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

Alex Delalande (1, 2) , Quentin Merigot (1)

1 LMO - Laboratoire de Mathématiques d'Orsay
2 DATASHAPE - Understanding the Shape of Data

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.

Domaines: Géométrie métrique [math.MG]; Analyse fonctionnelle [math.FA]

Alex Delalande, Quentin Merigot. Quantitative Stability of Optimal Transport Maps under Variations of the Target Measure. 2021. ⟨hal-03164147v1⟩