mlsanityData to 14 Oct 2026

Achieving Approximate Symmetry Is Exponentially Easier than Exact Symmetry

arXiv 2512.11855 ↗First posted 5 Dec 2025cs.LG cs.AIcited by 7

Not on the map. Why?

Abstract (arXiv)

Enforcing exact symmetry in machine learning models often yields significant gains in scientific applications, serving as a powerful inductive bias. However, recent work suggests that relying on approximate symmetry can offer greater flexibility and robustness. Despite promising empirical evidence, there has been little theoretical understanding, and in particular, a direct comparison between exact and approximate symmetry is missing from the literature. In this paper, we initiate this study by asking: What is the cost of enforcing exact versus approximate symmetry? To address this question, we introduce averaging complexity, a framework for quantifying the cost of enforcing symmetry via averaging. Our main result is an exponential separation: under standard conditions, exact symmetry requires linear averaging complexity, whereas approximate symmetry can be attained with only logarithmic complexity in the group size. To the best of our knowledge, this provides the first theoretical separation of these two cases, formally justifying why approximate symmetry may be preferable in practice. Beyond this, our tools and techniques may be of independent interest for the broader study of symmetries in machine learning.

Key facts. Cited by 7 papers in the map (data through 14 Oct 2026). Builds on 47 of its 75 references that are in the map. First posted on arXiv 5 Dec 2025.

Builds on

47 of 75 references.

References · 47

  1. 2017Deep Setscited by 1,421
  2. 2021Geometric Deep Learning: Grids, Groups, Graphs, Geodesics, and Gaugescited by 1,167
  3. 2016Group Equivariant Convolutional Networkscited by 1,165
  4. 2021E(n) Equivariant Graph Neural Networkscited by 965
  5. 2018Invariant and Equivariant Graph Networkscited by 346
  6. 2016Steerable CNNscited by 305
  7. 2022Equiformer: Equivariant Graph Attention Transformer for 3D Atomistic Graphscited by 264
  8. 2021Frame Averaging for Invariant and Equivariant Network Designcited by 168
  9. 2020Incorporating Symmetry into Deep Dynamics Models for Improved Generalizationcited by 126
  10. 2020Lorentz Group Equivariant Neural Network for Particle Physicscited by 121
  11. 2022Equivariance with Learned Canonicalization Functionscited by 109
  12. 2018A Kernel Theory of Modern Data Augmentationcited by 106
35 more
  1. 2022Approximately Equivariant Networks for Imperfectly Symmetric Dynamicscited by 97
  2. 2022SO(2)-Equivariant Reinforcement Learningcited by 84
  3. 2021Automatic Symmetry Discovery with Lie Algebra Convolutional Networkcited by 83
  4. 2021Residual Pathway Priors for Soft Equivariance Constraintscited by 75
  5. 2022Theory for Equivariant Quantum Neural Networkscited by 62
  6. 2023Approximation-Generalization Trade-offs under (Approximate) Group Equivariancecited by 46
  7. 2021Learning Partial Equivariances from Datacited by 45
  8. 2024Equivariant Frames and the Impossibility of Continuous Canonicalizationcited by 43
  9. 2023Generative Adversarial Symmetry Discoverycited by 42
  10. 2022Relaxing Equivariance Constraints with Non-stationary Continuous Filterscited by 41
  11. 2021Provably Strict Generalisation Benefit for Invariance in Kernel Methodscited by 30
  12. 2023Latent Space Symmetry Discoverycited by 29
  13. 2023The Exact Sample Complexity Gain from Invariances for Kernel Regressioncited by 28
  14. 2024A Canonicalization Perspective on Invariant and Equivariant Learningcited by 27
  15. 2024Lie Algebra Canonicalization: Equivariant Neural Operators under arbitrary Lie Groupscited by 24
  16. 2023Learning Layer-wise Equivariances Automatically using Gradientscited by 23
  17. 2024Improving Equivariant Model Training via Constraint Relaxationcited by 23
  18. 2023Discovering Symmetry Breaking in Physical Systems with Relaxed Group Convolutioncited by 20
  19. 2021Frame Averaging for Equivariant Shape Space Learningcited by 19
  20. 2024Equivariance via Minimal Frame Averaging for More Symmetries and Efficiencycited by 19
  21. 2023Regularizing Towards Soft Equivariance Under Mixed Symmetriescited by 16
  22. 2023Approximately Equivariant Graph Networkscited by 16
  23. 2024Symmetry Discovery Beyond Affine Transformationscited by 15
  24. 2023Sample Complexity of Probability Divergences under Group Symmetrycited by 14
  25. 2024On the hardness of learning under symmetriescited by 13
  26. 2023Sample Complexity Bounds for Estimating Probability Divergences under Invariancescited by 12
  27. 2023Generalized equivalences between subsampling and ridge regularizationcited by 10
  28. 2023Almost Equivariance via Lie Algebra Convolutionscited by 10
  29. 2024Approximate Equivariance in Reinforcement Learningcited by 9
  30. 2024Approximately Equivariant Neural Processescited by 8
  31. 2025A Tale of Two Symmetries: Exploring the Loss Landscape of Equivariant Modelscited by 8
  32. 2024Symmetry-Based Structured Matrices for Efficient Approximately Equivariant Networkscited by 7
  33. 2025Learning with Exact Invariances in Polynomial Timecited by 6
  34. 2024Approximately-symmetric neural networks for quantum spin liquidscited by 5
  35. 2024Relaxed Rotational Equivariance via G-Biases in Visioncited by 4

Built on by

Cited by year (2026 to 14 Oct)

Citing papers per year: 2026: 7.

7 papers.

  1. 2026Data Augmentation: A Fourier Analysis Perspectivecited by 4
  2. 2026Adaptive Symmetry Discovery for Dynamical System Identificationcited by 3
  3. 2026Efficient Learning and Symmetry Discovery under Exact Invariancescited by 3
  4. 2026Quantitative Approximation Rates for Group Equivariant Learningcited by 1
  5. 2026Sparse Data Augmentation for Optimization with Provable Guaranteescited by 1
  6. 2026Group Invariant Spectral Embeddingcited by 0
  7. 2026Dimension-Free Scaling Laws for Size Generalization: An Exponential Separationcited by 0