# G-RepsNet: A Fast and General Construction of Equivariant Networks for Arbitrary Matrix Groups

arXiv: https://arxiv.org/abs/2402.15413 · HTML: https://mlsanity.com/paper/2402.15413/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 27 of its 33 references that are in the map.
- First posted on arXiv 23 Feb 2024.

## Builds on (most cited first, 27 of 27)

- [Neural Message Passing for Quantum Chemistry](https://mlsanity.com/paper/1704.01212/) (2017)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [Group Equivariant Convolutional Networks](https://mlsanity.com/paper/1602.07576/) (2016)
- [E(n) Equivariant Graph Neural Networks](https://mlsanity.com/paper/2102.09844/) (2021)
- [Tensor field networks: Rotation- and translation-equivariant neural networks for 3D point clouds](https://mlsanity.com/paper/1802.08219/) (2018)
- [SE(3)-Transformers: 3D Roto-Translation Equivariant Attention Networks](https://mlsanity.com/paper/2006.10503/) (2020)
- [General E(2)-Equivariant Steerable CNNs](https://mlsanity.com/paper/1911.08251/) (2019)
- [Steerable CNNs](https://mlsanity.com/paper/1612.08498/) (2016)
- [Vector Neurons: A General Framework for SO(3)-Equivariant Networks](https://mlsanity.com/paper/2104.12229/) (2021)
- [On the Generalization of Equivariance and Convolution in Neural Networks to the Action of Compact Groups](https://mlsanity.com/paper/1802.03690/) (2018)
- [Harmonic Networks: Deep Translation and Rotation Equivariance](https://mlsanity.com/paper/1612.04642/) (2016)
- [Geometric and Physical Quantities Improve E(3) Equivariant Message Passing](https://mlsanity.com/paper/2110.02905/) (2021)
- [A Practical Method for Constructing Equivariant Multilayer Perceptrons for Arbitrary Matrix Groups](https://mlsanity.com/paper/2104.09459/) (2021)
- [Frame Averaging for Invariant and Equivariant Network Design](https://mlsanity.com/paper/2110.03336/) (2021)
- [MDP Homomorphic Networks: Group Symmetries in Reinforcement Learning](https://mlsanity.com/paper/2006.16908/) (2020)
- [Equivariance Through Parameter-Sharing](https://mlsanity.com/paper/1702.08389/) (2017)
- [Universal approximations of invariant maps by neural networks](https://mlsanity.com/paper/1804.10306/) (2018)
- [Scalars are universal: Equivariant machine learning, structured like classical physics](https://mlsanity.com/paper/2106.06610/) (2021)
- [Lorentz Group Equivariant Neural Network for Particle Physics](https://mlsanity.com/paper/2006.04780/) (2020)
- [Equivariant Flows: sampling configurations for multi-body systems with symmetric energies](https://mlsanity.com/paper/1910.00753/) (2019)
- [Covariant Compositional Networks For Learning Graphs](https://mlsanity.com/paper/1801.02144/) (2018)
- [Group Equivariant Fourier Neural Operators for Partial Differential Equations](https://mlsanity.com/paper/2306.05697/) (2023)
- [Equivariant Adaptation of Large Pretrained Models](https://mlsanity.com/paper/2310.01647/) (2023)
- [Learning Probabilistic Symmetrization for Architecture Agnostic Equivariance](https://mlsanity.com/paper/2306.02866/) (2023)
- [Equi-Tuning: Group Equivariant Fine-Tuning of Pretrained Models](https://mlsanity.com/paper/2210.06475/) (2022)
- [Efficient Equivariant Transfer Learning from Pretrained Models](https://mlsanity.com/paper/2305.09900/) (2023)
- [Robust Perception through Equivariance](https://mlsanity.com/paper/2212.06079/) (2022)

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

- [Interpretable Discovery of One-parameter Subgroups: A Modular Framework for Elliptical, Hyperbolic, and Parabolic Symmetries](https://mlsanity.com/paper/2509.22219/) (2025)

## Abstract (arXiv)

Group equivariance is a strong inductive bias useful in a wide range of deep learning tasks. However, constructing efficient equivariant networks for general groups and domains is difficult. Recent work by Finzi et al. (2021) directly solves the equivariance constraint for arbitrary matrix groups to obtain equivariant MLPs (EMLPs). But this method does not scale well and scaling is crucial in deep learning. Here, we introduce Group Representation Networks (G-RepsNets), a lightweight equivariant network for arbitrary matrix groups with features represented using tensor polynomials. The key intuition for our design is that using tensor representations in the hidden layers of a neural network along with simple inexpensive tensor operations can lead to expressive universal equivariant networks. We find G-RepsNet to be competitive to EMLP on several tasks with group symmetries such as O(5), O(1, 3), and O(3) with scalars, vectors, and second-order tensors as data types. On image classification tasks, we find that G-RepsNet using second-order representations is competitive and often even outperforms sophisticated state-of-the-art equivariant models such as GCNNs (Cohen & Welling, 2016a) and E(2)-CNNs (Weiler & Cesa, 2019). To further illustrate the generality of our approach, we show that G-RepsNet is competitive to G-FNO (Helwig et al., 2023) and EGNN (Satorras et al., 2021) on N-body predictions and solving PDEs, respectively, while being efficient.