In mathematics, the Wasserstein distance or Kantorovich–Rubinstein metric is a distance function defined between probability distributions on a given metric space M {\displaystyle M} . It is named after Leonid Vaseršteĭn. Intuitively, if each distribution is viewed as a unit amount of earth (soil) 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 earth that needs to be moved times the mean distance it has to be moved. This problem was first formalised by Gaspard Monge in 1781. Because of this analogy, the metric is known in computer science as the earth mover's distance.
Definition Let ( M , d ) {\displaystyle (M,d)} be a metric space that is a Polish space. For p ∈ [ 1 , + ∞ ] {\displaystyle p\in [1,+\infty ]} , the Wasserstein p {\displaystyle p} -distance between two probability measures μ {\displaystyle \mu } and ν {\displaystyle \nu } on M {\displaystyle M} with finite p {\displaystyle p} -moments is
W p ( μ , ν ) = inf γ ∈ Γ ( μ , ν ) ( E ( x , y ) ∼ γ d ( x , y ) p ) 1 p , {\displaystyle W_{p}(\mu ,\nu )=\inf _{\gamma \in \Gamma (\mu ,\nu )}\left(\mathbf {E} _{(x,y)\sim \gamma }d(x,y)^{p}\right)^{\frac {1}{p}},}
where Γ ( μ , ν ) {\displaystyle \Gamma (\mu ,\nu )} is the set of all couplings of μ {\displaystyle \mu } and ν {\displaystyle \nu } ; W ∞ ( μ , ν ) {\displaystyle W_{\infty }(\mu ,\nu )} is defined to be
lim p → + ∞ W p ( μ , ν ) {\displaystyle \lim _{p\rightarrow +\infty }W_{p}(\mu ,\nu )}
and corresponds to a supremum norm. Here, a coupling γ {\displaystyle \gamma } is a joint probability measure on M × M {\displaystyle M\times M} whose marginals are μ {\displaystyle \mu } and ν {\displaystyle \nu } on the first and second factors, respectively. This means that for all measurable A ⊂ M {\displaystyle A\subset M} , it fulfills
γ ( A × M ) = μ ( A ) {\displaystyle \gamma (A\times M)=\mu (A)} and
γ ( M × A ) = ν ( A ) {\displaystyle \gamma (M\times A)=\nu (A)} . The case of p = ∞ {\displaystyle p=\infty } is special: W ∞ ( μ , ν ) = lim p → + ∞ W p ( μ , ν ) = inf γ ∈ Γ ( μ , ν ) γ - e s s u p d ( x , y ) , {\displaystyle W_{\infty }(\mu ,\nu )=\lim _{p\rightarrow +\infty }W_{p}(\mu ,\nu )=\inf _{\gamma \in \Gamma (\mu ,\nu )}\gamma \operatorname {-essup} d(x,y),} where γ - e s s u p d ( x , y ) {\displaystyle \gamma \operatorname {-essup} d(x,y)} denotes the essential supremum of d ( x , y ) {\displaystyle d(x,y)} with respect to measure γ {\displaystyle \gamma } . The metric space (P∞(M), W∞) is complete if (M, d) is separable and complete. Here, P∞ is the space of all probability measures with bounded support.
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