Sharp optimality for high-dimensional covariance testing under sparse signals

This paper considers one-sample testing of a high-dimensional covariance matrix by deriving the detection boundary as a function of the signal sparsity and signal strength under the sparse alternative hypotheses. It first shows that the optimal detection boundary for testing sparse means is the minimax detection lower boundary for testing the covariance matrix. A multilevel thresholding test is proposed and is shown to be able to attain the detection lower boundary over a substantial range of the sparsity parameter, implying that the multilevel thresholding test is sharp optimal in the minimax sense over the range. The asymptotic distribution of the multilevel thresholding statistic for covariance matrices is derived under both Gaussian and non-Gaussian distributions by developing a novel U-statistic decomposition in conjunction with the matrix blocking and the coupling techniques to handle the complex dependence among the elements of the sample covariance matrix. The superiority in the detection boundary of the multilevel thresholding test over the existing tests is also demonstrated.