Finite Sample Approximations of Exact and Entropic Wasserstein Distances Between Covariance Operators and Gaussian Processes

This work studies finite sample approximations of the exact and entropic regularized Wasserstein distances between centered Gaussian processes and, more generally, covariance operators functional random processes. We first show that these distances/divergences are fully represented by reproducing kernel Hilbert space (RKHS) cross-covariance associated with corresponding functions. Using this representation, we Sinkhorn divergence two can be consistently efficiently estimated from their normalized finite-dimensional matrices, or alternatively, operators. Consequently, leads to a consistent efficient algorithm for estimating samples generated For fixed regularization parameter, convergence rates {\it dimension-independent} same order as those Hilbert-Schmidt distance. If at least one RKHS is finite-dimensional, obtain dimension-dependent} complexity distance