{"id":"https://openalex.org/W7134060027","doi":"https://doi.org/10.48550/arxiv.2603.04625","title":"K-Means as a Radial Basis function Network: a Variational and Gradient-based Equivalence","display_name":"K-Means as a Radial Basis function Network: a Variational and Gradient-based Equivalence","publication_year":2026,"publication_date":"2026-03-04","ids":{"openalex":"https://openalex.org/W7134060027","doi":"https://doi.org/10.48550/arxiv.2603.04625"},"language":null,"primary_location":{"id":"pmh:doi:10.48550/arxiv.2603.04625","is_oa":true,"landing_page_url":null,"pdf_url":null,"source":{"id":"https://openalex.org/S4406922384","display_name":"Open MIND","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"repository"},"license":"publisher-specific-oa","license_id":"https://openalex.org/licenses/publisher-specific-oa","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Article"},"type":"preprint","indexed_in":["datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":null,"any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5128254496","display_name":"Felipe de Jesus Felix Arredondo","orcid":null},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"Arredondo, Felipe de Jesus Felix","raw_affiliation_strings":[],"affiliations":[]},{"author_position":"middle","author":{"id":"https://openalex.org/A5029339230","display_name":"Alejandro Ucan","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Ucan-Puc, Alejandro","raw_affiliation_strings":[],"affiliations":[]},{"author_position":"last","author":{"id":"https://openalex.org/A5128254780","display_name":"Carlos Astengo Noguez","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Noguez, Carlos Astengo","raw_affiliation_strings":[],"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":3,"corresponding_author_ids":["https://openalex.org/A5128254496"],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":null,"has_fulltext":false,"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/T11612","display_name":"Stochastic Gradient Optimization Techniques","score":0.64410001039505,"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/T11612","display_name":"Stochastic Gradient Optimization Techniques","score":0.64410001039505,"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/T12303","display_name":"Tensor decomposition and applications","score":0.027400000020861626,"subfield":{"id":"https://openalex.org/subfields/2605","display_name":"Computational Mathematics"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T10057","display_name":"Face and Expression Recognition","score":0.021400000900030136,"subfield":{"id":"https://openalex.org/subfields/1707","display_name":"Computer Vision and Pattern Recognition"},"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/radial-basis-function","display_name":"Radial basis function","score":0.5539000034332275},{"id":"https://openalex.org/keywords/equivalence","display_name":"Equivalence (formal languages)","score":0.5335000157356262},{"id":"https://openalex.org/keywords/differentiable-function","display_name":"Differentiable function","score":0.5060999989509583},{"id":"https://openalex.org/keywords/embedding","display_name":"Embedding","score":0.5034000277519226},{"id":"https://openalex.org/keywords/centroid","display_name":"Centroid","score":0.44190001487731934},{"id":"https://openalex.org/keywords/artificial-neural-network","display_name":"Artificial neural network","score":0.4214000105857849},{"id":"https://openalex.org/keywords/monotone-polygon","display_name":"Monotone polygon","score":0.4212999939918518},{"id":"https://openalex.org/keywords/basis-function","display_name":"Basis function","score":0.359499990940094},{"id":"https://openalex.org/keywords/lipschitz-continuity","display_name":"Lipschitz continuity","score":0.357699990272522}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6248000264167786},{"id":"https://openalex.org/C98856871","wikidata":"https://www.wikidata.org/wiki/Q1588488","display_name":"Radial basis function","level":3,"score":0.5539000034332275},{"id":"https://openalex.org/C2780069185","wikidata":"https://www.wikidata.org/wiki/Q7977945","display_name":"Equivalence (formal languages)","level":2,"score":0.5335000157356262},{"id":"https://openalex.org/C202615002","wikidata":"https://www.wikidata.org/wiki/Q783507","display_name":"Differentiable function","level":2,"score":0.5060999989509583},{"id":"https://openalex.org/C41608201","wikidata":"https://www.wikidata.org/wiki/Q980509","display_name":"Embedding","level":2,"score":0.5034000277519226},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.4966999888420105},{"id":"https://openalex.org/C146599234","wikidata":"https://www.wikidata.org/wiki/Q511093","display_name":"Centroid","level":2,"score":0.44190001487731934},{"id":"https://openalex.org/C50644808","wikidata":"https://www.wikidata.org/wiki/Q192776","display_name":"Artificial neural network","level":2,"score":0.4214000105857849},{"id":"https://openalex.org/C2834757","wikidata":"https://www.wikidata.org/wiki/Q4925424","display_name":"Monotone polygon","level":2,"score":0.4212999939918518},{"id":"https://openalex.org/C5917680","wikidata":"https://www.wikidata.org/wiki/Q2621825","display_name":"Basis function","level":2,"score":0.359499990940094},{"id":"https://openalex.org/C22324862","wikidata":"https://www.wikidata.org/wiki/Q652707","display_name":"Lipschitz continuity","level":2,"score":0.357699990272522},{"id":"https://openalex.org/C126255220","wikidata":"https://www.wikidata.org/wiki/Q141495","display_name":"Mathematical optimization","level":1,"score":0.35760000348091125},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.35749998688697815},{"id":"https://openalex.org/C164660894","wikidata":"https://www.wikidata.org/wiki/Q2037833","display_name":"Piecewise","level":2,"score":0.3508000075817108},{"id":"https://openalex.org/C2777303404","wikidata":"https://www.wikidata.org/wiki/Q759757","display_name":"Convergence (economics)","level":2,"score":0.3416000008583069},{"id":"https://openalex.org/C90119067","wikidata":"https://www.wikidata.org/wiki/Q43260","display_name":"Polynomial","level":2,"score":0.32820001244544983},{"id":"https://openalex.org/C102205669","wikidata":"https://www.wikidata.org/wiki/Q191690","display_name":"Partition of unity","level":3,"score":0.32359999418258667},{"id":"https://openalex.org/C178009071","wikidata":"https://www.wikidata.org/wiki/Q93344","display_name":"Trigonometric functions","level":2,"score":0.31700000166893005},{"id":"https://openalex.org/C132917294","wikidata":"https://www.wikidata.org/wiki/Q2679684","display_name":"Radial basis function network","level":4,"score":0.298799991607666},{"id":"https://openalex.org/C61445026","wikidata":"https://www.wikidata.org/wiki/Q217608","display_name":"Fixed point","level":2,"score":0.290800005197525},{"id":"https://openalex.org/C14036430","wikidata":"https://www.wikidata.org/wiki/Q3736076","display_name":"Function (biology)","level":2,"score":0.28630000352859497},{"id":"https://openalex.org/C80269521","wikidata":"https://www.wikidata.org/wiki/Q7280269","display_name":"Radial function","level":2,"score":0.2669000029563904},{"id":"https://openalex.org/C182016430","wikidata":"https://www.wikidata.org/wiki/Q17029399","display_name":"Hierarchical RBF","level":4,"score":0.26570001244544983},{"id":"https://openalex.org/C45374587","wikidata":"https://www.wikidata.org/wiki/Q12525525","display_name":"Computation","level":2,"score":0.26460000872612},{"id":"https://openalex.org/C189237950","wikidata":"https://www.wikidata.org/wiki/Q2500758","display_name":"Stationary point","level":2,"score":0.26190000772476196},{"id":"https://openalex.org/C147764199","wikidata":"https://www.wikidata.org/wiki/Q6865248","display_name":"Minification","level":2,"score":0.259799987077713},{"id":"https://openalex.org/C204241405","wikidata":"https://www.wikidata.org/wiki/Q461499","display_name":"Transformation (genetics)","level":3,"score":0.25850000977516174},{"id":"https://openalex.org/C42812","wikidata":"https://www.wikidata.org/wiki/Q1082910","display_name":"Partition (number theory)","level":2,"score":0.25600001215934753},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.25369998812675476}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:doi:10.48550/arxiv.2603.04625","is_oa":true,"landing_page_url":null,"pdf_url":null,"source":{"id":"https://openalex.org/S4406922384","display_name":"Open MIND","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"repository"},"license":"publisher-specific-oa","license_id":"https://openalex.org/licenses/publisher-specific-oa","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Article"},{"id":"doi:10.48550/arxiv.2603.04625","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2603.04625","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"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":null,"is_accepted":false,"is_published":null,"raw_source_name":null,"raw_type":"article"}],"best_oa_location":{"id":"pmh:doi:10.48550/arxiv.2603.04625","is_oa":true,"landing_page_url":null,"pdf_url":null,"source":{"id":"https://openalex.org/S4406922384","display_name":"Open MIND","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"repository"},"license":"publisher-specific-oa","license_id":"https://openalex.org/licenses/publisher-specific-oa","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Article"},"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":0,"referenced_works":[],"related_works":[],"abstract_inverted_index":{"This":[0],"work":[1],"establishes":[2],"a":[3,36,151,163],"rigorous":[4],"variational":[5],"and":[6,14,30,75,123,142],"gradient-based":[7,62],"equivalence":[8],"between":[9,120],"the":[10,27,43,48,52,61,65,69,81,85,89,93,98,109,117,137],"classical":[11],"K-Means":[12,28,49,71,127,159],"algorithm":[13],"differentiable":[15,168],"Radial":[16],"Basis":[17],"Function":[18],"(RBF)":[19],"neural":[20],"networks":[21],"with":[22],"smooth":[23,37],"responsibilities.":[24],"By":[25],"reparameterizing":[26],"objective":[29,45],"embedding":[31],"its":[32],"distortion":[33],"functional":[34],"into":[35,132],"weighted":[38],"loss,":[39],"we":[40,96],"prove":[41],"that":[42,60],"RBF":[44,66,156],"$\u0393$-converges":[46],"to":[47,128],"solution":[50],"as":[51],"temperature":[53],"parameter":[54],"$\u03c3$":[55],"vanishes.":[56],"We":[57],"further":[58],"demonstrate":[59],"updates":[63],"of":[64,88,100,140,154],"centers":[67],"recover":[68],"exact":[70],"centroid":[72],"update":[73],"rule":[74],"induce":[76],"identical":[77],"training":[78],"trajectories":[79],"in":[80,92],"limit.":[82],"To":[83],"address":[84],"numerical":[86],"instability":[87],"Softmax":[90],"transformation":[91],"low-temperature":[94],"regime,":[95],"propose":[97],"integration":[99],"Entmax-1.5,":[101],"which":[102],"ensures":[103],"stable":[104],"polynomial":[105],"convergence":[106],"while":[107],"preserving":[108],"underlying":[110],"Voronoi":[111],"partition":[112],"structure.":[113],"These":[114],"results":[115],"bridge":[116],"conceptual":[118],"gap":[119],"discrete":[121],"partitioning":[122],"continuous":[124],"optimization,":[125],"enabling":[126],"be":[129],"embedded":[130],"directly":[131],"deep":[133],"learning":[134],"architectures":[135],"for":[136,166],"joint":[138],"optimization":[139],"representations":[141],"clusters.":[143],"Empirical":[144],"validation":[145],"across":[146],"diverse":[147],"synthetic":[148],"geometries":[149],"confirms":[150],"monotone":[152],"collapse":[153],"soft":[155],"centroids":[157],"toward":[158],"fixed":[160],"points,":[161],"providing":[162],"unified":[164],"framework":[165],"end-to-end":[167],"clustering.":[169]},"counts_by_year":[],"updated_date":"2026-04-04T16:13:02.066488","created_date":"2026-03-07T00:00:00"}
