{"id":"https://openalex.org/W3000236721","doi":"https://doi.org/10.1137/20m134695x","title":"Deep Network Approximation for Smooth Functions","display_name":"Deep Network Approximation for Smooth Functions","publication_year":2021,"publication_date":"2021-01-01","ids":{"openalex":"https://openalex.org/W3000236721","doi":"https://doi.org/10.1137/20m134695x","mag":"3000236721"},"language":"en","primary_location":{"id":"doi:10.1137/20m134695x","is_oa":false,"landing_page_url":"https://doi.org/10.1137/20m134695x","pdf_url":null,"source":{"id":"https://openalex.org/S74141806","display_name":"SIAM Journal on Mathematical Analysis","issn_l":"0036-1410","issn":["0036-1410","1095-7154"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320508","host_organization_name":"Society for Industrial and Applied Mathematics","host_organization_lineage":["https://openalex.org/P4310320508"],"host_organization_lineage_names":["Society for Industrial and Applied Mathematics"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"SIAM Journal on Mathematical Analysis","raw_type":"journal-article"},"type":"article","indexed_in":["arxiv","crossref","datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://arxiv.org/pdf/2001.03040","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5061472917","display_name":"Jianfeng Lu","orcid":"https://orcid.org/0000-0001-6255-5165"},"institutions":[{"id":"https://openalex.org/I170897317","display_name":"Duke University","ror":"https://ror.org/00py81415","country_code":"US","type":"education","lineage":["https://openalex.org/I170897317"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Jianfeng Lu","raw_affiliation_strings":["Duke University#TAB#"],"raw_orcid":"https://orcid.org/0000-0001-6255-5165","affiliations":[{"raw_affiliation_string":"Duke University#TAB#","institution_ids":["https://openalex.org/I170897317"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5068391881","display_name":"Zuowei Shen","orcid":"https://orcid.org/0000-0002-1861-1884"},"institutions":[{"id":"https://openalex.org/I165932596","display_name":"National University of Singapore","ror":"https://ror.org/01tgyzw49","country_code":"SG","type":"education","lineage":["https://openalex.org/I165932596"]}],"countries":["SG"],"is_corresponding":false,"raw_author_name":"Zuowei Shen","raw_affiliation_strings":["National University of Singapore,"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"National University of Singapore,","institution_ids":["https://openalex.org/I165932596"]}]},{"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":false,"raw_author_name":"Haizhao Yang","raw_affiliation_strings":["PURDUE UNIVERSITY"],"raw_orcid":"https://orcid.org/0000-0002-8408-1754","affiliations":[{"raw_affiliation_string":"PURDUE UNIVERSITY","institution_ids":["https://openalex.org/I219193219"]}]},{"author_position":"last","author":{"id":null,"display_name":"Shijun Zhang","orcid":null},"institutions":[{"id":"https://openalex.org/I165932596","display_name":"National University of Singapore","ror":"https://ror.org/01tgyzw49","country_code":"SG","type":"education","lineage":["https://openalex.org/I165932596"]}],"countries":["SG"],"is_corresponding":false,"raw_author_name":"Shijun Zhang","raw_affiliation_strings":["National University of Singapore,"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"National University of Singapore,","institution_ids":["https://openalex.org/I165932596"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":3,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":6.536,"has_fulltext":false,"cited_by_count":55,"citation_normalized_percentile":{"value":0.98536603,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":94,"max":100},"biblio":{"volume":"53","issue":"5","first_page":"5465","last_page":"5506"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11245","display_name":"Advanced Numerical Analysis Techniques","score":0.9994000196456909,"subfield":{"id":"https://openalex.org/subfields/2206","display_name":"Computational Mechanics"},"field":{"id":"https://openalex.org/fields/22","display_name":"Engineering"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T11245","display_name":"Advanced Numerical Analysis Techniques","score":0.9994000196456909,"subfield":{"id":"https://openalex.org/subfields/2206","display_name":"Computational Mechanics"},"field":{"id":"https://openalex.org/fields/22","display_name":"Engineering"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T10996","display_name":"Computational Geometry and Mesh Generation","score":0.9904999732971191,"subfield":{"id":"https://openalex.org/subfields/1704","display_name":"Computer Graphics and Computer-Aided Design"},"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/T12404","display_name":"Mathematical Approximation and Integration","score":0.9904999732971191,"subfield":{"id":"https://openalex.org/subfields/2612","display_name":"Numerical Analysis"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.552686333656311},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.5276766419410706},{"id":"https://openalex.org/keywords/approximation-error","display_name":"Approximation error","score":0.4778296947479248},{"id":"https://openalex.org/keywords/approximations-of-\u03c0","display_name":"Approximations of \u03c0","score":0.42217424511909485},{"id":"https://openalex.org/keywords/characterization","display_name":"Characterization (materials science)","score":0.4104193449020386},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.3251979947090149},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.3085533082485199},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.2600237727165222},{"id":"https://openalex.org/keywords/optics","display_name":"Optics","score":0.07327574491500854}],"concepts":[{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.552686333656311},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.5276766419410706},{"id":"https://openalex.org/C122383733","wikidata":"https://www.wikidata.org/wiki/Q865920","display_name":"Approximation error","level":2,"score":0.4778296947479248},{"id":"https://openalex.org/C193386753","wikidata":"https://www.wikidata.org/wiki/Q1130396","display_name":"Approximations of \u03c0","level":2,"score":0.42217424511909485},{"id":"https://openalex.org/C2780841128","wikidata":"https://www.wikidata.org/wiki/Q5073781","display_name":"Characterization (materials science)","level":2,"score":0.4104193449020386},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.3251979947090149},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.3085533082485199},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.2600237727165222},{"id":"https://openalex.org/C120665830","wikidata":"https://www.wikidata.org/wiki/Q14620","display_name":"Optics","level":1,"score":0.07327574491500854}],"mesh":[],"locations_count":4,"locations":[{"id":"doi:10.1137/20m134695x","is_oa":false,"landing_page_url":"https://doi.org/10.1137/20m134695x","pdf_url":null,"source":{"id":"https://openalex.org/S74141806","display_name":"SIAM Journal on Mathematical Analysis","issn_l":"0036-1410","issn":["0036-1410","1095-7154"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320508","host_organization_name":"Society for Industrial and Applied Mathematics","host_organization_lineage":["https://openalex.org/P4310320508"],"host_organization_lineage_names":["Society for Industrial and Applied Mathematics"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"SIAM Journal on Mathematical Analysis","raw_type":"journal-article"},{"id":"pmh:oai:arXiv.org:2001.03040","is_oa":true,"landing_page_url":"http://arxiv.org/abs/2001.03040","pdf_url":"https://arxiv.org/pdf/2001.03040","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"},{"id":"mag:3000236721","is_oa":true,"landing_page_url":"http://export.arxiv.org/pdf/2001.03040","pdf_url":null,"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":"arXiv (Cornell University)","raw_type":null},{"id":"doi:10.48550/arxiv.2001.03040","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2001.03040","pdf_url":null,"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":null,"is_accepted":false,"is_published":null,"raw_source_name":null,"raw_type":"article-journal"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:2001.03040","is_oa":true,"landing_page_url":"http://arxiv.org/abs/2001.03040","pdf_url":"https://arxiv.org/pdf/2001.03040","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":[{"id":"https://openalex.org/G1459758584","display_name":null,"funder_award_id":"DMS-1945029","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"},{"id":"https://openalex.org/G1788604635","display_name":null,"funder_award_id":"DMS-1415939","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"},{"id":"https://openalex.org/G6520894996","display_name":null,"funder_award_id":"DMS-201228","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"},{"id":"https://openalex.org/G7525462396","display_name":null,"funder_award_id":"CCF-1934964","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"}],"funders":[{"id":"https://openalex.org/F4320306076","display_name":"National Science Foundation","ror":"https://ror.org/021nxhr62"}],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":42,"referenced_works":["https://openalex.org/W1542886316","https://openalex.org/W1988115241","https://openalex.org/W1990261069","https://openalex.org/W2089947415","https://openalex.org/W2103496339","https://openalex.org/W2115881141","https://openalex.org/W2129192653","https://openalex.org/W2137983211","https://openalex.org/W2161388792","https://openalex.org/W2163605009","https://openalex.org/W2166116275","https://openalex.org/W2528305538","https://openalex.org/W2550848904","https://openalex.org/W2596625124","https://openalex.org/W2809090039","https://openalex.org/W2810756260","https://openalex.org/W2886881190","https://openalex.org/W2889756710","https://openalex.org/W2895992958","https://openalex.org/W2911867426","https://openalex.org/W2920266844","https://openalex.org/W2924785619","https://openalex.org/W2946840143","https://openalex.org/W2949883708","https://openalex.org/W2962761333","https://openalex.org/W2962949242","https://openalex.org/W2963146412","https://openalex.org/W2963220074","https://openalex.org/W2963626582","https://openalex.org/W2965192329","https://openalex.org/W2969750612","https://openalex.org/W2970330753","https://openalex.org/W2970755399","https://openalex.org/W2992906545","https://openalex.org/W3009390838","https://openalex.org/W3031010353","https://openalex.org/W3085710185","https://openalex.org/W3099710307","https://openalex.org/W3101996726","https://openalex.org/W3122402965","https://openalex.org/W3146803896","https://openalex.org/W3161792008"],"related_works":["https://openalex.org/W2528305538","https://openalex.org/W2166116275","https://openalex.org/W2103496339","https://openalex.org/W2963146412","https://openalex.org/W3101996726","https://openalex.org/W2787302432","https://openalex.org/W3146803896","https://openalex.org/W1988115241","https://openalex.org/W3099710307","https://openalex.org/W2920266844","https://openalex.org/W2992906545","https://openalex.org/W3102511045","https://openalex.org/W3031010353","https://openalex.org/W2550848904","https://openalex.org/W3027734040","https://openalex.org/W2965192329","https://openalex.org/W2899283552","https://openalex.org/W2886881190","https://openalex.org/W2760972773","https://openalex.org/W2158581396"],"abstract_inverted_index":{"This":[0],"paper":[1],"establishes":[2],"the":[3,94],"(nearly)":[4],"optimal":[5,85],"approximation":[6,51,86],"error":[7,52,87],"characterization":[8],"of":[9,21,43,70],"deep":[10,40,60,67],"rectified":[11],"linear":[12],"unit":[13],"(ReLU)":[14],"networks":[15,42,69],"for":[16,100],"smooth":[17],"functions":[18],"in":[19,93],"terms":[20],"both":[22],"width":[23,44,71,102],"and":[24,46,58,74,103,108],"depth":[25,47,75,104],"simultaneously.":[26],"To":[27],"that":[28,33,66,96],"end,":[29],"we":[30,64],"first":[31],"prove":[32],"multivariate":[34],"polynomials":[35],"can":[36,78],"be":[37],"approximated":[38],"by":[39,106],"ReLU":[41,61,68],"$\\mathcal{O}(N)$":[45],"$\\mathcal{O}(L)$":[48],"with":[49,82],"an":[50],"$\\mathcal{O}(N^{-L})$.":[53],"Through":[54],"local":[55],"Taylor":[56],"expansions":[57],"their":[59],"network":[62],"approximations,":[63],"show":[65],"$\\mathcal{O}(N\\ln":[72],"N)$":[73],"$\\mathcal{O}(L\\ln":[76],"L)$":[77],"approximate":[79],"$f\\in":[80],"C^s([0,1]^d)$":[81],"a":[83],"nearly":[84],"$\\mathcal{O}(\\|f\\|_{C^s([0,1]^d)}N^{-2s/d}L^{-2s/d})$.":[88],"Our":[89],"estimate":[90],"is":[91,98],"non-asymptotic":[92],"sense":[95],"it":[97],"valid":[99],"arbitrary":[101],"specified":[105],"$N\\in\\mathbb{N}^+$":[107],"$L\\in\\mathbb{N}^+$,":[109],"respectively.":[110]},"counts_by_year":[{"year":2023,"cited_by_count":2},{"year":2022,"cited_by_count":9},{"year":2021,"cited_by_count":32},{"year":2020,"cited_by_count":12}],"updated_date":"2026-07-15T18:14:33.161393","created_date":"2025-10-10T00:00:00"}