  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.
