Kantorovich metric
In mathematics, the Wasserstein (or Vaserstein) metric is a distance function defined between probability distributions on a given metric space M.
Intuitively, if each distribution is viewed as a unit amount of "dirt" piled on M, the metric is the minimum "cost" of turning one pile into the other, which is assumed to be the amount of dirt that needs to be moved times the distance it has to be moved. Because of this analogy, the metric is known in computer science as the earth mover's distance.
The name "Wasserstein distance" was coined by R. L. Dobrushin in 1970, after the Russian mathematician Leonid Vaseršteĭn who introduced the concept in 1969. Most English-language publications use the German spelling "Wasserstein" (attributed to the name "Vaserstein" being of German origin).
Let (M, d) be a metric space for which every probability measure on M is a Radon measure (a so-called Radon space). For p ≥ 1, let Pp(M) denote the collection of all probability measures μ on M with finite pth moment: for some x0 in M,
Then the pth Wasserstein distance between two probability measures μ and ν in Pp(M) is defined as
where Γ(μ, ν) denotes the collection of all measures on M × M with marginals μ and ν on the first and second factors respectively. (The set Γ(μ, ν) is also called the set of all couplings of μ and ν.)
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