{"id":"https://openalex.org/W2963227338","doi":"https://doi.org/10.1137/15m1014425","title":"Convergence Properties of the Randomized Extended Gauss--Seidel and Kaczmarz Methods","display_name":"Convergence Properties of the Randomized Extended Gauss--Seidel and Kaczmarz Methods","publication_year":2015,"publication_date":"2015-01-01","ids":{"openalex":"https://openalex.org/W2963227338","doi":"https://doi.org/10.1137/15m1014425","mag":"2963227338"},"language":"en","primary_location":{"id":"doi:10.1137/15m1014425","is_oa":false,"landing_page_url":"https://doi.org/10.1137/15m1014425","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/A5103201778","display_name":"Anna Ma","orcid":"https://orcid.org/0000-0002-0099-053X"},"institutions":[{"id":"https://openalex.org/I136722135","display_name":"Claremont Graduate University","ror":"https://ror.org/0157pnt69","country_code":"US","type":"education","lineage":["https://openalex.org/I136722135"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Anna Ma","raw_affiliation_strings":["Claremont Graduate University;"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Claremont Graduate University;","institution_ids":["https://openalex.org/I136722135"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5024153474","display_name":"Deanna Needell","orcid":"https://orcid.org/0000-0002-8058-8638"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Deanna Needell","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"last","author":{"id":"https://openalex.org/A5032389695","display_name":"Aaditya Ramdas","orcid":"https://orcid.org/0000-0003-0497-311X"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Aaditya Ramdas","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":13.7563,"has_fulltext":false,"cited_by_count":220,"citation_normalized_percentile":{"value":0.98748805,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":94,"max":100},"biblio":{"volume":"36","issue":"4","first_page":"1590","last_page":"1604"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11612","display_name":"Stochastic Gradient Optimization Techniques","score":0.9998999834060669,"subfield":{"id":"https://openalex.org/subfields/1702","display_name":"Artificial Intelligence"},"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/T11612","display_name":"Stochastic Gradient Optimization Techniques","score":0.9998999834060669,"subfield":{"id":"https://openalex.org/subfields/1702","display_name":"Artificial Intelligence"},"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":0.9980999827384949,"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/T12056","display_name":"Markov Chains and Monte Carlo Methods","score":0.9979000091552734,"subfield":{"id":"https://openalex.org/subfields/2613","display_name":"Statistics and Probability"},"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/mathematics","display_name":"Mathematics","score":0.8508559465408325},{"id":"https://openalex.org/keywords/overdetermined-system","display_name":"Overdetermined system","score":0.8419222831726074},{"id":"https://openalex.org/keywords/underdetermined-system","display_name":"Underdetermined system","score":0.7411776185035706},{"id":"https://openalex.org/keywords/gauss\u2013seidel-method","display_name":"Gauss\u2013Seidel method","score":0.7203412055969238},{"id":"https://openalex.org/keywords/convergence","display_name":"Convergence (economics)","score":0.5762364864349365},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.5011250972747803},{"id":"https://openalex.org/keywords/norm","display_name":"Norm (philosophy)","score":0.4287121593952179},{"id":"https://openalex.org/keywords/iterative-method","display_name":"Iterative method","score":0.400614470243454},{"id":"https://openalex.org/keywords/algebra-over-a-field","display_name":"Algebra over a field","score":0.3600883483886719},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.30754464864730835},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.26816028356552124}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8508559465408325},{"id":"https://openalex.org/C81901731","wikidata":"https://www.wikidata.org/wiki/Q331348","display_name":"Overdetermined system","level":2,"score":0.8419222831726074},{"id":"https://openalex.org/C179690561","wikidata":"https://www.wikidata.org/wiki/Q4316110","display_name":"Underdetermined system","level":2,"score":0.7411776185035706},{"id":"https://openalex.org/C3828260","wikidata":"https://www.wikidata.org/wiki/Q1069090","display_name":"Gauss\u2013Seidel method","level":3,"score":0.7203412055969238},{"id":"https://openalex.org/C2777303404","wikidata":"https://www.wikidata.org/wiki/Q759757","display_name":"Convergence (economics)","level":2,"score":0.5762364864349365},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.5011250972747803},{"id":"https://openalex.org/C191795146","wikidata":"https://www.wikidata.org/wiki/Q3878446","display_name":"Norm (philosophy)","level":2,"score":0.4287121593952179},{"id":"https://openalex.org/C159694833","wikidata":"https://www.wikidata.org/wiki/Q2321565","display_name":"Iterative method","level":2,"score":0.400614470243454},{"id":"https://openalex.org/C136119220","wikidata":"https://www.wikidata.org/wiki/Q1000660","display_name":"Algebra over a field","level":2,"score":0.3600883483886719},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.30754464864730835},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.26816028356552124},{"id":"https://openalex.org/C162324750","wikidata":"https://www.wikidata.org/wiki/Q8134","display_name":"Economics","level":0,"score":0.0},{"id":"https://openalex.org/C50522688","wikidata":"https://www.wikidata.org/wiki/Q189833","display_name":"Economic growth","level":1,"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/C199539241","wikidata":"https://www.wikidata.org/wiki/Q7748","display_name":"Law","level":1,"score":0.0}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.1137/15m1014425","is_oa":false,"landing_page_url":"https://doi.org/10.1137/15m1014425","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:scholarship.claremont.edu:cmc_fac_pub-1426","is_oa":false,"landing_page_url":"https://scholarship.claremont.edu/cmc_fac_pub/420","pdf_url":null,"source":{"id":"https://openalex.org/S4306401604","display_name":"Scholarship @ Claremont (The Claremont Colleges)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I872248612","host_organization_name":"Claremont Colleges","host_organization_lineage":["https://openalex.org/I872248612"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"CMC Faculty Publications and Research","raw_type":"text"}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":24,"referenced_works":["https://openalex.org/W1530115438","https://openalex.org/W1976661283","https://openalex.org/W1977769089","https://openalex.org/W1988844573","https://openalex.org/W1990919278","https://openalex.org/W2014513111","https://openalex.org/W2032357669","https://openalex.org/W2033242948","https://openalex.org/W2036835849","https://openalex.org/W2043324428","https://openalex.org/W2048305372","https://openalex.org/W2059044337","https://openalex.org/W2066157585","https://openalex.org/W2075350383","https://openalex.org/W2091826725","https://openalex.org/W2095984592","https://openalex.org/W2117686388","https://openalex.org/W2121485651","https://openalex.org/W2135294617","https://openalex.org/W2144932012","https://openalex.org/W2167820643","https://openalex.org/W2949745249","https://openalex.org/W3102157992","https://openalex.org/W3126008680"],"related_works":["https://openalex.org/W2041382836","https://openalex.org/W138713320","https://openalex.org/W4382519919","https://openalex.org/W3085447172","https://openalex.org/W3012385847","https://openalex.org/W4362680359","https://openalex.org/W2381883028","https://openalex.org/W2537671543","https://openalex.org/W67695323","https://openalex.org/W206356891"],"abstract_inverted_index":{"The":[0],"Kaczmarz":[1,38],"and":[2,32,48,76,98,141,149,163],"Gauss--Seidel":[3,58,118,134],"methods":[4,23,65,148,168],"both":[5,64,90,102,147,167],"solve":[6],"a":[7,28,52,86,110,161],"linear":[8],"system":[9,71,156],"$\\boldsymbol{X}{\\boldsymbol{\\beta}}":[10],"=":[11],"\\boldsymbol{y}$":[12],"by":[13,27],"iteratively":[14],"refining":[15],"the":[16,36,45,56,61,70,122,127,131,143],"solution":[17,125],"estimate.":[18],"Recent":[19],"interest":[20],"in":[21,42,126,153],"these":[22,95],"has":[24],"been":[25],"sparked":[26],"proof":[29,111],"of":[30,63,89,116,146,166],"Strohmer":[31],"Vershynin":[33],"which":[34],"shows":[35],"randomized":[37,57,117,133],"method":[39],"converges":[40,119],"linearly":[41,120],"expectation":[43],"to":[44,121,136],"solution.":[46],"Lewis":[47],"Leventhal":[49],"then":[50],"proved":[51],"similar":[53],"result":[54],"for":[55,94],"algorithm.":[59],"However,":[60],"behavior":[62],"depends":[66],"heavily":[67],"on":[68],"whether":[69,77],"is":[72,79,169],"underdetermined":[73,128],"or":[74,81],"overdetermined,":[75],"it":[78],"consistent":[80],"not.":[82],"Here":[83],"we":[84,107],"provide":[85,109],"unified":[87],"theory":[88,165],"methods,":[91],"their":[92,150],"variants":[93,152],"different":[96],"settings,":[97],"draw":[99],"connections":[100],"between":[101],"approaches.":[103],"In":[104],"doing":[105],"so,":[106],"also":[108],"that":[112],"an":[113],"extended":[114,151],"version":[115],"least":[123],"norm":[124],"case":[129],"(where":[130],"usual":[132],"fails":[135],"converge).":[137],"We":[138],"detail":[139],"analytically":[140],"empirically":[142],"convergence":[144],"properties":[145],"all":[154],"possible":[155],"settings.":[157],"With":[158],"this":[159],"result,":[160],"complete":[162],"rigorous":[164],"furnished.":[170]},"counts_by_year":[{"year":2026,"cited_by_count":12},{"year":2025,"cited_by_count":19},{"year":2024,"cited_by_count":32},{"year":2023,"cited_by_count":24},{"year":2022,"cited_by_count":22},{"year":2021,"cited_by_count":42},{"year":2020,"cited_by_count":29},{"year":2019,"cited_by_count":13},{"year":2018,"cited_by_count":11},{"year":2017,"cited_by_count":10},{"year":2016,"cited_by_count":4},{"year":2015,"cited_by_count":2}],"updated_date":"2026-08-06T08:24:18.245995","created_date":"2025-10-10T00:00:00"}