Papers › Manifold learning in Wasserstein space

Manifold learning in Wasserstein space

14 Nov 2023arXiv:2311.08549archive 2025-07-28

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 𝒫_(a.c.)(Ω) with Ω a compact and convex subset of ℝᵈ, metrized with the Wasserstein-2 distance 𝕎. We begin by introducing a construction of submanifolds Λ in 𝒫_(a.c.)(Ω) equipped with metric 𝕎_Λ, 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,𝕎_Λ) can be learned from samples {λᵢ}ᵢ₌₁ᴺ of Λ and pairwise extrinsic Wasserstein distances 𝕎 on 𝒫_(a.c.)(Ω) only. In particular, we show that the metric space (Λ,𝕎_Λ) 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.

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