# Tensor tree learns hidden relational structures in data to construct generative models

arXiv: https://arxiv.org/abs/2408.10669 · HTML: https://mlsanity.com/paper/2408.10669/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 12 of its 36 references that are in the map.
- First posted on arXiv 20 Aug 2024.

## Builds on (most cited first, 12 of 12)

- [Learning both Weights and Connections for Efficient Neural Networks](https://mlsanity.com/paper/1506.02626/) (2015)
- [Tensorizing Neural Networks](https://mlsanity.com/paper/1509.06569/) (2015)
- [Differentiable Learning of Quantum Circuit Born Machine](https://mlsanity.com/paper/1804.04168/) (2018)
- [Unsupervised Generative Modeling Using Matrix Product States](https://mlsanity.com/paper/1709.01662/) (2017)
- [The Born Supremacy: Quantum Advantage and Training of an Ising Born Machine](https://mlsanity.com/paper/1904.02214/) (2019)
- [Tree Tensor Networks for Generative Modeling](https://mlsanity.com/paper/1901.02217/) (2019)
- [Cold Case: The Lost MNIST Digits](https://mlsanity.com/paper/1905.10498/) (2019)
- [The Recovery of Causal Poly-Trees from Statistical Data](https://mlsanity.com/paper/1304.2736/) (2013)
- [Variational inference with a quantum computer](https://mlsanity.com/paper/2103.06720/) (2021)
- [A Framework for Demonstrating Practical Quantum Advantage: Racing Quantum against Classical Generative Models](https://mlsanity.com/paper/2303.15626/) (2023)
- [Probabilistic Graphical Models and Tensor Networks: A Hybrid Framework](https://mlsanity.com/paper/2106.15666/) (2021)
- [Combining Reinforcement Learning and Tensor Networks, with an Application to Dynamical Large Deviations](https://mlsanity.com/paper/2209.14089/) (2022)

## Built on by (most cited first, 3 of 3)

- [Plastic tensor networks for interpretable generative modeling](https://mlsanity.com/paper/2504.06722/) (2025)
- [Trainable Quantum Neural Network for Multiclass Image Classification with the Power of Pre-trained Tree Tensor Networks](https://mlsanity.com/paper/2504.14995/) (2025)
- [Entanglement is Half the Story: Post-Selection vs. Partial Traces](https://mlsanity.com/paper/2605.02385/) (2026)

## Abstract (arXiv)

Based on the tensor tree network with the Born machine framework, we propose a general method for constructing a generative model by expressing the target distribution function as the amplitude of the quantum wave function represented by a tensor tree. The key idea is dynamically optimizing the tree structure that minimizes the bond mutual information. The proposed method offers enhanced performance and uncovers hidden relational structures in the target data. We illustrate potential practical applications with four examples: (i) random patterns, (ii) QMNIST handwritten digits, (iii) Bayesian networks, and (iv) the pattern of stock price fluctuation pattern in S&P500. In (i) and (ii), the strongly correlated variables were concentrated near the center of the network; in (iii), the causality pattern was identified; and in (iv), a structure corresponding to the eleven sectors emerged.