{"id":"https://openalex.org/W2905923365","doi":"https://doi.org/10.1109/tnnls.2018.2884909","title":"Modified Gram\u2013Schmidt Method-Based Variable Projection Algorithm for Separable Nonlinear Models","display_name":"Modified Gram\u2013Schmidt Method-Based Variable Projection Algorithm for Separable Nonlinear Models","publication_year":2018,"publication_date":"2018-12-25","ids":{"openalex":"https://openalex.org/W2905923365","doi":"https://doi.org/10.1109/tnnls.2018.2884909","mag":"2905923365","pmid":"https://pubmed.ncbi.nlm.nih.gov/30596588"},"language":"en","primary_location":{"id":"doi:10.1109/tnnls.2018.2884909","is_oa":false,"landing_page_url":"https://doi.org/10.1109/tnnls.2018.2884909","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/A5002763479","display_name":"Guangyong Chen","orcid":"https://orcid.org/0000-0003-2088-9188"},"institutions":[{"id":"https://openalex.org/I1327237609","display_name":"Ministry of Education of the People's Republic of China","ror":"https://ror.org/01mv9t934","country_code":"CN","type":"funder","lineage":["https://openalex.org/I1327237609","https://openalex.org/I4210127390"]}],"countries":["CN"],"is_corresponding":true,"raw_author_name":"Guang-Yong Chen","raw_affiliation_strings":["Key Laboratory of Spatial Data Mining & Information Sharing, Ministry of Education, Fuzhou, China"],"affiliations":[{"raw_affiliation_string":"Key Laboratory of Spatial Data Mining & Information Sharing, Ministry of Education, Fuzhou, China","institution_ids":["https://openalex.org/I1327237609"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5047001937","display_name":"Min Gan","orcid":"https://orcid.org/0000-0002-2756-0054"},"institutions":[{"id":"https://openalex.org/I1327237609","display_name":"Ministry of Education of the People's Republic of China","ror":"https://ror.org/01mv9t934","country_code":"CN","type":"funder","lineage":["https://openalex.org/I1327237609","https://openalex.org/I4210127390"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Min Gan","raw_affiliation_strings":["Key Laboratory of Spatial Data Mining & Information Sharing, Ministry of Education, Fuzhou, China"],"affiliations":[{"raw_affiliation_string":"Key Laboratory of Spatial Data Mining & Information Sharing, Ministry of Education, Fuzhou, China","institution_ids":["https://openalex.org/I1327237609"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5100637638","display_name":"Feng Ding","orcid":"https://orcid.org/0000-0002-2721-2025"},"institutions":[{"id":"https://openalex.org/I143413998","display_name":"Qingdao University of Science and Technology","ror":"https://ror.org/041j8js14","country_code":"CN","type":"education","lineage":["https://openalex.org/I143413998"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Feng Ding","raw_affiliation_strings":["College of Automation and Electronic Engineering, Qingdao University of Science and Technology, Qingdao, China"],"affiliations":[{"raw_affiliation_string":"College of Automation and Electronic Engineering, Qingdao University of Science and Technology, Qingdao, China","institution_ids":["https://openalex.org/I143413998"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5100643265","display_name":"C. L. Philip Chen","orcid":"https://orcid.org/0000-0001-5451-7230"},"institutions":[{"id":"https://openalex.org/I204512498","display_name":"University of Macau","ror":"https://ror.org/01r4q9n85","country_code":"MO","type":"education","lineage":["https://openalex.org/I204512498"]}],"countries":["MO"],"is_corresponding":false,"raw_author_name":"C. L. Philip Chen","raw_affiliation_strings":["Department of Computer and Information Science, University of Macau, Macau, China"],"affiliations":[{"raw_affiliation_string":"Department of Computer and Information Science, University of Macau, Macau, China","institution_ids":["https://openalex.org/I204512498"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":4,"corresponding_author_ids":["https://openalex.org/A5002763479"],"corresponding_institution_ids":["https://openalex.org/I1327237609"],"apc_list":null,"apc_paid":null,"fwci":8.5874,"has_fulltext":false,"cited_by_count":68,"citation_normalized_percentile":{"value":0.98488364,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":90,"max":100},"biblio":{"volume":"30","issue":"8","first_page":"2410","last_page":"2418"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11447","display_name":"Blind Source Separation Techniques","score":0.9962000250816345,"subfield":{"id":"https://openalex.org/subfields/1711","display_name":"Signal Processing"},"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/T11447","display_name":"Blind Source Separation Techniques","score":0.9962000250816345,"subfield":{"id":"https://openalex.org/subfields/1711","display_name":"Signal Processing"},"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/T11233","display_name":"Advanced Adaptive Filtering Techniques","score":0.9923999905586243,"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/T10792","display_name":"Matrix Theory and Algorithms","score":0.9901999831199646,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/jacobian-matrix-and-determinant","display_name":"Jacobian matrix and determinant","score":0.8475397825241089},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.7059695720672607},{"id":"https://openalex.org/keywords/robustness","display_name":"Robustness (evolution)","score":0.6505451202392578},{"id":"https://openalex.org/keywords/nonlinear-system","display_name":"Nonlinear system","score":0.5902571082115173},{"id":"https://openalex.org/keywords/separable-space","display_name":"Separable space","score":0.5741252303123474},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.5541460514068604},{"id":"https://openalex.org/keywords/gramian-matrix","display_name":"Gramian matrix","score":0.5153350234031677},{"id":"https://openalex.org/keywords/dykstras-projection-algorithm","display_name":"Dykstra's projection algorithm","score":0.4610971212387085},{"id":"https://openalex.org/keywords/variable-elimination","display_name":"Variable elimination","score":0.4602119028568268},{"id":"https://openalex.org/keywords/projection","display_name":"Projection (relational algebra)","score":0.45105990767478943},{"id":"https://openalex.org/keywords/mathematical-optimization","display_name":"Mathematical optimization","score":0.3364560604095459},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.3312748074531555},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.1619332730770111},{"id":"https://openalex.org/keywords/artificial-intelligence","display_name":"Artificial intelligence","score":0.14336621761322021},{"id":"https://openalex.org/keywords/eigenvalues-and-eigenvectors","display_name":"Eigenvalues and eigenvectors","score":0.0993226170539856}],"concepts":[{"id":"https://openalex.org/C200331156","wikidata":"https://www.wikidata.org/wiki/Q506041","display_name":"Jacobian matrix and determinant","level":2,"score":0.8475397825241089},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.7059695720672607},{"id":"https://openalex.org/C63479239","wikidata":"https://www.wikidata.org/wiki/Q7353546","display_name":"Robustness (evolution)","level":3,"score":0.6505451202392578},{"id":"https://openalex.org/C158622935","wikidata":"https://www.wikidata.org/wiki/Q660848","display_name":"Nonlinear system","level":2,"score":0.5902571082115173},{"id":"https://openalex.org/C70710897","wikidata":"https://www.wikidata.org/wiki/Q680081","display_name":"Separable space","level":2,"score":0.5741252303123474},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.5541460514068604},{"id":"https://openalex.org/C77246614","wikidata":"https://www.wikidata.org/wiki/Q1409400","display_name":"Gramian matrix","level":3,"score":0.5153350234031677},{"id":"https://openalex.org/C202426404","wikidata":"https://www.wikidata.org/wiki/Q5318686","display_name":"Dykstra's projection algorithm","level":2,"score":0.4610971212387085},{"id":"https://openalex.org/C169272836","wikidata":"https://www.wikidata.org/wiki/Q5668307","display_name":"Variable elimination","level":3,"score":0.4602119028568268},{"id":"https://openalex.org/C57493831","wikidata":"https://www.wikidata.org/wiki/Q3134666","display_name":"Projection (relational algebra)","level":2,"score":0.45105990767478943},{"id":"https://openalex.org/C126255220","wikidata":"https://www.wikidata.org/wiki/Q141495","display_name":"Mathematical optimization","level":1,"score":0.3364560604095459},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.3312748074531555},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.1619332730770111},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.14336621761322021},{"id":"https://openalex.org/C158693339","wikidata":"https://www.wikidata.org/wiki/Q190524","display_name":"Eigenvalues and eigenvectors","level":2,"score":0.0993226170539856},{"id":"https://openalex.org/C104317684","wikidata":"https://www.wikidata.org/wiki/Q7187","display_name":"Gene","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/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.0},{"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/C55493867","wikidata":"https://www.wikidata.org/wiki/Q7094","display_name":"Biochemistry","level":1,"score":0.0},{"id":"https://openalex.org/C185592680","wikidata":"https://www.wikidata.org/wiki/Q2329","display_name":"Chemistry","level":0,"score":0.0},{"id":"https://openalex.org/C2776214188","wikidata":"https://www.wikidata.org/wiki/Q408386","display_name":"Inference","level":2,"score":0.0}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.1109/tnnls.2018.2884909","is_oa":false,"landing_page_url":"https://doi.org/10.1109/tnnls.2018.2884909","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:30596588","is_oa":false,"landing_page_url":"https://pubmed.ncbi.nlm.nih.gov/30596588","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/G3749562358","display_name":null,"funder_award_id":"61673155","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G653700208","display_name":null,"funder_award_id":"61751202","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G7611596349","display_name":null,"funder_award_id":"61751205","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G8161862873","display_name":null,"funder_award_id":"6157254","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":41,"referenced_works":["https://openalex.org/W1523505889","https://openalex.org/W1850361076","https://openalex.org/W1967150189","https://openalex.org/W1977280501","https://openalex.org/W1998332562","https://openalex.org/W2018089423","https://openalex.org/W2018496583","https://openalex.org/W2018617609","https://openalex.org/W2040763577","https://openalex.org/W2042297527","https://openalex.org/W2053277578","https://openalex.org/W2065288681","https://openalex.org/W2074221952","https://openalex.org/W2076284036","https://openalex.org/W2090636411","https://openalex.org/W2091849600","https://openalex.org/W2101791369","https://openalex.org/W2115666412","https://openalex.org/W2147243899","https://openalex.org/W2148352317","https://openalex.org/W2169457105","https://openalex.org/W2207983105","https://openalex.org/W2217490359","https://openalex.org/W2342078803","https://openalex.org/W2417406719","https://openalex.org/W2579665834","https://openalex.org/W2616577182","https://openalex.org/W2739833408","https://openalex.org/W2761183543","https://openalex.org/W2763015773","https://openalex.org/W2792051776","https://openalex.org/W2792800544","https://openalex.org/W2795713901","https://openalex.org/W2804206910","https://openalex.org/W2809169308","https://openalex.org/W2810167473","https://openalex.org/W2883369662","https://openalex.org/W2900774998","https://openalex.org/W3031689935","https://openalex.org/W4240369198","https://openalex.org/W6677377439"],"related_works":["https://openalex.org/W4390355884","https://openalex.org/W2007405763","https://openalex.org/W3136087161","https://openalex.org/W1486373823","https://openalex.org/W2119578520","https://openalex.org/W2053762185","https://openalex.org/W2913611334","https://openalex.org/W1984283682","https://openalex.org/W1965257389","https://openalex.org/W2913338282"],"abstract_inverted_index":{"Separable":[0],"nonlinear":[1,90],"models":[2],"are":[3],"very":[4],"common":[5],"in":[6],"various":[7,35],"research":[8],"fields,":[9],"such":[10,28],"as":[11],"machine":[12],"learning":[13],"and":[14,113],"system":[15],"identification.":[16],"The":[17],"variable":[18],"projection":[19],"(VP)":[20],"approach":[21],"is":[22,86],"efficient":[23],"for":[24,88],"the":[25,45,50,54,63,66,73,83,98,105,111,116],"optimization":[26],"of":[27,53,58,65,82,100,104,115],"models.":[29],"In":[30,69,93],"this":[31],"paper,":[32],"we":[33,48,96],"study":[34],"VP":[36,67,84,106,120],"algorithms":[37],"based":[38,71],"on":[39,72],"different":[40,102],"matrix":[41,56],"decompositions.":[42],"Compared":[43],"with":[44],"previous":[46],"method,":[47,77],"use":[49],"analytical":[51],"expression":[52],"Jacobian":[55],"instead":[57],"finite":[59],"differences.":[60],"This":[61],"improves":[62],"efficiency":[64,112],"algorithms.":[68],"particular,":[70],"modified":[74],"Gram-Schmidt":[75],"(MGS)":[76],"a":[78],"more":[79],"robust":[80],"implementation":[81],"algorithm":[85],"introduced":[87],"separable":[89],"least-squares":[91],"problems.":[92],"numerical":[94],"experiments,":[95],"compare":[97],"performance":[99],"five":[101],"implementations":[103],"algorithm.":[107,121],"Numerical":[108],"results":[109],"show":[110],"robustness":[114],"proposed":[117],"MGS":[118],"method-based":[119]},"counts_by_year":[{"year":2025,"cited_by_count":4},{"year":2024,"cited_by_count":1},{"year":2023,"cited_by_count":4},{"year":2022,"cited_by_count":7},{"year":2021,"cited_by_count":13},{"year":2020,"cited_by_count":12},{"year":2019,"cited_by_count":26},{"year":2018,"cited_by_count":1}],"updated_date":"2025-11-06T03:46:38.306776","created_date":"2025-10-10T00:00:00"}