{"id":"https://openalex.org/W4309184205","doi":"https://doi.org/10.1137/19m1299074","title":"Revisiting Landscape Analysis in Deep Neural Networks: Eliminating Decreasing Paths to Infinity","display_name":"Revisiting Landscape Analysis in Deep Neural Networks: Eliminating Decreasing Paths to Infinity","publication_year":2022,"publication_date":"2022-11-17","ids":{"openalex":"https://openalex.org/W4309184205","doi":"https://doi.org/10.1137/19m1299074"},"language":"en","primary_location":{"id":"doi:10.1137/19m1299074","is_oa":false,"landing_page_url":"https://doi.org/10.1137/19m1299074","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/A5083787469","display_name":"Shiyu Liang","orcid":"https://orcid.org/0009-0003-2917-2033"},"institutions":[{"id":"https://openalex.org/I157725225","display_name":"University of Illinois Urbana-Champaign","ror":"https://ror.org/047426m28","country_code":"US","type":"education","lineage":["https://openalex.org/I157725225"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Shiyu Liang","raw_affiliation_strings":["University of Illinois at Urbana-Champaign"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"University of Illinois at Urbana-Champaign","institution_ids":["https://openalex.org/I157725225"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5028963874","display_name":"Ruoyu Sun","orcid":"https://orcid.org/0000-0003-2487-5322"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Ruoyu Sun","raw_affiliation_strings":[],"raw_orcid":"https://orcid.org/0000-0003-2487-5322","affiliations":[]},{"author_position":"last","author":{"id":"https://openalex.org/A5078518595","display_name":"R. Srikant","orcid":"https://orcid.org/0000-0003-1483-5204"},"institutions":[{"id":"https://openalex.org/I157725225","display_name":"University of Illinois Urbana-Champaign","ror":"https://ror.org/047426m28","country_code":"US","type":"education","lineage":["https://openalex.org/I157725225"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"R. Srikant","raw_affiliation_strings":["University of Illinois at Urbana-Champaign"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"University of Illinois at Urbana-Champaign","institution_ids":["https://openalex.org/I157725225"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":0.4762,"has_fulltext":false,"cited_by_count":4,"citation_normalized_percentile":{"value":0.56159201,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":96},"biblio":{"volume":"32","issue":"4","first_page":"2797","last_page":"2827"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10500","display_name":"Sparse and Compressive Sensing Techniques","score":0.9916999936103821,"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":0.9916999936103821,"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/T11612","display_name":"Stochastic Gradient Optimization Techniques","score":0.9807000160217285,"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/T11206","display_name":"Model Reduction and Neural Networks","score":0.965399980545044,"subfield":{"id":"https://openalex.org/subfields/3109","display_name":"Statistical and Nonlinear Physics"},"field":{"id":"https://openalex.org/fields/31","display_name":"Physics and Astronomy"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/maxima-and-minima","display_name":"Maxima and minima","score":0.8715991377830505},{"id":"https://openalex.org/keywords/infinity","display_name":"Infinity","score":0.721455991268158},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.6911220550537109},{"id":"https://openalex.org/keywords/lipschitz-continuity","display_name":"Lipschitz continuity","score":0.6499310731887817},{"id":"https://openalex.org/keywords/euclidean-space","display_name":"Euclidean space","score":0.5883866548538208},{"id":"https://openalex.org/keywords/artificial-neural-network","display_name":"Artificial neural network","score":0.5397876501083374},{"id":"https://openalex.org/keywords/saddle-point","display_name":"Saddle point","score":0.5197767615318298},{"id":"https://openalex.org/keywords/gradient-descent","display_name":"Gradient descent","score":0.5121660232543945},{"id":"https://openalex.org/keywords/euclidean-geometry","display_name":"Euclidean geometry","score":0.5089099407196045},{"id":"https://openalex.org/keywords/function","display_name":"Function (biology)","score":0.4698447585105896},{"id":"https://openalex.org/keywords/set","display_name":"Set (abstract data type)","score":0.4524296224117279},{"id":"https://openalex.org/keywords/mathematical-optimization","display_name":"Mathematical optimization","score":0.40192025899887085},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.2899095416069031},{"id":"https://openalex.org/keywords/artificial-intelligence","display_name":"Artificial intelligence","score":0.2649431824684143},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.24745601415634155},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.2033877670764923},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.16756793856620789},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.11213478446006775}],"concepts":[{"id":"https://openalex.org/C186633575","wikidata":"https://www.wikidata.org/wiki/Q845060","display_name":"Maxima and minima","level":2,"score":0.8715991377830505},{"id":"https://openalex.org/C7321624","wikidata":"https://www.wikidata.org/wiki/Q205","display_name":"Infinity","level":2,"score":0.721455991268158},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6911220550537109},{"id":"https://openalex.org/C22324862","wikidata":"https://www.wikidata.org/wiki/Q652707","display_name":"Lipschitz continuity","level":2,"score":0.6499310731887817},{"id":"https://openalex.org/C186450821","wikidata":"https://www.wikidata.org/wiki/Q17295","display_name":"Euclidean space","level":2,"score":0.5883866548538208},{"id":"https://openalex.org/C50644808","wikidata":"https://www.wikidata.org/wiki/Q192776","display_name":"Artificial neural network","level":2,"score":0.5397876501083374},{"id":"https://openalex.org/C2681867","wikidata":"https://www.wikidata.org/wiki/Q690935","display_name":"Saddle point","level":2,"score":0.5197767615318298},{"id":"https://openalex.org/C153258448","wikidata":"https://www.wikidata.org/wiki/Q1199743","display_name":"Gradient descent","level":3,"score":0.5121660232543945},{"id":"https://openalex.org/C129782007","wikidata":"https://www.wikidata.org/wiki/Q162886","display_name":"Euclidean geometry","level":2,"score":0.5089099407196045},{"id":"https://openalex.org/C14036430","wikidata":"https://www.wikidata.org/wiki/Q3736076","display_name":"Function (biology)","level":2,"score":0.4698447585105896},{"id":"https://openalex.org/C177264268","wikidata":"https://www.wikidata.org/wiki/Q1514741","display_name":"Set (abstract data type)","level":2,"score":0.4524296224117279},{"id":"https://openalex.org/C126255220","wikidata":"https://www.wikidata.org/wiki/Q141495","display_name":"Mathematical optimization","level":1,"score":0.40192025899887085},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.2899095416069031},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.2649431824684143},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.24745601415634155},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.2033877670764923},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.16756793856620789},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.11213478446006775},{"id":"https://openalex.org/C199360897","wikidata":"https://www.wikidata.org/wiki/Q9143","display_name":"Programming language","level":1,"score":0.0},{"id":"https://openalex.org/C86803240","wikidata":"https://www.wikidata.org/wiki/Q420","display_name":"Biology","level":0,"score":0.0},{"id":"https://openalex.org/C78458016","wikidata":"https://www.wikidata.org/wiki/Q840400","display_name":"Evolutionary biology","level":1,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1137/19m1299074","is_oa":false,"landing_page_url":"https://doi.org/10.1137/19m1299074","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":[],"awards":[],"funders":[],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":19,"referenced_works":["https://openalex.org/W569478347","https://openalex.org/W1540764732","https://openalex.org/W1583837637","https://openalex.org/W1726806267","https://openalex.org/W2142949395","https://openalex.org/W2144730813","https://openalex.org/W2194775991","https://openalex.org/W2283023501","https://openalex.org/W2559655401","https://openalex.org/W2604272474","https://openalex.org/W2612690371","https://openalex.org/W2746420172","https://openalex.org/W2951985090","https://openalex.org/W2963417959","https://openalex.org/W2963446712","https://openalex.org/W2999905431","https://openalex.org/W3113425034","https://openalex.org/W3122701085","https://openalex.org/W4244633107"],"related_works":["https://openalex.org/W2306553012","https://openalex.org/W3185235544","https://openalex.org/W4251417359","https://openalex.org/W2335198763","https://openalex.org/W4288346661","https://openalex.org/W2947416323","https://openalex.org/W1989888202","https://openalex.org/W2978731891","https://openalex.org/W2602921706","https://openalex.org/W3022588902"],"abstract_inverted_index":{"Traditional":[0],"landscape":[1],"analysis":[2],"of":[3,119,151,163],"deep":[4,121],"neural":[5,80,122],"networks":[6,81,123],"aims":[7],"to":[8,26,56,58,65,95,110,139,146,173],"show":[9,147],"that":[10,28,47,87,148],"no":[11,131,136],"suboptimal":[12],"local":[13,39,133],"minima":[14,134],"exist":[15],"in":[16],"some":[17],"appropriate":[18,125],"sense.":[19],"From":[20],"this,":[21],"one":[22,84],"may":[23,154],"be":[24,98,155,158],"tempted":[25],"conclude":[27],"descent":[29,54],"algorithms":[30],"which":[31,68,153],"escape":[32],"saddle":[33],"points":[34],"will":[35],"reach":[36],"a":[37,53,116,164,169,174],"good":[38],"minimum.":[40],"However,":[41],"basic":[42],"optimization":[43],"theory":[44],"tell":[45],"us":[46],"it":[48],"is":[49,74,145],"also":[50],"possible":[51],"for":[52,78,115],"algorithm":[55],"diverge":[57],"infinity":[59],"if":[60],"there":[61,82],"are":[62],"paths":[63,94,138],"leading":[64],"infinity,":[66],"along":[67],"the":[69,106,127,149,161],"loss":[70,128],"function":[71,129],"decreases.":[72],"It":[73],"not":[75],"clear":[76],"whether":[77],"nonlinear":[79],"exists":[83],"setting":[85],"such":[86],"\u201cno":[88,92],"bad":[89,132],"local-min\u201d":[90],"and":[91,135,178],"decreasing":[93,137],"infinity\u201d":[96],"can":[97,157],"simultaneously":[99],"achieved.":[100],"In":[101],"this":[102,111],"paper,":[103],"we":[104],"give":[105],"first":[107],"positive":[108],"answer":[109],"question.":[112],"More":[113],"specifically,":[114],"large":[117],"class":[118],"overparameterized":[120],"with":[124],"regularizers,":[126,152],"has":[130,180],"infinity.":[140],"The":[141],"key":[142],"mathematical":[143],"trick":[144],"set":[150],"undesirable,":[156],"viewed":[159],"as":[160],"image":[162],"Lipschitz":[165],"continuous":[166],"mapping":[167],"from":[168],"lower-dimensional":[170],"Euclidean":[171,176],"space":[172,177],"higher-dimensional":[175],"thus":[179],"zero":[181],"measure.":[182]},"counts_by_year":[{"year":2025,"cited_by_count":1},{"year":2024,"cited_by_count":2},{"year":2020,"cited_by_count":1}],"updated_date":"2025-11-06T03:46:38.306776","created_date":"2025-10-10T00:00:00"}
