Sparse Inverse Covariance Selection with Non-Convex Regularizations

Sparse inverse covariance selection is a powerful tool for estimating sparse graphs in statistical learning, whose mathematical model is to maximize the regularized log-likelihood function. In this thesis, we consider introducing some non-convex regularizers to this problem. In particular, we first propose several non-convex regularized maximum likelihood estimation models. Using the specific structure of those regularizers, we then develop a DC programming approach for solving these models. We show that each subproblem of this approach is a weighted graphical lasso problem, and we propose a warm-started alternating direction method of multipliers to solve each subproblem. In addition, we propose a decomposition scheme by extending the exact covariance thresholding technique for graphical lasso to our general non-convex model, which enables us to solve large-scale problems efficiently. Finally, we compare the performance of our approach with some existing approaches on both randomly generated and real-life instances, and report some promising computational results.