# SymmetryLens: Unsupervised Symmetry Learning via Locality and Density Preservation

arXiv: https://arxiv.org/abs/2410.05232 · HTML: https://mlsanity.com/paper/2410.05232/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 16 of its 33 references that are in the map.
- First posted on arXiv 7 Oct 2024.

## Builds on (most cited first, 16 of 16)

- [Generative Adversarial Networks](https://mlsanity.com/paper/2203.00667/) (2022)
- [Geometric Deep Learning: Grids, Groups, Graphs, Geodesics, and Gauges](https://mlsanity.com/paper/2104.13478/) (2021)
- [Group Equivariant Convolutional Networks](https://mlsanity.com/paper/1602.07576/) (2016)
- [Hamiltonian Neural Networks](https://mlsanity.com/paper/1906.01563/) (2019)
- [Automatic Symmetry Discovery with Lie Algebra Convolutional Network](https://mlsanity.com/paper/2109.07103/) (2021)
- [Meta-Learning Symmetries by Reparameterization](https://mlsanity.com/paper/2007.02933/) (2020)
- [Learning Partial Equivariances from Data](https://mlsanity.com/paper/2110.10211/) (2021)
- [Generative Adversarial Symmetry Discovery](https://mlsanity.com/paper/2302.00236/) (2023)
- [Detecting Symmetries with Neural Networks](https://mlsanity.com/paper/2003.13679/) (2020)
- [Latent Space Symmetry Discovery](https://mlsanity.com/paper/2310.00105/) (2023)
- [Noether Networks: Meta-Learning Useful Conserved Quantities](https://mlsanity.com/paper/2112.03321/) (2021)
- [Symmetry-Based Representations for Artificial and Biological General Intelligence](https://mlsanity.com/paper/2203.09250/) (2022)
- [Deep Learning Symmetries and Their Lie Groups, Algebras, and Subalgebras from First Principles](https://mlsanity.com/paper/2301.05638/) (2023)
- [A Differential Entropy Estimator for Training Neural Networks](https://mlsanity.com/paper/2202.06618/) (2022)
- [Machine learning a manifold](https://mlsanity.com/paper/2112.07673/) (2021)
- [Accelerated Discovery of Machine-Learned Symmetries: Deriving the Exceptional Lie Groups G2, F4 and E6](https://mlsanity.com/paper/2307.04891/) (2023)

## Built on by (most cited first, 1 of 1)

- [Blind Recovery of Latent Domains via Unsupervised Symmetry Discovery](https://mlsanity.com/paper/2606.17782/) (2026)

## Abstract (arXiv)

We develop a new unsupervised symmetry learning method that starts with raw data and provides the minimal generator of an underlying Lie group of symmetries, together with a symmetry-equivariant representation of the data, which turns the hidden symmetry into an explicit one. The method is able to learn the pixel translation operator from a dataset with only an approximate translation symmetry and can learn quite different types of symmetries that are not apparent to the naked eye. The method is based on the formulation of an information-theoretic loss function that measures both the degree of symmetry of a dataset under a candidate symmetry generator and a proposed notion of locality of the samples, which is coupled to symmetry. We demonstrate that this coupling between symmetry and locality, together with an optimization technique developed for entropy estimation, results in a stable system that provides reproducible results.