A Graph Learning Algorithm Based On Gaussian Markov Random Fields And Minimax Concave Penalty

This paper presents a graph learning framework to produce sparse and accurate graphs from network data. While our formulation is inspired by the graphical lasso, a key difference is the use of a nonconvex alternative of the $\ell_1$ norm as well as a quadratic term to ensure overall convexity. Specifically, the weakly-convex minimax concave penalty (MCP) is used, which is given by subtracting the Huber function from the $\ell_1$ norm, inducing a less-biased sparse solution than $\ell_1$. In our framework, the graph Laplacian is represented by a linear transform of the vector corresponding to its upper triangular part. Via a reformulation relying on the Moreau decomposition, the problem can be solved by the primal-dual splitting method. An admissible choice of parameters for provable convergence is presented. Numerical examples show that the proposed method significantly outperforms its $\ell_1$-based counterpart for sparse grid graphs.