{"id":"https://openalex.org/W1998269045","doi":"https://doi.org/10.1137/07070471x","title":"Relative-Error $CUR$ Matrix Decompositions","display_name":"Relative-Error $CUR$ Matrix Decompositions","publication_year":2008,"publication_date":"2008-01-01","ids":{"openalex":"https://openalex.org/W1998269045","doi":"https://doi.org/10.1137/07070471x","mag":"1998269045"},"language":"en","primary_location":{"id":"doi:10.1137/07070471x","is_oa":false,"landing_page_url":"https://doi.org/10.1137/07070471x","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/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":["drinep@cs.rpi.edu#TAB#"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"drinep@cs.rpi.edu#TAB#","institution_ids":[]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5033006662","display_name":"Michael W. Mahoney","orcid":"https://orcid.org/0000-0001-7920-4652"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Michael W. Mahoney","raw_affiliation_strings":["mahoney@yahoo-inc.com"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"mahoney@yahoo-inc.com","institution_ids":[]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5045502987","display_name":"S. Muthukrishnan","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"S. Muthukrishnan","raw_affiliation_strings":["muthu@google.com#TAB#"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"muthu@google.com#TAB#","institution_ids":[]}]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":0,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":17.5259,"has_fulltext":false,"cited_by_count":479,"citation_normalized_percentile":{"value":0.99539284,"is_in_top_1_percent":true,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":99,"max":100},"biblio":{"volume":"30","issue":"2","first_page":"844","last_page":"881"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10500","display_name":"Sparse and Compressive Sensing Techniques","score":0.9998000264167786,"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"}},"topics":[{"id":"https://openalex.org/T10500","display_name":"Sparse and Compressive Sensing Techniques","score":0.9998000264167786,"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/T12303","display_name":"Tensor decomposition and applications","score":0.9995999932289124,"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/T11321","display_name":"Error Correcting Code Techniques","score":0.9965000152587891,"subfield":{"id":"https://openalex.org/subfields/1705","display_name":"Computer Networks and Communications"},"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":0.8123952150344849},{"id":"https://openalex.org/keywords/row","display_name":"Row","score":0.7156037092208862},{"id":"https://openalex.org/keywords/matrix-norm","display_name":"Matrix norm","score":0.6460450887680054},{"id":"https://openalex.org/keywords/matrix","display_name":"Matrix (chemical analysis)","score":0.636114239692688},{"id":"https://openalex.org/keywords/inverse","display_name":"Inverse","score":0.5808687210083008},{"id":"https://openalex.org/keywords/row-and-column-spaces","display_name":"Row and column spaces","score":0.5757175087928772},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.5742177963256836},{"id":"https://openalex.org/keywords/singular-value-decomposition","display_name":"Singular value decomposition","score":0.5601645112037659},{"id":"https://openalex.org/keywords/rank","display_name":"Rank (graph theory)","score":0.5415788292884827},{"id":"https://openalex.org/keywords/intersection","display_name":"Intersection (aeronautics)","score":0.4500957131385803},{"id":"https://openalex.org/keywords/singular-value","display_name":"Singular value","score":0.4267566502094269},{"id":"https://openalex.org/keywords/linear-algebra","display_name":"Linear algebra","score":0.4136436879634857},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.36231255531311035},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.33141466975212097},{"id":"https://openalex.org/keywords/eigenvalues-and-eigenvectors","display_name":"Eigenvalues and eigenvectors","score":0.15865889191627502},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.10967102646827698}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8123952150344849},{"id":"https://openalex.org/C135598885","wikidata":"https://www.wikidata.org/wiki/Q1366302","display_name":"Row","level":2,"score":0.7156037092208862},{"id":"https://openalex.org/C92207270","wikidata":"https://www.wikidata.org/wiki/Q939253","display_name":"Matrix norm","level":3,"score":0.6460450887680054},{"id":"https://openalex.org/C106487976","wikidata":"https://www.wikidata.org/wiki/Q685816","display_name":"Matrix (chemical analysis)","level":2,"score":0.636114239692688},{"id":"https://openalex.org/C207467116","wikidata":"https://www.wikidata.org/wiki/Q4385666","display_name":"Inverse","level":2,"score":0.5808687210083008},{"id":"https://openalex.org/C104140500","wikidata":"https://www.wikidata.org/wiki/Q2088159","display_name":"Row and column spaces","level":3,"score":0.5757175087928772},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.5742177963256836},{"id":"https://openalex.org/C22789450","wikidata":"https://www.wikidata.org/wiki/Q420904","display_name":"Singular value decomposition","level":2,"score":0.5601645112037659},{"id":"https://openalex.org/C164226766","wikidata":"https://www.wikidata.org/wiki/Q7293202","display_name":"Rank (graph theory)","level":2,"score":0.5415788292884827},{"id":"https://openalex.org/C64543145","wikidata":"https://www.wikidata.org/wiki/Q162942","display_name":"Intersection (aeronautics)","level":2,"score":0.4500957131385803},{"id":"https://openalex.org/C109282560","wikidata":"https://www.wikidata.org/wiki/Q4166054","display_name":"Singular value","level":3,"score":0.4267566502094269},{"id":"https://openalex.org/C139352143","wikidata":"https://www.wikidata.org/wiki/Q82571","display_name":"Linear algebra","level":2,"score":0.4136436879634857},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.36231255531311035},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.33141466975212097},{"id":"https://openalex.org/C158693339","wikidata":"https://www.wikidata.org/wiki/Q190524","display_name":"Eigenvalues and eigenvectors","level":2,"score":0.15865889191627502},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.10967102646827698},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","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/C77088390","wikidata":"https://www.wikidata.org/wiki/Q8513","display_name":"Database","level":1,"score":0.0},{"id":"https://openalex.org/C146978453","wikidata":"https://www.wikidata.org/wiki/Q3798668","display_name":"Aerospace engineering","level":1,"score":0.0},{"id":"https://openalex.org/C127413603","wikidata":"https://www.wikidata.org/wiki/Q11023","display_name":"Engineering","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/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"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":2,"locations":[{"id":"doi:10.1137/07070471x","is_oa":false,"landing_page_url":"https://doi.org/10.1137/07070471x","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:10.1.1.144.3011","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.144.3011","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://cs-www.cs.yale.edu/homes/mmahoney/pubs/lowrank_relerr_SIMAX.pdf","raw_type":"text"}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":48,"referenced_works":["https://openalex.org/W658559791","https://openalex.org/W1818277846","https://openalex.org/W1967134148","https://openalex.org/W1969720662","https://openalex.org/W1970576574","https://openalex.org/W1972547249","https://openalex.org/W1979750072","https://openalex.org/W1981313592","https://openalex.org/W1989626455","https://openalex.org/W1990044052","https://openalex.org/W2003690406","https://openalex.org/W2008751576","https://openalex.org/W2009193081","https://openalex.org/W2016182223","https://openalex.org/W2019432911","https://openalex.org/W2035885121","https://openalex.org/W2040884411","https://openalex.org/W2042465463","https://openalex.org/W2043804332","https://openalex.org/W2045390367","https://openalex.org/W2055576996","https://openalex.org/W2065060195","https://openalex.org/W2076204387","https://openalex.org/W2078465266","https://openalex.org/W2103318769","https://openalex.org/W2107866839","https://openalex.org/W2114220113","https://openalex.org/W2116810533","https://openalex.org/W2117354486","https://openalex.org/W2118965172","https://openalex.org/W2119233169","https://openalex.org/W2128858663","https://openalex.org/W2132582398","https://openalex.org/W2147927736","https://openalex.org/W2150102617","https://openalex.org/W2152356156","https://openalex.org/W2152402969","https://openalex.org/W2153934661","https://openalex.org/W2156722294","https://openalex.org/W2160840682","https://openalex.org/W2185319426","https://openalex.org/W2217809488","https://openalex.org/W2610857016","https://openalex.org/W2798707604","https://openalex.org/W2798909945","https://openalex.org/W3046452920","https://openalex.org/W4231187174","https://openalex.org/W4285719527"],"related_works":["https://openalex.org/W2313440505","https://openalex.org/W3123744736","https://openalex.org/W1505848319","https://openalex.org/W2137691148","https://openalex.org/W4281399881","https://openalex.org/W2905020035","https://openalex.org/W3166006430","https://openalex.org/W4281971614","https://openalex.org/W2049180840","https://openalex.org/W3121299875"],"abstract_inverted_index":{"Many":[0],"data":[1,74],"analysis":[2],"applications":[3],"deal":[4],"with":[5,183],"large":[6],"matrices":[7,157],"and":[8,28,33,54,76,106,118,146,154,163,168,217,234,245,253],"involve":[9],"approximating":[10],"the":[11,26,31,42,46,73,85,131,205,213,221,225,248,254],"matrix":[12,56,104,226],"using":[13],"a":[14,65,107,171,239],"small":[15,66],"number":[16,67,229],"of":[17,25,30,41,45,64,68,72,84,135,160,166,174,215,224,230,232,236],"\u201ccomponents.\u201d":[18],"Typically,":[19],"these":[20],"components":[21],"are":[22,34,59,92,144,156],"linear":[23],"combinations":[24],"rows":[27,71,235],"columns":[29,69,162,231],"matrix,":[32,75],"thus":[35],"difficult":[36],"to":[37,80],"interpret":[38],"in":[39,62,82,242],"terms":[40,63,83],"original":[43,86],"features":[44],"input":[47,99],"data.":[48,87],"In":[49,111,137],"this":[50],"paper,":[51],"we":[52,119,147,180],"propose":[53],"study":[55],"approximations":[57],"that":[58,158,182],"explicitly":[60],"expressed":[61],"and/or":[70],"thereby":[77],"more":[78],"amenable":[79],"interpretation":[81],"Our":[88],"main":[89],"algorithmic":[90],"results":[91],"two":[93],"randomized":[94],"algorithms":[95],"which":[96],"take":[97],"as":[98],"an":[100],"m":[101],"\u00d7":[102],"n":[103],"A":[105,121,149,192],"rank":[108],"parameter":[109],"k.":[110],"our":[112,138],"first":[113],"algorithm,":[114,179],"C":[115,128,233],"is":[116,130,170,204,220,238],"chosen,":[117,145],"let":[120,148],"\u2032":[122,150,193],"=":[123,151],"CC":[124],"+":[125,129,197],"A,":[126,167,216],"where":[127,202,218],"Moore\u2013Penrose":[132],"generalized":[133,172],"inverse":[134,173],"C.":[136],"second":[139],"algorithm":[140],"C,":[141],"U,":[142],"R":[143,155,237],"CUR.":[152],"(C":[153],"consist":[159],"actual":[161],"rows,":[164],"respectively,":[165],"U":[169],"their":[175],"intersection.)":[176],"For":[177],"each":[178],"show":[181],"probability":[184],"at":[185],"least":[186],"1":[187],"\u2212":[188,191,200],"\u03b4,":[189],"\u2016A":[190,199],"\u2016F":[194],"\u2264":[195],"(1":[196],"\u025b)":[198],"Ak\u2016F,":[201],"Ak":[203],"\u201cbest":[206],"\u201d":[207],"rank-k":[208],"approximation":[209],"provided":[210],"by":[211],"truncating":[212],"SVD":[214],"\u2016X\u2016F":[219],"Frobenius":[222],"norm":[223],"X.":[227],"The":[228],"low-degree":[240],"polynomial":[241],"k,":[243],"1/\u025b,":[244],"log(1/\u03b4).":[246],"Both":[247],"Numerical":[249],"Linear":[250],"Algebra":[251],"community":[252,258],"Theoretical":[255],"Computer":[256],"Science":[257],"have":[259],"studied":[260],"variants":[261]},"counts_by_year":[{"year":2026,"cited_by_count":7},{"year":2025,"cited_by_count":25},{"year":2024,"cited_by_count":35},{"year":2023,"cited_by_count":30},{"year":2022,"cited_by_count":22},{"year":2021,"cited_by_count":32},{"year":2020,"cited_by_count":40},{"year":2019,"cited_by_count":28},{"year":2018,"cited_by_count":31},{"year":2017,"cited_by_count":36},{"year":2016,"cited_by_count":44},{"year":2015,"cited_by_count":40},{"year":2014,"cited_by_count":27},{"year":2013,"cited_by_count":19},{"year":2012,"cited_by_count":15}],"updated_date":"2025-11-06T03:46:38.306776","created_date":"2025-10-10T00:00:00"}
