Duality of Graphical Models and Tensor Networks
This work bridges two research areas, potentially enabling cross-fertilization of algorithms and insights, though it is incremental in nature.
The paper establishes a duality between tensor networks and undirected graphical models with discrete variables, showing that tensor hypernetworks correspond to dual hypergraph models and translating concepts like marginalization to contraction.
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.