mlsanityData to 14 Oct 2026

Geometric Deep Learning and Equivariant Neural Networks

arXiv 2105.13926 ↗First posted 28 May 2021cs.LG cs.CV hep-thcited by 65

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Abstract (arXiv)

We survey the mathematical foundations of geometric deep learning, focusing on group equivariant and gauge equivariant neural networks. We develop gauge equivariant convolutional neural networks on arbitrary manifolds M using principal bundles with structure group K and equivariant maps between sections of associated vector bundles. We also discuss group equivariant neural networks for homogeneous spaces M=G/K, which are instead equivariant with respect to the global symmetry G on M. Group equivariant layers can be interpreted as intertwiners between induced representations of G, and we show their relation to gauge equivariant convolutional layers. We analyze several applications of this formalism, including semantic segmentation and object detection networks. We also discuss the case of spherical networks in great detail, corresponding to the case M=S2=SO(3)/SO(2). Here we emphasize the use of Fourier analysis involving Wigner matrices, spherical harmonics and Clebsch-Gordan coefficients for G=SO(3), illustrating the power of representation theory for deep learning.

Key facts. Cited by 65 papers in the map (data through 14 Oct 2026). Builds on 57 of its 84 references that are in the map. First posted on arXiv 28 May 2021.

Builds on

57 of 84 references.

References · 57

  1. 2017Attention Is All You Needcited by 60,361
  2. 2016The Cityscapes Dataset for Semantic Urban Scene Understandingcited by 4,444
  3. 2021Geometric Deep Learning: Grids, Groups, Graphs, Geodesics, and Gaugescited by 1,167
  4. 2016Group Equivariant Convolutional Networkscited by 1,165
  5. 2016Geometric deep learning: going beyond Euclidean datacited by 1,048
  6. 2018Tensor field networks: Rotation- and translation-equivariant neural networks for 3D point cloudscited by 746
  7. 2019Hamiltonian Neural Networkscited by 712
  8. 2018Spherical CNNscited by 456
  9. 2019General E(2)-Equivariant Steerable CNNscited by 446
  10. 2020Lagrangian Neural Networkscited by 398
  11. 20183D Steerable CNNs: Learning Rotationally Equivariant Features in Volumetric Datacited by 354
  12. 2018On the Generalization of Equivariance and Convolution in Neural Networks to the Action of Compact Groupscited by 300
38 more
  1. 2016Harmonic Networks: Deep Translation and Rotation Equivariancecited by 295
  2. 2019Gauge Equivariant Convolutional Networks and the Icosahedral CNNcited by 267
  3. 2017Learning SO(3) Equivariant Representations with Spherical CNNscited by 248
  4. 2018A General Theory of Equivariant CNNs on Homogeneous Spacescited by 228
  5. 2017Learning Steerable Filters for Rotation Equivariant CNNscited by 194
  6. 2021A Practical Method for Constructing Equivariant Multilayer Perceptrons for Arbitrary Matrix Groupscited by 187
  7. 2018Clebsch-Gordan Nets: a Fully Fourier Space Spherical Convolutional Neural Networkcited by 171
  8. 2021E(n) Equivariant Normalizing Flowscited by 165
  9. 2019Hamiltonian Generative Networkscited by 137
  10. 2018The Best of Both Worlds: Combining Recent Advances in Neural Machine Translationcited by 102
  11. 2016Rotation equivariant vector field networkscited by 99
  12. 2018Roto-Translation Covariant Convolutional Networks for Medical Image Analysiscited by 92
  13. 2020Gauge Equivariant Mesh CNNs: Anisotropic convolutions on geometric graphscited by 92
  14. 2016Exploiting Cyclic Symmetry in Convolutional Neural Networkscited by 91
  15. 2021Provably Strict Generalisation Benefit for Equivariant Modelscited by 91
  16. 2019Spherical CNNs on Unstructured Gridscited by 79
  17. 2020DeepSphere: a graph-based spherical CNNcited by 66
  18. 2020Attentive Group Equivariant Convolutional Networkscited by 63
  19. 2018DeepSphere: Efficient spherical Convolutional Neural Network with HEALPix sampling for cosmological applicationscited by 62
  20. 2018CubeNet: Equivariance to 3D Rotation and Translationcited by 59
  21. 2020Spin-Weighted Spherical CNNscited by 50
  22. 2020A Wigner-Eckart Theorem for Group Equivariant Convolution Kernelscited by 46
  23. 2020Lattice gauge equivariant convolutional neural networkscited by 45
  24. 2020CNNs on Surfaces using Rotation-Equivariant Featurescited by 43
  25. 2018N-body Networks: a Covariant Hierarchical Neural Network Architecture for Learning Atomic Potentialscited by 42
  26. 20183D G-CNNs for Pulmonary Nodule Detectioncited by 36
  27. 2020Efficient Generalized Spherical CNNscited by 29
  28. 2020Gauge equivariant neural networks for quantum lattice gauge theoriescited by 28
  29. 2018HexaConvcited by 27
  30. 2020Theoretical Aspects of Group Equivariant Neural Networkscited by 25
  31. 2018Towards a topological-geometrical theory of group equivariant non-expansive operators for data analysis and machine learningcited by 24
  32. 2017Learning Spherical Convolution for Fast Features from 360deg Imagerycited by 22
  33. 2017SalNet360: Saliency Maps for omni-directional images with CNNcited by 21
  34. 2021Homogeneous vector bundles and G-equivariant convolutional neural networkscited by 21
  35. 2021Rotation-Equivariant Deep Learning for Diffusion MRIcited by 20
  36. 2020Group Equivariant Generative Adversarial Networkscited by 19
  37. 2021PDO-eS2CNNs: Partial Differential Operator Based Equivariant Spherical CNNscited by 19
  38. 2021Noether: The More Things Change, the More Stay the Samecited by 19

50 most cited of 57 All 57 (JSON)

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Cited by year (2026 to 14 Oct)

Citing papers per year: 2021: 3; 2022: 10; 2023: 8; 2024: 12; 2025: 16; 2026: 16.

65 papers.

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  1. 2022Neural Conservation Laws: A Divergence-Free Perspectivecited by 70
  2. 2025Symmetry in Neural Network Parameter Spacescited by 36
  3. 2023The Exact Sample Complexity Gain from Invariances for Kernel Regressioncited by 28
  4. 2022A case for using rotation invariant features in state of the art feature matcherscited by 27
  5. 2022Equivariance versus Augmentation for Spherical Imagescited by 24
  6. 2023HEAL-SWIN: A Vision Transformer On The Spherecited by 20
  7. 2024Emergent Equivariance in Deep Ensemblescited by 18
  8. 2021ZZ-Net: A Universal Rotation Equivariant Architecture for 2D Point Cloudscited by 15
  9. 2021DeltaConv: Anisotropic Operators for Geometric Deep Learning on Point Cloudscited by 10
  10. 2025Flopping for FLOPs: Leveraging equivariance for computational efficiencycited by 8
  11. 2026Structure-Preserving Learning Improves Geometry Generalization in Neural PDEscited by 8
  12. 2024Understanding the Role of Equivariance in Self-supervised Learningcited by 7
38 more
  1. 2023Geometrical aspects of lattice gauge equivariant convolutional neural networkscited by 6
  2. 2022Generalized Permutants and Graph GENEOscited by 5
  3. 2023Unified theory for joint covariance properties under geometric image transformations for spatio-temporal receptive fields according to the generalized Gaussian derivative model for visual receptive fieldscited by 5
  4. 2025Mathematical Foundation of Interpretable Equivariant Surrogate Modelscited by 5
  5. 2025AtlasD: Automatic Local Symmetry Discoverycited by 5
  6. 2021Revisiting consistency for semi-supervised semantic segmentationcited by 4
  7. 2022Variational inference of fractional Brownian motion with linear computational complexitycited by 4
  8. 2024Equivariant Neural Tangent Kernelscited by 4
  9. 2024A novel approach to graph distinction through GENEOs and permutantscited by 4
  10. 2024R2Det: Exploring Relaxed Rotation Equivariance in 2D object detectioncited by 4
  11. 2025Lecture Notes on Normalizing Flows for Lattice Quantum Field Theoriescited by 4
  12. 2024Equivariant neural networks and piecewise linear representation theorycited by 3
  13. 2024Scale generalisation properties of extended scale-covariant and scale-invariant Gaussian derivative networks on image datasets with spatial scaling variationscited by 3
  14. 2025Adaptive Canonicalization with Application to Invariant Anisotropic Geometric Networkscited by 3
  15. 2026Disentangling Dynamical Systems: Causal Representation Learning Meets Local Sparse Attentioncited by 3
  16. 2022Long Scale Error Control in Low Light Image and Video Enhancement Using Equivariancecited by 2
  17. 2022A Feedforward Unitary Equivariant Neural Networkcited by 2
  18. 2024Equivariant Manifold Neural ODEs and Differential Invariantscited by 2
  19. 2025Learning Chern Numbers of Topological Insulators with Gauge Equivariant Neural Networkscited by 2
  20. 2025SO(3)-Equivariant Neural Networks for Learning from Scalar and Vector Fields on Spherescited by 2
  21. 2025SESaMo: Symmetry-Enforcing Stochastic Modulation for Normalizing Flowscited by 2
  22. 2026Rethinking Input Domains in Physics-Informed Neural Networks via Geometric Compactification Mappingscited by 2
  23. 2026The Geometry of Polynomial Group Convolutional Neural Networkscited by 2
  24. 2023Computing equivariant matrices on homogeneous spaces for Geometric Deep Learning and Automorphic Lie Algebrascited by 1
  25. 2023Influence of the Geometry of the world model on Curiosity Based Explorationcited by 1
  26. 2023A topological model for partial equivariance in deep learning and data analysiscited by 1
  27. 2024Manifold GCN: Diffusion-based Convolutional Neural Network for Manifold-valued Graphscited by 1
  28. 2024Geometric Generative Models based on Morphological Equivariant PDEs and GANscited by 1
  29. 2024Contrastive Learning Via Equivariant Representationcited by 1
  30. 2025SVarM: Linear Support Varifold Machines for Classification and Regression on Geometric Datacited by 1
  31. 2025RotaTouille: Rotation Equivariant Deep Learning for Contourscited by 1
  32. 2026How unconstrained machine-learning models learn physical symmetriescited by 1
  33. 2022SELTO: Sample-Efficient Learned Topology Optimizationcited by 0
  34. 2022In Search of Projectively Equivariant Networkscited by 0
  35. 2022Characterizing 4-string contact interaction using machine learningcited by 0
  36. 2023Deep Metric Tensor Regularized Policy Gradientcited by 0
  37. 2024Sampling Foundational Transformer: A Theoretical Perspectivecited by 0
  38. 2025CrystalFramer: Rethinking the Role of Frames for SE(3)-Invariant Crystal Structure Modelingcited by 0

50 most cited of 65 All 65 (JSON)