In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, the
Wasserstein distance or
Kantorovich–
Rubinstein metric is a
distance function defined between
probability distributions on a given
metric space
In mathematics, a metric space is a Set (mathematics), set together with a notion of ''distance'' between its Element (mathematics), elements, usually called point (geometry), points. The distance is measured by a function (mathematics), functi ...
. It is named after
Leonid Vaseršteĭn.
Intuitively, if each distribution is viewed as a unit amount of earth (soil) piled on ''
'', 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
Gaspard Monge, Comte de Péluse (; 9 May 1746 – 28 July 1818) was a French mathematician, commonly presented as the inventor of descriptive geometry, (the mathematical basis of) technical drawing, and the father of differential geometry. Dur ...
in 1781. Because of this analogy, the metric is known in
computer science
Computer science is the study of computation, information, and automation. Computer science spans Theoretical computer science, theoretical disciplines (such as algorithms, theory of computation, and information theory) to Applied science, ...
as the
earth mover's distance.
The name "Wasserstein distance" was coined by
R. L. Dobrushin in 1970, after learning of it in the work of
Leonid Vaseršteĭn on Markov processes describing large systems of automata (Russian, 1969). However the metric was first defined by
Leonid Kantorovich in ''The Mathematical Method of Production Planning and Organization'' (Russian original 1939) in the context of optimal transport planning of goods and materials. Some scholars thus encourage use of the terms "Kantorovich metric" and "Kantorovich distance". Most
English-language publications use the
German spelling "Wasserstein" (attributed to the name "Vaseršteĭn" () being of
Yiddish
Yiddish, historically Judeo-German, is a West Germanic language historically spoken by Ashkenazi Jews. It originated in 9th-century Central Europe, and provided the nascent Ashkenazi community with a vernacular based on High German fused with ...
origin).
Definition
Let
be a
metric space
In mathematics, a metric space is a Set (mathematics), set together with a notion of ''distance'' between its Element (mathematics), elements, usually called point (geometry), points. The distance is measured by a function (mathematics), functi ...
that is a
Polish space. For