Computing loop corrections by message passing

TL;DR

This paper introduces a message-passing algorithm for computing loop corrections in the Ising model, leveraging spanning trees to approximate effects of loopy interactions and providing bounds for free energy.

Contribution

It presents a novel approximation algorithm that handles both short and long loops in loopy graphs using message passing on spanning trees.

Findings

Algorithm provides accurate loop corrections for the Ising model.

Can compute upper and lower bounds for free energy.

Effective in dealing with complex loopy interaction graphs.

Abstract

Any spanning tree in a loopy interaction graph can be used for communicating the effect of the loopy interactions by introducing messages that are passed along the edges in the spanning tree. This defines an exact mapping of the problem on the loopy interaction graph onto an extended problem on a tree interaction graph, where the thermodynamic quantities can be computed by a message-passing algorithm based on the Bethe equations. We propose an approximation loop correction algorithm for the Ising model relying on the above representation of the problem. The algorithm deals at the same time with the short and long loops, and can be used to obtain upper and lower bounds for the free energy.