{"id":"https://openalex.org/W4403747153","doi":"https://doi.org/10.48550/arxiv.2409.12335","title":"Bridging the Gap Between Approximation and Learning via Optimal Approximation by ReLU MLPs of Maximal Regularity","display_name":"Bridging the Gap Between Approximation and Learning via Optimal Approximation by ReLU MLPs of Maximal Regularity","publication_year":2024,"publication_date":"2024-09-18","ids":{"openalex":"https://openalex.org/W4403747153","doi":"https://doi.org/10.48550/arxiv.2409.12335"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2409.12335","doi":null,"is_oa":true,"landing_page_url":"http://arxiv.org/abs/2409.12335","pdf_url":"https://arxiv.org/pdf/2409.12335","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":"cc-by","license_id":"https://openalex.org/licenses/cc-by","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/2409.12335","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5111220129","display_name":"Ruiyang Hong","orcid":null,"observed_orcids":[]},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Hong, Ruiyang","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"last","author":{"id":"https://openalex.org/A5036113771","display_name":"Anastasis Kratsios","orcid":"https://orcid.org/0000-0001-6791-3371","observed_orcids":["https://orcid.org/0000-0001-6791-3371"]},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Kratsios, Anastasis","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":0,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":null,"has_fulltext":true,"cited_by_count":19,"citation_normalized_percentile":null,"cited_by_percentile_year":null,"biblio":{"volume":null,"issue":null,"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":0.888700008392334,"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":0.888700008392334,"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":0.08630000054836273,"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/T12072","display_name":"Machine Learning and Algorithms","score":0.007300000172108412,"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":0.8830000162124634},{"id":"https://openalex.org/keywords/universal-approximation","display_name":"universal approximation","score":0.6159999966621399},{"id":"https://openalex.org/keywords/generalization","display_name":"generalization","score":0.6309999823570251},{"id":"https://openalex.org/keywords/holder-functions","display_name":"H\u00f6lder functions","score":0.8009999990463257},{"id":"https://openalex.org/keywords/sample-complexity","display_name":"sample complexity","score":0.8339999914169312},{"id":"https://openalex.org/keywords/uniformly-continuous-functions","display_name":"uniformly continuous functions","score":0.7670000195503235},{"id":"https://openalex.org/keywords/statistical-learning-theory","display_name":"statistical learning theory","score":0.3370000123977661}],"concepts":[{"id":"https://openalex.org/C174348530","wikidata":"https://www.wikidata.org/wiki/Q188635","display_name":"Bridging (networking)","level":2,"score":0.8608590364456177},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.36363938450813293},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.32814541459083557},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.3280022442340851},{"id":"https://openalex.org/C31258907","wikidata":"https://www.wikidata.org/wiki/Q1301371","display_name":"Computer network","level":1,"score":0.0}],"mesh":[],"locations_count":3,"locations":[{"id":"pmh:oai:arXiv.org:2409.12335","doi":null,"is_oa":true,"landing_page_url":"http://arxiv.org/abs/2409.12335","pdf_url":"https://arxiv.org/pdf/2409.12335","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":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},{"id":"pmh:oai:macsphere.mcmaster.ca:11375/30992","doi":null,"is_oa":true,"landing_page_url":"http://hdl.handle.net/11375/30992","pdf_url":"https://macsphere.mcmaster.ca/bitstreams/2d598755-ff9c-41b2-9682-db6bebae7bc7/download","source":{"id":"https://openalex.org/S4306402139","display_name":"MacSphere (McMaster University)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I98251732","host_organization_name":"McMaster University","host_organization_lineage":["https://openalex.org/I98251732"],"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":"Thesis"},{"id":"doi:10.48550/arxiv.2409.12335","doi":"https://doi.org/10.48550/arxiv.2409.12335","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2409.12335","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":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Preprint"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:2409.12335","doi":null,"is_oa":true,"landing_page_url":"http://arxiv.org/abs/2409.12335","pdf_url":"https://arxiv.org/pdf/2409.12335","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":"cc-by","license_id":"https://openalex.org/licenses/cc-by","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/G6938737173","display_name":"Geometric Deep Learning with Applications to Stochastic Finance","funder_award_id":"rgpin-2023-04482","funder_id":"https://openalex.org/F4320334593","funder_display_name":"Natural Sciences and Engineering Research Council of Canada"}],"funders":[{"id":"https://openalex.org/F4320334593","display_name":"Natural Sciences and Engineering Research Council of Canada","ror":"https://ror.org/01h531d29"}],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W4403747153.pdf","grobid_xml":"https://content.openalex.org/works/W4403747153.grobid-xml"},"referenced_works_count":9,"referenced_works":["https://openalex.org/W2785678896","https://openalex.org/W4353113517","https://openalex.org/W4379919393","https://openalex.org/W4389814128","https://openalex.org/W4399695990","https://openalex.org/W4400342125","https://openalex.org/W4402963196","https://openalex.org/W4405174070","https://openalex.org/W7163236216"],"related_works":["https://openalex.org/W4403624348","https://openalex.org/W7222959755","https://openalex.org/W4385431227","https://openalex.org/W3135609499","https://openalex.org/W3012168927","https://openalex.org/W3183349676","https://openalex.org/W4394671854","https://openalex.org/W2938185552","https://openalex.org/W2755888403","https://openalex.org/W2951316532"],"abstract_inverted_index":{"The":[0,17],"foundations":[1],"of":[2,12,62,94,143,178,185,225],"deep":[3,46],"learning":[4,15,47],"are":[5,50,100,105],"supported":[6],"by":[7,44,88],"the":[8,28,135,164,223,228,245],"seemingly":[9],"opposing":[10],"perspectives":[11],"approximation":[13],"or":[14],"theory.":[16],"former":[18],"advocates":[19],"for":[20],"large/expressive":[21],"models":[22],"that":[23,32,49,65,110,202,219,240],"need":[24],"not":[25,226],"generalize,":[26],"while":[27],"latter":[29],"considers":[30],"classes":[31],"generalize":[33],"but":[34,73,231],"may":[35],"be":[36,40,71,119],"too":[37],"small/constrained":[38],"to":[39,70,76,85,116,121],"universal":[41,72],"approximators.":[42],"Motivated":[43],"real-world":[45],"implementations":[48],"both":[51,67],"expressive":[52],"and":[53,104,140,152,155,166],"statistically":[54,106],"reliable,":[55],"we":[56],"ask:":[57],"\"Is":[58],"there":[59],"a":[60,82,90,122,129,181,199,213],"class":[61,93,177],"neural":[63,241],"networks":[64,242],"is":[66],"large":[68],"enough":[69,75],"structured":[74,92],"generalize?\"":[77],"This":[78],"paper":[79],"constructively":[80],"provides":[81],"positive":[83],"answer":[84],"this":[86,197],"question":[87],"identifying":[89],"highly":[91],"ReLU":[95,132],"multilayer":[96],"perceptions":[97],"(MLPs),":[98],"which":[99,169,217],"optimal":[101],"function":[102,113],"approximators":[103],"well-behaved.":[107],"We":[108,195],"show":[109],"any":[111,233],"$(L,\u03b1)$-H\u00f6lder":[112],"from":[114],"$[0,1]^d$":[115,127],"$[-n,n]$":[117],"can":[118,243],"approximated":[120],"uniform":[123],"$\\mathcal{O}(1/n)$":[124],"error":[125],"on":[126,249],"with":[128,134,148,212],"sparsely":[130],"connected":[131],"MLP":[133],"same":[136],"H\u00f6lder":[137,229],"exponent":[138],"$\u03b1$":[139],"coefficient":[141],"$L$,":[142],"width":[144],"$\\mathcal{O}(dn^{d/\u03b1})$,":[145],"depth":[146],"$\\mathcal{O}(\\log(d))$,":[147],"$\\mathcal{O}(dn^{d/\u03b1})$":[149],"nonzero":[150],"parameters,":[151],"whose":[153],"weights":[154],"biases":[156],"take":[157],"values":[158],"in":[159,163],"$\\{0,\\pm":[160],"1/2\\}$":[161],"except":[162],"first":[165],"last":[167],"layers":[168],"instead":[170],"have":[171],"magnitude":[172],"at-most":[173],"$n$.":[174],"Further,":[175],"our":[176,220],"MLPs":[179],"achieves":[180],"near-optimal":[182],"sample":[183],"complexity":[184],"$\\mathcal{O}(\\log(N)/\\sqrt{N})$":[186],"when":[187],"given":[188],"$N$":[189],"i.i.d.":[190],"normalized":[191],"sub-Gaussian":[192],"training":[193],"samples.":[194],"achieve":[196],"through":[198],"new":[200,214],"construction":[201,221],"perfectly":[203],"fits":[204],"together":[205],"linear":[206],"pieces":[207],"using":[208],"Kuhn":[209],"triangulations,":[210],"along":[211],"proof":[215],"technique":[216],"shows":[218],"preserves":[222],"regularity":[224],"only":[227],"functions,":[230],"also":[232],"uniformly":[234],"continuous":[235],"function.":[236],"Our":[237],"results":[238],"imply":[239],"solve":[244],"McShane":[246],"extension":[247],"problem":[248],"suitable":[250],"finite":[251],"sets.":[252]},"counts_by_year":[{"year":2026,"cited_by_count":10},{"year":2025,"cited_by_count":7},{"year":2024,"cited_by_count":2}],"updated_date":"2026-11-07T11:25:07.637180","created_date":"2024-10-25T00:00:00"}