mlsanityData to 14 Oct 2026

On the Generalization of Equivariance and Convolution in Neural Networks to the Action of Compact Groups

arXiv 1802.03690 ↗First posted 11 Feb 2018stat.ML cs.LGcited by 300

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Abstract (arXiv)

Convolutional neural networks have been extremely successful in the image recognition domain because they ensure equivariance to translations. There have been many recent attempts to generalize this framework to other domains, including graphs and data lying on manifolds. In this paper we give a rigorous, theoretical treatment of convolution and equivariance in neural networks with respect to not just translations, but the action of any compact group. Our main result is to prove that (given some natural constraints) convolutional structure is not just a sufficient, but also a necessary condition for equivariance to the action of a compact group. Our exposition makes use of concepts from representation theory and noncommutative harmonic analysis and derives new generalized convolution formulae.

Key facts. Cited by 300 papers in the map (data through 14 Oct 2026). First posted on arXiv 11 Feb 2018.

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Cited by year (2026 to 14 Oct)

Citing papers per year: 2021: 56; 2022: 60; 2023: 55; 2024: 42; 2025: 47; 2026: 40.

300 papers.

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  1. 2021CLIPort: What and Where Pathways for Robotic Manipulationcited by 622
  2. 2022Scientific Machine Learning through Physics-Informed Neural Networks: Where we are and What's nextcited by 608
  3. 2022MACE: Higher Order Equivariant Message Passing Neural Networks for Fast and Accurate Force Fieldscited by 474
  4. 2021Equivariant message passing for the prediction of tensorial properties and molecular spectracited by 441
  5. 2021Geometric and Physical Quantities Improve E(3) Equivariant Message Passingcited by 228
  6. 2021Equivariant Subgraph Aggregation Networkscited by 154
  7. 2021Geometric Deep Learning on Molecular Representationscited by 113
  8. 2024FlowMM: Generating Materials with Riemannian Flow Matchingcited by 109
  9. 2023Equivariant Architectures for Learning in Deep Weight Spacescited by 107
  10. 2022Approximately Equivariant Networks for Imperfectly Symmetric Dynamicscited by 97
  11. 2022Clifford Neural Layers for PDE Modelingcited by 92
  12. 2021Provably Strict Generalisation Benefit for Equivariant Modelscited by 91
38 more
  1. 2021Automatic Symmetry Discovery with Lie Algebra Convolutional Networkcited by 83
  2. 2021Is Attention Better Than Matrix Decomposition?cited by 77
  3. 2021SE(3) Equivariant Graph Neural Networks with Complete Local Framescited by 76
  4. 2023Permutation Equivariant Neural Functionalscited by 70
  5. 2021Geometric Deep Learning and Equivariant Neural Networkscited by 65
  6. 2022Theory for Equivariant Quantum Neural Networkscited by 62
  7. 2022Theory of Graph Neural Networks: Representation and Learningcited by 61
  8. 2021Robust Equivariant Imaging: a fully unsupervised framework for learning to image from noisy and partial measurementscited by 56
  9. 2022Group-Invariant Quantum Machine Learningcited by 56
  10. 2023Clifford Group Equivariant Neural Networkscited by 55
  11. 2022Learning Symmetric Embeddings for Equivariant World Modelscited by 54
  12. 2021Lossy Compression for Lossless Predictioncited by 52
  13. 2022Theoretical Guarantees for Permutation-Equivariant Quantum Neural Networkscited by 52
  14. 2023Neural Functional Transformerscited by 52
  15. 2023Approximation-Generalization Trade-offs under (Approximate) Group Equivariancecited by 46
  16. 2021Physics-Guided Deep Learning for Dynamical Systems: A Surveycited by 44
  17. 2023Generative Adversarial Symmetry Discoverycited by 42
  18. 2022Relaxing Equivariance Constraints with Non-stationary Continuous Filterscited by 41
  19. 2023Fast, Expressive SE(n) Equivariant Networks through Weight-Sharing in Position-Orientation Spacecited by 41
  20. 2022Representation Theory for Geometric Quantum Machine Learningcited by 40
  21. 2021Autobahn: Automorphism-based Graph Neural Netscited by 39
  22. 2022Unsupervised Learning of Group Invariant and Equivariant Representationscited by 39
  23. 2021Transporting Causal Mechanisms for Unsupervised Domain Adaptationcited by 38
  24. 2021Ab-Initio Potential Energy Surfaces by Pairing GNNs with Neural Wave Functionscited by 37
  25. 2023EquivAct: SIM(3)-Equivariant Visuomotor Policies beyond Rigid Object Manipulationcited by 37
  26. 2021DISCO: accurate Discrete Scale Convolutionscited by 36
  27. 2021Category Theory in Machine Learningcited by 36
  28. 2021Top-N: Equivariant set and graph generation without exchangeabilitycited by 36
  29. 2025Symmetry in Neural Network Parameter Spacescited by 36
  30. 2023Equivariant Deep Weight Space Alignmentcited by 35
  31. 2021Skip-Convolutions for Efficient Video Processingcited by 32
  32. 2021Provably Strict Generalisation Benefit for Invariance in Kernel Methodscited by 30
  33. 2021Symmetry-via-Duality: Invariant Neural Network Densities from Parameter-Space Correlatorscited by 29
  34. 2021Speeding up Learning Quantum States through Group Equivariant Convolutional Quantum Ans"atzecited by 29
  35. 2022Data-driven emergence of convolutional structure in neural networkscited by 29
  36. 2023Self-Supervised Learning with Lie Symmetries for Partial Differential Equationscited by 29
  37. 2023Latent Space Symmetry Discoverycited by 29
  38. 2022A case for using rotation invariant features in state of the art feature matcherscited by 27

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