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

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.