{"id":"https://openalex.org/W3120786534","doi":"https://doi.org/10.1109/tit.2022.3213607","title":"Boundary Conditions for Linear Exit Time Gradient Trajectories Around Saddle Points: Analysis and Algorithm","display_name":"Boundary Conditions for Linear Exit Time Gradient Trajectories Around Saddle Points: Analysis and Algorithm","publication_year":2022,"publication_date":"2022-10-10","ids":{"openalex":"https://openalex.org/W3120786534","doi":"https://doi.org/10.1109/tit.2022.3213607","mag":"3120786534"},"language":"en","primary_location":{"id":"doi:10.1109/tit.2022.3213607","is_oa":false,"landing_page_url":"https://doi.org/10.1109/tit.2022.3213607","pdf_url":null,"source":{"id":"https://openalex.org/S4502562","display_name":"IEEE Transactions on Information Theory","issn_l":"0018-9448","issn":["0018-9448","1557-9654"],"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 Information Theory","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/A5061333650","display_name":"Rishabh Dixit","orcid":"https://orcid.org/0000-0002-2649-7524"},"institutions":[{"id":"https://openalex.org/I102322142","display_name":"Rutgers, The State University of New Jersey","ror":"https://ror.org/05vt9qd57","country_code":"US","type":"education","lineage":["https://openalex.org/I102322142"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Rishabh Dixit","raw_affiliation_strings":["Department of Electrical and Computer Engineering, Rutgers University, New Brunswick, NJ, USA"],"raw_orcid":"https://orcid.org/0000-0002-2649-7524","affiliations":[{"raw_affiliation_string":"Department of Electrical and Computer Engineering, Rutgers University, New Brunswick, NJ, USA","institution_ids":["https://openalex.org/I102322142"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5034370705","display_name":"Mert G\u00fcrb\u00fczbalaban","orcid":"https://orcid.org/0000-0002-0575-2450"},"institutions":[{"id":"https://openalex.org/I102322142","display_name":"Rutgers, The State University of New Jersey","ror":"https://ror.org/05vt9qd57","country_code":"US","type":"education","lineage":["https://openalex.org/I102322142"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Mert G\u00fcrb\u00fczbalaban","raw_affiliation_strings":["Departments of Electrical & Computer Engineering, Management Science and Information Systems, and Statistics, Rutgers University, New Brunswick, NJ, USA"],"raw_orcid":"https://orcid.org/0000-0002-0575-2450","affiliations":[{"raw_affiliation_string":"Departments of Electrical & Computer Engineering, Management Science and Information Systems, and Statistics, Rutgers University, New Brunswick, NJ, USA","institution_ids":["https://openalex.org/I102322142"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5028718006","display_name":"Waheed U. Bajwa","orcid":"https://orcid.org/0000-0003-4406-5263"},"institutions":[{"id":"https://openalex.org/I102322142","display_name":"Rutgers, The State University of New Jersey","ror":"https://ror.org/05vt9qd57","country_code":"US","type":"education","lineage":["https://openalex.org/I102322142"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Waheed U. Bajwa","raw_affiliation_strings":["Departments of Electrical & Computer Engineering and Statistics, Rutgers University, New Brunswick, NJ, USA"],"raw_orcid":"https://orcid.org/0000-0003-4406-5263","affiliations":[{"raw_affiliation_string":"Departments of Electrical & Computer Engineering and Statistics, Rutgers University, New Brunswick, NJ, USA","institution_ids":["https://openalex.org/I102322142"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":[],"corresponding_institution_ids":["https://openalex.org/I102322142"],"apc_list":null,"apc_paid":null,"fwci":0.3126,"has_fulltext":false,"cited_by_count":4,"citation_normalized_percentile":{"value":0.44089537,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":97},"biblio":{"volume":"69","issue":"4","first_page":"2556","last_page":"2602"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10963","display_name":"Advanced Optimization Algorithms Research","score":0.9994000196456909,"subfield":{"id":"https://openalex.org/subfields/2612","display_name":"Numerical Analysis"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T10963","display_name":"Advanced Optimization Algorithms Research","score":0.9994000196456909,"subfield":{"id":"https://openalex.org/subfields/2612","display_name":"Numerical Analysis"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T10500","display_name":"Sparse and Compressive Sensing Techniques","score":0.9991000294685364,"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":0.9984999895095825,"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/saddle-point","display_name":"Saddle point","score":0.7907378673553467},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.6225729584693909},{"id":"https://openalex.org/keywords/saddle","display_name":"Saddle","score":0.603603720664978},{"id":"https://openalex.org/keywords/gradient-descent","display_name":"Gradient descent","score":0.5731266736984253},{"id":"https://openalex.org/keywords/boundary","display_name":"Boundary (topology)","score":0.5705902576446533},{"id":"https://openalex.org/keywords/curvature","display_name":"Curvature","score":0.5252294540405273},{"id":"https://openalex.org/keywords/rate-of-convergence","display_name":"Rate of convergence","score":0.503128707408905},{"id":"https://openalex.org/keywords/convergence","display_name":"Convergence (economics)","score":0.49553096294403076},{"id":"https://openalex.org/keywords/mathematical-optimization","display_name":"Mathematical optimization","score":0.46287089586257935},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.4526502788066864},{"id":"https://openalex.org/keywords/convex-function","display_name":"Convex function","score":0.41654932498931885},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.38673901557922363},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.2707270383834839},{"id":"https://openalex.org/keywords/regular-polygon","display_name":"Regular polygon","score":0.22298258543014526},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.18682390451431274},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.169906884431839},{"id":"https://openalex.org/keywords/key","display_name":"Key (lock)","score":0.09960731863975525}],"concepts":[{"id":"https://openalex.org/C2681867","wikidata":"https://www.wikidata.org/wiki/Q690935","display_name":"Saddle point","level":2,"score":0.7907378673553467},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6225729584693909},{"id":"https://openalex.org/C2777127463","wikidata":"https://www.wikidata.org/wiki/Q10862618","display_name":"Saddle","level":2,"score":0.603603720664978},{"id":"https://openalex.org/C153258448","wikidata":"https://www.wikidata.org/wiki/Q1199743","display_name":"Gradient descent","level":3,"score":0.5731266736984253},{"id":"https://openalex.org/C62354387","wikidata":"https://www.wikidata.org/wiki/Q875399","display_name":"Boundary (topology)","level":2,"score":0.5705902576446533},{"id":"https://openalex.org/C195065555","wikidata":"https://www.wikidata.org/wiki/Q214881","display_name":"Curvature","level":2,"score":0.5252294540405273},{"id":"https://openalex.org/C57869625","wikidata":"https://www.wikidata.org/wiki/Q1783502","display_name":"Rate of convergence","level":3,"score":0.503128707408905},{"id":"https://openalex.org/C2777303404","wikidata":"https://www.wikidata.org/wiki/Q759757","display_name":"Convergence (economics)","level":2,"score":0.49553096294403076},{"id":"https://openalex.org/C126255220","wikidata":"https://www.wikidata.org/wiki/Q141495","display_name":"Mathematical optimization","level":1,"score":0.46287089586257935},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.4526502788066864},{"id":"https://openalex.org/C145446738","wikidata":"https://www.wikidata.org/wiki/Q319913","display_name":"Convex function","level":3,"score":0.41654932498931885},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.38673901557922363},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.2707270383834839},{"id":"https://openalex.org/C112680207","wikidata":"https://www.wikidata.org/wiki/Q714886","display_name":"Regular polygon","level":2,"score":0.22298258543014526},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.18682390451431274},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.169906884431839},{"id":"https://openalex.org/C26517878","wikidata":"https://www.wikidata.org/wiki/Q228039","display_name":"Key (lock)","level":2,"score":0.09960731863975525},{"id":"https://openalex.org/C162324750","wikidata":"https://www.wikidata.org/wiki/Q8134","display_name":"Economics","level":0,"score":0.0},{"id":"https://openalex.org/C50644808","wikidata":"https://www.wikidata.org/wiki/Q192776","display_name":"Artificial neural network","level":2,"score":0.0},{"id":"https://openalex.org/C119857082","wikidata":"https://www.wikidata.org/wiki/Q2539","display_name":"Machine learning","level":1,"score":0.0},{"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/C38652104","wikidata":"https://www.wikidata.org/wiki/Q3510521","display_name":"Computer security","level":1,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1109/tit.2022.3213607","is_oa":false,"landing_page_url":"https://doi.org/10.1109/tit.2022.3213607","pdf_url":null,"source":{"id":"https://openalex.org/S4502562","display_name":"IEEE Transactions on Information Theory","issn_l":"0018-9448","issn":["0018-9448","1557-9654"],"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 Information Theory","raw_type":"journal-article"}],"best_oa_location":null,"sustainable_development_goals":[{"id":"https://metadata.un.org/sdg/11","score":0.6800000071525574,"display_name":"Sustainable cities and communities"}],"awards":[{"id":"https://openalex.org/G2408348303","display_name":null,"funder_award_id":"CCF-1814888","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"},{"id":"https://openalex.org/G3855261351","display_name":null,"funder_award_id":"N00014-21-1-2244","funder_id":"https://openalex.org/F4320337345","funder_display_name":"Office of Naval Research"},{"id":"https://openalex.org/G5213350891","display_name":null,"funder_award_id":"CCF-1907658","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"},{"id":"https://openalex.org/G6077994957","display_name":null,"funder_award_id":"N660011824020","funder_id":"https://openalex.org/F4320332180","funder_display_name":"Defense Advanced Research Projects Agency"},{"id":"https://openalex.org/G7146920611","display_name":null,"funder_award_id":"DMS-2053485","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"},{"id":"https://openalex.org/G7410136991","display_name":null,"funder_award_id":"W911NF-17-1-0546","funder_id":"https://openalex.org/F4320338281","funder_display_name":"Army Research Office"},{"id":"https://openalex.org/G8421618691","display_name":null,"funder_award_id":"CCF-1910110","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"},{"id":"https://openalex.org/G8556298469","display_name":null,"funder_award_id":"W911NF-21-1-0301","funder_id":"https://openalex.org/F4320338281","funder_display_name":"Army Research Office"},{"id":"https://openalex.org/G8694854942","display_name":null,"funder_award_id":"CCF-1453073","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/F4320332180","display_name":"Defense Advanced Research Projects Agency","ror":"https://ror.org/02caytj08"},{"id":"https://openalex.org/F4320337345","display_name":"Office of Naval Research","ror":"https://ror.org/00rk2pe57"},{"id":"https://openalex.org/F4320338281","display_name":"Army Research Office","ror":"https://ror.org/05epdh915"}],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":79,"referenced_works":["https://openalex.org/W181053014","https://openalex.org/W597535747","https://openalex.org/W1242194915","https://openalex.org/W1509448450","https://openalex.org/W1516681311","https://openalex.org/W1560130085","https://openalex.org/W1632601927","https://openalex.org/W1967138577","https://openalex.org/W1978065031","https://openalex.org/W1988720110","https://openalex.org/W2009941369","https://openalex.org/W2021361347","https://openalex.org/W2026258334","https://openalex.org/W2031103608","https://openalex.org/W2068728923","https://openalex.org/W2075920484","https://openalex.org/W2076680677","https://openalex.org/W2083115202","https://openalex.org/W2086867325","https://openalex.org/W2097417531","https://openalex.org/W2103111465","https://openalex.org/W2131329059","https://openalex.org/W2142717543","https://openalex.org/W2283214199","https://openalex.org/W2316564661","https://openalex.org/W2546420264","https://openalex.org/W2567576169","https://openalex.org/W2595897327","https://openalex.org/W2621920791","https://openalex.org/W2751090697","https://openalex.org/W2766900638","https://openalex.org/W2768503813","https://openalex.org/W2768927505","https://openalex.org/W2790867397","https://openalex.org/W2890842400","https://openalex.org/W2912134372","https://openalex.org/W2915116423","https://openalex.org/W2963012205","https://openalex.org/W2963092340","https://openalex.org/W2963129420","https://openalex.org/W2963248893","https://openalex.org/W2963349772","https://openalex.org/W2963416883","https://openalex.org/W2963487351","https://openalex.org/W2963687412","https://openalex.org/W2963926425","https://openalex.org/W2966465943","https://openalex.org/W2989128237","https://openalex.org/W3014542904","https://openalex.org/W3016499390","https://openalex.org/W3029830352","https://openalex.org/W3030881449","https://openalex.org/W3041093287","https://openalex.org/W3100522009","https://openalex.org/W3141595720","https://openalex.org/W3143596294","https://openalex.org/W4205370808","https://openalex.org/W4235625704","https://openalex.org/W4248613038","https://openalex.org/W4292230442","https://openalex.org/W6637461557","https://openalex.org/W6680012447","https://openalex.org/W6680611266","https://openalex.org/W6694931997","https://openalex.org/W6720643467","https://openalex.org/W6734280532","https://openalex.org/W6738336337","https://openalex.org/W6743395824","https://openalex.org/W6744060662","https://openalex.org/W6745998338","https://openalex.org/W6746050020","https://openalex.org/W6746089872","https://openalex.org/W6746152743","https://openalex.org/W6749200166","https://openalex.org/W6754488330","https://openalex.org/W6754512606","https://openalex.org/W6758302211","https://openalex.org/W6776511717","https://openalex.org/W6803548909"],"related_works":["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","https://openalex.org/W2094891943","https://openalex.org/W4240399920"],"abstract_inverted_index":{"Gradient-related":[0],"first-order":[1,35,49],"methods":[2,36,51],"have":[3],"become":[4],"the":[5,29,38,65,97,119,127,134,148,156,192,209,230,233],"workhorse":[6],"of":[7,13,28,31,34,41,48,56,75,102,107,115,122,129,155,191,199,232],"large-scale":[8],"numerical":[9],"optimization":[10,58],"problems.":[11],"Many":[12],"these":[14,42,87,103,123,130,144],"problems":[15,59],"involve":[16],"nonconvex":[17,57],"objective":[18],"functions":[19],"with":[20],"multiple":[21],"saddle":[22,80],"points,":[23],"which":[24,86,169,195],"necessitates":[25],"an":[26,174],"understanding":[27],"behavior":[30],"discrete":[32,50,76],"trajectories":[33,78,88,104,131,181],"within":[37,64],"geometrical":[39,66],"landscape":[40],"functions.":[43],"This":[44],"paper":[45,149],"concerns":[46],"convergence":[47,200],"to":[52,132,178,201,228],"a":[53,152,171,202,217,223],"local":[54,203],"minimum":[55],"that":[60,110],"comprise":[61],"strict-saddle":[62,91,108,137,184],"points":[63,109,138],"landscape.":[67],"To":[68],"this":[69],"end,":[70],"it":[71],"focuses":[72],"on":[73,143,216],"analysis":[74,190],"gradient":[77,158],"around":[79,136],"neighborhoods,":[81],"derives":[82],"sufficient":[83],"conditions":[84],"under":[85],"can":[89,182],"escape":[90,183],"neighborhoods":[92,106,135,185],"in":[93,105,186,208],"linear":[94,187],"time,":[95],"explores":[96],"contractive":[98],"and":[99,125,146],"expansive":[100],"dynamics":[101],"are":[111,213],"characterized":[112],"by":[113],"gradients":[114],"moderate":[116],"magnitude,":[117],"characterizes":[118],"non-curving":[120],"nature":[121],"trajectories,":[124],"highlights":[126],"inability":[128],"re-enter":[133],"after":[139],"exiting":[140],"them.":[141],"Based":[142],"insights":[145],"analyses,":[147],"then":[150,214],"proposes":[151],"simple":[153],"variant":[154],"vanilla":[157],"descent":[159],"algorithm,":[160,168,194],"termed":[161],"Curvature":[162],"Conditioned":[163],"Regularized":[164],"Gradient":[165],"Descent":[166],"(CCRGD)":[167],"utilizes":[170],"check":[172],"for":[173],"initial":[175],"boundary":[176],"condition":[177],"ensure":[179],"its":[180,197],"time.":[188],"Convergence":[189],"CCRGD":[193],"includes":[196],"rate":[198],"minimum,":[204],"is":[205],"also":[206],"presented":[207],"paper.":[210],"Numerical":[211],"experiments":[212],"provided":[215],"test":[218],"function":[219],"as":[220,222],"well":[221],"low-rank":[224],"matrix":[225],"factorization":[226],"problem":[227],"evaluate":[229],"efficacy":[231],"proposed":[234],"algorithm.":[235]},"counts_by_year":[{"year":2026,"cited_by_count":1},{"year":2022,"cited_by_count":1},{"year":2021,"cited_by_count":1},{"year":2020,"cited_by_count":1}],"updated_date":"2025-11-06T03:46:38.306776","created_date":"2025-10-10T00:00:00"}