{"id":"https://openalex.org/W2057450058","doi":"https://doi.org/10.1145/1944345.1944349","title":"Randomized algorithms for estimating the trace of an implicit symmetric positive semi-definite matrix","display_name":"Randomized algorithms for estimating the trace of an implicit symmetric positive semi-definite matrix","publication_year":2011,"publication_date":"2011-04-01","ids":{"openalex":"https://openalex.org/W2057450058","doi":"https://doi.org/10.1145/1944345.1944349","mag":"2057450058"},"language":"en","primary_location":{"id":"doi:10.1145/1944345.1944349","is_oa":false,"landing_page_url":"https://doi.org/10.1145/1944345.1944349","pdf_url":null,"source":{"id":"https://openalex.org/S118992489","display_name":"Journal of the ACM","issn_l":"0004-5411","issn":["0004-5411","1557-735X"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319798","host_organization_name":"Association for Computing Machinery","host_organization_lineage":["https://openalex.org/P4310319798"],"host_organization_lineage_names":["Association for Computing Machinery"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Journal of the ACM","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/A5072151198","display_name":"Haim Avron","orcid":"https://orcid.org/0000-0002-1688-9030"},"institutions":[{"id":"https://openalex.org/I1341412227","display_name":"IBM (United States)","ror":"https://ror.org/05hh8d621","country_code":"US","type":"company","lineage":["https://openalex.org/I1341412227"]},{"id":"https://openalex.org/I16391192","display_name":"Tel Aviv University","ror":"https://ror.org/04mhzgx49","country_code":"IL","type":"education","lineage":["https://openalex.org/I16391192"]}],"countries":["IL","US"],"is_corresponding":false,"raw_author_name":"Haim Avron","raw_affiliation_strings":["Tel-Aviv University, Tel-Aviv and IBM T.J. Watson Research Center, Yorktown Heights, NY","Tel-Aviv University, Tel-Aviv and IBM T.J. Watson Research Center, Yorktown Heights, NY#TAB#"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Tel-Aviv University, Tel-Aviv and IBM T.J. Watson Research Center, Yorktown Heights, NY","institution_ids":["https://openalex.org/I1341412227"]},{"raw_affiliation_string":"Tel-Aviv University, Tel-Aviv and IBM T.J. Watson Research Center, Yorktown Heights, NY#TAB#","institution_ids":["https://openalex.org/I16391192"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5056808449","display_name":"Sivan Toledo","orcid":"https://orcid.org/0000-0002-9524-7115"},"institutions":[{"id":"https://openalex.org/I16391192","display_name":"Tel Aviv University","ror":"https://ror.org/04mhzgx49","country_code":"IL","type":"education","lineage":["https://openalex.org/I16391192"]}],"countries":["IL"],"is_corresponding":false,"raw_author_name":"Sivan Toledo","raw_affiliation_strings":["Tel-Aviv University, Tel-Aviv"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Tel-Aviv University, Tel-Aviv","institution_ids":["https://openalex.org/I16391192"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":2,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":6.1449,"has_fulltext":false,"cited_by_count":280,"citation_normalized_percentile":{"value":0.96902714,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":98,"max":100},"biblio":{"volume":"58","issue":"2","first_page":"1","last_page":"34"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11716","display_name":"Random Matrices and Applications","score":0.9993000030517578,"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"}},"topics":[{"id":"https://openalex.org/T11716","display_name":"Random Matrices and Applications","score":0.9993000030517578,"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"}},{"id":"https://openalex.org/T10500","display_name":"Sparse and Compressive Sensing Techniques","score":0.9991999864578247,"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/T11612","display_name":"Stochastic Gradient Optimization Techniques","score":0.9973999857902527,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/estimator","display_name":"Estimator","score":0.8501704335212708},{"id":"https://openalex.org/keywords/trace","display_name":"TRACE (psycholinguistics)","score":0.7818889617919922},{"id":"https://openalex.org/keywords/randomness","display_name":"Randomness","score":0.6715856194496155},{"id":"https://openalex.org/keywords/rank","display_name":"Rank (graph theory)","score":0.635936975479126},{"id":"https://openalex.org/keywords/matrix","display_name":"Matrix (chemical analysis)","score":0.575652539730072},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.5462751388549805},{"id":"https://openalex.org/keywords/variance","display_name":"Variance (accounting)","score":0.5352863073348999},{"id":"https://openalex.org/keywords/positive-definite-matrix","display_name":"Positive-definite matrix","score":0.5258905291557312},{"id":"https://openalex.org/keywords/convergence","display_name":"Convergence (economics)","score":0.5257495045661926},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.4432866871356964},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.33322665095329285},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.3175045847892761},{"id":"https://openalex.org/keywords/statistics","display_name":"Statistics","score":0.302437961101532}],"concepts":[{"id":"https://openalex.org/C185429906","wikidata":"https://www.wikidata.org/wiki/Q1130160","display_name":"Estimator","level":2,"score":0.8501704335212708},{"id":"https://openalex.org/C75291252","wikidata":"https://www.wikidata.org/wiki/Q1315756","display_name":"TRACE (psycholinguistics)","level":2,"score":0.7818889617919922},{"id":"https://openalex.org/C125112378","wikidata":"https://www.wikidata.org/wiki/Q176640","display_name":"Randomness","level":2,"score":0.6715856194496155},{"id":"https://openalex.org/C164226766","wikidata":"https://www.wikidata.org/wiki/Q7293202","display_name":"Rank (graph theory)","level":2,"score":0.635936975479126},{"id":"https://openalex.org/C106487976","wikidata":"https://www.wikidata.org/wiki/Q685816","display_name":"Matrix (chemical analysis)","level":2,"score":0.575652539730072},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.5462751388549805},{"id":"https://openalex.org/C196083921","wikidata":"https://www.wikidata.org/wiki/Q7915758","display_name":"Variance (accounting)","level":2,"score":0.5352863073348999},{"id":"https://openalex.org/C49712288","wikidata":"https://www.wikidata.org/wiki/Q77601250","display_name":"Positive-definite matrix","level":3,"score":0.5258905291557312},{"id":"https://openalex.org/C2777303404","wikidata":"https://www.wikidata.org/wiki/Q759757","display_name":"Convergence (economics)","level":2,"score":0.5257495045661926},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.4432866871356964},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.33322665095329285},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.3175045847892761},{"id":"https://openalex.org/C105795698","wikidata":"https://www.wikidata.org/wiki/Q12483","display_name":"Statistics","level":1,"score":0.302437961101532},{"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/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.0},{"id":"https://openalex.org/C41895202","wikidata":"https://www.wikidata.org/wiki/Q8162","display_name":"Linguistics","level":1,"score":0.0},{"id":"https://openalex.org/C121955636","wikidata":"https://www.wikidata.org/wiki/Q4116214","display_name":"Accounting","level":1,"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/C159985019","wikidata":"https://www.wikidata.org/wiki/Q181790","display_name":"Composite material","level":1,"score":0.0},{"id":"https://openalex.org/C144133560","wikidata":"https://www.wikidata.org/wiki/Q4830453","display_name":"Business","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/C158693339","wikidata":"https://www.wikidata.org/wiki/Q190524","display_name":"Eigenvalues and eigenvectors","level":2,"score":0.0},{"id":"https://openalex.org/C138885662","wikidata":"https://www.wikidata.org/wiki/Q5891","display_name":"Philosophy","level":0,"score":0.0},{"id":"https://openalex.org/C162324750","wikidata":"https://www.wikidata.org/wiki/Q8134","display_name":"Economics","level":0,"score":0.0}],"mesh":[],"locations_count":4,"locations":[{"id":"doi:10.1145/1944345.1944349","is_oa":false,"landing_page_url":"https://doi.org/10.1145/1944345.1944349","pdf_url":null,"source":{"id":"https://openalex.org/S118992489","display_name":"Journal of the ACM","issn_l":"0004-5411","issn":["0004-5411","1557-735X"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319798","host_organization_name":"Association for Computing Machinery","host_organization_lineage":["https://openalex.org/P4310319798"],"host_organization_lineage_names":["Association for Computing Machinery"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Journal of the ACM","raw_type":"journal-article"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.163.6194","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.163.6194","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://www.cs.tau.ac.il/~haima/trace-v0.pdf","raw_type":"text"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.380.9436","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.380.9436","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://www.cs.tau.ac.il/~stoledo/Bib/Pubs/trace3.pdf","raw_type":"text"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.381.3534","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.381.3534","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://www.cs.tau.ac.il/~stoledo/Bib/Pubs/randomized-trace.pdf","raw_type":"text"}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[{"id":"https://openalex.org/G3847479357","display_name":null,"funder_award_id":"1045/09","funder_id":"https://openalex.org/F4320307762","funder_display_name":"International Business Machines Corporation"}],"funders":[{"id":"https://openalex.org/F4320307762","display_name":"International Business Machines Corporation","ror":"https://ror.org/05hh8d621"}],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":25,"referenced_works":["https://openalex.org/W1551441240","https://openalex.org/W1964233058","https://openalex.org/W1990457366","https://openalex.org/W2008776938","https://openalex.org/W2011773541","https://openalex.org/W2025614228","https://openalex.org/W2038011353","https://openalex.org/W2044880891","https://openalex.org/W2053171205","https://openalex.org/W2060090483","https://openalex.org/W2064975723","https://openalex.org/W2073737032","https://openalex.org/W2093813380","https://openalex.org/W2094557820","https://openalex.org/W2108214478","https://openalex.org/W2110556347","https://openalex.org/W2120595041","https://openalex.org/W2130152237","https://openalex.org/W2142449924","https://openalex.org/W2152402969","https://openalex.org/W2587926013","https://openalex.org/W2751862591","https://openalex.org/W3003669404","https://openalex.org/W4230818032","https://openalex.org/W4244155866"],"related_works":["https://openalex.org/W3034924094","https://openalex.org/W3094954546","https://openalex.org/W1488708774","https://openalex.org/W1982811510","https://openalex.org/W4391100477","https://openalex.org/W2402189625","https://openalex.org/W4327779705","https://openalex.org/W4310560702","https://openalex.org/W1515038401","https://openalex.org/W4294621063"],"abstract_inverted_index":{"We":[0,136,150,172],"analyze":[1,101,111],"the":[2,18,35,47,102,105,112,126,130,148,156,160,170,213],"convergence":[3],"of":[4,20,52,84,104,114],"randomized":[5],"trace":[6,19,191],"estimators.":[7,107,215],"Starting":[8],"at":[9,123,133],"1989,":[10],"several":[11],"algorithms":[12,72],"have":[13],"been":[14],"proposed":[15],"for":[16,46,187],"estimating":[17],"a":[21,184,204],"matrix":[22],"by":[23],"1/M\u03a3":[24],"i":[25,29,32,37,49,58,62,65],"=1":[26,59],"M":[27,60,116],"z":[28,36,48,61,92,96],"T":[30,63,93],"Az":[31,64,94],",":[33],"where":[34],"are":[38,73,141],"random":[39],"vectors;":[40],"different":[41,44,106],"estimators":[42,157,166],"use":[43],"distributions":[45],"s,":[50],"all":[51,212],"which":[53,78],"lead":[54],"to":[55,118,178,194],"E":[56],"(1/M\u03a3":[57],")":[66],"=":[67],"trace(":[68],"A":[69,85],").":[70],"These":[71],"useful":[74,144],"in":[75,77,129,145,197],"applications":[76,146],"there":[79],"is":[80,132,183,203],"no":[81],"explicit":[82],"representation":[83],"but":[86],"rather":[87],"an":[88],"efficient":[89],"method":[90],"compute":[91],"given":[95],".":[97],"Existing":[98],"results":[99],"only":[100],"variance":[103],"In":[108],"contrast,":[109],"we":[110],"number":[113],"samples":[115],"required":[117],"guarantee":[119],"that":[120,138,152,164,207],"with":[121],"probability":[122],"least":[124],"1-\u03b4,":[125],"relative":[127],"error":[128],"estimate":[131],"most":[134],"\u03f5.":[135],"argue":[137],"such":[139],"bounds":[140,154],"much":[142],"more":[143],"than":[147,159,211],"variance.":[149],"found":[151],"these":[153],"rank":[155],"differently":[158],"variance;":[161],"this":[162,179],"suggests":[163],"minimum-variance":[165],"may":[167],"not":[168],"be":[169,195],"best.":[171],"also":[173],"make":[174],"two":[175],"additional":[176],"contributions":[177],"area.":[180],"The":[181,201],"first":[182],"specialized":[185],"bound":[186],"projection":[188],"matrices,":[189],"whose":[190],"(rank)":[192],"needs":[193],"computed":[196],"electronic":[198],"structure":[199],"calculations.":[200],"second":[202],"new":[205],"estimator":[206],"uses":[208],"less":[209],"randomness":[210],"existing":[214]},"counts_by_year":[{"year":2026,"cited_by_count":8},{"year":2025,"cited_by_count":24},{"year":2024,"cited_by_count":26},{"year":2023,"cited_by_count":26},{"year":2022,"cited_by_count":19},{"year":2021,"cited_by_count":49},{"year":2020,"cited_by_count":32},{"year":2019,"cited_by_count":19},{"year":2018,"cited_by_count":12},{"year":2017,"cited_by_count":15},{"year":2016,"cited_by_count":13},{"year":2015,"cited_by_count":14},{"year":2014,"cited_by_count":7},{"year":2013,"cited_by_count":6},{"year":2012,"cited_by_count":6}],"updated_date":"2026-06-28T08:01:55.173337","created_date":"2025-10-10T00:00:00"}