{"id":"https://openalex.org/W2963622172","doi":"https://doi.org/10.1007/s10851-019-00889-w","title":"The Global Optimization Geometry of Shallow Linear Neural Networks","display_name":"The Global Optimization Geometry of Shallow Linear Neural Networks","publication_year":2019,"publication_date":"2019-05-31","ids":{"openalex":"https://openalex.org/W2963622172","doi":"https://doi.org/10.1007/s10851-019-00889-w","mag":"2963622172"},"language":"en","primary_location":{"id":"doi:10.1007/s10851-019-00889-w","is_oa":false,"landing_page_url":"https://doi.org/10.1007/s10851-019-00889-w","pdf_url":null,"source":{"id":"https://openalex.org/S66221708","display_name":"Journal of Mathematical Imaging and Vision","issn_l":"0924-9907","issn":["0924-9907","1573-7683"],"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":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Journal of Mathematical Imaging and Vision","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/A5011989964","display_name":"Zhihui Zhu","orcid":"https://orcid.org/0000-0002-3856-0375"},"institutions":[{"id":"https://openalex.org/I145311948","display_name":"Johns Hopkins University","ror":"https://ror.org/00za53h95","country_code":"US","type":"education","lineage":["https://openalex.org/I145311948"]}],"countries":["US"],"is_corresponding":true,"raw_author_name":"Zhihui Zhu","raw_affiliation_strings":["Mathematical Institute for Data Science, Johns Hopkins University, Baltimore, USA"],"raw_orcid":"https://orcid.org/0000-0002-3856-0375","affiliations":[{"raw_affiliation_string":"Mathematical Institute for Data Science, Johns Hopkins University, Baltimore, USA","institution_ids":["https://openalex.org/I145311948"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5025123281","display_name":"Daniel Soudry","orcid":"https://orcid.org/0000-0001-9368-6352"},"institutions":[{"id":"https://openalex.org/I174306211","display_name":"Technion \u2013 Israel Institute of Technology","ror":"https://ror.org/03qryx823","country_code":"IL","type":"education","lineage":["https://openalex.org/I174306211"]}],"countries":["IL"],"is_corresponding":false,"raw_author_name":"Daniel Soudry","raw_affiliation_strings":["Department of Electrical Engineering, Technion, Israel Institute of Technology, Haifa, Israel"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Electrical Engineering, Technion, Israel Institute of Technology, Haifa, Israel","institution_ids":["https://openalex.org/I174306211"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5005913897","display_name":"Yonina C. Eldar","orcid":"https://orcid.org/0000-0003-4358-5304"},"institutions":[{"id":"https://openalex.org/I53964585","display_name":"Weizmann Institute of Science","ror":"https://ror.org/0316ej306","country_code":"IL","type":"education","lineage":["https://openalex.org/I53964585"]}],"countries":["IL"],"is_corresponding":false,"raw_author_name":"Yonina C. Eldar","raw_affiliation_strings":["Department of Mathematics and Computer Science, Weizmann Institute of Science, Rehovot, Israel"],"raw_orcid":"https://orcid.org/0000-0003-4358-5304","affiliations":[{"raw_affiliation_string":"Department of Mathematics and Computer Science, Weizmann Institute of Science, Rehovot, Israel","institution_ids":["https://openalex.org/I53964585"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5065207980","display_name":"Michael B. Wakin","orcid":"https://orcid.org/0000-0002-2165-4586"},"institutions":[{"id":"https://openalex.org/I167576493","display_name":"Colorado School of Mines","ror":"https://ror.org/04raf6v53","country_code":"US","type":"education","lineage":["https://openalex.org/I167576493"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Michael B. Wakin","raw_affiliation_strings":["Department of Electrical Engineering, Colorado School of Mines, Golden, USA"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Electrical Engineering, Colorado School of Mines, Golden, USA","institution_ids":["https://openalex.org/I167576493"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":4,"corresponding_author_ids":["https://openalex.org/A5011989964"],"corresponding_institution_ids":["https://openalex.org/I145311948"],"apc_list":{"value":2780,"currency":"USD","value_usd":2780},"apc_paid":null,"fwci":3.2843,"has_fulltext":false,"cited_by_count":27,"citation_normalized_percentile":{"value":0.92712146,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":89,"max":98},"biblio":{"volume":"62","issue":"3","first_page":"279","last_page":"292"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11245","display_name":"Advanced Numerical Analysis Techniques","score":0.9980999827384949,"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/T11245","display_name":"Advanced Numerical Analysis Techniques","score":0.9980999827384949,"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":0.9977999925613403,"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/T10500","display_name":"Sparse and Compressive Sensing Techniques","score":0.9973000288009644,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/maxima-and-minima","display_name":"Maxima and minima","score":0.821450412273407},{"id":"https://openalex.org/keywords/hessian-matrix","display_name":"Hessian matrix","score":0.7660696506500244},{"id":"https://openalex.org/keywords/saddle-point","display_name":"Saddle point","score":0.7646615505218506},{"id":"https://openalex.org/keywords/artificial-neural-network","display_name":"Artificial neural network","score":0.7131341695785522},{"id":"https://openalex.org/keywords/gradient-descent","display_name":"Gradient descent","score":0.6428943872451782},{"id":"https://openalex.org/keywords/curvature","display_name":"Curvature","score":0.5900301337242126},{"id":"https://openalex.org/keywords/spurious-relationship","display_name":"Spurious relationship","score":0.5822182893753052},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.539914608001709},{"id":"https://openalex.org/keywords/convergence","display_name":"Convergence (economics)","score":0.5220460891723633},{"id":"https://openalex.org/keywords/eigenvalues-and-eigenvectors","display_name":"Eigenvalues and eigenvectors","score":0.5015077590942383},{"id":"https://openalex.org/keywords/mathematical-optimization","display_name":"Mathematical optimization","score":0.4673937261104584},{"id":"https://openalex.org/keywords/saddle","display_name":"Saddle","score":0.4563804268836975},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.439410924911499},{"id":"https://openalex.org/keywords/global-optimization","display_name":"Global optimization","score":0.43542203307151794},{"id":"https://openalex.org/keywords/method-of-steepest-descent","display_name":"Method of steepest descent","score":0.4290940761566162},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.3847615718841553},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.3784703016281128},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.3369637727737427},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.20676574110984802},{"id":"https://openalex.org/keywords/artificial-intelligence","display_name":"Artificial intelligence","score":0.16829246282577515}],"concepts":[{"id":"https://openalex.org/C186633575","wikidata":"https://www.wikidata.org/wiki/Q845060","display_name":"Maxima and minima","level":2,"score":0.821450412273407},{"id":"https://openalex.org/C203616005","wikidata":"https://www.wikidata.org/wiki/Q620495","display_name":"Hessian matrix","level":2,"score":0.7660696506500244},{"id":"https://openalex.org/C2681867","wikidata":"https://www.wikidata.org/wiki/Q690935","display_name":"Saddle point","level":2,"score":0.7646615505218506},{"id":"https://openalex.org/C50644808","wikidata":"https://www.wikidata.org/wiki/Q192776","display_name":"Artificial neural network","level":2,"score":0.7131341695785522},{"id":"https://openalex.org/C153258448","wikidata":"https://www.wikidata.org/wiki/Q1199743","display_name":"Gradient descent","level":3,"score":0.6428943872451782},{"id":"https://openalex.org/C195065555","wikidata":"https://www.wikidata.org/wiki/Q214881","display_name":"Curvature","level":2,"score":0.5900301337242126},{"id":"https://openalex.org/C97256817","wikidata":"https://www.wikidata.org/wiki/Q1462316","display_name":"Spurious relationship","level":2,"score":0.5822182893753052},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.539914608001709},{"id":"https://openalex.org/C2777303404","wikidata":"https://www.wikidata.org/wiki/Q759757","display_name":"Convergence (economics)","level":2,"score":0.5220460891723633},{"id":"https://openalex.org/C158693339","wikidata":"https://www.wikidata.org/wiki/Q190524","display_name":"Eigenvalues and eigenvectors","level":2,"score":0.5015077590942383},{"id":"https://openalex.org/C126255220","wikidata":"https://www.wikidata.org/wiki/Q141495","display_name":"Mathematical optimization","level":1,"score":0.4673937261104584},{"id":"https://openalex.org/C2777127463","wikidata":"https://www.wikidata.org/wiki/Q10862618","display_name":"Saddle","level":2,"score":0.4563804268836975},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.439410924911499},{"id":"https://openalex.org/C164752517","wikidata":"https://www.wikidata.org/wiki/Q5570875","display_name":"Global optimization","level":2,"score":0.43542203307151794},{"id":"https://openalex.org/C158847443","wikidata":"https://www.wikidata.org/wiki/Q1997812","display_name":"Method of steepest descent","level":2,"score":0.4290940761566162},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.3847615718841553},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.3784703016281128},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.3369637727737427},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.20676574110984802},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.16829246282577515},{"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/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"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/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","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}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1007/s10851-019-00889-w","is_oa":false,"landing_page_url":"https://doi.org/10.1007/s10851-019-00889-w","pdf_url":null,"source":{"id":"https://openalex.org/S66221708","display_name":"Journal of Mathematical Imaging and Vision","issn_l":"0924-9907","issn":["0924-9907","1573-7683"],"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":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Journal of Mathematical Imaging and Vision","raw_type":"journal-article"}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[{"id":"https://openalex.org/G1845771761","display_name":null,"funder_award_id":"No. 31/1031","funder_id":"https://openalex.org/F4320322252","funder_display_name":"Israel Science Foundation"},{"id":"https://openalex.org/G4395949833","display_name":null,"funder_award_id":"CAREER grant CCF\u20131149225","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"},{"id":"https://openalex.org/G7097378344","display_name":null,"funder_award_id":"ONR/SPAWAR contract N660011824020","funder_id":"https://openalex.org/F4320332180","funder_display_name":"Defense Advanced Research Projects Agency"},{"id":"https://openalex.org/G8993061120","display_name":null,"funder_award_id":"CCF\u20131409261","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"}],"funders":[{"id":"https://openalex.org/F4320306076","display_name":"National Science Foundation","ror":"https://ror.org/021nxhr62"},{"id":"https://openalex.org/F4320322252","display_name":"Israel Science Foundation","ror":"https://ror.org/04sazxf24"},{"id":"https://openalex.org/F4320332180","display_name":"Defense Advanced Research Projects Agency","ror":"https://ror.org/02caytj08"}],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":49,"referenced_works":["https://openalex.org/W147998453","https://openalex.org/W1697075315","https://openalex.org/W1971581014","https://openalex.org/W2009941369","https://openalex.org/W2041684917","https://openalex.org/W2078626246","https://openalex.org/W2103496339","https://openalex.org/W2128865943","https://openalex.org/W2137983211","https://openalex.org/W2219727625","https://openalex.org/W2220485312","https://openalex.org/W2283023501","https://openalex.org/W2396019715","https://openalex.org/W2405479119","https://openalex.org/W2474090883","https://openalex.org/W2505159980","https://openalex.org/W2528005577","https://openalex.org/W2592246480","https://openalex.org/W2592651140","https://openalex.org/W2593709294","https://openalex.org/W2601846890","https://openalex.org/W2609037894","https://openalex.org/W2610857016","https://openalex.org/W2619204584","https://openalex.org/W2746420172","https://openalex.org/W2789612570","https://openalex.org/W2804413060","https://openalex.org/W2808376087","https://openalex.org/W2889835342","https://openalex.org/W2919115771","https://openalex.org/W2950383522","https://openalex.org/W2962970675","https://openalex.org/W2963014255","https://openalex.org/W2963446085","https://openalex.org/W3100774664","https://openalex.org/W3102722370","https://openalex.org/W3103457637","https://openalex.org/W3105952791","https://openalex.org/W3113425034","https://openalex.org/W3121926921","https://openalex.org/W3146803896","https://openalex.org/W4229706427","https://openalex.org/W4238508979","https://openalex.org/W6600103761","https://openalex.org/W6600224582","https://openalex.org/W6600763685","https://openalex.org/W6602740733","https://openalex.org/W6603505682","https://openalex.org/W6610042523"],"related_works":["https://openalex.org/W1989888202","https://openalex.org/W1981914421","https://openalex.org/W4236459141","https://openalex.org/W4205304778","https://openalex.org/W2020252434","https://openalex.org/W73248859","https://openalex.org/W2584253892","https://openalex.org/W2350324449","https://openalex.org/W119381072","https://openalex.org/W2034033896"],"abstract_inverted_index":null,"counts_by_year":[{"year":2026,"cited_by_count":1},{"year":2025,"cited_by_count":2},{"year":2024,"cited_by_count":3},{"year":2023,"cited_by_count":2},{"year":2022,"cited_by_count":1},{"year":2021,"cited_by_count":7},{"year":2020,"cited_by_count":6},{"year":2019,"cited_by_count":5}],"updated_date":"2026-07-23T08:03:31.855105","created_date":"2025-10-10T00:00:00"}