Manifold learning in Wasserstein space
Keaton Hamm Caroline Moosmüller Bernhard Schmitzer Matthew Thorpe
This paper aims at building the theoretical foundations for manifold learning
algorithms in the space of absolutely continuous probability measures on a
compact and convex subset of $\mathbb{R}^d$, metrized with the Wasserstein-2
distance $W$. We begin by introducing a natural construction of submanifolds
$\Lambda$ of probability measures equipped with metric $W_\Lambda$, the
geodesic restriction of $W$ 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
$(\Lambda,W_{\Lambda})$ can be learned from samples $\{\lambda_i\}_{i=1}^N$ of
$\Lambda$ and pairwise extrinsic Wasserstein distances $W$ only. In particular,
we show that the metric space $(\Lambda,W_{\Lambda})$ 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.