# From probabilistic graphical models to generalized tensor networks for supervised learning

arXiv: https://arxiv.org/abs/1806.05964 · HTML: https://mlsanity.com/paper/1806.05964/

## Key facts

- Cited by 32 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 15 Jun 2018.

## Built on by (most cited first, 30 of 32)

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- [How to Square Tensor Networks and Circuits Without Squaring Them](https://mlsanity.com/paper/2512.17090/) (2025)
- [The Bayesian Method of Tensor Networks](https://mlsanity.com/paper/2101.00245/) (2021)
- [Generalization and Overfitting in Matrix Product State Machine Learning Architectures](https://mlsanity.com/paper/2208.04372/) (2022)
- [Differentiable Programming of Isometric Tensor Networks](https://mlsanity.com/paper/2110.03898/) (2021)
- [Neural Tangent Kernel of Matrix Product States: Convergence and Applications](https://mlsanity.com/paper/2111.14046/) (2021)
- [Quantum-inspired tensor network for Earth science](https://mlsanity.com/paper/2301.07528/) (2023)
- [On Faster Marginalization with Squared Circuits via Orthonormalization](https://mlsanity.com/paper/2412.07883/) (2024)
- [Lower and Upper Bounds on the VC-Dimension of Tensor Network Models](https://mlsanity.com/paper/2106.11827/) (2021)
- [Quantum-Inspired Tensor Neural Networks for Option Pricing](https://mlsanity.com/paper/2212.14076/) (2022)
- [Dirac-Equation Signal Processing: Physics Boosts Topological Machine Learning](https://mlsanity.com/paper/2412.05132/) (2024)
- [Advances in Machine Learning: Where Can Quantum Techniques Help?](https://mlsanity.com/paper/2507.08379/) (2025)
- [Quantum-Inspired Robust and Scalable SAR Object Classification](https://mlsanity.com/paper/2604.25755/) (2026)
- [Stack operation of tensor networks](https://mlsanity.com/paper/2203.16338/) (2022)
- [Application of Tensor Neural Networks to Pricing Bermudan Swaptions](https://mlsanity.com/paper/2304.09750/) (2023)
- [Joint Bayesian Parameter and Model Order Estimation for Low-Rank Probability Mass Tensors](https://mlsanity.com/paper/2410.06329/) (2024)
- [Explaining Anomalies with Tensor Networks](https://mlsanity.com/paper/2505.03911/) (2025)
- [Tensor-Augmented Convolutional Neural Networks: Enhancing Expressivity with Generic Tensor Kernels](https://mlsanity.com/paper/2604.08072/) (2026)
- [Quantum-inspired tensor networks in machine learning models](https://mlsanity.com/paper/2604.14287/) (2026)

## Abstract (arXiv)

Tensor networks have found a wide use in a variety of applications in physics and computer science, recently leading to both theoretical insights as well as practical algorithms in machine learning. In this work we explore the connection between tensor networks and probabilistic graphical models, and show that it motivates the definition of generalized tensor networks where information from a tensor can be copied and reused in other parts of the network. We discuss the relationship between generalized tensor network architectures used in quantum physics, such as string-bond states, and architectures commonly used in machine learning. We provide an algorithm to train these networks in a supervised-learning context and show that they overcome the limitations of regular tensor networks in higher dimensions, while keeping the computation efficient. A method to combine neural networks and tensor networks as part of a common deep learning architecture is also introduced. We benchmark our algorithm for several generalized tensor network architectures on the task of classifying images and sounds, and show that they outperform previously introduced tensor-network algorithms. The models we consider also have a natural implementation on a quantum computer and may guide the development of near-term quantum machine learning architectures.