Graph Zeta Function in the Bethe Free Energy and Loopy Belief Propagation

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.

• Type: Journal
• Year: 2010
• Where: CORR
• Authors: Yusuke Watanabe, Kenji Fukumizu
• Keywords: Bethe Free Energy, CORR 2010, Education, Fixed Point, Loopy Belief Propagation