[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-82270-en":3,"doc-seo-82270-105":29,"detail-sidebar-cat-0-en-105":95},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":20,"is_deleted":4,"is_public":20,"is_downloadable":20,"audit_status":20,"page_count":21,"language":22,"language_code":23,"site_id":24,"html_lang":23,"table_of_contents":25,"faqs":26,"seo_title":13,"seo_description":14,"update_tm":27,"read_time":28},82270,962075114101,"Seraphina","https://ap-avatar.wpscdn.com/avatar/e000253a75eb197efd?x-image-process=image/resize,m_fixed,w_180,h_180&k=1780044092746381165",8,"Research & Report","Influence Diagnostics in High-dimensional M-estimation: Precise Asymptotics","Leave-one-out influence measures how strongly a training point affects a statistical model by quantifying the change in test error or fitted parameters when that point is removed. While low-dimensional asymptotics are well studied, high-dimensional settings make individual influences depend on all other samples. For convex M-estimation with Gaussian design in the regime n ≍ d, the influence values across the training set converge to a precisely characterized limiting measure. The results show that highly influential samples concentrate near the decision boundary, motivating links to data-selection based active learning heuristics.","arXiv :2607 .09250v1 [ stat .ML] 10 Jul 2026  \nInfluence Diagnostics in High-dimensional M-estimation:  \nPrecise Asymptotics  \nHugo Cui  \nUniversité Paris-Saclay, CNRS, Laboratoire de mathématiques d’Orsay, 91405, Orsay, France  \nAbstract  \nThe impact of a given training point on a statistical model is classically measured through its leave-one-out influence, which quantifies the effect of its removal from the training set on the model accuracy. While the statistics of leave-one-out influences are well understood in the low-dimensional, large sample limit n ! ∞ , d = O(1), they become more intricate in high dimensions, as the influence of a given sample develops non-trivial dependencies on all other training samples. For convex M-estimation under Gaussian design, in the high-dimensional limit n ≍ d, we show that the distribution of the influences across the training set converges to a limiting measure which we sharply characterize. Building on these results, we provide evidence that influential samples tend to lie close to the decision boundary, thereby making contact with a standard data selection heuristic inactive learning.  \nContents  \n1 Introduction 2  \n1.1 Related works ..................................... 3  \n2 High-dimensional asymptotics of leave-one-out influences 5  \n2.1 ERM under Gaussian design ............................. 5  \n2.2 Influence metrics ................................... 6  \n2.3 Asymptotics of leave-one-out influences ....................... 7  \n2.4 Main technical results ................................ 8  \n2.5 Discussion ....................................... 13  \n3 Consequences for active learning 15  \n3.1 Related works on data selection ........................... 15  \n3.2 Conditional influence ................................. 17  \n4 Conclusion 19  \nA Assumptions and reminders 27  \nA.1 Assumptions ..................................... 27  \nA.2 Notations ....................................... 28  \nA.3 Leave-one-out approximation results ........................ 31  \nB Concentration of summary statistics 33  \nB.1 Pointwise concentration of functional statistics .................. 33  \nB.2 Concentration of Cauchy integrals ......................... 40  \nC Deterministic equivalents 50  \nC.1 Asymptotic residual distribution .......................... 50  \nC.2 Computation of the deterministic equivalents ................... 56  \nD Convergence of influence distributions 69  \nD.1 Comments on the risk normalization ........................ 69  \nD.2 Proof of Theorem 2.1 ................................. 72  \nD.3 Proof of Proposition 2.2 ............................... 74  \nE Auxiliary lemmas 76  \nF Extensions 78  \nG Details on numerical experiments 80  \n1 Introduction  \nHow does a given training sample ultimately shape the predictions of a statistical model? Given a dataset D = {xi , yi }i∈JnK with n covariate/label pairs, the relevance of the i−th sample (xi , yi) may be most naturally captured by its leave-one-out influence IFi , defined as the difference in test error or model parameters when training on D\\ (xi , yi) (from which the sample was deleted), rather than D. The leave-one-out influence thus isolates the impact of a sample on the trained model, affording a highly interpretable measure of importance. Perhaps then unsurprisingly, influences constitute a fundamental explainability metric at the confluence of several fields. Initially formalized in the seminal works of Hampel [1974], Cook [1979], Cook and Weisberg [1982, 1980], Hampel et al. [1986] as a diagnostic to probe model robustness or identify outliers, influence diagnostics are also used for data selection [Ting and Brochu, 2018] as a natural proxy for informativeness. In applied machine learning, influence metrics are leveraged to explicate black-box neural network predictions [Koh and Liang, 2017 , Barshan et al. , 2020] across abroad section of modern deep learning, from diffusion models [Mlodozeniec et al. , 2025] to large language","cbCaivz3KE5hhENm","https://ap.wps.com/l/cbCaivz3KE5hhENm","pdf",1215842,1,81,"English","en",105,"# Introduction\n## Related works\n# High-dimensional asymptotics of leave-one-out influences\n## ERM under Gaussian design\n## Influence metrics\n## Asymptotics of leave-one-out influences\n## Main technical results\n## Discussion\n# Consequences for active learning\n## Related works on data selection\n## Conditional influence\n# Conclusion","[{\"question\":\"What does leave-one-out influence quantify in this work?\",\"answer\":\"It quantifies the change in test error or model parameters when a single training point is removed from the training set.\"},{\"question\":\"Why do leave-one-out influences become difficult to analyze in high dimensions?\",\"answer\":\"When d is comparable to n, the influence of one sample develops non-trivial dependencies on the other training samples, breaking classical low-dimensional convergence behavior.\"},{\"question\":\"What is proven for convex M-estimation with Gaussian design when n ≍ d?\",\"answer\":\"The distribution of leave-one-out influences over the training set converges weakly to a limiting distribution, which the paper characterizes sharply.\"},{\"question\":\"How do the findings connect to active learning and data selection?\",\"answer\":\"The analysis provides evidence that influential samples tend to lie close to the decision boundary, making an influence-related data selection heuristic for active learning effective.\"}]",1784179293,204,{"code":4,"msg":30,"data":31},"ok",{"site_id":24,"language":23,"slug":32,"title":13,"keywords":33,"description":14,"schema_data":34,"social_meta":90,"head_meta":92,"extra_data":94,"updated_unix":27},"influence-diagnostics-in-high-dimensional-m-estimation-precise-asymptotics","",{"@graph":35,"@context":89},[36,53,68],{"@type":37,"itemListElement":38},"BreadcrumbList",[39,43,47,50],{"item":40,"name":41,"@type":42,"position":20},"https://docshare.wps.com","Home","ListItem",{"item":44,"name":45,"@type":42,"position":46},"https://docshare.wps.com/document/","Document",2,{"item":48,"name":12,"@type":42,"position":49},"https://docshare.wps.com/document/research-report/",3,{"item":51,"name":13,"@type":42,"position":52},"https://docshare.wps.com/document/influence-diagnostics-in-high-dimensional-m-estimation-precise-asymptotics/82270/",4,{"url":51,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":23,"description":14,"dateModified":61,"datePublished":62,"encodingFormat":60,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":40,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-17","2026-07-16",true,{"@type":65,"interactionType":66,"userInteractionCount":20},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71,77,81,85],{"name":72,"@type":73,"acceptedAnswer":74},"What does leave-one-out influence quantify in this work?","Question",{"text":75,"@type":76},"It quantifies the change in test error or model parameters when a single training point is removed from the training set.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"Why do leave-one-out influences become difficult to analyze in high dimensions?",{"text":80,"@type":76},"When d is comparable to n, the influence of one sample develops non-trivial dependencies on the other training samples, breaking classical low-dimensional convergence behavior.",{"name":82,"@type":73,"acceptedAnswer":83},"What is proven for convex M-estimation with Gaussian design when n ≍ d?",{"text":84,"@type":76},"The distribution of leave-one-out influences over the training set converges weakly to a limiting distribution, which the paper characterizes sharply.",{"name":86,"@type":73,"acceptedAnswer":87},"How do the findings connect to active learning and data selection?",{"text":88,"@type":76},"The analysis provides evidence that influential samples tend to lie close to the decision boundary, making an influence-related data selection heuristic for active learning effective.","https://schema.org",{"og:url":51,"og:type":91,"og:title":13,"og:site_name":58,"og:description":14},"article",{"robots":93,"canonical":51},"index,follow",{"doc_id":7,"site_id":24},{"code":4,"msg":5,"data":96},[97,101,105,109,114,119,124,127,132,135,139],{"id":20,"doc_module":4,"doc_module_name":45,"category_name":98,"show_sort_weight":99,"slug":100},"Story & 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