Convergence analysis of reweighted sum-product algorithms

Markov random fields are designed to represent structured dependencies among large collections of random variables, and are well-suited to capture the structure of real-world signals. Many fundamental tasks in signal processing (e.g., smoothing, denoising, segmentation etc.) require efficient methods for computing (approximate) marginal probabilities over subsets of nodes in the graph. The marginalization problem, though solvable in linear time for graphs without cycles, is computationally intractable for general graphs with cycles. This intractability motivates the use of approximate "message-passing" algorithms. This paper studies the convergence and stability properties of the family of reweighted sum-product algorithms, a generalization of the widely used sum-product or belief propagation algorithm, in which messages are adjusted with graph-dependent weights. For pairwise Markov random fields, we derive various conditions that are sufficient to ensure convergence, and also provide bounds on the geometric convergence rates. When specialized to the ordinary sum-product algorithm, these results provide strengthening of previous analyses. We prove that some of our conditions are necessary and sufficient for subclasses of homogeneous models, but not for general models. The experimental simulations on various classes of graphs validate our theoretical results.