An inverse problem in optimal transport on closed Riemannian manifolds

TL;DR

This paper investigates how to recover the Riemannian metric of a closed manifold from optimal transport maps using the squared Riemann distance, establishing conditions for unique determination up to scale.

Contribution

It demonstrates that the Riemannian metric can be uniquely recovered up to a constant factor from optimal transport maps with squared Riemann distance on closed manifolds.

Findings

The metric can be uniquely determined up to a multiplicative constant.

Optimal transport maps encode sufficient information to recover the metric.

The result applies to compact closed Riemannian manifolds.

Abstract

We consider the problem of recovering the Riemannian metric on a compact closed manifold from the optimal transport maps when the underlying cost function is the squared Riemann distance. We show that the metric can be uniquely determined up to a multiplicative constant.