{"id":"https://openalex.org/W2405242020","doi":"https://doi.org/10.1007/s00037-016-0144-9","title":"Matrix rigidity of random Toeplitz matrices","display_name":"Matrix rigidity of random Toeplitz matrices","publication_year":2016,"publication_date":"2016-06-10","ids":{"openalex":"https://openalex.org/W2405242020","doi":"https://doi.org/10.1007/s00037-016-0144-9","mag":"2405242020"},"language":"en","primary_location":{"id":"doi:10.1007/s00037-016-0144-9","is_oa":false,"landing_page_url":"https://doi.org/10.1007/s00037-016-0144-9","pdf_url":null,"source":{"id":"https://openalex.org/S193995496","display_name":"Computational Complexity","issn_l":"1016-3328","issn":["1016-3328","1420-8954"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320186","host_organization_name":"Birkh\u00e4user","host_organization_lineage":["https://openalex.org/P4310320186","https://openalex.org/P4310319900"],"host_organization_lineage_names":["Birkh\u00e4user","Springer Science+Business Media"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"computational complexity","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/A5013698877","display_name":"Oded Goldreich","orcid":"https://orcid.org/0000-0002-4329-135X"},"institutions":[{"id":"https://openalex.org/I53964585","display_name":"Weizmann Institute of Science","ror":"https://ror.org/0316ej306","country_code":"IL","type":"education","lineage":["https://openalex.org/I53964585"]}],"countries":["IL"],"is_corresponding":false,"raw_author_name":"Oded Goldreich","raw_affiliation_strings":["Weizmann Institute of Science, Rehovot, Israel","Weizmann institute of science;"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Weizmann Institute of Science, Rehovot, Israel","institution_ids":["https://openalex.org/I53964585"]},{"raw_affiliation_string":"Weizmann institute of science;","institution_ids":["https://openalex.org/I53964585"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5049425955","display_name":"Avishay Tal","orcid":"https://orcid.org/0000-0002-0375-6554"},"institutions":[{"id":"https://openalex.org/I40036882","display_name":"Institute for Advanced Study","ror":"https://ror.org/00f809463","country_code":"US","type":"facility","lineage":["https://openalex.org/I40036882"]}],"countries":["US"],"is_corresponding":true,"raw_author_name":"Avishay Tal","raw_affiliation_strings":["Institute for Advanced Study, Princeton, NJ, USA","Institute for Advanced Study"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Institute for Advanced Study, Princeton, NJ, USA","institution_ids":["https://openalex.org/I40036882"]},{"raw_affiliation_string":"Institute for Advanced Study","institution_ids":[]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":2,"corresponding_author_ids":["https://openalex.org/A5049425955"],"corresponding_institution_ids":["https://openalex.org/I40036882"],"apc_list":{"value":2890,"currency":"USD","value_usd":2890},"apc_paid":null,"fwci":0.0,"has_fulltext":false,"cited_by_count":0,"citation_normalized_percentile":{"value":0.02591416,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":null,"biblio":{"volume":"27","issue":"2","first_page":"305","last_page":"104"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10682","display_name":"Quantum Computing Algorithms and Architecture","score":0.9986000061035156,"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/T10682","display_name":"Quantum Computing Algorithms and Architecture","score":0.9986000061035156,"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/T12002","display_name":"Computability, Logic, AI Algorithms","score":0.9979000091552734,"subfield":{"id":"https://openalex.org/subfields/1703","display_name":"Computational Theory and Mathematics"},"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/T11797","display_name":"graph theory and CDMA systems","score":0.9968000054359436,"subfield":{"id":"https://openalex.org/subfields/2208","display_name":"Electrical and Electronic Engineering"},"field":{"id":"https://openalex.org/fields/22","display_name":"Engineering"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/toeplitz-matrix","display_name":"Toeplitz matrix","score":0.7581734657287598},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.5774495005607605},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.46888113021850586},{"id":"https://openalex.org/keywords/random-matrix","display_name":"Random matrix","score":0.4554247260093689},{"id":"https://openalex.org/keywords/rigidity","display_name":"Rigidity (electromagnetism)","score":0.45336514711380005},{"id":"https://openalex.org/keywords/matrix","display_name":"Matrix (chemical analysis)","score":0.4329540431499481},{"id":"https://openalex.org/keywords/matrix-analysis","display_name":"Matrix analysis","score":0.41081446409225464},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.3688089847564697},{"id":"https://openalex.org/keywords/algebra-over-a-field","display_name":"Algebra over a field","score":0.36225879192352295},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.1596938967704773},{"id":"https://openalex.org/keywords/materials-science","display_name":"Materials science","score":0.10434132814407349},{"id":"https://openalex.org/keywords/quantum-mechanics","display_name":"Quantum mechanics","score":0.07836279273033142},{"id":"https://openalex.org/keywords/eigenvalues-and-eigenvectors","display_name":"Eigenvalues and eigenvectors","score":0.04602557420730591}],"concepts":[{"id":"https://openalex.org/C147710293","wikidata":"https://www.wikidata.org/wiki/Q849428","display_name":"Toeplitz matrix","level":2,"score":0.7581734657287598},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.5774495005607605},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.46888113021850586},{"id":"https://openalex.org/C64812099","wikidata":"https://www.wikidata.org/wiki/Q176604","display_name":"Random matrix","level":3,"score":0.4554247260093689},{"id":"https://openalex.org/C160343418","wikidata":"https://www.wikidata.org/wiki/Q185256","display_name":"Rigidity (electromagnetism)","level":2,"score":0.45336514711380005},{"id":"https://openalex.org/C106487976","wikidata":"https://www.wikidata.org/wiki/Q685816","display_name":"Matrix (chemical analysis)","level":2,"score":0.4329540431499481},{"id":"https://openalex.org/C149488123","wikidata":"https://www.wikidata.org/wiki/Q17098730","display_name":"Matrix analysis","level":3,"score":0.41081446409225464},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.3688089847564697},{"id":"https://openalex.org/C136119220","wikidata":"https://www.wikidata.org/wiki/Q1000660","display_name":"Algebra over a field","level":2,"score":0.36225879192352295},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.1596938967704773},{"id":"https://openalex.org/C192562407","wikidata":"https://www.wikidata.org/wiki/Q228736","display_name":"Materials science","level":0,"score":0.10434132814407349},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.07836279273033142},{"id":"https://openalex.org/C158693339","wikidata":"https://www.wikidata.org/wiki/Q190524","display_name":"Eigenvalues and eigenvectors","level":2,"score":0.04602557420730591},{"id":"https://openalex.org/C159985019","wikidata":"https://www.wikidata.org/wiki/Q181790","display_name":"Composite material","level":1,"score":0.0}],"mesh":[],"locations_count":3,"locations":[{"id":"doi:10.1007/s00037-016-0144-9","is_oa":false,"landing_page_url":"https://doi.org/10.1007/s00037-016-0144-9","pdf_url":null,"source":{"id":"https://openalex.org/S193995496","display_name":"Computational Complexity","issn_l":"1016-3328","issn":["1016-3328","1420-8954"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320186","host_organization_name":"Birkh\u00e4user","host_organization_lineage":["https://openalex.org/P4310320186","https://openalex.org/P4310319900"],"host_organization_lineage_names":["Birkh\u00e4user","Springer Science+Business Media"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"computational complexity","raw_type":"journal-article"},{"id":"doi:10.1145/2897518.2897633","is_oa":false,"landing_page_url":"https://doi.org/10.1145/2897518.2897633","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Proceedings of the forty-eighth annual ACM symposium on Theory of Computing","raw_type":"proceedings-article"},{"id":"mag:2405242020","is_oa":false,"landing_page_url":"http://eccc.hpi-web.de/report/2015/079","pdf_url":null,"source":null,"license":null,"license_id":null,"version":null,"is_accepted":false,"is_published":null,"raw_source_name":null,"raw_type":null}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":19,"referenced_works":["https://openalex.org/W1530008367","https://openalex.org/W1968408352","https://openalex.org/W1979175600","https://openalex.org/W2006393845","https://openalex.org/W2008671178","https://openalex.org/W2019807548","https://openalex.org/W2049136377","https://openalex.org/W2064269488","https://openalex.org/W2137442467","https://openalex.org/W2163137752","https://openalex.org/W2165293955","https://openalex.org/W2408001118","https://openalex.org/W2413166304","https://openalex.org/W2952824848","https://openalex.org/W2988890240","https://openalex.org/W3184185275","https://openalex.org/W4252948584","https://openalex.org/W6610200006","https://openalex.org/W6713390907"],"related_works":["https://openalex.org/W3033458414","https://openalex.org/W2564242734","https://openalex.org/W1990092857","https://openalex.org/W3160083777","https://openalex.org/W1574691718","https://openalex.org/W2310381493","https://openalex.org/W2255223375","https://openalex.org/W2135562106","https://openalex.org/W2963193394","https://openalex.org/W3106139507","https://openalex.org/W2806426664","https://openalex.org/W2054776055","https://openalex.org/W2547308177","https://openalex.org/W1897762504","https://openalex.org/W1993433103","https://openalex.org/W2963497122","https://openalex.org/W3015477701","https://openalex.org/W2188133584","https://openalex.org/W3082126129","https://openalex.org/W3102706553"],"abstract_inverted_index":{"A":[0,2,13],"matrix":[1,17,166],"is":[3,79],"said":[4],"to":[5,155],"have":[6,50],"rigidity":[7,51],"s":[8,24],"for":[9,43,54,65,81,138,191],"rank":[10,19,55],"r":[11,20],"if":[12,148],"differs":[14],"from":[15,171],"any":[16,192],"of":[18,37,132,157,181,213],"on":[21,129],"more":[22],"than":[23],"entries.":[25],"We":[26,140,194],"prove":[27,142],"that":[28,78,88,143,164],"random":[29,44],"n-by-n":[30],"Toeplitz":[31],"matrices":[32,36],"over":[33,73],"$${\\mathbb{F}_{2}}$$":[34],"(i.e.,":[35],"the":[38,74,89,108,130,133,149,186,209],"form":[39],"$${A_{i,j}":[40],"=":[41,67,100],"a_{i-j}}$$":[42],"bits":[45],"$${a_{-(n-1)},":[46],"\\ldots,":[47],"a_{n-1}}$$":[48],")":[49],"$${\\Omega(n^3/(r^2\\log":[52],"n))}$$":[53],"$${r":[56,66],"\\ge":[57],"\\sqrt{n}}$$":[58],",":[59,72,179],"with":[60,204],"high":[61],"probability.":[62],"This":[63],"improves,":[64],"o(n/\\log":[68],"n":[69],"\\log\\log":[70],"n)}$$":[71],"$${\\Omega(\\frac{n^2}{r}":[75],"\\cdot\\log(\\frac{n}{r}))}$$":[76],"bound":[77,128,190],"known":[80,187],"many":[82],"explicit":[83,90,201],"matrices.":[84],"Our":[85],"result":[86],"implies":[87],"trilinear":[91],"$${[n]\\times":[92],"[n]":[93],"\\times":[94],"[2n]}$$":[95],"function":[96],"defined":[97],"by":[98,113],"$${F(x,y,z)":[99],"\\sum_{i,j}{x_i":[101],"y_j":[102],"z_{i+j}}}$$":[103],"has":[104,145,176],"complexity":[105,146,177],"$${\\Omega(n^{3/5})}$$":[106],"in":[107],"multilinear":[109,150],"circuit":[110],"model":[111],"suggested":[112],"Goldreich":[114],"and":[115],"Wigderson":[116],"(Electron":[117],"Colloq":[118],"Comput":[119],"Complex":[120],"20:43,":[121],"2013),":[122],"which":[123],"yields":[124],"an":[125,200],"$${\\exp(n^{3/5})}$$":[126],"lower":[127,206],"size":[131],"so-called":[134],"canonical":[135],"depth-three":[136],"circuits":[137,151],"F.":[139],"also":[141],"F":[144],"$${\\tilde{\\Omega}(n^{2/3})}$$":[147,178],"are":[152,169],"further":[153],"restricted":[154],"be":[156],"depth":[158,182],"2.":[159],"In":[160],"addition,":[161],"we":[162],"show":[163],"a":[165,172],"whose":[167],"entries":[168],"sampled":[170],"$${2^{-n}}$$":[173],"-biased":[174],"distribution":[175],"regardless":[180],"restrictions,":[183],"almost":[184],"matching":[185],"$${O(n^{2/3})}$$":[188],"upper":[189],"matrix.":[193],"turn":[195],"this":[196],"randomized":[197],"construction":[198,203,212],"into":[199],"4-linear":[202],"similar":[205],"bounds,":[207],"using":[208],"quadratic":[210],"small-biased":[211],"Mossel":[214],"et":[215],"al.":[216],"(Random":[217],"Struct":[218],"Algorithms":[219],"29(1):56\u201381,":[220],"2006).":[221]},"counts_by_year":[],"updated_date":"2026-07-30T17:31:21.811387","created_date":"2025-10-10T00:00:00"}