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

Yusuke Watanabe; Kenji Fukumizu

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arXiv · 1002.3307

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

Yusuke Watanabe·Kenji Fukumizu

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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.

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BibTeX RIS

Yusuke Watanabe, Kenji Fukumizu. 2010-02-17. Graph Zeta Function in the Bethe Free Energy and Loopy Belief Propagation. https://arxiv.org/abs/1002.3307

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