1-Wasserstein Metric and Generalizations
Let \(\mathcal{M}\) denote the space of all Radon measures on \(\mathbb{R}^d\) with finite mass. Moreover, let \(\mathcal{M}^p\) denote the space of Radon measures with finite \(p^{th}\) moment, that is, \(\int_{\mathbb{R}^d} |x|^p d\mu(x) < \infty\). Then, for \(\mu,\nu \in \mathcal{M}^p\) with \(|\mu|=|\nu|\) the \(p\)-Wasserstein distance is defined as :\(W_p(\mu,\nu) = ( |\mu| \underset{ \pi \in \Gamma(\mu,\nu) }{ \inf } \int_{\mathbb{R}^d \times \mathbb{R}^d } |x-y|^p d\pi(x,y))^{1/p}\) where \(\Gamma(\mu,\nu)\) denotes the set of all transport plans from \(\mu\) to \(\nu\). Note that, if we restrict \(\mu\) and \(\nu\) to be probability measures, then the \(p\)-Wasserstein distance can be seen as a special case of the Kantorovich Problem, where \(c(x_1,x_2) = |x_1-x_2|^p\). Furthermore, if we let \(\mathcal{P}^p\) denote the space of probability measures on \(\mathbb{R}^d\) with finite \(p^{th}\) moment, then \(W_p\) defines a metric on \(\mathcal{P}^p\).
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 \(\mu\) and \(\nu\). In this case, there is unit cost \(a >0\) for the addition or removal of mass from both \(\mu\) and \(\nu\), whereas the transport cost of mass between \(\mu\) and \(\nu\) stays the same with the classical Kantorovich Problem; multiplied with some rate \(b >0\) .
Note that, under this definition \({W}_p^{a,b}\) defines a metric on \(\mathcal{M}\), and \((\mathcal{M},{W}_p^{a,b})\) is a complete metric space.. Furthermore, this generalization allows one to extend the \(1\)-Wasserstein distance to the signed Radon measures as well. Let us denote the space of all signed Radon measures on \(\mathbb{R}^d\) with finite mass with \(\mathcal{M}^s(\mathbb{R}^d)\).
Moreover, if we let \(||\mu||^{a,b} = \mathbb{W}_1^{a,b}(\mu,0) = W_1^{a,b}(\mu_+ , \mu_- )\), then \((\mathcal{M}^s,||\mu||^{a,b})\) is a normed vector space. However, as opposed to the completeness of \((\mathcal{M},{W}_p^{a,b})\) , \((\mathcal{M}^s,||\mu||^{a,b})\) fails to be a Banach space.
Duality
As a special case of the Kantorovich Dual Problem when \(c(x_1,x_2) = |x_1-x_2|\), \(1\)-Wasserstein metric has the following dual characterization.
- Theorem. Let Lip\((\mathbb{R}^d)\) denote the space of all Lipschitz functions on \(\mathbb{R}^d\), and let \(||\varphi||_{\text{Lip}} = \underset{x\neq y}{\sup} \frac{\varphi(x)-\varphi(y)}{|x-y|}\). Then,
-
\(W_1(\mu,\nu) = \sup \{ \int_{\mathbb{R}^d} \varphi d(\mu-\nu) : \varphi \in \mathcal{C}^0_c, \text{ } ||\varphi||_{\text{Lip}} \leq 1 \}\). In a similar spirit, this duality result can be extended for generalized \(1\)-Wasserstein distances \(W^{1,1}_1\) and \(\mathbb{W}^{1,1}_1\) as well, where \(a\) and \(b\) are taken as 1. For measures with unequal mass, when the additional constraint \(||\varphi||_\infty \leq 1\) is imposed on the test functions, it holds that
:\(W^{1,1}_1(\mu,\nu) = \sup \{ \int_{\mathbb{R}^d} \varphi d(\mu-\nu) : \varphi \in \mathcal{C}^0_c, \text{ } ||\varphi||_\infty \leq 1, \text{ } ||\varphi||_{\text{Lip}} \leq 1 \}\).
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.
:\(\mathbb{W}^{1,1}_1(\mu,\nu) = \sup \{ \int_{\mathbb{R}^d} \varphi d(\mu-\nu) : \varphi \in \mathcal{C}^0_b, \text{ } ||\varphi||_\infty \leq 1, \text{ } ||\varphi||_{\text{Lip}} \leq 1 \}\).
References
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).
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