On Convergence Conditions of Gaussian Belief Propagation

In order to compute the marginal probability density function (PDF) with Gaussian belief propagation (BP), it is important to know whether it will converge in advance. By describing the message-passing process of Gaussian BP on the pairwise factor graph as a set of updating functions, the necessary and sufficient convergence condition of beliefs in synchronous Gaussian BP is first derived under a newly proposed initialization set. The proposed initialization set is proved to be largest among all currently known sets. Then, the necessary and sufficient convergence condition of beliefs in damped Gaussian BP is developed, with the allowable range of damping factor explicitly established. The results theoretically confirm the extensively reported conjecture that damping is helpful to improve the convergence of Gaussian BP. Under totally asynchronous scheduling, a sufficient convergence condition of beliefs is also derived for the same proposed initialization set. Relationships between the proposed convergence conditions and existing ones are established analytically. At last, numerical examples are presented to corroborate the established theories.