Estimating 2-Sinkhorn Divergence between Gaussian Processes from Finite-Dimensional Marginals

TL;DR

This paper investigates how to estimate the 2-Sinkhorn divergence between Gaussian processes using finite-dimensional marginals, demonstrating almost sure convergence and a dimension-free error rate dependent on regularization and sample size.

Contribution

It provides theoretical guarantees for the convergence and error bounds of Sinkhorn divergence estimation between Gaussian processes from finite marginals.

Findings

Almost sure convergence of the divergence estimate.

Error scales as (\u03b5^{-1} n^{-1/2}) with sample size n.

Error bound is dimension-free.

Abstract

\emph{Optimal Transport} (OT) has emerged as an important computational tool in machine learning and computer vision, providing a geometrical framework for studying probability measures. OT unfortunately suffers from the curse of dimensionality and requires regularization for practical computations, of which the \emph{entropic regularization} is a popular choice, which can be 'unbiased', resulting in a \emph{Sinkhorn divergence}. In this work, we study the convergence of estimating the 2-Sinkhorn divergence between \emph{Gaussian processes} (GPs) using their finite-dimensional marginal distributions. We show almost sure convergence of the divergence when the marginals are sampled according to some base measure. Furthermore, we show that using $n$ marginals the estimation error of the divergence scales in a dimension-free way as $\mathcal{O}\left(\epsilon^ {-1}n^{-\frac{1}{2}}\right)$, where $\epsilon$ is the magnitude of entropic regularization.