Quadratic-form optimal transport

• 成果类型: Article; Early Access
• 署名作者: Wang, Ruodu; Zhang, Zhenyuan
• 署名单位: University of Waterloo; Stanford University
• 刊物名称: MATHEMATICAL PROGRAMMING
• ISSN/ISSBN: 0025-5610
• DOI: 10.1007/s10107-025-02282-5
• 发表日期: 2025
• 关键词: regularized optimal transport; Assignment problems

摘要

We introduce the framework of quadratic-form optimal transport (QOT), whose transport cost has the form integral integral cd pi circle times 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.