# Encoding Involutory Invariances in Neural Networks

arXiv: https://arxiv.org/abs/2106.12891 · HTML: https://mlsanity.com/paper/2106.12891/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 19 of its 38 references that are in the map.
- First posted on arXiv 7 Jun 2021.

## Builds on (most cited first, 19 of 19)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Fashion-MNIST: a Novel Image Dataset for Benchmarking Machine Learning Algorithms](https://mlsanity.com/paper/1708.07747/) (2017)
- [Deep Sets](https://mlsanity.com/paper/1703.06114/) (2017)
- [Group Equivariant Convolutional Networks](https://mlsanity.com/paper/1602.07576/) (2016)
- [Hamiltonian Neural Networks](https://mlsanity.com/paper/1906.01563/) (2019)
- [General E(2)-Equivariant Steerable CNNs](https://mlsanity.com/paper/1911.08251/) (2019)
- [Physics Informed Deep Learning (Part I): Data-driven Solutions of Nonlinear Partial Differential Equations](https://mlsanity.com/paper/1711.10561/) (2017)
- [On the Generalization of Equivariance and Convolution in Neural Networks to the Action of Compact Groups](https://mlsanity.com/paper/1802.03690/) (2018)
- [Learning SO(3) Equivariant Representations with Spherical CNNs](https://mlsanity.com/paper/1711.06721/) (2017)
- [Universal Approximation with Deep Narrow Networks](https://mlsanity.com/paper/1905.08539/) (2019)
- [Incorporating Symmetry into Deep Dynamics Models for Improved Generalization](https://mlsanity.com/paper/2002.03061/) (2020)
- [Lorentz Group Equivariant Neural Network for Particle Physics](https://mlsanity.com/paper/2006.04780/) (2020)
- [Neural Networks Fail to Learn Periodic Functions and How to Fix It](https://mlsanity.com/paper/2006.08195/) (2020)
- [On Learning Sets of Symmetric Elements](https://mlsanity.com/paper/2002.08599/) (2020)
- [Physical Symmetries Embedded in Neural Networks](https://mlsanity.com/paper/1904.08991/) (2019)
- [Symmetry meets AI](https://mlsanity.com/paper/2103.06115/) (2021)
- [Streaming Classification of Variable Stars](https://mlsanity.com/paper/1912.02235/) (2019)
- [Beyond permutation equivariance in graph networks](https://mlsanity.com/paper/2103.14066/) (2021)
- [A Swarm Variant for the Schr"odinger Solver](https://mlsanity.com/paper/2104.04795/) (2021)

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

- [Addressing imperfect symmetry: A novel symmetry-learning actor-critic extension](https://mlsanity.com/paper/2309.02711/) (2023)

## Abstract (arXiv)

In certain situations, neural networks are trained upon data that obey underlying symmetries. However, the predictions do not respect the symmetries exactly unless embedded in the network structure. In this work, we introduce architectures that embed a special kind of symmetry namely, invariance with respect to involutory linear/affine transformations up to parity p=± 1. We provide rigorous theorems to show that the proposed network ensures such an invariance and present qualitative arguments for a special universal approximation theorem. An adaption of our techniques to CNN tasks for datasets with inherent horizontal/vertical reflection symmetry is demonstrated. Extensive experiments indicate that the proposed model outperforms baseline feed-forward and physics-informed neural networks while identically respecting the underlying symmetry.