Convergence rate analysis of Gaussian belief propagation for Markov networks

Gaussian belief propagation algorithm (GaBP) is one of the most important distributed algorithms in signal processing and statistical learning involving Markov networks. It is well known that the algorithm correctly computes marginal density functions from a high dimensional joint density function over a Markov network in a finite number of iterations when the underlying Gaussian graph is acyclic. It is also known more recently that the algorithm produces correct marginal means asymptotically for cyclic Gaussian graphs under the condition of walk summability (or generalised diagonal dominance). This paper extends this convergence result further by showing that the convergence is exponential under the generalised diagonal dominance condition, and provides a simple bound for the convergence rate. Our results are derived by combining the known walk summability approach for asymptotic convergence analysis with the control systems approach for stability analysis.