{"id":"https://openalex.org/W2080883935","doi":"https://doi.org/10.1137/s1064827501396464","title":"The Procrustes Problem for Orthogonal Kronecker Products","display_name":"The Procrustes Problem for Orthogonal Kronecker Products","publication_year":2003,"publication_date":"2003-01-01","ids":{"openalex":"https://openalex.org/W2080883935","doi":"https://doi.org/10.1137/s1064827501396464","mag":"2080883935"},"language":"en","primary_location":{"id":"doi:10.1137/s1064827501396464","is_oa":false,"landing_page_url":"https://doi.org/10.1137/s1064827501396464","pdf_url":null,"source":{"id":"https://openalex.org/S165512578","display_name":"SIAM Journal on Scientific Computing","issn_l":"1064-8275","issn":["1064-8275","1095-7197"],"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 Scientific Computing","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/A5003056989","display_name":"Adam W. Boja\u0144czyk","orcid":"https://orcid.org/0000-0002-5657-1465"},"institutions":[{"id":"https://openalex.org/I205783295","display_name":"Cornell University","ror":"https://ror.org/05bnh6r87","country_code":"US","type":"education","lineage":["https://openalex.org/I205783295"]}],"countries":["US"],"is_corresponding":true,"raw_author_name":"Adam W. Bojanczyk","raw_affiliation_strings":["Cornell University"],"affiliations":[{"raw_affiliation_string":"Cornell University","institution_ids":["https://openalex.org/I205783295"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5065074557","display_name":"Adam Lutoborski","orcid":null},"institutions":[{"id":"https://openalex.org/I530967","display_name":"Toronto Metropolitan University","ror":"https://ror.org/05g13zd79","country_code":"CA","type":"education","lineage":["https://openalex.org/I530967"]}],"countries":["CA"],"is_corresponding":false,"raw_author_name":"Adam Lutoborski","raw_affiliation_strings":["Department of Mathematics"],"affiliations":[{"raw_affiliation_string":"Department of Mathematics","institution_ids":["https://openalex.org/I530967"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":2,"corresponding_author_ids":["https://openalex.org/A5003056989"],"corresponding_institution_ids":["https://openalex.org/I205783295"],"apc_list":null,"apc_paid":null,"fwci":0.0,"has_fulltext":false,"cited_by_count":3,"citation_normalized_percentile":{"value":0.16679749,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":90,"max":94},"biblio":{"volume":"25","issue":"1","first_page":"148","last_page":"163"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10792","display_name":"Matrix Theory and Algorithms","score":0.9958999752998352,"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"}},"topics":[{"id":"https://openalex.org/T10792","display_name":"Matrix Theory and Algorithms","score":0.9958999752998352,"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/T11447","display_name":"Blind Source Separation Techniques","score":0.9878000020980835,"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/T10963","display_name":"Advanced Optimization Algorithms Research","score":0.9848999977111816,"subfield":{"id":"https://openalex.org/subfields/2612","display_name":"Numerical Analysis"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/kronecker-product","display_name":"Kronecker product","score":0.8617559671401978},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.8584396839141846},{"id":"https://openalex.org/keywords/kronecker-delta","display_name":"Kronecker delta","score":0.8453794717788696},{"id":"https://openalex.org/keywords/matrix-norm","display_name":"Matrix norm","score":0.621565580368042},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.551697313785553},{"id":"https://openalex.org/keywords/orthogonal-matrix","display_name":"Orthogonal matrix","score":0.5290101170539856},{"id":"https://openalex.org/keywords/product","display_name":"Product (mathematics)","score":0.5241996049880981},{"id":"https://openalex.org/keywords/norm","display_name":"Norm (philosophy)","score":0.455355167388916},{"id":"https://openalex.org/keywords/procrustes-analysis","display_name":"Procrustes analysis","score":0.4238443374633789},{"id":"https://openalex.org/keywords/matrix","display_name":"Matrix (chemical analysis)","score":0.42099520564079285},{"id":"https://openalex.org/keywords/convergence","display_name":"Convergence (economics)","score":0.41033440828323364},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.3235176205635071},{"id":"https://openalex.org/keywords/algebra-over-a-field","display_name":"Algebra over a field","score":0.3209701180458069},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.28374844789505005},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.15162497758865356},{"id":"https://openalex.org/keywords/orthogonal-basis","display_name":"Orthogonal basis","score":0.12951073050498962},{"id":"https://openalex.org/keywords/eigenvalues-and-eigenvectors","display_name":"Eigenvalues and eigenvectors","score":0.07339602708816528}],"concepts":[{"id":"https://openalex.org/C46030957","wikidata":"https://www.wikidata.org/wiki/Q1238125","display_name":"Kronecker product","level":3,"score":0.8617559671401978},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8584396839141846},{"id":"https://openalex.org/C39482219","wikidata":"https://www.wikidata.org/wiki/Q192826","display_name":"Kronecker delta","level":2,"score":0.8453794717788696},{"id":"https://openalex.org/C92207270","wikidata":"https://www.wikidata.org/wiki/Q939253","display_name":"Matrix norm","level":3,"score":0.621565580368042},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.551697313785553},{"id":"https://openalex.org/C44292817","wikidata":"https://www.wikidata.org/wiki/Q333871","display_name":"Orthogonal matrix","level":3,"score":0.5290101170539856},{"id":"https://openalex.org/C90673727","wikidata":"https://www.wikidata.org/wiki/Q901718","display_name":"Product (mathematics)","level":2,"score":0.5241996049880981},{"id":"https://openalex.org/C191795146","wikidata":"https://www.wikidata.org/wiki/Q3878446","display_name":"Norm (philosophy)","level":2,"score":0.455355167388916},{"id":"https://openalex.org/C2778585274","wikidata":"https://www.wikidata.org/wiki/Q2845240","display_name":"Procrustes analysis","level":2,"score":0.4238443374633789},{"id":"https://openalex.org/C106487976","wikidata":"https://www.wikidata.org/wiki/Q685816","display_name":"Matrix (chemical analysis)","level":2,"score":0.42099520564079285},{"id":"https://openalex.org/C2777303404","wikidata":"https://www.wikidata.org/wiki/Q759757","display_name":"Convergence (economics)","level":2,"score":0.41033440828323364},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.3235176205635071},{"id":"https://openalex.org/C136119220","wikidata":"https://www.wikidata.org/wiki/Q1000660","display_name":"Algebra over a field","level":2,"score":0.3209701180458069},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.28374844789505005},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.15162497758865356},{"id":"https://openalex.org/C187064257","wikidata":"https://www.wikidata.org/wiki/Q3306808","display_name":"Orthogonal basis","level":2,"score":0.12951073050498962},{"id":"https://openalex.org/C158693339","wikidata":"https://www.wikidata.org/wiki/Q190524","display_name":"Eigenvalues and eigenvectors","level":2,"score":0.07339602708816528},{"id":"https://openalex.org/C17744445","wikidata":"https://www.wikidata.org/wiki/Q36442","display_name":"Political science","level":0,"score":0.0},{"id":"https://openalex.org/C199539241","wikidata":"https://www.wikidata.org/wiki/Q7748","display_name":"Law","level":1,"score":0.0},{"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/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"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/C159985019","wikidata":"https://www.wikidata.org/wiki/Q181790","display_name":"Composite material","level":1,"score":0.0},{"id":"https://openalex.org/C50522688","wikidata":"https://www.wikidata.org/wiki/Q189833","display_name":"Economic growth","level":1,"score":0.0},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1137/s1064827501396464","is_oa":false,"landing_page_url":"https://doi.org/10.1137/s1064827501396464","pdf_url":null,"source":{"id":"https://openalex.org/S165512578","display_name":"SIAM Journal on Scientific Computing","issn_l":"1064-8275","issn":["1064-8275","1095-7197"],"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 Scientific Computing","raw_type":"journal-article"}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":10,"referenced_works":["https://openalex.org/W1968969130","https://openalex.org/W1976423768","https://openalex.org/W1981745143","https://openalex.org/W1988026710","https://openalex.org/W2011994564","https://openalex.org/W2045512849","https://openalex.org/W2073337492","https://openalex.org/W2076390783","https://openalex.org/W2105186495","https://openalex.org/W2136489582"],"related_works":["https://openalex.org/W123417539","https://openalex.org/W2079407403","https://openalex.org/W2060299328","https://openalex.org/W4375957393","https://openalex.org/W4233239985","https://openalex.org/W2117336295","https://openalex.org/W3102722572","https://openalex.org/W2090103374","https://openalex.org/W2080883935","https://openalex.org/W194966468"],"abstract_inverted_index":{"The":[0],"Procrustes":[1],"problem":[2],"for":[3,26,44,48],"orthogonal":[4,28],"Kronecker":[5,54],"products":[6],"is":[7],"considered.":[8],"Given":[9],"matrices":[10,30],"$A\\in":[11],"\\mathbb{R}^{n^2\\times":[12,15],"k^2}$,":[13],"$T\\in":[14],"n^2},$":[16],"$n\\ge":[17],"k$,":[18],"we":[19],"minimize":[20],"the":[21,49],"Frobenius":[22],"norm":[23],"$\\|T(Q\\otimes":[24],"Q)-A\\|$":[25],"all":[27],"Stiefel":[29],"$Q\\in":[31],"\\mathbb{R}^{n\\times":[32],"k}$,":[33],"$Q^TQ=I_{k}$.":[34],"We":[35],"introduce":[36],"and":[37,40,53],"implement":[38],"left":[39],"right":[41],"relaxation":[42],"methods":[43],"minimization.":[45],"Numerical":[46],"results":[47],"data":[50],"in":[51],"general":[52],"product":[55],"form":[56],"are":[57],"given":[58],"to":[59],"illustrate":[60],"convergence":[61],"of":[62],"both":[63],"methods.":[64]},"counts_by_year":[{"year":2016,"cited_by_count":1}],"updated_date":"2025-11-06T03:46:38.306776","created_date":"2025-10-10T00:00:00"}
