# Generative Learning of Continuous Data by Tensor Networks

arXiv: https://arxiv.org/abs/2310.20498 · HTML: https://mlsanity.com/paper/2310.20498/

## Key facts

- Cited by 9 papers in the map (data through 14 Oct 2026).
- Builds on 26 of its 74 references that are in the map.
- First posted on arXiv 31 Oct 2023.

## Builds on (most cited first, 26 of 26)

- [On the Expressive Power of Deep Learning: A Tensor Analysis](https://mlsanity.com/paper/1509.05009/) (2015)
- [Unsupervised Generative Modeling Using Matrix Product States](https://mlsanity.com/paper/1709.01662/) (2017)
- [Towards Quantum Machine Learning with Tensor Networks](https://mlsanity.com/paper/1803.11537/) (2018)
- [Tree Tensor Networks for Generative Modeling](https://mlsanity.com/paper/1901.02217/) (2019)
- [TT-Rec: Tensor Train Compression for Deep Learning Recommendation Models](https://mlsanity.com/paper/2101.11714/) (2021)
- [Expressive power of tensor-network factorizations for probabilistic modeling, with applications from hidden Markov models to quantum machine learning](https://mlsanity.com/paper/1907.03741/) (2019)
- [Tensor Networks for Dimensionality Reduction and Large-Scale Optimizations. Part 2 Applications and Future Perspectives](https://mlsanity.com/paper/1708.09165/) (2017)
- [Tensor Methods in Computer Vision and Deep Learning](https://mlsanity.com/paper/2107.03436/) (2021)
- [Quantum Entanglement in Deep Learning Architectures](https://mlsanity.com/paper/1803.09780/) (2018)
- [Supervised Learning with Quantum-Inspired Tensor Networks](https://mlsanity.com/paper/1605.05775/) (2016)
- [Information Perspective to Probabilistic Modeling: Boltzmann Machines versus Born Machines](https://mlsanity.com/paper/1712.04144/) (2017)
- [From probabilistic graphical models to generalized tensor networks for supervised learning](https://mlsanity.com/paper/1806.05964/) (2018)
- [Expressive power of recurrent neural networks](https://mlsanity.com/paper/1711.00811/) (2017)
- [Tensor-Train Density Estimation](https://mlsanity.com/paper/2108.00089/) (2021)
- [Tensor networks for unsupervised machine learning](https://mlsanity.com/paper/2106.12974/) (2021)
- [Entanglement and Tensor Networks for Supervised Image Classification](https://mlsanity.com/paper/2007.06082/) (2020)
- [A Multi-Scale Tensor Network Architecture for Classification and Regression](https://mlsanity.com/paper/2001.08286/) (2020)
- [Tensor networks and efficient descriptions of classical data](https://mlsanity.com/paper/2103.06872/) (2021)
- [Generative modeling via tensor train sketching](https://mlsanity.com/paper/2202.11788/) (2022)
- [Quantum Tensor Networks, Stochastic Processes, and Weighted Automata](https://mlsanity.com/paper/2010.10653/) (2020)
- [A quantum-inspired tensor network method for constrained combinatorial optimization problems](https://mlsanity.com/paper/2203.15246/) (2022)
- [Approximation Theory of Tree Tensor Networks: Tensorized Univariate Functions -- Part I](https://mlsanity.com/paper/2007.00118/) (2020)
- [Quantized Fourier and Polynomial Features for more Expressive Tensor Network Models](https://mlsanity.com/paper/2309.05436/) (2023)
- [Interaction Decompositions for Tensor Network Regression](https://mlsanity.com/paper/2208.06029/) (2022)
- [Approximation Theory of Tree Tensor Networks: Tensorized Multivariate Functions](https://mlsanity.com/paper/2101.11932/) (2021)
- [Distributive Pre-Training of Generative Modeling Using Matrix-Product States](https://mlsanity.com/paper/2306.14787/) (2023)

## Built on by (most cited first, 9 of 9)

- [Tensor Network for Anomaly Detection in the Latent Space of Proton Collision Events at the LHC](https://mlsanity.com/paper/2506.00102/) (2025)
- [How to Square Tensor Networks and Circuits Without Squaring Them](https://mlsanity.com/paper/2512.17090/) (2025)
- [Using matrix-product states for time-series machine learning](https://mlsanity.com/paper/2412.15826/) (2024)
- [Regularized second-order optimization of tensor-network Born machines](https://mlsanity.com/paper/2501.18691/) (2025)
- [Comb Tensor Networks vs. Matrix Product States: Enhanced Efficiency in High-Dimensional Spaces](https://mlsanity.com/paper/2412.06857/) (2024)
- [Explaining Anomalies with Tensor Networks](https://mlsanity.com/paper/2505.03911/) (2025)
- [Quantum-inspired tensor networks in machine learning models](https://mlsanity.com/paper/2604.14287/) (2026)
- [On the Approximation Complexity of Matrix Product Operator (MPO) Born Machines](https://mlsanity.com/paper/2605.11471/) (2026)
- [QuanVI: Score-based Variational Inference via Quantum Maximally Mixed States](https://mlsanity.com/paper/2609.39164/) (2026)

## Abstract (arXiv)

Beyond their origin in modeling many-body quantum systems, tensor networks have emerged as a promising class of models for solving machine learning problems, notably in unsupervised generative learning. While possessing many desirable features arising from their quantum-inspired nature, tensor network generative models have previously been largely restricted to binary or categorical data, limiting their utility in real-world modeling problems. We overcome this by introducing a new family of tensor network generative models for continuous data, which are capable of learning from distributions containing continuous random variables. We develop our method in the setting of matrix product states, first deriving a universal expressivity theorem proving the ability of this model family to approximate any reasonably smooth probability density function with arbitrary precision. We then benchmark the performance of this model on several synthetic and real-world datasets, finding that the model learns and generalizes well on distributions of continuous and discrete variables. We develop methods for modeling different data domains, and introduce a trainable compression layer which is found to increase model performance given limited memory or computational resources. Overall, our methods give important theoretical and empirical evidence of the efficacy of quantum-inspired methods for the rapidly growing field of generative learning.