Covariance regularization by thresholding

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 $(\log p)/n\to 0$, 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.

Published in at http://dx.doi.org/10.1214/08-AOS600 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org)

References in corpus (6)

• Regularized estimation of large covariance matrices
• Sparse permutation invariant covariance estimation
• Network exploration via the adaptive LASSO and SCAD penalties
• Operator norm consistent estimation of large-dimensional sparse covariance matrices
• Sparse Principal Components Analysis
• Sparse estimation of large covariance matrices via a nested Lasso penalty

Cited by in corpus (3)

• Sparse permutation invariant covariance estimation
• Operator norm consistent estimation of large-dimensional sparse covariance matrices
• Structural shrinkage of nonparametric spectral estimators for multivariate time series