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General E(2)-Equivariant Steerable CNNs

arXiv 1911.08251 ↗First posted 19 Nov 2019cs.CV cs.LG eess.IVcited by 446

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Abstract (arXiv)

The big empirical success of group equivariant networks has led in recent years to the sprouting of a great variety of equivariant network architectures. A particular focus has thereby been on rotation and reflection equivariant CNNs for planar images. Here we give a general description of E(2)-equivariant convolutions in the framework of Steerable CNNs. The theory of Steerable CNNs thereby yields constraints on the convolution kernels which depend on group representations describing the transformation laws of feature spaces. We show that these constraints for arbitrary group representations can be reduced to constraints under irreducible representations. A general solution of the kernel space constraint is given for arbitrary representations of the Euclidean group E(2) and its subgroups. We implement a wide range of previously proposed and entirely new equivariant network architectures and extensively compare their performances. E(2)-steerable convolutions are further shown to yield remarkable gains on CIFAR-10, CIFAR-100 and STL-10 when used as a drop-in replacement for non-equivariant convolutions.

Key facts. Cited by 446 papers in the map (data through 14 Oct 2026). First posted on arXiv 19 Nov 2019.

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Citing papers per year: 2021: 62; 2022: 80; 2023: 85; 2024: 69; 2025: 81; 2026: 69.

446 papers.

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  1. 2021Alias-Free Generative Adversarial Networkscited by 1,184
  2. 2021E(n) Equivariant Graph Neural Networkscited by 965
  3. 2021Geometric and Physical Quantities Improve E(3) Equivariant Message Passingcited by 228
  4. 2021ReDet: A Rotation-equivariant Detector for Aerial Object Detectioncited by 203
  5. 2021A Practical Method for Constructing Equivariant Multilayer Perceptrons for Arbitrary Matrix Groupscited by 187
  6. 2022Approximately Equivariant Networks for Imperfectly Symmetric Dynamicscited by 97
  7. 2022Clifford Neural Layers for PDE Modelingcited by 92
  8. 2025EQ-VAE: Equivariance Regularized Latent Space for Improved Generative Image Modelingcited by 89
  9. 2021Automatic Symmetry Discovery with Lie Algebra Convolutional Networkcited by 83
  10. 2021Independent SE(3)-Equivariant Models for End-to-End Rigid Protein Dockingcited by 82
  11. 2021Is Attention Better Than Matrix Decomposition?cited by 77
  12. 2021SE(3) Equivariant Graph Neural Networks with Complete Local Framescited by 76
38 more
  1. 2022PAC-Bayes Compression Bounds So Tight That They Can Explain Generalizationcited by 76
  2. 2021Residual Pathway Priors for Soft Equivariance Constraintscited by 75
  3. 2021Geometric Deep Learning and Equivariant Neural Networkscited by 65
  4. 2022When Physics Meets Machine Learning: A Survey of Physics-Informed Machine Learningcited by 65
  5. 2021Visual Representation Learning Does Not Generalize Strongly Within the Same Domaincited by 61
  6. 2023Clifford Group Equivariant Neural Networkscited by 55
  7. 2022Learning Symmetric Embeddings for Equivariant World Modelscited by 54
  8. 2022The Lie Derivative for Measuring Learned Equivariancecited by 54
  9. 2022Transformation-Equivariant 3D Object Detection for Autonomous Drivingcited by 52
  10. 2023Equivariant Similarity for Vision-Language Foundation Modelscited by 52
  11. 2023Adaptive Rotated Convolution for Rotated Object Detectioncited by 51
  12. 2022Equivariant Transporter Networkcited by 46
  13. 2021Learning Partial Equivariances from Datacited by 45
  14. 2023Group Equivariant Fourier Neural Operators for Partial Differential Equationscited by 45
  15. 2021Physics-Guided Deep Learning for Dynamical Systems: A Surveycited by 44
  16. 2023Generative Adversarial Symmetry Discoverycited by 42
  17. 2023Equivariant Adaptation of Large Pretrained Modelscited by 41
  18. 2023Fast, Expressive SE(n) Equivariant Networks through Weight-Sharing in Position-Orientation Spacecited by 41
  19. 2022How Much Data Are Augmentations Worth? An Investigation into Scaling Laws, Invariance, and Implicit Regularizationcited by 40
  20. 2022Unsupervised Learning of Group Invariant and Equivariant Representationscited by 39
  21. 2022Learning Physical Dynamics with Subequivariant Graph Neural Networkscited by 39
  22. 2023BEVPlace: Learning LiDAR-based Place Recognition using Bird's Eye View Imagescited by 39
  23. 2021DISCO: accurate Discrete Scale Convolutionscited by 36
  24. 2023EDGI: Equivariant Diffusion for Planning with Embodied Agentscited by 36
  25. 2025Symmetry in Neural Network Parameter Spacescited by 36
  26. 2022Imaging with Equivariant Deep Learningcited by 35
  27. 2021Revisiting Consistency Regularization for Semi-Supervised Learningcited by 34
  28. 2022The Surprising Effectiveness of Equivariant Models in Domains with Latent Symmetrycited by 33
  29. 2024Fourier Transporter: Bi-Equivariant Robotic Manipulation in 3Dcited by 33
  30. 2022LieGG: Studying Learned Lie Group Generatorscited by 31
  31. 2025LEAPS: A discrete neural sampler via locally equivariant networkscited by 31
  32. 2022Guaranteed Conservation of Momentum for Learning Particle-based Fluid Dynamicscited by 30
  33. 2023A Survey on Deep Learning-Based Monocular Spacecraft Pose Estimation: Current State, Limitations and Prospectscited by 30
  34. 2021Equivariant neural networks for inverse problemscited by 29
  35. 2023Latent Space Symmetry Discoverycited by 29
  36. 2022INO: Invariant Neural Operators for Learning Complex Physical Systems with Momentum Conservationcited by 28
  37. 2022A case for using rotation invariant features in state of the art feature matcherscited by 27
  38. 2023E(2)-Equivariant Vision Transformercited by 27

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