On the 1-Wasserstein Distance between Location-Scale Distributions and the Effect of Differential Privacy

Understand

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.