Piecewise linear approximation of empirical distributions under a Wasserstein distance constraint

Big data applications and Monte Carlo simulation results can nowadays easily contain data sets in the size of millions of entries. We consider the situation when the information on a large univariate data set or sample needs to be preserved, stored or transferred. We suggest an algorithm to approximate a univariate empirical distribution through a piecewise linear distribution which requires significantly less memory to store. The approximation is chosen in a computationally efficient manner, such that it preserves the mean, and its Wasserstein distance to the empirical distribution is sufficiently small.