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Linear discriminant analysis has been widely studied in data mining and pattern recognition. However, when performing the eigen-decomposition on the matrix pair (within-class scatter matrix and between-class scatter matrix) in some cases, one can find that there exist some degenerated eigenvalues, thereby resulting in indistinguishability of information from the eigen-subspace corresponding to some degenerated eigenvalue. In order to address this problem, we revisit linear discriminant analysis in this paper and propose a stable and effective algorithm for linear discriminant analysis in terms of an optimization criterion. By discussing the properties of the optimization criterion, we find that the eigenvectors in some eigen-subspaces may be indistinguishable if the degenerated eigenvalue occurs. Inspired from the idea of the maximum margin criterion (MMC), we embed MMC into the eigen-subspace corresponding to the degenerated eigenvalue to exploit discriminability of the eigenvectors in the eigen-subspace. Since the proposed algorithm can deal with the degenerated case of eigenvalues, it not only handles the small-sample-size problem but also enables us to select projection vectors from the null space of the between-class scatter matrix. Extensive experiments on several face images and microarray data sets are conducted to evaluate the proposed algorithm in terms of the classification performance, and experimental results show that our method has smaller standard deviations than other methods in most cases.

A Complete Subspace Analysis of Linear Discriminant Analysis and Its Robust Implementation