Partition Functions of Normal Factor Graphs

TL;DR

This paper introduces normal factor graphs (NFGs) as a graphical framework for representing partition functions, unifying various mathematical transformations and invariances in a novel, intuitive way.

Contribution

It formally defines NFGs for sums of products, highlighting their invariance properties and connecting them to Fourier, reparameterization, loop calculus, and Legendre transforms.

Findings

NFGs provide a new graphical representation for partition functions.

Certain NFG modifications leave the partition function invariant.

Unifies multiple mathematical transformations under a common framework.

Abstract

One of the most common types of functions in mathematics, physics, and engineering is a sum of products, sometimes called a partition function. After "normalization," a sum of products has a natural graphical representation, called a normal factor graph (NFG), in which vertices represent factors, edges represent internal variables, and half-edges represent the external variables of the partition function. In physics, so-called trace diagrams share similar features. We believe that the conceptual framework of representing sums of products as partition functions of NFGs is an important and intuitive paradigm that, surprisingly, does not seem to have been introduced explicitly in the previous factor graph literature. Of particular interest are NFG modifications that leave the partition function invariant. A simple subclass of such NFG modifications offers a unifying view of the Fourier transform, tree-based reparameterization, loop calculus, and the Legendre transform.