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

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 we are interested in finding \emph{all} fixed points. In this work, we formulate BP as a set of polynomial equations, the solutions of which correspond to the BP fixed points. We apply the numerical polynomial-homotopy-continuation (NPHC) method to solve such systems. It is commonly believed that uniqueness of BP fixed points implies convergence to this fixed point. Contrary to this conjecture, 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 of the exact marginal distribution.