# Tensor network to learn the wavefunction of data

arXiv: https://arxiv.org/abs/2111.08014 · HTML: https://mlsanity.com/paper/2111.08014/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 20 of its 31 references that are in the map.
- First posted on arXiv 15 Nov 2021.

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

- [Scaling Laws for Neural Language Models](https://mlsanity.com/paper/2001.08361/) (2020)
- [Understanding deep learning requires rethinking generalization](https://mlsanity.com/paper/1611.03530/) (2016)
- [Exploring Generalization in Deep Learning](https://mlsanity.com/paper/1706.08947/) (2017)
- [Explaining Neural Scaling Laws](https://mlsanity.com/paper/2102.06701/) (2021)
- [Estimating the intrinsic dimension of datasets by a minimal neighborhood information](https://mlsanity.com/paper/1803.06992/) (2018)
- [Generalization in Deep Learning](https://mlsanity.com/paper/1710.05468/) (2017)
- [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)
- [Cold Case: The Lost MNIST Digits](https://mlsanity.com/paper/1905.10498/) (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)
- [A Neural Scaling Law from the Dimension of the Data Manifold](https://mlsanity.com/paper/2004.10802/) (2020)
- [Quantum Entanglement in Deep Learning Architectures](https://mlsanity.com/paper/1803.09780/) (2018)
- [TensorNetwork for Machine Learning](https://mlsanity.com/paper/1906.06329/) (2019)
- [Deep Learning and Quantum Entanglement: Fundamental Connections with Implications to Network Design](https://mlsanity.com/paper/1704.01552/) (2017)
- [Entanglement and Tensor Networks for Supervised Image Classification](https://mlsanity.com/paper/2007.06082/) (2020)
- [Tensor networks and efficient descriptions of classical data](https://mlsanity.com/paper/2103.06872/) (2021)
- [Tensor network language model](https://mlsanity.com/paper/1710.10248/) (2017)
- [Language as a matrix product state](https://mlsanity.com/paper/1711.01416/) (2017)
- [Tangent-Space Gradient Optimization of Tensor Network for Machine Learning](https://mlsanity.com/paper/2001.04029/) (2020)
- [Entangled q-Convolutional Neural Nets](https://mlsanity.com/paper/2103.11785/) (2021)

## Built on by (most cited first, 4 of 4)

- [Grokking phase transitions in learning local rules with gradient descent](https://mlsanity.com/paper/2210.15435/) (2022)
- [Machine learning with tree tensor networks, CP rank constraints, and tensor dropout](https://mlsanity.com/paper/2305.19440/) (2023)
- [What Makes Data Suitable for a Locally Connected Neural Network? A Necessary and Sufficient Condition Based on Quantum Entanglement](https://mlsanity.com/paper/2303.11249/) (2023)
- [Deep tensor networks with matrix product operators](https://mlsanity.com/paper/2209.09098/) (2022)

## Abstract (arXiv)

How many different ways are there to handwrite digit 3? To quantify this question imagine extending a dataset of handwritten digits MNIST by sampling additional images until they start repeating. We call the collection of all resulting images of digit 3 the "full set." To study the properties of the full set we introduce a tensor network architecture which simultaneously accomplishes both classification (discrimination) and sampling tasks. Qualitatively, our trained network represents the indicator function of the full set. It therefore can be used to characterize the data itself. We illustrate that by studying the full sets associated with the digits of MNIST. Using quantum mechanical interpretation of our network we characterize the full set by calculating its entanglement entropy. We also study its geometric properties such as mean Hamming distance, effective dimension, and size. The latter answers the question above -- the total number of black and white threes written MNIST style is 272.