\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.

翻译：\ emph{ optimal Transport} (OT) 已经成为机器学习和计算机视觉中一个重要的计算工具,为研究概率计量提供了几何框架。 不幸的是, OT 受到维度的诅咒, 并要求对实际计算进行规范化, 其中 \ emph{ entoprotic manization} 是流行的选择, 可能是“ 不受偏向的 ”, 导致左( epsilon} { 1}n}\\\ frac{1\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\