A new approach to Cholesky-based covariance regularization in high dimensions

TL;DR

This paper introduces a novel regression-based Cholesky factorization approach for covariance matrix estimation in high-dimensional settings, ensuring positive definiteness and computational efficiency.

Contribution

It proposes a new covariance regularization method based on Cholesky factor regression interpretation, providing positive definite estimators with comparable cost to existing methods.

Findings

Produces positive definite banded covariance estimators

Establishes theoretical links between banded Cholesky factors and inverse covariance

Demonstrates competitive performance in simulations and real data

Abstract

In this paper we propose a new regression interpretation of the Cholesky factor of the covariance matrix, as opposed to the well known regression interpretation of the Cholesky factor of the inverse covariance, which leads to a new class of regularized covariance estimators suitable for high-dimensional problems. Regularizing the Cholesky factor of the covariance via this regression interpretation always results in a positive definite estimator. In particular, one can obtain a positive definite banded estimator of the covariance matrix at the same computational cost as the popular banded estimator proposed by Bickel and Levina (2008b), which is not guaranteed to be positive definite. We also establish theoretical connections between banding Cholesky factors of the covariance matrix and its inverse and constrained maximum likelihood estimation under the banding constraint, and compare the numerical performance of several methods in simulations and on a sonar data example.