We provide an exact expressions for the 1-Wasserstein distance between independent location-scale distributions. The expressions are represented using location and scale parameters and special functions such as the standard Gaussian CDF or the Gamma function. Specifically, we find that the 1-Wasserstein distance between independent univariate location-scale distributions is equivalent to the mean of a folded distribution within the same family whose underlying location and scale are equal to the difference of the locations and scales of the original distributions. A new linear upper bound on the 1-Wasserstein distance is presented and the asymptotic bounds of the 1-Wasserstein distance are detailed in the Gaussian case. The effect of differential privacy using the Laplace and Gaussian mechanisms on the 1-Wasserstein distance is studied using the closed-form expressions and bounds.

翻译：我们给出了独立位置尺度分布之间1-Wasserstein距离的精确表达式。这些表达式通过位置和尺度参数以及特殊函数（如标准高斯累积分布函数或伽马函数）来表示。具体而言，我们发现独立单变量位置尺度分布之间的1-Wasserstein距离等价于同一分布族内一个折叠分布的均值，该折叠分布的基础位置和尺度分别等于原始分布的位置差和尺度差。本文提出了1-Wasserstein距离的一个新线性上界，并在高斯情形下详细给出了1-Wasserstein距离的渐近界。最后，利用闭式表达式和界限，研究了采用拉普拉斯机制和高斯机制的差分隐私对1-Wasserstein距离的影响。