{"id":"https://openalex.org/W1992781176","doi":"https://doi.org/10.1145/2213977.2214035","title":"Reconstruction of depth-4 multilinear circuits with top fan-in 2","display_name":"Reconstruction of depth-4 multilinear circuits with top fan-in 2","publication_year":2012,"publication_date":"2012-05-19","ids":{"openalex":"https://openalex.org/W1992781176","doi":"https://doi.org/10.1145/2213977.2214035","mag":"1992781176"},"language":"en","primary_location":{"id":"doi:10.1145/2213977.2214035","is_oa":false,"landing_page_url":"https://doi.org/10.1145/2213977.2214035","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":["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/A5001183875","display_name":"Ankit Gupta","orcid":"https://orcid.org/0000-0002-5628-3623"},"institutions":[{"id":"https://openalex.org/I4210124949","display_name":"Microsoft Research (India)","ror":"https://ror.org/02w7f3w92","country_code":"IN","type":"company","lineage":["https://openalex.org/I1290206253","https://openalex.org/I4210124949"]},{"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"]}],"countries":["IN","US"],"is_corresponding":true,"raw_author_name":"Ankit Gupta","raw_affiliation_strings":["Microsoft Research India, Bangalore, India","Microsoft Research india, Bangalore, India#TAB#"],"affiliations":[{"raw_affiliation_string":"Microsoft Research India, Bangalore, India","institution_ids":["https://openalex.org/I4210124949"]},{"raw_affiliation_string":"Microsoft Research india, Bangalore, India#TAB#","institution_ids":["https://openalex.org/I1290206253"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5079701530","display_name":"Neeraj Kayal","orcid":null},"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/I4210124949","display_name":"Microsoft Research (India)","ror":"https://ror.org/02w7f3w92","country_code":"IN","type":"company","lineage":["https://openalex.org/I1290206253","https://openalex.org/I4210124949"]}],"countries":["IN","US"],"is_corresponding":false,"raw_author_name":"Neeraj Kayal","raw_affiliation_strings":["Microsoft Research India, Bangalore, India","Microsoft Research india, Bangalore, India#TAB#"],"affiliations":[{"raw_affiliation_string":"Microsoft Research India, Bangalore, India","institution_ids":["https://openalex.org/I4210124949"]},{"raw_affiliation_string":"Microsoft Research india, Bangalore, India#TAB#","institution_ids":["https://openalex.org/I1290206253"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5058727227","display_name":"Satya Lokam","orcid":null},"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/I4210124949","display_name":"Microsoft Research (India)","ror":"https://ror.org/02w7f3w92","country_code":"IN","type":"company","lineage":["https://openalex.org/I1290206253","https://openalex.org/I4210124949"]}],"countries":["IN","US"],"is_corresponding":false,"raw_author_name":"Satya Lokam","raw_affiliation_strings":["Microsoft Research India, Bangalore, India","Microsoft Research india, Bangalore, India#TAB#"],"affiliations":[{"raw_affiliation_string":"Microsoft Research India, Bangalore, India","institution_ids":["https://openalex.org/I4210124949"]},{"raw_affiliation_string":"Microsoft Research india, Bangalore, India#TAB#","institution_ids":["https://openalex.org/I1290206253"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":3,"corresponding_author_ids":["https://openalex.org/A5001183875"],"corresponding_institution_ids":["https://openalex.org/I1290206253","https://openalex.org/I4210124949"],"apc_list":null,"apc_paid":null,"fwci":2.0732,"has_fulltext":false,"cited_by_count":15,"citation_normalized_percentile":{"value":0.865237,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":99},"biblio":{"volume":null,"issue":null,"first_page":"625","last_page":"642"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10720","display_name":"Complexity and Algorithms in Graphs","score":0.9998999834060669,"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"}},"topics":[{"id":"https://openalex.org/T10720","display_name":"Complexity and Algorithms in Graphs","score":0.9998999834060669,"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":0.9995999932289124,"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/T10237","display_name":"Cryptography and Data Security","score":0.9987999796867371,"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/multilinear-map","display_name":"Multilinear map","score":0.9664185643196106},{"id":"https://openalex.org/keywords/electronic-circuit","display_name":"Electronic circuit","score":0.7415809631347656},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.5657252073287964},{"id":"https://openalex.org/keywords/bounded-function","display_name":"Bounded function","score":0.562985897064209},{"id":"https://openalex.org/keywords/polynomial","display_name":"Polynomial","score":0.5406138300895691},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.4455554485321045},{"id":"https://openalex.org/keywords/fan-in","display_name":"Fan-in","score":0.4427308440208435},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.41625893115997314},{"id":"https://openalex.org/keywords/arithmetic","display_name":"Arithmetic","score":0.38674718141555786},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.36899328231811523},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.130202978849411},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.11165249347686768},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.06226611137390137}],"concepts":[{"id":"https://openalex.org/C84392682","wikidata":"https://www.wikidata.org/wiki/Q1952404","display_name":"Multilinear map","level":2,"score":0.9664185643196106},{"id":"https://openalex.org/C134146338","wikidata":"https://www.wikidata.org/wiki/Q1815901","display_name":"Electronic circuit","level":2,"score":0.7415809631347656},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.5657252073287964},{"id":"https://openalex.org/C34388435","wikidata":"https://www.wikidata.org/wiki/Q2267362","display_name":"Bounded function","level":2,"score":0.562985897064209},{"id":"https://openalex.org/C90119067","wikidata":"https://www.wikidata.org/wiki/Q43260","display_name":"Polynomial","level":2,"score":0.5406138300895691},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.4455554485321045},{"id":"https://openalex.org/C179431463","wikidata":"https://www.wikidata.org/wiki/Q1760638","display_name":"Fan-in","level":2,"score":0.4427308440208435},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.41625893115997314},{"id":"https://openalex.org/C94375191","wikidata":"https://www.wikidata.org/wiki/Q11205","display_name":"Arithmetic","level":1,"score":0.38674718141555786},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.36899328231811523},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.130202978849411},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.11165249347686768},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.06226611137390137},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.0}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.1145/2213977.2214035","is_oa":false,"landing_page_url":"https://doi.org/10.1145/2213977.2214035","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:CiteSeerX.psu:10.1.1.954.1519","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.954.1519","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://research.microsoft.com/pubs/158918/spsp2%20reconstruction%20full.pdf","raw_type":"text"}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":34,"referenced_works":["https://openalex.org/W64656188","https://openalex.org/W639289183","https://openalex.org/W1503731768","https://openalex.org/W1518341038","https://openalex.org/W1571321110","https://openalex.org/W1599026900","https://openalex.org/W1896605895","https://openalex.org/W1967348711","https://openalex.org/W1968245619","https://openalex.org/W1974321539","https://openalex.org/W1996798784","https://openalex.org/W1998853590","https://openalex.org/W2017779408","https://openalex.org/W2018202033","https://openalex.org/W2023541349","https://openalex.org/W2026944800","https://openalex.org/W2050570613","https://openalex.org/W2055425281","https://openalex.org/W2067476801","https://openalex.org/W2080132708","https://openalex.org/W2106221905","https://openalex.org/W2109995168","https://openalex.org/W2121894367","https://openalex.org/W2128510017","https://openalex.org/W2137149976","https://openalex.org/W2140994271","https://openalex.org/W2148132824","https://openalex.org/W2172394696","https://openalex.org/W2172695686","https://openalex.org/W2195982063","https://openalex.org/W2611339529","https://openalex.org/W2740986359","https://openalex.org/W3192502741","https://openalex.org/W6635055964"],"related_works":["https://openalex.org/W4318719034","https://openalex.org/W37958683","https://openalex.org/W64890674","https://openalex.org/W2023541349","https://openalex.org/W295046251","https://openalex.org/W2150144243","https://openalex.org/W2024334120","https://openalex.org/W2403586276","https://openalex.org/W2100207144","https://openalex.org/W2913743598"],"abstract_inverted_index":{"We":[0],"present":[1],"a":[2,29,36,124],"randomized":[3],"algorithm":[4,23],"for":[5,66,79,86,134],"reconstructing":[6],"multilinear":[7,11,37,47],"\u03a3\u03a0\u03a3\u03a0(2)":[8,38,48],"circuits,":[9],"i.e.,":[10],"depth-4":[12,70],"circuits":[13,82,89,96,99],"with":[14,100,109],"fan-in":[15,102],"2":[16],"at":[17],"the":[18,62,117],"top":[19,101],"+":[20],"gate.":[21],"The":[22],"is":[24,61],"given":[25],"blackbox":[26],"access":[27],"to":[28,74],"polynomial":[30,125],"f":[31],"\u2208":[32],"F[x1,...,xn]":[33],"computable":[34],"by":[35],"circuit":[39],"of":[40,69,120],"size":[41],"s":[42],"and":[43,54,107,129],"outputs":[44],"an":[45],"equivalent":[46],"circuit,":[49],"runs":[50],"in":[51],"time":[52],"poly(n,s),":[53],"works":[55],"over":[56],"any":[57,67],"field":[58],"F.":[59],"This":[60],"first":[63],"reconstruction":[64,77],"result":[65],"model":[68],"arithmetic":[71,88,98],"circuits.":[72],"Prior":[73],"our":[75],"work,":[76],"results":[78],"bounded":[80],"depth":[81],"were":[83],"known":[84],"only":[85],"depth-2":[87],"(Klivans":[90],"&":[91,112],"Spielman,":[92],"STOC":[93,105],"2001),":[94],"\u03a3\u03a0\u03a3(2)":[95],"(depth-3":[97],"2)":[103],"(Shpilka,":[104],"2007),":[106],"\u03a3\u03a0\u03a3(k)":[108],"k=O(1)":[110],"(Karnin":[111],"Shpilka,":[113],"CCC":[114],"2009).":[115],"Moreover,":[116],"running":[118],"times":[119],"these":[121],"algorithms":[122],"have":[123],"dependence":[126],"on":[127],"|F|":[128],"hence":[130],"do":[131],"not":[132],"work":[133],"infinite":[135],"fields":[136],"such":[137],"as":[138],"Q.":[139]},"counts_by_year":[{"year":2026,"cited_by_count":1},{"year":2021,"cited_by_count":3},{"year":2020,"cited_by_count":1},{"year":2018,"cited_by_count":1},{"year":2017,"cited_by_count":2},{"year":2016,"cited_by_count":1},{"year":2015,"cited_by_count":1},{"year":2014,"cited_by_count":4},{"year":2013,"cited_by_count":1}],"updated_date":"2026-04-05T17:49:38.594831","created_date":"2025-10-10T00:00:00"}
