{"id":"https://openalex.org/W2590939752","doi":"https://doi.org/10.1137/16m110770x","title":"On the Largest Multilinear Singular Values of Higher-Order Tensors","display_name":"On the Largest Multilinear Singular Values of Higher-Order Tensors","publication_year":2017,"publication_date":"2017-01-01","ids":{"openalex":"https://openalex.org/W2590939752","doi":"https://doi.org/10.1137/16m110770x","mag":"2590939752"},"language":"en","primary_location":{"id":"doi:10.1137/16m110770x","is_oa":false,"landing_page_url":"https://doi.org/10.1137/16m110770x","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":["arxiv","crossref"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://arxiv.org/pdf/1612.03751","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5021106574","display_name":"Ignat Domanov","orcid":"https://orcid.org/0000-0001-7605-4177"},"institutions":[{"id":"https://openalex.org/I99464096","display_name":"KU Leuven","ror":"https://ror.org/05f950310","country_code":"BE","type":"education","lineage":["https://openalex.org/I99464096"]}],"countries":["BE"],"is_corresponding":false,"raw_author_name":"Ignat Domanov","raw_affiliation_strings":["Group Science, Engineering and Technology, KU Leuven -Kulak, E. Sabbelaan 53, 8500 Kortrijk, Belgium"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Group Science, Engineering and Technology, KU Leuven -Kulak, E. Sabbelaan 53, 8500 Kortrijk, Belgium","institution_ids":["https://openalex.org/I99464096"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5084820167","display_name":"Alwin Stegeman","orcid":"https://orcid.org/0000-0003-3385-2560"},"institutions":[{"id":"https://openalex.org/I99464096","display_name":"KU Leuven","ror":"https://ror.org/05f950310","country_code":"BE","type":"education","lineage":["https://openalex.org/I99464096"]}],"countries":["BE"],"is_corresponding":false,"raw_author_name":"Alwin Stegeman","raw_affiliation_strings":["Group Science, Engineering and Technology, KU Leuven -Kulak, E. Sabbelaan 53, 8500 Kortrijk, Belgium"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Group Science, Engineering and Technology, KU Leuven -Kulak, E. Sabbelaan 53, 8500 Kortrijk, Belgium","institution_ids":["https://openalex.org/I99464096"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5009006026","display_name":"Lieven De Lathauwer","orcid":"https://orcid.org/0000-0001-5562-5014"},"institutions":[{"id":"https://openalex.org/I99464096","display_name":"KU Leuven","ror":"https://ror.org/05f950310","country_code":"BE","type":"education","lineage":["https://openalex.org/I99464096"]}],"countries":["BE"],"is_corresponding":false,"raw_author_name":"Lieven De Lathauwer","raw_affiliation_strings":["Group Science, Engineering and Technology, KU Leuven -Kulak, E. Sabbelaan 53, 8500 Kortrijk, Belgium"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Group Science, Engineering and Technology, KU Leuven -Kulak, E. Sabbelaan 53, 8500 Kortrijk, Belgium","institution_ids":["https://openalex.org/I99464096"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":[],"corresponding_institution_ids":["https://openalex.org/I99464096"],"apc_list":null,"apc_paid":null,"fwci":0.442,"has_fulltext":false,"cited_by_count":4,"citation_normalized_percentile":{"value":0.54063018,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":96},"biblio":{"volume":"38","issue":"4","first_page":"1434","last_page":"1453"},"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/T11304","display_name":"Advanced Neuroimaging Techniques and Applications","score":0.925000011920929,"subfield":{"id":"https://openalex.org/subfields/2741","display_name":"Radiology, Nuclear Medicine and Imaging"},"field":{"id":"https://openalex.org/fields/27","display_name":"Medicine"},"domain":{"id":"https://openalex.org/domains/4","display_name":"Health Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/multilinear-map","display_name":"Multilinear map","score":0.9173834323883057},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.7838340401649475},{"id":"https://openalex.org/keywords/multilinear-algebra","display_name":"Multilinear algebra","score":0.727541446685791},{"id":"https://openalex.org/keywords/order","display_name":"Order (exchange)","score":0.6775594353675842},{"id":"https://openalex.org/keywords/singular-value","display_name":"Singular value","score":0.5473775863647461},{"id":"https://openalex.org/keywords/tensor","display_name":"Tensor (intrinsic definition)","score":0.4863153398036957},{"id":"https://openalex.org/keywords/algebra-over-a-field","display_name":"Algebra over a field","score":0.4409915804862976},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.43002626299858093},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.3284534811973572},{"id":"https://openalex.org/keywords/eigenvalues-and-eigenvectors","display_name":"Eigenvalues and eigenvectors","score":0.18668660521507263}],"concepts":[{"id":"https://openalex.org/C84392682","wikidata":"https://www.wikidata.org/wiki/Q1952404","display_name":"Multilinear map","level":2,"score":0.9173834323883057},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7838340401649475},{"id":"https://openalex.org/C60321788","wikidata":"https://www.wikidata.org/wiki/Q1197190","display_name":"Multilinear algebra","level":5,"score":0.727541446685791},{"id":"https://openalex.org/C182306322","wikidata":"https://www.wikidata.org/wiki/Q1779371","display_name":"Order (exchange)","level":2,"score":0.6775594353675842},{"id":"https://openalex.org/C109282560","wikidata":"https://www.wikidata.org/wiki/Q4166054","display_name":"Singular value","level":3,"score":0.5473775863647461},{"id":"https://openalex.org/C155281189","wikidata":"https://www.wikidata.org/wiki/Q3518150","display_name":"Tensor (intrinsic definition)","level":2,"score":0.4863153398036957},{"id":"https://openalex.org/C136119220","wikidata":"https://www.wikidata.org/wiki/Q1000660","display_name":"Algebra over a field","level":2,"score":0.4409915804862976},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.43002626299858093},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.3284534811973572},{"id":"https://openalex.org/C158693339","wikidata":"https://www.wikidata.org/wiki/Q190524","display_name":"Eigenvalues and eigenvectors","level":2,"score":0.18668660521507263},{"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/C169171071","wikidata":"https://www.wikidata.org/wiki/Q5449241","display_name":"Filtered algebra","level":3,"score":0.0},{"id":"https://openalex.org/C138354692","wikidata":"https://www.wikidata.org/wiki/Q1231309","display_name":"Division algebra","level":4,"score":0.0},{"id":"https://openalex.org/C162324750","wikidata":"https://www.wikidata.org/wiki/Q8134","display_name":"Economics","level":0,"score":0.0},{"id":"https://openalex.org/C10138342","wikidata":"https://www.wikidata.org/wiki/Q43015","display_name":"Finance","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}],"mesh":[],"locations_count":3,"locations":[{"id":"doi:10.1137/16m110770x","is_oa":false,"landing_page_url":"https://doi.org/10.1137/16m110770x","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:arXiv.org:1612.03751","is_oa":true,"landing_page_url":"http://arxiv.org/abs/1612.03751","pdf_url":"https://arxiv.org/pdf/1612.03751","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:lirias2repo.kuleuven.be:123456789/588422","is_oa":false,"landing_page_url":"https://lirias.kuleuven.be/bitstream/123456789/588422/3/16-167.pdf","pdf_url":null,"source":{"id":"https://openalex.org/S7407055369","display_name":"Lirias","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"SIAM Journal on Matrix Analysis and Applications, vol. 38 (4), Art.No. 4, (1434-1453)","raw_type":"info:eu-repo/semantics/article"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:1612.03751","is_oa":true,"landing_page_url":"http://arxiv.org/abs/1612.03751","pdf_url":"https://arxiv.org/pdf/1612.03751","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/G1198938275","display_name":null,"funder_award_id":"G.0881.14N","funder_id":"https://openalex.org/F4320321730","funder_display_name":"Fonds Wetenschappelijk Onderzoek"},{"id":"https://openalex.org/G8184458352","display_name":null,"funder_award_id":"G.0830.14N","funder_id":"https://openalex.org/F4320321730","funder_display_name":"Fonds Wetenschappelijk Onderzoek"},{"id":"https://openalex.org/G894225744","display_name":"Biomedical Data Fusion using Tensor based Blind Source Separation","funder_award_id":"339804","funder_id":"https://openalex.org/F4320338352","funder_display_name":"FP7 Ideas: European Research Council"}],"funders":[{"id":"https://openalex.org/F4320321730","display_name":"Fonds Wetenschappelijk Onderzoek","ror":"https://ror.org/03qtxy027"},{"id":"https://openalex.org/F4320322681","display_name":"Onderzoeksraad, KU Leuven","ror":"https://ror.org/05f950310"},{"id":"https://openalex.org/F4320338352","display_name":"FP7 Ideas: European Research Council","ror":"https://ror.org/0472cxd90"}],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":21,"referenced_works":["https://openalex.org/W227277429","https://openalex.org/W1664429083","https://openalex.org/W1963826206","https://openalex.org/W1978498923","https://openalex.org/W2013912476","https://openalex.org/W2019669853","https://openalex.org/W2024063662","https://openalex.org/W2024165284","https://openalex.org/W2024182011","https://openalex.org/W2056486673","https://openalex.org/W2119412403","https://openalex.org/W2120092158","https://openalex.org/W2161436879","https://openalex.org/W2165782098","https://openalex.org/W2469230926","https://openalex.org/W2554237793","https://openalex.org/W2583133131","https://openalex.org/W2587191609","https://openalex.org/W2962955087","https://openalex.org/W2980509148","https://openalex.org/W4246668994"],"related_works":["https://openalex.org/W2242909499","https://openalex.org/W2405717268","https://openalex.org/W2946637708","https://openalex.org/W3193555930","https://openalex.org/W1572092093","https://openalex.org/W1570745949","https://openalex.org/W2893527900","https://openalex.org/W4235893778","https://openalex.org/W2922481674","https://openalex.org/W2535617683"],"abstract_inverted_index":{"Let":[0],"$\\sigma_n$":[1],"denote":[2],"the":[3,31,44,77,114,159,178],"largest":[4,98],"mode-$n$":[5],"multilinear":[6,99,132,135,147,182],"singular":[7,100,148,183],"value":[8],"of":[9,131,144,158,161,174,181],"an":[10,92],"$I_1\\times\\dots":[11],"\\times":[12],"I_N$":[13],"tensor":[14,96],"$\\mathcal":[15,123],"T$.":[16],"We":[17,34,109,150],"prove":[18],"that":[19,37,58,112],"$":[20,27,60],"\\sigma_1^2+\\dots+\\sigma_{n-1}^2+\\sigma_{n+1}^2+\\dots+\\sigma_{N}^2\\leq":[21],"(N-2)\\|\\mathcal":[22],"T\\|^2":[23,63,68,73,118],"+":[24,64,69,74,119],"\\sigma_n^2,\\":[25],"n=1,\\dots,N,":[26],"where":[28],"$\\|\\cdot\\|$":[29],"denotes":[30],"Frobenius":[32],"norm.":[33],"also":[35,110],"show":[36,111],"at":[38,169],"least":[39,170],"for":[40,52],"third-order":[41],"cubic":[42],"tensors":[43,137],"inverse":[45],"problem":[46],"always":[47,90],"has":[48],"a":[49,129,141,152,171],"solution.":[50],"Namely,":[51],"each":[53],"$\\sigma_1$,":[54,105],"$\\sigma_2$,":[55,106],"and":[56,76,107,134,138,156],"$\\sigma_3$":[57],"satisfy":[59],"\\sigma_1^2+\\sigma_2^2\\leq":[61],"\\|\\mathcal":[62,67,72,117],"\\sigma_3^2,\\":[65],"\\sigma_1^2+\\sigma_3^2\\leq":[66],"\\sigma_2^2,\\":[70],"\\sigma_2^2+\\sigma_3^2\\leq":[71],"\\sigma_1^2,$":[75],"trivial":[78],"inequalities":[79],"$\\sigma_1\\geq":[80],"\\frac{1}{\\sqrt{n}}\\|\\mathcal":[81,84,87],"T\\|$,":[82,85,88],"$\\sigma_2\\geq":[83],"$\\sigma_3\\geq":[86],"there":[89],"exists":[91],"$n\\times":[93],"n\\times":[94],"n$":[95],"whose":[97],"values":[101,184],"are":[102,185],"equal":[103,127],"to":[104,128,167],"$\\sigma_3$.":[108],"if":[113],"equality":[115],"$\\sigma_1^2+\\sigma_2^2=":[116],"\\sigma_3^2$":[120],"holds,":[121],"then":[122],"T$":[124],"is":[125],"necessarily":[126],"sum":[130,160],"rank-$(L_1,1,L_1)$":[133],"rank-$(1,L_2,L_2)$":[136],"we":[139],"give":[140,168],"complete":[142],"description":[143],"all":[145],"its":[146],"values.":[149],"establish":[151],"connection":[153],"with":[154],"honeycombs":[155],"eigenvalues":[157],"two":[162],"Hermitian":[163],"matrices.":[164],"This":[165],"seems":[166],"partial":[172],"explanation":[173],"why":[175],"results":[176],"on":[177],"joint":[179],"distribution":[180],"scarce.":[186]},"counts_by_year":[{"year":2020,"cited_by_count":1},{"year":2019,"cited_by_count":2},{"year":2018,"cited_by_count":1}],"updated_date":"2026-07-12T07:33:03.550515","created_date":"2025-10-10T00:00:00"}