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Pré-Publication, Document De Travail Année : 2021

A Central Limit Theorem for Semidiscrete Wasserstein Distances

Un théorème de limite central pour les distances de Wasserstein semi-discrètes

Résumé

We prove a Central Limit Theorem for the empirical optimal transport cost, $\sqrt{\frac{nm}{n+m}}\{\mathcal{T}_c(P_n,Q_m)-\mathcal{T}_c(P,Q)\}$, in the semi discrete case, i.e when the distribution $P$ is supported in $N$ points, but without assumptions on $Q$. We show that the asymptotic distribution is the supremun of a centered Gaussian process, which is Gaussian under some additional conditions on the probability $Q$ and on the cost. Such results imply the central limit theorem for the $p$-Wassertein distance, for $p\geq 1$. This means that, for fixed $N$, the curse of dimensionality is avoided. To better understand the influence of such $N$, we provide bounds of $E|\mathcal{W}_1(P,Q_m)-\mathcal{W}_1(P,Q)|$ depending on $m$ and $N$. Finally, the semidiscrete framework provides a control on the second derivative of the dual formulation, which yields the first central limit theorem for the optimal transport potentials. The results are supported by simulations that help to visualize the given limits and bounds. We analyse also the cases where classical bootstrap works.
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Dates et versions

hal-03232450 , version 1 (21-05-2021)
hal-03232450 , version 2 (13-02-2022)
hal-03232450 , version 3 (16-02-2023)

Identifiants

Citer

Eustasio del Barrio, Alberto González-Sanz, Jean-Michel Loubes. A Central Limit Theorem for Semidiscrete Wasserstein Distances. 2021. ⟨hal-03232450v1⟩
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