Quantitative stability of optimal transport maps and linearization of the 2-Wasserstein space
Machine Learning
2022-05-05 v1 Machine Learning
Numerical Analysis
Metric Geometry
Numerical Analysis
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.
Keywords
Cite
@article{arxiv.1910.05954,
title = {Quantitative stability of optimal transport maps and linearization of the 2-Wasserstein space},
author = {Quentin Mérigot and Alex Delalande and Frédéric Chazal},
journal= {arXiv preprint arXiv:1910.05954},
year = {2022}
}
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21 pages