{"id":"https://openalex.org/W3094546928","doi":"https://doi.org/10.1515/mcma-2020-2076","title":"Gaussian variant of Freivalds\u2019 algorithm for efficient and reliable matrix product verification","display_name":"Gaussian variant of Freivalds\u2019 algorithm for efficient and reliable matrix product verification","publication_year":2020,"publication_date":"2020-10-08","ids":{"openalex":"https://openalex.org/W3094546928","doi":"https://doi.org/10.1515/mcma-2020-2076","mag":"3094546928"},"language":"en","primary_location":{"id":"doi:10.1515/mcma-2020-2076","is_oa":false,"landing_page_url":"https://doi.org/10.1515/mcma-2020-2076","pdf_url":null,"source":{"id":"https://openalex.org/S142132501","display_name":"Monte Carlo Methods and Applications","issn_l":"0929-9629","issn":["0929-9629","1569-3961"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310313990","host_organization_name":"De Gruyter","host_organization_lineage":["https://openalex.org/P4310313990"],"host_organization_lineage_names":["De Gruyter"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Monte Carlo Methods 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/A5032700028","display_name":"Hao Ji","orcid":"https://orcid.org/0000-0001-8303-1491"},"institutions":[{"id":"https://openalex.org/I98947143","display_name":"California State Polytechnic University","ror":"https://ror.org/05by5hm18","country_code":"US","type":"education","lineage":["https://openalex.org/I98947143"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Hao Ji","raw_affiliation_strings":["Department of Computer Science , California State Polytechnic University Pomona , Pomona , CA 91768 , USA"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Computer Science , California State Polytechnic University Pomona , Pomona , CA 91768 , USA","institution_ids":["https://openalex.org/I98947143"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5071387306","display_name":"Michael Mascagni","orcid":"https://orcid.org/0000-0003-3058-4580"},"institutions":[{"id":"https://openalex.org/I103163165","display_name":"Florida State University","ror":"https://ror.org/05g3dte14","country_code":"US","type":"education","lineage":["https://openalex.org/I103163165"]},{"id":"https://openalex.org/I1321296531","display_name":"National Institute of Standards and Technology","ror":"https://ror.org/05xpvk416","country_code":"US","type":"funder","lineage":["https://openalex.org/I1321296531","https://openalex.org/I1343035065"]},{"id":"https://openalex.org/I4210126846","display_name":"Information Technology Laboratory","ror":"https://ror.org/0440c3437","country_code":"US","type":"government","lineage":["https://openalex.org/I1321296531","https://openalex.org/I1343035065","https://openalex.org/I4210126846"]}],"countries":["US"],"is_corresponding":true,"raw_author_name":"Michael Mascagni","raw_affiliation_strings":["Department of Computer Science , Florida State University , Tallahassee , FL 32306-4530; and Applied and Computational Mathematics Division, Information Technology Laboratory, National Institute of Standards & Technology, ITL, Gaithersburg, MD 20899-8910 , USA","Department of Computer Science, Florida State University, Tallahassee, FL 32306-4530"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Computer Science , Florida State University , Tallahassee , FL 32306-4530; and Applied and Computational Mathematics Division, Information Technology Laboratory, National Institute of Standards & Technology, ITL, Gaithersburg, MD 20899-8910 , USA","institution_ids":["https://openalex.org/I4210126846","https://openalex.org/I1321296531","https://openalex.org/I103163165"]},{"raw_affiliation_string":"Department of Computer Science, Florida State University, Tallahassee, FL 32306-4530","institution_ids":["https://openalex.org/I103163165"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5079316675","display_name":"Yaohang Li","orcid":"https://orcid.org/0000-0003-0178-1876"},"institutions":[{"id":"https://openalex.org/I81365321","display_name":"Old Dominion University","ror":"https://ror.org/04zjtrb98","country_code":"US","type":"education","lineage":["https://openalex.org/I81365321"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Yaohang Li","raw_affiliation_strings":["Department of Computer Science , Old Dominion University , Norfolk , VA 23529 , USA"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Computer Science , Old Dominion University , Norfolk , VA 23529 , USA","institution_ids":["https://openalex.org/I81365321"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":3,"corresponding_author_ids":["https://openalex.org/A5071387306"],"corresponding_institution_ids":["https://openalex.org/I103163165","https://openalex.org/I1321296531","https://openalex.org/I4210126846"],"apc_list":null,"apc_paid":null,"fwci":0.272,"has_fulltext":false,"cited_by_count":3,"citation_normalized_percentile":{"value":0.65305738,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":91,"max":96},"biblio":{"volume":"26","issue":"4","first_page":"273","last_page":"284"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11612","display_name":"Stochastic Gradient Optimization Techniques","score":0.9994000196456909,"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":0.9994000196456909,"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/T10682","display_name":"Quantum Computing Algorithms and Architecture","score":0.9977999925613403,"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/T12056","display_name":"Markov Chains and Monte Carlo Methods","score":0.9948999881744385,"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/matrix-multiplication","display_name":"Matrix multiplication","score":0.5925793051719666},{"id":"https://openalex.org/keywords/product","display_name":"Product (mathematics)","score":0.5716073513031006},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.550983726978302},{"id":"https://openalex.org/keywords/correctness","display_name":"Correctness","score":0.5264468789100647},{"id":"https://openalex.org/keywords/matrix","display_name":"Matrix (chemical analysis)","score":0.49980902671813965},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.459076851606369},{"id":"https://openalex.org/keywords/gaussian","display_name":"Gaussian","score":0.4426965117454529},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.41103190183639526},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.3711085915565491},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.19403454661369324},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.07608091831207275},{"id":"https://openalex.org/keywords/materials-science","display_name":"Materials science","score":0.05610957741737366}],"concepts":[{"id":"https://openalex.org/C17349429","wikidata":"https://www.wikidata.org/wiki/Q1049914","display_name":"Matrix multiplication","level":3,"score":0.5925793051719666},{"id":"https://openalex.org/C90673727","wikidata":"https://www.wikidata.org/wiki/Q901718","display_name":"Product (mathematics)","level":2,"score":0.5716073513031006},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.550983726978302},{"id":"https://openalex.org/C55439883","wikidata":"https://www.wikidata.org/wiki/Q360812","display_name":"Correctness","level":2,"score":0.5264468789100647},{"id":"https://openalex.org/C106487976","wikidata":"https://www.wikidata.org/wiki/Q685816","display_name":"Matrix (chemical analysis)","level":2,"score":0.49980902671813965},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.459076851606369},{"id":"https://openalex.org/C163716315","wikidata":"https://www.wikidata.org/wiki/Q901177","display_name":"Gaussian","level":2,"score":0.4426965117454529},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.41103190183639526},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.3711085915565491},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.19403454661369324},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.07608091831207275},{"id":"https://openalex.org/C192562407","wikidata":"https://www.wikidata.org/wiki/Q228736","display_name":"Materials science","level":0,"score":0.05610957741737366},{"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/C84114770","wikidata":"https://www.wikidata.org/wiki/Q46344","display_name":"Quantum","level":2,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1515/mcma-2020-2076","is_oa":false,"landing_page_url":"https://doi.org/10.1515/mcma-2020-2076","pdf_url":null,"source":{"id":"https://openalex.org/S142132501","display_name":"Monte Carlo Methods and Applications","issn_l":"0929-9629","issn":["0929-9629","1569-3961"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310313990","host_organization_name":"De Gruyter","host_organization_lineage":["https://openalex.org/P4310313990"],"host_organization_lineage_names":["De Gruyter"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Monte Carlo Methods and Applications","raw_type":"journal-article"}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[{"id":"https://openalex.org/G3327866426","display_name":null,"funder_award_id":"1066471","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":false,"pdf":false},"content_urls":null,"referenced_works_count":42,"referenced_works":["https://openalex.org/W47137029","https://openalex.org/W1549438933","https://openalex.org/W1630587103","https://openalex.org/W1690582634","https://openalex.org/W1826883764","https://openalex.org/W1970576574","https://openalex.org/W1975980877","https://openalex.org/W1980480935","https://openalex.org/W1989579242","https://openalex.org/W2003690406","https://openalex.org/W2006384005","https://openalex.org/W2009063162","https://openalex.org/W2019807548","https://openalex.org/W2028581488","https://openalex.org/W2038073775","https://openalex.org/W2040256453","https://openalex.org/W2042465463","https://openalex.org/W2055892937","https://openalex.org/W2057711597","https://openalex.org/W2061397794","https://openalex.org/W2072072075","https://openalex.org/W2076889753","https://openalex.org/W2080063509","https://openalex.org/W2080083189","https://openalex.org/W2083606889","https://openalex.org/W2083613288","https://openalex.org/W2100970777","https://openalex.org/W2103071101","https://openalex.org/W2117756735","https://openalex.org/W2118582701","https://openalex.org/W2122249806","https://openalex.org/W2125768532","https://openalex.org/W2128564847","https://openalex.org/W2154962986","https://openalex.org/W2159161022","https://openalex.org/W2162669222","https://openalex.org/W2169213530","https://openalex.org/W2172243610","https://openalex.org/W2963108114","https://openalex.org/W3006591805","https://openalex.org/W4293005804","https://openalex.org/W4362230038"],"related_works":["https://openalex.org/W2404647514","https://openalex.org/W1667647204","https://openalex.org/W4247536566","https://openalex.org/W2018477250","https://openalex.org/W3119814709","https://openalex.org/W4241418540","https://openalex.org/W1508895727","https://openalex.org/W2883805925","https://openalex.org/W4200572757","https://openalex.org/W4285148873"],"abstract_inverted_index":{"Abstract":[0],"In":[1,361],"this":[2,263],"article,":[3],"we":[4,107,247,257],"consider":[5],"the":[6,11,33,56,65,98,121,145,205,261,265,267,280,306,309,312,316],"general":[7],"problem":[8],"of":[9,13,60,90,115,151,157,185,207,238,240,282,308],"checking":[10],"correctness":[12],"matrix":[14,61,314],"multiplication.":[15],"Given":[16],"three":[17],"<m:math":[18,39,67,123,162,190,215,288],"xmlns:m=\"http://www.w3.org/1998/Math/MathML\">":[19,40,68,124,163,191,216,289],"<m:mrow>":[20,41,42,69,125,164,167,194,217,218],"<m:mi>n</m:mi>":[21,23,170],"<m:mo>\u00d7</m:mo>":[22,44,71,127,220],"</m:mrow>":[24,46,49,73,129,174,175,197,222,225],"</m:math>":[25,50,74,130,176,199,226,294],"n\\times":[26],"n":[27],"matrices":[28,92],"\ud835\udc34,":[29],"\ud835\udc35":[30],"and":[31,63,96,142,251,315,351],"\ud835\udc36,":[32,78],"goal":[34],"is":[35,83,101,153,161,189,204,299],"to":[36,103,111,342],"verify":[37],"that":[38,156,260,269,331],"<m:mi>A</m:mi>":[43,70,126,219],"<m:mi>B</m:mi>":[45,72,128,221],"<m:mo>=</m:mo>":[47],"<m:mi>C</m:mi>":[48,224],"A\\times":[51,75,131,227],"B=C":[52],"without":[53],"carrying":[54],"out":[55],"computationally":[57],"costly":[58],"operations":[59],"multiplication":[62],"comparing":[64,144],"product":[66,122],"B":[76,132],"with":[77,155,243,335,371],"term":[79],"by":[80,119,275],"term.":[81],"This":[82],"especially":[84],"important":[85],"when":[86,97,214],"some":[87],"or":[88],"all":[89],"these":[91],"are":[93,332],"very":[94,301,366],"large,":[95],"computing":[99,359],"environment":[100],"prone":[102],"soft":[104],"errors.":[105],"Here":[106],"extend":[108],"Freivalds\u2019":[109,116,158,181],"algorithm":[110],"a":[112,138,186,233,283,300],"Gaussian":[113,139],"Variant":[114],"Algorithm":[117],"(GVFA)":[118],"projecting":[120],"as":[133,135,287],"well":[134],"\ud835\udc36":[136],"onto":[137],"random":[140],"vector":[141],"then":[143],"resulting":[146],"vectors.":[147],"The":[148],"computational":[149],"complexity":[150],"GVFA":[152,231,270,277,338,363],"consistent":[154],"algorithm,":[159,182],"which":[160],"<m:mi>O</m:mi>":[165],"<m:mo>\u2062</m:mo>":[166],"<m:mo>(</m:mo>":[168],"<m:msup>":[169,192,290],"<m:mn>2</m:mn>":[171,193],"</m:msup>":[172,198,293],"<m:mo>)</m:mo>":[173],"O(n^{2})":[177],".":[178],"However,":[179],"unlike":[180],"whose":[183],"probability":[184,268,281],"false":[187,234,272,284],"positive":[188,235,285],"<m:mo>-</m:mo>":[195],"<m:mi>k</m:mi>":[196,292],"2^{-k}":[200],",":[201,230,296],"where":[202,297],"\ud835\udc58":[203,278],"number":[206],"iterations,":[208],"our":[209,255],"theoretical":[210],"analysis":[211],"shows":[212],"that,":[213],"<m:mo>\u2260</m:mo>":[223],"B\\neq":[228],"C":[229],"produces":[232],"on":[236,305,311,357,369],"set":[237],"inputs":[239],"measure":[241],"zero":[242],"exact":[244],"arithmetic.":[245],"When":[246],"introduce":[248],"round-off":[249],"error":[250],"floating-point":[252,319],"arithmetic":[253,317],"into":[254],"analysis,":[256],"can":[258,339,353,364],"show":[259],"larger":[262],"error,":[264],"higher":[266],"avoids":[271],"positives.":[273],"Moreover,":[274],"iterating":[276],"times,":[279],"decreases":[286],"<m:mi>p</m:mi>":[291],"p^{k}":[295],"\ud835\udc5d":[298],"small":[302],"value":[303],"depending":[304],"nature":[307],"fault":[310,329,345],"result":[313],"system\u2019s":[318],"precision.":[320],"Unlike":[321],"deterministic":[322],"algorithms,":[323],"there":[324],"do":[325],"not":[326],"exist":[327],"any":[328],"patterns":[330],"completely":[333],"undetectable":[334],"GVFA.":[336],"Thus":[337],"be":[340,354,365],"used":[341],"provide":[343],"efficient":[344],"tolerance":[346],"in":[347],"numerical":[348],"linear":[349],"algebra,":[350],"it":[352],"efficiently":[355,367],"implemented":[356,368],"modern":[358],"architectures.":[360],"particular,":[362],"architectures":[370],"hardware":[372],"support":[373],"for":[374],"fused":[375],"multiply-add":[376],"operations.":[377]},"counts_by_year":[{"year":2025,"cited_by_count":1},{"year":2023,"cited_by_count":2}],"updated_date":"2026-05-21T06:26:12.895304","created_date":"2020-10-29T00:00:00"}