{"id":"https://openalex.org/W7214947789","doi":"https://doi.org/10.48550/arxiv.2609.37657","title":"The double descent and Runge phenomena in overparametrized polynomial interpolation","display_name":"The double descent and Runge phenomena in overparametrized polynomial interpolation","publication_year":2026,"publication_date":"2026-09-29","ids":{"openalex":"https://openalex.org/W7214947789","doi":"https://doi.org/10.48550/arxiv.2609.37657"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2609.37657","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.37657","pdf_url":"https://arxiv.org/pdf/2609.37657","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":null,"license_id":null,"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/2609.37657","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5031965789","display_name":"Jason Wein","orcid":null,"observed_orcids":[]},"institutions":[{"id":"https://openalex.org/I170201317","display_name":"University of Pittsburgh","ror":"https://ror.org/01an3r305","country_code":"US","type":"education","lineage":["https://openalex.org/I170201317"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Wein, Jason","raw_affiliation_strings":["Department of Mathematics , University of Pittsburgh , Pittsburgh , PA 15213"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics , University of Pittsburgh , Pittsburgh , PA 15213","institution_ids":["https://openalex.org/I170201317"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5024215146","display_name":"Stephan Wojtowytsch","orcid":"https://orcid.org/0000-0003-3766-5332","observed_orcids":["https://orcid.org/0000-0003-3766-5332"]},"institutions":[{"id":"https://openalex.org/I170201317","display_name":"University of Pittsburgh","ror":"https://ror.org/01an3r305","country_code":"US","type":"education","lineage":["https://openalex.org/I170201317"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Wojtowytsch, Stephan","raw_affiliation_strings":["Department of Mathematics , University of Pittsburgh , Pittsburgh , PA 15213"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics , University of Pittsburgh , Pittsburgh , PA 15213","institution_ids":["https://openalex.org/I170201317"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":[],"corresponding_institution_ids":["https://openalex.org/I170201317"],"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/T11191","display_name":"Mathematical functions and polynomials","score":0.8223000168800354,"subfield":{"id":"https://openalex.org/subfields/2604","display_name":"Applied Mathematics"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T11191","display_name":"Mathematical functions and polynomials","score":0.8223000168800354,"subfield":{"id":"https://openalex.org/subfields/2604","display_name":"Applied 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/T10320","display_name":"Neural Networks and Applications","score":0.01889999955892563,"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/T12404","display_name":"Mathematical Approximation and Integration","score":0.01860000006854534,"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/double-descent","display_name":"double descent","score":0.9380000233650208},{"id":"https://openalex.org/keywords/runge-phenomenon","display_name":"Runge phenomenon","score":0.9950000047683716},{"id":"https://openalex.org/keywords/polynomial-interpolation","display_name":"polynomial interpolation","score":0.9100000262260437},{"id":"https://openalex.org/keywords/overparameterization","display_name":"overparameterization","score":0.8100000023841858},{"id":"https://openalex.org/keywords/monomial-basis","display_name":"monomial basis","score":0.8640000224113464},{"id":"https://openalex.org/keywords/chebyshev-basis","display_name":"Chebyshev basis","score":0.9789999723434448},{"id":"https://openalex.org/keywords/l2-norm-minimization","display_name":"L2 norm minimization","score":0.8119999766349792},{"id":"https://openalex.org/keywords/l1-norm-minimization","display_name":"l1-norm minimization","score":0.9929999709129333},{"id":"https://openalex.org/keywords/equidistant-nodes","display_name":"equidistant nodes","score":0.7699999809265137},{"id":"https://openalex.org/keywords/chebyshev-nodes","display_name":"Chebyshev nodes","score":0.8500000238418579}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8104000091552734},{"id":"https://openalex.org/C171326582","wikidata":"https://www.wikidata.org/wiki/Q1052508","display_name":"Chebyshev nodes","level":3,"score":0.7590000033378601},{"id":"https://openalex.org/C88080468","wikidata":"https://www.wikidata.org/wiki/Q637347","display_name":"Polynomial interpolation","level":4,"score":0.6729999780654907},{"id":"https://openalex.org/C137800194","wikidata":"https://www.wikidata.org/wiki/Q11713455","display_name":"Interpolation (computer graphics)","level":3,"score":0.6287000179290771},{"id":"https://openalex.org/C158245278","wikidata":"https://www.wikidata.org/wiki/Q4386982","display_name":"Equidistant","level":2,"score":0.6147000193595886},{"id":"https://openalex.org/C111458787","wikidata":"https://www.wikidata.org/wiki/Q215405","display_name":"Legendre polynomials","level":2,"score":0.5309000015258789},{"id":"https://openalex.org/C90119067","wikidata":"https://www.wikidata.org/wiki/Q43260","display_name":"Polynomial","level":2,"score":0.46950000524520874},{"id":"https://openalex.org/C129785596","wikidata":"https://www.wikidata.org/wiki/Q619511","display_name":"Chebyshev polynomials","level":2,"score":0.4618000090122223},{"id":"https://openalex.org/C171836373","wikidata":"https://www.wikidata.org/wiki/Q2266329","display_name":"Linear interpolation","level":3,"score":0.4569000005722046},{"id":"https://openalex.org/C46132437","wikidata":"https://www.wikidata.org/wiki/Q6901742","display_name":"Monomial basis","level":3,"score":0.4560999870300293},{"id":"https://openalex.org/C31447003","wikidata":"https://www.wikidata.org/wiki/Q545002","display_name":"Spline interpolation","level":3,"score":0.45249998569488525},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.44940000772476196},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.40059998631477356},{"id":"https://openalex.org/C101044782","wikidata":"https://www.wikidata.org/wiki/Q6787887","display_name":"Matrix polynomial","level":3,"score":0.39340001344680786},{"id":"https://openalex.org/C12426560","wikidata":"https://www.wikidata.org/wiki/Q189569","display_name":"Basis (linear algebra)","level":2,"score":0.35830000042915344},{"id":"https://openalex.org/C21424316","wikidata":"https://www.wikidata.org/wiki/Q718621","display_name":"Chebyshev filter","level":2,"score":0.35679998993873596},{"id":"https://openalex.org/C11252640","wikidata":"https://www.wikidata.org/wiki/Q243723","display_name":"Monomial","level":2,"score":0.3528999984264374},{"id":"https://openalex.org/C2776637919","wikidata":"https://www.wikidata.org/wiki/Q624380","display_name":"Descent (aeronautics)","level":2,"score":0.352400004863739},{"id":"https://openalex.org/C28537468","wikidata":"https://www.wikidata.org/wiki/Q464356","display_name":"Trigonometric interpolation","level":5,"score":0.3237999975681305},{"id":"https://openalex.org/C97395012","wikidata":"https://www.wikidata.org/wiki/Q1422356","display_name":"Reciprocal polynomial","level":4,"score":0.3109999895095825},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.3102000057697296},{"id":"https://openalex.org/C153258448","wikidata":"https://www.wikidata.org/wiki/Q1199743","display_name":"Gradient descent","level":3,"score":0.29829999804496765},{"id":"https://openalex.org/C25878781","wikidata":"https://www.wikidata.org/wiki/Q1473607","display_name":"Degree of a polynomial","level":3,"score":0.2962999939918518},{"id":"https://openalex.org/C45025165","wikidata":"https://www.wikidata.org/wiki/Q13424738","display_name":"Stable polynomial","level":5,"score":0.29440000653266907},{"id":"https://openalex.org/C153262748","wikidata":"https://www.wikidata.org/wiki/Q17098449","display_name":"Legendre function","level":3,"score":0.27639999985694885},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.27219998836517334},{"id":"https://openalex.org/C145242015","wikidata":"https://www.wikidata.org/wiki/Q774123","display_name":"Approximation theory","level":2,"score":0.2612999975681305},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.25589999556541443},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.25429999828338623}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2609.37657","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.37657","pdf_url":"https://arxiv.org/pdf/2609.37657","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":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},{"id":"doi:10.48550/arxiv.2609.37657","doi":"https://doi.org/10.48550/arxiv.2609.37657","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2609.37657","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":null,"license_id":null,"version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Preprint"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:2609.37657","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.37657","pdf_url":"https://arxiv.org/pdf/2609.37657","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":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},"sustainable_development_goals":[],"study_designs":[],"awards":[],"funders":[{"id":"https://openalex.org/F4320310174","display_name":"University of Pittsburgh","ror":"https://ror.org/01an3r305"}],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7214947789.pdf","grobid_xml":"https://content.openalex.org/works/W7214947789.grobid-xml"},"referenced_works_count":5,"referenced_works":["https://openalex.org/W2992906545","https://openalex.org/W3032534482","https://openalex.org/W3116218254","https://openalex.org/W4402963196","https://openalex.org/W7164884728"],"related_works":["https://openalex.org/W2963010479","https://openalex.org/W4210757071","https://openalex.org/W7165546608","https://openalex.org/W2089171658","https://openalex.org/W4367843873","https://openalex.org/W7011831843","https://openalex.org/W2506066191","https://openalex.org/W7058543035","https://openalex.org/W2560682138","https://openalex.org/W2581752838"],"abstract_inverted_index":{"The":[0],"Runge":[1],"phenomenon":[2,16],"in":[3,17,28,43,54],"polynomial":[4,26,31],"interpolation":[5,27],"is":[6],"often":[7],"considered":[8],"a":[9],"classical":[10],"analogue":[11],"of":[12,73,77],"the":[13,44,48,55,74],"double":[14],"descent":[15],"machine":[18],"learning.":[19],"In":[20],"this":[21],"note,":[22],"we":[23],"explore":[24],"overparameterized":[25],"three":[29],"popular":[30],"bases:":[32],"Monomial,":[33],"Chebyshev":[34,65],"and":[35,64],"Legendre":[36],"basis":[37],"with":[38],"coefficients":[39],"that":[40],"are":[41,71],"minimal":[42,53],"$\\ell^2$-norm":[45],"(and,":[46],"for":[47,62],"monomial":[49],"basis,":[50],"also":[51],"those":[52],"$\\ell^1$-norm).":[56],"We":[57],"present":[58],"our":[59],"results":[60,70],"primarily":[61],"equidistant":[63],"data":[66],"points,":[67],"but":[68],"many":[69],"independent":[72],"exact":[75],"form":[76],"sampling.":[78]},"counts_by_year":[],"updated_date":"2026-11-07T11:25:07.637180","created_date":"2026-10-01T00:00:00"}