mlsanityData to 7 Oct 2026

Convergence and Concentration of Empirical Measures under Wasserstein Distance in Unbounded Functional Spaces

arXiv 1804.10556 ↗First posted 27 Apr 2018math.ST stat.ML stat.THcited by 41

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Abstract (arXiv)

We provide upper bounds of the expected Wasserstein distance between a probability measure and its empirical version, generalizing recent results for finite dimensional Euclidean spaces and bounded functional spaces. Such a generalization can cover Euclidean spaces with large dimensionality, with the optimal dependence on the dimensionality. Our method also covers the important case of Gaussian processes in separable Hilbert spaces, with rate-optimal upper bounds for functional data distributions whose coordinates decay geometrically or polynomially. Moreover, our bounds of the expected value can be combined with mean-concentration results to yield improved exponential tail probability bounds for the Wasserstein error of empirical measures under Bernstein-type or log Sobolev-type conditions.

Key facts. Cited by 41 papers in the map (data through 7 Oct 2026). First posted on arXiv 27 Apr 2018.

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Citing papers per year: 2021: 14; 2022: 5; 2023: 9; 2024: 3; 2025: 6; 2026: 4.

41 papers.

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  1. 2021Plugin Estimation of Smooth Optimal Transport Mapscited by 67
  2. 2023Scalable Optimal Transport Methods in Machine Learning: A Contemporary Surveycited by 27
  3. 2022A Finite-Particle Convergence Rate for Stein Variational Gradient Descentcited by 22
  4. 2021On the capacity of deep generative networks for approximating distributionscited by 19
  5. 2021Wasserstein Generative Adversarial Uncertainty Quantification in Physics-Informed Neural Networkscited by 16
  6. 2023Outlier-Robust Wasserstein DROcited by 16
  7. 2021Smooth p-Wasserstein Distance: Structure, Empirical Approximation, and Statistical Applicationscited by 15
  8. 2023Towards Understanding the Dynamics of Gaussian-Stein Variational Gradient Descentcited by 13
  9. 2023Bridging Distributionally Robust Learning and Offline RL: An Approach to Mitigate Distribution Shift and Partial Data Coveragecited by 12
  10. 2023Sample Complexity Bounds for Estimating Probability Divergences under Invariancescited by 11
  11. 2021Entropy Regularized Optimal Transport Independence Criterioncited by 9
  12. 2021Convergence of Gaussian-smoothed optimal transport distance with sub-gamma distributions and dependent samplescited by 8
29 more
  1. 2021Estimation and Quantization of Expected Persistence Diagramscited by 8
  2. 2021Costs and Benefits of Fair Regressioncited by 8
  3. 2021Dimension Reduction for Fr'echet Regressioncited by 8
  4. 2022Nonlinear Sufficient Dimension Reduction for Distribution-on-Distribution Regressioncited by 8
  5. 2023Robust Estimation under the Wasserstein Distancecited by 8
  6. 2021Functional optimal transport: map estimation and domain adaptation for functional datacited by 7
  7. 2022Shedding a PAC-Bayesian Light on Adaptive Sliced-Wasserstein Distancescited by 7
  8. 2023Provably Fast Finite Particle Variants of SVGD via Virtual Particle Stochastic Approximationcited by 6
  9. 2023A Framework for Fast and Stable Representations of Multiparameter Persistent Homology Decompositionscited by 6
  10. 2021Online Estimation and Optimization of Utility-Based Shortfall Riskcited by 5
  11. 2021Bounding Wasserstein distance with couplingscited by 5
  12. 2024Robust Distribution Learning with Local and Global Adversarial Corruptionscited by 5
  13. 2021Statistical Regeneration Guarantees of the Wasserstein Autoencoder with Latent Space Consistencycited by 3
  14. 2025Simultaneously Solving Infinitely Many LQ Mean Field Games In Hilbert Spaces: The Power of Neural Operatorscited by 3
  15. 2023Inference via robust optimal transportation: theory and methodscited by 2
  16. 2025Private Evolution Convergescited by 2
  17. 2021Limit Distribution Theory for the Smooth 1-Wasserstein Distance with Applicationscited by 1
  18. 2022Strong posterior contraction rates via Wasserstein dynamicscited by 1
  19. 2022Learning Distributions by Generative Adversarial Networks: Approximation and Generalizationcited by 1
  20. 2024Max-sliced Wasserstein concentration and uniform ratio bounds of empirical measures on RKHScited by 1
  21. 2024To Switch or Not to Switch? Balanced Policy Switching in Offline Reinforcement Learningcited by 1
  22. 2025Statistical Inference for Optimal Transport Maps: Recent Advances and Perspectivescited by 1
  23. 2025DAIL: Beyond Task Ambiguity for Language-Conditioned Reinforcement Learningcited by 1
  24. 2026Distribution-free two-sample testing with blurred total variation distancecited by 1
  25. 2025Dimension Agnostic Testing of Survey Data Credibility through the Lens of Regressioncited by 0
  26. 2025Estimation of Stochastic Optimal Transport Mapscited by 0
  27. 2026Efficient Distribution Learning with Error Bounds in Wasserstein Distancecited by 0
  28. 2026Generating DDPM-based Samples from Tilted Distributionscited by 0
  29. 2026Wasserstein Filtering: A Sample Selection Method for Robust Distribution Learningcited by 0