Quantitative stability of optimal transport maps under variations of the target measure

Alex Delalande, Quentin Mérigot

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-H{ö}lder 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-H{ö}lder equivalent to the 2-Wasserstein distance on such sets, justifiying its use in applications.