Gauge-free cluster variational method by maximal messages and moment matching

摘要

We present a new implementation of the Cluster Variational Method (CVM) as amessage passing algorithm. The kind of message passing algorithms used for CVM,usually named Generalized Belief Propagation, are a generalization of theBelief Propagation algorithm in the same way that CVM is a generalization ofthe Bethe approximation for estimating the partition function. However, theconnection between fixed points of GBP and the extremal points of the CVMfree-energy is usually not a one-to-one correspondence, because of theexistence of a gauge transformation involving the GBP messages. Our contribution is twofold. Firstly we propose a new way of definingmessages (fields) in a generic CVM approximation, such that messages arrive ona given region from all its ancestors, and not only from its direct parents, asin the standard Parent-to-Child GBP. We call this approach maximal messages.Secondly we focus on the case of binary variables, re-interpreting the messagesas fields enforcing the consistency between the moments of the local (marginal)probability distributions. We provide a precise rule to enforce allconsistencies, avoiding any redundancy, that would otherwise lead to a gaugetransformation on the messages. This moment matching method is gauge free, i.e.it guarantees that the resulting GBP is not gauge invariant. We apply our maximal messages and moment matching GBP to obtain an analyticalexpression for the critical temperature of the Ising model in generaldimensions at the level of plaquette-CVM. The values obtained outperform Betheestimates, and are comparable with loop corrected Belief Propagation equations.The method allows for a straightforward generalization to disordered systems.