# Theoretical Aspects of Group Equivariant Neural Networks

arXiv: https://arxiv.org/abs/2004.05154 · HTML: https://mlsanity.com/paper/2004.05154/

## Key facts

- Cited by 25 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 10 Apr 2020.

## Built on by (most cited first, 25 of 25)

- [Geometric Deep Learning and Equivariant Neural Networks](https://mlsanity.com/paper/2105.13926/) (2021)
- [Geometrically Equivariant Graph Neural Networks: A Survey](https://mlsanity.com/paper/2202.07230/) (2022)
- [Clifford Group Equivariant Neural Networks](https://mlsanity.com/paper/2305.11141/) (2023)
- [EDGI: Equivariant Diffusion for Planning with Embodied Agents](https://mlsanity.com/paper/2303.12410/) (2023)
- [Equivariance versus Augmentation for Spherical Images](https://mlsanity.com/paper/2202.03990/) (2022)
- [Scalable and Equivariant Spherical CNNs by Discrete-Continuous (DISCO) Convolutions](https://mlsanity.com/paper/2209.13603/) (2022)
- [Homogeneous vector bundles and G-equivariant convolutional neural networks](https://mlsanity.com/paper/2105.05400/) (2021)
- [Geometry-Complete Perceptron Networks for 3D Molecular Graphs](https://mlsanity.com/paper/2211.02504/) (2022)
- [RiEMann: Near Real-Time SE(3)-Equivariant Robot Manipulation without Point Cloud Segmentation](https://mlsanity.com/paper/2403.19460/) (2024)
- [Diffusion Generative Modeling on Lie Group Representations](https://mlsanity.com/paper/2502.02513/) (2025)
- [SO(2)-Equivariant Downwash Models for Close Proximity Flight](https://mlsanity.com/paper/2305.18983/) (2023)
- [A Unified Framework for Discovering Discrete Symmetries](https://mlsanity.com/paper/2309.02898/) (2023)
- [A Comparison Between Invariant and Equivariant Classical and Quantum Graph Neural Networks](https://mlsanity.com/paper/2311.18672/) (2023)
- [Steerable Equivariant Representation Learning](https://mlsanity.com/paper/2302.11349/) (2023)
- [Unifying O(3) Equivariant Neural Networks Design with Tensor-Network Formalism](https://mlsanity.com/paper/2211.07482/) (2022)
- [Equivariant neural networks and piecewise linear representation theory](https://mlsanity.com/paper/2408.00949/) (2024)
- [Deep Neural Networks with Efficient Guaranteed Invariances](https://mlsanity.com/paper/2303.01567/) (2023)
- [Neural Discovery of Permutation Subgroups](https://mlsanity.com/paper/2309.05352/) (2023)
- [Current Symmetry Group Equivariant Convolution Frameworks for Representation Learning](https://mlsanity.com/paper/2409.07327/) (2024)
- [Breaking the Symmetry: Resolving Symmetry Ambiguities in Equivariant Neural Networks](https://mlsanity.com/paper/2210.16646/) (2022)
- [Equivalence Between SE(3) Equivariant Networks via Steerable Kernels and Group Convolution](https://mlsanity.com/paper/2211.15903/) (2022)
- [On Uncertainty Calibration for Equivariant Functions](https://mlsanity.com/paper/2510.21691/) (2025)
- [Neural Vector Tomography for Reconstructing a Magnetization Vector Field](https://mlsanity.com/paper/2412.09927/) (2024)
- [Smoothness Errors in Dynamics Models and How to Avoid Them](https://mlsanity.com/paper/2602.05352/) (2026)
- [Representation Redundancy and Structural Complexity in Finite-Field Inversion](https://mlsanity.com/paper/2609.04583/) (2026)

## Abstract (arXiv)

Group equivariant neural networks have been explored in the past few years and are interesting from theoretical and practical standpoints. They leverage concepts from group representation theory, non-commutative harmonic analysis and differential geometry that do not often appear in machine learning. In practice, they have been shown to reduce sample and model complexity, notably in challenging tasks where input transformations such as arbitrary rotations are present. We begin this work with an exposition of group representation theory and the machinery necessary to define and evaluate integrals and convolutions on groups. Then, we show applications to recent SO(3) and SE(3) equivariant networks, namely the Spherical CNNs, Clebsch-Gordan Networks, and 3D Steerable CNNs. We proceed to discuss two recent theoretical results. The first, by Kondor and Trivedi (ICML'18), shows that a neural network is group equivariant if and only if it has a convolutional structure. The second, by Cohen et al. (NeurIPS'19), generalizes the first to a larger class of networks, with feature maps as fields on homogeneous spaces.