Wasserstein distance

Definition

Suppose a metric space $(X,d)$, which is a Polish space (i.e. homeomorphic to a separable complete metric space).

Suppose probability distributions

Then, the $p$-Wasserstein distance is #incomplete

Notes

• it is a distance function between probability distributions on a metric space
• used in transport theory, Monge-Kantorovich problem

References

1. https://en.wikipedia.org/wiki/Wasserstein_metric
2. N. Guillen and R. McCann, “Five lectures on optimal transportation: Geometry, regularity and applications,” Nov. 12, 2010, arXiv: arXiv:1011.2911. doi: 10.48550/arXiv.1011.2911.
3. https://ncatlab.org/nlab/show/Wasserstein+metric