{"id":"https://openalex.org/W4312095238","doi":"https://doi.org/10.1162/neco_a_01558","title":"Optimal Quadratic Binding for Relational Reasoning in Vector Symbolic Neural Architectures","display_name":"Optimal Quadratic Binding for Relational Reasoning in Vector Symbolic Neural Architectures","publication_year":2022,"publication_date":"2022-12-22","ids":{"openalex":"https://openalex.org/W4312095238","doi":"https://doi.org/10.1162/neco_a_01558","pmid":"https://pubmed.ncbi.nlm.nih.gov/36543330"},"language":"en","primary_location":{"id":"doi:10.1162/neco_a_01558","is_oa":false,"landing_page_url":"https://doi.org/10.1162/neco_a_01558","pdf_url":null,"source":{"id":"https://openalex.org/S207023548","display_name":"Neural Computation","issn_l":"0899-7667","issn":["0899-7667","1530-888X"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310315718","host_organization_name":"The MIT Press","host_organization_lineage":["https://openalex.org/P4310315718","https://openalex.org/P4310316440"],"host_organization_lineage_names":["The MIT Press","Massachusetts Institute of Technology"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Neural Computation","raw_type":"journal-article"},"type":"article","indexed_in":["crossref","pubmed"],"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/A5091301888","display_name":"Naoki Hiratani","orcid":"https://orcid.org/0000-0002-8568-2033"},"institutions":[{"id":"https://openalex.org/I136199984","display_name":"Harvard University","ror":"https://ror.org/03vek6s52","country_code":"US","type":"education","lineage":["https://openalex.org/I136199984"]}],"countries":["US"],"is_corresponding":true,"raw_author_name":"Naoki Hiratani","raw_affiliation_strings":["Center for Brain Science, Harvard University, Cambridge MA 02138, U.S.A. n.hiratani@gmail.com"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Center for Brain Science, Harvard University, Cambridge MA 02138, U.S.A. n.hiratani@gmail.com","institution_ids":["https://openalex.org/I136199984"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5073266403","display_name":"Haim Sompolinsky","orcid":"https://orcid.org/0000-0002-0322-0629"},"institutions":[{"id":"https://openalex.org/I197251160","display_name":"Hebrew University of Jerusalem","ror":"https://ror.org/03qxff017","country_code":"IL","type":"education","lineage":["https://openalex.org/I197251160"]},{"id":"https://openalex.org/I2801851002","display_name":"Harvard University Press","ror":"https://ror.org/006v7bf86","country_code":"US","type":"other","lineage":["https://openalex.org/I136199984","https://openalex.org/I2801851002"]}],"countries":["IL","US"],"is_corresponding":true,"raw_author_name":"Haim Sompolinsky","raw_affiliation_strings":["Center for Brain Science, Harvard University, Cambridge MA 02138, U.S.A","Edmond and Lily Safra Center for Brain Sciences, Hebrew University, Jerusalem 91904, Israel haim@fiz.huji.ac.il"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Center for Brain Science, Harvard University, Cambridge MA 02138, U.S.A","institution_ids":["https://openalex.org/I2801851002"]},{"raw_affiliation_string":"Edmond and Lily Safra Center for Brain Sciences, Hebrew University, Jerusalem 91904, Israel haim@fiz.huji.ac.il","institution_ids":["https://openalex.org/I197251160"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":3,"corresponding_author_ids":["https://openalex.org/A5073266403","https://openalex.org/A5091301888"],"corresponding_institution_ids":["https://openalex.org/I136199984","https://openalex.org/I197251160","https://openalex.org/I2801851002"],"apc_list":null,"apc_paid":null,"fwci":0.4492,"has_fulltext":false,"cited_by_count":7,"citation_normalized_percentile":{"value":0.56901302,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":90,"max":97},"biblio":{"volume":"35","issue":"2","first_page":"105","last_page":"155"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T12808","display_name":"Ferroelectric and Negative Capacitance Devices","score":0.988099992275238,"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"}},"topics":[{"id":"https://openalex.org/T12808","display_name":"Ferroelectric and Negative Capacitance Devices","score":0.988099992275238,"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"}},{"id":"https://openalex.org/T10465","display_name":"Neurobiology of Language and Bilingualism","score":0.9868000149726868,"subfield":{"id":"https://openalex.org/subfields/2805","display_name":"Cognitive Neuroscience"},"field":{"id":"https://openalex.org/fields/28","display_name":"Neuroscience"},"domain":{"id":"https://openalex.org/domains/1","display_name":"Life Sciences"}},{"id":"https://openalex.org/T10320","display_name":"Neural Networks and Applications","score":0.9854000210762024,"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/quadratic-equation","display_name":"Quadratic equation","score":0.5212996006011963},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.4771149456501007},{"id":"https://openalex.org/keywords/representation","display_name":"Representation (politics)","score":0.46454882621765137},{"id":"https://openalex.org/keywords/artificial-neural-network","display_name":"Artificial neural network","score":0.4643875062465668},{"id":"https://openalex.org/keywords/theoretical-computer-science","display_name":"Theoretical computer science","score":0.4613964259624481},{"id":"https://openalex.org/keywords/matrix","display_name":"Matrix (chemical analysis)","score":0.44544875621795654},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.3826371729373932},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.3756040334701538},{"id":"https://openalex.org/keywords/artificial-intelligence","display_name":"Artificial intelligence","score":0.36306536197662354}],"concepts":[{"id":"https://openalex.org/C129844170","wikidata":"https://www.wikidata.org/wiki/Q41299","display_name":"Quadratic equation","level":2,"score":0.5212996006011963},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.4771149456501007},{"id":"https://openalex.org/C2776359362","wikidata":"https://www.wikidata.org/wiki/Q2145286","display_name":"Representation (politics)","level":3,"score":0.46454882621765137},{"id":"https://openalex.org/C50644808","wikidata":"https://www.wikidata.org/wiki/Q192776","display_name":"Artificial neural network","level":2,"score":0.4643875062465668},{"id":"https://openalex.org/C80444323","wikidata":"https://www.wikidata.org/wiki/Q2878974","display_name":"Theoretical computer science","level":1,"score":0.4613964259624481},{"id":"https://openalex.org/C106487976","wikidata":"https://www.wikidata.org/wiki/Q685816","display_name":"Matrix (chemical analysis)","level":2,"score":0.44544875621795654},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.3826371729373932},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.3756040334701538},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.36306536197662354},{"id":"https://openalex.org/C94625758","wikidata":"https://www.wikidata.org/wiki/Q7163","display_name":"Politics","level":2,"score":0.0},{"id":"https://openalex.org/C192562407","wikidata":"https://www.wikidata.org/wiki/Q228736","display_name":"Materials science","level":0,"score":0.0},{"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/C199539241","wikidata":"https://www.wikidata.org/wiki/Q7748","display_name":"Law","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}],"mesh":[{"descriptor_ui":"D001921","descriptor_name":"Brain","qualifier_ui":null,"qualifier_name":null,"is_major_topic":true},{"descriptor_ui":"D001921","descriptor_name":"Brain","qualifier_ui":null,"qualifier_name":null,"is_major_topic":true},{"descriptor_ui":"D001921","descriptor_name":"Brain","qualifier_ui":null,"qualifier_name":null,"is_major_topic":true},{"descriptor_ui":"D001921","descriptor_name":"Brain","qualifier_ui":null,"qualifier_name":null,"is_major_topic":true},{"descriptor_ui":"D007802","descriptor_name":"Language","qualifier_ui":null,"qualifier_name":null,"is_major_topic":false},{"descriptor_ui":"D007802","descriptor_name":"Language","qualifier_ui":null,"qualifier_name":null,"is_major_topic":false},{"descriptor_ui":"D007802","descriptor_name":"Language","qualifier_ui":null,"qualifier_name":null,"is_major_topic":false},{"descriptor_ui":"D007802","descriptor_name":"Language","qualifier_ui":null,"qualifier_name":null,"is_major_topic":false},{"descriptor_ui":"D011340","descriptor_name":"Problem Solving","qualifier_ui":null,"qualifier_name":null,"is_major_topic":true}],"locations_count":2,"locations":[{"id":"doi:10.1162/neco_a_01558","is_oa":false,"landing_page_url":"https://doi.org/10.1162/neco_a_01558","pdf_url":null,"source":{"id":"https://openalex.org/S207023548","display_name":"Neural Computation","issn_l":"0899-7667","issn":["0899-7667","1530-888X"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310315718","host_organization_name":"The MIT Press","host_organization_lineage":["https://openalex.org/P4310315718","https://openalex.org/P4310316440"],"host_organization_lineage_names":["The MIT Press","Massachusetts Institute of Technology"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Neural Computation","raw_type":"journal-article"},{"id":"pmid:36543330","is_oa":false,"landing_page_url":"https://pubmed.ncbi.nlm.nih.gov/36543330","pdf_url":null,"source":{"id":"https://openalex.org/S4306525036","display_name":"PubMed","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I1299303238","host_organization_name":"National Institutes of Health","host_organization_lineage":["https://openalex.org/I1299303238"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Neural computation","raw_type":"Journal Article"}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":50,"referenced_works":["https://openalex.org/W1488378957","https://openalex.org/W1512291242","https://openalex.org/W1529533208","https://openalex.org/W1615576556","https://openalex.org/W1996797695","https://openalex.org/W2003033403","https://openalex.org/W2007105730","https://openalex.org/W2013494846","https://openalex.org/W2016708835","https://openalex.org/W2033708181","https://openalex.org/W2042588857","https://openalex.org/W2050189255","https://openalex.org/W2070862086","https://openalex.org/W2104436836","https://openalex.org/W2157306293","https://openalex.org/W2166667242","https://openalex.org/W2167876429","https://openalex.org/W2579555219","https://openalex.org/W2604272474","https://openalex.org/W2809202707","https://openalex.org/W2921115127","https://openalex.org/W2949972983","https://openalex.org/W2963650979","https://openalex.org/W2963907629","https://openalex.org/W2990138404","https://openalex.org/W2992685560","https://openalex.org/W3093755763","https://openalex.org/W3094277857","https://openalex.org/W3103493959","https://openalex.org/W3140196993","https://openalex.org/W3168260338","https://openalex.org/W3169139819","https://openalex.org/W3196972418","https://openalex.org/W3214202924","https://openalex.org/W4205199576","https://openalex.org/W4214766902","https://openalex.org/W4234572080","https://openalex.org/W4235693166","https://openalex.org/W4236965008","https://openalex.org/W4287554891","https://openalex.org/W4301409532","https://openalex.org/W6632432105","https://openalex.org/W6636444753","https://openalex.org/W6640773114","https://openalex.org/W6678846912","https://openalex.org/W6731265811","https://openalex.org/W6743068658","https://openalex.org/W6752160822","https://openalex.org/W6800725549","https://openalex.org/W6804155940"],"related_works":["https://openalex.org/W2062195135","https://openalex.org/W2795079307","https://openalex.org/W2391251536","https://openalex.org/W2793058541","https://openalex.org/W2362198218","https://openalex.org/W1983629434","https://openalex.org/W1982750869","https://openalex.org/W2019521278","https://openalex.org/W1984922432","https://openalex.org/W2113077220"],"abstract_inverted_index":{"Binding":[0],"operation":[1],"is":[2,46,96],"fundamental":[3],"to":[4,40,152],"many":[5],"cognitive":[6,10],"processes,":[7,20],"such":[8,24],"as":[9,25,175,177],"map":[11],"formation,":[12],"relational":[13,92],"reasoning,":[14],"and":[15,27,30,34,36,180],"language":[16],"comprehension.":[17],"In":[18],"these":[19,125],"two":[21],"different":[22],"modalities,":[23],"location":[26],"objects,":[28],"events":[29],"their":[31,37],"contextual":[32],"cues,":[33],"words":[35],"roles,":[38],"need":[39],"be":[41],"bound":[42,65,167],"together,":[43],"but":[44],"little":[45],"known":[47,134],"about":[48],"the":[49,80,97,178,199],"underlying":[50],"neural":[51,193],"mechanisms.":[52],"Previous":[53],"work":[54],"has":[55],"introduced":[56],"a":[57,102,110,128,137,148,163,170],"binding":[58,106,149,173,183,196],"model":[59],"based":[60,108],"on":[61,75,109],"quadratic":[62,86,172],"functions":[63],"of":[64,71,85,105,113,119,140,147,166,195],"pairs,":[66,168],"followed":[67],"by":[68],"vector":[69],"summation":[70],"multiple":[72],"pairs.":[73],"Based":[74],"this":[76,153],"framework,":[77],"we":[78],"address":[79],"following":[81],"questions:":[82],"Which":[83],"classes":[84],"matrices":[87,107,126],"are":[88,142,162],"optimal":[89],"for":[90],"decoding":[91],"structures?":[93],"And":[94],"what":[95],"resultant":[98],"accuracy?":[99],"We":[100,122,156],"introduce":[101],"new":[103,189],"class":[104],"matrix":[111],"representation":[112],"octonion":[114,154,179],"algebra,":[115],"an":[116],"eight-dimensional":[117],"extension":[118],"complex":[120],"numbers.":[121],"show":[123,158],"that":[124,159],"enable":[127],"more":[129],"accurate":[130],"unbinding":[131],"than":[132],"previously":[133,181],"methods":[135],"when":[136,160],"small":[138],"number":[139,165],"pairs":[141],"present.":[143],"Moreover,":[144],"numerical":[145],"optimization":[146],"operator":[150],"converges":[151],"binding.":[155],"also":[157],"there":[161],"large":[164],"however,":[169],"random":[171],"performs,":[174],"well":[176],"proposed":[182],"methods.":[184],"This":[185],"study":[186],"thus":[187],"provides":[188],"insight":[190],"into":[191],"potential":[192],"mechanisms":[194],"operations":[197],"in":[198],"brain.":[200]},"counts_by_year":[{"year":2026,"cited_by_count":1},{"year":2024,"cited_by_count":1},{"year":2023,"cited_by_count":3},{"year":2022,"cited_by_count":2}],"updated_date":"2026-08-21T09:56:20.448147","created_date":"2025-10-10T00:00:00"}
