Learning Sparse Graph with Minimax Concave Penalty under Gaussian Markov Random Fields
This work addresses the need for more interpretable and accurate sparse graph learning in Gaussian Markov random fields, though it is incremental as it builds on the graphical lasso framework.
The paper tackles the problem of learning sparse graphs from data by introducing a convex-analytic framework that uses a nonconvex minimax concave penalty to reduce estimation bias compared to the ℓ₁ norm, resulting in significantly better performance than existing methods with reasonable computational time.
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.