The Masked Sample Covariance Estimator: An Analysis via Matrix Concentration Inequalities

TL;DR

This paper analyzes the masked sample covariance estimator using matrix concentration inequalities, demonstrating improved sample complexity bounds for estimating structured covariance matrices, especially Gaussian banded matrices.

Contribution

It provides a new theoretical analysis of masked covariance estimation via matrix concentration inequalities, improving sample complexity bounds for Gaussian banded matrices.

Findings

For Gaussian distributions, n = O(B log^2 p) samples suffice for accurate estimation.

The analysis extends to general distributions with at least four moments.

The results show qualitative improvements over previous bounds.

Abstract

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 reweighted to reflect an a priori assessment of its importance. This paper provides a short analysis of the masked sample covariance estimator by means of a matrix concentration inequality. The main result applies to general distributions with at least four moments. 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.