{"id":"https://openalex.org/W2005577322","doi":"https://doi.org/10.1137/080731347","title":"Riemannian Metric and Geometric Mean for Positive Semidefinite Matrices of Fixed Rank","display_name":"Riemannian Metric and Geometric Mean for Positive Semidefinite Matrices of Fixed Rank","publication_year":2009,"publication_date":"2009-08-28","ids":{"openalex":"https://openalex.org/W2005577322","doi":"https://doi.org/10.1137/080731347","mag":"2005577322"},"language":"en","primary_location":{"id":"doi:10.1137/080731347","is_oa":false,"landing_page_url":"https://doi.org/10.1137/080731347","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":true,"oa_status":"green","oa_url":"https://orbi.uliege.be/handle/2268/19669","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5037120106","display_name":"Silv\u00e8re Bonnabel","orcid":"https://orcid.org/0000-0002-6001-7766"},"institutions":[{"id":"https://openalex.org/I4210136403","display_name":"Montefiore Hospital","ror":"https://ror.org/033z99h42","country_code":"GB","type":"healthcare","lineage":["https://openalex.org/I4210136403"]}],"countries":["GB"],"is_corresponding":false,"raw_author_name":"Silv\u00e8re Bonnabel","raw_affiliation_strings":["bonnabel@montefiore.ulg.ac.be and r.sepulchre@ulg.ac.be#TAB#"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"bonnabel@montefiore.ulg.ac.be and r.sepulchre@ulg.ac.be#TAB#","institution_ids":["https://openalex.org/I4210136403"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5067642477","display_name":"Rodolphe Sepulchre","orcid":"https://orcid.org/0000-0002-7047-3124"},"institutions":[{"id":"https://openalex.org/I157674565","display_name":"University of Li\u00e8ge","ror":"https://ror.org/00afp2z80","country_code":"BE","type":"education","lineage":["https://openalex.org/I157674565"]}],"countries":["BE"],"is_corresponding":false,"raw_author_name":"Rodolphe Sepulchre","raw_affiliation_strings":["Universit\u00e9 de Li\u00e8ge - ULi\u00e8ge > D\u00e9p. d'\u00e9lectric., \u00e9lectron. et informat. (Inst.Montefiore) > Syst\u00e8mes et mod\u00e9lisation >  >"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Universit\u00e9 de Li\u00e8ge - ULi\u00e8ge > D\u00e9p. d'\u00e9lectric., \u00e9lectron. et informat. (Inst.Montefiore) > Syst\u00e8mes et mod\u00e9lisation >  >","institution_ids":["https://openalex.org/I157674565"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":2,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":7.9591,"has_fulltext":true,"cited_by_count":137,"citation_normalized_percentile":{"value":0.9822759,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":94,"max":100},"biblio":{"volume":"31","issue":"3","first_page":"1055","last_page":"1070"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11564","display_name":"Mathematical Inequalities and Applications","score":0.9915000200271606,"subfield":{"id":"https://openalex.org/subfields/2604","display_name":"Applied Mathematics"},"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/T11564","display_name":"Mathematical Inequalities and Applications","score":0.9915000200271606,"subfield":{"id":"https://openalex.org/subfields/2604","display_name":"Applied Mathematics"},"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":0.9907000064849854,"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":0.9889000058174133,"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/mathematics","display_name":"Mathematics","score":0.8865156173706055},{"id":"https://openalex.org/keywords/riemannian-geometry","display_name":"Riemannian geometry","score":0.6229222416877747},{"id":"https://openalex.org/keywords/positive-definite-matrix","display_name":"Positive-definite matrix","score":0.6096478700637817},{"id":"https://openalex.org/keywords/metric","display_name":"Metric (unit)","score":0.5726037621498108},{"id":"https://openalex.org/keywords/invariant","display_name":"Invariant (physics)","score":0.5456314086914062},{"id":"https://openalex.org/keywords/fundamental-theorem-of-riemannian-geometry","display_name":"Fundamental theorem of Riemannian geometry","score":0.5410879850387573},{"id":"https://openalex.org/keywords/rank","display_name":"Rank (graph theory)","score":0.5197846293449402},{"id":"https://openalex.org/keywords/weighted-geometric-mean","display_name":"Weighted geometric mean","score":0.5158047676086426},{"id":"https://openalex.org/keywords/quotient","display_name":"Quotient","score":0.5029529929161072},{"id":"https://openalex.org/keywords/metric-space","display_name":"Metric space","score":0.4573875963687897},{"id":"https://openalex.org/keywords/fisher-information-metric","display_name":"Fisher information metric","score":0.4272739589214325},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.4148983955383301},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.3836054503917694},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.32507848739624023},{"id":"https://openalex.org/keywords/injective-metric-space","display_name":"Injective metric space","score":0.2785346210002899},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.2685784101486206},{"id":"https://openalex.org/keywords/geometric-mean","display_name":"Geometric mean","score":0.25229373574256897},{"id":"https://openalex.org/keywords/ricci-curvature","display_name":"Ricci curvature","score":0.13179725408554077}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8865156173706055},{"id":"https://openalex.org/C181104567","wikidata":"https://www.wikidata.org/wiki/Q761383","display_name":"Riemannian geometry","level":2,"score":0.6229222416877747},{"id":"https://openalex.org/C49712288","wikidata":"https://www.wikidata.org/wiki/Q77601250","display_name":"Positive-definite matrix","level":3,"score":0.6096478700637817},{"id":"https://openalex.org/C176217482","wikidata":"https://www.wikidata.org/wiki/Q860554","display_name":"Metric (unit)","level":2,"score":0.5726037621498108},{"id":"https://openalex.org/C190470478","wikidata":"https://www.wikidata.org/wiki/Q2370229","display_name":"Invariant (physics)","level":2,"score":0.5456314086914062},{"id":"https://openalex.org/C4306028","wikidata":"https://www.wikidata.org/wiki/Q2185349","display_name":"Fundamental theorem of Riemannian geometry","level":4,"score":0.5410879850387573},{"id":"https://openalex.org/C164226766","wikidata":"https://www.wikidata.org/wiki/Q7293202","display_name":"Rank (graph theory)","level":2,"score":0.5197846293449402},{"id":"https://openalex.org/C117656884","wikidata":"https://www.wikidata.org/wiki/Q2625569","display_name":"Weighted geometric mean","level":3,"score":0.5158047676086426},{"id":"https://openalex.org/C199422724","wikidata":"https://www.wikidata.org/wiki/Q41118","display_name":"Quotient","level":2,"score":0.5029529929161072},{"id":"https://openalex.org/C198043062","wikidata":"https://www.wikidata.org/wiki/Q180953","display_name":"Metric space","level":2,"score":0.4573875963687897},{"id":"https://openalex.org/C127104897","wikidata":"https://www.wikidata.org/wiki/Q5454858","display_name":"Fisher information metric","level":4,"score":0.4272739589214325},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.4148983955383301},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.3836054503917694},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.32507848739624023},{"id":"https://openalex.org/C177547787","wikidata":"https://www.wikidata.org/wiki/Q6034319","display_name":"Injective metric space","level":3,"score":0.2785346210002899},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.2685784101486206},{"id":"https://openalex.org/C40222840","wikidata":"https://www.wikidata.org/wiki/Q185049","display_name":"Geometric mean","level":2,"score":0.25229373574256897},{"id":"https://openalex.org/C12089564","wikidata":"https://www.wikidata.org/wiki/Q1195879","display_name":"Ricci curvature","level":3,"score":0.13179725408554077},{"id":"https://openalex.org/C162324750","wikidata":"https://www.wikidata.org/wiki/Q8134","display_name":"Economics","level":0,"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/C21547014","wikidata":"https://www.wikidata.org/wiki/Q1423657","display_name":"Operations management","level":1,"score":0.0},{"id":"https://openalex.org/C195065555","wikidata":"https://www.wikidata.org/wiki/Q214881","display_name":"Curvature","level":2,"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/C37914503","wikidata":"https://www.wikidata.org/wiki/Q156495","display_name":"Mathematical physics","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}],"mesh":[],"locations_count":5,"locations":[{"id":"doi:10.1137/080731347","is_oa":false,"landing_page_url":"https://doi.org/10.1137/080731347","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:CiteSeerX.psu:10.1.1.707.5785","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.707.5785","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"http://orbi.ulg.ac.be/bitstream/2268/19669/1/publishedversion.pdf","raw_type":"text"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.763.5950","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.763.5950","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://arxiv.org/pdf/0807.4462.pdf","raw_type":"text"},{"id":"pmh:oai:generic.eprints.org:577483","is_oa":false,"landing_page_url":"http://publications.eng.cam.ac.uk/577483/","pdf_url":null,"source":{"id":"https://openalex.org/S4406922847","display_name":"Cambridge University Engineering Department Publications Database","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"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"},{"id":"pmh:oai:orbi.ulg.ac.be:2268/19669","is_oa":true,"landing_page_url":"https://orbi.uliege.be/handle/2268/19669","pdf_url":null,"source":{"id":"https://openalex.org/S4306400651","display_name":"Open Repository and Bibliography (University of Li\u00e8ge)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I157674565","host_organization_name":"University of Li\u00e8ge","host_organization_lineage":["https://openalex.org/I157674565"],"host_organization_lineage_names":[],"type":"repository"},"license":"other-oa","license_id":"https://openalex.org/licenses/other-oa","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"SIAM Journal on Matrix Analysis and Applications, 31 (3), 1055-1070 (2009)","raw_type":"http://purl.org/coar/resource_type/c_6501"}],"best_oa_location":{"id":"pmh:oai:orbi.ulg.ac.be:2268/19669","is_oa":true,"landing_page_url":"https://orbi.uliege.be/handle/2268/19669","pdf_url":null,"source":{"id":"https://openalex.org/S4306400651","display_name":"Open Repository and Bibliography (University of Li\u00e8ge)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I157674565","host_organization_name":"University of Li\u00e8ge","host_organization_lineage":["https://openalex.org/I157674565"],"host_organization_lineage_names":[],"type":"repository"},"license":"other-oa","license_id":"https://openalex.org/licenses/other-oa","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"SIAM Journal on Matrix Analysis and Applications, 31 (3), 1055-1070 (2009)","raw_type":"http://purl.org/coar/resource_type/c_6501"},"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":35,"referenced_works":["https://openalex.org/W10800830","https://openalex.org/W88077589","https://openalex.org/W382151588","https://openalex.org/W1511670827","https://openalex.org/W1525296983","https://openalex.org/W1537145174","https://openalex.org/W1542652324","https://openalex.org/W1590757293","https://openalex.org/W1804110266","https://openalex.org/W1969002377","https://openalex.org/W1983104971","https://openalex.org/W1983496390","https://openalex.org/W1986964250","https://openalex.org/W1993087762","https://openalex.org/W1996955012","https://openalex.org/W1997441999","https://openalex.org/W1997870124","https://openalex.org/W2022416612","https://openalex.org/W2032236594","https://openalex.org/W2045512849","https://openalex.org/W2057807012","https://openalex.org/W2059640215","https://openalex.org/W2068787860","https://openalex.org/W2087640086","https://openalex.org/W2118443009","https://openalex.org/W2136517446","https://openalex.org/W2147631863","https://openalex.org/W2154059477","https://openalex.org/W2159788726","https://openalex.org/W2296319761","https://openalex.org/W2798909945","https://openalex.org/W3104122060","https://openalex.org/W3105926974","https://openalex.org/W3210795861","https://openalex.org/W4206551434"],"related_works":["https://openalex.org/W614130005","https://openalex.org/W4291625303","https://openalex.org/W4312090883","https://openalex.org/W2737579912","https://openalex.org/W1597014246","https://openalex.org/W4298160146","https://openalex.org/W4249424264","https://openalex.org/W596279361","https://openalex.org/W2099515730","https://openalex.org/W2953347060"],"abstract_inverted_index":{"Abstract.":[0],"This":[1],"paper":[2],"introduces":[3],"a":[4,24,90,108],"new":[5],"metric":[6,20,57],"and":[7,38,55,71,111],"mean":[8,98],"on":[9,94],"the":[10,31,35,39,56,77,100,103],"set":[11],"of":[12,16,34,76,107],"positive":[13,36],"semidefinite":[14],"matrices":[15],"fixed-rank.":[17],"The":[18,43,96],"proposed":[19],"is":[21,52,58,81,112],"derived":[22],"from":[23],"well-chosen":[25],"Riemannian":[26,45],"quotient":[27],"geometry":[28,33],"that":[29,65,83],"generalizes":[30],"reductive":[32],"cone":[37],"associated":[40,78],"natural":[41],"metric.":[42],"resulting":[44],"space":[46],"has":[47],"strong":[48],"geometrical":[49],"properties:":[50],"it":[51],"geodesically":[53],"com-plete,":[54],"invariant":[59],"with":[60],"respect":[61],"to":[62,114],"all":[63],"transformations":[64],"preserve":[66],"angles":[67],"(orthogonal":[68],"transformations,":[69],"scalings,":[70],"pseudoinversion).":[72],"A":[73],"meaningful":[74],"approximation":[75],"Rie-mannian":[79],"distance":[80],"proposed,":[82],"can":[84],"be":[85],"efficiently":[86],"numerically":[87],"computed":[88],"via":[89],"simple":[91],"algorithm":[92],"based":[93],"SVD.":[95],"induced":[97],"preserves":[99],"rank,":[101],"possesses":[102],"most":[104],"desirable":[105],"characteristics":[106],"geometric":[109],"mean,":[110],"easy":[113],"compute.":[115]},"counts_by_year":[{"year":2026,"cited_by_count":1},{"year":2025,"cited_by_count":10},{"year":2024,"cited_by_count":5},{"year":2023,"cited_by_count":13},{"year":2022,"cited_by_count":7},{"year":2021,"cited_by_count":22},{"year":2020,"cited_by_count":15},{"year":2019,"cited_by_count":12},{"year":2018,"cited_by_count":6},{"year":2017,"cited_by_count":7},{"year":2016,"cited_by_count":2},{"year":2015,"cited_by_count":4},{"year":2014,"cited_by_count":2},{"year":2013,"cited_by_count":9},{"year":2012,"cited_by_count":8}],"updated_date":"2026-07-15T18:14:33.161393","created_date":"2025-10-10T00:00:00"}
