Generalized Discriminant Analysis: A Matrix Exponential Approach

Linear discriminant analysis (LDA) is well known as a powerful tool for discriminant analysis. In the case of a small training data set, however, it cannot directly be applied to high-dimensional data. This case is the so-called small-sample-size or undersampled problem. In this paper, we propose an exponential discriminant analysis (EDA) technique to overcome the undersampled problem. The advantages of EDA are that, compared with principal component analysis (PCA) $+$ LDA, the EDA method can extract the most discriminant information that was contained in the null space of a within-class scatter matrix, and compared with another LDA extension, i.e., null-space LDA (NLDA), the discriminant information that was contained in the non-null space of the within-class scatter matrix is not discarded. Furthermore, EDA is equivalent to transforming original data into a new space by distance diffusion mapping, and then, LDA is applied in such a new space. As a result of diffusion mapping, the margin between different classes is enlarged, which is helpful in improving classification accuracy. Comparisons of experimental results on different data sets are given with respect to existing LDA extensions, including PCA $+$ LDA, LDA via generalized singular value decomposition, regularized LDA, NLDA, and LDA via QR decomposition, which demonstrate the effectiveness of the proposed EDA method.