mlsanityData to 14 Oct 2026

Sample Complexity of Probability Divergences under Group Symmetry

arXiv 2302.01915 ↗First posted 3 Feb 2023math.ST stat.ML stat.THcited by 14

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Abstract (arXiv)

We rigorously quantify the improvement in the sample complexity of variational divergence estimations for group-invariant distributions. In the cases of the Wasserstein-1 metric and the Lipschitz-regularized α-divergences, the reduction of sample complexity is proportional to the group size if the group is finite. In addition to the published version at ICML 2023, our proof indeed has included the case when the group is infinite such as compact Lie groups, the convergence rate can be further improved and depends on the intrinsic dimension of the fundamental domain characterized by the scaling of its covering number. Our approach is different from that in [Tahmasebi & Jegelka, ICML 2024] and our work also applies to asymmetric divergences, such as the Lipschitz-regularized α-divergences. For the maximum mean discrepancy (MMD), the improvement of sample complexity is more nuanced, as it depends on not only the group size but also the choice of kernel. Numerical simulations verify our theories.

Key facts. Cited by 14 papers in the map (data through 14 Oct 2026). Builds on 27 of its 72 references that are in the map. First posted on arXiv 3 Feb 2023.

Builds on

27 of 72 references.

References · 27

  1. 2017Improved Training of Wasserstein GANscited by 2,235
  2. 2016Group Equivariant Convolutional Networkscited by 1,165
  3. 2017Spectrally-normalized margin bounds for neural networkscited by 686
  4. 2018MINE: Mutual Information Neural Estimationcited by 683
  5. 2016f-GAN: Training Generative Neural Samplers using Variational Divergence Minimizationcited by 506
  6. 2019General E(2)-Equivariant Steerable CNNscited by 446
  7. 2017Wasserstein Auto-Encoderscited by 373
  8. 2020Differentiable Augmentation for Data-Efficient GAN Trainingcited by 363
  9. 2018A General Theory of Equivariant CNNs on Homogeneous Spacescited by 228
  10. 2021E(n) Equivariant Normalizing Flowscited by 165
  11. 2019Graph Normalizing Flowscited by 104
  12. 2020Sampling using SU(N) gauge equivariant flowscited by 84
15 more
  1. 2019Equivariant Flows: sampling configurations for multi-body systems with symmetric energiescited by 57
  2. 2014Efficient Estimation of Mutual Information for Strongly Dependent Variablescited by 46
  3. 2013Equitability, mutual information, and the maximal information coefficientcited by 45
  4. 2016Generalization Error of Invariant Classifierscited by 42
  5. 2019Equivariant Hamiltonian Flowscited by 39
  6. 2016Large-Scale Kernel Methods for Independence Testingcited by 32
  7. 2021Neural Estimation of Statistical Divergencescited by 25
  8. 2020Scalable Normalizing Flows for Permutation Invariant Densitiescited by 22
  9. 2020Variational Representations and Neural Network Estimation of R'enyi Divergencescited by 20
  10. 2020Group Equivariant Generative Adversarial Networkscited by 19
  11. 2022Structure-preserving GANscited by 17
  12. 2020Optimizing Variational Representations of Divergences and Accelerating their Statistical Estimationcited by 15
  13. 2021Smooth p-Wasserstein Distance: Structure, Empirical Approximation, and Statistical Applicationscited by 15
  14. 2021Kernel Two-Sample Tests for Manifold Datacited by 8
  15. 2022Function-space regularized R'enyi divergencescited by 7

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Cited by year (2026 to 14 Oct)

Citing papers per year: 2023: 2; 2024: 5; 2025: 4; 2026: 3.

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  1. 2023Sample Complexity Bounds for Estimating Probability Divergences under Invariancescited by 12
  2. 2025Achieving Approximate Symmetry Is Exponentially Easier than Exact Symmetrycited by 7
  3. 2025Learning with Exact Invariances in Polynomial Timecited by 6
  4. 2023Statistical Guarantees of Group-Invariant GANscited by 4
  5. 2024Robust Generative Learning with Lipschitz-Regularized α-Divergences Allows Minimal Assumptions on Target Distributionscited by 4
  6. 2024Nonlinear denoising score matching for enhanced learning of structured distributionscited by 3
  7. 2024Robustness and Structure Preservation in Flow-Based Generative Models via Wasserstein Path-Space Divergencescited by 3
  8. 2026Efficient Learning and Symmetry Discovery under Exact Invariancescited by 3
  9. 2024Statistical Error Bounds for GANs with Nonlinear Objective Functionalscited by 2
  10. 2024Robust training of implicit generative models for multivariate and heavy-tailed distributions with an invariant statistical losscited by 2
  11. 2025Bias Detection via Maximum Subgroup Discrepancycited by 2
  12. 2025Statistical Learning Guarantees for Group-Invariant Barron Functionscited by 1
2 more
  1. 2026Sparse Data Augmentation for Optimization with Provable Guaranteescited by 1
  2. 2026Dimension-Free Scaling Laws for Size Generalization: An Exponential Separationcited by 0