Quadratic-form Optimal Transport

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.