Asymptotic Generalization Bound of Fisher's Linear Discriminant Analysis
This work addresses a theoretical gap in statistical pattern recognition by extending FLDA analysis to high-dimensional settings, offering insights for researchers in machine learning and statistics, though it is incremental as it builds on classical results.
The paper tackles the limitations of Fisher's linear discriminant analysis (FLDA) by deriving an asymptotic generalization bound using random matrix theory, applicable when dimensionality and sample size increase proportionally, and provides a quantitative description in terms of the ratio γ = D/N and population discrimination power, leading to an upper bound on generalization error for binary classification.
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 size $N$ are proportionally large; 2) it does not provide a quantitative description on how the generalization ability of FLDA is affected by $D$ and $N$. In this paper, we present an asymptotic generalization analysis of FLDA based on random matrix theory, in a setting where both $D$ and $N$ increase and $D/N\longrightarrowγ\in[0,1)$. The obtained lower bound of the generalization discrimination power 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 $γ=D/N$ and the population discrimination power. Besides, the discrimination power bound also leads to an upper bound on the generalization error of binary-classification with FLDA.