Seonglae ChoWasserstein Space
Wasserstein manifold = the Wasserstein space endowed with (and analyzed via) a Riemannian-geometric structure; it is not a strict manifold in the usual sense, but rather a formal Riemannian manifold.
This work aims to overcome a limitation of existing linearized optimal transport frameworks. Namely, that they only provide accurate approximations on certain flat subsets—by developing learning theory on more general curved submanifolds.
The key construction defines a submanifold via (i) a bi-Lipschitz embedding from a compact smooth Riemannian manifold into , and (ii) deformation velocity fields . An intrinsic distance on is introduced through the geodesic restriction , which restricts geodesics of to lie in . As a main result, Proposition 2.13 proves a global constant bound . Theorem 2.14 establishes a local linearization error bound , yielding in Wasserstein space an analogue of the (1.4)-type bound for Riemannian submanifolds, and showing that the linearized OT approximation is correct at leading order. Finally, it is shown that from samples in and only the pairwise extrinsic Wasserstein distances, one can construct a graph that asymptotically recovers in the sense of Gromov–Wasserstein convergence.
arxiv.org
https://arxiv.org/pdf/2311.08549