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 Wasserstein is a surname and can refer to:
* Abraham Wasserstein (1921–1995), a German-born British and Israeli classicist
* Bernard Wasserstein (born 1948), a British historian
* Bruce Wasserstein (1947–2009), an American former investment b ...
distance or
Kantorovich
Leonid Vitalyevich Kantorovich (, ; 19 January 19127 April 1986) was a Soviet mathematician and economist, known for his theory and development of techniques for the optimal allocation of resources. He is regarded as the founder of linear programm ...
–
Rubinstein Rubinstein is a surname of Ashkenazi Jews. It comes from German and Yiddish, where it means "ruby-stone". Notable persons named Rubinstein include:
A–E
* Akiba Rubinstein (1880–1961), Polish chess grandmaster
* Amnon Rubinstein (1931-2024), I ...
metric is a
distance function
In mathematics, a metric space is a set together with a notion of ''distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are a general setting fo ...
defined between
probability distributions
In probability theory and statistics, a probability distribution is a function that gives the probabilities of occurrence of possible events for an experiment. It is a mathematical description of a random phenomenon in terms of its sample spac ...
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
Leonid Nisonovich Vaserstein () is a Russian- American mathematician, currently Professor of Mathematics at Penn State University. His research is focused on algebra and dynamical systems. He is well known for providing a simple proof of the Quil ...
.
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
In computer science, the earth mover's distance (EMD) is a measure of dissimilarity between two frequency distributions, densities, or measures, over a metric space ''D''.
Informally, if the distributions are interpreted as two different ways of ...
.
The name "Wasserstein distance" was coined by
R. L. Dobrushin in 1970, after learning of it in the work of
Leonid Vaseršteĭn
Leonid Nisonovich Vaserstein () is a Russian- American mathematician, currently Professor of Mathematics at Penn State University. His research is focused on algebra and dynamical systems. He is well known for providing a simple proof of the Quil ...
on Markov processes describing large systems of automata (Russian, 1969). However the metric was first defined by
Leonid Kantorovich
Leonid Vitalyevich Kantorovich (, ; 19 January 19127 April 1986) was a Soviet mathematician and economist, known for his theory and development of techniques for the optimal allocation of resources. He is regarded as the founder of linear programm ...
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
German(s) may refer to:
* Germany, the country of the Germans and German things
**Germania (Roman era)
* Germans, citizens of Germany, people of German ancestry, or native speakers of the German language
** For citizenship in Germany, see also Ge ...
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
In the mathematical discipline of general topology, a Polish space is a separable space, separable Completely metrizable space, completely metrizable topological space; that is, a space homeomorphic to a Complete space, complete metric space that h ...
. For