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Quantitative Stability of Optimal Transport Maps under Variations of the Target Measure

Alex Delalande 1, 2 Quentin Merigot 1
2 DATASHAPE - Understanding the Shape of Data
CRISAM - Inria Sophia Antipolis - Méditerranée , Inria Saclay - Ile de France
Abstract : This work studies the quantitative stability of the quadratic optimal transport map between a fixed probability density ρ and a probability measure µ on R^d , which we denote Tµ. Assuming that the source density ρ is bounded from above and below on a compact convex set, we prove that the map µ → Tµ is bi-Hölder continuous on large families of probability measures, such as the set of probability measures whose moment of order p > d is bounded by some constant. These stability estimates show that the linearized optimal transport metric W2,ρ(µ, ν) = Tµ − Tν L 2 (ρ,R d) is bi-Hölder equivalent to the 2-Wasserstein distance on such sets, justifiying its use in applications.
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Preprints, Working Papers, ...
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https://hal.archives-ouvertes.fr/hal-03164147
Contributor : Alex Delalande Connect in order to contact the contributor
Submitted on : Tuesday, March 9, 2021 - 4:46:01 PM
Last modification on : Friday, April 30, 2021 - 9:52:40 AM
Long-term archiving on: : Thursday, June 10, 2021 - 7:28:42 PM

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  • HAL Id : hal-03164147, version 1
  • ARXIV : 2103.05934

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Alex Delalande, Quentin Merigot. Quantitative Stability of Optimal Transport Maps under Variations of the Target Measure. 2021. ⟨hal-03164147⟩

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