{"id":"https://openalex.org/W7212307332","doi":"https://doi.org/10.48550/arxiv.2609.11173","title":"Hierarchical Clustering Can Jointly Satisfy Richness, Consistency, and Scale Invariance","display_name":"Hierarchical Clustering Can Jointly Satisfy Richness, Consistency, and Scale Invariance","publication_year":2026,"publication_date":"2026-09-10","ids":{"openalex":"https://openalex.org/W7212307332","doi":"https://doi.org/10.48550/arxiv.2609.11173"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2609.11173","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.11173","pdf_url":"https://arxiv.org/pdf/2609.11173","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},"volume":null,"type":"preprint","is_traditional":true,"indexed_in":["arxiv","datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://arxiv.org/pdf/2609.11173","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5092082421","display_name":"Daichi Kuroda","orcid":"https://orcid.org/0009-0004-3467-5177","observed_orcids":["https://orcid.org/0009-0004-3467-5177"]},"institutions":[{"id":"https://openalex.org/I5124864","display_name":"\u00c9cole Polytechnique F\u00e9d\u00e9rale de Lausanne","ror":"https://ror.org/02s376052","country_code":"CH","type":"education","lineage":["https://openalex.org/I2799323385","https://openalex.org/I5124864"]}],"countries":["CH"],"is_corresponding":false,"raw_author_name":"Kuroda, Daichi","raw_affiliation_strings":["School of Computer and Communication Sciences \u00c9cole Polytechnique F\u00e9d\u00e9rale de Lausanne (EPFL) Lausanne , 1015 , Switzerland"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"School of Computer and Communication Sciences \u00c9cole Polytechnique F\u00e9d\u00e9rale de Lausanne (EPFL) Lausanne , 1015 , Switzerland","institution_ids":["https://openalex.org/I5124864"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5020004829","display_name":"Maximilien Dreveton","orcid":"https://orcid.org/0000-0001-6613-0615","observed_orcids":["https://orcid.org/0000-0001-6613-0615"]},"institutions":[{"id":"https://openalex.org/I1294671590","display_name":"Centre National de la Recherche Scientifique","ror":"https://ror.org/02feahw73","country_code":"FR","type":"government","lineage":["https://openalex.org/I1294671590"]},{"id":"https://openalex.org/I4210144844","display_name":"Laboratoire d\u2019Analyse et de Math\u00e9matiques Appliqu\u00e9es","ror":"https://ror.org/0581g8849","country_code":"FR","type":"facility","lineage":["https://openalex.org/I1294671590","https://openalex.org/I197681013","https://openalex.org/I2800365227","https://openalex.org/I4210141950","https://openalex.org/I4210144844","https://openalex.org/I4210154111"]},{"id":"https://openalex.org/I4210154111","display_name":"Universit\u00e9 Gustave Eiffel","ror":"https://ror.org/03x42jk29","country_code":"FR","type":"education","lineage":["https://openalex.org/I2800365227","https://openalex.org/I4210154111"]}],"countries":["FR"],"is_corresponding":false,"raw_author_name":"Dreveton, Maximilien","raw_affiliation_strings":["LAMA , UMR-CNRS 8050 , Universit\u00e9 Gustave Eiffel 5 Bd Descartes , 77454 Marne-la-Vall\u00e9e , France"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"LAMA , UMR-CNRS 8050 , Universit\u00e9 Gustave Eiffel 5 Bd Descartes , 77454 Marne-la-Vall\u00e9e , France","institution_ids":["https://openalex.org/I1294671590","https://openalex.org/I4210144844","https://openalex.org/I4210154111"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5014508720","display_name":"Matthias Grossglauser","orcid":"https://orcid.org/0000-0002-3031-1438","observed_orcids":["https://orcid.org/0000-0002-3031-1438"]},"institutions":[{"id":"https://openalex.org/I1294671590","display_name":"Centre National de la Recherche Scientifique","ror":"https://ror.org/02feahw73","country_code":"FR","type":"government","lineage":["https://openalex.org/I1294671590"]},{"id":"https://openalex.org/I4210144844","display_name":"Laboratoire d\u2019Analyse et de Math\u00e9matiques Appliqu\u00e9es","ror":"https://ror.org/0581g8849","country_code":"FR","type":"facility","lineage":["https://openalex.org/I1294671590","https://openalex.org/I197681013","https://openalex.org/I2800365227","https://openalex.org/I4210141950","https://openalex.org/I4210144844","https://openalex.org/I4210154111"]},{"id":"https://openalex.org/I4210154111","display_name":"Universit\u00e9 Gustave Eiffel","ror":"https://ror.org/03x42jk29","country_code":"FR","type":"education","lineage":["https://openalex.org/I2800365227","https://openalex.org/I4210154111"]},{"id":"https://openalex.org/I5124864","display_name":"\u00c9cole Polytechnique F\u00e9d\u00e9rale de Lausanne","ror":"https://ror.org/02s376052","country_code":"CH","type":"education","lineage":["https://openalex.org/I2799323385","https://openalex.org/I5124864"]}],"countries":["CH","FR"],"is_corresponding":false,"raw_author_name":"Grossglauser, Matthias","raw_affiliation_strings":["LAMA , UMR-CNRS 8050 , Universit\u00e9 Gustave Eiffel 5 Bd Descartes , 77454 Marne-la-Vall\u00e9e , France","School of Computer and Communication Sciences \u00c9cole Polytechnique F\u00e9d\u00e9rale de Lausanne (EPFL) Lausanne , 1015 , Switzerland"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"LAMA , UMR-CNRS 8050 , Universit\u00e9 Gustave Eiffel 5 Bd Descartes , 77454 Marne-la-Vall\u00e9e , France","institution_ids":["https://openalex.org/I1294671590","https://openalex.org/I4210144844","https://openalex.org/I4210154111"]},{"raw_affiliation_string":"School of Computer and Communication Sciences \u00c9cole Polytechnique F\u00e9d\u00e9rale de Lausanne (EPFL) Lausanne , 1015 , Switzerland","institution_ids":["https://openalex.org/I5124864"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5067285971","display_name":"Patrick Thiran","orcid":"https://orcid.org/0009-0001-8176-6854","observed_orcids":["https://orcid.org/0009-0001-8176-6854"]},"institutions":[{"id":"https://openalex.org/I5124864","display_name":"\u00c9cole Polytechnique F\u00e9d\u00e9rale de Lausanne","ror":"https://ror.org/02s376052","country_code":"CH","type":"education","lineage":["https://openalex.org/I2799323385","https://openalex.org/I5124864"]}],"countries":["CH"],"is_corresponding":false,"raw_author_name":"Thiran, Patrick","raw_affiliation_strings":["School of Computer and Communication Sciences \u00c9cole Polytechnique F\u00e9d\u00e9rale de Lausanne (EPFL) Lausanne , 1015 , Switzerland"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"School of Computer and Communication Sciences \u00c9cole Polytechnique F\u00e9d\u00e9rale de Lausanne (EPFL) Lausanne , 1015 , Switzerland","institution_ids":["https://openalex.org/I5124864"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":4,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":null,"has_fulltext":true,"cited_by_count":0,"citation_normalized_percentile":null,"cited_by_percentile_year":null,"biblio":{"volume":null,"issue":null,"first_page":null,"last_page":null},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10637","display_name":"Advanced Clustering Algorithms Research","score":1.0,"subfield":{"id":"https://openalex.org/subfields/1702","display_name":"Artificial Intelligence"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T10637","display_name":"Advanced Clustering Algorithms Research","score":1.0,"subfield":{"id":"https://openalex.org/subfields/1702","display_name":"Artificial Intelligence"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T13373","display_name":"Data Mining and Machine Learning Applications","score":0.0,"subfield":{"id":"https://openalex.org/subfields/1710","display_name":"Information Systems"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T12536","display_name":"Topological and Geometric Data Analysis","score":0.0,"subfield":{"id":"https://openalex.org/subfields/1703","display_name":"Computational Theory and Mathematics"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/hierarchical-clustering","display_name":"hierarchical clustering","score":0.9990000128746033},{"id":"https://openalex.org/keywords/scale-invariance","display_name":"scale invariance","score":0.9070000052452087},{"id":"https://openalex.org/keywords/consistency-axiom","display_name":"consistency axiom","score":0.9959999918937683},{"id":"https://openalex.org/keywords/single-linkage","display_name":"single linkage","score":0.8479999899864197},{"id":"https://openalex.org/keywords/refinement-relations","display_name":"refinement relations","score":0.6050000190734863},{"id":"https://openalex.org/keywords/partial-orders","display_name":"partially ordered sets","score":0.7680000066757202},{"id":"https://openalex.org/keywords/cluster-definition","display_name":"cluster definition","score":0.4399999976158142}],"concepts":[{"id":"https://openalex.org/C167729594","wikidata":"https://www.wikidata.org/wiki/Q17736","display_name":"Axiom","level":2,"score":0.7304999828338623},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.70169997215271},{"id":"https://openalex.org/C31170391","wikidata":"https://www.wikidata.org/wiki/Q188619","display_name":"Hierarchy","level":2,"score":0.6880000233650208},{"id":"https://openalex.org/C2776261394","wikidata":"https://www.wikidata.org/wiki/Q315562","display_name":"Impossibility","level":2,"score":0.6707000136375427},{"id":"https://openalex.org/C73555534","wikidata":"https://www.wikidata.org/wiki/Q622825","display_name":"Cluster analysis","level":2,"score":0.6525999903678894},{"id":"https://openalex.org/C184898388","wikidata":"https://www.wikidata.org/wiki/Q1435712","display_name":"Pairwise comparison","level":2,"score":0.5317999720573425},{"id":"https://openalex.org/C2777212361","wikidata":"https://www.wikidata.org/wiki/Q5127848","display_name":"Class (philosophy)","level":2,"score":0.49399998784065247},{"id":"https://openalex.org/C92835128","wikidata":"https://www.wikidata.org/wiki/Q1277447","display_name":"Hierarchical clustering","level":3,"score":0.4869999885559082},{"id":"https://openalex.org/C125773388","wikidata":"https://www.wikidata.org/wiki/Q792542","display_name":"Axiomatic system","level":3,"score":0.4494999945163727},{"id":"https://openalex.org/C177264268","wikidata":"https://www.wikidata.org/wiki/Q1514741","display_name":"Set (abstract data type)","level":2,"score":0.4302999973297119},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.4120999872684479},{"id":"https://openalex.org/C17212007","wikidata":"https://www.wikidata.org/wiki/Q5511111","display_name":"Fuzzy clustering","level":3,"score":0.3702999949455261},{"id":"https://openalex.org/C80444323","wikidata":"https://www.wikidata.org/wiki/Q2878974","display_name":"Theoretical computer science","level":1,"score":0.36500000953674316},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.36320000886917114},{"id":"https://openalex.org/C2778755073","wikidata":"https://www.wikidata.org/wiki/Q10858537","display_name":"Scale (ratio)","level":2,"score":0.3479999899864197},{"id":"https://openalex.org/C27964816","wikidata":"https://www.wikidata.org/wiki/Q5164359","display_name":"Constrained clustering","level":5,"score":0.32589998841285706},{"id":"https://openalex.org/C25343380","wikidata":"https://www.wikidata.org/wiki/Q277521","display_name":"Relation (database)","level":2,"score":0.3009999990463257},{"id":"https://openalex.org/C2776502983","wikidata":"https://www.wikidata.org/wiki/Q690182","display_name":"Contrast (vision)","level":2,"score":0.29499998688697815},{"id":"https://openalex.org/C65180967","wikidata":"https://www.wikidata.org/wiki/Q130901","display_name":"Binary relation","level":2,"score":0.29420000314712524},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.29409998655319214},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.29109999537467957},{"id":"https://openalex.org/C162392398","wikidata":"https://www.wikidata.org/wiki/Q272404","display_name":"Finite set","level":2,"score":0.2808000147342682},{"id":"https://openalex.org/C2780801425","wikidata":"https://www.wikidata.org/wiki/Q5164392","display_name":"Construct (python library)","level":2,"score":0.2752000093460083},{"id":"https://openalex.org/C2780841128","wikidata":"https://www.wikidata.org/wiki/Q5073781","display_name":"Characterization (materials science)","level":2,"score":0.2678999900817871},{"id":"https://openalex.org/C22047676","wikidata":"https://www.wikidata.org/wiki/Q898680","display_name":"Clustering coefficient","level":3,"score":0.25690001249313354}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2609.11173","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.11173","pdf_url":"https://arxiv.org/pdf/2609.11173","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},{"id":"doi:10.48550/arxiv.2609.11173","doi":"https://doi.org/10.48550/arxiv.2609.11173","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2609.11173","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Preprint"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:2609.11173","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.11173","pdf_url":"https://arxiv.org/pdf/2609.11173","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},"sustainable_development_goals":[],"study_designs":[],"awards":[],"funders":[],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7212307332.pdf","grobid_xml":"https://content.openalex.org/works/W7212307332.grobid-xml"},"referenced_works_count":1,"referenced_works":["https://openalex.org/W7106767144"],"related_works":["https://openalex.org/W2999484166","https://openalex.org/W2009393773","https://openalex.org/W7096850019","https://openalex.org/W3009022873","https://openalex.org/W2159452463","https://openalex.org/W7157160011","https://openalex.org/W2020492796","https://openalex.org/W2111004121","https://openalex.org/W291782994","https://openalex.org/W7099774375"],"abstract_inverted_index":{"Despite":[0],"its":[1],"ubiquity,":[2],"clustering":[3,25,66,84],"lacks":[4],"a":[5,12,51,55,106,134,174,189],"universally":[6],"accepted":[7],"definition":[8],"of":[9,71,109,115,140,176,184],"what":[10],"is":[11,50,168],"cluster.":[13],"Kleinberg's":[14],"Impossibility":[15],"Theorem":[16],"formalizes":[17],"this":[18,39,44,166],"difficulty":[19],"by":[20,121,127],"showing":[21],"that":[22,125],"no":[23,148],"flat":[24,65],"method":[26,172],"can":[27],"simultaneously":[28],"satisfy":[29],"three":[30],"natural":[31],"axioms:":[32],"scale":[33],"invariance,":[34],"richness,":[35],"and":[36,105,151,180],"consistency.":[37],"In":[38,77],"paper,":[40],"we":[41,90],"ask":[42],"whether":[43],"impossibility":[45],"persists":[46],"when":[47],"the":[48,64,68,118,128,138],"output":[49],"hierarchy":[52,119,175],"rather":[53],"than":[54],"single":[56,110],"partition.":[57],"We":[58,93],"show":[59],"that,":[60],"in":[61],"contrast":[62],"to":[63],"setting,":[67],"hierarchical":[69,83],"analog":[70],"these":[72,87],"axioms":[73],"are":[74],"jointly":[75],"satisfiable.":[76],"fact,":[78],"there":[79],"exist":[80],"uncountably":[81,153],"many":[82,154],"methods":[85,100,186],"satisfying":[86],"axioms,":[88],"which":[89],"call":[91],"admissible.":[92],"explicitly":[94],"construct":[95],"several":[96],"admissible":[97,116,141,163,171,185],"methods,":[98,117],"including":[99],"based":[101],"on":[102,137],"well-separated":[103,178],"clusters":[104],"non-binary":[107],"version":[108],"linkage.":[111],"For":[112],"certain":[113],"pairs":[114],"produced":[120,126],"one":[122],"always":[123],"refines":[124],"other.":[129],"This":[130,143],"refinement":[131],"relation":[132],"defines":[133],"partial":[135],"order":[136],"class":[139],"methods.":[142,164],"partially":[144],"ordered":[145],"set":[146],"has":[147],"greatest":[149],"element":[150],"contains":[152,173],"pairwise":[155],"incompatible":[156],"maximal":[157],"elements,":[158],"revealing":[159],"substantial":[160],"diversity":[161,167],"among":[162],"Nevertheless,":[165],"constrained:":[169],"every":[170,181],"sufficiently":[177],"clusters,":[179],"finite":[182],"collection":[183],"shares":[187],"such":[188],"nontrivial":[190],"common":[191],"backbone.":[192]},"counts_by_year":[],"updated_date":"2026-11-07T11:25:07.637180","created_date":"2026-09-12T00:00:00"}
