{"id":"https://openalex.org/W4287323484","doi":"https://doi.org/10.1109/tnnls.2022.3204319","title":"Spurious Local Minima are Common for Deep Neural Networks With Piecewise Linear Activations","display_name":"Spurious Local Minima are Common for Deep Neural Networks With Piecewise Linear Activations","publication_year":2022,"publication_date":"2022-09-20","ids":{"openalex":"https://openalex.org/W4287323484","doi":"https://doi.org/10.1109/tnnls.2022.3204319","pmid":"https://pubmed.ncbi.nlm.nih.gov/36126034"},"language":"en","primary_location":{"id":"doi:10.1109/tnnls.2022.3204319","is_oa":false,"landing_page_url":"https://doi.org/10.1109/tnnls.2022.3204319","pdf_url":null,"source":{"id":"https://openalex.org/S4210175523","display_name":"IEEE Transactions on Neural Networks and Learning Systems","issn_l":"2162-237X","issn":["2162-237X","2162-2388"],"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 Neural Networks and Learning Systems","raw_type":"journal-article"},"type":"article","indexed_in":["crossref","pubmed"],"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/A5042428983","display_name":"Bo Liu","orcid":"https://orcid.org/0000-0002-3482-6930"},"institutions":[{"id":"https://openalex.org/I37796252","display_name":"Beijing University of Technology","ror":"https://ror.org/037b1pp87","country_code":"CN","type":"education","lineage":["https://openalex.org/I37796252"]}],"countries":["CN"],"is_corresponding":true,"raw_author_name":"Bo Liu","raw_affiliation_strings":["Faculty of Information Technology, College of Computer Science, Beijing University of Technology, Beijing, China"],"raw_orcid":"https://orcid.org/0000-0002-3482-6930","affiliations":[{"raw_affiliation_string":"Faculty of Information Technology, College of Computer Science, Beijing University of Technology, Beijing, China","institution_ids":["https://openalex.org/I37796252"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5042428983"],"corresponding_institution_ids":["https://openalex.org/I37796252"],"apc_list":null,"apc_paid":null,"fwci":0.9291,"has_fulltext":false,"cited_by_count":10,"citation_normalized_percentile":{"value":0.78065694,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":98},"biblio":{"volume":"35","issue":"4","first_page":"5382","last_page":"5394"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11612","display_name":"Stochastic Gradient Optimization Techniques","score":0.9994999766349792,"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/T11612","display_name":"Stochastic Gradient Optimization Techniques","score":0.9994999766349792,"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/T11689","display_name":"Adversarial Robustness in Machine Learning","score":0.9993000030517578,"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/T10036","display_name":"Advanced Neural Network Applications","score":0.9976000189781189,"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/maxima-and-minima","display_name":"Maxima and minima","score":0.9115451574325562},{"id":"https://openalex.org/keywords/spurious-relationship","display_name":"Spurious relationship","score":0.8450930118560791},{"id":"https://openalex.org/keywords/piecewise-linear-function","display_name":"Piecewise linear function","score":0.7493762373924255},{"id":"https://openalex.org/keywords/piecewise","display_name":"Piecewise","score":0.6427522301673889},{"id":"https://openalex.org/keywords/maximization","display_name":"Maximization","score":0.5393621325492859},{"id":"https://openalex.org/keywords/piecewise-linear-manifold","display_name":"Piecewise linear manifold","score":0.5324004292488098},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.5302298665046692},{"id":"https://openalex.org/keywords/artificial-neural-network","display_name":"Artificial neural network","score":0.48513421416282654},{"id":"https://openalex.org/keywords/minification","display_name":"Minification","score":0.47363370656967163},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.44670307636260986},{"id":"https://openalex.org/keywords/function","display_name":"Function (biology)","score":0.4457821249961853},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.37106919288635254},{"id":"https://openalex.org/keywords/mathematical-optimization","display_name":"Mathematical optimization","score":0.3616282641887665},{"id":"https://openalex.org/keywords/artificial-intelligence","display_name":"Artificial intelligence","score":0.31536030769348145},{"id":"https://openalex.org/keywords/machine-learning","display_name":"Machine learning","score":0.1693999469280243},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.07174080610275269},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.0688241720199585}],"concepts":[{"id":"https://openalex.org/C186633575","wikidata":"https://www.wikidata.org/wiki/Q845060","display_name":"Maxima and minima","level":2,"score":0.9115451574325562},{"id":"https://openalex.org/C97256817","wikidata":"https://www.wikidata.org/wiki/Q1462316","display_name":"Spurious relationship","level":2,"score":0.8450930118560791},{"id":"https://openalex.org/C17095337","wikidata":"https://www.wikidata.org/wiki/Q2375229","display_name":"Piecewise linear function","level":2,"score":0.7493762373924255},{"id":"https://openalex.org/C164660894","wikidata":"https://www.wikidata.org/wiki/Q2037833","display_name":"Piecewise","level":2,"score":0.6427522301673889},{"id":"https://openalex.org/C2776330181","wikidata":"https://www.wikidata.org/wiki/Q18358244","display_name":"Maximization","level":2,"score":0.5393621325492859},{"id":"https://openalex.org/C184175843","wikidata":"https://www.wikidata.org/wiki/Q7191425","display_name":"Piecewise linear manifold","level":3,"score":0.5324004292488098},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.5302298665046692},{"id":"https://openalex.org/C50644808","wikidata":"https://www.wikidata.org/wiki/Q192776","display_name":"Artificial neural network","level":2,"score":0.48513421416282654},{"id":"https://openalex.org/C147764199","wikidata":"https://www.wikidata.org/wiki/Q6865248","display_name":"Minification","level":2,"score":0.47363370656967163},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.44670307636260986},{"id":"https://openalex.org/C14036430","wikidata":"https://www.wikidata.org/wiki/Q3736076","display_name":"Function (biology)","level":2,"score":0.4457821249961853},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.37106919288635254},{"id":"https://openalex.org/C126255220","wikidata":"https://www.wikidata.org/wiki/Q141495","display_name":"Mathematical optimization","level":1,"score":0.3616282641887665},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.31536030769348145},{"id":"https://openalex.org/C119857082","wikidata":"https://www.wikidata.org/wiki/Q2539","display_name":"Machine learning","level":1,"score":0.1693999469280243},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.07174080610275269},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.0688241720199585},{"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":2,"locations":[{"id":"doi:10.1109/tnnls.2022.3204319","is_oa":false,"landing_page_url":"https://doi.org/10.1109/tnnls.2022.3204319","pdf_url":null,"source":{"id":"https://openalex.org/S4210175523","display_name":"IEEE Transactions on Neural Networks and Learning Systems","issn_l":"2162-237X","issn":["2162-237X","2162-2388"],"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 Neural Networks and Learning Systems","raw_type":"journal-article"},{"id":"pmid:36126034","is_oa":false,"landing_page_url":"https://pubmed.ncbi.nlm.nih.gov/36126034","pdf_url":null,"source":{"id":"https://openalex.org/S4306525036","display_name":"PubMed","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I1299303238","host_organization_name":"National Institutes of Health","host_organization_lineage":["https://openalex.org/I1299303238"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"IEEE transactions on neural networks and learning systems","raw_type":null}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[{"id":"https://openalex.org/G8055380421","display_name":null,"funder_award_id":"61806013","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":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":101,"referenced_works":["https://openalex.org/W639708223","https://openalex.org/W994053854","https://openalex.org/W1623819170","https://openalex.org/W1677182931","https://openalex.org/W1980475044","https://openalex.org/W2065787176","https://openalex.org/W2078626246","https://openalex.org/W2089947415","https://openalex.org/W2395611524","https://openalex.org/W2510991776","https://openalex.org/W2514392868","https://openalex.org/W2593380010","https://openalex.org/W2603221039","https://openalex.org/W2618003483","https://openalex.org/W2766371994","https://openalex.org/W2808958252","https://openalex.org/W2908206262","https://openalex.org/W2912330043","https://openalex.org/W2935604199","https://openalex.org/W2963417959","https://openalex.org/W2964010725","https://openalex.org/W2964232029","https://openalex.org/W2981407587","https://openalex.org/W3001454063","https://openalex.org/W3083948783","https://openalex.org/W3086769137","https://openalex.org/W3116968400","https://openalex.org/W3133170446","https://openalex.org/W3134476582","https://openalex.org/W3193726607","https://openalex.org/W4287978155","https://openalex.org/W4293379522","https://openalex.org/W4295724548","https://openalex.org/W4297688150","https://openalex.org/W4299879281","https://openalex.org/W4301222678","https://openalex.org/W6637373629","https://openalex.org/W6638926925","https://openalex.org/W6639736602","https://openalex.org/W6681804681","https://openalex.org/W6683595889","https://openalex.org/W6684191040","https://openalex.org/W6690388216","https://openalex.org/W6712153814","https://openalex.org/W6712446393","https://openalex.org/W6713348437","https://openalex.org/W6726353972","https://openalex.org/W6730621733","https://openalex.org/W6731278065","https://openalex.org/W6734185594","https://openalex.org/W6734523390","https://openalex.org/W6734728544","https://openalex.org/W6738310427","https://openalex.org/W6738373677","https://openalex.org/W6738685885","https://openalex.org/W6739166439","https://openalex.org/W6741416024","https://openalex.org/W6744882388","https://openalex.org/W6745448519","https://openalex.org/W6745505307","https://openalex.org/W6745652151","https://openalex.org/W6746713790","https://openalex.org/W6747381837","https://openalex.org/W6747847215","https://openalex.org/W6747991573","https://openalex.org/W6748173244","https://openalex.org/W6748318183","https://openalex.org/W6748748765","https://openalex.org/W6748920792","https://openalex.org/W6749856262","https://openalex.org/W6750998746","https://openalex.org/W6751860949","https://openalex.org/W6752495264","https://openalex.org/W6752544856","https://openalex.org/W6754682547","https://openalex.org/W6754903749","https://openalex.org/W6755212716","https://openalex.org/W6755287785","https://openalex.org/W6756001544","https://openalex.org/W6756091659","https://openalex.org/W6756120399","https://openalex.org/W6756122414","https://openalex.org/W6757628700","https://openalex.org/W6757928074","https://openalex.org/W6758955634","https://openalex.org/W6759039803","https://openalex.org/W6759083344","https://openalex.org/W6763744544","https://openalex.org/W6764243271","https://openalex.org/W6765959321","https://openalex.org/W6768667450","https://openalex.org/W6771472091","https://openalex.org/W6773606474","https://openalex.org/W6774374006","https://openalex.org/W6779024865","https://openalex.org/W6779426705","https://openalex.org/W6781006748","https://openalex.org/W6783894668","https://openalex.org/W6787648603","https://openalex.org/W6789049765","https://openalex.org/W6790668849"],"related_works":["https://openalex.org/W3136579521","https://openalex.org/W2021599769","https://openalex.org/W1523179148","https://openalex.org/W2943700667","https://openalex.org/W2950383522","https://openalex.org/W2619539882","https://openalex.org/W2043586353","https://openalex.org/W57070652","https://openalex.org/W1989138816","https://openalex.org/W3133655088"],"abstract_inverted_index":{"In":[0],"this":[1],"article,":[2],"theoretically,":[3],"it":[4],"is":[5,133],"shown":[6],"that":[7,30],"spurious":[8,44,109],"local":[9,45,110],"minima":[10,46],"are":[11,39,114,152],"common":[12],"for":[13],"deep":[14,52],"fully":[15,53],"connected":[16,54],"networks":[17,22,55,147],"and":[18,28,56,69,124,160],"average-pooling":[19],"convolutional":[20],"neural":[21],"(CNNs)":[23],"with":[24,58,92,119,148],"piecewise":[25,59,65,126,149],"linear":[26,35,60,66,127,144,150,158],"activations":[27,61,151],"datasets":[29],"cannot":[31],"be":[32],"fit":[33,78],"by":[34],"models.":[36],"Motivating":[37],"examples":[38],"given":[40],"to":[41,105,134,155],"explain":[42],"why":[43],"exist:":[47],"each":[48],"output":[49,75],"neuron":[50],"of":[51,72,81,101,108,143],"CNNs":[57],"produces":[62],"a":[63,136],"continuous":[64,121],"(CPWL)":[67],"function,":[68],"different":[70,93,99],"pieces":[71,159],"the":[73,86,106,162],"CPWL":[74,94,137],"can":[76],"optimally":[77],"disjoint":[79],"groups":[80],"data":[82,90],"samples":[83,91],"when":[84],"minimizing":[85],"empirical":[87,102],"risk.":[88],"Fitting":[89],"functions":[95,123],"usually":[96],"results":[97,113],"in":[98,116],"levels":[100],"risk,":[103],"leading":[104],"prevalence":[107],"minima.":[111],"The":[112,129],"proved":[115],"general":[117,125],"settings":[118],"arbitrary":[120],"loss":[122],"activations.":[128],"main":[130],"proof":[131],"technique":[132],"represent":[135],"function":[138],"as":[139],"maximization":[140,163],"over":[141,164],"minimization":[142,165],"pieces.":[145],"Deep":[146],"then":[153],"constructed":[154],"produce":[156],"these":[157],"implement":[161],"operation.":[166]},"counts_by_year":[{"year":2026,"cited_by_count":2},{"year":2025,"cited_by_count":3},{"year":2023,"cited_by_count":4},{"year":2022,"cited_by_count":1}],"updated_date":"2026-07-29T09:40:50.615796","created_date":"2025-10-10T00:00:00"}
