{"id":"https://openalex.org/W1481507124","doi":"https://doi.org/10.1137/120872516","title":"Diagonal and Low-Rank Matrix Decompositions, Correlation Matrices, and Ellipsoid Fitting","display_name":"Diagonal and Low-Rank Matrix Decompositions, Correlation Matrices, and Ellipsoid Fitting","publication_year":2012,"publication_date":"2012-01-01","ids":{"openalex":"https://openalex.org/W1481507124","doi":"https://doi.org/10.1137/120872516","mag":"1481507124"},"language":"en","primary_location":{"id":"doi:10.1137/120872516","is_oa":false,"landing_page_url":"https://doi.org/10.1137/120872516","pdf_url":null,"source":{"id":"https://openalex.org/S16958353","display_name":"SIAM Journal on Matrix Analysis and Applications","issn_l":"0895-4798","issn":["0895-4798","1095-7162"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320508","host_organization_name":"Society for Industrial and Applied Mathematics","host_organization_lineage":["https://openalex.org/P4310320508"],"host_organization_lineage_names":["Society for Industrial and Applied Mathematics"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"SIAM Journal on Matrix Analysis and Applications","raw_type":"journal-article"},"type":"article","indexed_in":["arxiv","crossref"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://arxiv.org/pdf/1204.1220","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":null,"display_name":"J. Saunderson","orcid":null},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"J. Saunderson","raw_affiliation_strings":[],"affiliations":[]},{"author_position":"middle","author":{"id":null,"display_name":"V. Chandrasekaran","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"V. Chandrasekaran","raw_affiliation_strings":[],"affiliations":[]},{"author_position":"middle","author":{"id":null,"display_name":"P. A. Parrilo","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"P. A. Parrilo","raw_affiliation_strings":[],"affiliations":[]},{"author_position":"last","author":{"id":null,"display_name":"A. S. Willsky","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"A. S. Willsky","raw_affiliation_strings":[],"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":4,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":3.6345,"has_fulltext":false,"cited_by_count":32,"citation_normalized_percentile":{"value":0.91974908,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":89,"max":99},"biblio":{"volume":"33","issue":"4","first_page":"1395","last_page":"1416"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10963","display_name":"Advanced Optimization Algorithms Research","score":0.375,"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"}},"topics":[{"id":"https://openalex.org/T10963","display_name":"Advanced Optimization Algorithms Research","score":0.375,"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":0.1062999963760376,"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/T10500","display_name":"Sparse and Compressive Sensing Techniques","score":0.10499999672174454,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/ellipsoid","display_name":"Ellipsoid","score":0.5983999967575073},{"id":"https://openalex.org/keywords/matrix","display_name":"Matrix (chemical analysis)","score":0.5785999894142151},{"id":"https://openalex.org/keywords/diagonal","display_name":"Diagonal","score":0.5637000203132629},{"id":"https://openalex.org/keywords/positive-definite-matrix","display_name":"Positive-definite matrix","score":0.48510000109672546},{"id":"https://openalex.org/keywords/convex-set","display_name":"Convex set","score":0.46869999170303345},{"id":"https://openalex.org/keywords/trace","display_name":"TRACE (psycholinguistics)","score":0.4417000114917755},{"id":"https://openalex.org/keywords/simple","display_name":"Simple (philosophy)","score":0.435699999332428},{"id":"https://openalex.org/keywords/regular-polygon","display_name":"Regular polygon","score":0.43389999866485596},{"id":"https://openalex.org/keywords/convex-optimization","display_name":"Convex optimization","score":0.4133000075817108},{"id":"https://openalex.org/keywords/diagonal-matrix","display_name":"Diagonal matrix","score":0.4099000096321106}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8223999738693237},{"id":"https://openalex.org/C57489055","wikidata":"https://www.wikidata.org/wiki/Q190046","display_name":"Ellipsoid","level":2,"score":0.5983999967575073},{"id":"https://openalex.org/C106487976","wikidata":"https://www.wikidata.org/wiki/Q685816","display_name":"Matrix (chemical analysis)","level":2,"score":0.5785999894142151},{"id":"https://openalex.org/C130367717","wikidata":"https://www.wikidata.org/wiki/Q189791","display_name":"Diagonal","level":2,"score":0.5637000203132629},{"id":"https://openalex.org/C49712288","wikidata":"https://www.wikidata.org/wiki/Q77601250","display_name":"Positive-definite matrix","level":3,"score":0.48510000109672546},{"id":"https://openalex.org/C49870271","wikidata":"https://www.wikidata.org/wiki/Q193657","display_name":"Convex set","level":4,"score":0.46869999170303345},{"id":"https://openalex.org/C75291252","wikidata":"https://www.wikidata.org/wiki/Q1315756","display_name":"TRACE (psycholinguistics)","level":2,"score":0.4417000114917755},{"id":"https://openalex.org/C2780586882","wikidata":"https://www.wikidata.org/wiki/Q7520643","display_name":"Simple (philosophy)","level":2,"score":0.435699999332428},{"id":"https://openalex.org/C112680207","wikidata":"https://www.wikidata.org/wiki/Q714886","display_name":"Regular polygon","level":2,"score":0.43389999866485596},{"id":"https://openalex.org/C157972887","wikidata":"https://www.wikidata.org/wiki/Q463359","display_name":"Convex optimization","level":3,"score":0.4133000075817108},{"id":"https://openalex.org/C113313756","wikidata":"https://www.wikidata.org/wiki/Q332791","display_name":"Diagonal matrix","level":3,"score":0.4099000096321106},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.4090000092983246},{"id":"https://openalex.org/C185142706","wikidata":"https://www.wikidata.org/wiki/Q1134404","display_name":"Covariance matrix","level":2,"score":0.4009000062942505},{"id":"https://openalex.org/C173801870","wikidata":"https://www.wikidata.org/wiki/Q201413","display_name":"Heuristic","level":2,"score":0.39590001106262207},{"id":"https://openalex.org/C32834561","wikidata":"https://www.wikidata.org/wiki/Q660730","display_name":"Subspace topology","level":2,"score":0.3950999975204468},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.39160001277923584},{"id":"https://openalex.org/C42355184","wikidata":"https://www.wikidata.org/wiki/Q1361088","display_name":"Matrix decomposition","level":3,"score":0.3833000063896179},{"id":"https://openalex.org/C54848796","wikidata":"https://www.wikidata.org/wiki/Q339011","display_name":"Symmetric matrix","level":3,"score":0.3610999882221222},{"id":"https://openalex.org/C12362212","wikidata":"https://www.wikidata.org/wiki/Q728435","display_name":"Linear subspace","level":2,"score":0.3452000021934509},{"id":"https://openalex.org/C85817219","wikidata":"https://www.wikidata.org/wiki/Q884772","display_name":"Block matrix","level":3,"score":0.3450999855995178},{"id":"https://openalex.org/C177264268","wikidata":"https://www.wikidata.org/wiki/Q1514741","display_name":"Set (abstract data type)","level":2,"score":0.3418999910354614},{"id":"https://openalex.org/C126255220","wikidata":"https://www.wikidata.org/wiki/Q141495","display_name":"Mathematical optimization","level":1,"score":0.33649998903274536},{"id":"https://openalex.org/C79187972","wikidata":"https://www.wikidata.org/wiki/Q2654791","display_name":"Conic optimization","level":5,"score":0.32850000262260437},{"id":"https://openalex.org/C137836250","wikidata":"https://www.wikidata.org/wiki/Q984063","display_name":"Optimization problem","level":2,"score":0.31029999256134033},{"id":"https://openalex.org/C84545080","wikidata":"https://www.wikidata.org/wiki/Q1147936","display_name":"Condition number","level":3,"score":0.30399999022483826},{"id":"https://openalex.org/C60761296","wikidata":"https://www.wikidata.org/wiki/Q651480","display_name":"Main diagonal","level":3,"score":0.3005000054836273},{"id":"https://openalex.org/C90199385","wikidata":"https://www.wikidata.org/wiki/Q6692777","display_name":"Low-rank approximation","level":3,"score":0.28029999136924744},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.2799000144004822},{"id":"https://openalex.org/C69044650","wikidata":"https://www.wikidata.org/wiki/Q2739329","display_name":"Square matrix","level":4,"score":0.2770000100135803},{"id":"https://openalex.org/C101901036","wikidata":"https://www.wikidata.org/wiki/Q2269096","display_name":"Semidefinite programming","level":2,"score":0.27630001306533813},{"id":"https://openalex.org/C60033838","wikidata":"https://www.wikidata.org/wiki/Q1751685","display_name":"Solution set","level":3,"score":0.2750999927520752},{"id":"https://openalex.org/C206194317","wikidata":"https://www.wikidata.org/wiki/Q1138624","display_name":"Convex hull","level":3,"score":0.2655999958515167},{"id":"https://openalex.org/C133589720","wikidata":"https://www.wikidata.org/wiki/Q6787891","display_name":"Matrix splitting","level":5,"score":0.25949999690055847},{"id":"https://openalex.org/C56372850","wikidata":"https://www.wikidata.org/wiki/Q1050404","display_name":"Sparse matrix","level":3,"score":0.25529998540878296},{"id":"https://openalex.org/C33343441","wikidata":"https://www.wikidata.org/wiki/Q806350","display_name":"Band matrix","level":5,"score":0.2524000108242035}],"mesh":[],"locations_count":3,"locations":[{"id":"doi:10.1137/120872516","is_oa":false,"landing_page_url":"https://doi.org/10.1137/120872516","pdf_url":null,"source":{"id":"https://openalex.org/S16958353","display_name":"SIAM Journal on Matrix Analysis and Applications","issn_l":"0895-4798","issn":["0895-4798","1095-7162"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320508","host_organization_name":"Society for Industrial and Applied Mathematics","host_organization_lineage":["https://openalex.org/P4310320508"],"host_organization_lineage_names":["Society for Industrial and Applied Mathematics"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"SIAM Journal on Matrix Analysis and Applications","raw_type":"journal-article"},{"id":"pmh:oai:arXiv.org:1204.1220","is_oa":true,"landing_page_url":"http://arxiv.org/abs/1204.1220","pdf_url":"https://arxiv.org/pdf/1204.1220","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":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},{"id":"pmh:oai:authors.library.caltech.edu:34685","is_oa":false,"landing_page_url":"https://authors.library.caltech.edu/34685/","pdf_url":null,"source":{"id":"https://openalex.org/S4306402161","display_name":"CaltechAUTHORS (California Institute of Technology)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I122411786","host_organization_name":"California Institute of Technology","host_organization_lineage":["https://openalex.org/I122411786"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"","raw_type":"Article"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:1204.1220","is_oa":true,"landing_page_url":"http://arxiv.org/abs/1204.1220","pdf_url":"https://arxiv.org/pdf/1204.1220","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":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":26,"referenced_works":["https://openalex.org/W1968135884","https://openalex.org/W1968398095","https://openalex.org/W1968890997","https://openalex.org/W1973488313","https://openalex.org/W1974145659","https://openalex.org/W1976962893","https://openalex.org/W1985123706","https://openalex.org/W2003753589","https://openalex.org/W2014079391","https://openalex.org/W2041626758","https://openalex.org/W2044502250","https://openalex.org/W2049633694","https://openalex.org/W2058497129","https://openalex.org/W2071921508","https://openalex.org/W2077661892","https://openalex.org/W2091560152","https://openalex.org/W2099641086","https://openalex.org/W2118408802","https://openalex.org/W2118550318","https://openalex.org/W2133487567","https://openalex.org/W2145962650","https://openalex.org/W2162654459","https://openalex.org/W2164693112","https://openalex.org/W2208136017","https://openalex.org/W2611328865","https://openalex.org/W2912522929"],"related_works":[],"abstract_inverted_index":{"In":[0],"this":[1],"paper":[2],"we":[3,65,148],"establish":[4,149],"links":[5],"between,":[6],"and":[7,34,50,90,204],"new":[8,195],"results":[9],"for,":[10],"three":[11,143],"problems":[12,144],"that":[13,26,137,158,213],"are":[14,145],"not":[15],"usually":[16],"considered":[17],"together.":[18],"The":[19,62,106],"first":[20],"is":[21,67,109,125,216],"a":[22,38,79,96,110,126,139,150,155,180,194,205,209,217],"matrix":[23,39,49,163,177],"decomposition":[24],"problem":[25,64,108],"arises":[27],"in":[28,98,138],"areas":[29],"such":[30],"as":[31,42,84,185],"statistical":[32],"modeling":[33],"signal":[35],"processing:":[36],"given":[37,114],"$X$":[40,58],"formed":[41],"the":[43,70,74,85,133,198],"sum":[44],"of":[45,73,76,197,200,211],"an":[46,51],"unknown":[47,52],"diagonal":[48,176],"low-rank":[53],"positive":[54,161],"semidefinite":[55,162],"matrix,":[56],"decompose":[57],"into":[59],"these":[60,142],"constituents.":[61],"second":[63],"consider":[66],"to":[68,103,193],"determine":[69,122],"facial":[71,93],"structure":[72,199],"set":[75,81,210],"correlation":[77,202],"matrices,":[78],"convex":[80,88,181],"also":[82],"known":[83,184],"elliptope.":[86],"This":[87,190],"body,":[89],"particularly":[91],"its":[92],"structure,":[94],"plays":[95],"role":[97],"applications":[99],"from":[100,172],"combinatorial":[101],"optimization":[102],"mathematical":[104],"finance.":[105],"third":[107],"basic":[111],"geometric":[112],"question:":[113],"points":[115,212],"$v_1,v_2,\\ldots,v_n\\in":[116],"\\mathbb{R}^k$":[117],"(where":[118],"$n":[119],">":[120],"k$)":[121],"whether":[123],"there":[124,215],"centered":[127,218],"ellipsoid":[128,219],"passing":[129,220],"exactly":[130],"through":[131,221],"all":[132],"points.":[134],"We":[135],"show":[136],"precise":[140],"sense":[141],"equivalent.":[146],"Furthermore":[147],"simple":[151,206],"sufficient":[152],"condition":[153,207],"on":[154,208],"subspace":[156],"$\\mathcal{U}$":[157,168],"ensures":[159,214],"any":[160,175],"$L$":[164],"with":[165],"column":[166],"space":[167],"can":[169],"be":[170],"recovered":[171],"$D+L$":[173],"for":[174],"$D$":[178],"using":[179],"optimization-based":[182],"heuristic":[183],"minimum":[186],"trace":[187],"factor":[188],"analysis.":[189],"result":[191],"leads":[192],"understanding":[196],"rank-deficient":[201],"matrices":[203],"them.":[222]},"counts_by_year":[{"year":2026,"cited_by_count":1},{"year":2025,"cited_by_count":2},{"year":2024,"cited_by_count":4},{"year":2023,"cited_by_count":2},{"year":2022,"cited_by_count":1},{"year":2020,"cited_by_count":1},{"year":2019,"cited_by_count":4},{"year":2018,"cited_by_count":1},{"year":2017,"cited_by_count":6},{"year":2016,"cited_by_count":2},{"year":2015,"cited_by_count":3},{"year":2014,"cited_by_count":4},{"year":2013,"cited_by_count":1}],"updated_date":"2026-03-20T23:20:44.827607","created_date":"2016-06-24T00:00:00"}