{"id":"https://openalex.org/W2169457105","doi":"https://doi.org/10.1109/tcyb.2013.2267893","title":"An Efficient Variable Projection Formulation for Separable Nonlinear Least Squares Problems","display_name":"An Efficient Variable Projection Formulation for Separable Nonlinear Least Squares Problems","publication_year":2013,"publication_date":"2013-07-03","ids":{"openalex":"https://openalex.org/W2169457105","doi":"https://doi.org/10.1109/tcyb.2013.2267893","mag":"2169457105","pmid":"https://pubmed.ncbi.nlm.nih.gov/23846514"},"language":"en","primary_location":{"id":"doi:10.1109/tcyb.2013.2267893","is_oa":false,"landing_page_url":"https://doi.org/10.1109/tcyb.2013.2267893","pdf_url":null,"source":{"id":"https://openalex.org/S4210191041","display_name":"IEEE Transactions on Cybernetics","issn_l":"2168-2267","issn":["2168-2267","2168-2275"],"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 Cybernetics","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/A5047001937","display_name":"Min Gan","orcid":"https://orcid.org/0000-0002-2756-0054"},"institutions":[{"id":"https://openalex.org/I16365422","display_name":"Hefei University of Technology","ror":"https://ror.org/02czkny70","country_code":"CN","type":"education","lineage":["https://openalex.org/I16365422"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Min Gan","raw_affiliation_strings":["School of Electrical Engineering and Automation, Hefei University of Technology, Hefei, China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"School of Electrical Engineering and Automation, Hefei University of Technology, Hefei, China","institution_ids":["https://openalex.org/I16365422"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5090216201","display_name":"Han\u2010Xiong Li","orcid":"https://orcid.org/0000-0002-0707-5940"},"institutions":[{"id":"https://openalex.org/I168719708","display_name":"City University of Hong Kong","ror":"https://ror.org/03q8dnn23","country_code":"HK","type":"education","lineage":["https://openalex.org/I168719708"]}],"countries":["HK"],"is_corresponding":false,"raw_author_name":"Han-Xiong Li","raw_affiliation_strings":["Department of Systems Engineering and Engineering Management, City University of Hong Kong, Kowloon Tong, Hong Kong"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Systems Engineering and Engineering Management, City University of Hong Kong, Kowloon Tong, Hong Kong","institution_ids":["https://openalex.org/I168719708"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":2,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":{"value":2645,"currency":"USD","value_usd":2645},"apc_paid":null,"fwci":3.2218,"has_fulltext":false,"cited_by_count":45,"citation_normalized_percentile":{"value":0.92426271,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":89,"max":99},"biblio":{"volume":"44","issue":"5","first_page":"707","last_page":"711"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10792","display_name":"Matrix Theory and Algorithms","score":0.9972000122070312,"subfield":{"id":"https://openalex.org/subfields/1703","display_name":"Computational Theory and Mathematics"},"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/T10792","display_name":"Matrix Theory and Algorithms","score":0.9972000122070312,"subfield":{"id":"https://openalex.org/subfields/1703","display_name":"Computational Theory and Mathematics"},"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/T13487","display_name":"Statistical and numerical algorithms","score":0.9968000054359436,"subfield":{"id":"https://openalex.org/subfields/2604","display_name":"Applied Mathematics"},"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/T11233","display_name":"Advanced Adaptive Filtering Techniques","score":0.9926000237464905,"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/projection","display_name":"Projection (relational algebra)","score":0.7123385071754456},{"id":"https://openalex.org/keywords/nonlinear-system","display_name":"Nonlinear system","score":0.685965895652771},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.6547133922576904},{"id":"https://openalex.org/keywords/non-linear-least-squares","display_name":"Non-linear least squares","score":0.6517586708068848},{"id":"https://openalex.org/keywords/variable","display_name":"Variable (mathematics)","score":0.6066172122955322},{"id":"https://openalex.org/keywords/least-squares-function-approximation","display_name":"Least-squares function approximation","score":0.5755894184112549},{"id":"https://openalex.org/keywords/separable-space","display_name":"Separable space","score":0.5364447832107544},{"id":"https://openalex.org/keywords/mathematical-optimization","display_name":"Mathematical optimization","score":0.5272017121315002},{"id":"https://openalex.org/keywords/matrix","display_name":"Matrix (chemical analysis)","score":0.49852490425109863},{"id":"https://openalex.org/keywords/dykstras-projection-algorithm","display_name":"Dykstra's projection algorithm","score":0.4974539577960968},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.44704607129096985},{"id":"https://openalex.org/keywords/matrix-decomposition","display_name":"Matrix decomposition","score":0.412533164024353},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.3529558777809143},{"id":"https://openalex.org/keywords/estimation-theory","display_name":"Estimation theory","score":0.11869308352470398},{"id":"https://openalex.org/keywords/statistics","display_name":"Statistics","score":0.05542406439781189}],"concepts":[{"id":"https://openalex.org/C57493831","wikidata":"https://www.wikidata.org/wiki/Q3134666","display_name":"Projection (relational algebra)","level":2,"score":0.7123385071754456},{"id":"https://openalex.org/C158622935","wikidata":"https://www.wikidata.org/wiki/Q660848","display_name":"Nonlinear system","level":2,"score":0.685965895652771},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6547133922576904},{"id":"https://openalex.org/C45923927","wikidata":"https://www.wikidata.org/wiki/Q3319230","display_name":"Non-linear least squares","level":3,"score":0.6517586708068848},{"id":"https://openalex.org/C182365436","wikidata":"https://www.wikidata.org/wiki/Q50701","display_name":"Variable (mathematics)","level":2,"score":0.6066172122955322},{"id":"https://openalex.org/C9936470","wikidata":"https://www.wikidata.org/wiki/Q6510405","display_name":"Least-squares function approximation","level":3,"score":0.5755894184112549},{"id":"https://openalex.org/C70710897","wikidata":"https://www.wikidata.org/wiki/Q680081","display_name":"Separable space","level":2,"score":0.5364447832107544},{"id":"https://openalex.org/C126255220","wikidata":"https://www.wikidata.org/wiki/Q141495","display_name":"Mathematical optimization","level":1,"score":0.5272017121315002},{"id":"https://openalex.org/C106487976","wikidata":"https://www.wikidata.org/wiki/Q685816","display_name":"Matrix (chemical analysis)","level":2,"score":0.49852490425109863},{"id":"https://openalex.org/C202426404","wikidata":"https://www.wikidata.org/wiki/Q5318686","display_name":"Dykstra's projection algorithm","level":2,"score":0.4974539577960968},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.44704607129096985},{"id":"https://openalex.org/C42355184","wikidata":"https://www.wikidata.org/wiki/Q1361088","display_name":"Matrix decomposition","level":3,"score":0.412533164024353},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.3529558777809143},{"id":"https://openalex.org/C167928553","wikidata":"https://www.wikidata.org/wiki/Q1376021","display_name":"Estimation theory","level":2,"score":0.11869308352470398},{"id":"https://openalex.org/C105795698","wikidata":"https://www.wikidata.org/wiki/Q12483","display_name":"Statistics","level":1,"score":0.05542406439781189},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.0},{"id":"https://openalex.org/C192562407","wikidata":"https://www.wikidata.org/wiki/Q228736","display_name":"Materials science","level":0,"score":0.0},{"id":"https://openalex.org/C185429906","wikidata":"https://www.wikidata.org/wiki/Q1130160","display_name":"Estimator","level":2,"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/C158693339","wikidata":"https://www.wikidata.org/wiki/Q190524","display_name":"Eigenvalues and eigenvectors","level":2,"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/C159985019","wikidata":"https://www.wikidata.org/wiki/Q181790","display_name":"Composite material","level":1,"score":0.0}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.1109/tcyb.2013.2267893","is_oa":false,"landing_page_url":"https://doi.org/10.1109/tcyb.2013.2267893","pdf_url":null,"source":{"id":"https://openalex.org/S4210191041","display_name":"IEEE Transactions on Cybernetics","issn_l":"2168-2267","issn":["2168-2267","2168-2275"],"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 Cybernetics","raw_type":"journal-article"},{"id":"pmid:23846514","is_oa":false,"landing_page_url":"https://pubmed.ncbi.nlm.nih.gov/23846514","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 cybernetics","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":14,"referenced_works":["https://openalex.org/W119734876","https://openalex.org/W1523505889","https://openalex.org/W1850361076","https://openalex.org/W1980770037","https://openalex.org/W2018089423","https://openalex.org/W2076284036","https://openalex.org/W2090636411","https://openalex.org/W2101164579","https://openalex.org/W2128787017","https://openalex.org/W2132208889","https://openalex.org/W2139957756","https://openalex.org/W2171904630","https://openalex.org/W2236623899","https://openalex.org/W2260859552"],"related_works":["https://openalex.org/W2026673180","https://openalex.org/W2063973645","https://openalex.org/W1959514624","https://openalex.org/W200101083","https://openalex.org/W2939592218","https://openalex.org/W2359835307","https://openalex.org/W327962130","https://openalex.org/W4380487384","https://openalex.org/W1997338499","https://openalex.org/W4378552425"],"abstract_inverted_index":{"We":[0],"consider":[1],"in":[2,12,116],"this":[3],"paper":[4],"a":[5,20,59],"class":[6],"of":[7,23,35,70,78,88],"nonlinear":[8,24,40,47],"least":[9,41],"squares":[10,42],"problems":[11,43],"which":[13],"the":[14,31,36,39,46,51,71,76,79,103,110],"model":[15],"can":[16],"be":[17,83],"represented":[18],"as":[19],"linear":[21,32],"combination":[22],"functions.":[25],"The":[26,68,91],"variable":[27,52,61],"projection":[28,53,62],"algorithm":[29,54,93],"projects":[30],"parameters":[33],"out":[34],"problem,":[37],"leaving":[38],"involving":[44],"only":[45],"parameters.":[48],"To":[49],"implement":[50],"more":[55],"efficiently,":[56],"we":[57],"propose":[58],"new":[60,104],"functional":[63],"based":[64],"on":[65],"matrix":[66,81],"decomposition.":[67],"advantage":[69],"proposed":[72,111],"formulation":[73],"is":[74,98],"that":[75,109],"size":[77],"decomposed":[80],"may":[82],"much":[84],"smaller":[85],"than":[86],"those":[87],"previous":[89],"ones.":[90],"Levenberg-Marquardt":[92],"using":[94],"finite":[95],"difference":[96],"method":[97],"then":[99],"applied":[100],"to":[101],"minimize":[102],"criterion.":[105],"Numerical":[106],"results":[107],"show":[108],"approach":[112],"achieves":[113],"significant":[114],"reduction":[115],"computing":[117],"time.":[118]},"counts_by_year":[{"year":2026,"cited_by_count":1},{"year":2025,"cited_by_count":3},{"year":2024,"cited_by_count":3},{"year":2023,"cited_by_count":3},{"year":2022,"cited_by_count":4},{"year":2021,"cited_by_count":3},{"year":2020,"cited_by_count":7},{"year":2019,"cited_by_count":6},{"year":2018,"cited_by_count":3},{"year":2017,"cited_by_count":3},{"year":2016,"cited_by_count":2},{"year":2015,"cited_by_count":1},{"year":2014,"cited_by_count":6}],"updated_date":"2026-08-21T09:56:20.448147","created_date":"2025-10-10T00:00:00"}