A Combinatorial Algorithm for the Semi-Discrete Optimal Transport Problem

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.