Distances and Riemannian metrics for multivariate spectral densities

TL;DR

This paper introduces divergence measures and Riemannian metrics for multivariate spectral densities, linking geometric structures with spectral analysis and providing explicit formulas for geodesics and distances.

Contribution

It develops a new class of divergence measures for spectral densities and explores their geometric properties, including explicit geodesic formulas for a specific case.

Findings

Distances are quadratic for infinitesimally close spectra

Explicit geodesic formulas are derived for a particular case

Connection to Fisher-Rao metric is established

Abstract

We first introduce a class of divergence measures between power spectral density matrices. These are derived by comparing the suitability of different models in the context of optimal prediction. Distances between "infinitesimally close" power spectra are quadratic, and hence, they induce a differential-geometric structure. We study the corresponding Riemannian metrics and, for a particular case, provide explicit formulae for the corresponding geodesics and geodesic distances. The close connection between the geometry of power spectra and the geometry of the Fisher-Rao metric is noted.