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Conference Papers Year : 2020

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

Quentin Merigot
Alex Delalande
Frédéric Chazal

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.

Dates and versions

hal-02315998 , version 1 (15-10-2019)

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Quentin Merigot, Alex Delalande, Frédéric Chazal. Quantitative stability of optimal transport maps and linearization of the 2-Wasserstein space. AISTATS 2020 - 23rd International Conference on Artificial Intelligence and Statistics, Aug 2020, Palermo /Online, Italy. pp.3186-3196. ⟨hal-02315998⟩
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