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Clebsch-Gordan Nets: a Fully Fourier Space Spherical Convolutional Neural Network

arXiv 1806.09231 ↗First posted 24 Jun 2018stat.ML cs.LGcited by 168

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Abstract (arXiv)

Recent work by Cohen et al. has achieved state-of-the-art results for learning spherical images in a rotation invariant way by using ideas from group representation theory and noncommutative harmonic analysis. In this paper we propose a generalization of this work that generally exhibits improved performace, but from an implementation point of view is actually simpler. An unusual feature of the proposed architecture is that it uses the Clebsch--Gordan transform as its only source of nonlinearity, thus avoiding repeated forward and backward Fourier transforms. The underlying ideas of the paper generalize to constructing neural networks that are invariant to the action of other compact groups.

Key facts. Cited by 168 papers in the map (data through 7 Oct 2026). First posted on arXiv 24 Jun 2018.

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Citing papers per year: 2021: 35; 2022: 44; 2023: 32; 2024: 22; 2025: 22; 2026: 13.

168 papers.

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  1. 2021E(3)-Equivariant Graph Neural Networks for Data-Efficient and Accurate Interatomic Potentialscited by 701
  2. 2021GemNet: Universal Directional Graph Neural Networks for Moleculescited by 342
  3. 2021Vector Neurons: A General Framework for SO(3)-Equivariant Networkscited by 304
  4. 2022Equiformer: Equivariant Graph Attention Transformer for 3D Atomistic Graphscited by 258
  5. 2022Learning Local Equivariant Representations for Large-Scale Atomistic Dynamicscited by 254
  6. 2021Geometric and Physical Quantities Improve E(3) Equivariant Message Passingcited by 225
  7. 2023EquiformerV2: Improved Equivariant Transformer for Scaling to Higher-Degree Representationscited by 201
  8. 2022Graph neural networks for materials science and chemistrycited by 181
  9. 2022e3nn: Euclidean Neural Networkscited by 171
  10. 2021Geometric Deep Learning on Molecular Representationscited by 113
  11. 2023Reducing SO(3) Convolutions to SO(2) for Efficient Equivariant GNNscited by 113
  12. 2021Automatic Symmetry Discovery with Lie Algebra Convolutional Networkcited by 82
38 more
  1. 2025Adjoint Sampling: Highly Scalable Diffusion Samplers via Adjoint Matchingcited by 78
  2. 2021SE(3) Equivariant Graph Neural Networks with Complete Local Framescited by 75
  3. 2021SE(3)-equivariant prediction of molecular wavefunctions and electronic densitiescited by 67
  4. 2021Geometric Deep Learning and Equivariant Neural Networkscited by 64
  5. 2022Theory for Equivariant Quantum Neural Networkscited by 60
  6. 2022Spherical Channels for Modeling Atomic Interactionscited by 47
  7. 2022Invariance Learning in Deep Neural Networks with Differentiable Laplace Approximationscited by 42
  8. 2022Relaxing Equivariance Constraints with Non-stationary Continuous Filterscited by 41
  9. 2022Representation Theory for Geometric Quantum Machine Learningcited by 40
  10. 2023Fast, Expressive SE(n) Equivariant Networks through Weight-Sharing in Position-Orientation Spacecited by 40
  11. 2021Autobahn: Automorphism-based Graph Neural Netscited by 39
  12. 2022Low Dimensional Invariant Embeddings for Universal Geometric Learningcited by 38
  13. 2021Machine-learning hidden symmetriescited by 31
  14. 2021Speeding up Learning Quantum States through Group Equivariant Convolutional Quantum Ans"atzecited by 29
  15. 2021Symmetry-via-Duality: Invariant Neural Network Densities from Parameter-Space Correlatorscited by 28
  16. 2021Informing Geometric Deep Learning with Electronic Interactions to Accelerate Quantum Chemistrycited by 27
  17. 2021Equivariant graph neural networks for fast electron density estimation of molecules, liquids, and solidscited by 24
  18. 2022Equivariance versus Augmentation for Spherical Imagescited by 24
  19. 2022Symmetry Group Equivariant Architectures for Physicscited by 23
  20. 2022Scalable and Equivariant Spherical CNNs by Discrete-Continuous (DISCO) Convolutionscited by 23
  21. 2024Improving Equivariant Model Training via Constraint Relaxationcited by 22
  22. 2021Homogeneous vector bundles and G-equivariant convolutional neural networkscited by 21
  23. 2022What is an equivariant neural network?cited by 21
  24. 2023Scaling Spherical CNNscited by 21
  25. 2021Rotation-Equivariant Deep Learning for Diffusion MRIcited by 20
  26. 2021Topological Obstructions to Autoencodingcited by 20
  27. 2022Unified Fourier-based Kernel and Nonlinearity Design for Equivariant Networks on Homogeneous Spacescited by 20
  28. 2024Beyond Euclid: An Illustrated Guide to Modern Machine Learning with Geometric, Topological, and Algebraic Structurescited by 20
  29. 2021PDO-eS2CNNs: Partial Differential Operator Based Equivariant Spherical CNNscited by 19
  30. 2022Flattening-Net: Deep Regular 2D Representation for 3D Point Cloud Analysiscited by 19
  31. 2025An Efficient Sparse Kernel Generator for O(3)-Equivariant Deep Networkscited by 19
  32. 2021Scattering Networks on the Sphere for Scalable and Rotationally Equivariant Spherical CNNscited by 18
  33. 2021Implicit Bias of Linear Equivariant Networkscited by 18
  34. 2022Equivariance Discovery by Learned Parameter-Sharingcited by 18
  35. 2022Learnable Polyphase Sampling for Shift Invariant and Equivariant Convolutional Networkscited by 18
  36. 2023HEAL-SWIN: A Vision Transformer On The Spherecited by 18
  37. 2024Generalizing Denoising to Non-Equilibrium Structures Improves Equivariant Force Fieldscited by 18
  38. 2022VectorAdam for Rotation Equivariant Geometry Optimizationcited by 17

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