Fast and Accurate Approximations of the Optimal Transport in Semi-Discrete and Discrete Settings
Given a $d$-dimensional continuous (resp. discrete) probability distribution $μ$ and a discrete distribution $ν$, the semi-discrete (resp. discrete) Optimal Transport (OT) problem asks for computing a minimum-cost plan to transport mass from $μ$ to $ν$; we assume $n$ to be the size of the support of the discrete distributions, and we assume we have access to an oracle outputting the mass of $μ$ inside a constant-complexity region in $O(1)$ time. In this paper, we present three approximation algorithms for the OT problem. (i) Semi-discrete additive approximation: For any $ε>0$, we present an algorithm that computes a semi-discrete transport plan with $ε$-additive error in $n^{O(d)}\log\frac{C_{\max}}ε$ time; here, $C_{\max}$ is the diameter of the supports of $μ$ and $ν$. (ii) Semi-discrete relative approximation: For any $ε>0$, we present an algorithm that computes a $(1+ε)$-approximate semi-discrete transport plan in $nε^{-O(d)}\log(n)\log^{O(d)}(\log n)$ time; here, we assume the ground distance is any $L_p$ norm. (iii) Discrete relative approximation: For any $ε>0$, we present a Monte-Carlo $(1+ε)$-approximation algorithm that computes a transport plan under any $L_p$ norm in $nε^{-O(d)}\log(n)\log^{O(d)}(\log n)$ time; here, we assume that the spread of the supports of $μ$ and $ν$ is polynomially bounded.