Sufficient conditions for convergence of the Sum-Product Algorithm

TL;DR

This paper introduces new polynomial-time verifiable conditions that ensure the convergence of the Sum-Product Algorithm to a unique fixed point across arbitrary factor graphs, including those with zeros in factors.

Contribution

The authors derive novel, broadly applicable convergence conditions for the Sum-Product Algorithm that improve upon existing bounds and incorporate local evidence and interaction types.

Findings

New convergence conditions are polynomial in the number of variables.

Conditions are valid for arbitrary factor graphs with zeros in factors.

Empirical results show the new bounds outperform existing ones.

Abstract

We derive novel conditions that guarantee convergence of the Sum-Product algorithm (also known as Loopy Belief Propagation or simply Belief Propagation) to a unique fixed point, irrespective of the initial messages. The computational complexity of the conditions is polynomial in the number of variables. In contrast with previously existing conditions, our results are directly applicable to arbitrary factor graphs (with discrete variables) and are shown to be valid also in the case of factors containing zeros, under some additional conditions. We compare our bounds with existing ones, numerically and, if possible, analytically. For binary variables with pairwise interactions, we derive sufficient conditions that take into account local evidence (i.e., single variable factors) and the type of pair interactions (attractive or repulsive). It is shown empirically that this bound outperforms existing bounds.