Covariance matrix estimation in high dimensions is a fundamental problem in machine learning and signal processing. A common structural assumption used to mitigate the challenges posed by high dimensionality is sparsity, which posits that most variable pairs exhibit negligible correlations. In this paper, we revisit the classical problem of positive definite sparse covariance estimation (PDSCE) introduced by Rothman (2012). Unlike many earlier approaches, this formulation incorporates a logarithmic barrier, which guarantees that the resulting covariance estimator is positive definite and thereby ensures the well-posedness of the estimation problem. However, the inclusion of the logarithmic barrier also leads to nontrivial optimization difficulties. To overcome these difficulties, we propose a dual proximal gradient method (DPGM) for solving the PDSCE problem. In contrast to existing primal-space approaches, DPGM operates directly in the dual space. This dual perspective provides several key advantages. First, DPGM significantly reduces computational costs, because positive definiteness is preserved automatically and no iterative subproblem solvers are required. Second, compared with primal optimization algorithms, DPGM offers stronger theoretical guarantees, including principled step size selection and improved iteration complexity. Extensive numerical experiments demonstrate that DPGM consistently outperforms existing methods, which confirms its effectiveness and scalability for high-dimensional sparse covariance estimation.

Covariance matrix estimation is a fundamental problem in multivariate data analysis, which becomes particularly challenging in high-dimensional settings due to the curse of dimensionality. To enhance estimation accuracy, structural regularization is often imposed on the precision matrix (the inverse covariance matrix) for covariance selection. In this paper, we study covariance selection in a distributed setting, where data is spread across a network of agents. We formulate the problem as a Gaussian maximum likelihood estimation problem with structural penalties and propose a novel algorithmic framework called NetGGM. Unlike existing methods that rely on a central coordinator, NetGGM operates in a fully decentralized manner with low computational complexity. We provide theoretical guarantees showing that NetGGM converges linearly to the global optimum while ensuring consensus among agents. Numerical experiments validate its convergence properties and demonstrate that it outperforms state-of-the-art methods in precision matrix estimation.