mlsanityData to 14 Oct 2026

Machine learning and invariant theory

arXiv 2209.14991 ↗First posted 29 Sep 2022stat.ML cs.LGcited by 13

Not on the map. Why?

Abstract (arXiv)

Inspired by constraints from physical law, equivariant machine learning restricts the learning to a hypothesis class where all the functions are equivariant with respect to some group action. Irreducible representations or invariant theory are typically used to parameterize the space of such functions. In this article, we introduce the topic and explain a couple of methods to explicitly parameterize equivariant functions that are being used in machine learning applications. In particular, we explicate a general procedure, attributed to Malgrange, to express all polynomial maps between linear spaces that are equivariant under the action of a group G, given a characterization of the invariant polynomials on a bigger space. The method also parametrizes smooth equivariant maps in the case that G is a compact Lie group.

Key facts. Cited by 13 papers in the map (data through 14 Oct 2026). Builds on 47 of its 74 references that are in the map. First posted on arXiv 29 Sep 2022.

Builds on

47 of 74 references.

References · 47

  1. 2018Neural Ordinary Differential Equationscited by 3,779
  2. 2017Neural Message Passing for Quantum Chemistrycited by 3,487
  3. 2016Understanding deep learning requires rethinking generalizationcited by 1,674
  4. 2017Deep Setscited by 1,421
  5. 2016Group Equivariant Convolutional Networkscited by 1,165
  6. 2013Spectral Networks and Locally Connected Networks on Graphscited by 1,102
  7. 2021E(n) Equivariant Graph Neural Networkscited by 965
  8. 2015Convolutional Networks on Graphs for Learning Molecular Fingerprintscited by 754
  9. 2018Tensor field networks: Rotation- and translation-equivariant neural networks for 3D point cloudscited by 746
  10. 2021E(3)-Equivariant Graph Neural Networks for Data-Efficient and Accurate Interatomic Potentialscited by 708
  11. 2020SE(3)-Transformers: 3D Roto-Translation Equivariant Attention Networkscited by 607
  12. 2018Invariant and Equivariant Graph Networkscited by 346
35 more
  1. 2021GemNet: Universal Directional Graph Neural Networks for Moleculescited by 343
  2. 2019Gauge Equivariant Convolutional Networks and the Icosahedral CNNcited by 267
  3. 2018A General Theory of Equivariant CNNs on Homogeneous Spacescited by 228
  4. 2022Sign and Basis Invariant Networks for Spectral Graph Representation Learningcited by 179
  5. 2022e3nn: Euclidean Neural Networkscited by 174
  6. 2019On the equivalence between graph isomorphism testing and function approximation with GNNscited by 172
  7. 2018Clebsch-Gordan Nets: a Fully Fourier Space Spherical Convolutional Neural Networkcited by 171
  8. 2019On the Universality of Invariant Networkscited by 152
  9. 2022EquiBind: Geometric Deep Learning for Drug Binding Structure Predictioncited by 144
  10. 2017Equivariance Through Parameter-Sharingcited by 141
  11. 2018Universal approximations of invariant maps by neural networkscited by 127
  12. 2021Scalars are universal: Equivariant machine learning, structured like classical physicscited by 127
  13. 2020Incorporating Symmetry into Deep Dynamics Models for Improved Generalizationcited by 126
  14. 2020Lorentz Group Equivariant Neural Network for Particle Physicscited by 121
  15. 2019Hamiltonian Graph Networks with ODE Integratorscited by 108
  16. 2022Approximately Equivariant Networks for Imperfectly Symmetric Dynamicscited by 97
  17. 2021Provably Strict Generalisation Benefit for Equivariant Modelscited by 91
  18. 2021Learning with invariances in random features and kernel modelscited by 89
  19. 2020On the Universality of Rotation Equivariant Point Cloud Networkscited by 80
  20. 2021Coordinate Independent Convolutional Networks -- Isometry and Gauge Equivariant Convolutions on Riemannian Manifoldscited by 73
  21. 2020Relevance of Rotationally Equivariant Convolutions for Predicting Molecular Propertiescited by 51
  22. 2018N-body Networks: a Covariant Hierarchical Neural Network Architecture for Learning Atomic Potentialscited by 42
  23. 2021Autobahn: Automorphism-based Graph Neural Netscited by 39
  24. 2022Low Dimensional Invariant Embeddings for Universal Geometric Learningcited by 38
  25. 2021Provably Strict Generalisation Benefit for Invariance in Kernel Methodscited by 30
  26. 2022Equivariance versus Augmentation for Spherical Imagescited by 24
  27. 2022Dimensionally Consistent Learning with Buckingham Picited by 22
  28. 2021Implicit Bias of Linear Equivariant Networkscited by 19
  29. 2022Group-invariant max filteringcited by 18
  30. 2022Dimensionless machine learning: Imposing exact units equivariancecited by 17
  31. 2022Data Augmentation vs. Equivariant Networks: A Theory of Generalization on Dynamics Forecastingcited by 17
  32. 2021ZZ-Net: A Universal Rotation Equivariant Architecture for 2D Point Cloudscited by 15
  33. 2021A simple equivariant machine learning method for dynamics based on scalarscited by 6
  34. 2022Predicting the Thermal Sunyaev-Zel'dovich Field using Modular and Equivariant Set-Based Neural Networkscited by 6
  35. 2021Invariant polynomials and machine learningcited by 4

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Cited by year (2026 to 14 Oct)

Citing papers per year: 2023: 2; 2024: 5; 2025: 2; 2026: 4.

13 papers.

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  1. 2023A Hitchhiker's Guide to Geometric GNNs for 3D Atomic Systemscited by 68
  2. 2024Learning on LoRAs: GL-Equivariant Processing of Low-Rank Weight Spaces for Large Finetuned Modelscited by 22
  3. 2023Towards fully covariant machine learningcited by 10
  4. 2024Symmetry-Based Structured Matrices for Efficient Approximately Equivariant Networkscited by 7
  5. 2024Prospective Learning: Learning for a Dynamic Futurecited by 4
  6. 2025On Universality Classes of Equivariant Networkscited by 4
  7. 2024Tensor learning with orthogonal, Lorentz, and symplectic symmetriescited by 3
  8. 2024Stability of sorting based embeddingscited by 3
  9. 2025Constructing 3D Rotational Invariance and Equivariance with Symmetric Tensor Networkscited by 0
  10. 2026Generalized Reduction to the Isotropy for Flexible Equivariant Neural Fieldscited by 0
  11. 2026Exact Quotients of Fresnel-Kummer Surfaces and Certified Biaxial Refractioncited by 0
  12. 2026Fundamental Limits of Transferability and Equivariance in Algebraic Signal Models I: Finite Dimensionscited by 0
1 more
  1. 2026Warm-starting PDE solvers with any-dimensional machine learningcited by 0