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<!DOCTYPE html> <html lang="en"> <head> <meta content="text/html; charset=utf-8" http-equiv="content-type"/> <title>PersonaX: A Recommendation Agent-Oriented User Modeling Framework for Long Behavior Sequence</title>  <meta content="width=device-width, initial-scale=1, shrink-to-fit=no" name="viewport"/> <link href="https://cdn.jsdelivr.net/npm/bootstrap@5.3.0/dist/css/bootstrap.min.css" rel="stylesheet" type="text/css"/> <link href="/static/browse/0.3.4/css/ar5iv.0.7.9.min.css" rel="stylesheet" type="text/css"/> <link href="/static/browse/0.3.4/css/ar5iv-fonts.0.7.9.min.css" rel="stylesheet" type="text/css"/> <link href="/static/browse/0.3.4/css/latexml_styles.css" rel="stylesheet" type="text/css"/> <script src="https://cdn.jsdelivr.net/npm/bootstrap@5.3.0/dist/js/bootstrap.bundle.min.js"></script> <script src="https://cdnjs.cloudflare.com/ajax/libs/html2canvas/1.3.3/html2canvas.min.js"></script> <script src="/static/browse/0.3.4/js/addons_new.js"></script> <script src="/static/browse/0.3.4/js/feedbackOverlay.js"></script> <base href="/html/2503.02398v1/"/></head> <body> <nav class="ltx_page_navbar"> <nav class="ltx_TOC"> <ol class="ltx_toclist"> <li class="ltx_tocentry ltx_tocentry_section"><a class="ltx_ref" href="https://arxiv.org/html/2503.02398v1#S1" title="In PersonaX: A Recommendation Agent-Oriented User Modeling Framework for Long Behavior Sequence"><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/2503.02398v1#S2" title="In PersonaX: A Recommendation Agent-Oriented User Modeling Framework for Long Behavior Sequence"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">2 </span>Preliminary</span></a> <ol class="ltx_toclist ltx_toclist_section"> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.02398v1#S2.SS1" title="In 2 Preliminary ‣ PersonaX: A Recommendation Agent-Oriented User Modeling Framework for Long Behavior Sequence"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">2.1 </span>User Modeling</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.02398v1#S2.SS2" title="In 2 Preliminary ‣ PersonaX: A Recommendation Agent-Oriented User Modeling Framework for Long Behavior Sequence"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">2.2 </span>Sub-Behavior Sequence (SBS) Selection.</span></a></li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_section"> <a class="ltx_ref" href="https://arxiv.org/html/2503.02398v1#S3" title="In PersonaX: A Recommendation Agent-Oriented User Modeling Framework for Long Behavior Sequence"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">3 </span>Method</span></a> <ol class="ltx_toclist ltx_toclist_section"> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.02398v1#S3.SS1" title="In 3 Method ‣ PersonaX: A Recommendation Agent-Oriented User Modeling Framework for Long Behavior Sequence"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">3.1 </span>Behavior Clustering</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.02398v1#S3.SS2" title="In 3 Method ‣ PersonaX: A Recommendation Agent-Oriented User Modeling Framework for Long Behavior Sequence"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">3.2 </span>Sampling Budget Allocation</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.02398v1#S3.SS3" title="In 3 Method ‣ PersonaX: A Recommendation Agent-Oriented User Modeling Framework for Long Behavior Sequence"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">3.3 </span>In-Cluster Selection</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.02398v1#S3.SS4" title="In 3 Method ‣ PersonaX: A Recommendation Agent-Oriented User Modeling Framework for Long Behavior Sequence"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">3.4 </span>Offline Profiling and Online Selection</span></a></li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_section"><a class="ltx_ref" href="https://arxiv.org/html/2503.02398v1#S4" title="In PersonaX: A Recommendation Agent-Oriented User Modeling Framework for Long Behavior Sequence"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">4 </span>Efficiency Analysis</span></a></li> <li class="ltx_tocentry ltx_tocentry_section"> <a class="ltx_ref" href="https://arxiv.org/html/2503.02398v1#S5" title="In PersonaX: A Recommendation Agent-Oriented User Modeling Framework for Long Behavior Sequence"><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/2503.02398v1#S5.SS1" title="In 5 Experiments ‣ PersonaX: A Recommendation Agent-Oriented User Modeling Framework for Long Behavior Sequence"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">5.1 </span>Experimental Setup</span></a> <ol class="ltx_toclist ltx_toclist_subsection"> <li class="ltx_tocentry ltx_tocentry_subsubsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.02398v1#S5.SS1.SSS1" title="In 5.1 Experimental Setup ‣ 5 Experiments ‣ PersonaX: A Recommendation Agent-Oriented User Modeling Framework for Long Behavior Sequence"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">5.1.1 </span>Datasets</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsubsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.02398v1#S5.SS1.SSS2" title="In 5.1 Experimental Setup ‣ 5 Experiments ‣ PersonaX: A Recommendation Agent-Oriented User Modeling Framework for Long Behavior Sequence"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">5.1.2 </span>Evaluation</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsubsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.02398v1#S5.SS1.SSS3" title="In 5.1 Experimental Setup ‣ 5 Experiments ‣ PersonaX: A Recommendation Agent-Oriented User Modeling Framework for Long Behavior Sequence"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">5.1.3 </span>Baseline Comparison</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsubsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.02398v1#S5.SS1.SSS4" title="In 5.1 Experimental Setup ‣ 5 Experiments ‣ PersonaX: A Recommendation Agent-Oriented User Modeling Framework for Long Behavior Sequence"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">5.1.4 </span>Backbone Agent Recommendation</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsubsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.02398v1#S5.SS1.SSS5" title="In 5.1 Experimental Setup ‣ 5 Experiments ‣ PersonaX: A Recommendation Agent-Oriented User Modeling Framework for Long Behavior Sequence"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">5.1.5 </span>Implementation Details</span></a></li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.02398v1#S5.SS2" title="In 5 Experiments ‣ PersonaX: A Recommendation Agent-Oriented User Modeling Framework for Long Behavior Sequence"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">5.2 </span>Performance Evaluation (RQ 1)</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.02398v1#S5.SS3" title="In 5 Experiments ‣ PersonaX: A Recommendation Agent-Oriented User Modeling Framework for Long Behavior Sequence"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">5.3 </span>Sampling Size Investigation (RQ 2)</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.02398v1#S5.SS4" title="In 5 Experiments ‣ PersonaX: A Recommendation Agent-Oriented User Modeling Framework for Long Behavior Sequence"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">5.4 </span>Hyper-parameter Analysis (RQ3)</span></a></li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_section"> <a class="ltx_ref" href="https://arxiv.org/html/2503.02398v1#S6" title="In PersonaX: A Recommendation Agent-Oriented User Modeling Framework for Long Behavior Sequence"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">6 </span>Related Works</span></a> <ol class="ltx_toclist ltx_toclist_section"> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.02398v1#S6.SS1" title="In 6 Related Works ‣ PersonaX: A Recommendation Agent-Oriented User Modeling Framework for Long Behavior Sequence"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">6.1 </span>Large Language Model for User Modeling</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.02398v1#S6.SS2" title="In 6 Related Works ‣ PersonaX: A Recommendation Agent-Oriented User Modeling Framework for Long Behavior Sequence"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">6.2 </span>Personalized Agents</span></a></li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_section"><a class="ltx_ref" href="https://arxiv.org/html/2503.02398v1#S7" title="In PersonaX: A Recommendation Agent-Oriented User Modeling Framework for Long Behavior Sequence"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">7 </span>Conclusion</span></a></li> <li class="ltx_tocentry ltx_tocentry_appendix"><a class="ltx_ref" href="https://arxiv.org/html/2503.02398v1#A1" title="In PersonaX: A Recommendation Agent-Oriented User Modeling Framework for Long Behavior Sequence"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">A </span>Datasets</span></a></li> <li class="ltx_tocentry ltx_tocentry_appendix"><a class="ltx_ref" href="https://arxiv.org/html/2503.02398v1#A2" title="In PersonaX: A Recommendation Agent-Oriented User Modeling Framework for Long Behavior Sequence"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">B </span>Backbone Methods</span></a></li> <li class="ltx_tocentry ltx_tocentry_appendix"><a class="ltx_ref" href="https://arxiv.org/html/2503.02398v1#A3" title="In PersonaX: A Recommendation Agent-Oriented User Modeling Framework for Long Behavior Sequence"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">C </span>Hyper-parameter Analysis and Sampling Process Visualization</span></a></li> <li class="ltx_tocentry ltx_tocentry_appendix"> <a class="ltx_ref" href="https://arxiv.org/html/2503.02398v1#A4" title="In PersonaX: A Recommendation Agent-Oriented User Modeling Framework for Long Behavior Sequence"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">D </span>Details about In-Cluter Selection</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/2503.02398v1#A4.SS1" title="In Appendix D Details about In-Cluter Selection ‣ PersonaX: A Recommendation Agent-Oriented User Modeling Framework for Long Behavior Sequence"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">D.1 </span>Prototypicality and Diversity Scoring</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.02398v1#A4.SS2" title="In Appendix D Details about In-Cluter Selection ‣ PersonaX: A Recommendation Agent-Oriented User Modeling Framework for Long Behavior Sequence"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">D.2 </span>Design Rationale</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.02398v1#A4.SS3" title="In Appendix D Details about In-Cluter Selection ‣ PersonaX: A Recommendation Agent-Oriented User Modeling Framework for Long Behavior Sequence"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">D.3 </span>Broader Implications</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.02398v1#A4.SS4" title="In Appendix D Details about In-Cluter Selection ‣ PersonaX: A Recommendation Agent-Oriented User Modeling Framework for Long Behavior Sequence"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">D.4 </span>Visualization Explanation</span></a></li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_appendix"><a class="ltx_ref" href="https://arxiv.org/html/2503.02398v1#A5" title="In PersonaX: A Recommendation Agent-Oriented User Modeling Framework for Long Behavior Sequence"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">E </span>Case Study</span></a></li> <li class="ltx_tocentry ltx_tocentry_appendix"><a class="ltx_ref" href="https://arxiv.org/html/2503.02398v1#A6" title="In PersonaX: A Recommendation Agent-Oriented User Modeling Framework for Long Behavior Sequence"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">F </span>Prompt Templates</span></a></li> </ol></nav> </nav> <div class="ltx_page_main"> <div class="ltx_page_content"> <article class="ltx_document"> <h1 class="ltx_title ltx_title_document">PersonaX: A Recommendation Agent-Oriented User Modeling Framework for Long Behavior Sequence</h1> <div class="ltx_authors"> <span class="ltx_creator ltx_role_author"> <span class="ltx_personname"> <span class="ltx_text ltx_font_bold" id="id2.1.id1">Yunxiao Shi<sup class="ltx_sup" id="id2.1.id1.1">1</sup></span>, <span class="ltx_text ltx_font_bold" id="id3.2.id2">Wujiang Xu<sup class="ltx_sup" id="id3.2.id2.1">2</sup></span>, <span class="ltx_text ltx_font_bold" id="id4.3.id3">Zeqi Zhang<sup class="ltx_sup" id="id4.3.id3.1">1</sup></span>, <br class="ltx_break"/><span class="ltx_text ltx_font_bold" id="id5.4.id4">Xing Zi<sup class="ltx_sup" id="id5.4.id4.1">1</sup></span>, <span class="ltx_text ltx_font_bold" id="id6.5.id5">Qiang Wu<sup class="ltx_sup" id="id6.5.id5.1">1</sup></span>, <span class="ltx_text ltx_font_bold" id="id7.6.id6">Min Xu<sup class="ltx_sup" id="id7.6.id6.1">1</sup></span> <br class="ltx_break"/> <br class="ltx_break"/><sup class="ltx_sup" id="id8.7.id7">1</sup>University of Technology Sydney, <sup class="ltx_sup" id="id9.8.id8">2</sup>Rutgers University </span><span class="ltx_author_notes">Corresponding author with email: Min.Xu@uts.edu.au.</span></span> </div> <div class="ltx_abstract"> <h6 class="ltx_title ltx_title_abstract">Abstract</h6> <p class="ltx_p" id="id1.1">Recommendation agents leverage large language models for user modeling (LLM-UM) to construct textual personas, guiding alignment with real users. However, existing LLM-UM methods struggle with long user-generated content (UGC) due to context limitations and performance degradation. To address this, sampling strategies prioritize relevance or recency are often applied, yet they inevitably neglect the diverse user interests embedded within the discarded behaviors, resulting in incomplete modeling and degraded profiling quality. Furthermore, relevance-based sampling requires real-time retrieval, forcing the user modeling process to operate online, which introduces significant latency overhead. In this paper, we propose PersonaX, an agent-agnostic LLM-UM framework that tackles these challenges through sub-behavior sequence (SBS) selection and offline multi-persona construction. PersonaX extracts compact SBS segments offline to capture diverse user interests, generating fine-grained textual personas that are cached for efficient online retrieval. This approach ensures that the user persona used for prompting remains highly relevant to the current context, while eliminating the need for online user modeling. For SBS selection, we ensure both efficiency (length <math alttext="<5" class="ltx_Math" display="inline" id="id1.1.m1.1"><semantics id="id1.1.m1.1a"><mrow id="id1.1.m1.1.1" xref="id1.1.m1.1.1.cmml"><mi id="id1.1.m1.1.1.2" xref="id1.1.m1.1.1.2.cmml"></mi><mo id="id1.1.m1.1.1.1" xref="id1.1.m1.1.1.1.cmml"><</mo><mn id="id1.1.m1.1.1.3" xref="id1.1.m1.1.1.3.cmml">5</mn></mrow><annotation-xml encoding="MathML-Content" id="id1.1.m1.1b"><apply id="id1.1.m1.1.1.cmml" xref="id1.1.m1.1.1"><lt id="id1.1.m1.1.1.1.cmml" xref="id1.1.m1.1.1.1"></lt><csymbol cd="latexml" id="id1.1.m1.1.1.2.cmml" xref="id1.1.m1.1.1.2">absent</csymbol><cn id="id1.1.m1.1.1.3.cmml" type="integer" xref="id1.1.m1.1.1.3">5</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="id1.1.m1.1c"><5</annotation><annotation encoding="application/x-llamapun" id="id1.1.m1.1d">< 5</annotation></semantics></math>) and high representational quality by balancing prototypicality and diversity within the sampled data. Extensive experiments validate the effectiveness and versatility of PersonaX in high-quality user profiling. Utilizing only 30–50% of the behavioral data with a sequence length of 480, integrating PersonaX with AgentCF yields an absolute performance improvement of 3–11%, while integration with Agent4Rec results in a gain of 10–50%. PersonaX as an agent-agnostic framework, sets a new benchmark for scalable user modeling, paving the way for more accurate and efficient LLM-driven recommendation agents <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>https://github.com/Ancientshi/PersonaX</span></span></span>.</p> </div> <div class="ltx_para ltx_noindent" id="p1"> <div class="ltx_block ltx_align_bottom" id="p1.1"> <p class="ltx_p" id="p1.1.1"><span class="ltx_text ltx_font_bold" id="p1.1.1.1">PersonaX: A Recommendation Agent-Oriented User Modeling Framework for Long Behavior Sequence</span></p> <br class="ltx_break ltx_centering"/> <p class="ltx_p ltx_align_center" id="p1.1.2" style="width:433.6pt;"><span class="ltx_text ltx_inline-block" id="p1.1.2.1" style="width:0.0pt;"> <span class="ltx_tabular ltx_align_top" id="p1.1.2.1.1"> <span class="ltx_tbody"> <span class="ltx_tr" id="p1.1.2.1.1.1.1"> <span class="ltx_td ltx_align_center" id="p1.1.2.1.1.1.1.1"><span class="ltx_text ltx_font_bold" id="p1.1.2.1.1.1.1.1.1"> Yunxiao Shi<sup class="ltx_sup" id="p1.1.2.1.1.1.1.1.1.1">1</sup>, Wujiang Xu<sup class="ltx_sup" id="p1.1.2.1.1.1.1.1.1.2">2</sup>, Zeqi Zhang<sup class="ltx_sup" id="p1.1.2.1.1.1.1.1.1.3">1</sup>,</span></span></span> <span class="ltx_tr" id="p1.1.2.1.1.2.2"> <span class="ltx_td ltx_align_center" id="p1.1.2.1.1.2.2.1"><span class="ltx_text ltx_font_bold" id="p1.1.2.1.1.2.2.1.1">Xing Zi<sup class="ltx_sup" id="p1.1.2.1.1.2.2.1.1.1">1</sup></span>, <span class="ltx_text ltx_font_bold" id="p1.1.2.1.1.2.2.1.2">Qiang Wu<sup class="ltx_sup" id="p1.1.2.1.1.2.2.1.2.1">1</sup></span>, <span class="ltx_text ltx_font_bold" id="p1.1.2.1.1.2.2.1.3">Min Xu<sup class="ltx_sup" id="p1.1.2.1.1.2.2.1.3.1">1</sup><span class="ltx_note ltx_role_thanks" id="p1.1.2.1.1.2.2.1.3.2"><sup class="ltx_note_mark">†</sup><span class="ltx_note_outer"><span class="ltx_note_content"><sup class="ltx_note_mark">†</sup><span class="ltx_note_type">thanks: </span><span class="ltx_text ltx_font_medium" id="p1.1.2.1.1.2.2.1.3.2.1">Corresponding author with email: Min.Xu@uts.edu.au.</span></span></span></span></span></span></span> <span class="ltx_tr" id="p1.1.2.1.1.3.3"> <span class="ltx_td ltx_align_center" id="p1.1.2.1.1.3.3.1"><sup class="ltx_sup" id="p1.1.2.1.1.3.3.1.1">1</sup>University of Technology Sydney, <sup class="ltx_sup" id="p1.1.2.1.1.3.3.1.2">2</sup>Rutgers University</span></span> </span> </span></span></p> <br class="ltx_break ltx_centering"/> </div> </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">User modeling (UM) <cite class="ltx_cite ltx_citemacro_cite">Tan and Jiang (<a class="ltx_ref" href="https://arxiv.org/html/2503.02398v1#bib.bib44" title="">2023</a>)</cite> methods extract implicit user persona traits from User-Generated Content (UGC), such as behavioral sequences, to construct meaningful representations that support instructional-based agent recommendations <cite class="ltx_cite ltx_citemacro_cite">Petruzzelli et al. (<a class="ltx_ref" href="https://arxiv.org/html/2503.02398v1#bib.bib26" title="">2024</a>); Zhang et al. (<a class="ltx_ref" href="https://arxiv.org/html/2503.02398v1#bib.bib56" title="">2024a</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.02398v1#bib.bib57" title="">b</a>)</cite>. Traditional NLP techniques such as Bag-of-Words (BoW), Latent Dirichlet Allocation (LDA) have been foundational in UM area <cite class="ltx_cite ltx_citemacro_cite">Harris (<a class="ltx_ref" href="https://arxiv.org/html/2503.02398v1#bib.bib11" title="">1954</a>); Blei et al. (<a class="ltx_ref" href="https://arxiv.org/html/2503.02398v1#bib.bib3" title="">2003</a>); Mikolov et al. (<a class="ltx_ref" href="https://arxiv.org/html/2503.02398v1#bib.bib22" title="">2013</a>); Vaswani (<a class="ltx_ref" href="https://arxiv.org/html/2503.02398v1#bib.bib47" title="">2017</a>); Sarzynska-Wawer et al. (<a class="ltx_ref" href="https://arxiv.org/html/2503.02398v1#bib.bib37" title="">2021</a>)</cite>. Recent advances in large language models (LLMs) <cite class="ltx_cite ltx_citemacro_cite">Brown et al. (<a class="ltx_ref" href="https://arxiv.org/html/2503.02398v1#bib.bib5" title="">2020</a>); Achiam et al. (<a class="ltx_ref" href="https://arxiv.org/html/2503.02398v1#bib.bib1" title="">2023</a>); Du et al. (<a class="ltx_ref" href="https://arxiv.org/html/2503.02398v1#bib.bib9" title="">2021</a>); Bai et al. (<a class="ltx_ref" href="https://arxiv.org/html/2503.02398v1#bib.bib2" title="">2023</a>)</cite> have opened new frontiers in semantic understanding. As a result, LLM-based user modeling (LLM-UM) approaches are gaining increasing research attention for their ability to capture nuanced and latent user personas from UGC.</p> </div> <div class="ltx_para" id="S1.p2"> <p class="ltx_p" id="S1.p2.1">A fundamental approach, known as <span class="ltx_text ltx_font_bold" id="S1.p2.1.1">Behavior Encoding</span>, leverages a user’s historical behavior sequence (BS) for profiling <cite class="ltx_cite ltx_citemacro_cite">Pi et al. (<a class="ltx_ref" href="https://arxiv.org/html/2503.02398v1#bib.bib28" title="">2020</a>)</cite>. This is achieved by encoding BS as demonstration examples within prompts, enabling LLM-driven agents to generalize from these examples and generate personalized outputs <cite class="ltx_cite ltx_citemacro_cite">Dai et al. (<a class="ltx_ref" href="https://arxiv.org/html/2503.02398v1#bib.bib8" title="">2023</a>); Liu et al. (<a class="ltx_ref" href="https://arxiv.org/html/2503.02398v1#bib.bib17" title="">2023</a>)</cite>. Beyond this straightforward methodology, Richardson et al. <cite class="ltx_cite ltx_citemacro_cite">Richardson et al. (<a class="ltx_ref" href="https://arxiv.org/html/2503.02398v1#bib.bib34" title="">2023</a>)</cite> demonstrated that summarizing core preference signals from extensive interactions enhances the personalization performance of LLMs. Techniques such as ONCE <cite class="ltx_cite ltx_citemacro_cite">Liu et al. (<a class="ltx_ref" href="https://arxiv.org/html/2503.02398v1#bib.bib18" title="">2024</a>)</cite>, Agent4Rec <cite class="ltx_cite ltx_citemacro_cite">Zhang et al. (<a class="ltx_ref" href="https://arxiv.org/html/2503.02398v1#bib.bib56" title="">2024a</a>)</cite>, and RecAgent <cite class="ltx_cite ltx_citemacro_cite">Wang et al. (<a class="ltx_ref" href="https://arxiv.org/html/2503.02398v1#bib.bib48" title="">2024</a>)</cite> utilize LLMs to distill behavioral data into concise textual personas that encapsulate user preferences—a process we refer to as <span class="ltx_text ltx_font_bold" id="S1.p2.1.2">Behavior Summarization</span>. Further extending this paradigm, methods like AgentCF <cite class="ltx_cite ltx_citemacro_cite">Zhang et al. (<a class="ltx_ref" href="https://arxiv.org/html/2503.02398v1#bib.bib57" title="">2024b</a>)</cite> and RecAgent <cite class="ltx_cite ltx_citemacro_cite">Wang et al. (<a class="ltx_ref" href="https://arxiv.org/html/2503.02398v1#bib.bib48" title="">2024</a>)</cite> adopt a <span class="ltx_text ltx_font_bold" id="S1.p2.1.3">Behavior Reflection</span> approach, employing reflection mechanisms <cite class="ltx_cite ltx_citemacro_cite">Cheng et al. (<a class="ltx_ref" href="https://arxiv.org/html/2503.02398v1#bib.bib7" title="">2023</a>); Zhao et al. (<a class="ltx_ref" href="https://arxiv.org/html/2503.02398v1#bib.bib58" title="">2024a</a>); Shi et al. (<a class="ltx_ref" href="https://arxiv.org/html/2503.02398v1#bib.bib39" title="">2024a</a>)</cite> on the behavior sequence to iteratively refine user personas. By evaluating the effectiveness of recommendations and dynamically adjusting the persona, these methods progressively align user profiles with the evolving nature of user behaviors.</p> </div> <div class="ltx_para" id="S1.p3"> <p class="ltx_p" id="S1.p3.1">However, modeling user personas from extensive UGC (e.g., long behavior sequences) presents critical challenges for LLM-UM: (1) <span class="ltx_text ltx_font_bold" id="S1.p3.1.1">Input limitations</span> – LLMs’ inherent input length constraints struggle to accommodate real-world recommendation systems’ extensive historical behavior data; (2) <span class="ltx_text ltx_font_bold" id="S1.p3.1.2">Mid-content oversight</span> – The "lost in the middle" phenomenon <cite class="ltx_cite ltx_citemacro_cite">Zhao et al. (<a class="ltx_ref" href="https://arxiv.org/html/2503.02398v1#bib.bib59" title="">2024b</a>); Shi et al. (<a class="ltx_ref" href="https://arxiv.org/html/2503.02398v1#bib.bib40" title="">2024b</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.02398v1#bib.bib41" title="">c</a>); Borgeaud et al. (<a class="ltx_ref" href="https://arxiv.org/html/2503.02398v1#bib.bib4" title="">2022</a>); Lewis et al. (<a class="ltx_ref" href="https://arxiv.org/html/2503.02398v1#bib.bib16" title="">2020</a>)</cite> causes LLMs to frequently overlook the middle context information, degrading preference understanding; (3) <span class="ltx_text ltx_font_bold" id="S1.p3.1.3">Online efficiency demands</span> – Current modeling approaches are typically tightly coupled with online inference. However, the time-consuming nature of long sequences leads to significant inefficiencies and high latency.</p> </div> <figure class="ltx_figure" id="S1.F1"><img alt="Refer to caption" class="ltx_graphics ltx_centering ltx_img_landscape" height="249" id="S1.F1.g1" src="x1.png" width="415"/> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure">Figure 1: </span>Visualization of different sampling strategies. <span class="ltx_text ltx_font_bold" id="S1.F1.3.1">A</span> illustrates the behavior distribution, clustering results, and selected/unselected samples. <span class="ltx_text ltx_font_bold" id="S1.F1.4.2">B</span> presents the allocation of selection budgets across clusters at a 50% selection ratio.</figcaption> </figure> <div class="ltx_para" id="S1.p4"> <p class="ltx_p" id="S1.p4.1">Sampling techniques help handle long user behavior sequences by processing Sub-Behavior Sequences (SBS) in LLM-UM approaches. <span class="ltx_text ltx_font_bold" id="S1.p4.1.1">Recent sampling</span> methods <cite class="ltx_cite ltx_citemacro_cite">Hou et al. (<a class="ltx_ref" href="https://arxiv.org/html/2503.02398v1#bib.bib12" title="">2024</a>); Zhang et al. (<a class="ltx_ref" href="https://arxiv.org/html/2503.02398v1#bib.bib57" title="">2024b</a>)</cite> truncate sequences temporally, focusing on short-term interests while overlooking long-term preferences, leading to incomplete user profiling and reduced recommendation quality. <span class="ltx_text ltx_font_bold" id="S1.p4.1.2">Relevance sampling</span> <cite class="ltx_cite ltx_citemacro_cite">Salemi et al. (<a class="ltx_ref" href="https://arxiv.org/html/2503.02398v1#bib.bib35" title="">2024</a>); Zhang et al. (<a class="ltx_ref" href="https://arxiv.org/html/2503.02398v1#bib.bib56" title="">2024a</a>); Zhou et al. (<a class="ltx_ref" href="https://arxiv.org/html/2503.02398v1#bib.bib62" title="">2024</a>)</cite> performs well but introduces online latency, as it selects relevant behaviors and models user personas before prompting agent to recommend.</p> </div> <div class="ltx_para" id="S1.p5"> <p class="ltx_p" id="S1.p5.1">These limitations reveal a fundamental tension: existing sampling strategies either sacrifice behavioral completeness for efficiency or compromise responsiveness for accuracy. How can we develop a novel frameworks that simultaneously makes LLM-UM methods achieve efficient processing of long behavior sequences and robust construction of comprehensive user personas? This challenge forms the core motivation for our work.</p> </div> <div class="ltx_para" id="S1.p6"> <p class="ltx_p" id="S1.p6.1">To address this challenge, we introduce PersonaX, a novel LLM-UM framework for user modeling from long behavior sequences. PersonaX extracts representative short sub-sequences (SBS) from the full interaction history, each capturing distinct aspects of user interests. These SBS are processed offline to construct fine-grained, multi-dimensional user personas, which are then integrated into downstream recommendation systems during online inference. PersonaX prioritizes sampling quality over quantity, using a small fraction of data to generate compact yet informative SBS (length < 5). This approach reduces the model’s focus on irrelevant or noisy samples, overcoming common issues such as input length constraints and mid-content oversight. Unlike relevance-based methods, PersonaX operates entirely offline, eliminating online inference latency. Furthermore, it offers a persistent user representation, negating the need for frequent updates, unlike recent-based sampling methods.</p> </div> <div class="ltx_para" id="S1.p7"> <p class="ltx_p" id="S1.p7.1">In summary, our contributions are threefold. (1) <span class="ltx_text ltx_font_bold" id="S1.p7.1.1">Core Behavior Selection</span>: We propose an innovative strategy for selecting SBS that balances prototypicality and diversity. This method produces compact, high-quality SBS while only 30–50% of the data utility ratio. (2) <span class="ltx_text ltx_font_bold" id="S1.p7.1.2">PersonaX Framework</span>: PersonaX is a cutting-edge framework designed for long behavior sequences, enhancing the performance and inference efficiency of existing agent recommendation methods. (3) <span class="ltx_text ltx_font_bold" id="S1.p7.1.3">Extensive Validation</span>: PersonaX is validated on two LLM-UM methods: Reflection and Summarization methods, across four long-sequence datasets. The results show significant improvements in ranking accuracy (3–11% or 10-50%) and online inference efficiency (50% reduction) for recommendation agents like AgentCF and Agent4Rec.</p> </div> </section> <section class="ltx_section" id="S2"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">2 </span>Preliminary</h2> <section class="ltx_subsection" id="S2.SS1"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">2.1 </span>User Modeling</h3> <div class="ltx_para" id="S2.SS1.p1"> <p class="ltx_p" id="S2.SS1.p1.10">Let <math alttext="\mathcal{S}=\{(I_{1},L_{1}),(I_{2},L_{2}),\dots,(I_{n},L_{n})\}" class="ltx_Math" display="inline" id="S2.SS1.p1.1.m1.4"><semantics id="S2.SS1.p1.1.m1.4a"><mrow id="S2.SS1.p1.1.m1.4.4" xref="S2.SS1.p1.1.m1.4.4.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p1.1.m1.4.4.5" xref="S2.SS1.p1.1.m1.4.4.5.cmml">𝒮</mi><mo id="S2.SS1.p1.1.m1.4.4.4" xref="S2.SS1.p1.1.m1.4.4.4.cmml">=</mo><mrow 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id="S2.SS1.p1.1.m1.4.4.3.3.3.2.2.3" xref="S2.SS1.p1.1.m1.4.4.3.3.3.2.2.3.cmml">n</mi></msub><mo id="S2.SS1.p1.1.m1.4.4.3.3.3.2.5" stretchy="false" xref="S2.SS1.p1.1.m1.4.4.3.3.3.3.cmml">)</mo></mrow><mo id="S2.SS1.p1.1.m1.4.4.3.3.8" stretchy="false" xref="S2.SS1.p1.1.m1.4.4.3.4.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p1.1.m1.4b"><apply id="S2.SS1.p1.1.m1.4.4.cmml" xref="S2.SS1.p1.1.m1.4.4"><eq id="S2.SS1.p1.1.m1.4.4.4.cmml" xref="S2.SS1.p1.1.m1.4.4.4"></eq><ci id="S2.SS1.p1.1.m1.4.4.5.cmml" xref="S2.SS1.p1.1.m1.4.4.5">𝒮</ci><set id="S2.SS1.p1.1.m1.4.4.3.4.cmml" xref="S2.SS1.p1.1.m1.4.4.3.3"><interval closure="open" id="S2.SS1.p1.1.m1.2.2.1.1.1.3.cmml" xref="S2.SS1.p1.1.m1.2.2.1.1.1.2"><apply id="S2.SS1.p1.1.m1.2.2.1.1.1.1.1.cmml" xref="S2.SS1.p1.1.m1.2.2.1.1.1.1.1"><csymbol cd="ambiguous" id="S2.SS1.p1.1.m1.2.2.1.1.1.1.1.1.cmml" xref="S2.SS1.p1.1.m1.2.2.1.1.1.1.1">subscript</csymbol><ci id="S2.SS1.p1.1.m1.2.2.1.1.1.1.1.2.cmml" xref="S2.SS1.p1.1.m1.2.2.1.1.1.1.1.2">𝐼</ci><cn id="S2.SS1.p1.1.m1.2.2.1.1.1.1.1.3.cmml" type="integer" xref="S2.SS1.p1.1.m1.2.2.1.1.1.1.1.3">1</cn></apply><apply id="S2.SS1.p1.1.m1.2.2.1.1.1.2.2.cmml" xref="S2.SS1.p1.1.m1.2.2.1.1.1.2.2"><csymbol cd="ambiguous" id="S2.SS1.p1.1.m1.2.2.1.1.1.2.2.1.cmml" xref="S2.SS1.p1.1.m1.2.2.1.1.1.2.2">subscript</csymbol><ci id="S2.SS1.p1.1.m1.2.2.1.1.1.2.2.2.cmml" xref="S2.SS1.p1.1.m1.2.2.1.1.1.2.2.2">𝐿</ci><cn id="S2.SS1.p1.1.m1.2.2.1.1.1.2.2.3.cmml" type="integer" xref="S2.SS1.p1.1.m1.2.2.1.1.1.2.2.3">1</cn></apply></interval><interval closure="open" id="S2.SS1.p1.1.m1.3.3.2.2.2.3.cmml" xref="S2.SS1.p1.1.m1.3.3.2.2.2.2"><apply id="S2.SS1.p1.1.m1.3.3.2.2.2.1.1.cmml" xref="S2.SS1.p1.1.m1.3.3.2.2.2.1.1"><csymbol cd="ambiguous" id="S2.SS1.p1.1.m1.3.3.2.2.2.1.1.1.cmml" xref="S2.SS1.p1.1.m1.3.3.2.2.2.1.1">subscript</csymbol><ci id="S2.SS1.p1.1.m1.3.3.2.2.2.1.1.2.cmml" xref="S2.SS1.p1.1.m1.3.3.2.2.2.1.1.2">𝐼</ci><cn id="S2.SS1.p1.1.m1.3.3.2.2.2.1.1.3.cmml" type="integer" xref="S2.SS1.p1.1.m1.3.3.2.2.2.1.1.3">2</cn></apply><apply id="S2.SS1.p1.1.m1.3.3.2.2.2.2.2.cmml" xref="S2.SS1.p1.1.m1.3.3.2.2.2.2.2"><csymbol cd="ambiguous" id="S2.SS1.p1.1.m1.3.3.2.2.2.2.2.1.cmml" xref="S2.SS1.p1.1.m1.3.3.2.2.2.2.2">subscript</csymbol><ci id="S2.SS1.p1.1.m1.3.3.2.2.2.2.2.2.cmml" xref="S2.SS1.p1.1.m1.3.3.2.2.2.2.2.2">𝐿</ci><cn id="S2.SS1.p1.1.m1.3.3.2.2.2.2.2.3.cmml" type="integer" xref="S2.SS1.p1.1.m1.3.3.2.2.2.2.2.3">2</cn></apply></interval><ci id="S2.SS1.p1.1.m1.1.1.cmml" xref="S2.SS1.p1.1.m1.1.1">…</ci><interval closure="open" id="S2.SS1.p1.1.m1.4.4.3.3.3.3.cmml" xref="S2.SS1.p1.1.m1.4.4.3.3.3.2"><apply id="S2.SS1.p1.1.m1.4.4.3.3.3.1.1.cmml" xref="S2.SS1.p1.1.m1.4.4.3.3.3.1.1"><csymbol cd="ambiguous" id="S2.SS1.p1.1.m1.4.4.3.3.3.1.1.1.cmml" xref="S2.SS1.p1.1.m1.4.4.3.3.3.1.1">subscript</csymbol><ci id="S2.SS1.p1.1.m1.4.4.3.3.3.1.1.2.cmml" xref="S2.SS1.p1.1.m1.4.4.3.3.3.1.1.2">𝐼</ci><ci id="S2.SS1.p1.1.m1.4.4.3.3.3.1.1.3.cmml" xref="S2.SS1.p1.1.m1.4.4.3.3.3.1.1.3">𝑛</ci></apply><apply id="S2.SS1.p1.1.m1.4.4.3.3.3.2.2.cmml" xref="S2.SS1.p1.1.m1.4.4.3.3.3.2.2"><csymbol cd="ambiguous" id="S2.SS1.p1.1.m1.4.4.3.3.3.2.2.1.cmml" xref="S2.SS1.p1.1.m1.4.4.3.3.3.2.2">subscript</csymbol><ci id="S2.SS1.p1.1.m1.4.4.3.3.3.2.2.2.cmml" xref="S2.SS1.p1.1.m1.4.4.3.3.3.2.2.2">𝐿</ci><ci id="S2.SS1.p1.1.m1.4.4.3.3.3.2.2.3.cmml" xref="S2.SS1.p1.1.m1.4.4.3.3.3.2.2.3">𝑛</ci></apply></interval></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p1.1.m1.4c">\mathcal{S}=\{(I_{1},L_{1}),(I_{2},L_{2}),\dots,(I_{n},L_{n})\}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p1.1.m1.4d">caligraphic_S = { ( italic_I start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_L start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) , ( italic_I start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , italic_L start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) , … , ( italic_I start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT , italic_L start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ) }</annotation></semantics></math> denotes a user’s historical behavior sequence of length <math alttext="n" class="ltx_Math" display="inline" id="S2.SS1.p1.2.m2.1"><semantics id="S2.SS1.p1.2.m2.1a"><mi id="S2.SS1.p1.2.m2.1.1" xref="S2.SS1.p1.2.m2.1.1.cmml">n</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.p1.2.m2.1b"><ci id="S2.SS1.p1.2.m2.1.1.cmml" xref="S2.SS1.p1.2.m2.1.1">𝑛</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p1.2.m2.1c">n</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p1.2.m2.1d">italic_n</annotation></semantics></math>, where <math alttext="I_{i}" class="ltx_Math" display="inline" id="S2.SS1.p1.3.m3.1"><semantics id="S2.SS1.p1.3.m3.1a"><msub id="S2.SS1.p1.3.m3.1.1" xref="S2.SS1.p1.3.m3.1.1.cmml"><mi id="S2.SS1.p1.3.m3.1.1.2" xref="S2.SS1.p1.3.m3.1.1.2.cmml">I</mi><mi id="S2.SS1.p1.3.m3.1.1.3" xref="S2.SS1.p1.3.m3.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S2.SS1.p1.3.m3.1b"><apply id="S2.SS1.p1.3.m3.1.1.cmml" xref="S2.SS1.p1.3.m3.1.1"><csymbol cd="ambiguous" id="S2.SS1.p1.3.m3.1.1.1.cmml" xref="S2.SS1.p1.3.m3.1.1">subscript</csymbol><ci id="S2.SS1.p1.3.m3.1.1.2.cmml" xref="S2.SS1.p1.3.m3.1.1.2">𝐼</ci><ci id="S2.SS1.p1.3.m3.1.1.3.cmml" xref="S2.SS1.p1.3.m3.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p1.3.m3.1c">I_{i}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p1.3.m3.1d">italic_I start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> represents the <math alttext="i" class="ltx_Math" display="inline" id="S2.SS1.p1.4.m4.1"><semantics id="S2.SS1.p1.4.m4.1a"><mi id="S2.SS1.p1.4.m4.1.1" xref="S2.SS1.p1.4.m4.1.1.cmml">i</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.p1.4.m4.1b"><ci id="S2.SS1.p1.4.m4.1.1.cmml" xref="S2.SS1.p1.4.m4.1.1">𝑖</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p1.4.m4.1c">i</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p1.4.m4.1d">italic_i</annotation></semantics></math>-th interacted item and <math alttext="L_{i}\in\{0,1\}" class="ltx_Math" display="inline" id="S2.SS1.p1.5.m5.2"><semantics id="S2.SS1.p1.5.m5.2a"><mrow id="S2.SS1.p1.5.m5.2.3" xref="S2.SS1.p1.5.m5.2.3.cmml"><msub id="S2.SS1.p1.5.m5.2.3.2" xref="S2.SS1.p1.5.m5.2.3.2.cmml"><mi id="S2.SS1.p1.5.m5.2.3.2.2" xref="S2.SS1.p1.5.m5.2.3.2.2.cmml">L</mi><mi id="S2.SS1.p1.5.m5.2.3.2.3" xref="S2.SS1.p1.5.m5.2.3.2.3.cmml">i</mi></msub><mo id="S2.SS1.p1.5.m5.2.3.1" xref="S2.SS1.p1.5.m5.2.3.1.cmml">∈</mo><mrow id="S2.SS1.p1.5.m5.2.3.3.2" xref="S2.SS1.p1.5.m5.2.3.3.1.cmml"><mo id="S2.SS1.p1.5.m5.2.3.3.2.1" stretchy="false" xref="S2.SS1.p1.5.m5.2.3.3.1.cmml">{</mo><mn id="S2.SS1.p1.5.m5.1.1" xref="S2.SS1.p1.5.m5.1.1.cmml">0</mn><mo id="S2.SS1.p1.5.m5.2.3.3.2.2" xref="S2.SS1.p1.5.m5.2.3.3.1.cmml">,</mo><mn id="S2.SS1.p1.5.m5.2.2" xref="S2.SS1.p1.5.m5.2.2.cmml">1</mn><mo id="S2.SS1.p1.5.m5.2.3.3.2.3" stretchy="false" xref="S2.SS1.p1.5.m5.2.3.3.1.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p1.5.m5.2b"><apply id="S2.SS1.p1.5.m5.2.3.cmml" xref="S2.SS1.p1.5.m5.2.3"><in id="S2.SS1.p1.5.m5.2.3.1.cmml" xref="S2.SS1.p1.5.m5.2.3.1"></in><apply id="S2.SS1.p1.5.m5.2.3.2.cmml" xref="S2.SS1.p1.5.m5.2.3.2"><csymbol cd="ambiguous" id="S2.SS1.p1.5.m5.2.3.2.1.cmml" xref="S2.SS1.p1.5.m5.2.3.2">subscript</csymbol><ci id="S2.SS1.p1.5.m5.2.3.2.2.cmml" xref="S2.SS1.p1.5.m5.2.3.2.2">𝐿</ci><ci id="S2.SS1.p1.5.m5.2.3.2.3.cmml" xref="S2.SS1.p1.5.m5.2.3.2.3">𝑖</ci></apply><set id="S2.SS1.p1.5.m5.2.3.3.1.cmml" xref="S2.SS1.p1.5.m5.2.3.3.2"><cn id="S2.SS1.p1.5.m5.1.1.cmml" type="integer" xref="S2.SS1.p1.5.m5.1.1">0</cn><cn id="S2.SS1.p1.5.m5.2.2.cmml" type="integer" xref="S2.SS1.p1.5.m5.2.2">1</cn></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p1.5.m5.2c">L_{i}\in\{0,1\}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p1.5.m5.2d">italic_L start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∈ { 0 , 1 }</annotation></semantics></math> indicates the corresponding interaction label (<math alttext="0" class="ltx_Math" display="inline" id="S2.SS1.p1.6.m6.1"><semantics id="S2.SS1.p1.6.m6.1a"><mn id="S2.SS1.p1.6.m6.1.1" xref="S2.SS1.p1.6.m6.1.1.cmml">0</mn><annotation-xml encoding="MathML-Content" id="S2.SS1.p1.6.m6.1b"><cn id="S2.SS1.p1.6.m6.1.1.cmml" type="integer" xref="S2.SS1.p1.6.m6.1.1">0</cn></annotation-xml></semantics></math> for dislike and <math alttext="1" class="ltx_Math" display="inline" id="S2.SS1.p1.7.m7.1"><semantics id="S2.SS1.p1.7.m7.1a"><mn id="S2.SS1.p1.7.m7.1.1" xref="S2.SS1.p1.7.m7.1.1.cmml">1</mn><annotation-xml encoding="MathML-Content" id="S2.SS1.p1.7.m7.1b"><cn id="S2.SS1.p1.7.m7.1.1.cmml" type="integer" xref="S2.SS1.p1.7.m7.1.1">1</cn></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p1.7.m7.1c">1</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p1.7.m7.1d">1</annotation></semantics></math> for like). We define the task of user modeling is to construct a precise and representative user persona <math alttext="\mathcal{P}(\mathcal{S})" class="ltx_Math" display="inline" id="S2.SS1.p1.8.m8.1"><semantics id="S2.SS1.p1.8.m8.1a"><mrow id="S2.SS1.p1.8.m8.1.2" xref="S2.SS1.p1.8.m8.1.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p1.8.m8.1.2.2" xref="S2.SS1.p1.8.m8.1.2.2.cmml">𝒫</mi><mo id="S2.SS1.p1.8.m8.1.2.1" xref="S2.SS1.p1.8.m8.1.2.1.cmml">⁢</mo><mrow id="S2.SS1.p1.8.m8.1.2.3.2" xref="S2.SS1.p1.8.m8.1.2.cmml"><mo id="S2.SS1.p1.8.m8.1.2.3.2.1" stretchy="false" xref="S2.SS1.p1.8.m8.1.2.cmml">(</mo><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p1.8.m8.1.1" xref="S2.SS1.p1.8.m8.1.1.cmml">𝒮</mi><mo id="S2.SS1.p1.8.m8.1.2.3.2.2" stretchy="false" xref="S2.SS1.p1.8.m8.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p1.8.m8.1b"><apply id="S2.SS1.p1.8.m8.1.2.cmml" xref="S2.SS1.p1.8.m8.1.2"><times id="S2.SS1.p1.8.m8.1.2.1.cmml" xref="S2.SS1.p1.8.m8.1.2.1"></times><ci id="S2.SS1.p1.8.m8.1.2.2.cmml" xref="S2.SS1.p1.8.m8.1.2.2">𝒫</ci><ci id="S2.SS1.p1.8.m8.1.1.cmml" xref="S2.SS1.p1.8.m8.1.1">𝒮</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p1.8.m8.1c">\mathcal{P}(\mathcal{S})</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p1.8.m8.1d">caligraphic_P ( caligraphic_S )</annotation></semantics></math> by leveraging the historical behavioral data <math alttext="\mathcal{S}" class="ltx_Math" display="inline" id="S2.SS1.p1.9.m9.1"><semantics id="S2.SS1.p1.9.m9.1a"><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p1.9.m9.1.1" xref="S2.SS1.p1.9.m9.1.1.cmml">𝒮</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.p1.9.m9.1b"><ci id="S2.SS1.p1.9.m9.1.1.cmml" xref="S2.SS1.p1.9.m9.1.1">𝒮</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p1.9.m9.1c">\mathcal{S}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p1.9.m9.1d">caligraphic_S</annotation></semantics></math>, where <math alttext="\mathcal{P}(\cdot)" class="ltx_Math" display="inline" id="S2.SS1.p1.10.m10.1"><semantics id="S2.SS1.p1.10.m10.1a"><mrow id="S2.SS1.p1.10.m10.1.2" xref="S2.SS1.p1.10.m10.1.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p1.10.m10.1.2.2" xref="S2.SS1.p1.10.m10.1.2.2.cmml">𝒫</mi><mo id="S2.SS1.p1.10.m10.1.2.1" xref="S2.SS1.p1.10.m10.1.2.1.cmml">⁢</mo><mrow id="S2.SS1.p1.10.m10.1.2.3.2" xref="S2.SS1.p1.10.m10.1.2.cmml"><mo id="S2.SS1.p1.10.m10.1.2.3.2.1" stretchy="false" xref="S2.SS1.p1.10.m10.1.2.cmml">(</mo><mo id="S2.SS1.p1.10.m10.1.1" lspace="0em" rspace="0em" xref="S2.SS1.p1.10.m10.1.1.cmml">⋅</mo><mo id="S2.SS1.p1.10.m10.1.2.3.2.2" stretchy="false" xref="S2.SS1.p1.10.m10.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p1.10.m10.1b"><apply id="S2.SS1.p1.10.m10.1.2.cmml" xref="S2.SS1.p1.10.m10.1.2"><times id="S2.SS1.p1.10.m10.1.2.1.cmml" xref="S2.SS1.p1.10.m10.1.2.1"></times><ci id="S2.SS1.p1.10.m10.1.2.2.cmml" xref="S2.SS1.p1.10.m10.1.2.2">𝒫</ci><ci id="S2.SS1.p1.10.m10.1.1.cmml" xref="S2.SS1.p1.10.m10.1.1">⋅</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p1.10.m10.1c">\mathcal{P}(\cdot)</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p1.10.m10.1d">caligraphic_P ( ⋅ )</annotation></semantics></math> is a user modeling method (e.g., Summarization and Reflection). The learned user persona should capture the implicit preference patterns underlying interactions, enabling augmentation for downstream instructional agent recommendation.</p> </div> </section> <section class="ltx_subsection" id="S2.SS2"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">2.2 </span>Sub-Behavior Sequence (SBS) Selection.</h3> <div class="ltx_para" id="S2.SS2.p1"> <p class="ltx_p" id="S2.SS2.p1.7">To tackle the challenge of LLM-UM struggling with analyzing long UGC, sampling methods are often employed on the full historical sequence <math alttext="\mathcal{S}" class="ltx_Math" display="inline" id="S2.SS2.p1.1.m1.1"><semantics id="S2.SS2.p1.1.m1.1a"><mi class="ltx_font_mathcaligraphic" id="S2.SS2.p1.1.m1.1.1" xref="S2.SS2.p1.1.m1.1.1.cmml">𝒮</mi><annotation-xml encoding="MathML-Content" id="S2.SS2.p1.1.m1.1b"><ci id="S2.SS2.p1.1.m1.1.1.cmml" xref="S2.SS2.p1.1.m1.1.1">𝒮</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p1.1.m1.1c">\mathcal{S}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p1.1.m1.1d">caligraphic_S</annotation></semantics></math>. These methods aim to extract a Sub-Behavior Sequence (SBS) that retains the most essential information necessary for accurate user profiling while significantly reducing sequence length. 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id="S2.SS2.p1.2.m2.4.4.3.3.3.3.cmml" xref="S2.SS2.p1.2.m2.4.4.3.3.3.3">𝑘</ci></apply></set></apply><apply id="S2.SS2.p1.2.m2.4.4c.cmml" xref="S2.SS2.p1.2.m2.4.4"><subset id="S2.SS2.p1.2.m2.4.4.7.cmml" xref="S2.SS2.p1.2.m2.4.4.7"></subset><share href="https://arxiv.org/html/2503.02398v1#S2.SS2.p1.2.m2.4.4.3.cmml" id="S2.SS2.p1.2.m2.4.4d.cmml" xref="S2.SS2.p1.2.m2.4.4"></share><ci id="S2.SS2.p1.2.m2.4.4.8.cmml" xref="S2.SS2.p1.2.m2.4.4.8">𝒮</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p1.2.m2.4c">\mathcal{S}^{*}=\{\hat{I}_{1},\hat{I}_{2},\dots,\hat{I}_{k}\}\subseteq\mathcal% {S}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p1.2.m2.4d">caligraphic_S start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT = { over^ start_ARG italic_I end_ARG start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , over^ start_ARG italic_I end_ARG start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , … , over^ start_ARG italic_I end_ARG start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT } ⊆ caligraphic_S</annotation></semantics></math> denote the SBS of length <math alttext="k" class="ltx_Math" display="inline" id="S2.SS2.p1.3.m3.1"><semantics id="S2.SS2.p1.3.m3.1a"><mi id="S2.SS2.p1.3.m3.1.1" xref="S2.SS2.p1.3.m3.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="S2.SS2.p1.3.m3.1b"><ci id="S2.SS2.p1.3.m3.1.1.cmml" xref="S2.SS2.p1.3.m3.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p1.3.m3.1c">k</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p1.3.m3.1d">italic_k</annotation></semantics></math> (<math alttext="k\ll n" class="ltx_Math" display="inline" id="S2.SS2.p1.4.m4.1"><semantics id="S2.SS2.p1.4.m4.1a"><mrow id="S2.SS2.p1.4.m4.1.1" xref="S2.SS2.p1.4.m4.1.1.cmml"><mi id="S2.SS2.p1.4.m4.1.1.2" xref="S2.SS2.p1.4.m4.1.1.2.cmml">k</mi><mo id="S2.SS2.p1.4.m4.1.1.1" xref="S2.SS2.p1.4.m4.1.1.1.cmml">≪</mo><mi id="S2.SS2.p1.4.m4.1.1.3" xref="S2.SS2.p1.4.m4.1.1.3.cmml">n</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p1.4.m4.1b"><apply id="S2.SS2.p1.4.m4.1.1.cmml" xref="S2.SS2.p1.4.m4.1.1"><csymbol cd="latexml" id="S2.SS2.p1.4.m4.1.1.1.cmml" xref="S2.SS2.p1.4.m4.1.1.1">much-less-than</csymbol><ci id="S2.SS2.p1.4.m4.1.1.2.cmml" xref="S2.SS2.p1.4.m4.1.1.2">𝑘</ci><ci id="S2.SS2.p1.4.m4.1.1.3.cmml" xref="S2.SS2.p1.4.m4.1.1.3">𝑛</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p1.4.m4.1c">k\ll n</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p1.4.m4.1d">italic_k ≪ italic_n</annotation></semantics></math>), where <math alttext="\hat{I}_{i}" class="ltx_Math" display="inline" id="S2.SS2.p1.5.m5.1"><semantics id="S2.SS2.p1.5.m5.1a"><msub id="S2.SS2.p1.5.m5.1.1" xref="S2.SS2.p1.5.m5.1.1.cmml"><mover accent="true" id="S2.SS2.p1.5.m5.1.1.2" xref="S2.SS2.p1.5.m5.1.1.2.cmml"><mi id="S2.SS2.p1.5.m5.1.1.2.2" xref="S2.SS2.p1.5.m5.1.1.2.2.cmml">I</mi><mo id="S2.SS2.p1.5.m5.1.1.2.1" xref="S2.SS2.p1.5.m5.1.1.2.1.cmml">^</mo></mover><mi id="S2.SS2.p1.5.m5.1.1.3" xref="S2.SS2.p1.5.m5.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S2.SS2.p1.5.m5.1b"><apply id="S2.SS2.p1.5.m5.1.1.cmml" xref="S2.SS2.p1.5.m5.1.1"><csymbol cd="ambiguous" id="S2.SS2.p1.5.m5.1.1.1.cmml" xref="S2.SS2.p1.5.m5.1.1">subscript</csymbol><apply id="S2.SS2.p1.5.m5.1.1.2.cmml" xref="S2.SS2.p1.5.m5.1.1.2"><ci id="S2.SS2.p1.5.m5.1.1.2.1.cmml" xref="S2.SS2.p1.5.m5.1.1.2.1">^</ci><ci id="S2.SS2.p1.5.m5.1.1.2.2.cmml" xref="S2.SS2.p1.5.m5.1.1.2.2">𝐼</ci></apply><ci id="S2.SS2.p1.5.m5.1.1.3.cmml" xref="S2.SS2.p1.5.m5.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p1.5.m5.1c">\hat{I}_{i}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p1.5.m5.1d">over^ start_ARG italic_I end_ARG start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> represents the <math alttext="i" class="ltx_Math" display="inline" id="S2.SS2.p1.6.m6.1"><semantics id="S2.SS2.p1.6.m6.1a"><mi id="S2.SS2.p1.6.m6.1.1" xref="S2.SS2.p1.6.m6.1.1.cmml">i</mi><annotation-xml encoding="MathML-Content" id="S2.SS2.p1.6.m6.1b"><ci id="S2.SS2.p1.6.m6.1.1.cmml" xref="S2.SS2.p1.6.m6.1.1">𝑖</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p1.6.m6.1c">i</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p1.6.m6.1d">italic_i</annotation></semantics></math>-th selected behavior. The selection ratio, <math alttext="\frac{k}{n}" class="ltx_Math" display="inline" id="S2.SS2.p1.7.m7.1"><semantics id="S2.SS2.p1.7.m7.1a"><mfrac id="S2.SS2.p1.7.m7.1.1" xref="S2.SS2.p1.7.m7.1.1.cmml"><mi id="S2.SS2.p1.7.m7.1.1.2" xref="S2.SS2.p1.7.m7.1.1.2.cmml">k</mi><mi id="S2.SS2.p1.7.m7.1.1.3" xref="S2.SS2.p1.7.m7.1.1.3.cmml">n</mi></mfrac><annotation-xml encoding="MathML-Content" id="S2.SS2.p1.7.m7.1b"><apply id="S2.SS2.p1.7.m7.1.1.cmml" xref="S2.SS2.p1.7.m7.1.1"><divide id="S2.SS2.p1.7.m7.1.1.1.cmml" xref="S2.SS2.p1.7.m7.1.1"></divide><ci id="S2.SS2.p1.7.m7.1.1.2.cmml" xref="S2.SS2.p1.7.m7.1.1.2">𝑘</ci><ci id="S2.SS2.p1.7.m7.1.1.3.cmml" xref="S2.SS2.p1.7.m7.1.1.3">𝑛</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p1.7.m7.1c">\frac{k}{n}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p1.7.m7.1d">divide start_ARG italic_k end_ARG start_ARG italic_n end_ARG</annotation></semantics></math>, quantifies the compression achieved.</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>Method</h2> <section class="ltx_subsection" id="S3.SS1"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">3.1 </span>Behavior Clustering</h3> <div class="ltx_para" id="S3.SS1.p1"> <p class="ltx_p" id="S3.SS1.p1.6">We employ hierarchical clustering to group items based on user interest similarity, treating each cluster as a cohesive analysis unit. A language embedding model <math alttext="\mathbf{E}(\cdot)" class="ltx_Math" display="inline" id="S3.SS1.p1.1.m1.1"><semantics id="S3.SS1.p1.1.m1.1a"><mrow id="S3.SS1.p1.1.m1.1.2" xref="S3.SS1.p1.1.m1.1.2.cmml"><mi id="S3.SS1.p1.1.m1.1.2.2" xref="S3.SS1.p1.1.m1.1.2.2.cmml">𝐄</mi><mo id="S3.SS1.p1.1.m1.1.2.1" xref="S3.SS1.p1.1.m1.1.2.1.cmml">⁢</mo><mrow id="S3.SS1.p1.1.m1.1.2.3.2" xref="S3.SS1.p1.1.m1.1.2.cmml"><mo id="S3.SS1.p1.1.m1.1.2.3.2.1" stretchy="false" xref="S3.SS1.p1.1.m1.1.2.cmml">(</mo><mo id="S3.SS1.p1.1.m1.1.1" lspace="0em" rspace="0em" xref="S3.SS1.p1.1.m1.1.1.cmml">⋅</mo><mo id="S3.SS1.p1.1.m1.1.2.3.2.2" stretchy="false" xref="S3.SS1.p1.1.m1.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p1.1.m1.1b"><apply id="S3.SS1.p1.1.m1.1.2.cmml" xref="S3.SS1.p1.1.m1.1.2"><times id="S3.SS1.p1.1.m1.1.2.1.cmml" xref="S3.SS1.p1.1.m1.1.2.1"></times><ci id="S3.SS1.p1.1.m1.1.2.2.cmml" xref="S3.SS1.p1.1.m1.1.2.2">𝐄</ci><ci id="S3.SS1.p1.1.m1.1.1.cmml" xref="S3.SS1.p1.1.m1.1.1">⋅</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p1.1.m1.1c">\mathbf{E}(\cdot)</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p1.1.m1.1d">bold_E ( ⋅ )</annotation></semantics></math>, such as BGE Embedding <cite class="ltx_cite ltx_citemacro_cite">Chen et al. (<a class="ltx_ref" href="https://arxiv.org/html/2503.02398v1#bib.bib6" title="">2024</a>)</cite> or EasyRec <cite class="ltx_cite ltx_citemacro_cite">Ren and Huang (<a class="ltx_ref" href="https://arxiv.org/html/2503.02398v1#bib.bib33" title="">2024</a>)</cite>, encodes each item <math alttext="I_{i}" class="ltx_Math" display="inline" id="S3.SS1.p1.2.m2.1"><semantics id="S3.SS1.p1.2.m2.1a"><msub id="S3.SS1.p1.2.m2.1.1" xref="S3.SS1.p1.2.m2.1.1.cmml"><mi id="S3.SS1.p1.2.m2.1.1.2" xref="S3.SS1.p1.2.m2.1.1.2.cmml">I</mi><mi id="S3.SS1.p1.2.m2.1.1.3" xref="S3.SS1.p1.2.m2.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS1.p1.2.m2.1b"><apply id="S3.SS1.p1.2.m2.1.1.cmml" xref="S3.SS1.p1.2.m2.1.1"><csymbol cd="ambiguous" id="S3.SS1.p1.2.m2.1.1.1.cmml" xref="S3.SS1.p1.2.m2.1.1">subscript</csymbol><ci id="S3.SS1.p1.2.m2.1.1.2.cmml" xref="S3.SS1.p1.2.m2.1.1.2">𝐼</ci><ci id="S3.SS1.p1.2.m2.1.1.3.cmml" xref="S3.SS1.p1.2.m2.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p1.2.m2.1c">I_{i}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p1.2.m2.1d">italic_I start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> into a dense vector <math alttext="\mathbf{e}_{i}" class="ltx_Math" display="inline" id="S3.SS1.p1.3.m3.1"><semantics id="S3.SS1.p1.3.m3.1a"><msub id="S3.SS1.p1.3.m3.1.1" xref="S3.SS1.p1.3.m3.1.1.cmml"><mi id="S3.SS1.p1.3.m3.1.1.2" xref="S3.SS1.p1.3.m3.1.1.2.cmml">𝐞</mi><mi id="S3.SS1.p1.3.m3.1.1.3" xref="S3.SS1.p1.3.m3.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS1.p1.3.m3.1b"><apply id="S3.SS1.p1.3.m3.1.1.cmml" xref="S3.SS1.p1.3.m3.1.1"><csymbol cd="ambiguous" id="S3.SS1.p1.3.m3.1.1.1.cmml" xref="S3.SS1.p1.3.m3.1.1">subscript</csymbol><ci id="S3.SS1.p1.3.m3.1.1.2.cmml" xref="S3.SS1.p1.3.m3.1.1.2">𝐞</ci><ci id="S3.SS1.p1.3.m3.1.1.3.cmml" xref="S3.SS1.p1.3.m3.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p1.3.m3.1c">\mathbf{e}_{i}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p1.3.m3.1d">bold_e start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math>. Let <math alttext="\mathcal{E}=\{\mathbf{e}_{1},\mathbf{e}_{2},\dots,\mathbf{e}_{n}\}" class="ltx_Math" display="inline" id="S3.SS1.p1.4.m4.4"><semantics id="S3.SS1.p1.4.m4.4a"><mrow id="S3.SS1.p1.4.m4.4.4" xref="S3.SS1.p1.4.m4.4.4.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.SS1.p1.4.m4.4.4.5" xref="S3.SS1.p1.4.m4.4.4.5.cmml">ℰ</mi><mo id="S3.SS1.p1.4.m4.4.4.4" xref="S3.SS1.p1.4.m4.4.4.4.cmml">=</mo><mrow id="S3.SS1.p1.4.m4.4.4.3.3" xref="S3.SS1.p1.4.m4.4.4.3.4.cmml"><mo id="S3.SS1.p1.4.m4.4.4.3.3.4" stretchy="false" xref="S3.SS1.p1.4.m4.4.4.3.4.cmml">{</mo><msub id="S3.SS1.p1.4.m4.2.2.1.1.1" xref="S3.SS1.p1.4.m4.2.2.1.1.1.cmml"><mi id="S3.SS1.p1.4.m4.2.2.1.1.1.2" xref="S3.SS1.p1.4.m4.2.2.1.1.1.2.cmml">𝐞</mi><mn id="S3.SS1.p1.4.m4.2.2.1.1.1.3" xref="S3.SS1.p1.4.m4.2.2.1.1.1.3.cmml">1</mn></msub><mo id="S3.SS1.p1.4.m4.4.4.3.3.5" xref="S3.SS1.p1.4.m4.4.4.3.4.cmml">,</mo><msub id="S3.SS1.p1.4.m4.3.3.2.2.2" xref="S3.SS1.p1.4.m4.3.3.2.2.2.cmml"><mi id="S3.SS1.p1.4.m4.3.3.2.2.2.2" xref="S3.SS1.p1.4.m4.3.3.2.2.2.2.cmml">𝐞</mi><mn id="S3.SS1.p1.4.m4.3.3.2.2.2.3" xref="S3.SS1.p1.4.m4.3.3.2.2.2.3.cmml">2</mn></msub><mo id="S3.SS1.p1.4.m4.4.4.3.3.6" xref="S3.SS1.p1.4.m4.4.4.3.4.cmml">,</mo><mi id="S3.SS1.p1.4.m4.1.1" mathvariant="normal" xref="S3.SS1.p1.4.m4.1.1.cmml">…</mi><mo id="S3.SS1.p1.4.m4.4.4.3.3.7" xref="S3.SS1.p1.4.m4.4.4.3.4.cmml">,</mo><msub id="S3.SS1.p1.4.m4.4.4.3.3.3" xref="S3.SS1.p1.4.m4.4.4.3.3.3.cmml"><mi id="S3.SS1.p1.4.m4.4.4.3.3.3.2" xref="S3.SS1.p1.4.m4.4.4.3.3.3.2.cmml">𝐞</mi><mi id="S3.SS1.p1.4.m4.4.4.3.3.3.3" xref="S3.SS1.p1.4.m4.4.4.3.3.3.3.cmml">n</mi></msub><mo id="S3.SS1.p1.4.m4.4.4.3.3.8" stretchy="false" xref="S3.SS1.p1.4.m4.4.4.3.4.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p1.4.m4.4b"><apply id="S3.SS1.p1.4.m4.4.4.cmml" xref="S3.SS1.p1.4.m4.4.4"><eq id="S3.SS1.p1.4.m4.4.4.4.cmml" xref="S3.SS1.p1.4.m4.4.4.4"></eq><ci id="S3.SS1.p1.4.m4.4.4.5.cmml" xref="S3.SS1.p1.4.m4.4.4.5">ℰ</ci><set id="S3.SS1.p1.4.m4.4.4.3.4.cmml" xref="S3.SS1.p1.4.m4.4.4.3.3"><apply id="S3.SS1.p1.4.m4.2.2.1.1.1.cmml" xref="S3.SS1.p1.4.m4.2.2.1.1.1"><csymbol cd="ambiguous" id="S3.SS1.p1.4.m4.2.2.1.1.1.1.cmml" xref="S3.SS1.p1.4.m4.2.2.1.1.1">subscript</csymbol><ci id="S3.SS1.p1.4.m4.2.2.1.1.1.2.cmml" xref="S3.SS1.p1.4.m4.2.2.1.1.1.2">𝐞</ci><cn id="S3.SS1.p1.4.m4.2.2.1.1.1.3.cmml" type="integer" xref="S3.SS1.p1.4.m4.2.2.1.1.1.3">1</cn></apply><apply id="S3.SS1.p1.4.m4.3.3.2.2.2.cmml" xref="S3.SS1.p1.4.m4.3.3.2.2.2"><csymbol cd="ambiguous" id="S3.SS1.p1.4.m4.3.3.2.2.2.1.cmml" xref="S3.SS1.p1.4.m4.3.3.2.2.2">subscript</csymbol><ci id="S3.SS1.p1.4.m4.3.3.2.2.2.2.cmml" xref="S3.SS1.p1.4.m4.3.3.2.2.2.2">𝐞</ci><cn id="S3.SS1.p1.4.m4.3.3.2.2.2.3.cmml" type="integer" xref="S3.SS1.p1.4.m4.3.3.2.2.2.3">2</cn></apply><ci id="S3.SS1.p1.4.m4.1.1.cmml" xref="S3.SS1.p1.4.m4.1.1">…</ci><apply id="S3.SS1.p1.4.m4.4.4.3.3.3.cmml" xref="S3.SS1.p1.4.m4.4.4.3.3.3"><csymbol cd="ambiguous" id="S3.SS1.p1.4.m4.4.4.3.3.3.1.cmml" xref="S3.SS1.p1.4.m4.4.4.3.3.3">subscript</csymbol><ci id="S3.SS1.p1.4.m4.4.4.3.3.3.2.cmml" xref="S3.SS1.p1.4.m4.4.4.3.3.3.2">𝐞</ci><ci id="S3.SS1.p1.4.m4.4.4.3.3.3.3.cmml" xref="S3.SS1.p1.4.m4.4.4.3.3.3.3">𝑛</ci></apply></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p1.4.m4.4c">\mathcal{E}=\{\mathbf{e}_{1},\mathbf{e}_{2},\dots,\mathbf{e}_{n}\}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p1.4.m4.4d">caligraphic_E = { bold_e start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , bold_e start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , … , bold_e start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT }</annotation></semantics></math> represent the item embeddings from the user’s interaction history. Pairwise similarity is measured via Euclidean distance: <math alttext="d(\mathbf{e}_{i},\mathbf{e}_{j})=\|\mathbf{e}_{i}-\mathbf{e}_{j}\|_{2}" class="ltx_Math" display="inline" id="S3.SS1.p1.5.m5.3"><semantics id="S3.SS1.p1.5.m5.3a"><mrow id="S3.SS1.p1.5.m5.3.3" xref="S3.SS1.p1.5.m5.3.3.cmml"><mrow id="S3.SS1.p1.5.m5.2.2.2" xref="S3.SS1.p1.5.m5.2.2.2.cmml"><mi id="S3.SS1.p1.5.m5.2.2.2.4" xref="S3.SS1.p1.5.m5.2.2.2.4.cmml">d</mi><mo id="S3.SS1.p1.5.m5.2.2.2.3" xref="S3.SS1.p1.5.m5.2.2.2.3.cmml">⁢</mo><mrow id="S3.SS1.p1.5.m5.2.2.2.2.2" xref="S3.SS1.p1.5.m5.2.2.2.2.3.cmml"><mo id="S3.SS1.p1.5.m5.2.2.2.2.2.3" stretchy="false" xref="S3.SS1.p1.5.m5.2.2.2.2.3.cmml">(</mo><msub id="S3.SS1.p1.5.m5.1.1.1.1.1.1" xref="S3.SS1.p1.5.m5.1.1.1.1.1.1.cmml"><mi id="S3.SS1.p1.5.m5.1.1.1.1.1.1.2" xref="S3.SS1.p1.5.m5.1.1.1.1.1.1.2.cmml">𝐞</mi><mi id="S3.SS1.p1.5.m5.1.1.1.1.1.1.3" xref="S3.SS1.p1.5.m5.1.1.1.1.1.1.3.cmml">i</mi></msub><mo id="S3.SS1.p1.5.m5.2.2.2.2.2.4" xref="S3.SS1.p1.5.m5.2.2.2.2.3.cmml">,</mo><msub id="S3.SS1.p1.5.m5.2.2.2.2.2.2" xref="S3.SS1.p1.5.m5.2.2.2.2.2.2.cmml"><mi id="S3.SS1.p1.5.m5.2.2.2.2.2.2.2" xref="S3.SS1.p1.5.m5.2.2.2.2.2.2.2.cmml">𝐞</mi><mi id="S3.SS1.p1.5.m5.2.2.2.2.2.2.3" xref="S3.SS1.p1.5.m5.2.2.2.2.2.2.3.cmml">j</mi></msub><mo id="S3.SS1.p1.5.m5.2.2.2.2.2.5" stretchy="false" xref="S3.SS1.p1.5.m5.2.2.2.2.3.cmml">)</mo></mrow></mrow><mo id="S3.SS1.p1.5.m5.3.3.4" xref="S3.SS1.p1.5.m5.3.3.4.cmml">=</mo><msub id="S3.SS1.p1.5.m5.3.3.3" xref="S3.SS1.p1.5.m5.3.3.3.cmml"><mrow id="S3.SS1.p1.5.m5.3.3.3.1.1" xref="S3.SS1.p1.5.m5.3.3.3.1.2.cmml"><mo id="S3.SS1.p1.5.m5.3.3.3.1.1.2" stretchy="false" xref="S3.SS1.p1.5.m5.3.3.3.1.2.1.cmml">‖</mo><mrow id="S3.SS1.p1.5.m5.3.3.3.1.1.1" xref="S3.SS1.p1.5.m5.3.3.3.1.1.1.cmml"><msub id="S3.SS1.p1.5.m5.3.3.3.1.1.1.2" xref="S3.SS1.p1.5.m5.3.3.3.1.1.1.2.cmml"><mi id="S3.SS1.p1.5.m5.3.3.3.1.1.1.2.2" xref="S3.SS1.p1.5.m5.3.3.3.1.1.1.2.2.cmml">𝐞</mi><mi id="S3.SS1.p1.5.m5.3.3.3.1.1.1.2.3" xref="S3.SS1.p1.5.m5.3.3.3.1.1.1.2.3.cmml">i</mi></msub><mo id="S3.SS1.p1.5.m5.3.3.3.1.1.1.1" xref="S3.SS1.p1.5.m5.3.3.3.1.1.1.1.cmml">−</mo><msub id="S3.SS1.p1.5.m5.3.3.3.1.1.1.3" xref="S3.SS1.p1.5.m5.3.3.3.1.1.1.3.cmml"><mi id="S3.SS1.p1.5.m5.3.3.3.1.1.1.3.2" xref="S3.SS1.p1.5.m5.3.3.3.1.1.1.3.2.cmml">𝐞</mi><mi id="S3.SS1.p1.5.m5.3.3.3.1.1.1.3.3" xref="S3.SS1.p1.5.m5.3.3.3.1.1.1.3.3.cmml">j</mi></msub></mrow><mo id="S3.SS1.p1.5.m5.3.3.3.1.1.3" stretchy="false" xref="S3.SS1.p1.5.m5.3.3.3.1.2.1.cmml">‖</mo></mrow><mn id="S3.SS1.p1.5.m5.3.3.3.3" xref="S3.SS1.p1.5.m5.3.3.3.3.cmml">2</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p1.5.m5.3b"><apply id="S3.SS1.p1.5.m5.3.3.cmml" xref="S3.SS1.p1.5.m5.3.3"><eq id="S3.SS1.p1.5.m5.3.3.4.cmml" xref="S3.SS1.p1.5.m5.3.3.4"></eq><apply id="S3.SS1.p1.5.m5.2.2.2.cmml" xref="S3.SS1.p1.5.m5.2.2.2"><times id="S3.SS1.p1.5.m5.2.2.2.3.cmml" xref="S3.SS1.p1.5.m5.2.2.2.3"></times><ci id="S3.SS1.p1.5.m5.2.2.2.4.cmml" xref="S3.SS1.p1.5.m5.2.2.2.4">𝑑</ci><interval closure="open" id="S3.SS1.p1.5.m5.2.2.2.2.3.cmml" xref="S3.SS1.p1.5.m5.2.2.2.2.2"><apply id="S3.SS1.p1.5.m5.1.1.1.1.1.1.cmml" xref="S3.SS1.p1.5.m5.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S3.SS1.p1.5.m5.1.1.1.1.1.1.1.cmml" xref="S3.SS1.p1.5.m5.1.1.1.1.1.1">subscript</csymbol><ci id="S3.SS1.p1.5.m5.1.1.1.1.1.1.2.cmml" xref="S3.SS1.p1.5.m5.1.1.1.1.1.1.2">𝐞</ci><ci id="S3.SS1.p1.5.m5.1.1.1.1.1.1.3.cmml" xref="S3.SS1.p1.5.m5.1.1.1.1.1.1.3">𝑖</ci></apply><apply id="S3.SS1.p1.5.m5.2.2.2.2.2.2.cmml" xref="S3.SS1.p1.5.m5.2.2.2.2.2.2"><csymbol cd="ambiguous" id="S3.SS1.p1.5.m5.2.2.2.2.2.2.1.cmml" xref="S3.SS1.p1.5.m5.2.2.2.2.2.2">subscript</csymbol><ci id="S3.SS1.p1.5.m5.2.2.2.2.2.2.2.cmml" xref="S3.SS1.p1.5.m5.2.2.2.2.2.2.2">𝐞</ci><ci id="S3.SS1.p1.5.m5.2.2.2.2.2.2.3.cmml" xref="S3.SS1.p1.5.m5.2.2.2.2.2.2.3">𝑗</ci></apply></interval></apply><apply id="S3.SS1.p1.5.m5.3.3.3.cmml" xref="S3.SS1.p1.5.m5.3.3.3"><csymbol cd="ambiguous" id="S3.SS1.p1.5.m5.3.3.3.2.cmml" xref="S3.SS1.p1.5.m5.3.3.3">subscript</csymbol><apply id="S3.SS1.p1.5.m5.3.3.3.1.2.cmml" xref="S3.SS1.p1.5.m5.3.3.3.1.1"><csymbol cd="latexml" id="S3.SS1.p1.5.m5.3.3.3.1.2.1.cmml" xref="S3.SS1.p1.5.m5.3.3.3.1.1.2">norm</csymbol><apply id="S3.SS1.p1.5.m5.3.3.3.1.1.1.cmml" xref="S3.SS1.p1.5.m5.3.3.3.1.1.1"><minus id="S3.SS1.p1.5.m5.3.3.3.1.1.1.1.cmml" xref="S3.SS1.p1.5.m5.3.3.3.1.1.1.1"></minus><apply id="S3.SS1.p1.5.m5.3.3.3.1.1.1.2.cmml" xref="S3.SS1.p1.5.m5.3.3.3.1.1.1.2"><csymbol cd="ambiguous" id="S3.SS1.p1.5.m5.3.3.3.1.1.1.2.1.cmml" xref="S3.SS1.p1.5.m5.3.3.3.1.1.1.2">subscript</csymbol><ci id="S3.SS1.p1.5.m5.3.3.3.1.1.1.2.2.cmml" xref="S3.SS1.p1.5.m5.3.3.3.1.1.1.2.2">𝐞</ci><ci id="S3.SS1.p1.5.m5.3.3.3.1.1.1.2.3.cmml" xref="S3.SS1.p1.5.m5.3.3.3.1.1.1.2.3">𝑖</ci></apply><apply id="S3.SS1.p1.5.m5.3.3.3.1.1.1.3.cmml" xref="S3.SS1.p1.5.m5.3.3.3.1.1.1.3"><csymbol cd="ambiguous" id="S3.SS1.p1.5.m5.3.3.3.1.1.1.3.1.cmml" xref="S3.SS1.p1.5.m5.3.3.3.1.1.1.3">subscript</csymbol><ci id="S3.SS1.p1.5.m5.3.3.3.1.1.1.3.2.cmml" xref="S3.SS1.p1.5.m5.3.3.3.1.1.1.3.2">𝐞</ci><ci id="S3.SS1.p1.5.m5.3.3.3.1.1.1.3.3.cmml" xref="S3.SS1.p1.5.m5.3.3.3.1.1.1.3.3">𝑗</ci></apply></apply></apply><cn id="S3.SS1.p1.5.m5.3.3.3.3.cmml" type="integer" xref="S3.SS1.p1.5.m5.3.3.3.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p1.5.m5.3c">d(\mathbf{e}_{i},\mathbf{e}_{j})=\|\mathbf{e}_{i}-\mathbf{e}_{j}\|_{2}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p1.5.m5.3d">italic_d ( bold_e start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , bold_e start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ) = ∥ bold_e start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT - bold_e start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ∥ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math>, denoted as <math alttext="d_{i,j}" class="ltx_Math" display="inline" id="S3.SS1.p1.6.m6.2"><semantics id="S3.SS1.p1.6.m6.2a"><msub id="S3.SS1.p1.6.m6.2.3" xref="S3.SS1.p1.6.m6.2.3.cmml"><mi id="S3.SS1.p1.6.m6.2.3.2" xref="S3.SS1.p1.6.m6.2.3.2.cmml">d</mi><mrow id="S3.SS1.p1.6.m6.2.2.2.4" xref="S3.SS1.p1.6.m6.2.2.2.3.cmml"><mi id="S3.SS1.p1.6.m6.1.1.1.1" xref="S3.SS1.p1.6.m6.1.1.1.1.cmml">i</mi><mo id="S3.SS1.p1.6.m6.2.2.2.4.1" xref="S3.SS1.p1.6.m6.2.2.2.3.cmml">,</mo><mi id="S3.SS1.p1.6.m6.2.2.2.2" xref="S3.SS1.p1.6.m6.2.2.2.2.cmml">j</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="S3.SS1.p1.6.m6.2b"><apply id="S3.SS1.p1.6.m6.2.3.cmml" xref="S3.SS1.p1.6.m6.2.3"><csymbol cd="ambiguous" id="S3.SS1.p1.6.m6.2.3.1.cmml" xref="S3.SS1.p1.6.m6.2.3">subscript</csymbol><ci id="S3.SS1.p1.6.m6.2.3.2.cmml" xref="S3.SS1.p1.6.m6.2.3.2">𝑑</ci><list id="S3.SS1.p1.6.m6.2.2.2.3.cmml" xref="S3.SS1.p1.6.m6.2.2.2.4"><ci id="S3.SS1.p1.6.m6.1.1.1.1.cmml" xref="S3.SS1.p1.6.m6.1.1.1.1">𝑖</ci><ci id="S3.SS1.p1.6.m6.2.2.2.2.cmml" xref="S3.SS1.p1.6.m6.2.2.2.2">𝑗</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p1.6.m6.2c">d_{i,j}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p1.6.m6.2d">italic_d start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S3.SS1.p2"> <p class="ltx_p" id="S3.SS1.p2.5">Clustering is controlled by a distance threshold <math alttext="\tau" class="ltx_Math" display="inline" id="S3.SS1.p2.1.m1.1"><semantics id="S3.SS1.p2.1.m1.1a"><mi id="S3.SS1.p2.1.m1.1.1" xref="S3.SS1.p2.1.m1.1.1.cmml">τ</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.p2.1.m1.1b"><ci id="S3.SS1.p2.1.m1.1.1.cmml" xref="S3.SS1.p2.1.m1.1.1">𝜏</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p2.1.m1.1c">\tau</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p2.1.m1.1d">italic_τ</annotation></semantics></math>, which restricts the maximum intra-cluster distance while preventing merges between clusters with inter-cluster distances below <math alttext="\tau" class="ltx_Math" display="inline" id="S3.SS1.p2.2.m2.1"><semantics id="S3.SS1.p2.2.m2.1a"><mi id="S3.SS1.p2.2.m2.1.1" xref="S3.SS1.p2.2.m2.1.1.cmml">τ</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.p2.2.m2.1b"><ci id="S3.SS1.p2.2.m2.1.1.cmml" xref="S3.SS1.p2.2.m2.1.1">𝜏</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p2.2.m2.1c">\tau</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p2.2.m2.1d">italic_τ</annotation></semantics></math>. The resulting clusters <math alttext="\mathcal{C}=\{c_{1},c_{2},\dots,c_{m}\}" class="ltx_Math" display="inline" id="S3.SS1.p2.3.m3.4"><semantics id="S3.SS1.p2.3.m3.4a"><mrow id="S3.SS1.p2.3.m3.4.4" xref="S3.SS1.p2.3.m3.4.4.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.SS1.p2.3.m3.4.4.5" xref="S3.SS1.p2.3.m3.4.4.5.cmml">𝒞</mi><mo id="S3.SS1.p2.3.m3.4.4.4" xref="S3.SS1.p2.3.m3.4.4.4.cmml">=</mo><mrow id="S3.SS1.p2.3.m3.4.4.3.3" xref="S3.SS1.p2.3.m3.4.4.3.4.cmml"><mo id="S3.SS1.p2.3.m3.4.4.3.3.4" stretchy="false" xref="S3.SS1.p2.3.m3.4.4.3.4.cmml">{</mo><msub id="S3.SS1.p2.3.m3.2.2.1.1.1" xref="S3.SS1.p2.3.m3.2.2.1.1.1.cmml"><mi id="S3.SS1.p2.3.m3.2.2.1.1.1.2" xref="S3.SS1.p2.3.m3.2.2.1.1.1.2.cmml">c</mi><mn id="S3.SS1.p2.3.m3.2.2.1.1.1.3" xref="S3.SS1.p2.3.m3.2.2.1.1.1.3.cmml">1</mn></msub><mo id="S3.SS1.p2.3.m3.4.4.3.3.5" xref="S3.SS1.p2.3.m3.4.4.3.4.cmml">,</mo><msub id="S3.SS1.p2.3.m3.3.3.2.2.2" xref="S3.SS1.p2.3.m3.3.3.2.2.2.cmml"><mi id="S3.SS1.p2.3.m3.3.3.2.2.2.2" xref="S3.SS1.p2.3.m3.3.3.2.2.2.2.cmml">c</mi><mn id="S3.SS1.p2.3.m3.3.3.2.2.2.3" xref="S3.SS1.p2.3.m3.3.3.2.2.2.3.cmml">2</mn></msub><mo id="S3.SS1.p2.3.m3.4.4.3.3.6" xref="S3.SS1.p2.3.m3.4.4.3.4.cmml">,</mo><mi id="S3.SS1.p2.3.m3.1.1" mathvariant="normal" xref="S3.SS1.p2.3.m3.1.1.cmml">…</mi><mo id="S3.SS1.p2.3.m3.4.4.3.3.7" xref="S3.SS1.p2.3.m3.4.4.3.4.cmml">,</mo><msub id="S3.SS1.p2.3.m3.4.4.3.3.3" xref="S3.SS1.p2.3.m3.4.4.3.3.3.cmml"><mi id="S3.SS1.p2.3.m3.4.4.3.3.3.2" xref="S3.SS1.p2.3.m3.4.4.3.3.3.2.cmml">c</mi><mi id="S3.SS1.p2.3.m3.4.4.3.3.3.3" xref="S3.SS1.p2.3.m3.4.4.3.3.3.3.cmml">m</mi></msub><mo id="S3.SS1.p2.3.m3.4.4.3.3.8" stretchy="false" xref="S3.SS1.p2.3.m3.4.4.3.4.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p2.3.m3.4b"><apply id="S3.SS1.p2.3.m3.4.4.cmml" xref="S3.SS1.p2.3.m3.4.4"><eq id="S3.SS1.p2.3.m3.4.4.4.cmml" xref="S3.SS1.p2.3.m3.4.4.4"></eq><ci id="S3.SS1.p2.3.m3.4.4.5.cmml" xref="S3.SS1.p2.3.m3.4.4.5">𝒞</ci><set id="S3.SS1.p2.3.m3.4.4.3.4.cmml" xref="S3.SS1.p2.3.m3.4.4.3.3"><apply id="S3.SS1.p2.3.m3.2.2.1.1.1.cmml" xref="S3.SS1.p2.3.m3.2.2.1.1.1"><csymbol cd="ambiguous" id="S3.SS1.p2.3.m3.2.2.1.1.1.1.cmml" xref="S3.SS1.p2.3.m3.2.2.1.1.1">subscript</csymbol><ci id="S3.SS1.p2.3.m3.2.2.1.1.1.2.cmml" xref="S3.SS1.p2.3.m3.2.2.1.1.1.2">𝑐</ci><cn id="S3.SS1.p2.3.m3.2.2.1.1.1.3.cmml" type="integer" xref="S3.SS1.p2.3.m3.2.2.1.1.1.3">1</cn></apply><apply id="S3.SS1.p2.3.m3.3.3.2.2.2.cmml" xref="S3.SS1.p2.3.m3.3.3.2.2.2"><csymbol cd="ambiguous" id="S3.SS1.p2.3.m3.3.3.2.2.2.1.cmml" xref="S3.SS1.p2.3.m3.3.3.2.2.2">subscript</csymbol><ci id="S3.SS1.p2.3.m3.3.3.2.2.2.2.cmml" xref="S3.SS1.p2.3.m3.3.3.2.2.2.2">𝑐</ci><cn id="S3.SS1.p2.3.m3.3.3.2.2.2.3.cmml" type="integer" xref="S3.SS1.p2.3.m3.3.3.2.2.2.3">2</cn></apply><ci id="S3.SS1.p2.3.m3.1.1.cmml" xref="S3.SS1.p2.3.m3.1.1">…</ci><apply id="S3.SS1.p2.3.m3.4.4.3.3.3.cmml" xref="S3.SS1.p2.3.m3.4.4.3.3.3"><csymbol cd="ambiguous" id="S3.SS1.p2.3.m3.4.4.3.3.3.1.cmml" xref="S3.SS1.p2.3.m3.4.4.3.3.3">subscript</csymbol><ci id="S3.SS1.p2.3.m3.4.4.3.3.3.2.cmml" xref="S3.SS1.p2.3.m3.4.4.3.3.3.2">𝑐</ci><ci id="S3.SS1.p2.3.m3.4.4.3.3.3.3.cmml" xref="S3.SS1.p2.3.m3.4.4.3.3.3.3">𝑚</ci></apply></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p2.3.m3.4c">\mathcal{C}=\{c_{1},c_{2},\dots,c_{m}\}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p2.3.m3.4d">caligraphic_C = { italic_c start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_c start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , … , italic_c start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT }</annotation></semantics></math> satisfy Intra-cluster constraint: <math alttext="\forall c\in\mathcal{C},\forall I_{i},I_{j}\in c,d_{i,j}<\tau" class="ltx_Math" display="inline" id="S3.SS1.p2.4.m4.5"><semantics id="S3.SS1.p2.4.m4.5a"><mrow id="S3.SS1.p2.4.m4.5.5.2" xref="S3.SS1.p2.4.m4.5.5.3.cmml"><mrow id="S3.SS1.p2.4.m4.4.4.1.1" xref="S3.SS1.p2.4.m4.4.4.1.1.cmml"><mrow id="S3.SS1.p2.4.m4.4.4.1.1.3" xref="S3.SS1.p2.4.m4.4.4.1.1.3.cmml"><mo id="S3.SS1.p2.4.m4.4.4.1.1.3.1" rspace="0.167em" xref="S3.SS1.p2.4.m4.4.4.1.1.3.1.cmml">∀</mo><mi id="S3.SS1.p2.4.m4.4.4.1.1.3.2" xref="S3.SS1.p2.4.m4.4.4.1.1.3.2.cmml">c</mi></mrow><mo id="S3.SS1.p2.4.m4.4.4.1.1.2" xref="S3.SS1.p2.4.m4.4.4.1.1.2.cmml">∈</mo><mrow id="S3.SS1.p2.4.m4.4.4.1.1.1.1" xref="S3.SS1.p2.4.m4.4.4.1.1.1.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.SS1.p2.4.m4.3.3" xref="S3.SS1.p2.4.m4.3.3.cmml">𝒞</mi><mo id="S3.SS1.p2.4.m4.4.4.1.1.1.1.2" xref="S3.SS1.p2.4.m4.4.4.1.1.1.2.cmml">,</mo><mrow id="S3.SS1.p2.4.m4.4.4.1.1.1.1.1" xref="S3.SS1.p2.4.m4.4.4.1.1.1.1.1.cmml"><mo id="S3.SS1.p2.4.m4.4.4.1.1.1.1.1.1" rspace="0.167em" xref="S3.SS1.p2.4.m4.4.4.1.1.1.1.1.1.cmml">∀</mo><msub id="S3.SS1.p2.4.m4.4.4.1.1.1.1.1.2" xref="S3.SS1.p2.4.m4.4.4.1.1.1.1.1.2.cmml"><mi id="S3.SS1.p2.4.m4.4.4.1.1.1.1.1.2.2" xref="S3.SS1.p2.4.m4.4.4.1.1.1.1.1.2.2.cmml">I</mi><mi 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id="S3.SS1.p2.4.m4.5.5.2.2.2.2.cmml" xref="S3.SS1.p2.4.m4.5.5.2.2.2.2"><lt id="S3.SS1.p2.4.m4.5.5.2.2.2.2.1.cmml" xref="S3.SS1.p2.4.m4.5.5.2.2.2.2.1"></lt><apply id="S3.SS1.p2.4.m4.5.5.2.2.2.2.2.cmml" xref="S3.SS1.p2.4.m4.5.5.2.2.2.2.2"><csymbol cd="ambiguous" id="S3.SS1.p2.4.m4.5.5.2.2.2.2.2.1.cmml" xref="S3.SS1.p2.4.m4.5.5.2.2.2.2.2">subscript</csymbol><ci id="S3.SS1.p2.4.m4.5.5.2.2.2.2.2.2.cmml" xref="S3.SS1.p2.4.m4.5.5.2.2.2.2.2.2">𝑑</ci><list id="S3.SS1.p2.4.m4.2.2.2.3.cmml" xref="S3.SS1.p2.4.m4.2.2.2.4"><ci id="S3.SS1.p2.4.m4.1.1.1.1.cmml" xref="S3.SS1.p2.4.m4.1.1.1.1">𝑖</ci><ci id="S3.SS1.p2.4.m4.2.2.2.2.cmml" xref="S3.SS1.p2.4.m4.2.2.2.2">𝑗</ci></list></apply><ci id="S3.SS1.p2.4.m4.5.5.2.2.2.2.3.cmml" xref="S3.SS1.p2.4.m4.5.5.2.2.2.2.3">𝜏</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p2.4.m4.5c">\forall c\in\mathcal{C},\forall I_{i},I_{j}\in c,d_{i,j}<\tau</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p2.4.m4.5d">∀ italic_c ∈ caligraphic_C , ∀ italic_I start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , italic_I start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ∈ italic_c , italic_d start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT < italic_τ</annotation></semantics></math> and Inter-cluster constraint: <math alttext="\forall c_{p},c_{q}\in\mathcal{C},c_{p}\neq c_{q},d(c_{p},c_{q})\geq\tau" class="ltx_Math" display="inline" id="S3.SS1.p2.5.m5.2"><semantics id="S3.SS1.p2.5.m5.2a"><mrow id="S3.SS1.p2.5.m5.2.2.2" xref="S3.SS1.p2.5.m5.2.2.3.cmml"><mrow id="S3.SS1.p2.5.m5.1.1.1.1" xref="S3.SS1.p2.5.m5.1.1.1.1.cmml"><mrow id="S3.SS1.p2.5.m5.1.1.1.1.2.2" xref="S3.SS1.p2.5.m5.1.1.1.1.2.3.cmml"><mrow id="S3.SS1.p2.5.m5.1.1.1.1.1.1.1" xref="S3.SS1.p2.5.m5.1.1.1.1.1.1.1.cmml"><mo id="S3.SS1.p2.5.m5.1.1.1.1.1.1.1.1" rspace="0.167em" xref="S3.SS1.p2.5.m5.1.1.1.1.1.1.1.1.cmml">∀</mo><msub id="S3.SS1.p2.5.m5.1.1.1.1.1.1.1.2" xref="S3.SS1.p2.5.m5.1.1.1.1.1.1.1.2.cmml"><mi 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xref="S3.SS1.p2.5.m5.2.2.2.2.2.2.2.3.cmml">⁢</mo><mrow id="S3.SS1.p2.5.m5.2.2.2.2.2.2.2.2.2" xref="S3.SS1.p2.5.m5.2.2.2.2.2.2.2.2.3.cmml"><mo id="S3.SS1.p2.5.m5.2.2.2.2.2.2.2.2.2.3" stretchy="false" xref="S3.SS1.p2.5.m5.2.2.2.2.2.2.2.2.3.cmml">(</mo><msub id="S3.SS1.p2.5.m5.2.2.2.2.2.2.1.1.1.1" xref="S3.SS1.p2.5.m5.2.2.2.2.2.2.1.1.1.1.cmml"><mi id="S3.SS1.p2.5.m5.2.2.2.2.2.2.1.1.1.1.2" xref="S3.SS1.p2.5.m5.2.2.2.2.2.2.1.1.1.1.2.cmml">c</mi><mi id="S3.SS1.p2.5.m5.2.2.2.2.2.2.1.1.1.1.3" xref="S3.SS1.p2.5.m5.2.2.2.2.2.2.1.1.1.1.3.cmml">p</mi></msub><mo id="S3.SS1.p2.5.m5.2.2.2.2.2.2.2.2.2.4" xref="S3.SS1.p2.5.m5.2.2.2.2.2.2.2.2.3.cmml">,</mo><msub id="S3.SS1.p2.5.m5.2.2.2.2.2.2.2.2.2.2" xref="S3.SS1.p2.5.m5.2.2.2.2.2.2.2.2.2.2.cmml"><mi id="S3.SS1.p2.5.m5.2.2.2.2.2.2.2.2.2.2.2" xref="S3.SS1.p2.5.m5.2.2.2.2.2.2.2.2.2.2.2.cmml">c</mi><mi id="S3.SS1.p2.5.m5.2.2.2.2.2.2.2.2.2.2.3" xref="S3.SS1.p2.5.m5.2.2.2.2.2.2.2.2.2.2.3.cmml">q</mi></msub><mo id="S3.SS1.p2.5.m5.2.2.2.2.2.2.2.2.2.5" 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start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT ≠ italic_c start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT , italic_d ( italic_c start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT , italic_c start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ) ≥ italic_τ</annotation></semantics></math>.</p> </div> </section> <section class="ltx_subsection" id="S3.SS2"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">3.2 </span>Sampling Budget Allocation</h3> <div class="ltx_para" id="S3.SS2.p1"> <p class="ltx_p" id="S3.SS2.p1.1">Given a finite budget <math alttext="k" class="ltx_Math" display="inline" id="S3.SS2.p1.1.m1.1"><semantics id="S3.SS2.p1.1.m1.1a"><mi id="S3.SS2.p1.1.m1.1.1" xref="S3.SS2.p1.1.m1.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="S3.SS2.p1.1.m1.1b"><ci id="S3.SS2.p1.1.m1.1.1.cmml" xref="S3.SS2.p1.1.m1.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p1.1.m1.1c">k</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p1.1.m1.1d">italic_k</annotation></semantics></math> for sampling historical behaviors, we propose a Sampling Budget Allocation Strategy to distribute this budget across clusters. The algorithm dynamically adjusts allocation based on cluster size distribution, ensuring that smaller clusters are given sufficient attention while preventing larger clusters from dominating the selection process. This promotes a balanced distribution of selected samples, preserving the diversity of sampled behaviors and maintaining a representative coverage of the data <cite class="ltx_cite ltx_citemacro_cite">Zheng et al. (<a class="ltx_ref" href="https://arxiv.org/html/2503.02398v1#bib.bib61" title="">2023</a>)</cite>.</p> </div> <div class="ltx_para" id="S3.SS2.p2"> <p class="ltx_p" id="S3.SS2.p2.4">The strategy first sorts clusters by size in ascending order. Each cluster is initially assigned an average allocation <math alttext="q" class="ltx_Math" display="inline" id="S3.SS2.p2.1.m1.1"><semantics id="S3.SS2.p2.1.m1.1a"><mi id="S3.SS2.p2.1.m1.1.1" xref="S3.SS2.p2.1.m1.1.1.cmml">q</mi><annotation-xml encoding="MathML-Content" id="S3.SS2.p2.1.m1.1b"><ci id="S3.SS2.p2.1.m1.1.1.cmml" xref="S3.SS2.p2.1.m1.1.1">𝑞</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p2.1.m1.1c">q</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p2.1.m1.1d">italic_q</annotation></semantics></math>. If a cluster’s size is smaller than <math alttext="q" class="ltx_Math" display="inline" id="S3.SS2.p2.2.m2.1"><semantics id="S3.SS2.p2.2.m2.1a"><mi id="S3.SS2.p2.2.m2.1.1" xref="S3.SS2.p2.2.m2.1.1.cmml">q</mi><annotation-xml encoding="MathML-Content" id="S3.SS2.p2.2.m2.1b"><ci id="S3.SS2.p2.2.m2.1.1.cmml" xref="S3.SS2.p2.2.m2.1.1">𝑞</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p2.2.m2.1c">q</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p2.2.m2.1d">italic_q</annotation></semantics></math>, it receives its exact size, and <math alttext="q" class="ltx_Math" display="inline" id="S3.SS2.p2.3.m3.1"><semantics id="S3.SS2.p2.3.m3.1a"><mi id="S3.SS2.p2.3.m3.1.1" xref="S3.SS2.p2.3.m3.1.1.cmml">q</mi><annotation-xml encoding="MathML-Content" id="S3.SS2.p2.3.m3.1b"><ci id="S3.SS2.p2.3.m3.1.1.cmml" xref="S3.SS2.p2.3.m3.1.1">𝑞</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p2.3.m3.1c">q</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p2.3.m3.1d">italic_q</annotation></semantics></math> is recalculated based on the remaining quota. Otherwise, the cluster is allocated <math alttext="q" class="ltx_Math" display="inline" id="S3.SS2.p2.4.m4.1"><semantics id="S3.SS2.p2.4.m4.1a"><mi id="S3.SS2.p2.4.m4.1.1" xref="S3.SS2.p2.4.m4.1.1.cmml">q</mi><annotation-xml encoding="MathML-Content" id="S3.SS2.p2.4.m4.1b"><ci id="S3.SS2.p2.4.m4.1.1.cmml" xref="S3.SS2.p2.4.m4.1.1">𝑞</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p2.4.m4.1c">q</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p2.4.m4.1d">italic_q</annotation></semantics></math>. This process repeats iteratively until the entire budget is assigned. Algorithm <a class="ltx_ref" href="https://arxiv.org/html/2503.02398v1#alg1" title="Algorithm 1 ‣ 3.2 Sampling Budget Allocation ‣ 3 Method ‣ PersonaX: A Recommendation Agent-Oriented User Modeling Framework for Long Behavior Sequence"><span class="ltx_text ltx_ref_tag">1</span></a> details the method, and Figure <a class="ltx_ref" href="https://arxiv.org/html/2503.02398v1#S1.F1" title="Figure 1 ‣ 1 Introduction ‣ PersonaX: A Recommendation Agent-Oriented User Modeling Framework for Long Behavior Sequence"><span class="ltx_text ltx_ref_tag">1</span></a>.B illustrates an example, where smaller clusters are fully allocated first, and the remaining budget is evenly distributed among larger clusters.</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> Sampling Budget Allocation</figcaption> <div 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xref="alg1.l1.m2.4.4.3.4.cmml">{</mo><msub id="alg1.l1.m2.2.2.1.1.1" xref="alg1.l1.m2.2.2.1.1.1.cmml"><mi id="alg1.l1.m2.2.2.1.1.1.2" xref="alg1.l1.m2.2.2.1.1.1.2.cmml">s</mi><mn id="alg1.l1.m2.2.2.1.1.1.3" xref="alg1.l1.m2.2.2.1.1.1.3.cmml">1</mn></msub><mo id="alg1.l1.m2.4.4.3.3.5" xref="alg1.l1.m2.4.4.3.4.cmml">,</mo><msub id="alg1.l1.m2.3.3.2.2.2" xref="alg1.l1.m2.3.3.2.2.2.cmml"><mi id="alg1.l1.m2.3.3.2.2.2.2" xref="alg1.l1.m2.3.3.2.2.2.2.cmml">s</mi><mn id="alg1.l1.m2.3.3.2.2.2.3" xref="alg1.l1.m2.3.3.2.2.2.3.cmml">2</mn></msub><mo id="alg1.l1.m2.4.4.3.3.6" xref="alg1.l1.m2.4.4.3.4.cmml">,</mo><mi id="alg1.l1.m2.1.1" mathvariant="normal" xref="alg1.l1.m2.1.1.cmml">…</mi><mo id="alg1.l1.m2.4.4.3.3.7" xref="alg1.l1.m2.4.4.3.4.cmml">,</mo><msub id="alg1.l1.m2.4.4.3.3.3" xref="alg1.l1.m2.4.4.3.3.3.cmml"><mi id="alg1.l1.m2.4.4.3.3.3.2" xref="alg1.l1.m2.4.4.3.3.3.2.cmml">s</mi><mi id="alg1.l1.m2.4.4.3.3.3.3" xref="alg1.l1.m2.4.4.3.3.3.3.cmml">m</mi></msub><mo id="alg1.l1.m2.4.4.3.3.8" stretchy="false" xref="alg1.l1.m2.4.4.3.4.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="alg1.l1.m2.4b"><apply id="alg1.l1.m2.4.4.cmml" xref="alg1.l1.m2.4.4"><eq id="alg1.l1.m2.4.4.4.cmml" xref="alg1.l1.m2.4.4.4"></eq><ci id="alg1.l1.m2.4.4.5.cmml" xref="alg1.l1.m2.4.4.5">𝐬</ci><set id="alg1.l1.m2.4.4.3.4.cmml" xref="alg1.l1.m2.4.4.3.3"><apply id="alg1.l1.m2.2.2.1.1.1.cmml" xref="alg1.l1.m2.2.2.1.1.1"><csymbol cd="ambiguous" id="alg1.l1.m2.2.2.1.1.1.1.cmml" xref="alg1.l1.m2.2.2.1.1.1">subscript</csymbol><ci id="alg1.l1.m2.2.2.1.1.1.2.cmml" xref="alg1.l1.m2.2.2.1.1.1.2">𝑠</ci><cn id="alg1.l1.m2.2.2.1.1.1.3.cmml" type="integer" xref="alg1.l1.m2.2.2.1.1.1.3">1</cn></apply><apply id="alg1.l1.m2.3.3.2.2.2.cmml" xref="alg1.l1.m2.3.3.2.2.2"><csymbol cd="ambiguous" id="alg1.l1.m2.3.3.2.2.2.1.cmml" xref="alg1.l1.m2.3.3.2.2.2">subscript</csymbol><ci id="alg1.l1.m2.3.3.2.2.2.2.cmml" xref="alg1.l1.m2.3.3.2.2.2.2">𝑠</ci><cn id="alg1.l1.m2.3.3.2.2.2.3.cmml" type="integer" xref="alg1.l1.m2.3.3.2.2.2.3">2</cn></apply><ci id="alg1.l1.m2.1.1.cmml" xref="alg1.l1.m2.1.1">…</ci><apply id="alg1.l1.m2.4.4.3.3.3.cmml" xref="alg1.l1.m2.4.4.3.3.3"><csymbol cd="ambiguous" id="alg1.l1.m2.4.4.3.3.3.1.cmml" xref="alg1.l1.m2.4.4.3.3.3">subscript</csymbol><ci id="alg1.l1.m2.4.4.3.3.3.2.cmml" xref="alg1.l1.m2.4.4.3.3.3.2">𝑠</ci><ci id="alg1.l1.m2.4.4.3.3.3.3.cmml" xref="alg1.l1.m2.4.4.3.3.3.3">𝑚</ci></apply></set></apply></annotation-xml><annotation encoding="application/x-tex" id="alg1.l1.m2.4c">\mathbf{s}=\{s_{1},s_{2},\dots,s_{m}\}</annotation><annotation encoding="application/x-llamapun" id="alg1.l1.m2.4d">bold_s = { italic_s start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_s start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , … , italic_s start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT }</annotation></semantics></math> where <math alttext="s_{i}=|c_{i}|" class="ltx_Math" display="inline" id="alg1.l1.m3.1"><semantics id="alg1.l1.m3.1a"><mrow id="alg1.l1.m3.1.1" xref="alg1.l1.m3.1.1.cmml"><msub id="alg1.l1.m3.1.1.3" xref="alg1.l1.m3.1.1.3.cmml"><mi id="alg1.l1.m3.1.1.3.2" xref="alg1.l1.m3.1.1.3.2.cmml">s</mi><mi id="alg1.l1.m3.1.1.3.3" xref="alg1.l1.m3.1.1.3.3.cmml">i</mi></msub><mo id="alg1.l1.m3.1.1.2" xref="alg1.l1.m3.1.1.2.cmml">=</mo><mrow id="alg1.l1.m3.1.1.1.1" xref="alg1.l1.m3.1.1.1.2.cmml"><mo id="alg1.l1.m3.1.1.1.1.2" stretchy="false" xref="alg1.l1.m3.1.1.1.2.1.cmml">|</mo><msub id="alg1.l1.m3.1.1.1.1.1" xref="alg1.l1.m3.1.1.1.1.1.cmml"><mi id="alg1.l1.m3.1.1.1.1.1.2" xref="alg1.l1.m3.1.1.1.1.1.2.cmml">c</mi><mi id="alg1.l1.m3.1.1.1.1.1.3" xref="alg1.l1.m3.1.1.1.1.1.3.cmml">i</mi></msub><mo id="alg1.l1.m3.1.1.1.1.3" stretchy="false" xref="alg1.l1.m3.1.1.1.2.1.cmml">|</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="alg1.l1.m3.1b"><apply id="alg1.l1.m3.1.1.cmml" xref="alg1.l1.m3.1.1"><eq id="alg1.l1.m3.1.1.2.cmml" xref="alg1.l1.m3.1.1.2"></eq><apply id="alg1.l1.m3.1.1.3.cmml" xref="alg1.l1.m3.1.1.3"><csymbol cd="ambiguous" id="alg1.l1.m3.1.1.3.1.cmml" xref="alg1.l1.m3.1.1.3">subscript</csymbol><ci id="alg1.l1.m3.1.1.3.2.cmml" xref="alg1.l1.m3.1.1.3.2">𝑠</ci><ci id="alg1.l1.m3.1.1.3.3.cmml" xref="alg1.l1.m3.1.1.3.3">𝑖</ci></apply><apply id="alg1.l1.m3.1.1.1.2.cmml" xref="alg1.l1.m3.1.1.1.1"><abs id="alg1.l1.m3.1.1.1.2.1.cmml" xref="alg1.l1.m3.1.1.1.1.2"></abs><apply id="alg1.l1.m3.1.1.1.1.1.cmml" xref="alg1.l1.m3.1.1.1.1.1"><csymbol cd="ambiguous" id="alg1.l1.m3.1.1.1.1.1.1.cmml" xref="alg1.l1.m3.1.1.1.1.1">subscript</csymbol><ci id="alg1.l1.m3.1.1.1.1.1.2.cmml" xref="alg1.l1.m3.1.1.1.1.1.2">𝑐</ci><ci id="alg1.l1.m3.1.1.1.1.1.3.cmml" xref="alg1.l1.m3.1.1.1.1.1.3">𝑖</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="alg1.l1.m3.1c">s_{i}=|c_{i}|</annotation><annotation encoding="application/x-llamapun" id="alg1.l1.m3.1d">italic_s start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = | italic_c start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT |</annotation></semantics></math>, Total sampling budget <math alttext="k" class="ltx_Math" display="inline" id="alg1.l1.m4.1"><semantics id="alg1.l1.m4.1a"><mi id="alg1.l1.m4.1.1" xref="alg1.l1.m4.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="alg1.l1.m4.1b"><ci id="alg1.l1.m4.1.1.cmml" xref="alg1.l1.m4.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="alg1.l1.m4.1c">k</annotation><annotation encoding="application/x-llamapun" id="alg1.l1.m4.1d">italic_k</annotation></semantics></math> </div> <div class="ltx_listingline" id="alg1.l2"> <span class="ltx_tag ltx_tag_listingline">2:</span><span class="ltx_text ltx_font_bold" id="alg1.l2.1">Output:</span> Allocation list <math alttext="\mathbf{A}=\{a_{1},a_{2},\dots,a_{m}\}" class="ltx_Math" display="inline" id="alg1.l2.m1.4"><semantics id="alg1.l2.m1.4a"><mrow id="alg1.l2.m1.4.4" xref="alg1.l2.m1.4.4.cmml"><mi id="alg1.l2.m1.4.4.5" xref="alg1.l2.m1.4.4.5.cmml">𝐀</mi><mo id="alg1.l2.m1.4.4.4" xref="alg1.l2.m1.4.4.4.cmml">=</mo><mrow id="alg1.l2.m1.4.4.3.3" xref="alg1.l2.m1.4.4.3.4.cmml"><mo id="alg1.l2.m1.4.4.3.3.4" stretchy="false" xref="alg1.l2.m1.4.4.3.4.cmml">{</mo><msub id="alg1.l2.m1.2.2.1.1.1" xref="alg1.l2.m1.2.2.1.1.1.cmml"><mi id="alg1.l2.m1.2.2.1.1.1.2" xref="alg1.l2.m1.2.2.1.1.1.2.cmml">a</mi><mn id="alg1.l2.m1.2.2.1.1.1.3" xref="alg1.l2.m1.2.2.1.1.1.3.cmml">1</mn></msub><mo id="alg1.l2.m1.4.4.3.3.5" xref="alg1.l2.m1.4.4.3.4.cmml">,</mo><msub id="alg1.l2.m1.3.3.2.2.2" xref="alg1.l2.m1.3.3.2.2.2.cmml"><mi id="alg1.l2.m1.3.3.2.2.2.2" xref="alg1.l2.m1.3.3.2.2.2.2.cmml">a</mi><mn id="alg1.l2.m1.3.3.2.2.2.3" xref="alg1.l2.m1.3.3.2.2.2.3.cmml">2</mn></msub><mo id="alg1.l2.m1.4.4.3.3.6" xref="alg1.l2.m1.4.4.3.4.cmml">,</mo><mi id="alg1.l2.m1.1.1" mathvariant="normal" xref="alg1.l2.m1.1.1.cmml">…</mi><mo id="alg1.l2.m1.4.4.3.3.7" xref="alg1.l2.m1.4.4.3.4.cmml">,</mo><msub id="alg1.l2.m1.4.4.3.3.3" xref="alg1.l2.m1.4.4.3.3.3.cmml"><mi id="alg1.l2.m1.4.4.3.3.3.2" xref="alg1.l2.m1.4.4.3.3.3.2.cmml">a</mi><mi id="alg1.l2.m1.4.4.3.3.3.3" xref="alg1.l2.m1.4.4.3.3.3.3.cmml">m</mi></msub><mo id="alg1.l2.m1.4.4.3.3.8" stretchy="false" xref="alg1.l2.m1.4.4.3.4.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="alg1.l2.m1.4b"><apply id="alg1.l2.m1.4.4.cmml" xref="alg1.l2.m1.4.4"><eq id="alg1.l2.m1.4.4.4.cmml" xref="alg1.l2.m1.4.4.4"></eq><ci id="alg1.l2.m1.4.4.5.cmml" xref="alg1.l2.m1.4.4.5">𝐀</ci><set id="alg1.l2.m1.4.4.3.4.cmml" xref="alg1.l2.m1.4.4.3.3"><apply id="alg1.l2.m1.2.2.1.1.1.cmml" xref="alg1.l2.m1.2.2.1.1.1"><csymbol cd="ambiguous" id="alg1.l2.m1.2.2.1.1.1.1.cmml" xref="alg1.l2.m1.2.2.1.1.1">subscript</csymbol><ci id="alg1.l2.m1.2.2.1.1.1.2.cmml" xref="alg1.l2.m1.2.2.1.1.1.2">𝑎</ci><cn id="alg1.l2.m1.2.2.1.1.1.3.cmml" type="integer" xref="alg1.l2.m1.2.2.1.1.1.3">1</cn></apply><apply id="alg1.l2.m1.3.3.2.2.2.cmml" xref="alg1.l2.m1.3.3.2.2.2"><csymbol cd="ambiguous" id="alg1.l2.m1.3.3.2.2.2.1.cmml" xref="alg1.l2.m1.3.3.2.2.2">subscript</csymbol><ci id="alg1.l2.m1.3.3.2.2.2.2.cmml" xref="alg1.l2.m1.3.3.2.2.2.2">𝑎</ci><cn id="alg1.l2.m1.3.3.2.2.2.3.cmml" type="integer" xref="alg1.l2.m1.3.3.2.2.2.3">2</cn></apply><ci id="alg1.l2.m1.1.1.cmml" xref="alg1.l2.m1.1.1">…</ci><apply id="alg1.l2.m1.4.4.3.3.3.cmml" xref="alg1.l2.m1.4.4.3.3.3"><csymbol cd="ambiguous" id="alg1.l2.m1.4.4.3.3.3.1.cmml" xref="alg1.l2.m1.4.4.3.3.3">subscript</csymbol><ci id="alg1.l2.m1.4.4.3.3.3.2.cmml" xref="alg1.l2.m1.4.4.3.3.3.2">𝑎</ci><ci id="alg1.l2.m1.4.4.3.3.3.3.cmml" xref="alg1.l2.m1.4.4.3.3.3.3">𝑚</ci></apply></set></apply></annotation-xml><annotation encoding="application/x-tex" id="alg1.l2.m1.4c">\mathbf{A}=\{a_{1},a_{2},\dots,a_{m}\}</annotation><annotation encoding="application/x-llamapun" id="alg1.l2.m1.4d">bold_A = { italic_a start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_a start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , … , italic_a start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT }</annotation></semantics></math> </div> <div class="ltx_listingline" id="alg1.l3"> <span class="ltx_tag ltx_tag_listingline">3:</span><span class="ltx_text ltx_font_bold" id="alg1.l3.1">function</span> <span class="ltx_text ltx_font_smallcaps" id="alg1.l3.2">AllocateBudget</span>(<math alttext="\mathbf{C},\mathbf{s},k" class="ltx_Math" display="inline" id="alg1.l3.m1.3"><semantics id="alg1.l3.m1.3a"><mrow id="alg1.l3.m1.3.4.2" xref="alg1.l3.m1.3.4.1.cmml"><mi id="alg1.l3.m1.1.1" xref="alg1.l3.m1.1.1.cmml">𝐂</mi><mo id="alg1.l3.m1.3.4.2.1" xref="alg1.l3.m1.3.4.1.cmml">,</mo><mi id="alg1.l3.m1.2.2" xref="alg1.l3.m1.2.2.cmml">𝐬</mi><mo id="alg1.l3.m1.3.4.2.2" xref="alg1.l3.m1.3.4.1.cmml">,</mo><mi id="alg1.l3.m1.3.3" xref="alg1.l3.m1.3.3.cmml">k</mi></mrow><annotation-xml encoding="MathML-Content" id="alg1.l3.m1.3b"><list id="alg1.l3.m1.3.4.1.cmml" xref="alg1.l3.m1.3.4.2"><ci id="alg1.l3.m1.1.1.cmml" xref="alg1.l3.m1.1.1">𝐂</ci><ci id="alg1.l3.m1.2.2.cmml" xref="alg1.l3.m1.2.2">𝐬</ci><ci id="alg1.l3.m1.3.3.cmml" xref="alg1.l3.m1.3.3">𝑘</ci></list></annotation-xml><annotation encoding="application/x-tex" id="alg1.l3.m1.3c">\mathbf{C},\mathbf{s},k</annotation><annotation encoding="application/x-llamapun" id="alg1.l3.m1.3d">bold_C , bold_s , italic_k</annotation></semantics></math>) </div> <div class="ltx_listingline" id="alg1.l4"> <span class="ltx_tag ltx_tag_listingline">4:</span> Sort <math alttext="\mathbf{s}" class="ltx_Math" display="inline" id="alg1.l4.m1.1"><semantics id="alg1.l4.m1.1a"><mi id="alg1.l4.m1.1.1" xref="alg1.l4.m1.1.1.cmml">𝐬</mi><annotation-xml encoding="MathML-Content" id="alg1.l4.m1.1b"><ci id="alg1.l4.m1.1.1.cmml" xref="alg1.l4.m1.1.1">𝐬</ci></annotation-xml><annotation encoding="application/x-tex" id="alg1.l4.m1.1c">\mathbf{s}</annotation><annotation encoding="application/x-llamapun" id="alg1.l4.m1.1d">bold_s</annotation></semantics></math> in ascending order and obtain sorted indices <math alttext="\mathbf{I}" class="ltx_Math" display="inline" id="alg1.l4.m2.1"><semantics id="alg1.l4.m2.1a"><mi id="alg1.l4.m2.1.1" xref="alg1.l4.m2.1.1.cmml">𝐈</mi><annotation-xml encoding="MathML-Content" id="alg1.l4.m2.1b"><ci id="alg1.l4.m2.1.1.cmml" xref="alg1.l4.m2.1.1">𝐈</ci></annotation-xml><annotation encoding="application/x-tex" id="alg1.l4.m2.1c">\mathbf{I}</annotation><annotation encoding="application/x-llamapun" id="alg1.l4.m2.1d">bold_I</annotation></semantics></math> </div> <div class="ltx_listingline" id="alg1.l5"> <span class="ltx_tag ltx_tag_listingline">5:</span> Initialize allocation <math alttext="\mathbf{A}\leftarrow[0,0,\dots,0]" class="ltx_Math" display="inline" id="alg1.l5.m1.4"><semantics id="alg1.l5.m1.4a"><mrow id="alg1.l5.m1.4.5" xref="alg1.l5.m1.4.5.cmml"><mi id="alg1.l5.m1.4.5.2" xref="alg1.l5.m1.4.5.2.cmml">𝐀</mi><mo id="alg1.l5.m1.4.5.1" stretchy="false" xref="alg1.l5.m1.4.5.1.cmml">←</mo><mrow id="alg1.l5.m1.4.5.3.2" xref="alg1.l5.m1.4.5.3.1.cmml"><mo id="alg1.l5.m1.4.5.3.2.1" stretchy="false" xref="alg1.l5.m1.4.5.3.1.cmml">[</mo><mn id="alg1.l5.m1.1.1" xref="alg1.l5.m1.1.1.cmml">0</mn><mo id="alg1.l5.m1.4.5.3.2.2" xref="alg1.l5.m1.4.5.3.1.cmml">,</mo><mn id="alg1.l5.m1.2.2" xref="alg1.l5.m1.2.2.cmml">0</mn><mo id="alg1.l5.m1.4.5.3.2.3" xref="alg1.l5.m1.4.5.3.1.cmml">,</mo><mi id="alg1.l5.m1.3.3" mathvariant="normal" xref="alg1.l5.m1.3.3.cmml">…</mi><mo id="alg1.l5.m1.4.5.3.2.4" xref="alg1.l5.m1.4.5.3.1.cmml">,</mo><mn id="alg1.l5.m1.4.4" xref="alg1.l5.m1.4.4.cmml">0</mn><mo id="alg1.l5.m1.4.5.3.2.5" stretchy="false" xref="alg1.l5.m1.4.5.3.1.cmml">]</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="alg1.l5.m1.4b"><apply id="alg1.l5.m1.4.5.cmml" xref="alg1.l5.m1.4.5"><ci id="alg1.l5.m1.4.5.1.cmml" xref="alg1.l5.m1.4.5.1">←</ci><ci id="alg1.l5.m1.4.5.2.cmml" xref="alg1.l5.m1.4.5.2">𝐀</ci><list id="alg1.l5.m1.4.5.3.1.cmml" xref="alg1.l5.m1.4.5.3.2"><cn id="alg1.l5.m1.1.1.cmml" type="integer" xref="alg1.l5.m1.1.1">0</cn><cn id="alg1.l5.m1.2.2.cmml" type="integer" xref="alg1.l5.m1.2.2">0</cn><ci id="alg1.l5.m1.3.3.cmml" xref="alg1.l5.m1.3.3">…</ci><cn id="alg1.l5.m1.4.4.cmml" type="integer" xref="alg1.l5.m1.4.4">0</cn></list></apply></annotation-xml><annotation encoding="application/x-tex" id="alg1.l5.m1.4c">\mathbf{A}\leftarrow[0,0,\dots,0]</annotation><annotation encoding="application/x-llamapun" id="alg1.l5.m1.4d">bold_A ← [ 0 , 0 , … , 0 ]</annotation></semantics></math> </div> <div class="ltx_listingline" id="alg1.l6"> <span class="ltx_tag ltx_tag_listingline">6:</span> Remaining budget <math alttext="B\leftarrow k" class="ltx_Math" display="inline" id="alg1.l6.m1.1"><semantics id="alg1.l6.m1.1a"><mrow id="alg1.l6.m1.1.1" xref="alg1.l6.m1.1.1.cmml"><mi id="alg1.l6.m1.1.1.2" xref="alg1.l6.m1.1.1.2.cmml">B</mi><mo id="alg1.l6.m1.1.1.1" stretchy="false" xref="alg1.l6.m1.1.1.1.cmml">←</mo><mi id="alg1.l6.m1.1.1.3" xref="alg1.l6.m1.1.1.3.cmml">k</mi></mrow><annotation-xml encoding="MathML-Content" id="alg1.l6.m1.1b"><apply id="alg1.l6.m1.1.1.cmml" xref="alg1.l6.m1.1.1"><ci id="alg1.l6.m1.1.1.1.cmml" xref="alg1.l6.m1.1.1.1">←</ci><ci id="alg1.l6.m1.1.1.2.cmml" xref="alg1.l6.m1.1.1.2">𝐵</ci><ci id="alg1.l6.m1.1.1.3.cmml" xref="alg1.l6.m1.1.1.3">𝑘</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="alg1.l6.m1.1c">B\leftarrow k</annotation><annotation encoding="application/x-llamapun" id="alg1.l6.m1.1d">italic_B ← italic_k</annotation></semantics></math> </div> <div class="ltx_listingline" id="alg1.l7"> <span class="ltx_tag ltx_tag_listingline">7:</span> <span class="ltx_text ltx_font_bold" id="alg1.l7.1">for</span> each cluster <math alttext="i" class="ltx_Math" display="inline" id="alg1.l7.m1.1"><semantics id="alg1.l7.m1.1a"><mi id="alg1.l7.m1.1.1" xref="alg1.l7.m1.1.1.cmml">i</mi><annotation-xml encoding="MathML-Content" id="alg1.l7.m1.1b"><ci id="alg1.l7.m1.1.1.cmml" xref="alg1.l7.m1.1.1">𝑖</ci></annotation-xml><annotation encoding="application/x-tex" id="alg1.l7.m1.1c">i</annotation><annotation encoding="application/x-llamapun" id="alg1.l7.m1.1d">italic_i</annotation></semantics></math> in sorted order <span class="ltx_text ltx_font_bold" id="alg1.l7.2">do</span> </div> <div class="ltx_listingline" id="alg1.l8"> <span class="ltx_tag ltx_tag_listingline">8:</span> <math alttext="r\leftarrow" class="ltx_Math" display="inline" id="alg1.l8.m1.1"><semantics id="alg1.l8.m1.1a"><mrow id="alg1.l8.m1.1.1" xref="alg1.l8.m1.1.1.cmml"><mi id="alg1.l8.m1.1.1.2" xref="alg1.l8.m1.1.1.2.cmml">r</mi><mo id="alg1.l8.m1.1.1.1" stretchy="false" xref="alg1.l8.m1.1.1.1.cmml">←</mo><mi id="alg1.l8.m1.1.1.3" xref="alg1.l8.m1.1.1.3.cmml"></mi></mrow><annotation-xml encoding="MathML-Content" id="alg1.l8.m1.1b"><apply id="alg1.l8.m1.1.1.cmml" xref="alg1.l8.m1.1.1"><ci id="alg1.l8.m1.1.1.1.cmml" xref="alg1.l8.m1.1.1.1">←</ci><ci id="alg1.l8.m1.1.1.2.cmml" xref="alg1.l8.m1.1.1.2">𝑟</ci><csymbol cd="latexml" id="alg1.l8.m1.1.1.3.cmml" xref="alg1.l8.m1.1.1.3">absent</csymbol></apply></annotation-xml><annotation encoding="application/x-tex" id="alg1.l8.m1.1c">r\leftarrow</annotation><annotation encoding="application/x-llamapun" id="alg1.l8.m1.1d">italic_r ←</annotation></semantics></math> number of remaining clusters </div> <div class="ltx_listingline" id="alg1.l9"> <span class="ltx_tag ltx_tag_listingline">9:</span> <math alttext="q\leftarrow B\mathbin{//}r" class="ltx_Math" display="inline" id="alg1.l9.m1.1"><semantics id="alg1.l9.m1.1a"><mrow id="alg1.l9.m1.1.1" xref="alg1.l9.m1.1.1.cmml"><mi id="alg1.l9.m1.1.1.2" xref="alg1.l9.m1.1.1.2.cmml">q</mi><mo id="alg1.l9.m1.1.1.1" stretchy="false" xref="alg1.l9.m1.1.1.1.cmml">←</mo><mrow id="alg1.l9.m1.1.1.3" xref="alg1.l9.m1.1.1.3.cmml"><mi id="alg1.l9.m1.1.1.3.2" xref="alg1.l9.m1.1.1.3.2.cmml">B</mi><mo class="ltx_mathvariant_italic" id="alg1.l9.m1.1.1.3.1" lspace="0.222em" mathvariant="italic" rspace="0.222em" xref="alg1.l9.m1.1.1.3.1.cmml">//</mo><mi id="alg1.l9.m1.1.1.3.3" xref="alg1.l9.m1.1.1.3.3.cmml">r</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="alg1.l9.m1.1b"><apply id="alg1.l9.m1.1.1.cmml" xref="alg1.l9.m1.1.1"><ci id="alg1.l9.m1.1.1.1.cmml" xref="alg1.l9.m1.1.1.1">←</ci><ci id="alg1.l9.m1.1.1.2.cmml" xref="alg1.l9.m1.1.1.2">𝑞</ci><apply id="alg1.l9.m1.1.1.3.cmml" xref="alg1.l9.m1.1.1.3"><ci id="alg1.l9.m1.1.1.3.1.cmml" xref="alg1.l9.m1.1.1.3.1">italic-//</ci><ci id="alg1.l9.m1.1.1.3.2.cmml" xref="alg1.l9.m1.1.1.3.2">𝐵</ci><ci id="alg1.l9.m1.1.1.3.3.cmml" xref="alg1.l9.m1.1.1.3.3">𝑟</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="alg1.l9.m1.1c">q\leftarrow B\mathbin{//}r</annotation><annotation encoding="application/x-llamapun" id="alg1.l9.m1.1d">italic_q ← italic_B italic_// italic_r</annotation></semantics></math> <span class="ltx_text" id="alg1.l9.1" style="float:right;"><math alttext="\triangleright" class="ltx_Math" display="inline" id="alg1.l9.1.m1.1"><semantics id="alg1.l9.1.m1.1a"><mo id="alg1.l9.1.m1.1.1" xref="alg1.l9.1.m1.1.1.cmml">▷</mo><annotation-xml encoding="MathML-Content" id="alg1.l9.1.m1.1b"><ci id="alg1.l9.1.m1.1.1.cmml" xref="alg1.l9.1.m1.1.1">▷</ci></annotation-xml><annotation encoding="application/x-tex" id="alg1.l9.1.m1.1c">\triangleright</annotation><annotation encoding="application/x-llamapun" id="alg1.l9.1.m1.1d">▷</annotation></semantics></math> Average allocation per remaining cluster </span> </div> <div class="ltx_listingline" id="alg1.l10"> <span class="ltx_tag ltx_tag_listingline">10:</span> <math alttext="a_{i}\leftarrow\min(s_{i},q)" class="ltx_Math" display="inline" id="alg1.l10.m1.3"><semantics id="alg1.l10.m1.3a"><mrow id="alg1.l10.m1.3.3" xref="alg1.l10.m1.3.3.cmml"><msub id="alg1.l10.m1.3.3.3" xref="alg1.l10.m1.3.3.3.cmml"><mi id="alg1.l10.m1.3.3.3.2" xref="alg1.l10.m1.3.3.3.2.cmml">a</mi><mi id="alg1.l10.m1.3.3.3.3" xref="alg1.l10.m1.3.3.3.3.cmml">i</mi></msub><mo id="alg1.l10.m1.3.3.2" stretchy="false" xref="alg1.l10.m1.3.3.2.cmml">←</mo><mrow id="alg1.l10.m1.3.3.1.1" xref="alg1.l10.m1.3.3.1.2.cmml"><mi id="alg1.l10.m1.1.1" xref="alg1.l10.m1.1.1.cmml">min</mi><mo id="alg1.l10.m1.3.3.1.1a" xref="alg1.l10.m1.3.3.1.2.cmml">⁡</mo><mrow id="alg1.l10.m1.3.3.1.1.1" xref="alg1.l10.m1.3.3.1.2.cmml"><mo id="alg1.l10.m1.3.3.1.1.1.2" stretchy="false" xref="alg1.l10.m1.3.3.1.2.cmml">(</mo><msub id="alg1.l10.m1.3.3.1.1.1.1" xref="alg1.l10.m1.3.3.1.1.1.1.cmml"><mi id="alg1.l10.m1.3.3.1.1.1.1.2" xref="alg1.l10.m1.3.3.1.1.1.1.2.cmml">s</mi><mi id="alg1.l10.m1.3.3.1.1.1.1.3" xref="alg1.l10.m1.3.3.1.1.1.1.3.cmml">i</mi></msub><mo id="alg1.l10.m1.3.3.1.1.1.3" xref="alg1.l10.m1.3.3.1.2.cmml">,</mo><mi id="alg1.l10.m1.2.2" xref="alg1.l10.m1.2.2.cmml">q</mi><mo id="alg1.l10.m1.3.3.1.1.1.4" stretchy="false" xref="alg1.l10.m1.3.3.1.2.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="alg1.l10.m1.3b"><apply id="alg1.l10.m1.3.3.cmml" xref="alg1.l10.m1.3.3"><ci id="alg1.l10.m1.3.3.2.cmml" xref="alg1.l10.m1.3.3.2">←</ci><apply id="alg1.l10.m1.3.3.3.cmml" xref="alg1.l10.m1.3.3.3"><csymbol cd="ambiguous" id="alg1.l10.m1.3.3.3.1.cmml" xref="alg1.l10.m1.3.3.3">subscript</csymbol><ci id="alg1.l10.m1.3.3.3.2.cmml" xref="alg1.l10.m1.3.3.3.2">𝑎</ci><ci id="alg1.l10.m1.3.3.3.3.cmml" xref="alg1.l10.m1.3.3.3.3">𝑖</ci></apply><apply id="alg1.l10.m1.3.3.1.2.cmml" xref="alg1.l10.m1.3.3.1.1"><min id="alg1.l10.m1.1.1.cmml" xref="alg1.l10.m1.1.1"></min><apply id="alg1.l10.m1.3.3.1.1.1.1.cmml" xref="alg1.l10.m1.3.3.1.1.1.1"><csymbol cd="ambiguous" id="alg1.l10.m1.3.3.1.1.1.1.1.cmml" xref="alg1.l10.m1.3.3.1.1.1.1">subscript</csymbol><ci id="alg1.l10.m1.3.3.1.1.1.1.2.cmml" xref="alg1.l10.m1.3.3.1.1.1.1.2">𝑠</ci><ci id="alg1.l10.m1.3.3.1.1.1.1.3.cmml" xref="alg1.l10.m1.3.3.1.1.1.1.3">𝑖</ci></apply><ci id="alg1.l10.m1.2.2.cmml" xref="alg1.l10.m1.2.2">𝑞</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="alg1.l10.m1.3c">a_{i}\leftarrow\min(s_{i},q)</annotation><annotation encoding="application/x-llamapun" id="alg1.l10.m1.3d">italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ← roman_min ( italic_s start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , italic_q )</annotation></semantics></math> <span class="ltx_text" id="alg1.l10.2" style="float:right;"><math alttext="\triangleright" class="ltx_Math" display="inline" id="alg1.l10.1.m1.1"><semantics id="alg1.l10.1.m1.1a"><mo id="alg1.l10.1.m1.1.1" xref="alg1.l10.1.m1.1.1.cmml">▷</mo><annotation-xml encoding="MathML-Content" id="alg1.l10.1.m1.1b"><ci id="alg1.l10.1.m1.1.1.cmml" xref="alg1.l10.1.m1.1.1">▷</ci></annotation-xml><annotation encoding="application/x-tex" id="alg1.l10.1.m1.1c">\triangleright</annotation><annotation encoding="application/x-llamapun" id="alg1.l10.1.m1.1d">▷</annotation></semantics></math> Allocate min of cluster size or <math alttext="q" class="ltx_Math" display="inline" id="alg1.l10.2.m2.1"><semantics id="alg1.l10.2.m2.1a"><mi id="alg1.l10.2.m2.1.1" xref="alg1.l10.2.m2.1.1.cmml">q</mi><annotation-xml encoding="MathML-Content" id="alg1.l10.2.m2.1b"><ci id="alg1.l10.2.m2.1.1.cmml" xref="alg1.l10.2.m2.1.1">𝑞</ci></annotation-xml><annotation encoding="application/x-tex" id="alg1.l10.2.m2.1c">q</annotation><annotation encoding="application/x-llamapun" id="alg1.l10.2.m2.1d">italic_q</annotation></semantics></math> </span> </div> <div class="ltx_listingline" id="alg1.l11"> <span class="ltx_tag ltx_tag_listingline">11:</span> <math alttext="B\leftarrow B-a_{i}" class="ltx_Math" display="inline" id="alg1.l11.m1.1"><semantics id="alg1.l11.m1.1a"><mrow id="alg1.l11.m1.1.1" xref="alg1.l11.m1.1.1.cmml"><mi id="alg1.l11.m1.1.1.2" xref="alg1.l11.m1.1.1.2.cmml">B</mi><mo id="alg1.l11.m1.1.1.1" stretchy="false" xref="alg1.l11.m1.1.1.1.cmml">←</mo><mrow id="alg1.l11.m1.1.1.3" xref="alg1.l11.m1.1.1.3.cmml"><mi id="alg1.l11.m1.1.1.3.2" xref="alg1.l11.m1.1.1.3.2.cmml">B</mi><mo id="alg1.l11.m1.1.1.3.1" xref="alg1.l11.m1.1.1.3.1.cmml">−</mo><msub id="alg1.l11.m1.1.1.3.3" xref="alg1.l11.m1.1.1.3.3.cmml"><mi id="alg1.l11.m1.1.1.3.3.2" xref="alg1.l11.m1.1.1.3.3.2.cmml">a</mi><mi id="alg1.l11.m1.1.1.3.3.3" xref="alg1.l11.m1.1.1.3.3.3.cmml">i</mi></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="alg1.l11.m1.1b"><apply id="alg1.l11.m1.1.1.cmml" xref="alg1.l11.m1.1.1"><ci id="alg1.l11.m1.1.1.1.cmml" xref="alg1.l11.m1.1.1.1">←</ci><ci id="alg1.l11.m1.1.1.2.cmml" xref="alg1.l11.m1.1.1.2">𝐵</ci><apply id="alg1.l11.m1.1.1.3.cmml" xref="alg1.l11.m1.1.1.3"><minus id="alg1.l11.m1.1.1.3.1.cmml" xref="alg1.l11.m1.1.1.3.1"></minus><ci id="alg1.l11.m1.1.1.3.2.cmml" xref="alg1.l11.m1.1.1.3.2">𝐵</ci><apply id="alg1.l11.m1.1.1.3.3.cmml" xref="alg1.l11.m1.1.1.3.3"><csymbol cd="ambiguous" id="alg1.l11.m1.1.1.3.3.1.cmml" xref="alg1.l11.m1.1.1.3.3">subscript</csymbol><ci id="alg1.l11.m1.1.1.3.3.2.cmml" xref="alg1.l11.m1.1.1.3.3.2">𝑎</ci><ci id="alg1.l11.m1.1.1.3.3.3.cmml" xref="alg1.l11.m1.1.1.3.3.3">𝑖</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="alg1.l11.m1.1c">B\leftarrow B-a_{i}</annotation><annotation encoding="application/x-llamapun" id="alg1.l11.m1.1d">italic_B ← italic_B - italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> <span class="ltx_text" id="alg1.l11.1" style="float:right;"><math alttext="\triangleright" class="ltx_Math" display="inline" id="alg1.l11.1.m1.1"><semantics id="alg1.l11.1.m1.1a"><mo id="alg1.l11.1.m1.1.1" xref="alg1.l11.1.m1.1.1.cmml">▷</mo><annotation-xml encoding="MathML-Content" id="alg1.l11.1.m1.1b"><ci id="alg1.l11.1.m1.1.1.cmml" xref="alg1.l11.1.m1.1.1">▷</ci></annotation-xml><annotation encoding="application/x-tex" id="alg1.l11.1.m1.1c">\triangleright</annotation><annotation encoding="application/x-llamapun" id="alg1.l11.1.m1.1d">▷</annotation></semantics></math> Update remaining budget </span> </div> <div class="ltx_listingline" id="alg1.l12"> <span class="ltx_tag ltx_tag_listingline">12:</span> <span class="ltx_text ltx_font_bold" id="alg1.l12.1">end</span> <span class="ltx_text ltx_font_bold" id="alg1.l12.2">for</span> </div> <div class="ltx_listingline" id="alg1.l13"> <span class="ltx_tag ltx_tag_listingline">13:</span> <span class="ltx_text ltx_font_bold" id="alg1.l13.2">while</span> <math alttext="B>0" class="ltx_Math" display="inline" id="alg1.l13.m1.1"><semantics id="alg1.l13.m1.1a"><mrow id="alg1.l13.m1.1.1" xref="alg1.l13.m1.1.1.cmml"><mi id="alg1.l13.m1.1.1.2" xref="alg1.l13.m1.1.1.2.cmml">B</mi><mo id="alg1.l13.m1.1.1.1" xref="alg1.l13.m1.1.1.1.cmml">></mo><mn id="alg1.l13.m1.1.1.3" xref="alg1.l13.m1.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="alg1.l13.m1.1b"><apply id="alg1.l13.m1.1.1.cmml" xref="alg1.l13.m1.1.1"><gt id="alg1.l13.m1.1.1.1.cmml" xref="alg1.l13.m1.1.1.1"></gt><ci id="alg1.l13.m1.1.1.2.cmml" xref="alg1.l13.m1.1.1.2">𝐵</ci><cn id="alg1.l13.m1.1.1.3.cmml" type="integer" xref="alg1.l13.m1.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="alg1.l13.m1.1c">B>0</annotation><annotation encoding="application/x-llamapun" id="alg1.l13.m1.1d">italic_B > 0</annotation></semantics></math> <span class="ltx_text ltx_font_bold" id="alg1.l13.3">do</span> <span class="ltx_text" id="alg1.l13.1" style="float:right;"><math alttext="\triangleright" class="ltx_Math" display="inline" id="alg1.l13.1.m1.1"><semantics id="alg1.l13.1.m1.1a"><mo id="alg1.l13.1.m1.1.1" xref="alg1.l13.1.m1.1.1.cmml">▷</mo><annotation-xml encoding="MathML-Content" id="alg1.l13.1.m1.1b"><ci id="alg1.l13.1.m1.1.1.cmml" xref="alg1.l13.1.m1.1.1">▷</ci></annotation-xml><annotation encoding="application/x-tex" id="alg1.l13.1.m1.1c">\triangleright</annotation><annotation encoding="application/x-llamapun" id="alg1.l13.1.m1.1d">▷</annotation></semantics></math> Distribute any remaining budget </span> </div> <div class="ltx_listingline" id="alg1.l14"> <span class="ltx_tag ltx_tag_listingline">14:</span> <span class="ltx_text ltx_font_bold" id="alg1.l14.1">for</span> each cluster <math alttext="i" class="ltx_Math" display="inline" id="alg1.l14.m1.1"><semantics id="alg1.l14.m1.1a"><mi id="alg1.l14.m1.1.1" xref="alg1.l14.m1.1.1.cmml">i</mi><annotation-xml encoding="MathML-Content" id="alg1.l14.m1.1b"><ci id="alg1.l14.m1.1.1.cmml" xref="alg1.l14.m1.1.1">𝑖</ci></annotation-xml><annotation encoding="application/x-tex" id="alg1.l14.m1.1c">i</annotation><annotation encoding="application/x-llamapun" id="alg1.l14.m1.1d">italic_i</annotation></semantics></math> <span class="ltx_text ltx_font_bold" id="alg1.l14.2">in sorted order</span> <span class="ltx_text ltx_font_bold" id="alg1.l14.3">if</span> <math alttext="a_{i}<s_{i}" class="ltx_Math" display="inline" id="alg1.l14.m2.1"><semantics id="alg1.l14.m2.1a"><mrow id="alg1.l14.m2.1.1" xref="alg1.l14.m2.1.1.cmml"><msub id="alg1.l14.m2.1.1.2" xref="alg1.l14.m2.1.1.2.cmml"><mi id="alg1.l14.m2.1.1.2.2" xref="alg1.l14.m2.1.1.2.2.cmml">a</mi><mi id="alg1.l14.m2.1.1.2.3" xref="alg1.l14.m2.1.1.2.3.cmml">i</mi></msub><mo id="alg1.l14.m2.1.1.1" xref="alg1.l14.m2.1.1.1.cmml"><</mo><msub id="alg1.l14.m2.1.1.3" xref="alg1.l14.m2.1.1.3.cmml"><mi id="alg1.l14.m2.1.1.3.2" xref="alg1.l14.m2.1.1.3.2.cmml">s</mi><mi id="alg1.l14.m2.1.1.3.3" xref="alg1.l14.m2.1.1.3.3.cmml">i</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="alg1.l14.m2.1b"><apply id="alg1.l14.m2.1.1.cmml" xref="alg1.l14.m2.1.1"><lt id="alg1.l14.m2.1.1.1.cmml" xref="alg1.l14.m2.1.1.1"></lt><apply id="alg1.l14.m2.1.1.2.cmml" xref="alg1.l14.m2.1.1.2"><csymbol cd="ambiguous" id="alg1.l14.m2.1.1.2.1.cmml" xref="alg1.l14.m2.1.1.2">subscript</csymbol><ci id="alg1.l14.m2.1.1.2.2.cmml" xref="alg1.l14.m2.1.1.2.2">𝑎</ci><ci id="alg1.l14.m2.1.1.2.3.cmml" xref="alg1.l14.m2.1.1.2.3">𝑖</ci></apply><apply id="alg1.l14.m2.1.1.3.cmml" xref="alg1.l14.m2.1.1.3"><csymbol cd="ambiguous" id="alg1.l14.m2.1.1.3.1.cmml" xref="alg1.l14.m2.1.1.3">subscript</csymbol><ci id="alg1.l14.m2.1.1.3.2.cmml" xref="alg1.l14.m2.1.1.3.2">𝑠</ci><ci id="alg1.l14.m2.1.1.3.3.cmml" xref="alg1.l14.m2.1.1.3.3">𝑖</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="alg1.l14.m2.1c">a_{i}<s_{i}</annotation><annotation encoding="application/x-llamapun" id="alg1.l14.m2.1d">italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT < italic_s start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> <span class="ltx_text ltx_font_bold" id="alg1.l14.4">do</span> </div> <div class="ltx_listingline" id="alg1.l15"> <span class="ltx_tag ltx_tag_listingline">15:</span> <math alttext="a_{i}\leftarrow a_{i}+1" class="ltx_Math" display="inline" id="alg1.l15.m1.1"><semantics id="alg1.l15.m1.1a"><mrow id="alg1.l15.m1.1.1" xref="alg1.l15.m1.1.1.cmml"><msub id="alg1.l15.m1.1.1.2" xref="alg1.l15.m1.1.1.2.cmml"><mi id="alg1.l15.m1.1.1.2.2" xref="alg1.l15.m1.1.1.2.2.cmml">a</mi><mi id="alg1.l15.m1.1.1.2.3" xref="alg1.l15.m1.1.1.2.3.cmml">i</mi></msub><mo id="alg1.l15.m1.1.1.1" stretchy="false" xref="alg1.l15.m1.1.1.1.cmml">←</mo><mrow id="alg1.l15.m1.1.1.3" xref="alg1.l15.m1.1.1.3.cmml"><msub id="alg1.l15.m1.1.1.3.2" xref="alg1.l15.m1.1.1.3.2.cmml"><mi id="alg1.l15.m1.1.1.3.2.2" xref="alg1.l15.m1.1.1.3.2.2.cmml">a</mi><mi id="alg1.l15.m1.1.1.3.2.3" xref="alg1.l15.m1.1.1.3.2.3.cmml">i</mi></msub><mo id="alg1.l15.m1.1.1.3.1" xref="alg1.l15.m1.1.1.3.1.cmml">+</mo><mn id="alg1.l15.m1.1.1.3.3" xref="alg1.l15.m1.1.1.3.3.cmml">1</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="alg1.l15.m1.1b"><apply id="alg1.l15.m1.1.1.cmml" xref="alg1.l15.m1.1.1"><ci id="alg1.l15.m1.1.1.1.cmml" xref="alg1.l15.m1.1.1.1">←</ci><apply id="alg1.l15.m1.1.1.2.cmml" xref="alg1.l15.m1.1.1.2"><csymbol cd="ambiguous" id="alg1.l15.m1.1.1.2.1.cmml" xref="alg1.l15.m1.1.1.2">subscript</csymbol><ci id="alg1.l15.m1.1.1.2.2.cmml" xref="alg1.l15.m1.1.1.2.2">𝑎</ci><ci id="alg1.l15.m1.1.1.2.3.cmml" xref="alg1.l15.m1.1.1.2.3">𝑖</ci></apply><apply id="alg1.l15.m1.1.1.3.cmml" xref="alg1.l15.m1.1.1.3"><plus id="alg1.l15.m1.1.1.3.1.cmml" xref="alg1.l15.m1.1.1.3.1"></plus><apply id="alg1.l15.m1.1.1.3.2.cmml" xref="alg1.l15.m1.1.1.3.2"><csymbol cd="ambiguous" id="alg1.l15.m1.1.1.3.2.1.cmml" xref="alg1.l15.m1.1.1.3.2">subscript</csymbol><ci id="alg1.l15.m1.1.1.3.2.2.cmml" xref="alg1.l15.m1.1.1.3.2.2">𝑎</ci><ci id="alg1.l15.m1.1.1.3.2.3.cmml" xref="alg1.l15.m1.1.1.3.2.3">𝑖</ci></apply><cn id="alg1.l15.m1.1.1.3.3.cmml" type="integer" xref="alg1.l15.m1.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="alg1.l15.m1.1c">a_{i}\leftarrow a_{i}+1</annotation><annotation encoding="application/x-llamapun" id="alg1.l15.m1.1d">italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ← italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT + 1</annotation></semantics></math> </div> <div class="ltx_listingline" id="alg1.l16"> <span class="ltx_tag ltx_tag_listingline">16:</span> <math alttext="B\leftarrow B-1" class="ltx_Math" display="inline" id="alg1.l16.m1.1"><semantics id="alg1.l16.m1.1a"><mrow id="alg1.l16.m1.1.1" xref="alg1.l16.m1.1.1.cmml"><mi id="alg1.l16.m1.1.1.2" xref="alg1.l16.m1.1.1.2.cmml">B</mi><mo id="alg1.l16.m1.1.1.1" stretchy="false" xref="alg1.l16.m1.1.1.1.cmml">←</mo><mrow id="alg1.l16.m1.1.1.3" xref="alg1.l16.m1.1.1.3.cmml"><mi id="alg1.l16.m1.1.1.3.2" xref="alg1.l16.m1.1.1.3.2.cmml">B</mi><mo id="alg1.l16.m1.1.1.3.1" xref="alg1.l16.m1.1.1.3.1.cmml">−</mo><mn id="alg1.l16.m1.1.1.3.3" xref="alg1.l16.m1.1.1.3.3.cmml">1</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="alg1.l16.m1.1b"><apply id="alg1.l16.m1.1.1.cmml" xref="alg1.l16.m1.1.1"><ci id="alg1.l16.m1.1.1.1.cmml" xref="alg1.l16.m1.1.1.1">←</ci><ci id="alg1.l16.m1.1.1.2.cmml" xref="alg1.l16.m1.1.1.2">𝐵</ci><apply id="alg1.l16.m1.1.1.3.cmml" xref="alg1.l16.m1.1.1.3"><minus id="alg1.l16.m1.1.1.3.1.cmml" xref="alg1.l16.m1.1.1.3.1"></minus><ci id="alg1.l16.m1.1.1.3.2.cmml" xref="alg1.l16.m1.1.1.3.2">𝐵</ci><cn id="alg1.l16.m1.1.1.3.3.cmml" type="integer" xref="alg1.l16.m1.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="alg1.l16.m1.1c">B\leftarrow B-1</annotation><annotation encoding="application/x-llamapun" id="alg1.l16.m1.1d">italic_B ← italic_B - 1</annotation></semantics></math> </div> <div class="ltx_listingline" id="alg1.l17"> <span class="ltx_tag ltx_tag_listingline">17:</span> <span class="ltx_text ltx_font_bold" id="alg1.l17.1">if</span> <math alttext="B=0" class="ltx_Math" display="inline" id="alg1.l17.m1.1"><semantics id="alg1.l17.m1.1a"><mrow id="alg1.l17.m1.1.1" xref="alg1.l17.m1.1.1.cmml"><mi id="alg1.l17.m1.1.1.2" xref="alg1.l17.m1.1.1.2.cmml">B</mi><mo id="alg1.l17.m1.1.1.1" xref="alg1.l17.m1.1.1.1.cmml">=</mo><mn id="alg1.l17.m1.1.1.3" xref="alg1.l17.m1.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="alg1.l17.m1.1b"><apply id="alg1.l17.m1.1.1.cmml" xref="alg1.l17.m1.1.1"><eq id="alg1.l17.m1.1.1.1.cmml" xref="alg1.l17.m1.1.1.1"></eq><ci id="alg1.l17.m1.1.1.2.cmml" xref="alg1.l17.m1.1.1.2">𝐵</ci><cn id="alg1.l17.m1.1.1.3.cmml" type="integer" xref="alg1.l17.m1.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="alg1.l17.m1.1c">B=0</annotation><annotation encoding="application/x-llamapun" id="alg1.l17.m1.1d">italic_B = 0</annotation></semantics></math> <span class="ltx_text ltx_font_bold" id="alg1.l17.2">then</span> <span class="ltx_text ltx_font_bold" id="alg1.l17.3">break</span> </div> <div class="ltx_listingline" id="alg1.l18"> <span class="ltx_tag ltx_tag_listingline">18:</span> <span class="ltx_text ltx_font_bold" id="alg1.l18.1">end</span> <span class="ltx_text ltx_font_bold" id="alg1.l18.2">if</span> </div> <div class="ltx_listingline" id="alg1.l19"> <span class="ltx_tag ltx_tag_listingline">19:</span> <span class="ltx_text ltx_font_bold" id="alg1.l19.1">end</span> <span class="ltx_text ltx_font_bold" id="alg1.l19.2">for</span> </div> <div class="ltx_listingline" id="alg1.l20"> <span class="ltx_tag ltx_tag_listingline">20:</span> <span class="ltx_text ltx_font_bold" id="alg1.l20.1">end</span> <span class="ltx_text ltx_font_bold" id="alg1.l20.2">while</span> </div> <div class="ltx_listingline" id="alg1.l21"> <span class="ltx_tag ltx_tag_listingline">21:</span> Restore original order for <math alttext="\mathbf{A}" class="ltx_Math" display="inline" id="alg1.l21.m1.1"><semantics id="alg1.l21.m1.1a"><mi id="alg1.l21.m1.1.1" xref="alg1.l21.m1.1.1.cmml">𝐀</mi><annotation-xml encoding="MathML-Content" id="alg1.l21.m1.1b"><ci id="alg1.l21.m1.1.1.cmml" xref="alg1.l21.m1.1.1">𝐀</ci></annotation-xml><annotation encoding="application/x-tex" id="alg1.l21.m1.1c">\mathbf{A}</annotation><annotation encoding="application/x-llamapun" id="alg1.l21.m1.1d">bold_A</annotation></semantics></math> using <math alttext="\mathbf{I}" class="ltx_Math" display="inline" id="alg1.l21.m2.1"><semantics id="alg1.l21.m2.1a"><mi id="alg1.l21.m2.1.1" xref="alg1.l21.m2.1.1.cmml">𝐈</mi><annotation-xml encoding="MathML-Content" id="alg1.l21.m2.1b"><ci id="alg1.l21.m2.1.1.cmml" xref="alg1.l21.m2.1.1">𝐈</ci></annotation-xml><annotation encoding="application/x-tex" id="alg1.l21.m2.1c">\mathbf{I}</annotation><annotation encoding="application/x-llamapun" id="alg1.l21.m2.1d">bold_I</annotation></semantics></math> </div> <div class="ltx_listingline" id="alg1.l22"> <span class="ltx_tag ltx_tag_listingline">22:</span> <span class="ltx_text ltx_font_bold" id="alg1.l22.1">return</span> <math alttext="\mathbf{A}" class="ltx_Math" display="inline" id="alg1.l22.m1.1"><semantics id="alg1.l22.m1.1a"><mi id="alg1.l22.m1.1.1" xref="alg1.l22.m1.1.1.cmml">𝐀</mi><annotation-xml encoding="MathML-Content" id="alg1.l22.m1.1b"><ci id="alg1.l22.m1.1.1.cmml" xref="alg1.l22.m1.1.1">𝐀</ci></annotation-xml><annotation encoding="application/x-tex" id="alg1.l22.m1.1c">\mathbf{A}</annotation><annotation encoding="application/x-llamapun" id="alg1.l22.m1.1d">bold_A</annotation></semantics></math> </div> <div class="ltx_listingline" id="alg1.l23"> <span class="ltx_tag ltx_tag_listingline">23:</span><span class="ltx_text ltx_font_bold" id="alg1.l23.1">end</span> <span class="ltx_text ltx_font_bold" id="alg1.l23.2">function</span> </div> </div> </figure> <figure class="ltx_float ltx_float_algorithm ltx_framed ltx_framed_top" id="alg2"> <figcaption class="ltx_caption"><span class="ltx_tag ltx_tag_float"><span class="ltx_text ltx_font_bold" id="alg2.2.1.1">Algorithm 2</span> </span> In-Cluster Selection</figcaption> <div class="ltx_listing ltx_listing" id="alg2.3"> <div class="ltx_listingline" id="alg2.l1"> <span class="ltx_tag ltx_tag_listingline">1:</span><span class="ltx_text ltx_font_bold" id="alg2.l1.1">Input:</span> Cluster <math alttext="c_{i}=\{I_{1},I_{2},\dots,I_{n_{i}}\}" class="ltx_Math" display="inline" id="alg2.l1.m1.4"><semantics id="alg2.l1.m1.4a"><mrow id="alg2.l1.m1.4.4" xref="alg2.l1.m1.4.4.cmml"><msub id="alg2.l1.m1.4.4.5" xref="alg2.l1.m1.4.4.5.cmml"><mi id="alg2.l1.m1.4.4.5.2" xref="alg2.l1.m1.4.4.5.2.cmml">c</mi><mi id="alg2.l1.m1.4.4.5.3" xref="alg2.l1.m1.4.4.5.3.cmml">i</mi></msub><mo id="alg2.l1.m1.4.4.4" xref="alg2.l1.m1.4.4.4.cmml">=</mo><mrow id="alg2.l1.m1.4.4.3.3" xref="alg2.l1.m1.4.4.3.4.cmml"><mo id="alg2.l1.m1.4.4.3.3.4" stretchy="false" xref="alg2.l1.m1.4.4.3.4.cmml">{</mo><msub id="alg2.l1.m1.2.2.1.1.1" xref="alg2.l1.m1.2.2.1.1.1.cmml"><mi id="alg2.l1.m1.2.2.1.1.1.2" xref="alg2.l1.m1.2.2.1.1.1.2.cmml">I</mi><mn id="alg2.l1.m1.2.2.1.1.1.3" xref="alg2.l1.m1.2.2.1.1.1.3.cmml">1</mn></msub><mo id="alg2.l1.m1.4.4.3.3.5" xref="alg2.l1.m1.4.4.3.4.cmml">,</mo><msub id="alg2.l1.m1.3.3.2.2.2" xref="alg2.l1.m1.3.3.2.2.2.cmml"><mi id="alg2.l1.m1.3.3.2.2.2.2" xref="alg2.l1.m1.3.3.2.2.2.2.cmml">I</mi><mn id="alg2.l1.m1.3.3.2.2.2.3" xref="alg2.l1.m1.3.3.2.2.2.3.cmml">2</mn></msub><mo id="alg2.l1.m1.4.4.3.3.6" xref="alg2.l1.m1.4.4.3.4.cmml">,</mo><mi id="alg2.l1.m1.1.1" mathvariant="normal" xref="alg2.l1.m1.1.1.cmml">…</mi><mo id="alg2.l1.m1.4.4.3.3.7" xref="alg2.l1.m1.4.4.3.4.cmml">,</mo><msub id="alg2.l1.m1.4.4.3.3.3" xref="alg2.l1.m1.4.4.3.3.3.cmml"><mi id="alg2.l1.m1.4.4.3.3.3.2" xref="alg2.l1.m1.4.4.3.3.3.2.cmml">I</mi><msub id="alg2.l1.m1.4.4.3.3.3.3" xref="alg2.l1.m1.4.4.3.3.3.3.cmml"><mi id="alg2.l1.m1.4.4.3.3.3.3.2" xref="alg2.l1.m1.4.4.3.3.3.3.2.cmml">n</mi><mi id="alg2.l1.m1.4.4.3.3.3.3.3" xref="alg2.l1.m1.4.4.3.3.3.3.3.cmml">i</mi></msub></msub><mo id="alg2.l1.m1.4.4.3.3.8" stretchy="false" xref="alg2.l1.m1.4.4.3.4.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="alg2.l1.m1.4b"><apply id="alg2.l1.m1.4.4.cmml" xref="alg2.l1.m1.4.4"><eq id="alg2.l1.m1.4.4.4.cmml" xref="alg2.l1.m1.4.4.4"></eq><apply id="alg2.l1.m1.4.4.5.cmml" xref="alg2.l1.m1.4.4.5"><csymbol cd="ambiguous" id="alg2.l1.m1.4.4.5.1.cmml" xref="alg2.l1.m1.4.4.5">subscript</csymbol><ci id="alg2.l1.m1.4.4.5.2.cmml" xref="alg2.l1.m1.4.4.5.2">𝑐</ci><ci id="alg2.l1.m1.4.4.5.3.cmml" xref="alg2.l1.m1.4.4.5.3">𝑖</ci></apply><set id="alg2.l1.m1.4.4.3.4.cmml" xref="alg2.l1.m1.4.4.3.3"><apply id="alg2.l1.m1.2.2.1.1.1.cmml" xref="alg2.l1.m1.2.2.1.1.1"><csymbol cd="ambiguous" id="alg2.l1.m1.2.2.1.1.1.1.cmml" xref="alg2.l1.m1.2.2.1.1.1">subscript</csymbol><ci id="alg2.l1.m1.2.2.1.1.1.2.cmml" xref="alg2.l1.m1.2.2.1.1.1.2">𝐼</ci><cn id="alg2.l1.m1.2.2.1.1.1.3.cmml" type="integer" xref="alg2.l1.m1.2.2.1.1.1.3">1</cn></apply><apply id="alg2.l1.m1.3.3.2.2.2.cmml" xref="alg2.l1.m1.3.3.2.2.2"><csymbol cd="ambiguous" id="alg2.l1.m1.3.3.2.2.2.1.cmml" xref="alg2.l1.m1.3.3.2.2.2">subscript</csymbol><ci id="alg2.l1.m1.3.3.2.2.2.2.cmml" xref="alg2.l1.m1.3.3.2.2.2.2">𝐼</ci><cn id="alg2.l1.m1.3.3.2.2.2.3.cmml" type="integer" xref="alg2.l1.m1.3.3.2.2.2.3">2</cn></apply><ci id="alg2.l1.m1.1.1.cmml" xref="alg2.l1.m1.1.1">…</ci><apply id="alg2.l1.m1.4.4.3.3.3.cmml" xref="alg2.l1.m1.4.4.3.3.3"><csymbol cd="ambiguous" id="alg2.l1.m1.4.4.3.3.3.1.cmml" xref="alg2.l1.m1.4.4.3.3.3">subscript</csymbol><ci id="alg2.l1.m1.4.4.3.3.3.2.cmml" xref="alg2.l1.m1.4.4.3.3.3.2">𝐼</ci><apply id="alg2.l1.m1.4.4.3.3.3.3.cmml" xref="alg2.l1.m1.4.4.3.3.3.3"><csymbol cd="ambiguous" id="alg2.l1.m1.4.4.3.3.3.3.1.cmml" xref="alg2.l1.m1.4.4.3.3.3.3">subscript</csymbol><ci id="alg2.l1.m1.4.4.3.3.3.3.2.cmml" xref="alg2.l1.m1.4.4.3.3.3.3.2">𝑛</ci><ci id="alg2.l1.m1.4.4.3.3.3.3.3.cmml" xref="alg2.l1.m1.4.4.3.3.3.3.3">𝑖</ci></apply></apply></set></apply></annotation-xml><annotation encoding="application/x-tex" id="alg2.l1.m1.4c">c_{i}=\{I_{1},I_{2},\dots,I_{n_{i}}\}</annotation><annotation encoding="application/x-llamapun" id="alg2.l1.m1.4d">italic_c start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = { italic_I start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_I start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , … , italic_I start_POSTSUBSCRIPT italic_n start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUBSCRIPT }</annotation></semantics></math>, centroid <math alttext="\mathbf{\mu}_{i}" class="ltx_Math" display="inline" id="alg2.l1.m2.1"><semantics id="alg2.l1.m2.1a"><msub id="alg2.l1.m2.1.1" xref="alg2.l1.m2.1.1.cmml"><mi id="alg2.l1.m2.1.1.2" xref="alg2.l1.m2.1.1.2.cmml">μ</mi><mi id="alg2.l1.m2.1.1.3" xref="alg2.l1.m2.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="alg2.l1.m2.1b"><apply id="alg2.l1.m2.1.1.cmml" xref="alg2.l1.m2.1.1"><csymbol cd="ambiguous" id="alg2.l1.m2.1.1.1.cmml" xref="alg2.l1.m2.1.1">subscript</csymbol><ci id="alg2.l1.m2.1.1.2.cmml" xref="alg2.l1.m2.1.1.2">𝜇</ci><ci id="alg2.l1.m2.1.1.3.cmml" xref="alg2.l1.m2.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="alg2.l1.m2.1c">\mathbf{\mu}_{i}</annotation><annotation encoding="application/x-llamapun" id="alg2.l1.m2.1d">italic_μ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math>, selection size <math alttext="a_{i}" class="ltx_Math" display="inline" id="alg2.l1.m3.1"><semantics id="alg2.l1.m3.1a"><msub id="alg2.l1.m3.1.1" xref="alg2.l1.m3.1.1.cmml"><mi id="alg2.l1.m3.1.1.2" xref="alg2.l1.m3.1.1.2.cmml">a</mi><mi id="alg2.l1.m3.1.1.3" xref="alg2.l1.m3.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="alg2.l1.m3.1b"><apply id="alg2.l1.m3.1.1.cmml" xref="alg2.l1.m3.1.1"><csymbol cd="ambiguous" id="alg2.l1.m3.1.1.1.cmml" xref="alg2.l1.m3.1.1">subscript</csymbol><ci id="alg2.l1.m3.1.1.2.cmml" xref="alg2.l1.m3.1.1.2">𝑎</ci><ci id="alg2.l1.m3.1.1.3.cmml" xref="alg2.l1.m3.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="alg2.l1.m3.1c">a_{i}</annotation><annotation encoding="application/x-llamapun" id="alg2.l1.m3.1d">italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math>, weights <math alttext="w_{p}" class="ltx_Math" display="inline" id="alg2.l1.m4.1"><semantics id="alg2.l1.m4.1a"><msub id="alg2.l1.m4.1.1" xref="alg2.l1.m4.1.1.cmml"><mi id="alg2.l1.m4.1.1.2" xref="alg2.l1.m4.1.1.2.cmml">w</mi><mi id="alg2.l1.m4.1.1.3" xref="alg2.l1.m4.1.1.3.cmml">p</mi></msub><annotation-xml encoding="MathML-Content" id="alg2.l1.m4.1b"><apply id="alg2.l1.m4.1.1.cmml" xref="alg2.l1.m4.1.1"><csymbol cd="ambiguous" id="alg2.l1.m4.1.1.1.cmml" xref="alg2.l1.m4.1.1">subscript</csymbol><ci id="alg2.l1.m4.1.1.2.cmml" xref="alg2.l1.m4.1.1.2">𝑤</ci><ci id="alg2.l1.m4.1.1.3.cmml" xref="alg2.l1.m4.1.1.3">𝑝</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="alg2.l1.m4.1c">w_{p}</annotation><annotation encoding="application/x-llamapun" id="alg2.l1.m4.1d">italic_w start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="w_{d}" class="ltx_Math" display="inline" id="alg2.l1.m5.1"><semantics id="alg2.l1.m5.1a"><msub id="alg2.l1.m5.1.1" xref="alg2.l1.m5.1.1.cmml"><mi id="alg2.l1.m5.1.1.2" xref="alg2.l1.m5.1.1.2.cmml">w</mi><mi id="alg2.l1.m5.1.1.3" xref="alg2.l1.m5.1.1.3.cmml">d</mi></msub><annotation-xml encoding="MathML-Content" id="alg2.l1.m5.1b"><apply id="alg2.l1.m5.1.1.cmml" xref="alg2.l1.m5.1.1"><csymbol cd="ambiguous" id="alg2.l1.m5.1.1.1.cmml" xref="alg2.l1.m5.1.1">subscript</csymbol><ci id="alg2.l1.m5.1.1.2.cmml" xref="alg2.l1.m5.1.1.2">𝑤</ci><ci id="alg2.l1.m5.1.1.3.cmml" xref="alg2.l1.m5.1.1.3">𝑑</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="alg2.l1.m5.1c">w_{d}</annotation><annotation encoding="application/x-llamapun" id="alg2.l1.m5.1d">italic_w start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT</annotation></semantics></math>. </div> <div class="ltx_listingline" id="alg2.l2"> <span class="ltx_tag ltx_tag_listingline">2:</span><span class="ltx_text ltx_font_bold" id="alg2.l2.1">Output:</span> Sub-Behavior Sequence <math alttext="S_{i}^{*}" class="ltx_Math" display="inline" id="alg2.l2.m1.1"><semantics id="alg2.l2.m1.1a"><msubsup id="alg2.l2.m1.1.1" xref="alg2.l2.m1.1.1.cmml"><mi id="alg2.l2.m1.1.1.2.2" xref="alg2.l2.m1.1.1.2.2.cmml">S</mi><mi id="alg2.l2.m1.1.1.2.3" xref="alg2.l2.m1.1.1.2.3.cmml">i</mi><mo id="alg2.l2.m1.1.1.3" xref="alg2.l2.m1.1.1.3.cmml">∗</mo></msubsup><annotation-xml encoding="MathML-Content" id="alg2.l2.m1.1b"><apply id="alg2.l2.m1.1.1.cmml" xref="alg2.l2.m1.1.1"><csymbol cd="ambiguous" id="alg2.l2.m1.1.1.1.cmml" xref="alg2.l2.m1.1.1">superscript</csymbol><apply id="alg2.l2.m1.1.1.2.cmml" xref="alg2.l2.m1.1.1"><csymbol cd="ambiguous" id="alg2.l2.m1.1.1.2.1.cmml" xref="alg2.l2.m1.1.1">subscript</csymbol><ci id="alg2.l2.m1.1.1.2.2.cmml" xref="alg2.l2.m1.1.1.2.2">𝑆</ci><ci id="alg2.l2.m1.1.1.2.3.cmml" xref="alg2.l2.m1.1.1.2.3">𝑖</ci></apply><times id="alg2.l2.m1.1.1.3.cmml" xref="alg2.l2.m1.1.1.3"></times></apply></annotation-xml><annotation encoding="application/x-tex" id="alg2.l2.m1.1c">S_{i}^{*}</annotation><annotation encoding="application/x-llamapun" id="alg2.l2.m1.1d">italic_S start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math>. </div> <div class="ltx_listingline" id="alg2.l3"> <span class="ltx_tag ltx_tag_listingline">3:</span><span class="ltx_text ltx_font_bold" id="alg2.l3.1">function</span> <span class="ltx_text ltx_font_smallcaps" id="alg2.l3.2">DynamicSelect</span>(<math alttext="c_{i},\mathbf{\mu}_{i},a_{i},w_{p},w_{d}" class="ltx_Math" display="inline" id="alg2.l3.m1.5"><semantics id="alg2.l3.m1.5a"><mrow id="alg2.l3.m1.5.5.5" xref="alg2.l3.m1.5.5.6.cmml"><msub id="alg2.l3.m1.1.1.1.1" xref="alg2.l3.m1.1.1.1.1.cmml"><mi id="alg2.l3.m1.1.1.1.1.2" xref="alg2.l3.m1.1.1.1.1.2.cmml">c</mi><mi id="alg2.l3.m1.1.1.1.1.3" xref="alg2.l3.m1.1.1.1.1.3.cmml">i</mi></msub><mo id="alg2.l3.m1.5.5.5.6" xref="alg2.l3.m1.5.5.6.cmml">,</mo><msub id="alg2.l3.m1.2.2.2.2" xref="alg2.l3.m1.2.2.2.2.cmml"><mi id="alg2.l3.m1.2.2.2.2.2" xref="alg2.l3.m1.2.2.2.2.2.cmml">μ</mi><mi id="alg2.l3.m1.2.2.2.2.3" xref="alg2.l3.m1.2.2.2.2.3.cmml">i</mi></msub><mo id="alg2.l3.m1.5.5.5.7" xref="alg2.l3.m1.5.5.6.cmml">,</mo><msub id="alg2.l3.m1.3.3.3.3" xref="alg2.l3.m1.3.3.3.3.cmml"><mi id="alg2.l3.m1.3.3.3.3.2" xref="alg2.l3.m1.3.3.3.3.2.cmml">a</mi><mi id="alg2.l3.m1.3.3.3.3.3" xref="alg2.l3.m1.3.3.3.3.3.cmml">i</mi></msub><mo id="alg2.l3.m1.5.5.5.8" xref="alg2.l3.m1.5.5.6.cmml">,</mo><msub id="alg2.l3.m1.4.4.4.4" xref="alg2.l3.m1.4.4.4.4.cmml"><mi id="alg2.l3.m1.4.4.4.4.2" xref="alg2.l3.m1.4.4.4.4.2.cmml">w</mi><mi id="alg2.l3.m1.4.4.4.4.3" xref="alg2.l3.m1.4.4.4.4.3.cmml">p</mi></msub><mo id="alg2.l3.m1.5.5.5.9" xref="alg2.l3.m1.5.5.6.cmml">,</mo><msub id="alg2.l3.m1.5.5.5.5" xref="alg2.l3.m1.5.5.5.5.cmml"><mi id="alg2.l3.m1.5.5.5.5.2" xref="alg2.l3.m1.5.5.5.5.2.cmml">w</mi><mi id="alg2.l3.m1.5.5.5.5.3" xref="alg2.l3.m1.5.5.5.5.3.cmml">d</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="alg2.l3.m1.5b"><list id="alg2.l3.m1.5.5.6.cmml" xref="alg2.l3.m1.5.5.5"><apply id="alg2.l3.m1.1.1.1.1.cmml" xref="alg2.l3.m1.1.1.1.1"><csymbol cd="ambiguous" id="alg2.l3.m1.1.1.1.1.1.cmml" xref="alg2.l3.m1.1.1.1.1">subscript</csymbol><ci id="alg2.l3.m1.1.1.1.1.2.cmml" xref="alg2.l3.m1.1.1.1.1.2">𝑐</ci><ci id="alg2.l3.m1.1.1.1.1.3.cmml" xref="alg2.l3.m1.1.1.1.1.3">𝑖</ci></apply><apply id="alg2.l3.m1.2.2.2.2.cmml" xref="alg2.l3.m1.2.2.2.2"><csymbol cd="ambiguous" id="alg2.l3.m1.2.2.2.2.1.cmml" xref="alg2.l3.m1.2.2.2.2">subscript</csymbol><ci id="alg2.l3.m1.2.2.2.2.2.cmml" xref="alg2.l3.m1.2.2.2.2.2">𝜇</ci><ci id="alg2.l3.m1.2.2.2.2.3.cmml" xref="alg2.l3.m1.2.2.2.2.3">𝑖</ci></apply><apply id="alg2.l3.m1.3.3.3.3.cmml" xref="alg2.l3.m1.3.3.3.3"><csymbol cd="ambiguous" id="alg2.l3.m1.3.3.3.3.1.cmml" xref="alg2.l3.m1.3.3.3.3">subscript</csymbol><ci id="alg2.l3.m1.3.3.3.3.2.cmml" xref="alg2.l3.m1.3.3.3.3.2">𝑎</ci><ci id="alg2.l3.m1.3.3.3.3.3.cmml" xref="alg2.l3.m1.3.3.3.3.3">𝑖</ci></apply><apply id="alg2.l3.m1.4.4.4.4.cmml" xref="alg2.l3.m1.4.4.4.4"><csymbol cd="ambiguous" id="alg2.l3.m1.4.4.4.4.1.cmml" xref="alg2.l3.m1.4.4.4.4">subscript</csymbol><ci id="alg2.l3.m1.4.4.4.4.2.cmml" xref="alg2.l3.m1.4.4.4.4.2">𝑤</ci><ci id="alg2.l3.m1.4.4.4.4.3.cmml" xref="alg2.l3.m1.4.4.4.4.3">𝑝</ci></apply><apply id="alg2.l3.m1.5.5.5.5.cmml" xref="alg2.l3.m1.5.5.5.5"><csymbol cd="ambiguous" id="alg2.l3.m1.5.5.5.5.1.cmml" xref="alg2.l3.m1.5.5.5.5">subscript</csymbol><ci id="alg2.l3.m1.5.5.5.5.2.cmml" xref="alg2.l3.m1.5.5.5.5.2">𝑤</ci><ci id="alg2.l3.m1.5.5.5.5.3.cmml" xref="alg2.l3.m1.5.5.5.5.3">𝑑</ci></apply></list></annotation-xml><annotation encoding="application/x-tex" id="alg2.l3.m1.5c">c_{i},\mathbf{\mu}_{i},a_{i},w_{p},w_{d}</annotation><annotation encoding="application/x-llamapun" id="alg2.l3.m1.5d">italic_c start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , italic_μ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , italic_w start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT , italic_w start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT</annotation></semantics></math>) </div> <div class="ltx_listingline" id="alg2.l4"> <span class="ltx_tag ltx_tag_listingline">4:</span> Initialize <math alttext="c_{i}^{*}\leftarrow\emptyset" class="ltx_Math" display="inline" id="alg2.l4.m1.1"><semantics id="alg2.l4.m1.1a"><mrow id="alg2.l4.m1.1.1" xref="alg2.l4.m1.1.1.cmml"><msubsup id="alg2.l4.m1.1.1.2" xref="alg2.l4.m1.1.1.2.cmml"><mi id="alg2.l4.m1.1.1.2.2.2" xref="alg2.l4.m1.1.1.2.2.2.cmml">c</mi><mi id="alg2.l4.m1.1.1.2.2.3" xref="alg2.l4.m1.1.1.2.2.3.cmml">i</mi><mo id="alg2.l4.m1.1.1.2.3" xref="alg2.l4.m1.1.1.2.3.cmml">∗</mo></msubsup><mo id="alg2.l4.m1.1.1.1" stretchy="false" xref="alg2.l4.m1.1.1.1.cmml">←</mo><mi id="alg2.l4.m1.1.1.3" mathvariant="normal" xref="alg2.l4.m1.1.1.3.cmml">∅</mi></mrow><annotation-xml encoding="MathML-Content" id="alg2.l4.m1.1b"><apply id="alg2.l4.m1.1.1.cmml" xref="alg2.l4.m1.1.1"><ci id="alg2.l4.m1.1.1.1.cmml" xref="alg2.l4.m1.1.1.1">←</ci><apply id="alg2.l4.m1.1.1.2.cmml" xref="alg2.l4.m1.1.1.2"><csymbol cd="ambiguous" id="alg2.l4.m1.1.1.2.1.cmml" xref="alg2.l4.m1.1.1.2">superscript</csymbol><apply id="alg2.l4.m1.1.1.2.2.cmml" xref="alg2.l4.m1.1.1.2"><csymbol cd="ambiguous" id="alg2.l4.m1.1.1.2.2.1.cmml" xref="alg2.l4.m1.1.1.2">subscript</csymbol><ci id="alg2.l4.m1.1.1.2.2.2.cmml" xref="alg2.l4.m1.1.1.2.2.2">𝑐</ci><ci id="alg2.l4.m1.1.1.2.2.3.cmml" xref="alg2.l4.m1.1.1.2.2.3">𝑖</ci></apply><times id="alg2.l4.m1.1.1.2.3.cmml" xref="alg2.l4.m1.1.1.2.3"></times></apply><emptyset id="alg2.l4.m1.1.1.3.cmml" xref="alg2.l4.m1.1.1.3"></emptyset></apply></annotation-xml><annotation encoding="application/x-tex" id="alg2.l4.m1.1c">c_{i}^{*}\leftarrow\emptyset</annotation><annotation encoding="application/x-llamapun" id="alg2.l4.m1.1d">italic_c start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ← ∅</annotation></semantics></math> </div> <div class="ltx_listingline" id="alg2.l5"> <span class="ltx_tag ltx_tag_listingline">5:</span> Compute item embeddings <math alttext="\mathbf{E}(c_{i})=\{\mathbf{e}_{1},\mathbf{e}_{2},\dots,\mathbf{e}_{n_{i}}\}" class="ltx_Math" display="inline" id="alg2.l5.m1.5"><semantics id="alg2.l5.m1.5a"><mrow id="alg2.l5.m1.5.5" xref="alg2.l5.m1.5.5.cmml"><mrow id="alg2.l5.m1.2.2.1" xref="alg2.l5.m1.2.2.1.cmml"><mi id="alg2.l5.m1.2.2.1.3" xref="alg2.l5.m1.2.2.1.3.cmml">𝐄</mi><mo id="alg2.l5.m1.2.2.1.2" xref="alg2.l5.m1.2.2.1.2.cmml">⁢</mo><mrow id="alg2.l5.m1.2.2.1.1.1" xref="alg2.l5.m1.2.2.1.1.1.1.cmml"><mo id="alg2.l5.m1.2.2.1.1.1.2" stretchy="false" xref="alg2.l5.m1.2.2.1.1.1.1.cmml">(</mo><msub id="alg2.l5.m1.2.2.1.1.1.1" xref="alg2.l5.m1.2.2.1.1.1.1.cmml"><mi id="alg2.l5.m1.2.2.1.1.1.1.2" xref="alg2.l5.m1.2.2.1.1.1.1.2.cmml">c</mi><mi id="alg2.l5.m1.2.2.1.1.1.1.3" xref="alg2.l5.m1.2.2.1.1.1.1.3.cmml">i</mi></msub><mo id="alg2.l5.m1.2.2.1.1.1.3" stretchy="false" xref="alg2.l5.m1.2.2.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="alg2.l5.m1.5.5.5" xref="alg2.l5.m1.5.5.5.cmml">=</mo><mrow id="alg2.l5.m1.5.5.4.3" xref="alg2.l5.m1.5.5.4.4.cmml"><mo id="alg2.l5.m1.5.5.4.3.4" stretchy="false" xref="alg2.l5.m1.5.5.4.4.cmml">{</mo><msub id="alg2.l5.m1.3.3.2.1.1" xref="alg2.l5.m1.3.3.2.1.1.cmml"><mi id="alg2.l5.m1.3.3.2.1.1.2" xref="alg2.l5.m1.3.3.2.1.1.2.cmml">𝐞</mi><mn id="alg2.l5.m1.3.3.2.1.1.3" xref="alg2.l5.m1.3.3.2.1.1.3.cmml">1</mn></msub><mo id="alg2.l5.m1.5.5.4.3.5" xref="alg2.l5.m1.5.5.4.4.cmml">,</mo><msub id="alg2.l5.m1.4.4.3.2.2" xref="alg2.l5.m1.4.4.3.2.2.cmml"><mi id="alg2.l5.m1.4.4.3.2.2.2" xref="alg2.l5.m1.4.4.3.2.2.2.cmml">𝐞</mi><mn id="alg2.l5.m1.4.4.3.2.2.3" xref="alg2.l5.m1.4.4.3.2.2.3.cmml">2</mn></msub><mo id="alg2.l5.m1.5.5.4.3.6" xref="alg2.l5.m1.5.5.4.4.cmml">,</mo><mi id="alg2.l5.m1.1.1" mathvariant="normal" xref="alg2.l5.m1.1.1.cmml">…</mi><mo id="alg2.l5.m1.5.5.4.3.7" xref="alg2.l5.m1.5.5.4.4.cmml">,</mo><msub id="alg2.l5.m1.5.5.4.3.3" xref="alg2.l5.m1.5.5.4.3.3.cmml"><mi id="alg2.l5.m1.5.5.4.3.3.2" xref="alg2.l5.m1.5.5.4.3.3.2.cmml">𝐞</mi><msub id="alg2.l5.m1.5.5.4.3.3.3" xref="alg2.l5.m1.5.5.4.3.3.3.cmml"><mi id="alg2.l5.m1.5.5.4.3.3.3.2" xref="alg2.l5.m1.5.5.4.3.3.3.2.cmml">n</mi><mi id="alg2.l5.m1.5.5.4.3.3.3.3" xref="alg2.l5.m1.5.5.4.3.3.3.3.cmml">i</mi></msub></msub><mo id="alg2.l5.m1.5.5.4.3.8" stretchy="false" xref="alg2.l5.m1.5.5.4.4.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="alg2.l5.m1.5b"><apply id="alg2.l5.m1.5.5.cmml" xref="alg2.l5.m1.5.5"><eq id="alg2.l5.m1.5.5.5.cmml" xref="alg2.l5.m1.5.5.5"></eq><apply id="alg2.l5.m1.2.2.1.cmml" xref="alg2.l5.m1.2.2.1"><times id="alg2.l5.m1.2.2.1.2.cmml" xref="alg2.l5.m1.2.2.1.2"></times><ci id="alg2.l5.m1.2.2.1.3.cmml" xref="alg2.l5.m1.2.2.1.3">𝐄</ci><apply id="alg2.l5.m1.2.2.1.1.1.1.cmml" xref="alg2.l5.m1.2.2.1.1.1"><csymbol cd="ambiguous" id="alg2.l5.m1.2.2.1.1.1.1.1.cmml" xref="alg2.l5.m1.2.2.1.1.1">subscript</csymbol><ci id="alg2.l5.m1.2.2.1.1.1.1.2.cmml" xref="alg2.l5.m1.2.2.1.1.1.1.2">𝑐</ci><ci id="alg2.l5.m1.2.2.1.1.1.1.3.cmml" xref="alg2.l5.m1.2.2.1.1.1.1.3">𝑖</ci></apply></apply><set id="alg2.l5.m1.5.5.4.4.cmml" xref="alg2.l5.m1.5.5.4.3"><apply id="alg2.l5.m1.3.3.2.1.1.cmml" xref="alg2.l5.m1.3.3.2.1.1"><csymbol cd="ambiguous" id="alg2.l5.m1.3.3.2.1.1.1.cmml" xref="alg2.l5.m1.3.3.2.1.1">subscript</csymbol><ci id="alg2.l5.m1.3.3.2.1.1.2.cmml" xref="alg2.l5.m1.3.3.2.1.1.2">𝐞</ci><cn id="alg2.l5.m1.3.3.2.1.1.3.cmml" type="integer" xref="alg2.l5.m1.3.3.2.1.1.3">1</cn></apply><apply id="alg2.l5.m1.4.4.3.2.2.cmml" xref="alg2.l5.m1.4.4.3.2.2"><csymbol cd="ambiguous" id="alg2.l5.m1.4.4.3.2.2.1.cmml" xref="alg2.l5.m1.4.4.3.2.2">subscript</csymbol><ci id="alg2.l5.m1.4.4.3.2.2.2.cmml" xref="alg2.l5.m1.4.4.3.2.2.2">𝐞</ci><cn id="alg2.l5.m1.4.4.3.2.2.3.cmml" type="integer" xref="alg2.l5.m1.4.4.3.2.2.3">2</cn></apply><ci id="alg2.l5.m1.1.1.cmml" xref="alg2.l5.m1.1.1">…</ci><apply id="alg2.l5.m1.5.5.4.3.3.cmml" xref="alg2.l5.m1.5.5.4.3.3"><csymbol cd="ambiguous" id="alg2.l5.m1.5.5.4.3.3.1.cmml" xref="alg2.l5.m1.5.5.4.3.3">subscript</csymbol><ci id="alg2.l5.m1.5.5.4.3.3.2.cmml" xref="alg2.l5.m1.5.5.4.3.3.2">𝐞</ci><apply id="alg2.l5.m1.5.5.4.3.3.3.cmml" xref="alg2.l5.m1.5.5.4.3.3.3"><csymbol cd="ambiguous" id="alg2.l5.m1.5.5.4.3.3.3.1.cmml" xref="alg2.l5.m1.5.5.4.3.3.3">subscript</csymbol><ci id="alg2.l5.m1.5.5.4.3.3.3.2.cmml" xref="alg2.l5.m1.5.5.4.3.3.3.2">𝑛</ci><ci id="alg2.l5.m1.5.5.4.3.3.3.3.cmml" xref="alg2.l5.m1.5.5.4.3.3.3.3">𝑖</ci></apply></apply></set></apply></annotation-xml><annotation encoding="application/x-tex" id="alg2.l5.m1.5c">\mathbf{E}(c_{i})=\{\mathbf{e}_{1},\mathbf{e}_{2},\dots,\mathbf{e}_{n_{i}}\}</annotation><annotation encoding="application/x-llamapun" id="alg2.l5.m1.5d">bold_E ( italic_c start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) = { bold_e start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , bold_e start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , … , bold_e start_POSTSUBSCRIPT italic_n start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUBSCRIPT }</annotation></semantics></math>, where <math alttext="\mathbf{e}_{j}" class="ltx_Math" display="inline" id="alg2.l5.m2.1"><semantics id="alg2.l5.m2.1a"><msub id="alg2.l5.m2.1.1" xref="alg2.l5.m2.1.1.cmml"><mi id="alg2.l5.m2.1.1.2" xref="alg2.l5.m2.1.1.2.cmml">𝐞</mi><mi id="alg2.l5.m2.1.1.3" xref="alg2.l5.m2.1.1.3.cmml">j</mi></msub><annotation-xml encoding="MathML-Content" id="alg2.l5.m2.1b"><apply id="alg2.l5.m2.1.1.cmml" xref="alg2.l5.m2.1.1"><csymbol cd="ambiguous" id="alg2.l5.m2.1.1.1.cmml" xref="alg2.l5.m2.1.1">subscript</csymbol><ci id="alg2.l5.m2.1.1.2.cmml" xref="alg2.l5.m2.1.1.2">𝐞</ci><ci id="alg2.l5.m2.1.1.3.cmml" xref="alg2.l5.m2.1.1.3">𝑗</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="alg2.l5.m2.1c">\mathbf{e}_{j}</annotation><annotation encoding="application/x-llamapun" id="alg2.l5.m2.1d">bold_e start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT</annotation></semantics></math> is the embedding of item <math alttext="I_{j}\in c_{i}" class="ltx_Math" display="inline" id="alg2.l5.m3.1"><semantics id="alg2.l5.m3.1a"><mrow id="alg2.l5.m3.1.1" xref="alg2.l5.m3.1.1.cmml"><msub id="alg2.l5.m3.1.1.2" xref="alg2.l5.m3.1.1.2.cmml"><mi id="alg2.l5.m3.1.1.2.2" xref="alg2.l5.m3.1.1.2.2.cmml">I</mi><mi id="alg2.l5.m3.1.1.2.3" xref="alg2.l5.m3.1.1.2.3.cmml">j</mi></msub><mo id="alg2.l5.m3.1.1.1" xref="alg2.l5.m3.1.1.1.cmml">∈</mo><msub id="alg2.l5.m3.1.1.3" xref="alg2.l5.m3.1.1.3.cmml"><mi id="alg2.l5.m3.1.1.3.2" xref="alg2.l5.m3.1.1.3.2.cmml">c</mi><mi id="alg2.l5.m3.1.1.3.3" xref="alg2.l5.m3.1.1.3.3.cmml">i</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="alg2.l5.m3.1b"><apply id="alg2.l5.m3.1.1.cmml" xref="alg2.l5.m3.1.1"><in id="alg2.l5.m3.1.1.1.cmml" xref="alg2.l5.m3.1.1.1"></in><apply id="alg2.l5.m3.1.1.2.cmml" xref="alg2.l5.m3.1.1.2"><csymbol cd="ambiguous" id="alg2.l5.m3.1.1.2.1.cmml" xref="alg2.l5.m3.1.1.2">subscript</csymbol><ci id="alg2.l5.m3.1.1.2.2.cmml" xref="alg2.l5.m3.1.1.2.2">𝐼</ci><ci id="alg2.l5.m3.1.1.2.3.cmml" xref="alg2.l5.m3.1.1.2.3">𝑗</ci></apply><apply id="alg2.l5.m3.1.1.3.cmml" xref="alg2.l5.m3.1.1.3"><csymbol cd="ambiguous" id="alg2.l5.m3.1.1.3.1.cmml" xref="alg2.l5.m3.1.1.3">subscript</csymbol><ci id="alg2.l5.m3.1.1.3.2.cmml" xref="alg2.l5.m3.1.1.3.2">𝑐</ci><ci id="alg2.l5.m3.1.1.3.3.cmml" xref="alg2.l5.m3.1.1.3.3">𝑖</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="alg2.l5.m3.1c">I_{j}\in c_{i}</annotation><annotation encoding="application/x-llamapun" id="alg2.l5.m3.1d">italic_I start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ∈ italic_c start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math>. </div> <div class="ltx_listingline" id="alg2.l6"> <span class="ltx_tag ltx_tag_listingline">6:</span> Select the initial item: <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="\mathbf{e}_{\text{init}}=\arg\min_{\mathbf{e}_{j}\in\mathbf{E}(c_{i})}d(% \mathbf{e}_{j},\mathbf{\mu}_{i})" class="ltx_Math" display="block" id="S3.Ex1.m1.3"><semantics id="S3.Ex1.m1.3a"><mrow id="S3.Ex1.m1.3.3" xref="S3.Ex1.m1.3.3.cmml"><msub id="S3.Ex1.m1.3.3.4" xref="S3.Ex1.m1.3.3.4.cmml"><mi id="S3.Ex1.m1.3.3.4.2" xref="S3.Ex1.m1.3.3.4.2.cmml">𝐞</mi><mtext id="S3.Ex1.m1.3.3.4.3" xref="S3.Ex1.m1.3.3.4.3a.cmml">init</mtext></msub><mo id="S3.Ex1.m1.3.3.3" xref="S3.Ex1.m1.3.3.3.cmml">=</mo><mrow id="S3.Ex1.m1.3.3.2" xref="S3.Ex1.m1.3.3.2.cmml"><mrow id="S3.Ex1.m1.3.3.2.4" xref="S3.Ex1.m1.3.3.2.4.cmml"><mi id="S3.Ex1.m1.3.3.2.4.1" xref="S3.Ex1.m1.3.3.2.4.1.cmml">arg</mi><mo id="S3.Ex1.m1.3.3.2.4a" lspace="0.167em" xref="S3.Ex1.m1.3.3.2.4.cmml">⁡</mo><mrow id="S3.Ex1.m1.3.3.2.4.2" xref="S3.Ex1.m1.3.3.2.4.2.cmml"><munder id="S3.Ex1.m1.3.3.2.4.2.1" xref="S3.Ex1.m1.3.3.2.4.2.1.cmml"><mi id="S3.Ex1.m1.3.3.2.4.2.1.2" xref="S3.Ex1.m1.3.3.2.4.2.1.2.cmml">min</mi><mrow id="S3.Ex1.m1.1.1.1" xref="S3.Ex1.m1.1.1.1.cmml"><msub id="S3.Ex1.m1.1.1.1.3" xref="S3.Ex1.m1.1.1.1.3.cmml"><mi id="S3.Ex1.m1.1.1.1.3.2" xref="S3.Ex1.m1.1.1.1.3.2.cmml">𝐞</mi><mi id="S3.Ex1.m1.1.1.1.3.3" xref="S3.Ex1.m1.1.1.1.3.3.cmml">j</mi></msub><mo id="S3.Ex1.m1.1.1.1.2" xref="S3.Ex1.m1.1.1.1.2.cmml">∈</mo><mrow id="S3.Ex1.m1.1.1.1.1" xref="S3.Ex1.m1.1.1.1.1.cmml"><mi id="S3.Ex1.m1.1.1.1.1.3" xref="S3.Ex1.m1.1.1.1.1.3.cmml">𝐄</mi><mo id="S3.Ex1.m1.1.1.1.1.2" xref="S3.Ex1.m1.1.1.1.1.2.cmml">⁢</mo><mrow id="S3.Ex1.m1.1.1.1.1.1.1" xref="S3.Ex1.m1.1.1.1.1.1.1.1.cmml"><mo id="S3.Ex1.m1.1.1.1.1.1.1.2" stretchy="false" xref="S3.Ex1.m1.1.1.1.1.1.1.1.cmml">(</mo><msub id="S3.Ex1.m1.1.1.1.1.1.1.1" xref="S3.Ex1.m1.1.1.1.1.1.1.1.cmml"><mi id="S3.Ex1.m1.1.1.1.1.1.1.1.2" xref="S3.Ex1.m1.1.1.1.1.1.1.1.2.cmml">c</mi><mi id="S3.Ex1.m1.1.1.1.1.1.1.1.3" xref="S3.Ex1.m1.1.1.1.1.1.1.1.3.cmml">i</mi></msub><mo id="S3.Ex1.m1.1.1.1.1.1.1.3" stretchy="false" xref="S3.Ex1.m1.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow></munder><mo id="S3.Ex1.m1.3.3.2.4.2a" lspace="0.167em" xref="S3.Ex1.m1.3.3.2.4.2.cmml">⁡</mo><mi id="S3.Ex1.m1.3.3.2.4.2.2" xref="S3.Ex1.m1.3.3.2.4.2.2.cmml">d</mi></mrow></mrow><mo id="S3.Ex1.m1.3.3.2.3" xref="S3.Ex1.m1.3.3.2.3.cmml">⁢</mo><mrow id="S3.Ex1.m1.3.3.2.2.2" xref="S3.Ex1.m1.3.3.2.2.3.cmml"><mo id="S3.Ex1.m1.3.3.2.2.2.3" stretchy="false" xref="S3.Ex1.m1.3.3.2.2.3.cmml">(</mo><msub id="S3.Ex1.m1.2.2.1.1.1.1" xref="S3.Ex1.m1.2.2.1.1.1.1.cmml"><mi id="S3.Ex1.m1.2.2.1.1.1.1.2" xref="S3.Ex1.m1.2.2.1.1.1.1.2.cmml">𝐞</mi><mi id="S3.Ex1.m1.2.2.1.1.1.1.3" xref="S3.Ex1.m1.2.2.1.1.1.1.3.cmml">j</mi></msub><mo id="S3.Ex1.m1.3.3.2.2.2.4" xref="S3.Ex1.m1.3.3.2.2.3.cmml">,</mo><msub id="S3.Ex1.m1.3.3.2.2.2.2" xref="S3.Ex1.m1.3.3.2.2.2.2.cmml"><mi id="S3.Ex1.m1.3.3.2.2.2.2.2" xref="S3.Ex1.m1.3.3.2.2.2.2.2.cmml">μ</mi><mi id="S3.Ex1.m1.3.3.2.2.2.2.3" xref="S3.Ex1.m1.3.3.2.2.2.2.3.cmml">i</mi></msub><mo id="S3.Ex1.m1.3.3.2.2.2.5" stretchy="false" xref="S3.Ex1.m1.3.3.2.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Ex1.m1.3b"><apply id="S3.Ex1.m1.3.3.cmml" xref="S3.Ex1.m1.3.3"><eq id="S3.Ex1.m1.3.3.3.cmml" xref="S3.Ex1.m1.3.3.3"></eq><apply id="S3.Ex1.m1.3.3.4.cmml" xref="S3.Ex1.m1.3.3.4"><csymbol cd="ambiguous" id="S3.Ex1.m1.3.3.4.1.cmml" xref="S3.Ex1.m1.3.3.4">subscript</csymbol><ci id="S3.Ex1.m1.3.3.4.2.cmml" xref="S3.Ex1.m1.3.3.4.2">𝐞</ci><ci id="S3.Ex1.m1.3.3.4.3a.cmml" xref="S3.Ex1.m1.3.3.4.3"><mtext id="S3.Ex1.m1.3.3.4.3.cmml" mathsize="70%" xref="S3.Ex1.m1.3.3.4.3">init</mtext></ci></apply><apply id="S3.Ex1.m1.3.3.2.cmml" xref="S3.Ex1.m1.3.3.2"><times id="S3.Ex1.m1.3.3.2.3.cmml" xref="S3.Ex1.m1.3.3.2.3"></times><apply id="S3.Ex1.m1.3.3.2.4.cmml" xref="S3.Ex1.m1.3.3.2.4"><arg id="S3.Ex1.m1.3.3.2.4.1.cmml" xref="S3.Ex1.m1.3.3.2.4.1"></arg><apply id="S3.Ex1.m1.3.3.2.4.2.cmml" xref="S3.Ex1.m1.3.3.2.4.2"><apply id="S3.Ex1.m1.3.3.2.4.2.1.cmml" xref="S3.Ex1.m1.3.3.2.4.2.1"><csymbol cd="ambiguous" id="S3.Ex1.m1.3.3.2.4.2.1.1.cmml" xref="S3.Ex1.m1.3.3.2.4.2.1">subscript</csymbol><min id="S3.Ex1.m1.3.3.2.4.2.1.2.cmml" xref="S3.Ex1.m1.3.3.2.4.2.1.2"></min><apply id="S3.Ex1.m1.1.1.1.cmml" xref="S3.Ex1.m1.1.1.1"><in id="S3.Ex1.m1.1.1.1.2.cmml" xref="S3.Ex1.m1.1.1.1.2"></in><apply id="S3.Ex1.m1.1.1.1.3.cmml" xref="S3.Ex1.m1.1.1.1.3"><csymbol cd="ambiguous" id="S3.Ex1.m1.1.1.1.3.1.cmml" xref="S3.Ex1.m1.1.1.1.3">subscript</csymbol><ci id="S3.Ex1.m1.1.1.1.3.2.cmml" xref="S3.Ex1.m1.1.1.1.3.2">𝐞</ci><ci id="S3.Ex1.m1.1.1.1.3.3.cmml" xref="S3.Ex1.m1.1.1.1.3.3">𝑗</ci></apply><apply id="S3.Ex1.m1.1.1.1.1.cmml" xref="S3.Ex1.m1.1.1.1.1"><times id="S3.Ex1.m1.1.1.1.1.2.cmml" xref="S3.Ex1.m1.1.1.1.1.2"></times><ci id="S3.Ex1.m1.1.1.1.1.3.cmml" xref="S3.Ex1.m1.1.1.1.1.3">𝐄</ci><apply id="S3.Ex1.m1.1.1.1.1.1.1.1.cmml" xref="S3.Ex1.m1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S3.Ex1.m1.1.1.1.1.1.1.1.1.cmml" xref="S3.Ex1.m1.1.1.1.1.1.1">subscript</csymbol><ci id="S3.Ex1.m1.1.1.1.1.1.1.1.2.cmml" xref="S3.Ex1.m1.1.1.1.1.1.1.1.2">𝑐</ci><ci id="S3.Ex1.m1.1.1.1.1.1.1.1.3.cmml" xref="S3.Ex1.m1.1.1.1.1.1.1.1.3">𝑖</ci></apply></apply></apply></apply><ci id="S3.Ex1.m1.3.3.2.4.2.2.cmml" xref="S3.Ex1.m1.3.3.2.4.2.2">𝑑</ci></apply></apply><interval closure="open" id="S3.Ex1.m1.3.3.2.2.3.cmml" xref="S3.Ex1.m1.3.3.2.2.2"><apply id="S3.Ex1.m1.2.2.1.1.1.1.cmml" xref="S3.Ex1.m1.2.2.1.1.1.1"><csymbol cd="ambiguous" id="S3.Ex1.m1.2.2.1.1.1.1.1.cmml" xref="S3.Ex1.m1.2.2.1.1.1.1">subscript</csymbol><ci id="S3.Ex1.m1.2.2.1.1.1.1.2.cmml" xref="S3.Ex1.m1.2.2.1.1.1.1.2">𝐞</ci><ci id="S3.Ex1.m1.2.2.1.1.1.1.3.cmml" xref="S3.Ex1.m1.2.2.1.1.1.1.3">𝑗</ci></apply><apply id="S3.Ex1.m1.3.3.2.2.2.2.cmml" xref="S3.Ex1.m1.3.3.2.2.2.2"><csymbol cd="ambiguous" id="S3.Ex1.m1.3.3.2.2.2.2.1.cmml" xref="S3.Ex1.m1.3.3.2.2.2.2">subscript</csymbol><ci id="S3.Ex1.m1.3.3.2.2.2.2.2.cmml" xref="S3.Ex1.m1.3.3.2.2.2.2.2">𝜇</ci><ci id="S3.Ex1.m1.3.3.2.2.2.2.3.cmml" xref="S3.Ex1.m1.3.3.2.2.2.2.3">𝑖</ci></apply></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Ex1.m1.3c">\mathbf{e}_{\text{init}}=\arg\min_{\mathbf{e}_{j}\in\mathbf{E}(c_{i})}d(% \mathbf{e}_{j},\mathbf{\mu}_{i})</annotation><annotation encoding="application/x-llamapun" id="S3.Ex1.m1.3d">bold_e start_POSTSUBSCRIPT init end_POSTSUBSCRIPT = roman_arg roman_min start_POSTSUBSCRIPT bold_e start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ∈ bold_E ( italic_c start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) end_POSTSUBSCRIPT italic_d ( bold_e start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT , italic_μ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> </div> <div class="ltx_listingline" id="alg2.l7"> <span class="ltx_tag ltx_tag_listingline">7:</span> Update <math alttext="c_{i}^{*}\leftarrow c_{i}^{*}\cup\{\mathbf{e}_{\text{init}}\}" class="ltx_Math" display="inline" id="alg2.l7.m1.1"><semantics id="alg2.l7.m1.1a"><mrow id="alg2.l7.m1.1.1" xref="alg2.l7.m1.1.1.cmml"><msubsup id="alg2.l7.m1.1.1.3" xref="alg2.l7.m1.1.1.3.cmml"><mi id="alg2.l7.m1.1.1.3.2.2" xref="alg2.l7.m1.1.1.3.2.2.cmml">c</mi><mi id="alg2.l7.m1.1.1.3.2.3" xref="alg2.l7.m1.1.1.3.2.3.cmml">i</mi><mo id="alg2.l7.m1.1.1.3.3" xref="alg2.l7.m1.1.1.3.3.cmml">∗</mo></msubsup><mo id="alg2.l7.m1.1.1.2" stretchy="false" xref="alg2.l7.m1.1.1.2.cmml">←</mo><mrow id="alg2.l7.m1.1.1.1" xref="alg2.l7.m1.1.1.1.cmml"><msubsup id="alg2.l7.m1.1.1.1.3" xref="alg2.l7.m1.1.1.1.3.cmml"><mi id="alg2.l7.m1.1.1.1.3.2.2" xref="alg2.l7.m1.1.1.1.3.2.2.cmml">c</mi><mi id="alg2.l7.m1.1.1.1.3.2.3" xref="alg2.l7.m1.1.1.1.3.2.3.cmml">i</mi><mo id="alg2.l7.m1.1.1.1.3.3" xref="alg2.l7.m1.1.1.1.3.3.cmml">∗</mo></msubsup><mo id="alg2.l7.m1.1.1.1.2" xref="alg2.l7.m1.1.1.1.2.cmml">∪</mo><mrow id="alg2.l7.m1.1.1.1.1.1" xref="alg2.l7.m1.1.1.1.1.2.cmml"><mo id="alg2.l7.m1.1.1.1.1.1.2" stretchy="false" xref="alg2.l7.m1.1.1.1.1.2.cmml">{</mo><msub id="alg2.l7.m1.1.1.1.1.1.1" xref="alg2.l7.m1.1.1.1.1.1.1.cmml"><mi id="alg2.l7.m1.1.1.1.1.1.1.2" xref="alg2.l7.m1.1.1.1.1.1.1.2.cmml">𝐞</mi><mtext id="alg2.l7.m1.1.1.1.1.1.1.3" xref="alg2.l7.m1.1.1.1.1.1.1.3a.cmml">init</mtext></msub><mo id="alg2.l7.m1.1.1.1.1.1.3" stretchy="false" xref="alg2.l7.m1.1.1.1.1.2.cmml">}</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="alg2.l7.m1.1b"><apply id="alg2.l7.m1.1.1.cmml" xref="alg2.l7.m1.1.1"><ci id="alg2.l7.m1.1.1.2.cmml" xref="alg2.l7.m1.1.1.2">←</ci><apply id="alg2.l7.m1.1.1.3.cmml" xref="alg2.l7.m1.1.1.3"><csymbol cd="ambiguous" id="alg2.l7.m1.1.1.3.1.cmml" xref="alg2.l7.m1.1.1.3">superscript</csymbol><apply id="alg2.l7.m1.1.1.3.2.cmml" xref="alg2.l7.m1.1.1.3"><csymbol cd="ambiguous" id="alg2.l7.m1.1.1.3.2.1.cmml" xref="alg2.l7.m1.1.1.3">subscript</csymbol><ci id="alg2.l7.m1.1.1.3.2.2.cmml" xref="alg2.l7.m1.1.1.3.2.2">𝑐</ci><ci id="alg2.l7.m1.1.1.3.2.3.cmml" xref="alg2.l7.m1.1.1.3.2.3">𝑖</ci></apply><times id="alg2.l7.m1.1.1.3.3.cmml" xref="alg2.l7.m1.1.1.3.3"></times></apply><apply id="alg2.l7.m1.1.1.1.cmml" xref="alg2.l7.m1.1.1.1"><union id="alg2.l7.m1.1.1.1.2.cmml" xref="alg2.l7.m1.1.1.1.2"></union><apply id="alg2.l7.m1.1.1.1.3.cmml" xref="alg2.l7.m1.1.1.1.3"><csymbol cd="ambiguous" id="alg2.l7.m1.1.1.1.3.1.cmml" xref="alg2.l7.m1.1.1.1.3">superscript</csymbol><apply id="alg2.l7.m1.1.1.1.3.2.cmml" xref="alg2.l7.m1.1.1.1.3"><csymbol cd="ambiguous" id="alg2.l7.m1.1.1.1.3.2.1.cmml" xref="alg2.l7.m1.1.1.1.3">subscript</csymbol><ci id="alg2.l7.m1.1.1.1.3.2.2.cmml" xref="alg2.l7.m1.1.1.1.3.2.2">𝑐</ci><ci id="alg2.l7.m1.1.1.1.3.2.3.cmml" xref="alg2.l7.m1.1.1.1.3.2.3">𝑖</ci></apply><times id="alg2.l7.m1.1.1.1.3.3.cmml" xref="alg2.l7.m1.1.1.1.3.3"></times></apply><set id="alg2.l7.m1.1.1.1.1.2.cmml" xref="alg2.l7.m1.1.1.1.1.1"><apply id="alg2.l7.m1.1.1.1.1.1.1.cmml" xref="alg2.l7.m1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="alg2.l7.m1.1.1.1.1.1.1.1.cmml" xref="alg2.l7.m1.1.1.1.1.1.1">subscript</csymbol><ci id="alg2.l7.m1.1.1.1.1.1.1.2.cmml" xref="alg2.l7.m1.1.1.1.1.1.1.2">𝐞</ci><ci id="alg2.l7.m1.1.1.1.1.1.1.3a.cmml" xref="alg2.l7.m1.1.1.1.1.1.1.3"><mtext id="alg2.l7.m1.1.1.1.1.1.1.3.cmml" mathsize="70%" xref="alg2.l7.m1.1.1.1.1.1.1.3">init</mtext></ci></apply></set></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="alg2.l7.m1.1c">c_{i}^{*}\leftarrow c_{i}^{*}\cup\{\mathbf{e}_{\text{init}}\}</annotation><annotation encoding="application/x-llamapun" id="alg2.l7.m1.1d">italic_c start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ← italic_c start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ∪ { bold_e start_POSTSUBSCRIPT init end_POSTSUBSCRIPT }</annotation></semantics></math> and <math alttext="\mathbf{E}(c_{i})\leftarrow\mathbf{E}(c_{i})\setminus\{\mathbf{e}_{\text{init}}\}" class="ltx_Math" display="inline" id="alg2.l7.m2.3"><semantics id="alg2.l7.m2.3a"><mrow id="alg2.l7.m2.3.3" xref="alg2.l7.m2.3.3.cmml"><mrow id="alg2.l7.m2.1.1.1" xref="alg2.l7.m2.1.1.1.cmml"><mi id="alg2.l7.m2.1.1.1.3" xref="alg2.l7.m2.1.1.1.3.cmml">𝐄</mi><mo id="alg2.l7.m2.1.1.1.2" xref="alg2.l7.m2.1.1.1.2.cmml">⁢</mo><mrow id="alg2.l7.m2.1.1.1.1.1" xref="alg2.l7.m2.1.1.1.1.1.1.cmml"><mo id="alg2.l7.m2.1.1.1.1.1.2" stretchy="false" xref="alg2.l7.m2.1.1.1.1.1.1.cmml">(</mo><msub id="alg2.l7.m2.1.1.1.1.1.1" xref="alg2.l7.m2.1.1.1.1.1.1.cmml"><mi id="alg2.l7.m2.1.1.1.1.1.1.2" xref="alg2.l7.m2.1.1.1.1.1.1.2.cmml">c</mi><mi id="alg2.l7.m2.1.1.1.1.1.1.3" xref="alg2.l7.m2.1.1.1.1.1.1.3.cmml">i</mi></msub><mo id="alg2.l7.m2.1.1.1.1.1.3" stretchy="false" xref="alg2.l7.m2.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="alg2.l7.m2.3.3.4" stretchy="false" xref="alg2.l7.m2.3.3.4.cmml">←</mo><mrow id="alg2.l7.m2.3.3.3" xref="alg2.l7.m2.3.3.3.cmml"><mrow id="alg2.l7.m2.2.2.2.1" xref="alg2.l7.m2.2.2.2.1.cmml"><mi id="alg2.l7.m2.2.2.2.1.3" xref="alg2.l7.m2.2.2.2.1.3.cmml">𝐄</mi><mo id="alg2.l7.m2.2.2.2.1.2" xref="alg2.l7.m2.2.2.2.1.2.cmml">⁢</mo><mrow id="alg2.l7.m2.2.2.2.1.1.1" xref="alg2.l7.m2.2.2.2.1.1.1.1.cmml"><mo id="alg2.l7.m2.2.2.2.1.1.1.2" stretchy="false" xref="alg2.l7.m2.2.2.2.1.1.1.1.cmml">(</mo><msub id="alg2.l7.m2.2.2.2.1.1.1.1" xref="alg2.l7.m2.2.2.2.1.1.1.1.cmml"><mi id="alg2.l7.m2.2.2.2.1.1.1.1.2" xref="alg2.l7.m2.2.2.2.1.1.1.1.2.cmml">c</mi><mi id="alg2.l7.m2.2.2.2.1.1.1.1.3" xref="alg2.l7.m2.2.2.2.1.1.1.1.3.cmml">i</mi></msub><mo id="alg2.l7.m2.2.2.2.1.1.1.3" stretchy="false" xref="alg2.l7.m2.2.2.2.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="alg2.l7.m2.3.3.3.3" xref="alg2.l7.m2.3.3.3.3.cmml">∖</mo><mrow id="alg2.l7.m2.3.3.3.2.1" xref="alg2.l7.m2.3.3.3.2.2.cmml"><mo id="alg2.l7.m2.3.3.3.2.1.2" stretchy="false" xref="alg2.l7.m2.3.3.3.2.2.cmml">{</mo><msub id="alg2.l7.m2.3.3.3.2.1.1" xref="alg2.l7.m2.3.3.3.2.1.1.cmml"><mi id="alg2.l7.m2.3.3.3.2.1.1.2" xref="alg2.l7.m2.3.3.3.2.1.1.2.cmml">𝐞</mi><mtext id="alg2.l7.m2.3.3.3.2.1.1.3" xref="alg2.l7.m2.3.3.3.2.1.1.3a.cmml">init</mtext></msub><mo id="alg2.l7.m2.3.3.3.2.1.3" stretchy="false" xref="alg2.l7.m2.3.3.3.2.2.cmml">}</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="alg2.l7.m2.3b"><apply id="alg2.l7.m2.3.3.cmml" xref="alg2.l7.m2.3.3"><ci id="alg2.l7.m2.3.3.4.cmml" xref="alg2.l7.m2.3.3.4">←</ci><apply id="alg2.l7.m2.1.1.1.cmml" xref="alg2.l7.m2.1.1.1"><times id="alg2.l7.m2.1.1.1.2.cmml" xref="alg2.l7.m2.1.1.1.2"></times><ci id="alg2.l7.m2.1.1.1.3.cmml" xref="alg2.l7.m2.1.1.1.3">𝐄</ci><apply id="alg2.l7.m2.1.1.1.1.1.1.cmml" xref="alg2.l7.m2.1.1.1.1.1"><csymbol cd="ambiguous" id="alg2.l7.m2.1.1.1.1.1.1.1.cmml" xref="alg2.l7.m2.1.1.1.1.1">subscript</csymbol><ci id="alg2.l7.m2.1.1.1.1.1.1.2.cmml" xref="alg2.l7.m2.1.1.1.1.1.1.2">𝑐</ci><ci id="alg2.l7.m2.1.1.1.1.1.1.3.cmml" xref="alg2.l7.m2.1.1.1.1.1.1.3">𝑖</ci></apply></apply><apply id="alg2.l7.m2.3.3.3.cmml" xref="alg2.l7.m2.3.3.3"><setdiff id="alg2.l7.m2.3.3.3.3.cmml" xref="alg2.l7.m2.3.3.3.3"></setdiff><apply id="alg2.l7.m2.2.2.2.1.cmml" xref="alg2.l7.m2.2.2.2.1"><times id="alg2.l7.m2.2.2.2.1.2.cmml" xref="alg2.l7.m2.2.2.2.1.2"></times><ci id="alg2.l7.m2.2.2.2.1.3.cmml" xref="alg2.l7.m2.2.2.2.1.3">𝐄</ci><apply id="alg2.l7.m2.2.2.2.1.1.1.1.cmml" xref="alg2.l7.m2.2.2.2.1.1.1"><csymbol cd="ambiguous" id="alg2.l7.m2.2.2.2.1.1.1.1.1.cmml" xref="alg2.l7.m2.2.2.2.1.1.1">subscript</csymbol><ci id="alg2.l7.m2.2.2.2.1.1.1.1.2.cmml" xref="alg2.l7.m2.2.2.2.1.1.1.1.2">𝑐</ci><ci id="alg2.l7.m2.2.2.2.1.1.1.1.3.cmml" xref="alg2.l7.m2.2.2.2.1.1.1.1.3">𝑖</ci></apply></apply><set id="alg2.l7.m2.3.3.3.2.2.cmml" xref="alg2.l7.m2.3.3.3.2.1"><apply id="alg2.l7.m2.3.3.3.2.1.1.cmml" xref="alg2.l7.m2.3.3.3.2.1.1"><csymbol cd="ambiguous" id="alg2.l7.m2.3.3.3.2.1.1.1.cmml" xref="alg2.l7.m2.3.3.3.2.1.1">subscript</csymbol><ci id="alg2.l7.m2.3.3.3.2.1.1.2.cmml" xref="alg2.l7.m2.3.3.3.2.1.1.2">𝐞</ci><ci id="alg2.l7.m2.3.3.3.2.1.1.3a.cmml" xref="alg2.l7.m2.3.3.3.2.1.1.3"><mtext id="alg2.l7.m2.3.3.3.2.1.1.3.cmml" mathsize="70%" xref="alg2.l7.m2.3.3.3.2.1.1.3">init</mtext></ci></apply></set></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="alg2.l7.m2.3c">\mathbf{E}(c_{i})\leftarrow\mathbf{E}(c_{i})\setminus\{\mathbf{e}_{\text{init}}\}</annotation><annotation encoding="application/x-llamapun" id="alg2.l7.m2.3d">bold_E ( italic_c start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) ← bold_E ( italic_c start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) ∖ { bold_e start_POSTSUBSCRIPT init end_POSTSUBSCRIPT }</annotation></semantics></math>. </div> <div class="ltx_listingline" id="alg2.l8"> <span class="ltx_tag ltx_tag_listingline">8:</span> <span class="ltx_text ltx_font_bold" id="alg2.l8.1">while</span> <math alttext="|c_{i}^{*}|<a_{i}" class="ltx_Math" display="inline" id="alg2.l8.m1.1"><semantics id="alg2.l8.m1.1a"><mrow id="alg2.l8.m1.1.1" xref="alg2.l8.m1.1.1.cmml"><mrow id="alg2.l8.m1.1.1.1.1" xref="alg2.l8.m1.1.1.1.2.cmml"><mo id="alg2.l8.m1.1.1.1.1.2" stretchy="false" xref="alg2.l8.m1.1.1.1.2.1.cmml">|</mo><msubsup id="alg2.l8.m1.1.1.1.1.1" xref="alg2.l8.m1.1.1.1.1.1.cmml"><mi id="alg2.l8.m1.1.1.1.1.1.2.2" xref="alg2.l8.m1.1.1.1.1.1.2.2.cmml">c</mi><mi id="alg2.l8.m1.1.1.1.1.1.2.3" xref="alg2.l8.m1.1.1.1.1.1.2.3.cmml">i</mi><mo id="alg2.l8.m1.1.1.1.1.1.3" xref="alg2.l8.m1.1.1.1.1.1.3.cmml">∗</mo></msubsup><mo id="alg2.l8.m1.1.1.1.1.3" stretchy="false" xref="alg2.l8.m1.1.1.1.2.1.cmml">|</mo></mrow><mo id="alg2.l8.m1.1.1.2" xref="alg2.l8.m1.1.1.2.cmml"><</mo><msub id="alg2.l8.m1.1.1.3" xref="alg2.l8.m1.1.1.3.cmml"><mi id="alg2.l8.m1.1.1.3.2" xref="alg2.l8.m1.1.1.3.2.cmml">a</mi><mi id="alg2.l8.m1.1.1.3.3" xref="alg2.l8.m1.1.1.3.3.cmml">i</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="alg2.l8.m1.1b"><apply id="alg2.l8.m1.1.1.cmml" xref="alg2.l8.m1.1.1"><lt id="alg2.l8.m1.1.1.2.cmml" xref="alg2.l8.m1.1.1.2"></lt><apply id="alg2.l8.m1.1.1.1.2.cmml" xref="alg2.l8.m1.1.1.1.1"><abs id="alg2.l8.m1.1.1.1.2.1.cmml" xref="alg2.l8.m1.1.1.1.1.2"></abs><apply id="alg2.l8.m1.1.1.1.1.1.cmml" xref="alg2.l8.m1.1.1.1.1.1"><csymbol cd="ambiguous" id="alg2.l8.m1.1.1.1.1.1.1.cmml" xref="alg2.l8.m1.1.1.1.1.1">superscript</csymbol><apply id="alg2.l8.m1.1.1.1.1.1.2.cmml" xref="alg2.l8.m1.1.1.1.1.1"><csymbol cd="ambiguous" id="alg2.l8.m1.1.1.1.1.1.2.1.cmml" xref="alg2.l8.m1.1.1.1.1.1">subscript</csymbol><ci id="alg2.l8.m1.1.1.1.1.1.2.2.cmml" xref="alg2.l8.m1.1.1.1.1.1.2.2">𝑐</ci><ci id="alg2.l8.m1.1.1.1.1.1.2.3.cmml" xref="alg2.l8.m1.1.1.1.1.1.2.3">𝑖</ci></apply><times id="alg2.l8.m1.1.1.1.1.1.3.cmml" xref="alg2.l8.m1.1.1.1.1.1.3"></times></apply></apply><apply id="alg2.l8.m1.1.1.3.cmml" xref="alg2.l8.m1.1.1.3"><csymbol cd="ambiguous" id="alg2.l8.m1.1.1.3.1.cmml" xref="alg2.l8.m1.1.1.3">subscript</csymbol><ci id="alg2.l8.m1.1.1.3.2.cmml" xref="alg2.l8.m1.1.1.3.2">𝑎</ci><ci id="alg2.l8.m1.1.1.3.3.cmml" xref="alg2.l8.m1.1.1.3.3">𝑖</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="alg2.l8.m1.1c">|c_{i}^{*}|<a_{i}</annotation><annotation encoding="application/x-llamapun" id="alg2.l8.m1.1d">| italic_c start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT | < italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> <span class="ltx_text ltx_font_bold" id="alg2.l8.2">do</span> </div> <div class="ltx_listingline" id="alg2.l9"> <span class="ltx_tag ltx_tag_listingline">9:</span> <span class="ltx_text ltx_font_bold" id="alg2.l9.1">Compute Marginal Gains:</span> </div> <div class="ltx_listingline" id="alg2.l10"> <span class="ltx_tag ltx_tag_listingline">10:</span> <span class="ltx_text ltx_font_bold" id="alg2.l10.1">for all</span> <math alttext="\mathbf{e}_{j}\in\mathbf{E}(c_{i})" class="ltx_Math" display="inline" id="alg2.l10.m1.1"><semantics id="alg2.l10.m1.1a"><mrow id="alg2.l10.m1.1.1" xref="alg2.l10.m1.1.1.cmml"><msub id="alg2.l10.m1.1.1.3" xref="alg2.l10.m1.1.1.3.cmml"><mi id="alg2.l10.m1.1.1.3.2" xref="alg2.l10.m1.1.1.3.2.cmml">𝐞</mi><mi id="alg2.l10.m1.1.1.3.3" xref="alg2.l10.m1.1.1.3.3.cmml">j</mi></msub><mo id="alg2.l10.m1.1.1.2" xref="alg2.l10.m1.1.1.2.cmml">∈</mo><mrow id="alg2.l10.m1.1.1.1" xref="alg2.l10.m1.1.1.1.cmml"><mi id="alg2.l10.m1.1.1.1.3" xref="alg2.l10.m1.1.1.1.3.cmml">𝐄</mi><mo id="alg2.l10.m1.1.1.1.2" xref="alg2.l10.m1.1.1.1.2.cmml">⁢</mo><mrow id="alg2.l10.m1.1.1.1.1.1" xref="alg2.l10.m1.1.1.1.1.1.1.cmml"><mo id="alg2.l10.m1.1.1.1.1.1.2" stretchy="false" xref="alg2.l10.m1.1.1.1.1.1.1.cmml">(</mo><msub id="alg2.l10.m1.1.1.1.1.1.1" xref="alg2.l10.m1.1.1.1.1.1.1.cmml"><mi id="alg2.l10.m1.1.1.1.1.1.1.2" xref="alg2.l10.m1.1.1.1.1.1.1.2.cmml">c</mi><mi 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id="alg2.l10.m1.1.1.1.1.1.1.1.cmml" xref="alg2.l10.m1.1.1.1.1.1">subscript</csymbol><ci id="alg2.l10.m1.1.1.1.1.1.1.2.cmml" xref="alg2.l10.m1.1.1.1.1.1.1.2">𝑐</ci><ci id="alg2.l10.m1.1.1.1.1.1.1.3.cmml" xref="alg2.l10.m1.1.1.1.1.1.1.3">𝑖</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="alg2.l10.m1.1c">\mathbf{e}_{j}\in\mathbf{E}(c_{i})</annotation><annotation encoding="application/x-llamapun" id="alg2.l10.m1.1d">bold_e start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ∈ bold_E ( italic_c start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT )</annotation></semantics></math> <span class="ltx_text ltx_font_bold" id="alg2.l10.2">do</span> </div> <div class="ltx_listingline" id="alg2.l11"> <span class="ltx_tag ltx_tag_listingline">11:</span> Compute prototypicality gain: <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="g_{p}(\mathbf{e}_{j})=\frac{w_{p}}{1+d(\mathbf{e}_{j},\mathbf{\mu}_{i})}" class="ltx_Math" display="block" id="S3.Ex2.m1.3"><semantics id="S3.Ex2.m1.3a"><mrow id="S3.Ex2.m1.3.3" xref="S3.Ex2.m1.3.3.cmml"><mrow id="S3.Ex2.m1.3.3.1" xref="S3.Ex2.m1.3.3.1.cmml"><msub id="S3.Ex2.m1.3.3.1.3" xref="S3.Ex2.m1.3.3.1.3.cmml"><mi id="S3.Ex2.m1.3.3.1.3.2" xref="S3.Ex2.m1.3.3.1.3.2.cmml">g</mi><mi id="S3.Ex2.m1.3.3.1.3.3" xref="S3.Ex2.m1.3.3.1.3.3.cmml">p</mi></msub><mo id="S3.Ex2.m1.3.3.1.2" xref="S3.Ex2.m1.3.3.1.2.cmml">⁢</mo><mrow id="S3.Ex2.m1.3.3.1.1.1" xref="S3.Ex2.m1.3.3.1.1.1.1.cmml"><mo id="S3.Ex2.m1.3.3.1.1.1.2" stretchy="false" xref="S3.Ex2.m1.3.3.1.1.1.1.cmml">(</mo><msub id="S3.Ex2.m1.3.3.1.1.1.1" xref="S3.Ex2.m1.3.3.1.1.1.1.cmml"><mi id="S3.Ex2.m1.3.3.1.1.1.1.2" xref="S3.Ex2.m1.3.3.1.1.1.1.2.cmml">𝐞</mi><mi id="S3.Ex2.m1.3.3.1.1.1.1.3" xref="S3.Ex2.m1.3.3.1.1.1.1.3.cmml">j</mi></msub><mo id="S3.Ex2.m1.3.3.1.1.1.3" stretchy="false" xref="S3.Ex2.m1.3.3.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S3.Ex2.m1.3.3.2" xref="S3.Ex2.m1.3.3.2.cmml">=</mo><mfrac id="S3.Ex2.m1.2.2" xref="S3.Ex2.m1.2.2.cmml"><msub id="S3.Ex2.m1.2.2.4" xref="S3.Ex2.m1.2.2.4.cmml"><mi id="S3.Ex2.m1.2.2.4.2" xref="S3.Ex2.m1.2.2.4.2.cmml">w</mi><mi id="S3.Ex2.m1.2.2.4.3" xref="S3.Ex2.m1.2.2.4.3.cmml">p</mi></msub><mrow id="S3.Ex2.m1.2.2.2" xref="S3.Ex2.m1.2.2.2.cmml"><mn id="S3.Ex2.m1.2.2.2.4" xref="S3.Ex2.m1.2.2.2.4.cmml">1</mn><mo id="S3.Ex2.m1.2.2.2.3" xref="S3.Ex2.m1.2.2.2.3.cmml">+</mo><mrow id="S3.Ex2.m1.2.2.2.2" xref="S3.Ex2.m1.2.2.2.2.cmml"><mi id="S3.Ex2.m1.2.2.2.2.4" xref="S3.Ex2.m1.2.2.2.2.4.cmml">d</mi><mo id="S3.Ex2.m1.2.2.2.2.3" xref="S3.Ex2.m1.2.2.2.2.3.cmml">⁢</mo><mrow id="S3.Ex2.m1.2.2.2.2.2.2" xref="S3.Ex2.m1.2.2.2.2.2.3.cmml"><mo id="S3.Ex2.m1.2.2.2.2.2.2.3" stretchy="false" xref="S3.Ex2.m1.2.2.2.2.2.3.cmml">(</mo><msub id="S3.Ex2.m1.1.1.1.1.1.1.1" xref="S3.Ex2.m1.1.1.1.1.1.1.1.cmml"><mi id="S3.Ex2.m1.1.1.1.1.1.1.1.2" xref="S3.Ex2.m1.1.1.1.1.1.1.1.2.cmml">𝐞</mi><mi id="S3.Ex2.m1.1.1.1.1.1.1.1.3" xref="S3.Ex2.m1.1.1.1.1.1.1.1.3.cmml">j</mi></msub><mo id="S3.Ex2.m1.2.2.2.2.2.2.4" xref="S3.Ex2.m1.2.2.2.2.2.3.cmml">,</mo><msub id="S3.Ex2.m1.2.2.2.2.2.2.2" xref="S3.Ex2.m1.2.2.2.2.2.2.2.cmml"><mi id="S3.Ex2.m1.2.2.2.2.2.2.2.2" xref="S3.Ex2.m1.2.2.2.2.2.2.2.2.cmml">μ</mi><mi id="S3.Ex2.m1.2.2.2.2.2.2.2.3" xref="S3.Ex2.m1.2.2.2.2.2.2.2.3.cmml">i</mi></msub><mo id="S3.Ex2.m1.2.2.2.2.2.2.5" stretchy="false" xref="S3.Ex2.m1.2.2.2.2.2.3.cmml">)</mo></mrow></mrow></mrow></mfrac></mrow><annotation-xml encoding="MathML-Content" id="S3.Ex2.m1.3b"><apply id="S3.Ex2.m1.3.3.cmml" xref="S3.Ex2.m1.3.3"><eq id="S3.Ex2.m1.3.3.2.cmml" xref="S3.Ex2.m1.3.3.2"></eq><apply id="S3.Ex2.m1.3.3.1.cmml" xref="S3.Ex2.m1.3.3.1"><times id="S3.Ex2.m1.3.3.1.2.cmml" xref="S3.Ex2.m1.3.3.1.2"></times><apply id="S3.Ex2.m1.3.3.1.3.cmml" xref="S3.Ex2.m1.3.3.1.3"><csymbol cd="ambiguous" id="S3.Ex2.m1.3.3.1.3.1.cmml" xref="S3.Ex2.m1.3.3.1.3">subscript</csymbol><ci id="S3.Ex2.m1.3.3.1.3.2.cmml" xref="S3.Ex2.m1.3.3.1.3.2">𝑔</ci><ci id="S3.Ex2.m1.3.3.1.3.3.cmml" xref="S3.Ex2.m1.3.3.1.3.3">𝑝</ci></apply><apply id="S3.Ex2.m1.3.3.1.1.1.1.cmml" xref="S3.Ex2.m1.3.3.1.1.1"><csymbol cd="ambiguous" id="S3.Ex2.m1.3.3.1.1.1.1.1.cmml" xref="S3.Ex2.m1.3.3.1.1.1">subscript</csymbol><ci id="S3.Ex2.m1.3.3.1.1.1.1.2.cmml" xref="S3.Ex2.m1.3.3.1.1.1.1.2">𝐞</ci><ci id="S3.Ex2.m1.3.3.1.1.1.1.3.cmml" xref="S3.Ex2.m1.3.3.1.1.1.1.3">𝑗</ci></apply></apply><apply id="S3.Ex2.m1.2.2.cmml" xref="S3.Ex2.m1.2.2"><divide id="S3.Ex2.m1.2.2.3.cmml" xref="S3.Ex2.m1.2.2"></divide><apply id="S3.Ex2.m1.2.2.4.cmml" xref="S3.Ex2.m1.2.2.4"><csymbol cd="ambiguous" id="S3.Ex2.m1.2.2.4.1.cmml" xref="S3.Ex2.m1.2.2.4">subscript</csymbol><ci id="S3.Ex2.m1.2.2.4.2.cmml" xref="S3.Ex2.m1.2.2.4.2">𝑤</ci><ci id="S3.Ex2.m1.2.2.4.3.cmml" xref="S3.Ex2.m1.2.2.4.3">𝑝</ci></apply><apply id="S3.Ex2.m1.2.2.2.cmml" xref="S3.Ex2.m1.2.2.2"><plus id="S3.Ex2.m1.2.2.2.3.cmml" xref="S3.Ex2.m1.2.2.2.3"></plus><cn id="S3.Ex2.m1.2.2.2.4.cmml" type="integer" xref="S3.Ex2.m1.2.2.2.4">1</cn><apply id="S3.Ex2.m1.2.2.2.2.cmml" xref="S3.Ex2.m1.2.2.2.2"><times id="S3.Ex2.m1.2.2.2.2.3.cmml" xref="S3.Ex2.m1.2.2.2.2.3"></times><ci id="S3.Ex2.m1.2.2.2.2.4.cmml" xref="S3.Ex2.m1.2.2.2.2.4">𝑑</ci><interval closure="open" id="S3.Ex2.m1.2.2.2.2.2.3.cmml" xref="S3.Ex2.m1.2.2.2.2.2.2"><apply id="S3.Ex2.m1.1.1.1.1.1.1.1.cmml" xref="S3.Ex2.m1.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S3.Ex2.m1.1.1.1.1.1.1.1.1.cmml" xref="S3.Ex2.m1.1.1.1.1.1.1.1">subscript</csymbol><ci id="S3.Ex2.m1.1.1.1.1.1.1.1.2.cmml" xref="S3.Ex2.m1.1.1.1.1.1.1.1.2">𝐞</ci><ci id="S3.Ex2.m1.1.1.1.1.1.1.1.3.cmml" xref="S3.Ex2.m1.1.1.1.1.1.1.1.3">𝑗</ci></apply><apply id="S3.Ex2.m1.2.2.2.2.2.2.2.cmml" xref="S3.Ex2.m1.2.2.2.2.2.2.2"><csymbol cd="ambiguous" id="S3.Ex2.m1.2.2.2.2.2.2.2.1.cmml" xref="S3.Ex2.m1.2.2.2.2.2.2.2">subscript</csymbol><ci id="S3.Ex2.m1.2.2.2.2.2.2.2.2.cmml" xref="S3.Ex2.m1.2.2.2.2.2.2.2.2">𝜇</ci><ci id="S3.Ex2.m1.2.2.2.2.2.2.2.3.cmml" xref="S3.Ex2.m1.2.2.2.2.2.2.2.3">𝑖</ci></apply></interval></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Ex2.m1.3c">g_{p}(\mathbf{e}_{j})=\frac{w_{p}}{1+d(\mathbf{e}_{j},\mathbf{\mu}_{i})}</annotation><annotation encoding="application/x-llamapun" id="S3.Ex2.m1.3d">italic_g start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT ( bold_e start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ) = divide start_ARG italic_w start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT end_ARG start_ARG 1 + italic_d ( bold_e start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT , italic_μ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) end_ARG</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> </div> <div class="ltx_listingline" id="alg2.l12"> <span class="ltx_tag ltx_tag_listingline">12:</span> Compute diversity gain: <table class="ltx_equation ltx_eqn_table" id="S3.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="g_{d}(\mathbf{e}_{j})=\frac{2w_{d}}{c_{i}}\sum_{\mathbf{e}_{b}\in c_{i}^{*}}d(% \mathbf{e}_{j},\mathbf{e}_{b})" class="ltx_Math" display="block" id="S3.Ex3.m1.3"><semantics id="S3.Ex3.m1.3a"><mrow id="S3.Ex3.m1.3.3" xref="S3.Ex3.m1.3.3.cmml"><mrow id="S3.Ex3.m1.1.1.1" xref="S3.Ex3.m1.1.1.1.cmml"><msub id="S3.Ex3.m1.1.1.1.3" xref="S3.Ex3.m1.1.1.1.3.cmml"><mi id="S3.Ex3.m1.1.1.1.3.2" xref="S3.Ex3.m1.1.1.1.3.2.cmml">g</mi><mi id="S3.Ex3.m1.1.1.1.3.3" xref="S3.Ex3.m1.1.1.1.3.3.cmml">d</mi></msub><mo id="S3.Ex3.m1.1.1.1.2" xref="S3.Ex3.m1.1.1.1.2.cmml">⁢</mo><mrow id="S3.Ex3.m1.1.1.1.1.1" xref="S3.Ex3.m1.1.1.1.1.1.1.cmml"><mo id="S3.Ex3.m1.1.1.1.1.1.2" stretchy="false" xref="S3.Ex3.m1.1.1.1.1.1.1.cmml">(</mo><msub id="S3.Ex3.m1.1.1.1.1.1.1" 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encoding="application/x-tex" id="S3.Ex3.m1.3c">g_{d}(\mathbf{e}_{j})=\frac{2w_{d}}{c_{i}}\sum_{\mathbf{e}_{b}\in c_{i}^{*}}d(% \mathbf{e}_{j},\mathbf{e}_{b})</annotation><annotation encoding="application/x-llamapun" id="S3.Ex3.m1.3d">italic_g start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT ( bold_e start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ) = divide start_ARG 2 italic_w start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT end_ARG start_ARG italic_c start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_ARG ∑ start_POSTSUBSCRIPT bold_e start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT ∈ italic_c start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT italic_d ( bold_e start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT , bold_e start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> </div> <div class="ltx_listingline" id="alg2.l13"> <span class="ltx_tag ltx_tag_listingline">13:</span> <span class="ltx_text ltx_font_bold" id="alg2.l13.1">end</span> <span class="ltx_text ltx_font_bold" id="alg2.l13.2">for</span> </div> <div class="ltx_listingline" id="alg2.l14"> <span class="ltx_tag ltx_tag_listingline">14:</span> <span class="ltx_text ltx_font_bold" id="alg2.l14.1">Evaluate Selection Priority:</span> </div> <div class="ltx_listingline" id="alg2.l15"> <span class="ltx_tag ltx_tag_listingline">15:</span> Identify the item maximizing the combined gain: <table class="ltx_equation ltx_eqn_table" id="S3.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="\mathbf{e}_{j}^{*}=\arg\max_{\mathbf{e}_{j}\in\mathbf{E}(c_{i})}\left(g_{p}(% \mathbf{e}_{j})+g_{d}(\mathbf{e}_{j})\right)" class="ltx_Math" display="block" id="S3.Ex4.m1.3"><semantics id="S3.Ex4.m1.3a"><mrow id="S3.Ex4.m1.3.3" 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end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT = roman_arg roman_max start_POSTSUBSCRIPT bold_e start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ∈ bold_E ( italic_c start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) end_POSTSUBSCRIPT ( italic_g start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT ( bold_e start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ) + italic_g start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT ( bold_e start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ) )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> </div> <div class="ltx_listingline" id="alg2.l16"> <span class="ltx_tag ltx_tag_listingline">16:</span> Update <math alttext="c_{i}^{*}\leftarrow c_{i}^{*}\cup\{\mathbf{e}_{j}^{*}\}" class="ltx_Math" display="inline" id="alg2.l16.m1.1"><semantics id="alg2.l16.m1.1a"><mrow id="alg2.l16.m1.1.1" xref="alg2.l16.m1.1.1.cmml"><msubsup id="alg2.l16.m1.1.1.3" xref="alg2.l16.m1.1.1.3.cmml"><mi 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xref="alg2.l16.m1.1.1.1.1.1.1.2.2.cmml">𝐞</mi><mi id="alg2.l16.m1.1.1.1.1.1.1.2.3" xref="alg2.l16.m1.1.1.1.1.1.1.2.3.cmml">j</mi><mo id="alg2.l16.m1.1.1.1.1.1.1.3" xref="alg2.l16.m1.1.1.1.1.1.1.3.cmml">∗</mo></msubsup><mo id="alg2.l16.m1.1.1.1.1.1.3" stretchy="false" xref="alg2.l16.m1.1.1.1.1.2.cmml">}</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="alg2.l16.m1.1b"><apply id="alg2.l16.m1.1.1.cmml" xref="alg2.l16.m1.1.1"><ci id="alg2.l16.m1.1.1.2.cmml" xref="alg2.l16.m1.1.1.2">←</ci><apply id="alg2.l16.m1.1.1.3.cmml" xref="alg2.l16.m1.1.1.3"><csymbol cd="ambiguous" id="alg2.l16.m1.1.1.3.1.cmml" xref="alg2.l16.m1.1.1.3">superscript</csymbol><apply id="alg2.l16.m1.1.1.3.2.cmml" xref="alg2.l16.m1.1.1.3"><csymbol cd="ambiguous" id="alg2.l16.m1.1.1.3.2.1.cmml" xref="alg2.l16.m1.1.1.3">subscript</csymbol><ci id="alg2.l16.m1.1.1.3.2.2.cmml" xref="alg2.l16.m1.1.1.3.2.2">𝑐</ci><ci id="alg2.l16.m1.1.1.3.2.3.cmml" xref="alg2.l16.m1.1.1.3.2.3">𝑖</ci></apply><times id="alg2.l16.m1.1.1.3.3.cmml" xref="alg2.l16.m1.1.1.3.3"></times></apply><apply id="alg2.l16.m1.1.1.1.cmml" xref="alg2.l16.m1.1.1.1"><union id="alg2.l16.m1.1.1.1.2.cmml" xref="alg2.l16.m1.1.1.1.2"></union><apply id="alg2.l16.m1.1.1.1.3.cmml" xref="alg2.l16.m1.1.1.1.3"><csymbol cd="ambiguous" id="alg2.l16.m1.1.1.1.3.1.cmml" xref="alg2.l16.m1.1.1.1.3">superscript</csymbol><apply id="alg2.l16.m1.1.1.1.3.2.cmml" xref="alg2.l16.m1.1.1.1.3"><csymbol cd="ambiguous" id="alg2.l16.m1.1.1.1.3.2.1.cmml" xref="alg2.l16.m1.1.1.1.3">subscript</csymbol><ci id="alg2.l16.m1.1.1.1.3.2.2.cmml" xref="alg2.l16.m1.1.1.1.3.2.2">𝑐</ci><ci id="alg2.l16.m1.1.1.1.3.2.3.cmml" xref="alg2.l16.m1.1.1.1.3.2.3">𝑖</ci></apply><times id="alg2.l16.m1.1.1.1.3.3.cmml" xref="alg2.l16.m1.1.1.1.3.3"></times></apply><set id="alg2.l16.m1.1.1.1.1.2.cmml" xref="alg2.l16.m1.1.1.1.1.1"><apply id="alg2.l16.m1.1.1.1.1.1.1.cmml" xref="alg2.l16.m1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="alg2.l16.m1.1.1.1.1.1.1.1.cmml" xref="alg2.l16.m1.1.1.1.1.1.1">superscript</csymbol><apply id="alg2.l16.m1.1.1.1.1.1.1.2.cmml" xref="alg2.l16.m1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="alg2.l16.m1.1.1.1.1.1.1.2.1.cmml" xref="alg2.l16.m1.1.1.1.1.1.1">subscript</csymbol><ci id="alg2.l16.m1.1.1.1.1.1.1.2.2.cmml" xref="alg2.l16.m1.1.1.1.1.1.1.2.2">𝐞</ci><ci id="alg2.l16.m1.1.1.1.1.1.1.2.3.cmml" xref="alg2.l16.m1.1.1.1.1.1.1.2.3">𝑗</ci></apply><times id="alg2.l16.m1.1.1.1.1.1.1.3.cmml" xref="alg2.l16.m1.1.1.1.1.1.1.3"></times></apply></set></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="alg2.l16.m1.1c">c_{i}^{*}\leftarrow c_{i}^{*}\cup\{\mathbf{e}_{j}^{*}\}</annotation><annotation encoding="application/x-llamapun" id="alg2.l16.m1.1d">italic_c start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ← italic_c start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ∪ { bold_e start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT }</annotation></semantics></math> and <math alttext="\mathbf{E}(c_{i})\leftarrow\mathbf{E}(c_{i})\setminus\{\mathbf{e}_{j}^{*}\}" class="ltx_Math" display="inline" id="alg2.l16.m2.3"><semantics id="alg2.l16.m2.3a"><mrow id="alg2.l16.m2.3.3" xref="alg2.l16.m2.3.3.cmml"><mrow id="alg2.l16.m2.1.1.1" xref="alg2.l16.m2.1.1.1.cmml"><mi id="alg2.l16.m2.1.1.1.3" xref="alg2.l16.m2.1.1.1.3.cmml">𝐄</mi><mo id="alg2.l16.m2.1.1.1.2" xref="alg2.l16.m2.1.1.1.2.cmml">⁢</mo><mrow id="alg2.l16.m2.1.1.1.1.1" xref="alg2.l16.m2.1.1.1.1.1.1.cmml"><mo id="alg2.l16.m2.1.1.1.1.1.2" stretchy="false" xref="alg2.l16.m2.1.1.1.1.1.1.cmml">(</mo><msub id="alg2.l16.m2.1.1.1.1.1.1" xref="alg2.l16.m2.1.1.1.1.1.1.cmml"><mi id="alg2.l16.m2.1.1.1.1.1.1.2" xref="alg2.l16.m2.1.1.1.1.1.1.2.cmml">c</mi><mi id="alg2.l16.m2.1.1.1.1.1.1.3" xref="alg2.l16.m2.1.1.1.1.1.1.3.cmml">i</mi></msub><mo id="alg2.l16.m2.1.1.1.1.1.3" stretchy="false" xref="alg2.l16.m2.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="alg2.l16.m2.3.3.4" stretchy="false" xref="alg2.l16.m2.3.3.4.cmml">←</mo><mrow id="alg2.l16.m2.3.3.3" xref="alg2.l16.m2.3.3.3.cmml"><mrow id="alg2.l16.m2.2.2.2.1" xref="alg2.l16.m2.2.2.2.1.cmml"><mi id="alg2.l16.m2.2.2.2.1.3" xref="alg2.l16.m2.2.2.2.1.3.cmml">𝐄</mi><mo id="alg2.l16.m2.2.2.2.1.2" xref="alg2.l16.m2.2.2.2.1.2.cmml">⁢</mo><mrow id="alg2.l16.m2.2.2.2.1.1.1" xref="alg2.l16.m2.2.2.2.1.1.1.1.cmml"><mo id="alg2.l16.m2.2.2.2.1.1.1.2" stretchy="false" xref="alg2.l16.m2.2.2.2.1.1.1.1.cmml">(</mo><msub id="alg2.l16.m2.2.2.2.1.1.1.1" xref="alg2.l16.m2.2.2.2.1.1.1.1.cmml"><mi id="alg2.l16.m2.2.2.2.1.1.1.1.2" xref="alg2.l16.m2.2.2.2.1.1.1.1.2.cmml">c</mi><mi id="alg2.l16.m2.2.2.2.1.1.1.1.3" xref="alg2.l16.m2.2.2.2.1.1.1.1.3.cmml">i</mi></msub><mo id="alg2.l16.m2.2.2.2.1.1.1.3" stretchy="false" xref="alg2.l16.m2.2.2.2.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="alg2.l16.m2.3.3.3.3" xref="alg2.l16.m2.3.3.3.3.cmml">∖</mo><mrow id="alg2.l16.m2.3.3.3.2.1" xref="alg2.l16.m2.3.3.3.2.2.cmml"><mo id="alg2.l16.m2.3.3.3.2.1.2" stretchy="false" xref="alg2.l16.m2.3.3.3.2.2.cmml">{</mo><msubsup id="alg2.l16.m2.3.3.3.2.1.1" xref="alg2.l16.m2.3.3.3.2.1.1.cmml"><mi id="alg2.l16.m2.3.3.3.2.1.1.2.2" xref="alg2.l16.m2.3.3.3.2.1.1.2.2.cmml">𝐞</mi><mi id="alg2.l16.m2.3.3.3.2.1.1.2.3" xref="alg2.l16.m2.3.3.3.2.1.1.2.3.cmml">j</mi><mo id="alg2.l16.m2.3.3.3.2.1.1.3" xref="alg2.l16.m2.3.3.3.2.1.1.3.cmml">∗</mo></msubsup><mo id="alg2.l16.m2.3.3.3.2.1.3" stretchy="false" xref="alg2.l16.m2.3.3.3.2.2.cmml">}</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="alg2.l16.m2.3b"><apply id="alg2.l16.m2.3.3.cmml" xref="alg2.l16.m2.3.3"><ci id="alg2.l16.m2.3.3.4.cmml" xref="alg2.l16.m2.3.3.4">←</ci><apply id="alg2.l16.m2.1.1.1.cmml" xref="alg2.l16.m2.1.1.1"><times id="alg2.l16.m2.1.1.1.2.cmml" xref="alg2.l16.m2.1.1.1.2"></times><ci id="alg2.l16.m2.1.1.1.3.cmml" xref="alg2.l16.m2.1.1.1.3">𝐄</ci><apply id="alg2.l16.m2.1.1.1.1.1.1.cmml" xref="alg2.l16.m2.1.1.1.1.1"><csymbol cd="ambiguous" id="alg2.l16.m2.1.1.1.1.1.1.1.cmml" xref="alg2.l16.m2.1.1.1.1.1">subscript</csymbol><ci id="alg2.l16.m2.1.1.1.1.1.1.2.cmml" xref="alg2.l16.m2.1.1.1.1.1.1.2">𝑐</ci><ci id="alg2.l16.m2.1.1.1.1.1.1.3.cmml" xref="alg2.l16.m2.1.1.1.1.1.1.3">𝑖</ci></apply></apply><apply id="alg2.l16.m2.3.3.3.cmml" xref="alg2.l16.m2.3.3.3"><setdiff id="alg2.l16.m2.3.3.3.3.cmml" xref="alg2.l16.m2.3.3.3.3"></setdiff><apply id="alg2.l16.m2.2.2.2.1.cmml" xref="alg2.l16.m2.2.2.2.1"><times id="alg2.l16.m2.2.2.2.1.2.cmml" xref="alg2.l16.m2.2.2.2.1.2"></times><ci id="alg2.l16.m2.2.2.2.1.3.cmml" xref="alg2.l16.m2.2.2.2.1.3">𝐄</ci><apply id="alg2.l16.m2.2.2.2.1.1.1.1.cmml" xref="alg2.l16.m2.2.2.2.1.1.1"><csymbol cd="ambiguous" id="alg2.l16.m2.2.2.2.1.1.1.1.1.cmml" xref="alg2.l16.m2.2.2.2.1.1.1">subscript</csymbol><ci id="alg2.l16.m2.2.2.2.1.1.1.1.2.cmml" xref="alg2.l16.m2.2.2.2.1.1.1.1.2">𝑐</ci><ci id="alg2.l16.m2.2.2.2.1.1.1.1.3.cmml" xref="alg2.l16.m2.2.2.2.1.1.1.1.3">𝑖</ci></apply></apply><set id="alg2.l16.m2.3.3.3.2.2.cmml" xref="alg2.l16.m2.3.3.3.2.1"><apply id="alg2.l16.m2.3.3.3.2.1.1.cmml" xref="alg2.l16.m2.3.3.3.2.1.1"><csymbol cd="ambiguous" id="alg2.l16.m2.3.3.3.2.1.1.1.cmml" xref="alg2.l16.m2.3.3.3.2.1.1">superscript</csymbol><apply id="alg2.l16.m2.3.3.3.2.1.1.2.cmml" xref="alg2.l16.m2.3.3.3.2.1.1"><csymbol cd="ambiguous" id="alg2.l16.m2.3.3.3.2.1.1.2.1.cmml" xref="alg2.l16.m2.3.3.3.2.1.1">subscript</csymbol><ci id="alg2.l16.m2.3.3.3.2.1.1.2.2.cmml" xref="alg2.l16.m2.3.3.3.2.1.1.2.2">𝐞</ci><ci id="alg2.l16.m2.3.3.3.2.1.1.2.3.cmml" xref="alg2.l16.m2.3.3.3.2.1.1.2.3">𝑗</ci></apply><times id="alg2.l16.m2.3.3.3.2.1.1.3.cmml" xref="alg2.l16.m2.3.3.3.2.1.1.3"></times></apply></set></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="alg2.l16.m2.3c">\mathbf{E}(c_{i})\leftarrow\mathbf{E}(c_{i})\setminus\{\mathbf{e}_{j}^{*}\}</annotation><annotation encoding="application/x-llamapun" id="alg2.l16.m2.3d">bold_E ( italic_c start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) ← bold_E ( italic_c start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) ∖ { bold_e start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT }</annotation></semantics></math>. </div> <div class="ltx_listingline" id="alg2.l17"> <span class="ltx_tag ltx_tag_listingline">17:</span> <span class="ltx_text ltx_font_bold" id="alg2.l17.1">end</span> <span class="ltx_text ltx_font_bold" id="alg2.l17.2">while</span>Chronologically sort <math alttext="c_{i}^{*}" class="ltx_Math" display="inline" id="alg2.l17.m1.1"><semantics id="alg2.l17.m1.1a"><msubsup id="alg2.l17.m1.1.1" xref="alg2.l17.m1.1.1.cmml"><mi id="alg2.l17.m1.1.1.2.2" xref="alg2.l17.m1.1.1.2.2.cmml">c</mi><mi id="alg2.l17.m1.1.1.2.3" xref="alg2.l17.m1.1.1.2.3.cmml">i</mi><mo id="alg2.l17.m1.1.1.3" xref="alg2.l17.m1.1.1.3.cmml">∗</mo></msubsup><annotation-xml encoding="MathML-Content" id="alg2.l17.m1.1b"><apply id="alg2.l17.m1.1.1.cmml" xref="alg2.l17.m1.1.1"><csymbol cd="ambiguous" id="alg2.l17.m1.1.1.1.cmml" xref="alg2.l17.m1.1.1">superscript</csymbol><apply id="alg2.l17.m1.1.1.2.cmml" xref="alg2.l17.m1.1.1"><csymbol cd="ambiguous" id="alg2.l17.m1.1.1.2.1.cmml" xref="alg2.l17.m1.1.1">subscript</csymbol><ci id="alg2.l17.m1.1.1.2.2.cmml" xref="alg2.l17.m1.1.1.2.2">𝑐</ci><ci id="alg2.l17.m1.1.1.2.3.cmml" xref="alg2.l17.m1.1.1.2.3">𝑖</ci></apply><times id="alg2.l17.m1.1.1.3.cmml" xref="alg2.l17.m1.1.1.3"></times></apply></annotation-xml><annotation encoding="application/x-tex" id="alg2.l17.m1.1c">c_{i}^{*}</annotation><annotation encoding="application/x-llamapun" id="alg2.l17.m1.1d">italic_c start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math> to get <math alttext="S_{i}^{*}" class="ltx_Math" display="inline" id="alg2.l17.m2.1"><semantics id="alg2.l17.m2.1a"><msubsup id="alg2.l17.m2.1.1" xref="alg2.l17.m2.1.1.cmml"><mi id="alg2.l17.m2.1.1.2.2" xref="alg2.l17.m2.1.1.2.2.cmml">S</mi><mi id="alg2.l17.m2.1.1.2.3" xref="alg2.l17.m2.1.1.2.3.cmml">i</mi><mo id="alg2.l17.m2.1.1.3" xref="alg2.l17.m2.1.1.3.cmml">∗</mo></msubsup><annotation-xml encoding="MathML-Content" id="alg2.l17.m2.1b"><apply id="alg2.l17.m2.1.1.cmml" xref="alg2.l17.m2.1.1"><csymbol cd="ambiguous" id="alg2.l17.m2.1.1.1.cmml" xref="alg2.l17.m2.1.1">superscript</csymbol><apply id="alg2.l17.m2.1.1.2.cmml" xref="alg2.l17.m2.1.1"><csymbol cd="ambiguous" id="alg2.l17.m2.1.1.2.1.cmml" xref="alg2.l17.m2.1.1">subscript</csymbol><ci id="alg2.l17.m2.1.1.2.2.cmml" xref="alg2.l17.m2.1.1.2.2">𝑆</ci><ci id="alg2.l17.m2.1.1.2.3.cmml" xref="alg2.l17.m2.1.1.2.3">𝑖</ci></apply><times id="alg2.l17.m2.1.1.3.cmml" xref="alg2.l17.m2.1.1.3"></times></apply></annotation-xml><annotation encoding="application/x-tex" id="alg2.l17.m2.1c">S_{i}^{*}</annotation><annotation encoding="application/x-llamapun" id="alg2.l17.m2.1d">italic_S start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math>. </div> <div class="ltx_listingline" id="alg2.l18"> <span class="ltx_tag ltx_tag_listingline">18:</span> <span class="ltx_text ltx_font_bold" id="alg2.l18.1">return</span> <math alttext="S_{i}^{*}" class="ltx_Math" display="inline" id="alg2.l18.m1.1"><semantics id="alg2.l18.m1.1a"><msubsup id="alg2.l18.m1.1.1" xref="alg2.l18.m1.1.1.cmml"><mi id="alg2.l18.m1.1.1.2.2" xref="alg2.l18.m1.1.1.2.2.cmml">S</mi><mi id="alg2.l18.m1.1.1.2.3" xref="alg2.l18.m1.1.1.2.3.cmml">i</mi><mo id="alg2.l18.m1.1.1.3" xref="alg2.l18.m1.1.1.3.cmml">∗</mo></msubsup><annotation-xml encoding="MathML-Content" id="alg2.l18.m1.1b"><apply id="alg2.l18.m1.1.1.cmml" xref="alg2.l18.m1.1.1"><csymbol cd="ambiguous" id="alg2.l18.m1.1.1.1.cmml" xref="alg2.l18.m1.1.1">superscript</csymbol><apply id="alg2.l18.m1.1.1.2.cmml" xref="alg2.l18.m1.1.1"><csymbol cd="ambiguous" id="alg2.l18.m1.1.1.2.1.cmml" xref="alg2.l18.m1.1.1">subscript</csymbol><ci id="alg2.l18.m1.1.1.2.2.cmml" xref="alg2.l18.m1.1.1.2.2">𝑆</ci><ci id="alg2.l18.m1.1.1.2.3.cmml" xref="alg2.l18.m1.1.1.2.3">𝑖</ci></apply><times id="alg2.l18.m1.1.1.3.cmml" xref="alg2.l18.m1.1.1.3"></times></apply></annotation-xml><annotation encoding="application/x-tex" id="alg2.l18.m1.1c">S_{i}^{*}</annotation><annotation encoding="application/x-llamapun" id="alg2.l18.m1.1d">italic_S start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math> </div> <div class="ltx_listingline" id="alg2.l19"> <span class="ltx_tag ltx_tag_listingline">19:</span><span class="ltx_text ltx_font_bold" id="alg2.l19.1">end</span> <span class="ltx_text ltx_font_bold" id="alg2.l19.2">function</span> </div> </div> </figure> </section> <section class="ltx_subsection" id="S3.SS3"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">3.3 </span>In-Cluster Selection</h3> <div class="ltx_para" id="S3.SS3.p1"> <p class="ltx_p" id="S3.SS3.p1.4">After partitioning user behaviors into semantically coherent clusters and each cluster is allocated with a sampling quota, we are to select a representative subset from each cluster. Data selection methods that greedily choose items closest to the cluster centroid (e.g., <cite class="ltx_cite ltx_citemacro_cite">Welling (<a class="ltx_ref" href="https://arxiv.org/html/2503.02398v1#bib.bib49" title="">2009</a>); Rebuffi et al. (<a class="ltx_ref" href="https://arxiv.org/html/2503.02398v1#bib.bib32" title="">2017</a>); Sorscher et al. (<a class="ltx_ref" href="https://arxiv.org/html/2503.02398v1#bib.bib42" title="">2022</a>)</cite>) yield overly homogeneous user profiling, while boundary-focused strategies (e.g., <cite class="ltx_cite ltx_citemacro_cite">Paul et al. (<a class="ltx_ref" href="https://arxiv.org/html/2503.02398v1#bib.bib25" title="">2021</a>); Toneva et al. (<a class="ltx_ref" href="https://arxiv.org/html/2503.02398v1#bib.bib45" title="">2019</a>)</cite>) risk overemphasizing diversity at the expense of prototypical patterns. To address these issues, we introduce a sampling strategy that balances prototypicality and diversity within each cluster. For a cluster <math alttext="c_{i}" class="ltx_Math" display="inline" id="S3.SS3.p1.1.m1.1"><semantics id="S3.SS3.p1.1.m1.1a"><msub id="S3.SS3.p1.1.m1.1.1" xref="S3.SS3.p1.1.m1.1.1.cmml"><mi id="S3.SS3.p1.1.m1.1.1.2" xref="S3.SS3.p1.1.m1.1.1.2.cmml">c</mi><mi id="S3.SS3.p1.1.m1.1.1.3" xref="S3.SS3.p1.1.m1.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS3.p1.1.m1.1b"><apply id="S3.SS3.p1.1.m1.1.1.cmml" xref="S3.SS3.p1.1.m1.1.1"><csymbol cd="ambiguous" id="S3.SS3.p1.1.m1.1.1.1.cmml" xref="S3.SS3.p1.1.m1.1.1">subscript</csymbol><ci id="S3.SS3.p1.1.m1.1.1.2.cmml" xref="S3.SS3.p1.1.m1.1.1.2">𝑐</ci><ci id="S3.SS3.p1.1.m1.1.1.3.cmml" xref="S3.SS3.p1.1.m1.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p1.1.m1.1c">c_{i}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p1.1.m1.1d">italic_c start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math>, its centroid is computed as <math 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xref="S3.SS3.p1.2.m2.2.2.1.1.1.2"></times><ci id="S3.SS3.p1.2.m2.2.2.1.1.1.3.cmml" xref="S3.SS3.p1.2.m2.2.2.1.1.1.3">𝐄</ci><apply id="S3.SS3.p1.2.m2.2.2.1.1.1.1.1.1.cmml" xref="S3.SS3.p1.2.m2.2.2.1.1.1.1.1"><csymbol cd="ambiguous" id="S3.SS3.p1.2.m2.2.2.1.1.1.1.1.1.1.cmml" xref="S3.SS3.p1.2.m2.2.2.1.1.1.1.1">subscript</csymbol><ci id="S3.SS3.p1.2.m2.2.2.1.1.1.1.1.1.2.cmml" xref="S3.SS3.p1.2.m2.2.2.1.1.1.1.1.1.2">𝐼</ci><ci id="S3.SS3.p1.2.m2.2.2.1.1.1.1.1.1.3.cmml" xref="S3.SS3.p1.2.m2.2.2.1.1.1.1.1.1.3">𝑗</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p1.2.m2.2c">\mathbf{\mu}_{i}=\frac{1}{|c_{i}|}\sum_{I_{j}\in c_{i}}\mathbf{E}(I_{j})</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p1.2.m2.2d">italic_μ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = divide start_ARG 1 end_ARG start_ARG | italic_c start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT | end_ARG ∑ start_POSTSUBSCRIPT italic_I start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ∈ italic_c start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUBSCRIPT bold_E ( italic_I start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT )</annotation></semantics></math>. Let <math alttext="c_{i}^{*}" class="ltx_Math" display="inline" id="S3.SS3.p1.3.m3.1"><semantics id="S3.SS3.p1.3.m3.1a"><msubsup id="S3.SS3.p1.3.m3.1.1" xref="S3.SS3.p1.3.m3.1.1.cmml"><mi id="S3.SS3.p1.3.m3.1.1.2.2" xref="S3.SS3.p1.3.m3.1.1.2.2.cmml">c</mi><mi id="S3.SS3.p1.3.m3.1.1.2.3" xref="S3.SS3.p1.3.m3.1.1.2.3.cmml">i</mi><mo id="S3.SS3.p1.3.m3.1.1.3" xref="S3.SS3.p1.3.m3.1.1.3.cmml">∗</mo></msubsup><annotation-xml encoding="MathML-Content" id="S3.SS3.p1.3.m3.1b"><apply id="S3.SS3.p1.3.m3.1.1.cmml" xref="S3.SS3.p1.3.m3.1.1"><csymbol cd="ambiguous" id="S3.SS3.p1.3.m3.1.1.1.cmml" xref="S3.SS3.p1.3.m3.1.1">superscript</csymbol><apply id="S3.SS3.p1.3.m3.1.1.2.cmml" xref="S3.SS3.p1.3.m3.1.1"><csymbol cd="ambiguous" id="S3.SS3.p1.3.m3.1.1.2.1.cmml" xref="S3.SS3.p1.3.m3.1.1">subscript</csymbol><ci id="S3.SS3.p1.3.m3.1.1.2.2.cmml" xref="S3.SS3.p1.3.m3.1.1.2.2">𝑐</ci><ci id="S3.SS3.p1.3.m3.1.1.2.3.cmml" xref="S3.SS3.p1.3.m3.1.1.2.3">𝑖</ci></apply><times id="S3.SS3.p1.3.m3.1.1.3.cmml" xref="S3.SS3.p1.3.m3.1.1.3"></times></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p1.3.m3.1c">c_{i}^{*}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p1.3.m3.1d">italic_c start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math> denote the selected subset from <math alttext="c_{i}" class="ltx_Math" display="inline" id="S3.SS3.p1.4.m4.1"><semantics id="S3.SS3.p1.4.m4.1a"><msub id="S3.SS3.p1.4.m4.1.1" xref="S3.SS3.p1.4.m4.1.1.cmml"><mi id="S3.SS3.p1.4.m4.1.1.2" xref="S3.SS3.p1.4.m4.1.1.2.cmml">c</mi><mi id="S3.SS3.p1.4.m4.1.1.3" xref="S3.SS3.p1.4.m4.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS3.p1.4.m4.1b"><apply id="S3.SS3.p1.4.m4.1.1.cmml" xref="S3.SS3.p1.4.m4.1.1"><csymbol cd="ambiguous" id="S3.SS3.p1.4.m4.1.1.1.cmml" xref="S3.SS3.p1.4.m4.1.1">subscript</csymbol><ci id="S3.SS3.p1.4.m4.1.1.2.cmml" xref="S3.SS3.p1.4.m4.1.1.2">𝑐</ci><ci id="S3.SS3.p1.4.m4.1.1.3.cmml" xref="S3.SS3.p1.4.m4.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p1.4.m4.1c">c_{i}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p1.4.m4.1d">italic_c start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math>. Our goal is to maximize both the similarity of selected items to the centroid and the diversity among them:</p> </div> <div class="ltx_para" id="S3.SS3.p2"> <table class="ltx_equationgroup ltx_eqn_table" id="S3.Ex5"> <tbody id="S3.Ex5X"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\max_{c_{i}^{*}}\Biggl{(}" class="ltx_math_unparsed" display="inline" id="S3.Ex5X.2.1.1.m1.1"><semantics id="S3.Ex5X.2.1.1.m1.1a"><mrow id="S3.Ex5X.2.1.1.m1.1b"><munder id="S3.Ex5X.2.1.1.m1.1.1"><mi id="S3.Ex5X.2.1.1.m1.1.1.2">max</mi><msubsup id="S3.Ex5X.2.1.1.m1.1.1.3"><mi id="S3.Ex5X.2.1.1.m1.1.1.3.2.2">c</mi><mi id="S3.Ex5X.2.1.1.m1.1.1.3.2.3">i</mi><mo id="S3.Ex5X.2.1.1.m1.1.1.3.3">∗</mo></msubsup></munder><mo id="S3.Ex5X.2.1.1.m1.1.2" maxsize="260%" minsize="260%">(</mo></mrow><annotation encoding="application/x-tex" id="S3.Ex5X.2.1.1.m1.1c">\displaystyle\max_{c_{i}^{*}}\Biggl{(}</annotation><annotation encoding="application/x-llamapun" id="S3.Ex5X.2.1.1.m1.1d">roman_max start_POSTSUBSCRIPT italic_c start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT (</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle w_{p}\cdot\sum_{I_{j}\in c_{i}^{*}}\frac{1}{1+d(\mathbf{e}_{j},% \mathbf{\mu}_{i})}+w_{d}\cdot\frac{2}{a_{i}}\sum_{\begin{subarray}{c}I_{a},I_{% b}\in c_{i}^{*}\\ a\neq b\end{subarray}}d(\mathbf{e}_{a},\mathbf{e}_{b})\Biggr{)}" class="ltx_math_unparsed" display="inline" id="S3.Ex5X.3.2.2.m1.2"><semantics id="S3.Ex5X.3.2.2.m1.2a"><mrow id="S3.Ex5X.3.2.2.m1.2b"><msub id="S3.Ex5X.3.2.2.m1.2.3"><mi id="S3.Ex5X.3.2.2.m1.2.3.2">w</mi><mi id="S3.Ex5X.3.2.2.m1.2.3.3">p</mi></msub><mo id="S3.Ex5X.3.2.2.m1.2.4" lspace="0.222em" rspace="0.222em">⋅</mo><mstyle displaystyle="true" id="S3.Ex5X.3.2.2.m1.2.5"><munder id="S3.Ex5X.3.2.2.m1.2.5a"><mo id="S3.Ex5X.3.2.2.m1.2.5.2" movablelimits="false">∑</mo><mrow id="S3.Ex5X.3.2.2.m1.2.5.3"><msub id="S3.Ex5X.3.2.2.m1.2.5.3.2"><mi id="S3.Ex5X.3.2.2.m1.2.5.3.2.2">I</mi><mi id="S3.Ex5X.3.2.2.m1.2.5.3.2.3">j</mi></msub><mo id="S3.Ex5X.3.2.2.m1.2.5.3.1">∈</mo><msubsup id="S3.Ex5X.3.2.2.m1.2.5.3.3"><mi id="S3.Ex5X.3.2.2.m1.2.5.3.3.2.2">c</mi><mi id="S3.Ex5X.3.2.2.m1.2.5.3.3.2.3">i</mi><mo id="S3.Ex5X.3.2.2.m1.2.5.3.3.3">∗</mo></msubsup></mrow></munder></mstyle><mstyle displaystyle="true" id="S3.Ex5X.3.2.2.m1.2.2"><mfrac id="S3.Ex5X.3.2.2.m1.2.2a"><mn id="S3.Ex5X.3.2.2.m1.2.2.4">1</mn><mrow id="S3.Ex5X.3.2.2.m1.2.2.2"><mn id="S3.Ex5X.3.2.2.m1.2.2.2.4">1</mn><mo id="S3.Ex5X.3.2.2.m1.2.2.2.3">+</mo><mrow id="S3.Ex5X.3.2.2.m1.2.2.2.2"><mi id="S3.Ex5X.3.2.2.m1.2.2.2.2.4">d</mi><mo id="S3.Ex5X.3.2.2.m1.2.2.2.2.3">⁢</mo><mrow id="S3.Ex5X.3.2.2.m1.2.2.2.2.2.2"><mo id="S3.Ex5X.3.2.2.m1.2.2.2.2.2.2.3" stretchy="false">(</mo><msub id="S3.Ex5X.3.2.2.m1.1.1.1.1.1.1.1"><mi id="S3.Ex5X.3.2.2.m1.1.1.1.1.1.1.1.2">𝐞</mi><mi id="S3.Ex5X.3.2.2.m1.1.1.1.1.1.1.1.3">j</mi></msub><mo id="S3.Ex5X.3.2.2.m1.2.2.2.2.2.2.4">,</mo><msub id="S3.Ex5X.3.2.2.m1.2.2.2.2.2.2.2"><mi id="S3.Ex5X.3.2.2.m1.2.2.2.2.2.2.2.2">μ</mi><mi id="S3.Ex5X.3.2.2.m1.2.2.2.2.2.2.2.3">i</mi></msub><mo id="S3.Ex5X.3.2.2.m1.2.2.2.2.2.2.5" stretchy="false">)</mo></mrow></mrow></mrow></mfrac></mstyle><mo id="S3.Ex5X.3.2.2.m1.2.6">+</mo><msub id="S3.Ex5X.3.2.2.m1.2.7"><mi id="S3.Ex5X.3.2.2.m1.2.7.2">w</mi><mi id="S3.Ex5X.3.2.2.m1.2.7.3">d</mi></msub><mo id="S3.Ex5X.3.2.2.m1.2.8" lspace="0.222em" rspace="0.222em">⋅</mo><mstyle displaystyle="true" id="S3.Ex5X.3.2.2.m1.2.9"><mfrac id="S3.Ex5X.3.2.2.m1.2.9a"><mn id="S3.Ex5X.3.2.2.m1.2.9.2">2</mn><msub id="S3.Ex5X.3.2.2.m1.2.9.3"><mi id="S3.Ex5X.3.2.2.m1.2.9.3.2">a</mi><mi id="S3.Ex5X.3.2.2.m1.2.9.3.3">i</mi></msub></mfrac></mstyle><mstyle displaystyle="true" id="S3.Ex5X.3.2.2.m1.2.10"><munder id="S3.Ex5X.3.2.2.m1.2.10a"><mo id="S3.Ex5X.3.2.2.m1.2.10.2" movablelimits="false">∑</mo><mtable id="S3.Ex5.m1.1.1.1.1.1.1.1.1.1.1.mf" rowspacing="0pt"><mtr id="S3.Ex5.m1.1.1.1.1.1.1.1.1.1.1.mfa"><mtd id="S3.Ex5.m1.1.1.1.1.1.1.1.1.1.1.mfb"><mrow id="S3.Ex5.m1.1.1.1.1.1.1.1.1.1.1.mf.2.2.2.2"><mrow id="S3.Ex5.m1.1.1.1.1.1.1.1.1.1.1.mf.2.2.2.2.2.2"><msub id="S3.Ex5.m1.1.1.1.1.1.1.1.1.1.1.mf.1.1.1.1.1.1.1"><mi id="S3.Ex5.m1.1.1.1.1.1.1.1.1.1.1.mf.1.1.1.1.1.1.1.2">I</mi><mi id="S3.Ex5.m1.1.1.1.1.1.1.1.1.1.1.mf.1.1.1.1.1.1.1.3">a</mi></msub><mo id="S3.Ex5.m1.1.1.1.1.1.1.1.1.1.1.mf.2.2.2.2.2.2.3">,</mo><msub id="S3.Ex5.m1.1.1.1.1.1.1.1.1.1.1.mf.2.2.2.2.2.2.2"><mi id="S3.Ex5.m1.1.1.1.1.1.1.1.1.1.1.mf.2.2.2.2.2.2.2.2">I</mi><mi id="S3.Ex5.m1.1.1.1.1.1.1.1.1.1.1.mf.2.2.2.2.2.2.2.3">b</mi></msub></mrow><mo id="S3.Ex5.m1.1.1.1.1.1.1.1.1.1.1.mf.2.2.2.2.3">∈</mo><msubsup id="S3.Ex5.m1.1.1.1.1.1.1.1.1.1.1.mf.2.2.2.2.4"><mi id="S3.Ex5.m1.1.1.1.1.1.1.1.1.1.1.mf.2.2.2.2.4.2.2">c</mi><mi id="S3.Ex5.m1.1.1.1.1.1.1.1.1.1.1.mf.2.2.2.2.4.2.3">i</mi><mo id="S3.Ex5.m1.1.1.1.1.1.1.1.1.1.1.mf.2.2.2.2.4.3">∗</mo></msubsup></mrow></mtd></mtr><mtr id="S3.Ex5.m1.1.1.1.1.1.1.1.1.1.1.mfc"><mtd id="S3.Ex5.m1.1.1.1.1.1.1.1.1.1.1.mfd"><mrow id="S3.Ex5.m1.1.1.1.1.1.1.1.1.1.1.mf.3.1.1"><mi id="S3.Ex5.m1.1.1.1.1.1.1.1.1.1.1.mf.3.1.1.2">a</mi><mo id="S3.Ex5.m1.1.1.1.1.1.1.1.1.1.1.mf.3.1.1.1">≠</mo><mi id="S3.Ex5.m1.1.1.1.1.1.1.1.1.1.1.mf.3.1.1.3">b</mi></mrow></mtd></mtr></mtable></munder></mstyle><mi id="S3.Ex5X.3.2.2.m1.2.11">d</mi><mrow id="S3.Ex5X.3.2.2.m1.2.12"><mo id="S3.Ex5X.3.2.2.m1.2.12.1" stretchy="false">(</mo><msub id="S3.Ex5X.3.2.2.m1.2.12.2"><mi id="S3.Ex5X.3.2.2.m1.2.12.2.2">𝐞</mi><mi id="S3.Ex5X.3.2.2.m1.2.12.2.3">a</mi></msub><mo id="S3.Ex5X.3.2.2.m1.2.12.3">,</mo><msub id="S3.Ex5X.3.2.2.m1.2.12.4"><mi id="S3.Ex5X.3.2.2.m1.2.12.4.2">𝐞</mi><mi id="S3.Ex5X.3.2.2.m1.2.12.4.3">b</mi></msub><mo id="S3.Ex5X.3.2.2.m1.2.12.5" stretchy="false">)</mo></mrow><mo id="S3.Ex5X.3.2.2.m1.2.13" maxsize="260%" minsize="260%">)</mo></mrow><annotation encoding="application/x-tex" id="S3.Ex5X.3.2.2.m1.2c">\displaystyle w_{p}\cdot\sum_{I_{j}\in c_{i}^{*}}\frac{1}{1+d(\mathbf{e}_{j},% \mathbf{\mu}_{i})}+w_{d}\cdot\frac{2}{a_{i}}\sum_{\begin{subarray}{c}I_{a},I_{% b}\in c_{i}^{*}\\ a\neq b\end{subarray}}d(\mathbf{e}_{a},\mathbf{e}_{b})\Biggr{)}</annotation><annotation encoding="application/x-llamapun" id="S3.Ex5X.3.2.2.m1.2d">italic_w start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT ⋅ ∑ start_POSTSUBSCRIPT italic_I start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ∈ italic_c start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT divide start_ARG 1 end_ARG start_ARG 1 + italic_d ( bold_e start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT , italic_μ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) end_ARG + italic_w start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT ⋅ divide start_ARG 2 end_ARG start_ARG italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_ARG ∑ start_POSTSUBSCRIPT start_ARG start_ROW start_CELL italic_I start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT , italic_I start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT ∈ italic_c start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT end_CELL end_ROW start_ROW start_CELL italic_a ≠ italic_b end_CELL end_ROW end_ARG end_POSTSUBSCRIPT italic_d ( bold_e start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT , bold_e start_POSTSUBSCRIPT italic_b 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.SS3.p3"> <p class="ltx_p" id="S3.SS3.p3.4">Here, <math alttext="w_{p}=\alpha^{-10}" class="ltx_Math" display="inline" id="S3.SS3.p3.1.m1.1"><semantics id="S3.SS3.p3.1.m1.1a"><mrow id="S3.SS3.p3.1.m1.1.1" xref="S3.SS3.p3.1.m1.1.1.cmml"><msub id="S3.SS3.p3.1.m1.1.1.2" xref="S3.SS3.p3.1.m1.1.1.2.cmml"><mi id="S3.SS3.p3.1.m1.1.1.2.2" xref="S3.SS3.p3.1.m1.1.1.2.2.cmml">w</mi><mi id="S3.SS3.p3.1.m1.1.1.2.3" xref="S3.SS3.p3.1.m1.1.1.2.3.cmml">p</mi></msub><mo id="S3.SS3.p3.1.m1.1.1.1" xref="S3.SS3.p3.1.m1.1.1.1.cmml">=</mo><msup id="S3.SS3.p3.1.m1.1.1.3" xref="S3.SS3.p3.1.m1.1.1.3.cmml"><mi id="S3.SS3.p3.1.m1.1.1.3.2" xref="S3.SS3.p3.1.m1.1.1.3.2.cmml">α</mi><mrow id="S3.SS3.p3.1.m1.1.1.3.3" xref="S3.SS3.p3.1.m1.1.1.3.3.cmml"><mo id="S3.SS3.p3.1.m1.1.1.3.3a" xref="S3.SS3.p3.1.m1.1.1.3.3.cmml">−</mo><mn id="S3.SS3.p3.1.m1.1.1.3.3.2" xref="S3.SS3.p3.1.m1.1.1.3.3.2.cmml">10</mn></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="S3.SS3.p3.1.m1.1b"><apply id="S3.SS3.p3.1.m1.1.1.cmml" xref="S3.SS3.p3.1.m1.1.1"><eq id="S3.SS3.p3.1.m1.1.1.1.cmml" xref="S3.SS3.p3.1.m1.1.1.1"></eq><apply id="S3.SS3.p3.1.m1.1.1.2.cmml" xref="S3.SS3.p3.1.m1.1.1.2"><csymbol cd="ambiguous" id="S3.SS3.p3.1.m1.1.1.2.1.cmml" xref="S3.SS3.p3.1.m1.1.1.2">subscript</csymbol><ci id="S3.SS3.p3.1.m1.1.1.2.2.cmml" xref="S3.SS3.p3.1.m1.1.1.2.2">𝑤</ci><ci id="S3.SS3.p3.1.m1.1.1.2.3.cmml" xref="S3.SS3.p3.1.m1.1.1.2.3">𝑝</ci></apply><apply id="S3.SS3.p3.1.m1.1.1.3.cmml" xref="S3.SS3.p3.1.m1.1.1.3"><csymbol cd="ambiguous" id="S3.SS3.p3.1.m1.1.1.3.1.cmml" xref="S3.SS3.p3.1.m1.1.1.3">superscript</csymbol><ci id="S3.SS3.p3.1.m1.1.1.3.2.cmml" xref="S3.SS3.p3.1.m1.1.1.3.2">𝛼</ci><apply id="S3.SS3.p3.1.m1.1.1.3.3.cmml" xref="S3.SS3.p3.1.m1.1.1.3.3"><minus id="S3.SS3.p3.1.m1.1.1.3.3.1.cmml" xref="S3.SS3.p3.1.m1.1.1.3.3"></minus><cn id="S3.SS3.p3.1.m1.1.1.3.3.2.cmml" type="integer" xref="S3.SS3.p3.1.m1.1.1.3.3.2">10</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p3.1.m1.1c">w_{p}=\alpha^{-10}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p3.1.m1.1d">italic_w start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT = italic_α start_POSTSUPERSCRIPT - 10 end_POSTSUPERSCRIPT</annotation></semantics></math> and <math alttext="w_{d}=1-w_{p}" class="ltx_Math" display="inline" id="S3.SS3.p3.2.m2.1"><semantics id="S3.SS3.p3.2.m2.1a"><mrow id="S3.SS3.p3.2.m2.1.1" xref="S3.SS3.p3.2.m2.1.1.cmml"><msub id="S3.SS3.p3.2.m2.1.1.2" xref="S3.SS3.p3.2.m2.1.1.2.cmml"><mi id="S3.SS3.p3.2.m2.1.1.2.2" xref="S3.SS3.p3.2.m2.1.1.2.2.cmml">w</mi><mi id="S3.SS3.p3.2.m2.1.1.2.3" xref="S3.SS3.p3.2.m2.1.1.2.3.cmml">d</mi></msub><mo id="S3.SS3.p3.2.m2.1.1.1" xref="S3.SS3.p3.2.m2.1.1.1.cmml">=</mo><mrow id="S3.SS3.p3.2.m2.1.1.3" xref="S3.SS3.p3.2.m2.1.1.3.cmml"><mn id="S3.SS3.p3.2.m2.1.1.3.2" xref="S3.SS3.p3.2.m2.1.1.3.2.cmml">1</mn><mo id="S3.SS3.p3.2.m2.1.1.3.1" xref="S3.SS3.p3.2.m2.1.1.3.1.cmml">−</mo><msub id="S3.SS3.p3.2.m2.1.1.3.3" xref="S3.SS3.p3.2.m2.1.1.3.3.cmml"><mi id="S3.SS3.p3.2.m2.1.1.3.3.2" xref="S3.SS3.p3.2.m2.1.1.3.3.2.cmml">w</mi><mi id="S3.SS3.p3.2.m2.1.1.3.3.3" xref="S3.SS3.p3.2.m2.1.1.3.3.3.cmml">p</mi></msub></mrow></mrow><annotation-xml 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id="S3.SS3.p3.2.m2.1.1.3.3.3.cmml" xref="S3.SS3.p3.2.m2.1.1.3.3.3">𝑝</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p3.2.m2.1c">w_{d}=1-w_{p}</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p3.2.m2.1d">italic_w start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT = 1 - italic_w start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT</annotation></semantics></math>, with the hyperparameter <math alttext="\alpha" class="ltx_Math" display="inline" id="S3.SS3.p3.3.m3.1"><semantics id="S3.SS3.p3.3.m3.1a"><mi id="S3.SS3.p3.3.m3.1.1" xref="S3.SS3.p3.3.m3.1.1.cmml">α</mi><annotation-xml encoding="MathML-Content" id="S3.SS3.p3.3.m3.1b"><ci id="S3.SS3.p3.3.m3.1.1.cmml" xref="S3.SS3.p3.3.m3.1.1">𝛼</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p3.3.m3.1c">\alpha</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p3.3.m3.1d">italic_α</annotation></semantics></math> tuning the trade-off: values near 1.001 approximate centroid selection, while values around 1.4 approach boundary selection. Empirically, <math alttext="\alpha" class="ltx_Math" display="inline" id="S3.SS3.p3.4.m4.1"><semantics id="S3.SS3.p3.4.m4.1a"><mi id="S3.SS3.p3.4.m4.1.1" xref="S3.SS3.p3.4.m4.1.1.cmml">α</mi><annotation-xml encoding="MathML-Content" id="S3.SS3.p3.4.m4.1b"><ci id="S3.SS3.p3.4.m4.1.1.cmml" xref="S3.SS3.p3.4.m4.1.1">𝛼</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p3.4.m4.1c">\alpha</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p3.4.m4.1d">italic_α</annotation></semantics></math> is typically set between 1.06 and 1.08 (see Section <a class="ltx_ref" href="https://arxiv.org/html/2503.02398v1#S5.SS4" title="5.4 Hyper-parameter Analysis (RQ3) ‣ 5 Experiments ‣ PersonaX: A Recommendation Agent-Oriented User Modeling Framework for Long Behavior Sequence"><span class="ltx_text ltx_ref_tag">5.4</span></a>). We frame the selection as discrete optimization problem and using a Greedy Selection algorithm (Algorithm <a class="ltx_ref" href="https://arxiv.org/html/2503.02398v1#alg2" title="Algorithm 2 ‣ 3.2 Sampling Budget Allocation ‣ 3 Method ‣ PersonaX: A Recommendation Agent-Oriented User Modeling Framework for Long Behavior Sequence"><span class="ltx_text ltx_ref_tag">2</span></a>) to solve it, which iteratively selects the element with the highest marginal gain in the objective function. A visual explanation of the selection algorithm is provided in Appendix <a class="ltx_ref" href="https://arxiv.org/html/2503.02398v1#A4" title="Appendix D Details about In-Cluter Selection ‣ PersonaX: A Recommendation Agent-Oriented User Modeling Framework for Long Behavior Sequence"><span class="ltx_text ltx_ref_tag">D</span></a>.</p> </div> <figure class="ltx_figure" id="S3.F2"><img alt="Refer to caption" class="ltx_graphics ltx_centering ltx_img_landscape" height="227" id="S3.F2.g1" src="x2.png" width="401"/> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure">Figure 2: </span>Online time cost analysis.</figcaption> </figure> </section> <section class="ltx_subsection" id="S3.SS4"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">3.4 </span>Offline Profiling and Online Selection</h3> <div class="ltx_para" id="S3.SS4.p1"> <p class="ltx_p" id="S3.SS4.p1.3">After selecting representative SBS, PersonaX continue to construct persona offline. Given a selected behavior subset <math alttext="c_{i}^{*}" class="ltx_Math" display="inline" id="S3.SS4.p1.1.m1.1"><semantics id="S3.SS4.p1.1.m1.1a"><msubsup id="S3.SS4.p1.1.m1.1.1" xref="S3.SS4.p1.1.m1.1.1.cmml"><mi id="S3.SS4.p1.1.m1.1.1.2.2" xref="S3.SS4.p1.1.m1.1.1.2.2.cmml">c</mi><mi id="S3.SS4.p1.1.m1.1.1.2.3" xref="S3.SS4.p1.1.m1.1.1.2.3.cmml">i</mi><mo id="S3.SS4.p1.1.m1.1.1.3" xref="S3.SS4.p1.1.m1.1.1.3.cmml">∗</mo></msubsup><annotation-xml encoding="MathML-Content" id="S3.SS4.p1.1.m1.1b"><apply id="S3.SS4.p1.1.m1.1.1.cmml" xref="S3.SS4.p1.1.m1.1.1"><csymbol cd="ambiguous" id="S3.SS4.p1.1.m1.1.1.1.cmml" xref="S3.SS4.p1.1.m1.1.1">superscript</csymbol><apply id="S3.SS4.p1.1.m1.1.1.2.cmml" xref="S3.SS4.p1.1.m1.1.1"><csymbol cd="ambiguous" id="S3.SS4.p1.1.m1.1.1.2.1.cmml" xref="S3.SS4.p1.1.m1.1.1">subscript</csymbol><ci id="S3.SS4.p1.1.m1.1.1.2.2.cmml" xref="S3.SS4.p1.1.m1.1.1.2.2">𝑐</ci><ci id="S3.SS4.p1.1.m1.1.1.2.3.cmml" xref="S3.SS4.p1.1.m1.1.1.2.3">𝑖</ci></apply><times id="S3.SS4.p1.1.m1.1.1.3.cmml" xref="S3.SS4.p1.1.m1.1.1.3"></times></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.p1.1.m1.1c">c_{i}^{*}</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.p1.1.m1.1d">italic_c start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT</annotation></semantics></math>, we generate a corresponding persona <math alttext="p_{i}=\mathcal{P}(c_{i}^{*})" class="ltx_Math" display="inline" id="S3.SS4.p1.2.m2.1"><semantics id="S3.SS4.p1.2.m2.1a"><mrow id="S3.SS4.p1.2.m2.1.1" xref="S3.SS4.p1.2.m2.1.1.cmml"><msub id="S3.SS4.p1.2.m2.1.1.3" xref="S3.SS4.p1.2.m2.1.1.3.cmml"><mi id="S3.SS4.p1.2.m2.1.1.3.2" xref="S3.SS4.p1.2.m2.1.1.3.2.cmml">p</mi><mi id="S3.SS4.p1.2.m2.1.1.3.3" xref="S3.SS4.p1.2.m2.1.1.3.3.cmml">i</mi></msub><mo id="S3.SS4.p1.2.m2.1.1.2" xref="S3.SS4.p1.2.m2.1.1.2.cmml">=</mo><mrow id="S3.SS4.p1.2.m2.1.1.1" xref="S3.SS4.p1.2.m2.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.SS4.p1.2.m2.1.1.1.3" xref="S3.SS4.p1.2.m2.1.1.1.3.cmml">𝒫</mi><mo id="S3.SS4.p1.2.m2.1.1.1.2" xref="S3.SS4.p1.2.m2.1.1.1.2.cmml">⁢</mo><mrow id="S3.SS4.p1.2.m2.1.1.1.1.1" xref="S3.SS4.p1.2.m2.1.1.1.1.1.1.cmml"><mo id="S3.SS4.p1.2.m2.1.1.1.1.1.2" stretchy="false" xref="S3.SS4.p1.2.m2.1.1.1.1.1.1.cmml">(</mo><msubsup id="S3.SS4.p1.2.m2.1.1.1.1.1.1" xref="S3.SS4.p1.2.m2.1.1.1.1.1.1.cmml"><mi id="S3.SS4.p1.2.m2.1.1.1.1.1.1.2.2" xref="S3.SS4.p1.2.m2.1.1.1.1.1.1.2.2.cmml">c</mi><mi id="S3.SS4.p1.2.m2.1.1.1.1.1.1.2.3" xref="S3.SS4.p1.2.m2.1.1.1.1.1.1.2.3.cmml">i</mi><mo id="S3.SS4.p1.2.m2.1.1.1.1.1.1.3" xref="S3.SS4.p1.2.m2.1.1.1.1.1.1.3.cmml">∗</mo></msubsup><mo id="S3.SS4.p1.2.m2.1.1.1.1.1.3" stretchy="false" xref="S3.SS4.p1.2.m2.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS4.p1.2.m2.1b"><apply id="S3.SS4.p1.2.m2.1.1.cmml" xref="S3.SS4.p1.2.m2.1.1"><eq id="S3.SS4.p1.2.m2.1.1.2.cmml" xref="S3.SS4.p1.2.m2.1.1.2"></eq><apply id="S3.SS4.p1.2.m2.1.1.3.cmml" xref="S3.SS4.p1.2.m2.1.1.3"><csymbol cd="ambiguous" id="S3.SS4.p1.2.m2.1.1.3.1.cmml" xref="S3.SS4.p1.2.m2.1.1.3">subscript</csymbol><ci id="S3.SS4.p1.2.m2.1.1.3.2.cmml" xref="S3.SS4.p1.2.m2.1.1.3.2">𝑝</ci><ci id="S3.SS4.p1.2.m2.1.1.3.3.cmml" xref="S3.SS4.p1.2.m2.1.1.3.3">𝑖</ci></apply><apply id="S3.SS4.p1.2.m2.1.1.1.cmml" xref="S3.SS4.p1.2.m2.1.1.1"><times id="S3.SS4.p1.2.m2.1.1.1.2.cmml" xref="S3.SS4.p1.2.m2.1.1.1.2"></times><ci id="S3.SS4.p1.2.m2.1.1.1.3.cmml" xref="S3.SS4.p1.2.m2.1.1.1.3">𝒫</ci><apply id="S3.SS4.p1.2.m2.1.1.1.1.1.1.cmml" xref="S3.SS4.p1.2.m2.1.1.1.1.1"><csymbol cd="ambiguous" id="S3.SS4.p1.2.m2.1.1.1.1.1.1.1.cmml" xref="S3.SS4.p1.2.m2.1.1.1.1.1">superscript</csymbol><apply id="S3.SS4.p1.2.m2.1.1.1.1.1.1.2.cmml" xref="S3.SS4.p1.2.m2.1.1.1.1.1"><csymbol cd="ambiguous" id="S3.SS4.p1.2.m2.1.1.1.1.1.1.2.1.cmml" xref="S3.SS4.p1.2.m2.1.1.1.1.1">subscript</csymbol><ci id="S3.SS4.p1.2.m2.1.1.1.1.1.1.2.2.cmml" xref="S3.SS4.p1.2.m2.1.1.1.1.1.1.2.2">𝑐</ci><ci id="S3.SS4.p1.2.m2.1.1.1.1.1.1.2.3.cmml" xref="S3.SS4.p1.2.m2.1.1.1.1.1.1.2.3">𝑖</ci></apply><times id="S3.SS4.p1.2.m2.1.1.1.1.1.1.3.cmml" xref="S3.SS4.p1.2.m2.1.1.1.1.1.1.3"></times></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.p1.2.m2.1c">p_{i}=\mathcal{P}(c_{i}^{*})</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.p1.2.m2.1d">italic_p start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = caligraphic_P ( italic_c start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT )</annotation></semantics></math>. To ensure contextually relevant recommendations, PersonaX retrieves the most pertinent persona snippet <math alttext="P_{\text{selected}}" class="ltx_Math" display="inline" id="S3.SS4.p1.3.m3.1"><semantics id="S3.SS4.p1.3.m3.1a"><msub id="S3.SS4.p1.3.m3.1.1" xref="S3.SS4.p1.3.m3.1.1.cmml"><mi id="S3.SS4.p1.3.m3.1.1.2" xref="S3.SS4.p1.3.m3.1.1.2.cmml">P</mi><mtext id="S3.SS4.p1.3.m3.1.1.3" xref="S3.SS4.p1.3.m3.1.1.3a.cmml">selected</mtext></msub><annotation-xml encoding="MathML-Content" id="S3.SS4.p1.3.m3.1b"><apply id="S3.SS4.p1.3.m3.1.1.cmml" xref="S3.SS4.p1.3.m3.1.1"><csymbol cd="ambiguous" id="S3.SS4.p1.3.m3.1.1.1.cmml" xref="S3.SS4.p1.3.m3.1.1">subscript</csymbol><ci id="S3.SS4.p1.3.m3.1.1.2.cmml" xref="S3.SS4.p1.3.m3.1.1.2">𝑃</ci><ci id="S3.SS4.p1.3.m3.1.1.3a.cmml" xref="S3.SS4.p1.3.m3.1.1.3"><mtext id="S3.SS4.p1.3.m3.1.1.3.cmml" mathsize="70%" xref="S3.SS4.p1.3.m3.1.1.3">selected</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS4.p1.3.m3.1c">P_{\text{selected}}</annotation><annotation encoding="application/x-llamapun" id="S3.SS4.p1.3.m3.1d">italic_P start_POSTSUBSCRIPT selected end_POSTSUBSCRIPT</annotation></semantics></math> online, which is integrated into prompt templates to instruct agent recommendation.</p> </div> <figure class="ltx_table" id="S3.T1"> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_table">Table 1: </span>Time complexity analysis. Cluster, A.1 and A.2 refers to clustering method used in Section <a class="ltx_ref" href="https://arxiv.org/html/2503.02398v1#S3.SS1" title="3.1 Behavior Clustering ‣ 3 Method ‣ PersonaX: A Recommendation Agent-Oriented User Modeling Framework for Long Behavior Sequence"><span class="ltx_text ltx_ref_tag">3.1</span></a>, Algorithms <a class="ltx_ref" href="https://arxiv.org/html/2503.02398v1#alg1" title="Algorithm 1 ‣ 3.2 Sampling Budget Allocation ‣ 3 Method ‣ PersonaX: A Recommendation Agent-Oriented User Modeling Framework for Long Behavior Sequence"><span class="ltx_text ltx_ref_tag">1</span></a> and <a class="ltx_ref" href="https://arxiv.org/html/2503.02398v1#alg2" title="Algorithm 2 ‣ 3.2 Sampling Budget Allocation ‣ 3 Method ‣ PersonaX: A Recommendation Agent-Oriented User Modeling Framework for Long Behavior Sequence"><span class="ltx_text ltx_ref_tag">2</span></a>, respectively.</figcaption> <table class="ltx_tabular ltx_centering ltx_align_middle" id="S3.T1.8"> <tbody class="ltx_tbody"> <tr class="ltx_tr" id="S3.T1.8.9.1"> <td class="ltx_td ltx_align_center ltx_border_tt" id="S3.T1.8.9.1.1">LLM-UM</td> <td class="ltx_td ltx_align_center ltx_border_tt" id="S3.T1.8.9.1.2">Sampling</td> <td class="ltx_td ltx_align_center ltx_border_tt" id="S3.T1.8.9.1.3">Offline</td> <td class="ltx_td ltx_align_center ltx_border_tt" id="S3.T1.8.9.1.4">Online</td> </tr> <tr class="ltx_tr" id="S3.T1.1.1"> <td class="ltx_td ltx_align_center ltx_border_t" id="S3.T1.1.1.2" rowspan="2"><span class="ltx_text" id="S3.T1.1.1.2.1">Reflection</span></td> <td class="ltx_td ltx_align_center ltx_border_t" id="S3.T1.1.1.3">Recent</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S3.T1.1.1.4">\</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S3.T1.1.1.1"><math alttext="O(N_{I}T+2kT+1)" class="ltx_Math" display="inline" id="S3.T1.1.1.1.m1.1"><semantics id="S3.T1.1.1.1.m1.1a"><mrow id="S3.T1.1.1.1.m1.1.1" xref="S3.T1.1.1.1.m1.1.1.cmml"><mi id="S3.T1.1.1.1.m1.1.1.3" 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italic_C italic_T + italic_n italic_d + Cluster + A.1 + A.2 ) )</annotation></semantics></math></td> <td class="ltx_td ltx_align_center ltx_border_bb ltx_border_t" id="S3.T1.8.8.2"><math alttext="N_{I}O(T+C+d)" class="ltx_Math" display="inline" id="S3.T1.8.8.2.m1.1"><semantics id="S3.T1.8.8.2.m1.1a"><mrow id="S3.T1.8.8.2.m1.1.1" xref="S3.T1.8.8.2.m1.1.1.cmml"><msub id="S3.T1.8.8.2.m1.1.1.3" xref="S3.T1.8.8.2.m1.1.1.3.cmml"><mi id="S3.T1.8.8.2.m1.1.1.3.2" xref="S3.T1.8.8.2.m1.1.1.3.2.cmml">N</mi><mi id="S3.T1.8.8.2.m1.1.1.3.3" xref="S3.T1.8.8.2.m1.1.1.3.3.cmml">I</mi></msub><mo id="S3.T1.8.8.2.m1.1.1.2" xref="S3.T1.8.8.2.m1.1.1.2.cmml">⁢</mo><mi id="S3.T1.8.8.2.m1.1.1.4" xref="S3.T1.8.8.2.m1.1.1.4.cmml">O</mi><mo id="S3.T1.8.8.2.m1.1.1.2a" xref="S3.T1.8.8.2.m1.1.1.2.cmml">⁢</mo><mrow id="S3.T1.8.8.2.m1.1.1.1.1" xref="S3.T1.8.8.2.m1.1.1.1.1.1.cmml"><mo id="S3.T1.8.8.2.m1.1.1.1.1.2" stretchy="false" xref="S3.T1.8.8.2.m1.1.1.1.1.1.cmml">(</mo><mrow id="S3.T1.8.8.2.m1.1.1.1.1.1" 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xref="S3.T1.8.8.2.m1.1.1.3.2">𝑁</ci><ci id="S3.T1.8.8.2.m1.1.1.3.3.cmml" xref="S3.T1.8.8.2.m1.1.1.3.3">𝐼</ci></apply><ci id="S3.T1.8.8.2.m1.1.1.4.cmml" xref="S3.T1.8.8.2.m1.1.1.4">𝑂</ci><apply id="S3.T1.8.8.2.m1.1.1.1.1.1.cmml" xref="S3.T1.8.8.2.m1.1.1.1.1"><plus id="S3.T1.8.8.2.m1.1.1.1.1.1.1.cmml" xref="S3.T1.8.8.2.m1.1.1.1.1.1.1"></plus><ci id="S3.T1.8.8.2.m1.1.1.1.1.1.2.cmml" xref="S3.T1.8.8.2.m1.1.1.1.1.1.2">𝑇</ci><ci id="S3.T1.8.8.2.m1.1.1.1.1.1.3.cmml" xref="S3.T1.8.8.2.m1.1.1.1.1.1.3">𝐶</ci><ci id="S3.T1.8.8.2.m1.1.1.1.1.1.4.cmml" xref="S3.T1.8.8.2.m1.1.1.1.1.1.4">𝑑</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.T1.8.8.2.m1.1c">N_{I}O(T+C+d)</annotation><annotation encoding="application/x-llamapun" id="S3.T1.8.8.2.m1.1d">italic_N start_POSTSUBSCRIPT italic_I end_POSTSUBSCRIPT italic_O ( italic_T + italic_C + italic_d )</annotation></semantics></math></td> </tr> </tbody> </table> </figure> </section> </section> <section class="ltx_section" id="S4"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">4 </span>Efficiency Analysis</h2> <div class="ltx_para" id="S4.p1"> <p class="ltx_p" id="S4.p1.9">Recent sampling operates in constant time <math alttext="O(1)" class="ltx_Math" display="inline" id="S4.p1.1.m1.1"><semantics id="S4.p1.1.m1.1a"><mrow id="S4.p1.1.m1.1.2" xref="S4.p1.1.m1.1.2.cmml"><mi id="S4.p1.1.m1.1.2.2" xref="S4.p1.1.m1.1.2.2.cmml">O</mi><mo id="S4.p1.1.m1.1.2.1" xref="S4.p1.1.m1.1.2.1.cmml">⁢</mo><mrow id="S4.p1.1.m1.1.2.3.2" xref="S4.p1.1.m1.1.2.cmml"><mo id="S4.p1.1.m1.1.2.3.2.1" stretchy="false" xref="S4.p1.1.m1.1.2.cmml">(</mo><mn id="S4.p1.1.m1.1.1" xref="S4.p1.1.m1.1.1.cmml">1</mn><mo id="S4.p1.1.m1.1.2.3.2.2" stretchy="false" xref="S4.p1.1.m1.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.p1.1.m1.1b"><apply id="S4.p1.1.m1.1.2.cmml" xref="S4.p1.1.m1.1.2"><times id="S4.p1.1.m1.1.2.1.cmml" xref="S4.p1.1.m1.1.2.1"></times><ci id="S4.p1.1.m1.1.2.2.cmml" xref="S4.p1.1.m1.1.2.2">𝑂</ci><cn id="S4.p1.1.m1.1.1.cmml" type="integer" xref="S4.p1.1.m1.1.1">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p1.1.m1.1c">O(1)</annotation><annotation encoding="application/x-llamapun" id="S4.p1.1.m1.1d">italic_O ( 1 )</annotation></semantics></math>. Let <math alttext="O(d)" class="ltx_Math" display="inline" id="S4.p1.2.m2.1"><semantics id="S4.p1.2.m2.1a"><mrow id="S4.p1.2.m2.1.2" xref="S4.p1.2.m2.1.2.cmml"><mi id="S4.p1.2.m2.1.2.2" xref="S4.p1.2.m2.1.2.2.cmml">O</mi><mo id="S4.p1.2.m2.1.2.1" xref="S4.p1.2.m2.1.2.1.cmml">⁢</mo><mrow id="S4.p1.2.m2.1.2.3.2" xref="S4.p1.2.m2.1.2.cmml"><mo id="S4.p1.2.m2.1.2.3.2.1" stretchy="false" xref="S4.p1.2.m2.1.2.cmml">(</mo><mi id="S4.p1.2.m2.1.1" xref="S4.p1.2.m2.1.1.cmml">d</mi><mo id="S4.p1.2.m2.1.2.3.2.2" stretchy="false" xref="S4.p1.2.m2.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.p1.2.m2.1b"><apply id="S4.p1.2.m2.1.2.cmml" xref="S4.p1.2.m2.1.2"><times id="S4.p1.2.m2.1.2.1.cmml" xref="S4.p1.2.m2.1.2.1"></times><ci id="S4.p1.2.m2.1.2.2.cmml" xref="S4.p1.2.m2.1.2.2">𝑂</ci><ci id="S4.p1.2.m2.1.1.cmml" xref="S4.p1.2.m2.1.1">𝑑</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p1.2.m2.1c">O(d)</annotation><annotation encoding="application/x-llamapun" id="S4.p1.2.m2.1d">italic_O ( italic_d )</annotation></semantics></math> denotes the time required to encode an item into an embedding vector, and <math alttext="O(n\log k)" class="ltx_Math" display="inline" id="S4.p1.3.m3.1"><semantics id="S4.p1.3.m3.1a"><mrow id="S4.p1.3.m3.1.1" xref="S4.p1.3.m3.1.1.cmml"><mi id="S4.p1.3.m3.1.1.3" xref="S4.p1.3.m3.1.1.3.cmml">O</mi><mo id="S4.p1.3.m3.1.1.2" xref="S4.p1.3.m3.1.1.2.cmml">⁢</mo><mrow id="S4.p1.3.m3.1.1.1.1" xref="S4.p1.3.m3.1.1.1.1.1.cmml"><mo id="S4.p1.3.m3.1.1.1.1.2" stretchy="false" xref="S4.p1.3.m3.1.1.1.1.1.cmml">(</mo><mrow id="S4.p1.3.m3.1.1.1.1.1" xref="S4.p1.3.m3.1.1.1.1.1.cmml"><mi id="S4.p1.3.m3.1.1.1.1.1.2" xref="S4.p1.3.m3.1.1.1.1.1.2.cmml">n</mi><mo id="S4.p1.3.m3.1.1.1.1.1.1" lspace="0.167em" xref="S4.p1.3.m3.1.1.1.1.1.1.cmml">⁢</mo><mrow id="S4.p1.3.m3.1.1.1.1.1.3" xref="S4.p1.3.m3.1.1.1.1.1.3.cmml"><mi id="S4.p1.3.m3.1.1.1.1.1.3.1" xref="S4.p1.3.m3.1.1.1.1.1.3.1.cmml">log</mi><mo id="S4.p1.3.m3.1.1.1.1.1.3a" lspace="0.167em" xref="S4.p1.3.m3.1.1.1.1.1.3.cmml">⁡</mo><mi id="S4.p1.3.m3.1.1.1.1.1.3.2" xref="S4.p1.3.m3.1.1.1.1.1.3.2.cmml">k</mi></mrow></mrow><mo id="S4.p1.3.m3.1.1.1.1.3" stretchy="false" xref="S4.p1.3.m3.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.p1.3.m3.1b"><apply id="S4.p1.3.m3.1.1.cmml" xref="S4.p1.3.m3.1.1"><times id="S4.p1.3.m3.1.1.2.cmml" xref="S4.p1.3.m3.1.1.2"></times><ci id="S4.p1.3.m3.1.1.3.cmml" xref="S4.p1.3.m3.1.1.3">𝑂</ci><apply id="S4.p1.3.m3.1.1.1.1.1.cmml" xref="S4.p1.3.m3.1.1.1.1"><times id="S4.p1.3.m3.1.1.1.1.1.1.cmml" xref="S4.p1.3.m3.1.1.1.1.1.1"></times><ci id="S4.p1.3.m3.1.1.1.1.1.2.cmml" xref="S4.p1.3.m3.1.1.1.1.1.2">𝑛</ci><apply id="S4.p1.3.m3.1.1.1.1.1.3.cmml" xref="S4.p1.3.m3.1.1.1.1.1.3"><log id="S4.p1.3.m3.1.1.1.1.1.3.1.cmml" xref="S4.p1.3.m3.1.1.1.1.1.3.1"></log><ci id="S4.p1.3.m3.1.1.1.1.1.3.2.cmml" xref="S4.p1.3.m3.1.1.1.1.1.3.2">𝑘</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p1.3.m3.1c">O(n\log k)</annotation><annotation encoding="application/x-llamapun" id="S4.p1.3.m3.1d">italic_O ( italic_n roman_log italic_k )</annotation></semantics></math> represents the complexity of selecting the top <math alttext="k" class="ltx_Math" display="inline" id="S4.p1.4.m4.1"><semantics id="S4.p1.4.m4.1a"><mi id="S4.p1.4.m4.1.1" xref="S4.p1.4.m4.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="S4.p1.4.m4.1b"><ci id="S4.p1.4.m4.1.1.cmml" xref="S4.p1.4.m4.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.p1.4.m4.1c">k</annotation><annotation encoding="application/x-llamapun" id="S4.p1.4.m4.1d">italic_k</annotation></semantics></math> most relevant items from <math alttext="n" class="ltx_Math" display="inline" id="S4.p1.5.m5.1"><semantics id="S4.p1.5.m5.1a"><mi id="S4.p1.5.m5.1.1" xref="S4.p1.5.m5.1.1.cmml">n</mi><annotation-xml encoding="MathML-Content" id="S4.p1.5.m5.1b"><ci id="S4.p1.5.m5.1.1.cmml" xref="S4.p1.5.m5.1.1">𝑛</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.p1.5.m5.1c">n</annotation><annotation encoding="application/x-llamapun" id="S4.p1.5.m5.1d">italic_n</annotation></semantics></math> items. Thus Relevance sampling has a time complexity of <math alttext="O(nd+n\log k)" class="ltx_Math" display="inline" id="S4.p1.6.m6.1"><semantics id="S4.p1.6.m6.1a"><mrow id="S4.p1.6.m6.1.1" xref="S4.p1.6.m6.1.1.cmml"><mi id="S4.p1.6.m6.1.1.3" xref="S4.p1.6.m6.1.1.3.cmml">O</mi><mo id="S4.p1.6.m6.1.1.2" xref="S4.p1.6.m6.1.1.2.cmml">⁢</mo><mrow id="S4.p1.6.m6.1.1.1.1" xref="S4.p1.6.m6.1.1.1.1.1.cmml"><mo id="S4.p1.6.m6.1.1.1.1.2" stretchy="false" xref="S4.p1.6.m6.1.1.1.1.1.cmml">(</mo><mrow id="S4.p1.6.m6.1.1.1.1.1" xref="S4.p1.6.m6.1.1.1.1.1.cmml"><mrow id="S4.p1.6.m6.1.1.1.1.1.2" xref="S4.p1.6.m6.1.1.1.1.1.2.cmml"><mi id="S4.p1.6.m6.1.1.1.1.1.2.2" xref="S4.p1.6.m6.1.1.1.1.1.2.2.cmml">n</mi><mo id="S4.p1.6.m6.1.1.1.1.1.2.1" xref="S4.p1.6.m6.1.1.1.1.1.2.1.cmml">⁢</mo><mi id="S4.p1.6.m6.1.1.1.1.1.2.3" xref="S4.p1.6.m6.1.1.1.1.1.2.3.cmml">d</mi></mrow><mo id="S4.p1.6.m6.1.1.1.1.1.1" xref="S4.p1.6.m6.1.1.1.1.1.1.cmml">+</mo><mrow id="S4.p1.6.m6.1.1.1.1.1.3" xref="S4.p1.6.m6.1.1.1.1.1.3.cmml"><mi id="S4.p1.6.m6.1.1.1.1.1.3.2" xref="S4.p1.6.m6.1.1.1.1.1.3.2.cmml">n</mi><mo id="S4.p1.6.m6.1.1.1.1.1.3.1" lspace="0.167em" xref="S4.p1.6.m6.1.1.1.1.1.3.1.cmml">⁢</mo><mrow id="S4.p1.6.m6.1.1.1.1.1.3.3" xref="S4.p1.6.m6.1.1.1.1.1.3.3.cmml"><mi id="S4.p1.6.m6.1.1.1.1.1.3.3.1" xref="S4.p1.6.m6.1.1.1.1.1.3.3.1.cmml">log</mi><mo id="S4.p1.6.m6.1.1.1.1.1.3.3a" lspace="0.167em" xref="S4.p1.6.m6.1.1.1.1.1.3.3.cmml">⁡</mo><mi id="S4.p1.6.m6.1.1.1.1.1.3.3.2" xref="S4.p1.6.m6.1.1.1.1.1.3.3.2.cmml">k</mi></mrow></mrow></mrow><mo id="S4.p1.6.m6.1.1.1.1.3" stretchy="false" xref="S4.p1.6.m6.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.p1.6.m6.1b"><apply id="S4.p1.6.m6.1.1.cmml" xref="S4.p1.6.m6.1.1"><times id="S4.p1.6.m6.1.1.2.cmml" xref="S4.p1.6.m6.1.1.2"></times><ci id="S4.p1.6.m6.1.1.3.cmml" xref="S4.p1.6.m6.1.1.3">𝑂</ci><apply id="S4.p1.6.m6.1.1.1.1.1.cmml" xref="S4.p1.6.m6.1.1.1.1"><plus id="S4.p1.6.m6.1.1.1.1.1.1.cmml" xref="S4.p1.6.m6.1.1.1.1.1.1"></plus><apply id="S4.p1.6.m6.1.1.1.1.1.2.cmml" xref="S4.p1.6.m6.1.1.1.1.1.2"><times id="S4.p1.6.m6.1.1.1.1.1.2.1.cmml" xref="S4.p1.6.m6.1.1.1.1.1.2.1"></times><ci id="S4.p1.6.m6.1.1.1.1.1.2.2.cmml" xref="S4.p1.6.m6.1.1.1.1.1.2.2">𝑛</ci><ci id="S4.p1.6.m6.1.1.1.1.1.2.3.cmml" xref="S4.p1.6.m6.1.1.1.1.1.2.3">𝑑</ci></apply><apply id="S4.p1.6.m6.1.1.1.1.1.3.cmml" xref="S4.p1.6.m6.1.1.1.1.1.3"><times id="S4.p1.6.m6.1.1.1.1.1.3.1.cmml" xref="S4.p1.6.m6.1.1.1.1.1.3.1"></times><ci id="S4.p1.6.m6.1.1.1.1.1.3.2.cmml" xref="S4.p1.6.m6.1.1.1.1.1.3.2">𝑛</ci><apply id="S4.p1.6.m6.1.1.1.1.1.3.3.cmml" xref="S4.p1.6.m6.1.1.1.1.1.3.3"><log id="S4.p1.6.m6.1.1.1.1.1.3.3.1.cmml" xref="S4.p1.6.m6.1.1.1.1.1.3.3.1"></log><ci id="S4.p1.6.m6.1.1.1.1.1.3.3.2.cmml" xref="S4.p1.6.m6.1.1.1.1.1.3.3.2">𝑘</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p1.6.m6.1c">O(nd+n\log k)</annotation><annotation encoding="application/x-llamapun" id="S4.p1.6.m6.1d">italic_O ( italic_n italic_d + italic_n roman_log italic_k )</annotation></semantics></math>. For SBS sampling applied in PersonaX, we use <math alttext="O(\text{Cluster}+\text{Alg.1}+\text{Alg.2})" class="ltx_Math" display="inline" id="S4.p1.7.m7.1"><semantics id="S4.p1.7.m7.1a"><mrow id="S4.p1.7.m7.1.1" xref="S4.p1.7.m7.1.1.cmml"><mi id="S4.p1.7.m7.1.1.3" xref="S4.p1.7.m7.1.1.3.cmml">O</mi><mo id="S4.p1.7.m7.1.1.2" xref="S4.p1.7.m7.1.1.2.cmml">⁢</mo><mrow id="S4.p1.7.m7.1.1.1.1" xref="S4.p1.7.m7.1.1.1.1.1.cmml"><mo id="S4.p1.7.m7.1.1.1.1.2" stretchy="false" xref="S4.p1.7.m7.1.1.1.1.1.cmml">(</mo><mrow id="S4.p1.7.m7.1.1.1.1.1" xref="S4.p1.7.m7.1.1.1.1.1.cmml"><mtext id="S4.p1.7.m7.1.1.1.1.1.2" xref="S4.p1.7.m7.1.1.1.1.1.2a.cmml">Cluster</mtext><mo id="S4.p1.7.m7.1.1.1.1.1.1" xref="S4.p1.7.m7.1.1.1.1.1.1.cmml">+</mo><mtext id="S4.p1.7.m7.1.1.1.1.1.3" xref="S4.p1.7.m7.1.1.1.1.1.3a.cmml">Alg.1</mtext><mo id="S4.p1.7.m7.1.1.1.1.1.1a" xref="S4.p1.7.m7.1.1.1.1.1.1.cmml">+</mo><mtext id="S4.p1.7.m7.1.1.1.1.1.4" xref="S4.p1.7.m7.1.1.1.1.1.4a.cmml">Alg.2</mtext></mrow><mo id="S4.p1.7.m7.1.1.1.1.3" stretchy="false" xref="S4.p1.7.m7.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.p1.7.m7.1b"><apply id="S4.p1.7.m7.1.1.cmml" xref="S4.p1.7.m7.1.1"><times id="S4.p1.7.m7.1.1.2.cmml" xref="S4.p1.7.m7.1.1.2"></times><ci id="S4.p1.7.m7.1.1.3.cmml" xref="S4.p1.7.m7.1.1.3">𝑂</ci><apply id="S4.p1.7.m7.1.1.1.1.1.cmml" xref="S4.p1.7.m7.1.1.1.1"><plus id="S4.p1.7.m7.1.1.1.1.1.1.cmml" xref="S4.p1.7.m7.1.1.1.1.1.1"></plus><ci id="S4.p1.7.m7.1.1.1.1.1.2a.cmml" xref="S4.p1.7.m7.1.1.1.1.1.2"><mtext id="S4.p1.7.m7.1.1.1.1.1.2.cmml" xref="S4.p1.7.m7.1.1.1.1.1.2">Cluster</mtext></ci><ci id="S4.p1.7.m7.1.1.1.1.1.3a.cmml" xref="S4.p1.7.m7.1.1.1.1.1.3"><mtext id="S4.p1.7.m7.1.1.1.1.1.3.cmml" xref="S4.p1.7.m7.1.1.1.1.1.3">Alg.1</mtext></ci><ci id="S4.p1.7.m7.1.1.1.1.1.4a.cmml" xref="S4.p1.7.m7.1.1.1.1.1.4"><mtext id="S4.p1.7.m7.1.1.1.1.1.4.cmml" xref="S4.p1.7.m7.1.1.1.1.1.4">Alg.2</mtext></ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p1.7.m7.1c">O(\text{Cluster}+\text{Alg.1}+\text{Alg.2})</annotation><annotation encoding="application/x-llamapun" id="S4.p1.7.m7.1d">italic_O ( Cluster + Alg.1 + Alg.2 )</annotation></semantics></math> represent the time cost for process we depict from Section <a class="ltx_ref" href="https://arxiv.org/html/2503.02398v1#S3.SS1" title="3.1 Behavior Clustering ‣ 3 Method ‣ PersonaX: A Recommendation Agent-Oriented User Modeling Framework for Long Behavior Sequence"><span class="ltx_text ltx_ref_tag">3.1</span></a> to <a class="ltx_ref" href="https://arxiv.org/html/2503.02398v1#S3.SS3" title="3.3 In-Cluster Selection ‣ 3 Method ‣ PersonaX: A Recommendation Agent-Oriented User Modeling Framework for Long Behavior Sequence"><span class="ltx_text ltx_ref_tag">3.3</span></a>. Let <math alttext="O(T)" class="ltx_Math" display="inline" id="S4.p1.8.m8.1"><semantics id="S4.p1.8.m8.1a"><mrow id="S4.p1.8.m8.1.2" xref="S4.p1.8.m8.1.2.cmml"><mi id="S4.p1.8.m8.1.2.2" xref="S4.p1.8.m8.1.2.2.cmml">O</mi><mo id="S4.p1.8.m8.1.2.1" xref="S4.p1.8.m8.1.2.1.cmml">⁢</mo><mrow id="S4.p1.8.m8.1.2.3.2" xref="S4.p1.8.m8.1.2.cmml"><mo id="S4.p1.8.m8.1.2.3.2.1" stretchy="false" xref="S4.p1.8.m8.1.2.cmml">(</mo><mi id="S4.p1.8.m8.1.1" xref="S4.p1.8.m8.1.1.cmml">T</mi><mo id="S4.p1.8.m8.1.2.3.2.2" stretchy="false" xref="S4.p1.8.m8.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.p1.8.m8.1b"><apply id="S4.p1.8.m8.1.2.cmml" xref="S4.p1.8.m8.1.2"><times id="S4.p1.8.m8.1.2.1.cmml" xref="S4.p1.8.m8.1.2.1"></times><ci id="S4.p1.8.m8.1.2.2.cmml" xref="S4.p1.8.m8.1.2.2">𝑂</ci><ci id="S4.p1.8.m8.1.1.cmml" xref="S4.p1.8.m8.1.1">𝑇</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p1.8.m8.1c">O(T)</annotation><annotation encoding="application/x-llamapun" id="S4.p1.8.m8.1d">italic_O ( italic_T )</annotation></semantics></math> represent the time complexity of an API request to LLMs, and <math alttext="N_{I}" class="ltx_Math" display="inline" id="S4.p1.9.m9.1"><semantics id="S4.p1.9.m9.1a"><msub id="S4.p1.9.m9.1.1" xref="S4.p1.9.m9.1.1.cmml"><mi id="S4.p1.9.m9.1.1.2" xref="S4.p1.9.m9.1.1.2.cmml">N</mi><mi id="S4.p1.9.m9.1.1.3" xref="S4.p1.9.m9.1.1.3.cmml">I</mi></msub><annotation-xml encoding="MathML-Content" id="S4.p1.9.m9.1b"><apply id="S4.p1.9.m9.1.1.cmml" xref="S4.p1.9.m9.1.1"><csymbol cd="ambiguous" id="S4.p1.9.m9.1.1.1.cmml" xref="S4.p1.9.m9.1.1">subscript</csymbol><ci id="S4.p1.9.m9.1.1.2.cmml" xref="S4.p1.9.m9.1.1.2">𝑁</ci><ci id="S4.p1.9.m9.1.1.3.cmml" xref="S4.p1.9.m9.1.1.3">𝐼</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p1.9.m9.1c">N_{I}</annotation><annotation encoding="application/x-llamapun" id="S4.p1.9.m9.1d">italic_N start_POSTSUBSCRIPT italic_I end_POSTSUBSCRIPT</annotation></semantics></math> denotes the number of items inferred per user. We perform an analysis of the time complexity during both offline and online stages associated with two LLM-UM approaches—Reflection and Summarization—combined with Recent and Relevance sampling strategies. Additionally, we evaluate our proposed PersonaX for comparison. The results are summarized in Table <a class="ltx_ref" href="https://arxiv.org/html/2503.02398v1#S3.T1" title="Table 1 ‣ 3.4 Offline Profiling and Online Selection ‣ 3 Method ‣ PersonaX: A Recommendation Agent-Oriented User Modeling Framework for Long Behavior Sequence"><span class="ltx_text ltx_ref_tag">1</span></a>.</p> </div> <div class="ltx_para" id="S4.p2"> <p class="ltx_p" id="S4.p2.13">The primary contributors to time consumption are <math alttext="T" class="ltx_Math" display="inline" id="S4.p2.1.m1.1"><semantics id="S4.p2.1.m1.1a"><mi id="S4.p2.1.m1.1.1" xref="S4.p2.1.m1.1.1.cmml">T</mi><annotation-xml encoding="MathML-Content" id="S4.p2.1.m1.1b"><ci id="S4.p2.1.m1.1.1.cmml" xref="S4.p2.1.m1.1.1">𝑇</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.p2.1.m1.1c">T</annotation><annotation encoding="application/x-llamapun" id="S4.p2.1.m1.1d">italic_T</annotation></semantics></math> and <math alttext="d" class="ltx_Math" display="inline" id="S4.p2.2.m2.1"><semantics id="S4.p2.2.m2.1a"><mi id="S4.p2.2.m2.1.1" xref="S4.p2.2.m2.1.1.cmml">d</mi><annotation-xml encoding="MathML-Content" id="S4.p2.2.m2.1b"><ci id="S4.p2.2.m2.1.1.cmml" xref="S4.p2.2.m2.1.1">𝑑</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.p2.2.m2.1c">d</annotation><annotation encoding="application/x-llamapun" id="S4.p2.2.m2.1d">italic_d</annotation></semantics></math>, while <math alttext="O(C)" class="ltx_Math" display="inline" id="S4.p2.3.m3.1"><semantics id="S4.p2.3.m3.1a"><mrow id="S4.p2.3.m3.1.2" xref="S4.p2.3.m3.1.2.cmml"><mi id="S4.p2.3.m3.1.2.2" xref="S4.p2.3.m3.1.2.2.cmml">O</mi><mo id="S4.p2.3.m3.1.2.1" xref="S4.p2.3.m3.1.2.1.cmml">⁢</mo><mrow id="S4.p2.3.m3.1.2.3.2" xref="S4.p2.3.m3.1.2.cmml"><mo id="S4.p2.3.m3.1.2.3.2.1" stretchy="false" xref="S4.p2.3.m3.1.2.cmml">(</mo><mi id="S4.p2.3.m3.1.1" xref="S4.p2.3.m3.1.1.cmml">C</mi><mo id="S4.p2.3.m3.1.2.3.2.2" stretchy="false" xref="S4.p2.3.m3.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.p2.3.m3.1b"><apply id="S4.p2.3.m3.1.2.cmml" xref="S4.p2.3.m3.1.2"><times id="S4.p2.3.m3.1.2.1.cmml" xref="S4.p2.3.m3.1.2.1"></times><ci id="S4.p2.3.m3.1.2.2.cmml" xref="S4.p2.3.m3.1.2.2">𝑂</ci><ci id="S4.p2.3.m3.1.1.cmml" xref="S4.p2.3.m3.1.1">𝐶</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p2.3.m3.1c">O(C)</annotation><annotation encoding="application/x-llamapun" id="S4.p2.3.m3.1d">italic_O ( italic_C )</annotation></semantics></math>, <math alttext="O(n\log k)" class="ltx_Math" display="inline" id="S4.p2.4.m4.1"><semantics id="S4.p2.4.m4.1a"><mrow id="S4.p2.4.m4.1.1" xref="S4.p2.4.m4.1.1.cmml"><mi id="S4.p2.4.m4.1.1.3" xref="S4.p2.4.m4.1.1.3.cmml">O</mi><mo id="S4.p2.4.m4.1.1.2" xref="S4.p2.4.m4.1.1.2.cmml">⁢</mo><mrow id="S4.p2.4.m4.1.1.1.1" xref="S4.p2.4.m4.1.1.1.1.1.cmml"><mo id="S4.p2.4.m4.1.1.1.1.2" stretchy="false" 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xref="S4.p2.4.m4.1.1.1.1"><times id="S4.p2.4.m4.1.1.1.1.1.1.cmml" xref="S4.p2.4.m4.1.1.1.1.1.1"></times><ci id="S4.p2.4.m4.1.1.1.1.1.2.cmml" xref="S4.p2.4.m4.1.1.1.1.1.2">𝑛</ci><apply id="S4.p2.4.m4.1.1.1.1.1.3.cmml" xref="S4.p2.4.m4.1.1.1.1.1.3"><log id="S4.p2.4.m4.1.1.1.1.1.3.1.cmml" xref="S4.p2.4.m4.1.1.1.1.1.3.1"></log><ci id="S4.p2.4.m4.1.1.1.1.1.3.2.cmml" xref="S4.p2.4.m4.1.1.1.1.1.3.2">𝑘</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p2.4.m4.1c">O(n\log k)</annotation><annotation encoding="application/x-llamapun" id="S4.p2.4.m4.1d">italic_O ( italic_n roman_log italic_k )</annotation></semantics></math>, <math alttext="O(\text{Cluster+A.1+A.2})" class="ltx_Math" display="inline" id="S4.p2.5.m5.1"><semantics id="S4.p2.5.m5.1a"><mrow id="S4.p2.5.m5.1.2" xref="S4.p2.5.m5.1.2.cmml"><mi id="S4.p2.5.m5.1.2.2" xref="S4.p2.5.m5.1.2.2.cmml">O</mi><mo id="S4.p2.5.m5.1.2.1" xref="S4.p2.5.m5.1.2.1.cmml">⁢</mo><mrow id="S4.p2.5.m5.1.2.3.2" xref="S4.p2.5.m5.1.1a.cmml"><mo id="S4.p2.5.m5.1.2.3.2.1" stretchy="false" xref="S4.p2.5.m5.1.1a.cmml">(</mo><mtext id="S4.p2.5.m5.1.1" xref="S4.p2.5.m5.1.1.cmml">Cluster+A.1+A.2</mtext><mo id="S4.p2.5.m5.1.2.3.2.2" stretchy="false" xref="S4.p2.5.m5.1.1a.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.p2.5.m5.1b"><apply id="S4.p2.5.m5.1.2.cmml" xref="S4.p2.5.m5.1.2"><times id="S4.p2.5.m5.1.2.1.cmml" xref="S4.p2.5.m5.1.2.1"></times><ci id="S4.p2.5.m5.1.2.2.cmml" xref="S4.p2.5.m5.1.2.2">𝑂</ci><ci id="S4.p2.5.m5.1.1a.cmml" xref="S4.p2.5.m5.1.2.3.2"><mtext id="S4.p2.5.m5.1.1.cmml" xref="S4.p2.5.m5.1.1">Cluster+A.1+A.2</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p2.5.m5.1c">O(\text{Cluster+A.1+A.2})</annotation><annotation encoding="application/x-llamapun" id="S4.p2.5.m5.1d">italic_O ( Cluster+A.1+A.2 )</annotation></semantics></math> in ranking, clustering, and sampling are negligible. The Reflection approach exhibits significant time overhead, especially with relevance sampling, which introduces a quadratic dependency of <math alttext="N_{I}^{2}T" class="ltx_Math" display="inline" id="S4.p2.6.m6.1"><semantics id="S4.p2.6.m6.1a"><mrow id="S4.p2.6.m6.1.1" xref="S4.p2.6.m6.1.1.cmml"><msubsup id="S4.p2.6.m6.1.1.2" xref="S4.p2.6.m6.1.1.2.cmml"><mi id="S4.p2.6.m6.1.1.2.2.2" xref="S4.p2.6.m6.1.1.2.2.2.cmml">N</mi><mi id="S4.p2.6.m6.1.1.2.2.3" xref="S4.p2.6.m6.1.1.2.2.3.cmml">I</mi><mn id="S4.p2.6.m6.1.1.2.3" xref="S4.p2.6.m6.1.1.2.3.cmml">2</mn></msubsup><mo id="S4.p2.6.m6.1.1.1" xref="S4.p2.6.m6.1.1.1.cmml">⁢</mo><mi id="S4.p2.6.m6.1.1.3" xref="S4.p2.6.m6.1.1.3.cmml">T</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.p2.6.m6.1b"><apply id="S4.p2.6.m6.1.1.cmml" xref="S4.p2.6.m6.1.1"><times id="S4.p2.6.m6.1.1.1.cmml" xref="S4.p2.6.m6.1.1.1"></times><apply id="S4.p2.6.m6.1.1.2.cmml" xref="S4.p2.6.m6.1.1.2"><csymbol cd="ambiguous" id="S4.p2.6.m6.1.1.2.1.cmml" xref="S4.p2.6.m6.1.1.2">superscript</csymbol><apply id="S4.p2.6.m6.1.1.2.2.cmml" xref="S4.p2.6.m6.1.1.2"><csymbol cd="ambiguous" id="S4.p2.6.m6.1.1.2.2.1.cmml" xref="S4.p2.6.m6.1.1.2">subscript</csymbol><ci id="S4.p2.6.m6.1.1.2.2.2.cmml" xref="S4.p2.6.m6.1.1.2.2.2">𝑁</ci><ci id="S4.p2.6.m6.1.1.2.2.3.cmml" xref="S4.p2.6.m6.1.1.2.2.3">𝐼</ci></apply><cn id="S4.p2.6.m6.1.1.2.3.cmml" type="integer" xref="S4.p2.6.m6.1.1.2.3">2</cn></apply><ci id="S4.p2.6.m6.1.1.3.cmml" xref="S4.p2.6.m6.1.1.3">𝑇</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p2.6.m6.1c">N_{I}^{2}T</annotation><annotation encoding="application/x-llamapun" id="S4.p2.6.m6.1d">italic_N start_POSTSUBSCRIPT italic_I end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_T</annotation></semantics></math>. In contrast, other methods maintain linear complexity, <math alttext="N_{I}T" class="ltx_Math" display="inline" id="S4.p2.7.m7.1"><semantics id="S4.p2.7.m7.1a"><mrow id="S4.p2.7.m7.1.1" xref="S4.p2.7.m7.1.1.cmml"><msub id="S4.p2.7.m7.1.1.2" xref="S4.p2.7.m7.1.1.2.cmml"><mi id="S4.p2.7.m7.1.1.2.2" xref="S4.p2.7.m7.1.1.2.2.cmml">N</mi><mi id="S4.p2.7.m7.1.1.2.3" xref="S4.p2.7.m7.1.1.2.3.cmml">I</mi></msub><mo id="S4.p2.7.m7.1.1.1" xref="S4.p2.7.m7.1.1.1.cmml">⁢</mo><mi id="S4.p2.7.m7.1.1.3" xref="S4.p2.7.m7.1.1.3.cmml">T</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.p2.7.m7.1b"><apply id="S4.p2.7.m7.1.1.cmml" xref="S4.p2.7.m7.1.1"><times id="S4.p2.7.m7.1.1.1.cmml" xref="S4.p2.7.m7.1.1.1"></times><apply id="S4.p2.7.m7.1.1.2.cmml" xref="S4.p2.7.m7.1.1.2"><csymbol cd="ambiguous" id="S4.p2.7.m7.1.1.2.1.cmml" xref="S4.p2.7.m7.1.1.2">subscript</csymbol><ci id="S4.p2.7.m7.1.1.2.2.cmml" xref="S4.p2.7.m7.1.1.2.2">𝑁</ci><ci id="S4.p2.7.m7.1.1.2.3.cmml" xref="S4.p2.7.m7.1.1.2.3">𝐼</ci></apply><ci id="S4.p2.7.m7.1.1.3.cmml" xref="S4.p2.7.m7.1.1.3">𝑇</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p2.7.m7.1c">N_{I}T</annotation><annotation encoding="application/x-llamapun" id="S4.p2.7.m7.1d">italic_N start_POSTSUBSCRIPT italic_I end_POSTSUBSCRIPT italic_T</annotation></semantics></math>. Assuming <math alttext="n=500" class="ltx_Math" display="inline" id="S4.p2.8.m8.1"><semantics id="S4.p2.8.m8.1a"><mrow id="S4.p2.8.m8.1.1" xref="S4.p2.8.m8.1.1.cmml"><mi id="S4.p2.8.m8.1.1.2" xref="S4.p2.8.m8.1.1.2.cmml">n</mi><mo id="S4.p2.8.m8.1.1.1" xref="S4.p2.8.m8.1.1.1.cmml">=</mo><mn id="S4.p2.8.m8.1.1.3" xref="S4.p2.8.m8.1.1.3.cmml">500</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.p2.8.m8.1b"><apply id="S4.p2.8.m8.1.1.cmml" xref="S4.p2.8.m8.1.1"><eq id="S4.p2.8.m8.1.1.1.cmml" xref="S4.p2.8.m8.1.1.1"></eq><ci id="S4.p2.8.m8.1.1.2.cmml" xref="S4.p2.8.m8.1.1.2">𝑛</ci><cn id="S4.p2.8.m8.1.1.3.cmml" type="integer" xref="S4.p2.8.m8.1.1.3">500</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p2.8.m8.1c">n=500</annotation><annotation encoding="application/x-llamapun" id="S4.p2.8.m8.1d">italic_n = 500</annotation></semantics></math>, <math alttext="C=20" class="ltx_Math" display="inline" id="S4.p2.9.m9.1"><semantics id="S4.p2.9.m9.1a"><mrow id="S4.p2.9.m9.1.1" xref="S4.p2.9.m9.1.1.cmml"><mi id="S4.p2.9.m9.1.1.2" xref="S4.p2.9.m9.1.1.2.cmml">C</mi><mo id="S4.p2.9.m9.1.1.1" xref="S4.p2.9.m9.1.1.1.cmml">=</mo><mn id="S4.p2.9.m9.1.1.3" xref="S4.p2.9.m9.1.1.3.cmml">20</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.p2.9.m9.1b"><apply id="S4.p2.9.m9.1.1.cmml" xref="S4.p2.9.m9.1.1"><eq id="S4.p2.9.m9.1.1.1.cmml" xref="S4.p2.9.m9.1.1.1"></eq><ci id="S4.p2.9.m9.1.1.2.cmml" xref="S4.p2.9.m9.1.1.2">𝐶</ci><cn id="S4.p2.9.m9.1.1.3.cmml" type="integer" xref="S4.p2.9.m9.1.1.3">20</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p2.9.m9.1c">C=20</annotation><annotation encoding="application/x-llamapun" id="S4.p2.9.m9.1d">italic_C = 20</annotation></semantics></math>, <math alttext="T=3" class="ltx_Math" display="inline" id="S4.p2.10.m10.1"><semantics id="S4.p2.10.m10.1a"><mrow id="S4.p2.10.m10.1.1" xref="S4.p2.10.m10.1.1.cmml"><mi id="S4.p2.10.m10.1.1.2" xref="S4.p2.10.m10.1.1.2.cmml">T</mi><mo id="S4.p2.10.m10.1.1.1" xref="S4.p2.10.m10.1.1.1.cmml">=</mo><mn id="S4.p2.10.m10.1.1.3" xref="S4.p2.10.m10.1.1.3.cmml">3</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.p2.10.m10.1b"><apply id="S4.p2.10.m10.1.1.cmml" xref="S4.p2.10.m10.1.1"><eq id="S4.p2.10.m10.1.1.1.cmml" xref="S4.p2.10.m10.1.1.1"></eq><ci id="S4.p2.10.m10.1.1.2.cmml" xref="S4.p2.10.m10.1.1.2">𝑇</ci><cn id="S4.p2.10.m10.1.1.3.cmml" type="integer" xref="S4.p2.10.m10.1.1.3">3</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p2.10.m10.1c">T=3</annotation><annotation encoding="application/x-llamapun" id="S4.p2.10.m10.1d">italic_T = 3</annotation></semantics></math>, <math alttext="d=1" class="ltx_Math" display="inline" id="S4.p2.11.m11.1"><semantics id="S4.p2.11.m11.1a"><mrow id="S4.p2.11.m11.1.1" xref="S4.p2.11.m11.1.1.cmml"><mi id="S4.p2.11.m11.1.1.2" xref="S4.p2.11.m11.1.1.2.cmml">d</mi><mo id="S4.p2.11.m11.1.1.1" xref="S4.p2.11.m11.1.1.1.cmml">=</mo><mn id="S4.p2.11.m11.1.1.3" xref="S4.p2.11.m11.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.p2.11.m11.1b"><apply id="S4.p2.11.m11.1.1.cmml" xref="S4.p2.11.m11.1.1"><eq id="S4.p2.11.m11.1.1.1.cmml" xref="S4.p2.11.m11.1.1.1"></eq><ci id="S4.p2.11.m11.1.1.2.cmml" xref="S4.p2.11.m11.1.1.2">𝑑</ci><cn id="S4.p2.11.m11.1.1.3.cmml" type="integer" xref="S4.p2.11.m11.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p2.11.m11.1c">d=1</annotation><annotation encoding="application/x-llamapun" id="S4.p2.11.m11.1d">italic_d = 1</annotation></semantics></math>, and <math alttext="k=10" class="ltx_Math" display="inline" id="S4.p2.12.m12.1"><semantics id="S4.p2.12.m12.1a"><mrow id="S4.p2.12.m12.1.1" xref="S4.p2.12.m12.1.1.cmml"><mi id="S4.p2.12.m12.1.1.2" xref="S4.p2.12.m12.1.1.2.cmml">k</mi><mo id="S4.p2.12.m12.1.1.1" xref="S4.p2.12.m12.1.1.1.cmml">=</mo><mn id="S4.p2.12.m12.1.1.3" xref="S4.p2.12.m12.1.1.3.cmml">10</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.p2.12.m12.1b"><apply id="S4.p2.12.m12.1.1.cmml" xref="S4.p2.12.m12.1.1"><eq id="S4.p2.12.m12.1.1.1.cmml" xref="S4.p2.12.m12.1.1.1"></eq><ci id="S4.p2.12.m12.1.1.2.cmml" xref="S4.p2.12.m12.1.1.2">𝑘</ci><cn id="S4.p2.12.m12.1.1.3.cmml" type="integer" xref="S4.p2.12.m12.1.1.3">10</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p2.12.m12.1c">k=10</annotation><annotation encoding="application/x-llamapun" id="S4.p2.12.m12.1d">italic_k = 10</annotation></semantics></math>, and varying <math alttext="N_{I}" class="ltx_Math" display="inline" id="S4.p2.13.m13.1"><semantics id="S4.p2.13.m13.1a"><msub id="S4.p2.13.m13.1.1" xref="S4.p2.13.m13.1.1.cmml"><mi id="S4.p2.13.m13.1.1.2" xref="S4.p2.13.m13.1.1.2.cmml">N</mi><mi id="S4.p2.13.m13.1.1.3" xref="S4.p2.13.m13.1.1.3.cmml">I</mi></msub><annotation-xml encoding="MathML-Content" id="S4.p2.13.m13.1b"><apply id="S4.p2.13.m13.1.1.cmml" xref="S4.p2.13.m13.1.1"><csymbol cd="ambiguous" id="S4.p2.13.m13.1.1.1.cmml" xref="S4.p2.13.m13.1.1">subscript</csymbol><ci id="S4.p2.13.m13.1.1.2.cmml" xref="S4.p2.13.m13.1.1.2">𝑁</ci><ci id="S4.p2.13.m13.1.1.3.cmml" xref="S4.p2.13.m13.1.1.3">𝐼</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p2.13.m13.1c">N_{I}</annotation><annotation encoding="application/x-llamapun" id="S4.p2.13.m13.1d">italic_N start_POSTSUBSCRIPT italic_I end_POSTSUBSCRIPT</annotation></semantics></math> over 100, 500, and 1000. Figure <a class="ltx_ref" href="https://arxiv.org/html/2503.02398v1#S3.F2" title="Figure 2 ‣ 3.3 In-Cluster Selection ‣ 3 Method ‣ PersonaX: A Recommendation Agent-Oriented User Modeling Framework for Long Behavior Sequence"><span class="ltx_text ltx_ref_tag">2</span></a> presents a visualized analysis of online time consumption. We backbone the LLM-UM method as summarization, then we compared PersonaX with Recent and Relevance sampling strategies. The results show that PersonaX halves the computational time of Summarization + Relevance and matches the efficiency of Summarization + Recent.</p> </div> </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">In this section, we are to address these research questions (<span class="ltx_text ltx_font_bold" id="S5.p1.1.1">RQs</span>):</p> </div> <div class="ltx_para" id="S5.p2"> <p class="ltx_p" id="S5.p2.1"><math alttext="\bullet" class="ltx_Math" display="inline" id="S5.p2.1.m1.1"><semantics id="S5.p2.1.m1.1a"><mo id="S5.p2.1.m1.1.1" xref="S5.p2.1.m1.1.1.cmml">∙</mo><annotation-xml encoding="MathML-Content" id="S5.p2.1.m1.1b"><ci id="S5.p2.1.m1.1.1.cmml" xref="S5.p2.1.m1.1.1">∙</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.p2.1.m1.1c">\bullet</annotation><annotation encoding="application/x-llamapun" id="S5.p2.1.m1.1d">∙</annotation></semantics></math> <span class="ltx_text ltx_font_bold" id="S5.p2.1.1">RQ1</span>: How does PersonaX improve downstream agent recommendation, and how the performance compared with baseline approaches?</p> </div> <div class="ltx_para" id="S5.p3"> <p class="ltx_p" id="S5.p3.1"><math alttext="\bullet" class="ltx_Math" display="inline" id="S5.p3.1.m1.1"><semantics id="S5.p3.1.m1.1a"><mo id="S5.p3.1.m1.1.1" xref="S5.p3.1.m1.1.1.cmml">∙</mo><annotation-xml encoding="MathML-Content" id="S5.p3.1.m1.1b"><ci id="S5.p3.1.m1.1.1.cmml" xref="S5.p3.1.m1.1.1">∙</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.p3.1.m1.1c">\bullet</annotation><annotation encoding="application/x-llamapun" id="S5.p3.1.m1.1d">∙</annotation></semantics></math> <span class="ltx_text ltx_font_bold" id="S5.p3.1.1">RQ2</span>: How does the sampling size of historical behaviors affect the efficacy of user modeling?</p> </div> <div class="ltx_para" id="S5.p4"> <p class="ltx_p" id="S5.p4.1"><math alttext="\bullet" class="ltx_Math" display="inline" id="S5.p4.1.m1.1"><semantics id="S5.p4.1.m1.1a"><mo id="S5.p4.1.m1.1.1" xref="S5.p4.1.m1.1.1.cmml">∙</mo><annotation-xml encoding="MathML-Content" id="S5.p4.1.m1.1b"><ci id="S5.p4.1.m1.1.1.cmml" xref="S5.p4.1.m1.1.1">∙</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.p4.1.m1.1c">\bullet</annotation><annotation encoding="application/x-llamapun" id="S5.p4.1.m1.1d">∙</annotation></semantics></math> <span class="ltx_text ltx_font_bold" id="S5.p4.1.1">RQ3:</span> How sensitive is our method to hyper-parameter settings, and how can optimal parameters be chosen?</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>Experimental Setup</h3> <section class="ltx_subsubsection" id="S5.SS1.SSS1"> <h4 class="ltx_title ltx_title_subsubsection"> <span class="ltx_tag ltx_tag_subsubsection">5.1.1 </span>Datasets</h4> <div class="ltx_para" id="S5.SS1.SSS1.p1"> <p class="ltx_p" id="S5.SS1.SSS1.p1.3">We evaluate on two widely used subsets of the Amazon review dataset <cite class="ltx_cite ltx_citemacro_cite">Ni et al. (<a class="ltx_ref" href="https://arxiv.org/html/2503.02398v1#bib.bib23" title="">2019</a>)</cite>: <span class="ltx_text ltx_font_italic" id="S5.SS1.SSS1.p1.3.1">CDs and Vinyl</span> and <span class="ltx_text ltx_font_italic" id="S5.SS1.SSS1.p1.3.2">Books</span>. For the CDs dataset, similar to the settings in <cite class="ltx_cite ltx_citemacro_cite">Zhang et al. (<a class="ltx_ref" href="https://arxiv.org/html/2503.02398v1#bib.bib57" title="">2024b</a>)</cite>, we consider two variants, <math alttext="\texttt{CDs}_{\texttt{50}}" class="ltx_Math" display="inline" id="S5.SS1.SSS1.p1.1.m1.1"><semantics id="S5.SS1.SSS1.p1.1.m1.1a"><msub id="S5.SS1.SSS1.p1.1.m1.1.1" xref="S5.SS1.SSS1.p1.1.m1.1.1.cmml"><mtext class="ltx_mathvariant_monospace" id="S5.SS1.SSS1.p1.1.m1.1.1.2" xref="S5.SS1.SSS1.p1.1.m1.1.1.2a.cmml">CDs</mtext><mtext class="ltx_mathvariant_monospace" id="S5.SS1.SSS1.p1.1.m1.1.1.3" xref="S5.SS1.SSS1.p1.1.m1.1.1.3a.cmml">50</mtext></msub><annotation-xml encoding="MathML-Content" id="S5.SS1.SSS1.p1.1.m1.1b"><apply id="S5.SS1.SSS1.p1.1.m1.1.1.cmml" xref="S5.SS1.SSS1.p1.1.m1.1.1"><csymbol cd="ambiguous" id="S5.SS1.SSS1.p1.1.m1.1.1.1.cmml" xref="S5.SS1.SSS1.p1.1.m1.1.1">subscript</csymbol><ci id="S5.SS1.SSS1.p1.1.m1.1.1.2a.cmml" xref="S5.SS1.SSS1.p1.1.m1.1.1.2"><mtext class="ltx_mathvariant_monospace" id="S5.SS1.SSS1.p1.1.m1.1.1.2.cmml" xref="S5.SS1.SSS1.p1.1.m1.1.1.2">CDs</mtext></ci><ci id="S5.SS1.SSS1.p1.1.m1.1.1.3a.cmml" xref="S5.SS1.SSS1.p1.1.m1.1.1.3"><mtext class="ltx_mathvariant_monospace" id="S5.SS1.SSS1.p1.1.m1.1.1.3.cmml" mathsize="70%" xref="S5.SS1.SSS1.p1.1.m1.1.1.3">50</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS1.SSS1.p1.1.m1.1c">\texttt{CDs}_{\texttt{50}}</annotation><annotation encoding="application/x-llamapun" id="S5.SS1.SSS1.p1.1.m1.1d">CDs start_POSTSUBSCRIPT 50 end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="\texttt{CDs}_{\texttt{200}}" class="ltx_Math" display="inline" id="S5.SS1.SSS1.p1.2.m2.1"><semantics id="S5.SS1.SSS1.p1.2.m2.1a"><msub id="S5.SS1.SSS1.p1.2.m2.1.1" xref="S5.SS1.SSS1.p1.2.m2.1.1.cmml"><mtext class="ltx_mathvariant_monospace" id="S5.SS1.SSS1.p1.2.m2.1.1.2" xref="S5.SS1.SSS1.p1.2.m2.1.1.2a.cmml">CDs</mtext><mtext class="ltx_mathvariant_monospace" id="S5.SS1.SSS1.p1.2.m2.1.1.3" xref="S5.SS1.SSS1.p1.2.m2.1.1.3a.cmml">200</mtext></msub><annotation-xml encoding="MathML-Content" id="S5.SS1.SSS1.p1.2.m2.1b"><apply id="S5.SS1.SSS1.p1.2.m2.1.1.cmml" xref="S5.SS1.SSS1.p1.2.m2.1.1"><csymbol cd="ambiguous" id="S5.SS1.SSS1.p1.2.m2.1.1.1.cmml" xref="S5.SS1.SSS1.p1.2.m2.1.1">subscript</csymbol><ci id="S5.SS1.SSS1.p1.2.m2.1.1.2a.cmml" xref="S5.SS1.SSS1.p1.2.m2.1.1.2"><mtext class="ltx_mathvariant_monospace" id="S5.SS1.SSS1.p1.2.m2.1.1.2.cmml" xref="S5.SS1.SSS1.p1.2.m2.1.1.2">CDs</mtext></ci><ci id="S5.SS1.SSS1.p1.2.m2.1.1.3a.cmml" xref="S5.SS1.SSS1.p1.2.m2.1.1.3"><mtext class="ltx_mathvariant_monospace" id="S5.SS1.SSS1.p1.2.m2.1.1.3.cmml" mathsize="70%" xref="S5.SS1.SSS1.p1.2.m2.1.1.3">200</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS1.SSS1.p1.2.m2.1c">\texttt{CDs}_{\texttt{200}}</annotation><annotation encoding="application/x-llamapun" id="S5.SS1.SSS1.p1.2.m2.1d">CDs start_POSTSUBSCRIPT 200 end_POSTSUBSCRIPT</annotation></semantics></math>, which have average user interaction sequence lengths of 50 and 200, respectively. For the Books dataset, rather than restricting each user’s interactions to 20 items as in <cite class="ltx_cite ltx_citemacro_cite">Zhang et al. (<a class="ltx_ref" href="https://arxiv.org/html/2503.02398v1#bib.bib56" title="">2024a</a>)</cite>, we adopt the approach outlined in <cite class="ltx_cite ltx_citemacro_cite">Pi et al. (<a class="ltx_ref" href="https://arxiv.org/html/2503.02398v1#bib.bib27" title="">2019</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.02398v1#bib.bib28" title="">2020</a>)</cite> to construct longer sequences, resulting in <math alttext="\texttt{Books}_{\texttt{480}}" class="ltx_Math" display="inline" id="S5.SS1.SSS1.p1.3.m3.1"><semantics id="S5.SS1.SSS1.p1.3.m3.1a"><msub id="S5.SS1.SSS1.p1.3.m3.1.1" xref="S5.SS1.SSS1.p1.3.m3.1.1.cmml"><mtext class="ltx_mathvariant_monospace" id="S5.SS1.SSS1.p1.3.m3.1.1.2" xref="S5.SS1.SSS1.p1.3.m3.1.1.2a.cmml">Books</mtext><mtext class="ltx_mathvariant_monospace" id="S5.SS1.SSS1.p1.3.m3.1.1.3" xref="S5.SS1.SSS1.p1.3.m3.1.1.3a.cmml">480</mtext></msub><annotation-xml encoding="MathML-Content" id="S5.SS1.SSS1.p1.3.m3.1b"><apply id="S5.SS1.SSS1.p1.3.m3.1.1.cmml" xref="S5.SS1.SSS1.p1.3.m3.1.1"><csymbol cd="ambiguous" id="S5.SS1.SSS1.p1.3.m3.1.1.1.cmml" xref="S5.SS1.SSS1.p1.3.m3.1.1">subscript</csymbol><ci id="S5.SS1.SSS1.p1.3.m3.1.1.2a.cmml" xref="S5.SS1.SSS1.p1.3.m3.1.1.2"><mtext class="ltx_mathvariant_monospace" id="S5.SS1.SSS1.p1.3.m3.1.1.2.cmml" xref="S5.SS1.SSS1.p1.3.m3.1.1.2">Books</mtext></ci><ci id="S5.SS1.SSS1.p1.3.m3.1.1.3a.cmml" xref="S5.SS1.SSS1.p1.3.m3.1.1.3"><mtext class="ltx_mathvariant_monospace" id="S5.SS1.SSS1.p1.3.m3.1.1.3.cmml" mathsize="70%" xref="S5.SS1.SSS1.p1.3.m3.1.1.3">480</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS1.SSS1.p1.3.m3.1c">\texttt{Books}_{\texttt{480}}</annotation><annotation encoding="application/x-llamapun" id="S5.SS1.SSS1.p1.3.m3.1d">Books start_POSTSUBSCRIPT 480 end_POSTSUBSCRIPT</annotation></semantics></math>. A more detailed description, statistical analysis, and reproducibility are provided in Appendix <a class="ltx_ref" href="https://arxiv.org/html/2503.02398v1#A1" title="Appendix A Datasets ‣ PersonaX: A Recommendation Agent-Oriented User Modeling Framework for Long Behavior Sequence"><span class="ltx_text ltx_ref_tag">A</span></a>.</p> </div> </section> <section class="ltx_subsubsection" id="S5.SS1.SSS2"> <h4 class="ltx_title ltx_title_subsubsection"> <span class="ltx_tag ltx_tag_subsubsection">5.1.2 </span>Evaluation</h4> <div class="ltx_para" id="S5.SS1.SSS2.p1"> <p class="ltx_p" id="S5.SS1.SSS2.p1.1">We utilize all the interaction data except the most recent one to construct the user’s behavior history <cite class="ltx_cite ltx_citemacro_cite">Kang and McAuley (<a class="ltx_ref" href="https://arxiv.org/html/2503.02398v1#bib.bib14" title="">2018</a>)</cite>. And the most recent interaction is reserved for testing. We randomly sample 9 negative items and combine them with the positive item, converting these 10 items into textual descriptions to form the candidate set. For evaluation metric, we adopt the typical top-<math alttext="N" class="ltx_Math" display="inline" id="S5.SS1.SSS2.p1.1.m1.1"><semantics id="S5.SS1.SSS2.p1.1.m1.1a"><mi id="S5.SS1.SSS2.p1.1.m1.1.1" xref="S5.SS1.SSS2.p1.1.m1.1.1.cmml">N</mi><annotation-xml encoding="MathML-Content" id="S5.SS1.SSS2.p1.1.m1.1b"><ci id="S5.SS1.SSS2.p1.1.m1.1.1.cmml" xref="S5.SS1.SSS2.p1.1.m1.1.1">𝑁</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.SS1.SSS2.p1.1.m1.1c">N</annotation><annotation encoding="application/x-llamapun" id="S5.SS1.SSS2.p1.1.m1.1d">italic_N</annotation></semantics></math> metrics hit rate (HR@{1, 5}), normalized discounted cumulative gain (NDCG@{5}) <cite class="ltx_cite ltx_citemacro_citep">(Järvelin and Kekäläinen, <a class="ltx_ref" href="https://arxiv.org/html/2503.02398v1#bib.bib13" title="">2002</a>)</cite> and Mean Reciprocal Rank (MRR@{10}) <cite class="ltx_cite ltx_citemacro_cite">Sarwar et al. (<a class="ltx_ref" href="https://arxiv.org/html/2503.02398v1#bib.bib36" title="">2001</a>)</cite>. For all evaluation metrics in our experiments, higher values indicate better performance. Also, an intuitive case study is provided in Appendix <a class="ltx_ref" href="https://arxiv.org/html/2503.02398v1#A5" title="Appendix E Case Study ‣ PersonaX: A Recommendation Agent-Oriented User Modeling Framework for Long Behavior Sequence"><span class="ltx_text ltx_ref_tag">E</span></a>.</p> </div> <figure class="ltx_table" id="S5.T2"> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_table">Table 2: </span>Performance comparison study. <math alttext="\text{AgentCF}_{\text{B}}" class="ltx_Math" display="inline" id="S5.T2.3.m1.1"><semantics id="S5.T2.3.m1.1b"><msub id="S5.T2.3.m1.1.1" xref="S5.T2.3.m1.1.1.cmml"><mtext class="ltx_mathvariant_italic" id="S5.T2.3.m1.1.1.2" xref="S5.T2.3.m1.1.1.2a.cmml">AgentCF</mtext><mtext class="ltx_mathvariant_italic" id="S5.T2.3.m1.1.1.3" xref="S5.T2.3.m1.1.1.3a.cmml">B</mtext></msub><annotation-xml encoding="MathML-Content" id="S5.T2.3.m1.1c"><apply id="S5.T2.3.m1.1.1.cmml" xref="S5.T2.3.m1.1.1"><csymbol cd="ambiguous" 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start_POSTSUBSCRIPT B+R end_POSTSUBSCRIPT</annotation></semantics></math></span> applies the Reflection+Relevance. <span class="ltx_text ltx_framed ltx_framed_underline" id="S5.T2.9.2">Agent4Rec</span> applies the Summarization+Full.</figcaption> <div class="ltx_inline-block ltx_align_center ltx_transformed_outer" id="S5.T2.7" style="width:433.6pt;height:126.3pt;vertical-align:-0.0pt;"><span class="ltx_transformed_inner" style="transform:translate(-92.3pt,26.9pt) scale(0.701475945306652,0.701475945306652) ;"> <table class="ltx_tabular ltx_guessed_headers ltx_align_middle" id="S5.T2.7.3"> <thead class="ltx_thead"> <tr class="ltx_tr" id="S5.T2.7.3.4.1"> <th class="ltx_td ltx_align_left ltx_th ltx_th_column ltx_th_row ltx_border_r ltx_border_t" id="S5.T2.7.3.4.1.1">LLM-UM</th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_r ltx_border_t" colspan="4" id="S5.T2.7.3.4.1.2">Reflection</th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_t" colspan="8" id="S5.T2.7.3.4.1.3">Summarization</th> </tr> <tr class="ltx_tr" id="S5.T2.7.3.3"> <th class="ltx_td ltx_align_left ltx_th ltx_th_column ltx_th_row ltx_border_r ltx_border_t" id="S5.T2.7.3.3.4">Datasets</th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_r ltx_border_t" colspan="4" id="S5.T2.5.1.1.1"><math alttext="\texttt{CDs}_{\texttt{50}}" class="ltx_Math" display="inline" id="S5.T2.5.1.1.1.m1.1"><semantics id="S5.T2.5.1.1.1.m1.1a"><msub id="S5.T2.5.1.1.1.m1.1.1" xref="S5.T2.5.1.1.1.m1.1.1.cmml"><mtext class="ltx_mathvariant_monospace" id="S5.T2.5.1.1.1.m1.1.1.2" xref="S5.T2.5.1.1.1.m1.1.1.2a.cmml">CDs</mtext><mtext class="ltx_mathvariant_monospace" id="S5.T2.5.1.1.1.m1.1.1.3" xref="S5.T2.5.1.1.1.m1.1.1.3a.cmml">50</mtext></msub><annotation-xml encoding="MathML-Content" id="S5.T2.5.1.1.1.m1.1b"><apply id="S5.T2.5.1.1.1.m1.1.1.cmml" xref="S5.T2.5.1.1.1.m1.1.1"><csymbol cd="ambiguous" id="S5.T2.5.1.1.1.m1.1.1.1.cmml" xref="S5.T2.5.1.1.1.m1.1.1">subscript</csymbol><ci id="S5.T2.5.1.1.1.m1.1.1.2a.cmml" xref="S5.T2.5.1.1.1.m1.1.1.2"><mtext class="ltx_mathvariant_monospace" id="S5.T2.5.1.1.1.m1.1.1.2.cmml" xref="S5.T2.5.1.1.1.m1.1.1.2">CDs</mtext></ci><ci id="S5.T2.5.1.1.1.m1.1.1.3a.cmml" xref="S5.T2.5.1.1.1.m1.1.1.3"><mtext class="ltx_mathvariant_monospace" id="S5.T2.5.1.1.1.m1.1.1.3.cmml" mathsize="70%" xref="S5.T2.5.1.1.1.m1.1.1.3">50</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.T2.5.1.1.1.m1.1c">\texttt{CDs}_{\texttt{50}}</annotation><annotation encoding="application/x-llamapun" id="S5.T2.5.1.1.1.m1.1d">CDs start_POSTSUBSCRIPT 50 end_POSTSUBSCRIPT</annotation></semantics></math></th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_r ltx_border_t" colspan="4" id="S5.T2.6.2.2.2"><math alttext="\texttt{CDs}_{\texttt{200}}" class="ltx_Math" display="inline" id="S5.T2.6.2.2.2.m1.1"><semantics id="S5.T2.6.2.2.2.m1.1a"><msub id="S5.T2.6.2.2.2.m1.1.1" xref="S5.T2.6.2.2.2.m1.1.1.cmml"><mtext class="ltx_mathvariant_monospace" id="S5.T2.6.2.2.2.m1.1.1.2" xref="S5.T2.6.2.2.2.m1.1.1.2a.cmml">CDs</mtext><mtext class="ltx_mathvariant_monospace" id="S5.T2.6.2.2.2.m1.1.1.3" xref="S5.T2.6.2.2.2.m1.1.1.3a.cmml">200</mtext></msub><annotation-xml encoding="MathML-Content" id="S5.T2.6.2.2.2.m1.1b"><apply id="S5.T2.6.2.2.2.m1.1.1.cmml" xref="S5.T2.6.2.2.2.m1.1.1"><csymbol cd="ambiguous" id="S5.T2.6.2.2.2.m1.1.1.1.cmml" xref="S5.T2.6.2.2.2.m1.1.1">subscript</csymbol><ci id="S5.T2.6.2.2.2.m1.1.1.2a.cmml" xref="S5.T2.6.2.2.2.m1.1.1.2"><mtext class="ltx_mathvariant_monospace" id="S5.T2.6.2.2.2.m1.1.1.2.cmml" xref="S5.T2.6.2.2.2.m1.1.1.2">CDs</mtext></ci><ci id="S5.T2.6.2.2.2.m1.1.1.3a.cmml" xref="S5.T2.6.2.2.2.m1.1.1.3"><mtext class="ltx_mathvariant_monospace" id="S5.T2.6.2.2.2.m1.1.1.3.cmml" mathsize="70%" xref="S5.T2.6.2.2.2.m1.1.1.3">200</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.T2.6.2.2.2.m1.1c">\texttt{CDs}_{\texttt{200}}</annotation><annotation encoding="application/x-llamapun" id="S5.T2.6.2.2.2.m1.1d">CDs start_POSTSUBSCRIPT 200 end_POSTSUBSCRIPT</annotation></semantics></math></th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_t" colspan="4" id="S5.T2.7.3.3.3"><math alttext="\texttt{Books}_{\texttt{480}}" class="ltx_Math" display="inline" id="S5.T2.7.3.3.3.m1.1"><semantics id="S5.T2.7.3.3.3.m1.1a"><msub id="S5.T2.7.3.3.3.m1.1.1" xref="S5.T2.7.3.3.3.m1.1.1.cmml"><mtext class="ltx_mathvariant_monospace" id="S5.T2.7.3.3.3.m1.1.1.2" xref="S5.T2.7.3.3.3.m1.1.1.2a.cmml">Books</mtext><mtext class="ltx_mathvariant_monospace" id="S5.T2.7.3.3.3.m1.1.1.3" xref="S5.T2.7.3.3.3.m1.1.1.3a.cmml">480</mtext></msub><annotation-xml encoding="MathML-Content" id="S5.T2.7.3.3.3.m1.1b"><apply id="S5.T2.7.3.3.3.m1.1.1.cmml" xref="S5.T2.7.3.3.3.m1.1.1"><csymbol cd="ambiguous" id="S5.T2.7.3.3.3.m1.1.1.1.cmml" xref="S5.T2.7.3.3.3.m1.1.1">subscript</csymbol><ci id="S5.T2.7.3.3.3.m1.1.1.2a.cmml" xref="S5.T2.7.3.3.3.m1.1.1.2"><mtext class="ltx_mathvariant_monospace" id="S5.T2.7.3.3.3.m1.1.1.2.cmml" xref="S5.T2.7.3.3.3.m1.1.1.2">Books</mtext></ci><ci id="S5.T2.7.3.3.3.m1.1.1.3a.cmml" xref="S5.T2.7.3.3.3.m1.1.1.3"><mtext class="ltx_mathvariant_monospace" id="S5.T2.7.3.3.3.m1.1.1.3.cmml" mathsize="70%" xref="S5.T2.7.3.3.3.m1.1.1.3">480</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.T2.7.3.3.3.m1.1c">\texttt{Books}_{\texttt{480}}</annotation><annotation encoding="application/x-llamapun" id="S5.T2.7.3.3.3.m1.1d">Books start_POSTSUBSCRIPT 480 end_POSTSUBSCRIPT</annotation></semantics></math></th> </tr> <tr class="ltx_tr" id="S5.T2.7.3.5.2"> <th class="ltx_td ltx_align_left ltx_th ltx_th_column ltx_th_row ltx_border_r ltx_border_t" id="S5.T2.7.3.5.2.1">Metrics</th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_t" id="S5.T2.7.3.5.2.2">Hit@1</th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_t" id="S5.T2.7.3.5.2.3">Hit@5</th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_t" id="S5.T2.7.3.5.2.4">NDCG@5</th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_r ltx_border_t" id="S5.T2.7.3.5.2.5">MRR@10</th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_t" id="S5.T2.7.3.5.2.6">Hit@1</th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_t" id="S5.T2.7.3.5.2.7">Hit@5</th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_t" id="S5.T2.7.3.5.2.8">NDCG@5</th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_r ltx_border_t" id="S5.T2.7.3.5.2.9">MRR@10</th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_t" id="S5.T2.7.3.5.2.10">Hit@1</th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_t" id="S5.T2.7.3.5.2.11">Hit@5</th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_t" id="S5.T2.7.3.5.2.12">NDCG@5</th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_t" id="S5.T2.7.3.5.2.13">MRR@10</th> </tr> </thead> <tbody class="ltx_tbody"> <tr class="ltx_tr" id="S5.T2.7.3.6.1"> <th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_r ltx_border_t" id="S5.T2.7.3.6.1.1">Full</th> <td class="ltx_td ltx_align_center ltx_border_t" id="S5.T2.7.3.6.1.2">19.00</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S5.T2.7.3.6.1.3">66.00</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S5.T2.7.3.6.1.4">42.56</td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S5.T2.7.3.6.1.5">39.38</td> <td class="ltx_td ltx_align_center ltx_border_t" id="S5.T2.7.3.6.1.6"><span class="ltx_text ltx_framed ltx_framed_underline" id="S5.T2.7.3.6.1.6.1">36.00</span></td> <td class="ltx_td ltx_align_center ltx_border_t" id="S5.T2.7.3.6.1.7"><span class="ltx_text ltx_framed ltx_framed_underline" id="S5.T2.7.3.6.1.7.1">67.00</span></td> <td class="ltx_td ltx_align_center ltx_border_t" id="S5.T2.7.3.6.1.8"><span class="ltx_text ltx_framed ltx_framed_underline" id="S5.T2.7.3.6.1.8.1">51.75</span></td> <td class="ltx_td ltx_align_center ltx_border_r ltx_border_t" id="S5.T2.7.3.6.1.9"><span class="ltx_text ltx_framed ltx_framed_underline" id="S5.T2.7.3.6.1.9.1">50.78</span></td> <td class="ltx_td ltx_align_center ltx_border_t" id="S5.T2.7.3.6.1.10"><span class="ltx_text ltx_framed ltx_framed_underline" id="S5.T2.7.3.6.1.10.1">19.00</span></td> <td class="ltx_td ltx_align_center ltx_border_t" id="S5.T2.7.3.6.1.11"><span class="ltx_text ltx_framed ltx_framed_underline" id="S5.T2.7.3.6.1.11.1">50.00</span></td> <td class="ltx_td ltx_align_center ltx_border_t" id="S5.T2.7.3.6.1.12"><span class="ltx_text ltx_framed ltx_framed_underline" id="S5.T2.7.3.6.1.12.1">34.59</span></td> <td class="ltx_td ltx_align_center ltx_border_t" id="S5.T2.7.3.6.1.13"><span class="ltx_text ltx_framed ltx_framed_underline" id="S5.T2.7.3.6.1.13.1">35.76</span></td> </tr> <tr class="ltx_tr" id="S5.T2.7.3.7.2"> <th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_r" id="S5.T2.7.3.7.2.1">Random</th> <td class="ltx_td ltx_align_center" id="S5.T2.7.3.7.2.2">31.00</td> <td class="ltx_td ltx_align_center" id="S5.T2.7.3.7.2.3">67.00</td> <td class="ltx_td ltx_align_center" id="S5.T2.7.3.7.2.4">49.18</td> <td class="ltx_td ltx_align_center ltx_border_r" id="S5.T2.7.3.7.2.5">47.74</td> <td class="ltx_td ltx_align_center" id="S5.T2.7.3.7.2.6">36.00</td> <td class="ltx_td ltx_align_center" id="S5.T2.7.3.7.2.7">68.00</td> <td class="ltx_td ltx_align_center" id="S5.T2.7.3.7.2.8">51.26</td> <td class="ltx_td ltx_align_center ltx_border_r" id="S5.T2.7.3.7.2.9">50.24</td> <td class="ltx_td ltx_align_center" id="S5.T2.7.3.7.2.10">33.00</td> <td class="ltx_td ltx_align_center" id="S5.T2.7.3.7.2.11">73.00</td> <td class="ltx_td ltx_align_center" id="S5.T2.7.3.7.2.12">53.59</td> <td class="ltx_td ltx_align_center" id="S5.T2.7.3.7.2.13">50.50</td> </tr> <tr class="ltx_tr" id="S5.T2.7.3.8.3"> <th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_r" id="S5.T2.7.3.8.3.1">Recent</th> <td class="ltx_td ltx_align_center" id="S5.T2.7.3.8.3.2"><span class="ltx_text ltx_font_italic" id="S5.T2.7.3.8.3.2.1">34.00</span></td> <td class="ltx_td ltx_align_center" id="S5.T2.7.3.8.3.3"><span class="ltx_text ltx_font_italic" id="S5.T2.7.3.8.3.3.1">69.00</span></td> <td class="ltx_td ltx_align_center" id="S5.T2.7.3.8.3.4"><span class="ltx_text ltx_font_italic" id="S5.T2.7.3.8.3.4.1">50.69</span></td> <td class="ltx_td ltx_align_center ltx_border_r" id="S5.T2.7.3.8.3.5"><span class="ltx_text ltx_font_italic" id="S5.T2.7.3.8.3.5.1">49.31</span></td> <td class="ltx_td ltx_align_center" id="S5.T2.7.3.8.3.6">39.00</td> <td class="ltx_td ltx_align_center" id="S5.T2.7.3.8.3.7">68.00</td> <td class="ltx_td ltx_align_center" id="S5.T2.7.3.8.3.8">53.89</td> <td class="ltx_td ltx_align_center ltx_border_r" id="S5.T2.7.3.8.3.9">53.34</td> <td class="ltx_td ltx_align_center" id="S5.T2.7.3.8.3.10">35.00</td> <td class="ltx_td ltx_align_center" id="S5.T2.7.3.8.3.11">74.00</td> <td class="ltx_td ltx_align_center" id="S5.T2.7.3.8.3.12">55.23</td> <td class="ltx_td ltx_align_center" id="S5.T2.7.3.8.3.13">52.76</td> </tr> <tr class="ltx_tr" id="S5.T2.7.3.9.4"> <th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_r" id="S5.T2.7.3.9.4.1">Relevance</th> <td class="ltx_td ltx_align_center" id="S5.T2.7.3.9.4.2"><span class="ltx_text ltx_font_italic ltx_framed ltx_framed_underline" id="S5.T2.7.3.9.4.2.1">40.00</span></td> <td class="ltx_td ltx_align_center" id="S5.T2.7.3.9.4.3"><span class="ltx_text ltx_font_italic ltx_framed ltx_framed_underline" id="S5.T2.7.3.9.4.3.1">69.00</span></td> <td class="ltx_td ltx_align_center" id="S5.T2.7.3.9.4.4"><span class="ltx_text ltx_font_italic ltx_framed ltx_framed_underline" id="S5.T2.7.3.9.4.4.1">54.97</span></td> <td class="ltx_td ltx_align_center ltx_border_r" id="S5.T2.7.3.9.4.5"><span class="ltx_text ltx_font_italic ltx_framed ltx_framed_underline" id="S5.T2.7.3.9.4.5.1">54.47</span></td> <td class="ltx_td ltx_align_center" id="S5.T2.7.3.9.4.6">51.00</td> <td class="ltx_td ltx_align_center" id="S5.T2.7.3.9.4.7">73.00</td> <td class="ltx_td ltx_align_center" id="S5.T2.7.3.9.4.8">61.73</td> <td class="ltx_td ltx_align_center ltx_border_r" id="S5.T2.7.3.9.4.9">61.98</td> <td class="ltx_td ltx_align_center" id="S5.T2.7.3.9.4.10">61.00</td> <td class="ltx_td ltx_align_center" id="S5.T2.7.3.9.4.11">80.00</td> <td class="ltx_td ltx_align_center" id="S5.T2.7.3.9.4.12">71.50</td> <td class="ltx_td ltx_align_center" id="S5.T2.7.3.9.4.13">71.86</td> </tr> <tr class="ltx_tr" id="S5.T2.7.3.10.5"> <th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_r" id="S5.T2.7.3.10.5.1">Centroid</th> <td class="ltx_td ltx_align_center" id="S5.T2.7.3.10.5.2">43.00</td> <td class="ltx_td ltx_align_center" id="S5.T2.7.3.10.5.3">66.00</td> <td class="ltx_td ltx_align_center" id="S5.T2.7.3.10.5.4">55.21</td> <td class="ltx_td ltx_align_center ltx_border_r" id="S5.T2.7.3.10.5.5">55.91</td> <td class="ltx_td ltx_align_center" id="S5.T2.7.3.10.5.6">42.00</td> <td class="ltx_td ltx_align_center" id="S5.T2.7.3.10.5.7">70.00</td> <td class="ltx_td ltx_align_center" id="S5.T2.7.3.10.5.8">57.07</td> <td class="ltx_td ltx_align_center ltx_border_r" id="S5.T2.7.3.10.5.9">56.53</td> <td class="ltx_td ltx_align_center" id="S5.T2.7.3.10.5.10">60.00</td> <td class="ltx_td ltx_align_center" id="S5.T2.7.3.10.5.11">81.00</td> <td class="ltx_td ltx_align_center" id="S5.T2.7.3.10.5.12">71.61</td> <td class="ltx_td ltx_align_center" id="S5.T2.7.3.10.5.13">70.67</td> </tr> <tr class="ltx_tr" id="S5.T2.7.3.11.6"> <th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_r" id="S5.T2.7.3.11.6.1">Boundary</th> <td class="ltx_td ltx_align_center" id="S5.T2.7.3.11.6.2">42.00</td> <td class="ltx_td ltx_align_center" id="S5.T2.7.3.11.6.3">68.00</td> <td class="ltx_td ltx_align_center" id="S5.T2.7.3.11.6.4">55.85</td> <td class="ltx_td ltx_align_center ltx_border_r" id="S5.T2.7.3.11.6.5">55.73</td> <td class="ltx_td ltx_align_center" id="S5.T2.7.3.11.6.6">48.00</td> <td class="ltx_td ltx_align_center" id="S5.T2.7.3.11.6.7">66.00</td> <td class="ltx_td ltx_align_center" id="S5.T2.7.3.11.6.8">57.13</td> <td class="ltx_td ltx_align_center ltx_border_r" id="S5.T2.7.3.11.6.9">58.71</td> <td class="ltx_td ltx_align_center" id="S5.T2.7.3.11.6.10">58.00</td> <td class="ltx_td ltx_align_center" id="S5.T2.7.3.11.6.11">80.00</td> <td class="ltx_td ltx_align_center" id="S5.T2.7.3.11.6.12">70.38</td> <td class="ltx_td ltx_align_center" id="S5.T2.7.3.11.6.13">69.55</td> </tr> <tr class="ltx_tr" id="S5.T2.7.3.12.7"> <th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_b ltx_border_r" id="S5.T2.7.3.12.7.1">Ours</th> <td class="ltx_td ltx_align_center ltx_border_b" id="S5.T2.7.3.12.7.2"><span class="ltx_text ltx_font_bold" id="S5.T2.7.3.12.7.2.1" style="color:#BF0040;">45.00</span></td> <td class="ltx_td ltx_align_center ltx_border_b" id="S5.T2.7.3.12.7.3"><span class="ltx_text ltx_font_bold" id="S5.T2.7.3.12.7.3.1" style="color:#BF0040;">72.00</span></td> <td class="ltx_td ltx_align_center ltx_border_b" id="S5.T2.7.3.12.7.4"><span class="ltx_text ltx_font_bold" id="S5.T2.7.3.12.7.4.1" style="color:#BF0040;">57.34</span></td> <td class="ltx_td ltx_align_center ltx_border_b ltx_border_r" id="S5.T2.7.3.12.7.5"><span class="ltx_text ltx_font_bold" id="S5.T2.7.3.12.7.5.1" style="color:#BF0040;">58.38</span></td> <td class="ltx_td ltx_align_center ltx_border_b" id="S5.T2.7.3.12.7.6"><span class="ltx_text ltx_font_bold" id="S5.T2.7.3.12.7.6.1" style="color:#BF0040;">55.00</span></td> <td class="ltx_td ltx_align_center ltx_border_b" id="S5.T2.7.3.12.7.7"><span class="ltx_text ltx_font_bold" id="S5.T2.7.3.12.7.7.1" style="color:#BF0040;">75.00</span></td> <td class="ltx_td ltx_align_center ltx_border_b" id="S5.T2.7.3.12.7.8"><span class="ltx_text ltx_font_bold" id="S5.T2.7.3.12.7.8.1" style="color:#BF0040;">64.56</span></td> <td class="ltx_td ltx_align_center ltx_border_b ltx_border_r" id="S5.T2.7.3.12.7.9"><span class="ltx_text ltx_font_bold" id="S5.T2.7.3.12.7.9.1" style="color:#BF0040;">65.06</span></td> <td class="ltx_td ltx_align_center ltx_border_b" id="S5.T2.7.3.12.7.10"><span class="ltx_text ltx_font_bold" id="S5.T2.7.3.12.7.10.1" style="color:#BF0040;">65.00</span></td> <td class="ltx_td ltx_align_center ltx_border_b" id="S5.T2.7.3.12.7.11"><span class="ltx_text ltx_font_bold" id="S5.T2.7.3.12.7.11.1" style="color:#BF0040;">83.00</span></td> <td class="ltx_td ltx_align_center ltx_border_b" id="S5.T2.7.3.12.7.12"><span class="ltx_text ltx_font_bold" id="S5.T2.7.3.12.7.12.1" style="color:#BF0040;">74.26</span></td> <td class="ltx_td ltx_align_center ltx_border_b" id="S5.T2.7.3.12.7.13"><span class="ltx_text ltx_font_bold" id="S5.T2.7.3.12.7.13.1" style="color:#BF0040;">73.22</span></td> </tr> </tbody> </table> </span></div> </figure> </section> <section class="ltx_subsubsection" id="S5.SS1.SSS3"> <h4 class="ltx_title ltx_title_subsubsection"> <span class="ltx_tag ltx_tag_subsubsection">5.1.3 </span>Baseline Comparison</h4> <div class="ltx_para" id="S5.SS1.SSS3.p1"> <p class="ltx_p" id="S5.SS1.SSS3.p1.2">We integrate the two LLM-UM methods—Reflection and Summarization—with various cutting-edge sampling strategies, expanding the range of user modeling approaches to enhance our comparison. The behavior sequence sampling methods considered are as follows: (1) Full <cite class="ltx_cite ltx_citemacro_cite">Zhang et al. (<a class="ltx_ref" href="https://arxiv.org/html/2503.02398v1#bib.bib56" title="">2024a</a>)</cite>: Using complete user behavior sequence. (2) Recent <cite class="ltx_cite ltx_citemacro_cite">Zhang et al. (<a class="ltx_ref" href="https://arxiv.org/html/2503.02398v1#bib.bib57" title="">2024b</a>)</cite>: Selecting the most recent behaviors to capture the user’s short-term preferences. (3) Relevance <cite class="ltx_cite ltx_citemacro_cite">Zhang et al. (<a class="ltx_ref" href="https://arxiv.org/html/2503.02398v1#bib.bib57" title="">2024b</a>); Pi et al. (<a class="ltx_ref" href="https://arxiv.org/html/2503.02398v1#bib.bib28" title="">2020</a>)</cite>: Retrieving the subset of behaviors most pertinent to the recommendation scenario from the user’s long-term preferences. (4) Random <cite class="ltx_cite ltx_citemacro_cite">Guo et al. (<a class="ltx_ref" href="https://arxiv.org/html/2503.02398v1#bib.bib10" title="">2022</a>); Prabhu et al. (<a class="ltx_ref" href="https://arxiv.org/html/2503.02398v1#bib.bib29" title="">2020</a>)</cite>: Randomly selecting a portion of behaviors, it is a robust and effective sampling method. (5) Centroid Selection <cite class="ltx_cite ltx_citemacro_cite">Welling (<a class="ltx_ref" href="https://arxiv.org/html/2503.02398v1#bib.bib49" title="">2009</a>); Rebuffi et al. (<a class="ltx_ref" href="https://arxiv.org/html/2503.02398v1#bib.bib32" title="">2017</a>); Sorscher et al. (<a class="ltx_ref" href="https://arxiv.org/html/2503.02398v1#bib.bib42" title="">2022</a>)</cite>: As outlined in Section <a class="ltx_ref" href="https://arxiv.org/html/2503.02398v1#S3.SS3" title="3.3 In-Cluster Selection ‣ 3 Method ‣ PersonaX: A Recommendation Agent-Oriented User Modeling Framework for Long Behavior Sequence"><span class="ltx_text ltx_ref_tag">3.3</span></a>, we configure <math alttext="\alpha=1.001" class="ltx_Math" display="inline" id="S5.SS1.SSS3.p1.1.m1.1"><semantics id="S5.SS1.SSS3.p1.1.m1.1a"><mrow id="S5.SS1.SSS3.p1.1.m1.1.1" xref="S5.SS1.SSS3.p1.1.m1.1.1.cmml"><mi id="S5.SS1.SSS3.p1.1.m1.1.1.2" xref="S5.SS1.SSS3.p1.1.m1.1.1.2.cmml">α</mi><mo id="S5.SS1.SSS3.p1.1.m1.1.1.1" xref="S5.SS1.SSS3.p1.1.m1.1.1.1.cmml">=</mo><mn id="S5.SS1.SSS3.p1.1.m1.1.1.3" xref="S5.SS1.SSS3.p1.1.m1.1.1.3.cmml">1.001</mn></mrow><annotation-xml encoding="MathML-Content" id="S5.SS1.SSS3.p1.1.m1.1b"><apply id="S5.SS1.SSS3.p1.1.m1.1.1.cmml" xref="S5.SS1.SSS3.p1.1.m1.1.1"><eq id="S5.SS1.SSS3.p1.1.m1.1.1.1.cmml" xref="S5.SS1.SSS3.p1.1.m1.1.1.1"></eq><ci id="S5.SS1.SSS3.p1.1.m1.1.1.2.cmml" xref="S5.SS1.SSS3.p1.1.m1.1.1.2">𝛼</ci><cn id="S5.SS1.SSS3.p1.1.m1.1.1.3.cmml" type="float" xref="S5.SS1.SSS3.p1.1.m1.1.1.3">1.001</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS1.SSS3.p1.1.m1.1c">\alpha=1.001</annotation><annotation encoding="application/x-llamapun" id="S5.SS1.SSS3.p1.1.m1.1d">italic_α = 1.001</annotation></semantics></math> in Algorithm <a class="ltx_ref" href="https://arxiv.org/html/2503.02398v1#alg2" title="Algorithm 2 ‣ 3.2 Sampling Budget Allocation ‣ 3 Method ‣ PersonaX: A Recommendation Agent-Oriented User Modeling Framework for Long Behavior Sequence"><span class="ltx_text ltx_ref_tag">2</span></a>. This configuration prioritizes the selection of samples that are closest to the cluster centroid, effectively capturing the most prototypical data points within the cluster. (6) Boundary Selection <cite class="ltx_cite ltx_citemacro_cite">Paul et al. (<a class="ltx_ref" href="https://arxiv.org/html/2503.02398v1#bib.bib25" title="">2021</a>); Toneva et al. (<a class="ltx_ref" href="https://arxiv.org/html/2503.02398v1#bib.bib45" title="">2019</a>)</cite>: As detailed in Section <a class="ltx_ref" href="https://arxiv.org/html/2503.02398v1#S3.SS3" title="3.3 In-Cluster Selection ‣ 3 Method ‣ PersonaX: A Recommendation Agent-Oriented User Modeling Framework for Long Behavior Sequence"><span class="ltx_text ltx_ref_tag">3.3</span></a>, we set <math alttext="\alpha=1.4" class="ltx_Math" display="inline" id="S5.SS1.SSS3.p1.2.m2.1"><semantics id="S5.SS1.SSS3.p1.2.m2.1a"><mrow id="S5.SS1.SSS3.p1.2.m2.1.1" xref="S5.SS1.SSS3.p1.2.m2.1.1.cmml"><mi id="S5.SS1.SSS3.p1.2.m2.1.1.2" xref="S5.SS1.SSS3.p1.2.m2.1.1.2.cmml">α</mi><mo id="S5.SS1.SSS3.p1.2.m2.1.1.1" xref="S5.SS1.SSS3.p1.2.m2.1.1.1.cmml">=</mo><mn id="S5.SS1.SSS3.p1.2.m2.1.1.3" xref="S5.SS1.SSS3.p1.2.m2.1.1.3.cmml">1.4</mn></mrow><annotation-xml encoding="MathML-Content" id="S5.SS1.SSS3.p1.2.m2.1b"><apply id="S5.SS1.SSS3.p1.2.m2.1.1.cmml" xref="S5.SS1.SSS3.p1.2.m2.1.1"><eq id="S5.SS1.SSS3.p1.2.m2.1.1.1.cmml" xref="S5.SS1.SSS3.p1.2.m2.1.1.1"></eq><ci id="S5.SS1.SSS3.p1.2.m2.1.1.2.cmml" xref="S5.SS1.SSS3.p1.2.m2.1.1.2">𝛼</ci><cn id="S5.SS1.SSS3.p1.2.m2.1.1.3.cmml" type="float" xref="S5.SS1.SSS3.p1.2.m2.1.1.3">1.4</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS1.SSS3.p1.2.m2.1c">\alpha=1.4</annotation><annotation encoding="application/x-llamapun" id="S5.SS1.SSS3.p1.2.m2.1d">italic_α = 1.4</annotation></semantics></math> in Algorithm <a class="ltx_ref" href="https://arxiv.org/html/2503.02398v1#alg2" title="Algorithm 2 ‣ 3.2 Sampling Budget Allocation ‣ 3 Method ‣ PersonaX: A Recommendation Agent-Oriented User Modeling Framework for Long Behavior Sequence"><span class="ltx_text ltx_ref_tag">2</span></a>. Under this setting, the algorithm selects samples located at the cluster boundary and emphasizes the diversity coverage.</p> </div> <div class="ltx_para" id="S5.SS1.SSS3.p2"> <p class="ltx_p" id="S5.SS1.SSS3.p2.3">In this setting, to benchmark PersonaX against prior works, including <span class="ltx_text ltx_markedasmath" id="S5.SS1.SSS3.p2.3.1">AgentCF</span><span class="ltx_text ltx_markedasmath" id="S5.SS1.SSS3.p2.3.2">B</span>, which combines Reflection with Recent sampling, <span class="ltx_text ltx_markedasmath" id="S5.SS1.SSS3.p2.3.3">AgentCF</span><span class="ltx_text ltx_markedasmath" id="S5.SS1.SSS3.p2.3.4">B+R</span>, which integrates Reflection with Relevance sampling, and <span class="ltx_text ltx_markedasmath" id="S5.SS1.SSS3.p2.3.5">Agent4Rec</span>, which employs Summarization with Full sampling.</p> </div> </section> <section class="ltx_subsubsection" id="S5.SS1.SSS4"> <h4 class="ltx_title ltx_title_subsubsection"> <span class="ltx_tag ltx_tag_subsubsection">5.1.4 </span>Backbone Agent Recommendation</h4> <div class="ltx_para" id="S5.SS1.SSS4.p1"> <p class="ltx_p" id="S5.SS1.SSS4.p1.1">To assess the effectiveness of PersonaX in improving downstream agent recommendation performance, we select two state-of-the-art methods. <span class="ltx_text ltx_font_bold" id="S5.SS1.SSS4.p1.1.1">AgentCF</span> <cite class="ltx_cite ltx_citemacro_cite">Zhang et al. (<a class="ltx_ref" href="https://arxiv.org/html/2503.02398v1#bib.bib57" title="">2024b</a>)</cite> models user personas using a <span class="ltx_text ltx_framed ltx_framed_underline" id="S5.SS1.SSS4.p1.1.2">Reflection</span> mechanism, while <span class="ltx_text ltx_font_bold" id="S5.SS1.SSS4.p1.1.3">Agent4Rec</span> <cite class="ltx_cite ltx_citemacro_cite">Zhang et al. (<a class="ltx_ref" href="https://arxiv.org/html/2503.02398v1#bib.bib56" title="">2024a</a>)</cite> captures users’ unique preferences through a <span class="ltx_text ltx_framed ltx_framed_underline" id="S5.SS1.SSS4.p1.1.4">Summarization</span> method. Further details on the foundational methods can be found in Appendix <a class="ltx_ref" href="https://arxiv.org/html/2503.02398v1#A2" title="Appendix B Backbone Methods ‣ PersonaX: A Recommendation Agent-Oriented User Modeling Framework for Long Behavior Sequence"><span class="ltx_text ltx_ref_tag">B</span></a>.</p> </div> </section> <section class="ltx_subsubsection" id="S5.SS1.SSS5"> <h4 class="ltx_title ltx_title_subsubsection"> <span class="ltx_tag ltx_tag_subsubsection">5.1.5 </span>Implementation Details</h4> <div class="ltx_para" id="S5.SS1.SSS5.p1"> <p class="ltx_p" id="S5.SS1.SSS5.p1.16">We applied AgentCF to <math alttext="\texttt{CDs}_{\texttt{50}}" class="ltx_Math" display="inline" id="S5.SS1.SSS5.p1.1.m1.1"><semantics id="S5.SS1.SSS5.p1.1.m1.1a"><msub id="S5.SS1.SSS5.p1.1.m1.1.1" xref="S5.SS1.SSS5.p1.1.m1.1.1.cmml"><mtext class="ltx_mathvariant_monospace" id="S5.SS1.SSS5.p1.1.m1.1.1.2" xref="S5.SS1.SSS5.p1.1.m1.1.1.2a.cmml">CDs</mtext><mtext class="ltx_mathvariant_monospace" id="S5.SS1.SSS5.p1.1.m1.1.1.3" xref="S5.SS1.SSS5.p1.1.m1.1.1.3a.cmml">50</mtext></msub><annotation-xml encoding="MathML-Content" id="S5.SS1.SSS5.p1.1.m1.1b"><apply id="S5.SS1.SSS5.p1.1.m1.1.1.cmml" xref="S5.SS1.SSS5.p1.1.m1.1.1"><csymbol cd="ambiguous" id="S5.SS1.SSS5.p1.1.m1.1.1.1.cmml" xref="S5.SS1.SSS5.p1.1.m1.1.1">subscript</csymbol><ci id="S5.SS1.SSS5.p1.1.m1.1.1.2a.cmml" xref="S5.SS1.SSS5.p1.1.m1.1.1.2"><mtext class="ltx_mathvariant_monospace" id="S5.SS1.SSS5.p1.1.m1.1.1.2.cmml" xref="S5.SS1.SSS5.p1.1.m1.1.1.2">CDs</mtext></ci><ci id="S5.SS1.SSS5.p1.1.m1.1.1.3a.cmml" xref="S5.SS1.SSS5.p1.1.m1.1.1.3"><mtext class="ltx_mathvariant_monospace" id="S5.SS1.SSS5.p1.1.m1.1.1.3.cmml" mathsize="70%" xref="S5.SS1.SSS5.p1.1.m1.1.1.3">50</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS1.SSS5.p1.1.m1.1c">\texttt{CDs}_{\texttt{50}}</annotation><annotation encoding="application/x-llamapun" id="S5.SS1.SSS5.p1.1.m1.1d">CDs start_POSTSUBSCRIPT 50 end_POSTSUBSCRIPT</annotation></semantics></math>, and Agent4Rec for <math alttext="\texttt{CDs}_{\texttt{200}}" class="ltx_Math" display="inline" id="S5.SS1.SSS5.p1.2.m2.1"><semantics id="S5.SS1.SSS5.p1.2.m2.1a"><msub id="S5.SS1.SSS5.p1.2.m2.1.1" xref="S5.SS1.SSS5.p1.2.m2.1.1.cmml"><mtext class="ltx_mathvariant_monospace" id="S5.SS1.SSS5.p1.2.m2.1.1.2" xref="S5.SS1.SSS5.p1.2.m2.1.1.2a.cmml">CDs</mtext><mtext class="ltx_mathvariant_monospace" id="S5.SS1.SSS5.p1.2.m2.1.1.3" xref="S5.SS1.SSS5.p1.2.m2.1.1.3a.cmml">200</mtext></msub><annotation-xml encoding="MathML-Content" id="S5.SS1.SSS5.p1.2.m2.1b"><apply id="S5.SS1.SSS5.p1.2.m2.1.1.cmml" xref="S5.SS1.SSS5.p1.2.m2.1.1"><csymbol cd="ambiguous" id="S5.SS1.SSS5.p1.2.m2.1.1.1.cmml" xref="S5.SS1.SSS5.p1.2.m2.1.1">subscript</csymbol><ci id="S5.SS1.SSS5.p1.2.m2.1.1.2a.cmml" xref="S5.SS1.SSS5.p1.2.m2.1.1.2"><mtext class="ltx_mathvariant_monospace" id="S5.SS1.SSS5.p1.2.m2.1.1.2.cmml" xref="S5.SS1.SSS5.p1.2.m2.1.1.2">CDs</mtext></ci><ci id="S5.SS1.SSS5.p1.2.m2.1.1.3a.cmml" xref="S5.SS1.SSS5.p1.2.m2.1.1.3"><mtext class="ltx_mathvariant_monospace" id="S5.SS1.SSS5.p1.2.m2.1.1.3.cmml" mathsize="70%" xref="S5.SS1.SSS5.p1.2.m2.1.1.3">200</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS1.SSS5.p1.2.m2.1c">\texttt{CDs}_{\texttt{200}}</annotation><annotation encoding="application/x-llamapun" id="S5.SS1.SSS5.p1.2.m2.1d">CDs start_POSTSUBSCRIPT 200 end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="\texttt{Books}_{\texttt{480}}" class="ltx_Math" display="inline" id="S5.SS1.SSS5.p1.3.m3.1"><semantics id="S5.SS1.SSS5.p1.3.m3.1a"><msub id="S5.SS1.SSS5.p1.3.m3.1.1" xref="S5.SS1.SSS5.p1.3.m3.1.1.cmml"><mtext class="ltx_mathvariant_monospace" id="S5.SS1.SSS5.p1.3.m3.1.1.2" xref="S5.SS1.SSS5.p1.3.m3.1.1.2a.cmml">Books</mtext><mtext class="ltx_mathvariant_monospace" id="S5.SS1.SSS5.p1.3.m3.1.1.3" xref="S5.SS1.SSS5.p1.3.m3.1.1.3a.cmml">480</mtext></msub><annotation-xml encoding="MathML-Content" id="S5.SS1.SSS5.p1.3.m3.1b"><apply id="S5.SS1.SSS5.p1.3.m3.1.1.cmml" xref="S5.SS1.SSS5.p1.3.m3.1.1"><csymbol cd="ambiguous" id="S5.SS1.SSS5.p1.3.m3.1.1.1.cmml" xref="S5.SS1.SSS5.p1.3.m3.1.1">subscript</csymbol><ci id="S5.SS1.SSS5.p1.3.m3.1.1.2a.cmml" xref="S5.SS1.SSS5.p1.3.m3.1.1.2"><mtext class="ltx_mathvariant_monospace" id="S5.SS1.SSS5.p1.3.m3.1.1.2.cmml" xref="S5.SS1.SSS5.p1.3.m3.1.1.2">Books</mtext></ci><ci id="S5.SS1.SSS5.p1.3.m3.1.1.3a.cmml" xref="S5.SS1.SSS5.p1.3.m3.1.1.3"><mtext class="ltx_mathvariant_monospace" id="S5.SS1.SSS5.p1.3.m3.1.1.3.cmml" mathsize="70%" xref="S5.SS1.SSS5.p1.3.m3.1.1.3">480</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS1.SSS5.p1.3.m3.1c">\texttt{Books}_{\texttt{480}}</annotation><annotation encoding="application/x-llamapun" id="S5.SS1.SSS5.p1.3.m3.1d">Books start_POSTSUBSCRIPT 480 end_POSTSUBSCRIPT</annotation></semantics></math>. 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We also ensured that each cluster sampled at least one behavior by enforcing <math alttext="k=\min(m,n\cdot\text{ratio})" class="ltx_Math" display="inline" id="S5.SS1.SSS5.p1.10.m10.3"><semantics id="S5.SS1.SSS5.p1.10.m10.3a"><mrow id="S5.SS1.SSS5.p1.10.m10.3.3" xref="S5.SS1.SSS5.p1.10.m10.3.3.cmml"><mi id="S5.SS1.SSS5.p1.10.m10.3.3.3" xref="S5.SS1.SSS5.p1.10.m10.3.3.3.cmml">k</mi><mo id="S5.SS1.SSS5.p1.10.m10.3.3.2" xref="S5.SS1.SSS5.p1.10.m10.3.3.2.cmml">=</mo><mrow id="S5.SS1.SSS5.p1.10.m10.3.3.1.1" xref="S5.SS1.SSS5.p1.10.m10.3.3.1.2.cmml"><mi id="S5.SS1.SSS5.p1.10.m10.1.1" xref="S5.SS1.SSS5.p1.10.m10.1.1.cmml">min</mi><mo id="S5.SS1.SSS5.p1.10.m10.3.3.1.1a" xref="S5.SS1.SSS5.p1.10.m10.3.3.1.2.cmml">⁡</mo><mrow id="S5.SS1.SSS5.p1.10.m10.3.3.1.1.1" xref="S5.SS1.SSS5.p1.10.m10.3.3.1.2.cmml"><mo id="S5.SS1.SSS5.p1.10.m10.3.3.1.1.1.2" stretchy="false" xref="S5.SS1.SSS5.p1.10.m10.3.3.1.2.cmml">(</mo><mi id="S5.SS1.SSS5.p1.10.m10.2.2" xref="S5.SS1.SSS5.p1.10.m10.2.2.cmml">m</mi><mo id="S5.SS1.SSS5.p1.10.m10.3.3.1.1.1.3" xref="S5.SS1.SSS5.p1.10.m10.3.3.1.2.cmml">,</mo><mrow id="S5.SS1.SSS5.p1.10.m10.3.3.1.1.1.1" xref="S5.SS1.SSS5.p1.10.m10.3.3.1.1.1.1.cmml"><mi id="S5.SS1.SSS5.p1.10.m10.3.3.1.1.1.1.2" xref="S5.SS1.SSS5.p1.10.m10.3.3.1.1.1.1.2.cmml">n</mi><mo id="S5.SS1.SSS5.p1.10.m10.3.3.1.1.1.1.1" lspace="0.222em" rspace="0.222em" xref="S5.SS1.SSS5.p1.10.m10.3.3.1.1.1.1.1.cmml">⋅</mo><mtext id="S5.SS1.SSS5.p1.10.m10.3.3.1.1.1.1.3" xref="S5.SS1.SSS5.p1.10.m10.3.3.1.1.1.1.3a.cmml">ratio</mtext></mrow><mo id="S5.SS1.SSS5.p1.10.m10.3.3.1.1.1.4" stretchy="false" xref="S5.SS1.SSS5.p1.10.m10.3.3.1.2.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.SS1.SSS5.p1.10.m10.3b"><apply id="S5.SS1.SSS5.p1.10.m10.3.3.cmml" xref="S5.SS1.SSS5.p1.10.m10.3.3"><eq id="S5.SS1.SSS5.p1.10.m10.3.3.2.cmml" xref="S5.SS1.SSS5.p1.10.m10.3.3.2"></eq><ci id="S5.SS1.SSS5.p1.10.m10.3.3.3.cmml" xref="S5.SS1.SSS5.p1.10.m10.3.3.3">𝑘</ci><apply id="S5.SS1.SSS5.p1.10.m10.3.3.1.2.cmml" xref="S5.SS1.SSS5.p1.10.m10.3.3.1.1"><min id="S5.SS1.SSS5.p1.10.m10.1.1.cmml" xref="S5.SS1.SSS5.p1.10.m10.1.1"></min><ci id="S5.SS1.SSS5.p1.10.m10.2.2.cmml" xref="S5.SS1.SSS5.p1.10.m10.2.2">𝑚</ci><apply id="S5.SS1.SSS5.p1.10.m10.3.3.1.1.1.1.cmml" xref="S5.SS1.SSS5.p1.10.m10.3.3.1.1.1.1"><ci id="S5.SS1.SSS5.p1.10.m10.3.3.1.1.1.1.1.cmml" xref="S5.SS1.SSS5.p1.10.m10.3.3.1.1.1.1.1">⋅</ci><ci id="S5.SS1.SSS5.p1.10.m10.3.3.1.1.1.1.2.cmml" xref="S5.SS1.SSS5.p1.10.m10.3.3.1.1.1.1.2">𝑛</ci><ci id="S5.SS1.SSS5.p1.10.m10.3.3.1.1.1.1.3a.cmml" xref="S5.SS1.SSS5.p1.10.m10.3.3.1.1.1.1.3"><mtext id="S5.SS1.SSS5.p1.10.m10.3.3.1.1.1.1.3.cmml" xref="S5.SS1.SSS5.p1.10.m10.3.3.1.1.1.1.3">ratio</mtext></ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS1.SSS5.p1.10.m10.3c">k=\min(m,n\cdot\text{ratio})</annotation><annotation encoding="application/x-llamapun" id="S5.SS1.SSS5.p1.10.m10.3d">italic_k = roman_min ( italic_m , italic_n ⋅ ratio )</annotation></semantics></math>. To evaluate the performance of the baseline methods, we varied the hyper-parameter selection ratio across different ranges for each dataset. Specifically, for <math alttext="\texttt{CDs}_{\texttt{50}}" class="ltx_Math" display="inline" id="S5.SS1.SSS5.p1.11.m11.1"><semantics id="S5.SS1.SSS5.p1.11.m11.1a"><msub id="S5.SS1.SSS5.p1.11.m11.1.1" xref="S5.SS1.SSS5.p1.11.m11.1.1.cmml"><mtext class="ltx_mathvariant_monospace" id="S5.SS1.SSS5.p1.11.m11.1.1.2" xref="S5.SS1.SSS5.p1.11.m11.1.1.2a.cmml">CDs</mtext><mtext class="ltx_mathvariant_monospace" id="S5.SS1.SSS5.p1.11.m11.1.1.3" xref="S5.SS1.SSS5.p1.11.m11.1.1.3a.cmml">50</mtext></msub><annotation-xml encoding="MathML-Content" id="S5.SS1.SSS5.p1.11.m11.1b"><apply id="S5.SS1.SSS5.p1.11.m11.1.1.cmml" xref="S5.SS1.SSS5.p1.11.m11.1.1"><csymbol cd="ambiguous" id="S5.SS1.SSS5.p1.11.m11.1.1.1.cmml" xref="S5.SS1.SSS5.p1.11.m11.1.1">subscript</csymbol><ci id="S5.SS1.SSS5.p1.11.m11.1.1.2a.cmml" xref="S5.SS1.SSS5.p1.11.m11.1.1.2"><mtext class="ltx_mathvariant_monospace" id="S5.SS1.SSS5.p1.11.m11.1.1.2.cmml" xref="S5.SS1.SSS5.p1.11.m11.1.1.2">CDs</mtext></ci><ci id="S5.SS1.SSS5.p1.11.m11.1.1.3a.cmml" xref="S5.SS1.SSS5.p1.11.m11.1.1.3"><mtext class="ltx_mathvariant_monospace" id="S5.SS1.SSS5.p1.11.m11.1.1.3.cmml" mathsize="70%" xref="S5.SS1.SSS5.p1.11.m11.1.1.3">50</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS1.SSS5.p1.11.m11.1c">\texttt{CDs}_{\texttt{50}}</annotation><annotation encoding="application/x-llamapun" id="S5.SS1.SSS5.p1.11.m11.1d">CDs start_POSTSUBSCRIPT 50 end_POSTSUBSCRIPT</annotation></semantics></math>, the selection ratio was chosen from <math alttext="\{0.02,0.06,0.08,0.1,0.16,0.2,0.3\}" class="ltx_Math" display="inline" id="S5.SS1.SSS5.p1.12.m12.7"><semantics id="S5.SS1.SSS5.p1.12.m12.7a"><mrow id="S5.SS1.SSS5.p1.12.m12.7.8.2" xref="S5.SS1.SSS5.p1.12.m12.7.8.1.cmml"><mo id="S5.SS1.SSS5.p1.12.m12.7.8.2.1" stretchy="false" xref="S5.SS1.SSS5.p1.12.m12.7.8.1.cmml">{</mo><mn id="S5.SS1.SSS5.p1.12.m12.1.1" xref="S5.SS1.SSS5.p1.12.m12.1.1.cmml">0.02</mn><mo id="S5.SS1.SSS5.p1.12.m12.7.8.2.2" xref="S5.SS1.SSS5.p1.12.m12.7.8.1.cmml">,</mo><mn id="S5.SS1.SSS5.p1.12.m12.2.2" xref="S5.SS1.SSS5.p1.12.m12.2.2.cmml">0.06</mn><mo id="S5.SS1.SSS5.p1.12.m12.7.8.2.3" xref="S5.SS1.SSS5.p1.12.m12.7.8.1.cmml">,</mo><mn id="S5.SS1.SSS5.p1.12.m12.3.3" xref="S5.SS1.SSS5.p1.12.m12.3.3.cmml">0.08</mn><mo id="S5.SS1.SSS5.p1.12.m12.7.8.2.4" xref="S5.SS1.SSS5.p1.12.m12.7.8.1.cmml">,</mo><mn id="S5.SS1.SSS5.p1.12.m12.4.4" xref="S5.SS1.SSS5.p1.12.m12.4.4.cmml">0.1</mn><mo id="S5.SS1.SSS5.p1.12.m12.7.8.2.5" xref="S5.SS1.SSS5.p1.12.m12.7.8.1.cmml">,</mo><mn id="S5.SS1.SSS5.p1.12.m12.5.5" xref="S5.SS1.SSS5.p1.12.m12.5.5.cmml">0.16</mn><mo id="S5.SS1.SSS5.p1.12.m12.7.8.2.6" xref="S5.SS1.SSS5.p1.12.m12.7.8.1.cmml">,</mo><mn id="S5.SS1.SSS5.p1.12.m12.6.6" xref="S5.SS1.SSS5.p1.12.m12.6.6.cmml">0.2</mn><mo id="S5.SS1.SSS5.p1.12.m12.7.8.2.7" xref="S5.SS1.SSS5.p1.12.m12.7.8.1.cmml">,</mo><mn id="S5.SS1.SSS5.p1.12.m12.7.7" xref="S5.SS1.SSS5.p1.12.m12.7.7.cmml">0.3</mn><mo id="S5.SS1.SSS5.p1.12.m12.7.8.2.8" stretchy="false" xref="S5.SS1.SSS5.p1.12.m12.7.8.1.cmml">}</mo></mrow><annotation-xml encoding="MathML-Content" id="S5.SS1.SSS5.p1.12.m12.7b"><set id="S5.SS1.SSS5.p1.12.m12.7.8.1.cmml" xref="S5.SS1.SSS5.p1.12.m12.7.8.2"><cn id="S5.SS1.SSS5.p1.12.m12.1.1.cmml" type="float" xref="S5.SS1.SSS5.p1.12.m12.1.1">0.02</cn><cn id="S5.SS1.SSS5.p1.12.m12.2.2.cmml" type="float" xref="S5.SS1.SSS5.p1.12.m12.2.2">0.06</cn><cn id="S5.SS1.SSS5.p1.12.m12.3.3.cmml" type="float" xref="S5.SS1.SSS5.p1.12.m12.3.3">0.08</cn><cn id="S5.SS1.SSS5.p1.12.m12.4.4.cmml" type="float" xref="S5.SS1.SSS5.p1.12.m12.4.4">0.1</cn><cn id="S5.SS1.SSS5.p1.12.m12.5.5.cmml" type="float" xref="S5.SS1.SSS5.p1.12.m12.5.5">0.16</cn><cn id="S5.SS1.SSS5.p1.12.m12.6.6.cmml" type="float" xref="S5.SS1.SSS5.p1.12.m12.6.6">0.2</cn><cn id="S5.SS1.SSS5.p1.12.m12.7.7.cmml" type="float" xref="S5.SS1.SSS5.p1.12.m12.7.7">0.3</cn></set></annotation-xml><annotation encoding="application/x-tex" id="S5.SS1.SSS5.p1.12.m12.7c">\{0.02,0.06,0.08,0.1,0.16,0.2,0.3\}</annotation><annotation encoding="application/x-llamapun" id="S5.SS1.SSS5.p1.12.m12.7d">{ 0.02 , 0.06 , 0.08 , 0.1 , 0.16 , 0.2 , 0.3 }</annotation></semantics></math>. Similarly, for <math alttext="\texttt{CDs}_{\texttt{200}}" class="ltx_Math" display="inline" id="S5.SS1.SSS5.p1.13.m13.1"><semantics id="S5.SS1.SSS5.p1.13.m13.1a"><msub id="S5.SS1.SSS5.p1.13.m13.1.1" xref="S5.SS1.SSS5.p1.13.m13.1.1.cmml"><mtext class="ltx_mathvariant_monospace" id="S5.SS1.SSS5.p1.13.m13.1.1.2" xref="S5.SS1.SSS5.p1.13.m13.1.1.2a.cmml">CDs</mtext><mtext class="ltx_mathvariant_monospace" id="S5.SS1.SSS5.p1.13.m13.1.1.3" xref="S5.SS1.SSS5.p1.13.m13.1.1.3a.cmml">200</mtext></msub><annotation-xml encoding="MathML-Content" id="S5.SS1.SSS5.p1.13.m13.1b"><apply id="S5.SS1.SSS5.p1.13.m13.1.1.cmml" xref="S5.SS1.SSS5.p1.13.m13.1.1"><csymbol cd="ambiguous" id="S5.SS1.SSS5.p1.13.m13.1.1.1.cmml" xref="S5.SS1.SSS5.p1.13.m13.1.1">subscript</csymbol><ci id="S5.SS1.SSS5.p1.13.m13.1.1.2a.cmml" xref="S5.SS1.SSS5.p1.13.m13.1.1.2"><mtext class="ltx_mathvariant_monospace" id="S5.SS1.SSS5.p1.13.m13.1.1.2.cmml" xref="S5.SS1.SSS5.p1.13.m13.1.1.2">CDs</mtext></ci><ci id="S5.SS1.SSS5.p1.13.m13.1.1.3a.cmml" xref="S5.SS1.SSS5.p1.13.m13.1.1.3"><mtext class="ltx_mathvariant_monospace" id="S5.SS1.SSS5.p1.13.m13.1.1.3.cmml" mathsize="70%" xref="S5.SS1.SSS5.p1.13.m13.1.1.3">200</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS1.SSS5.p1.13.m13.1c">\texttt{CDs}_{\texttt{200}}</annotation><annotation encoding="application/x-llamapun" id="S5.SS1.SSS5.p1.13.m13.1d">CDs start_POSTSUBSCRIPT 200 end_POSTSUBSCRIPT</annotation></semantics></math>, it ranged over <math alttext="\{0.005,0.01,0.02,0.03,0.05,0.08,0.1\}" class="ltx_Math" display="inline" id="S5.SS1.SSS5.p1.14.m14.7"><semantics id="S5.SS1.SSS5.p1.14.m14.7a"><mrow id="S5.SS1.SSS5.p1.14.m14.7.8.2" xref="S5.SS1.SSS5.p1.14.m14.7.8.1.cmml"><mo id="S5.SS1.SSS5.p1.14.m14.7.8.2.1" stretchy="false" xref="S5.SS1.SSS5.p1.14.m14.7.8.1.cmml">{</mo><mn id="S5.SS1.SSS5.p1.14.m14.1.1" xref="S5.SS1.SSS5.p1.14.m14.1.1.cmml">0.005</mn><mo id="S5.SS1.SSS5.p1.14.m14.7.8.2.2" xref="S5.SS1.SSS5.p1.14.m14.7.8.1.cmml">,</mo><mn id="S5.SS1.SSS5.p1.14.m14.2.2" xref="S5.SS1.SSS5.p1.14.m14.2.2.cmml">0.01</mn><mo id="S5.SS1.SSS5.p1.14.m14.7.8.2.3" xref="S5.SS1.SSS5.p1.14.m14.7.8.1.cmml">,</mo><mn id="S5.SS1.SSS5.p1.14.m14.3.3" xref="S5.SS1.SSS5.p1.14.m14.3.3.cmml">0.02</mn><mo id="S5.SS1.SSS5.p1.14.m14.7.8.2.4" xref="S5.SS1.SSS5.p1.14.m14.7.8.1.cmml">,</mo><mn id="S5.SS1.SSS5.p1.14.m14.4.4" xref="S5.SS1.SSS5.p1.14.m14.4.4.cmml">0.03</mn><mo id="S5.SS1.SSS5.p1.14.m14.7.8.2.5" xref="S5.SS1.SSS5.p1.14.m14.7.8.1.cmml">,</mo><mn id="S5.SS1.SSS5.p1.14.m14.5.5" xref="S5.SS1.SSS5.p1.14.m14.5.5.cmml">0.05</mn><mo id="S5.SS1.SSS5.p1.14.m14.7.8.2.6" xref="S5.SS1.SSS5.p1.14.m14.7.8.1.cmml">,</mo><mn id="S5.SS1.SSS5.p1.14.m14.6.6" xref="S5.SS1.SSS5.p1.14.m14.6.6.cmml">0.08</mn><mo id="S5.SS1.SSS5.p1.14.m14.7.8.2.7" xref="S5.SS1.SSS5.p1.14.m14.7.8.1.cmml">,</mo><mn id="S5.SS1.SSS5.p1.14.m14.7.7" xref="S5.SS1.SSS5.p1.14.m14.7.7.cmml">0.1</mn><mo id="S5.SS1.SSS5.p1.14.m14.7.8.2.8" stretchy="false" xref="S5.SS1.SSS5.p1.14.m14.7.8.1.cmml">}</mo></mrow><annotation-xml encoding="MathML-Content" id="S5.SS1.SSS5.p1.14.m14.7b"><set id="S5.SS1.SSS5.p1.14.m14.7.8.1.cmml" xref="S5.SS1.SSS5.p1.14.m14.7.8.2"><cn id="S5.SS1.SSS5.p1.14.m14.1.1.cmml" type="float" xref="S5.SS1.SSS5.p1.14.m14.1.1">0.005</cn><cn id="S5.SS1.SSS5.p1.14.m14.2.2.cmml" type="float" xref="S5.SS1.SSS5.p1.14.m14.2.2">0.01</cn><cn id="S5.SS1.SSS5.p1.14.m14.3.3.cmml" type="float" xref="S5.SS1.SSS5.p1.14.m14.3.3">0.02</cn><cn id="S5.SS1.SSS5.p1.14.m14.4.4.cmml" type="float" xref="S5.SS1.SSS5.p1.14.m14.4.4">0.03</cn><cn id="S5.SS1.SSS5.p1.14.m14.5.5.cmml" type="float" xref="S5.SS1.SSS5.p1.14.m14.5.5">0.05</cn><cn id="S5.SS1.SSS5.p1.14.m14.6.6.cmml" type="float" xref="S5.SS1.SSS5.p1.14.m14.6.6">0.08</cn><cn id="S5.SS1.SSS5.p1.14.m14.7.7.cmml" type="float" xref="S5.SS1.SSS5.p1.14.m14.7.7">0.1</cn></set></annotation-xml><annotation encoding="application/x-tex" id="S5.SS1.SSS5.p1.14.m14.7c">\{0.005,0.01,0.02,0.03,0.05,0.08,0.1\}</annotation><annotation encoding="application/x-llamapun" id="S5.SS1.SSS5.p1.14.m14.7d">{ 0.005 , 0.01 , 0.02 , 0.03 , 0.05 , 0.08 , 0.1 }</annotation></semantics></math>, and for <math alttext="\texttt{Books}_{\texttt{480}}" class="ltx_Math" display="inline" id="S5.SS1.SSS5.p1.15.m15.1"><semantics id="S5.SS1.SSS5.p1.15.m15.1a"><msub id="S5.SS1.SSS5.p1.15.m15.1.1" xref="S5.SS1.SSS5.p1.15.m15.1.1.cmml"><mtext class="ltx_mathvariant_monospace" id="S5.SS1.SSS5.p1.15.m15.1.1.2" xref="S5.SS1.SSS5.p1.15.m15.1.1.2a.cmml">Books</mtext><mtext class="ltx_mathvariant_monospace" id="S5.SS1.SSS5.p1.15.m15.1.1.3" xref="S5.SS1.SSS5.p1.15.m15.1.1.3a.cmml">480</mtext></msub><annotation-xml encoding="MathML-Content" id="S5.SS1.SSS5.p1.15.m15.1b"><apply id="S5.SS1.SSS5.p1.15.m15.1.1.cmml" xref="S5.SS1.SSS5.p1.15.m15.1.1"><csymbol cd="ambiguous" id="S5.SS1.SSS5.p1.15.m15.1.1.1.cmml" xref="S5.SS1.SSS5.p1.15.m15.1.1">subscript</csymbol><ci id="S5.SS1.SSS5.p1.15.m15.1.1.2a.cmml" xref="S5.SS1.SSS5.p1.15.m15.1.1.2"><mtext class="ltx_mathvariant_monospace" id="S5.SS1.SSS5.p1.15.m15.1.1.2.cmml" xref="S5.SS1.SSS5.p1.15.m15.1.1.2">Books</mtext></ci><ci id="S5.SS1.SSS5.p1.15.m15.1.1.3a.cmml" xref="S5.SS1.SSS5.p1.15.m15.1.1.3"><mtext class="ltx_mathvariant_monospace" id="S5.SS1.SSS5.p1.15.m15.1.1.3.cmml" mathsize="70%" xref="S5.SS1.SSS5.p1.15.m15.1.1.3">480</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS1.SSS5.p1.15.m15.1c">\texttt{Books}_{\texttt{480}}</annotation><annotation encoding="application/x-llamapun" id="S5.SS1.SSS5.p1.15.m15.1d">Books start_POSTSUBSCRIPT 480 end_POSTSUBSCRIPT</annotation></semantics></math>, the selection ratio spanned <math alttext="\{0.002,0.005,0.008,0.011,0.014\}" class="ltx_Math" display="inline" id="S5.SS1.SSS5.p1.16.m16.5"><semantics id="S5.SS1.SSS5.p1.16.m16.5a"><mrow id="S5.SS1.SSS5.p1.16.m16.5.6.2" xref="S5.SS1.SSS5.p1.16.m16.5.6.1.cmml"><mo 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encoding="MathML-Content" id="S5.SS1.SSS5.p1.16.m16.5b"><set id="S5.SS1.SSS5.p1.16.m16.5.6.1.cmml" xref="S5.SS1.SSS5.p1.16.m16.5.6.2"><cn id="S5.SS1.SSS5.p1.16.m16.1.1.cmml" type="float" xref="S5.SS1.SSS5.p1.16.m16.1.1">0.002</cn><cn id="S5.SS1.SSS5.p1.16.m16.2.2.cmml" type="float" xref="S5.SS1.SSS5.p1.16.m16.2.2">0.005</cn><cn id="S5.SS1.SSS5.p1.16.m16.3.3.cmml" type="float" xref="S5.SS1.SSS5.p1.16.m16.3.3">0.008</cn><cn id="S5.SS1.SSS5.p1.16.m16.4.4.cmml" type="float" xref="S5.SS1.SSS5.p1.16.m16.4.4">0.011</cn><cn id="S5.SS1.SSS5.p1.16.m16.5.5.cmml" type="float" xref="S5.SS1.SSS5.p1.16.m16.5.5">0.014</cn></set></annotation-xml><annotation encoding="application/x-tex" id="S5.SS1.SSS5.p1.16.m16.5c">\{0.002,0.005,0.008,0.011,0.014\}</annotation><annotation encoding="application/x-llamapun" id="S5.SS1.SSS5.p1.16.m16.5d">{ 0.002 , 0.005 , 0.008 , 0.011 , 0.014 }</annotation></semantics></math>. The prompt templates are provided in Appendix <a class="ltx_ref" href="https://arxiv.org/html/2503.02398v1#A6" title="Appendix F Prompt Templates ‣ PersonaX: A Recommendation Agent-Oriented User Modeling Framework for Long Behavior Sequence"><span class="ltx_text ltx_ref_tag">F</span></a>.</p> </div> <figure class="ltx_figure" id="S5.F3"><img alt="Refer to caption" class="ltx_graphics ltx_centering ltx_img_landscape" height="215" id="S5.F3.g1" src="x3.png" width="858"/> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure">Figure 3: </span>Analysis of the impact of sampling size on user modeling.</figcaption> </figure> </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>Performance Evaluation (RQ 1)</h3> <div class="ltx_para" id="S5.SS2.p1"> <p class="ltx_p" id="S5.SS2.p1.1">Key observations and insights from Tables <a class="ltx_ref" href="https://arxiv.org/html/2503.02398v1#S5.T3" title="Table 3 ‣ 5.2 Performance Evaluation (RQ 1) ‣ 5 Experiments ‣ PersonaX: A Recommendation Agent-Oriented User Modeling Framework for Long Behavior Sequence"><span class="ltx_text ltx_ref_tag">3</span></a> highlight the robustness and effectiveness of our proposed method across various agent recommendation approaches, datasets, and evaluation metrics. PersonaX consistently outperforms the Full approach under any level of data resource utilization, even in scenarios where PersonaX achieves its least favorable results. Notably, on the <math alttext="\texttt{Books}_{\texttt{480}}" class="ltx_Math" display="inline" id="S5.SS2.p1.1.m1.1"><semantics id="S5.SS2.p1.1.m1.1a"><msub id="S5.SS2.p1.1.m1.1.1" xref="S5.SS2.p1.1.m1.1.1.cmml"><mtext class="ltx_mathvariant_monospace" id="S5.SS2.p1.1.m1.1.1.2" xref="S5.SS2.p1.1.m1.1.1.2a.cmml">Books</mtext><mtext class="ltx_mathvariant_monospace" id="S5.SS2.p1.1.m1.1.1.3" xref="S5.SS2.p1.1.m1.1.1.3a.cmml">480</mtext></msub><annotation-xml encoding="MathML-Content" id="S5.SS2.p1.1.m1.1b"><apply id="S5.SS2.p1.1.m1.1.1.cmml" xref="S5.SS2.p1.1.m1.1.1"><csymbol cd="ambiguous" id="S5.SS2.p1.1.m1.1.1.1.cmml" xref="S5.SS2.p1.1.m1.1.1">subscript</csymbol><ci id="S5.SS2.p1.1.m1.1.1.2a.cmml" xref="S5.SS2.p1.1.m1.1.1.2"><mtext class="ltx_mathvariant_monospace" id="S5.SS2.p1.1.m1.1.1.2.cmml" xref="S5.SS2.p1.1.m1.1.1.2">Books</mtext></ci><ci id="S5.SS2.p1.1.m1.1.1.3a.cmml" xref="S5.SS2.p1.1.m1.1.1.3"><mtext class="ltx_mathvariant_monospace" id="S5.SS2.p1.1.m1.1.1.3.cmml" mathsize="70%" xref="S5.SS2.p1.1.m1.1.1.3">480</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.p1.1.m1.1c">\texttt{Books}_{\texttt{480}}</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.p1.1.m1.1d">Books start_POSTSUBSCRIPT 480 end_POSTSUBSCRIPT</annotation></semantics></math> dataset, which features longer behavior sequences, our method achieves significant improvements over the Full methods. This phenomenon highlights the shortcomings of existing agent recommendation methods in handling long behavior sequences, but PersonaX fills this critical research gap.</p> </div> <div class="ltx_para" id="S5.SS2.p2"> <p class="ltx_p" id="S5.SS2.p2.2">Table <a class="ltx_ref" href="https://arxiv.org/html/2503.02398v1#S5.T2" title="Table 2 ‣ 5.1.2 Evaluation ‣ 5.1 Experimental Setup ‣ 5 Experiments ‣ PersonaX: A Recommendation Agent-Oriented User Modeling Framework for Long Behavior Sequence"><span class="ltx_text ltx_ref_tag">2</span></a> reports the best MRR@10, highlighting PersonaX’s performance advantages over baselines. Our approach demonstrates substantial improvements over the widely adopted and strong baseline method, Relevance. For example, on the <math alttext="\texttt{CDs}_{\texttt{50}}" class="ltx_Math" display="inline" id="S5.SS2.p2.1.m1.1"><semantics id="S5.SS2.p2.1.m1.1a"><msub id="S5.SS2.p2.1.m1.1.1" xref="S5.SS2.p2.1.m1.1.1.cmml"><mtext class="ltx_mathvariant_monospace" id="S5.SS2.p2.1.m1.1.1.2" xref="S5.SS2.p2.1.m1.1.1.2a.cmml">CDs</mtext><mtext class="ltx_mathvariant_monospace" id="S5.SS2.p2.1.m1.1.1.3" xref="S5.SS2.p2.1.m1.1.1.3a.cmml">50</mtext></msub><annotation-xml encoding="MathML-Content" id="S5.SS2.p2.1.m1.1b"><apply id="S5.SS2.p2.1.m1.1.1.cmml" xref="S5.SS2.p2.1.m1.1.1"><csymbol cd="ambiguous" id="S5.SS2.p2.1.m1.1.1.1.cmml" xref="S5.SS2.p2.1.m1.1.1">subscript</csymbol><ci id="S5.SS2.p2.1.m1.1.1.2a.cmml" xref="S5.SS2.p2.1.m1.1.1.2"><mtext class="ltx_mathvariant_monospace" id="S5.SS2.p2.1.m1.1.1.2.cmml" xref="S5.SS2.p2.1.m1.1.1.2">CDs</mtext></ci><ci id="S5.SS2.p2.1.m1.1.1.3a.cmml" xref="S5.SS2.p2.1.m1.1.1.3"><mtext class="ltx_mathvariant_monospace" id="S5.SS2.p2.1.m1.1.1.3.cmml" mathsize="70%" xref="S5.SS2.p2.1.m1.1.1.3">50</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.p2.1.m1.1c">\texttt{CDs}_{\texttt{50}}</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.p2.1.m1.1d">CDs start_POSTSUBSCRIPT 50 end_POSTSUBSCRIPT</annotation></semantics></math> dataset, our method achieves a Hit@1 score of 45.00, significantly exceeding the 40.00 obtained by Relevance. Similarly, we observe the suboptimal performance of the Centroid and Boundary methods, particularly on <math alttext="\texttt{CDs}_{\texttt{200}}" class="ltx_Math" display="inline" id="S5.SS2.p2.2.m2.1"><semantics id="S5.SS2.p2.2.m2.1a"><msub id="S5.SS2.p2.2.m2.1.1" xref="S5.SS2.p2.2.m2.1.1.cmml"><mtext class="ltx_mathvariant_monospace" id="S5.SS2.p2.2.m2.1.1.2" xref="S5.SS2.p2.2.m2.1.1.2a.cmml">CDs</mtext><mtext class="ltx_mathvariant_monospace" id="S5.SS2.p2.2.m2.1.1.3" xref="S5.SS2.p2.2.m2.1.1.3a.cmml">200</mtext></msub><annotation-xml encoding="MathML-Content" id="S5.SS2.p2.2.m2.1b"><apply id="S5.SS2.p2.2.m2.1.1.cmml" xref="S5.SS2.p2.2.m2.1.1"><csymbol cd="ambiguous" id="S5.SS2.p2.2.m2.1.1.1.cmml" xref="S5.SS2.p2.2.m2.1.1">subscript</csymbol><ci id="S5.SS2.p2.2.m2.1.1.2a.cmml" xref="S5.SS2.p2.2.m2.1.1.2"><mtext class="ltx_mathvariant_monospace" id="S5.SS2.p2.2.m2.1.1.2.cmml" xref="S5.SS2.p2.2.m2.1.1.2">CDs</mtext></ci><ci id="S5.SS2.p2.2.m2.1.1.3a.cmml" xref="S5.SS2.p2.2.m2.1.1.3"><mtext class="ltx_mathvariant_monospace" id="S5.SS2.p2.2.m2.1.1.3.cmml" mathsize="70%" xref="S5.SS2.p2.2.m2.1.1.3">200</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS2.p2.2.m2.1c">\texttt{CDs}_{\texttt{200}}</annotation><annotation encoding="application/x-llamapun" id="S5.SS2.p2.2.m2.1d">CDs start_POSTSUBSCRIPT 200 end_POSTSUBSCRIPT</annotation></semantics></math>. Upon analysis, we attribute the underperformance of the Centroid method to its tendency to sample overly homogeneous information, which results in overly simplistic and narrow user personas. While the Boundary method ensures sample diversity, an excessive focus on diversity can dilute the representation of typical user persona characteristics. In contrast, our method consistently delivers superior and stable performance, highlighting the effectiveness of balancing prototypicality and diversity. This equilibrium enables our approach to capture nuanced user personas with greater precision, establishing it as a robust and versatile solution for user modeling.</p> </div> <figure class="ltx_table" id="S5.T3"> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_table">Table 3: </span>Performance of PersonaX at different selection ratios. We highlight <span class="ltx_text ltx_font_bold" id="S5.T3.6.1" style="color:#BF0040;">best performance</span>, and the <span class="ltx_text ltx_font_bold" id="S5.T3.7.2" style="color:#FC8A6A;">worst performance</span>.</figcaption><div class="ltx_flex_figure"> <div class="ltx_flex_cell ltx_flex_size_1"> <p class="ltx_p ltx_figure_panel ltx_align_center" id="S5.T3.1"><span class="ltx_text ltx_font_bold" id="S5.T3.1.1">Reflection on <math alttext="\texttt{CDs}_{\texttt{50}}" class="ltx_Math" display="inline" id="S5.T3.1.1.m1.1"><semantics id="S5.T3.1.1.m1.1a"><msub id="S5.T3.1.1.m1.1.1" xref="S5.T3.1.1.m1.1.1.cmml"><mtext class="ltx_mathvariant_monospace" id="S5.T3.1.1.m1.1.1.2" xref="S5.T3.1.1.m1.1.1.2a.cmml">CDs</mtext><mtext class="ltx_mathvariant_monospace" id="S5.T3.1.1.m1.1.1.3" xref="S5.T3.1.1.m1.1.1.3a.cmml">50</mtext></msub><annotation-xml encoding="MathML-Content" id="S5.T3.1.1.m1.1b"><apply id="S5.T3.1.1.m1.1.1.cmml" xref="S5.T3.1.1.m1.1.1"><csymbol cd="ambiguous" id="S5.T3.1.1.m1.1.1.1.cmml" xref="S5.T3.1.1.m1.1.1">subscript</csymbol><ci id="S5.T3.1.1.m1.1.1.2a.cmml" xref="S5.T3.1.1.m1.1.1.2"><mtext class="ltx_mathvariant_monospace" id="S5.T3.1.1.m1.1.1.2.cmml" xref="S5.T3.1.1.m1.1.1.2">CDs</mtext></ci><ci id="S5.T3.1.1.m1.1.1.3a.cmml" xref="S5.T3.1.1.m1.1.1.3"><mtext class="ltx_mathvariant_monospace" id="S5.T3.1.1.m1.1.1.3.cmml" mathsize="70%" xref="S5.T3.1.1.m1.1.1.3">50</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.T3.1.1.m1.1c">\texttt{CDs}_{\texttt{50}}</annotation><annotation encoding="application/x-llamapun" id="S5.T3.1.1.m1.1d">CDs start_POSTSUBSCRIPT 50 end_POSTSUBSCRIPT</annotation></semantics></math></span> <br class="ltx_break"/> <span class="ltx_tabular ltx_guessed_headers ltx_align_middle" id="S5.T3.1.2"> <span class="ltx_thead"> <span class="ltx_tr" id="S5.T3.1.2.1.1"> <span class="ltx_td ltx_align_left ltx_th ltx_th_column ltx_th_row ltx_border_r ltx_border_t" id="S5.T3.1.2.1.1.1">Ratio</span> <span class="ltx_td ltx_align_left ltx_th ltx_th_column ltx_th_row ltx_border_r ltx_border_t" id="S5.T3.1.2.1.1.2">#SBS</span> <span class="ltx_td ltx_align_left ltx_th ltx_th_column ltx_border_t" id="S5.T3.1.2.1.1.3">HR@1</span> <span class="ltx_td ltx_align_left ltx_th ltx_th_column ltx_border_t" id="S5.T3.1.2.1.1.4">HR@5</span> <span class="ltx_td ltx_align_left ltx_th ltx_th_column ltx_border_t" id="S5.T3.1.2.1.1.5">NDCG@5</span> <span class="ltx_td ltx_align_left ltx_th ltx_th_column ltx_border_t" id="S5.T3.1.2.1.1.6">MRR</span></span> </span> <span class="ltx_tbody"> <span class="ltx_tr" id="S5.T3.1.2.2.1"> <span class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_r ltx_border_t" id="S5.T3.1.2.2.1.1">100</span> <span class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_r ltx_border_t" id="S5.T3.1.2.2.1.2">5.56</span> <span class="ltx_td ltx_align_left ltx_border_t" id="S5.T3.1.2.2.1.3">41.00</span> <span class="ltx_td ltx_align_left ltx_border_t" id="S5.T3.1.2.2.1.4">67.00</span> <span class="ltx_td ltx_align_left ltx_border_t" id="S5.T3.1.2.2.1.5"><span class="ltx_text ltx_font_bold" id="S5.T3.1.2.2.1.5.1" style="color:#FC8A6A;">54.67</span></span> <span class="ltx_td ltx_align_left ltx_border_t" id="S5.T3.1.2.2.1.6">54.67</span></span> <span class="ltx_tr" id="S5.T3.1.2.3.2"> <span class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_r" id="S5.T3.1.2.3.2.1">90</span> <span class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_r" id="S5.T3.1.2.3.2.2">4.69</span> <span class="ltx_td ltx_align_left" id="S5.T3.1.2.3.2.3">42.00</span> <span class="ltx_td ltx_align_left" id="S5.T3.1.2.3.2.4">69.00</span> <span class="ltx_td ltx_align_left" id="S5.T3.1.2.3.2.5">55.66</span> <span class="ltx_td ltx_align_left" id="S5.T3.1.2.3.2.6">55.22</span></span> <span class="ltx_tr" id="S5.T3.1.2.4.3"> <span class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_r" id="S5.T3.1.2.4.3.1">70</span> <span class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_r" id="S5.T3.1.2.4.3.2">3.52</span> <span class="ltx_td ltx_align_left" id="S5.T3.1.2.4.3.3"><span class="ltx_text ltx_font_bold" id="S5.T3.1.2.4.3.3.1" style="color:#FC8A6A;">39.00</span></span> <span class="ltx_td ltx_align_left" id="S5.T3.1.2.4.3.4">70.00</span> <span class="ltx_td ltx_align_left" id="S5.T3.1.2.4.3.5">54.95</span> <span class="ltx_td ltx_align_left" id="S5.T3.1.2.4.3.6"><span class="ltx_text ltx_font_bold" id="S5.T3.1.2.4.3.6.1" style="color:#FC8A6A;">53.50</span></span></span> <span class="ltx_tr" id="S5.T3.1.2.5.4"> <span class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_r" id="S5.T3.1.2.5.4.1">50</span> <span class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_r" id="S5.T3.1.2.5.4.2">2.88</span> <span class="ltx_td ltx_align_left" id="S5.T3.1.2.5.4.3">41.00</span> <span class="ltx_td ltx_align_left" id="S5.T3.1.2.5.4.4">67.00</span> <span class="ltx_td ltx_align_left" id="S5.T3.1.2.5.4.5">54.69</span> <span class="ltx_td ltx_align_left" id="S5.T3.1.2.5.4.6">55.08</span></span> <span class="ltx_tr" id="S5.T3.1.2.6.5"> <span class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_r" id="S5.T3.1.2.6.5.1">30</span> <span class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_r" id="S5.T3.1.2.6.5.2">1.83</span> <span class="ltx_td ltx_align_left" id="S5.T3.1.2.6.5.3"><span class="ltx_text ltx_font_bold" id="S5.T3.1.2.6.5.3.1" style="color:#BF0040;">45.00</span></span> <span class="ltx_td ltx_align_left" id="S5.T3.1.2.6.5.4"><span class="ltx_text ltx_font_bold" id="S5.T3.1.2.6.5.4.1" style="color:#BF0040;">72.00</span></span> <span class="ltx_td ltx_align_left" id="S5.T3.1.2.6.5.5"><span class="ltx_text ltx_font_bold" id="S5.T3.1.2.6.5.5.1" style="color:#BF0040;">57.34</span></span> <span class="ltx_td ltx_align_left" id="S5.T3.1.2.6.5.6"><span class="ltx_text ltx_font_bold" id="S5.T3.1.2.6.5.6.1" style="color:#BF0040;">58.38</span></span></span> <span class="ltx_tr" id="S5.T3.1.2.7.6"> <span class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_b ltx_border_r" id="S5.T3.1.2.7.6.1">10</span> <span class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_b ltx_border_r" id="S5.T3.1.2.7.6.2">1.0</span> <span class="ltx_td ltx_align_left ltx_border_b" id="S5.T3.1.2.7.6.3">42.00</span> <span class="ltx_td ltx_align_left ltx_border_b" id="S5.T3.1.2.7.6.4"><span class="ltx_text ltx_font_bold" id="S5.T3.1.2.7.6.4.1" style="color:#FC8A6A;">66.00</span></span> <span class="ltx_td ltx_align_left ltx_border_b" id="S5.T3.1.2.7.6.5">56.07</span> <span class="ltx_td ltx_align_left ltx_border_b" id="S5.T3.1.2.7.6.6">55.25</span></span> </span> </span></p> </div> <div class="ltx_flex_break"></div> <div class="ltx_flex_cell ltx_flex_size_1"> <p class="ltx_p ltx_figure_panel ltx_align_center" id="S5.T3.8"><span class="ltx_rule" style="width:100%;height:1px;background:black;display:inline-block;"> </span><span class="ltx_rule" style="width:100%;height:1px;background:black;display:inline-block;"> </span></p> </div> <div class="ltx_flex_break"></div> <div class="ltx_flex_cell ltx_flex_size_1"> <p class="ltx_p ltx_figure_panel ltx_align_center" id="S5.T3.2"><span class="ltx_text ltx_font_bold" id="S5.T3.2.1">Summarization on <math alttext="\texttt{CDs}_{\texttt{200}}" class="ltx_Math" display="inline" id="S5.T3.2.1.m1.1"><semantics id="S5.T3.2.1.m1.1a"><msub id="S5.T3.2.1.m1.1.1" xref="S5.T3.2.1.m1.1.1.cmml"><mtext class="ltx_mathvariant_monospace" id="S5.T3.2.1.m1.1.1.2" xref="S5.T3.2.1.m1.1.1.2a.cmml">CDs</mtext><mtext class="ltx_mathvariant_monospace" id="S5.T3.2.1.m1.1.1.3" xref="S5.T3.2.1.m1.1.1.3a.cmml">200</mtext></msub><annotation-xml encoding="MathML-Content" id="S5.T3.2.1.m1.1b"><apply id="S5.T3.2.1.m1.1.1.cmml" xref="S5.T3.2.1.m1.1.1"><csymbol cd="ambiguous" id="S5.T3.2.1.m1.1.1.1.cmml" xref="S5.T3.2.1.m1.1.1">subscript</csymbol><ci id="S5.T3.2.1.m1.1.1.2a.cmml" xref="S5.T3.2.1.m1.1.1.2"><mtext class="ltx_mathvariant_monospace" id="S5.T3.2.1.m1.1.1.2.cmml" xref="S5.T3.2.1.m1.1.1.2">CDs</mtext></ci><ci id="S5.T3.2.1.m1.1.1.3a.cmml" xref="S5.T3.2.1.m1.1.1.3"><mtext class="ltx_mathvariant_monospace" id="S5.T3.2.1.m1.1.1.3.cmml" mathsize="70%" xref="S5.T3.2.1.m1.1.1.3">200</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.T3.2.1.m1.1c">\texttt{CDs}_{\texttt{200}}</annotation><annotation encoding="application/x-llamapun" id="S5.T3.2.1.m1.1d">CDs start_POSTSUBSCRIPT 200 end_POSTSUBSCRIPT</annotation></semantics></math></span> <br class="ltx_break"/> <span class="ltx_tabular ltx_guessed_headers ltx_align_middle" id="S5.T3.2.2"> <span class="ltx_thead"> <span class="ltx_tr" id="S5.T3.2.2.1.1"> <span class="ltx_td ltx_align_left ltx_th ltx_th_column ltx_th_row ltx_border_r ltx_border_t" id="S5.T3.2.2.1.1.1">Ratio</span> <span class="ltx_td ltx_align_left ltx_th ltx_th_column ltx_th_row ltx_border_r ltx_border_t" id="S5.T3.2.2.1.1.2">#SBS</span> <span class="ltx_td ltx_align_left ltx_th ltx_th_column ltx_border_t" id="S5.T3.2.2.1.1.3">HR@1</span> <span class="ltx_td ltx_align_left ltx_th ltx_th_column ltx_border_t" id="S5.T3.2.2.1.1.4">HR@5</span> <span class="ltx_td ltx_align_left ltx_th ltx_th_column ltx_border_t" id="S5.T3.2.2.1.1.5">NDCG@5</span> <span class="ltx_td ltx_align_left ltx_th ltx_th_column ltx_border_t" id="S5.T3.2.2.1.1.6">MRR</span></span> </span> <span class="ltx_tbody"> <span class="ltx_tr" id="S5.T3.2.2.2.1"> <span class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_r ltx_border_t" id="S5.T3.2.2.2.1.1">100</span> <span class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_r ltx_border_t" id="S5.T3.2.2.2.1.2">8.15</span> <span class="ltx_td ltx_align_left ltx_border_t" id="S5.T3.2.2.2.1.3"><span class="ltx_text ltx_font_bold" id="S5.T3.2.2.2.1.3.1" style="color:#FC8A6A;">43.00</span></span> <span class="ltx_td ltx_align_left ltx_border_t" id="S5.T3.2.2.2.1.4"><span class="ltx_text ltx_font_bold" id="S5.T3.2.2.2.1.4.1" style="color:#FC8A6A;">68.00</span></span> <span class="ltx_td ltx_align_left ltx_border_t" id="S5.T3.2.2.2.1.5"><span class="ltx_text ltx_font_bold" id="S5.T3.2.2.2.1.5.1" style="color:#FC8A6A;">56.85</span></span> <span class="ltx_td ltx_align_left ltx_border_t" id="S5.T3.2.2.2.1.6"><span class="ltx_text ltx_font_bold" id="S5.T3.2.2.2.1.6.1" style="color:#FC8A6A;">57.07</span></span></span> <span class="ltx_tr" id="S5.T3.2.2.3.2"> <span class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_r" id="S5.T3.2.2.3.2.1">90</span> <span class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_r" id="S5.T3.2.2.3.2.2">7.19</span> <span class="ltx_td ltx_align_left" id="S5.T3.2.2.3.2.3">49.00</span> <span class="ltx_td ltx_align_left" id="S5.T3.2.2.3.2.4">70.00</span> <span class="ltx_td ltx_align_left" id="S5.T3.2.2.3.2.5">59.66</span> <span class="ltx_td ltx_align_left" id="S5.T3.2.2.3.2.6">59.95</span></span> <span class="ltx_tr" id="S5.T3.2.2.4.3"> <span class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_r" id="S5.T3.2.2.4.3.1">70</span> <span class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_r" id="S5.T3.2.2.4.3.2">5.48</span> <span class="ltx_td ltx_align_left" id="S5.T3.2.2.4.3.3">47.00</span> <span class="ltx_td ltx_align_left" id="S5.T3.2.2.4.3.4">71.00</span> <span class="ltx_td ltx_align_left" id="S5.T3.2.2.4.3.5">60.54</span> <span class="ltx_td ltx_align_left" id="S5.T3.2.2.4.3.6">60.54</span></span> <span class="ltx_tr" id="S5.T3.2.2.5.4"> <span class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_r" id="S5.T3.2.2.5.4.1">50</span> <span class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_r" id="S5.T3.2.2.5.4.2">3.59</span> <span class="ltx_td ltx_align_left" id="S5.T3.2.2.5.4.3"><span class="ltx_text ltx_font_bold" id="S5.T3.2.2.5.4.3.1" style="color:#BF0040;">55.00</span></span> <span class="ltx_td ltx_align_left" id="S5.T3.2.2.5.4.4"><span class="ltx_text ltx_font_bold" id="S5.T3.2.2.5.4.4.1" style="color:#BF0040;">75.00</span></span> <span class="ltx_td ltx_align_left" id="S5.T3.2.2.5.4.5"><span class="ltx_text ltx_font_bold" id="S5.T3.2.2.5.4.5.1" style="color:#BF0040;">64.56</span></span> <span class="ltx_td ltx_align_left" id="S5.T3.2.2.5.4.6"><span class="ltx_text ltx_font_bold" id="S5.T3.2.2.5.4.6.1" style="color:#BF0040;">65.06</span></span></span> <span class="ltx_tr" id="S5.T3.2.2.6.5"> <span class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_r" id="S5.T3.2.2.6.5.1">30</span> <span class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_r" id="S5.T3.2.2.6.5.2">2.3</span> <span class="ltx_td ltx_align_left" id="S5.T3.2.2.6.5.3">51.00</span> <span class="ltx_td ltx_align_left" id="S5.T3.2.2.6.5.4">73.00</span> <span class="ltx_td ltx_align_left" id="S5.T3.2.2.6.5.5">62.45</span> <span class="ltx_td ltx_align_left" id="S5.T3.2.2.6.5.6">62.42</span></span> <span class="ltx_tr" id="S5.T3.2.2.7.6"> <span class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_b ltx_border_r" id="S5.T3.2.2.7.6.1">10</span> <span class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_b ltx_border_r" id="S5.T3.2.2.7.6.2">1.0</span> <span class="ltx_td ltx_align_left ltx_border_b" id="S5.T3.2.2.7.6.3">47.00</span> <span class="ltx_td ltx_align_left ltx_border_b" id="S5.T3.2.2.7.6.4">72.00</span> <span class="ltx_td ltx_align_left ltx_border_b" id="S5.T3.2.2.7.6.5">61.91</span> <span class="ltx_td ltx_align_left ltx_border_b" id="S5.T3.2.2.7.6.6">60.99</span></span> </span> </span></p> </div> <div class="ltx_flex_break"></div> <div class="ltx_flex_cell ltx_flex_size_1"> <p class="ltx_p ltx_figure_panel ltx_align_center" id="S5.T3.9"><span class="ltx_rule" style="width:100%;height:1px;background:black;display:inline-block;"> </span><span class="ltx_rule" style="width:100%;height:1px;background:black;display:inline-block;"> </span></p> </div> <div class="ltx_flex_break"></div> <div class="ltx_flex_cell ltx_flex_size_1"> <p class="ltx_p ltx_figure_panel ltx_align_center" id="S5.T3.3"><span class="ltx_text ltx_font_bold" id="S5.T3.3.1">Summarization on <math alttext="\texttt{Books}_{\texttt{480}}" class="ltx_Math" display="inline" id="S5.T3.3.1.m1.1"><semantics id="S5.T3.3.1.m1.1a"><msub id="S5.T3.3.1.m1.1.1" xref="S5.T3.3.1.m1.1.1.cmml"><mtext class="ltx_mathvariant_monospace" id="S5.T3.3.1.m1.1.1.2" xref="S5.T3.3.1.m1.1.1.2a.cmml">Books</mtext><mtext class="ltx_mathvariant_monospace" id="S5.T3.3.1.m1.1.1.3" xref="S5.T3.3.1.m1.1.1.3a.cmml">480</mtext></msub><annotation-xml encoding="MathML-Content" id="S5.T3.3.1.m1.1b"><apply id="S5.T3.3.1.m1.1.1.cmml" xref="S5.T3.3.1.m1.1.1"><csymbol cd="ambiguous" id="S5.T3.3.1.m1.1.1.1.cmml" xref="S5.T3.3.1.m1.1.1">subscript</csymbol><ci id="S5.T3.3.1.m1.1.1.2a.cmml" xref="S5.T3.3.1.m1.1.1.2"><mtext class="ltx_mathvariant_monospace" id="S5.T3.3.1.m1.1.1.2.cmml" xref="S5.T3.3.1.m1.1.1.2">Books</mtext></ci><ci id="S5.T3.3.1.m1.1.1.3a.cmml" xref="S5.T3.3.1.m1.1.1.3"><mtext class="ltx_mathvariant_monospace" id="S5.T3.3.1.m1.1.1.3.cmml" mathsize="70%" xref="S5.T3.3.1.m1.1.1.3">480</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.T3.3.1.m1.1c">\texttt{Books}_{\texttt{480}}</annotation><annotation encoding="application/x-llamapun" id="S5.T3.3.1.m1.1d">Books start_POSTSUBSCRIPT 480 end_POSTSUBSCRIPT</annotation></semantics></math></span> <br class="ltx_break"/> <span class="ltx_tabular ltx_guessed_headers ltx_align_middle" id="S5.T3.3.2"> <span class="ltx_thead"> <span class="ltx_tr" id="S5.T3.3.2.1.1"> <span class="ltx_td ltx_align_left ltx_th ltx_th_column ltx_th_row ltx_border_r ltx_border_t" id="S5.T3.3.2.1.1.1">Ratio</span> <span class="ltx_td ltx_align_left ltx_th ltx_th_column ltx_th_row ltx_border_r ltx_border_t" id="S5.T3.3.2.1.1.2">#SBS</span> <span class="ltx_td ltx_align_left ltx_th ltx_th_column ltx_border_t" id="S5.T3.3.2.1.1.3">HR@1</span> <span class="ltx_td ltx_align_left ltx_th ltx_th_column ltx_border_t" id="S5.T3.3.2.1.1.4">HR@5</span> <span class="ltx_td ltx_align_left ltx_th ltx_th_column ltx_border_t" id="S5.T3.3.2.1.1.5">NDCG@5</span> <span class="ltx_td ltx_align_left ltx_th ltx_th_column ltx_border_t" id="S5.T3.3.2.1.1.6">MRR</span></span> </span> <span class="ltx_tbody"> <span class="ltx_tr" id="S5.T3.3.2.2.1"> <span class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_r ltx_border_t" id="S5.T3.3.2.2.1.1">100</span> <span class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_r ltx_border_t" id="S5.T3.3.2.2.1.2">15.35</span> <span class="ltx_td ltx_align_left ltx_border_t" id="S5.T3.3.2.2.1.3">61.00</span> <span class="ltx_td ltx_align_left ltx_border_t" id="S5.T3.3.2.2.1.4">83.00</span> <span class="ltx_td ltx_align_left ltx_border_t" id="S5.T3.3.2.2.1.5">73.56</span> <span class="ltx_td ltx_align_left ltx_border_t" id="S5.T3.3.2.2.1.6">72.18</span></span> <span class="ltx_tr" id="S5.T3.3.2.3.2"> <span class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_r" id="S5.T3.3.2.3.2.1">90</span> <span class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_r" id="S5.T3.3.2.3.2.2">11.74</span> <span class="ltx_td ltx_align_left" id="S5.T3.3.2.3.2.3"><span class="ltx_text ltx_font_bold" id="S5.T3.3.2.3.2.3.1" style="color:#FC8A6A;">59.00</span></span> <span class="ltx_td ltx_align_left" id="S5.T3.3.2.3.2.4"><span class="ltx_text ltx_font_bold" id="S5.T3.3.2.3.2.4.1" style="color:#FC8A6A;">80.00</span></span> <span class="ltx_td ltx_align_left" id="S5.T3.3.2.3.2.5"><span class="ltx_text ltx_font_bold" id="S5.T3.3.2.3.2.5.1" style="color:#FC8A6A;">71.36</span></span> <span class="ltx_td ltx_align_left" id="S5.T3.3.2.3.2.6"><span class="ltx_text ltx_font_bold" id="S5.T3.3.2.3.2.6.1" style="color:#FC8A6A;">71.70</span></span></span> <span class="ltx_tr" id="S5.T3.3.2.4.3"> <span class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_r" id="S5.T3.3.2.4.3.1">70</span> <span class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_r" id="S5.T3.3.2.4.3.2">8.41</span> <span class="ltx_td ltx_align_left" id="S5.T3.3.2.4.3.3">64.00</span> <span class="ltx_td ltx_align_left" id="S5.T3.3.2.4.3.4">81.00</span> <span class="ltx_td ltx_align_left" id="S5.T3.3.2.4.3.5">72.55</span> <span class="ltx_td ltx_align_left" id="S5.T3.3.2.4.3.6">72.62</span></span> <span class="ltx_tr" id="S5.T3.3.2.5.4"> <span class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_r" id="S5.T3.3.2.5.4.1">50</span> <span class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_r" id="S5.T3.3.2.5.4.2">4.2</span> <span class="ltx_td ltx_align_left" id="S5.T3.3.2.5.4.3"><span class="ltx_text ltx_font_bold" id="S5.T3.3.2.5.4.3.1" style="color:#BF0040;">65.00</span></span> <span class="ltx_td ltx_align_left" id="S5.T3.3.2.5.4.4"><span class="ltx_text ltx_font_bold" id="S5.T3.3.2.5.4.4.1" style="color:#BF0040;">83.00</span></span> <span class="ltx_td ltx_align_left" id="S5.T3.3.2.5.4.5"><span class="ltx_text ltx_font_bold" id="S5.T3.3.2.5.4.5.1" style="color:#BF0040;">74.26</span></span> <span class="ltx_td ltx_align_left" id="S5.T3.3.2.5.4.6"><span class="ltx_text ltx_font_bold" id="S5.T3.3.2.5.4.6.1" style="color:#BF0040;">73.22</span></span></span> <span class="ltx_tr" id="S5.T3.3.2.6.5"> <span class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_r" id="S5.T3.3.2.6.5.1">30</span> <span class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_r" id="S5.T3.3.2.6.5.2">1.82</span> <span class="ltx_td ltx_align_left" id="S5.T3.3.2.6.5.3">64.00</span> <span class="ltx_td ltx_align_left" id="S5.T3.3.2.6.5.4">82.00</span> <span class="ltx_td ltx_align_left" id="S5.T3.3.2.6.5.5">73.68</span> <span class="ltx_td ltx_align_left" id="S5.T3.3.2.6.5.6">72.14</span></span> <span class="ltx_tr" id="S5.T3.3.2.7.6"> <span class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_b ltx_border_r" id="S5.T3.3.2.7.6.1">10</span> <span class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_b ltx_border_r" id="S5.T3.3.2.7.6.2">1.0</span> <span class="ltx_td ltx_align_left ltx_border_b" id="S5.T3.3.2.7.6.3">63.00</span> <span class="ltx_td ltx_align_left ltx_border_b" id="S5.T3.3.2.7.6.4">83.00</span> <span class="ltx_td ltx_align_left ltx_border_b" id="S5.T3.3.2.7.6.5">72.90</span> <span class="ltx_td ltx_align_left ltx_border_b" id="S5.T3.3.2.7.6.6">71.75</span></span> </span> </span></p> </div> </div> </figure> </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>Sampling Size Investigation (RQ 2)</h3> <div class="ltx_para" id="S5.SS3.p1"> <p class="ltx_p" id="S5.SS3.p1.1">Understanding the influence of sequence length of SBS on the efficacy of user modeling is a pivotal research question. Traditional recommendation systems have largely relied on long-sequence modeling strategies, such as SIM <cite class="ltx_cite ltx_citemacro_cite">Pi et al. (<a class="ltx_ref" href="https://arxiv.org/html/2503.02398v1#bib.bib28" title="">2020</a>)</cite>, which, when applied to datasets like Amazon Books, typically sample 10 interactions to approximate short-term behavioral patterns and 90 interactions for long-term modeling. However, in the context of LLM-UM, prior works such as AgentCF and Agent4Rec have yet to conduct a systematic investigation into the effect of sequence length on user modeling performance.</p> </div> <div class="ltx_para" id="S5.SS3.p2"> <p class="ltx_p" id="S5.SS3.p2.3">To address this gap, we first conduct analysis on PersonaX. As shown in Tables <a class="ltx_ref" href="https://arxiv.org/html/2503.02398v1#S5.T3" title="Table 3 ‣ 5.2 Performance Evaluation (RQ 1) ‣ 5 Experiments ‣ PersonaX: A Recommendation Agent-Oriented User Modeling Framework for Long Behavior Sequence"><span class="ltx_text ltx_ref_tag">3</span></a>, the results indicate that performance generally peaks at intermediate selection ratios or short SBS lengths. For instance, 30% selection ratio for <math alttext="\texttt{CDs}_{\texttt{50}}" class="ltx_Math" display="inline" id="S5.SS3.p2.1.m1.1"><semantics id="S5.SS3.p2.1.m1.1a"><msub id="S5.SS3.p2.1.m1.1.1" xref="S5.SS3.p2.1.m1.1.1.cmml"><mtext class="ltx_mathvariant_monospace" id="S5.SS3.p2.1.m1.1.1.2" xref="S5.SS3.p2.1.m1.1.1.2a.cmml">CDs</mtext><mtext class="ltx_mathvariant_monospace" id="S5.SS3.p2.1.m1.1.1.3" xref="S5.SS3.p2.1.m1.1.1.3a.cmml">50</mtext></msub><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">subscript</csymbol><ci id="S5.SS3.p2.1.m1.1.1.2a.cmml" xref="S5.SS3.p2.1.m1.1.1.2"><mtext class="ltx_mathvariant_monospace" id="S5.SS3.p2.1.m1.1.1.2.cmml" xref="S5.SS3.p2.1.m1.1.1.2">CDs</mtext></ci><ci id="S5.SS3.p2.1.m1.1.1.3a.cmml" xref="S5.SS3.p2.1.m1.1.1.3"><mtext class="ltx_mathvariant_monospace" id="S5.SS3.p2.1.m1.1.1.3.cmml" mathsize="70%" xref="S5.SS3.p2.1.m1.1.1.3">50</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS3.p2.1.m1.1c">\texttt{CDs}_{\texttt{50}}</annotation><annotation encoding="application/x-llamapun" id="S5.SS3.p2.1.m1.1d">CDs start_POSTSUBSCRIPT 50 end_POSTSUBSCRIPT</annotation></semantics></math> and 50% for both <math alttext="\texttt{CDs}_{\texttt{200}}" class="ltx_Math" display="inline" id="S5.SS3.p2.2.m2.1"><semantics id="S5.SS3.p2.2.m2.1a"><msub id="S5.SS3.p2.2.m2.1.1" xref="S5.SS3.p2.2.m2.1.1.cmml"><mtext class="ltx_mathvariant_monospace" id="S5.SS3.p2.2.m2.1.1.2" xref="S5.SS3.p2.2.m2.1.1.2a.cmml">CDs</mtext><mtext class="ltx_mathvariant_monospace" id="S5.SS3.p2.2.m2.1.1.3" xref="S5.SS3.p2.2.m2.1.1.3a.cmml">200</mtext></msub><annotation-xml encoding="MathML-Content" id="S5.SS3.p2.2.m2.1b"><apply id="S5.SS3.p2.2.m2.1.1.cmml" xref="S5.SS3.p2.2.m2.1.1"><csymbol cd="ambiguous" id="S5.SS3.p2.2.m2.1.1.1.cmml" xref="S5.SS3.p2.2.m2.1.1">subscript</csymbol><ci id="S5.SS3.p2.2.m2.1.1.2a.cmml" xref="S5.SS3.p2.2.m2.1.1.2"><mtext class="ltx_mathvariant_monospace" id="S5.SS3.p2.2.m2.1.1.2.cmml" xref="S5.SS3.p2.2.m2.1.1.2">CDs</mtext></ci><ci id="S5.SS3.p2.2.m2.1.1.3a.cmml" xref="S5.SS3.p2.2.m2.1.1.3"><mtext class="ltx_mathvariant_monospace" id="S5.SS3.p2.2.m2.1.1.3.cmml" mathsize="70%" xref="S5.SS3.p2.2.m2.1.1.3">200</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS3.p2.2.m2.1c">\texttt{CDs}_{\texttt{200}}</annotation><annotation encoding="application/x-llamapun" id="S5.SS3.p2.2.m2.1d">CDs start_POSTSUBSCRIPT 200 end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="\texttt{Books}_{\texttt{480}}" class="ltx_Math" display="inline" id="S5.SS3.p2.3.m3.1"><semantics id="S5.SS3.p2.3.m3.1a"><msub id="S5.SS3.p2.3.m3.1.1" xref="S5.SS3.p2.3.m3.1.1.cmml"><mtext class="ltx_mathvariant_monospace" id="S5.SS3.p2.3.m3.1.1.2" xref="S5.SS3.p2.3.m3.1.1.2a.cmml">Books</mtext><mtext class="ltx_mathvariant_monospace" id="S5.SS3.p2.3.m3.1.1.3" xref="S5.SS3.p2.3.m3.1.1.3a.cmml">480</mtext></msub><annotation-xml encoding="MathML-Content" id="S5.SS3.p2.3.m3.1b"><apply id="S5.SS3.p2.3.m3.1.1.cmml" xref="S5.SS3.p2.3.m3.1.1"><csymbol cd="ambiguous" id="S5.SS3.p2.3.m3.1.1.1.cmml" xref="S5.SS3.p2.3.m3.1.1">subscript</csymbol><ci id="S5.SS3.p2.3.m3.1.1.2a.cmml" xref="S5.SS3.p2.3.m3.1.1.2"><mtext class="ltx_mathvariant_monospace" id="S5.SS3.p2.3.m3.1.1.2.cmml" xref="S5.SS3.p2.3.m3.1.1.2">Books</mtext></ci><ci id="S5.SS3.p2.3.m3.1.1.3a.cmml" xref="S5.SS3.p2.3.m3.1.1.3"><mtext class="ltx_mathvariant_monospace" id="S5.SS3.p2.3.m3.1.1.3.cmml" mathsize="70%" xref="S5.SS3.p2.3.m3.1.1.3">480</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS3.p2.3.m3.1c">\texttt{Books}_{\texttt{480}}</annotation><annotation encoding="application/x-llamapun" id="S5.SS3.p2.3.m3.1d">Books start_POSTSUBSCRIPT 480 end_POSTSUBSCRIPT</annotation></semantics></math>. We further examined the performance of three sampling strategies—Random, Recent, and Relevance—under varying sampling sizes, as illustrated in Figure <a class="ltx_ref" href="https://arxiv.org/html/2503.02398v1#S5.F3" title="Figure 3 ‣ 5.1.5 Implementation Details ‣ 5.1 Experimental Setup ‣ 5 Experiments ‣ PersonaX: A Recommendation Agent-Oriented User Modeling Framework for Long Behavior Sequence"><span class="ltx_text ltx_ref_tag">3</span></a>, finding that while initial increases in sampling size improve performance, oversampling eventually leads to performance deterioration. The optimal sampling size varies across datasets. Specifically, for the Relevance method, the ideal size is approximately 3, while the Recent method demonstrates heightened sensitivity to dataset characteristics, with the most recent single behavior often yielding strong results. For the Random method, a sampling size of around 5 is most effective.</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>Hyper-parameter Analysis (RQ3)</h3> <div class="ltx_para" id="S5.SS4.p1"> <p class="ltx_p" id="S5.SS4.p1.8">This section delves into the influence of the hyperparameters <math alttext="\tau" class="ltx_Math" display="inline" id="S5.SS4.p1.1.m1.1"><semantics id="S5.SS4.p1.1.m1.1a"><mi id="S5.SS4.p1.1.m1.1.1" xref="S5.SS4.p1.1.m1.1.1.cmml">τ</mi><annotation-xml encoding="MathML-Content" id="S5.SS4.p1.1.m1.1b"><ci id="S5.SS4.p1.1.m1.1.1.cmml" xref="S5.SS4.p1.1.m1.1.1">𝜏</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.SS4.p1.1.m1.1c">\tau</annotation><annotation encoding="application/x-llamapun" id="S5.SS4.p1.1.m1.1d">italic_τ</annotation></semantics></math> and <math alttext="\alpha" class="ltx_Math" display="inline" id="S5.SS4.p1.2.m2.1"><semantics id="S5.SS4.p1.2.m2.1a"><mi id="S5.SS4.p1.2.m2.1.1" xref="S5.SS4.p1.2.m2.1.1.cmml">α</mi><annotation-xml encoding="MathML-Content" id="S5.SS4.p1.2.m2.1b"><ci id="S5.SS4.p1.2.m2.1.1.cmml" xref="S5.SS4.p1.2.m2.1.1">𝛼</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.SS4.p1.2.m2.1c">\alpha</annotation><annotation encoding="application/x-llamapun" id="S5.SS4.p1.2.m2.1d">italic_α</annotation></semantics></math> on the performance of PersonaX, as they play pivotal roles in shaping the hierarchical clustering and in-cluster behavior selection processes. Specifically, <math alttext="\tau" class="ltx_Math" display="inline" id="S5.SS4.p1.3.m3.1"><semantics id="S5.SS4.p1.3.m3.1a"><mi id="S5.SS4.p1.3.m3.1.1" xref="S5.SS4.p1.3.m3.1.1.cmml">τ</mi><annotation-xml encoding="MathML-Content" id="S5.SS4.p1.3.m3.1b"><ci id="S5.SS4.p1.3.m3.1.1.cmml" xref="S5.SS4.p1.3.m3.1.1">𝜏</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.SS4.p1.3.m3.1c">\tau</annotation><annotation encoding="application/x-llamapun" id="S5.SS4.p1.3.m3.1d">italic_τ</annotation></semantics></math> dictates the granularity of the hierarchical clustering. A larger <math alttext="\tau" class="ltx_Math" display="inline" id="S5.SS4.p1.4.m4.1"><semantics id="S5.SS4.p1.4.m4.1a"><mi id="S5.SS4.p1.4.m4.1.1" xref="S5.SS4.p1.4.m4.1.1.cmml">τ</mi><annotation-xml encoding="MathML-Content" id="S5.SS4.p1.4.m4.1b"><ci id="S5.SS4.p1.4.m4.1.1.cmml" xref="S5.SS4.p1.4.m4.1.1">𝜏</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.SS4.p1.4.m4.1c">\tau</annotation><annotation encoding="application/x-llamapun" id="S5.SS4.p1.4.m4.1d">italic_τ</annotation></semantics></math> value yields coarser clusters, encompassing a broader spectrum of behavioral samples with potentially greater divergence from the cluster centroid. In contrast, a smaller <math alttext="\tau" class="ltx_Math" display="inline" id="S5.SS4.p1.5.m5.1"><semantics id="S5.SS4.p1.5.m5.1a"><mi id="S5.SS4.p1.5.m5.1.1" xref="S5.SS4.p1.5.m5.1.1.cmml">τ</mi><annotation-xml encoding="MathML-Content" id="S5.SS4.p1.5.m5.1b"><ci id="S5.SS4.p1.5.m5.1.1.cmml" xref="S5.SS4.p1.5.m5.1.1">𝜏</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.SS4.p1.5.m5.1c">\tau</annotation><annotation encoding="application/x-llamapun" id="S5.SS4.p1.5.m5.1d">italic_τ</annotation></semantics></math> enforces a more stringent clustering criterion, resulting in finer-grained clusters characterized by higher intra-cluster homogeneity. On the other hand, <math alttext="\alpha" class="ltx_Math" display="inline" id="S5.SS4.p1.6.m6.1"><semantics id="S5.SS4.p1.6.m6.1a"><mi id="S5.SS4.p1.6.m6.1.1" xref="S5.SS4.p1.6.m6.1.1.cmml">α</mi><annotation-xml encoding="MathML-Content" id="S5.SS4.p1.6.m6.1b"><ci id="S5.SS4.p1.6.m6.1.1.cmml" xref="S5.SS4.p1.6.m6.1.1">𝛼</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.SS4.p1.6.m6.1c">\alpha</annotation><annotation encoding="application/x-llamapun" id="S5.SS4.p1.6.m6.1d">italic_α</annotation></semantics></math> modulates the balance between prototypicality and diversity during the in-cluster behavior selection stage. A higher <math alttext="\alpha" class="ltx_Math" display="inline" id="S5.SS4.p1.7.m7.1"><semantics id="S5.SS4.p1.7.m7.1a"><mi id="S5.SS4.p1.7.m7.1.1" xref="S5.SS4.p1.7.m7.1.1.cmml">α</mi><annotation-xml encoding="MathML-Content" id="S5.SS4.p1.7.m7.1b"><ci id="S5.SS4.p1.7.m7.1.1.cmml" xref="S5.SS4.p1.7.m7.1.1">𝛼</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.SS4.p1.7.m7.1c">\alpha</annotation><annotation encoding="application/x-llamapun" id="S5.SS4.p1.7.m7.1d">italic_α</annotation></semantics></math> amplifies the preference for diversity. Conversely, a lower <math alttext="\alpha" class="ltx_Math" display="inline" id="S5.SS4.p1.8.m8.1"><semantics id="S5.SS4.p1.8.m8.1a"><mi id="S5.SS4.p1.8.m8.1.1" xref="S5.SS4.p1.8.m8.1.1.cmml">α</mi><annotation-xml encoding="MathML-Content" id="S5.SS4.p1.8.m8.1b"><ci id="S5.SS4.p1.8.m8.1.1.cmml" xref="S5.SS4.p1.8.m8.1.1">𝛼</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.SS4.p1.8.m8.1c">\alpha</annotation><annotation encoding="application/x-llamapun" id="S5.SS4.p1.8.m8.1d">italic_α</annotation></semantics></math> emphasizes prototypicality, favoring samples that close to cluster centroid. Our empirical analysis, as illustrated in Figure <a class="ltx_ref" href="https://arxiv.org/html/2503.02398v1#S5.F4" title="Figure 4 ‣ 5.4 Hyper-parameter Analysis (RQ3) ‣ 5 Experiments ‣ PersonaX: A Recommendation Agent-Oriented User Modeling Framework for Long Behavior Sequence"><span class="ltx_text ltx_ref_tag">4</span></a>, uncovers nuanced patterns in how these hyperparameters influence the model’s overall performance.</p> </div> <figure class="ltx_figure" id="S5.F4"><img alt="Refer to caption" class="ltx_graphics ltx_centering ltx_img_landscape" height="188" id="S5.F4.g1" src="x4.png" width="401"/> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure">Figure 4: </span>Impact of <math alttext="\tau" class="ltx_Math" display="inline" id="S5.F4.3.m1.1"><semantics id="S5.F4.3.m1.1b"><mi id="S5.F4.3.m1.1.1" xref="S5.F4.3.m1.1.1.cmml">τ</mi><annotation-xml encoding="MathML-Content" id="S5.F4.3.m1.1c"><ci id="S5.F4.3.m1.1.1.cmml" xref="S5.F4.3.m1.1.1">𝜏</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.F4.3.m1.1d">\tau</annotation><annotation encoding="application/x-llamapun" id="S5.F4.3.m1.1e">italic_τ</annotation></semantics></math> and <math alttext="\alpha" class="ltx_Math" display="inline" id="S5.F4.4.m2.1"><semantics id="S5.F4.4.m2.1b"><mi id="S5.F4.4.m2.1.1" xref="S5.F4.4.m2.1.1.cmml">α</mi><annotation-xml encoding="MathML-Content" id="S5.F4.4.m2.1c"><ci id="S5.F4.4.m2.1.1.cmml" xref="S5.F4.4.m2.1.1">𝛼</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.F4.4.m2.1d">\alpha</annotation><annotation encoding="application/x-llamapun" id="S5.F4.4.m2.1e">italic_α</annotation></semantics></math> on PersonaX.</figcaption> </figure> <div class="ltx_para" id="S5.SS4.p2"> <p class="ltx_p" id="S5.SS4.p2.8">Key findings include: (1) At low selection ratios (e.g., <math alttext="0.1" class="ltx_Math" display="inline" id="S5.SS4.p2.1.m1.1"><semantics id="S5.SS4.p2.1.m1.1a"><mn 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centroid-focused sampling; (2) Higher <math alttext="\alpha" class="ltx_Math" display="inline" id="S5.SS4.p2.3.m3.1"><semantics id="S5.SS4.p2.3.m3.1a"><mi id="S5.SS4.p2.3.m3.1.1" xref="S5.SS4.p2.3.m3.1.1.cmml">α</mi><annotation-xml encoding="MathML-Content" id="S5.SS4.p2.3.m3.1b"><ci id="S5.SS4.p2.3.m3.1.1.cmml" xref="S5.SS4.p2.3.m3.1.1">𝛼</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.SS4.p2.3.m3.1c">\alpha</annotation><annotation encoding="application/x-llamapun" id="S5.SS4.p2.3.m3.1d">italic_α</annotation></semantics></math> values (e.g., 1.06–1.08) significantly improve performance at large ratios (0.5–0.9) underscores the efficacy of incorporating diverse samples; (3) Optimal <math alttext="\tau" class="ltx_Math" display="inline" id="S5.SS4.p2.4.m4.1"><semantics id="S5.SS4.p2.4.m4.1a"><mi id="S5.SS4.p2.4.m4.1.1" xref="S5.SS4.p2.4.m4.1.1.cmml">τ</mi><annotation-xml encoding="MathML-Content" id="S5.SS4.p2.4.m4.1b"><ci id="S5.SS4.p2.4.m4.1.1.cmml" xref="S5.SS4.p2.4.m4.1.1">𝜏</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.SS4.p2.4.m4.1c">\tau</annotation><annotation encoding="application/x-llamapun" id="S5.SS4.p2.4.m4.1d">italic_τ</annotation></semantics></math> depends on cluster scope (e.g., constrained cluster scope with <math alttext="\tau=0.5" class="ltx_Math" display="inline" id="S5.SS4.p2.5.m5.1"><semantics id="S5.SS4.p2.5.m5.1a"><mrow id="S5.SS4.p2.5.m5.1.1" xref="S5.SS4.p2.5.m5.1.1.cmml"><mi id="S5.SS4.p2.5.m5.1.1.2" xref="S5.SS4.p2.5.m5.1.1.2.cmml">τ</mi><mo id="S5.SS4.p2.5.m5.1.1.1" xref="S5.SS4.p2.5.m5.1.1.1.cmml">=</mo><mn id="S5.SS4.p2.5.m5.1.1.3" xref="S5.SS4.p2.5.m5.1.1.3.cmml">0.5</mn></mrow><annotation-xml encoding="MathML-Content" id="S5.SS4.p2.5.m5.1b"><apply id="S5.SS4.p2.5.m5.1.1.cmml" xref="S5.SS4.p2.5.m5.1.1"><eq id="S5.SS4.p2.5.m5.1.1.1.cmml" xref="S5.SS4.p2.5.m5.1.1.1"></eq><ci id="S5.SS4.p2.5.m5.1.1.2.cmml" xref="S5.SS4.p2.5.m5.1.1.2">𝜏</ci><cn id="S5.SS4.p2.5.m5.1.1.3.cmml" type="float" xref="S5.SS4.p2.5.m5.1.1.3">0.5</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS4.p2.5.m5.1c">\tau=0.5</annotation><annotation encoding="application/x-llamapun" id="S5.SS4.p2.5.m5.1d">italic_τ = 0.5</annotation></semantics></math> pairs with <math alttext="\alpha=1.08" class="ltx_Math" display="inline" id="S5.SS4.p2.6.m6.1"><semantics id="S5.SS4.p2.6.m6.1a"><mrow id="S5.SS4.p2.6.m6.1.1" xref="S5.SS4.p2.6.m6.1.1.cmml"><mi id="S5.SS4.p2.6.m6.1.1.2" xref="S5.SS4.p2.6.m6.1.1.2.cmml">α</mi><mo id="S5.SS4.p2.6.m6.1.1.1" xref="S5.SS4.p2.6.m6.1.1.1.cmml">=</mo><mn id="S5.SS4.p2.6.m6.1.1.3" xref="S5.SS4.p2.6.m6.1.1.3.cmml">1.08</mn></mrow><annotation-xml encoding="MathML-Content" id="S5.SS4.p2.6.m6.1b"><apply id="S5.SS4.p2.6.m6.1.1.cmml" xref="S5.SS4.p2.6.m6.1.1"><eq id="S5.SS4.p2.6.m6.1.1.1.cmml" xref="S5.SS4.p2.6.m6.1.1.1"></eq><ci id="S5.SS4.p2.6.m6.1.1.2.cmml" xref="S5.SS4.p2.6.m6.1.1.2">𝛼</ci><cn id="S5.SS4.p2.6.m6.1.1.3.cmml" type="float" xref="S5.SS4.p2.6.m6.1.1.3">1.08</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS4.p2.6.m6.1c">\alpha=1.08</annotation><annotation encoding="application/x-llamapun" id="S5.SS4.p2.6.m6.1d">italic_α = 1.08</annotation></semantics></math>, while the broader cluster scope with <math alttext="\tau=0.7" class="ltx_Math" display="inline" id="S5.SS4.p2.7.m7.1"><semantics id="S5.SS4.p2.7.m7.1a"><mrow id="S5.SS4.p2.7.m7.1.1" xref="S5.SS4.p2.7.m7.1.1.cmml"><mi id="S5.SS4.p2.7.m7.1.1.2" xref="S5.SS4.p2.7.m7.1.1.2.cmml">τ</mi><mo id="S5.SS4.p2.7.m7.1.1.1" xref="S5.SS4.p2.7.m7.1.1.1.cmml">=</mo><mn id="S5.SS4.p2.7.m7.1.1.3" xref="S5.SS4.p2.7.m7.1.1.3.cmml">0.7</mn></mrow><annotation-xml encoding="MathML-Content" id="S5.SS4.p2.7.m7.1b"><apply id="S5.SS4.p2.7.m7.1.1.cmml" xref="S5.SS4.p2.7.m7.1.1"><eq id="S5.SS4.p2.7.m7.1.1.1.cmml" xref="S5.SS4.p2.7.m7.1.1.1"></eq><ci id="S5.SS4.p2.7.m7.1.1.2.cmml" xref="S5.SS4.p2.7.m7.1.1.2">𝜏</ci><cn id="S5.SS4.p2.7.m7.1.1.3.cmml" type="float" xref="S5.SS4.p2.7.m7.1.1.3">0.7</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS4.p2.7.m7.1c">\tau=0.7</annotation><annotation encoding="application/x-llamapun" id="S5.SS4.p2.7.m7.1d">italic_τ = 0.7</annotation></semantics></math> favors <math alttext="\alpha=1.06" class="ltx_Math" display="inline" id="S5.SS4.p2.8.m8.1"><semantics id="S5.SS4.p2.8.m8.1a"><mrow id="S5.SS4.p2.8.m8.1.1" xref="S5.SS4.p2.8.m8.1.1.cmml"><mi id="S5.SS4.p2.8.m8.1.1.2" xref="S5.SS4.p2.8.m8.1.1.2.cmml">α</mi><mo id="S5.SS4.p2.8.m8.1.1.1" xref="S5.SS4.p2.8.m8.1.1.1.cmml">=</mo><mn id="S5.SS4.p2.8.m8.1.1.3" xref="S5.SS4.p2.8.m8.1.1.3.cmml">1.06</mn></mrow><annotation-xml encoding="MathML-Content" id="S5.SS4.p2.8.m8.1b"><apply id="S5.SS4.p2.8.m8.1.1.cmml" xref="S5.SS4.p2.8.m8.1.1"><eq id="S5.SS4.p2.8.m8.1.1.1.cmml" xref="S5.SS4.p2.8.m8.1.1.1"></eq><ci id="S5.SS4.p2.8.m8.1.1.2.cmml" xref="S5.SS4.p2.8.m8.1.1.2">𝛼</ci><cn id="S5.SS4.p2.8.m8.1.1.3.cmml" type="float" xref="S5.SS4.p2.8.m8.1.1.3">1.06</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.SS4.p2.8.m8.1c">\alpha=1.06</annotation><annotation encoding="application/x-llamapun" id="S5.SS4.p2.8.m8.1d">italic_α = 1.06</annotation></semantics></math> avoid overemphasizing highly diverse samples). (4) PersonaX exhibits robustness, with worst-case performance (71.6) nearly matching the best performance of the relevance baseline (71.86). In Appendix <a class="ltx_ref" href="https://arxiv.org/html/2503.02398v1#A3" title="Appendix C Hyper-parameter Analysis and Sampling Process Visualization ‣ PersonaX: A Recommendation Agent-Oriented User Modeling Framework for Long Behavior Sequence"><span class="ltx_text ltx_ref_tag">C</span></a>, we present a detailed illustration alongside a visualization analysis of the sampling process.</p> </div> </section> </section> <section class="ltx_section" id="S6"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">6 </span>Related Works</h2> <section class="ltx_subsection" id="S6.SS1"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">6.1 </span>Large Language Model for User Modeling</h3> <div class="ltx_para" id="S6.SS1.p1"> <p class="ltx_p" id="S6.SS1.p1.1">User Modeling (UM) aims to extract valuable insights and patterns from user-generated content (UGC), and Large Language Models (LLMs) excel in characterizing user personalities and discerning preferences. Leveraging LLMs for UM has gained increasing attention, and the generated textual personas can be applied to downstream personalization tasks <cite class="ltx_cite ltx_citemacro_cite">Xu et al. (<a class="ltx_ref" href="https://arxiv.org/html/2503.02398v1#bib.bib53" title="">2024a</a>); Mei and Zhang (<a class="ltx_ref" href="https://arxiv.org/html/2503.02398v1#bib.bib21" title="">2023</a>); Xu et al. (<a class="ltx_ref" href="https://arxiv.org/html/2503.02398v1#bib.bib50" title="">2023</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.02398v1#bib.bib54" title="">2024b</a>)</cite>. For example, ONCE <cite class="ltx_cite ltx_citemacro_cite">Liu et al. (<a class="ltx_ref" href="https://arxiv.org/html/2503.02398v1#bib.bib18" title="">2024</a>)</cite> utilizes ChatGPT to infer users’ preferred topics and regions, enhancing click-through rate prediction with these generated profiles. Kang et al. <cite class="ltx_cite ltx_citemacro_cite">Kang et al. (<a class="ltx_ref" href="https://arxiv.org/html/2503.02398v1#bib.bib15" title="">2023</a>)</cite> enable LLMs to comprehend user preferences from behavior history to predict user ratings. LLMRec <cite class="ltx_cite ltx_citemacro_cite">Lyu et al. (<a class="ltx_ref" href="https://arxiv.org/html/2503.02398v1#bib.bib20" title="">2024</a>)</cite> identifies limitations in directly using raw item descriptions, which often fail to capture the subtle nuances of user preferences. To address this, it employs four distinct text enrichment strategies to enhance the input and improve recommendation performance. LLMRank <cite class="ltx_cite ltx_citemacro_cite">Hou et al. (<a class="ltx_ref" href="https://arxiv.org/html/2503.02398v1#bib.bib12" title="">2024</a>)</cite> introduces specialized prompting and bootstrapping techniques that incorporate user interaction histories, effectively aligning with user intent. Moreover, two prominent strategies—Summarization and Reflection—have been widely adopted in leading agent recommendation frameworks, such as Agent4Rec <cite class="ltx_cite ltx_citemacro_cite">Zhang et al. (<a class="ltx_ref" href="https://arxiv.org/html/2503.02398v1#bib.bib56" title="">2024a</a>)</cite>, RecAgent <cite class="ltx_cite ltx_citemacro_cite">Wang et al. (<a class="ltx_ref" href="https://arxiv.org/html/2503.02398v1#bib.bib48" title="">2024</a>)</cite>, and AgentCF <cite class="ltx_cite ltx_citemacro_cite">Zhang et al. (<a class="ltx_ref" href="https://arxiv.org/html/2503.02398v1#bib.bib57" title="">2024b</a>)</cite>. Summarization focuses on distilling user behaviors, while reflection emphasizes iterative learning from interactions.</p> </div> <div class="ltx_para" id="S6.SS1.p2"> <p class="ltx_p" id="S6.SS1.p2.1">However, no research has focused on the performance of LLMs when handling extensive UGC, nor has any LLM-UM method been proficient at efficiently and accurately modeling user personas from long behavior sequences. We are the first to address this gap and introduce PersonaX.</p> </div> </section> <section class="ltx_subsection" id="S6.SS2"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">6.2 </span>Personalized Agents</h3> <div class="ltx_para" id="S6.SS2.p1"> <p class="ltx_p" id="S6.SS2.p1.1">LLM-driven agents have gained prominence for their autonomous decision-making, tool utilization <cite class="ltx_cite ltx_citemacro_cite">Yang et al. (<a class="ltx_ref" href="https://arxiv.org/html/2503.02398v1#bib.bib55" title="">2023</a>); Qin et al. (<a class="ltx_ref" href="https://arxiv.org/html/2503.02398v1#bib.bib30" title="">2023</a>); Xu et al. (<a class="ltx_ref" href="https://arxiv.org/html/2503.02398v1#bib.bib51" title="">2025a</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.02398v1#bib.bib52" title="">b</a>)</cite>, and adaptive intelligence. Recent advances enable personalized agents through encoded personalities <cite class="ltx_cite ltx_citemacro_cite">Rao et al. (<a class="ltx_ref" href="https://arxiv.org/html/2503.02398v1#bib.bib31" title="">2023</a>)</cite>, backgrounds, and behavioral traits in prompts. Such persona-driven frameworks enhance user engagement through human-like interactions <cite class="ltx_cite ltx_citemacro_cite">Sun et al. (<a class="ltx_ref" href="https://arxiv.org/html/2503.02398v1#bib.bib43" title="">2024</a>)</cite>, with applications like CharacterAgent <cite class="ltx_cite ltx_citemacro_cite">Shao et al. (<a class="ltx_ref" href="https://arxiv.org/html/2503.02398v1#bib.bib38" title="">2023</a>)</cite> demonstrating consistent persona emulation of historical figures for immersive simulations. The personalization of agent also enable the simulations of social dynamics <cite class="ltx_cite ltx_citemacro_cite">Park et al. (<a class="ltx_ref" href="https://arxiv.org/html/2503.02398v1#bib.bib24" title="">2023</a>)</cite>, competition <cite class="ltx_cite ltx_citemacro_cite"><a class="ltx_ref" href="https://arxiv.org/html/2503.02398v1#bib.bib60" title="">Zhao et al. </a></cite>, and collaboration <cite class="ltx_cite ltx_citemacro_cite">Tran et al. (<a class="ltx_ref" href="https://arxiv.org/html/2503.02398v1#bib.bib46" title="">2025</a>)</cite>.</p> </div> <div class="ltx_para" id="S6.SS2.p2"> <p class="ltx_p" id="S6.SS2.p2.1">However, recommendation agents face distinct challenges: Unlike predefined personas, user preferences in recommendation contexts are implicit and behaviorally embedded rather than verbally expressed. This creates alignment difficulties between agent decisions and users’ latent preferences. The primary objective of PersonaX is to develop a highly accurate and realistic user modeling method, enabling instruction-based agents to consistently simulate and align with the decision-making behaviors of the users they surrogate.</p> </div> </section> </section> <section class="ltx_section" id="S7"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">7 </span>Conclusion</h2> <div class="ltx_para" id="S7.p1"> <p class="ltx_p" id="S7.p1.1">In this study, we present PersonaX, an innovative LLM-UM framework oriented for agent recommendation specially designed for processing long user behavior sequences. PersonaX utilizes only 30%-50% of the user’s historical behavior data and strategically select high-quality sub-behavior sequences of short length (often <math alttext="<5" class="ltx_Math" display="inline" id="S7.p1.1.m1.1"><semantics id="S7.p1.1.m1.1a"><mrow id="S7.p1.1.m1.1.1" xref="S7.p1.1.m1.1.1.cmml"><mi id="S7.p1.1.m1.1.1.2" xref="S7.p1.1.m1.1.1.2.cmml"></mi><mo id="S7.p1.1.m1.1.1.1" xref="S7.p1.1.m1.1.1.1.cmml"><</mo><mn id="S7.p1.1.m1.1.1.3" xref="S7.p1.1.m1.1.1.3.cmml">5</mn></mrow><annotation-xml encoding="MathML-Content" id="S7.p1.1.m1.1b"><apply id="S7.p1.1.m1.1.1.cmml" xref="S7.p1.1.m1.1.1"><lt id="S7.p1.1.m1.1.1.1.cmml" xref="S7.p1.1.m1.1.1.1"></lt><csymbol cd="latexml" id="S7.p1.1.m1.1.1.2.cmml" xref="S7.p1.1.m1.1.1.2">absent</csymbol><cn id="S7.p1.1.m1.1.1.3.cmml" type="integer" xref="S7.p1.1.m1.1.1.3">5</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S7.p1.1.m1.1c"><5</annotation><annotation encoding="application/x-llamapun" id="S7.p1.1.m1.1d">< 5</annotation></semantics></math>) for generating broad spectrum of persona snippets offline. When PersonaX integrated into existing agent recommendation methods, such as AgentCF and Agent4Rec, PersonaX delivers substantial performance gains—ranging from 3% to 11% over AgentCF, and an impressive 10% to 50% improvement over Agent4Rec. In terms of online efficiency, PersonaX outperforms the best baseline method, Summarization + Relevance Sampling, by halving the computation time. We believe that PersonaX significantly facilitate the agent recommendation in predictive accuracy and inference efficiency.</p> </div> </section> <section class="ltx_section" id="Sx1"> <h2 class="ltx_title ltx_title_section">Limitations</h2> <div class="ltx_para" id="Sx1.p1"> <p class="ltx_p" id="Sx1.p1.1">While PersonaX effectively tackles the challenge of modeling user behavior over extended sequences in LLM-based user modeling, its performance in real-world streaming data scenarios remains unexplored. This presents a promising opportunity for future enhancements. A fundamental characteristic of PersonaX lies in its offline generation of multiple personas, capturing diverse aspects of user preferences. This design facilitates long-horizon modeling, where personas encapsulate user interests over extended periods and maintain their effectiveness for prolonged use, surpassing approaches (e.g., AgentCF) that depend on recent-sampling strategies and require frequent profile updates. However, an exciting direction for future work involves exploring the optimal duration for which these precomputed personas retain their efficacy in online deployment. Understanding the dynamics of performance degradation over time can inform strategies for adaptive persona updates. Thus, an unresolved work is the integration of an incremental learning mechanism within PersonaX to continuously refine and update user representations, which presents a compelling opportunity to enhance its responsiveness and robustness to evolving user interests.</p> </div> </section> <section class="ltx_section" id="Sx2"> <h2 class="ltx_title ltx_title_section">Ethics</h2> <div class="ltx_para" id="Sx2.p1"> <p class="ltx_p" id="Sx2.p1.1">Our study models user profiles based on historical behavioral data. We use publicly available datasets collected under standard ethical protocols and strictly adhere to their intended research use. PersonaX is designed solely for academic purposes, ensuring compliance with data access conditions. Our datasets are either pre-anonymized. We prevent re-identification and check for offensive content, ensuring responsible and unbiased profiling. 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Association for Computing Machinery. </span> </li> </ul> </section> <div class="ltx_pagination ltx_role_newpage"></div> <nav class="ltx_TOC ltx_list_toc ltx_toc_toc"><h6 class="ltx_title ltx_title_contents">Contents</h6> <ol class="ltx_toclist"> <li class="ltx_tocentry ltx_tocentry_section"><a class="ltx_ref" href="https://arxiv.org/html/2503.02398v1#S1" title="In PersonaX: A Recommendation Agent-Oriented User Modeling Framework for Long Behavior Sequence"><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/2503.02398v1#S2" title="In PersonaX: A Recommendation Agent-Oriented User Modeling Framework for Long Behavior Sequence"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">2 </span>Preliminary</span></a> <ol class="ltx_toclist ltx_toclist_section"> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.02398v1#S2.SS1" title="In 2 Preliminary ‣ PersonaX: A Recommendation Agent-Oriented User Modeling Framework for Long Behavior Sequence"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">2.1 </span>User Modeling</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.02398v1#S2.SS2" title="In 2 Preliminary ‣ PersonaX: A Recommendation Agent-Oriented User Modeling Framework for Long Behavior Sequence"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">2.2 </span>Sub-Behavior Sequence (SBS) Selection.</span></a></li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_section"> <a class="ltx_ref" href="https://arxiv.org/html/2503.02398v1#S3" title="In PersonaX: A Recommendation Agent-Oriented User Modeling Framework for Long Behavior Sequence"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">3 </span>Method</span></a> <ol class="ltx_toclist ltx_toclist_section"> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.02398v1#S3.SS1" title="In 3 Method ‣ PersonaX: A Recommendation Agent-Oriented User Modeling Framework for Long Behavior Sequence"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">3.1 </span>Behavior Clustering</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.02398v1#S3.SS2" title="In 3 Method ‣ PersonaX: A Recommendation Agent-Oriented User Modeling Framework for Long Behavior Sequence"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">3.2 </span>Sampling Budget Allocation</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.02398v1#S3.SS3" title="In 3 Method ‣ PersonaX: A Recommendation Agent-Oriented User Modeling Framework for Long Behavior Sequence"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">3.3 </span>In-Cluster Selection</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.02398v1#S3.SS4" title="In 3 Method ‣ PersonaX: A Recommendation Agent-Oriented User Modeling Framework for Long Behavior Sequence"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">3.4 </span>Offline Profiling and Online Selection</span></a></li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_section"><a class="ltx_ref" href="https://arxiv.org/html/2503.02398v1#S4" title="In PersonaX: A Recommendation Agent-Oriented User Modeling Framework for Long Behavior Sequence"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">4 </span>Efficiency Analysis</span></a></li> <li class="ltx_tocentry ltx_tocentry_section"> <a class="ltx_ref" href="https://arxiv.org/html/2503.02398v1#S5" title="In PersonaX: A Recommendation Agent-Oriented User Modeling Framework for Long Behavior Sequence"><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/2503.02398v1#S5.SS1" title="In 5 Experiments ‣ PersonaX: A Recommendation Agent-Oriented User Modeling Framework for Long Behavior Sequence"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">5.1 </span>Experimental Setup</span></a> <ol class="ltx_toclist ltx_toclist_subsection"> <li class="ltx_tocentry ltx_tocentry_subsubsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.02398v1#S5.SS1.SSS1" title="In 5.1 Experimental Setup ‣ 5 Experiments ‣ PersonaX: A Recommendation Agent-Oriented User Modeling Framework for Long Behavior Sequence"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">5.1.1 </span>Datasets</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsubsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.02398v1#S5.SS1.SSS2" title="In 5.1 Experimental Setup ‣ 5 Experiments ‣ PersonaX: A Recommendation Agent-Oriented User Modeling Framework for Long Behavior Sequence"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">5.1.2 </span>Evaluation</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsubsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.02398v1#S5.SS1.SSS3" title="In 5.1 Experimental Setup ‣ 5 Experiments ‣ PersonaX: A Recommendation Agent-Oriented User Modeling Framework for Long Behavior Sequence"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">5.1.3 </span>Baseline Comparison</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsubsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.02398v1#S5.SS1.SSS4" title="In 5.1 Experimental Setup ‣ 5 Experiments ‣ PersonaX: A Recommendation Agent-Oriented User Modeling Framework for Long Behavior Sequence"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">5.1.4 </span>Backbone Agent Recommendation</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsubsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.02398v1#S5.SS1.SSS5" title="In 5.1 Experimental Setup ‣ 5 Experiments ‣ PersonaX: A Recommendation Agent-Oriented User Modeling Framework for Long Behavior Sequence"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">5.1.5 </span>Implementation Details</span></a></li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.02398v1#S5.SS2" title="In 5 Experiments ‣ PersonaX: A Recommendation Agent-Oriented User Modeling Framework for Long Behavior Sequence"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">5.2 </span>Performance Evaluation (RQ 1)</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.02398v1#S5.SS3" title="In 5 Experiments ‣ PersonaX: A Recommendation Agent-Oriented User Modeling Framework for Long Behavior Sequence"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">5.3 </span>Sampling Size Investigation (RQ 2)</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.02398v1#S5.SS4" title="In 5 Experiments ‣ PersonaX: A Recommendation Agent-Oriented User Modeling Framework for Long Behavior Sequence"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">5.4 </span>Hyper-parameter Analysis (RQ3)</span></a></li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_section"> <a class="ltx_ref" href="https://arxiv.org/html/2503.02398v1#S6" title="In PersonaX: A Recommendation Agent-Oriented User Modeling Framework for Long Behavior Sequence"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">6 </span>Related Works</span></a> <ol class="ltx_toclist ltx_toclist_section"> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.02398v1#S6.SS1" title="In 6 Related Works ‣ PersonaX: A Recommendation Agent-Oriented User Modeling Framework for Long Behavior Sequence"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">6.1 </span>Large Language Model for User Modeling</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.02398v1#S6.SS2" title="In 6 Related Works ‣ PersonaX: A Recommendation Agent-Oriented User Modeling Framework for Long Behavior Sequence"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">6.2 </span>Personalized Agents</span></a></li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_section"><a class="ltx_ref" href="https://arxiv.org/html/2503.02398v1#S7" title="In PersonaX: A Recommendation Agent-Oriented User Modeling Framework for Long Behavior Sequence"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">7 </span>Conclusion</span></a></li> <li class="ltx_tocentry ltx_tocentry_appendix"><a class="ltx_ref" href="https://arxiv.org/html/2503.02398v1#A1" title="In PersonaX: A Recommendation Agent-Oriented User Modeling Framework for Long Behavior Sequence"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">A </span>Datasets</span></a></li> <li class="ltx_tocentry ltx_tocentry_appendix"><a class="ltx_ref" href="https://arxiv.org/html/2503.02398v1#A2" title="In PersonaX: A Recommendation Agent-Oriented User Modeling Framework for Long Behavior Sequence"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">B </span>Backbone Methods</span></a></li> <li class="ltx_tocentry ltx_tocentry_appendix"><a class="ltx_ref" href="https://arxiv.org/html/2503.02398v1#A3" title="In PersonaX: A Recommendation Agent-Oriented User Modeling Framework for Long Behavior Sequence"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">C </span>Hyper-parameter Analysis and Sampling Process Visualization</span></a></li> <li class="ltx_tocentry ltx_tocentry_appendix"> <a class="ltx_ref" href="https://arxiv.org/html/2503.02398v1#A4" title="In PersonaX: A Recommendation Agent-Oriented User Modeling Framework for Long Behavior Sequence"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">D </span>Details about In-Cluter Selection</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/2503.02398v1#A4.SS1" title="In Appendix D Details about In-Cluter Selection ‣ PersonaX: A Recommendation Agent-Oriented User Modeling Framework for Long Behavior Sequence"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">D.1 </span>Prototypicality and Diversity Scoring</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.02398v1#A4.SS2" title="In Appendix D Details about In-Cluter Selection ‣ PersonaX: A Recommendation Agent-Oriented User Modeling Framework for Long Behavior Sequence"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">D.2 </span>Design Rationale</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.02398v1#A4.SS3" title="In Appendix D Details about In-Cluter Selection ‣ PersonaX: A Recommendation Agent-Oriented User Modeling Framework for Long Behavior Sequence"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">D.3 </span>Broader Implications</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.02398v1#A4.SS4" title="In Appendix D Details about In-Cluter Selection ‣ PersonaX: A Recommendation Agent-Oriented User Modeling Framework for Long Behavior Sequence"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">D.4 </span>Visualization Explanation</span></a></li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_appendix"><a class="ltx_ref" href="https://arxiv.org/html/2503.02398v1#A5" title="In PersonaX: A Recommendation Agent-Oriented User Modeling Framework for Long Behavior Sequence"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">E </span>Case Study</span></a></li> <li class="ltx_tocentry ltx_tocentry_appendix"><a class="ltx_ref" href="https://arxiv.org/html/2503.02398v1#A6" title="In PersonaX: A Recommendation Agent-Oriented User Modeling Framework for Long Behavior Sequence"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">F </span>Prompt Templates</span></a></li> </ol></nav> <section class="ltx_section" id="Sx3"> <h2 class="ltx_title ltx_title_section">APPENDIX</h2> <div class="ltx_pagination ltx_role_newpage"></div> <figure class="ltx_table" id="A0.T4"> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_table">Table 4: </span>Summary of preprocessed subset statistics. "Avg.L" represents the average length of user behavior sequences.</figcaption> <table class="ltx_tabular ltx_centering ltx_guessed_headers ltx_align_middle" id="A0.T4.3"> <thead class="ltx_thead"> <tr class="ltx_tr" id="A0.T4.3.4.1"> <th class="ltx_td ltx_align_left ltx_th ltx_th_column ltx_th_row ltx_border_tt" id="A0.T4.3.4.1.1"><span class="ltx_text ltx_font_bold" id="A0.T4.3.4.1.1.1">Subsets</span></th> <th class="ltx_td ltx_align_left ltx_th ltx_th_column ltx_th_row ltx_border_tt" id="A0.T4.3.4.1.2"><span class="ltx_text ltx_font_bold" id="A0.T4.3.4.1.2.1">#Users</span></th> <th class="ltx_td ltx_align_left ltx_th ltx_th_column ltx_border_tt" id="A0.T4.3.4.1.3"><span class="ltx_text ltx_font_bold" id="A0.T4.3.4.1.3.1">#Items</span></th> <th class="ltx_td ltx_align_left ltx_th ltx_th_column ltx_border_tt" id="A0.T4.3.4.1.4"><span class="ltx_text ltx_font_bold" id="A0.T4.3.4.1.4.1">#Inters</span></th> <th class="ltx_td ltx_align_left ltx_th ltx_th_column ltx_border_tt" id="A0.T4.3.4.1.5"><span class="ltx_text ltx_font_bold" id="A0.T4.3.4.1.5.1">Sparsity</span></th> <th class="ltx_td ltx_align_left ltx_th ltx_th_column ltx_border_tt" id="A0.T4.3.4.1.6"><span class="ltx_text ltx_font_bold" id="A0.T4.3.4.1.6.1">Avg.L</span></th> </tr> </thead> <tbody class="ltx_tbody"> <tr class="ltx_tr" id="A0.T4.1.1"> <th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_t" id="A0.T4.1.1.1"><math alttext="\texttt{CDs}_{\texttt{50}}" class="ltx_Math" display="inline" id="A0.T4.1.1.1.m1.1"><semantics id="A0.T4.1.1.1.m1.1a"><msub id="A0.T4.1.1.1.m1.1.1" xref="A0.T4.1.1.1.m1.1.1.cmml"><mtext class="ltx_mathvariant_monospace" id="A0.T4.1.1.1.m1.1.1.2" xref="A0.T4.1.1.1.m1.1.1.2a.cmml">CDs</mtext><mtext class="ltx_mathvariant_monospace" id="A0.T4.1.1.1.m1.1.1.3" xref="A0.T4.1.1.1.m1.1.1.3a.cmml">50</mtext></msub><annotation-xml encoding="MathML-Content" id="A0.T4.1.1.1.m1.1b"><apply id="A0.T4.1.1.1.m1.1.1.cmml" xref="A0.T4.1.1.1.m1.1.1"><csymbol cd="ambiguous" id="A0.T4.1.1.1.m1.1.1.1.cmml" xref="A0.T4.1.1.1.m1.1.1">subscript</csymbol><ci id="A0.T4.1.1.1.m1.1.1.2a.cmml" xref="A0.T4.1.1.1.m1.1.1.2"><mtext class="ltx_mathvariant_monospace" id="A0.T4.1.1.1.m1.1.1.2.cmml" xref="A0.T4.1.1.1.m1.1.1.2">CDs</mtext></ci><ci id="A0.T4.1.1.1.m1.1.1.3a.cmml" xref="A0.T4.1.1.1.m1.1.1.3"><mtext class="ltx_mathvariant_monospace" id="A0.T4.1.1.1.m1.1.1.3.cmml" mathsize="70%" xref="A0.T4.1.1.1.m1.1.1.3">50</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A0.T4.1.1.1.m1.1c">\texttt{CDs}_{\texttt{50}}</annotation><annotation encoding="application/x-llamapun" id="A0.T4.1.1.1.m1.1d">CDs start_POSTSUBSCRIPT 50 end_POSTSUBSCRIPT</annotation></semantics></math></th> <th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_t" id="A0.T4.1.1.2">100</th> <td class="ltx_td ltx_align_left ltx_border_t" id="A0.T4.1.1.3">4,899</td> <td class="ltx_td ltx_align_left ltx_border_t" id="A0.T4.1.1.4">5,000</td> <td class="ltx_td ltx_align_left ltx_border_t" id="A0.T4.1.1.5">98.97%</td> <td class="ltx_td ltx_align_left ltx_border_t" id="A0.T4.1.1.6">50.00</td> </tr> <tr class="ltx_tr" id="A0.T4.2.2"> <th class="ltx_td ltx_align_left ltx_th ltx_th_row" id="A0.T4.2.2.1"><math alttext="\texttt{CDs}_{\texttt{200}}" class="ltx_Math" display="inline" id="A0.T4.2.2.1.m1.1"><semantics id="A0.T4.2.2.1.m1.1a"><msub id="A0.T4.2.2.1.m1.1.1" xref="A0.T4.2.2.1.m1.1.1.cmml"><mtext class="ltx_mathvariant_monospace" id="A0.T4.2.2.1.m1.1.1.2" xref="A0.T4.2.2.1.m1.1.1.2a.cmml">CDs</mtext><mtext class="ltx_mathvariant_monospace" id="A0.T4.2.2.1.m1.1.1.3" xref="A0.T4.2.2.1.m1.1.1.3a.cmml">200</mtext></msub><annotation-xml encoding="MathML-Content" id="A0.T4.2.2.1.m1.1b"><apply id="A0.T4.2.2.1.m1.1.1.cmml" xref="A0.T4.2.2.1.m1.1.1"><csymbol cd="ambiguous" id="A0.T4.2.2.1.m1.1.1.1.cmml" xref="A0.T4.2.2.1.m1.1.1">subscript</csymbol><ci id="A0.T4.2.2.1.m1.1.1.2a.cmml" xref="A0.T4.2.2.1.m1.1.1.2"><mtext class="ltx_mathvariant_monospace" id="A0.T4.2.2.1.m1.1.1.2.cmml" xref="A0.T4.2.2.1.m1.1.1.2">CDs</mtext></ci><ci id="A0.T4.2.2.1.m1.1.1.3a.cmml" xref="A0.T4.2.2.1.m1.1.1.3"><mtext class="ltx_mathvariant_monospace" id="A0.T4.2.2.1.m1.1.1.3.cmml" mathsize="70%" xref="A0.T4.2.2.1.m1.1.1.3">200</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A0.T4.2.2.1.m1.1c">\texttt{CDs}_{\texttt{200}}</annotation><annotation encoding="application/x-llamapun" id="A0.T4.2.2.1.m1.1d">CDs start_POSTSUBSCRIPT 200 end_POSTSUBSCRIPT</annotation></semantics></math></th> <th class="ltx_td ltx_align_left ltx_th ltx_th_row" id="A0.T4.2.2.2">1000</th> <td class="ltx_td ltx_align_left" id="A0.T4.2.2.3">101,902</td> <td class="ltx_td ltx_align_left" id="A0.T4.2.2.4">200,336</td> <td class="ltx_td ltx_align_left" id="A0.T4.2.2.5">99.80%</td> <td class="ltx_td ltx_align_left" id="A0.T4.2.2.6">200.34</td> </tr> <tr class="ltx_tr" id="A0.T4.3.3"> <th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_bb" id="A0.T4.3.3.1"><math alttext="\texttt{Books}_{\texttt{480}}" class="ltx_Math" display="inline" id="A0.T4.3.3.1.m1.1"><semantics id="A0.T4.3.3.1.m1.1a"><msub id="A0.T4.3.3.1.m1.1.1" xref="A0.T4.3.3.1.m1.1.1.cmml"><mtext class="ltx_mathvariant_monospace" id="A0.T4.3.3.1.m1.1.1.2" xref="A0.T4.3.3.1.m1.1.1.2a.cmml">Books</mtext><mtext class="ltx_mathvariant_monospace" id="A0.T4.3.3.1.m1.1.1.3" xref="A0.T4.3.3.1.m1.1.1.3a.cmml">480</mtext></msub><annotation-xml encoding="MathML-Content" id="A0.T4.3.3.1.m1.1b"><apply id="A0.T4.3.3.1.m1.1.1.cmml" xref="A0.T4.3.3.1.m1.1.1"><csymbol cd="ambiguous" id="A0.T4.3.3.1.m1.1.1.1.cmml" xref="A0.T4.3.3.1.m1.1.1">subscript</csymbol><ci id="A0.T4.3.3.1.m1.1.1.2a.cmml" xref="A0.T4.3.3.1.m1.1.1.2"><mtext class="ltx_mathvariant_monospace" id="A0.T4.3.3.1.m1.1.1.2.cmml" xref="A0.T4.3.3.1.m1.1.1.2">Books</mtext></ci><ci id="A0.T4.3.3.1.m1.1.1.3a.cmml" xref="A0.T4.3.3.1.m1.1.1.3"><mtext class="ltx_mathvariant_monospace" id="A0.T4.3.3.1.m1.1.1.3.cmml" mathsize="70%" xref="A0.T4.3.3.1.m1.1.1.3">480</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A0.T4.3.3.1.m1.1c">\texttt{Books}_{\texttt{480}}</annotation><annotation encoding="application/x-llamapun" id="A0.T4.3.3.1.m1.1d">Books start_POSTSUBSCRIPT 480 end_POSTSUBSCRIPT</annotation></semantics></math></th> <th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_bb" id="A0.T4.3.3.2">1000</th> <td class="ltx_td ltx_align_left ltx_border_bb" id="A0.T4.3.3.3">222,539</td> <td class="ltx_td ltx_align_left ltx_border_bb" id="A0.T4.3.3.4">481,455</td> <td class="ltx_td ltx_align_left ltx_border_bb" id="A0.T4.3.3.5">99.78%</td> <td class="ltx_td ltx_align_left ltx_border_bb" id="A0.T4.3.3.6">481.46</td> </tr> </tbody> </table> </figure> </section> <section class="ltx_appendix" id="A1"> <h2 class="ltx_title ltx_title_appendix"> <span class="ltx_tag ltx_tag_appendix">Appendix A </span>Datasets</h2> <div class="ltx_para" id="A1.p1"> <p class="ltx_p" id="A1.p1.1">In this appendix, we provide a detailed description of the dataset construction and statistics.</p> </div> <div class="ltx_para" id="A1.p2"> <p class="ltx_p" id="A1.p2.3">Building on prior studies such as AgentCF <cite class="ltx_cite ltx_citemacro_cite">Zhang et al. (<a class="ltx_ref" href="https://arxiv.org/html/2503.02398v1#bib.bib57" title="">2024b</a>)</cite>, Agent4Rec <cite class="ltx_cite ltx_citemacro_cite">Zhang et al. (<a class="ltx_ref" href="https://arxiv.org/html/2503.02398v1#bib.bib56" title="">2024a</a>)</cite>, and EasyRec <cite class="ltx_cite ltx_citemacro_cite">Ren and Huang (<a class="ltx_ref" href="https://arxiv.org/html/2503.02398v1#bib.bib33" title="">2024</a>)</cite>, we evaluate our proposed method using two widely adopted subsets of the Amazon review dataset <cite class="ltx_cite ltx_citemacro_cite">Ni et al. (<a class="ltx_ref" href="https://arxiv.org/html/2503.02398v1#bib.bib23" title="">2019</a>)</cite>: <span class="ltx_text ltx_font_italic" id="A1.p2.3.1">CDs and Vinyl</span> and <span class="ltx_text ltx_font_italic" id="A1.p2.3.2">Books</span>. For the <span class="ltx_text ltx_markedasmath ltx_font_typewriter" id="A1.p2.3.3">CDs</span> dataset, we construct <math alttext="\texttt{CDs}_{\texttt{50}}" class="ltx_Math" display="inline" id="A1.p2.2.m2.1"><semantics id="A1.p2.2.m2.1a"><msub id="A1.p2.2.m2.1.1" xref="A1.p2.2.m2.1.1.cmml"><mtext class="ltx_mathvariant_monospace" id="A1.p2.2.m2.1.1.2" xref="A1.p2.2.m2.1.1.2a.cmml">CDs</mtext><mtext class="ltx_mathvariant_monospace" id="A1.p2.2.m2.1.1.3" xref="A1.p2.2.m2.1.1.3a.cmml">50</mtext></msub><annotation-xml encoding="MathML-Content" id="A1.p2.2.m2.1b"><apply id="A1.p2.2.m2.1.1.cmml" xref="A1.p2.2.m2.1.1"><csymbol cd="ambiguous" id="A1.p2.2.m2.1.1.1.cmml" xref="A1.p2.2.m2.1.1">subscript</csymbol><ci id="A1.p2.2.m2.1.1.2a.cmml" xref="A1.p2.2.m2.1.1.2"><mtext class="ltx_mathvariant_monospace" id="A1.p2.2.m2.1.1.2.cmml" xref="A1.p2.2.m2.1.1.2">CDs</mtext></ci><ci id="A1.p2.2.m2.1.1.3a.cmml" xref="A1.p2.2.m2.1.1.3"><mtext class="ltx_mathvariant_monospace" id="A1.p2.2.m2.1.1.3.cmml" mathsize="70%" xref="A1.p2.2.m2.1.1.3">50</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.p2.2.m2.1c">\texttt{CDs}_{\texttt{50}}</annotation><annotation encoding="application/x-llamapun" id="A1.p2.2.m2.1d">CDs start_POSTSUBSCRIPT 50 end_POSTSUBSCRIPT</annotation></semantics></math>, and <math alttext="\texttt{CDs}_{\texttt{200}}" class="ltx_Math" display="inline" id="A1.p2.3.m3.1"><semantics id="A1.p2.3.m3.1a"><msub id="A1.p2.3.m3.1.1" xref="A1.p2.3.m3.1.1.cmml"><mtext class="ltx_mathvariant_monospace" id="A1.p2.3.m3.1.1.2" xref="A1.p2.3.m3.1.1.2a.cmml">CDs</mtext><mtext class="ltx_mathvariant_monospace" id="A1.p2.3.m3.1.1.3" xref="A1.p2.3.m3.1.1.3a.cmml">200</mtext></msub><annotation-xml encoding="MathML-Content" id="A1.p2.3.m3.1b"><apply id="A1.p2.3.m3.1.1.cmml" xref="A1.p2.3.m3.1.1"><csymbol cd="ambiguous" id="A1.p2.3.m3.1.1.1.cmml" xref="A1.p2.3.m3.1.1">subscript</csymbol><ci id="A1.p2.3.m3.1.1.2a.cmml" xref="A1.p2.3.m3.1.1.2"><mtext class="ltx_mathvariant_monospace" id="A1.p2.3.m3.1.1.2.cmml" xref="A1.p2.3.m3.1.1.2">CDs</mtext></ci><ci id="A1.p2.3.m3.1.1.3a.cmml" xref="A1.p2.3.m3.1.1.3"><mtext class="ltx_mathvariant_monospace" id="A1.p2.3.m3.1.1.3.cmml" mathsize="70%" xref="A1.p2.3.m3.1.1.3">200</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.p2.3.m3.1c">\texttt{CDs}_{\texttt{200}}</annotation><annotation encoding="application/x-llamapun" id="A1.p2.3.m3.1d">CDs start_POSTSUBSCRIPT 200 end_POSTSUBSCRIPT</annotation></semantics></math>, with average user interaction sequence lengths of 50 and 200, respectively. These settings are similar as those used in AgentCF <cite class="ltx_cite ltx_citemacro_cite">Zhang et al. (<a class="ltx_ref" href="https://arxiv.org/html/2503.02398v1#bib.bib57" title="">2024b</a>)</cite>.</p> </div> <div class="ltx_para" id="A1.p3"> <p class="ltx_p" id="A1.p3.2">For the <span class="ltx_text ltx_markedasmath ltx_font_typewriter" id="A1.p3.2.1">Books</span> dataset, departing from the approach of Agent4Rec which limits each user’s interactions to 20 items, we follow the guidelines of <cite class="ltx_cite ltx_citemacro_cite">Pi et al. (<a class="ltx_ref" href="https://arxiv.org/html/2503.02398v1#bib.bib27" title="">2019</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.02398v1#bib.bib28" title="">2020</a>)</cite> to construct longer interaction sequences. Specifically, we create <math alttext="\texttt{Books}_{\texttt{480}}" class="ltx_Math" display="inline" id="A1.p3.2.m2.1"><semantics id="A1.p3.2.m2.1a"><msub id="A1.p3.2.m2.1.1" xref="A1.p3.2.m2.1.1.cmml"><mtext class="ltx_mathvariant_monospace" id="A1.p3.2.m2.1.1.2" xref="A1.p3.2.m2.1.1.2a.cmml">Books</mtext><mtext class="ltx_mathvariant_monospace" id="A1.p3.2.m2.1.1.3" xref="A1.p3.2.m2.1.1.3a.cmml">480</mtext></msub><annotation-xml encoding="MathML-Content" id="A1.p3.2.m2.1b"><apply id="A1.p3.2.m2.1.1.cmml" xref="A1.p3.2.m2.1.1"><csymbol cd="ambiguous" id="A1.p3.2.m2.1.1.1.cmml" xref="A1.p3.2.m2.1.1">subscript</csymbol><ci id="A1.p3.2.m2.1.1.2a.cmml" xref="A1.p3.2.m2.1.1.2"><mtext class="ltx_mathvariant_monospace" id="A1.p3.2.m2.1.1.2.cmml" xref="A1.p3.2.m2.1.1.2">Books</mtext></ci><ci id="A1.p3.2.m2.1.1.3a.cmml" xref="A1.p3.2.m2.1.1.3"><mtext class="ltx_mathvariant_monospace" id="A1.p3.2.m2.1.1.3.cmml" mathsize="70%" xref="A1.p3.2.m2.1.1.3">480</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.p3.2.m2.1c">\texttt{Books}_{\texttt{480}}</annotation><annotation encoding="application/x-llamapun" id="A1.p3.2.m2.1d">Books start_POSTSUBSCRIPT 480 end_POSTSUBSCRIPT</annotation></semantics></math>, with average sequence lengths of 480, respectively. Detailed statistics for these datasets are provided in Table <a class="ltx_ref" href="https://arxiv.org/html/2503.02398v1#A0.T4" title="Table 4 ‣ APPENDIX ‣ PersonaX: A Recommendation Agent-Oriented User Modeling Framework for Long Behavior Sequence"><span class="ltx_text ltx_ref_tag">4</span></a>.</p> </div> <div class="ltx_para" id="A1.p4"> <p class="ltx_p" id="A1.p4.1">Due to the high computational cost and expense associated with API calls for GPT-4o-mini, we conduct each experiment only once per dataset to ensure feasibility within a reasonable budget. This approach is common in agent recommendation studies <cite class="ltx_cite ltx_citemacro_cite">Zhang et al. (<a class="ltx_ref" href="https://arxiv.org/html/2503.02398v1#bib.bib56" title="">2024a</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.02398v1#bib.bib57" title="">b</a>); Wang et al. (<a class="ltx_ref" href="https://arxiv.org/html/2503.02398v1#bib.bib48" title="">2024</a>); Luo et al. (<a class="ltx_ref" href="https://arxiv.org/html/2503.02398v1#bib.bib19" title="">2023</a>)</cite> and large-scale recommendation system research. Moreover, the larger number of users (1000) in our study enhances the reliability of the experimental results.</p> </div> </section> <section class="ltx_appendix" id="A2"> <h2 class="ltx_title ltx_title_appendix"> <span class="ltx_tag ltx_tag_appendix">Appendix B </span>Backbone Methods</h2> <div class="ltx_para" id="A2.p1"> <p class="ltx_p" id="A2.p1.1">We provide a detailed description of the backbone methods used for validation.</p> </div> <div class="ltx_para" id="A2.p2"> <p class="ltx_p" id="A2.p2.1"><span class="ltx_text ltx_font_bold" id="A2.p2.1.1">AgentCF <cite class="ltx_cite ltx_citemacro_cite">Zhang et al. (<a class="ltx_ref" href="https://arxiv.org/html/2503.02398v1#bib.bib57" title="">2024b</a>)</cite></span> employs a reflective mechanism to model user personas. In the original framework, both the user profile and item profile are dynamically updated. In our implementation, the item profile is textually represented by concatenating the item’s fields, while the user profile is initially set to "<span class="ltx_text ltx_font_typewriter" id="A2.p2.1.2">Currently Unknown</span>" and is iteratively refined through continuous reflection. Furthermore, for the downstream recommendation ranking task in AgentCF, we replace the original LLM-based ranking with the EasyRec framework <cite class="ltx_cite ltx_citemacro_cite">Ren and Huang (<a class="ltx_ref" href="https://arxiv.org/html/2503.02398v1#bib.bib33" title="">2024</a>)</cite>. EasyRec is the first large language embedding model specifically designed for recommendation. It aligns textual semantic spaces with collaborative behavioral signals, enabling recommendation tasks to rely solely on textual instructions (e.g., user preference descriptions and item profiles) while achieving performance comparable to traditional state-of-the-art models. Leveraging EasyRec for point-wise ranking is more experimentally efficient, accurate, and robust compared with LLMs.</p> </div> <div class="ltx_para" id="A2.p3"> <p class="ltx_p" id="A2.p3.1"><span class="ltx_text ltx_font_bold" id="A2.p3.1.1">Agent4Rec <cite class="ltx_cite ltx_citemacro_cite">Zhang et al. (<a class="ltx_ref" href="https://arxiv.org/html/2503.02398v1#bib.bib56" title="">2024a</a>)</cite></span> maintains an agent profile comprising two key components: social traits and unique tastes. In our implementation, we streamline the process by focusing solely on capturing diverse user interests through the construction of unique tastes, thus simplifying experimentation. To achieve this, we adopt the summarization method from the original work, which distills user preferences from their behavioral sequences. Additionally, we replace the original rating prediction task in the Agent4Rec framework with a ranking task.</p> </div> <figure class="ltx_figure" id="A2.F5"><img alt="Refer to caption" class="ltx_graphics ltx_centering ltx_img_landscape" height="285" id="A2.F5.g1" src="x5.png" width="775"/> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure">Figure 5: </span>Sampling process for a user in <math alttext="\texttt{Books}_{\texttt{480}}" class="ltx_Math" display="inline" id="A2.F5.2.m1.1"><semantics id="A2.F5.2.m1.1b"><msub id="A2.F5.2.m1.1.1" xref="A2.F5.2.m1.1.1.cmml"><mtext class="ltx_mathvariant_monospace" id="A2.F5.2.m1.1.1.2" xref="A2.F5.2.m1.1.1.2a.cmml">Books</mtext><mtext class="ltx_mathvariant_monospace" id="A2.F5.2.m1.1.1.3" xref="A2.F5.2.m1.1.1.3a.cmml">480</mtext></msub><annotation-xml encoding="MathML-Content" id="A2.F5.2.m1.1c"><apply id="A2.F5.2.m1.1.1.cmml" xref="A2.F5.2.m1.1.1"><csymbol cd="ambiguous" id="A2.F5.2.m1.1.1.1.cmml" xref="A2.F5.2.m1.1.1">subscript</csymbol><ci id="A2.F5.2.m1.1.1.2a.cmml" xref="A2.F5.2.m1.1.1.2"><mtext class="ltx_mathvariant_monospace" id="A2.F5.2.m1.1.1.2.cmml" xref="A2.F5.2.m1.1.1.2">Books</mtext></ci><ci id="A2.F5.2.m1.1.1.3a.cmml" xref="A2.F5.2.m1.1.1.3"><mtext class="ltx_mathvariant_monospace" id="A2.F5.2.m1.1.1.3.cmml" mathsize="70%" xref="A2.F5.2.m1.1.1.3">480</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.F5.2.m1.1d">\texttt{Books}_{\texttt{480}}</annotation><annotation encoding="application/x-llamapun" id="A2.F5.2.m1.1e">Books start_POSTSUBSCRIPT 480 end_POSTSUBSCRIPT</annotation></semantics></math> with a 50% selection ratio. Points are color-coded and outlined. Non-transparent points signify data selected, whereas transparent points delineate behaviors not sampled. A offers a holistic perspective on the user’s comprehensive behavior distribution, capturing the full extent of engagement patterns. BẼ presents parts of behaviors distributions and sampling process under varying configurations of hyper-parameters. Triangles denote the centroids of the clusters.</figcaption> </figure> </section> <section class="ltx_appendix" id="A3"> <h2 class="ltx_title ltx_title_appendix"> <span class="ltx_tag ltx_tag_appendix">Appendix C </span>Hyper-parameter Analysis and Sampling Process Visualization</h2> <div class="ltx_para" id="A3.p1"> <p class="ltx_p" id="A3.p1.8">This section delves into the influence of the hyperparameters <math alttext="\tau" class="ltx_Math" display="inline" id="A3.p1.1.m1.1"><semantics id="A3.p1.1.m1.1a"><mi id="A3.p1.1.m1.1.1" xref="A3.p1.1.m1.1.1.cmml">τ</mi><annotation-xml encoding="MathML-Content" id="A3.p1.1.m1.1b"><ci id="A3.p1.1.m1.1.1.cmml" xref="A3.p1.1.m1.1.1">𝜏</ci></annotation-xml><annotation encoding="application/x-tex" id="A3.p1.1.m1.1c">\tau</annotation><annotation encoding="application/x-llamapun" id="A3.p1.1.m1.1d">italic_τ</annotation></semantics></math> and <math alttext="\alpha" class="ltx_Math" display="inline" id="A3.p1.2.m2.1"><semantics id="A3.p1.2.m2.1a"><mi id="A3.p1.2.m2.1.1" xref="A3.p1.2.m2.1.1.cmml">α</mi><annotation-xml encoding="MathML-Content" id="A3.p1.2.m2.1b"><ci id="A3.p1.2.m2.1.1.cmml" xref="A3.p1.2.m2.1.1">𝛼</ci></annotation-xml><annotation encoding="application/x-tex" id="A3.p1.2.m2.1c">\alpha</annotation><annotation encoding="application/x-llamapun" id="A3.p1.2.m2.1d">italic_α</annotation></semantics></math> on the performance of PersonaX, as they play pivotal roles in shaping the hierarchical clustering and in-cluster behavior selection processes. Specifically, <math alttext="\tau" class="ltx_Math" display="inline" id="A3.p1.3.m3.1"><semantics id="A3.p1.3.m3.1a"><mi id="A3.p1.3.m3.1.1" xref="A3.p1.3.m3.1.1.cmml">τ</mi><annotation-xml encoding="MathML-Content" id="A3.p1.3.m3.1b"><ci id="A3.p1.3.m3.1.1.cmml" xref="A3.p1.3.m3.1.1">𝜏</ci></annotation-xml><annotation encoding="application/x-tex" id="A3.p1.3.m3.1c">\tau</annotation><annotation encoding="application/x-llamapun" id="A3.p1.3.m3.1d">italic_τ</annotation></semantics></math> dictates the granularity of the hierarchical clustering. A larger <math alttext="\tau" class="ltx_Math" display="inline" id="A3.p1.4.m4.1"><semantics id="A3.p1.4.m4.1a"><mi id="A3.p1.4.m4.1.1" xref="A3.p1.4.m4.1.1.cmml">τ</mi><annotation-xml encoding="MathML-Content" id="A3.p1.4.m4.1b"><ci id="A3.p1.4.m4.1.1.cmml" xref="A3.p1.4.m4.1.1">𝜏</ci></annotation-xml><annotation encoding="application/x-tex" id="A3.p1.4.m4.1c">\tau</annotation><annotation encoding="application/x-llamapun" id="A3.p1.4.m4.1d">italic_τ</annotation></semantics></math> value yields coarser clusters, encompassing a broader spectrum of behavioral samples with potentially greater divergence from the cluster centroid. In contrast, a smaller <math alttext="\tau" class="ltx_Math" display="inline" id="A3.p1.5.m5.1"><semantics id="A3.p1.5.m5.1a"><mi id="A3.p1.5.m5.1.1" xref="A3.p1.5.m5.1.1.cmml">τ</mi><annotation-xml encoding="MathML-Content" id="A3.p1.5.m5.1b"><ci id="A3.p1.5.m5.1.1.cmml" xref="A3.p1.5.m5.1.1">𝜏</ci></annotation-xml><annotation encoding="application/x-tex" id="A3.p1.5.m5.1c">\tau</annotation><annotation encoding="application/x-llamapun" id="A3.p1.5.m5.1d">italic_τ</annotation></semantics></math> enforces a more stringent clustering criterion, resulting in finer-grained clusters characterized by higher intra-cluster homogeneity. On the other hand, <math alttext="\alpha" class="ltx_Math" display="inline" id="A3.p1.6.m6.1"><semantics id="A3.p1.6.m6.1a"><mi id="A3.p1.6.m6.1.1" xref="A3.p1.6.m6.1.1.cmml">α</mi><annotation-xml encoding="MathML-Content" id="A3.p1.6.m6.1b"><ci id="A3.p1.6.m6.1.1.cmml" xref="A3.p1.6.m6.1.1">𝛼</ci></annotation-xml><annotation encoding="application/x-tex" id="A3.p1.6.m6.1c">\alpha</annotation><annotation encoding="application/x-llamapun" id="A3.p1.6.m6.1d">italic_α</annotation></semantics></math> modulates the balance between prototypicality and diversity during the in-cluster behavior selection stage. A higher <math alttext="\alpha" class="ltx_Math" display="inline" id="A3.p1.7.m7.1"><semantics id="A3.p1.7.m7.1a"><mi id="A3.p1.7.m7.1.1" xref="A3.p1.7.m7.1.1.cmml">α</mi><annotation-xml encoding="MathML-Content" id="A3.p1.7.m7.1b"><ci id="A3.p1.7.m7.1.1.cmml" xref="A3.p1.7.m7.1.1">𝛼</ci></annotation-xml><annotation encoding="application/x-tex" id="A3.p1.7.m7.1c">\alpha</annotation><annotation encoding="application/x-llamapun" id="A3.p1.7.m7.1d">italic_α</annotation></semantics></math> amplifies the preference for selecting behavior samples further from the cluster centroid, thereby enhancing diversity within the cluster. Conversely, a lower <math alttext="\alpha" class="ltx_Math" display="inline" id="A3.p1.8.m8.1"><semantics id="A3.p1.8.m8.1a"><mi id="A3.p1.8.m8.1.1" xref="A3.p1.8.m8.1.1.cmml">α</mi><annotation-xml encoding="MathML-Content" id="A3.p1.8.m8.1b"><ci id="A3.p1.8.m8.1.1.cmml" xref="A3.p1.8.m8.1.1">𝛼</ci></annotation-xml><annotation encoding="application/x-tex" id="A3.p1.8.m8.1c">\alpha</annotation><annotation encoding="application/x-llamapun" id="A3.p1.8.m8.1d">italic_α</annotation></semantics></math> emphasizes prototypicality, favoring samples that closely align with the cluster centroid. Our empirical analysis, as illustrated in Figure <a class="ltx_ref" href="https://arxiv.org/html/2503.02398v1#S5.F4" title="Figure 4 ‣ 5.4 Hyper-parameter Analysis (RQ3) ‣ 5 Experiments ‣ PersonaX: A Recommendation Agent-Oriented User Modeling Framework for Long Behavior Sequence"><span class="ltx_text ltx_ref_tag">4</span></a>, uncovers nuanced patterns in how these hyperparameters influence the model’s overall performance.</p> </div> <div class="ltx_para" id="A3.p2"> <p class="ltx_p" id="A3.p2.8"><span class="ltx_text ltx_font_bold" id="A3.p2.8.1">1. Performance at Low Ratios:</span> Across <math alttext="\tau" class="ltx_Math" display="inline" id="A3.p2.1.m1.1"><semantics id="A3.p2.1.m1.1a"><mi id="A3.p2.1.m1.1.1" xref="A3.p2.1.m1.1.1.cmml">τ</mi><annotation-xml encoding="MathML-Content" id="A3.p2.1.m1.1b"><ci id="A3.p2.1.m1.1.1.cmml" xref="A3.p2.1.m1.1.1">𝜏</ci></annotation-xml><annotation encoding="application/x-tex" id="A3.p2.1.m1.1c">\tau</annotation><annotation encoding="application/x-llamapun" id="A3.p2.1.m1.1d">italic_τ</annotation></semantics></math> and <math alttext="\alpha" class="ltx_Math" display="inline" id="A3.p2.2.m2.1"><semantics id="A3.p2.2.m2.1a"><mi id="A3.p2.2.m2.1.1" xref="A3.p2.2.m2.1.1.cmml">α</mi><annotation-xml encoding="MathML-Content" id="A3.p2.2.m2.1b"><ci id="A3.p2.2.m2.1.1.cmml" xref="A3.p2.2.m2.1.1">𝛼</ci></annotation-xml><annotation encoding="application/x-tex" id="A3.p2.2.m2.1c">\alpha</annotation><annotation encoding="application/x-llamapun" id="A3.p2.2.m2.1d">italic_α</annotation></semantics></math> configurations, the performances at lower ratios (e.g., <math alttext="0.1" class="ltx_Math" display="inline" id="A3.p2.3.m3.1"><semantics id="A3.p2.3.m3.1a"><mn id="A3.p2.3.m3.1.1" xref="A3.p2.3.m3.1.1.cmml">0.1</mn><annotation-xml encoding="MathML-Content" id="A3.p2.3.m3.1b"><cn id="A3.p2.3.m3.1.1.cmml" type="float" xref="A3.p2.3.m3.1.1">0.1</cn></annotation-xml><annotation encoding="application/x-tex" id="A3.p2.3.m3.1c">0.1</annotation><annotation encoding="application/x-llamapun" id="A3.p2.3.m3.1d">0.1</annotation></semantics></math>, <math alttext="0.3" class="ltx_Math" display="inline" id="A3.p2.4.m4.1"><semantics id="A3.p2.4.m4.1a"><mn id="A3.p2.4.m4.1.1" xref="A3.p2.4.m4.1.1.cmml">0.3</mn><annotation-xml encoding="MathML-Content" id="A3.p2.4.m4.1b"><cn id="A3.p2.4.m4.1.1.cmml" type="float" xref="A3.p2.4.m4.1.1">0.3</cn></annotation-xml><annotation encoding="application/x-tex" id="A3.p2.4.m4.1c">0.3</annotation><annotation encoding="application/x-llamapun" id="A3.p2.4.m4.1d">0.3</annotation></semantics></math>) remain similar. This is because the selected samples at low ratios primarily originate near the cluster centroid, regardless of the diversity adjustment imposed by <math alttext="\alpha" class="ltx_Math" display="inline" id="A3.p2.5.m5.1"><semantics id="A3.p2.5.m5.1a"><mi id="A3.p2.5.m5.1.1" xref="A3.p2.5.m5.1.1.cmml">α</mi><annotation-xml encoding="MathML-Content" id="A3.p2.5.m5.1b"><ci id="A3.p2.5.m5.1.1.cmml" xref="A3.p2.5.m5.1.1">𝛼</ci></annotation-xml><annotation encoding="application/x-tex" id="A3.p2.5.m5.1c">\alpha</annotation><annotation encoding="application/x-llamapun" id="A3.p2.5.m5.1d">italic_α</annotation></semantics></math>. Slightly superior performance of <math alttext="\tau=0.5" class="ltx_Math" display="inline" id="A3.p2.6.m6.1"><semantics id="A3.p2.6.m6.1a"><mrow id="A3.p2.6.m6.1.1" xref="A3.p2.6.m6.1.1.cmml"><mi id="A3.p2.6.m6.1.1.2" xref="A3.p2.6.m6.1.1.2.cmml">τ</mi><mo id="A3.p2.6.m6.1.1.1" xref="A3.p2.6.m6.1.1.1.cmml">=</mo><mn id="A3.p2.6.m6.1.1.3" xref="A3.p2.6.m6.1.1.3.cmml">0.5</mn></mrow><annotation-xml encoding="MathML-Content" id="A3.p2.6.m6.1b"><apply id="A3.p2.6.m6.1.1.cmml" xref="A3.p2.6.m6.1.1"><eq id="A3.p2.6.m6.1.1.1.cmml" xref="A3.p2.6.m6.1.1.1"></eq><ci id="A3.p2.6.m6.1.1.2.cmml" xref="A3.p2.6.m6.1.1.2">𝜏</ci><cn id="A3.p2.6.m6.1.1.3.cmml" type="float" xref="A3.p2.6.m6.1.1.3">0.5</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.p2.6.m6.1c">\tau=0.5</annotation><annotation encoding="application/x-llamapun" id="A3.p2.6.m6.1d">italic_τ = 0.5</annotation></semantics></math> compared to <math alttext="\tau=0.7" class="ltx_Math" display="inline" id="A3.p2.7.m7.1"><semantics id="A3.p2.7.m7.1a"><mrow id="A3.p2.7.m7.1.1" xref="A3.p2.7.m7.1.1.cmml"><mi id="A3.p2.7.m7.1.1.2" xref="A3.p2.7.m7.1.1.2.cmml">τ</mi><mo id="A3.p2.7.m7.1.1.1" xref="A3.p2.7.m7.1.1.1.cmml">=</mo><mn id="A3.p2.7.m7.1.1.3" xref="A3.p2.7.m7.1.1.3.cmml">0.7</mn></mrow><annotation-xml encoding="MathML-Content" id="A3.p2.7.m7.1b"><apply id="A3.p2.7.m7.1.1.cmml" xref="A3.p2.7.m7.1.1"><eq id="A3.p2.7.m7.1.1.1.cmml" xref="A3.p2.7.m7.1.1.1"></eq><ci id="A3.p2.7.m7.1.1.2.cmml" xref="A3.p2.7.m7.1.1.2">𝜏</ci><cn id="A3.p2.7.m7.1.1.3.cmml" type="float" xref="A3.p2.7.m7.1.1.3">0.7</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.p2.7.m7.1c">\tau=0.7</annotation><annotation encoding="application/x-llamapun" id="A3.p2.7.m7.1d">italic_τ = 0.7</annotation></semantics></math> at these ratios is attributed to the finer clustering granularity of <math alttext="\tau=0.5" class="ltx_Math" display="inline" id="A3.p2.8.m8.1"><semantics id="A3.p2.8.m8.1a"><mrow id="A3.p2.8.m8.1.1" xref="A3.p2.8.m8.1.1.cmml"><mi id="A3.p2.8.m8.1.1.2" xref="A3.p2.8.m8.1.1.2.cmml">τ</mi><mo id="A3.p2.8.m8.1.1.1" xref="A3.p2.8.m8.1.1.1.cmml">=</mo><mn id="A3.p2.8.m8.1.1.3" xref="A3.p2.8.m8.1.1.3.cmml">0.5</mn></mrow><annotation-xml encoding="MathML-Content" id="A3.p2.8.m8.1b"><apply id="A3.p2.8.m8.1.1.cmml" xref="A3.p2.8.m8.1.1"><eq id="A3.p2.8.m8.1.1.1.cmml" xref="A3.p2.8.m8.1.1.1"></eq><ci id="A3.p2.8.m8.1.1.2.cmml" xref="A3.p2.8.m8.1.1.2">𝜏</ci><cn id="A3.p2.8.m8.1.1.3.cmml" type="float" xref="A3.p2.8.m8.1.1.3">0.5</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.p2.8.m8.1c">\tau=0.5</annotation><annotation encoding="application/x-llamapun" id="A3.p2.8.m8.1d">italic_τ = 0.5</annotation></semantics></math>, which ensures that selected samples exhibit higher prototypicality.</p> </div> <div class="ltx_para" id="A3.p3"> <p class="ltx_p" id="A3.p3.6"><span class="ltx_text ltx_font_bold" id="A3.p3.2.2">2. Performance at High Ratios (<math alttext="0.5" class="ltx_Math" display="inline" id="A3.p3.1.1.m1.1"><semantics id="A3.p3.1.1.m1.1a"><mn id="A3.p3.1.1.m1.1.1" xref="A3.p3.1.1.m1.1.1.cmml">0.5</mn><annotation-xml encoding="MathML-Content" id="A3.p3.1.1.m1.1b"><cn id="A3.p3.1.1.m1.1.1.cmml" type="float" xref="A3.p3.1.1.m1.1.1">0.5</cn></annotation-xml><annotation encoding="application/x-tex" id="A3.p3.1.1.m1.1c">0.5</annotation><annotation encoding="application/x-llamapun" id="A3.p3.1.1.m1.1d">0.5</annotation></semantics></math>–<math alttext="0.9" class="ltx_Math" display="inline" id="A3.p3.2.2.m2.1"><semantics id="A3.p3.2.2.m2.1a"><mn id="A3.p3.2.2.m2.1.1" xref="A3.p3.2.2.m2.1.1.cmml">0.9</mn><annotation-xml encoding="MathML-Content" id="A3.p3.2.2.m2.1b"><cn id="A3.p3.2.2.m2.1.1.cmml" type="float" xref="A3.p3.2.2.m2.1.1">0.9</cn></annotation-xml><annotation encoding="application/x-tex" id="A3.p3.2.2.m2.1c">0.9</annotation><annotation encoding="application/x-llamapun" id="A3.p3.2.2.m2.1d">0.9</annotation></semantics></math>):</span> At higher ratios, configurations with larger <math alttext="\alpha" class="ltx_Math" display="inline" id="A3.p3.3.m1.1"><semantics id="A3.p3.3.m1.1a"><mi id="A3.p3.3.m1.1.1" xref="A3.p3.3.m1.1.1.cmml">α</mi><annotation-xml encoding="MathML-Content" id="A3.p3.3.m1.1b"><ci id="A3.p3.3.m1.1.1.cmml" xref="A3.p3.3.m1.1.1">𝛼</ci></annotation-xml><annotation encoding="application/x-tex" id="A3.p3.3.m1.1c">\alpha</annotation><annotation encoding="application/x-llamapun" id="A3.p3.3.m1.1d">italic_α</annotation></semantics></math> values (e.g., <math alttext="\alpha=1.06,1.08" class="ltx_Math" display="inline" id="A3.p3.4.m2.2"><semantics id="A3.p3.4.m2.2a"><mrow id="A3.p3.4.m2.2.3" xref="A3.p3.4.m2.2.3.cmml"><mi id="A3.p3.4.m2.2.3.2" xref="A3.p3.4.m2.2.3.2.cmml">α</mi><mo id="A3.p3.4.m2.2.3.1" xref="A3.p3.4.m2.2.3.1.cmml">=</mo><mrow id="A3.p3.4.m2.2.3.3.2" xref="A3.p3.4.m2.2.3.3.1.cmml"><mn id="A3.p3.4.m2.1.1" xref="A3.p3.4.m2.1.1.cmml">1.06</mn><mo id="A3.p3.4.m2.2.3.3.2.1" xref="A3.p3.4.m2.2.3.3.1.cmml">,</mo><mn id="A3.p3.4.m2.2.2" xref="A3.p3.4.m2.2.2.cmml">1.08</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="A3.p3.4.m2.2b"><apply id="A3.p3.4.m2.2.3.cmml" xref="A3.p3.4.m2.2.3"><eq id="A3.p3.4.m2.2.3.1.cmml" xref="A3.p3.4.m2.2.3.1"></eq><ci id="A3.p3.4.m2.2.3.2.cmml" xref="A3.p3.4.m2.2.3.2">𝛼</ci><list id="A3.p3.4.m2.2.3.3.1.cmml" xref="A3.p3.4.m2.2.3.3.2"><cn id="A3.p3.4.m2.1.1.cmml" type="float" xref="A3.p3.4.m2.1.1">1.06</cn><cn id="A3.p3.4.m2.2.2.cmml" type="float" xref="A3.p3.4.m2.2.2">1.08</cn></list></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.p3.4.m2.2c">\alpha=1.06,1.08</annotation><annotation encoding="application/x-llamapun" id="A3.p3.4.m2.2d">italic_α = 1.06 , 1.08</annotation></semantics></math>) outperform their smaller-<math alttext="\alpha" class="ltx_Math" display="inline" id="A3.p3.5.m3.1"><semantics id="A3.p3.5.m3.1a"><mi id="A3.p3.5.m3.1.1" xref="A3.p3.5.m3.1.1.cmml">α</mi><annotation-xml encoding="MathML-Content" id="A3.p3.5.m3.1b"><ci id="A3.p3.5.m3.1.1.cmml" xref="A3.p3.5.m3.1.1">𝛼</ci></annotation-xml><annotation encoding="application/x-tex" id="A3.p3.5.m3.1c">\alpha</annotation><annotation encoding="application/x-llamapun" id="A3.p3.5.m3.1d">italic_α</annotation></semantics></math> counterparts (e.g., <math alttext="\alpha=1.01,1.04" class="ltx_Math" display="inline" id="A3.p3.6.m4.2"><semantics id="A3.p3.6.m4.2a"><mrow id="A3.p3.6.m4.2.3" xref="A3.p3.6.m4.2.3.cmml"><mi id="A3.p3.6.m4.2.3.2" xref="A3.p3.6.m4.2.3.2.cmml">α</mi><mo id="A3.p3.6.m4.2.3.1" xref="A3.p3.6.m4.2.3.1.cmml">=</mo><mrow id="A3.p3.6.m4.2.3.3.2" xref="A3.p3.6.m4.2.3.3.1.cmml"><mn id="A3.p3.6.m4.1.1" xref="A3.p3.6.m4.1.1.cmml">1.01</mn><mo id="A3.p3.6.m4.2.3.3.2.1" xref="A3.p3.6.m4.2.3.3.1.cmml">,</mo><mn id="A3.p3.6.m4.2.2" xref="A3.p3.6.m4.2.2.cmml">1.04</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="A3.p3.6.m4.2b"><apply id="A3.p3.6.m4.2.3.cmml" xref="A3.p3.6.m4.2.3"><eq id="A3.p3.6.m4.2.3.1.cmml" xref="A3.p3.6.m4.2.3.1"></eq><ci id="A3.p3.6.m4.2.3.2.cmml" xref="A3.p3.6.m4.2.3.2">𝛼</ci><list id="A3.p3.6.m4.2.3.3.1.cmml" xref="A3.p3.6.m4.2.3.3.2"><cn id="A3.p3.6.m4.1.1.cmml" type="float" xref="A3.p3.6.m4.1.1">1.01</cn><cn id="A3.p3.6.m4.2.2.cmml" type="float" xref="A3.p3.6.m4.2.2">1.04</cn></list></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.p3.6.m4.2c">\alpha=1.01,1.04</annotation><annotation encoding="application/x-llamapun" id="A3.p3.6.m4.2d">italic_α = 1.01 , 1.04</annotation></semantics></math>). This highlights the efficacy of the in-cluster selection strategy: after a core set of prototypical samples is chosen, incorporating more diverse samples significantly enhances performance. The inclusion of diversity helps capture broader behavioral patterns, leading to improved generalization.</p> </div> <div class="ltx_para" id="A3.p4"> <p class="ltx_p" id="A3.p4.10"><span class="ltx_text ltx_font_bold" id="A3.p4.10.1">3. Trade-offs in Specific Settings:</span> A nuanced behavior is observed in the interaction between <math alttext="\tau" class="ltx_Math" display="inline" id="A3.p4.1.m1.1"><semantics id="A3.p4.1.m1.1a"><mi id="A3.p4.1.m1.1.1" xref="A3.p4.1.m1.1.1.cmml">τ</mi><annotation-xml encoding="MathML-Content" id="A3.p4.1.m1.1b"><ci id="A3.p4.1.m1.1.1.cmml" xref="A3.p4.1.m1.1.1">𝜏</ci></annotation-xml><annotation encoding="application/x-tex" id="A3.p4.1.m1.1c">\tau</annotation><annotation encoding="application/x-llamapun" id="A3.p4.1.m1.1d">italic_τ</annotation></semantics></math> and <math alttext="\alpha" class="ltx_Math" display="inline" id="A3.p4.2.m2.1"><semantics id="A3.p4.2.m2.1a"><mi id="A3.p4.2.m2.1.1" xref="A3.p4.2.m2.1.1.cmml">α</mi><annotation-xml encoding="MathML-Content" id="A3.p4.2.m2.1b"><ci id="A3.p4.2.m2.1.1.cmml" xref="A3.p4.2.m2.1.1">𝛼</ci></annotation-xml><annotation encoding="application/x-tex" id="A3.p4.2.m2.1c">\alpha</annotation><annotation encoding="application/x-llamapun" id="A3.p4.2.m2.1d">italic_α</annotation></semantics></math>. For <math alttext="\tau=0.5" class="ltx_Math" display="inline" id="A3.p4.3.m3.1"><semantics id="A3.p4.3.m3.1a"><mrow id="A3.p4.3.m3.1.1" xref="A3.p4.3.m3.1.1.cmml"><mi id="A3.p4.3.m3.1.1.2" xref="A3.p4.3.m3.1.1.2.cmml">τ</mi><mo id="A3.p4.3.m3.1.1.1" xref="A3.p4.3.m3.1.1.1.cmml">=</mo><mn id="A3.p4.3.m3.1.1.3" xref="A3.p4.3.m3.1.1.3.cmml">0.5</mn></mrow><annotation-xml encoding="MathML-Content" id="A3.p4.3.m3.1b"><apply id="A3.p4.3.m3.1.1.cmml" xref="A3.p4.3.m3.1.1"><eq id="A3.p4.3.m3.1.1.1.cmml" xref="A3.p4.3.m3.1.1.1"></eq><ci id="A3.p4.3.m3.1.1.2.cmml" xref="A3.p4.3.m3.1.1.2">𝜏</ci><cn id="A3.p4.3.m3.1.1.3.cmml" type="float" xref="A3.p4.3.m3.1.1.3">0.5</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.p4.3.m3.1c">\tau=0.5</annotation><annotation encoding="application/x-llamapun" id="A3.p4.3.m3.1d">italic_τ = 0.5</annotation></semantics></math>, <math alttext="\alpha=1.08" class="ltx_Math" display="inline" id="A3.p4.4.m4.1"><semantics id="A3.p4.4.m4.1a"><mrow id="A3.p4.4.m4.1.1" xref="A3.p4.4.m4.1.1.cmml"><mi id="A3.p4.4.m4.1.1.2" xref="A3.p4.4.m4.1.1.2.cmml">α</mi><mo id="A3.p4.4.m4.1.1.1" xref="A3.p4.4.m4.1.1.1.cmml">=</mo><mn id="A3.p4.4.m4.1.1.3" xref="A3.p4.4.m4.1.1.3.cmml">1.08</mn></mrow><annotation-xml encoding="MathML-Content" id="A3.p4.4.m4.1b"><apply id="A3.p4.4.m4.1.1.cmml" xref="A3.p4.4.m4.1.1"><eq id="A3.p4.4.m4.1.1.1.cmml" xref="A3.p4.4.m4.1.1.1"></eq><ci id="A3.p4.4.m4.1.1.2.cmml" xref="A3.p4.4.m4.1.1.2">𝛼</ci><cn id="A3.p4.4.m4.1.1.3.cmml" type="float" xref="A3.p4.4.m4.1.1.3">1.08</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.p4.4.m4.1c">\alpha=1.08</annotation><annotation encoding="application/x-llamapun" id="A3.p4.4.m4.1d">italic_α = 1.08</annotation></semantics></math> performs better than <math alttext="\alpha=1.06" class="ltx_Math" display="inline" id="A3.p4.5.m5.1"><semantics id="A3.p4.5.m5.1a"><mrow id="A3.p4.5.m5.1.1" xref="A3.p4.5.m5.1.1.cmml"><mi id="A3.p4.5.m5.1.1.2" xref="A3.p4.5.m5.1.1.2.cmml">α</mi><mo id="A3.p4.5.m5.1.1.1" xref="A3.p4.5.m5.1.1.1.cmml">=</mo><mn id="A3.p4.5.m5.1.1.3" xref="A3.p4.5.m5.1.1.3.cmml">1.06</mn></mrow><annotation-xml encoding="MathML-Content" id="A3.p4.5.m5.1b"><apply id="A3.p4.5.m5.1.1.cmml" xref="A3.p4.5.m5.1.1"><eq id="A3.p4.5.m5.1.1.1.cmml" xref="A3.p4.5.m5.1.1.1"></eq><ci id="A3.p4.5.m5.1.1.2.cmml" xref="A3.p4.5.m5.1.1.2">𝛼</ci><cn id="A3.p4.5.m5.1.1.3.cmml" type="float" xref="A3.p4.5.m5.1.1.3">1.06</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.p4.5.m5.1c">\alpha=1.06</annotation><annotation encoding="application/x-llamapun" id="A3.p4.5.m5.1d">italic_α = 1.06</annotation></semantics></math>, suggesting that in scenarios where the cluster scope is relatively constrained, the diversity of samples becomes pivotal, necessitating a higher <math alttext="\alpha" class="ltx_Math" display="inline" id="A3.p4.6.m6.1"><semantics id="A3.p4.6.m6.1a"><mi id="A3.p4.6.m6.1.1" xref="A3.p4.6.m6.1.1.cmml">α</mi><annotation-xml encoding="MathML-Content" id="A3.p4.6.m6.1b"><ci id="A3.p4.6.m6.1.1.cmml" xref="A3.p4.6.m6.1.1">𝛼</ci></annotation-xml><annotation encoding="application/x-tex" id="A3.p4.6.m6.1c">\alpha</annotation><annotation encoding="application/x-llamapun" id="A3.p4.6.m6.1d">italic_α</annotation></semantics></math> to effectively prioritize and capture heterogeneous behaviors. For <math alttext="\tau=0.7" class="ltx_Math" display="inline" id="A3.p4.7.m7.1"><semantics id="A3.p4.7.m7.1a"><mrow id="A3.p4.7.m7.1.1" xref="A3.p4.7.m7.1.1.cmml"><mi id="A3.p4.7.m7.1.1.2" xref="A3.p4.7.m7.1.1.2.cmml">τ</mi><mo id="A3.p4.7.m7.1.1.1" xref="A3.p4.7.m7.1.1.1.cmml">=</mo><mn id="A3.p4.7.m7.1.1.3" xref="A3.p4.7.m7.1.1.3.cmml">0.7</mn></mrow><annotation-xml encoding="MathML-Content" id="A3.p4.7.m7.1b"><apply id="A3.p4.7.m7.1.1.cmml" xref="A3.p4.7.m7.1.1"><eq id="A3.p4.7.m7.1.1.1.cmml" xref="A3.p4.7.m7.1.1.1"></eq><ci id="A3.p4.7.m7.1.1.2.cmml" xref="A3.p4.7.m7.1.1.2">𝜏</ci><cn id="A3.p4.7.m7.1.1.3.cmml" type="float" xref="A3.p4.7.m7.1.1.3">0.7</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.p4.7.m7.1c">\tau=0.7</annotation><annotation encoding="application/x-llamapun" id="A3.p4.7.m7.1d">italic_τ = 0.7</annotation></semantics></math>, <math alttext="\alpha=1.06" class="ltx_Math" display="inline" id="A3.p4.8.m8.1"><semantics id="A3.p4.8.m8.1a"><mrow id="A3.p4.8.m8.1.1" xref="A3.p4.8.m8.1.1.cmml"><mi id="A3.p4.8.m8.1.1.2" xref="A3.p4.8.m8.1.1.2.cmml">α</mi><mo id="A3.p4.8.m8.1.1.1" xref="A3.p4.8.m8.1.1.1.cmml">=</mo><mn id="A3.p4.8.m8.1.1.3" xref="A3.p4.8.m8.1.1.3.cmml">1.06</mn></mrow><annotation-xml encoding="MathML-Content" id="A3.p4.8.m8.1b"><apply id="A3.p4.8.m8.1.1.cmml" xref="A3.p4.8.m8.1.1"><eq id="A3.p4.8.m8.1.1.1.cmml" xref="A3.p4.8.m8.1.1.1"></eq><ci id="A3.p4.8.m8.1.1.2.cmml" xref="A3.p4.8.m8.1.1.2">𝛼</ci><cn id="A3.p4.8.m8.1.1.3.cmml" type="float" xref="A3.p4.8.m8.1.1.3">1.06</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.p4.8.m8.1c">\alpha=1.06</annotation><annotation encoding="application/x-llamapun" id="A3.p4.8.m8.1d">italic_α = 1.06</annotation></semantics></math> outperforms <math alttext="\alpha=1.08" class="ltx_Math" display="inline" id="A3.p4.9.m9.1"><semantics id="A3.p4.9.m9.1a"><mrow id="A3.p4.9.m9.1.1" xref="A3.p4.9.m9.1.1.cmml"><mi id="A3.p4.9.m9.1.1.2" xref="A3.p4.9.m9.1.1.2.cmml">α</mi><mo id="A3.p4.9.m9.1.1.1" xref="A3.p4.9.m9.1.1.1.cmml">=</mo><mn id="A3.p4.9.m9.1.1.3" xref="A3.p4.9.m9.1.1.3.cmml">1.08</mn></mrow><annotation-xml encoding="MathML-Content" id="A3.p4.9.m9.1b"><apply id="A3.p4.9.m9.1.1.cmml" xref="A3.p4.9.m9.1.1"><eq id="A3.p4.9.m9.1.1.1.cmml" xref="A3.p4.9.m9.1.1.1"></eq><ci id="A3.p4.9.m9.1.1.2.cmml" xref="A3.p4.9.m9.1.1.2">𝛼</ci><cn id="A3.p4.9.m9.1.1.3.cmml" type="float" xref="A3.p4.9.m9.1.1.3">1.08</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.p4.9.m9.1c">\alpha=1.08</annotation><annotation encoding="application/x-llamapun" id="A3.p4.9.m9.1d">italic_α = 1.08</annotation></semantics></math>, as the broader cluster scope with <math alttext="\alpha=1.08" class="ltx_Math" display="inline" id="A3.p4.10.m10.1"><semantics id="A3.p4.10.m10.1a"><mrow id="A3.p4.10.m10.1.1" xref="A3.p4.10.m10.1.1.cmml"><mi id="A3.p4.10.m10.1.1.2" xref="A3.p4.10.m10.1.1.2.cmml">α</mi><mo id="A3.p4.10.m10.1.1.1" xref="A3.p4.10.m10.1.1.1.cmml">=</mo><mn id="A3.p4.10.m10.1.1.3" xref="A3.p4.10.m10.1.1.3.cmml">1.08</mn></mrow><annotation-xml encoding="MathML-Content" id="A3.p4.10.m10.1b"><apply id="A3.p4.10.m10.1.1.cmml" xref="A3.p4.10.m10.1.1"><eq id="A3.p4.10.m10.1.1.1.cmml" xref="A3.p4.10.m10.1.1.1"></eq><ci id="A3.p4.10.m10.1.1.2.cmml" xref="A3.p4.10.m10.1.1.2">𝛼</ci><cn id="A3.p4.10.m10.1.1.3.cmml" type="float" xref="A3.p4.10.m10.1.1.3">1.08</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.p4.10.m10.1c">\alpha=1.08</annotation><annotation encoding="application/x-llamapun" id="A3.p4.10.m10.1d">italic_α = 1.08</annotation></semantics></math> potentially overemphasizes highly diverse samples, leading to a slight degradation in overall performance. This interplay underscores the importance of balancing cluster granularity and diversity during sample selection.</p> </div> <div class="ltx_para" id="A3.p5"> <p class="ltx_p" id="A3.p5.3"><span class="ltx_text ltx_font_bold" id="A3.p5.3.1">4. Parameter Robustness:</span> Our framework demonstrates robust performance across a wide range of hyper-parameter settings. For instance, the worst best performance (<math alttext="71.6" class="ltx_Math" display="inline" id="A3.p5.1.m1.1"><semantics id="A3.p5.1.m1.1a"><mn id="A3.p5.1.m1.1.1" xref="A3.p5.1.m1.1.1.cmml">71.6</mn><annotation-xml encoding="MathML-Content" id="A3.p5.1.m1.1b"><cn id="A3.p5.1.m1.1.1.cmml" type="float" xref="A3.p5.1.m1.1.1">71.6</cn></annotation-xml><annotation encoding="application/x-tex" id="A3.p5.1.m1.1c">71.6</annotation><annotation encoding="application/x-llamapun" id="A3.p5.1.m1.1d">71.6</annotation></semantics></math>) achieved with <math alttext="\tau=0.7,\alpha=1.04" class="ltx_Math" display="inline" id="A3.p5.2.m2.2"><semantics id="A3.p5.2.m2.2a"><mrow id="A3.p5.2.m2.2.2.2" xref="A3.p5.2.m2.2.2.3.cmml"><mrow id="A3.p5.2.m2.1.1.1.1" xref="A3.p5.2.m2.1.1.1.1.cmml"><mi id="A3.p5.2.m2.1.1.1.1.2" xref="A3.p5.2.m2.1.1.1.1.2.cmml">τ</mi><mo id="A3.p5.2.m2.1.1.1.1.1" xref="A3.p5.2.m2.1.1.1.1.1.cmml">=</mo><mn id="A3.p5.2.m2.1.1.1.1.3" xref="A3.p5.2.m2.1.1.1.1.3.cmml">0.7</mn></mrow><mo id="A3.p5.2.m2.2.2.2.3" xref="A3.p5.2.m2.2.2.3a.cmml">,</mo><mrow id="A3.p5.2.m2.2.2.2.2" xref="A3.p5.2.m2.2.2.2.2.cmml"><mi id="A3.p5.2.m2.2.2.2.2.2" xref="A3.p5.2.m2.2.2.2.2.2.cmml">α</mi><mo id="A3.p5.2.m2.2.2.2.2.1" xref="A3.p5.2.m2.2.2.2.2.1.cmml">=</mo><mn id="A3.p5.2.m2.2.2.2.2.3" xref="A3.p5.2.m2.2.2.2.2.3.cmml">1.04</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="A3.p5.2.m2.2b"><apply id="A3.p5.2.m2.2.2.3.cmml" xref="A3.p5.2.m2.2.2.2"><csymbol cd="ambiguous" id="A3.p5.2.m2.2.2.3a.cmml" xref="A3.p5.2.m2.2.2.2.3">formulae-sequence</csymbol><apply id="A3.p5.2.m2.1.1.1.1.cmml" xref="A3.p5.2.m2.1.1.1.1"><eq id="A3.p5.2.m2.1.1.1.1.1.cmml" xref="A3.p5.2.m2.1.1.1.1.1"></eq><ci id="A3.p5.2.m2.1.1.1.1.2.cmml" xref="A3.p5.2.m2.1.1.1.1.2">𝜏</ci><cn id="A3.p5.2.m2.1.1.1.1.3.cmml" type="float" xref="A3.p5.2.m2.1.1.1.1.3">0.7</cn></apply><apply id="A3.p5.2.m2.2.2.2.2.cmml" xref="A3.p5.2.m2.2.2.2.2"><eq id="A3.p5.2.m2.2.2.2.2.1.cmml" xref="A3.p5.2.m2.2.2.2.2.1"></eq><ci id="A3.p5.2.m2.2.2.2.2.2.cmml" xref="A3.p5.2.m2.2.2.2.2.2">𝛼</ci><cn id="A3.p5.2.m2.2.2.2.2.3.cmml" type="float" xref="A3.p5.2.m2.2.2.2.2.3">1.04</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A3.p5.2.m2.2c">\tau=0.7,\alpha=1.04</annotation><annotation encoding="application/x-llamapun" id="A3.p5.2.m2.2d">italic_τ = 0.7 , italic_α = 1.04</annotation></semantics></math> is only marginally lower than the best performance of the relevance baseline (<math alttext="71.86" class="ltx_Math" display="inline" id="A3.p5.3.m3.1"><semantics id="A3.p5.3.m3.1a"><mn id="A3.p5.3.m3.1.1" xref="A3.p5.3.m3.1.1.cmml">71.86</mn><annotation-xml encoding="MathML-Content" id="A3.p5.3.m3.1b"><cn id="A3.p5.3.m3.1.1.cmml" type="float" xref="A3.p5.3.m3.1.1">71.86</cn></annotation-xml><annotation encoding="application/x-tex" id="A3.p5.3.m3.1c">71.86</annotation><annotation encoding="application/x-llamapun" id="A3.p5.3.m3.1d">71.86</annotation></semantics></math>). This indicates that our method remains effective without being overly sensitive to hyper-parameter adjustments.</p> </div> <div class="ltx_para" id="A3.p6"> <p class="ltx_p" id="A3.p6.2">To provide an intuitive analysis of the sampling process, we conducted a visualization study, as illustrated in Figure <a class="ltx_ref" href="https://arxiv.org/html/2503.02398v1#A2.F5" title="Figure 5 ‣ Appendix B Backbone Methods ‣ PersonaX: A Recommendation Agent-Oriented User Modeling Framework for Long Behavior Sequence"><span class="ltx_text ltx_ref_tag">5</span></a>. From Figure <a class="ltx_ref" href="https://arxiv.org/html/2503.02398v1#A2.F5" title="Figure 5 ‣ Appendix B Backbone Methods ‣ PersonaX: A Recommendation Agent-Oriented User Modeling Framework for Long Behavior Sequence"><span class="ltx_text ltx_ref_tag">5</span></a>.A, it is evident that smaller clusters are preferentially allocated an adequate sampling quota compared to larger ones. This observation underscores the efficacy of the proposed Algorithm <a class="ltx_ref" href="https://arxiv.org/html/2503.02398v1#alg1" title="Algorithm 1 ‣ 3.2 Sampling Budget Allocation ‣ 3 Method ‣ PersonaX: A Recommendation Agent-Oriented User Modeling Framework for Long Behavior Sequence"><span class="ltx_text ltx_ref_tag">1</span></a>, which strategically prioritizes smaller clusters to ensure sufficient sampling. By adopting this approach, the algorithm effectively preserves the user’s diverse interests, including long-tail preferences, even under constrained sampling resources. The comparisons between Figure <a class="ltx_ref" href="https://arxiv.org/html/2503.02398v1#A2.F5" title="Figure 5 ‣ Appendix B Backbone Methods ‣ PersonaX: A Recommendation Agent-Oriented User Modeling Framework for Long Behavior Sequence"><span class="ltx_text ltx_ref_tag">5</span></a>.B and Figure <a class="ltx_ref" href="https://arxiv.org/html/2503.02398v1#A2.F5" title="Figure 5 ‣ Appendix B Backbone Methods ‣ PersonaX: A Recommendation Agent-Oriented User Modeling Framework for Long Behavior Sequence"><span class="ltx_text ltx_ref_tag">5</span></a>.C, as well as Figure <a class="ltx_ref" href="https://arxiv.org/html/2503.02398v1#A2.F5" title="Figure 5 ‣ Appendix B Backbone Methods ‣ PersonaX: A Recommendation Agent-Oriented User Modeling Framework for Long Behavior Sequence"><span class="ltx_text ltx_ref_tag">5</span></a>.D and Figure <a class="ltx_ref" href="https://arxiv.org/html/2503.02398v1#A2.F5" title="Figure 5 ‣ Appendix B Backbone Methods ‣ PersonaX: A Recommendation Agent-Oriented User Modeling Framework for Long Behavior Sequence"><span class="ltx_text ltx_ref_tag">5</span></a>.E, highlight the impact of <math alttext="\alpha" class="ltx_Math" display="inline" id="A3.p6.1.m1.1"><semantics id="A3.p6.1.m1.1a"><mi id="A3.p6.1.m1.1.1" xref="A3.p6.1.m1.1.1.cmml">α</mi><annotation-xml encoding="MathML-Content" id="A3.p6.1.m1.1b"><ci id="A3.p6.1.m1.1.1.cmml" xref="A3.p6.1.m1.1.1">𝛼</ci></annotation-xml><annotation encoding="application/x-tex" id="A3.p6.1.m1.1c">\alpha</annotation><annotation encoding="application/x-llamapun" id="A3.p6.1.m1.1d">italic_α</annotation></semantics></math>. Specifically, smaller <math alttext="\alpha" class="ltx_Math" display="inline" id="A3.p6.2.m2.1"><semantics id="A3.p6.2.m2.1a"><mi id="A3.p6.2.m2.1.1" xref="A3.p6.2.m2.1.1.cmml">α</mi><annotation-xml encoding="MathML-Content" id="A3.p6.2.m2.1b"><ci id="A3.p6.2.m2.1.1.cmml" xref="A3.p6.2.m2.1.1">𝛼</ci></annotation-xml><annotation encoding="application/x-tex" id="A3.p6.2.m2.1c">\alpha</annotation><annotation encoding="application/x-llamapun" id="A3.p6.2.m2.1d">italic_α</annotation></semantics></math> values tend to focus the sample selection closer to the cluster centroids. Furthermore, the comparisons between Figure <a class="ltx_ref" href="https://arxiv.org/html/2503.02398v1#A2.F5" title="Figure 5 ‣ Appendix B Backbone Methods ‣ PersonaX: A Recommendation Agent-Oriented User Modeling Framework for Long Behavior Sequence"><span class="ltx_text ltx_ref_tag">5</span></a>.B and Figure <a class="ltx_ref" href="https://arxiv.org/html/2503.02398v1#A2.F5" title="Figure 5 ‣ Appendix B Backbone Methods ‣ PersonaX: A Recommendation Agent-Oriented User Modeling Framework for Long Behavior Sequence"><span class="ltx_text ltx_ref_tag">5</span></a>.D, and between Figure <a class="ltx_ref" href="https://arxiv.org/html/2503.02398v1#A2.F5" title="Figure 5 ‣ Appendix B Backbone Methods ‣ PersonaX: A Recommendation Agent-Oriented User Modeling Framework for Long Behavior Sequence"><span class="ltx_text ltx_ref_tag">5</span></a>.C and Figure <a class="ltx_ref" href="https://arxiv.org/html/2503.02398v1#A2.F5" title="Figure 5 ‣ Appendix B Backbone Methods ‣ PersonaX: A Recommendation Agent-Oriented User Modeling Framework for Long Behavior Sequence"><span class="ltx_text ltx_ref_tag">5</span></a>.E, demonstrate that a a more granular clustering can constrain Algorithm <a class="ltx_ref" href="https://arxiv.org/html/2503.02398v1#alg2" title="Algorithm 2 ‣ 3.2 Sampling Budget Allocation ‣ 3 Method ‣ PersonaX: A Recommendation Agent-Oriented User Modeling Framework for Long Behavior Sequence"><span class="ltx_text ltx_ref_tag">2</span></a> from selecting samples that deviate excessively from the cluster centroids. This constraint mitigates potential performance degradation caused by overemphasis on unrelated samples.</p> </div> <div class="ltx_para" id="A3.p7"> <p class="ltx_p" id="A3.p7.4">The experimental findings and visualization analysis suggest that both <math alttext="\tau" class="ltx_Math" display="inline" id="A3.p7.1.m1.1"><semantics id="A3.p7.1.m1.1a"><mi id="A3.p7.1.m1.1.1" xref="A3.p7.1.m1.1.1.cmml">τ</mi><annotation-xml encoding="MathML-Content" id="A3.p7.1.m1.1b"><ci id="A3.p7.1.m1.1.1.cmml" xref="A3.p7.1.m1.1.1">𝜏</ci></annotation-xml><annotation encoding="application/x-tex" id="A3.p7.1.m1.1c">\tau</annotation><annotation encoding="application/x-llamapun" id="A3.p7.1.m1.1d">italic_τ</annotation></semantics></math> and <math alttext="\alpha" class="ltx_Math" display="inline" id="A3.p7.2.m2.1"><semantics id="A3.p7.2.m2.1a"><mi id="A3.p7.2.m2.1.1" xref="A3.p7.2.m2.1.1.cmml">α</mi><annotation-xml encoding="MathML-Content" id="A3.p7.2.m2.1b"><ci id="A3.p7.2.m2.1.1.cmml" xref="A3.p7.2.m2.1.1">𝛼</ci></annotation-xml><annotation encoding="application/x-tex" id="A3.p7.2.m2.1c">\alpha</annotation><annotation encoding="application/x-llamapun" id="A3.p7.2.m2.1d">italic_α</annotation></semantics></math> require empirical tuning to identify optimal configurations. We recommended a balance between prototypicality and diversity, for example a larger <math alttext="\alpha" class="ltx_Math" display="inline" id="A3.p7.3.m3.1"><semantics id="A3.p7.3.m3.1a"><mi id="A3.p7.3.m3.1.1" xref="A3.p7.3.m3.1.1.cmml">α</mi><annotation-xml encoding="MathML-Content" id="A3.p7.3.m3.1b"><ci id="A3.p7.3.m3.1.1.cmml" xref="A3.p7.3.m3.1.1">𝛼</ci></annotation-xml><annotation encoding="application/x-tex" id="A3.p7.3.m3.1c">\alpha</annotation><annotation encoding="application/x-llamapun" id="A3.p7.3.m3.1d">italic_α</annotation></semantics></math> values combined with appropriately tuned small <math alttext="\tau" class="ltx_Math" display="inline" id="A3.p7.4.m4.1"><semantics id="A3.p7.4.m4.1a"><mi id="A3.p7.4.m4.1.1" xref="A3.p7.4.m4.1.1.cmml">τ</mi><annotation-xml encoding="MathML-Content" id="A3.p7.4.m4.1b"><ci id="A3.p7.4.m4.1.1.cmml" xref="A3.p7.4.m4.1.1">𝜏</ci></annotation-xml><annotation encoding="application/x-tex" id="A3.p7.4.m4.1c">\tau</annotation><annotation encoding="application/x-llamapun" id="A3.p7.4.m4.1d">italic_τ</annotation></semantics></math>.</p> </div> </section> <section class="ltx_appendix" id="A4"> <h2 class="ltx_title ltx_title_appendix"> <span class="ltx_tag ltx_tag_appendix">Appendix D </span>Details about In-Cluter Selection</h2> <div class="ltx_para" id="A4.p1"> <p class="ltx_p" id="A4.p1.1">In this section, we delve into the mechanisms governing sample selection by proposing a principled scoring system to evaluate the prototypicality and diversity of candidate samples. The scoring mechanism is derived from two complementary perspectives: prototypicality</p> <table class="ltx_equation ltx_eqn_table" id="A4.Ex6"> <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="\frac{1}{1+d(\mathbf{e}_{j},\mathbf{\mu}_{i})}" class="ltx_Math" display="block" id="A4.Ex6.m1.2"><semantics id="A4.Ex6.m1.2a"><mfrac id="A4.Ex6.m1.2.2" xref="A4.Ex6.m1.2.2.cmml"><mn id="A4.Ex6.m1.2.2.4" xref="A4.Ex6.m1.2.2.4.cmml">1</mn><mrow id="A4.Ex6.m1.2.2.2" xref="A4.Ex6.m1.2.2.2.cmml"><mn id="A4.Ex6.m1.2.2.2.4" xref="A4.Ex6.m1.2.2.2.4.cmml">1</mn><mo id="A4.Ex6.m1.2.2.2.3" xref="A4.Ex6.m1.2.2.2.3.cmml">+</mo><mrow id="A4.Ex6.m1.2.2.2.2" xref="A4.Ex6.m1.2.2.2.2.cmml"><mi id="A4.Ex6.m1.2.2.2.2.4" xref="A4.Ex6.m1.2.2.2.2.4.cmml">d</mi><mo id="A4.Ex6.m1.2.2.2.2.3" xref="A4.Ex6.m1.2.2.2.2.3.cmml">⁢</mo><mrow id="A4.Ex6.m1.2.2.2.2.2.2" 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ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\frac{2}{a_{i}}\sum_{\begin{subarray}{c}I_{a},I_{b}\in c_{i}^{*}\\ a\neq b\end{subarray}}d(\mathbf{e}_{a},\mathbf{e}_{b})" class="ltx_Math" display="block" id="A4.Ex7.m1.3"><semantics id="A4.Ex7.m1.3a"><mrow id="A4.Ex7.m1.3.3" xref="A4.Ex7.m1.3.3.cmml"><mfrac id="A4.Ex7.m1.3.3.4" xref="A4.Ex7.m1.3.3.4.cmml"><mn id="A4.Ex7.m1.3.3.4.2" xref="A4.Ex7.m1.3.3.4.2.cmml">2</mn><msub id="A4.Ex7.m1.3.3.4.3" xref="A4.Ex7.m1.3.3.4.3.cmml"><mi id="A4.Ex7.m1.3.3.4.3.2" xref="A4.Ex7.m1.3.3.4.3.2.cmml">a</mi><mi id="A4.Ex7.m1.3.3.4.3.3" xref="A4.Ex7.m1.3.3.4.3.3.cmml">i</mi></msub></mfrac><mo id="A4.Ex7.m1.3.3.3" xref="A4.Ex7.m1.3.3.3.cmml">⁢</mo><mrow id="A4.Ex7.m1.3.3.2" xref="A4.Ex7.m1.3.3.2.cmml"><munder id="A4.Ex7.m1.3.3.2.3" xref="A4.Ex7.m1.3.3.2.3.cmml"><mo id="A4.Ex7.m1.3.3.2.3.2" movablelimits="false" xref="A4.Ex7.m1.3.3.2.3.2.cmml">∑</mo><mtable 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encoding="application/x-llamapun" id="A4.Ex7.m1.3d">divide start_ARG 2 end_ARG start_ARG italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_ARG ∑ start_POSTSUBSCRIPT start_ARG start_ROW start_CELL italic_I start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT , italic_I start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT ∈ italic_c start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT end_CELL end_ROW start_ROW start_CELL italic_a ≠ italic_b end_CELL end_ROW end_ARG end_POSTSUBSCRIPT italic_d ( bold_e start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT , bold_e start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="A4.p1.3">, which quantifies the extent to which the selected samples span a broader spectrum of the data distribution.</p> </div> <section class="ltx_subsection" id="A4.SS1"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">D.1 </span>Prototypicality and Diversity Scoring</h3> <div class="ltx_para" id="A4.SS1.p1"> <p class="ltx_p" id="A4.SS1.p1.1">From the formulation below,</p> <table class="ltx_equationgroup ltx_eqn_table" id="A4.Ex8"> <tbody id="A4.Ex8X"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\max_{c_{i}^{*}}\Biggl{(}" class="ltx_math_unparsed" display="inline" id="A4.Ex8X.2.1.1.m1.1"><semantics id="A4.Ex8X.2.1.1.m1.1a"><mrow id="A4.Ex8X.2.1.1.m1.1b"><munder id="A4.Ex8X.2.1.1.m1.1.1"><mi id="A4.Ex8X.2.1.1.m1.1.1.2">max</mi><msubsup id="A4.Ex8X.2.1.1.m1.1.1.3"><mi id="A4.Ex8X.2.1.1.m1.1.1.3.2.2">c</mi><mi id="A4.Ex8X.2.1.1.m1.1.1.3.2.3">i</mi><mo id="A4.Ex8X.2.1.1.m1.1.1.3.3">∗</mo></msubsup></munder><mo id="A4.Ex8X.2.1.1.m1.1.2" maxsize="260%" minsize="260%">(</mo></mrow><annotation encoding="application/x-tex" id="A4.Ex8X.2.1.1.m1.1c">\displaystyle\max_{c_{i}^{*}}\Biggl{(}</annotation><annotation encoding="application/x-llamapun" id="A4.Ex8X.2.1.1.m1.1d">roman_max start_POSTSUBSCRIPT italic_c start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT (</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle w_{p}\cdot\sum_{I_{j}\in c_{i}^{*}}\frac{1}{1+d(\mathbf{e}_{j},% \mathbf{\mu}_{i})}+w_{d}\cdot\frac{2}{a_{i}}\sum_{\begin{subarray}{c}I_{a},I_{% b}\in c_{i}^{*}\\ a\neq b\end{subarray}}d(\mathbf{e}_{a},\mathbf{e}_{b})\Biggr{)}" class="ltx_math_unparsed" display="inline" id="A4.Ex8X.3.2.2.m1.2"><semantics id="A4.Ex8X.3.2.2.m1.2a"><mrow id="A4.Ex8X.3.2.2.m1.2b"><msub id="A4.Ex8X.3.2.2.m1.2.3"><mi id="A4.Ex8X.3.2.2.m1.2.3.2">w</mi><mi id="A4.Ex8X.3.2.2.m1.2.3.3">p</mi></msub><mo id="A4.Ex8X.3.2.2.m1.2.4" lspace="0.222em" rspace="0.222em">⋅</mo><mstyle displaystyle="true" 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id="A4.Ex8X.3.2.2.m1.2.13" maxsize="260%" minsize="260%">)</mo></mrow><annotation encoding="application/x-tex" id="A4.Ex8X.3.2.2.m1.2c">\displaystyle w_{p}\cdot\sum_{I_{j}\in c_{i}^{*}}\frac{1}{1+d(\mathbf{e}_{j},% \mathbf{\mu}_{i})}+w_{d}\cdot\frac{2}{a_{i}}\sum_{\begin{subarray}{c}I_{a},I_{% b}\in c_{i}^{*}\\ a\neq b\end{subarray}}d(\mathbf{e}_{a},\mathbf{e}_{b})\Biggr{)}</annotation><annotation encoding="application/x-llamapun" id="A4.Ex8X.3.2.2.m1.2d">italic_w start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT ⋅ ∑ start_POSTSUBSCRIPT italic_I start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ∈ italic_c start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT divide start_ARG 1 end_ARG start_ARG 1 + italic_d ( bold_e start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT , italic_μ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) end_ARG + italic_w start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT ⋅ divide start_ARG 2 end_ARG start_ARG italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_ARG ∑ start_POSTSUBSCRIPT start_ARG start_ROW start_CELL italic_I start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT , italic_I start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT ∈ italic_c start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT end_CELL end_ROW start_ROW start_CELL italic_a ≠ italic_b end_CELL end_ROW end_ARG end_POSTSUBSCRIPT italic_d ( bold_e start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT , bold_e start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT ) )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="A4.SS1.p1.2">it is evident that the prototypicality score exhibits an inverse relationship with the distance between a sample and the center of its cluster. As a sample moves further from the cluster centroid, its prototypicality diminishes proportionally, reflecting its reduced ability to represent the typical characteristics of the cluster. The diversity score considers the pairwise distances between the candidate sample and the samples already selected. This ensures that the inclusion of a new sample enriches the diversity of the chosen subset by discouraging redundancy.</p> </div> <div class="ltx_para" id="A4.SS1.p2"> <p class="ltx_p" id="A4.SS1.p2.6">To compute the diversity score, we employ the scaling factor <math alttext="2/a_{i}" class="ltx_Math" display="inline" id="A4.SS1.p2.1.m1.1"><semantics id="A4.SS1.p2.1.m1.1a"><mrow id="A4.SS1.p2.1.m1.1.1" xref="A4.SS1.p2.1.m1.1.1.cmml"><mn id="A4.SS1.p2.1.m1.1.1.2" xref="A4.SS1.p2.1.m1.1.1.2.cmml">2</mn><mo id="A4.SS1.p2.1.m1.1.1.1" xref="A4.SS1.p2.1.m1.1.1.1.cmml">/</mo><msub id="A4.SS1.p2.1.m1.1.1.3" xref="A4.SS1.p2.1.m1.1.1.3.cmml"><mi id="A4.SS1.p2.1.m1.1.1.3.2" xref="A4.SS1.p2.1.m1.1.1.3.2.cmml">a</mi><mi id="A4.SS1.p2.1.m1.1.1.3.3" xref="A4.SS1.p2.1.m1.1.1.3.3.cmml">i</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="A4.SS1.p2.1.m1.1b"><apply id="A4.SS1.p2.1.m1.1.1.cmml" xref="A4.SS1.p2.1.m1.1.1"><divide id="A4.SS1.p2.1.m1.1.1.1.cmml" xref="A4.SS1.p2.1.m1.1.1.1"></divide><cn id="A4.SS1.p2.1.m1.1.1.2.cmml" type="integer" xref="A4.SS1.p2.1.m1.1.1.2">2</cn><apply id="A4.SS1.p2.1.m1.1.1.3.cmml" xref="A4.SS1.p2.1.m1.1.1.3"><csymbol cd="ambiguous" id="A4.SS1.p2.1.m1.1.1.3.1.cmml" xref="A4.SS1.p2.1.m1.1.1.3">subscript</csymbol><ci id="A4.SS1.p2.1.m1.1.1.3.2.cmml" xref="A4.SS1.p2.1.m1.1.1.3.2">𝑎</ci><ci id="A4.SS1.p2.1.m1.1.1.3.3.cmml" xref="A4.SS1.p2.1.m1.1.1.3.3">𝑖</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.SS1.p2.1.m1.1c">2/a_{i}</annotation><annotation encoding="application/x-llamapun" id="A4.SS1.p2.1.m1.1d">2 / italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math>. 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By adopting <math alttext="2/a_{i}" class="ltx_Math" display="inline" id="A4.SS1.p2.3.m3.1"><semantics id="A4.SS1.p2.3.m3.1a"><mrow id="A4.SS1.p2.3.m3.1.1" xref="A4.SS1.p2.3.m3.1.1.cmml"><mn id="A4.SS1.p2.3.m3.1.1.2" xref="A4.SS1.p2.3.m3.1.1.2.cmml">2</mn><mo id="A4.SS1.p2.3.m3.1.1.1" xref="A4.SS1.p2.3.m3.1.1.1.cmml">/</mo><msub id="A4.SS1.p2.3.m3.1.1.3" xref="A4.SS1.p2.3.m3.1.1.3.cmml"><mi id="A4.SS1.p2.3.m3.1.1.3.2" xref="A4.SS1.p2.3.m3.1.1.3.2.cmml">a</mi><mi id="A4.SS1.p2.3.m3.1.1.3.3" xref="A4.SS1.p2.3.m3.1.1.3.3.cmml">i</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="A4.SS1.p2.3.m3.1b"><apply id="A4.SS1.p2.3.m3.1.1.cmml" xref="A4.SS1.p2.3.m3.1.1"><divide id="A4.SS1.p2.3.m3.1.1.1.cmml" xref="A4.SS1.p2.3.m3.1.1.1"></divide><cn id="A4.SS1.p2.3.m3.1.1.2.cmml" type="integer" xref="A4.SS1.p2.3.m3.1.1.2">2</cn><apply id="A4.SS1.p2.3.m3.1.1.3.cmml" xref="A4.SS1.p2.3.m3.1.1.3"><csymbol cd="ambiguous" id="A4.SS1.p2.3.m3.1.1.3.1.cmml" xref="A4.SS1.p2.3.m3.1.1.3">subscript</csymbol><ci id="A4.SS1.p2.3.m3.1.1.3.2.cmml" xref="A4.SS1.p2.3.m3.1.1.3.2">𝑎</ci><ci id="A4.SS1.p2.3.m3.1.1.3.3.cmml" xref="A4.SS1.p2.3.m3.1.1.3.3">𝑖</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.SS1.p2.3.m3.1c">2/a_{i}</annotation><annotation encoding="application/x-llamapun" id="A4.SS1.p2.3.m3.1d">2 / italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math>, we deliberately amplify the influence of diversity as <math alttext="a_{i}" class="ltx_Math" display="inline" id="A4.SS1.p2.4.m4.1"><semantics id="A4.SS1.p2.4.m4.1a"><msub id="A4.SS1.p2.4.m4.1.1" xref="A4.SS1.p2.4.m4.1.1.cmml"><mi id="A4.SS1.p2.4.m4.1.1.2" xref="A4.SS1.p2.4.m4.1.1.2.cmml">a</mi><mi id="A4.SS1.p2.4.m4.1.1.3" xref="A4.SS1.p2.4.m4.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="A4.SS1.p2.4.m4.1b"><apply id="A4.SS1.p2.4.m4.1.1.cmml" xref="A4.SS1.p2.4.m4.1.1"><csymbol cd="ambiguous" id="A4.SS1.p2.4.m4.1.1.1.cmml" xref="A4.SS1.p2.4.m4.1.1">subscript</csymbol><ci id="A4.SS1.p2.4.m4.1.1.2.cmml" xref="A4.SS1.p2.4.m4.1.1.2">𝑎</ci><ci id="A4.SS1.p2.4.m4.1.1.3.cmml" xref="A4.SS1.p2.4.m4.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.SS1.p2.4.m4.1c">a_{i}</annotation><annotation encoding="application/x-llamapun" id="A4.SS1.p2.4.m4.1d">italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> increases, thereby prioritizing the inclusion of diverse samples in scenarios where a cluster is allocated enough sampling budget. This design reflects an underlying intent: as <math alttext="a_{i}" class="ltx_Math" display="inline" id="A4.SS1.p2.5.m5.1"><semantics id="A4.SS1.p2.5.m5.1a"><msub id="A4.SS1.p2.5.m5.1.1" xref="A4.SS1.p2.5.m5.1.1.cmml"><mi id="A4.SS1.p2.5.m5.1.1.2" xref="A4.SS1.p2.5.m5.1.1.2.cmml">a</mi><mi id="A4.SS1.p2.5.m5.1.1.3" xref="A4.SS1.p2.5.m5.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="A4.SS1.p2.5.m5.1b"><apply id="A4.SS1.p2.5.m5.1.1.cmml" xref="A4.SS1.p2.5.m5.1.1"><csymbol cd="ambiguous" id="A4.SS1.p2.5.m5.1.1.1.cmml" xref="A4.SS1.p2.5.m5.1.1">subscript</csymbol><ci id="A4.SS1.p2.5.m5.1.1.2.cmml" xref="A4.SS1.p2.5.m5.1.1.2">𝑎</ci><ci id="A4.SS1.p2.5.m5.1.1.3.cmml" xref="A4.SS1.p2.5.m5.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.SS1.p2.5.m5.1c">a_{i}</annotation><annotation encoding="application/x-llamapun" id="A4.SS1.p2.5.m5.1d">italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> grows, the system places greater emphasis on diversity to ensure comprehensive coverage of the data distribution. Conversely, when <math alttext="a_{i}" class="ltx_Math" display="inline" id="A4.SS1.p2.6.m6.1"><semantics id="A4.SS1.p2.6.m6.1a"><msub id="A4.SS1.p2.6.m6.1.1" xref="A4.SS1.p2.6.m6.1.1.cmml"><mi id="A4.SS1.p2.6.m6.1.1.2" xref="A4.SS1.p2.6.m6.1.1.2.cmml">a</mi><mi id="A4.SS1.p2.6.m6.1.1.3" xref="A4.SS1.p2.6.m6.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="A4.SS1.p2.6.m6.1b"><apply id="A4.SS1.p2.6.m6.1.1.cmml" xref="A4.SS1.p2.6.m6.1.1"><csymbol cd="ambiguous" id="A4.SS1.p2.6.m6.1.1.1.cmml" xref="A4.SS1.p2.6.m6.1.1">subscript</csymbol><ci id="A4.SS1.p2.6.m6.1.1.2.cmml" xref="A4.SS1.p2.6.m6.1.1.2">𝑎</ci><ci id="A4.SS1.p2.6.m6.1.1.3.cmml" xref="A4.SS1.p2.6.m6.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.SS1.p2.6.m6.1c">a_{i}</annotation><annotation encoding="application/x-llamapun" id="A4.SS1.p2.6.m6.1d">italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> is small, prototypicality takes precedence, directing attention toward selecting samples that are most representative of their respective clusters.</p> </div> </section> <section class="ltx_subsection" id="A4.SS2"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">D.2 </span>Design Rationale</h3> <div class="ltx_para" id="A4.SS2.p1"> <p class="ltx_p" id="A4.SS2.p1.1">The decision to amplify diversity dynamically aligns with our broader goal of achieving a balanced and adaptive sample selection process. By coupling prototypicality with diversity in this manner, we address two critical challenges in data selection:</p> </div> <div class="ltx_para" id="A4.SS2.p2"> <p class="ltx_p" id="A4.SS2.p2.1">1. <span class="ltx_text ltx_font_bold" id="A4.SS2.p2.1.1">Representative Sampling</span>: When the sample pool is sparse, selecting highly prototypical samples ensures that the chosen subset faithfully captures the core characteristics of the data clusters. This is particularly crucial in tasks where the representativeness of the selected data has a direct impact on model performance, such as user profiling or content recommendation.</p> </div> <div class="ltx_para" id="A4.SS2.p3"> <p class="ltx_p" id="A4.SS2.p3.1">2. <span class="ltx_text ltx_font_bold" id="A4.SS2.p3.1.1">Comprehensive Coverage</span>: In cases where the candidate pool is dense, diversity becomes increasingly important to avoid redundancy and to capture the subtle variations within the data distribution. By amplifying diversity when <math alttext="a_{i}" class="ltx_Math" display="inline" id="A4.SS2.p3.1.m1.1"><semantics id="A4.SS2.p3.1.m1.1a"><msub id="A4.SS2.p3.1.m1.1.1" xref="A4.SS2.p3.1.m1.1.1.cmml"><mi id="A4.SS2.p3.1.m1.1.1.2" xref="A4.SS2.p3.1.m1.1.1.2.cmml">a</mi><mi id="A4.SS2.p3.1.m1.1.1.3" xref="A4.SS2.p3.1.m1.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="A4.SS2.p3.1.m1.1b"><apply id="A4.SS2.p3.1.m1.1.1.cmml" xref="A4.SS2.p3.1.m1.1.1"><csymbol cd="ambiguous" id="A4.SS2.p3.1.m1.1.1.1.cmml" xref="A4.SS2.p3.1.m1.1.1">subscript</csymbol><ci id="A4.SS2.p3.1.m1.1.1.2.cmml" xref="A4.SS2.p3.1.m1.1.1.2">𝑎</ci><ci id="A4.SS2.p3.1.m1.1.1.3.cmml" xref="A4.SS2.p3.1.m1.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.SS2.p3.1.m1.1c">a_{i}</annotation><annotation encoding="application/x-llamapun" id="A4.SS2.p3.1.m1.1d">italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> is large, our scoring mechanism ensures that the selected subset spans the breadth of the distribution, enabling downstream models to generalize better across diverse scenarios.</p> </div> <figure class="ltx_figure" id="A4.F6"><img alt="Refer to caption" class="ltx_graphics ltx_centering ltx_img_landscape" height="265" id="A4.F6.g1" src="x6.png" width="346"/> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure">Figure 6: </span>Trade off between <math alttext="w_{p}" class="ltx_Math" display="inline" id="A4.F6.4.m1.1"><semantics id="A4.F6.4.m1.1b"><msub id="A4.F6.4.m1.1.1" xref="A4.F6.4.m1.1.1.cmml"><mi id="A4.F6.4.m1.1.1.2" xref="A4.F6.4.m1.1.1.2.cmml">w</mi><mi id="A4.F6.4.m1.1.1.3" xref="A4.F6.4.m1.1.1.3.cmml">p</mi></msub><annotation-xml encoding="MathML-Content" id="A4.F6.4.m1.1c"><apply id="A4.F6.4.m1.1.1.cmml" xref="A4.F6.4.m1.1.1"><csymbol cd="ambiguous" id="A4.F6.4.m1.1.1.1.cmml" xref="A4.F6.4.m1.1.1">subscript</csymbol><ci id="A4.F6.4.m1.1.1.2.cmml" xref="A4.F6.4.m1.1.1.2">𝑤</ci><ci id="A4.F6.4.m1.1.1.3.cmml" xref="A4.F6.4.m1.1.1.3">𝑝</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.F6.4.m1.1d">w_{p}</annotation><annotation encoding="application/x-llamapun" id="A4.F6.4.m1.1e">italic_w start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="w_{d}" class="ltx_Math" display="inline" id="A4.F6.5.m2.1"><semantics id="A4.F6.5.m2.1b"><msub id="A4.F6.5.m2.1.1" xref="A4.F6.5.m2.1.1.cmml"><mi id="A4.F6.5.m2.1.1.2" xref="A4.F6.5.m2.1.1.2.cmml">w</mi><mi id="A4.F6.5.m2.1.1.3" xref="A4.F6.5.m2.1.1.3.cmml">d</mi></msub><annotation-xml encoding="MathML-Content" id="A4.F6.5.m2.1c"><apply id="A4.F6.5.m2.1.1.cmml" xref="A4.F6.5.m2.1.1"><csymbol cd="ambiguous" id="A4.F6.5.m2.1.1.1.cmml" xref="A4.F6.5.m2.1.1">subscript</csymbol><ci id="A4.F6.5.m2.1.1.2.cmml" xref="A4.F6.5.m2.1.1.2">𝑤</ci><ci id="A4.F6.5.m2.1.1.3.cmml" xref="A4.F6.5.m2.1.1.3">𝑑</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.F6.5.m2.1d">w_{d}</annotation><annotation encoding="application/x-llamapun" id="A4.F6.5.m2.1e">italic_w start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT</annotation></semantics></math> at different settings of <math alttext="\alpha" class="ltx_Math" display="inline" id="A4.F6.6.m3.1"><semantics id="A4.F6.6.m3.1b"><mi id="A4.F6.6.m3.1.1" xref="A4.F6.6.m3.1.1.cmml">α</mi><annotation-xml encoding="MathML-Content" id="A4.F6.6.m3.1c"><ci id="A4.F6.6.m3.1.1.cmml" xref="A4.F6.6.m3.1.1">𝛼</ci></annotation-xml><annotation encoding="application/x-tex" id="A4.F6.6.m3.1d">\alpha</annotation><annotation encoding="application/x-llamapun" id="A4.F6.6.m3.1e">italic_α</annotation></semantics></math>.</figcaption> </figure> </section> <section class="ltx_subsection" id="A4.SS3"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">D.3 </span>Broader Implications</h3> <div class="ltx_para" id="A4.SS3.p1"> <p class="ltx_p" id="A4.SS3.p1.1">The proposed scoring framework introduces a novel perspective on balancing representativeness and diversity in data selection. By dynamically modulating the influence of diversity based on the local sample density, our approach strikes a principled balance between selecting typical and atypical samples. This adaptability is particularly valuable in data-centric applications, where sample selection directly affects the quality of downstream tasks, such as dataset pruning, user interest modeling, and few-shot learning.</p> </div> </section> <section class="ltx_subsection" id="A4.SS4"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">D.4 </span>Visualization Explanation</h3> <div class="ltx_para" id="A4.SS4.p1"> <p class="ltx_p" id="A4.SS4.p1.8">Figure <a class="ltx_ref" href="https://arxiv.org/html/2503.02398v1#A4.F6" title="Figure 6 ‣ D.2 Design Rationale ‣ Appendix D Details about In-Cluter Selection ‣ PersonaX: A Recommendation Agent-Oriented User Modeling Framework for Long Behavior Sequence"><span class="ltx_text ltx_ref_tag">6</span></a> shows the trade-off between <math alttext="w_{p}" class="ltx_Math" display="inline" id="A4.SS4.p1.1.m1.1"><semantics id="A4.SS4.p1.1.m1.1a"><msub id="A4.SS4.p1.1.m1.1.1" xref="A4.SS4.p1.1.m1.1.1.cmml"><mi id="A4.SS4.p1.1.m1.1.1.2" xref="A4.SS4.p1.1.m1.1.1.2.cmml">w</mi><mi id="A4.SS4.p1.1.m1.1.1.3" xref="A4.SS4.p1.1.m1.1.1.3.cmml">p</mi></msub><annotation-xml encoding="MathML-Content" id="A4.SS4.p1.1.m1.1b"><apply id="A4.SS4.p1.1.m1.1.1.cmml" xref="A4.SS4.p1.1.m1.1.1"><csymbol cd="ambiguous" id="A4.SS4.p1.1.m1.1.1.1.cmml" xref="A4.SS4.p1.1.m1.1.1">subscript</csymbol><ci id="A4.SS4.p1.1.m1.1.1.2.cmml" xref="A4.SS4.p1.1.m1.1.1.2">𝑤</ci><ci id="A4.SS4.p1.1.m1.1.1.3.cmml" xref="A4.SS4.p1.1.m1.1.1.3">𝑝</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.SS4.p1.1.m1.1c">w_{p}</annotation><annotation encoding="application/x-llamapun" id="A4.SS4.p1.1.m1.1d">italic_w start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="w_{d}" class="ltx_Math" display="inline" id="A4.SS4.p1.2.m2.1"><semantics id="A4.SS4.p1.2.m2.1a"><msub id="A4.SS4.p1.2.m2.1.1" xref="A4.SS4.p1.2.m2.1.1.cmml"><mi id="A4.SS4.p1.2.m2.1.1.2" xref="A4.SS4.p1.2.m2.1.1.2.cmml">w</mi><mi id="A4.SS4.p1.2.m2.1.1.3" xref="A4.SS4.p1.2.m2.1.1.3.cmml">d</mi></msub><annotation-xml encoding="MathML-Content" id="A4.SS4.p1.2.m2.1b"><apply id="A4.SS4.p1.2.m2.1.1.cmml" xref="A4.SS4.p1.2.m2.1.1"><csymbol cd="ambiguous" id="A4.SS4.p1.2.m2.1.1.1.cmml" xref="A4.SS4.p1.2.m2.1.1">subscript</csymbol><ci id="A4.SS4.p1.2.m2.1.1.2.cmml" xref="A4.SS4.p1.2.m2.1.1.2">𝑤</ci><ci id="A4.SS4.p1.2.m2.1.1.3.cmml" xref="A4.SS4.p1.2.m2.1.1.3">𝑑</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.SS4.p1.2.m2.1c">w_{d}</annotation><annotation encoding="application/x-llamapun" id="A4.SS4.p1.2.m2.1d">italic_w start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT</annotation></semantics></math> across different settings of <math alttext="\alpha" class="ltx_Math" display="inline" id="A4.SS4.p1.3.m3.1"><semantics id="A4.SS4.p1.3.m3.1a"><mi id="A4.SS4.p1.3.m3.1.1" xref="A4.SS4.p1.3.m3.1.1.cmml">α</mi><annotation-xml encoding="MathML-Content" id="A4.SS4.p1.3.m3.1b"><ci id="A4.SS4.p1.3.m3.1.1.cmml" xref="A4.SS4.p1.3.m3.1.1">𝛼</ci></annotation-xml><annotation encoding="application/x-tex" id="A4.SS4.p1.3.m3.1c">\alpha</annotation><annotation encoding="application/x-llamapun" id="A4.SS4.p1.3.m3.1d">italic_α</annotation></semantics></math>. As observed in the figure, when <math alttext="\alpha" class="ltx_Math" display="inline" id="A4.SS4.p1.4.m4.1"><semantics id="A4.SS4.p1.4.m4.1a"><mi id="A4.SS4.p1.4.m4.1.1" xref="A4.SS4.p1.4.m4.1.1.cmml">α</mi><annotation-xml encoding="MathML-Content" id="A4.SS4.p1.4.m4.1b"><ci id="A4.SS4.p1.4.m4.1.1.cmml" xref="A4.SS4.p1.4.m4.1.1">𝛼</ci></annotation-xml><annotation encoding="application/x-tex" id="A4.SS4.p1.4.m4.1c">\alpha</annotation><annotation encoding="application/x-llamapun" id="A4.SS4.p1.4.m4.1d">italic_α</annotation></semantics></math> is small, <math alttext="w_{p}" class="ltx_Math" display="inline" id="A4.SS4.p1.5.m5.1"><semantics id="A4.SS4.p1.5.m5.1a"><msub id="A4.SS4.p1.5.m5.1.1" xref="A4.SS4.p1.5.m5.1.1.cmml"><mi id="A4.SS4.p1.5.m5.1.1.2" xref="A4.SS4.p1.5.m5.1.1.2.cmml">w</mi><mi id="A4.SS4.p1.5.m5.1.1.3" xref="A4.SS4.p1.5.m5.1.1.3.cmml">p</mi></msub><annotation-xml encoding="MathML-Content" id="A4.SS4.p1.5.m5.1b"><apply id="A4.SS4.p1.5.m5.1.1.cmml" xref="A4.SS4.p1.5.m5.1.1"><csymbol cd="ambiguous" id="A4.SS4.p1.5.m5.1.1.1.cmml" xref="A4.SS4.p1.5.m5.1.1">subscript</csymbol><ci id="A4.SS4.p1.5.m5.1.1.2.cmml" xref="A4.SS4.p1.5.m5.1.1.2">𝑤</ci><ci id="A4.SS4.p1.5.m5.1.1.3.cmml" xref="A4.SS4.p1.5.m5.1.1.3">𝑝</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.SS4.p1.5.m5.1c">w_{p}</annotation><annotation encoding="application/x-llamapun" id="A4.SS4.p1.5.m5.1d">italic_w start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT</annotation></semantics></math> dominates the sampling process, leading to the selection of samples near the cluster center. These samples are prototypical and reflect the representative thematic interests of the cluster. As <math alttext="\alpha" class="ltx_Math" display="inline" id="A4.SS4.p1.6.m6.1"><semantics id="A4.SS4.p1.6.m6.1a"><mi id="A4.SS4.p1.6.m6.1.1" xref="A4.SS4.p1.6.m6.1.1.cmml">α</mi><annotation-xml encoding="MathML-Content" id="A4.SS4.p1.6.m6.1b"><ci id="A4.SS4.p1.6.m6.1.1.cmml" xref="A4.SS4.p1.6.m6.1.1">𝛼</ci></annotation-xml><annotation encoding="application/x-tex" id="A4.SS4.p1.6.m6.1c">\alpha</annotation><annotation encoding="application/x-llamapun" id="A4.SS4.p1.6.m6.1d">italic_α</annotation></semantics></math> increases, <math alttext="w_{d}" class="ltx_Math" display="inline" id="A4.SS4.p1.7.m7.1"><semantics id="A4.SS4.p1.7.m7.1a"><msub id="A4.SS4.p1.7.m7.1.1" xref="A4.SS4.p1.7.m7.1.1.cmml"><mi id="A4.SS4.p1.7.m7.1.1.2" xref="A4.SS4.p1.7.m7.1.1.2.cmml">w</mi><mi id="A4.SS4.p1.7.m7.1.1.3" xref="A4.SS4.p1.7.m7.1.1.3.cmml">d</mi></msub><annotation-xml encoding="MathML-Content" id="A4.SS4.p1.7.m7.1b"><apply id="A4.SS4.p1.7.m7.1.1.cmml" xref="A4.SS4.p1.7.m7.1.1"><csymbol cd="ambiguous" id="A4.SS4.p1.7.m7.1.1.1.cmml" xref="A4.SS4.p1.7.m7.1.1">subscript</csymbol><ci id="A4.SS4.p1.7.m7.1.1.2.cmml" xref="A4.SS4.p1.7.m7.1.1.2">𝑤</ci><ci id="A4.SS4.p1.7.m7.1.1.3.cmml" xref="A4.SS4.p1.7.m7.1.1.3">𝑑</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.SS4.p1.7.m7.1c">w_{d}</annotation><annotation encoding="application/x-llamapun" id="A4.SS4.p1.7.m7.1d">italic_w start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT</annotation></semantics></math> becomes more prominent, and <math alttext="w_{p}" class="ltx_Math" display="inline" id="A4.SS4.p1.8.m8.1"><semantics id="A4.SS4.p1.8.m8.1a"><msub id="A4.SS4.p1.8.m8.1.1" xref="A4.SS4.p1.8.m8.1.1.cmml"><mi id="A4.SS4.p1.8.m8.1.1.2" xref="A4.SS4.p1.8.m8.1.1.2.cmml">w</mi><mi id="A4.SS4.p1.8.m8.1.1.3" xref="A4.SS4.p1.8.m8.1.1.3.cmml">p</mi></msub><annotation-xml encoding="MathML-Content" id="A4.SS4.p1.8.m8.1b"><apply id="A4.SS4.p1.8.m8.1.1.cmml" xref="A4.SS4.p1.8.m8.1.1"><csymbol cd="ambiguous" id="A4.SS4.p1.8.m8.1.1.1.cmml" xref="A4.SS4.p1.8.m8.1.1">subscript</csymbol><ci id="A4.SS4.p1.8.m8.1.1.2.cmml" xref="A4.SS4.p1.8.m8.1.1.2">𝑤</ci><ci id="A4.SS4.p1.8.m8.1.1.3.cmml" xref="A4.SS4.p1.8.m8.1.1.3">𝑝</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.SS4.p1.8.m8.1c">w_{p}</annotation><annotation encoding="application/x-llamapun" id="A4.SS4.p1.8.m8.1d">italic_w start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT</annotation></semantics></math> approaches 0, causing the sampling process to prioritize diverse samples in order to enhance generalization.</p> </div> <figure class="ltx_figure" id="A4.F7"><img alt="Refer to caption" class="ltx_graphics ltx_centering ltx_img_square" height="398" id="A4.F7.g1" src="x7.png" width="401"/> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure">Figure 7: </span>Dynamics of In-Cluster sample selection: We set <math alttext="\alpha=0.7" class="ltx_Math" display="inline" id="A4.F7.3.m1.1"><semantics id="A4.F7.3.m1.1b"><mrow id="A4.F7.3.m1.1.1" xref="A4.F7.3.m1.1.1.cmml"><mi id="A4.F7.3.m1.1.1.2" xref="A4.F7.3.m1.1.1.2.cmml">α</mi><mo id="A4.F7.3.m1.1.1.1" xref="A4.F7.3.m1.1.1.1.cmml">=</mo><mn id="A4.F7.3.m1.1.1.3" xref="A4.F7.3.m1.1.1.3.cmml">0.7</mn></mrow><annotation-xml encoding="MathML-Content" id="A4.F7.3.m1.1c"><apply id="A4.F7.3.m1.1.1.cmml" xref="A4.F7.3.m1.1.1"><eq id="A4.F7.3.m1.1.1.1.cmml" xref="A4.F7.3.m1.1.1.1"></eq><ci id="A4.F7.3.m1.1.1.2.cmml" xref="A4.F7.3.m1.1.1.2">𝛼</ci><cn id="A4.F7.3.m1.1.1.3.cmml" type="float" xref="A4.F7.3.m1.1.1.3">0.7</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A4.F7.3.m1.1d">\alpha=0.7</annotation><annotation encoding="application/x-llamapun" id="A4.F7.3.m1.1e">italic_α = 0.7</annotation></semantics></math>, with the samples distributed within the range of <math alttext="[-2,2]" class="ltx_Math" display="inline" id="A4.F7.4.m2.2"><semantics id="A4.F7.4.m2.2b"><mrow id="A4.F7.4.m2.2.2.1" xref="A4.F7.4.m2.2.2.2.cmml"><mo id="A4.F7.4.m2.2.2.1.2" stretchy="false" xref="A4.F7.4.m2.2.2.2.cmml">[</mo><mrow id="A4.F7.4.m2.2.2.1.1" xref="A4.F7.4.m2.2.2.1.1.cmml"><mo id="A4.F7.4.m2.2.2.1.1b" xref="A4.F7.4.m2.2.2.1.1.cmml">−</mo><mn id="A4.F7.4.m2.2.2.1.1.2" xref="A4.F7.4.m2.2.2.1.1.2.cmml">2</mn></mrow><mo id="A4.F7.4.m2.2.2.1.3" xref="A4.F7.4.m2.2.2.2.cmml">,</mo><mn id="A4.F7.4.m2.1.1" xref="A4.F7.4.m2.1.1.cmml">2</mn><mo id="A4.F7.4.m2.2.2.1.4" stretchy="false" xref="A4.F7.4.m2.2.2.2.cmml">]</mo></mrow><annotation-xml encoding="MathML-Content" id="A4.F7.4.m2.2c"><interval closure="closed" id="A4.F7.4.m2.2.2.2.cmml" xref="A4.F7.4.m2.2.2.1"><apply id="A4.F7.4.m2.2.2.1.1.cmml" xref="A4.F7.4.m2.2.2.1.1"><minus id="A4.F7.4.m2.2.2.1.1.1.cmml" xref="A4.F7.4.m2.2.2.1.1"></minus><cn id="A4.F7.4.m2.2.2.1.1.2.cmml" type="integer" xref="A4.F7.4.m2.2.2.1.1.2">2</cn></apply><cn id="A4.F7.4.m2.1.1.cmml" type="integer" xref="A4.F7.4.m2.1.1">2</cn></interval></annotation-xml><annotation encoding="application/x-tex" id="A4.F7.4.m2.2d">[-2,2]</annotation><annotation encoding="application/x-llamapun" id="A4.F7.4.m2.2e">[ - 2 , 2 ]</annotation></semantics></math>.</figcaption> </figure> <div class="ltx_para" id="A4.SS4.p2"> <p class="ltx_p" id="A4.SS4.p2.1">Figure <a class="ltx_ref" href="https://arxiv.org/html/2503.02398v1#A4.F7" title="Figure 7 ‣ D.4 Visualization Explanation ‣ Appendix D Details about In-Cluter Selection ‣ PersonaX: A Recommendation Agent-Oriented User Modeling Framework for Long Behavior Sequence"><span class="ltx_text ltx_ref_tag">7</span></a> presents a dynamic visualization of the sampling process in Algorithm <a class="ltx_ref" href="https://arxiv.org/html/2503.02398v1#alg2" title="Algorithm 2 ‣ 3.2 Sampling Budget Allocation ‣ 3 Method ‣ PersonaX: A Recommendation Agent-Oriented User Modeling Framework for Long Behavior Sequence"><span class="ltx_text ltx_ref_tag">2</span></a>. As illustrated, the algorithm iteratively selects samples by jointly optimizing for both prototypicality and diversity, thereby maximizing the combined gain. This approach stands in contrast to conventional data selection methods, which often exhibit a unimodal bias—either favoring simple, centrally clustered, and highly representative samples <cite class="ltx_cite ltx_citemacro_cite">Welling (<a class="ltx_ref" href="https://arxiv.org/html/2503.02398v1#bib.bib49" title="">2009</a>); Rebuffi et al. (<a class="ltx_ref" href="https://arxiv.org/html/2503.02398v1#bib.bib32" title="">2017</a>); Sorscher et al. (<a class="ltx_ref" href="https://arxiv.org/html/2503.02398v1#bib.bib42" title="">2022</a>)</cite> or prioritizing difficult, outlier samples with strong generalization potential <cite class="ltx_cite ltx_citemacro_cite">Paul et al. (<a class="ltx_ref" href="https://arxiv.org/html/2503.02398v1#bib.bib25" title="">2021</a>); Toneva et al. (<a class="ltx_ref" href="https://arxiv.org/html/2503.02398v1#bib.bib45" title="">2019</a>)</cite>.</p> </div> <div class="ltx_para" id="A4.SS4.p3"> <p class="ltx_p" id="A4.SS4.p3.3">Empirical analysis of the hyperparameter <math alttext="\alpha" class="ltx_Math" display="inline" id="A4.SS4.p3.1.m1.1"><semantics id="A4.SS4.p3.1.m1.1a"><mi id="A4.SS4.p3.1.m1.1.1" xref="A4.SS4.p3.1.m1.1.1.cmml">α</mi><annotation-xml encoding="MathML-Content" id="A4.SS4.p3.1.m1.1b"><ci id="A4.SS4.p3.1.m1.1.1.cmml" xref="A4.SS4.p3.1.m1.1.1">𝛼</ci></annotation-xml><annotation encoding="application/x-tex" id="A4.SS4.p3.1.m1.1c">\alpha</annotation><annotation encoding="application/x-llamapun" id="A4.SS4.p3.1.m1.1d">italic_α</annotation></semantics></math>, which governs the trade-off between prototypicality and diversity, reveals a practical range of <math alttext="1.06" class="ltx_Math" display="inline" id="A4.SS4.p3.2.m2.1"><semantics id="A4.SS4.p3.2.m2.1a"><mn id="A4.SS4.p3.2.m2.1.1" xref="A4.SS4.p3.2.m2.1.1.cmml">1.06</mn><annotation-xml encoding="MathML-Content" id="A4.SS4.p3.2.m2.1b"><cn id="A4.SS4.p3.2.m2.1.1.cmml" type="float" xref="A4.SS4.p3.2.m2.1.1">1.06</cn></annotation-xml><annotation encoding="application/x-tex" id="A4.SS4.p3.2.m2.1c">1.06</annotation><annotation encoding="application/x-llamapun" id="A4.SS4.p3.2.m2.1d">1.06</annotation></semantics></math>–<math alttext="1.08" class="ltx_Math" display="inline" id="A4.SS4.p3.3.m3.1"><semantics id="A4.SS4.p3.3.m3.1a"><mn id="A4.SS4.p3.3.m3.1.1" xref="A4.SS4.p3.3.m3.1.1.cmml">1.08</mn><annotation-xml encoding="MathML-Content" id="A4.SS4.p3.3.m3.1b"><cn id="A4.SS4.p3.3.m3.1.1.cmml" type="float" xref="A4.SS4.p3.3.m3.1.1">1.08</cn></annotation-xml><annotation encoding="application/x-tex" id="A4.SS4.p3.3.m3.1c">1.08</annotation><annotation encoding="application/x-llamapun" id="A4.SS4.p3.3.m3.1d">1.08</annotation></semantics></math>. Within this regime, PersonaX often firstly selects a minimal set of prototypical samples and then shifting its focus toward maximizing sample diversity. We believe this is because of the superior few-shot generalization capabilities of LLMs. These models inherently require fewer prototypical instances to capture core user interests, thereby shifting their emphasis toward diverse sample acquisition to further enhance generalization.</p> </div> <figure class="ltx_table" id="A4.T5"> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_table">Table 5: </span>Intuitive comparison of user personas modeled by different methods. Specifically, (A) Relevance sampling (length 3), (B) Recent sampling (length 10), and (C) PersonaX (selection ratio 30%).</figcaption> <table class="ltx_tabular ltx_centering ltx_guessed_headers ltx_align_middle" id="A4.T5.1"> <thead class="ltx_thead"> <tr class="ltx_tr" id="A4.T5.1.1.1"> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_th_row ltx_border_l ltx_border_r ltx_border_t" id="A4.T5.1.1.1.1"><span class="ltx_text ltx_font_bold" id="A4.T5.1.1.1.1.1">Method</span></th> <th class="ltx_td ltx_align_justify ltx_align_top ltx_th ltx_th_column ltx_border_r ltx_border_t" id="A4.T5.1.1.1.2"> <span class="ltx_inline-block ltx_align_top" id="A4.T5.1.1.1.2.1"> <span class="ltx_p" id="A4.T5.1.1.1.2.1.1" style="width:216.8pt;"><span class="ltx_text ltx_font_bold" id="A4.T5.1.1.1.2.1.1.1">User Persona</span></span> </span> </th> <th class="ltx_td ltx_align_justify ltx_align_top ltx_th ltx_th_column ltx_border_r ltx_border_t" id="A4.T5.1.1.1.3"> <span class="ltx_inline-block ltx_align_top" id="A4.T5.1.1.1.3.1"> <span class="ltx_p" id="A4.T5.1.1.1.3.1.1" style="width:180.7pt;"><span class="ltx_text ltx_font_bold" id="A4.T5.1.1.1.3.1.1.1">o3-mini’s Reasoning Evaluation</span></span> </span> </th> </tr> </thead> <tbody class="ltx_tbody"> <tr class="ltx_tr" id="A4.T5.1.2.1"> <th class="ltx_td ltx_align_center ltx_th ltx_th_row ltx_border_l ltx_border_r ltx_border_t" id="A4.T5.1.2.1.1">A</th> <td class="ltx_td ltx_align_justify ltx_align_top ltx_border_r ltx_border_t" id="A4.T5.1.2.1.2"> <span class="ltx_inline-block ltx_align_top" id="A4.T5.1.2.1.2.1"> <span class="ltx_p" id="A4.T5.1.2.1.2.1.1" style="width:216.8pt;">I prefer Rock and Progressive Rock music, particularly works by artists like Trey Anastasio. I enjoy albums that feature new songs and live performances, as they provide a dynamic listening experience. I appreciate behind-the-scenes content that adds depth to the music experience. I also enjoy Contemporary Folk music, especially works by artists like Steve Winwood, as they can offer a fresh take on musical compositions. I dislike R&B and Soul music, particularly collections that focus on remixes of older tracks, as I find them less engaging. I prefer original compositions and fresh takes on music rather than revisiting past favorites. Additionally, I am not particularly interested in World Music, especially live performances that do not align with my preferred genres.</span> </span> </td> <td class="ltx_td ltx_align_justify ltx_align_top ltx_border_r ltx_border_t" id="A4.T5.1.2.1.3"> <span class="ltx_inline-block ltx_align_top" id="A4.T5.1.2.1.3.1"> <span class="ltx_p" id="A4.T5.1.2.1.3.1.1" style="width:180.7pt;">This description is detailed and rich in information, but it introduces multiple music genres which may dilute the focus on the user’s core preferences.</span> </span> </td> </tr> <tr class="ltx_tr" id="A4.T5.1.3.2"> <th class="ltx_td ltx_align_center ltx_th ltx_th_row ltx_border_l ltx_border_r ltx_border_t" id="A4.T5.1.3.2.1">B</th> <td class="ltx_td ltx_align_justify ltx_align_top ltx_border_r ltx_border_t" id="A4.T5.1.3.2.2"> <span class="ltx_inline-block ltx_align_top" id="A4.T5.1.3.2.2.1"> <span class="ltx_p" id="A4.T5.1.3.2.2.1.1" style="width:216.8pt;">I prefer Classic Rock and Arena Rock music, particularly albums that showcase strong melodies and engaging performances, such as ’Chickenfoot’ by Chickenfoot. I enjoy music that has a fun, energetic vibe and often features collaborative efforts from talented musicians. I appreciate emotional ballads and songs that reflect depth and complexity in lyrics and composition. I dislike Blues Rock that focuses on traditional guitar work and may lack the innovative sounds I seek, as exemplified by ’Smokestacks, Broom Dusters & Hoochie Coochie Men’ by Micky Moody, which I find less appealing due to its more conventional approach.</span> </span> </td> <td class="ltx_td ltx_align_justify ltx_align_top ltx_border_r ltx_border_t" id="A4.T5.1.3.2.3"> <span class="ltx_inline-block ltx_align_top" id="A4.T5.1.3.2.3.1"> <span class="ltx_p" id="A4.T5.1.3.2.3.1.1" style="width:180.7pt;">This description focuses on a subset of rock music—Classic and Arena Rock—which contrasts with the broader rock and progressive preferences seen in the other descriptions. It is detailed but less aligned with the core focus compared to C.</span> </span> </td> </tr> <tr class="ltx_tr" id="A4.T5.1.4.3"> <th class="ltx_td ltx_align_center ltx_th ltx_th_row ltx_border_b ltx_border_l ltx_border_r ltx_border_t" id="A4.T5.1.4.3.1">C</th> <td class="ltx_td ltx_align_justify ltx_align_top ltx_border_b ltx_border_r ltx_border_t" id="A4.T5.1.4.3.2"> <span class="ltx_inline-block ltx_align_top" id="A4.T5.1.4.3.2.1"> <span class="ltx_p" id="A4.T5.1.4.3.2.1.1" style="width:216.8pt;">I prefer rock and progressive music, particularly works by notable artists like Trey Anastasio. I enjoy albums that offer a collection of new songs, especially those that include additional content such as live performances and behind-the-scenes footage. I dislike pop and dance music, particularly generic albums that lack depth or a compelling narrative. I appreciate immersive listening experiences that connect me to the artist’s journey and creative process.</span> </span> </td> <td class="ltx_td ltx_align_justify ltx_align_top ltx_border_b ltx_border_r ltx_border_t" id="A4.T5.1.4.3.3"> <span class="ltx_inline-block ltx_align_top" id="A4.T5.1.4.3.3.1"> <span class="ltx_p" id="A4.T5.1.4.3.3.1.1" style="width:180.7pt;">This description is the most concise and focused, effectively capturing the user’s core interests—new material, live performances, and behind-the-scenes insights—without extraneous details, making it the highest quality among the three.</span> </span> </td> </tr> </tbody> </table> </figure> <figure class="ltx_table" id="A4.T6"> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_table">Table 6: </span>Quantitative Evaluation of User Persona Modeling Methods</figcaption> <table class="ltx_tabular ltx_centering ltx_guessed_headers ltx_align_middle" id="A4.T6.1"> <thead class="ltx_thead"> <tr class="ltx_tr" id="A4.T6.1.1.1"> <th class="ltx_td ltx_align_left ltx_th ltx_th_column ltx_th_row ltx_border_tt" id="A4.T6.1.1.1.1"><span class="ltx_text ltx_font_bold" id="A4.T6.1.1.1.1.1">Method</span></th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_tt" id="A4.T6.1.1.1.2"><span class="ltx_text ltx_font_bold" id="A4.T6.1.1.1.2.1">NDCG@1</span></th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_tt" id="A4.T6.1.1.1.3"><span class="ltx_text ltx_font_bold" id="A4.T6.1.1.1.3.1">NDCG@5</span></th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_tt" id="A4.T6.1.1.1.4"><span class="ltx_text ltx_font_bold" id="A4.T6.1.1.1.4.1">NDCG@10</span></th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_tt" id="A4.T6.1.1.1.5"><span class="ltx_text ltx_font_bold" id="A4.T6.1.1.1.5.1">Hit@1</span></th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_tt" id="A4.T6.1.1.1.6"><span class="ltx_text ltx_font_bold" id="A4.T6.1.1.1.6.1">Hit@5</span></th> <th class="ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_tt" id="A4.T6.1.1.1.7"><span class="ltx_text ltx_font_bold" id="A4.T6.1.1.1.7.1">Hit@10</span></th> <th class="ltx_td ltx_nopad_r ltx_align_center ltx_th ltx_th_column ltx_border_tt" id="A4.T6.1.1.1.8"><span class="ltx_text ltx_font_bold" id="A4.T6.1.1.1.8.1">MRR</span></th> </tr> </thead> <tbody class="ltx_tbody"> <tr class="ltx_tr" id="A4.T6.1.2.1"> <th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_t" id="A4.T6.1.2.1.1">A</th> <td class="ltx_td ltx_align_center ltx_border_t" id="A4.T6.1.2.1.2">0.00</td> <td class="ltx_td ltx_align_center ltx_border_t" id="A4.T6.1.2.1.3">0.42</td> <td class="ltx_td ltx_align_center ltx_border_t" id="A4.T6.1.2.1.4">0.54</td> <td class="ltx_td ltx_align_center ltx_border_t" id="A4.T6.1.2.1.5">0.00</td> <td class="ltx_td ltx_align_center ltx_border_t" id="A4.T6.1.2.1.6">0.67</td> <td class="ltx_td ltx_align_center ltx_border_t" id="A4.T6.1.2.1.7">1.00</td> <td class="ltx_td ltx_nopad_r ltx_align_center ltx_border_t" id="A4.T6.1.2.1.8">0.39</td> </tr> <tr class="ltx_tr" id="A4.T6.1.3.2"> <th class="ltx_td ltx_align_left ltx_th ltx_th_row" id="A4.T6.1.3.2.1">B</th> <td class="ltx_td ltx_align_center" id="A4.T6.1.3.2.2">0.00</td> <td class="ltx_td ltx_align_center" id="A4.T6.1.3.2.3">0.54</td> <td class="ltx_td ltx_align_center" id="A4.T6.1.3.2.4">0.54</td> <td class="ltx_td ltx_align_center" id="A4.T6.1.3.2.5">0.00</td> <td class="ltx_td ltx_align_center" id="A4.T6.1.3.2.6">1.00</td> <td class="ltx_td ltx_align_center" id="A4.T6.1.3.2.7">1.00</td> <td class="ltx_td ltx_nopad_r ltx_align_center" id="A4.T6.1.3.2.8">0.39</td> </tr> <tr class="ltx_tr" id="A4.T6.1.4.3"> <th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_bb" id="A4.T6.1.4.3.1">C</th> <td class="ltx_td ltx_align_center ltx_border_bb" id="A4.T6.1.4.3.2">0.33</td> <td class="ltx_td ltx_align_center ltx_border_bb" id="A4.T6.1.4.3.3">0.71</td> <td class="ltx_td ltx_align_center ltx_border_bb" id="A4.T6.1.4.3.4">0.71</td> <td class="ltx_td ltx_align_center ltx_border_bb" id="A4.T6.1.4.3.5">0.33</td> <td class="ltx_td ltx_align_center ltx_border_bb" id="A4.T6.1.4.3.6">1.00</td> <td class="ltx_td ltx_align_center ltx_border_bb" id="A4.T6.1.4.3.7">1.00</td> <td class="ltx_td ltx_nopad_r ltx_align_center ltx_border_bb" id="A4.T6.1.4.3.8">0.61</td> </tr> </tbody> </table> </figure> </section> </section> <section class="ltx_appendix" id="A5"> <h2 class="ltx_title ltx_title_appendix"> <span class="ltx_tag ltx_tag_appendix">Appendix E </span>Case Study</h2> <div class="ltx_para" id="A5.p1"> <p class="ltx_p" id="A5.p1.1">In this section, we present a case study comparing user personas modeled using Relevance, Recent, and PersonaX methods, with the backbone LLM-UM approach fixed as Reflection. The dataset used is <math alttext="\texttt{CDs}_{50}" class="ltx_Math" display="inline" id="A5.p1.1.m1.1"><semantics id="A5.p1.1.m1.1a"><msub id="A5.p1.1.m1.1.1" xref="A5.p1.1.m1.1.1.cmml"><mtext class="ltx_mathvariant_monospace" id="A5.p1.1.m1.1.1.2" xref="A5.p1.1.m1.1.1.2a.cmml">CDs</mtext><mn id="A5.p1.1.m1.1.1.3" xref="A5.p1.1.m1.1.1.3.cmml">50</mn></msub><annotation-xml encoding="MathML-Content" id="A5.p1.1.m1.1b"><apply id="A5.p1.1.m1.1.1.cmml" xref="A5.p1.1.m1.1.1"><csymbol cd="ambiguous" id="A5.p1.1.m1.1.1.1.cmml" xref="A5.p1.1.m1.1.1">subscript</csymbol><ci id="A5.p1.1.m1.1.1.2a.cmml" xref="A5.p1.1.m1.1.1.2"><mtext class="ltx_mathvariant_monospace" id="A5.p1.1.m1.1.1.2.cmml" xref="A5.p1.1.m1.1.1.2">CDs</mtext></ci><cn id="A5.p1.1.m1.1.1.3.cmml" type="integer" xref="A5.p1.1.m1.1.1.3">50</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A5.p1.1.m1.1c">\texttt{CDs}_{50}</annotation><annotation encoding="application/x-llamapun" id="A5.p1.1.m1.1d">CDs start_POSTSUBSCRIPT 50 end_POSTSUBSCRIPT</annotation></semantics></math>, with the User ID <span class="ltx_text ltx_font_typewriter" id="A5.p1.1.1">A2NQUGGYM0DBM1</span>. The results are summarized in Table <a class="ltx_ref" href="https://arxiv.org/html/2503.02398v1#A4.T5" title="Table 5 ‣ D.4 Visualization Explanation ‣ Appendix D Details about In-Cluter Selection ‣ PersonaX: A Recommendation Agent-Oriented User Modeling Framework for Long Behavior Sequence"><span class="ltx_text ltx_ref_tag">5</span></a>. We evaluate their quality by OpenAI’s o3-mini, using its reasoning capabilities in an LLM-As-Judge framework. The evaluation indicated that Model C had the highest modeling quality <span class="ltx_note ltx_role_footnote" id="footnote2"><sup class="ltx_note_mark">2</sup><span class="ltx_note_outer"><span class="ltx_note_content"><sup class="ltx_note_mark">2</sup><span class="ltx_tag ltx_tag_note">2</span>Repeated inquiries occasionally resulted in A being rated higher, with the justification that A offered a more comprehensive view. However, this comprehensiveness came at the cost of interest modeling that was more diffuse and less precise.</span></span></span>. The explanation provided was that C demonstrated superior descriptive quality, capturing the user’s core preferences for rock and progressive music with concise and precise language. It also emphasized the user’s interest in new releases, live performances, and behind-the-scenes content, while avoiding extraneous information misaligned with primary interests. In contrast, Model A, while rich in information, introduced a broader range of music styles that diluted focus, and Model B predominantly emphasized an alternative style of rock, leading to inconsistencies with the other descriptions.</p> </div> <div class="ltx_para" id="A5.p2"> <p class="ltx_p" id="A5.p2.1">We conducted three rounds of quantitative evaluations on the ranking task, each comprising one positive item alongside nigh negative items. As shown in Table <a class="ltx_ref" href="https://arxiv.org/html/2503.02398v1#A4.T6" title="Table 6 ‣ D.4 Visualization Explanation ‣ Appendix D Details about In-Cluter Selection ‣ PersonaX: A Recommendation Agent-Oriented User Modeling Framework for Long Behavior Sequence"><span class="ltx_text ltx_ref_tag">6</span></a>, Method C achieved the highest performance, followed by Method B, while Method A exhibited the poorest performance.</p> </div> </section> <section class="ltx_appendix" id="A6"> <h2 class="ltx_title ltx_title_appendix"> <span class="ltx_tag ltx_tag_appendix">Appendix F </span>Prompt Templates</h2> <div class="ltx_para" id="A6.p1"> <p class="ltx_p" id="A6.p1.1">We present the prompt templates used in AgentCF, as shown in Figure <a class="ltx_ref" href="https://arxiv.org/html/2503.02398v1#A6.F8" title="Figure 8 ‣ Appendix F Prompt Templates ‣ PersonaX: A Recommendation Agent-Oriented User Modeling Framework for Long Behavior Sequence"><span class="ltx_text ltx_ref_tag">8</span></a> and Figure <a class="ltx_ref" href="https://arxiv.org/html/2503.02398v1#A6.F9" title="Figure 9 ‣ Appendix F Prompt Templates ‣ PersonaX: A Recommendation Agent-Oriented User Modeling Framework for Long Behavior Sequence"><span class="ltx_text ltx_ref_tag">9</span></a>, and those employed in Agent4Rec, depicted in Figure <a class="ltx_ref" href="https://arxiv.org/html/2503.02398v1#A6.F10" title="Figure 10 ‣ Appendix F Prompt Templates ‣ PersonaX: A Recommendation Agent-Oriented User Modeling Framework for Long Behavior Sequence"><span class="ltx_text ltx_ref_tag">10</span></a>.</p> </div> <figure class="ltx_figure" id="A6.F8"> <p class="ltx_p ltx_align_center" id="A6.F8.1"><span class="ltx_text" id="A6.F8.1.1"><svg class="ltx_picture" height="444.38" id="A6.F8.1.1.pic1" overflow="visible" version="1.1" width="600"><g fill="#000000" stroke="#000000" stroke-width="0.4pt" transform="translate(0,444.38) matrix(1 0 0 -1 0 0)"><g fill="#98450E" fill-opacity="1.0"><path d="M 0 4.63 L 0 439.75 C 0 442.3 2.07 444.38 4.63 444.38 L 595.37 444.38 C 597.93 444.38 600 442.3 600 439.75 L 600 4.63 C 600 2.07 597.93 0 595.37 0 L 4.63 0 C 2.07 0 0 2.07 0 4.63 Z" style="stroke:none"></path></g><g fill="#FEF9F5" fill-opacity="1.0"><path d="M 0.69 4.63 L 0.69 422.82 L 599.31 422.82 L 599.31 4.63 C 599.31 2.45 597.55 0.69 595.37 0.69 L 4.63 0.69 C 2.45 0.69 0.69 2.45 0.69 4.63 Z" style="stroke:none"></path></g><g fill-opacity="1.0" transform="matrix(1.0 0.0 0.0 1.0 20.38 427.45)"><foreignobject color="#FFFFFF" height="12.3" overflow="visible" transform="matrix(1 0 0 -1 0 16.6)" width="559.25"> <span class="ltx_inline-block ltx_minipage ltx_align_bottom" id="A6.F8.1.1.pic1.1.1.1.1.1" style="width:404.2pt;"> <span class="ltx_p" id="A6.F8.1.1.pic1.1.1.1.1.1.1"><span class="ltx_text ltx_font_bold" id="A6.F8.1.1.pic1.1.1.1.1.1.1.1">Prompt Template for Forward Inference Process of AgentCF</span></span> </span></foreignobject></g><g fill-opacity="1.0" transform="matrix(1.0 0.0 0.0 1.0 20.38 12.5)"><foreignobject color="#000000" height="398.51" overflow="visible" transform="matrix(1 0 0 -1 0 16.6)" width="559.25"> <span class="ltx_inline-block ltx_minipage ltx_align_bottom" id="A6.F8.1.1.pic1.2.2.2.1.1" style="width:404.2pt;"> <span class="ltx_p" id="A6.F8.1.1.pic1.2.2.2.1.1.1"><span class="ltx_text ltx_font_bold" id="A6.F8.1.1.pic1.2.2.2.1.1.1.1">Task:</span> We provide a user’s personal profile in [User Profile], which includes the user’s preferences, dislikes, and other relevant information. You need play the role of the user. And we also provide two candidate items, A and B, with their features in [Item Feature]. You need to choice between the two item candidates based on your profile and the features of the items. Furthermore, you must articulate why you’ve chosen that particular item while rejecting the other.</span> <span class="ltx_p" id="A6.F8.1.1.pic1.2.2.2.1.1.2"><span class="ltx_text ltx_font_bold" id="A6.F8.1.1.pic1.2.2.2.1.1.2.1">User Profile:</span> {profile}</span> <span class="ltx_p" id="A6.F8.1.1.pic1.2.2.2.1.1.3"><span class="ltx_text ltx_font_bold" id="A6.F8.1.1.pic1.2.2.2.1.1.3.1">Item Feature:</span> Item A: {item a} Item B: {item b}</span> <span class="ltx_p" id="A6.F8.1.1.pic1.2.2.2.1.1.4"><span class="ltx_text ltx_font_bold" id="A6.F8.1.1.pic1.2.2.2.1.1.4.1">Steps to Follow:</span></span> <span class="ltx_p" id="A6.F8.1.1.pic1.2.2.2.1.1.5">1. Extract your preferences and dislikes from your self-introduction.</span> <span class="ltx_p" id="A6.F8.1.1.pic1.2.2.2.1.1.6">2. Evaluate the two candidate in light of your preferences and dislikes. Make your choice by considering the correlation between your preferences/dislikes and the features of the candidates.</span> <span class="ltx_p" id="A6.F8.1.1.pic1.2.2.2.1.1.7">3. Explain why you made such choices, from the perspective of the relationship between your preferences/dislikes and the features of these candidate items.</span> <span class="ltx_p" id="A6.F8.1.1.pic1.2.2.2.1.1.8"><span class="ltx_text ltx_font_bold" id="A6.F8.1.1.pic1.2.2.2.1.1.8.1">Important Notes:</span></span> <span class="ltx_p" id="A6.F8.1.1.pic1.2.2.2.1.1.9">1. Your output should strictly be in the following format: Chosen Item: Item A or Item B Explanation: Your detailed rationale behind your choice and reasons for rejecting the other item.</span> <span class="ltx_p" id="A6.F8.1.1.pic1.2.2.2.1.1.10">2. When identifying user’s likes and dislikes, do not fabricate them! If your [User Profile] doesn’t specify any relevant preferences or dislikes, use common knowledge to inform your decision.</span> <span class="ltx_p" id="A6.F8.1.1.pic1.2.2.2.1.1.11">3. You **must** choose one of these two candidates, and **cannot** choose both.</span> <span class="ltx_p" id="A6.F8.1.1.pic1.2.2.2.1.1.12">4. Your explanation needs to be comprehensive and specific. Your reasoning should delve into the finer attributes of the items.</span> <span class="ltx_p" id="A6.F8.1.1.pic1.2.2.2.1.1.13">5. Base your explanation on facts. For instance, if your self-introduction doesn’t reveal any specific preferences or dislikes, justify your decision using available or common knowledge.</span> <span class="ltx_p" id="A6.F8.1.1.pic1.2.2.2.1.1.14">6. Please ignore the effect of Item position and length, they do not affect your decision.</span> <span class="ltx_p" id="A6.F8.1.1.pic1.2.2.2.1.1.15"><span class="ltx_text ltx_font_bold" id="A6.F8.1.1.pic1.2.2.2.1.1.15.1">Response Example:</span> <span class="ltx_text ltx_font_italic" id="A6.F8.1.1.pic1.2.2.2.1.1.15.2">Chosen Item: Item A Explanation: I chose Item A because… <br class="ltx_break"/></span></span> </span></foreignobject></g></g></svg> </span></p> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure">Figure 8: </span>Prompt template for the forward process of AgentCF to predict one user potentially liked item between a positive one and a negarive one.</figcaption> </figure> <figure class="ltx_figure" id="A6.F9"> <p class="ltx_p ltx_align_center" id="A6.F9.1"><span class="ltx_text" id="A6.F9.1.1"><svg class="ltx_picture" height="460.98" id="A6.F9.1.1.pic1" overflow="visible" version="1.1" width="600"><g fill="#000000" stroke="#000000" stroke-width="0.4pt" transform="translate(0,460.98) matrix(1 0 0 -1 0 0)"><g fill="#98450E" fill-opacity="1.0"><path d="M 0 4.63 L 0 456.35 C 0 458.91 2.07 460.98 4.63 460.98 L 595.37 460.98 C 597.93 460.98 600 458.91 600 456.35 L 600 4.63 C 600 2.07 597.93 0 595.37 0 L 4.63 0 C 2.07 0 0 2.07 0 4.63 Z" style="stroke:none"></path></g><g fill="#FEF9F5" fill-opacity="1.0"><path d="M 0.69 4.63 L 0.69 439.42 L 599.31 439.42 L 599.31 4.63 C 599.31 2.45 597.55 0.69 595.37 0.69 L 4.63 0.69 C 2.45 0.69 0.69 2.45 0.69 4.63 Z" style="stroke:none"></path></g><g fill-opacity="1.0" transform="matrix(1.0 0.0 0.0 1.0 20.38 444.05)"><foreignobject color="#FFFFFF" height="12.3" overflow="visible" transform="matrix(1 0 0 -1 0 16.6)" width="559.25"> <span class="ltx_inline-block ltx_minipage ltx_align_bottom" id="A6.F9.1.1.pic1.1.1.1.1.1" style="width:404.2pt;"> <span class="ltx_p" id="A6.F9.1.1.pic1.1.1.1.1.1.1"><span class="ltx_text ltx_font_bold" id="A6.F9.1.1.pic1.1.1.1.1.1.1.1">Prompt Template for Backward Reflection Process of AgentCF</span></span> </span></foreignobject></g><g fill-opacity="1.0" transform="matrix(1.0 0.0 0.0 1.0 20.38 12.5)"><foreignobject color="#000000" height="415.11" overflow="visible" transform="matrix(1 0 0 -1 0 16.6)" width="559.25"> <span class="ltx_inline-block ltx_minipage ltx_align_bottom" id="A6.F9.1.1.pic1.2.2.2.1.1" style="width:404.2pt;"> <span class="ltx_p" id="A6.F9.1.1.pic1.2.2.2.1.1.1"><span class="ltx_text ltx_font_bold" id="A6.F9.1.1.pic1.2.2.2.1.1.1.1">Background:</span> We provide a user’s personal profile in [User Profile], which includes the user’s preferences, dislikes, and other relevant information. You need play the role of the user. Recently, you considered choosing one more prefered Item from two candidates. The features of these two candidate are provided in [Item Feature]. And your choice and explanation is in [Choice and Explanation], which reveals your previous judgment for these two candidates.</span> <span class="ltx_p" id="A6.F9.1.1.pic1.2.2.2.1.1.2"><span class="ltx_text ltx_font_bold" id="A6.F9.1.1.pic1.2.2.2.1.1.2.1">User Profile:</span> {profile}</span> <span class="ltx_p" id="A6.F9.1.1.pic1.2.2.2.1.1.3"><span class="ltx_text ltx_font_bold" id="A6.F9.1.1.pic1.2.2.2.1.1.3.1">Item Feature:</span> Item A: {item a} Item B: {item b}</span> <span class="ltx_p" id="A6.F9.1.1.pic1.2.2.2.1.1.4"><span class="ltx_text ltx_font_bold" id="A6.F9.1.1.pic1.2.2.2.1.1.4.1">Choice and Explanation:</span> {response}</span> <span class="ltx_p" id="A6.F9.1.1.pic1.2.2.2.1.1.5"><span class="ltx_text ltx_font_bold" id="A6.F9.1.1.pic1.2.2.2.1.1.5.1">Task:</span> However, The user in the real world actually prefer to choose Item B, and reject the Item A that you initially chose. This indicates that you made an incorrect choice, the [Choice and Explanation] was mistaken. Therefore, you need to reflect and update [User Profile].</span> <span class="ltx_p" id="A6.F9.1.1.pic1.2.2.2.1.1.6"><span class="ltx_text ltx_font_bold" id="A6.F9.1.1.pic1.2.2.2.1.1.6.1">Steps to Follow:</span></span> <span class="ltx_p" id="A6.F9.1.1.pic1.2.2.2.1.1.7">1. Analyze the misconceptions in your previous [Choice and Explanation] about your preferences and dislikes, as recorded in your explanation, and correct these mistakes.</span> <span class="ltx_p" id="A6.F9.1.1.pic1.2.2.2.1.1.8">2. Explore your new preferences based on the Item B you really enjoy, and determine your dislikes based on the Item a you truly don’t enjoy.</span> <span class="ltx_p" id="A6.F9.1.1.pic1.2.2.2.1.1.9">3. Summarize your past preferences and dislikes from your previous [User Profile]. Combine your newfound preferences and dislikes with your past ones. Filter and remove any conflicting or repetitive parts in your past [User Profile] that contradict your current preferences and dislikes.</span> <span class="ltx_p" id="A6.F9.1.1.pic1.2.2.2.1.1.10">4. Generate a update profile use the following format:</span> <span class="ltx_p" id="A6.F9.1.1.pic1.2.2.2.1.1.11">My updated profile: {Please write your updated profile here}</span> <span class="ltx_p" id="A6.F9.1.1.pic1.2.2.2.1.1.12"><span class="ltx_text ltx_font_bold" id="A6.F9.1.1.pic1.2.2.2.1.1.12.1">Important Notes:</span></span> <span class="ltx_p" id="A6.F9.1.1.pic1.2.2.2.1.1.13">1. Keep your updated profile under 180 words.</span> <span class="ltx_p" id="A6.F9.1.1.pic1.2.2.2.1.1.14">2. Any overall assessments or summarization in your profile are forbidden.</span> <span class="ltx_p" id="A6.F9.1.1.pic1.2.2.2.1.1.15">3. Your updated profile should only describe the features of items you prefer or dislike, without mentioning your wrong choice or your thinking process in updating your profile.</span> <span class="ltx_p" id="A6.F9.1.1.pic1.2.2.2.1.1.16">4. Your profile should be specific and personalized. Any preferences and dislikes that cannot distinguish you from others are not worth recording.</span> <span class="ltx_p" id="A6.F9.1.1.pic1.2.2.2.1.1.17"><span class="ltx_text ltx_font_bold" id="A6.F9.1.1.pic1.2.2.2.1.1.17.1">Response Example:</span> <span class="ltx_text ltx_font_italic" id="A6.F9.1.1.pic1.2.2.2.1.1.17.2">My updated profile: I … <br class="ltx_break"/></span></span> </span></foreignobject></g></g></svg> </span></p> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure">Figure 9: </span>Prompt template for the backward process of AgentCF to apply the reflect mechanism for updating user profile.</figcaption> </figure> <figure class="ltx_figure" id="A6.F10"> <p class="ltx_p ltx_align_center" id="A6.F10.1"><span class="ltx_text" id="A6.F10.1.1"><svg class="ltx_picture" height="373.65" id="A6.F10.1.1.pic1" overflow="visible" version="1.1" width="600"><g fill="#000000" stroke="#000000" stroke-width="0.4pt" transform="translate(0,373.65) matrix(1 0 0 -1 0 0)"><g fill="#98450E" fill-opacity="1.0"><path d="M 0 4.63 L 0 369.03 C 0 371.58 2.07 373.65 4.63 373.65 L 595.37 373.65 C 597.93 373.65 600 371.58 600 369.03 L 600 4.63 C 600 2.07 597.93 0 595.37 0 L 4.63 0 C 2.07 0 0 2.07 0 4.63 Z" style="stroke:none"></path></g><g fill="#FEF9F5" fill-opacity="1.0"><path d="M 0.69 4.63 L 0.69 352.1 L 599.31 352.1 L 599.31 4.63 C 599.31 2.45 597.55 0.69 595.37 0.69 L 4.63 0.69 C 2.45 0.69 0.69 2.45 0.69 4.63 Z" style="stroke:none"></path></g><g fill-opacity="1.0" transform="matrix(1.0 0.0 0.0 1.0 20.38 356.73)"><foreignobject color="#FFFFFF" height="12.3" overflow="visible" transform="matrix(1 0 0 -1 0 16.6)" width="559.25"> <span class="ltx_inline-block ltx_minipage ltx_align_bottom" id="A6.F10.1.1.pic1.1.1.1.1.1" style="width:404.2pt;"> <span class="ltx_p" id="A6.F10.1.1.pic1.1.1.1.1.1.1"><span class="ltx_text ltx_font_bold" id="A6.F10.1.1.pic1.1.1.1.1.1.1.1">Prompt Template for Summarization Process of Agent4Rec</span></span> </span></foreignobject></g><g fill-opacity="1.0" transform="matrix(1.0 0.0 0.0 1.0 20.38 12.5)"><foreignobject color="#000000" height="327.78" overflow="visible" transform="matrix(1 0 0 -1 0 16.6)" width="559.25"> <span class="ltx_inline-block ltx_minipage ltx_align_bottom" id="A6.F10.1.1.pic1.2.2.2.1.1" style="width:404.2pt;"> <span class="ltx_p" id="A6.F10.1.1.pic1.2.2.2.1.1.1"><span class="ltx_text ltx_font_bold" id="A6.F10.1.1.pic1.2.2.2.1.1.1.1">Task:</span> We provide a user’s personal profile in [User Profile], which includes the user’s preferences and other relevant information. Additionally, we provide a sequence of liked items in [Sequence Item Profile] that the user has interacted with. Your task is to analyze these items in the context of the user’s existing profile and produce an updated profile that reflects any new preferences, or insights inferred from the user’s interactions with these items.</span> <span class="ltx_p" id="A6.F10.1.1.pic1.2.2.2.1.1.2"><span class="ltx_text ltx_font_bold" id="A6.F10.1.1.pic1.2.2.2.1.1.2.1">User Profile:</span> {profile}</span> <span class="ltx_p" id="A6.F10.1.1.pic1.2.2.2.1.1.3"><span class="ltx_text ltx_font_bold" id="A6.F10.1.1.pic1.2.2.2.1.1.3.1">Sequence Item Profile:</span> {sequence item profile}</span> <span class="ltx_p" id="A6.F10.1.1.pic1.2.2.2.1.1.4"><span class="ltx_text ltx_font_bold" id="A6.F10.1.1.pic1.2.2.2.1.1.4.1">Steps to Follow:</span></span> <span class="ltx_p" id="A6.F10.1.1.pic1.2.2.2.1.1.5">1. Carefully review the user’s existing profile to understand their stated preferences and dislikes.</span> <span class="ltx_p" id="A6.F10.1.1.pic1.2.2.2.1.1.6">2. Analyze the features of the items in the provided sequence, noting any common themes, attributes, or patterns.</span> <span class="ltx_p" id="A6.F10.1.1.pic1.2.2.2.1.1.7">3. Identify any new preferences that can be inferred from the user’s interactions with these items.</span> <span class="ltx_p" id="A6.F10.1.1.pic1.2.2.2.1.1.8">4. Summarize and update the user’s profile by incorporating the new insights, adding new preferences or dislikes, and highlighting any changes or developments in the user’s tastes. Important Notes</span> <span class="ltx_p" id="A6.F10.1.1.pic1.2.2.2.1.1.9">5. Your output should strictly be in the following format: Summarization: {Your updated profile.}</span> <span class="ltx_p" id="A6.F10.1.1.pic1.2.2.2.1.1.10">6. Do not contradict the user’s existing preferences unless there is clear evidence from the sequence items that their tastes have changed.</span> <span class="ltx_p" id="A6.F10.1.1.pic1.2.2.2.1.1.11">7. Base your summary on facts and logical inferences drawn from the items in the sequence.</span> <span class="ltx_p" id="A6.F10.1.1.pic1.2.2.2.1.1.12">8. Be comprehensive and specific in your summarization, focusing on the finer attributes and features of the items that relate to the user’s preferences.</span> <span class="ltx_p" id="A6.F10.1.1.pic1.2.2.2.1.1.13">9. Avoid fabricating any information not supported by the user’s profile or the sequence items.</span> <span class="ltx_p" id="A6.F10.1.1.pic1.2.2.2.1.1.14"><span class="ltx_text ltx_font_bold" id="A6.F10.1.1.pic1.2.2.2.1.1.14.1">Response Example:</span> Summarization: You’ve developed interest in ….</span> </span></foreignobject></g></g></svg> </span></p> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure">Figure 10: </span>Prompt template of Agent4Rec to apply the summarization mechanism for distilling user profile.</figcaption> </figure> <div class="ltx_pagination ltx_role_newpage"></div> </section> </article> </div> <footer class="ltx_page_footer"> <div class="ltx_page_logo">Generated on Tue Mar 4 08:43:25 2025 by <a class="ltx_LaTeXML_logo" href="http://dlmf.nist.gov/LaTeXML/"><span style="letter-spacing:-0.2em; margin-right:0.1em;">L<span class="ltx_font_smallcaps" style="position:relative; bottom:2.2pt;">a</span>T<span class="ltx_font_smallcaps" style="font-size:120%;position:relative; bottom:-0.2ex;">e</span></span><span style="font-size:90%; position:relative; bottom:-0.2ex;">XML</span><img alt="Mascot Sammy" src="data:image/png;base64,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"/></a> </div></footer> </div> </body> </html>