{"id":"https://openalex.org/W3027452332","doi":"https://doi.org/10.1007/s10208-021-09525-9","title":"On Randomized Trace Estimates for Indefinite Matrices with an Application to Determinants","display_name":"On Randomized Trace Estimates for Indefinite Matrices with an Application to Determinants","publication_year":2021,"publication_date":"2021-07-09","ids":{"openalex":"https://openalex.org/W3027452332","doi":"https://doi.org/10.1007/s10208-021-09525-9","mag":"3027452332"},"language":"en","primary_location":{"id":"doi:10.1007/s10208-021-09525-9","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s10208-021-09525-9","pdf_url":"https://link.springer.com/content/pdf/10.1007/s10208-021-09525-9.pdf","source":{"id":"https://openalex.org/S151639445","display_name":"Foundations of Computational Mathematics","issn_l":"1615-3375","issn":["1615-3375","1615-3383"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319900","host_organization_name":"Springer Science+Business Media","host_organization_lineage":["https://openalex.org/P4310319900","https://openalex.org/P4310319965"],"host_organization_lineage_names":["Springer Science+Business Media","Springer Nature"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Foundations of Computational Mathematics","raw_type":"journal-article"},"type":"preprint","indexed_in":["arxiv","crossref","datacite"],"open_access":{"is_oa":true,"oa_status":"hybrid","oa_url":"https://link.springer.com/content/pdf/10.1007/s10208-021-09525-9.pdf","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5080876002","display_name":"Alice Cortinovis","orcid":"https://orcid.org/0000-0001-6917-5106"},"institutions":[{"id":"https://openalex.org/I5124864","display_name":"\u00c9cole Polytechnique F\u00e9d\u00e9rale de Lausanne","ror":"https://ror.org/02s376052","country_code":"CH","type":"education","lineage":["https://openalex.org/I2799323385","https://openalex.org/I5124864"]}],"countries":["CH"],"is_corresponding":true,"raw_author_name":"Alice Cortinovis","raw_affiliation_strings":["Institute of Mathematics, EPF Lausanne, 1015, Lausanne, Switzerland","\u00c9cole Polytechnique F\u00e9d\u00e9rale de Lausanne,"],"affiliations":[{"raw_affiliation_string":"Institute of Mathematics, EPF Lausanne, 1015, Lausanne, Switzerland","institution_ids":["https://openalex.org/I5124864"]},{"raw_affiliation_string":"\u00c9cole Polytechnique F\u00e9d\u00e9rale de Lausanne,","institution_ids":["https://openalex.org/I5124864"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5084601962","display_name":"Daniel Kre\u00dfner","orcid":"https://orcid.org/0000-0003-3369-2958"},"institutions":[{"id":"https://openalex.org/I5124864","display_name":"\u00c9cole Polytechnique F\u00e9d\u00e9rale de Lausanne","ror":"https://ror.org/02s376052","country_code":"CH","type":"education","lineage":["https://openalex.org/I2799323385","https://openalex.org/I5124864"]}],"countries":["CH"],"is_corresponding":false,"raw_author_name":"Daniel Kressner","raw_affiliation_strings":["Institute of Mathematics, EPF Lausanne, 1015, Lausanne, Switzerland","\u00c9cole Polytechnique F\u00e9d\u00e9rale de Lausanne,"],"affiliations":[{"raw_affiliation_string":"Institute of Mathematics, EPF Lausanne, 1015, Lausanne, Switzerland","institution_ids":["https://openalex.org/I5124864"]},{"raw_affiliation_string":"\u00c9cole Polytechnique F\u00e9d\u00e9rale de Lausanne,","institution_ids":["https://openalex.org/I5124864"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":2,"corresponding_author_ids":["https://openalex.org/A5080876002"],"corresponding_institution_ids":["https://openalex.org/I5124864"],"apc_list":{"value":2390,"currency":"EUR","value_usd":2990},"apc_paid":{"value":2390,"currency":"EUR","value_usd":2990},"fwci":2.1295,"has_fulltext":true,"cited_by_count":9,"citation_normalized_percentile":{"value":0.88114635,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":90,"max":98},"biblio":{"volume":"22","issue":"3","first_page":"875","last_page":"903"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T12056","display_name":"Markov Chains and Monte Carlo Methods","score":0.9984999895095825,"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/T12056","display_name":"Markov Chains and Monte Carlo Methods","score":0.9984999895095825,"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":0.9983000159263611,"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/T12072","display_name":"Machine Learning and Algorithms","score":0.9973000288009644,"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/trace","display_name":"TRACE (psycholinguistics)","score":0.842248797416687},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.6324378252029419},{"id":"https://openalex.org/keywords/matrix","display_name":"Matrix (chemical analysis)","score":0.6081457734107971},{"id":"https://openalex.org/keywords/multivariate-random-variable","display_name":"Multivariate random variable","score":0.5837971568107605},{"id":"https://openalex.org/keywords/randomized-algorithm","display_name":"Randomized algorithm","score":0.5576032400131226},{"id":"https://openalex.org/keywords/random-matrix","display_name":"Random matrix","score":0.5265552997589111},{"id":"https://openalex.org/keywords/logarithm","display_name":"Logarithm","score":0.5011184215545654},{"id":"https://openalex.org/keywords/positive-definite-matrix","display_name":"Positive-definite matrix","score":0.4723801612854004},{"id":"https://openalex.org/keywords/gaussian","display_name":"Gaussian","score":0.46889060735702515},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.3672328293323517},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.3514021337032318},{"id":"https://openalex.org/keywords/statistics","display_name":"Statistics","score":0.3224080801010132},{"id":"https://openalex.org/keywords/random-variable","display_name":"Random variable","score":0.23603367805480957},{"id":"https://openalex.org/keywords/eigenvalues-and-eigenvectors","display_name":"Eigenvalues and eigenvectors","score":0.12048879265785217},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.0968160629272461}],"concepts":[{"id":"https://openalex.org/C75291252","wikidata":"https://www.wikidata.org/wiki/Q1315756","display_name":"TRACE (psycholinguistics)","level":2,"score":0.842248797416687},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6324378252029419},{"id":"https://openalex.org/C106487976","wikidata":"https://www.wikidata.org/wiki/Q685816","display_name":"Matrix (chemical analysis)","level":2,"score":0.6081457734107971},{"id":"https://openalex.org/C138405894","wikidata":"https://www.wikidata.org/wiki/Q3179949","display_name":"Multivariate random variable","level":3,"score":0.5837971568107605},{"id":"https://openalex.org/C128669082","wikidata":"https://www.wikidata.org/wiki/Q583461","display_name":"Randomized algorithm","level":2,"score":0.5576032400131226},{"id":"https://openalex.org/C64812099","wikidata":"https://www.wikidata.org/wiki/Q176604","display_name":"Random matrix","level":3,"score":0.5265552997589111},{"id":"https://openalex.org/C39927690","wikidata":"https://www.wikidata.org/wiki/Q11197","display_name":"Logarithm","level":2,"score":0.5011184215545654},{"id":"https://openalex.org/C49712288","wikidata":"https://www.wikidata.org/wiki/Q77601250","display_name":"Positive-definite matrix","level":3,"score":0.4723801612854004},{"id":"https://openalex.org/C163716315","wikidata":"https://www.wikidata.org/wiki/Q901177","display_name":"Gaussian","level":2,"score":0.46889060735702515},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.3672328293323517},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.3514021337032318},{"id":"https://openalex.org/C105795698","wikidata":"https://www.wikidata.org/wiki/Q12483","display_name":"Statistics","level":1,"score":0.3224080801010132},{"id":"https://openalex.org/C122123141","wikidata":"https://www.wikidata.org/wiki/Q176623","display_name":"Random variable","level":2,"score":0.23603367805480957},{"id":"https://openalex.org/C158693339","wikidata":"https://www.wikidata.org/wiki/Q190524","display_name":"Eigenvalues and eigenvectors","level":2,"score":0.12048879265785217},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.0968160629272461},{"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/C192562407","wikidata":"https://www.wikidata.org/wiki/Q228736","display_name":"Materials science","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/C138885662","wikidata":"https://www.wikidata.org/wiki/Q5891","display_name":"Philosophy","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}],"mesh":[],"locations_count":5,"locations":[{"id":"doi:10.1007/s10208-021-09525-9","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s10208-021-09525-9","pdf_url":"https://link.springer.com/content/pdf/10.1007/s10208-021-09525-9.pdf","source":{"id":"https://openalex.org/S151639445","display_name":"Foundations of Computational Mathematics","issn_l":"1615-3375","issn":["1615-3375","1615-3383"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319900","host_organization_name":"Springer Science+Business Media","host_organization_lineage":["https://openalex.org/P4310319900","https://openalex.org/P4310319965"],"host_organization_lineage_names":["Springer Science+Business Media","Springer Nature"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Foundations of Computational Mathematics","raw_type":"journal-article"},{"id":"pmh:oai:arXiv.org:2005.10009","is_oa":true,"landing_page_url":"http://arxiv.org/abs/2005.10009","pdf_url":"https://arxiv.org/pdf/2005.10009","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":"pmh:oai:infoscience.epfl.ch:287507","is_oa":true,"landing_page_url":"http://infoscience.epfl.ch/record/287507","pdf_url":"https://infoscience.epfl.ch/record/287507/files/s10208-021-09525-9.pdf","source":{"id":"https://openalex.org/S4306400487","display_name":"Infoscience (Ecole Polytechnique F\u00e9d\u00e9rale de Lausanne)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by-nc-nd","license_id":"https://openalex.org/licenses/cc-by-nc-nd","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"","raw_type":"research article"},{"id":"mag:3027452332","is_oa":true,"landing_page_url":"http://export.arxiv.org/pdf/2005.10009","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.2005.10009","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2005.10009","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":"doi:10.1007/s10208-021-09525-9","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s10208-021-09525-9","pdf_url":"https://link.springer.com/content/pdf/10.1007/s10208-021-09525-9.pdf","source":{"id":"https://openalex.org/S151639445","display_name":"Foundations of Computational Mathematics","issn_l":"1615-3375","issn":["1615-3375","1615-3383"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319900","host_organization_name":"Springer Science+Business Media","host_organization_lineage":["https://openalex.org/P4310319900","https://openalex.org/P4310319965"],"host_organization_lineage_names":["Springer Science+Business Media","Springer Nature"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Foundations of Computational Mathematics","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[],"funders":[{"id":"https://openalex.org/F4320320915","display_name":"\u00c9cole Polytechnique F\u00e9d\u00e9rale de Lausanne","ror":"https://ror.org/02s376052"},{"id":"https://openalex.org/F4320320924","display_name":"Schweizerischer Nationalfonds zur F\u00f6rderung der Wissenschaftlichen Forschung","ror":"https://ror.org/00yjd3n13"}],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W3027452332.pdf","grobid_xml":"https://content.openalex.org/works/W3027452332.grobid-xml"},"referenced_works_count":45,"referenced_works":["https://openalex.org/W143004564","https://openalex.org/W603018635","https://openalex.org/W1491884104","https://openalex.org/W1510593638","https://openalex.org/W1560153690","https://openalex.org/W1561236277","https://openalex.org/W1570557693","https://openalex.org/W1820075673","https://openalex.org/W1964718108","https://openalex.org/W1990457366","https://openalex.org/W1991085742","https://openalex.org/W2007755560","https://openalex.org/W2008389020","https://openalex.org/W2012501405","https://openalex.org/W2035080386","https://openalex.org/W2057450058","https://openalex.org/W2072864444","https://openalex.org/W2080872718","https://openalex.org/W2081060015","https://openalex.org/W2089113956","https://openalex.org/W2102802796","https://openalex.org/W2130152237","https://openalex.org/W2135912864","https://openalex.org/W2342372809","https://openalex.org/W2547245280","https://openalex.org/W2607802485","https://openalex.org/W2750001921","https://openalex.org/W2761580201","https://openalex.org/W2764280570","https://openalex.org/W2768220934","https://openalex.org/W2804648164","https://openalex.org/W2893995718","https://openalex.org/W2952402828","https://openalex.org/W2963190151","https://openalex.org/W2963262327","https://openalex.org/W2963586518","https://openalex.org/W2964350355","https://openalex.org/W3005345825","https://openalex.org/W3011465071","https://openalex.org/W3105113144","https://openalex.org/W3138124916","https://openalex.org/W3141126219","https://openalex.org/W4244155866","https://openalex.org/W4245577611","https://openalex.org/W6618416959"],"related_works":["https://openalex.org/W3105113144","https://openalex.org/W1990457366","https://openalex.org/W3016589770","https://openalex.org/W2057450058","https://openalex.org/W2398301322","https://openalex.org/W3141126219","https://openalex.org/W2764280570","https://openalex.org/W2547245280","https://openalex.org/W1983656984","https://openalex.org/W2218248567","https://openalex.org/W2522906795","https://openalex.org/W1824950334","https://openalex.org/W2073393212","https://openalex.org/W2601695767","https://openalex.org/W2129566246","https://openalex.org/W2951019947","https://openalex.org/W2161687405","https://openalex.org/W2771719469","https://openalex.org/W2914588611","https://openalex.org/W1968135884"],"abstract_inverted_index":{"Abstract":[0],"Randomized":[1],"trace":[2,13,86,108,182,191],"estimation":[3,77],"is":[4,39,58,145,199,206],"a":[5,15,32,45,64,137],"popular":[6],"and":[7,164,213],"well-studied":[8],"technique":[9],"that":[10,66],"approximates":[11],"the":[12,21,59,95,131,146,178,185,190,202],"of":[14,23,31,47,84,133,148,180,192],"large-scale":[16],"matrix":[17,203],"B":[18,38,53,113,128,149,195],"by":[19,136,162],"computing":[20],"average":[22],"$$x^T":[24],"Bx$$":[25],"<mml:math":[26],"xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"><mml:mrow><mml:msup><mml:mi>x</mml:mi><mml:mi>T</mml:mi></mml:msup><mml:mi>B</mml:mi><mml:mi>x</mml:mi></mml:mrow></mml:math>":[27],"for":[28,61,72,78,106,126,152,171,188,208],"many":[29],"samples":[30,135],"random":[33,118,226],"vector":[34],"X":[35],".":[36,54,150],"Often,":[37],"symmetric":[40],"positive":[41],"definite":[42],"(SPD)":[43],"but":[44,223],"number":[46,132],"applications":[48],"give":[49],"rise":[50],"to":[51,111,201,218],"indefinite":[52,112,127],"Most":[55],"notably,":[56],"this":[57,99],"case":[60],"log-determinant":[62,209],"estimation,":[63],"task":[65],"features":[67],"prominently":[68],"in":[69,74],"statistical":[70],"learning,":[71],"instance":[73],"maximum":[75],"likelihood":[76],"Gaussian":[79,117,225],"process":[80],"regression.":[81],"The":[82],"analysis":[83],"randomized":[85,107,181],"estimates,":[87],"including":[88],"tail":[89,104],"bounds,":[90],"has":[91],"mostly":[92],"focused":[93],"on":[94],"SPD":[96,154],"case.":[97],"In":[98],"work,":[100],"we":[101],"derive":[102],"new":[103],"bounds":[105,121],"estimates":[109,183],"applied":[110],"with":[114,184],"Rademacher":[115,172,222],"or":[116,140],"vectors.":[119,173,227],"These":[120],"significantly":[122],"improve":[123,212],"existing":[124,160,216],"results":[125],",":[129],"reducing":[130],"required":[134],"factor":[138],"n":[139,144],"even":[141],"more,":[142],"where":[143],"size":[147],"Even":[151],"an":[153,159,215],"matrix,":[155],"our":[156],"work":[157,175],"improves":[158],"result":[161],"Roosta-Khorasani":[163],"Ascher":[165],"(Found":[166],"Comput":[167],"Math,":[168],"15(5):1187\u20131212,":[169],"2015)":[170],"This":[174],"also":[176,224],"analyzes":[177],"combination":[179],"Lanczos":[186],"method":[187],"approximating":[189],"f":[193],"(":[194],").":[196],"Particular":[197],"attention":[198],"paid":[200],"logarithm,":[204],"which":[205],"needed":[207],"estimation.":[210],"We":[211],"extend":[214],"result,":[217],"not":[219],"only":[220],"cover":[221]},"counts_by_year":[{"year":2024,"cited_by_count":1},{"year":2022,"cited_by_count":2},{"year":2021,"cited_by_count":6}],"updated_date":"2026-03-20T23:20:44.827607","created_date":"2025-10-10T00:00:00"}