Estimating 2-Sinkhorn Divergence between Gaussian Processes from Finite-Dimensional Marginals
This work provides a theoretical understanding of estimating Sinkhorn divergence for Gaussian processes, which is relevant for researchers working with optimal transport and Gaussian processes.
This paper investigates the estimation of 2-Sinkhorn divergence between Gaussian processes using finite-dimensional marginal distributions. It demonstrates almost sure convergence of the divergence when marginals are sampled according to a base measure, achieving an estimation error scaling as O(ε^(-1)n^(-1/2)) with n marginals.
\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 $\mathcal{O}\left(ε^ {-1}n^{-\frac{1}{2}}\right)$, where $ε$ is the magnitude of entropic regularization.