Duality of Graphical Models and Tensor Networks

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.