Sliced optimal transport: is it a suitable replacement?

Abstract

We introduce a one-parameter family of metrics on the space of Borel probability measures on Euclidean space with finite $p$th moment for $1\leq p <\infty$, called the $\textit{sliced Monge--Kantorovich metrics}$, which include the sliced Wasserstein and max-sliced Wasserstein metrics. We then show that these are complete, separable metric spaces that are topologically equivalent to the classical Monge--Kantorovich metrics and these metrics have a dual representation. However, we also prove these sliced metrics are $\textit{not}$ bi-Lipschitz equivalent to the classical ones in most cases, and also the spaces are (except for an endpoint case) $\textit{not}$ geodesic. The completeness, duality, and non-geodesicness are new even in the sliced and max-sliced Wasserstein cases, and non bi-Lipschitz equivalence is only known for a few specific cases. In particular this indicates that sliced and max-sliced Wasserstein metrics are not suitable direct replacements for the classical Monge--Kantorovich metrics in problems where the specific metric or geodesic structure are critical.