# A General Theory of Correct, Incorrect, and Extrinsic Equivariance

arXiv: https://arxiv.org/abs/2303.04745 · HTML: https://mlsanity.com/paper/2303.04745/

## Key facts

- Cited by 19 papers in the map (data through 14 Oct 2026).
- Builds on 48 of its 65 references that are in the map.
- First posted on arXiv 8 Mar 2023.

## Builds on (most cited first, 30 of 48)

- [Group Equivariant Convolutional Networks](https://mlsanity.com/paper/1602.07576/) (2016)
- [General E(2)-Equivariant Steerable CNNs](https://mlsanity.com/paper/1911.08251/) (2019)
- [Steerable CNNs](https://mlsanity.com/paper/1612.08498/) (2016)
- [On the Generalization of Equivariance and Convolution in Neural Networks to the Action of Compact Groups](https://mlsanity.com/paper/1802.03690/) (2018)
- [Cormorant: Covariant Molecular Neural Networks](https://mlsanity.com/paper/1906.04015/) (2019)
- [Generalizing Convolutional Neural Networks for Equivariance to Lie Groups on Arbitrary Continuous Data](https://mlsanity.com/paper/2002.12880/) (2020)
- [Understanding image representations by measuring their equivariance and equivalence](https://mlsanity.com/paper/1411.5908/) (2014)
- [A General Theory of Equivariant CNNs on Homogeneous Spaces](https://mlsanity.com/paper/1811.02017/) (2018)
- [Neural Descriptor Fields: SE(3)-Equivariant Object Representations for Manipulation](https://mlsanity.com/paper/2112.05124/) (2021)
- [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)
- [Universal approximations of invariant maps by neural networks](https://mlsanity.com/paper/1804.10306/) (2018)
- [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)
- [Geometric Deep Learning on Molecular Representations](https://mlsanity.com/paper/2107.12375/) (2021)
- [On Learning Sets of Symmetric Elements](https://mlsanity.com/paper/2002.08599/) (2020)
- [Provably Strict Generalisation Benefit for Equivariant Models](https://mlsanity.com/paper/2102.10333/) (2021)
- [SO(2)-Equivariant Reinforcement Learning](https://mlsanity.com/paper/2203.04439/) (2022)
- [Automatic Symmetry Discovery with Lie Algebra Convolutional Network](https://mlsanity.com/paper/2109.07103/) (2021)
- [On the Benefits of Invariance in Neural Networks](https://mlsanity.com/paper/2005.00178/) (2020)
- [Meta-Learning Symmetries by Reparameterization](https://mlsanity.com/paper/2007.02933/) (2020)
- [Equivariant Q Learning in Spatial Action Spaces](https://mlsanity.com/paper/2110.15443/) (2021)
- [Equivariant Descriptor Fields: SE(3)-Equivariant Energy-Based Models for End-to-End Visual Robotic Manipulation Learning](https://mlsanity.com/paper/2206.08321/) (2022)
- [Sample Efficient Grasp Learning Using Equivariant Models](https://mlsanity.com/paper/2202.09468/) (2022)
- [SE(3)-Equivariant Relational Rearrangement with Neural Descriptor Fields](https://mlsanity.com/paper/2211.09786/) (2022)
- [TAX-Pose: Task-Specific Cross-Pose Estimation for Robot Manipulation](https://mlsanity.com/paper/2211.09325/) (2022)
- [Equivariant Transporter Network](https://mlsanity.com/paper/2202.09400/) (2022)
- [Leveraging SE(3) Equivariance for Self-Supervised Category-Level Object Pose Estimation](https://mlsanity.com/paper/2111.00190/) (2021)
- [Generalization Error of Invariant Classifiers](https://mlsanity.com/paper/1610.04574/) (2016)
- [On-Robot Learning With Equivariant Models](https://mlsanity.com/paper/2203.04923/) (2022)

## Built on by (most cited first, 19 of 19)

- [Equivariant Diffusion Policy](https://mlsanity.com/paper/2407.01812/) (2024)
- [Diffusion-EDFs: Bi-equivariant Denoising Generative Modeling on SE(3) for Visual Robotic Manipulation](https://mlsanity.com/paper/2309.02685/) (2023)
- [Discovering Symmetry Breaking in Physical Systems with Relaxed Group Convolution](https://mlsanity.com/paper/2310.02299/) (2023)
- [Equivariant Reinforcement Learning under Partial Observability](https://mlsanity.com/paper/2408.14336/) (2024)
- [Almost Equivariance via Lie Algebra Convolutions](https://mlsanity.com/paper/2310.13164/) (2023)
- [Symmetry-aware Reinforcement Learning for Robotic Assembly under Partial Observability with a Soft Wrist](https://mlsanity.com/paper/2402.18002/) (2024)
- [Approximate Equivariance in Reinforcement Learning](https://mlsanity.com/paper/2411.04225/) (2024)
- [A Tale of Two Symmetries: Exploring the Loss Landscape of Equivariant Models](https://mlsanity.com/paper/2506.02269/) (2025)
- [SE(3)-Equivariant Diffusion Policy in Spherical Fourier Space](https://mlsanity.com/paper/2507.01723/) (2025)
- [AtlasD: Automatic Local Symmetry Discovery](https://mlsanity.com/paper/2504.10777/) (2025)
- [Group Downsampling with Equivariant Anti-aliasing](https://mlsanity.com/paper/2504.17258/) (2025)
- [Partially Equivariant Reinforcement Learning in Symmetry-Breaking Environments](https://mlsanity.com/paper/2512.00915/) (2025)
- [Learning Polynomial Problems with SL(2,ℝ) Equivariance](https://mlsanity.com/paper/2312.02146/) (2023)
- [PEnGUiN: Partially Equivariant Graph NeUral Networks for Sample Efficient MARL](https://mlsanity.com/paper/2503.15615/) (2025)
- [Optimal Symmetries in Binary Classification](https://mlsanity.com/paper/2408.08823/) (2024)
- [On Uncertainty Calibration for Equivariant Functions](https://mlsanity.com/paper/2510.21691/) (2025)
- [Smoothness Errors in Dynamics Models and How to Avoid Them](https://mlsanity.com/paper/2602.05352/) (2026)
- [Tunable Soft Equivariance with Guarantees](https://mlsanity.com/paper/2603.26657/) (2026)
- [Measuring Learned Equivariance: An Information Theoretic Approach](https://mlsanity.com/paper/2610.17269/) (2026)

## Abstract (arXiv)

Although equivariant machine learning has proven effective at many tasks, success depends heavily on the assumption that the ground truth function is symmetric over the entire domain matching the symmetry in an equivariant neural network. A missing piece in the equivariant learning literature is the analysis of equivariant networks when symmetry exists only partially in the domain. In this work, we present a general theory for such a situation. We propose pointwise definitions of correct, incorrect, and extrinsic equivariance, which allow us to quantify continuously the degree of each type of equivariance a function displays. We then study the impact of various degrees of incorrect or extrinsic symmetry on model error. We prove error lower bounds for invariant or equivariant networks in classification or regression settings with partially incorrect symmetry. We also analyze the potentially harmful effects of extrinsic equivariance. Experiments validate these results in three different environments.