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

• URL: http://arxiv.org/abs/2102.03267v1
• Date: Fri, 5 Feb 2021 16:17:55 GMT
• Title: Estimating 2-Sinkhorn Divergence between Gaussian Processes from Finite-Dimensional Marginals
• Authors: Anton Mallasto
• Abstract summary: We study the convergence of estimating the 2-Sinkhorn divergence between emphGaussian 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.
• Score: 4.416484585765028
• License: http://creativecommons.org/licenses/by/4.0/
• 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.

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