Distributional Convergence of the Sliced Wasserstein Process
Distributional Convergence of the Sliced Wasserstein Process
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发表时间：
2022-06
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通讯作者：
Jiaqi Xi;Jonathan Niles-Weed
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作者：
Jiaqi Xi;Jonathan Niles-Weed
由于在高维背景下计算Wasserstein距离的统计和计算挑战，机器学习研究人员基于计算度量的一维投影之间的距离定义了修改的Wasserstein距离。对如何汇总这些预计距离的不同选择（平均、随机抽样、最大化）会产生不同的距离，需要不同的统计分析。我们定义了\n {切片Wasserstein过程}，一个随机过程定义的经验Wasserstein距离的经验概率措施的所有一维子空间的投影之间，并证明了这个过程的一致分布极限定理。其结果是，我们得到了一个统一的框架，证明分布极限结果的基础上，所有Wasserstein距离的一维投影。我们说明了这些结果的一些例子中，没有分布的限制是以前已知的。
Motivated by the statistical and computational challenges of computing Wasserstein distances in high-dimensional contexts, machine learning researchers have defined modified Wasserstein distances based on computing distances between one-dimensional projections of the measures. Different choices of how to aggregate these projected distances (averaging, random sampling, maximizing) give rise to different distances, requiring different statistical analyses. We define the \emph{Sliced Wasserstein Process}, a stochastic process defined by the empirical Wasserstein distance between projections of empirical probability measures to all one-dimensional subspaces, and prove a uniform distributional limit theorem for this process. As a result, we obtain a unified framework in which to prove distributional limit results for all Wasserstein distances based on one-dimensional projections. We illustrate these results on a number of examples where no distributional limits were previously known.