Manifold learning in Wasserstein space

TL;DR

This paper develops a theoretical framework for manifold learning in the Wasserstein space of probability measures, introducing submanifolds, and demonstrating how to recover their structure and tangent spaces from samples using optimal transport.

Contribution

It introduces a novel construction of submanifolds in Wasserstein space and provides methods to learn their structure and tangent spaces from sample data.

Findings

Submanifolds in Wasserstein space can be constructed with local linearization properties.

The latent manifold structure can be asymptotically recovered from pairwise Wasserstein distances.

Tangent spaces can be approximated via spectral analysis of covariance operators.

Abstract

This paper aims at building the theoretical foundations for manifold learning algorithms in the space of absolutely continuous probability measures $\mathcal{P}_{\mathrm{a.c.}}(\Omega)$ with $\Omega$ 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 $\Lambda$ in $\mathcal{P}_{\mathrm{a.c.}}(\Omega)$ equipped with metric $\mathbb{W}_\Lambda$, the geodesic restriction of $\mathbb{W}$ 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 $\mathbb{R}^d$. We then show how the latent manifold structure of $(\Lambda,\mathbb{W}_{\Lambda})$ can be learned from samples $\{\lambda_i\}_{i=1}^N$ of $\Lambda$ and pairwise extrinsic Wasserstein distances $\mathbb{W}$ on $\mathcal{P}_{\mathrm{a.c.}}(\Omega)$ only. In particular, we show that the metric space $(\Lambda,\mathbb{W}_{\Lambda})$ can be asymptotically recovered in the sense of Gromov--Wasserstein from a graph with nodes $\{\lambda_i\}_{i=1}^N$ and edge weights $W(\lambda_i,\lambda_j)$. In addition, we demonstrate how the tangent space at a sample $\lambda$ can be asymptotically recovered via spectral analysis of a suitable ``covariance operator'' using optimal transport maps from $\lambda$ to sufficiently close and diverse samples $\{\lambda_i\}_{i=1}^N$. The paper closes with some explicit constructions of submanifolds $\Lambda$ and numerical examples on the recovery of tangent spaces through spectral analysis.