{"id":"https://openalex.org/W3130599955","doi":"https://doi.org/10.1137/21m1443881","title":"A Higher-Order Generalized Singular Value Decomposition for Rank-Deficient Matrices","display_name":"A Higher-Order Generalized Singular Value Decomposition for Rank-Deficient Matrices","publication_year":2023,"publication_date":"2023-07-21","ids":{"openalex":"https://openalex.org/W3130599955","doi":"https://doi.org/10.1137/21m1443881","mag":"3130599955"},"language":"en","primary_location":{"id":"doi:10.1137/21m1443881","is_oa":false,"landing_page_url":"https://doi.org/10.1137/21m1443881","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/A5068717775","display_name":"Idris Kempf","orcid":"https://orcid.org/0000-0002-9304-2773"},"institutions":[{"id":"https://openalex.org/I40120149","display_name":"University of Oxford","ror":"https://ror.org/052gg0110","country_code":"GB","type":"education","lineage":["https://openalex.org/I40120149"]}],"countries":["GB"],"is_corresponding":true,"raw_author_name":"Idris Kempf","raw_affiliation_strings":["Department of Engineering Science, University of Oxford, Parks Road, Oxford, OX1 3PJ, UK"],"raw_orcid":"https://orcid.org/0000-0002-9304-2773","affiliations":[{"raw_affiliation_string":"Department of Engineering Science, University of Oxford, Parks Road, Oxford, OX1 3PJ, UK","institution_ids":["https://openalex.org/I40120149"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5001694225","display_name":"Paul J. Goulart","orcid":"https://orcid.org/0000-0002-0456-4124"},"institutions":[{"id":"https://openalex.org/I40120149","display_name":"University of Oxford","ror":"https://ror.org/052gg0110","country_code":"GB","type":"education","lineage":["https://openalex.org/I40120149"]}],"countries":["GB"],"is_corresponding":false,"raw_author_name":"Paul J. Goulart","raw_affiliation_strings":["Department of Engineering Science, University of Oxford, Parks Road, Oxford, OX1 3PJ, UK"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Engineering Science, University of Oxford, Parks Road, Oxford, OX1 3PJ, UK","institution_ids":["https://openalex.org/I40120149"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5038299471","display_name":"Stephen Duncan","orcid":null},"institutions":[{"id":"https://openalex.org/I40120149","display_name":"University of Oxford","ror":"https://ror.org/052gg0110","country_code":"GB","type":"education","lineage":["https://openalex.org/I40120149"]}],"countries":["GB"],"is_corresponding":false,"raw_author_name":"Stephen R. Duncan","raw_affiliation_strings":["Department of Engineering Science, University of Oxford, Parks Road, Oxford, OX1 3PJ, UK"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Engineering Science, University of Oxford, Parks Road, Oxford, OX1 3PJ, UK","institution_ids":["https://openalex.org/I40120149"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":3,"corresponding_author_ids":["https://openalex.org/A5068717775"],"corresponding_institution_ids":["https://openalex.org/I40120149"],"apc_list":null,"apc_paid":null,"fwci":1.2464,"has_fulltext":false,"cited_by_count":7,"citation_normalized_percentile":{"value":0.7603796,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":90,"max":98},"biblio":{"volume":"44","issue":"3","first_page":"1047","last_page":"1072"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T12303","display_name":"Tensor decomposition and applications","score":0.9968000054359436,"subfield":{"id":"https://openalex.org/subfields/2605","display_name":"Computational 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/T12303","display_name":"Tensor decomposition and applications","score":0.9968000054359436,"subfield":{"id":"https://openalex.org/subfields/2605","display_name":"Computational 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/T10222","display_name":"Genomics and Chromatin Dynamics","score":0.9775999784469604,"subfield":{"id":"https://openalex.org/subfields/1312","display_name":"Molecular Biology"},"field":{"id":"https://openalex.org/fields/13","display_name":"Biochemistry, Genetics and Molecular Biology"},"domain":{"id":"https://openalex.org/domains/1","display_name":"Life Sciences"}},{"id":"https://openalex.org/T10885","display_name":"Gene expression and cancer classification","score":0.9711999893188477,"subfield":{"id":"https://openalex.org/subfields/1312","display_name":"Molecular Biology"},"field":{"id":"https://openalex.org/fields/13","display_name":"Biochemistry, Genetics and Molecular Biology"},"domain":{"id":"https://openalex.org/domains/1","display_name":"Life Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/singular-value-decomposition","display_name":"Singular value decomposition","score":0.7727679014205933},{"id":"https://openalex.org/keywords/linear-subspace","display_name":"Linear subspace","score":0.7074266076087952},{"id":"https://openalex.org/keywords/rank","display_name":"Rank (graph theory)","score":0.7039033770561218},{"id":"https://openalex.org/keywords/matrix-decomposition","display_name":"Matrix decomposition","score":0.5315254330635071},{"id":"https://openalex.org/keywords/matrix","display_name":"Matrix (chemical analysis)","score":0.5090412497520447},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.4818366765975952},{"id":"https://openalex.org/keywords/factorization","display_name":"Factorization","score":0.4682527482509613},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.4299640953540802},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.34578001499176025},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.3056254982948303},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.11600100994110107},{"id":"https://openalex.org/keywords/eigenvalues-and-eigenvectors","display_name":"Eigenvalues and eigenvectors","score":0.11059632897377014}],"concepts":[{"id":"https://openalex.org/C22789450","wikidata":"https://www.wikidata.org/wiki/Q420904","display_name":"Singular value decomposition","level":2,"score":0.7727679014205933},{"id":"https://openalex.org/C12362212","wikidata":"https://www.wikidata.org/wiki/Q728435","display_name":"Linear subspace","level":2,"score":0.7074266076087952},{"id":"https://openalex.org/C164226766","wikidata":"https://www.wikidata.org/wiki/Q7293202","display_name":"Rank (graph theory)","level":2,"score":0.7039033770561218},{"id":"https://openalex.org/C42355184","wikidata":"https://www.wikidata.org/wiki/Q1361088","display_name":"Matrix decomposition","level":3,"score":0.5315254330635071},{"id":"https://openalex.org/C106487976","wikidata":"https://www.wikidata.org/wiki/Q685816","display_name":"Matrix (chemical analysis)","level":2,"score":0.5090412497520447},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.4818366765975952},{"id":"https://openalex.org/C187834632","wikidata":"https://www.wikidata.org/wiki/Q188804","display_name":"Factorization","level":2,"score":0.4682527482509613},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.4299640953540802},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.34578001499176025},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.3056254982948303},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.11600100994110107},{"id":"https://openalex.org/C158693339","wikidata":"https://www.wikidata.org/wiki/Q190524","display_name":"Eigenvalues and eigenvectors","level":2,"score":0.11059632897377014},{"id":"https://openalex.org/C192562407","wikidata":"https://www.wikidata.org/wiki/Q228736","display_name":"Materials science","level":0,"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/C159985019","wikidata":"https://www.wikidata.org/wiki/Q181790","display_name":"Composite material","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}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.1137/21m1443881","is_oa":false,"landing_page_url":"https://doi.org/10.1137/21m1443881","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:ora.ox.ac.uk:uuid:218bce04-a2f5-4b43-af60-ba3eab79d896","is_oa":false,"landing_page_url":"https://ora.ox.ac.uk/objects/uuid:218bce04-a2f5-4b43-af60-ba3eab79d896","pdf_url":null,"source":{"id":"https://openalex.org/S4306402636","display_name":"Oxford University Research Archive (ORA) (University of Oxford)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I40120149","host_organization_name":"University of Oxford","host_organization_lineage":["https://openalex.org/I40120149"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Symplectic Elements","raw_type":"Journal article"}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[],"funders":[{"id":"https://openalex.org/F4320313096","display_name":"Diamond Light Source","ror":"https://ror.org/05etxs293"},{"id":"https://openalex.org/F4320334627","display_name":"Engineering and Physical Sciences Research Council","ror":"https://ror.org/0439y7842"}],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":31,"referenced_works":["https://openalex.org/W1539811621","https://openalex.org/W1575147392","https://openalex.org/W1976384891","https://openalex.org/W1977271127","https://openalex.org/W1985852691","https://openalex.org/W1997782219","https://openalex.org/W2013912476","https://openalex.org/W2022679851","https://openalex.org/W2030813272","https://openalex.org/W2038784978","https://openalex.org/W2039333245","https://openalex.org/W2060705109","https://openalex.org/W2069954540","https://openalex.org/W2099388546","https://openalex.org/W2103537693","https://openalex.org/W2121739212","https://openalex.org/W2125624993","https://openalex.org/W2128985829","https://openalex.org/W2582365613","https://openalex.org/W2798909945","https://openalex.org/W2915098601","https://openalex.org/W2951881727","https://openalex.org/W2955229320","https://openalex.org/W3039567237","https://openalex.org/W3102869303","https://openalex.org/W3119760816","https://openalex.org/W3127717271","https://openalex.org/W3129878065","https://openalex.org/W4252112222","https://openalex.org/W4300657400","https://openalex.org/W4312258136"],"related_works":["https://openalex.org/W2654090051","https://openalex.org/W2782904003","https://openalex.org/W2118633810","https://openalex.org/W2150953077","https://openalex.org/W4226434912","https://openalex.org/W2002598339","https://openalex.org/W1995410415","https://openalex.org/W2133814741","https://openalex.org/W3108343813","https://openalex.org/W4312481107"],"abstract_inverted_index":{"The":[0,41],"higher-order":[1,130],"generalized":[2],"singular":[3],"value":[4],"decomposition":[5,132],"(HO-GSVD)":[6],"is":[7,135],"a":[8,68],"matrix":[9,84],"factorization":[10],"technique":[11],"that":[12,30,56,73,93,117],"extends":[13,74],"the":[14,59,71,83,94,97,106,129,139,144],"GSVD":[15],"to":[16,26,77,101,111,120,128,138,150],"&lt;i&gt;N&lt;/i&gt;":[17,45],"\u2265":[18],"2":[19],"data":[20,79],"matrices":[21,46,60,80,151],"and":[22,170],"can":[23],"be":[24],"used":[25],"identify":[27,115],"common":[28,109],"subspaces":[29,110],"are":[31,118],"shared":[32],"across":[33],"multiple":[34],"large-scale":[35],"datasets":[36],"with":[37,152],"different":[38],"row":[39],"dimensions.":[40],"standard":[42,145],"HO-GSVD":[43,72,99,146],"factors":[44],"&lt;i&gt;A&lt;/i&gt;&lt;sub&gt;&lt;i&gt;i&lt;/i&gt;&lt;/sub&gt;":[47,51,61,87],"\u2208":[48],"\u211d&lt;sup&gt;&lt;i&gt;m&lt;/i&gt;&lt;sub&gt;&lt;i&gt;i&lt;/i&gt;&lt;/sub&gt;\u00d7&lt;i&gt;n&lt;/i&gt;&lt;/sup&gt;":[49],"as":[50,162],"=":[52],"&lt;i&gt;U&lt;sub&gt;i&lt;/sub&gt;&lt;/i&gt;\u2211&lt;sub&gt;&lt;i&gt;i&lt;/i&gt;&lt;/sub&gt;&lt;i&gt;V&lt;/i&gt;&lt;sup&gt;T&lt;/sup&gt;":[53],"but":[54],"requires":[55],"each":[57],"of":[58,70,85,96,108,143],"has":[62,88],"full":[63,89],"column":[64],"rank.":[65],"We":[66,104,123],"propose":[67],"modification":[69],"its":[75,148],"applicability":[76],"rank-deficient":[78],"&lt;i&gt;A&lt;/i&gt;&lt;sub&gt;&lt;i&gt;i&lt;/i&gt;&lt;/sub&gt;.":[81,122],"If":[82],"stacked":[86],"rank,":[90],"we":[91],"show":[92],"properties":[95],"original":[98],"extend":[100,105,125],"our":[102,126],"approach.":[103],"notion":[107],"isolated":[112],"subspaces,":[113],"which":[114,134],"features":[116],"unique":[119],"one":[121],"also":[124],"results":[127],"cosine-sine":[131],"(HO-CSD),":[133],"closely":[136],"related":[137],"HO-GSVD.":[140],"Our":[141],"extension":[142],"allows":[147],"application":[149],"&lt;i&gt;m&lt;sub&gt;i&lt;/sub&gt;&lt;/i&gt;":[153],"&lt;":[154,159],"&lt;i&gt;n&lt;/i&gt;":[155],"or":[156],"rank":[157],"(&lt;i&gt;A&lt;/i&gt;&lt;sub&gt;&lt;i&gt;i&lt;/i&gt;&lt;/sub&gt;)":[158],"&lt;i&gt;n&lt;/i&gt;,":[160],"such":[161],"those":[163],"encountered":[164],"in":[165],"bioinformatics,":[166],"neuroscience,":[167],"control":[168],"theory,":[169],"classification":[171],"problems.":[172]},"counts_by_year":[{"year":2025,"cited_by_count":4},{"year":2024,"cited_by_count":2},{"year":2022,"cited_by_count":1}],"updated_date":"2025-11-06T03:46:38.306776","created_date":"2025-10-10T00:00:00"}
