[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-0-en-105":3,"doc-seo-156005-105":59,"doc-detail-156005-en":131},{"code":4,"msg":5,"data":6},0,"success",[7,13,18,23,28,33,38,43,48,51,55],{"id":8,"doc_module":4,"doc_module_name":9,"category_name":10,"show_sort_weight":11,"slug":12},1,"Document","Story & Novel",90,"story-novel",{"id":14,"doc_module":4,"doc_module_name":9,"category_name":15,"show_sort_weight":16,"slug":17},2,"Literature",80,"literature",{"id":19,"doc_module":4,"doc_module_name":9,"category_name":20,"show_sort_weight":21,"slug":22},4,"Exam",70,"exam",{"id":24,"doc_module":4,"doc_module_name":9,"category_name":25,"show_sort_weight":26,"slug":27},5,"Comic",60,"comic",{"id":29,"doc_module":4,"doc_module_name":9,"category_name":30,"show_sort_weight":31,"slug":32},6,"Technology",50,"technology",{"id":34,"doc_module":4,"doc_module_name":9,"category_name":35,"show_sort_weight":36,"slug":37},7,"Healthcare",40,"healthcare",{"id":39,"doc_module":4,"doc_module_name":9,"category_name":40,"show_sort_weight":41,"slug":42},8,"Research & Report",30,"research-report",{"id":44,"doc_module":4,"doc_module_name":9,"category_name":45,"show_sort_weight":46,"slug":47},9,"Religion & Spirituality",20,"religion-spirituality",{"id":46,"doc_module":4,"doc_module_name":9,"category_name":49,"show_sort_weight":46,"slug":50},"World Cup","world-cup",{"id":52,"doc_module":4,"doc_module_name":9,"category_name":53,"show_sort_weight":52,"slug":54},10,"Lifestyle","lifestyle",{"id":56,"doc_module":4,"doc_module_name":9,"category_name":57,"show_sort_weight":24,"slug":58},19,"General","general",{"code":4,"msg":60,"data":61},"ok",{"site_id":62,"language":63,"slug":64,"title":65,"keywords":66,"description":67,"schema_data":68,"social_meta":124,"head_meta":126,"extra_data":128,"updated_unix":130},105,"en","clustering-redemption-beyond-the-impossibility-of-kleinbergs-axioms","Clustering Redemption - Beyond the Impossibility of Kleinberg's Axioms","","Kleinberg defined three axioms for clustering and proved that no clustering function can satisfy them all simultaneously. The consistency axiom requires invariance under transformations that strengthen similarity within true clusters and weaken it across clusters. This work addresses failure cases where the correct number of clusters changes. The authors modify consistency to hold only when the cluster count remains fixed, using cost functions to infer the appropriate number of clusters. They prove that single linkage satisfies the revised axioms and show that under well-clusterable input, k-means can also satisfy them, supporting objective-driven algorithm design.",{"@graph":69,"@context":123},[70,84,106],{"@type":71,"itemListElement":72},"BreadcrumbList",[73,77,79,82],{"item":74,"name":75,"@type":76,"position":8},"https://docshare.wps.com","Home","ListItem",{"item":78,"name":9,"@type":76,"position":14},"https://docshare.wps.com/document/",{"item":80,"name":40,"@type":76,"position":81},"https://docshare.wps.com/document/research-report/",3,{"item":83,"name":65,"@type":76,"position":19},"https://docshare.wps.com/document/clustering-redemption-beyond-the-impossibility-of-kleinbergs-axioms/156005/",{"url":83,"name":65,"@type":85,"image":86,"author":91,"headline":65,"publisher":94,"fileFormat":97,"inLanguage":63,"description":67,"dateModified":98,"datePublished":99,"encodingFormat":97,"isAccessibleForFree":100,"interactionStatistic":101},"DigitalDocument",{"url":87,"@type":88,"width":89,"height":90},"https://docshare.wps.com/thumbnails/clustering-redemption-beyond-the-impossibility-of-kleinbergs-axioms/156005.png","ImageObject",300,407,{"name":92,"@type":93},"Skyler","Person",{"url":74,"name":95,"@type":96},"DocShare","Organization","application/pdf","2026-10-29","2026-08-28",true,{"@type":102,"interactionType":103,"userInteractionCount":105},"InteractionCounter",{"@type":104},"ViewAction",14,{"@type":107,"mainEntity":108},"FAQPage",[109,115,119],{"name":110,"@type":111,"acceptedAnswer":112},"What does Kleinberg’s impossibility result state about clustering axioms?","Question",{"text":113,"@type":114},"It shows there is no clustering function that simultaneously satisfies scale invariance, richness, and consistency for all inputs and dissimilarity transformations.","Answer",{"name":116,"@type":111,"acceptedAnswer":117},"How do the authors modify the consistency axiom in this paper?",{"text":118,"@type":114},"They require consistency only when the number of clusters remains unchanged, instead of demanding invariance even when the “correct” cluster count varies after a helpful transformation.",{"name":120,"@type":111,"acceptedAnswer":121},"Which clustering methods does the paper show satisfy the modified axioms?",{"text":122,"@type":114},"The paper proves that single linkage satisfies the modified axioms, and it argues that for well-clusterable inputs some popular procedures like k-means can also satisfy them.","https://schema.org",{"og:url":83,"og:type":125,"og:title":65,"og:site_name":95,"og:description":67},"article",{"robots":127,"canonical":83},"index,follow",{"doc_id":129,"site_id":62},156005,1787942597,{"code":4,"msg":5,"data":132},{"doc_id":129,"user_id":133,"nickname":92,"user_avatar":134,"doc_module":4,"category_id":39,"category_name":40,"doc_title":65,"doc_description":67,"doc_content":135,"file_id":136,"file_url":137,"file_type":138,"file_size":139,"view_count":105,"is_deleted":4,"is_public":8,"is_downloadable":8,"audit_status":8,"page_count":52,"language":140,"language_code":63,"site_id":62,"html_lang":63,"table_of_contents":141,"faqs":142,"seo_title":143,"seo_description":67,"update_tm":130,"read_time":144,"preview_page_count":4},2336464648746,"https://ap-avatar.wpscdn.com/davatar_276721f389ce27ea32af1340a28f341c","Clustering Redemption–Beyond the Impossibility of Kleinberg's Axioms  \nVincent Cohen-Addad􀀆  \nSorbonne Universités, UPMC Univ Paris 06, CNRS, LIP6  \n[vincent.cohen-addad@lip6.fr](vincent.cohen-addad@lip6.fr)  \nVarun Kanade: University of Oxford [varunk@cs.ox.ac.uk](varunk@cs.ox.ac.uk)  \nFrederik Mallmann-Trenn;  \nMIT [mallmann@mit.edu](mallmann@mit.edu)  \nAbstract  \nKleinberg [20] stated three axioms that any clustering procedure should satisfy and showed there is no clustering procedure that simultaneously satisfies all three. Oneof these, called the consistency axiom, requires that when the data is modified in a helpful way, i.e. if points in the same cluster are made more similar and those indifferent ones made less similar, the algorithm should output the same clustering. To circumvent this impossibility result, research has focused on considering clustering procedures that have a clustering quality measure (or a cost) and showing that a modification of Kleinberg's axioms that takes cost into account lead to feasible clustering procedures. In this work, we take a different approach, based on the observation that the consistency axiom fails to be satisfied when the “correct”  \nnumber of clusters changes. We modify this axiom by making use of cost functions to determine the correct number of clusters, and require that consistency holds only if the number of clusters remains unchanged. We show that single linkage satisfies the modified axioms, and if the input is well-clusterable, some popular procedures such as k-means also satisfy the axioms, taking a step towards explaining the success of these objective functions for guiding the design of algorithms.  \n1 Introduction  \nIn a highly influential paper, Kleinberg [20] showed that clustering is impossible in the following sense: there exists no clustering function, i.e. a function that takes a point-set and a pairwise dis-similarity function4 defined on them as input, and outputs a partition of the point-set, that simultaneously fulfills three simple and “reasonable” axioms—scale invariance, richness and consistency. Scale invariance requires that scaling all the dis-similarities by the same positive number should not change the output partition. Richness requires that for any partition of the point-set, there should be a way to define pairwise dis-similarities such that the clustering function will produce said partition as output. Finally, consistency requires the following: if a clustering function outputsa certain partition of a point-set, given a certain dis-similarity function, then applying this clustering function to a transformed dis-similarity function that makes points within the same part more similar and points in different parts less similar, should yield the same partition.  \nWhile seemingly very natural in the context of clustering, the last of these axioms, consistency, is somewhat questionable as has been discussed by researchers over the years (see e.g. [30] and  \n􀀆 Ce projet a bénéficié d'une aide de l'État gérée par l'Agence Nationale de la Recherche au titre du Programme FOCAL portant la référence suivante : ANR-18-CE40-0004-01 .  \n:This work was supported in part by the Alan Turing Institute through the EPSRC grant EP/N510129/1 .  \n;This work was supported in part by NSF Award Numbers CCF-1461559, CCF-0939370, and CCF-1810758 .  \n4We use dis-similarity rather than distance, as for the most part we don't require the point-set and the associated dis-similarity function to form a metric space.  \n32nd Conference on Neural Information Processing Systems (NeurIPS 2018), Montréal, Canada.  \nreferences therein) . Consider a dataset with a natural clustering consisting of k parts. Kleinberg's consistency axiom allows a transformation of the dis-similarity function by which one cluster may be subdivided into two subclusters, such that points in the same subcluster are very similar to each other, but sufficiently dis-similar to points from the other sub-cluster. The transforme","cbCaij4dKBiY5w3i","https://ap.wps.com/l/cbCaij4dKBiY5w3i","pdf",480569,"English","# Abstract\n# Introduction","[{\"question\":\"What does Kleinberg’s impossibility result state about clustering axioms?\",\"answer\":\"It shows there is no clustering function that simultaneously satisfies scale invariance, richness, and consistency for all inputs and dissimilarity transformations.\"},{\"question\":\"How do the authors modify the consistency axiom in this paper?\",\"answer\":\"They require consistency only when the number of clusters remains unchanged, instead of demanding invariance even when the “correct” cluster count varies after a helpful transformation.\"},{\"question\":\"Which clustering methods does the paper show satisfy the modified axioms?\",\"answer\":\"The paper proves that single linkage satisfies the modified axioms, and it argues that for well-clusterable inputs some popular procedures like k-means can also satisfy them.\"}]","Clustering Redemption - Beyond the Impossibility of Kleinberg's Axioms | PDF",25]