A Direct Approach for Sparse Quadratic Discriminant Analysis

TL;DR

This paper introduces DA-QDA, a new method for high-dimensional quadratic discriminant analysis that directly estimates key discriminant quantities, achieves consistency under sparsity, and is computationally efficient.

Contribution

The paper proposes DA-QDA, a novel direct estimation approach for high-dimensional QDA that is both theoretically consistent and computationally faster than existing methods.

Findings

Estimates quadratic interactions and linear indices accurately under sparsity.

Misclassification rate converges to Bayes optimal even in high dimensions.

Algorithm based on ADMM outperforms competitors in speed.

Abstract

Quadratic discriminant analysis (QDA) is a standard tool for classification due to its simplicity and flexibility. Because the number of its parameters scales quadratically with the number of the variables, QDA is not practical, however, when the dimensionality is relatively large. To address this, we propose a novel procedure named DA-QDA for QDA in analyzing high-dimensional data. Formulated in a simple and coherent framework, DA-QDA aims to directly estimate the key quantities in the Bayes discriminant function including quadratic interactions and a linear index of the variables for classification. Under appropriate sparsity assumptions, we establish consistency results for estimating the interactions and the linear index, and further demonstrate that the misclassification rate of our procedure converges to the optimal Bayes risk, even when the dimensionality is exponentially high with respect to the sample size. An efficient algorithm based on the alternating direction method of multipliers (ADMM) is developed for finding interactions, which is much faster than its competitor in the literature. The promising performance of DA-QDA is illustrated via extensive simulation studies and the analysis of four real datasets.