# Equivariant and Steerable Neural Networks: A review with special emphasis on the symmetric group

arXiv: https://arxiv.org/abs/2301.03019 · HTML: https://mlsanity.com/paper/2301.03019/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 16 of its 30 references that are in the map.
- First posted on arXiv 8 Jan 2023.

## Builds on (most cited first, 16 of 16)

- [Group Equivariant Convolutional Networks](https://mlsanity.com/paper/1602.07576/) (2016)
- [Dynamic Routing Between Capsules](https://mlsanity.com/paper/1710.09829/) (2017)
- [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)
- [Harmonic Networks: Deep Translation and Rotation Equivariance](https://mlsanity.com/paper/1612.04642/) (2016)
- [Gauge Equivariant Convolutional Networks and the Icosahedral CNN](https://mlsanity.com/paper/1902.04615/) (2019)
- [A General Theory of Equivariant CNNs on Homogeneous Spaces](https://mlsanity.com/paper/1811.02017/) (2018)
- [Universal approximations of invariant maps by neural networks](https://mlsanity.com/paper/1804.10306/) (2018)
- [Exploiting Cyclic Symmetry in Convolutional Neural Networks](https://mlsanity.com/paper/1602.02660/) (2016)
- [Rotation-invariant convolutional neural networks for galaxy morphology prediction](https://mlsanity.com/paper/1503.07077/) (2015)
- [Deep Roto-Translation Scattering for Object Classification](https://mlsanity.com/paper/1412.8659/) (2014)
- [Equivalence of approximation by convolutional neural networks and fully-connected networks](https://mlsanity.com/paper/1809.00973/) (2018)
- [Locally Scale-Invariant Convolutional Neural Networks](https://mlsanity.com/paper/1412.5104/) (2014)
- [Intertwiners between Induced Representations (with Applications to the Theory of Equivariant Neural Networks)](https://mlsanity.com/paper/1803.10743/) (2018)
- [Universal Approximation Theorem for Equivariant Maps by Group CNNs](https://mlsanity.com/paper/2012.13882/) (2020)

## Built on by (most cited first, 2 of 2)

- [Equivariant Neural Networks for Force-Field Models of Lattice Systems](https://mlsanity.com/paper/2601.04104/) (2026)
- [Adaptive aggregation of Monte Carlo augmented decomposed filters for efficient group-equivariant convolutional neural network](https://mlsanity.com/paper/2305.10110/) (2023)

## Abstract (arXiv)

Convolutional neural networks revolutionized computer vision and natrual language processing. Their efficiency, as compared to fully connected neural networks, has its origin in the architecture, where convolutions reflect the translation invariance in space and time in pattern or speech recognition tasks. Recently, Cohen and Welling have put this in the broader perspective of invariance under symmetry groups, which leads to the concept of group equivaiant neural networks and more generally steerable neural networks. In this article, we review the architecture of such networks including equivariant layers and filter banks, activation with capsules and group pooling. We apply this formalism to the symmetric group, for which we work out a number of details on representations and capsules that are not found in the literature.