# The Surprising Effectiveness of Equivariant Models in Domains with Latent Symmetry

arXiv: https://arxiv.org/abs/2211.09231 · HTML: https://mlsanity.com/paper/2211.09231/

## Key facts

- Cited by 33 papers in the map (data through 14 Oct 2026).
- Builds on 36 of its 44 references that are in the map.
- First posted on arXiv 16 Nov 2022.

## Builds on (most cited first, 30 of 36)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Representation Learning with Contrastive Predictive Coding](https://mlsanity.com/paper/1807.03748/) (2018)
- [Soft Actor-Critic: Off-Policy Maximum Entropy Deep Reinforcement Learning with a Stochastic Actor](https://mlsanity.com/paper/1801.01290/) (2018)
- [Group Equivariant Convolutional Networks](https://mlsanity.com/paper/1602.07576/) (2016)
- [Prioritized Experience Replay](https://mlsanity.com/paper/1511.05952/) (2015)
- [CURL: Contrastive Unsupervised Representations for Reinforcement Learning](https://mlsanity.com/paper/2004.04136/) (2020)
- [Image Augmentation Is All You Need: Regularizing Deep Reinforcement Learning from Pixels](https://mlsanity.com/paper/2004.13649/) (2020)
- [Reinforcement Learning with Augmented Data](https://mlsanity.com/paper/2004.14990/) (2020)
- [Spherical CNNs](https://mlsanity.com/paper/1801.10130/) (2018)
- [General E(2)-Equivariant Steerable CNNs](https://mlsanity.com/paper/1911.08251/) (2019)
- [dm_control: Software and Tasks for Continuous Control](https://mlsanity.com/paper/2006.12983/) (2020)
- [Mastering Visual Continuous Control: Improved Data-Augmented Reinforcement Learning](https://mlsanity.com/paper/2107.09645/) (2021)
- [Steerable CNNs](https://mlsanity.com/paper/1612.08498/) (2016)
- [Cormorant: Covariant Molecular Neural Networks](https://mlsanity.com/paper/1906.04015/) (2019)
- [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)
- [MDP Homomorphic Networks: Group Symmetries in Reinforcement Learning](https://mlsanity.com/paper/2006.16908/) (2020)
- [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)
- [On Learning Sets of Symmetric Elements](https://mlsanity.com/paper/2002.08599/) (2020)
- [Gauge Equivariant Mesh CNNs: Anisotropic convolutions on geometric graphs](https://mlsanity.com/paper/2003.05425/) (2020)
- [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 Universality of Rotation Equivariant Point Cloud Networks](https://mlsanity.com/paper/2010.02449/) (2020)
- [Meta-Learning Symmetries by Reparameterization](https://mlsanity.com/paper/2007.02933/) (2020)
- [Explorations in Homeomorphic Variational Auto-Encoding](https://mlsanity.com/paper/1807.04689/) (2018)
- [Equivariant Q Learning in Spatial Action Spaces](https://mlsanity.com/paper/2110.15443/) (2021)
- [Learning Symmetric Embeddings for Equivariant World Models](https://mlsanity.com/paper/2204.11371/) (2022)
- [Sample Efficient Grasp Learning Using Equivariant Models](https://mlsanity.com/paper/2202.09468/) (2022)
- [Equivariant Transporter Network](https://mlsanity.com/paper/2202.09400/) (2022)

## Built on by (most cited first, 30 of 33)

- [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)
- [Latent Space Symmetry Discovery](https://mlsanity.com/paper/2310.00105/) (2023)
- [A General Theory of Correct, Incorrect, and Extrinsic Equivariance](https://mlsanity.com/paper/2303.04745/) (2023)
- [Deep SE(3)-Equivariant Geometric Reasoning for Precise Placement Tasks](https://mlsanity.com/paper/2404.13478/) (2024)
- [Imagination Policy: Using Generative Point Cloud Models for Learning Manipulation Policies](https://mlsanity.com/paper/2406.11740/) (2024)
- [Equivariant Reinforcement Learning under Partial Observability](https://mlsanity.com/paper/2408.14336/) (2024)
- [Two by Two: Learning Multi-Task Pairwise Objects Assembly for Generalizable Robot Manipulation](https://mlsanity.com/paper/2504.06961/) (2025)
- [SE(3)-Equivariant Robot Learning and Control: A Tutorial Survey](https://mlsanity.com/paper/2503.09829/) (2025)
- [Equivariant Symmetry Breaking Sets](https://mlsanity.com/paper/2402.02681/) (2024)
- [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)
- [EqNIO: Subequivariant Neural Inertial Odometry](https://mlsanity.com/paper/2408.06321/) (2024)
- [SO(2)-Equivariant Downwash Models for Close Proximity Flight](https://mlsanity.com/paper/2305.18983/) (2023)
- [Understanding when Dynamics-Invariant Data Augmentations Benefit Model-Free Reinforcement Learning Updates](https://mlsanity.com/paper/2310.17786/) (2023)
- [Equivariant Single View Pose Prediction Via Induced and Restricted Representations](https://mlsanity.com/paper/2307.03704/) (2023)
- [Equivariant Offline Reinforcement Learning](https://mlsanity.com/paper/2406.13961/) (2024)
- [3D Equivariant Visuomotor Policy Learning via Spherical Projection](https://mlsanity.com/paper/2505.16969/) (2025)
- [Equivariant Ensembles and Regularization for Reinforcement Learning in Map-based Path Planning](https://mlsanity.com/paper/2403.12856/) (2024)
- [SiT: Symmetry-Invariant Transformers for Generalisation in Reinforcement Learning](https://mlsanity.com/paper/2406.15025/) (2024)
- [Algebras of actions in an agent's representations of the world](https://mlsanity.com/paper/2310.01536/) (2023)
- [Learning Polynomial Problems with SL(2,ℝ) Equivariance](https://mlsanity.com/paper/2312.02146/) (2023)
- [Segment as You Wish -- Free-Form Language-Based Segmentation for Medical Images](https://mlsanity.com/paper/2410.12831/) (2024)
- [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)
- [Symmetries-enhanced Multi-Agent Reinforcement Learning](https://mlsanity.com/paper/2501.01136/) (2025)
- [On Uncertainty Calibration for Equivariant Functions](https://mlsanity.com/paper/2510.21691/) (2025)
- [REViT: Roto-reflection Equivariant Convolutional Vision Transformer](https://mlsanity.com/paper/2606.25318/) (2026)
- [Time Reversal Symmetry for Efficient Robotic Manipulations in Deep Reinforcement Learning](https://mlsanity.com/paper/2505.13925/) (2025)
- [Group-Invariant Unsupervised Skill Discovery: Symmetry-aware Skill Representations for Generalizable Behavior](https://mlsanity.com/paper/2601.14000/) (2026)

## Abstract (arXiv)

Extensive work has demonstrated that equivariant neural networks can significantly improve sample efficiency and generalization by enforcing an inductive bias in the network architecture. These applications typically assume that the domain symmetry is fully described by explicit transformations of the model inputs and outputs. However, many real-life applications contain only latent or partial symmetries which cannot be easily described by simple transformations of the input. In these cases, it is necessary to learn symmetry in the environment instead of imposing it mathematically on the network architecture. We discover, surprisingly, that imposing equivariance constraints that do not exactly match the domain symmetry is very helpful in learning the true symmetry in the environment. We differentiate between extrinsic and incorrect symmetry constraints and show that while imposing incorrect symmetry can impede the model's performance, imposing extrinsic symmetry can actually improve performance. We demonstrate that an equivariant model can significantly outperform non-equivariant methods on domains with latent symmetries both in supervised learning and in reinforcement learning for robotic manipulation and control problems.