{"id":"https://openalex.org/W4414531469","doi":"https://doi.org/10.1007/s10957-025-02833-y","title":"Majorization-Minimization Bregman Proximal Gradient Algorithms for NMF with the Kullback\u2013Leibler Divergence","display_name":"Majorization-Minimization Bregman Proximal Gradient Algorithms for NMF with the Kullback\u2013Leibler Divergence","publication_year":2025,"publication_date":"2025-09-26","ids":{"openalex":"https://openalex.org/W4414531469","doi":"https://doi.org/10.1007/s10957-025-02833-y"},"language":"en","primary_location":{"id":"doi:10.1007/s10957-025-02833-y","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s10957-025-02833-y","pdf_url":"https://link.springer.com/content/pdf/10.1007/s10957-025-02833-y.pdf","source":{"id":"https://openalex.org/S52430896","display_name":"Journal of Optimization Theory and Applications","issn_l":"0022-3239","issn":["0022-3239","1573-2878"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319900","host_organization_name":"Springer Science+Business Media","host_organization_lineage":["https://openalex.org/P4310319900","https://openalex.org/P4310319965"],"host_organization_lineage_names":["Springer Science+Business Media","Springer Nature"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Journal of Optimization Theory and Applications","raw_type":"journal-article"},"type":"article","indexed_in":["crossref"],"open_access":{"is_oa":true,"oa_status":"hybrid","oa_url":"https://link.springer.com/content/pdf/10.1007/s10957-025-02833-y.pdf","any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5042046434","display_name":"Shota Takahashi","orcid":"https://orcid.org/0000-0003-3405-7163"},"institutions":[{"id":"https://openalex.org/I74801974","display_name":"The University of Tokyo","ror":"https://ror.org/057zh3y96","country_code":"JP","type":"education","lineage":["https://openalex.org/I74801974"]}],"countries":["JP"],"is_corresponding":true,"raw_author_name":"Shota Takahashi","raw_affiliation_strings":["Graduate School of Information Science and Technology, The University of Tokyo, 7-3-1 Hongo, 113-8656, Bunkyo, Tokyo, Japan"],"raw_orcid":"https://orcid.org/0000-0003-3405-7163","affiliations":[{"raw_affiliation_string":"Graduate School of Information Science and Technology, The University of Tokyo, 7-3-1 Hongo, 113-8656, Bunkyo, Tokyo, Japan","institution_ids":["https://openalex.org/I74801974"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5000782060","display_name":"Mirai Tanaka","orcid":"https://orcid.org/0000-0003-3588-5585"},"institutions":[{"id":"https://openalex.org/I200475212","display_name":"The Graduate University for Advanced Studies, SOKENDAI","ror":"https://ror.org/0516ah480","country_code":"JP","type":"education","lineage":["https://openalex.org/I200475212"]},{"id":"https://openalex.org/I4210134673","display_name":"The Institute of Statistical Mathematics","ror":"https://ror.org/03jcejr58","country_code":"JP","type":"facility","lineage":["https://openalex.org/I1319490839","https://openalex.org/I4210134673","https://openalex.org/I4210158934"]}],"countries":["JP"],"is_corresponding":false,"raw_author_name":"Mirai Tanaka","raw_affiliation_strings":["Graduate Institute for Advanced Studies, Shonankokusaimura, Hayama, Miura, 240-0115, Kanagawa, Japan","The Institute of Statistical Mathematics, 10-3 Midori-cho, 190-8562, Tachikawa, Tokyo, Japan"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Graduate Institute for Advanced Studies, Shonankokusaimura, Hayama, Miura, 240-0115, Kanagawa, Japan","institution_ids":["https://openalex.org/I200475212"]},{"raw_affiliation_string":"The Institute of Statistical Mathematics, 10-3 Midori-cho, 190-8562, Tachikawa, Tokyo, Japan","institution_ids":["https://openalex.org/I4210134673"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5041203451","display_name":"Shiro Ikeda","orcid":"https://orcid.org/0000-0002-2462-1448"},"institutions":[{"id":"https://openalex.org/I200475212","display_name":"The Graduate University for Advanced Studies, SOKENDAI","ror":"https://ror.org/0516ah480","country_code":"JP","type":"education","lineage":["https://openalex.org/I200475212"]},{"id":"https://openalex.org/I4210134673","display_name":"The Institute of Statistical Mathematics","ror":"https://ror.org/03jcejr58","country_code":"JP","type":"facility","lineage":["https://openalex.org/I1319490839","https://openalex.org/I4210134673","https://openalex.org/I4210158934"]}],"countries":["JP"],"is_corresponding":false,"raw_author_name":"Shiro Ikeda","raw_affiliation_strings":["Graduate Institute for Advanced Studies, Shonankokusaimura, Hayama, Miura, 240-0115, Kanagawa, Japan","The Institute of Statistical Mathematics, 10-3 Midori-cho, 190-8562, Tachikawa, Tokyo, Japan"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Graduate Institute for Advanced Studies, Shonankokusaimura, Hayama, Miura, 240-0115, Kanagawa, Japan","institution_ids":["https://openalex.org/I200475212"]},{"raw_affiliation_string":"The Institute of Statistical Mathematics, 10-3 Midori-cho, 190-8562, Tachikawa, Tokyo, Japan","institution_ids":["https://openalex.org/I4210134673"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":3,"corresponding_author_ids":["https://openalex.org/A5042046434"],"corresponding_institution_ids":["https://openalex.org/I74801974"],"apc_list":{"value":2390,"currency":"EUR","value_usd":2990},"apc_paid":{"value":2390,"currency":"EUR","value_usd":2990},"fwci":0.0,"has_fulltext":true,"cited_by_count":0,"citation_normalized_percentile":{"value":0.26648124,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":null,"biblio":{"volume":"208","issue":"1","first_page":null,"last_page":null},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10500","display_name":"Sparse and Compressive Sensing Techniques","score":0.9998000264167786,"subfield":{"id":"https://openalex.org/subfields/2206","display_name":"Computational Mechanics"},"field":{"id":"https://openalex.org/fields/22","display_name":"Engineering"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T10500","display_name":"Sparse and Compressive Sensing Techniques","score":0.9998000264167786,"subfield":{"id":"https://openalex.org/subfields/2206","display_name":"Computational Mechanics"},"field":{"id":"https://openalex.org/fields/22","display_name":"Engineering"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T11205","display_name":"Numerical methods in inverse problems","score":0.996399998664856,"subfield":{"id":"https://openalex.org/subfields/2610","display_name":"Mathematical Physics"},"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/T10963","display_name":"Advanced Optimization Algorithms Research","score":0.9908999800682068,"subfield":{"id":"https://openalex.org/subfields/2612","display_name":"Numerical Analysis"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/bregman-divergence","display_name":"Bregman divergence","score":0.8888000249862671},{"id":"https://openalex.org/keywords/non-negative-matrix-factorization","display_name":"Non-negative matrix factorization","score":0.7544000148773193},{"id":"https://openalex.org/keywords/separable-space","display_name":"Separable space","score":0.5705999732017517},{"id":"https://openalex.org/keywords/sequence","display_name":"Sequence (biology)","score":0.5422000288963318},{"id":"https://openalex.org/keywords/stationary-point","display_name":"Stationary point","score":0.5152000188827515},{"id":"https://openalex.org/keywords/function","display_name":"Function (biology)","score":0.49380001425743103},{"id":"https://openalex.org/keywords/theory-of-computation","display_name":"Theory of computation","score":0.4821000099182129},{"id":"https://openalex.org/keywords/convex-function","display_name":"Convex function","score":0.4000000059604645},{"id":"https://openalex.org/keywords/variable","display_name":"Variable (mathematics)","score":0.39640000462532043}],"concepts":[{"id":"https://openalex.org/C149073432","wikidata":"https://www.wikidata.org/wiki/Q4960382","display_name":"Bregman divergence","level":2,"score":0.8888000249862671},{"id":"https://openalex.org/C152671427","wikidata":"https://www.wikidata.org/wiki/Q10843505","display_name":"Non-negative matrix factorization","level":4,"score":0.7544000148773193},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7182999849319458},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.7146000266075134},{"id":"https://openalex.org/C70710897","wikidata":"https://www.wikidata.org/wiki/Q680081","display_name":"Separable space","level":2,"score":0.5705999732017517},{"id":"https://openalex.org/C2778112365","wikidata":"https://www.wikidata.org/wiki/Q3511065","display_name":"Sequence (biology)","level":2,"score":0.5422000288963318},{"id":"https://openalex.org/C189237950","wikidata":"https://www.wikidata.org/wiki/Q2500758","display_name":"Stationary point","level":2,"score":0.5152000188827515},{"id":"https://openalex.org/C14036430","wikidata":"https://www.wikidata.org/wiki/Q3736076","display_name":"Function (biology)","level":2,"score":0.49380001425743103},{"id":"https://openalex.org/C24858836","wikidata":"https://www.wikidata.org/wiki/Q844718","display_name":"Theory of computation","level":2,"score":0.4821000099182129},{"id":"https://openalex.org/C145446738","wikidata":"https://www.wikidata.org/wiki/Q319913","display_name":"Convex function","level":3,"score":0.4000000059604645},{"id":"https://openalex.org/C182365436","wikidata":"https://www.wikidata.org/wiki/Q50701","display_name":"Variable (mathematics)","level":2,"score":0.39640000462532043},{"id":"https://openalex.org/C207390915","wikidata":"https://www.wikidata.org/wiki/Q1230525","display_name":"Divergence (linguistics)","level":2,"score":0.3953000009059906},{"id":"https://openalex.org/C106487976","wikidata":"https://www.wikidata.org/wiki/Q685816","display_name":"Matrix (chemical analysis)","level":2,"score":0.38659998774528503},{"id":"https://openalex.org/C159694833","wikidata":"https://www.wikidata.org/wiki/Q2321565","display_name":"Iterative method","level":2,"score":0.3666999936103821},{"id":"https://openalex.org/C132459708","wikidata":"https://www.wikidata.org/wiki/Q744069","display_name":"Extrapolation","level":2,"score":0.3628999888896942},{"id":"https://openalex.org/C126255220","wikidata":"https://www.wikidata.org/wiki/Q141495","display_name":"Mathematical optimization","level":1,"score":0.32120001316070557},{"id":"https://openalex.org/C72134830","wikidata":"https://www.wikidata.org/wiki/Q5166524","display_name":"Convexity","level":2,"score":0.3138999938964844},{"id":"https://openalex.org/C104267543","wikidata":"https://www.wikidata.org/wiki/Q208163","display_name":"Signal processing","level":3,"score":0.30649998784065247},{"id":"https://openalex.org/C63548660","wikidata":"https://www.wikidata.org/wiki/Q858223","display_name":"Limit point","level":2,"score":0.3018999993801117},{"id":"https://openalex.org/C189950617","wikidata":"https://www.wikidata.org/wiki/Q937228","display_name":"Property (philosophy)","level":2,"score":0.2930000126361847},{"id":"https://openalex.org/C94940","wikidata":"https://www.wikidata.org/wiki/Q652941","display_name":"Hermitian matrix","level":2,"score":0.2922999858856201},{"id":"https://openalex.org/C115680565","wikidata":"https://www.wikidata.org/wiki/Q5977448","display_name":"Gradient method","level":2,"score":0.28790000081062317},{"id":"https://openalex.org/C42355184","wikidata":"https://www.wikidata.org/wiki/Q1361088","display_name":"Matrix decomposition","level":3,"score":0.2847999930381775},{"id":"https://openalex.org/C81793267","wikidata":"https://www.wikidata.org/wiki/Q7180962","display_name":"Phase retrieval","level":3,"score":0.27090001106262207},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.26420000195503235},{"id":"https://openalex.org/C103278499","wikidata":"https://www.wikidata.org/wiki/Q254465","display_name":"Similarity (geometry)","level":3,"score":0.2621999979019165},{"id":"https://openalex.org/C2777303404","wikidata":"https://www.wikidata.org/wiki/Q759757","display_name":"Convergence (economics)","level":2,"score":0.2621000111103058},{"id":"https://openalex.org/C153258448","wikidata":"https://www.wikidata.org/wiki/Q1199743","display_name":"Gradient descent","level":3,"score":0.2583000063896179}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1007/s10957-025-02833-y","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s10957-025-02833-y","pdf_url":"https://link.springer.com/content/pdf/10.1007/s10957-025-02833-y.pdf","source":{"id":"https://openalex.org/S52430896","display_name":"Journal of Optimization Theory and Applications","issn_l":"0022-3239","issn":["0022-3239","1573-2878"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319900","host_organization_name":"Springer Science+Business Media","host_organization_lineage":["https://openalex.org/P4310319900","https://openalex.org/P4310319965"],"host_organization_lineage_names":["Springer Science+Business Media","Springer Nature"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Journal of Optimization Theory and Applications","raw_type":"journal-article"}],"best_oa_location":{"id":"doi:10.1007/s10957-025-02833-y","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s10957-025-02833-y","pdf_url":"https://link.springer.com/content/pdf/10.1007/s10957-025-02833-y.pdf","source":{"id":"https://openalex.org/S52430896","display_name":"Journal of Optimization Theory and Applications","issn_l":"0022-3239","issn":["0022-3239","1573-2878"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319900","host_organization_name":"Springer Science+Business Media","host_organization_lineage":["https://openalex.org/P4310319900","https://openalex.org/P4310319965"],"host_organization_lineage_names":["Springer Science+Business Media","Springer Nature"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Journal of Optimization Theory and Applications","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[{"id":"https://openalex.org/G6028204286","display_name":null,"funder_award_id":"JP20H01951","funder_id":"https://openalex.org/F4320334764","funder_display_name":"Japan Society for the Promotion of Science"},{"id":"https://openalex.org/G6443275218","display_name":null,"funder_award_id":"JP25K21156","funder_id":"https://openalex.org/F4320334764","funder_display_name":"Japan Society for the Promotion of Science"},{"id":"https://openalex.org/G6614977947","display_name":null,"funder_award_id":"JP23K19953","funder_id":"https://openalex.org/F4320334764","funder_display_name":"Japan Society for the Promotion of Science"},{"id":"https://openalex.org/G679356420","display_name":null,"funder_award_id":"JP23H01642","funder_id":"https://openalex.org/F4320334764","funder_display_name":"Japan Society for the Promotion of Science"}],"funders":[{"id":"https://openalex.org/F4320322832","display_name":"University of Tokyo","ror":"https://ror.org/057zh3y96"},{"id":"https://openalex.org/F4320334764","display_name":"Japan Society for the Promotion of Science","ror":"https://ror.org/00hhkn466"}],"has_content":{"grobid_xml":false,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W4414531469.pdf"},"referenced_works_count":45,"referenced_works":["https://openalex.org/W205960364","https://openalex.org/W1965555277","https://openalex.org/W1998576973","https://openalex.org/W2001096515","https://openalex.org/W2012642013","https://openalex.org/W2019569173","https://openalex.org/W2021361347","https://openalex.org/W2022416612","https://openalex.org/W2022541060","https://openalex.org/W2022712430","https://openalex.org/W2033468335","https://openalex.org/W2038497950","https://openalex.org/W2048742402","https://openalex.org/W2100556411","https://openalex.org/W2124172487","https://openalex.org/W2129732816","https://openalex.org/W2153562582","https://openalex.org/W2168660030","https://openalex.org/W2265744594","https://openalex.org/W2535496140","https://openalex.org/W2551153860","https://openalex.org/W2763081248","https://openalex.org/W2791154158","https://openalex.org/W2948172096","https://openalex.org/W2963967579","https://openalex.org/W2969687526","https://openalex.org/W3091430646","https://openalex.org/W3098464223","https://openalex.org/W3100202987","https://openalex.org/W3100208521","https://openalex.org/W3114089334","https://openalex.org/W3160314930","https://openalex.org/W3161616175","https://openalex.org/W3173819510","https://openalex.org/W4285483548","https://openalex.org/W4292241382","https://openalex.org/W4311173715","https://openalex.org/W4313420686","https://openalex.org/W4313593186","https://openalex.org/W4313680291","https://openalex.org/W4366823049","https://openalex.org/W4393156988","https://openalex.org/W4398187116","https://openalex.org/W4404469989","https://openalex.org/W4413177609"],"related_works":[],"abstract_inverted_index":{"Abstract":[0],"Nonnegative":[1],"matrix":[2,20],"factorization":[3],"(NMF)":[4],"is":[5],"a":[6,17,66,74,116,141,149],"popular":[7],"method":[8],"in":[9,132],"machine":[10],"learning":[11],"and":[12,39,54,80,108,126,169],"signal":[13],"processing":[14],"to":[15,44,73,82,148],"decompose":[16],"given":[18],"nonnegative":[19,23],"into":[21],"two":[22,95],"matrices.":[24],"In":[25,156],"this":[26,136],"paper,":[27],"we":[28,63,138,159],"propose":[29],"new":[30],"algorithms,":[31],"called":[32],"majorization-minimization":[33],"Bregman":[34,118],"proximal":[35],"gradient":[36],"algorithm":[37],"(MMBPG)":[38],"MMBPG":[40,79,107],"with":[41,115,163],"extrapolation":[42],"(MMBPGe)":[43],"solve":[45],"NMF.":[46,87,155],"These":[47],"iterative":[48],"algorithms":[49,102,162,165],"minimize":[50],"the":[51,60,83,122],"objective":[52],"function":[53,57],"its":[55,129],"potential":[56],"monotonically.":[58],"Assuming":[59],"Kurdyka\u2013\u0141ojasiewicz":[61],"property,":[62],"establish":[64],"that":[65,120,127,140],"sequence":[67,142],"generated":[68,143],"by":[69,144],"MMBPG(e)":[70,145],"globally":[71,146],"converges":[72,147],"stationary":[75],"point.":[76],"We":[77],"apply":[78],"MMBPGe":[81,109],"Kullback\u2013Leibler":[84],"(KL)":[85],"divergence-based":[86],"While":[88],"most":[89],"existing":[90,164],"KL-based":[91,111,154],"NMF":[92,112],"methods":[93],"update":[94,103],"blocks":[96],"or":[97],"each":[98],"variable":[99],"alternately,":[100],"our":[101],"all":[104],"variables":[105],"simultaneously.":[106],"for":[110],"are":[113],"equipped":[114],"separable":[117],"distance":[119],"satisfies":[121],"smooth":[123],"adaptable":[124],"property":[125],"makes":[128],"subproblem":[130],"solvable":[131],"closed":[133],"form.":[134],"Using":[135],"fact,":[137],"guarantee":[139],"Karush\u2013Kuhn\u2013Tucker":[150],"(KKT)":[151],"point":[152],"of":[153],"numerical":[157],"experiments,":[158],"compare":[160],"proposed":[161],"on":[166],"synthetic":[167],"data":[168],"real-world":[170],"data.":[171]},"counts_by_year":[],"updated_date":"2026-04-02T13:48:15.688549","created_date":"2025-10-10T00:00:00"}
