Efficient Computation of ℓ1 Regularized Estimates in Gaussian Graphical Models

In this article, I propose an effcient algorithm to compute ℓ1 regularized maximum likelihood estimates in the Gaussian graphical model. These estimators, recently proposed in an earlier article by Yuan and Lin, conduct parameter estimation and model selection simultaneously and have been shown to enjoy nice properties in both large and finite samples. To compute the estimates, however, can be very challenging in practice because of the high dimensionality and positive definiteness constraint on the covariance matrix. Taking advantage of the recent advance in semidefinite programming, Yuan and Lin suggested a sophisticated interior-point algorithm to solve the optimization problem. Although a polynomial time algorithm, the optimization technique is known not to be scalable for high-dimensional problems. Alternatively, this article shows that the estimates can be computed by iteratively solving a sequence of ℓ1 regularized quadratic programs. By effectively exploiting the sparsity of the graphical structure, I propose a new algorithm that can be applied to problems of larger scale. When combined with a path-following strategy, the new algorithm can be used to efficiently approximate the entire solution path of the ℓ1 regularized maximum likelihood estimates, which also facilitates the choice of tuning parameter. I demonstrate the efficacy and usefulness of the proposed algorithm on a few simulations and real datasets.