Wasserstein metric
From Wikipedia, the free encyclopedia
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 a German-influenced transliteration of the Cyrillic lettering. Following Ambrosio, Gigli & Savaré, this article acknowledges that the "Vasershtein" spelling is more correct, but that 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.
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[edit] 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 , let denote the collection of all probability measures on M with finite pth moment.
Then the pth Wasserstein distance between two probability measures is defined as
where Γ(μ,ν) denotes the collection of all measures on with marginals μ and ν on the first and second factors respectively.
The above distance is usually denoted Wp(μ,ν) (typically among authors who prefer the "Wasserstein" spelling) or (typically among authors who prefer the "Vasershtein" spelling).
It can be shown that Wp satisfies all the axioms of a metric on .
[edit] Properties
[edit] Dual representation of W1
The following dual representation of W1 is a special case of the 1958 duality theorem of Kantorovich and Rubinstein (1958): when μ,ν have bounded support,
where Lip(f) denotes the minimal Lipschitz constant for f.
Compare this with the definition of the Radon metric:
If the metric d is bounded by some constant C, then
and so convergence in the Radon metric (also known as strong convergence) implies convergence in the Wasserstein metric, but not vice versa.
[edit] Separability and completeness
For any , the metric space is separable, and is complete if (M,d) is separable and complete.
[edit] See also
[edit] References
- Ambrosio, L., Gigli, N. & Savaré, G. (2005). Gradient Flows in Metric Spaces and in the Space of Probability Measures. ETH Zürich, Birkhäuser Verlag, Basel. ISBN 3-764-32428-7.
- Rueshendorff, L., "Wasserstein metric" SpringerLink Encyclopaedia of Mathematics (2001)