{"id":"https://openalex.org/W4379513944","doi":"https://doi.org/10.1137/22m1489125","title":"Euclidean Representation of Low-Rank Matrices and Its Geometric Properties","display_name":"Euclidean Representation of Low-Rank Matrices and Its Geometric Properties","publication_year":2023,"publication_date":"2023-06-06","ids":{"openalex":"https://openalex.org/W4379513944","doi":"https://doi.org/10.1137/22m1489125"},"language":"en","primary_location":{"id":"doi:10.1137/22m1489125","is_oa":false,"landing_page_url":"http://dx.doi.org/10.1137/22m1489125","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/A5044364993","display_name":"Fangzheng Xie","orcid":"https://orcid.org/0000-0003-2436-9542"},"institutions":[{"id":"https://openalex.org/I4210119109","display_name":"Indiana University Bloomington","ror":"https://ror.org/02k40bc56","country_code":"US","type":"education","lineage":["https://openalex.org/I4210119109","https://openalex.org/I592451"]}],"countries":["US"],"is_corresponding":true,"raw_author_name":"Fangzheng Xie","raw_affiliation_strings":["Department of Statistics, Indiana University, Bloomington, IN 47408 USA"],"raw_orcid":"https://orcid.org/0000-0003-2436-9542","affiliations":[{"raw_affiliation_string":"Department of Statistics, Indiana University, Bloomington, IN 47408 USA","institution_ids":["https://openalex.org/I4210119109"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5044364993"],"corresponding_institution_ids":["https://openalex.org/I4210119109"],"apc_list":null,"apc_paid":null,"fwci":0.2071,"has_fulltext":false,"cited_by_count":1,"citation_normalized_percentile":{"value":0.44431508,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":90,"max":94},"biblio":{"volume":"44","issue":"2","first_page":"822","last_page":"866"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10500","display_name":"Sparse and Compressive Sensing Techniques","score":0.9991000294685364,"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"}},"topics":[{"id":"https://openalex.org/T10500","display_name":"Sparse and Compressive Sensing Techniques","score":0.9991000294685364,"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"}},{"id":"https://openalex.org/T12303","display_name":"Tensor decomposition and applications","score":0.9851999878883362,"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":0.9569000005722046,"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/mathematics","display_name":"Mathematics","score":0.809486985206604},{"id":"https://openalex.org/keywords/rank","display_name":"Rank (graph theory)","score":0.6195998787879944},{"id":"https://openalex.org/keywords/linear-subspace","display_name":"Linear subspace","score":0.5779795050621033},{"id":"https://openalex.org/keywords/euclidean-geometry","display_name":"Euclidean geometry","score":0.5738633275032043},{"id":"https://openalex.org/keywords/orthonormal-basis","display_name":"Orthonormal basis","score":0.5599499940872192},{"id":"https://openalex.org/keywords/matrix-analysis","display_name":"Matrix analysis","score":0.5587128400802612},{"id":"https://openalex.org/keywords/grassmannian","display_name":"Grassmannian","score":0.5570863485336304},{"id":"https://openalex.org/keywords/matrix","display_name":"Matrix (chemical analysis)","score":0.5126206278800964},{"id":"https://openalex.org/keywords/euclidean-distance","display_name":"Euclidean distance","score":0.4999065399169922},{"id":"https://openalex.org/keywords/multiplicative-function","display_name":"Multiplicative function","score":0.4778006672859192},{"id":"https://openalex.org/keywords/euclidean-distance-matrix","display_name":"Euclidean distance matrix","score":0.4614715576171875},{"id":"https://openalex.org/keywords/matrix-norm","display_name":"Matrix norm","score":0.4371337592601776},{"id":"https://openalex.org/keywords/orthogonal-matrix","display_name":"Orthogonal matrix","score":0.43701618909835815},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.42758089303970337},{"id":"https://openalex.org/keywords/manifold","display_name":"Manifold (fluid mechanics)","score":0.42084455490112305},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.33422836661338806},{"id":"https://openalex.org/keywords/orthogonal-basis","display_name":"Orthogonal basis","score":0.26534730195999146},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.20698639750480652},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.1647416651248932},{"id":"https://openalex.org/keywords/eigenvalues-and-eigenvectors","display_name":"Eigenvalues and eigenvectors","score":0.13651350140571594}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.809486985206604},{"id":"https://openalex.org/C164226766","wikidata":"https://www.wikidata.org/wiki/Q7293202","display_name":"Rank (graph theory)","level":2,"score":0.6195998787879944},{"id":"https://openalex.org/C12362212","wikidata":"https://www.wikidata.org/wiki/Q728435","display_name":"Linear subspace","level":2,"score":0.5779795050621033},{"id":"https://openalex.org/C129782007","wikidata":"https://www.wikidata.org/wiki/Q162886","display_name":"Euclidean geometry","level":2,"score":0.5738633275032043},{"id":"https://openalex.org/C5806529","wikidata":"https://www.wikidata.org/wiki/Q2365325","display_name":"Orthonormal basis","level":2,"score":0.5599499940872192},{"id":"https://openalex.org/C149488123","wikidata":"https://www.wikidata.org/wiki/Q17098730","display_name":"Matrix analysis","level":3,"score":0.5587128400802612},{"id":"https://openalex.org/C162929932","wikidata":"https://www.wikidata.org/wiki/Q129638","display_name":"Grassmannian","level":2,"score":0.5570863485336304},{"id":"https://openalex.org/C106487976","wikidata":"https://www.wikidata.org/wiki/Q685816","display_name":"Matrix (chemical analysis)","level":2,"score":0.5126206278800964},{"id":"https://openalex.org/C120174047","wikidata":"https://www.wikidata.org/wiki/Q847073","display_name":"Euclidean distance","level":2,"score":0.4999065399169922},{"id":"https://openalex.org/C42747912","wikidata":"https://www.wikidata.org/wiki/Q1048447","display_name":"Multiplicative function","level":2,"score":0.4778006672859192},{"id":"https://openalex.org/C171558263","wikidata":"https://www.wikidata.org/wiki/Q5406122","display_name":"Euclidean distance matrix","level":3,"score":0.4614715576171875},{"id":"https://openalex.org/C92207270","wikidata":"https://www.wikidata.org/wiki/Q939253","display_name":"Matrix norm","level":3,"score":0.4371337592601776},{"id":"https://openalex.org/C44292817","wikidata":"https://www.wikidata.org/wiki/Q333871","display_name":"Orthogonal matrix","level":3,"score":0.43701618909835815},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.42758089303970337},{"id":"https://openalex.org/C529865628","wikidata":"https://www.wikidata.org/wiki/Q1790740","display_name":"Manifold (fluid mechanics)","level":2,"score":0.42084455490112305},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.33422836661338806},{"id":"https://openalex.org/C187064257","wikidata":"https://www.wikidata.org/wiki/Q3306808","display_name":"Orthogonal basis","level":2,"score":0.26534730195999146},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.20698639750480652},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.1647416651248932},{"id":"https://openalex.org/C158693339","wikidata":"https://www.wikidata.org/wiki/Q190524","display_name":"Eigenvalues and eigenvectors","level":2,"score":0.13651350140571594},{"id":"https://openalex.org/C127413603","wikidata":"https://www.wikidata.org/wiki/Q11023","display_name":"Engineering","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/C159985019","wikidata":"https://www.wikidata.org/wiki/Q181790","display_name":"Composite material","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/C78519656","wikidata":"https://www.wikidata.org/wiki/Q101333","display_name":"Mechanical engineering","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:10.1137/22m1489125","is_oa":false,"landing_page_url":"http://dx.doi.org/10.1137/22m1489125","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"}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":64,"referenced_works":["https://openalex.org/W131753375","https://openalex.org/W658559791","https://openalex.org/W1520752838","https://openalex.org/W1848620603","https://openalex.org/W1966156785","https://openalex.org/W1966553486","https://openalex.org/W1967184357","https://openalex.org/W1970377488","https://openalex.org/W1970453577","https://openalex.org/W1975900269","https://openalex.org/W1978494516","https://openalex.org/W1987349051","https://openalex.org/W1990407445","https://openalex.org/W1992132942","https://openalex.org/W1996023858","https://openalex.org/W2008472376","https://openalex.org/W2035076036","https://openalex.org/W2040373108","https://openalex.org/W2040957249","https://openalex.org/W2045512849","https://openalex.org/W2045547832","https://openalex.org/W2046164006","https://openalex.org/W2051623559","https://openalex.org/W2070777494","https://openalex.org/W2075295044","https://openalex.org/W2087142053","https://openalex.org/W2088911135","https://openalex.org/W2092398999","https://openalex.org/W2099781564","https://openalex.org/W2125295547","https://openalex.org/W2128132991","https://openalex.org/W2134332047","https://openalex.org/W2137126448","https://openalex.org/W2142907115","https://openalex.org/W2147631863","https://openalex.org/W2167430018","https://openalex.org/W2266012714","https://openalex.org/W2611328865","https://openalex.org/W2619970632","https://openalex.org/W2789400297","https://openalex.org/W2962740067","https://openalex.org/W2963224548","https://openalex.org/W2963350409","https://openalex.org/W2963732630","https://openalex.org/W2964044082","https://openalex.org/W2964056747","https://openalex.org/W2964066048","https://openalex.org/W2966291568","https://openalex.org/W3003561462","https://openalex.org/W3038586155","https://openalex.org/W3043617248","https://openalex.org/W3098616514","https://openalex.org/W3101741532","https://openalex.org/W3101788906","https://openalex.org/W3101873331","https://openalex.org/W3102974460","https://openalex.org/W3104227803","https://openalex.org/W3105790468","https://openalex.org/W3106209397","https://openalex.org/W3165011705","https://openalex.org/W4205293427","https://openalex.org/W4206033904","https://openalex.org/W4255333822","https://openalex.org/W4388513805"],"related_works":["https://openalex.org/W194966468","https://openalex.org/W2081926135","https://openalex.org/W4312058650","https://openalex.org/W3135432081","https://openalex.org/W2352624089","https://openalex.org/W3198150939","https://openalex.org/W2355451819","https://openalex.org/W144046499","https://openalex.org/W1560376696","https://openalex.org/W2053254742"],"abstract_inverted_index":{"In":[0],"this":[1],"paper,":[2],"we":[3,16],"propose":[4],"a":[5,18,123],"novel":[6],"and":[7,22,30,41,159],"user-friendly":[8],"Euclidean":[9,68,94,115],"representation":[10,116],"framework":[11,144],"for":[12,25],"low-rank":[13,33,50,61,137,165],"matrices.":[14,138,166],"Correspondingly,":[15],"establish":[17,109],"collection":[19,134],"of":[20,32,112,135,141,164],"technical":[21],"theoretical":[23],"tools":[24],"analyzing":[26],"the":[27,37,42,48,57,67,71,82,85,93,103,110,113,129,133,142,149,153,156,160],"intrinsic":[28],"perturbation":[29,162],"geometry":[31],"matrices":[34,62],"in":[35],"which":[36],"underlying":[38],"referential":[39,83],"matrix":[40,44,51],"perturbed":[43],"both":[45],"live":[46],"on":[47,152],"same":[49],"manifold.":[52],"It":[53],"turns":[54],"out":[55],"that":[56,120],"Frobenius":[58],"distance":[59,69,87,95],"between":[60,70,88,96],"is":[63,90],"locally":[64,80],"equivalent":[65,91],"to":[66,92],"corresponding":[72],"representing":[73],"vectors.":[74],"Our":[75],"analyses":[76],"also":[77],"show":[78],"that,":[79],"around":[81],"matrix,":[84],"sine-theta":[86],"subspaces":[89],"two":[97],"appropriately":[98],"selected":[99],"orthonormal":[100],"basis,":[101],"circumventing":[102],"orthogonal":[104],"Procrustes":[105],"analysis.":[106],"We":[107],"further":[108],"regularity":[111],"proposed":[114,143],"function":[117],"by":[118],"showing":[119],"it":[121],"has":[122],"nondegenerate":[124],"Fr\u00e9chet":[125],"derivative,":[126],"thereby":[127],"establishing":[128],"manifold":[130],"structure":[131,151],"over":[132],"all":[136],"Two":[139],"applications":[140],"are":[145],"showcased,":[146],"namely,":[147],"constructing":[148],"Riemannian":[150],"Grassmannian":[154],"using":[155],"Cayley":[157],"parameterization":[158],"multiplicative":[161],"analysis":[163]},"counts_by_year":[{"year":2024,"cited_by_count":1}],"updated_date":"2025-12-24T23:09:58.560324","created_date":"2025-10-10T00:00:00"}