# Learning symmetries in datasets

arXiv: https://arxiv.org/abs/2504.05174 · HTML: https://mlsanity.com/paper/2504.05174/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 15 of its 26 references that are in the map.
- First posted on arXiv 7 Apr 2025.

## Builds on (most cited first, 15 of 15)

- [Group Equivariant Convolutional Networks](https://mlsanity.com/paper/1602.07576/) (2016)
- [Challenging Common Assumptions in the Unsupervised Learning of Disentangled Representations](https://mlsanity.com/paper/1811.12359/) (2018)
- [InfoGAN: Interpretable Representation Learning by Information Maximizing Generative Adversarial Nets](https://mlsanity.com/paper/1606.03657/) (2016)
- [Deep Learning and the Information Bottleneck Principle](https://mlsanity.com/paper/1503.02406/) (2015)
- [Disentangling by Factorising](https://mlsanity.com/paper/1802.05983/) (2018)
- [Isolating Sources of Disentanglement in Variational Autoencoders](https://mlsanity.com/paper/1802.04942/) (2018)
- [A Hierarchical Latent Vector Model for Learning Long-Term Structure in Music](https://mlsanity.com/paper/1803.05428/) (2018)
- [Discovering physical concepts with neural networks](https://mlsanity.com/paper/1807.10300/) (2018)
- [Explorations in Homeomorphic Variational Auto-Encoding](https://mlsanity.com/paper/1807.04689/) (2018)
- [Latent Space Symmetry Discovery](https://mlsanity.com/paper/2310.00105/) (2023)
- [PELICAN: Permutation Equivariant and Lorentz Invariant or Covariant Aggregator Network for Particle Physics](https://mlsanity.com/paper/2211.00454/) (2022)
- [Better Latent Spaces for Better Autoencoders](https://mlsanity.com/paper/2104.08291/) (2021)
- [Lorentz group equivariant autoencoders](https://mlsanity.com/paper/2212.07347/) (2022)
- [Symmetry meets AI](https://mlsanity.com/paper/2103.06115/) (2021)
- [Exploring how a Generative AI interprets music](https://mlsanity.com/paper/2308.00015/) (2023)

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

- [Latent Spectroscopy: Posterior Collapse as a Feature](https://mlsanity.com/paper/2605.22691/) (2026)

## Abstract (arXiv)

We investigate how symmetries present in datasets affect the structure of the latent space learned by Variational Autoencoders (VAEs). By training VAEs on data originating from simple mechanical systems and particle collisions, we analyze the organization of the latent space through a relevance measure that identifies the most meaningful latent directions. We show that when symmetries or approximate symmetries are present, the VAE self-organizes its latent space, effectively compressing the data along a reduced number of latent variables. This behavior captures the intrinsic dimensionality determined by the symmetry constraints and reveals hidden relations among the features. Furthermore, we provide a theoretical analysis of a simple toy model, demonstrating how, under idealized conditions, the latent space aligns with the symmetry directions of the data manifold. We illustrate these findings with examples ranging from two-dimensional datasets with O(2) symmetry to realistic datasets from electron-positron and proton-proton collisions. Our results highlight the potential of unsupervised generative models to expose underlying structures in data and offer a novel approach to symmetry discovery without explicit supervision.