Covariance regularization by thresholding

TL;DR

This paper introduces a method for regularizing covariance matrices through hard thresholding, providing consistency results, explicit rates, and practical threshold selection, with applications to climate data.

Contribution

It presents a novel thresholding approach for covariance regularization with theoretical guarantees and a practical resampling scheme for threshold selection.

Findings

Thresholded covariance estimates are consistent under sparsity and Gaussianity.

Explicit convergence rates are derived for the thresholded estimator.

Simulation and climate data examples demonstrate the method's effectiveness.

Abstract

This paper considers regularizing a covariance matrix of $p$ variables estimated from $n$ observations, by hard thresholding. We show that the thresholded estimate is consistent in the operator norm as long as the true covariance matrix is sparse in a suitable sense, the variables are Gaussian or sub-Gaussian, and $(\log p)/n\to0$, and obtain explicit rates. The results are uniform over families of covariance matrices which satisfy a fairly natural notion of sparsity. We discuss an intuitive resampling scheme for threshold selection and prove a general cross-validation result that justifies this approach. We also compare thresholding to other covariance estimators in simulations and on an example from climate data.