On general matrix exponential discriminant analysis methods for high dimensionality reduction

Abstract

Recently, some matrix exponential-based discriminant analysis methods were proposed for high dimensionality reduction. It has been shown that they often have more discriminant power than the corresponding discriminant analysis methods. However, one has to solve some large-scale matrix exponential eigenvalue problems which constitutes the bottleneck in this type of methods. The main contribution of this paper is twofold: First, we propose a framework of fast implementation on general matrix exponential-based discriminant analysis methods. The key is to equivalently transform large-scale matrix computation problems into much smaller ones. On the other hand, it was mentioned in Wang et al. (IEEE Trans Image Process 23:920–930, 2014) that the exponential model is more reliable than the original one and suppresses the sensitivity to pertubations. However, the interpretation is only heuristic, and to the best of our knowledge, there is no theoretical justification for reliability and stability of the matrix discriminant analysis methods. To fill in this gap, the second contribution of our work is to provide stability analysis for the fast exponential discriminant analysis method from a theoretical point of view. Numerical experiments illustrate the numerical behavior of the proposed algorithm, and demonstrate that our algorithm is more stable than many state-of-the-art algorithms for high dimensionality reduction.