{"id":"https://openalex.org/W2003891054","doi":"https://doi.org/10.1109/tit.2014.2333734","title":"Quaternion Reproducing Kernel Hilbert Spaces: Existence and Uniqueness Conditions","display_name":"Quaternion Reproducing Kernel Hilbert Spaces: Existence and Uniqueness Conditions","publication_year":2014,"publication_date":"2014-06-30","ids":{"openalex":"https://openalex.org/W2003891054","doi":"https://doi.org/10.1109/tit.2014.2333734","mag":"2003891054"},"language":"en","primary_location":{"id":"doi:10.1109/tit.2014.2333734","is_oa":true,"landing_page_url":"https://doi.org/10.1109/tit.2014.2333734","pdf_url":"https://ieeexplore.ieee.org/ielx7/18/6878505/06846303.pdf","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":"cc-by","license_id":"https://openalex.org/licenses/cc-by","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":true,"oa_status":"hybrid","oa_url":"https://ieeexplore.ieee.org/ielx7/18/6878505/06846303.pdf","any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5020822083","display_name":"Felipe Tobar","orcid":"https://orcid.org/0000-0003-2486-3583"},"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":true,"raw_author_name":"Felipe A. Tobar","raw_affiliation_strings":["Department of Electrical and Electronic Engineering, Imperial College London, London, U.K","Dept of Electrical and Electronic Engineering, Imperial College London, London, U.K"],"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":"Dept of Electrical and Electronic Engineering, Imperial College London, London, U.K","institution_ids":[]}]},{"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","Dept of Electrical and Electronic Engineering, Imperial College London, London, U.K"],"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":"Dept of Electrical and Electronic Engineering, Imperial College London, London, U.K","institution_ids":[]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":2,"corresponding_author_ids":["https://openalex.org/A5020822083"],"corresponding_institution_ids":["https://openalex.org/I47508984"],"apc_list":null,"apc_paid":{"value":1526,"currency":"EUR","value_usd":1645},"fwci":5.3726,"has_fulltext":true,"cited_by_count":68,"citation_normalized_percentile":{"value":0.95951435,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":89,"max":100},"biblio":{"volume":"60","issue":"9","first_page":"5736","last_page":"5749"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T12740","display_name":"Gait Recognition and Analysis","score":0.9983000159263611,"subfield":{"id":"https://openalex.org/subfields/2204","display_name":"Biomedical Engineering"},"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/T12740","display_name":"Gait Recognition and Analysis","score":0.9983000159263611,"subfield":{"id":"https://openalex.org/subfields/2204","display_name":"Biomedical Engineering"},"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/T11325","display_name":"Inertial Sensor and Navigation","score":0.9959999918937683,"subfield":{"id":"https://openalex.org/subfields/2202","display_name":"Aerospace Engineering"},"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/T10326","display_name":"Indoor and Outdoor Localization Technologies","score":0.9947999715805054,"subfield":{"id":"https://openalex.org/subfields/2208","display_name":"Electrical and Electronic Engineering"},"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/quaternion","display_name":"Quaternion","score":0.8990570902824402},{"id":"https://openalex.org/keywords/reproducing-kernel-hilbert-space","display_name":"Reproducing kernel Hilbert space","score":0.8591217994689941},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.724955677986145},{"id":"https://openalex.org/keywords/kernel","display_name":"Kernel (algebra)","score":0.6762030124664307},{"id":"https://openalex.org/keywords/hilbert-space","display_name":"Hilbert space","score":0.5235393047332764},{"id":"https://openalex.org/keywords/uniqueness","display_name":"Uniqueness","score":0.41967257857322693},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.39136800169944763},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.3740292191505432},{"id":"https://openalex.org/keywords/algebra-over-a-field","display_name":"Algebra over a field","score":0.34822791814804077},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.30589574575424194},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.27324628829956055},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.10173317790031433}],"concepts":[{"id":"https://openalex.org/C200127275","wikidata":"https://www.wikidata.org/wiki/Q173853","display_name":"Quaternion","level":2,"score":0.8990570902824402},{"id":"https://openalex.org/C80884492","wikidata":"https://www.wikidata.org/wiki/Q3345678","display_name":"Reproducing kernel Hilbert space","level":3,"score":0.8591217994689941},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.724955677986145},{"id":"https://openalex.org/C74193536","wikidata":"https://www.wikidata.org/wiki/Q574844","display_name":"Kernel (algebra)","level":2,"score":0.6762030124664307},{"id":"https://openalex.org/C62799726","wikidata":"https://www.wikidata.org/wiki/Q190056","display_name":"Hilbert space","level":2,"score":0.5235393047332764},{"id":"https://openalex.org/C2777021972","wikidata":"https://www.wikidata.org/wiki/Q22976830","display_name":"Uniqueness","level":2,"score":0.41967257857322693},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.39136800169944763},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.3740292191505432},{"id":"https://openalex.org/C136119220","wikidata":"https://www.wikidata.org/wiki/Q1000660","display_name":"Algebra over a field","level":2,"score":0.34822791814804077},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.30589574575424194},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.27324628829956055},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.10173317790031433}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1109/tit.2014.2333734","is_oa":true,"landing_page_url":"https://doi.org/10.1109/tit.2014.2333734","pdf_url":"https://ieeexplore.ieee.org/ielx7/18/6878505/06846303.pdf","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":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"IEEE Transactions on Information Theory","raw_type":"journal-article"}],"best_oa_location":{"id":"doi:10.1109/tit.2014.2333734","is_oa":true,"landing_page_url":"https://doi.org/10.1109/tit.2014.2333734","pdf_url":"https://ieeexplore.ieee.org/ielx7/18/6878505/06846303.pdf","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":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"IEEE Transactions on Information Theory","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[{"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"}],"funders":[{"id":"https://openalex.org/F4320334627","display_name":"Engineering and Physical Sciences Research Council","ror":"https://ror.org/0439y7842"}],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W2003891054.pdf","grobid_xml":"https://content.openalex.org/works/W2003891054.grobid-xml"},"referenced_works_count":64,"referenced_works":["https://openalex.org/W107110140","https://openalex.org/W142384176","https://openalex.org/W576428146","https://openalex.org/W632146489","https://openalex.org/W1526146785","https://openalex.org/W1530699444","https://openalex.org/W1560724230","https://openalex.org/W1563088657","https://openalex.org/W1636503046","https://openalex.org/W1755117326","https://openalex.org/W1862043434","https://openalex.org/W1879453234","https://openalex.org/W1974549223","https://openalex.org/W1975681165","https://openalex.org/W1982175152","https://openalex.org/W1986280275","https://openalex.org/W1987129039","https://openalex.org/W1989286669","https://openalex.org/W1992774275","https://openalex.org/W2001543294","https://openalex.org/W2014158063","https://openalex.org/W2027467343","https://openalex.org/W2031863147","https://openalex.org/W2040575700","https://openalex.org/W2062144489","https://openalex.org/W2069222481","https://openalex.org/W2087347434","https://openalex.org/W2090550497","https://openalex.org/W2098210464","https://openalex.org/W2098985067","https://openalex.org/W2109743529","https://openalex.org/W2114765937","https://openalex.org/W2121069620","https://openalex.org/W2126186068","https://openalex.org/W2134893878","https://openalex.org/W2136003379","https://openalex.org/W2137226992","https://openalex.org/W2139320579","https://openalex.org/W2141817643","https://openalex.org/W2143426320","https://openalex.org/W2144761325","https://openalex.org/W2153290280","https://openalex.org/W2154558852","https://openalex.org/W2157910179","https://openalex.org/W2161397505","https://openalex.org/W2165867509","https://openalex.org/W2167932108","https://openalex.org/W2170575305","https://openalex.org/W2329415387","https://openalex.org/W2476841360","https://openalex.org/W2537383145","https://openalex.org/W3141604666","https://openalex.org/W3183392806","https://openalex.org/W3195149063","https://openalex.org/W4230031267","https://openalex.org/W4232280359","https://openalex.org/W4292101331","https://openalex.org/W6631556622","https://openalex.org/W6636630834","https://openalex.org/W6637962894","https://openalex.org/W6639126400","https://openalex.org/W6680532697","https://openalex.org/W6680819885","https://openalex.org/W6728889068"],"related_works":["https://openalex.org/W2091669036","https://openalex.org/W2374023116","https://openalex.org/W4299118182","https://openalex.org/W4285290579","https://openalex.org/W1551479102","https://openalex.org/W2003891054","https://openalex.org/W2293919958","https://openalex.org/W3134083315","https://openalex.org/W1986863144","https://openalex.org/W2896419989"],"abstract_inverted_index":{"The":[0,74,101],"existence":[1],"and":[2,58,82,86,98,113,132,140],"uniqueness":[3],"conditions":[4],"of":[5,24,36,54,94,105],"quaternion":[6,38,50,79,138],"reproducing":[7],"kernel":[8,26,66,80,119],"Hilbert":[9,39],"spaces":[10],"(QRKHS)":[11],"are":[12,61],"established":[13,91],"in":[14,92],"order":[15],"to":[16,45,49],"provide":[17],"a":[18,33],"mathematical":[19],"foundation":[20],"for":[21],"the":[22,55,77,83,106,144],"development":[23],"quaternion-valued":[25,71,108],"learning":[27],"algorithms.":[28],"This":[29],"is":[30,89,116],"achieved":[31],"through":[32,118],"rigorous":[34],"account":[35],"left":[37],"spaces,":[40],"which":[41],"makes":[42],"it":[43],"possible":[44],"generalise":[46],"standard":[47],"RKHS":[48],"RKHS.":[51],"Quaternion":[52],"versions":[53],"Riesz":[56],"representation":[57],"Moore-Aronszajn":[59],"theorems":[60],"next":[62],"introduced,":[63],"thus":[64],"underpinning":[65],"estimation":[67,103],"algorithms":[68],"operating":[69],"on":[70,124],"feature":[72],"spaces.":[73],"difference":[75],"between":[76],"proposed":[78],"concept":[81],"existing":[84],"real":[85,125],"vector":[87],"approaches":[88],"also":[90],"terms":[93],"both":[95],"theoretical":[96],"advantages":[97],"computational":[99],"complexity.":[100],"enhanced":[102],"ability":[104],"so-introduced":[107],"kernels":[109,142],"over":[110],"their":[111],"real-":[112],"vector-valued":[114],"counterparts":[115],"validated":[117],"ridge":[120],"regression":[121],"applications.":[122],"Simulations":[123],"world":[126],"3D":[127],"inertial":[128],"body":[129],"sensor":[130],"data":[131],"nonlinear":[133],"channel":[134],"equalisation":[135],"using":[136],"novel":[137],"cubic":[139],"Gaussian":[141],"support":[143],"approach.":[145]},"counts_by_year":[{"year":2026,"cited_by_count":3},{"year":2025,"cited_by_count":4},{"year":2024,"cited_by_count":4},{"year":2023,"cited_by_count":3},{"year":2022,"cited_by_count":4},{"year":2021,"cited_by_count":4},{"year":2020,"cited_by_count":3},{"year":2019,"cited_by_count":10},{"year":2018,"cited_by_count":5},{"year":2017,"cited_by_count":9},{"year":2016,"cited_by_count":10},{"year":2015,"cited_by_count":8},{"year":2014,"cited_by_count":1}],"updated_date":"2026-03-27T14:29:43.386196","created_date":"2025-10-10T00:00:00"}