# Tree Tensor Networks for Generative Modeling

arXiv: https://arxiv.org/abs/1901.02217 · HTML: https://mlsanity.com/paper/1901.02217/

## Key facts

- Cited by 64 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 8 Jan 2019.

## Built on by (most cited first, 30 of 64)

- [Variational Quantum Reinforcement Learning via Evolutionary Optimization](https://mlsanity.com/paper/2109.00540/) (2021)
- [Subtractive Mixture Models via Squaring: Representation and Learning](https://mlsanity.com/paper/2310.00724/) (2023)
- [Trainability barriers and opportunities in quantum generative modeling](https://mlsanity.com/paper/2305.02881/) (2023)
- [Tensor Networks Meet Neural Networks: A Survey and Future Perspectives](https://mlsanity.com/paper/2302.09019/) (2023)
- [Grokking phase transitions in learning local rules with gradient descent](https://mlsanity.com/paper/2210.15435/) (2022)
- [What is the Relationship between Tensor Factorizations and Circuits (and How Can We Exploit it)?](https://mlsanity.com/paper/2409.07953/) (2024)
- [Tensor networks for unsupervised machine learning](https://mlsanity.com/paper/2106.12974/) (2021)
- [Sum of Squares Circuits](https://mlsanity.com/paper/2408.11778/) (2024)
- [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)
- [Machine learning with tree tensor networks, CP rank constraints, and tensor dropout](https://mlsanity.com/paper/2305.19440/) (2023)
- [On the Relationship Between Monotone and Squared Probabilistic Circuits](https://mlsanity.com/paper/2408.00876/) (2024)
- [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-Inspired Tensor Neural Networks for Partial Differential Equations](https://mlsanity.com/paper/2208.02235/) (2022)
- [Generative Learning of Continuous Data by Tensor Networks](https://mlsanity.com/paper/2310.20498/) (2023)
- [Fast and Expressive Multi-Byte Prediction with Probabilistic Circuits](https://mlsanity.com/paper/2511.11346/) (2025)
- [Improvements to Gradient Descent Methods for Quantum Tensor Network Machine Learning](https://mlsanity.com/paper/2203.03366/) (2022)
- [Unsupervised Recognition of Informative Features via Tensor Network Machine Learning and Quantum Entanglement Variations](https://mlsanity.com/paper/2207.06031/) (2022)
- [A Matrix Product State Model for Simultaneous Classification and Generation](https://mlsanity.com/paper/2406.17441/) (2024)
- [Discrete generative diffusion models without stochastic differential equations: a tensor network approach](https://mlsanity.com/paper/2407.11133/) (2024)
- [Generative Modeling via Tree Tensor Network States](https://mlsanity.com/paper/2209.01341/) (2022)
- [Deep Tree Tensor Networks](https://mlsanity.com/paper/2502.09928/) (2025)
- [How to Square Tensor Networks and Circuits Without Squaring Them](https://mlsanity.com/paper/2512.17090/) (2025)
- [Tensor network to learn the wavefunction of data](https://mlsanity.com/paper/2111.08014/) (2021)
- [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)
- [Tensorization is a powerful but underexplored tool for compression and interpretability of neural networks](https://mlsanity.com/paper/2505.20132/) (2025)
- [Many-body Approximation for Non-negative Tensors](https://mlsanity.com/paper/2209.15338/) (2022)

## Abstract (arXiv)

Matrix product states (MPS), a tensor network designed for one-dimensional quantum systems, has been recently proposed for generative modeling of natural data (such as images) in terms of `Born machine'. However, the exponential decay of correlation in MPS restricts its representation power heavily for modeling complex data such as natural images. In this work, we push forward the effort of applying tensor networks to machine learning by employing the Tree Tensor Network (TTN) which exhibits balanced performance in expressibility and efficient training and sampling. We design the tree tensor network to utilize the 2-dimensional prior of the natural images and develop sweeping learning and sampling algorithms which can be efficiently implemented utilizing Graphical Processing Units (GPU). We apply our model to random binary patterns and the binary MNIST datasets of handwritten digits. We show that TTN is superior to MPS for generative modeling in keeping correlation of pixels in natural images, as well as giving better log-likelihood scores in standard datasets of handwritten digits. We also compare its performance with state-of-the-art generative models such as the Variational AutoEncoders, Restricted Boltzmann machines, and PixelCNN. Finally, we discuss the future development of Tensor Network States in machine learning problems.