{"id":"https://openalex.org/W7171982207","doi":"https://doi.org/10.1007/s10957-026-03070-7","title":"Recovery Bounds for Cardinality Regularized Optimization Problem","display_name":"Recovery Bounds for Cardinality Regularized Optimization Problem","publication_year":2026,"publication_date":"2026-07-31","ids":{"openalex":"https://openalex.org/W7171982207","doi":"https://doi.org/10.1007/s10957-026-03070-7"},"language":"en","primary_location":{"id":"doi:10.1007/s10957-026-03070-7","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s10957-026-03070-7","pdf_url":"https://link.springer.com/content/pdf/10.1007/s10957-026-03070-7.pdf","source":{"id":"https://openalex.org/S52430896","display_name":"Journal of Optimization Theory and Applications","issn_l":"0022-3239","issn":["0022-3239","1573-2878"],"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":"Journal of Optimization Theory and Applications","raw_type":"journal-article"},"type":"article","indexed_in":["crossref"],"open_access":{"is_oa":true,"oa_status":"hybrid","oa_url":"https://link.springer.com/content/pdf/10.1007/s10957-026-03070-7.pdf","any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5076706599","display_name":"Carisa Kwok Wai Yu","orcid":"https://orcid.org/0000-0002-0787-5382"},"institutions":[{"id":"https://openalex.org/I47605537","display_name":"Hang Seng University of Hong Kong","ror":"https://ror.org/04fa64g55","country_code":"HK","type":"education","lineage":["https://openalex.org/I47605537"]},{"id":"https://openalex.org/I889458895","display_name":"University of Hong Kong","ror":"https://ror.org/02zhqgq86","country_code":"HK","type":"education","lineage":["https://openalex.org/I889458895"]}],"countries":["HK"],"is_corresponding":false,"raw_author_name":"Carisa Kwok Wai Yu","raw_affiliation_strings":["Department of Mathematics, Statistics and Insurance, The Hang Seng University of Hong Kong, Shatin, Hong Kong, People\u2019s Republic of China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics, Statistics and Insurance, The Hang Seng University of Hong Kong, Shatin, Hong Kong, People\u2019s Republic of China","institution_ids":["https://openalex.org/I47605537","https://openalex.org/I889458895"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5101868817","display_name":"Yaohua Hu","orcid":"https://orcid.org/0000-0001-6412-5804"},"institutions":[{"id":"https://openalex.org/I180726961","display_name":"Shenzhen University","ror":"https://ror.org/01vy4gh70","country_code":"CN","type":"education","lineage":["https://openalex.org/I180726961"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Yaohua Hu","raw_affiliation_strings":["School of Mathematical Sciences, Shenzhen University, Shenzhen, 518060, People\u2019s Republic of China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"School of Mathematical Sciences, Shenzhen University, Shenzhen, 518060, People\u2019s Republic of China","institution_ids":["https://openalex.org/I180726961"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5144103115","display_name":"Minghua Li","orcid":null},"institutions":[{"id":"https://openalex.org/I158842170","display_name":"Chongqing University","ror":"https://ror.org/023rhb549","country_code":"CN","type":"education","lineage":["https://openalex.org/I158842170"]},{"id":"https://openalex.org/I4210110609","display_name":"Chongqing University of Arts and Sciences","ror":"https://ror.org/01rcvq140","country_code":"CN","type":"education","lineage":["https://openalex.org/I4210110609"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Minghua Li","raw_affiliation_strings":["School of Mathematics and Artificial Intelligence, Chongqing Key Laboratory of Statistical Optimization and Complex Data, Chongqing University of Arts and Sciences, Yongchuan, Chongqing, 402160, People\u2019s Republic of China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"School of Mathematics and Artificial Intelligence, Chongqing Key Laboratory of Statistical Optimization and Complex Data, Chongqing University of Arts and Sciences, Yongchuan, Chongqing, 402160, People\u2019s Republic of China","institution_ids":["https://openalex.org/I158842170","https://openalex.org/I4210110609"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5024523994","display_name":"Xiaoqi Yang","orcid":"https://orcid.org/0000-0002-5583-4032"},"institutions":[{"id":"https://openalex.org/I14243506","display_name":"Hong Kong Polytechnic University","ror":"https://ror.org/0030zas98","country_code":"HK","type":"education","lineage":["https://openalex.org/I14243506"]}],"countries":["HK"],"is_corresponding":true,"raw_author_name":"Xiaoqi Yang","raw_affiliation_strings":["Department of Applied Mathematics, The Hong Kong Polytechnic University, Kowloon, Hong Kong, People\u2019s Republic of China"],"raw_orcid":"https://orcid.org/0000-0002-5583-4032","affiliations":[{"raw_affiliation_string":"Department of Applied Mathematics, The Hong Kong Polytechnic University, Kowloon, Hong Kong, People\u2019s Republic of China","institution_ids":["https://openalex.org/I14243506"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":6,"corresponding_author_ids":["https://openalex.org/A5024523994"],"corresponding_institution_ids":["https://openalex.org/I14243506"],"apc_list":{"value":3190,"currency":"USD","value_usd":3190},"apc_paid":{"value":3190,"currency":"USD","value_usd":3190},"fwci":0.0,"has_fulltext":true,"cited_by_count":0,"citation_normalized_percentile":{"value":0.77520979,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":null,"biblio":{"volume":"210","issue":"2","first_page":null,"last_page":null},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10500","display_name":"Sparse and Compressive Sensing Techniques","score":0.6959999799728394,"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":0.6959999799728394,"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/T10545","display_name":"Optimization and Variational Analysis","score":0.23739999532699585,"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"}},{"id":"https://openalex.org/T11612","display_name":"Stochastic Gradient Optimization Techniques","score":0.020600000396370888,"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/regularization","display_name":"Regularization (linguistics)","score":0.6446999907493591},{"id":"https://openalex.org/keywords/eigenvalues-and-eigenvectors","display_name":"Eigenvalues and eigenvectors","score":0.6150000095367432},{"id":"https://openalex.org/keywords/theory-of-computation","display_name":"Theory of computation","score":0.5659000277519226},{"id":"https://openalex.org/keywords/optimization-problem","display_name":"Optimization problem","score":0.5633000135421753},{"id":"https://openalex.org/keywords/oracle","display_name":"Oracle","score":0.42179998755455017},{"id":"https://openalex.org/keywords/property","display_name":"Property (philosophy)","score":0.41679999232292175},{"id":"https://openalex.org/keywords/cardinality","display_name":"Cardinality (data modeling)","score":0.4081999957561493},{"id":"https://openalex.org/keywords/sparse-approximation","display_name":"Sparse approximation","score":0.39100000262260437}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7404999732971191},{"id":"https://openalex.org/C2776135515","wikidata":"https://www.wikidata.org/wiki/Q17143721","display_name":"Regularization (linguistics)","level":2,"score":0.6446999907493591},{"id":"https://openalex.org/C158693339","wikidata":"https://www.wikidata.org/wiki/Q190524","display_name":"Eigenvalues and eigenvectors","level":2,"score":0.6150000095367432},{"id":"https://openalex.org/C24858836","wikidata":"https://www.wikidata.org/wiki/Q844718","display_name":"Theory of computation","level":2,"score":0.5659000277519226},{"id":"https://openalex.org/C137836250","wikidata":"https://www.wikidata.org/wiki/Q984063","display_name":"Optimization problem","level":2,"score":0.5633000135421753},{"id":"https://openalex.org/C126255220","wikidata":"https://www.wikidata.org/wiki/Q141495","display_name":"Mathematical optimization","level":1,"score":0.4530999958515167},{"id":"https://openalex.org/C55166926","wikidata":"https://www.wikidata.org/wiki/Q2892946","display_name":"Oracle","level":2,"score":0.42179998755455017},{"id":"https://openalex.org/C189950617","wikidata":"https://www.wikidata.org/wiki/Q937228","display_name":"Property (philosophy)","level":2,"score":0.41679999232292175},{"id":"https://openalex.org/C87117476","wikidata":"https://www.wikidata.org/wiki/Q362383","display_name":"Cardinality (data modeling)","level":2,"score":0.4081999957561493},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.4034999907016754},{"id":"https://openalex.org/C124066611","wikidata":"https://www.wikidata.org/wiki/Q28684319","display_name":"Sparse approximation","level":2,"score":0.39100000262260437},{"id":"https://openalex.org/C2777303404","wikidata":"https://www.wikidata.org/wiki/Q759757","display_name":"Convergence (economics)","level":2,"score":0.3790000081062317},{"id":"https://openalex.org/C56372850","wikidata":"https://www.wikidata.org/wiki/Q1050404","display_name":"Sparse matrix","level":3,"score":0.3285999894142151},{"id":"https://openalex.org/C135252773","wikidata":"https://www.wikidata.org/wiki/Q1567213","display_name":"Inverse problem","level":2,"score":0.31130000948905945},{"id":"https://openalex.org/C177264268","wikidata":"https://www.wikidata.org/wiki/Q1514741","display_name":"Set (abstract data type)","level":2,"score":0.29179999232292175},{"id":"https://openalex.org/C99217422","wikidata":"https://www.wikidata.org/wiki/Q4867576","display_name":"Basis pursuit","level":4,"score":0.28780001401901245},{"id":"https://openalex.org/C164752517","wikidata":"https://www.wikidata.org/wiki/Q5570875","display_name":"Global optimization","level":2,"score":0.2754000127315521},{"id":"https://openalex.org/C77553402","wikidata":"https://www.wikidata.org/wiki/Q13222579","display_name":"Upper and lower bounds","level":2,"score":0.2689000070095062},{"id":"https://openalex.org/C124851039","wikidata":"https://www.wikidata.org/wiki/Q2665459","display_name":"Compressed sensing","level":2,"score":0.2651999890804291},{"id":"https://openalex.org/C60033838","wikidata":"https://www.wikidata.org/wiki/Q1751685","display_name":"Solution set","level":3,"score":0.26429998874664307},{"id":"https://openalex.org/C2776436953","wikidata":"https://www.wikidata.org/wiki/Q5163215","display_name":"Consistency (knowledge bases)","level":2,"score":0.25209999084472656},{"id":"https://openalex.org/C101901036","wikidata":"https://www.wikidata.org/wiki/Q2269096","display_name":"Semidefinite programming","level":2,"score":0.25189998745918274}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1007/s10957-026-03070-7","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s10957-026-03070-7","pdf_url":"https://link.springer.com/content/pdf/10.1007/s10957-026-03070-7.pdf","source":{"id":"https://openalex.org/S52430896","display_name":"Journal of Optimization Theory and Applications","issn_l":"0022-3239","issn":["0022-3239","1573-2878"],"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":"Journal of Optimization Theory and Applications","raw_type":"journal-article"}],"best_oa_location":{"id":"doi:10.1007/s10957-026-03070-7","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s10957-026-03070-7","pdf_url":"https://link.springer.com/content/pdf/10.1007/s10957-026-03070-7.pdf","source":{"id":"https://openalex.org/S52430896","display_name":"Journal of Optimization Theory and Applications","issn_l":"0022-3239","issn":["0022-3239","1573-2878"],"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":"Journal of Optimization Theory and Applications","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[{"id":"https://openalex.org/G6779693019","display_name":null,"funder_award_id":"12222112","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"}],"funders":[{"id":"https://openalex.org/F4320321001","display_name":"National Natural Science Foundation of China","ror":"https://ror.org/01h0zpd94"}],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7171982207.pdf","grobid_xml":"https://content.openalex.org/works/W7171982207.grobid-xml"},"referenced_works_count":39,"referenced_works":["https://openalex.org/W1560240297","https://openalex.org/W1973397215","https://openalex.org/W1980454827","https://openalex.org/W1985621639","https://openalex.org/W2004886822","https://openalex.org/W2006262045","https://openalex.org/W2020740117","https://openalex.org/W2025666718","https://openalex.org/W2038082462","https://openalex.org/W2045756756","https://openalex.org/W2050834445","https://openalex.org/W2056201402","https://openalex.org/W2065030431","https://openalex.org/W2077964948","https://openalex.org/W2078204800","https://openalex.org/W2084978503","https://openalex.org/W2100556411","https://openalex.org/W2109842006","https://openalex.org/W2115706991","https://openalex.org/W2116581043","https://openalex.org/W2129131372","https://openalex.org/W2135046866","https://openalex.org/W2138019504","https://openalex.org/W2138265962","https://openalex.org/W2147631685","https://openalex.org/W2163232332","https://openalex.org/W2595512563","https://openalex.org/W2598498858","https://openalex.org/W2922398374","https://openalex.org/W2954920938","https://openalex.org/W2963322354","https://openalex.org/W2963579802","https://openalex.org/W3021333344","https://openalex.org/W3118396380","https://openalex.org/W3191576853","https://openalex.org/W4296483470","https://openalex.org/W4392507228","https://openalex.org/W4400482253","https://openalex.org/W4406594005"],"related_works":[],"abstract_inverted_index":{"Abstract":[0],"Mixed":[1],"sparse":[2,21,45,60,112,119],"optimization":[3,46,113],"has":[4],"been":[5],"extensively":[6],"applied":[7],"in":[8,15,18,69,168],"the":[9,20,25,36,57,65,70,77,88,101,105,116,126,137,140,143,152],"modeling":[10],"of":[11,59,64,104,118,125,128,139,142,170],"many":[12],"important":[13],"problems":[14],"various":[16],"disciplines,":[17],"which":[19],"structure":[22],"appears":[23],"as":[24,160],"inter-group":[26],"and":[27,47,72,81,95,176],"intra-group":[28],"manners":[29],"simultaneously.":[30],"In":[31,52],"this":[32],"paper,":[33],"we":[34,54,86],"consider":[35],"$$\\ell":[37,106,144,153],"_0$$":[38,107],"\u2113":[39,108,146,155],"0":[40,109,158,165],"regularization":[41,150,159],"problem":[42,114],"for":[43,100],"mixed":[44,111],"investigate":[48],"its":[49],"consistency":[50],"theory.":[51],"particular,":[53],"first":[55],"introduce":[56],"notions":[58],"eigenvalue":[61,83,120],"conditions,":[62],"one":[63],"weakest":[66],"regularity":[67,93,171],"conditions":[68],"literature,":[71],"discuss":[73],"their":[74],"relations":[75],"with":[76],"uniquely":[78],"solvable":[79],"property":[80,90],"restricted":[82],"conditions.":[84],"Then":[85],"establish":[87],"oracle":[89],"without":[91],"any":[92],"condition":[94],"provide":[96],"a":[97],"recovery":[98,177],"bound":[99],"global":[102,173],"solutions":[103],"regularized":[110],"under":[115],"assumption":[117],"condition.":[121],"Moreover,":[122],"by":[123],"virtue":[124],"notion":[127],"epi-convergence,":[129],"an":[130],"asymptotic":[131],"analysis":[132],"is":[133],"provided":[134],"to":[135,151],"advance":[136],"understanding":[138],"convergence":[141],"_{2,p}$$":[145],"2":[147,156],",":[148,157,167],"p":[149,163],"_{2,0}$$":[154],"$$p\\rightarrow":[161],"0_+$$":[162],"\u2192":[164],"+":[166],"terms":[169],"condition,":[172],"solution":[174],"set":[175],"bound,":[178],"respectively.":[179]},"counts_by_year":[],"updated_date":"2026-08-24T07:32:12.397491","created_date":"2026-08-01T00:00:00"}