{"id":"https://openalex.org/W4200134383","doi":"https://doi.org/10.1142/s0219691321500582","title":"A note on the applications of one primary function in deep neural networks","display_name":"A note on the applications of one primary function in deep neural networks","publication_year":2021,"publication_date":"2021-12-04","ids":{"openalex":"https://openalex.org/W4200134383","doi":"https://doi.org/10.1142/s0219691321500582"},"language":"en","primary_location":{"id":"doi:10.1142/s0219691321500582","is_oa":false,"landing_page_url":"https://doi.org/10.1142/s0219691321500582","pdf_url":null,"source":{"id":"https://openalex.org/S56986848","display_name":"International Journal of Wavelets Multiresolution and Information Processing","issn_l":"0219-6913","issn":["0219-6913","1793-690X"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319815","host_organization_name":"World Scientific","host_organization_lineage":["https://openalex.org/P4310319815"],"host_organization_lineage_names":["World Scientific"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"International Journal of Wavelets, Multiresolution and Information Processing","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/A5018410320","display_name":"Hengjie Chen","orcid":"https://orcid.org/0000-0002-9143-9945"},"institutions":[{"id":"https://openalex.org/I1328775524","display_name":"Zhejiang Sci-Tech University","ror":"https://ror.org/03893we55","country_code":"CN","type":"education","lineage":["https://openalex.org/I1328775524"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Hengjie Chen","raw_affiliation_strings":["Department of Mathematical Sciences, Zhejiang Sci-Tech University, Hangzhou 310018, P. R. China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematical Sciences, Zhejiang Sci-Tech University, Hangzhou 310018, P. R. China","institution_ids":["https://openalex.org/I1328775524"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5100664776","display_name":"Zhong Li","orcid":"https://orcid.org/0000-0002-1767-1519"},"institutions":[{"id":"https://openalex.org/I1328775524","display_name":"Zhejiang Sci-Tech University","ror":"https://ror.org/03893we55","country_code":"CN","type":"education","lineage":["https://openalex.org/I1328775524"]}],"countries":["CN"],"is_corresponding":true,"raw_author_name":"Zhong Li","raw_affiliation_strings":["Department of Mathematical Sciences, Zhejiang Sci-Tech University, Hangzhou 310018, P. R. China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematical Sciences, Zhejiang Sci-Tech University, Hangzhou 310018, P. R. China","institution_ids":["https://openalex.org/I1328775524"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":2,"corresponding_author_ids":["https://openalex.org/A5100664776"],"corresponding_institution_ids":["https://openalex.org/I1328775524"],"apc_list":null,"apc_paid":null,"fwci":0.0,"has_fulltext":false,"cited_by_count":1,"citation_normalized_percentile":{"value":0.18982864,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":91,"max":95},"biblio":{"volume":"20","issue":"04","first_page":null,"last_page":null},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10320","display_name":"Neural Networks and Applications","score":0.9998000264167786,"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":0.9998000264167786,"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":0.9976000189781189,"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":0.9872999787330627,"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/equidistant","display_name":"Equidistant","score":0.6735572814941406},{"id":"https://openalex.org/keywords/function","display_name":"Function (biology)","score":0.579434335231781},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.576468825340271},{"id":"https://openalex.org/keywords/unit-interval","display_name":"Unit interval","score":0.5366491675376892},{"id":"https://openalex.org/keywords/interval","display_name":"Interval (graph theory)","score":0.5304526686668396},{"id":"https://openalex.org/keywords/distribution","display_name":"Distribution (mathematics)","score":0.5043290853500366},{"id":"https://openalex.org/keywords/integer","display_name":"Integer (computer science)","score":0.48459064960479736},{"id":"https://openalex.org/keywords/constant","display_name":"Constant (computer programming)","score":0.4647952914237976},{"id":"https://openalex.org/keywords/artificial-neural-network","display_name":"Artificial neural network","score":0.45605039596557617},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.4173322319984436},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.40865814685821533},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.2713235318660736},{"id":"https://openalex.org/keywords/artificial-intelligence","display_name":"Artificial intelligence","score":0.20559772849082947},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.19486135244369507},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.10373467206954956}],"concepts":[{"id":"https://openalex.org/C158245278","wikidata":"https://www.wikidata.org/wiki/Q4386982","display_name":"Equidistant","level":2,"score":0.6735572814941406},{"id":"https://openalex.org/C14036430","wikidata":"https://www.wikidata.org/wiki/Q3736076","display_name":"Function (biology)","level":2,"score":0.579434335231781},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.576468825340271},{"id":"https://openalex.org/C128394911","wikidata":"https://www.wikidata.org/wiki/Q1987578","display_name":"Unit interval","level":2,"score":0.5366491675376892},{"id":"https://openalex.org/C2778067643","wikidata":"https://www.wikidata.org/wiki/Q166507","display_name":"Interval (graph theory)","level":2,"score":0.5304526686668396},{"id":"https://openalex.org/C110121322","wikidata":"https://www.wikidata.org/wiki/Q865811","display_name":"Distribution (mathematics)","level":2,"score":0.5043290853500366},{"id":"https://openalex.org/C97137487","wikidata":"https://www.wikidata.org/wiki/Q729138","display_name":"Integer (computer science)","level":2,"score":0.48459064960479736},{"id":"https://openalex.org/C2777027219","wikidata":"https://www.wikidata.org/wiki/Q1284190","display_name":"Constant (computer programming)","level":2,"score":0.4647952914237976},{"id":"https://openalex.org/C50644808","wikidata":"https://www.wikidata.org/wiki/Q192776","display_name":"Artificial neural network","level":2,"score":0.45605039596557617},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.4173322319984436},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.40865814685821533},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.2713235318660736},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.20559772849082947},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.19486135244369507},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.10373467206954956},{"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/C86803240","wikidata":"https://www.wikidata.org/wiki/Q420","display_name":"Biology","level":0,"score":0.0},{"id":"https://openalex.org/C78458016","wikidata":"https://www.wikidata.org/wiki/Q840400","display_name":"Evolutionary biology","level":1,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1142/s0219691321500582","is_oa":false,"landing_page_url":"https://doi.org/10.1142/s0219691321500582","pdf_url":null,"source":{"id":"https://openalex.org/S56986848","display_name":"International Journal of Wavelets Multiresolution and Information Processing","issn_l":"0219-6913","issn":["0219-6913","1793-690X"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319815","host_organization_name":"World Scientific","host_organization_lineage":["https://openalex.org/P4310319815"],"host_organization_lineage_names":["World Scientific"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"International Journal of Wavelets, Multiresolution and Information Processing","raw_type":"journal-article"}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[{"id":"https://openalex.org/G3156292133","display_name":null,"funder_award_id":"12171434","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"}],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":28,"referenced_works":["https://openalex.org/W355521715","https://openalex.org/W1971735090","https://openalex.org/W1975824488","https://openalex.org/W1983776716","https://openalex.org/W2002023932","https://openalex.org/W2023750212","https://openalex.org/W2038266964","https://openalex.org/W2041793405","https://openalex.org/W2057700945","https://openalex.org/W2103496339","https://openalex.org/W2139923370","https://openalex.org/W2157890204","https://openalex.org/W2158581396","https://openalex.org/W2166116275","https://openalex.org/W2168420370","https://openalex.org/W2267573953","https://openalex.org/W2528305538","https://openalex.org/W2597851033","https://openalex.org/W2724635206","https://openalex.org/W2886881190","https://openalex.org/W2919115771","https://openalex.org/W2924785619","https://openalex.org/W2963172624","https://openalex.org/W2963626582","https://openalex.org/W3105890793","https://openalex.org/W3108043981","https://openalex.org/W4240805545","https://openalex.org/W4255975175"],"related_works":["https://openalex.org/W2055230676","https://openalex.org/W2373708349","https://openalex.org/W2370379095","https://openalex.org/W2117630634","https://openalex.org/W2347490332","https://openalex.org/W1604481637","https://openalex.org/W39381033","https://openalex.org/W2242060590","https://openalex.org/W2184232393","https://openalex.org/W2226528826"],"abstract_inverted_index":{"By":[0,74],"applying":[1],"fundamental":[2],"mathematical":[3],"knowledge,":[4],"this":[5,162],"paper":[6,163],"proves":[7],"that":[8,26,85,132],"the":[9,24,27,30,48,63,86,108,118,129,138,149],"function":[10,31,51,87,176],"[Formula:":[11,20,44,71,88,104,114,123,177],"see":[12,21,45,72,89,105,115,124,178],"text]":[13,22,46,73,90,106,116,125,179],"is":[14,57,122],"an":[15,78],"integer":[16],"no":[17],"less":[18],"than":[19],"has":[23,133],"property":[25],"difference":[28],"between":[29],"value":[32],"of":[33,36,50,53,65,120,152],"middle":[34],"point":[35],"arbitrarily":[37],"two":[38,55],"adjacent":[39],"equidistant":[40,109],"distribution":[41,110],"nodes":[42,56,66,111],"on":[43,62,107,128],"and":[47,68,117,137,147,156,174,180,184],"mean":[49],"values":[52],"these":[54],"a":[58,95,144],"constant":[59],"depending":[60],"only":[61,69],"number":[64],"if":[67,70],"them,":[75],"we":[76,142],"establish":[77],"important":[79],"result":[80,131],"about":[81],"deep":[82,96,145,166],"neural":[83],"networks":[84],"can":[91],"be":[92],"interpolated":[93],"by":[94],"Rectified":[97],"Linear":[98],"Unit":[99],"(ReLU)":[100],"network":[101,146,167],"with":[102,168],"depth":[103],"in":[112],"interval":[113],"error":[119,150],"approximation":[121,153],"Then":[126],"based":[127],"main":[130],"just":[134],"been":[135],"proven":[136],"Chebyshev":[139],"orthogonal":[140],"polynomials,":[141],"construct":[143],"give":[148],"estimate":[151],"to":[154],"polynomials":[155],"continuous":[157],"functions,":[158],"respectively.":[159],"In":[160],"addition,":[161],"constructs":[164],"one":[165],"local":[169],"sparse":[170],"connections,":[171],"shared":[172],"weights":[173],"activation":[175],"discusses":[181],"its":[182],"density":[183],"complexity.":[185]},"counts_by_year":[{"year":2025,"cited_by_count":1}],"updated_date":"2026-05-21T06:26:12.895304","created_date":"2025-10-10T00:00:00"}
