Density estimation of multivariate samples using Wasserstein distance

Density estimation is a central topic in statistics and a fundamental task of machine learning. In this paper, we present an algorithm for approximating multivariate empirical densities with a piecewise constant distribution defined on a hyperrectangular-shaped partition of the domain. The piecewise constant distribution is constructed through a hierarchical bisection scheme, such that locally, the sample cannot be statistically distinguished from a uniform distribution. The Wasserstein distance has been used to measure the uniformity of the sample data points lying in each partition element. Since the resulting density estimator requires significantly less memory to be stored, it can be used in a situation where the information contained in a multivariate sample needs to be preserved, transferred or analysed.