The Masked Sample Covariance Estimator: An Analysis via the Matrix Laplace Transform

Covariance estimation becomes challenging in the regime where the number p of variables outstrips the number n of samples available to construct the estimate. One way to circumvent this problem is to assume that the covariance matrix is nearly sparse and to focus on estimating only the significant entries. To analyze this approach, Levina and Vershynin (2011) introduce a formalism called masked covariance estimation, where each entry of the sample covariance estimator is reweighed to reflect an a priori assessment of its importance. This paper provides a new analysis of the masked sample covariance estimator based on the matrix Laplace transform method. The main result applies to general subgaussian distributions. Specialized to the case of a Gaussian distribution, the theory offers qualitative improvements over earlier work. For example, the new results show that n = O(B log ^2 p) samples suffice to estimate a banded covariance matrix with bandwidth B up to a relative spectral-norm error, in contrast to the sample complexity n = O(B log ^5 p) obtained by Levina and Vershynin.