Abstract The total variation distance between probability measures cannot be bounded by the Wasserstein metric in general. If we consider sufficiently smooth probability densities, however, it is possible to bound the total variation by a power of the Wasserstein distance. We provide a sharp upper bound which depends on the Sobolev norms of the densities involved.

中文翻译：

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Wasserstein 平滑密度总变异的上限

摘要 一般而言，概率度量之间的总变异距离不能受 Wasserstein 度量的限制。但是，如果我们考虑足够平滑的概率密度，则可以通过 Wasserstein 距离的幂来限制总变化。我们提供了一个尖锐的上限，该上限取决于所涉及的密度的 Sobolev 范数。