Efficient Robust Graph Learning Based on Minimax Concave Penalty and $\gamma$-Cross Entropy

This paper presents an efficient robust method to learn sparse graphs from contaminated data. Specifically, the convex-analytic approach using the minimax concave penalty is formulated using the so-called $\gamma$-lasso which exploits the $\gamma-$ cross entropy. We devise a weighting technique which designs the data weights based on the $\ell_{1}$ distance in addition to the Mahalanobis distance for avoiding possible failures of outlier rejection due to the combinatorial graph Laplacian structure. Numerical examples show that the proposed method significantly outperforms $\gamma$-lasso and tlasso as well as the existing non-robust graph learning methods in contaminated situations.