{"id":"https://openalex.org/W3048707547","doi":"https://doi.org/10.1162/neco_a_01364","title":"Deep Network With Approximation Error Being Reciprocal of Width to Power of Square Root of Depth","display_name":"Deep Network With Approximation Error Being Reciprocal of Width to Power of Square Root of Depth","publication_year":2021,"publication_date":"2021-01-01","ids":{"openalex":"https://openalex.org/W3048707547","doi":"https://doi.org/10.1162/neco_a_01364","mag":"3048707547","pmid":"https://pubmed.ncbi.nlm.nih.gov/33513325"},"language":"en","primary_location":{"id":"doi:10.1162/neco_a_01364","is_oa":false,"landing_page_url":"https://doi.org/10.1162/neco_a_01364","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"],"host_organization_lineage_names":["The MIT Press"],"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":["arxiv","crossref","pubmed"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://arxiv.org/pdf/2006.12231","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5068391881","display_name":"Zuowei Shen","orcid":"https://orcid.org/0000-0002-1861-1884"},"institutions":[{"id":"https://openalex.org/I219193219","display_name":"Purdue University West Lafayette","ror":"https://ror.org/02dqehb95","country_code":"US","type":"education","lineage":["https://openalex.org/I219193219"]}],"countries":["US"],"is_corresponding":true,"raw_author_name":"Zuowei Shen","raw_affiliation_strings":["Department of Mathematics, Purdue University, West Lafayette, IN 47907, USA, haizhao@purdue.edu"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics, Purdue University, West Lafayette, IN 47907, USA, haizhao@purdue.edu","institution_ids":["https://openalex.org/I219193219"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5079602544","display_name":"Haizhao Yang","orcid":"https://orcid.org/0000-0002-8408-1754"},"institutions":[{"id":"https://openalex.org/I219193219","display_name":"Purdue University West Lafayette","ror":"https://ror.org/02dqehb95","country_code":"US","type":"education","lineage":["https://openalex.org/I219193219"]}],"countries":["US"],"is_corresponding":true,"raw_author_name":"Haizhao Yang","raw_affiliation_strings":["Department of Mathematics, Purdue University, West Lafayette, IN 47907, USA, haizhao@purdue.edu"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics, Purdue University, West Lafayette, IN 47907, USA, haizhao@purdue.edu","institution_ids":["https://openalex.org/I219193219"]}]},{"author_position":"last","author":{"id":null,"display_name":"Shijun Zhang","orcid":null},"institutions":[{"id":"https://openalex.org/I219193219","display_name":"Purdue University West Lafayette","ror":"https://ror.org/02dqehb95","country_code":"US","type":"education","lineage":["https://openalex.org/I219193219"]}],"countries":["US"],"is_corresponding":true,"raw_author_name":"Shijun Zhang","raw_affiliation_strings":["Department of Mathematics, Purdue University, West Lafayette, IN 47907, USA, zhangshijun@u.nus.edu"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics, Purdue University, West Lafayette, IN 47907, USA, zhangshijun@u.nus.edu","institution_ids":["https://openalex.org/I219193219"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5068391881","https://openalex.org/A5079602544"],"corresponding_institution_ids":["https://openalex.org/I219193219"],"apc_list":null,"apc_paid":null,"fwci":5.4593,"has_fulltext":false,"cited_by_count":56,"citation_normalized_percentile":{"value":0.96300632,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":94,"max":100},"biblio":{"volume":"33","issue":"4","first_page":"1005","last_page":"1036"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10320","display_name":"Neural Networks and Applications","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/T10320","display_name":"Neural Networks and Applications","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/T12676","display_name":"Machine Learning and ELM","score":0.9930999875068665,"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/T10057","display_name":"Face and Expression Recognition","score":0.9911999702453613,"subfield":{"id":"https://openalex.org/subfields/1707","display_name":"Computer Vision and Pattern Recognition"},"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/modulus-of-continuity","display_name":"Modulus of continuity","score":0.7378595471382141},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.6530385613441467},{"id":"https://openalex.org/keywords/reciprocal","display_name":"Reciprocal","score":0.6145670413970947},{"id":"https://openalex.org/keywords/function","display_name":"Function (biology)","score":0.5601319074630737},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.5556282997131348},{"id":"https://openalex.org/keywords/square-root","display_name":"Square root","score":0.5212126970291138},{"id":"https://openalex.org/keywords/approximation-error","display_name":"Approximation error","score":0.48520997166633606},{"id":"https://openalex.org/keywords/curse-of-dimensionality","display_name":"Curse of dimensionality","score":0.42893078923225403},{"id":"https://openalex.org/keywords/constant","display_name":"Constant (computer programming)","score":0.4273245930671692},{"id":"https://openalex.org/keywords/square","display_name":"Square (algebra)","score":0.4180677831172943},{"id":"https://openalex.org/keywords/continuous-function","display_name":"Continuous function (set theory)","score":0.4127585291862488},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.39490652084350586},{"id":"https://openalex.org/keywords/type","display_name":"Type (biology)","score":0.21637645363807678},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.15231400728225708},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.114501953125},{"id":"https://openalex.org/keywords/statistics","display_name":"Statistics","score":0.07706114649772644}],"concepts":[{"id":"https://openalex.org/C3098717","wikidata":"https://www.wikidata.org/wiki/Q1144872","display_name":"Modulus of continuity","level":3,"score":0.7378595471382141},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6530385613441467},{"id":"https://openalex.org/C2777742833","wikidata":"https://www.wikidata.org/wiki/Q1964083","display_name":"Reciprocal","level":2,"score":0.6145670413970947},{"id":"https://openalex.org/C14036430","wikidata":"https://www.wikidata.org/wiki/Q3736076","display_name":"Function (biology)","level":2,"score":0.5601319074630737},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.5556282997131348},{"id":"https://openalex.org/C11577676","wikidata":"https://www.wikidata.org/wiki/Q134237","display_name":"Square root","level":2,"score":0.5212126970291138},{"id":"https://openalex.org/C122383733","wikidata":"https://www.wikidata.org/wiki/Q865920","display_name":"Approximation error","level":2,"score":0.48520997166633606},{"id":"https://openalex.org/C111030470","wikidata":"https://www.wikidata.org/wiki/Q1430460","display_name":"Curse of dimensionality","level":2,"score":0.42893078923225403},{"id":"https://openalex.org/C2777027219","wikidata":"https://www.wikidata.org/wiki/Q1284190","display_name":"Constant (computer programming)","level":2,"score":0.4273245930671692},{"id":"https://openalex.org/C135692309","wikidata":"https://www.wikidata.org/wiki/Q111124","display_name":"Square (algebra)","level":2,"score":0.4180677831172943},{"id":"https://openalex.org/C2778199601","wikidata":"https://www.wikidata.org/wiki/Q5165476","display_name":"Continuous function (set theory)","level":3,"score":0.4127585291862488},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.39490652084350586},{"id":"https://openalex.org/C2777299769","wikidata":"https://www.wikidata.org/wiki/Q3707858","display_name":"Type (biology)","level":2,"score":0.21637645363807678},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.15231400728225708},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.114501953125},{"id":"https://openalex.org/C105795698","wikidata":"https://www.wikidata.org/wiki/Q12483","display_name":"Statistics","level":1,"score":0.07706114649772644},{"id":"https://openalex.org/C138885662","wikidata":"https://www.wikidata.org/wiki/Q5891","display_name":"Philosophy","level":0,"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/C199360897","wikidata":"https://www.wikidata.org/wiki/Q9143","display_name":"Programming language","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/C41895202","wikidata":"https://www.wikidata.org/wiki/Q8162","display_name":"Linguistics","level":1,"score":0.0},{"id":"https://openalex.org/C18903297","wikidata":"https://www.wikidata.org/wiki/Q7150","display_name":"Ecology","level":1,"score":0.0}],"mesh":[],"locations_count":3,"locations":[{"id":"doi:10.1162/neco_a_01364","is_oa":false,"landing_page_url":"https://doi.org/10.1162/neco_a_01364","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"],"host_organization_lineage_names":["The MIT Press"],"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:33513325","is_oa":false,"landing_page_url":"https://pubmed.ncbi.nlm.nih.gov/33513325","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":null},{"id":"pmh:oai:arXiv.org:2006.12231","is_oa":true,"landing_page_url":"http://arxiv.org/abs/2006.12231","pdf_url":"https://arxiv.org/pdf/2006.12231","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"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"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:2006.12231","is_oa":true,"landing_page_url":"http://arxiv.org/abs/2006.12231","pdf_url":"https://arxiv.org/pdf/2006.12231","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"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":[],"awards":[],"funders":[],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":89,"referenced_works":["https://openalex.org/W1990261069","https://openalex.org/W2024060531","https://openalex.org/W2062368570","https://openalex.org/W2081059029","https://openalex.org/W2103496339","https://openalex.org/W2115881141","https://openalex.org/W2137983211","https://openalex.org/W2151029520","https://openalex.org/W2152195021","https://openalex.org/W2152593035","https://openalex.org/W2166116275","https://openalex.org/W2171074980","https://openalex.org/W2242818861","https://openalex.org/W2340142742","https://openalex.org/W2524428287","https://openalex.org/W2528305538","https://openalex.org/W2533570097","https://openalex.org/W2543580944","https://openalex.org/W2550848904","https://openalex.org/W2596625124","https://openalex.org/W2787302432","https://openalex.org/W2798742790","https://openalex.org/W2809090039","https://openalex.org/W2810756260","https://openalex.org/W2885273747","https://openalex.org/W2886881190","https://openalex.org/W2891788052","https://openalex.org/W2895992958","https://openalex.org/W2904250082","https://openalex.org/W2907087576","https://openalex.org/W2917375922","https://openalex.org/W2920266844","https://openalex.org/W2923130667","https://openalex.org/W2924785619","https://openalex.org/W2946840143","https://openalex.org/W2949883708","https://openalex.org/W2950533765","https://openalex.org/W2956024278","https://openalex.org/W2962761333","https://openalex.org/W2962949242","https://openalex.org/W2963082137","https://openalex.org/W2963146412","https://openalex.org/W2963172624","https://openalex.org/W2963220074","https://openalex.org/W2963239103","https://openalex.org/W2963626582","https://openalex.org/W2965067322","https://openalex.org/W2965192329","https://openalex.org/W2969750612","https://openalex.org/W2970330753","https://openalex.org/W2970755399","https://openalex.org/W2973531092","https://openalex.org/W2975935587","https://openalex.org/W2992906545","https://openalex.org/W2995681639","https://openalex.org/W2996168800","https://openalex.org/W2999905431","https://openalex.org/W2999942072","https://openalex.org/W3000236721","https://openalex.org/W3009390838","https://openalex.org/W3028473598","https://openalex.org/W3030043499","https://openalex.org/W3031010353","https://openalex.org/W3034803983","https://openalex.org/W3037386592","https://openalex.org/W3044387630","https://openalex.org/W3089015099","https://openalex.org/W3099710307","https://openalex.org/W3099745170","https://openalex.org/W3101996726","https://openalex.org/W3104183394","https://openalex.org/W3105504331","https://openalex.org/W3122402965","https://openalex.org/W3146803896","https://openalex.org/W3146938414","https://openalex.org/W3161792008","https://openalex.org/W3204236384","https://openalex.org/W4242989628","https://openalex.org/W4250503569","https://openalex.org/W4251483105","https://openalex.org/W4253824286","https://openalex.org/W4289293816","https://openalex.org/W4298647576","https://openalex.org/W4378009079","https://openalex.org/W6727208969","https://openalex.org/W6748651261","https://openalex.org/W6755463368","https://openalex.org/W6758902322","https://openalex.org/W6760138846"],"related_works":["https://openalex.org/W3136430826","https://openalex.org/W2906791148","https://openalex.org/W2105365571","https://openalex.org/W2162250345","https://openalex.org/W2063893713","https://openalex.org/W2019223705","https://openalex.org/W2097376833","https://openalex.org/W2264263601","https://openalex.org/W2358621376","https://openalex.org/W2348550508"],"abstract_inverted_index":{"A":[0],"new":[1,98],"network":[2,9],"with":[3,12,41,56,82],"super-approximation":[4],"power":[5,109],"is":[6,10,92,117,136],"introduced.":[7],"This":[8],"built":[11],"Floor":[13],"(\u230ax\u230b)":[14],"or":[15],"ReLU":[16],"(max{0,x})":[17],"activation":[18],"function":[19,52,78,141],"in":[20,107,132],"each":[21],"neuron;":[22],"hence,":[23],"we":[24,36],"call":[25],"such":[26],"networks":[27,40,101],"Floor-ReLU":[28,39],"networks.":[29],"For":[30],"any":[31],"hyperparameters":[32],"N\u2208N+":[33],"and":[34,44,63,69,144],"L\u2208N+,":[35],"show":[37],"that":[38],"width":[42],"max{d,5N+13}":[43],"depth":[45],"64dL+3":[46],"can":[47],"uniformly":[48],"approximate":[49],"a":[50,83,95,140],"H\u00f6lder":[51,67,122],"f":[53,79],"on":[54,80],"[0,1]d":[55,81],"an":[57,75],"approximation":[58,90,108,134],"error":[59],"3\u03bbd\u03b1/2N-\u03b1L,":[60],"where":[61],"\u03b1\u2208(0,1]":[62],"\u03bb":[64],"are":[65],"the":[66,88,103,111,126,150],"order":[68],"constant,":[70],"respectively.":[71],"More":[72],"generally":[73],"for":[74,121],"arbitrary":[76],"continuous":[77,123],"modulus":[84,151],"of":[85,100,105,113,142,147,152],"continuity":[86],"\u03c9f(\u00b7),":[87],"constructive":[89],"rate":[91,135],"\u03c9f(dN-L)+2\u03c9f(d)N-L.":[93],"As":[94],"consequence,":[96],"this":[97],"class":[99],"overcomes":[102],"curse":[104],"dimensionality":[106],"when":[110],"variation":[112],"\u03c9f(r)":[114],"as":[115],"r\u21920":[116],"moderate":[118],"(e.g.,":[119],"\u03c9f(r)\u2272r\u03b1":[120],"functions),":[124],"since":[125],"major":[127],"term":[128],"to":[129],"be":[130],"considered":[131],"our":[133],"essentially":[137],"d":[138,148],"times":[139],"N":[143],"L":[145],"independent":[146],"within":[149],"continuity.":[153]},"counts_by_year":[{"year":2026,"cited_by_count":3},{"year":2025,"cited_by_count":11},{"year":2024,"cited_by_count":4},{"year":2023,"cited_by_count":10},{"year":2022,"cited_by_count":9},{"year":2021,"cited_by_count":17},{"year":2020,"cited_by_count":2}],"updated_date":"2026-05-21T06:26:12.895304","created_date":"2025-10-10T00:00:00"}
