Phase Transitions for High-Dimensional Quadratic Discriminant Analysis with Rare and Weak Signals

Consider a two-class classification problem where we observe samples $(X_i, Y_i)$ for i = 1, ..., n, $X_i \in \mathcal{R}^p$ and $Y_i \in \{0, 1\}$. Given $Y_i = k$, $X_i$ is assumed to follow a multivariate normal distribution with mean $\mu_k \in \mathcal{R}^k$ and covariance matrix $\Sigma_k$, $k=0,1$. Supposing a new sample $X$ from the same mixture is observed, our goal is to estimate its class label $Y$. The difficulty lies in the rarity and weakness of the differences in the mean vector and in the covariance matrices. By incorporating the quadratic terms $\Omega_k=\Sigma^{-1}_k$ from the two classes, we formulate the likelihood-based classification as a Quadratic Discriminant Analysis (QDA) problem. Hence, we propose the QDA classification method with the feature-selection step. Compared with recent work on the linear case (LDA) with $\Omega_k$ assumed to be the same, the current setting is much more general. The numerical results from real datasets support our theories and demonstrate the necessity and superiority of using QDA over LDA for classification under the rare and weak model. We set up a rare and weak model for both the mean vector and the precision matrix. With the model parameters, we clearly depict the boundary separating the region of successful classification from the region of unsuccessful classification of the newly proposed QDA with a feature-selection method, for the two cases that $\mu_k$ is either known or unknown. We also explore the region of successful classification of the QDA approach when both $\mu_k$ and $\Omega_k$ are unknown. The results again suggest that the quadratic term has a major influence over the LDA for the classification decision and classification accuracy.