In this talk, we propose a variant of ISOMAP we call Wasserstein Isometric Mapping (Wassmap), a parameter-free nonlinear dimensionality reduction technique that provides solutions to some drawbacks in existing global nonlinear dimensionality reduction algorithms in imaging applications. Wassmap represents images via probability measures in Wasserstein space, then uses pairwise quadratic Wasserstein distances

between the associated measures to produce a low-dimensional, approximately isometric embedding.

We show that the algorithm is able to exactly recover parameters of some image manifolds including

those generated by translations or dilations of a fixed generating measure. Additionally, we show

that a discrete version of the algorithm retrieves parameters from manifolds generated from discrete

measures by providing a theoretical bridge to transfer recovery results from functional data to discrete

data. Testing of the proposed algorithms on various image data manifolds show that Wassmap yields

good embeddings compared with other global techniques.