Quantitative stability of optimal transport maps and linearization of the 2-Wasserstein space
This work addresses the challenge of applying standard learning algorithms to measure data by linearizing the Wasserstein space, which is incremental as it builds on existing optimal transport theory.
The paper tackles the problem of embedding probability measures into a Hilbert space using optimal transport maps from a reference density, which linearizes the 2-Wasserstein space and enables machine learning algorithms on measure data. The main result shows that this embedding is (bi-)Hölder continuous under specific conditions, providing dimension-independent stability for optimal transport maps.
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.