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

2026-04-08

AI Summary

This study addresses the challenges of positive definiteness and scalability in estimating covariance matrices driven by high-dimensional covariates. The authors propose a covariate-dependent Cholesky decomposition framework that models the covariance matrix as a function of individual covariates through varying-coefficient sequential regressions. For the first time, a joint sparsity structure is introduced into the Cholesky factors, ensuring both positive definiteness and adaptability to high-dimensional settings. By integrating modified Cholesky decomposition, joint sparsity regularization, and a block coordinate descent algorithm, the method achieves an ℓ₂ convergence rate, as established by theoretical analysis. Numerical experiments and an application to gene co-expression networks in brain cancer demonstrate that the approach yields accurate and stable covariance estimates in high-dimensional scenarios.

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.

Problem

• covariance regression
• positive definiteness
• high-dimensional
• Cholesky decomposition
• covariate-dependent

Innovation

• covariate-dependent Cholesky decomposition
• high-dimensional covariance regression
• joint sparsity
• positive definite covariance
• varying-coefficient model