On Convergence Conditions of Gaussian Belief Propagation

Abstract

In order to compute the marginal probability density function (PDF) with Gaussian belief propagation (BP), it is impor- tant 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 pro- posed initialization set is proved to be largest among all currently known sets. Then, the necessary and sufficient convergence con- dition of beliefs in damped Gaussian BP is developed, with the allowable range of damping factor explicitly established. The re- sults 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 initializa- tion set. Relationships between the proposed convergence condi- tions and existing ones are established analytically. At last, numer- ical examples are presented to corroborate the established theories.