Comparison between W<sub>2</sub> distance and Ḣ<sup>−1</sup> norm, and Localization of Wasserstein distance

Peyre, Rémi

doi:10.1051/cocv/2017050

Comparison between W₂ distance and Ḣ⁻¹ norm, and Localization of Wasserstein distance

Peyre, Rémi

ESAIM: Control, Optimisation and Calculus of Variations, Tome 24 (2018) no. 4, pp. 1489-1501

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Résumé

It is well known that the quadratic Wasserstein distance $W_{2} (\cdot, \cdot)$ is formally equivalent, for infinitesimally small perturbations, to some weighted $H^{- 1}$ homogeneous Sobolev norm. In this article I show that this equivalence can be integrated to get non-asymptotic comparison results between these distances. Then I give an application of these results to prove that the $W_{2}$ distance exhibits some localization phenomenon: if $μ$ and $ν$ are measures on $ℝ^{n}$ and $φ : ℝ^{n} \to ℝ_{+}$ is some bump function with compact support, then under mild hypotheses, you can bound above the Wasserstein distance between $φ \cdot μ$ and $φ \cdot ν$ by an explicit multiple of $W_{2} (μ, ν)$ .

MR Zbl

DOI : 10.1051/cocv/2017050

Classification : 49Q20, 28A75, 46E35
Keywords: Wasserstein distance, homogeneous Sobolev norm, localization

@article{COCV_2018__24_4_1489_0,
     author = {Peyre, R\'emi},
     title = {Comparison between {W\protect\textsubscript{2}} distance and Ḣ\protect\textsuperscript{\ensuremath{-}1} norm, and {Localization} of {Wasserstein} distance},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     pages = {1489--1501},
     year = {2018},
     publisher = {EDP-Sciences},
     volume = {24},
     number = {4},
     doi = {10.1051/cocv/2017050},
     zbl = {1415.49031},
     mrnumber = {3922440},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.1051/cocv/2017050/}
}

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Peyre, Rémi. Comparison between W₂ distance and Ḣ⁻¹ norm, and Localization of Wasserstein distance. ESAIM: Control, Optimisation and Calculus of Variations, Tome 24 (2018) no. 4, pp. 1489-1501. doi: 10.1051/cocv/2017050

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