2022/01/03 by Nenna, Luca, Pass, Brendan
#Analysis of PDEs (math.AP) #FOS: Mathematics #Numerical Analysis (math.NA) #Optimization and Control (math.OC) #Probability (math.PR)
paper · doi:10.48550/arxiv.2201.00875
We develop the theory of a metric, which we call the ν-based Wasserstein metric and denote by Wν, on the set of probability measures \mathcal P(X) on a domain X ⊆ ℝm. This metric is based on a slight refinement of the notion of generalized geodesics with respect to a base measure ν and is relevant in particular for the case when ν is singular with respect to m-dimensional Lebesgue measure; it is also closely related to the concept of linearized optimal transport. The ν-based Wasserstein metric is defined in terms of an iterated variational problem involving optimal transport to ν; we also characterize it in terms of integrations of classical Wasserstein distance between the conditional probabilities and through limits of certain multi-marginal optimal transport problems. As we vary the base measure ν, the ν-based Wasserstein metric interpolates between the usual quadratic Wasserstein distance and a metric associated with the uniquely defined generalized geodesics obtained when ν is sufficiently regular. When ν concentrates on a lower dimensional submanifold of ℝm, we prove that the variational problem in the definition of the ν-based Wasserstein distance has a unique solution. We establish geodesic convexity of the usual class of functionals and of the set of source measures μ such that optimal transport between μ and ν satisfies a strengthening of the generalized nestedness condition introduced in \citeMcCannPass20.We finally introduce a slight variant of the dual metric mentioned above in order to prove convergence of an iterative scheme to solve a variational problem arising in game theory.