Convergence of Gaussian-smoothed optimal transport distance with sub-gamma distributions and dependent samples
Convergence of Gaussian-smoothed optimal transport distance with sub-gamma distributions and dependent samples
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发表时间：
2021-02
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通讯作者：
Yixing Zhang;Xiuyuan Cheng;G. Reeves
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作者：
Yixing Zhang;Xiuyuan Cheng;G. Reeves
Goldfeld等人最近提出的高斯平滑的最佳运输（GOT）框架在估计中缩放到高维度，并为熵正则化提供了替代方案。本文提供了融合保证，可在更一般的设置下估算GOT距离。对于高斯平滑的$ p $ -wasserstein距离，$ d $尺寸，我们的结果只需要一刻大于$ d + 2p $。对于亚伽马分布的特殊情况，我们量化了对尺寸$ d $的依赖性，并建立了相对于比例参数的相变。我们还证明了依赖样品的收敛性，仅需要对通过内核空间特征图的协方差测量的样品的成对依赖性条件。我们分析的关键步骤是表明，GOT距离是由内核最大平均差异（MMD）距离的家庭主导的，其核心取决于成本函数以及高斯平滑的量。该见解为GOT框架提供了进一步的解释性，并引入了具有理想属性的一类内核MMD距离。理论结果由数值实验支持。
The Gaussian-smoothed optimal transport (GOT) framework, recently proposed by Goldfeld et al., scales to high dimensions in estimation and provides an alternative to entropy regularization. This paper provides convergence guarantees for estimating the GOT distance under more general settings. For the Gaussian-smoothed $p$-Wasserstein distance in $d$ dimensions, our results require only the existence of a moment greater than $d + 2p$. For the special case of sub-gamma distributions, we quantify the dependence on the dimension $d$ and establish a phase transition with respect to the scale parameter. We also prove convergence for dependent samples, only requiring a condition on the pairwise dependence of the samples measured by the covariance of the feature map of a kernel space. A key step in our analysis is to show that the GOT distance is dominated by a family of kernel maximum mean discrepancy (MMD) distances with a kernel that depends on the cost function as well as the amount of Gaussian smoothing. This insight provides further interpretability for the GOT framework and also introduces a class of kernel MMD distances with desirable properties. The theoretical results are supported by numerical experiments.