Greedy Matching in Optimal Transport with concave cost

TL;DR

This paper studies a simple greedy matching algorithm for optimal transport problems with concave costs, proving its effectiveness theoretically for certain parameters and empirically showing near-optimal results especially with highly concave costs.

Contribution

It introduces and analyzes a greedy matching algorithm for optimal transport with concave costs, providing theoretical guarantees and empirical evidence of its near-optimal performance.

Findings

The greedy algorithm is effective when the cost function is $d(x,y)^p$ with $0 < p < 1/2$.

Empirical results show the algorithm's solutions are close to optimal, especially with more concave costs.

The approach offers a simple, scalable method for certain optimal transport problems.

Abstract

We consider the optimal transport problem between a set of $n$ red points and a set of $n$ blue points subject to a concave cost function such as $c(x,y) = \|x-y\|^{p}$ for $0< p < 1$. Our focus is on a particularly simple matching algorithm: match the closest red and blue point, remove them both and repeat. We prove that it provides good results in any metric space $(X,d)$ when the cost function is $c(x,y) = d(x,y)^{p}$ with $0 < p < 1/2$. Empirically, the algorithm produces results that are remarkably close to optimal -- especially as the cost function gets more concave; this suggests that greedy matching may be a good toy model for Optimal Transport for very concave transport cost.