  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.
