{"id":"https://openalex.org/W1855615675","doi":"https://doi.org/10.1109/tnnls.2015.2440473","title":"Optimization in Quaternion Dynamic Systems: Gradient, Hessian, and Learning Algorithms","display_name":"Optimization in Quaternion Dynamic Systems: Gradient, Hessian, and Learning Algorithms","publication_year":2015,"publication_date":"2015-06-16","ids":{"openalex":"https://openalex.org/W1855615675","doi":"https://doi.org/10.1109/tnnls.2015.2440473","mag":"1855615675","pmid":"https://pubmed.ncbi.nlm.nih.gov/26087504"},"language":"en","primary_location":{"id":"doi:10.1109/tnnls.2015.2440473","is_oa":false,"landing_page_url":"https://doi.org/10.1109/tnnls.2015.2440473","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":["arxiv","crossref","pubmed"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://arxiv.org/pdf/1406.3587","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5012826227","display_name":"Dongpo Xu","orcid":"https://orcid.org/0000-0002-9663-9743"},"institutions":[{"id":"https://openalex.org/I184983240","display_name":"Northeast Normal University","ror":"https://ror.org/02rkvz144","country_code":"CN","type":"education","lineage":["https://openalex.org/I184983240"]},{"id":"https://openalex.org/I47508984","display_name":"Imperial College London","ror":"https://ror.org/041kmwe10","country_code":"GB","type":"education","lineage":["https://openalex.org/I47508984"]},{"id":"https://openalex.org/I151727225","display_name":"Harbin Engineering University","ror":"https://ror.org/03x80pn82","country_code":"CN","type":"education","lineage":["https://openalex.org/I151727225"]}],"countries":["CN","GB"],"is_corresponding":true,"raw_author_name":"Dongpo Xu","raw_affiliation_strings":["College of Science, Harbin Engineering University, Harbin, China","Department of Electrical and Electronic Engineering, Imperial College London, London, U.K","School of Mathematics and Statistics, Northeast Normal University, Changchun, China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"College of Science, Harbin Engineering University, Harbin, China","institution_ids":["https://openalex.org/I151727225"]},{"raw_affiliation_string":"Department of Electrical and Electronic Engineering, Imperial College London, London, U.K","institution_ids":["https://openalex.org/I47508984"]},{"raw_affiliation_string":"School of Mathematics and Statistics, Northeast Normal University, Changchun, China","institution_ids":["https://openalex.org/I184983240"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5005242197","display_name":"Yili Xia","orcid":"https://orcid.org/0000-0002-4402-8131"},"institutions":[{"id":"https://openalex.org/I76569877","display_name":"Southeast University","ror":"https://ror.org/04ct4d772","country_code":"CN","type":"education","lineage":["https://openalex.org/I76569877"]},{"id":"https://openalex.org/I47508984","display_name":"Imperial College London","ror":"https://ror.org/041kmwe10","country_code":"GB","type":"education","lineage":["https://openalex.org/I47508984"]}],"countries":["CN","GB"],"is_corresponding":false,"raw_author_name":"Yili Xia","raw_affiliation_strings":["Department of Electrical and Electronic Engineering, Imperial College London, London, U.K","School of Information Science and Engineering, Southeast University, Nanjing, China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Electrical and Electronic Engineering, Imperial College London, London, U.K","institution_ids":["https://openalex.org/I47508984"]},{"raw_affiliation_string":"School of Information Science and Engineering, Southeast University, Nanjing, China","institution_ids":["https://openalex.org/I76569877"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5103001848","display_name":"Danilo P. Mandic","orcid":"https://orcid.org/0000-0001-8432-3963"},"institutions":[{"id":"https://openalex.org/I47508984","display_name":"Imperial College London","ror":"https://ror.org/041kmwe10","country_code":"GB","type":"education","lineage":["https://openalex.org/I47508984"]}],"countries":["GB"],"is_corresponding":false,"raw_author_name":"Danilo P. Mandic","raw_affiliation_strings":["Department of Electrical and Electronic Engineering, Imperial College London, London, U.K"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Electrical and Electronic Engineering, Imperial College London, London, U.K","institution_ids":["https://openalex.org/I47508984"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":3,"corresponding_author_ids":["https://openalex.org/A5012826227"],"corresponding_institution_ids":["https://openalex.org/I151727225","https://openalex.org/I184983240","https://openalex.org/I47508984"],"apc_list":null,"apc_paid":null,"fwci":15.7544,"has_fulltext":false,"cited_by_count":133,"citation_normalized_percentile":{"value":0.99456961,"is_in_top_1_percent":true,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":90,"max":100},"biblio":{"volume":"27","issue":"2","first_page":"249","last_page":"261"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T12037","display_name":"Algebraic and Geometric Analysis","score":0.9994000196456909,"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"}},"topics":[{"id":"https://openalex.org/T12037","display_name":"Algebraic and Geometric Analysis","score":0.9994000196456909,"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/T11210","display_name":"Mathematical Analysis and Transform Methods","score":0.9904999732971191,"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/T10792","display_name":"Matrix Theory and Algorithms","score":0.9794999957084656,"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/hessian-matrix","display_name":"Hessian matrix","score":0.8212363719940186},{"id":"https://openalex.org/keywords/quaternion","display_name":"Quaternion","score":0.6215211749076843},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.5304293632507324},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.5139302611351013},{"id":"https://openalex.org/keywords/mathematical-optimization","display_name":"Mathematical optimization","score":0.3647070825099945},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.3521430492401123},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.286583811044693},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.08280262351036072}],"concepts":[{"id":"https://openalex.org/C203616005","wikidata":"https://www.wikidata.org/wiki/Q620495","display_name":"Hessian matrix","level":2,"score":0.8212363719940186},{"id":"https://openalex.org/C200127275","wikidata":"https://www.wikidata.org/wiki/Q173853","display_name":"Quaternion","level":2,"score":0.6215211749076843},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.5304293632507324},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.5139302611351013},{"id":"https://openalex.org/C126255220","wikidata":"https://www.wikidata.org/wiki/Q141495","display_name":"Mathematical optimization","level":1,"score":0.3647070825099945},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.3521430492401123},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.286583811044693},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.08280262351036072}],"mesh":[],"locations_count":4,"locations":[{"id":"doi:10.1109/tnnls.2015.2440473","is_oa":false,"landing_page_url":"https://doi.org/10.1109/tnnls.2015.2440473","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:26087504","is_oa":false,"landing_page_url":"https://pubmed.ncbi.nlm.nih.gov/26087504","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},{"id":"pmh:oai:arXiv.org:1406.3587","is_oa":true,"landing_page_url":"http://arxiv.org/abs/1406.3587","pdf_url":"https://arxiv.org/pdf/1406.3587","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:spiral.imperial.ac.uk:10044/1/43619","is_oa":false,"landing_page_url":"http://hdl.handle.net/10044/1/43619","pdf_url":null,"source":{"id":"https://openalex.org/S4306401396","display_name":"Spiral (Imperial College London)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I47508984","host_organization_name":"Imperial College London","host_organization_lineage":["https://openalex.org/I47508984"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"261","raw_type":"Journal Article"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:1406.3587","is_oa":true,"landing_page_url":"http://arxiv.org/abs/1406.3587","pdf_url":"https://arxiv.org/pdf/1406.3587","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"},"sustainable_development_goals":[],"awards":[{"id":"https://openalex.org/G3589703592","display_name":null,"funder_award_id":"61401094","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G4139520870","display_name":null,"funder_award_id":"61301202","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G4473062908","display_name":null,"funder_award_id":"20122304120028","funder_id":"https://openalex.org/F4320336024","funder_display_name":"Specialized Research Fund for the Doctoral Program of Higher Education of China"},{"id":"https://openalex.org/G756866442","display_name":null,"funder_award_id":"EP/H026266/1","funder_id":"https://openalex.org/F4320334627","funder_display_name":"Engineering and Physical Sciences Research Council"},{"id":"https://openalex.org/G7616653276","display_name":null,"funder_award_id":"BK20140645","funder_id":"https://openalex.org/F4320322769","funder_display_name":"Natural Science Foundation of Jiangsu Province"}],"funders":[{"id":"https://openalex.org/F4320321001","display_name":"National Natural Science Foundation of China","ror":"https://ror.org/01h0zpd94"},{"id":"https://openalex.org/F4320322769","display_name":"Natural Science Foundation of Jiangsu Province","ror":"https://ror.org/01h0zpd94"},{"id":"https://openalex.org/F4320334627","display_name":"Engineering and Physical Sciences Research Council","ror":"https://ror.org/0439y7842"},{"id":"https://openalex.org/F4320336024","display_name":"Specialized Research Fund for the Doctoral Program of Higher Education of China","ror":null}],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":59,"referenced_works":["https://openalex.org/W107110140","https://openalex.org/W142384176","https://openalex.org/W632146489","https://openalex.org/W1492221128","https://openalex.org/W1542938076","https://openalex.org/W1700035086","https://openalex.org/W1802825888","https://openalex.org/W1921661266","https://openalex.org/W1989757000","https://openalex.org/W1991017496","https://openalex.org/W1999542175","https://openalex.org/W2003891054","https://openalex.org/W2007715285","https://openalex.org/W2017931222","https://openalex.org/W2023334490","https://openalex.org/W2029030701","https://openalex.org/W2030147677","https://openalex.org/W2033772417","https://openalex.org/W2040575700","https://openalex.org/W2043978450","https://openalex.org/W2058399474","https://openalex.org/W2062144489","https://openalex.org/W2065234731","https://openalex.org/W2067639877","https://openalex.org/W2069222481","https://openalex.org/W2082542187","https://openalex.org/W2090550497","https://openalex.org/W2094646141","https://openalex.org/W2099057098","https://openalex.org/W2108704185","https://openalex.org/W2114388374","https://openalex.org/W2114765937","https://openalex.org/W2118369318","https://openalex.org/W2119870052","https://openalex.org/W2124171390","https://openalex.org/W2124348301","https://openalex.org/W2130071409","https://openalex.org/W2134563598","https://openalex.org/W2137135002","https://openalex.org/W2161397505","https://openalex.org/W2164452510","https://openalex.org/W2168858784","https://openalex.org/W2169571074","https://openalex.org/W2170400088","https://openalex.org/W2170646498","https://openalex.org/W2293451440","https://openalex.org/W2327214751","https://openalex.org/W2476841360","https://openalex.org/W2489292218","https://openalex.org/W2605543710","https://openalex.org/W3125910865","https://openalex.org/W3140025309","https://openalex.org/W4213332888","https://openalex.org/W4232809274","https://openalex.org/W4251652949","https://openalex.org/W4289784446","https://openalex.org/W4293775970","https://openalex.org/W6637501666","https://openalex.org/W6967614468"],"related_works":["https://openalex.org/W4391375266","https://openalex.org/W2611031068","https://openalex.org/W1704347466","https://openalex.org/W4283017538","https://openalex.org/W3127350447","https://openalex.org/W1996936972","https://openalex.org/W3108700312","https://openalex.org/W1545275724","https://openalex.org/W2802707792","https://openalex.org/W4225472102"],"abstract_inverted_index":{"The":[0,165],"optimization":[1,81,156],"of":[2,6,28,36,48,60,79,123,135,152,162,186],"real":[3,34,95,132],"scalar":[4],"functions":[5,35],"quaternion":[7,37,61,86,106,126,137,155,196],"variables,":[8],"such":[9],"as":[10,97,178],"the":[11,26,29,46,67,85,91,94,104,108,113,124,136,160,173,179,187],"mean":[12],"square":[13],"error":[14],"or":[15],"array":[16],"output":[17],"power,":[18],"underpins":[19],"many":[20],"practical":[21],"applications.":[22],"Solutions":[23],"typically":[24],"require":[25],"calculation":[27],"gradient":[30,62,127,138],"and":[31,63,115,118,128,139,157,182,199],"Hessian.":[32],"However,":[33],"variables":[38],"are":[39,43,145,190],"essentially":[40],"nonanalytic,":[41],"which":[42],"prohibitive":[44],"to":[45,142,171],"development":[47],"quaternion-valued":[49],"learning":[50,163],"systems.":[51],"To":[52],"address":[53],"this":[54],"issue,":[55],"we":[56],"propose":[57],"new":[58,150],"definitions":[59],"Hessian,":[64],"based":[65],"on":[66],"novel":[68,125],"generalized":[69],"Hamilton-real":[70],"(GHR)":[71],"calculus,":[72],"thus":[73],"making":[74],"a":[75,120,149],"possible":[76],"efficient":[77],"derivation":[78],"general":[80],"algorithms":[82],"directly":[83],"in":[84,154,195],"field,":[87],"rather":[88],"than":[89],"using":[90],"isomorphism":[92],"with":[93,130],"domain,":[96],"is":[98,169],"current":[99],"practice.":[100],"In":[101],"addition,":[102],"unlike":[103],"existing":[105],"gradients,":[107],"GHR":[109,167],"calculus":[110,168],"allows":[111],"for":[112,119],"product":[114],"chain":[116],"rule,":[117],"one-to-one":[121],"correspondence":[122],"Hessian":[129,140],"their":[131],"counterparts.":[133],"Properties":[134],"relevant":[141],"numerical":[143],"applications":[144],"also":[146],"introduced,":[147],"opening":[148],"avenue":[151],"research":[153],"greatly":[158],"simplified":[159],"derivations":[161],"algorithms.":[164,184],"proposed":[166,188],"shown":[170],"yield":[172],"same":[174],"generic":[175],"algorithm":[176],"forms":[177],"corresponding":[180],"real-":[181],"complex-valued":[183],"Advantages":[185],"framework":[189],"illuminated":[191],"over":[192],"illustrative":[193],"simulations":[194],"signal":[197],"processing":[198],"neural":[200],"networks.":[201]},"counts_by_year":[{"year":2026,"cited_by_count":6},{"year":2025,"cited_by_count":12},{"year":2024,"cited_by_count":13},{"year":2023,"cited_by_count":15},{"year":2022,"cited_by_count":15},{"year":2021,"cited_by_count":6},{"year":2020,"cited_by_count":17},{"year":2019,"cited_by_count":18},{"year":2018,"cited_by_count":9},{"year":2017,"cited_by_count":12},{"year":2016,"cited_by_count":6},{"year":2015,"cited_by_count":3},{"year":2014,"cited_by_count":1}],"updated_date":"2026-06-03T09:05:47.796612","created_date":"2025-10-10T00:00:00"}