Ex16.m1.1.1.1.1.2.2.1.1.1.1" xref="S2.Ex16.m1.1.1.1.1.2.2.1.1.1.1.cmml"><mo id="S2.Ex16.m1.1.1.1.1.2.2.1.1.1.1a" xref="S2.Ex16.m1.1.1.1.1.2.2.1.1.1.1.cmml">−</mo><mrow id="S2.Ex16.m1.1.1.1.1.2.2.1.1.1.1.3" xref="S2.Ex16.m1.1.1.1.1.2.2.1.1.1.1.3.cmml"><mrow id="S2.Ex16.m1.1.1.1.1.2.2.1.1.1.1.3.2" xref="S2.Ex16.m1.1.1.1.1.2.2.1.1.1.1.3.2.cmml"><mn id="S2.Ex16.m1.1.1.1.1.2.2.1.1.1.1.3.2.2" xref="S2.Ex16.m1.1.1.1.1.2.2.1.1.1.1.3.2.2.cmml">1</mn><mo id="S2.Ex16.m1.1.1.1.1.2.2.1.1.1.1.3.2.1" xref="S2.Ex16.m1.1.1.1.1.2.2.1.1.1.1.3.2.1.cmml">/</mo><mn id="S2.Ex16.m1.1.1.1.1.2.2.1.1.1.1.3.2.3" xref="S2.Ex16.m1.1.1.1.1.2.2.1.1.1.1.3.2.3.cmml">2</mn></mrow><mo id="S2.Ex16.m1.1.1.1.1.2.2.1.1.1.1.3.1" xref="S2.Ex16.m1.1.1.1.1.2.2.1.1.1.1.3.1.cmml">⁢</mo><mrow id="S2.Ex16.m1.1.1.1.1.2.2.1.1.1.1.3.3.2" xref="S2.Ex16.m1.1.1.1.1.2.2.1.1.1.1.3.cmml"><mo id="S2.Ex16.m1.1.1.1.1.2.2.1.1.1.1.3.3.2.1" stretchy="false" xref="S2.Ex16.m1.1.1.1.1.2.2.1.1.1.1.3.cmml">(</mo><mi id="S2.Ex16.m1.1.1.1.1.2.2.1.1.1.1.1" xref="S2.Ex16.m1.1.1.1.1.2.2.1.1.1.1.1.cmml">t</mi><mo id="S2.Ex16.m1.1.1.1.1.2.2.1.1.1.1.3.3.2.2" stretchy="false" xref="S2.Ex16.m1.1.1.1.1.2.2.1.1.1.1.3.cmml">)</mo></mrow></mrow></mrow></msubsup><mo id="S2.Ex16.m1.1.1.1.1.2.2.1.1a" lspace="0.167em" xref="S2.Ex16.m1.1.1.1.1.2.2.1.1.cmml">⁡</mo><msub id="S2.Ex16.m1.1.1.1.1.2.2.1.1.3" xref="S2.Ex16.m1.1.1.1.1.2.2.1.1.3.cmml"><mi id="S2.Ex16.m1.1.1.1.1.2.2.1.1.3.2" xref="S2.Ex16.m1.1.1.1.1.2.2.1.1.3.2.cmml">𝐲</mi><mn id="S2.Ex16.m1.1.1.1.1.2.2.1.1.3.3" xref="S2.Ex16.m1.1.1.1.1.2.2.1.1.3.3.cmml">2</mn></msub></mrow></mtd></mtr><mtr id="S2.Ex16.m1.1.1.1.1e" xref="S2.Ex16.m1.1.1.1.1.cmml"><mtd id="S2.Ex16.m1.1.1.1.1f" xref="S2.Ex16.m1.1.1.1.1.cmml"><mi id="S2.Ex16.m1.1.1.1.1.4.1.1" mathvariant="normal" xref="S2.Ex16.m1.1.1.1.1.4.1.1.cmml">⋮</mi></mtd></mtr><mtr id="S2.Ex16.m1.1.1.1.1g" xref="S2.Ex16.m1.1.1.1.1.cmml"><mtd id="S2.Ex16.m1.1.1.1.1h" xref="S2.Ex16.m1.1.1.1.1.cmml"><mrow id="S2.Ex16.m1.1.1.1.1.3.3.1.1" xref="S2.Ex16.m1.1.1.1.1.3.3.1.1.cmml"><msubsup id="S2.Ex16.m1.1.1.1.1.3.3.1.1.2" xref="S2.Ex16.m1.1.1.1.1.3.3.1.1.2.cmml"><mi id="S2.Ex16.m1.1.1.1.1.3.3.1.1.2.2.2" xref="S2.Ex16.m1.1.1.1.1.3.3.1.1.2.2.2.cmml">𝚲</mi><mi id="S2.Ex16.m1.1.1.1.1.3.3.1.1.2.2.3" xref="S2.Ex16.m1.1.1.1.1.3.3.1.1.2.2.3.cmml">n</mi><mrow id="S2.Ex16.m1.1.1.1.1.3.3.1.1.1.1" xref="S2.Ex16.m1.1.1.1.1.3.3.1.1.1.1.cmml"><mo id="S2.Ex16.m1.1.1.1.1.3.3.1.1.1.1a" xref="S2.Ex16.m1.1.1.1.1.3.3.1.1.1.1.cmml">−</mo><mrow id="S2.Ex16.m1.1.1.1.1.3.3.1.1.1.1.3" xref="S2.Ex16.m1.1.1.1.1.3.3.1.1.1.1.3.cmml"><mrow id="S2.Ex16.m1.1.1.1.1.3.3.1.1.1.1.3.2" xref="S2.Ex16.m1.1.1.1.1.3.3.1.1.1.1.3.2.cmml"><mn id="S2.Ex16.m1.1.1.1.1.3.3.1.1.1.1.3.2.2" xref="S2.Ex16.m1.1.1.1.1.3.3.1.1.1.1.3.2.2.cmml">1</mn><mo id="S2.Ex16.m1.1.1.1.1.3.3.1.1.1.1.3.2.1" xref="S2.Ex16.m1.1.1.1.1.3.3.1.1.1.1.3.2.1.cmml">/</mo><mn id="S2.Ex16.m1.1.1.1.1.3.3.1.1.1.1.3.2.3" xref="S2.Ex16.m1.1.1.1.1.3.3.1.1.1.1.3.2.3.cmml">2</mn></mrow><mo id="S2.Ex16.m1.1.1.1.1.3.3.1.1.1.1.3.1" xref="S2.Ex16.m1.1.1.1.1.3.3.1.1.1.1.3.1.cmml">⁢</mo><mrow id="S2.Ex16.m1.1.1.1.1.3.3.1.1.1.1.3.3.2" xref="S2.Ex16.m1.1.1.1.1.3.3.1.1.1.1.3.cmml"><mo id="S2.Ex16.m1.1.1.1.1.3.3.1.1.1.1.3.3.2.1" stretchy="false" xref="S2.Ex16.m1.1.1.1.1.3.3.1.1.1.1.3.cmml">(</mo><mi id="S2.Ex16.m1.1.1.1.1.3.3.1.1.1.1.1" xref="S2.Ex16.m1.1.1.1.1.3.3.1.1.1.1.1.cmml">t</mi><mo id="S2.Ex16.m1.1.1.1.1.3.3.1.1.1.1.3.3.2.2" stretchy="false" xref="S2.Ex16.m1.1.1.1.1.3.3.1.1.1.1.3.cmml">)</mo></mrow></mrow></mrow></msubsup><mo id="S2.Ex16.m1.1.1.1.1.3.3.1.1a" lspace="0.167em" xref="S2.Ex16.m1.1.1.1.1.3.3.1.1.cmml">⁡</mo><msub id="S2.Ex16.m1.1.1.1.1.3.3.1.1.3" xref="S2.Ex16.m1.1.1.1.1.3.3.1.1.3.cmml"><mi id="S2.Ex16.m1.1.1.1.1.3.3.1.1.3.2" xref="S2.Ex16.m1.1.1.1.1.3.3.1.1.3.2.cmml">𝐲</mi><mi id="S2.Ex16.m1.1.1.1.1.3.3.1.1.3.3" xref="S2.Ex16.m1.1.1.1.1.3.3.1.1.3.3.cmml">n</mi></msub></mrow></mtd></mtr></mtable><mo id="S2.Ex16.m1.1.1.3.2" xref="S2.Ex16.m1.1.1.2.1.cmml">]</mo></mrow></mrow><mo id="S2.Ex16.m1.3.3.1.1.2.3" rspace="1.167em" xref="S2.Ex16.m1.3.3.1.1.3a.cmml">,</mo><mrow id="S2.Ex16.m1.3.3.1.1.2.2" xref="S2.Ex16.m1.3.3.1.1.2.2.cmml"><mover accent="true" id="S2.Ex16.m1.3.3.1.1.2.2.2" xref="S2.Ex16.m1.3.3.1.1.2.2.2.cmml"><mi id="S2.Ex16.m1.3.3.1.1.2.2.2.2" xref="S2.Ex16.m1.3.3.1.1.2.2.2.2.cmml">𝐗</mi><mo id="S2.Ex16.m1.3.3.1.1.2.2.2.1" xref="S2.Ex16.m1.3.3.1.1.2.2.2.1.cmml">˘</mo></mover><mo id="S2.Ex16.m1.3.3.1.1.2.2.1" xref="S2.Ex16.m1.3.3.1.1.2.2.1.cmml">=</mo><mrow id="S2.Ex16.m1.2.2.3" xref="S2.Ex16.m1.2.2.2.cmml"><mo id="S2.Ex16.m1.2.2.3.1" xref="S2.Ex16.m1.2.2.2.1.cmml">[</mo><mtable displaystyle="true" id="S2.Ex16.m1.2.2.1.1" rowspacing="0pt" xref="S2.Ex16.m1.2.2.1.1.cmml"><mtr id="S2.Ex16.m1.2.2.1.1a" xref="S2.Ex16.m1.2.2.1.1.cmml"><mtd id="S2.Ex16.m1.2.2.1.1b" xref="S2.Ex16.m1.2.2.1.1.cmml"><mrow id="S2.Ex16.m1.2.2.1.1.1.1.1.1" xref="S2.Ex16.m1.2.2.1.1.1.1.1.1.cmml"><msubsup id="S2.Ex16.m1.2.2.1.1.1.1.1.1.2" xref="S2.Ex16.m1.2.2.1.1.1.1.1.1.2.cmml"><mi id="S2.Ex16.m1.2.2.1.1.1.1.1.1.2.2.2" xref="S2.Ex16.m1.2.2.1.1.1.1.1.1.2.2.2.cmml">𝚲</mi><mn id="S2.Ex16.m1.2.2.1.1.1.1.1.1.2.2.3" xref="S2.Ex16.m1.2.2.1.1.1.1.1.1.2.2.3.cmml">1</mn><mrow id="S2.Ex16.m1.2.2.1.1.1.1.1.1.1.1" xref="S2.Ex16.m1.2.2.1.1.1.1.1.1.1.1.cmml"><mo id="S2.Ex16.m1.2.2.1.1.1.1.1.1.1.1a" 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id="S2.Ex16.m1.3c">\breve{\operatorname{\mathbf{y}}}=\begin{bmatrix}\operatorname{{\mathbf{% \Lambda}}}_{1}^{-1/2(t)}\operatorname{\mathbf{y}}_{1}\vspace{0.4mm}\\ \operatorname{{\mathbf{\Lambda}}}_{2}^{-1/2(t)}\operatorname{\mathbf{y}}_{2}% \vspace{0.4mm}\\ \vdots\vspace{0.4mm}\\ \operatorname{{\mathbf{\Lambda}}}_{n}^{-1/2(t)}\operatorname{\mathbf{y}}_{n}% \end{bmatrix},\quad\breve{\operatorname{\mathbf{X}}}=\begin{bmatrix}% \operatorname{{\mathbf{\Lambda}}}_{1}^{-1/2(t)}\operatorname{\mathbf{X}}_{1}% \vspace{0.4mm}\\ \operatorname{{\mathbf{\Lambda}}}_{2}^{-1/2(t)}\operatorname{\mathbf{X}}_{2}% \vspace{0.4mm}\\ \vdots\vspace{0.4mm}\\ \operatorname{{\mathbf{\Lambda}}}_{n}^{-1/2(t)}\operatorname{\mathbf{X}}_{n}% \end{bmatrix}.</annotation><annotation encoding="application/x-llamapun" id="S2.Ex16.m1.3d">over˘ start_ARG bold_y end_ARG = [ start_ARG start_ROW start_CELL bold_Λ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT - 1 / 2 ( italic_t ) end_POSTSUPERSCRIPT bold_y start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_CELL end_ROW start_ROW start_CELL bold_Λ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT - 1 / 2 ( italic_t ) end_POSTSUPERSCRIPT bold_y start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_CELL end_ROW start_ROW start_CELL ⋮ end_CELL end_ROW start_ROW start_CELL bold_Λ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT - 1 / 2 ( italic_t ) end_POSTSUPERSCRIPT bold_y start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_CELL end_ROW end_ARG ] , over˘ start_ARG bold_X end_ARG = [ start_ARG start_ROW start_CELL bold_Λ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT - 1 / 2 ( italic_t ) end_POSTSUPERSCRIPT bold_X start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_CELL end_ROW start_ROW start_CELL bold_Λ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT - 1 / 2 ( italic_t ) end_POSTSUPERSCRIPT bold_X start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_CELL end_ROW start_ROW start_CELL ⋮ end_CELL end_ROW start_ROW start_CELL bold_Λ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT - 1 / 2 ( italic_t ) end_POSTSUPERSCRIPT bold_X start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_CELL end_ROW end_ARG ] .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S2.SS1a.p1.31">Then, maximizing <math alttext="\log f_{\operatorname{\mathbf{y}}|\operatorname{{\bm{\theta}}}}(\operatorname{% \mathbf{y}})" class="ltx_Math" display="inline" id="S2.SS1a.p1.20.m1.1"><semantics id="S2.SS1a.p1.20.m1.1a"><mrow id="S2.SS1a.p1.20.m1.1.2" xref="S2.SS1a.p1.20.m1.1.2.cmml"><mrow id="S2.SS1a.p1.20.m1.1.2.2" xref="S2.SS1a.p1.20.m1.1.2.2.cmml"><mi id="S2.SS1a.p1.20.m1.1.2.2.1" xref="S2.SS1a.p1.20.m1.1.2.2.1.cmml">log</mi><mo id="S2.SS1a.p1.20.m1.1.2.2a" lspace="0.167em" xref="S2.SS1a.p1.20.m1.1.2.2.cmml">⁡</mo><msub id="S2.SS1a.p1.20.m1.1.2.2.2" xref="S2.SS1a.p1.20.m1.1.2.2.2.cmml"><mi id="S2.SS1a.p1.20.m1.1.2.2.2.2" xref="S2.SS1a.p1.20.m1.1.2.2.2.2.cmml">f</mi><mrow id="S2.SS1a.p1.20.m1.1.2.2.2.3" xref="S2.SS1a.p1.20.m1.1.2.2.2.3.cmml"><mi id="S2.SS1a.p1.20.m1.1.2.2.2.3.2" xref="S2.SS1a.p1.20.m1.1.2.2.2.3.2.cmml">𝐲</mi><mo fence="false" id="S2.SS1a.p1.20.m1.1.2.2.2.3.1" xref="S2.SS1a.p1.20.m1.1.2.2.2.3.1.cmml">|</mo><mi id="S2.SS1a.p1.20.m1.1.2.2.2.3.3" xref="S2.SS1a.p1.20.m1.1.2.2.2.3.3.cmml">𝜽</mi></mrow></msub></mrow><mo id="S2.SS1a.p1.20.m1.1.2.1" xref="S2.SS1a.p1.20.m1.1.2.1.cmml">⁢</mo><mrow id="S2.SS1a.p1.20.m1.1.2.3.2" xref="S2.SS1a.p1.20.m1.1.2.cmml"><mo id="S2.SS1a.p1.20.m1.1.2.3.2.1" stretchy="false" xref="S2.SS1a.p1.20.m1.1.2.cmml">(</mo><mi id="S2.SS1a.p1.20.m1.1.1" xref="S2.SS1a.p1.20.m1.1.1.cmml">𝐲</mi><mo id="S2.SS1a.p1.20.m1.1.2.3.2.2" stretchy="false" xref="S2.SS1a.p1.20.m1.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1a.p1.20.m1.1b"><apply id="S2.SS1a.p1.20.m1.1.2.cmml" xref="S2.SS1a.p1.20.m1.1.2"><times id="S2.SS1a.p1.20.m1.1.2.1.cmml" xref="S2.SS1a.p1.20.m1.1.2.1"></times><apply id="S2.SS1a.p1.20.m1.1.2.2.cmml" xref="S2.SS1a.p1.20.m1.1.2.2"><log id="S2.SS1a.p1.20.m1.1.2.2.1.cmml" xref="S2.SS1a.p1.20.m1.1.2.2.1"></log><apply id="S2.SS1a.p1.20.m1.1.2.2.2.cmml" xref="S2.SS1a.p1.20.m1.1.2.2.2"><csymbol cd="ambiguous" id="S2.SS1a.p1.20.m1.1.2.2.2.1.cmml" xref="S2.SS1a.p1.20.m1.1.2.2.2">subscript</csymbol><ci id="S2.SS1a.p1.20.m1.1.2.2.2.2.cmml" xref="S2.SS1a.p1.20.m1.1.2.2.2.2">𝑓</ci><apply id="S2.SS1a.p1.20.m1.1.2.2.2.3.cmml" xref="S2.SS1a.p1.20.m1.1.2.2.2.3"><csymbol cd="latexml" id="S2.SS1a.p1.20.m1.1.2.2.2.3.1.cmml" xref="S2.SS1a.p1.20.m1.1.2.2.2.3.1">conditional</csymbol><ci id="S2.SS1a.p1.20.m1.1.2.2.2.3.2.cmml" xref="S2.SS1a.p1.20.m1.1.2.2.2.3.2">𝐲</ci><ci id="S2.SS1a.p1.20.m1.1.2.2.2.3.3.cmml" xref="S2.SS1a.p1.20.m1.1.2.2.2.3.3">𝜽</ci></apply></apply></apply><ci id="S2.SS1a.p1.20.m1.1.1.cmml" xref="S2.SS1a.p1.20.m1.1.1">𝐲</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1a.p1.20.m1.1c">\log f_{\operatorname{\mathbf{y}}|\operatorname{{\bm{\theta}}}}(\operatorname{% \mathbf{y}})</annotation><annotation encoding="application/x-llamapun" id="S2.SS1a.p1.20.m1.1d">roman_log italic_f start_POSTSUBSCRIPT bold_y | bold_italic_θ end_POSTSUBSCRIPT ( bold_y )</annotation></semantics></math> with respect to <math alttext="\operatorname{\mathbf{b}}" class="ltx_Math" display="inline" id="S2.SS1a.p1.21.m2.1"><semantics id="S2.SS1a.p1.21.m2.1a"><mi id="S2.SS1a.p1.21.m2.1.1" xref="S2.SS1a.p1.21.m2.1.1.cmml">𝐛</mi><annotation-xml encoding="MathML-Content" id="S2.SS1a.p1.21.m2.1b"><ci id="S2.SS1a.p1.21.m2.1.1.cmml" xref="S2.SS1a.p1.21.m2.1.1">𝐛</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1a.p1.21.m2.1c">\operatorname{\mathbf{b}}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1a.p1.21.m2.1d">bold_b</annotation></semantics></math> while conditioning on <math alttext="\operatorname{\mathbf{L}}_{1}^{(t)}" class="ltx_Math" display="inline" id="S2.SS1a.p1.22.m3.1"><semantics id="S2.SS1a.p1.22.m3.1a"><msubsup id="S2.SS1a.p1.22.m3.1.2" xref="S2.SS1a.p1.22.m3.1.2.cmml"><mi id="S2.SS1a.p1.22.m3.1.2.2.2" xref="S2.SS1a.p1.22.m3.1.2.2.2.cmml">𝐋</mi><mn id="S2.SS1a.p1.22.m3.1.2.2.3" xref="S2.SS1a.p1.22.m3.1.2.2.3.cmml">1</mn><mrow id="S2.SS1a.p1.22.m3.1.1.1.3" xref="S2.SS1a.p1.22.m3.1.2.cmml"><mo id="S2.SS1a.p1.22.m3.1.1.1.3.1" stretchy="false" xref="S2.SS1a.p1.22.m3.1.2.cmml">(</mo><mi id="S2.SS1a.p1.22.m3.1.1.1.1" xref="S2.SS1a.p1.22.m3.1.1.1.1.cmml">t</mi><mo id="S2.SS1a.p1.22.m3.1.1.1.3.2" stretchy="false" xref="S2.SS1a.p1.22.m3.1.2.cmml">)</mo></mrow></msubsup><annotation-xml encoding="MathML-Content" id="S2.SS1a.p1.22.m3.1b"><apply id="S2.SS1a.p1.22.m3.1.2.cmml" xref="S2.SS1a.p1.22.m3.1.2"><csymbol cd="ambiguous" id="S2.SS1a.p1.22.m3.1.2.1.cmml" xref="S2.SS1a.p1.22.m3.1.2">superscript</csymbol><apply id="S2.SS1a.p1.22.m3.1.2.2.cmml" xref="S2.SS1a.p1.22.m3.1.2"><csymbol cd="ambiguous" id="S2.SS1a.p1.22.m3.1.2.2.1.cmml" xref="S2.SS1a.p1.22.m3.1.2">subscript</csymbol><ci id="S2.SS1a.p1.22.m3.1.2.2.2.cmml" xref="S2.SS1a.p1.22.m3.1.2.2.2">𝐋</ci><cn id="S2.SS1a.p1.22.m3.1.2.2.3.cmml" type="integer" xref="S2.SS1a.p1.22.m3.1.2.2.3">1</cn></apply><ci id="S2.SS1a.p1.22.m3.1.1.1.1.cmml" xref="S2.SS1a.p1.22.m3.1.1.1.1">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1a.p1.22.m3.1c">\operatorname{\mathbf{L}}_{1}^{(t)}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1a.p1.22.m3.1d">bold_L start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_t ) end_POSTSUPERSCRIPT</annotation></semantics></math> and <math alttext="\operatorname{\mathbf{L}}_{2}^{(t)}" class="ltx_Math" display="inline" id="S2.SS1a.p1.23.m4.1"><semantics id="S2.SS1a.p1.23.m4.1a"><msubsup id="S2.SS1a.p1.23.m4.1.2" xref="S2.SS1a.p1.23.m4.1.2.cmml"><mi id="S2.SS1a.p1.23.m4.1.2.2.2" xref="S2.SS1a.p1.23.m4.1.2.2.2.cmml">𝐋</mi><mn id="S2.SS1a.p1.23.m4.1.2.2.3" xref="S2.SS1a.p1.23.m4.1.2.2.3.cmml">2</mn><mrow id="S2.SS1a.p1.23.m4.1.1.1.3" xref="S2.SS1a.p1.23.m4.1.2.cmml"><mo id="S2.SS1a.p1.23.m4.1.1.1.3.1" stretchy="false" xref="S2.SS1a.p1.23.m4.1.2.cmml">(</mo><mi id="S2.SS1a.p1.23.m4.1.1.1.1" xref="S2.SS1a.p1.23.m4.1.1.1.1.cmml">t</mi><mo id="S2.SS1a.p1.23.m4.1.1.1.3.2" stretchy="false" xref="S2.SS1a.p1.23.m4.1.2.cmml">)</mo></mrow></msubsup><annotation-xml encoding="MathML-Content" id="S2.SS1a.p1.23.m4.1b"><apply id="S2.SS1a.p1.23.m4.1.2.cmml" xref="S2.SS1a.p1.23.m4.1.2"><csymbol cd="ambiguous" id="S2.SS1a.p1.23.m4.1.2.1.cmml" xref="S2.SS1a.p1.23.m4.1.2">superscript</csymbol><apply id="S2.SS1a.p1.23.m4.1.2.2.cmml" xref="S2.SS1a.p1.23.m4.1.2"><csymbol cd="ambiguous" id="S2.SS1a.p1.23.m4.1.2.2.1.cmml" xref="S2.SS1a.p1.23.m4.1.2">subscript</csymbol><ci id="S2.SS1a.p1.23.m4.1.2.2.2.cmml" xref="S2.SS1a.p1.23.m4.1.2.2.2">𝐋</ci><cn id="S2.SS1a.p1.23.m4.1.2.2.3.cmml" type="integer" xref="S2.SS1a.p1.23.m4.1.2.2.3">2</cn></apply><ci id="S2.SS1a.p1.23.m4.1.1.1.1.cmml" xref="S2.SS1a.p1.23.m4.1.1.1.1">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1a.p1.23.m4.1c">\operatorname{\mathbf{L}}_{2}^{(t)}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1a.p1.23.m4.1d">bold_L start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_t ) end_POSTSUPERSCRIPT</annotation></semantics></math> amounts to minimizing the quadratic form <math alttext="(\breve{\operatorname{\mathbf{y}}}-\breve{\operatorname{\mathbf{X}}}% \operatorname{\mathbf{b}})^{\top}(\breve{\operatorname{\mathbf{y}}}-\breve{% \operatorname{\mathbf{X}}}\operatorname{\mathbf{b}})" class="ltx_Math" display="inline" id="S2.SS1a.p1.24.m5.2"><semantics id="S2.SS1a.p1.24.m5.2a"><mrow id="S2.SS1a.p1.24.m5.2.2" xref="S2.SS1a.p1.24.m5.2.2.cmml"><msup id="S2.SS1a.p1.24.m5.1.1.1" xref="S2.SS1a.p1.24.m5.1.1.1.cmml"><mrow id="S2.SS1a.p1.24.m5.1.1.1.1.1" xref="S2.SS1a.p1.24.m5.1.1.1.1.1.1.cmml"><mo id="S2.SS1a.p1.24.m5.1.1.1.1.1.2" stretchy="false" xref="S2.SS1a.p1.24.m5.1.1.1.1.1.1.cmml">(</mo><mrow id="S2.SS1a.p1.24.m5.1.1.1.1.1.1" xref="S2.SS1a.p1.24.m5.1.1.1.1.1.1.cmml"><mover accent="true" id="S2.SS1a.p1.24.m5.1.1.1.1.1.1.2" xref="S2.SS1a.p1.24.m5.1.1.1.1.1.1.2.cmml"><mi id="S2.SS1a.p1.24.m5.1.1.1.1.1.1.2.2" xref="S2.SS1a.p1.24.m5.1.1.1.1.1.1.2.2.cmml">𝐲</mi><mo id="S2.SS1a.p1.24.m5.1.1.1.1.1.1.2.1" xref="S2.SS1a.p1.24.m5.1.1.1.1.1.1.2.1.cmml">˘</mo></mover><mo id="S2.SS1a.p1.24.m5.1.1.1.1.1.1.1" lspace="0em" xref="S2.SS1a.p1.24.m5.1.1.1.1.1.1.1.cmml">−</mo><mrow id="S2.SS1a.p1.24.m5.1.1.1.1.1.1.3" xref="S2.SS1a.p1.24.m5.1.1.1.1.1.1.3.cmml"><mover accent="true" id="S2.SS1a.p1.24.m5.1.1.1.1.1.1.3.2" xref="S2.SS1a.p1.24.m5.1.1.1.1.1.1.3.2.cmml"><mi id="S2.SS1a.p1.24.m5.1.1.1.1.1.1.3.2.2" xref="S2.SS1a.p1.24.m5.1.1.1.1.1.1.3.2.2.cmml">𝐗</mi><mo id="S2.SS1a.p1.24.m5.1.1.1.1.1.1.3.2.1" xref="S2.SS1a.p1.24.m5.1.1.1.1.1.1.3.2.1.cmml">˘</mo></mover><mo id="S2.SS1a.p1.24.m5.1.1.1.1.1.1.3.1" lspace="0.167em" xref="S2.SS1a.p1.24.m5.1.1.1.1.1.1.3.1.cmml">⁢</mo><mi id="S2.SS1a.p1.24.m5.1.1.1.1.1.1.3.3" xref="S2.SS1a.p1.24.m5.1.1.1.1.1.1.3.3.cmml">𝐛</mi></mrow></mrow><mo id="S2.SS1a.p1.24.m5.1.1.1.1.1.3" stretchy="false" xref="S2.SS1a.p1.24.m5.1.1.1.1.1.1.cmml">)</mo></mrow><mo id="S2.SS1a.p1.24.m5.1.1.1.3" xref="S2.SS1a.p1.24.m5.1.1.1.3.cmml">⊤</mo></msup><mo id="S2.SS1a.p1.24.m5.2.2.3" xref="S2.SS1a.p1.24.m5.2.2.3.cmml">⁢</mo><mrow id="S2.SS1a.p1.24.m5.2.2.2.1" xref="S2.SS1a.p1.24.m5.2.2.2.1.1.cmml"><mo id="S2.SS1a.p1.24.m5.2.2.2.1.2" stretchy="false" xref="S2.SS1a.p1.24.m5.2.2.2.1.1.cmml">(</mo><mrow id="S2.SS1a.p1.24.m5.2.2.2.1.1" xref="S2.SS1a.p1.24.m5.2.2.2.1.1.cmml"><mover accent="true" id="S2.SS1a.p1.24.m5.2.2.2.1.1.2" xref="S2.SS1a.p1.24.m5.2.2.2.1.1.2.cmml"><mi id="S2.SS1a.p1.24.m5.2.2.2.1.1.2.2" xref="S2.SS1a.p1.24.m5.2.2.2.1.1.2.2.cmml">𝐲</mi><mo id="S2.SS1a.p1.24.m5.2.2.2.1.1.2.1" xref="S2.SS1a.p1.24.m5.2.2.2.1.1.2.1.cmml">˘</mo></mover><mo id="S2.SS1a.p1.24.m5.2.2.2.1.1.1" lspace="0em" xref="S2.SS1a.p1.24.m5.2.2.2.1.1.1.cmml">−</mo><mrow id="S2.SS1a.p1.24.m5.2.2.2.1.1.3" xref="S2.SS1a.p1.24.m5.2.2.2.1.1.3.cmml"><mover accent="true" id="S2.SS1a.p1.24.m5.2.2.2.1.1.3.2" xref="S2.SS1a.p1.24.m5.2.2.2.1.1.3.2.cmml"><mi id="S2.SS1a.p1.24.m5.2.2.2.1.1.3.2.2" xref="S2.SS1a.p1.24.m5.2.2.2.1.1.3.2.2.cmml">𝐗</mi><mo id="S2.SS1a.p1.24.m5.2.2.2.1.1.3.2.1" xref="S2.SS1a.p1.24.m5.2.2.2.1.1.3.2.1.cmml">˘</mo></mover><mo id="S2.SS1a.p1.24.m5.2.2.2.1.1.3.1" lspace="0.167em" xref="S2.SS1a.p1.24.m5.2.2.2.1.1.3.1.cmml">⁢</mo><mi id="S2.SS1a.p1.24.m5.2.2.2.1.1.3.3" xref="S2.SS1a.p1.24.m5.2.2.2.1.1.3.3.cmml">𝐛</mi></mrow></mrow><mo id="S2.SS1a.p1.24.m5.2.2.2.1.3" stretchy="false" xref="S2.SS1a.p1.24.m5.2.2.2.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1a.p1.24.m5.2b"><apply id="S2.SS1a.p1.24.m5.2.2.cmml" xref="S2.SS1a.p1.24.m5.2.2"><times id="S2.SS1a.p1.24.m5.2.2.3.cmml" xref="S2.SS1a.p1.24.m5.2.2.3"></times><apply id="S2.SS1a.p1.24.m5.1.1.1.cmml" xref="S2.SS1a.p1.24.m5.1.1.1"><csymbol cd="ambiguous" id="S2.SS1a.p1.24.m5.1.1.1.2.cmml" xref="S2.SS1a.p1.24.m5.1.1.1">superscript</csymbol><apply id="S2.SS1a.p1.24.m5.1.1.1.1.1.1.cmml" xref="S2.SS1a.p1.24.m5.1.1.1.1.1"><minus id="S2.SS1a.p1.24.m5.1.1.1.1.1.1.1.cmml" xref="S2.SS1a.p1.24.m5.1.1.1.1.1.1.1"></minus><apply id="S2.SS1a.p1.24.m5.1.1.1.1.1.1.2.cmml" xref="S2.SS1a.p1.24.m5.1.1.1.1.1.1.2"><ci id="S2.SS1a.p1.24.m5.1.1.1.1.1.1.2.1.cmml" xref="S2.SS1a.p1.24.m5.1.1.1.1.1.1.2.1">˘</ci><ci id="S2.SS1a.p1.24.m5.1.1.1.1.1.1.2.2.cmml" xref="S2.SS1a.p1.24.m5.1.1.1.1.1.1.2.2">𝐲</ci></apply><apply id="S2.SS1a.p1.24.m5.1.1.1.1.1.1.3.cmml" xref="S2.SS1a.p1.24.m5.1.1.1.1.1.1.3"><times id="S2.SS1a.p1.24.m5.1.1.1.1.1.1.3.1.cmml" xref="S2.SS1a.p1.24.m5.1.1.1.1.1.1.3.1"></times><apply id="S2.SS1a.p1.24.m5.1.1.1.1.1.1.3.2.cmml" xref="S2.SS1a.p1.24.m5.1.1.1.1.1.1.3.2"><ci id="S2.SS1a.p1.24.m5.1.1.1.1.1.1.3.2.1.cmml" xref="S2.SS1a.p1.24.m5.1.1.1.1.1.1.3.2.1">˘</ci><ci id="S2.SS1a.p1.24.m5.1.1.1.1.1.1.3.2.2.cmml" xref="S2.SS1a.p1.24.m5.1.1.1.1.1.1.3.2.2">𝐗</ci></apply><ci id="S2.SS1a.p1.24.m5.1.1.1.1.1.1.3.3.cmml" xref="S2.SS1a.p1.24.m5.1.1.1.1.1.1.3.3">𝐛</ci></apply></apply><csymbol cd="latexml" id="S2.SS1a.p1.24.m5.1.1.1.3.cmml" xref="S2.SS1a.p1.24.m5.1.1.1.3">top</csymbol></apply><apply id="S2.SS1a.p1.24.m5.2.2.2.1.1.cmml" xref="S2.SS1a.p1.24.m5.2.2.2.1"><minus id="S2.SS1a.p1.24.m5.2.2.2.1.1.1.cmml" xref="S2.SS1a.p1.24.m5.2.2.2.1.1.1"></minus><apply id="S2.SS1a.p1.24.m5.2.2.2.1.1.2.cmml" xref="S2.SS1a.p1.24.m5.2.2.2.1.1.2"><ci id="S2.SS1a.p1.24.m5.2.2.2.1.1.2.1.cmml" xref="S2.SS1a.p1.24.m5.2.2.2.1.1.2.1">˘</ci><ci id="S2.SS1a.p1.24.m5.2.2.2.1.1.2.2.cmml" xref="S2.SS1a.p1.24.m5.2.2.2.1.1.2.2">𝐲</ci></apply><apply id="S2.SS1a.p1.24.m5.2.2.2.1.1.3.cmml" xref="S2.SS1a.p1.24.m5.2.2.2.1.1.3"><times id="S2.SS1a.p1.24.m5.2.2.2.1.1.3.1.cmml" xref="S2.SS1a.p1.24.m5.2.2.2.1.1.3.1"></times><apply id="S2.SS1a.p1.24.m5.2.2.2.1.1.3.2.cmml" xref="S2.SS1a.p1.24.m5.2.2.2.1.1.3.2"><ci id="S2.SS1a.p1.24.m5.2.2.2.1.1.3.2.1.cmml" xref="S2.SS1a.p1.24.m5.2.2.2.1.1.3.2.1">˘</ci><ci id="S2.SS1a.p1.24.m5.2.2.2.1.1.3.2.2.cmml" xref="S2.SS1a.p1.24.m5.2.2.2.1.1.3.2.2">𝐗</ci></apply><ci id="S2.SS1a.p1.24.m5.2.2.2.1.1.3.3.cmml" xref="S2.SS1a.p1.24.m5.2.2.2.1.1.3.3">𝐛</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1a.p1.24.m5.2c">(\breve{\operatorname{\mathbf{y}}}-\breve{\operatorname{\mathbf{X}}}% \operatorname{\mathbf{b}})^{\top}(\breve{\operatorname{\mathbf{y}}}-\breve{% \operatorname{\mathbf{X}}}\operatorname{\mathbf{b}})</annotation><annotation encoding="application/x-llamapun" id="S2.SS1a.p1.24.m5.2d">( over˘ start_ARG bold_y end_ARG - over˘ start_ARG bold_X end_ARG bold_b ) start_POSTSUPERSCRIPT ⊤ end_POSTSUPERSCRIPT ( over˘ start_ARG bold_y end_ARG - over˘ start_ARG bold_X end_ARG bold_b )</annotation></semantics></math> with respect to <math alttext="\operatorname{\mathbf{b}}" class="ltx_Math" display="inline" id="S2.SS1a.p1.25.m6.1"><semantics id="S2.SS1a.p1.25.m6.1a"><mi id="S2.SS1a.p1.25.m6.1.1" xref="S2.SS1a.p1.25.m6.1.1.cmml">𝐛</mi><annotation-xml encoding="MathML-Content" id="S2.SS1a.p1.25.m6.1b"><ci id="S2.SS1a.p1.25.m6.1.1.cmml" xref="S2.SS1a.p1.25.m6.1.1">𝐛</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1a.p1.25.m6.1c">\operatorname{\mathbf{b}}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1a.p1.25.m6.1d">bold_b</annotation></semantics></math>. We estimate <math alttext="\operatorname{\mathbf{B}}^{(t+1)}" class="ltx_Math" display="inline" id="S2.SS1a.p1.26.m7.1"><semantics id="S2.SS1a.p1.26.m7.1a"><msup id="S2.SS1a.p1.26.m7.1.2" xref="S2.SS1a.p1.26.m7.1.2.cmml"><mi id="S2.SS1a.p1.26.m7.1.2.2" xref="S2.SS1a.p1.26.m7.1.2.2.cmml">𝐁</mi><mrow id="S2.SS1a.p1.26.m7.1.1.1.1" xref="S2.SS1a.p1.26.m7.1.1.1.1.1.cmml"><mo id="S2.SS1a.p1.26.m7.1.1.1.1.2" stretchy="false" xref="S2.SS1a.p1.26.m7.1.1.1.1.1.cmml">(</mo><mrow id="S2.SS1a.p1.26.m7.1.1.1.1.1" xref="S2.SS1a.p1.26.m7.1.1.1.1.1.cmml"><mi id="S2.SS1a.p1.26.m7.1.1.1.1.1.2" xref="S2.SS1a.p1.26.m7.1.1.1.1.1.2.cmml">t</mi><mo id="S2.SS1a.p1.26.m7.1.1.1.1.1.1" xref="S2.SS1a.p1.26.m7.1.1.1.1.1.1.cmml">+</mo><mn id="S2.SS1a.p1.26.m7.1.1.1.1.1.3" xref="S2.SS1a.p1.26.m7.1.1.1.1.1.3.cmml">1</mn></mrow><mo id="S2.SS1a.p1.26.m7.1.1.1.1.3" stretchy="false" xref="S2.SS1a.p1.26.m7.1.1.1.1.1.cmml">)</mo></mrow></msup><annotation-xml encoding="MathML-Content" id="S2.SS1a.p1.26.m7.1b"><apply id="S2.SS1a.p1.26.m7.1.2.cmml" xref="S2.SS1a.p1.26.m7.1.2"><csymbol cd="ambiguous" id="S2.SS1a.p1.26.m7.1.2.1.cmml" xref="S2.SS1a.p1.26.m7.1.2">superscript</csymbol><ci id="S2.SS1a.p1.26.m7.1.2.2.cmml" xref="S2.SS1a.p1.26.m7.1.2.2">𝐁</ci><apply id="S2.SS1a.p1.26.m7.1.1.1.1.1.cmml" xref="S2.SS1a.p1.26.m7.1.1.1.1"><plus id="S2.SS1a.p1.26.m7.1.1.1.1.1.1.cmml" xref="S2.SS1a.p1.26.m7.1.1.1.1.1.1"></plus><ci id="S2.SS1a.p1.26.m7.1.1.1.1.1.2.cmml" xref="S2.SS1a.p1.26.m7.1.1.1.1.1.2">𝑡</ci><cn id="S2.SS1a.p1.26.m7.1.1.1.1.1.3.cmml" type="integer" xref="S2.SS1a.p1.26.m7.1.1.1.1.1.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1a.p1.26.m7.1c">\operatorname{\mathbf{B}}^{(t+1)}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1a.p1.26.m7.1d">bold_B start_POSTSUPERSCRIPT ( italic_t + 1 ) end_POSTSUPERSCRIPT</annotation></semantics></math> and <math alttext="\tau^{2(t+1)}" class="ltx_Math" display="inline" id="S2.SS1a.p1.27.m8.1"><semantics id="S2.SS1a.p1.27.m8.1a"><msup id="S2.SS1a.p1.27.m8.1.2" xref="S2.SS1a.p1.27.m8.1.2.cmml"><mi id="S2.SS1a.p1.27.m8.1.2.2" xref="S2.SS1a.p1.27.m8.1.2.2.cmml">τ</mi><mrow id="S2.SS1a.p1.27.m8.1.1.1" xref="S2.SS1a.p1.27.m8.1.1.1.cmml"><mn id="S2.SS1a.p1.27.m8.1.1.1.3" xref="S2.SS1a.p1.27.m8.1.1.1.3.cmml">2</mn><mo id="S2.SS1a.p1.27.m8.1.1.1.2" xref="S2.SS1a.p1.27.m8.1.1.1.2.cmml">⁢</mo><mrow id="S2.SS1a.p1.27.m8.1.1.1.1.1" xref="S2.SS1a.p1.27.m8.1.1.1.1.1.1.cmml"><mo id="S2.SS1a.p1.27.m8.1.1.1.1.1.2" stretchy="false" xref="S2.SS1a.p1.27.m8.1.1.1.1.1.1.cmml">(</mo><mrow id="S2.SS1a.p1.27.m8.1.1.1.1.1.1" xref="S2.SS1a.p1.27.m8.1.1.1.1.1.1.cmml"><mi id="S2.SS1a.p1.27.m8.1.1.1.1.1.1.2" xref="S2.SS1a.p1.27.m8.1.1.1.1.1.1.2.cmml">t</mi><mo id="S2.SS1a.p1.27.m8.1.1.1.1.1.1.1" xref="S2.SS1a.p1.27.m8.1.1.1.1.1.1.1.cmml">+</mo><mn id="S2.SS1a.p1.27.m8.1.1.1.1.1.1.3" xref="S2.SS1a.p1.27.m8.1.1.1.1.1.1.3.cmml">1</mn></mrow><mo id="S2.SS1a.p1.27.m8.1.1.1.1.1.3" stretchy="false" xref="S2.SS1a.p1.27.m8.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></msup><annotation-xml encoding="MathML-Content" id="S2.SS1a.p1.27.m8.1b"><apply id="S2.SS1a.p1.27.m8.1.2.cmml" xref="S2.SS1a.p1.27.m8.1.2"><csymbol cd="ambiguous" id="S2.SS1a.p1.27.m8.1.2.1.cmml" xref="S2.SS1a.p1.27.m8.1.2">superscript</csymbol><ci id="S2.SS1a.p1.27.m8.1.2.2.cmml" xref="S2.SS1a.p1.27.m8.1.2.2">𝜏</ci><apply id="S2.SS1a.p1.27.m8.1.1.1.cmml" xref="S2.SS1a.p1.27.m8.1.1.1"><times id="S2.SS1a.p1.27.m8.1.1.1.2.cmml" xref="S2.SS1a.p1.27.m8.1.1.1.2"></times><cn id="S2.SS1a.p1.27.m8.1.1.1.3.cmml" type="integer" xref="S2.SS1a.p1.27.m8.1.1.1.3">2</cn><apply id="S2.SS1a.p1.27.m8.1.1.1.1.1.1.cmml" xref="S2.SS1a.p1.27.m8.1.1.1.1.1"><plus id="S2.SS1a.p1.27.m8.1.1.1.1.1.1.1.cmml" xref="S2.SS1a.p1.27.m8.1.1.1.1.1.1.1"></plus><ci id="S2.SS1a.p1.27.m8.1.1.1.1.1.1.2.cmml" xref="S2.SS1a.p1.27.m8.1.1.1.1.1.1.2">𝑡</ci><cn id="S2.SS1a.p1.27.m8.1.1.1.1.1.1.3.cmml" type="integer" xref="S2.SS1a.p1.27.m8.1.1.1.1.1.1.3">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1a.p1.27.m8.1c">\tau^{2(t+1)}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1a.p1.27.m8.1d">italic_τ start_POSTSUPERSCRIPT 2 ( italic_t + 1 ) end_POSTSUPERSCRIPT</annotation></semantics></math> simultaneously by adding the scaled lasso penalty terms to this quadratic form as in Equation (8) in the main manuscript to enforce sparsity in <math alttext="\operatorname{\mathbf{B}}^{(t+1)}" class="ltx_Math" display="inline" id="S2.SS1a.p1.28.m9.1"><semantics id="S2.SS1a.p1.28.m9.1a"><msup id="S2.SS1a.p1.28.m9.1.2" xref="S2.SS1a.p1.28.m9.1.2.cmml"><mi id="S2.SS1a.p1.28.m9.1.2.2" xref="S2.SS1a.p1.28.m9.1.2.2.cmml">𝐁</mi><mrow id="S2.SS1a.p1.28.m9.1.1.1.1" xref="S2.SS1a.p1.28.m9.1.1.1.1.1.cmml"><mo id="S2.SS1a.p1.28.m9.1.1.1.1.2" stretchy="false" xref="S2.SS1a.p1.28.m9.1.1.1.1.1.cmml">(</mo><mrow id="S2.SS1a.p1.28.m9.1.1.1.1.1" xref="S2.SS1a.p1.28.m9.1.1.1.1.1.cmml"><mi id="S2.SS1a.p1.28.m9.1.1.1.1.1.2" xref="S2.SS1a.p1.28.m9.1.1.1.1.1.2.cmml">t</mi><mo id="S2.SS1a.p1.28.m9.1.1.1.1.1.1" xref="S2.SS1a.p1.28.m9.1.1.1.1.1.1.cmml">+</mo><mn id="S2.SS1a.p1.28.m9.1.1.1.1.1.3" xref="S2.SS1a.p1.28.m9.1.1.1.1.1.3.cmml">1</mn></mrow><mo id="S2.SS1a.p1.28.m9.1.1.1.1.3" stretchy="false" xref="S2.SS1a.p1.28.m9.1.1.1.1.1.cmml">)</mo></mrow></msup><annotation-xml encoding="MathML-Content" id="S2.SS1a.p1.28.m9.1b"><apply id="S2.SS1a.p1.28.m9.1.2.cmml" xref="S2.SS1a.p1.28.m9.1.2"><csymbol cd="ambiguous" id="S2.SS1a.p1.28.m9.1.2.1.cmml" xref="S2.SS1a.p1.28.m9.1.2">superscript</csymbol><ci id="S2.SS1a.p1.28.m9.1.2.2.cmml" xref="S2.SS1a.p1.28.m9.1.2.2">𝐁</ci><apply id="S2.SS1a.p1.28.m9.1.1.1.1.1.cmml" xref="S2.SS1a.p1.28.m9.1.1.1.1"><plus id="S2.SS1a.p1.28.m9.1.1.1.1.1.1.cmml" xref="S2.SS1a.p1.28.m9.1.1.1.1.1.1"></plus><ci id="S2.SS1a.p1.28.m9.1.1.1.1.1.2.cmml" xref="S2.SS1a.p1.28.m9.1.1.1.1.1.2">𝑡</ci><cn id="S2.SS1a.p1.28.m9.1.1.1.1.1.3.cmml" type="integer" xref="S2.SS1a.p1.28.m9.1.1.1.1.1.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1a.p1.28.m9.1c">\operatorname{\mathbf{B}}^{(t+1)}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1a.p1.28.m9.1d">bold_B start_POSTSUPERSCRIPT ( italic_t + 1 ) end_POSTSUPERSCRIPT</annotation></semantics></math> and reduce the bias in estimating <math alttext="\tau^{2(t+1)}" class="ltx_Math" display="inline" id="S2.SS1a.p1.29.m10.1"><semantics id="S2.SS1a.p1.29.m10.1a"><msup id="S2.SS1a.p1.29.m10.1.2" xref="S2.SS1a.p1.29.m10.1.2.cmml"><mi id="S2.SS1a.p1.29.m10.1.2.2" xref="S2.SS1a.p1.29.m10.1.2.2.cmml">τ</mi><mrow id="S2.SS1a.p1.29.m10.1.1.1" xref="S2.SS1a.p1.29.m10.1.1.1.cmml"><mn id="S2.SS1a.p1.29.m10.1.1.1.3" xref="S2.SS1a.p1.29.m10.1.1.1.3.cmml">2</mn><mo id="S2.SS1a.p1.29.m10.1.1.1.2" xref="S2.SS1a.p1.29.m10.1.1.1.2.cmml">⁢</mo><mrow id="S2.SS1a.p1.29.m10.1.1.1.1.1" xref="S2.SS1a.p1.29.m10.1.1.1.1.1.1.cmml"><mo id="S2.SS1a.p1.29.m10.1.1.1.1.1.2" stretchy="false" xref="S2.SS1a.p1.29.m10.1.1.1.1.1.1.cmml">(</mo><mrow id="S2.SS1a.p1.29.m10.1.1.1.1.1.1" xref="S2.SS1a.p1.29.m10.1.1.1.1.1.1.cmml"><mi id="S2.SS1a.p1.29.m10.1.1.1.1.1.1.2" xref="S2.SS1a.p1.29.m10.1.1.1.1.1.1.2.cmml">t</mi><mo id="S2.SS1a.p1.29.m10.1.1.1.1.1.1.1" xref="S2.SS1a.p1.29.m10.1.1.1.1.1.1.1.cmml">+</mo><mn id="S2.SS1a.p1.29.m10.1.1.1.1.1.1.3" xref="S2.SS1a.p1.29.m10.1.1.1.1.1.1.3.cmml">1</mn></mrow><mo id="S2.SS1a.p1.29.m10.1.1.1.1.1.3" stretchy="false" xref="S2.SS1a.p1.29.m10.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></msup><annotation-xml encoding="MathML-Content" id="S2.SS1a.p1.29.m10.1b"><apply id="S2.SS1a.p1.29.m10.1.2.cmml" xref="S2.SS1a.p1.29.m10.1.2"><csymbol cd="ambiguous" id="S2.SS1a.p1.29.m10.1.2.1.cmml" xref="S2.SS1a.p1.29.m10.1.2">superscript</csymbol><ci id="S2.SS1a.p1.29.m10.1.2.2.cmml" xref="S2.SS1a.p1.29.m10.1.2.2">𝜏</ci><apply id="S2.SS1a.p1.29.m10.1.1.1.cmml" xref="S2.SS1a.p1.29.m10.1.1.1"><times id="S2.SS1a.p1.29.m10.1.1.1.2.cmml" xref="S2.SS1a.p1.29.m10.1.1.1.2"></times><cn id="S2.SS1a.p1.29.m10.1.1.1.3.cmml" type="integer" xref="S2.SS1a.p1.29.m10.1.1.1.3">2</cn><apply id="S2.SS1a.p1.29.m10.1.1.1.1.1.1.cmml" xref="S2.SS1a.p1.29.m10.1.1.1.1.1"><plus id="S2.SS1a.p1.29.m10.1.1.1.1.1.1.1.cmml" xref="S2.SS1a.p1.29.m10.1.1.1.1.1.1.1"></plus><ci id="S2.SS1a.p1.29.m10.1.1.1.1.1.1.2.cmml" xref="S2.SS1a.p1.29.m10.1.1.1.1.1.1.2">𝑡</ci><cn id="S2.SS1a.p1.29.m10.1.1.1.1.1.1.3.cmml" type="integer" xref="S2.SS1a.p1.29.m10.1.1.1.1.1.1.3">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1a.p1.29.m10.1c">\tau^{2(t+1)}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1a.p1.29.m10.1d">italic_τ start_POSTSUPERSCRIPT 2 ( italic_t + 1 ) end_POSTSUPERSCRIPT</annotation></semantics></math>. We also update <math alttext="\operatorname{{\bm{\theta}}}^{(t)}" class="ltx_Math" display="inline" id="S2.SS1a.p1.30.m11.1"><semantics id="S2.SS1a.p1.30.m11.1a"><msup id="S2.SS1a.p1.30.m11.1.2" xref="S2.SS1a.p1.30.m11.1.2.cmml"><mi id="S2.SS1a.p1.30.m11.1.2.2" xref="S2.SS1a.p1.30.m11.1.2.2.cmml">𝜽</mi><mrow id="S2.SS1a.p1.30.m11.1.1.1.3" xref="S2.SS1a.p1.30.m11.1.2.cmml"><mo id="S2.SS1a.p1.30.m11.1.1.1.3.1" stretchy="false" xref="S2.SS1a.p1.30.m11.1.2.cmml">(</mo><mi id="S2.SS1a.p1.30.m11.1.1.1.1" xref="S2.SS1a.p1.30.m11.1.1.1.1.cmml">t</mi><mo id="S2.SS1a.p1.30.m11.1.1.1.3.2" stretchy="false" xref="S2.SS1a.p1.30.m11.1.2.cmml">)</mo></mrow></msup><annotation-xml encoding="MathML-Content" id="S2.SS1a.p1.30.m11.1b"><apply id="S2.SS1a.p1.30.m11.1.2.cmml" xref="S2.SS1a.p1.30.m11.1.2"><csymbol cd="ambiguous" id="S2.SS1a.p1.30.m11.1.2.1.cmml" xref="S2.SS1a.p1.30.m11.1.2">superscript</csymbol><ci id="S2.SS1a.p1.30.m11.1.2.2.cmml" xref="S2.SS1a.p1.30.m11.1.2.2">𝜽</ci><ci id="S2.SS1a.p1.30.m11.1.1.1.1.cmml" xref="S2.SS1a.p1.30.m11.1.1.1.1">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1a.p1.30.m11.1c">\operatorname{{\bm{\theta}}}^{(t)}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1a.p1.30.m11.1d">bold_italic_θ start_POSTSUPERSCRIPT ( italic_t ) end_POSTSUPERSCRIPT</annotation></semantics></math> to <math alttext="\operatorname{{\bm{\theta}}}^{(t+1/3)}=(\operatorname{\mathbf{B}}^{(t+1)},% \operatorname{\mathbf{L}}_{1}^{(t)},\operatorname{\mathbf{L}}_{2}^{(t)},\tau^{% 2(t+1)})" class="ltx_Math" display="inline" id="S2.SS1a.p1.31.m12.9"><semantics id="S2.SS1a.p1.31.m12.9a"><mrow id="S2.SS1a.p1.31.m12.9.9" xref="S2.SS1a.p1.31.m12.9.9.cmml"><msup id="S2.SS1a.p1.31.m12.9.9.6" xref="S2.SS1a.p1.31.m12.9.9.6.cmml"><mi id="S2.SS1a.p1.31.m12.9.9.6.2" xref="S2.SS1a.p1.31.m12.9.9.6.2.cmml">𝜽</mi><mrow id="S2.SS1a.p1.31.m12.1.1.1.1" xref="S2.SS1a.p1.31.m12.1.1.1.1.1.cmml"><mo id="S2.SS1a.p1.31.m12.1.1.1.1.2" 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encoding="application/x-tex" id="S2.SS1a.p1.31.m12.9c">\operatorname{{\bm{\theta}}}^{(t+1/3)}=(\operatorname{\mathbf{B}}^{(t+1)},% \operatorname{\mathbf{L}}_{1}^{(t)},\operatorname{\mathbf{L}}_{2}^{(t)},\tau^{% 2(t+1)})</annotation><annotation encoding="application/x-llamapun" id="S2.SS1a.p1.31.m12.9d">bold_italic_θ start_POSTSUPERSCRIPT ( italic_t + 1 / 3 ) end_POSTSUPERSCRIPT = ( bold_B start_POSTSUPERSCRIPT ( italic_t + 1 ) end_POSTSUPERSCRIPT , bold_L start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_t ) end_POSTSUPERSCRIPT , bold_L start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_t ) end_POSTSUPERSCRIPT , italic_τ start_POSTSUPERSCRIPT 2 ( italic_t + 1 ) end_POSTSUPERSCRIPT )</annotation></semantics></math>.</p> </div> </section> <section class="ltx_subsection" id="S2.SS2a"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">2.2 </span>Derivation of <math alttext="\operatorname{\mathbf{L}}_{1}^{(t+1)}" 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encoding="application/x-tex" id="S2.SS2a.p1.3.m3.3c">(\operatorname{\mathbf{y}}_{i},\operatorname{\mathbf{c}}_{i(k)})</annotation><annotation encoding="application/x-llamapun" id="S2.SS2a.p1.3.m3.3d">( bold_y start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , bold_c start_POSTSUBSCRIPT italic_i ( italic_k ) end_POSTSUBSCRIPT )</annotation></semantics></math>, for <math alttext="i=1,\ldots,n" class="ltx_Math" display="inline" id="S2.SS2a.p1.4.m4.3"><semantics id="S2.SS2a.p1.4.m4.3a"><mrow id="S2.SS2a.p1.4.m4.3.4" xref="S2.SS2a.p1.4.m4.3.4.cmml"><mi id="S2.SS2a.p1.4.m4.3.4.2" xref="S2.SS2a.p1.4.m4.3.4.2.cmml">i</mi><mo id="S2.SS2a.p1.4.m4.3.4.1" xref="S2.SS2a.p1.4.m4.3.4.1.cmml">=</mo><mrow id="S2.SS2a.p1.4.m4.3.4.3.2" xref="S2.SS2a.p1.4.m4.3.4.3.1.cmml"><mn id="S2.SS2a.p1.4.m4.1.1" xref="S2.SS2a.p1.4.m4.1.1.cmml">1</mn><mo id="S2.SS2a.p1.4.m4.3.4.3.2.1" xref="S2.SS2a.p1.4.m4.3.4.3.1.cmml">,</mo><mi id="S2.SS2a.p1.4.m4.2.2" mathvariant="normal" xref="S2.SS2a.p1.4.m4.2.2.cmml">…</mi><mo 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class="ltx_Math" display="inline" id="S2.SS2a.p1.5.m5.2"><semantics id="S2.SS2a.p1.5.m5.2a"><mrow id="S2.SS2a.p1.5.m5.2.3" xref="S2.SS2a.p1.5.m5.2.3.cmml"><mi id="S2.SS2a.p1.5.m5.2.3.2" xref="S2.SS2a.p1.5.m5.2.3.2.cmml">k</mi><mo id="S2.SS2a.p1.5.m5.2.3.1" xref="S2.SS2a.p1.5.m5.2.3.1.cmml">=</mo><mrow id="S2.SS2a.p1.5.m5.2.3.3.2" xref="S2.SS2a.p1.5.m5.2.3.3.1.cmml"><mn id="S2.SS2a.p1.5.m5.1.1" xref="S2.SS2a.p1.5.m5.1.1.cmml">1</mn><mo id="S2.SS2a.p1.5.m5.2.3.3.2.1" xref="S2.SS2a.p1.5.m5.2.3.3.1.cmml">,</mo><mn id="S2.SS2a.p1.5.m5.2.2" xref="S2.SS2a.p1.5.m5.2.2.cmml">2</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2a.p1.5.m5.2b"><apply id="S2.SS2a.p1.5.m5.2.3.cmml" xref="S2.SS2a.p1.5.m5.2.3"><eq id="S2.SS2a.p1.5.m5.2.3.1.cmml" xref="S2.SS2a.p1.5.m5.2.3.1"></eq><ci id="S2.SS2a.p1.5.m5.2.3.2.cmml" xref="S2.SS2a.p1.5.m5.2.3.2">𝑘</ci><list id="S2.SS2a.p1.5.m5.2.3.3.1.cmml" xref="S2.SS2a.p1.5.m5.2.3.3.2"><cn id="S2.SS2a.p1.5.m5.1.1.cmml" type="integer" xref="S2.SS2a.p1.5.m5.1.1">1</cn><cn id="S2.SS2a.p1.5.m5.2.2.cmml" type="integer" xref="S2.SS2a.p1.5.m5.2.2">2</cn></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2a.p1.5.m5.2c">k=1,2</annotation><annotation encoding="application/x-llamapun" id="S2.SS2a.p1.5.m5.2d">italic_k = 1 , 2</annotation></semantics></math>; see Equations (9), (10), and (11) in the main manuscript. The AECM algorithm then maximizes <math alttext="\operatorname{\mathbb{E}}[\log f_{\operatorname{\mathbf{y}}|\operatorname{% \mathbf{c}},\operatorname{{\bm{\theta}}}}(\operatorname{\mathbf{y}})|% \operatorname{\mathbf{y}},\operatorname{{\bm{\theta}}}^{(t+1/3)}]" class="ltx_Math" display="inline" id="S2.SS2a.p1.6.m6.7"><semantics id="S2.SS2a.p1.6.m6.7a"><mrow id="S2.SS2a.p1.6.m6.7.7.1" xref="S2.SS2a.p1.6.m6.7.7.2.cmml"><mi id="S2.SS2a.p1.6.m6.6.6" xref="S2.SS2a.p1.6.m6.6.6.cmml">𝔼</mi><mo id="S2.SS2a.p1.6.m6.7.7.1a" xref="S2.SS2a.p1.6.m6.7.7.2.cmml">⁡</mo><mrow id="S2.SS2a.p1.6.m6.7.7.1.1" xref="S2.SS2a.p1.6.m6.7.7.2.cmml"><mo id="S2.SS2a.p1.6.m6.7.7.1.1.2" stretchy="false" xref="S2.SS2a.p1.6.m6.7.7.2.cmml">[</mo><mrow id="S2.SS2a.p1.6.m6.7.7.1.1.1" xref="S2.SS2a.p1.6.m6.7.7.1.1.1.cmml"><mrow id="S2.SS2a.p1.6.m6.7.7.1.1.1.3" xref="S2.SS2a.p1.6.m6.7.7.1.1.1.3.cmml"><mrow id="S2.SS2a.p1.6.m6.7.7.1.1.1.3.2" xref="S2.SS2a.p1.6.m6.7.7.1.1.1.3.2.cmml"><mi id="S2.SS2a.p1.6.m6.7.7.1.1.1.3.2.1" xref="S2.SS2a.p1.6.m6.7.7.1.1.1.3.2.1.cmml">log</mi><mo id="S2.SS2a.p1.6.m6.7.7.1.1.1.3.2a" lspace="0.167em" xref="S2.SS2a.p1.6.m6.7.7.1.1.1.3.2.cmml">⁡</mo><msub id="S2.SS2a.p1.6.m6.7.7.1.1.1.3.2.2" xref="S2.SS2a.p1.6.m6.7.7.1.1.1.3.2.2.cmml"><mi id="S2.SS2a.p1.6.m6.7.7.1.1.1.3.2.2.2" xref="S2.SS2a.p1.6.m6.7.7.1.1.1.3.2.2.2.cmml">f</mi><mrow id="S2.SS2a.p1.6.m6.2.2.2" xref="S2.SS2a.p1.6.m6.2.2.2.cmml"><mi id="S2.SS2a.p1.6.m6.2.2.2.4" xref="S2.SS2a.p1.6.m6.2.2.2.4.cmml">𝐲</mi><mo fence="false" id="S2.SS2a.p1.6.m6.2.2.2.3" xref="S2.SS2a.p1.6.m6.2.2.2.3.cmml">|</mo><mrow id="S2.SS2a.p1.6.m6.2.2.2.5.2" xref="S2.SS2a.p1.6.m6.2.2.2.5.1.cmml"><mi id="S2.SS2a.p1.6.m6.1.1.1.1" xref="S2.SS2a.p1.6.m6.1.1.1.1.cmml">𝐜</mi><mo id="S2.SS2a.p1.6.m6.2.2.2.5.2.1" xref="S2.SS2a.p1.6.m6.2.2.2.5.1.cmml">,</mo><mi id="S2.SS2a.p1.6.m6.2.2.2.2" xref="S2.SS2a.p1.6.m6.2.2.2.2.cmml">𝜽</mi></mrow></mrow></msub></mrow><mo id="S2.SS2a.p1.6.m6.7.7.1.1.1.3.1" xref="S2.SS2a.p1.6.m6.7.7.1.1.1.3.1.cmml">⁢</mo><mrow 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stretchy="false" xref="S2.SS2a.p1.6.m6.3.3.1.1.1.cmml">(</mo><mrow id="S2.SS2a.p1.6.m6.3.3.1.1.1" xref="S2.SS2a.p1.6.m6.3.3.1.1.1.cmml"><mi id="S2.SS2a.p1.6.m6.3.3.1.1.1.2" xref="S2.SS2a.p1.6.m6.3.3.1.1.1.2.cmml">t</mi><mo id="S2.SS2a.p1.6.m6.3.3.1.1.1.1" xref="S2.SS2a.p1.6.m6.3.3.1.1.1.1.cmml">+</mo><mrow id="S2.SS2a.p1.6.m6.3.3.1.1.1.3" xref="S2.SS2a.p1.6.m6.3.3.1.1.1.3.cmml"><mn id="S2.SS2a.p1.6.m6.3.3.1.1.1.3.2" xref="S2.SS2a.p1.6.m6.3.3.1.1.1.3.2.cmml">1</mn><mo id="S2.SS2a.p1.6.m6.3.3.1.1.1.3.1" xref="S2.SS2a.p1.6.m6.3.3.1.1.1.3.1.cmml">/</mo><mn id="S2.SS2a.p1.6.m6.3.3.1.1.1.3.3" xref="S2.SS2a.p1.6.m6.3.3.1.1.1.3.3.cmml">3</mn></mrow></mrow><mo id="S2.SS2a.p1.6.m6.3.3.1.1.3" stretchy="false" xref="S2.SS2a.p1.6.m6.3.3.1.1.1.cmml">)</mo></mrow></msup></mrow></mrow><mo id="S2.SS2a.p1.6.m6.7.7.1.1.3" stretchy="false" xref="S2.SS2a.p1.6.m6.7.7.2.cmml">]</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2a.p1.6.m6.7b"><apply id="S2.SS2a.p1.6.m6.7.7.2.cmml" xref="S2.SS2a.p1.6.m6.7.7.1"><ci id="S2.SS2a.p1.6.m6.6.6.cmml" xref="S2.SS2a.p1.6.m6.6.6">𝔼</ci><apply id="S2.SS2a.p1.6.m6.7.7.1.1.1.cmml" xref="S2.SS2a.p1.6.m6.7.7.1.1.1"><csymbol cd="latexml" id="S2.SS2a.p1.6.m6.7.7.1.1.1.2.cmml" xref="S2.SS2a.p1.6.m6.7.7.1.1.1.2">conditional</csymbol><apply id="S2.SS2a.p1.6.m6.7.7.1.1.1.3.cmml" xref="S2.SS2a.p1.6.m6.7.7.1.1.1.3"><times id="S2.SS2a.p1.6.m6.7.7.1.1.1.3.1.cmml" xref="S2.SS2a.p1.6.m6.7.7.1.1.1.3.1"></times><apply id="S2.SS2a.p1.6.m6.7.7.1.1.1.3.2.cmml" xref="S2.SS2a.p1.6.m6.7.7.1.1.1.3.2"><log id="S2.SS2a.p1.6.m6.7.7.1.1.1.3.2.1.cmml" xref="S2.SS2a.p1.6.m6.7.7.1.1.1.3.2.1"></log><apply id="S2.SS2a.p1.6.m6.7.7.1.1.1.3.2.2.cmml" xref="S2.SS2a.p1.6.m6.7.7.1.1.1.3.2.2"><csymbol cd="ambiguous" id="S2.SS2a.p1.6.m6.7.7.1.1.1.3.2.2.1.cmml" xref="S2.SS2a.p1.6.m6.7.7.1.1.1.3.2.2">subscript</csymbol><ci id="S2.SS2a.p1.6.m6.7.7.1.1.1.3.2.2.2.cmml" xref="S2.SS2a.p1.6.m6.7.7.1.1.1.3.2.2.2">𝑓</ci><apply id="S2.SS2a.p1.6.m6.2.2.2.cmml" xref="S2.SS2a.p1.6.m6.2.2.2"><csymbol cd="latexml" id="S2.SS2a.p1.6.m6.2.2.2.3.cmml" xref="S2.SS2a.p1.6.m6.2.2.2.3">conditional</csymbol><ci id="S2.SS2a.p1.6.m6.2.2.2.4.cmml" xref="S2.SS2a.p1.6.m6.2.2.2.4">𝐲</ci><list id="S2.SS2a.p1.6.m6.2.2.2.5.1.cmml" xref="S2.SS2a.p1.6.m6.2.2.2.5.2"><ci id="S2.SS2a.p1.6.m6.1.1.1.1.cmml" xref="S2.SS2a.p1.6.m6.1.1.1.1">𝐜</ci><ci id="S2.SS2a.p1.6.m6.2.2.2.2.cmml" xref="S2.SS2a.p1.6.m6.2.2.2.2">𝜽</ci></list></apply></apply></apply><ci id="S2.SS2a.p1.6.m6.4.4.cmml" xref="S2.SS2a.p1.6.m6.4.4">𝐲</ci></apply><list id="S2.SS2a.p1.6.m6.7.7.1.1.1.1.2.cmml" xref="S2.SS2a.p1.6.m6.7.7.1.1.1.1.1"><ci id="S2.SS2a.p1.6.m6.5.5.cmml" xref="S2.SS2a.p1.6.m6.5.5">𝐲</ci><apply id="S2.SS2a.p1.6.m6.7.7.1.1.1.1.1.1.cmml" xref="S2.SS2a.p1.6.m6.7.7.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S2.SS2a.p1.6.m6.7.7.1.1.1.1.1.1.1.cmml" xref="S2.SS2a.p1.6.m6.7.7.1.1.1.1.1.1">superscript</csymbol><ci id="S2.SS2a.p1.6.m6.7.7.1.1.1.1.1.1.2.cmml" xref="S2.SS2a.p1.6.m6.7.7.1.1.1.1.1.1.2">𝜽</ci><apply id="S2.SS2a.p1.6.m6.3.3.1.1.1.cmml" xref="S2.SS2a.p1.6.m6.3.3.1.1"><plus id="S2.SS2a.p1.6.m6.3.3.1.1.1.1.cmml" xref="S2.SS2a.p1.6.m6.3.3.1.1.1.1"></plus><ci id="S2.SS2a.p1.6.m6.3.3.1.1.1.2.cmml" xref="S2.SS2a.p1.6.m6.3.3.1.1.1.2">𝑡</ci><apply id="S2.SS2a.p1.6.m6.3.3.1.1.1.3.cmml" xref="S2.SS2a.p1.6.m6.3.3.1.1.1.3"><divide id="S2.SS2a.p1.6.m6.3.3.1.1.1.3.1.cmml" xref="S2.SS2a.p1.6.m6.3.3.1.1.1.3.1"></divide><cn id="S2.SS2a.p1.6.m6.3.3.1.1.1.3.2.cmml" type="integer" xref="S2.SS2a.p1.6.m6.3.3.1.1.1.3.2">1</cn><cn id="S2.SS2a.p1.6.m6.3.3.1.1.1.3.3.cmml" type="integer" xref="S2.SS2a.p1.6.m6.3.3.1.1.1.3.3">3</cn></apply></apply></apply></list></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2a.p1.6.m6.7c">\operatorname{\mathbb{E}}[\log f_{\operatorname{\mathbf{y}}|\operatorname{% \mathbf{c}},\operatorname{{\bm{\theta}}}}(\operatorname{\mathbf{y}})|% \operatorname{\mathbf{y}},\operatorname{{\bm{\theta}}}^{(t+1/3)}]</annotation><annotation encoding="application/x-llamapun" id="S2.SS2a.p1.6.m6.7d">blackboard_E [ roman_log italic_f start_POSTSUBSCRIPT bold_y | bold_c , bold_italic_θ end_POSTSUBSCRIPT ( bold_y ) | bold_y , bold_italic_θ start_POSTSUPERSCRIPT ( italic_t + 1 / 3 ) end_POSTSUPERSCRIPT ]</annotation></semantics></math> with respect to <math alttext="\operatorname{\mathbf{L}}_{1}" class="ltx_Math" display="inline" id="S2.SS2a.p1.7.m7.1"><semantics id="S2.SS2a.p1.7.m7.1a"><msub id="S2.SS2a.p1.7.m7.1.1" xref="S2.SS2a.p1.7.m7.1.1.cmml"><mi id="S2.SS2a.p1.7.m7.1.1.2" xref="S2.SS2a.p1.7.m7.1.1.2.cmml">𝐋</mi><mn id="S2.SS2a.p1.7.m7.1.1.3" xref="S2.SS2a.p1.7.m7.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S2.SS2a.p1.7.m7.1b"><apply id="S2.SS2a.p1.7.m7.1.1.cmml" xref="S2.SS2a.p1.7.m7.1.1"><csymbol cd="ambiguous" id="S2.SS2a.p1.7.m7.1.1.1.cmml" xref="S2.SS2a.p1.7.m7.1.1">subscript</csymbol><ci id="S2.SS2a.p1.7.m7.1.1.2.cmml" xref="S2.SS2a.p1.7.m7.1.1.2">𝐋</ci><cn id="S2.SS2a.p1.7.m7.1.1.3.cmml" type="integer" xref="S2.SS2a.p1.7.m7.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2a.p1.7.m7.1c">\operatorname{\mathbf{L}}_{1}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2a.p1.7.m7.1d">bold_L start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math>, where <math alttext="\log f_{\operatorname{\mathbf{y}}|\operatorname{\mathbf{c}},\operatorname{{\bm% {\theta}}}}(\operatorname{\mathbf{y}})|\operatorname{\mathbf{y}},\operatorname% {{\bm{\theta}}}^{(t+1/3)}" class="ltx_Math" display="inline" id="S2.SS2a.p1.8.m8.6"><semantics id="S2.SS2a.p1.8.m8.6a"><mrow id="S2.SS2a.p1.8.m8.6.6" xref="S2.SS2a.p1.8.m8.6.6.cmml"><mrow id="S2.SS2a.p1.8.m8.6.6.3" xref="S2.SS2a.p1.8.m8.6.6.3.cmml"><mrow id="S2.SS2a.p1.8.m8.6.6.3.2" xref="S2.SS2a.p1.8.m8.6.6.3.2.cmml"><mi id="S2.SS2a.p1.8.m8.6.6.3.2.1" 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id="S2.SS2a.p1.8.m8.3.3.1.1.1.3.2.cmml" type="integer" xref="S2.SS2a.p1.8.m8.3.3.1.1.1.3.2">1</cn><cn id="S2.SS2a.p1.8.m8.3.3.1.1.1.3.3.cmml" type="integer" xref="S2.SS2a.p1.8.m8.3.3.1.1.1.3.3">3</cn></apply></apply></apply></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2a.p1.8.m8.6c">\log f_{\operatorname{\mathbf{y}}|\operatorname{\mathbf{c}},\operatorname{{\bm% {\theta}}}}(\operatorname{\mathbf{y}})|\operatorname{\mathbf{y}},\operatorname% {{\bm{\theta}}}^{(t+1/3)}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2a.p1.8.m8.6d">roman_log italic_f start_POSTSUBSCRIPT bold_y | bold_c , bold_italic_θ end_POSTSUBSCRIPT ( bold_y ) | bold_y , bold_italic_θ start_POSTSUPERSCRIPT ( italic_t + 1 / 3 ) end_POSTSUPERSCRIPT</annotation></semantics></math> is the complete data log likelihood in Equation of (6) of the main manuscript; see also Section 3 and Equation (8) in the main manuscript.</p> </div> <div class="ltx_para" id="S2.SS2a.p2"> <p class="ltx_p" id="S2.SS2a.p2.8">Consider the estimation of <math alttext="\operatorname{\mathbf{L}}_{1}" class="ltx_Math" display="inline" id="S2.SS2a.p2.1.m1.1"><semantics id="S2.SS2a.p2.1.m1.1a"><msub id="S2.SS2a.p2.1.m1.1.1" xref="S2.SS2a.p2.1.m1.1.1.cmml"><mi id="S2.SS2a.p2.1.m1.1.1.2" xref="S2.SS2a.p2.1.m1.1.1.2.cmml">𝐋</mi><mn id="S2.SS2a.p2.1.m1.1.1.3" xref="S2.SS2a.p2.1.m1.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S2.SS2a.p2.1.m1.1b"><apply id="S2.SS2a.p2.1.m1.1.1.cmml" xref="S2.SS2a.p2.1.m1.1.1"><csymbol cd="ambiguous" id="S2.SS2a.p2.1.m1.1.1.1.cmml" xref="S2.SS2a.p2.1.m1.1.1">subscript</csymbol><ci id="S2.SS2a.p2.1.m1.1.1.2.cmml" xref="S2.SS2a.p2.1.m1.1.1.2">𝐋</ci><cn id="S2.SS2a.p2.1.m1.1.1.3.cmml" type="integer" xref="S2.SS2a.p2.1.m1.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2a.p2.1.m1.1c">\operatorname{\mathbf{L}}_{1}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2a.p2.1.m1.1d">bold_L start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> given <math alttext="\operatorname{{\bm{\theta}}}^{(t+1/3)}" class="ltx_Math" display="inline" id="S2.SS2a.p2.2.m2.1"><semantics id="S2.SS2a.p2.2.m2.1a"><msup id="S2.SS2a.p2.2.m2.1.2" xref="S2.SS2a.p2.2.m2.1.2.cmml"><mi id="S2.SS2a.p2.2.m2.1.2.2" xref="S2.SS2a.p2.2.m2.1.2.2.cmml">𝜽</mi><mrow id="S2.SS2a.p2.2.m2.1.1.1.1" xref="S2.SS2a.p2.2.m2.1.1.1.1.1.cmml"><mo id="S2.SS2a.p2.2.m2.1.1.1.1.2" stretchy="false" xref="S2.SS2a.p2.2.m2.1.1.1.1.1.cmml">(</mo><mrow id="S2.SS2a.p2.2.m2.1.1.1.1.1" xref="S2.SS2a.p2.2.m2.1.1.1.1.1.cmml"><mi id="S2.SS2a.p2.2.m2.1.1.1.1.1.2" xref="S2.SS2a.p2.2.m2.1.1.1.1.1.2.cmml">t</mi><mo id="S2.SS2a.p2.2.m2.1.1.1.1.1.1" xref="S2.SS2a.p2.2.m2.1.1.1.1.1.1.cmml">+</mo><mrow id="S2.SS2a.p2.2.m2.1.1.1.1.1.3" xref="S2.SS2a.p2.2.m2.1.1.1.1.1.3.cmml"><mn id="S2.SS2a.p2.2.m2.1.1.1.1.1.3.2" xref="S2.SS2a.p2.2.m2.1.1.1.1.1.3.2.cmml">1</mn><mo id="S2.SS2a.p2.2.m2.1.1.1.1.1.3.1" xref="S2.SS2a.p2.2.m2.1.1.1.1.1.3.1.cmml">/</mo><mn id="S2.SS2a.p2.2.m2.1.1.1.1.1.3.3" xref="S2.SS2a.p2.2.m2.1.1.1.1.1.3.3.cmml">3</mn></mrow></mrow><mo id="S2.SS2a.p2.2.m2.1.1.1.1.3" stretchy="false" xref="S2.SS2a.p2.2.m2.1.1.1.1.1.cmml">)</mo></mrow></msup><annotation-xml encoding="MathML-Content" id="S2.SS2a.p2.2.m2.1b"><apply id="S2.SS2a.p2.2.m2.1.2.cmml" xref="S2.SS2a.p2.2.m2.1.2"><csymbol cd="ambiguous" id="S2.SS2a.p2.2.m2.1.2.1.cmml" xref="S2.SS2a.p2.2.m2.1.2">superscript</csymbol><ci id="S2.SS2a.p2.2.m2.1.2.2.cmml" xref="S2.SS2a.p2.2.m2.1.2.2">𝜽</ci><apply id="S2.SS2a.p2.2.m2.1.1.1.1.1.cmml" xref="S2.SS2a.p2.2.m2.1.1.1.1"><plus id="S2.SS2a.p2.2.m2.1.1.1.1.1.1.cmml" xref="S2.SS2a.p2.2.m2.1.1.1.1.1.1"></plus><ci id="S2.SS2a.p2.2.m2.1.1.1.1.1.2.cmml" xref="S2.SS2a.p2.2.m2.1.1.1.1.1.2">𝑡</ci><apply id="S2.SS2a.p2.2.m2.1.1.1.1.1.3.cmml" xref="S2.SS2a.p2.2.m2.1.1.1.1.1.3"><divide id="S2.SS2a.p2.2.m2.1.1.1.1.1.3.1.cmml" xref="S2.SS2a.p2.2.m2.1.1.1.1.1.3.1"></divide><cn id="S2.SS2a.p2.2.m2.1.1.1.1.1.3.2.cmml" type="integer" xref="S2.SS2a.p2.2.m2.1.1.1.1.1.3.2">1</cn><cn id="S2.SS2a.p2.2.m2.1.1.1.1.1.3.3.cmml" type="integer" xref="S2.SS2a.p2.2.m2.1.1.1.1.1.3.3">3</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2a.p2.2.m2.1c">\operatorname{{\bm{\theta}}}^{(t+1/3)}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2a.p2.2.m2.1d">bold_italic_θ start_POSTSUPERSCRIPT ( italic_t + 1 / 3 ) end_POSTSUPERSCRIPT</annotation></semantics></math> in the second cycle of the AECM algorithm. The E step in the second cycle computes the conditional expectation <math alttext="\operatorname{\mathbb{E}}[\log f_{\operatorname{\mathbf{y}}|\operatorname{% \mathbf{c}},\operatorname{{\bm{\theta}}}}(\operatorname{\mathbf{y}})|% \operatorname{\mathbf{y}},\operatorname{{\bm{\theta}}}^{(t+1/3)}]" class="ltx_Math" display="inline" id="S2.SS2a.p2.3.m3.7"><semantics id="S2.SS2a.p2.3.m3.7a"><mrow id="S2.SS2a.p2.3.m3.7.7.1" xref="S2.SS2a.p2.3.m3.7.7.2.cmml"><mi id="S2.SS2a.p2.3.m3.6.6" xref="S2.SS2a.p2.3.m3.6.6.cmml">𝔼</mi><mo id="S2.SS2a.p2.3.m3.7.7.1a" xref="S2.SS2a.p2.3.m3.7.7.2.cmml">⁡</mo><mrow id="S2.SS2a.p2.3.m3.7.7.1.1" xref="S2.SS2a.p2.3.m3.7.7.2.cmml"><mo id="S2.SS2a.p2.3.m3.7.7.1.1.2" stretchy="false" xref="S2.SS2a.p2.3.m3.7.7.2.cmml">[</mo><mrow id="S2.SS2a.p2.3.m3.7.7.1.1.1" xref="S2.SS2a.p2.3.m3.7.7.1.1.1.cmml"><mrow id="S2.SS2a.p2.3.m3.7.7.1.1.1.3" xref="S2.SS2a.p2.3.m3.7.7.1.1.1.3.cmml"><mrow id="S2.SS2a.p2.3.m3.7.7.1.1.1.3.2" xref="S2.SS2a.p2.3.m3.7.7.1.1.1.3.2.cmml"><mi 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id="S2.SS2a.p2.3.m3.7.7.1.1.1.1.1.1.2.cmml" xref="S2.SS2a.p2.3.m3.7.7.1.1.1.1.1.1.2">𝜽</ci><apply id="S2.SS2a.p2.3.m3.3.3.1.1.1.cmml" xref="S2.SS2a.p2.3.m3.3.3.1.1"><plus id="S2.SS2a.p2.3.m3.3.3.1.1.1.1.cmml" xref="S2.SS2a.p2.3.m3.3.3.1.1.1.1"></plus><ci id="S2.SS2a.p2.3.m3.3.3.1.1.1.2.cmml" xref="S2.SS2a.p2.3.m3.3.3.1.1.1.2">𝑡</ci><apply id="S2.SS2a.p2.3.m3.3.3.1.1.1.3.cmml" xref="S2.SS2a.p2.3.m3.3.3.1.1.1.3"><divide id="S2.SS2a.p2.3.m3.3.3.1.1.1.3.1.cmml" xref="S2.SS2a.p2.3.m3.3.3.1.1.1.3.1"></divide><cn id="S2.SS2a.p2.3.m3.3.3.1.1.1.3.2.cmml" type="integer" xref="S2.SS2a.p2.3.m3.3.3.1.1.1.3.2">1</cn><cn id="S2.SS2a.p2.3.m3.3.3.1.1.1.3.3.cmml" type="integer" xref="S2.SS2a.p2.3.m3.3.3.1.1.1.3.3">3</cn></apply></apply></apply></list></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2a.p2.3.m3.7c">\operatorname{\mathbb{E}}[\log f_{\operatorname{\mathbf{y}}|\operatorname{% \mathbf{c}},\operatorname{{\bm{\theta}}}}(\operatorname{\mathbf{y}})|% \operatorname{\mathbf{y}},\operatorname{{\bm{\theta}}}^{(t+1/3)}]</annotation><annotation encoding="application/x-llamapun" id="S2.SS2a.p2.3.m3.7d">blackboard_E [ roman_log italic_f start_POSTSUBSCRIPT bold_y | bold_c , bold_italic_θ end_POSTSUBSCRIPT ( bold_y ) | bold_y , bold_italic_θ start_POSTSUPERSCRIPT ( italic_t + 1 / 3 ) end_POSTSUPERSCRIPT ]</annotation></semantics></math>. The CM step in the second cycle maximizes this objective with respect to <math alttext="\operatorname{\mathbf{L}}_{1}" class="ltx_Math" display="inline" id="S2.SS2a.p2.4.m4.1"><semantics id="S2.SS2a.p2.4.m4.1a"><msub id="S2.SS2a.p2.4.m4.1.1" xref="S2.SS2a.p2.4.m4.1.1.cmml"><mi id="S2.SS2a.p2.4.m4.1.1.2" xref="S2.SS2a.p2.4.m4.1.1.2.cmml">𝐋</mi><mn id="S2.SS2a.p2.4.m4.1.1.3" xref="S2.SS2a.p2.4.m4.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S2.SS2a.p2.4.m4.1b"><apply id="S2.SS2a.p2.4.m4.1.1.cmml" xref="S2.SS2a.p2.4.m4.1.1"><csymbol cd="ambiguous" id="S2.SS2a.p2.4.m4.1.1.1.cmml" xref="S2.SS2a.p2.4.m4.1.1">subscript</csymbol><ci id="S2.SS2a.p2.4.m4.1.1.2.cmml" xref="S2.SS2a.p2.4.m4.1.1.2">𝐋</ci><cn id="S2.SS2a.p2.4.m4.1.1.3.cmml" type="integer" xref="S2.SS2a.p2.4.m4.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2a.p2.4.m4.1c">\operatorname{\mathbf{L}}_{1}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2a.p2.4.m4.1d">bold_L start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math>. This is equivalent to maximizing <math alttext="\operatorname{\mathbb{E}}[\log f_{\tilde{\operatorname{\mathbf{y}}}|% \operatorname{\mathbf{c}},\operatorname{{\bm{\theta}}}}(\tilde{\operatorname{% \mathbf{y}}})|\tilde{\operatorname{\mathbf{y}}},\operatorname{{\bm{\theta}}}^{% (t+1/3)}]" class="ltx_Math" display="inline" id="S2.SS2a.p2.5.m5.7"><semantics id="S2.SS2a.p2.5.m5.7a"><mrow id="S2.SS2a.p2.5.m5.7.7.1" xref="S2.SS2a.p2.5.m5.7.7.2.cmml"><mi id="S2.SS2a.p2.5.m5.6.6" xref="S2.SS2a.p2.5.m5.6.6.cmml">𝔼</mi><mo id="S2.SS2a.p2.5.m5.7.7.1a" xref="S2.SS2a.p2.5.m5.7.7.2.cmml">⁡</mo><mrow id="S2.SS2a.p2.5.m5.7.7.1.1" xref="S2.SS2a.p2.5.m5.7.7.2.cmml"><mo id="S2.SS2a.p2.5.m5.7.7.1.1.2" stretchy="false" xref="S2.SS2a.p2.5.m5.7.7.2.cmml">[</mo><mrow id="S2.SS2a.p2.5.m5.7.7.1.1.1" xref="S2.SS2a.p2.5.m5.7.7.1.1.1.cmml"><mrow id="S2.SS2a.p2.5.m5.7.7.1.1.1.3" xref="S2.SS2a.p2.5.m5.7.7.1.1.1.3.cmml"><mrow id="S2.SS2a.p2.5.m5.7.7.1.1.1.3.2" xref="S2.SS2a.p2.5.m5.7.7.1.1.1.3.2.cmml"><mi 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\mathbf{y}}})|\tilde{\operatorname{\mathbf{y}}},\operatorname{{\bm{\theta}}}^{% (t+1/3)}]</annotation><annotation encoding="application/x-llamapun" id="S2.SS2a.p2.5.m5.7d">blackboard_E [ roman_log italic_f start_POSTSUBSCRIPT over~ start_ARG bold_y end_ARG | bold_c , bold_italic_θ end_POSTSUBSCRIPT ( over~ start_ARG bold_y end_ARG ) | over~ start_ARG bold_y end_ARG , bold_italic_θ start_POSTSUPERSCRIPT ( italic_t + 1 / 3 ) end_POSTSUPERSCRIPT ]</annotation></semantics></math> with respect to <math alttext="\operatorname{\mathbf{L}}_{1}" class="ltx_Math" display="inline" id="S2.SS2a.p2.6.m6.1"><semantics id="S2.SS2a.p2.6.m6.1a"><msub id="S2.SS2a.p2.6.m6.1.1" xref="S2.SS2a.p2.6.m6.1.1.cmml"><mi id="S2.SS2a.p2.6.m6.1.1.2" xref="S2.SS2a.p2.6.m6.1.1.2.cmml">𝐋</mi><mn id="S2.SS2a.p2.6.m6.1.1.3" xref="S2.SS2a.p2.6.m6.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S2.SS2a.p2.6.m6.1b"><apply id="S2.SS2a.p2.6.m6.1.1.cmml" xref="S2.SS2a.p2.6.m6.1.1"><csymbol cd="ambiguous" 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xref="S2.SS2a.p2.7.m7.1.2.3.1"></minus><ci id="S2.SS2a.p2.7.m7.1.2.3.2.cmml" xref="S2.SS2a.p2.7.m7.1.2.3.2">𝐲</ci><apply id="S2.SS2a.p2.7.m7.1.2.3.3.cmml" xref="S2.SS2a.p2.7.m7.1.2.3.3"><ci id="S2.SS2a.p2.7.m7.1.2.3.3.1.cmml" xref="S2.SS2a.p2.7.m7.1.2.3.3.1">𝐗</ci><apply id="S2.SS2a.p2.7.m7.1.2.3.3.2.cmml" xref="S2.SS2a.p2.7.m7.1.2.3.3.2"><csymbol cd="ambiguous" id="S2.SS2a.p2.7.m7.1.2.3.3.2.1.cmml" xref="S2.SS2a.p2.7.m7.1.2.3.3.2">superscript</csymbol><ci id="S2.SS2a.p2.7.m7.1.2.3.3.2.2.cmml" xref="S2.SS2a.p2.7.m7.1.2.3.3.2.2">𝐛</ci><apply id="S2.SS2a.p2.7.m7.1.1.1.1.1.cmml" xref="S2.SS2a.p2.7.m7.1.1.1.1"><plus id="S2.SS2a.p2.7.m7.1.1.1.1.1.1.cmml" xref="S2.SS2a.p2.7.m7.1.1.1.1.1.1"></plus><ci id="S2.SS2a.p2.7.m7.1.1.1.1.1.2.cmml" xref="S2.SS2a.p2.7.m7.1.1.1.1.1.2">𝑡</ci><cn id="S2.SS2a.p2.7.m7.1.1.1.1.1.3.cmml" type="integer" xref="S2.SS2a.p2.7.m7.1.1.1.1.1.3">1</cn></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2a.p2.7.m7.1c">\tilde{\operatorname{\mathbf{y}}}=\operatorname{\mathbf{y}}-\operatorname{% \mathbf{X}}\operatorname{\mathbf{b}}^{(t+1)}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2a.p2.7.m7.1d">over~ start_ARG bold_y end_ARG = bold_y - bold_X bold_b start_POSTSUPERSCRIPT ( italic_t + 1 ) end_POSTSUPERSCRIPT</annotation></semantics></math>. The complete data log likelihood of <math alttext="(\tilde{\operatorname{\mathbf{y}}},\operatorname{\mathbf{c}})" class="ltx_Math" display="inline" id="S2.SS2a.p2.8.m8.2"><semantics id="S2.SS2a.p2.8.m8.2a"><mrow id="S2.SS2a.p2.8.m8.2.3.2" xref="S2.SS2a.p2.8.m8.2.3.1.cmml"><mo id="S2.SS2a.p2.8.m8.2.3.2.1" stretchy="false" xref="S2.SS2a.p2.8.m8.2.3.1.cmml">(</mo><mover accent="true" id="S2.SS2a.p2.8.m8.1.1" xref="S2.SS2a.p2.8.m8.1.1.cmml"><mi id="S2.SS2a.p2.8.m8.1.1.2" xref="S2.SS2a.p2.8.m8.1.1.2.cmml">𝐲</mi><mo id="S2.SS2a.p2.8.m8.1.1.1" xref="S2.SS2a.p2.8.m8.1.1.1.cmml">~</mo></mover><mo id="S2.SS2a.p2.8.m8.2.3.2.2" xref="S2.SS2a.p2.8.m8.2.3.1.cmml">,</mo><mi id="S2.SS2a.p2.8.m8.2.2" xref="S2.SS2a.p2.8.m8.2.2.cmml">𝐜</mi><mo id="S2.SS2a.p2.8.m8.2.3.2.3" stretchy="false" xref="S2.SS2a.p2.8.m8.2.3.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2a.p2.8.m8.2b"><interval closure="open" id="S2.SS2a.p2.8.m8.2.3.1.cmml" xref="S2.SS2a.p2.8.m8.2.3.2"><apply id="S2.SS2a.p2.8.m8.1.1.cmml" xref="S2.SS2a.p2.8.m8.1.1"><ci id="S2.SS2a.p2.8.m8.1.1.1.cmml" xref="S2.SS2a.p2.8.m8.1.1.1">~</ci><ci id="S2.SS2a.p2.8.m8.1.1.2.cmml" xref="S2.SS2a.p2.8.m8.1.1.2">𝐲</ci></apply><ci id="S2.SS2a.p2.8.m8.2.2.cmml" xref="S2.SS2a.p2.8.m8.2.2">𝐜</ci></interval></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2a.p2.8.m8.2c">(\tilde{\operatorname{\mathbf{y}}},\operatorname{\mathbf{c}})</annotation><annotation encoding="application/x-llamapun" id="S2.SS2a.p2.8.m8.2d">( over~ start_ARG bold_y end_ARG , bold_c )</annotation></semantics></math> is</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="S3.EGx22"> <tbody id="S2.E5a"><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\log f_{\tilde{\operatorname{\mathbf{y}}}|\operatorname{\mathbf{c% }},\operatorname{{\bm{\theta}}}}(\tilde{\operatorname{\mathbf{y}}})=-\frac{N}{% 2}\log 2\pi\tau^{2}-\frac{1}{2\tau^{2}}\sum_{i=1}^{n}(\tilde{\operatorname{% \mathbf{y}}}_{i}-\operatorname{\mathbf{Z}}_{i(2)}\operatorname{\mathbf{L}}_{(2% )}\operatorname{\mathbf{c}}_{i(2)})^{\top}(\tilde{\operatorname{\mathbf{y}}}_{% i}-\operatorname{\mathbf{Z}}_{i(2)}\operatorname{\mathbf{L}}_{(2)}% \operatorname{\mathbf{c}}_{i(2)})," class="ltx_Math" display="inline" id="S2.E5a.m1.10"><semantics id="S2.E5a.m1.10a"><mrow id="S2.E5a.m1.10.10.1" xref="S2.E5a.m1.10.10.1.1.cmml"><mrow id="S2.E5a.m1.10.10.1.1" xref="S2.E5a.m1.10.10.1.1.cmml"><mrow id="S2.E5a.m1.10.10.1.1.4" xref="S2.E5a.m1.10.10.1.1.4.cmml"><mrow id="S2.E5a.m1.10.10.1.1.4.2" xref="S2.E5a.m1.10.10.1.1.4.2.cmml"><mi id="S2.E5a.m1.10.10.1.1.4.2.1" xref="S2.E5a.m1.10.10.1.1.4.2.1.cmml">log</mi><mo id="S2.E5a.m1.10.10.1.1.4.2a" lspace="0.167em" xref="S2.E5a.m1.10.10.1.1.4.2.cmml">⁡</mo><msub id="S2.E5a.m1.10.10.1.1.4.2.2" xref="S2.E5a.m1.10.10.1.1.4.2.2.cmml"><mi id="S2.E5a.m1.10.10.1.1.4.2.2.2" xref="S2.E5a.m1.10.10.1.1.4.2.2.2.cmml">f</mi><mrow id="S2.E5a.m1.2.2.2" xref="S2.E5a.m1.2.2.2.cmml"><mover accent="true" id="S2.E5a.m1.2.2.2.4" xref="S2.E5a.m1.2.2.2.4.cmml"><mi id="S2.E5a.m1.2.2.2.4.2" xref="S2.E5a.m1.2.2.2.4.2.cmml">𝐲</mi><mo id="S2.E5a.m1.2.2.2.4.1" xref="S2.E5a.m1.2.2.2.4.1.cmml">~</mo></mover><mo fence="false" id="S2.E5a.m1.2.2.2.3" xref="S2.E5a.m1.2.2.2.3.cmml">|</mo><mrow id="S2.E5a.m1.2.2.2.5.2" xref="S2.E5a.m1.2.2.2.5.1.cmml"><mi id="S2.E5a.m1.1.1.1.1" xref="S2.E5a.m1.1.1.1.1.cmml">𝐜</mi><mo id="S2.E5a.m1.2.2.2.5.2.1" xref="S2.E5a.m1.2.2.2.5.1.cmml">,</mo><mi id="S2.E5a.m1.2.2.2.2" xref="S2.E5a.m1.2.2.2.2.cmml">𝜽</mi></mrow></mrow></msub></mrow><mo id="S2.E5a.m1.10.10.1.1.4.1" xref="S2.E5a.m1.10.10.1.1.4.1.cmml">⁢</mo><mrow id="S2.E5a.m1.10.10.1.1.4.3.2" xref="S2.E5a.m1.9.9.cmml"><mo id="S2.E5a.m1.10.10.1.1.4.3.2.1" stretchy="false" xref="S2.E5a.m1.9.9.cmml">(</mo><mover accent="true" 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xref="S2.E5a.m1.10.10.1.1.2.4.2.2.3.cmml">2</mn></mfrac></mstyle><mo id="S2.E5a.m1.10.10.1.1.2.4.2.1" lspace="0.167em" xref="S2.E5a.m1.10.10.1.1.2.4.2.1.cmml">⁢</mo><mrow id="S2.E5a.m1.10.10.1.1.2.4.2.3" xref="S2.E5a.m1.10.10.1.1.2.4.2.3.cmml"><mi id="S2.E5a.m1.10.10.1.1.2.4.2.3.1" xref="S2.E5a.m1.10.10.1.1.2.4.2.3.1.cmml">log</mi><mo id="S2.E5a.m1.10.10.1.1.2.4.2.3a" lspace="0.167em" xref="S2.E5a.m1.10.10.1.1.2.4.2.3.cmml">⁡</mo><mrow id="S2.E5a.m1.10.10.1.1.2.4.2.3.2" xref="S2.E5a.m1.10.10.1.1.2.4.2.3.2.cmml"><mn id="S2.E5a.m1.10.10.1.1.2.4.2.3.2.2" xref="S2.E5a.m1.10.10.1.1.2.4.2.3.2.2.cmml">2</mn><mo id="S2.E5a.m1.10.10.1.1.2.4.2.3.2.1" xref="S2.E5a.m1.10.10.1.1.2.4.2.3.2.1.cmml">⁢</mo><mi id="S2.E5a.m1.10.10.1.1.2.4.2.3.2.3" xref="S2.E5a.m1.10.10.1.1.2.4.2.3.2.3.cmml">π</mi><mo id="S2.E5a.m1.10.10.1.1.2.4.2.3.2.1a" xref="S2.E5a.m1.10.10.1.1.2.4.2.3.2.1.cmml">⁢</mo><msup id="S2.E5a.m1.10.10.1.1.2.4.2.3.2.4" xref="S2.E5a.m1.10.10.1.1.2.4.2.3.2.4.cmml"><mi 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xref="S2.E5a.m1.10.10.1.1.2.2.4.3.3.2.cmml">τ</mi><mn id="S2.E5a.m1.10.10.1.1.2.2.4.3.3.3" xref="S2.E5a.m1.10.10.1.1.2.2.4.3.3.3.cmml">2</mn></msup></mrow></mfrac></mstyle><mo id="S2.E5a.m1.10.10.1.1.2.2.3" xref="S2.E5a.m1.10.10.1.1.2.2.3.cmml">⁢</mo><mrow id="S2.E5a.m1.10.10.1.1.2.2.2" xref="S2.E5a.m1.10.10.1.1.2.2.2.cmml"><mstyle displaystyle="true" id="S2.E5a.m1.10.10.1.1.2.2.2.3" xref="S2.E5a.m1.10.10.1.1.2.2.2.3.cmml"><munderover id="S2.E5a.m1.10.10.1.1.2.2.2.3a" xref="S2.E5a.m1.10.10.1.1.2.2.2.3.cmml"><mo id="S2.E5a.m1.10.10.1.1.2.2.2.3.2.2" movablelimits="false" xref="S2.E5a.m1.10.10.1.1.2.2.2.3.2.2.cmml">∑</mo><mrow id="S2.E5a.m1.10.10.1.1.2.2.2.3.2.3" xref="S2.E5a.m1.10.10.1.1.2.2.2.3.2.3.cmml"><mi id="S2.E5a.m1.10.10.1.1.2.2.2.3.2.3.2" xref="S2.E5a.m1.10.10.1.1.2.2.2.3.2.3.2.cmml">i</mi><mo id="S2.E5a.m1.10.10.1.1.2.2.2.3.2.3.1" xref="S2.E5a.m1.10.10.1.1.2.2.2.3.2.3.1.cmml">=</mo><mn id="S2.E5a.m1.10.10.1.1.2.2.2.3.2.3.3" xref="S2.E5a.m1.10.10.1.1.2.2.2.3.2.3.3.cmml">1</mn></mrow><mi id="S2.E5a.m1.10.10.1.1.2.2.2.3.3" xref="S2.E5a.m1.10.10.1.1.2.2.2.3.3.cmml">n</mi></munderover></mstyle><mrow id="S2.E5a.m1.10.10.1.1.2.2.2.2" xref="S2.E5a.m1.10.10.1.1.2.2.2.2.cmml"><msup id="S2.E5a.m1.10.10.1.1.1.1.1.1.1" xref="S2.E5a.m1.10.10.1.1.1.1.1.1.1.cmml"><mrow id="S2.E5a.m1.10.10.1.1.1.1.1.1.1.1.1" xref="S2.E5a.m1.10.10.1.1.1.1.1.1.1.1.1.1.cmml"><mo id="S2.E5a.m1.10.10.1.1.1.1.1.1.1.1.1.2" stretchy="false" xref="S2.E5a.m1.10.10.1.1.1.1.1.1.1.1.1.1.cmml">(</mo><mrow id="S2.E5a.m1.10.10.1.1.1.1.1.1.1.1.1.1" xref="S2.E5a.m1.10.10.1.1.1.1.1.1.1.1.1.1.cmml"><msub id="S2.E5a.m1.10.10.1.1.1.1.1.1.1.1.1.1.2" xref="S2.E5a.m1.10.10.1.1.1.1.1.1.1.1.1.1.2.cmml"><mover accent="true" id="S2.E5a.m1.10.10.1.1.1.1.1.1.1.1.1.1.2.2" xref="S2.E5a.m1.10.10.1.1.1.1.1.1.1.1.1.1.2.2.cmml"><mi id="S2.E5a.m1.10.10.1.1.1.1.1.1.1.1.1.1.2.2.2" xref="S2.E5a.m1.10.10.1.1.1.1.1.1.1.1.1.1.2.2.2.cmml">𝐲</mi><mo 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}},\operatorname{{\bm{\theta}}}}(\tilde{\operatorname{\mathbf{y}}})=-\frac{N}{% 2}\log 2\pi\tau^{2}-\frac{1}{2\tau^{2}}\sum_{i=1}^{n}(\tilde{\operatorname{% \mathbf{y}}}_{i}-\operatorname{\mathbf{Z}}_{i(2)}\operatorname{\mathbf{L}}_{(2% )}\operatorname{\mathbf{c}}_{i(2)})^{\top}(\tilde{\operatorname{\mathbf{y}}}_{% i}-\operatorname{\mathbf{Z}}_{i(2)}\operatorname{\mathbf{L}}_{(2)}% \operatorname{\mathbf{c}}_{i(2)}),</annotation><annotation encoding="application/x-llamapun" id="S2.E5a.m1.10d">roman_log italic_f start_POSTSUBSCRIPT over~ start_ARG bold_y end_ARG | bold_c , bold_italic_θ end_POSTSUBSCRIPT ( over~ start_ARG bold_y end_ARG ) = - divide start_ARG italic_N end_ARG start_ARG 2 end_ARG roman_log 2 italic_π italic_τ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT - divide start_ARG 1 end_ARG start_ARG 2 italic_τ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG ∑ start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT ( over~ start_ARG bold_y end_ARG start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT - bold_Z start_POSTSUBSCRIPT italic_i ( 2 ) end_POSTSUBSCRIPT bold_L start_POSTSUBSCRIPT ( 2 ) end_POSTSUBSCRIPT bold_c start_POSTSUBSCRIPT italic_i ( 2 ) end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT ⊤ end_POSTSUPERSCRIPT ( over~ start_ARG bold_y end_ARG start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT - bold_Z start_POSTSUBSCRIPT italic_i ( 2 ) end_POSTSUBSCRIPT bold_L start_POSTSUBSCRIPT ( 2 ) end_POSTSUBSCRIPT bold_c start_POSTSUBSCRIPT italic_i ( 2 ) 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">(5)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S2.SS2a.p2.12">where, as defined in Section 2.1 of the main manuscript, <math alttext="\operatorname{\mathbf{Z}}_{i(2)}" class="ltx_Math" display="inline" id="S2.SS2a.p2.9.m1.1"><semantics id="S2.SS2a.p2.9.m1.1a"><msub id="S2.SS2a.p2.9.m1.1.2" xref="S2.SS2a.p2.9.m1.1.2.cmml"><mi id="S2.SS2a.p2.9.m1.1.2.2" xref="S2.SS2a.p2.9.m1.1.2.2.cmml">𝐙</mi><mrow id="S2.SS2a.p2.9.m1.1.1.1" xref="S2.SS2a.p2.9.m1.1.1.1.cmml"><mi id="S2.SS2a.p2.9.m1.1.1.1.3" xref="S2.SS2a.p2.9.m1.1.1.1.3.cmml">i</mi><mo id="S2.SS2a.p2.9.m1.1.1.1.2" xref="S2.SS2a.p2.9.m1.1.1.1.2.cmml">⁢</mo><mrow id="S2.SS2a.p2.9.m1.1.1.1.4.2" xref="S2.SS2a.p2.9.m1.1.1.1.cmml"><mo id="S2.SS2a.p2.9.m1.1.1.1.4.2.1" stretchy="false" xref="S2.SS2a.p2.9.m1.1.1.1.cmml">(</mo><mn id="S2.SS2a.p2.9.m1.1.1.1.1" xref="S2.SS2a.p2.9.m1.1.1.1.1.cmml">2</mn><mo id="S2.SS2a.p2.9.m1.1.1.1.4.2.2" stretchy="false" xref="S2.SS2a.p2.9.m1.1.1.1.cmml">)</mo></mrow></mrow></msub><annotation-xml encoding="MathML-Content" id="S2.SS2a.p2.9.m1.1b"><apply id="S2.SS2a.p2.9.m1.1.2.cmml" xref="S2.SS2a.p2.9.m1.1.2"><csymbol cd="ambiguous" id="S2.SS2a.p2.9.m1.1.2.1.cmml" xref="S2.SS2a.p2.9.m1.1.2">subscript</csymbol><ci id="S2.SS2a.p2.9.m1.1.2.2.cmml" xref="S2.SS2a.p2.9.m1.1.2.2">𝐙</ci><apply id="S2.SS2a.p2.9.m1.1.1.1.cmml" xref="S2.SS2a.p2.9.m1.1.1.1"><times id="S2.SS2a.p2.9.m1.1.1.1.2.cmml" xref="S2.SS2a.p2.9.m1.1.1.1.2"></times><ci id="S2.SS2a.p2.9.m1.1.1.1.3.cmml" xref="S2.SS2a.p2.9.m1.1.1.1.3">𝑖</ci><cn id="S2.SS2a.p2.9.m1.1.1.1.1.cmml" type="integer" xref="S2.SS2a.p2.9.m1.1.1.1.1">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2a.p2.9.m1.1c">\operatorname{\mathbf{Z}}_{i(2)}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2a.p2.9.m1.1d">bold_Z start_POSTSUBSCRIPT italic_i ( 2 ) end_POSTSUBSCRIPT</annotation></semantics></math> is the matrix whose <math alttext="j" class="ltx_Math" display="inline" id="S2.SS2a.p2.10.m2.1"><semantics id="S2.SS2a.p2.10.m2.1a"><mi id="S2.SS2a.p2.10.m2.1.1" xref="S2.SS2a.p2.10.m2.1.1.cmml">j</mi><annotation-xml encoding="MathML-Content" id="S2.SS2a.p2.10.m2.1b"><ci id="S2.SS2a.p2.10.m2.1.1.cmml" 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xref="S2.SS2a.p2.11.m3.5.5.2.2.3.3.3">2</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2a.p2.11.m3.5c">\operatorname{\mathbf{z}}_{ij(2)}=\text{vec}(\operatorname{\mathbf{Z}}_{ij(2)}% )=\text{vec}(\operatorname{\mathbf{Z}}_{ij}^{\top}),\operatorname{\mathbf{L}}_% {(2)}=\operatorname{\mathbf{L}}_{1}\otimes\operatorname{\mathbf{L}}_{2}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2a.p2.11.m3.5d">bold_z start_POSTSUBSCRIPT italic_i italic_j ( 2 ) end_POSTSUBSCRIPT = vec ( bold_Z start_POSTSUBSCRIPT italic_i italic_j ( 2 ) end_POSTSUBSCRIPT ) = vec ( bold_Z start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ⊤ end_POSTSUPERSCRIPT ) , bold_L start_POSTSUBSCRIPT ( 2 ) end_POSTSUBSCRIPT = bold_L start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ⊗ bold_L start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math> and <math 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id="S2.SS2a.p2.12.m4.2c">\operatorname{\mathbf{c}}_{i(2)}=\text{vec}(\operatorname{\mathbf{C}}_{i}^{% \top})</annotation><annotation encoding="application/x-llamapun" id="S2.SS2a.p2.12.m4.2d">bold_c start_POSTSUBSCRIPT italic_i ( 2 ) end_POSTSUBSCRIPT = vec ( bold_C start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ⊤ end_POSTSUPERSCRIPT )</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S2.SS2a.p3"> <p class="ltx_p" id="S2.SS2a.p3.2">We derive the loss function in the E step of the second cycle. For <math alttext="i=1\ldots n" class="ltx_Math" display="inline" id="S2.SS2a.p3.1.m1.1"><semantics id="S2.SS2a.p3.1.m1.1a"><mrow id="S2.SS2a.p3.1.m1.1.1" xref="S2.SS2a.p3.1.m1.1.1.cmml"><mi id="S2.SS2a.p3.1.m1.1.1.2" xref="S2.SS2a.p3.1.m1.1.1.2.cmml">i</mi><mo id="S2.SS2a.p3.1.m1.1.1.1" xref="S2.SS2a.p3.1.m1.1.1.1.cmml">=</mo><mrow id="S2.SS2a.p3.1.m1.1.1.3" xref="S2.SS2a.p3.1.m1.1.1.3.cmml"><mn id="S2.SS2a.p3.1.m1.1.1.3.2" xref="S2.SS2a.p3.1.m1.1.1.3.2.cmml">1</mn><mo id="S2.SS2a.p3.1.m1.1.1.3.1" xref="S2.SS2a.p3.1.m1.1.1.3.1.cmml">⁢</mo><mi id="S2.SS2a.p3.1.m1.1.1.3.3" mathvariant="normal" xref="S2.SS2a.p3.1.m1.1.1.3.3.cmml">…</mi><mo id="S2.SS2a.p3.1.m1.1.1.3.1a" xref="S2.SS2a.p3.1.m1.1.1.3.1.cmml">⁢</mo><mi id="S2.SS2a.p3.1.m1.1.1.3.4" xref="S2.SS2a.p3.1.m1.1.1.3.4.cmml">n</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2a.p3.1.m1.1b"><apply id="S2.SS2a.p3.1.m1.1.1.cmml" xref="S2.SS2a.p3.1.m1.1.1"><eq id="S2.SS2a.p3.1.m1.1.1.1.cmml" xref="S2.SS2a.p3.1.m1.1.1.1"></eq><ci id="S2.SS2a.p3.1.m1.1.1.2.cmml" xref="S2.SS2a.p3.1.m1.1.1.2">𝑖</ci><apply id="S2.SS2a.p3.1.m1.1.1.3.cmml" xref="S2.SS2a.p3.1.m1.1.1.3"><times id="S2.SS2a.p3.1.m1.1.1.3.1.cmml" xref="S2.SS2a.p3.1.m1.1.1.3.1"></times><cn id="S2.SS2a.p3.1.m1.1.1.3.2.cmml" type="integer" xref="S2.SS2a.p3.1.m1.1.1.3.2">1</cn><ci id="S2.SS2a.p3.1.m1.1.1.3.3.cmml" xref="S2.SS2a.p3.1.m1.1.1.3.3">…</ci><ci id="S2.SS2a.p3.1.m1.1.1.3.4.cmml" xref="S2.SS2a.p3.1.m1.1.1.3.4">𝑛</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2a.p3.1.m1.1c">i=1\ldots n</annotation><annotation encoding="application/x-llamapun" id="S2.SS2a.p3.1.m1.1d">italic_i = 1 … italic_n</annotation></semantics></math>, the joint distribution of <math alttext="(\tilde{\operatorname{\mathbf{y}}}_{i},\operatorname{\mathbf{c}}_{i(2)})" class="ltx_Math" display="inline" id="S2.SS2a.p3.2.m2.3"><semantics id="S2.SS2a.p3.2.m2.3a"><mrow id="S2.SS2a.p3.2.m2.3.3.2" 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xref="S2.SS2a.p3.2.m2.2.2.1.1.2.2">𝐲</ci></apply><ci id="S2.SS2a.p3.2.m2.2.2.1.1.3.cmml" xref="S2.SS2a.p3.2.m2.2.2.1.1.3">𝑖</ci></apply><apply id="S2.SS2a.p3.2.m2.3.3.2.2.cmml" xref="S2.SS2a.p3.2.m2.3.3.2.2"><csymbol cd="ambiguous" id="S2.SS2a.p3.2.m2.3.3.2.2.1.cmml" xref="S2.SS2a.p3.2.m2.3.3.2.2">subscript</csymbol><ci id="S2.SS2a.p3.2.m2.3.3.2.2.2.cmml" xref="S2.SS2a.p3.2.m2.3.3.2.2.2">𝐜</ci><apply id="S2.SS2a.p3.2.m2.1.1.1.cmml" xref="S2.SS2a.p3.2.m2.1.1.1"><times id="S2.SS2a.p3.2.m2.1.1.1.2.cmml" xref="S2.SS2a.p3.2.m2.1.1.1.2"></times><ci id="S2.SS2a.p3.2.m2.1.1.1.3.cmml" xref="S2.SS2a.p3.2.m2.1.1.1.3">𝑖</ci><cn id="S2.SS2a.p3.2.m2.1.1.1.1.cmml" type="integer" xref="S2.SS2a.p3.2.m2.1.1.1.1">2</cn></apply></apply></interval></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2a.p3.2.m2.3c">(\tilde{\operatorname{\mathbf{y}}}_{i},\operatorname{\mathbf{c}}_{i(2)})</annotation><annotation encoding="application/x-llamapun" id="S2.SS2a.p3.2.m2.3d">( over~ start_ARG bold_y end_ARG start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , bold_c start_POSTSUBSCRIPT italic_i ( 2 ) end_POSTSUBSCRIPT )</annotation></semantics></math> is</p> <table class="ltx_equation ltx_eqn_table" id="S2.Ex17"> <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="\begin{bmatrix}\tilde{\operatorname{\mathbf{y}}}_{i}\vspace{1mm}\\ \operatorname{\mathbf{c}}_{i(2)}\end{bmatrix}\sim N_{m_{i}+S_{1}S_{2}}\left(% \begin{bmatrix}\mathbf{0}\\ \mathbf{0}\end{bmatrix},\tau^{2}\begin{bmatrix}\operatorname{\mathbf{Z}}_{i(2)% }\operatorname{\mathbf{L}}_{(2)}\operatorname{\mathbf{L}}_{(2)}^{\top}% \operatorname{\mathbf{Z}}_{i(2)}^{\top}+\operatorname{\mathbf{I}}_{m_{i}}&% \operatorname{\mathbf{Z}}_{i(2)}\operatorname{\mathbf{L}}_{(2)}\vspace{1mm}\\ \operatorname{\mathbf{L}}_{(2)}^{\top}\operatorname{\mathbf{Z}}_{i(2)}^{\top}&% \operatorname{\mathbf{I}}_{S_{1}S_{2}}\end{bmatrix}\right)." class="ltx_Math" display="block" id="S2.Ex17.m1.4"><semantics id="S2.Ex17.m1.4a"><mrow id="S2.Ex17.m1.4.4.1" xref="S2.Ex17.m1.4.4.1.1.cmml"><mrow id="S2.Ex17.m1.4.4.1.1" xref="S2.Ex17.m1.4.4.1.1.cmml"><mrow id="S2.Ex17.m1.1.1.3" xref="S2.Ex17.m1.1.1.2.cmml"><mo id="S2.Ex17.m1.1.1.3.1" xref="S2.Ex17.m1.1.1.2.1.cmml">[</mo><mtable displaystyle="true" id="S2.Ex17.m1.1.1.1.1" rowspacing="0pt" xref="S2.Ex17.m1.1.1.1.1.cmml"><mtr id="S2.Ex17.m1.1.1.1.1a" xref="S2.Ex17.m1.1.1.1.1.cmml"><mtd id="S2.Ex17.m1.1.1.1.1b" xref="S2.Ex17.m1.1.1.1.1.cmml"><msub id="S2.Ex17.m1.1.1.1.1.2.1.1" xref="S2.Ex17.m1.1.1.1.1.2.1.1.cmml"><mover accent="true" id="S2.Ex17.m1.1.1.1.1.2.1.1.2" xref="S2.Ex17.m1.1.1.1.1.2.1.1.2.cmml"><mi id="S2.Ex17.m1.1.1.1.1.2.1.1.2.2" xref="S2.Ex17.m1.1.1.1.1.2.1.1.2.2.cmml">𝐲</mi><mo id="S2.Ex17.m1.1.1.1.1.2.1.1.2.1" xref="S2.Ex17.m1.1.1.1.1.2.1.1.2.1.cmml">~</mo></mover><mi id="S2.Ex17.m1.1.1.1.1.2.1.1.3" xref="S2.Ex17.m1.1.1.1.1.2.1.1.3.cmml">i</mi></msub></mtd></mtr><mtr id="S2.Ex17.m1.1.1.1.1c" xref="S2.Ex17.m1.1.1.1.1.cmml"><mtd id="S2.Ex17.m1.1.1.1.1d" xref="S2.Ex17.m1.1.1.1.1.cmml"><msub id="S2.Ex17.m1.1.1.1.1.1.1.1.1" xref="S2.Ex17.m1.1.1.1.1.1.1.1.1.cmml"><mi id="S2.Ex17.m1.1.1.1.1.1.1.1.1.3" xref="S2.Ex17.m1.1.1.1.1.1.1.1.1.3.cmml">𝐜</mi><mrow id="S2.Ex17.m1.1.1.1.1.1.1.1.1.1.1" xref="S2.Ex17.m1.1.1.1.1.1.1.1.1.1.1.cmml"><mi id="S2.Ex17.m1.1.1.1.1.1.1.1.1.1.1.3" xref="S2.Ex17.m1.1.1.1.1.1.1.1.1.1.1.3.cmml">i</mi><mo id="S2.Ex17.m1.1.1.1.1.1.1.1.1.1.1.2" xref="S2.Ex17.m1.1.1.1.1.1.1.1.1.1.1.2.cmml">⁢</mo><mrow id="S2.Ex17.m1.1.1.1.1.1.1.1.1.1.1.4.2" xref="S2.Ex17.m1.1.1.1.1.1.1.1.1.1.1.cmml"><mo id="S2.Ex17.m1.1.1.1.1.1.1.1.1.1.1.4.2.1" stretchy="false" xref="S2.Ex17.m1.1.1.1.1.1.1.1.1.1.1.cmml">(</mo><mn id="S2.Ex17.m1.1.1.1.1.1.1.1.1.1.1.1" xref="S2.Ex17.m1.1.1.1.1.1.1.1.1.1.1.1.cmml">2</mn><mo id="S2.Ex17.m1.1.1.1.1.1.1.1.1.1.1.4.2.2" stretchy="false" 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id="S2.Ex17.m1.3.3.1.1.8.8.3.1.3.3.3.cmml" type="integer" xref="S2.Ex17.m1.3.3.1.1.8.8.3.1.3.3.3">2</cn></apply></apply></apply></matrixrow></matrix></apply></apply></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Ex17.m1.4c">\begin{bmatrix}\tilde{\operatorname{\mathbf{y}}}_{i}\vspace{1mm}\\ \operatorname{\mathbf{c}}_{i(2)}\end{bmatrix}\sim N_{m_{i}+S_{1}S_{2}}\left(% \begin{bmatrix}\mathbf{0}\\ \mathbf{0}\end{bmatrix},\tau^{2}\begin{bmatrix}\operatorname{\mathbf{Z}}_{i(2)% }\operatorname{\mathbf{L}}_{(2)}\operatorname{\mathbf{L}}_{(2)}^{\top}% \operatorname{\mathbf{Z}}_{i(2)}^{\top}+\operatorname{\mathbf{I}}_{m_{i}}&% \operatorname{\mathbf{Z}}_{i(2)}\operatorname{\mathbf{L}}_{(2)}\vspace{1mm}\\ \operatorname{\mathbf{L}}_{(2)}^{\top}\operatorname{\mathbf{Z}}_{i(2)}^{\top}&% \operatorname{\mathbf{I}}_{S_{1}S_{2}}\end{bmatrix}\right).</annotation><annotation encoding="application/x-llamapun" id="S2.Ex17.m1.4d">[ start_ARG start_ROW start_CELL over~ start_ARG bold_y end_ARG start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_CELL end_ROW start_ROW start_CELL bold_c start_POSTSUBSCRIPT italic_i ( 2 ) end_POSTSUBSCRIPT end_CELL end_ROW end_ARG ] ∼ italic_N start_POSTSUBSCRIPT italic_m start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT + italic_S start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_S start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( [ start_ARG start_ROW start_CELL bold_0 end_CELL end_ROW start_ROW start_CELL bold_0 end_CELL end_ROW end_ARG ] , italic_τ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT [ start_ARG start_ROW start_CELL bold_Z start_POSTSUBSCRIPT italic_i ( 2 ) end_POSTSUBSCRIPT bold_L start_POSTSUBSCRIPT ( 2 ) end_POSTSUBSCRIPT bold_L start_POSTSUBSCRIPT ( 2 ) end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ⊤ end_POSTSUPERSCRIPT bold_Z start_POSTSUBSCRIPT italic_i ( 2 ) end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ⊤ end_POSTSUPERSCRIPT + bold_I start_POSTSUBSCRIPT italic_m start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUBSCRIPT end_CELL start_CELL bold_Z start_POSTSUBSCRIPT italic_i ( 2 ) end_POSTSUBSCRIPT bold_L start_POSTSUBSCRIPT ( 2 ) end_POSTSUBSCRIPT end_CELL end_ROW start_ROW start_CELL bold_L start_POSTSUBSCRIPT ( 2 ) end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ⊤ end_POSTSUPERSCRIPT bold_Z start_POSTSUBSCRIPT italic_i ( 2 ) end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ⊤ end_POSTSUPERSCRIPT end_CELL start_CELL bold_I start_POSTSUBSCRIPT italic_S start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_S start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT end_CELL end_ROW end_ARG ] ) .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S2.SS2a.p3.7">Using the analytic form of the conditional distribution of <math alttext="\operatorname{\mathbf{c}}_{i(2)}" class="ltx_Math" display="inline" id="S2.SS2a.p3.3.m1.1"><semantics id="S2.SS2a.p3.3.m1.1a"><msub id="S2.SS2a.p3.3.m1.1.2" xref="S2.SS2a.p3.3.m1.1.2.cmml"><mi id="S2.SS2a.p3.3.m1.1.2.2" xref="S2.SS2a.p3.3.m1.1.2.2.cmml">𝐜</mi><mrow id="S2.SS2a.p3.3.m1.1.1.1" xref="S2.SS2a.p3.3.m1.1.1.1.cmml"><mi id="S2.SS2a.p3.3.m1.1.1.1.3" xref="S2.SS2a.p3.3.m1.1.1.1.3.cmml">i</mi><mo id="S2.SS2a.p3.3.m1.1.1.1.2" xref="S2.SS2a.p3.3.m1.1.1.1.2.cmml">⁢</mo><mrow id="S2.SS2a.p3.3.m1.1.1.1.4.2" xref="S2.SS2a.p3.3.m1.1.1.1.cmml"><mo id="S2.SS2a.p3.3.m1.1.1.1.4.2.1" stretchy="false" xref="S2.SS2a.p3.3.m1.1.1.1.cmml">(</mo><mn id="S2.SS2a.p3.3.m1.1.1.1.1" xref="S2.SS2a.p3.3.m1.1.1.1.1.cmml">2</mn><mo id="S2.SS2a.p3.3.m1.1.1.1.4.2.2" stretchy="false" xref="S2.SS2a.p3.3.m1.1.1.1.cmml">)</mo></mrow></mrow></msub><annotation-xml encoding="MathML-Content" id="S2.SS2a.p3.3.m1.1b"><apply id="S2.SS2a.p3.3.m1.1.2.cmml" xref="S2.SS2a.p3.3.m1.1.2"><csymbol cd="ambiguous" id="S2.SS2a.p3.3.m1.1.2.1.cmml" xref="S2.SS2a.p3.3.m1.1.2">subscript</csymbol><ci id="S2.SS2a.p3.3.m1.1.2.2.cmml" xref="S2.SS2a.p3.3.m1.1.2.2">𝐜</ci><apply id="S2.SS2a.p3.3.m1.1.1.1.cmml" xref="S2.SS2a.p3.3.m1.1.1.1"><times id="S2.SS2a.p3.3.m1.1.1.1.2.cmml" xref="S2.SS2a.p3.3.m1.1.1.1.2"></times><ci id="S2.SS2a.p3.3.m1.1.1.1.3.cmml" xref="S2.SS2a.p3.3.m1.1.1.1.3">𝑖</ci><cn id="S2.SS2a.p3.3.m1.1.1.1.1.cmml" type="integer" xref="S2.SS2a.p3.3.m1.1.1.1.1">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2a.p3.3.m1.1c">\operatorname{\mathbf{c}}_{i(2)}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2a.p3.3.m1.1d">bold_c start_POSTSUBSCRIPT italic_i ( 2 ) end_POSTSUBSCRIPT</annotation></semantics></math> given <math alttext="\tilde{\operatorname{\mathbf{y}}}_{i}" class="ltx_Math" display="inline" id="S2.SS2a.p3.4.m2.1"><semantics id="S2.SS2a.p3.4.m2.1a"><msub id="S2.SS2a.p3.4.m2.1.1" xref="S2.SS2a.p3.4.m2.1.1.cmml"><mover accent="true" id="S2.SS2a.p3.4.m2.1.1.2" xref="S2.SS2a.p3.4.m2.1.1.2.cmml"><mi id="S2.SS2a.p3.4.m2.1.1.2.2" xref="S2.SS2a.p3.4.m2.1.1.2.2.cmml">𝐲</mi><mo id="S2.SS2a.p3.4.m2.1.1.2.1" xref="S2.SS2a.p3.4.m2.1.1.2.1.cmml">~</mo></mover><mi id="S2.SS2a.p3.4.m2.1.1.3" xref="S2.SS2a.p3.4.m2.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S2.SS2a.p3.4.m2.1b"><apply id="S2.SS2a.p3.4.m2.1.1.cmml" xref="S2.SS2a.p3.4.m2.1.1"><csymbol cd="ambiguous" id="S2.SS2a.p3.4.m2.1.1.1.cmml" xref="S2.SS2a.p3.4.m2.1.1">subscript</csymbol><apply id="S2.SS2a.p3.4.m2.1.1.2.cmml" xref="S2.SS2a.p3.4.m2.1.1.2"><ci id="S2.SS2a.p3.4.m2.1.1.2.1.cmml" xref="S2.SS2a.p3.4.m2.1.1.2.1">~</ci><ci id="S2.SS2a.p3.4.m2.1.1.2.2.cmml" xref="S2.SS2a.p3.4.m2.1.1.2.2">𝐲</ci></apply><ci id="S2.SS2a.p3.4.m2.1.1.3.cmml" xref="S2.SS2a.p3.4.m2.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2a.p3.4.m2.1c">\tilde{\operatorname{\mathbf{y}}}_{i}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2a.p3.4.m2.1d">over~ start_ARG bold_y end_ARG start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> in (<a class="ltx_ref" href="https://arxiv.org/html/2503.13782v1#S2.E5a" title="In 2.2 Derivation of 𝐋₁^(𝑡+1) and 𝐋₂^(𝑡+1) ‣ 2 Derivation of the AECM Algorithm Updates ‣ Regularized Parameter Estimation in Mixed Model Trace Regression"><span class="ltx_text ltx_ref_tag">5</span></a>), gives the loss function <math alttext="-\operatorname{\mathcal{Q}}_{(1)}" class="ltx_Math" display="inline" id="S2.SS2a.p3.5.m3.1"><semantics id="S2.SS2a.p3.5.m3.1a"><mrow id="S2.SS2a.p3.5.m3.1.2" xref="S2.SS2a.p3.5.m3.1.2.cmml"><mo id="S2.SS2a.p3.5.m3.1.2a" rspace="0.167em" xref="S2.SS2a.p3.5.m3.1.2.cmml">−</mo><msub id="S2.SS2a.p3.5.m3.1.2.2" xref="S2.SS2a.p3.5.m3.1.2.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS2a.p3.5.m3.1.2.2.2" xref="S2.SS2a.p3.5.m3.1.2.2.2.cmml">𝒬</mi><mrow id="S2.SS2a.p3.5.m3.1.1.1.3" xref="S2.SS2a.p3.5.m3.1.2.2.cmml"><mo id="S2.SS2a.p3.5.m3.1.1.1.3.1" stretchy="false" xref="S2.SS2a.p3.5.m3.1.2.2.cmml">(</mo><mn id="S2.SS2a.p3.5.m3.1.1.1.1" xref="S2.SS2a.p3.5.m3.1.1.1.1.cmml">1</mn><mo id="S2.SS2a.p3.5.m3.1.1.1.3.2" stretchy="false" xref="S2.SS2a.p3.5.m3.1.2.2.cmml">)</mo></mrow></msub></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2a.p3.5.m3.1b"><apply id="S2.SS2a.p3.5.m3.1.2.cmml" xref="S2.SS2a.p3.5.m3.1.2"><minus id="S2.SS2a.p3.5.m3.1.2.1.cmml" xref="S2.SS2a.p3.5.m3.1.2"></minus><apply id="S2.SS2a.p3.5.m3.1.2.2.cmml" xref="S2.SS2a.p3.5.m3.1.2.2"><csymbol cd="ambiguous" id="S2.SS2a.p3.5.m3.1.2.2.1.cmml" xref="S2.SS2a.p3.5.m3.1.2.2">subscript</csymbol><ci id="S2.SS2a.p3.5.m3.1.2.2.2.cmml" xref="S2.SS2a.p3.5.m3.1.2.2.2">𝒬</ci><cn id="S2.SS2a.p3.5.m3.1.1.1.1.cmml" type="integer" xref="S2.SS2a.p3.5.m3.1.1.1.1">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2a.p3.5.m3.1c">-\operatorname{\mathcal{Q}}_{(1)}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2a.p3.5.m3.1d">- caligraphic_Q start_POSTSUBSCRIPT ( 1 ) end_POSTSUBSCRIPT</annotation></semantics></math> for estimating <math alttext="\operatorname{\mathbf{L}}_{1}" class="ltx_Math" display="inline" id="S2.SS2a.p3.6.m4.1"><semantics id="S2.SS2a.p3.6.m4.1a"><msub id="S2.SS2a.p3.6.m4.1.1" xref="S2.SS2a.p3.6.m4.1.1.cmml"><mi id="S2.SS2a.p3.6.m4.1.1.2" xref="S2.SS2a.p3.6.m4.1.1.2.cmml">𝐋</mi><mn id="S2.SS2a.p3.6.m4.1.1.3" xref="S2.SS2a.p3.6.m4.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S2.SS2a.p3.6.m4.1b"><apply id="S2.SS2a.p3.6.m4.1.1.cmml" xref="S2.SS2a.p3.6.m4.1.1"><csymbol cd="ambiguous" id="S2.SS2a.p3.6.m4.1.1.1.cmml" xref="S2.SS2a.p3.6.m4.1.1">subscript</csymbol><ci id="S2.SS2a.p3.6.m4.1.1.2.cmml" xref="S2.SS2a.p3.6.m4.1.1.2">𝐋</ci><cn id="S2.SS2a.p3.6.m4.1.1.3.cmml" type="integer" xref="S2.SS2a.p3.6.m4.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2a.p3.6.m4.1c">\operatorname{\mathbf{L}}_{1}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2a.p3.6.m4.1d">bold_L start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math>. Specifically, the negative of the conditional expectation of the term in (<a class="ltx_ref" href="https://arxiv.org/html/2503.13782v1#S2.E5a" title="In 2.2 Derivation of 𝐋₁^(𝑡+1) and 𝐋₂^(𝑡+1) ‣ 2 Derivation of the AECM Algorithm Updates ‣ Regularized Parameter Estimation in Mixed Model Trace Regression"><span class="ltx_text ltx_ref_tag">5</span></a>) that depends on <math alttext="\operatorname{\mathbf{L}}_{1}" class="ltx_Math" display="inline" id="S2.SS2a.p3.7.m5.1"><semantics id="S2.SS2a.p3.7.m5.1a"><msub id="S2.SS2a.p3.7.m5.1.1" xref="S2.SS2a.p3.7.m5.1.1.cmml"><mi id="S2.SS2a.p3.7.m5.1.1.2" xref="S2.SS2a.p3.7.m5.1.1.2.cmml">𝐋</mi><mn id="S2.SS2a.p3.7.m5.1.1.3" xref="S2.SS2a.p3.7.m5.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S2.SS2a.p3.7.m5.1b"><apply id="S2.SS2a.p3.7.m5.1.1.cmml" xref="S2.SS2a.p3.7.m5.1.1"><csymbol cd="ambiguous" id="S2.SS2a.p3.7.m5.1.1.1.cmml" xref="S2.SS2a.p3.7.m5.1.1">subscript</csymbol><ci id="S2.SS2a.p3.7.m5.1.1.2.cmml" xref="S2.SS2a.p3.7.m5.1.1.2">𝐋</ci><cn id="S2.SS2a.p3.7.m5.1.1.3.cmml" type="integer" xref="S2.SS2a.p3.7.m5.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2a.p3.7.m5.1c">\operatorname{\mathbf{L}}_{1}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2a.p3.7.m5.1d">bold_L start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> is</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="S3.EGx23"> <tbody id="S2.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-\operatorname{\mathcal{Q}}_{(1)}(\operatorname{\mathbf{L}}_{1})=% \operatorname{\mathbb{E}}\left[\sum_{i=1}^{n}(\operatorname{\mathbf{c}}_{i(2)}% ^{\top}\operatorname{\mathbf{L}}_{(2)}^{\top}\operatorname{\mathbf{Z}}_{i(2)}^% {\top}\operatorname{\mathbf{Z}}_{i(2)}\operatorname{\mathbf{L}}_{(2)}% \operatorname{\mathbf{c}}_{i(2)}-2\tilde{\operatorname{\mathbf{y}}}_{i}^{\top}% \operatorname{\mathbf{Z}}_{i(2)}\operatorname{\mathbf{L}}_{(2)}\operatorname{% \mathbf{c}}_{i(2)})|\tilde{\operatorname{\mathbf{y}}},\operatorname{{\bm{% \theta}}}^{(t+1/3)}\right]," class="ltx_Math" display="inline" id="S2.Ex18.m1.14"><semantics id="S2.Ex18.m1.14a"><mrow id="S2.Ex18.m1.14.14.1" xref="S2.Ex18.m1.14.14.1.1.cmml"><mrow id="S2.Ex18.m1.14.14.1.1" xref="S2.Ex18.m1.14.14.1.1.cmml"><mrow id="S2.Ex18.m1.14.14.1.1.2" xref="S2.Ex18.m1.14.14.1.1.2.cmml"><mo id="S2.Ex18.m1.14.14.1.1.2a" rspace="0.167em" xref="S2.Ex18.m1.14.14.1.1.2.cmml">−</mo><mrow id="S2.Ex18.m1.14.14.1.1.2.2.2" xref="S2.Ex18.m1.14.14.1.1.2.2.3.cmml"><msub id="S2.Ex18.m1.14.14.1.1.1.1.1.1" xref="S2.Ex18.m1.14.14.1.1.1.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.Ex18.m1.14.14.1.1.1.1.1.1.2" xref="S2.Ex18.m1.14.14.1.1.1.1.1.1.2.cmml">𝒬</mi><mrow id="S2.Ex18.m1.1.1.1.3" xref="S2.Ex18.m1.14.14.1.1.1.1.1.1.cmml"><mo id="S2.Ex18.m1.1.1.1.3.1" stretchy="false" xref="S2.Ex18.m1.14.14.1.1.1.1.1.1.cmml">(</mo><mn id="S2.Ex18.m1.1.1.1.1" xref="S2.Ex18.m1.1.1.1.1.cmml">1</mn><mo id="S2.Ex18.m1.1.1.1.3.2" stretchy="false" xref="S2.Ex18.m1.14.14.1.1.1.1.1.1.cmml">)</mo></mrow></msub><mo id="S2.Ex18.m1.14.14.1.1.2.2.2a" xref="S2.Ex18.m1.14.14.1.1.2.2.3.cmml">⁡</mo><mrow id="S2.Ex18.m1.14.14.1.1.2.2.2.2" xref="S2.Ex18.m1.14.14.1.1.2.2.3.cmml"><mo id="S2.Ex18.m1.14.14.1.1.2.2.2.2.2" stretchy="false" xref="S2.Ex18.m1.14.14.1.1.2.2.3.cmml">(</mo><msub id="S2.Ex18.m1.14.14.1.1.2.2.2.2.1" xref="S2.Ex18.m1.14.14.1.1.2.2.2.2.1.cmml"><mi id="S2.Ex18.m1.14.14.1.1.2.2.2.2.1.2" xref="S2.Ex18.m1.14.14.1.1.2.2.2.2.1.2.cmml">𝐋</mi><mn id="S2.Ex18.m1.14.14.1.1.2.2.2.2.1.3" xref="S2.Ex18.m1.14.14.1.1.2.2.2.2.1.3.cmml">1</mn></msub><mo id="S2.Ex18.m1.14.14.1.1.2.2.2.2.3" stretchy="false" xref="S2.Ex18.m1.14.14.1.1.2.2.3.cmml">)</mo></mrow></mrow></mrow><mo id="S2.Ex18.m1.14.14.1.1.4" xref="S2.Ex18.m1.14.14.1.1.4.cmml">=</mo><mrow id="S2.Ex18.m1.14.14.1.1.3.1" xref="S2.Ex18.m1.14.14.1.1.3.2.cmml"><mi id="S2.Ex18.m1.13.13" xref="S2.Ex18.m1.13.13.cmml">𝔼</mi><mo id="S2.Ex18.m1.14.14.1.1.3.1a" xref="S2.Ex18.m1.14.14.1.1.3.2.cmml">⁡</mo><mrow id="S2.Ex18.m1.14.14.1.1.3.1.1" xref="S2.Ex18.m1.14.14.1.1.3.2.cmml"><mo id="S2.Ex18.m1.14.14.1.1.3.1.1.2" xref="S2.Ex18.m1.14.14.1.1.3.2.cmml">[</mo><mrow id="S2.Ex18.m1.14.14.1.1.3.1.1.1" xref="S2.Ex18.m1.14.14.1.1.3.1.1.1.cmml"><mrow id="S2.Ex18.m1.14.14.1.1.3.1.1.1.1" xref="S2.Ex18.m1.14.14.1.1.3.1.1.1.1.cmml"><mstyle displaystyle="true" id="S2.Ex18.m1.14.14.1.1.3.1.1.1.1.2" xref="S2.Ex18.m1.14.14.1.1.3.1.1.1.1.2.cmml"><munderover id="S2.Ex18.m1.14.14.1.1.3.1.1.1.1.2a" xref="S2.Ex18.m1.14.14.1.1.3.1.1.1.1.2.cmml"><mo id="S2.Ex18.m1.14.14.1.1.3.1.1.1.1.2.2.2" movablelimits="false" xref="S2.Ex18.m1.14.14.1.1.3.1.1.1.1.2.2.2.cmml">∑</mo><mrow id="S2.Ex18.m1.14.14.1.1.3.1.1.1.1.2.2.3" xref="S2.Ex18.m1.14.14.1.1.3.1.1.1.1.2.2.3.cmml"><mi id="S2.Ex18.m1.14.14.1.1.3.1.1.1.1.2.2.3.2" xref="S2.Ex18.m1.14.14.1.1.3.1.1.1.1.2.2.3.2.cmml">i</mi><mo id="S2.Ex18.m1.14.14.1.1.3.1.1.1.1.2.2.3.1" xref="S2.Ex18.m1.14.14.1.1.3.1.1.1.1.2.2.3.1.cmml">=</mo><mn id="S2.Ex18.m1.14.14.1.1.3.1.1.1.1.2.2.3.3" xref="S2.Ex18.m1.14.14.1.1.3.1.1.1.1.2.2.3.3.cmml">1</mn></mrow><mi id="S2.Ex18.m1.14.14.1.1.3.1.1.1.1.2.3" xref="S2.Ex18.m1.14.14.1.1.3.1.1.1.1.2.3.cmml">n</mi></munderover></mstyle><mrow id="S2.Ex18.m1.14.14.1.1.3.1.1.1.1.1.1" xref="S2.Ex18.m1.14.14.1.1.3.1.1.1.1.1.1.1.cmml"><mo id="S2.Ex18.m1.14.14.1.1.3.1.1.1.1.1.1.2" stretchy="false" xref="S2.Ex18.m1.14.14.1.1.3.1.1.1.1.1.1.1.cmml">(</mo><mrow id="S2.Ex18.m1.14.14.1.1.3.1.1.1.1.1.1.1" xref="S2.Ex18.m1.14.14.1.1.3.1.1.1.1.1.1.1.cmml"><mrow id="S2.Ex18.m1.14.14.1.1.3.1.1.1.1.1.1.1.2" xref="S2.Ex18.m1.14.14.1.1.3.1.1.1.1.1.1.1.2.cmml"><msubsup id="S2.Ex18.m1.14.14.1.1.3.1.1.1.1.1.1.1.2.1" xref="S2.Ex18.m1.14.14.1.1.3.1.1.1.1.1.1.1.2.1.cmml"><mi id="S2.Ex18.m1.14.14.1.1.3.1.1.1.1.1.1.1.2.1.2.2" xref="S2.Ex18.m1.14.14.1.1.3.1.1.1.1.1.1.1.2.1.2.2.cmml">𝐜</mi><mrow id="S2.Ex18.m1.2.2.1" xref="S2.Ex18.m1.2.2.1.cmml"><mi id="S2.Ex18.m1.2.2.1.3" xref="S2.Ex18.m1.2.2.1.3.cmml">i</mi><mo id="S2.Ex18.m1.2.2.1.2" xref="S2.Ex18.m1.2.2.1.2.cmml">⁢</mo><mrow id="S2.Ex18.m1.2.2.1.4.2" xref="S2.Ex18.m1.2.2.1.cmml"><mo id="S2.Ex18.m1.2.2.1.4.2.1" stretchy="false" xref="S2.Ex18.m1.2.2.1.cmml">(</mo><mn id="S2.Ex18.m1.2.2.1.1" xref="S2.Ex18.m1.2.2.1.1.cmml">2</mn><mo id="S2.Ex18.m1.2.2.1.4.2.2" stretchy="false" xref="S2.Ex18.m1.2.2.1.cmml">)</mo></mrow></mrow><mo id="S2.Ex18.m1.14.14.1.1.3.1.1.1.1.1.1.1.2.1.3" xref="S2.Ex18.m1.14.14.1.1.3.1.1.1.1.1.1.1.2.1.3.cmml">⊤</mo></msubsup><mo id="S2.Ex18.m1.14.14.1.1.3.1.1.1.1.1.1.1.2a" lspace="0.167em" xref="S2.Ex18.m1.14.14.1.1.3.1.1.1.1.1.1.1.2.cmml">⁡</mo><mrow id="S2.Ex18.m1.14.14.1.1.3.1.1.1.1.1.1.1.2.2" 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\theta}}}^{(t+1/3)}\right],</annotation><annotation encoding="application/x-llamapun" id="S2.Ex18.m1.14d">- caligraphic_Q start_POSTSUBSCRIPT ( 1 ) end_POSTSUBSCRIPT ( bold_L start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) = blackboard_E [ ∑ start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT ( bold_c start_POSTSUBSCRIPT italic_i ( 2 ) end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ⊤ end_POSTSUPERSCRIPT bold_L start_POSTSUBSCRIPT ( 2 ) end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ⊤ end_POSTSUPERSCRIPT bold_Z start_POSTSUBSCRIPT italic_i ( 2 ) end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ⊤ end_POSTSUPERSCRIPT bold_Z start_POSTSUBSCRIPT italic_i ( 2 ) end_POSTSUBSCRIPT bold_L start_POSTSUBSCRIPT ( 2 ) end_POSTSUBSCRIPT bold_c start_POSTSUBSCRIPT italic_i ( 2 ) end_POSTSUBSCRIPT - 2 over~ start_ARG bold_y end_ARG start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ⊤ end_POSTSUPERSCRIPT bold_Z start_POSTSUBSCRIPT italic_i ( 2 ) end_POSTSUBSCRIPT bold_L start_POSTSUBSCRIPT ( 2 ) end_POSTSUBSCRIPT bold_c start_POSTSUBSCRIPT italic_i ( 2 ) end_POSTSUBSCRIPT ) | over~ start_ARG bold_y end_ARG , bold_italic_θ start_POSTSUPERSCRIPT ( italic_t + 1 / 3 ) end_POSTSUPERSCRIPT ] ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S2.SS2a.p3.11">and maximizing <math alttext="\operatorname{\mathbb{E}}[\log f_{\tilde{\operatorname{\mathbf{y}}}|% \operatorname{\mathbf{c}},\operatorname{{\bm{\theta}}}}(\tilde{\operatorname{% \mathbf{y}}})|\tilde{\operatorname{\mathbf{y}}},\operatorname{{\bm{\theta}}}^{% (t+1/3)}]" class="ltx_Math" display="inline" id="S2.SS2a.p3.8.m1.7"><semantics id="S2.SS2a.p3.8.m1.7a"><mrow id="S2.SS2a.p3.8.m1.7.7.1" xref="S2.SS2a.p3.8.m1.7.7.2.cmml"><mi id="S2.SS2a.p3.8.m1.6.6" xref="S2.SS2a.p3.8.m1.6.6.cmml">𝔼</mi><mo id="S2.SS2a.p3.8.m1.7.7.1a" xref="S2.SS2a.p3.8.m1.7.7.2.cmml">⁡</mo><mrow id="S2.SS2a.p3.8.m1.7.7.1.1" 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xref="S2.SS2a.p3.8.m1.5.5.1">~</ci><ci id="S2.SS2a.p3.8.m1.5.5.2.cmml" xref="S2.SS2a.p3.8.m1.5.5.2">𝐲</ci></apply><apply id="S2.SS2a.p3.8.m1.7.7.1.1.1.1.1.1.cmml" xref="S2.SS2a.p3.8.m1.7.7.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S2.SS2a.p3.8.m1.7.7.1.1.1.1.1.1.1.cmml" xref="S2.SS2a.p3.8.m1.7.7.1.1.1.1.1.1">superscript</csymbol><ci id="S2.SS2a.p3.8.m1.7.7.1.1.1.1.1.1.2.cmml" xref="S2.SS2a.p3.8.m1.7.7.1.1.1.1.1.1.2">𝜽</ci><apply id="S2.SS2a.p3.8.m1.3.3.1.1.1.cmml" xref="S2.SS2a.p3.8.m1.3.3.1.1"><plus id="S2.SS2a.p3.8.m1.3.3.1.1.1.1.cmml" xref="S2.SS2a.p3.8.m1.3.3.1.1.1.1"></plus><ci id="S2.SS2a.p3.8.m1.3.3.1.1.1.2.cmml" xref="S2.SS2a.p3.8.m1.3.3.1.1.1.2">𝑡</ci><apply id="S2.SS2a.p3.8.m1.3.3.1.1.1.3.cmml" xref="S2.SS2a.p3.8.m1.3.3.1.1.1.3"><divide id="S2.SS2a.p3.8.m1.3.3.1.1.1.3.1.cmml" xref="S2.SS2a.p3.8.m1.3.3.1.1.1.3.1"></divide><cn id="S2.SS2a.p3.8.m1.3.3.1.1.1.3.2.cmml" type="integer" xref="S2.SS2a.p3.8.m1.3.3.1.1.1.3.2">1</cn><cn id="S2.SS2a.p3.8.m1.3.3.1.1.1.3.3.cmml" type="integer" xref="S2.SS2a.p3.8.m1.3.3.1.1.1.3.3">3</cn></apply></apply></apply></list></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2a.p3.8.m1.7c">\operatorname{\mathbb{E}}[\log f_{\tilde{\operatorname{\mathbf{y}}}|% \operatorname{\mathbf{c}},\operatorname{{\bm{\theta}}}}(\tilde{\operatorname{% \mathbf{y}}})|\tilde{\operatorname{\mathbf{y}}},\operatorname{{\bm{\theta}}}^{% (t+1/3)}]</annotation><annotation encoding="application/x-llamapun" id="S2.SS2a.p3.8.m1.7d">blackboard_E [ roman_log italic_f start_POSTSUBSCRIPT over~ start_ARG bold_y end_ARG | bold_c , bold_italic_θ end_POSTSUBSCRIPT ( over~ start_ARG bold_y end_ARG ) | over~ start_ARG bold_y end_ARG , bold_italic_θ start_POSTSUPERSCRIPT ( italic_t + 1 / 3 ) end_POSTSUPERSCRIPT ]</annotation></semantics></math> with respect to <math alttext="\operatorname{\mathbf{L}}_{1}" class="ltx_Math" display="inline" id="S2.SS2a.p3.9.m2.1"><semantics id="S2.SS2a.p3.9.m2.1a"><msub id="S2.SS2a.p3.9.m2.1.1" xref="S2.SS2a.p3.9.m2.1.1.cmml"><mi id="S2.SS2a.p3.9.m2.1.1.2" xref="S2.SS2a.p3.9.m2.1.1.2.cmml">𝐋</mi><mn id="S2.SS2a.p3.9.m2.1.1.3" xref="S2.SS2a.p3.9.m2.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S2.SS2a.p3.9.m2.1b"><apply id="S2.SS2a.p3.9.m2.1.1.cmml" xref="S2.SS2a.p3.9.m2.1.1"><csymbol cd="ambiguous" id="S2.SS2a.p3.9.m2.1.1.1.cmml" xref="S2.SS2a.p3.9.m2.1.1">subscript</csymbol><ci id="S2.SS2a.p3.9.m2.1.1.2.cmml" xref="S2.SS2a.p3.9.m2.1.1.2">𝐋</ci><cn id="S2.SS2a.p3.9.m2.1.1.3.cmml" type="integer" xref="S2.SS2a.p3.9.m2.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2a.p3.9.m2.1c">\operatorname{\mathbf{L}}_{1}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2a.p3.9.m2.1d">bold_L start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> amounts to minimizing <math alttext="-\operatorname{\mathcal{Q}}_{(1)}(\operatorname{\mathbf{L}}_{1})" class="ltx_Math" display="inline" id="S2.SS2a.p3.10.m3.3"><semantics id="S2.SS2a.p3.10.m3.3a"><mrow id="S2.SS2a.p3.10.m3.3.3" xref="S2.SS2a.p3.10.m3.3.3.cmml"><mo id="S2.SS2a.p3.10.m3.3.3a" rspace="0.167em" xref="S2.SS2a.p3.10.m3.3.3.cmml">−</mo><mrow id="S2.SS2a.p3.10.m3.3.3.2.2" xref="S2.SS2a.p3.10.m3.3.3.2.3.cmml"><msub id="S2.SS2a.p3.10.m3.2.2.1.1.1" xref="S2.SS2a.p3.10.m3.2.2.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS2a.p3.10.m3.2.2.1.1.1.2" xref="S2.SS2a.p3.10.m3.2.2.1.1.1.2.cmml">𝒬</mi><mrow id="S2.SS2a.p3.10.m3.1.1.1.3" xref="S2.SS2a.p3.10.m3.2.2.1.1.1.cmml"><mo id="S2.SS2a.p3.10.m3.1.1.1.3.1" stretchy="false" xref="S2.SS2a.p3.10.m3.2.2.1.1.1.cmml">(</mo><mn id="S2.SS2a.p3.10.m3.1.1.1.1" xref="S2.SS2a.p3.10.m3.1.1.1.1.cmml">1</mn><mo id="S2.SS2a.p3.10.m3.1.1.1.3.2" stretchy="false" xref="S2.SS2a.p3.10.m3.2.2.1.1.1.cmml">)</mo></mrow></msub><mo id="S2.SS2a.p3.10.m3.3.3.2.2a" xref="S2.SS2a.p3.10.m3.3.3.2.3.cmml">⁡</mo><mrow id="S2.SS2a.p3.10.m3.3.3.2.2.2" xref="S2.SS2a.p3.10.m3.3.3.2.3.cmml"><mo id="S2.SS2a.p3.10.m3.3.3.2.2.2.2" stretchy="false" xref="S2.SS2a.p3.10.m3.3.3.2.3.cmml">(</mo><msub id="S2.SS2a.p3.10.m3.3.3.2.2.2.1" xref="S2.SS2a.p3.10.m3.3.3.2.2.2.1.cmml"><mi id="S2.SS2a.p3.10.m3.3.3.2.2.2.1.2" xref="S2.SS2a.p3.10.m3.3.3.2.2.2.1.2.cmml">𝐋</mi><mn id="S2.SS2a.p3.10.m3.3.3.2.2.2.1.3" xref="S2.SS2a.p3.10.m3.3.3.2.2.2.1.3.cmml">1</mn></msub><mo id="S2.SS2a.p3.10.m3.3.3.2.2.2.3" stretchy="false" xref="S2.SS2a.p3.10.m3.3.3.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2a.p3.10.m3.3b"><apply id="S2.SS2a.p3.10.m3.3.3.cmml" xref="S2.SS2a.p3.10.m3.3.3"><minus id="S2.SS2a.p3.10.m3.3.3.3.cmml" xref="S2.SS2a.p3.10.m3.3.3"></minus><apply id="S2.SS2a.p3.10.m3.3.3.2.3.cmml" xref="S2.SS2a.p3.10.m3.3.3.2.2"><apply id="S2.SS2a.p3.10.m3.2.2.1.1.1.cmml" xref="S2.SS2a.p3.10.m3.2.2.1.1.1"><csymbol cd="ambiguous" id="S2.SS2a.p3.10.m3.2.2.1.1.1.1.cmml" xref="S2.SS2a.p3.10.m3.2.2.1.1.1">subscript</csymbol><ci 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alttext="\operatorname{\mathbf{L}}_{1}" class="ltx_Math" display="inline" id="S2.SS2a.p3.11.m4.1"><semantics id="S2.SS2a.p3.11.m4.1a"><msub id="S2.SS2a.p3.11.m4.1.1" xref="S2.SS2a.p3.11.m4.1.1.cmml"><mi id="S2.SS2a.p3.11.m4.1.1.2" xref="S2.SS2a.p3.11.m4.1.1.2.cmml">𝐋</mi><mn id="S2.SS2a.p3.11.m4.1.1.3" xref="S2.SS2a.p3.11.m4.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S2.SS2a.p3.11.m4.1b"><apply id="S2.SS2a.p3.11.m4.1.1.cmml" xref="S2.SS2a.p3.11.m4.1.1"><csymbol cd="ambiguous" id="S2.SS2a.p3.11.m4.1.1.1.cmml" xref="S2.SS2a.p3.11.m4.1.1">subscript</csymbol><ci id="S2.SS2a.p3.11.m4.1.1.2.cmml" xref="S2.SS2a.p3.11.m4.1.1.2">𝐋</ci><cn id="S2.SS2a.p3.11.m4.1.1.3.cmml" type="integer" xref="S2.SS2a.p3.11.m4.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2a.p3.11.m4.1c">\operatorname{\mathbf{L}}_{1}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2a.p3.11.m4.1d">bold_L start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math>. 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xref="S2.SS2a.p3.12.1.1.1.m1.2.3"><csymbol cd="ambiguous" id="S2.SS2a.p3.12.1.1.1.m1.2.3.2.1.cmml" xref="S2.SS2a.p3.12.1.1.1.m1.2.3">subscript</csymbol><ci id="S2.SS2a.p3.12.1.1.1.m1.2.3.2.2.cmml" xref="S2.SS2a.p3.12.1.1.1.m1.2.3.2.2">𝝁</ci><apply id="S2.SS2a.p3.12.1.1.1.m1.1.1.1.cmml" xref="S2.SS2a.p3.12.1.1.1.m1.1.1.1"><times id="S2.SS2a.p3.12.1.1.1.m1.1.1.1.2.cmml" xref="S2.SS2a.p3.12.1.1.1.m1.1.1.1.2"></times><ci id="S2.SS2a.p3.12.1.1.1.m1.1.1.1.3.cmml" xref="S2.SS2a.p3.12.1.1.1.m1.1.1.1.3">𝑖</ci><cn id="S2.SS2a.p3.12.1.1.1.m1.1.1.1.1.cmml" type="integer" xref="S2.SS2a.p3.12.1.1.1.m1.1.1.1.1">2</cn></apply></apply><ci id="S2.SS2a.p3.12.1.1.1.m1.2.2.1.1.cmml" xref="S2.SS2a.p3.12.1.1.1.m1.2.2.1.1">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2a.p3.12.1.1.1.m1.2c">\operatorname{{\bm{\mu}}}_{i(2)}^{(t)}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2a.p3.12.1.1.1.m1.2d">bold_italic_μ start_POSTSUBSCRIPT italic_i ( 2 ) end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_t ) end_POSTSUPERSCRIPT</annotation></semantics></math></td> <td class="ltx_td ltx_align_center" id="S2.SS2a.p3.13.2.2.2"><math alttext="=" class="ltx_Math" display="inline" id="S2.SS2a.p3.13.2.2.2.m1.1"><semantics id="S2.SS2a.p3.13.2.2.2.m1.1a"><mo id="S2.SS2a.p3.13.2.2.2.m1.1.1" xref="S2.SS2a.p3.13.2.2.2.m1.1.1.cmml">=</mo><annotation-xml encoding="MathML-Content" id="S2.SS2a.p3.13.2.2.2.m1.1b"><eq id="S2.SS2a.p3.13.2.2.2.m1.1.1.cmml" xref="S2.SS2a.p3.13.2.2.2.m1.1.1"></eq></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2a.p3.13.2.2.2.m1.1c">=</annotation><annotation encoding="application/x-llamapun" id="S2.SS2a.p3.13.2.2.2.m1.1d">=</annotation></semantics></math></td> <td class="ltx_td ltx_align_right" id="S2.SS2a.p3.14.3.3.3"><math alttext="\operatorname{\mathbb{E}}(\operatorname{\mathbf{c}}_{i(2)}|\tilde{% \operatorname{\mathbf{y}}}_{i},\operatorname{{\bm{\theta}}}^{(t+1/3)})" class="ltx_Math" display="inline" 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id="S2.SS2a.p3.14.3.3.3.m1.4c">\operatorname{\mathbb{E}}(\operatorname{\mathbf{c}}_{i(2)}|\tilde{% \operatorname{\mathbf{y}}}_{i},\operatorname{{\bm{\theta}}}^{(t+1/3)})</annotation><annotation encoding="application/x-llamapun" id="S2.SS2a.p3.14.3.3.3.m1.4d">blackboard_E ( bold_c start_POSTSUBSCRIPT italic_i ( 2 ) end_POSTSUBSCRIPT | over~ start_ARG bold_y end_ARG start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , bold_italic_θ start_POSTSUPERSCRIPT ( italic_t + 1 / 3 ) end_POSTSUPERSCRIPT )</annotation></semantics></math></td> <td class="ltx_td ltx_align_center" id="S2.SS2a.p3.15.4.4.4"><math alttext="=" class="ltx_Math" display="inline" id="S2.SS2a.p3.15.4.4.4.m1.1"><semantics id="S2.SS2a.p3.15.4.4.4.m1.1a"><mo id="S2.SS2a.p3.15.4.4.4.m1.1.1" xref="S2.SS2a.p3.15.4.4.4.m1.1.1.cmml">=</mo><annotation-xml encoding="MathML-Content" id="S2.SS2a.p3.15.4.4.4.m1.1b"><eq id="S2.SS2a.p3.15.4.4.4.m1.1.1.cmml" xref="S2.SS2a.p3.15.4.4.4.m1.1.1"></eq></annotation-xml><annotation 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id="S2.SS2a.p3.16.5.5.5.m1.9.13.2.2">𝐲</mi><mo id="S2.SS2a.p3.16.5.5.5.m1.9.13.2.1">~</mo></mover><mi id="S2.SS2a.p3.16.5.5.5.m1.9.13.3">i</mi></msub></mrow><annotation encoding="application/x-tex" id="S2.SS2a.p3.16.5.5.5.m1.9c">\operatorname{\mathbf{L}}_{(2)}^{(t)\top}\operatorname{\mathbf{Z}}_{i(2)}^{% \top}(\operatorname{\mathbf{Z}}_{i(2)}\operatorname{\mathbf{L}}_{(2)}^{(t)}% \operatorname{\mathbf{L}}_{(2)}^{(t)\top}\operatorname{\mathbf{Z}}_{i(2)}^{% \top}+\operatorname{\mathbf{I}}_{m_{i}})^{-1}\tilde{\operatorname{\mathbf{y}}}% _{i}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2a.p3.16.5.5.5.m1.9d">bold_L start_POSTSUBSCRIPT ( 2 ) end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_t ) ⊤ end_POSTSUPERSCRIPT bold_Z start_POSTSUBSCRIPT italic_i ( 2 ) end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ⊤ end_POSTSUPERSCRIPT ( bold_Z start_POSTSUBSCRIPT italic_i ( 2 ) end_POSTSUBSCRIPT bold_L start_POSTSUBSCRIPT ( 2 ) end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_t ) 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xref="S2.SS2a.p3.19.8.8.3.m1.2.2.1.1.1.2">𝑡</ci><apply id="S2.SS2a.p3.19.8.8.3.m1.2.2.1.1.1.3.cmml" xref="S2.SS2a.p3.19.8.8.3.m1.2.2.1.1.1.3"><divide id="S2.SS2a.p3.19.8.8.3.m1.2.2.1.1.1.3.1.cmml" xref="S2.SS2a.p3.19.8.8.3.m1.2.2.1.1.1.3.1"></divide><cn id="S2.SS2a.p3.19.8.8.3.m1.2.2.1.1.1.3.2.cmml" type="integer" xref="S2.SS2a.p3.19.8.8.3.m1.2.2.1.1.1.3.2">1</cn><cn id="S2.SS2a.p3.19.8.8.3.m1.2.2.1.1.1.3.3.cmml" type="integer" xref="S2.SS2a.p3.19.8.8.3.m1.2.2.1.1.1.3.3">3</cn></apply></apply></apply></list></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2a.p3.19.8.8.3.m1.3c">\text{Cov}(\operatorname{\mathbf{c}}_{i(2)}|\tilde{\operatorname{\mathbf{y}}}_% {i},\operatorname{{\bm{\theta}}}^{(t+1/3)})</annotation><annotation encoding="application/x-llamapun" id="S2.SS2a.p3.19.8.8.3.m1.3d">Cov ( bold_c start_POSTSUBSCRIPT italic_i ( 2 ) end_POSTSUBSCRIPT | over~ start_ARG bold_y end_ARG start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , bold_italic_θ start_POSTSUPERSCRIPT ( italic_t + 1 / 3 ) end_POSTSUPERSCRIPT )</annotation></semantics></math></td> <td class="ltx_td ltx_align_center" id="S2.SS2a.p3.20.9.9.4"><math alttext="=" class="ltx_Math" display="inline" id="S2.SS2a.p3.20.9.9.4.m1.1"><semantics id="S2.SS2a.p3.20.9.9.4.m1.1a"><mo id="S2.SS2a.p3.20.9.9.4.m1.1.1" xref="S2.SS2a.p3.20.9.9.4.m1.1.1.cmml">=</mo><annotation-xml encoding="MathML-Content" id="S2.SS2a.p3.20.9.9.4.m1.1b"><eq id="S2.SS2a.p3.20.9.9.4.m1.1.1.cmml" xref="S2.SS2a.p3.20.9.9.4.m1.1.1"></eq></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2a.p3.20.9.9.4.m1.1c">=</annotation><annotation encoding="application/x-llamapun" id="S2.SS2a.p3.20.9.9.4.m1.1d">=</annotation></semantics></math></td> <td class="ltx_td ltx_align_left" id="S2.SS2a.p3.21.10.10.5"> <math alttext="\tau^{2(t+1)}(\operatorname{\mathbf{I}}_{S_{1}S_{2}}+\operatorname{\mathbf{L}}% _{(2)}^{(t)\top}\operatorname{\mathbf{Z}}_{i(2)}^{\top}\operatorname{\mathbf{Z% 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italic_t ) ⊤ end_POSTSUPERSCRIPT</annotation></semantics></math>.</td> </tr> </tbody> </table> <p class="ltx_p" id="S2.SS2a.p3.27">Then, we can reexpress <math alttext="-\operatorname{\mathcal{Q}}_{(1)}(\operatorname{\mathbf{L}}_{1})" class="ltx_Math" display="inline" id="S2.SS2a.p3.27.m1.3"><semantics id="S2.SS2a.p3.27.m1.3a"><mrow id="S2.SS2a.p3.27.m1.3.3" xref="S2.SS2a.p3.27.m1.3.3.cmml"><mo id="S2.SS2a.p3.27.m1.3.3a" rspace="0.167em" xref="S2.SS2a.p3.27.m1.3.3.cmml">−</mo><mrow id="S2.SS2a.p3.27.m1.3.3.2.2" xref="S2.SS2a.p3.27.m1.3.3.2.3.cmml"><msub id="S2.SS2a.p3.27.m1.2.2.1.1.1" xref="S2.SS2a.p3.27.m1.2.2.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS2a.p3.27.m1.2.2.1.1.1.2" xref="S2.SS2a.p3.27.m1.2.2.1.1.1.2.cmml">𝒬</mi><mrow id="S2.SS2a.p3.27.m1.1.1.1.3" xref="S2.SS2a.p3.27.m1.2.2.1.1.1.cmml"><mo id="S2.SS2a.p3.27.m1.1.1.1.3.1" stretchy="false" xref="S2.SS2a.p3.27.m1.2.2.1.1.1.cmml">(</mo><mn id="S2.SS2a.p3.27.m1.1.1.1.1" xref="S2.SS2a.p3.27.m1.1.1.1.1.cmml">1</mn><mo id="S2.SS2a.p3.27.m1.1.1.1.3.2" stretchy="false" xref="S2.SS2a.p3.27.m1.2.2.1.1.1.cmml">)</mo></mrow></msub><mo id="S2.SS2a.p3.27.m1.3.3.2.2a" xref="S2.SS2a.p3.27.m1.3.3.2.3.cmml">⁡</mo><mrow id="S2.SS2a.p3.27.m1.3.3.2.2.2" xref="S2.SS2a.p3.27.m1.3.3.2.3.cmml"><mo id="S2.SS2a.p3.27.m1.3.3.2.2.2.2" stretchy="false" xref="S2.SS2a.p3.27.m1.3.3.2.3.cmml">(</mo><msub id="S2.SS2a.p3.27.m1.3.3.2.2.2.1" xref="S2.SS2a.p3.27.m1.3.3.2.2.2.1.cmml"><mi id="S2.SS2a.p3.27.m1.3.3.2.2.2.1.2" xref="S2.SS2a.p3.27.m1.3.3.2.2.2.1.2.cmml">𝐋</mi><mn id="S2.SS2a.p3.27.m1.3.3.2.2.2.1.3" xref="S2.SS2a.p3.27.m1.3.3.2.2.2.1.3.cmml">1</mn></msub><mo id="S2.SS2a.p3.27.m1.3.3.2.2.2.3" stretchy="false" xref="S2.SS2a.p3.27.m1.3.3.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2a.p3.27.m1.3b"><apply id="S2.SS2a.p3.27.m1.3.3.cmml" xref="S2.SS2a.p3.27.m1.3.3"><minus id="S2.SS2a.p3.27.m1.3.3.3.cmml" xref="S2.SS2a.p3.27.m1.3.3"></minus><apply id="S2.SS2a.p3.27.m1.3.3.2.3.cmml" xref="S2.SS2a.p3.27.m1.3.3.2.2"><apply id="S2.SS2a.p3.27.m1.2.2.1.1.1.cmml" xref="S2.SS2a.p3.27.m1.2.2.1.1.1"><csymbol cd="ambiguous" id="S2.SS2a.p3.27.m1.2.2.1.1.1.1.cmml" xref="S2.SS2a.p3.27.m1.2.2.1.1.1">subscript</csymbol><ci id="S2.SS2a.p3.27.m1.2.2.1.1.1.2.cmml" xref="S2.SS2a.p3.27.m1.2.2.1.1.1.2">𝒬</ci><cn id="S2.SS2a.p3.27.m1.1.1.1.1.cmml" type="integer" xref="S2.SS2a.p3.27.m1.1.1.1.1">1</cn></apply><apply id="S2.SS2a.p3.27.m1.3.3.2.2.2.1.cmml" xref="S2.SS2a.p3.27.m1.3.3.2.2.2.1"><csymbol cd="ambiguous" id="S2.SS2a.p3.27.m1.3.3.2.2.2.1.1.cmml" xref="S2.SS2a.p3.27.m1.3.3.2.2.2.1">subscript</csymbol><ci id="S2.SS2a.p3.27.m1.3.3.2.2.2.1.2.cmml" xref="S2.SS2a.p3.27.m1.3.3.2.2.2.1.2">𝐋</ci><cn id="S2.SS2a.p3.27.m1.3.3.2.2.2.1.3.cmml" type="integer" xref="S2.SS2a.p3.27.m1.3.3.2.2.2.1.3">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2a.p3.27.m1.3c">-\operatorname{\mathcal{Q}}_{(1)}(\operatorname{\mathbf{L}}_{1})</annotation><annotation encoding="application/x-llamapun" id="S2.SS2a.p3.27.m1.3d">- caligraphic_Q start_POSTSUBSCRIPT ( 1 ) end_POSTSUBSCRIPT ( bold_L start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT )</annotation></semantics></math> as</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="S3.EGx24"> <tbody id="S2.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-\operatorname{\mathcal{Q}}_{(1)}(\operatorname{\mathbf{L}}_{1})" class="ltx_Math" display="inline" id="S2.Ex19.m1.3"><semantics id="S2.Ex19.m1.3a"><mrow id="S2.Ex19.m1.3.3" xref="S2.Ex19.m1.3.3.cmml"><mo id="S2.Ex19.m1.3.3a" rspace="0.167em" xref="S2.Ex19.m1.3.3.cmml">−</mo><mrow id="S2.Ex19.m1.3.3.2.2" xref="S2.Ex19.m1.3.3.2.3.cmml"><msub id="S2.Ex19.m1.2.2.1.1.1" xref="S2.Ex19.m1.2.2.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.Ex19.m1.2.2.1.1.1.2" xref="S2.Ex19.m1.2.2.1.1.1.2.cmml">𝒬</mi><mrow id="S2.Ex19.m1.1.1.1.3" xref="S2.Ex19.m1.2.2.1.1.1.cmml"><mo id="S2.Ex19.m1.1.1.1.3.1" stretchy="false" xref="S2.Ex19.m1.2.2.1.1.1.cmml">(</mo><mn id="S2.Ex19.m1.1.1.1.1" xref="S2.Ex19.m1.1.1.1.1.cmml">1</mn><mo id="S2.Ex19.m1.1.1.1.3.2" stretchy="false" xref="S2.Ex19.m1.2.2.1.1.1.cmml">)</mo></mrow></msub><mo id="S2.Ex19.m1.3.3.2.2a" xref="S2.Ex19.m1.3.3.2.3.cmml">⁡</mo><mrow id="S2.Ex19.m1.3.3.2.2.2" xref="S2.Ex19.m1.3.3.2.3.cmml"><mo id="S2.Ex19.m1.3.3.2.2.2.2" stretchy="false" xref="S2.Ex19.m1.3.3.2.3.cmml">(</mo><msub id="S2.Ex19.m1.3.3.2.2.2.1" xref="S2.Ex19.m1.3.3.2.2.2.1.cmml"><mi id="S2.Ex19.m1.3.3.2.2.2.1.2" xref="S2.Ex19.m1.3.3.2.2.2.1.2.cmml">𝐋</mi><mn id="S2.Ex19.m1.3.3.2.2.2.1.3" xref="S2.Ex19.m1.3.3.2.2.2.1.3.cmml">1</mn></msub><mo id="S2.Ex19.m1.3.3.2.2.2.3" stretchy="false" xref="S2.Ex19.m1.3.3.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.Ex19.m1.3b"><apply id="S2.Ex19.m1.3.3.cmml" xref="S2.Ex19.m1.3.3"><minus id="S2.Ex19.m1.3.3.3.cmml" xref="S2.Ex19.m1.3.3"></minus><apply id="S2.Ex19.m1.3.3.2.3.cmml" xref="S2.Ex19.m1.3.3.2.2"><apply id="S2.Ex19.m1.2.2.1.1.1.cmml" xref="S2.Ex19.m1.2.2.1.1.1"><csymbol cd="ambiguous" id="S2.Ex19.m1.2.2.1.1.1.1.cmml" xref="S2.Ex19.m1.2.2.1.1.1">subscript</csymbol><ci id="S2.Ex19.m1.2.2.1.1.1.2.cmml" xref="S2.Ex19.m1.2.2.1.1.1.2">𝒬</ci><cn id="S2.Ex19.m1.1.1.1.1.cmml" type="integer" xref="S2.Ex19.m1.1.1.1.1">1</cn></apply><apply id="S2.Ex19.m1.3.3.2.2.2.1.cmml" xref="S2.Ex19.m1.3.3.2.2.2.1"><csymbol cd="ambiguous" id="S2.Ex19.m1.3.3.2.2.2.1.1.cmml" xref="S2.Ex19.m1.3.3.2.2.2.1">subscript</csymbol><ci id="S2.Ex19.m1.3.3.2.2.2.1.2.cmml" xref="S2.Ex19.m1.3.3.2.2.2.1.2">𝐋</ci><cn id="S2.Ex19.m1.3.3.2.2.2.1.3.cmml" type="integer" xref="S2.Ex19.m1.3.3.2.2.2.1.3">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Ex19.m1.3c">\displaystyle-\operatorname{\mathcal{Q}}_{(1)}(\operatorname{\mathbf{L}}_{1})</annotation><annotation encoding="application/x-llamapun" id="S2.Ex19.m1.3d">- caligraphic_Q start_POSTSUBSCRIPT ( 1 ) end_POSTSUBSCRIPT ( bold_L start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT )</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle=\operatorname{\mathbb{E}}\left\{\sum_{i=1}^{n}\left[\text{tr}(% \operatorname{\mathbf{L}}_{(2)}^{\top}\operatorname{\mathbf{Z}}_{i(2)}^{\top}% \operatorname{\mathbf{Z}}_{i(2)}\operatorname{\mathbf{L}}_{(2)}\operatorname{% \mathbf{c}}_{i(2)}\operatorname{\mathbf{c}}_{i(2)}^{\top})-2\tilde{% \operatorname{\mathbf{y}}}_{i}^{\top}\operatorname{\mathbf{Z}}_{i(2)}% \operatorname{\mathbf{L}}_{(2)}\operatorname{\mathbf{c}}_{i(2)}\right]|\tilde{% \operatorname{\mathbf{y}}},\operatorname{{\bm{\theta}}}^{(t+1/3)}\right\}" class="ltx_Math" display="inline" id="S2.Ex19.m2.13"><semantics id="S2.Ex19.m2.13a"><mrow id="S2.Ex19.m2.13.13" xref="S2.Ex19.m2.13.13.cmml"><mi id="S2.Ex19.m2.13.13.3" xref="S2.Ex19.m2.13.13.3.cmml"></mi><mo id="S2.Ex19.m2.13.13.2" xref="S2.Ex19.m2.13.13.2.cmml">=</mo><mrow id="S2.Ex19.m2.13.13.1.1" xref="S2.Ex19.m2.13.13.1.2.cmml"><mi id="S2.Ex19.m2.12.12" xref="S2.Ex19.m2.12.12.cmml">𝔼</mi><mo id="S2.Ex19.m2.13.13.1.1a" xref="S2.Ex19.m2.13.13.1.2.cmml">⁡</mo><mrow id="S2.Ex19.m2.13.13.1.1.1" xref="S2.Ex19.m2.13.13.1.2.cmml"><mo id="S2.Ex19.m2.13.13.1.1.1.2" xref="S2.Ex19.m2.13.13.1.2.cmml">{</mo><mrow id="S2.Ex19.m2.13.13.1.1.1.1" xref="S2.Ex19.m2.13.13.1.1.1.1.cmml"><mrow id="S2.Ex19.m2.13.13.1.1.1.1.1" xref="S2.Ex19.m2.13.13.1.1.1.1.1.cmml"><mstyle displaystyle="true" id="S2.Ex19.m2.13.13.1.1.1.1.1.2" xref="S2.Ex19.m2.13.13.1.1.1.1.1.2.cmml"><munderover id="S2.Ex19.m2.13.13.1.1.1.1.1.2a" 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xref="S2.Ex19.m2.13.13.1.1.1.1.1.1.1.1.1.1.1.1.1.cmml">(</mo><mn id="S2.Ex19.m2.1.1.1.1" xref="S2.Ex19.m2.1.1.1.1.cmml">2</mn><mo id="S2.Ex19.m2.1.1.1.3.2" stretchy="false" xref="S2.Ex19.m2.13.13.1.1.1.1.1.1.1.1.1.1.1.1.1.cmml">)</mo></mrow><mo id="S2.Ex19.m2.13.13.1.1.1.1.1.1.1.1.1.1.1.1.1.3" xref="S2.Ex19.m2.13.13.1.1.1.1.1.1.1.1.1.1.1.1.1.3.cmml">⊤</mo></msubsup><mo id="S2.Ex19.m2.13.13.1.1.1.1.1.1.1.1.1.1.1.1a" lspace="0.167em" xref="S2.Ex19.m2.13.13.1.1.1.1.1.1.1.1.1.1.1.1.cmml">⁡</mo><mrow id="S2.Ex19.m2.13.13.1.1.1.1.1.1.1.1.1.1.1.1.2" xref="S2.Ex19.m2.13.13.1.1.1.1.1.1.1.1.1.1.1.1.2.cmml"><msubsup id="S2.Ex19.m2.13.13.1.1.1.1.1.1.1.1.1.1.1.1.2.1" xref="S2.Ex19.m2.13.13.1.1.1.1.1.1.1.1.1.1.1.1.2.1.cmml"><mi id="S2.Ex19.m2.13.13.1.1.1.1.1.1.1.1.1.1.1.1.2.1.2.2" xref="S2.Ex19.m2.13.13.1.1.1.1.1.1.1.1.1.1.1.1.2.1.2.2.cmml">𝐙</mi><mrow id="S2.Ex19.m2.2.2.1" xref="S2.Ex19.m2.2.2.1.cmml"><mi id="S2.Ex19.m2.2.2.1.3" xref="S2.Ex19.m2.2.2.1.3.cmml">i</mi><mo id="S2.Ex19.m2.2.2.1.2" xref="S2.Ex19.m2.2.2.1.2.cmml">⁢</mo><mrow id="S2.Ex19.m2.2.2.1.4.2" xref="S2.Ex19.m2.2.2.1.cmml"><mo id="S2.Ex19.m2.2.2.1.4.2.1" stretchy="false" xref="S2.Ex19.m2.2.2.1.cmml">(</mo><mn id="S2.Ex19.m2.2.2.1.1" xref="S2.Ex19.m2.2.2.1.1.cmml">2</mn><mo id="S2.Ex19.m2.2.2.1.4.2.2" stretchy="false" xref="S2.Ex19.m2.2.2.1.cmml">)</mo></mrow></mrow><mo id="S2.Ex19.m2.13.13.1.1.1.1.1.1.1.1.1.1.1.1.2.1.3" xref="S2.Ex19.m2.13.13.1.1.1.1.1.1.1.1.1.1.1.1.2.1.3.cmml">⊤</mo></msubsup><mo id="S2.Ex19.m2.13.13.1.1.1.1.1.1.1.1.1.1.1.1.2a" lspace="0.167em" xref="S2.Ex19.m2.13.13.1.1.1.1.1.1.1.1.1.1.1.1.2.cmml">⁡</mo><mrow id="S2.Ex19.m2.13.13.1.1.1.1.1.1.1.1.1.1.1.1.2.2" xref="S2.Ex19.m2.13.13.1.1.1.1.1.1.1.1.1.1.1.1.2.2.cmml"><msub id="S2.Ex19.m2.13.13.1.1.1.1.1.1.1.1.1.1.1.1.2.2.1" xref="S2.Ex19.m2.13.13.1.1.1.1.1.1.1.1.1.1.1.1.2.2.1.cmml"><mi id="S2.Ex19.m2.13.13.1.1.1.1.1.1.1.1.1.1.1.1.2.2.1.2" xref="S2.Ex19.m2.13.13.1.1.1.1.1.1.1.1.1.1.1.1.2.2.1.2.cmml">𝐙</mi><mrow id="S2.Ex19.m2.3.3.1" xref="S2.Ex19.m2.3.3.1.cmml"><mi id="S2.Ex19.m2.3.3.1.3" xref="S2.Ex19.m2.3.3.1.3.cmml">i</mi><mo id="S2.Ex19.m2.3.3.1.2" xref="S2.Ex19.m2.3.3.1.2.cmml">⁢</mo><mrow id="S2.Ex19.m2.3.3.1.4.2" xref="S2.Ex19.m2.3.3.1.cmml"><mo id="S2.Ex19.m2.3.3.1.4.2.1" stretchy="false" xref="S2.Ex19.m2.3.3.1.cmml">(</mo><mn id="S2.Ex19.m2.3.3.1.1" xref="S2.Ex19.m2.3.3.1.1.cmml">2</mn><mo id="S2.Ex19.m2.3.3.1.4.2.2" stretchy="false" xref="S2.Ex19.m2.3.3.1.cmml">)</mo></mrow></mrow></msub><mo id="S2.Ex19.m2.13.13.1.1.1.1.1.1.1.1.1.1.1.1.2.2a" lspace="0.167em" xref="S2.Ex19.m2.13.13.1.1.1.1.1.1.1.1.1.1.1.1.2.2.cmml">⁡</mo><mrow id="S2.Ex19.m2.13.13.1.1.1.1.1.1.1.1.1.1.1.1.2.2.2" xref="S2.Ex19.m2.13.13.1.1.1.1.1.1.1.1.1.1.1.1.2.2.2.cmml"><msub id="S2.Ex19.m2.13.13.1.1.1.1.1.1.1.1.1.1.1.1.2.2.2.1" xref="S2.Ex19.m2.13.13.1.1.1.1.1.1.1.1.1.1.1.1.2.2.2.1.cmml"><mi id="S2.Ex19.m2.13.13.1.1.1.1.1.1.1.1.1.1.1.1.2.2.2.1.2" xref="S2.Ex19.m2.13.13.1.1.1.1.1.1.1.1.1.1.1.1.2.2.2.1.2.cmml">𝐋</mi><mrow id="S2.Ex19.m2.4.4.1.3" xref="S2.Ex19.m2.13.13.1.1.1.1.1.1.1.1.1.1.1.1.2.2.2.1.cmml"><mo id="S2.Ex19.m2.4.4.1.3.1" stretchy="false" xref="S2.Ex19.m2.13.13.1.1.1.1.1.1.1.1.1.1.1.1.2.2.2.1.cmml">(</mo><mn id="S2.Ex19.m2.4.4.1.1" xref="S2.Ex19.m2.4.4.1.1.cmml">2</mn><mo id="S2.Ex19.m2.4.4.1.3.2" stretchy="false" xref="S2.Ex19.m2.13.13.1.1.1.1.1.1.1.1.1.1.1.1.2.2.2.1.cmml">)</mo></mrow></msub><mo id="S2.Ex19.m2.13.13.1.1.1.1.1.1.1.1.1.1.1.1.2.2.2a" lspace="0.167em" xref="S2.Ex19.m2.13.13.1.1.1.1.1.1.1.1.1.1.1.1.2.2.2.cmml">⁡</mo><mrow id="S2.Ex19.m2.13.13.1.1.1.1.1.1.1.1.1.1.1.1.2.2.2.2" xref="S2.Ex19.m2.13.13.1.1.1.1.1.1.1.1.1.1.1.1.2.2.2.2.cmml"><msub id="S2.Ex19.m2.13.13.1.1.1.1.1.1.1.1.1.1.1.1.2.2.2.2.1" xref="S2.Ex19.m2.13.13.1.1.1.1.1.1.1.1.1.1.1.1.2.2.2.2.1.cmml"><mi id="S2.Ex19.m2.13.13.1.1.1.1.1.1.1.1.1.1.1.1.2.2.2.2.1.2" xref="S2.Ex19.m2.13.13.1.1.1.1.1.1.1.1.1.1.1.1.2.2.2.2.1.2.cmml">𝐜</mi><mrow id="S2.Ex19.m2.5.5.1" xref="S2.Ex19.m2.5.5.1.cmml"><mi id="S2.Ex19.m2.5.5.1.3" xref="S2.Ex19.m2.5.5.1.3.cmml">i</mi><mo id="S2.Ex19.m2.5.5.1.2" xref="S2.Ex19.m2.5.5.1.2.cmml">⁢</mo><mrow id="S2.Ex19.m2.5.5.1.4.2" xref="S2.Ex19.m2.5.5.1.cmml"><mo id="S2.Ex19.m2.5.5.1.4.2.1" stretchy="false" xref="S2.Ex19.m2.5.5.1.cmml">(</mo><mn id="S2.Ex19.m2.5.5.1.1" xref="S2.Ex19.m2.5.5.1.1.cmml">2</mn><mo id="S2.Ex19.m2.5.5.1.4.2.2" stretchy="false" xref="S2.Ex19.m2.5.5.1.cmml">)</mo></mrow></mrow></msub><mo id="S2.Ex19.m2.13.13.1.1.1.1.1.1.1.1.1.1.1.1.2.2.2.2a" lspace="0.167em" xref="S2.Ex19.m2.13.13.1.1.1.1.1.1.1.1.1.1.1.1.2.2.2.2.cmml">⁡</mo><msubsup id="S2.Ex19.m2.13.13.1.1.1.1.1.1.1.1.1.1.1.1.2.2.2.2.2" xref="S2.Ex19.m2.13.13.1.1.1.1.1.1.1.1.1.1.1.1.2.2.2.2.2.cmml"><mi id="S2.Ex19.m2.13.13.1.1.1.1.1.1.1.1.1.1.1.1.2.2.2.2.2.2.2" xref="S2.Ex19.m2.13.13.1.1.1.1.1.1.1.1.1.1.1.1.2.2.2.2.2.2.2.cmml">𝐜</mi><mrow id="S2.Ex19.m2.6.6.1" xref="S2.Ex19.m2.6.6.1.cmml"><mi id="S2.Ex19.m2.6.6.1.3" xref="S2.Ex19.m2.6.6.1.3.cmml">i</mi><mo id="S2.Ex19.m2.6.6.1.2" xref="S2.Ex19.m2.6.6.1.2.cmml">⁢</mo><mrow id="S2.Ex19.m2.6.6.1.4.2" xref="S2.Ex19.m2.6.6.1.cmml"><mo id="S2.Ex19.m2.6.6.1.4.2.1" stretchy="false" xref="S2.Ex19.m2.6.6.1.cmml">(</mo><mn id="S2.Ex19.m2.6.6.1.1" xref="S2.Ex19.m2.6.6.1.1.cmml">2</mn><mo id="S2.Ex19.m2.6.6.1.4.2.2" stretchy="false" xref="S2.Ex19.m2.6.6.1.cmml">)</mo></mrow></mrow><mo id="S2.Ex19.m2.13.13.1.1.1.1.1.1.1.1.1.1.1.1.2.2.2.2.2.3" xref="S2.Ex19.m2.13.13.1.1.1.1.1.1.1.1.1.1.1.1.2.2.2.2.2.3.cmml">⊤</mo></msubsup></mrow></mrow></mrow></mrow></mrow><mo id="S2.Ex19.m2.13.13.1.1.1.1.1.1.1.1.1.1.1.3" stretchy="false" xref="S2.Ex19.m2.13.13.1.1.1.1.1.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S2.Ex19.m2.13.13.1.1.1.1.1.1.1.1.2" xref="S2.Ex19.m2.13.13.1.1.1.1.1.1.1.1.2.cmml">−</mo><mrow id="S2.Ex19.m2.13.13.1.1.1.1.1.1.1.1.3" xref="S2.Ex19.m2.13.13.1.1.1.1.1.1.1.1.3.cmml"><mn id="S2.Ex19.m2.13.13.1.1.1.1.1.1.1.1.3.2" xref="S2.Ex19.m2.13.13.1.1.1.1.1.1.1.1.3.2.cmml">2</mn><mo id="S2.Ex19.m2.13.13.1.1.1.1.1.1.1.1.3.1" lspace="0.167em" xref="S2.Ex19.m2.13.13.1.1.1.1.1.1.1.1.3.1.cmml">⁢</mo><msubsup id="S2.Ex19.m2.13.13.1.1.1.1.1.1.1.1.3.3" xref="S2.Ex19.m2.13.13.1.1.1.1.1.1.1.1.3.3.cmml"><mover accent="true" id="S2.Ex19.m2.13.13.1.1.1.1.1.1.1.1.3.3.2.2" xref="S2.Ex19.m2.13.13.1.1.1.1.1.1.1.1.3.3.2.2.cmml"><mi id="S2.Ex19.m2.13.13.1.1.1.1.1.1.1.1.3.3.2.2.2" xref="S2.Ex19.m2.13.13.1.1.1.1.1.1.1.1.3.3.2.2.2.cmml">𝐲</mi><mo id="S2.Ex19.m2.13.13.1.1.1.1.1.1.1.1.3.3.2.2.1" xref="S2.Ex19.m2.13.13.1.1.1.1.1.1.1.1.3.3.2.2.1.cmml">~</mo></mover><mi id="S2.Ex19.m2.13.13.1.1.1.1.1.1.1.1.3.3.2.3" xref="S2.Ex19.m2.13.13.1.1.1.1.1.1.1.1.3.3.2.3.cmml">i</mi><mo id="S2.Ex19.m2.13.13.1.1.1.1.1.1.1.1.3.3.3" xref="S2.Ex19.m2.13.13.1.1.1.1.1.1.1.1.3.3.3.cmml">⊤</mo></msubsup><mo id="S2.Ex19.m2.13.13.1.1.1.1.1.1.1.1.3.1a" lspace="0.167em" xref="S2.Ex19.m2.13.13.1.1.1.1.1.1.1.1.3.1.cmml">⁢</mo><mrow id="S2.Ex19.m2.13.13.1.1.1.1.1.1.1.1.3.4" xref="S2.Ex19.m2.13.13.1.1.1.1.1.1.1.1.3.4.cmml"><msub id="S2.Ex19.m2.13.13.1.1.1.1.1.1.1.1.3.4.1" xref="S2.Ex19.m2.13.13.1.1.1.1.1.1.1.1.3.4.1.cmml"><mi id="S2.Ex19.m2.13.13.1.1.1.1.1.1.1.1.3.4.1.2" xref="S2.Ex19.m2.13.13.1.1.1.1.1.1.1.1.3.4.1.2.cmml">𝐙</mi><mrow id="S2.Ex19.m2.7.7.1" xref="S2.Ex19.m2.7.7.1.cmml"><mi id="S2.Ex19.m2.7.7.1.3" xref="S2.Ex19.m2.7.7.1.3.cmml">i</mi><mo id="S2.Ex19.m2.7.7.1.2" xref="S2.Ex19.m2.7.7.1.2.cmml">⁢</mo><mrow id="S2.Ex19.m2.7.7.1.4.2" xref="S2.Ex19.m2.7.7.1.cmml"><mo id="S2.Ex19.m2.7.7.1.4.2.1" stretchy="false" xref="S2.Ex19.m2.7.7.1.cmml">(</mo><mn id="S2.Ex19.m2.7.7.1.1" xref="S2.Ex19.m2.7.7.1.1.cmml">2</mn><mo id="S2.Ex19.m2.7.7.1.4.2.2" stretchy="false" xref="S2.Ex19.m2.7.7.1.cmml">)</mo></mrow></mrow></msub><mo id="S2.Ex19.m2.13.13.1.1.1.1.1.1.1.1.3.4a" lspace="0.167em" xref="S2.Ex19.m2.13.13.1.1.1.1.1.1.1.1.3.4.cmml">⁡</mo><mrow id="S2.Ex19.m2.13.13.1.1.1.1.1.1.1.1.3.4.2" xref="S2.Ex19.m2.13.13.1.1.1.1.1.1.1.1.3.4.2.cmml"><msub id="S2.Ex19.m2.13.13.1.1.1.1.1.1.1.1.3.4.2.1" xref="S2.Ex19.m2.13.13.1.1.1.1.1.1.1.1.3.4.2.1.cmml"><mi id="S2.Ex19.m2.13.13.1.1.1.1.1.1.1.1.3.4.2.1.2" xref="S2.Ex19.m2.13.13.1.1.1.1.1.1.1.1.3.4.2.1.2.cmml">𝐋</mi><mrow id="S2.Ex19.m2.8.8.1.3" xref="S2.Ex19.m2.13.13.1.1.1.1.1.1.1.1.3.4.2.1.cmml"><mo id="S2.Ex19.m2.8.8.1.3.1" stretchy="false" xref="S2.Ex19.m2.13.13.1.1.1.1.1.1.1.1.3.4.2.1.cmml">(</mo><mn id="S2.Ex19.m2.8.8.1.1" xref="S2.Ex19.m2.8.8.1.1.cmml">2</mn><mo id="S2.Ex19.m2.8.8.1.3.2" stretchy="false" xref="S2.Ex19.m2.13.13.1.1.1.1.1.1.1.1.3.4.2.1.cmml">)</mo></mrow></msub><mo id="S2.Ex19.m2.13.13.1.1.1.1.1.1.1.1.3.4.2a" lspace="0.167em" xref="S2.Ex19.m2.13.13.1.1.1.1.1.1.1.1.3.4.2.cmml">⁡</mo><msub id="S2.Ex19.m2.13.13.1.1.1.1.1.1.1.1.3.4.2.2" xref="S2.Ex19.m2.13.13.1.1.1.1.1.1.1.1.3.4.2.2.cmml"><mi id="S2.Ex19.m2.13.13.1.1.1.1.1.1.1.1.3.4.2.2.2" xref="S2.Ex19.m2.13.13.1.1.1.1.1.1.1.1.3.4.2.2.2.cmml">𝐜</mi><mrow id="S2.Ex19.m2.9.9.1" xref="S2.Ex19.m2.9.9.1.cmml"><mi id="S2.Ex19.m2.9.9.1.3" 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id="S2.Ex19.m2.10.10.1.1.1.3.2.cmml" type="integer" xref="S2.Ex19.m2.10.10.1.1.1.3.2">1</cn><cn id="S2.Ex19.m2.10.10.1.1.1.3.3.cmml" type="integer" xref="S2.Ex19.m2.10.10.1.1.1.3.3">3</cn></apply></apply></apply></list></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Ex19.m2.13c">\displaystyle=\operatorname{\mathbb{E}}\left\{\sum_{i=1}^{n}\left[\text{tr}(% \operatorname{\mathbf{L}}_{(2)}^{\top}\operatorname{\mathbf{Z}}_{i(2)}^{\top}% \operatorname{\mathbf{Z}}_{i(2)}\operatorname{\mathbf{L}}_{(2)}\operatorname{% \mathbf{c}}_{i(2)}\operatorname{\mathbf{c}}_{i(2)}^{\top})-2\tilde{% \operatorname{\mathbf{y}}}_{i}^{\top}\operatorname{\mathbf{Z}}_{i(2)}% \operatorname{\mathbf{L}}_{(2)}\operatorname{\mathbf{c}}_{i(2)}\right]|\tilde{% \operatorname{\mathbf{y}}},\operatorname{{\bm{\theta}}}^{(t+1/3)}\right\}</annotation><annotation encoding="application/x-llamapun" id="S2.Ex19.m2.13d">= blackboard_E { ∑ start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT [ tr ( bold_L start_POSTSUBSCRIPT ( 2 ) end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ⊤ end_POSTSUPERSCRIPT bold_Z start_POSTSUBSCRIPT italic_i ( 2 ) end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ⊤ end_POSTSUPERSCRIPT bold_Z start_POSTSUBSCRIPT italic_i ( 2 ) end_POSTSUBSCRIPT bold_L start_POSTSUBSCRIPT ( 2 ) end_POSTSUBSCRIPT bold_c start_POSTSUBSCRIPT italic_i ( 2 ) end_POSTSUBSCRIPT bold_c start_POSTSUBSCRIPT italic_i ( 2 ) end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ⊤ end_POSTSUPERSCRIPT ) - 2 over~ start_ARG bold_y end_ARG start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ⊤ end_POSTSUPERSCRIPT bold_Z start_POSTSUBSCRIPT italic_i ( 2 ) end_POSTSUBSCRIPT bold_L start_POSTSUBSCRIPT ( 2 ) end_POSTSUBSCRIPT bold_c start_POSTSUBSCRIPT italic_i ( 2 ) end_POSTSUBSCRIPT ] | over~ start_ARG bold_y end_ARG , bold_italic_θ start_POSTSUPERSCRIPT ( italic_t + 1 / 3 ) end_POSTSUPERSCRIPT }</annotation></semantics></math></td> <td 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encoding="application/x-tex" id="S2.Ex20.m1.11c">\displaystyle=\sum_{i=1}^{n}\left[\text{tr}(\operatorname{\mathbf{L}}_{(2)}^{% \top}\operatorname{\mathbf{Z}}_{i(2)}^{\top}\operatorname{\mathbf{Z}}_{i(2)}% \operatorname{\mathbf{L}}_{(2)}\operatorname{{\mathbf{\Gamma}}}_{i(2)}^{(t)})-% 2\tilde{\operatorname{\mathbf{y}}}_{i}^{\top}\operatorname{\mathbf{Z}}_{i(2)}% \operatorname{\mathbf{L}}_{(2)}\operatorname{{\bm{\mu}}}_{i(2)}^{(t)}\right]</annotation><annotation encoding="application/x-llamapun" id="S2.Ex20.m1.11d">= ∑ start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT [ tr ( bold_L start_POSTSUBSCRIPT ( 2 ) end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ⊤ end_POSTSUPERSCRIPT bold_Z start_POSTSUBSCRIPT italic_i ( 2 ) end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ⊤ end_POSTSUPERSCRIPT bold_Z start_POSTSUBSCRIPT italic_i ( 2 ) end_POSTSUBSCRIPT bold_L start_POSTSUBSCRIPT ( 2 ) end_POSTSUBSCRIPT bold_Γ start_POSTSUBSCRIPT italic_i ( 2 ) 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\operatorname{\mathbf{L}}_{(2)}^{\top}\operatorname{\mathbf{Z}}_{i(2)}^{\top})% -2\operatorname{{\bm{\mu}}}_{i(2)}^{(t)\top}\operatorname{\mathbf{L}}_{(2)}^{% \top}\operatorname{\mathbf{Z}}_{i(2)}^{\top}\tilde{\operatorname{\mathbf{y}}}_% {i}\right]." class="ltx_Math" display="inline" id="S2.Ex21.m1.11"><semantics id="S2.Ex21.m1.11a"><mrow id="S2.Ex21.m1.11.11.1" xref="S2.Ex21.m1.11.11.1.1.cmml"><mrow id="S2.Ex21.m1.11.11.1.1" xref="S2.Ex21.m1.11.11.1.1.cmml"><mi id="S2.Ex21.m1.11.11.1.1.3" xref="S2.Ex21.m1.11.11.1.1.3.cmml"></mi><mo id="S2.Ex21.m1.11.11.1.1.2" xref="S2.Ex21.m1.11.11.1.1.2.cmml">=</mo><mrow id="S2.Ex21.m1.11.11.1.1.1" xref="S2.Ex21.m1.11.11.1.1.1.cmml"><mstyle displaystyle="true" id="S2.Ex21.m1.11.11.1.1.1.2" xref="S2.Ex21.m1.11.11.1.1.1.2.cmml"><munderover id="S2.Ex21.m1.11.11.1.1.1.2a" xref="S2.Ex21.m1.11.11.1.1.1.2.cmml"><mo id="S2.Ex21.m1.11.11.1.1.1.2.2.2" movablelimits="false" xref="S2.Ex21.m1.11.11.1.1.1.2.2.2.cmml">∑</mo><mrow id="S2.Ex21.m1.11.11.1.1.1.2.2.3" 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xref="S2.Ex21.m1.11.11.1.1.1.1.1.1.1.1.1.1.cmml"><mo id="S2.Ex21.m1.11.11.1.1.1.1.1.1.1.1.1.2" stretchy="false" xref="S2.Ex21.m1.11.11.1.1.1.1.1.1.1.1.1.1.cmml">(</mo><mrow id="S2.Ex21.m1.11.11.1.1.1.1.1.1.1.1.1.1" xref="S2.Ex21.m1.11.11.1.1.1.1.1.1.1.1.1.1.cmml"><msub id="S2.Ex21.m1.11.11.1.1.1.1.1.1.1.1.1.1.1" xref="S2.Ex21.m1.11.11.1.1.1.1.1.1.1.1.1.1.1.cmml"><mi id="S2.Ex21.m1.11.11.1.1.1.1.1.1.1.1.1.1.1.2" xref="S2.Ex21.m1.11.11.1.1.1.1.1.1.1.1.1.1.1.2.cmml">𝐙</mi><mrow id="S2.Ex21.m1.1.1.1" xref="S2.Ex21.m1.1.1.1.cmml"><mi id="S2.Ex21.m1.1.1.1.3" xref="S2.Ex21.m1.1.1.1.3.cmml">i</mi><mo id="S2.Ex21.m1.1.1.1.2" xref="S2.Ex21.m1.1.1.1.2.cmml">⁢</mo><mrow id="S2.Ex21.m1.1.1.1.4.2" xref="S2.Ex21.m1.1.1.1.cmml"><mo id="S2.Ex21.m1.1.1.1.4.2.1" stretchy="false" xref="S2.Ex21.m1.1.1.1.cmml">(</mo><mn id="S2.Ex21.m1.1.1.1.1" xref="S2.Ex21.m1.1.1.1.1.cmml">2</mn><mo id="S2.Ex21.m1.1.1.1.4.2.2" stretchy="false" xref="S2.Ex21.m1.1.1.1.cmml">)</mo></mrow></mrow></msub><mo id="S2.Ex21.m1.11.11.1.1.1.1.1.1.1.1.1.1a" lspace="0.167em" xref="S2.Ex21.m1.11.11.1.1.1.1.1.1.1.1.1.1.cmml">⁡</mo><mrow id="S2.Ex21.m1.11.11.1.1.1.1.1.1.1.1.1.1.2" xref="S2.Ex21.m1.11.11.1.1.1.1.1.1.1.1.1.1.2.cmml"><msub id="S2.Ex21.m1.11.11.1.1.1.1.1.1.1.1.1.1.2.1" xref="S2.Ex21.m1.11.11.1.1.1.1.1.1.1.1.1.1.2.1.cmml"><mi id="S2.Ex21.m1.11.11.1.1.1.1.1.1.1.1.1.1.2.1.2" xref="S2.Ex21.m1.11.11.1.1.1.1.1.1.1.1.1.1.2.1.2.cmml">𝐋</mi><mrow id="S2.Ex21.m1.2.2.1.3" xref="S2.Ex21.m1.11.11.1.1.1.1.1.1.1.1.1.1.2.1.cmml"><mo id="S2.Ex21.m1.2.2.1.3.1" stretchy="false" xref="S2.Ex21.m1.11.11.1.1.1.1.1.1.1.1.1.1.2.1.cmml">(</mo><mn id="S2.Ex21.m1.2.2.1.1" xref="S2.Ex21.m1.2.2.1.1.cmml">2</mn><mo id="S2.Ex21.m1.2.2.1.3.2" stretchy="false" xref="S2.Ex21.m1.11.11.1.1.1.1.1.1.1.1.1.1.2.1.cmml">)</mo></mrow></msub><mo id="S2.Ex21.m1.11.11.1.1.1.1.1.1.1.1.1.1.2a" lspace="0.167em" xref="S2.Ex21.m1.11.11.1.1.1.1.1.1.1.1.1.1.2.cmml">⁡</mo><mrow id="S2.Ex21.m1.11.11.1.1.1.1.1.1.1.1.1.1.2.2" xref="S2.Ex21.m1.11.11.1.1.1.1.1.1.1.1.1.1.2.2.cmml"><msubsup id="S2.Ex21.m1.11.11.1.1.1.1.1.1.1.1.1.1.2.2.1" xref="S2.Ex21.m1.11.11.1.1.1.1.1.1.1.1.1.1.2.2.1.cmml"><mi id="S2.Ex21.m1.11.11.1.1.1.1.1.1.1.1.1.1.2.2.1.2.2" xref="S2.Ex21.m1.11.11.1.1.1.1.1.1.1.1.1.1.2.2.1.2.2.cmml">𝚪</mi><mrow id="S2.Ex21.m1.3.3.1" xref="S2.Ex21.m1.3.3.1.cmml"><mi id="S2.Ex21.m1.3.3.1.3" xref="S2.Ex21.m1.3.3.1.3.cmml">i</mi><mo id="S2.Ex21.m1.3.3.1.2" xref="S2.Ex21.m1.3.3.1.2.cmml">⁢</mo><mrow id="S2.Ex21.m1.3.3.1.4.2" xref="S2.Ex21.m1.3.3.1.cmml"><mo id="S2.Ex21.m1.3.3.1.4.2.1" stretchy="false" xref="S2.Ex21.m1.3.3.1.cmml">(</mo><mn id="S2.Ex21.m1.3.3.1.1" xref="S2.Ex21.m1.3.3.1.1.cmml">2</mn><mo id="S2.Ex21.m1.3.3.1.4.2.2" stretchy="false" xref="S2.Ex21.m1.3.3.1.cmml">)</mo></mrow></mrow><mrow id="S2.Ex21.m1.4.4.1.3" xref="S2.Ex21.m1.11.11.1.1.1.1.1.1.1.1.1.1.2.2.1.cmml"><mo id="S2.Ex21.m1.4.4.1.3.1" stretchy="false" xref="S2.Ex21.m1.11.11.1.1.1.1.1.1.1.1.1.1.2.2.1.cmml">(</mo><mi id="S2.Ex21.m1.4.4.1.1" xref="S2.Ex21.m1.4.4.1.1.cmml">t</mi><mo id="S2.Ex21.m1.4.4.1.3.2" stretchy="false" xref="S2.Ex21.m1.11.11.1.1.1.1.1.1.1.1.1.1.2.2.1.cmml">)</mo></mrow></msubsup><mo id="S2.Ex21.m1.11.11.1.1.1.1.1.1.1.1.1.1.2.2a" lspace="0.167em" xref="S2.Ex21.m1.11.11.1.1.1.1.1.1.1.1.1.1.2.2.cmml">⁡</mo><mrow id="S2.Ex21.m1.11.11.1.1.1.1.1.1.1.1.1.1.2.2.2" xref="S2.Ex21.m1.11.11.1.1.1.1.1.1.1.1.1.1.2.2.2.cmml"><msubsup id="S2.Ex21.m1.11.11.1.1.1.1.1.1.1.1.1.1.2.2.2.1" xref="S2.Ex21.m1.11.11.1.1.1.1.1.1.1.1.1.1.2.2.2.1.cmml"><mi id="S2.Ex21.m1.11.11.1.1.1.1.1.1.1.1.1.1.2.2.2.1.2.2" xref="S2.Ex21.m1.11.11.1.1.1.1.1.1.1.1.1.1.2.2.2.1.2.2.cmml">𝐋</mi><mrow id="S2.Ex21.m1.5.5.1.3" xref="S2.Ex21.m1.11.11.1.1.1.1.1.1.1.1.1.1.2.2.2.1.cmml"><mo id="S2.Ex21.m1.5.5.1.3.1" stretchy="false" xref="S2.Ex21.m1.11.11.1.1.1.1.1.1.1.1.1.1.2.2.2.1.cmml">(</mo><mn id="S2.Ex21.m1.5.5.1.1" xref="S2.Ex21.m1.5.5.1.1.cmml">2</mn><mo id="S2.Ex21.m1.5.5.1.3.2" stretchy="false" xref="S2.Ex21.m1.11.11.1.1.1.1.1.1.1.1.1.1.2.2.2.1.cmml">)</mo></mrow><mo id="S2.Ex21.m1.11.11.1.1.1.1.1.1.1.1.1.1.2.2.2.1.3" xref="S2.Ex21.m1.11.11.1.1.1.1.1.1.1.1.1.1.2.2.2.1.3.cmml">⊤</mo></msubsup><mo id="S2.Ex21.m1.11.11.1.1.1.1.1.1.1.1.1.1.2.2.2a" lspace="0.167em" xref="S2.Ex21.m1.11.11.1.1.1.1.1.1.1.1.1.1.2.2.2.cmml">⁡</mo><msubsup id="S2.Ex21.m1.11.11.1.1.1.1.1.1.1.1.1.1.2.2.2.2" xref="S2.Ex21.m1.11.11.1.1.1.1.1.1.1.1.1.1.2.2.2.2.cmml"><mi id="S2.Ex21.m1.11.11.1.1.1.1.1.1.1.1.1.1.2.2.2.2.2.2" xref="S2.Ex21.m1.11.11.1.1.1.1.1.1.1.1.1.1.2.2.2.2.2.2.cmml">𝐙</mi><mrow id="S2.Ex21.m1.6.6.1" xref="S2.Ex21.m1.6.6.1.cmml"><mi id="S2.Ex21.m1.6.6.1.3" xref="S2.Ex21.m1.6.6.1.3.cmml">i</mi><mo id="S2.Ex21.m1.6.6.1.2" xref="S2.Ex21.m1.6.6.1.2.cmml">⁢</mo><mrow id="S2.Ex21.m1.6.6.1.4.2" xref="S2.Ex21.m1.6.6.1.cmml"><mo id="S2.Ex21.m1.6.6.1.4.2.1" stretchy="false" xref="S2.Ex21.m1.6.6.1.cmml">(</mo><mn id="S2.Ex21.m1.6.6.1.1" xref="S2.Ex21.m1.6.6.1.1.cmml">2</mn><mo id="S2.Ex21.m1.6.6.1.4.2.2" stretchy="false" xref="S2.Ex21.m1.6.6.1.cmml">)</mo></mrow></mrow><mo id="S2.Ex21.m1.11.11.1.1.1.1.1.1.1.1.1.1.2.2.2.2.3" xref="S2.Ex21.m1.11.11.1.1.1.1.1.1.1.1.1.1.2.2.2.2.3.cmml">⊤</mo></msubsup></mrow></mrow></mrow></mrow><mo id="S2.Ex21.m1.11.11.1.1.1.1.1.1.1.1.1.3" stretchy="false" xref="S2.Ex21.m1.11.11.1.1.1.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S2.Ex21.m1.11.11.1.1.1.1.1.1.2" xref="S2.Ex21.m1.11.11.1.1.1.1.1.1.2.cmml">−</mo><mrow id="S2.Ex21.m1.11.11.1.1.1.1.1.1.3" xref="S2.Ex21.m1.11.11.1.1.1.1.1.1.3.cmml"><mn id="S2.Ex21.m1.11.11.1.1.1.1.1.1.3.2" xref="S2.Ex21.m1.11.11.1.1.1.1.1.1.3.2.cmml">2</mn><mo id="S2.Ex21.m1.11.11.1.1.1.1.1.1.3.1" lspace="0.167em" xref="S2.Ex21.m1.11.11.1.1.1.1.1.1.3.1.cmml">⁢</mo><mrow id="S2.Ex21.m1.11.11.1.1.1.1.1.1.3.3" xref="S2.Ex21.m1.11.11.1.1.1.1.1.1.3.3.cmml"><msubsup id="S2.Ex21.m1.11.11.1.1.1.1.1.1.3.3.1" xref="S2.Ex21.m1.11.11.1.1.1.1.1.1.3.3.1.cmml"><mi id="S2.Ex21.m1.11.11.1.1.1.1.1.1.3.3.1.2.2" xref="S2.Ex21.m1.11.11.1.1.1.1.1.1.3.3.1.2.2.cmml">𝝁</mi><mrow id="S2.Ex21.m1.7.7.1" xref="S2.Ex21.m1.7.7.1.cmml"><mi id="S2.Ex21.m1.7.7.1.3" xref="S2.Ex21.m1.7.7.1.3.cmml">i</mi><mo id="S2.Ex21.m1.7.7.1.2" xref="S2.Ex21.m1.7.7.1.2.cmml">⁢</mo><mrow id="S2.Ex21.m1.7.7.1.4.2" xref="S2.Ex21.m1.7.7.1.cmml"><mo id="S2.Ex21.m1.7.7.1.4.2.1" stretchy="false" xref="S2.Ex21.m1.7.7.1.cmml">(</mo><mn id="S2.Ex21.m1.7.7.1.1" xref="S2.Ex21.m1.7.7.1.1.cmml">2</mn><mo id="S2.Ex21.m1.7.7.1.4.2.2" stretchy="false" xref="S2.Ex21.m1.7.7.1.cmml">)</mo></mrow></mrow><mrow id="S2.Ex21.m1.8.8.1" xref="S2.Ex21.m1.8.8.1.cmml"><mrow id="S2.Ex21.m1.8.8.1.3.2" xref="S2.Ex21.m1.8.8.1.cmml"><mo id="S2.Ex21.m1.8.8.1.3.2.1" stretchy="false" xref="S2.Ex21.m1.8.8.1.cmml">(</mo><mi id="S2.Ex21.m1.8.8.1.1" xref="S2.Ex21.m1.8.8.1.1.cmml">t</mi><mo id="S2.Ex21.m1.8.8.1.3.2.2" stretchy="false" xref="S2.Ex21.m1.8.8.1.cmml">)</mo></mrow><mo id="S2.Ex21.m1.8.8.1.4" lspace="0.222em" 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xref="S2.Ex21.m1.11.11.1.1.1.1.1.1.3.3.2.2.2.2.1">~</ci><ci id="S2.Ex21.m1.11.11.1.1.1.1.1.1.3.3.2.2.2.2.2.cmml" xref="S2.Ex21.m1.11.11.1.1.1.1.1.1.3.3.2.2.2.2.2">𝐲</ci></apply><ci id="S2.Ex21.m1.11.11.1.1.1.1.1.1.3.3.2.2.2.3.cmml" xref="S2.Ex21.m1.11.11.1.1.1.1.1.1.3.3.2.2.2.3">𝑖</ci></apply></apply></apply></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Ex21.m1.11c">\displaystyle=\sum_{i=1}^{n}\left[\text{tr}(\operatorname{\mathbf{Z}}_{i(2)}% \operatorname{\mathbf{L}}_{(2)}\operatorname{{\mathbf{\Gamma}}}_{i(2)}^{(t)}% \operatorname{\mathbf{L}}_{(2)}^{\top}\operatorname{\mathbf{Z}}_{i(2)}^{\top})% -2\operatorname{{\bm{\mu}}}_{i(2)}^{(t)\top}\operatorname{\mathbf{L}}_{(2)}^{% \top}\operatorname{\mathbf{Z}}_{i(2)}^{\top}\tilde{\operatorname{\mathbf{y}}}_% {i}\right].</annotation><annotation encoding="application/x-llamapun" id="S2.Ex21.m1.11d">= ∑ start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT [ tr ( bold_Z start_POSTSUBSCRIPT italic_i ( 2 ) end_POSTSUBSCRIPT bold_L start_POSTSUBSCRIPT ( 2 ) end_POSTSUBSCRIPT bold_Γ start_POSTSUBSCRIPT italic_i ( 2 ) end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_t ) end_POSTSUPERSCRIPT bold_L start_POSTSUBSCRIPT ( 2 ) end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ⊤ end_POSTSUPERSCRIPT bold_Z start_POSTSUBSCRIPT italic_i ( 2 ) end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ⊤ end_POSTSUPERSCRIPT ) - 2 bold_italic_μ start_POSTSUBSCRIPT italic_i ( 2 ) end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_t ) ⊤ end_POSTSUPERSCRIPT bold_L start_POSTSUBSCRIPT ( 2 ) end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ⊤ end_POSTSUPERSCRIPT bold_Z start_POSTSUBSCRIPT italic_i ( 2 ) end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ⊤ end_POSTSUPERSCRIPT over~ start_ARG bold_y end_ARG start_POSTSUBSCRIPT italic_i 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">(6)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S2.SS2a.p3.29">We simplify this expression by reexpressing the first component of this sum, <math alttext="\text{tr}(\operatorname{\mathbf{Z}}_{i(2)}\operatorname{\mathbf{L}}_{(2)}% \operatorname{{\mathbf{\Gamma}}}_{i(2)}^{(t)}\operatorname{\mathbf{L}}_{(2)}^{% \top}\operatorname{\mathbf{Z}}_{i(2)}^{\top})" class="ltx_Math" display="inline" id="S2.SS2a.p3.28.m1.7"><semantics id="S2.SS2a.p3.28.m1.7a"><mrow id="S2.SS2a.p3.28.m1.7.7" xref="S2.SS2a.p3.28.m1.7.7.cmml"><mtext id="S2.SS2a.p3.28.m1.7.7.3" xref="S2.SS2a.p3.28.m1.7.7.3a.cmml">tr</mtext><mo id="S2.SS2a.p3.28.m1.7.7.2" xref="S2.SS2a.p3.28.m1.7.7.2.cmml">⁢</mo><mrow id="S2.SS2a.p3.28.m1.7.7.1.1" xref="S2.SS2a.p3.28.m1.7.7.1.1.1.cmml"><mo id="S2.SS2a.p3.28.m1.7.7.1.1.2" stretchy="false" xref="S2.SS2a.p3.28.m1.7.7.1.1.1.cmml">(</mo><mrow id="S2.SS2a.p3.28.m1.7.7.1.1.1" 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start_POSTSUBSCRIPT italic_i ( 2 ) end_POSTSUBSCRIPT bold_L start_POSTSUBSCRIPT ( 2 ) end_POSTSUBSCRIPT bold_Γ start_POSTSUBSCRIPT italic_i ( 2 ) end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_t ) end_POSTSUPERSCRIPT bold_L start_POSTSUBSCRIPT ( 2 ) end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ⊤ end_POSTSUPERSCRIPT bold_Z start_POSTSUBSCRIPT italic_i ( 2 ) end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ⊤ end_POSTSUPERSCRIPT )</annotation></semantics></math>. If <math alttext="\operatorname{\mathbf{l}}_{1}=\text{vec}(\operatorname{\mathbf{L}}_{1})" class="ltx_Math" display="inline" id="S2.SS2a.p3.29.m2.1"><semantics id="S2.SS2a.p3.29.m2.1a"><mrow id="S2.SS2a.p3.29.m2.1.1" xref="S2.SS2a.p3.29.m2.1.1.cmml"><msub id="S2.SS2a.p3.29.m2.1.1.3" xref="S2.SS2a.p3.29.m2.1.1.3.cmml"><mi id="S2.SS2a.p3.29.m2.1.1.3.2" xref="S2.SS2a.p3.29.m2.1.1.3.2.cmml">𝐥</mi><mn id="S2.SS2a.p3.29.m2.1.1.3.3" xref="S2.SS2a.p3.29.m2.1.1.3.3.cmml">1</mn></msub><mo id="S2.SS2a.p3.29.m2.1.1.2" xref="S2.SS2a.p3.29.m2.1.1.2.cmml">=</mo><mrow id="S2.SS2a.p3.29.m2.1.1.1" xref="S2.SS2a.p3.29.m2.1.1.1.cmml"><mtext id="S2.SS2a.p3.29.m2.1.1.1.3" xref="S2.SS2a.p3.29.m2.1.1.1.3a.cmml">vec</mtext><mo id="S2.SS2a.p3.29.m2.1.1.1.2" xref="S2.SS2a.p3.29.m2.1.1.1.2.cmml">⁢</mo><mrow id="S2.SS2a.p3.29.m2.1.1.1.1.1" xref="S2.SS2a.p3.29.m2.1.1.1.1.1.1.cmml"><mo id="S2.SS2a.p3.29.m2.1.1.1.1.1.2" stretchy="false" xref="S2.SS2a.p3.29.m2.1.1.1.1.1.1.cmml">(</mo><msub 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xref="S2.SS2a.p3.29.m2.1.1.1"><times id="S2.SS2a.p3.29.m2.1.1.1.2.cmml" xref="S2.SS2a.p3.29.m2.1.1.1.2"></times><ci id="S2.SS2a.p3.29.m2.1.1.1.3a.cmml" xref="S2.SS2a.p3.29.m2.1.1.1.3"><mtext id="S2.SS2a.p3.29.m2.1.1.1.3.cmml" xref="S2.SS2a.p3.29.m2.1.1.1.3">vec</mtext></ci><apply id="S2.SS2a.p3.29.m2.1.1.1.1.1.1.cmml" xref="S2.SS2a.p3.29.m2.1.1.1.1.1"><csymbol cd="ambiguous" id="S2.SS2a.p3.29.m2.1.1.1.1.1.1.1.cmml" xref="S2.SS2a.p3.29.m2.1.1.1.1.1">subscript</csymbol><ci id="S2.SS2a.p3.29.m2.1.1.1.1.1.1.2.cmml" xref="S2.SS2a.p3.29.m2.1.1.1.1.1.1.2">𝐋</ci><cn id="S2.SS2a.p3.29.m2.1.1.1.1.1.1.3.cmml" type="integer" xref="S2.SS2a.p3.29.m2.1.1.1.1.1.1.3">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2a.p3.29.m2.1c">\operatorname{\mathbf{l}}_{1}=\text{vec}(\operatorname{\mathbf{L}}_{1})</annotation><annotation encoding="application/x-llamapun" id="S2.SS2a.p3.29.m2.1d">bold_l start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = vec ( bold_L start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT )</annotation></semantics></math>, then</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="S3.EGx25"> <tbody id="S2.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_right ltx_eqn_cell"><math alttext="\displaystyle\operatorname{\mathbf{L}}_{(2)}^{\top}\operatorname{\mathbf{Z}}_{% i(2)}^{\top}" class="ltx_Math" display="inline" id="S2.Ex22.m1.2"><semantics id="S2.Ex22.m1.2a"><mrow id="S2.Ex22.m1.2.3" xref="S2.Ex22.m1.2.3.cmml"><msubsup id="S2.Ex22.m1.2.3.1" xref="S2.Ex22.m1.2.3.1.cmml"><mi id="S2.Ex22.m1.2.3.1.2.2" xref="S2.Ex22.m1.2.3.1.2.2.cmml">𝐋</mi><mrow id="S2.Ex22.m1.1.1.1.3" xref="S2.Ex22.m1.2.3.1.cmml"><mo id="S2.Ex22.m1.1.1.1.3.1" stretchy="false" xref="S2.Ex22.m1.2.3.1.cmml">(</mo><mn id="S2.Ex22.m1.1.1.1.1" xref="S2.Ex22.m1.1.1.1.1.cmml">2</mn><mo id="S2.Ex22.m1.1.1.1.3.2" stretchy="false" xref="S2.Ex22.m1.2.3.1.cmml">)</mo></mrow><mo id="S2.Ex22.m1.2.3.1.3" xref="S2.Ex22.m1.2.3.1.3.cmml">⊤</mo></msubsup><mo id="S2.Ex22.m1.2.3a" lspace="0.167em" xref="S2.Ex22.m1.2.3.cmml">⁡</mo><msubsup id="S2.Ex22.m1.2.3.2" xref="S2.Ex22.m1.2.3.2.cmml"><mi id="S2.Ex22.m1.2.3.2.2.2" xref="S2.Ex22.m1.2.3.2.2.2.cmml">𝐙</mi><mrow id="S2.Ex22.m1.2.2.1" xref="S2.Ex22.m1.2.2.1.cmml"><mi id="S2.Ex22.m1.2.2.1.3" xref="S2.Ex22.m1.2.2.1.3.cmml">i</mi><mo id="S2.Ex22.m1.2.2.1.2" xref="S2.Ex22.m1.2.2.1.2.cmml">⁢</mo><mrow id="S2.Ex22.m1.2.2.1.4.2" xref="S2.Ex22.m1.2.2.1.cmml"><mo id="S2.Ex22.m1.2.2.1.4.2.1" stretchy="false" xref="S2.Ex22.m1.2.2.1.cmml">(</mo><mn id="S2.Ex22.m1.2.2.1.1" xref="S2.Ex22.m1.2.2.1.1.cmml">2</mn><mo id="S2.Ex22.m1.2.2.1.4.2.2" stretchy="false" xref="S2.Ex22.m1.2.2.1.cmml">)</mo></mrow></mrow><mo id="S2.Ex22.m1.2.3.2.3" xref="S2.Ex22.m1.2.3.2.3.cmml">⊤</mo></msubsup></mrow><annotation-xml encoding="MathML-Content" id="S2.Ex22.m1.2b"><apply id="S2.Ex22.m1.2.3.cmml" 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alttext="\displaystyle=\operatorname{\mathbf{L}}_{(2)}^{\top}\begin{bmatrix}% \operatorname{\mathbf{z}}_{i1(2)}&\operatorname{\mathbf{z}}_{i2(2)}&\ldots&% \operatorname{\mathbf{z}}_{im_{i}(2)}\end{bmatrix}" class="ltx_Math" display="inline" id="S2.Ex22.m2.2"><semantics id="S2.Ex22.m2.2a"><mrow id="S2.Ex22.m2.2.3" xref="S2.Ex22.m2.2.3.cmml"><mi id="S2.Ex22.m2.2.3.2" xref="S2.Ex22.m2.2.3.2.cmml"></mi><mo id="S2.Ex22.m2.2.3.1" xref="S2.Ex22.m2.2.3.1.cmml">=</mo><mrow id="S2.Ex22.m2.2.3.3" xref="S2.Ex22.m2.2.3.3.cmml"><msubsup id="S2.Ex22.m2.2.3.3.1" xref="S2.Ex22.m2.2.3.3.1.cmml"><mi id="S2.Ex22.m2.2.3.3.1.2.2" xref="S2.Ex22.m2.2.3.3.1.2.2.cmml">𝐋</mi><mrow id="S2.Ex22.m2.2.2.1.3" xref="S2.Ex22.m2.2.3.3.1.cmml"><mo id="S2.Ex22.m2.2.2.1.3.1" stretchy="false" xref="S2.Ex22.m2.2.3.3.1.cmml">(</mo><mn id="S2.Ex22.m2.2.2.1.1" xref="S2.Ex22.m2.2.2.1.1.cmml">2</mn><mo id="S2.Ex22.m2.2.2.1.3.2" stretchy="false" xref="S2.Ex22.m2.2.3.3.1.cmml">)</mo></mrow><mo id="S2.Ex22.m2.2.3.3.1.3" xref="S2.Ex22.m2.2.3.3.1.3.cmml">⊤</mo></msubsup><mo id="S2.Ex22.m2.2.3.3a" xref="S2.Ex22.m2.2.3.3.cmml">⁡</mo><mrow id="S2.Ex22.m2.1.1a.3" xref="S2.Ex22.m2.1.1a.2.cmml"><mo id="S2.Ex22.m2.1.1a.3.1" xref="S2.Ex22.m2.1.1a.2.1.cmml">[</mo><mtable columnspacing="5pt" id="S2.Ex22.m2.1.1.1.1" xref="S2.Ex22.m2.1.1.1.1.cmml"><mtr id="S2.Ex22.m2.1.1.1.1a" xref="S2.Ex22.m2.1.1.1.1.cmml"><mtd id="S2.Ex22.m2.1.1.1.1b" xref="S2.Ex22.m2.1.1.1.1.cmml"><msub id="S2.Ex22.m2.1.1.1.1.1.1.1.1" xref="S2.Ex22.m2.1.1.1.1.1.1.1.1.cmml"><mi id="S2.Ex22.m2.1.1.1.1.1.1.1.1.3" xref="S2.Ex22.m2.1.1.1.1.1.1.1.1.3.cmml">𝐳</mi><mrow id="S2.Ex22.m2.1.1.1.1.1.1.1.1.1.1" xref="S2.Ex22.m2.1.1.1.1.1.1.1.1.1.1.cmml"><mi id="S2.Ex22.m2.1.1.1.1.1.1.1.1.1.1.3" xref="S2.Ex22.m2.1.1.1.1.1.1.1.1.1.1.3.cmml">i</mi><mo id="S2.Ex22.m2.1.1.1.1.1.1.1.1.1.1.2" xref="S2.Ex22.m2.1.1.1.1.1.1.1.1.1.1.2.cmml">⁢</mo><mn id="S2.Ex22.m2.1.1.1.1.1.1.1.1.1.1.4" xref="S2.Ex22.m2.1.1.1.1.1.1.1.1.1.1.4.cmml">1</mn><mo 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id="S2.Ex23.m1.1c">\displaystyle=\begin{bmatrix}(\operatorname{\mathbf{L}}_{1}^{\top}\otimes% \operatorname{\mathbf{L}}_{2}^{\top})\operatorname{\mathbf{z}}_{i1(2)}&(% \operatorname{\mathbf{L}}_{1}^{\top}\otimes\operatorname{\mathbf{L}}_{2}^{\top% })\operatorname{\mathbf{z}}_{i2(2)}&\ldots&(\operatorname{\mathbf{L}}_{1}^{% \top}\otimes\operatorname{\mathbf{L}}_{2}^{\top})\operatorname{\mathbf{z}}_{im% _{i}(2)}\end{bmatrix}</annotation><annotation encoding="application/x-llamapun" id="S2.Ex23.m1.1d">= [ start_ARG start_ROW start_CELL ( bold_L start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ⊤ end_POSTSUPERSCRIPT ⊗ bold_L start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ⊤ end_POSTSUPERSCRIPT ) bold_z start_POSTSUBSCRIPT italic_i 1 ( 2 ) end_POSTSUBSCRIPT end_CELL start_CELL ( bold_L start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ⊤ end_POSTSUPERSCRIPT ⊗ bold_L start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ⊤ end_POSTSUPERSCRIPT ) bold_z start_POSTSUBSCRIPT italic_i 2 ( 2 ) end_POSTSUBSCRIPT end_CELL start_CELL … end_CELL start_CELL ( bold_L start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ⊤ end_POSTSUPERSCRIPT ⊗ bold_L start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ⊤ end_POSTSUPERSCRIPT ) bold_z start_POSTSUBSCRIPT italic_i italic_m start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( 2 ) end_POSTSUBSCRIPT end_CELL end_ROW end_ARG ]</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="S2.Ex24"><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=\begin{bmatrix}\text{vec}(\operatorname{\mathbf{L}}_{2}^{\top}% \operatorname{\mathbf{Z}}_{i1(2)}\operatorname{\mathbf{L}}_{1})&\text{vec}(% \operatorname{\mathbf{L}}_{2}^{\top}\operatorname{\mathbf{Z}}_{i2(2)}% 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xref="S2.Ex24.m1.1.1.1.1.6.6.6.2.2.1.1.2.2">subscript</csymbol><ci id="S2.Ex24.m1.1.1.1.1.6.6.6.2.2.1.1.2.2.2.cmml" xref="S2.Ex24.m1.1.1.1.1.6.6.6.2.2.1.1.2.2.2">𝐋</ci><cn id="S2.Ex24.m1.1.1.1.1.6.6.6.2.2.1.1.2.2.3.cmml" type="integer" xref="S2.Ex24.m1.1.1.1.1.6.6.6.2.2.1.1.2.2.3">1</cn></apply></apply></apply></apply></matrixrow></matrix></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Ex24.m1.1c">\displaystyle=\begin{bmatrix}\text{vec}(\operatorname{\mathbf{L}}_{2}^{\top}% \operatorname{\mathbf{Z}}_{i1(2)}\operatorname{\mathbf{L}}_{1})&\text{vec}(% \operatorname{\mathbf{L}}_{2}^{\top}\operatorname{\mathbf{Z}}_{i2(2)}% \operatorname{\mathbf{L}}_{1})&\ldots&\text{vec}(\operatorname{\mathbf{L}}_{2}% ^{\top}\operatorname{\mathbf{Z}}_{im_{i}(2)}\operatorname{\mathbf{L}}_{1})\end% {bmatrix}</annotation><annotation encoding="application/x-llamapun" id="S2.Ex24.m1.1d">= [ start_ARG start_ROW start_CELL vec ( bold_L start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ⊤ end_POSTSUPERSCRIPT bold_Z start_POSTSUBSCRIPT italic_i 1 ( 2 ) end_POSTSUBSCRIPT bold_L start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) end_CELL start_CELL vec ( bold_L start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ⊤ end_POSTSUPERSCRIPT bold_Z start_POSTSUBSCRIPT italic_i 2 ( 2 ) end_POSTSUBSCRIPT bold_L start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) end_CELL start_CELL … end_CELL start_CELL vec ( bold_L start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ⊤ end_POSTSUPERSCRIPT bold_Z start_POSTSUBSCRIPT italic_i italic_m start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( 2 ) end_POSTSUBSCRIPT bold_L start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) end_CELL end_ROW end_ARG ]</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="S2.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_eqn_cell"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle=\begin{bmatrix}\text{vec}(\operatorname{\mathbf{L}}_{2}^{\top}% \operatorname{\mathbf{Z}}_{i1(1)}^{\top}\operatorname{\mathbf{L}}_{1})&\text{% vec}(\operatorname{\mathbf{L}}_{2}^{\top}\operatorname{\mathbf{Z}}_{i2(1)}^{% \top}\operatorname{\mathbf{L}}_{1})&\ldots&\text{vec}(\operatorname{\mathbf{L}% }_{2}^{\top}\operatorname{\mathbf{Z}}_{im_{i}(1)}^{\top}\operatorname{\mathbf{% L}}_{1})\end{bmatrix}," class="ltx_Math" display="inline" id="S2.Ex25.m1.2"><semantics id="S2.Ex25.m1.2a"><mrow id="S2.Ex25.m1.2.2.1" xref="S2.Ex25.m1.2.2.1.1.cmml"><mrow id="S2.Ex25.m1.2.2.1.1" xref="S2.Ex25.m1.2.2.1.1.cmml"><mi id="S2.Ex25.m1.2.2.1.1.2" xref="S2.Ex25.m1.2.2.1.1.2.cmml"></mi><mo id="S2.Ex25.m1.2.2.1.1.1" xref="S2.Ex25.m1.2.2.1.1.1.cmml">=</mo><mrow id="S2.Ex25.m1.1.1a.3" xref="S2.Ex25.m1.1.1a.2.cmml"><mo id="S2.Ex25.m1.1.1a.3.1" xref="S2.Ex25.m1.1.1a.2.1.cmml">[</mo><mtable columnspacing="5pt" id="S2.Ex25.m1.1.1.1.1" xref="S2.Ex25.m1.1.1.1.1.cmml"><mtr id="S2.Ex25.m1.1.1.1.1a" xref="S2.Ex25.m1.1.1.1.1.cmml"><mtd id="S2.Ex25.m1.1.1.1.1b" xref="S2.Ex25.m1.1.1.1.1.cmml"><mrow id="S2.Ex25.m1.1.1.1.1.2.2.2.2" xref="S2.Ex25.m1.1.1.1.1.2.2.2.2.cmml"><mtext id="S2.Ex25.m1.1.1.1.1.2.2.2.2.4" xref="S2.Ex25.m1.1.1.1.1.2.2.2.2.4a.cmml">vec</mtext><mo id="S2.Ex25.m1.1.1.1.1.2.2.2.2.3" xref="S2.Ex25.m1.1.1.1.1.2.2.2.2.3.cmml">⁢</mo><mrow id="S2.Ex25.m1.1.1.1.1.2.2.2.2.2.1" xref="S2.Ex25.m1.1.1.1.1.2.2.2.2.2.1.1.cmml"><mo id="S2.Ex25.m1.1.1.1.1.2.2.2.2.2.1.2" stretchy="false" xref="S2.Ex25.m1.1.1.1.1.2.2.2.2.2.1.1.cmml">(</mo><mrow id="S2.Ex25.m1.1.1.1.1.2.2.2.2.2.1.1" xref="S2.Ex25.m1.1.1.1.1.2.2.2.2.2.1.1.cmml"><msubsup id="S2.Ex25.m1.1.1.1.1.2.2.2.2.2.1.1.1" xref="S2.Ex25.m1.1.1.1.1.2.2.2.2.2.1.1.1.cmml"><mi id="S2.Ex25.m1.1.1.1.1.2.2.2.2.2.1.1.1.2.2" xref="S2.Ex25.m1.1.1.1.1.2.2.2.2.2.1.1.1.2.2.cmml">𝐋</mi><mn id="S2.Ex25.m1.1.1.1.1.2.2.2.2.2.1.1.1.2.3" xref="S2.Ex25.m1.1.1.1.1.2.2.2.2.2.1.1.1.2.3.cmml">2</mn><mo id="S2.Ex25.m1.1.1.1.1.2.2.2.2.2.1.1.1.3" xref="S2.Ex25.m1.1.1.1.1.2.2.2.2.2.1.1.1.3.cmml">⊤</mo></msubsup><mo id="S2.Ex25.m1.1.1.1.1.2.2.2.2.2.1.1a" lspace="0.167em" xref="S2.Ex25.m1.1.1.1.1.2.2.2.2.2.1.1.cmml">⁡</mo><mrow id="S2.Ex25.m1.1.1.1.1.2.2.2.2.2.1.1.2" xref="S2.Ex25.m1.1.1.1.1.2.2.2.2.2.1.1.2.cmml"><msubsup id="S2.Ex25.m1.1.1.1.1.2.2.2.2.2.1.1.2.1" xref="S2.Ex25.m1.1.1.1.1.2.2.2.2.2.1.1.2.1.cmml"><mi id="S2.Ex25.m1.1.1.1.1.2.2.2.2.2.1.1.2.1.2.2" xref="S2.Ex25.m1.1.1.1.1.2.2.2.2.2.1.1.2.1.2.2.cmml">𝐙</mi><mrow id="S2.Ex25.m1.1.1.1.1.1.1.1.1.1.1" xref="S2.Ex25.m1.1.1.1.1.1.1.1.1.1.1.cmml"><mi id="S2.Ex25.m1.1.1.1.1.1.1.1.1.1.1.3" xref="S2.Ex25.m1.1.1.1.1.1.1.1.1.1.1.3.cmml">i</mi><mo id="S2.Ex25.m1.1.1.1.1.1.1.1.1.1.1.2" xref="S2.Ex25.m1.1.1.1.1.1.1.1.1.1.1.2.cmml">⁢</mo><mn id="S2.Ex25.m1.1.1.1.1.1.1.1.1.1.1.4" xref="S2.Ex25.m1.1.1.1.1.1.1.1.1.1.1.4.cmml">1</mn><mo id="S2.Ex25.m1.1.1.1.1.1.1.1.1.1.1.2a" xref="S2.Ex25.m1.1.1.1.1.1.1.1.1.1.1.2.cmml">⁢</mo><mrow id="S2.Ex25.m1.1.1.1.1.1.1.1.1.1.1.5.2" xref="S2.Ex25.m1.1.1.1.1.1.1.1.1.1.1.cmml"><mo id="S2.Ex25.m1.1.1.1.1.1.1.1.1.1.1.5.2.1" stretchy="false" xref="S2.Ex25.m1.1.1.1.1.1.1.1.1.1.1.cmml">(</mo><mn id="S2.Ex25.m1.1.1.1.1.1.1.1.1.1.1.1" xref="S2.Ex25.m1.1.1.1.1.1.1.1.1.1.1.1.cmml">1</mn><mo id="S2.Ex25.m1.1.1.1.1.1.1.1.1.1.1.5.2.2" stretchy="false" xref="S2.Ex25.m1.1.1.1.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S2.Ex25.m1.1.1.1.1.2.2.2.2.2.1.1.2.1.3" xref="S2.Ex25.m1.1.1.1.1.2.2.2.2.2.1.1.2.1.3.cmml">⊤</mo></msubsup><mo id="S2.Ex25.m1.1.1.1.1.2.2.2.2.2.1.1.2a" lspace="0.167em" xref="S2.Ex25.m1.1.1.1.1.2.2.2.2.2.1.1.2.cmml">⁡</mo><msub id="S2.Ex25.m1.1.1.1.1.2.2.2.2.2.1.1.2.2" xref="S2.Ex25.m1.1.1.1.1.2.2.2.2.2.1.1.2.2.cmml"><mi id="S2.Ex25.m1.1.1.1.1.2.2.2.2.2.1.1.2.2.2" xref="S2.Ex25.m1.1.1.1.1.2.2.2.2.2.1.1.2.2.2.cmml">𝐋</mi><mn id="S2.Ex25.m1.1.1.1.1.2.2.2.2.2.1.1.2.2.3" xref="S2.Ex25.m1.1.1.1.1.2.2.2.2.2.1.1.2.2.3.cmml">1</mn></msub></mrow></mrow><mo id="S2.Ex25.m1.1.1.1.1.2.2.2.2.2.1.3" stretchy="false" xref="S2.Ex25.m1.1.1.1.1.2.2.2.2.2.1.1.cmml">)</mo></mrow></mrow></mtd><mtd id="S2.Ex25.m1.1.1.1.1c" xref="S2.Ex25.m1.1.1.1.1.cmml"><mrow id="S2.Ex25.m1.1.1.1.1.4.4.4.2" xref="S2.Ex25.m1.1.1.1.1.4.4.4.2.cmml"><mtext id="S2.Ex25.m1.1.1.1.1.4.4.4.2.4" xref="S2.Ex25.m1.1.1.1.1.4.4.4.2.4a.cmml">vec</mtext><mo id="S2.Ex25.m1.1.1.1.1.4.4.4.2.3" xref="S2.Ex25.m1.1.1.1.1.4.4.4.2.3.cmml">⁢</mo><mrow id="S2.Ex25.m1.1.1.1.1.4.4.4.2.2.1" xref="S2.Ex25.m1.1.1.1.1.4.4.4.2.2.1.1.cmml"><mo id="S2.Ex25.m1.1.1.1.1.4.4.4.2.2.1.2" stretchy="false" xref="S2.Ex25.m1.1.1.1.1.4.4.4.2.2.1.1.cmml">(</mo><mrow id="S2.Ex25.m1.1.1.1.1.4.4.4.2.2.1.1" xref="S2.Ex25.m1.1.1.1.1.4.4.4.2.2.1.1.cmml"><msubsup id="S2.Ex25.m1.1.1.1.1.4.4.4.2.2.1.1.1" xref="S2.Ex25.m1.1.1.1.1.4.4.4.2.2.1.1.1.cmml"><mi id="S2.Ex25.m1.1.1.1.1.4.4.4.2.2.1.1.1.2.2" xref="S2.Ex25.m1.1.1.1.1.4.4.4.2.2.1.1.1.2.2.cmml">𝐋</mi><mn id="S2.Ex25.m1.1.1.1.1.4.4.4.2.2.1.1.1.2.3" xref="S2.Ex25.m1.1.1.1.1.4.4.4.2.2.1.1.1.2.3.cmml">2</mn><mo id="S2.Ex25.m1.1.1.1.1.4.4.4.2.2.1.1.1.3" xref="S2.Ex25.m1.1.1.1.1.4.4.4.2.2.1.1.1.3.cmml">⊤</mo></msubsup><mo id="S2.Ex25.m1.1.1.1.1.4.4.4.2.2.1.1a" lspace="0.167em" xref="S2.Ex25.m1.1.1.1.1.4.4.4.2.2.1.1.cmml">⁡</mo><mrow id="S2.Ex25.m1.1.1.1.1.4.4.4.2.2.1.1.2" xref="S2.Ex25.m1.1.1.1.1.4.4.4.2.2.1.1.2.cmml"><msubsup id="S2.Ex25.m1.1.1.1.1.4.4.4.2.2.1.1.2.1" xref="S2.Ex25.m1.1.1.1.1.4.4.4.2.2.1.1.2.1.cmml"><mi id="S2.Ex25.m1.1.1.1.1.4.4.4.2.2.1.1.2.1.2.2" xref="S2.Ex25.m1.1.1.1.1.4.4.4.2.2.1.1.2.1.2.2.cmml">𝐙</mi><mrow id="S2.Ex25.m1.1.1.1.1.3.3.3.1.1.1" xref="S2.Ex25.m1.1.1.1.1.3.3.3.1.1.1.cmml"><mi id="S2.Ex25.m1.1.1.1.1.3.3.3.1.1.1.3" xref="S2.Ex25.m1.1.1.1.1.3.3.3.1.1.1.3.cmml">i</mi><mo id="S2.Ex25.m1.1.1.1.1.3.3.3.1.1.1.2" xref="S2.Ex25.m1.1.1.1.1.3.3.3.1.1.1.2.cmml">⁢</mo><mn id="S2.Ex25.m1.1.1.1.1.3.3.3.1.1.1.4" xref="S2.Ex25.m1.1.1.1.1.3.3.3.1.1.1.4.cmml">2</mn><mo id="S2.Ex25.m1.1.1.1.1.3.3.3.1.1.1.2a" xref="S2.Ex25.m1.1.1.1.1.3.3.3.1.1.1.2.cmml">⁢</mo><mrow id="S2.Ex25.m1.1.1.1.1.3.3.3.1.1.1.5.2" xref="S2.Ex25.m1.1.1.1.1.3.3.3.1.1.1.cmml"><mo id="S2.Ex25.m1.1.1.1.1.3.3.3.1.1.1.5.2.1" stretchy="false" xref="S2.Ex25.m1.1.1.1.1.3.3.3.1.1.1.cmml">(</mo><mn id="S2.Ex25.m1.1.1.1.1.3.3.3.1.1.1.1" xref="S2.Ex25.m1.1.1.1.1.3.3.3.1.1.1.1.cmml">1</mn><mo id="S2.Ex25.m1.1.1.1.1.3.3.3.1.1.1.5.2.2" stretchy="false" xref="S2.Ex25.m1.1.1.1.1.3.3.3.1.1.1.cmml">)</mo></mrow></mrow><mo id="S2.Ex25.m1.1.1.1.1.4.4.4.2.2.1.1.2.1.3" xref="S2.Ex25.m1.1.1.1.1.4.4.4.2.2.1.1.2.1.3.cmml">⊤</mo></msubsup><mo id="S2.Ex25.m1.1.1.1.1.4.4.4.2.2.1.1.2a" lspace="0.167em" xref="S2.Ex25.m1.1.1.1.1.4.4.4.2.2.1.1.2.cmml">⁡</mo><msub id="S2.Ex25.m1.1.1.1.1.4.4.4.2.2.1.1.2.2" xref="S2.Ex25.m1.1.1.1.1.4.4.4.2.2.1.1.2.2.cmml"><mi id="S2.Ex25.m1.1.1.1.1.4.4.4.2.2.1.1.2.2.2" xref="S2.Ex25.m1.1.1.1.1.4.4.4.2.2.1.1.2.2.2.cmml">𝐋</mi><mn 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id="S2.Ex25.m1.1.1.1.1.6.6.6.2.2.1.1.2.2.2.cmml" xref="S2.Ex25.m1.1.1.1.1.6.6.6.2.2.1.1.2.2.2">𝐋</ci><cn id="S2.Ex25.m1.1.1.1.1.6.6.6.2.2.1.1.2.2.3.cmml" type="integer" xref="S2.Ex25.m1.1.1.1.1.6.6.6.2.2.1.1.2.2.3">1</cn></apply></apply></apply></apply></matrixrow></matrix></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Ex25.m1.2c">\displaystyle=\begin{bmatrix}\text{vec}(\operatorname{\mathbf{L}}_{2}^{\top}% \operatorname{\mathbf{Z}}_{i1(1)}^{\top}\operatorname{\mathbf{L}}_{1})&\text{% vec}(\operatorname{\mathbf{L}}_{2}^{\top}\operatorname{\mathbf{Z}}_{i2(1)}^{% \top}\operatorname{\mathbf{L}}_{1})&\ldots&\text{vec}(\operatorname{\mathbf{L}% }_{2}^{\top}\operatorname{\mathbf{Z}}_{im_{i}(1)}^{\top}\operatorname{\mathbf{% L}}_{1})\end{bmatrix},</annotation><annotation encoding="application/x-llamapun" id="S2.Ex25.m1.2d">= [ start_ARG start_ROW start_CELL vec ( bold_L start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ⊤ end_POSTSUPERSCRIPT bold_Z start_POSTSUBSCRIPT italic_i 1 ( 1 ) end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ⊤ end_POSTSUPERSCRIPT bold_L start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) end_CELL start_CELL vec ( bold_L start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ⊤ end_POSTSUPERSCRIPT bold_Z start_POSTSUBSCRIPT italic_i 2 ( 1 ) end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ⊤ end_POSTSUPERSCRIPT bold_L start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) end_CELL start_CELL … end_CELL start_CELL vec ( bold_L start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ⊤ end_POSTSUPERSCRIPT bold_Z start_POSTSUBSCRIPT italic_i italic_m start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( 1 ) end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ⊤ end_POSTSUPERSCRIPT bold_L start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) end_CELL end_ROW end_ARG ] ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="S2.Ex26"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td 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id="S2.Ex26.m1.5.5.1.1.cmml" type="integer" xref="S2.Ex26.m1.5.5.1.1">2</cn></apply><csymbol cd="latexml" id="S2.Ex26.m1.7.7.1.1.1.2.2.2.1.3.cmml" xref="S2.Ex26.m1.7.7.1.1.1.2.2.2.1.3">top</csymbol></apply><apply id="S2.Ex26.m1.7.7.1.1.1.2.2.2.2.cmml" xref="S2.Ex26.m1.7.7.1.1.1.2.2.2.2"><csymbol cd="ambiguous" id="S2.Ex26.m1.7.7.1.1.1.2.2.2.2.1.cmml" xref="S2.Ex26.m1.7.7.1.1.1.2.2.2.2">superscript</csymbol><apply id="S2.Ex26.m1.7.7.1.1.1.2.2.2.2.2.cmml" xref="S2.Ex26.m1.7.7.1.1.1.2.2.2.2"><csymbol cd="ambiguous" id="S2.Ex26.m1.7.7.1.1.1.2.2.2.2.2.1.cmml" xref="S2.Ex26.m1.7.7.1.1.1.2.2.2.2">subscript</csymbol><ci id="S2.Ex26.m1.7.7.1.1.1.2.2.2.2.2.2.cmml" xref="S2.Ex26.m1.7.7.1.1.1.2.2.2.2.2.2">𝐙</ci><apply id="S2.Ex26.m1.6.6.1.cmml" xref="S2.Ex26.m1.6.6.1"><times id="S2.Ex26.m1.6.6.1.2.cmml" xref="S2.Ex26.m1.6.6.1.2"></times><ci id="S2.Ex26.m1.6.6.1.3.cmml" xref="S2.Ex26.m1.6.6.1.3">𝑖</ci><cn id="S2.Ex26.m1.6.6.1.1.cmml" type="integer" xref="S2.Ex26.m1.6.6.1.1">2</cn></apply></apply><csymbol cd="latexml" id="S2.Ex26.m1.7.7.1.1.1.2.2.2.2.3.cmml" xref="S2.Ex26.m1.7.7.1.1.1.2.2.2.2.3">top</csymbol></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Ex26.m1.7c">\displaystyle\text{tr}(\operatorname{\mathbf{Z}}_{i(2)}\operatorname{\mathbf{L% }}_{(2)}\operatorname{{\mathbf{\Gamma}}}_{i(2)}^{(t)}\operatorname{\mathbf{L}}% _{(2)}^{\top}\operatorname{\mathbf{Z}}_{i(2)}^{\top})</annotation><annotation encoding="application/x-llamapun" id="S2.Ex26.m1.7d">tr ( bold_Z start_POSTSUBSCRIPT italic_i ( 2 ) end_POSTSUBSCRIPT bold_L start_POSTSUBSCRIPT ( 2 ) end_POSTSUBSCRIPT bold_Γ start_POSTSUBSCRIPT italic_i ( 2 ) end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_t ) end_POSTSUPERSCRIPT bold_L start_POSTSUBSCRIPT ( 2 ) end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ⊤ end_POSTSUPERSCRIPT bold_Z start_POSTSUBSCRIPT italic_i ( 2 ) end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ⊤ end_POSTSUPERSCRIPT )</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle=\sum_{j=1}^{m_{i}}\text{vec}(\operatorname{\mathbf{L}}_{2}^{\top% }\operatorname{\mathbf{Z}}_{ij(1)}^{\top}\operatorname{\mathbf{L}}_{1})^{\top}% \operatorname{{\mathbf{\Gamma}}}_{i(2)}^{(t)}\text{vec}(\operatorname{\mathbf{% L}}_{2}^{\top}\operatorname{\mathbf{Z}}_{ij(1)}^{\top}\operatorname{\mathbf{L}% }_{1})" class="ltx_Math" display="inline" id="S2.Ex26.m2.6"><semantics id="S2.Ex26.m2.6a"><mrow id="S2.Ex26.m2.6.6" xref="S2.Ex26.m2.6.6.cmml"><mi id="S2.Ex26.m2.6.6.4" xref="S2.Ex26.m2.6.6.4.cmml"></mi><mo id="S2.Ex26.m2.6.6.3" xref="S2.Ex26.m2.6.6.3.cmml">=</mo><mrow id="S2.Ex26.m2.6.6.2" xref="S2.Ex26.m2.6.6.2.cmml"><mstyle displaystyle="true" id="S2.Ex26.m2.6.6.2.3" xref="S2.Ex26.m2.6.6.2.3.cmml"><munderover id="S2.Ex26.m2.6.6.2.3a" xref="S2.Ex26.m2.6.6.2.3.cmml"><mo id="S2.Ex26.m2.6.6.2.3.2.2" movablelimits="false" 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xref="S2.Ex26.m2.6.6.2.2.2.1.1.1.3">top</csymbol></apply><apply id="S2.Ex26.m2.6.6.2.2.2.1.1.2.cmml" xref="S2.Ex26.m2.6.6.2.2.2.1.1.2"><apply id="S2.Ex26.m2.6.6.2.2.2.1.1.2.1.cmml" xref="S2.Ex26.m2.6.6.2.2.2.1.1.2.1"><csymbol cd="ambiguous" id="S2.Ex26.m2.6.6.2.2.2.1.1.2.1.1.cmml" xref="S2.Ex26.m2.6.6.2.2.2.1.1.2.1">superscript</csymbol><apply id="S2.Ex26.m2.6.6.2.2.2.1.1.2.1.2.cmml" xref="S2.Ex26.m2.6.6.2.2.2.1.1.2.1"><csymbol cd="ambiguous" id="S2.Ex26.m2.6.6.2.2.2.1.1.2.1.2.1.cmml" xref="S2.Ex26.m2.6.6.2.2.2.1.1.2.1">subscript</csymbol><ci id="S2.Ex26.m2.6.6.2.2.2.1.1.2.1.2.2.cmml" xref="S2.Ex26.m2.6.6.2.2.2.1.1.2.1.2.2">𝐙</ci><apply id="S2.Ex26.m2.4.4.1.cmml" xref="S2.Ex26.m2.4.4.1"><times id="S2.Ex26.m2.4.4.1.2.cmml" xref="S2.Ex26.m2.4.4.1.2"></times><ci id="S2.Ex26.m2.4.4.1.3.cmml" xref="S2.Ex26.m2.4.4.1.3">𝑖</ci><ci id="S2.Ex26.m2.4.4.1.4.cmml" xref="S2.Ex26.m2.4.4.1.4">𝑗</ci><cn id="S2.Ex26.m2.4.4.1.1.cmml" type="integer" xref="S2.Ex26.m2.4.4.1.1">1</cn></apply></apply><csymbol cd="latexml" id="S2.Ex26.m2.6.6.2.2.2.1.1.2.1.3.cmml" xref="S2.Ex26.m2.6.6.2.2.2.1.1.2.1.3">top</csymbol></apply><apply id="S2.Ex26.m2.6.6.2.2.2.1.1.2.2.cmml" xref="S2.Ex26.m2.6.6.2.2.2.1.1.2.2"><csymbol cd="ambiguous" id="S2.Ex26.m2.6.6.2.2.2.1.1.2.2.1.cmml" xref="S2.Ex26.m2.6.6.2.2.2.1.1.2.2">subscript</csymbol><ci id="S2.Ex26.m2.6.6.2.2.2.1.1.2.2.2.cmml" xref="S2.Ex26.m2.6.6.2.2.2.1.1.2.2.2">𝐋</ci><cn id="S2.Ex26.m2.6.6.2.2.2.1.1.2.2.3.cmml" type="integer" xref="S2.Ex26.m2.6.6.2.2.2.1.1.2.2.3">1</cn></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Ex26.m2.6c">\displaystyle=\sum_{j=1}^{m_{i}}\text{vec}(\operatorname{\mathbf{L}}_{2}^{\top% }\operatorname{\mathbf{Z}}_{ij(1)}^{\top}\operatorname{\mathbf{L}}_{1})^{\top}% \operatorname{{\mathbf{\Gamma}}}_{i(2)}^{(t)}\text{vec}(\operatorname{\mathbf{% L}}_{2}^{\top}\operatorname{\mathbf{Z}}_{ij(1)}^{\top}\operatorname{\mathbf{L}% }_{1})</annotation><annotation encoding="application/x-llamapun" id="S2.Ex26.m2.6d">= ∑ start_POSTSUBSCRIPT italic_j = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_m start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUPERSCRIPT vec ( bold_L start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ⊤ end_POSTSUPERSCRIPT bold_Z start_POSTSUBSCRIPT italic_i italic_j ( 1 ) end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ⊤ end_POSTSUPERSCRIPT bold_L start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT ⊤ end_POSTSUPERSCRIPT bold_Γ start_POSTSUBSCRIPT italic_i ( 2 ) end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_t ) end_POSTSUPERSCRIPT vec ( bold_L start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ⊤ end_POSTSUPERSCRIPT bold_Z start_POSTSUBSCRIPT italic_i italic_j ( 1 ) end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ⊤ end_POSTSUPERSCRIPT bold_L start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="S2.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_eqn_cell"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle=\sum_{j=1}^{m_{i}}\text{vec}(\operatorname{\mathbf{L}}_{1})^{% \top}(\operatorname{\mathbf{I}}_{S_{1}}\otimes\operatorname{\mathbf{Z}}_{ij(1)% }\operatorname{\mathbf{L}}_{2})\operatorname{{\mathbf{\Gamma}}}_{i(2)}^{(t)}(% \operatorname{\mathbf{I}}_{S_{1}}\otimes\operatorname{\mathbf{L}}_{2}^{\top}% \operatorname{\mathbf{Z}}_{ij(1)}^{\top})\text{vec}(\operatorname{\mathbf{L}}_% {1})" class="ltx_Math" display="inline" id="S2.Ex27.m1.9"><semantics id="S2.Ex27.m1.9a"><mrow id="S2.Ex27.m1.9.9" xref="S2.Ex27.m1.9.9.cmml"><mi id="S2.Ex27.m1.9.9.7" xref="S2.Ex27.m1.9.9.7.cmml"></mi><mo id="S2.Ex27.m1.9.9.6" xref="S2.Ex27.m1.9.9.6.cmml">=</mo><mrow id="S2.Ex27.m1.9.9.5" xref="S2.Ex27.m1.9.9.5.cmml"><mstyle displaystyle="true" id="S2.Ex27.m1.9.9.5.6" 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id="S2.Ex27.m1.9.9.5.5.5.1.1.3.cmml" type="integer" xref="S2.Ex27.m1.9.9.5.5.5.1.1.3">1</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Ex27.m1.9c">\displaystyle=\sum_{j=1}^{m_{i}}\text{vec}(\operatorname{\mathbf{L}}_{1})^{% \top}(\operatorname{\mathbf{I}}_{S_{1}}\otimes\operatorname{\mathbf{Z}}_{ij(1)% }\operatorname{\mathbf{L}}_{2})\operatorname{{\mathbf{\Gamma}}}_{i(2)}^{(t)}(% \operatorname{\mathbf{I}}_{S_{1}}\otimes\operatorname{\mathbf{L}}_{2}^{\top}% \operatorname{\mathbf{Z}}_{ij(1)}^{\top})\text{vec}(\operatorname{\mathbf{L}}_% {1})</annotation><annotation encoding="application/x-llamapun" id="S2.Ex27.m1.9d">= ∑ start_POSTSUBSCRIPT italic_j = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_m start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUPERSCRIPT vec ( bold_L start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT ⊤ end_POSTSUPERSCRIPT ( bold_I start_POSTSUBSCRIPT italic_S start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ⊗ bold_Z start_POSTSUBSCRIPT italic_i italic_j ( 1 ) end_POSTSUBSCRIPT bold_L start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) bold_Γ start_POSTSUBSCRIPT italic_i ( 2 ) end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_t ) end_POSTSUPERSCRIPT ( bold_I start_POSTSUBSCRIPT italic_S start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ⊗ bold_L start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ⊤ end_POSTSUPERSCRIPT bold_Z start_POSTSUBSCRIPT italic_i italic_j ( 1 ) end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ⊤ end_POSTSUPERSCRIPT ) vec ( bold_L start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="S2.Ex28"><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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encoding="application/x-llamapun" id="S2.Ex28.m1.5d">= bold_l start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ⊤ end_POSTSUPERSCRIPT [ ∑ start_POSTSUBSCRIPT italic_j = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_m start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUPERSCRIPT ( bold_I start_POSTSUBSCRIPT italic_S start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ⊗ bold_Z start_POSTSUBSCRIPT italic_i italic_j ( 1 ) end_POSTSUBSCRIPT bold_L start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) bold_Γ start_POSTSUBSCRIPT italic_i ( 2 ) end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_t ) end_POSTSUPERSCRIPT ( bold_I start_POSTSUBSCRIPT italic_S start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ⊗ bold_L start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ⊤ end_POSTSUPERSCRIPT bold_Z start_POSTSUBSCRIPT italic_i italic_j ( 1 ) end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ⊤ end_POSTSUPERSCRIPT ) ] bold_l start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S2.SS2a.p3.31">Next, we pull <math alttext="\operatorname{\mathbf{L}}_{1}" class="ltx_Math" display="inline" id="S2.SS2a.p3.30.m1.1"><semantics id="S2.SS2a.p3.30.m1.1a"><msub id="S2.SS2a.p3.30.m1.1.1" xref="S2.SS2a.p3.30.m1.1.1.cmml"><mi id="S2.SS2a.p3.30.m1.1.1.2" xref="S2.SS2a.p3.30.m1.1.1.2.cmml">𝐋</mi><mn id="S2.SS2a.p3.30.m1.1.1.3" xref="S2.SS2a.p3.30.m1.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S2.SS2a.p3.30.m1.1b"><apply id="S2.SS2a.p3.30.m1.1.1.cmml" xref="S2.SS2a.p3.30.m1.1.1"><csymbol cd="ambiguous" id="S2.SS2a.p3.30.m1.1.1.1.cmml" xref="S2.SS2a.p3.30.m1.1.1">subscript</csymbol><ci id="S2.SS2a.p3.30.m1.1.1.2.cmml" xref="S2.SS2a.p3.30.m1.1.1.2">𝐋</ci><cn id="S2.SS2a.p3.30.m1.1.1.3.cmml" type="integer" xref="S2.SS2a.p3.30.m1.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2a.p3.30.m1.1c">\operatorname{\mathbf{L}}_{1}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2a.p3.30.m1.1d">bold_L start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> out of the second component of (<a class="ltx_ref" href="https://arxiv.org/html/2503.13782v1#S2.Ex21" title="In 2.2 Derivation of 𝐋₁^(𝑡+1) and 𝐋₂^(𝑡+1) ‣ 2 Derivation of the AECM Algorithm Updates ‣ Regularized Parameter Estimation in Mixed Model Trace Regression"><span class="ltx_text ltx_ref_tag">6</span></a>), <math alttext="\operatorname{{\bm{\mu}}}_{i(2)}^{(t)\top}\operatorname{\mathbf{L}}_{(2)}^{% \top}\operatorname{\mathbf{Z}}_{i(2)}^{\top}\tilde{\operatorname{\mathbf{y}}}_% {i}" class="ltx_Math" display="inline" id="S2.SS2a.p3.31.m2.4"><semantics id="S2.SS2a.p3.31.m2.4a"><mrow id="S2.SS2a.p3.31.m2.4.5" xref="S2.SS2a.p3.31.m2.4.5.cmml"><msubsup id="S2.SS2a.p3.31.m2.4.5.1" xref="S2.SS2a.p3.31.m2.4.5.1.cmml"><mi id="S2.SS2a.p3.31.m2.4.5.1.2.2" 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xref="S2.SS2a.p3.31.m2.4.5.2.2.2.2.1">~</ci><ci id="S2.SS2a.p3.31.m2.4.5.2.2.2.2.2.cmml" xref="S2.SS2a.p3.31.m2.4.5.2.2.2.2.2">𝐲</ci></apply><ci id="S2.SS2a.p3.31.m2.4.5.2.2.2.3.cmml" xref="S2.SS2a.p3.31.m2.4.5.2.2.2.3">𝑖</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2a.p3.31.m2.4c">\operatorname{{\bm{\mu}}}_{i(2)}^{(t)\top}\operatorname{\mathbf{L}}_{(2)}^{% \top}\operatorname{\mathbf{Z}}_{i(2)}^{\top}\tilde{\operatorname{\mathbf{y}}}_% {i}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2a.p3.31.m2.4d">bold_italic_μ start_POSTSUBSCRIPT italic_i ( 2 ) end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_t ) ⊤ end_POSTSUPERSCRIPT bold_L start_POSTSUBSCRIPT ( 2 ) end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ⊤ end_POSTSUPERSCRIPT bold_Z start_POSTSUBSCRIPT italic_i ( 2 ) end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ⊤ end_POSTSUPERSCRIPT over~ start_ARG bold_y end_ARG start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> to obtain that</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="S3.EGx26"> <tbody id="S2.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\operatorname{\mathbf{Z}}_{i(2)}^{\top}\tilde{\operatorname{% \mathbf{y}}}_{i}" class="ltx_Math" display="inline" id="S2.Ex29.m1.1"><semantics id="S2.Ex29.m1.1a"><mrow id="S2.Ex29.m1.1.2" xref="S2.Ex29.m1.1.2.cmml"><msubsup id="S2.Ex29.m1.1.2.1" xref="S2.Ex29.m1.1.2.1.cmml"><mi id="S2.Ex29.m1.1.2.1.2.2" xref="S2.Ex29.m1.1.2.1.2.2.cmml">𝐙</mi><mrow id="S2.Ex29.m1.1.1.1" xref="S2.Ex29.m1.1.1.1.cmml"><mi id="S2.Ex29.m1.1.1.1.3" xref="S2.Ex29.m1.1.1.1.3.cmml">i</mi><mo id="S2.Ex29.m1.1.1.1.2" xref="S2.Ex29.m1.1.1.1.2.cmml">⁢</mo><mrow id="S2.Ex29.m1.1.1.1.4.2" xref="S2.Ex29.m1.1.1.1.cmml"><mo id="S2.Ex29.m1.1.1.1.4.2.1" stretchy="false" xref="S2.Ex29.m1.1.1.1.cmml">(</mo><mn id="S2.Ex29.m1.1.1.1.1" xref="S2.Ex29.m1.1.1.1.1.cmml">2</mn><mo id="S2.Ex29.m1.1.1.1.4.2.2" stretchy="false" xref="S2.Ex29.m1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S2.Ex29.m1.1.2.1.3" xref="S2.Ex29.m1.1.2.1.3.cmml">⊤</mo></msubsup><mo id="S2.Ex29.m1.1.2a" lspace="0.167em" xref="S2.Ex29.m1.1.2.cmml">⁡</mo><msub id="S2.Ex29.m1.1.2.2" xref="S2.Ex29.m1.1.2.2.cmml"><mover accent="true" id="S2.Ex29.m1.1.2.2.2" xref="S2.Ex29.m1.1.2.2.2.cmml"><mi id="S2.Ex29.m1.1.2.2.2.2" xref="S2.Ex29.m1.1.2.2.2.2.cmml">𝐲</mi><mo id="S2.Ex29.m1.1.2.2.2.1" xref="S2.Ex29.m1.1.2.2.2.1.cmml">~</mo></mover><mi id="S2.Ex29.m1.1.2.2.3" xref="S2.Ex29.m1.1.2.2.3.cmml">i</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S2.Ex29.m1.1b"><apply id="S2.Ex29.m1.1.2.cmml" xref="S2.Ex29.m1.1.2"><apply id="S2.Ex29.m1.1.2.1.cmml" xref="S2.Ex29.m1.1.2.1"><csymbol cd="ambiguous" id="S2.Ex29.m1.1.2.1.1.cmml" xref="S2.Ex29.m1.1.2.1">superscript</csymbol><apply 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id="S2.Ex29.m1.1.2.2.3.cmml" xref="S2.Ex29.m1.1.2.2.3">𝑖</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Ex29.m1.1c">\displaystyle\operatorname{\mathbf{Z}}_{i(2)}^{\top}\tilde{\operatorname{% \mathbf{y}}}_{i}</annotation><annotation encoding="application/x-llamapun" id="S2.Ex29.m1.1d">bold_Z start_POSTSUBSCRIPT italic_i ( 2 ) end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ⊤ end_POSTSUPERSCRIPT over~ start_ARG bold_y end_ARG start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle=\sum_{j=1}^{m_{i}}\tilde{y}_{ij}\operatorname{\mathbf{z}}_{ij(2)}," class="ltx_Math" display="inline" id="S2.Ex29.m2.2"><semantics id="S2.Ex29.m2.2a"><mrow id="S2.Ex29.m2.2.2.1" xref="S2.Ex29.m2.2.2.1.1.cmml"><mrow id="S2.Ex29.m2.2.2.1.1" xref="S2.Ex29.m2.2.2.1.1.cmml"><mi id="S2.Ex29.m2.2.2.1.1.2" xref="S2.Ex29.m2.2.2.1.1.2.cmml"></mi><mo id="S2.Ex29.m2.2.2.1.1.1" 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id="S2.Ex29.m2.2c">\displaystyle=\sum_{j=1}^{m_{i}}\tilde{y}_{ij}\operatorname{\mathbf{z}}_{ij(2)},</annotation><annotation encoding="application/x-llamapun" id="S2.Ex29.m2.2d">= ∑ start_POSTSUBSCRIPT italic_j = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_m start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUPERSCRIPT over~ start_ARG italic_y end_ARG start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT bold_z start_POSTSUBSCRIPT italic_i italic_j ( 2 ) end_POSTSUBSCRIPT ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="S2.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\operatorname{\mathbf{L}}_{(2)}^{\top}\operatorname{\mathbf{Z}}_{% i(2)}^{\top}\tilde{\operatorname{\mathbf{y}}}_{i}" class="ltx_Math" display="inline" id="S2.Ex30.m1.2"><semantics 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id="S2.Ex30.m1.2c">\displaystyle\operatorname{\mathbf{L}}_{(2)}^{\top}\operatorname{\mathbf{Z}}_{% i(2)}^{\top}\tilde{\operatorname{\mathbf{y}}}_{i}</annotation><annotation encoding="application/x-llamapun" id="S2.Ex30.m1.2d">bold_L start_POSTSUBSCRIPT ( 2 ) end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ⊤ end_POSTSUPERSCRIPT bold_Z start_POSTSUBSCRIPT italic_i ( 2 ) end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ⊤ end_POSTSUPERSCRIPT over~ start_ARG bold_y end_ARG start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle=(\operatorname{\mathbf{L}}_{1}^{\top}\otimes\operatorname{% \mathbf{L}}_{2}^{\top})\sum_{j=1}^{m_{i}}\tilde{y}_{ij}\operatorname{\mathbf{z% }}_{ij(2)}" class="ltx_Math" display="inline" id="S2.Ex30.m2.2"><semantics id="S2.Ex30.m2.2a"><mrow id="S2.Ex30.m2.2.2" xref="S2.Ex30.m2.2.2.cmml"><mi id="S2.Ex30.m2.2.2.3" xref="S2.Ex30.m2.2.2.3.cmml"></mi><mo id="S2.Ex30.m2.2.2.2" 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xref="S2.Ex31.m1.1.1.1.3">𝑖</ci><ci id="S2.Ex31.m1.1.1.1.4.cmml" xref="S2.Ex31.m1.1.1.1.4">𝑗</ci><cn id="S2.Ex31.m1.1.1.1.1.cmml" type="integer" xref="S2.Ex31.m1.1.1.1.1">2</cn></apply></apply></apply></apply></apply><apply id="S2.Ex31.m1.2.2.1.1.1.1.4.cmml" xref="S2.Ex31.m1.2.2.1.1.1.1.4"><csymbol cd="ambiguous" id="S2.Ex31.m1.2.2.1.1.1.1.4.1.cmml" xref="S2.Ex31.m1.2.2.1.1.1.1.4">subscript</csymbol><ci id="S2.Ex31.m1.2.2.1.1.1.1.4.2.cmml" xref="S2.Ex31.m1.2.2.1.1.1.1.4.2">𝐋</ci><cn id="S2.Ex31.m1.2.2.1.1.1.1.4.3.cmml" type="integer" xref="S2.Ex31.m1.2.2.1.1.1.1.4.3">1</cn></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Ex31.m1.2c">\displaystyle=\text{vec}\left[\operatorname{\mathbf{L}}_{2}^{\top}(\sum_{j=1}^% {m_{i}}\tilde{y}_{ij}\operatorname{\mathbf{Z}}_{ij(2)})\operatorname{\mathbf{L% }}_{1}\right]</annotation><annotation encoding="application/x-llamapun" id="S2.Ex31.m1.2d">= vec [ bold_L start_POSTSUBSCRIPT 2 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xref="S2.Ex32.m1.2.2.1.1.1.1.4.3">1</cn></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Ex32.m1.2c">\displaystyle=\text{vec}\left[\operatorname{\mathbf{L}}_{2}^{\top}(\sum_{j=1}^% {m_{i}}\tilde{y}_{ij}\operatorname{\mathbf{Z}}_{ij(1)}^{\top})\operatorname{% \mathbf{L}}_{1}\right]</annotation><annotation encoding="application/x-llamapun" id="S2.Ex32.m1.2d">= vec [ bold_L start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ⊤ end_POSTSUPERSCRIPT ( ∑ start_POSTSUBSCRIPT italic_j = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_m start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUPERSCRIPT over~ start_ARG italic_y end_ARG start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT bold_Z start_POSTSUBSCRIPT italic_i italic_j ( 1 ) end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ⊤ end_POSTSUPERSCRIPT ) bold_L start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ]</annotation></semantics></math></td> <td class="ltx_eqn_cell 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xref="S2.Ex34.m1.4.5.2.2.2.2"><ci id="S2.Ex34.m1.4.5.2.2.2.2.1.cmml" xref="S2.Ex34.m1.4.5.2.2.2.2.1">~</ci><ci id="S2.Ex34.m1.4.5.2.2.2.2.2.cmml" xref="S2.Ex34.m1.4.5.2.2.2.2.2">𝐲</ci></apply><ci id="S2.Ex34.m1.4.5.2.2.2.3.cmml" xref="S2.Ex34.m1.4.5.2.2.2.3">𝑖</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Ex34.m1.4c">\displaystyle\operatorname{{\bm{\mu}}}_{i(2)}^{(t)\top}\operatorname{\mathbf{L% }}_{(2)}^{\top}\operatorname{\mathbf{Z}}_{i(2)}^{\top}\tilde{\operatorname{% \mathbf{y}}}_{i}</annotation><annotation encoding="application/x-llamapun" id="S2.Ex34.m1.4d">bold_italic_μ start_POSTSUBSCRIPT italic_i ( 2 ) end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_t ) ⊤ end_POSTSUPERSCRIPT bold_L start_POSTSUBSCRIPT ( 2 ) end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ⊤ end_POSTSUPERSCRIPT bold_Z start_POSTSUBSCRIPT italic_i ( 2 ) end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ⊤ end_POSTSUPERSCRIPT over~ start_ARG bold_y end_ARG start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle=\operatorname{{\bm{\mu}}}_{i(2)}^{(t)\top}\left[\operatorname{% \mathbf{I}}_{S_{1}}\otimes\operatorname{\mathbf{L}}_{2}^{\top}(\sum_{j=1}^{m_{% i}}\tilde{y}_{ij}\operatorname{\mathbf{Z}}_{ij(1)}^{\top})\right]\operatorname% {\mathbf{l}}_{1}." class="ltx_Math" display="inline" id="S2.Ex34.m2.4"><semantics id="S2.Ex34.m2.4a"><mrow id="S2.Ex34.m2.4.4.1" xref="S2.Ex34.m2.4.4.1.1.cmml"><mrow id="S2.Ex34.m2.4.4.1.1" xref="S2.Ex34.m2.4.4.1.1.cmml"><mi id="S2.Ex34.m2.4.4.1.1.4" xref="S2.Ex34.m2.4.4.1.1.4.cmml"></mi><mo id="S2.Ex34.m2.4.4.1.1.3" xref="S2.Ex34.m2.4.4.1.1.3.cmml">=</mo><mrow id="S2.Ex34.m2.4.4.1.1.2" xref="S2.Ex34.m2.4.4.1.1.2.cmml"><mrow id="S2.Ex34.m2.4.4.1.1.2.2.2" xref="S2.Ex34.m2.4.4.1.1.2.2.3.cmml"><msubsup id="S2.Ex34.m2.4.4.1.1.1.1.1.1" xref="S2.Ex34.m2.4.4.1.1.1.1.1.1.cmml"><mi id="S2.Ex34.m2.4.4.1.1.1.1.1.1.2.2" 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id="S2.Ex34.m2.4.4.1.1.2.4.3.cmml" type="integer" xref="S2.Ex34.m2.4.4.1.1.2.4.3">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Ex34.m2.4c">\displaystyle=\operatorname{{\bm{\mu}}}_{i(2)}^{(t)\top}\left[\operatorname{% \mathbf{I}}_{S_{1}}\otimes\operatorname{\mathbf{L}}_{2}^{\top}(\sum_{j=1}^{m_{% i}}\tilde{y}_{ij}\operatorname{\mathbf{Z}}_{ij(1)}^{\top})\right]\operatorname% {\mathbf{l}}_{1}.</annotation><annotation encoding="application/x-llamapun" id="S2.Ex34.m2.4d">= bold_italic_μ start_POSTSUBSCRIPT italic_i ( 2 ) end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_t ) ⊤ end_POSTSUPERSCRIPT [ bold_I start_POSTSUBSCRIPT italic_S start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ⊗ bold_L start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ⊤ end_POSTSUPERSCRIPT ( ∑ start_POSTSUBSCRIPT italic_j = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_m start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUPERSCRIPT over~ start_ARG italic_y end_ARG start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT bold_Z start_POSTSUBSCRIPT italic_i italic_j ( 1 ) end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ⊤ end_POSTSUPERSCRIPT ) ] bold_l start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S2.SS2a.p3.36">When estimating <math alttext="\operatorname{\mathbf{L}}_{1}" class="ltx_Math" display="inline" id="S2.SS2a.p3.32.m1.1"><semantics id="S2.SS2a.p3.32.m1.1a"><msub id="S2.SS2a.p3.32.m1.1.1" xref="S2.SS2a.p3.32.m1.1.1.cmml"><mi id="S2.SS2a.p3.32.m1.1.1.2" xref="S2.SS2a.p3.32.m1.1.1.2.cmml">𝐋</mi><mn id="S2.SS2a.p3.32.m1.1.1.3" xref="S2.SS2a.p3.32.m1.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S2.SS2a.p3.32.m1.1b"><apply id="S2.SS2a.p3.32.m1.1.1.cmml" xref="S2.SS2a.p3.32.m1.1.1"><csymbol cd="ambiguous" id="S2.SS2a.p3.32.m1.1.1.1.cmml" xref="S2.SS2a.p3.32.m1.1.1">subscript</csymbol><ci id="S2.SS2a.p3.32.m1.1.1.2.cmml" xref="S2.SS2a.p3.32.m1.1.1.2">𝐋</ci><cn id="S2.SS2a.p3.32.m1.1.1.3.cmml" type="integer" xref="S2.SS2a.p3.32.m1.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2a.p3.32.m1.1c">\operatorname{\mathbf{L}}_{1}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2a.p3.32.m1.1d">bold_L start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math>, we condition on <math alttext="\operatorname{\mathbf{L}}_{2}^{(t)}" class="ltx_Math" display="inline" id="S2.SS2a.p3.33.m2.1"><semantics id="S2.SS2a.p3.33.m2.1a"><msubsup id="S2.SS2a.p3.33.m2.1.2" xref="S2.SS2a.p3.33.m2.1.2.cmml"><mi id="S2.SS2a.p3.33.m2.1.2.2.2" xref="S2.SS2a.p3.33.m2.1.2.2.2.cmml">𝐋</mi><mn id="S2.SS2a.p3.33.m2.1.2.2.3" xref="S2.SS2a.p3.33.m2.1.2.2.3.cmml">2</mn><mrow id="S2.SS2a.p3.33.m2.1.1.1.3" xref="S2.SS2a.p3.33.m2.1.2.cmml"><mo id="S2.SS2a.p3.33.m2.1.1.1.3.1" stretchy="false" xref="S2.SS2a.p3.33.m2.1.2.cmml">(</mo><mi id="S2.SS2a.p3.33.m2.1.1.1.1" xref="S2.SS2a.p3.33.m2.1.1.1.1.cmml">t</mi><mo id="S2.SS2a.p3.33.m2.1.1.1.3.2" stretchy="false" xref="S2.SS2a.p3.33.m2.1.2.cmml">)</mo></mrow></msubsup><annotation-xml encoding="MathML-Content" id="S2.SS2a.p3.33.m2.1b"><apply id="S2.SS2a.p3.33.m2.1.2.cmml" xref="S2.SS2a.p3.33.m2.1.2"><csymbol cd="ambiguous" id="S2.SS2a.p3.33.m2.1.2.1.cmml" xref="S2.SS2a.p3.33.m2.1.2">superscript</csymbol><apply id="S2.SS2a.p3.33.m2.1.2.2.cmml" xref="S2.SS2a.p3.33.m2.1.2"><csymbol cd="ambiguous" id="S2.SS2a.p3.33.m2.1.2.2.1.cmml" xref="S2.SS2a.p3.33.m2.1.2">subscript</csymbol><ci id="S2.SS2a.p3.33.m2.1.2.2.2.cmml" xref="S2.SS2a.p3.33.m2.1.2.2.2">𝐋</ci><cn id="S2.SS2a.p3.33.m2.1.2.2.3.cmml" type="integer" xref="S2.SS2a.p3.33.m2.1.2.2.3">2</cn></apply><ci id="S2.SS2a.p3.33.m2.1.1.1.1.cmml" xref="S2.SS2a.p3.33.m2.1.1.1.1">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2a.p3.33.m2.1c">\operatorname{\mathbf{L}}_{2}^{(t)}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2a.p3.33.m2.1d">bold_L start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_t ) end_POSTSUPERSCRIPT</annotation></semantics></math>, thus we replace <math alttext="\operatorname{\mathbf{L}}_{2}" class="ltx_Math" display="inline" id="S2.SS2a.p3.34.m3.1"><semantics id="S2.SS2a.p3.34.m3.1a"><msub id="S2.SS2a.p3.34.m3.1.1" xref="S2.SS2a.p3.34.m3.1.1.cmml"><mi id="S2.SS2a.p3.34.m3.1.1.2" xref="S2.SS2a.p3.34.m3.1.1.2.cmml">𝐋</mi><mn id="S2.SS2a.p3.34.m3.1.1.3" xref="S2.SS2a.p3.34.m3.1.1.3.cmml">2</mn></msub><annotation-xml encoding="MathML-Content" id="S2.SS2a.p3.34.m3.1b"><apply id="S2.SS2a.p3.34.m3.1.1.cmml" xref="S2.SS2a.p3.34.m3.1.1"><csymbol cd="ambiguous" id="S2.SS2a.p3.34.m3.1.1.1.cmml" xref="S2.SS2a.p3.34.m3.1.1">subscript</csymbol><ci id="S2.SS2a.p3.34.m3.1.1.2.cmml" xref="S2.SS2a.p3.34.m3.1.1.2">𝐋</ci><cn id="S2.SS2a.p3.34.m3.1.1.3.cmml" type="integer" xref="S2.SS2a.p3.34.m3.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2a.p3.34.m3.1c">\operatorname{\mathbf{L}}_{2}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2a.p3.34.m3.1d">bold_L start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math> with <math alttext="\operatorname{\mathbf{L}}_{2}^{(t)}" class="ltx_Math" display="inline" id="S2.SS2a.p3.35.m4.1"><semantics id="S2.SS2a.p3.35.m4.1a"><msubsup id="S2.SS2a.p3.35.m4.1.2" xref="S2.SS2a.p3.35.m4.1.2.cmml"><mi id="S2.SS2a.p3.35.m4.1.2.2.2" xref="S2.SS2a.p3.35.m4.1.2.2.2.cmml">𝐋</mi><mn id="S2.SS2a.p3.35.m4.1.2.2.3" xref="S2.SS2a.p3.35.m4.1.2.2.3.cmml">2</mn><mrow id="S2.SS2a.p3.35.m4.1.1.1.3" xref="S2.SS2a.p3.35.m4.1.2.cmml"><mo id="S2.SS2a.p3.35.m4.1.1.1.3.1" stretchy="false" xref="S2.SS2a.p3.35.m4.1.2.cmml">(</mo><mi id="S2.SS2a.p3.35.m4.1.1.1.1" xref="S2.SS2a.p3.35.m4.1.1.1.1.cmml">t</mi><mo id="S2.SS2a.p3.35.m4.1.1.1.3.2" stretchy="false" xref="S2.SS2a.p3.35.m4.1.2.cmml">)</mo></mrow></msubsup><annotation-xml encoding="MathML-Content" id="S2.SS2a.p3.35.m4.1b"><apply id="S2.SS2a.p3.35.m4.1.2.cmml" xref="S2.SS2a.p3.35.m4.1.2"><csymbol cd="ambiguous" id="S2.SS2a.p3.35.m4.1.2.1.cmml" xref="S2.SS2a.p3.35.m4.1.2">superscript</csymbol><apply id="S2.SS2a.p3.35.m4.1.2.2.cmml" xref="S2.SS2a.p3.35.m4.1.2"><csymbol cd="ambiguous" id="S2.SS2a.p3.35.m4.1.2.2.1.cmml" xref="S2.SS2a.p3.35.m4.1.2">subscript</csymbol><ci id="S2.SS2a.p3.35.m4.1.2.2.2.cmml" xref="S2.SS2a.p3.35.m4.1.2.2.2">𝐋</ci><cn id="S2.SS2a.p3.35.m4.1.2.2.3.cmml" type="integer" xref="S2.SS2a.p3.35.m4.1.2.2.3">2</cn></apply><ci id="S2.SS2a.p3.35.m4.1.1.1.1.cmml" xref="S2.SS2a.p3.35.m4.1.1.1.1">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2a.p3.35.m4.1c">\operatorname{\mathbf{L}}_{2}^{(t)}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2a.p3.35.m4.1d">bold_L start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_t ) end_POSTSUPERSCRIPT</annotation></semantics></math> in our expression for <math alttext="-\operatorname{\mathcal{Q}}_{(1)}(\operatorname{\mathbf{L}}_{1})" class="ltx_Math" display="inline" id="S2.SS2a.p3.36.m5.3"><semantics id="S2.SS2a.p3.36.m5.3a"><mrow id="S2.SS2a.p3.36.m5.3.3" xref="S2.SS2a.p3.36.m5.3.3.cmml"><mo id="S2.SS2a.p3.36.m5.3.3a" rspace="0.167em" xref="S2.SS2a.p3.36.m5.3.3.cmml">−</mo><mrow id="S2.SS2a.p3.36.m5.3.3.2.2" xref="S2.SS2a.p3.36.m5.3.3.2.3.cmml"><msub id="S2.SS2a.p3.36.m5.2.2.1.1.1" xref="S2.SS2a.p3.36.m5.2.2.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS2a.p3.36.m5.2.2.1.1.1.2" xref="S2.SS2a.p3.36.m5.2.2.1.1.1.2.cmml">𝒬</mi><mrow id="S2.SS2a.p3.36.m5.1.1.1.3" xref="S2.SS2a.p3.36.m5.2.2.1.1.1.cmml"><mo id="S2.SS2a.p3.36.m5.1.1.1.3.1" stretchy="false" xref="S2.SS2a.p3.36.m5.2.2.1.1.1.cmml">(</mo><mn id="S2.SS2a.p3.36.m5.1.1.1.1" xref="S2.SS2a.p3.36.m5.1.1.1.1.cmml">1</mn><mo id="S2.SS2a.p3.36.m5.1.1.1.3.2" stretchy="false" xref="S2.SS2a.p3.36.m5.2.2.1.1.1.cmml">)</mo></mrow></msub><mo id="S2.SS2a.p3.36.m5.3.3.2.2a" xref="S2.SS2a.p3.36.m5.3.3.2.3.cmml">⁡</mo><mrow id="S2.SS2a.p3.36.m5.3.3.2.2.2" xref="S2.SS2a.p3.36.m5.3.3.2.3.cmml"><mo id="S2.SS2a.p3.36.m5.3.3.2.2.2.2" stretchy="false" xref="S2.SS2a.p3.36.m5.3.3.2.3.cmml">(</mo><msub id="S2.SS2a.p3.36.m5.3.3.2.2.2.1" xref="S2.SS2a.p3.36.m5.3.3.2.2.2.1.cmml"><mi id="S2.SS2a.p3.36.m5.3.3.2.2.2.1.2" xref="S2.SS2a.p3.36.m5.3.3.2.2.2.1.2.cmml">𝐋</mi><mn id="S2.SS2a.p3.36.m5.3.3.2.2.2.1.3" xref="S2.SS2a.p3.36.m5.3.3.2.2.2.1.3.cmml">1</mn></msub><mo id="S2.SS2a.p3.36.m5.3.3.2.2.2.3" stretchy="false" xref="S2.SS2a.p3.36.m5.3.3.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2a.p3.36.m5.3b"><apply id="S2.SS2a.p3.36.m5.3.3.cmml" xref="S2.SS2a.p3.36.m5.3.3"><minus id="S2.SS2a.p3.36.m5.3.3.3.cmml" xref="S2.SS2a.p3.36.m5.3.3"></minus><apply id="S2.SS2a.p3.36.m5.3.3.2.3.cmml" xref="S2.SS2a.p3.36.m5.3.3.2.2"><apply id="S2.SS2a.p3.36.m5.2.2.1.1.1.cmml" xref="S2.SS2a.p3.36.m5.2.2.1.1.1"><csymbol cd="ambiguous" id="S2.SS2a.p3.36.m5.2.2.1.1.1.1.cmml" xref="S2.SS2a.p3.36.m5.2.2.1.1.1">subscript</csymbol><ci id="S2.SS2a.p3.36.m5.2.2.1.1.1.2.cmml" xref="S2.SS2a.p3.36.m5.2.2.1.1.1.2">𝒬</ci><cn id="S2.SS2a.p3.36.m5.1.1.1.1.cmml" type="integer" xref="S2.SS2a.p3.36.m5.1.1.1.1">1</cn></apply><apply id="S2.SS2a.p3.36.m5.3.3.2.2.2.1.cmml" xref="S2.SS2a.p3.36.m5.3.3.2.2.2.1"><csymbol cd="ambiguous" id="S2.SS2a.p3.36.m5.3.3.2.2.2.1.1.cmml" xref="S2.SS2a.p3.36.m5.3.3.2.2.2.1">subscript</csymbol><ci id="S2.SS2a.p3.36.m5.3.3.2.2.2.1.2.cmml" xref="S2.SS2a.p3.36.m5.3.3.2.2.2.1.2">𝐋</ci><cn id="S2.SS2a.p3.36.m5.3.3.2.2.2.1.3.cmml" type="integer" xref="S2.SS2a.p3.36.m5.3.3.2.2.2.1.3">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2a.p3.36.m5.3c">-\operatorname{\mathcal{Q}}_{(1)}(\operatorname{\mathbf{L}}_{1})</annotation><annotation encoding="application/x-llamapun" id="S2.SS2a.p3.36.m5.3d">- caligraphic_Q start_POSTSUBSCRIPT ( 1 ) end_POSTSUBSCRIPT ( bold_L start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT )</annotation></semantics></math>. Let</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="S3.EGx27"> <tbody id="S2.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_right ltx_eqn_cell"><math alttext="\displaystyle\operatorname{\mathbf{H}}_{(1)}^{(t)}" class="ltx_Math" display="inline" id="S2.Ex35.m1.2"><semantics id="S2.Ex35.m1.2a"><msubsup id="S2.Ex35.m1.2.3" xref="S2.Ex35.m1.2.3.cmml"><mi id="S2.Ex35.m1.2.3.2.2" xref="S2.Ex35.m1.2.3.2.2.cmml">𝐇</mi><mrow id="S2.Ex35.m1.1.1.1.3" xref="S2.Ex35.m1.2.3.cmml"><mo id="S2.Ex35.m1.1.1.1.3.1" stretchy="false" xref="S2.Ex35.m1.2.3.cmml">(</mo><mn id="S2.Ex35.m1.1.1.1.1" xref="S2.Ex35.m1.1.1.1.1.cmml">1</mn><mo id="S2.Ex35.m1.1.1.1.3.2" stretchy="false" xref="S2.Ex35.m1.2.3.cmml">)</mo></mrow><mrow id="S2.Ex35.m1.2.2.1.3" xref="S2.Ex35.m1.2.3.cmml"><mo id="S2.Ex35.m1.2.2.1.3.1" stretchy="false" xref="S2.Ex35.m1.2.3.cmml">(</mo><mi id="S2.Ex35.m1.2.2.1.1" xref="S2.Ex35.m1.2.2.1.1.cmml">t</mi><mo id="S2.Ex35.m1.2.2.1.3.2" stretchy="false" xref="S2.Ex35.m1.2.3.cmml">)</mo></mrow></msubsup><annotation-xml encoding="MathML-Content" id="S2.Ex35.m1.2b"><apply id="S2.Ex35.m1.2.3.cmml" xref="S2.Ex35.m1.2.3"><csymbol cd="ambiguous" id="S2.Ex35.m1.2.3.1.cmml" xref="S2.Ex35.m1.2.3">superscript</csymbol><apply id="S2.Ex35.m1.2.3.2.cmml" xref="S2.Ex35.m1.2.3"><csymbol cd="ambiguous" id="S2.Ex35.m1.2.3.2.1.cmml" xref="S2.Ex35.m1.2.3">subscript</csymbol><ci id="S2.Ex35.m1.2.3.2.2.cmml" xref="S2.Ex35.m1.2.3.2.2">𝐇</ci><cn id="S2.Ex35.m1.1.1.1.1.cmml" type="integer" xref="S2.Ex35.m1.1.1.1.1">1</cn></apply><ci id="S2.Ex35.m1.2.2.1.1.cmml" xref="S2.Ex35.m1.2.2.1.1">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Ex35.m1.2c">\displaystyle\operatorname{\mathbf{H}}_{(1)}^{(t)}</annotation><annotation encoding="application/x-llamapun" id="S2.Ex35.m1.2d">bold_H start_POSTSUBSCRIPT ( 1 ) end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_t ) end_POSTSUPERSCRIPT</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle=\sum_{i=1}^{n}\sum_{j=1}^{m_{i}}(\operatorname{\mathbf{I}}_{S_{1% }}\otimes\operatorname{\mathbf{Z}}_{ij(1)}\operatorname{\mathbf{L}}_{2}^{(t)})% \operatorname{{\mathbf{\Gamma}}}_{i(2)}^{(t)}(\operatorname{\mathbf{I}}_{S_{1}% }\otimes\operatorname{\mathbf{L}}_{2}^{(t)\top}\operatorname{\mathbf{Z}}_{ij(1% )}^{\top})," class="ltx_Math" display="inline" id="S2.Ex35.m2.7"><semantics id="S2.Ex35.m2.7a"><mrow id="S2.Ex35.m2.7.7.1" xref="S2.Ex35.m2.7.7.1.1.cmml"><mrow id="S2.Ex35.m2.7.7.1.1" xref="S2.Ex35.m2.7.7.1.1.cmml"><mi id="S2.Ex35.m2.7.7.1.1.5" xref="S2.Ex35.m2.7.7.1.1.5.cmml"></mi><mo id="S2.Ex35.m2.7.7.1.1.4" xref="S2.Ex35.m2.7.7.1.1.4.cmml">=</mo><mrow id="S2.Ex35.m2.7.7.1.1.3" xref="S2.Ex35.m2.7.7.1.1.3.cmml"><mstyle displaystyle="true" id="S2.Ex35.m2.7.7.1.1.3.4" xref="S2.Ex35.m2.7.7.1.1.3.4.cmml"><munderover id="S2.Ex35.m2.7.7.1.1.3.4a" xref="S2.Ex35.m2.7.7.1.1.3.4.cmml"><mo id="S2.Ex35.m2.7.7.1.1.3.4.2.2" movablelimits="false" xref="S2.Ex35.m2.7.7.1.1.3.4.2.2.cmml">∑</mo><mrow id="S2.Ex35.m2.7.7.1.1.3.4.2.3" xref="S2.Ex35.m2.7.7.1.1.3.4.2.3.cmml"><mi id="S2.Ex35.m2.7.7.1.1.3.4.2.3.2" xref="S2.Ex35.m2.7.7.1.1.3.4.2.3.2.cmml">i</mi><mo id="S2.Ex35.m2.7.7.1.1.3.4.2.3.1" xref="S2.Ex35.m2.7.7.1.1.3.4.2.3.1.cmml">=</mo><mn id="S2.Ex35.m2.7.7.1.1.3.4.2.3.3" xref="S2.Ex35.m2.7.7.1.1.3.4.2.3.3.cmml">1</mn></mrow><mi id="S2.Ex35.m2.7.7.1.1.3.4.3" xref="S2.Ex35.m2.7.7.1.1.3.4.3.cmml">n</mi></munderover></mstyle><mrow id="S2.Ex35.m2.7.7.1.1.3.3" xref="S2.Ex35.m2.7.7.1.1.3.3.cmml"><mstyle displaystyle="true" id="S2.Ex35.m2.7.7.1.1.3.3.4" xref="S2.Ex35.m2.7.7.1.1.3.3.4.cmml"><munderover id="S2.Ex35.m2.7.7.1.1.3.3.4a" xref="S2.Ex35.m2.7.7.1.1.3.3.4.cmml"><mo id="S2.Ex35.m2.7.7.1.1.3.3.4.2.2" movablelimits="false" xref="S2.Ex35.m2.7.7.1.1.3.3.4.2.2.cmml">∑</mo><mrow id="S2.Ex35.m2.7.7.1.1.3.3.4.2.3" 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}\otimes\operatorname{\mathbf{L}}_{2}^{(t)\top}\operatorname{\mathbf{Z}}_{ij(1% )}^{\top}),</annotation><annotation encoding="application/x-llamapun" id="S2.Ex35.m2.7d">= ∑ start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT ∑ start_POSTSUBSCRIPT italic_j = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_m start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUPERSCRIPT ( bold_I start_POSTSUBSCRIPT italic_S start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ⊗ bold_Z start_POSTSUBSCRIPT italic_i italic_j ( 1 ) end_POSTSUBSCRIPT bold_L start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_t ) end_POSTSUPERSCRIPT ) bold_Γ start_POSTSUBSCRIPT italic_i ( 2 ) end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_t ) end_POSTSUPERSCRIPT ( bold_I start_POSTSUBSCRIPT italic_S start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ⊗ bold_L start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_t ) ⊤ end_POSTSUPERSCRIPT bold_Z start_POSTSUBSCRIPT italic_i italic_j ( 1 ) end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ⊤ end_POSTSUPERSCRIPT ) ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="S2.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_right ltx_eqn_cell"><math alttext="\displaystyle\operatorname{\mathbf{g}}_{(1)}^{(t)}" class="ltx_Math" display="inline" id="S2.Ex36.m1.2"><semantics id="S2.Ex36.m1.2a"><msubsup id="S2.Ex36.m1.2.3" xref="S2.Ex36.m1.2.3.cmml"><mi id="S2.Ex36.m1.2.3.2.2" xref="S2.Ex36.m1.2.3.2.2.cmml">𝐠</mi><mrow id="S2.Ex36.m1.1.1.1.3" xref="S2.Ex36.m1.2.3.cmml"><mo id="S2.Ex36.m1.1.1.1.3.1" stretchy="false" xref="S2.Ex36.m1.2.3.cmml">(</mo><mn id="S2.Ex36.m1.1.1.1.1" xref="S2.Ex36.m1.1.1.1.1.cmml">1</mn><mo id="S2.Ex36.m1.1.1.1.3.2" stretchy="false" xref="S2.Ex36.m1.2.3.cmml">)</mo></mrow><mrow id="S2.Ex36.m1.2.2.1.3" xref="S2.Ex36.m1.2.3.cmml"><mo id="S2.Ex36.m1.2.2.1.3.1" stretchy="false" xref="S2.Ex36.m1.2.3.cmml">(</mo><mi id="S2.Ex36.m1.2.2.1.1" xref="S2.Ex36.m1.2.2.1.1.cmml">t</mi><mo id="S2.Ex36.m1.2.2.1.3.2" stretchy="false" xref="S2.Ex36.m1.2.3.cmml">)</mo></mrow></msubsup><annotation-xml encoding="MathML-Content" id="S2.Ex36.m1.2b"><apply id="S2.Ex36.m1.2.3.cmml" xref="S2.Ex36.m1.2.3"><csymbol cd="ambiguous" id="S2.Ex36.m1.2.3.1.cmml" xref="S2.Ex36.m1.2.3">superscript</csymbol><apply id="S2.Ex36.m1.2.3.2.cmml" xref="S2.Ex36.m1.2.3"><csymbol cd="ambiguous" id="S2.Ex36.m1.2.3.2.1.cmml" xref="S2.Ex36.m1.2.3">subscript</csymbol><ci id="S2.Ex36.m1.2.3.2.2.cmml" xref="S2.Ex36.m1.2.3.2.2">𝐠</ci><cn id="S2.Ex36.m1.1.1.1.1.cmml" type="integer" xref="S2.Ex36.m1.1.1.1.1">1</cn></apply><ci id="S2.Ex36.m1.2.2.1.1.cmml" xref="S2.Ex36.m1.2.2.1.1">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Ex36.m1.2c">\displaystyle\operatorname{\mathbf{g}}_{(1)}^{(t)}</annotation><annotation encoding="application/x-llamapun" id="S2.Ex36.m1.2d">bold_g start_POSTSUBSCRIPT ( 1 ) end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_t ) end_POSTSUPERSCRIPT</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle=\sum_{i=1}^{n}\left[\operatorname{\mathbf{I}}_{S_{1}}\otimes(% \sum_{j=1}^{m_{i}}\tilde{y}_{ij}\operatorname{\mathbf{Z}}_{ij(1)})% \operatorname{\mathbf{L}}_{2}^{(t)}\right]\operatorname{{\bm{\mu}}}_{i(2)}^{(t% )}," class="ltx_Math" display="inline" id="S2.Ex36.m2.5"><semantics id="S2.Ex36.m2.5a"><mrow id="S2.Ex36.m2.5.5.1" xref="S2.Ex36.m2.5.5.1.1.cmml"><mrow id="S2.Ex36.m2.5.5.1.1" xref="S2.Ex36.m2.5.5.1.1.cmml"><mi id="S2.Ex36.m2.5.5.1.1.3" xref="S2.Ex36.m2.5.5.1.1.3.cmml"></mi><mo id="S2.Ex36.m2.5.5.1.1.2" xref="S2.Ex36.m2.5.5.1.1.2.cmml">=</mo><mrow id="S2.Ex36.m2.5.5.1.1.1" xref="S2.Ex36.m2.5.5.1.1.1.cmml"><mstyle 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\operatorname{\mathbf{L}}_{2}^{(t)}\right]\operatorname{{\bm{\mu}}}_{i(2)}^{(t% )},</annotation><annotation encoding="application/x-llamapun" id="S2.Ex36.m2.5d">= ∑ start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT [ bold_I start_POSTSUBSCRIPT italic_S start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ⊗ ( ∑ start_POSTSUBSCRIPT italic_j = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_m start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUPERSCRIPT over~ start_ARG italic_y end_ARG start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT bold_Z start_POSTSUBSCRIPT italic_i italic_j ( 1 ) end_POSTSUBSCRIPT ) bold_L start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_t ) end_POSTSUPERSCRIPT ] bold_italic_μ start_POSTSUBSCRIPT italic_i ( 2 ) end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_t ) end_POSTSUPERSCRIPT ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S2.SS2a.p3.37">then <math alttext="-\operatorname{\mathcal{Q}}_{(1)}(\operatorname{\mathbf{L}}_{1})=\operatorname% {\mathbf{l}}_{1}^{\top}\operatorname{\mathbf{H}}_{(1)}^{(t)}\operatorname{% \mathbf{l}}_{1}-2\operatorname{\mathbf{g}}_{(1)}^{(t)\top}\operatorname{% \mathbf{l}}_{1}" class="ltx_Math" display="inline" id="S2.SS2a.p3.37.m1.7"><semantics id="S2.SS2a.p3.37.m1.7a"><mrow id="S2.SS2a.p3.37.m1.7.7" xref="S2.SS2a.p3.37.m1.7.7.cmml"><mrow id="S2.SS2a.p3.37.m1.7.7.2" xref="S2.SS2a.p3.37.m1.7.7.2.cmml"><mo id="S2.SS2a.p3.37.m1.7.7.2a" rspace="0.167em" xref="S2.SS2a.p3.37.m1.7.7.2.cmml">−</mo><mrow id="S2.SS2a.p3.37.m1.7.7.2.2.2" xref="S2.SS2a.p3.37.m1.7.7.2.2.3.cmml"><msub id="S2.SS2a.p3.37.m1.6.6.1.1.1.1" xref="S2.SS2a.p3.37.m1.6.6.1.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS2a.p3.37.m1.6.6.1.1.1.1.2" xref="S2.SS2a.p3.37.m1.6.6.1.1.1.1.2.cmml">𝒬</mi><mrow id="S2.SS2a.p3.37.m1.1.1.1.3" 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xref="S2.SS2a.p3.37.m1.7.7.4.3.3.2.2">𝐥</ci><cn id="S2.SS2a.p3.37.m1.7.7.4.3.3.2.3.cmml" type="integer" xref="S2.SS2a.p3.37.m1.7.7.4.3.3.2.3">1</cn></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2a.p3.37.m1.7c">-\operatorname{\mathcal{Q}}_{(1)}(\operatorname{\mathbf{L}}_{1})=\operatorname% {\mathbf{l}}_{1}^{\top}\operatorname{\mathbf{H}}_{(1)}^{(t)}\operatorname{% \mathbf{l}}_{1}-2\operatorname{\mathbf{g}}_{(1)}^{(t)\top}\operatorname{% \mathbf{l}}_{1}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2a.p3.37.m1.7d">- caligraphic_Q start_POSTSUBSCRIPT ( 1 ) end_POSTSUBSCRIPT ( bold_L start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) = bold_l start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ⊤ end_POSTSUPERSCRIPT bold_H start_POSTSUBSCRIPT ( 1 ) end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_t ) end_POSTSUPERSCRIPT bold_l start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT - 2 bold_g start_POSTSUBSCRIPT ( 1 ) end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_t ) ⊤ end_POSTSUPERSCRIPT bold_l start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math>, which is expressed as the quadratic form</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="S3.EGx28"> <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-\operatorname{\mathcal{Q}}_{(1)}(\operatorname{\mathbf{L}}_{1})% \propto(\operatorname{\mathbf{H}}_{(1)}^{-1/2(t)}\operatorname{\mathbf{g}}_{(1% )}^{(t)}-\operatorname{\mathbf{H}}_{(1)}^{1/2(t)\top}\operatorname{\mathbf{l}}% _{1})^{\top}(\operatorname{\mathbf{H}}_{(1)}^{-1/2(t)}\operatorname{\mathbf{g}% }_{(1)}^{(t)}-\operatorname{\mathbf{H}}_{(1)}^{1/2(t)\top}\operatorname{% \mathbf{l}}_{1})," class="ltx_Math" display="inline" id="S2.E7.m1.14"><semantics id="S2.E7.m1.14a"><mrow id="S2.E7.m1.14.14.1" 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id="S2.E7.m1.14c">\displaystyle-\operatorname{\mathcal{Q}}_{(1)}(\operatorname{\mathbf{L}}_{1})% \propto(\operatorname{\mathbf{H}}_{(1)}^{-1/2(t)}\operatorname{\mathbf{g}}_{(1% )}^{(t)}-\operatorname{\mathbf{H}}_{(1)}^{1/2(t)\top}\operatorname{\mathbf{l}}% _{1})^{\top}(\operatorname{\mathbf{H}}_{(1)}^{-1/2(t)}\operatorname{\mathbf{g}% }_{(1)}^{(t)}-\operatorname{\mathbf{H}}_{(1)}^{1/2(t)\top}\operatorname{% \mathbf{l}}_{1}),</annotation><annotation encoding="application/x-llamapun" id="S2.E7.m1.14d">- caligraphic_Q start_POSTSUBSCRIPT ( 1 ) end_POSTSUBSCRIPT ( bold_L start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) ∝ ( bold_H start_POSTSUBSCRIPT ( 1 ) end_POSTSUBSCRIPT start_POSTSUPERSCRIPT - 1 / 2 ( italic_t ) end_POSTSUPERSCRIPT bold_g start_POSTSUBSCRIPT ( 1 ) end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_t ) end_POSTSUPERSCRIPT - bold_H start_POSTSUBSCRIPT ( 1 ) end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 1 / 2 ( italic_t ) ⊤ end_POSTSUPERSCRIPT bold_l start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT ⊤ end_POSTSUPERSCRIPT ( bold_H start_POSTSUBSCRIPT ( 1 ) end_POSTSUBSCRIPT start_POSTSUPERSCRIPT - 1 / 2 ( italic_t ) end_POSTSUPERSCRIPT bold_g start_POSTSUBSCRIPT ( 1 ) end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_t ) end_POSTSUPERSCRIPT - bold_H start_POSTSUBSCRIPT ( 1 ) end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 1 / 2 ( italic_t ) ⊤ end_POSTSUPERSCRIPT bold_l start_POSTSUBSCRIPT 1 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">(7)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S2.SS2a.p3.48">where <math alttext="\operatorname{\mathbf{H}}_{(1)}^{1/2(t)}" class="ltx_Math" display="inline" id="S2.SS2a.p3.38.m1.2"><semantics id="S2.SS2a.p3.38.m1.2a"><msubsup id="S2.SS2a.p3.38.m1.2.3" xref="S2.SS2a.p3.38.m1.2.3.cmml"><mi 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xref="S2.SS2a.p3.38.m1.2.2.1.2"></times><apply id="S2.SS2a.p3.38.m1.2.2.1.3.cmml" xref="S2.SS2a.p3.38.m1.2.2.1.3"><divide id="S2.SS2a.p3.38.m1.2.2.1.3.1.cmml" xref="S2.SS2a.p3.38.m1.2.2.1.3.1"></divide><cn id="S2.SS2a.p3.38.m1.2.2.1.3.2.cmml" type="integer" xref="S2.SS2a.p3.38.m1.2.2.1.3.2">1</cn><cn id="S2.SS2a.p3.38.m1.2.2.1.3.3.cmml" type="integer" xref="S2.SS2a.p3.38.m1.2.2.1.3.3">2</cn></apply><ci id="S2.SS2a.p3.38.m1.2.2.1.1.cmml" xref="S2.SS2a.p3.38.m1.2.2.1.1">𝑡</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2a.p3.38.m1.2c">\operatorname{\mathbf{H}}_{(1)}^{1/2(t)}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2a.p3.38.m1.2d">bold_H start_POSTSUBSCRIPT ( 1 ) end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 1 / 2 ( italic_t ) end_POSTSUPERSCRIPT</annotation></semantics></math> is any matrix such that <math alttext="\operatorname{\mathbf{H}}_{(1)}^{1/2(t)}\operatorname{\mathbf{H}}_{(1)}^{1/2(t% 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id="S2.SS2a.p3.39.m2.6c">\operatorname{\mathbf{H}}_{(1)}^{1/2(t)}\operatorname{\mathbf{H}}_{(1)}^{1/2(t% )\top}=\operatorname{\mathbf{H}}_{(1)}^{(t)}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2a.p3.39.m2.6d">bold_H start_POSTSUBSCRIPT ( 1 ) end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 1 / 2 ( italic_t ) end_POSTSUPERSCRIPT bold_H start_POSTSUBSCRIPT ( 1 ) end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 1 / 2 ( italic_t ) ⊤ end_POSTSUPERSCRIPT = bold_H start_POSTSUBSCRIPT ( 1 ) end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_t ) end_POSTSUPERSCRIPT</annotation></semantics></math> and <math alttext="\operatorname{\mathbf{H}}_{(1)}^{-1/2(t)}" class="ltx_Math" display="inline" id="S2.SS2a.p3.40.m3.2"><semantics id="S2.SS2a.p3.40.m3.2a"><msubsup id="S2.SS2a.p3.40.m3.2.3" xref="S2.SS2a.p3.40.m3.2.3.cmml"><mi id="S2.SS2a.p3.40.m3.2.3.2.2" xref="S2.SS2a.p3.40.m3.2.3.2.2.cmml">𝐇</mi><mrow id="S2.SS2a.p3.40.m3.1.1.1.3" xref="S2.SS2a.p3.40.m3.2.3.cmml"><mo id="S2.SS2a.p3.40.m3.1.1.1.3.1" stretchy="false" xref="S2.SS2a.p3.40.m3.2.3.cmml">(</mo><mn id="S2.SS2a.p3.40.m3.1.1.1.1" xref="S2.SS2a.p3.40.m3.1.1.1.1.cmml">1</mn><mo id="S2.SS2a.p3.40.m3.1.1.1.3.2" stretchy="false" xref="S2.SS2a.p3.40.m3.2.3.cmml">)</mo></mrow><mrow id="S2.SS2a.p3.40.m3.2.2.1" xref="S2.SS2a.p3.40.m3.2.2.1.cmml"><mo id="S2.SS2a.p3.40.m3.2.2.1a" xref="S2.SS2a.p3.40.m3.2.2.1.cmml">−</mo><mrow id="S2.SS2a.p3.40.m3.2.2.1.3" xref="S2.SS2a.p3.40.m3.2.2.1.3.cmml"><mrow id="S2.SS2a.p3.40.m3.2.2.1.3.2" xref="S2.SS2a.p3.40.m3.2.2.1.3.2.cmml"><mn id="S2.SS2a.p3.40.m3.2.2.1.3.2.2" xref="S2.SS2a.p3.40.m3.2.2.1.3.2.2.cmml">1</mn><mo id="S2.SS2a.p3.40.m3.2.2.1.3.2.1" xref="S2.SS2a.p3.40.m3.2.2.1.3.2.1.cmml">/</mo><mn id="S2.SS2a.p3.40.m3.2.2.1.3.2.3" xref="S2.SS2a.p3.40.m3.2.2.1.3.2.3.cmml">2</mn></mrow><mo id="S2.SS2a.p3.40.m3.2.2.1.3.1" xref="S2.SS2a.p3.40.m3.2.2.1.3.1.cmml">⁢</mo><mrow id="S2.SS2a.p3.40.m3.2.2.1.3.3.2" xref="S2.SS2a.p3.40.m3.2.2.1.3.cmml"><mo id="S2.SS2a.p3.40.m3.2.2.1.3.3.2.1" stretchy="false" xref="S2.SS2a.p3.40.m3.2.2.1.3.cmml">(</mo><mi id="S2.SS2a.p3.40.m3.2.2.1.1" xref="S2.SS2a.p3.40.m3.2.2.1.1.cmml">t</mi><mo id="S2.SS2a.p3.40.m3.2.2.1.3.3.2.2" stretchy="false" xref="S2.SS2a.p3.40.m3.2.2.1.3.cmml">)</mo></mrow></mrow></mrow></msubsup><annotation-xml encoding="MathML-Content" id="S2.SS2a.p3.40.m3.2b"><apply id="S2.SS2a.p3.40.m3.2.3.cmml" xref="S2.SS2a.p3.40.m3.2.3"><csymbol cd="ambiguous" id="S2.SS2a.p3.40.m3.2.3.1.cmml" xref="S2.SS2a.p3.40.m3.2.3">superscript</csymbol><apply id="S2.SS2a.p3.40.m3.2.3.2.cmml" xref="S2.SS2a.p3.40.m3.2.3"><csymbol cd="ambiguous" id="S2.SS2a.p3.40.m3.2.3.2.1.cmml" xref="S2.SS2a.p3.40.m3.2.3">subscript</csymbol><ci id="S2.SS2a.p3.40.m3.2.3.2.2.cmml" xref="S2.SS2a.p3.40.m3.2.3.2.2">𝐇</ci><cn id="S2.SS2a.p3.40.m3.1.1.1.1.cmml" type="integer" xref="S2.SS2a.p3.40.m3.1.1.1.1">1</cn></apply><apply id="S2.SS2a.p3.40.m3.2.2.1.cmml" xref="S2.SS2a.p3.40.m3.2.2.1"><minus id="S2.SS2a.p3.40.m3.2.2.1.2.cmml" xref="S2.SS2a.p3.40.m3.2.2.1"></minus><apply id="S2.SS2a.p3.40.m3.2.2.1.3.cmml" xref="S2.SS2a.p3.40.m3.2.2.1.3"><times id="S2.SS2a.p3.40.m3.2.2.1.3.1.cmml" xref="S2.SS2a.p3.40.m3.2.2.1.3.1"></times><apply id="S2.SS2a.p3.40.m3.2.2.1.3.2.cmml" xref="S2.SS2a.p3.40.m3.2.2.1.3.2"><divide id="S2.SS2a.p3.40.m3.2.2.1.3.2.1.cmml" xref="S2.SS2a.p3.40.m3.2.2.1.3.2.1"></divide><cn id="S2.SS2a.p3.40.m3.2.2.1.3.2.2.cmml" type="integer" xref="S2.SS2a.p3.40.m3.2.2.1.3.2.2">1</cn><cn id="S2.SS2a.p3.40.m3.2.2.1.3.2.3.cmml" type="integer" xref="S2.SS2a.p3.40.m3.2.2.1.3.2.3">2</cn></apply><ci id="S2.SS2a.p3.40.m3.2.2.1.1.cmml" xref="S2.SS2a.p3.40.m3.2.2.1.1">𝑡</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2a.p3.40.m3.2c">\operatorname{\mathbf{H}}_{(1)}^{-1/2(t)}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2a.p3.40.m3.2d">bold_H start_POSTSUBSCRIPT ( 1 ) end_POSTSUBSCRIPT start_POSTSUPERSCRIPT - 1 / 2 ( italic_t ) end_POSTSUPERSCRIPT</annotation></semantics></math> is any generalized inverse of <math alttext="\operatorname{\mathbf{H}}_{(1)}^{1/2(t)}" class="ltx_Math" display="inline" id="S2.SS2a.p3.41.m4.2"><semantics id="S2.SS2a.p3.41.m4.2a"><msubsup id="S2.SS2a.p3.41.m4.2.3" xref="S2.SS2a.p3.41.m4.2.3.cmml"><mi id="S2.SS2a.p3.41.m4.2.3.2.2" xref="S2.SS2a.p3.41.m4.2.3.2.2.cmml">𝐇</mi><mrow id="S2.SS2a.p3.41.m4.1.1.1.3" xref="S2.SS2a.p3.41.m4.2.3.cmml"><mo id="S2.SS2a.p3.41.m4.1.1.1.3.1" stretchy="false" xref="S2.SS2a.p3.41.m4.2.3.cmml">(</mo><mn id="S2.SS2a.p3.41.m4.1.1.1.1" xref="S2.SS2a.p3.41.m4.1.1.1.1.cmml">1</mn><mo id="S2.SS2a.p3.41.m4.1.1.1.3.2" stretchy="false" xref="S2.SS2a.p3.41.m4.2.3.cmml">)</mo></mrow><mrow id="S2.SS2a.p3.41.m4.2.2.1" xref="S2.SS2a.p3.41.m4.2.2.1.cmml"><mrow id="S2.SS2a.p3.41.m4.2.2.1.3" xref="S2.SS2a.p3.41.m4.2.2.1.3.cmml"><mn id="S2.SS2a.p3.41.m4.2.2.1.3.2" xref="S2.SS2a.p3.41.m4.2.2.1.3.2.cmml">1</mn><mo id="S2.SS2a.p3.41.m4.2.2.1.3.1" xref="S2.SS2a.p3.41.m4.2.2.1.3.1.cmml">/</mo><mn id="S2.SS2a.p3.41.m4.2.2.1.3.3" xref="S2.SS2a.p3.41.m4.2.2.1.3.3.cmml">2</mn></mrow><mo id="S2.SS2a.p3.41.m4.2.2.1.2" xref="S2.SS2a.p3.41.m4.2.2.1.2.cmml">⁢</mo><mrow id="S2.SS2a.p3.41.m4.2.2.1.4.2" xref="S2.SS2a.p3.41.m4.2.2.1.cmml"><mo id="S2.SS2a.p3.41.m4.2.2.1.4.2.1" stretchy="false" xref="S2.SS2a.p3.41.m4.2.2.1.cmml">(</mo><mi id="S2.SS2a.p3.41.m4.2.2.1.1" xref="S2.SS2a.p3.41.m4.2.2.1.1.cmml">t</mi><mo id="S2.SS2a.p3.41.m4.2.2.1.4.2.2" stretchy="false" xref="S2.SS2a.p3.41.m4.2.2.1.cmml">)</mo></mrow></mrow></msubsup><annotation-xml encoding="MathML-Content" id="S2.SS2a.p3.41.m4.2b"><apply id="S2.SS2a.p3.41.m4.2.3.cmml" xref="S2.SS2a.p3.41.m4.2.3"><csymbol cd="ambiguous" id="S2.SS2a.p3.41.m4.2.3.1.cmml" xref="S2.SS2a.p3.41.m4.2.3">superscript</csymbol><apply id="S2.SS2a.p3.41.m4.2.3.2.cmml" xref="S2.SS2a.p3.41.m4.2.3"><csymbol cd="ambiguous" id="S2.SS2a.p3.41.m4.2.3.2.1.cmml" xref="S2.SS2a.p3.41.m4.2.3">subscript</csymbol><ci id="S2.SS2a.p3.41.m4.2.3.2.2.cmml" xref="S2.SS2a.p3.41.m4.2.3.2.2">𝐇</ci><cn id="S2.SS2a.p3.41.m4.1.1.1.1.cmml" type="integer" xref="S2.SS2a.p3.41.m4.1.1.1.1">1</cn></apply><apply id="S2.SS2a.p3.41.m4.2.2.1.cmml" xref="S2.SS2a.p3.41.m4.2.2.1"><times id="S2.SS2a.p3.41.m4.2.2.1.2.cmml" xref="S2.SS2a.p3.41.m4.2.2.1.2"></times><apply id="S2.SS2a.p3.41.m4.2.2.1.3.cmml" xref="S2.SS2a.p3.41.m4.2.2.1.3"><divide id="S2.SS2a.p3.41.m4.2.2.1.3.1.cmml" xref="S2.SS2a.p3.41.m4.2.2.1.3.1"></divide><cn id="S2.SS2a.p3.41.m4.2.2.1.3.2.cmml" type="integer" xref="S2.SS2a.p3.41.m4.2.2.1.3.2">1</cn><cn id="S2.SS2a.p3.41.m4.2.2.1.3.3.cmml" type="integer" xref="S2.SS2a.p3.41.m4.2.2.1.3.3">2</cn></apply><ci id="S2.SS2a.p3.41.m4.2.2.1.1.cmml" xref="S2.SS2a.p3.41.m4.2.2.1.1">𝑡</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2a.p3.41.m4.2c">\operatorname{\mathbf{H}}_{(1)}^{1/2(t)}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2a.p3.41.m4.2d">bold_H start_POSTSUBSCRIPT ( 1 ) end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 1 / 2 ( italic_t ) end_POSTSUPERSCRIPT</annotation></semantics></math>. It can be shown that <math alttext="\operatorname{\mathbf{g}}_{(1)}^{(t)}" class="ltx_Math" display="inline" id="S2.SS2a.p3.42.m5.2"><semantics id="S2.SS2a.p3.42.m5.2a"><msubsup id="S2.SS2a.p3.42.m5.2.3" xref="S2.SS2a.p3.42.m5.2.3.cmml"><mi id="S2.SS2a.p3.42.m5.2.3.2.2" xref="S2.SS2a.p3.42.m5.2.3.2.2.cmml">𝐠</mi><mrow id="S2.SS2a.p3.42.m5.1.1.1.3" xref="S2.SS2a.p3.42.m5.2.3.cmml"><mo id="S2.SS2a.p3.42.m5.1.1.1.3.1" stretchy="false" xref="S2.SS2a.p3.42.m5.2.3.cmml">(</mo><mn id="S2.SS2a.p3.42.m5.1.1.1.1" xref="S2.SS2a.p3.42.m5.1.1.1.1.cmml">1</mn><mo id="S2.SS2a.p3.42.m5.1.1.1.3.2" stretchy="false" xref="S2.SS2a.p3.42.m5.2.3.cmml">)</mo></mrow><mrow id="S2.SS2a.p3.42.m5.2.2.1.3" xref="S2.SS2a.p3.42.m5.2.3.cmml"><mo id="S2.SS2a.p3.42.m5.2.2.1.3.1" stretchy="false" xref="S2.SS2a.p3.42.m5.2.3.cmml">(</mo><mi id="S2.SS2a.p3.42.m5.2.2.1.1" xref="S2.SS2a.p3.42.m5.2.2.1.1.cmml">t</mi><mo id="S2.SS2a.p3.42.m5.2.2.1.3.2" stretchy="false" xref="S2.SS2a.p3.42.m5.2.3.cmml">)</mo></mrow></msubsup><annotation-xml encoding="MathML-Content" id="S2.SS2a.p3.42.m5.2b"><apply id="S2.SS2a.p3.42.m5.2.3.cmml" xref="S2.SS2a.p3.42.m5.2.3"><csymbol cd="ambiguous" id="S2.SS2a.p3.42.m5.2.3.1.cmml" xref="S2.SS2a.p3.42.m5.2.3">superscript</csymbol><apply id="S2.SS2a.p3.42.m5.2.3.2.cmml" xref="S2.SS2a.p3.42.m5.2.3"><csymbol cd="ambiguous" id="S2.SS2a.p3.42.m5.2.3.2.1.cmml" xref="S2.SS2a.p3.42.m5.2.3">subscript</csymbol><ci id="S2.SS2a.p3.42.m5.2.3.2.2.cmml" xref="S2.SS2a.p3.42.m5.2.3.2.2">𝐠</ci><cn id="S2.SS2a.p3.42.m5.1.1.1.1.cmml" type="integer" xref="S2.SS2a.p3.42.m5.1.1.1.1">1</cn></apply><ci id="S2.SS2a.p3.42.m5.2.2.1.1.cmml" xref="S2.SS2a.p3.42.m5.2.2.1.1">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2a.p3.42.m5.2c">\operatorname{\mathbf{g}}_{(1)}^{(t)}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2a.p3.42.m5.2d">bold_g start_POSTSUBSCRIPT ( 1 ) end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_t ) end_POSTSUPERSCRIPT</annotation></semantics></math> is in the column space of <math alttext="\operatorname{\mathbf{H}}_{(1)}^{(t)}" class="ltx_Math" display="inline" id="S2.SS2a.p3.43.m6.2"><semantics id="S2.SS2a.p3.43.m6.2a"><msubsup id="S2.SS2a.p3.43.m6.2.3" xref="S2.SS2a.p3.43.m6.2.3.cmml"><mi id="S2.SS2a.p3.43.m6.2.3.2.2" xref="S2.SS2a.p3.43.m6.2.3.2.2.cmml">𝐇</mi><mrow id="S2.SS2a.p3.43.m6.1.1.1.3" xref="S2.SS2a.p3.43.m6.2.3.cmml"><mo id="S2.SS2a.p3.43.m6.1.1.1.3.1" stretchy="false" xref="S2.SS2a.p3.43.m6.2.3.cmml">(</mo><mn id="S2.SS2a.p3.43.m6.1.1.1.1" xref="S2.SS2a.p3.43.m6.1.1.1.1.cmml">1</mn><mo id="S2.SS2a.p3.43.m6.1.1.1.3.2" stretchy="false" xref="S2.SS2a.p3.43.m6.2.3.cmml">)</mo></mrow><mrow id="S2.SS2a.p3.43.m6.2.2.1.3" xref="S2.SS2a.p3.43.m6.2.3.cmml"><mo id="S2.SS2a.p3.43.m6.2.2.1.3.1" stretchy="false" xref="S2.SS2a.p3.43.m6.2.3.cmml">(</mo><mi id="S2.SS2a.p3.43.m6.2.2.1.1" xref="S2.SS2a.p3.43.m6.2.2.1.1.cmml">t</mi><mo id="S2.SS2a.p3.43.m6.2.2.1.3.2" stretchy="false" xref="S2.SS2a.p3.43.m6.2.3.cmml">)</mo></mrow></msubsup><annotation-xml encoding="MathML-Content" id="S2.SS2a.p3.43.m6.2b"><apply id="S2.SS2a.p3.43.m6.2.3.cmml" xref="S2.SS2a.p3.43.m6.2.3"><csymbol cd="ambiguous" id="S2.SS2a.p3.43.m6.2.3.1.cmml" xref="S2.SS2a.p3.43.m6.2.3">superscript</csymbol><apply id="S2.SS2a.p3.43.m6.2.3.2.cmml" xref="S2.SS2a.p3.43.m6.2.3"><csymbol cd="ambiguous" id="S2.SS2a.p3.43.m6.2.3.2.1.cmml" xref="S2.SS2a.p3.43.m6.2.3">subscript</csymbol><ci id="S2.SS2a.p3.43.m6.2.3.2.2.cmml" xref="S2.SS2a.p3.43.m6.2.3.2.2">𝐇</ci><cn id="S2.SS2a.p3.43.m6.1.1.1.1.cmml" type="integer" xref="S2.SS2a.p3.43.m6.1.1.1.1">1</cn></apply><ci id="S2.SS2a.p3.43.m6.2.2.1.1.cmml" xref="S2.SS2a.p3.43.m6.2.2.1.1">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2a.p3.43.m6.2c">\operatorname{\mathbf{H}}_{(1)}^{(t)}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2a.p3.43.m6.2d">bold_H start_POSTSUBSCRIPT ( 1 ) end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_t ) end_POSTSUPERSCRIPT</annotation></semantics></math>, and thus minimizing <math alttext="-\operatorname{\mathcal{Q}}_{(1)}(\operatorname{\mathbf{L}}_{1})" class="ltx_Math" display="inline" id="S2.SS2a.p3.44.m7.3"><semantics id="S2.SS2a.p3.44.m7.3a"><mrow id="S2.SS2a.p3.44.m7.3.3" xref="S2.SS2a.p3.44.m7.3.3.cmml"><mo id="S2.SS2a.p3.44.m7.3.3a" rspace="0.167em" xref="S2.SS2a.p3.44.m7.3.3.cmml">−</mo><mrow id="S2.SS2a.p3.44.m7.3.3.2.2" xref="S2.SS2a.p3.44.m7.3.3.2.3.cmml"><msub id="S2.SS2a.p3.44.m7.2.2.1.1.1" xref="S2.SS2a.p3.44.m7.2.2.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS2a.p3.44.m7.2.2.1.1.1.2" xref="S2.SS2a.p3.44.m7.2.2.1.1.1.2.cmml">𝒬</mi><mrow id="S2.SS2a.p3.44.m7.1.1.1.3" xref="S2.SS2a.p3.44.m7.2.2.1.1.1.cmml"><mo id="S2.SS2a.p3.44.m7.1.1.1.3.1" stretchy="false" xref="S2.SS2a.p3.44.m7.2.2.1.1.1.cmml">(</mo><mn id="S2.SS2a.p3.44.m7.1.1.1.1" xref="S2.SS2a.p3.44.m7.1.1.1.1.cmml">1</mn><mo id="S2.SS2a.p3.44.m7.1.1.1.3.2" stretchy="false" xref="S2.SS2a.p3.44.m7.2.2.1.1.1.cmml">)</mo></mrow></msub><mo id="S2.SS2a.p3.44.m7.3.3.2.2a" xref="S2.SS2a.p3.44.m7.3.3.2.3.cmml">⁡</mo><mrow id="S2.SS2a.p3.44.m7.3.3.2.2.2" xref="S2.SS2a.p3.44.m7.3.3.2.3.cmml"><mo id="S2.SS2a.p3.44.m7.3.3.2.2.2.2" stretchy="false" xref="S2.SS2a.p3.44.m7.3.3.2.3.cmml">(</mo><msub id="S2.SS2a.p3.44.m7.3.3.2.2.2.1" xref="S2.SS2a.p3.44.m7.3.3.2.2.2.1.cmml"><mi id="S2.SS2a.p3.44.m7.3.3.2.2.2.1.2" xref="S2.SS2a.p3.44.m7.3.3.2.2.2.1.2.cmml">𝐋</mi><mn id="S2.SS2a.p3.44.m7.3.3.2.2.2.1.3" xref="S2.SS2a.p3.44.m7.3.3.2.2.2.1.3.cmml">1</mn></msub><mo id="S2.SS2a.p3.44.m7.3.3.2.2.2.3" stretchy="false" xref="S2.SS2a.p3.44.m7.3.3.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2a.p3.44.m7.3b"><apply id="S2.SS2a.p3.44.m7.3.3.cmml" xref="S2.SS2a.p3.44.m7.3.3"><minus id="S2.SS2a.p3.44.m7.3.3.3.cmml" xref="S2.SS2a.p3.44.m7.3.3"></minus><apply id="S2.SS2a.p3.44.m7.3.3.2.3.cmml" xref="S2.SS2a.p3.44.m7.3.3.2.2"><apply id="S2.SS2a.p3.44.m7.2.2.1.1.1.cmml" xref="S2.SS2a.p3.44.m7.2.2.1.1.1"><csymbol cd="ambiguous" id="S2.SS2a.p3.44.m7.2.2.1.1.1.1.cmml" xref="S2.SS2a.p3.44.m7.2.2.1.1.1">subscript</csymbol><ci id="S2.SS2a.p3.44.m7.2.2.1.1.1.2.cmml" xref="S2.SS2a.p3.44.m7.2.2.1.1.1.2">𝒬</ci><cn id="S2.SS2a.p3.44.m7.1.1.1.1.cmml" type="integer" xref="S2.SS2a.p3.44.m7.1.1.1.1">1</cn></apply><apply id="S2.SS2a.p3.44.m7.3.3.2.2.2.1.cmml" xref="S2.SS2a.p3.44.m7.3.3.2.2.2.1"><csymbol cd="ambiguous" id="S2.SS2a.p3.44.m7.3.3.2.2.2.1.1.cmml" xref="S2.SS2a.p3.44.m7.3.3.2.2.2.1">subscript</csymbol><ci id="S2.SS2a.p3.44.m7.3.3.2.2.2.1.2.cmml" xref="S2.SS2a.p3.44.m7.3.3.2.2.2.1.2">𝐋</ci><cn id="S2.SS2a.p3.44.m7.3.3.2.2.2.1.3.cmml" type="integer" xref="S2.SS2a.p3.44.m7.3.3.2.2.2.1.3">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2a.p3.44.m7.3c">-\operatorname{\mathcal{Q}}_{(1)}(\operatorname{\mathbf{L}}_{1})</annotation><annotation encoding="application/x-llamapun" id="S2.SS2a.p3.44.m7.3d">- caligraphic_Q start_POSTSUBSCRIPT ( 1 ) end_POSTSUBSCRIPT ( bold_L start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT )</annotation></semantics></math> with respect to <math alttext="\operatorname{\mathbf{l}}_{1}" class="ltx_Math" display="inline" id="S2.SS2a.p3.45.m8.1"><semantics id="S2.SS2a.p3.45.m8.1a"><msub id="S2.SS2a.p3.45.m8.1.1" xref="S2.SS2a.p3.45.m8.1.1.cmml"><mi id="S2.SS2a.p3.45.m8.1.1.2" xref="S2.SS2a.p3.45.m8.1.1.2.cmml">𝐥</mi><mn id="S2.SS2a.p3.45.m8.1.1.3" xref="S2.SS2a.p3.45.m8.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S2.SS2a.p3.45.m8.1b"><apply id="S2.SS2a.p3.45.m8.1.1.cmml" xref="S2.SS2a.p3.45.m8.1.1"><csymbol cd="ambiguous" id="S2.SS2a.p3.45.m8.1.1.1.cmml" xref="S2.SS2a.p3.45.m8.1.1">subscript</csymbol><ci id="S2.SS2a.p3.45.m8.1.1.2.cmml" xref="S2.SS2a.p3.45.m8.1.1.2">𝐥</ci><cn id="S2.SS2a.p3.45.m8.1.1.3.cmml" type="integer" xref="S2.SS2a.p3.45.m8.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2a.p3.45.m8.1c">\operatorname{\mathbf{l}}_{1}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2a.p3.45.m8.1d">bold_l start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> is equivalent to minimizing (<a class="ltx_ref" href="https://arxiv.org/html/2503.13782v1#S2.E7" title="In 2.2 Derivation of 𝐋₁^(𝑡+1) and 𝐋₂^(𝑡+1) ‣ 2 Derivation of the AECM Algorithm Updates ‣ Regularized Parameter Estimation in Mixed Model Trace Regression"><span class="ltx_text ltx_ref_tag">7</span></a>) with respect to <math alttext="\operatorname{\mathbf{l}}_{1}" class="ltx_Math" display="inline" id="S2.SS2a.p3.46.m9.1"><semantics id="S2.SS2a.p3.46.m9.1a"><msub id="S2.SS2a.p3.46.m9.1.1" xref="S2.SS2a.p3.46.m9.1.1.cmml"><mi id="S2.SS2a.p3.46.m9.1.1.2" xref="S2.SS2a.p3.46.m9.1.1.2.cmml">𝐥</mi><mn id="S2.SS2a.p3.46.m9.1.1.3" xref="S2.SS2a.p3.46.m9.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S2.SS2a.p3.46.m9.1b"><apply id="S2.SS2a.p3.46.m9.1.1.cmml" xref="S2.SS2a.p3.46.m9.1.1"><csymbol cd="ambiguous" id="S2.SS2a.p3.46.m9.1.1.1.cmml" xref="S2.SS2a.p3.46.m9.1.1">subscript</csymbol><ci id="S2.SS2a.p3.46.m9.1.1.2.cmml" xref="S2.SS2a.p3.46.m9.1.1.2">𝐥</ci><cn id="S2.SS2a.p3.46.m9.1.1.3.cmml" type="integer" xref="S2.SS2a.p3.46.m9.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2a.p3.46.m9.1c">\operatorname{\mathbf{l}}_{1}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2a.p3.46.m9.1d">bold_l start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math>. This statement remains true even when <math alttext="\operatorname{\mathbf{H}}_{(1)}^{(t)}" class="ltx_Math" display="inline" id="S2.SS2a.p3.47.m10.2"><semantics id="S2.SS2a.p3.47.m10.2a"><msubsup id="S2.SS2a.p3.47.m10.2.3" xref="S2.SS2a.p3.47.m10.2.3.cmml"><mi id="S2.SS2a.p3.47.m10.2.3.2.2" xref="S2.SS2a.p3.47.m10.2.3.2.2.cmml">𝐇</mi><mrow id="S2.SS2a.p3.47.m10.1.1.1.3" xref="S2.SS2a.p3.47.m10.2.3.cmml"><mo id="S2.SS2a.p3.47.m10.1.1.1.3.1" stretchy="false" xref="S2.SS2a.p3.47.m10.2.3.cmml">(</mo><mn id="S2.SS2a.p3.47.m10.1.1.1.1" xref="S2.SS2a.p3.47.m10.1.1.1.1.cmml">1</mn><mo id="S2.SS2a.p3.47.m10.1.1.1.3.2" stretchy="false" xref="S2.SS2a.p3.47.m10.2.3.cmml">)</mo></mrow><mrow id="S2.SS2a.p3.47.m10.2.2.1.3" xref="S2.SS2a.p3.47.m10.2.3.cmml"><mo id="S2.SS2a.p3.47.m10.2.2.1.3.1" stretchy="false" xref="S2.SS2a.p3.47.m10.2.3.cmml">(</mo><mi id="S2.SS2a.p3.47.m10.2.2.1.1" xref="S2.SS2a.p3.47.m10.2.2.1.1.cmml">t</mi><mo id="S2.SS2a.p3.47.m10.2.2.1.3.2" stretchy="false" xref="S2.SS2a.p3.47.m10.2.3.cmml">)</mo></mrow></msubsup><annotation-xml encoding="MathML-Content" id="S2.SS2a.p3.47.m10.2b"><apply id="S2.SS2a.p3.47.m10.2.3.cmml" xref="S2.SS2a.p3.47.m10.2.3"><csymbol cd="ambiguous" id="S2.SS2a.p3.47.m10.2.3.1.cmml" xref="S2.SS2a.p3.47.m10.2.3">superscript</csymbol><apply id="S2.SS2a.p3.47.m10.2.3.2.cmml" xref="S2.SS2a.p3.47.m10.2.3"><csymbol cd="ambiguous" id="S2.SS2a.p3.47.m10.2.3.2.1.cmml" xref="S2.SS2a.p3.47.m10.2.3">subscript</csymbol><ci id="S2.SS2a.p3.47.m10.2.3.2.2.cmml" xref="S2.SS2a.p3.47.m10.2.3.2.2">𝐇</ci><cn id="S2.SS2a.p3.47.m10.1.1.1.1.cmml" type="integer" xref="S2.SS2a.p3.47.m10.1.1.1.1">1</cn></apply><ci id="S2.SS2a.p3.47.m10.2.2.1.1.cmml" xref="S2.SS2a.p3.47.m10.2.2.1.1">𝑡</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2a.p3.47.m10.2c">\operatorname{\mathbf{H}}_{(1)}^{(t)}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2a.p3.47.m10.2d">bold_H start_POSTSUBSCRIPT ( 1 ) end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_t ) end_POSTSUPERSCRIPT</annotation></semantics></math> is a singular matrix. We further add to (<a class="ltx_ref" href="https://arxiv.org/html/2503.13782v1#S2.E7" title="In 2.2 Derivation of 𝐋₁^(𝑡+1) and 𝐋₂^(𝑡+1) ‣ 2 Derivation of the AECM Algorithm Updates ‣ Regularized Parameter Estimation in Mixed Model Trace Regression"><span class="ltx_text ltx_ref_tag">7</span></a>) a group lasso penalty as in Equation (10) in the main manuscript to enforce rank constraints on our estimate of <math alttext="\operatorname{\mathbf{L}}_{1}" class="ltx_Math" display="inline" id="S2.SS2a.p3.48.m11.1"><semantics id="S2.SS2a.p3.48.m11.1a"><msub id="S2.SS2a.p3.48.m11.1.1" xref="S2.SS2a.p3.48.m11.1.1.cmml"><mi id="S2.SS2a.p3.48.m11.1.1.2" xref="S2.SS2a.p3.48.m11.1.1.2.cmml">𝐋</mi><mn id="S2.SS2a.p3.48.m11.1.1.3" xref="S2.SS2a.p3.48.m11.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S2.SS2a.p3.48.m11.1b"><apply id="S2.SS2a.p3.48.m11.1.1.cmml" xref="S2.SS2a.p3.48.m11.1.1"><csymbol cd="ambiguous" id="S2.SS2a.p3.48.m11.1.1.1.cmml" xref="S2.SS2a.p3.48.m11.1.1">subscript</csymbol><ci id="S2.SS2a.p3.48.m11.1.1.2.cmml" xref="S2.SS2a.p3.48.m11.1.1.2">𝐋</ci><cn id="S2.SS2a.p3.48.m11.1.1.3.cmml" type="integer" xref="S2.SS2a.p3.48.m11.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2a.p3.48.m11.1c">\operatorname{\mathbf{L}}_{1}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2a.p3.48.m11.1d">bold_L start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math>. This completes the derivation of the objective in Equation (10) of the main manuscript.</p> </div> <div class="ltx_para" id="S2.SS2a.p4"> <p class="ltx_p" id="S2.SS2a.p4.2">The objective for estimation <math alttext="\operatorname{\mathbf{L}}_{2}^{(t+1)}" class="ltx_Math" display="inline" id="S2.SS2a.p4.1.m1.1"><semantics id="S2.SS2a.p4.1.m1.1a"><msubsup id="S2.SS2a.p4.1.m1.1.2" xref="S2.SS2a.p4.1.m1.1.2.cmml"><mi id="S2.SS2a.p4.1.m1.1.2.2.2" xref="S2.SS2a.p4.1.m1.1.2.2.2.cmml">𝐋</mi><mn id="S2.SS2a.p4.1.m1.1.2.2.3" xref="S2.SS2a.p4.1.m1.1.2.2.3.cmml">2</mn><mrow id="S2.SS2a.p4.1.m1.1.1.1.1" xref="S2.SS2a.p4.1.m1.1.1.1.1.1.cmml"><mo id="S2.SS2a.p4.1.m1.1.1.1.1.2" stretchy="false" xref="S2.SS2a.p4.1.m1.1.1.1.1.1.cmml">(</mo><mrow id="S2.SS2a.p4.1.m1.1.1.1.1.1" xref="S2.SS2a.p4.1.m1.1.1.1.1.1.cmml"><mi id="S2.SS2a.p4.1.m1.1.1.1.1.1.2" xref="S2.SS2a.p4.1.m1.1.1.1.1.1.2.cmml">t</mi><mo id="S2.SS2a.p4.1.m1.1.1.1.1.1.1" xref="S2.SS2a.p4.1.m1.1.1.1.1.1.1.cmml">+</mo><mn id="S2.SS2a.p4.1.m1.1.1.1.1.1.3" xref="S2.SS2a.p4.1.m1.1.1.1.1.1.3.cmml">1</mn></mrow><mo id="S2.SS2a.p4.1.m1.1.1.1.1.3" stretchy="false" xref="S2.SS2a.p4.1.m1.1.1.1.1.1.cmml">)</mo></mrow></msubsup><annotation-xml encoding="MathML-Content" id="S2.SS2a.p4.1.m1.1b"><apply id="S2.SS2a.p4.1.m1.1.2.cmml" xref="S2.SS2a.p4.1.m1.1.2"><csymbol cd="ambiguous" id="S2.SS2a.p4.1.m1.1.2.1.cmml" xref="S2.SS2a.p4.1.m1.1.2">superscript</csymbol><apply id="S2.SS2a.p4.1.m1.1.2.2.cmml" xref="S2.SS2a.p4.1.m1.1.2"><csymbol cd="ambiguous" id="S2.SS2a.p4.1.m1.1.2.2.1.cmml" xref="S2.SS2a.p4.1.m1.1.2">subscript</csymbol><ci id="S2.SS2a.p4.1.m1.1.2.2.2.cmml" xref="S2.SS2a.p4.1.m1.1.2.2.2">𝐋</ci><cn id="S2.SS2a.p4.1.m1.1.2.2.3.cmml" type="integer" xref="S2.SS2a.p4.1.m1.1.2.2.3">2</cn></apply><apply id="S2.SS2a.p4.1.m1.1.1.1.1.1.cmml" xref="S2.SS2a.p4.1.m1.1.1.1.1"><plus id="S2.SS2a.p4.1.m1.1.1.1.1.1.1.cmml" xref="S2.SS2a.p4.1.m1.1.1.1.1.1.1"></plus><ci id="S2.SS2a.p4.1.m1.1.1.1.1.1.2.cmml" xref="S2.SS2a.p4.1.m1.1.1.1.1.1.2">𝑡</ci><cn id="S2.SS2a.p4.1.m1.1.1.1.1.1.3.cmml" type="integer" xref="S2.SS2a.p4.1.m1.1.1.1.1.1.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2a.p4.1.m1.1c">\operatorname{\mathbf{L}}_{2}^{(t+1)}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2a.p4.1.m1.1d">bold_L start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_t + 1 ) end_POSTSUPERSCRIPT</annotation></semantics></math> in Equation (11) of the main manuscript is obtained by swapping indices 1 and 2 in the derivation above for <math alttext="\operatorname{\mathbf{L}}_{1}" class="ltx_Math" display="inline" id="S2.SS2a.p4.2.m2.1"><semantics id="S2.SS2a.p4.2.m2.1a"><msub id="S2.SS2a.p4.2.m2.1.1" xref="S2.SS2a.p4.2.m2.1.1.cmml"><mi id="S2.SS2a.p4.2.m2.1.1.2" xref="S2.SS2a.p4.2.m2.1.1.2.cmml">𝐋</mi><mn id="S2.SS2a.p4.2.m2.1.1.3" xref="S2.SS2a.p4.2.m2.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S2.SS2a.p4.2.m2.1b"><apply id="S2.SS2a.p4.2.m2.1.1.cmml" xref="S2.SS2a.p4.2.m2.1.1"><csymbol cd="ambiguous" id="S2.SS2a.p4.2.m2.1.1.1.cmml" xref="S2.SS2a.p4.2.m2.1.1">subscript</csymbol><ci id="S2.SS2a.p4.2.m2.1.1.2.cmml" xref="S2.SS2a.p4.2.m2.1.1.2">𝐋</ci><cn id="S2.SS2a.p4.2.m2.1.1.3.cmml" type="integer" xref="S2.SS2a.p4.2.m2.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2a.p4.2.m2.1c">\operatorname{\mathbf{L}}_{1}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2a.p4.2.m2.1d">bold_L start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> updates.</p> </div> </section> </section> <section class="ltx_section" id="S3a"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">3 </span>MMTR Misspecification Plots</h2> <div class="ltx_para" id="S3a.p1"> <p class="ltx_p" id="S3a.p1.4">We present additional plots for each of the simulation scenarios described in Section 4.2 showing how the relative errors in estimating <math alttext="\operatorname{{\mathbf{\Lambda}}}" class="ltx_Math" display="inline" id="S3a.p1.1.m1.1"><semantics id="S3a.p1.1.m1.1a"><mi id="S3a.p1.1.m1.1.1" xref="S3a.p1.1.m1.1.1.cmml">𝚲</mi><annotation-xml encoding="MathML-Content" id="S3a.p1.1.m1.1b"><ci id="S3a.p1.1.m1.1.1.cmml" xref="S3a.p1.1.m1.1.1">𝚲</ci></annotation-xml><annotation encoding="application/x-tex" id="S3a.p1.1.m1.1c">\operatorname{{\mathbf{\Lambda}}}</annotation><annotation encoding="application/x-llamapun" id="S3a.p1.1.m1.1d">bold_Λ</annotation></semantics></math> by MMTR change depending on the GEE’s equicorrelation model parameter <math alttext="\alpha" class="ltx_Math" display="inline" id="S3a.p1.2.m2.1"><semantics id="S3a.p1.2.m2.1a"><mi id="S3a.p1.2.m2.1.1" xref="S3a.p1.2.m2.1.1.cmml">α</mi><annotation-xml encoding="MathML-Content" id="S3a.p1.2.m2.1b"><ci id="S3a.p1.2.m2.1.1.cmml" xref="S3a.p1.2.m2.1.1">𝛼</ci></annotation-xml><annotation encoding="application/x-tex" id="S3a.p1.2.m2.1c">\alpha</annotation><annotation encoding="application/x-llamapun" id="S3a.p1.2.m2.1d">italic_α</annotation></semantics></math> (Figures <a class="ltx_ref" href="https://arxiv.org/html/2503.13782v1#S3.F1" title="Figure 1 ‣ 3 MMTR Misspecification Plots ‣ Regularized Parameter Estimation in Mixed Model Trace Regression"><span class="ltx_text ltx_ref_tag">1</span></a> and <a class="ltx_ref" href="https://arxiv.org/html/2503.13782v1#S3.F2" title="Figure 2 ‣ 3 MMTR Misspecification Plots ‣ Regularized Parameter Estimation in Mixed Model Trace Regression"><span class="ltx_text ltx_ref_tag">2</span></a>). The <math alttext="\alpha" class="ltx_Math" display="inline" id="S3a.p1.3.m3.1"><semantics id="S3a.p1.3.m3.1a"><mi id="S3a.p1.3.m3.1.1" xref="S3a.p1.3.m3.1.1.cmml">α</mi><annotation-xml encoding="MathML-Content" id="S3a.p1.3.m3.1b"><ci id="S3a.p1.3.m3.1.1.cmml" xref="S3a.p1.3.m3.1.1">𝛼</ci></annotation-xml><annotation encoding="application/x-tex" id="S3a.p1.3.m3.1c">\alpha</annotation><annotation encoding="application/x-llamapun" id="S3a.p1.3.m3.1d">italic_α</annotation></semantics></math> term in the GEE’s equicorrelation model determines the correlation between all pairs of responses within each group of observations. The greater the <math alttext="\alpha" class="ltx_Math" display="inline" id="S3a.p1.4.m4.1"><semantics id="S3a.p1.4.m4.1a"><mi id="S3a.p1.4.m4.1.1" xref="S3a.p1.4.m4.1.1.cmml">α</mi><annotation-xml encoding="MathML-Content" id="S3a.p1.4.m4.1b"><ci id="S3a.p1.4.m4.1.1.cmml" xref="S3a.p1.4.m4.1.1">𝛼</ci></annotation-xml><annotation encoding="application/x-tex" id="S3a.p1.4.m4.1c">\alpha</annotation><annotation encoding="application/x-llamapun" id="S3a.p1.4.m4.1d">italic_α</annotation></semantics></math> value, the more misspecified the model is to the MMTR setting, which assumes the covariance between pairs of responses in each group to be a function of that group’s random-effects covariates. These plots demonstrate that the more misspecified the correlation structure is from a random-effects setting, the worse MMTR will perform in estimating the true covariance matrix.</p> </div> <figure class="ltx_figure" id="S3.F1"><img alt="Refer to caption" class="ltx_graphics ltx_img_landscape" height="465" id="S3.F1.g1" src="extracted/6288390/misspec_fixed_y_scales.png" width="698"/> <figcaption class="ltx_caption"><span class="ltx_tag ltx_tag_figure">Figure 1: </span>Relative errors in estimating <math alttext="\operatorname{{\mathbf{\Lambda}}}" class="ltx_Math" display="inline" id="S3.F1.3.m1.1"><semantics id="S3.F1.3.m1.1b"><mi id="S3.F1.3.m1.1.1" xref="S3.F1.3.m1.1.1.cmml">𝚲</mi><annotation-xml encoding="MathML-Content" id="S3.F1.3.m1.1c"><ci id="S3.F1.3.m1.1.1.cmml" xref="S3.F1.3.m1.1.1">𝚲</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.F1.3.m1.1d">\operatorname{{\mathbf{\Lambda}}}</annotation><annotation encoding="application/x-llamapun" id="S3.F1.3.m1.1e">bold_Λ</annotation></semantics></math> by MMTR against <math alttext="\alpha" class="ltx_Math" display="inline" id="S3.F1.4.m2.1"><semantics id="S3.F1.4.m2.1b"><mi id="S3.F1.4.m2.1.1" xref="S3.F1.4.m2.1.1.cmml">α</mi><annotation-xml encoding="MathML-Content" id="S3.F1.4.m2.1c"><ci id="S3.F1.4.m2.1.1.cmml" xref="S3.F1.4.m2.1.1">𝛼</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.F1.4.m2.1d">\alpha</annotation><annotation encoding="application/x-llamapun" id="S3.F1.4.m2.1e">italic_α</annotation></semantics></math>. All plots here have the same range on the y axis.</figcaption> </figure> <figure class="ltx_figure" id="S3.F2"><img alt="Refer to caption" class="ltx_graphics ltx_img_landscape" height="465" id="S3.F2.g1" src="extracted/6288390/misspec_diff_y_scales.png" width="698"/> <figcaption class="ltx_caption"><span class="ltx_tag ltx_tag_figure">Figure 2: </span>Relative errors in estimating <math alttext="\operatorname{{\mathbf{\Lambda}}}" class="ltx_Math" display="inline" id="S3.F2.3.m1.1"><semantics id="S3.F2.3.m1.1b"><mi id="S3.F2.3.m1.1.1" xref="S3.F2.3.m1.1.1.cmml">𝚲</mi><annotation-xml encoding="MathML-Content" id="S3.F2.3.m1.1c"><ci id="S3.F2.3.m1.1.1.cmml" xref="S3.F2.3.m1.1.1">𝚲</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.F2.3.m1.1d">\operatorname{{\mathbf{\Lambda}}}</annotation><annotation encoding="application/x-llamapun" id="S3.F2.3.m1.1e">bold_Λ</annotation></semantics></math> by MMTR against <math alttext="\alpha" class="ltx_Math" display="inline" id="S3.F2.4.m2.1"><semantics id="S3.F2.4.m2.1b"><mi id="S3.F2.4.m2.1.1" xref="S3.F2.4.m2.1.1.cmml">α</mi><annotation-xml encoding="MathML-Content" id="S3.F2.4.m2.1c"><ci id="S3.F2.4.m2.1.1.cmml" xref="S3.F2.4.m2.1.1">𝛼</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.F2.4.m2.1d">\alpha</annotation><annotation encoding="application/x-llamapun" id="S3.F2.4.m2.1e">italic_α</annotation></semantics></math>. These plots are zoomed in from Figure <a class="ltx_ref" href="https://arxiv.org/html/2503.13782v1#S3.F1" title="Figure 1 ‣ 3 MMTR Misspecification Plots ‣ Regularized Parameter Estimation in Mixed Model Trace Regression"><span class="ltx_text ltx_ref_tag">1</span></a>.</figcaption> </figure> <div class="ltx_pagination ltx_role_newpage"></div> </section> </article> </div> <footer class="ltx_page_footer"> <div class="ltx_page_logo">Generated on Tue Mar 18 00:05:48 2025 by <a class="ltx_LaTeXML_logo" href="http://dlmf.nist.gov/LaTeXML/"><span style="letter-spacing:-0.2em; margin-right:0.1em;">L<span class="ltx_font_smallcaps" style="position:relative; bottom:2.2pt;">a</span>T<span class="ltx_font_smallcaps" style="font-size:120%;position:relative; bottom:-0.2ex;">e</span></span><span style="font-size:90%; position:relative; bottom:-0.2ex;">XML</span><img alt="Mascot Sammy" src="data:image/png;base64,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"/></a> </div></footer> </div> </body> </html>