Convergence Behavior of Belief Propagation: Estimating Regions of Attraction via Lyapunov Functions

In this work, we estimate the regions of attraction for belief propagation. This extends existing stability analysis and provides initial message values for which belief propagation is guaranteed to converge. Our approach utilizes the theory of Lyapunov functions that, however, does not readily yield useful regions of attraction. Therefore, we utilize polynomial sum-of-squares relaxations and provide an algorithm that computes valid Lyapunov functions. This admits a novel way of studying the solution space of belief propagation. Finally, we apply our approach to small-scale models and discuss the effect of the potentials on the regions of attraction.