Fixed Points of Belief Propagation -- An Analysis via Polynomial Homotopy Continuation

TL;DR

This paper introduces a polynomial formulation of belief propagation fixed points and employs a numerical homotopy continuation method to find all solutions, revealing insights into convergence and uniqueness in graphical models.

Contribution

It formulates BP fixed points as polynomial equations and applies NPHC to find all solutions, providing new understanding of convergence and fixed point structure.

Findings

NPHC finds all BP fixed points in tested models.

Structure influences the number of fixed points and convergence.

Unique fixed points do not guarantee convergence.

Abstract

Belief propagation (BP) is an iterative method to perform approximate inference on arbitrary graphical models. Whether BP converges and if the solution is a unique fixed point depends on both the structure and the parametrization of the model. To understand this dependence it is interesting to find \emph{all} fixed points. In this work, we formulate a set of polynomial equations, the solutions of which correspond to BP fixed points. To solve such a nonlinear system we present the numerical polynomial-homotopy-continuation (NPHC) method. Experiments on binary Ising models and on error-correcting codes show how our method is capable of obtaining all BP fixed points. On Ising models with fixed parameters we show how the structure influences both the number of fixed points and the convergence properties. We further asses the accuracy of the marginals and weighted combinations thereof. Weighting marginals with their respective partition function increases the accuracy in all experiments. Contrary to the conjecture that uniqueness of BP fixed points implies convergence, we find graphs for which BP fails to converge, even though a unique fixed point exists. Moreover, we show that this fixed point gives a good approximation, and the NPHC method is able to obtain this fixed point.