Numerical Assessment of Wasserstein and KL Metrics: Case Study on Normal and Cauchy Distributions

The objective of this article is to assess properties of Wasserstein metric and Kullback-Leibler divergence when working with samples of normal and Cauchy distributions, varying its dimension and sample size. These distributions are chosen to illustrate the contrasting behaviour of light- and heavy-tailed behaviour. A pair of samples are generated for each random vector. As both samples come from the same distribution, their distance measures are expected to converge to zero as the sample size increases. Verifying whether this convergence actually occurs is one of the objectives of this paper. Each metric is assessed by its convergence behaviour regarding both types of distributions.