Structural Effect and Spectral Enhancement of High-Dimensional Regularized Linear Discriminant Analysis

TL;DR

This paper analyzes how data structure influences high-dimensional regularized LDA performance, introduces SEDA to optimize data structure via spectral adjustment, and demonstrates improved accuracy through theoretical and empirical results.

Contribution

It provides a non-asymptotic misclassification rate approximation, proposes the SEDA algorithm with spectral adjustment, and develops new eigenvector theory for high-dimensional LDA.

Findings

SEDA outperforms existing LDA methods in accuracy

Spectral adjustment improves classification and dimensionality reduction

Theoretical analysis guides optimal parameter selection

Abstract

Regularized linear discriminant analysis (RLDA) is a widely used tool for classification and dimensionality reduction, but its performance in high-dimensional scenarios is inconsistent. Existing theoretical analyses of RLDA often lack clear insight into how data structure affects classification performance. To address this issue, we derive a non-asymptotic approximation of the misclassification rate and thus analyze the structural effect and structural adjustment strategies of RLDA. Based on this, we propose the Spectral Enhanced Discriminant Analysis (SEDA) algorithm, which optimizes the data structure by adjusting the spiked eigenvalues of the population covariance matrix. By developing a new theoretical result on eigenvectors in random matrix theory, we derive an asymptotic approximation on the misclassification rate of SEDA. The bias correction algorithm and parameter selection strategy are then obtained. Experiments on synthetic and real datasets show that SEDA achieves higher classification accuracy and dimensionality reduction compared to existing LDA methods.