A Cluster-Cumulant Expansion at the Fixed Points of Belief Propagation

TL;DR

The paper introduces a new cluster-cumulant expansion (CCE) method based on belief propagation fixed points, which is versatile, improves accuracy over loop-series, and avoids convergence issues of belief propagation.

Contribution

It presents a novel CCE method that extends to arbitrary state spaces and generalized belief propagation, improving accuracy and computational stability over existing loop-series approaches.

Findings

CCE improves upon loop-series accuracy

It is applicable to arbitrary state spaces

It avoids convergence issues of belief propagation

Abstract

We introduce a new cluster-cumulant expansion (CCE) based on the fixed points of iterative belief propagation (IBP). This expansion is similar in spirit to the loop-series (LS) recently introduced in [1]. However, in contrast to the latter, the CCE enjoys the following important qualities: 1) it is defined for arbitrary state spaces 2) it is easily extended to fixed points of generalized belief propagation (GBP), 3) disconnected groups of variables will not contribute to the CCE and 4) the accuracy of the expansion empirically improves upon that of the LS. The CCE is based on the same M\"obius transform as the Kikuchi approximation, but unlike GBP does not require storing the beliefs of the GBP-clusters nor does it suffer from convergence issues during belief updating.