Local stability of Belief Propagation algorithm with multiple fixed points

TL;DR

This paper establishes a new sufficient condition for the local stability of belief propagation fixed points, linking graph structure and beliefs, which explains its better performance on sparse graphs.

Contribution

It introduces a novel sufficient condition for local stability of belief propagation fixed points based on graph structure and beliefs.

Findings

Belief propagation's stability depends on graph sparsity.

The new condition explains improved performance on sparse graphs.

Fixed points' stability can be characterized by the derived condition.

Abstract

A number of problems in statistical physics and computer science can be expressed as the computation of marginal probabilities over a Markov random field. Belief propagation, an iterative message-passing algorithm, computes exactly such marginals when the underlying graph is a tree. But it has gained its popularity as an efficient way to approximate them in the more general case, even if it can exhibits multiple fixed points and is not guaranteed to converge. In this paper, we express a new sufficient condition for local stability of a belief propagation fixed point in terms of the graph structure and the beliefs values at the fixed point. This gives credence to the usual understanding that Belief Propagation performs better on sparse graphs.