February 2024 Central limit theorems for semi-discrete Wasserstein distances
Eustasio del Barrio, Alberto González Sanz, Jean-Michel Loubes
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Bernoulli 30(1): 554-580 (February 2024). DOI: 10.3150/23-BEJ1608

Abstract

We prove a Central Limit Theorem for the empirical optimal transport cost, nmn+m{Tc(Pn,Qm)Tc(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 sup 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 p1. 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|Wpp(P,Qm)Wpp(P,Q)| depending on m and N. Finally, the semi-discrete framework provides a control on the second derivative of the dual formulation, which yields the first central limit theorem for the optimal transport potentials and Laguerre cells. The results are supported by simulations that help to visualize the given limits and bounds. We analyse also the cases where classical bootstrap works.

Funding Statement

The research of Eustasio del Barrio and Alberto González Sanz is partially supported by grant PID2021-128314NB-I00 funded by MCIN/AEI/ 10.13039/501100011033/FEDER, UE. The research of Alberto González Sanz and Jean-Michel Loubes is partially supported by the AI Interdisciplinary Institute ANITI, which is funded by the French “Investing for the Future – PIA3” program under the Grant agreement ANR-19-PI3A-0004.

Acknowledgements

The authors would like to thank Luis-Alberto Rodríguez for showing us the paper Cárcamo, Cuevas and Rodríguez (2020), which is key for the proof of Theorem 2.4, and the anonymous reviewers, for encouraging us to extend the results of the paper and providing us useful references.

Citation

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Eustasio del Barrio. Alberto González Sanz. Jean-Michel Loubes. "Central limit theorems for semi-discrete Wasserstein distances." Bernoulli 30 (1) 554 - 580, February 2024. https://doi.org/10.3150/23-BEJ1608

Information

Received: 1 October 2022; Published: February 2024
First available in Project Euclid: 8 November 2023

MathSciNet: MR4665589
zbMATH: 07788895
Digital Object Identifier: 10.3150/23-BEJ1608

Keywords: central limit theorem , Laguerre cells , Optimal transport , optimal transport potentials , semi-discrete optimal transport

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Vol.30 • No. 1 • February 2024
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