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

Graph Zeta Function in the Bethe Free Energy and Loopy Belief Propagation
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发表时间：
2009-12
期刊：
ArXiv
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通讯作者：
Yusuke Watanabe;K. Fukumizu
Yusuke Watanabe;K. Fukumizu
中科院分区：
其他
文献类型：
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作者：
Yusuke Watanabe;K. Fukumizu

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通过建立一个将Bethe自由能的黑森函数与边Zeta函数联系起来的公式，我们提出了一种分析Loopy信念传播(LBP)的新方法。该公式对LBP有许多理论上的影响。给出了Bethe自由能的Hessian为正定的一个充分条件，从而证明了具有多个圈的图是非凸性的。该公式阐明了LBP不动点的局部稳定性与Bethe自由能的局部极小值之间的关系。我们还提出了一种新的判定LBP不动点唯一性的方法，并给出了不同的唯一性条件。
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.