Penalized Likelihood Approach for Graph Learning in the Presence of Outliers

Graph learning is an important problem in the field of graph signal processing. However, the data available in real-world applications are often contaminated with outliers, which makes the application of traditional methods challenging. In this paper, we address this problem by developing an algorithm that effectively learns the graph Laplacian matrix from node signals corrupted by outliers. Specifically, we maximize the penalized log-likelihood of the uncorrupted data, where the penalty is chosen via the false discovery rate (FDR) principle, with respect to both the number of outliers and their locations, as well as the precision matrix of the data under the graph Laplacian constraints. To illustrate the robustness to outliers, we compare our method with two state-of-the-art graph learning methods, one that considers outliers in the data and one that does not, using different performance metrics. Our findings demonstrate that the proposed method efficiently detects the number and positions of outliers and accurately learns the graph in their presence.