# Generative modeling with projected entangled-pair states

arXiv: https://arxiv.org/abs/2202.08177 · HTML: https://mlsanity.com/paper/2202.08177/

## Key facts

- Cited by 16 papers in the map (data through 14 Oct 2026).
- Builds on 20 of its 41 references that are in the map.
- First posted on arXiv 16 Feb 2022.

## Builds on (most cited first, 20 of 20)

- [UMAP: Uniform Manifold Approximation and Projection for Dimension Reduction](https://mlsanity.com/paper/1802.03426/) (2018)
- [Pixel Recurrent Neural Networks](https://mlsanity.com/paper/1601.06759/) (2016)
- [Unsupervised Generative Modeling Using Matrix Product States](https://mlsanity.com/paper/1709.01662/) (2017)
- [Tree Tensor Networks for Generative Modeling](https://mlsanity.com/paper/1901.02217/) (2019)
- [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)
- [Learning Relevant Features of Data with Multi-scale Tensor Networks](https://mlsanity.com/paper/1801.00315/) (2017)
- [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)
- [Supervised Learning with Projected Entangled Pair States](https://mlsanity.com/paper/2009.09932/) (2020)
- [Locally Masked Convolution for Autoregressive Models](https://mlsanity.com/paper/2006.12486/) (2020)
- [Tensor networks for unsupervised machine learning](https://mlsanity.com/paper/2106.12974/) (2021)
- [Modeling Sequences with Quantum States: A Look Under the Hood](https://mlsanity.com/paper/1910.07425/) (2019)
- [Entanglement and Tensor Networks for Supervised Image Classification](https://mlsanity.com/paper/2007.06082/) (2020)
- [Hybrid Tensor Decomposition in Neural Network Compression](https://mlsanity.com/paper/2006.15938/) (2020)
- [Tensor networks and efficient descriptions of classical data](https://mlsanity.com/paper/2103.06872/) (2021)
- [Probabilistic Modeling with Matrix Product States](https://mlsanity.com/paper/1902.06888/) (2019)
- [Boltzmann machines as two-dimensional tensor networks](https://mlsanity.com/paper/2105.04130/) (2021)
- [Matrix Product State for Feature Extraction of Higher-Order Tensors](https://mlsanity.com/paper/1503.00516/) (2015)
- [Matrix Product State for Higher-Order Tensor Compression and Classification](https://mlsanity.com/paper/1609.04541/) (2016)
- [Iterative Neural Autoregressive Distribution Estimator (NADE-k)](https://mlsanity.com/paper/1406.1485/) (2014)

## Built on by (most cited first, 16 of 16)

- [Trainability barriers and opportunities in quantum generative modeling](https://mlsanity.com/paper/2305.02881/) (2023)
- [Machine learning with tree tensor networks, CP rank constraints, and tensor dropout](https://mlsanity.com/paper/2305.19440/) (2023)
- [Quantum-Inspired Machine Learning: a Survey](https://mlsanity.com/paper/2308.11269/) (2023)
- [A Matrix Product State Model for Simultaneous Classification and Generation](https://mlsanity.com/paper/2406.17441/) (2024)
- [TensorKrowch: Smooth integration of tensor networks in machine learning](https://mlsanity.com/paper/2306.08595/) (2023)
- [High-dimensional density estimation with tensorizing flow](https://mlsanity.com/paper/2212.00759/) (2022)
- [Tensorization of neural networks for improved privacy and interpretability](https://mlsanity.com/paper/2501.06300/) (2025)
- [Quantum-inspired Techniques in Tensor Networks for Industrial Contexts](https://mlsanity.com/paper/2404.11277/) (2024)
- [No-Free-Lunch Theories for Tensor-Network Machine Learning Models](https://mlsanity.com/paper/2412.05674/) (2024)
- [Efficient Probabilistic Tensor Networks](https://mlsanity.com/paper/2510.00382/) (2025)
- [Tensor Methods for Language Models: From Token Representation to Training, Adaptation, Inference, Compression, and Interpretability](https://mlsanity.com/paper/2608.30505/) (2026)
- [Stack operation of tensor networks](https://mlsanity.com/paper/2203.16338/) (2022)
- [Intelligent diagnostic scheme for lung cancer screening with Raman spectra data by tensor network machine learning](https://mlsanity.com/paper/2303.06340/) (2023)
- [T'ecnicas Quantum-Inspired en Tensor Networks para Contextos Industriales](https://mlsanity.com/paper/2404.17645/) (2024)
- [Efficient Generative Modeling with Unitary Matrix Product States Using Riemannian Optimization](https://mlsanity.com/paper/2603.12026/) (2026)
- [When AI meets quantum information: A comprehensive review](https://mlsanity.com/paper/2607.00365/) (2026)

## Abstract (arXiv)

We argue and demonstrate that projected entangled-pair states (PEPS) outperform matrix product states significantly for the task of generative modeling of datasets with an intrinsic two-dimensional structure such as images. Our approach builds on a recently introduced algorithm for sampling PEPS, which allows for the efficient optimization and sampling of the distributions.