Quantile optimization in semidiscrete optimal transport

TL;DR

This paper introduces a novel quantile-based objective in semidiscrete optimal transport, providing a complete characterization of optimal plans and efficient computational methods, with applications to geographical partitioning.

Contribution

It is the first to study quantile minimization in optimal transport and develops simulation-based algorithms for the semidiscrete case.

Findings

Derived a complete characterization of optimal plans for quantile objectives.

Developed efficient simulation-based methods for computation.

Revealed a new geometric structure in geographical partitioning applications.

Abstract

Optimal transport is the problem of designing a joint distribution for two random variables with fixed marginals. In virtually the entire literature on this topic, the objective is to minimize expected cost. This paper is the first to study a variant in which the goal is to minimize a quantile of the cost, rather than the mean. For the semidiscrete setting, where one distribution is continuous and the other is discrete, we derive a complete characterization of the optimal transport plan and develop simulation-based methods to efficiently compute it. One particularly novel aspect of our approach is the efficient computation of a tie-breaking rule that preserves marginal distributions. In the context of geographical partitioning problems, the optimal plan is shown to produce a novel geometric structure.