Manifold learning in Wasserstein space

Abstract

This paper aims at building the theoretical foundations for manifold learning algorithms in the space of absolutely continuous probability measures $\mathcal{P}_{\mathrm{a.c.}}(Ω)$ with $Ω$ a compact and convex subset of $\mathbb{R}^d$, metrized with the Wasserstein-2 distance $\mathbb{W}$. We begin by introducing a construction of submanifolds $Λ$ in $\mathcal{P}_{\mathrm{a.c.}}(Ω)$ equipped with metric $\mathbb{W}_Λ$, the geodesic restriction of $\mathbb{W}$ to $Λ$. 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 $\mathbb{R}^d$. We then show how the latent manifold structure of $(Λ,\mathbb{W}_Λ)$ can be learned from samples $\{λ_i\}_{i=1}^N$ of $Λ$ and pairwise extrinsic Wasserstein distances $\mathbb{W}$ on $\mathcal{P}_{\mathrm{a.c.}}(Ω)$ only. In particular, we show that the metric space $(Λ,\mathbb{W}_Λ)$ can be asymptotically recovered in the sense of Gromov--Wasserstein from a graph with nodes $\{λ_i\}_{i=1}^N$ and edge weights $W(λ_i,λ_j)$. 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 $\{λ_i\}_{i=1}^N$. The paper closes with some explicit constructions of submanifolds $Λ$ and numerical examples on the recovery of tangent spaces through spectral analysis.