# Symmetry Discovery for Different Data Types

arXiv: https://arxiv.org/abs/2410.09841 · HTML: https://mlsanity.com/paper/2410.09841/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- Builds on 26 of its 34 references that are in the map.
- First posted on arXiv 13 Oct 2024.

## Builds on (most cited first, 26 of 26)

- [Semi-Supervised Classification with Graph Convolutional Networks](https://mlsanity.com/paper/1609.02907/) (2016)
- [Group Equivariant Convolutional Networks](https://mlsanity.com/paper/1602.07576/) (2016)
- [E(n) Equivariant Graph Neural Networks](https://mlsanity.com/paper/2102.09844/) (2021)
- [Hamiltonian Neural Networks](https://mlsanity.com/paper/1906.01563/) (2019)
- [General E(2)-Equivariant Steerable CNNs](https://mlsanity.com/paper/1911.08251/) (2019)
- [3D Steerable CNNs: Learning Rotationally Equivariant Features in Volumetric Data](https://mlsanity.com/paper/1807.02547/) (2018)
- [Steerable CNNs](https://mlsanity.com/paper/1612.08498/) (2016)
- [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)
- [Adan: Adaptive Nesterov Momentum Algorithm for Faster Optimizing Deep Models](https://mlsanity.com/paper/2208.06677/) (2022)
- [Predicting Neural Network Accuracy from Weights](https://mlsanity.com/paper/2002.11448/) (2020)
- [Equivariant Architectures for Learning in Deep Weight Spaces](https://mlsanity.com/paper/2301.12780/) (2023)
- [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)
- [Classifying the classifier: dissecting the weight space of neural networks](https://mlsanity.com/paper/2002.05688/) (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)
- [PDO-eConvs: Partial Differential Operator Based Equivariant Convolutions](https://mlsanity.com/paper/2007.10408/) (2020)
- [LieGG: Studying Learned Lie Group Generators](https://mlsanity.com/paper/2210.04345/) (2022)
- [Diffeomorphic Counterfactuals with Generative Models](https://mlsanity.com/paper/2206.05075/) (2022)
- [Learning Layer-wise Equivariances Automatically using Gradients](https://mlsanity.com/paper/2310.06131/) (2023)
- [Enabling equivariance for arbitrary Lie groups](https://mlsanity.com/paper/2111.08251/) (2021)
- [PDO-eS2CNNs: Partial Differential Operator Based Equivariant Spherical CNNs](https://mlsanity.com/paper/2104.03584/) (2021)
- [A Generative Model of Symmetry Transformations](https://mlsanity.com/paper/2403.01946/) (2024)
- [PDO-s3DCNNs: Partial Differential Operator Based Steerable 3D CNNs](https://mlsanity.com/paper/2208.03720/) (2022)

## Built on by (most cited first, 8 of 8)

- [Explicit Discovery of Nonlinear Symmetries from Dynamic Data](https://mlsanity.com/paper/2510.01855/) (2025)
- [Discovering Symmetry Groups with Flow Matching](https://mlsanity.com/paper/2512.20043/) (2025)
- [Adaptive Symmetry Discovery for Dynamical System Identification](https://mlsanity.com/paper/2608.08091/) (2026)
- [Efficient Learning and Symmetry Discovery under Exact Invariances](https://mlsanity.com/paper/2609.07031/) (2026)
- [Exact Symmetry as Algebra: Group-Structured Tensors that Enforce Physical Selection Rules](https://mlsanity.com/paper/2605.20440/) (2026)
- [Compositional Symmetry as Compression: Lie Pseudogroup Structure in Algorithmic Agents](https://mlsanity.com/paper/2510.10586/) (2025)
- [Observable- and Positional-Encoding-Dependent Symmetry Readout from Neural Network Weights](https://mlsanity.com/paper/2607.03108/) (2026)
- [LieDiscover: Adaptive Symbolic Library Construction for Explicit Open-form Symmetry Discovery](https://mlsanity.com/paper/2609.33663/) (2026)

## Abstract (arXiv)

Equivariant neural networks incorporate symmetries into their architecture, achieving higher generalization performance. However, constructing equivariant neural networks typically requires prior knowledge of data types and symmetries, which is difficult to achieve in most tasks. In this paper, we propose LieSD, a method for discovering symmetries via trained neural networks which approximate the input-output mappings of the tasks. It characterizes equivariance and invariance (a special case of equivariance) of continuous groups using Lie algebra and directly solves the Lie algebra space through the inputs, outputs, and gradients of the trained neural network. Then, we extend the method to make it applicable to multi-channel data and tensor data, respectively. We validate the performance of LieSD on tasks with symmetries such as the two-body problem, the moment of inertia matrix prediction, and top quark tagging. Compared with the baseline, LieSD can accurately determine the number of Lie algebra bases without the need for expensive group sampling. Furthermore, LieSD can perform well on non-uniform datasets, whereas methods based on GANs fail.