Quantitative stability of optimal transport maps under variations of the target measure
Alex Delalande, Quentin Mérigot
Abstract
We study the quantitative stability of the quadratic optimal transport map between a fixed probability density ρ and a probability measure μ on Rd, 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‖L2(ρ,Rd) is bi-Hölder equivalent to the 2-Wasserstein distance on such sets, justifying its use in applications.
Topics & Concepts
MathematicsProbability measureBounded functionMeasure (data warehouse)Metric (unit)Stability (learning theory)Wasserstein metricMoment (physics)Regular polygonQuadratic equationSet (abstract data type)Applied mathematicsMathematical analysisCombinatoricsGeometryMachine learningProgramming languageEconomicsClassical mechanicsDatabasePhysicsComputer scienceOperations managementGeometric Analysis and Curvature FlowsGeometry and complex manifoldsPoint processes and geometric inequalities