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arXiv · 1910.05954

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

Quentin Mérigot·Alex Delalande·Frédéric Chazal
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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-)Hölder continuous, when the reference density is uniform over a convex set, and can be equivalently phrased as a dimension-independent Hölder-stability results for optimal transport maps.

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Quentin Mérigot, Alex Delalande, Frédéric Chazal. 2019-10-14. Quantitative stability of optimal transport maps and linearization of the 2-Wasserstein space. https://arxiv.org/abs/1910.05954

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