{"id":"https://openalex.org/W1992426838","doi":"https://doi.org/10.1137/110837711","title":"Third-Order Tensors as Operators on Matrices: A Theoretical and Computational Framework with Applications in Imaging","display_name":"Third-Order Tensors as Operators on Matrices: A Theoretical and Computational Framework with Applications in Imaging","publication_year":2013,"publication_date":"2013-01-01","ids":{"openalex":"https://openalex.org/W1992426838","doi":"https://doi.org/10.1137/110837711","mag":"1992426838"},"language":"en","primary_location":{"id":"doi:10.1137/110837711","is_oa":false,"landing_page_url":"https://doi.org/10.1137/110837711","pdf_url":null,"source":{"id":"https://openalex.org/S16958353","display_name":"SIAM Journal on Matrix Analysis and Applications","issn_l":"0895-4798","issn":["0895-4798","1095-7162"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320508","host_organization_name":"Society for Industrial and Applied Mathematics","host_organization_lineage":["https://openalex.org/P4310320508"],"host_organization_lineage_names":["Society for Industrial and Applied Mathematics"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"SIAM Journal on Matrix Analysis and Applications","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/A5041916316","display_name":"Misha E. Kilmer","orcid":"https://orcid.org/0000-0003-4249-4742"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Misha E. Kilmer","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"middle","author":{"id":"https://openalex.org/A5065566251","display_name":"Karen Braman","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Karen Braman","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"middle","author":{"id":"https://openalex.org/A5100619560","display_name":"Ning Hao","orcid":"https://orcid.org/0000-0002-3104-5370"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Ning Hao","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"last","author":{"id":"https://openalex.org/A5109424224","display_name":"Randy C. Hoover","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Randy C. Hoover","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":0,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":5.5435,"has_fulltext":false,"cited_by_count":1208,"citation_normalized_percentile":{"value":0.96892655,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":94,"max":100},"biblio":{"volume":"34","issue":"1","first_page":"148","last_page":"172"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T12303","display_name":"Tensor decomposition and applications","score":1.0,"subfield":{"id":"https://openalex.org/subfields/2605","display_name":"Computational 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/T12303","display_name":"Tensor decomposition and applications","score":1.0,"subfield":{"id":"https://openalex.org/subfields/2605","display_name":"Computational 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":0.9901000261306763,"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/T10500","display_name":"Sparse and Compressive Sensing Techniques","score":0.9825999736785889,"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/mathematics","display_name":"Mathematics","score":0.7612800002098083},{"id":"https://openalex.org/keywords/linear-algebra","display_name":"Linear algebra","score":0.6076390743255615},{"id":"https://openalex.org/keywords/krylov-subspace","display_name":"Krylov subspace","score":0.5760250091552734},{"id":"https://openalex.org/keywords/tensor-product","display_name":"Tensor product","score":0.5552534461021423},{"id":"https://openalex.org/keywords/algebra-over-a-field","display_name":"Algebra over a field","score":0.5441102385520935},{"id":"https://openalex.org/keywords/orthogonalization","display_name":"Orthogonalization","score":0.5173258781433105},{"id":"https://openalex.org/keywords/tensor","display_name":"Tensor (intrinsic definition)","score":0.5063053369522095},{"id":"https://openalex.org/keywords/orthogonality","display_name":"Orthogonality","score":0.5044375658035278},{"id":"https://openalex.org/keywords/kernel","display_name":"Kernel (algebra)","score":0.4927099049091339},{"id":"https://openalex.org/keywords/bilinear-form","display_name":"Bilinear form","score":0.4752350151538849},{"id":"https://openalex.org/keywords/numerical-linear-algebra","display_name":"Numerical linear algebra","score":0.4654218852519989},{"id":"https://openalex.org/keywords/outer-product","display_name":"Outer product","score":0.45377853512763977},{"id":"https://openalex.org/keywords/deblurring","display_name":"Deblurring","score":0.4529196619987488},{"id":"https://openalex.org/keywords/orthogonal-complement","display_name":"Orthogonal complement","score":0.4472973644733429},{"id":"https://openalex.org/keywords/matrix","display_name":"Matrix (chemical analysis)","score":0.41681498289108276},{"id":"https://openalex.org/keywords/bilinear-interpolation","display_name":"Bilinear interpolation","score":0.41317445039749146},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.3960148096084595},{"id":"https://openalex.org/keywords/linear-system","display_name":"Linear system","score":0.36333733797073364},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.3425472676753998},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.3256748914718628},{"id":"https://openalex.org/keywords/image-processing","display_name":"Image processing","score":0.3014594316482544},{"id":"https://openalex.org/keywords/subspace-topology","display_name":"Subspace topology","score":0.25690844655036926},{"id":"https://openalex.org/keywords/image","display_name":"Image (mathematics)","score":0.24865660071372986},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.1902475357055664},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.13191288709640503},{"id":"https://openalex.org/keywords/artificial-intelligence","display_name":"Artificial intelligence","score":0.10230904817581177},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.09934777021408081}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7612800002098083},{"id":"https://openalex.org/C139352143","wikidata":"https://www.wikidata.org/wiki/Q82571","display_name":"Linear algebra","level":2,"score":0.6076390743255615},{"id":"https://openalex.org/C147060835","wikidata":"https://www.wikidata.org/wiki/Q1757151","display_name":"Krylov subspace","level":3,"score":0.5760250091552734},{"id":"https://openalex.org/C51255310","wikidata":"https://www.wikidata.org/wiki/Q1163016","display_name":"Tensor product","level":2,"score":0.5552534461021423},{"id":"https://openalex.org/C136119220","wikidata":"https://www.wikidata.org/wiki/Q1000660","display_name":"Algebra over a field","level":2,"score":0.5441102385520935},{"id":"https://openalex.org/C47559304","wikidata":"https://www.wikidata.org/wiki/Q1702189","display_name":"Orthogonalization","level":2,"score":0.5173258781433105},{"id":"https://openalex.org/C155281189","wikidata":"https://www.wikidata.org/wiki/Q3518150","display_name":"Tensor (intrinsic definition)","level":2,"score":0.5063053369522095},{"id":"https://openalex.org/C17137986","wikidata":"https://www.wikidata.org/wiki/Q215067","display_name":"Orthogonality","level":2,"score":0.5044375658035278},{"id":"https://openalex.org/C74193536","wikidata":"https://www.wikidata.org/wiki/Q574844","display_name":"Kernel (algebra)","level":2,"score":0.4927099049091339},{"id":"https://openalex.org/C8828549","wikidata":"https://www.wikidata.org/wiki/Q837924","display_name":"Bilinear form","level":2,"score":0.4752350151538849},{"id":"https://openalex.org/C163834973","wikidata":"https://www.wikidata.org/wiki/Q2004891","display_name":"Numerical linear algebra","level":3,"score":0.4654218852519989},{"id":"https://openalex.org/C180623205","wikidata":"https://www.wikidata.org/wiki/Q1268589","display_name":"Outer product","level":3,"score":0.45377853512763977},{"id":"https://openalex.org/C2777693668","wikidata":"https://www.wikidata.org/wiki/Q25053743","display_name":"Deblurring","level":5,"score":0.4529196619987488},{"id":"https://openalex.org/C154399039","wikidata":"https://www.wikidata.org/wiki/Q1780921","display_name":"Orthogonal complement","level":3,"score":0.4472973644733429},{"id":"https://openalex.org/C106487976","wikidata":"https://www.wikidata.org/wiki/Q685816","display_name":"Matrix (chemical analysis)","level":2,"score":0.41681498289108276},{"id":"https://openalex.org/C205203396","wikidata":"https://www.wikidata.org/wiki/Q612143","display_name":"Bilinear interpolation","level":2,"score":0.41317445039749146},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.3960148096084595},{"id":"https://openalex.org/C6802819","wikidata":"https://www.wikidata.org/wiki/Q1072174","display_name":"Linear system","level":2,"score":0.36333733797073364},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.3425472676753998},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.3256748914718628},{"id":"https://openalex.org/C9417928","wikidata":"https://www.wikidata.org/wiki/Q1070689","display_name":"Image processing","level":3,"score":0.3014594316482544},{"id":"https://openalex.org/C32834561","wikidata":"https://www.wikidata.org/wiki/Q660730","display_name":"Subspace topology","level":2,"score":0.25690844655036926},{"id":"https://openalex.org/C115961682","wikidata":"https://www.wikidata.org/wiki/Q860623","display_name":"Image (mathematics)","level":2,"score":0.24865660071372986},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.1902475357055664},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.13191288709640503},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.10230904817581177},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.09934777021408081},{"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/C159985019","wikidata":"https://www.wikidata.org/wiki/Q181790","display_name":"Composite material","level":1,"score":0.0},{"id":"https://openalex.org/C106430172","wikidata":"https://www.wikidata.org/wiki/Q6002272","display_name":"Image restoration","level":4,"score":0.0},{"id":"https://openalex.org/C105795698","wikidata":"https://www.wikidata.org/wiki/Q12483","display_name":"Statistics","level":1,"score":0.0}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.1137/110837711","is_oa":false,"landing_page_url":"https://doi.org/10.1137/110837711","pdf_url":null,"source":{"id":"https://openalex.org/S16958353","display_name":"SIAM Journal on Matrix Analysis and Applications","issn_l":"0895-4798","issn":["0895-4798","1095-7162"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320508","host_organization_name":"Society for Industrial and Applied Mathematics","host_organization_lineage":["https://openalex.org/P4310320508"],"host_organization_lineage_names":["Society for Industrial and Applied Mathematics"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"SIAM Journal on Matrix Analysis and Applications","raw_type":"journal-article"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.386.1248","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.386.1248","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://www.cs.tufts.edu/tech_reports/reports/2011-01/report.pdf","raw_type":"text"}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":26,"referenced_works":["https://openalex.org/W1576347883","https://openalex.org/W1919986261","https://openalex.org/W1963826206","https://openalex.org/W1970563156","https://openalex.org/W2000215628","https://openalex.org/W2006808740","https://openalex.org/W2013912476","https://openalex.org/W2024165284","https://openalex.org/W2024266080","https://openalex.org/W2043571470","https://openalex.org/W2049878767","https://openalex.org/W2057538344","https://openalex.org/W2070028074","https://openalex.org/W2098693229","https://openalex.org/W2101641981","https://openalex.org/W2104027686","https://openalex.org/W2113185518","https://openalex.org/W2113213865","https://openalex.org/W2113722075","https://openalex.org/W2121739212","https://openalex.org/W2145019972","https://openalex.org/W2149183719","https://openalex.org/W2160551639","https://openalex.org/W2162521467","https://openalex.org/W2162604195","https://openalex.org/W2522038268"],"related_works":["https://openalex.org/W1991111524","https://openalex.org/W2023964056","https://openalex.org/W2015109384","https://openalex.org/W126085529","https://openalex.org/W2150643146","https://openalex.org/W2886879564","https://openalex.org/W2592290387","https://openalex.org/W4299440441","https://openalex.org/W2560154353","https://openalex.org/W4311726891"],"abstract_inverted_index":{"Recent":[0],"work":[1],"by":[2,98,162],"Kilmer":[3],"and":[4,13,64,77,81,92,102,105,151,176,204],"Martin":[5],"[Linear":[6,15],"Algebra":[7,16],"Appl.,":[8,17],"435":[9],"(2011),":[10],"pp.":[11,20],"641--658]":[12],"Braman":[14],"433":[18],"(2010),":[19],"1241--1253]":[21],"provides":[22],"a":[23,52,95,99,196,206],"setting":[24],"in":[25,158,187],"which":[26,58,65],"the":[27,56,83,90,103,106,109,119,146,159,171,191],"familiar":[28],"tools":[29],"of":[30,71,79,85,94,108,111,121,136,148],"linear":[31],"algebra":[32],"can":[33],"be":[34],"extended":[35],"to":[36,67,88,118,144,154],"better":[37],"understand":[38],"third-order":[39,100],"tensors.":[40],"Continuing":[41],"along":[42],"this":[43,45],"vein,":[44],"paper":[46],"investigates":[47],"further":[48],"implications":[49],"including":[50],"(1)":[51],"bilinear":[53],"operator":[54],"on":[55],"matrices":[57],"is":[59],"nearly":[60],"an":[61,127,133],"inner":[62],"product":[63],"leads":[66,142],"definitions":[68],"for":[69,131,165,201],"length":[70],"matrices,":[72,76,80],"angle":[73],"between":[74],"two":[75,185],"orthogonality":[78],"(2)":[82],"use":[84],"t-linear":[86],"combinations":[87],"characterize":[89],"range":[91],"kernel":[93],"mapping":[96],"defined":[97,157],"tensor":[101,149,155],"t-product":[104],"quantification":[107],"dimensions":[110],"those":[112],"sets.":[113],"These":[114],"theoretical":[115,139],"results":[116],"lead":[117],"study":[120],"orthogonal":[122,134,192],"projections":[123],"as":[124,126,170],"well":[125],"effective":[128],"Gram--Schmidt":[129],"process":[130],"producing":[132],"basis":[135],"matrices.":[137],"The":[138],"framework":[140],"also":[141],"us":[143],"consider":[145],"notion":[147],"polynomials":[150],"their":[152],"relation":[153],"eigentuples":[156],"recent":[160],"article":[161],"Braman.":[163],"Implications":[164],"extending":[166],"basic":[167],"algorithms":[168],"such":[169],"power":[172],"method,":[173],"QR":[174],"iteration,":[175],"Krylov":[177],"subspace":[178],"methods":[179],"are":[180],"discussed.":[181],"We":[182],"conclude":[183],"with":[184],"examples":[186],"image":[188,208],"processing:":[189],"using":[190],"elements":[193],"generated":[194],"via":[195],"Golub--Kahan":[197],"iterative":[198],"bidiagonalization":[199],"scheme":[200],"object":[202],"recognition":[203],"solving":[205],"regularized":[207],"deblurring":[209],"problem.":[210]},"counts_by_year":[{"year":2026,"cited_by_count":77},{"year":2025,"cited_by_count":174},{"year":2024,"cited_by_count":163},{"year":2023,"cited_by_count":147},{"year":2022,"cited_by_count":154},{"year":2021,"cited_by_count":140},{"year":2020,"cited_by_count":113},{"year":2019,"cited_by_count":90},{"year":2018,"cited_by_count":58},{"year":2017,"cited_by_count":38},{"year":2016,"cited_by_count":25},{"year":2015,"cited_by_count":16},{"year":2014,"cited_by_count":5},{"year":2013,"cited_by_count":6},{"year":2012,"cited_by_count":2}],"updated_date":"2026-07-18T07:39:51.176621","created_date":"2025-10-10T00:00:00"}