A Spatial-Sign based Direct Approach for High Dimensional Sparse Quadratic Discriminant Analysis

Abstract: In this paper, we study the problem of high-dimensional sparse quadratic discriminant analysis (QDA). We propose a novel classification method, termed SSQDA, which is constructed via constrained convex optimization based on the sample spatial median and spatial sign covariance matrix under the assumption of an elliptically symmetric distribution. The proposed classifier is shown to achieve the optimal convergence rate over a broad class of parameter spaces, up to a logarithmic factor. Extensive simulation studies and real data applications demonstrate that SSQDA is both robust and efficient, particularly in the presence of heavy-tailed distributions, highlighting its practical advantages in high-dimensional classification tasks.