Wassmap: Wasserstein Isometric Mapping for Image Manifold Learning

TL;DR

Wassmap is a novel nonlinear dimensionality reduction method that uses Wasserstein space to produce isometric embeddings of image manifolds, effectively recovering parameters for various transformations and outperforming existing techniques.

Contribution

This paper introduces Wassmap, a new approach leveraging Wasserstein geometry for image manifold learning, addressing limitations of existing methods with theoretical recovery guarantees.

Findings

Wassmap can exactly recover parameters of certain image manifolds.

The discrete version of Wassmap effectively retrieves parameters from discrete measures.

Wassmap outperforms other global and local embedding techniques in experiments.

Abstract

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.