# Differentiable Programming of Isometric Tensor Networks

arXiv: https://arxiv.org/abs/2110.03898 · HTML: https://mlsanity.com/paper/2110.03898/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 20 of its 84 references that are in the map.
- First posted on arXiv 8 Oct 2021.

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

- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [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)
- [Efficient Riemannian Optimization on the Stiefel Manifold via the Cayley Transform](https://mlsanity.com/paper/2002.01113/) (2020)
- [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)
- [Neural Network Renormalization Group](https://mlsanity.com/paper/1802.02840/) (2018)
- [Enhancing Generative Models via Quantum Correlations](https://mlsanity.com/paper/2101.08354/) (2021)
- [Mutual Information, Neural Networks and the Renormalization Group](https://mlsanity.com/paper/1704.06279/) (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)
- [TensorNetwork for Machine Learning](https://mlsanity.com/paper/1906.06329/) (2019)
- [Generative Tensor Network Classification Model for Supervised Machine Learning](https://mlsanity.com/paper/1903.10742/) (2019)
- [Entanglement and Tensor Networks for Supervised Image Classification](https://mlsanity.com/paper/2007.06082/) (2020)
- [A Framework of Constraint Preserving Update Schemes for Optimization on Stiefel Manifold](https://mlsanity.com/paper/1301.0172/) (2013)
- [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)
- [RG-Flow: A hierarchical and explainable flow model based on renormalization group and sparse prior](https://mlsanity.com/paper/2010.00029/) (2020)
- [Adaptive Learning of Tensor Network Structures](https://mlsanity.com/paper/2008.05437/) (2020)
- [Number-State Preserving Tensor Networks as Classifiers for Supervised Learning](https://mlsanity.com/paper/1905.06352/) (2019)
- [Learning Non-linear Wavelet Transformation via Normalizing Flow](https://mlsanity.com/paper/2101.11306/) (2021)

## Built on by (most cited first, 2 of 2)

- [Tensor networks for quantum machine learning](https://mlsanity.com/paper/2303.11735/) (2023)
- [Automatically Differentiable Nonlinear Tensor Networks (ADNTNs) for Exponential Parameter Compression of Deep Neural Networks](https://mlsanity.com/paper/2606.00130/) (2026)

## Abstract (arXiv)

Differentiable programming is a new programming paradigm which enables large scale optimization through automatic calculation of gradients also known as auto-differentiation. This concept emerges from deep learning, and has also been generalized to tensor network optimizations. Here, we extend the differentiable programming to tensor networks with isometric constraints with applications to multiscale entanglement renormalization ansatz (MERA) and tensor network renormalization (TNR). By introducing several gradient-based optimization methods for the isometric tensor network and comparing with Evenbly-Vidal method, we show that auto-differentiation has a better performance for both stability and accuracy. We numerically tested our methods on 1D critical quantum Ising spin chain and 2D classical Ising model. We calculate the ground state energy for the 1D quantum model and internal energy for the classical model, and scaling dimensions of scaling operators and find they all agree with the theory well.