Riemannian Distances between Covariance Operators and Gaussian Processes

Abstract

In thisworkwe study several recently formulated Riemannian distances between infinite-dimensional positive definite Hilbert-Schmidt operators in the context of covariance operators associated with functional random processes. Specifically, we focus on the affine-invariant Riemannian and Log-Hilbert-Schmidt distances and the family of Alpha Procrustes distances, which include both the Bures-Wasserstein and Log-Hilbert-Schmidt distances as special cases. In particular, we present finitedimensional approximations of the infinite-dimensional distances and show their convergence to the exact distances. The theoretical formulation is illustrated with numerical experiments on covariance operators of Gaussian processes.