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

arXiv: https://arxiv.org/abs/1804.10556 · HTML: https://mlsanity.com/paper/1804.10556/

## Key facts

- Cited by 41 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 27 Apr 2018.

## Built on by (most cited first, 30 of 41)

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- [Towards Understanding the Dynamics of Gaussian-Stein Variational Gradient Descent](https://mlsanity.com/paper/2305.14076/) (2023)
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- [Nonlinear Sufficient Dimension Reduction for Distribution-on-Distribution Regression](https://mlsanity.com/paper/2207.04613/) (2022)
- [Robust Estimation under the Wasserstein Distance](https://mlsanity.com/paper/2302.01237/) (2023)
- [Functional optimal transport: map estimation and domain adaptation for functional data](https://mlsanity.com/paper/2102.03895/) (2021)
- [Shedding a PAC-Bayesian Light on Adaptive Sliced-Wasserstein Distances](https://mlsanity.com/paper/2206.03230/) (2022)
- [A Framework for Fast and Stable Representations of Multiparameter Persistent Homology Decompositions](https://mlsanity.com/paper/2306.11170/) (2023)
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- [Online Estimation and Optimization of Utility-Based Shortfall Risk](https://mlsanity.com/paper/2111.08805/) (2021)
- [Bounding Wasserstein distance with couplings](https://mlsanity.com/paper/2112.03152/) (2021)
- [Robust Distribution Learning with Local and Global Adversarial Corruptions](https://mlsanity.com/paper/2406.06509/) (2024)
- [Statistical Regeneration Guarantees of the Wasserstein Autoencoder with Latent Space Consistency](https://mlsanity.com/paper/2110.03995/) (2021)
- [Simultaneously Solving Infinitely Many LQ Mean Field Games In Hilbert Spaces: The Power of Neural Operators](https://mlsanity.com/paper/2510.20017/) (2025)
- [Inference via robust optimal transportation: theory and methods](https://mlsanity.com/paper/2301.06297/) (2023)
- [Private Evolution Converges](https://mlsanity.com/paper/2506.08312/) (2025)
- [Limit Distribution Theory for the Smooth 1-Wasserstein Distance with Applications](https://mlsanity.com/paper/2107.13494/) (2021)
- [Strong posterior contraction rates via Wasserstein dynamics](https://mlsanity.com/paper/2203.10754/) (2022)

## 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.