  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.
