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

arXiv: https://arxiv.org/abs/1605.06451 · HTML: https://mlsanity.com/paper/1605.06451/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 20 May 2016.

## Built on by (most cited first, 4 of 4)

- [Understanding the Behavior of Belief Propagation](https://mlsanity.com/paper/2209.05464/) (2022)
- [Structural Optimization of Factor Graphs for Symbol Detection via Continuous Clustering and Machine Learning](https://mlsanity.com/paper/2211.11406/) (2022)
- [Minima and Critical Points of the Bethe Free Energy Are Invariant Under Deformation Retractions of Factor Graphs](https://mlsanity.com/paper/2510.05380/) (2025)
- [Local Message Passing on Frustrated Systems](https://mlsanity.com/paper/2306.01494/) (2023)

## Abstract (arXiv)

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 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.