Optimal Transport between Gaussian random fields

TL;DR

This paper explores the optimal transport problem between Gaussian stationary random fields, revealing that the solution involves a weighted Hellinger distance between their spectral densities, which defines a geodesic on the spectral density manifold.

Contribution

It introduces a novel connection between optimal transport and spectral density geometry for Gaussian random fields, including a geodesic characterization.

Findings

Optimal transport corresponds to a weighted Hellinger distance.

The distance defines a geodesic on the spectral density manifold.

Results apply to both aperiodic and periodic Gaussian fields.

Abstract

We consider the optimal transport problem between zero mean Gaussian stationary random fields both in the aperiodic and periodic case. We show that the solution corresponds to a weighted Hellinger distance between the multivariate and multidimensional power spectral densities of the random fields. Then, we show that such a distance defines a geodesic, which depends on the weight function, on the manifold of the multivariate and multidimensional power spectral densities.