Distributed Graph Learning From Smooth Data: A Bayesian Framework

The emerging field of graph learning, which aims to learn reasonable graph structures from data, plays a vital role in Graph Signal Processing (GSP) and finds applications in various data processing domains. However, the existing approaches have primarily focused on learning deterministic graphs, and thus are not suitable for applications involving topological stochasticity, such as epidemiological models. In this paper, we develop a hierarchical Bayesian model for graph learning problem. Specifically, the generative model of smooth signals is formulated by transforming the graph topology into self-expressiveness coefficients and incorporating individual noise for each vertex. Tailored probability distributions are imposed on each edge to characterize the valid graph topology constraints along with edge-level probabilistic information. Building upon this, we derive the Bayesian Graph Learning (BGL) approach to efficiently estimate the graph structure in a distributed manner. In particular, based on the specific probabilistic dependencies, we derive a series of message passing rules by a mixture of Generalized Approximate Message Passing (GAMP) message and Belief Propagation (BP) message to iteratively approximate the posterior probabilities. Numerical experiments with both artificial and real data demonstrate that BGL learns more accurate graph structures and enhances machine learning tasks compared to state-of-the-art methods.