{"id":"https://openalex.org/W1592122239","doi":"https://doi.org/10.1007/3-540-62685-9_19","title":"Vapnik-Chervonenkis dimension of recurrent neural networks","display_name":"Vapnik-Chervonenkis dimension of recurrent neural networks","publication_year":1997,"publication_date":"1997-01-01","ids":{"openalex":"https://openalex.org/W1592122239","doi":"https://doi.org/10.1007/3-540-62685-9_19","mag":"1592122239"},"language":"en","primary_location":{"id":"doi:10.1007/3-540-62685-9_19","is_oa":false,"landing_page_url":"https://doi.org/10.1007/3-540-62685-9_19","pdf_url":null,"source":{"id":"https://openalex.org/S106296714","display_name":"Lecture notes in computer science","issn_l":"0302-9743","issn":["0302-9743","1611-3349"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319900","host_organization_name":"Springer Science+Business Media","host_organization_lineage":["https://openalex.org/P4310319900","https://openalex.org/P4310319965"],"host_organization_lineage_names":["Springer Science+Business Media","Springer Nature"],"type":"book series"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Lecture Notes in Computer Science","raw_type":"book-chapter"},"type":"book-chapter","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":false,"raw_author_name":"Pascal Koiran","raw_affiliation_strings":["Laboratoire de l'Informatique du Parall\u00e9lisme Ecole Normale Sup\u00e9rieure de Lyon, 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, 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"]},{"id":"https://openalex.org/I2799411422","display_name":"Center for Discrete Mathematics and Theoretical Computer Science","ror":"https://ror.org/00k551w06","country_code":"US","type":"facility","lineage":["https://openalex.org/I102322142","https://openalex.org/I118347220","https://openalex.org/I1283103587","https://openalex.org/I1311060795","https://openalex.org/I20089843","https://openalex.org/I2799411422","https://openalex.org/I4210107353"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Eduardo D. Sontag","raw_affiliation_strings":["Department of Mathematics, Rutgers University, 08903, New Brunswick, NJ, USA","DIMACS - Center for Discrete Mathematics and Theoretical Computer Science [Rutgers] (DIMACS Center/CoRE Building/4th Floor Rutgers University 96 Frelinghuysen Road Piscataway, NJ 08854-8018 - United States)"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics, Rutgers University, 08903, New Brunswick, NJ, USA","institution_ids":["https://openalex.org/I102322142"]},{"raw_affiliation_string":"DIMACS - Center for Discrete Mathematics and Theoretical Computer Science [Rutgers] (DIMACS Center/CoRE Building/4th Floor Rutgers University 96 Frelinghuysen Road Piscataway, NJ 08854-8018 - United States)","institution_ids":["https://openalex.org/I2799411422","https://openalex.org/I102322142"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":2,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":{"value":5000,"currency":"EUR","value_usd":5392},"apc_paid":null,"fwci":2.15,"has_fulltext":false,"cited_by_count":19,"citation_normalized_percentile":{"value":0.87074007,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":90,"max":97},"biblio":{"volume":null,"issue":null,"first_page":"223","last_page":"237"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10320","display_name":"Neural Networks and Applications","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"}},"topics":[{"id":"https://openalex.org/T10320","display_name":"Neural Networks and Applications","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/T10820","display_name":"Fuzzy Logic and Control Systems","score":0.9922000169754028,"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/T11206","display_name":"Model Reduction and Neural Networks","score":0.9660999774932861,"subfield":{"id":"https://openalex.org/subfields/3109","display_name":"Statistical and Nonlinear Physics"},"field":{"id":"https://openalex.org/fields/31","display_name":"Physics and Astronomy"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/dimension","display_name":"Dimension (graph theory)","score":0.7134670615196228},{"id":"https://openalex.org/keywords/feed-forward","display_name":"Feed forward","score":0.6571760177612305},{"id":"https://openalex.org/keywords/piecewise","display_name":"Piecewise","score":0.6196841597557068},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.6100846529006958},{"id":"https://openalex.org/keywords/artificial-neural-network","display_name":"Artificial neural network","score":0.6080184578895569},{"id":"https://openalex.org/keywords/feedforward-neural-network","display_name":"Feedforward neural network","score":0.5870535373687744},{"id":"https://openalex.org/keywords/activation-function","display_name":"Activation function","score":0.5810969471931458},{"id":"https://openalex.org/keywords/polynomial","display_name":"Polynomial","score":0.5438050627708435},{"id":"https://openalex.org/keywords/multiplicative-function","display_name":"Multiplicative function","score":0.5372661352157593},{"id":"https://openalex.org/keywords/recurrent-neural-network","display_name":"Recurrent neural network","score":0.5095253586769104},{"id":"https://openalex.org/keywords/sigmoid-function","display_name":"Sigmoid function","score":0.4960411489009857},{"id":"https://openalex.org/keywords/sequence","display_name":"Sequence (biology)","score":0.48198097944259644},{"id":"https://openalex.org/keywords/function","display_name":"Function (biology)","score":0.4463391602039337},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.3654860854148865},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.3342773914337158},{"id":"https://openalex.org/keywords/artificial-intelligence","display_name":"Artificial intelligence","score":0.21977034211158752},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.18218016624450684},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.057469069957733154}],"concepts":[{"id":"https://openalex.org/C33676613","wikidata":"https://www.wikidata.org/wiki/Q13415176","display_name":"Dimension (graph theory)","level":2,"score":0.7134670615196228},{"id":"https://openalex.org/C38858127","wikidata":"https://www.wikidata.org/wiki/Q5441228","display_name":"Feed forward","level":2,"score":0.6571760177612305},{"id":"https://openalex.org/C164660894","wikidata":"https://www.wikidata.org/wiki/Q2037833","display_name":"Piecewise","level":2,"score":0.6196841597557068},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.6100846529006958},{"id":"https://openalex.org/C50644808","wikidata":"https://www.wikidata.org/wiki/Q192776","display_name":"Artificial neural network","level":2,"score":0.6080184578895569},{"id":"https://openalex.org/C47702885","wikidata":"https://www.wikidata.org/wiki/Q5441227","display_name":"Feedforward neural network","level":3,"score":0.5870535373687744},{"id":"https://openalex.org/C38365724","wikidata":"https://www.wikidata.org/wiki/Q4677469","display_name":"Activation function","level":3,"score":0.5810969471931458},{"id":"https://openalex.org/C90119067","wikidata":"https://www.wikidata.org/wiki/Q43260","display_name":"Polynomial","level":2,"score":0.5438050627708435},{"id":"https://openalex.org/C42747912","wikidata":"https://www.wikidata.org/wiki/Q1048447","display_name":"Multiplicative function","level":2,"score":0.5372661352157593},{"id":"https://openalex.org/C147168706","wikidata":"https://www.wikidata.org/wiki/Q1457734","display_name":"Recurrent neural network","level":3,"score":0.5095253586769104},{"id":"https://openalex.org/C81388566","wikidata":"https://www.wikidata.org/wiki/Q526668","display_name":"Sigmoid function","level":3,"score":0.4960411489009857},{"id":"https://openalex.org/C2778112365","wikidata":"https://www.wikidata.org/wiki/Q3511065","display_name":"Sequence (biology)","level":2,"score":0.48198097944259644},{"id":"https://openalex.org/C14036430","wikidata":"https://www.wikidata.org/wiki/Q3736076","display_name":"Function (biology)","level":2,"score":0.4463391602039337},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.3654860854148865},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.3342773914337158},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.21977034211158752},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.18218016624450684},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.057469069957733154},{"id":"https://openalex.org/C54355233","wikidata":"https://www.wikidata.org/wiki/Q7162","display_name":"Genetics","level":1,"score":0.0},{"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/C133731056","wikidata":"https://www.wikidata.org/wiki/Q4917288","display_name":"Control engineering","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/C127413603","wikidata":"https://www.wikidata.org/wiki/Q11023","display_name":"Engineering","level":0,"score":0.0}],"mesh":[],"locations_count":7,"locations":[{"id":"doi:10.1007/3-540-62685-9_19","is_oa":false,"landing_page_url":"https://doi.org/10.1007/3-540-62685-9_19","pdf_url":null,"source":{"id":"https://openalex.org/S106296714","display_name":"Lecture notes in computer science","issn_l":"0302-9743","issn":["0302-9743","1611-3349"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319900","host_organization_name":"Springer Science+Business Media","host_organization_lineage":["https://openalex.org/P4310319900","https://openalex.org/P4310319965"],"host_organization_lineage_names":["Springer Science+Business Media","Springer Nature"],"type":"book series"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Lecture Notes in Computer Science","raw_type":"book-chapter"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.50.7595","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.50.7595","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://www.math.rutgers.edu/~sontag/FTP_DIR/koiran-dam.ps.gz","raw_type":"text"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.53.5530","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.53.5530","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"ftp://dimacs.rutgers.edu/pub/dimacs/TechnicalReports/TechReports/1996/96-56.ps.gz","raw_type":"text"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.57.3578","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.57.3578","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://www.math.rutgers.edu/~sontag/FTP_DIR/97eurocolt.ps.gz","raw_type":"text"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.75.3126","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.75.3126","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://www.math.rutgers.edu/~sontag/FTP_DIR/koiran-dam.pdf","raw_type":"text"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.86.113","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.86.113","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://www.math.rutgers.edu/pub/sontag/97eurocolt.pdf","raw_type":"text"},{"id":"pmh:oai:HAL:hal-04655363v1","is_oa":false,"landing_page_url":"https://hal.science/hal-04655363","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":"EuroCOLT 1997, Mar 1997, Jerusalem, Israel. pp.223-237, ⟨10.1007/3-540-62685-9_19⟩","raw_type":"Conference papers"}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":20,"referenced_works":["https://openalex.org/W1484739396","https://openalex.org/W1575881144","https://openalex.org/W1964826896","https://openalex.org/W1994530392","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/W2139906513","https://openalex.org/W2154952480","https://openalex.org/W2155020618","https://openalex.org/W2156582723","https://openalex.org/W2165758113","https://openalex.org/W2728988083","https://openalex.org/W4213416190","https://openalex.org/W6663103956","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":2025,"cited_by_count":1},{"year":2024,"cited_by_count":2},{"year":2019,"cited_by_count":1},{"year":2018,"cited_by_count":3},{"year":2012,"cited_by_count":1}],"updated_date":"2026-06-11T09:08:48.828518","created_date":"2025-10-10T00:00:00"}