  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.
