Sliced Wasserstein Geodesics and Equivalence Wasserstein and Sliced Wasserstein metrics

TL;DR

This paper introduces a new family of sliced Wasserstein geodesics that differ from standard Wasserstein geodesics, demonstrating the non-equivalence of these metrics and exploring their geometric and computational implications.

Contribution

It presents the first explicit construction of sliced Wasserstein geodesics that are not Wasserstein geodesics, proving the non-equivalence of these metrics in higher dimensions.

Findings

Sliced Wasserstein geodesics can be H"older continuous with respect to Wasserstein geodesics.

The paper provides a direct proof of non-equivalence of the metrics in dimensions greater than 2.

The geometric structure of sliced Wasserstein space differs significantly from Wasserstein space.

Abstract

This paper will introduce a family of sliced Wasserstein geodesics which are not standard Wasserstein geodesics, objects yet to be discovered in the literature. These objects exhibit how the geometric structure of the Sliced Wasserstein space differs from the Wasserstein space, and provides a simple example of how solving the barycenter and gradient flow problems change when moving between these metrics. Some of these geodesics will only be H\"older continuous with respect to the Wasserstein metric and thus will provide a direct proof that Sliced-Wasserstein and regular Wasserstein metrics are not equivalent. Previous proofs of this were done for various cases in [2] and [5]. This paper, not only provides a direct proof, but also fills in gaps showing these metrics not equivalent in dimensions greater than 2.