Document Type

Journal Article

Department/Unit

Department of Mathematics

Title

Fast algorithms for the generalized foley–sammon discriminant analysis

Language

English

Abstract

Linear discriminant analysis (LDA) is one of the most popular approaches for feature extraction and dimension reduction to overcome the curse of the dimensionality of the highdimensional data in many applications of data mining, machine learning, and bioinformatics. In this paper, we made two main contributions to an important LDA scheme, the generalized Foley-Sammon transform (GFST) [Foley and Sammon, IEEE Trans. Comput., 24 (1975), pp. 281-289; Guo et al., Pattern Recognition Lett., 24 (2003), pp. 147-158] or a trace ratio model [Wang et al., Proceedings of the International Conference on Computer Vision and Pattern Recognition, 2007, pp. 1-8] and its regularized GFST (RGFST), which handles the undersampled problem that involves small samples size n, but with high number of features N (N > n) and arises frequently in many modern applications. Our first main result is to establish an equivalent reduced model for the RGFST which effectively improves the computational overhead. The iteration method proposed by Wang et al. is applied to solve the GFST or the reduced RGFST. It has been proven by Wang et al. that this iteration converges globally and fast convergence was observed numerically, but there is no theoretical analysis on the convergence rate thus far. Our second main contribution completes this important and missing piece by proving the quadratic convergence even under two kinds of inexact computations. Practical implementations, including computational complexity and storage requirements, are also discussed. Our experimental results on several real world data sets indicate the efficiency of the algorithm and the advantages of the GFST model in classification. Copyright © 2010 Society for Industrial and Applied Mathematics.

Keywords

Dimension reduction, Foley-Sammon Transform, Global convergence, Linear discriminant analysis, Quadratic convergence, Regularization

Publication Date

2010

Source Publication Title

SIAM Journal on Matrix Analysis and Applications

Volume

31

Issue

4

Start Page

1584

End Page

1605

Publisher

Society for Industrial and Applied Mathematics

DOI

10.1137/080720863

Link to Publisher's Edition

http://dx.doi.org/10.1137/080720863

ISSN (print)

08954798

ISSN (electronic)

10957162

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