Lie Group Methods for Optimization with Orthogonality Constraints
Identifieur interne : 001072 ( Main/Corpus ); précédent : 001071; suivant : 001073Lie Group Methods for Optimization with Orthogonality Constraints
Auteurs : D. PlumbleySource :
- Lecture Notes in Computer Science [ 0302-9743 ] ; 2004.
Abstract
Abstract: Optimization of a cost function J(W) under an orthogonality constraint WW T =I is a common requirement for ICA methods. In this paper, we will review the use of Lie group methods to perform this constrained optimization. Instead of searching in the space of n× n matrices W, we will introduce the concept of the Lie group SO(n) of orthogonal matrices, and the corresponding Lie algebraso(n). Using so(n) for our coordinates, we can multiplicatively update W by a rotation matrix R so that W′=RW always remains orthogonal. Steepest descent and conjugate gradient algorithms can be used in this framework.
Url:
DOI: 10.1007/978-3-540-30110-3_157
Links to Exploration step
ISTEX:FE2C407982B78EAB754C83671C0DF725862957C5Le document en format XML
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Plumbley</name><affiliation><mods:affiliation>Department of Electronic Engineering, Queen Mary University of London, Mile End Road, E1 4NS, London, UK</mods:affiliation></affiliation><affiliation><mods:affiliation>E-mail: mark.plumbley@elec.qmul.ac.uk</mods:affiliation></affiliation></author></analytic><monogr></monogr><series><title level="s">Lecture Notes in Computer Science</title><imprint><date>2004</date></imprint><idno type="ISSN">0302-9743</idno><idno type="eISSN">1611-3349</idno><idno type="ISSN">0302-9743</idno></series><idno type="istex">FE2C407982B78EAB754C83671C0DF725862957C5</idno><idno type="DOI">10.1007/978-3-540-30110-3_157</idno><idno type="ChapterID">Chap157</idno><idno type="ChapterID">157</idno></biblStruct></sourceDesc><seriesStmt><idno type="ISSN">0302-9743</idno></seriesStmt></fileDesc><profileDesc><textClass></textClass><langUsage><language ident="en">en</language></langUsage></profileDesc></teiHeader><front><div type="abstract" xml:lang="en">Abstract: Optimization of a cost function J(W) under an orthogonality constraint WW T =I is a common requirement for ICA methods. 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In this paper, we will review the use of Lie group methods to perform this constrained optimization. Instead of searching in the space of n× n matrices W, we will introduce the concept of the Lie group SO(n) of orthogonal matrices, and the corresponding Lie algebraso(n). Using so(n) for our coordinates, we can multiplicatively update W by a rotation matrix R so that W′=RW always remains orthogonal. 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MediaType="eBook" NumberingDepth="2" NumberingStyle="ContentOnly" OutputMedium="All" TocLevels="0"><BookID>978-3-540-30110-3</BookID><BookTitle>Independent Component Analysis and Blind Signal Separation</BookTitle><BookSubTitle>Fifth International Conference, ICA 2004, Granada, Spain, September 22-24, 2004. Proceedings</BookSubTitle><BookVolumeNumber>3195</BookVolumeNumber><BookSequenceNumber>3195</BookSequenceNumber><BookDOI>10.1007/b100528</BookDOI><BookTitleID>112470</BookTitleID><BookPrintISBN>978-3-540-23056-4</BookPrintISBN><BookElectronicISBN>978-3-540-30110-3</BookElectronicISBN><BookChapterCount>158</BookChapterCount><BookCopyright><CopyrightHolderName>Springer-Verlag Berlin Heidelberg</CopyrightHolderName><CopyrightYear>2004</CopyrightYear></BookCopyright><BookSubjectGroup><BookSubject Code="I" Type="Primary">Computer Science</BookSubject><BookSubject Code="I13030" Priority="1" Type="Secondary">Special Purpose and Application-Based Systems</BookSubject><BookSubject Code="I16021" Priority="2" Type="Secondary">Algorithm Analysis and Problem Complexity</BookSubject><BookSubject Code="I16013" Priority="3" Type="Secondary">Computation by Abstract Devices</BookSubject><BookSubject Code="I15041" Priority="4" Type="Secondary">Coding and Information Theory</BookSubject><BookSubject Code="S12008" Priority="5" Type="Secondary">Statistics and Computing/Statistics Programs</BookSubject><BookSubject Code="T24051" Priority="6" Type="Secondary">Signal, Image and Speech Processing</BookSubject><SubjectCollection Code="SUCO11645">Computer Science</SubjectCollection></BookSubjectGroup><BookContext><SeriesID>558</SeriesID></BookContext></BookInfo><BookHeader><EditorGroup><Editor AffiliationIDS="Aff16"><EditorName DisplayOrder="Western"><GivenName>Carlos</GivenName><GivenName>G.</GivenName><FamilyName>Puntonet</FamilyName></EditorName><Contact><Email>carlos@atc.ugr.es</Email></Contact></Editor><Editor AffiliationIDS="Aff17"><EditorName DisplayOrder="Western"><GivenName>Alberto</GivenName><FamilyName>Prieto</FamilyName></EditorName><Contact><Email>aprieto@ugr.es</Email></Contact></Editor><Affiliation ID="Aff16"><OrgDivision>Dept. of Architecture and Computer Technology</OrgDivision><OrgName>University of Granada</OrgName><OrgAddress><Country>Spain</Country></OrgAddress></Affiliation><Affiliation ID="Aff17"><OrgDivision>Department of Computer Architecture and Computer Technology</OrgDivision><OrgName>University of Granada</OrgName><OrgAddress><Country> </Country></OrgAddress></Affiliation></EditorGroup></BookHeader><Part ID="Part9"><PartInfo TocLevels="0"><PartID>9</PartID><PartSequenceNumber>9</PartSequenceNumber><PartTitle>Invited Contributions</PartTitle><PartChapterCount>2</PartChapterCount><PartContext><SeriesID>558</SeriesID><BookTitle>Independent Component Analysis and Blind Signal Separation</BookTitle></PartContext></PartInfo><Chapter ID="Chap157" Language="En"><ChapterInfo ChapterType="OriginalPaper" ContainsESM="No" NumberingDepth="2" NumberingStyle="ContentOnly" TocLevels="0"><ChapterID>157</ChapterID><ChapterDOI>10.1007/978-3-540-30110-3_157</ChapterDOI><ChapterSequenceNumber>157</ChapterSequenceNumber><ChapterTitle Language="En">Lie Group Methods for Optimization with Orthogonality Constraints</ChapterTitle><ChapterFirstPage>1245</ChapterFirstPage><ChapterLastPage>1252</ChapterLastPage><ChapterCopyright><CopyrightHolderName>Springer-Verlag Berlin Heidelberg</CopyrightHolderName><CopyrightYear>2004</CopyrightYear></ChapterCopyright><ChapterGrants Type="Regular"><MetadataGrant Grant="OpenAccess"></MetadataGrant><AbstractGrant Grant="OpenAccess"></AbstractGrant><BodyPDFGrant Grant="Restricted"></BodyPDFGrant><BodyHTMLGrant Grant="Restricted"></BodyHTMLGrant><BibliographyGrant Grant="Restricted"></BibliographyGrant><ESMGrant Grant="Restricted"></ESMGrant></ChapterGrants><ChapterContext><SeriesID>558</SeriesID><PartID>9</PartID><BookID>978-3-540-30110-3</BookID><BookTitle>Independent Component Analysis and Blind Signal Separation</BookTitle></ChapterContext></ChapterInfo><ChapterHeader><AuthorGroup><Author AffiliationIDS="Aff18"><AuthorName DisplayOrder="Western"><GivenName>Mark</GivenName><GivenName>D.</GivenName><FamilyName>Plumbley</FamilyName></AuthorName><Contact><Email>mark.plumbley@elec.qmul.ac.uk</Email></Contact></Author><Affiliation ID="Aff18"><OrgDivision>Department of Electronic Engineering</OrgDivision><OrgName>Queen Mary University of London</OrgName><OrgAddress><Street>Mile End Road</Street><City>London</City><Postcode>E1 4NS</Postcode><Country>UK</Country></OrgAddress></Affiliation></AuthorGroup><Abstract ID="Abs1" Language="En"><Heading>Abstract</Heading><Para>Optimization of a cost function <Emphasis Type="Italic">J</Emphasis>(<Emphasis Type="Bold">W</Emphasis>) under an orthogonality constraint <Emphasis Type="Bold">WW</Emphasis><Superscript><Emphasis Type="Italic">T</Emphasis></Superscript>=<Emphasis Type="Bold">I</Emphasis> is a common requirement for ICA methods. In this paper, we will review the use of <Emphasis Type="Italic">Lie group</Emphasis> methods to perform this constrained optimization. Instead of searching in the space of <Emphasis Type="Italic">n</Emphasis>× <Emphasis Type="Italic">n</Emphasis> matrices <Emphasis Type="Bold">W</Emphasis>, we will introduce the concept of the Lie group SO(<Emphasis Type="Italic">n</Emphasis>) of orthogonal matrices, and the corresponding <Emphasis Type="Italic">Lie algebraso</Emphasis>(<Emphasis Type="Italic">n</Emphasis>). Using <Emphasis Type="Italic">so</Emphasis>(<Emphasis Type="Italic">n</Emphasis>) for our coordinates, we can multiplicatively update <Emphasis Type="Bold">W</Emphasis> by a rotation matrix <Emphasis Type="Bold">R</Emphasis> so that <Emphasis Type="Bold">W</Emphasis>′=<Emphasis Type="Bold">RW</Emphasis> always remains orthogonal. Steepest descent and conjugate gradient algorithms can be used in this framework.</Para></Abstract></ChapterHeader><NoBody></NoBody></Chapter></Part></Book></Series></Publisher></istex:document></istex:metadataXml><mods version="3.6"><titleInfo lang="en"><title>Lie Group Methods for Optimization with Orthogonality Constraints</title></titleInfo><titleInfo type="alternative" contentType="CDATA" lang="en"><title>Lie Group Methods for Optimization with Orthogonality Constraints</title></titleInfo><name type="personal"><namePart type="given">Mark</namePart><namePart type="given">D.</namePart><namePart type="family">Plumbley</namePart><affiliation>Department of Electronic Engineering, Queen Mary University of London, Mile End Road, E1 4NS, London, UK</affiliation><affiliation>E-mail: mark.plumbley@elec.qmul.ac.uk</affiliation><role><roleTerm type="text">author</roleTerm></role></name><typeOfResource>text</typeOfResource><genre type="conference [research-article]" displayLabel="OriginalPaper"></genre><originInfo><publisher>Springer Berlin Heidelberg</publisher><place><placeTerm type="text">Berlin, Heidelberg</placeTerm></place><dateIssued encoding="w3cdtf">2004</dateIssued><copyrightDate encoding="w3cdtf">2004</copyrightDate></originInfo><language><languageTerm type="code" authority="rfc3066">en</languageTerm><languageTerm type="code" authority="iso639-2b">eng</languageTerm></language><physicalDescription><internetMediaType>text/html</internetMediaType></physicalDescription><abstract lang="en">Abstract: Optimization of a cost function J(W) under an orthogonality constraint WW T =I is a common requirement for ICA methods. In this paper, we will review the use of Lie group methods to perform this constrained optimization. Instead of searching in the space of n× n matrices W, we will introduce the concept of the Lie group SO(n) of orthogonal matrices, and the corresponding Lie algebraso(n). Using so(n) for our coordinates, we can multiplicatively update W by a rotation matrix R so that W′=RW always remains orthogonal. Steepest descent and conjugate gradient algorithms can be used in this framework.</abstract><relatedItem type="host"><titleInfo><title>Independent Component Analysis and Blind Signal Separation</title><subTitle>Fifth International Conference, ICA 2004, Granada, Spain, September 22-24, 2004. Proceedings</subTitle></titleInfo><name type="personal"><namePart type="given">Carlos</namePart><namePart type="given">G.</namePart><namePart type="family">Puntonet</namePart><affiliation>Dept. of Architecture and Computer Technology, University of Granada, Spain</affiliation><affiliation>E-mail: carlos@atc.ugr.es</affiliation><role><roleTerm type="text">editor</roleTerm></role></name><name type="personal"><namePart type="given">Alberto</namePart><namePart type="family">Prieto</namePart><affiliation>Department of Computer Architecture and Computer Technology, University of Granada</affiliation><affiliation>E-mail: aprieto@ugr.es</affiliation><role><roleTerm type="text">editor</roleTerm></role></name><genre type="Book Series" displayLabel="Proceedings"></genre><originInfo><copyrightDate encoding="w3cdtf">2004</copyrightDate><issuance>monographic</issuance></originInfo><subject><genre>Book Subject Collection</genre><topic authority="SpringerSubjectCodes" authorityURI="SUCO11645">Computer Science</topic></subject><subject><genre>Book Subject Group</genre><topic authority="SpringerSubjectCodes" authorityURI="I">Computer Science</topic><topic authority="SpringerSubjectCodes" authorityURI="I13030">Special Purpose and Application-Based Systems</topic><topic authority="SpringerSubjectCodes" authorityURI="I16021">Algorithm Analysis and Problem Complexity</topic><topic authority="SpringerSubjectCodes" authorityURI="I16013">Computation by Abstract Devices</topic><topic authority="SpringerSubjectCodes" authorityURI="I15041">Coding and Information Theory</topic><topic authority="SpringerSubjectCodes" authorityURI="S12008">Statistics and Computing/Statistics Programs</topic><topic authority="SpringerSubjectCodes" authorityURI="T24051">Signal, Image and Speech Processing</topic></subject><identifier type="DOI">10.1007/b100528</identifier><identifier type="ISBN">978-3-540-23056-4</identifier><identifier type="eISBN">978-3-540-30110-3</identifier><identifier type="ISSN">0302-9743</identifier><identifier type="eISSN">1611-3349</identifier><identifier type="BookTitleID">112470</identifier><identifier type="BookID">978-3-540-30110-3</identifier><identifier type="BookChapterCount">158</identifier><identifier type="BookVolumeNumber">3195</identifier><identifier type="BookSequenceNumber">3195</identifier><identifier type="PartChapterCount">2</identifier><part><date>2004</date><detail type="part"><title>Invited Contributions</title></detail><detail type="volume"><number>3195</number><caption>vol.</caption></detail><extent unit="pages"><start>1245</start><end>1252</end></extent></part><recordInfo><recordOrigin>Springer-Verlag Berlin Heidelberg, 2004</recordOrigin></recordInfo></relatedItem><relatedItem type="series"><titleInfo><title>Lecture Notes in Computer Science</title></titleInfo><name type="personal"><namePart type="given">David</namePart><namePart type="family">Hutchison</namePart><affiliation>Lancaster University, UK</affiliation><role><roleTerm type="text">editor</roleTerm></role></name><name type="personal"><namePart type="given">Takeo</namePart><namePart type="family">Kanade</namePart><affiliation>Carnegie Mellon University, Pittsburgh, PA, USA</affiliation><role><roleTerm type="text">editor</roleTerm></role></name><name type="personal"><namePart type="given">Josef</namePart><namePart type="family">Kittler</namePart><affiliation>University of Surrey, Guildford, UK</affiliation><role><roleTerm type="text">editor</roleTerm></role></name><name type="personal"><namePart type="given">Jon</namePart><namePart type="given">M.</namePart><namePart type="family">Kleinberg</namePart><affiliation>Cornell University, Ithaca, NY, USA</affiliation><role><roleTerm type="text">editor</roleTerm></role></name><name type="personal"><namePart type="given">Friedemann</namePart><namePart type="family">Mattern</namePart><affiliation>ETH Zurich, Switzerland</affiliation><role><roleTerm type="text">editor</roleTerm></role></name><name type="personal"><namePart type="given">John</namePart><namePart type="given">C.</namePart><namePart type="family">Mitchell</namePart><affiliation>Stanford University, CA, USA</affiliation><role><roleTerm type="text">editor</roleTerm></role></name><name type="personal"><namePart type="given">Moni</namePart><namePart type="family">Naor</namePart><affiliation>Weizmann Institute of Science, Rehovot, Israel</affiliation><role><roleTerm type="text">editor</roleTerm></role></name><name type="personal"><namePart type="given">Oscar</namePart><namePart type="family">Nierstrasz</namePart><affiliation>University of Bern, Switzerland</affiliation><role><roleTerm type="text">editor</roleTerm></role></name><name type="personal"><namePart type="given">C.</namePart><namePart type="family">Pandu Rangan</namePart><affiliation>Indian Institute of Technology, Madras, India</affiliation><role><roleTerm type="text">editor</roleTerm></role></name><name type="personal"><namePart type="given">Bernhard</namePart><namePart type="family">Steffen</namePart><affiliation>University of Dortmund, Germany</affiliation><role><roleTerm type="text">editor</roleTerm></role></name><name type="personal"><namePart type="given">Madhu</namePart><namePart type="family">Sudan</namePart><affiliation>Massachusetts Institute of Technology, MA, USA</affiliation><role><roleTerm type="text">editor</roleTerm></role></name><name type="personal"><namePart type="given">Demetri</namePart><namePart type="family">Terzopoulos</namePart><affiliation>New York University, NY, USA</affiliation><role><roleTerm type="text">editor</roleTerm></role></name><name type="personal"><namePart type="given">Dough</namePart><namePart type="family">Tygar</namePart><affiliation>University of California, Berkeley, CA, USA</affiliation><role><roleTerm type="text">editor</roleTerm></role></name><name type="personal"><namePart type="given">Moshe</namePart><namePart 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