Manifold learning in Wasserstein space

Code

• github.com/bernhard-schmitzer/w2manifolds
  OfficialIn papernone★ 2

Abstract

This paper aims at building the theoretical foundations for manifold learning algorithms in the space of absolutely continuous probability measures P_a.c.() with a compact and convex subset of R^d, metrized with the Wasserstein-2 distance W. We begin by introducing a construction of submanifolds in P_a.c.() equipped with metric W_, the geodesic restriction of W to . 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 R^d. We then show how the latent manifold structure of (,W_) can be learned from samples \_i\_i=1^N of and pairwise extrinsic Wasserstein distances W on P_a.c.() only. In particular, we show that the metric space (,W_) can be asymptotically recovered in the sense of Gromov--Wasserstein from a graph with nodes \_i\_i=1^N and edge weights W(_i,_j). In addition, we demonstrate how the tangent space at a sample can be asymptotically recovered via spectral analysis of a suitable ``covariance operator'' using optimal transport maps from to sufficiently close and diverse samples \_i\_i=1^N. The paper closes with some explicit constructions of submanifolds and numerical examples on the recovery of tangent spaces through spectral analysis.

Reproductions