  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.
