{"id":"https://openalex.org/W2024254345","doi":"https://doi.org/10.1109/tit.2015.2424238","title":"Sample Complexity of Dictionary Learning and Other Matrix Factorizations","display_name":"Sample Complexity of Dictionary Learning and Other Matrix Factorizations","publication_year":2015,"publication_date":"2015-04-17","ids":{"openalex":"https://openalex.org/W2024254345","doi":"https://doi.org/10.1109/tit.2015.2424238","mag":"2024254345"},"language":"en","primary_location":{"id":"doi:10.1109/tit.2015.2424238","is_oa":false,"landing_page_url":"https://doi.org/10.1109/tit.2015.2424238","pdf_url":null,"source":{"id":"https://openalex.org/S4502562","display_name":"IEEE Transactions on Information Theory","issn_l":"0018-9448","issn":["0018-9448","1557-9654"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319808","host_organization_name":"Institute of Electrical and Electronics Engineers","host_organization_lineage":["https://openalex.org/P4310319808"],"host_organization_lineage_names":["Institute of Electrical and Electronics Engineers"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"IEEE Transactions on Information Theory","raw_type":"journal-article"},"type":"article","indexed_in":["arxiv","crossref"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://arxiv.org/pdf/1312.3790","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5041230403","display_name":"R\u00e9mi Gribonval","orcid":"https://orcid.org/0000-0002-9450-8125"},"institutions":[{"id":"https://openalex.org/I2802519937","display_name":"Institut de Recherche en Informatique et Syst\u00e8mes Al\u00e9atoires","ror":"https://ror.org/00myn0z94","country_code":"FR","type":"facility","lineage":["https://openalex.org/I1294671590","https://openalex.org/I1294671590","https://openalex.org/I1326498283","https://openalex.org/I205703379","https://openalex.org/I2802204017","https://openalex.org/I2802519937","https://openalex.org/I28221208","https://openalex.org/I4210127572","https://openalex.org/I4210159245","https://openalex.org/I56067802"]},{"id":"https://openalex.org/I1294671590","display_name":"Centre National de la Recherche Scientifique","ror":"https://ror.org/02feahw73","country_code":"FR","type":"government","lineage":["https://openalex.org/I1294671590"]}],"countries":["FR"],"is_corresponding":true,"raw_author_name":"Remi Gribonval","raw_affiliation_strings":["Institut de Recherche en Syst\u00e8mes Al\u00e9atoires (Inria & CNRS), Rennes","PANAMA - Parcimonie et Nouveaux Algorithmes pour le Signal et la Mod\u00e9lisation Audio (Campus de Beaulieu 35042 Rennes cedex - France)"],"affiliations":[{"raw_affiliation_string":"Institut de Recherche en Syst\u00e8mes Al\u00e9atoires (Inria & CNRS), Rennes","institution_ids":["https://openalex.org/I2802519937","https://openalex.org/I1294671590"]},{"raw_affiliation_string":"PANAMA - Parcimonie et Nouveaux Algorithmes pour le Signal et la Mod\u00e9lisation Audio (Campus de Beaulieu 35042 Rennes cedex - France)","institution_ids":[]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5033851429","display_name":"Rodolphe Jenatton","orcid":null},"institutions":[{"id":"https://openalex.org/I1326498283","display_name":"Institut national de recherche en sciences et technologies du num\u00e9rique","ror":"https://ror.org/02kvxyf05","country_code":"FR","type":"government","lineage":["https://openalex.org/I1326498283"]},{"id":"https://openalex.org/I4210103330","display_name":"\u00c9cole Normale Sup\u00e9rieure","ror":"https://ror.org/01dp7jr64","country_code":"BI","type":"education","lineage":["https://openalex.org/I4210103330"]},{"id":"https://openalex.org/I142476485","display_name":"\u00c9cole Polytechnique","ror":"https://ror.org/05hy3tk52","country_code":"FR","type":"education","lineage":["https://openalex.org/I142476485","https://openalex.org/I4210145102"]},{"id":"https://openalex.org/I1294671590","display_name":"Centre National de la Recherche Scientifique","ror":"https://ror.org/02feahw73","country_code":"FR","type":"government","lineage":["https://openalex.org/I1294671590"]},{"id":"https://openalex.org/I4210107641","display_name":"Centre de Math\u00e9matiques Appliqu\u00e9es de l'\u00c9cole polytechnique","ror":"https://ror.org/012e1xn46","country_code":"FR","type":"facility","lineage":["https://openalex.org/I1294671590","https://openalex.org/I1294671590","https://openalex.org/I1326498283","https://openalex.org/I142476485","https://openalex.org/I4210107641","https://openalex.org/I4210141950","https://openalex.org/I4210145102"]},{"id":"https://openalex.org/I4210089985","display_name":"Amazon (Germany)","ror":"https://ror.org/00b9ktm87","country_code":"DE","type":"company","lineage":["https://openalex.org/I1311688040","https://openalex.org/I4210089985"]}],"countries":["BI","DE","FR"],"is_corresponding":false,"raw_author_name":"Rodolphe Jenatton","raw_affiliation_strings":["Amazon Development Center Germany, Berlin, Germany","Centre de Math\u00e9matiques Appliqu\u00e9es (CMAP, UMR CNRS 7641), Ecole Polytechnique, Palaiseau, France","Laboratoire d\u2019Informatique, Ecole Normale Superieure, Paris, France","CMAP - Centre de Math\u00e9matiques Appliqu\u00e9es - Ecole Polytechnique (\u00c9cole Polytechnique Route de Saclay 91128 Palaiseau Cedex - France)","SIERRA - Statistical Machine Learning and Parsimony (INRIA 23 avenue d'Italie 75013 Paris - France)"],"affiliations":[{"raw_affiliation_string":"Amazon Development Center Germany, Berlin, Germany","institution_ids":["https://openalex.org/I4210089985"]},{"raw_affiliation_string":"Centre de Math\u00e9matiques Appliqu\u00e9es (CMAP, UMR CNRS 7641), Ecole Polytechnique, Palaiseau, France","institution_ids":["https://openalex.org/I4210107641","https://openalex.org/I142476485","https://openalex.org/I1294671590"]},{"raw_affiliation_string":"Laboratoire d\u2019Informatique, Ecole Normale Superieure, Paris, France","institution_ids":["https://openalex.org/I4210103330"]},{"raw_affiliation_string":"CMAP - Centre de Math\u00e9matiques Appliqu\u00e9es - Ecole Polytechnique (\u00c9cole Polytechnique Route de Saclay 91128 Palaiseau Cedex - France)","institution_ids":["https://openalex.org/I4210107641","https://openalex.org/I142476485"]},{"raw_affiliation_string":"SIERRA - Statistical Machine Learning and Parsimony (INRIA 23 avenue d'Italie 75013 Paris - France)","institution_ids":["https://openalex.org/I1326498283"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5001483226","display_name":"Francis Bach","orcid":"https://orcid.org/0000-0001-8644-1058"},"institutions":[{"id":"https://openalex.org/I29607241","display_name":"\u00c9cole Normale Sup\u00e9rieure - PSL","ror":"https://ror.org/05a0dhs15","country_code":"FR","type":"other","lineage":["https://openalex.org/I2746051580","https://openalex.org/I29607241"]},{"id":"https://openalex.org/I1326498283","display_name":"Institut national de recherche en sciences et technologies du num\u00e9rique","ror":"https://ror.org/02kvxyf05","country_code":"FR","type":"government","lineage":["https://openalex.org/I1326498283"]}],"countries":["FR"],"is_corresponding":false,"raw_author_name":"Francis Bach","raw_affiliation_strings":["Laboratoire d\u2019Informatique, \u00c9cole Normale Sup\u00e9rieure, Paris, France","LIENS - Laboratoire d'informatique de l'\u00e9cole normale sup\u00e9rieure (45 Rue d'Ulm 75230 PARIS CEDEX 05 - France)","SIERRA - Statistical Machine Learning and Parsimony (INRIA 23 avenue d'Italie 75013 Paris - France)"],"affiliations":[{"raw_affiliation_string":"Laboratoire d\u2019Informatique, \u00c9cole Normale Sup\u00e9rieure, Paris, France","institution_ids":["https://openalex.org/I29607241"]},{"raw_affiliation_string":"LIENS - Laboratoire d'informatique de l'\u00e9cole normale sup\u00e9rieure (45 Rue d'Ulm 75230 PARIS CEDEX 05 - France)","institution_ids":["https://openalex.org/I29607241"]},{"raw_affiliation_string":"SIERRA - Statistical Machine Learning and Parsimony (INRIA 23 avenue d'Italie 75013 Paris - France)","institution_ids":["https://openalex.org/I1326498283"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5055847035","display_name":"Martin Kleinsteuber","orcid":"https://orcid.org/0000-0002-4323-9260"},"institutions":[{"id":"https://openalex.org/I62916508","display_name":"Technical University of Munich","ror":"https://ror.org/02kkvpp62","country_code":"DE","type":"education","lineage":["https://openalex.org/I62916508"]}],"countries":["DE"],"is_corresponding":false,"raw_author_name":"Martin Kleinsteuber","raw_affiliation_strings":["Department of Electrical and Computer Engineering, Technische Universit\u00e4t M\u00fcnchen, Munich, Germany","TUM - Technische Universit\u00e4t Munchen - Technical University Munich - Universit\u00e9 Technique de Munich (Arcisstrasse 21, D- 80333 M\u00fcnchen - Germany)"],"affiliations":[{"raw_affiliation_string":"Department of Electrical and Computer Engineering, Technische Universit\u00e4t M\u00fcnchen, Munich, Germany","institution_ids":["https://openalex.org/I62916508"]},{"raw_affiliation_string":"TUM - Technische Universit\u00e4t Munchen - Technical University Munich - Universit\u00e9 Technique de Munich (Arcisstrasse 21, D- 80333 M\u00fcnchen - Germany)","institution_ids":["https://openalex.org/I62916508"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5065943708","display_name":"Matthias Seibert","orcid":null},"institutions":[{"id":"https://openalex.org/I62916508","display_name":"Technical University of Munich","ror":"https://ror.org/02kkvpp62","country_code":"DE","type":"education","lineage":["https://openalex.org/I62916508"]}],"countries":["DE"],"is_corresponding":false,"raw_author_name":"Matthias Seibert","raw_affiliation_strings":["Department of Electrical and Computer Engineering, Technische Universit\u00e4t M\u00fcnchen, Munich, Germany","TUM - Technische Universit\u00e4t Munchen - Technical University Munich - Universit\u00e9 Technique de Munich (Arcisstrasse 21, D- 80333 M\u00fcnchen - Germany)"],"affiliations":[{"raw_affiliation_string":"Department of Electrical and Computer Engineering, Technische Universit\u00e4t M\u00fcnchen, Munich, Germany","institution_ids":["https://openalex.org/I62916508"]},{"raw_affiliation_string":"TUM - Technische Universit\u00e4t Munchen - Technical University Munich - Universit\u00e9 Technique de Munich (Arcisstrasse 21, D- 80333 M\u00fcnchen - Germany)","institution_ids":["https://openalex.org/I62916508"]}]}],"institutions":[],"countries_distinct_count":3,"institutions_distinct_count":5,"corresponding_author_ids":["https://openalex.org/A5041230403"],"corresponding_institution_ids":["https://openalex.org/I1294671590","https://openalex.org/I2802519937"],"apc_list":null,"apc_paid":null,"fwci":14.2443,"has_fulltext":false,"cited_by_count":69,"citation_normalized_percentile":{"value":0.98819734,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":94,"max":100},"biblio":{"volume":"61","issue":"6","first_page":"3469","last_page":"3486"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10181","display_name":"Natural Language Processing Techniques","score":0.902899980545044,"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/T10181","display_name":"Natural Language Processing Techniques","score":0.902899980545044,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.6040101051330566},{"id":"https://openalex.org/keywords/sample-complexity","display_name":"Sample complexity","score":0.5981301665306091},{"id":"https://openalex.org/keywords/dictionary-learning","display_name":"Dictionary learning","score":0.5234838724136353},{"id":"https://openalex.org/keywords/matrix-algebra","display_name":"Matrix algebra","score":0.47974884510040283},{"id":"https://openalex.org/keywords/sample","display_name":"Sample (material)","score":0.4702562093734741},{"id":"https://openalex.org/keywords/matrix","display_name":"Matrix (chemical analysis)","score":0.43525442481040955},{"id":"https://openalex.org/keywords/matrix-decomposition","display_name":"Matrix decomposition","score":0.4133490324020386},{"id":"https://openalex.org/keywords/artificial-intelligence","display_name":"Artificial intelligence","score":0.3862399458885193},{"id":"https://openalex.org/keywords/mathematics-education","display_name":"Mathematics education","score":0.3826916813850403},{"id":"https://openalex.org/keywords/theoretical-computer-science","display_name":"Theoretical computer science","score":0.34277456998825073},{"id":"https://openalex.org/keywords/data-science","display_name":"Data science","score":0.3278296887874603},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.22557517886161804},{"id":"https://openalex.org/keywords/sparse-approximation","display_name":"Sparse approximation","score":0.0939379632472992}],"concepts":[{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.6040101051330566},{"id":"https://openalex.org/C2778445095","wikidata":"https://www.wikidata.org/wiki/Q18354077","display_name":"Sample complexity","level":2,"score":0.5981301665306091},{"id":"https://openalex.org/C2988886741","wikidata":"https://www.wikidata.org/wiki/Q25304494","display_name":"Dictionary learning","level":3,"score":0.5234838724136353},{"id":"https://openalex.org/C2988995629","wikidata":"https://www.wikidata.org/wiki/Q2915729","display_name":"Matrix algebra","level":3,"score":0.47974884510040283},{"id":"https://openalex.org/C198531522","wikidata":"https://www.wikidata.org/wiki/Q485146","display_name":"Sample (material)","level":2,"score":0.4702562093734741},{"id":"https://openalex.org/C106487976","wikidata":"https://www.wikidata.org/wiki/Q685816","display_name":"Matrix (chemical analysis)","level":2,"score":0.43525442481040955},{"id":"https://openalex.org/C42355184","wikidata":"https://www.wikidata.org/wiki/Q1361088","display_name":"Matrix decomposition","level":3,"score":0.4133490324020386},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.3862399458885193},{"id":"https://openalex.org/C145420912","wikidata":"https://www.wikidata.org/wiki/Q853077","display_name":"Mathematics education","level":1,"score":0.3826916813850403},{"id":"https://openalex.org/C80444323","wikidata":"https://www.wikidata.org/wiki/Q2878974","display_name":"Theoretical computer science","level":1,"score":0.34277456998825073},{"id":"https://openalex.org/C2522767166","wikidata":"https://www.wikidata.org/wiki/Q2374463","display_name":"Data science","level":1,"score":0.3278296887874603},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.22557517886161804},{"id":"https://openalex.org/C124066611","wikidata":"https://www.wikidata.org/wiki/Q28684319","display_name":"Sparse approximation","level":2,"score":0.0939379632472992},{"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/C159985019","wikidata":"https://www.wikidata.org/wiki/Q181790","display_name":"Composite material","level":1,"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/C158693339","wikidata":"https://www.wikidata.org/wiki/Q190524","display_name":"Eigenvalues and eigenvectors","level":2,"score":0.0},{"id":"https://openalex.org/C185592680","wikidata":"https://www.wikidata.org/wiki/Q2329","display_name":"Chemistry","level":0,"score":0.0},{"id":"https://openalex.org/C43617362","wikidata":"https://www.wikidata.org/wiki/Q170050","display_name":"Chromatography","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}],"mesh":[],"locations_count":4,"locations":[{"id":"doi:10.1109/tit.2015.2424238","is_oa":false,"landing_page_url":"https://doi.org/10.1109/tit.2015.2424238","pdf_url":null,"source":{"id":"https://openalex.org/S4502562","display_name":"IEEE Transactions on Information Theory","issn_l":"0018-9448","issn":["0018-9448","1557-9654"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319808","host_organization_name":"Institute of Electrical and Electronics Engineers","host_organization_lineage":["https://openalex.org/P4310319808"],"host_organization_lineage_names":["Institute of Electrical and Electronics Engineers"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"IEEE Transactions on Information Theory","raw_type":"journal-article"},{"id":"pmh:oai:arXiv.org:1312.3790","is_oa":true,"landing_page_url":"http://arxiv.org/abs/1312.3790","pdf_url":"https://arxiv.org/pdf/1312.3790","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":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"","raw_type":"text"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.748.1796","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.748.1796","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://arxiv.org/pdf/1312.3790.pdf","raw_type":"text"},{"id":"pmh:oai:HAL:hal-00918142v3","is_oa":true,"landing_page_url":"https://inria.hal.science/hal-00918142","pdf_url":null,"source":{"id":"https://openalex.org/S4306402512","display_name":"HAL (Le Centre pour la Communication Scientifique Directe)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I1294671590","host_organization_name":"Centre National de la Recherche Scientifique","host_organization_lineage":["https://openalex.org/I1294671590"],"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":"IEEE Transactions on Information Theory, 2015, 61 (6), pp.3469-3486. &#x27E8;10.1109/TIT.2015.2424238&#x27E9;","raw_type":"Journal articles"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:1312.3790","is_oa":true,"landing_page_url":"http://arxiv.org/abs/1312.3790","pdf_url":"https://arxiv.org/pdf/1312.3790","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":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"","raw_type":"text"},"sustainable_development_goals":[{"display_name":"Quality Education","score":0.7200000286102295,"id":"https://metadata.un.org/sdg/4"}],"awards":[{"id":"https://openalex.org/G1850372982","display_name":null,"funder_award_id":"KL 2189/8-1","funder_id":"https://openalex.org/F4320320879","funder_display_name":"Deutsche Forschungsgemeinschaft"},{"id":"https://openalex.org/G3044931183","display_name":null,"funder_award_id":"225913","funder_id":"https://openalex.org/F4320334697","funder_display_name":"Directorate-General for Information Society and Media"},{"id":"https://openalex.org/G5704530987","display_name":null,"funder_award_id":"ERC-StG-2011-277906","funder_id":"https://openalex.org/F4320334678","funder_display_name":"European Research Council"}],"funders":[{"id":"https://openalex.org/F4320320879","display_name":"Deutsche Forschungsgemeinschaft","ror":"https://ror.org/018mejw64"},{"id":"https://openalex.org/F4320334678","display_name":"European Research Council","ror":"https://ror.org/0472cxd90"},{"id":"https://openalex.org/F4320334697","display_name":"Directorate-General for Information Society and Media","ror":null}],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":78,"referenced_works":["https://openalex.org/W18355306","https://openalex.org/W228380312","https://openalex.org/W1482235099","https://openalex.org/W1539012881","https://openalex.org/W1576854369","https://openalex.org/W1589014993","https://openalex.org/W1591116419","https://openalex.org/W1634005169","https://openalex.org/W1890834058","https://openalex.org/W1963826206","https://openalex.org/W1973890885","https://openalex.org/W1999174140","https://openalex.org/W2004544971","https://openalex.org/W2008740515","https://openalex.org/W2013912476","https://openalex.org/W2034683677","https://openalex.org/W2045512849","https://openalex.org/W2047278710","https://openalex.org/W2050744541","https://openalex.org/W2052044664","https://openalex.org/W2080892236","https://openalex.org/W2091475238","https://openalex.org/W2098290597","https://openalex.org/W2099321050","https://openalex.org/W2105464873","https://openalex.org/W2106334384","https://openalex.org/W2108687351","https://openalex.org/W2109999391","https://openalex.org/W2110323469","https://openalex.org/W2111187851","https://openalex.org/W2112447569","https://openalex.org/W2114609025","https://openalex.org/W2115429828","https://openalex.org/W2115755118","https://openalex.org/W2119385818","https://openalex.org/W2124623336","https://openalex.org/W2127114523","https://openalex.org/W2131172946","https://openalex.org/W2135029798","https://openalex.org/W2137216203","https://openalex.org/W2139095575","https://openalex.org/W2139943676","https://openalex.org/W2142766915","https://openalex.org/W2142940228","https://openalex.org/W2145096794","https://openalex.org/W2156117587","https://openalex.org/W2160547390","https://openalex.org/W2161065057","https://openalex.org/W2163398148","https://openalex.org/W2167349346","https://openalex.org/W2296399167","https://openalex.org/W2406524659","https://openalex.org/W2950190315","https://openalex.org/W2962959718","https://openalex.org/W2963641291","https://openalex.org/W2965497096","https://openalex.org/W3103870504","https://openalex.org/W3104629729","https://openalex.org/W3124790153","https://openalex.org/W3143596294","https://openalex.org/W4235713725","https://openalex.org/W4247201904","https://openalex.org/W4250589301","https://openalex.org/W4300824154","https://openalex.org/W6600760533","https://openalex.org/W6632282575","https://openalex.org/W6635134988","https://openalex.org/W6659258362","https://openalex.org/W6675992869","https://openalex.org/W6676454372","https://openalex.org/W6676634215","https://openalex.org/W6676727762","https://openalex.org/W6677107155","https://openalex.org/W6677725119","https://openalex.org/W6679168169","https://openalex.org/W6680012447","https://openalex.org/W6680640683","https://openalex.org/W6712659126"],"related_works":["https://openalex.org/W2964179321","https://openalex.org/W4293399083","https://openalex.org/W977514099","https://openalex.org/W2214934461","https://openalex.org/W2999552052","https://openalex.org/W3103945479","https://openalex.org/W2803930412","https://openalex.org/W4299839289","https://openalex.org/W2908946645","https://openalex.org/W2069570686"],"abstract_inverted_index":{"Many":[0],"modern":[1],"tools":[2],"in":[3,65],"machine":[4],"learning":[5],"and":[6,22],"signal":[7],"processing,":[8],"such":[9,113],"as":[10],"sparse":[11],"dictionary":[12],"learning,":[13],"principal":[14],"component":[15],"analysis,":[16],"non-negative":[17],"matrix":[18,31,147,173],"factorization,":[19],"K-means":[20],"clustering,":[21],"so":[23],"on,":[24],"rely":[25],"on":[26,126,139],"the":[27,42,49,53,56,73,92,97,105,108,120,127,140,165,171],"factorization":[28,148,174],"of":[29,52,59,78,107,117,119,143,167],"a":[30,38,136],"obtained":[32],"by":[33,67],"concatenating":[34],"high-dimensional":[35],"vectors":[36],"from":[37,96],"training":[39,60],"collection.":[40,75],"While":[41],"idealized":[43],"task":[44],"would":[45],"be":[46],"to":[47,82,87,156,164],"optimize":[48],"expected":[50,98],"quality":[51],"factors":[54,128],"over":[55,72],"underlying":[57],"distribution":[58],"vectors,":[61],"it":[62],"is":[63,81],"achieved":[64],"practice":[66],"minimizing":[68],"an":[69],"empirical":[70,93,109],"average":[71,94],"considered":[74,172],"The":[76,115,150],"focus":[77],"this":[79],"paper":[80],"provide":[83],"sample":[84,141],"complexity":[85,142],"estimates":[86],"uniformly":[88],"control":[89],"how":[90],"much":[91],"deviates":[95],"cost":[99],"function.":[100],"Standard":[101],"arguments":[102],"imply":[103],"that":[104],"performance":[106],"predictor":[110],"also":[111],"exhibit":[112],"guarantees.":[114],"level":[116],"genericity":[118],"approach":[121],"encompasses":[122],"several":[123,144],"possible":[124],"constraints":[125],"(tensor":[129],"product":[130],"structure,":[131],"shift-invariance,":[132],"sparsity...),":[133],"thus":[134],"providing":[135],"unified":[137],"perspective":[138],"widely":[145],"used":[146],"schemes.":[149],"derived":[151],"generalization":[152],"bounds":[153],"behave":[154],"proportional":[155],"(log":[157],"(n)/n)":[158],"<sup":[159],"xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"":[160],"xmlns:xlink=\"http://www.w3.org/1999/xlink\">1/2</sup>":[161],"with":[162],"respect":[163],"number":[166],"samples":[168],"n":[169],"for":[170],"techniques.":[175]},"counts_by_year":[{"year":2025,"cited_by_count":3},{"year":2024,"cited_by_count":3},{"year":2023,"cited_by_count":2},{"year":2022,"cited_by_count":4},{"year":2021,"cited_by_count":3},{"year":2020,"cited_by_count":7},{"year":2019,"cited_by_count":10},{"year":2018,"cited_by_count":9},{"year":2017,"cited_by_count":4},{"year":2016,"cited_by_count":13},{"year":2015,"cited_by_count":6},{"year":2014,"cited_by_count":5}],"updated_date":"2026-04-21T08:09:41.155169","created_date":"2025-10-10T00:00:00"}
