{"id":"https://openalex.org/W2963097630","doi":"https://doi.org/10.48550/arxiv.1709.02540","title":"The Expressive Power of Neural Networks: A View from the Width","display_name":"The Expressive Power of Neural Networks: A View from the Width","publication_year":2017,"publication_date":"2017-09-08","ids":{"openalex":"https://openalex.org/W2963097630","doi":"https://doi.org/10.48550/arxiv.1709.02540","mag":"2963097630"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:1709.02540","doi":null,"is_oa":true,"landing_page_url":"http://arxiv.org/abs/1709.02540","pdf_url":"https://arxiv.org/pdf/1709.02540","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},"volume":null,"type":"preprint","is_traditional":true,"indexed_in":["arxiv","datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://arxiv.org/pdf/1709.02540","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5100388729","display_name":"Lu Zhou","orcid":"https://orcid.org/0009-0009-8174-0919","observed_orcids":["https://orcid.org/0009-0009-8174-0919"]},"institutions":[{"id":"https://openalex.org/I20231570","display_name":"Peking University","ror":"https://ror.org/02v51f717","country_code":"CN","type":"education","lineage":["https://openalex.org/I20231570"]},{"id":"https://openalex.org/I4210096250","display_name":"Beijing Institute of Big Data Research","ror":"https://ror.org/00s1sz824","country_code":"CN","type":"facility","lineage":["https://openalex.org/I20231570","https://openalex.org/I37796252","https://openalex.org/I4210096250"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Lu, Zhou","raw_affiliation_strings":[", Department of Mathematics , Peking University 2 , Key Laboratory of Machine Perception , MOE , School of EECS , Peking University 3 , Center for Data Science , Peking University , Beijing Institute of Big Data Research","Department of Mathematics, Peking University and Center for Data Science, Peking University, Beijing Institute of Big Data Research#TAB#"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":", Department of Mathematics , Peking University 2 , Key Laboratory of Machine Perception , MOE , School of EECS , Peking University 3 , Center for Data Science , Peking University , Beijing Institute of Big Data Research","institution_ids":["https://openalex.org/I20231570","https://openalex.org/I4210096250"]},{"raw_affiliation_string":"Department of Mathematics, Peking University and Center for Data Science, Peking University, Beijing Institute of Big Data Research#TAB#","institution_ids":["https://openalex.org/I20231570","https://openalex.org/I4210096250"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5006777225","display_name":"Hongming Pu","orcid":null,"observed_orcids":[]},"institutions":[{"id":"https://openalex.org/I20231570","display_name":"Peking University","ror":"https://ror.org/02v51f717","country_code":"CN","type":"education","lineage":["https://openalex.org/I20231570"]},{"id":"https://openalex.org/I4210096250","display_name":"Beijing Institute of Big Data Research","ror":"https://ror.org/00s1sz824","country_code":"CN","type":"facility","lineage":["https://openalex.org/I20231570","https://openalex.org/I37796252","https://openalex.org/I4210096250"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Pu, Hongming","raw_affiliation_strings":[", Department of Mathematics , Peking University 2 , Key Laboratory of Machine Perception , MOE , School of EECS , Peking University 3 , Center for Data Science , Peking University , Beijing Institute of Big Data Research","Department of Mathematics  Peking University"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":", Department of Mathematics , Peking University 2 , Key Laboratory of Machine Perception , MOE , School of EECS , Peking University 3 , Center for Data Science , Peking University , Beijing Institute of Big Data Research","institution_ids":["https://openalex.org/I20231570","https://openalex.org/I4210096250"]},{"raw_affiliation_string":"Department of Mathematics  Peking University","institution_ids":["https://openalex.org/I20231570"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5103069699","display_name":"Feicheng Wang","orcid":"https://orcid.org/0000-0002-9268-554X","observed_orcids":["https://orcid.org/0000-0002-9268-554X"]},"institutions":[{"id":"https://openalex.org/I20231570","display_name":"Peking University","ror":"https://ror.org/02v51f717","country_code":"CN","type":"education","lineage":["https://openalex.org/I20231570"]},{"id":"https://openalex.org/I4210096250","display_name":"Beijing Institute of Big Data Research","ror":"https://ror.org/00s1sz824","country_code":"CN","type":"facility","lineage":["https://openalex.org/I20231570","https://openalex.org/I37796252","https://openalex.org/I4210096250"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Wang, Feicheng","raw_affiliation_strings":[", Department of Mathematics , Peking University 2 , Key Laboratory of Machine Perception , MOE , School of EECS , Peking University 3 , Center for Data Science , Peking University , Beijing Institute of Big Data Research","Department of Mathematics, Peking University and Center for Data Science, Peking University, Beijing Institute of Big Data Research#TAB#"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":", Department of Mathematics , Peking University 2 , Key Laboratory of Machine Perception , MOE , School of EECS , Peking University 3 , Center for Data Science , Peking University , Beijing Institute of Big Data Research","institution_ids":["https://openalex.org/I20231570","https://openalex.org/I4210096250"]},{"raw_affiliation_string":"Department of Mathematics, Peking University and Center for Data Science, Peking University, Beijing Institute of Big Data Research#TAB#","institution_ids":["https://openalex.org/I20231570","https://openalex.org/I4210096250"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5101417955","display_name":"Zhiqiang Hu","orcid":"https://orcid.org/0000-0003-0830-815X","observed_orcids":["https://orcid.org/0000-0003-0830-815X"]},"institutions":[{"id":"https://openalex.org/I20231570","display_name":"Peking University","ror":"https://ror.org/02v51f717","country_code":"CN","type":"education","lineage":["https://openalex.org/I20231570"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Hu, Zhiqiang","raw_affiliation_strings":["Long Beach , CA , USA","Key Laboratory of Machine Perception, MOE, School of EECS, Peking University"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Long Beach , CA , USA","institution_ids":[]},{"raw_affiliation_string":"Key Laboratory of Machine Perception, MOE, School of EECS, Peking University","institution_ids":["https://openalex.org/I20231570"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5100406711","display_name":"Liwei Wang","orcid":"https://orcid.org/0000-0001-9970-8604","observed_orcids":["https://orcid.org/0000-0001-9970-8604","https://orcid.org/0009-0000-8695-6238"]},"institutions":[{"id":"https://openalex.org/I20231570","display_name":"Peking University","ror":"https://ror.org/02v51f717","country_code":"CN","type":"education","lineage":["https://openalex.org/I20231570"]},{"id":"https://openalex.org/I4210096250","display_name":"Beijing Institute of Big Data Research","ror":"https://ror.org/00s1sz824","country_code":"CN","type":"facility","lineage":["https://openalex.org/I20231570","https://openalex.org/I37796252","https://openalex.org/I4210096250"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Wang, Liwei","raw_affiliation_strings":["Long Beach , CA , USA","Key Laboratory of Machine Perception, MOE, School of EECS, Peking University and Center for Data Science, Peking University, Beijing Institute of Big Data Research#TAB#"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Long Beach , CA , USA","institution_ids":[]},{"raw_affiliation_string":"Key Laboratory of Machine Perception, MOE, School of EECS, Peking University and Center for Data Science, Peking University, Beijing Institute of Big Data Research#TAB#","institution_ids":["https://openalex.org/I20231570","https://openalex.org/I4210096250"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":2,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":null,"has_fulltext":true,"cited_by_count":647,"citation_normalized_percentile":null,"cited_by_percentile_year":null,"biblio":{"volume":"30","issue":null,"first_page":"6232","last_page":"6240"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10320","display_name":"Neural Networks and Applications","score":0.975600004196167,"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":0.975600004196167,"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/T11612","display_name":"Stochastic Gradient Optimization Techniques","score":0.016300000250339508,"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/T12676","display_name":"Machine Learning and ELM","score":0.0017000000225380063,"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/relu-neural-networks","display_name":"ReLU networks","score":0.9120000004768372},{"id":"https://openalex.org/keywords/universal-approximation-theorem","display_name":"universal approximation theorem","score":0.9169999957084656},{"id":"https://openalex.org/keywords/phase-transitions","display_name":"phase transitions","score":0.7300000190734863}],"concepts":[{"id":"https://openalex.org/C77553402","wikidata":"https://www.wikidata.org/wiki/Q13222579","display_name":"Upper and lower bounds","level":2,"score":0.7014349699020386},{"id":"https://openalex.org/C34388435","wikidata":"https://www.wikidata.org/wiki/Q2267362","display_name":"Bounded function","level":2,"score":0.6962685585021973},{"id":"https://openalex.org/C50644808","wikidata":"https://www.wikidata.org/wiki/Q192776","display_name":"Artificial neural network","level":2,"score":0.6531070470809937},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.5622457265853882},{"id":"https://openalex.org/C33676613","wikidata":"https://www.wikidata.org/wiki/Q13415176","display_name":"Dimension (graph theory)","level":2,"score":0.5306151509284973},{"id":"https://openalex.org/C90119067","wikidata":"https://www.wikidata.org/wiki/Q43260","display_name":"Polynomial","level":2,"score":0.5293470025062561},{"id":"https://openalex.org/C151376022","wikidata":"https://www.wikidata.org/wiki/Q168698","display_name":"Exponential function","level":2,"score":0.49220481514930725},{"id":"https://openalex.org/C184720557","wikidata":"https://www.wikidata.org/wiki/Q7825049","display_name":"Topology (electrical circuits)","level":2,"score":0.408990740776062},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.372404009103775},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.3583822250366211},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.2295551300048828},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.18228337168693542},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.11404648423194885}],"mesh":[],"locations_count":3,"locations":[{"id":"pmh:oai:arXiv.org:1709.02540","doi":null,"is_oa":true,"landing_page_url":"http://arxiv.org/abs/1709.02540","pdf_url":"https://arxiv.org/pdf/1709.02540","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},{"id":"doi:10.48550/arxiv.1709.02540","doi":"https://doi.org/10.48550/arxiv.1709.02540","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.1709.02540","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Preprint"},{"id":"mag:2963097630","doi":null,"is_oa":false,"landing_page_url":"https://papers.nips.cc/paper/2017/file/32cbf687880eb1674a07bf717761dd3a-Paper.pdf","pdf_url":null,"source":{"id":"https://openalex.org/S4363606243","display_name":"neural information processing systems","issn_l":"1049-5258","issn":["1049-5258"],"is_oa":false,"is_in_doaj":false,"is_core":true,"listed_in":["cwts-core","norway-2"],"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"conference"},"license":null,"license_id":null,"version":null,"is_accepted":false,"is_published":false,"raw_source_name":"Neural Information Processing Systems","raw_type":null}],"best_oa_location":{"id":"pmh:oai:arXiv.org:1709.02540","doi":null,"is_oa":true,"landing_page_url":"http://arxiv.org/abs/1709.02540","pdf_url":"https://arxiv.org/pdf/1709.02540","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},"sustainable_development_goals":[],"study_designs":[],"awards":[{"id":"https://openalex.org/G3410860813","display_name":null,"funder_award_id":"61573026","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G3503326317","display_name":null,"funder_award_id":"2015CB352502","funder_id":"https://openalex.org/F4320335777","funder_display_name":"National Key Research and Development Program of China"},{"id":"https://openalex.org/G839750106","display_name":null,"funder_award_id":"2015CB352502","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"}],"funders":[{"id":"https://openalex.org/F4320321001","display_name":"National Natural Science Foundation of China","ror":"https://ror.org/01h0zpd94"},{"id":"https://openalex.org/F4320324787","display_name":"Peking University","ror":"https://ror.org/02v51f717"},{"id":"https://openalex.org/F4320335777","display_name":"National Key Research and Development Program of China","ror":null}],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W2963097630.pdf","grobid_xml":"https://content.openalex.org/works/W2963097630.grobid-xml"},"referenced_works_count":15,"referenced_works":["https://openalex.org/W1686810756","https://openalex.org/W1852909287","https://openalex.org/W1971735090","https://openalex.org/W2103496339","https://openalex.org/W2137983211","https://openalex.org/W2141473882","https://openalex.org/W2163605009","https://openalex.org/W2194775991","https://openalex.org/W2281746805","https://openalex.org/W2528305538","https://openalex.org/W2533570097","https://openalex.org/W2596625124","https://openalex.org/W2608609325","https://openalex.org/W2962742960","https://openalex.org/W2963241012"],"related_works":["https://openalex.org/W4226052297","https://openalex.org/W4379251926","https://openalex.org/W3120371886","https://openalex.org/W3099132019","https://openalex.org/W3005526090","https://openalex.org/W4389942372","https://openalex.org/W3101563662","https://openalex.org/W2766099245","https://openalex.org/W3087545942","https://openalex.org/W3124006463"],"abstract_inverted_index":{"The":[0],"expressive":[1],"power":[2],"of":[3,21,24,37,104,110,118,147,173,240],"neural":[4,38,111],"networks":[5,47,120,153,160,175,204],"is":[6,70,131,163,186,232],"important":[7],"for":[8,61,79,237],"understanding":[9],"deep":[10,119],"learning.":[11],"Most":[12],"existing":[13],"works":[14,100],"consider":[15],"this":[16,28],"problem":[17],"from":[18],"the":[19,22,35,71,102,108,141,145,194,208,238],"view":[20],"depth":[23,105,185,231],"a":[25,57,80,95,190,212],"network.":[26],"In":[27],"paper,":[29],"we":[30,139,197],"study":[31],"how":[32],"width":[33,236],"affects":[34],"expressiveness":[36,239],"networks.":[39,112,242],"Classical":[40],"results":[41,225],"state":[42],"that":[43,154,169,202,230],"depth-bounded":[44],"(e.g.":[45],"depth-$2$)":[46],"with":[48,221],"suitable":[49],"activation":[50],"functions":[51,85],"are":[52,74,116],"universal":[53,58,75],"approximators.":[54,76],"We":[55,167],"show":[56,168],"approximation":[59],"theorem":[60],"width-bounded":[62],"ReLU":[63,66,91,148,241],"networks:":[64,149],"width-$(n+4)$":[65],"networks,":[67,92],"where":[68],"$n$":[69],"input":[72],"dimension,":[73],"Moreover,":[77],"except":[78],"measure":[81],"zero":[82],"set,":[83],"all":[84],"cannot":[86,122,155,177],"be":[87,123,156,178],"approximated":[88],"by":[89,106,125,158,180,199,211],"width-$n$":[90],"which":[93,121,176],"exhibits":[94],"phase":[96],"transition.":[97],"Several":[98],"recent":[99],"demonstrate":[101,198],"benefits":[103],"proving":[107],"depth-efficiency":[109],"That":[113],"is,":[114],"there":[115,151,170],"classes":[117,172],"realized":[124,157,179],"any":[126,181],"shallow":[127,219],"network":[128,183,220],"whose":[129,161,184,205],"size":[130,162,206],"no":[132,187],"more":[133,188,227,233],"than":[134,189,235],"an":[135],"exponential":[136],"bound.":[137,192],"Here":[138],"pose":[140],"dual":[142],"question":[143],"on":[144],"width-efficiency":[146],"Are":[150],"wide":[152,174,217],"narrow":[159,182,203],"not":[164],"substantially":[165],"larger?":[166],"exist":[171],"polynomial":[191,209],"On":[193],"other":[195],"hand,":[196],"extensive":[200],"experiments":[201],"exceed":[207],"bound":[210],"constant":[213],"factor":[214],"can":[215],"approximate":[216],"and":[218],"high":[222],"accuracy.":[223],"Our":[224],"provide":[226],"comprehensive":[228],"evidence":[229],"effective":[234]},"counts_by_year":[{"year":2026,"cited_by_count":11},{"year":2025,"cited_by_count":48},{"year":2024,"cited_by_count":58},{"year":2023,"cited_by_count":66},{"year":2022,"cited_by_count":104},{"year":2021,"cited_by_count":137},{"year":2020,"cited_by_count":149},{"year":2019,"cited_by_count":58},{"year":2018,"cited_by_count":15},{"year":2017,"cited_by_count":1}],"updated_date":"2026-11-08T12:12:28.789561","created_date":"2019-07-30T00:00:00"}