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 in- spired by the graphical lasso, a key difference is the use of a noncon- vex alternative of the ℓ1 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 func- tion from the ℓ1 norm, inducing a less-biased sparse solution than ℓ1. In our framework, the graph Laplacian is represented by a lin- ear transform of the vector corresponding to its upper triangular part. Via a reformulation relying on the Moreau decomposition, the prob- lem can be solved by the primal-dual splitting method. An admis- sible choice of parameters for provable convergence is presented. Numerical examples show that the proposed method significantly outperforms its ℓ1-based counterpart for sparse grid graphs.