1-Wasserstein metric and generalizations

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Let denote the space of all Radon measures on with finite mass. Moreover, let denote the space of Radon measures with finite moment, that is, . Then, for with the -Wasserstein distance is defined as[1]

where denotes the set of all transport plans from to . Note that, if we restrict and to be probability measures, then the -Wasserstein distance can be seen as a special case of the Kantorovich Problem, where . Furthermore, if we let denote the space of probability measures on with finite moment, then defines a metric on .

Measures with unequal mass and signed measures

The classical Wasserstein distance can be generalized for measures with unequal mass by allowing the addition and removal of mass to and .[1] In this case, there is unit cost for the addition or removal of mass from both and , whereas the transport cost of mass between and stays the same with the classical Kantorovich Problem; multiplied with some rate .

Definition.[1] For some and , the generalized Wasserstein distance is given by

where is the classical -Wasserstein distance for measures with equal mass.

Note that, under this definition defines a metric on , and is a complete metric space.[1]. Furthermore, this generalization allows one to extend the -Wasserstein distance to the signed Radon measures as well. Let us denote the space of all signed Radon measures on with finite mass with .

Definition.[2] For some , the generalized Wasserstein distance for signed measures is given by

where are any measures in such that and .

Moreover, if we let , then is a normed vector space. However, as opposed to the completeness of , fails to be a Banach space.[2]


As a special case of the Kantorovich Dual Problem when , -Wasserstein metric has the following dual characterization.

Theorem.[3] Let Lip denote the space of all Lipschitz functions on , and let . Then,


In a similar spirit, this duality result can be extended for generalized -Wasserstein distances and as well, where and are taken as 1. For measures with unequal mass, when the additional constraint is imposed on the test functions, it holds that[4]


In terms of the generalized 1-Wasserstein metric for signed measures, when we relax the compact support condition on the test functions we obtain the equivalent duality characterization as follows.[2]