Positive Definite Sparse Covariance Estimation via Dual Space Optimization

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.