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

Abstract

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 ℓ <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</inf> 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 ℓ <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</inf> norm, inducing a less-biased sparse solution than ℓ <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</inf> . 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 ℓ <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</inf> -based counterpart for sparse grid graphs.