mlsanityData to 7 Oct 2026

Equivariance with Learned Canonicalization Functions

arXiv 2211.06489 ↗First posted 11 Nov 2022cs.LG cs.AIcited by 106

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Abstract (arXiv)

Symmetry-based neural networks often constrain the architecture in order to achieve invariance or equivariance to a group of transformations. In this paper, we propose an alternative that avoids this architectural constraint by learning to produce canonical representations of the data. These canonicalization functions can readily be plugged into non-equivariant backbone architectures. We offer explicit ways to implement them for some groups of interest. We show that this approach enjoys universality while providing interpretable insights. Our main hypothesis, supported by our empirical results, is that learning a small neural network to perform canonicalization is better than using predefined heuristics. Our experiments show that learning the canonicalization function is competitive with existing techniques for learning equivariant functions across many tasks, including image classification, N-body dynamics prediction, point cloud classification and part segmentation, while being faster across the board.

Key facts. Cited by 106 papers in the map (data through 7 Oct 2026). Builds on 55 of its 79 references that are in the map. First posted on arXiv 11 Nov 2022.

Builds on

55 of 79 references.

References · 55

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38 more
  1. 2021Geometric Deep Learning: Grids, Groups, Graphs, Geodesics, and Gaugescited by 1,131
  2. 2021E(n) Equivariant Graph Neural Networkscited by 951
  3. 2018Tensor field networks: Rotation- and translation-equivariant neural networks for 3D point cloudscited by 735
  4. 2020SE(3)-Transformers: 3D Roto-Translation Equivariant Attention Networkscited by 602
  5. 2016Input Convex Neural Networkscited by 407
  6. 2018Neural Relational Inference for Interacting Systemscited by 339
  7. 2021Vector Neurons: A General Framework for SO(3)-Equivariant Networkscited by 304
  8. 2017Learning SO(3) Equivariant Representations with Spherical CNNscited by 244
  9. 2019Kornia: an Open Source Differentiable Computer Vision Library for PyTorchcited by 231
  10. 2021Efficient and Modular Implicit Differentiationcited by 199
  11. 2020Fast Differentiable Sorting and Rankingcited by 196
  12. 2021A Practical Method for Constructing Equivariant Multilayer Perceptrons for Arbitrary Matrix Groupscited by 185
  13. 2018Learning Latent Permutations with Gumbel-Sinkhorn Networkscited by 172
  14. 2021Frame Averaging for Invariant and Equivariant Network Designcited by 166
  15. 2018Point Convolutional Neural Networks by Extension Operatorscited by 154
  16. 2021Investigating Bi-Level Optimization for Learning and Vision from a Unified Perspective: A Survey and Beyondcited by 151
  17. 2017Equivariance Through Parameter-Sharingcited by 139
  18. 2020MDP Homomorphic Networks: Group Symmetries in Reinforcement Learningcited by 133
  19. 2018Universal approximations of invariant maps by neural networkscited by 127
  20. 2021Scalars are universal: Equivariant machine learning, structured like classical physicscited by 127
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  22. 2016On Differentiating Parameterized Argmin and Argmax Problems with Application to Bi-level Optimizationcited by 108
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  26. 2019Rotation Invariant Convolutions for 3D Point Clouds Deep Learningcited by 80
  27. 2021SE(3) Equivariant Graph Neural Networks with Complete Local Framescited by 75
  28. 2020A Rotation-Invariant Framework for Deep Point Cloud Analysiscited by 62
  29. 2019Effective Rotation-invariant Point CNN with Spherical Harmonics kernelscited by 57
  30. 2019On Universal Equivariant Set Networkscited by 45
  31. 2022Unsupervised Learning of Group Invariant and Equivariant Representationscited by 38
  32. 2019Equivariant Transformer Networkscited by 35
  33. 2022Equivariant Point Cloud Analysis via Learning Orientations for Message Passingcited by 32
  34. 2020Finding Symmetry Breaking Order Parameters with Euclidean Neural Networkscited by 30
  35. 2021Equivariant neural networks for inverse problemscited by 29
  36. 2020Global Context Aware Convolutions for 3D Point Cloud Understandingcited by 26
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50 most cited of 55 All 55 (JSON)

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Cited by year (2026 to 7 Oct)

Citing papers per year: 2023: 16; 2024: 29; 2025: 35; 2026: 26.

106 papers.

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  1. 2023MiDi: Mixed Graph and 3D Denoising Diffusion for Molecule Generationcited by 96
  2. 2023A Hitchhiker's Guide to Geometric GNNs for 3D Atomic Systemscited by 65
  3. 2024Orb: A Fast, Scalable Neural Network Potentialcited by 63
  4. 2025SymmCD: Symmetry-Preserving Crystal Generation with Diffusion Modelscited by 48
  5. 2024Equivariant Frames and the Impossibility of Continuous Canonicalizationcited by 41
  6. 2023Equivariant Adaptation of Large Pretrained Modelscited by 40
  7. 2023Learning Probabilistic Symmetrization for Architecture Agnostic Equivariancecited by 25
  8. 2024A Canonicalization Perspective on Invariant and Equivariant Learningcited by 25
  9. 2023Symmetry Breaking and Equivariant Neural Networkscited by 24
  10. 2024Scale Equivariant Graph Metanetworkscited by 23
  11. 2024Lie Algebra Canonicalization: Equivariant Neural Operators under arbitrary Lie Groupscited by 23
  12. 2023Laplacian Canonization: A Minimalist Approach to Sign and Basis Invariant Spectral Embeddingcited by 21
38 more
  1. 2024Learning on LoRAs: GL-Equivariant Processing of Low-Rank Weight Spaces for Large Finetuned Modelscited by 21
  2. 2024Emergent Equivariance in Deep Ensemblescited by 18
  3. 2023Expressive Sign Equivariant Networks for Spectral Geometric Learningcited by 15
  4. 2025Improving Equivariant Networks with Probabilistic Symmetry Breakingcited by 15
  5. 2024Symmetry-Informed Governing Equation Discoverycited by 14
  6. 2024Beyond Canonicalization: How Tensorial Messages Improve Equivariant Message Passingcited by 13
  7. 2024Relaxed Equivariance via Multitask Learningcited by 13
  8. 2025Lorentz Local Canonicalization: How to Make Any Network Lorentz-Equivariantcited by 12
  9. 2023Efficient Equivariant Transfer Learning from Pretrained Modelscited by 11
  10. 2023Towards fully covariant machine learningcited by 10
  11. 2023Using Multiple Vector Channels Improves E(n)-Equivariant Graph Neural Networkscited by 10
  12. 2024Weisfeiler Leman for Euclidean Equivariant Machine Learningcited by 10
  13. 2024A Generative Model of Symmetry Transformationscited by 10
  14. 2024Equivariant Symmetry Breaking Setscited by 9
  15. 2025Symmetry-Aware Bayesian Flow Networks for Crystal Generationcited by 9
  16. 2023Latent Field Discovery In Interacting Dynamical Systems With Neural Fieldscited by 7
  17. 2024Tilt your Head: Activating the Hidden Spatial-Invariance of Classifierscited by 7
  18. 2024Improved Canonicalization for Model Agnostic Equivariancecited by 7
  19. 2024Equivariant Denoisers Cannot Copy Graphs: Align Your Graph Diffusion Modelscited by 7
  20. 2025Achieving Approximate Symmetry Is Exponentially Easier than Exact Symmetrycited by 6
  21. 2024An intuitive multi-frequency feature representation for SO(3)-equivariant networkscited by 5
  22. 2024Approximately-symmetric neural networks for quantum spin liquidscited by 5
  23. 2025Learning with Exact Invariances in Polynomial Timecited by 5
  24. 2025Shifting the Paradigm: A Diffeomorphism Between Time Series Data Manifolds for Achieving Shift-Invariancy in Deep Learningcited by 5
  25. 2025Test-Time Canonicalization by Foundation Models for Robust Perceptioncited by 5
  26. 2025Learning Inter-Atomic Potentials without Explicit Equivariancecited by 5
  27. 2026Rethinking Diffusion Models with Symmetries through Canonicalization with Applications to Molecular Graph Generationcited by 5
  28. 2023Learning Symmetrization for Equivariance with Orbit Distance Minimizationcited by 4
  29. 2025Symmetry-Aware Generative Modeling through Learned Canonicalizationcited by 4
  30. 2025Spectral State Space Model for Rotation-Invariant Visual Representation Learningcited by 4
  31. 2025On Universality Classes of Equivariant Networkscited by 4
  32. 2025Scaling Laws and Symmetry, Evidence from Neural Force Fieldscited by 4
  33. 2025Equivariant Neural Networks for General Linear Symmetries on Lie Algebrascited by 4
  34. 2023Equivariant geometric convolutions for emulation of dynamical systemscited by 3
  35. 2024Tensor learning with orthogonal, Lorentz, and symplectic symmetriescited by 3
  36. 2025Zero-Shot Visual Generalization in Robot Manipulationcited by 3
  37. 2025Quick ViTs: Speeding up Vision Transformers through Equivariancecited by 3
  38. 2025Equivariant Flow Matching for Symmetry-Breaking Bifurcation Problemscited by 3

50 most cited of 106 All 106 (JSON)