Fast transport optimization for Monge costs on the circle

TL;DR

This paper develops a fast algorithm for optimal transport problems on the circle with Monge costs, leveraging a new theoretical framework based on locally optimal plans and convex cost functions.

Contribution

It introduces a novel notion of locally optimal transport plans inspired by weak KAM theory and applies it to create an efficient approximation algorithm for circular transport problems.

Findings

The algorithm approximates the optimal cost within epsilon in O(N |log epsilon|) time.

Exact solutions are achievable in O(N log M) operations when masses are multiples of 1/M.

The theory characterizes locally optimal plans as conjugate to shifts and convex in a shift parameter.

Abstract

Consider the problem of optimally matching two measures on the circle, or equivalently two periodic measures on the real line, and suppose the cost of matching two points satisfies the Monge condition. We introduce a notion of locally optimal transport plan, motivated by the weak KAM (Aubry-Mather) theory, and show that all locally optimal transport plans are conjugate to shifts and that the cost of a locally optimal transport plan is a convex function of a shift parameter. This theory is applied to a transportation problem arising in image processing: for two sets of point masses on the circle, both of which have the same total mass, find an optimal transport plan with respect to a given cost function satisfying the Monge condition. In the circular case the sorting strategy fails to provide a unique candidate solution and a naive approach requires a quadratic number of operations. For the case of $N$ real-valued point masses we present an O(N |log epsilon|) algorithm that approximates the optimal cost within epsilon; when all masses are integer multiples of 1/M, the algorithm gives an exact solution in O(N log M) operations.