# Quadratic-form Optimal Transport

arXiv: https://arxiv.org/abs/2501.04658 · HTML: https://mlsanity.com/paper/2501.04658/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 8 Jan 2025.

## Built on by (most cited first, 3 of 3)

- [Coupled Flow Matching](https://mlsanity.com/paper/2510.23015/) (2025)
- [Dynamic Generalized Gromov-Wasserstein Optimal Transport](https://mlsanity.com/paper/2609.20008/) (2026)
- [A Unified Dual Method for Matching Problems](https://mlsanity.com/paper/2609.39339/) (2026)

## Abstract (arXiv)

We introduce the framework of quadratic-form optimal transport (QOT), whose transport cost has the form iint c,dπ otimesdπ for some coupling π 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.