Manifold learning in Wasserstein space

This paper aims at building the theoretical foundations for manifold learning algorithms in the space of absolutely continuous probability measures Pa.c.(Omega) with Omega a compact and convex subset of ℝd, metrized with the Wasserstein-2 distance mathbbW. We begin by introducing a construction of submanifolds Lambda in Pa.c.(Omega) equipped with metric mathbbW_Lambda, the geodesic restriction of mathbbW to Lambda. In contrast to other constructions, these submanifolds are not necessarily flat, but still allow for local linearizations in a similar fashion to Riemannian submanifolds of ℝd. We then show how the latent manifold structure of (Lambda,mathbbWLambda) can be learned from samples λii=1N of Lambda and pairwise extrinsic Wasserstein distances mathbbW on Pa.c.(Omega) only. In particular, we show that the metric space (Lambda,mathbbWLambda) can be asymptotically recovered in the sense of Gromov--Wasserstein from a graph with nodes λii=1N and edge weights W(λi,λj). In addition, we demonstrate how the tangent space at a sample λ can be asymptotically recovered via spectral analysis of a suitable ``covariance operator'' using optimal transport maps from λ to sufficiently close and diverse samples λii=1N. The paper closes with some explicit constructions of submanifolds Lambda and numerical examples on the recovery of tangent spaces through spectral analysis.