# Spectral Discovery of Continuous Symmetries via Generalized Fourier Transforms

arXiv: https://arxiv.org/abs/2603.07299 · HTML: https://mlsanity.com/paper/2603.07299/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 22 of its 28 references that are in the map.
- First posted on arXiv 7 Mar 2026.

## Builds on (most cited first, 22 of 22)

- [Deep Sets](https://mlsanity.com/paper/1703.06114/) (2017)
- [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)
- [General E(2)-Equivariant Steerable CNNs](https://mlsanity.com/paper/1911.08251/) (2019)
- [DiffDock: Diffusion Steps, Twists, and Turns for Molecular Docking](https://mlsanity.com/paper/2210.01776/) (2022)
- [On the Generalization of Equivariance and Convolution in Neural Networks to the Action of Compact Groups](https://mlsanity.com/paper/1802.03690/) (2018)
- [Generalizing Convolutional Neural Networks for Equivariance to Lie Groups on Arbitrary Continuous Data](https://mlsanity.com/paper/2002.12880/) (2020)
- [A General Theory of Equivariant CNNs on Homogeneous Spaces](https://mlsanity.com/paper/1811.02017/) (2018)
- [Learning Steerable Filters for Rotation Equivariant CNNs](https://mlsanity.com/paper/1711.07289/) (2017)
- [A Practical Method for Constructing Equivariant Multilayer Perceptrons for Arbitrary Matrix Groups](https://mlsanity.com/paper/2104.09459/) (2021)
- [Clebsch-Gordan Nets: a Fully Fourier Space Spherical Convolutional Neural Network](https://mlsanity.com/paper/1806.09231/) (2018)
- [EquiBind: Geometric Deep Learning for Drug Binding Structure Prediction](https://mlsanity.com/paper/2202.05146/) (2022)
- [Learning Invariances in Neural Networks](https://mlsanity.com/paper/2010.11882/) (2020)
- [Generative Adversarial Symmetry Discovery](https://mlsanity.com/paper/2302.00236/) (2023)
- [Learning the Irreducible Representations of Commutative Lie Groups](https://mlsanity.com/paper/1402.4437/) (2014)
- [LieGG: Studying Learned Lie Group Generators](https://mlsanity.com/paper/2210.04345/) (2022)
- [Latent Space Symmetry Discovery](https://mlsanity.com/paper/2310.00105/) (2023)
- [Bispectral Neural Networks](https://mlsanity.com/paper/2209.03416/) (2022)
- [Learning Infinitesimal Generators of Continuous Symmetries from Data](https://mlsanity.com/paper/2410.21853/) (2024)
- [Symmetry Discovery Beyond Affine Transformations](https://mlsanity.com/paper/2406.03619/) (2024)
- [Disentangling images with Lie group transformations and sparse coding](https://mlsanity.com/paper/2012.12071/) (2020)
- [EquiCPI: SE(3)-Equivariant Geometric Deep Learning for Structure-Aware Prediction of Compound-Protein Interactions](https://mlsanity.com/paper/2504.04654/) (2025)

## Abstract (arXiv)

Continuous symmetries are fundamental to many scientific and learning problems, yet they are often unknown a priori. Existing symmetry discovery approaches typically search directly in the space of transformation generators or rely on learned augmentation schemes. We propose a fundamentally different perspective based on spectral structure. We introduce a framework for discovering continuous one-parameter subgroups using the Generalized Fourier Transform (GFT). Our central observation is that invariance to a subgroup induces structured sparsity in the spectral decomposition of a function across irreducible representations. Instead of optimizing over generators, we detect symmetries by identifying this induced sparsity pattern in the spectral domain. We develop symmetry detection procedures on maximal tori, where the GFT reduces to multi-dimensional Fourier analysis through their irreducible representations. Across structured tasks, including the double pendulum and top quark tagging, we demonstrate that spectral sparsity reliably reveals one-parameter symmetries. These results position spectral analysis as a principled and interpretable alternative to generator-based symmetry discovery.