mlsanityData to 7 Oct 2026

Sample Complexity Bounds for Estimating Probability Divergences under Invariances

arXiv 2311.02868 ↗First posted 6 Nov 2023cs.LGcited by 11

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Abstract (arXiv)

Group-invariant probability distributions appear in many data-generative models in machine learning, such as graphs, point clouds, and images. In practice, one often needs to estimate divergences between such distributions. In this work, we study how the inherent invariances, with respect to any smooth action of a Lie group on a manifold, improve sample complexity when estimating the 1-Wasserstein distance, the Sobolev Integral Probability Metrics (Sobolev IPMs), the Maximum Mean Discrepancy (MMD), and also the complexity of the density estimation problem (in the L2 and L^∞ distance). Our results indicate a two-fold gain: (1) reducing the sample complexity by a multiplicative factor corresponding to the group size (for finite groups) or the normalized volume of the quotient space (for groups of positive dimension); (2) improving the exponent in the convergence rate (for groups of positive dimension). These results are completely new for groups of positive dimension and extend recent bounds for finite group actions.

Key facts. Cited by 11 papers in the map (data through 7 Oct 2026). Builds on 23 of its 56 references that are in the map. First posted on arXiv 6 Nov 2023.

Builds on

23 of 56 references.

References · 23

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  3. 2019Cormorant: Covariant Molecular Neural Networkscited by 262
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11 more
  1. 2018Nonparametric Density Estimation under Adversarial Lossescited by 26
  2. 2021Measuring Generalization with Optimal Transportcited by 24
  3. 2019Nonparametric Density Estimation & Convergence Rates for GANs under Besov IPM Lossescited by 22
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  5. 2021Intrinsic Dimension Estimation Using Wasserstein Distancescited by 20
  6. 2020Group Equivariant Generative Adversarial Networkscited by 19
  7. 2022Structure-preserving GANscited by 17
  8. 2023Sample Complexity of Probability Divergences under Group Symmetrycited by 13
  9. 2019Estimating Certain Integral Probability Metric (IPM) is as Hard as Estimating under the IPMcited by 11
  10. 2019(q,p)-Wasserstein GANs: Comparing Ground Metrics for Wasserstein GANscited by 10
  11. 2020A Review on Modern Computational Optimal Transport Methods with Applications in Biomedical Researchcited by 8

Built on by

Cited by year (2026 to 7 Oct)

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

11 papers.

  1. 2024SemlaFlow -- Efficient 3D Molecular Generation with Latent Attention and Equivariant Flow Matchingcited by 40
  2. 2025Achieving Approximate Symmetry Is Exponentially Easier than Exact Symmetrycited by 6
  3. 2025Learning with Exact Invariances in Polynomial Timecited by 5
  4. 2024Robust Generative Learning with Lipschitz-Regularized α-Divergences Allows Minimal Assumptions on Target Distributionscited by 4
  5. 2024Robustness and Structure Preservation in Flow-Based Generative Models via Wasserstein Path-Space Divergencescited by 3
  6. 2026Error Analysis of Bayesian Inverse Problems with Generative Priorscited by 2
  7. 2026Adaptive Symmetry Discovery for Dynamical System Identificationcited by 2
  8. 2026Efficient Learning and Symmetry Discovery under Exact Invariancescited by 2
  9. 2025Statistical Learning Guarantees for Group-Invariant Barron Functionscited by 1
  10. 2026GEPC: Group-Equivariant Posterior Consistency for Out-of-Distribution Detection in Diffusion Modelscited by 1
  11. 2026Sparse Data Augmentation for Optimization with Provable Guaranteescited by 0