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Harmonic Networks: Deep Translation and Rotation Equivariance

arXiv 1612.04642 ↗First posted 14 Dec 2016cs.CV cs.LG stat.MLcited by 293

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Abstract (arXiv)

Translating or rotating an input image should not affect the results of many computer vision tasks. Convolutional neural networks (CNNs) are already translation equivariant: input image translations produce proportionate feature map translations. This is not the case for rotations. Global rotation equivariance is typically sought through data augmentation, but patch-wise equivariance is more difficult. We present Harmonic Networks or H-Nets, a CNN exhibiting equivariance to patch-wise translation and 360-rotation. We achieve this by replacing regular CNN filters with circular harmonics, returning a maximal response and orientation for every receptive field patch. H-Nets use a rich, parameter-efficient and low computational complexity representation, and we show that deep feature maps within the network encode complicated rotational invariants. We demonstrate that our layers are general enough to be used in conjunction with the latest architectures and techniques, such as deep supervision and batch normalization. We also achieve state-of-the-art classification on rotated-MNIST, and competitive results on other benchmark challenges.

Key facts. Cited by 293 papers in the map (data through 7 Oct 2026). First posted on arXiv 14 Dec 2016.

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Cited by year (2026 to 7 Oct)

Citing papers per year: 2021: 72; 2022: 67; 2023: 58; 2024: 34; 2025: 37; 2026: 25.

293 papers.

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  1. 2021Alias-Free Generative Adversarial Networkscited by 1,181
  2. 2022Equiformer: Equivariant Graph Attention Transformer for 3D Atomistic Graphscited by 258
  3. 2021Geometric and Physical Quantities Improve E(3) Equivariant Message Passingcited by 225
  4. 2021ReDet: A Rotation-equivariant Detector for Aerial Object Detectioncited by 202
  5. 2023EquiformerV2: Improved Equivariant Transformer for Scaling to Higher-Degree Representationscited by 201
  6. 2021Frame Averaging for Invariant and Equivariant Network Designcited by 166
  7. 2021No-Reference Image Quality Assessment via Transformers, Relative Ranking, and Self-Consistencycited by 165
  8. 2023Reducing SO(3) Convolutions to SO(2) for Efficient Equivariant GNNscited by 113
  9. 2023EqMotion: Equivariant Multi-agent Motion Prediction with Invariant Interaction Reasoningcited by 97
  10. 2022Approximately Equivariant Networks for Imperfectly Symmetric Dynamicscited by 95
  11. 2022Clifford Neural Layers for PDE Modelingcited by 91
  12. 2021Automatic Symmetry Discovery with Lie Algebra Convolutional Networkcited by 82
38 more
  1. 2021Equivariant Point Network for 3D Point Cloud Analysiscited by 81
  2. 2022MetaShift: A Dataset of Datasets for Evaluating Contextual Distribution Shifts and Training Conflictscited by 81
  3. 2021SE(3) Equivariant Graph Neural Networks with Complete Local Framescited by 75
  4. 2021Residual Pathway Priors for Soft Equivariance Constraintscited by 74
  5. 2021You Only Hypothesize Once: Point Cloud Registration with Rotation-equivariant Descriptorscited by 68
  6. 2021SE(3)-equivariant prediction of molecular wavefunctions and electronic densitiescited by 67
  7. 2021Geometric Deep Learning and Equivariant Neural Networkscited by 64
  8. 2022Learning Symmetric Embeddings for Equivariant World Modelscited by 54
  9. 2023Clifford Group Equivariant Neural Networkscited by 54
  10. 2022The Lie Derivative for Measuring Learned Equivariancecited by 51
  11. 2022DEVIANT: Depth EquiVarIAnt NeTwork for Monocular 3D Object Detectioncited by 48
  12. 2021Learning Partial Equivariances from Datacited by 44
  13. 2021Time-Equivariant Contrastive Video Representation Learningcited by 44
  14. 2022Accurate 3-DoF Camera Geo-Localization via Ground-to-Satellite Image Matchingcited by 44
  15. 2021Physics-Guided Deep Learning for Dynamical Systems: A Surveycited by 43
  16. 2022Invariance Learning in Deep Neural Networks with Differentiable Laplace Approximationscited by 42
  17. 2023Group Equivariant Fourier Neural Operators for Partial Differential Equationscited by 42
  18. 2022Relaxing Equivariance Constraints with Non-stationary Continuous Filterscited by 41
  19. 2022Unsupervised Learning of Group Invariant and Equivariant Representationscited by 38
  20. 2022Expansion and Shrinkage of Localization for Weakly-Supervised Semantic Segmentationcited by 37
  21. 2021DISCO: accurate Discrete Scale Convolutionscited by 35
  22. 2022GLASS: Global to Local Attention for Scene-Text Spottingcited by 34
  23. 2022Robust (Controlled) Table-to-Text Generation with Structure-Aware Equivariance Learningcited by 32
  24. 2021Equivariant neural networks for inverse problemscited by 29
  25. 2021Symmetry-via-Duality: Invariant Neural Network Densities from Parameter-Space Correlatorscited by 28
  26. 2022Sample Efficiency of Data Augmentation Consistency Regularizationcited by 28
  27. 2022LieGG: Studying Learned Lie Group Generatorscited by 28
  28. 2021Commutative Lie Group VAE for Disentanglement Learningcited by 27
  29. 2023E(2)-Equivariant Vision Transformercited by 27
  30. 2021Resolution learning in deep convolutional networks using scale-space theorycited by 26
  31. 2021Fourier Series Expansion Based Filter Parametrization for Equivariant Convolutionscited by 26
  32. 2022Self-Supervised Equivariant Learning for Oriented Keypoint Detectioncited by 26
  33. 2022VN-Transformer: Rotation-Equivariant Attention for Vector Neuronscited by 26
  34. 2021Multi-Agent MDP Homomorphic Networkscited by 24
  35. 2021Roto-translated Local Coordinate Frames For Interacting Dynamical Systemscited by 24
  36. 2022Equivariant Networks for Crystal Structurescited by 24
  37. 2023Learning Rotation-Equivariant Features for Visual Correspondencecited by 23
  38. 2023Learning Layer-wise Equivariances Automatically using Gradientscited by 23

50 most cited of 293 All 293 (JSON)