# Sliced optimal transport: is it a suitable replacement?

arXiv: https://arxiv.org/abs/2311.15874 · HTML: https://mlsanity.com/paper/2311.15874/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 27 Nov 2023.

## Built on by (most cited first, 1 of 1)

- [Convergence Rates for Distribution Matching with Sliced Optimal Transport](https://mlsanity.com/paper/2602.10691/) (2026)

## Abstract (arXiv)

We introduce a one-parameter family of metrics on the space of Borel probability measures on Euclidean space with finite pth moment for 1≤ p <∞, called the textitsliced 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 textitnot bi-Lipschitz equivalent to the classical ones in most cases, and also the spaces are (except for an endpoint case) textitnot 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.