{"id":"https://openalex.org/W2042896810","doi":"https://doi.org/10.1090/s0025-5718-2010-02450-0","title":"Least-squares approximation by elements from matrix orbits achieved by gradient flows on compact lie groups","display_name":"Least-squares approximation by elements from matrix orbits achieved by gradient flows on compact lie groups","publication_year":2010,"publication_date":"2010-12-13","ids":{"openalex":"https://openalex.org/W2042896810","doi":"https://doi.org/10.1090/s0025-5718-2010-02450-0","mag":"2042896810"},"language":"en","primary_location":{"id":"doi:10.1090/s0025-5718-2010-02450-0","is_oa":true,"landing_page_url":"https://doi.org/10.1090/s0025-5718-2010-02450-0","pdf_url":"https://www.ams.org/mcom/2011-80-275/S0025-5718-2010-02450-0/S0025-5718-2010-02450-0.pdf","source":{"id":"https://openalex.org/S102439543","display_name":"Mathematics of Computation","issn_l":"0025-5718","issn":["0025-5718","1088-6842"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310315719","host_organization_name":"American Mathematical Society","host_organization_lineage":["https://openalex.org/P4310315719"],"host_organization_lineage_names":["American Mathematical Society"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Mathematics of Computation","raw_type":"journal-article"},"type":"article","indexed_in":["arxiv","crossref"],"open_access":{"is_oa":true,"oa_status":"bronze","oa_url":"https://www.ams.org/mcom/2011-80-275/S0025-5718-2010-02450-0/S0025-5718-2010-02450-0.pdf","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":null,"display_name":"Chi-Kwong Li","orcid":null},"institutions":[{"id":"https://openalex.org/I16285277","display_name":"William & Mary","ror":"https://ror.org/03hsf0573","country_code":"US","type":"education","lineage":["https://openalex.org/I16285277"]},{"id":"https://openalex.org/I267592682","display_name":"Williams (United States)","ror":"https://ror.org/007zhvp17","country_code":"US","type":"company","lineage":["https://openalex.org/I267592682"]}],"countries":["US"],"is_corresponding":true,"raw_author_name":"Chi-Kwong Li","raw_affiliation_strings":["Department of Mathematics, College of William and Mary, Williamsburg, Virginia 23187"],"affiliations":[{"raw_affiliation_string":"Department of Mathematics, College of William and Mary, Williamsburg, Virginia 23187","institution_ids":["https://openalex.org/I16285277","https://openalex.org/I267592682"]}]},{"author_position":"middle","author":{"id":null,"display_name":"Yiu-Tung Poon","orcid":null},"institutions":[{"id":"https://openalex.org/I173911158","display_name":"Iowa State University","ror":"https://ror.org/04rswrd78","country_code":"US","type":"education","lineage":["https://openalex.org/I173911158"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Yiu-Tung Poon","raw_affiliation_strings":["Department of Mathematics, Iowa State University, Ames, Iowa 50051"],"affiliations":[{"raw_affiliation_string":"Department of Mathematics, Iowa State University, Ames, Iowa 50051","institution_ids":["https://openalex.org/I173911158"]}]},{"author_position":"last","author":{"id":null,"display_name":"Thomas Schulte-Herbr\u00fcggen","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Thomas Schulte-Herbr\u00fcggen","raw_affiliation_strings":[],"affiliations":[]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":3,"corresponding_author_ids":[],"corresponding_institution_ids":["https://openalex.org/I16285277","https://openalex.org/I267592682"],"apc_list":null,"apc_paid":null,"fwci":0.4664,"has_fulltext":true,"cited_by_count":4,"citation_normalized_percentile":{"value":0.73411893,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":90,"max":94},"biblio":{"volume":"80","issue":"275","first_page":"1601","last_page":"1621"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11612","display_name":"Stochastic Gradient Optimization Techniques","score":0.3871000111103058,"subfield":{"id":"https://openalex.org/subfields/1702","display_name":"Artificial Intelligence"},"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/T11612","display_name":"Stochastic Gradient Optimization Techniques","score":0.3871000111103058,"subfield":{"id":"https://openalex.org/subfields/1702","display_name":"Artificial Intelligence"},"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/T10792","display_name":"Matrix Theory and Algorithms","score":0.1307000070810318,"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/T10963","display_name":"Advanced Optimization Algorithms Research","score":0.08209999650716782,"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/unitary-matrix","display_name":"Unitary matrix","score":0.7228999733924866},{"id":"https://openalex.org/keywords/unitary-state","display_name":"Unitary state","score":0.6639000177383423},{"id":"https://openalex.org/keywords/congruence-relation","display_name":"Congruence relation","score":0.6579999923706055},{"id":"https://openalex.org/keywords/equivalence-relation","display_name":"Equivalence relation","score":0.6051999926567078},{"id":"https://openalex.org/keywords/circular-ensemble","display_name":"Circular ensemble","score":0.5566999912261963},{"id":"https://openalex.org/keywords/matrix","display_name":"Matrix (chemical analysis)","score":0.5473999977111816},{"id":"https://openalex.org/keywords/lie-group","display_name":"Lie group","score":0.5368000268936157},{"id":"https://openalex.org/keywords/equivalence","display_name":"Equivalence (formal languages)","score":0.5076000094413757},{"id":"https://openalex.org/keywords/euclidean-geometry","display_name":"Euclidean geometry","score":0.4927000105381012},{"id":"https://openalex.org/keywords/matrix-function","display_name":"Matrix function","score":0.3824999928474426}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.9024999737739563},{"id":"https://openalex.org/C96214315","wikidata":"https://www.wikidata.org/wiki/Q727103","display_name":"Unitary matrix","level":3,"score":0.7228999733924866},{"id":"https://openalex.org/C67820243","wikidata":"https://www.wikidata.org/wiki/Q179164","display_name":"Unitary state","level":2,"score":0.6639000177383423},{"id":"https://openalex.org/C9973445","wikidata":"https://www.wikidata.org/wiki/Q8349849","display_name":"Congruence relation","level":2,"score":0.6579999923706055},{"id":"https://openalex.org/C156103551","wikidata":"https://www.wikidata.org/wiki/Q130998","display_name":"Equivalence relation","level":2,"score":0.6051999926567078},{"id":"https://openalex.org/C11902944","wikidata":"https://www.wikidata.org/wiki/Q5121662","display_name":"Circular ensemble","level":4,"score":0.5566999912261963},{"id":"https://openalex.org/C106487976","wikidata":"https://www.wikidata.org/wiki/Q685816","display_name":"Matrix (chemical analysis)","level":2,"score":0.5473999977111816},{"id":"https://openalex.org/C187915474","wikidata":"https://www.wikidata.org/wiki/Q622679","display_name":"Lie group","level":2,"score":0.5368000268936157},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.5254999995231628},{"id":"https://openalex.org/C2780069185","wikidata":"https://www.wikidata.org/wiki/Q7977945","display_name":"Equivalence (formal languages)","level":2,"score":0.5076000094413757},{"id":"https://openalex.org/C129782007","wikidata":"https://www.wikidata.org/wiki/Q162886","display_name":"Euclidean geometry","level":2,"score":0.4927000105381012},{"id":"https://openalex.org/C4263655","wikidata":"https://www.wikidata.org/wiki/Q2699958","display_name":"Matrix function","level":4,"score":0.3824999928474426},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.3815999925136566},{"id":"https://openalex.org/C196644772","wikidata":"https://www.wikidata.org/wiki/Q3884964","display_name":"Orbit (dynamics)","level":2,"score":0.3553999960422516},{"id":"https://openalex.org/C94940","wikidata":"https://www.wikidata.org/wiki/Q652941","display_name":"Hermitian matrix","level":2,"score":0.33660000562667847},{"id":"https://openalex.org/C99072794","wikidata":"https://www.wikidata.org/wiki/Q1500617","display_name":"Unitary group","level":3,"score":0.3361999988555908},{"id":"https://openalex.org/C54848796","wikidata":"https://www.wikidata.org/wiki/Q339011","display_name":"Symmetric matrix","level":3,"score":0.32179999351501465},{"id":"https://openalex.org/C132074034","wikidata":"https://www.wikidata.org/wiki/Q154210","display_name":"Congruence (geometry)","level":2,"score":0.31839999556541443},{"id":"https://openalex.org/C186450821","wikidata":"https://www.wikidata.org/wiki/Q17295","display_name":"Euclidean space","level":2,"score":0.3077999949455261},{"id":"https://openalex.org/C2988995629","wikidata":"https://www.wikidata.org/wiki/Q2915729","display_name":"Matrix algebra","level":3,"score":0.29670000076293945},{"id":"https://openalex.org/C51568863","wikidata":"https://www.wikidata.org/wiki/Q664495","display_name":"Lie algebra","level":2,"score":0.29490000009536743},{"id":"https://openalex.org/C171558263","wikidata":"https://www.wikidata.org/wiki/Q5406122","display_name":"Euclidean distance matrix","level":3,"score":0.2937999963760376},{"id":"https://openalex.org/C2988629283","wikidata":"https://www.wikidata.org/wiki/Q44337","display_name":"Complex matrix","level":2,"score":0.28619998693466187},{"id":"https://openalex.org/C22365015","wikidata":"https://www.wikidata.org/wiki/Q1003162","display_name":"Simple Lie group","level":3,"score":0.28200000524520874},{"id":"https://openalex.org/C32955637","wikidata":"https://www.wikidata.org/wiki/Q684363","display_name":"Special unitary group","level":2,"score":0.2815000116825104},{"id":"https://openalex.org/C175380299","wikidata":"https://www.wikidata.org/wiki/Q6852339","display_name":"Normal matrix","level":3,"score":0.27970001101493835},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.27639999985694885},{"id":"https://openalex.org/C136119220","wikidata":"https://www.wikidata.org/wiki/Q1000660","display_name":"Algebra over a field","level":2,"score":0.2736000120639801},{"id":"https://openalex.org/C2776122010","wikidata":"https://www.wikidata.org/wiki/Q10359539","display_name":"Unitary representation","level":3,"score":0.26910001039505005},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.2685000002384186},{"id":"https://openalex.org/C69044650","wikidata":"https://www.wikidata.org/wiki/Q2739329","display_name":"Square matrix","level":4,"score":0.2624000012874603},{"id":"https://openalex.org/C145242015","wikidata":"https://www.wikidata.org/wiki/Q774123","display_name":"Approximation theory","level":2,"score":0.25049999356269836}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.1090/s0025-5718-2010-02450-0","is_oa":true,"landing_page_url":"https://doi.org/10.1090/s0025-5718-2010-02450-0","pdf_url":"https://www.ams.org/mcom/2011-80-275/S0025-5718-2010-02450-0/S0025-5718-2010-02450-0.pdf","source":{"id":"https://openalex.org/S102439543","display_name":"Mathematics of Computation","issn_l":"0025-5718","issn":["0025-5718","1088-6842"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310315719","host_organization_name":"American Mathematical Society","host_organization_lineage":["https://openalex.org/P4310315719"],"host_organization_lineage_names":["American Mathematical Society"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Mathematics of Computation","raw_type":"journal-article"},{"id":"pmh:oai:arXiv.org:0812.1817","is_oa":true,"landing_page_url":"http://arxiv.org/abs/0812.1817","pdf_url":"https://arxiv.org/pdf/0812.1817","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"}],"best_oa_location":{"id":"doi:10.1090/s0025-5718-2010-02450-0","is_oa":true,"landing_page_url":"https://doi.org/10.1090/s0025-5718-2010-02450-0","pdf_url":"https://www.ams.org/mcom/2011-80-275/S0025-5718-2010-02450-0/S0025-5718-2010-02450-0.pdf","source":{"id":"https://openalex.org/S102439543","display_name":"Mathematics of Computation","issn_l":"0025-5718","issn":["0025-5718","1088-6842"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310315719","host_organization_name":"American Mathematical Society","host_organization_lineage":["https://openalex.org/P4310315719"],"host_organization_lineage_names":["American Mathematical Society"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Mathematics of Computation","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[],"funders":[{"id":"https://openalex.org/F4320306076","display_name":"National Science Foundation","ror":"https://ror.org/021nxhr62"}],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W2042896810.pdf","grobid_xml":"https://content.openalex.org/works/W2042896810.grobid-xml"},"referenced_works_count":0,"referenced_works":[],"related_works":[],"abstract_inverted_index":{"Let":[0],"<inline-formula":[1,34,75,98],"content-type=\"math/mathml\">":[2,35,76,99,166],"<mml:math":[3,36,77,100,168],"xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"":[4,37,78,101,169],"alttext=\"upper":[5,38,79,102],"S":[6,103,113,198],"left-parenthesis":[7,104,114,199],"upper":[8,105,112,115,118,173,179,182,186,191,197,200,213],"A":[9,80,106,116,187,201],"right-parenthesis\">":[10,121],"<mml:semantics>":[11,40,82,122,217],"<mml:mrow>":[12,123,218],"<mml:mi>S</mml:mi>":[13,124,138,279],"<mml:mo":[14,17,125,131,139,145,219,226,235,265,280,286,304],"stretchy=\"false\">(</mml:mo>":[15,126,140,281],"<mml:mi>A</mml:mi>":[16,41,84,128,142,258,283],"stretchy=\"false\">)</mml:mo>":[18,132,146,287],"</mml:mrow>":[19,147,229,239,269,307,310],"<mml:annotation":[20,42,87,148,311],"encoding=\"application/x-tex\">S(A)</mml:annotation>":[21],"</mml:semantics>":[22,44,89,153,333],"</mml:math>":[23,45,90,154,334],"</inline-formula>":[24,46,91,155],"denote":[25],"the":[26,68,93,157,161,339],"orbit":[27],"of":[28,71,95,159,338],"a":[29,48,72],"complex":[30],"or":[31],"real":[32],"matrix":[33,74],"A\">":[39],"encoding=\"application/x-tex\">A</mml:annotation>":[43],"under":[47],"certain":[49],"equivalence":[50],"relation":[51],"such":[52],"as":[53],"unitary":[54,56,58],"similarity,":[55],"equivalence,":[57],"congruences":[59],"etc.":[60],"Efficient":[61],"gradient-flow":[62],"algorithms":[63],"are":[64,347],"constructed":[65],"to":[66,341],"determine":[67],"best":[69],"approximation":[70],"given":[73],"0\">":[81],"<mml:msub>":[83,127,141,241,250,257,272,282],"<mml:mn>0</mml:mn>":[85,259],"</mml:msub>":[86,130,144,244,253,260,275,285],"encoding=\"application/x-tex\">A_0</mml:annotation>":[88],"by":[92],"sum":[94],"matrices":[96],"in":[97,156],"1":[107,175,209],"right-parenthesis":[108,205],"comma":[109,111,206,210,212],"ellipsis":[110,211],"Subscript":[117,181,193,202],"N":[119,183,214],"Baseline":[120,184,195,204],"<mml:mn>1</mml:mn>":[129,243,293],"<mml:mo>,</mml:mo>":[133,137,288,294,298],"<mml:mo>":[134,246,254,276,295],"\u2026":[135,296],"</mml:mo>":[136,248,256,278,297],"<mml:mi>N</mml:mi>":[143,252,299],"encoding=\"application/x-tex\">S(A_1),":[149],"\\dots":[150,329],",":[151,330],"S(A_N)</mml:annotation>":[152],"sense":[158],"finding":[160],"Euclidean":[162],"least-squares":[163],"distance":[164],"<disp-formula":[165],"\\[":[167],"alttext=\"min":[170],"left-brace":[171],"double-vertical-bar":[172,189],"X":[174,180,192],"plus":[176,178],"midline-horizontal-ellipsis":[177],"minus":[185],"0":[188],"colon":[190],"j":[194,203,207,326],"element-of":[196],"equals":[208],"right-brace":[215],"period\">":[216],"movablelimits=\"true\"":[220],"form=\"prefix\">min</mml:mo>":[221],"<mml:mstyle":[222,231,261,300],"scriptlevel=\"0\">":[223,232,262,301],"<mml:mrow":[224,233,263,302],"class=\"MJX-TeXAtom-ORD\">":[225,234,264,303],"maxsize=\"1.623em\"":[227,305],"minsize=\"1.623em\">{</mml:mo>":[228],"</mml:mstyle>":[230,240,270,308],"symmetric=\"true\"":[236,266],"maxsize=\"1.2em\"":[237,267],"minsize=\"1.2em\">\u2016</mml:mo>":[238,268],"<mml:mi>X</mml:mi>":[242,251,273],"<mml:mo>+</mml:mo>":[245,249],"\u22ef":[247],"\u2212":[255],"<mml:mo>:</mml:mo>":[271],"<mml:mi>j</mml:mi>":[274,284,291],"\u2208":[277],"<mml:mtext>":[289],"</mml:mtext>":[290],"<mml:mo>=</mml:mo>":[292],"minsize=\"1.623em\">}</mml:mo>":[306],"<mml:mo>.</mml:mo>":[309],"encoding=\"application/x-tex\">\\min":[312],"\\Big":[313],"\\{\\big":[314],"\\|X_1+":[315],"\\cdots":[316],"+":[317],"X_N":[318],"-":[319],"A_0\\big":[320],"\\|:":[321],"X_j":[322],"\\in":[323],"S(A_j),":[324],"\\":[325],"=":[327],"1,":[328],"N\\Big":[331],"\\}.</mml:annotation>":[332],"\\]":[335],"</disp-formula>":[336],"Connections":[337],"results":[340],"different":[342],"pure":[343],"and":[344],"applied":[345],"areas":[346],"discussed.":[348]},"counts_by_year":[{"year":2019,"cited_by_count":1},{"year":2018,"cited_by_count":1},{"year":2017,"cited_by_count":1},{"year":2012,"cited_by_count":1}],"updated_date":"2026-04-22T06:01:30.510260","created_date":"2016-06-24T00:00:00"}
