mlsanityData to 7 Oct 2026

Clifford Group Equivariant Neural Networks

arXiv 2305.11141 ↗First posted 18 May 2023cs.LG cs.AIcited by 54

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Abstract (arXiv)

We introduce Clifford Group Equivariant Neural Networks: a novel approach for constructing O(n)- and E(n)-equivariant models. We identify and study the textitClifford group, a subgroup inside the Clifford algebra tailored to achieve several favorable properties. Primarily, the group's action forms an orthogonal automorphism that extends beyond the typical vector space to the entire Clifford algebra while respecting the multivector grading. This leads to several non-equivalent subrepresentations corresponding to the multivector decomposition. Furthermore, we prove that the action respects not just the vector space structure of the Clifford algebra but also its multiplicative structure, i.e., the geometric product. These findings imply that every polynomial in multivectors, An advantage worth mentioning is that we obtain expressive layers that can elegantly generalize to inner-product spaces of any dimension. We demonstrate, notably from a single core implementation, state-of-the-art performance on several distinct tasks, including a three-dimensional n-body experiment, a four-dimensional Lorentz-equivariant high-energy physics experiment, and a five-dimensional convex hull experiment.

Key facts. Cited by 54 papers in the map (data through 7 Oct 2026). Builds on 45 of its 105 references that are in the map. First posted on arXiv 18 May 2023.

Builds on

45 of 105 references.

References · 45

  1. 2019PyTorch: An Imperative Style, High-Performance Deep Learning Librarycited by 20,160
  2. 2017Neural Message Passing for Quantum Chemistrycited by 3,460
  3. 2016Aggregated Residual Transformations for Deep Neural Networkscited by 2,540
  4. 2017Deep Setscited by 1,394
  5. 2016Group Equivariant Convolutional Networkscited by 1,154
  6. 2021E(n) Equivariant Graph Neural Networkscited by 951
  7. 2018Tensor field networks: Rotation- and translation-equivariant neural networks for 3D point cloudscited by 735
  8. 2021E(3)-Equivariant Graph Neural Networks for Data-Efficient and Accurate Interatomic Potentialscited by 701
  9. 2020SE(3)-Transformers: 3D Roto-Translation Equivariant Attention Networkscited by 602
  10. 2020Directional Message Passing for Molecular Graphscited by 556
  11. 2019Revisiting Point Cloud Classification: A New Benchmark Dataset and Classification Model on Real-World Datacited by 532
  12. 2018Spherical CNNscited by 452
33 more
  1. 2019General E(2)-Equivariant Steerable CNNscited by 441
  2. 2021Equivariant message passing for the prediction of tensorial properties and molecular spectracited by 438
  3. 20183D Steerable CNNs: Learning Rotationally Equivariant Features in Volumetric Datacited by 348
  4. 2021GemNet: Universal Directional Graph Neural Networks for Moleculescited by 342
  5. 2018Neural Relational Inference for Interacting Systemscited by 339
  6. 2020Learning from Protein Structure with Geometric Vector Perceptronscited by 308
  7. 2021Vector Neurons: A General Framework for SO(3)-Equivariant Networkscited by 304
  8. 2018On the Generalization of Equivariance and Convolution in Neural Networks to the Action of Compact Groupscited by 296
  9. 2016Harmonic Networks: Deep Translation and Rotation Equivariancecited by 293
  10. 2019Cormorant: Covariant Molecular Neural Networkscited by 262
  11. 2020Generalizing Convolutional Neural Networks for Equivariance to Lie Groups on Arbitrary Continuous Datacited by 248
  12. 2020Equivariant Flows: Exact Likelihood Generative Learning for Symmetric Densitiescited by 248
  13. 2020GEOM: Energy-annotated molecular conformations for property prediction and molecular generationcited by 238
  14. 2021Geometric and Physical Quantities Improve E(3) Equivariant Message Passingcited by 225
  15. 2021A Practical Method for Constructing Equivariant Multilayer Perceptrons for Arbitrary Matrix Groupscited by 185
  16. 2022TorchMD-NET: Equivariant Transformers for Neural Network based Molecular Potentialscited by 179
  17. 2020Lorentz Group Equivariant Neural Network for Particle Physicscited by 120
  18. 2019B-Spline CNNs on Lie Groupscited by 97
  19. 2022Clifford Neural Layers for PDE Modelingcited by 91
  20. 2018Energy Flow Networks: Deep Sets for Particle Jetscited by 83
  21. 2020ATOM3D: Tasks On Molecules in Three Dimensionscited by 73
  22. 2021Coordinate Independent Convolutional Networks -- Isometry and Gauge Equivariant Convolutions on Riemannian Manifoldscited by 72
  23. 2022Equivariant Graph Mechanics Networks with Constraintscited by 67
  24. 2022Geometrically Equivariant Graph Neural Networks: A Surveycited by 55
  25. 2019Quaternion Convolutional Neural Networkscited by 48
  26. 2020An Introduction to Electrocatalyst Design using Machine Learning for Renewable Energy Storagecited by 45
  27. 2018Quaternion Recurrent Neural Networkscited by 41
  28. 2023Geometric Clifford Algebra Networkscited by 37
  29. 2023Rigid Body Flows for Sampling Molecular Crystal Structurescited by 27
  30. 2020Theoretical Aspects of Group Equivariant Neural Networkscited by 25
  31. 2021Graded Symmetry Groups: Plane and Simplecited by 11
  32. 2020Embed Me If You Can: A Geometric Perceptroncited by 10
  33. 2021Geometric Algebra Attention Networks for Small Point Cloudscited by 10

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Cited by year (2026 to 7 Oct)

Citing papers per year: 2023: 8; 2024: 21; 2025: 15; 2026: 10.

54 papers.

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  1. 2023A Hitchhiker's Guide to Geometric GNNs for 3D Atomic Systemscited by 65
  2. 2024Does equivariance matter at scale?cited by 49
  3. 2023Geometric Algebra Transformercited by 46
  4. 2023Fast, Expressive SE(n) Equivariant Networks through Weight-Sharing in Position-Orientation Spacecited by 40
  5. 2024Lorentz-Equivariant Geometric Algebra Transformers for High-Energy Physicscited by 36
  6. 2024A Lorentz-Equivariant Transformer for All of the LHCcited by 27
  7. 2024Clifford-Steerable Convolutional Neural Networkscited by 24
  8. 2024Clifford Group Equivariant Simplicial Message Passing Networkscited by 22
  9. 2024E(n) Equivariant Topological Neural Networkscited by 22
  10. 2024Equivariance via Minimal Frame Averaging for More Symmetries and Efficiencycited by 18
  11. 2023MatSciML: A Broad, Multi-Task Benchmark for Solid-State Materials Modelingcited by 14
  12. 2023Learning Lagrangian Fluid Mechanics with E(3)-Equivariant Graph Neural Networkscited by 12
38 more
  1. 2023Euclidean, Projective, Conformal: Choosing a Geometric Algebra for Equivariant Transformerscited by 12
  2. 2024LaB-GATr: geometric algebra transformers for large biomedical surface and volume meshescited by 12
  3. 2025Lorentz Local Canonicalization: How to Make Any Network Lorentz-Equivariantcited by 12
  4. 2024Generating Highly Designable Proteins with Geometric Algebra Flow Matchingcited by 7
  5. 2025GAGrasp: Geometric Algebra Diffusion for Dexterous Graspingcited by 7
  6. 2025STFlow: Data-Coupled Flow Matching for Geometric Trajectory Simulationcited by 7
  7. 2024SE(3)-Hyena Operator for Scalable Equivariant Learningcited by 6
  8. 2025Mathematical Foundation of Interpretable Equivariant Surrogate Modelscited by 5
  9. 2025Latent Mixture of Symmetries for Sample-Efficient Dynamic Learningcited by 5
  10. 2024Multivector Neurons: Better and Faster O(n)-Equivariant Clifford Graph Neural Networkscited by 4
  11. 2025Governing Equation Discovery from Data Based on Differential Invariantscited by 4
  12. 2025Geometric Hyena Networks for Large-scale Equivariant Learningcited by 4
  13. 2025Equivariant Neural Networks for General Linear Symmetries on Lie Algebrascited by 4
  14. 2023PCN: A Deep Learning Approach to Jet Tagging Utilizing Novel Graph Construction Methods and Chebyshev Graph Convolutionscited by 3
  15. 2024ILPO-NET: Network for the invariant recognition of arbitrary volumetric patterns in 3Dcited by 3
  16. 2024E(n) Equivariant Message Passing Cellular Networkscited by 3
  17. 2025Hybrid Diffusion Policies with Projective Geometric Algebra for Efficient Robot Manipulation Learningcited by 3
  18. 2024Current Symmetry Group Equivariant Convolution Frameworks for Representation Learningcited by 2
  19. 2024Towards Stable, Globally Expressive Graph Representations with Laplacian Eigenvectorscited by 2
  20. 2025CP2: Leveraging Geometry for Conformal Prediction via Canonicalizationcited by 2
  21. 2024Image and Point-cloud Classification for Jet Analysis in High-Energy Physics: A surveycited by 1
  22. 2024On the Fourier analysis in the SO(3) space : EquiLoPO Networkcited by 1
  23. 2024Geometric Algebra Meets Large Language Models: Instruction-Based Transformations of Separate Meshes in 3D, Interactive and Controllable Scenescited by 1
  24. 2024Optimal Symmetries in Binary Classificationcited by 1
  25. 2026Versor: A Geometric Sequence Architecturecited by 1
  26. 2026Adaptive Domain Models: Bayesian Evolution, Warm Rotation, and Principled Training for Geometric and Neuromorphic AIcited by 1
  27. 2026Geometric Algebra Meets Cartesian Tensors: Higher-Order Equivariance for Interatomic Potentialscited by 1
  28. 2023Compact Matrix Quantum Group Equivariant Neural Networkscited by 0
  29. 2024Metric Learning for Clifford Group Equivariant Neural Networkscited by 0
  30. 2024STAResNet: a Network in Spacetime Algebra to solve Maxwell's PDEscited by 0
  31. 2025FoGE: Fock Space inspired encoding for graph promptingcited by 0
  32. 2025Single-Core Superscalar Optimization of Clifford Neural Layerscited by 0
  33. 2025Conditional Clifford-Steerable CNNs for PDE Modelingcited by 0
  34. 2025Learning Group Actions In Disentangled Latent Image Representationscited by 0
  35. 2025Soft Geometric Inductive Bias for Object Centric Dynamicscited by 0
  36. 2026On Lie Groups Preserving Subspaces of Degenerate Clifford Algebrascited by 0
  37. 2026Clifford Kolmogorov-Arnold Networkscited by 0
  38. 2026Equivariant Latent Alignment via Flow Matching under Group Symmetriescited by 0

50 most cited of 54 All 54 (JSON)