In mathematics, the Wasserstein distance or Kantorovich–Rubinstein metric is a distance function defined between probability distributions on a given metric space . 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 in 1781. Because of this analogy, the metric is known in computer 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" (Васерштейн) being of German origin).
Let be a metric space that is a Radon space. For , the Wasserstein -distance between two probability measures and on with finite -moments is
where is the set of all couplings of and . A coupling is a joint probability measure on whose marginals are and on the first and second factors, respectively. That is,
One way to understand the above definition is to consider the optimal transport problem. That is, for a distribution of mass on a space , we wish to transport the mass in such a way that it is transformed into the distribution on the same space; transforming the 'pile of earth' to the pile . This problem only makes sense if the pile to be created has the same mass as the pile to be moved; therefore without loss of generality assume that and are probability distributions containing a total mass of 1.
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In mathematics and economics, transportation theory or transport theory is a name given to the study of optimal transportation and allocation of resources. The problem was formalized by the French mathematician Gaspard Monge in 1781. In the 1920s A.N. Tolstoi was one of the first to study the transportation problem mathematically. In 1930, in the collection Transportation Planning Volume I for the National Commissariat of Transportation of the Soviet Union, he published a paper "Methods of Finding the Minimal Kilometrage in Cargo-transportation in space".
Introduces the course on optimal transport, covering historical background, concepts of push forward measures, transport maps, and the Kantorovich problem.
The first part is devoted to Monge and Kantorovitch problems, discussing the existence and the properties of the optimal plan. The second part introduces the Wasserstein distance on measures and devel
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