Quantitative stability of optimal transport maps and linearization of the 2-Wasserstein space
Proceedings of the Twenty Third International Conference on Artificial Intelligence and Statistics, PMLR 108:3186-3196, 2020.
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 is shown to be bi-Hölder continuous. It enables the direct use of generic supervised and unsupervised learning algorithms on measure data consistently w.r.t. the Wasserstein geometry.