|Publication ahead of print|
|Published online||24 October 2018|
Comparison between W2 distance and Ḣ−1 norm, and Localization of Wasserstein distance★
Institut Élie Cartan de Lorraine, Campus Aiguillettes,
1 boulevard des Aiguillettes, BP 70239,
Vandœuvre-lès-Nancy Cedex, France
* Corresponding author: firstname.lastname@example.org
It is well known that the quadratic Wasserstein distance W2(⋅, ⋅) 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 W2 distance exhibits some localization phenomenon: if μ and ν are measures on ℝn and ϕ: ℝn → ℝ+ is some bump function with compact support, then under mild hypotheses, you can bound above the Wasserstein distance between ϕ ⋅ μ and ϕ ⋅ ν by an explicit multiple of W2(μ, ν).
Mathematics Subject Classification: 49Q20 / 28A75 / 46E35
Key words: Wasserstein distance / homogeneous Sobolev norm / localization
© EDP Sciences, SMAI 2018
Current usage metrics show cumulative count of Article Views (full-text article views including HTML views, PDF and ePub downloads, according to the available data) and Abstracts Views on Vision4Press platform.
Data correspond to usage on the plateform after 2015. The current usage metrics is available 48-96 hours after online publication and is updated daily on week days.
Initial download of the metrics may take a while.