Optimal Transport between Gaussian Stationary Processes

TL;DR

This paper explores the optimal transport problem for Gaussian stationary processes, revealing that the solution corresponds to a weighted Hellinger distance between spectral densities and proposing a spectral estimation method based on this insight.

Contribution

It establishes a connection between optimal transport and spectral distances for Gaussian processes and introduces a spectral estimation approach using this distance.

Findings

Solution corresponds to a weighted Hellinger distance

Proposes a spectral estimation method based on this distance

Provides theoretical insights into transport between Gaussian processes

Abstract

We consider the optimal transport problem between multivariate Gaussian stationary stochastic processes. The transportation effort is the variance of the filtered discrepancy process. The main contribution of this technical note is to show that the corresponding solution leads to a weighted Hellinger distance between multivariate power spectral densities. Then, we propose a spectral estimation approach in the case of indirect measurements which is based on this distance.