A covariate-dependent Cholesky decomposition for high-dimensional covariance regression

TL;DR

This paper introduces a covariate-dependent Cholesky decomposition framework for high-dimensional covariance matrix estimation, ensuring positive definiteness while incorporating covariate effects.

Contribution

It extends the modified Cholesky decomposition to model covariate effects on covariance matrices with a joint sparsity structure for high-dimensional data.

Findings

The proposed method guarantees positive definiteness of covariance estimators.

It demonstrates effective estimation in high-dimensional gene expression data.

Numerical experiments validate the method's accuracy and applicability.

Abstract

Estimation of covariance matrices is a fundamental problem in multivariate statistics. Recently, growing efforts have focused on incorporating covariate effects into these matrices, facilitating subject-specific estimation. Despite these advances, guaranteeing the positive definiteness of the resulting estimators remains a challenging problem. In this paper, we present a new varying-coefficient sequential regression framework that extends the modified Cholesky decomposition to model the positive definite covariance matrix as a function of subject-level covariates. To handle high-dimensional responses and covariates, we impose a joint sparsity structure that simultaneously promotes sparsity in both the covariate effects and the entries in the Cholesky factors that are modulated by these covariates. We approach parameter estimation with a blockwise coordinate descent algorithm, and investigate the $\ell_2$ convergence rate of the estimated parameters. The efficacy of the proposed method is demonstrated through numerical experiments and an application to a gene co-expression network study with brain cancer patients.