Positive-definite regularized estimation for high-dimensional covariance on scalar regression

Covariance is an important measure of marginal dependence among variables. However, heterogeneity in subject covariances and regression models for high-dimensional covariance matrices is not well studied. Compared to regression analysis for conditional means, modeling high-dimensional covariances is much more challenging due to the large set of free parameters and the intrinsic positive-definite property that puts constraints on the regression parameters. In this paper, we propose a regularized estimation method for the regression coefficients of covariances under sufficient and necessary constraints for the positive definiteness of the conditional average covariance matrices given covariates. The proposed estimator satisfies the sparsity and positive-definite properties simultaneously. An alternating direction method of multipliers (ADMM) algorithm is proposed to solve the constrained and regularized optimization problem. We show the convergence of the proposed ADMM algorithm and derive the convergence rates of the proposed estimators for the regression coefficients and the heterogeneous covariances. The proposed method is evaluated by simulation studies, and its practical application is demonstrated by a case study on brain connectivity.