mlsanityData to 13 Oct 2026

Vector Neurons: A General Framework for SO(3)-Equivariant Networks

arXiv 2104.12229 ↗First posted 25 Apr 2021cs.CVcited by 305

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Abstract (arXiv)

Invariance and equivariance to the rotation group have been widely discussed in the 3D deep learning community for pointclouds. Yet most proposed methods either use complex mathematical tools that may limit their accessibility, or are tied to specific input data types and network architectures. In this paper, we introduce a general framework built on top of what we call Vector Neuron representations for creating SO(3)-equivariant neural networks for pointcloud processing. Extending neurons from 1D scalars to 3D vectors, our vector neurons enable a simple mapping of SO(3) actions to latent spaces thereby providing a framework for building equivariance in common neural operations -- including linear layers, non-linearities, pooling, and normalizations. Due to their simplicity, vector neurons are versatile and, as we demonstrate, can be incorporated into diverse network architecture backbones, allowing them to process geometry inputs in arbitrary poses. Despite its simplicity, our method performs comparably well in accuracy and generalization with other more complex and specialized state-of-the-art methods on classification and segmentation tasks. We also show for the first time a rotation equivariant reconstruction network.

Key facts. Cited by 305 papers in the map (data through 13 Oct 2026). Builds on 38 of its 46 references that are in the map. First posted on arXiv 25 Apr 2021.

Builds on

38 of 46 references.

References · 38

  1. 2016Layer Normalizationcited by 5,624
  2. 2016PointNet: Deep Learning on Point Sets for 3D Classification and Segmentationcited by 5,231
  3. 2017PointNet++: Deep Hierarchical Feature Learning on Point Sets in a Metric Spacecited by 4,389
  4. 2015ShapeNet: An Information-Rich 3D Model Repositorycited by 2,889
  5. 2018Dynamic Graph CNN for Learning on Point Cloudscited by 2,222
  6. 2018Occupancy Networks: Learning 3D Reconstruction in Function Spacecited by 2,039
  7. 2015Spatial Transformer Networkscited by 1,425
  8. 2016Instance Normalization: The Missing Ingredient for Fast Stylizationcited by 1,193
  9. 2021E(n) Equivariant Graph Neural Networkscited by 957
  10. 2018PointCNN: Convolution On X-Transformed Pointscited by 764
  11. 2018Tensor field networks: Rotation- and translation-equivariant neural networks for 3D point cloudscited by 739
  12. 2020SE(3)-Transformers: 3D Roto-Translation Equivariant Attention Networkscited by 603
26 more
  1. 20163D-R2N2: A Unified Approach for Single and Multi-view 3D Object Reconstructioncited by 587
  2. 2019Normalized Object Coordinate Space for Category-Level 6D Object Pose and Size Estimationcited by 492
  3. 2018Spherical CNNscited by 452
  4. 20183D Steerable CNNs: Learning Rotationally Equivariant Features in Volumetric Datacited by 350
  5. 2019Cormorant: Covariant Molecular Neural Networkscited by 263
  6. 2017Learning SO(3) Equivariant Representations with Spherical CNNscited by 246
  7. 2018Clebsch-Gordan Nets: a Fully Fourier Space Spherical Convolutional Neural Networkcited by 170
  8. 2019Category-Level Articulated Object Pose Estimationcited by 158
  9. 2018Point Convolutional Neural Networks by Extension Operatorscited by 154
  10. 2019NASA: Neural Articulated Shape Approximationcited by 149
  11. 2019ShellNet: Efficient Point Cloud Convolutional Neural Networks using Concentric Shells Statisticscited by 104
  12. 2020On Learning Sets of Symmetric Elementscited by 97
  13. 2019Rotation Invariant Convolutions for 3D Point Clouds Deep Learningcited by 80
  14. 2020On the Universality of Rotation Equivariant Point Cloud Networkscited by 80
  15. 2018Unsupervised Geometry-Aware Representation for 3D Human Pose Estimationcited by 73
  16. 2020A Rotation-Invariant Framework for Deep Point Cloud Analysiscited by 62
  17. 2019Effective Rotation-invariant Point CNN with Spherical Harmonics kernelscited by 57
  18. 2020Equivariant Neural Renderingcited by 48
  19. 2020A Wigner-Eckart Theorem for Group Equivariant Convolution Kernelscited by 46
  20. 2019Equivariant Multi-View Networkscited by 44
  21. 2017Interpretable Transformations with Encoder-Decoder Networkscited by 33
  22. 20193D-Rotation-Equivariant Quaternion Neural Networkscited by 30
  23. 2020Canonical Capsules: Self-Supervised Capsules in Canonical Posecited by 30
  24. 2020Global Context Aware Convolutions for 3D Point Cloud Understandingcited by 26
  25. 2019Rotation Invariant Point Cloud Classification: Where Local Geometry Meets Global Topologycited by 11
  26. 2019ACNe: Attentive Context Normalization for Robust Permutation-Equivariant Learningcited by 9

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Cited by year (2026 to 13 Oct)

Citing papers per year: 2021: 19; 2022: 58; 2023: 69; 2024: 68; 2025: 50; 2026: 41.

305 papers.

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  1. 2021Neural Fields in Visual Computing and Beyondcited by 539
  2. 2021Geometric and Physical Quantities Improve E(3) Equivariant Message Passingcited by 225
  3. 2021Neural Descriptor Fields: SE(3)-Equivariant Object Representations for Manipulationcited by 206
  4. 2022Pocket2Mol: Efficient Molecular Sampling Based on 3D Protein Pocketscited by 182
  5. 2021Frame Averaging for Invariant and Equivariant Network Designcited by 166
  6. 2022SE(3)-DiffusionFields: Learning smooth cost functions for joint grasp and motion optimization through diffusioncited by 125
  7. 2023Reducing SO(3) Convolutions to SO(2) for Efficient Equivariant GNNscited by 114
  8. 2022Equivariance with Learned Canonicalization Functionscited by 106
  9. 2023Neural Kernel Surface Reconstructioncited by 102
  10. 2023EqMotion: Equivariant Multi-agent Motion Prediction with Invariant Interaction Reasoningcited by 97
  11. 2021SE(3) Equivariant Graph Neural Networks with Complete Local Framescited by 75
  12. 2024EquiBot: SIM(3)-Equivariant Diffusion Policy for Generalizable and Data Efficient Learningcited by 73
38 more
  1. 2022Benchmarking and Analyzing Point Cloud Classification under Corruptionscited by 70
  2. 2021You Only Hypothesize Once: Point Cloud Registration with Rotation-equivariant Descriptorscited by 68
  3. 2023LEAP: Liberate Sparse-view 3D Modeling from Camera Posescited by 65
  4. 2022Theseus: A Library for Differentiable Nonlinear Optimizationcited by 64
  5. 2021Neural Fields as Learnable Kernels for 3D Reconstructioncited by 58
  6. 2023Learning Neural Constitutive Laws From Motion Observations for Generalizable PDE Dynamicscited by 56
  7. 2022Equivariant Descriptor Fields: SE(3)-Equivariant Energy-Based Models for End-to-End Visual Robotic Manipulation Learningcited by 55
  8. 2023Clifford Group Equivariant Neural Networkscited by 54
  9. 2022Transformation-Equivariant 3D Object Detection for Autonomous Drivingcited by 52
  10. 2022SE(3)-Equivariant Relational Rearrangement with Neural Descriptor Fieldscited by 51
  11. 2023Diffusion-EDFs: Bi-equivariant Denoising Generative Modeling on SE(3) for Visual Robotic Manipulationcited by 51
  12. 2022TAX-Pose: Task-Specific Cross-Pose Estimation for Robot Manipulationcited by 47
  13. 2021Leveraging SE(3) Equivariance for Self-Supervised Category-Level Object Pose Estimationcited by 45
  14. 2023Equivariant Adaptation of Large Pretrained Modelscited by 40
  15. 2025Diffusion Models for Robotic Manipulation: A Surveycited by 40
  16. 2023BEVPlace: Learning LiDAR-based Place Recognition using Bird's Eye View Imagescited by 39
  17. 2023Smooth, exact rotational symmetrization for deep learning on point cloudscited by 39
  18. 2022Low Dimensional Invariant Embeddings for Universal Geometric Learningcited by 38
  19. 2023EquivAct: SIM(3)-Equivariant Visuomotor Policies beyond Rigid Object Manipulationcited by 37
  20. 2022E3Bind: An End-to-End Equivariant Network for Protein-Ligand Dockingcited by 35
  21. 2024Enabling Efficient Equivariant Operations in the Fourier Basis via Gaunt Tensor Productscited by 35
  22. 2022Edge Grasp Network: A Graph-Based SE(3)-invariant Approach to Grasp Detectioncited by 34
  23. 20223DLinker: An E(3) Equivariant Variational Autoencoder for Molecular Linker Designcited by 33
  24. 2024Fourier Transporter: Bi-Equivariant Robotic Manipulation in 3Dcited by 33
  25. 2022Equivariant Point Cloud Analysis via Learning Orientations for Message Passingcited by 32
  26. 2022Generalised Implicit Neural Representationscited by 31
  27. 2022Frame Mining: a Free Lunch for Learning Robotic Manipulation from 3D Point Cloudscited by 31
  28. 2022Contrastive Representation Learning for 3D Protein Structurescited by 30
  29. 2022Mesh Neural Networks for SE(3)-Equivariant Hemodynamics Estimation on the Artery Wallcited by 29
  30. 2023Shelving, Stacking, Hanging: Relational Pose Diffusion for Multi-modal Rearrangementcited by 29
  31. 2022USEEK: Unsupervised SE(3)-Equivariant 3D Keypoints for Generalizable Manipulationcited by 27
  32. 2023Leveraging SE(3) Equivariance for Learning 3D Geometric Shape Assemblycited by 27
  33. 2022SE(3)-Equivariant Attention Networks for Shape Reconstruction in Function Spacecited by 26
  34. 2022VN-Transformer: Rotation-Equivariant Attention for Vector Neuronscited by 26
  35. 2022RIGA: Rotation-Invariant and Globally-Aware Descriptors for Point Cloud Registrationcited by 25
  36. 2023Learning Subpocket Prototypes for Generalizable Structure-based Drug Designcited by 25
  37. 2023Learning Probabilistic Symmetrization for Architecture Agnostic Equivariancecited by 25
  38. 2024ActionFlow: Equivariant, Accurate, and Efficient Policies with Spatially Symmetric Flow Matchingcited by 25

50 most cited of 305 All 305 (JSON)