# Equivariant and Invariant Reynolds Networks

arXiv: https://arxiv.org/abs/2110.08092 · HTML: https://mlsanity.com/paper/2110.08092/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 24 of its 31 references that are in the map.
- First posted on arXiv 15 Oct 2021.

## Builds on (most cited first, 24 of 24)

- [Semi-Supervised Classification with Graph Convolutional Networks](https://mlsanity.com/paper/1609.02907/) (2016)
- [PointNet: Deep Learning on Point Sets for 3D Classification and Segmentation](https://mlsanity.com/paper/1612.00593/) (2016)
- [Convolutional Neural Networks on Graphs with Fast Localized Spectral Filtering](https://mlsanity.com/paper/1606.09375/) (2016)
- [Deep Sets](https://mlsanity.com/paper/1703.06114/) (2017)
- [Spectral Networks and Locally Connected Networks on Graphs](https://mlsanity.com/paper/1312.6203/) (2013)
- [Provably Powerful Graph Networks](https://mlsanity.com/paper/1905.11136/) (2019)
- [Invariant and Equivariant Graph Networks](https://mlsanity.com/paper/1812.09902/) (2018)
- [Deep Convolutional Networks on Graph-Structured Data](https://mlsanity.com/paper/1506.05163/) (2015)
- [CayleyNets: Graph Convolutional Neural Networks with Complex Rational Spectral Filters](https://mlsanity.com/paper/1705.07664/) (2017)
- [Universal Invariant and Equivariant Graph Neural Networks](https://mlsanity.com/paper/1905.04943/) (2019)
- [On the equivalence between graph isomorphism testing and function approximation with GNNs](https://mlsanity.com/paper/1905.12560/) (2019)
- [On the Universality of Invariant Networks](https://mlsanity.com/paper/1901.09342/) (2019)
- [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)
- [On Learning Sets of Symmetric Elements](https://mlsanity.com/paper/2002.08599/) (2020)
- [Deep Models of Interactions Across Sets](https://mlsanity.com/paper/1803.02879/) (2018)
- [Covariant Compositional Networks For Learning Graphs](https://mlsanity.com/paper/1801.02144/) (2018)
- [On Universal Equivariant Set Networks](https://mlsanity.com/paper/1910.02421/) (2019)
- [Graph Homomorphism Convolution](https://mlsanity.com/paper/2005.01214/) (2020)
- [A Simple Proof of the Universality of Invariant/Equivariant Graph Neural Networks](https://mlsanity.com/paper/1910.03802/) (2019)
- [Neural Networks for Learning Counterfactual G-Invariances from Single Environments](https://mlsanity.com/paper/2104.10105/) (2021)
- [A Computationally Efficient Neural Network Invariant to the Action of Symmetry Subgroups](https://mlsanity.com/paper/2002.07528/) (2020)
- [Equivariant Entity-Relationship Networks](https://mlsanity.com/paper/1903.09033/) (2019)

## Built on by (most cited first, 4 of 4)

- [Learning Probabilistic Symmetrization for Architecture Agnostic Equivariance](https://mlsanity.com/paper/2306.02866/) (2023)
- [Generalizing Neural Human Fitting to Unseen Poses With Articulated SE(3) Equivariance](https://mlsanity.com/paper/2304.10528/) (2023)
- [LoFi: Neural Local Fields for Scalable Image Reconstruction](https://mlsanity.com/paper/2411.04995/) (2024)
- [Equivariant Splitting: Self-supervised learning from incomplete data](https://mlsanity.com/paper/2510.00929/) (2025)

## Abstract (arXiv)

Invariant and equivariant networks are useful in learning data with symmetry, including images, sets, point clouds, and graphs. In this paper, we consider invariant and equivariant networks for symmetries of finite groups. Invariant and equivariant networks have been constructed by various researchers using Reynolds operators. However, Reynolds operators are computationally expensive when the order of the group is large because they use the sum over the whole group, which poses an implementation difficulty. To overcome this difficulty, we consider representing the Reynolds operator as a sum over a subset instead of a sum over the whole group. We call such a subset a Reynolds design, and an operator defined by a sum over a Reynolds design a reductive Reynolds operator. For example, in the case of a graph with n nodes, the computational complexity of the reductive Reynolds operator is reduced to O(n2), while the computational complexity of the Reynolds operator is O(n!). We construct learning models based on the reductive Reynolds operator called equivariant and invariant Reynolds networks (ReyNets) and prove that they have universal approximation property. Reynolds designs for equivariant ReyNets are derived from combinatorial observations with Young diagrams, while Reynolds designs for invariant ReyNets are derived from invariants called Reynolds dimensions defined on the set of invariant polynomials. Numerical experiments show that the performance of our models is comparable to state-of-the-art methods.