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<!DOCTYPE html> <html lang="en"> <head> <meta content="text/html; charset=utf-8" http-equiv="content-type"/> <title>Integrating Planning into Single-Turn Long-Form Text Generation</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/2410.06203v1/"/></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/2410.06203v1#S1" title="In Integrating Planning into Single-Turn Long-Form Text Generation"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">1 </span>Introduction</span></a></li> <li class="ltx_tocentry ltx_tocentry_section"> <a class="ltx_ref" href="https://arxiv.org/html/2410.06203v1#S2" title="In Integrating Planning into Single-Turn Long-Form Text Generation"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">2 </span>Related work</span></a> <ol class="ltx_toclist ltx_toclist_section"> <li class="ltx_tocentry ltx_tocentry_paragraph"><a class="ltx_ref" href="https://arxiv.org/html/2410.06203v1#S2.SS0.SSS0.Px1" title="In 2 Related work ‣ Integrating Planning into Single-Turn Long-Form Text Generation"><span class="ltx_text ltx_ref_title">Planning.</span></a></li> <li class="ltx_tocentry ltx_tocentry_paragraph"><a class="ltx_ref" href="https://arxiv.org/html/2410.06203v1#S2.SS0.SSS0.Px2" title="In 2 Related work ‣ Integrating Planning into Single-Turn Long-Form Text Generation"><span class="ltx_text ltx_ref_title">Long-form text generation.</span></a></li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_section"> <a class="ltx_ref" href="https://arxiv.org/html/2410.06203v1#S3" title="In Integrating Planning into Single-Turn Long-Form Text Generation"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">3 </span>Problem Setup</span></a> <ol class="ltx_toclist ltx_toclist_section"> <li class="ltx_tocentry ltx_tocentry_paragraph"><a class="ltx_ref" href="https://arxiv.org/html/2410.06203v1#S3.SS0.SSS0.Px1" title="In 3 Problem Setup ‣ Integrating Planning into Single-Turn Long-Form Text Generation"><span class="ltx_text ltx_ref_title">Long-form text generation.</span></a></li> <li class="ltx_tocentry ltx_tocentry_paragraph"><a class="ltx_ref" href="https://arxiv.org/html/2410.06203v1#S3.SS0.SSS0.Px2" title="In 3 Problem Setup ‣ Integrating Planning into Single-Turn Long-Form Text Generation"><span class="ltx_text ltx_ref_title">Intermediate steps.</span></a></li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_section"> <a class="ltx_ref" href="https://arxiv.org/html/2410.06203v1#S4" title="In Integrating Planning into Single-Turn Long-Form Text Generation"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">4 </span>Methodology</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/2410.06203v1#S4.SS1" title="In 4 Methodology ‣ Integrating Planning into Single-Turn Long-Form Text Generation"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">4.1 </span>Writing Plan as Intermediate Steps</span></a> <ol class="ltx_toclist ltx_toclist_subsection"> <li class="ltx_tocentry ltx_tocentry_paragraph"><a class="ltx_ref" href="https://arxiv.org/html/2410.06203v1#S4.SS1.SSS0.Px1" title="In 4.1 Writing Plan as Intermediate Steps ‣ 4 Methodology ‣ Integrating Planning into Single-Turn Long-Form Text Generation"><span class="ltx_text ltx_ref_title">Document Summary</span></a></li> <li class="ltx_tocentry ltx_tocentry_paragraph"><a class="ltx_ref" href="https://arxiv.org/html/2410.06203v1#S4.SS1.SSS0.Px2" title="In 4.1 Writing Plan as Intermediate Steps ‣ 4 Methodology ‣ Integrating Planning into Single-Turn Long-Form Text Generation"><span class="ltx_text ltx_ref_title">Document Outline</span></a></li> <li class="ltx_tocentry ltx_tocentry_paragraph"><a class="ltx_ref" href="https://arxiv.org/html/2410.06203v1#S4.SS1.SSS0.Px3" title="In 4.1 Writing Plan as Intermediate Steps ‣ 4 Methodology ‣ Integrating Planning into Single-Turn Long-Form Text Generation"><span class="ltx_text ltx_ref_title">Document Key Information</span></a></li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2410.06203v1#S4.SS2" title="In 4 Methodology ‣ Integrating Planning into Single-Turn Long-Form Text Generation"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">4.2 </span>Structured Intermediate Steps Construction</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"> <a class="ltx_ref" href="https://arxiv.org/html/2410.06203v1#S4.SS3" title="In 4 Methodology ‣ Integrating Planning into Single-Turn Long-Form Text Generation"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">4.3 </span>Fine-tuning</span></a> <ol class="ltx_toclist ltx_toclist_subsection"> <li class="ltx_tocentry ltx_tocentry_paragraph"><a class="ltx_ref" href="https://arxiv.org/html/2410.06203v1#S4.SS3.SSS0.Px1" title="In 4.3 Fine-tuning ‣ 4 Methodology ‣ Integrating Planning into Single-Turn Long-Form Text Generation"><span class="ltx_text ltx_ref_title">Input to output with intermediate steps (<math alttext="x_{i}\rightarrow{\mathbf{z}}_{i}\oplus y_{i}" class="ltx_Math" display="inline"><semantics><mrow><msub><mi>x</mi><mi>i</mi></msub><mo stretchy="false">→</mo><mrow><msub><mi>𝐳</mi><mi>i</mi></msub><mo>⊕</mo><msub><mi>y</mi><mi>i</mi></msub></mrow></mrow><annotation-xml encoding="MathML-Content"><apply><ci>→</ci><apply><csymbol cd="ambiguous">subscript</csymbol><ci>𝑥</ci><ci>𝑖</ci></apply><apply><csymbol cd="latexml">direct-sum</csymbol><apply><csymbol cd="ambiguous">subscript</csymbol><ci>𝐳</ci><ci>𝑖</ci></apply><apply><csymbol cd="ambiguous">subscript</csymbol><ci>𝑦</ci><ci>𝑖</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex">x_{i}\rightarrow{\mathbf{z}}_{i}\oplus y_{i}</annotation><annotation encoding="application/x-llamapun">italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT → bold_z start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ⊕ italic_y start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math>)</span></a></li> <li class="ltx_tocentry ltx_tocentry_paragraph"><a class="ltx_ref" href="https://arxiv.org/html/2410.06203v1#S4.SS3.SSS0.Px2" title="In 4.3 Fine-tuning ‣ 4 Methodology ‣ Integrating Planning into Single-Turn Long-Form Text Generation"><span class="ltx_text ltx_ref_title">Input and intermediate steps to output (<math alttext="x_{i}\oplus{\mathbf{z}}_{i}\rightarrow y_{i}" class="ltx_Math" display="inline"><semantics><mrow><mrow><msub><mi>x</mi><mi>i</mi></msub><mo>⊕</mo><msub><mi>𝐳</mi><mi>i</mi></msub></mrow><mo stretchy="false">→</mo><msub><mi>y</mi><mi>i</mi></msub></mrow><annotation-xml encoding="MathML-Content"><apply><ci>→</ci><apply><csymbol cd="latexml">direct-sum</csymbol><apply><csymbol cd="ambiguous">subscript</csymbol><ci>𝑥</ci><ci>𝑖</ci></apply><apply><csymbol cd="ambiguous">subscript</csymbol><ci>𝐳</ci><ci>𝑖</ci></apply></apply><apply><csymbol cd="ambiguous">subscript</csymbol><ci>𝑦</ci><ci>𝑖</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex">x_{i}\oplus{\mathbf{z}}_{i}\rightarrow y_{i}</annotation><annotation encoding="application/x-llamapun">italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ⊕ bold_z start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT → italic_y start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math>)</span></a></li> </ol> </li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_section"> <a class="ltx_ref" href="https://arxiv.org/html/2410.06203v1#S5" title="In Integrating Planning into Single-Turn Long-Form Text Generation"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">5 </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/2410.06203v1#S5.SS1" title="In 5 Experiments ‣ Integrating Planning into Single-Turn Long-Form Text Generation"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">5.1 </span>Benchmark Datasets</span></a> <ol class="ltx_toclist ltx_toclist_subsection"> <li class="ltx_tocentry ltx_tocentry_paragraph"><a class="ltx_ref" href="https://arxiv.org/html/2410.06203v1#S5.SS1.SSS0.Px1" title="In 5.1 Benchmark Datasets ‣ 5 Experiments ‣ Integrating Planning into Single-Turn Long-Form Text Generation"><span class="ltx_text ltx_ref_title">SciNews Dataset.</span></a></li> <li class="ltx_tocentry ltx_tocentry_paragraph"><a class="ltx_ref" href="https://arxiv.org/html/2410.06203v1#S5.SS1.SSS0.Px2" title="In 5.1 Benchmark Datasets ‣ 5 Experiments ‣ Integrating Planning into Single-Turn Long-Form Text Generation"><span class="ltx_text ltx_ref_title">Wikipedia Dataset.</span></a></li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_subsection"> <a class="ltx_ref" href="https://arxiv.org/html/2410.06203v1#S5.SS2" title="In 5 Experiments ‣ Integrating Planning into Single-Turn Long-Form Text Generation"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">5.2 </span>Evaluation Metrics</span></a> <ol class="ltx_toclist ltx_toclist_subsection"> <li class="ltx_tocentry ltx_tocentry_paragraph"><a class="ltx_ref" href="https://arxiv.org/html/2410.06203v1#S5.SS2.SSS0.Px1" title="In 5.2 Evaluation Metrics ‣ 5 Experiments ‣ Integrating Planning into Single-Turn Long-Form Text Generation"><span class="ltx_text ltx_ref_title">Offline metrics.</span></a></li> <li class="ltx_tocentry ltx_tocentry_paragraph"><a class="ltx_ref" href="https://arxiv.org/html/2410.06203v1#S5.SS2.SSS0.Px2" title="In 5.2 Evaluation Metrics ‣ 5 Experiments ‣ Integrating Planning into Single-Turn Long-Form Text Generation"><span class="ltx_text ltx_ref_title">Human and auto SxS.</span></a></li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2410.06203v1#S5.SS3" title="In 5 Experiments ‣ Integrating Planning into Single-Turn Long-Form Text Generation"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">5.3 </span>Experiment setup</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"> <a class="ltx_ref" href="https://arxiv.org/html/2410.06203v1#S5.SS4" title="In 5 Experiments ‣ Integrating Planning into Single-Turn Long-Form Text Generation"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">5.4 </span>Results</span></a> <ol class="ltx_toclist ltx_toclist_subsection"> <li class="ltx_tocentry ltx_tocentry_paragraph"><a class="ltx_ref" href="https://arxiv.org/html/2410.06203v1#S5.SS4.SSS0.Px1" title="In 5.4 Results ‣ 5 Experiments ‣ Integrating Planning into Single-Turn Long-Form Text Generation"><span class="ltx_text ltx_ref_title">Overall comparison.</span></a></li> <li class="ltx_tocentry ltx_tocentry_paragraph"><a class="ltx_ref" href="https://arxiv.org/html/2410.06203v1#S5.SS4.SSS0.Px2" title="In 5.4 Results ‣ 5 Experiments ‣ Integrating Planning into Single-Turn Long-Form Text Generation"><span class="ltx_text ltx_ref_title">Comparison of different training and inference recipes.</span></a></li> <li class="ltx_tocentry ltx_tocentry_paragraph"><a class="ltx_ref" href="https://arxiv.org/html/2410.06203v1#S5.SS4.SSS0.Px3" title="In 5.4 Results ‣ 5 Experiments ‣ Integrating Planning into Single-Turn Long-Form Text Generation"><span class="ltx_text ltx_ref_title">Single-turn vs. multi-turn.</span></a></li> <li class="ltx_tocentry ltx_tocentry_paragraph"><a class="ltx_ref" href="https://arxiv.org/html/2410.06203v1#S5.SS4.SSS0.Px4" title="In 5.4 Results ‣ 5 Experiments ‣ Integrating Planning into Single-Turn Long-Form Text Generation"><span class="ltx_text ltx_ref_title">Human and auto SxS results.</span></a></li> <li class="ltx_tocentry ltx_tocentry_paragraph"><a class="ltx_ref" href="https://arxiv.org/html/2410.06203v1#S5.SS4.SSS0.Px5" title="In 5.4 Results ‣ 5 Experiments ‣ Integrating Planning into Single-Turn Long-Form Text Generation"><span class="ltx_text ltx_ref_title">Alternative base model: Gemini 1.5 Flash.</span></a></li> </ol> </li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_section"><a class="ltx_ref" href="https://arxiv.org/html/2410.06203v1#S6" title="In Integrating Planning into Single-Turn Long-Form Text Generation"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">6 </span>Conclusions</span></a></li> <li class="ltx_tocentry ltx_tocentry_section"><a class="ltx_ref" href="https://arxiv.org/html/2410.06203v1#S7" title="In Integrating Planning into Single-Turn Long-Form Text Generation"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">7 </span>Limitations</span></a></li> <li class="ltx_tocentry ltx_tocentry_section"><a class="ltx_ref" href="https://arxiv.org/html/2410.06203v1#S8" title="In Integrating Planning into Single-Turn Long-Form Text Generation"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">8 </span>Ethics Statement</span></a></li> <li class="ltx_tocentry ltx_tocentry_appendix"> <a class="ltx_ref" href="https://arxiv.org/html/2410.06203v1#A1" title="In Integrating Planning into Single-Turn Long-Form Text Generation"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">A </span>Prompts</span></a> <ol class="ltx_toclist ltx_toclist_appendix"> <li class="ltx_tocentry ltx_tocentry_subsection"> <a class="ltx_ref" href="https://arxiv.org/html/2410.06203v1#A1.SS1" title="In Appendix A Prompts ‣ Integrating Planning into Single-Turn Long-Form Text Generation"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">A.1 </span>SciNews</span></a> <ol class="ltx_toclist ltx_toclist_subsection"> <li class="ltx_tocentry ltx_tocentry_paragraph"><a class="ltx_ref" href="https://arxiv.org/html/2410.06203v1#A1.SS1.SSS0.Px1" title="In A.1 SciNews ‣ Appendix A Prompts ‣ Integrating Planning into Single-Turn Long-Form Text Generation"><span class="ltx_text ltx_ref_title">Only generating the full article (<math alttext="x_{i}\rightarrow y_{i}" class="ltx_Math" display="inline"><semantics><mrow><msub><mi>x</mi><mi>i</mi></msub><mo stretchy="false">→</mo><msub><mi>y</mi><mi>i</mi></msub></mrow><annotation-xml encoding="MathML-Content"><apply><ci>→</ci><apply><csymbol cd="ambiguous">subscript</csymbol><ci>𝑥</ci><ci>𝑖</ci></apply><apply><csymbol cd="ambiguous">subscript</csymbol><ci>𝑦</ci><ci>𝑖</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex">x_{i}\rightarrow y_{i}</annotation><annotation encoding="application/x-llamapun">italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT → italic_y start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math>).</span></a></li> <li class="ltx_tocentry ltx_tocentry_paragraph"><a class="ltx_ref" href="https://arxiv.org/html/2410.06203v1#A1.SS1.SSS0.Px2" title="In A.1 SciNews ‣ Appendix A Prompts ‣ Integrating Planning into Single-Turn Long-Form Text Generation"><span class="ltx_text ltx_ref_title">Generating the intermediate steps and the full article (<math alttext="x_{i}\rightarrow{\mathbf{z}}_{i}\oplus y_{i}" class="ltx_Math" display="inline"><semantics><mrow><msub><mi>x</mi><mi>i</mi></msub><mo stretchy="false">→</mo><mrow><msub><mi>𝐳</mi><mi>i</mi></msub><mo>⊕</mo><msub><mi>y</mi><mi>i</mi></msub></mrow></mrow><annotation-xml encoding="MathML-Content"><apply><ci>→</ci><apply><csymbol cd="ambiguous">subscript</csymbol><ci>𝑥</ci><ci>𝑖</ci></apply><apply><csymbol cd="latexml">direct-sum</csymbol><apply><csymbol cd="ambiguous">subscript</csymbol><ci>𝐳</ci><ci>𝑖</ci></apply><apply><csymbol cd="ambiguous">subscript</csymbol><ci>𝑦</ci><ci>𝑖</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex">x_{i}\rightarrow{\mathbf{z}}_{i}\oplus y_{i}</annotation><annotation encoding="application/x-llamapun">italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT → bold_z start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ⊕ italic_y start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math>).</span></a></li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_subsection"> <a class="ltx_ref" href="https://arxiv.org/html/2410.06203v1#A1.SS2" title="In Appendix A Prompts ‣ Integrating Planning into Single-Turn Long-Form Text Generation"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">A.2 </span>Wikipedia</span></a> <ol class="ltx_toclist ltx_toclist_subsection"> <li class="ltx_tocentry ltx_tocentry_paragraph"><a class="ltx_ref" href="https://arxiv.org/html/2410.06203v1#A1.SS2.SSS0.Px1" title="In A.2 Wikipedia ‣ Appendix A Prompts ‣ Integrating Planning into Single-Turn Long-Form Text Generation"><span class="ltx_text ltx_ref_title">Only generating the full article (<math alttext="x_{i}\rightarrow y_{i}" class="ltx_Math" display="inline"><semantics><mrow><msub><mi>x</mi><mi>i</mi></msub><mo stretchy="false">→</mo><msub><mi>y</mi><mi>i</mi></msub></mrow><annotation-xml encoding="MathML-Content"><apply><ci>→</ci><apply><csymbol cd="ambiguous">subscript</csymbol><ci>𝑥</ci><ci>𝑖</ci></apply><apply><csymbol cd="ambiguous">subscript</csymbol><ci>𝑦</ci><ci>𝑖</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex">x_{i}\rightarrow y_{i}</annotation><annotation encoding="application/x-llamapun">italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT → italic_y start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math>).</span></a></li> <li class="ltx_tocentry ltx_tocentry_paragraph"><a class="ltx_ref" href="https://arxiv.org/html/2410.06203v1#A1.SS2.SSS0.Px2" title="In A.2 Wikipedia ‣ Appendix A Prompts ‣ Integrating Planning into Single-Turn Long-Form Text Generation"><span class="ltx_text ltx_ref_title">Generating the intermediate steps and the full article (<math alttext="x_{i}\rightarrow{\mathbf{z}}_{i}\oplus y_{i}" class="ltx_Math" display="inline"><semantics><mrow><msub><mi>x</mi><mi>i</mi></msub><mo stretchy="false">→</mo><mrow><msub><mi>𝐳</mi><mi>i</mi></msub><mo>⊕</mo><msub><mi>y</mi><mi>i</mi></msub></mrow></mrow><annotation-xml encoding="MathML-Content"><apply><ci>→</ci><apply><csymbol cd="ambiguous">subscript</csymbol><ci>𝑥</ci><ci>𝑖</ci></apply><apply><csymbol cd="latexml">direct-sum</csymbol><apply><csymbol cd="ambiguous">subscript</csymbol><ci>𝐳</ci><ci>𝑖</ci></apply><apply><csymbol cd="ambiguous">subscript</csymbol><ci>𝑦</ci><ci>𝑖</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex">x_{i}\rightarrow{\mathbf{z}}_{i}\oplus y_{i}</annotation><annotation encoding="application/x-llamapun">italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT → bold_z start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ⊕ italic_y start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math>).</span></a></li> </ol> </li> </ol> </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_title_document">Integrating Planning into Single-Turn Long-Form Text Generation</h1> <div class="ltx_authors"> <span class="ltx_creator ltx_role_author"> <span class="ltx_personname">Yi Liang<span class="ltx_note ltx_role_footnote" id="footnote1"><sup class="ltx_note_mark">1</sup><span class="ltx_note_outer"><span class="ltx_note_content"><sup class="ltx_note_mark">1</sup><span class="ltx_tag ltx_tag_note">1</span>Contributed equally.</span></span></span> </span><span class="ltx_author_notes"> <span class="ltx_contact ltx_role_affiliation">Google DeepMind, {yiliang,wuyou,hlz,chenlili,jmshen,yilingjia,zhenqin,sumitsanghai,xuanhui,bemike}@google.com </span></span></span> <span class="ltx_creator ltx_role_author"> <span class="ltx_personname">You Wu<sup class="ltx_sup" id="id1.1.id1">*</sup> </span><span class="ltx_author_notes"> <span class="ltx_contact ltx_role_affiliation">Google DeepMind, {yiliang,wuyou,hlz,chenlili,jmshen,yilingjia,zhenqin,sumitsanghai,xuanhui,bemike}@google.com </span></span></span> <span class="ltx_creator ltx_role_author"> <span class="ltx_personname">Honglei Zhuang </span><span class="ltx_author_notes"> <span class="ltx_contact ltx_role_affiliation">Google DeepMind, {yiliang,wuyou,hlz,chenlili,jmshen,yilingjia,zhenqin,sumitsanghai,xuanhui,bemike}@google.com </span></span></span> <span class="ltx_creator ltx_role_author"> <span class="ltx_personname">Li Chen </span><span class="ltx_author_notes"> <span class="ltx_contact ltx_role_affiliation">Google DeepMind, {yiliang,wuyou,hlz,chenlili,jmshen,yilingjia,zhenqin,sumitsanghai,xuanhui,bemike}@google.com </span></span></span> <span class="ltx_creator ltx_role_author"> <span class="ltx_personname">Jiaming Shen </span><span class="ltx_author_notes"> <span class="ltx_contact ltx_role_affiliation">Google DeepMind, {yiliang,wuyou,hlz,chenlili,jmshen,yilingjia,zhenqin,sumitsanghai,xuanhui,bemike}@google.com </span></span></span> <span class="ltx_creator ltx_role_author"> <span class="ltx_personname">Yiling Jia </span><span class="ltx_author_notes"> <span class="ltx_contact ltx_role_affiliation">Google DeepMind, {yiliang,wuyou,hlz,chenlili,jmshen,yilingjia,zhenqin,sumitsanghai,xuanhui,bemike}@google.com </span></span></span> <span class="ltx_creator ltx_role_author"> <span class="ltx_personname">Zhen Qin </span><span class="ltx_author_notes"> <span class="ltx_contact ltx_role_affiliation">Google DeepMind, {yiliang,wuyou,hlz,chenlili,jmshen,yilingjia,zhenqin,sumitsanghai,xuanhui,bemike}@google.com </span></span></span> <span class="ltx_creator ltx_role_author"> <span class="ltx_personname">Sumit Sanghai </span><span class="ltx_author_notes"> <span class="ltx_contact ltx_role_affiliation">Google DeepMind, {yiliang,wuyou,hlz,chenlili,jmshen,yilingjia,zhenqin,sumitsanghai,xuanhui,bemike}@google.com </span></span></span> <span class="ltx_creator ltx_role_author"> <span class="ltx_personname">Xuanhui Wang </span><span class="ltx_author_notes"> <span class="ltx_contact ltx_role_affiliation">Google DeepMind, {yiliang,wuyou,hlz,chenlili,jmshen,yilingjia,zhenqin,sumitsanghai,xuanhui,bemike}@google.com </span></span></span> <span class="ltx_creator ltx_role_author"> <span class="ltx_personname">Carl Yang </span><span class="ltx_author_notes"> <span class="ltx_contact ltx_role_affiliation">Emory University, j.carlyang@emory.edu </span></span></span> <span class="ltx_creator ltx_role_author"> <span class="ltx_personname">Michael Bendersky </span><span class="ltx_author_notes"> <span class="ltx_contact ltx_role_affiliation">Google DeepMind, {yiliang,wuyou,hlz,chenlili,jmshen,yilingjia,zhenqin,sumitsanghai,xuanhui,bemike}@google.com </span></span></span> </div> <div class="ltx_abstract"> <h6 class="ltx_title ltx_title_abstract">Abstract</h6> <p class="ltx_p" id="id2.id1">Generating high-quality, in-depth textual documents, such as academic papers, news articles, Wikipedia entries, and books, remains a significant challenge for Large Language Models (LLMs). In this paper, we propose to use planning to generate long form content. To achieve our goal, we generate intermediate steps via an auxiliary task that teaches the LLM to plan, reason and structure before generating the final text. Our main novelty lies in a single auxiliary task that does not require multiple rounds of prompting or planning. To overcome the scarcity of training data for these intermediate steps, we leverage LLMs to generate synthetic intermediate writing data such as outlines, key information and summaries from existing full articles. Our experiments demonstrate on two datasets from different domains, namely the scientific news dataset <span class="ltx_text ltx_font_italic" id="id2.id1.1">SciNews</span> and Wikipedia datasets in <span class="ltx_text ltx_font_italic" id="id2.id1.2">KILT-Wiki</span> and <span class="ltx_text ltx_font_italic" id="id2.id1.3">FreshWiki</span>, that LLMs fine-tuned with the auxiliary task generate higher quality documents. We observed +2.5% improvement in ROUGE-Lsum, and a strong 3.60 overall win/loss ratio via human SxS evaluation, with clear wins in organization, relevance, and verifiability.</p> </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">Large language models (LLMs) have achieved remarkable progress in various text generation tasks, ranging from creative writing to summarization and dialogue generation <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2410.06203v1#bib.bib11" title="">11</a>, <a class="ltx_ref" href="https://arxiv.org/html/2410.06203v1#bib.bib8" title="">8</a>]</cite>. However, generating high-quality, coherent, and substantive long-form documents, such as academic papers, news articles, and books, remains a significant challenge <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2410.06203v1#bib.bib23" title="">23</a>, <a class="ltx_ref" href="https://arxiv.org/html/2410.06203v1#bib.bib24" title="">24</a>]</cite>. Existing work mostly tackles this challenge by sequentially prompting LLMs to write a small segment of the document at each call and integrating the outputs into the final long-form document <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2410.06203v1#bib.bib24" title="">24</a>, <a class="ltx_ref" href="https://arxiv.org/html/2410.06203v1#bib.bib33" title="">33</a>, <a class="ltx_ref" href="https://arxiv.org/html/2410.06203v1#bib.bib19" title="">19</a>]</cite>. Such systems usually require additional components to ensure the coherence or consistency of the entire document <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2410.06203v1#bib.bib4" title="">4</a>, <a class="ltx_ref" href="https://arxiv.org/html/2410.06203v1#bib.bib30" title="">30</a>]</cite>.</p> </div> <div class="ltx_para" id="S1.p2"> <p class="ltx_p" id="S1.p2.1">In this paper, we present a novel approach that directly fine-tunes LLMs to generate long-form documents in one single call. This approach enables us to fully leverage the token-level attention mechanism in the decoding process, thereby ensuring the coherence and consistency of the generated document. It also significantly reduces the complexity of the system during serving and deployment to write long-form documents. </p> </div> <div class="ltx_para" id="S1.p3"> <p class="ltx_p" id="S1.p3.1">Writing high-quality long-form documents often involves a pre-writing stage <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2410.06203v1#bib.bib18" title="">18</a>]</cite> where authors outline the structure, develop key arguments and plan for the overall flow of the document. This additional stage enables the writers to simplify the tasks into manageable sub-tasks, similar to the idea of Chain-of-Thought (CoT) in multi-step reasoning <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2410.06203v1#bib.bib26" title="">26</a>]</cite>. In fact, this stage is often included in the design of multi-stage long-form document writing systems <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2410.06203v1#bib.bib33" title="">33</a>, <a class="ltx_ref" href="https://arxiv.org/html/2410.06203v1#bib.bib32" title="">32</a>, <a class="ltx_ref" href="https://arxiv.org/html/2410.06203v1#bib.bib19" title="">19</a>]</cite> through multiple rounds of prompting.</p> </div> <div class="ltx_para" id="S1.p4"> <p class="ltx_p" id="S1.p4.1">Motivated by this idea, we propose to integrate this pre-writing stage into our development of the single-turn LLM writer. Specifically, we introduce a series of auxiliary training tasks to endow LLMs with the skills to plan and structure long-form documents before generating the final full article. For example, one auxiliary task could involve providing the LLM with the writing context as input and expecting it to produce an outline with key insights as the output. Another auxiliary task can present the LLM with the writing context and an outline, with the goal of generating the complete article. We argue that fine-tuning the LLM writer with a mixture of writing tasks, coupled with guidance at varying levels of granularity, can enhance the model’s ability to produce long-form documents that are inherently well-structured and coherent. </p> </div> <div class="ltx_para" id="S1.p5"> <p class="ltx_p" id="S1.p5.1">While it is relatively easy to obtain sufficiently large corpora of full articles for supervised fine-tuning, obtaining intermediate writings such as article outline and key points directly from human writers is considerably more challenging as these are typically not well documented and made public. To address this, we leverage the few-shot capabilities of LLMs to generate synthetic intermediate writings from full articles, along with the original document structure when available. Note that, generating a concise summary, excerpt, or outline from a full-length, detailed article is much easier than doing the reverse. Therefore, it becomes rather manageable to create abundant intermediate-writing data for the purpose of fine-tuning LLMs towards learning to plan for writing full articles.</p> </div> <div class="ltx_para" id="S1.p6"> <p class="ltx_p" id="S1.p6.1">Our experiments on multiple datasets demonstrate that LLM writers trained with the auxiliary tasks generate higher quality long-form documents in a single pass, even when the final inference task does not prompt the model to produce intermediate planning steps.</p> </div> <div class="ltx_para" id="S1.p7"> <p class="ltx_p" id="S1.p7.1">Our main contributions are summarized as follows:</p> <ul class="ltx_itemize" id="S1.I1"> <li class="ltx_item" id="S1.I1.i1" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">•</span> <div class="ltx_para" id="S1.I1.i1.p1"> <p class="ltx_p" id="S1.I1.i1.p1.1">We propose a novel approach that directly fine-tunes LLMs to generate the entire long-form document in a single pass, simplifying the generation process and enhancing coherence.</p> </div> </li> <li class="ltx_item" id="S1.I1.i2" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">•</span> <div class="ltx_para" id="S1.I1.i2.p1"> <p class="ltx_p" id="S1.I1.i2.p1.1">Inspired by human writing practices, the proposed framework incorporates the pre-writing stages by introducing auxiliary training tasks that teach the LLMs to plan and structure documents before generating the final text.</p> </div> </li> <li class="ltx_item" id="S1.I1.i3" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">•</span> <div class="ltx_para" id="S1.I1.i3.p1"> <p class="ltx_p" id="S1.I1.i3.p1.1">To overcome the challenge of limited training data for intermediate writing steps, we leverage LLMs’ ability to generate synthetic summaries, outlines, and key information from existing full articles. This innovative approach unlocks a vast new source of training data for LLM planning.</p> </div> </li> <li class="ltx_item" id="S1.I1.i4" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">•</span> <div class="ltx_para" id="S1.I1.i4.p1"> <p class="ltx_p" id="S1.I1.i4.p1.1">Our extensive experimental results demonstrated the effectiveness of the proposed approach on multiple datasets, showing that LLMs fine-tuned with the auxiliary tasks produce higher quality, more coherent long-form documents in a single pass. </p> </div> </li> </ul> </div> </section> <section class="ltx_section" id="S2"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">2 </span>Related work</h2> <section class="ltx_paragraph" id="S2.SS0.SSS0.Px1"> <h4 class="ltx_title ltx_title_paragraph">Planning.</h4> <div class="ltx_para" id="S2.SS0.SSS0.Px1.p1"> <p class="ltx_p" id="S2.SS0.SSS0.Px1.p1.1">Our work contributes to the field of planning in long-form text generation. Humans typically simplify complex tasks into manageable subtasks, a method mirrored in recent approaches employing large language models (LLMs) for planning. Techniques such as Chain of Thought (CoT) <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2410.06203v1#bib.bib26" title="">26</a>]</cite>, Zero-shot-CoT <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2410.06203v1#bib.bib9" title="">9</a>]</cite>, Self-consistent CoT (CoT-SC) <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2410.06203v1#bib.bib25" title="">25</a>]</cite> guide LLMs through sequential reasoning by utilizing intermediate reasoning steps. More advanced methods, like Tree of Thoughts (ToT) <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2410.06203v1#bib.bib34" title="">34</a>]</cite>, GoT <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2410.06203v1#bib.bib3" title="">3</a>]</cite> enhance these strategies with tree-like and graph-based reasoning structures, respectively. Additional methods, including RePrompting <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2410.06203v1#bib.bib31" title="">31</a>]</cite> and ReWOO <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2410.06203v1#bib.bib27" title="">27</a>]</cite> ensure planning accuracy by integrating observational data and using corrective prompting HuggingGPT <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2410.06203v1#bib.bib21" title="">21</a>]</cite> further decomposes tasks into sub-goals solved through iterative LLM interactions, contrasting the one-shot approach of CoT and Zero-shotCoT. Despite these innovations, specialized zero-shot plan generation for specific domains remains to be challenging, addressed by models like LLM+P <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2410.06203v1#bib.bib12" title="">12</a>]</cite> , LLM-DP <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2410.06203v1#bib.bib5" title="">5</a>]</cite>, and CO-LLM <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2410.06203v1#bib.bib35" title="">35</a>]</cite>. Built upon previous works, our approach trains models specifically to enhance planning within the domain of long-form text generation.</p> </div> </section> <section class="ltx_paragraph" id="S2.SS0.SSS0.Px2"> <h4 class="ltx_title ltx_title_paragraph">Long-form text generation.</h4> <div class="ltx_para" id="S2.SS0.SSS0.Px2.p1"> <p class="ltx_p" id="S2.SS0.SSS0.Px2.p1.1">Long-form text generation and question answering (LFQA), which require maintaining coherence over extended texts, remain highly challenging for large language models (LLMs), as evidenced by numerous studies <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2410.06203v1#bib.bib7" title="">7</a>, <a class="ltx_ref" href="https://arxiv.org/html/2410.06203v1#bib.bib28" title="">28</a>, <a class="ltx_ref" href="https://arxiv.org/html/2410.06203v1#bib.bib10" title="">10</a>, <a class="ltx_ref" href="https://arxiv.org/html/2410.06203v1#bib.bib15" title="">15</a>, <a class="ltx_ref" href="https://arxiv.org/html/2410.06203v1#bib.bib22" title="">22</a>]</cite>. <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2410.06203v1#bib.bib24" title="">24</a>]</cite> introduced a progressive generation technique utilizing pretrained language models to enhance the creation of extended narratives. <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2410.06203v1#bib.bib30" title="">30</a>]</cite> explored discourse-aware neural extractive text summarization, essential for maintaining logical flow and thematic consistency in long documents. There have also been efforts to generate Wikipedia articles, as documented by <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2410.06203v1#bib.bib2" title="">2</a>, <a class="ltx_ref" href="https://arxiv.org/html/2410.06203v1#bib.bib14" title="">14</a>, <a class="ltx_ref" href="https://arxiv.org/html/2410.06203v1#bib.bib13" title="">13</a>]</cite>, along with recent advancements by <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2410.06203v1#bib.bib6" title="">6</a>]</cite>. <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2410.06203v1#bib.bib1" title="">1</a>]</cite> developed the Imitate Retrieve-Paraphrase framework to enhance expository writing at the paragraph level, particularly focusing on the integration of information from diverse sources. Our research introduces a novel method that leverages organic long documents to create intermediate text generation plans. This approach trains models to enhance their abilities not only in generating text but also in generating and adhering to these structured plans, thereby distinguishing our method from previous work. The use of the Retrieval-Augmented Generation (RAG) technique is outside the scope of our current discussion, although our method is compatible with RAG applications.</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>Problem Setup</h2> <figure class="ltx_figure" id="S3.F1"><img alt="Refer to caption" class="ltx_graphics ltx_centering ltx_img_landscape" height="178" id="S3.F1.g1" src="extracted/5908544/figures/training_and_inference.png" width="598"/> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure">Figure 1: </span>Input/output of training and inference tasks.</figcaption> </figure> <section class="ltx_paragraph" id="S3.SS0.SSS0.Px1"> <h4 class="ltx_title ltx_title_paragraph">Long-form text generation.</h4> <div class="ltx_para" id="S3.SS0.SSS0.Px1.p1"> <p class="ltx_p" id="S3.SS0.SSS0.Px1.p1.3">Given an input context <math alttext="x_{i}" class="ltx_Math" display="inline" id="S3.SS0.SSS0.Px1.p1.1.m1.1"><semantics id="S3.SS0.SSS0.Px1.p1.1.m1.1a"><msub id="S3.SS0.SSS0.Px1.p1.1.m1.1.1" xref="S3.SS0.SSS0.Px1.p1.1.m1.1.1.cmml"><mi id="S3.SS0.SSS0.Px1.p1.1.m1.1.1.2" xref="S3.SS0.SSS0.Px1.p1.1.m1.1.1.2.cmml">x</mi><mi id="S3.SS0.SSS0.Px1.p1.1.m1.1.1.3" xref="S3.SS0.SSS0.Px1.p1.1.m1.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS0.SSS0.Px1.p1.1.m1.1b"><apply id="S3.SS0.SSS0.Px1.p1.1.m1.1.1.cmml" xref="S3.SS0.SSS0.Px1.p1.1.m1.1.1"><csymbol cd="ambiguous" id="S3.SS0.SSS0.Px1.p1.1.m1.1.1.1.cmml" xref="S3.SS0.SSS0.Px1.p1.1.m1.1.1">subscript</csymbol><ci id="S3.SS0.SSS0.Px1.p1.1.m1.1.1.2.cmml" xref="S3.SS0.SSS0.Px1.p1.1.m1.1.1.2">𝑥</ci><ci id="S3.SS0.SSS0.Px1.p1.1.m1.1.1.3.cmml" xref="S3.SS0.SSS0.Px1.p1.1.m1.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS0.SSS0.Px1.p1.1.m1.1c">x_{i}</annotation><annotation encoding="application/x-llamapun" id="S3.SS0.SSS0.Px1.p1.1.m1.1d">italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math>, which can either be the source academic paper for SciNews generation or a sentence prompt (e.g., “generate wiki page about {topic}” for Wikipedia page generation), the objective is to fine-tune an LLM to generate a long-form article <math alttext="y_{i}" class="ltx_Math" display="inline" id="S3.SS0.SSS0.Px1.p1.2.m2.1"><semantics id="S3.SS0.SSS0.Px1.p1.2.m2.1a"><msub id="S3.SS0.SSS0.Px1.p1.2.m2.1.1" xref="S3.SS0.SSS0.Px1.p1.2.m2.1.1.cmml"><mi id="S3.SS0.SSS0.Px1.p1.2.m2.1.1.2" xref="S3.SS0.SSS0.Px1.p1.2.m2.1.1.2.cmml">y</mi><mi id="S3.SS0.SSS0.Px1.p1.2.m2.1.1.3" xref="S3.SS0.SSS0.Px1.p1.2.m2.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS0.SSS0.Px1.p1.2.m2.1b"><apply id="S3.SS0.SSS0.Px1.p1.2.m2.1.1.cmml" xref="S3.SS0.SSS0.Px1.p1.2.m2.1.1"><csymbol cd="ambiguous" id="S3.SS0.SSS0.Px1.p1.2.m2.1.1.1.cmml" xref="S3.SS0.SSS0.Px1.p1.2.m2.1.1">subscript</csymbol><ci id="S3.SS0.SSS0.Px1.p1.2.m2.1.1.2.cmml" xref="S3.SS0.SSS0.Px1.p1.2.m2.1.1.2">𝑦</ci><ci id="S3.SS0.SSS0.Px1.p1.2.m2.1.1.3.cmml" xref="S3.SS0.SSS0.Px1.p1.2.m2.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS0.SSS0.Px1.p1.2.m2.1c">y_{i}</annotation><annotation encoding="application/x-llamapun" id="S3.SS0.SSS0.Px1.p1.2.m2.1d">italic_y start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> from <math alttext="x_{i}" class="ltx_Math" display="inline" id="S3.SS0.SSS0.Px1.p1.3.m3.1"><semantics id="S3.SS0.SSS0.Px1.p1.3.m3.1a"><msub id="S3.SS0.SSS0.Px1.p1.3.m3.1.1" xref="S3.SS0.SSS0.Px1.p1.3.m3.1.1.cmml"><mi id="S3.SS0.SSS0.Px1.p1.3.m3.1.1.2" xref="S3.SS0.SSS0.Px1.p1.3.m3.1.1.2.cmml">x</mi><mi id="S3.SS0.SSS0.Px1.p1.3.m3.1.1.3" xref="S3.SS0.SSS0.Px1.p1.3.m3.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS0.SSS0.Px1.p1.3.m3.1b"><apply id="S3.SS0.SSS0.Px1.p1.3.m3.1.1.cmml" xref="S3.SS0.SSS0.Px1.p1.3.m3.1.1"><csymbol cd="ambiguous" id="S3.SS0.SSS0.Px1.p1.3.m3.1.1.1.cmml" xref="S3.SS0.SSS0.Px1.p1.3.m3.1.1">subscript</csymbol><ci id="S3.SS0.SSS0.Px1.p1.3.m3.1.1.2.cmml" xref="S3.SS0.SSS0.Px1.p1.3.m3.1.1.2">𝑥</ci><ci id="S3.SS0.SSS0.Px1.p1.3.m3.1.1.3.cmml" xref="S3.SS0.SSS0.Px1.p1.3.m3.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS0.SSS0.Px1.p1.3.m3.1c">x_{i}</annotation><annotation encoding="application/x-llamapun" id="S3.SS0.SSS0.Px1.p1.3.m3.1d">italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S3.SS0.SSS0.Px1.p2"> <p class="ltx_p" id="S3.SS0.SSS0.Px1.p2.2">The dataset <math alttext="D" class="ltx_Math" display="inline" id="S3.SS0.SSS0.Px1.p2.1.m1.1"><semantics id="S3.SS0.SSS0.Px1.p2.1.m1.1a"><mi id="S3.SS0.SSS0.Px1.p2.1.m1.1.1" xref="S3.SS0.SSS0.Px1.p2.1.m1.1.1.cmml">D</mi><annotation-xml encoding="MathML-Content" id="S3.SS0.SSS0.Px1.p2.1.m1.1b"><ci id="S3.SS0.SSS0.Px1.p2.1.m1.1.1.cmml" xref="S3.SS0.SSS0.Px1.p2.1.m1.1.1">𝐷</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS0.SSS0.Px1.p2.1.m1.1c">D</annotation><annotation encoding="application/x-llamapun" id="S3.SS0.SSS0.Px1.p2.1.m1.1d">italic_D</annotation></semantics></math> used for fine-tuning is structured to align with this objective. It consists of pairs of input contexts and their corresponding final documents. Formally, the dataset <math alttext="D" class="ltx_Math" display="inline" id="S3.SS0.SSS0.Px1.p2.2.m2.1"><semantics id="S3.SS0.SSS0.Px1.p2.2.m2.1a"><mi id="S3.SS0.SSS0.Px1.p2.2.m2.1.1" xref="S3.SS0.SSS0.Px1.p2.2.m2.1.1.cmml">D</mi><annotation-xml encoding="MathML-Content" id="S3.SS0.SSS0.Px1.p2.2.m2.1b"><ci id="S3.SS0.SSS0.Px1.p2.2.m2.1.1.cmml" xref="S3.SS0.SSS0.Px1.p2.2.m2.1.1">𝐷</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS0.SSS0.Px1.p2.2.m2.1c">D</annotation><annotation encoding="application/x-llamapun" id="S3.SS0.SSS0.Px1.p2.2.m2.1d">italic_D</annotation></semantics></math> is defined as:</p> <table class="ltx_equation ltx_eqn_table" id="S3.Ex1"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="D=\{(x_{1},y_{1}),(x_{2},y_{2}),\ldots,(x_{n},y_{n})\}" class="ltx_Math" display="block" id="S3.Ex1.m1.4"><semantics id="S3.Ex1.m1.4a"><mrow 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id="S3.Ex1.m1.3.3.2.2.2.2.2.3.cmml" type="integer" xref="S3.Ex1.m1.3.3.2.2.2.2.2.3">2</cn></apply></interval><ci id="S3.Ex1.m1.1.1.cmml" xref="S3.Ex1.m1.1.1">…</ci><interval closure="open" id="S3.Ex1.m1.4.4.3.3.3.3.cmml" xref="S3.Ex1.m1.4.4.3.3.3.2"><apply id="S3.Ex1.m1.4.4.3.3.3.1.1.cmml" xref="S3.Ex1.m1.4.4.3.3.3.1.1"><csymbol cd="ambiguous" id="S3.Ex1.m1.4.4.3.3.3.1.1.1.cmml" xref="S3.Ex1.m1.4.4.3.3.3.1.1">subscript</csymbol><ci id="S3.Ex1.m1.4.4.3.3.3.1.1.2.cmml" xref="S3.Ex1.m1.4.4.3.3.3.1.1.2">𝑥</ci><ci id="S3.Ex1.m1.4.4.3.3.3.1.1.3.cmml" xref="S3.Ex1.m1.4.4.3.3.3.1.1.3">𝑛</ci></apply><apply id="S3.Ex1.m1.4.4.3.3.3.2.2.cmml" xref="S3.Ex1.m1.4.4.3.3.3.2.2"><csymbol cd="ambiguous" id="S3.Ex1.m1.4.4.3.3.3.2.2.1.cmml" xref="S3.Ex1.m1.4.4.3.3.3.2.2">subscript</csymbol><ci id="S3.Ex1.m1.4.4.3.3.3.2.2.2.cmml" xref="S3.Ex1.m1.4.4.3.3.3.2.2.2">𝑦</ci><ci id="S3.Ex1.m1.4.4.3.3.3.2.2.3.cmml" xref="S3.Ex1.m1.4.4.3.3.3.2.2.3">𝑛</ci></apply></interval></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Ex1.m1.4c">D=\{(x_{1},y_{1}),(x_{2},y_{2}),\ldots,(x_{n},y_{n})\}</annotation><annotation encoding="application/x-llamapun" id="S3.Ex1.m1.4d">italic_D = { ( italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_y start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) , ( italic_x start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , italic_y start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) , … , ( italic_x start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT , italic_y start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ) }</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S3.SS0.SSS0.Px1.p2.5">where each tuple <math alttext="(x_{i},y_{i})" class="ltx_Math" display="inline" id="S3.SS0.SSS0.Px1.p2.3.m1.2"><semantics id="S3.SS0.SSS0.Px1.p2.3.m1.2a"><mrow id="S3.SS0.SSS0.Px1.p2.3.m1.2.2.2" xref="S3.SS0.SSS0.Px1.p2.3.m1.2.2.3.cmml"><mo id="S3.SS0.SSS0.Px1.p2.3.m1.2.2.2.3" stretchy="false" xref="S3.SS0.SSS0.Px1.p2.3.m1.2.2.3.cmml">(</mo><msub id="S3.SS0.SSS0.Px1.p2.3.m1.1.1.1.1" xref="S3.SS0.SSS0.Px1.p2.3.m1.1.1.1.1.cmml"><mi id="S3.SS0.SSS0.Px1.p2.3.m1.1.1.1.1.2" xref="S3.SS0.SSS0.Px1.p2.3.m1.1.1.1.1.2.cmml">x</mi><mi id="S3.SS0.SSS0.Px1.p2.3.m1.1.1.1.1.3" xref="S3.SS0.SSS0.Px1.p2.3.m1.1.1.1.1.3.cmml">i</mi></msub><mo id="S3.SS0.SSS0.Px1.p2.3.m1.2.2.2.4" xref="S3.SS0.SSS0.Px1.p2.3.m1.2.2.3.cmml">,</mo><msub id="S3.SS0.SSS0.Px1.p2.3.m1.2.2.2.2" xref="S3.SS0.SSS0.Px1.p2.3.m1.2.2.2.2.cmml"><mi id="S3.SS0.SSS0.Px1.p2.3.m1.2.2.2.2.2" xref="S3.SS0.SSS0.Px1.p2.3.m1.2.2.2.2.2.cmml">y</mi><mi id="S3.SS0.SSS0.Px1.p2.3.m1.2.2.2.2.3" xref="S3.SS0.SSS0.Px1.p2.3.m1.2.2.2.2.3.cmml">i</mi></msub><mo id="S3.SS0.SSS0.Px1.p2.3.m1.2.2.2.5" stretchy="false" xref="S3.SS0.SSS0.Px1.p2.3.m1.2.2.3.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.SS0.SSS0.Px1.p2.3.m1.2b"><interval closure="open" id="S3.SS0.SSS0.Px1.p2.3.m1.2.2.3.cmml" xref="S3.SS0.SSS0.Px1.p2.3.m1.2.2.2"><apply id="S3.SS0.SSS0.Px1.p2.3.m1.1.1.1.1.cmml" xref="S3.SS0.SSS0.Px1.p2.3.m1.1.1.1.1"><csymbol cd="ambiguous" id="S3.SS0.SSS0.Px1.p2.3.m1.1.1.1.1.1.cmml" xref="S3.SS0.SSS0.Px1.p2.3.m1.1.1.1.1">subscript</csymbol><ci id="S3.SS0.SSS0.Px1.p2.3.m1.1.1.1.1.2.cmml" xref="S3.SS0.SSS0.Px1.p2.3.m1.1.1.1.1.2">𝑥</ci><ci id="S3.SS0.SSS0.Px1.p2.3.m1.1.1.1.1.3.cmml" xref="S3.SS0.SSS0.Px1.p2.3.m1.1.1.1.1.3">𝑖</ci></apply><apply id="S3.SS0.SSS0.Px1.p2.3.m1.2.2.2.2.cmml" xref="S3.SS0.SSS0.Px1.p2.3.m1.2.2.2.2"><csymbol cd="ambiguous" id="S3.SS0.SSS0.Px1.p2.3.m1.2.2.2.2.1.cmml" xref="S3.SS0.SSS0.Px1.p2.3.m1.2.2.2.2">subscript</csymbol><ci id="S3.SS0.SSS0.Px1.p2.3.m1.2.2.2.2.2.cmml" xref="S3.SS0.SSS0.Px1.p2.3.m1.2.2.2.2.2">𝑦</ci><ci id="S3.SS0.SSS0.Px1.p2.3.m1.2.2.2.2.3.cmml" xref="S3.SS0.SSS0.Px1.p2.3.m1.2.2.2.2.3">𝑖</ci></apply></interval></annotation-xml><annotation encoding="application/x-tex" id="S3.SS0.SSS0.Px1.p2.3.m1.2c">(x_{i},y_{i})</annotation><annotation encoding="application/x-llamapun" id="S3.SS0.SSS0.Px1.p2.3.m1.2d">( italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , italic_y start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT )</annotation></semantics></math> represents an input context <math alttext="x_{i}" class="ltx_Math" display="inline" id="S3.SS0.SSS0.Px1.p2.4.m2.1"><semantics id="S3.SS0.SSS0.Px1.p2.4.m2.1a"><msub id="S3.SS0.SSS0.Px1.p2.4.m2.1.1" xref="S3.SS0.SSS0.Px1.p2.4.m2.1.1.cmml"><mi id="S3.SS0.SSS0.Px1.p2.4.m2.1.1.2" xref="S3.SS0.SSS0.Px1.p2.4.m2.1.1.2.cmml">x</mi><mi id="S3.SS0.SSS0.Px1.p2.4.m2.1.1.3" xref="S3.SS0.SSS0.Px1.p2.4.m2.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS0.SSS0.Px1.p2.4.m2.1b"><apply id="S3.SS0.SSS0.Px1.p2.4.m2.1.1.cmml" xref="S3.SS0.SSS0.Px1.p2.4.m2.1.1"><csymbol cd="ambiguous" id="S3.SS0.SSS0.Px1.p2.4.m2.1.1.1.cmml" xref="S3.SS0.SSS0.Px1.p2.4.m2.1.1">subscript</csymbol><ci id="S3.SS0.SSS0.Px1.p2.4.m2.1.1.2.cmml" xref="S3.SS0.SSS0.Px1.p2.4.m2.1.1.2">𝑥</ci><ci id="S3.SS0.SSS0.Px1.p2.4.m2.1.1.3.cmml" xref="S3.SS0.SSS0.Px1.p2.4.m2.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS0.SSS0.Px1.p2.4.m2.1c">x_{i}</annotation><annotation encoding="application/x-llamapun" id="S3.SS0.SSS0.Px1.p2.4.m2.1d">italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> and the final document <math alttext="y_{i}" class="ltx_Math" display="inline" id="S3.SS0.SSS0.Px1.p2.5.m3.1"><semantics id="S3.SS0.SSS0.Px1.p2.5.m3.1a"><msub id="S3.SS0.SSS0.Px1.p2.5.m3.1.1" xref="S3.SS0.SSS0.Px1.p2.5.m3.1.1.cmml"><mi id="S3.SS0.SSS0.Px1.p2.5.m3.1.1.2" xref="S3.SS0.SSS0.Px1.p2.5.m3.1.1.2.cmml">y</mi><mi id="S3.SS0.SSS0.Px1.p2.5.m3.1.1.3" xref="S3.SS0.SSS0.Px1.p2.5.m3.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS0.SSS0.Px1.p2.5.m3.1b"><apply id="S3.SS0.SSS0.Px1.p2.5.m3.1.1.cmml" xref="S3.SS0.SSS0.Px1.p2.5.m3.1.1"><csymbol cd="ambiguous" id="S3.SS0.SSS0.Px1.p2.5.m3.1.1.1.cmml" xref="S3.SS0.SSS0.Px1.p2.5.m3.1.1">subscript</csymbol><ci id="S3.SS0.SSS0.Px1.p2.5.m3.1.1.2.cmml" xref="S3.SS0.SSS0.Px1.p2.5.m3.1.1.2">𝑦</ci><ci id="S3.SS0.SSS0.Px1.p2.5.m3.1.1.3.cmml" xref="S3.SS0.SSS0.Px1.p2.5.m3.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS0.SSS0.Px1.p2.5.m3.1c">y_{i}</annotation><annotation encoding="application/x-llamapun" id="S3.SS0.SSS0.Px1.p2.5.m3.1d">italic_y start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math>.</p> </div> </section> <section class="ltx_paragraph" id="S3.SS0.SSS0.Px2"> <h4 class="ltx_title ltx_title_paragraph">Intermediate steps.</h4> <div class="ltx_para" id="S3.SS0.SSS0.Px2.p1"> <p class="ltx_p" id="S3.SS0.SSS0.Px2.p1.6">To reduce the cognitive burden on LLM when generating a full article directly from the input context, intermediate steps are often introduced, gradually leading from the input context to the final full article. In particular, for each input context <math alttext="x_{i}" class="ltx_Math" display="inline" id="S3.SS0.SSS0.Px2.p1.1.m1.1"><semantics id="S3.SS0.SSS0.Px2.p1.1.m1.1a"><msub id="S3.SS0.SSS0.Px2.p1.1.m1.1.1" xref="S3.SS0.SSS0.Px2.p1.1.m1.1.1.cmml"><mi id="S3.SS0.SSS0.Px2.p1.1.m1.1.1.2" xref="S3.SS0.SSS0.Px2.p1.1.m1.1.1.2.cmml">x</mi><mi id="S3.SS0.SSS0.Px2.p1.1.m1.1.1.3" xref="S3.SS0.SSS0.Px2.p1.1.m1.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS0.SSS0.Px2.p1.1.m1.1b"><apply id="S3.SS0.SSS0.Px2.p1.1.m1.1.1.cmml" xref="S3.SS0.SSS0.Px2.p1.1.m1.1.1"><csymbol cd="ambiguous" id="S3.SS0.SSS0.Px2.p1.1.m1.1.1.1.cmml" xref="S3.SS0.SSS0.Px2.p1.1.m1.1.1">subscript</csymbol><ci id="S3.SS0.SSS0.Px2.p1.1.m1.1.1.2.cmml" xref="S3.SS0.SSS0.Px2.p1.1.m1.1.1.2">𝑥</ci><ci id="S3.SS0.SSS0.Px2.p1.1.m1.1.1.3.cmml" xref="S3.SS0.SSS0.Px2.p1.1.m1.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS0.SSS0.Px2.p1.1.m1.1c">x_{i}</annotation><annotation encoding="application/x-llamapun" id="S3.SS0.SSS0.Px2.p1.1.m1.1d">italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> and its corresponding article <math alttext="y_{i}" class="ltx_Math" display="inline" id="S3.SS0.SSS0.Px2.p1.2.m2.1"><semantics id="S3.SS0.SSS0.Px2.p1.2.m2.1a"><msub id="S3.SS0.SSS0.Px2.p1.2.m2.1.1" xref="S3.SS0.SSS0.Px2.p1.2.m2.1.1.cmml"><mi id="S3.SS0.SSS0.Px2.p1.2.m2.1.1.2" xref="S3.SS0.SSS0.Px2.p1.2.m2.1.1.2.cmml">y</mi><mi id="S3.SS0.SSS0.Px2.p1.2.m2.1.1.3" xref="S3.SS0.SSS0.Px2.p1.2.m2.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS0.SSS0.Px2.p1.2.m2.1b"><apply id="S3.SS0.SSS0.Px2.p1.2.m2.1.1.cmml" xref="S3.SS0.SSS0.Px2.p1.2.m2.1.1"><csymbol cd="ambiguous" id="S3.SS0.SSS0.Px2.p1.2.m2.1.1.1.cmml" xref="S3.SS0.SSS0.Px2.p1.2.m2.1.1">subscript</csymbol><ci id="S3.SS0.SSS0.Px2.p1.2.m2.1.1.2.cmml" xref="S3.SS0.SSS0.Px2.p1.2.m2.1.1.2">𝑦</ci><ci id="S3.SS0.SSS0.Px2.p1.2.m2.1.1.3.cmml" xref="S3.SS0.SSS0.Px2.p1.2.m2.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS0.SSS0.Px2.p1.2.m2.1c">y_{i}</annotation><annotation encoding="application/x-llamapun" id="S3.SS0.SSS0.Px2.p1.2.m2.1d">italic_y start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math>, the intermediate steps <math alttext="{\mathbf{z}}_{i}" class="ltx_Math" display="inline" id="S3.SS0.SSS0.Px2.p1.3.m3.1"><semantics id="S3.SS0.SSS0.Px2.p1.3.m3.1a"><msub id="S3.SS0.SSS0.Px2.p1.3.m3.1.1" xref="S3.SS0.SSS0.Px2.p1.3.m3.1.1.cmml"><mi id="S3.SS0.SSS0.Px2.p1.3.m3.1.1.2" xref="S3.SS0.SSS0.Px2.p1.3.m3.1.1.2.cmml">𝐳</mi><mi id="S3.SS0.SSS0.Px2.p1.3.m3.1.1.3" xref="S3.SS0.SSS0.Px2.p1.3.m3.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS0.SSS0.Px2.p1.3.m3.1b"><apply id="S3.SS0.SSS0.Px2.p1.3.m3.1.1.cmml" xref="S3.SS0.SSS0.Px2.p1.3.m3.1.1"><csymbol cd="ambiguous" id="S3.SS0.SSS0.Px2.p1.3.m3.1.1.1.cmml" xref="S3.SS0.SSS0.Px2.p1.3.m3.1.1">subscript</csymbol><ci id="S3.SS0.SSS0.Px2.p1.3.m3.1.1.2.cmml" xref="S3.SS0.SSS0.Px2.p1.3.m3.1.1.2">𝐳</ci><ci id="S3.SS0.SSS0.Px2.p1.3.m3.1.1.3.cmml" xref="S3.SS0.SSS0.Px2.p1.3.m3.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS0.SSS0.Px2.p1.3.m3.1c">{\mathbf{z}}_{i}</annotation><annotation encoding="application/x-llamapun" id="S3.SS0.SSS0.Px2.p1.3.m3.1d">bold_z start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> can either be a set of distinct pieces of information (e.g., summary, outline, key information) or a sequence of information where each element <math alttext="(j+1)" class="ltx_Math" display="inline" id="S3.SS0.SSS0.Px2.p1.4.m4.1"><semantics id="S3.SS0.SSS0.Px2.p1.4.m4.1a"><mrow id="S3.SS0.SSS0.Px2.p1.4.m4.1.1.1" xref="S3.SS0.SSS0.Px2.p1.4.m4.1.1.1.1.cmml"><mo id="S3.SS0.SSS0.Px2.p1.4.m4.1.1.1.2" stretchy="false" xref="S3.SS0.SSS0.Px2.p1.4.m4.1.1.1.1.cmml">(</mo><mrow id="S3.SS0.SSS0.Px2.p1.4.m4.1.1.1.1" xref="S3.SS0.SSS0.Px2.p1.4.m4.1.1.1.1.cmml"><mi id="S3.SS0.SSS0.Px2.p1.4.m4.1.1.1.1.2" xref="S3.SS0.SSS0.Px2.p1.4.m4.1.1.1.1.2.cmml">j</mi><mo id="S3.SS0.SSS0.Px2.p1.4.m4.1.1.1.1.1" xref="S3.SS0.SSS0.Px2.p1.4.m4.1.1.1.1.1.cmml">+</mo><mn id="S3.SS0.SSS0.Px2.p1.4.m4.1.1.1.1.3" xref="S3.SS0.SSS0.Px2.p1.4.m4.1.1.1.1.3.cmml">1</mn></mrow><mo id="S3.SS0.SSS0.Px2.p1.4.m4.1.1.1.3" stretchy="false" xref="S3.SS0.SSS0.Px2.p1.4.m4.1.1.1.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.SS0.SSS0.Px2.p1.4.m4.1b"><apply id="S3.SS0.SSS0.Px2.p1.4.m4.1.1.1.1.cmml" xref="S3.SS0.SSS0.Px2.p1.4.m4.1.1.1"><plus id="S3.SS0.SSS0.Px2.p1.4.m4.1.1.1.1.1.cmml" xref="S3.SS0.SSS0.Px2.p1.4.m4.1.1.1.1.1"></plus><ci id="S3.SS0.SSS0.Px2.p1.4.m4.1.1.1.1.2.cmml" xref="S3.SS0.SSS0.Px2.p1.4.m4.1.1.1.1.2">𝑗</ci><cn id="S3.SS0.SSS0.Px2.p1.4.m4.1.1.1.1.3.cmml" type="integer" xref="S3.SS0.SSS0.Px2.p1.4.m4.1.1.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS0.SSS0.Px2.p1.4.m4.1c">(j+1)</annotation><annotation encoding="application/x-llamapun" id="S3.SS0.SSS0.Px2.p1.4.m4.1d">( italic_j + 1 )</annotation></semantics></math>-th is dependent on the preceding element <math alttext="j" class="ltx_Math" display="inline" id="S3.SS0.SSS0.Px2.p1.5.m5.1"><semantics id="S3.SS0.SSS0.Px2.p1.5.m5.1a"><mi id="S3.SS0.SSS0.Px2.p1.5.m5.1.1" xref="S3.SS0.SSS0.Px2.p1.5.m5.1.1.cmml">j</mi><annotation-xml encoding="MathML-Content" id="S3.SS0.SSS0.Px2.p1.5.m5.1b"><ci id="S3.SS0.SSS0.Px2.p1.5.m5.1.1.cmml" xref="S3.SS0.SSS0.Px2.p1.5.m5.1.1">𝑗</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS0.SSS0.Px2.p1.5.m5.1c">j</annotation><annotation encoding="application/x-llamapun" id="S3.SS0.SSS0.Px2.p1.5.m5.1d">italic_j</annotation></semantics></math>. We denote <math alttext="{\mathbf{z}}_{i}" class="ltx_Math" display="inline" id="S3.SS0.SSS0.Px2.p1.6.m6.1"><semantics id="S3.SS0.SSS0.Px2.p1.6.m6.1a"><msub id="S3.SS0.SSS0.Px2.p1.6.m6.1.1" xref="S3.SS0.SSS0.Px2.p1.6.m6.1.1.cmml"><mi id="S3.SS0.SSS0.Px2.p1.6.m6.1.1.2" xref="S3.SS0.SSS0.Px2.p1.6.m6.1.1.2.cmml">𝐳</mi><mi id="S3.SS0.SSS0.Px2.p1.6.m6.1.1.3" xref="S3.SS0.SSS0.Px2.p1.6.m6.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS0.SSS0.Px2.p1.6.m6.1b"><apply id="S3.SS0.SSS0.Px2.p1.6.m6.1.1.cmml" xref="S3.SS0.SSS0.Px2.p1.6.m6.1.1"><csymbol cd="ambiguous" id="S3.SS0.SSS0.Px2.p1.6.m6.1.1.1.cmml" xref="S3.SS0.SSS0.Px2.p1.6.m6.1.1">subscript</csymbol><ci id="S3.SS0.SSS0.Px2.p1.6.m6.1.1.2.cmml" xref="S3.SS0.SSS0.Px2.p1.6.m6.1.1.2">𝐳</ci><ci id="S3.SS0.SSS0.Px2.p1.6.m6.1.1.3.cmml" xref="S3.SS0.SSS0.Px2.p1.6.m6.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS0.SSS0.Px2.p1.6.m6.1c">{\mathbf{z}}_{i}</annotation><annotation encoding="application/x-llamapun" id="S3.SS0.SSS0.Px2.p1.6.m6.1d">bold_z start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> as:</p> <table class="ltx_equation ltx_eqn_table" id="S3.Ex2"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="{\mathbf{z}}_{i}=\{z_{i1},z_{i2},\ldots,z_{ik}\}" class="ltx_Math" display="block" id="S3.Ex2.m1.4"><semantics id="S3.Ex2.m1.4a"><mrow id="S3.Ex2.m1.4.4" xref="S3.Ex2.m1.4.4.cmml"><msub id="S3.Ex2.m1.4.4.5" xref="S3.Ex2.m1.4.4.5.cmml"><mi id="S3.Ex2.m1.4.4.5.2" xref="S3.Ex2.m1.4.4.5.2.cmml">𝐳</mi><mi id="S3.Ex2.m1.4.4.5.3" xref="S3.Ex2.m1.4.4.5.3.cmml">i</mi></msub><mo id="S3.Ex2.m1.4.4.4" xref="S3.Ex2.m1.4.4.4.cmml">=</mo><mrow id="S3.Ex2.m1.4.4.3.3" xref="S3.Ex2.m1.4.4.3.4.cmml"><mo id="S3.Ex2.m1.4.4.3.3.4" stretchy="false" xref="S3.Ex2.m1.4.4.3.4.cmml">{</mo><msub id="S3.Ex2.m1.2.2.1.1.1" xref="S3.Ex2.m1.2.2.1.1.1.cmml"><mi id="S3.Ex2.m1.2.2.1.1.1.2" xref="S3.Ex2.m1.2.2.1.1.1.2.cmml">z</mi><mrow id="S3.Ex2.m1.2.2.1.1.1.3" xref="S3.Ex2.m1.2.2.1.1.1.3.cmml"><mi id="S3.Ex2.m1.2.2.1.1.1.3.2" xref="S3.Ex2.m1.2.2.1.1.1.3.2.cmml">i</mi><mo id="S3.Ex2.m1.2.2.1.1.1.3.1" xref="S3.Ex2.m1.2.2.1.1.1.3.1.cmml">⁢</mo><mn id="S3.Ex2.m1.2.2.1.1.1.3.3" xref="S3.Ex2.m1.2.2.1.1.1.3.3.cmml">1</mn></mrow></msub><mo id="S3.Ex2.m1.4.4.3.3.5" xref="S3.Ex2.m1.4.4.3.4.cmml">,</mo><msub id="S3.Ex2.m1.3.3.2.2.2" xref="S3.Ex2.m1.3.3.2.2.2.cmml"><mi id="S3.Ex2.m1.3.3.2.2.2.2" xref="S3.Ex2.m1.3.3.2.2.2.2.cmml">z</mi><mrow id="S3.Ex2.m1.3.3.2.2.2.3" xref="S3.Ex2.m1.3.3.2.2.2.3.cmml"><mi id="S3.Ex2.m1.3.3.2.2.2.3.2" xref="S3.Ex2.m1.3.3.2.2.2.3.2.cmml">i</mi><mo id="S3.Ex2.m1.3.3.2.2.2.3.1" xref="S3.Ex2.m1.3.3.2.2.2.3.1.cmml">⁢</mo><mn id="S3.Ex2.m1.3.3.2.2.2.3.3" xref="S3.Ex2.m1.3.3.2.2.2.3.3.cmml">2</mn></mrow></msub><mo id="S3.Ex2.m1.4.4.3.3.6" xref="S3.Ex2.m1.4.4.3.4.cmml">,</mo><mi id="S3.Ex2.m1.1.1" mathvariant="normal" xref="S3.Ex2.m1.1.1.cmml">…</mi><mo id="S3.Ex2.m1.4.4.3.3.7" xref="S3.Ex2.m1.4.4.3.4.cmml">,</mo><msub id="S3.Ex2.m1.4.4.3.3.3" xref="S3.Ex2.m1.4.4.3.3.3.cmml"><mi id="S3.Ex2.m1.4.4.3.3.3.2" xref="S3.Ex2.m1.4.4.3.3.3.2.cmml">z</mi><mrow id="S3.Ex2.m1.4.4.3.3.3.3" xref="S3.Ex2.m1.4.4.3.3.3.3.cmml"><mi id="S3.Ex2.m1.4.4.3.3.3.3.2" xref="S3.Ex2.m1.4.4.3.3.3.3.2.cmml">i</mi><mo id="S3.Ex2.m1.4.4.3.3.3.3.1" xref="S3.Ex2.m1.4.4.3.3.3.3.1.cmml">⁢</mo><mi id="S3.Ex2.m1.4.4.3.3.3.3.3" xref="S3.Ex2.m1.4.4.3.3.3.3.3.cmml">k</mi></mrow></msub><mo id="S3.Ex2.m1.4.4.3.3.8" stretchy="false" xref="S3.Ex2.m1.4.4.3.4.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Ex2.m1.4b"><apply id="S3.Ex2.m1.4.4.cmml" xref="S3.Ex2.m1.4.4"><eq id="S3.Ex2.m1.4.4.4.cmml" xref="S3.Ex2.m1.4.4.4"></eq><apply id="S3.Ex2.m1.4.4.5.cmml" xref="S3.Ex2.m1.4.4.5"><csymbol cd="ambiguous" id="S3.Ex2.m1.4.4.5.1.cmml" xref="S3.Ex2.m1.4.4.5">subscript</csymbol><ci id="S3.Ex2.m1.4.4.5.2.cmml" xref="S3.Ex2.m1.4.4.5.2">𝐳</ci><ci id="S3.Ex2.m1.4.4.5.3.cmml" xref="S3.Ex2.m1.4.4.5.3">𝑖</ci></apply><set id="S3.Ex2.m1.4.4.3.4.cmml" xref="S3.Ex2.m1.4.4.3.3"><apply id="S3.Ex2.m1.2.2.1.1.1.cmml" xref="S3.Ex2.m1.2.2.1.1.1"><csymbol cd="ambiguous" id="S3.Ex2.m1.2.2.1.1.1.1.cmml" xref="S3.Ex2.m1.2.2.1.1.1">subscript</csymbol><ci id="S3.Ex2.m1.2.2.1.1.1.2.cmml" xref="S3.Ex2.m1.2.2.1.1.1.2">𝑧</ci><apply id="S3.Ex2.m1.2.2.1.1.1.3.cmml" xref="S3.Ex2.m1.2.2.1.1.1.3"><times id="S3.Ex2.m1.2.2.1.1.1.3.1.cmml" xref="S3.Ex2.m1.2.2.1.1.1.3.1"></times><ci id="S3.Ex2.m1.2.2.1.1.1.3.2.cmml" xref="S3.Ex2.m1.2.2.1.1.1.3.2">𝑖</ci><cn id="S3.Ex2.m1.2.2.1.1.1.3.3.cmml" type="integer" xref="S3.Ex2.m1.2.2.1.1.1.3.3">1</cn></apply></apply><apply id="S3.Ex2.m1.3.3.2.2.2.cmml" xref="S3.Ex2.m1.3.3.2.2.2"><csymbol cd="ambiguous" id="S3.Ex2.m1.3.3.2.2.2.1.cmml" xref="S3.Ex2.m1.3.3.2.2.2">subscript</csymbol><ci id="S3.Ex2.m1.3.3.2.2.2.2.cmml" xref="S3.Ex2.m1.3.3.2.2.2.2">𝑧</ci><apply id="S3.Ex2.m1.3.3.2.2.2.3.cmml" xref="S3.Ex2.m1.3.3.2.2.2.3"><times id="S3.Ex2.m1.3.3.2.2.2.3.1.cmml" xref="S3.Ex2.m1.3.3.2.2.2.3.1"></times><ci id="S3.Ex2.m1.3.3.2.2.2.3.2.cmml" xref="S3.Ex2.m1.3.3.2.2.2.3.2">𝑖</ci><cn id="S3.Ex2.m1.3.3.2.2.2.3.3.cmml" type="integer" xref="S3.Ex2.m1.3.3.2.2.2.3.3">2</cn></apply></apply><ci id="S3.Ex2.m1.1.1.cmml" xref="S3.Ex2.m1.1.1">…</ci><apply id="S3.Ex2.m1.4.4.3.3.3.cmml" xref="S3.Ex2.m1.4.4.3.3.3"><csymbol cd="ambiguous" id="S3.Ex2.m1.4.4.3.3.3.1.cmml" xref="S3.Ex2.m1.4.4.3.3.3">subscript</csymbol><ci id="S3.Ex2.m1.4.4.3.3.3.2.cmml" xref="S3.Ex2.m1.4.4.3.3.3.2">𝑧</ci><apply id="S3.Ex2.m1.4.4.3.3.3.3.cmml" xref="S3.Ex2.m1.4.4.3.3.3.3"><times id="S3.Ex2.m1.4.4.3.3.3.3.1.cmml" xref="S3.Ex2.m1.4.4.3.3.3.3.1"></times><ci id="S3.Ex2.m1.4.4.3.3.3.3.2.cmml" xref="S3.Ex2.m1.4.4.3.3.3.3.2">𝑖</ci><ci id="S3.Ex2.m1.4.4.3.3.3.3.3.cmml" xref="S3.Ex2.m1.4.4.3.3.3.3.3">𝑘</ci></apply></apply></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Ex2.m1.4c">{\mathbf{z}}_{i}=\{z_{i1},z_{i2},\ldots,z_{ik}\}</annotation><annotation encoding="application/x-llamapun" id="S3.Ex2.m1.4d">bold_z start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = { italic_z start_POSTSUBSCRIPT italic_i 1 end_POSTSUBSCRIPT , italic_z start_POSTSUBSCRIPT italic_i 2 end_POSTSUBSCRIPT , … , italic_z start_POSTSUBSCRIPT italic_i italic_k end_POSTSUBSCRIPT }</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> </div> <div class="ltx_para" id="S3.SS0.SSS0.Px2.p2"> <p class="ltx_p" id="S3.SS0.SSS0.Px2.p2.4">The overall idea is that the auxiliary information provided at these intermediate steps <math alttext="{\mathbf{z}}_{i}" class="ltx_Math" display="inline" id="S3.SS0.SSS0.Px2.p2.1.m1.1"><semantics id="S3.SS0.SSS0.Px2.p2.1.m1.1a"><msub id="S3.SS0.SSS0.Px2.p2.1.m1.1.1" xref="S3.SS0.SSS0.Px2.p2.1.m1.1.1.cmml"><mi id="S3.SS0.SSS0.Px2.p2.1.m1.1.1.2" xref="S3.SS0.SSS0.Px2.p2.1.m1.1.1.2.cmml">𝐳</mi><mi id="S3.SS0.SSS0.Px2.p2.1.m1.1.1.3" xref="S3.SS0.SSS0.Px2.p2.1.m1.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS0.SSS0.Px2.p2.1.m1.1b"><apply id="S3.SS0.SSS0.Px2.p2.1.m1.1.1.cmml" xref="S3.SS0.SSS0.Px2.p2.1.m1.1.1"><csymbol cd="ambiguous" id="S3.SS0.SSS0.Px2.p2.1.m1.1.1.1.cmml" xref="S3.SS0.SSS0.Px2.p2.1.m1.1.1">subscript</csymbol><ci id="S3.SS0.SSS0.Px2.p2.1.m1.1.1.2.cmml" xref="S3.SS0.SSS0.Px2.p2.1.m1.1.1.2">𝐳</ci><ci id="S3.SS0.SSS0.Px2.p2.1.m1.1.1.3.cmml" xref="S3.SS0.SSS0.Px2.p2.1.m1.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS0.SSS0.Px2.p2.1.m1.1c">{\mathbf{z}}_{i}</annotation><annotation encoding="application/x-llamapun" id="S3.SS0.SSS0.Px2.p2.1.m1.1d">bold_z start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> can effectively guide the LLM to gradually approach the final target article <math alttext="y_{i}" class="ltx_Math" display="inline" id="S3.SS0.SSS0.Px2.p2.2.m2.1"><semantics id="S3.SS0.SSS0.Px2.p2.2.m2.1a"><msub id="S3.SS0.SSS0.Px2.p2.2.m2.1.1" xref="S3.SS0.SSS0.Px2.p2.2.m2.1.1.cmml"><mi id="S3.SS0.SSS0.Px2.p2.2.m2.1.1.2" xref="S3.SS0.SSS0.Px2.p2.2.m2.1.1.2.cmml">y</mi><mi id="S3.SS0.SSS0.Px2.p2.2.m2.1.1.3" xref="S3.SS0.SSS0.Px2.p2.2.m2.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS0.SSS0.Px2.p2.2.m2.1b"><apply id="S3.SS0.SSS0.Px2.p2.2.m2.1.1.cmml" xref="S3.SS0.SSS0.Px2.p2.2.m2.1.1"><csymbol cd="ambiguous" id="S3.SS0.SSS0.Px2.p2.2.m2.1.1.1.cmml" xref="S3.SS0.SSS0.Px2.p2.2.m2.1.1">subscript</csymbol><ci id="S3.SS0.SSS0.Px2.p2.2.m2.1.1.2.cmml" xref="S3.SS0.SSS0.Px2.p2.2.m2.1.1.2">𝑦</ci><ci id="S3.SS0.SSS0.Px2.p2.2.m2.1.1.3.cmml" xref="S3.SS0.SSS0.Px2.p2.2.m2.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS0.SSS0.Px2.p2.2.m2.1c">y_{i}</annotation><annotation encoding="application/x-llamapun" id="S3.SS0.SSS0.Px2.p2.2.m2.1d">italic_y start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> from the potentially abstract or noisy input context <math alttext="x_{i}" class="ltx_Math" display="inline" id="S3.SS0.SSS0.Px2.p2.3.m3.1"><semantics id="S3.SS0.SSS0.Px2.p2.3.m3.1a"><msub id="S3.SS0.SSS0.Px2.p2.3.m3.1.1" xref="S3.SS0.SSS0.Px2.p2.3.m3.1.1.cmml"><mi id="S3.SS0.SSS0.Px2.p2.3.m3.1.1.2" xref="S3.SS0.SSS0.Px2.p2.3.m3.1.1.2.cmml">x</mi><mi id="S3.SS0.SSS0.Px2.p2.3.m3.1.1.3" xref="S3.SS0.SSS0.Px2.p2.3.m3.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS0.SSS0.Px2.p2.3.m3.1b"><apply id="S3.SS0.SSS0.Px2.p2.3.m3.1.1.cmml" xref="S3.SS0.SSS0.Px2.p2.3.m3.1.1"><csymbol cd="ambiguous" id="S3.SS0.SSS0.Px2.p2.3.m3.1.1.1.cmml" xref="S3.SS0.SSS0.Px2.p2.3.m3.1.1">subscript</csymbol><ci id="S3.SS0.SSS0.Px2.p2.3.m3.1.1.2.cmml" xref="S3.SS0.SSS0.Px2.p2.3.m3.1.1.2">𝑥</ci><ci id="S3.SS0.SSS0.Px2.p2.3.m3.1.1.3.cmml" xref="S3.SS0.SSS0.Px2.p2.3.m3.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS0.SSS0.Px2.p2.3.m3.1c">x_{i}</annotation><annotation encoding="application/x-llamapun" id="S3.SS0.SSS0.Px2.p2.3.m3.1d">italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math>. There are multiple ways to instantiate the concrete content of <math alttext="{\mathbf{z}}_{i}" class="ltx_Math" display="inline" id="S3.SS0.SSS0.Px2.p2.4.m4.1"><semantics id="S3.SS0.SSS0.Px2.p2.4.m4.1a"><msub id="S3.SS0.SSS0.Px2.p2.4.m4.1.1" xref="S3.SS0.SSS0.Px2.p2.4.m4.1.1.cmml"><mi id="S3.SS0.SSS0.Px2.p2.4.m4.1.1.2" xref="S3.SS0.SSS0.Px2.p2.4.m4.1.1.2.cmml">𝐳</mi><mi id="S3.SS0.SSS0.Px2.p2.4.m4.1.1.3" xref="S3.SS0.SSS0.Px2.p2.4.m4.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS0.SSS0.Px2.p2.4.m4.1b"><apply id="S3.SS0.SSS0.Px2.p2.4.m4.1.1.cmml" xref="S3.SS0.SSS0.Px2.p2.4.m4.1.1"><csymbol cd="ambiguous" id="S3.SS0.SSS0.Px2.p2.4.m4.1.1.1.cmml" xref="S3.SS0.SSS0.Px2.p2.4.m4.1.1">subscript</csymbol><ci id="S3.SS0.SSS0.Px2.p2.4.m4.1.1.2.cmml" xref="S3.SS0.SSS0.Px2.p2.4.m4.1.1.2">𝐳</ci><ci id="S3.SS0.SSS0.Px2.p2.4.m4.1.1.3.cmml" xref="S3.SS0.SSS0.Px2.p2.4.m4.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS0.SSS0.Px2.p2.4.m4.1c">{\mathbf{z}}_{i}</annotation><annotation encoding="application/x-llamapun" id="S3.SS0.SSS0.Px2.p2.4.m4.1d">bold_z start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math>. For example:</p> </div> <div class="ltx_para" id="S3.SS0.SSS0.Px2.p3"> <ol class="ltx_enumerate" id="S3.I1"> <li class="ltx_item" id="S3.I1.i1" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">1.</span> <div class="ltx_para" id="S3.I1.i1.p1"> <p class="ltx_p" id="S3.I1.i1.p1.4"><span class="ltx_text ltx_font_bold" id="S3.I1.i1.p1.4.1">Linear:</span> One can decompose the full article writing task into writing one section at a time. In this approach, the first intermediate step <math alttext="z_{i1}" class="ltx_Math" display="inline" id="S3.I1.i1.p1.1.m1.1"><semantics id="S3.I1.i1.p1.1.m1.1a"><msub id="S3.I1.i1.p1.1.m1.1.1" xref="S3.I1.i1.p1.1.m1.1.1.cmml"><mi id="S3.I1.i1.p1.1.m1.1.1.2" xref="S3.I1.i1.p1.1.m1.1.1.2.cmml">z</mi><mrow id="S3.I1.i1.p1.1.m1.1.1.3" xref="S3.I1.i1.p1.1.m1.1.1.3.cmml"><mi id="S3.I1.i1.p1.1.m1.1.1.3.2" xref="S3.I1.i1.p1.1.m1.1.1.3.2.cmml">i</mi><mo id="S3.I1.i1.p1.1.m1.1.1.3.1" xref="S3.I1.i1.p1.1.m1.1.1.3.1.cmml">⁢</mo><mn id="S3.I1.i1.p1.1.m1.1.1.3.3" xref="S3.I1.i1.p1.1.m1.1.1.3.3.cmml">1</mn></mrow></msub><annotation-xml encoding="MathML-Content" id="S3.I1.i1.p1.1.m1.1b"><apply id="S3.I1.i1.p1.1.m1.1.1.cmml" xref="S3.I1.i1.p1.1.m1.1.1"><csymbol cd="ambiguous" id="S3.I1.i1.p1.1.m1.1.1.1.cmml" xref="S3.I1.i1.p1.1.m1.1.1">subscript</csymbol><ci id="S3.I1.i1.p1.1.m1.1.1.2.cmml" xref="S3.I1.i1.p1.1.m1.1.1.2">𝑧</ci><apply id="S3.I1.i1.p1.1.m1.1.1.3.cmml" xref="S3.I1.i1.p1.1.m1.1.1.3"><times id="S3.I1.i1.p1.1.m1.1.1.3.1.cmml" xref="S3.I1.i1.p1.1.m1.1.1.3.1"></times><ci id="S3.I1.i1.p1.1.m1.1.1.3.2.cmml" xref="S3.I1.i1.p1.1.m1.1.1.3.2">𝑖</ci><cn id="S3.I1.i1.p1.1.m1.1.1.3.3.cmml" type="integer" xref="S3.I1.i1.p1.1.m1.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I1.i1.p1.1.m1.1c">z_{i1}</annotation><annotation encoding="application/x-llamapun" id="S3.I1.i1.p1.1.m1.1d">italic_z start_POSTSUBSCRIPT italic_i 1 end_POSTSUBSCRIPT</annotation></semantics></math> would be the first section; the <math alttext="j" class="ltx_Math" display="inline" id="S3.I1.i1.p1.2.m2.1"><semantics id="S3.I1.i1.p1.2.m2.1a"><mi id="S3.I1.i1.p1.2.m2.1.1" xref="S3.I1.i1.p1.2.m2.1.1.cmml">j</mi><annotation-xml encoding="MathML-Content" id="S3.I1.i1.p1.2.m2.1b"><ci id="S3.I1.i1.p1.2.m2.1.1.cmml" xref="S3.I1.i1.p1.2.m2.1.1">𝑗</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.I1.i1.p1.2.m2.1c">j</annotation><annotation encoding="application/x-llamapun" id="S3.I1.i1.p1.2.m2.1d">italic_j</annotation></semantics></math>-th intermediate step <math alttext="z_{ij}" class="ltx_Math" display="inline" id="S3.I1.i1.p1.3.m3.1"><semantics id="S3.I1.i1.p1.3.m3.1a"><msub id="S3.I1.i1.p1.3.m3.1.1" xref="S3.I1.i1.p1.3.m3.1.1.cmml"><mi id="S3.I1.i1.p1.3.m3.1.1.2" xref="S3.I1.i1.p1.3.m3.1.1.2.cmml">z</mi><mrow id="S3.I1.i1.p1.3.m3.1.1.3" xref="S3.I1.i1.p1.3.m3.1.1.3.cmml"><mi id="S3.I1.i1.p1.3.m3.1.1.3.2" xref="S3.I1.i1.p1.3.m3.1.1.3.2.cmml">i</mi><mo id="S3.I1.i1.p1.3.m3.1.1.3.1" xref="S3.I1.i1.p1.3.m3.1.1.3.1.cmml">⁢</mo><mi id="S3.I1.i1.p1.3.m3.1.1.3.3" xref="S3.I1.i1.p1.3.m3.1.1.3.3.cmml">j</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="S3.I1.i1.p1.3.m3.1b"><apply id="S3.I1.i1.p1.3.m3.1.1.cmml" xref="S3.I1.i1.p1.3.m3.1.1"><csymbol cd="ambiguous" id="S3.I1.i1.p1.3.m3.1.1.1.cmml" xref="S3.I1.i1.p1.3.m3.1.1">subscript</csymbol><ci id="S3.I1.i1.p1.3.m3.1.1.2.cmml" xref="S3.I1.i1.p1.3.m3.1.1.2">𝑧</ci><apply id="S3.I1.i1.p1.3.m3.1.1.3.cmml" xref="S3.I1.i1.p1.3.m3.1.1.3"><times id="S3.I1.i1.p1.3.m3.1.1.3.1.cmml" xref="S3.I1.i1.p1.3.m3.1.1.3.1"></times><ci id="S3.I1.i1.p1.3.m3.1.1.3.2.cmml" xref="S3.I1.i1.p1.3.m3.1.1.3.2">𝑖</ci><ci id="S3.I1.i1.p1.3.m3.1.1.3.3.cmml" xref="S3.I1.i1.p1.3.m3.1.1.3.3">𝑗</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I1.i1.p1.3.m3.1c">z_{ij}</annotation><annotation encoding="application/x-llamapun" id="S3.I1.i1.p1.3.m3.1d">italic_z start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT</annotation></semantics></math> would be concatenating one more section to the <math alttext="(j-1)" class="ltx_Math" display="inline" id="S3.I1.i1.p1.4.m4.1"><semantics id="S3.I1.i1.p1.4.m4.1a"><mrow id="S3.I1.i1.p1.4.m4.1.1.1" xref="S3.I1.i1.p1.4.m4.1.1.1.1.cmml"><mo id="S3.I1.i1.p1.4.m4.1.1.1.2" stretchy="false" xref="S3.I1.i1.p1.4.m4.1.1.1.1.cmml">(</mo><mrow id="S3.I1.i1.p1.4.m4.1.1.1.1" xref="S3.I1.i1.p1.4.m4.1.1.1.1.cmml"><mi id="S3.I1.i1.p1.4.m4.1.1.1.1.2" xref="S3.I1.i1.p1.4.m4.1.1.1.1.2.cmml">j</mi><mo id="S3.I1.i1.p1.4.m4.1.1.1.1.1" xref="S3.I1.i1.p1.4.m4.1.1.1.1.1.cmml">−</mo><mn id="S3.I1.i1.p1.4.m4.1.1.1.1.3" xref="S3.I1.i1.p1.4.m4.1.1.1.1.3.cmml">1</mn></mrow><mo id="S3.I1.i1.p1.4.m4.1.1.1.3" stretchy="false" xref="S3.I1.i1.p1.4.m4.1.1.1.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.I1.i1.p1.4.m4.1b"><apply id="S3.I1.i1.p1.4.m4.1.1.1.1.cmml" xref="S3.I1.i1.p1.4.m4.1.1.1"><minus id="S3.I1.i1.p1.4.m4.1.1.1.1.1.cmml" xref="S3.I1.i1.p1.4.m4.1.1.1.1.1"></minus><ci id="S3.I1.i1.p1.4.m4.1.1.1.1.2.cmml" xref="S3.I1.i1.p1.4.m4.1.1.1.1.2">𝑗</ci><cn id="S3.I1.i1.p1.4.m4.1.1.1.1.3.cmml" type="integer" xref="S3.I1.i1.p1.4.m4.1.1.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I1.i1.p1.4.m4.1c">(j-1)</annotation><annotation encoding="application/x-llamapun" id="S3.I1.i1.p1.4.m4.1d">( italic_j - 1 )</annotation></semantics></math>-th draft.</p> </div> </li> <li class="ltx_item" id="S3.I1.i2" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">2.</span> <div class="ltx_para" id="S3.I1.i2.p1"> <p class="ltx_p" id="S3.I1.i2.p1.1"><span class="ltx_text ltx_font_bold" id="S3.I1.i2.p1.1.1">Top-down:</span> Another common strategy is to use intermediate steps from the most abstract outline to gradually more detailed content. For example, the first intermediate step <math alttext="z_{i1}" class="ltx_Math" display="inline" id="S3.I1.i2.p1.1.m1.1"><semantics id="S3.I1.i2.p1.1.m1.1a"><msub id="S3.I1.i2.p1.1.m1.1.1" xref="S3.I1.i2.p1.1.m1.1.1.cmml"><mi id="S3.I1.i2.p1.1.m1.1.1.2" xref="S3.I1.i2.p1.1.m1.1.1.2.cmml">z</mi><mrow id="S3.I1.i2.p1.1.m1.1.1.3" xref="S3.I1.i2.p1.1.m1.1.1.3.cmml"><mi id="S3.I1.i2.p1.1.m1.1.1.3.2" xref="S3.I1.i2.p1.1.m1.1.1.3.2.cmml">i</mi><mo id="S3.I1.i2.p1.1.m1.1.1.3.1" xref="S3.I1.i2.p1.1.m1.1.1.3.1.cmml">⁢</mo><mn id="S3.I1.i2.p1.1.m1.1.1.3.3" xref="S3.I1.i2.p1.1.m1.1.1.3.3.cmml">1</mn></mrow></msub><annotation-xml encoding="MathML-Content" id="S3.I1.i2.p1.1.m1.1b"><apply id="S3.I1.i2.p1.1.m1.1.1.cmml" xref="S3.I1.i2.p1.1.m1.1.1"><csymbol cd="ambiguous" id="S3.I1.i2.p1.1.m1.1.1.1.cmml" xref="S3.I1.i2.p1.1.m1.1.1">subscript</csymbol><ci id="S3.I1.i2.p1.1.m1.1.1.2.cmml" xref="S3.I1.i2.p1.1.m1.1.1.2">𝑧</ci><apply id="S3.I1.i2.p1.1.m1.1.1.3.cmml" xref="S3.I1.i2.p1.1.m1.1.1.3"><times id="S3.I1.i2.p1.1.m1.1.1.3.1.cmml" xref="S3.I1.i2.p1.1.m1.1.1.3.1"></times><ci id="S3.I1.i2.p1.1.m1.1.1.3.2.cmml" xref="S3.I1.i2.p1.1.m1.1.1.3.2">𝑖</ci><cn id="S3.I1.i2.p1.1.m1.1.1.3.3.cmml" type="integer" xref="S3.I1.i2.p1.1.m1.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I1.i2.p1.1.m1.1c">z_{i1}</annotation><annotation encoding="application/x-llamapun" id="S3.I1.i2.p1.1.m1.1d">italic_z start_POSTSUBSCRIPT italic_i 1 end_POSTSUBSCRIPT</annotation></semantics></math> can be an abstract outline only with the title of each section of the article. Each subsequent intermediate step would gradually elaborate on the content within each section. This kind of intermediate steps are not usually readily available, and may need to be constructed from the full article, which we will describe later in this paper.</p> </div> </li> </ol> </div> <div class="ltx_para" id="S3.SS0.SSS0.Px2.p4"> <p class="ltx_p" id="S3.SS0.SSS0.Px2.p4.10">Note that many existing works aim to developing multi-turn LLM writers, which involves a series of tasks such as generating <math alttext="z_{i1}" class="ltx_Math" display="inline" id="S3.SS0.SSS0.Px2.p4.1.m1.1"><semantics id="S3.SS0.SSS0.Px2.p4.1.m1.1a"><msub id="S3.SS0.SSS0.Px2.p4.1.m1.1.1" xref="S3.SS0.SSS0.Px2.p4.1.m1.1.1.cmml"><mi id="S3.SS0.SSS0.Px2.p4.1.m1.1.1.2" xref="S3.SS0.SSS0.Px2.p4.1.m1.1.1.2.cmml">z</mi><mrow id="S3.SS0.SSS0.Px2.p4.1.m1.1.1.3" xref="S3.SS0.SSS0.Px2.p4.1.m1.1.1.3.cmml"><mi id="S3.SS0.SSS0.Px2.p4.1.m1.1.1.3.2" xref="S3.SS0.SSS0.Px2.p4.1.m1.1.1.3.2.cmml">i</mi><mo id="S3.SS0.SSS0.Px2.p4.1.m1.1.1.3.1" xref="S3.SS0.SSS0.Px2.p4.1.m1.1.1.3.1.cmml">⁢</mo><mn id="S3.SS0.SSS0.Px2.p4.1.m1.1.1.3.3" xref="S3.SS0.SSS0.Px2.p4.1.m1.1.1.3.3.cmml">1</mn></mrow></msub><annotation-xml encoding="MathML-Content" id="S3.SS0.SSS0.Px2.p4.1.m1.1b"><apply id="S3.SS0.SSS0.Px2.p4.1.m1.1.1.cmml" xref="S3.SS0.SSS0.Px2.p4.1.m1.1.1"><csymbol cd="ambiguous" id="S3.SS0.SSS0.Px2.p4.1.m1.1.1.1.cmml" xref="S3.SS0.SSS0.Px2.p4.1.m1.1.1">subscript</csymbol><ci id="S3.SS0.SSS0.Px2.p4.1.m1.1.1.2.cmml" xref="S3.SS0.SSS0.Px2.p4.1.m1.1.1.2">𝑧</ci><apply id="S3.SS0.SSS0.Px2.p4.1.m1.1.1.3.cmml" xref="S3.SS0.SSS0.Px2.p4.1.m1.1.1.3"><times id="S3.SS0.SSS0.Px2.p4.1.m1.1.1.3.1.cmml" xref="S3.SS0.SSS0.Px2.p4.1.m1.1.1.3.1"></times><ci id="S3.SS0.SSS0.Px2.p4.1.m1.1.1.3.2.cmml" xref="S3.SS0.SSS0.Px2.p4.1.m1.1.1.3.2">𝑖</ci><cn id="S3.SS0.SSS0.Px2.p4.1.m1.1.1.3.3.cmml" type="integer" xref="S3.SS0.SSS0.Px2.p4.1.m1.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS0.SSS0.Px2.p4.1.m1.1c">z_{i1}</annotation><annotation encoding="application/x-llamapun" id="S3.SS0.SSS0.Px2.p4.1.m1.1d">italic_z start_POSTSUBSCRIPT italic_i 1 end_POSTSUBSCRIPT</annotation></semantics></math> from <math alttext="x_{i}" class="ltx_Math" display="inline" id="S3.SS0.SSS0.Px2.p4.2.m2.1"><semantics id="S3.SS0.SSS0.Px2.p4.2.m2.1a"><msub id="S3.SS0.SSS0.Px2.p4.2.m2.1.1" xref="S3.SS0.SSS0.Px2.p4.2.m2.1.1.cmml"><mi id="S3.SS0.SSS0.Px2.p4.2.m2.1.1.2" xref="S3.SS0.SSS0.Px2.p4.2.m2.1.1.2.cmml">x</mi><mi id="S3.SS0.SSS0.Px2.p4.2.m2.1.1.3" xref="S3.SS0.SSS0.Px2.p4.2.m2.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS0.SSS0.Px2.p4.2.m2.1b"><apply id="S3.SS0.SSS0.Px2.p4.2.m2.1.1.cmml" xref="S3.SS0.SSS0.Px2.p4.2.m2.1.1"><csymbol cd="ambiguous" id="S3.SS0.SSS0.Px2.p4.2.m2.1.1.1.cmml" xref="S3.SS0.SSS0.Px2.p4.2.m2.1.1">subscript</csymbol><ci id="S3.SS0.SSS0.Px2.p4.2.m2.1.1.2.cmml" xref="S3.SS0.SSS0.Px2.p4.2.m2.1.1.2">𝑥</ci><ci id="S3.SS0.SSS0.Px2.p4.2.m2.1.1.3.cmml" xref="S3.SS0.SSS0.Px2.p4.2.m2.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS0.SSS0.Px2.p4.2.m2.1c">x_{i}</annotation><annotation encoding="application/x-llamapun" id="S3.SS0.SSS0.Px2.p4.2.m2.1d">italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math>, then <math alttext="z_{ij}" class="ltx_Math" display="inline" id="S3.SS0.SSS0.Px2.p4.3.m3.1"><semantics id="S3.SS0.SSS0.Px2.p4.3.m3.1a"><msub id="S3.SS0.SSS0.Px2.p4.3.m3.1.1" xref="S3.SS0.SSS0.Px2.p4.3.m3.1.1.cmml"><mi id="S3.SS0.SSS0.Px2.p4.3.m3.1.1.2" xref="S3.SS0.SSS0.Px2.p4.3.m3.1.1.2.cmml">z</mi><mrow id="S3.SS0.SSS0.Px2.p4.3.m3.1.1.3" xref="S3.SS0.SSS0.Px2.p4.3.m3.1.1.3.cmml"><mi id="S3.SS0.SSS0.Px2.p4.3.m3.1.1.3.2" xref="S3.SS0.SSS0.Px2.p4.3.m3.1.1.3.2.cmml">i</mi><mo id="S3.SS0.SSS0.Px2.p4.3.m3.1.1.3.1" xref="S3.SS0.SSS0.Px2.p4.3.m3.1.1.3.1.cmml">⁢</mo><mi id="S3.SS0.SSS0.Px2.p4.3.m3.1.1.3.3" xref="S3.SS0.SSS0.Px2.p4.3.m3.1.1.3.3.cmml">j</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="S3.SS0.SSS0.Px2.p4.3.m3.1b"><apply id="S3.SS0.SSS0.Px2.p4.3.m3.1.1.cmml" xref="S3.SS0.SSS0.Px2.p4.3.m3.1.1"><csymbol cd="ambiguous" id="S3.SS0.SSS0.Px2.p4.3.m3.1.1.1.cmml" xref="S3.SS0.SSS0.Px2.p4.3.m3.1.1">subscript</csymbol><ci id="S3.SS0.SSS0.Px2.p4.3.m3.1.1.2.cmml" xref="S3.SS0.SSS0.Px2.p4.3.m3.1.1.2">𝑧</ci><apply id="S3.SS0.SSS0.Px2.p4.3.m3.1.1.3.cmml" xref="S3.SS0.SSS0.Px2.p4.3.m3.1.1.3"><times id="S3.SS0.SSS0.Px2.p4.3.m3.1.1.3.1.cmml" xref="S3.SS0.SSS0.Px2.p4.3.m3.1.1.3.1"></times><ci id="S3.SS0.SSS0.Px2.p4.3.m3.1.1.3.2.cmml" xref="S3.SS0.SSS0.Px2.p4.3.m3.1.1.3.2">𝑖</ci><ci id="S3.SS0.SSS0.Px2.p4.3.m3.1.1.3.3.cmml" xref="S3.SS0.SSS0.Px2.p4.3.m3.1.1.3.3">𝑗</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS0.SSS0.Px2.p4.3.m3.1c">z_{ij}</annotation><annotation encoding="application/x-llamapun" id="S3.SS0.SSS0.Px2.p4.3.m3.1d">italic_z start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT</annotation></semantics></math> from <math alttext="z_{i(j-1)}" class="ltx_Math" display="inline" id="S3.SS0.SSS0.Px2.p4.4.m4.1"><semantics id="S3.SS0.SSS0.Px2.p4.4.m4.1a"><msub id="S3.SS0.SSS0.Px2.p4.4.m4.1.2" xref="S3.SS0.SSS0.Px2.p4.4.m4.1.2.cmml"><mi id="S3.SS0.SSS0.Px2.p4.4.m4.1.2.2" xref="S3.SS0.SSS0.Px2.p4.4.m4.1.2.2.cmml">z</mi><mrow id="S3.SS0.SSS0.Px2.p4.4.m4.1.1.1" xref="S3.SS0.SSS0.Px2.p4.4.m4.1.1.1.cmml"><mi id="S3.SS0.SSS0.Px2.p4.4.m4.1.1.1.3" xref="S3.SS0.SSS0.Px2.p4.4.m4.1.1.1.3.cmml">i</mi><mo id="S3.SS0.SSS0.Px2.p4.4.m4.1.1.1.2" xref="S3.SS0.SSS0.Px2.p4.4.m4.1.1.1.2.cmml">⁢</mo><mrow id="S3.SS0.SSS0.Px2.p4.4.m4.1.1.1.1.1" xref="S3.SS0.SSS0.Px2.p4.4.m4.1.1.1.1.1.1.cmml"><mo id="S3.SS0.SSS0.Px2.p4.4.m4.1.1.1.1.1.2" stretchy="false" xref="S3.SS0.SSS0.Px2.p4.4.m4.1.1.1.1.1.1.cmml">(</mo><mrow id="S3.SS0.SSS0.Px2.p4.4.m4.1.1.1.1.1.1" xref="S3.SS0.SSS0.Px2.p4.4.m4.1.1.1.1.1.1.cmml"><mi id="S3.SS0.SSS0.Px2.p4.4.m4.1.1.1.1.1.1.2" xref="S3.SS0.SSS0.Px2.p4.4.m4.1.1.1.1.1.1.2.cmml">j</mi><mo id="S3.SS0.SSS0.Px2.p4.4.m4.1.1.1.1.1.1.1" xref="S3.SS0.SSS0.Px2.p4.4.m4.1.1.1.1.1.1.1.cmml">−</mo><mn id="S3.SS0.SSS0.Px2.p4.4.m4.1.1.1.1.1.1.3" xref="S3.SS0.SSS0.Px2.p4.4.m4.1.1.1.1.1.1.3.cmml">1</mn></mrow><mo id="S3.SS0.SSS0.Px2.p4.4.m4.1.1.1.1.1.3" stretchy="false" xref="S3.SS0.SSS0.Px2.p4.4.m4.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></msub><annotation-xml encoding="MathML-Content" id="S3.SS0.SSS0.Px2.p4.4.m4.1b"><apply id="S3.SS0.SSS0.Px2.p4.4.m4.1.2.cmml" xref="S3.SS0.SSS0.Px2.p4.4.m4.1.2"><csymbol cd="ambiguous" id="S3.SS0.SSS0.Px2.p4.4.m4.1.2.1.cmml" xref="S3.SS0.SSS0.Px2.p4.4.m4.1.2">subscript</csymbol><ci id="S3.SS0.SSS0.Px2.p4.4.m4.1.2.2.cmml" xref="S3.SS0.SSS0.Px2.p4.4.m4.1.2.2">𝑧</ci><apply id="S3.SS0.SSS0.Px2.p4.4.m4.1.1.1.cmml" xref="S3.SS0.SSS0.Px2.p4.4.m4.1.1.1"><times id="S3.SS0.SSS0.Px2.p4.4.m4.1.1.1.2.cmml" xref="S3.SS0.SSS0.Px2.p4.4.m4.1.1.1.2"></times><ci id="S3.SS0.SSS0.Px2.p4.4.m4.1.1.1.3.cmml" xref="S3.SS0.SSS0.Px2.p4.4.m4.1.1.1.3">𝑖</ci><apply id="S3.SS0.SSS0.Px2.p4.4.m4.1.1.1.1.1.1.cmml" xref="S3.SS0.SSS0.Px2.p4.4.m4.1.1.1.1.1"><minus id="S3.SS0.SSS0.Px2.p4.4.m4.1.1.1.1.1.1.1.cmml" xref="S3.SS0.SSS0.Px2.p4.4.m4.1.1.1.1.1.1.1"></minus><ci id="S3.SS0.SSS0.Px2.p4.4.m4.1.1.1.1.1.1.2.cmml" xref="S3.SS0.SSS0.Px2.p4.4.m4.1.1.1.1.1.1.2">𝑗</ci><cn id="S3.SS0.SSS0.Px2.p4.4.m4.1.1.1.1.1.1.3.cmml" type="integer" xref="S3.SS0.SSS0.Px2.p4.4.m4.1.1.1.1.1.1.3">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS0.SSS0.Px2.p4.4.m4.1c">z_{i(j-1)}</annotation><annotation encoding="application/x-llamapun" id="S3.SS0.SSS0.Px2.p4.4.m4.1d">italic_z start_POSTSUBSCRIPT italic_i ( italic_j - 1 ) end_POSTSUBSCRIPT</annotation></semantics></math>. However, this is not the focus of our paper. In this paper, as shown in Figure <a class="ltx_ref" href="https://arxiv.org/html/2410.06203v1#S3.F1" title="Figure 1 ‣ 3 Problem Setup ‣ Integrating Planning into Single-Turn Long-Form Text Generation"><span class="ltx_text ltx_ref_tag">1</span></a>, we introduced intermediate steps to create different training tasks for fine-tuning LLMs. Our final objective remains to generate the full article <math alttext="y_{i}" class="ltx_Math" display="inline" id="S3.SS0.SSS0.Px2.p4.5.m5.1"><semantics id="S3.SS0.SSS0.Px2.p4.5.m5.1a"><msub id="S3.SS0.SSS0.Px2.p4.5.m5.1.1" xref="S3.SS0.SSS0.Px2.p4.5.m5.1.1.cmml"><mi id="S3.SS0.SSS0.Px2.p4.5.m5.1.1.2" xref="S3.SS0.SSS0.Px2.p4.5.m5.1.1.2.cmml">y</mi><mi id="S3.SS0.SSS0.Px2.p4.5.m5.1.1.3" xref="S3.SS0.SSS0.Px2.p4.5.m5.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS0.SSS0.Px2.p4.5.m5.1b"><apply id="S3.SS0.SSS0.Px2.p4.5.m5.1.1.cmml" xref="S3.SS0.SSS0.Px2.p4.5.m5.1.1"><csymbol cd="ambiguous" id="S3.SS0.SSS0.Px2.p4.5.m5.1.1.1.cmml" xref="S3.SS0.SSS0.Px2.p4.5.m5.1.1">subscript</csymbol><ci id="S3.SS0.SSS0.Px2.p4.5.m5.1.1.2.cmml" xref="S3.SS0.SSS0.Px2.p4.5.m5.1.1.2">𝑦</ci><ci id="S3.SS0.SSS0.Px2.p4.5.m5.1.1.3.cmml" xref="S3.SS0.SSS0.Px2.p4.5.m5.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS0.SSS0.Px2.p4.5.m5.1c">y_{i}</annotation><annotation encoding="application/x-llamapun" id="S3.SS0.SSS0.Px2.p4.5.m5.1d">italic_y start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> from the input context <math alttext="x_{i}" class="ltx_Math" display="inline" id="S3.SS0.SSS0.Px2.p4.6.m6.1"><semantics id="S3.SS0.SSS0.Px2.p4.6.m6.1a"><msub id="S3.SS0.SSS0.Px2.p4.6.m6.1.1" xref="S3.SS0.SSS0.Px2.p4.6.m6.1.1.cmml"><mi id="S3.SS0.SSS0.Px2.p4.6.m6.1.1.2" xref="S3.SS0.SSS0.Px2.p4.6.m6.1.1.2.cmml">x</mi><mi id="S3.SS0.SSS0.Px2.p4.6.m6.1.1.3" xref="S3.SS0.SSS0.Px2.p4.6.m6.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS0.SSS0.Px2.p4.6.m6.1b"><apply id="S3.SS0.SSS0.Px2.p4.6.m6.1.1.cmml" xref="S3.SS0.SSS0.Px2.p4.6.m6.1.1"><csymbol cd="ambiguous" id="S3.SS0.SSS0.Px2.p4.6.m6.1.1.1.cmml" xref="S3.SS0.SSS0.Px2.p4.6.m6.1.1">subscript</csymbol><ci id="S3.SS0.SSS0.Px2.p4.6.m6.1.1.2.cmml" xref="S3.SS0.SSS0.Px2.p4.6.m6.1.1.2">𝑥</ci><ci id="S3.SS0.SSS0.Px2.p4.6.m6.1.1.3.cmml" xref="S3.SS0.SSS0.Px2.p4.6.m6.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS0.SSS0.Px2.p4.6.m6.1c">x_{i}</annotation><annotation encoding="application/x-llamapun" id="S3.SS0.SSS0.Px2.p4.6.m6.1d">italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> in a single turn. During inference, the input is always only the input context <math alttext="x_{i}" class="ltx_Math" display="inline" id="S3.SS0.SSS0.Px2.p4.7.m7.1"><semantics id="S3.SS0.SSS0.Px2.p4.7.m7.1a"><msub id="S3.SS0.SSS0.Px2.p4.7.m7.1.1" xref="S3.SS0.SSS0.Px2.p4.7.m7.1.1.cmml"><mi id="S3.SS0.SSS0.Px2.p4.7.m7.1.1.2" xref="S3.SS0.SSS0.Px2.p4.7.m7.1.1.2.cmml">x</mi><mi id="S3.SS0.SSS0.Px2.p4.7.m7.1.1.3" xref="S3.SS0.SSS0.Px2.p4.7.m7.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS0.SSS0.Px2.p4.7.m7.1b"><apply id="S3.SS0.SSS0.Px2.p4.7.m7.1.1.cmml" xref="S3.SS0.SSS0.Px2.p4.7.m7.1.1"><csymbol cd="ambiguous" id="S3.SS0.SSS0.Px2.p4.7.m7.1.1.1.cmml" xref="S3.SS0.SSS0.Px2.p4.7.m7.1.1">subscript</csymbol><ci id="S3.SS0.SSS0.Px2.p4.7.m7.1.1.2.cmml" xref="S3.SS0.SSS0.Px2.p4.7.m7.1.1.2">𝑥</ci><ci id="S3.SS0.SSS0.Px2.p4.7.m7.1.1.3.cmml" xref="S3.SS0.SSS0.Px2.p4.7.m7.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS0.SSS0.Px2.p4.7.m7.1c">x_{i}</annotation><annotation encoding="application/x-llamapun" id="S3.SS0.SSS0.Px2.p4.7.m7.1d">italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math>. The model can either generate only the article <math alttext="y_{i}" class="ltx_Math" display="inline" id="S3.SS0.SSS0.Px2.p4.8.m8.1"><semantics id="S3.SS0.SSS0.Px2.p4.8.m8.1a"><msub id="S3.SS0.SSS0.Px2.p4.8.m8.1.1" xref="S3.SS0.SSS0.Px2.p4.8.m8.1.1.cmml"><mi id="S3.SS0.SSS0.Px2.p4.8.m8.1.1.2" xref="S3.SS0.SSS0.Px2.p4.8.m8.1.1.2.cmml">y</mi><mi id="S3.SS0.SSS0.Px2.p4.8.m8.1.1.3" xref="S3.SS0.SSS0.Px2.p4.8.m8.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS0.SSS0.Px2.p4.8.m8.1b"><apply id="S3.SS0.SSS0.Px2.p4.8.m8.1.1.cmml" xref="S3.SS0.SSS0.Px2.p4.8.m8.1.1"><csymbol cd="ambiguous" id="S3.SS0.SSS0.Px2.p4.8.m8.1.1.1.cmml" xref="S3.SS0.SSS0.Px2.p4.8.m8.1.1">subscript</csymbol><ci id="S3.SS0.SSS0.Px2.p4.8.m8.1.1.2.cmml" xref="S3.SS0.SSS0.Px2.p4.8.m8.1.1.2">𝑦</ci><ci id="S3.SS0.SSS0.Px2.p4.8.m8.1.1.3.cmml" xref="S3.SS0.SSS0.Px2.p4.8.m8.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS0.SSS0.Px2.p4.8.m8.1c">y_{i}</annotation><annotation encoding="application/x-llamapun" id="S3.SS0.SSS0.Px2.p4.8.m8.1d">italic_y start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> or both the intermediate steps <math alttext="{\mathbf{z}}_{i}" class="ltx_Math" display="inline" id="S3.SS0.SSS0.Px2.p4.9.m9.1"><semantics id="S3.SS0.SSS0.Px2.p4.9.m9.1a"><msub id="S3.SS0.SSS0.Px2.p4.9.m9.1.1" xref="S3.SS0.SSS0.Px2.p4.9.m9.1.1.cmml"><mi id="S3.SS0.SSS0.Px2.p4.9.m9.1.1.2" xref="S3.SS0.SSS0.Px2.p4.9.m9.1.1.2.cmml">𝐳</mi><mi id="S3.SS0.SSS0.Px2.p4.9.m9.1.1.3" xref="S3.SS0.SSS0.Px2.p4.9.m9.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS0.SSS0.Px2.p4.9.m9.1b"><apply id="S3.SS0.SSS0.Px2.p4.9.m9.1.1.cmml" xref="S3.SS0.SSS0.Px2.p4.9.m9.1.1"><csymbol cd="ambiguous" id="S3.SS0.SSS0.Px2.p4.9.m9.1.1.1.cmml" xref="S3.SS0.SSS0.Px2.p4.9.m9.1.1">subscript</csymbol><ci id="S3.SS0.SSS0.Px2.p4.9.m9.1.1.2.cmml" xref="S3.SS0.SSS0.Px2.p4.9.m9.1.1.2">𝐳</ci><ci id="S3.SS0.SSS0.Px2.p4.9.m9.1.1.3.cmml" xref="S3.SS0.SSS0.Px2.p4.9.m9.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS0.SSS0.Px2.p4.9.m9.1c">{\mathbf{z}}_{i}</annotation><annotation encoding="application/x-llamapun" id="S3.SS0.SSS0.Px2.p4.9.m9.1d">bold_z start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> and the article <math alttext="y_{i}" class="ltx_Math" display="inline" id="S3.SS0.SSS0.Px2.p4.10.m10.1"><semantics id="S3.SS0.SSS0.Px2.p4.10.m10.1a"><msub id="S3.SS0.SSS0.Px2.p4.10.m10.1.1" xref="S3.SS0.SSS0.Px2.p4.10.m10.1.1.cmml"><mi id="S3.SS0.SSS0.Px2.p4.10.m10.1.1.2" xref="S3.SS0.SSS0.Px2.p4.10.m10.1.1.2.cmml">y</mi><mi id="S3.SS0.SSS0.Px2.p4.10.m10.1.1.3" xref="S3.SS0.SSS0.Px2.p4.10.m10.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS0.SSS0.Px2.p4.10.m10.1b"><apply id="S3.SS0.SSS0.Px2.p4.10.m10.1.1.cmml" xref="S3.SS0.SSS0.Px2.p4.10.m10.1.1"><csymbol cd="ambiguous" id="S3.SS0.SSS0.Px2.p4.10.m10.1.1.1.cmml" xref="S3.SS0.SSS0.Px2.p4.10.m10.1.1">subscript</csymbol><ci id="S3.SS0.SSS0.Px2.p4.10.m10.1.1.2.cmml" xref="S3.SS0.SSS0.Px2.p4.10.m10.1.1.2">𝑦</ci><ci id="S3.SS0.SSS0.Px2.p4.10.m10.1.1.3.cmml" xref="S3.SS0.SSS0.Px2.p4.10.m10.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS0.SSS0.Px2.p4.10.m10.1c">y_{i}</annotation><annotation encoding="application/x-llamapun" id="S3.SS0.SSS0.Px2.p4.10.m10.1d">italic_y start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math>, as long as the article can be easily extracted from the complete model output through post-processing.</p> </div> </section> </section> <section class="ltx_section" id="S4"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">4 </span>Methodology</h2> <div class="ltx_para" id="S4.p1"> <p class="ltx_p" id="S4.p1.2">Our proposed framework addresses the challenge of generating long-form text with LLMs by utilizing the inherent structure of documents. Specifically, we construct intermediate steps <math alttext="{\mathbf{z}}_{i}" class="ltx_Math" display="inline" id="S4.p1.1.m1.1"><semantics id="S4.p1.1.m1.1a"><msub id="S4.p1.1.m1.1.1" xref="S4.p1.1.m1.1.1.cmml"><mi id="S4.p1.1.m1.1.1.2" xref="S4.p1.1.m1.1.1.2.cmml">𝐳</mi><mi id="S4.p1.1.m1.1.1.3" xref="S4.p1.1.m1.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S4.p1.1.m1.1b"><apply id="S4.p1.1.m1.1.1.cmml" xref="S4.p1.1.m1.1.1"><csymbol cd="ambiguous" id="S4.p1.1.m1.1.1.1.cmml" xref="S4.p1.1.m1.1.1">subscript</csymbol><ci id="S4.p1.1.m1.1.1.2.cmml" xref="S4.p1.1.m1.1.1.2">𝐳</ci><ci id="S4.p1.1.m1.1.1.3.cmml" xref="S4.p1.1.m1.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p1.1.m1.1c">{\mathbf{z}}_{i}</annotation><annotation encoding="application/x-llamapun" id="S4.p1.1.m1.1d">bold_z start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> based on this structure to guide the generation process. The key insight is that the inherent structure, such as an article’s outline, provides crucial guidance for organizing the generated article <math alttext="y_{i}" class="ltx_Math" display="inline" id="S4.p1.2.m2.1"><semantics id="S4.p1.2.m2.1a"><msub id="S4.p1.2.m2.1.1" xref="S4.p1.2.m2.1.1.cmml"><mi id="S4.p1.2.m2.1.1.2" xref="S4.p1.2.m2.1.1.2.cmml">y</mi><mi id="S4.p1.2.m2.1.1.3" xref="S4.p1.2.m2.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S4.p1.2.m2.1b"><apply id="S4.p1.2.m2.1.1.cmml" xref="S4.p1.2.m2.1.1"><csymbol cd="ambiguous" id="S4.p1.2.m2.1.1.1.cmml" xref="S4.p1.2.m2.1.1">subscript</csymbol><ci id="S4.p1.2.m2.1.1.2.cmml" xref="S4.p1.2.m2.1.1.2">𝑦</ci><ci id="S4.p1.2.m2.1.1.3.cmml" xref="S4.p1.2.m2.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p1.2.m2.1c">y_{i}</annotation><annotation encoding="application/x-llamapun" id="S4.p1.2.m2.1d">italic_y start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S4.p2"> <p class="ltx_p" id="S4.p2.2">Within this framework, we construct the intermediate steps <math alttext="{\mathbf{z}}_{i}" class="ltx_Math" display="inline" id="S4.p2.1.m1.1"><semantics id="S4.p2.1.m1.1a"><msub id="S4.p2.1.m1.1.1" xref="S4.p2.1.m1.1.1.cmml"><mi id="S4.p2.1.m1.1.1.2" xref="S4.p2.1.m1.1.1.2.cmml">𝐳</mi><mi id="S4.p2.1.m1.1.1.3" xref="S4.p2.1.m1.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S4.p2.1.m1.1b"><apply id="S4.p2.1.m1.1.1.cmml" xref="S4.p2.1.m1.1.1"><csymbol cd="ambiguous" id="S4.p2.1.m1.1.1.1.cmml" xref="S4.p2.1.m1.1.1">subscript</csymbol><ci id="S4.p2.1.m1.1.1.2.cmml" xref="S4.p2.1.m1.1.1.2">𝐳</ci><ci id="S4.p2.1.m1.1.1.3.cmml" xref="S4.p2.1.m1.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p2.1.m1.1c">{\mathbf{z}}_{i}</annotation><annotation encoding="application/x-llamapun" id="S4.p2.1.m1.1d">bold_z start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> by extracting the implicit structural information inherent in a formulated document and then constructing a hierarchical writing plan accordingly. These intermediate steps then serve as a foundation for fine-tuning the LLM, focusing on tasks that emphasize structural understanding and plan interpretation. At inference time, the fine-tuned LLM, equipped with enhanced structural and plan interpretation capabilities, generates the final long-form text <math alttext="y_{i}" class="ltx_Math" display="inline" id="S4.p2.2.m2.1"><semantics id="S4.p2.2.m2.1a"><msub id="S4.p2.2.m2.1.1" xref="S4.p2.2.m2.1.1.cmml"><mi id="S4.p2.2.m2.1.1.2" xref="S4.p2.2.m2.1.1.2.cmml">y</mi><mi id="S4.p2.2.m2.1.1.3" xref="S4.p2.2.m2.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S4.p2.2.m2.1b"><apply id="S4.p2.2.m2.1.1.cmml" xref="S4.p2.2.m2.1.1"><csymbol cd="ambiguous" id="S4.p2.2.m2.1.1.1.cmml" xref="S4.p2.2.m2.1.1">subscript</csymbol><ci id="S4.p2.2.m2.1.1.2.cmml" xref="S4.p2.2.m2.1.1.2">𝑦</ci><ci id="S4.p2.2.m2.1.1.3.cmml" xref="S4.p2.2.m2.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p2.2.m2.1c">y_{i}</annotation><annotation encoding="application/x-llamapun" id="S4.p2.2.m2.1d">italic_y start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> in a single pass. The specific components of this framework are discussed in the following sections.</p> </div> <section class="ltx_subsection" id="S4.SS1"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">4.1 </span>Writing Plan as Intermediate Steps</h3> <div class="ltx_para" id="S4.SS1.p1"> <p class="ltx_p" id="S4.SS1.p1.1">Writing high-quality long-form documents typically begins with a pre-writing phase, where authors establish the structural framework, develop key arguments, and formulate the overall trajectory of the narrative. In this work, we define the pre-writing stage as a series of intermediate steps essential for generating long-form documents. Specifically, without loss of generality, we concentrate on three distinct types of intermediate steps: document summary, document outline, and document key information.</p> </div> <section class="ltx_paragraph" id="S4.SS1.SSS0.Px1"> <h4 class="ltx_title ltx_title_paragraph">Document Summary</h4> <div class="ltx_para" id="S4.SS1.SSS0.Px1.p1"> <p class="ltx_p" id="S4.SS1.SSS0.Px1.p1.1">A document summary is a concise representation that captures the core message of a document, summarizing key points, themes, or arguments while omitting peripheral details. It provides a quick overview, enabling readers to grasp the essential content without reading the entire document.</p> </div> </section> <section class="ltx_paragraph" id="S4.SS1.SSS0.Px2"> <h4 class="ltx_title ltx_title_paragraph">Document Outline</h4> <div class="ltx_para" id="S4.SS1.SSS0.Px2.p1"> <p class="ltx_p" id="S4.SS1.SSS0.Px2.p1.1">A document outline represents the hierarchical structure of a document. It reveals the organization and flow of ideas, the relationships between sections, and the key points within the document.</p> </div> </section> <section class="ltx_paragraph" id="S4.SS1.SSS0.Px3"> <h4 class="ltx_title ltx_title_paragraph">Document Key Information</h4> <div class="ltx_para" id="S4.SS1.SSS0.Px3.p1"> <p class="ltx_p" id="S4.SS1.SSS0.Px3.p1.1">Document key information includes the most crucial facts, findings, or insights within a document. It typically represents the core knowledge that the document aims to convey or support.</p> </div> </section> </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>Structured Intermediate Steps Construction</h3> <figure class="ltx_figure" id="S4.F2"><img alt="Refer to caption" class="ltx_graphics ltx_centering ltx_img_landscape" height="177" id="S4.F2.g1" src="extracted/5908544/figures/intermediate_construction.png" width="598"/> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure">Figure 2: </span>Constructing intermediate steps.</figcaption> </figure> <div class="ltx_para" id="S4.SS2.p1"> <p class="ltx_p" id="S4.SS2.p1.1">The intermediate steps <math alttext="{\mathbf{z}}_{i}" class="ltx_Math" display="inline" id="S4.SS2.p1.1.m1.1"><semantics id="S4.SS2.p1.1.m1.1a"><msub id="S4.SS2.p1.1.m1.1.1" xref="S4.SS2.p1.1.m1.1.1.cmml"><mi id="S4.SS2.p1.1.m1.1.1.2" xref="S4.SS2.p1.1.m1.1.1.2.cmml">𝐳</mi><mi id="S4.SS2.p1.1.m1.1.1.3" xref="S4.SS2.p1.1.m1.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS2.p1.1.m1.1b"><apply id="S4.SS2.p1.1.m1.1.1.cmml" xref="S4.SS2.p1.1.m1.1.1"><csymbol cd="ambiguous" id="S4.SS2.p1.1.m1.1.1.1.cmml" xref="S4.SS2.p1.1.m1.1.1">subscript</csymbol><ci id="S4.SS2.p1.1.m1.1.1.2.cmml" xref="S4.SS2.p1.1.m1.1.1.2">𝐳</ci><ci id="S4.SS2.p1.1.m1.1.1.3.cmml" xref="S4.SS2.p1.1.m1.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p1.1.m1.1c">{\mathbf{z}}_{i}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p1.1.m1.1d">bold_z start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> involved in the creating long-form documents provide valuable structural guidance for the generation process. However, this structured information is not always explicitly available, nor is the alignment between the structure and the final document. As discussed above, authors typically establish these intermediate steps when writing long-form documents. Therefore, we posit that such implicit structural information is already embedded within the final document itself.</p> </div> <div class="ltx_para" id="S4.SS2.p2"> <p class="ltx_p" id="S4.SS2.p2.1">Building on this observation, we propose a method to extract the intermediate steps from established documents <math alttext="y_{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">y</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">𝑦</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">y_{i}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p2.1.m1.1d">italic_y start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math>, such as news articles and Wikipedia entries. Leveraging recent advances in LLMs, particularly their impressive few-shot learning capabilities, we aim to synthetically generate these intermediate steps directly from the documents. This approach captures the implicit structural information without relying on explicitly provided intermediate steps. Furthermore, it can generate multiple intermediate steps for a single document, thus enabling exploration of various organizational strategies. Lastly, our method provides a consistent and scalable solution for generating intermediate steps, making it well-suited for large-scale document processing. Figure <a class="ltx_ref" href="https://arxiv.org/html/2410.06203v1#S4.F2" title="Figure 2 ‣ 4.2 Structured Intermediate Steps Construction ‣ 4 Methodology ‣ Integrating Planning into Single-Turn Long-Form Text Generation"><span class="ltx_text ltx_ref_tag">2</span></a> summarizes the process of our proposed process to construct intermediate steps from the article.</p> </div> <div class="ltx_para" id="S4.SS2.p3"> <p class="ltx_p" id="S4.SS2.p3.6">To extract the intermediate steps from an article <math alttext="y_{i}" class="ltx_Math" display="inline" id="S4.SS2.p3.1.m1.1"><semantics id="S4.SS2.p3.1.m1.1a"><msub id="S4.SS2.p3.1.m1.1.1" xref="S4.SS2.p3.1.m1.1.1.cmml"><mi id="S4.SS2.p3.1.m1.1.1.2" xref="S4.SS2.p3.1.m1.1.1.2.cmml">y</mi><mi id="S4.SS2.p3.1.m1.1.1.3" xref="S4.SS2.p3.1.m1.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS2.p3.1.m1.1b"><apply id="S4.SS2.p3.1.m1.1.1.cmml" xref="S4.SS2.p3.1.m1.1.1"><csymbol cd="ambiguous" id="S4.SS2.p3.1.m1.1.1.1.cmml" 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id="S4.SS2.p3.2.m2.1d">italic_K</annotation></semantics></math> intermediate step candidates <math alttext="\{\mathbf{z}_{i}^{k}\}_{k=1}^{K}" class="ltx_Math" display="inline" id="S4.SS2.p3.3.m3.1"><semantics id="S4.SS2.p3.3.m3.1a"><msubsup id="S4.SS2.p3.3.m3.1.1" xref="S4.SS2.p3.3.m3.1.1.cmml"><mrow id="S4.SS2.p3.3.m3.1.1.1.1.1" xref="S4.SS2.p3.3.m3.1.1.1.1.2.cmml"><mo id="S4.SS2.p3.3.m3.1.1.1.1.1.2" stretchy="false" xref="S4.SS2.p3.3.m3.1.1.1.1.2.cmml">{</mo><msubsup id="S4.SS2.p3.3.m3.1.1.1.1.1.1" xref="S4.SS2.p3.3.m3.1.1.1.1.1.1.cmml"><mi id="S4.SS2.p3.3.m3.1.1.1.1.1.1.2.2" xref="S4.SS2.p3.3.m3.1.1.1.1.1.1.2.2.cmml">𝐳</mi><mi id="S4.SS2.p3.3.m3.1.1.1.1.1.1.2.3" xref="S4.SS2.p3.3.m3.1.1.1.1.1.1.2.3.cmml">i</mi><mi id="S4.SS2.p3.3.m3.1.1.1.1.1.1.3" xref="S4.SS2.p3.3.m3.1.1.1.1.1.1.3.cmml">k</mi></msubsup><mo id="S4.SS2.p3.3.m3.1.1.1.1.1.3" stretchy="false" xref="S4.SS2.p3.3.m3.1.1.1.1.2.cmml">}</mo></mrow><mrow id="S4.SS2.p3.3.m3.1.1.1.3" xref="S4.SS2.p3.3.m3.1.1.1.3.cmml"><mi 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id="S4.SS2.p3.3.m3.1c">\{\mathbf{z}_{i}^{k}\}_{k=1}^{K}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p3.3.m3.1d">{ bold_z start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT } start_POSTSUBSCRIPT italic_k = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_K end_POSTSUPERSCRIPT</annotation></semantics></math>, where <math alttext="{\mathbf{z}}_{i}" 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">𝐳</mi><mi id="S4.SS2.p3.4.m4.1.1.3" xref="S4.SS2.p3.4.m4.1.1.3.cmml">i</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" 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Despite LLM’s impressive summarization capabilities, they can sometimes produce hallucination information. To mitigate this, we select the best intermediate step from <math alttext="K" class="ltx_Math" display="inline" id="S4.SS2.p3.5.m5.1"><semantics id="S4.SS2.p3.5.m5.1a"><mi id="S4.SS2.p3.5.m5.1.1" xref="S4.SS2.p3.5.m5.1.1.cmml">K</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.p3.5.m5.1b"><ci id="S4.SS2.p3.5.m5.1.1.cmml" xref="S4.SS2.p3.5.m5.1.1">𝐾</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p3.5.m5.1c">K</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p3.5.m5.1d">italic_K</annotation></semantics></math> candidates using a designed scoring function that evaluate coherence and completeness with respect to the document <math alttext="y_{i}" class="ltx_Math" display="inline" id="S4.SS2.p3.6.m6.1"><semantics id="S4.SS2.p3.6.m6.1a"><msub id="S4.SS2.p3.6.m6.1.1" xref="S4.SS2.p3.6.m6.1.1.cmml"><mi id="S4.SS2.p3.6.m6.1.1.2" xref="S4.SS2.p3.6.m6.1.1.2.cmml">y</mi><mi id="S4.SS2.p3.6.m6.1.1.3" xref="S4.SS2.p3.6.m6.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS2.p3.6.m6.1b"><apply id="S4.SS2.p3.6.m6.1.1.cmml" xref="S4.SS2.p3.6.m6.1.1"><csymbol cd="ambiguous" id="S4.SS2.p3.6.m6.1.1.1.cmml" xref="S4.SS2.p3.6.m6.1.1">subscript</csymbol><ci id="S4.SS2.p3.6.m6.1.1.2.cmml" xref="S4.SS2.p3.6.m6.1.1.2">𝑦</ci><ci id="S4.SS2.p3.6.m6.1.1.3.cmml" xref="S4.SS2.p3.6.m6.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p3.6.m6.1c">y_{i}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p3.6.m6.1d">italic_y start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math>. The candidate that demonstrates the highest overall quality is chosen.</p> </div> <div class="ltx_para" id="S4.SS2.p4"> <p class="ltx_p" id="S4.SS2.p4.1">The construction of the intermediate steps is detailed in Algorithm <a class="ltx_ref" href="https://arxiv.org/html/2410.06203v1#alg1" title="Algorithm 1 ‣ 4.2 Structured Intermediate Steps Construction ‣ 4 Methodology ‣ Integrating Planning into Single-Turn Long-Form Text Generation"><span class="ltx_text ltx_ref_tag">1</span></a>. Take the document summary as an example, we first normalize the word and paragraph counts relative to the input text. We then apply a sinusoidal scoring function, <math alttext="\mathbf{g}(\cdot)" class="ltx_Math" display="inline" id="S4.SS2.p4.1.m1.1"><semantics id="S4.SS2.p4.1.m1.1a"><mrow id="S4.SS2.p4.1.m1.1.2" xref="S4.SS2.p4.1.m1.1.2.cmml"><mi id="S4.SS2.p4.1.m1.1.2.2" xref="S4.SS2.p4.1.m1.1.2.2.cmml">𝐠</mi><mo id="S4.SS2.p4.1.m1.1.2.1" xref="S4.SS2.p4.1.m1.1.2.1.cmml">⁢</mo><mrow id="S4.SS2.p4.1.m1.1.2.3.2" xref="S4.SS2.p4.1.m1.1.2.cmml"><mo id="S4.SS2.p4.1.m1.1.2.3.2.1" stretchy="false" xref="S4.SS2.p4.1.m1.1.2.cmml">(</mo><mo id="S4.SS2.p4.1.m1.1.1" lspace="0em" rspace="0em" xref="S4.SS2.p4.1.m1.1.1.cmml">⋅</mo><mo id="S4.SS2.p4.1.m1.1.2.3.2.2" stretchy="false" xref="S4.SS2.p4.1.m1.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p4.1.m1.1b"><apply id="S4.SS2.p4.1.m1.1.2.cmml" xref="S4.SS2.p4.1.m1.1.2"><times id="S4.SS2.p4.1.m1.1.2.1.cmml" xref="S4.SS2.p4.1.m1.1.2.1"></times><ci id="S4.SS2.p4.1.m1.1.2.2.cmml" xref="S4.SS2.p4.1.m1.1.2.2">𝐠</ci><ci id="S4.SS2.p4.1.m1.1.1.cmml" xref="S4.SS2.p4.1.m1.1.1">⋅</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p4.1.m1.1c">\mathbf{g}(\cdot)</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.p4.1.m1.1d">bold_g ( ⋅ )</annotation></semantics></math> in the Algorithm <a class="ltx_ref" href="https://arxiv.org/html/2410.06203v1#alg1" title="Algorithm 1 ‣ 4.2 Structured Intermediate Steps Construction ‣ 4 Methodology ‣ Integrating Planning into Single-Turn Long-Form Text Generation"><span class="ltx_text ltx_ref_tag">1</span></a>, derived from empirical observations, to assess the adherence to desirable length and structure. Additionally, we incorporate an entailment score using a hallucination model, measuring the bi-directional logical coherence between the summary and the original news article. In our experiment, we use the hallucination model introduced in <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2410.06203v1#bib.bib20" title="">20</a>]</cite>. This entailment score assessed both the extent to which the summary accurately reflected the news (no hallucination) and the degree to which the summary covered the key information in the news. These individual scores (word/paragraph count, entailment) were combined multiplicatively to produce a final score for each summary, allowing us to select the highest-scoring option as the optimal summary.</p> </div> <figure class="ltx_float ltx_float_algorithm ltx_framed ltx_framed_top" id="alg1"> <figcaption class="ltx_caption"><span class="ltx_tag ltx_tag_float"><span class="ltx_text ltx_font_bold" id="alg1.2.1.1">Algorithm 1</span> </span> Constructing intermediate steps data from organic long-form text data</figcaption> <div class="ltx_listing ltx_listing" id="alg1.3"> <div class="ltx_listingline" id="alg1.l1"> <span class="ltx_text ltx_font_bold" id="alg1.l1.1">Input: </span> Organic long-form text <math alttext="\mathcal{Y}" class="ltx_Math" display="inline" id="alg1.l1.m1.1"><semantics id="alg1.l1.m1.1a"><mi class="ltx_font_mathcaligraphic" id="alg1.l1.m1.1.1" xref="alg1.l1.m1.1.1.cmml">𝒴</mi><annotation-xml encoding="MathML-Content" id="alg1.l1.m1.1b"><ci id="alg1.l1.m1.1.1.cmml" xref="alg1.l1.m1.1.1">𝒴</ci></annotation-xml><annotation encoding="application/x-tex" id="alg1.l1.m1.1c">\mathcal{Y}</annotation><annotation encoding="application/x-llamapun" id="alg1.l1.m1.1d">caligraphic_Y</annotation></semantics></math> </div> <div class="ltx_listingline" id="alg1.l2"> <span class="ltx_text ltx_font_bold" id="alg1.l2.1">Output: </span> Intermediate steps </div> <div class="ltx_listingline" id="alg1.l3"> <span class="ltx_text ltx_font_bold" id="alg1.l3.1">for</span> <math alttext="y_{i}" class="ltx_Math" display="inline" id="alg1.l3.m1.1"><semantics id="alg1.l3.m1.1a"><msub id="alg1.l3.m1.1.1" xref="alg1.l3.m1.1.1.cmml"><mi id="alg1.l3.m1.1.1.2" xref="alg1.l3.m1.1.1.2.cmml">y</mi><mi id="alg1.l3.m1.1.1.3" xref="alg1.l3.m1.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="alg1.l3.m1.1b"><apply id="alg1.l3.m1.1.1.cmml" xref="alg1.l3.m1.1.1"><csymbol cd="ambiguous" id="alg1.l3.m1.1.1.1.cmml" xref="alg1.l3.m1.1.1">subscript</csymbol><ci id="alg1.l3.m1.1.1.2.cmml" xref="alg1.l3.m1.1.1.2">𝑦</ci><ci id="alg1.l3.m1.1.1.3.cmml" xref="alg1.l3.m1.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="alg1.l3.m1.1c">y_{i}</annotation><annotation encoding="application/x-llamapun" id="alg1.l3.m1.1d">italic_y start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> in <math alttext="\mathcal{Y}" class="ltx_Math" display="inline" id="alg1.l3.m2.1"><semantics id="alg1.l3.m2.1a"><mi class="ltx_font_mathcaligraphic" id="alg1.l3.m2.1.1" xref="alg1.l3.m2.1.1.cmml">𝒴</mi><annotation-xml encoding="MathML-Content" id="alg1.l3.m2.1b"><ci id="alg1.l3.m2.1.1.cmml" xref="alg1.l3.m2.1.1">𝒴</ci></annotation-xml><annotation encoding="application/x-tex" id="alg1.l3.m2.1c">\mathcal{Y}</annotation><annotation encoding="application/x-llamapun" id="alg1.l3.m2.1d">caligraphic_Y</annotation></semantics></math> <span class="ltx_text ltx_font_bold" id="alg1.l3.2">do</span> </div> <div class="ltx_listingline" id="alg1.l4"> <math alttext="\{{\mathbf{z}}_{i}^{k}\}_{k=1}^{K}=\text{few\_shot\_llm}(y_{i})" class="ltx_Math" display="inline" id="alg1.l4.m1.2"><semantics id="alg1.l4.m1.2a"><mrow id="alg1.l4.m1.2.2" xref="alg1.l4.m1.2.2.cmml"><msubsup id="alg1.l4.m1.1.1.1" xref="alg1.l4.m1.1.1.1.cmml"><mrow id="alg1.l4.m1.1.1.1.1.1.1" xref="alg1.l4.m1.1.1.1.1.1.2.cmml"><mo id="alg1.l4.m1.1.1.1.1.1.1.2" stretchy="false" xref="alg1.l4.m1.1.1.1.1.1.2.cmml">{</mo><msubsup id="alg1.l4.m1.1.1.1.1.1.1.1" xref="alg1.l4.m1.1.1.1.1.1.1.1.cmml"><mi id="alg1.l4.m1.1.1.1.1.1.1.1.2.2" xref="alg1.l4.m1.1.1.1.1.1.1.1.2.2.cmml">𝐳</mi><mi id="alg1.l4.m1.1.1.1.1.1.1.1.2.3" xref="alg1.l4.m1.1.1.1.1.1.1.1.2.3.cmml">i</mi><mi id="alg1.l4.m1.1.1.1.1.1.1.1.3" xref="alg1.l4.m1.1.1.1.1.1.1.1.3.cmml">k</mi></msubsup><mo id="alg1.l4.m1.1.1.1.1.1.1.3" stretchy="false" xref="alg1.l4.m1.1.1.1.1.1.2.cmml">}</mo></mrow><mrow id="alg1.l4.m1.1.1.1.1.3" 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encoding="application/x-llamapun" id="alg1.l5.m2.1d">{ bold_z start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT } start_POSTSUBSCRIPT italic_k = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_K end_POSTSUPERSCRIPT</annotation></semantics></math> <span class="ltx_text ltx_font_bold" id="alg1.l5.2">do</span> </div> <div class="ltx_listingline" id="alg1.l6"> <math alttext="\text{WordRatio}_{k}=\text{WordCount}({\mathbf{z}}_{i}^{k})/\text{WordCount}(y% _{i})" class="ltx_Math" display="inline" id="alg1.l6.m1.2"><semantics id="alg1.l6.m1.2a"><mrow id="alg1.l6.m1.2.2" xref="alg1.l6.m1.2.2.cmml"><msub id="alg1.l6.m1.2.2.4" xref="alg1.l6.m1.2.2.4.cmml"><mtext id="alg1.l6.m1.2.2.4.2" xref="alg1.l6.m1.2.2.4.2a.cmml">WordRatio</mtext><mi id="alg1.l6.m1.2.2.4.3" xref="alg1.l6.m1.2.2.4.3.cmml">k</mi></msub><mo id="alg1.l6.m1.2.2.3" xref="alg1.l6.m1.2.2.3.cmml">=</mo><mrow id="alg1.l6.m1.2.2.2" xref="alg1.l6.m1.2.2.2.cmml"><mrow 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xref="alg1.l6.m1.2.2.2.2.1.1.3">𝑖</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="alg1.l6.m1.2c">\text{WordRatio}_{k}=\text{WordCount}({\mathbf{z}}_{i}^{k})/\text{WordCount}(y% _{i})</annotation><annotation encoding="application/x-llamapun" id="alg1.l6.m1.2d">WordRatio start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT = WordCount ( bold_z start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT ) / WordCount ( italic_y start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT )</annotation></semantics></math> </div> <div class="ltx_listingline" id="alg1.l7"> <math alttext="\text{SentenceRatio}_{k}=\text{SentenceCount}({\mathbf{z}}_{i}^{k})/\text{% SentenceCount}(y_{i})" class="ltx_Math" display="inline" id="alg1.l7.m1.2"><semantics id="alg1.l7.m1.2a"><mrow id="alg1.l7.m1.2.2" xref="alg1.l7.m1.2.2.cmml"><msub id="alg1.l7.m1.2.2.4" xref="alg1.l7.m1.2.2.4.cmml"><mtext id="alg1.l7.m1.2.2.4.2" 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id="alg1.l7.m1.2.2.2.2.1.1.1.cmml" xref="alg1.l7.m1.2.2.2.2.1">subscript</csymbol><ci id="alg1.l7.m1.2.2.2.2.1.1.2.cmml" xref="alg1.l7.m1.2.2.2.2.1.1.2">𝑦</ci><ci id="alg1.l7.m1.2.2.2.2.1.1.3.cmml" xref="alg1.l7.m1.2.2.2.2.1.1.3">𝑖</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="alg1.l7.m1.2c">\text{SentenceRatio}_{k}=\text{SentenceCount}({\mathbf{z}}_{i}^{k})/\text{% SentenceCount}(y_{i})</annotation><annotation encoding="application/x-llamapun" id="alg1.l7.m1.2d">SentenceRatio start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT = SentenceCount ( bold_z start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT ) / SentenceCount ( italic_y start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT )</annotation></semantics></math> </div> <div class="ltx_listingline" id="alg1.l8"> <math alttext="\text{LengthScore}_{k}=\mathbf{g}(\text{WordRatio}_{k},\text{SentenceRatio}_{k})" class="ltx_Math" display="inline" 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id="alg1.l13.2">for</span> </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>Fine-tuning</h3> <div class="ltx_para" id="S4.SS3.p1"> <p class="ltx_p" id="S4.SS3.p1.2">To empower the LLM with the ability to utilize structural information during long-form text generation, we propose a fine-tuning approach that goes beyond the conventional input-to-output (e.g., <math alttext="x_{i}\rightarrow y_{i}" 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"><msub id="S4.SS3.p1.1.m1.1.1.2" xref="S4.SS3.p1.1.m1.1.1.2.cmml"><mi id="S4.SS3.p1.1.m1.1.1.2.2" xref="S4.SS3.p1.1.m1.1.1.2.2.cmml">x</mi><mi id="S4.SS3.p1.1.m1.1.1.2.3" xref="S4.SS3.p1.1.m1.1.1.2.3.cmml">i</mi></msub><mo id="S4.SS3.p1.1.m1.1.1.1" stretchy="false" xref="S4.SS3.p1.1.m1.1.1.1.cmml">→</mo><msub 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id="S4.SS3.p1.1.m1.1.1.3.3.cmml" xref="S4.SS3.p1.1.m1.1.1.3.3">𝑖</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p1.1.m1.1c">x_{i}\rightarrow y_{i}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p1.1.m1.1d">italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT → italic_y start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math>) paradigm. Our method leverages intermediates steps <math alttext="{\mathbf{z}}" class="ltx_Math" display="inline" id="S4.SS3.p1.2.m2.1"><semantics id="S4.SS3.p1.2.m2.1a"><mi id="S4.SS3.p1.2.m2.1.1" xref="S4.SS3.p1.2.m2.1.1.cmml">𝐳</mi><annotation-xml encoding="MathML-Content" id="S4.SS3.p1.2.m2.1b"><ci id="S4.SS3.p1.2.m2.1.1.cmml" xref="S4.SS3.p1.2.m2.1.1">𝐳</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p1.2.m2.1c">{\mathbf{z}}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p1.2.m2.1d">bold_z</annotation></semantics></math>, which encapsulate the structural essence of the desired output.</p> </div> <div class="ltx_para" id="S4.SS3.p2"> <p class="ltx_p" id="S4.SS3.p2.1">Specifically, we introduce two auxiliary fine-tuning tasks in addition to the standard <math alttext="x_{i}\rightarrow y_{i}" class="ltx_Math" display="inline" id="S4.SS3.p2.1.m1.1"><semantics id="S4.SS3.p2.1.m1.1a"><mrow id="S4.SS3.p2.1.m1.1.1" xref="S4.SS3.p2.1.m1.1.1.cmml"><msub 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xref="S4.SS3.p2.1.m1.1.1.2.3">𝑖</ci></apply><apply id="S4.SS3.p2.1.m1.1.1.3.cmml" xref="S4.SS3.p2.1.m1.1.1.3"><csymbol cd="ambiguous" id="S4.SS3.p2.1.m1.1.1.3.1.cmml" xref="S4.SS3.p2.1.m1.1.1.3">subscript</csymbol><ci id="S4.SS3.p2.1.m1.1.1.3.2.cmml" xref="S4.SS3.p2.1.m1.1.1.3.2">𝑦</ci><ci id="S4.SS3.p2.1.m1.1.1.3.3.cmml" xref="S4.SS3.p2.1.m1.1.1.3.3">𝑖</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p2.1.m1.1c">x_{i}\rightarrow y_{i}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.p2.1.m1.1d">italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT → italic_y start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> task.</p> </div> <section class="ltx_paragraph" id="S4.SS3.SSS0.Px1"> <h4 class="ltx_title ltx_title_paragraph">Input to output with intermediate steps (<math alttext="x_{i}\rightarrow{\mathbf{z}}_{i}\oplus y_{i}" class="ltx_Math" display="inline" id="S4.SS3.SSS0.Px1.1.m1.1"><semantics 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id="S4.SS3.SSS0.Px1.1.m1.1e">italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT → bold_z start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ⊕ italic_y start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math>)</h4> <div class="ltx_para" id="S4.SS3.SSS0.Px1.p1"> <p class="ltx_p" id="S4.SS3.SSS0.Px1.p1.2">: This task trains the LLM to generate both the final long-form text <math alttext="y_{i}" class="ltx_Math" display="inline" id="S4.SS3.SSS0.Px1.p1.1.m1.1"><semantics id="S4.SS3.SSS0.Px1.p1.1.m1.1a"><msub id="S4.SS3.SSS0.Px1.p1.1.m1.1.1" xref="S4.SS3.SSS0.Px1.p1.1.m1.1.1.cmml"><mi id="S4.SS3.SSS0.Px1.p1.1.m1.1.1.2" xref="S4.SS3.SSS0.Px1.p1.1.m1.1.1.2.cmml">y</mi><mi id="S4.SS3.SSS0.Px1.p1.1.m1.1.1.3" xref="S4.SS3.SSS0.Px1.p1.1.m1.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS3.SSS0.Px1.p1.1.m1.1b"><apply id="S4.SS3.SSS0.Px1.p1.1.m1.1.1.cmml" xref="S4.SS3.SSS0.Px1.p1.1.m1.1.1"><csymbol cd="ambiguous" id="S4.SS3.SSS0.Px1.p1.1.m1.1.1.1.cmml" xref="S4.SS3.SSS0.Px1.p1.1.m1.1.1">subscript</csymbol><ci id="S4.SS3.SSS0.Px1.p1.1.m1.1.1.2.cmml" xref="S4.SS3.SSS0.Px1.p1.1.m1.1.1.2">𝑦</ci><ci id="S4.SS3.SSS0.Px1.p1.1.m1.1.1.3.cmml" xref="S4.SS3.SSS0.Px1.p1.1.m1.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.SSS0.Px1.p1.1.m1.1c">y_{i}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.SSS0.Px1.p1.1.m1.1d">italic_y start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> and the corresponding intermediate steps <math alttext="{\mathbf{z}}_{i}" class="ltx_Math" display="inline" id="S4.SS3.SSS0.Px1.p1.2.m2.1"><semantics id="S4.SS3.SSS0.Px1.p1.2.m2.1a"><msub id="S4.SS3.SSS0.Px1.p1.2.m2.1.1" xref="S4.SS3.SSS0.Px1.p1.2.m2.1.1.cmml"><mi id="S4.SS3.SSS0.Px1.p1.2.m2.1.1.2" xref="S4.SS3.SSS0.Px1.p1.2.m2.1.1.2.cmml">𝐳</mi><mi id="S4.SS3.SSS0.Px1.p1.2.m2.1.1.3" xref="S4.SS3.SSS0.Px1.p1.2.m2.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS3.SSS0.Px1.p1.2.m2.1b"><apply id="S4.SS3.SSS0.Px1.p1.2.m2.1.1.cmml" xref="S4.SS3.SSS0.Px1.p1.2.m2.1.1"><csymbol cd="ambiguous" id="S4.SS3.SSS0.Px1.p1.2.m2.1.1.1.cmml" xref="S4.SS3.SSS0.Px1.p1.2.m2.1.1">subscript</csymbol><ci id="S4.SS3.SSS0.Px1.p1.2.m2.1.1.2.cmml" xref="S4.SS3.SSS0.Px1.p1.2.m2.1.1.2">𝐳</ci><ci id="S4.SS3.SSS0.Px1.p1.2.m2.1.1.3.cmml" xref="S4.SS3.SSS0.Px1.p1.2.m2.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.SSS0.Px1.p1.2.m2.1c">{\mathbf{z}}_{i}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.SSS0.Px1.p1.2.m2.1d">bold_z start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math>. It encourages the model to internalize the relationship between content and structure, fostering a deeper understanding of how different elements of a document contribute to its overall organization. For this task, the training dataset is,</p> <table class="ltx_equation ltx_eqn_table" id="S4.Ex3"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="D=\{(x_{1},{\mathbf{z}}_{1}\oplus y_{1}),(x_{2},{\mathbf{z}}_{2}\oplus y_{2}),% \ldots,(x_{n},{\mathbf{z}}_{n}\oplus y_{n})\}" class="ltx_Math" display="block" id="S4.Ex3.m1.4"><semantics id="S4.Ex3.m1.4a"><mrow id="S4.Ex3.m1.4.4" xref="S4.Ex3.m1.4.4.cmml"><mi id="S4.Ex3.m1.4.4.5" xref="S4.Ex3.m1.4.4.5.cmml">D</mi><mo id="S4.Ex3.m1.4.4.4" xref="S4.Ex3.m1.4.4.4.cmml">=</mo><mrow id="S4.Ex3.m1.4.4.3.3" xref="S4.Ex3.m1.4.4.3.4.cmml"><mo id="S4.Ex3.m1.4.4.3.3.4" stretchy="false" xref="S4.Ex3.m1.4.4.3.4.cmml">{</mo><mrow id="S4.Ex3.m1.2.2.1.1.1.2" xref="S4.Ex3.m1.2.2.1.1.1.3.cmml"><mo id="S4.Ex3.m1.2.2.1.1.1.2.3" stretchy="false" xref="S4.Ex3.m1.2.2.1.1.1.3.cmml">(</mo><msub id="S4.Ex3.m1.2.2.1.1.1.1.1" 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start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , bold_z start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ⊕ italic_y start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) , … , ( italic_x start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT , bold_z start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ⊕ italic_y start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ) }</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> </div> </section> <section class="ltx_paragraph" id="S4.SS3.SSS0.Px2"> <h4 class="ltx_title ltx_title_paragraph">Input and intermediate steps to output (<math alttext="x_{i}\oplus{\mathbf{z}}_{i}\rightarrow y_{i}" class="ltx_Math" display="inline" id="S4.SS3.SSS0.Px2.1.m1.1"><semantics id="S4.SS3.SSS0.Px2.1.m1.1b"><mrow id="S4.SS3.SSS0.Px2.1.m1.1.1" xref="S4.SS3.SSS0.Px2.1.m1.1.1.cmml"><mrow id="S4.SS3.SSS0.Px2.1.m1.1.1.2" xref="S4.SS3.SSS0.Px2.1.m1.1.1.2.cmml"><msub id="S4.SS3.SSS0.Px2.1.m1.1.1.2.2" xref="S4.SS3.SSS0.Px2.1.m1.1.1.2.2.cmml"><mi 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id="S4.SS3.SSS0.Px2.1.m1.1.1.cmml" xref="S4.SS3.SSS0.Px2.1.m1.1.1"><ci id="S4.SS3.SSS0.Px2.1.m1.1.1.1.cmml" xref="S4.SS3.SSS0.Px2.1.m1.1.1.1">→</ci><apply id="S4.SS3.SSS0.Px2.1.m1.1.1.2.cmml" xref="S4.SS3.SSS0.Px2.1.m1.1.1.2"><csymbol cd="latexml" id="S4.SS3.SSS0.Px2.1.m1.1.1.2.1.cmml" xref="S4.SS3.SSS0.Px2.1.m1.1.1.2.1">direct-sum</csymbol><apply id="S4.SS3.SSS0.Px2.1.m1.1.1.2.2.cmml" xref="S4.SS3.SSS0.Px2.1.m1.1.1.2.2"><csymbol cd="ambiguous" id="S4.SS3.SSS0.Px2.1.m1.1.1.2.2.1.cmml" xref="S4.SS3.SSS0.Px2.1.m1.1.1.2.2">subscript</csymbol><ci id="S4.SS3.SSS0.Px2.1.m1.1.1.2.2.2.cmml" xref="S4.SS3.SSS0.Px2.1.m1.1.1.2.2.2">𝑥</ci><ci id="S4.SS3.SSS0.Px2.1.m1.1.1.2.2.3.cmml" xref="S4.SS3.SSS0.Px2.1.m1.1.1.2.2.3">𝑖</ci></apply><apply id="S4.SS3.SSS0.Px2.1.m1.1.1.2.3.cmml" xref="S4.SS3.SSS0.Px2.1.m1.1.1.2.3"><csymbol cd="ambiguous" id="S4.SS3.SSS0.Px2.1.m1.1.1.2.3.1.cmml" xref="S4.SS3.SSS0.Px2.1.m1.1.1.2.3">subscript</csymbol><ci id="S4.SS3.SSS0.Px2.1.m1.1.1.2.3.2.cmml" xref="S4.SS3.SSS0.Px2.1.m1.1.1.2.3.2">𝐳</ci><ci id="S4.SS3.SSS0.Px2.1.m1.1.1.2.3.3.cmml" xref="S4.SS3.SSS0.Px2.1.m1.1.1.2.3.3">𝑖</ci></apply></apply><apply id="S4.SS3.SSS0.Px2.1.m1.1.1.3.cmml" xref="S4.SS3.SSS0.Px2.1.m1.1.1.3"><csymbol cd="ambiguous" id="S4.SS3.SSS0.Px2.1.m1.1.1.3.1.cmml" xref="S4.SS3.SSS0.Px2.1.m1.1.1.3">subscript</csymbol><ci id="S4.SS3.SSS0.Px2.1.m1.1.1.3.2.cmml" xref="S4.SS3.SSS0.Px2.1.m1.1.1.3.2">𝑦</ci><ci id="S4.SS3.SSS0.Px2.1.m1.1.1.3.3.cmml" xref="S4.SS3.SSS0.Px2.1.m1.1.1.3.3">𝑖</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.SSS0.Px2.1.m1.1d">x_{i}\oplus{\mathbf{z}}_{i}\rightarrow y_{i}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.SSS0.Px2.1.m1.1e">italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ⊕ bold_z start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT → italic_y start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math>)</h4> <div class="ltx_para" id="S4.SS3.SSS0.Px2.p1"> <p class="ltx_p" id="S4.SS3.SSS0.Px2.p1.5">: In this task, the LLM is provided with both the input <math alttext="x" class="ltx_Math" display="inline" id="S4.SS3.SSS0.Px2.p1.1.m1.1"><semantics id="S4.SS3.SSS0.Px2.p1.1.m1.1a"><mi id="S4.SS3.SSS0.Px2.p1.1.m1.1.1" xref="S4.SS3.SSS0.Px2.p1.1.m1.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="S4.SS3.SSS0.Px2.p1.1.m1.1b"><ci id="S4.SS3.SSS0.Px2.p1.1.m1.1.1.cmml" xref="S4.SS3.SSS0.Px2.p1.1.m1.1.1">𝑥</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.SSS0.Px2.p1.1.m1.1c">x</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.SSS0.Px2.p1.1.m1.1d">italic_x</annotation></semantics></math> and the synthetically generated intermediate steps <math alttext="{\mathbf{z}}_{i}" class="ltx_Math" display="inline" id="S4.SS3.SSS0.Px2.p1.2.m2.1"><semantics id="S4.SS3.SSS0.Px2.p1.2.m2.1a"><msub id="S4.SS3.SSS0.Px2.p1.2.m2.1.1" xref="S4.SS3.SSS0.Px2.p1.2.m2.1.1.cmml"><mi 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The model is then trained to generate the final long-form text <math alttext="y_{i}" class="ltx_Math" display="inline" id="S4.SS3.SSS0.Px2.p1.3.m3.1"><semantics id="S4.SS3.SSS0.Px2.p1.3.m3.1a"><msub id="S4.SS3.SSS0.Px2.p1.3.m3.1.1" xref="S4.SS3.SSS0.Px2.p1.3.m3.1.1.cmml"><mi id="S4.SS3.SSS0.Px2.p1.3.m3.1.1.2" xref="S4.SS3.SSS0.Px2.p1.3.m3.1.1.2.cmml">y</mi><mi id="S4.SS3.SSS0.Px2.p1.3.m3.1.1.3" xref="S4.SS3.SSS0.Px2.p1.3.m3.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS3.SSS0.Px2.p1.3.m3.1b"><apply id="S4.SS3.SSS0.Px2.p1.3.m3.1.1.cmml" xref="S4.SS3.SSS0.Px2.p1.3.m3.1.1"><csymbol cd="ambiguous" id="S4.SS3.SSS0.Px2.p1.3.m3.1.1.1.cmml" xref="S4.SS3.SSS0.Px2.p1.3.m3.1.1">subscript</csymbol><ci id="S4.SS3.SSS0.Px2.p1.3.m3.1.1.2.cmml" xref="S4.SS3.SSS0.Px2.p1.3.m3.1.1.2">𝑦</ci><ci id="S4.SS3.SSS0.Px2.p1.3.m3.1.1.3.cmml" xref="S4.SS3.SSS0.Px2.p1.3.m3.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.SSS0.Px2.p1.3.m3.1c">y_{i}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.SSS0.Px2.p1.3.m3.1d">italic_y start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math>, conditioned on the structural guidance provided by <math alttext="{\mathbf{z}}" class="ltx_Math" display="inline" id="S4.SS3.SSS0.Px2.p1.4.m4.1"><semantics id="S4.SS3.SSS0.Px2.p1.4.m4.1a"><mi id="S4.SS3.SSS0.Px2.p1.4.m4.1.1" xref="S4.SS3.SSS0.Px2.p1.4.m4.1.1.cmml">𝐳</mi><annotation-xml encoding="MathML-Content" id="S4.SS3.SSS0.Px2.p1.4.m4.1b"><ci id="S4.SS3.SSS0.Px2.p1.4.m4.1.1.cmml" xref="S4.SS3.SSS0.Px2.p1.4.m4.1.1">𝐳</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.SSS0.Px2.p1.4.m4.1c">{\mathbf{z}}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.SSS0.Px2.p1.4.m4.1d">bold_z</annotation></semantics></math>. Here, <math alttext="{\mathbf{z}}" class="ltx_Math" display="inline" id="S4.SS3.SSS0.Px2.p1.5.m5.1"><semantics id="S4.SS3.SSS0.Px2.p1.5.m5.1a"><mi id="S4.SS3.SSS0.Px2.p1.5.m5.1.1" xref="S4.SS3.SSS0.Px2.p1.5.m5.1.1.cmml">𝐳</mi><annotation-xml encoding="MathML-Content" id="S4.SS3.SSS0.Px2.p1.5.m5.1b"><ci id="S4.SS3.SSS0.Px2.p1.5.m5.1.1.cmml" xref="S4.SS3.SSS0.Px2.p1.5.m5.1.1">𝐳</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.SSS0.Px2.p1.5.m5.1c">{\mathbf{z}}</annotation><annotation encoding="application/x-llamapun" id="S4.SS3.SSS0.Px2.p1.5.m5.1d">bold_z</annotation></semantics></math> acts as a blueprint for generating the long-form text, helping the model maintain focus and generate more coherent and structured output. In this task, the training dataset is:</p> <table class="ltx_equation ltx_eqn_table" id="S4.Ex4"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="D=\{(x_{1}\oplus{\mathbf{z}}_{1},y_{1}),(x_{2}\oplus{\mathbf{z}}_{2},y_{2}),% \ldots,(x_{n}\oplus{\mathbf{z}}_{n},\oplus y_{n})\}" class="ltx_Math" display="block" id="S4.Ex4.m1.4"><semantics id="S4.Ex4.m1.4a"><mrow id="S4.Ex4.m1.4.4" xref="S4.Ex4.m1.4.4.cmml"><mi id="S4.Ex4.m1.4.4.5" xref="S4.Ex4.m1.4.4.5.cmml">D</mi><mo id="S4.Ex4.m1.4.4.4" xref="S4.Ex4.m1.4.4.4.cmml">=</mo><mrow id="S4.Ex4.m1.4.4.3.3" xref="S4.Ex4.m1.4.4.3.4.cmml"><mo id="S4.Ex4.m1.4.4.3.3.4" stretchy="false" xref="S4.Ex4.m1.4.4.3.4.cmml">{</mo><mrow id="S4.Ex4.m1.2.2.1.1.1.2" xref="S4.Ex4.m1.2.2.1.1.1.3.cmml"><mo id="S4.Ex4.m1.2.2.1.1.1.2.3" stretchy="false" xref="S4.Ex4.m1.2.2.1.1.1.3.cmml">(</mo><mrow id="S4.Ex4.m1.2.2.1.1.1.1.1" 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id="S4.Ex4.m1.4c">D=\{(x_{1}\oplus{\mathbf{z}}_{1},y_{1}),(x_{2}\oplus{\mathbf{z}}_{2},y_{2}),% \ldots,(x_{n}\oplus{\mathbf{z}}_{n},\oplus y_{n})\}</annotation><annotation encoding="application/x-llamapun" id="S4.Ex4.m1.4d">italic_D = { ( italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ⊕ bold_z start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_y start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) , ( italic_x start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ⊕ bold_z start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , italic_y start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) , … , ( italic_x start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ⊕ bold_z start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT , ⊕ italic_y start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ) }</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> </div> <div class="ltx_para" id="S4.SS3.SSS0.Px2.p2"> <p class="ltx_p" id="S4.SS3.SSS0.Px2.p2.2">The two tasks complement each other by focusing on different aspects of the planned generation process. 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By incorporating these auxiliary tasks, our fine-tuning process aims to improve the LLM’s capability of generating long-form text that is not only informative but also well-structured. </p> </div> </section> </section> </section> <section class="ltx_section" id="S5"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">5 </span>Experiments</h2> <div class="ltx_para" id="S5.p1"> <p class="ltx_p" id="S5.p1.1">We conduct extensive experiments on multiple datasets with different setups to validate the effectiveness of our proposed methods.</p> </div> <section class="ltx_subsection" id="S5.SS1"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">5.1 </span>Benchmark Datasets</h3> <div class="ltx_para" id="S5.SS1.p1"> <p class="ltx_p" id="S5.SS1.p1.1">To validate the effectiveness of our planned generation method, we conducted experiments using the SciNews dataset and Wikipedia dataset.</p> </div> <section class="ltx_paragraph" id="S5.SS1.SSS0.Px1"> <h4 class="ltx_title ltx_title_paragraph">SciNews Dataset.</h4> <div class="ltx_para" id="S5.SS1.SSS0.Px1.p1"> <p class="ltx_p" id="S5.SS1.SSS0.Px1.p1.2">SciNews dataset <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2410.06203v1#bib.bib17" title="">17</a>]</cite> is developed to facilitate the task of compiling academic publications into scientific news reports. This dataset provides a parallel collection of academic publications and their corresponding scientific news reports across 9 disciplines, including Medicine, Biology, Physics, and Chemistry. It comprises 41,872 samples, with each academic publication averaging 7,760.90 tokens and each news report averaging 694 tokens. In this dataset, we use each academic publication as the input <math alttext="x_{i}" class="ltx_Math" display="inline" id="S5.SS1.SSS0.Px1.p1.1.m1.1"><semantics id="S5.SS1.SSS0.Px1.p1.1.m1.1a"><msub id="S5.SS1.SSS0.Px1.p1.1.m1.1.1" xref="S5.SS1.SSS0.Px1.p1.1.m1.1.1.cmml"><mi id="S5.SS1.SSS0.Px1.p1.1.m1.1.1.2" xref="S5.SS1.SSS0.Px1.p1.1.m1.1.1.2.cmml">x</mi><mi id="S5.SS1.SSS0.Px1.p1.1.m1.1.1.3" xref="S5.SS1.SSS0.Px1.p1.1.m1.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S5.SS1.SSS0.Px1.p1.1.m1.1b"><apply id="S5.SS1.SSS0.Px1.p1.1.m1.1.1.cmml" xref="S5.SS1.SSS0.Px1.p1.1.m1.1.1"><csymbol cd="ambiguous" id="S5.SS1.SSS0.Px1.p1.1.m1.1.1.1.cmml" xref="S5.SS1.SSS0.Px1.p1.1.m1.1.1">subscript</csymbol><ci id="S5.SS1.SSS0.Px1.p1.1.m1.1.1.2.cmml" xref="S5.SS1.SSS0.Px1.p1.1.m1.1.1.2">𝑥</ci><ci id="S5.SS1.SSS0.Px1.p1.1.m1.1.1.3.cmml" xref="S5.SS1.SSS0.Px1.p1.1.m1.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS1.SSS0.Px1.p1.1.m1.1c">x_{i}</annotation><annotation encoding="application/x-llamapun" id="S5.SS1.SSS0.Px1.p1.1.m1.1d">italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> and its corresponding news report as <math alttext="y_{i}" class="ltx_Math" display="inline" id="S5.SS1.SSS0.Px1.p1.2.m2.1"><semantics id="S5.SS1.SSS0.Px1.p1.2.m2.1a"><msub id="S5.SS1.SSS0.Px1.p1.2.m2.1.1" xref="S5.SS1.SSS0.Px1.p1.2.m2.1.1.cmml"><mi id="S5.SS1.SSS0.Px1.p1.2.m2.1.1.2" xref="S5.SS1.SSS0.Px1.p1.2.m2.1.1.2.cmml">y</mi><mi id="S5.SS1.SSS0.Px1.p1.2.m2.1.1.3" xref="S5.SS1.SSS0.Px1.p1.2.m2.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S5.SS1.SSS0.Px1.p1.2.m2.1b"><apply id="S5.SS1.SSS0.Px1.p1.2.m2.1.1.cmml" xref="S5.SS1.SSS0.Px1.p1.2.m2.1.1"><csymbol cd="ambiguous" id="S5.SS1.SSS0.Px1.p1.2.m2.1.1.1.cmml" xref="S5.SS1.SSS0.Px1.p1.2.m2.1.1">subscript</csymbol><ci id="S5.SS1.SSS0.Px1.p1.2.m2.1.1.2.cmml" xref="S5.SS1.SSS0.Px1.p1.2.m2.1.1.2">𝑦</ci><ci id="S5.SS1.SSS0.Px1.p1.2.m2.1.1.3.cmml" xref="S5.SS1.SSS0.Px1.p1.2.m2.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS1.SSS0.Px1.p1.2.m2.1c">y_{i}</annotation><annotation encoding="application/x-llamapun" id="S5.SS1.SSS0.Px1.p1.2.m2.1d">italic_y start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math>. We randomly sample 33,497 items as the training split, 1,000 items for validation and 200 items as the evaluation split.</p> </div> </section> <section class="ltx_paragraph" id="S5.SS1.SSS0.Px2"> <h4 class="ltx_title ltx_title_paragraph">Wikipedia Dataset.</h4> <div class="ltx_para" id="S5.SS1.SSS0.Px2.p1"> <p class="ltx_p" id="S5.SS1.SSS0.Px2.p1.1">We use the Wikipedia corpus from the KILT benchmark dataset <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2410.06203v1#bib.bib16" title="">16</a>]</cite> to construct the training and validation splits. This benchmark dataset is sourced from a snapshot of Wikipedia taken on August 1, 2019, and contains 5.9 million articles. The evaluation split is constructed from the FreshWiki corpus <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2410.06203v1#bib.bib19" title="">19</a>]</cite>, which consists of 100 high-quality Wikipedia page with most edits for each month from February 2022 to September 2023. Both dataset includes the full text of Wikipedia pages, along with descriptions and titles. We filtered both corpus by removing articles with less than 1,000 words and those lacking any structured sections.</p> </div> <div class="ltx_para" id="S5.SS1.SSS0.Px2.p2"> <p class="ltx_p" id="S5.SS1.SSS0.Px2.p2.2">For each Wikipedia article <math alttext="y_{i}" class="ltx_Math" display="inline" id="S5.SS1.SSS0.Px2.p2.1.m1.1"><semantics id="S5.SS1.SSS0.Px2.p2.1.m1.1a"><msub id="S5.SS1.SSS0.Px2.p2.1.m1.1.1" xref="S5.SS1.SSS0.Px2.p2.1.m1.1.1.cmml"><mi id="S5.SS1.SSS0.Px2.p2.1.m1.1.1.2" xref="S5.SS1.SSS0.Px2.p2.1.m1.1.1.2.cmml">y</mi><mi id="S5.SS1.SSS0.Px2.p2.1.m1.1.1.3" xref="S5.SS1.SSS0.Px2.p2.1.m1.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S5.SS1.SSS0.Px2.p2.1.m1.1b"><apply id="S5.SS1.SSS0.Px2.p2.1.m1.1.1.cmml" xref="S5.SS1.SSS0.Px2.p2.1.m1.1.1"><csymbol cd="ambiguous" id="S5.SS1.SSS0.Px2.p2.1.m1.1.1.1.cmml" xref="S5.SS1.SSS0.Px2.p2.1.m1.1.1">subscript</csymbol><ci id="S5.SS1.SSS0.Px2.p2.1.m1.1.1.2.cmml" xref="S5.SS1.SSS0.Px2.p2.1.m1.1.1.2">𝑦</ci><ci id="S5.SS1.SSS0.Px2.p2.1.m1.1.1.3.cmml" xref="S5.SS1.SSS0.Px2.p2.1.m1.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS1.SSS0.Px2.p2.1.m1.1c">y_{i}</annotation><annotation encoding="application/x-llamapun" id="S5.SS1.SSS0.Px2.p2.1.m1.1d">italic_y start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math>, we formulated the input context <math alttext="x_{i}" class="ltx_Math" display="inline" id="S5.SS1.SSS0.Px2.p2.2.m2.1"><semantics id="S5.SS1.SSS0.Px2.p2.2.m2.1a"><msub id="S5.SS1.SSS0.Px2.p2.2.m2.1.1" xref="S5.SS1.SSS0.Px2.p2.2.m2.1.1.cmml"><mi id="S5.SS1.SSS0.Px2.p2.2.m2.1.1.2" xref="S5.SS1.SSS0.Px2.p2.2.m2.1.1.2.cmml">x</mi><mi id="S5.SS1.SSS0.Px2.p2.2.m2.1.1.3" xref="S5.SS1.SSS0.Px2.p2.2.m2.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S5.SS1.SSS0.Px2.p2.2.m2.1b"><apply id="S5.SS1.SSS0.Px2.p2.2.m2.1.1.cmml" xref="S5.SS1.SSS0.Px2.p2.2.m2.1.1"><csymbol cd="ambiguous" id="S5.SS1.SSS0.Px2.p2.2.m2.1.1.1.cmml" xref="S5.SS1.SSS0.Px2.p2.2.m2.1.1">subscript</csymbol><ci id="S5.SS1.SSS0.Px2.p2.2.m2.1.1.2.cmml" xref="S5.SS1.SSS0.Px2.p2.2.m2.1.1.2">𝑥</ci><ci id="S5.SS1.SSS0.Px2.p2.2.m2.1.1.3.cmml" xref="S5.SS1.SSS0.Px2.p2.2.m2.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS1.SSS0.Px2.p2.2.m2.1c">x_{i}</annotation><annotation encoding="application/x-llamapun" id="S5.SS1.SSS0.Px2.p2.2.m2.1d">italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> as a one sentence prompt:</p> <blockquote class="ltx_quote" id="S5.SS1.SSS0.Px2.p2.3"> <p class="ltx_p" id="S5.SS1.SSS0.Px2.p2.3.1"><em class="ltx_emph ltx_font_italic" id="S5.SS1.SSS0.Px2.p2.3.1.1">Generate a comprehensive Wikipedia page about the specified topic.</em> <em class="ltx_emph ltx_font_italic" id="S5.SS1.SSS0.Px2.p2.3.1.2">topic: [entity]</em></p> </blockquote> </div> <div class="ltx_para" id="S5.SS1.SSS0.Px2.p3"> <p class="ltx_p" id="S5.SS1.SSS0.Px2.p3.1">We randomly sampled 1,900 items from the KILT corpus as the training split, 232 items for the validation split, and used all the 98 filtered items from the FreshWiki corpus as the evaluation split.</p> </div> </section> </section> <section class="ltx_subsection" id="S5.SS2"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">5.2 </span>Evaluation Metrics</h3> <section class="ltx_paragraph" id="S5.SS2.SSS0.Px1"> <h4 class="ltx_title ltx_title_paragraph">Offline metrics.</h4> <div class="ltx_para" id="S5.SS2.SSS0.Px1.p1"> <p class="ltx_p" id="S5.SS2.SSS0.Px1.p1.1">To assess the effectiveness of the proposed method, we utilized the F1 scores of ROUGE-1 (R1), ROUGE-2 (R2), ROUGE-L (RL), and ROUGE-Lsum (RLsum) metrics for this evaluation.</p> </div> </section> <section class="ltx_paragraph" id="S5.SS2.SSS0.Px2"> <h4 class="ltx_title ltx_title_paragraph">Human and auto SxS.</h4> <div class="ltx_para" id="S5.SS2.SSS0.Px2.p1"> <p class="ltx_p" id="S5.SS2.SSS0.Px2.p1.1">In addition to these ROUGE metrics, and in line with the concept of critical evaluation <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2410.06203v1#bib.bib29" title="">29</a>]</cite>, we also conducted SxS evaluations, both by human expert raters and by an LLM evaluator. For the LLM evaluator, we employed a more capable LLM (Gemini Ultra) as an automatic SxS rater to compare the generated articles from two methods. For both human and auto SxS, raters are asked to rate which one of two generated articles is better. The base and the test sides are randomly flipped to avoid potential bias towards one side. We calculate and report the win rate and W/L ratio of our proposed method compared against the baseline.</p> </div> </section> </section> <section class="ltx_subsection" id="S5.SS3"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">5.3 </span>Experiment setup</h3> <div class="ltx_para" id="S5.SS3.p1"> <p class="ltx_p" id="S5.SS3.p1.1">We use Gemini Pro model for both our baselines and all of our fine-tuning experiments. For experiments on the SciNews dataset, we limit the input sequence length to the model to be no more than around 16k tokens, and the output sequence length to be no more than 4k tokens. For experiments on the Wikipedia dataset, we limit the input sequence length to be no more than 1k tokens as the input context in this data set is much shorter. The output sequence length limit is set to 6k tokens.</p> </div> <div class="ltx_para" id="S5.SS3.p2"> <p class="ltx_p" id="S5.SS3.p2.1">All the fine-tuning experiments are conducted on TPU. The batch size is set to 16 and 32 respectively for SciNews and Wikipedia datasets. The maximum learning rate for all experiments is set as <math alttext="10^{-5}" class="ltx_Math" display="inline" id="S5.SS3.p2.1.m1.1"><semantics id="S5.SS3.p2.1.m1.1a"><msup id="S5.SS3.p2.1.m1.1.1" xref="S5.SS3.p2.1.m1.1.1.cmml"><mn id="S5.SS3.p2.1.m1.1.1.2" xref="S5.SS3.p2.1.m1.1.1.2.cmml">10</mn><mrow id="S5.SS3.p2.1.m1.1.1.3" xref="S5.SS3.p2.1.m1.1.1.3.cmml"><mo id="S5.SS3.p2.1.m1.1.1.3a" xref="S5.SS3.p2.1.m1.1.1.3.cmml">−</mo><mn id="S5.SS3.p2.1.m1.1.1.3.2" xref="S5.SS3.p2.1.m1.1.1.3.2.cmml">5</mn></mrow></msup><annotation-xml encoding="MathML-Content" id="S5.SS3.p2.1.m1.1b"><apply id="S5.SS3.p2.1.m1.1.1.cmml" xref="S5.SS3.p2.1.m1.1.1"><csymbol cd="ambiguous" id="S5.SS3.p2.1.m1.1.1.1.cmml" xref="S5.SS3.p2.1.m1.1.1">superscript</csymbol><cn id="S5.SS3.p2.1.m1.1.1.2.cmml" type="integer" xref="S5.SS3.p2.1.m1.1.1.2">10</cn><apply id="S5.SS3.p2.1.m1.1.1.3.cmml" xref="S5.SS3.p2.1.m1.1.1.3"><minus id="S5.SS3.p2.1.m1.1.1.3.1.cmml" xref="S5.SS3.p2.1.m1.1.1.3"></minus><cn id="S5.SS3.p2.1.m1.1.1.3.2.cmml" type="integer" xref="S5.SS3.p2.1.m1.1.1.3.2">5</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS3.p2.1.m1.1c">10^{-5}</annotation><annotation encoding="application/x-llamapun" id="S5.SS3.p2.1.m1.1d">10 start_POSTSUPERSCRIPT - 5 end_POSTSUPERSCRIPT</annotation></semantics></math>.</p> </div> </section> <section class="ltx_subsection" id="S5.SS4"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">5.4 </span>Results</h3> <section class="ltx_paragraph" id="S5.SS4.SSS0.Px1"> <h4 class="ltx_title ltx_title_paragraph">Overall comparison.</h4> <div class="ltx_para" id="S5.SS4.SSS0.Px1.p1"> <p class="ltx_p" id="S5.SS4.SSS0.Px1.p1.1">Table <a class="ltx_ref" href="https://arxiv.org/html/2410.06203v1#S5.T1" title="Table 1 ‣ Overall comparison. ‣ 5.4 Results ‣ 5 Experiments ‣ Integrating Planning into Single-Turn Long-Form Text Generation"><span class="ltx_text ltx_ref_tag">1</span></a> shows the results of the following models and prompt settings:</p> </div> <div class="ltx_para" id="S5.SS4.SSS0.Px1.p2"> <ol class="ltx_enumerate" id="S5.I1"> <li class="ltx_item" id="S5.I1.i1" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">1.</span> <div class="ltx_para" id="S5.I1.i1.p1"> <p class="ltx_p" id="S5.I1.i1.p1.1"><span class="ltx_text ltx_font_bold" id="S5.I1.i1.p1.1.1">Zero-shot (ZS):</span> Directly using the LLM without fine-tuning. The input context <math alttext="x_{i}" class="ltx_Math" display="inline" id="S5.I1.i1.p1.1.m1.1"><semantics id="S5.I1.i1.p1.1.m1.1a"><msub id="S5.I1.i1.p1.1.m1.1.1" xref="S5.I1.i1.p1.1.m1.1.1.cmml"><mi id="S5.I1.i1.p1.1.m1.1.1.2" xref="S5.I1.i1.p1.1.m1.1.1.2.cmml">x</mi><mi id="S5.I1.i1.p1.1.m1.1.1.3" xref="S5.I1.i1.p1.1.m1.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S5.I1.i1.p1.1.m1.1b"><apply id="S5.I1.i1.p1.1.m1.1.1.cmml" xref="S5.I1.i1.p1.1.m1.1.1"><csymbol cd="ambiguous" id="S5.I1.i1.p1.1.m1.1.1.1.cmml" xref="S5.I1.i1.p1.1.m1.1.1">subscript</csymbol><ci id="S5.I1.i1.p1.1.m1.1.1.2.cmml" xref="S5.I1.i1.p1.1.m1.1.1.2">𝑥</ci><ci id="S5.I1.i1.p1.1.m1.1.1.3.cmml" xref="S5.I1.i1.p1.1.m1.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I1.i1.p1.1.m1.1c">x_{i}</annotation><annotation encoding="application/x-llamapun" id="S5.I1.i1.p1.1.m1.1d">italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> to the LLM is the same as the our proposed methods.</p> </div> </li> <li class="ltx_item" id="S5.I1.i2" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">2.</span> <div class="ltx_para" id="S5.I1.i2.p1"> <p class="ltx_p" id="S5.I1.i2.p1.3"><span class="ltx_text ltx_font_bold" id="S5.I1.i2.p1.3.1">Fine-tuning without intermediate steps (FT w/o I):</span> Only fine-tune the LLM with the input context <math alttext="x_{i}" class="ltx_Math" display="inline" id="S5.I1.i2.p1.1.m1.1"><semantics id="S5.I1.i2.p1.1.m1.1a"><msub id="S5.I1.i2.p1.1.m1.1.1" xref="S5.I1.i2.p1.1.m1.1.1.cmml"><mi id="S5.I1.i2.p1.1.m1.1.1.2" xref="S5.I1.i2.p1.1.m1.1.1.2.cmml">x</mi><mi id="S5.I1.i2.p1.1.m1.1.1.3" xref="S5.I1.i2.p1.1.m1.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S5.I1.i2.p1.1.m1.1b"><apply id="S5.I1.i2.p1.1.m1.1.1.cmml" xref="S5.I1.i2.p1.1.m1.1.1"><csymbol cd="ambiguous" id="S5.I1.i2.p1.1.m1.1.1.1.cmml" xref="S5.I1.i2.p1.1.m1.1.1">subscript</csymbol><ci id="S5.I1.i2.p1.1.m1.1.1.2.cmml" xref="S5.I1.i2.p1.1.m1.1.1.2">𝑥</ci><ci id="S5.I1.i2.p1.1.m1.1.1.3.cmml" xref="S5.I1.i2.p1.1.m1.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I1.i2.p1.1.m1.1c">x_{i}</annotation><annotation encoding="application/x-llamapun" id="S5.I1.i2.p1.1.m1.1d">italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> as input and the output article <math alttext="y_{i}" class="ltx_Math" display="inline" id="S5.I1.i2.p1.2.m2.1"><semantics id="S5.I1.i2.p1.2.m2.1a"><msub id="S5.I1.i2.p1.2.m2.1.1" xref="S5.I1.i2.p1.2.m2.1.1.cmml"><mi id="S5.I1.i2.p1.2.m2.1.1.2" xref="S5.I1.i2.p1.2.m2.1.1.2.cmml">y</mi><mi id="S5.I1.i2.p1.2.m2.1.1.3" xref="S5.I1.i2.p1.2.m2.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S5.I1.i2.p1.2.m2.1b"><apply id="S5.I1.i2.p1.2.m2.1.1.cmml" xref="S5.I1.i2.p1.2.m2.1.1"><csymbol cd="ambiguous" id="S5.I1.i2.p1.2.m2.1.1.1.cmml" xref="S5.I1.i2.p1.2.m2.1.1">subscript</csymbol><ci id="S5.I1.i2.p1.2.m2.1.1.2.cmml" xref="S5.I1.i2.p1.2.m2.1.1.2">𝑦</ci><ci id="S5.I1.i2.p1.2.m2.1.1.3.cmml" xref="S5.I1.i2.p1.2.m2.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I1.i2.p1.2.m2.1c">y_{i}</annotation><annotation encoding="application/x-llamapun" id="S5.I1.i2.p1.2.m2.1d">italic_y start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> as output without using any intermediate steps <math alttext="{\mathbf{z}}_{i}" class="ltx_Math" display="inline" id="S5.I1.i2.p1.3.m3.1"><semantics id="S5.I1.i2.p1.3.m3.1a"><msub id="S5.I1.i2.p1.3.m3.1.1" xref="S5.I1.i2.p1.3.m3.1.1.cmml"><mi id="S5.I1.i2.p1.3.m3.1.1.2" xref="S5.I1.i2.p1.3.m3.1.1.2.cmml">𝐳</mi><mi id="S5.I1.i2.p1.3.m3.1.1.3" xref="S5.I1.i2.p1.3.m3.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S5.I1.i2.p1.3.m3.1b"><apply id="S5.I1.i2.p1.3.m3.1.1.cmml" xref="S5.I1.i2.p1.3.m3.1.1"><csymbol cd="ambiguous" id="S5.I1.i2.p1.3.m3.1.1.1.cmml" xref="S5.I1.i2.p1.3.m3.1.1">subscript</csymbol><ci id="S5.I1.i2.p1.3.m3.1.1.2.cmml" xref="S5.I1.i2.p1.3.m3.1.1.2">𝐳</ci><ci id="S5.I1.i2.p1.3.m3.1.1.3.cmml" xref="S5.I1.i2.p1.3.m3.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I1.i2.p1.3.m3.1c">{\mathbf{z}}_{i}</annotation><annotation encoding="application/x-llamapun" id="S5.I1.i2.p1.3.m3.1d">bold_z start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math>.</p> </div> </li> <li class="ltx_item" id="S5.I1.i3" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">3.</span> <div class="ltx_para" id="S5.I1.i3.p1"> <p class="ltx_p" id="S5.I1.i3.p1.10"><span class="ltx_text ltx_font_bold" id="S5.I1.i3.p1.10.1">Fine-tuning with intermediate steps (FT w/ I):</span> Fine-tune the LLM with the a mixture of two training tasks: generating the full article <math alttext="y_{i}" class="ltx_Math" display="inline" id="S5.I1.i3.p1.1.m1.1"><semantics id="S5.I1.i3.p1.1.m1.1a"><msub id="S5.I1.i3.p1.1.m1.1.1" xref="S5.I1.i3.p1.1.m1.1.1.cmml"><mi id="S5.I1.i3.p1.1.m1.1.1.2" xref="S5.I1.i3.p1.1.m1.1.1.2.cmml">y</mi><mi id="S5.I1.i3.p1.1.m1.1.1.3" xref="S5.I1.i3.p1.1.m1.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S5.I1.i3.p1.1.m1.1b"><apply id="S5.I1.i3.p1.1.m1.1.1.cmml" xref="S5.I1.i3.p1.1.m1.1.1"><csymbol cd="ambiguous" id="S5.I1.i3.p1.1.m1.1.1.1.cmml" xref="S5.I1.i3.p1.1.m1.1.1">subscript</csymbol><ci id="S5.I1.i3.p1.1.m1.1.1.2.cmml" xref="S5.I1.i3.p1.1.m1.1.1.2">𝑦</ci><ci id="S5.I1.i3.p1.1.m1.1.1.3.cmml" xref="S5.I1.i3.p1.1.m1.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I1.i3.p1.1.m1.1c">y_{i}</annotation><annotation encoding="application/x-llamapun" id="S5.I1.i3.p1.1.m1.1d">italic_y start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> from the input context <math alttext="x_{i}" class="ltx_Math" display="inline" id="S5.I1.i3.p1.2.m2.1"><semantics id="S5.I1.i3.p1.2.m2.1a"><msub id="S5.I1.i3.p1.2.m2.1.1" xref="S5.I1.i3.p1.2.m2.1.1.cmml"><mi id="S5.I1.i3.p1.2.m2.1.1.2" xref="S5.I1.i3.p1.2.m2.1.1.2.cmml">x</mi><mi id="S5.I1.i3.p1.2.m2.1.1.3" xref="S5.I1.i3.p1.2.m2.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S5.I1.i3.p1.2.m2.1b"><apply id="S5.I1.i3.p1.2.m2.1.1.cmml" xref="S5.I1.i3.p1.2.m2.1.1"><csymbol cd="ambiguous" id="S5.I1.i3.p1.2.m2.1.1.1.cmml" xref="S5.I1.i3.p1.2.m2.1.1">subscript</csymbol><ci id="S5.I1.i3.p1.2.m2.1.1.2.cmml" xref="S5.I1.i3.p1.2.m2.1.1.2">𝑥</ci><ci id="S5.I1.i3.p1.2.m2.1.1.3.cmml" xref="S5.I1.i3.p1.2.m2.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I1.i3.p1.2.m2.1c">x_{i}</annotation><annotation encoding="application/x-llamapun" id="S5.I1.i3.p1.2.m2.1d">italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> (<math alttext="x_{i}\rightarrow y_{i}" class="ltx_Math" display="inline" id="S5.I1.i3.p1.3.m3.1"><semantics id="S5.I1.i3.p1.3.m3.1a"><mrow id="S5.I1.i3.p1.3.m3.1.1" xref="S5.I1.i3.p1.3.m3.1.1.cmml"><msub id="S5.I1.i3.p1.3.m3.1.1.2" xref="S5.I1.i3.p1.3.m3.1.1.2.cmml"><mi id="S5.I1.i3.p1.3.m3.1.1.2.2" xref="S5.I1.i3.p1.3.m3.1.1.2.2.cmml">x</mi><mi id="S5.I1.i3.p1.3.m3.1.1.2.3" xref="S5.I1.i3.p1.3.m3.1.1.2.3.cmml">i</mi></msub><mo id="S5.I1.i3.p1.3.m3.1.1.1" stretchy="false" xref="S5.I1.i3.p1.3.m3.1.1.1.cmml">→</mo><msub id="S5.I1.i3.p1.3.m3.1.1.3" xref="S5.I1.i3.p1.3.m3.1.1.3.cmml"><mi id="S5.I1.i3.p1.3.m3.1.1.3.2" xref="S5.I1.i3.p1.3.m3.1.1.3.2.cmml">y</mi><mi id="S5.I1.i3.p1.3.m3.1.1.3.3" xref="S5.I1.i3.p1.3.m3.1.1.3.3.cmml">i</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S5.I1.i3.p1.3.m3.1b"><apply id="S5.I1.i3.p1.3.m3.1.1.cmml" xref="S5.I1.i3.p1.3.m3.1.1"><ci id="S5.I1.i3.p1.3.m3.1.1.1.cmml" xref="S5.I1.i3.p1.3.m3.1.1.1">→</ci><apply id="S5.I1.i3.p1.3.m3.1.1.2.cmml" xref="S5.I1.i3.p1.3.m3.1.1.2"><csymbol cd="ambiguous" id="S5.I1.i3.p1.3.m3.1.1.2.1.cmml" xref="S5.I1.i3.p1.3.m3.1.1.2">subscript</csymbol><ci id="S5.I1.i3.p1.3.m3.1.1.2.2.cmml" xref="S5.I1.i3.p1.3.m3.1.1.2.2">𝑥</ci><ci id="S5.I1.i3.p1.3.m3.1.1.2.3.cmml" xref="S5.I1.i3.p1.3.m3.1.1.2.3">𝑖</ci></apply><apply id="S5.I1.i3.p1.3.m3.1.1.3.cmml" xref="S5.I1.i3.p1.3.m3.1.1.3"><csymbol cd="ambiguous" id="S5.I1.i3.p1.3.m3.1.1.3.1.cmml" xref="S5.I1.i3.p1.3.m3.1.1.3">subscript</csymbol><ci id="S5.I1.i3.p1.3.m3.1.1.3.2.cmml" xref="S5.I1.i3.p1.3.m3.1.1.3.2">𝑦</ci><ci id="S5.I1.i3.p1.3.m3.1.1.3.3.cmml" xref="S5.I1.i3.p1.3.m3.1.1.3.3">𝑖</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I1.i3.p1.3.m3.1c">x_{i}\rightarrow y_{i}</annotation><annotation encoding="application/x-llamapun" id="S5.I1.i3.p1.3.m3.1d">italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT → italic_y start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math>), and generating all the intermediate steps <math alttext="{\mathbf{z}}_{i}" class="ltx_Math" display="inline" id="S5.I1.i3.p1.4.m4.1"><semantics id="S5.I1.i3.p1.4.m4.1a"><msub id="S5.I1.i3.p1.4.m4.1.1" xref="S5.I1.i3.p1.4.m4.1.1.cmml"><mi id="S5.I1.i3.p1.4.m4.1.1.2" xref="S5.I1.i3.p1.4.m4.1.1.2.cmml">𝐳</mi><mi id="S5.I1.i3.p1.4.m4.1.1.3" xref="S5.I1.i3.p1.4.m4.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S5.I1.i3.p1.4.m4.1b"><apply id="S5.I1.i3.p1.4.m4.1.1.cmml" xref="S5.I1.i3.p1.4.m4.1.1"><csymbol cd="ambiguous" id="S5.I1.i3.p1.4.m4.1.1.1.cmml" xref="S5.I1.i3.p1.4.m4.1.1">subscript</csymbol><ci id="S5.I1.i3.p1.4.m4.1.1.2.cmml" xref="S5.I1.i3.p1.4.m4.1.1.2">𝐳</ci><ci id="S5.I1.i3.p1.4.m4.1.1.3.cmml" xref="S5.I1.i3.p1.4.m4.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I1.i3.p1.4.m4.1c">{\mathbf{z}}_{i}</annotation><annotation encoding="application/x-llamapun" id="S5.I1.i3.p1.4.m4.1d">bold_z start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> and the full article <math alttext="y_{i}" class="ltx_Math" display="inline" id="S5.I1.i3.p1.5.m5.1"><semantics id="S5.I1.i3.p1.5.m5.1a"><msub id="S5.I1.i3.p1.5.m5.1.1" xref="S5.I1.i3.p1.5.m5.1.1.cmml"><mi id="S5.I1.i3.p1.5.m5.1.1.2" xref="S5.I1.i3.p1.5.m5.1.1.2.cmml">y</mi><mi id="S5.I1.i3.p1.5.m5.1.1.3" xref="S5.I1.i3.p1.5.m5.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S5.I1.i3.p1.5.m5.1b"><apply id="S5.I1.i3.p1.5.m5.1.1.cmml" xref="S5.I1.i3.p1.5.m5.1.1"><csymbol cd="ambiguous" id="S5.I1.i3.p1.5.m5.1.1.1.cmml" xref="S5.I1.i3.p1.5.m5.1.1">subscript</csymbol><ci id="S5.I1.i3.p1.5.m5.1.1.2.cmml" xref="S5.I1.i3.p1.5.m5.1.1.2">𝑦</ci><ci id="S5.I1.i3.p1.5.m5.1.1.3.cmml" xref="S5.I1.i3.p1.5.m5.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I1.i3.p1.5.m5.1c">y_{i}</annotation><annotation encoding="application/x-llamapun" id="S5.I1.i3.p1.5.m5.1d">italic_y start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> from the input context <math alttext="x_{i}" class="ltx_Math" display="inline" id="S5.I1.i3.p1.6.m6.1"><semantics id="S5.I1.i3.p1.6.m6.1a"><msub id="S5.I1.i3.p1.6.m6.1.1" xref="S5.I1.i3.p1.6.m6.1.1.cmml"><mi id="S5.I1.i3.p1.6.m6.1.1.2" xref="S5.I1.i3.p1.6.m6.1.1.2.cmml">x</mi><mi id="S5.I1.i3.p1.6.m6.1.1.3" xref="S5.I1.i3.p1.6.m6.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S5.I1.i3.p1.6.m6.1b"><apply id="S5.I1.i3.p1.6.m6.1.1.cmml" xref="S5.I1.i3.p1.6.m6.1.1"><csymbol cd="ambiguous" id="S5.I1.i3.p1.6.m6.1.1.1.cmml" xref="S5.I1.i3.p1.6.m6.1.1">subscript</csymbol><ci id="S5.I1.i3.p1.6.m6.1.1.2.cmml" xref="S5.I1.i3.p1.6.m6.1.1.2">𝑥</ci><ci id="S5.I1.i3.p1.6.m6.1.1.3.cmml" xref="S5.I1.i3.p1.6.m6.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I1.i3.p1.6.m6.1c">x_{i}</annotation><annotation encoding="application/x-llamapun" id="S5.I1.i3.p1.6.m6.1d">italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> (<math alttext="x_{i}\rightarrow{\mathbf{z}}_{i}\oplus y_{i}" class="ltx_Math" display="inline" id="S5.I1.i3.p1.7.m7.1"><semantics id="S5.I1.i3.p1.7.m7.1a"><mrow id="S5.I1.i3.p1.7.m7.1.1" xref="S5.I1.i3.p1.7.m7.1.1.cmml"><msub id="S5.I1.i3.p1.7.m7.1.1.2" xref="S5.I1.i3.p1.7.m7.1.1.2.cmml"><mi id="S5.I1.i3.p1.7.m7.1.1.2.2" xref="S5.I1.i3.p1.7.m7.1.1.2.2.cmml">x</mi><mi id="S5.I1.i3.p1.7.m7.1.1.2.3" xref="S5.I1.i3.p1.7.m7.1.1.2.3.cmml">i</mi></msub><mo id="S5.I1.i3.p1.7.m7.1.1.1" stretchy="false" xref="S5.I1.i3.p1.7.m7.1.1.1.cmml">→</mo><mrow id="S5.I1.i3.p1.7.m7.1.1.3" xref="S5.I1.i3.p1.7.m7.1.1.3.cmml"><msub id="S5.I1.i3.p1.7.m7.1.1.3.2" xref="S5.I1.i3.p1.7.m7.1.1.3.2.cmml"><mi id="S5.I1.i3.p1.7.m7.1.1.3.2.2" xref="S5.I1.i3.p1.7.m7.1.1.3.2.2.cmml">𝐳</mi><mi id="S5.I1.i3.p1.7.m7.1.1.3.2.3" xref="S5.I1.i3.p1.7.m7.1.1.3.2.3.cmml">i</mi></msub><mo id="S5.I1.i3.p1.7.m7.1.1.3.1" xref="S5.I1.i3.p1.7.m7.1.1.3.1.cmml">⊕</mo><msub id="S5.I1.i3.p1.7.m7.1.1.3.3" xref="S5.I1.i3.p1.7.m7.1.1.3.3.cmml"><mi id="S5.I1.i3.p1.7.m7.1.1.3.3.2" xref="S5.I1.i3.p1.7.m7.1.1.3.3.2.cmml">y</mi><mi id="S5.I1.i3.p1.7.m7.1.1.3.3.3" xref="S5.I1.i3.p1.7.m7.1.1.3.3.3.cmml">i</mi></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.I1.i3.p1.7.m7.1b"><apply id="S5.I1.i3.p1.7.m7.1.1.cmml" xref="S5.I1.i3.p1.7.m7.1.1"><ci id="S5.I1.i3.p1.7.m7.1.1.1.cmml" xref="S5.I1.i3.p1.7.m7.1.1.1">→</ci><apply id="S5.I1.i3.p1.7.m7.1.1.2.cmml" xref="S5.I1.i3.p1.7.m7.1.1.2"><csymbol cd="ambiguous" id="S5.I1.i3.p1.7.m7.1.1.2.1.cmml" xref="S5.I1.i3.p1.7.m7.1.1.2">subscript</csymbol><ci id="S5.I1.i3.p1.7.m7.1.1.2.2.cmml" xref="S5.I1.i3.p1.7.m7.1.1.2.2">𝑥</ci><ci id="S5.I1.i3.p1.7.m7.1.1.2.3.cmml" xref="S5.I1.i3.p1.7.m7.1.1.2.3">𝑖</ci></apply><apply id="S5.I1.i3.p1.7.m7.1.1.3.cmml" xref="S5.I1.i3.p1.7.m7.1.1.3"><csymbol cd="latexml" id="S5.I1.i3.p1.7.m7.1.1.3.1.cmml" xref="S5.I1.i3.p1.7.m7.1.1.3.1">direct-sum</csymbol><apply id="S5.I1.i3.p1.7.m7.1.1.3.2.cmml" xref="S5.I1.i3.p1.7.m7.1.1.3.2"><csymbol cd="ambiguous" id="S5.I1.i3.p1.7.m7.1.1.3.2.1.cmml" xref="S5.I1.i3.p1.7.m7.1.1.3.2">subscript</csymbol><ci id="S5.I1.i3.p1.7.m7.1.1.3.2.2.cmml" xref="S5.I1.i3.p1.7.m7.1.1.3.2.2">𝐳</ci><ci id="S5.I1.i3.p1.7.m7.1.1.3.2.3.cmml" xref="S5.I1.i3.p1.7.m7.1.1.3.2.3">𝑖</ci></apply><apply id="S5.I1.i3.p1.7.m7.1.1.3.3.cmml" xref="S5.I1.i3.p1.7.m7.1.1.3.3"><csymbol cd="ambiguous" id="S5.I1.i3.p1.7.m7.1.1.3.3.1.cmml" xref="S5.I1.i3.p1.7.m7.1.1.3.3">subscript</csymbol><ci id="S5.I1.i3.p1.7.m7.1.1.3.3.2.cmml" xref="S5.I1.i3.p1.7.m7.1.1.3.3.2">𝑦</ci><ci id="S5.I1.i3.p1.7.m7.1.1.3.3.3.cmml" xref="S5.I1.i3.p1.7.m7.1.1.3.3.3">𝑖</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I1.i3.p1.7.m7.1c">x_{i}\rightarrow{\mathbf{z}}_{i}\oplus y_{i}</annotation><annotation encoding="application/x-llamapun" id="S5.I1.i3.p1.7.m7.1d">italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT → bold_z start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ⊕ italic_y start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math>). The instruction prompt from the two tasks are different to ensure the LLM can differentiate these tasks. See Section <a class="ltx_ref" href="https://arxiv.org/html/2410.06203v1#S4.SS3" title="4.3 Fine-tuning ‣ 4 Methodology ‣ Integrating Planning into Single-Turn Long-Form Text Generation"><span class="ltx_text ltx_ref_tag">4.3</span></a> for more details. During inference the LLM is prompted with input context <math alttext="x_{i}" class="ltx_Math" display="inline" id="S5.I1.i3.p1.8.m8.1"><semantics id="S5.I1.i3.p1.8.m8.1a"><msub id="S5.I1.i3.p1.8.m8.1.1" xref="S5.I1.i3.p1.8.m8.1.1.cmml"><mi id="S5.I1.i3.p1.8.m8.1.1.2" xref="S5.I1.i3.p1.8.m8.1.1.2.cmml">x</mi><mi id="S5.I1.i3.p1.8.m8.1.1.3" xref="S5.I1.i3.p1.8.m8.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S5.I1.i3.p1.8.m8.1b"><apply id="S5.I1.i3.p1.8.m8.1.1.cmml" xref="S5.I1.i3.p1.8.m8.1.1"><csymbol cd="ambiguous" id="S5.I1.i3.p1.8.m8.1.1.1.cmml" xref="S5.I1.i3.p1.8.m8.1.1">subscript</csymbol><ci id="S5.I1.i3.p1.8.m8.1.1.2.cmml" xref="S5.I1.i3.p1.8.m8.1.1.2">𝑥</ci><ci id="S5.I1.i3.p1.8.m8.1.1.3.cmml" xref="S5.I1.i3.p1.8.m8.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I1.i3.p1.8.m8.1c">x_{i}</annotation><annotation encoding="application/x-llamapun" id="S5.I1.i3.p1.8.m8.1d">italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> to only output the full article <math alttext="y_{i}" class="ltx_Math" display="inline" id="S5.I1.i3.p1.9.m9.1"><semantics id="S5.I1.i3.p1.9.m9.1a"><msub id="S5.I1.i3.p1.9.m9.1.1" xref="S5.I1.i3.p1.9.m9.1.1.cmml"><mi id="S5.I1.i3.p1.9.m9.1.1.2" xref="S5.I1.i3.p1.9.m9.1.1.2.cmml">y</mi><mi id="S5.I1.i3.p1.9.m9.1.1.3" xref="S5.I1.i3.p1.9.m9.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S5.I1.i3.p1.9.m9.1b"><apply id="S5.I1.i3.p1.9.m9.1.1.cmml" xref="S5.I1.i3.p1.9.m9.1.1"><csymbol cd="ambiguous" id="S5.I1.i3.p1.9.m9.1.1.1.cmml" xref="S5.I1.i3.p1.9.m9.1.1">subscript</csymbol><ci id="S5.I1.i3.p1.9.m9.1.1.2.cmml" xref="S5.I1.i3.p1.9.m9.1.1.2">𝑦</ci><ci id="S5.I1.i3.p1.9.m9.1.1.3.cmml" xref="S5.I1.i3.p1.9.m9.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I1.i3.p1.9.m9.1c">y_{i}</annotation><annotation encoding="application/x-llamapun" id="S5.I1.i3.p1.9.m9.1d">italic_y start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> (<math alttext="x_{i}\rightarrow y_{i}" class="ltx_Math" display="inline" id="S5.I1.i3.p1.10.m10.1"><semantics id="S5.I1.i3.p1.10.m10.1a"><mrow id="S5.I1.i3.p1.10.m10.1.1" xref="S5.I1.i3.p1.10.m10.1.1.cmml"><msub id="S5.I1.i3.p1.10.m10.1.1.2" xref="S5.I1.i3.p1.10.m10.1.1.2.cmml"><mi id="S5.I1.i3.p1.10.m10.1.1.2.2" xref="S5.I1.i3.p1.10.m10.1.1.2.2.cmml">x</mi><mi id="S5.I1.i3.p1.10.m10.1.1.2.3" xref="S5.I1.i3.p1.10.m10.1.1.2.3.cmml">i</mi></msub><mo id="S5.I1.i3.p1.10.m10.1.1.1" stretchy="false" xref="S5.I1.i3.p1.10.m10.1.1.1.cmml">→</mo><msub id="S5.I1.i3.p1.10.m10.1.1.3" xref="S5.I1.i3.p1.10.m10.1.1.3.cmml"><mi id="S5.I1.i3.p1.10.m10.1.1.3.2" xref="S5.I1.i3.p1.10.m10.1.1.3.2.cmml">y</mi><mi id="S5.I1.i3.p1.10.m10.1.1.3.3" xref="S5.I1.i3.p1.10.m10.1.1.3.3.cmml">i</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S5.I1.i3.p1.10.m10.1b"><apply id="S5.I1.i3.p1.10.m10.1.1.cmml" xref="S5.I1.i3.p1.10.m10.1.1"><ci id="S5.I1.i3.p1.10.m10.1.1.1.cmml" xref="S5.I1.i3.p1.10.m10.1.1.1">→</ci><apply id="S5.I1.i3.p1.10.m10.1.1.2.cmml" xref="S5.I1.i3.p1.10.m10.1.1.2"><csymbol cd="ambiguous" id="S5.I1.i3.p1.10.m10.1.1.2.1.cmml" xref="S5.I1.i3.p1.10.m10.1.1.2">subscript</csymbol><ci id="S5.I1.i3.p1.10.m10.1.1.2.2.cmml" xref="S5.I1.i3.p1.10.m10.1.1.2.2">𝑥</ci><ci id="S5.I1.i3.p1.10.m10.1.1.2.3.cmml" xref="S5.I1.i3.p1.10.m10.1.1.2.3">𝑖</ci></apply><apply id="S5.I1.i3.p1.10.m10.1.1.3.cmml" xref="S5.I1.i3.p1.10.m10.1.1.3"><csymbol cd="ambiguous" id="S5.I1.i3.p1.10.m10.1.1.3.1.cmml" xref="S5.I1.i3.p1.10.m10.1.1.3">subscript</csymbol><ci id="S5.I1.i3.p1.10.m10.1.1.3.2.cmml" xref="S5.I1.i3.p1.10.m10.1.1.3.2">𝑦</ci><ci id="S5.I1.i3.p1.10.m10.1.1.3.3.cmml" xref="S5.I1.i3.p1.10.m10.1.1.3.3">𝑖</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I1.i3.p1.10.m10.1c">x_{i}\rightarrow y_{i}</annotation><annotation encoding="application/x-llamapun" id="S5.I1.i3.p1.10.m10.1d">italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT → italic_y start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math>).</p> </div> </li> <li class="ltx_item" id="S5.I1.i4" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">4.</span> <div class="ltx_para" id="S5.I1.i4.p1"> <p class="ltx_p" id="S5.I1.i4.p1.4"><span class="ltx_text ltx_font_bold" id="S5.I1.i4.p1.4.1">Fine-tuning and inference with intermediate steps (FT w/ I, Output w/ I):</span> Similar to the fine-tuning with intermediate steps method above, except that during inference the LLM is prompted with input context <math alttext="x_{i}" class="ltx_Math" display="inline" id="S5.I1.i4.p1.1.m1.1"><semantics id="S5.I1.i4.p1.1.m1.1a"><msub id="S5.I1.i4.p1.1.m1.1.1" xref="S5.I1.i4.p1.1.m1.1.1.cmml"><mi id="S5.I1.i4.p1.1.m1.1.1.2" xref="S5.I1.i4.p1.1.m1.1.1.2.cmml">x</mi><mi id="S5.I1.i4.p1.1.m1.1.1.3" xref="S5.I1.i4.p1.1.m1.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S5.I1.i4.p1.1.m1.1b"><apply id="S5.I1.i4.p1.1.m1.1.1.cmml" xref="S5.I1.i4.p1.1.m1.1.1"><csymbol cd="ambiguous" id="S5.I1.i4.p1.1.m1.1.1.1.cmml" xref="S5.I1.i4.p1.1.m1.1.1">subscript</csymbol><ci id="S5.I1.i4.p1.1.m1.1.1.2.cmml" xref="S5.I1.i4.p1.1.m1.1.1.2">𝑥</ci><ci id="S5.I1.i4.p1.1.m1.1.1.3.cmml" xref="S5.I1.i4.p1.1.m1.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I1.i4.p1.1.m1.1c">x_{i}</annotation><annotation encoding="application/x-llamapun" id="S5.I1.i4.p1.1.m1.1d">italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> to generate both intermediate steps <math alttext="{\mathbf{z}}_{i}" class="ltx_Math" display="inline" id="S5.I1.i4.p1.2.m2.1"><semantics id="S5.I1.i4.p1.2.m2.1a"><msub id="S5.I1.i4.p1.2.m2.1.1" xref="S5.I1.i4.p1.2.m2.1.1.cmml"><mi id="S5.I1.i4.p1.2.m2.1.1.2" xref="S5.I1.i4.p1.2.m2.1.1.2.cmml">𝐳</mi><mi id="S5.I1.i4.p1.2.m2.1.1.3" xref="S5.I1.i4.p1.2.m2.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S5.I1.i4.p1.2.m2.1b"><apply id="S5.I1.i4.p1.2.m2.1.1.cmml" xref="S5.I1.i4.p1.2.m2.1.1"><csymbol cd="ambiguous" id="S5.I1.i4.p1.2.m2.1.1.1.cmml" xref="S5.I1.i4.p1.2.m2.1.1">subscript</csymbol><ci id="S5.I1.i4.p1.2.m2.1.1.2.cmml" xref="S5.I1.i4.p1.2.m2.1.1.2">𝐳</ci><ci id="S5.I1.i4.p1.2.m2.1.1.3.cmml" xref="S5.I1.i4.p1.2.m2.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I1.i4.p1.2.m2.1c">{\mathbf{z}}_{i}</annotation><annotation encoding="application/x-llamapun" id="S5.I1.i4.p1.2.m2.1d">bold_z start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> and the full article <math alttext="y_{i}" class="ltx_Math" display="inline" id="S5.I1.i4.p1.3.m3.1"><semantics id="S5.I1.i4.p1.3.m3.1a"><msub id="S5.I1.i4.p1.3.m3.1.1" xref="S5.I1.i4.p1.3.m3.1.1.cmml"><mi id="S5.I1.i4.p1.3.m3.1.1.2" xref="S5.I1.i4.p1.3.m3.1.1.2.cmml">y</mi><mi id="S5.I1.i4.p1.3.m3.1.1.3" xref="S5.I1.i4.p1.3.m3.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S5.I1.i4.p1.3.m3.1b"><apply id="S5.I1.i4.p1.3.m3.1.1.cmml" xref="S5.I1.i4.p1.3.m3.1.1"><csymbol cd="ambiguous" id="S5.I1.i4.p1.3.m3.1.1.1.cmml" xref="S5.I1.i4.p1.3.m3.1.1">subscript</csymbol><ci id="S5.I1.i4.p1.3.m3.1.1.2.cmml" xref="S5.I1.i4.p1.3.m3.1.1.2">𝑦</ci><ci id="S5.I1.i4.p1.3.m3.1.1.3.cmml" xref="S5.I1.i4.p1.3.m3.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I1.i4.p1.3.m3.1c">y_{i}</annotation><annotation encoding="application/x-llamapun" id="S5.I1.i4.p1.3.m3.1d">italic_y start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> (<math alttext="x_{i}\rightarrow{\mathbf{z}}_{i}\oplus y_{i}" class="ltx_Math" display="inline" id="S5.I1.i4.p1.4.m4.1"><semantics id="S5.I1.i4.p1.4.m4.1a"><mrow id="S5.I1.i4.p1.4.m4.1.1" xref="S5.I1.i4.p1.4.m4.1.1.cmml"><msub id="S5.I1.i4.p1.4.m4.1.1.2" 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xref="S5.I1.i4.p1.4.m4.1.1.3.2.3">𝑖</ci></apply><apply id="S5.I1.i4.p1.4.m4.1.1.3.3.cmml" xref="S5.I1.i4.p1.4.m4.1.1.3.3"><csymbol cd="ambiguous" id="S5.I1.i4.p1.4.m4.1.1.3.3.1.cmml" xref="S5.I1.i4.p1.4.m4.1.1.3.3">subscript</csymbol><ci id="S5.I1.i4.p1.4.m4.1.1.3.3.2.cmml" xref="S5.I1.i4.p1.4.m4.1.1.3.3.2">𝑦</ci><ci id="S5.I1.i4.p1.4.m4.1.1.3.3.3.cmml" xref="S5.I1.i4.p1.4.m4.1.1.3.3.3">𝑖</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I1.i4.p1.4.m4.1c">x_{i}\rightarrow{\mathbf{z}}_{i}\oplus y_{i}</annotation><annotation encoding="application/x-llamapun" id="S5.I1.i4.p1.4.m4.1d">italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT → bold_z start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ⊕ italic_y start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math>). Only the full article is extracted from the generated output for evaluation.</p> </div> </li> </ol> </div> <figure class="ltx_table" id="S5.T1"> <figcaption class="ltx_caption"><span class="ltx_tag ltx_tag_table">Table 1: </span>Performance of different methods with various training and prompt setups</figcaption> <p class="ltx_p ltx_align_center" id="S5.T1.4"><span class="ltx_text" id="S5.T1.4.4"> <span class="ltx_tabular ltx_guessed_headers ltx_align_middle" id="S5.T1.4.4.4"> <span class="ltx_thead"> <span class="ltx_tr" id="S5.T1.4.4.4.4"> <span class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_th_row ltx_border_r ltx_border_t" id="S5.T1.4.4.4.4.5"><span class="ltx_text ltx_font_bold" id="S5.T1.4.4.4.4.5.1">Dataset</span></span> <span class="ltx_td ltx_align_left ltx_th ltx_th_column ltx_th_row ltx_border_r ltx_border_t" id="S5.T1.4.4.4.4.6"><span class="ltx_text ltx_font_bold" id="S5.T1.4.4.4.4.6.1">Methods</span></span> <span class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_r ltx_border_t" id="S5.T1.1.1.1.1.1"><span class="ltx_text ltx_font_bold" id="S5.T1.1.1.1.1.1.1">R1<sub class="ltx_sub" id="S5.T1.1.1.1.1.1.1.1"><span class="ltx_text ltx_font_medium ltx_font_italic" id="S5.T1.1.1.1.1.1.1.1.1">F1</span></sub></span></span> <span class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_r ltx_border_t" id="S5.T1.2.2.2.2.2"><span class="ltx_text ltx_font_bold" id="S5.T1.2.2.2.2.2.1">R2<sub class="ltx_sub" id="S5.T1.2.2.2.2.2.1.1"><span class="ltx_text ltx_font_medium ltx_font_italic" id="S5.T1.2.2.2.2.2.1.1.1">F1</span></sub></span></span> <span class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_r ltx_border_t" id="S5.T1.3.3.3.3.3"><span class="ltx_text ltx_font_bold" id="S5.T1.3.3.3.3.3.1">RL<sub class="ltx_sub" id="S5.T1.3.3.3.3.3.1.1"><span class="ltx_text ltx_font_medium ltx_font_italic" id="S5.T1.3.3.3.3.3.1.1.1">F1</span></sub></span></span> <span class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_t" id="S5.T1.4.4.4.4.4"><span class="ltx_text ltx_font_bold" id="S5.T1.4.4.4.4.4.1">RLsum<sub class="ltx_sub" id="S5.T1.4.4.4.4.4.1.1"><span class="ltx_text ltx_font_medium ltx_font_italic" id="S5.T1.4.4.4.4.4.1.1.1">F1</span></sub></span></span></span> </span> <span class="ltx_tbody"> <span class="ltx_tr" id="S5.T1.4.4.4.5.1"> <span class="ltx_td ltx_align_center ltx_th ltx_th_row ltx_border_r ltx_border_t ltx_rowspan ltx_rowspan_4" id="S5.T1.4.4.4.5.1.1"><span class="ltx_text" id="S5.T1.4.4.4.5.1.1.1">SciNews</span></span> <span class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_r ltx_border_t" id="S5.T1.4.4.4.5.1.2">ZS</span> <span class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S5.T1.4.4.4.5.1.3">37.16</span> <span class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S5.T1.4.4.4.5.1.4">10.03</span> <span class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S5.T1.4.4.4.5.1.5">16.30</span> <span class="ltx_td ltx_align_center ltx_border_t" id="S5.T1.4.4.4.5.1.6">35.03</span></span> <span class="ltx_tr" id="S5.T1.4.4.4.6.2"> <span class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_r" id="S5.T1.4.4.4.6.2.1">FT w/o I</span> <span class="ltx_td ltx_align_center ltx_border_r" id="S5.T1.4.4.4.6.2.2">46.66</span> <span class="ltx_td ltx_align_center ltx_border_r" id="S5.T1.4.4.4.6.2.3">13.39</span> <span class="ltx_td ltx_align_center ltx_border_r" id="S5.T1.4.4.4.6.2.4">19.26</span> <span class="ltx_td ltx_align_center" id="S5.T1.4.4.4.6.2.5">44.14</span></span> <span class="ltx_tr" id="S5.T1.4.4.4.7.3"> <span class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_r" id="S5.T1.4.4.4.7.3.1">FT w/ I</span> <span class="ltx_td ltx_align_center ltx_border_r" id="S5.T1.4.4.4.7.3.2">48.63</span> <span class="ltx_td ltx_align_center ltx_border_r" id="S5.T1.4.4.4.7.3.3">14.39</span> <span class="ltx_td ltx_align_center ltx_border_r" id="S5.T1.4.4.4.7.3.4">19.57</span> <span class="ltx_td ltx_align_center" id="S5.T1.4.4.4.7.3.5">45.92</span></span> <span class="ltx_tr" id="S5.T1.4.4.4.8.4"> <span class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_r" id="S5.T1.4.4.4.8.4.1">FT w/ I, Output w/ I</span> <span class="ltx_td ltx_align_center ltx_border_r" id="S5.T1.4.4.4.8.4.2"><span class="ltx_text ltx_font_bold" id="S5.T1.4.4.4.8.4.2.1">49.39</span></span> <span class="ltx_td ltx_align_center ltx_border_r" id="S5.T1.4.4.4.8.4.3"><span class="ltx_text ltx_font_bold" id="S5.T1.4.4.4.8.4.3.1">14.69</span></span> <span class="ltx_td ltx_align_center ltx_border_r" id="S5.T1.4.4.4.8.4.4"><span class="ltx_text ltx_font_bold" id="S5.T1.4.4.4.8.4.4.1">19.68</span></span> <span class="ltx_td ltx_align_center" id="S5.T1.4.4.4.8.4.5"><span class="ltx_text ltx_font_bold" id="S5.T1.4.4.4.8.4.5.1">46.61</span></span></span> <span class="ltx_tr" id="S5.T1.4.4.4.9.5"> <span class="ltx_td ltx_align_center ltx_th ltx_th_row ltx_border_b ltx_border_r ltx_border_t ltx_rowspan ltx_rowspan_4" id="S5.T1.4.4.4.9.5.1"><span class="ltx_text" id="S5.T1.4.4.4.9.5.1.1">Wikipedia</span></span> <span class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_r ltx_border_t" id="S5.T1.4.4.4.9.5.2">ZS</span> <span class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S5.T1.4.4.4.9.5.3">28.51</span> <span class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S5.T1.4.4.4.9.5.4">10.65</span> <span class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S5.T1.4.4.4.9.5.5">13.91</span> <span class="ltx_td ltx_align_center ltx_border_t" id="S5.T1.4.4.4.9.5.6">27.52</span></span> <span class="ltx_tr" id="S5.T1.4.4.4.10.6"> <span class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_r" id="S5.T1.4.4.4.10.6.1">FT w/o I</span> <span class="ltx_td ltx_align_center ltx_border_r" id="S5.T1.4.4.4.10.6.2">42.61</span> <span class="ltx_td ltx_align_center ltx_border_r" id="S5.T1.4.4.4.10.6.3">13.69</span> <span class="ltx_td ltx_align_center ltx_border_r" id="S5.T1.4.4.4.10.6.4">16.30</span> <span class="ltx_td ltx_align_center" id="S5.T1.4.4.4.10.6.5">41.28</span></span> <span class="ltx_tr" id="S5.T1.4.4.4.11.7"> <span class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_r" id="S5.T1.4.4.4.11.7.1">FT w/ I</span> <span class="ltx_td ltx_align_center ltx_border_r" id="S5.T1.4.4.4.11.7.2">45.65</span> <span class="ltx_td ltx_align_center ltx_border_r" id="S5.T1.4.4.4.11.7.3">15.21</span> <span class="ltx_td ltx_align_center ltx_border_r" id="S5.T1.4.4.4.11.7.4">17.44</span> <span class="ltx_td ltx_align_center" id="S5.T1.4.4.4.11.7.5">44.23</span></span> <span class="ltx_tr" id="S5.T1.4.4.4.12.8"> <span class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_b ltx_border_r" id="S5.T1.4.4.4.12.8.1">FT w/ I, Output w/ I</span> <span class="ltx_td ltx_align_center ltx_border_b ltx_border_r" id="S5.T1.4.4.4.12.8.2"><span class="ltx_text ltx_font_bold" id="S5.T1.4.4.4.12.8.2.1">47.07</span></span> <span class="ltx_td ltx_align_center ltx_border_b ltx_border_r" id="S5.T1.4.4.4.12.8.3"><span class="ltx_text ltx_font_bold" id="S5.T1.4.4.4.12.8.3.1">15.72</span></span> <span class="ltx_td ltx_align_center ltx_border_b ltx_border_r" id="S5.T1.4.4.4.12.8.4"><span class="ltx_text ltx_font_bold" id="S5.T1.4.4.4.12.8.4.1">17.72</span></span> <span class="ltx_td ltx_align_center ltx_border_b" id="S5.T1.4.4.4.12.8.5"><span class="ltx_text ltx_font_bold" id="S5.T1.4.4.4.12.8.5.1">45.67</span></span></span> </span> </span></span></p> </figure> <div class="ltx_para" id="S5.SS4.SSS0.Px1.p3"> <p class="ltx_p" id="S5.SS4.SSS0.Px1.p3.1">Compared with the zero-shot baseline ZS, fine-tuning LLM, even without the intermediate steps, significantly improved the ROUGE scores. Moreover, when LLM is fine-tuned with the mixture training data which incorporated the task of generating intermediate steps, the performance is further improved on both datasets. Specifically, on both datasets, the R1 score improved by more than +2.7-4.4% when compared to the FT w/o I baseline, the R2 score by more than +1.3-2.0%, and the RLsum score by more than +2.5-4.4%.</p> </div> <div class="ltx_para" id="S5.SS4.SSS0.Px1.p4"> <p class="ltx_p" id="S5.SS4.SSS0.Px1.p4.1">These results demonstrate that the inclusion of the generating intermediate steps task during fine-tuning significantly enhances long-form text generation by LLMs. The improvements across various metrics and both datasets highlight the robustness of the proposed methods.</p> </div> <div class="ltx_para" id="S5.SS4.SSS0.Px1.p5"> <p class="ltx_p" id="S5.SS4.SSS0.Px1.p5.1">We also observe that prompting the LLM to output the intermediate steps <math alttext="{\mathbf{z}}_{i}" class="ltx_Math" display="inline" id="S5.SS4.SSS0.Px1.p5.1.m1.1"><semantics id="S5.SS4.SSS0.Px1.p5.1.m1.1a"><msub id="S5.SS4.SSS0.Px1.p5.1.m1.1.1" xref="S5.SS4.SSS0.Px1.p5.1.m1.1.1.cmml"><mi id="S5.SS4.SSS0.Px1.p5.1.m1.1.1.2" xref="S5.SS4.SSS0.Px1.p5.1.m1.1.1.2.cmml">𝐳</mi><mi id="S5.SS4.SSS0.Px1.p5.1.m1.1.1.3" xref="S5.SS4.SSS0.Px1.p5.1.m1.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S5.SS4.SSS0.Px1.p5.1.m1.1b"><apply id="S5.SS4.SSS0.Px1.p5.1.m1.1.1.cmml" xref="S5.SS4.SSS0.Px1.p5.1.m1.1.1"><csymbol cd="ambiguous" id="S5.SS4.SSS0.Px1.p5.1.m1.1.1.1.cmml" xref="S5.SS4.SSS0.Px1.p5.1.m1.1.1">subscript</csymbol><ci id="S5.SS4.SSS0.Px1.p5.1.m1.1.1.2.cmml" xref="S5.SS4.SSS0.Px1.p5.1.m1.1.1.2">𝐳</ci><ci id="S5.SS4.SSS0.Px1.p5.1.m1.1.1.3.cmml" xref="S5.SS4.SSS0.Px1.p5.1.m1.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS4.SSS0.Px1.p5.1.m1.1c">{\mathbf{z}}_{i}</annotation><annotation encoding="application/x-llamapun" id="S5.SS4.SSS0.Px1.p5.1.m1.1d">bold_z start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> during inference (FT w/I, Output W/I) achieves better performance when prompting the LLM to only output the article (FT w/I). This is expected as explicitly writing down the planning process during the inference can help the model to generate more structured article, similar to the usage of chain-of-thought prompting for reasoning tasks <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2410.06203v1#bib.bib26" title="">26</a>]</cite>.</p> </div> </section> <section class="ltx_paragraph" id="S5.SS4.SSS0.Px2"> <h4 class="ltx_title ltx_title_paragraph">Comparison of different training and inference recipes.</h4> <div class="ltx_para" id="S5.SS4.SSS0.Px2.p1"> <p class="ltx_p" id="S5.SS4.SSS0.Px2.p1.5">We also conduct a comparison to understand whether different combinations of training data mixtures and inference tasks lead to different effectiveness. Each mixture contains one or multiple training tasks: directly generating the article from input context <math alttext="x_{i}\rightarrow y_{i}" class="ltx_Math" display="inline" id="S5.SS4.SSS0.Px2.p1.1.m1.1"><semantics id="S5.SS4.SSS0.Px2.p1.1.m1.1a"><mrow id="S5.SS4.SSS0.Px2.p1.1.m1.1.1" xref="S5.SS4.SSS0.Px2.p1.1.m1.1.1.cmml"><msub id="S5.SS4.SSS0.Px2.p1.1.m1.1.1.2" xref="S5.SS4.SSS0.Px2.p1.1.m1.1.1.2.cmml"><mi id="S5.SS4.SSS0.Px2.p1.1.m1.1.1.2.2" xref="S5.SS4.SSS0.Px2.p1.1.m1.1.1.2.2.cmml">x</mi><mi id="S5.SS4.SSS0.Px2.p1.1.m1.1.1.2.3" xref="S5.SS4.SSS0.Px2.p1.1.m1.1.1.2.3.cmml">i</mi></msub><mo id="S5.SS4.SSS0.Px2.p1.1.m1.1.1.1" stretchy="false" xref="S5.SS4.SSS0.Px2.p1.1.m1.1.1.1.cmml">→</mo><msub id="S5.SS4.SSS0.Px2.p1.1.m1.1.1.3" xref="S5.SS4.SSS0.Px2.p1.1.m1.1.1.3.cmml"><mi id="S5.SS4.SSS0.Px2.p1.1.m1.1.1.3.2" xref="S5.SS4.SSS0.Px2.p1.1.m1.1.1.3.2.cmml">y</mi><mi id="S5.SS4.SSS0.Px2.p1.1.m1.1.1.3.3" xref="S5.SS4.SSS0.Px2.p1.1.m1.1.1.3.3.cmml">i</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S5.SS4.SSS0.Px2.p1.1.m1.1b"><apply id="S5.SS4.SSS0.Px2.p1.1.m1.1.1.cmml" xref="S5.SS4.SSS0.Px2.p1.1.m1.1.1"><ci id="S5.SS4.SSS0.Px2.p1.1.m1.1.1.1.cmml" xref="S5.SS4.SSS0.Px2.p1.1.m1.1.1.1">→</ci><apply id="S5.SS4.SSS0.Px2.p1.1.m1.1.1.2.cmml" xref="S5.SS4.SSS0.Px2.p1.1.m1.1.1.2"><csymbol cd="ambiguous" id="S5.SS4.SSS0.Px2.p1.1.m1.1.1.2.1.cmml" xref="S5.SS4.SSS0.Px2.p1.1.m1.1.1.2">subscript</csymbol><ci id="S5.SS4.SSS0.Px2.p1.1.m1.1.1.2.2.cmml" xref="S5.SS4.SSS0.Px2.p1.1.m1.1.1.2.2">𝑥</ci><ci id="S5.SS4.SSS0.Px2.p1.1.m1.1.1.2.3.cmml" xref="S5.SS4.SSS0.Px2.p1.1.m1.1.1.2.3">𝑖</ci></apply><apply id="S5.SS4.SSS0.Px2.p1.1.m1.1.1.3.cmml" xref="S5.SS4.SSS0.Px2.p1.1.m1.1.1.3"><csymbol cd="ambiguous" id="S5.SS4.SSS0.Px2.p1.1.m1.1.1.3.1.cmml" xref="S5.SS4.SSS0.Px2.p1.1.m1.1.1.3">subscript</csymbol><ci id="S5.SS4.SSS0.Px2.p1.1.m1.1.1.3.2.cmml" xref="S5.SS4.SSS0.Px2.p1.1.m1.1.1.3.2">𝑦</ci><ci id="S5.SS4.SSS0.Px2.p1.1.m1.1.1.3.3.cmml" xref="S5.SS4.SSS0.Px2.p1.1.m1.1.1.3.3">𝑖</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS4.SSS0.Px2.p1.1.m1.1c">x_{i}\rightarrow y_{i}</annotation><annotation encoding="application/x-llamapun" id="S5.SS4.SSS0.Px2.p1.1.m1.1d">italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT → italic_y start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math>; generating both the intermediate steps and the article from the input context <math alttext="x_{i}\rightarrow{\mathbf{z}}_{i}\oplus y_{i}" class="ltx_Math" display="inline" id="S5.SS4.SSS0.Px2.p1.2.m2.1"><semantics id="S5.SS4.SSS0.Px2.p1.2.m2.1a"><mrow id="S5.SS4.SSS0.Px2.p1.2.m2.1.1" xref="S5.SS4.SSS0.Px2.p1.2.m2.1.1.cmml"><msub id="S5.SS4.SSS0.Px2.p1.2.m2.1.1.2" xref="S5.SS4.SSS0.Px2.p1.2.m2.1.1.2.cmml"><mi id="S5.SS4.SSS0.Px2.p1.2.m2.1.1.2.2" xref="S5.SS4.SSS0.Px2.p1.2.m2.1.1.2.2.cmml">x</mi><mi id="S5.SS4.SSS0.Px2.p1.2.m2.1.1.2.3" xref="S5.SS4.SSS0.Px2.p1.2.m2.1.1.2.3.cmml">i</mi></msub><mo id="S5.SS4.SSS0.Px2.p1.2.m2.1.1.1" stretchy="false" xref="S5.SS4.SSS0.Px2.p1.2.m2.1.1.1.cmml">→</mo><mrow id="S5.SS4.SSS0.Px2.p1.2.m2.1.1.3" xref="S5.SS4.SSS0.Px2.p1.2.m2.1.1.3.cmml"><msub id="S5.SS4.SSS0.Px2.p1.2.m2.1.1.3.2" xref="S5.SS4.SSS0.Px2.p1.2.m2.1.1.3.2.cmml"><mi id="S5.SS4.SSS0.Px2.p1.2.m2.1.1.3.2.2" xref="S5.SS4.SSS0.Px2.p1.2.m2.1.1.3.2.2.cmml">𝐳</mi><mi id="S5.SS4.SSS0.Px2.p1.2.m2.1.1.3.2.3" xref="S5.SS4.SSS0.Px2.p1.2.m2.1.1.3.2.3.cmml">i</mi></msub><mo id="S5.SS4.SSS0.Px2.p1.2.m2.1.1.3.1" xref="S5.SS4.SSS0.Px2.p1.2.m2.1.1.3.1.cmml">⊕</mo><msub id="S5.SS4.SSS0.Px2.p1.2.m2.1.1.3.3" xref="S5.SS4.SSS0.Px2.p1.2.m2.1.1.3.3.cmml"><mi id="S5.SS4.SSS0.Px2.p1.2.m2.1.1.3.3.2" xref="S5.SS4.SSS0.Px2.p1.2.m2.1.1.3.3.2.cmml">y</mi><mi id="S5.SS4.SSS0.Px2.p1.2.m2.1.1.3.3.3" xref="S5.SS4.SSS0.Px2.p1.2.m2.1.1.3.3.3.cmml">i</mi></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.SS4.SSS0.Px2.p1.2.m2.1b"><apply id="S5.SS4.SSS0.Px2.p1.2.m2.1.1.cmml" xref="S5.SS4.SSS0.Px2.p1.2.m2.1.1"><ci id="S5.SS4.SSS0.Px2.p1.2.m2.1.1.1.cmml" xref="S5.SS4.SSS0.Px2.p1.2.m2.1.1.1">→</ci><apply id="S5.SS4.SSS0.Px2.p1.2.m2.1.1.2.cmml" xref="S5.SS4.SSS0.Px2.p1.2.m2.1.1.2"><csymbol cd="ambiguous" id="S5.SS4.SSS0.Px2.p1.2.m2.1.1.2.1.cmml" xref="S5.SS4.SSS0.Px2.p1.2.m2.1.1.2">subscript</csymbol><ci id="S5.SS4.SSS0.Px2.p1.2.m2.1.1.2.2.cmml" xref="S5.SS4.SSS0.Px2.p1.2.m2.1.1.2.2">𝑥</ci><ci id="S5.SS4.SSS0.Px2.p1.2.m2.1.1.2.3.cmml" xref="S5.SS4.SSS0.Px2.p1.2.m2.1.1.2.3">𝑖</ci></apply><apply id="S5.SS4.SSS0.Px2.p1.2.m2.1.1.3.cmml" xref="S5.SS4.SSS0.Px2.p1.2.m2.1.1.3"><csymbol cd="latexml" id="S5.SS4.SSS0.Px2.p1.2.m2.1.1.3.1.cmml" xref="S5.SS4.SSS0.Px2.p1.2.m2.1.1.3.1">direct-sum</csymbol><apply id="S5.SS4.SSS0.Px2.p1.2.m2.1.1.3.2.cmml" xref="S5.SS4.SSS0.Px2.p1.2.m2.1.1.3.2"><csymbol cd="ambiguous" id="S5.SS4.SSS0.Px2.p1.2.m2.1.1.3.2.1.cmml" xref="S5.SS4.SSS0.Px2.p1.2.m2.1.1.3.2">subscript</csymbol><ci id="S5.SS4.SSS0.Px2.p1.2.m2.1.1.3.2.2.cmml" xref="S5.SS4.SSS0.Px2.p1.2.m2.1.1.3.2.2">𝐳</ci><ci id="S5.SS4.SSS0.Px2.p1.2.m2.1.1.3.2.3.cmml" xref="S5.SS4.SSS0.Px2.p1.2.m2.1.1.3.2.3">𝑖</ci></apply><apply id="S5.SS4.SSS0.Px2.p1.2.m2.1.1.3.3.cmml" xref="S5.SS4.SSS0.Px2.p1.2.m2.1.1.3.3"><csymbol cd="ambiguous" id="S5.SS4.SSS0.Px2.p1.2.m2.1.1.3.3.1.cmml" xref="S5.SS4.SSS0.Px2.p1.2.m2.1.1.3.3">subscript</csymbol><ci id="S5.SS4.SSS0.Px2.p1.2.m2.1.1.3.3.2.cmml" xref="S5.SS4.SSS0.Px2.p1.2.m2.1.1.3.3.2">𝑦</ci><ci id="S5.SS4.SSS0.Px2.p1.2.m2.1.1.3.3.3.cmml" xref="S5.SS4.SSS0.Px2.p1.2.m2.1.1.3.3.3">𝑖</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS4.SSS0.Px2.p1.2.m2.1c">x_{i}\rightarrow{\mathbf{z}}_{i}\oplus y_{i}</annotation><annotation encoding="application/x-llamapun" id="S5.SS4.SSS0.Px2.p1.2.m2.1d">italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT → bold_z start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ⊕ italic_y start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math>; and generating the article from the input context and the intermediate steps: <math alttext="x_{i}\oplus{\mathbf{z}}_{i}\rightarrow y_{i}" class="ltx_Math" display="inline" id="S5.SS4.SSS0.Px2.p1.3.m3.1"><semantics id="S5.SS4.SSS0.Px2.p1.3.m3.1a"><mrow id="S5.SS4.SSS0.Px2.p1.3.m3.1.1" xref="S5.SS4.SSS0.Px2.p1.3.m3.1.1.cmml"><mrow id="S5.SS4.SSS0.Px2.p1.3.m3.1.1.2" xref="S5.SS4.SSS0.Px2.p1.3.m3.1.1.2.cmml"><msub id="S5.SS4.SSS0.Px2.p1.3.m3.1.1.2.2" xref="S5.SS4.SSS0.Px2.p1.3.m3.1.1.2.2.cmml"><mi id="S5.SS4.SSS0.Px2.p1.3.m3.1.1.2.2.2" xref="S5.SS4.SSS0.Px2.p1.3.m3.1.1.2.2.2.cmml">x</mi><mi id="S5.SS4.SSS0.Px2.p1.3.m3.1.1.2.2.3" xref="S5.SS4.SSS0.Px2.p1.3.m3.1.1.2.2.3.cmml">i</mi></msub><mo id="S5.SS4.SSS0.Px2.p1.3.m3.1.1.2.1" xref="S5.SS4.SSS0.Px2.p1.3.m3.1.1.2.1.cmml">⊕</mo><msub id="S5.SS4.SSS0.Px2.p1.3.m3.1.1.2.3" xref="S5.SS4.SSS0.Px2.p1.3.m3.1.1.2.3.cmml"><mi id="S5.SS4.SSS0.Px2.p1.3.m3.1.1.2.3.2" xref="S5.SS4.SSS0.Px2.p1.3.m3.1.1.2.3.2.cmml">𝐳</mi><mi id="S5.SS4.SSS0.Px2.p1.3.m3.1.1.2.3.3" xref="S5.SS4.SSS0.Px2.p1.3.m3.1.1.2.3.3.cmml">i</mi></msub></mrow><mo id="S5.SS4.SSS0.Px2.p1.3.m3.1.1.1" stretchy="false" xref="S5.SS4.SSS0.Px2.p1.3.m3.1.1.1.cmml">→</mo><msub id="S5.SS4.SSS0.Px2.p1.3.m3.1.1.3" xref="S5.SS4.SSS0.Px2.p1.3.m3.1.1.3.cmml"><mi id="S5.SS4.SSS0.Px2.p1.3.m3.1.1.3.2" xref="S5.SS4.SSS0.Px2.p1.3.m3.1.1.3.2.cmml">y</mi><mi id="S5.SS4.SSS0.Px2.p1.3.m3.1.1.3.3" xref="S5.SS4.SSS0.Px2.p1.3.m3.1.1.3.3.cmml">i</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S5.SS4.SSS0.Px2.p1.3.m3.1b"><apply id="S5.SS4.SSS0.Px2.p1.3.m3.1.1.cmml" xref="S5.SS4.SSS0.Px2.p1.3.m3.1.1"><ci id="S5.SS4.SSS0.Px2.p1.3.m3.1.1.1.cmml" xref="S5.SS4.SSS0.Px2.p1.3.m3.1.1.1">→</ci><apply id="S5.SS4.SSS0.Px2.p1.3.m3.1.1.2.cmml" xref="S5.SS4.SSS0.Px2.p1.3.m3.1.1.2"><csymbol cd="latexml" id="S5.SS4.SSS0.Px2.p1.3.m3.1.1.2.1.cmml" xref="S5.SS4.SSS0.Px2.p1.3.m3.1.1.2.1">direct-sum</csymbol><apply id="S5.SS4.SSS0.Px2.p1.3.m3.1.1.2.2.cmml" xref="S5.SS4.SSS0.Px2.p1.3.m3.1.1.2.2"><csymbol cd="ambiguous" id="S5.SS4.SSS0.Px2.p1.3.m3.1.1.2.2.1.cmml" xref="S5.SS4.SSS0.Px2.p1.3.m3.1.1.2.2">subscript</csymbol><ci id="S5.SS4.SSS0.Px2.p1.3.m3.1.1.2.2.2.cmml" xref="S5.SS4.SSS0.Px2.p1.3.m3.1.1.2.2.2">𝑥</ci><ci id="S5.SS4.SSS0.Px2.p1.3.m3.1.1.2.2.3.cmml" xref="S5.SS4.SSS0.Px2.p1.3.m3.1.1.2.2.3">𝑖</ci></apply><apply id="S5.SS4.SSS0.Px2.p1.3.m3.1.1.2.3.cmml" xref="S5.SS4.SSS0.Px2.p1.3.m3.1.1.2.3"><csymbol cd="ambiguous" id="S5.SS4.SSS0.Px2.p1.3.m3.1.1.2.3.1.cmml" xref="S5.SS4.SSS0.Px2.p1.3.m3.1.1.2.3">subscript</csymbol><ci id="S5.SS4.SSS0.Px2.p1.3.m3.1.1.2.3.2.cmml" xref="S5.SS4.SSS0.Px2.p1.3.m3.1.1.2.3.2">𝐳</ci><ci id="S5.SS4.SSS0.Px2.p1.3.m3.1.1.2.3.3.cmml" xref="S5.SS4.SSS0.Px2.p1.3.m3.1.1.2.3.3">𝑖</ci></apply></apply><apply id="S5.SS4.SSS0.Px2.p1.3.m3.1.1.3.cmml" xref="S5.SS4.SSS0.Px2.p1.3.m3.1.1.3"><csymbol cd="ambiguous" id="S5.SS4.SSS0.Px2.p1.3.m3.1.1.3.1.cmml" xref="S5.SS4.SSS0.Px2.p1.3.m3.1.1.3">subscript</csymbol><ci id="S5.SS4.SSS0.Px2.p1.3.m3.1.1.3.2.cmml" xref="S5.SS4.SSS0.Px2.p1.3.m3.1.1.3.2">𝑦</ci><ci id="S5.SS4.SSS0.Px2.p1.3.m3.1.1.3.3.cmml" xref="S5.SS4.SSS0.Px2.p1.3.m3.1.1.3.3">𝑖</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS4.SSS0.Px2.p1.3.m3.1c">x_{i}\oplus{\mathbf{z}}_{i}\rightarrow y_{i}</annotation><annotation encoding="application/x-llamapun" id="S5.SS4.SSS0.Px2.p1.3.m3.1d">italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ⊕ bold_z start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT → italic_y start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math>. We also test different inference tasks, including only prompting the LLM to output the entire article (<math alttext="x_{i}\rightarrow y_{i}" class="ltx_Math" display="inline" id="S5.SS4.SSS0.Px2.p1.4.m4.1"><semantics id="S5.SS4.SSS0.Px2.p1.4.m4.1a"><mrow id="S5.SS4.SSS0.Px2.p1.4.m4.1.1" xref="S5.SS4.SSS0.Px2.p1.4.m4.1.1.cmml"><msub id="S5.SS4.SSS0.Px2.p1.4.m4.1.1.2" xref="S5.SS4.SSS0.Px2.p1.4.m4.1.1.2.cmml"><mi id="S5.SS4.SSS0.Px2.p1.4.m4.1.1.2.2" xref="S5.SS4.SSS0.Px2.p1.4.m4.1.1.2.2.cmml">x</mi><mi id="S5.SS4.SSS0.Px2.p1.4.m4.1.1.2.3" xref="S5.SS4.SSS0.Px2.p1.4.m4.1.1.2.3.cmml">i</mi></msub><mo id="S5.SS4.SSS0.Px2.p1.4.m4.1.1.1" stretchy="false" xref="S5.SS4.SSS0.Px2.p1.4.m4.1.1.1.cmml">→</mo><msub id="S5.SS4.SSS0.Px2.p1.4.m4.1.1.3" xref="S5.SS4.SSS0.Px2.p1.4.m4.1.1.3.cmml"><mi id="S5.SS4.SSS0.Px2.p1.4.m4.1.1.3.2" xref="S5.SS4.SSS0.Px2.p1.4.m4.1.1.3.2.cmml">y</mi><mi id="S5.SS4.SSS0.Px2.p1.4.m4.1.1.3.3" xref="S5.SS4.SSS0.Px2.p1.4.m4.1.1.3.3.cmml">i</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S5.SS4.SSS0.Px2.p1.4.m4.1b"><apply id="S5.SS4.SSS0.Px2.p1.4.m4.1.1.cmml" xref="S5.SS4.SSS0.Px2.p1.4.m4.1.1"><ci id="S5.SS4.SSS0.Px2.p1.4.m4.1.1.1.cmml" xref="S5.SS4.SSS0.Px2.p1.4.m4.1.1.1">→</ci><apply id="S5.SS4.SSS0.Px2.p1.4.m4.1.1.2.cmml" xref="S5.SS4.SSS0.Px2.p1.4.m4.1.1.2"><csymbol cd="ambiguous" id="S5.SS4.SSS0.Px2.p1.4.m4.1.1.2.1.cmml" xref="S5.SS4.SSS0.Px2.p1.4.m4.1.1.2">subscript</csymbol><ci id="S5.SS4.SSS0.Px2.p1.4.m4.1.1.2.2.cmml" xref="S5.SS4.SSS0.Px2.p1.4.m4.1.1.2.2">𝑥</ci><ci id="S5.SS4.SSS0.Px2.p1.4.m4.1.1.2.3.cmml" xref="S5.SS4.SSS0.Px2.p1.4.m4.1.1.2.3">𝑖</ci></apply><apply id="S5.SS4.SSS0.Px2.p1.4.m4.1.1.3.cmml" xref="S5.SS4.SSS0.Px2.p1.4.m4.1.1.3"><csymbol cd="ambiguous" id="S5.SS4.SSS0.Px2.p1.4.m4.1.1.3.1.cmml" xref="S5.SS4.SSS0.Px2.p1.4.m4.1.1.3">subscript</csymbol><ci id="S5.SS4.SSS0.Px2.p1.4.m4.1.1.3.2.cmml" xref="S5.SS4.SSS0.Px2.p1.4.m4.1.1.3.2">𝑦</ci><ci id="S5.SS4.SSS0.Px2.p1.4.m4.1.1.3.3.cmml" xref="S5.SS4.SSS0.Px2.p1.4.m4.1.1.3.3">𝑖</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS4.SSS0.Px2.p1.4.m4.1c">x_{i}\rightarrow y_{i}</annotation><annotation encoding="application/x-llamapun" id="S5.SS4.SSS0.Px2.p1.4.m4.1d">italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT → italic_y start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math>) or prompting the LLM to output both the intermediate steps and the article (<math alttext="x_{i}\rightarrow{\mathbf{z}}_{i}\oplus y_{i}" class="ltx_Math" display="inline" id="S5.SS4.SSS0.Px2.p1.5.m5.1"><semantics id="S5.SS4.SSS0.Px2.p1.5.m5.1a"><mrow id="S5.SS4.SSS0.Px2.p1.5.m5.1.1" xref="S5.SS4.SSS0.Px2.p1.5.m5.1.1.cmml"><msub id="S5.SS4.SSS0.Px2.p1.5.m5.1.1.2" xref="S5.SS4.SSS0.Px2.p1.5.m5.1.1.2.cmml"><mi id="S5.SS4.SSS0.Px2.p1.5.m5.1.1.2.2" xref="S5.SS4.SSS0.Px2.p1.5.m5.1.1.2.2.cmml">x</mi><mi id="S5.SS4.SSS0.Px2.p1.5.m5.1.1.2.3" xref="S5.SS4.SSS0.Px2.p1.5.m5.1.1.2.3.cmml">i</mi></msub><mo id="S5.SS4.SSS0.Px2.p1.5.m5.1.1.1" stretchy="false" xref="S5.SS4.SSS0.Px2.p1.5.m5.1.1.1.cmml">→</mo><mrow id="S5.SS4.SSS0.Px2.p1.5.m5.1.1.3" xref="S5.SS4.SSS0.Px2.p1.5.m5.1.1.3.cmml"><msub id="S5.SS4.SSS0.Px2.p1.5.m5.1.1.3.2" xref="S5.SS4.SSS0.Px2.p1.5.m5.1.1.3.2.cmml"><mi id="S5.SS4.SSS0.Px2.p1.5.m5.1.1.3.2.2" xref="S5.SS4.SSS0.Px2.p1.5.m5.1.1.3.2.2.cmml">𝐳</mi><mi id="S5.SS4.SSS0.Px2.p1.5.m5.1.1.3.2.3" xref="S5.SS4.SSS0.Px2.p1.5.m5.1.1.3.2.3.cmml">i</mi></msub><mo id="S5.SS4.SSS0.Px2.p1.5.m5.1.1.3.1" xref="S5.SS4.SSS0.Px2.p1.5.m5.1.1.3.1.cmml">⊕</mo><msub id="S5.SS4.SSS0.Px2.p1.5.m5.1.1.3.3" xref="S5.SS4.SSS0.Px2.p1.5.m5.1.1.3.3.cmml"><mi id="S5.SS4.SSS0.Px2.p1.5.m5.1.1.3.3.2" xref="S5.SS4.SSS0.Px2.p1.5.m5.1.1.3.3.2.cmml">y</mi><mi id="S5.SS4.SSS0.Px2.p1.5.m5.1.1.3.3.3" xref="S5.SS4.SSS0.Px2.p1.5.m5.1.1.3.3.3.cmml">i</mi></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.SS4.SSS0.Px2.p1.5.m5.1b"><apply id="S5.SS4.SSS0.Px2.p1.5.m5.1.1.cmml" xref="S5.SS4.SSS0.Px2.p1.5.m5.1.1"><ci id="S5.SS4.SSS0.Px2.p1.5.m5.1.1.1.cmml" xref="S5.SS4.SSS0.Px2.p1.5.m5.1.1.1">→</ci><apply id="S5.SS4.SSS0.Px2.p1.5.m5.1.1.2.cmml" xref="S5.SS4.SSS0.Px2.p1.5.m5.1.1.2"><csymbol cd="ambiguous" id="S5.SS4.SSS0.Px2.p1.5.m5.1.1.2.1.cmml" xref="S5.SS4.SSS0.Px2.p1.5.m5.1.1.2">subscript</csymbol><ci id="S5.SS4.SSS0.Px2.p1.5.m5.1.1.2.2.cmml" xref="S5.SS4.SSS0.Px2.p1.5.m5.1.1.2.2">𝑥</ci><ci id="S5.SS4.SSS0.Px2.p1.5.m5.1.1.2.3.cmml" xref="S5.SS4.SSS0.Px2.p1.5.m5.1.1.2.3">𝑖</ci></apply><apply id="S5.SS4.SSS0.Px2.p1.5.m5.1.1.3.cmml" xref="S5.SS4.SSS0.Px2.p1.5.m5.1.1.3"><csymbol cd="latexml" id="S5.SS4.SSS0.Px2.p1.5.m5.1.1.3.1.cmml" xref="S5.SS4.SSS0.Px2.p1.5.m5.1.1.3.1">direct-sum</csymbol><apply id="S5.SS4.SSS0.Px2.p1.5.m5.1.1.3.2.cmml" xref="S5.SS4.SSS0.Px2.p1.5.m5.1.1.3.2"><csymbol cd="ambiguous" id="S5.SS4.SSS0.Px2.p1.5.m5.1.1.3.2.1.cmml" xref="S5.SS4.SSS0.Px2.p1.5.m5.1.1.3.2">subscript</csymbol><ci id="S5.SS4.SSS0.Px2.p1.5.m5.1.1.3.2.2.cmml" xref="S5.SS4.SSS0.Px2.p1.5.m5.1.1.3.2.2">𝐳</ci><ci id="S5.SS4.SSS0.Px2.p1.5.m5.1.1.3.2.3.cmml" xref="S5.SS4.SSS0.Px2.p1.5.m5.1.1.3.2.3">𝑖</ci></apply><apply id="S5.SS4.SSS0.Px2.p1.5.m5.1.1.3.3.cmml" xref="S5.SS4.SSS0.Px2.p1.5.m5.1.1.3.3"><csymbol cd="ambiguous" id="S5.SS4.SSS0.Px2.p1.5.m5.1.1.3.3.1.cmml" xref="S5.SS4.SSS0.Px2.p1.5.m5.1.1.3.3">subscript</csymbol><ci id="S5.SS4.SSS0.Px2.p1.5.m5.1.1.3.3.2.cmml" xref="S5.SS4.SSS0.Px2.p1.5.m5.1.1.3.3.2">𝑦</ci><ci id="S5.SS4.SSS0.Px2.p1.5.m5.1.1.3.3.3.cmml" xref="S5.SS4.SSS0.Px2.p1.5.m5.1.1.3.3.3">𝑖</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS4.SSS0.Px2.p1.5.m5.1c">x_{i}\rightarrow{\mathbf{z}}_{i}\oplus y_{i}</annotation><annotation encoding="application/x-llamapun" id="S5.SS4.SSS0.Px2.p1.5.m5.1d">italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT → bold_z start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ⊕ italic_y start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math>).</p> </div> <figure class="ltx_table" id="S5.T2"> <figcaption class="ltx_caption"><span class="ltx_tag ltx_tag_table">Table 2: </span>Performance of models with different training mixtures during fine-tuning.</figcaption> <p class="ltx_p ltx_align_center" id="S5.T2.36"><span class="ltx_text" id="S5.T2.36.36"> <span class="ltx_tabular ltx_guessed_headers ltx_align_middle" id="S5.T2.36.36.36"> <span class="ltx_thead"> <span class="ltx_tr" id="S5.T2.4.4.4.4"> <span class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_th_row ltx_border_r ltx_border_t" id="S5.T2.4.4.4.4.5"><span class="ltx_text ltx_font_bold" id="S5.T2.4.4.4.4.5.1">Dataset</span></span> <span class="ltx_td ltx_align_left ltx_th ltx_th_column ltx_th_row ltx_border_r ltx_border_t" id="S5.T2.4.4.4.4.6"><span class="ltx_text ltx_font_bold" id="S5.T2.4.4.4.4.6.1">Training Mixture</span></span> <span class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_th_row ltx_border_r ltx_border_t" id="S5.T2.4.4.4.4.7"><span class="ltx_text ltx_font_bold" id="S5.T2.4.4.4.4.7.1">Inference</span></span> <span class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_r ltx_border_t" id="S5.T2.1.1.1.1.1"><span class="ltx_text ltx_font_bold" id="S5.T2.1.1.1.1.1.1">R1<sub class="ltx_sub" id="S5.T2.1.1.1.1.1.1.1"><span class="ltx_text ltx_font_medium ltx_font_italic" id="S5.T2.1.1.1.1.1.1.1.1">F1</span></sub></span></span> <span class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_r ltx_border_t" id="S5.T2.2.2.2.2.2"><span class="ltx_text ltx_font_bold" id="S5.T2.2.2.2.2.2.1">R2<sub class="ltx_sub" id="S5.T2.2.2.2.2.2.1.1"><span class="ltx_text ltx_font_medium ltx_font_italic" id="S5.T2.2.2.2.2.2.1.1.1">F1</span></sub></span></span> <span class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_r ltx_border_t" id="S5.T2.3.3.3.3.3"><span class="ltx_text ltx_font_bold" id="S5.T2.3.3.3.3.3.1">RL<sub class="ltx_sub" id="S5.T2.3.3.3.3.3.1.1"><span class="ltx_text ltx_font_medium ltx_font_italic" id="S5.T2.3.3.3.3.3.1.1.1">F1</span></sub></span></span> <span class="ltx_td ltx_nopad_r ltx_align_center ltx_th ltx_th_column ltx_border_t" id="S5.T2.4.4.4.4.4"><span class="ltx_text ltx_font_bold" id="S5.T2.4.4.4.4.4.1">RLsum<sub class="ltx_sub" id="S5.T2.4.4.4.4.4.1.1"><span class="ltx_text ltx_font_medium ltx_font_italic" id="S5.T2.4.4.4.4.4.1.1.1">F1</span></sub></span></span></span> </span> <span class="ltx_tbody"> <span class="ltx_tr" id="S5.T2.6.6.6.6"> <span class="ltx_td ltx_align_center ltx_th ltx_th_row ltx_border_r ltx_border_t ltx_rowspan ltx_rowspan_5" id="S5.T2.6.6.6.6.3"><span class="ltx_text" id="S5.T2.6.6.6.6.3.1">SciNews</span></span> <span class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_r ltx_border_t" id="S5.T2.5.5.5.5.1"><math alttext="x_{i}\rightarrow y_{i}" class="ltx_Math" display="inline" id="S5.T2.5.5.5.5.1.m1.1"><semantics id="S5.T2.5.5.5.5.1.m1.1a"><mrow id="S5.T2.5.5.5.5.1.m1.1.1" xref="S5.T2.5.5.5.5.1.m1.1.1.cmml"><msub id="S5.T2.5.5.5.5.1.m1.1.1.2" xref="S5.T2.5.5.5.5.1.m1.1.1.2.cmml"><mi id="S5.T2.5.5.5.5.1.m1.1.1.2.2" xref="S5.T2.5.5.5.5.1.m1.1.1.2.2.cmml">x</mi><mi id="S5.T2.5.5.5.5.1.m1.1.1.2.3" xref="S5.T2.5.5.5.5.1.m1.1.1.2.3.cmml">i</mi></msub><mo id="S5.T2.5.5.5.5.1.m1.1.1.1" stretchy="false" xref="S5.T2.5.5.5.5.1.m1.1.1.1.cmml">→</mo><msub id="S5.T2.5.5.5.5.1.m1.1.1.3" xref="S5.T2.5.5.5.5.1.m1.1.1.3.cmml"><mi id="S5.T2.5.5.5.5.1.m1.1.1.3.2" xref="S5.T2.5.5.5.5.1.m1.1.1.3.2.cmml">y</mi><mi id="S5.T2.5.5.5.5.1.m1.1.1.3.3" xref="S5.T2.5.5.5.5.1.m1.1.1.3.3.cmml">i</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S5.T2.5.5.5.5.1.m1.1b"><apply id="S5.T2.5.5.5.5.1.m1.1.1.cmml" xref="S5.T2.5.5.5.5.1.m1.1.1"><ci id="S5.T2.5.5.5.5.1.m1.1.1.1.cmml" xref="S5.T2.5.5.5.5.1.m1.1.1.1">→</ci><apply id="S5.T2.5.5.5.5.1.m1.1.1.2.cmml" xref="S5.T2.5.5.5.5.1.m1.1.1.2"><csymbol cd="ambiguous" id="S5.T2.5.5.5.5.1.m1.1.1.2.1.cmml" xref="S5.T2.5.5.5.5.1.m1.1.1.2">subscript</csymbol><ci id="S5.T2.5.5.5.5.1.m1.1.1.2.2.cmml" xref="S5.T2.5.5.5.5.1.m1.1.1.2.2">𝑥</ci><ci id="S5.T2.5.5.5.5.1.m1.1.1.2.3.cmml" xref="S5.T2.5.5.5.5.1.m1.1.1.2.3">𝑖</ci></apply><apply id="S5.T2.5.5.5.5.1.m1.1.1.3.cmml" xref="S5.T2.5.5.5.5.1.m1.1.1.3"><csymbol cd="ambiguous" id="S5.T2.5.5.5.5.1.m1.1.1.3.1.cmml" xref="S5.T2.5.5.5.5.1.m1.1.1.3">subscript</csymbol><ci id="S5.T2.5.5.5.5.1.m1.1.1.3.2.cmml" xref="S5.T2.5.5.5.5.1.m1.1.1.3.2">𝑦</ci><ci id="S5.T2.5.5.5.5.1.m1.1.1.3.3.cmml" xref="S5.T2.5.5.5.5.1.m1.1.1.3.3">𝑖</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.T2.5.5.5.5.1.m1.1c">x_{i}\rightarrow y_{i}</annotation><annotation encoding="application/x-llamapun" id="S5.T2.5.5.5.5.1.m1.1d">italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT → italic_y start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math></span> <span class="ltx_td ltx_align_center ltx_th ltx_th_row ltx_border_r ltx_border_t" id="S5.T2.6.6.6.6.2"><math alttext="x_{i}\rightarrow y_{i}" class="ltx_Math" display="inline" id="S5.T2.6.6.6.6.2.m1.1"><semantics id="S5.T2.6.6.6.6.2.m1.1a"><mrow id="S5.T2.6.6.6.6.2.m1.1.1" xref="S5.T2.6.6.6.6.2.m1.1.1.cmml"><msub id="S5.T2.6.6.6.6.2.m1.1.1.2" xref="S5.T2.6.6.6.6.2.m1.1.1.2.cmml"><mi id="S5.T2.6.6.6.6.2.m1.1.1.2.2" xref="S5.T2.6.6.6.6.2.m1.1.1.2.2.cmml">x</mi><mi id="S5.T2.6.6.6.6.2.m1.1.1.2.3" xref="S5.T2.6.6.6.6.2.m1.1.1.2.3.cmml">i</mi></msub><mo id="S5.T2.6.6.6.6.2.m1.1.1.1" stretchy="false" xref="S5.T2.6.6.6.6.2.m1.1.1.1.cmml">→</mo><msub id="S5.T2.6.6.6.6.2.m1.1.1.3" xref="S5.T2.6.6.6.6.2.m1.1.1.3.cmml"><mi id="S5.T2.6.6.6.6.2.m1.1.1.3.2" xref="S5.T2.6.6.6.6.2.m1.1.1.3.2.cmml">y</mi><mi id="S5.T2.6.6.6.6.2.m1.1.1.3.3" xref="S5.T2.6.6.6.6.2.m1.1.1.3.3.cmml">i</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S5.T2.6.6.6.6.2.m1.1b"><apply id="S5.T2.6.6.6.6.2.m1.1.1.cmml" xref="S5.T2.6.6.6.6.2.m1.1.1"><ci id="S5.T2.6.6.6.6.2.m1.1.1.1.cmml" xref="S5.T2.6.6.6.6.2.m1.1.1.1">→</ci><apply id="S5.T2.6.6.6.6.2.m1.1.1.2.cmml" xref="S5.T2.6.6.6.6.2.m1.1.1.2"><csymbol cd="ambiguous" id="S5.T2.6.6.6.6.2.m1.1.1.2.1.cmml" xref="S5.T2.6.6.6.6.2.m1.1.1.2">subscript</csymbol><ci id="S5.T2.6.6.6.6.2.m1.1.1.2.2.cmml" xref="S5.T2.6.6.6.6.2.m1.1.1.2.2">𝑥</ci><ci id="S5.T2.6.6.6.6.2.m1.1.1.2.3.cmml" xref="S5.T2.6.6.6.6.2.m1.1.1.2.3">𝑖</ci></apply><apply id="S5.T2.6.6.6.6.2.m1.1.1.3.cmml" xref="S5.T2.6.6.6.6.2.m1.1.1.3"><csymbol cd="ambiguous" id="S5.T2.6.6.6.6.2.m1.1.1.3.1.cmml" xref="S5.T2.6.6.6.6.2.m1.1.1.3">subscript</csymbol><ci id="S5.T2.6.6.6.6.2.m1.1.1.3.2.cmml" xref="S5.T2.6.6.6.6.2.m1.1.1.3.2">𝑦</ci><ci id="S5.T2.6.6.6.6.2.m1.1.1.3.3.cmml" xref="S5.T2.6.6.6.6.2.m1.1.1.3.3">𝑖</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.T2.6.6.6.6.2.m1.1c">x_{i}\rightarrow y_{i}</annotation><annotation encoding="application/x-llamapun" id="S5.T2.6.6.6.6.2.m1.1d">italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT → italic_y start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math></span> <span class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S5.T2.6.6.6.6.4">46.66</span> <span class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S5.T2.6.6.6.6.5">13.39</span> <span class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S5.T2.6.6.6.6.6">19.26</span> <span class="ltx_td ltx_nopad_r ltx_align_center ltx_border_t" id="S5.T2.6.6.6.6.7">44.14</span></span> <span class="ltx_tr" id="S5.T2.8.8.8.8"> <span class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_r" id="S5.T2.7.7.7.7.1"><math alttext="x_{i}\rightarrow{\mathbf{z}}_{i}\oplus y_{i}" class="ltx_Math" display="inline" id="S5.T2.7.7.7.7.1.m1.1"><semantics id="S5.T2.7.7.7.7.1.m1.1a"><mrow id="S5.T2.7.7.7.7.1.m1.1.1" xref="S5.T2.7.7.7.7.1.m1.1.1.cmml"><msub id="S5.T2.7.7.7.7.1.m1.1.1.2" xref="S5.T2.7.7.7.7.1.m1.1.1.2.cmml"><mi id="S5.T2.7.7.7.7.1.m1.1.1.2.2" xref="S5.T2.7.7.7.7.1.m1.1.1.2.2.cmml">x</mi><mi id="S5.T2.7.7.7.7.1.m1.1.1.2.3" xref="S5.T2.7.7.7.7.1.m1.1.1.2.3.cmml">i</mi></msub><mo id="S5.T2.7.7.7.7.1.m1.1.1.1" stretchy="false" xref="S5.T2.7.7.7.7.1.m1.1.1.1.cmml">→</mo><mrow id="S5.T2.7.7.7.7.1.m1.1.1.3" xref="S5.T2.7.7.7.7.1.m1.1.1.3.cmml"><msub id="S5.T2.7.7.7.7.1.m1.1.1.3.2" xref="S5.T2.7.7.7.7.1.m1.1.1.3.2.cmml"><mi id="S5.T2.7.7.7.7.1.m1.1.1.3.2.2" xref="S5.T2.7.7.7.7.1.m1.1.1.3.2.2.cmml">𝐳</mi><mi id="S5.T2.7.7.7.7.1.m1.1.1.3.2.3" xref="S5.T2.7.7.7.7.1.m1.1.1.3.2.3.cmml">i</mi></msub><mo id="S5.T2.7.7.7.7.1.m1.1.1.3.1" xref="S5.T2.7.7.7.7.1.m1.1.1.3.1.cmml">⊕</mo><msub id="S5.T2.7.7.7.7.1.m1.1.1.3.3" xref="S5.T2.7.7.7.7.1.m1.1.1.3.3.cmml"><mi id="S5.T2.7.7.7.7.1.m1.1.1.3.3.2" xref="S5.T2.7.7.7.7.1.m1.1.1.3.3.2.cmml">y</mi><mi id="S5.T2.7.7.7.7.1.m1.1.1.3.3.3" xref="S5.T2.7.7.7.7.1.m1.1.1.3.3.3.cmml">i</mi></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.T2.7.7.7.7.1.m1.1b"><apply id="S5.T2.7.7.7.7.1.m1.1.1.cmml" xref="S5.T2.7.7.7.7.1.m1.1.1"><ci id="S5.T2.7.7.7.7.1.m1.1.1.1.cmml" xref="S5.T2.7.7.7.7.1.m1.1.1.1">→</ci><apply id="S5.T2.7.7.7.7.1.m1.1.1.2.cmml" xref="S5.T2.7.7.7.7.1.m1.1.1.2"><csymbol cd="ambiguous" id="S5.T2.7.7.7.7.1.m1.1.1.2.1.cmml" xref="S5.T2.7.7.7.7.1.m1.1.1.2">subscript</csymbol><ci id="S5.T2.7.7.7.7.1.m1.1.1.2.2.cmml" xref="S5.T2.7.7.7.7.1.m1.1.1.2.2">𝑥</ci><ci id="S5.T2.7.7.7.7.1.m1.1.1.2.3.cmml" xref="S5.T2.7.7.7.7.1.m1.1.1.2.3">𝑖</ci></apply><apply id="S5.T2.7.7.7.7.1.m1.1.1.3.cmml" xref="S5.T2.7.7.7.7.1.m1.1.1.3"><csymbol cd="latexml" id="S5.T2.7.7.7.7.1.m1.1.1.3.1.cmml" xref="S5.T2.7.7.7.7.1.m1.1.1.3.1">direct-sum</csymbol><apply id="S5.T2.7.7.7.7.1.m1.1.1.3.2.cmml" xref="S5.T2.7.7.7.7.1.m1.1.1.3.2"><csymbol cd="ambiguous" id="S5.T2.7.7.7.7.1.m1.1.1.3.2.1.cmml" xref="S5.T2.7.7.7.7.1.m1.1.1.3.2">subscript</csymbol><ci id="S5.T2.7.7.7.7.1.m1.1.1.3.2.2.cmml" xref="S5.T2.7.7.7.7.1.m1.1.1.3.2.2">𝐳</ci><ci id="S5.T2.7.7.7.7.1.m1.1.1.3.2.3.cmml" xref="S5.T2.7.7.7.7.1.m1.1.1.3.2.3">𝑖</ci></apply><apply id="S5.T2.7.7.7.7.1.m1.1.1.3.3.cmml" xref="S5.T2.7.7.7.7.1.m1.1.1.3.3"><csymbol cd="ambiguous" id="S5.T2.7.7.7.7.1.m1.1.1.3.3.1.cmml" xref="S5.T2.7.7.7.7.1.m1.1.1.3.3">subscript</csymbol><ci id="S5.T2.7.7.7.7.1.m1.1.1.3.3.2.cmml" xref="S5.T2.7.7.7.7.1.m1.1.1.3.3.2">𝑦</ci><ci id="S5.T2.7.7.7.7.1.m1.1.1.3.3.3.cmml" xref="S5.T2.7.7.7.7.1.m1.1.1.3.3.3">𝑖</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.T2.7.7.7.7.1.m1.1c">x_{i}\rightarrow{\mathbf{z}}_{i}\oplus y_{i}</annotation><annotation encoding="application/x-llamapun" id="S5.T2.7.7.7.7.1.m1.1d">italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT → bold_z start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ⊕ italic_y start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math></span> <span class="ltx_td ltx_align_center ltx_th ltx_th_row ltx_border_r" id="S5.T2.8.8.8.8.2"><math alttext="x_{i}\rightarrow y_{i}" class="ltx_Math" display="inline" id="S5.T2.8.8.8.8.2.m1.1"><semantics id="S5.T2.8.8.8.8.2.m1.1a"><mrow id="S5.T2.8.8.8.8.2.m1.1.1" xref="S5.T2.8.8.8.8.2.m1.1.1.cmml"><msub id="S5.T2.8.8.8.8.2.m1.1.1.2" xref="S5.T2.8.8.8.8.2.m1.1.1.2.cmml"><mi id="S5.T2.8.8.8.8.2.m1.1.1.2.2" xref="S5.T2.8.8.8.8.2.m1.1.1.2.2.cmml">x</mi><mi id="S5.T2.8.8.8.8.2.m1.1.1.2.3" xref="S5.T2.8.8.8.8.2.m1.1.1.2.3.cmml">i</mi></msub><mo id="S5.T2.8.8.8.8.2.m1.1.1.1" stretchy="false" xref="S5.T2.8.8.8.8.2.m1.1.1.1.cmml">→</mo><msub id="S5.T2.8.8.8.8.2.m1.1.1.3" xref="S5.T2.8.8.8.8.2.m1.1.1.3.cmml"><mi id="S5.T2.8.8.8.8.2.m1.1.1.3.2" xref="S5.T2.8.8.8.8.2.m1.1.1.3.2.cmml">y</mi><mi id="S5.T2.8.8.8.8.2.m1.1.1.3.3" xref="S5.T2.8.8.8.8.2.m1.1.1.3.3.cmml">i</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S5.T2.8.8.8.8.2.m1.1b"><apply id="S5.T2.8.8.8.8.2.m1.1.1.cmml" xref="S5.T2.8.8.8.8.2.m1.1.1"><ci id="S5.T2.8.8.8.8.2.m1.1.1.1.cmml" xref="S5.T2.8.8.8.8.2.m1.1.1.1">→</ci><apply id="S5.T2.8.8.8.8.2.m1.1.1.2.cmml" xref="S5.T2.8.8.8.8.2.m1.1.1.2"><csymbol cd="ambiguous" id="S5.T2.8.8.8.8.2.m1.1.1.2.1.cmml" xref="S5.T2.8.8.8.8.2.m1.1.1.2">subscript</csymbol><ci id="S5.T2.8.8.8.8.2.m1.1.1.2.2.cmml" xref="S5.T2.8.8.8.8.2.m1.1.1.2.2">𝑥</ci><ci id="S5.T2.8.8.8.8.2.m1.1.1.2.3.cmml" xref="S5.T2.8.8.8.8.2.m1.1.1.2.3">𝑖</ci></apply><apply id="S5.T2.8.8.8.8.2.m1.1.1.3.cmml" xref="S5.T2.8.8.8.8.2.m1.1.1.3"><csymbol cd="ambiguous" id="S5.T2.8.8.8.8.2.m1.1.1.3.1.cmml" xref="S5.T2.8.8.8.8.2.m1.1.1.3">subscript</csymbol><ci id="S5.T2.8.8.8.8.2.m1.1.1.3.2.cmml" xref="S5.T2.8.8.8.8.2.m1.1.1.3.2">𝑦</ci><ci id="S5.T2.8.8.8.8.2.m1.1.1.3.3.cmml" xref="S5.T2.8.8.8.8.2.m1.1.1.3.3">𝑖</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.T2.8.8.8.8.2.m1.1c">x_{i}\rightarrow y_{i}</annotation><annotation encoding="application/x-llamapun" id="S5.T2.8.8.8.8.2.m1.1d">italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT → italic_y start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math></span> <span class="ltx_td ltx_align_center ltx_border_r" id="S5.T2.8.8.8.8.3">48.94</span> <span class="ltx_td ltx_align_center ltx_border_r" id="S5.T2.8.8.8.8.4">14.28</span> <span class="ltx_td ltx_align_center ltx_border_r" id="S5.T2.8.8.8.8.5">19.48</span> <span class="ltx_td ltx_nopad_r ltx_align_center" id="S5.T2.8.8.8.8.6">45.54</span></span> <span class="ltx_tr" id="S5.T2.10.10.10.10"> <span class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_r" id="S5.T2.9.9.9.9.1"><math alttext="x_{i}\rightarrow{\mathbf{z}}_{i}\oplus y_{i}" class="ltx_Math" display="inline" id="S5.T2.9.9.9.9.1.m1.1"><semantics id="S5.T2.9.9.9.9.1.m1.1a"><mrow id="S5.T2.9.9.9.9.1.m1.1.1" xref="S5.T2.9.9.9.9.1.m1.1.1.cmml"><msub id="S5.T2.9.9.9.9.1.m1.1.1.2" xref="S5.T2.9.9.9.9.1.m1.1.1.2.cmml"><mi id="S5.T2.9.9.9.9.1.m1.1.1.2.2" xref="S5.T2.9.9.9.9.1.m1.1.1.2.2.cmml">x</mi><mi id="S5.T2.9.9.9.9.1.m1.1.1.2.3" xref="S5.T2.9.9.9.9.1.m1.1.1.2.3.cmml">i</mi></msub><mo id="S5.T2.9.9.9.9.1.m1.1.1.1" stretchy="false" xref="S5.T2.9.9.9.9.1.m1.1.1.1.cmml">→</mo><mrow id="S5.T2.9.9.9.9.1.m1.1.1.3" xref="S5.T2.9.9.9.9.1.m1.1.1.3.cmml"><msub id="S5.T2.9.9.9.9.1.m1.1.1.3.2" xref="S5.T2.9.9.9.9.1.m1.1.1.3.2.cmml"><mi id="S5.T2.9.9.9.9.1.m1.1.1.3.2.2" xref="S5.T2.9.9.9.9.1.m1.1.1.3.2.2.cmml">𝐳</mi><mi id="S5.T2.9.9.9.9.1.m1.1.1.3.2.3" xref="S5.T2.9.9.9.9.1.m1.1.1.3.2.3.cmml">i</mi></msub><mo id="S5.T2.9.9.9.9.1.m1.1.1.3.1" xref="S5.T2.9.9.9.9.1.m1.1.1.3.1.cmml">⊕</mo><msub id="S5.T2.9.9.9.9.1.m1.1.1.3.3" xref="S5.T2.9.9.9.9.1.m1.1.1.3.3.cmml"><mi id="S5.T2.9.9.9.9.1.m1.1.1.3.3.2" xref="S5.T2.9.9.9.9.1.m1.1.1.3.3.2.cmml">y</mi><mi id="S5.T2.9.9.9.9.1.m1.1.1.3.3.3" xref="S5.T2.9.9.9.9.1.m1.1.1.3.3.3.cmml">i</mi></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.T2.9.9.9.9.1.m1.1b"><apply id="S5.T2.9.9.9.9.1.m1.1.1.cmml" xref="S5.T2.9.9.9.9.1.m1.1.1"><ci id="S5.T2.9.9.9.9.1.m1.1.1.1.cmml" xref="S5.T2.9.9.9.9.1.m1.1.1.1">→</ci><apply id="S5.T2.9.9.9.9.1.m1.1.1.2.cmml" xref="S5.T2.9.9.9.9.1.m1.1.1.2"><csymbol cd="ambiguous" id="S5.T2.9.9.9.9.1.m1.1.1.2.1.cmml" xref="S5.T2.9.9.9.9.1.m1.1.1.2">subscript</csymbol><ci id="S5.T2.9.9.9.9.1.m1.1.1.2.2.cmml" xref="S5.T2.9.9.9.9.1.m1.1.1.2.2">𝑥</ci><ci id="S5.T2.9.9.9.9.1.m1.1.1.2.3.cmml" xref="S5.T2.9.9.9.9.1.m1.1.1.2.3">𝑖</ci></apply><apply id="S5.T2.9.9.9.9.1.m1.1.1.3.cmml" xref="S5.T2.9.9.9.9.1.m1.1.1.3"><csymbol cd="latexml" id="S5.T2.9.9.9.9.1.m1.1.1.3.1.cmml" xref="S5.T2.9.9.9.9.1.m1.1.1.3.1">direct-sum</csymbol><apply id="S5.T2.9.9.9.9.1.m1.1.1.3.2.cmml" xref="S5.T2.9.9.9.9.1.m1.1.1.3.2"><csymbol cd="ambiguous" id="S5.T2.9.9.9.9.1.m1.1.1.3.2.1.cmml" xref="S5.T2.9.9.9.9.1.m1.1.1.3.2">subscript</csymbol><ci id="S5.T2.9.9.9.9.1.m1.1.1.3.2.2.cmml" xref="S5.T2.9.9.9.9.1.m1.1.1.3.2.2">𝐳</ci><ci id="S5.T2.9.9.9.9.1.m1.1.1.3.2.3.cmml" xref="S5.T2.9.9.9.9.1.m1.1.1.3.2.3">𝑖</ci></apply><apply id="S5.T2.9.9.9.9.1.m1.1.1.3.3.cmml" xref="S5.T2.9.9.9.9.1.m1.1.1.3.3"><csymbol cd="ambiguous" id="S5.T2.9.9.9.9.1.m1.1.1.3.3.1.cmml" xref="S5.T2.9.9.9.9.1.m1.1.1.3.3">subscript</csymbol><ci id="S5.T2.9.9.9.9.1.m1.1.1.3.3.2.cmml" xref="S5.T2.9.9.9.9.1.m1.1.1.3.3.2">𝑦</ci><ci id="S5.T2.9.9.9.9.1.m1.1.1.3.3.3.cmml" xref="S5.T2.9.9.9.9.1.m1.1.1.3.3.3">𝑖</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.T2.9.9.9.9.1.m1.1c">x_{i}\rightarrow{\mathbf{z}}_{i}\oplus y_{i}</annotation><annotation encoding="application/x-llamapun" id="S5.T2.9.9.9.9.1.m1.1d">italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT → bold_z start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ⊕ italic_y start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math></span> <span class="ltx_td ltx_align_center ltx_th ltx_th_row ltx_border_r" id="S5.T2.10.10.10.10.2"><math alttext="x_{i}\rightarrow{\mathbf{z}}_{i}\oplus y_{i}" class="ltx_Math" display="inline" id="S5.T2.10.10.10.10.2.m1.1"><semantics id="S5.T2.10.10.10.10.2.m1.1a"><mrow id="S5.T2.10.10.10.10.2.m1.1.1" xref="S5.T2.10.10.10.10.2.m1.1.1.cmml"><msub id="S5.T2.10.10.10.10.2.m1.1.1.2" xref="S5.T2.10.10.10.10.2.m1.1.1.2.cmml"><mi id="S5.T2.10.10.10.10.2.m1.1.1.2.2" xref="S5.T2.10.10.10.10.2.m1.1.1.2.2.cmml">x</mi><mi id="S5.T2.10.10.10.10.2.m1.1.1.2.3" xref="S5.T2.10.10.10.10.2.m1.1.1.2.3.cmml">i</mi></msub><mo id="S5.T2.10.10.10.10.2.m1.1.1.1" stretchy="false" xref="S5.T2.10.10.10.10.2.m1.1.1.1.cmml">→</mo><mrow id="S5.T2.10.10.10.10.2.m1.1.1.3" xref="S5.T2.10.10.10.10.2.m1.1.1.3.cmml"><msub id="S5.T2.10.10.10.10.2.m1.1.1.3.2" xref="S5.T2.10.10.10.10.2.m1.1.1.3.2.cmml"><mi id="S5.T2.10.10.10.10.2.m1.1.1.3.2.2" xref="S5.T2.10.10.10.10.2.m1.1.1.3.2.2.cmml">𝐳</mi><mi id="S5.T2.10.10.10.10.2.m1.1.1.3.2.3" xref="S5.T2.10.10.10.10.2.m1.1.1.3.2.3.cmml">i</mi></msub><mo id="S5.T2.10.10.10.10.2.m1.1.1.3.1" xref="S5.T2.10.10.10.10.2.m1.1.1.3.1.cmml">⊕</mo><msub id="S5.T2.10.10.10.10.2.m1.1.1.3.3" xref="S5.T2.10.10.10.10.2.m1.1.1.3.3.cmml"><mi id="S5.T2.10.10.10.10.2.m1.1.1.3.3.2" xref="S5.T2.10.10.10.10.2.m1.1.1.3.3.2.cmml">y</mi><mi id="S5.T2.10.10.10.10.2.m1.1.1.3.3.3" xref="S5.T2.10.10.10.10.2.m1.1.1.3.3.3.cmml">i</mi></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.T2.10.10.10.10.2.m1.1b"><apply id="S5.T2.10.10.10.10.2.m1.1.1.cmml" xref="S5.T2.10.10.10.10.2.m1.1.1"><ci id="S5.T2.10.10.10.10.2.m1.1.1.1.cmml" xref="S5.T2.10.10.10.10.2.m1.1.1.1">→</ci><apply id="S5.T2.10.10.10.10.2.m1.1.1.2.cmml" xref="S5.T2.10.10.10.10.2.m1.1.1.2"><csymbol cd="ambiguous" id="S5.T2.10.10.10.10.2.m1.1.1.2.1.cmml" xref="S5.T2.10.10.10.10.2.m1.1.1.2">subscript</csymbol><ci id="S5.T2.10.10.10.10.2.m1.1.1.2.2.cmml" xref="S5.T2.10.10.10.10.2.m1.1.1.2.2">𝑥</ci><ci id="S5.T2.10.10.10.10.2.m1.1.1.2.3.cmml" xref="S5.T2.10.10.10.10.2.m1.1.1.2.3">𝑖</ci></apply><apply id="S5.T2.10.10.10.10.2.m1.1.1.3.cmml" xref="S5.T2.10.10.10.10.2.m1.1.1.3"><csymbol cd="latexml" id="S5.T2.10.10.10.10.2.m1.1.1.3.1.cmml" xref="S5.T2.10.10.10.10.2.m1.1.1.3.1">direct-sum</csymbol><apply id="S5.T2.10.10.10.10.2.m1.1.1.3.2.cmml" xref="S5.T2.10.10.10.10.2.m1.1.1.3.2"><csymbol cd="ambiguous" id="S5.T2.10.10.10.10.2.m1.1.1.3.2.1.cmml" xref="S5.T2.10.10.10.10.2.m1.1.1.3.2">subscript</csymbol><ci id="S5.T2.10.10.10.10.2.m1.1.1.3.2.2.cmml" xref="S5.T2.10.10.10.10.2.m1.1.1.3.2.2">𝐳</ci><ci id="S5.T2.10.10.10.10.2.m1.1.1.3.2.3.cmml" xref="S5.T2.10.10.10.10.2.m1.1.1.3.2.3">𝑖</ci></apply><apply id="S5.T2.10.10.10.10.2.m1.1.1.3.3.cmml" xref="S5.T2.10.10.10.10.2.m1.1.1.3.3"><csymbol cd="ambiguous" id="S5.T2.10.10.10.10.2.m1.1.1.3.3.1.cmml" xref="S5.T2.10.10.10.10.2.m1.1.1.3.3">subscript</csymbol><ci id="S5.T2.10.10.10.10.2.m1.1.1.3.3.2.cmml" xref="S5.T2.10.10.10.10.2.m1.1.1.3.3.2">𝑦</ci><ci id="S5.T2.10.10.10.10.2.m1.1.1.3.3.3.cmml" xref="S5.T2.10.10.10.10.2.m1.1.1.3.3.3">𝑖</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.T2.10.10.10.10.2.m1.1c">x_{i}\rightarrow{\mathbf{z}}_{i}\oplus y_{i}</annotation><annotation encoding="application/x-llamapun" id="S5.T2.10.10.10.10.2.m1.1d">italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT → bold_z start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ⊕ italic_y start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math></span> <span class="ltx_td ltx_align_center ltx_border_r" id="S5.T2.10.10.10.10.3">49.02</span> <span class="ltx_td ltx_align_center ltx_border_r" id="S5.T2.10.10.10.10.4">14.50</span> <span class="ltx_td ltx_align_center ltx_border_r" id="S5.T2.10.10.10.10.5">19.61</span> <span class="ltx_td ltx_nopad_r ltx_align_center" id="S5.T2.10.10.10.10.6">46.24</span></span> <span class="ltx_tr" id="S5.T2.13.13.13.13"> <span class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_r" id="S5.T2.12.12.12.12.2"><math alttext="x_{i}\rightarrow y_{i}" class="ltx_Math" display="inline" id="S5.T2.11.11.11.11.1.m1.1"><semantics id="S5.T2.11.11.11.11.1.m1.1a"><mrow id="S5.T2.11.11.11.11.1.m1.1.1" xref="S5.T2.11.11.11.11.1.m1.1.1.cmml"><msub id="S5.T2.11.11.11.11.1.m1.1.1.2" xref="S5.T2.11.11.11.11.1.m1.1.1.2.cmml"><mi id="S5.T2.11.11.11.11.1.m1.1.1.2.2" xref="S5.T2.11.11.11.11.1.m1.1.1.2.2.cmml">x</mi><mi id="S5.T2.11.11.11.11.1.m1.1.1.2.3" xref="S5.T2.11.11.11.11.1.m1.1.1.2.3.cmml">i</mi></msub><mo id="S5.T2.11.11.11.11.1.m1.1.1.1" stretchy="false" xref="S5.T2.11.11.11.11.1.m1.1.1.1.cmml">→</mo><msub id="S5.T2.11.11.11.11.1.m1.1.1.3" xref="S5.T2.11.11.11.11.1.m1.1.1.3.cmml"><mi id="S5.T2.11.11.11.11.1.m1.1.1.3.2" xref="S5.T2.11.11.11.11.1.m1.1.1.3.2.cmml">y</mi><mi id="S5.T2.11.11.11.11.1.m1.1.1.3.3" xref="S5.T2.11.11.11.11.1.m1.1.1.3.3.cmml">i</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S5.T2.11.11.11.11.1.m1.1b"><apply id="S5.T2.11.11.11.11.1.m1.1.1.cmml" xref="S5.T2.11.11.11.11.1.m1.1.1"><ci id="S5.T2.11.11.11.11.1.m1.1.1.1.cmml" xref="S5.T2.11.11.11.11.1.m1.1.1.1">→</ci><apply id="S5.T2.11.11.11.11.1.m1.1.1.2.cmml" xref="S5.T2.11.11.11.11.1.m1.1.1.2"><csymbol cd="ambiguous" id="S5.T2.11.11.11.11.1.m1.1.1.2.1.cmml" xref="S5.T2.11.11.11.11.1.m1.1.1.2">subscript</csymbol><ci id="S5.T2.11.11.11.11.1.m1.1.1.2.2.cmml" xref="S5.T2.11.11.11.11.1.m1.1.1.2.2">𝑥</ci><ci id="S5.T2.11.11.11.11.1.m1.1.1.2.3.cmml" xref="S5.T2.11.11.11.11.1.m1.1.1.2.3">𝑖</ci></apply><apply id="S5.T2.11.11.11.11.1.m1.1.1.3.cmml" xref="S5.T2.11.11.11.11.1.m1.1.1.3"><csymbol cd="ambiguous" id="S5.T2.11.11.11.11.1.m1.1.1.3.1.cmml" xref="S5.T2.11.11.11.11.1.m1.1.1.3">subscript</csymbol><ci id="S5.T2.11.11.11.11.1.m1.1.1.3.2.cmml" xref="S5.T2.11.11.11.11.1.m1.1.1.3.2">𝑦</ci><ci id="S5.T2.11.11.11.11.1.m1.1.1.3.3.cmml" xref="S5.T2.11.11.11.11.1.m1.1.1.3.3">𝑖</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.T2.11.11.11.11.1.m1.1c">x_{i}\rightarrow y_{i}</annotation><annotation encoding="application/x-llamapun" id="S5.T2.11.11.11.11.1.m1.1d">italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT → italic_y start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math>; <math alttext="x_{i}\rightarrow{\mathbf{z}}_{i}\oplus y_{i}" class="ltx_Math" display="inline" id="S5.T2.12.12.12.12.2.m2.1"><semantics id="S5.T2.12.12.12.12.2.m2.1a"><mrow id="S5.T2.12.12.12.12.2.m2.1.1" xref="S5.T2.12.12.12.12.2.m2.1.1.cmml"><msub id="S5.T2.12.12.12.12.2.m2.1.1.2" xref="S5.T2.12.12.12.12.2.m2.1.1.2.cmml"><mi id="S5.T2.12.12.12.12.2.m2.1.1.2.2" xref="S5.T2.12.12.12.12.2.m2.1.1.2.2.cmml">x</mi><mi id="S5.T2.12.12.12.12.2.m2.1.1.2.3" xref="S5.T2.12.12.12.12.2.m2.1.1.2.3.cmml">i</mi></msub><mo id="S5.T2.12.12.12.12.2.m2.1.1.1" stretchy="false" xref="S5.T2.12.12.12.12.2.m2.1.1.1.cmml">→</mo><mrow id="S5.T2.12.12.12.12.2.m2.1.1.3" xref="S5.T2.12.12.12.12.2.m2.1.1.3.cmml"><msub id="S5.T2.12.12.12.12.2.m2.1.1.3.2" xref="S5.T2.12.12.12.12.2.m2.1.1.3.2.cmml"><mi id="S5.T2.12.12.12.12.2.m2.1.1.3.2.2" xref="S5.T2.12.12.12.12.2.m2.1.1.3.2.2.cmml">𝐳</mi><mi id="S5.T2.12.12.12.12.2.m2.1.1.3.2.3" xref="S5.T2.12.12.12.12.2.m2.1.1.3.2.3.cmml">i</mi></msub><mo id="S5.T2.12.12.12.12.2.m2.1.1.3.1" xref="S5.T2.12.12.12.12.2.m2.1.1.3.1.cmml">⊕</mo><msub id="S5.T2.12.12.12.12.2.m2.1.1.3.3" xref="S5.T2.12.12.12.12.2.m2.1.1.3.3.cmml"><mi id="S5.T2.12.12.12.12.2.m2.1.1.3.3.2" xref="S5.T2.12.12.12.12.2.m2.1.1.3.3.2.cmml">y</mi><mi id="S5.T2.12.12.12.12.2.m2.1.1.3.3.3" xref="S5.T2.12.12.12.12.2.m2.1.1.3.3.3.cmml">i</mi></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.T2.12.12.12.12.2.m2.1b"><apply id="S5.T2.12.12.12.12.2.m2.1.1.cmml" xref="S5.T2.12.12.12.12.2.m2.1.1"><ci id="S5.T2.12.12.12.12.2.m2.1.1.1.cmml" xref="S5.T2.12.12.12.12.2.m2.1.1.1">→</ci><apply id="S5.T2.12.12.12.12.2.m2.1.1.2.cmml" xref="S5.T2.12.12.12.12.2.m2.1.1.2"><csymbol cd="ambiguous" id="S5.T2.12.12.12.12.2.m2.1.1.2.1.cmml" xref="S5.T2.12.12.12.12.2.m2.1.1.2">subscript</csymbol><ci id="S5.T2.12.12.12.12.2.m2.1.1.2.2.cmml" xref="S5.T2.12.12.12.12.2.m2.1.1.2.2">𝑥</ci><ci id="S5.T2.12.12.12.12.2.m2.1.1.2.3.cmml" xref="S5.T2.12.12.12.12.2.m2.1.1.2.3">𝑖</ci></apply><apply id="S5.T2.12.12.12.12.2.m2.1.1.3.cmml" xref="S5.T2.12.12.12.12.2.m2.1.1.3"><csymbol cd="latexml" id="S5.T2.12.12.12.12.2.m2.1.1.3.1.cmml" xref="S5.T2.12.12.12.12.2.m2.1.1.3.1">direct-sum</csymbol><apply id="S5.T2.12.12.12.12.2.m2.1.1.3.2.cmml" xref="S5.T2.12.12.12.12.2.m2.1.1.3.2"><csymbol cd="ambiguous" id="S5.T2.12.12.12.12.2.m2.1.1.3.2.1.cmml" xref="S5.T2.12.12.12.12.2.m2.1.1.3.2">subscript</csymbol><ci id="S5.T2.12.12.12.12.2.m2.1.1.3.2.2.cmml" xref="S5.T2.12.12.12.12.2.m2.1.1.3.2.2">𝐳</ci><ci id="S5.T2.12.12.12.12.2.m2.1.1.3.2.3.cmml" xref="S5.T2.12.12.12.12.2.m2.1.1.3.2.3">𝑖</ci></apply><apply id="S5.T2.12.12.12.12.2.m2.1.1.3.3.cmml" xref="S5.T2.12.12.12.12.2.m2.1.1.3.3"><csymbol cd="ambiguous" id="S5.T2.12.12.12.12.2.m2.1.1.3.3.1.cmml" xref="S5.T2.12.12.12.12.2.m2.1.1.3.3">subscript</csymbol><ci id="S5.T2.12.12.12.12.2.m2.1.1.3.3.2.cmml" xref="S5.T2.12.12.12.12.2.m2.1.1.3.3.2">𝑦</ci><ci id="S5.T2.12.12.12.12.2.m2.1.1.3.3.3.cmml" xref="S5.T2.12.12.12.12.2.m2.1.1.3.3.3">𝑖</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.T2.12.12.12.12.2.m2.1c">x_{i}\rightarrow{\mathbf{z}}_{i}\oplus y_{i}</annotation><annotation encoding="application/x-llamapun" id="S5.T2.12.12.12.12.2.m2.1d">italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT → bold_z start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ⊕ italic_y start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math></span> <span class="ltx_td ltx_align_center ltx_th ltx_th_row ltx_border_r" id="S5.T2.13.13.13.13.3"><math alttext="x_{i}\rightarrow y_{i}" class="ltx_Math" display="inline" id="S5.T2.13.13.13.13.3.m1.1"><semantics id="S5.T2.13.13.13.13.3.m1.1a"><mrow id="S5.T2.13.13.13.13.3.m1.1.1" xref="S5.T2.13.13.13.13.3.m1.1.1.cmml"><msub id="S5.T2.13.13.13.13.3.m1.1.1.2" xref="S5.T2.13.13.13.13.3.m1.1.1.2.cmml"><mi id="S5.T2.13.13.13.13.3.m1.1.1.2.2" xref="S5.T2.13.13.13.13.3.m1.1.1.2.2.cmml">x</mi><mi id="S5.T2.13.13.13.13.3.m1.1.1.2.3" xref="S5.T2.13.13.13.13.3.m1.1.1.2.3.cmml">i</mi></msub><mo id="S5.T2.13.13.13.13.3.m1.1.1.1" stretchy="false" xref="S5.T2.13.13.13.13.3.m1.1.1.1.cmml">→</mo><msub id="S5.T2.13.13.13.13.3.m1.1.1.3" xref="S5.T2.13.13.13.13.3.m1.1.1.3.cmml"><mi id="S5.T2.13.13.13.13.3.m1.1.1.3.2" xref="S5.T2.13.13.13.13.3.m1.1.1.3.2.cmml">y</mi><mi id="S5.T2.13.13.13.13.3.m1.1.1.3.3" xref="S5.T2.13.13.13.13.3.m1.1.1.3.3.cmml">i</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S5.T2.13.13.13.13.3.m1.1b"><apply id="S5.T2.13.13.13.13.3.m1.1.1.cmml" xref="S5.T2.13.13.13.13.3.m1.1.1"><ci id="S5.T2.13.13.13.13.3.m1.1.1.1.cmml" xref="S5.T2.13.13.13.13.3.m1.1.1.1">→</ci><apply id="S5.T2.13.13.13.13.3.m1.1.1.2.cmml" xref="S5.T2.13.13.13.13.3.m1.1.1.2"><csymbol cd="ambiguous" id="S5.T2.13.13.13.13.3.m1.1.1.2.1.cmml" xref="S5.T2.13.13.13.13.3.m1.1.1.2">subscript</csymbol><ci id="S5.T2.13.13.13.13.3.m1.1.1.2.2.cmml" xref="S5.T2.13.13.13.13.3.m1.1.1.2.2">𝑥</ci><ci id="S5.T2.13.13.13.13.3.m1.1.1.2.3.cmml" xref="S5.T2.13.13.13.13.3.m1.1.1.2.3">𝑖</ci></apply><apply id="S5.T2.13.13.13.13.3.m1.1.1.3.cmml" xref="S5.T2.13.13.13.13.3.m1.1.1.3"><csymbol cd="ambiguous" id="S5.T2.13.13.13.13.3.m1.1.1.3.1.cmml" xref="S5.T2.13.13.13.13.3.m1.1.1.3">subscript</csymbol><ci id="S5.T2.13.13.13.13.3.m1.1.1.3.2.cmml" xref="S5.T2.13.13.13.13.3.m1.1.1.3.2">𝑦</ci><ci id="S5.T2.13.13.13.13.3.m1.1.1.3.3.cmml" xref="S5.T2.13.13.13.13.3.m1.1.1.3.3">𝑖</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.T2.13.13.13.13.3.m1.1c">x_{i}\rightarrow y_{i}</annotation><annotation encoding="application/x-llamapun" id="S5.T2.13.13.13.13.3.m1.1d">italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT → italic_y start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math></span> <span class="ltx_td ltx_align_center ltx_border_r" id="S5.T2.13.13.13.13.4">48.63</span> <span class="ltx_td ltx_align_center ltx_border_r" id="S5.T2.13.13.13.13.5">14.39</span> <span class="ltx_td ltx_align_center ltx_border_r" id="S5.T2.13.13.13.13.6">19.57</span> <span class="ltx_td ltx_nopad_r ltx_align_center" id="S5.T2.13.13.13.13.7">45.92</span></span> <span class="ltx_tr" id="S5.T2.16.16.16.16"> <span class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_r" id="S5.T2.15.15.15.15.2"><math alttext="x_{i}\rightarrow y_{i}" class="ltx_Math" display="inline" id="S5.T2.14.14.14.14.1.m1.1"><semantics id="S5.T2.14.14.14.14.1.m1.1a"><mrow id="S5.T2.14.14.14.14.1.m1.1.1" xref="S5.T2.14.14.14.14.1.m1.1.1.cmml"><msub id="S5.T2.14.14.14.14.1.m1.1.1.2" xref="S5.T2.14.14.14.14.1.m1.1.1.2.cmml"><mi id="S5.T2.14.14.14.14.1.m1.1.1.2.2" xref="S5.T2.14.14.14.14.1.m1.1.1.2.2.cmml">x</mi><mi id="S5.T2.14.14.14.14.1.m1.1.1.2.3" xref="S5.T2.14.14.14.14.1.m1.1.1.2.3.cmml">i</mi></msub><mo id="S5.T2.14.14.14.14.1.m1.1.1.1" stretchy="false" xref="S5.T2.14.14.14.14.1.m1.1.1.1.cmml">→</mo><msub id="S5.T2.14.14.14.14.1.m1.1.1.3" xref="S5.T2.14.14.14.14.1.m1.1.1.3.cmml"><mi id="S5.T2.14.14.14.14.1.m1.1.1.3.2" xref="S5.T2.14.14.14.14.1.m1.1.1.3.2.cmml">y</mi><mi id="S5.T2.14.14.14.14.1.m1.1.1.3.3" xref="S5.T2.14.14.14.14.1.m1.1.1.3.3.cmml">i</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S5.T2.14.14.14.14.1.m1.1b"><apply id="S5.T2.14.14.14.14.1.m1.1.1.cmml" xref="S5.T2.14.14.14.14.1.m1.1.1"><ci id="S5.T2.14.14.14.14.1.m1.1.1.1.cmml" xref="S5.T2.14.14.14.14.1.m1.1.1.1">→</ci><apply id="S5.T2.14.14.14.14.1.m1.1.1.2.cmml" xref="S5.T2.14.14.14.14.1.m1.1.1.2"><csymbol cd="ambiguous" id="S5.T2.14.14.14.14.1.m1.1.1.2.1.cmml" xref="S5.T2.14.14.14.14.1.m1.1.1.2">subscript</csymbol><ci id="S5.T2.14.14.14.14.1.m1.1.1.2.2.cmml" xref="S5.T2.14.14.14.14.1.m1.1.1.2.2">𝑥</ci><ci id="S5.T2.14.14.14.14.1.m1.1.1.2.3.cmml" xref="S5.T2.14.14.14.14.1.m1.1.1.2.3">𝑖</ci></apply><apply id="S5.T2.14.14.14.14.1.m1.1.1.3.cmml" xref="S5.T2.14.14.14.14.1.m1.1.1.3"><csymbol cd="ambiguous" id="S5.T2.14.14.14.14.1.m1.1.1.3.1.cmml" xref="S5.T2.14.14.14.14.1.m1.1.1.3">subscript</csymbol><ci id="S5.T2.14.14.14.14.1.m1.1.1.3.2.cmml" xref="S5.T2.14.14.14.14.1.m1.1.1.3.2">𝑦</ci><ci id="S5.T2.14.14.14.14.1.m1.1.1.3.3.cmml" xref="S5.T2.14.14.14.14.1.m1.1.1.3.3">𝑖</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.T2.14.14.14.14.1.m1.1c">x_{i}\rightarrow y_{i}</annotation><annotation encoding="application/x-llamapun" id="S5.T2.14.14.14.14.1.m1.1d">italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT → italic_y start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math>; <math alttext="x_{i}\rightarrow{\mathbf{z}}_{i}\oplus y_{i}" class="ltx_Math" display="inline" id="S5.T2.15.15.15.15.2.m2.1"><semantics id="S5.T2.15.15.15.15.2.m2.1a"><mrow id="S5.T2.15.15.15.15.2.m2.1.1" xref="S5.T2.15.15.15.15.2.m2.1.1.cmml"><msub id="S5.T2.15.15.15.15.2.m2.1.1.2" xref="S5.T2.15.15.15.15.2.m2.1.1.2.cmml"><mi id="S5.T2.15.15.15.15.2.m2.1.1.2.2" xref="S5.T2.15.15.15.15.2.m2.1.1.2.2.cmml">x</mi><mi id="S5.T2.15.15.15.15.2.m2.1.1.2.3" xref="S5.T2.15.15.15.15.2.m2.1.1.2.3.cmml">i</mi></msub><mo id="S5.T2.15.15.15.15.2.m2.1.1.1" stretchy="false" xref="S5.T2.15.15.15.15.2.m2.1.1.1.cmml">→</mo><mrow id="S5.T2.15.15.15.15.2.m2.1.1.3" xref="S5.T2.15.15.15.15.2.m2.1.1.3.cmml"><msub id="S5.T2.15.15.15.15.2.m2.1.1.3.2" xref="S5.T2.15.15.15.15.2.m2.1.1.3.2.cmml"><mi id="S5.T2.15.15.15.15.2.m2.1.1.3.2.2" xref="S5.T2.15.15.15.15.2.m2.1.1.3.2.2.cmml">𝐳</mi><mi id="S5.T2.15.15.15.15.2.m2.1.1.3.2.3" xref="S5.T2.15.15.15.15.2.m2.1.1.3.2.3.cmml">i</mi></msub><mo id="S5.T2.15.15.15.15.2.m2.1.1.3.1" xref="S5.T2.15.15.15.15.2.m2.1.1.3.1.cmml">⊕</mo><msub id="S5.T2.15.15.15.15.2.m2.1.1.3.3" xref="S5.T2.15.15.15.15.2.m2.1.1.3.3.cmml"><mi id="S5.T2.15.15.15.15.2.m2.1.1.3.3.2" xref="S5.T2.15.15.15.15.2.m2.1.1.3.3.2.cmml">y</mi><mi id="S5.T2.15.15.15.15.2.m2.1.1.3.3.3" xref="S5.T2.15.15.15.15.2.m2.1.1.3.3.3.cmml">i</mi></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.T2.15.15.15.15.2.m2.1b"><apply id="S5.T2.15.15.15.15.2.m2.1.1.cmml" xref="S5.T2.15.15.15.15.2.m2.1.1"><ci id="S5.T2.15.15.15.15.2.m2.1.1.1.cmml" xref="S5.T2.15.15.15.15.2.m2.1.1.1">→</ci><apply id="S5.T2.15.15.15.15.2.m2.1.1.2.cmml" xref="S5.T2.15.15.15.15.2.m2.1.1.2"><csymbol cd="ambiguous" id="S5.T2.15.15.15.15.2.m2.1.1.2.1.cmml" xref="S5.T2.15.15.15.15.2.m2.1.1.2">subscript</csymbol><ci id="S5.T2.15.15.15.15.2.m2.1.1.2.2.cmml" xref="S5.T2.15.15.15.15.2.m2.1.1.2.2">𝑥</ci><ci id="S5.T2.15.15.15.15.2.m2.1.1.2.3.cmml" xref="S5.T2.15.15.15.15.2.m2.1.1.2.3">𝑖</ci></apply><apply id="S5.T2.15.15.15.15.2.m2.1.1.3.cmml" xref="S5.T2.15.15.15.15.2.m2.1.1.3"><csymbol cd="latexml" id="S5.T2.15.15.15.15.2.m2.1.1.3.1.cmml" xref="S5.T2.15.15.15.15.2.m2.1.1.3.1">direct-sum</csymbol><apply id="S5.T2.15.15.15.15.2.m2.1.1.3.2.cmml" xref="S5.T2.15.15.15.15.2.m2.1.1.3.2"><csymbol cd="ambiguous" id="S5.T2.15.15.15.15.2.m2.1.1.3.2.1.cmml" xref="S5.T2.15.15.15.15.2.m2.1.1.3.2">subscript</csymbol><ci id="S5.T2.15.15.15.15.2.m2.1.1.3.2.2.cmml" xref="S5.T2.15.15.15.15.2.m2.1.1.3.2.2">𝐳</ci><ci id="S5.T2.15.15.15.15.2.m2.1.1.3.2.3.cmml" xref="S5.T2.15.15.15.15.2.m2.1.1.3.2.3">𝑖</ci></apply><apply id="S5.T2.15.15.15.15.2.m2.1.1.3.3.cmml" xref="S5.T2.15.15.15.15.2.m2.1.1.3.3"><csymbol cd="ambiguous" id="S5.T2.15.15.15.15.2.m2.1.1.3.3.1.cmml" xref="S5.T2.15.15.15.15.2.m2.1.1.3.3">subscript</csymbol><ci id="S5.T2.15.15.15.15.2.m2.1.1.3.3.2.cmml" xref="S5.T2.15.15.15.15.2.m2.1.1.3.3.2">𝑦</ci><ci id="S5.T2.15.15.15.15.2.m2.1.1.3.3.3.cmml" xref="S5.T2.15.15.15.15.2.m2.1.1.3.3.3">𝑖</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.T2.15.15.15.15.2.m2.1c">x_{i}\rightarrow{\mathbf{z}}_{i}\oplus y_{i}</annotation><annotation encoding="application/x-llamapun" id="S5.T2.15.15.15.15.2.m2.1d">italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT → bold_z start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ⊕ italic_y start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math></span> <span class="ltx_td ltx_align_center ltx_th ltx_th_row ltx_border_r" id="S5.T2.16.16.16.16.3"><math alttext="x_{i}\rightarrow{\mathbf{z}}_{i}\oplus y_{i}" class="ltx_Math" display="inline" id="S5.T2.16.16.16.16.3.m1.1"><semantics id="S5.T2.16.16.16.16.3.m1.1a"><mrow id="S5.T2.16.16.16.16.3.m1.1.1" xref="S5.T2.16.16.16.16.3.m1.1.1.cmml"><msub id="S5.T2.16.16.16.16.3.m1.1.1.2" xref="S5.T2.16.16.16.16.3.m1.1.1.2.cmml"><mi id="S5.T2.16.16.16.16.3.m1.1.1.2.2" xref="S5.T2.16.16.16.16.3.m1.1.1.2.2.cmml">x</mi><mi id="S5.T2.16.16.16.16.3.m1.1.1.2.3" xref="S5.T2.16.16.16.16.3.m1.1.1.2.3.cmml">i</mi></msub><mo id="S5.T2.16.16.16.16.3.m1.1.1.1" stretchy="false" xref="S5.T2.16.16.16.16.3.m1.1.1.1.cmml">→</mo><mrow id="S5.T2.16.16.16.16.3.m1.1.1.3" xref="S5.T2.16.16.16.16.3.m1.1.1.3.cmml"><msub id="S5.T2.16.16.16.16.3.m1.1.1.3.2" xref="S5.T2.16.16.16.16.3.m1.1.1.3.2.cmml"><mi id="S5.T2.16.16.16.16.3.m1.1.1.3.2.2" xref="S5.T2.16.16.16.16.3.m1.1.1.3.2.2.cmml">𝐳</mi><mi id="S5.T2.16.16.16.16.3.m1.1.1.3.2.3" xref="S5.T2.16.16.16.16.3.m1.1.1.3.2.3.cmml">i</mi></msub><mo id="S5.T2.16.16.16.16.3.m1.1.1.3.1" xref="S5.T2.16.16.16.16.3.m1.1.1.3.1.cmml">⊕</mo><msub id="S5.T2.16.16.16.16.3.m1.1.1.3.3" xref="S5.T2.16.16.16.16.3.m1.1.1.3.3.cmml"><mi id="S5.T2.16.16.16.16.3.m1.1.1.3.3.2" xref="S5.T2.16.16.16.16.3.m1.1.1.3.3.2.cmml">y</mi><mi id="S5.T2.16.16.16.16.3.m1.1.1.3.3.3" xref="S5.T2.16.16.16.16.3.m1.1.1.3.3.3.cmml">i</mi></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.T2.16.16.16.16.3.m1.1b"><apply id="S5.T2.16.16.16.16.3.m1.1.1.cmml" xref="S5.T2.16.16.16.16.3.m1.1.1"><ci id="S5.T2.16.16.16.16.3.m1.1.1.1.cmml" xref="S5.T2.16.16.16.16.3.m1.1.1.1">→</ci><apply id="S5.T2.16.16.16.16.3.m1.1.1.2.cmml" xref="S5.T2.16.16.16.16.3.m1.1.1.2"><csymbol cd="ambiguous" id="S5.T2.16.16.16.16.3.m1.1.1.2.1.cmml" xref="S5.T2.16.16.16.16.3.m1.1.1.2">subscript</csymbol><ci id="S5.T2.16.16.16.16.3.m1.1.1.2.2.cmml" xref="S5.T2.16.16.16.16.3.m1.1.1.2.2">𝑥</ci><ci id="S5.T2.16.16.16.16.3.m1.1.1.2.3.cmml" xref="S5.T2.16.16.16.16.3.m1.1.1.2.3">𝑖</ci></apply><apply id="S5.T2.16.16.16.16.3.m1.1.1.3.cmml" xref="S5.T2.16.16.16.16.3.m1.1.1.3"><csymbol cd="latexml" id="S5.T2.16.16.16.16.3.m1.1.1.3.1.cmml" xref="S5.T2.16.16.16.16.3.m1.1.1.3.1">direct-sum</csymbol><apply id="S5.T2.16.16.16.16.3.m1.1.1.3.2.cmml" xref="S5.T2.16.16.16.16.3.m1.1.1.3.2"><csymbol cd="ambiguous" id="S5.T2.16.16.16.16.3.m1.1.1.3.2.1.cmml" xref="S5.T2.16.16.16.16.3.m1.1.1.3.2">subscript</csymbol><ci id="S5.T2.16.16.16.16.3.m1.1.1.3.2.2.cmml" xref="S5.T2.16.16.16.16.3.m1.1.1.3.2.2">𝐳</ci><ci id="S5.T2.16.16.16.16.3.m1.1.1.3.2.3.cmml" xref="S5.T2.16.16.16.16.3.m1.1.1.3.2.3">𝑖</ci></apply><apply id="S5.T2.16.16.16.16.3.m1.1.1.3.3.cmml" xref="S5.T2.16.16.16.16.3.m1.1.1.3.3"><csymbol cd="ambiguous" id="S5.T2.16.16.16.16.3.m1.1.1.3.3.1.cmml" xref="S5.T2.16.16.16.16.3.m1.1.1.3.3">subscript</csymbol><ci id="S5.T2.16.16.16.16.3.m1.1.1.3.3.2.cmml" xref="S5.T2.16.16.16.16.3.m1.1.1.3.3.2">𝑦</ci><ci id="S5.T2.16.16.16.16.3.m1.1.1.3.3.3.cmml" xref="S5.T2.16.16.16.16.3.m1.1.1.3.3.3">𝑖</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.T2.16.16.16.16.3.m1.1c">x_{i}\rightarrow{\mathbf{z}}_{i}\oplus y_{i}</annotation><annotation encoding="application/x-llamapun" id="S5.T2.16.16.16.16.3.m1.1d">italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT → bold_z start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ⊕ italic_y start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math></span> <span class="ltx_td ltx_align_center ltx_border_r" id="S5.T2.16.16.16.16.4"><span class="ltx_text ltx_font_bold" id="S5.T2.16.16.16.16.4.1">49.39</span></span> <span class="ltx_td ltx_align_center ltx_border_r" id="S5.T2.16.16.16.16.5"><span class="ltx_text ltx_font_bold" id="S5.T2.16.16.16.16.5.1">14.69</span></span> <span class="ltx_td ltx_align_center ltx_border_r" id="S5.T2.16.16.16.16.6"><span class="ltx_text ltx_font_bold" id="S5.T2.16.16.16.16.6.1">19.68</span></span> <span class="ltx_td ltx_nopad_r ltx_align_center" id="S5.T2.16.16.16.16.7"><span class="ltx_text ltx_font_bold" id="S5.T2.16.16.16.16.7.1">46.61</span></span></span> <span class="ltx_tr" id="S5.T2.20.20.20.20"> <span class="ltx_td ltx_th ltx_th_row ltx_border_r" id="S5.T2.20.20.20.20.5"></span> <span class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_r" id="S5.T2.19.19.19.19.3"><math alttext="x_{i}\rightarrow y_{i}" class="ltx_Math" display="inline" id="S5.T2.17.17.17.17.1.m1.1"><semantics id="S5.T2.17.17.17.17.1.m1.1a"><mrow id="S5.T2.17.17.17.17.1.m1.1.1" xref="S5.T2.17.17.17.17.1.m1.1.1.cmml"><msub id="S5.T2.17.17.17.17.1.m1.1.1.2" xref="S5.T2.17.17.17.17.1.m1.1.1.2.cmml"><mi id="S5.T2.17.17.17.17.1.m1.1.1.2.2" xref="S5.T2.17.17.17.17.1.m1.1.1.2.2.cmml">x</mi><mi id="S5.T2.17.17.17.17.1.m1.1.1.2.3" xref="S5.T2.17.17.17.17.1.m1.1.1.2.3.cmml">i</mi></msub><mo id="S5.T2.17.17.17.17.1.m1.1.1.1" stretchy="false" xref="S5.T2.17.17.17.17.1.m1.1.1.1.cmml">→</mo><msub id="S5.T2.17.17.17.17.1.m1.1.1.3" xref="S5.T2.17.17.17.17.1.m1.1.1.3.cmml"><mi id="S5.T2.17.17.17.17.1.m1.1.1.3.2" xref="S5.T2.17.17.17.17.1.m1.1.1.3.2.cmml">y</mi><mi id="S5.T2.17.17.17.17.1.m1.1.1.3.3" xref="S5.T2.17.17.17.17.1.m1.1.1.3.3.cmml">i</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S5.T2.17.17.17.17.1.m1.1b"><apply id="S5.T2.17.17.17.17.1.m1.1.1.cmml" xref="S5.T2.17.17.17.17.1.m1.1.1"><ci id="S5.T2.17.17.17.17.1.m1.1.1.1.cmml" xref="S5.T2.17.17.17.17.1.m1.1.1.1">→</ci><apply id="S5.T2.17.17.17.17.1.m1.1.1.2.cmml" xref="S5.T2.17.17.17.17.1.m1.1.1.2"><csymbol cd="ambiguous" id="S5.T2.17.17.17.17.1.m1.1.1.2.1.cmml" xref="S5.T2.17.17.17.17.1.m1.1.1.2">subscript</csymbol><ci id="S5.T2.17.17.17.17.1.m1.1.1.2.2.cmml" xref="S5.T2.17.17.17.17.1.m1.1.1.2.2">𝑥</ci><ci id="S5.T2.17.17.17.17.1.m1.1.1.2.3.cmml" xref="S5.T2.17.17.17.17.1.m1.1.1.2.3">𝑖</ci></apply><apply id="S5.T2.17.17.17.17.1.m1.1.1.3.cmml" xref="S5.T2.17.17.17.17.1.m1.1.1.3"><csymbol cd="ambiguous" id="S5.T2.17.17.17.17.1.m1.1.1.3.1.cmml" xref="S5.T2.17.17.17.17.1.m1.1.1.3">subscript</csymbol><ci id="S5.T2.17.17.17.17.1.m1.1.1.3.2.cmml" xref="S5.T2.17.17.17.17.1.m1.1.1.3.2">𝑦</ci><ci id="S5.T2.17.17.17.17.1.m1.1.1.3.3.cmml" xref="S5.T2.17.17.17.17.1.m1.1.1.3.3">𝑖</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.T2.17.17.17.17.1.m1.1c">x_{i}\rightarrow y_{i}</annotation><annotation encoding="application/x-llamapun" id="S5.T2.17.17.17.17.1.m1.1d">italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT → italic_y start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math>; <math alttext="x_{i}\rightarrow{\mathbf{z}}_{i}\oplus y_{i}" class="ltx_Math" display="inline" id="S5.T2.18.18.18.18.2.m2.1"><semantics id="S5.T2.18.18.18.18.2.m2.1a"><mrow id="S5.T2.18.18.18.18.2.m2.1.1" xref="S5.T2.18.18.18.18.2.m2.1.1.cmml"><msub id="S5.T2.18.18.18.18.2.m2.1.1.2" xref="S5.T2.18.18.18.18.2.m2.1.1.2.cmml"><mi id="S5.T2.18.18.18.18.2.m2.1.1.2.2" xref="S5.T2.18.18.18.18.2.m2.1.1.2.2.cmml">x</mi><mi id="S5.T2.18.18.18.18.2.m2.1.1.2.3" xref="S5.T2.18.18.18.18.2.m2.1.1.2.3.cmml">i</mi></msub><mo id="S5.T2.18.18.18.18.2.m2.1.1.1" stretchy="false" xref="S5.T2.18.18.18.18.2.m2.1.1.1.cmml">→</mo><mrow id="S5.T2.18.18.18.18.2.m2.1.1.3" xref="S5.T2.18.18.18.18.2.m2.1.1.3.cmml"><msub id="S5.T2.18.18.18.18.2.m2.1.1.3.2" xref="S5.T2.18.18.18.18.2.m2.1.1.3.2.cmml"><mi id="S5.T2.18.18.18.18.2.m2.1.1.3.2.2" xref="S5.T2.18.18.18.18.2.m2.1.1.3.2.2.cmml">𝐳</mi><mi id="S5.T2.18.18.18.18.2.m2.1.1.3.2.3" xref="S5.T2.18.18.18.18.2.m2.1.1.3.2.3.cmml">i</mi></msub><mo id="S5.T2.18.18.18.18.2.m2.1.1.3.1" xref="S5.T2.18.18.18.18.2.m2.1.1.3.1.cmml">⊕</mo><msub id="S5.T2.18.18.18.18.2.m2.1.1.3.3" xref="S5.T2.18.18.18.18.2.m2.1.1.3.3.cmml"><mi id="S5.T2.18.18.18.18.2.m2.1.1.3.3.2" xref="S5.T2.18.18.18.18.2.m2.1.1.3.3.2.cmml">y</mi><mi id="S5.T2.18.18.18.18.2.m2.1.1.3.3.3" xref="S5.T2.18.18.18.18.2.m2.1.1.3.3.3.cmml">i</mi></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.T2.18.18.18.18.2.m2.1b"><apply id="S5.T2.18.18.18.18.2.m2.1.1.cmml" xref="S5.T2.18.18.18.18.2.m2.1.1"><ci id="S5.T2.18.18.18.18.2.m2.1.1.1.cmml" xref="S5.T2.18.18.18.18.2.m2.1.1.1">→</ci><apply id="S5.T2.18.18.18.18.2.m2.1.1.2.cmml" xref="S5.T2.18.18.18.18.2.m2.1.1.2"><csymbol cd="ambiguous" id="S5.T2.18.18.18.18.2.m2.1.1.2.1.cmml" xref="S5.T2.18.18.18.18.2.m2.1.1.2">subscript</csymbol><ci id="S5.T2.18.18.18.18.2.m2.1.1.2.2.cmml" xref="S5.T2.18.18.18.18.2.m2.1.1.2.2">𝑥</ci><ci id="S5.T2.18.18.18.18.2.m2.1.1.2.3.cmml" xref="S5.T2.18.18.18.18.2.m2.1.1.2.3">𝑖</ci></apply><apply id="S5.T2.18.18.18.18.2.m2.1.1.3.cmml" xref="S5.T2.18.18.18.18.2.m2.1.1.3"><csymbol cd="latexml" id="S5.T2.18.18.18.18.2.m2.1.1.3.1.cmml" xref="S5.T2.18.18.18.18.2.m2.1.1.3.1">direct-sum</csymbol><apply id="S5.T2.18.18.18.18.2.m2.1.1.3.2.cmml" xref="S5.T2.18.18.18.18.2.m2.1.1.3.2"><csymbol cd="ambiguous" id="S5.T2.18.18.18.18.2.m2.1.1.3.2.1.cmml" xref="S5.T2.18.18.18.18.2.m2.1.1.3.2">subscript</csymbol><ci id="S5.T2.18.18.18.18.2.m2.1.1.3.2.2.cmml" xref="S5.T2.18.18.18.18.2.m2.1.1.3.2.2">𝐳</ci><ci id="S5.T2.18.18.18.18.2.m2.1.1.3.2.3.cmml" xref="S5.T2.18.18.18.18.2.m2.1.1.3.2.3">𝑖</ci></apply><apply id="S5.T2.18.18.18.18.2.m2.1.1.3.3.cmml" xref="S5.T2.18.18.18.18.2.m2.1.1.3.3"><csymbol cd="ambiguous" id="S5.T2.18.18.18.18.2.m2.1.1.3.3.1.cmml" xref="S5.T2.18.18.18.18.2.m2.1.1.3.3">subscript</csymbol><ci id="S5.T2.18.18.18.18.2.m2.1.1.3.3.2.cmml" xref="S5.T2.18.18.18.18.2.m2.1.1.3.3.2">𝑦</ci><ci id="S5.T2.18.18.18.18.2.m2.1.1.3.3.3.cmml" xref="S5.T2.18.18.18.18.2.m2.1.1.3.3.3">𝑖</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.T2.18.18.18.18.2.m2.1c">x_{i}\rightarrow{\mathbf{z}}_{i}\oplus y_{i}</annotation><annotation encoding="application/x-llamapun" id="S5.T2.18.18.18.18.2.m2.1d">italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT → bold_z start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ⊕ italic_y start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math>; <math alttext="x_{i}\oplus{\mathbf{z}}_{i}\rightarrow y_{i}" class="ltx_Math" display="inline" id="S5.T2.19.19.19.19.3.m3.1"><semantics id="S5.T2.19.19.19.19.3.m3.1a"><mrow id="S5.T2.19.19.19.19.3.m3.1.1" xref="S5.T2.19.19.19.19.3.m3.1.1.cmml"><mrow id="S5.T2.19.19.19.19.3.m3.1.1.2" xref="S5.T2.19.19.19.19.3.m3.1.1.2.cmml"><msub id="S5.T2.19.19.19.19.3.m3.1.1.2.2" xref="S5.T2.19.19.19.19.3.m3.1.1.2.2.cmml"><mi id="S5.T2.19.19.19.19.3.m3.1.1.2.2.2" xref="S5.T2.19.19.19.19.3.m3.1.1.2.2.2.cmml">x</mi><mi id="S5.T2.19.19.19.19.3.m3.1.1.2.2.3" xref="S5.T2.19.19.19.19.3.m3.1.1.2.2.3.cmml">i</mi></msub><mo id="S5.T2.19.19.19.19.3.m3.1.1.2.1" xref="S5.T2.19.19.19.19.3.m3.1.1.2.1.cmml">⊕</mo><msub id="S5.T2.19.19.19.19.3.m3.1.1.2.3" xref="S5.T2.19.19.19.19.3.m3.1.1.2.3.cmml"><mi id="S5.T2.19.19.19.19.3.m3.1.1.2.3.2" xref="S5.T2.19.19.19.19.3.m3.1.1.2.3.2.cmml">𝐳</mi><mi id="S5.T2.19.19.19.19.3.m3.1.1.2.3.3" xref="S5.T2.19.19.19.19.3.m3.1.1.2.3.3.cmml">i</mi></msub></mrow><mo id="S5.T2.19.19.19.19.3.m3.1.1.1" stretchy="false" xref="S5.T2.19.19.19.19.3.m3.1.1.1.cmml">→</mo><msub id="S5.T2.19.19.19.19.3.m3.1.1.3" xref="S5.T2.19.19.19.19.3.m3.1.1.3.cmml"><mi id="S5.T2.19.19.19.19.3.m3.1.1.3.2" xref="S5.T2.19.19.19.19.3.m3.1.1.3.2.cmml">y</mi><mi id="S5.T2.19.19.19.19.3.m3.1.1.3.3" xref="S5.T2.19.19.19.19.3.m3.1.1.3.3.cmml">i</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S5.T2.19.19.19.19.3.m3.1b"><apply id="S5.T2.19.19.19.19.3.m3.1.1.cmml" xref="S5.T2.19.19.19.19.3.m3.1.1"><ci id="S5.T2.19.19.19.19.3.m3.1.1.1.cmml" xref="S5.T2.19.19.19.19.3.m3.1.1.1">→</ci><apply id="S5.T2.19.19.19.19.3.m3.1.1.2.cmml" xref="S5.T2.19.19.19.19.3.m3.1.1.2"><csymbol cd="latexml" id="S5.T2.19.19.19.19.3.m3.1.1.2.1.cmml" xref="S5.T2.19.19.19.19.3.m3.1.1.2.1">direct-sum</csymbol><apply id="S5.T2.19.19.19.19.3.m3.1.1.2.2.cmml" xref="S5.T2.19.19.19.19.3.m3.1.1.2.2"><csymbol cd="ambiguous" id="S5.T2.19.19.19.19.3.m3.1.1.2.2.1.cmml" xref="S5.T2.19.19.19.19.3.m3.1.1.2.2">subscript</csymbol><ci id="S5.T2.19.19.19.19.3.m3.1.1.2.2.2.cmml" xref="S5.T2.19.19.19.19.3.m3.1.1.2.2.2">𝑥</ci><ci id="S5.T2.19.19.19.19.3.m3.1.1.2.2.3.cmml" xref="S5.T2.19.19.19.19.3.m3.1.1.2.2.3">𝑖</ci></apply><apply id="S5.T2.19.19.19.19.3.m3.1.1.2.3.cmml" xref="S5.T2.19.19.19.19.3.m3.1.1.2.3"><csymbol cd="ambiguous" id="S5.T2.19.19.19.19.3.m3.1.1.2.3.1.cmml" xref="S5.T2.19.19.19.19.3.m3.1.1.2.3">subscript</csymbol><ci id="S5.T2.19.19.19.19.3.m3.1.1.2.3.2.cmml" xref="S5.T2.19.19.19.19.3.m3.1.1.2.3.2">𝐳</ci><ci id="S5.T2.19.19.19.19.3.m3.1.1.2.3.3.cmml" xref="S5.T2.19.19.19.19.3.m3.1.1.2.3.3">𝑖</ci></apply></apply><apply id="S5.T2.19.19.19.19.3.m3.1.1.3.cmml" xref="S5.T2.19.19.19.19.3.m3.1.1.3"><csymbol cd="ambiguous" id="S5.T2.19.19.19.19.3.m3.1.1.3.1.cmml" xref="S5.T2.19.19.19.19.3.m3.1.1.3">subscript</csymbol><ci id="S5.T2.19.19.19.19.3.m3.1.1.3.2.cmml" xref="S5.T2.19.19.19.19.3.m3.1.1.3.2">𝑦</ci><ci id="S5.T2.19.19.19.19.3.m3.1.1.3.3.cmml" xref="S5.T2.19.19.19.19.3.m3.1.1.3.3">𝑖</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.T2.19.19.19.19.3.m3.1c">x_{i}\oplus{\mathbf{z}}_{i}\rightarrow y_{i}</annotation><annotation encoding="application/x-llamapun" id="S5.T2.19.19.19.19.3.m3.1d">italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ⊕ bold_z start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT → italic_y start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math></span> <span class="ltx_td ltx_align_center ltx_th ltx_th_row ltx_border_r" id="S5.T2.20.20.20.20.4"><math alttext="x_{i}\rightarrow y_{i}" class="ltx_Math" display="inline" id="S5.T2.20.20.20.20.4.m1.1"><semantics id="S5.T2.20.20.20.20.4.m1.1a"><mrow id="S5.T2.20.20.20.20.4.m1.1.1" xref="S5.T2.20.20.20.20.4.m1.1.1.cmml"><msub id="S5.T2.20.20.20.20.4.m1.1.1.2" xref="S5.T2.20.20.20.20.4.m1.1.1.2.cmml"><mi id="S5.T2.20.20.20.20.4.m1.1.1.2.2" xref="S5.T2.20.20.20.20.4.m1.1.1.2.2.cmml">x</mi><mi id="S5.T2.20.20.20.20.4.m1.1.1.2.3" xref="S5.T2.20.20.20.20.4.m1.1.1.2.3.cmml">i</mi></msub><mo id="S5.T2.20.20.20.20.4.m1.1.1.1" stretchy="false" xref="S5.T2.20.20.20.20.4.m1.1.1.1.cmml">→</mo><msub id="S5.T2.20.20.20.20.4.m1.1.1.3" xref="S5.T2.20.20.20.20.4.m1.1.1.3.cmml"><mi id="S5.T2.20.20.20.20.4.m1.1.1.3.2" xref="S5.T2.20.20.20.20.4.m1.1.1.3.2.cmml">y</mi><mi id="S5.T2.20.20.20.20.4.m1.1.1.3.3" xref="S5.T2.20.20.20.20.4.m1.1.1.3.3.cmml">i</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S5.T2.20.20.20.20.4.m1.1b"><apply id="S5.T2.20.20.20.20.4.m1.1.1.cmml" xref="S5.T2.20.20.20.20.4.m1.1.1"><ci id="S5.T2.20.20.20.20.4.m1.1.1.1.cmml" xref="S5.T2.20.20.20.20.4.m1.1.1.1">→</ci><apply id="S5.T2.20.20.20.20.4.m1.1.1.2.cmml" xref="S5.T2.20.20.20.20.4.m1.1.1.2"><csymbol cd="ambiguous" id="S5.T2.20.20.20.20.4.m1.1.1.2.1.cmml" xref="S5.T2.20.20.20.20.4.m1.1.1.2">subscript</csymbol><ci id="S5.T2.20.20.20.20.4.m1.1.1.2.2.cmml" xref="S5.T2.20.20.20.20.4.m1.1.1.2.2">𝑥</ci><ci id="S5.T2.20.20.20.20.4.m1.1.1.2.3.cmml" xref="S5.T2.20.20.20.20.4.m1.1.1.2.3">𝑖</ci></apply><apply id="S5.T2.20.20.20.20.4.m1.1.1.3.cmml" xref="S5.T2.20.20.20.20.4.m1.1.1.3"><csymbol cd="ambiguous" id="S5.T2.20.20.20.20.4.m1.1.1.3.1.cmml" xref="S5.T2.20.20.20.20.4.m1.1.1.3">subscript</csymbol><ci id="S5.T2.20.20.20.20.4.m1.1.1.3.2.cmml" xref="S5.T2.20.20.20.20.4.m1.1.1.3.2">𝑦</ci><ci id="S5.T2.20.20.20.20.4.m1.1.1.3.3.cmml" xref="S5.T2.20.20.20.20.4.m1.1.1.3.3">𝑖</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.T2.20.20.20.20.4.m1.1c">x_{i}\rightarrow y_{i}</annotation><annotation encoding="application/x-llamapun" id="S5.T2.20.20.20.20.4.m1.1d">italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT → italic_y start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math></span> <span class="ltx_td ltx_align_center ltx_border_r" id="S5.T2.20.20.20.20.6">47.99</span> <span class="ltx_td ltx_align_center ltx_border_r" id="S5.T2.20.20.20.20.7">14.21</span> <span class="ltx_td ltx_align_center ltx_border_r" id="S5.T2.20.20.20.20.8">19.42</span> <span class="ltx_td ltx_nopad_r ltx_align_center" id="S5.T2.20.20.20.20.9">45.33</span></span> <span class="ltx_tr" id="S5.T2.22.22.22.22"> <span class="ltx_td ltx_align_center ltx_th ltx_th_row ltx_border_r ltx_border_t ltx_rowspan ltx_rowspan_5" id="S5.T2.22.22.22.22.3"><span class="ltx_text" id="S5.T2.22.22.22.22.3.1">Wikipedia</span></span> <span class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_r ltx_border_t" id="S5.T2.21.21.21.21.1"><math alttext="x_{i}\rightarrow y_{i}" class="ltx_Math" display="inline" id="S5.T2.21.21.21.21.1.m1.1"><semantics id="S5.T2.21.21.21.21.1.m1.1a"><mrow id="S5.T2.21.21.21.21.1.m1.1.1" xref="S5.T2.21.21.21.21.1.m1.1.1.cmml"><msub id="S5.T2.21.21.21.21.1.m1.1.1.2" xref="S5.T2.21.21.21.21.1.m1.1.1.2.cmml"><mi id="S5.T2.21.21.21.21.1.m1.1.1.2.2" xref="S5.T2.21.21.21.21.1.m1.1.1.2.2.cmml">x</mi><mi id="S5.T2.21.21.21.21.1.m1.1.1.2.3" xref="S5.T2.21.21.21.21.1.m1.1.1.2.3.cmml">i</mi></msub><mo id="S5.T2.21.21.21.21.1.m1.1.1.1" stretchy="false" xref="S5.T2.21.21.21.21.1.m1.1.1.1.cmml">→</mo><msub id="S5.T2.21.21.21.21.1.m1.1.1.3" xref="S5.T2.21.21.21.21.1.m1.1.1.3.cmml"><mi id="S5.T2.21.21.21.21.1.m1.1.1.3.2" xref="S5.T2.21.21.21.21.1.m1.1.1.3.2.cmml">y</mi><mi id="S5.T2.21.21.21.21.1.m1.1.1.3.3" xref="S5.T2.21.21.21.21.1.m1.1.1.3.3.cmml">i</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S5.T2.21.21.21.21.1.m1.1b"><apply id="S5.T2.21.21.21.21.1.m1.1.1.cmml" xref="S5.T2.21.21.21.21.1.m1.1.1"><ci id="S5.T2.21.21.21.21.1.m1.1.1.1.cmml" xref="S5.T2.21.21.21.21.1.m1.1.1.1">→</ci><apply id="S5.T2.21.21.21.21.1.m1.1.1.2.cmml" xref="S5.T2.21.21.21.21.1.m1.1.1.2"><csymbol cd="ambiguous" id="S5.T2.21.21.21.21.1.m1.1.1.2.1.cmml" xref="S5.T2.21.21.21.21.1.m1.1.1.2">subscript</csymbol><ci id="S5.T2.21.21.21.21.1.m1.1.1.2.2.cmml" xref="S5.T2.21.21.21.21.1.m1.1.1.2.2">𝑥</ci><ci id="S5.T2.21.21.21.21.1.m1.1.1.2.3.cmml" xref="S5.T2.21.21.21.21.1.m1.1.1.2.3">𝑖</ci></apply><apply id="S5.T2.21.21.21.21.1.m1.1.1.3.cmml" xref="S5.T2.21.21.21.21.1.m1.1.1.3"><csymbol cd="ambiguous" id="S5.T2.21.21.21.21.1.m1.1.1.3.1.cmml" xref="S5.T2.21.21.21.21.1.m1.1.1.3">subscript</csymbol><ci id="S5.T2.21.21.21.21.1.m1.1.1.3.2.cmml" xref="S5.T2.21.21.21.21.1.m1.1.1.3.2">𝑦</ci><ci id="S5.T2.21.21.21.21.1.m1.1.1.3.3.cmml" xref="S5.T2.21.21.21.21.1.m1.1.1.3.3">𝑖</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.T2.21.21.21.21.1.m1.1c">x_{i}\rightarrow y_{i}</annotation><annotation encoding="application/x-llamapun" id="S5.T2.21.21.21.21.1.m1.1d">italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT → italic_y start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math></span> <span class="ltx_td ltx_align_center ltx_th ltx_th_row ltx_border_r ltx_border_t" id="S5.T2.22.22.22.22.2"><math alttext="x_{i}\rightarrow y_{i}" class="ltx_Math" display="inline" id="S5.T2.22.22.22.22.2.m1.1"><semantics id="S5.T2.22.22.22.22.2.m1.1a"><mrow id="S5.T2.22.22.22.22.2.m1.1.1" xref="S5.T2.22.22.22.22.2.m1.1.1.cmml"><msub id="S5.T2.22.22.22.22.2.m1.1.1.2" xref="S5.T2.22.22.22.22.2.m1.1.1.2.cmml"><mi id="S5.T2.22.22.22.22.2.m1.1.1.2.2" xref="S5.T2.22.22.22.22.2.m1.1.1.2.2.cmml">x</mi><mi id="S5.T2.22.22.22.22.2.m1.1.1.2.3" xref="S5.T2.22.22.22.22.2.m1.1.1.2.3.cmml">i</mi></msub><mo id="S5.T2.22.22.22.22.2.m1.1.1.1" stretchy="false" xref="S5.T2.22.22.22.22.2.m1.1.1.1.cmml">→</mo><msub id="S5.T2.22.22.22.22.2.m1.1.1.3" xref="S5.T2.22.22.22.22.2.m1.1.1.3.cmml"><mi id="S5.T2.22.22.22.22.2.m1.1.1.3.2" xref="S5.T2.22.22.22.22.2.m1.1.1.3.2.cmml">y</mi><mi id="S5.T2.22.22.22.22.2.m1.1.1.3.3" xref="S5.T2.22.22.22.22.2.m1.1.1.3.3.cmml">i</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S5.T2.22.22.22.22.2.m1.1b"><apply id="S5.T2.22.22.22.22.2.m1.1.1.cmml" xref="S5.T2.22.22.22.22.2.m1.1.1"><ci id="S5.T2.22.22.22.22.2.m1.1.1.1.cmml" xref="S5.T2.22.22.22.22.2.m1.1.1.1">→</ci><apply id="S5.T2.22.22.22.22.2.m1.1.1.2.cmml" xref="S5.T2.22.22.22.22.2.m1.1.1.2"><csymbol cd="ambiguous" id="S5.T2.22.22.22.22.2.m1.1.1.2.1.cmml" xref="S5.T2.22.22.22.22.2.m1.1.1.2">subscript</csymbol><ci id="S5.T2.22.22.22.22.2.m1.1.1.2.2.cmml" xref="S5.T2.22.22.22.22.2.m1.1.1.2.2">𝑥</ci><ci id="S5.T2.22.22.22.22.2.m1.1.1.2.3.cmml" xref="S5.T2.22.22.22.22.2.m1.1.1.2.3">𝑖</ci></apply><apply id="S5.T2.22.22.22.22.2.m1.1.1.3.cmml" xref="S5.T2.22.22.22.22.2.m1.1.1.3"><csymbol cd="ambiguous" id="S5.T2.22.22.22.22.2.m1.1.1.3.1.cmml" xref="S5.T2.22.22.22.22.2.m1.1.1.3">subscript</csymbol><ci id="S5.T2.22.22.22.22.2.m1.1.1.3.2.cmml" xref="S5.T2.22.22.22.22.2.m1.1.1.3.2">𝑦</ci><ci id="S5.T2.22.22.22.22.2.m1.1.1.3.3.cmml" xref="S5.T2.22.22.22.22.2.m1.1.1.3.3">𝑖</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.T2.22.22.22.22.2.m1.1c">x_{i}\rightarrow y_{i}</annotation><annotation encoding="application/x-llamapun" id="S5.T2.22.22.22.22.2.m1.1d">italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT → italic_y start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math></span> <span class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S5.T2.22.22.22.22.4">42.61</span> <span class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S5.T2.22.22.22.22.5">13.69</span> <span class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S5.T2.22.22.22.22.6">16.30</span> <span class="ltx_td ltx_nopad_r ltx_align_center ltx_border_t" id="S5.T2.22.22.22.22.7">41.28</span></span> <span class="ltx_tr" id="S5.T2.24.24.24.24"> <span class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_r" id="S5.T2.23.23.23.23.1"><math alttext="x_{i}\rightarrow{\mathbf{z}}_{i}\oplus y_{i}" class="ltx_Math" display="inline" id="S5.T2.23.23.23.23.1.m1.1"><semantics id="S5.T2.23.23.23.23.1.m1.1a"><mrow id="S5.T2.23.23.23.23.1.m1.1.1" xref="S5.T2.23.23.23.23.1.m1.1.1.cmml"><msub id="S5.T2.23.23.23.23.1.m1.1.1.2" xref="S5.T2.23.23.23.23.1.m1.1.1.2.cmml"><mi id="S5.T2.23.23.23.23.1.m1.1.1.2.2" xref="S5.T2.23.23.23.23.1.m1.1.1.2.2.cmml">x</mi><mi id="S5.T2.23.23.23.23.1.m1.1.1.2.3" xref="S5.T2.23.23.23.23.1.m1.1.1.2.3.cmml">i</mi></msub><mo id="S5.T2.23.23.23.23.1.m1.1.1.1" stretchy="false" xref="S5.T2.23.23.23.23.1.m1.1.1.1.cmml">→</mo><mrow id="S5.T2.23.23.23.23.1.m1.1.1.3" xref="S5.T2.23.23.23.23.1.m1.1.1.3.cmml"><msub id="S5.T2.23.23.23.23.1.m1.1.1.3.2" xref="S5.T2.23.23.23.23.1.m1.1.1.3.2.cmml"><mi id="S5.T2.23.23.23.23.1.m1.1.1.3.2.2" xref="S5.T2.23.23.23.23.1.m1.1.1.3.2.2.cmml">𝐳</mi><mi id="S5.T2.23.23.23.23.1.m1.1.1.3.2.3" xref="S5.T2.23.23.23.23.1.m1.1.1.3.2.3.cmml">i</mi></msub><mo id="S5.T2.23.23.23.23.1.m1.1.1.3.1" xref="S5.T2.23.23.23.23.1.m1.1.1.3.1.cmml">⊕</mo><msub id="S5.T2.23.23.23.23.1.m1.1.1.3.3" xref="S5.T2.23.23.23.23.1.m1.1.1.3.3.cmml"><mi id="S5.T2.23.23.23.23.1.m1.1.1.3.3.2" xref="S5.T2.23.23.23.23.1.m1.1.1.3.3.2.cmml">y</mi><mi id="S5.T2.23.23.23.23.1.m1.1.1.3.3.3" xref="S5.T2.23.23.23.23.1.m1.1.1.3.3.3.cmml">i</mi></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.T2.23.23.23.23.1.m1.1b"><apply id="S5.T2.23.23.23.23.1.m1.1.1.cmml" xref="S5.T2.23.23.23.23.1.m1.1.1"><ci id="S5.T2.23.23.23.23.1.m1.1.1.1.cmml" xref="S5.T2.23.23.23.23.1.m1.1.1.1">→</ci><apply id="S5.T2.23.23.23.23.1.m1.1.1.2.cmml" xref="S5.T2.23.23.23.23.1.m1.1.1.2"><csymbol cd="ambiguous" id="S5.T2.23.23.23.23.1.m1.1.1.2.1.cmml" xref="S5.T2.23.23.23.23.1.m1.1.1.2">subscript</csymbol><ci id="S5.T2.23.23.23.23.1.m1.1.1.2.2.cmml" xref="S5.T2.23.23.23.23.1.m1.1.1.2.2">𝑥</ci><ci id="S5.T2.23.23.23.23.1.m1.1.1.2.3.cmml" xref="S5.T2.23.23.23.23.1.m1.1.1.2.3">𝑖</ci></apply><apply id="S5.T2.23.23.23.23.1.m1.1.1.3.cmml" xref="S5.T2.23.23.23.23.1.m1.1.1.3"><csymbol cd="latexml" id="S5.T2.23.23.23.23.1.m1.1.1.3.1.cmml" xref="S5.T2.23.23.23.23.1.m1.1.1.3.1">direct-sum</csymbol><apply id="S5.T2.23.23.23.23.1.m1.1.1.3.2.cmml" xref="S5.T2.23.23.23.23.1.m1.1.1.3.2"><csymbol cd="ambiguous" id="S5.T2.23.23.23.23.1.m1.1.1.3.2.1.cmml" xref="S5.T2.23.23.23.23.1.m1.1.1.3.2">subscript</csymbol><ci id="S5.T2.23.23.23.23.1.m1.1.1.3.2.2.cmml" xref="S5.T2.23.23.23.23.1.m1.1.1.3.2.2">𝐳</ci><ci id="S5.T2.23.23.23.23.1.m1.1.1.3.2.3.cmml" xref="S5.T2.23.23.23.23.1.m1.1.1.3.2.3">𝑖</ci></apply><apply id="S5.T2.23.23.23.23.1.m1.1.1.3.3.cmml" xref="S5.T2.23.23.23.23.1.m1.1.1.3.3"><csymbol cd="ambiguous" id="S5.T2.23.23.23.23.1.m1.1.1.3.3.1.cmml" xref="S5.T2.23.23.23.23.1.m1.1.1.3.3">subscript</csymbol><ci id="S5.T2.23.23.23.23.1.m1.1.1.3.3.2.cmml" xref="S5.T2.23.23.23.23.1.m1.1.1.3.3.2">𝑦</ci><ci id="S5.T2.23.23.23.23.1.m1.1.1.3.3.3.cmml" xref="S5.T2.23.23.23.23.1.m1.1.1.3.3.3">𝑖</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.T2.23.23.23.23.1.m1.1c">x_{i}\rightarrow{\mathbf{z}}_{i}\oplus y_{i}</annotation><annotation encoding="application/x-llamapun" id="S5.T2.23.23.23.23.1.m1.1d">italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT → bold_z start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ⊕ italic_y start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math></span> <span class="ltx_td ltx_align_center ltx_th ltx_th_row ltx_border_r" id="S5.T2.24.24.24.24.2"><math alttext="x_{i}\rightarrow y_{i}" class="ltx_Math" display="inline" id="S5.T2.24.24.24.24.2.m1.1"><semantics id="S5.T2.24.24.24.24.2.m1.1a"><mrow id="S5.T2.24.24.24.24.2.m1.1.1" xref="S5.T2.24.24.24.24.2.m1.1.1.cmml"><msub id="S5.T2.24.24.24.24.2.m1.1.1.2" xref="S5.T2.24.24.24.24.2.m1.1.1.2.cmml"><mi id="S5.T2.24.24.24.24.2.m1.1.1.2.2" xref="S5.T2.24.24.24.24.2.m1.1.1.2.2.cmml">x</mi><mi id="S5.T2.24.24.24.24.2.m1.1.1.2.3" xref="S5.T2.24.24.24.24.2.m1.1.1.2.3.cmml">i</mi></msub><mo id="S5.T2.24.24.24.24.2.m1.1.1.1" stretchy="false" xref="S5.T2.24.24.24.24.2.m1.1.1.1.cmml">→</mo><msub id="S5.T2.24.24.24.24.2.m1.1.1.3" xref="S5.T2.24.24.24.24.2.m1.1.1.3.cmml"><mi id="S5.T2.24.24.24.24.2.m1.1.1.3.2" xref="S5.T2.24.24.24.24.2.m1.1.1.3.2.cmml">y</mi><mi id="S5.T2.24.24.24.24.2.m1.1.1.3.3" xref="S5.T2.24.24.24.24.2.m1.1.1.3.3.cmml">i</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S5.T2.24.24.24.24.2.m1.1b"><apply id="S5.T2.24.24.24.24.2.m1.1.1.cmml" xref="S5.T2.24.24.24.24.2.m1.1.1"><ci id="S5.T2.24.24.24.24.2.m1.1.1.1.cmml" xref="S5.T2.24.24.24.24.2.m1.1.1.1">→</ci><apply id="S5.T2.24.24.24.24.2.m1.1.1.2.cmml" xref="S5.T2.24.24.24.24.2.m1.1.1.2"><csymbol cd="ambiguous" id="S5.T2.24.24.24.24.2.m1.1.1.2.1.cmml" xref="S5.T2.24.24.24.24.2.m1.1.1.2">subscript</csymbol><ci id="S5.T2.24.24.24.24.2.m1.1.1.2.2.cmml" xref="S5.T2.24.24.24.24.2.m1.1.1.2.2">𝑥</ci><ci id="S5.T2.24.24.24.24.2.m1.1.1.2.3.cmml" xref="S5.T2.24.24.24.24.2.m1.1.1.2.3">𝑖</ci></apply><apply id="S5.T2.24.24.24.24.2.m1.1.1.3.cmml" xref="S5.T2.24.24.24.24.2.m1.1.1.3"><csymbol cd="ambiguous" id="S5.T2.24.24.24.24.2.m1.1.1.3.1.cmml" xref="S5.T2.24.24.24.24.2.m1.1.1.3">subscript</csymbol><ci id="S5.T2.24.24.24.24.2.m1.1.1.3.2.cmml" xref="S5.T2.24.24.24.24.2.m1.1.1.3.2">𝑦</ci><ci id="S5.T2.24.24.24.24.2.m1.1.1.3.3.cmml" xref="S5.T2.24.24.24.24.2.m1.1.1.3.3">𝑖</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.T2.24.24.24.24.2.m1.1c">x_{i}\rightarrow y_{i}</annotation><annotation encoding="application/x-llamapun" id="S5.T2.24.24.24.24.2.m1.1d">italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT → italic_y start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math></span> <span class="ltx_td ltx_align_center ltx_border_r" id="S5.T2.24.24.24.24.3">35.49</span> <span class="ltx_td ltx_align_center ltx_border_r" id="S5.T2.24.24.24.24.4">12.58</span> <span class="ltx_td ltx_align_center ltx_border_r" id="S5.T2.24.24.24.24.5">15.06</span> <span class="ltx_td ltx_nopad_r ltx_align_center" id="S5.T2.24.24.24.24.6">34.28</span></span> <span class="ltx_tr" id="S5.T2.26.26.26.26"> <span class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_r" id="S5.T2.25.25.25.25.1"><math alttext="x_{i}\rightarrow{\mathbf{z}}_{i}\oplus y_{i}" class="ltx_Math" display="inline" id="S5.T2.25.25.25.25.1.m1.1"><semantics id="S5.T2.25.25.25.25.1.m1.1a"><mrow id="S5.T2.25.25.25.25.1.m1.1.1" xref="S5.T2.25.25.25.25.1.m1.1.1.cmml"><msub id="S5.T2.25.25.25.25.1.m1.1.1.2" xref="S5.T2.25.25.25.25.1.m1.1.1.2.cmml"><mi id="S5.T2.25.25.25.25.1.m1.1.1.2.2" xref="S5.T2.25.25.25.25.1.m1.1.1.2.2.cmml">x</mi><mi id="S5.T2.25.25.25.25.1.m1.1.1.2.3" xref="S5.T2.25.25.25.25.1.m1.1.1.2.3.cmml">i</mi></msub><mo id="S5.T2.25.25.25.25.1.m1.1.1.1" stretchy="false" xref="S5.T2.25.25.25.25.1.m1.1.1.1.cmml">→</mo><mrow id="S5.T2.25.25.25.25.1.m1.1.1.3" xref="S5.T2.25.25.25.25.1.m1.1.1.3.cmml"><msub id="S5.T2.25.25.25.25.1.m1.1.1.3.2" xref="S5.T2.25.25.25.25.1.m1.1.1.3.2.cmml"><mi id="S5.T2.25.25.25.25.1.m1.1.1.3.2.2" xref="S5.T2.25.25.25.25.1.m1.1.1.3.2.2.cmml">𝐳</mi><mi id="S5.T2.25.25.25.25.1.m1.1.1.3.2.3" xref="S5.T2.25.25.25.25.1.m1.1.1.3.2.3.cmml">i</mi></msub><mo id="S5.T2.25.25.25.25.1.m1.1.1.3.1" xref="S5.T2.25.25.25.25.1.m1.1.1.3.1.cmml">⊕</mo><msub id="S5.T2.25.25.25.25.1.m1.1.1.3.3" xref="S5.T2.25.25.25.25.1.m1.1.1.3.3.cmml"><mi id="S5.T2.25.25.25.25.1.m1.1.1.3.3.2" xref="S5.T2.25.25.25.25.1.m1.1.1.3.3.2.cmml">y</mi><mi id="S5.T2.25.25.25.25.1.m1.1.1.3.3.3" xref="S5.T2.25.25.25.25.1.m1.1.1.3.3.3.cmml">i</mi></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.T2.25.25.25.25.1.m1.1b"><apply id="S5.T2.25.25.25.25.1.m1.1.1.cmml" xref="S5.T2.25.25.25.25.1.m1.1.1"><ci id="S5.T2.25.25.25.25.1.m1.1.1.1.cmml" xref="S5.T2.25.25.25.25.1.m1.1.1.1">→</ci><apply id="S5.T2.25.25.25.25.1.m1.1.1.2.cmml" xref="S5.T2.25.25.25.25.1.m1.1.1.2"><csymbol cd="ambiguous" id="S5.T2.25.25.25.25.1.m1.1.1.2.1.cmml" xref="S5.T2.25.25.25.25.1.m1.1.1.2">subscript</csymbol><ci id="S5.T2.25.25.25.25.1.m1.1.1.2.2.cmml" xref="S5.T2.25.25.25.25.1.m1.1.1.2.2">𝑥</ci><ci id="S5.T2.25.25.25.25.1.m1.1.1.2.3.cmml" xref="S5.T2.25.25.25.25.1.m1.1.1.2.3">𝑖</ci></apply><apply id="S5.T2.25.25.25.25.1.m1.1.1.3.cmml" xref="S5.T2.25.25.25.25.1.m1.1.1.3"><csymbol cd="latexml" id="S5.T2.25.25.25.25.1.m1.1.1.3.1.cmml" xref="S5.T2.25.25.25.25.1.m1.1.1.3.1">direct-sum</csymbol><apply id="S5.T2.25.25.25.25.1.m1.1.1.3.2.cmml" xref="S5.T2.25.25.25.25.1.m1.1.1.3.2"><csymbol cd="ambiguous" id="S5.T2.25.25.25.25.1.m1.1.1.3.2.1.cmml" xref="S5.T2.25.25.25.25.1.m1.1.1.3.2">subscript</csymbol><ci id="S5.T2.25.25.25.25.1.m1.1.1.3.2.2.cmml" xref="S5.T2.25.25.25.25.1.m1.1.1.3.2.2">𝐳</ci><ci id="S5.T2.25.25.25.25.1.m1.1.1.3.2.3.cmml" xref="S5.T2.25.25.25.25.1.m1.1.1.3.2.3">𝑖</ci></apply><apply id="S5.T2.25.25.25.25.1.m1.1.1.3.3.cmml" xref="S5.T2.25.25.25.25.1.m1.1.1.3.3"><csymbol cd="ambiguous" id="S5.T2.25.25.25.25.1.m1.1.1.3.3.1.cmml" xref="S5.T2.25.25.25.25.1.m1.1.1.3.3">subscript</csymbol><ci id="S5.T2.25.25.25.25.1.m1.1.1.3.3.2.cmml" xref="S5.T2.25.25.25.25.1.m1.1.1.3.3.2">𝑦</ci><ci id="S5.T2.25.25.25.25.1.m1.1.1.3.3.3.cmml" xref="S5.T2.25.25.25.25.1.m1.1.1.3.3.3">𝑖</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.T2.25.25.25.25.1.m1.1c">x_{i}\rightarrow{\mathbf{z}}_{i}\oplus y_{i}</annotation><annotation encoding="application/x-llamapun" id="S5.T2.25.25.25.25.1.m1.1d">italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT → bold_z start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ⊕ italic_y start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math></span> <span class="ltx_td ltx_align_center ltx_th ltx_th_row ltx_border_r" id="S5.T2.26.26.26.26.2"><math alttext="x_{i}\rightarrow{\mathbf{z}}_{i}\oplus y_{i}" class="ltx_Math" display="inline" id="S5.T2.26.26.26.26.2.m1.1"><semantics id="S5.T2.26.26.26.26.2.m1.1a"><mrow id="S5.T2.26.26.26.26.2.m1.1.1" xref="S5.T2.26.26.26.26.2.m1.1.1.cmml"><msub id="S5.T2.26.26.26.26.2.m1.1.1.2" xref="S5.T2.26.26.26.26.2.m1.1.1.2.cmml"><mi id="S5.T2.26.26.26.26.2.m1.1.1.2.2" xref="S5.T2.26.26.26.26.2.m1.1.1.2.2.cmml">x</mi><mi id="S5.T2.26.26.26.26.2.m1.1.1.2.3" xref="S5.T2.26.26.26.26.2.m1.1.1.2.3.cmml">i</mi></msub><mo id="S5.T2.26.26.26.26.2.m1.1.1.1" stretchy="false" xref="S5.T2.26.26.26.26.2.m1.1.1.1.cmml">→</mo><mrow id="S5.T2.26.26.26.26.2.m1.1.1.3" xref="S5.T2.26.26.26.26.2.m1.1.1.3.cmml"><msub id="S5.T2.26.26.26.26.2.m1.1.1.3.2" xref="S5.T2.26.26.26.26.2.m1.1.1.3.2.cmml"><mi id="S5.T2.26.26.26.26.2.m1.1.1.3.2.2" xref="S5.T2.26.26.26.26.2.m1.1.1.3.2.2.cmml">𝐳</mi><mi id="S5.T2.26.26.26.26.2.m1.1.1.3.2.3" xref="S5.T2.26.26.26.26.2.m1.1.1.3.2.3.cmml">i</mi></msub><mo id="S5.T2.26.26.26.26.2.m1.1.1.3.1" xref="S5.T2.26.26.26.26.2.m1.1.1.3.1.cmml">⊕</mo><msub id="S5.T2.26.26.26.26.2.m1.1.1.3.3" xref="S5.T2.26.26.26.26.2.m1.1.1.3.3.cmml"><mi id="S5.T2.26.26.26.26.2.m1.1.1.3.3.2" xref="S5.T2.26.26.26.26.2.m1.1.1.3.3.2.cmml">y</mi><mi id="S5.T2.26.26.26.26.2.m1.1.1.3.3.3" xref="S5.T2.26.26.26.26.2.m1.1.1.3.3.3.cmml">i</mi></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.T2.26.26.26.26.2.m1.1b"><apply id="S5.T2.26.26.26.26.2.m1.1.1.cmml" xref="S5.T2.26.26.26.26.2.m1.1.1"><ci id="S5.T2.26.26.26.26.2.m1.1.1.1.cmml" xref="S5.T2.26.26.26.26.2.m1.1.1.1">→</ci><apply id="S5.T2.26.26.26.26.2.m1.1.1.2.cmml" xref="S5.T2.26.26.26.26.2.m1.1.1.2"><csymbol cd="ambiguous" id="S5.T2.26.26.26.26.2.m1.1.1.2.1.cmml" xref="S5.T2.26.26.26.26.2.m1.1.1.2">subscript</csymbol><ci id="S5.T2.26.26.26.26.2.m1.1.1.2.2.cmml" xref="S5.T2.26.26.26.26.2.m1.1.1.2.2">𝑥</ci><ci id="S5.T2.26.26.26.26.2.m1.1.1.2.3.cmml" xref="S5.T2.26.26.26.26.2.m1.1.1.2.3">𝑖</ci></apply><apply id="S5.T2.26.26.26.26.2.m1.1.1.3.cmml" xref="S5.T2.26.26.26.26.2.m1.1.1.3"><csymbol cd="latexml" id="S5.T2.26.26.26.26.2.m1.1.1.3.1.cmml" xref="S5.T2.26.26.26.26.2.m1.1.1.3.1">direct-sum</csymbol><apply id="S5.T2.26.26.26.26.2.m1.1.1.3.2.cmml" xref="S5.T2.26.26.26.26.2.m1.1.1.3.2"><csymbol cd="ambiguous" id="S5.T2.26.26.26.26.2.m1.1.1.3.2.1.cmml" xref="S5.T2.26.26.26.26.2.m1.1.1.3.2">subscript</csymbol><ci id="S5.T2.26.26.26.26.2.m1.1.1.3.2.2.cmml" xref="S5.T2.26.26.26.26.2.m1.1.1.3.2.2">𝐳</ci><ci id="S5.T2.26.26.26.26.2.m1.1.1.3.2.3.cmml" xref="S5.T2.26.26.26.26.2.m1.1.1.3.2.3">𝑖</ci></apply><apply id="S5.T2.26.26.26.26.2.m1.1.1.3.3.cmml" xref="S5.T2.26.26.26.26.2.m1.1.1.3.3"><csymbol cd="ambiguous" id="S5.T2.26.26.26.26.2.m1.1.1.3.3.1.cmml" xref="S5.T2.26.26.26.26.2.m1.1.1.3.3">subscript</csymbol><ci id="S5.T2.26.26.26.26.2.m1.1.1.3.3.2.cmml" xref="S5.T2.26.26.26.26.2.m1.1.1.3.3.2">𝑦</ci><ci id="S5.T2.26.26.26.26.2.m1.1.1.3.3.3.cmml" xref="S5.T2.26.26.26.26.2.m1.1.1.3.3.3">𝑖</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.T2.26.26.26.26.2.m1.1c">x_{i}\rightarrow{\mathbf{z}}_{i}\oplus y_{i}</annotation><annotation encoding="application/x-llamapun" id="S5.T2.26.26.26.26.2.m1.1d">italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT → bold_z start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ⊕ italic_y start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math></span> <span class="ltx_td ltx_align_center ltx_border_r" id="S5.T2.26.26.26.26.3">44.04</span> <span class="ltx_td ltx_align_center ltx_border_r" id="S5.T2.26.26.26.26.4">14.39</span> <span class="ltx_td ltx_align_center ltx_border_r" id="S5.T2.26.26.26.26.5">16.98</span> <span class="ltx_td ltx_nopad_r ltx_align_center" id="S5.T2.26.26.26.26.6">42.69</span></span> <span class="ltx_tr" id="S5.T2.29.29.29.29"> <span class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_r" id="S5.T2.28.28.28.28.2"><math alttext="x_{i}\rightarrow y_{i}" class="ltx_Math" display="inline" id="S5.T2.27.27.27.27.1.m1.1"><semantics id="S5.T2.27.27.27.27.1.m1.1a"><mrow id="S5.T2.27.27.27.27.1.m1.1.1" xref="S5.T2.27.27.27.27.1.m1.1.1.cmml"><msub id="S5.T2.27.27.27.27.1.m1.1.1.2" xref="S5.T2.27.27.27.27.1.m1.1.1.2.cmml"><mi id="S5.T2.27.27.27.27.1.m1.1.1.2.2" xref="S5.T2.27.27.27.27.1.m1.1.1.2.2.cmml">x</mi><mi id="S5.T2.27.27.27.27.1.m1.1.1.2.3" xref="S5.T2.27.27.27.27.1.m1.1.1.2.3.cmml">i</mi></msub><mo id="S5.T2.27.27.27.27.1.m1.1.1.1" stretchy="false" xref="S5.T2.27.27.27.27.1.m1.1.1.1.cmml">→</mo><msub id="S5.T2.27.27.27.27.1.m1.1.1.3" xref="S5.T2.27.27.27.27.1.m1.1.1.3.cmml"><mi id="S5.T2.27.27.27.27.1.m1.1.1.3.2" xref="S5.T2.27.27.27.27.1.m1.1.1.3.2.cmml">y</mi><mi id="S5.T2.27.27.27.27.1.m1.1.1.3.3" xref="S5.T2.27.27.27.27.1.m1.1.1.3.3.cmml">i</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S5.T2.27.27.27.27.1.m1.1b"><apply id="S5.T2.27.27.27.27.1.m1.1.1.cmml" xref="S5.T2.27.27.27.27.1.m1.1.1"><ci id="S5.T2.27.27.27.27.1.m1.1.1.1.cmml" xref="S5.T2.27.27.27.27.1.m1.1.1.1">→</ci><apply id="S5.T2.27.27.27.27.1.m1.1.1.2.cmml" xref="S5.T2.27.27.27.27.1.m1.1.1.2"><csymbol cd="ambiguous" id="S5.T2.27.27.27.27.1.m1.1.1.2.1.cmml" xref="S5.T2.27.27.27.27.1.m1.1.1.2">subscript</csymbol><ci id="S5.T2.27.27.27.27.1.m1.1.1.2.2.cmml" xref="S5.T2.27.27.27.27.1.m1.1.1.2.2">𝑥</ci><ci id="S5.T2.27.27.27.27.1.m1.1.1.2.3.cmml" xref="S5.T2.27.27.27.27.1.m1.1.1.2.3">𝑖</ci></apply><apply id="S5.T2.27.27.27.27.1.m1.1.1.3.cmml" xref="S5.T2.27.27.27.27.1.m1.1.1.3"><csymbol cd="ambiguous" id="S5.T2.27.27.27.27.1.m1.1.1.3.1.cmml" xref="S5.T2.27.27.27.27.1.m1.1.1.3">subscript</csymbol><ci id="S5.T2.27.27.27.27.1.m1.1.1.3.2.cmml" xref="S5.T2.27.27.27.27.1.m1.1.1.3.2">𝑦</ci><ci id="S5.T2.27.27.27.27.1.m1.1.1.3.3.cmml" xref="S5.T2.27.27.27.27.1.m1.1.1.3.3">𝑖</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.T2.27.27.27.27.1.m1.1c">x_{i}\rightarrow y_{i}</annotation><annotation encoding="application/x-llamapun" id="S5.T2.27.27.27.27.1.m1.1d">italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT → italic_y start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math>; <math alttext="x_{i}\rightarrow{\mathbf{z}}_{i}\oplus y_{i}" class="ltx_Math" display="inline" id="S5.T2.28.28.28.28.2.m2.1"><semantics id="S5.T2.28.28.28.28.2.m2.1a"><mrow id="S5.T2.28.28.28.28.2.m2.1.1" xref="S5.T2.28.28.28.28.2.m2.1.1.cmml"><msub id="S5.T2.28.28.28.28.2.m2.1.1.2" xref="S5.T2.28.28.28.28.2.m2.1.1.2.cmml"><mi id="S5.T2.28.28.28.28.2.m2.1.1.2.2" xref="S5.T2.28.28.28.28.2.m2.1.1.2.2.cmml">x</mi><mi id="S5.T2.28.28.28.28.2.m2.1.1.2.3" xref="S5.T2.28.28.28.28.2.m2.1.1.2.3.cmml">i</mi></msub><mo id="S5.T2.28.28.28.28.2.m2.1.1.1" stretchy="false" xref="S5.T2.28.28.28.28.2.m2.1.1.1.cmml">→</mo><mrow id="S5.T2.28.28.28.28.2.m2.1.1.3" xref="S5.T2.28.28.28.28.2.m2.1.1.3.cmml"><msub id="S5.T2.28.28.28.28.2.m2.1.1.3.2" xref="S5.T2.28.28.28.28.2.m2.1.1.3.2.cmml"><mi id="S5.T2.28.28.28.28.2.m2.1.1.3.2.2" xref="S5.T2.28.28.28.28.2.m2.1.1.3.2.2.cmml">𝐳</mi><mi id="S5.T2.28.28.28.28.2.m2.1.1.3.2.3" xref="S5.T2.28.28.28.28.2.m2.1.1.3.2.3.cmml">i</mi></msub><mo id="S5.T2.28.28.28.28.2.m2.1.1.3.1" xref="S5.T2.28.28.28.28.2.m2.1.1.3.1.cmml">⊕</mo><msub id="S5.T2.28.28.28.28.2.m2.1.1.3.3" xref="S5.T2.28.28.28.28.2.m2.1.1.3.3.cmml"><mi id="S5.T2.28.28.28.28.2.m2.1.1.3.3.2" xref="S5.T2.28.28.28.28.2.m2.1.1.3.3.2.cmml">y</mi><mi id="S5.T2.28.28.28.28.2.m2.1.1.3.3.3" xref="S5.T2.28.28.28.28.2.m2.1.1.3.3.3.cmml">i</mi></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.T2.28.28.28.28.2.m2.1b"><apply id="S5.T2.28.28.28.28.2.m2.1.1.cmml" xref="S5.T2.28.28.28.28.2.m2.1.1"><ci id="S5.T2.28.28.28.28.2.m2.1.1.1.cmml" xref="S5.T2.28.28.28.28.2.m2.1.1.1">→</ci><apply id="S5.T2.28.28.28.28.2.m2.1.1.2.cmml" xref="S5.T2.28.28.28.28.2.m2.1.1.2"><csymbol cd="ambiguous" id="S5.T2.28.28.28.28.2.m2.1.1.2.1.cmml" xref="S5.T2.28.28.28.28.2.m2.1.1.2">subscript</csymbol><ci id="S5.T2.28.28.28.28.2.m2.1.1.2.2.cmml" xref="S5.T2.28.28.28.28.2.m2.1.1.2.2">𝑥</ci><ci id="S5.T2.28.28.28.28.2.m2.1.1.2.3.cmml" xref="S5.T2.28.28.28.28.2.m2.1.1.2.3">𝑖</ci></apply><apply id="S5.T2.28.28.28.28.2.m2.1.1.3.cmml" xref="S5.T2.28.28.28.28.2.m2.1.1.3"><csymbol cd="latexml" id="S5.T2.28.28.28.28.2.m2.1.1.3.1.cmml" xref="S5.T2.28.28.28.28.2.m2.1.1.3.1">direct-sum</csymbol><apply id="S5.T2.28.28.28.28.2.m2.1.1.3.2.cmml" xref="S5.T2.28.28.28.28.2.m2.1.1.3.2"><csymbol cd="ambiguous" id="S5.T2.28.28.28.28.2.m2.1.1.3.2.1.cmml" xref="S5.T2.28.28.28.28.2.m2.1.1.3.2">subscript</csymbol><ci id="S5.T2.28.28.28.28.2.m2.1.1.3.2.2.cmml" xref="S5.T2.28.28.28.28.2.m2.1.1.3.2.2">𝐳</ci><ci id="S5.T2.28.28.28.28.2.m2.1.1.3.2.3.cmml" xref="S5.T2.28.28.28.28.2.m2.1.1.3.2.3">𝑖</ci></apply><apply id="S5.T2.28.28.28.28.2.m2.1.1.3.3.cmml" xref="S5.T2.28.28.28.28.2.m2.1.1.3.3"><csymbol cd="ambiguous" id="S5.T2.28.28.28.28.2.m2.1.1.3.3.1.cmml" xref="S5.T2.28.28.28.28.2.m2.1.1.3.3">subscript</csymbol><ci id="S5.T2.28.28.28.28.2.m2.1.1.3.3.2.cmml" xref="S5.T2.28.28.28.28.2.m2.1.1.3.3.2">𝑦</ci><ci id="S5.T2.28.28.28.28.2.m2.1.1.3.3.3.cmml" xref="S5.T2.28.28.28.28.2.m2.1.1.3.3.3">𝑖</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.T2.28.28.28.28.2.m2.1c">x_{i}\rightarrow{\mathbf{z}}_{i}\oplus y_{i}</annotation><annotation encoding="application/x-llamapun" id="S5.T2.28.28.28.28.2.m2.1d">italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT → bold_z start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ⊕ italic_y start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math></span> <span class="ltx_td ltx_align_center ltx_th ltx_th_row ltx_border_r" id="S5.T2.29.29.29.29.3"><math alttext="x_{i}\rightarrow y_{i}" class="ltx_Math" display="inline" id="S5.T2.29.29.29.29.3.m1.1"><semantics id="S5.T2.29.29.29.29.3.m1.1a"><mrow id="S5.T2.29.29.29.29.3.m1.1.1" xref="S5.T2.29.29.29.29.3.m1.1.1.cmml"><msub id="S5.T2.29.29.29.29.3.m1.1.1.2" xref="S5.T2.29.29.29.29.3.m1.1.1.2.cmml"><mi id="S5.T2.29.29.29.29.3.m1.1.1.2.2" xref="S5.T2.29.29.29.29.3.m1.1.1.2.2.cmml">x</mi><mi id="S5.T2.29.29.29.29.3.m1.1.1.2.3" xref="S5.T2.29.29.29.29.3.m1.1.1.2.3.cmml">i</mi></msub><mo id="S5.T2.29.29.29.29.3.m1.1.1.1" stretchy="false" xref="S5.T2.29.29.29.29.3.m1.1.1.1.cmml">→</mo><msub id="S5.T2.29.29.29.29.3.m1.1.1.3" xref="S5.T2.29.29.29.29.3.m1.1.1.3.cmml"><mi id="S5.T2.29.29.29.29.3.m1.1.1.3.2" xref="S5.T2.29.29.29.29.3.m1.1.1.3.2.cmml">y</mi><mi id="S5.T2.29.29.29.29.3.m1.1.1.3.3" xref="S5.T2.29.29.29.29.3.m1.1.1.3.3.cmml">i</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S5.T2.29.29.29.29.3.m1.1b"><apply id="S5.T2.29.29.29.29.3.m1.1.1.cmml" xref="S5.T2.29.29.29.29.3.m1.1.1"><ci id="S5.T2.29.29.29.29.3.m1.1.1.1.cmml" xref="S5.T2.29.29.29.29.3.m1.1.1.1">→</ci><apply id="S5.T2.29.29.29.29.3.m1.1.1.2.cmml" xref="S5.T2.29.29.29.29.3.m1.1.1.2"><csymbol cd="ambiguous" id="S5.T2.29.29.29.29.3.m1.1.1.2.1.cmml" xref="S5.T2.29.29.29.29.3.m1.1.1.2">subscript</csymbol><ci id="S5.T2.29.29.29.29.3.m1.1.1.2.2.cmml" xref="S5.T2.29.29.29.29.3.m1.1.1.2.2">𝑥</ci><ci id="S5.T2.29.29.29.29.3.m1.1.1.2.3.cmml" xref="S5.T2.29.29.29.29.3.m1.1.1.2.3">𝑖</ci></apply><apply id="S5.T2.29.29.29.29.3.m1.1.1.3.cmml" xref="S5.T2.29.29.29.29.3.m1.1.1.3"><csymbol cd="ambiguous" id="S5.T2.29.29.29.29.3.m1.1.1.3.1.cmml" xref="S5.T2.29.29.29.29.3.m1.1.1.3">subscript</csymbol><ci id="S5.T2.29.29.29.29.3.m1.1.1.3.2.cmml" xref="S5.T2.29.29.29.29.3.m1.1.1.3.2">𝑦</ci><ci id="S5.T2.29.29.29.29.3.m1.1.1.3.3.cmml" xref="S5.T2.29.29.29.29.3.m1.1.1.3.3">𝑖</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.T2.29.29.29.29.3.m1.1c">x_{i}\rightarrow y_{i}</annotation><annotation encoding="application/x-llamapun" id="S5.T2.29.29.29.29.3.m1.1d">italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT → italic_y start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math></span> <span class="ltx_td ltx_align_center ltx_border_r" id="S5.T2.29.29.29.29.4">45.65</span> <span class="ltx_td ltx_align_center ltx_border_r" id="S5.T2.29.29.29.29.5">15.21</span> <span class="ltx_td ltx_align_center ltx_border_r" id="S5.T2.29.29.29.29.6">17.44</span> <span class="ltx_td ltx_nopad_r ltx_align_center" id="S5.T2.29.29.29.29.7">44.23</span></span> <span class="ltx_tr" id="S5.T2.32.32.32.32"> <span class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_r" id="S5.T2.31.31.31.31.2"><math alttext="x_{i}\rightarrow y_{i}" class="ltx_Math" display="inline" id="S5.T2.30.30.30.30.1.m1.1"><semantics id="S5.T2.30.30.30.30.1.m1.1a"><mrow id="S5.T2.30.30.30.30.1.m1.1.1" xref="S5.T2.30.30.30.30.1.m1.1.1.cmml"><msub id="S5.T2.30.30.30.30.1.m1.1.1.2" xref="S5.T2.30.30.30.30.1.m1.1.1.2.cmml"><mi id="S5.T2.30.30.30.30.1.m1.1.1.2.2" xref="S5.T2.30.30.30.30.1.m1.1.1.2.2.cmml">x</mi><mi id="S5.T2.30.30.30.30.1.m1.1.1.2.3" xref="S5.T2.30.30.30.30.1.m1.1.1.2.3.cmml">i</mi></msub><mo id="S5.T2.30.30.30.30.1.m1.1.1.1" stretchy="false" xref="S5.T2.30.30.30.30.1.m1.1.1.1.cmml">→</mo><msub id="S5.T2.30.30.30.30.1.m1.1.1.3" xref="S5.T2.30.30.30.30.1.m1.1.1.3.cmml"><mi id="S5.T2.30.30.30.30.1.m1.1.1.3.2" xref="S5.T2.30.30.30.30.1.m1.1.1.3.2.cmml">y</mi><mi id="S5.T2.30.30.30.30.1.m1.1.1.3.3" xref="S5.T2.30.30.30.30.1.m1.1.1.3.3.cmml">i</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S5.T2.30.30.30.30.1.m1.1b"><apply id="S5.T2.30.30.30.30.1.m1.1.1.cmml" xref="S5.T2.30.30.30.30.1.m1.1.1"><ci id="S5.T2.30.30.30.30.1.m1.1.1.1.cmml" xref="S5.T2.30.30.30.30.1.m1.1.1.1">→</ci><apply id="S5.T2.30.30.30.30.1.m1.1.1.2.cmml" xref="S5.T2.30.30.30.30.1.m1.1.1.2"><csymbol cd="ambiguous" id="S5.T2.30.30.30.30.1.m1.1.1.2.1.cmml" xref="S5.T2.30.30.30.30.1.m1.1.1.2">subscript</csymbol><ci id="S5.T2.30.30.30.30.1.m1.1.1.2.2.cmml" xref="S5.T2.30.30.30.30.1.m1.1.1.2.2">𝑥</ci><ci id="S5.T2.30.30.30.30.1.m1.1.1.2.3.cmml" xref="S5.T2.30.30.30.30.1.m1.1.1.2.3">𝑖</ci></apply><apply id="S5.T2.30.30.30.30.1.m1.1.1.3.cmml" xref="S5.T2.30.30.30.30.1.m1.1.1.3"><csymbol cd="ambiguous" id="S5.T2.30.30.30.30.1.m1.1.1.3.1.cmml" xref="S5.T2.30.30.30.30.1.m1.1.1.3">subscript</csymbol><ci id="S5.T2.30.30.30.30.1.m1.1.1.3.2.cmml" xref="S5.T2.30.30.30.30.1.m1.1.1.3.2">𝑦</ci><ci id="S5.T2.30.30.30.30.1.m1.1.1.3.3.cmml" xref="S5.T2.30.30.30.30.1.m1.1.1.3.3">𝑖</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.T2.30.30.30.30.1.m1.1c">x_{i}\rightarrow y_{i}</annotation><annotation encoding="application/x-llamapun" id="S5.T2.30.30.30.30.1.m1.1d">italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT → italic_y start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math>; <math alttext="x_{i}\rightarrow{\mathbf{z}}_{i}\oplus y_{i}" class="ltx_Math" display="inline" id="S5.T2.31.31.31.31.2.m2.1"><semantics id="S5.T2.31.31.31.31.2.m2.1a"><mrow id="S5.T2.31.31.31.31.2.m2.1.1" xref="S5.T2.31.31.31.31.2.m2.1.1.cmml"><msub id="S5.T2.31.31.31.31.2.m2.1.1.2" xref="S5.T2.31.31.31.31.2.m2.1.1.2.cmml"><mi id="S5.T2.31.31.31.31.2.m2.1.1.2.2" xref="S5.T2.31.31.31.31.2.m2.1.1.2.2.cmml">x</mi><mi id="S5.T2.31.31.31.31.2.m2.1.1.2.3" xref="S5.T2.31.31.31.31.2.m2.1.1.2.3.cmml">i</mi></msub><mo id="S5.T2.31.31.31.31.2.m2.1.1.1" stretchy="false" xref="S5.T2.31.31.31.31.2.m2.1.1.1.cmml">→</mo><mrow id="S5.T2.31.31.31.31.2.m2.1.1.3" xref="S5.T2.31.31.31.31.2.m2.1.1.3.cmml"><msub id="S5.T2.31.31.31.31.2.m2.1.1.3.2" xref="S5.T2.31.31.31.31.2.m2.1.1.3.2.cmml"><mi id="S5.T2.31.31.31.31.2.m2.1.1.3.2.2" xref="S5.T2.31.31.31.31.2.m2.1.1.3.2.2.cmml">𝐳</mi><mi id="S5.T2.31.31.31.31.2.m2.1.1.3.2.3" xref="S5.T2.31.31.31.31.2.m2.1.1.3.2.3.cmml">i</mi></msub><mo id="S5.T2.31.31.31.31.2.m2.1.1.3.1" xref="S5.T2.31.31.31.31.2.m2.1.1.3.1.cmml">⊕</mo><msub id="S5.T2.31.31.31.31.2.m2.1.1.3.3" xref="S5.T2.31.31.31.31.2.m2.1.1.3.3.cmml"><mi id="S5.T2.31.31.31.31.2.m2.1.1.3.3.2" xref="S5.T2.31.31.31.31.2.m2.1.1.3.3.2.cmml">y</mi><mi id="S5.T2.31.31.31.31.2.m2.1.1.3.3.3" xref="S5.T2.31.31.31.31.2.m2.1.1.3.3.3.cmml">i</mi></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.T2.31.31.31.31.2.m2.1b"><apply id="S5.T2.31.31.31.31.2.m2.1.1.cmml" xref="S5.T2.31.31.31.31.2.m2.1.1"><ci id="S5.T2.31.31.31.31.2.m2.1.1.1.cmml" xref="S5.T2.31.31.31.31.2.m2.1.1.1">→</ci><apply id="S5.T2.31.31.31.31.2.m2.1.1.2.cmml" xref="S5.T2.31.31.31.31.2.m2.1.1.2"><csymbol cd="ambiguous" id="S5.T2.31.31.31.31.2.m2.1.1.2.1.cmml" xref="S5.T2.31.31.31.31.2.m2.1.1.2">subscript</csymbol><ci id="S5.T2.31.31.31.31.2.m2.1.1.2.2.cmml" xref="S5.T2.31.31.31.31.2.m2.1.1.2.2">𝑥</ci><ci id="S5.T2.31.31.31.31.2.m2.1.1.2.3.cmml" xref="S5.T2.31.31.31.31.2.m2.1.1.2.3">𝑖</ci></apply><apply id="S5.T2.31.31.31.31.2.m2.1.1.3.cmml" xref="S5.T2.31.31.31.31.2.m2.1.1.3"><csymbol cd="latexml" id="S5.T2.31.31.31.31.2.m2.1.1.3.1.cmml" xref="S5.T2.31.31.31.31.2.m2.1.1.3.1">direct-sum</csymbol><apply id="S5.T2.31.31.31.31.2.m2.1.1.3.2.cmml" xref="S5.T2.31.31.31.31.2.m2.1.1.3.2"><csymbol cd="ambiguous" id="S5.T2.31.31.31.31.2.m2.1.1.3.2.1.cmml" xref="S5.T2.31.31.31.31.2.m2.1.1.3.2">subscript</csymbol><ci id="S5.T2.31.31.31.31.2.m2.1.1.3.2.2.cmml" xref="S5.T2.31.31.31.31.2.m2.1.1.3.2.2">𝐳</ci><ci id="S5.T2.31.31.31.31.2.m2.1.1.3.2.3.cmml" xref="S5.T2.31.31.31.31.2.m2.1.1.3.2.3">𝑖</ci></apply><apply id="S5.T2.31.31.31.31.2.m2.1.1.3.3.cmml" xref="S5.T2.31.31.31.31.2.m2.1.1.3.3"><csymbol cd="ambiguous" id="S5.T2.31.31.31.31.2.m2.1.1.3.3.1.cmml" xref="S5.T2.31.31.31.31.2.m2.1.1.3.3">subscript</csymbol><ci id="S5.T2.31.31.31.31.2.m2.1.1.3.3.2.cmml" xref="S5.T2.31.31.31.31.2.m2.1.1.3.3.2">𝑦</ci><ci id="S5.T2.31.31.31.31.2.m2.1.1.3.3.3.cmml" xref="S5.T2.31.31.31.31.2.m2.1.1.3.3.3">𝑖</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.T2.31.31.31.31.2.m2.1c">x_{i}\rightarrow{\mathbf{z}}_{i}\oplus y_{i}</annotation><annotation encoding="application/x-llamapun" id="S5.T2.31.31.31.31.2.m2.1d">italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT → bold_z start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ⊕ italic_y start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math></span> <span class="ltx_td ltx_align_center ltx_th ltx_th_row ltx_border_r" id="S5.T2.32.32.32.32.3"><math alttext="x_{i}\rightarrow{\mathbf{z}}_{i}\oplus y_{i}" class="ltx_Math" display="inline" id="S5.T2.32.32.32.32.3.m1.1"><semantics id="S5.T2.32.32.32.32.3.m1.1a"><mrow id="S5.T2.32.32.32.32.3.m1.1.1" xref="S5.T2.32.32.32.32.3.m1.1.1.cmml"><msub id="S5.T2.32.32.32.32.3.m1.1.1.2" xref="S5.T2.32.32.32.32.3.m1.1.1.2.cmml"><mi id="S5.T2.32.32.32.32.3.m1.1.1.2.2" xref="S5.T2.32.32.32.32.3.m1.1.1.2.2.cmml">x</mi><mi id="S5.T2.32.32.32.32.3.m1.1.1.2.3" xref="S5.T2.32.32.32.32.3.m1.1.1.2.3.cmml">i</mi></msub><mo id="S5.T2.32.32.32.32.3.m1.1.1.1" stretchy="false" xref="S5.T2.32.32.32.32.3.m1.1.1.1.cmml">→</mo><mrow id="S5.T2.32.32.32.32.3.m1.1.1.3" xref="S5.T2.32.32.32.32.3.m1.1.1.3.cmml"><msub id="S5.T2.32.32.32.32.3.m1.1.1.3.2" xref="S5.T2.32.32.32.32.3.m1.1.1.3.2.cmml"><mi id="S5.T2.32.32.32.32.3.m1.1.1.3.2.2" xref="S5.T2.32.32.32.32.3.m1.1.1.3.2.2.cmml">𝐳</mi><mi id="S5.T2.32.32.32.32.3.m1.1.1.3.2.3" xref="S5.T2.32.32.32.32.3.m1.1.1.3.2.3.cmml">i</mi></msub><mo id="S5.T2.32.32.32.32.3.m1.1.1.3.1" xref="S5.T2.32.32.32.32.3.m1.1.1.3.1.cmml">⊕</mo><msub id="S5.T2.32.32.32.32.3.m1.1.1.3.3" xref="S5.T2.32.32.32.32.3.m1.1.1.3.3.cmml"><mi id="S5.T2.32.32.32.32.3.m1.1.1.3.3.2" xref="S5.T2.32.32.32.32.3.m1.1.1.3.3.2.cmml">y</mi><mi id="S5.T2.32.32.32.32.3.m1.1.1.3.3.3" xref="S5.T2.32.32.32.32.3.m1.1.1.3.3.3.cmml">i</mi></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.T2.32.32.32.32.3.m1.1b"><apply id="S5.T2.32.32.32.32.3.m1.1.1.cmml" xref="S5.T2.32.32.32.32.3.m1.1.1"><ci id="S5.T2.32.32.32.32.3.m1.1.1.1.cmml" xref="S5.T2.32.32.32.32.3.m1.1.1.1">→</ci><apply id="S5.T2.32.32.32.32.3.m1.1.1.2.cmml" xref="S5.T2.32.32.32.32.3.m1.1.1.2"><csymbol cd="ambiguous" id="S5.T2.32.32.32.32.3.m1.1.1.2.1.cmml" xref="S5.T2.32.32.32.32.3.m1.1.1.2">subscript</csymbol><ci id="S5.T2.32.32.32.32.3.m1.1.1.2.2.cmml" xref="S5.T2.32.32.32.32.3.m1.1.1.2.2">𝑥</ci><ci id="S5.T2.32.32.32.32.3.m1.1.1.2.3.cmml" xref="S5.T2.32.32.32.32.3.m1.1.1.2.3">𝑖</ci></apply><apply id="S5.T2.32.32.32.32.3.m1.1.1.3.cmml" xref="S5.T2.32.32.32.32.3.m1.1.1.3"><csymbol cd="latexml" id="S5.T2.32.32.32.32.3.m1.1.1.3.1.cmml" xref="S5.T2.32.32.32.32.3.m1.1.1.3.1">direct-sum</csymbol><apply id="S5.T2.32.32.32.32.3.m1.1.1.3.2.cmml" xref="S5.T2.32.32.32.32.3.m1.1.1.3.2"><csymbol cd="ambiguous" id="S5.T2.32.32.32.32.3.m1.1.1.3.2.1.cmml" xref="S5.T2.32.32.32.32.3.m1.1.1.3.2">subscript</csymbol><ci id="S5.T2.32.32.32.32.3.m1.1.1.3.2.2.cmml" xref="S5.T2.32.32.32.32.3.m1.1.1.3.2.2">𝐳</ci><ci id="S5.T2.32.32.32.32.3.m1.1.1.3.2.3.cmml" xref="S5.T2.32.32.32.32.3.m1.1.1.3.2.3">𝑖</ci></apply><apply id="S5.T2.32.32.32.32.3.m1.1.1.3.3.cmml" xref="S5.T2.32.32.32.32.3.m1.1.1.3.3"><csymbol cd="ambiguous" id="S5.T2.32.32.32.32.3.m1.1.1.3.3.1.cmml" xref="S5.T2.32.32.32.32.3.m1.1.1.3.3">subscript</csymbol><ci id="S5.T2.32.32.32.32.3.m1.1.1.3.3.2.cmml" xref="S5.T2.32.32.32.32.3.m1.1.1.3.3.2">𝑦</ci><ci id="S5.T2.32.32.32.32.3.m1.1.1.3.3.3.cmml" xref="S5.T2.32.32.32.32.3.m1.1.1.3.3.3">𝑖</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.T2.32.32.32.32.3.m1.1c">x_{i}\rightarrow{\mathbf{z}}_{i}\oplus y_{i}</annotation><annotation encoding="application/x-llamapun" id="S5.T2.32.32.32.32.3.m1.1d">italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT → bold_z start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ⊕ italic_y start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math></span> <span class="ltx_td ltx_align_center ltx_border_r" id="S5.T2.32.32.32.32.4"><span class="ltx_text ltx_font_bold" id="S5.T2.32.32.32.32.4.1">47.07</span></span> <span class="ltx_td ltx_align_center ltx_border_r" id="S5.T2.32.32.32.32.5"><span class="ltx_text ltx_font_bold" id="S5.T2.32.32.32.32.5.1">15.72</span></span> <span class="ltx_td ltx_align_center ltx_border_r" id="S5.T2.32.32.32.32.6"><span class="ltx_text ltx_font_bold" id="S5.T2.32.32.32.32.6.1">17.72</span></span> <span class="ltx_td ltx_nopad_r ltx_align_center" id="S5.T2.32.32.32.32.7"><span class="ltx_text ltx_font_bold" id="S5.T2.32.32.32.32.7.1">45.67</span></span></span> <span class="ltx_tr" id="S5.T2.36.36.36.36"> <span class="ltx_td ltx_th ltx_th_row ltx_border_b ltx_border_r" id="S5.T2.36.36.36.36.5"></span> <span class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_b ltx_border_r" id="S5.T2.35.35.35.35.3"><math alttext="x_{i}\rightarrow y_{i}" class="ltx_Math" display="inline" id="S5.T2.33.33.33.33.1.m1.1"><semantics id="S5.T2.33.33.33.33.1.m1.1a"><mrow id="S5.T2.33.33.33.33.1.m1.1.1" xref="S5.T2.33.33.33.33.1.m1.1.1.cmml"><msub id="S5.T2.33.33.33.33.1.m1.1.1.2" xref="S5.T2.33.33.33.33.1.m1.1.1.2.cmml"><mi id="S5.T2.33.33.33.33.1.m1.1.1.2.2" xref="S5.T2.33.33.33.33.1.m1.1.1.2.2.cmml">x</mi><mi id="S5.T2.33.33.33.33.1.m1.1.1.2.3" xref="S5.T2.33.33.33.33.1.m1.1.1.2.3.cmml">i</mi></msub><mo id="S5.T2.33.33.33.33.1.m1.1.1.1" stretchy="false" xref="S5.T2.33.33.33.33.1.m1.1.1.1.cmml">→</mo><msub id="S5.T2.33.33.33.33.1.m1.1.1.3" xref="S5.T2.33.33.33.33.1.m1.1.1.3.cmml"><mi id="S5.T2.33.33.33.33.1.m1.1.1.3.2" xref="S5.T2.33.33.33.33.1.m1.1.1.3.2.cmml">y</mi><mi id="S5.T2.33.33.33.33.1.m1.1.1.3.3" xref="S5.T2.33.33.33.33.1.m1.1.1.3.3.cmml">i</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S5.T2.33.33.33.33.1.m1.1b"><apply id="S5.T2.33.33.33.33.1.m1.1.1.cmml" xref="S5.T2.33.33.33.33.1.m1.1.1"><ci id="S5.T2.33.33.33.33.1.m1.1.1.1.cmml" xref="S5.T2.33.33.33.33.1.m1.1.1.1">→</ci><apply id="S5.T2.33.33.33.33.1.m1.1.1.2.cmml" xref="S5.T2.33.33.33.33.1.m1.1.1.2"><csymbol cd="ambiguous" id="S5.T2.33.33.33.33.1.m1.1.1.2.1.cmml" xref="S5.T2.33.33.33.33.1.m1.1.1.2">subscript</csymbol><ci id="S5.T2.33.33.33.33.1.m1.1.1.2.2.cmml" xref="S5.T2.33.33.33.33.1.m1.1.1.2.2">𝑥</ci><ci id="S5.T2.33.33.33.33.1.m1.1.1.2.3.cmml" xref="S5.T2.33.33.33.33.1.m1.1.1.2.3">𝑖</ci></apply><apply id="S5.T2.33.33.33.33.1.m1.1.1.3.cmml" xref="S5.T2.33.33.33.33.1.m1.1.1.3"><csymbol cd="ambiguous" id="S5.T2.33.33.33.33.1.m1.1.1.3.1.cmml" xref="S5.T2.33.33.33.33.1.m1.1.1.3">subscript</csymbol><ci id="S5.T2.33.33.33.33.1.m1.1.1.3.2.cmml" xref="S5.T2.33.33.33.33.1.m1.1.1.3.2">𝑦</ci><ci id="S5.T2.33.33.33.33.1.m1.1.1.3.3.cmml" xref="S5.T2.33.33.33.33.1.m1.1.1.3.3">𝑖</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.T2.33.33.33.33.1.m1.1c">x_{i}\rightarrow y_{i}</annotation><annotation encoding="application/x-llamapun" id="S5.T2.33.33.33.33.1.m1.1d">italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT → italic_y start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math>; <math alttext="x_{i}\rightarrow{\mathbf{z}}_{i}\oplus y_{i}" class="ltx_Math" display="inline" id="S5.T2.34.34.34.34.2.m2.1"><semantics id="S5.T2.34.34.34.34.2.m2.1a"><mrow id="S5.T2.34.34.34.34.2.m2.1.1" xref="S5.T2.34.34.34.34.2.m2.1.1.cmml"><msub id="S5.T2.34.34.34.34.2.m2.1.1.2" xref="S5.T2.34.34.34.34.2.m2.1.1.2.cmml"><mi id="S5.T2.34.34.34.34.2.m2.1.1.2.2" xref="S5.T2.34.34.34.34.2.m2.1.1.2.2.cmml">x</mi><mi id="S5.T2.34.34.34.34.2.m2.1.1.2.3" xref="S5.T2.34.34.34.34.2.m2.1.1.2.3.cmml">i</mi></msub><mo id="S5.T2.34.34.34.34.2.m2.1.1.1" stretchy="false" xref="S5.T2.34.34.34.34.2.m2.1.1.1.cmml">→</mo><mrow id="S5.T2.34.34.34.34.2.m2.1.1.3" xref="S5.T2.34.34.34.34.2.m2.1.1.3.cmml"><msub id="S5.T2.34.34.34.34.2.m2.1.1.3.2" xref="S5.T2.34.34.34.34.2.m2.1.1.3.2.cmml"><mi id="S5.T2.34.34.34.34.2.m2.1.1.3.2.2" xref="S5.T2.34.34.34.34.2.m2.1.1.3.2.2.cmml">𝐳</mi><mi id="S5.T2.34.34.34.34.2.m2.1.1.3.2.3" xref="S5.T2.34.34.34.34.2.m2.1.1.3.2.3.cmml">i</mi></msub><mo id="S5.T2.34.34.34.34.2.m2.1.1.3.1" xref="S5.T2.34.34.34.34.2.m2.1.1.3.1.cmml">⊕</mo><msub id="S5.T2.34.34.34.34.2.m2.1.1.3.3" xref="S5.T2.34.34.34.34.2.m2.1.1.3.3.cmml"><mi id="S5.T2.34.34.34.34.2.m2.1.1.3.3.2" xref="S5.T2.34.34.34.34.2.m2.1.1.3.3.2.cmml">y</mi><mi id="S5.T2.34.34.34.34.2.m2.1.1.3.3.3" xref="S5.T2.34.34.34.34.2.m2.1.1.3.3.3.cmml">i</mi></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.T2.34.34.34.34.2.m2.1b"><apply id="S5.T2.34.34.34.34.2.m2.1.1.cmml" xref="S5.T2.34.34.34.34.2.m2.1.1"><ci id="S5.T2.34.34.34.34.2.m2.1.1.1.cmml" xref="S5.T2.34.34.34.34.2.m2.1.1.1">→</ci><apply id="S5.T2.34.34.34.34.2.m2.1.1.2.cmml" xref="S5.T2.34.34.34.34.2.m2.1.1.2"><csymbol cd="ambiguous" id="S5.T2.34.34.34.34.2.m2.1.1.2.1.cmml" xref="S5.T2.34.34.34.34.2.m2.1.1.2">subscript</csymbol><ci id="S5.T2.34.34.34.34.2.m2.1.1.2.2.cmml" xref="S5.T2.34.34.34.34.2.m2.1.1.2.2">𝑥</ci><ci id="S5.T2.34.34.34.34.2.m2.1.1.2.3.cmml" xref="S5.T2.34.34.34.34.2.m2.1.1.2.3">𝑖</ci></apply><apply id="S5.T2.34.34.34.34.2.m2.1.1.3.cmml" xref="S5.T2.34.34.34.34.2.m2.1.1.3"><csymbol cd="latexml" id="S5.T2.34.34.34.34.2.m2.1.1.3.1.cmml" xref="S5.T2.34.34.34.34.2.m2.1.1.3.1">direct-sum</csymbol><apply id="S5.T2.34.34.34.34.2.m2.1.1.3.2.cmml" xref="S5.T2.34.34.34.34.2.m2.1.1.3.2"><csymbol cd="ambiguous" id="S5.T2.34.34.34.34.2.m2.1.1.3.2.1.cmml" xref="S5.T2.34.34.34.34.2.m2.1.1.3.2">subscript</csymbol><ci id="S5.T2.34.34.34.34.2.m2.1.1.3.2.2.cmml" xref="S5.T2.34.34.34.34.2.m2.1.1.3.2.2">𝐳</ci><ci id="S5.T2.34.34.34.34.2.m2.1.1.3.2.3.cmml" xref="S5.T2.34.34.34.34.2.m2.1.1.3.2.3">𝑖</ci></apply><apply id="S5.T2.34.34.34.34.2.m2.1.1.3.3.cmml" xref="S5.T2.34.34.34.34.2.m2.1.1.3.3"><csymbol cd="ambiguous" id="S5.T2.34.34.34.34.2.m2.1.1.3.3.1.cmml" xref="S5.T2.34.34.34.34.2.m2.1.1.3.3">subscript</csymbol><ci id="S5.T2.34.34.34.34.2.m2.1.1.3.3.2.cmml" xref="S5.T2.34.34.34.34.2.m2.1.1.3.3.2">𝑦</ci><ci id="S5.T2.34.34.34.34.2.m2.1.1.3.3.3.cmml" xref="S5.T2.34.34.34.34.2.m2.1.1.3.3.3">𝑖</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.T2.34.34.34.34.2.m2.1c">x_{i}\rightarrow{\mathbf{z}}_{i}\oplus y_{i}</annotation><annotation encoding="application/x-llamapun" id="S5.T2.34.34.34.34.2.m2.1d">italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT → bold_z start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ⊕ italic_y start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math>; <math alttext="x_{i}\oplus{\mathbf{z}}_{i}\rightarrow y_{i}" class="ltx_Math" display="inline" id="S5.T2.35.35.35.35.3.m3.1"><semantics id="S5.T2.35.35.35.35.3.m3.1a"><mrow id="S5.T2.35.35.35.35.3.m3.1.1" xref="S5.T2.35.35.35.35.3.m3.1.1.cmml"><mrow id="S5.T2.35.35.35.35.3.m3.1.1.2" xref="S5.T2.35.35.35.35.3.m3.1.1.2.cmml"><msub id="S5.T2.35.35.35.35.3.m3.1.1.2.2" xref="S5.T2.35.35.35.35.3.m3.1.1.2.2.cmml"><mi id="S5.T2.35.35.35.35.3.m3.1.1.2.2.2" xref="S5.T2.35.35.35.35.3.m3.1.1.2.2.2.cmml">x</mi><mi id="S5.T2.35.35.35.35.3.m3.1.1.2.2.3" xref="S5.T2.35.35.35.35.3.m3.1.1.2.2.3.cmml">i</mi></msub><mo id="S5.T2.35.35.35.35.3.m3.1.1.2.1" xref="S5.T2.35.35.35.35.3.m3.1.1.2.1.cmml">⊕</mo><msub id="S5.T2.35.35.35.35.3.m3.1.1.2.3" xref="S5.T2.35.35.35.35.3.m3.1.1.2.3.cmml"><mi id="S5.T2.35.35.35.35.3.m3.1.1.2.3.2" xref="S5.T2.35.35.35.35.3.m3.1.1.2.3.2.cmml">𝐳</mi><mi id="S5.T2.35.35.35.35.3.m3.1.1.2.3.3" xref="S5.T2.35.35.35.35.3.m3.1.1.2.3.3.cmml">i</mi></msub></mrow><mo id="S5.T2.35.35.35.35.3.m3.1.1.1" stretchy="false" xref="S5.T2.35.35.35.35.3.m3.1.1.1.cmml">→</mo><msub id="S5.T2.35.35.35.35.3.m3.1.1.3" xref="S5.T2.35.35.35.35.3.m3.1.1.3.cmml"><mi id="S5.T2.35.35.35.35.3.m3.1.1.3.2" xref="S5.T2.35.35.35.35.3.m3.1.1.3.2.cmml">y</mi><mi id="S5.T2.35.35.35.35.3.m3.1.1.3.3" xref="S5.T2.35.35.35.35.3.m3.1.1.3.3.cmml">i</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S5.T2.35.35.35.35.3.m3.1b"><apply id="S5.T2.35.35.35.35.3.m3.1.1.cmml" xref="S5.T2.35.35.35.35.3.m3.1.1"><ci id="S5.T2.35.35.35.35.3.m3.1.1.1.cmml" xref="S5.T2.35.35.35.35.3.m3.1.1.1">→</ci><apply id="S5.T2.35.35.35.35.3.m3.1.1.2.cmml" xref="S5.T2.35.35.35.35.3.m3.1.1.2"><csymbol cd="latexml" id="S5.T2.35.35.35.35.3.m3.1.1.2.1.cmml" xref="S5.T2.35.35.35.35.3.m3.1.1.2.1">direct-sum</csymbol><apply id="S5.T2.35.35.35.35.3.m3.1.1.2.2.cmml" xref="S5.T2.35.35.35.35.3.m3.1.1.2.2"><csymbol cd="ambiguous" id="S5.T2.35.35.35.35.3.m3.1.1.2.2.1.cmml" xref="S5.T2.35.35.35.35.3.m3.1.1.2.2">subscript</csymbol><ci id="S5.T2.35.35.35.35.3.m3.1.1.2.2.2.cmml" xref="S5.T2.35.35.35.35.3.m3.1.1.2.2.2">𝑥</ci><ci id="S5.T2.35.35.35.35.3.m3.1.1.2.2.3.cmml" xref="S5.T2.35.35.35.35.3.m3.1.1.2.2.3">𝑖</ci></apply><apply id="S5.T2.35.35.35.35.3.m3.1.1.2.3.cmml" xref="S5.T2.35.35.35.35.3.m3.1.1.2.3"><csymbol cd="ambiguous" id="S5.T2.35.35.35.35.3.m3.1.1.2.3.1.cmml" xref="S5.T2.35.35.35.35.3.m3.1.1.2.3">subscript</csymbol><ci id="S5.T2.35.35.35.35.3.m3.1.1.2.3.2.cmml" xref="S5.T2.35.35.35.35.3.m3.1.1.2.3.2">𝐳</ci><ci id="S5.T2.35.35.35.35.3.m3.1.1.2.3.3.cmml" xref="S5.T2.35.35.35.35.3.m3.1.1.2.3.3">𝑖</ci></apply></apply><apply id="S5.T2.35.35.35.35.3.m3.1.1.3.cmml" xref="S5.T2.35.35.35.35.3.m3.1.1.3"><csymbol cd="ambiguous" id="S5.T2.35.35.35.35.3.m3.1.1.3.1.cmml" xref="S5.T2.35.35.35.35.3.m3.1.1.3">subscript</csymbol><ci id="S5.T2.35.35.35.35.3.m3.1.1.3.2.cmml" xref="S5.T2.35.35.35.35.3.m3.1.1.3.2">𝑦</ci><ci id="S5.T2.35.35.35.35.3.m3.1.1.3.3.cmml" xref="S5.T2.35.35.35.35.3.m3.1.1.3.3">𝑖</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.T2.35.35.35.35.3.m3.1c">x_{i}\oplus{\mathbf{z}}_{i}\rightarrow y_{i}</annotation><annotation encoding="application/x-llamapun" id="S5.T2.35.35.35.35.3.m3.1d">italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ⊕ bold_z start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT → italic_y start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math></span> <span class="ltx_td ltx_align_center ltx_th ltx_th_row ltx_border_b ltx_border_r" id="S5.T2.36.36.36.36.4"><math alttext="x_{i}\rightarrow y_{i}" class="ltx_Math" display="inline" id="S5.T2.36.36.36.36.4.m1.1"><semantics id="S5.T2.36.36.36.36.4.m1.1a"><mrow id="S5.T2.36.36.36.36.4.m1.1.1" xref="S5.T2.36.36.36.36.4.m1.1.1.cmml"><msub id="S5.T2.36.36.36.36.4.m1.1.1.2" xref="S5.T2.36.36.36.36.4.m1.1.1.2.cmml"><mi id="S5.T2.36.36.36.36.4.m1.1.1.2.2" xref="S5.T2.36.36.36.36.4.m1.1.1.2.2.cmml">x</mi><mi id="S5.T2.36.36.36.36.4.m1.1.1.2.3" xref="S5.T2.36.36.36.36.4.m1.1.1.2.3.cmml">i</mi></msub><mo id="S5.T2.36.36.36.36.4.m1.1.1.1" stretchy="false" xref="S5.T2.36.36.36.36.4.m1.1.1.1.cmml">→</mo><msub id="S5.T2.36.36.36.36.4.m1.1.1.3" xref="S5.T2.36.36.36.36.4.m1.1.1.3.cmml"><mi id="S5.T2.36.36.36.36.4.m1.1.1.3.2" xref="S5.T2.36.36.36.36.4.m1.1.1.3.2.cmml">y</mi><mi id="S5.T2.36.36.36.36.4.m1.1.1.3.3" xref="S5.T2.36.36.36.36.4.m1.1.1.3.3.cmml">i</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S5.T2.36.36.36.36.4.m1.1b"><apply id="S5.T2.36.36.36.36.4.m1.1.1.cmml" xref="S5.T2.36.36.36.36.4.m1.1.1"><ci id="S5.T2.36.36.36.36.4.m1.1.1.1.cmml" xref="S5.T2.36.36.36.36.4.m1.1.1.1">→</ci><apply id="S5.T2.36.36.36.36.4.m1.1.1.2.cmml" xref="S5.T2.36.36.36.36.4.m1.1.1.2"><csymbol cd="ambiguous" id="S5.T2.36.36.36.36.4.m1.1.1.2.1.cmml" xref="S5.T2.36.36.36.36.4.m1.1.1.2">subscript</csymbol><ci id="S5.T2.36.36.36.36.4.m1.1.1.2.2.cmml" xref="S5.T2.36.36.36.36.4.m1.1.1.2.2">𝑥</ci><ci id="S5.T2.36.36.36.36.4.m1.1.1.2.3.cmml" xref="S5.T2.36.36.36.36.4.m1.1.1.2.3">𝑖</ci></apply><apply id="S5.T2.36.36.36.36.4.m1.1.1.3.cmml" xref="S5.T2.36.36.36.36.4.m1.1.1.3"><csymbol cd="ambiguous" id="S5.T2.36.36.36.36.4.m1.1.1.3.1.cmml" xref="S5.T2.36.36.36.36.4.m1.1.1.3">subscript</csymbol><ci id="S5.T2.36.36.36.36.4.m1.1.1.3.2.cmml" xref="S5.T2.36.36.36.36.4.m1.1.1.3.2">𝑦</ci><ci id="S5.T2.36.36.36.36.4.m1.1.1.3.3.cmml" xref="S5.T2.36.36.36.36.4.m1.1.1.3.3">𝑖</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.T2.36.36.36.36.4.m1.1c">x_{i}\rightarrow y_{i}</annotation><annotation encoding="application/x-llamapun" id="S5.T2.36.36.36.36.4.m1.1d">italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT → italic_y start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math></span> <span class="ltx_td ltx_align_center ltx_border_b ltx_border_r" id="S5.T2.36.36.36.36.6">45.31</span> <span class="ltx_td ltx_align_center ltx_border_b ltx_border_r" id="S5.T2.36.36.36.36.7">14.80</span> <span class="ltx_td ltx_align_center ltx_border_b ltx_border_r" id="S5.T2.36.36.36.36.8">17.10</span> <span class="ltx_td ltx_nopad_r ltx_align_center ltx_border_b" id="S5.T2.36.36.36.36.9">43.92</span></span> </span> </span></span></p> </figure> <div class="ltx_para" id="S5.SS4.SSS0.Px2.p2"> <p class="ltx_p" id="S5.SS4.SSS0.Px2.p2.3">Table <a class="ltx_ref" href="https://arxiv.org/html/2410.06203v1#S5.T2" title="Table 2 ‣ Comparison of different training and inference recipes. ‣ 5.4 Results ‣ 5 Experiments ‣ Integrating Planning into Single-Turn Long-Form Text Generation"><span class="ltx_text ltx_ref_tag">2</span></a> illustrates the comparison results. The results show that models fine-tuned on the training mixture with intermediate steps consistently outperform models only fine-tuned to generate the article directly. On both datasets, the best performance is achieved when the model is fine-tuned on the mixture of <math alttext="x_{i}\rightarrow y_{i}" class="ltx_Math" display="inline" id="S5.SS4.SSS0.Px2.p2.1.m1.1"><semantics id="S5.SS4.SSS0.Px2.p2.1.m1.1a"><mrow id="S5.SS4.SSS0.Px2.p2.1.m1.1.1" xref="S5.SS4.SSS0.Px2.p2.1.m1.1.1.cmml"><msub id="S5.SS4.SSS0.Px2.p2.1.m1.1.1.2" xref="S5.SS4.SSS0.Px2.p2.1.m1.1.1.2.cmml"><mi id="S5.SS4.SSS0.Px2.p2.1.m1.1.1.2.2" xref="S5.SS4.SSS0.Px2.p2.1.m1.1.1.2.2.cmml">x</mi><mi id="S5.SS4.SSS0.Px2.p2.1.m1.1.1.2.3" xref="S5.SS4.SSS0.Px2.p2.1.m1.1.1.2.3.cmml">i</mi></msub><mo id="S5.SS4.SSS0.Px2.p2.1.m1.1.1.1" stretchy="false" xref="S5.SS4.SSS0.Px2.p2.1.m1.1.1.1.cmml">→</mo><msub id="S5.SS4.SSS0.Px2.p2.1.m1.1.1.3" xref="S5.SS4.SSS0.Px2.p2.1.m1.1.1.3.cmml"><mi id="S5.SS4.SSS0.Px2.p2.1.m1.1.1.3.2" xref="S5.SS4.SSS0.Px2.p2.1.m1.1.1.3.2.cmml">y</mi><mi id="S5.SS4.SSS0.Px2.p2.1.m1.1.1.3.3" xref="S5.SS4.SSS0.Px2.p2.1.m1.1.1.3.3.cmml">i</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S5.SS4.SSS0.Px2.p2.1.m1.1b"><apply id="S5.SS4.SSS0.Px2.p2.1.m1.1.1.cmml" xref="S5.SS4.SSS0.Px2.p2.1.m1.1.1"><ci id="S5.SS4.SSS0.Px2.p2.1.m1.1.1.1.cmml" xref="S5.SS4.SSS0.Px2.p2.1.m1.1.1.1">→</ci><apply id="S5.SS4.SSS0.Px2.p2.1.m1.1.1.2.cmml" xref="S5.SS4.SSS0.Px2.p2.1.m1.1.1.2"><csymbol cd="ambiguous" id="S5.SS4.SSS0.Px2.p2.1.m1.1.1.2.1.cmml" xref="S5.SS4.SSS0.Px2.p2.1.m1.1.1.2">subscript</csymbol><ci id="S5.SS4.SSS0.Px2.p2.1.m1.1.1.2.2.cmml" xref="S5.SS4.SSS0.Px2.p2.1.m1.1.1.2.2">𝑥</ci><ci id="S5.SS4.SSS0.Px2.p2.1.m1.1.1.2.3.cmml" xref="S5.SS4.SSS0.Px2.p2.1.m1.1.1.2.3">𝑖</ci></apply><apply id="S5.SS4.SSS0.Px2.p2.1.m1.1.1.3.cmml" xref="S5.SS4.SSS0.Px2.p2.1.m1.1.1.3"><csymbol cd="ambiguous" id="S5.SS4.SSS0.Px2.p2.1.m1.1.1.3.1.cmml" xref="S5.SS4.SSS0.Px2.p2.1.m1.1.1.3">subscript</csymbol><ci id="S5.SS4.SSS0.Px2.p2.1.m1.1.1.3.2.cmml" xref="S5.SS4.SSS0.Px2.p2.1.m1.1.1.3.2">𝑦</ci><ci id="S5.SS4.SSS0.Px2.p2.1.m1.1.1.3.3.cmml" xref="S5.SS4.SSS0.Px2.p2.1.m1.1.1.3.3">𝑖</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS4.SSS0.Px2.p2.1.m1.1c">x_{i}\rightarrow y_{i}</annotation><annotation encoding="application/x-llamapun" id="S5.SS4.SSS0.Px2.p2.1.m1.1d">italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT → italic_y start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> task and the <math alttext="x_{i}\rightarrow{\mathbf{z}}_{i}\oplus y_{i}" class="ltx_Math" display="inline" id="S5.SS4.SSS0.Px2.p2.2.m2.1"><semantics id="S5.SS4.SSS0.Px2.p2.2.m2.1a"><mrow id="S5.SS4.SSS0.Px2.p2.2.m2.1.1" xref="S5.SS4.SSS0.Px2.p2.2.m2.1.1.cmml"><msub id="S5.SS4.SSS0.Px2.p2.2.m2.1.1.2" xref="S5.SS4.SSS0.Px2.p2.2.m2.1.1.2.cmml"><mi id="S5.SS4.SSS0.Px2.p2.2.m2.1.1.2.2" xref="S5.SS4.SSS0.Px2.p2.2.m2.1.1.2.2.cmml">x</mi><mi id="S5.SS4.SSS0.Px2.p2.2.m2.1.1.2.3" xref="S5.SS4.SSS0.Px2.p2.2.m2.1.1.2.3.cmml">i</mi></msub><mo id="S5.SS4.SSS0.Px2.p2.2.m2.1.1.1" stretchy="false" 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intermediate steps and the article (<math alttext="x_{i}\rightarrow{\mathbf{z}}_{i}\oplus y_{i}" class="ltx_Math" display="inline" id="S5.SS4.SSS0.Px2.p2.3.m3.1"><semantics id="S5.SS4.SSS0.Px2.p2.3.m3.1a"><mrow id="S5.SS4.SSS0.Px2.p2.3.m3.1.1" xref="S5.SS4.SSS0.Px2.p2.3.m3.1.1.cmml"><msub id="S5.SS4.SSS0.Px2.p2.3.m3.1.1.2" xref="S5.SS4.SSS0.Px2.p2.3.m3.1.1.2.cmml"><mi id="S5.SS4.SSS0.Px2.p2.3.m3.1.1.2.2" xref="S5.SS4.SSS0.Px2.p2.3.m3.1.1.2.2.cmml">x</mi><mi id="S5.SS4.SSS0.Px2.p2.3.m3.1.1.2.3" xref="S5.SS4.SSS0.Px2.p2.3.m3.1.1.2.3.cmml">i</mi></msub><mo id="S5.SS4.SSS0.Px2.p2.3.m3.1.1.1" stretchy="false" xref="S5.SS4.SSS0.Px2.p2.3.m3.1.1.1.cmml">→</mo><mrow id="S5.SS4.SSS0.Px2.p2.3.m3.1.1.3" xref="S5.SS4.SSS0.Px2.p2.3.m3.1.1.3.cmml"><msub id="S5.SS4.SSS0.Px2.p2.3.m3.1.1.3.2" xref="S5.SS4.SSS0.Px2.p2.3.m3.1.1.3.2.cmml"><mi id="S5.SS4.SSS0.Px2.p2.3.m3.1.1.3.2.2" xref="S5.SS4.SSS0.Px2.p2.3.m3.1.1.3.2.2.cmml">𝐳</mi><mi id="S5.SS4.SSS0.Px2.p2.3.m3.1.1.3.2.3" xref="S5.SS4.SSS0.Px2.p2.3.m3.1.1.3.2.3.cmml">i</mi></msub><mo id="S5.SS4.SSS0.Px2.p2.3.m3.1.1.3.1" xref="S5.SS4.SSS0.Px2.p2.3.m3.1.1.3.1.cmml">⊕</mo><msub id="S5.SS4.SSS0.Px2.p2.3.m3.1.1.3.3" xref="S5.SS4.SSS0.Px2.p2.3.m3.1.1.3.3.cmml"><mi id="S5.SS4.SSS0.Px2.p2.3.m3.1.1.3.3.2" xref="S5.SS4.SSS0.Px2.p2.3.m3.1.1.3.3.2.cmml">y</mi><mi id="S5.SS4.SSS0.Px2.p2.3.m3.1.1.3.3.3" xref="S5.SS4.SSS0.Px2.p2.3.m3.1.1.3.3.3.cmml">i</mi></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.SS4.SSS0.Px2.p2.3.m3.1b"><apply id="S5.SS4.SSS0.Px2.p2.3.m3.1.1.cmml" xref="S5.SS4.SSS0.Px2.p2.3.m3.1.1"><ci id="S5.SS4.SSS0.Px2.p2.3.m3.1.1.1.cmml" xref="S5.SS4.SSS0.Px2.p2.3.m3.1.1.1">→</ci><apply id="S5.SS4.SSS0.Px2.p2.3.m3.1.1.2.cmml" xref="S5.SS4.SSS0.Px2.p2.3.m3.1.1.2"><csymbol cd="ambiguous" id="S5.SS4.SSS0.Px2.p2.3.m3.1.1.2.1.cmml" xref="S5.SS4.SSS0.Px2.p2.3.m3.1.1.2">subscript</csymbol><ci id="S5.SS4.SSS0.Px2.p2.3.m3.1.1.2.2.cmml" xref="S5.SS4.SSS0.Px2.p2.3.m3.1.1.2.2">𝑥</ci><ci id="S5.SS4.SSS0.Px2.p2.3.m3.1.1.2.3.cmml" xref="S5.SS4.SSS0.Px2.p2.3.m3.1.1.2.3">𝑖</ci></apply><apply id="S5.SS4.SSS0.Px2.p2.3.m3.1.1.3.cmml" xref="S5.SS4.SSS0.Px2.p2.3.m3.1.1.3"><csymbol cd="latexml" id="S5.SS4.SSS0.Px2.p2.3.m3.1.1.3.1.cmml" xref="S5.SS4.SSS0.Px2.p2.3.m3.1.1.3.1">direct-sum</csymbol><apply id="S5.SS4.SSS0.Px2.p2.3.m3.1.1.3.2.cmml" xref="S5.SS4.SSS0.Px2.p2.3.m3.1.1.3.2"><csymbol cd="ambiguous" id="S5.SS4.SSS0.Px2.p2.3.m3.1.1.3.2.1.cmml" xref="S5.SS4.SSS0.Px2.p2.3.m3.1.1.3.2">subscript</csymbol><ci id="S5.SS4.SSS0.Px2.p2.3.m3.1.1.3.2.2.cmml" xref="S5.SS4.SSS0.Px2.p2.3.m3.1.1.3.2.2">𝐳</ci><ci id="S5.SS4.SSS0.Px2.p2.3.m3.1.1.3.2.3.cmml" xref="S5.SS4.SSS0.Px2.p2.3.m3.1.1.3.2.3">𝑖</ci></apply><apply id="S5.SS4.SSS0.Px2.p2.3.m3.1.1.3.3.cmml" xref="S5.SS4.SSS0.Px2.p2.3.m3.1.1.3.3"><csymbol cd="ambiguous" id="S5.SS4.SSS0.Px2.p2.3.m3.1.1.3.3.1.cmml" xref="S5.SS4.SSS0.Px2.p2.3.m3.1.1.3.3">subscript</csymbol><ci id="S5.SS4.SSS0.Px2.p2.3.m3.1.1.3.3.2.cmml" xref="S5.SS4.SSS0.Px2.p2.3.m3.1.1.3.3.2">𝑦</ci><ci id="S5.SS4.SSS0.Px2.p2.3.m3.1.1.3.3.3.cmml" xref="S5.SS4.SSS0.Px2.p2.3.m3.1.1.3.3.3">𝑖</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS4.SSS0.Px2.p2.3.m3.1c">x_{i}\rightarrow{\mathbf{z}}_{i}\oplus y_{i}</annotation><annotation encoding="application/x-llamapun" id="S5.SS4.SSS0.Px2.p2.3.m3.1d">italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT → bold_z start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ⊕ italic_y start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math>) during inference.</p> </div> <div class="ltx_para" id="S5.SS4.SSS0.Px2.p3"> <p class="ltx_p" id="S5.SS4.SSS0.Px2.p3.3">We also experiment with the mixture of three tasks on the Wikipedia data set: the vanilla <math alttext="x_{i}\rightarrow y_{i}" class="ltx_Math" display="inline" id="S5.SS4.SSS0.Px2.p3.1.m1.1"><semantics id="S5.SS4.SSS0.Px2.p3.1.m1.1a"><mrow id="S5.SS4.SSS0.Px2.p3.1.m1.1.1" xref="S5.SS4.SSS0.Px2.p3.1.m1.1.1.cmml"><msub id="S5.SS4.SSS0.Px2.p3.1.m1.1.1.2" xref="S5.SS4.SSS0.Px2.p3.1.m1.1.1.2.cmml"><mi id="S5.SS4.SSS0.Px2.p3.1.m1.1.1.2.2" xref="S5.SS4.SSS0.Px2.p3.1.m1.1.1.2.2.cmml">x</mi><mi id="S5.SS4.SSS0.Px2.p3.1.m1.1.1.2.3" xref="S5.SS4.SSS0.Px2.p3.1.m1.1.1.2.3.cmml">i</mi></msub><mo id="S5.SS4.SSS0.Px2.p3.1.m1.1.1.1" stretchy="false" xref="S5.SS4.SSS0.Px2.p3.1.m1.1.1.1.cmml">→</mo><msub id="S5.SS4.SSS0.Px2.p3.1.m1.1.1.3" xref="S5.SS4.SSS0.Px2.p3.1.m1.1.1.3.cmml"><mi id="S5.SS4.SSS0.Px2.p3.1.m1.1.1.3.2" xref="S5.SS4.SSS0.Px2.p3.1.m1.1.1.3.2.cmml">y</mi><mi id="S5.SS4.SSS0.Px2.p3.1.m1.1.1.3.3" xref="S5.SS4.SSS0.Px2.p3.1.m1.1.1.3.3.cmml">i</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S5.SS4.SSS0.Px2.p3.1.m1.1b"><apply id="S5.SS4.SSS0.Px2.p3.1.m1.1.1.cmml" xref="S5.SS4.SSS0.Px2.p3.1.m1.1.1"><ci id="S5.SS4.SSS0.Px2.p3.1.m1.1.1.1.cmml" xref="S5.SS4.SSS0.Px2.p3.1.m1.1.1.1">→</ci><apply id="S5.SS4.SSS0.Px2.p3.1.m1.1.1.2.cmml" xref="S5.SS4.SSS0.Px2.p3.1.m1.1.1.2"><csymbol cd="ambiguous" id="S5.SS4.SSS0.Px2.p3.1.m1.1.1.2.1.cmml" xref="S5.SS4.SSS0.Px2.p3.1.m1.1.1.2">subscript</csymbol><ci id="S5.SS4.SSS0.Px2.p3.1.m1.1.1.2.2.cmml" xref="S5.SS4.SSS0.Px2.p3.1.m1.1.1.2.2">𝑥</ci><ci id="S5.SS4.SSS0.Px2.p3.1.m1.1.1.2.3.cmml" xref="S5.SS4.SSS0.Px2.p3.1.m1.1.1.2.3">𝑖</ci></apply><apply id="S5.SS4.SSS0.Px2.p3.1.m1.1.1.3.cmml" xref="S5.SS4.SSS0.Px2.p3.1.m1.1.1.3"><csymbol cd="ambiguous" id="S5.SS4.SSS0.Px2.p3.1.m1.1.1.3.1.cmml" xref="S5.SS4.SSS0.Px2.p3.1.m1.1.1.3">subscript</csymbol><ci id="S5.SS4.SSS0.Px2.p3.1.m1.1.1.3.2.cmml" xref="S5.SS4.SSS0.Px2.p3.1.m1.1.1.3.2">𝑦</ci><ci id="S5.SS4.SSS0.Px2.p3.1.m1.1.1.3.3.cmml" xref="S5.SS4.SSS0.Px2.p3.1.m1.1.1.3.3">𝑖</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS4.SSS0.Px2.p3.1.m1.1c">x_{i}\rightarrow y_{i}</annotation><annotation encoding="application/x-llamapun" id="S5.SS4.SSS0.Px2.p3.1.m1.1d">italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT → italic_y start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> task, the <math alttext="x_{i}\rightarrow{\mathbf{z}}_{i}\oplus y_{i}" class="ltx_Math" display="inline" id="S5.SS4.SSS0.Px2.p3.2.m2.1"><semantics id="S5.SS4.SSS0.Px2.p3.2.m2.1a"><mrow id="S5.SS4.SSS0.Px2.p3.2.m2.1.1" xref="S5.SS4.SSS0.Px2.p3.2.m2.1.1.cmml"><msub id="S5.SS4.SSS0.Px2.p3.2.m2.1.1.2" xref="S5.SS4.SSS0.Px2.p3.2.m2.1.1.2.cmml"><mi id="S5.SS4.SSS0.Px2.p3.2.m2.1.1.2.2" xref="S5.SS4.SSS0.Px2.p3.2.m2.1.1.2.2.cmml">x</mi><mi id="S5.SS4.SSS0.Px2.p3.2.m2.1.1.2.3" xref="S5.SS4.SSS0.Px2.p3.2.m2.1.1.2.3.cmml">i</mi></msub><mo id="S5.SS4.SSS0.Px2.p3.2.m2.1.1.1" stretchy="false" xref="S5.SS4.SSS0.Px2.p3.2.m2.1.1.1.cmml">→</mo><mrow id="S5.SS4.SSS0.Px2.p3.2.m2.1.1.3" xref="S5.SS4.SSS0.Px2.p3.2.m2.1.1.3.cmml"><msub id="S5.SS4.SSS0.Px2.p3.2.m2.1.1.3.2" xref="S5.SS4.SSS0.Px2.p3.2.m2.1.1.3.2.cmml"><mi id="S5.SS4.SSS0.Px2.p3.2.m2.1.1.3.2.2" xref="S5.SS4.SSS0.Px2.p3.2.m2.1.1.3.2.2.cmml">𝐳</mi><mi id="S5.SS4.SSS0.Px2.p3.2.m2.1.1.3.2.3" xref="S5.SS4.SSS0.Px2.p3.2.m2.1.1.3.2.3.cmml">i</mi></msub><mo id="S5.SS4.SSS0.Px2.p3.2.m2.1.1.3.1" xref="S5.SS4.SSS0.Px2.p3.2.m2.1.1.3.1.cmml">⊕</mo><msub id="S5.SS4.SSS0.Px2.p3.2.m2.1.1.3.3" xref="S5.SS4.SSS0.Px2.p3.2.m2.1.1.3.3.cmml"><mi id="S5.SS4.SSS0.Px2.p3.2.m2.1.1.3.3.2" xref="S5.SS4.SSS0.Px2.p3.2.m2.1.1.3.3.2.cmml">y</mi><mi id="S5.SS4.SSS0.Px2.p3.2.m2.1.1.3.3.3" xref="S5.SS4.SSS0.Px2.p3.2.m2.1.1.3.3.3.cmml">i</mi></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.SS4.SSS0.Px2.p3.2.m2.1b"><apply id="S5.SS4.SSS0.Px2.p3.2.m2.1.1.cmml" xref="S5.SS4.SSS0.Px2.p3.2.m2.1.1"><ci id="S5.SS4.SSS0.Px2.p3.2.m2.1.1.1.cmml" xref="S5.SS4.SSS0.Px2.p3.2.m2.1.1.1">→</ci><apply id="S5.SS4.SSS0.Px2.p3.2.m2.1.1.2.cmml" xref="S5.SS4.SSS0.Px2.p3.2.m2.1.1.2"><csymbol cd="ambiguous" id="S5.SS4.SSS0.Px2.p3.2.m2.1.1.2.1.cmml" xref="S5.SS4.SSS0.Px2.p3.2.m2.1.1.2">subscript</csymbol><ci id="S5.SS4.SSS0.Px2.p3.2.m2.1.1.2.2.cmml" xref="S5.SS4.SSS0.Px2.p3.2.m2.1.1.2.2">𝑥</ci><ci id="S5.SS4.SSS0.Px2.p3.2.m2.1.1.2.3.cmml" xref="S5.SS4.SSS0.Px2.p3.2.m2.1.1.2.3">𝑖</ci></apply><apply id="S5.SS4.SSS0.Px2.p3.2.m2.1.1.3.cmml" xref="S5.SS4.SSS0.Px2.p3.2.m2.1.1.3"><csymbol cd="latexml" id="S5.SS4.SSS0.Px2.p3.2.m2.1.1.3.1.cmml" xref="S5.SS4.SSS0.Px2.p3.2.m2.1.1.3.1">direct-sum</csymbol><apply id="S5.SS4.SSS0.Px2.p3.2.m2.1.1.3.2.cmml" xref="S5.SS4.SSS0.Px2.p3.2.m2.1.1.3.2"><csymbol cd="ambiguous" id="S5.SS4.SSS0.Px2.p3.2.m2.1.1.3.2.1.cmml" xref="S5.SS4.SSS0.Px2.p3.2.m2.1.1.3.2">subscript</csymbol><ci id="S5.SS4.SSS0.Px2.p3.2.m2.1.1.3.2.2.cmml" xref="S5.SS4.SSS0.Px2.p3.2.m2.1.1.3.2.2">𝐳</ci><ci id="S5.SS4.SSS0.Px2.p3.2.m2.1.1.3.2.3.cmml" xref="S5.SS4.SSS0.Px2.p3.2.m2.1.1.3.2.3">𝑖</ci></apply><apply id="S5.SS4.SSS0.Px2.p3.2.m2.1.1.3.3.cmml" xref="S5.SS4.SSS0.Px2.p3.2.m2.1.1.3.3"><csymbol cd="ambiguous" id="S5.SS4.SSS0.Px2.p3.2.m2.1.1.3.3.1.cmml" xref="S5.SS4.SSS0.Px2.p3.2.m2.1.1.3.3">subscript</csymbol><ci id="S5.SS4.SSS0.Px2.p3.2.m2.1.1.3.3.2.cmml" xref="S5.SS4.SSS0.Px2.p3.2.m2.1.1.3.3.2">𝑦</ci><ci id="S5.SS4.SSS0.Px2.p3.2.m2.1.1.3.3.3.cmml" xref="S5.SS4.SSS0.Px2.p3.2.m2.1.1.3.3.3">𝑖</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS4.SSS0.Px2.p3.2.m2.1c">x_{i}\rightarrow{\mathbf{z}}_{i}\oplus y_{i}</annotation><annotation encoding="application/x-llamapun" id="S5.SS4.SSS0.Px2.p3.2.m2.1d">italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT → bold_z start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ⊕ italic_y start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> task which generates the intermediate steps, and the <math alttext="x_{i}\oplus{\mathbf{z}}_{i}\rightarrow y_{i}" class="ltx_Math" display="inline" id="S5.SS4.SSS0.Px2.p3.3.m3.1"><semantics id="S5.SS4.SSS0.Px2.p3.3.m3.1a"><mrow id="S5.SS4.SSS0.Px2.p3.3.m3.1.1" xref="S5.SS4.SSS0.Px2.p3.3.m3.1.1.cmml"><mrow id="S5.SS4.SSS0.Px2.p3.3.m3.1.1.2" xref="S5.SS4.SSS0.Px2.p3.3.m3.1.1.2.cmml"><msub id="S5.SS4.SSS0.Px2.p3.3.m3.1.1.2.2" xref="S5.SS4.SSS0.Px2.p3.3.m3.1.1.2.2.cmml"><mi id="S5.SS4.SSS0.Px2.p3.3.m3.1.1.2.2.2" xref="S5.SS4.SSS0.Px2.p3.3.m3.1.1.2.2.2.cmml">x</mi><mi id="S5.SS4.SSS0.Px2.p3.3.m3.1.1.2.2.3" xref="S5.SS4.SSS0.Px2.p3.3.m3.1.1.2.2.3.cmml">i</mi></msub><mo id="S5.SS4.SSS0.Px2.p3.3.m3.1.1.2.1" xref="S5.SS4.SSS0.Px2.p3.3.m3.1.1.2.1.cmml">⊕</mo><msub id="S5.SS4.SSS0.Px2.p3.3.m3.1.1.2.3" xref="S5.SS4.SSS0.Px2.p3.3.m3.1.1.2.3.cmml"><mi id="S5.SS4.SSS0.Px2.p3.3.m3.1.1.2.3.2" xref="S5.SS4.SSS0.Px2.p3.3.m3.1.1.2.3.2.cmml">𝐳</mi><mi id="S5.SS4.SSS0.Px2.p3.3.m3.1.1.2.3.3" xref="S5.SS4.SSS0.Px2.p3.3.m3.1.1.2.3.3.cmml">i</mi></msub></mrow><mo id="S5.SS4.SSS0.Px2.p3.3.m3.1.1.1" stretchy="false" xref="S5.SS4.SSS0.Px2.p3.3.m3.1.1.1.cmml">→</mo><msub id="S5.SS4.SSS0.Px2.p3.3.m3.1.1.3" xref="S5.SS4.SSS0.Px2.p3.3.m3.1.1.3.cmml"><mi 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However, further adding the third task does not seem to bring extra performance improvement. This might suggest that writing an article given the outlines is a relatively trivial task for the underlying LLM in our experiments. Further fine-tuning the LLM on this task does not bring additional insight for the model to write better article.</p> </div> <figure class="ltx_table" id="S5.T3"> <figcaption class="ltx_caption"><span class="ltx_tag ltx_tag_table">Table 3: </span>Single-turn vs. multi-turn inference on SciNews.</figcaption> <p class="ltx_p ltx_align_center" id="S5.T3.10"><span class="ltx_text" id="S5.T3.10.10"> <span class="ltx_tabular ltx_guessed_headers ltx_align_middle" id="S5.T3.10.10.10"> <span class="ltx_thead"> <span class="ltx_tr" id="S5.T3.4.4.4.4"> <span class="ltx_td ltx_align_left ltx_th ltx_th_column ltx_th_row ltx_border_r ltx_border_t" id="S5.T3.4.4.4.4.5"><span class="ltx_text ltx_font_bold" id="S5.T3.4.4.4.4.5.1">Training Mixture</span></span> <span class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_th_row ltx_border_r ltx_border_t" id="S5.T3.4.4.4.4.6"><span class="ltx_text ltx_font_bold" id="S5.T3.4.4.4.4.6.1">Inference</span></span> <span class="ltx_td ltx_align_center ltx_th 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id="S5.T3.7.7.7.7.3.m1.1"><semantics id="S5.T3.7.7.7.7.3.m1.1a"><mrow id="S5.T3.7.7.7.7.3.m1.1.1" xref="S5.T3.7.7.7.7.3.m1.1.1.cmml"><msub id="S5.T3.7.7.7.7.3.m1.1.1.2" xref="S5.T3.7.7.7.7.3.m1.1.1.2.cmml"><mi id="S5.T3.7.7.7.7.3.m1.1.1.2.2" xref="S5.T3.7.7.7.7.3.m1.1.1.2.2.cmml">x</mi><mi id="S5.T3.7.7.7.7.3.m1.1.1.2.3" xref="S5.T3.7.7.7.7.3.m1.1.1.2.3.cmml">i</mi></msub><mo id="S5.T3.7.7.7.7.3.m1.1.1.1" stretchy="false" xref="S5.T3.7.7.7.7.3.m1.1.1.1.cmml">→</mo><mrow id="S5.T3.7.7.7.7.3.m1.1.1.3" xref="S5.T3.7.7.7.7.3.m1.1.1.3.cmml"><msub id="S5.T3.7.7.7.7.3.m1.1.1.3.2" xref="S5.T3.7.7.7.7.3.m1.1.1.3.2.cmml"><mi id="S5.T3.7.7.7.7.3.m1.1.1.3.2.2" xref="S5.T3.7.7.7.7.3.m1.1.1.3.2.2.cmml">𝐳</mi><mi id="S5.T3.7.7.7.7.3.m1.1.1.3.2.3" xref="S5.T3.7.7.7.7.3.m1.1.1.3.2.3.cmml">i</mi></msub><mo id="S5.T3.7.7.7.7.3.m1.1.1.3.1" xref="S5.T3.7.7.7.7.3.m1.1.1.3.1.cmml">⊕</mo><msub id="S5.T3.7.7.7.7.3.m1.1.1.3.3" xref="S5.T3.7.7.7.7.3.m1.1.1.3.3.cmml"><mi id="S5.T3.7.7.7.7.3.m1.1.1.3.3.2" xref="S5.T3.7.7.7.7.3.m1.1.1.3.3.2.cmml">y</mi><mi id="S5.T3.7.7.7.7.3.m1.1.1.3.3.3" xref="S5.T3.7.7.7.7.3.m1.1.1.3.3.3.cmml">i</mi></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.T3.7.7.7.7.3.m1.1b"><apply id="S5.T3.7.7.7.7.3.m1.1.1.cmml" xref="S5.T3.7.7.7.7.3.m1.1.1"><ci id="S5.T3.7.7.7.7.3.m1.1.1.1.cmml" xref="S5.T3.7.7.7.7.3.m1.1.1.1">→</ci><apply id="S5.T3.7.7.7.7.3.m1.1.1.2.cmml" xref="S5.T3.7.7.7.7.3.m1.1.1.2"><csymbol cd="ambiguous" id="S5.T3.7.7.7.7.3.m1.1.1.2.1.cmml" xref="S5.T3.7.7.7.7.3.m1.1.1.2">subscript</csymbol><ci id="S5.T3.7.7.7.7.3.m1.1.1.2.2.cmml" xref="S5.T3.7.7.7.7.3.m1.1.1.2.2">𝑥</ci><ci id="S5.T3.7.7.7.7.3.m1.1.1.2.3.cmml" xref="S5.T3.7.7.7.7.3.m1.1.1.2.3">𝑖</ci></apply><apply id="S5.T3.7.7.7.7.3.m1.1.1.3.cmml" xref="S5.T3.7.7.7.7.3.m1.1.1.3"><csymbol cd="latexml" id="S5.T3.7.7.7.7.3.m1.1.1.3.1.cmml" xref="S5.T3.7.7.7.7.3.m1.1.1.3.1">direct-sum</csymbol><apply id="S5.T3.7.7.7.7.3.m1.1.1.3.2.cmml" xref="S5.T3.7.7.7.7.3.m1.1.1.3.2"><csymbol cd="ambiguous" id="S5.T3.7.7.7.7.3.m1.1.1.3.2.1.cmml" xref="S5.T3.7.7.7.7.3.m1.1.1.3.2">subscript</csymbol><ci id="S5.T3.7.7.7.7.3.m1.1.1.3.2.2.cmml" xref="S5.T3.7.7.7.7.3.m1.1.1.3.2.2">𝐳</ci><ci id="S5.T3.7.7.7.7.3.m1.1.1.3.2.3.cmml" xref="S5.T3.7.7.7.7.3.m1.1.1.3.2.3">𝑖</ci></apply><apply id="S5.T3.7.7.7.7.3.m1.1.1.3.3.cmml" xref="S5.T3.7.7.7.7.3.m1.1.1.3.3"><csymbol cd="ambiguous" id="S5.T3.7.7.7.7.3.m1.1.1.3.3.1.cmml" xref="S5.T3.7.7.7.7.3.m1.1.1.3.3">subscript</csymbol><ci id="S5.T3.7.7.7.7.3.m1.1.1.3.3.2.cmml" xref="S5.T3.7.7.7.7.3.m1.1.1.3.3.2">𝑦</ci><ci id="S5.T3.7.7.7.7.3.m1.1.1.3.3.3.cmml" xref="S5.T3.7.7.7.7.3.m1.1.1.3.3.3">𝑖</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.T3.7.7.7.7.3.m1.1c">x_{i}\rightarrow{\mathbf{z}}_{i}\oplus y_{i}</annotation><annotation encoding="application/x-llamapun" id="S5.T3.7.7.7.7.3.m1.1d">italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT → bold_z start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ⊕ italic_y start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math></span> <span class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S5.T3.7.7.7.7.4"><span class="ltx_text ltx_font_bold" id="S5.T3.7.7.7.7.4.1">49.39</span></span> <span class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S5.T3.7.7.7.7.5"><span class="ltx_text ltx_font_bold" id="S5.T3.7.7.7.7.5.1">14.69</span></span> <span class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S5.T3.7.7.7.7.6"><span class="ltx_text ltx_font_bold" id="S5.T3.7.7.7.7.6.1">19.68</span></span> <span class="ltx_td ltx_nopad_r ltx_align_center ltx_border_t" id="S5.T3.7.7.7.7.7"><span class="ltx_text ltx_font_bold" id="S5.T3.7.7.7.7.7.1">46.61</span></span></span> <span class="ltx_tr" id="S5.T3.10.10.10.10"> <span class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_b ltx_border_r" id="S5.T3.9.9.9.9.2"><math alttext="x_{i}\rightarrow y_{i}" class="ltx_Math" display="inline" id="S5.T3.8.8.8.8.1.m1.1"><semantics id="S5.T3.8.8.8.8.1.m1.1a"><mrow id="S5.T3.8.8.8.8.1.m1.1.1" xref="S5.T3.8.8.8.8.1.m1.1.1.cmml"><msub id="S5.T3.8.8.8.8.1.m1.1.1.2" xref="S5.T3.8.8.8.8.1.m1.1.1.2.cmml"><mi id="S5.T3.8.8.8.8.1.m1.1.1.2.2" xref="S5.T3.8.8.8.8.1.m1.1.1.2.2.cmml">x</mi><mi id="S5.T3.8.8.8.8.1.m1.1.1.2.3" xref="S5.T3.8.8.8.8.1.m1.1.1.2.3.cmml">i</mi></msub><mo id="S5.T3.8.8.8.8.1.m1.1.1.1" stretchy="false" xref="S5.T3.8.8.8.8.1.m1.1.1.1.cmml">→</mo><msub id="S5.T3.8.8.8.8.1.m1.1.1.3" xref="S5.T3.8.8.8.8.1.m1.1.1.3.cmml"><mi id="S5.T3.8.8.8.8.1.m1.1.1.3.2" xref="S5.T3.8.8.8.8.1.m1.1.1.3.2.cmml">y</mi><mi id="S5.T3.8.8.8.8.1.m1.1.1.3.3" xref="S5.T3.8.8.8.8.1.m1.1.1.3.3.cmml">i</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S5.T3.8.8.8.8.1.m1.1b"><apply id="S5.T3.8.8.8.8.1.m1.1.1.cmml" xref="S5.T3.8.8.8.8.1.m1.1.1"><ci id="S5.T3.8.8.8.8.1.m1.1.1.1.cmml" xref="S5.T3.8.8.8.8.1.m1.1.1.1">→</ci><apply id="S5.T3.8.8.8.8.1.m1.1.1.2.cmml" xref="S5.T3.8.8.8.8.1.m1.1.1.2"><csymbol cd="ambiguous" id="S5.T3.8.8.8.8.1.m1.1.1.2.1.cmml" xref="S5.T3.8.8.8.8.1.m1.1.1.2">subscript</csymbol><ci id="S5.T3.8.8.8.8.1.m1.1.1.2.2.cmml" xref="S5.T3.8.8.8.8.1.m1.1.1.2.2">𝑥</ci><ci id="S5.T3.8.8.8.8.1.m1.1.1.2.3.cmml" xref="S5.T3.8.8.8.8.1.m1.1.1.2.3">𝑖</ci></apply><apply id="S5.T3.8.8.8.8.1.m1.1.1.3.cmml" xref="S5.T3.8.8.8.8.1.m1.1.1.3"><csymbol cd="ambiguous" id="S5.T3.8.8.8.8.1.m1.1.1.3.1.cmml" xref="S5.T3.8.8.8.8.1.m1.1.1.3">subscript</csymbol><ci id="S5.T3.8.8.8.8.1.m1.1.1.3.2.cmml" xref="S5.T3.8.8.8.8.1.m1.1.1.3.2">𝑦</ci><ci id="S5.T3.8.8.8.8.1.m1.1.1.3.3.cmml" xref="S5.T3.8.8.8.8.1.m1.1.1.3.3">𝑖</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.T3.8.8.8.8.1.m1.1c">x_{i}\rightarrow y_{i}</annotation><annotation encoding="application/x-llamapun" id="S5.T3.8.8.8.8.1.m1.1d">italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT → italic_y start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math>; <math alttext="x_{i}\rightarrow{\mathbf{z}}_{i}\oplus y_{i}" class="ltx_Math" display="inline" id="S5.T3.9.9.9.9.2.m2.1"><semantics id="S5.T3.9.9.9.9.2.m2.1a"><mrow id="S5.T3.9.9.9.9.2.m2.1.1" xref="S5.T3.9.9.9.9.2.m2.1.1.cmml"><msub id="S5.T3.9.9.9.9.2.m2.1.1.2" xref="S5.T3.9.9.9.9.2.m2.1.1.2.cmml"><mi id="S5.T3.9.9.9.9.2.m2.1.1.2.2" xref="S5.T3.9.9.9.9.2.m2.1.1.2.2.cmml">x</mi><mi id="S5.T3.9.9.9.9.2.m2.1.1.2.3" xref="S5.T3.9.9.9.9.2.m2.1.1.2.3.cmml">i</mi></msub><mo id="S5.T3.9.9.9.9.2.m2.1.1.1" stretchy="false" xref="S5.T3.9.9.9.9.2.m2.1.1.1.cmml">→</mo><mrow id="S5.T3.9.9.9.9.2.m2.1.1.3" xref="S5.T3.9.9.9.9.2.m2.1.1.3.cmml"><msub id="S5.T3.9.9.9.9.2.m2.1.1.3.2" xref="S5.T3.9.9.9.9.2.m2.1.1.3.2.cmml"><mi id="S5.T3.9.9.9.9.2.m2.1.1.3.2.2" xref="S5.T3.9.9.9.9.2.m2.1.1.3.2.2.cmml">𝐳</mi><mi id="S5.T3.9.9.9.9.2.m2.1.1.3.2.3" xref="S5.T3.9.9.9.9.2.m2.1.1.3.2.3.cmml">i</mi></msub><mo id="S5.T3.9.9.9.9.2.m2.1.1.3.1" xref="S5.T3.9.9.9.9.2.m2.1.1.3.1.cmml">⊕</mo><msub id="S5.T3.9.9.9.9.2.m2.1.1.3.3" xref="S5.T3.9.9.9.9.2.m2.1.1.3.3.cmml"><mi id="S5.T3.9.9.9.9.2.m2.1.1.3.3.2" xref="S5.T3.9.9.9.9.2.m2.1.1.3.3.2.cmml">y</mi><mi id="S5.T3.9.9.9.9.2.m2.1.1.3.3.3" xref="S5.T3.9.9.9.9.2.m2.1.1.3.3.3.cmml">i</mi></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.T3.9.9.9.9.2.m2.1b"><apply id="S5.T3.9.9.9.9.2.m2.1.1.cmml" xref="S5.T3.9.9.9.9.2.m2.1.1"><ci id="S5.T3.9.9.9.9.2.m2.1.1.1.cmml" xref="S5.T3.9.9.9.9.2.m2.1.1.1">→</ci><apply id="S5.T3.9.9.9.9.2.m2.1.1.2.cmml" xref="S5.T3.9.9.9.9.2.m2.1.1.2"><csymbol cd="ambiguous" id="S5.T3.9.9.9.9.2.m2.1.1.2.1.cmml" xref="S5.T3.9.9.9.9.2.m2.1.1.2">subscript</csymbol><ci id="S5.T3.9.9.9.9.2.m2.1.1.2.2.cmml" xref="S5.T3.9.9.9.9.2.m2.1.1.2.2">𝑥</ci><ci id="S5.T3.9.9.9.9.2.m2.1.1.2.3.cmml" xref="S5.T3.9.9.9.9.2.m2.1.1.2.3">𝑖</ci></apply><apply id="S5.T3.9.9.9.9.2.m2.1.1.3.cmml" xref="S5.T3.9.9.9.9.2.m2.1.1.3"><csymbol cd="latexml" 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xref="S5.T3.10.10.10.10.3.m1.2.2.2.2.3.3">𝑖</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.T3.10.10.10.10.3.m1.2c">x_{i}\rightarrow{\mathbf{z}}_{i},x_{i}\oplus{\mathbf{z}}_{i}\rightarrow y_{i}</annotation><annotation encoding="application/x-llamapun" id="S5.T3.10.10.10.10.3.m1.2d">italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT → bold_z start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ⊕ bold_z start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT → italic_y start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math></span> <span class="ltx_td ltx_align_center ltx_border_b ltx_border_r" id="S5.T3.10.10.10.10.4">46.76</span> <span class="ltx_td ltx_align_center ltx_border_b ltx_border_r" id="S5.T3.10.10.10.10.5">14.68</span> <span class="ltx_td ltx_align_center ltx_border_b ltx_border_r" id="S5.T3.10.10.10.10.6">19.43</span> <span class="ltx_td ltx_nopad_r ltx_align_center ltx_border_b" id="S5.T3.10.10.10.10.7">43.73</span></span> </span> </span></span></p> </figure> </section> <section class="ltx_paragraph" id="S5.SS4.SSS0.Px3"> <h4 class="ltx_title ltx_title_paragraph">Single-turn vs. multi-turn.</h4> <div class="ltx_para" id="S5.SS4.SSS0.Px3.p1"> <p class="ltx_p" id="S5.SS4.SSS0.Px3.p1.1">In addition to identifying the best recipe of training mixture and <span class="ltx_text ltx_font_italic" id="S5.SS4.SSS0.Px3.p1.1.1">single-turn</span> inference, we also compare the most competitive training recipe in Table <a class="ltx_ref" href="https://arxiv.org/html/2410.06203v1#S5.T2" title="Table 2 ‣ Comparison of different training and inference recipes. ‣ 5.4 Results ‣ 5 Experiments ‣ Integrating Planning into Single-Turn Long-Form Text Generation"><span class="ltx_text ltx_ref_tag">2</span></a> against multi-turn inference on SciNews. Results in Table <a class="ltx_ref" href="https://arxiv.org/html/2410.06203v1#S5.T3" title="Table 3 ‣ Comparison of different training and inference recipes. ‣ 5.4 Results ‣ 5 Experiments ‣ Integrating Planning into Single-Turn Long-Form Text Generation"><span class="ltx_text ltx_ref_tag">3</span></a> show that single-turn inference notably outperforms multi-turn inference.</p> </div> <figure class="ltx_table" id="S5.T4"> <figcaption class="ltx_caption"><span class="ltx_tag ltx_tag_table">Table 4: </span>Human SxS results comparing the proposed FT w/ I method to FT w/o I baseline </figcaption> <p class="ltx_p ltx_align_center" id="S5.T4.1"><span class="ltx_text" id="S5.T4.1.1"> <span class="ltx_tabular ltx_guessed_headers ltx_align_middle" id="S5.T4.1.1.1"> <span class="ltx_thead"> <span class="ltx_tr" id="S5.T4.1.1.1.1.1"> <span class="ltx_td ltx_align_left ltx_th ltx_th_column ltx_th_row ltx_border_r ltx_border_t" id="S5.T4.1.1.1.1.1.1"><span class="ltx_text ltx_font_bold" id="S5.T4.1.1.1.1.1.1.1">Dataset</span></span> <span class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_r ltx_border_t" id="S5.T4.1.1.1.1.1.2"><span class="ltx_text ltx_font_bold" id="S5.T4.1.1.1.1.1.2.1">Overall W/L</span></span> <span class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_r ltx_border_t" id="S5.T4.1.1.1.1.1.3"><span class="ltx_text ltx_font_bold" id="S5.T4.1.1.1.1.1.3.1">Coherence & Organization</span></span> <span class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_r ltx_border_t" id="S5.T4.1.1.1.1.1.4"><span class="ltx_text ltx_font_bold" id="S5.T4.1.1.1.1.1.4.1">Relevance & Focus</span></span> <span class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_t" id="S5.T4.1.1.1.1.1.5"><span class="ltx_text ltx_font_bold" id="S5.T4.1.1.1.1.1.5.1">Verifiability</span></span></span> </span> <span class="ltx_tbody"> <span class="ltx_tr" id="S5.T4.1.1.1.2.1"> <span class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_r ltx_border_t" id="S5.T4.1.1.1.2.1.1"><span class="ltx_text" id="S5.T4.1.1.1.2.1.1.1">SciNews</span></span> <span class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S5.T4.1.1.1.2.1.2">3.60</span> <span class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S5.T4.1.1.1.2.1.3">4.25</span> <span class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S5.T4.1.1.1.2.1.4">3.00</span> <span class="ltx_td ltx_align_center ltx_border_t" id="S5.T4.1.1.1.2.1.5">7.75</span></span> <span class="ltx_tr" id="S5.T4.1.1.1.3.2"> <span class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_b ltx_border_r ltx_border_t" id="S5.T4.1.1.1.3.2.1"><span class="ltx_text" id="S5.T4.1.1.1.3.2.1.1">Wikipedia</span></span> <span class="ltx_td ltx_align_center ltx_border_b ltx_border_r ltx_border_t" id="S5.T4.1.1.1.3.2.2">1.56</span> <span class="ltx_td ltx_align_center ltx_border_b ltx_border_r ltx_border_t" id="S5.T4.1.1.1.3.2.3">1.10</span> <span class="ltx_td ltx_align_center ltx_border_b ltx_border_r ltx_border_t" id="S5.T4.1.1.1.3.2.4">1.38</span> <span class="ltx_td ltx_align_center ltx_border_b ltx_border_t" id="S5.T4.1.1.1.3.2.5">1.40</span></span> </span> </span></span></p> </figure> </section> <section class="ltx_paragraph" id="S5.SS4.SSS0.Px4"> <h4 class="ltx_title ltx_title_paragraph">Human and auto SxS results.</h4> <div class="ltx_para" id="S5.SS4.SSS0.Px4.p1"> <p class="ltx_p" id="S5.SS4.SSS0.Px4.p1.1">To further validate the effectiveness of our methods, we conducted two sets of side-by-side ratings, one by expert human raters and one by LLM autorater, to compare the outputs of our methods against those of the baseline.</p> </div> <div class="ltx_para" id="S5.SS4.SSS0.Px4.p2"> <p class="ltx_p" id="S5.SS4.SSS0.Px4.p2.5">For the SciNews dataset, we select the FT w/I method using the training mixture of <math alttext="x_{i}\rightarrow y_{i}" class="ltx_Math" display="inline" id="S5.SS4.SSS0.Px4.p2.1.m1.1"><semantics 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In both cases, the baseline is the vanilla FT w/o I method without using intermediate steps.</p> </div> <div class="ltx_para" id="S5.SS4.SSS0.Px4.p3"> <p class="ltx_p" id="S5.SS4.SSS0.Px4.p3.1">For human SxS, we asked expert human raters to rate 50 items of each data set. They were asked to compare the two outputs for each input item (“paper body” for SciNews, “topic” for Wikipedia) in three of the criteria defined in <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2410.06203v1#bib.bib19" title="">19</a>]</cite> that are most relevant for our task, namely <span class="ltx_text ltx_font_italic" id="S5.SS4.SSS0.Px4.p3.1.1">coherence and organization</span>, <span class="ltx_text ltx_font_italic" id="S5.SS4.SSS0.Px4.p3.1.2">relevance and focus</span>, and <span class="ltx_text ltx_font_italic" id="S5.SS4.SSS0.Px4.p3.1.3">verifiability</span>, and to provide an overall assessment. The results are presented in Table <a class="ltx_ref" href="https://arxiv.org/html/2410.06203v1#S5.T4" title="Table 4 ‣ Single-turn vs. multi-turn. ‣ 5.4 Results ‣ 5 Experiments ‣ Integrating Planning into Single-Turn Long-Form Text Generation"><span class="ltx_text ltx_ref_tag">4</span></a>. Our methods show strong win against the baseline on SciNews, both in the overall quality and in each of the three criteria. On Wikipedia, we also observed a positive result in all criteria and overall.</p> </div> <div class="ltx_para" id="S5.SS4.SSS0.Px4.p4"> <p class="ltx_p" id="S5.SS4.SSS0.Px4.p4.1">Besides Human SxS evaluation, we also employed the automated LLM SxS as additional validation, on 200 samples from SciNews and all 100 samples of FreshWiki. The results show that the FT w/I method achieves W/L ratio of 2.85 and 1.20 on SciNews and Wikipedia datasets, respectively, both larger than 1. </p> </div> <figure class="ltx_table" id="S5.T5"> <figcaption class="ltx_caption"><span class="ltx_tag ltx_tag_table">Table 5: </span>Performance of different strategies on SciNews with Gemini 1.5 Flash as the base model.</figcaption> <p class="ltx_p ltx_align_center" id="S5.T5.13"><span class="ltx_text" id="S5.T5.13.13"> <span class="ltx_tabular ltx_guessed_headers ltx_align_middle" id="S5.T5.13.13.13"> <span class="ltx_thead"> <span class="ltx_tr" id="S5.T5.4.4.4.4"> <span class="ltx_td ltx_align_left ltx_th ltx_th_column ltx_th_row ltx_border_r ltx_border_t" id="S5.T5.4.4.4.4.5"><span class="ltx_text ltx_font_bold" id="S5.T5.4.4.4.4.5.1">Training Mixture</span></span> <span class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_th_row ltx_border_r ltx_border_t" id="S5.T5.4.4.4.4.6"><span class="ltx_text ltx_font_bold" id="S5.T5.4.4.4.4.6.1">Inference</span></span> <span class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_r ltx_border_t" id="S5.T5.1.1.1.1.1"><span class="ltx_text ltx_font_bold" id="S5.T5.1.1.1.1.1.1">R1<sub class="ltx_sub" id="S5.T5.1.1.1.1.1.1.1"><span class="ltx_text ltx_font_medium ltx_font_italic" id="S5.T5.1.1.1.1.1.1.1.1">F1</span></sub></span></span> <span class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_r ltx_border_t" id="S5.T5.2.2.2.2.2"><span class="ltx_text ltx_font_bold" id="S5.T5.2.2.2.2.2.1">R2<sub class="ltx_sub" id="S5.T5.2.2.2.2.2.1.1"><span class="ltx_text ltx_font_medium ltx_font_italic" id="S5.T5.2.2.2.2.2.1.1.1">F1</span></sub></span></span> <span class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_r ltx_border_t" id="S5.T5.3.3.3.3.3"><span class="ltx_text ltx_font_bold" id="S5.T5.3.3.3.3.3.1">RL<sub class="ltx_sub" id="S5.T5.3.3.3.3.3.1.1"><span class="ltx_text ltx_font_medium ltx_font_italic" id="S5.T5.3.3.3.3.3.1.1.1">F1</span></sub></span></span> <span class="ltx_td ltx_nopad_r ltx_align_center ltx_th ltx_th_column ltx_border_t" id="S5.T5.4.4.4.4.4"><span class="ltx_text ltx_font_bold" id="S5.T5.4.4.4.4.4.1">RLsum<sub class="ltx_sub" id="S5.T5.4.4.4.4.4.1.1"><span class="ltx_text ltx_font_medium ltx_font_italic" id="S5.T5.4.4.4.4.4.1.1.1">F1</span></sub></span></span></span> </span> <span class="ltx_tbody"> <span class="ltx_tr" id="S5.T5.5.5.5.5"> <span class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_r ltx_border_t" id="S5.T5.5.5.5.5.2">zero-shot</span> <span class="ltx_td ltx_align_center ltx_th ltx_th_row ltx_border_r ltx_border_t" id="S5.T5.5.5.5.5.1"><math alttext="x_{i}\rightarrow y_{i}" class="ltx_Math" display="inline" id="S5.T5.5.5.5.5.1.m1.1"><semantics id="S5.T5.5.5.5.5.1.m1.1a"><mrow id="S5.T5.5.5.5.5.1.m1.1.1" xref="S5.T5.5.5.5.5.1.m1.1.1.cmml"><msub id="S5.T5.5.5.5.5.1.m1.1.1.2" xref="S5.T5.5.5.5.5.1.m1.1.1.2.cmml"><mi id="S5.T5.5.5.5.5.1.m1.1.1.2.2" xref="S5.T5.5.5.5.5.1.m1.1.1.2.2.cmml">x</mi><mi id="S5.T5.5.5.5.5.1.m1.1.1.2.3" xref="S5.T5.5.5.5.5.1.m1.1.1.2.3.cmml">i</mi></msub><mo id="S5.T5.5.5.5.5.1.m1.1.1.1" stretchy="false" xref="S5.T5.5.5.5.5.1.m1.1.1.1.cmml">→</mo><msub id="S5.T5.5.5.5.5.1.m1.1.1.3" xref="S5.T5.5.5.5.5.1.m1.1.1.3.cmml"><mi id="S5.T5.5.5.5.5.1.m1.1.1.3.2" xref="S5.T5.5.5.5.5.1.m1.1.1.3.2.cmml">y</mi><mi id="S5.T5.5.5.5.5.1.m1.1.1.3.3" xref="S5.T5.5.5.5.5.1.m1.1.1.3.3.cmml">i</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S5.T5.5.5.5.5.1.m1.1b"><apply id="S5.T5.5.5.5.5.1.m1.1.1.cmml" xref="S5.T5.5.5.5.5.1.m1.1.1"><ci id="S5.T5.5.5.5.5.1.m1.1.1.1.cmml" xref="S5.T5.5.5.5.5.1.m1.1.1.1">→</ci><apply id="S5.T5.5.5.5.5.1.m1.1.1.2.cmml" xref="S5.T5.5.5.5.5.1.m1.1.1.2"><csymbol cd="ambiguous" id="S5.T5.5.5.5.5.1.m1.1.1.2.1.cmml" xref="S5.T5.5.5.5.5.1.m1.1.1.2">subscript</csymbol><ci id="S5.T5.5.5.5.5.1.m1.1.1.2.2.cmml" xref="S5.T5.5.5.5.5.1.m1.1.1.2.2">𝑥</ci><ci id="S5.T5.5.5.5.5.1.m1.1.1.2.3.cmml" xref="S5.T5.5.5.5.5.1.m1.1.1.2.3">𝑖</ci></apply><apply id="S5.T5.5.5.5.5.1.m1.1.1.3.cmml" xref="S5.T5.5.5.5.5.1.m1.1.1.3"><csymbol cd="ambiguous" id="S5.T5.5.5.5.5.1.m1.1.1.3.1.cmml" xref="S5.T5.5.5.5.5.1.m1.1.1.3">subscript</csymbol><ci id="S5.T5.5.5.5.5.1.m1.1.1.3.2.cmml" xref="S5.T5.5.5.5.5.1.m1.1.1.3.2">𝑦</ci><ci id="S5.T5.5.5.5.5.1.m1.1.1.3.3.cmml" xref="S5.T5.5.5.5.5.1.m1.1.1.3.3">𝑖</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.T5.5.5.5.5.1.m1.1c">x_{i}\rightarrow y_{i}</annotation><annotation encoding="application/x-llamapun" id="S5.T5.5.5.5.5.1.m1.1d">italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT → italic_y start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math></span> <span class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S5.T5.5.5.5.5.3">43.93</span> <span class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S5.T5.5.5.5.5.4">11.61</span> <span class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S5.T5.5.5.5.5.5">17.87</span> <span class="ltx_td ltx_nopad_r ltx_align_center ltx_border_t" id="S5.T5.5.5.5.5.6">41.22</span></span> <span class="ltx_tr" id="S5.T5.6.6.6.6"> <span class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_r" id="S5.T5.6.6.6.6.2">zero-shot</span> <span class="ltx_td ltx_align_center ltx_th ltx_th_row ltx_border_r" id="S5.T5.6.6.6.6.1"><math alttext="x_{i}\rightarrow{\mathbf{z}}_{i}\oplus y_{i}" class="ltx_Math" display="inline" id="S5.T5.6.6.6.6.1.m1.1"><semantics id="S5.T5.6.6.6.6.1.m1.1a"><mrow id="S5.T5.6.6.6.6.1.m1.1.1" xref="S5.T5.6.6.6.6.1.m1.1.1.cmml"><msub id="S5.T5.6.6.6.6.1.m1.1.1.2" xref="S5.T5.6.6.6.6.1.m1.1.1.2.cmml"><mi id="S5.T5.6.6.6.6.1.m1.1.1.2.2" xref="S5.T5.6.6.6.6.1.m1.1.1.2.2.cmml">x</mi><mi id="S5.T5.6.6.6.6.1.m1.1.1.2.3" xref="S5.T5.6.6.6.6.1.m1.1.1.2.3.cmml">i</mi></msub><mo id="S5.T5.6.6.6.6.1.m1.1.1.1" stretchy="false" xref="S5.T5.6.6.6.6.1.m1.1.1.1.cmml">→</mo><mrow id="S5.T5.6.6.6.6.1.m1.1.1.3" xref="S5.T5.6.6.6.6.1.m1.1.1.3.cmml"><msub id="S5.T5.6.6.6.6.1.m1.1.1.3.2" xref="S5.T5.6.6.6.6.1.m1.1.1.3.2.cmml"><mi id="S5.T5.6.6.6.6.1.m1.1.1.3.2.2" xref="S5.T5.6.6.6.6.1.m1.1.1.3.2.2.cmml">𝐳</mi><mi id="S5.T5.6.6.6.6.1.m1.1.1.3.2.3" xref="S5.T5.6.6.6.6.1.m1.1.1.3.2.3.cmml">i</mi></msub><mo id="S5.T5.6.6.6.6.1.m1.1.1.3.1" xref="S5.T5.6.6.6.6.1.m1.1.1.3.1.cmml">⊕</mo><msub id="S5.T5.6.6.6.6.1.m1.1.1.3.3" xref="S5.T5.6.6.6.6.1.m1.1.1.3.3.cmml"><mi id="S5.T5.6.6.6.6.1.m1.1.1.3.3.2" xref="S5.T5.6.6.6.6.1.m1.1.1.3.3.2.cmml">y</mi><mi id="S5.T5.6.6.6.6.1.m1.1.1.3.3.3" xref="S5.T5.6.6.6.6.1.m1.1.1.3.3.3.cmml">i</mi></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.T5.6.6.6.6.1.m1.1b"><apply id="S5.T5.6.6.6.6.1.m1.1.1.cmml" xref="S5.T5.6.6.6.6.1.m1.1.1"><ci id="S5.T5.6.6.6.6.1.m1.1.1.1.cmml" xref="S5.T5.6.6.6.6.1.m1.1.1.1">→</ci><apply id="S5.T5.6.6.6.6.1.m1.1.1.2.cmml" xref="S5.T5.6.6.6.6.1.m1.1.1.2"><csymbol cd="ambiguous" id="S5.T5.6.6.6.6.1.m1.1.1.2.1.cmml" xref="S5.T5.6.6.6.6.1.m1.1.1.2">subscript</csymbol><ci id="S5.T5.6.6.6.6.1.m1.1.1.2.2.cmml" xref="S5.T5.6.6.6.6.1.m1.1.1.2.2">𝑥</ci><ci id="S5.T5.6.6.6.6.1.m1.1.1.2.3.cmml" xref="S5.T5.6.6.6.6.1.m1.1.1.2.3">𝑖</ci></apply><apply id="S5.T5.6.6.6.6.1.m1.1.1.3.cmml" xref="S5.T5.6.6.6.6.1.m1.1.1.3"><csymbol cd="latexml" id="S5.T5.6.6.6.6.1.m1.1.1.3.1.cmml" xref="S5.T5.6.6.6.6.1.m1.1.1.3.1">direct-sum</csymbol><apply id="S5.T5.6.6.6.6.1.m1.1.1.3.2.cmml" xref="S5.T5.6.6.6.6.1.m1.1.1.3.2"><csymbol cd="ambiguous" id="S5.T5.6.6.6.6.1.m1.1.1.3.2.1.cmml" xref="S5.T5.6.6.6.6.1.m1.1.1.3.2">subscript</csymbol><ci id="S5.T5.6.6.6.6.1.m1.1.1.3.2.2.cmml" xref="S5.T5.6.6.6.6.1.m1.1.1.3.2.2">𝐳</ci><ci id="S5.T5.6.6.6.6.1.m1.1.1.3.2.3.cmml" xref="S5.T5.6.6.6.6.1.m1.1.1.3.2.3">𝑖</ci></apply><apply id="S5.T5.6.6.6.6.1.m1.1.1.3.3.cmml" xref="S5.T5.6.6.6.6.1.m1.1.1.3.3"><csymbol cd="ambiguous" id="S5.T5.6.6.6.6.1.m1.1.1.3.3.1.cmml" xref="S5.T5.6.6.6.6.1.m1.1.1.3.3">subscript</csymbol><ci id="S5.T5.6.6.6.6.1.m1.1.1.3.3.2.cmml" xref="S5.T5.6.6.6.6.1.m1.1.1.3.3.2">𝑦</ci><ci id="S5.T5.6.6.6.6.1.m1.1.1.3.3.3.cmml" xref="S5.T5.6.6.6.6.1.m1.1.1.3.3.3">𝑖</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.T5.6.6.6.6.1.m1.1c">x_{i}\rightarrow{\mathbf{z}}_{i}\oplus y_{i}</annotation><annotation encoding="application/x-llamapun" id="S5.T5.6.6.6.6.1.m1.1d">italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT → bold_z start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ⊕ italic_y start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math></span> <span class="ltx_td ltx_align_center ltx_border_r" id="S5.T5.6.6.6.6.3">40.62</span> <span class="ltx_td ltx_align_center ltx_border_r" id="S5.T5.6.6.6.6.4">10.39</span> <span class="ltx_td ltx_align_center ltx_border_r" id="S5.T5.6.6.6.6.5">17.20</span> <span class="ltx_td ltx_nopad_r ltx_align_center" id="S5.T5.6.6.6.6.6">38.18</span></span> <span class="ltx_tr" id="S5.T5.8.8.8.8"> <span class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_r" id="S5.T5.7.7.7.7.1"><math alttext="x_{i}\rightarrow y_{i}" class="ltx_Math" display="inline" id="S5.T5.7.7.7.7.1.m1.1"><semantics id="S5.T5.7.7.7.7.1.m1.1a"><mrow id="S5.T5.7.7.7.7.1.m1.1.1" xref="S5.T5.7.7.7.7.1.m1.1.1.cmml"><msub id="S5.T5.7.7.7.7.1.m1.1.1.2" xref="S5.T5.7.7.7.7.1.m1.1.1.2.cmml"><mi id="S5.T5.7.7.7.7.1.m1.1.1.2.2" xref="S5.T5.7.7.7.7.1.m1.1.1.2.2.cmml">x</mi><mi id="S5.T5.7.7.7.7.1.m1.1.1.2.3" xref="S5.T5.7.7.7.7.1.m1.1.1.2.3.cmml">i</mi></msub><mo id="S5.T5.7.7.7.7.1.m1.1.1.1" stretchy="false" xref="S5.T5.7.7.7.7.1.m1.1.1.1.cmml">→</mo><msub id="S5.T5.7.7.7.7.1.m1.1.1.3" xref="S5.T5.7.7.7.7.1.m1.1.1.3.cmml"><mi id="S5.T5.7.7.7.7.1.m1.1.1.3.2" xref="S5.T5.7.7.7.7.1.m1.1.1.3.2.cmml">y</mi><mi id="S5.T5.7.7.7.7.1.m1.1.1.3.3" xref="S5.T5.7.7.7.7.1.m1.1.1.3.3.cmml">i</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S5.T5.7.7.7.7.1.m1.1b"><apply id="S5.T5.7.7.7.7.1.m1.1.1.cmml" xref="S5.T5.7.7.7.7.1.m1.1.1"><ci id="S5.T5.7.7.7.7.1.m1.1.1.1.cmml" xref="S5.T5.7.7.7.7.1.m1.1.1.1">→</ci><apply id="S5.T5.7.7.7.7.1.m1.1.1.2.cmml" xref="S5.T5.7.7.7.7.1.m1.1.1.2"><csymbol cd="ambiguous" id="S5.T5.7.7.7.7.1.m1.1.1.2.1.cmml" xref="S5.T5.7.7.7.7.1.m1.1.1.2">subscript</csymbol><ci id="S5.T5.7.7.7.7.1.m1.1.1.2.2.cmml" xref="S5.T5.7.7.7.7.1.m1.1.1.2.2">𝑥</ci><ci id="S5.T5.7.7.7.7.1.m1.1.1.2.3.cmml" xref="S5.T5.7.7.7.7.1.m1.1.1.2.3">𝑖</ci></apply><apply id="S5.T5.7.7.7.7.1.m1.1.1.3.cmml" xref="S5.T5.7.7.7.7.1.m1.1.1.3"><csymbol cd="ambiguous" id="S5.T5.7.7.7.7.1.m1.1.1.3.1.cmml" xref="S5.T5.7.7.7.7.1.m1.1.1.3">subscript</csymbol><ci id="S5.T5.7.7.7.7.1.m1.1.1.3.2.cmml" xref="S5.T5.7.7.7.7.1.m1.1.1.3.2">𝑦</ci><ci id="S5.T5.7.7.7.7.1.m1.1.1.3.3.cmml" xref="S5.T5.7.7.7.7.1.m1.1.1.3.3">𝑖</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.T5.7.7.7.7.1.m1.1c">x_{i}\rightarrow y_{i}</annotation><annotation encoding="application/x-llamapun" id="S5.T5.7.7.7.7.1.m1.1d">italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT → italic_y start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math></span> <span class="ltx_td ltx_align_center ltx_th ltx_th_row ltx_border_r" id="S5.T5.8.8.8.8.2"><math alttext="x_{i}\rightarrow y_{i}" class="ltx_Math" display="inline" id="S5.T5.8.8.8.8.2.m1.1"><semantics id="S5.T5.8.8.8.8.2.m1.1a"><mrow id="S5.T5.8.8.8.8.2.m1.1.1" xref="S5.T5.8.8.8.8.2.m1.1.1.cmml"><msub id="S5.T5.8.8.8.8.2.m1.1.1.2" xref="S5.T5.8.8.8.8.2.m1.1.1.2.cmml"><mi id="S5.T5.8.8.8.8.2.m1.1.1.2.2" xref="S5.T5.8.8.8.8.2.m1.1.1.2.2.cmml">x</mi><mi id="S5.T5.8.8.8.8.2.m1.1.1.2.3" xref="S5.T5.8.8.8.8.2.m1.1.1.2.3.cmml">i</mi></msub><mo id="S5.T5.8.8.8.8.2.m1.1.1.1" stretchy="false" xref="S5.T5.8.8.8.8.2.m1.1.1.1.cmml">→</mo><msub id="S5.T5.8.8.8.8.2.m1.1.1.3" xref="S5.T5.8.8.8.8.2.m1.1.1.3.cmml"><mi id="S5.T5.8.8.8.8.2.m1.1.1.3.2" xref="S5.T5.8.8.8.8.2.m1.1.1.3.2.cmml">y</mi><mi id="S5.T5.8.8.8.8.2.m1.1.1.3.3" xref="S5.T5.8.8.8.8.2.m1.1.1.3.3.cmml">i</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S5.T5.8.8.8.8.2.m1.1b"><apply id="S5.T5.8.8.8.8.2.m1.1.1.cmml" xref="S5.T5.8.8.8.8.2.m1.1.1"><ci id="S5.T5.8.8.8.8.2.m1.1.1.1.cmml" xref="S5.T5.8.8.8.8.2.m1.1.1.1">→</ci><apply id="S5.T5.8.8.8.8.2.m1.1.1.2.cmml" xref="S5.T5.8.8.8.8.2.m1.1.1.2"><csymbol cd="ambiguous" id="S5.T5.8.8.8.8.2.m1.1.1.2.1.cmml" xref="S5.T5.8.8.8.8.2.m1.1.1.2">subscript</csymbol><ci id="S5.T5.8.8.8.8.2.m1.1.1.2.2.cmml" xref="S5.T5.8.8.8.8.2.m1.1.1.2.2">𝑥</ci><ci id="S5.T5.8.8.8.8.2.m1.1.1.2.3.cmml" xref="S5.T5.8.8.8.8.2.m1.1.1.2.3">𝑖</ci></apply><apply id="S5.T5.8.8.8.8.2.m1.1.1.3.cmml" xref="S5.T5.8.8.8.8.2.m1.1.1.3"><csymbol cd="ambiguous" id="S5.T5.8.8.8.8.2.m1.1.1.3.1.cmml" xref="S5.T5.8.8.8.8.2.m1.1.1.3">subscript</csymbol><ci id="S5.T5.8.8.8.8.2.m1.1.1.3.2.cmml" xref="S5.T5.8.8.8.8.2.m1.1.1.3.2">𝑦</ci><ci id="S5.T5.8.8.8.8.2.m1.1.1.3.3.cmml" xref="S5.T5.8.8.8.8.2.m1.1.1.3.3">𝑖</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.T5.8.8.8.8.2.m1.1c">x_{i}\rightarrow y_{i}</annotation><annotation encoding="application/x-llamapun" id="S5.T5.8.8.8.8.2.m1.1d">italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT → italic_y start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math></span> <span class="ltx_td ltx_align_center ltx_border_r" id="S5.T5.8.8.8.8.3">45.38</span> <span class="ltx_td ltx_align_center ltx_border_r" id="S5.T5.8.8.8.8.4">12.11</span> <span class="ltx_td ltx_align_center ltx_border_r" id="S5.T5.8.8.8.8.5">18.00</span> <span class="ltx_td ltx_nopad_r ltx_align_center" id="S5.T5.8.8.8.8.6">42.62</span></span> <span class="ltx_tr" id="S5.T5.10.10.10.10"> <span class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_r" id="S5.T5.9.9.9.9.1"><math alttext="x_{i}\rightarrow y_{i};x_{i}\rightarrow{\mathbf{z}}_{i}\oplus y_{i}" class="ltx_Math" display="inline" id="S5.T5.9.9.9.9.1.m1.2"><semantics id="S5.T5.9.9.9.9.1.m1.2a"><mrow id="S5.T5.9.9.9.9.1.m1.2.2.2" xref="S5.T5.9.9.9.9.1.m1.2.2.3.cmml"><mrow id="S5.T5.9.9.9.9.1.m1.1.1.1.1" xref="S5.T5.9.9.9.9.1.m1.1.1.1.1.cmml"><msub id="S5.T5.9.9.9.9.1.m1.1.1.1.1.2" xref="S5.T5.9.9.9.9.1.m1.1.1.1.1.2.cmml"><mi id="S5.T5.9.9.9.9.1.m1.1.1.1.1.2.2" xref="S5.T5.9.9.9.9.1.m1.1.1.1.1.2.2.cmml">x</mi><mi id="S5.T5.9.9.9.9.1.m1.1.1.1.1.2.3" xref="S5.T5.9.9.9.9.1.m1.1.1.1.1.2.3.cmml">i</mi></msub><mo id="S5.T5.9.9.9.9.1.m1.1.1.1.1.1" stretchy="false" xref="S5.T5.9.9.9.9.1.m1.1.1.1.1.1.cmml">→</mo><msub id="S5.T5.9.9.9.9.1.m1.1.1.1.1.3" xref="S5.T5.9.9.9.9.1.m1.1.1.1.1.3.cmml"><mi id="S5.T5.9.9.9.9.1.m1.1.1.1.1.3.2" xref="S5.T5.9.9.9.9.1.m1.1.1.1.1.3.2.cmml">y</mi><mi id="S5.T5.9.9.9.9.1.m1.1.1.1.1.3.3" xref="S5.T5.9.9.9.9.1.m1.1.1.1.1.3.3.cmml">i</mi></msub></mrow><mo id="S5.T5.9.9.9.9.1.m1.2.2.2.3" xref="S5.T5.9.9.9.9.1.m1.2.2.3a.cmml">;</mo><mrow id="S5.T5.9.9.9.9.1.m1.2.2.2.2" xref="S5.T5.9.9.9.9.1.m1.2.2.2.2.cmml"><msub id="S5.T5.9.9.9.9.1.m1.2.2.2.2.2" xref="S5.T5.9.9.9.9.1.m1.2.2.2.2.2.cmml"><mi id="S5.T5.9.9.9.9.1.m1.2.2.2.2.2.2" xref="S5.T5.9.9.9.9.1.m1.2.2.2.2.2.2.cmml">x</mi><mi id="S5.T5.9.9.9.9.1.m1.2.2.2.2.2.3" xref="S5.T5.9.9.9.9.1.m1.2.2.2.2.2.3.cmml">i</mi></msub><mo id="S5.T5.9.9.9.9.1.m1.2.2.2.2.1" stretchy="false" xref="S5.T5.9.9.9.9.1.m1.2.2.2.2.1.cmml">→</mo><mrow id="S5.T5.9.9.9.9.1.m1.2.2.2.2.3" xref="S5.T5.9.9.9.9.1.m1.2.2.2.2.3.cmml"><msub id="S5.T5.9.9.9.9.1.m1.2.2.2.2.3.2" xref="S5.T5.9.9.9.9.1.m1.2.2.2.2.3.2.cmml"><mi id="S5.T5.9.9.9.9.1.m1.2.2.2.2.3.2.2" xref="S5.T5.9.9.9.9.1.m1.2.2.2.2.3.2.2.cmml">𝐳</mi><mi id="S5.T5.9.9.9.9.1.m1.2.2.2.2.3.2.3" xref="S5.T5.9.9.9.9.1.m1.2.2.2.2.3.2.3.cmml">i</mi></msub><mo id="S5.T5.9.9.9.9.1.m1.2.2.2.2.3.1" xref="S5.T5.9.9.9.9.1.m1.2.2.2.2.3.1.cmml">⊕</mo><msub id="S5.T5.9.9.9.9.1.m1.2.2.2.2.3.3" xref="S5.T5.9.9.9.9.1.m1.2.2.2.2.3.3.cmml"><mi id="S5.T5.9.9.9.9.1.m1.2.2.2.2.3.3.2" xref="S5.T5.9.9.9.9.1.m1.2.2.2.2.3.3.2.cmml">y</mi><mi id="S5.T5.9.9.9.9.1.m1.2.2.2.2.3.3.3" xref="S5.T5.9.9.9.9.1.m1.2.2.2.2.3.3.3.cmml">i</mi></msub></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.T5.9.9.9.9.1.m1.2b"><apply id="S5.T5.9.9.9.9.1.m1.2.2.3.cmml" xref="S5.T5.9.9.9.9.1.m1.2.2.2"><csymbol cd="ambiguous" id="S5.T5.9.9.9.9.1.m1.2.2.3a.cmml" xref="S5.T5.9.9.9.9.1.m1.2.2.2.3">formulae-sequence</csymbol><apply id="S5.T5.9.9.9.9.1.m1.1.1.1.1.cmml" xref="S5.T5.9.9.9.9.1.m1.1.1.1.1"><ci id="S5.T5.9.9.9.9.1.m1.1.1.1.1.1.cmml" xref="S5.T5.9.9.9.9.1.m1.1.1.1.1.1">→</ci><apply id="S5.T5.9.9.9.9.1.m1.1.1.1.1.2.cmml" xref="S5.T5.9.9.9.9.1.m1.1.1.1.1.2"><csymbol cd="ambiguous" id="S5.T5.9.9.9.9.1.m1.1.1.1.1.2.1.cmml" xref="S5.T5.9.9.9.9.1.m1.1.1.1.1.2">subscript</csymbol><ci id="S5.T5.9.9.9.9.1.m1.1.1.1.1.2.2.cmml" xref="S5.T5.9.9.9.9.1.m1.1.1.1.1.2.2">𝑥</ci><ci id="S5.T5.9.9.9.9.1.m1.1.1.1.1.2.3.cmml" xref="S5.T5.9.9.9.9.1.m1.1.1.1.1.2.3">𝑖</ci></apply><apply id="S5.T5.9.9.9.9.1.m1.1.1.1.1.3.cmml" xref="S5.T5.9.9.9.9.1.m1.1.1.1.1.3"><csymbol cd="ambiguous" id="S5.T5.9.9.9.9.1.m1.1.1.1.1.3.1.cmml" xref="S5.T5.9.9.9.9.1.m1.1.1.1.1.3">subscript</csymbol><ci id="S5.T5.9.9.9.9.1.m1.1.1.1.1.3.2.cmml" xref="S5.T5.9.9.9.9.1.m1.1.1.1.1.3.2">𝑦</ci><ci id="S5.T5.9.9.9.9.1.m1.1.1.1.1.3.3.cmml" xref="S5.T5.9.9.9.9.1.m1.1.1.1.1.3.3">𝑖</ci></apply></apply><apply id="S5.T5.9.9.9.9.1.m1.2.2.2.2.cmml" xref="S5.T5.9.9.9.9.1.m1.2.2.2.2"><ci id="S5.T5.9.9.9.9.1.m1.2.2.2.2.1.cmml" xref="S5.T5.9.9.9.9.1.m1.2.2.2.2.1">→</ci><apply id="S5.T5.9.9.9.9.1.m1.2.2.2.2.2.cmml" xref="S5.T5.9.9.9.9.1.m1.2.2.2.2.2"><csymbol cd="ambiguous" id="S5.T5.9.9.9.9.1.m1.2.2.2.2.2.1.cmml" xref="S5.T5.9.9.9.9.1.m1.2.2.2.2.2">subscript</csymbol><ci id="S5.T5.9.9.9.9.1.m1.2.2.2.2.2.2.cmml" xref="S5.T5.9.9.9.9.1.m1.2.2.2.2.2.2">𝑥</ci><ci id="S5.T5.9.9.9.9.1.m1.2.2.2.2.2.3.cmml" xref="S5.T5.9.9.9.9.1.m1.2.2.2.2.2.3">𝑖</ci></apply><apply id="S5.T5.9.9.9.9.1.m1.2.2.2.2.3.cmml" xref="S5.T5.9.9.9.9.1.m1.2.2.2.2.3"><csymbol cd="latexml" id="S5.T5.9.9.9.9.1.m1.2.2.2.2.3.1.cmml" xref="S5.T5.9.9.9.9.1.m1.2.2.2.2.3.1">direct-sum</csymbol><apply id="S5.T5.9.9.9.9.1.m1.2.2.2.2.3.2.cmml" xref="S5.T5.9.9.9.9.1.m1.2.2.2.2.3.2"><csymbol cd="ambiguous" id="S5.T5.9.9.9.9.1.m1.2.2.2.2.3.2.1.cmml" xref="S5.T5.9.9.9.9.1.m1.2.2.2.2.3.2">subscript</csymbol><ci id="S5.T5.9.9.9.9.1.m1.2.2.2.2.3.2.2.cmml" xref="S5.T5.9.9.9.9.1.m1.2.2.2.2.3.2.2">𝐳</ci><ci id="S5.T5.9.9.9.9.1.m1.2.2.2.2.3.2.3.cmml" xref="S5.T5.9.9.9.9.1.m1.2.2.2.2.3.2.3">𝑖</ci></apply><apply id="S5.T5.9.9.9.9.1.m1.2.2.2.2.3.3.cmml" xref="S5.T5.9.9.9.9.1.m1.2.2.2.2.3.3"><csymbol cd="ambiguous" id="S5.T5.9.9.9.9.1.m1.2.2.2.2.3.3.1.cmml" xref="S5.T5.9.9.9.9.1.m1.2.2.2.2.3.3">subscript</csymbol><ci id="S5.T5.9.9.9.9.1.m1.2.2.2.2.3.3.2.cmml" xref="S5.T5.9.9.9.9.1.m1.2.2.2.2.3.3.2">𝑦</ci><ci id="S5.T5.9.9.9.9.1.m1.2.2.2.2.3.3.3.cmml" xref="S5.T5.9.9.9.9.1.m1.2.2.2.2.3.3.3">𝑖</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.T5.9.9.9.9.1.m1.2c">x_{i}\rightarrow y_{i};x_{i}\rightarrow{\mathbf{z}}_{i}\oplus y_{i}</annotation><annotation encoding="application/x-llamapun" id="S5.T5.9.9.9.9.1.m1.2d">italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT → italic_y start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ; italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT → bold_z start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ⊕ italic_y start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math></span> <span class="ltx_td ltx_align_center ltx_th ltx_th_row ltx_border_r" id="S5.T5.10.10.10.10.2"><math alttext="x_{i}\rightarrow y_{i}" class="ltx_Math" display="inline" id="S5.T5.10.10.10.10.2.m1.1"><semantics id="S5.T5.10.10.10.10.2.m1.1a"><mrow id="S5.T5.10.10.10.10.2.m1.1.1" xref="S5.T5.10.10.10.10.2.m1.1.1.cmml"><msub id="S5.T5.10.10.10.10.2.m1.1.1.2" xref="S5.T5.10.10.10.10.2.m1.1.1.2.cmml"><mi id="S5.T5.10.10.10.10.2.m1.1.1.2.2" xref="S5.T5.10.10.10.10.2.m1.1.1.2.2.cmml">x</mi><mi id="S5.T5.10.10.10.10.2.m1.1.1.2.3" xref="S5.T5.10.10.10.10.2.m1.1.1.2.3.cmml">i</mi></msub><mo id="S5.T5.10.10.10.10.2.m1.1.1.1" stretchy="false" xref="S5.T5.10.10.10.10.2.m1.1.1.1.cmml">→</mo><msub id="S5.T5.10.10.10.10.2.m1.1.1.3" xref="S5.T5.10.10.10.10.2.m1.1.1.3.cmml"><mi id="S5.T5.10.10.10.10.2.m1.1.1.3.2" xref="S5.T5.10.10.10.10.2.m1.1.1.3.2.cmml">y</mi><mi id="S5.T5.10.10.10.10.2.m1.1.1.3.3" xref="S5.T5.10.10.10.10.2.m1.1.1.3.3.cmml">i</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S5.T5.10.10.10.10.2.m1.1b"><apply id="S5.T5.10.10.10.10.2.m1.1.1.cmml" xref="S5.T5.10.10.10.10.2.m1.1.1"><ci id="S5.T5.10.10.10.10.2.m1.1.1.1.cmml" xref="S5.T5.10.10.10.10.2.m1.1.1.1">→</ci><apply id="S5.T5.10.10.10.10.2.m1.1.1.2.cmml" xref="S5.T5.10.10.10.10.2.m1.1.1.2"><csymbol cd="ambiguous" id="S5.T5.10.10.10.10.2.m1.1.1.2.1.cmml" xref="S5.T5.10.10.10.10.2.m1.1.1.2">subscript</csymbol><ci id="S5.T5.10.10.10.10.2.m1.1.1.2.2.cmml" xref="S5.T5.10.10.10.10.2.m1.1.1.2.2">𝑥</ci><ci id="S5.T5.10.10.10.10.2.m1.1.1.2.3.cmml" xref="S5.T5.10.10.10.10.2.m1.1.1.2.3">𝑖</ci></apply><apply id="S5.T5.10.10.10.10.2.m1.1.1.3.cmml" xref="S5.T5.10.10.10.10.2.m1.1.1.3"><csymbol cd="ambiguous" id="S5.T5.10.10.10.10.2.m1.1.1.3.1.cmml" xref="S5.T5.10.10.10.10.2.m1.1.1.3">subscript</csymbol><ci id="S5.T5.10.10.10.10.2.m1.1.1.3.2.cmml" xref="S5.T5.10.10.10.10.2.m1.1.1.3.2">𝑦</ci><ci id="S5.T5.10.10.10.10.2.m1.1.1.3.3.cmml" xref="S5.T5.10.10.10.10.2.m1.1.1.3.3">𝑖</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.T5.10.10.10.10.2.m1.1c">x_{i}\rightarrow y_{i}</annotation><annotation encoding="application/x-llamapun" id="S5.T5.10.10.10.10.2.m1.1d">italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT → italic_y start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math></span> <span class="ltx_td ltx_align_center ltx_border_r" id="S5.T5.10.10.10.10.3">46.32</span> <span class="ltx_td ltx_align_center ltx_border_r" id="S5.T5.10.10.10.10.4">12.32</span> <span class="ltx_td ltx_align_center ltx_border_r" id="S5.T5.10.10.10.10.5">18.05</span> <span class="ltx_td ltx_nopad_r ltx_align_center" id="S5.T5.10.10.10.10.6">43.48</span></span> <span class="ltx_tr" id="S5.T5.13.13.13.13"> <span class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_b ltx_border_r" id="S5.T5.12.12.12.12.2"><math alttext="x_{i}\rightarrow y_{i}" class="ltx_Math" display="inline" id="S5.T5.11.11.11.11.1.m1.1"><semantics id="S5.T5.11.11.11.11.1.m1.1a"><mrow id="S5.T5.11.11.11.11.1.m1.1.1" xref="S5.T5.11.11.11.11.1.m1.1.1.cmml"><msub id="S5.T5.11.11.11.11.1.m1.1.1.2" xref="S5.T5.11.11.11.11.1.m1.1.1.2.cmml"><mi id="S5.T5.11.11.11.11.1.m1.1.1.2.2" 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end_POSTSUBSCRIPT → bold_z start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ⊕ italic_y start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math></span> <span class="ltx_td ltx_align_center ltx_border_b ltx_border_r" id="S5.T5.13.13.13.13.4"><span class="ltx_text ltx_font_bold" id="S5.T5.13.13.13.13.4.1">46.53</span></span> <span class="ltx_td ltx_align_center ltx_border_b ltx_border_r" id="S5.T5.13.13.13.13.5"><span class="ltx_text ltx_font_bold" id="S5.T5.13.13.13.13.5.1">12.45</span></span> <span class="ltx_td ltx_align_center ltx_border_b ltx_border_r" id="S5.T5.13.13.13.13.6"><span class="ltx_text ltx_font_bold" id="S5.T5.13.13.13.13.6.1">18.16</span></span> <span class="ltx_td ltx_nopad_r ltx_align_center ltx_border_b" id="S5.T5.13.13.13.13.7"><span class="ltx_text ltx_font_bold" id="S5.T5.13.13.13.13.7.1">43.77</span></span></span> </span> </span></span></p> </figure> </section> <section class="ltx_paragraph" id="S5.SS4.SSS0.Px5"> <h4 class="ltx_title ltx_title_paragraph">Alternative base model: Gemini 1.5 Flash.</h4> <div class="ltx_para" id="S5.SS4.SSS0.Px5.p1"> <p class="ltx_p" id="S5.SS4.SSS0.Px5.p1.1">To show that our method are applicable to the latest generation of LLMs, we also conducted smaller scale experiments (due to cost limitations) on Gemini 1.5 Flash as the base model. Table <a class="ltx_ref" href="https://arxiv.org/html/2410.06203v1#S5.T5" title="Table 5 ‣ Human and auto SxS results. ‣ 5.4 Results ‣ 5 Experiments ‣ Integrating Planning into Single-Turn Long-Form Text Generation"><span class="ltx_text ltx_ref_tag">5</span></a> show the results on SciNews, which corroborate the findings on Gemini 1.0 Pro shown in Table <a class="ltx_ref" href="https://arxiv.org/html/2410.06203v1#S5.T2" title="Table 2 ‣ Comparison of different training and inference recipes. ‣ 5.4 Results ‣ 5 Experiments ‣ Integrating Planning into Single-Turn Long-Form Text Generation"><span class="ltx_text ltx_ref_tag">2</span></a>.</p> </div> </section> </section> </section> <section class="ltx_section" id="S6"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">6 </span>Conclusions</h2> <div class="ltx_para" id="S6.p1"> <p class="ltx_p" id="S6.p1.1">In this paper, we explore to fine-tune LLM to write long-form articles in one single turn. We propose to include intermediate planning steps, such as starting with a concise summary, writing down the outlines and collecting some key information into the mixture of the training data. Noting that such intermediate steps are not available in most existing data sets, we propose to construct synthetic intermediate steps from existing full-length articles. We prompt an LLM to extract, shorten and summarize the article into a tree structure, where each level corresponds to an intermediate step. We fine-tune the LLM writer with different mixtures before prompt the LLM writer to write the full article. Our experiments on two data sets from different domains: SciNews and Wikipedia, verify that our proposed method can substantially boost the quality of the generated articles.</p> </div> <div class="ltx_para" id="S6.p2"> <p class="ltx_p" id="S6.p2.1">Our exploration creates a new paradigm in long-form text generation: instead of applying LLM iteratively to generate intermediate steps until the full article is obtained, we include the intermediate steps into a mixture of training data and directly fine-tune an LLM to generate all of them in one turn. While we only experiment with one specific way of constructing the intermediate steps (as tree-structured planning outlines), there are many other different possible ways to create the auxiliary training mixture. As modern LLMs support longer and longer input and output sequence lengths, we believe it is possible to build even more more sophisticated long-form LLM writers with embedded capabilities other than planning. For example, one can also fine-tune the LLM to perform fact-checking reasoning on-the-fly to improve the factuality of the generated article.</p> </div> </section> <section class="ltx_section" id="S7"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">7 </span>Limitations</h2> <div class="ltx_para" id="S7.p1"> <p class="ltx_p" id="S7.p1.1">In this work we focus on writing long-form articles in a totally closed-book setup, where the only external information is provided in the input context. However, a more realistic setting would be retrieval-augmented generation (RAG), which equips the LLMs with a retriever to external knowledge. Since we do not adopt a RAG setting, we also do not compare to other existing methods that adopt RAG (e.g., STORM <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2410.06203v1#bib.bib19" title="">19</a>]</cite>) While we do not adopt a RAG setting, we believe our work can be easily adapted to incorporate RAG to achieve better performance.</p> </div> </section> <section class="ltx_section" id="S8"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">8 </span>Ethics Statement</h2> <div class="ltx_para" id="S8.p1"> <p class="ltx_p" id="S8.p1.1">Our approach to incorporating planning steps into LLM training aims to significantly improve the quality of long-form text generation. This advancement has the potential to benefit various sectors by providing more coherent, accurate, and contextually rich content. We believe that the benefits of our work, including the enhancement of educational resources and the facilitation of better information dissemination, outweigh the minimal risks involved.</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_tag_bibitem">[1]</span> <span class="ltx_bibblock"> N. Balepur, J. Huang, and K. C.-C. Chang. </span> <span class="ltx_bibblock">Expository text generation: Imitate, retrieve, paraphrase. </span> <span class="ltx_bibblock"><span class="ltx_text ltx_font_italic" id="bib.bib1.1.1">arXiv preprint arXiv:2305.03276</span>, 2023. </span> </li> <li class="ltx_bibitem" id="bib.bib2"> <span class="ltx_tag ltx_tag_bibitem">[2]</span> <span class="ltx_bibblock"> S. Banerjee and P. Mitra. </span> <span class="ltx_bibblock">Wikiwrite: generating wikipedia articles automatically. </span> <span class="ltx_bibblock">In <span class="ltx_text ltx_font_italic" id="bib.bib2.1.1">Proceedings of the Twenty-Fifth International Joint Conference on Artificial Intelligence</span>, IJCAI’16, page 2740–2746, 2016. </span> </li> <li class="ltx_bibitem" id="bib.bib3"> <span class="ltx_tag ltx_tag_bibitem">[3]</span> <span class="ltx_bibblock"> M. Besta, N. Blach, A. Kubicek, R. Gerstenberger, M. Podstawski, L. Gianinazzi, J. Gajda, T. Lehmann, H. Niewiadomski, P. Nyczyk, et al. </span> <span class="ltx_bibblock">Graph of thoughts: Solving elaborate problems with large language models. </span> <span class="ltx_bibblock">In <span class="ltx_text ltx_font_italic" id="bib.bib3.1.1">Proceedings of the AAAI Conference on Artificial Intelligence</span>, volume 38, pages 17682–17690, 2024. </span> </li> <li class="ltx_bibitem" id="bib.bib4"> <span class="ltx_tag ltx_tag_bibitem">[4]</span> <span class="ltx_bibblock"> K. Cho, B. Van Merriënboer, C. Gulcehre, D. Bahdanau, F. Bougares, H. Schwenk, and Y. Bengio. </span> <span class="ltx_bibblock">Learning phrase representations using rnn encoder-decoder for statistical machine translation. </span> <span class="ltx_bibblock"><span class="ltx_text ltx_font_italic" id="bib.bib4.1.1">arXiv preprint arXiv:1406.1078</span>, 2014. </span> </li> <li class="ltx_bibitem" id="bib.bib5"> <span class="ltx_tag ltx_tag_bibitem">[5]</span> <span class="ltx_bibblock"> G. Dagan, F. Keller, and A. Lascarides. </span> <span class="ltx_bibblock">Dynamic planning with a llm. </span> <span class="ltx_bibblock"><span class="ltx_text ltx_font_italic" id="bib.bib5.1.1">arXiv preprint arXiv:2308.06391</span>, 2023. </span> </li> <li class="ltx_bibitem" id="bib.bib6"> <span class="ltx_tag ltx_tag_bibitem">[6]</span> <span class="ltx_bibblock"> A. Fan and C. Gardent. </span> <span class="ltx_bibblock">Generating full length wikipedia biographies: The impact of gender bias on the retrieval-based generation of women biographies. </span> <span class="ltx_bibblock"><span class="ltx_text ltx_font_italic" id="bib.bib6.1.1">arXiv preprint arXiv:2204.05879</span>, 2022. </span> </li> <li class="ltx_bibitem" id="bib.bib7"> <span class="ltx_tag ltx_tag_bibitem">[7]</span> <span class="ltx_bibblock"> A. Fan, Y. Jernite, E. Perez, D. Grangier, J. Weston, and M. Auli. </span> <span class="ltx_bibblock">Eli5: Long form question answering. </span> <span class="ltx_bibblock"><span class="ltx_text ltx_font_italic" id="bib.bib7.1.1">arXiv preprint arXiv:1907.09190</span>, 2019. </span> </li> <li class="ltx_bibitem" id="bib.bib8"> <span class="ltx_tag ltx_tag_bibitem">[8]</span> <span class="ltx_bibblock"> L. Gao and J. Callan. </span> <span class="ltx_bibblock">Unsupervised corpus aware language model pre-training for dense passage retrieval. </span> <span class="ltx_bibblock"><span class="ltx_text ltx_font_italic" id="bib.bib8.1.1">arXiv preprint arXiv:2108.05540</span>, 2021. </span> </li> <li class="ltx_bibitem" id="bib.bib9"> <span class="ltx_tag ltx_tag_bibitem">[9]</span> <span class="ltx_bibblock"> T. Kojima, S. S. Gu, M. Reid, Y. Matsuo, and Y. Iwasawa. </span> <span class="ltx_bibblock">Large language models are zero-shot reasoners. </span> <span class="ltx_bibblock">In S. Koyejo, S. Mohamed, A. Agarwal, D. Belgrave, K. Cho, and A. Oh, editors, <span class="ltx_text ltx_font_italic" id="bib.bib9.1.1">Advances in Neural Information Processing Systems</span>, volume 35, pages 22199–22213. Curran Associates, Inc., 2022. </span> </li> <li class="ltx_bibitem" id="bib.bib10"> <span class="ltx_tag ltx_tag_bibitem">[10]</span> <span class="ltx_bibblock"> K. Krishna, A. Roy, and M. Iyyer. </span> <span class="ltx_bibblock">Hurdles to progress in long-form question answering. </span> <span class="ltx_bibblock"><span class="ltx_text ltx_font_italic" id="bib.bib10.1.1">arXiv preprint arXiv:2103.06332</span>, 2021. </span> </li> <li class="ltx_bibitem" id="bib.bib11"> <span class="ltx_tag ltx_tag_bibitem">[11]</span> <span class="ltx_bibblock"> J. Li, T. Tang, W. X. Zhao, J.-Y. Nie, and J.-R. Wen. </span> <span class="ltx_bibblock">Pre-trained language models for text generation: A survey. </span> <span class="ltx_bibblock"><span class="ltx_text ltx_font_italic" id="bib.bib11.1.1">ACM Computing Surveys</span>, 56(9):1–39, 2024. </span> </li> <li class="ltx_bibitem" id="bib.bib12"> <span class="ltx_tag ltx_tag_bibitem">[12]</span> <span class="ltx_bibblock"> B. Liu, Y. Jiang, X. Zhang, Q. Liu, S. Zhang, J. Biswas, and P. Stone. </span> <span class="ltx_bibblock">Llm+ p: Empowering large language models with optimal planning proficiency. </span> <span class="ltx_bibblock"><span class="ltx_text ltx_font_italic" id="bib.bib12.1.1">arXiv preprint arXiv:2304.11477</span>, 2023. </span> </li> <li class="ltx_bibitem" id="bib.bib13"> <span class="ltx_tag ltx_tag_bibitem">[13]</span> <span class="ltx_bibblock"> P. J. Liu, M. Saleh, E. Pot, B. Goodrich, R. Sepassi, L. Kaiser, and N. Shazeer. </span> <span class="ltx_bibblock">Generating wikipedia by summarizing long sequences. </span> <span class="ltx_bibblock"><span class="ltx_text ltx_font_italic" id="bib.bib13.1.1">arXiv preprint arXiv:1801.10198</span>, 2018. </span> </li> <li class="ltx_bibitem" id="bib.bib14"> <span class="ltx_tag ltx_tag_bibitem">[14]</span> <span class="ltx_bibblock"> J. Minguillón, M. Lerga, E. Aibar, J. Lladós-Masllorens, and A. Meseguer-Artola. </span> <span class="ltx_bibblock">Semi-automatic generation of a corpus of wikipedia articles on science and technology. </span> <span class="ltx_bibblock"><span class="ltx_text ltx_font_italic" id="bib.bib14.1.1">Profesional de la información/Information Professional</span>, 26(5):995–1005, 2017. </span> </li> <li class="ltx_bibitem" id="bib.bib15"> <span class="ltx_tag ltx_tag_bibitem">[15]</span> <span class="ltx_bibblock"> R. Nakano, J. Hilton, S. Balaji, J. Wu, L. Ouyang, C. Kim, C. Hesse, S. Jain, V. Kosaraju, W. Saunders, et al. </span> <span class="ltx_bibblock">Webgpt: Browser-assisted question-answering with human feedback. </span> <span class="ltx_bibblock"><span class="ltx_text ltx_font_italic" id="bib.bib15.1.1">arXiv preprint arXiv:2112.09332</span>, 2021. </span> </li> <li class="ltx_bibitem" id="bib.bib16"> <span class="ltx_tag ltx_tag_bibitem">[16]</span> <span class="ltx_bibblock"> F. Petroni, A. Piktus, A. Fan, P. Lewis, M. Yazdani, N. D. Cao, J. Thorne, Y. Jernite, V. Plachouras, T. Rockt"aschel, and S. Riedel. </span> <span class="ltx_bibblock">KILT: a Benchmark for Knowledge Intensive Language Tasks. </span> <span class="ltx_bibblock">2020. </span> </li> <li class="ltx_bibitem" id="bib.bib17"> <span class="ltx_tag ltx_tag_bibitem">[17]</span> <span class="ltx_bibblock"> D. Pu, Y. Wang, J. Loy, and V. Demberg. </span> <span class="ltx_bibblock">Scinews: From scholarly complexities to public narratives – a dataset for scientific news report generation. </span> <span class="ltx_bibblock"><span class="ltx_text ltx_font_italic" id="bib.bib17.1.1">Department of Computer Science, Department of Language Science and Technology, Saarland Informatics Campus, Saarland University</span>, 2024. </span> </li> <li class="ltx_bibitem" id="bib.bib18"> <span class="ltx_tag ltx_tag_bibitem">[18]</span> <span class="ltx_bibblock"> D. G. Rohman. </span> <span class="ltx_bibblock">Pre-writing the stage of discovery in the writing process. </span> <span class="ltx_bibblock"><span class="ltx_text ltx_font_italic" id="bib.bib18.1.1">College composition and communication</span>, 16(2):106–112, 1965. </span> </li> <li class="ltx_bibitem" id="bib.bib19"> <span class="ltx_tag ltx_tag_bibitem">[19]</span> <span class="ltx_bibblock"> Y. Shao, Y. Jiang, T. A. Kanell, P. Xu, O. Khattab, and M. S. Lam. </span> <span class="ltx_bibblock">Assisting in writing wikipedia-like articles from scratch with large language models. </span> <span class="ltx_bibblock"><span class="ltx_text ltx_font_italic" id="bib.bib19.1.1">arXiv preprint arXiv:2402.14207</span>, 2024. </span> </li> <li class="ltx_bibitem" id="bib.bib20"> <span class="ltx_tag ltx_tag_bibitem">[20]</span> <span class="ltx_bibblock"> J. Shen, J. Liu, D. Finnie, N. Rahmati, M. Bendersky, and M. Najork. </span> <span class="ltx_bibblock">“why is this misleading?”: Detecting news headline hallucinations with explanations. </span> <span class="ltx_bibblock">In <span class="ltx_text ltx_font_italic" id="bib.bib20.1.1">Proceedings of the ACM Web Conference 2023</span>, pages 1662–1672, 2023. </span> </li> <li class="ltx_bibitem" id="bib.bib21"> <span class="ltx_tag ltx_tag_bibitem">[21]</span> <span class="ltx_bibblock"> Y. Shen, K. Song, X. Tan, D. Li, W. Lu, and Y. Zhuang. </span> <span class="ltx_bibblock">Hugginggpt: Solving ai tasks with chatgpt and its friends in hugging face. </span> <span class="ltx_bibblock"><span class="ltx_text ltx_font_italic" id="bib.bib21.1.1">Advances in Neural Information Processing Systems</span>, 36, 2024. </span> </li> <li class="ltx_bibitem" id="bib.bib22"> <span class="ltx_tag ltx_tag_bibitem">[22]</span> <span class="ltx_bibblock"> D. Su, X. Li, J. Zhang, L. Shang, X. Jiang, Q. Liu, and P. Fung. </span> <span class="ltx_bibblock">Read before generate! faithful long form question answering with machine reading. </span> <span class="ltx_bibblock"><span class="ltx_text ltx_font_italic" id="bib.bib22.1.1">arXiv preprint arXiv:2203.00343</span>, 2022. </span> </li> <li class="ltx_bibitem" id="bib.bib23"> <span class="ltx_tag ltx_tag_bibitem">[23]</span> <span class="ltx_bibblock"> S. Sun, K. Thai, and M. Iyyer. </span> <span class="ltx_bibblock">Chapterbreak: A challenge dataset for long-range language models. </span> <span class="ltx_bibblock"><span class="ltx_text ltx_font_italic" id="bib.bib23.1.1">arXiv preprint arXiv:2204.10878</span>, 2022. </span> </li> <li class="ltx_bibitem" id="bib.bib24"> <span class="ltx_tag ltx_tag_bibitem">[24]</span> <span class="ltx_bibblock"> B. Tan, Z. Yang, M. AI-Shedivat, E. P. Xing, and Z. Hu. </span> <span class="ltx_bibblock">Progressive generation of long text with pretrained language models. </span> <span class="ltx_bibblock"><span class="ltx_text ltx_font_italic" id="bib.bib24.1.1">arXiv preprint arXiv:2006.15720</span>, 2020. </span> </li> <li class="ltx_bibitem" id="bib.bib25"> <span class="ltx_tag ltx_tag_bibitem">[25]</span> <span class="ltx_bibblock"> X. Wang, J. Wei, D. Schuurmans, Q. V. Le, E. H. Chi, S. Narang, A. Chowdhery, and D. Zhou. </span> <span class="ltx_bibblock">Self-consistency improves chain of thought reasoning in language models. </span> <span class="ltx_bibblock">In <span class="ltx_text ltx_font_italic" id="bib.bib25.1.1">The Eleventh International Conference on Learning Representations</span>, 2022. </span> </li> <li class="ltx_bibitem" id="bib.bib26"> <span class="ltx_tag ltx_tag_bibitem">[26]</span> <span class="ltx_bibblock"> J. Wei, X. Wang, D. Schuurmans, M. Bosma, b. ichter, F. Xia, E. Chi, Q. V. Le, and D. Zhou. </span> <span class="ltx_bibblock">Chain-of-thought prompting elicits reasoning in large language models. </span> <span class="ltx_bibblock">In S. Koyejo, S. Mohamed, A. Agarwal, D. Belgrave, K. Cho, and A. Oh, editors, <span class="ltx_text ltx_font_italic" id="bib.bib26.1.1">Advances in Neural Information Processing Systems</span>, volume 35, pages 24824–24837. Curran Associates, Inc., 2022. </span> </li> <li class="ltx_bibitem" id="bib.bib27"> <span class="ltx_tag ltx_tag_bibitem">[27]</span> <span class="ltx_bibblock"> B. Xu, Z. Peng, B. Lei, S. Mukherjee, Y. Liu, and D. Xu. </span> <span class="ltx_bibblock">Rewoo: Decoupling reasoning from observations for efficient augmented language models. </span> <span class="ltx_bibblock"><span class="ltx_text ltx_font_italic" id="bib.bib27.1.1">arXiv preprint arXiv:2305.18323</span>, 2023. </span> </li> <li class="ltx_bibitem" id="bib.bib28"> <span class="ltx_tag ltx_tag_bibitem">[28]</span> <span class="ltx_bibblock"> F. Xu, Y. Song, M. Iyyer, and E. Choi. </span> <span class="ltx_bibblock">A critical evaluation of evaluations for long-form question answering. </span> <span class="ltx_bibblock"><span class="ltx_text ltx_font_italic" id="bib.bib28.1.1">arXiv preprint arXiv:2305.18201</span>, 2023. </span> </li> <li class="ltx_bibitem" id="bib.bib29"> <span class="ltx_tag ltx_tag_bibitem">[29]</span> <span class="ltx_bibblock"> F. Xu, Y. Song, M. Iyyer, and E. Choi. </span> <span class="ltx_bibblock">A critical evaluation of evaluations for long-form question answering. </span> <span class="ltx_bibblock">In <span class="ltx_text ltx_font_italic" id="bib.bib29.1.1">Association of Computational Linguistics</span>, 2023. </span> </li> <li class="ltx_bibitem" id="bib.bib30"> <span class="ltx_tag ltx_tag_bibitem">[30]</span> <span class="ltx_bibblock"> J. Xu, Z. Gan, Y. Cheng, and J. Liu. </span> <span class="ltx_bibblock">Discourse-aware neural extractive text summarization. </span> <span class="ltx_bibblock"><span class="ltx_text ltx_font_italic" id="bib.bib30.1.1">arXiv preprint arXiv:1910.14142</span>, 2019. </span> </li> <li class="ltx_bibitem" id="bib.bib31"> <span class="ltx_tag ltx_tag_bibitem">[31]</span> <span class="ltx_bibblock"> W. Xu, A. Banburski-Fahey, and N. Jojic. </span> <span class="ltx_bibblock">Reprompting: Automated chain-of-thought prompt inference through gibbs sampling. </span> <span class="ltx_bibblock"><span class="ltx_text ltx_font_italic" id="bib.bib31.1.1">arXiv preprint arXiv:2305.09993</span>, 2023. </span> </li> <li class="ltx_bibitem" id="bib.bib32"> <span class="ltx_tag ltx_tag_bibitem">[32]</span> <span class="ltx_bibblock"> K. Yang, D. Klein, N. Peng, and Y. Tian. </span> <span class="ltx_bibblock">Doc: Improving long story coherence with detailed outline control. </span> <span class="ltx_bibblock">In <span class="ltx_text ltx_font_italic" id="bib.bib32.1.1">Proceedings of the 61st Annual Meeting of the Association for Computational Linguistics (Volume 1: Long Papers)</span>, pages 3378–3465, 2023. </span> </li> <li class="ltx_bibitem" id="bib.bib33"> <span class="ltx_tag ltx_tag_bibitem">[33]</span> <span class="ltx_bibblock"> K. Yang, Y. Tian, N. Peng, and D. Klein. </span> <span class="ltx_bibblock">Re3: Generating longer stories with recursive reprompting and revision. </span> <span class="ltx_bibblock">In <span class="ltx_text ltx_font_italic" id="bib.bib33.1.1">Proceedings of the 2022 Conference on Empirical Methods in Natural Language Processing</span>, pages 4393–4479, 2022. </span> </li> <li class="ltx_bibitem" id="bib.bib34"> <span class="ltx_tag ltx_tag_bibitem">[34]</span> <span class="ltx_bibblock"> S. Yao, D. Yu, J. Zhao, I. Shafran, T. Griffiths, Y. Cao, and K. Narasimhan. </span> <span class="ltx_bibblock">Tree of thoughts: Deliberate problem solving with large language models. </span> <span class="ltx_bibblock"><span class="ltx_text ltx_font_italic" id="bib.bib34.1.1">Advances in Neural Information Processing Systems</span>, 36, 2024. </span> </li> <li class="ltx_bibitem" id="bib.bib35"> <span class="ltx_tag ltx_tag_bibitem">[35]</span> <span class="ltx_bibblock"> H. Zhang, W. Du, J. Shan, Q. Zhou, Y. Du, J. B. Tenenbaum, T. Shu, and C. Gan. </span> <span class="ltx_bibblock">Building cooperative embodied agents modularly with large language models. </span> <span class="ltx_bibblock"><span class="ltx_text ltx_font_italic" id="bib.bib35.1.1">arXiv preprint arXiv:2307.02485</span>, 2023. </span> </li> </ul> </section> <section class="ltx_appendix" id="A1"> <h2 class="ltx_title ltx_title_appendix"> <span class="ltx_tag ltx_tag_appendix">Appendix A </span>Prompts</h2> <div class="ltx_para" id="A1.p1"> <p class="ltx_p" id="A1.p1.1">In this section we provide the prompts we used for different tasks.</p> </div> <section class="ltx_subsection" id="A1.SS1"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">A.1 </span>SciNews</h3> <section class="ltx_paragraph" id="A1.SS1.SSS0.Px1"> <h4 class="ltx_title ltx_title_paragraph">Only generating the full article (<math alttext="x_{i}\rightarrow y_{i}" class="ltx_Math" display="inline" id="A1.SS1.SSS0.Px1.1.m1.1"><semantics id="A1.SS1.SSS0.Px1.1.m1.1b"><mrow id="A1.SS1.SSS0.Px1.1.m1.1.1" xref="A1.SS1.SSS0.Px1.1.m1.1.1.cmml"><msub id="A1.SS1.SSS0.Px1.1.m1.1.1.2" xref="A1.SS1.SSS0.Px1.1.m1.1.1.2.cmml"><mi id="A1.SS1.SSS0.Px1.1.m1.1.1.2.2" xref="A1.SS1.SSS0.Px1.1.m1.1.1.2.2.cmml">x</mi><mi id="A1.SS1.SSS0.Px1.1.m1.1.1.2.3" xref="A1.SS1.SSS0.Px1.1.m1.1.1.2.3.cmml">i</mi></msub><mo id="A1.SS1.SSS0.Px1.1.m1.1.1.1" stretchy="false" xref="A1.SS1.SSS0.Px1.1.m1.1.1.1.cmml">→</mo><msub id="A1.SS1.SSS0.Px1.1.m1.1.1.3" xref="A1.SS1.SSS0.Px1.1.m1.1.1.3.cmml"><mi id="A1.SS1.SSS0.Px1.1.m1.1.1.3.2" xref="A1.SS1.SSS0.Px1.1.m1.1.1.3.2.cmml">y</mi><mi id="A1.SS1.SSS0.Px1.1.m1.1.1.3.3" xref="A1.SS1.SSS0.Px1.1.m1.1.1.3.3.cmml">i</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="A1.SS1.SSS0.Px1.1.m1.1c"><apply id="A1.SS1.SSS0.Px1.1.m1.1.1.cmml" xref="A1.SS1.SSS0.Px1.1.m1.1.1"><ci id="A1.SS1.SSS0.Px1.1.m1.1.1.1.cmml" xref="A1.SS1.SSS0.Px1.1.m1.1.1.1">→</ci><apply id="A1.SS1.SSS0.Px1.1.m1.1.1.2.cmml" xref="A1.SS1.SSS0.Px1.1.m1.1.1.2"><csymbol cd="ambiguous" id="A1.SS1.SSS0.Px1.1.m1.1.1.2.1.cmml" xref="A1.SS1.SSS0.Px1.1.m1.1.1.2">subscript</csymbol><ci id="A1.SS1.SSS0.Px1.1.m1.1.1.2.2.cmml" xref="A1.SS1.SSS0.Px1.1.m1.1.1.2.2">𝑥</ci><ci id="A1.SS1.SSS0.Px1.1.m1.1.1.2.3.cmml" xref="A1.SS1.SSS0.Px1.1.m1.1.1.2.3">𝑖</ci></apply><apply id="A1.SS1.SSS0.Px1.1.m1.1.1.3.cmml" xref="A1.SS1.SSS0.Px1.1.m1.1.1.3"><csymbol cd="ambiguous" id="A1.SS1.SSS0.Px1.1.m1.1.1.3.1.cmml" xref="A1.SS1.SSS0.Px1.1.m1.1.1.3">subscript</csymbol><ci id="A1.SS1.SSS0.Px1.1.m1.1.1.3.2.cmml" xref="A1.SS1.SSS0.Px1.1.m1.1.1.3.2">𝑦</ci><ci id="A1.SS1.SSS0.Px1.1.m1.1.1.3.3.cmml" xref="A1.SS1.SSS0.Px1.1.m1.1.1.3.3">𝑖</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.SSS0.Px1.1.m1.1d">x_{i}\rightarrow y_{i}</annotation><annotation encoding="application/x-llamapun" id="A1.SS1.SSS0.Px1.1.m1.1e">italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT → italic_y start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math>).</h4> <div class="ltx_para" id="A1.SS1.SSS0.Px1.p1"> <p class="ltx_p" id="A1.SS1.SSS0.Px1.p1.1">Below we provide the prompt for</p> </div> <div class="ltx_para ltx_noindent" id="A1.SS1.SSS0.Px1.p2"> <svg class="ltx_picture" height="58" id="A1.SS1.SSS0.Px1.p2.pic1" overflow="visible" version="1.1" width="600"><g fill="#000000" stroke="#000000" stroke-width="0.4pt" transform="translate(0,58) matrix(1 0 0 -1 0 0)"><g fill="#404040" fill-opacity="1.0"><path d="M 0 5.91 L 0 52.09 C 0 55.36 2.64 58 5.91 58 L 594.09 58 C 597.36 58 600 55.36 600 52.09 L 600 5.91 C 600 2.64 597.36 0 594.09 0 L 5.91 0 C 2.64 0 0 2.64 0 5.91 Z" style="stroke:none"></path></g><g fill="#F2F2F2" fill-opacity="1.0"><path d="M 1.97 5.91 L 1.97 52.09 C 1.97 54.27 3.73 56.03 5.91 56.03 L 594.09 56.03 C 596.27 56.03 598.03 54.27 598.03 52.09 L 598.03 5.91 C 598.03 3.73 596.27 1.97 594.09 1.97 L 5.91 1.97 C 3.73 1.97 1.97 3.73 1.97 5.91 Z" style="stroke:none"></path></g><g fill-opacity="1.0" transform="matrix(1.0 0.0 0.0 1.0 21.65 13.78)"><foreignobject color="#000000" height="30.44" overflow="visible" transform="matrix(1 0 0 -1 0 16.6)" width="556.69"> <span class="ltx_inline-block ltx_minipage ltx_align_bottom" id="A1.SS1.SSS0.Px1.p2.pic1.1.1.1.1.1" style="width:402.3pt;"> <span class="ltx_p" id="A1.SS1.SSS0.Px1.p2.pic1.1.1.1.1.1.1">Given the body of the academic paper, generate a whole news article.</span> <span class="ltx_p" id="A1.SS1.SSS0.Px1.p2.pic1.1.1.1.1.1.2">Academic paper body: {input_paper}</span> </span></foreignobject></g></g></svg> </div> </section> <section class="ltx_paragraph" id="A1.SS1.SSS0.Px2"> <h4 class="ltx_title ltx_title_paragraph">Generating the intermediate steps and the full article (<math alttext="x_{i}\rightarrow{\mathbf{z}}_{i}\oplus y_{i}" class="ltx_Math" display="inline" id="A1.SS1.SSS0.Px2.1.m1.1"><semantics id="A1.SS1.SSS0.Px2.1.m1.1b"><mrow id="A1.SS1.SSS0.Px2.1.m1.1.1" xref="A1.SS1.SSS0.Px2.1.m1.1.1.cmml"><msub id="A1.SS1.SSS0.Px2.1.m1.1.1.2" xref="A1.SS1.SSS0.Px2.1.m1.1.1.2.cmml"><mi id="A1.SS1.SSS0.Px2.1.m1.1.1.2.2" xref="A1.SS1.SSS0.Px2.1.m1.1.1.2.2.cmml">x</mi><mi id="A1.SS1.SSS0.Px2.1.m1.1.1.2.3" xref="A1.SS1.SSS0.Px2.1.m1.1.1.2.3.cmml">i</mi></msub><mo id="A1.SS1.SSS0.Px2.1.m1.1.1.1" stretchy="false" xref="A1.SS1.SSS0.Px2.1.m1.1.1.1.cmml">→</mo><mrow id="A1.SS1.SSS0.Px2.1.m1.1.1.3" xref="A1.SS1.SSS0.Px2.1.m1.1.1.3.cmml"><msub id="A1.SS1.SSS0.Px2.1.m1.1.1.3.2" xref="A1.SS1.SSS0.Px2.1.m1.1.1.3.2.cmml"><mi id="A1.SS1.SSS0.Px2.1.m1.1.1.3.2.2" xref="A1.SS1.SSS0.Px2.1.m1.1.1.3.2.2.cmml">𝐳</mi><mi id="A1.SS1.SSS0.Px2.1.m1.1.1.3.2.3" xref="A1.SS1.SSS0.Px2.1.m1.1.1.3.2.3.cmml">i</mi></msub><mo id="A1.SS1.SSS0.Px2.1.m1.1.1.3.1" xref="A1.SS1.SSS0.Px2.1.m1.1.1.3.1.cmml">⊕</mo><msub id="A1.SS1.SSS0.Px2.1.m1.1.1.3.3" xref="A1.SS1.SSS0.Px2.1.m1.1.1.3.3.cmml"><mi id="A1.SS1.SSS0.Px2.1.m1.1.1.3.3.2" xref="A1.SS1.SSS0.Px2.1.m1.1.1.3.3.2.cmml">y</mi><mi id="A1.SS1.SSS0.Px2.1.m1.1.1.3.3.3" xref="A1.SS1.SSS0.Px2.1.m1.1.1.3.3.3.cmml">i</mi></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.SS1.SSS0.Px2.1.m1.1c"><apply id="A1.SS1.SSS0.Px2.1.m1.1.1.cmml" xref="A1.SS1.SSS0.Px2.1.m1.1.1"><ci id="A1.SS1.SSS0.Px2.1.m1.1.1.1.cmml" xref="A1.SS1.SSS0.Px2.1.m1.1.1.1">→</ci><apply id="A1.SS1.SSS0.Px2.1.m1.1.1.2.cmml" xref="A1.SS1.SSS0.Px2.1.m1.1.1.2"><csymbol cd="ambiguous" id="A1.SS1.SSS0.Px2.1.m1.1.1.2.1.cmml" xref="A1.SS1.SSS0.Px2.1.m1.1.1.2">subscript</csymbol><ci id="A1.SS1.SSS0.Px2.1.m1.1.1.2.2.cmml" xref="A1.SS1.SSS0.Px2.1.m1.1.1.2.2">𝑥</ci><ci id="A1.SS1.SSS0.Px2.1.m1.1.1.2.3.cmml" xref="A1.SS1.SSS0.Px2.1.m1.1.1.2.3">𝑖</ci></apply><apply id="A1.SS1.SSS0.Px2.1.m1.1.1.3.cmml" xref="A1.SS1.SSS0.Px2.1.m1.1.1.3"><csymbol cd="latexml" id="A1.SS1.SSS0.Px2.1.m1.1.1.3.1.cmml" xref="A1.SS1.SSS0.Px2.1.m1.1.1.3.1">direct-sum</csymbol><apply id="A1.SS1.SSS0.Px2.1.m1.1.1.3.2.cmml" xref="A1.SS1.SSS0.Px2.1.m1.1.1.3.2"><csymbol cd="ambiguous" id="A1.SS1.SSS0.Px2.1.m1.1.1.3.2.1.cmml" xref="A1.SS1.SSS0.Px2.1.m1.1.1.3.2">subscript</csymbol><ci id="A1.SS1.SSS0.Px2.1.m1.1.1.3.2.2.cmml" xref="A1.SS1.SSS0.Px2.1.m1.1.1.3.2.2">𝐳</ci><ci id="A1.SS1.SSS0.Px2.1.m1.1.1.3.2.3.cmml" xref="A1.SS1.SSS0.Px2.1.m1.1.1.3.2.3">𝑖</ci></apply><apply id="A1.SS1.SSS0.Px2.1.m1.1.1.3.3.cmml" xref="A1.SS1.SSS0.Px2.1.m1.1.1.3.3"><csymbol cd="ambiguous" id="A1.SS1.SSS0.Px2.1.m1.1.1.3.3.1.cmml" xref="A1.SS1.SSS0.Px2.1.m1.1.1.3.3">subscript</csymbol><ci id="A1.SS1.SSS0.Px2.1.m1.1.1.3.3.2.cmml" xref="A1.SS1.SSS0.Px2.1.m1.1.1.3.3.2">𝑦</ci><ci id="A1.SS1.SSS0.Px2.1.m1.1.1.3.3.3.cmml" xref="A1.SS1.SSS0.Px2.1.m1.1.1.3.3.3">𝑖</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS1.SSS0.Px2.1.m1.1d">x_{i}\rightarrow{\mathbf{z}}_{i}\oplus y_{i}</annotation><annotation encoding="application/x-llamapun" id="A1.SS1.SSS0.Px2.1.m1.1e">italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT → bold_z start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ⊕ italic_y start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math>).</h4> <div class="ltx_para" id="A1.SS1.SSS0.Px2.p1"> <p class="ltx_p" id="A1.SS1.SSS0.Px2.p1.1">Below we provide the prompt for</p> </div> <div class="ltx_para ltx_noindent" id="A1.SS1.SSS0.Px2.p2"> <svg class="ltx_picture" height="74.6" id="A1.SS1.SSS0.Px2.p2.pic1" overflow="visible" version="1.1" width="600"><g fill="#000000" stroke="#000000" stroke-width="0.4pt" transform="translate(0,74.6) matrix(1 0 0 -1 0 0)"><g fill="#404040" fill-opacity="1.0"><path d="M 0 5.91 L 0 68.7 C 0 71.96 2.64 74.6 5.91 74.6 L 594.09 74.6 C 597.36 74.6 600 71.96 600 68.7 L 600 5.91 C 600 2.64 597.36 0 594.09 0 L 5.91 0 C 2.64 0 0 2.64 0 5.91 Z" style="stroke:none"></path></g><g fill="#F2F2F2" fill-opacity="1.0"><path d="M 1.97 5.91 L 1.97 68.7 C 1.97 70.87 3.73 72.64 5.91 72.64 L 594.09 72.64 C 596.27 72.64 598.03 70.87 598.03 68.7 L 598.03 5.91 C 598.03 3.73 596.27 1.97 594.09 1.97 L 5.91 1.97 C 3.73 1.97 1.97 3.73 1.97 5.91 Z" style="stroke:none"></path></g><g fill-opacity="1.0" transform="matrix(1.0 0.0 0.0 1.0 21.65 13.78)"><foreignobject color="#000000" height="47.05" overflow="visible" transform="matrix(1 0 0 -1 0 16.6)" width="556.69"> <span class="ltx_inline-block ltx_minipage ltx_align_bottom" id="A1.SS1.SSS0.Px2.p2.pic1.1.1.1.1.1" style="width:402.3pt;"> <span class="ltx_p" id="A1.SS1.SSS0.Px2.p2.pic1.1.1.1.1.1.1">Given the academic paper’s full text, first generate a news article’s summary, the high-level outline and detailed key information snippets, then leverage those information to generate a complete news article with title and body.</span> <span class="ltx_p" id="A1.SS1.SSS0.Px2.p2.pic1.1.1.1.1.1.2">Academic paper body: {input_paper}</span> </span></foreignobject></g></g></svg> </div> </section> </section> <section class="ltx_subsection" id="A1.SS2"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">A.2 </span>Wikipedia</h3> <section class="ltx_paragraph" id="A1.SS2.SSS0.Px1"> <h4 class="ltx_title ltx_title_paragraph">Only generating the full article (<math alttext="x_{i}\rightarrow y_{i}" class="ltx_Math" display="inline" id="A1.SS2.SSS0.Px1.1.m1.1"><semantics id="A1.SS2.SSS0.Px1.1.m1.1b"><mrow id="A1.SS2.SSS0.Px1.1.m1.1.1" xref="A1.SS2.SSS0.Px1.1.m1.1.1.cmml"><msub id="A1.SS2.SSS0.Px1.1.m1.1.1.2" xref="A1.SS2.SSS0.Px1.1.m1.1.1.2.cmml"><mi id="A1.SS2.SSS0.Px1.1.m1.1.1.2.2" xref="A1.SS2.SSS0.Px1.1.m1.1.1.2.2.cmml">x</mi><mi id="A1.SS2.SSS0.Px1.1.m1.1.1.2.3" xref="A1.SS2.SSS0.Px1.1.m1.1.1.2.3.cmml">i</mi></msub><mo id="A1.SS2.SSS0.Px1.1.m1.1.1.1" stretchy="false" xref="A1.SS2.SSS0.Px1.1.m1.1.1.1.cmml">→</mo><msub id="A1.SS2.SSS0.Px1.1.m1.1.1.3" xref="A1.SS2.SSS0.Px1.1.m1.1.1.3.cmml"><mi id="A1.SS2.SSS0.Px1.1.m1.1.1.3.2" xref="A1.SS2.SSS0.Px1.1.m1.1.1.3.2.cmml">y</mi><mi id="A1.SS2.SSS0.Px1.1.m1.1.1.3.3" xref="A1.SS2.SSS0.Px1.1.m1.1.1.3.3.cmml">i</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="A1.SS2.SSS0.Px1.1.m1.1c"><apply id="A1.SS2.SSS0.Px1.1.m1.1.1.cmml" xref="A1.SS2.SSS0.Px1.1.m1.1.1"><ci id="A1.SS2.SSS0.Px1.1.m1.1.1.1.cmml" xref="A1.SS2.SSS0.Px1.1.m1.1.1.1">→</ci><apply id="A1.SS2.SSS0.Px1.1.m1.1.1.2.cmml" xref="A1.SS2.SSS0.Px1.1.m1.1.1.2"><csymbol cd="ambiguous" id="A1.SS2.SSS0.Px1.1.m1.1.1.2.1.cmml" xref="A1.SS2.SSS0.Px1.1.m1.1.1.2">subscript</csymbol><ci id="A1.SS2.SSS0.Px1.1.m1.1.1.2.2.cmml" xref="A1.SS2.SSS0.Px1.1.m1.1.1.2.2">𝑥</ci><ci id="A1.SS2.SSS0.Px1.1.m1.1.1.2.3.cmml" xref="A1.SS2.SSS0.Px1.1.m1.1.1.2.3">𝑖</ci></apply><apply id="A1.SS2.SSS0.Px1.1.m1.1.1.3.cmml" xref="A1.SS2.SSS0.Px1.1.m1.1.1.3"><csymbol cd="ambiguous" id="A1.SS2.SSS0.Px1.1.m1.1.1.3.1.cmml" xref="A1.SS2.SSS0.Px1.1.m1.1.1.3">subscript</csymbol><ci id="A1.SS2.SSS0.Px1.1.m1.1.1.3.2.cmml" xref="A1.SS2.SSS0.Px1.1.m1.1.1.3.2">𝑦</ci><ci id="A1.SS2.SSS0.Px1.1.m1.1.1.3.3.cmml" xref="A1.SS2.SSS0.Px1.1.m1.1.1.3.3">𝑖</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS2.SSS0.Px1.1.m1.1d">x_{i}\rightarrow y_{i}</annotation><annotation encoding="application/x-llamapun" id="A1.SS2.SSS0.Px1.1.m1.1e">italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT → italic_y start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math>).</h4> <div class="ltx_para" id="A1.SS2.SSS0.Px1.p1"> <p class="ltx_p" id="A1.SS2.SSS0.Px1.p1.1">Below we provide the prompt for</p> </div> <div class="ltx_para ltx_noindent" id="A1.SS2.SSS0.Px1.p2"> <svg class="ltx_picture" height="41.4" id="A1.SS2.SSS0.Px1.p2.pic1" overflow="visible" version="1.1" width="600"><g fill="#000000" stroke="#000000" stroke-width="0.4pt" transform="translate(0,41.4) matrix(1 0 0 -1 0 0)"><g fill="#404040" fill-opacity="1.0"><path d="M 0 5.91 L 0 35.49 C 0 38.75 2.64 41.4 5.91 41.4 L 594.09 41.4 C 597.36 41.4 600 38.75 600 35.49 L 600 5.91 C 600 2.64 597.36 0 594.09 0 L 5.91 0 C 2.64 0 0 2.64 0 5.91 Z" style="stroke:none"></path></g><g fill="#F2F2F2" fill-opacity="1.0"><path d="M 1.97 5.91 L 1.97 35.49 C 1.97 37.66 3.73 39.43 5.91 39.43 L 594.09 39.43 C 596.27 39.43 598.03 37.66 598.03 35.49 L 598.03 5.91 C 598.03 3.73 596.27 1.97 594.09 1.97 L 5.91 1.97 C 3.73 1.97 1.97 3.73 1.97 5.91 Z" style="stroke:none"></path></g><g fill-opacity="1.0" transform="matrix(1.0 0.0 0.0 1.0 21.65 13.78)"><foreignobject color="#000000" height="13.84" overflow="visible" transform="matrix(1 0 0 -1 0 16.6)" width="556.69"> <span class="ltx_inline-block ltx_minipage ltx_align_bottom" id="A1.SS2.SSS0.Px1.p2.pic1.1.1.1.1.1" style="width:402.3pt;"> <span class="ltx_p" id="A1.SS2.SSS0.Px1.p2.pic1.1.1.1.1.1.1">Generate a comprehensive Wikipedia page about the specified topic.</span> <span class="ltx_p" id="A1.SS2.SSS0.Px1.p2.pic1.1.1.1.1.1.2">Topic: {input_topic}</span> </span></foreignobject></g></g></svg> </div> </section> <section class="ltx_paragraph" id="A1.SS2.SSS0.Px2"> <h4 class="ltx_title ltx_title_paragraph">Generating the intermediate steps and the full article (<math alttext="x_{i}\rightarrow{\mathbf{z}}_{i}\oplus y_{i}" class="ltx_Math" display="inline" id="A1.SS2.SSS0.Px2.1.m1.1"><semantics id="A1.SS2.SSS0.Px2.1.m1.1b"><mrow id="A1.SS2.SSS0.Px2.1.m1.1.1" xref="A1.SS2.SSS0.Px2.1.m1.1.1.cmml"><msub id="A1.SS2.SSS0.Px2.1.m1.1.1.2" xref="A1.SS2.SSS0.Px2.1.m1.1.1.2.cmml"><mi id="A1.SS2.SSS0.Px2.1.m1.1.1.2.2" xref="A1.SS2.SSS0.Px2.1.m1.1.1.2.2.cmml">x</mi><mi id="A1.SS2.SSS0.Px2.1.m1.1.1.2.3" xref="A1.SS2.SSS0.Px2.1.m1.1.1.2.3.cmml">i</mi></msub><mo id="A1.SS2.SSS0.Px2.1.m1.1.1.1" stretchy="false" xref="A1.SS2.SSS0.Px2.1.m1.1.1.1.cmml">→</mo><mrow id="A1.SS2.SSS0.Px2.1.m1.1.1.3" xref="A1.SS2.SSS0.Px2.1.m1.1.1.3.cmml"><msub id="A1.SS2.SSS0.Px2.1.m1.1.1.3.2" xref="A1.SS2.SSS0.Px2.1.m1.1.1.3.2.cmml"><mi id="A1.SS2.SSS0.Px2.1.m1.1.1.3.2.2" xref="A1.SS2.SSS0.Px2.1.m1.1.1.3.2.2.cmml">𝐳</mi><mi id="A1.SS2.SSS0.Px2.1.m1.1.1.3.2.3" xref="A1.SS2.SSS0.Px2.1.m1.1.1.3.2.3.cmml">i</mi></msub><mo id="A1.SS2.SSS0.Px2.1.m1.1.1.3.1" xref="A1.SS2.SSS0.Px2.1.m1.1.1.3.1.cmml">⊕</mo><msub id="A1.SS2.SSS0.Px2.1.m1.1.1.3.3" xref="A1.SS2.SSS0.Px2.1.m1.1.1.3.3.cmml"><mi id="A1.SS2.SSS0.Px2.1.m1.1.1.3.3.2" xref="A1.SS2.SSS0.Px2.1.m1.1.1.3.3.2.cmml">y</mi><mi id="A1.SS2.SSS0.Px2.1.m1.1.1.3.3.3" xref="A1.SS2.SSS0.Px2.1.m1.1.1.3.3.3.cmml">i</mi></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.SS2.SSS0.Px2.1.m1.1c"><apply id="A1.SS2.SSS0.Px2.1.m1.1.1.cmml" xref="A1.SS2.SSS0.Px2.1.m1.1.1"><ci id="A1.SS2.SSS0.Px2.1.m1.1.1.1.cmml" xref="A1.SS2.SSS0.Px2.1.m1.1.1.1">→</ci><apply id="A1.SS2.SSS0.Px2.1.m1.1.1.2.cmml" xref="A1.SS2.SSS0.Px2.1.m1.1.1.2"><csymbol cd="ambiguous" id="A1.SS2.SSS0.Px2.1.m1.1.1.2.1.cmml" xref="A1.SS2.SSS0.Px2.1.m1.1.1.2">subscript</csymbol><ci id="A1.SS2.SSS0.Px2.1.m1.1.1.2.2.cmml" xref="A1.SS2.SSS0.Px2.1.m1.1.1.2.2">𝑥</ci><ci id="A1.SS2.SSS0.Px2.1.m1.1.1.2.3.cmml" xref="A1.SS2.SSS0.Px2.1.m1.1.1.2.3">𝑖</ci></apply><apply id="A1.SS2.SSS0.Px2.1.m1.1.1.3.cmml" xref="A1.SS2.SSS0.Px2.1.m1.1.1.3"><csymbol cd="latexml" id="A1.SS2.SSS0.Px2.1.m1.1.1.3.1.cmml" xref="A1.SS2.SSS0.Px2.1.m1.1.1.3.1">direct-sum</csymbol><apply id="A1.SS2.SSS0.Px2.1.m1.1.1.3.2.cmml" xref="A1.SS2.SSS0.Px2.1.m1.1.1.3.2"><csymbol cd="ambiguous" id="A1.SS2.SSS0.Px2.1.m1.1.1.3.2.1.cmml" xref="A1.SS2.SSS0.Px2.1.m1.1.1.3.2">subscript</csymbol><ci id="A1.SS2.SSS0.Px2.1.m1.1.1.3.2.2.cmml" xref="A1.SS2.SSS0.Px2.1.m1.1.1.3.2.2">𝐳</ci><ci id="A1.SS2.SSS0.Px2.1.m1.1.1.3.2.3.cmml" xref="A1.SS2.SSS0.Px2.1.m1.1.1.3.2.3">𝑖</ci></apply><apply id="A1.SS2.SSS0.Px2.1.m1.1.1.3.3.cmml" xref="A1.SS2.SSS0.Px2.1.m1.1.1.3.3"><csymbol cd="ambiguous" id="A1.SS2.SSS0.Px2.1.m1.1.1.3.3.1.cmml" xref="A1.SS2.SSS0.Px2.1.m1.1.1.3.3">subscript</csymbol><ci id="A1.SS2.SSS0.Px2.1.m1.1.1.3.3.2.cmml" xref="A1.SS2.SSS0.Px2.1.m1.1.1.3.3.2">𝑦</ci><ci id="A1.SS2.SSS0.Px2.1.m1.1.1.3.3.3.cmml" xref="A1.SS2.SSS0.Px2.1.m1.1.1.3.3.3">𝑖</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.SS2.SSS0.Px2.1.m1.1d">x_{i}\rightarrow{\mathbf{z}}_{i}\oplus y_{i}</annotation><annotation encoding="application/x-llamapun" id="A1.SS2.SSS0.Px2.1.m1.1e">italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT → bold_z start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ⊕ italic_y start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math>).</h4> <div class="ltx_para" id="A1.SS2.SSS0.Px2.p1"> <p class="ltx_p" id="A1.SS2.SSS0.Px2.p1.1">Below we provide the prompt for</p> </div> <div class="ltx_para ltx_noindent" id="A1.SS2.SSS0.Px2.p2"> <svg class="ltx_picture" height="91.21" id="A1.SS2.SSS0.Px2.p2.pic1" overflow="visible" version="1.1" width="600"><g fill="#000000" stroke="#000000" stroke-width="0.4pt" transform="translate(0,91.21) matrix(1 0 0 -1 0 0)"><g fill="#404040" fill-opacity="1.0"><path d="M 0 5.91 L 0 85.3 C 0 88.57 2.64 91.21 5.91 91.21 L 594.09 91.21 C 597.36 91.21 600 88.57 600 85.3 L 600 5.91 C 600 2.64 597.36 0 594.09 0 L 5.91 0 C 2.64 0 0 2.64 0 5.91 Z" style="stroke:none"></path></g><g fill="#F2F2F2" fill-opacity="1.0"><path d="M 1.97 5.91 L 1.97 85.3 C 1.97 87.48 3.73 89.24 5.91 89.24 L 594.09 89.24 C 596.27 89.24 598.03 87.48 598.03 85.3 L 598.03 5.91 C 598.03 3.73 596.27 1.97 594.09 1.97 L 5.91 1.97 C 3.73 1.97 1.97 3.73 1.97 5.91 Z" style="stroke:none"></path></g><g fill-opacity="1.0" transform="matrix(1.0 0.0 0.0 1.0 21.65 13.78)"><foreignobject color="#000000" height="63.65" overflow="visible" transform="matrix(1 0 0 -1 0 16.6)" width="556.69"> <span class="ltx_inline-block ltx_minipage ltx_align_bottom" id="A1.SS2.SSS0.Px2.p2.pic1.1.1.1.1.1" style="width:402.3pt;"> <span class="ltx_p" id="A1.SS2.SSS0.Px2.p2.pic1.1.1.1.1.1.1">Given a specific topic, you are asked to write a comprehensive Wikipedia page about this topic. Let’s write step by step. First generate a summary, a high-level outline and a list of detailed key information snippets. Then, follow the summary, high-level outline and detailed key information snippets, generate a Wikipedia page about this topic.</span> <span class="ltx_p" id="A1.SS2.SSS0.Px2.p2.pic1.1.1.1.1.1.2">Topic: {input_topic}</span> </span></foreignobject></g></g></svg> </div> <div class="ltx_pagination ltx_role_newpage"></div> </section> </section> </section> </article> </div> <footer class="ltx_page_footer"> <div class="ltx_page_logo">Generated on Tue Oct 8 17:01:38 2024 by <a class="ltx_LaTeXML_logo" href="http://dlmf.nist.gov/LaTeXML/"><span style="letter-spacing:-0.2em; margin-right:0.1em;">L<span 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>