Wasserstein metric

In mathematics, the Wasserstein or Kantorovich–Rubinstein metric or distance is a distance function defined between probability distributions on a given metric space M {displaystyle M} . In mathematics, the Wasserstein or Kantorovich–Rubinstein metric or distance is a distance function defined between probability distributions on a given metric space M {displaystyle M} . Intuitively, if each distribution is viewed as a unit amount of 'dirt' piled on M {displaystyle 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 mean 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 'Vaseršteĭn' being of German origin). Let ( M , d ) {displaystyle (M,d)} be a metric space for which every probability measure on M {displaystyle M} is a Radon measure (a so-called Radon space). For p ≥ 1 {displaystyle pgeq 1} , let P p ( M ) {displaystyle P_{p}(M)} denote the collection of all probability measures μ {displaystyle mu } on M {displaystyle M} with finite p th {displaystyle p^{ ext{th}}} moment. Then, there exists some x 0 {displaystyle x_{0}} in M {displaystyle M} such that: The p th {displaystyle p^{ ext{th}}} Wasserstein distance between two probability measures μ {displaystyle mu } and ν {displaystyle u } in P p ( M ) {displaystyle P_{p}(M)} is defined as where Γ ( μ , ν ) {displaystyle Gamma (mu , u )} denotes the collection of all measures on M × M {displaystyle M imes M} with marginals μ {displaystyle mu } and ν {displaystyle u } on the first and second factors respectively. (The set Γ ( μ , ν ) {displaystyle Gamma (mu , u )} is also called the set of all couplings of μ {displaystyle mu } and ν {displaystyle u } .) The above distance is usually denoted W p ( μ , ν ) {displaystyle W_{p}(mu , u )} (typically among authors who prefer the 'Wasserstein' spelling) or ℓ p ( μ , ν ) {displaystyle ell _{p}(mu , u )} (typically among authors who prefer the 'Vaserstein' spelling). The remainder of this article will use the W p {displaystyle W_{p}} notation.