Graphical Nonconvex Optimization for Optimal Estimation in Gaussian Graphical Models
This addresses the limitation of the graphical lasso in not achieving optimal convergence rates, offering a computationally tractable solution for statisticians and data scientists working with Gaussian graphical models, though it appears incremental as it builds on existing methods.
The paper tackles the problem of learning high-dimensional Gaussian graphical models, proposing a graphical nonconvex optimization method that achieves the oracle rate of convergence and outperforms other popular methods in numerical studies.
We consider the problem of learning high-dimensional Gaussian graphical models. The graphical lasso is one of the most popular methods for estimating Gaussian graphical models. However, it does not achieve the oracle rate of convergence. In this paper, we propose the graphical nonconvex optimization for optimal estimation in Gaussian graphical models, which is then approximated by a sequence of convex programs. Our proposal is computationally tractable and produces an estimator that achieves the oracle rate of convergence. The statistical error introduced by the sequential approximation using the convex programs are clearly demonstrated via a contraction property. The rate of convergence can be further improved using the notion of sparsity pattern. The proposed methodology is then extended to semiparametric graphical models. We show through numerical studies that the proposed estimator outperforms other popular methods for estimating Gaussian graphical models.