Learning Sparse Graph with Minimax Concave Penalty under Gaussian Markov Random Fields

TL;DR

This paper introduces a novel nonconvex penalty-based framework for learning sparse Gaussian Markov Random Fields, improving interpretability and reducing bias compared to traditional methods like graphical lasso.

Contribution

It proposes a convex-analytic approach using a minimax concave penalty for better sparse graph estimation, with efficient solution algorithms and improved performance over existing methods.

Findings

Outperforms existing graph learning methods in accuracy

Reduces estimation bias compared to $ ext{l}_1$ regularization

Maintains computational efficiency with primal-dual splitting

Abstract

This paper presents a convex-analytic framework to learn sparse graphs from data. While our problem formulation is inspired by an extension of the graphical lasso using the so-called combinatorial graph Laplacian framework, a key difference is the use of a nonconvex alternative to the $\ell_1$ norm to attain graphs with better interpretability. Specifically, we use the weakly-convex minimax concave penalty (the difference between the $\ell_1$ norm and the Huber function) which is known to yield sparse solutions with lower estimation bias than $\ell_1$ for regression problems. In our framework, the graph Laplacian is replaced in the optimization by a linear transform of the vector corresponding to its upper triangular part. Via a reformulation relying on Moreau's decomposition, we show that overall convexity is guaranteed by introducing a quadratic function to our cost function. The problem can be solved efficiently by the primal-dual splitting method, of which the admissible conditions for provable convergence are presented. Numerical examples show that the proposed method significantly outperforms the existing graph learning methods with reasonable CPU time.