Inexact Krylov Subspace Algorithms for Large Matrix Exponential Eigenproblem from Dimensionality Reduction

Abstract

Matrix exponential discriminant analysis (EDA) is a generalized discriminant analysis method based on matrix exponential. It can essentially overcome the intrinsic difficulty of small sample size problem that exists in the classical linear discriminant analysis (LDA). However, for data with high dimension, one has to solve a large matrix exponential eigenproblem in this method, and the time complexity is dominated by the computation of exponential of large matrices. In this paper, we propose two inexact Krylov subspace algorithms for solving the large matrix exponential eigenproblem effectively. The contribution of this work is threefold. First, we consider how to compute matrix exponential-vector products efficiently, which is the key step in the Krylov subspace method. Second, we compare the discriminant analysis criterion of EDA and that of LDA from a theoretical point of view. Third, we establish a relationship between the accuracy of the approximate eigenvectors and the distance to nearest neighbour classifier, and show why the matrix exponential eigenproblem can be solved approximately in practice. Numerical experiments on some real-world databases show superiority of our new algorithms over many state-of-the-art algorithms for face recognition.