File Download
 
Links for fulltext
(May Require Subscription)
 
Supplementary

Article: Characterization of all solutions for undersampled uncorrelated linear discriminant analysis problems
  • Basic View
  • Metadata View
  • XML View
TitleCharacterization of all solutions for undersampled uncorrelated linear discriminant analysis problems
 
AuthorsChu, D2
Goh, ST2
Hung, YS1
 
KeywordsData Dimensionality Reduction
QR Factorization
Uncorrelated Linear Discriminant Analysis
 
Issue Date2011
 
PublisherSociety for Industrial and Applied Mathematics. The Journal's web site is located at http://epubs.siam.org/sam-bin/dbq/toclist/SIMAX
 
CitationSIAM Journal On Matrix Analysis And Applications, 2011, v. 32 n. 3, p. 820-844 [How to Cite?]
DOI: http://dx.doi.org/10.1137/100792007
 
AbstractIn this paper the uncorrelated linear discriminant analysis (ULDA) for undersampled problems is studied. The main contributions of the present work include the following: (i) all solutions of the optimization problem used for establishing the ULDA are parameterized explicitly; (ii) the optimal solutions among all solutions of the corresponding optimization problem are characterized in terms of both the ratio of between-class distance to within-class distance and the maximum likelihood classification, and it is proved that these optimal solutions are exactly the solutions of the corresponding optimization problem with minimum Frobenius norm, also minimum nuclear norm; these properties provide a good mathematical justification for preferring the minimum-norm transformation over other possible solutions as the optimal transformation in ULDA; (iii) explicit necessary and sufficient conditions are provided to ensure that these minimal solutions lead to a larger ratio of between-class distance to within-class distance, thereby achieving larger discrimination in the reduced subspace than that in the original data space, and our numerical experiments show that these necessary and sufficient conditions hold true generally. Furthermore, a new and fast ULDA algorithm is developed, which is eigendecomposition-free and SVD-free, and its effectiveness is demonstrated by some real-world data sets. © 2011 Society for Industrial and Applied Mathematics.
 
ISSN0895-4798
2012 Impact Factor: 1.342
2012 SCImago Journal Rankings: 1.509
 
DOIhttp://dx.doi.org/10.1137/100792007
 
ISI Accession Number IDWOS:000295399200009
Funding AgencyGrant Number
NUSR-146-000-140-112
Funding Information:

The work of these authors was supported by NUS research grant R-146-000-140-112.

 
ReferencesReferences in Scopus
 
DC FieldValue
dc.contributor.authorChu, D
 
dc.contributor.authorGoh, ST
 
dc.contributor.authorHung, YS
 
dc.date.accessioned2012-08-08T08:34:47Z
 
dc.date.available2012-08-08T08:34:47Z
 
dc.date.issued2011
 
dc.description.abstractIn this paper the uncorrelated linear discriminant analysis (ULDA) for undersampled problems is studied. The main contributions of the present work include the following: (i) all solutions of the optimization problem used for establishing the ULDA are parameterized explicitly; (ii) the optimal solutions among all solutions of the corresponding optimization problem are characterized in terms of both the ratio of between-class distance to within-class distance and the maximum likelihood classification, and it is proved that these optimal solutions are exactly the solutions of the corresponding optimization problem with minimum Frobenius norm, also minimum nuclear norm; these properties provide a good mathematical justification for preferring the minimum-norm transformation over other possible solutions as the optimal transformation in ULDA; (iii) explicit necessary and sufficient conditions are provided to ensure that these minimal solutions lead to a larger ratio of between-class distance to within-class distance, thereby achieving larger discrimination in the reduced subspace than that in the original data space, and our numerical experiments show that these necessary and sufficient conditions hold true generally. Furthermore, a new and fast ULDA algorithm is developed, which is eigendecomposition-free and SVD-free, and its effectiveness is demonstrated by some real-world data sets. © 2011 Society for Industrial and Applied Mathematics.
 
dc.description.naturepublished_or_final_version
 
dc.identifier.citationSIAM Journal On Matrix Analysis And Applications, 2011, v. 32 n. 3, p. 820-844 [How to Cite?]
DOI: http://dx.doi.org/10.1137/100792007
 
dc.identifier.doihttp://dx.doi.org/10.1137/100792007
 
dc.identifier.epage844
 
dc.identifier.hkuros206381
 
dc.identifier.isiWOS:000295399200009
Funding AgencyGrant Number
NUSR-146-000-140-112
Funding Information:

The work of these authors was supported by NUS research grant R-146-000-140-112.

 
dc.identifier.issn0895-4798
2012 Impact Factor: 1.342
2012 SCImago Journal Rankings: 1.509
 
dc.identifier.issue3
 
dc.identifier.scopuseid_2-s2.0-80054044731
 
dc.identifier.spage820
 
dc.identifier.urihttp://hdl.handle.net/10722/155676
 
dc.identifier.volume32
 
dc.languageeng
 
dc.publisherSociety for Industrial and Applied Mathematics. The Journal's web site is located at http://epubs.siam.org/sam-bin/dbq/toclist/SIMAX
 
dc.publisher.placeUnited States
 
dc.relation.ispartofSIAM Journal on Matrix Analysis and Applications
 
dc.relation.referencesReferences in Scopus
 
dc.rightsSIAM Journal on Matrix Analysis and Applications. Copyright © Society for Industrial and Applied Mathematics
 
dc.rightsCreative Commons: Attribution 3.0 Hong Kong License
 
dc.subjectData Dimensionality Reduction
 
dc.subjectQR Factorization
 
dc.subjectUncorrelated Linear Discriminant Analysis
 
dc.titleCharacterization of all solutions for undersampled uncorrelated linear discriminant analysis problems
 
dc.typeArticle
 
<?xml encoding="utf-8" version="1.0"?>
<item><contributor.author>Chu, D</contributor.author>
<contributor.author>Goh, ST</contributor.author>
<contributor.author>Hung, YS</contributor.author>
<date.accessioned>2012-08-08T08:34:47Z</date.accessioned>
<date.available>2012-08-08T08:34:47Z</date.available>
<date.issued>2011</date.issued>
<identifier.citation>SIAM Journal On Matrix Analysis And Applications, 2011, v. 32 n. 3, p. 820-844</identifier.citation>
<identifier.issn>0895-4798</identifier.issn>
<identifier.uri>http://hdl.handle.net/10722/155676</identifier.uri>
<description.abstract>In this paper the uncorrelated linear discriminant analysis (ULDA) for undersampled problems is studied. The main contributions of the present work include the following: (i) all solutions of the optimization problem used for establishing the ULDA are parameterized explicitly; (ii) the optimal solutions among all solutions of the corresponding optimization problem are characterized in terms of both the ratio of between-class distance to within-class distance and the maximum likelihood classification, and it is proved that these optimal solutions are exactly the solutions of the corresponding optimization problem with minimum Frobenius norm, also minimum nuclear norm; these properties provide a good mathematical justification for preferring the minimum-norm transformation over other possible solutions as the optimal transformation in ULDA; (iii) explicit necessary and sufficient conditions are provided to ensure that these minimal solutions lead to a larger ratio of between-class distance to within-class distance, thereby achieving larger discrimination in the reduced subspace than that in the original data space, and our numerical experiments show that these necessary and sufficient conditions hold true generally. Furthermore, a new and fast ULDA algorithm is developed, which is eigendecomposition-free and SVD-free, and its effectiveness is demonstrated by some real-world data sets. &#169; 2011 Society for Industrial and Applied Mathematics.</description.abstract>
<language>eng</language>
<publisher>Society for Industrial and Applied Mathematics. The Journal&apos;s web site is located at http://epubs.siam.org/sam-bin/dbq/toclist/SIMAX</publisher>
<relation.ispartof>SIAM Journal on Matrix Analysis and Applications</relation.ispartof>
<rights>SIAM Journal on Matrix Analysis and Applications. Copyright &#169; Society for Industrial and Applied Mathematics</rights>
<rights>Creative Commons: Attribution 3.0 Hong Kong License</rights>
<subject>Data Dimensionality Reduction</subject>
<subject>QR Factorization</subject>
<subject>Uncorrelated Linear Discriminant Analysis</subject>
<title>Characterization of all solutions for undersampled uncorrelated linear discriminant analysis problems</title>
<type>Article</type>
<description.nature>published_or_final_version</description.nature>
<identifier.doi>10.1137/100792007</identifier.doi>
<identifier.scopus>eid_2-s2.0-80054044731</identifier.scopus>
<identifier.hkuros>206381</identifier.hkuros>
<relation.references>http://www.scopus.com/mlt/select.url?eid=2-s2.0-80054044731&amp;selection=ref&amp;src=s&amp;origin=recordpage</relation.references>
<identifier.volume>32</identifier.volume>
<identifier.issue>3</identifier.issue>
<identifier.spage>820</identifier.spage>
<identifier.epage>844</identifier.epage>
<identifier.isi>WOS:000295399200009</identifier.isi>
<publisher.place>United States</publisher.place>
<bitstream.url>http://hub.hku.hk/bitstream/10722/155676/1/Content.pdf</bitstream.url>
</item>
Author Affiliations
  1. The University of Hong Kong
  2. National University of Singapore