{"id":"https://openalex.org/W4412623262","doi":"https://doi.org/10.1109/tai.2025.3592162","title":"On Expressivity of Height in Neural Networks","display_name":"On Expressivity of Height in Neural Networks","publication_year":2025,"publication_date":"2025-07-24","ids":{"openalex":"https://openalex.org/W4412623262","doi":"https://doi.org/10.1109/tai.2025.3592162"},"language":"en","primary_location":{"id":"doi:10.1109/tai.2025.3592162","is_oa":false,"landing_page_url":"https://doi.org/10.1109/tai.2025.3592162","pdf_url":null,"source":{"id":"https://openalex.org/S4210169448","display_name":"IEEE Transactions on Artificial Intelligence","issn_l":"2691-4581","issn":["2691-4581"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319808","host_organization_name":"Institute of Electrical and Electronics Engineers","host_organization_lineage":["https://openalex.org/P4310319808"],"host_organization_lineage_names":["Institute of Electrical and Electronics Engineers"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"IEEE Transactions on Artificial Intelligence","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/A5018677035","display_name":"Fenglei Fan","orcid":"https://orcid.org/0000-0003-3691-5141"},"institutions":[{"id":"https://openalex.org/I168719708","display_name":"City University of Hong Kong","ror":"https://ror.org/03q8dnn23","country_code":"HK","type":"education","lineage":["https://openalex.org/I168719708"]}],"countries":["HK"],"is_corresponding":false,"raw_author_name":"Feng-Lei Fan","raw_affiliation_strings":["Frontier of Artificial Networks (FAN) Lab, Department of City University, The City University of Hong Kong, Hong Kong"],"raw_orcid":"https://orcid.org/0000-0003-3691-5141","affiliations":[{"raw_affiliation_string":"Frontier of Artificial Networks (FAN) Lab, Department of City University, The City University of Hong Kong, Hong Kong","institution_ids":["https://openalex.org/I168719708"]}]},{"author_position":"middle","author":{"id":null,"display_name":"Ze-Yu Li","orcid":"https://orcid.org/0009-0002-6576-3298"},"institutions":[{"id":"https://openalex.org/I177725633","display_name":"Chinese University of Hong Kong","ror":"https://ror.org/00t33hh48","country_code":"HK","type":"education","lineage":["https://openalex.org/I177725633"]}],"countries":["HK"],"is_corresponding":false,"raw_author_name":"Ze-Yu Li","raw_affiliation_strings":["Department of Mathematics, The Chinese University of Hong Kong, Hong Kong, Hong Kong","Department of Mathematics, The Chinese University of Hong Kong, Hong Kong"],"raw_orcid":"https://orcid.org/0009-0002-6576-3298","affiliations":[{"raw_affiliation_string":"Department of Mathematics, The Chinese University of Hong Kong, Hong Kong, Hong Kong","institution_ids":["https://openalex.org/I177725633"]},{"raw_affiliation_string":"Department of Mathematics, The Chinese University of Hong Kong, Hong Kong","institution_ids":["https://openalex.org/I177725633"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5100643588","display_name":"Huan Xiong","orcid":"https://orcid.org/0000-0001-5401-7834"},"institutions":[{"id":"https://openalex.org/I204983213","display_name":"Harbin Institute of Technology","ror":"https://ror.org/01yqg2h08","country_code":"CN","type":"education","lineage":["https://openalex.org/I204983213"]},{"id":"https://openalex.org/I4210161462","display_name":"Heilongjiang Institute of Technology","ror":"https://ror.org/05x0m9n95","country_code":"CN","type":"education","lineage":["https://openalex.org/I4210161462"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Huan Xiong","raw_affiliation_strings":["Institute of Advanced Mathematics, Harbin Institute of Technology, Harbin, China","Institute of Advanced Mathematics, Harbin Institute of Technology, Harbin, Heilongjiang Province, China"],"raw_orcid":"https://orcid.org/0000-0001-5401-7834","affiliations":[{"raw_affiliation_string":"Institute of Advanced Mathematics, Harbin Institute of Technology, Harbin, China","institution_ids":["https://openalex.org/I204983213"]},{"raw_affiliation_string":"Institute of Advanced Mathematics, Harbin Institute of Technology, Harbin, Heilongjiang Province, China","institution_ids":["https://openalex.org/I204983213","https://openalex.org/I4210161462"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5059443966","display_name":"Tieyong Zeng","orcid":"https://orcid.org/0000-0002-0688-202X"},"institutions":[{"id":"https://openalex.org/I177725633","display_name":"Chinese University of Hong Kong","ror":"https://ror.org/00t33hh48","country_code":"HK","type":"education","lineage":["https://openalex.org/I177725633"]}],"countries":["HK"],"is_corresponding":false,"raw_author_name":"Tieyong Zeng","raw_affiliation_strings":["Department of Mathematics, The Chinese University of Hong Kong, Hong Kong, Hong Kong","Department of Mathematics, The Chinese University of Hong Kong, Hong Kong"],"raw_orcid":"https://orcid.org/0000-0002-0688-202X","affiliations":[{"raw_affiliation_string":"Department of Mathematics, The Chinese University of Hong Kong, Hong Kong, Hong Kong","institution_ids":["https://openalex.org/I177725633"]},{"raw_affiliation_string":"Department of Mathematics, The Chinese University of Hong Kong, Hong Kong","institution_ids":["https://openalex.org/I177725633"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":4,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":0.0,"has_fulltext":false,"cited_by_count":0,"citation_normalized_percentile":{"value":0.06902688,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":null,"biblio":{"volume":"7","issue":"3","first_page":"1223","last_page":"1237"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10320","display_name":"Neural Networks and Applications","score":0.4934000074863434,"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":0.4934000074863434,"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/expressivity","display_name":"Expressivity","score":0.8731420636177063},{"id":"https://openalex.org/keywords/artificial-neural-network","display_name":"Artificial neural network","score":0.4702431261539459},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.4595377445220947},{"id":"https://openalex.org/keywords/artificial-intelligence","display_name":"Artificial intelligence","score":0.35226142406463623},{"id":"https://openalex.org/keywords/biology","display_name":"Biology","score":0.1717510223388672},{"id":"https://openalex.org/keywords/genetics","display_name":"Genetics","score":0.09168198704719543}],"concepts":[{"id":"https://openalex.org/C92811239","wikidata":"https://www.wikidata.org/wiki/Q20998670","display_name":"Expressivity","level":2,"score":0.8731420636177063},{"id":"https://openalex.org/C50644808","wikidata":"https://www.wikidata.org/wiki/Q192776","display_name":"Artificial neural network","level":2,"score":0.4702431261539459},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.4595377445220947},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.35226142406463623},{"id":"https://openalex.org/C86803240","wikidata":"https://www.wikidata.org/wiki/Q420","display_name":"Biology","level":0,"score":0.1717510223388672},{"id":"https://openalex.org/C54355233","wikidata":"https://www.wikidata.org/wiki/Q7162","display_name":"Genetics","level":1,"score":0.09168198704719543}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1109/tai.2025.3592162","is_oa":false,"landing_page_url":"https://doi.org/10.1109/tai.2025.3592162","pdf_url":null,"source":{"id":"https://openalex.org/S4210169448","display_name":"IEEE Transactions on Artificial Intelligence","issn_l":"2691-4581","issn":["2691-4581"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319808","host_organization_name":"Institute of Electrical and Electronics Engineers","host_organization_lineage":["https://openalex.org/P4310319808"],"host_organization_lineage_names":["Institute of Electrical and Electronics Engineers"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"IEEE Transactions on Artificial Intelligence","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":25,"referenced_works":["https://openalex.org/W1485981043","https://openalex.org/W2064675550","https://openalex.org/W2089947415","https://openalex.org/W2108598243","https://openalex.org/W2137983211","https://openalex.org/W2157331557","https://openalex.org/W2528305538","https://openalex.org/W2550848904","https://openalex.org/W2919115771","https://openalex.org/W2949883708","https://openalex.org/W2995896684","https://openalex.org/W2999571325","https://openalex.org/W3034421924","https://openalex.org/W3040743533","https://openalex.org/W3048707547","https://openalex.org/W3087545942","https://openalex.org/W3097724114","https://openalex.org/W3133816032","https://openalex.org/W3204236384","https://openalex.org/W3210425550","https://openalex.org/W4385482701","https://openalex.org/W4385567529","https://openalex.org/W4388399798","https://openalex.org/W4388953010","https://openalex.org/W4407577741"],"related_works":["https://openalex.org/W4391375266","https://openalex.org/W2899084033","https://openalex.org/W2748952813","https://openalex.org/W2048138016","https://openalex.org/W2408998960","https://openalex.org/W2739995301","https://openalex.org/W4285275252","https://openalex.org/W2144961711","https://openalex.org/W1460228849","https://openalex.org/W2067704480"],"abstract_inverted_index":{"In":[0],"this":[1,65],"work,":[2],"beyond":[3],"width":[4],"and":[5,84,168,207,219,231],"depth":[6,70],"in":[7,28,113,165],"a":[8,14,18,93,114,119,147,174],"traditional":[9,175],"neural":[10,15],"network,":[11],"we":[12],"augment":[13],"network":[16,96,149,177],"with":[17],"new":[19],"dimension":[20],"called":[21],"height":[22],"by":[23],"inserting":[24],"links":[25],"to":[26,35,43,173],"neurons":[27],"the":[29,36,52,88,143],"same":[30,89],"layer,":[31],"which":[32],"gives":[33],"rise":[34],"notion":[37],"of":[38,91,145],"height.":[39],"These":[40],"links,":[41,46],"referred":[42],"as":[44],"intra-layer":[45],"form":[47],"hierarchical":[48],"structures":[49],"that":[50,64,212],"enhance":[51],"network\u2019s":[53],"approximation":[54],"ability":[55],"(expressivity)":[56],"without":[57],"increasing":[58],"its":[59],"parameter":[60],"count.":[61],"We":[62],"show":[63],"3D":[66,94,148,213],"architecture":[67],"(width":[68,80],"\u00d7":[69,71,81,100,104,125,153,157,183],"height)":[72],"is":[73],"more":[74,111],"powerful":[75],"than":[76,118],"conventional":[77],"2D":[78,120,176],"networks":[79,214],"depth)":[82],"theoretically":[83],"empirically.":[85],"First,":[86],"given":[87],"number":[90],"neurons,":[92],"ReLU":[95],"(<italic":[97,122,150,178],"xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"":[98,102,106,123,127,131,138,151,155,159,179,181,185,189,194],"xmlns:xlink=\"http://www.w3.org/1999/xlink\">W</i>":[99,152],"<italic":[101,105,126,154,158,184],"xmlns:xlink=\"http://www.w3.org/1999/xlink\">K</i>":[103,156],"xmlns:xlink=\"http://www.w3.org/1999/xlink\">H</i>)":[107],"can":[108,161,215],"generate":[109],"exponentially":[110,171],"pieces":[112],"piecewise":[115],"linear":[116],"function":[117],"one":[121],"xmlns:xlink=\"http://www.w3.org/1999/xlink\">HW</i>":[124],"xmlns:xlink=\"http://www.w3.org/1999/xlink\">K</i>),":[128,186],"i.e.,":[129,187],"<inline-formula":[130,137,188,193],"xmlns:xlink=\"http://www.w3.org/1999/xlink\"><tex-math":[132,139,190,195],"notation=\"LaTeX\">$\\mathcal{O}((2^{H}":[133],"-":[134],"1)W)^{K})$</tex-math></inline-formula>":[135],"vs":[136,192],"notation=\"LaTeX\">$\\mathcal{O}((HW)^{K})$</tex-math></inline-formula>.":[140],"Next,":[141],"from":[142],"perspective":[144],"approximation,":[146],"xmlns:xlink=\"http://www.w3.org/1999/xlink\">W</i>)":[160],"achieve":[162],"higher":[163],"accuracy":[164],"approximating":[166],"polynomials":[167],"reduce":[169],"errors":[170],"compared":[172],"xmlns:xlink=\"http://www.w3.org/1999/xlink\">W</i><sup":[180],"xmlns:xlink=\"http://www.w3.org/1999/xlink\">2</sup>":[182],"notation=\"LaTeX\">$\\mathcal{O}(2^{-2W\\,K})$</tex-math></inline-formula>":[191],"notation=\"LaTeX\">$\\mathcal{O}(W^{-K})$</tex-math></inline-formula>.":[196],"Lastly,":[197],"numerical":[198],"experiments":[199],"on":[200],"5":[201],"synthetic":[202],"datasets,":[203,206],"15":[204],"tabular":[205],"3":[208],"image":[209],"benchmarks":[210],"verify":[211],"deliver":[216],"competitive":[217],"regression":[218],"classification":[220],"performance.":[221],"Our":[222],"code":[223],"has":[224],"been":[225],"released":[226],"for":[227],"readers\u2019":[228],"free":[229],"download":[230],"evaluation.":[232]},"counts_by_year":[],"updated_date":"2026-07-22T07:51:19.307946","created_date":"2025-10-10T00:00:00"}