Asymptotic Generalization Bound of Fisher's Linear Discriminant Analysis

Fisher's linear discriminant analysis (FLDA) is an important dimension reduction method in statistical pattern recognition. It has been shown that FLDA is asymptotically Bayes optimal under the homoscedastic Gaussian assumption. However, this classical result has the following two major limitations: 1) it holds only for a fixed dimensionality $D$, and thus does not apply when $D$ and the training sample number $N$ are proportionally large; 2) it does not provide a quantitative description on the performance of FLDA. In this paper, we present an asymptotic generalization analysis of FLDA based on random matrix theory in the setting where both $D$ and $N$ increase and $\lim D/N=\gamma\in[0,1)$. The obtained asymptotic generalization bound overcomes both limitations of the classical result, i.e., it is applicable when $D$ and $N$ are proportionally large and provides a quantitative description of the generalization ability of FLDA in terms of the ratio $D/N$ and the population discrimination power.