Duality of Graphical Models and Tensor Networks

TL;DR

This paper establishes a duality between tensor networks and undirected graphical models, revealing how concepts and algorithms in one domain correspond to those in the other, fostering cross-disciplinary insights.

Contribution

It introduces the concept of tensor hypernetworks on hypergraphs and demonstrates their exact correspondence to graphical models on dual hypergraphs, translating notions and algorithms between the two.

Findings

Belief propagation corresponds to tensor network contraction algorithms.

Marginalization in graphical models is dual to contraction in tensor networks.

Tensor hypernetworks on hypergraphs provide a new framework for understanding duality.

Abstract

In this article we show the duality between tensor networks and undirected graphical models with discrete variables. We study tensor networks on hypergraphs, which we call tensor hypernetworks. We show that the tensor hypernetwork on a hypergraph exactly corresponds to the graphical model given by the dual hypergraph. We translate various notions under duality. For example, marginalization in a graphical model is dual to contraction in the tensor network. Algorithms also translate under duality. We show that belief propagation corresponds to a known algorithm for tensor network contraction. This article is a reminder that the research areas of graphical models and tensor networks can benefit from interaction.