{"id":"https://openalex.org/W2188030589","doi":"https://doi.org/10.1109/tnse.2016.2516339","title":"A Mathematical Theory for Clustering in Metric Spaces","display_name":"A Mathematical Theory for Clustering in Metric Spaces","publication_year":2016,"publication_date":"2016-01-01","ids":{"openalex":"https://openalex.org/W2188030589","doi":"https://doi.org/10.1109/tnse.2016.2516339","mag":"2188030589"},"language":"en","primary_location":{"id":"doi:10.1109/tnse.2016.2516339","is_oa":false,"landing_page_url":"https://doi.org/10.1109/tnse.2016.2516339","pdf_url":null,"source":{"id":"https://openalex.org/S2484352698","display_name":"IEEE Transactions on Network Science and Engineering","issn_l":"2327-4697","issn":["2327-4697","2334-329X"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319808","host_organization_name":"Institute of Electrical and Electronics Engineers","host_organization_lineage":["https://openalex.org/P4310319808"],"host_organization_lineage_names":["Institute of Electrical and Electronics Engineers"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"IEEE Transactions on Network Science and Engineering","raw_type":"journal-article"},"type":"article","indexed_in":["arxiv","crossref"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://arxiv.org/pdf/1509.07755","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5016945609","display_name":"Cheng\u2010Shang Chang","orcid":"https://orcid.org/0000-0002-5386-4756"},"institutions":[{"id":"https://openalex.org/I25846049","display_name":"National Tsing Hua University","ror":"https://ror.org/00zdnkx70","country_code":"TW","type":"education","lineage":["https://openalex.org/I25846049"]}],"countries":["TW"],"is_corresponding":false,"raw_author_name":"Cheng-Shang Chang","raw_affiliation_strings":["Institute of Communications Engineering, National Tsing Hua University, Hsinchu, Taiwan, R.O.C"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Institute of Communications Engineering, National Tsing Hua University, Hsinchu, Taiwan, R.O.C","institution_ids":["https://openalex.org/I25846049"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5062720793","display_name":"Wanjiun Liao","orcid":"https://orcid.org/0000-0001-5396-8849"},"institutions":[{"id":"https://openalex.org/I16733864","display_name":"National Taiwan University","ror":"https://ror.org/05bqach95","country_code":"TW","type":"education","lineage":["https://openalex.org/I16733864"]}],"countries":["TW"],"is_corresponding":false,"raw_author_name":"Wanjiun Liao","raw_affiliation_strings":["Department of Electrical Engineering, National Taiwan University, Taipei, Taiwan, R.O.C"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Electrical Engineering, National Taiwan University, Taipei, Taiwan, R.O.C","institution_ids":["https://openalex.org/I16733864"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5100736724","display_name":"Yu\u2010Sheng Chen","orcid":"https://orcid.org/0000-0002-5990-0455"},"institutions":[{"id":"https://openalex.org/I16733864","display_name":"National Taiwan University","ror":"https://ror.org/05bqach95","country_code":"TW","type":"education","lineage":["https://openalex.org/I16733864"]}],"countries":["TW"],"is_corresponding":false,"raw_author_name":"Yu-Sheng Chen","raw_affiliation_strings":["Department of Electrical Engineering, National Taiwan University, Taipei, Taiwan, R.O.C"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Electrical Engineering, National Taiwan University, Taipei, Taiwan, R.O.C","institution_ids":["https://openalex.org/I16733864"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5071208937","display_name":"Li-Heng Liou","orcid":null},"institutions":[{"id":"https://openalex.org/I25846049","display_name":"National Tsing Hua University","ror":"https://ror.org/00zdnkx70","country_code":"TW","type":"education","lineage":["https://openalex.org/I25846049"]}],"countries":["TW"],"is_corresponding":false,"raw_author_name":"Li-Heng Liou","raw_affiliation_strings":["Institute of Communications Engineering, National Tsing Hua University, Hsinchu, Taiwan, R.O.C"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Institute of Communications Engineering, National Tsing Hua University, Hsinchu, Taiwan, R.O.C","institution_ids":["https://openalex.org/I25846049"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":2,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":7.0307,"has_fulltext":false,"cited_by_count":31,"citation_normalized_percentile":{"value":0.96526639,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":90,"max":99},"biblio":{"volume":"3","issue":"1","first_page":"2","last_page":"16"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10637","display_name":"Advanced Clustering Algorithms Research","score":0.9991999864578247,"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":0.9991999864578247,"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/T10064","display_name":"Complex Network Analysis Techniques","score":0.9979000091552734,"subfield":{"id":"https://openalex.org/subfields/3109","display_name":"Statistical and Nonlinear Physics"},"field":{"id":"https://openalex.org/fields/31","display_name":"Physics and Astronomy"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T11106","display_name":"Data Management and Algorithms","score":0.9878000020980835,"subfield":{"id":"https://openalex.org/subfields/1711","display_name":"Signal Processing"},"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/cluster-analysis","display_name":"Cluster analysis","score":0.6957510113716125},{"id":"https://openalex.org/keywords/hierarchical-clustering","display_name":"Hierarchical clustering","score":0.6312873363494873},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.5316373109817505},{"id":"https://openalex.org/keywords/measure","display_name":"Measure (data warehouse)","score":0.5000741481781006},{"id":"https://openalex.org/keywords/duality","display_name":"Duality (order theory)","score":0.4840068519115448},{"id":"https://openalex.org/keywords/single-linkage-clustering","display_name":"Single-linkage clustering","score":0.4692528247833252},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.46031472086906433},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.43354111909866333},{"id":"https://openalex.org/keywords/fuzzy-clustering","display_name":"Fuzzy clustering","score":0.42200326919555664},{"id":"https://openalex.org/keywords/correlation-clustering","display_name":"Correlation clustering","score":0.4110628664493561},{"id":"https://openalex.org/keywords/theoretical-computer-science","display_name":"Theoretical computer science","score":0.34141138195991516},{"id":"https://openalex.org/keywords/data-mining","display_name":"Data mining","score":0.33336013555526733},{"id":"https://openalex.org/keywords/canopy-clustering-algorithm","display_name":"Canopy clustering algorithm","score":0.29691141843795776},{"id":"https://openalex.org/keywords/artificial-intelligence","display_name":"Artificial intelligence","score":0.26758626103401184},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.1925249695777893}],"concepts":[{"id":"https://openalex.org/C73555534","wikidata":"https://www.wikidata.org/wiki/Q622825","display_name":"Cluster analysis","level":2,"score":0.6957510113716125},{"id":"https://openalex.org/C92835128","wikidata":"https://www.wikidata.org/wiki/Q1277447","display_name":"Hierarchical clustering","level":3,"score":0.6312873363494873},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.5316373109817505},{"id":"https://openalex.org/C2780009758","wikidata":"https://www.wikidata.org/wiki/Q6804172","display_name":"Measure (data warehouse)","level":2,"score":0.5000741481781006},{"id":"https://openalex.org/C2778023678","wikidata":"https://www.wikidata.org/wiki/Q554403","display_name":"Duality (order theory)","level":2,"score":0.4840068519115448},{"id":"https://openalex.org/C22648726","wikidata":"https://www.wikidata.org/wiki/Q7523744","display_name":"Single-linkage clustering","level":5,"score":0.4692528247833252},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.46031472086906433},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.43354111909866333},{"id":"https://openalex.org/C17212007","wikidata":"https://www.wikidata.org/wiki/Q5511111","display_name":"Fuzzy clustering","level":3,"score":0.42200326919555664},{"id":"https://openalex.org/C94641424","wikidata":"https://www.wikidata.org/wiki/Q5172845","display_name":"Correlation clustering","level":3,"score":0.4110628664493561},{"id":"https://openalex.org/C80444323","wikidata":"https://www.wikidata.org/wiki/Q2878974","display_name":"Theoretical computer science","level":1,"score":0.34141138195991516},{"id":"https://openalex.org/C124101348","wikidata":"https://www.wikidata.org/wiki/Q172491","display_name":"Data mining","level":1,"score":0.33336013555526733},{"id":"https://openalex.org/C104047586","wikidata":"https://www.wikidata.org/wiki/Q5033439","display_name":"Canopy clustering algorithm","level":4,"score":0.29691141843795776},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.26758626103401184},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.1925249695777893}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.1109/tnse.2016.2516339","is_oa":false,"landing_page_url":"https://doi.org/10.1109/tnse.2016.2516339","pdf_url":null,"source":{"id":"https://openalex.org/S2484352698","display_name":"IEEE Transactions on Network Science and Engineering","issn_l":"2327-4697","issn":["2327-4697","2334-329X"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319808","host_organization_name":"Institute of Electrical and Electronics Engineers","host_organization_lineage":["https://openalex.org/P4310319808"],"host_organization_lineage_names":["Institute of Electrical and Electronics Engineers"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"IEEE Transactions on Network Science and Engineering","raw_type":"journal-article"},{"id":"pmh:oai:arXiv.org:1509.07755","is_oa":true,"landing_page_url":"http://arxiv.org/abs/1509.07755","pdf_url":"https://arxiv.org/pdf/1509.07755","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,"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"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:1509.07755","is_oa":true,"landing_page_url":"http://arxiv.org/abs/1509.07755","pdf_url":"https://arxiv.org/pdf/1509.07755","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,"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":[],"awards":[{"id":"https://openalex.org/G4195754236","display_name":null,"funder_award_id":"NSC 102-2221-E-002-014-MY2","funder_id":"https://openalex.org/F4320321040","funder_display_name":"National Science Council"},{"id":"https://openalex.org/G8357184552","display_name":null,"funder_award_id":"102-2221-E-007-006-MY3","funder_id":"https://openalex.org/F4320321040","funder_display_name":"National Science Council"}],"funders":[{"id":"https://openalex.org/F4320321040","display_name":"National Science Council","ror":"https://ror.org/02kv4zf79"}],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":79,"referenced_works":["https://openalex.org/W115754844","https://openalex.org/W130123586","https://openalex.org/W229097380","https://openalex.org/W1673310716","https://openalex.org/W1736726159","https://openalex.org/W1828649173","https://openalex.org/W1856487264","https://openalex.org/W1914749871","https://openalex.org/W1943383135","https://openalex.org/W1967702193","https://openalex.org/W1970301364","https://openalex.org/W1992419399","https://openalex.org/W2006554089","https://openalex.org/W2011430131","https://openalex.org/W2023348178","https://openalex.org/W2023968567","https://openalex.org/W2024514015","https://openalex.org/W2026997470","https://openalex.org/W2041504706","https://openalex.org/W2044988896","https://openalex.org/W2050032036","https://openalex.org/W2060424759","https://openalex.org/W2060979493","https://openalex.org/W2066705617","https://openalex.org/W2073459066","https://openalex.org/W2075243501","https://openalex.org/W2083620785","https://openalex.org/W2089458547","https://openalex.org/W2095293504","https://openalex.org/W2096879261","https://openalex.org/W2099242680","https://openalex.org/W2101234009","https://openalex.org/W2103135182","https://openalex.org/W2108399535","https://openalex.org/W2109726592","https://openalex.org/W2111004121","https://openalex.org/W2116819994","https://openalex.org/W2121947440","https://openalex.org/W2124209874","https://openalex.org/W2126337883","https://openalex.org/W2131681506","https://openalex.org/W2132914434","https://openalex.org/W2135674549","https://openalex.org/W2135957668","https://openalex.org/W2140829309","https://openalex.org/W2146292423","https://openalex.org/W2150593711","https://openalex.org/W2155074104","https://openalex.org/W2159237780","https://openalex.org/W2165874743","https://openalex.org/W2168346701","https://openalex.org/W2179351199","https://openalex.org/W2204035083","https://openalex.org/W2420733993","https://openalex.org/W2798115135","https://openalex.org/W2914201910","https://openalex.org/W2962801026","https://openalex.org/W2963472676","https://openalex.org/W2963831924","https://openalex.org/W2964029874","https://openalex.org/W2999729612","https://openalex.org/W3099768174","https://openalex.org/W3101413764","https://openalex.org/W3138330851","https://openalex.org/W4229641819","https://openalex.org/W4236155230","https://openalex.org/W4252435334","https://openalex.org/W4298882835","https://openalex.org/W4312512934","https://openalex.org/W6605874804","https://openalex.org/W6637131181","https://openalex.org/W6638682866","https://openalex.org/W6640006209","https://openalex.org/W6668990524","https://openalex.org/W6675354045","https://openalex.org/W6676671362","https://openalex.org/W6679406828","https://openalex.org/W6680977128","https://openalex.org/W6684578312"],"related_works":["https://openalex.org/W2243299842","https://openalex.org/W2384052049","https://openalex.org/W2406185607","https://openalex.org/W2599570117","https://openalex.org/W2129417512","https://openalex.org/W2161927371","https://openalex.org/W4231226332","https://openalex.org/W2389934482","https://openalex.org/W3087769312","https://openalex.org/W4308301473"],"abstract_inverted_index":{"Clustering":[0],"is":[1,47,54,227,279],"one":[2,42],"of":[3,33,43,60,79,95,110,146,186,219,224,252,271],"the":[4,18,31,44,50,96,100,135,195,205,220,228,264,272],"most":[5,221],"fundamental":[6,45],"problems":[7],"in":[8,17,57,93,194,263],"data":[9,61,80,253],"analysis":[10],"and":[11,143,162,180,235],"it":[12,181],"has":[13,175],"been":[14,25],"studied":[15],"extensively":[16],"literature.":[19],"Though":[20],"many":[21],"clustering":[22,27,35,249],"algorithms":[23,36],"have":[24,129],"proposed,":[26],"theories":[28],"that":[29,112,128,208,280],"justify":[30],"use":[32],"these":[34],"are":[37,113,124],"still":[38],"unsatisfactory.":[39],"In":[40,63],"particular,":[41],"challenges":[46],"to":[48,70,115,198,244,287],"address":[49,71],"following":[51],"question:":[52,137],"What":[53],"a":[55,58,73,77,84,91,105,108,119,150,154,158,163,176,184,200,232,236,240,245,250,256,268,281],"cluster":[56,106],"set":[59,78,109,251],"points?":[62],"this":[64],"paper,":[65,196],"we":[66,103,121,140,156],"make":[67],"an":[68],"attempt":[69],"such":[72,118,149,153],"question":[74],"by":[75],"considering":[76],"points":[81,111,254],"associated":[82],"with":[83,183,255],"distance":[85,97,233],"measure":[86,234,283],"(metric).":[87],"We":[88,132],"first":[89],"propose":[90,157],"newcohesionmeasure":[92],"terms":[94],"measure.":[98,238,258],"Using":[99],"cohesion":[101,237,257,282],"measure,":[102],"define":[104],"as":[107,267],"cohesive":[114],"themselves.":[116],"For":[117,152],"definition,":[120],"show":[122],"there":[123],"various":[125],"equivalent":[126],"statements":[127],"intuitive":[130],"explanations.":[131],"then":[133],"consider":[134],"second":[136],"How":[138],"do":[139],"find":[141],"clusters":[142,147],"good":[144],"partitions":[145],"under":[148],"definition?":[151],"question,":[155],"hierarchical":[159,168,172],"agglomerative":[160,169,173],"algorithm":[161,174,193,213,247,261],"partitional":[164,189],"algorithm.":[165,203,275],"Unlike":[166,204],"standard":[167],"algorithms,":[170],"our":[171,225],"specific":[177],"stopping":[178],"criterion":[179],"stops":[182],"partition":[185],"clusters.":[187],"Our":[188],"algorithm,":[190],"called":[191],"the$K$-sets":[192],"appears":[197],"be":[199,288],"new":[201],"iterative":[202],"Lloyd":[206],"iteration":[207],"needs":[209],"two-step":[210],"minimization,":[211],"our$K$-sets":[212],"only":[214],"takes":[215],"one-step":[216],"minimization.":[217],"One":[218],"interesting":[222],"findings":[223],"paper":[226],"duality":[229,241],"result":[230,242],"between":[231],"Such":[239],"leads":[243],"dual$K$-sets":[246,260],"for":[248],"The":[259,276],"converges":[262],"same":[265],"way":[266],"sequential":[269],"version":[270],"classical":[273],"kernel$K$-means":[274],"key":[277],"difference":[278],"does":[284],"not":[285],"need":[286],"positive":[289],"semi-definite.":[290]},"counts_by_year":[{"year":2025,"cited_by_count":2},{"year":2024,"cited_by_count":3},{"year":2023,"cited_by_count":1},{"year":2022,"cited_by_count":2},{"year":2021,"cited_by_count":5},{"year":2020,"cited_by_count":5},{"year":2019,"cited_by_count":2},{"year":2018,"cited_by_count":4},{"year":2017,"cited_by_count":6},{"year":2016,"cited_by_count":1}],"updated_date":"2026-08-05T07:39:15.569665","created_date":"2025-10-10T00:00:00"}