{"id":"https://openalex.org/W4402574295","doi":"https://doi.org/10.1287/ijoo.2022.0030","title":"Nonconvex Factorization and Manifold Formulations Are Almost Equivalent in Low-Rank Matrix Optimization","display_name":"Nonconvex Factorization and Manifold Formulations Are Almost Equivalent in Low-Rank Matrix Optimization","publication_year":2024,"publication_date":"2024-09-17","ids":{"openalex":"https://openalex.org/W4402574295","doi":"https://doi.org/10.1287/ijoo.2022.0030"},"language":"en","primary_location":{"id":"doi:10.1287/ijoo.2022.0030","is_oa":true,"landing_page_url":"https://doi.org/10.1287/ijoo.2022.0030","pdf_url":null,"source":{"id":"https://openalex.org/S4210207407","display_name":"INFORMS Journal on Optimization","issn_l":"2575-1484","issn":["2575-1484","2575-1492"],"is_oa":true,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310315699","host_organization_name":"Institute for Operations Research and the Management Sciences","host_organization_lineage":["https://openalex.org/P4310315699"],"host_organization_lineage_names":["Institute for Operations Research and the Management Sciences"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"INFORMS Journal on Optimization","raw_type":"journal-article"},"type":"article","indexed_in":["crossref"],"open_access":{"is_oa":true,"oa_status":"diamond","oa_url":"https://doi.org/10.1287/ijoo.2022.0030","any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5048735624","display_name":"Yuetian Luo","orcid":"https://orcid.org/0000-0001-5145-3868"},"institutions":[{"id":"https://openalex.org/I40347166","display_name":"University of Chicago","ror":"https://ror.org/024mw5h28","country_code":"US","type":"education","lineage":["https://openalex.org/I40347166"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Yuetian Luo","raw_affiliation_strings":["Data Science Institute, University of Chicago, Chicago, Illinois 60637","Data Science Institute, University of Chicago, Chicago, Illinois 60637;"],"raw_orcid":"https://orcid.org/0000-0001-5145-3868","affiliations":[{"raw_affiliation_string":"Data Science Institute, University of Chicago, Chicago, Illinois 60637","institution_ids":["https://openalex.org/I40347166"]},{"raw_affiliation_string":"Data Science Institute, University of Chicago, Chicago, Illinois 60637;","institution_ids":["https://openalex.org/I40347166"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5100354513","display_name":"Xudong Li","orcid":"https://orcid.org/0000-0002-9275-1965"},"institutions":[{"id":"https://openalex.org/I24943067","display_name":"Fudan University","ror":"https://ror.org/013q1eq08","country_code":"CN","type":"education","lineage":["https://openalex.org/I24943067"]}],"countries":["CN"],"is_corresponding":true,"raw_author_name":"Xudong Li","raw_affiliation_strings":["School of Data Science, Fudan University, Shanghai 200433, China","School of Data Science, Fudan University, Shanghai 200433, China;"],"raw_orcid":"https://orcid.org/0000-0002-9275-1965","affiliations":[{"raw_affiliation_string":"School of Data Science, Fudan University, Shanghai 200433, China","institution_ids":["https://openalex.org/I24943067"]},{"raw_affiliation_string":"School of Data Science, Fudan University, Shanghai 200433, China;","institution_ids":["https://openalex.org/I24943067"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5025435311","display_name":"Anru R. Zhang","orcid":"https://orcid.org/0000-0002-8721-5252"},"institutions":[{"id":"https://openalex.org/I170897317","display_name":"Duke University","ror":"https://ror.org/00py81415","country_code":"US","type":"education","lineage":["https://openalex.org/I170897317"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Anru R. Zhang","raw_affiliation_strings":["Department of Biostatistics & Bioinformatics, Computer Science, Mathematics, and Statistical Science, Duke University, Durham, North Carolina 27710"],"raw_orcid":"https://orcid.org/0000-0002-8721-5252","affiliations":[{"raw_affiliation_string":"Department of Biostatistics & Bioinformatics, Computer Science, Mathematics, and Statistical Science, Duke University, Durham, North Carolina 27710","institution_ids":["https://openalex.org/I170897317"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":3,"corresponding_author_ids":["https://openalex.org/A5100354513"],"corresponding_institution_ids":["https://openalex.org/I24943067"],"apc_list":null,"apc_paid":null,"fwci":0.2527,"has_fulltext":false,"cited_by_count":1,"citation_normalized_percentile":{"value":0.47188333,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":90,"max":94},"biblio":{"volume":"7","issue":"1","first_page":"38","last_page":"60"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10500","display_name":"Sparse and Compressive Sensing Techniques","score":0.9991999864578247,"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":0.9991999864578247,"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/T10792","display_name":"Matrix Theory and Algorithms","score":0.9970999956130981,"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"}},{"id":"https://openalex.org/T11183","display_name":"Advanced X-ray Imaging Techniques","score":0.9894000291824341,"subfield":{"id":"https://openalex.org/subfields/3108","display_name":"Radiation"},"field":{"id":"https://openalex.org/fields/31","display_name":"Physics and Astronomy"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/rank","display_name":"Rank (graph theory)","score":0.6365243196487427},{"id":"https://openalex.org/keywords/manifold","display_name":"Manifold (fluid mechanics)","score":0.6150819659233093},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.6014052033424377},{"id":"https://openalex.org/keywords/factorization","display_name":"Factorization","score":0.5821996927261353},{"id":"https://openalex.org/keywords/matrix","display_name":"Matrix (chemical analysis)","score":0.5055053234100342},{"id":"https://openalex.org/keywords/mathematical-optimization","display_name":"Mathematical optimization","score":0.436541348695755},{"id":"https://openalex.org/keywords/matrix-decomposition","display_name":"Matrix decomposition","score":0.43001413345336914},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.3764762580394745},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.3292822241783142},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.21349933743476868},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.12411344051361084},{"id":"https://openalex.org/keywords/materials-science","display_name":"Materials science","score":0.0925397276878357},{"id":"https://openalex.org/keywords/engineering","display_name":"Engineering","score":0.07651162147521973}],"concepts":[{"id":"https://openalex.org/C164226766","wikidata":"https://www.wikidata.org/wiki/Q7293202","display_name":"Rank (graph theory)","level":2,"score":0.6365243196487427},{"id":"https://openalex.org/C529865628","wikidata":"https://www.wikidata.org/wiki/Q1790740","display_name":"Manifold (fluid mechanics)","level":2,"score":0.6150819659233093},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6014052033424377},{"id":"https://openalex.org/C187834632","wikidata":"https://www.wikidata.org/wiki/Q188804","display_name":"Factorization","level":2,"score":0.5821996927261353},{"id":"https://openalex.org/C106487976","wikidata":"https://www.wikidata.org/wiki/Q685816","display_name":"Matrix (chemical analysis)","level":2,"score":0.5055053234100342},{"id":"https://openalex.org/C126255220","wikidata":"https://www.wikidata.org/wiki/Q141495","display_name":"Mathematical optimization","level":1,"score":0.436541348695755},{"id":"https://openalex.org/C42355184","wikidata":"https://www.wikidata.org/wiki/Q1361088","display_name":"Matrix decomposition","level":3,"score":0.43001413345336914},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.3764762580394745},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.3292822241783142},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.21349933743476868},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.12411344051361084},{"id":"https://openalex.org/C192562407","wikidata":"https://www.wikidata.org/wiki/Q228736","display_name":"Materials science","level":0,"score":0.0925397276878357},{"id":"https://openalex.org/C127413603","wikidata":"https://www.wikidata.org/wiki/Q11023","display_name":"Engineering","level":0,"score":0.07651162147521973},{"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/C158693339","wikidata":"https://www.wikidata.org/wiki/Q190524","display_name":"Eigenvalues and eigenvectors","level":2,"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/C78519656","wikidata":"https://www.wikidata.org/wiki/Q101333","display_name":"Mechanical engineering","level":1,"score":0.0}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.1287/ijoo.2022.0030","is_oa":true,"landing_page_url":"https://doi.org/10.1287/ijoo.2022.0030","pdf_url":null,"source":{"id":"https://openalex.org/S4210207407","display_name":"INFORMS Journal on Optimization","issn_l":"2575-1484","issn":["2575-1484","2575-1492"],"is_oa":true,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310315699","host_organization_name":"Institute for Operations Research and the Management Sciences","host_organization_lineage":["https://openalex.org/P4310315699"],"host_organization_lineage_names":["Institute for Operations Research and the Management Sciences"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"INFORMS Journal on Optimization","raw_type":"journal-article"},{"id":"pmh:oai:RePEc:inm:orijoo:v:7:y:2025:i:1:p:38-60","is_oa":false,"landing_page_url":"http://doi.org/10.1287/ijoo.2022.0030","pdf_url":null,"source":{"id":"https://openalex.org/S4306401271","display_name":"RePEc: Research Papers in Economics","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I77793887","host_organization_name":"Federal Reserve Bank of St. Louis","host_organization_lineage":["https://openalex.org/I77793887"],"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":"article"}],"best_oa_location":{"id":"doi:10.1287/ijoo.2022.0030","is_oa":true,"landing_page_url":"https://doi.org/10.1287/ijoo.2022.0030","pdf_url":null,"source":{"id":"https://openalex.org/S4210207407","display_name":"INFORMS Journal on Optimization","issn_l":"2575-1484","issn":["2575-1484","2575-1492"],"is_oa":true,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310315699","host_organization_name":"Institute for Operations Research and the Management Sciences","host_organization_lineage":["https://openalex.org/P4310315699"],"host_organization_lineage_names":["Institute for Operations Research and the Management Sciences"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"INFORMS Journal on Optimization","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":56,"referenced_works":["https://openalex.org/W1551360398","https://openalex.org/W2007593159","https://openalex.org/W2037253724","https://openalex.org/W2042182457","https://openalex.org/W2045512849","https://openalex.org/W2058078260","https://openalex.org/W2060204507","https://openalex.org/W2078397124","https://openalex.org/W2110662303","https://openalex.org/W2117198760","https://openalex.org/W2133105246","https://openalex.org/W2138505091","https://openalex.org/W2142949395","https://openalex.org/W2145080587","https://openalex.org/W2277129572","https://openalex.org/W2283023501","https://openalex.org/W2402588523","https://openalex.org/W2542482481","https://openalex.org/W2592246480","https://openalex.org/W2768927505","https://openalex.org/W2789612570","https://openalex.org/W2795405496","https://openalex.org/W2892058114","https://openalex.org/W2904850325","https://openalex.org/W2912134372","https://openalex.org/W2931810883","https://openalex.org/W2962771824","https://openalex.org/W2963731874","https://openalex.org/W2963945345","https://openalex.org/W2964014592","https://openalex.org/W2964146881","https://openalex.org/W2964249688","https://openalex.org/W2964262188","https://openalex.org/W2969215180","https://openalex.org/W2993205200","https://openalex.org/W3003707928","https://openalex.org/W3004471367","https://openalex.org/W3014571813","https://openalex.org/W3037096902","https://openalex.org/W3043157417","https://openalex.org/W3092363548","https://openalex.org/W3100522009","https://openalex.org/W3105952791","https://openalex.org/W3106518847","https://openalex.org/W3113425034","https://openalex.org/W3118581654","https://openalex.org/W3119520928","https://openalex.org/W3152061570","https://openalex.org/W4205293427","https://openalex.org/W4213187536","https://openalex.org/W4231570791","https://openalex.org/W4248613038","https://openalex.org/W4323539420","https://openalex.org/W4367676999","https://openalex.org/W4378980949","https://openalex.org/W4383736860"],"related_works":["https://openalex.org/W2794559785","https://openalex.org/W1754499339","https://openalex.org/W2361403716","https://openalex.org/W1973739845","https://openalex.org/W119752240","https://openalex.org/W2904061867","https://openalex.org/W4289115725","https://openalex.org/W2967798957","https://openalex.org/W3144644423","https://openalex.org/W2024384195"],"abstract_inverted_index":{"In":[0,70,156],"this":[1,115],"paper,":[2],"we":[3,49,72,187],"consider":[4],"the":[5,10,32,54,65,75,96,100,104,110,117,123,139,154,157,162,184,195,202,215,234,245,257],"geometric":[6,85,119,136,179,205],"landscape":[7,97,120,163,180],"connection":[8,98,121,164],"of":[9,34,47,56,112,143,165,207,217,239,250],"widely":[11],"studied":[12],"manifold":[13,66,124,146],"and":[14,22,36,61,67,93,103,125,132,145,170,197,214,225,243,254,256,260],"factorization":[15,68,126,144,167,220],"formulations":[16,127,168],"in":[17,95,148,153,194,201,219],"low-rank":[18,149,159,211],"positive":[19],"semidefinite":[20],"(PSD)":[21],"general":[23,105,158],"matrix":[24,150,160,212],"optimization.":[25],"We":[26],"establish":[27,198],"a":[28,45,135],"sandwich":[29,76,185],"relation":[30,77],"on":[31,53,204],"spectrum":[33],"Riemannian":[35],"Euclidean":[37],"Hessians":[38],"at":[39,272],"first-order":[40],"stationary":[41,59],"points":[42],"(FOSPs).":[43],"As":[44],"result":[46],"that,":[48],"obtain":[50],"an":[51],"equivalence":[52],"set":[55],"FOSPs,":[57],"second-order":[58],"points,":[60],"strict":[62],"saddles":[63],"between":[64,122],"formulations.":[69],"addition,":[71],"show":[73],"that":[74],"can":[78],"be":[79],"used":[80],"to":[81,90,190],"transfer":[82],"more":[83],"quantitative":[84],"properties":[86],"from":[87,222],"one":[88],"formulation":[89],"another.":[91],"Similarities":[92],"differences":[94],"under":[99],"PSD":[101],"case":[102,106],"are":[107,188],"discussed.":[108],"To":[109],"best":[111],"our":[113],"knowledge,":[114],"is":[116,173,270],"first":[118],"for":[128,138],"handling":[129],"rank":[130],"constraints,":[131],"it":[133],"provides":[134],"explanation":[137],"similar":[140],"empirical":[141],"performance":[142],"approaches":[147],"optimization":[151],"observed":[152],"literature.":[155],"optimization,":[161,213],"two":[166],"(unregularized":[169],"regularized":[171],"ones)":[172],"also":[174],"provided.":[175],"By":[176],"applying":[177],"these":[178],"connections":[181],"(in":[182],"particular,":[183],"relation),":[186],"able":[189],"solve":[191],"unanswered":[192],"questions":[193],"literature":[196],"stronger":[199],"results":[200],"applications":[203],"analysis":[206],"phase":[208],"retrieval,":[209],"well-conditioned":[210],"role":[216],"regularization":[218],"arising":[221],"machine":[223],"learning":[224],"signal":[226],"processing.":[227],"Funding:":[228],"This":[229],"work":[230],"was":[231],"supported":[232],"by":[233],"National":[235,246],"Key":[236],"R&D":[237],"Program":[238,262],"China":[240,251],"[Grants":[241,252],"2020YFA0711900":[242],"2020YFA0711901],":[244],"Natural":[247],"Science":[248,259],"Foundation":[249],"12271107":[253],"62141407],":[255],"Shanghai":[258],"Technology":[261],"[Grant":[263],"21JC1400600].":[264],"Supplemental":[265],"Material:":[266],"The":[267],"online":[268],"appendix":[269],"available":[271],"https://doi.org/10.1287/ijoo.2022.0030":[273],".":[274]},"counts_by_year":[{"year":2024,"cited_by_count":1}],"updated_date":"2026-08-21T09:56:20.448147","created_date":"2025-10-10T00:00:00"}
