Article: Characterization of all solutions for undersampled uncorrelated linear discriminant analysis problems

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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
2011 Impact Factor: 1.368
2011 SCImago Journal Rankings: 0.069
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 Field
Value
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
2011 Impact Factor: 1.368
2011 SCImago Journal Rankings: 0.069
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
Author Affiliations
  1. The University of Hong Kong
  2. National University of Singapore