{"id":"https://openalex.org/W7164549776","doi":"https://doi.org/10.1109/msp.2026.3662660","title":"Computational Nonsmooth Analysis: Modern applications and recent developments [Special Issue on the Mathematics of Deep Learning]","display_name":"Computational Nonsmooth Analysis: Modern applications and recent developments [Special Issue on the Mathematics of Deep Learning]","publication_year":2026,"publication_date":"2026-05-01","ids":{"openalex":"https://openalex.org/W7164549776","doi":"https://doi.org/10.1109/msp.2026.3662660"},"language":null,"primary_location":{"id":"doi:10.1109/msp.2026.3662660","is_oa":false,"landing_page_url":"https://doi.org/10.1109/msp.2026.3662660","pdf_url":null,"source":{"id":"https://openalex.org/S120977877","display_name":"IEEE Signal Processing Magazine","issn_l":"1053-5888","issn":["1053-5888","1558-0792"],"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 Signal Processing Magazine","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/A5091356450","display_name":"Lai Tian","orcid":"https://orcid.org/0000-0002-0328-2651"},"institutions":[{"id":"https://openalex.org/I1174212","display_name":"University of Southern California","ror":"https://ror.org/03taz7m60","country_code":"US","type":"education","lineage":["https://openalex.org/I1174212"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Lai Tian","raw_affiliation_strings":["University of Southern California, Los Angeles, CA, USA"],"raw_orcid":"https://orcid.org/0000-0002-0328-2651","affiliations":[{"raw_affiliation_string":"University of Southern California, Los Angeles, CA, USA","institution_ids":["https://openalex.org/I1174212"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5105881332","display_name":"Anthony Man\u2013Cho So","orcid":"https://orcid.org/0000-0003-2588-7851"},"institutions":[{"id":"https://openalex.org/I177725633","display_name":"Chinese University of Hong Kong","ror":"https://ror.org/00t33hh48","country_code":"HK","type":"education","lineage":["https://openalex.org/I177725633"]}],"countries":["HK"],"is_corresponding":false,"raw_author_name":"Anthony Man-Cho So","raw_affiliation_strings":["Department of Systems Engineering and Engineering, Management of the Chinese University of Hong Kong (CUHK), Hong Kong, China"],"raw_orcid":"https://orcid.org/0000-0003-2588-7851","affiliations":[{"raw_affiliation_string":"Department of Systems Engineering and Engineering, Management of the Chinese University of Hong Kong (CUHK), Hong Kong, China","institution_ids":["https://openalex.org/I177725633"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":2,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":0.0,"has_fulltext":false,"cited_by_count":0,"citation_normalized_percentile":{"value":0.95069513,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":null,"biblio":{"volume":"43","issue":"3","first_page":"102","last_page":"114"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T12814","display_name":"Gaussian Processes and Bayesian Inference","score":0.046799998730421066,"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":0.046799998730421066,"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":0.04580000042915344,"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/T10775","display_name":"Generative Adversarial Networks and Image Synthesis","score":0.03739999979734421,"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/calculus","display_name":"Calculus (dental)","score":0.31630000472068787},{"id":"https://openalex.org/keywords/signal-processing","display_name":"Signal processing","score":0.2685999870300293},{"id":"https://openalex.org/keywords/computational-complexity-theory","display_name":"Computational complexity theory","score":0.26510000228881836},{"id":"https://openalex.org/keywords/algebra-over-a-field","display_name":"Algebra over a field","score":0.25519999861717224}],"concepts":[{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.5920000076293945},{"id":"https://openalex.org/C2777686260","wikidata":"https://www.wikidata.org/wiki/Q144037","display_name":"Calculus (dental)","level":2,"score":0.31630000472068787},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.30550000071525574},{"id":"https://openalex.org/C80444323","wikidata":"https://www.wikidata.org/wiki/Q2878974","display_name":"Theoretical computer science","level":1,"score":0.29589998722076416},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.2833000123500824},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.2773999869823456},{"id":"https://openalex.org/C104267543","wikidata":"https://www.wikidata.org/wiki/Q208163","display_name":"Signal processing","level":3,"score":0.2685999870300293},{"id":"https://openalex.org/C179799912","wikidata":"https://www.wikidata.org/wiki/Q205084","display_name":"Computational complexity theory","level":2,"score":0.26510000228881836},{"id":"https://openalex.org/C136119220","wikidata":"https://www.wikidata.org/wiki/Q1000660","display_name":"Algebra over a field","level":2,"score":0.25519999861717224},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.2451000064611435}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1109/msp.2026.3662660","is_oa":false,"landing_page_url":"https://doi.org/10.1109/msp.2026.3662660","pdf_url":null,"source":{"id":"https://openalex.org/S120977877","display_name":"IEEE Signal Processing Magazine","issn_l":"1053-5888","issn":["1053-5888","1558-0792"],"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 Signal Processing Magazine","raw_type":"journal-article"}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":27,"referenced_works":["https://openalex.org/W103882287","https://openalex.org/W1551360398","https://openalex.org/W1783840476","https://openalex.org/W1972393809","https://openalex.org/W2003212828","https://openalex.org/W2011835437","https://openalex.org/W2051651878","https://openalex.org/W2079135578","https://openalex.org/W2136953603","https://openalex.org/W2478429860","https://openalex.org/W2605585793","https://openalex.org/W2901838001","https://openalex.org/W2963190258","https://openalex.org/W2963534244","https://openalex.org/W2964210434","https://openalex.org/W3007070539","https://openalex.org/W3016321495","https://openalex.org/W3083575911","https://openalex.org/W3216451529","https://openalex.org/W4206529845","https://openalex.org/W4249513058","https://openalex.org/W4249667877","https://openalex.org/W4252698487","https://openalex.org/W4390637589","https://openalex.org/W4406151864","https://openalex.org/W4410034083","https://openalex.org/W7133238670"],"related_works":[],"abstract_inverted_index":{"Neural":[0],"networks":[1,34],"(NNs)":[2],"have":[3],"transformed":[4],"the":[5,68,94,113,119],"field":[6],"of":[7,28,32,64,73,80,96,123],"artificial":[8],"intelligence,":[9],"showcasing":[10],"remarkable":[11],"abilities":[12],"in":[13,52,67,99,110,126],"learning":[14,129],"complex":[15],"representations,":[16],"scaling":[17],"to":[18,86,115],"large":[19],"datasets,":[20],"and":[21,55,70,121,130],"driving":[22],"advancements":[23,66,98],"across":[24],"a":[25],"wide":[26],"range":[27],"applications.":[29,102],"The":[30],"training":[31],"these":[33,97],"often":[35],"involves":[36],"optimizing":[37],"functions":[38],"that":[39,83],"are":[40],"not":[41],"only":[42],"nonconvex":[43],"but":[44],"possibly":[45],"also":[46,92],"nonsmooth,":[47],"which":[48,116],"introduces":[49],"significant":[50],"challenges":[51],"algorithm":[53],"design":[54],"analysis.":[56],"In":[57],"this":[58,111],"article,":[59],"we":[60,84,104],"provide":[61],"an":[62,78],"overview":[63],"recent":[65],"theoretical":[69],"computational":[71],"aspects":[72],"nonsmooth":[74,89,124],"optimization,":[75],"thereby":[76],"highlighting":[77],"area":[79],"growing":[81],"importance":[82],"refer":[85],"as":[87],"\u201ccomputational":[88],"analysis.\u201d":[90],"We":[91],"demonstrate":[93],"relevance":[95],"some":[100,106],"representative":[101],"Lastly,":[103],"discuss":[105],"fundamental":[107],"open":[108],"questions":[109],"area,":[112],"answers":[114],"will":[117],"impact":[118],"use":[120],"understanding":[122],"models":[125],"modern":[127],"deep":[128],"signal":[131],"processing.":[132]},"counts_by_year":[],"updated_date":"2026-06-14T06:11:07.267592","created_date":"2026-06-13T00:00:00"}
