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Quantitative stability in optimal transport
Abstract
Optimal transport consists in sending a given source probability measure ? to a given target probability measure ? in an optimal way with respect to a certain cost. On bounded subsets of ℝ^d, if the cost is given by the squared Euclidean distance and if ? is absolutely continuous, there exists a unique optimal transport map from ? to ?. Optimal transport has been widely applied across various domains, notably in analysis, probability, statistics, geometry and optimization. In this talk, we provide a quantitative answer to the following stability question: if ? is perturbed, can the optimal transport map from ? to ? change significantly? The answer depends on the properties of the density ?. This question takes its roots in numerical optimal transport, and has found applications to other problems like the statistical estimation of optimal transport maps, the computation of Wasserstein barycenters, and the convergence of Sinkhorn's algorithm. The talk is based on joint works with Quentin Mérigot and Jun Kitagawa.
