# Boltzmann machines as two-dimensional tensor networks

arXiv: https://arxiv.org/abs/2105.04130 · HTML: https://mlsanity.com/paper/2105.04130/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- Builds on 4 of its 31 references that are in the map.
- First posted on arXiv 10 May 2021.

## Builds on (most cited first, 4 of 4)

- [Neural-Network Quantum States, String-Bond States, and Chiral Topological States](https://mlsanity.com/paper/1710.04045/) (2017)
- [Equivalence of restricted Boltzmann machines and tensor network states](https://mlsanity.com/paper/1701.04831/) (2017)
- [Advanced Mean Field Theory of Restricted Boltzmann Machine](https://mlsanity.com/paper/1502.00186/) (2015)
- [Training Restricted Boltzmann Machines via the Thouless-Anderson-Palmer Free Energy](https://mlsanity.com/paper/1506.02914/) (2015)

## Built on by (most cited first, 7 of 7)

- [Tensor networks for unsupervised machine learning](https://mlsanity.com/paper/2106.12974/) (2021)
- [Tensor networks for quantum machine learning](https://mlsanity.com/paper/2303.11735/) (2023)
- [Tensor networks for interpretable and efficient quantum-inspired machine learning](https://mlsanity.com/paper/2311.11258/) (2023)
- [Generative modeling with projected entangled-pair states](https://mlsanity.com/paper/2202.08177/) (2022)
- [SHAP Meets Tensor Networks: Provably Tractable Explanations with Parallelism](https://mlsanity.com/paper/2510.21599/) (2025)
- [Neural Tangent Kernel of Matrix Product States: Convergence and Applications](https://mlsanity.com/paper/2111.14046/) (2021)
- [Seeding neural network quantum states with tensor network states](https://mlsanity.com/paper/2506.23550/) (2025)

## Abstract (arXiv)

Restricted Boltzmann machines (RBM) and deep Boltzmann machines (DBM) are important models in machine learning, and recently found numerous applications in quantum many-body physics. We show that there are fundamental connections between them and tensor networks. In particular, we demonstrate that any RBM and DBM can be exactly represented as a two-dimensional tensor network. This representation gives an understanding of the expressive power of RBM and DBM using entanglement structures of the tensor networks, also provides an efficient tensor network contraction algorithm for the computing partition function of RBM and DBM. Using numerical experiments, we demonstrate that the proposed algorithm is much more accurate than the state-of-the-art machine learning methods in estimating the partition function of restricted Boltzmann machines and deep Boltzmann machines, and have potential applications in training deep Boltzmann machines for general machine learning tasks.