mlsanityData to 7 Oct 2026

Lie Neurons: Adjoint-Equivariant Neural Networks for Semisimple Lie Algebras

arXiv 2310.04521 ↗First posted 6 Oct 2023cs.LG cs.AIcited by 6

Not on the map. Why?

Abstract (arXiv)

This paper proposes an equivariant neural network that takes data in any semi-simple Lie algebra as input. The corresponding group acts on the Lie algebra as adjoint operations, making our proposed network adjoint-equivariant. Our framework generalizes the Vector Neurons, a simple SO(3)-equivariant network, from 3-D Euclidean space to Lie algebra spaces, building upon the invariance property of the Killing form. Furthermore, we propose novel Lie bracket layers and geometric channel mixing layers that extend the modeling capacity. Experiments are conducted for the mathfrakso(3), mathfraksl(3), and mathfraksp(4) Lie algebras on various tasks, including fitting equivariant and invariant functions, learning system dynamics, point cloud registration, and homography-based shape classification. Our proposed equivariant network shows wide applicability and competitive performance in various domains.

Key facts. Cited by 6 papers in the map (data through 7 Oct 2026). Builds on 41 of its 64 references that are in the map. First posted on arXiv 6 Oct 2023.

Builds on

41 of 64 references.

References · 41

  1. 2015Deep Residual Learning for Image Recognitioncited by 51,809
  2. 2018Neural Ordinary Differential Equationscited by 3,733
  3. 2016Group Equivariant Convolutional Networkscited by 1,154
  4. 2018Tensor field networks: Rotation- and translation-equivariant neural networks for 3D point cloudscited by 735
  5. 2021E(3)-Equivariant Graph Neural Networks for Data-Efficient and Accurate Interatomic Potentialscited by 701
  6. 2020SE(3)-Transformers: 3D Roto-Translation Equivariant Attention Networkscited by 602
  7. 2019General E(2)-Equivariant Steerable CNNscited by 441
  8. 20183D Steerable CNNs: Learning Rotationally Equivariant Features in Volumetric Datacited by 348
  9. 2018Invariant and Equivariant Graph Networkscited by 339
  10. 2021Vector Neurons: A General Framework for SO(3)-Equivariant Networkscited by 304
  11. 2016Steerable CNNscited by 298
  12. 2018On the Generalization of Equivariance and Convolution in Neural Networks to the Action of Compact Groupscited by 296
29 more
  1. 2016Harmonic Networks: Deep Translation and Rotation Equivariancecited by 293
  2. 2020Generalizing Convolutional Neural Networks for Equivariance to Lie Groups on Arbitrary Continuous Datacited by 248
  3. 2021Geometric and Physical Quantities Improve E(3) Equivariant Message Passingcited by 225
  4. 2018A General Theory of Equivariant CNNs on Homogeneous Spacescited by 223
  5. 2021A Practical Method for Constructing Equivariant Multilayer Perceptrons for Arbitrary Matrix Groupscited by 185
  6. 2022e3nn: Euclidean Neural Networkscited by 171
  7. 2019Deep Scale-spaces: Equivariance Over Scalecited by 113
  8. 2020LieTransformer: Equivariant self-attention for Lie Groupscited by 100
  9. 2019B-Spline CNNs on Lie Groupscited by 97
  10. 2021Equivariant Point Network for 3D Point Cloud Analysiscited by 81
  11. 2020A Rotation-Invariant Framework for Deep Point Cloud Analysiscited by 62
  12. 2018CubeNet: Equivariance to 3D Rotation and Translationcited by 59
  13. 2020A Wigner-Eckart Theorem for Group Equivariant Convolution Kernelscited by 46
  14. 20183D G-CNNs for Pulmonary Nodule Detectioncited by 36
  15. 2018HexaConvcited by 27
  16. 2022SE(3)-Equivariant Attention Networks for Shape Reconstruction in Function Spacecited by 26
  17. 2021Correspondence-Free Point Cloud Registration with SO(3)-Equivariant Implicit Shape Representationscited by 24
  18. 2021Exploiting Redundancy: Separable Group Convolutional Networks on Lie Groupscited by 22
  19. 2021Enabling equivariance for arbitrary Lie groupscited by 22
  20. 2022Unified Fourier-based Kernel and Nonlinearity Design for Equivariant Networks on Homogeneous Spacescited by 20
  21. 20234D Panoptic Segmentation as Invariant and Equivariant Field Predictioncited by 19
  22. 2017Convolutional Networks for Spherical Signalscited by 18
  23. 2022E2PN: Efficient SE(3)-Equivariant Point Networkcited by 17
  24. 2014The invariant extended Kalman filter as a stable observercited by 14
  25. 2021LARNet: Lie Algebra Residual Network for Face Recognitioncited by 14
  26. 2021Legged Robot State Estimation using Invariant Kalman Filtering and Learned Contact Eventscited by 12
  27. 2022Lie Algebraic Cost Function Design for Control on Lie Groupscited by 5
  28. 2018Invariant Smoothing on Lie Groupscited by 2
  29. 2021Equivariant filter (EqF): A general filter design for systems on homogeneous spacescited by 1

Built on by

Cited by year (2026 to 7 Oct)

Citing papers per year: 2025: 4; 2026: 2.

6 papers.

  1. 2025Equivariant Neural Networks for General Linear Symmetries on Lie Algebrascited by 4
  2. 2025AdS-GNN -- a Conformally Equivariant Graph Neural Networkcited by 2
  3. 2025Equi-RO: A 4D mmWave Radar Odometry via Equivariant Networkscited by 2
  4. 2025Cartan Networks: Group theoretical Hyperbolic Deep Learningcited by 0
  5. 2026mathbbSL(n) Representation Learning: An Intrinsic Mixed-Curvature Space with Higher Curvature Capacities and Deeper Order-Aware Compositioncited by 0
  6. 2026GINIO: A Geometric SO(3)-Equivariant Interface for Neural Inertial Odometrycited by 0