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Exploiting Cyclic Symmetry in Convolutional Neural Networks

arXiv 1602.02660 ↗First posted 8 Feb 2016cs.LG cs.CV cs.NEcited by 91

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Abstract (arXiv)

Many classes of images exhibit rotational symmetry. Convolutional neural networks are sometimes trained using data augmentation to exploit this, but they are still required to learn the rotation equivariance properties from the data. Encoding these properties into the network architecture, as we are already used to doing for translation equivariance by using convolutional layers, could result in a more efficient use of the parameter budget by relieving the model from learning them. We introduce four operations which can be inserted into neural network models as layers, and which can be combined to make these models partially equivariant to rotations. They also enable parameter sharing across different orientations. We evaluate the effect of these architectural modifications on three datasets which exhibit rotational symmetry and demonstrate improved performance with smaller models.

Key facts. Cited by 91 papers in the map (data through 7 Oct 2026). First posted on arXiv 8 Feb 2016.

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Citing papers per year: 2021: 31; 2022: 21; 2023: 17; 2024: 9; 2025: 9; 2026: 4.

91 papers.

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  1. 2021NeRV: Neural Representations for Videoscited by 255
  2. 2021E(n) Equivariant Normalizing Flowscited by 165
  3. 2022Approximately Equivariant Networks for Imperfectly Symmetric Dynamicscited by 95
  4. 2021SE(3) Equivariant Graph Neural Networks with Complete Local Framescited by 75
  5. 2023Pushing the Limits of Fewshot Anomaly Detection in Industry Vision: Graphcorecited by 75
  6. 2021Geometric Deep Learning and Equivariant Neural Networkscited by 64
  7. 2021Exploring Complementary Strengths of Invariant and Equivariant Representations for Few-Shot Learningcited by 56
  8. 2021Self-Contrastive Learning with Hard Negative Sampling for Self-supervised Point Cloud Learningcited by 52
  9. 2022DEVIANT: Depth EquiVarIAnt NeTwork for Monocular 3D Object Detectioncited by 48
  10. 2021Learning Partial Equivariances from Datacited by 44
  11. 2021Physics-Guided Deep Learning for Dynamical Systems: A Surveycited by 43
  12. 2022Unsupervised Learning of Group Invariant and Equivariant Representationscited by 38
38 more
  1. 2025InterAct: Advancing Large-Scale Versatile 3D Human-Object Interaction Generationcited by 31
  2. 2021Equivariant neural networks for inverse problemscited by 29
  3. 2021Commutative Lie Group VAE for Disentanglement Learningcited by 27
  4. 2022Symmetry Group Equivariant Architectures for Physicscited by 23
  5. 2024Lie Algebra Canonicalization: Equivariant Neural Operators under arbitrary Lie Groupscited by 23
  6. 2022A Theory of PAC Learnability under Transformation Invariancescited by 21
  7. 2022Maximum Class Separation as Inductive Bias in One Matrixcited by 21
  8. 2022RIC-CNN: Rotation-Invariant Coordinate Convolutional Neural Networkcited by 20
  9. 2023Discovering Symmetry Breaking in Physical Systems with Relaxed Group Convolutioncited by 19
  10. 2021Scale-invariant scale-channel networks: Deep networks that generalise to previously unseen scalescited by 17
  11. 2022A PAC-Bayesian Generalization Bound for Equivariant Networkscited by 17
  12. 2023Revisiting Data Augmentation for Rotational Invariance in Convolutional Neural Networkscited by 16
  13. 2021Group Equivariant Subsamplingcited by 15
  14. 2023What Affects Learned Equivariance in Deep Image Recognition Models?cited by 14
  15. 2021Relative stability toward diffeomorphisms indicates performance in deep netscited by 12
  16. 2021Symmetry reduction for deep reinforcement learning active control of chaotic spatiotemporal dynamicscited by 10
  17. 2021Learning Equivariant Energy Based Models with Equivariant Stein Variational Gradient Descentcited by 10
  18. 2022Learning with Capsules: A Surveycited by 10
  19. 2023Rotation Invariant Transformer for Recognizing Object in UAVscited by 10
  20. 2022Robust Perception through Equivariancecited by 9
  21. 2024A Characterization Theorem for Equivariant Networks with Point-wise Activationscited by 9
  22. 2022Analysis of (sub-)Riemannian PDE-G-CNNscited by 8
  23. 2025Flopping for FLOPs: Leveraging equivariance for computational efficiencycited by 8
  24. 2021Mesh-based graph convolutional neural networks for modeling materials with microstructurecited by 7
  25. 2021Universal approximation and model compression for radial neural networkscited by 7
  26. 2024How Deep Networks Learn Sparse and Hierarchical Data: the Sparse Random Hierarchy Modelcited by 7
  27. 2023PixelRNN: In-pixel Recurrent Neural Networks for End-to-end-optimized Perception with Neural Sensorscited by 6
  28. 2023E(2) Equivariant Neural Networks for Robust Galaxy Morphology Classificationcited by 6
  29. 2021Improving Sound Event Classification by Increasing Shift Invariance in Convolutional Neural Networkscited by 5
  30. 2023Steerable Equivariant Representation Learningcited by 5
  31. 2021Adjoint Rigid Transform Network: Task-conditioned Alignment of 3D Shapescited by 4
  32. 2022Analysis of convolutional neural network image classifiers in a rotationally symmetric modelcited by 4
  33. 2022Quiver neural networkscited by 4
  34. 2022Beyond CNNs: Exploiting Further Inherent Symmetries in Medical Image Segmentationcited by 4
  35. 2025Spectral State Space Model for Rotation-Invariant Visual Representation Learningcited by 4
  36. 2021Self-Supervised Multi-View Learning via Auto-Encoding 3D Transformationscited by 3
  37. 2021Self-Supervised Graph Representation Learning via Topology Transformationscited by 3
  38. 2021Designing Rotationally Invariant Neural Networks from PDEs and Variational Methodscited by 3

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