Quadratic-form Optimal Transport

Understand

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.