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Pré-publication, Document de travail

Lifted and geometric differentiability of the squared quadratic Wasserstein distance

Aurélien Alfonsi 1 Benjamin Jourdain 1, 2
2 MATHRISK - Mathematical Risk Handling
UPEM - Université Paris-Est Marne-la-Vallée, ENPC - École des Ponts ParisTech, Inria de Paris
Abstract : In this paper, we remark that any optimal coupling for the quadratic Wasserstein distance W22(μ,ν) between two probability measures μ and ν with finite second order moments on Rd is the composition of a martingale coupling with an optimal transport map T. We check the existence of an optimal coupling in which this map gives the unique optimal coupling between μ and T#μ. Next, we prove that σ↦W22(σ,ν) is differentiable at μ in the Lions~\cite{Lions} and the geometric senses iff there is a unique optimal coupling between μ and ν and this coupling is given by a map. Besides, we give a self-contained proof that mere Fréchet differentiability of a law invariant function F on L2(Ω,P;Rd) is enough for the Fréchet differential at X to be a measurable function of X.
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Pré-publication, Document de travail
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https://hal-enpc.archives-ouvertes.fr/hal-01934705
Contributeur : Aurélien Alfonsi <>
Soumis le : lundi 26 novembre 2018 - 10:57:47
Dernière modification le : mercredi 26 février 2020 - 19:06:17

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  • HAL Id : hal-01934705, version 1
  • ARXIV : 1811.07787

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Aurélien Alfonsi, Benjamin Jourdain. Lifted and geometric differentiability of the squared quadratic Wasserstein distance. 2018. ⟨hal-01934705⟩

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