Published on Mon Oct 14 2019

Quantitative stability of optimal transport maps and linearization of the 2-Wasserstein space

Quentin Mérigot, Alex Delalande, Frédéric Chazal

This work studies an explicit embedding of the set of probability measures into a Hilbert space. This embedding linearizes to some extent the 2-Wasserstein space. It enables the direct use of generic supervised and unsupervised learning algorithms.

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Abstract

This work studies an explicit embedding of the set of probability measures into a Hilbert space, defined using optimal transport maps from a reference probability density. This embedding linearizes to some extent the 2-Wasserstein space, and enables the direct use of generic supervised and unsupervised learning algorithms on measure data. Our main result is that the embedding is (bi-)H\"older continuous, when the reference density is uniform over a convex set, and can be equivalently phrased as a dimension-independent H\"older-stability results for optimal transport maps.