Codes and Expansions (CodEx) Seminar

Keaton Hamm (The University of Texas at Arlington):
Manifold Learning in Wasserstein Space

We study a framework for manifold learning in the Wasserstein space of probability measures with finite second moment. We model submanifolds of this space via bi-Lipschitz embeddings of some abstract parameter manifold. We consider the restriction of the Wasserstein metric to the submanifold and show that it is a metric on the submanifold. From the embedding, we obtain quantitative bounds for the difference of the submanifold metric and the ambient Wasserstein distance in small neighborhoods. Using these results we show that a finite sample of the manifold and its graph shortest path metric converge in the Gromov-Hausdorff sense to the continuous metric on the manifold. We also consider local tangent space approximations to the manifold in the spirit of local PCA.