Mean convergence rates for Gaussian-smoothed Wasserstein distances and classical Wasserstein distances
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| Format: | Preprint |
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2025
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| _version_ | 1866910009228525568 |
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| author | Cosso, Andrea Martini, Mattia Perelli, Laura |
| author_facet | Cosso, Andrea Martini, Mattia Perelli, Laura |
| contents | We establish upper bounds for the expected $p$-th power of the Gaussian-smoothed $p$-Wasserstein distance between a probability measure $μ$ and the corresponding empirical measure $μ_N$, whenever $μ$ has finite $q$-th moment for some $q>p$. This generalizes recent results that were valid only for $q>2p+2d$. We provide two distinct proofs of such a result. We also investigate the optimality of these bounds by establishing a lower bound of order $N^{-1/2-\varepsilon}$ for a probability measure possessing finite moments of all orders. Finally, we exploit a third upper bound for the Gaussian-smoothed $p$-Wasserstein distance to derive new convergence rates for the classical $p$-Wasserstein distance in the critical regime where $μ$ has finite $p$-th moment but infinite moments of order $q > p$, covering for instance the case of Zygmund classes $L^p(\log L)^α$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2504_17477 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Mean convergence rates for Gaussian-smoothed Wasserstein distances and classical Wasserstein distances Cosso, Andrea Martini, Mattia Perelli, Laura Probability 60B10, 60F25, 60G57 We establish upper bounds for the expected $p$-th power of the Gaussian-smoothed $p$-Wasserstein distance between a probability measure $μ$ and the corresponding empirical measure $μ_N$, whenever $μ$ has finite $q$-th moment for some $q>p$. This generalizes recent results that were valid only for $q>2p+2d$. We provide two distinct proofs of such a result. We also investigate the optimality of these bounds by establishing a lower bound of order $N^{-1/2-\varepsilon}$ for a probability measure possessing finite moments of all orders. Finally, we exploit a third upper bound for the Gaussian-smoothed $p$-Wasserstein distance to derive new convergence rates for the classical $p$-Wasserstein distance in the critical regime where $μ$ has finite $p$-th moment but infinite moments of order $q > p$, covering for instance the case of Zygmund classes $L^p(\log L)^α$. |
| title | Mean convergence rates for Gaussian-smoothed Wasserstein distances and classical Wasserstein distances |
| topic | Probability 60B10, 60F25, 60G57 |
| url | https://arxiv.org/abs/2504.17477 |