# Duality of Graphical Models and Tensor Networks

arXiv: https://arxiv.org/abs/1710.01437 · HTML: https://mlsanity.com/paper/1710.01437/

## Key facts

- Cited by 15 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 4 Oct 2017.

## Built on by (most cited first, 15 of 15)

- [What is the Relationship between Tensor Factorizations and Circuits (and How Can We Exploit it)?](https://mlsanity.com/paper/2409.07953/) (2024)
- [Tensor networks and efficient descriptions of classical data](https://mlsanity.com/paper/2103.06872/) (2021)
- [Probabilistic Graphical Models and Tensor Networks: A Hybrid Framework](https://mlsanity.com/paper/2106.15666/) (2021)
- [You Only Derive Once (YODO): Automatic Differentiation for Efficient Sensitivity Analysis in Bayesian Networks](https://mlsanity.com/paper/2206.08687/) (2022)
- [High-dimensional density estimation with tensorizing flow](https://mlsanity.com/paper/2212.00759/) (2022)
- [Probabilistic Inference in the Era of Tensor Networks and Differential Programming](https://mlsanity.com/paper/2405.14060/) (2024)
- [The YODO algorithm: An efficient computational framework for sensitivity analysis in Bayesian networks](https://mlsanity.com/paper/2302.00364/) (2023)
- [Probabilistic Regular Tree Priors for Scientific Symbolic Reasoning](https://mlsanity.com/paper/2306.08506/) (2023)
- [Spherical and Hyperbolic Toric Topology-Based Codes On Graph Embedding for Ising MRF Models: Classical and Quantum Topology Machine Learning](https://mlsanity.com/paper/2307.15778/) (2023)
- [Global Sensitivity Analysis of Uncertain Parameters in Bayesian Networks](https://mlsanity.com/paper/2406.05764/) (2024)
- [Tensor Network Estimation of Distribution Algorithms](https://mlsanity.com/paper/2412.19780/) (2024)
- [Global sensitivity analysis in probabilistic graphical models](https://mlsanity.com/paper/2110.03749/) (2021)
- [Combining Local Symmetry Exploitation and Reinforcement Learning for Optimised Probabilistic Inference -- A Work In Progress](https://mlsanity.com/paper/2503.08786/) (2025)
- [TT-FSI: Scalable Faithful Shapley Interactions via Tensor-Train](https://mlsanity.com/paper/2601.01903/) (2026)
- [On Probabilistic Inference Through Parametric Tensor Decomposition in Base Tensor Networks](https://mlsanity.com/paper/2609.23774/) (2026)

## Abstract (arXiv)

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.