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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.
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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^{[1]} (Russian, 1969). However the metric was first defined by Leonid Kantorovich in *The Mathematical Method of Production Planning and Organization*^{[2]} (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 German origin).

Let be a metric space for which every Borel probability measure on * * is a Radon measure (a so-called Radon space). For , let denote the collection of all probability measures on * * with finite moment, that is, there exists some in * * such that:

The ** Wasserstein distance** between two probability measures * * and in is defined as

where denotes the collection of all measures on with marginals * * and * * on the first and second factors respectively. (The set is also called the set of all **couplings** of * * and * *.)

The above distance is usually denoted (typically among authors who prefer the "Wasserstein" spelling) or (typically among authors who prefer the "Vaserstein" spelling). The remainder of this article will use the * * notation.

The Wasserstein metric may be equivalently defined by

where denotes the expected value of a random variable and the infimum is taken over all joint distributions of the random variables and with marginals * * and respectively.

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. Assume also that there is given some cost function

that gives the cost of transporting a unit mass from the point to the point . A transport plan to move into can be described by a function which gives the amount of mass to move from to . You can imagine the task as the need to move a pile of earth of shape to the hole in the ground of shape such that at the end, both the pile of earth and the hole in the ground completely vanish. In order for this plan to be meaningful, it must satisfy the following properties

That is, that the total mass moved *out of* an infinitesimal region around must be equal to and the total mass moved *into* a region around must be . This is equivalent to the requirement that be a joint probability distribution with marginals and . Thus, the infinitesimal mass transported from to is , and the cost of moving is , following the definition of the cost function. Therefore, the total cost of a transport plan is

The plan is not unique; the optimal transport plan is the plan with the minimal cost out of all possible transport plans. As mentioned, the requirement for a plan to be valid is that it is a joint distribution with marginals and ; letting denote the set of all such measures as in the first section, the cost of the optimal plan is

If the cost of a move is simply the distance between the two points, then the optimal cost is identical to the definition of the distance.

Let and be two degenerate distributions (i.e. Dirac delta distributions) located at points and in . There is only one possible coupling of these two measures, namely the point mass located at . Thus, using the usual absolute value function as the distance function on , for any , the -Wasserstein distance between and is

By similar reasoning, if and are point masses located at points and in , and we use the usual Euclidean norm on as the distance function, then

Let and be two non-degenerate Gaussian measures (i.e. normal distributions) on , with respective expected values and and symmetric positive semi-definite covariance matrices and . Then,^{[3]} with respect to the usual Euclidean norm on , the 2-Wasserstein distance between and is

This result generalises the earlier example of the Wasserstein distance between two point masses (at least in the case ), since a point mass can be regarded as a normal distribution with covariance matrix equal to zero, in which case the trace term disappears and only the term involving the Euclidean distance between the means remains.

Let be probability measures on , and denote their cumulative distribution functions by and . Then the transport problem has an analytic solution: Optimal transport preserves the order of probability mass elements, so the mass at quantile of moves to quantile of . Thus, the -Wasserstein distance between and is

where and are the quantile functions (inverse CDFs). In the case of , a change of variables leads to the formula

- .

The Wasserstein metric is a natural way to compare the probability distributions of two variables *X* and *Y*, where one variable is derived from the other by small, non-uniform perturbations (random or deterministic).

In computer science, for example, the metric *W*_{1} is widely used to compare discrete distributions, *e.g.* the color histograms of two digital images; see earth mover's distance for more details.

In their paper 'Wasserstein GAN', Arjovsky et al.^{[4]} use the Wasserstein-1 metric as a way to improve the original framework of Generative Adversarial Networks (GAN), to alleviate the vanishing gradient and the mode collapse issues. The special case of normal distributions is used in a Frechet Inception Distance.

The Wasserstein metric has a formal link with Procrustes analysis, with application to chirality measures,^{[5]} and to shape analysis.^{[6]}

In computational biology, Wasserstein metric can be used to compare between persistence diagrams of cytometry datasets.^{[7]}

The Wasserstein metric also has been used in inverse problems in geophysics.^{[8]}

It can be shown that *W*_{p} satisfies all the axioms of a metric on **P**_{p}(*M*). Furthermore, convergence with respect to *W*_{p} is equivalent to the usual weak convergence of measures plus convergence of the first *p*th moments.^{[9]}

The following dual representation of *W*_{1} is a special case of the duality theorem of Kantorovich and Rubinstein (1958): when *μ* and *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 (identical to **total variation convergence** when *M* is a Polish space) implies convergence in the Wasserstein metric, but not vice versa.

The following is an intuitive proof which skips over technical points. A fully rigorous proof is found in.^{[10]}

**Discrete case**: When is discrete, solving for the 1-Wasserstein distance is a problem in linear programming:

By carefully writing the above equations as matrix equations, we obtain its dual problem^{[11]}:

For the general case, the dual problem is found by converting sums to integrals:

Suppose you want to ship some coal from mines, distributed as , to factories, distributed as . The cost function of transport is . Now a shipper comes and offers to do the transport for you. You would pay him per coal for loading the coal at , and pay him per coal for unloading the coal at .

For you to accept the deal, the price schedule must satisfy . The Kantorovich duality states that the shipper can make a price schedule that makes you pay almost as much as you would ship yourself.

This result can be pressed further to yield:

**Theorem** (Kantorovich-Rubenstein duality) — When the probability space is a metric space, then
for any fixed ,

It suffices to prove the case of . Start with

Thus,

The two infimal convolution steps are visually clear when the probability space is .

For notational convenience, let denote the infimal convolution operation.

For the first step, where we used , plot out the curve of , then at each point, draw a cone of slope 1, and take the lower envelope of the cones as , as shown in the diagram, then cannot increase with slope larger than 1. Thus all its secants have slope .

For the second step, picture the infimal convolution , then if all secants of have slope at most 1, then the lower envelope of are just the cone-apices themselves, thus .

**1D Example**. When both are distributions on , then integration by parts give

Benamou & Brenier found a dual representation of by fluid mechanics, which allows efficient solution by convex optimization.^{[13]}^{[14]}

Given two probability distributions on with density , then

Under suitable assumptions, the Wasserstein distance of order two is Lipschitz equivalent to a negative-order homogeneous Sobolev norm.^{[15]} More precisely, if we take to be a connected Riemannian manifold equipped with a positive measure , then we may define for the seminorm

and for a signed measure on the dual norm

Then any two probability measures and on satisfy the upper bound

In the other direction, if and each have densities with respect to the standard volume measure on that are both bounded above some , and has non-negative Ricci curvature, then

For any *p* ≥ 1, the metric space (**P**_{p}(*M*), *W*_{p}) is separable, and is complete if (*M*, *d*) is separable and complete.^{[16]}

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