{"id":"https://openalex.org/W4284712609","doi":"https://doi.org/10.1007/s00332-022-09823-8","title":"Landscape Analysis for Shallow Neural Networks: Complete Classification of Critical Points for Affine Target Functions","display_name":"Landscape Analysis for Shallow Neural Networks: Complete Classification of Critical Points for Affine Target Functions","publication_year":2022,"publication_date":"2022-07-05","ids":{"openalex":"https://openalex.org/W4284712609","doi":"https://doi.org/10.1007/s00332-022-09823-8"},"language":"en","primary_location":{"id":"doi:10.1007/s00332-022-09823-8","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s00332-022-09823-8","pdf_url":"https://link.springer.com/content/pdf/10.1007/s00332-022-09823-8.pdf","source":{"id":"https://openalex.org/S131276044","display_name":"Journal of Nonlinear Science","issn_l":"0938-8974","issn":["0938-8974","1432-1467"],"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":"other-oa","license_id":"https://openalex.org/licenses/other-oa","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Journal of Nonlinear Science","raw_type":"journal-article"},"type":"article","indexed_in":["arxiv","crossref","datacite"],"open_access":{"is_oa":true,"oa_status":"hybrid","oa_url":"https://link.springer.com/content/pdf/10.1007/s00332-022-09823-8.pdf","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5002005113","display_name":"Patrick Cheridito","orcid":"https://orcid.org/0000-0001-9074-7295"},"institutions":[{"id":"https://openalex.org/I35440088","display_name":"ETH Zurich","ror":"https://ror.org/05a28rw58","country_code":"CH","type":"education","lineage":["https://openalex.org/I2799323385","https://openalex.org/I35440088"]}],"countries":["CH"],"is_corresponding":true,"raw_author_name":"Patrick Cheridito","raw_affiliation_strings":["Department of Mathematics, ETH Zurich, Zurich, Switzerland"],"affiliations":[{"raw_affiliation_string":"Department of Mathematics, ETH Zurich, Zurich, Switzerland","institution_ids":["https://openalex.org/I35440088"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5074382323","display_name":"Arnulf Jentzen","orcid":"https://orcid.org/0000-0002-9840-3339"},"institutions":[{"id":"https://openalex.org/I4210116924","display_name":"Chinese University of Hong Kong, Shenzhen","ror":"https://ror.org/02d5ks197","country_code":"CN","type":"education","lineage":["https://openalex.org/I177725633","https://openalex.org/I180726961","https://openalex.org/I4210116924"]},{"id":"https://openalex.org/I130238516","display_name":"University of Minnesota","ror":"https://ror.org/017zqws13","country_code":"US","type":"education","lineage":["https://openalex.org/I130238516"]},{"id":"https://openalex.org/I4210099586","display_name":"Shenzhen Research Institute of Big Data","ror":"https://ror.org/00z1gwf89","country_code":"CN","type":"education","lineage":["https://openalex.org/I4210099586"]}],"countries":["CN","US"],"is_corresponding":false,"raw_author_name":"Arnulf Jentzen","raw_affiliation_strings":["Applied Mathematics: Institute for Analysis and Numerics, Faculty of Mathematics and Computer Science, University of Mnster, Mnster, Germany","School of Data Science and Shenzhen Research Institute of Big Data, The Chinese University of Hong Kong, Shenzhen, China"],"affiliations":[{"raw_affiliation_string":"Applied Mathematics: Institute for Analysis and Numerics, Faculty of Mathematics and Computer Science, University of Mnster, Mnster, Germany","institution_ids":["https://openalex.org/I130238516"]},{"raw_affiliation_string":"School of Data Science and Shenzhen Research Institute of Big Data, The Chinese University of Hong Kong, Shenzhen, China","institution_ids":["https://openalex.org/I4210116924","https://openalex.org/I4210099586"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5001762952","display_name":"Florian Rossmannek","orcid":"https://orcid.org/0000-0001-5772-5086"},"institutions":[{"id":"https://openalex.org/I35440088","display_name":"ETH Zurich","ror":"https://ror.org/05a28rw58","country_code":"CH","type":"education","lineage":["https://openalex.org/I2799323385","https://openalex.org/I35440088"]}],"countries":["CH"],"is_corresponding":false,"raw_author_name":"Florian Rossmannek","raw_affiliation_strings":["Department of Mathematics, ETH Zurich, Zurich, Switzerland"],"affiliations":[{"raw_affiliation_string":"Department of Mathematics, ETH Zurich, Zurich, Switzerland","institution_ids":["https://openalex.org/I35440088"]}]}],"institutions":[],"countries_distinct_count":3,"institutions_distinct_count":3,"corresponding_author_ids":["https://openalex.org/A5002005113"],"corresponding_institution_ids":["https://openalex.org/I35440088"],"apc_list":{"value":2290,"currency":"EUR","value_usd":2890},"apc_paid":{"value":2290,"currency":"EUR","value_usd":2890},"fwci":0.9209,"has_fulltext":true,"cited_by_count":10,"citation_normalized_percentile":{"value":0.7275445,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":91,"max":97},"biblio":{"volume":"32","issue":"5","first_page":null,"last_page":null},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10502","display_name":"Advanced Memory and Neural Computing","score":0.9997000098228455,"subfield":{"id":"https://openalex.org/subfields/2208","display_name":"Electrical and Electronic Engineering"},"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/T10502","display_name":"Advanced Memory and Neural Computing","score":0.9997000098228455,"subfield":{"id":"https://openalex.org/subfields/2208","display_name":"Electrical and Electronic Engineering"},"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/T12808","display_name":"Ferroelectric and Negative Capacitance Devices","score":0.9991999864578247,"subfield":{"id":"https://openalex.org/subfields/2208","display_name":"Electrical and Electronic Engineering"},"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/T10320","display_name":"Neural Networks and Applications","score":0.998199999332428,"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/maxima-and-minima","display_name":"Maxima and minima","score":0.9345162510871887},{"id":"https://openalex.org/keywords/saddle-point","display_name":"Saddle point","score":0.7243288159370422},{"id":"https://openalex.org/keywords/affine-transformation","display_name":"Affine transformation","score":0.7202712297439575},{"id":"https://openalex.org/keywords/artificial-neural-network","display_name":"Artificial neural network","score":0.6465680599212646},{"id":"https://openalex.org/keywords/quadratic-equation","display_name":"Quadratic equation","score":0.634348452091217},{"id":"https://openalex.org/keywords/function","display_name":"Function (biology)","score":0.501204252243042},{"id":"https://openalex.org/keywords/saddle","display_name":"Saddle","score":0.47799152135849},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.46000176668167114},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.45240291953086853},{"id":"https://openalex.org/keywords/activation-function","display_name":"Activation function","score":0.4350816607475281},{"id":"https://openalex.org/keywords/quadratic-function","display_name":"Quadratic function","score":0.415442556142807},{"id":"https://openalex.org/keywords/topology","display_name":"Topology (electrical circuits)","score":0.4058426022529602},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.3567713797092438},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.3454129099845886},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.3037334084510803},{"id":"https://openalex.org/keywords/artificial-intelligence","display_name":"Artificial intelligence","score":0.24949705600738525},{"id":"https://openalex.org/keywords/mathematical-optimization","display_name":"Mathematical optimization","score":0.2021988034248352},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.1594824492931366},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.14213138818740845}],"concepts":[{"id":"https://openalex.org/C186633575","wikidata":"https://www.wikidata.org/wiki/Q845060","display_name":"Maxima and minima","level":2,"score":0.9345162510871887},{"id":"https://openalex.org/C2681867","wikidata":"https://www.wikidata.org/wiki/Q690935","display_name":"Saddle point","level":2,"score":0.7243288159370422},{"id":"https://openalex.org/C92757383","wikidata":"https://www.wikidata.org/wiki/Q382497","display_name":"Affine transformation","level":2,"score":0.7202712297439575},{"id":"https://openalex.org/C50644808","wikidata":"https://www.wikidata.org/wiki/Q192776","display_name":"Artificial neural network","level":2,"score":0.6465680599212646},{"id":"https://openalex.org/C129844170","wikidata":"https://www.wikidata.org/wiki/Q41299","display_name":"Quadratic equation","level":2,"score":0.634348452091217},{"id":"https://openalex.org/C14036430","wikidata":"https://www.wikidata.org/wiki/Q3736076","display_name":"Function (biology)","level":2,"score":0.501204252243042},{"id":"https://openalex.org/C2777127463","wikidata":"https://www.wikidata.org/wiki/Q10862618","display_name":"Saddle","level":2,"score":0.47799152135849},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.46000176668167114},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.45240291953086853},{"id":"https://openalex.org/C38365724","wikidata":"https://www.wikidata.org/wiki/Q4677469","display_name":"Activation function","level":3,"score":0.4350816607475281},{"id":"https://openalex.org/C166437778","wikidata":"https://www.wikidata.org/wiki/Q50695","display_name":"Quadratic function","level":3,"score":0.415442556142807},{"id":"https://openalex.org/C184720557","wikidata":"https://www.wikidata.org/wiki/Q7825049","display_name":"Topology (electrical circuits)","level":2,"score":0.4058426022529602},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.3567713797092438},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.3454129099845886},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.3037334084510803},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.24949705600738525},{"id":"https://openalex.org/C126255220","wikidata":"https://www.wikidata.org/wiki/Q141495","display_name":"Mathematical optimization","level":1,"score":0.2021988034248352},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.1594824492931366},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.14213138818740845},{"id":"https://openalex.org/C78458016","wikidata":"https://www.wikidata.org/wiki/Q840400","display_name":"Evolutionary biology","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}],"mesh":[],"locations_count":4,"locations":[{"id":"doi:10.1007/s00332-022-09823-8","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s00332-022-09823-8","pdf_url":"https://link.springer.com/content/pdf/10.1007/s00332-022-09823-8.pdf","source":{"id":"https://openalex.org/S131276044","display_name":"Journal of Nonlinear Science","issn_l":"0938-8974","issn":["0938-8974","1432-1467"],"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":"other-oa","license_id":"https://openalex.org/licenses/other-oa","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Journal of Nonlinear Science","raw_type":"journal-article"},{"id":"pmh:oai:arXiv.org:2103.10922","is_oa":true,"landing_page_url":"http://arxiv.org/abs/2103.10922","pdf_url":"https://arxiv.org/pdf/2103.10922","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},{"id":"pmh:oai:www.research-collection.ethz.ch:20.500.11850/557739","is_oa":true,"landing_page_url":"http://hdl.handle.net/20.500.11850/557739","pdf_url":null,"source":{"id":"https://openalex.org/S4306402302","display_name":"Repository for Publications and Research Data (ETH Zurich)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I35440088","host_organization_name":"ETH Zurich","host_organization_lineage":["https://openalex.org/I35440088"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Journal of Nonlinear Science, 32 (5)","raw_type":"info:eu-repo/semantics/publishedVersion"},{"id":"doi:10.3929/ethz-b-000557739","is_oa":true,"landing_page_url":"https://doi.org/10.3929/ethz-b-000557739","pdf_url":null,"source":{"id":"https://openalex.org/S7407051236","display_name":"ETH Z\u00fcrich Research Collection","issn_l":null,"issn":[],"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":null,"is_accepted":false,"is_published":null,"raw_source_name":null,"raw_type":"article-journal"}],"best_oa_location":{"id":"doi:10.1007/s00332-022-09823-8","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s00332-022-09823-8","pdf_url":"https://link.springer.com/content/pdf/10.1007/s00332-022-09823-8.pdf","source":{"id":"https://openalex.org/S131276044","display_name":"Journal of Nonlinear Science","issn_l":"0938-8974","issn":["0938-8974","1432-1467"],"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":"other-oa","license_id":"https://openalex.org/licenses/other-oa","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Journal of Nonlinear Science","raw_type":"journal-article"},"sustainable_development_goals":[{"id":"https://metadata.un.org/sdg/15","score":0.41999998688697815,"display_name":"Life in Land"}],"awards":[{"id":"https://openalex.org/G1177974802","display_name":null,"funder_award_id":"EXC 2044-390685587","funder_id":"https://openalex.org/F4320320879","funder_display_name":"Deutsche Forschungsgemeinschaft"},{"id":"https://openalex.org/G1652504919","display_name":null,"funder_award_id":"EXC 2044","funder_id":"https://openalex.org/F4320320879","funder_display_name":"Deutsche Forschungsgemeinschaft"},{"id":"https://openalex.org/G2110285270","display_name":null,"funder_award_id":"2044-390685587","funder_id":"https://openalex.org/F4320320879","funder_display_name":"Deutsche Forschungsgemeinschaft"},{"id":"https://openalex.org/G2817606592","display_name":null,"funder_award_id":"390685587","funder_id":"https://openalex.org/F4320320879","funder_display_name":"Deutsche Forschungsgemeinschaft"},{"id":"https://openalex.org/G3237484952","display_name":null,"funder_award_id":"Germany's Excellence Strategy EXC 2044-390685587","funder_id":"https://openalex.org/F4320320879","funder_display_name":"Deutsche Forschungsgemeinschaft"},{"id":"https://openalex.org/G5074471488","display_name":null,"funder_award_id":"Excellence Strategy EXC 2044-390685587","funder_id":"https://openalex.org/F4320320879","funder_display_name":"Deutsche Forschungsgemeinschaft"},{"id":"https://openalex.org/G5106512922","display_name":null,"funder_award_id":"Deutsche Forschungsgemeinschaft (DFG","funder_id":"https://openalex.org/F4320320879","funder_display_name":"Deutsche Forschungsgemeinschaft"},{"id":"https://openalex.org/G5246122527","display_name":null,"funder_award_id":"T00120220001","funder_id":"https://openalex.org/F4320331102","funder_display_name":"Shenzhen Research Institute of Big Data"},{"id":"https://openalex.org/G8330397625","display_name":null,"funder_award_id":"Funding","funder_id":"https://openalex.org/F4320320879","funder_display_name":"Deutsche Forschungsgemeinschaft"},{"id":"https://openalex.org/G870869636","display_name":null,"funder_award_id":"EXC 2044-39068558","funder_id":"https://openalex.org/F4320320879","funder_display_name":"Deutsche Forschungsgemeinschaft"}],"funders":[{"id":"https://openalex.org/F4320320879","display_name":"Deutsche Forschungsgemeinschaft","ror":"https://ror.org/018mejw64"},{"id":"https://openalex.org/F4320321652","display_name":"Eidgen\u00f6ssische Technische Hochschule Z\u00fcrich","ror":"https://ror.org/05a28rw58"},{"id":"https://openalex.org/F4320331102","display_name":"Shenzhen Research Institute of Big Data","ror":"https://ror.org/00z1gwf89"},{"id":"https://openalex.org/F4320333998","display_name":"Shenzhen Research Institute, City University of Hong Kong","ror":"https://ror.org/00xc0ma20"}],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W4284712609.pdf","grobid_xml":"https://content.openalex.org/works/W4284712609.grobid-xml"},"referenced_works_count":8,"referenced_works":["https://openalex.org/W1988485873","https://openalex.org/W2078626246","https://openalex.org/W2915116423","https://openalex.org/W2919115771","https://openalex.org/W2963417959","https://openalex.org/W3025750359","https://openalex.org/W3090857165","https://openalex.org/W3132264265"],"related_works":["https://openalex.org/W131971937","https://openalex.org/W1557737441","https://openalex.org/W1996762907","https://openalex.org/W2095858157","https://openalex.org/W2752635268","https://openalex.org/W2324860879","https://openalex.org/W2021599769","https://openalex.org/W3114201969","https://openalex.org/W4236459141","https://openalex.org/W2056823902"],"abstract_inverted_index":{"Abstract":[0],"In":[1,26,50,82],"this":[2],"paper,":[3],"we":[4,30,52,68],"analyze":[5],"the":[6,9,36,40,43,62,89,111],"landscape":[7],"of":[8,12,35,64,91,100,110,114],"true":[10],"loss":[11],"neural":[13],"networks":[14],"with":[15],"one":[16],"hidden":[17,115],"layer":[18],"and":[19,48,60,104],"ReLU,":[20,22],"leaky":[21,92],"or":[23,94],"quadratic":[24,95],"activation.":[25,96],"all":[27],"three":[28],"cases,":[29],"provide":[31],"a":[32,101,107],"complete":[33],"classification":[34],"critical":[37],"points":[38],"in":[39,88],"case":[41,90],"where":[42],"target":[44],"function":[45],"is":[46,99],"affine":[47],"one-dimensional.":[49],"particular,":[51,83],"show":[53],"that":[54,70,117],"there":[55],"exist":[56],"no":[57],"local":[58,72],"maxima":[59],"clarify":[61],"structure":[63],"saddle":[65],"points.":[66],"Moreover,":[67],"prove":[69],"non-global":[71],"minima":[73],"can":[74,118],"only":[75],"be":[76],"caused":[77],"by":[78],"\u2018dead\u2019":[79],"ReLU":[80,93],"neurons.":[81],"they":[84],"do":[85],"not":[86],"appear":[87],"Our":[97],"approach":[98],"combinatorial":[102],"nature":[103],"builds":[105],"on":[106],"careful":[108],"analysis":[109],"different":[112],"types":[113],"neurons":[116],"occur.":[119]},"counts_by_year":[{"year":2025,"cited_by_count":1},{"year":2024,"cited_by_count":3},{"year":2023,"cited_by_count":2},{"year":2022,"cited_by_count":4}],"updated_date":"2026-03-20T23:20:44.827607","created_date":"2025-10-10T00:00:00"}