Efficient Graph Laplacian Estimation by a Proximal Newton Approach

The Laplacian-constrained Gaussian Markov Random Field (LGMRF) is a common multivariate statistical model for learning a weighted sparse dependency graph from given data. This graph learning problem is formulated as a maximum likelihood estimation (MLE) of the precision matrix, subject to Laplacian structural constraints, with a sparsity-inducing penalty term. This paper aims to solve this learning problem accurately and efficiently. First, since the commonly-used $\ell_1$-norm penalty is less appropriate in this setting, we employ the nonconvex minimax concave penalty (MCP), which promotes sparse solutions with lower estimation bias. Second, as opposed to most existing first-order methods for this problem, we base our method on the second-order proximal Newton approach to obtain an efficient solver for large-scale networks. This approach is considered the most efficient for the related graphical LASSO problem and allows for several algorithmic features we exploit, such as using Conjugate Gradients, preconditioning, and splitting to active/free sets. Numerical experiments demonstrate the advantages of the proposed method in terms of \emph{both} computational complexity and graph learning accuracy compared to existing methods.