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

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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Dates and versions

hal-03164147 , version 1 (09-03-2021)
hal-03164147 , version 2 (08-03-2023)

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Alex Delalande, Quentin Merigot. Quantitative Stability of Optimal Transport Maps under Variations of the Target Measure. 2023. ⟨hal-03164147v2⟩
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