{"id":"https://openalex.org/W1670963965","doi":"https://doi.org/10.1145/2213977.2214006","title":"Hypercontractivity, sum-of-squares proofs, and their applications","display_name":"Hypercontractivity, sum-of-squares proofs, and their applications","publication_year":2012,"publication_date":"2012-05-19","ids":{"openalex":"https://openalex.org/W1670963965","doi":"https://doi.org/10.1145/2213977.2214006","mag":"1670963965"},"language":"en","primary_location":{"id":"doi:10.1145/2213977.2214006","is_oa":false,"landing_page_url":"https://doi.org/10.1145/2213977.2214006","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-fourth annual ACM symposium on Theory of computing","raw_type":"proceedings-article"},"type":"article","indexed_in":["arxiv","crossref","datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://arxiv.org/pdf/1205.4484","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5000871493","display_name":"Boaz Barak","orcid":"https://orcid.org/0000-0001-9724-4578"},"institutions":[{"id":"https://openalex.org/I1290206253","display_name":"Microsoft (United States)","ror":"https://ror.org/00d0nc645","country_code":"US","type":"company","lineage":["https://openalex.org/I1290206253"]},{"id":"https://openalex.org/I4400600948","display_name":"Microsoft Research New England (United States)","ror":"https://ror.org/010wkqn50","country_code":null,"type":"company","lineage":["https://openalex.org/I1290206253","https://openalex.org/I4400600948"]}],"countries":["US"],"is_corresponding":true,"raw_author_name":"Boaz Barak","raw_affiliation_strings":["Microsoft Research New England, Cambridge, MA, USA","[Microsoft Research New England, Cambridge, MA, USA]"],"affiliations":[{"raw_affiliation_string":"Microsoft Research New England, Cambridge, MA, USA","institution_ids":["https://openalex.org/I1290206253","https://openalex.org/I4400600948"]},{"raw_affiliation_string":"[Microsoft Research New England, Cambridge, MA, USA]","institution_ids":["https://openalex.org/I1290206253","https://openalex.org/I4400600948"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5022691107","display_name":"Fgsl Brand\u00e3o","orcid":null},"institutions":[{"id":"https://openalex.org/I110200422","display_name":"Universidade Federal de Minas Gerais","ror":"https://ror.org/0176yjw32","country_code":"BR","type":"education","lineage":["https://openalex.org/I110200422"]}],"countries":["BR"],"is_corresponding":false,"raw_author_name":"Fernando G.S.L. Brandao","raw_affiliation_strings":["Universidade Federal de Minas Gerais, Belo Horizonte, Brazil","( Universidade Federal de Minas Gerais Belo Horizonte Brazil )"],"affiliations":[{"raw_affiliation_string":"Universidade Federal de Minas Gerais, Belo Horizonte, Brazil","institution_ids":["https://openalex.org/I110200422"]},{"raw_affiliation_string":"( Universidade Federal de Minas Gerais Belo Horizonte Brazil )","institution_ids":["https://openalex.org/I110200422"]}]},{"author_position":"middle","author":{"id":null,"display_name":"Aram W. Harrow","orcid":null},"institutions":[{"id":"https://openalex.org/I201448701","display_name":"University of Washington","ror":"https://ror.org/00cvxb145","country_code":"US","type":"education","lineage":["https://openalex.org/I201448701"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Aram W. Harrow","raw_affiliation_strings":["University of Washington, Seattle, WA, USA","Univ. of Washington, Seattle, WA, USA"],"affiliations":[{"raw_affiliation_string":"University of Washington, Seattle, WA, USA","institution_ids":["https://openalex.org/I201448701"]},{"raw_affiliation_string":"Univ. of Washington, Seattle, WA, USA","institution_ids":["https://openalex.org/I201448701"]}]},{"author_position":"middle","author":{"id":null,"display_name":"Jonathan Kelner","orcid":null},"institutions":[{"id":"https://openalex.org/I63966007","display_name":"Massachusetts Institute of Technology","ror":"https://ror.org/042nb2s44","country_code":"US","type":"education","lineage":["https://openalex.org/I63966007"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Jonathan Kelner","raw_affiliation_strings":["Massachusetts Institute of Technology, Cambridge, MA, USA","Massachusetts Institute of Technology Cambridge, MA USA"],"affiliations":[{"raw_affiliation_string":"Massachusetts Institute of Technology, Cambridge, MA, USA","institution_ids":["https://openalex.org/I63966007"]},{"raw_affiliation_string":"Massachusetts Institute of Technology Cambridge, MA USA","institution_ids":["https://openalex.org/I63966007"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5089128021","display_name":"David Steurer","orcid":"https://orcid.org/0000-0002-3908-8180"},"institutions":[{"id":"https://openalex.org/I1290206253","display_name":"Microsoft (United States)","ror":"https://ror.org/00d0nc645","country_code":"US","type":"company","lineage":["https://openalex.org/I1290206253"]},{"id":"https://openalex.org/I4400600948","display_name":"Microsoft Research New England (United States)","ror":"https://ror.org/010wkqn50","country_code":null,"type":"company","lineage":["https://openalex.org/I1290206253","https://openalex.org/I4400600948"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"David Steurer","raw_affiliation_strings":["Microsoft Research New England, Cambridge, MA, USA","[Microsoft Research New England, Cambridge, MA, USA]"],"affiliations":[{"raw_affiliation_string":"Microsoft Research New England, Cambridge, MA, USA","institution_ids":["https://openalex.org/I1290206253","https://openalex.org/I4400600948"]},{"raw_affiliation_string":"[Microsoft Research New England, Cambridge, MA, USA]","institution_ids":["https://openalex.org/I1290206253","https://openalex.org/I4400600948"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5045927034","display_name":"Yuan Zhou","orcid":"https://orcid.org/0009-0008-1706-6539"},"institutions":[{"id":"https://openalex.org/I74973139","display_name":"Carnegie Mellon University","ror":"https://ror.org/05x2bcf33","country_code":"US","type":"education","lineage":["https://openalex.org/I74973139"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Yuan Zhou","raw_affiliation_strings":["Carnegie Melon University, Pittsburgh, PA, USA"],"affiliations":[{"raw_affiliation_string":"Carnegie Melon University, Pittsburgh, PA, USA","institution_ids":["https://openalex.org/I74973139"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":6,"corresponding_author_ids":["https://openalex.org/A5000871493"],"corresponding_institution_ids":["https://openalex.org/I1290206253","https://openalex.org/I4400600948"],"apc_list":null,"apc_paid":null,"fwci":0.0,"has_fulltext":false,"cited_by_count":6,"citation_normalized_percentile":{"value":0.17179563,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":90,"max":96},"biblio":{"volume":null,"issue":null,"first_page":"307","last_page":"326"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T13497","display_name":"Hermeneutics and Narrative Identity","score":0.9879000186920166,"subfield":{"id":"https://openalex.org/subfields/1211","display_name":"Philosophy"},"field":{"id":"https://openalex.org/fields/12","display_name":"Arts and Humanities"},"domain":{"id":"https://openalex.org/domains/2","display_name":"Social Sciences"}},"topics":[{"id":"https://openalex.org/T13497","display_name":"Hermeneutics and Narrative Identity","score":0.9879000186920166,"subfield":{"id":"https://openalex.org/subfields/1211","display_name":"Philosophy"},"field":{"id":"https://openalex.org/fields/12","display_name":"Arts and Humanities"},"domain":{"id":"https://openalex.org/domains/2","display_name":"Social Sciences"}},{"id":"https://openalex.org/T13695","display_name":"Aging, Elder Care, and Social Issues","score":0.9749000072479248,"subfield":{"id":"https://openalex.org/subfields/3600","display_name":"General Health Professions"},"field":{"id":"https://openalex.org/fields/36","display_name":"Health Professions"},"domain":{"id":"https://openalex.org/domains/4","display_name":"Health Sciences"}},{"id":"https://openalex.org/T13099","display_name":"Health, Medicine and Society","score":0.95660001039505,"subfield":{"id":"https://openalex.org/subfields/3600","display_name":"General Health Professions"},"field":{"id":"https://openalex.org/fields/36","display_name":"Health Professions"},"domain":{"id":"https://openalex.org/domains/4","display_name":"Health Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.7697468996047974},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.6313410401344299},{"id":"https://openalex.org/keywords/conjecture","display_name":"Conjecture","score":0.5647591948509216},{"id":"https://openalex.org/keywords/upper-and-lower-bounds","display_name":"Upper and lower bounds","score":0.5312544703483582},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.5182230472564697},{"id":"https://openalex.org/keywords/bounded-function","display_name":"Bounded function","score":0.4952773153781891},{"id":"https://openalex.org/keywords/explained-sum-of-squares","display_name":"Explained sum of squares","score":0.441714882850647},{"id":"https://openalex.org/keywords/adjacency-matrix","display_name":"Adjacency matrix","score":0.4341874122619629},{"id":"https://openalex.org/keywords/norm","display_name":"Norm (philosophy)","score":0.42516931891441345},{"id":"https://openalex.org/keywords/mathematical-proof","display_name":"Mathematical proof","score":0.4181390404701233},{"id":"https://openalex.org/keywords/graph","display_name":"Graph","score":0.3094559907913208}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7697468996047974},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.6313410401344299},{"id":"https://openalex.org/C2780990831","wikidata":"https://www.wikidata.org/wiki/Q319141","display_name":"Conjecture","level":2,"score":0.5647591948509216},{"id":"https://openalex.org/C77553402","wikidata":"https://www.wikidata.org/wiki/Q13222579","display_name":"Upper and lower bounds","level":2,"score":0.5312544703483582},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.5182230472564697},{"id":"https://openalex.org/C34388435","wikidata":"https://www.wikidata.org/wiki/Q2267362","display_name":"Bounded function","level":2,"score":0.4952773153781891},{"id":"https://openalex.org/C49847556","wikidata":"https://www.wikidata.org/wiki/Q3964631","display_name":"Explained sum of squares","level":2,"score":0.441714882850647},{"id":"https://openalex.org/C180356752","wikidata":"https://www.wikidata.org/wiki/Q727035","display_name":"Adjacency matrix","level":3,"score":0.4341874122619629},{"id":"https://openalex.org/C191795146","wikidata":"https://www.wikidata.org/wiki/Q3878446","display_name":"Norm (philosophy)","level":2,"score":0.42516931891441345},{"id":"https://openalex.org/C108710211","wikidata":"https://www.wikidata.org/wiki/Q11538","display_name":"Mathematical proof","level":2,"score":0.4181390404701233},{"id":"https://openalex.org/C132525143","wikidata":"https://www.wikidata.org/wiki/Q141488","display_name":"Graph","level":2,"score":0.3094559907913208},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.0},{"id":"https://openalex.org/C105795698","wikidata":"https://www.wikidata.org/wiki/Q12483","display_name":"Statistics","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/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"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":4,"locations":[{"id":"doi:10.1145/2213977.2214006","is_oa":false,"landing_page_url":"https://doi.org/10.1145/2213977.2214006","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-fourth annual ACM symposium on Theory of computing","raw_type":"proceedings-article"},{"id":"pmh:oai:arXiv.org:1205.4484","is_oa":true,"landing_page_url":"http://arxiv.org/abs/1205.4484","pdf_url":"https://arxiv.org/pdf/1205.4484","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:dspace.mit.edu:1721.1/80386","is_oa":true,"landing_page_url":"http://hdl.handle.net/1721.1/80386","pdf_url":null,"source":{"id":"https://openalex.org/S4306400425","display_name":"DSpace@MIT (Massachusetts Institute of Technology)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I63966007","host_organization_name":"Massachusetts Institute of Technology","host_organization_lineage":["https://openalex.org/I63966007"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by-nc-sa","license_id":"https://openalex.org/licenses/cc-by-nc-sa","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"arXiv","raw_type":"http://purl.org/eprint/type/JournalArticle"},{"id":"doi:10.48550/arxiv.1205.4484","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.1205.4484","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-journal"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:1205.4484","is_oa":true,"landing_page_url":"http://arxiv.org/abs/1205.4484","pdf_url":"https://arxiv.org/pdf/1205.4484","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"},"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":37,"referenced_works":["https://openalex.org/W1571037334","https://openalex.org/W1590432342","https://openalex.org/W1608352849","https://openalex.org/W1886138972","https://openalex.org/W1967184028","https://openalex.org/W1967344706","https://openalex.org/W1981566278","https://openalex.org/W2001334051","https://openalex.org/W2002025272","https://openalex.org/W2015573459","https://openalex.org/W2016990905","https://openalex.org/W2019404063","https://openalex.org/W2023144280","https://openalex.org/W2026298842","https://openalex.org/W2030369304","https://openalex.org/W2042134536","https://openalex.org/W2044630019","https://openalex.org/W2044949772","https://openalex.org/W2047642496","https://openalex.org/W2049143039","https://openalex.org/W2074447841","https://openalex.org/W2077397558","https://openalex.org/W2089105401","https://openalex.org/W2095315734","https://openalex.org/W2098350988","https://openalex.org/W2099677042","https://openalex.org/W2106264302","https://openalex.org/W2116123253","https://openalex.org/W2120358419","https://openalex.org/W2133263321","https://openalex.org/W2138689422","https://openalex.org/W2144126354","https://openalex.org/W2151395725","https://openalex.org/W2163925208","https://openalex.org/W2166536886","https://openalex.org/W2327680971","https://openalex.org/W6631279664"],"related_works":["https://openalex.org/W2028932986","https://openalex.org/W2109437253","https://openalex.org/W2951749524","https://openalex.org/W2148042319","https://openalex.org/W1859301526","https://openalex.org/W2095933180","https://openalex.org/W2123056342","https://openalex.org/W2950545914","https://openalex.org/W3000895154","https://openalex.org/W2114477570","https://openalex.org/W2314825336","https://openalex.org/W2093469244","https://openalex.org/W2597306538","https://openalex.org/W2899413771","https://openalex.org/W2607993745","https://openalex.org/W3203069180","https://openalex.org/W2953082259","https://openalex.org/W2964245737","https://openalex.org/W2040889609","https://openalex.org/W2226187912"],"abstract_inverted_index":{"We":[0,46,109,218],"study":[1,39],"the":[2,7,38,48,69,72,75,93,98,107,129,138,150,154,161,167,170,187,196,203,223,228,247,258,261,271,289,299],"computational":[3],"complexity":[4],"of":[5,10,40,74,78,106,128,137,140,153,169,178,191,205,232],"approximating":[6],"2-to-q":[8],"norm":[9,95,152,225,231,249,263],"linear":[11],"operators":[12],"(defined":[13],"as":[14,23,25,164,199,201],"|A|2->q":[15],"=":[16],"maxv\u2260":[17],"0|Av|q/|v|2)":[18],"for":[19,133,288,298],"q":[20,55],">":[21],"2,":[22],"well":[24,165,200],"connections":[26,238],"between":[27,221],"this":[28],"question":[29],"and":[30,37,66,173,226,284],"issues":[31],"arising":[32],"in":[33,120,257,279],"quantum":[34,240,290],"information":[35,241],"theory":[36],"Khot's":[41],"Unique":[42],"Games":[43],"Conjecture":[44,101],"(UGC).":[45],"show":[47,111,219],"following:":[49],"For":[50],"any":[51],"constant":[52],"even":[53],"integer":[54],"\u2265":[56],"4,":[57],"a":[58,62,87,89,103,114,125,233,235,267,294],"graph":[59],"G":[60],"is":[61,250],"small-set":[63],"expander":[64],"if":[65,68],"only":[67],"projector":[70,155],"into":[71],"span":[73],"top":[76],"eigenvectors":[77],"G's":[79],"adjacency":[80],"matrix":[81],"has":[82],"bounded":[83],"2->q":[84,94],"norm.":[85,301],"As":[86],"corollary,":[88],"good":[90,115,268],"approximation":[91,116,269,297],"to":[92,156,214,239,252,254],"will":[96],"refute":[97],"Small-Set":[99],"Expansion":[100],"---":[102],"close":[104],"variant":[105],"UGC.":[108],"also":[110],"that":[112,211],"such":[113],"can":[117,276],"be":[118,277],"obtained":[119],"exp(n2/q)":[121],"time,":[122],"thus":[123],"obtaining":[124],"different":[126],"proof":[127],"known":[130,213,286],"subexponential":[131],"algorithm":[132],"Small-Set-Expansion.":[134],"Constant":[135],"rounds":[136,194],"\"Sum":[139,204],"Squares\"":[141],"semidefinite":[142],"programing":[143],"hierarchy":[144,207],"certify":[145,166],"an":[146],"upper":[147,189],"bound":[148,190],"on":[149,186],"2->4":[151,224,248,262,300],"low":[157],"degree":[158],"polynomials":[159],"over":[160],"Boolean":[162],"cube,":[163],"unsatisfiability":[168],"\"noisy":[171],"cube\"":[172],"\"short":[174,197],"code\"":[175],"based":[176],"instances":[177],"Unique-Games":[179],"considered":[180],"by":[181],"prior":[182],"works.":[183],"This":[184],"improves":[185],"previous":[188],"exp(logO(1)":[192],"n)":[193],"(for":[195],"code\"),":[198],"separates":[202],"Squares\"/\"Lasserre\"":[206],"from":[208],"weaker":[209],"hierarchies":[210],"were":[212],"require":[215],"\u03c9(1)":[216],"rounds.":[217],"reductions":[220],"computing":[222,227],"injective":[229],"tensor":[230],"tensor,":[234],"problem":[236,292],"with":[237],"theory.":[242],"Three":[243],"corollaries":[244],"are:":[245],"(i)":[246],"NP-hard":[251],"approximate":[253],"precision":[255],"inverse-polynomial":[256],"dimension,":[259],"(ii)":[260],"does":[264],"not":[265],"have":[266],"(in":[270],"sense":[272],"above)":[273],"unless":[274],"3-SAT":[275],"solved":[278],"time":[280],"exp(\u221an":[281],"poly":[282],"log(n)),":[283],"(iii)":[285],"algorithms":[287],"separability":[291],"imply":[293],"non-trivial":[295],"additive":[296]},"counts_by_year":[{"year":2020,"cited_by_count":2},{"year":2018,"cited_by_count":1},{"year":2017,"cited_by_count":2},{"year":2016,"cited_by_count":1}],"updated_date":"2026-03-10T16:38:18.471706","created_date":"2025-10-10T00:00:00"}
