mlsanityData to 7 Oct 2026

Unified Fourier-based Kernel and Nonlinearity Design for Equivariant Networks on Homogeneous Spaces

arXiv 2206.08362 ↗First posted 16 Jun 2022cs.CVcited by 20

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Abstract (arXiv)

We introduce a unified framework for group equivariant networks on homogeneous spaces derived from a Fourier perspective. We consider tensor-valued feature fields, before and after a convolutional layer. We present a unified derivation of kernels via the Fourier domain by leveraging the sparsity of Fourier coefficients of the lifted feature fields. The sparsity emerges when the stabilizer subgroup of the homogeneous space is a compact Lie group. We further introduce a nonlinear activation, via an elementwise nonlinearity on the regular representation after lifting and projecting back to the field through an equivariant convolution. We show that other methods treating features as the Fourier coefficients in the stabilizer subgroup are special cases of our activation. Experiments on SO(3) and SE(3) show state-of-the-art performance in spherical vector field regression, point cloud classification, and molecular completion.

Key facts. Cited by 20 papers in the map (data through 7 Oct 2026). Builds on 35 of its 54 references that are in the map. First posted on arXiv 16 Jun 2022.

Builds on

35 of 54 references.

References · 35

  1. 2016Group Equivariant Convolutional Networkscited by 1,154
  2. 2021E(n) Equivariant Graph Neural Networkscited by 951
  3. 2018Tensor field networks: Rotation- and translation-equivariant neural networks for 3D point cloudscited by 735
  4. 2020SE(3)-Transformers: 3D Roto-Translation Equivariant Attention Networkscited by 602
  5. 2019General E(2)-Equivariant Steerable CNNscited by 441
  6. 20183D Steerable CNNs: Learning Rotationally Equivariant Features in Volumetric Datacited by 348
  7. 2018Invariant and Equivariant Graph Networkscited by 339
  8. 2021Vector Neurons: A General Framework for SO(3)-Equivariant Networkscited by 304
  9. 2018On the Generalization of Equivariance and Convolution in Neural Networks to the Action of Compact Groupscited by 296
  10. 2016Harmonic Networks: Deep Translation and Rotation Equivariancecited by 293
  11. 2019Cormorant: Covariant Molecular Neural Networkscited by 262
  12. 2020Generalizing Convolutional Neural Networks for Equivariance to Lie Groups on Arbitrary Continuous Datacited by 248
23 more
  1. 2017Learning SO(3) Equivariant Representations with Spherical CNNscited by 244
  2. 2018A General Theory of Equivariant CNNs on Homogeneous Spacescited by 223
  3. 2021A Practical Method for Constructing Equivariant Multilayer Perceptrons for Arbitrary Matrix Groupscited by 185
  4. 2019Universal Invariant and Equivariant Graph Neural Networkscited by 182
  5. 2018Clebsch-Gordan Nets: a Fully Fourier Space Spherical Convolutional Neural Networkcited by 168
  6. 2021Frame Averaging for Invariant and Equivariant Network Designcited by 166
  7. 2019B-Spline CNNs on Lie Groupscited by 97
  8. 2020On Learning Sets of Symmetric Elementscited by 97
  9. 2020Gauge Equivariant Mesh CNNs: Anisotropic convolutions on geometric graphscited by 90
  10. 2021Equivariant Point Network for 3D Point Cloud Analysiscited by 81
  11. 2019Rotation Invariant Convolutions for 3D Point Clouds Deep Learningcited by 80
  12. 2020On the Universality of Rotation Equivariant Point Cloud Networkscited by 80
  13. 2021Coordinate Independent Convolutional Networks -- Isometry and Gauge Equivariant Convolutions on Riemannian Manifoldscited by 72
  14. 2020Group Equivariant Stand-Alone Self-Attention For Visioncited by 63
  15. 2020A Rotation-Invariant Framework for Deep Point Cloud Analysiscited by 62
  16. 2018CubeNet: Equivariance to 3D Rotation and Translationcited by 59
  17. 2019Effective Rotation-invariant Point CNN with Spherical Harmonics kernelscited by 57
  18. 2020Spin-Weighted Spherical CNNscited by 50
  19. 2019Equivariant Multi-View Networkscited by 44
  20. 2020Efficient Generalized Spherical CNNscited by 29
  21. 2020Global Context Aware Convolutions for 3D Point Cloud Understandingcited by 26
  22. 2021Homogeneous vector bundles and G-equivariant convolutional neural networkscited by 21
  23. 2021Frame Averaging for Equivariant Shape Space Learningcited by 18

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Cited by year

Citing papers per year: 2022: 1; 2023: 9; 2024: 7; 2025: 3.

20 papers.

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  1. 2023Approximation-Generalization Trade-offs under (Approximate) Group Equivariancecited by 45
  2. 2023EquivAct: SIM(3)-Equivariant Visuomotor Policies beyond Rigid Object Manipulationcited by 37
  3. 2024Improving Equivariant Model Training via Constraint Relaxationcited by 22
  4. 2023Scaling Spherical CNNscited by 21
  5. 2023Banana: Banach Fixed-Point Network for Pointcloud Segmentation with Inter-Part Equivariancecited by 19
  6. 2023EFEM: Equivariant Neural Field Expectation Maximization for 3D Object Segmentation Without Scene Supervisioncited by 18
  7. 2023Building Neural Networks on Matrix Manifolds: A Gyrovector Space Approachcited by 18
  8. 2024EqNIO: Subequivariant Neural Inertial Odometrycited by 8
  9. 2024Symmetry-Based Structured Matrices for Efficient Approximately Equivariant Networkscited by 7
  10. 2023Equivariant Single View Pose Prediction Via Induced and Restricted Representationscited by 6
  11. 2023Lie Neurons: Adjoint-Equivariant Neural Networks for Semisimple Lie Algebrascited by 6
  12. 2024SE(3) Equivariant Ray Embeddings for Implicit Multi-View Depth Estimationcited by 6
8 more
  1. 20243D Equivariant Pose Regression via Direct Wigner-D Harmonics Predictioncited by 4
  2. 2025Equivariant Neural Networks for General Linear Symmetries on Lie Algebrascited by 4
  3. 2023Equivariant geometric convolutions for emulation of dynamical systemscited by 3
  4. 2024Object Dynamics Modeling with Hierarchical Point Cloud-based Representationscited by 3
  5. 2024Current Symmetry Group Equivariant Convolution Frameworks for Representation Learningcited by 2
  6. 2025Learning Generalizable Shape Completion with SIM(3) Equivariancecited by 2
  7. 2022Equivariant Light Field Convolution and Transformercited by 1
  8. 2025Hierarchical Direction Perception via Atomic Dot-Product Operators for Rotation-Invariant Point Clouds Learningcited by 0