High-dimensional covariance matrix estimation using a low-rank and diagonal decomposition

TL;DR

This paper introduces a new method for estimating high-dimensional covariance matrices by decomposing them into low-rank and diagonal parts, with proven consistency and practical algorithms demonstrated through simulations and financial data.

Contribution

It proposes a novel low-rank plus diagonal decomposition approach for covariance estimation, with theoretical consistency results and an efficient coordinate descent algorithm.

Findings

Estimator performs well in Kullback-Leibler loss on simulated data.

Method improves portfolio selection in stock return analysis.

Algorithm effectively recovers covariance structure in high dimensions.

Abstract

We study high-dimensional covariance/precision matrix estimation under the assumption that the covariance/precision matrix can be decomposed into a low-rank component L and a diagonal component D. The rank of L can either be chosen to be small or controlled by a penalty function. Under moderate conditions on the population covariance/precision matrix itself and on the penalty function, we prove some consistency results for our estimators. A blockwise coordinate descent algorithm, which iteratively updates L and D, is then proposed to obtain the estimator in practice. Finally, various numerical experiments are presented: using simulated data, we show that our estimator performs quite well in terms of the Kullback-Leibler loss; using stock return data, we show that our method can be applied to obtain enhanced solutions to the Markowitz portfolio selection problem.