{"id":"https://openalex.org/W2782047334","doi":"https://doi.org/10.1137/18m1163658","title":"Low-Rank Matrix Approximations Do Not Need a Singular Value Gap","display_name":"Low-Rank Matrix Approximations Do Not Need a Singular Value Gap","publication_year":2019,"publication_date":"2019-01-01","ids":{"openalex":"https://openalex.org/W2782047334","doi":"https://doi.org/10.1137/18m1163658","mag":"2782047334"},"language":"en","primary_location":{"id":"doi:10.1137/18m1163658","is_oa":false,"landing_page_url":"https://doi.org/10.1137/18m1163658","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":"preprint","indexed_in":["arxiv","crossref","datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://arxiv.org/pdf/1801.00670","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5081378199","display_name":"Petros Drineas","orcid":"https://orcid.org/0000-0003-1994-8670"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Petros Drineas","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"last","author":{"id":"https://openalex.org/A5036959652","display_name":"Ilse C. F. Ipsen","orcid":"https://orcid.org/0000-0001-5645-5854"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Ilse C. F. Ipsen","raw_affiliation_strings":[],"raw_orcid":"https://orcid.org/0000-0001-5645-5854","affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":2,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":0.2022,"has_fulltext":true,"cited_by_count":1,"citation_normalized_percentile":{"value":0.49829731,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":90,"max":94},"biblio":{"volume":"40","issue":"1","first_page":"299","last_page":"319"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10792","display_name":"Matrix Theory and Algorithms","score":0.9993000030517578,"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"}},"topics":[{"id":"https://openalex.org/T10792","display_name":"Matrix Theory and Algorithms","score":0.9993000030517578,"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":0.9980000257492065,"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/T13487","display_name":"Statistical and numerical algorithms","score":0.993399977684021,"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/singular-value","display_name":"Singular value","score":0.8388354778289795},{"id":"https://openalex.org/keywords/matrix-norm","display_name":"Matrix norm","score":0.7993872761726379},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.7174153327941895},{"id":"https://openalex.org/keywords/projector","display_name":"Projector","score":0.6877229809761047},{"id":"https://openalex.org/keywords/rank","display_name":"Rank (graph theory)","score":0.6589528322219849},{"id":"https://openalex.org/keywords/low-rank-approximation","display_name":"Low-rank approximation","score":0.6297712326049805},{"id":"https://openalex.org/keywords/multiplicative-function","display_name":"Multiplicative function","score":0.5617485046386719},{"id":"https://openalex.org/keywords/subspace-topology","display_name":"Subspace topology","score":0.540686309337616},{"id":"https://openalex.org/keywords/matrix","display_name":"Matrix (chemical analysis)","score":0.5348727703094482},{"id":"https://openalex.org/keywords/singular-value-decomposition","display_name":"Singular value decomposition","score":0.5294491648674011},{"id":"https://openalex.org/keywords/approximations-of-\u03c0","display_name":"Approximations of \u03c0","score":0.5114516019821167},{"id":"https://openalex.org/keywords/norm","display_name":"Norm (philosophy)","score":0.49531450867652893},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.3440697193145752},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.3328038454055786},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.3293302357196808},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.27082300186157227},{"id":"https://openalex.org/keywords/eigenvalues-and-eigenvectors","display_name":"Eigenvalues and eigenvectors","score":0.25700843334198},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.12072551250457764},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.10707986354827881}],"concepts":[{"id":"https://openalex.org/C109282560","wikidata":"https://www.wikidata.org/wiki/Q4166054","display_name":"Singular value","level":3,"score":0.8388354778289795},{"id":"https://openalex.org/C92207270","wikidata":"https://www.wikidata.org/wiki/Q939253","display_name":"Matrix norm","level":3,"score":0.7993872761726379},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7174153327941895},{"id":"https://openalex.org/C2776865275","wikidata":"https://www.wikidata.org/wiki/Q311666","display_name":"Projector","level":2,"score":0.6877229809761047},{"id":"https://openalex.org/C164226766","wikidata":"https://www.wikidata.org/wiki/Q7293202","display_name":"Rank (graph theory)","level":2,"score":0.6589528322219849},{"id":"https://openalex.org/C90199385","wikidata":"https://www.wikidata.org/wiki/Q6692777","display_name":"Low-rank approximation","level":3,"score":0.6297712326049805},{"id":"https://openalex.org/C42747912","wikidata":"https://www.wikidata.org/wiki/Q1048447","display_name":"Multiplicative function","level":2,"score":0.5617485046386719},{"id":"https://openalex.org/C32834561","wikidata":"https://www.wikidata.org/wiki/Q660730","display_name":"Subspace topology","level":2,"score":0.540686309337616},{"id":"https://openalex.org/C106487976","wikidata":"https://www.wikidata.org/wiki/Q685816","display_name":"Matrix (chemical analysis)","level":2,"score":0.5348727703094482},{"id":"https://openalex.org/C22789450","wikidata":"https://www.wikidata.org/wiki/Q420904","display_name":"Singular value decomposition","level":2,"score":0.5294491648674011},{"id":"https://openalex.org/C193386753","wikidata":"https://www.wikidata.org/wiki/Q1130396","display_name":"Approximations of \u03c0","level":2,"score":0.5114516019821167},{"id":"https://openalex.org/C191795146","wikidata":"https://www.wikidata.org/wiki/Q3878446","display_name":"Norm (philosophy)","level":2,"score":0.49531450867652893},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.3440697193145752},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.3328038454055786},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.3293302357196808},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.27082300186157227},{"id":"https://openalex.org/C158693339","wikidata":"https://www.wikidata.org/wiki/Q190524","display_name":"Eigenvalues and eigenvectors","level":2,"score":0.25700843334198},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.12072551250457764},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.10707986354827881},{"id":"https://openalex.org/C155281189","wikidata":"https://www.wikidata.org/wiki/Q3518150","display_name":"Tensor (intrinsic definition)","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/C199539241","wikidata":"https://www.wikidata.org/wiki/Q7748","display_name":"Law","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},{"id":"https://openalex.org/C17744445","wikidata":"https://www.wikidata.org/wiki/Q36442","display_name":"Political science","level":0,"score":0.0},{"id":"https://openalex.org/C120665830","wikidata":"https://www.wikidata.org/wiki/Q14620","display_name":"Optics","level":1,"score":0.0}],"mesh":[],"locations_count":5,"locations":[{"id":"doi:10.1137/18m1163658","is_oa":false,"landing_page_url":"https://doi.org/10.1137/18m1163658","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:1801.00670","is_oa":true,"landing_page_url":"http://arxiv.org/abs/1801.00670","pdf_url":"https://arxiv.org/pdf/1801.00670","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"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":"","raw_type":"text"},{"id":"pmh:oai:arXiv.org:1801.00670","is_oa":true,"landing_page_url":"https://arxiv.org/abs/1801.00670","pdf_url":"https://arxiv.org/pdf/1801.00670","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"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"},{"id":"mag:2782047334","is_oa":true,"landing_page_url":"https://arxiv.org/pdf/1801.00670","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"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":"arXiv (Cornell University)","raw_type":null},{"id":"doi:10.48550/arxiv.1801.00670","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.1801.00670","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"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":null,"is_accepted":false,"is_published":null,"raw_source_name":null,"raw_type":"article"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:1801.00670","is_oa":true,"landing_page_url":"http://arxiv.org/abs/1801.00670","pdf_url":"https://arxiv.org/pdf/1801.00670","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"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":"","raw_type":"text"},"sustainable_development_goals":[],"awards":[{"id":"https://openalex.org/G304234241","display_name":null,"funder_award_id":"IIS-1447283","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"},{"id":"https://openalex.org/G5719697544","display_name":"BIGDATA: F: DKA: Collaborative Research: Randomized Numerical Linear Algebra (RandNLA) for multi-linear and non-linear data","funder_award_id":"1447283","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"},{"id":"https://openalex.org/G8384358473","display_name":null,"funder_award_id":"IIS-1302231","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"},{"id":"https://openalex.org/G8794311462","display_name":"III: Medium: Mining petabytes of data using cloud computing and a massively parallel cyberinstrument","funder_award_id":"1302231","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"}],"funders":[{"id":"https://openalex.org/F4320306076","display_name":"National Science Foundation","ror":"https://ror.org/021nxhr62"}],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W2782047334.pdf","grobid_xml":"https://content.openalex.org/works/W2782047334.grobid-xml"},"referenced_works_count":24,"referenced_works":["https://openalex.org/W658559791","https://openalex.org/W1488435683","https://openalex.org/W1925032196","https://openalex.org/W1970377488","https://openalex.org/W1970576574","https://openalex.org/W1970950689","https://openalex.org/W1976798627","https://openalex.org/W1984492644","https://openalex.org/W1996522765","https://openalex.org/W2014042806","https://openalex.org/W2014802378","https://openalex.org/W2025626352","https://openalex.org/W2031215441","https://openalex.org/W2042465463","https://openalex.org/W2051555215","https://openalex.org/W2149380762","https://openalex.org/W2157988812","https://openalex.org/W2177697562","https://openalex.org/W2394933259","https://openalex.org/W2516218596","https://openalex.org/W2963441460","https://openalex.org/W2964214436","https://openalex.org/W3022909171","https://openalex.org/W3112410551"],"related_works":["https://openalex.org/W3017414697","https://openalex.org/W3185453068","https://openalex.org/W3109471745","https://openalex.org/W2737871803","https://openalex.org/W2745326244","https://openalex.org/W3197152554","https://openalex.org/W2953143557","https://openalex.org/W2411401818","https://openalex.org/W2901438688","https://openalex.org/W2003814119","https://openalex.org/W1963826850","https://openalex.org/W2962782032","https://openalex.org/W2077749929","https://openalex.org/W2964540803","https://openalex.org/W185640140","https://openalex.org/W2963676768","https://openalex.org/W3097787030","https://openalex.org/W2293965695","https://openalex.org/W2078303291","https://openalex.org/W2005361512"],"abstract_inverted_index":{"Low-rank":[0],"approximations":[1,118],"to":[2,58,79,87],"a":[3,15,51,66,109],"real":[4],"matrix":[5,16],"$\\mathbf{A}$":[6,83,105],"can":[7,26],"be":[8,27],"computed":[9],"from":[10,29],"$\\mathbf{Z}\\mathbf{Z}^T\\mathbf{A}$,":[11],"where":[12],"$\\mathbf{Z}$":[13],"is":[14,50,76],"with":[17],"orthonormal":[18],"columns,":[19],"and":[20,44,84,86,119],"the":[21,24,40,95,98,101],"accuracy":[22,96],"of":[23,32,94,97],"approximation":[25],"estimated":[28],"some":[30],"norm":[31],"$\\mathbf{A}-\\mathbf{Z}\\mathbf{Z}^T\\mathbf{A}$.":[33],"We":[34,70],"show":[35,72],"that":[36,73],"computing":[37],"$\\mathbf{A}-\\mathbf{Z}\\mathbf{Z}^T\\mathbf{A}$":[38],"in":[39,56,82,92],"two-norm,":[41],"Frobenius":[42],"norms,":[43],"more":[45],"generally":[46],"any":[47],"Schatten":[48],"$p$-norm":[49],"well-posed":[52],"mathematical":[53],"problem;":[54],"and,":[55],"contrast":[57],"dominant":[59],"subspace":[60,120],"computations,":[61],"it":[62],"does":[63,106],"not":[64],"require":[65],"singular":[67,110],"value":[68],"gap.":[69],"also":[71],"this":[74],"problem":[75],"well-conditioned":[77],"(insensitive)":[78],"additive":[80],"perturbations":[81,91],"$\\mathbf{Z}$,":[85],"dimension-changing":[88],"or":[89],"multiplicative":[90],"$\\mathbf{A}$---regardless":[93],"approximation.":[99],"For":[100],"special":[102],"case":[103],"when":[104],"indeed":[107],"have":[108],"values":[111],"gap,":[112],"connections":[113],"are":[114],"established":[115],"between":[116],"low-rank":[117],"angles.":[121]},"counts_by_year":[{"year":2019,"cited_by_count":1}],"updated_date":"2026-06-22T08:00:12.763002","created_date":"2025-10-10T00:00:00"}