On the dimension effect of regularized linear discriminant analysis

TL;DR

This paper analyzes how the dimension of data impacts the performance of LDA and RLDA classifiers in high-dimensional settings, deriving explicit asymptotic error formulas and proposing bias-corrected classifiers for improved accuracy.

Contribution

It provides explicit asymptotic misclassification error expressions for LDA and RLDA in high dimensions and introduces bias correction methods to enhance classifier performance.

Findings

Bias-corrected classifiers outperform standard LDA and RLDA.

Explicit formulas reveal the impact of data dimension on classification error.

Simulation results validate theoretical insights.

Abstract

This paper studies the dimension effect of the linear discriminant analysis (LDA) and the regularized linear discriminant analysis (RLDA) classifiers for large dimensional data where the observation dimension $p$ is of the same order as the sample size $n$. More specifically, built on properties of the Wishart distribution and recent results in random matrix theory, we derive explicit expressions for the asymptotic misclassification errors of LDA and RLDA respectively, from which we gain insights of how dimension affects the performance of classification and in what sense. Motivated by these results, we propose adjusted classifiers by correcting the bias brought by the unequal sample sizes. The bias-corrected LDA and RLDA classifiers are shown to have smaller misclassification rates than LDA and RLDA respectively. Several interesting examples are discussed in detail and the theoretical results on dimension effect are illustrated via extensive simulation studies.