Fixed Points of Gaussian Belief Propagation and Relation to Convergence

In general, Gaussian belief propagation (BP) is not guaranteed to converge in loopy graphs. But when Gaussian BP converges to a fixed point of its update equations, the exact means of the marginal distributions can be obtained. However, given a Gaussian graphical model, whether Gaussian BP would have fixed points is unknown. Moreover, the relation between the convergence of Gaussian BP and its fixed points is not clear. To answer these questions, the necessary and sufficient existence condition and an easily verifiable sufficient existence condition of fixed points of Gaussian BP are proposed. Conditioned on the existence of fixed points of Gaussian BP, the convergence conditions of outgoing messages' parameters are analyzed, where outgoing message denotes the message flowing from a variable node to a factor node on a factor graph. It is proved that outgoing messages' precisions (the reciprocal of variance) would converge if there exist fixed points for Gaussian BP. This provides an elegant interpretation of the convergence condition of outgoing messages' precisions. On the other hand, to guarantee the convergence of outgoing messages' means, a sufficient condition for Gaussian models with a single loop and a sufficient condition that can be checked without obtaining the converged outgoing messages' precisions are derived. The relations between the convergence conditions of outgoing messages' parameters and those of incoming messages' parameters are also revealed, making any convergence condition of outgoing messages valid for verifying the convergence of incoming messages. Numerical results are presented to corroborate the newly established theories.