Optimal Linear Classification via Eigenvalue Shrinkage: The Case of Additive Noise

In this paper, we consider the general problem of testing the mean of two high-dimensional distributions with a common, unknown covariance using a linear classifier. Traditionally such a classifier is formed from the sample covariance matrix of some given training data, but, as is well-known, the performance of this classifier is poor when the number of training data $n$ is not much larger than the data dimension $p$. We thus seek a covariance estimator to replace sample covariance. To account for the fact that $n$ and $p$ may be of comparable size, we adopt the "large-dimensional asymptotic model" in which $n$ and $p$ go to infinity in a fixed ratio. Under this assumption, we identify a covariance estimator that is detection-theoretic optimal within the general shrinkage class of C. Stein, and we give consistent estimates for the corresponding classifier's type-I and type-II errors.