{"id":"https://openalex.org/W2907726033","doi":"https://doi.org/10.1137/18m1234412","title":"The GSVD: Where are the Ellipses?, Matrix Trigonometry, and More","display_name":"The GSVD: Where are the Ellipses?, Matrix Trigonometry, and More","publication_year":2020,"publication_date":"2020-01-01","ids":{"openalex":"https://openalex.org/W2907726033","doi":"https://doi.org/10.1137/18m1234412","mag":"2907726033"},"language":"en","primary_location":{"id":"doi:10.1137/18m1234412","is_oa":false,"landing_page_url":"https://doi.org/10.1137/18m1234412","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/1901.00485","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":null,"display_name":"Alan Edelman","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Alan Edelman","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"last","author":{"id":null,"display_name":"Yuyang Wang","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Yuyang Wang","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":0,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":1.0995,"has_fulltext":false,"cited_by_count":9,"citation_normalized_percentile":{"value":0.72315424,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":98},"biblio":{"volume":"41","issue":"4","first_page":"1826","last_page":"1856"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T12417","display_name":"Morphological variations and asymmetry","score":0.33180001378059387,"subfield":{"id":"https://openalex.org/subfields/2608","display_name":"Geometry and Topology"},"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/T12417","display_name":"Morphological variations and asymmetry","score":0.33180001378059387,"subfield":{"id":"https://openalex.org/subfields/2608","display_name":"Geometry and Topology"},"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/T12549","display_name":"Image and Object Detection Techniques","score":0.1103999987244606,"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"}},{"id":"https://openalex.org/T13487","display_name":"Statistical and numerical algorithms","score":0.08089999854564667,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/tikhonov-regularization","display_name":"Tikhonov regularization","score":0.7993999719619751},{"id":"https://openalex.org/keywords/singular-value-decomposition","display_name":"Singular value decomposition","score":0.7757999897003174},{"id":"https://openalex.org/keywords/singular-value","display_name":"Singular value","score":0.6557999849319458},{"id":"https://openalex.org/keywords/context","display_name":"Context (archaeology)","score":0.5507000088691711},{"id":"https://openalex.org/keywords/singular-spectrum-analysis","display_name":"Singular spectrum analysis","score":0.4650999903678894},{"id":"https://openalex.org/keywords/ellipse","display_name":"Ellipse","score":0.4156000018119812},{"id":"https://openalex.org/keywords/matrix","display_name":"Matrix (chemical analysis)","score":0.3977000117301941},{"id":"https://openalex.org/keywords/regularization","display_name":"Regularization (linguistics)","score":0.38510000705718994}],"concepts":[{"id":"https://openalex.org/C152442038","wikidata":"https://www.wikidata.org/wiki/Q2778212","display_name":"Tikhonov regularization","level":3,"score":0.7993999719619751},{"id":"https://openalex.org/C22789450","wikidata":"https://www.wikidata.org/wiki/Q420904","display_name":"Singular value decomposition","level":2,"score":0.7757999897003174},{"id":"https://openalex.org/C109282560","wikidata":"https://www.wikidata.org/wiki/Q4166054","display_name":"Singular value","level":3,"score":0.6557999849319458},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.5526999831199646},{"id":"https://openalex.org/C2779343474","wikidata":"https://www.wikidata.org/wiki/Q3109175","display_name":"Context (archaeology)","level":2,"score":0.5507000088691711},{"id":"https://openalex.org/C136272165","wikidata":"https://www.wikidata.org/wiki/Q4048889","display_name":"Singular spectrum analysis","level":3,"score":0.4650999903678894},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.446399986743927},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.4226999878883362},{"id":"https://openalex.org/C74261601","wikidata":"https://www.wikidata.org/wiki/Q40112","display_name":"Ellipse","level":2,"score":0.4156000018119812},{"id":"https://openalex.org/C106487976","wikidata":"https://www.wikidata.org/wiki/Q685816","display_name":"Matrix (chemical analysis)","level":2,"score":0.3977000117301941},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.3889999985694885},{"id":"https://openalex.org/C2776135515","wikidata":"https://www.wikidata.org/wiki/Q17143721","display_name":"Regularization (linguistics)","level":2,"score":0.38510000705718994},{"id":"https://openalex.org/C69738355","wikidata":"https://www.wikidata.org/wiki/Q1228929","display_name":"Linear discriminant analysis","level":2,"score":0.37619999051094055},{"id":"https://openalex.org/C84545080","wikidata":"https://www.wikidata.org/wiki/Q1147936","display_name":"Condition number","level":3,"score":0.37400001287460327},{"id":"https://openalex.org/C42355184","wikidata":"https://www.wikidata.org/wiki/Q1361088","display_name":"Matrix decomposition","level":3,"score":0.2992999851703644},{"id":"https://openalex.org/C132459708","wikidata":"https://www.wikidata.org/wiki/Q744069","display_name":"Extrapolation","level":2,"score":0.2939000129699707},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.29030001163482666},{"id":"https://openalex.org/C135909967","wikidata":"https://www.wikidata.org/wiki/Q4338245","display_name":"Singular solution","level":2,"score":0.28439998626708984},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.27630001306533813},{"id":"https://openalex.org/C135252773","wikidata":"https://www.wikidata.org/wiki/Q1567213","display_name":"Inverse problem","level":2,"score":0.27000001072883606},{"id":"https://openalex.org/C2779193601","wikidata":"https://www.wikidata.org/wiki/Q20026918","display_name":"Mathematical theory","level":2,"score":0.27000001072883606},{"id":"https://openalex.org/C147764199","wikidata":"https://www.wikidata.org/wiki/Q6865248","display_name":"Minification","level":2,"score":0.2563999891281128},{"id":"https://openalex.org/C126255220","wikidata":"https://www.wikidata.org/wiki/Q141495","display_name":"Mathematical optimization","level":1,"score":0.2533999979496002}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.1137/18m1234412","is_oa":false,"landing_page_url":"https://doi.org/10.1137/18m1234412","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:1901.00485","is_oa":true,"landing_page_url":"http://arxiv.org/abs/1901.00485","pdf_url":"https://arxiv.org/pdf/1901.00485","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"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"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:1901.00485","is_oa":true,"landing_page_url":"http://arxiv.org/abs/1901.00485","pdf_url":"https://arxiv.org/pdf/1901.00485","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"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":21,"referenced_works":["https://openalex.org/W1902027874","https://openalex.org/W1965534165","https://openalex.org/W1977271127","https://openalex.org/W1985852691","https://openalex.org/W1995424939","https://openalex.org/W1997782219","https://openalex.org/W2022336648","https://openalex.org/W2027077102","https://openalex.org/W2031058636","https://openalex.org/W2036279667","https://openalex.org/W2045512849","https://openalex.org/W2062243225","https://openalex.org/W2071177772","https://openalex.org/W2099388546","https://openalex.org/W2103384443","https://openalex.org/W2103631031","https://openalex.org/W2119459069","https://openalex.org/W2129256542","https://openalex.org/W2140835403","https://openalex.org/W2165344682","https://openalex.org/W2889740987"],"related_works":[],"abstract_inverted_index":{"This":[0,49],"paper":[1,50],"provides":[2,42],"an":[3],"advanced":[4],"mathematical":[5,107],"theory":[6,81],"of":[7,21,70,174,201,214,222],"the":[8,19,22,40,46,56,67,96,113,118,121,127,141,163,172,177,212,220,223],"generalized":[9,58,97,147],"singular":[10,59,68,98,114],"value":[11],"decomposition":[12],"(GSVD)":[13],"and":[14,32,88,103,162,195,203],"its":[15,154],"applications.":[16,93],"We":[17,36,72,157],"explore":[18],"geometry":[20],"GSVD":[23,41,178,228],"providing":[24],"a":[25,30,33,52,101,106,146],"long-sought-for":[26],"picture":[27],"which":[28],"includes":[29],"horizontal":[31],"vertical":[34],"multiaxis.":[35],"further":[37],"propose":[38,139],"that":[39,78,124,140],"natural":[43,83,105,155,164,224],"coordinates":[44],"for":[45,84,91,109,117],"Grassmann":[47],"manifold.":[48],"proves":[51],"theorem":[53],"showing":[54],"how":[55],"finite":[57],"values":[60,69],"do":[61,63,116],"or":[62],"not":[64,131],"relate":[65],"to":[66,75,132],"$AB^\\dagger$.":[71],"then":[73],"turn":[74],"applications,":[76],"arguing":[77],"this":[79,159],"geometrical":[80],"is":[82,149],"understanding":[85],"existing":[86],"applications":[87,111,175],"recognizing":[89],"opportunities":[90],"new":[92],"In":[94,120],"particular":[95],"vectors":[99,115],"play":[100],"direct":[102],"as":[104,112,136,145,199],"role":[108],"certain":[110],"SVD.":[119],"same":[122],"way":[123],"experts":[125],"on":[126],"SVD":[128,134],"often":[129,143],"prefer":[130],"cast":[133,144,152],"problems":[135],"eigenproblems,":[137],"we":[138,218],"GSVD,":[142],"eigenproblem,":[148],"perhaps":[150],"best":[151],"in":[153,171,226],"setting.":[156],"illustrate":[158],"theoretical":[160],"approach":[161],"multiaxes":[165,225],"(with":[166],"labels":[167],"from":[168],"technical":[169],"domains)":[170],"context":[173],"where":[176],"arises:":[179],"Tikhonov":[180],"regularization":[181],"(unregularized":[182],"versus":[183,188,193,208],"regularized),":[184],"genome":[185],"reconstruction":[186],"(humans":[187],"yeast),":[189],"signal":[190],"processing":[191],"(signal":[192],"noise),":[194],"statistical":[196],"analysis":[197,200,205],"such":[198],"variance":[202],"discriminant":[204],"(between":[206],"clusters":[207],"within":[209],"clusters.)":[210],"With":[211],"aid":[213],"our":[215],"ellipse":[216],"figure,":[217],"encourage":[219],"labeling":[221],"any":[227],"problem.":[229]},"counts_by_year":[{"year":2026,"cited_by_count":2},{"year":2025,"cited_by_count":2},{"year":2024,"cited_by_count":2},{"year":2023,"cited_by_count":2},{"year":2022,"cited_by_count":1}],"updated_date":"2026-08-05T07:39:15.569665","created_date":"2019-01-11T00:00:00"}