{"id":"https://openalex.org/W7151794310","doi":"https://doi.org/10.48550/arxiv.2604.05635","title":"From Uniform to Learned Knots: A Study of Spline-Based Numerical Encodings for Tabular Deep Learning","display_name":"From Uniform to Learned Knots: A Study of Spline-Based Numerical Encodings for Tabular Deep Learning","publication_year":2026,"publication_date":"2026-04-07","ids":{"openalex":"https://openalex.org/W7151794310","doi":"https://doi.org/10.48550/arxiv.2604.05635"},"language":null,"primary_location":{"id":"doi:10.48550/arxiv.2604.05635","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2604.05635","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":false,"raw_source_name":null,"raw_type":"Preprint"},"type":"preprint","indexed_in":["datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://doi.org/10.48550/arxiv.2604.05635","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5133219501","display_name":"Manish Kumar","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Kumar, Manish","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"middle","author":{"id":"https://openalex.org/A5133158035","display_name":"Anton Frederik Thielmann","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Thielmann, Anton Frederik","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"middle","author":{"id":"https://openalex.org/A5133174961","display_name":"Christoph Weisser","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Weisser, Christoph","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"last","author":{"id":"https://openalex.org/A5016630281","display_name":"Benjamin S\u00e4fken","orcid":"https://orcid.org/0000-0003-4702-3333"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"S\u00e4fken, Benjamin","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":false,"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":0.1851000040769577,"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":0.1851000040769577,"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/T12303","display_name":"Tensor decomposition and applications","score":0.10279999673366547,"subfield":{"id":"https://openalex.org/subfields/2605","display_name":"Computational Mathematics"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T11435","display_name":"Polynomial and algebraic computation","score":0.06870000064373016,"subfield":{"id":"https://openalex.org/subfields/1703","display_name":"Computational Theory and Mathematics"},"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/preprocessor","display_name":"Preprocessor","score":0.7253999710083008},{"id":"https://openalex.org/keywords/deep-learning","display_name":"Deep learning","score":0.5717999935150146},{"id":"https://openalex.org/keywords/differentiable-function","display_name":"Differentiable function","score":0.5640000104904175},{"id":"https://openalex.org/keywords/encoding","display_name":"Encoding (memory)","score":0.4790000021457672},{"id":"https://openalex.org/keywords/focus","display_name":"Focus (optics)","score":0.45829999446868896},{"id":"https://openalex.org/keywords/knot","display_name":"Knot (papermaking)","score":0.40959998965263367},{"id":"https://openalex.org/keywords/regression","display_name":"Regression","score":0.36550000309944153}],"concepts":[{"id":"https://openalex.org/C34736171","wikidata":"https://www.wikidata.org/wiki/Q918333","display_name":"Preprocessor","level":2,"score":0.7253999710083008},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.5917999744415283},{"id":"https://openalex.org/C108583219","wikidata":"https://www.wikidata.org/wiki/Q197536","display_name":"Deep learning","level":2,"score":0.5717999935150146},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.5706999897956848},{"id":"https://openalex.org/C202615002","wikidata":"https://www.wikidata.org/wiki/Q783507","display_name":"Differentiable function","level":2,"score":0.5640000104904175},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.5605000257492065},{"id":"https://openalex.org/C125411270","wikidata":"https://www.wikidata.org/wiki/Q18653","display_name":"Encoding (memory)","level":2,"score":0.4790000021457672},{"id":"https://openalex.org/C192209626","wikidata":"https://www.wikidata.org/wiki/Q190909","display_name":"Focus (optics)","level":2,"score":0.45829999446868896},{"id":"https://openalex.org/C2779863119","wikidata":"https://www.wikidata.org/wiki/Q6423048","display_name":"Knot (papermaking)","level":2,"score":0.40959998965263367},{"id":"https://openalex.org/C83546350","wikidata":"https://www.wikidata.org/wiki/Q1139051","display_name":"Regression","level":2,"score":0.36550000309944153},{"id":"https://openalex.org/C119857082","wikidata":"https://www.wikidata.org/wiki/Q2539","display_name":"Machine learning","level":1,"score":0.364300012588501},{"id":"https://openalex.org/C153180895","wikidata":"https://www.wikidata.org/wiki/Q7148389","display_name":"Pattern recognition (psychology)","level":2,"score":0.3515999913215637},{"id":"https://openalex.org/C10390562","wikidata":"https://www.wikidata.org/wiki/Q581809","display_name":"Spline (mechanical)","level":2,"score":0.3398999869823456},{"id":"https://openalex.org/C2776359362","wikidata":"https://www.wikidata.org/wiki/Q2145286","display_name":"Representation (politics)","level":3,"score":0.337799996137619},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.3319999873638153},{"id":"https://openalex.org/C48753275","wikidata":"https://www.wikidata.org/wiki/Q11216","display_name":"Numerical analysis","level":2,"score":0.32580000162124634},{"id":"https://openalex.org/C80444323","wikidata":"https://www.wikidata.org/wiki/Q2878974","display_name":"Theoretical computer science","level":1,"score":0.29510000348091125},{"id":"https://openalex.org/C500300565","wikidata":"https://www.wikidata.org/wiki/Q925667","display_name":"Computer simulation","level":2,"score":0.2702000141143799},{"id":"https://openalex.org/C126255220","wikidata":"https://www.wikidata.org/wiki/Q141495","display_name":"Mathematical optimization","level":1,"score":0.2662999927997589},{"id":"https://openalex.org/C179518139","wikidata":"https://www.wikidata.org/wiki/Q5140297","display_name":"Coding (social sciences)","level":2,"score":0.2612999975681305},{"id":"https://openalex.org/C179799912","wikidata":"https://www.wikidata.org/wiki/Q205084","display_name":"Computational complexity theory","level":2,"score":0.2606000006198883},{"id":"https://openalex.org/C2982736386","wikidata":"https://www.wikidata.org/wiki/Q2539","display_name":"Statistical learning","level":2,"score":0.25619998574256897}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.48550/arxiv.2604.05635","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2604.05635","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":"Preprint"}],"best_oa_location":{"id":"doi:10.48550/arxiv.2604.05635","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2604.05635","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":false,"raw_source_name":null,"raw_type":"Preprint"},"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":0,"referenced_works":[],"related_works":[],"abstract_inverted_index":{"Numerical":[0],"preprocessing":[1,39,133],"remains":[2,44],"an":[3],"important":[4],"component":[5],"of":[6,13,36,100,115,141,240,247],"tabular":[7,41],"deep":[8,42],"learning,":[9],"where":[10],"the":[11,34,86,105,139,147,159,180,211,227],"representation":[12],"continuous":[14],"features":[15],"can":[16,206],"strongly":[17,145],"affect":[18],"downstream":[19],"performance.":[20],"Although":[21],"its":[22],"importance":[23],"is":[24,158],"well":[25,46],"established":[26],"for":[27,67,193,198,221],"classical":[28],"statistical":[29],"and":[30,74,82,118,124,127,151,185,195,223],"machine":[31],"learning":[32,43],"models,":[33],"role":[35],"explicit":[37],"numerical":[38,60,69,132,142,231],"in":[40,238,245],"less":[45],"understood.":[47],"In":[48],"this":[49,53],"work,":[50],"we":[51,89],"study":[52],"question":[54],"with":[55,104,188],"a":[56,91,112],"focus":[57],"on":[58,111,146,179],"spline-based":[59,165],"encodings.":[61],"We":[62,107,200],"investigate":[63],"three":[64],"spline":[65,181],"families":[66],"encoding":[68,156,173],"features,":[70],"namely":[71],"B-splines,":[72],"M-splines,":[73],"integrated":[75],"splines":[76],"(I-splines),":[77],"under":[78,210],"uniform,":[79],"quantile-based,":[80],"target-aware,":[81],"learnable-knot":[83,87,204],"placement.":[84],"For":[85,153,169],"variants,":[88],"use":[90],"differentiable":[92],"knot":[93,101],"parameterization":[94],"that":[95,138,203,230],"enables":[96],"stable":[97],"end-to-end":[98],"optimization":[99],"locations":[102],"jointly":[103],"backbone.":[106,152],"evaluate":[108],"these":[109],"encodings":[110,143,166,232],"diverse":[113],"collection":[114],"public":[116],"regression":[117],"classification":[119],"datasets":[120],"using":[121],"MLP,":[122],"ResNet,":[123],"FT-Transformer":[125],"backbones,":[126],"compare":[128],"them":[129],"against":[130],"common":[131],"baselines.":[134],"Our":[135],"results":[136,228],"show":[137,229],"effect":[140],"depends":[144,178],"task,":[148],"output":[149,186],"size,":[150,187],"classification,":[154],"piecewise-linear":[155],"(PLE)":[157],"most":[160],"robust":[161],"choice":[162],"overall,":[163],"while":[164],"remain":[167],"competitive.":[168],"regression,":[170],"no":[171],"single":[172],"dominates":[174],"uniformly.":[175],"Instead,":[176],"performance":[177],"family,":[182],"knot-placement":[183],"strategy,":[184],"larger":[189],"gains":[190],"typically":[191],"observed":[192],"MLP":[194],"ResNet":[196],"than":[197],"FT-Transformer.":[199],"further":[201],"find":[202],"variants":[205],"be":[207,234],"optimized":[208],"stably":[209],"proposed":[212],"parameterization,":[213],"but":[214,243],"may":[215],"substantially":[216],"increase":[217],"training":[218],"cost,":[219],"especially":[220],"M-spline":[222],"I-spline":[224],"expansions.":[225],"Overall,":[226],"should":[233],"assessed":[235],"not":[236],"only":[237],"terms":[239,246],"predictive":[241],"performance,":[242],"also":[244],"computational":[248],"overhead.":[249]},"counts_by_year":[],"updated_date":"2026-07-01T06:00:48.157686","created_date":"2026-04-09T00:00:00"}