Fast Optimal Transport through Sliced Wasserstein Generalized Geodesics

TL;DR

This paper introduces min-SWGG, a new fast proxy for Wasserstein distance based on one-dimensional projections, providing a computationally efficient method with theoretical guarantees and practical benefits in applications like shape matching and image colorization.

Contribution

The paper proposes min-SWGG, a novel Wasserstein distance proxy leveraging generalized geodesics and closed-form solutions, enabling fast computation and transport plan derivation.

Findings

min-SWGG is an upper bound of Wasserstein distance

It has complexity similar to Sliced-Wasserstein

Empirical results show benefits in shape matching and image colorization

Abstract

Wasserstein distance (WD) and the associated optimal transport plan have been proven useful in many applications where probability measures are at stake. In this paper, we propose a new proxy of the squared WD, coined min-SWGG, that is based on the transport map induced by an optimal one-dimensional projection of the two input distributions. We draw connections between min-SWGG and Wasserstein generalized geodesics in which the pivot measure is supported on a line. We notably provide a new closed form for the exact Wasserstein distance in the particular case of one of the distributions supported on a line allowing us to derive a fast computational scheme that is amenable to gradient descent optimization. We show that min-SWGG is an upper bound of WD and that it has a complexity similar to as Sliced-Wasserstein, with the additional feature of providing an associated transport plan. We also investigate some theoretical properties such as metricity, weak convergence, computational and topological properties. Empirical evidences support the benefits of min-SWGG in various contexts, from gradient flows, shape matching and image colorization, among others.