Manifold Learning in Wasserstein Space
Keaton Hamm, Caroline Moosmüller, Bernhard Schmitzer, Matthew Thorpe · SIAM Journal on Mathematical Analysis · 2025
Abstract. This paper aims at building the theoretical foundations for manifold learning algorithms in the space of absolutely continuous probability measures [Formula: see text] with [Formula: see text] a compact and convex subset of [Formula: see text], metrized with the Wasserstein-2 distance [Formula: see text]. We begin by introducing a construction of submanifolds [Formula: see text] in [Formula: see text] equipped with metric [Formula: see text], the geodesic restriction of [Formula: see text] to [Formula: see text]. 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 [Formula: see text]. We then show how the latent manifold structure of [Formula: see text] can be learned from samples [Formula: see text] of [Formula: see text] and pairwise extrinsic Wasserstein distances [Formula: see text] on [Formula: see text] only. In particular, we show that the metric space [Formula: see text] can be asymptotically recovered in the sense of Gromov–Wasserstein from a graph with nodes [Formula: see text] and edge weights [Formula: see text]. In addition, we demonstrate how the tangent space at a sample [Formula: see text] can be asymptotically recovered via spectral analysis of a suitable “covariance operator” using optimal transport maps from [Formula: see text] to sufficiently close and diverse samples [Formula: see text]. The paper closes with some explicit constructions of submanifolds [Formula: see text] and numerical examples on the recovery of tangent spaces through spectral analysis.