Wassmap: Wasserstein Isometric Mapping for Image Manifold Learning
This addresses limitations in existing nonlinear dimensionality reduction methods for imaging applications, though it appears to be an incremental improvement combining Wasserstein metrics with manifold learning.
The authors proposed Wassmap, a nonlinear dimensionality reduction technique for image manifolds that uses Wasserstein distances between probability measures to create approximately isometric embeddings. They demonstrated exact parameter recovery for translation/dilation manifolds and showed competitive performance against existing methods on various image datasets.
In this paper, we propose Wasserstein Isometric Mapping (Wassmap), a 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 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 and local techniques.