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

2023/04/28 by Chhachhi, Saurab, Teng, Fei · 3 citations
#Applications (stat.AP) #FOS: Computer and information sciences #FOS: Mathematics #Machine Learning (stat.ML) #Probability (math.PR)

paper · doi:10.48550/arxiv.2304.14869

Abstract

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.

Cited by

• Subbaths Cluster Dynamical Mean-Field Theory (SB-CDMFT)
• Optimal Adjustment and Combination of Independent Discrete p-Values
• Uncertainty-driven Embedding Convolution

Related

• Limit laws of the empirical Wasserstein distance: Gaussian distributions
• Sublinear Algorithms for Wasserstein and Total Variation Distances: Applications to Fairness and Privacy Auditing
• Why the 1-Wasserstein distance is the area between the two marginal CDFs
• Distances between nested densities and a measure of the impact of the\n prior in Bayesian statistics
• Wasserstein Statistics in One-dimensional Location-Scale Model