🤖 AI Summary
This paper addresses the challenging problem of jointly estimating high-dimensional sparse partial correlation and inverse covariance matrices. We propose a two-stage joint partial regression method: within a per-variable linear regression framework, we first embed both sparsity-inducing regularization and positive definite cone projection into a unified optimization objective, thereby simultaneously ensuring the positive definiteness and sparsity of the estimated matrix. The method is efficiently implemented via a proximal splitting algorithm. We establish theoretical guarantees showing that the inverse covariance estimator achieves the optimal convergence rate, while the partial correlation estimator attains a strictly tighter error bound than existing state-of-the-art methods. Extensive experiments on synthetic and real-world high-dimensional datasets demonstrate that our approach significantly outperforms mainstream competitors—including Graphical Lasso and CLIME—in both estimation accuracy and computational efficiency. This work provides a new paradigm for high-dimensional graphical model learning that combines strong theoretical foundations with practical effectiveness.
📝 Abstract
We present a new method for estimating high-dimensional sparse partial correlation and inverse covariance matrices, which exploits the connection between the inverse covariance matrix and linear regression. The method is a two-stage estimation method wherein each individual feature is regressed on all other features while positive semi-definiteness is enforced simultaneously. We provide statistical rates of convergence for the proposed method which match, and improve upon, the state-of-the-art for inverse covariance and partial correlation matrix estimation, respectively. We also propose an efficient proximal splitting algorithm for numerically computing the estimate. The effectiveness of the proposed method is demonstrated on both synthetic and real-world data.