{"id":"https://openalex.org/W4402502303","doi":"https://doi.org/10.48550/arxiv.2408.09624","title":"Attention is a smoothed cubic spline","display_name":"Attention is a smoothed cubic spline","publication_year":2024,"publication_date":"2024-08-19","ids":{"openalex":"https://openalex.org/W4402502303","doi":"https://doi.org/10.48550/arxiv.2408.09624"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2408.09624","is_oa":true,"landing_page_url":"http://arxiv.org/abs/2408.09624","pdf_url":"https://arxiv.org/pdf/2408.09624","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"},"type":"preprint","indexed_in":["arxiv","datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://arxiv.org/pdf/2408.09624","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5124324936","display_name":"Zehua Lai","orcid":null,"observed_orcids":[]},"institutions":[{"id":"https://openalex.org/I86519309","display_name":"The University of Texas at Austin","ror":"https://ror.org/00hj54h04","country_code":"US","type":"education","lineage":["https://openalex.org/I86519309"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Lai, Zehua","raw_affiliation_strings":["Department of Mathematics , University of Texas , Austin , TX 78712"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics , University of Texas , Austin , TX 78712","institution_ids":["https://openalex.org/I86519309"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5050794507","display_name":"Lek\u2010Heng Lim","orcid":"https://orcid.org/0000-0002-9808-0138","observed_orcids":["https://orcid.org/0000-0002-9808-0138"]},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Lim, Lek-Heng","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"last","author":{"id":"https://openalex.org/A5049705179","display_name":"Yucong Liu","orcid":null,"observed_orcids":[]},"institutions":[{"id":"https://openalex.org/I86519309","display_name":"The University of Texas at Austin","ror":"https://ror.org/00hj54h04","country_code":"US","type":"education","lineage":["https://openalex.org/I86519309"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Liu, Yucong","raw_affiliation_strings":["Department of Mathematics , University of Texas , Austin , TX 78712"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics , University of Texas , Austin , TX 78712","institution_ids":["https://openalex.org/I86519309"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":null,"has_fulltext":true,"cited_by_count":0,"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/T11245","display_name":"Advanced Numerical Analysis Techniques","score":0.9951000213623047,"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":0.9951000213623047,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/transformer-attention","display_name":"transformer attention","score":0.5619999766349792},{"id":"https://openalex.org/keywords/cubic-splines","display_name":"cubic splines","score":0.925000011920929},{"id":"https://openalex.org/keywords/spline-approximation","display_name":"spline approximation","score":0.515999972820282},{"id":"https://openalex.org/keywords/relu-activation-function","display_name":"ReLU activation","score":0.921999990940094},{"id":"https://openalex.org/keywords/encoder-decoder-attention","display_name":"encoder-decoder attention","score":0.7839999794960022},{"id":"https://openalex.org/keywords/masked-attention","display_name":"attention masking","score":0.7590000033378601},{"id":"https://openalex.org/keywords/smooth-activation-functions","display_name":"smooth activation functions","score":0.734000027179718},{"id":"https://openalex.org/keywords/softmax-function","display_name":"softmax","score":0.5440000295639038},{"id":"https://openalex.org/keywords/feedforward-neural-networks","display_name":"feedforward neural networks","score":0.5929999947547913},{"id":"https://openalex.org/keywords/approximation-theory","display_name":"approximation theory","score":0.41600000858306885}],"concepts":[{"id":"https://openalex.org/C107457265","wikidata":"https://www.wikidata.org/wiki/Q7546460","display_name":"Smoothing spline","level":4,"score":0.4500005841255188},{"id":"https://openalex.org/C10390562","wikidata":"https://www.wikidata.org/wiki/Q581809","display_name":"Spline (mechanical)","level":2,"score":0.43461665511131287},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.4296756982803345},{"id":"https://openalex.org/C84607441","wikidata":"https://www.wikidata.org/wiki/Q1537263","display_name":"Cubic Hermite spline","level":5,"score":0.4187623858451843},{"id":"https://openalex.org/C31447003","wikidata":"https://www.wikidata.org/wiki/Q545002","display_name":"Spline interpolation","level":3,"score":0.3854605257511139},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.3347013592720032},{"id":"https://openalex.org/C142362112","wikidata":"https://www.wikidata.org/wiki/Q735","display_name":"Art","level":0,"score":0.3210896849632263},{"id":"https://openalex.org/C105795698","wikidata":"https://www.wikidata.org/wiki/Q12483","display_name":"Statistics","level":1,"score":0.23443728685379028},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.20562750101089478},{"id":"https://openalex.org/C97355855","wikidata":"https://www.wikidata.org/wiki/Q11473","display_name":"Thermodynamics","level":1,"score":0.09691178798675537},{"id":"https://openalex.org/C205203396","wikidata":"https://www.wikidata.org/wiki/Q612143","display_name":"Bilinear interpolation","level":2,"score":0.0}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2408.09624","is_oa":true,"landing_page_url":"http://arxiv.org/abs/2408.09624","pdf_url":"https://arxiv.org/pdf/2408.09624","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":"doi:10.48550/arxiv.2408.09624","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2408.09624","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":null,"raw_source_name":null,"raw_type":"Preprint"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:2408.09624","is_oa":true,"landing_page_url":"http://arxiv.org/abs/2408.09624","pdf_url":"https://arxiv.org/pdf/2408.09624","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":[],"sustainable_development_goals_aurora":[],"study_designs":[],"awards":[],"funders":[],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W4402502303.pdf","grobid_xml":"https://content.openalex.org/works/W4402502303.grobid-xml"},"referenced_works_count":0,"referenced_works":[],"related_works":["https://openalex.org/W2117487173","https://openalex.org/W2106020793","https://openalex.org/W2380799314","https://openalex.org/W1975728046","https://openalex.org/W2510730013","https://openalex.org/W2154654948","https://openalex.org/W4250117113","https://openalex.org/W2070774473","https://openalex.org/W2102842245","https://openalex.org/W2018229429"],"abstract_inverted_index":{"We":[0],"highlight":[1],"a":[2,13,16,30,33,66,125,129,139,147],"perhaps":[3],"important":[4],"but":[5,26],"hitherto":[6],"unobserved":[7],"insight:":[8],"The":[9],"attention":[10,57,75],"module":[11],"in":[12,21,42,65,184,197],"transformer":[14,31,67,102,163,179],"is":[15,68,124,131,142,154],"smoothed":[17,140],"cubic":[18,60,78,106],"spline.":[19],"Viewed":[20],"this":[22,24,152],"manner,":[23],"mysterious":[25],"critical":[27],"component":[28,64],"of":[29,36,71,73,177,186,189],"becomes":[32],"natural":[34],"development":[35],"an":[37],"old":[38],"notion":[39],"deeply":[40],"entrenched":[41],"classical":[43],"approximation":[44],"theory.":[45],"More":[46],"precisely,":[47],"we":[48,111,159],"show":[49],"that":[50],"with":[51,146],"ReLU-activation,":[52],"attention,":[53,55],"masked":[54],"encoder-decoder":[56,95],"are":[58,105],"all":[59,88],"splines.":[61,109],"As":[62],"every":[63,122],"constructed":[69],"out":[70],"compositions":[72],"various":[74],"modules":[76],"(=":[77,85],"splines)":[79],"and":[80,94,99,150,193],"feed":[81],"forward":[82],"neural":[83],"networks":[84],"linear":[86],"splines),":[87],"its":[89],"components":[90],"--":[91,104],"encoder,":[92],"decoder,":[93],"blocks;":[96],"multilayered":[97],"encoders":[98],"decoders;":[100],"the":[101,113,117,161,175,178,190],"itself":[103],"or":[107],"higher-order":[108],"If":[110],"assume":[112],"Pierce-Birkhoff":[114],"conjecture,":[115],"then":[116],"converse":[118],"also":[119],"holds,":[120],"i.e.,":[121],"spline":[123,130],"ReLU-activated":[126],"encoder.":[127],"Since":[128],"generally":[132],"just":[133],"$C^2$,":[134],"one":[135,188],"way":[136],"to":[137,156],"obtain":[138],"$C^\\infty$-version":[141],"by":[143,166,180],"replacing":[144],"ReLU":[145],"smooth":[148],"activation;":[149],"if":[151],"activation":[153],"chosen":[155],"be":[157],"SoftMax,":[158],"recover":[160],"original":[162],"as":[164],"proposed":[165],"Vaswani":[167],"et":[168],"al.":[169],"This":[170],"insight":[171],"sheds":[172],"light":[173],"on":[174],"nature":[176],"casting":[181],"it":[182],"entirely":[183],"terms":[185],"splines,":[187],"best":[191],"known":[192],"thoroughly":[194],"understood":[195],"objects":[196],"applied":[198],"mathematics.":[199]},"counts_by_year":[],"updated_date":"2026-09-30T09:27:27.532063","created_date":"2025-10-10T00:00:00"}
