{"id":"https://openalex.org/W2119582979","doi":"https://doi.org/10.1137/040609902","title":"Least-Squares Covariance Matrix Adjustment","display_name":"Least-Squares Covariance Matrix Adjustment","publication_year":2005,"publication_date":"2005-01-01","ids":{"openalex":"https://openalex.org/W2119582979","doi":"https://doi.org/10.1137/040609902","mag":"2119582979"},"language":"en","primary_location":{"id":"doi:10.1137/040609902","is_oa":false,"landing_page_url":"https://doi.org/10.1137/040609902","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":["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/A5011176205","display_name":"Stephen Boyd","orcid":"https://orcid.org/0000-0001-8353-6000"},"institutions":[{"id":"https://openalex.org/I97018004","display_name":"Stanford University","ror":"https://ror.org/00f54p054","country_code":"US","type":"education","lineage":["https://openalex.org/I97018004"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Stephen Boyd","raw_affiliation_strings":["Department of Electrical Engineering, Stanford University, Stanford, CA 94305-9510"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Electrical Engineering, Stanford University, Stanford, CA 94305-9510","institution_ids":["https://openalex.org/I97018004"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5005130812","display_name":"Lin Xiao","orcid":"https://orcid.org/0000-0003-3172-3490"},"institutions":[{"id":"https://openalex.org/I122411786","display_name":"California Institute of Technology","ror":"https://ror.org/05dxps055","country_code":"US","type":"education","lineage":["https://openalex.org/I122411786"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Lin Xiao","raw_affiliation_strings":["Center for the Mathematics of Information, California Institute of Technology, Pasadena, CA 91125-9300"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Center for the Mathematics of Information, California Institute of Technology, Pasadena, CA 91125-9300","institution_ids":["https://openalex.org/I122411786"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":2,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":8.9231,"has_fulltext":false,"cited_by_count":128,"citation_normalized_percentile":{"value":0.98293781,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":94,"max":100},"biblio":{"volume":"27","issue":"2","first_page":"532","last_page":"546"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10500","display_name":"Sparse and Compressive Sensing Techniques","score":0.9997000098228455,"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":0.9997000098228455,"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":0.9983999729156494,"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/T11236","display_name":"Control Systems and Identification","score":0.994700014591217,"subfield":{"id":"https://openalex.org/subfields/2207","display_name":"Control and Systems Engineering"},"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/mathematics","display_name":"Mathematics","score":0.8599034547805786},{"id":"https://openalex.org/keywords/matrix","display_name":"Matrix (chemical analysis)","score":0.519207775592804},{"id":"https://openalex.org/keywords/covariance-matrix","display_name":"Covariance matrix","score":0.48314422369003296},{"id":"https://openalex.org/keywords/linear-least-squares","display_name":"Linear least squares","score":0.44796258211135864},{"id":"https://openalex.org/keywords/least-squares-function-approximation","display_name":"Least-squares function approximation","score":0.4408211410045624},{"id":"https://openalex.org/keywords/positive-definite-matrix","display_name":"Positive-definite matrix","score":0.4404899477958679},{"id":"https://openalex.org/keywords/estimation-of-covariance-matrices","display_name":"Estimation of covariance matrices","score":0.4259454309940338},{"id":"https://openalex.org/keywords/covariance","display_name":"Covariance","score":0.42289724946022034},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.4155641496181488},{"id":"https://openalex.org/keywords/matrix-norm","display_name":"Matrix norm","score":0.4154326021671295},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.37065356969833374},{"id":"https://openalex.org/keywords/mathematical-optimization","display_name":"Mathematical optimization","score":0.3459683954715729},{"id":"https://openalex.org/keywords/eigenvalues-and-eigenvectors","display_name":"Eigenvalues and eigenvectors","score":0.2686823606491089},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.15370580554008484},{"id":"https://openalex.org/keywords/statistics","display_name":"Statistics","score":0.14347457885742188},{"id":"https://openalex.org/keywords/linear-model","display_name":"Linear model","score":0.08311375975608826}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8599034547805786},{"id":"https://openalex.org/C106487976","wikidata":"https://www.wikidata.org/wiki/Q685816","display_name":"Matrix (chemical analysis)","level":2,"score":0.519207775592804},{"id":"https://openalex.org/C185142706","wikidata":"https://www.wikidata.org/wiki/Q1134404","display_name":"Covariance matrix","level":2,"score":0.48314422369003296},{"id":"https://openalex.org/C58663186","wikidata":"https://www.wikidata.org/wiki/Q6553462","display_name":"Linear least squares","level":3,"score":0.44796258211135864},{"id":"https://openalex.org/C9936470","wikidata":"https://www.wikidata.org/wiki/Q6510405","display_name":"Least-squares function approximation","level":3,"score":0.4408211410045624},{"id":"https://openalex.org/C49712288","wikidata":"https://www.wikidata.org/wiki/Q77601250","display_name":"Positive-definite matrix","level":3,"score":0.4404899477958679},{"id":"https://openalex.org/C180877172","wikidata":"https://www.wikidata.org/wiki/Q5401390","display_name":"Estimation of covariance matrices","level":3,"score":0.4259454309940338},{"id":"https://openalex.org/C178650346","wikidata":"https://www.wikidata.org/wiki/Q201984","display_name":"Covariance","level":2,"score":0.42289724946022034},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.4155641496181488},{"id":"https://openalex.org/C92207270","wikidata":"https://www.wikidata.org/wiki/Q939253","display_name":"Matrix norm","level":3,"score":0.4154326021671295},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.37065356969833374},{"id":"https://openalex.org/C126255220","wikidata":"https://www.wikidata.org/wiki/Q141495","display_name":"Mathematical optimization","level":1,"score":0.3459683954715729},{"id":"https://openalex.org/C158693339","wikidata":"https://www.wikidata.org/wiki/Q190524","display_name":"Eigenvalues and eigenvectors","level":2,"score":0.2686823606491089},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.15370580554008484},{"id":"https://openalex.org/C105795698","wikidata":"https://www.wikidata.org/wiki/Q12483","display_name":"Statistics","level":1,"score":0.14347457885742188},{"id":"https://openalex.org/C163175372","wikidata":"https://www.wikidata.org/wiki/Q3339222","display_name":"Linear model","level":2,"score":0.08311375975608826},{"id":"https://openalex.org/C185429906","wikidata":"https://www.wikidata.org/wiki/Q1130160","display_name":"Estimator","level":2,"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/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","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/C192562407","wikidata":"https://www.wikidata.org/wiki/Q228736","display_name":"Materials science","level":0,"score":0.0}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.1137/040609902","is_oa":false,"landing_page_url":"https://doi.org/10.1137/040609902","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:authors.library.caltech.edu:1382","is_oa":false,"landing_page_url":"https://authors.library.caltech.edu/1382/","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":null,"sustainable_development_goals":[{"display_name":"Reduced inequalities","id":"https://metadata.un.org/sdg/10","score":0.46000000834465027}],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":22,"referenced_works":["https://openalex.org/W66060913","https://openalex.org/W182881619","https://openalex.org/W1503425191","https://openalex.org/W1563653928","https://openalex.org/W1568288633","https://openalex.org/W1606198627","https://openalex.org/W1982165727","https://openalex.org/W2013243898","https://openalex.org/W2015263936","https://openalex.org/W2047917018","https://openalex.org/W2050875740","https://openalex.org/W2059586807","https://openalex.org/W2072863821","https://openalex.org/W2091560152","https://openalex.org/W2296319761","https://openalex.org/W2748138089","https://openalex.org/W2798766386","https://openalex.org/W2912522929","https://openalex.org/W4211092921","https://openalex.org/W4243539849","https://openalex.org/W4251656402","https://openalex.org/W4302155765"],"related_works":["https://openalex.org/W2168915192","https://openalex.org/W2921280830","https://openalex.org/W2887132723","https://openalex.org/W2572601863","https://openalex.org/W1670628120","https://openalex.org/W1976318097","https://openalex.org/W2756533552","https://openalex.org/W4286579657","https://openalex.org/W1974588588","https://openalex.org/W2028568336"],"abstract_inverted_index":{"We":[0],"consider":[1],"the":[2,6,20,60,67,75,107,135,143,182,190,219,228,237,241],"problem":[3,45,103,118,151],"of":[4,62,77,130,137,173,209,236],"finding":[5],"smallest":[7],"adjustment":[8,44,147,160,259],"to":[9,83,134,154,253],"a":[10,47,84,100,116,128,170,232],"given":[11,30],"symmetric":[12],"$n":[13],"\\times":[14],"n$":[15],"matrix,":[16],"as":[17,206,247],"measured":[18],"by":[19,181,194],"Euclidean":[21],"or":[22,124],"Frobenius":[23],"norm,":[24],"so":[25],"that":[26,119,224],"it":[27,106],"satisfies":[28],"some":[29],"linear":[31],"equalities":[32],"and":[33,35,51,104,127,139,202],"inequalities,":[34],"in":[36,66,142,225],"addition":[37],"is":[38,46,69,79,231],"positive":[39],"semidefinite.":[40],"This":[41,149],"least-squares":[42,109,145,158,257],"covariance":[43,146,159,258],"convex":[48],"optimization":[49],"problem,":[50,221],"can":[52,203],"be":[53,164,204],"efficiently":[54],"solved":[55,180],"using":[56,166],"standard":[57,167],"methods":[58],"when":[59,196,213],"number":[61,76,129,136,172],"variables":[63,78,132],"(i.e.,":[64],"entries":[65],"matrix)":[68],"modest,":[70],"say,":[71],"under":[72],"$1000$.":[73],"Since":[74],"$n(n+1)/2$,":[80],"this":[81,112,249],"corresponds":[82],"limit":[85],"around":[86],"$n=45$.":[87],"Malick":[88,195],"[{\\it":[89],"SIAM":[90],"J.":[91],"Matrix":[92],"Anal.\\":[93],"Appl.,}":[94],"26":[95],"(2005),":[96],"pp.":[97],"272--284]":[98],"studies":[99],"closely":[101],"related":[102],"calls":[105],"semidefinite":[108],"problem.":[110,148],"In":[111],"paper":[113],"we":[114,222],"formulate":[115],"dual":[117,150,183,186,191,211,220],"has":[120,244],"no":[121,199],"matrix":[122,125,243],"inequality":[123,140,200,216],"variables,":[126],"(scalar)":[131],"equal":[133],"equality":[138],"constraints":[141,201],"original":[144,238,242],"allows":[152,251],"us":[153,252],"solve":[155,254],"far":[156],"larger":[157],"problems":[161,175],"than":[162],"would":[163],"possible":[165],"methods.":[168],"Assuming":[169],"modest":[171],"constraints,":[174],"with":[176,189],"$n=1000$":[177],"are":[178,198,215],"readily":[179],"method.":[184],"The":[185],"method":[187,192,212],"coincides":[188],"proposed":[193],"there":[197,214],"obtained":[205],"an":[207],"extension":[208],"his":[210],"constraints.":[217],"Using":[218],"show":[223],"many":[226],"cases":[227],"optimal":[229],"solution":[230],"low":[233],"rank":[234],"update":[235],"matrix.":[239],"When":[240],"structure,":[245],"such":[246],"sparsity,":[248],"observation":[250],"very":[255],"large":[256],"problems.":[260]},"counts_by_year":[{"year":2026,"cited_by_count":2},{"year":2025,"cited_by_count":4},{"year":2024,"cited_by_count":2},{"year":2023,"cited_by_count":2},{"year":2022,"cited_by_count":4},{"year":2021,"cited_by_count":5},{"year":2020,"cited_by_count":5},{"year":2019,"cited_by_count":5},{"year":2018,"cited_by_count":6},{"year":2017,"cited_by_count":2},{"year":2016,"cited_by_count":8},{"year":2015,"cited_by_count":8},{"year":2014,"cited_by_count":14},{"year":2013,"cited_by_count":4},{"year":2012,"cited_by_count":9}],"updated_date":"2026-06-11T09:08:48.828518","created_date":"2025-10-10T00:00:00"}