Quadratic-form Optimal Transport

TL;DR

This paper introduces quadratic-form optimal transport (QOT), a new framework with unique mathematical properties and explicit solutions for various cost functions, expanding the scope of optimal transport theory.

Contribution

The paper develops the QOT framework, characterizes its properties, and introduces the diamond transport coupling with explicit solutions for a broad class of costs.

Findings

QOT has distinct mathematical structures from classic transport.

Explicit solutions are obtained for several cases, including diamond transport.

QOT encompasses applications like covariance, Kendall's tau, and Gromov--Wasserstein distance.

Abstract

We introduce the framework of quadratic-form optimal transport (QOT), whose transport cost has the form $\iint c\,\mathrm{d}\pi \otimes\mathrm{d}\pi$ for some coupling $\pi$ between two marginals. Interesting examples of quadratic-form transport cost and their optimization include inequality measurement, the variance of a bivariate function, covariance, Kendall's tau, the Gromov--Wasserstein distance, quadratic assignment problems, and quadratic regularization of classic optimal transport. QOT leads to substantially different mathematical structures compared to classic transport problems and many technical challenges. We illustrate the fundamental properties of QOT and provide several cases where explicit solutions are obtained. For a wide class of cost functions, including the rectangular cost functions, the QOT problem is solved by a new coupling called the diamond transport, whose copula is supported on a diamond in the unit square.