Wasserstein metric

In mathematics, the Wasserstein metric is a metric on the space of probability measures on a given metric space.

The Wasserstein distance is named for the Russian mathematician L.N. Vasershtein. The usage of "Wasserstein" can be attributed to the fact that the name "Vasershtein" is of Germanic origin, and that "Wasserstein" is the German spelling. "Vasershtein", on the other hand, is a transliteration into English of the Russian transliteration of the original German name. The "Wasserstein" spelling is more widespread in English-language publications.

It was first introduced by L.N. Vasershtein in 1969; R.L. Dobrushin coined the term "Wasserstein/Vasershtein distance" in 1970.

Definition
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 &ge; 1, let Pp(M) denote the collection of all probability measures &mu; on M with finite pth moment: for some x0 in M,


 * $$\int_{M} d(x, x_{0})^{p} \, \mathrm{d} \mu (x).$$

Then the pth Wasserstein distance between two probability measures &mu; and &nu; in Pp(M) is defined as


 * $$\left( \inf_{\gamma \in \Gamma (\mu, \nu)} \int_{M \times M} d(x, y)^{p} \, \mathrm{d} \gamma (x, y) \right)^{1/p},$$

where &Gamma;(&mu;, &nu;) denotes the collection of all measures on M &times; M with marginals &mu; and &nu; on the first and second factors respectively.

The above distance is usually denoted Wp(&mu;, &nu;) (typically among authors who prefer the "Wasserstein" spelling) or ℓp(&mu;, &nu;) (typically among authors who prefer the "Vasershtein" spelling). The remainder of this article will use the Wp notation.

The Wasserstein metric may be equivalently defined by


 * $$W_{p} (\mu, \nu)^{p} = \inf \mathbf{E} \big[ | X - Y |^{p} \big],$$

where E[Z] denotes the expected value of a random variable Z and the infimum is taken over all random variables X and Y with distributions &mu; and &nu; respectively.

Metric structure
It can be shown that Wp satisfies all the axioms of a metric on Pp(M). Convergence with respect to W2, for example, is equivalent to the usual weak convergence of measures plus convergence of second moments.

Dual representation of W1
The following dual representation of W1 is a special case of the duality theorem of Kantorovich and Rubinstein (1958): when &mu; and &nu; have bounded support,


 * $$W_{1} (\mu, \nu) = \sup \left\{ \left. \int_{M} f(x) \, \mathrm{d} (\mu - \nu) (x) \right| \mbox{continuous } f : M \to \mathbb{R}, \mathrm{Lip} (f) \leq 1 \right\},$$

where Lip(f) denotes the minimal Lipschitz constant for f.

Compare this with the definition of the Radon metric:


 * $$\rho (\mu, \nu) := \sup \left\{ \left. \int_{M} f(x) \, \mathrm{d} (\mu - \nu) (x) \right| \mbox{continuous } f : M \to [-1, 1] \subsetneq \mathbb{R} \right\}.$$

If the metric d is bounded by some constant C, then


 * $$2 W_{1} (\mu, \nu) \leq C \rho (\mu, \nu),$$

and so convergence in the Radon metric (also known as strong convergence) implies convergence in the Wasserstein metric, but not vice versa.

Separability and completeness
For any p &ge; 1, the metric space (Pp(M), Wp) is separable, and is complete if (M, d) is separable and complete.