Learning Graphs from Smooth Signals under Moment Uncertainty

We consider the problem of inferring the graph structure from a given set of smooth graph signals. The number of perceived graph signals is always finite and possibly noisy, thus the statistical properties of the data distribution is ambiguous. Traditional graph learning models do not take this distributional uncertainty into account, thus performance may be sensitive to different sets of data. In this paper, we propose a distributionally robust approach to graph learning, which incorporates the first and second moment uncertainty into the smooth graph learning model. Specifically, we cast our graph learning model as a minimax optimization problem, and further reformulate it as a nonconvex minimization problem with linear constraints. In our proposed formulation, we find a theoretical interpretation of the Laplacian regularizer, which is adopted in many existing works in an intuitive manner. Although the first moment uncertainty leads to an annoying square root term in the objective function, we prove that it enjoys the smoothness property with probability 1 over the entire constraint. We develop a efficient projected gradient descent (PGD) method and establish its global iterate convergence to a critical point. We conduct extensive experiments on both synthetic and real data to verify the effectiveness of our model and the efficiency of the PGD algorithm. Compared with the state-of-the-art smooth graph learning methods, our approach exhibits superior and more robust performance across different populations of signals in terms of various evaluation metrics.