{"id":"https://openalex.org/W2402751180","doi":"https://doi.org/10.4086/toc.2014.v010a002","title":"","display_name":"","publication_year":2014,"publication_date":"2014-01-01","ids":{"openalex":"https://openalex.org/W2402751180","doi":"https://doi.org/10.4086/toc.2014.v010a002","mag":"2402751180"},"language":"en","primary_location":{"id":"doi:10.4086/toc.2014.v010a002","is_oa":true,"landing_page_url":"https://doi.org/10.4086/toc.2014.v010a002","pdf_url":"https://theoryofcomputing.org/articles/v010a002/v010a002.pdf","source":{"id":"https://openalex.org/S142665794","display_name":"Theory of Computing","issn_l":"1557-2862","issn":["1557-2862"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310315690","host_organization_name":"University of Chicago","host_organization_lineage":["https://openalex.org/P4310315690"],"host_organization_lineage_names":["University of Chicago"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Theory of Computing","raw_type":"journal-article"},"type":"article","indexed_in":["crossref","doaj"],"open_access":{"is_oa":true,"oa_status":"diamond","oa_url":"https://theoryofcomputing.org/articles/v010a002/v010a002.pdf","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5054966819","display_name":"Ilias Diakonikolas","orcid":"https://orcid.org/0000-0002-5486-1856"},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"Ilias Diakonikolas","raw_affiliation_strings":[],"affiliations":[]},{"author_position":"middle","author":{"id":"https://openalex.org/A5014866889","display_name":"Rocco A. Servedio","orcid":"https://orcid.org/0000-0003-2407-543X"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Rocco A. Servedio","raw_affiliation_strings":[],"affiliations":[]},{"author_position":"middle","author":{"id":"https://openalex.org/A5029451171","display_name":"Li-Yang Tan","orcid":"https://orcid.org/0000-0002-1279-4384"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Li-Yang Tan","raw_affiliation_strings":[],"affiliations":[]},{"author_position":"last","author":{"id":"https://openalex.org/A5109311357","display_name":"Andrew Wan","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Andrew Wan","raw_affiliation_strings":[],"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":4,"corresponding_author_ids":["https://openalex.org/A5054966819"],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":3.9124,"has_fulltext":true,"cited_by_count":13,"citation_normalized_percentile":{"value":0.93659761,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":89,"max":97},"biblio":{"volume":"10","issue":"1","first_page":"27","last_page":"53"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10963","display_name":"Advanced Optimization Algorithms Research","score":0.9819999933242798,"subfield":{"id":"https://openalex.org/subfields/2612","display_name":"Numerical Analysis"},"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/T10963","display_name":"Advanced Optimization Algorithms Research","score":0.9819999933242798,"subfield":{"id":"https://openalex.org/subfields/2612","display_name":"Numerical Analysis"},"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/T10720","display_name":"Complexity and Algorithms in Graphs","score":0.9781000018119812,"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/T12072","display_name":"Machine Learning and Algorithms","score":0.9623000025749207,"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/mathematics","display_name":"Mathematics","score":0.8127537965774536},{"id":"https://openalex.org/keywords/degree","display_name":"Degree (music)","score":0.7361117005348206},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.725340723991394},{"id":"https://openalex.org/keywords/sign","display_name":"Sign (mathematics)","score":0.6588966846466064},{"id":"https://openalex.org/keywords/constant","display_name":"Constant (computer programming)","score":0.5782890915870667},{"id":"https://openalex.org/keywords/lemma","display_name":"Lemma (botany)","score":0.5744978189468384},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.5234838724136353},{"id":"https://openalex.org/keywords/logarithm","display_name":"Logarithm","score":0.5138552188873291},{"id":"https://openalex.org/keywords/integer","display_name":"Integer (computer science)","score":0.49747565388679504},{"id":"https://openalex.org/keywords/polynomial","display_name":"Polynomial","score":0.49457332491874695},{"id":"https://openalex.org/keywords/boolean-function","display_name":"Boolean function","score":0.4883992075920105},{"id":"https://openalex.org/keywords/fraction","display_name":"Fraction (chemistry)","score":0.4201822578907013},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.12244969606399536},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.06364840269088745},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.0631324052810669}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8127537965774536},{"id":"https://openalex.org/C2775997480","wikidata":"https://www.wikidata.org/wiki/Q586277","display_name":"Degree (music)","level":2,"score":0.7361117005348206},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.725340723991394},{"id":"https://openalex.org/C139676723","wikidata":"https://www.wikidata.org/wiki/Q1193832","display_name":"Sign (mathematics)","level":2,"score":0.6588966846466064},{"id":"https://openalex.org/C2777027219","wikidata":"https://www.wikidata.org/wiki/Q1284190","display_name":"Constant (computer programming)","level":2,"score":0.5782890915870667},{"id":"https://openalex.org/C2777759810","wikidata":"https://www.wikidata.org/wiki/Q149316","display_name":"Lemma (botany)","level":3,"score":0.5744978189468384},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.5234838724136353},{"id":"https://openalex.org/C39927690","wikidata":"https://www.wikidata.org/wiki/Q11197","display_name":"Logarithm","level":2,"score":0.5138552188873291},{"id":"https://openalex.org/C97137487","wikidata":"https://www.wikidata.org/wiki/Q729138","display_name":"Integer (computer science)","level":2,"score":0.49747565388679504},{"id":"https://openalex.org/C90119067","wikidata":"https://www.wikidata.org/wiki/Q43260","display_name":"Polynomial","level":2,"score":0.49457332491874695},{"id":"https://openalex.org/C187455244","wikidata":"https://www.wikidata.org/wiki/Q942353","display_name":"Boolean function","level":2,"score":0.4883992075920105},{"id":"https://openalex.org/C149629883","wikidata":"https://www.wikidata.org/wiki/Q660926","display_name":"Fraction (chemistry)","level":2,"score":0.4201822578907013},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.12244969606399536},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.06364840269088745},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.0631324052810669},{"id":"https://openalex.org/C24890656","wikidata":"https://www.wikidata.org/wiki/Q82811","display_name":"Acoustics","level":1,"score":0.0},{"id":"https://openalex.org/C185592680","wikidata":"https://www.wikidata.org/wiki/Q2329","display_name":"Chemistry","level":0,"score":0.0},{"id":"https://openalex.org/C199360897","wikidata":"https://www.wikidata.org/wiki/Q9143","display_name":"Programming language","level":1,"score":0.0},{"id":"https://openalex.org/C178790620","wikidata":"https://www.wikidata.org/wiki/Q11351","display_name":"Organic chemistry","level":1,"score":0.0},{"id":"https://openalex.org/C46757340","wikidata":"https://www.wikidata.org/wiki/Q43238","display_name":"Poaceae","level":2,"score":0.0},{"id":"https://openalex.org/C86803240","wikidata":"https://www.wikidata.org/wiki/Q420","display_name":"Biology","level":0,"score":0.0},{"id":"https://openalex.org/C18903297","wikidata":"https://www.wikidata.org/wiki/Q7150","display_name":"Ecology","level":1,"score":0.0}],"mesh":[],"locations_count":3,"locations":[{"id":"doi:10.4086/toc.2014.v010a002","is_oa":true,"landing_page_url":"https://doi.org/10.4086/toc.2014.v010a002","pdf_url":"https://theoryofcomputing.org/articles/v010a002/v010a002.pdf","source":{"id":"https://openalex.org/S142665794","display_name":"Theory of Computing","issn_l":"1557-2862","issn":["1557-2862"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310315690","host_organization_name":"University of Chicago","host_organization_lineage":["https://openalex.org/P4310315690"],"host_organization_lineage_names":["University of Chicago"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Theory of Computing","raw_type":"journal-article"},{"id":"pmh:oai:pure.ed.ac.uk:openaire/b4da644d-ff40-45a1-9a23-34cf517ab6db","is_oa":true,"landing_page_url":"https://www.research.ed.ac.uk/en/publications/b4da644d-ff40-45a1-9a23-34cf517ab6db","pdf_url":null,"source":{"id":"https://openalex.org/S4406922455","display_name":"Edinburgh Research Explorer","issn_l":null,"issn":null,"is_oa":false,"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","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Diakonikolas, I, Servedio, R A, Tan, L-Y & Wan, A 2014, 'A Regularity Lemma and Low-Weight Approximators for Low-Degree Polynomial Threshold Functions', Theory of Computing, vol. 10, no. 2, pp. 27-53. https://doi.org/10.4086/toc.2014.v010a002","raw_type":"info:eu-repo/semantics/publishedVersion"},{"id":"pmh:oai:pure.ed.ac.uk:publications/b4da644d-ff40-45a1-9a23-34cf517ab6db","is_oa":true,"landing_page_url":"http://www.theoryofcomputing.org/articles/v010a002","pdf_url":null,"source":{"id":"https://openalex.org/S4306400321","display_name":"Edinburgh Research Explorer (University of Edinburgh)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I98677209","host_organization_name":"University of Edinburgh","host_organization_lineage":["https://openalex.org/I98677209"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Diakonikolas, I, Servedio, R A, Tan, L-Y & Wan, A 2014, 'A Regularity Lemma and Low-Weight Approximators for Low-Degree Polynomial Threshold Functions', Theory of Computing, vol. 10, no. 2, pp. 27-53. https://doi.org/10.4086/toc.2014.v010a002","raw_type":"info:eu-repo/semantics/publishedVersion"}],"best_oa_location":{"id":"doi:10.4086/toc.2014.v010a002","is_oa":true,"landing_page_url":"https://doi.org/10.4086/toc.2014.v010a002","pdf_url":"https://theoryofcomputing.org/articles/v010a002/v010a002.pdf","source":{"id":"https://openalex.org/S142665794","display_name":"Theory of Computing","issn_l":"1557-2862","issn":["1557-2862"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310315690","host_organization_name":"University of Chicago","host_organization_lineage":["https://openalex.org/P4310315690"],"host_organization_lineage_names":["University of Chicago"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Theory of Computing","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[{"id":"https://openalex.org/G204445403","display_name":"CT-ISG: Cross-Leveraging Cryptography with Learning Theory","funder_award_id":"0716245","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"},{"id":"https://openalex.org/G4922521430","display_name":"AF: Small: Collaborative Research: The Polynomial Method for Learning","funder_award_id":"0915929","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"},{"id":"https://openalex.org/G5688190990","display_name":"Research in Games, Fixpoints, and Approximation","funder_award_id":"0728736","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"},{"id":"https://openalex.org/G5788850664","display_name":null,"funder_award_id":"HR0011-08-1-0069","funder_id":"https://openalex.org/F4320332180","funder_display_name":"Defense Advanced Research Projects Agency"},{"id":"https://openalex.org/G6093009578","display_name":null,"funder_award_id":"CCF-1115703","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"},{"id":"https://openalex.org/F4320323220","display_name":"Alexander S. Onassis Public Benefit Foundation","ror":"https://ror.org/017nq0d63"},{"id":"https://openalex.org/F4320332180","display_name":"Defense Advanced Research Projects Agency","ror":"https://ror.org/02caytj08"}],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W2402751180.pdf","grobid_xml":"https://content.openalex.org/works/W2402751180.grobid-xml"},"referenced_works_count":38,"referenced_works":["https://openalex.org/W24307859","https://openalex.org/W1497671620","https://openalex.org/W1499712944","https://openalex.org/W1542217987","https://openalex.org/W1988425761","https://openalex.org/W1997142544","https://openalex.org/W2005888001","https://openalex.org/W2042194938","https://openalex.org/W2060775054","https://openalex.org/W2063836706","https://openalex.org/W2078134499","https://openalex.org/W2078929687","https://openalex.org/W2083360043","https://openalex.org/W2097287345","https://openalex.org/W2097575812","https://openalex.org/W2098809394","https://openalex.org/W2102339446","https://openalex.org/W2103749128","https://openalex.org/W2106552140","https://openalex.org/W2116537669","https://openalex.org/W2138092283","https://openalex.org/W2148858749","https://openalex.org/W2156021398","https://openalex.org/W2166652312","https://openalex.org/W2172810949","https://openalex.org/W2179261203","https://openalex.org/W2327680971","https://openalex.org/W2568404849","https://openalex.org/W2610386774","https://openalex.org/W2611287588","https://openalex.org/W2885803187","https://openalex.org/W2914539165","https://openalex.org/W2951206707","https://openalex.org/W2964194804","https://openalex.org/W3100065552","https://openalex.org/W3102754027","https://openalex.org/W3151527484","https://openalex.org/W4226347946"],"related_works":["https://openalex.org/W3006682748","https://openalex.org/W1097487277","https://openalex.org/W2768610039","https://openalex.org/W4308265086","https://openalex.org/W2949964783","https://openalex.org/W2106889256","https://openalex.org/W2801357573","https://openalex.org/W2982363256","https://openalex.org/W2974000959","https://openalex.org/W3215977945"],"abstract_inverted_index":{"We":[0],"give":[1],"a":[2,32,51,55,69],"\"regularity":[3],"lemma\"":[4],"for":[5],"degree-d":[6,26],"polynomial":[7,66],"threshold":[8],"functions":[9],"(PTFs)":[10],"over":[11],"the":[12,42,59,65,73],"Boolean":[13],"cube":[14],"{-1,":[15],"1}":[16],"n":[17],".":[18],"Roughly":[19],"speaking,":[20],"this":[21],"result":[22],"shows":[23],"that":[24,38],"every":[25],"PTF":[27,53,56],"can":[28],"be":[29],"decomposed":[30],"into":[31],"constant":[33],"number":[34],"of":[35,41,61,72,76],"subfunctions":[36,43],"such":[37],"almost":[39],"all":[40],"are":[44],"close":[45],"to":[46],"being":[47],"regular":[48],"PTFs.":[49],"Here":[50],"\"regular\"":[52],"is":[54,68],"sign(p(x))":[57],"where":[58],"influence":[60,75],"each":[62],"variable":[63],"on":[64],"p(x)":[67],"small":[70],"fraction":[71],"total":[74],"p.":[77]},"counts_by_year":[{"year":2021,"cited_by_count":2},{"year":2019,"cited_by_count":1},{"year":2017,"cited_by_count":3},{"year":2015,"cited_by_count":2},{"year":2014,"cited_by_count":3},{"year":2013,"cited_by_count":1}],"updated_date":"2026-04-21T08:09:41.155169","created_date":"2016-06-24T00:00:00"}
