mlsanityData to 14 Oct 2026

Rotational Equivariance in Machine Learning: A Comprehensive Tutorial

arXiv 2608.31045 ↗First posted 31 Aug 2026cs.LGcited by 0

Not on the map. Why?

Abstract (arXiv)

Rotational symmetry is one of the most important structural principles in machine learning on 3D data. In applications ranging from physics and materials science to 3D computer vision, predictions should not depend on an arbitrary choice of coordinate frame. Rotational equivariance captures this requirement mathematically by enforcing that a rotation of the input induces a corresponding transformation of the model output. This tutorial provides a comprehensive introduction to rotational equivariance, starting from the physical and geometric intuition behind coordinate independence and building up the necessary machinery from geometric deep learning, group theory, and representation theory. We introduce message passing on Euclidean graphs, group actions and representations, spherical harmonics, Wigner matrices, tensor products, and Clebsch-Gordan decomposition, and explain how these ingredients give rise to modern equivariant architectures. We then survey the principal strategies for incorporating rotational equivariance in deep learning, including group convolutions, internal tensorial representations, and canonicalization-based methods, and discuss their practical strengths and limitations. The tutorial aims to lower the barrier to the subject by connecting the underlying mathematics to practical model design, by unifying ideas that are often expressed in different formal languages, and by helping practitioners choose among competing approaches through a clear discussion of their trade-offs.

Key facts. No citing papers in the map yet (data through 14 Oct 2026). Builds on 63 of its 85 references that are in the map. First posted on arXiv 31 Aug 2026.

Builds on

63 of 85 references.

References · 63

  1. 2017Attention Is All You Needcited by 60,361
  2. 2016PointNet: Deep Learning on Point Sets for 3D Classification and Segmentationcited by 5,248
  3. 2017PointNet++: Deep Hierarchical Feature Learning on Point Sets in a Metric Spacecited by 4,401
  4. 2021Switch Transformers: Scaling to Trillion Parameter Models with Simple and Efficient Sparsitycited by 3,073
  5. 2018Dynamic Graph CNN for Learning on Point Cloudscited by 2,223
  6. 2021Geometric Deep Learning: Grids, Groups, Graphs, Geodesics, and Gaugescited by 1,167
  7. 2016Group Equivariant Convolutional Networkscited by 1,165
  8. 2021E(n) Equivariant Graph Neural Networkscited by 965
  9. 2018Tensor field networks: Rotation- and translation-equivariant neural networks for 3D point cloudscited by 746
  10. 2021E(3)-Equivariant Graph Neural Networks for Data-Efficient and Accurate Interatomic Potentialscited by 708
  11. 2020SE(3)-Transformers: 3D Roto-Translation Equivariant Attention Networkscited by 607
  12. 2020Directional Message Passing for Molecular Graphscited by 561
38 more
  1. 2022MACE: Higher Order Equivariant Message Passing Neural Networks for Fast and Accurate Force Fieldscited by 474
  2. 2019General E(2)-Equivariant Steerable CNNscited by 446
  3. 20183D Steerable CNNs: Learning Rotationally Equivariant Features in Volumetric Datacited by 354
  4. 2019Relation-Shape Convolutional Neural Network for Point Cloud Analysiscited by 346
  5. 2021GemNet: Universal Directional Graph Neural Networks for Moleculescited by 343
  6. 2021Vector Neurons: A General Framework for SO(3)-Equivariant Networkscited by 305
  7. 2019Gauge Equivariant Convolutional Networks and the Icosahedral CNNcited by 267
  8. 2022Learning Local Equivariant Representations for Large-Scale Atomistic Dynamicscited by 256
  9. 2018A General Theory of Equivariant CNNs on Homogeneous Spacescited by 228
  10. 2021Geometric and Physical Quantities Improve E(3) Equivariant Message Passingcited by 228
  11. 2023EquiformerV2: Improved Equivariant Transformer for Scaling to Higher-Degree Representationscited by 204
  12. 2022e3nn: Euclidean Neural Networkscited by 174
  13. 2021Spherical Message Passing for 3D Graph Networkscited by 168
  14. 2021Frame Averaging for Invariant and Equivariant Network Designcited by 168
  15. 2023Reducing SO(3) Convolutions to SO(2) for Efficient Equivariant GNNscited by 117
  16. 2018Roto-Translation Covariant Convolutional Networks for Medical Image Analysiscited by 92
  17. 2020Gauge Equivariant Mesh CNNs: Anisotropic convolutions on geometric graphscited by 92
  18. 2025Learning Smooth and Expressive Interatomic Potentials for Physical Property Predictioncited by 84
  19. 2021SE(3) Equivariant Graph Neural Networks with Complete Local Framescited by 76
  20. 2023A Hitchhiker's Guide to Geometric GNNs for 3D Atomic Systemscited by 68
  21. 2020A Rotation-Invariant Framework for Deep Point Cloud Analysiscited by 62
  22. 2018CubeNet: Equivariance to 3D Rotation and Translationcited by 59
  23. 2023Swallowing the Bitter Pill: Simplified Scalable Conformer Generationcited by 56
  24. 2023Clifford Group Equivariant Neural Networkscited by 55
  25. 2023TensorNet: Cartesian Tensor Representations for Efficient Learning of Molecular Potentialscited by 54
  26. 2024Does equivariance matter at scale?cited by 52
  27. 2023Geometric Algebra Transformercited by 48
  28. 2022Spherical Channels for Modeling Atomic Interactionscited by 47
  29. 2023A new perspective on building efficient and expressive 3D equivariant graph neural networkscited by 47
  30. 2024Equivariant Frames and the Impossibility of Continuous Canonicalizationcited by 43
  31. 2019Quaternion Equivariant Capsule Networks for 3D Point Cloudscited by 41
  32. 2023Fast, Expressive SE(n) Equivariant Networks through Weight-Sharing in Position-Orientation Spacecited by 41
  33. 2024The Importance of Being Scalable: Improving the Speed and Accuracy of Neural Network Interatomic Potentials Across Chemical Domainscited by 40
  34. 2023Smooth, exact rotational symmetrization for deep learning on point cloudscited by 39
  35. 2024Lorentz-Equivariant Geometric Algebra Transformers for High-Energy Physicscited by 38
  36. 2022Equivariant Point Cloud Analysis via Learning Orientations for Message Passingcited by 32
  37. 2024Cartesian atomic cluster expansion for machine learning interatomic potentialscited by 31
  38. 2020Global Context Aware Convolutions for 3D Point Cloud Understandingcited by 26

50 most cited of 63 All 63 (JSON)

Built on by

Not cited yet.