{"id":"https://openalex.org/W3150868911","doi":"https://doi.org/10.1137/21m1408713","title":"Linearly Constrained Nonsmooth Optimization for Training Autoencoders","display_name":"Linearly Constrained Nonsmooth Optimization for Training Autoencoders","publication_year":2022,"publication_date":"2022-08-11","ids":{"openalex":"https://openalex.org/W3150868911","doi":"https://doi.org/10.1137/21m1408713","mag":"3150868911"},"language":"en","primary_location":{"id":"doi:10.1137/21m1408713","is_oa":false,"landing_page_url":"https://doi.org/10.1137/21m1408713","pdf_url":null,"source":{"id":"https://openalex.org/S928796702","display_name":"SIAM Journal on Optimization","issn_l":"1052-6234","issn":["1052-6234","1095-7189"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320508","host_organization_name":"Society for Industrial and Applied Mathematics","host_organization_lineage":["https://openalex.org/P4310320508"],"host_organization_lineage_names":["Society for Industrial and Applied Mathematics"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"SIAM Journal on Optimization","raw_type":"journal-article"},"type":"article","indexed_in":["crossref"],"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/A5100431792","display_name":"Wei Liu","orcid":"https://orcid.org/0000-0002-3865-8145"},"institutions":[{"id":"https://openalex.org/I4210120485","display_name":"Academy of Mathematics and Systems Science","ror":"https://ror.org/02jkmyk67","country_code":"CN","type":"facility","lineage":["https://openalex.org/I19820366","https://openalex.org/I4210120485"]},{"id":"https://openalex.org/I4210165038","display_name":"University of Chinese Academy of Sciences","ror":"https://ror.org/05qbk4x57","country_code":"CN","type":"education","lineage":["https://openalex.org/I19820366","https://openalex.org/I4210165038"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Wei Liu","raw_affiliation_strings":["State Key Laboratory of Scientific and Engineering Computing, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, and University of Chinese Academy of Sciences, China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"State Key Laboratory of Scientific and Engineering Computing, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, and University of Chinese Academy of Sciences, China","institution_ids":["https://openalex.org/I4210120485","https://openalex.org/I4210165038"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5100352246","display_name":"Xin Liu","orcid":"https://orcid.org/0000-0002-5705-0805"},"institutions":[{"id":"https://openalex.org/I4210120485","display_name":"Academy of Mathematics and Systems Science","ror":"https://ror.org/02jkmyk67","country_code":"CN","type":"facility","lineage":["https://openalex.org/I19820366","https://openalex.org/I4210120485"]},{"id":"https://openalex.org/I4210165038","display_name":"University of Chinese Academy of Sciences","ror":"https://ror.org/05qbk4x57","country_code":"CN","type":"education","lineage":["https://openalex.org/I19820366","https://openalex.org/I4210165038"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Xin Liu","raw_affiliation_strings":["State Key Laboratory of Scientific and Engineering Computing, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, and University of Chinese Academy of Sciences, China"],"raw_orcid":"https://orcid.org/0000-0002-5705-0805","affiliations":[{"raw_affiliation_string":"State Key Laboratory of Scientific and Engineering Computing, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, and University of Chinese Academy of Sciences, China","institution_ids":["https://openalex.org/I4210120485","https://openalex.org/I4210165038"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5100369445","display_name":"Xiaojun Chen","orcid":"https://orcid.org/0000-0002-6627-2637"},"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":false,"raw_author_name":"Xiaojun Chen","raw_affiliation_strings":["Department of Applied Mathematics, The Hong Kong Polytechnic University, Hong Kong, China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Applied Mathematics, The Hong Kong Polytechnic University, Hong Kong, China","institution_ids":["https://openalex.org/I14243506"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":3,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":0.9523,"has_fulltext":false,"cited_by_count":8,"citation_normalized_percentile":{"value":0.66290492,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":98},"biblio":{"volume":"32","issue":"3","first_page":"1931","last_page":"1957"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10500","display_name":"Sparse and Compressive Sensing Techniques","score":0.9976000189781189,"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":0.9976000189781189,"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":0.9945999979972839,"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/T10688","display_name":"Image and Signal Denoising Methods","score":0.9926999807357788,"subfield":{"id":"https://openalex.org/subfields/1707","display_name":"Computer Vision and Pattern Recognition"},"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/mathematics","display_name":"Mathematics","score":0.6927554607391357},{"id":"https://openalex.org/keywords/robustness","display_name":"Robustness (evolution)","score":0.6351574659347534},{"id":"https://openalex.org/keywords/smoothing","display_name":"Smoothing","score":0.6195101737976074},{"id":"https://openalex.org/keywords/bounded-function","display_name":"Bounded function","score":0.6184813976287842},{"id":"https://openalex.org/keywords/minification","display_name":"Minification","score":0.559108555316925},{"id":"https://openalex.org/keywords/mathematical-optimization","display_name":"Mathematical optimization","score":0.5496255159378052},{"id":"https://openalex.org/keywords/stationary-point","display_name":"Stationary point","score":0.5215854048728943},{"id":"https://openalex.org/keywords/norm","display_name":"Norm (philosophy)","score":0.4985518455505371},{"id":"https://openalex.org/keywords/regularization","display_name":"Regularization (linguistics)","score":0.49049729108810425},{"id":"https://openalex.org/keywords/convergence","display_name":"Convergence (economics)","score":0.4418522119522095},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.4319028854370117},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.32548290491104126},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.2953025698661804},{"id":"https://openalex.org/keywords/artificial-intelligence","display_name":"Artificial intelligence","score":0.26154953241348267}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6927554607391357},{"id":"https://openalex.org/C63479239","wikidata":"https://www.wikidata.org/wiki/Q7353546","display_name":"Robustness (evolution)","level":3,"score":0.6351574659347534},{"id":"https://openalex.org/C3770464","wikidata":"https://www.wikidata.org/wiki/Q775963","display_name":"Smoothing","level":2,"score":0.6195101737976074},{"id":"https://openalex.org/C34388435","wikidata":"https://www.wikidata.org/wiki/Q2267362","display_name":"Bounded function","level":2,"score":0.6184813976287842},{"id":"https://openalex.org/C147764199","wikidata":"https://www.wikidata.org/wiki/Q6865248","display_name":"Minification","level":2,"score":0.559108555316925},{"id":"https://openalex.org/C126255220","wikidata":"https://www.wikidata.org/wiki/Q141495","display_name":"Mathematical optimization","level":1,"score":0.5496255159378052},{"id":"https://openalex.org/C189237950","wikidata":"https://www.wikidata.org/wiki/Q2500758","display_name":"Stationary point","level":2,"score":0.5215854048728943},{"id":"https://openalex.org/C191795146","wikidata":"https://www.wikidata.org/wiki/Q3878446","display_name":"Norm (philosophy)","level":2,"score":0.4985518455505371},{"id":"https://openalex.org/C2776135515","wikidata":"https://www.wikidata.org/wiki/Q17143721","display_name":"Regularization (linguistics)","level":2,"score":0.49049729108810425},{"id":"https://openalex.org/C2777303404","wikidata":"https://www.wikidata.org/wiki/Q759757","display_name":"Convergence (economics)","level":2,"score":0.4418522119522095},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.4319028854370117},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.32548290491104126},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.2953025698661804},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.26154953241348267},{"id":"https://openalex.org/C50522688","wikidata":"https://www.wikidata.org/wiki/Q189833","display_name":"Economic growth","level":1,"score":0.0},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.0},{"id":"https://openalex.org/C55493867","wikidata":"https://www.wikidata.org/wiki/Q7094","display_name":"Biochemistry","level":1,"score":0.0},{"id":"https://openalex.org/C199539241","wikidata":"https://www.wikidata.org/wiki/Q7748","display_name":"Law","level":1,"score":0.0},{"id":"https://openalex.org/C105795698","wikidata":"https://www.wikidata.org/wiki/Q12483","display_name":"Statistics","level":1,"score":0.0},{"id":"https://openalex.org/C162324750","wikidata":"https://www.wikidata.org/wiki/Q8134","display_name":"Economics","level":0,"score":0.0},{"id":"https://openalex.org/C17744445","wikidata":"https://www.wikidata.org/wiki/Q36442","display_name":"Political science","level":0,"score":0.0},{"id":"https://openalex.org/C104317684","wikidata":"https://www.wikidata.org/wiki/Q7187","display_name":"Gene","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}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1137/21m1408713","is_oa":false,"landing_page_url":"https://doi.org/10.1137/21m1408713","pdf_url":null,"source":{"id":"https://openalex.org/S928796702","display_name":"SIAM Journal on Optimization","issn_l":"1052-6234","issn":["1052-6234","1095-7189"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320508","host_organization_name":"Society for Industrial and Applied Mathematics","host_organization_lineage":["https://openalex.org/P4310320508"],"host_organization_lineage_names":["Society for Industrial and Applied Mathematics"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"SIAM Journal on Optimization","raw_type":"journal-article"}],"best_oa_location":null,"sustainable_development_goals":[{"display_name":"Peace, Justice and strong institutions","id":"https://metadata.un.org/sdg/16","score":0.4300000071525574}],"awards":[{"id":"https://openalex.org/G335111166","display_name":null,"funder_award_id":"12125108","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G3907981520","display_name":null,"funder_award_id":"11991021","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G5781594238","display_name":"\u5206\u5e03\u5f0f\u4f18\u5316\u7684\u7406\u8bba\u4e0e\u7b97\u6cd5","funder_award_id":"11971466","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G735499983","display_name":null,"funder_award_id":"PolyU15300120","funder_id":"https://openalex.org/F4320321592","funder_display_name":"Research Grants Council, University Grants Committee"},{"id":"https://openalex.org/G953615589","display_name":null,"funder_award_id":"ZDBS-LY-7022","funder_id":"https://openalex.org/F4320321133","funder_display_name":"Chinese Academy of Sciences"}],"funders":[{"id":"https://openalex.org/F4320321001","display_name":"National Natural Science Foundation of China","ror":"https://ror.org/01h0zpd94"},{"id":"https://openalex.org/F4320321133","display_name":"Chinese Academy of Sciences","ror":"https://ror.org/034t30j35"},{"id":"https://openalex.org/F4320321592","display_name":"Research Grants Council, University Grants Committee","ror":"https://ror.org/00djwmt25"},{"id":"https://openalex.org/F4320322598","display_name":"Hong Kong Polytechnic University","ror":"https://ror.org/0030zas98"}],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":38,"referenced_works":["https://openalex.org/W6908809","https://openalex.org/W17525587","https://openalex.org/W1522301498","https://openalex.org/W1551360398","https://openalex.org/W1583837637","https://openalex.org/W1665214252","https://openalex.org/W1990615478","https://openalex.org/W1995341919","https://openalex.org/W2017257315","https://openalex.org/W2062227835","https://openalex.org/W2067547116","https://openalex.org/W2098000549","https://openalex.org/W2100495367","https://openalex.org/W2105482846","https://openalex.org/W2112796928","https://openalex.org/W2144513243","https://openalex.org/W2146502635","https://openalex.org/W2156387975","https://openalex.org/W2334285086","https://openalex.org/W2346438296","https://openalex.org/W2546302380","https://openalex.org/W2559997609","https://openalex.org/W2593390416","https://openalex.org/W2598912124","https://openalex.org/W2763583057","https://openalex.org/W2792643794","https://openalex.org/W2795073474","https://openalex.org/W2807697622","https://openalex.org/W2808678810","https://openalex.org/W2963055445","https://openalex.org/W2963906098","https://openalex.org/W2981921564","https://openalex.org/W2994927236","https://openalex.org/W3034315405","https://openalex.org/W3036100489","https://openalex.org/W3210165781","https://openalex.org/W4242472394","https://openalex.org/W4255783720"],"related_works":["https://openalex.org/W2383807498","https://openalex.org/W1978572805","https://openalex.org/W1997992934","https://openalex.org/W1987225439","https://openalex.org/W4238188170","https://openalex.org/W4213275102","https://openalex.org/W2151138761","https://openalex.org/W2283037149","https://openalex.org/W2944731954","https://openalex.org/W2989495362"],"abstract_inverted_index":{"A":[0,85],"regularized":[1,49,75],"minimization":[2,76],"model":[3,34,50,77],"with":[4,46,78],"$l_1$-norm":[5,79],"penalty":[6,33,80],"(RP)":[7],"is":[8,90,110],"introduced":[9],"for":[10,82],"training":[11,83],"the":[12,26,37,47,68,94,98,106,117,122,125],"autoencoders":[13],"that":[14,25,60],"belong":[15],"to":[16,92,100],"a":[17,56,72,101],"class":[18],"of":[19,67,97,105,124],"two-layer":[20],"neural":[21],"networks.":[22],"We":[23,54],"show":[24],"RP":[27,107],"can":[28],"act":[29],"as":[30,119,121],"an":[31],"exact":[32],"which":[35],"shares":[36],"same":[38],"global":[39,65],"minimizers,":[40,42],"local":[41],"and":[43,70,108],"d(irectional)-stationary":[44],"points":[45],"original":[48],"under":[51],"mild":[52],"conditions.":[53],"construct":[55],"bounded":[57],"box":[58],"region":[59],"contains":[61],"at":[62],"least":[63],"one":[64],"minimizer":[66],"RP,":[69],"propose":[71],"linearly":[73],"constrained":[74],"(LRP)":[81],"autoencoders.":[84],"smoothing":[86],"proximal":[87],"gradient":[88],"algorithm":[89,99],"designed":[91],"solve":[93],"LRP.":[95],"Convergence":[96],"generalized":[102],"d-stationary":[103],"point":[104],"LRP":[109],"delivered.":[111],"Comprehensive":[112],"numerical":[113],"experiments":[114],"convincingly":[115],"illustrate":[116],"efficiency":[118],"well":[120],"robustness":[123],"proposed":[126],"algorithm.":[127]},"counts_by_year":[{"year":2026,"cited_by_count":1},{"year":2025,"cited_by_count":4},{"year":2023,"cited_by_count":2},{"year":2021,"cited_by_count":1}],"updated_date":"2025-11-06T03:46:38.306776","created_date":"2025-10-10T00:00:00"}