{"id":"https://openalex.org/W1665378886","doi":"https://doi.org/10.1007/s11222-019-09886-w","title":"Hilbert space methods for reduced-rank Gaussian process regression","display_name":"Hilbert space methods for reduced-rank Gaussian process regression","publication_year":2019,"publication_date":"2019-08-05","ids":{"openalex":"https://openalex.org/W1665378886","doi":"https://doi.org/10.1007/s11222-019-09886-w","mag":"1665378886"},"language":"en","primary_location":{"id":"doi:10.1007/s11222-019-09886-w","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s11222-019-09886-w","pdf_url":"https://link.springer.com/content/pdf/10.1007/s11222-019-09886-w.pdf","source":{"id":"https://openalex.org/S5437875","display_name":"Statistics and Computing","issn_l":"0960-3174","issn":["0960-3174","1573-1375"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319900","host_organization_name":"Springer Science+Business Media","host_organization_lineage":["https://openalex.org/P4310319900","https://openalex.org/P4310319965"],"host_organization_lineage_names":["Springer Science+Business Media","Springer Nature"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Statistics and Computing","raw_type":"journal-article"},"type":"article","indexed_in":["arxiv","crossref"],"open_access":{"is_oa":true,"oa_status":"hybrid","oa_url":"https://link.springer.com/content/pdf/10.1007/s11222-019-09886-w.pdf","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":null,"display_name":"Arno Solin","orcid":"https://orcid.org/0000-0002-0958-7886"},"institutions":[{"id":"https://openalex.org/I9927081","display_name":"Aalto University","ror":"https://ror.org/020hwjq30","country_code":"FI","type":"education","lineage":["https://openalex.org/I9927081"]}],"countries":["FI"],"is_corresponding":true,"raw_author_name":"Arno Solin","raw_affiliation_strings":["Department of Computer Science, Aalto University, P.O. Box 15400, 00076, Aalto, Finland"],"raw_orcid":"https://orcid.org/0000-0002-0958-7886","affiliations":[{"raw_affiliation_string":"Department of Computer Science, Aalto University, P.O. Box 15400, 00076, Aalto, Finland","institution_ids":["https://openalex.org/I9927081"]}]},{"author_position":"last","author":{"id":null,"display_name":"Simo S\u00e4rkk\u00e4","orcid":"https://orcid.org/0000-0002-7031-9354"},"institutions":[{"id":"https://openalex.org/I9927081","display_name":"Aalto University","ror":"https://ror.org/020hwjq30","country_code":"FI","type":"education","lineage":["https://openalex.org/I9927081"]}],"countries":["FI"],"is_corresponding":false,"raw_author_name":"Simo S\u00e4rkk\u00e4","raw_affiliation_strings":["Department of Electrical Engineering and Automation, Aalto University, P.O. Box 12200, 00076, Aalto, Finland"],"raw_orcid":"https://orcid.org/0000-0002-7031-9354","affiliations":[{"raw_affiliation_string":"Department of Electrical Engineering and Automation, Aalto University, P.O. Box 12200, 00076, Aalto, Finland","institution_ids":["https://openalex.org/I9927081"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":[],"corresponding_institution_ids":["https://openalex.org/I9927081"],"apc_list":{"value":2890,"currency":"USD","value_usd":2890},"apc_paid":{"value":2890,"currency":"USD","value_usd":2890},"fwci":7.8039,"has_fulltext":true,"cited_by_count":136,"citation_normalized_percentile":{"value":0.97704543,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":90,"max":100},"biblio":{"volume":"30","issue":"2","first_page":"419","last_page":"446"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T12814","display_name":"Gaussian Processes and Bayesian Inference","score":0.9199000000953674,"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/T12814","display_name":"Gaussian Processes and Bayesian Inference","score":0.9199000000953674,"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":0.01759999990463257,"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/T10848","display_name":"Advanced Multi-Objective Optimization Algorithms","score":0.013000000268220901,"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/covariance-function","display_name":"Covariance function","score":0.6722000241279602},{"id":"https://openalex.org/keywords/covariance","display_name":"Covariance","score":0.6488999724388123},{"id":"https://openalex.org/keywords/covariance-operator","display_name":"Covariance operator","score":0.5698000192642212},{"id":"https://openalex.org/keywords/rational-quadratic-covariance-function","display_name":"Rational quadratic covariance function","score":0.5587999820709229},{"id":"https://openalex.org/keywords/gaussian-process","display_name":"Gaussian process","score":0.5090000033378601},{"id":"https://openalex.org/keywords/mat\u00e9rn-covariance-function","display_name":"Mat\u00e9rn covariance function","score":0.47749999165534973},{"id":"https://openalex.org/keywords/reproducing-kernel-hilbert-space","display_name":"Reproducing kernel Hilbert space","score":0.4562999904155731},{"id":"https://openalex.org/keywords/eigenfunction","display_name":"Eigenfunction","score":0.45339998602867126},{"id":"https://openalex.org/keywords/hilbert-space","display_name":"Hilbert space","score":0.43299999833106995}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8327000141143799},{"id":"https://openalex.org/C137250428","wikidata":"https://www.wikidata.org/wiki/Q5178897","display_name":"Covariance function","level":3,"score":0.6722000241279602},{"id":"https://openalex.org/C178650346","wikidata":"https://www.wikidata.org/wiki/Q201984","display_name":"Covariance","level":2,"score":0.6488999724388123},{"id":"https://openalex.org/C103692563","wikidata":"https://www.wikidata.org/wiki/Q5178900","display_name":"Covariance operator","level":3,"score":0.5698000192642212},{"id":"https://openalex.org/C148893098","wikidata":"https://www.wikidata.org/wiki/Q7295778","display_name":"Rational quadratic covariance function","level":5,"score":0.5587999820709229},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.5508000254631042},{"id":"https://openalex.org/C61326573","wikidata":"https://www.wikidata.org/wiki/Q1496376","display_name":"Gaussian process","level":3,"score":0.5090000033378601},{"id":"https://openalex.org/C118006245","wikidata":"https://www.wikidata.org/wiki/Q6792079","display_name":"Mat\u00e9rn covariance function","level":5,"score":0.47749999165534973},{"id":"https://openalex.org/C80884492","wikidata":"https://www.wikidata.org/wiki/Q3345678","display_name":"Reproducing kernel Hilbert space","level":3,"score":0.4562999904155731},{"id":"https://openalex.org/C128803854","wikidata":"https://www.wikidata.org/wiki/Q1307821","display_name":"Eigenfunction","level":3,"score":0.45339998602867126},{"id":"https://openalex.org/C62799726","wikidata":"https://www.wikidata.org/wiki/Q190056","display_name":"Hilbert space","level":2,"score":0.43299999833106995},{"id":"https://openalex.org/C180877172","wikidata":"https://www.wikidata.org/wiki/Q5401390","display_name":"Estimation of covariance matrices","level":3,"score":0.42640000581741333},{"id":"https://openalex.org/C104942944","wikidata":"https://www.wikidata.org/wiki/Q3434686","display_name":"Truncation error","level":2,"score":0.38920000195503235},{"id":"https://openalex.org/C5917680","wikidata":"https://www.wikidata.org/wiki/Q2621825","display_name":"Basis function","level":2,"score":0.3752000033855438},{"id":"https://openalex.org/C111030470","wikidata":"https://www.wikidata.org/wiki/Q1430460","display_name":"Curse of dimensionality","level":2,"score":0.36340001225471497},{"id":"https://openalex.org/C57869625","wikidata":"https://www.wikidata.org/wiki/Q1783502","display_name":"Rate of convergence","level":3,"score":0.3610999882221222},{"id":"https://openalex.org/C163716315","wikidata":"https://www.wikidata.org/wiki/Q901177","display_name":"Gaussian","level":2,"score":0.3467999994754791},{"id":"https://openalex.org/C164660894","wikidata":"https://www.wikidata.org/wiki/Q2037833","display_name":"Piecewise","level":2,"score":0.3125},{"id":"https://openalex.org/C186080144","wikidata":"https://www.wikidata.org/wiki/Q358733","display_name":"Series expansion","level":2,"score":0.310699999332428},{"id":"https://openalex.org/C126255220","wikidata":"https://www.wikidata.org/wiki/Q141495","display_name":"Mathematical optimization","level":1,"score":0.3052999973297119},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.2985000014305115},{"id":"https://openalex.org/C143724316","wikidata":"https://www.wikidata.org/wiki/Q312468","display_name":"Series (stratigraphy)","level":2,"score":0.28690001368522644},{"id":"https://openalex.org/C14036430","wikidata":"https://www.wikidata.org/wiki/Q3736076","display_name":"Function (biology)","level":2,"score":0.2773999869823456},{"id":"https://openalex.org/C126372606","wikidata":"https://www.wikidata.org/wiki/Q6503511","display_name":"Law of total covariance","level":5,"score":0.27709999680519104},{"id":"https://openalex.org/C106195933","wikidata":"https://www.wikidata.org/wiki/Q7847935","display_name":"Truncation (statistics)","level":2,"score":0.2768000066280365},{"id":"https://openalex.org/C51267290","wikidata":"https://www.wikidata.org/wiki/Q5527848","display_name":"Gaussian random field","level":4,"score":0.2718000113964081},{"id":"https://openalex.org/C158693339","wikidata":"https://www.wikidata.org/wiki/Q190524","display_name":"Eigenvalues and eigenvectors","level":2,"score":0.271699994802475},{"id":"https://openalex.org/C97937538","wikidata":"https://www.wikidata.org/wiki/Q199691","display_name":"Laplace transform","level":2,"score":0.27129998803138733},{"id":"https://openalex.org/C34388435","wikidata":"https://www.wikidata.org/wiki/Q2267362","display_name":"Bounded function","level":2,"score":0.258899986743927},{"id":"https://openalex.org/C207864730","wikidata":"https://www.wikidata.org/wiki/Q179467","display_name":"Fourier series","level":2,"score":0.25440001487731934}],"mesh":[],"locations_count":3,"locations":[{"id":"doi:10.1007/s11222-019-09886-w","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s11222-019-09886-w","pdf_url":"https://link.springer.com/content/pdf/10.1007/s11222-019-09886-w.pdf","source":{"id":"https://openalex.org/S5437875","display_name":"Statistics and Computing","issn_l":"0960-3174","issn":["0960-3174","1573-1375"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319900","host_organization_name":"Springer Science+Business Media","host_organization_lineage":["https://openalex.org/P4310319900","https://openalex.org/P4310319965"],"host_organization_lineage_names":["Springer Science+Business Media","Springer Nature"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Statistics and Computing","raw_type":"journal-article"},{"id":"pmh:oai:arXiv.org:1401.5508","is_oa":true,"landing_page_url":"http://arxiv.org/abs/1401.5508","pdf_url":"https://arxiv.org/pdf/1401.5508","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,"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":"pmh:oai:aaltodoc.aalto.fi:123456789/39797","is_oa":true,"landing_page_url":"https://aaltodoc.aalto.fi/handle/123456789/39797","pdf_url":null,"source":{"id":"https://openalex.org/S4306401662","display_name":"Aaltodoc (Aalto University)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I9927081","host_organization_name":"Aalto University","host_organization_lineage":["https://openalex.org/I9927081"],"host_organization_lineage_names":[],"type":"repository"},"license":"other-oa","license_id":"https://openalex.org/licenses/other-oa","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"info:eu-repo/semantics/article"}],"best_oa_location":{"id":"doi:10.1007/s11222-019-09886-w","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s11222-019-09886-w","pdf_url":"https://link.springer.com/content/pdf/10.1007/s11222-019-09886-w.pdf","source":{"id":"https://openalex.org/S5437875","display_name":"Statistics and Computing","issn_l":"0960-3174","issn":["0960-3174","1573-1375"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319900","host_organization_name":"Springer Science+Business Media","host_organization_lineage":["https://openalex.org/P4310319900","https://openalex.org/P4310319965"],"host_organization_lineage_names":["Springer Science+Business Media","Springer Nature"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Statistics and Computing","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[{"id":"https://openalex.org/G5304614257","display_name":null,"funder_award_id":"308640","funder_id":"https://openalex.org/F4320335063","funder_display_name":"Luonnontieteiden ja Tekniikan Tutkimuksen Toimikunta"}],"funders":[{"id":"https://openalex.org/F4320335063","display_name":"Luonnontieteiden ja Tekniikan Tutkimuksen Toimikunta","ror":"https://ror.org/05k73zm37"}],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W1665378886.pdf","grobid_xml":"https://content.openalex.org/works/W1665378886.grobid-xml"},"referenced_works_count":25,"referenced_works":["https://openalex.org/W61851564","https://openalex.org/W621546036","https://openalex.org/W1531455566","https://openalex.org/W1583837637","https://openalex.org/W1837874438","https://openalex.org/W1951659264","https://openalex.org/W1973310094","https://openalex.org/W1997321933","https://openalex.org/W1999479532","https://openalex.org/W2002355073","https://openalex.org/W2019130306","https://openalex.org/W2059448777","https://openalex.org/W2062231581","https://openalex.org/W2083795796","https://openalex.org/W2084501074","https://openalex.org/W2099994229","https://openalex.org/W2125840857","https://openalex.org/W2129564505","https://openalex.org/W2148474239","https://openalex.org/W2171810522","https://openalex.org/W2551943286","https://openalex.org/W4298876635","https://openalex.org/W4300475775","https://openalex.org/W6600872795","https://openalex.org/W6811789344"],"related_works":["https://openalex.org/W4287199989","https://openalex.org/W2094102389","https://openalex.org/W237018462","https://openalex.org/W2982538860","https://openalex.org/W1497639102","https://openalex.org/W2064551531","https://openalex.org/W3181645452","https://openalex.org/W1599659911","https://openalex.org/W3013816997","https://openalex.org/W3103364054"],"abstract_inverted_index":{"This":[0],"paper":[1],"proposes":[2],"a":[3,35,75,215],"novel":[4],"scheme":[5],"for":[6,111,120,180],"reduced-rank":[7],"Gaussian":[8,64],"process":[9],"regression.":[10],"The":[11,116,198,219],"method":[12,220],"is":[13,162,187,209,221],"based":[14],"on":[15],"an":[16,27,205,212],"approximate":[17,44],"series":[18],"expansion":[19,29,199],"of":[20,26,30,38,48,58,62,102,105,139,146,158,164,173,194],"the":[21,31,46,49,59,63,68,97,103,106,132,137,140,144,155,159,165,170,174,181,195],"covariance":[22,50,107,175,184],"function":[23,51,176,185],"in":[24,34],"terms":[25],"eigenfunction":[28],"Laplace":[32],"operator":[33],"compact":[36,141],"subset":[37,142],"\\(\\mathbb":[39],"{R}^d\\)":[40],".":[41],"On":[42],"this":[43],"eigenbasis,":[45],"eigenvalues":[47],"can":[52],"be":[53,72],"expressed":[54],"as":[55,79,211],"simple":[56],"functions":[57,91,99],"spectral":[60],"density":[61],"process,":[65],"which":[66,109,208],"allows":[67,110,119],"GP":[69],"inference":[70],"to":[71,149,201,223],"solved":[73],"under":[74],"computational":[76],"cost":[77],"scaling":[78],"\\(\\mathcal":[80,84],"{O}(nm^2)\\)":[81],"(initial)":[82],"and":[83,92,128,143,179,228,234],"{O}(m^3)\\)":[85],"(hyperparameter":[86],"learning)":[87],"with":[88,124,204,232],"m":[89],"basis":[90,98],"n":[93],"data":[94],"points.":[95],"Furthermore,":[96],"are":[100],"independent":[101,163],"parameters":[104],"function,":[108],"very":[112],"fast":[113],"hyperparameter":[114],"learning.":[115],"approach":[117],"also":[118,152],"rigorous":[121],"error":[122,161],"analysis":[123],"Hilbert":[125,202],"space":[126],"theory,":[127],"we":[129],"show":[130,153],"that":[131,154,169],"approximation":[133],"becomes":[134],"exact":[135],"when":[136],"size":[138],"number":[145],"eigenfunctions":[147],"go":[148],"infinity.":[150],"We":[151],"convergence":[156],"rate":[157],"truncation":[160],"input":[166,196,217],"dimensionality":[167],"provided":[168],"differentiability":[171],"order":[172],"increases":[177],"appropriately,":[178],"squared":[182],"exponential":[183],"it":[186],"always":[188],"bounded":[189],"by":[190],"\\({\\sim":[191],"}1/m\\)":[192],"regardless":[193],"dimensionality.":[197],"generalizes":[200],"spaces":[203],"inner":[206],"product":[207],"defined":[210],"integral":[213],"over":[214],"specified":[216],"density.":[218],"compared":[222],"previously":[224],"proposed":[225],"methods":[226],"theoretically":[227],"through":[229],"empirical":[230],"tests":[231],"simulated":[233],"real":[235],"data.":[236]},"counts_by_year":[{"year":2026,"cited_by_count":18},{"year":2025,"cited_by_count":30},{"year":2024,"cited_by_count":19},{"year":2023,"cited_by_count":19},{"year":2022,"cited_by_count":27},{"year":2021,"cited_by_count":12},{"year":2020,"cited_by_count":7},{"year":2019,"cited_by_count":2},{"year":2018,"cited_by_count":1},{"year":2016,"cited_by_count":1}],"updated_date":"2026-08-21T09:56:20.448147","created_date":"2016-06-24T00:00:00"}
