mlsanityData to 7 Oct 2026

A General Theory of Correct, Incorrect, and Extrinsic Equivariance

arXiv 2303.04745 ↗First posted 8 Mar 2023cs.LG stat.MLcited by 18

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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.

Key facts. Cited by 18 papers in the map (data through 7 Oct 2026). Builds on 48 of its 65 references that are in the map. First posted on arXiv 8 Mar 2023.

Builds on

48 of 65 references.

References · 48

  1. 2016Group Equivariant Convolutional Networkscited by 1,154
  2. 2019General E(2)-Equivariant Steerable CNNscited by 441
  3. 2016Steerable CNNscited by 298
  4. 2018On the Generalization of Equivariance and Convolution in Neural Networks to the Action of Compact Groupscited by 296
  5. 2019Cormorant: Covariant Molecular Neural Networkscited by 262
  6. 2020Generalizing Convolutional Neural Networks for Equivariance to Lie Groups on Arbitrary Continuous Datacited by 248
  7. 2014Understanding image representations by measuring their equivariance and equivalencecited by 240
  8. 2018A General Theory of Equivariant CNNs on Homogeneous Spacescited by 223
  9. 2021Neural Descriptor Fields: SE(3)-Equivariant Object Representations for Manipulationcited by 204
  10. 2019On the Universality of Invariant Networkscited by 151
  11. 2020MDP Homomorphic Networks: Group Symmetries in Reinforcement Learningcited by 133
  12. 2018Universal approximations of invariant maps by neural networkscited by 127
36 more
  1. 2020Incorporating Symmetry into Deep Dynamics Models for Improved Generalizationcited by 123
  2. 2020Lorentz Group Equivariant Neural Network for Particle Physicscited by 120
  3. 2021Geometric Deep Learning on Molecular Representationscited by 113
  4. 2020On Learning Sets of Symmetric Elementscited by 97
  5. 2021Provably Strict Generalisation Benefit for Equivariant Modelscited by 90
  6. 2021Automatic Symmetry Discovery with Lie Algebra Convolutional Networkcited by 82
  7. 2022SO(2)-Equivariant Reinforcement Learningcited by 79
  8. 2020On the Benefits of Invariance in Neural Networkscited by 78
  9. 2020Meta-Learning Symmetries by Reparameterizationcited by 75
  10. 2021Equivariant Q Learning in Spatial Action Spacescited by 61
  11. 2022Equivariant Descriptor Fields: SE(3)-Equivariant Energy-Based Models for End-to-End Visual Robotic Manipulation Learningcited by 54
  12. 2022Sample Efficient Grasp Learning Using Equivariant Modelscited by 52
  13. 2022SE(3)-Equivariant Relational Rearrangement with Neural Descriptor Fieldscited by 51
  14. 2022TAX-Pose: Task-Specific Cross-Pose Estimation for Robot Manipulationcited by 47
  15. 2021Leveraging SE(3) Equivariance for Self-Supervised Category-Level Object Pose Estimationcited by 45
  16. 2022Equivariant Transporter Networkcited by 44
  17. 2016Generalization Error of Invariant Classifierscited by 42
  18. 2022Unsupervised Learning of Group Invariant and Equivariant Representationscited by 38
  19. 2022On-Robot Learning With Equivariant Modelscited by 38
  20. 2019Improved Generalization Bounds of Group Invariant / Equivariant Deep Networks via Quotient Feature Spacescited by 34
  21. 2022Edge Grasp Network: A Graph-Based SE(3)-invariant Approach to Grasp Detectioncited by 34
  22. 2022The Surprising Effectiveness of Equivariant Models in Domains with Latent Symmetrycited by 32
  23. 2022LieGG: Studying Learned Lie Group Generatorscited by 28
  24. 2020Trajectory Prediction using Equivariant Continuous Convolutioncited by 26
  25. 2022SE(3)-Equivariant Attention Networks for Shape Reconstruction in Function Spacecited by 26
  26. 2023Self-Supervised Category-Level Articulated Object Pose Estimation with Part-Level SE(3) Equivariancecited by 26
  27. 2015Dreaming More Data: Class-dependent Distributions over Diffeomorphisms for Learned Data Augmentationcited by 24
  28. 2021CADDA: Class-wise Automatic Differentiable Data Augmentation for EEG Signalscited by 22
  29. 2022SEIL: Simulation-augmented Equivariant Imitation Learningcited by 22
  30. 2022Equivariant Representation Learning via Class-Pose Decompositioncited by 17
  31. 2022A PAC-Bayesian Generalization Bound for Equivariant Networkscited by 17
  32. 2023Image to Sphere: Learning Equivariant Features for Efficient Pose Predictioncited by 17
  33. 2022Learning Instance-Specific Augmentations by Capturing Local Invariancescited by 15
  34. 2022BulletArm: An Open-Source Robotic Manipulation Benchmark and Learning Frameworkcited by 9
  35. 2022Equivariance-aware Architectural Optimization of Neural Networkscited by 8
  36. 2022Integrating Symmetry into Differentiable Planning with Steerable Convolutionscited by 6

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Cited by year (2026 to 7 Oct)

Citing papers per year: 2023: 4; 2024: 5; 2025: 7; 2026: 2.

18 papers.

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  1. 2024Equivariant Diffusion Policycited by 61
  2. 2023Diffusion-EDFs: Bi-equivariant Denoising Generative Modeling on SE(3) for Visual Robotic Manipulationcited by 51
  3. 2023Discovering Symmetry Breaking in Physical Systems with Relaxed Group Convolutioncited by 19
  4. 2024Equivariant Reinforcement Learning under Partial Observabilitycited by 17
  5. 2023Almost Equivariance via Lie Algebra Convolutionscited by 10
  6. 2024Approximate Equivariance in Reinforcement Learningcited by 9
  7. 2025A Tale of Two Symmetries: Exploring the Loss Landscape of Equivariant Modelscited by 8
  8. 2024Symmetry-aware Reinforcement Learning for Robotic Assembly under Partial Observability with a Soft Wristcited by 7
  9. 2025SE(3)-Equivariant Diffusion Policy in Spherical Fourier Spacecited by 7
  10. 2025AtlasD: Automatic Local Symmetry Discoverycited by 4
  11. 2025Group Downsampling with Equivariant Anti-aliasingcited by 3
  12. 2025Partially Equivariant Reinforcement Learning in Symmetry-Breaking Environmentscited by 3
6 more
  1. 2023Learning Polynomial Problems with SL(2,ℝ) Equivariancecited by 2
  2. 2024Optimal Symmetries in Binary Classificationcited by 1
  3. 2025PEnGUiN: Partially Equivariant Graph NeUral Networks for Sample Efficient MARLcited by 1
  4. 2025On Uncertainty Calibration for Equivariant Functionscited by 1
  5. 2026Smoothness Errors in Dynamics Models and How to Avoid Themcited by 0
  6. 2026Tunable Soft Equivariance with Guaranteescited by 0