Graph Zeta Function in the Bethe Free Energy and Loopy Belief Propagation
Abstract
We propose a new approach to the analysis of Loopy Belief Propagation (LBP) by establishing a formula that connects the Hessian of the Bethe free energy with the edge zeta function. The formula has a number of theoretical implications on LBP. It is applied to give a sufficient condition that the Hessian of the Bethe free energy is positive definite, which shows non-convexity for graphs with multiple cycles. The formula clarifies the relation between the local stability of a fixed point of LBP and local minima of the Bethe free energy. We also propose a new approach to the uniqueness of LBP fixed point, and show various conditions of uniqueness.
Keywords
Cite
@article{arxiv.1002.3307,
title = {Graph Zeta Function in the Bethe Free Energy and Loopy Belief Propagation},
author = {Yusuke Watanabe and Kenji Fukumizu},
journal= {arXiv preprint arXiv:1002.3307},
year = {2010}
}
Comments
19 pages, Annual Conference on Neural Information Processing Systems (NIPS 2009), together with the supplementary material