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

Area of Science:

• Statistics
• Machine Learning
• Neuroscience

Background:

• Covariance measures marginal dependence but modeling high-dimensional, heterogeneous covariances is challenging.
• Existing methods struggle with the large parameter space and positive-definiteness constraints of covariance matrices.

Purpose of the Study:

• To propose a regularized estimation method for regression coefficients of covariances.
• To address constraints for positive definiteness in conditional average covariance matrices.
• To develop an estimator that simultaneously achieves sparsity and positive definiteness.

Main Methods:

• A regularized estimation method for regression coefficients of covariances.
• Incorporation of sufficient and necessary constraints for positive definiteness.
• An alternating direction method of multipliers (ADMM) algorithm to solve the optimization problem.

Main Results:

• The proposed estimator satisfies both sparsity and positive-definite properties.
• Convergence of the ADMM algorithm is demonstrated.
• Convergence rates for regression coefficients and heterogeneous covariances are derived.

Conclusions:

• The novel method effectively models high-dimensional, heterogeneous covariances.
• The ADMM algorithm provides a robust solution for the constrained optimization problem.
• The approach is validated through simulations and a brain connectivity case study.