A Convex Optimization Approach to High-Dimensional Sparse Quadratic Discriminant Analysis

TL;DR

This paper introduces a convex optimization method called SDAR for high-dimensional sparse Quadratic Discriminant Analysis, establishing its optimal convergence rates and demonstrating its effectiveness through simulations and real data analysis.

Contribution

The paper develops a rate-optimal convex optimization algorithm for high-dimensional sparse QDA, extending it to multi-group and Gaussian copula models.

Findings

SDAR achieves minimax optimal convergence rates.

Simulation studies show SDAR's strong numerical performance.

Application to cancer data illustrates practical utility.

Abstract

In this paper, we study high-dimensional sparse Quadratic Discriminant Analysis (QDA) and aim to establish the optimal convergence rates for the classification error. Minimax lower bounds are established to demonstrate the necessity of structural assumptions such as sparsity conditions on the discriminating direction and differential graph for the possible construction of consistent high-dimensional QDA rules. We then propose a classification algorithm called SDAR using constrained convex optimization under the sparsity assumptions. Both minimax upper and lower bounds are obtained and this classification rule is shown to be simultaneously rate optimal over a collection of parameter spaces, up to a logarithmic factor. Simulation studies demonstrate that SDAR performs well numerically. The algorithm is also illustrated through an analysis of prostate cancer data and colon tissue data. The methodology and theory developed for high-dimensional QDA for two groups in the Gaussian setting are also extended to multi-group classification and to classification under the Gaussian copula model.