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A General Theory of Equivariant CNNs on Homogeneous Spaces

arXiv 1811.02017 ↗First posted 5 Nov 2018cs.LG cs.AI cs.CGcited by 223

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Abstract (arXiv)

We present a general theory of Group equivariant Convolutional Neural Networks (G-CNNs) on homogeneous spaces such as Euclidean space and the sphere. Feature maps in these networks represent fields on a homogeneous base space, and layers are equivariant maps between spaces of fields. The theory enables a systematic classification of all existing G-CNNs in terms of their symmetry group, base space, and field type. We also consider a fundamental question: what is the most general kind of equivariant linear map between feature spaces (fields) of given types? Following Mackey, we show that such maps correspond one-to-one with convolutions using equivariant kernels, and characterize the space of such kernels.

Key facts. Cited by 223 papers in the map (data through 7 Oct 2026). First posted on arXiv 5 Nov 2018.

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Cited by year (2026 to 7 Oct)

Citing papers per year: 2021: 43; 2022: 50; 2023: 44; 2024: 31; 2025: 29; 2026: 26.

223 papers.

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  1. 2021GemNet: Universal Directional Graph Neural Networks for Moleculescited by 342
  2. 2021A Practical Method for Constructing Equivariant Multilayer Perceptrons for Arbitrary Matrix Groupscited by 185
  3. 2022Approximately Equivariant Networks for Imperfectly Symmetric Dynamicscited by 95
  4. 2022Clifford Neural Layers for PDE Modelingcited by 91
  5. 2021Provably Strict Generalisation Benefit for Equivariant Modelscited by 90
  6. 2021Automatic Symmetry Discovery with Lie Algebra Convolutional Networkcited by 82
  7. 2024Better than classical? The subtle art of benchmarking quantum machine learning modelscited by 82
  8. 2021SE(3) Equivariant Graph Neural Networks with Complete Local Framescited by 75
  9. 2021SE(3)-equivariant prediction of molecular wavefunctions and electronic densitiescited by 67
  10. 2021Geometric Deep Learning and Equivariant Neural Networkscited by 64
  11. 2022Theory for Equivariant Quantum Neural Networkscited by 60
  12. 2022On the Principles of Parsimony and Self-Consistency for the Emergence of Intelligencecited by 50
38 more
  1. 2023Approximation-Generalization Trade-offs under (Approximate) Group Equivariancecited by 45
  2. 2021Learning Partial Equivariances from Datacited by 44
  3. 2022Representation Theory for Geometric Quantum Machine Learningcited by 40
  4. 2023Generative Adversarial Symmetry Discoverycited by 40
  5. 2023Fast, Expressive SE(n) Equivariant Networks through Weight-Sharing in Position-Orientation Spacecited by 40
  6. 2021Autobahn: Automorphism-based Graph Neural Netscited by 39
  7. 2022Unsupervised Learning of Group Invariant and Equivariant Representationscited by 38
  8. 2023EquivAct: SIM(3)-Equivariant Visuomotor Policies beyond Rigid Object Manipulationcited by 37
  9. 2021Category Theory in Machine Learningcited by 36
  10. 2021Top-N: Equivariant set and graph generation without exchangeabilitycited by 36
  11. 2021DISCO: accurate Discrete Scale Convolutionscited by 35
  12. 2021Skip-Convolutions for Efficient Video Processingcited by 32
  13. 2022The Surprising Effectiveness of Equivariant Models in Domains with Latent Symmetrycited by 32
  14. 2024Fourier Transporter: Bi-Equivariant Robotic Manipulation in 3Dcited by 32
  15. 2024Position: Categorical Deep Learning is an Algebraic Theory of All Architecturescited by 32
  16. 2021Equivariant neural networks for inverse problemscited by 29
  17. 2021Provably Strict Generalisation Benefit for Invariance in Kernel Methodscited by 29
  18. 2023Latent Space Symmetry Discoverycited by 29
  19. 2023E(2)-Equivariant Vision Transformercited by 27
  20. 2021Fourier Series Expansion Based Filter Parametrization for Equivariant Convolutionscited by 26
  21. 2022Self-Supervised Equivariant Learning for Oriented Keypoint Detectioncited by 26
  22. 2021Grounding inductive biases in natural images:invariance stems from variations in datacited by 25
  23. 2022Equivariance versus Augmentation for Spherical Imagescited by 24
  24. 2022Equivariant Networks for Crystal Structurescited by 24
  25. 2024Clifford-Steerable Convolutional Neural Networkscited by 24
  26. 2022The Neural Process Family: Survey, Applications and Perspectivescited by 23
  27. 2023Learning Rotation-Equivariant Features for Visual Correspondencecited by 23
  28. 2023Learning Layer-wise Equivariances Automatically using Gradientscited by 23
  29. 2021Steerable Partial Differential Operators for Equivariant Neural Networkscited by 22
  30. 2021Understanding the Generalization Benefit of Model Invariance from a Data Perspectivecited by 22
  31. 2024E(n) Equivariant Topological Neural Networkscited by 22
  32. 2024Improving Equivariant Model Training via Constraint Relaxationcited by 22
  33. 2021Homogeneous vector bundles and G-equivariant convolutional neural networkscited by 21
  34. 2023Tangent Bundle Convolutional Learning: from Manifolds to Cellular Sheaves and Backcited by 21
  35. 2021Rotation-Equivariant Deep Learning for Diffusion MRIcited by 20
  36. 2022Unified Fourier-based Kernel and Nonlinearity Design for Equivariant Networks on Homogeneous Spacescited by 20
  37. 2023From Bricks to Bridges: Product of Invariances to Enhance Latent Space Communicationcited by 20
  38. 2024Beyond Euclid: An Illustrated Guide to Modern Machine Learning with Geometric, Topological, and Algebraic Structurescited by 20

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