{"id":"https://openalex.org/W2113484479","doi":"https://doi.org/10.5555/1577069.1755870","title":"When Is There a Representer Theorem? Vector Versus Matrix Regularizers","display_name":"When Is There a Representer Theorem? Vector Versus Matrix Regularizers","publication_year":2009,"publication_date":"2009-12-01","ids":{"openalex":"https://openalex.org/W2113484479","doi":"https://doi.org/10.5555/1577069.1755870","mag":"2113484479"},"language":"en","primary_location":{"id":"mag:2113484479","is_oa":false,"landing_page_url":"https://jmlr.csail.mit.edu/papers/volume10/argyriou09a/argyriou09a.pdf","pdf_url":null,"source":{"id":"https://openalex.org/S118988714","display_name":"Journal of Machine Learning Research","issn_l":"1532-4435","issn":["1532-4435","1533-7928"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310315718","host_organization_name":"The MIT Press","host_organization_lineage":["https://openalex.org/P4310315718"],"host_organization_lineage_names":["The MIT Press"],"type":"journal"},"license":null,"license_id":null,"version":null,"is_accepted":false,"is_published":false,"raw_source_name":"Journal of Machine Learning Research","raw_type":null},"type":"article","indexed_in":[],"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/A5068666260","display_name":"Andreas A. Argyriou","orcid":"https://orcid.org/0000-0003-2131-7114"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Andreas A. Argyriou","raw_affiliation_strings":[],"raw_orcid":"https://orcid.org/0000-0003-2131-7114","affiliations":[]},{"author_position":"middle","author":{"id":"https://openalex.org/A5002096458","display_name":"Charles A. Micchelli","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Charles A. Micchelli","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"last","author":{"id":"https://openalex.org/A5034260726","display_name":"Massimiliano Pontil","orcid":"https://orcid.org/0000-0001-9415-098X"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Massimiliano Pontil","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":9.719,"has_fulltext":false,"cited_by_count":133,"citation_normalized_percentile":{"value":0.9881411,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":91,"max":100},"biblio":{"volume":"10","issue":"87","first_page":"2507","last_page":"2529"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10500","display_name":"Sparse and Compressive Sensing Techniques","score":0.9998000264167786,"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/T10500","display_name":"Sparse and Compressive Sensing Techniques","score":0.9998000264167786,"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/T11205","display_name":"Numerical methods in inverse problems","score":0.9966999888420105,"subfield":{"id":"https://openalex.org/subfields/2610","display_name":"Mathematical Physics"},"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/T11236","display_name":"Control Systems and Identification","score":0.9965000152587891,"subfield":{"id":"https://openalex.org/subfields/2207","display_name":"Control and Systems Engineering"},"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/representer-theorem","display_name":"Representer theorem","score":0.781113862991333},{"id":"https://openalex.org/keywords/regularization","display_name":"Regularization (linguistics)","score":0.5858320593833923},{"id":"https://openalex.org/keywords/vector-valued-function","display_name":"Vector-valued function","score":0.5329253077507019},{"id":"https://openalex.org/keywords/matrix","display_name":"Matrix (chemical analysis)","score":0.5242053270339966},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.5044630765914917},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.49741509556770325},{"id":"https://openalex.org/keywords/differentiable-function","display_name":"Differentiable function","score":0.49398618936538696},{"id":"https://openalex.org/keywords/basis","display_name":"Basis (linear algebra)","score":0.46841540932655334},{"id":"https://openalex.org/keywords/class","display_name":"Class (philosophy)","score":0.460789293050766},{"id":"https://openalex.org/keywords/kernel","display_name":"Kernel (algebra)","score":0.4556560516357422},{"id":"https://openalex.org/keywords/context","display_name":"Context (archaeology)","score":0.44769492745399475},{"id":"https://openalex.org/keywords/matrix-norm","display_name":"Matrix norm","score":0.4415853023529053},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.4342513680458069},{"id":"https://openalex.org/keywords/mathematical-optimization","display_name":"Mathematical optimization","score":0.39878547191619873},{"id":"https://openalex.org/keywords/algebra-over-a-field","display_name":"Algebra over a field","score":0.3543804883956909},{"id":"https://openalex.org/keywords/support-vector-machine","display_name":"Support vector machine","score":0.32446926832199097},{"id":"https://openalex.org/keywords/artificial-intelligence","display_name":"Artificial intelligence","score":0.28927814960479736},{"id":"https://openalex.org/keywords/kernel-method","display_name":"Kernel method","score":0.27057093381881714},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.21611347794532776},{"id":"https://openalex.org/keywords/radial-basis-function-kernel","display_name":"Radial basis function kernel","score":0.14191514253616333},{"id":"https://openalex.org/keywords/eigenvalues-and-eigenvectors","display_name":"Eigenvalues and eigenvectors","score":0.10357648134231567}],"concepts":[{"id":"https://openalex.org/C172623408","wikidata":"https://www.wikidata.org/wiki/Q7314263","display_name":"Representer theorem","level":5,"score":0.781113862991333},{"id":"https://openalex.org/C2776135515","wikidata":"https://www.wikidata.org/wiki/Q17143721","display_name":"Regularization (linguistics)","level":2,"score":0.5858320593833923},{"id":"https://openalex.org/C2456702","wikidata":"https://www.wikidata.org/wiki/Q1375954","display_name":"Vector-valued function","level":2,"score":0.5329253077507019},{"id":"https://openalex.org/C106487976","wikidata":"https://www.wikidata.org/wiki/Q685816","display_name":"Matrix (chemical analysis)","level":2,"score":0.5242053270339966},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.5044630765914917},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.49741509556770325},{"id":"https://openalex.org/C202615002","wikidata":"https://www.wikidata.org/wiki/Q783507","display_name":"Differentiable function","level":2,"score":0.49398618936538696},{"id":"https://openalex.org/C12426560","wikidata":"https://www.wikidata.org/wiki/Q189569","display_name":"Basis (linear algebra)","level":2,"score":0.46841540932655334},{"id":"https://openalex.org/C2777212361","wikidata":"https://www.wikidata.org/wiki/Q5127848","display_name":"Class (philosophy)","level":2,"score":0.460789293050766},{"id":"https://openalex.org/C74193536","wikidata":"https://www.wikidata.org/wiki/Q574844","display_name":"Kernel (algebra)","level":2,"score":0.4556560516357422},{"id":"https://openalex.org/C2779343474","wikidata":"https://www.wikidata.org/wiki/Q3109175","display_name":"Context (archaeology)","level":2,"score":0.44769492745399475},{"id":"https://openalex.org/C92207270","wikidata":"https://www.wikidata.org/wiki/Q939253","display_name":"Matrix norm","level":3,"score":0.4415853023529053},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.4342513680458069},{"id":"https://openalex.org/C126255220","wikidata":"https://www.wikidata.org/wiki/Q141495","display_name":"Mathematical optimization","level":1,"score":0.39878547191619873},{"id":"https://openalex.org/C136119220","wikidata":"https://www.wikidata.org/wiki/Q1000660","display_name":"Algebra over a field","level":2,"score":0.3543804883956909},{"id":"https://openalex.org/C12267149","wikidata":"https://www.wikidata.org/wiki/Q282453","display_name":"Support vector machine","level":2,"score":0.32446926832199097},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.28927814960479736},{"id":"https://openalex.org/C122280245","wikidata":"https://www.wikidata.org/wiki/Q620622","display_name":"Kernel method","level":3,"score":0.27057093381881714},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.21611347794532776},{"id":"https://openalex.org/C75866337","wikidata":"https://www.wikidata.org/wiki/Q7280263","display_name":"Radial basis function kernel","level":4,"score":0.14191514253616333},{"id":"https://openalex.org/C158693339","wikidata":"https://www.wikidata.org/wiki/Q190524","display_name":"Eigenvalues and eigenvectors","level":2,"score":0.10357648134231567},{"id":"https://openalex.org/C86803240","wikidata":"https://www.wikidata.org/wiki/Q420","display_name":"Biology","level":0,"score":0.0},{"id":"https://openalex.org/C192562407","wikidata":"https://www.wikidata.org/wiki/Q228736","display_name":"Materials science","level":0,"score":0.0},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.0},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.0},{"id":"https://openalex.org/C159985019","wikidata":"https://www.wikidata.org/wiki/Q181790","display_name":"Composite material","level":1,"score":0.0},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.0},{"id":"https://openalex.org/C151730666","wikidata":"https://www.wikidata.org/wiki/Q7205","display_name":"Paleontology","level":1,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"mag:2113484479","is_oa":false,"landing_page_url":"https://jmlr.csail.mit.edu/papers/volume10/argyriou09a/argyriou09a.pdf","pdf_url":null,"source":{"id":"https://openalex.org/S118988714","display_name":"Journal of Machine Learning Research","issn_l":"1532-4435","issn":["1532-4435","1533-7928"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310315718","host_organization_name":"The MIT Press","host_organization_lineage":["https://openalex.org/P4310315718"],"host_organization_lineage_names":["The MIT Press"],"type":"journal"},"license":null,"license_id":null,"version":null,"is_accepted":false,"is_published":null,"raw_source_name":"Journal of Machine Learning Research","raw_type":null}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":52,"referenced_works":["https://openalex.org/W740415","https://openalex.org/W123755266","https://openalex.org/W173396443","https://openalex.org/W1181527464","https://openalex.org/W1484835511","https://openalex.org/W1488435683","https://openalex.org/W1510073064","https://openalex.org/W1540155273","https://openalex.org/W1541700983","https://openalex.org/W1562931283","https://openalex.org/W1569436870","https://openalex.org/W1605479404","https://openalex.org/W1639316992","https://openalex.org/W1966096622","https://openalex.org/W1972163814","https://openalex.org/W1981025032","https://openalex.org/W1986280275","https://openalex.org/W2002355073","https://openalex.org/W2047028564","https://openalex.org/W2047278710","https://openalex.org/W2054121219","https://openalex.org/W2065180801","https://openalex.org/W2080726496","https://openalex.org/W2087347434","https://openalex.org/W2101593431","https://openalex.org/W2110652811","https://openalex.org/W2115003579","https://openalex.org/W2116413942","https://openalex.org/W2121033924","https://openalex.org/W2121959576","https://openalex.org/W2122090912","https://openalex.org/W2123856749","https://openalex.org/W2130903752","https://openalex.org/W2138019504","https://openalex.org/W2144752499","https://openalex.org/W2146766088","https://openalex.org/W2149846618","https://openalex.org/W2151211319","https://openalex.org/W2151462928","https://openalex.org/W2156267734","https://openalex.org/W2156909104","https://openalex.org/W2163198669","https://openalex.org/W2165644552","https://openalex.org/W2165744911","https://openalex.org/W2170563643","https://openalex.org/W2202343345","https://openalex.org/W2604243882","https://openalex.org/W2610857016","https://openalex.org/W2611328865","https://openalex.org/W2912522929","https://openalex.org/W3119651796","https://openalex.org/W3147133440"],"related_works":["https://openalex.org/W1540155273","https://openalex.org/W1986280275","https://openalex.org/W2144752499","https://openalex.org/W1560724230","https://openalex.org/W2047278710","https://openalex.org/W2015904350","https://openalex.org/W2135046866","https://openalex.org/W2065180801","https://openalex.org/W2148603752","https://openalex.org/W2099720998","https://openalex.org/W1510073064","https://openalex.org/W2119187866","https://openalex.org/W2143104527","https://openalex.org/W2125085449","https://openalex.org/W2154389918","https://openalex.org/W2130903752","https://openalex.org/W2115003579","https://openalex.org/W2110652811","https://openalex.org/W1939941161","https://openalex.org/W1562931283"],"abstract_inverted_index":{"We":[0,82,123],"consider":[1],"a":[2,10,29,41,92,94,110,118,125],"general":[3,112],"class":[4,120,132],"of":[5,12,17,32,44,59,72,79,121,133,142,157],"regularization":[6,88],"methods":[7,61,89],"which":[8,90,96,115],"learn":[9,91],"vector":[11,39],"parameters":[13],"on":[14],"the":[15,26,33,37,45,52,57,70,73,77,100,154],"basis":[16,58],"linear":[18,42],"measurements.":[19],"It":[20],"is":[21,28,40,97],"well":[22],"known":[23,50],"that":[24],"if":[25],"regularizer":[27],"nondecreasing":[30,159],"function":[31],"L2":[34],"norm,":[35],"then":[36],"learned":[38],"combination":[43],"input":[46],"data.":[47],"This":[48],"result,":[49],"as":[51],"representer":[53,113],"theorem,":[54,114],"lies":[55],"at":[56],"kernel-based":[60],"in":[62,76],"machine":[63],"learning.":[64,104],"In":[65,105],"this":[66,106,131],"paper,":[67],"we":[68,108,137],"prove":[69],"necessity":[71],"above":[74],"condition,":[75],"case":[78],"differentiable":[80],"regularizers.":[81,122],"further":[83],"extend":[84],"our":[85],"analysis":[86,146],"to":[87,102],"matrix,":[93],"problem":[95],"motivated":[98],"by":[99],"application":[101],"multi-task":[103],"context,":[107],"study":[109],"more":[111],"holds":[116],"for":[117],"larger":[119],"provide":[124],"necessary":[126],"and":[127,136],"sufficient":[128],"condition":[129],"characterizing":[130],"matrix":[134,151,158],"regularizers":[135],"highlight":[138],"some":[139],"concrete":[140],"examples":[141],"practical":[143],"importance.":[144],"Our":[145],"uses":[147],"basic":[148],"principles":[149],"from":[150],"theory,":[152],"especially":[153],"useful":[155],"notion":[156],"functions.":[160]},"counts_by_year":[{"year":2025,"cited_by_count":1},{"year":2023,"cited_by_count":4},{"year":2022,"cited_by_count":3},{"year":2021,"cited_by_count":8},{"year":2020,"cited_by_count":15},{"year":2019,"cited_by_count":11},{"year":2018,"cited_by_count":10},{"year":2017,"cited_by_count":10},{"year":2016,"cited_by_count":9},{"year":2015,"cited_by_count":13},{"year":2014,"cited_by_count":12},{"year":2013,"cited_by_count":9},{"year":2012,"cited_by_count":16}],"updated_date":"2025-10-10T17:16:08.811792","created_date":"2025-10-10T00:00:00"}