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

TL;DR

This paper introduces a novel formula linking the Hessian of the Bethe free energy with the edge zeta function, providing insights into LBP stability, convexity, and uniqueness in graphs with cycles.

Contribution

It establishes a new theoretical connection between the Hessian of the Bethe free energy and the edge zeta function, advancing understanding of LBP behavior.

Findings

Provides a sufficient condition for the Hessian's positive definiteness

Shows non-convexity of Bethe free energy in graphs with multiple cycles

Proposes conditions for the uniqueness of LBP fixed points

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.