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On the Universality of Rotation Equivariant Point Cloud Networks

arXiv 2010.02449 ↗First posted 6 Oct 2020cs.LG cs.CGcited by 80

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Abstract (arXiv)

Learning functions on point clouds has applications in many fields, including computer vision, computer graphics, physics, and chemistry. Recently, there has been a growing interest in neural architectures that are invariant or equivariant to all three shape-preserving transformations of point clouds: translation, rotation, and permutation. In this paper, we present a first study of the approximation power of these architectures. We first derive two sufficient conditions for an equivariant architecture to have the universal approximation property, based on a novel characterization of the space of equivariant polynomials. We then use these conditions to show that two recently suggested models are universal, and for devising two other novel universal architectures.

Key facts. Cited by 80 papers in the map (data through 7 Oct 2026). First posted on arXiv 6 Oct 2020.

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Cited by year (2026 to 7 Oct)

Citing papers per year: 2021: 15; 2022: 12; 2023: 16; 2024: 17; 2025: 13; 2026: 7.

80 papers.

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  1. 2021GemNet: Universal Directional Graph Neural Networks for Moleculescited by 342
  2. 2021Vector Neurons: A General Framework for SO(3)-Equivariant Networkscited by 304
  3. 2021A Practical Method for Constructing Equivariant Multilayer Perceptrons for Arbitrary Matrix Groupscited by 185
  4. 2022Graph neural networks for materials science and chemistrycited by 181
  5. 2022e3nn: Euclidean Neural Networkscited by 171
  6. 2021Frame Averaging for Invariant and Equivariant Network Designcited by 166
  7. 2021Scalars are universal: Equivariant machine learning, structured like classical physicscited by 127
  8. 2023On the Expressive Power of Geometric Graph Neural Networkscited by 89
  9. 2021SE(3) Equivariant Graph Neural Networks with Complete Local Framescited by 75
  10. 2023A Hitchhiker's Guide to Geometric GNNs for 3D Atomic Systemscited by 65
  11. 2022Geometrically Equivariant Graph Neural Networks: A Surveycited by 55
  12. 2022Learning Symmetric Embeddings for Equivariant World Modelscited by 54
38 more
  1. 2023Ewald-based Long-Range Message Passing for Molecular Graphscited by 46
  2. 2023A new perspective on building efficient and expressive 3D equivariant graph neural networkscited by 46
  3. 2023Fast, Expressive SE(n) Equivariant Networks through Weight-Sharing in Position-Orientation Spacecited by 40
  4. 2022Low Dimensional Invariant Embeddings for Universal Geometric Learningcited by 38
  5. 2024Enabling Efficient Equivariant Operations in the Fourier Basis via Gaunt Tensor Productscited by 35
  6. 2022The Surprising Effectiveness of Equivariant Models in Domains with Latent Symmetrycited by 32
  7. 2024A Survey of Geometric Graph Neural Networks: Data Structures, Models and Applicationscited by 28
  8. 2023The Exact Sample Complexity Gain from Invariances for Kernel Regressioncited by 27
  9. 2021Directional Message Passing on Molecular Graphs via Synthetic Coordinatescited by 24
  10. 2023Is Distance Matrix Enough for Geometric Deep Learning?cited by 22
  11. 2023Symmetry-Informed Geometric Representation for Molecules, Proteins, and Crystalline Materialscited by 22
  12. 2023Explainable Equivariant Neural Networks for Particle Physics: PELICANcited by 22
  13. 2022Unified Fourier-based Kernel and Nonlinearity Design for Equivariant Networks on Homogeneous Spacescited by 20
  14. 2021Frame Averaging for Equivariant Shape Space Learningcited by 18
  15. 2024Are High-Degree Representations Really Unnecessary in Equivariant Graph Neural Networks?cited by 18
  16. 2025Equivariance Everywhere All At Once: A Recipe for Graph Foundation Modelscited by 17
  17. 2021Equivariant Networks for Pixelized Spherescited by 16
  18. 2021ZZ-Net: A Universal Rotation Equivariant Architecture for 2D Point Cloudscited by 15
  19. 2023Expressive Sign Equivariant Networks for Spectral Geometric Learningcited by 15
  20. 2022Graph Neural Network with Local Frame for Molecular Potential Energy Surfacecited by 14
  21. 2022Machine learning and invariant theorycited by 13
  22. 2025Probing Equivariance and Symmetry Breaking in Convolutional Networkscited by 12
  23. 2025E2Former: An Efficient and Equivariant Transformer with Linear-Scaling Tensor Productscited by 12
  24. 2021Equivariant geometric learning for digital rock physics: estimating formation factor and effective permeability tensors from Morse graphcited by 11
  25. 2022A Simple and Universal Rotation Equivariant Point-cloud Networkcited by 11
  26. 2024On the Completeness of Invariant Geometric Deep Learning Modelscited by 11
  27. 2023Complete Neural Networks for Complete Euclidean Graphscited by 10
  28. 2024Weisfeiler Leman for Euclidean Equivariant Machine Learningcited by 10
  29. 2025Learning (Approximately) Equivariant Networks via Constrained Optimizationcited by 10
  30. 2024A Characterization Theorem for Equivariant Networks with Point-wise Activationscited by 9
  31. 2024On the Expressive Power of Sparse Geometric MPNNscited by 7
  32. 2026Identifiable Equivariant Networks are Layerwise Equivariantcited by 7
  33. 2024Equivariant Graph Network Approximations of High-Degree Polynomials for Force Field Predictioncited by 6
  34. 2021RPR-Net: A Point Cloud-based Rotation-aware Large Scale Place Recognition Networkcited by 5
  35. 2023AI for Science: An Emerging Agendacited by 5
  36. 2023Learning Symmetrization for Equivariance with Orbit Distance Minimizationcited by 4
  37. 2024Graph as Point Setcited by 4
  38. 2025Rethinking Addressing in Language Models via Contexualized Equivariant Positional Encodingcited by 4

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