Fast and Accurate Approximations of the Optimal Transport in Semi-Discrete and Discrete Settings

Given a d-dimensional continuous (resp. discrete) probability distribution mu and a discrete distribution nu, the semi-discrete (resp. discrete) optimal transport (OT) problem asks for computing a minimum-cost plan to transport mass from mu to nu; we assume n to be the number of points in the support of the discrete distributions. In this paper, we present three approximation algorithms for the OT problem with strong provable guarantees. (i) Additive approximation for semi-discrete OT: For any parameter epsilon > 0, we present an algorithm that computes a semi-discrete transport plan (tau) over tilde with cost cent((tau) over tilde) <= cent(tau*) + epsilon in n(O(d)) log D/epsilon time; here, tau* is the optimal transport plan, D is the diameter of the supports of mu and nu, and we assume we have access to an oracle that outputs the mass of mu inside a constant-complexity region in O(1) time. Our algorithm works for several ground distances including the L-p-norm and the squared-Euclidean distance. (ii) Relative approximation for semi-discrete OT: For any parameter epsilon > 0, we present an algorithm that computes a semi-discrete transport plan (tau) over tilde with cost cent((tau) over tilde) <= (1 + epsilon)cent(tau*) in n log(n) . (epsilon(-1) log log n)(O(d)) expected time; here, tau* is the optimal transport plan, and we assume we have access to an oracle that outputs the mass of mu inside an orthogonal box in O(1) time, and the ground distance is any L-p norm. (iii) Relative approximation for discrete OT: For any parameter epsilon > 0, we present a Monte-Carlo algorithm that computes a transport plan (tau) over tilde with an expected cost cent((tau) over tilde) <= (1 + epsilon)cent(tau*) under any L-p norm in n log(n) . (epsilon(-1) log log n)(O(d)) time; here, tau* is an optimal transport plan and we assume that the spread of the supports of mu and nu is polynomially bounded.