Second-Order Sparse Sufficient Dimension Reduction with Applications to Quadratic Discriminant Analysis

Motivated by exploratory data analysis, sufficient dimension reduction (SDR) methods, especially inverse regression methods such as sliced inverse regression (SIR) and sliced averaged variance estimation (SAVE), have been central to multivariate analysis for more than three decades. Despite their popularity, the extension of these methods to high-dimensional settings remains challenging. This paper addresses the computational and theoretical limitations of the less explored second-order SDR methods in high dimensions. We introduce a novel approach for sparse subspace estimation that utilizes quadratic convex optimization and leverages the group structure of tensor parameters, achieving significant parameter reduction. The proposed two-step estimator achieves consistency in dimension selection, variable selection, and subspace estimation at a high convergence rate under mild conditions. The effectiveness and efficiency of the proposed method are further demonstrated through extensive simulation studies and real data examples. Additionally, the proposed sparse second-order SDR techniques are applied to quadratic discriminant analysis (QDA) problems and provide practitioners with a sparse projective classification method with theoretical guarantees and strong empirical performance.