{"id":"https://openalex.org/W3082179244","doi":"https://doi.org/10.1080/10618600.2020.1814788","title":"Shrinking the Covariance Matrix Using Convex Penalties on the Matrix-Log Transformation","display_name":"Shrinking the Covariance Matrix Using Convex Penalties on the Matrix-Log Transformation","publication_year":2020,"publication_date":"2020-09-01","ids":{"openalex":"https://openalex.org/W3082179244","doi":"https://doi.org/10.1080/10618600.2020.1814788","mag":"3082179244"},"language":"en","primary_location":{"id":"doi:10.1080/10618600.2020.1814788","is_oa":false,"landing_page_url":"https://doi.org/10.1080/10618600.2020.1814788","pdf_url":null,"source":{"id":"https://openalex.org/S76159266","display_name":"Journal of Computational and Graphical Statistics","issn_l":"1061-8600","issn":["1061-8600","1537-2715"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320547","host_organization_name":"Taylor & Francis","host_organization_lineage":["https://openalex.org/P4310320547"],"host_organization_lineage_names":["Taylor & Francis"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Journal of Computational and Graphical Statistics","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/A5030654505","display_name":"Mengxi Yi","orcid":"https://orcid.org/0000-0001-9592-959X"},"institutions":[{"id":"https://openalex.org/I146563203","display_name":"University of International Business and Economics","ror":"https://ror.org/05khqpb71","country_code":"CN","type":"education","lineage":["https://openalex.org/I146563203"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Mengxi Yi","raw_affiliation_strings":["School of Statistics, University of International Business and Economics, Beijing, China;;"],"affiliations":[{"raw_affiliation_string":"School of Statistics, University of International Business and Economics, Beijing, China;;","institution_ids":["https://openalex.org/I146563203"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5057170468","display_name":"David E. Tyler","orcid":"https://orcid.org/0000-0003-2040-6318"},"institutions":[{"id":"https://openalex.org/I102322142","display_name":"Rutgers, The State University of New Jersey","ror":"https://ror.org/05vt9qd57","country_code":"US","type":"education","lineage":["https://openalex.org/I102322142"]}],"countries":["US"],"is_corresponding":true,"raw_author_name":"David E. Tyler","raw_affiliation_strings":["Department of Statistics, Rutgers, The State University of New Jersey, Piscataway, NJ"],"affiliations":[{"raw_affiliation_string":"Department of Statistics, Rutgers, The State University of New Jersey, Piscataway, NJ","institution_ids":["https://openalex.org/I102322142"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":2,"corresponding_author_ids":["https://openalex.org/A5057170468"],"corresponding_institution_ids":["https://openalex.org/I102322142"],"apc_list":null,"apc_paid":null,"fwci":1.0798,"has_fulltext":false,"cited_by_count":9,"citation_normalized_percentile":{"value":0.79295225,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":90,"max":97},"biblio":{"volume":"30","issue":"2","first_page":"442","last_page":"451"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11871","display_name":"Advanced Statistical Methods and Models","score":0.9969000220298767,"subfield":{"id":"https://openalex.org/subfields/2613","display_name":"Statistics and Probability"},"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/T11871","display_name":"Advanced Statistical Methods and Models","score":0.9969000220298767,"subfield":{"id":"https://openalex.org/subfields/2613","display_name":"Statistics and Probability"},"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/T10500","display_name":"Sparse and Compressive Sensing Techniques","score":0.9951000213623047,"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/T10057","display_name":"Face and Expression Recognition","score":0.9882000088691711,"subfield":{"id":"https://openalex.org/subfields/1707","display_name":"Computer Vision and Pattern Recognition"},"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/mathematics","display_name":"Mathematics","score":0.7710350155830383},{"id":"https://openalex.org/keywords/covariance-matrix","display_name":"Covariance matrix","score":0.6208739280700684},{"id":"https://openalex.org/keywords/convexity","display_name":"Convexity","score":0.6198668479919434},{"id":"https://openalex.org/keywords/covariance","display_name":"Covariance","score":0.5906611680984497},{"id":"https://openalex.org/keywords/convex-optimization","display_name":"Convex optimization","score":0.5273880958557129},{"id":"https://openalex.org/keywords/matrix","display_name":"Matrix (chemical analysis)","score":0.5014829635620117},{"id":"https://openalex.org/keywords/regular-polygon","display_name":"Regular polygon","score":0.4574006199836731},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.43786200881004333},{"id":"https://openalex.org/keywords/estimation-of-covariance-matrices","display_name":"Estimation of covariance matrices","score":0.43143048882484436},{"id":"https://openalex.org/keywords/mathematical-optimization","display_name":"Mathematical optimization","score":0.41553381085395813},{"id":"https://openalex.org/keywords/convex-function","display_name":"Convex function","score":0.41429081559181213},{"id":"https://openalex.org/keywords/convex-analysis","display_name":"Convex analysis","score":0.41429057717323303},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.3910813629627228},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.2665623426437378},{"id":"https://openalex.org/keywords/statistics","display_name":"Statistics","score":0.16995921730995178}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7710350155830383},{"id":"https://openalex.org/C185142706","wikidata":"https://www.wikidata.org/wiki/Q1134404","display_name":"Covariance matrix","level":2,"score":0.6208739280700684},{"id":"https://openalex.org/C72134830","wikidata":"https://www.wikidata.org/wiki/Q5166524","display_name":"Convexity","level":2,"score":0.6198668479919434},{"id":"https://openalex.org/C178650346","wikidata":"https://www.wikidata.org/wiki/Q201984","display_name":"Covariance","level":2,"score":0.5906611680984497},{"id":"https://openalex.org/C157972887","wikidata":"https://www.wikidata.org/wiki/Q463359","display_name":"Convex optimization","level":3,"score":0.5273880958557129},{"id":"https://openalex.org/C106487976","wikidata":"https://www.wikidata.org/wiki/Q685816","display_name":"Matrix (chemical analysis)","level":2,"score":0.5014829635620117},{"id":"https://openalex.org/C112680207","wikidata":"https://www.wikidata.org/wiki/Q714886","display_name":"Regular polygon","level":2,"score":0.4574006199836731},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.43786200881004333},{"id":"https://openalex.org/C180877172","wikidata":"https://www.wikidata.org/wiki/Q5401390","display_name":"Estimation of covariance matrices","level":3,"score":0.43143048882484436},{"id":"https://openalex.org/C126255220","wikidata":"https://www.wikidata.org/wiki/Q141495","display_name":"Mathematical optimization","level":1,"score":0.41553381085395813},{"id":"https://openalex.org/C145446738","wikidata":"https://www.wikidata.org/wiki/Q319913","display_name":"Convex function","level":3,"score":0.41429081559181213},{"id":"https://openalex.org/C12108790","wikidata":"https://www.wikidata.org/wiki/Q2234833","display_name":"Convex analysis","level":4,"score":0.41429057717323303},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.3910813629627228},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.2665623426437378},{"id":"https://openalex.org/C105795698","wikidata":"https://www.wikidata.org/wiki/Q12483","display_name":"Statistics","level":1,"score":0.16995921730995178},{"id":"https://openalex.org/C162324750","wikidata":"https://www.wikidata.org/wiki/Q8134","display_name":"Economics","level":0,"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/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.0},{"id":"https://openalex.org/C106159729","wikidata":"https://www.wikidata.org/wiki/Q2294553","display_name":"Financial economics","level":1,"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":1,"locations":[{"id":"doi:10.1080/10618600.2020.1814788","is_oa":false,"landing_page_url":"https://doi.org/10.1080/10618600.2020.1814788","pdf_url":null,"source":{"id":"https://openalex.org/S76159266","display_name":"Journal of Computational and Graphical Statistics","issn_l":"1061-8600","issn":["1061-8600","1537-2715"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320547","host_organization_name":"Taylor & Francis","host_organization_lineage":["https://openalex.org/P4310320547"],"host_organization_lineage_names":["Taylor & Francis"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Journal of Computational and Graphical Statistics","raw_type":"journal-article"}],"best_oa_location":null,"sustainable_development_goals":[{"id":"https://metadata.un.org/sdg/16","score":0.4300000071525574,"display_name":"Peace, Justice and strong institutions"}],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":25,"referenced_works":["https://openalex.org/W88077589","https://openalex.org/W585133165","https://openalex.org/W1968756693","https://openalex.org/W1972276842","https://openalex.org/W1976990135","https://openalex.org/W1997314000","https://openalex.org/W2008330147","https://openalex.org/W2031398712","https://openalex.org/W2032236594","https://openalex.org/W2045391460","https://openalex.org/W2072207996","https://openalex.org/W2074763854","https://openalex.org/W2081746825","https://openalex.org/W2090963365","https://openalex.org/W2093608496","https://openalex.org/W2130351130","https://openalex.org/W2132555912","https://openalex.org/W2487020265","https://openalex.org/W2610862920","https://openalex.org/W2804234031","https://openalex.org/W3098365576","https://openalex.org/W3098880893","https://openalex.org/W3101788651","https://openalex.org/W4256170606","https://openalex.org/W4297757075"],"related_works":["https://openalex.org/W2921280830","https://openalex.org/W2572601863","https://openalex.org/W2887132723","https://openalex.org/W1670628120","https://openalex.org/W1976318097","https://openalex.org/W2756533552","https://openalex.org/W1583261817","https://openalex.org/W4286579657","https://openalex.org/W1974588588","https://openalex.org/W2118568436"],"abstract_inverted_index":{"For":[0],"q-dimensional":[1,171],"data,":[2],"penalized":[3,87,107,128,165],"versions":[4],"of":[5,37,70,73,125,152,155,178,181],"the":[6,13,24,35,38,71,121,126,153,163,176,179],"sample":[7,14],"covariance":[8,39,129],"matrix":[9],"are":[10,49,67,76,91,150,197],"important":[11],"when":[12,159],"size":[15],"is":[16,31,42,140],"small":[17],"or":[18],"modest":[19],"relative":[20],"to":[21,44,110,119,142,169],"q.":[22],"Since":[23],"negative":[25],"log-likelihood":[26],"under":[27],"multivariate":[28],"normal":[29],"sampling":[30],"convex":[32,51,77,93,113,172],"in":[33,52,78,83,94,98,114,138,144],"\u03a3\u22121,":[34],"inverse":[36],"matrix,":[40],"it":[41],"common":[43],"consider":[45],"additive":[46],"penalties":[47,65,148],"which":[48,66,149,183],"also":[50],"\u03a3\u22121.":[53,84,99],"More":[54,131],"recently,":[55],"Deng":[56],"and":[57,59,75,123],"Tsui":[58],"Yu":[60],"et":[61],"al.":[62],"have":[63],"proposed":[64],"strictly":[68],"functions":[69,151],"roots":[72,154,180],"\u03a3":[74,96,139,146],"log":[79,95,145],"\u03a3,":[80,182],"but":[81],"not":[82],"The":[85],"resulting":[86,164],"optimization":[88,108,166,173],"problems,":[89],"though,":[90],"neither":[92],"nor":[97],"In":[100,157],"this":[101,195],"article,":[102],"however,":[103],"we":[104,133],"show":[105,134],"these":[106],"problems":[109],"be":[111,186],"geodesically":[112],"\u03a3.":[115,156],"This":[116],"allows":[117],"us":[118],"establish":[120],"existence":[122],"uniqueness":[124],"corresponding":[127],"matrices.":[130],"generally,":[132],"that":[135],"geodesic":[136],"convexity":[137,143],"equivalent":[141],"for":[147,194],"addition,":[158],"using":[160],"such":[161],"penalties,":[162],"problem":[167,174],"reduces":[168],"a":[170],"on":[175],"logs":[177],"can":[184],"then":[185],"readily":[187],"solved":[188],"via":[189],"Newton\u2019s":[190],"algorithm.":[191],"Supplementary":[192],"materials":[193],"article":[196],"available":[198],"online.":[199]},"counts_by_year":[{"year":2025,"cited_by_count":2},{"year":2024,"cited_by_count":1},{"year":2023,"cited_by_count":3},{"year":2020,"cited_by_count":2},{"year":2019,"cited_by_count":1}],"updated_date":"2025-11-06T03:46:38.306776","created_date":"2025-10-10T00:00:00"}
