{"id":"https://openalex.org/W3119520928","doi":"https://doi.org/10.1109/tit.2021.3049171","title":"The Global Optimization Geometry of Low-Rank Matrix Optimization","display_name":"The Global Optimization Geometry of Low-Rank Matrix Optimization","publication_year":2021,"publication_date":"2021-01-05","ids":{"openalex":"https://openalex.org/W3119520928","doi":"https://doi.org/10.1109/tit.2021.3049171","mag":"3119520928"},"language":"en","primary_location":{"id":"doi:10.1109/tit.2021.3049171","is_oa":false,"landing_page_url":"https://doi.org/10.1109/tit.2021.3049171","pdf_url":null,"source":{"id":"https://openalex.org/S4502562","display_name":"IEEE Transactions on Information Theory","issn_l":"0018-9448","issn":["0018-9448","1557-9654"],"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 Information Theory","raw_type":"journal-article"},"type":"article","indexed_in":["crossref"],"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/A5011989964","display_name":"Zhihui Zhu","orcid":"https://orcid.org/0000-0002-3856-0375"},"institutions":[{"id":"https://openalex.org/I131651094","display_name":"University of Denver","ror":"https://ror.org/04w7skc03","country_code":"US","type":"education","lineage":["https://openalex.org/I131651094"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Zhihui Zhu","raw_affiliation_strings":["University of Denver, Denver, CO, USA"],"raw_orcid":"https://orcid.org/0000-0002-3856-0375","affiliations":[{"raw_affiliation_string":"University of Denver, Denver, CO, USA","institution_ids":["https://openalex.org/I131651094"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5033881291","display_name":"Qiuwei Li","orcid":"https://orcid.org/0000-0002-2306-6649"},"institutions":[{"id":"https://openalex.org/I161318765","display_name":"University of California, Los Angeles","ror":"https://ror.org/046rm7j60","country_code":"US","type":"education","lineage":["https://openalex.org/I161318765"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Qiuwei Li","raw_affiliation_strings":["University of California at Los Angeles, Los Angeles, CA, USA"],"raw_orcid":"https://orcid.org/0000-0002-2306-6649","affiliations":[{"raw_affiliation_string":"University of California at Los Angeles, Los Angeles, CA, USA","institution_ids":["https://openalex.org/I161318765"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5101848896","display_name":"Gongguo Tang","orcid":"https://orcid.org/0000-0002-7879-1338"},"institutions":[{"id":"https://openalex.org/I167576493","display_name":"Colorado School of Mines","ror":"https://ror.org/04raf6v53","country_code":"US","type":"education","lineage":["https://openalex.org/I167576493"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Gongguo Tang","raw_affiliation_strings":["Colorado School of Mines, Golden, CO, USA"],"raw_orcid":"https://orcid.org/0000-0002-7879-1338","affiliations":[{"raw_affiliation_string":"Colorado School of Mines, Golden, CO, USA","institution_ids":["https://openalex.org/I167576493"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5065207980","display_name":"Michael B. Wakin","orcid":"https://orcid.org/0000-0002-2165-4586"},"institutions":[{"id":"https://openalex.org/I167576493","display_name":"Colorado School of Mines","ror":"https://ror.org/04raf6v53","country_code":"US","type":"education","lineage":["https://openalex.org/I167576493"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Michael B. Wakin","raw_affiliation_strings":["Colorado School of Mines, Golden, CO, USA"],"raw_orcid":"https://orcid.org/0000-0002-2165-4586","affiliations":[{"raw_affiliation_string":"Colorado School of Mines, Golden, CO, USA","institution_ids":["https://openalex.org/I167576493"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":3,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":4.2543,"has_fulltext":false,"cited_by_count":39,"citation_normalized_percentile":{"value":0.95864635,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":97,"max":99},"biblio":{"volume":"67","issue":"2","first_page":"1308","last_page":"1331"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10500","display_name":"Sparse and Compressive Sensing Techniques","score":0.9998999834060669,"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":0.9998999834060669,"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/T10963","display_name":"Advanced Optimization Algorithms Research","score":0.9986000061035156,"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"}},{"id":"https://openalex.org/T10792","display_name":"Matrix Theory and Algorithms","score":0.9879000186920166,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/notation","display_name":"Notation","score":0.6775205135345459},{"id":"https://openalex.org/keywords/rank","display_name":"Rank (graph theory)","score":0.6493054628372192},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.47446781396865845},{"id":"https://openalex.org/keywords/algebra-over-a-field","display_name":"Algebra over a field","score":0.3882865905761719},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.3626072108745575},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.35544294118881226},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.3388251066207886},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.1849556565284729},{"id":"https://openalex.org/keywords/arithmetic","display_name":"Arithmetic","score":0.13090738654136658}],"concepts":[{"id":"https://openalex.org/C45357846","wikidata":"https://www.wikidata.org/wiki/Q2001982","display_name":"Notation","level":2,"score":0.6775205135345459},{"id":"https://openalex.org/C164226766","wikidata":"https://www.wikidata.org/wiki/Q7293202","display_name":"Rank (graph theory)","level":2,"score":0.6493054628372192},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.47446781396865845},{"id":"https://openalex.org/C136119220","wikidata":"https://www.wikidata.org/wiki/Q1000660","display_name":"Algebra over a field","level":2,"score":0.3882865905761719},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.3626072108745575},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.35544294118881226},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.3388251066207886},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.1849556565284729},{"id":"https://openalex.org/C94375191","wikidata":"https://www.wikidata.org/wiki/Q11205","display_name":"Arithmetic","level":1,"score":0.13090738654136658}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1109/tit.2021.3049171","is_oa":false,"landing_page_url":"https://doi.org/10.1109/tit.2021.3049171","pdf_url":null,"source":{"id":"https://openalex.org/S4502562","display_name":"IEEE Transactions on Information Theory","issn_l":"0018-9448","issn":["0018-9448","1557-9654"],"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 Information Theory","raw_type":"journal-article"}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[{"id":"https://openalex.org/G3745342036","display_name":null,"funder_award_id":"CAREER CCF-1149225","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"},{"id":"https://openalex.org/G4323043452","display_name":null,"funder_award_id":"CCF-1464205","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"},{"id":"https://openalex.org/G4737188276","display_name":null,"funder_award_id":"CCF-1409261","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"},{"id":"https://openalex.org/G6077994957","display_name":null,"funder_award_id":"N660011824020","funder_id":"https://openalex.org/F4320332180","funder_display_name":"Defense Advanced Research Projects Agency"},{"id":"https://openalex.org/G8305295122","display_name":"Collaborative Research: CIF: Small: Deep Sparse Models: Analysis and Algorithms","funder_award_id":"2008460","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"}],"funders":[{"id":"https://openalex.org/F4320306076","display_name":"National Science Foundation","ror":"https://ror.org/021nxhr62"},{"id":"https://openalex.org/F4320332180","display_name":"Defense Advanced Research Projects Agency","ror":"https://ror.org/02caytj08"}],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":87,"referenced_works":["https://openalex.org/W147998453","https://openalex.org/W1836708065","https://openalex.org/W1884015253","https://openalex.org/W2057069782","https://openalex.org/W2059283452","https://openalex.org/W2103972604","https://openalex.org/W2106005123","https://openalex.org/W2109521352","https://openalex.org/W2116950495","https://openalex.org/W2118550318","https://openalex.org/W2122090912","https://openalex.org/W2129131372","https://openalex.org/W2134618502","https://openalex.org/W2138505091","https://openalex.org/W2142949395","https://openalex.org/W2143075842","https://openalex.org/W2143238292","https://openalex.org/W2161374719","https://openalex.org/W2162451874","https://openalex.org/W2171253836","https://openalex.org/W2189224591","https://openalex.org/W2255205876","https://openalex.org/W2268674159","https://openalex.org/W2283023501","https://openalex.org/W2283214199","https://openalex.org/W2405479119","https://openalex.org/W2406099690","https://openalex.org/W2467441529","https://openalex.org/W2542482481","https://openalex.org/W2547949325","https://openalex.org/W2566608758","https://openalex.org/W2568381517","https://openalex.org/W2592246480","https://openalex.org/W2592651140","https://openalex.org/W2601846890","https://openalex.org/W2604130501","https://openalex.org/W2611328865","https://openalex.org/W2789612570","https://openalex.org/W2931810883","https://openalex.org/W2945117139","https://openalex.org/W2950201748","https://openalex.org/W2962970675","https://openalex.org/W2963014255","https://openalex.org/W2963092340","https://openalex.org/W2963099043","https://openalex.org/W2963130606","https://openalex.org/W2963357892","https://openalex.org/W2963381277","https://openalex.org/W2963404710","https://openalex.org/W2963464803","https://openalex.org/W2963731874","https://openalex.org/W2963865165","https://openalex.org/W2964106499","https://openalex.org/W2964156132","https://openalex.org/W2964200481","https://openalex.org/W2964204240","https://openalex.org/W2970821902","https://openalex.org/W3101302494","https://openalex.org/W3103457637","https://openalex.org/W3104289311","https://openalex.org/W3105489725","https://openalex.org/W3105952791","https://openalex.org/W3113425034","https://openalex.org/W4206519735","https://openalex.org/W4229706427","https://openalex.org/W4294328701","https://openalex.org/W4300829514","https://openalex.org/W6608805005","https://openalex.org/W6638615092","https://openalex.org/W6639384702","https://openalex.org/W6677671969","https://openalex.org/W6677837081","https://openalex.org/W6680025182","https://openalex.org/W6681238269","https://openalex.org/W6687403782","https://openalex.org/W6695367633","https://openalex.org/W6712153814","https://openalex.org/W6713600550","https://openalex.org/W6713869322","https://openalex.org/W6719853796","https://openalex.org/W6726972909","https://openalex.org/W6729126563","https://openalex.org/W6729214561","https://openalex.org/W6734280532","https://openalex.org/W6735923916","https://openalex.org/W6761899565","https://openalex.org/W6767396694"],"related_works":["https://openalex.org/W2504004674","https://openalex.org/W1595229445","https://openalex.org/W2498744856","https://openalex.org/W4390482104","https://openalex.org/W322408318","https://openalex.org/W149041114","https://openalex.org/W2963177394","https://openalex.org/W1965815883","https://openalex.org/W4313359513","https://openalex.org/W763418848"],"abstract_inverted_index":{"This":[0],"paper":[1],"considers":[2],"general":[3,10],"rank-constrained":[4],"optimization":[5,132,196,353],"problems":[6],"that":[7,33,141,240,273,348],"minimize":[8],"a":[9,191,199,252,349,361],"objective":[11,144,157,275],"function":[12,145,158,276],"<inline-formula":[13,25,53,61,72,80,88,96,112,120,208,216,224,232,241,255,300,318,333],"xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"":[14,26,54,62,73,81,89,97,113,121,209,217,225,233,242,256,301,319,334],"xmlns:xlink=\"http://www.w3.org/1999/xlink\">":[15,27,55,63,74,82,90,98,114,122,210,218,226,234,243,257,302,320,335],"<tex-math":[16,28,56,64,75,83,91,99,115,123,211,219,227,235,244,258,303,321,336],"notation=\"LaTeX\">${f}(":[17],"{X})$":[18],"</tex-math></inline-formula>":[19,31,59,70,78,86,94,102,118,126,214,222,230,238,250,262,309,327,342],"over":[20,108],"the":[21,41,48,109,130,135,142,147,155,185,195,266,274,278,289,296,314,329],"set":[22],"of":[23,134,170,198,277,351],"rectangular":[24],"notation=\"LaTeX\">${n}\\times":[29,92,212],"{m}$":[30],"matrices":[32,111,223],"have":[34],"rank":[35,42],"at":[36],"most":[37],"r.":[38],"To":[39],"tackle":[40],"constraint":[43],"and":[44,79,95,105,119,139,164,215,231,287,328],"also":[45,189,312],"to":[46,206,360],"reduce":[47],"computational":[49],"burden,":[50],"we":[51,204,271],"factorize":[52],"notation=\"LaTeX\">$":[57,65,76,84,116,124,228,236,245,259],"{X}$":[58],"into":[60],"{U}":[66,246],"{V}":[67,247],"^{\\mathrm":[68,248],"{T}}$":[69,249],"where":[71,203,299,317,332],"{U}$":[77,117,229],"{V}$":[85,125,237],"are":[87],"{r}$":[93,101,213,221,308,326,341],"notation=\"LaTeX\">${m}\\times":[100,220],"matrices,":[103],"respectively,":[104],"then":[106],"optimize":[107],"small":[110],".":[127,263,343],"We":[128,188],"characterize":[129],"global":[131,168,362],"geometry":[133,197],"nonconvex":[136],"factored":[137,186],"problem":[138,202,281],"show":[140,272],"corresponding":[143],"satisfies":[146,160],"robust":[148,267],"strict":[149,268,290],"saddle":[150,269,291],"property":[151,292],"as":[152,154,176,356],"long":[153],"original":[156],"f":[159],"restricted":[161],"strong":[162],"convexity":[163],"smoothness":[165],"properties,":[166],"ensuring":[167],"convergence":[169],"many":[171],"local":[172,285],"search":[173],"algorithms":[174,354],"(such":[175,355],"noisy":[177],"gradient":[178,357],"descent)":[179,358],"in":[180],"polynomial":[181],"time":[182],"for":[183,194,295,313],"solving":[184],"problem.":[187],"provide":[190],"comprehensive":[192],"analysis":[193],"matrix":[200,254,279],"factorization":[201,280],"aim":[205],"find":[207],"such":[239],"approximates":[251],"given":[253],"{X}^\\star":[260],"$":[261],"Aside":[264],"from":[265],"property,":[270],"has":[282],"no":[283],"spurious":[284],"minima":[286],"obeys":[288],"not":[293],"only":[294],"exact-parameterization":[297],"case":[298,316,331],"notation=\"LaTeX\">$\\mathrm":[304,322,337],"{rank}(":[305,323,338],"{X}^\\star)":[306,324,339],"=":[307],",":[310],"but":[311],"over-parameterization":[315],"&lt;":[325],"under-parameterization":[330],"&gt;":[340],"These":[344],"geometric":[345],"properties":[346],"imply":[347],"number":[350],"iterative":[352],"converge":[359],"solution":[363],"with":[364],"random":[365],"initialization.":[366]},"counts_by_year":[{"year":2026,"cited_by_count":3},{"year":2025,"cited_by_count":8},{"year":2024,"cited_by_count":7},{"year":2023,"cited_by_count":8},{"year":2022,"cited_by_count":9},{"year":2021,"cited_by_count":4}],"updated_date":"2026-07-17T09:13:05.818461","created_date":"2025-10-10T00:00:00"}
