Learning Graph Laplacian with MCP

TL;DR

This paper introduces a novel non-convex MCP penalty for learning graph Laplacians, employing an efficient inexact proximal DCA and semismooth Newton method, showing improved performance over existing methods.

Contribution

The paper develops a new algorithm combining proximal DCA and semismooth Newton methods for MCP-penalized graph Laplacian learning, with proven convergence and enhanced efficiency.

Findings

MCP promotes sparsity effectively in graph Laplacian learning.

The proposed method outperforms existing algorithms in efficiency and reliability.

Numerical experiments validate the effectiveness of the non-convex MCP penalty.

Abstract

We consider the problem of learning a graph under the Laplacian constraint with a non-convex penalty: minimax concave penalty (MCP). For solving the MCP penalized graphical model, we design an inexact proximal difference-of-convex algorithm (DCA) and prove its convergence to critical points. We note that each subproblem of the proximal DCA enjoys the nice property that the objective function in its dual problem is continuously differentiable with a semismooth gradient. Therefore, we apply an efficient semismooth Newton method to subproblems of the proximal DCA. Numerical experiments on various synthetic and real data sets demonstrate the effectiveness of the non-convex penalty MCP in promoting sparsity. Compared with the existing state-of-the-art method, our method is demonstrated to be more efficient and reliable for learning graph Laplacian with MCP.