{"id":"https://openalex.org/W2053955420","doi":"https://doi.org/10.1016/s0166-218x(98)00014-6","title":"Vapnik-Chervonenkis dimension of recurrent neural networks","display_name":"Vapnik-Chervonenkis dimension of recurrent neural networks","publication_year":1998,"publication_date":"1998-08-01","ids":{"openalex":"https://openalex.org/W2053955420","doi":"https://doi.org/10.1016/s0166-218x(98)00014-6","mag":"2053955420"},"language":"en","primary_location":{"id":"doi:10.1016/s0166-218x(98)00014-6","is_oa":false,"landing_page_url":"https://doi.org/10.1016/s0166-218x(98)00014-6","pdf_url":null,"source":{"id":"https://openalex.org/S171741597","display_name":"Discrete Applied Mathematics","issn_l":"0166-218X","issn":["0166-218X","1872-6771"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320990","host_organization_name":"Elsevier BV","host_organization_lineage":["https://openalex.org/P4310320990"],"host_organization_lineage_names":["Elsevier BV"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Discrete Applied Mathematics","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/A5105512963","display_name":"Pascal Koiran","orcid":"https://orcid.org/0000-0003-3405-7155"},"institutions":[{"id":"https://openalex.org/I113428412","display_name":"\u00c9cole Normale Sup\u00e9rieure de Lyon","ror":"https://ror.org/04zmssz18","country_code":"FR","type":"education","lineage":["https://openalex.org/I113428412","https://openalex.org/I203339264"]},{"id":"https://openalex.org/I1294671590","display_name":"Centre National de la Recherche Scientifique","ror":"https://ror.org/02feahw73","country_code":"FR","type":"government","lineage":["https://openalex.org/I1294671590"]},{"id":"https://openalex.org/I4210144566","display_name":"Laboratoire de l'Informatique du Parall\u00e9lisme","ror":"https://ror.org/04msnz457","country_code":"FR","type":"facility","lineage":["https://openalex.org/I100532134","https://openalex.org/I113428412","https://openalex.org/I1294671590","https://openalex.org/I1326498283","https://openalex.org/I203339264","https://openalex.org/I203339264","https://openalex.org/I4210144566"]}],"countries":["FR"],"is_corresponding":true,"raw_author_name":"Pascal Koiran","raw_affiliation_strings":["Laboratoire de l'Informatique du Parall\u00e9lisme, Ecole Normale Sup\u00e9rieure de Lyon \u2014 CNRS, 46 all\u00e9e d'Italie, 69364 Lyon Cedex 07, France","LIP - Laboratoire de l'Informatique du Parall\u00e9lisme (46 All\u00e9e d'Italie 69364 LYON CEDEX 07 - France)","MC2 - Mod\u00e8les de calcul, Complexit\u00e9, Combinatoire (ENS Lyon 46 all\u00e9e d'Italie 69364 Lyon Cedex 07 - France)"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Laboratoire de l'Informatique du Parall\u00e9lisme, Ecole Normale Sup\u00e9rieure de Lyon \u2014 CNRS, 46 all\u00e9e d'Italie, 69364 Lyon Cedex 07, France","institution_ids":["https://openalex.org/I4210144566","https://openalex.org/I113428412","https://openalex.org/I1294671590"]},{"raw_affiliation_string":"LIP - Laboratoire de l'Informatique du Parall\u00e9lisme (46 All\u00e9e d'Italie 69364 LYON CEDEX 07 - France)","institution_ids":["https://openalex.org/I4210144566","https://openalex.org/I113428412"]},{"raw_affiliation_string":"MC2 - Mod\u00e8les de calcul, Complexit\u00e9, Combinatoire (ENS Lyon 46 all\u00e9e d'Italie 69364 Lyon Cedex 07 - France)","institution_ids":["https://openalex.org/I113428412"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5006126782","display_name":"Eduardo D. Sontag","orcid":null},"institutions":[{"id":"https://openalex.org/I102322142","display_name":"Rutgers, The State University of New Jersey","ror":"https://ror.org/05vt9qd57","country_code":"US","type":"education","lineage":["https://openalex.org/I102322142"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Eduardo D. Sontag","raw_affiliation_strings":["Department of Mathematics, Rutgers University, New Brunswick, NJ 08903, USA","Department of Mathematics Rutgers University (United States)"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics, Rutgers University, New Brunswick, NJ 08903, USA","institution_ids":["https://openalex.org/I102322142"]},{"raw_affiliation_string":"Department of Mathematics Rutgers University (United States)","institution_ids":["https://openalex.org/I102322142"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":2,"corresponding_author_ids":["https://openalex.org/A5105512963"],"corresponding_institution_ids":["https://openalex.org/I113428412","https://openalex.org/I1294671590","https://openalex.org/I4210144566"],"apc_list":{"value":2750,"currency":"USD","value_usd":2750},"apc_paid":null,"fwci":6.0483,"has_fulltext":false,"cited_by_count":47,"citation_normalized_percentile":{"value":0.96322679,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":89,"max":99},"biblio":{"volume":"86","issue":"1","first_page":"63","last_page":"79"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T12072","display_name":"Machine Learning and Algorithms","score":0.9998999834060669,"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/T12072","display_name":"Machine Learning and Algorithms","score":0.9998999834060669,"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/T12535","display_name":"Machine Learning and Data Classification","score":0.9887999892234802,"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/T10720","display_name":"Complexity and Algorithms in Graphs","score":0.9807000160217285,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/dimension","display_name":"Dimension (graph theory)","score":0.7629972696304321},{"id":"https://openalex.org/keywords/polynomial","display_name":"Polynomial","score":0.6502848863601685},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.631969690322876},{"id":"https://openalex.org/keywords/piecewise","display_name":"Piecewise","score":0.614898145198822},{"id":"https://openalex.org/keywords/sigmoid-function","display_name":"Sigmoid function","score":0.5754493474960327},{"id":"https://openalex.org/keywords/activation-function","display_name":"Activation function","score":0.5570990443229675},{"id":"https://openalex.org/keywords/artificial-neural-network","display_name":"Artificial neural network","score":0.5395393371582031},{"id":"https://openalex.org/keywords/vc-dimension","display_name":"VC dimension","score":0.535313606262207},{"id":"https://openalex.org/keywords/feed-forward","display_name":"Feed forward","score":0.5340795516967773},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.5060469508171082},{"id":"https://openalex.org/keywords/feedforward-neural-network","display_name":"Feedforward neural network","score":0.48249387741088867},{"id":"https://openalex.org/keywords/multiplicative-function","display_name":"Multiplicative function","score":0.4592897593975067},{"id":"https://openalex.org/keywords/function","display_name":"Function (biology)","score":0.4432869255542755},{"id":"https://openalex.org/keywords/sequence","display_name":"Sequence (biology)","score":0.4158342480659485},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.36760610342025757},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.2893075942993164},{"id":"https://openalex.org/keywords/artificial-intelligence","display_name":"Artificial intelligence","score":0.16005143523216248},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.12039116024971008}],"concepts":[{"id":"https://openalex.org/C33676613","wikidata":"https://www.wikidata.org/wiki/Q13415176","display_name":"Dimension (graph theory)","level":2,"score":0.7629972696304321},{"id":"https://openalex.org/C90119067","wikidata":"https://www.wikidata.org/wiki/Q43260","display_name":"Polynomial","level":2,"score":0.6502848863601685},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.631969690322876},{"id":"https://openalex.org/C164660894","wikidata":"https://www.wikidata.org/wiki/Q2037833","display_name":"Piecewise","level":2,"score":0.614898145198822},{"id":"https://openalex.org/C81388566","wikidata":"https://www.wikidata.org/wiki/Q526668","display_name":"Sigmoid function","level":3,"score":0.5754493474960327},{"id":"https://openalex.org/C38365724","wikidata":"https://www.wikidata.org/wiki/Q4677469","display_name":"Activation function","level":3,"score":0.5570990443229675},{"id":"https://openalex.org/C50644808","wikidata":"https://www.wikidata.org/wiki/Q192776","display_name":"Artificial neural network","level":2,"score":0.5395393371582031},{"id":"https://openalex.org/C119322782","wikidata":"https://www.wikidata.org/wiki/Q2662236","display_name":"VC dimension","level":2,"score":0.535313606262207},{"id":"https://openalex.org/C38858127","wikidata":"https://www.wikidata.org/wiki/Q5441228","display_name":"Feed forward","level":2,"score":0.5340795516967773},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.5060469508171082},{"id":"https://openalex.org/C47702885","wikidata":"https://www.wikidata.org/wiki/Q5441227","display_name":"Feedforward neural network","level":3,"score":0.48249387741088867},{"id":"https://openalex.org/C42747912","wikidata":"https://www.wikidata.org/wiki/Q1048447","display_name":"Multiplicative function","level":2,"score":0.4592897593975067},{"id":"https://openalex.org/C14036430","wikidata":"https://www.wikidata.org/wiki/Q3736076","display_name":"Function (biology)","level":2,"score":0.4432869255542755},{"id":"https://openalex.org/C2778112365","wikidata":"https://www.wikidata.org/wiki/Q3511065","display_name":"Sequence (biology)","level":2,"score":0.4158342480659485},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.36760610342025757},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.2893075942993164},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.16005143523216248},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.12039116024971008},{"id":"https://openalex.org/C78458016","wikidata":"https://www.wikidata.org/wiki/Q840400","display_name":"Evolutionary biology","level":1,"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/C54355233","wikidata":"https://www.wikidata.org/wiki/Q7162","display_name":"Genetics","level":1,"score":0.0},{"id":"https://openalex.org/C127413603","wikidata":"https://www.wikidata.org/wiki/Q11023","display_name":"Engineering","level":0,"score":0.0},{"id":"https://openalex.org/C133731056","wikidata":"https://www.wikidata.org/wiki/Q4917288","display_name":"Control engineering","level":1,"score":0.0}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.1016/s0166-218x(98)00014-6","is_oa":false,"landing_page_url":"https://doi.org/10.1016/s0166-218x(98)00014-6","pdf_url":null,"source":{"id":"https://openalex.org/S171741597","display_name":"Discrete Applied Mathematics","issn_l":"0166-218X","issn":["0166-218X","1872-6771"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320990","host_organization_name":"Elsevier BV","host_organization_lineage":["https://openalex.org/P4310320990"],"host_organization_lineage_names":["Elsevier BV"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Discrete Applied Mathematics","raw_type":"journal-article"},{"id":"pmh:oai:HAL:hal-04649633v1","is_oa":false,"landing_page_url":"https://hal.science/hal-04649633","pdf_url":null,"source":{"id":"https://openalex.org/S4306402512","display_name":"HAL (Le Centre pour la Communication Scientifique Directe)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I1294671590","host_organization_name":"Centre National de la Recherche Scientifique","host_organization_lineage":["https://openalex.org/I1294671590"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Discrete Applied Mathematics, 1998, 86 (1), pp.63-79. &#x27E8;10.1016/S0166-218X(98)00014-6&#x27E9;","raw_type":"Journal articles"}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":22,"referenced_works":["https://openalex.org/W1484739396","https://openalex.org/W1575881144","https://openalex.org/W1964826896","https://openalex.org/W1988165767","https://openalex.org/W1994530392","https://openalex.org/W2006698588","https://openalex.org/W2019671617","https://openalex.org/W2031585281","https://openalex.org/W2050778826","https://openalex.org/W2050946712","https://openalex.org/W2067619114","https://openalex.org/W2116664051","https://openalex.org/W2128084896","https://openalex.org/W2137597499","https://openalex.org/W2154952480","https://openalex.org/W2155020618","https://openalex.org/W2156582723","https://openalex.org/W2165758113","https://openalex.org/W2728988083","https://openalex.org/W4249207878","https://openalex.org/W6677221974","https://openalex.org/W6680350967"],"related_works":["https://openalex.org/W3024979424","https://openalex.org/W4283785902","https://openalex.org/W3032499992","https://openalex.org/W2041004593","https://openalex.org/W3134817226","https://openalex.org/W2895192346","https://openalex.org/W3110577345","https://openalex.org/W3136579697","https://openalex.org/W2086501033","https://openalex.org/W2557823583"],"abstract_inverted_index":null,"counts_by_year":[{"year":2026,"cited_by_count":1},{"year":2025,"cited_by_count":2},{"year":2024,"cited_by_count":6},{"year":2023,"cited_by_count":2},{"year":2022,"cited_by_count":3},{"year":2021,"cited_by_count":1},{"year":2020,"cited_by_count":1},{"year":2019,"cited_by_count":6},{"year":2018,"cited_by_count":2},{"year":2017,"cited_by_count":1},{"year":2015,"cited_by_count":1},{"year":2012,"cited_by_count":2}],"updated_date":"2026-05-07T13:39:58.223016","created_date":"2025-10-10T00:00:00"}
