Estimation of Riemannian distances between covariance operators and Gaussian processes

TL;DR

This paper investigates two Riemannian distances between covariance operators of Gaussian processes, demonstrating their convergence and consistent estimation from finite samples, with applications in functional data analysis.

Contribution

It introduces a theoretical framework for estimating Riemannian distances between infinite-dimensional covariance operators with dimension-independent convergence.

Findings

Distances converge in Hilbert-Schmidt norm.

Distances can be estimated consistently from finite samples.

Numerical experiments validate the theoretical results.

Abstract

In this work we study two Riemannian distances between infinite-dimensional positive definite Hilbert-Schmidt operators, namely affine-invariant Riemannian and Log-Hilbert-Schmidt distances, in the context of covariance operators associated with functional stochastic processes, in particular Gaussian processes. Our first main results show that both distances converge in the Hilbert-Schmidt norm. Using concentration results for Hilbert space-valued random variables, we then show that both distances can be consistently and efficiently estimated from (i) sample covariance operators, (ii) finite, normalized covariance matrices, and (iii) finite samples generated by the given processes, all with dimension-independent convergence. Our theoretical analysis exploits extensively the methodology of reproducing kernel Hilbert space (RKHS) covariance and cross-covariance operators. The theoretical formulation is illustrated with numerical experiments on covariance operators of Gaussian processes.