High Dimensional Quadratic Discriminant Analysis: Optimality and Phase Transitions

Abstract

Consider a two-class classification problem where we observe samples $(X_i, Y_i)$ for i = 1, ..., n, $X_i \in R^p$ and $Y_i$ in {0, 1}. Given $Y_i = k$, $X_i$ is assumed to follow a multivariate normal distribution with mean $μ_k \in R^k$ and covariance matrix $Σ_k$, k=0,1. Supposing a new sample X from the same mixture is observed, our goal is to estimate its class label Y. Such a high-dimensional classification problem has been studied thoroughly when Sigma_0 = Sigma_1. However, the discussions over the case $Σ_0 \neq Σ_1$ are much less over the years. This paper presents the quadratic discriminant analysis (QDA) for the weak signals (QDAw) algorithm, and the QDA with feature selection (QDAfs) algorithm. QDAfs applies Partial Correlation Screening to estimate $\hatΩ_0$ and $\hatΩ_1$, and then applies a hard-thresholding on the diagonals of $\hatΩ_0 - \hatΩ_1$. QDAfs further includes the linear term $d^T X$, where d is achieved by a hard-thresholding on $\hatΩ_1\hatμ_1 - \hatΩ_0\hatμ_0$. We further propose the rare and weak model to model the signals in $Ω_0 - Ω_1$ and $μ_0 - μ_1$. Based on the signal weakness and sparsity in $μ_0 - μ_1$, we propose two ways to estimate labels: 1) QDAw for weak but dense signals; 2) QDAfs for relatively strong but sparse signals. We figure out the classification boundary on the 4-dim parameter space: 1) Region of possibility, where either QDAw or QDAfs will achieve a mis-classification error rate of 0; 2) Region of impossibility, where all classifiers will have a constant error rate. The numerical results from real datasets support our theories and demonstrate the necessity and superiority of using QDA over LDA for classification.