Adaptive estimation of covariance matrices via Cholesky decomposition

TL;DR

This paper introduces ChoSelect, a new method for estimating large covariance matrices using Cholesky decomposition, which adapts to sparsity and ordering of variables, and combines with Lasso for improved efficiency.

Contribution

The paper presents a novel covariance estimation procedure based on Cholesky factor sparsity, with theoretical guarantees and a combined Lasso approach for flexibility.

Findings

ChoSelect achieves non-asymptotic oracle inequalities.

The combined method with Lasso is consistent under weaker assumptions.

Numerical experiments demonstrate practical effectiveness.

Abstract

This paper studies the estimation of a large covariance matrix. We introduce a novel procedure called ChoSelect based on the Cholesky factor of the inverse covariance. This method uses a dimension reduction strategy by selecting the pattern of zero of the Cholesky factor. Alternatively, ChoSelect can be interpreted as a graph estimation procedure for directed Gaussian graphical models. Our approach is particularly relevant when the variables under study have a natural ordering (e.g. time series) or more generally when the Cholesky factor is approximately sparse. ChoSelect achieves non-asymptotic oracle inequalities with respect to the Kullback-Leibler entropy. Moreover, it satisfies various adaptive properties from a minimax point of view. We also introduce and study a two-stage procedure that combines ChoSelect with the Lasso. This last method enables the practitioner to choose his own trade-off between statistical efficiency and computational complexity. Moreover, it is consistent under weaker assumptions than the Lasso. The practical performances of the different procedures are assessed on numerical examples.