Optimal Discriminant Analysis in High-Dimensional Latent Factor Models

TL;DR

This paper introduces an optimal high-dimensional classification method based on principal components within a latent-variable model, providing theoretical guarantees and demonstrating superior empirical performance.

Contribution

It formulates a latent-variable model to justify PCA-based classification, deriving optimal convergence rates and offering a data-driven approach for selecting the number of components.

Findings

Derived explicit convergence rates for the PC-based classifier.

Proved the rates are minimax optimal up to logarithmic factors.

Demonstrated superior performance over existing methods in simulations and real data.

Abstract

In high-dimensional classification problems, a commonly used approach is to first project the high-dimensional features into a lower dimensional space, and base the classification on the resulting lower dimensional projections. In this paper, we formulate a latent-variable model with a hidden low-dimensional structure to justify this two-step procedure and to guide which projection to choose. We propose a computationally efficient classifier that takes certain principal components (PCs) of the observed features as projections, with the number of retained PCs selected in a data-driven way. A general theory is established for analyzing such two-step classifiers based on any projections. We derive explicit rates of convergence of the excess risk of the proposed PC-based classifier. The obtained rates are further shown to be optimal up to logarithmic factors in the minimax sense. Our theory allows the lower-dimension to grow with the sample size and is also valid even when the feature dimension (greatly) exceeds the sample size. Extensive simulations corroborate our theoretical findings. The proposed method also performs favorably relative to other existing discriminant methods on three real data examples.