Optimal Transport (OT, also known as the Wasserstein distance) is a popular metric for comparing probability distributions and has been successfully used in many machine-learning applications.In the semi-discrete $2$-Wasserstein problem, we wish to compute the cheapest way to transport all the mass from a continuous distribution $\mu$ to a discrete distribution $\nu$ in $\mathbb{R}^d$ for $d\ge 1$, where the cost of transporting unit mass between points $a$ and $b$ is $d(a,b)=||a-b||^2$. When both distributions are discrete, a simple combinatorial framework has been used to find the exact solution (see e.g.

Estimating properties of discrete distributions is a fundamental problem in statistical learning.

Estimating properties of discrete distributions is a fundamental problem in statistical learning.

We consider the fundamental learning problem of estimating properties of distributions over large domains.

In order to cope with this "curse of dimensionality," we

Modern statistical estimation is often performed in a distributed setting where each sample belongs to a single user who shares their data with a central server.

Modern statistical estimation is often performed in a distributed setting where each sample belongs to a single user who shares their data with a central server.

This field has experienced a surge in activity as researchers have produced numerous computational tools to analyze such data sets.