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 Rd, metrized with the Wasserstein-2
distance W. We begin by introducing a natural construction of submanifolds
Λ of probability measures equipped with metric WΛ, the
geodesic restriction of 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
Rd. We then show how the latent manifold structure of
(Λ,WΛ) can be learned from samples {λi}i=1N of
Λ and pairwise extrinsic Wasserstein distances W only. In particular,
we show that the metric space (Λ,WΛ) can be asymptotically
recovered in the sense of Gromov--Wasserstein from a graph with nodes
{λi}i=1N and edge weights W(λi,λj). 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 {λi}i=1N. The paper closes with some explicit
constructions of submanifolds Λ and numerical examples on the recovery
of tangent spaces through spectral analysis