{"id":"https://openalex.org/W2113474244","doi":"https://doi.org/10.5555/1953048.2078189","title":"Theoretical Analysis of Bayesian Matrix Factorization","display_name":"Theoretical Analysis of Bayesian Matrix Factorization","publication_year":2011,"publication_date":"2011-02-01","ids":{"openalex":"https://openalex.org/W2113474244","doi":"https://doi.org/10.5555/1953048.2078189","mag":"2113474244"},"language":"en","primary_location":{"id":"pmh:oai:CiteSeerX.psu:10.1.1.386.2780","doi":null,"is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.386.2780","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://sugiyama-www.cs.titech.ac.jp/~sugi/2011/VBMF.pdf","raw_type":"text"},"volume":null,"type":"article","is_traditional":true,"indexed_in":[],"open_access":{"is_oa":false,"oa_status":"closed","oa_url":null,"any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5034738514","display_name":"Shinichi Nakajima","orcid":"https://orcid.org/0000-0003-3970-4569","observed_orcids":["https://orcid.org/0000-0003-3970-4569"]},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Shinichi Nakajima","raw_affiliation_strings":[],"raw_orcid":"https://orcid.org/0000-0003-3970-4569","affiliations":[]},{"author_position":"last","author":{"id":"https://openalex.org/A5072744508","display_name":"Masashi Sugiyama","orcid":"https://orcid.org/0000-0001-6658-6743","observed_orcids":["https://orcid.org/0000-0001-6658-6743"]},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Masashi Sugiyama","raw_affiliation_strings":[],"raw_orcid":"https://orcid.org/0000-0001-6658-6743","affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":0,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":1.3505,"has_fulltext":false,"cited_by_count":60,"citation_normalized_percentile":{"value":0.74619289,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":88,"max":100},"biblio":{"volume":"12","issue":"79","first_page":"2583","last_page":"2648"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T12303","display_name":"Tensor decomposition and applications","score":0.14229999482631683,"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"}},"topics":[{"id":"https://openalex.org/T12303","display_name":"Tensor decomposition and applications","score":0.14229999482631683,"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/T12814","display_name":"Gaussian Processes and Bayesian Inference","score":0.13699999451637268,"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/T10500","display_name":"Sparse and Compressive Sensing Techniques","score":0.12430000305175781,"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/probabilistic-matrix-factorization","display_name":"probabilistic matrix factorization","score":0.9649999737739563},{"id":"https://openalex.org/keywords/non-identifiability","display_name":"non-identifiability","score":0.9539999961853027},{"id":"https://openalex.org/keywords/flat-prior","display_name":"flat prior","score":0.718999981880188},{"id":"https://openalex.org/keywords/empirical-bayes","display_name":"empirical Bayes","score":0.7039999961853027},{"id":"https://openalex.org/keywords/hyperparameter-learning","display_name":"hyperparameter learning","score":0.7730000019073486},{"id":"https://openalex.org/keywords/low-rank-estimation","display_name":"low-rank estimation","score":0.7590000033378601},{"id":"https://openalex.org/keywords/finite-sample-analysis","display_name":"finite-sample analysis","score":0.7549999952316284}],"concepts":[{"id":"https://openalex.org/C177769412","wikidata":"https://www.wikidata.org/wiki/Q278090","display_name":"Prior probability","level":3,"score":0.7780966758728027},{"id":"https://openalex.org/C180145272","wikidata":"https://www.wikidata.org/wiki/Q7504144","display_name":"Shrinkage","level":2,"score":0.6934968829154968},{"id":"https://openalex.org/C8642999","wikidata":"https://www.wikidata.org/wiki/Q4171168","display_name":"Hyperparameter","level":2,"score":0.5532066226005554},{"id":"https://openalex.org/C42355184","wikidata":"https://www.wikidata.org/wiki/Q1361088","display_name":"Matrix decomposition","level":3,"score":0.5096876621246338},{"id":"https://openalex.org/C107673813","wikidata":"https://www.wikidata.org/wiki/Q812534","display_name":"Bayesian probability","level":2,"score":0.4994997978210449},{"id":"https://openalex.org/C106487976","wikidata":"https://www.wikidata.org/wiki/Q685816","display_name":"Matrix (chemical analysis)","level":2,"score":0.4994237422943115},{"id":"https://openalex.org/C102592046","wikidata":"https://www.wikidata.org/wiki/Q7504144","display_name":"Shrinkage estimator","level":5,"score":0.4949151873588562},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.49247369170188904},{"id":"https://openalex.org/C160234255","wikidata":"https://www.wikidata.org/wiki/Q812535","display_name":"Bayesian inference","level":3,"score":0.48663219809532166},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.4691770374774933},{"id":"https://openalex.org/C191795146","wikidata":"https://www.wikidata.org/wiki/Q3878446","display_name":"Norm (philosophy)","level":2,"score":0.45831602811813354},{"id":"https://openalex.org/C185429906","wikidata":"https://www.wikidata.org/wiki/Q1130160","display_name":"Estimator","level":2,"score":0.45778384804725647},{"id":"https://openalex.org/C207201462","wikidata":"https://www.wikidata.org/wiki/Q182505","display_name":"Bayes' theorem","level":3,"score":0.4237288236618042},{"id":"https://openalex.org/C75291252","wikidata":"https://www.wikidata.org/wiki/Q1315756","display_name":"TRACE (psycholinguistics)","level":2,"score":0.41937369108200073},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.40722566843032837},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.39933231472969055},{"id":"https://openalex.org/C105795698","wikidata":"https://www.wikidata.org/wiki/Q12483","display_name":"Statistics","level":1,"score":0.2077612280845642},{"id":"https://openalex.org/C191393472","wikidata":"https://www.wikidata.org/wiki/Q15222032","display_name":"Bias of an estimator","level":4,"score":0.16356948018074036},{"id":"https://openalex.org/C158693339","wikidata":"https://www.wikidata.org/wiki/Q190524","display_name":"Eigenvalues and eigenvectors","level":2,"score":0.09284761548042297},{"id":"https://openalex.org/C192562407","wikidata":"https://www.wikidata.org/wiki/Q228736","display_name":"Materials science","level":0,"score":0.0},{"id":"https://openalex.org/C138885662","wikidata":"https://www.wikidata.org/wiki/Q5891","display_name":"Philosophy","level":0,"score":0.0},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.0},{"id":"https://openalex.org/C165646398","wikidata":"https://www.wikidata.org/wiki/Q3755281","display_name":"Minimum-variance unbiased estimator","level":3,"score":0.0},{"id":"https://openalex.org/C159985019","wikidata":"https://www.wikidata.org/wiki/Q181790","display_name":"Composite material","level":1,"score":0.0},{"id":"https://openalex.org/C199539241","wikidata":"https://www.wikidata.org/wiki/Q7748","display_name":"Law","level":1,"score":0.0},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.0},{"id":"https://openalex.org/C41895202","wikidata":"https://www.wikidata.org/wiki/Q8162","display_name":"Linguistics","level":1,"score":0.0},{"id":"https://openalex.org/C17744445","wikidata":"https://www.wikidata.org/wiki/Q36442","display_name":"Political science","level":0,"score":0.0}],"mesh":[],"locations_count":3,"locations":[{"id":"pmh:oai:CiteSeerX.psu:10.1.1.386.2780","doi":null,"is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.386.2780","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://sugiyama-www.cs.titech.ac.jp/~sugi/2011/VBMF.pdf","raw_type":"text"},{"id":"mag:2113474244","doi":null,"is_oa":false,"landing_page_url":"https://jmlr.org/papers/volume12/nakajima11a/nakajima11a.pdf","pdf_url":null,"source":{"id":"https://openalex.org/S118988714","display_name":"Journal of Machine Learning Research","issn_l":"1532-4435","issn":["1532-4435","1533-7928"],"is_oa":false,"is_in_doaj":false,"is_core":true,"listed_in":["anvur-class-a","anvur-scientific","ccf-a","cwts-core","dongbi-a","jufo-3","ki-jl-2","norway-2","poland-140","russia-white-list-1","vabb-shw"],"host_organization":"https://openalex.org/P4310315718","host_organization_name":"The MIT Press","host_organization_lineage":["https://openalex.org/P4310315718","https://openalex.org/P4310316440"],"host_organization_lineage_names":["The MIT Press","Massachusetts Institute of Technology"],"type":"journal"},"license":null,"license_id":null,"version":null,"is_accepted":false,"is_published":false,"raw_source_name":"Journal of Machine Learning Research","raw_type":null},{"id":"mag:3007091751","doi":"https://doi.org/10.5555/1953048.2078189","is_oa":false,"landing_page_url":"https://dl.acm.org/doi/10.5555/1953048.2078189","pdf_url":null,"source":{"id":"https://openalex.org/S118988714","display_name":"Journal of Machine Learning Research","issn_l":"1532-4435","issn":["1532-4435","1533-7928"],"is_oa":false,"is_in_doaj":false,"is_core":true,"listed_in":["anvur-class-a","anvur-scientific","ccf-a","cwts-core","dongbi-a","jufo-3","ki-jl-2","norway-2","poland-140","russia-white-list-1","vabb-shw"],"host_organization":"https://openalex.org/P4310315718","host_organization_name":"The MIT Press","host_organization_lineage":["https://openalex.org/P4310315718","https://openalex.org/P4310316440"],"host_organization_lineage_names":["The MIT Press","Massachusetts Institute of Technology"],"type":"journal"},"license":null,"license_id":null,"version":null,"is_accepted":false,"is_published":false,"raw_source_name":"Journal of Machine Learning Research","raw_type":null}],"best_oa_location":null,"sustainable_development_goals":[],"study_designs":[],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":54,"referenced_works":["https://openalex.org/W120633810","https://openalex.org/W1498183065","https://openalex.org/W1548802052","https://openalex.org/W1554544485","https://openalex.org/W1567512734","https://openalex.org/W1584444527","https://openalex.org/W1587573376","https://openalex.org/W1598266570","https://openalex.org/W1618600317","https://openalex.org/W1663973292","https://openalex.org/W1902027874","https://openalex.org/W1965555277","https://openalex.org/W1968908999","https://openalex.org/W1976618413","https://openalex.org/W1978102570","https://openalex.org/W1995466435","https://openalex.org/W1999047234","https://openalex.org/W2003471362","https://openalex.org/W2006309149","https://openalex.org/W2006681603","https://openalex.org/W2014927347","https://openalex.org/W2025341678","https://openalex.org/W2042204462","https://openalex.org/W2049401243","https://openalex.org/W2057122525","https://openalex.org/W2062125287","https://openalex.org/W2063698478","https://openalex.org/W2072857774","https://openalex.org/W2096602764","https://openalex.org/W2098385651","https://openalex.org/W2103972604","https://openalex.org/W2106139345","https://openalex.org/W2107455764","https://openalex.org/W2108090678","https://openalex.org/W2118034806","https://openalex.org/W2119752830","https://openalex.org/W2120217353","https://openalex.org/W2122090912","https://openalex.org/W2122759946","https://openalex.org/W2137225583","https://openalex.org/W2137245235","https://openalex.org/W2137902180","https://openalex.org/W2141623022","https://openalex.org/W2148522164","https://openalex.org/W2165395308","https://openalex.org/W2292751606","https://openalex.org/W2404400936","https://openalex.org/W2535123820","https://openalex.org/W2911546748","https://openalex.org/W3119651796","https://openalex.org/W3140968660","https://openalex.org/W3195084822","https://openalex.org/W3203384579","https://openalex.org/W4212863985"],"related_works":["https://openalex.org/W1568524747","https://openalex.org/W2119752830","https://openalex.org/W2167448284","https://openalex.org/W2127292230","https://openalex.org/W69540849","https://openalex.org/W2148970008","https://openalex.org/W2921374680","https://openalex.org/W2775495709","https://openalex.org/W7099525201","https://openalex.org/W2398250798"],"abstract_inverted_index":{"Recently,":[0],"variational":[1],"Bayesian":[2],"(VB)":[3],"techniques":[4],"have":[5],"been":[6],"applied":[7],"to":[8,14,85,125,160],"probabilistic":[9],"matrix":[10,30,133,143],"factorization":[11,31,134],"and":[12,59,144,187],"shown":[13,124],"perform":[15],"very":[16],"well":[17],"in":[18,50,183],"experiments.":[19],"In":[20],"this":[21,152],"paper,":[22,177],"we":[23,41,105,178],"theoretically":[24],"elucidate":[25],"properties":[26],"of":[27,37,46,131,193],"the":[28,38,51,54,60,86,102,111,129,132,138,141,171,176,184],"VB":[29,172],"(VBMF)":[32],"method.":[33],"Through":[34],"finite-sample":[35],"analysis":[36,159],"VBMF":[39,52],"estimator,":[40],"show":[42,106],"that":[43,110,136],"two":[44],"types":[45],"shrinkage":[47,58,76,95,113],"factors":[48],"exist":[49],"estimator:":[53],"positive-part":[55],"James-Stein":[56],"(PJS)":[57],"trace-norm":[61,75,94],"shrinkage,":[62],"both":[63],"acting":[64],"on":[65,170],"each":[66],"singular":[67],"component":[68],"separately":[69],"for":[70],"producing":[71],"low-rank":[72],"solutions.":[73],"The":[74],"is":[77,115,123,147,191],"simply":[78],"induced":[79,127],"by":[80,128],"non-flat":[81],"prior":[82],"information,":[83],"similarly":[84],"maximum":[87],"a":[88,107],"posteriori":[89],"(MAP)":[90],"approach.":[91],"Thus,":[92],"no":[93,180],"remains":[96],"when":[97],"priors":[98],"are":[99,166],"non-informative.":[100],"On":[101],"other":[103],"hand,":[104],"counter-intuitive":[108],"fact":[109],"PJS":[112],"factor":[114],"kept":[116],"activated":[117],"even":[118],"with":[119],"flat":[120],"priors.":[121],"This":[122],"be":[126],"non-identifiability":[130],"model,":[135],"is,":[137],"mapping":[139],"between":[140],"target":[142],"factorized":[145],"matrices":[146],"not":[148],"one-to-one.":[149],"We":[150,155],"call":[151],"model-induced":[153],"regularization.":[154],"further":[156],"extend":[157],"our":[158],"empirical":[161],"Bayes":[162],"scenarios":[163],"where":[164],"hyperparameters":[165],"also":[167],"learned":[168],"based":[169],"free":[173],"energy.":[174],"Throughout":[175],"assume":[179],"missing":[181],"entry":[182],"observed":[185],"matrix,":[186],"therefore":[188],"collaborative":[189],"filtering":[190],"out":[192],"scope.":[194]},"counts_by_year":[{"year":2025,"cited_by_count":1},{"year":2023,"cited_by_count":2},{"year":2022,"cited_by_count":1},{"year":2021,"cited_by_count":2},{"year":2020,"cited_by_count":1},{"year":2019,"cited_by_count":28},{"year":2018,"cited_by_count":5},{"year":2017,"cited_by_count":5},{"year":2016,"cited_by_count":1},{"year":2015,"cited_by_count":1},{"year":2014,"cited_by_count":3},{"year":2013,"cited_by_count":6},{"year":2012,"cited_by_count":4}],"updated_date":"2026-11-04T10:54:05.112534","created_date":"2016-06-24T00:00:00"}