Approximate inference on planar graphs using Loop Calculus and Belief Propagation

TL;DR

This paper presents an efficient algorithm for approximate inference on planar graphical models using loop calculus and Pfaffian representations, significantly improving accuracy and computational efficiency over previous methods.

Contribution

It introduces a novel truncation scheme and Pfaffian-based series representation for loop calculus on planar graphs, enhancing approximation accuracy and efficiency.

Findings

The first Pfaffian series term yields highly accurate approximations.

The proposed algorithm outperforms previous truncation schemes.

It is competitive with state-of-the-art inference methods.

Abstract

We introduce novel results for approximate inference on planar graphical models using the loop calculus framework. The loop calculus (Chertkov and Chernyak, 2006b) allows to express the exact partition function Z of a graphical model as a finite sum of terms that can be evaluated once the belief propagation (BP) solution is known. In general, full summation over all correction terms is intractable. We develop an algorithm for the approach presented in Chertkov et al. (2008) which represents an efficient truncation scheme on planar graphs and a new representation of the series in terms of Pfaffians of matrices. We analyze in detail both the loop series and the Pfaffian series for models with binary variables and pairwise interactions, and show that the first term of the Pfaffian series can provide very accurate approximations. The algorithm outperforms previous truncation schemes of the loop series and is competitive with other state-of-the-art methods for approximate inference.