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
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 . 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. 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.. 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. 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.
As a special case of the Kantorovich Dual Problem when , -Wasserstein metric has the following dual characterization.
Theorem. 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
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.
- ↑ 1.0 1.1 1.2 1.3 Piccoli, B., Rossi, F. Generalized Wasserstein Distance and its Application to Transport Equations with Source. Arch Rational Mech Anal 211, 335–358 (2014). https://doi.org/10.1007/s00205-013-0669-x
- ↑ 2.0 2.1 2.2 Piccoli, B., Rossi, F., Tournus M. A Wasserstein norm for signed measures, with application to nonlocal transport equation with source term. arXiv:1910.05105 (2019).
- ↑ C. Villani, Topics in Optimal Transportation, Chapter 1.
- ↑ Piccoli, B., Rossi, F. Piccoli, B., Rossi, F. On Properties of the Generalized Wasserstein Distance. Arch Rational Mech Anal 222, 1339–1365 (2016). https://doi.org/10.1007/s00205-016-1026-7