Sliced Wasserstein Geodesics and Equivalence Wasserstein and Sliced Wasserstein metrics

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.