Connes distance and optimal transport

TL;DR

This paper explores the connection between Connes spectral distance in noncommutative geometry and the Wasserstein distance in optimal transport, highlighting their relationship on manifolds and differences on discrete spaces.

Contribution

It clarifies the relationship between spectral and Wasserstein distances, providing examples that illustrate their similarities and differences in various settings.

Findings

Spectral and Wasserstein distances coincide on probability measures on Riemannian manifolds.

On discrete spaces, spectral distance between states differs from Wasserstein distance with spectral cost.

The paper offers insights into the geometric interpretation of noncommutative distances.

Abstract

We give a brief overview on the relation between Connes spectral distance in noncommutative geometry and the Wasserstein distance of order 1 in optimal transport. We first recall how these two distances coincide on the space of probability measures on a Riemannian manifold. Then we work out a simple example on a discrete space, showing that the spectral distance between arbitrary states does not coincide with the Wasserstein distance with cost the spectral distance between pure states.