KantorovichRubinstein

The Kantorovich-Rubinstein duality, also known as the optimal transport duality or the Wasserstein duality, is a fundamental result in the theory of optimal transport. It establishes a connection between the problem of finding the cheapest way to move a distribution of mass from one location to another and a related maximization problem.

In its simplest form, it states that the minimum cost of an optimal transport plan between two probability measures is equal to the maximum difference between the expected values of two corresponding Kantorovich potentials. These potentials are functions that satisfy certain constraints related to the cost function.

The duality is particularly significant because it transforms a potentially complex optimization problem (finding the optimal transport plan) into a more tractable one (finding the optimal potentials). This has wide-ranging applications in various fields, including economics, physics, machine learning, and statistics. For instance, it provides a theoretical foundation for the Wasserstein distance, a metric used to measure the difference between probability distributions, which is crucial in areas like generative modeling and domain adaptation. The development of this duality is largely attributed to the work of Leonid Kantorovich and later refined by Gennadiy Rubinstein.