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. 1. Introduction Numerous problems involve the comparison of point clouds, i.e. sets of points that lie in a metric space and for which the spatial distribution is of interest. Seeing the point clouds as discrete probability measures in a metric space, it is natural to compare them using Wasserstein distances defined by the optimal transport theory [37]. These distances have indeed been successfully used in a variety of applications in machine learning [11, 3, 25, 23, 19, 1] and in statistics [39, 12, 8, 35]. In the discrete setting, many efficient algorithms have been proposed to compute or approximate the Wasserstein distances, such as Sinkhorn-Knopp and auction algorithms - see [34] and references therein.