[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-84335-en":3,"doc-seo-84335-105":29,"detail-sidebar-cat-0-en-105":91},{"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},84335,1099514068365,"Aurelia","https://ap-avatar.wpscdn.com/avatar/10000253d8d9f28188e?_k=1776742907772140068",8,"Research & Report","An interpretable Good-Turing restart criterion for k-means++","The k-means++ algorithm is typically restarted a fixed, arbitrary number of times to avoid poor local optima, even though data sets differ widely in difficulty. This practice makes performance comparisons unreliable and wastes computation on easy cases while potentially under-allocating effort to hard ones. The proposed GTRC combines a Good–Turing estimate with an unconditional bound and a confidence-based bound, stopping when the probability of further improvement drops below ε. Across 36 data sets, GTRC matches or matches fixed well-chosen restart counts while adapting restart numbers in an interpretable, data-dependent way.","arXiv :2607 .08243v 1 [ cs .LG] 9 Jul 2026  \nAn interpretable Good–Turing restart criterion  \nfor k-means++  \nRenato Cordeiro de Amorim∗  \nAbstract  \nThe k-means++ algorithm is commonly restarted multiple times to avoid poor local optima, yet the number of restarts is almost always chosen arbitrarily and applied uniformly regardless of data set difficulty. This undermines any comparison relying on such a choice and wastes computation on easy data sets while potentially under-serving hard ones. We introduce GTRC, a restart criterion combining a Good-Turing estimate, a proven unconditional bound, and a confidence-based bound on the probability that a further restart would improve on the current result, stopping once this probability falls below a user-specified tolerance ε . Across 36 data sets, GTRC reached clustering quality competitive with well-chosen fixed restart counts, while the number of restarts used varied considerably and appropriately with data set difficulty, governed by an interpretable, data-dependent signal rather than a fixed rule. GTRC offers a principled and reportable alternative to fixing the number of k-means++ restartsin advance. Software: [https://github.com/RCdeAmorim/Good-Turing](https://github.com/RCdeAmorim/Good-Turing)Restart-Criterion.  \nKeywords: k-means++, restart criterion, Good-Turing estimation.  \n1 Introduction  \nClustering algorithms follow the unsupervised learning framework, and by consequence do not require labelled samples to learn from. Hence, such algorithms have become one of the main tools in exploratory data analysis, and have been applied to a number of areas such as bioinformatics, cybersecurity, quantitative finance, and computer vision, among others [1, 2, 3, 4, 5, 6] .  \nThe k-means++ algorithm [7] remains, arguably, the most popular clustering algorithm in use today [8] . Given a data set and a number of clusters k , kmeans++ produces a partition of X by iteratively minimising a within-cluster sum of squares objective. A well-known weakness of k-means++ is that it isnot guaranteed to reach the global minimum of its objective, as it is sensitive to  \n∗ School of Computer Science and Electronic Engineering, University of Essex, Colchester, [UK. r.amorim@essex.ac.uk](UK. r.amorim@essex.ac.uk)  \nthe choice of initial centroids (a weakness inherited from k-means, see Section 2.1) . To mitigate this, it is common practice to restart k-means++ multiple times and select as final partition that with the lowest objective value.  \nThis practice is so widespread that it has become the default behaviour in popular software packages such as scikit-learn [9] . However, the number of such restarts is invariably chosen arbitrarily. Some studies use ten restarts, others twenty, and others still one hundred. And potentially even more problematic: after the number of restarts is selected, it is usually applied to all data sets regardless of their structure. When a fixed number of restarts is applied indiscriminately across data sets of varying complexity, the resulting baseline is either wasteful on easy data sets or inadequate on hard ones, potentially undermining the validity of any comparison that relies on it.  \nTo our knowledge, there is no established method providing a principled basis for the choice of restarts. In this paper, we address this problem directly by introducing the Good–Turing restart criterion (GTRC), a theoretically-grounded, data-adaptive stopping rule with a clear probabilistic interpretation. It halts when the estimated probability that a further restart would improve on the best partition found so far falls below a user-specified tolerance. This makes the criterion not only adaptive to the complexity of each data set, but also transparent and interpretable.  \n2 Background  \nIn this section we review the two areas of prior work this paper draws on. We first summarise the k-means and k-means++ algorithms and the role of restarts in mitigating their sensitivity to initiali","cbCaijIOs8DanoWE","https://ap.wps.com/l/cbCaijIOs8DanoWE","pdf",514291,1,17,"English","en",105,"# Introduction\n# Background\n## Clustering\n## Good–Turing estimation\n# Restart criterion (GTRC)","[{\"question\":\"Why do existing k-means++ workflows use multiple restarts, and what problem does the paper identify with this practice?\",\"answer\":\"Multiple restarts are used to mitigate sensitivity to initialization and poor local optima. The paper argues that the restart count is usually arbitrary and applied uniformly across data sets, which can waste computation on easy sets and fail to adequately address hard ones, undermining comparisons.\"},{\"question\":\"What is GTRC and how does it decide when to stop restarting k-means++?\",\"answer\":\"GTRC (Good–Turing restart criterion) is a data-adaptive stopping rule. It combines a Good–Turing estimate with a proven unconditional bound and a confidence-based bound, and it stops once the estimated probability that another restart will improve the current best partition falls below a user tolerance ε.\"},{\"question\":\"How does the paper evaluate GTRC, and what are the main results?\",\"answer\":\"The method is tested across 36 data sets. GTRC achieves clustering quality competitive with fixed restart counts chosen well in advance, while using a number of restarts that varies appropriately with data set difficulty based on an interpretable, data-dependent signal.\"}]",1784194896,43,{"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":86,"head_meta":88,"extra_data":90,"updated_unix":27},"an-interpretable-good-turing-restart-criterion-for-k-means","",{"@graph":35,"@context":85},[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/an-interpretable-good-turing-restart-criterion-for-k-means/84335/",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],{"name":72,"@type":73,"acceptedAnswer":74},"Why do existing k-means++ workflows use multiple restarts, and what problem does the paper identify with this practice?","Question",{"text":75,"@type":76},"Multiple restarts are used to mitigate sensitivity to initialization and poor local optima. The paper argues that the restart count is usually arbitrary and applied uniformly across data sets, which can waste computation on easy sets and fail to adequately address hard ones, undermining comparisons.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"What is GTRC and how does it decide when to stop restarting k-means++?",{"text":80,"@type":76},"GTRC (Good–Turing restart criterion) is a data-adaptive stopping rule. It combines a Good–Turing estimate with a proven unconditional bound and a confidence-based bound, and it stops once the estimated probability that another restart will improve the current best partition falls below a user tolerance ε.",{"name":82,"@type":73,"acceptedAnswer":83},"How does the paper evaluate GTRC, and what are the main results?",{"text":84,"@type":76},"The method is tested across 36 data sets. GTRC achieves clustering quality competitive with fixed restart counts chosen well in advance, while using a number of restarts that varies appropriately with data set difficulty based on an interpretable, data-dependent signal.","https://schema.org",{"og:url":51,"og:type":87,"og:title":13,"og:site_name":58,"og:description":14},"article",{"robots":89,"canonical":51},"index,follow",{"doc_id":7,"site_id":24},{"code":4,"msg":5,"data":92},[93,97,101,105,110,115,120,123,128,131,135],{"id":20,"doc_module":4,"doc_module_name":45,"category_name":94,"show_sort_weight":95,"slug":96},"Story & Novel",90,"story-novel",{"id":46,"doc_module":4,"doc_module_name":45,"category_name":98,"show_sort_weight":99,"slug":100},"Literature",80,"literature",{"id":52,"doc_module":4,"doc_module_name":45,"category_name":102,"show_sort_weight":103,"slug":104},"Exam",70,"exam",{"id":106,"doc_module":4,"doc_module_name":45,"category_name":107,"show_sort_weight":108,"slug":109},5,"Comic",60,"comic",{"id":111,"doc_module":4,"doc_module_name":45,"category_name":112,"show_sort_weight":113,"slug":114},6,"Technology",50,"technology",{"id":116,"doc_module":4,"doc_module_name":45,"category_name":117,"show_sort_weight":118,"slug":119},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":45,"category_name":12,"show_sort_weight":121,"slug":122},30,"research-report",{"id":124,"doc_module":4,"doc_module_name":45,"category_name":125,"show_sort_weight":126,"slug":127},9,"Religion & Spirituality",20,"religion-spirituality",{"id":126,"doc_module":4,"doc_module_name":45,"category_name":129,"show_sort_weight":126,"slug":130},"World Cup","world-cup",{"id":132,"doc_module":4,"doc_module_name":45,"category_name":133,"show_sort_weight":132,"slug":134},10,"Lifestyle","lifestyle",{"id":136,"doc_module":4,"doc_module_name":45,"category_name":137,"show_sort_weight":106,"slug":138},19,"General","general"]