  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.
