2009
NeurIPS
NeurIPS 2009
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.
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- Artificial Intelligence and Mathematics & Optimization
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Hot Topic Early Bird
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Authors
Topics
Artificial Intelligence > Bayesian & Probabilistic > Probabilistic Modeling
Machine Learning > Optimization & Theory > Bayesian Inference
Machine Learning > Optimization & Theory > Stochastic Processes
Mathematics & Optimization > Mathematics > Graph Theory
Machine Learning > Bayesian & Probabilistic > Probabilistic Modeling
Machine Learning > Core Methods > Graphical Models
Machine Learning > Bayesian & Probabilistic > Variational Inference