The fast rate of convergence of the smooth adapted Wasserstein distance
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| Format: | Preprint |
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2025
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| _version_ | 1866918400634126336 |
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| author | Larsson, Martin Park, Jonghwa Wiesel, Johannes |
| author_facet | Larsson, Martin Park, Jonghwa Wiesel, Johannes |
| contents | Estimating a $d$-dimensional distribution $μ$ by the empirical measure $\hatμ_n$ of its samples is an important task in probability theory, statistics and machine learning. It is well known that $\mathbb{E}[\mathcal{W}_p(\hatμ_n, μ)]\lesssim n^{-1/d}$ for $d>2p$, where $\mathcal{W}_p$ denotes the $p$-Wasserstein metric. An effective tool to combat this curse of dimensionality is the smooth Wasserstein distance $\mathcal{W}^{(σ)}_p$, which measures the distance between two probability measures after having convolved them with isotropic Gaussian noise $\mathcal{N}(0,σ^2\text{I})$. In this paper we apply this smoothing technique to the adapted Wasserstein distance. We show that the smooth adapted Wasserstein distance $\mathcal{A}\mathcal{W}_p^{(σ)}$ achieves the fast rate of convergence $\mathbb{E}[\mathcal{A}\mathcal{W}_p^{(σ)}(\hatμ_n, μ)]\lesssim n^{-1/2}$, if $μ$ is subgaussian. This result follows from the surprising fact, that any subgaussian measure $μ$ convolved with a Gaussian distribution has locally Lipschitz kernels. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2503_10827 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The fast rate of convergence of the smooth adapted Wasserstein distance Larsson, Martin Park, Jonghwa Wiesel, Johannes Probability Estimating a $d$-dimensional distribution $μ$ by the empirical measure $\hatμ_n$ of its samples is an important task in probability theory, statistics and machine learning. It is well known that $\mathbb{E}[\mathcal{W}_p(\hatμ_n, μ)]\lesssim n^{-1/d}$ for $d>2p$, where $\mathcal{W}_p$ denotes the $p$-Wasserstein metric. An effective tool to combat this curse of dimensionality is the smooth Wasserstein distance $\mathcal{W}^{(σ)}_p$, which measures the distance between two probability measures after having convolved them with isotropic Gaussian noise $\mathcal{N}(0,σ^2\text{I})$. In this paper we apply this smoothing technique to the adapted Wasserstein distance. We show that the smooth adapted Wasserstein distance $\mathcal{A}\mathcal{W}_p^{(σ)}$ achieves the fast rate of convergence $\mathbb{E}[\mathcal{A}\mathcal{W}_p^{(σ)}(\hatμ_n, μ)]\lesssim n^{-1/2}$, if $μ$ is subgaussian. This result follows from the surprising fact, that any subgaussian measure $μ$ convolved with a Gaussian distribution has locally Lipschitz kernels. |
| title | The fast rate of convergence of the smooth adapted Wasserstein distance |
| topic | Probability |
| url | https://arxiv.org/abs/2503.10827 |