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 $b>0$ .

Definition. For some $a,b\in \mathbb {R} _{++}$ and $p\geq 1$, the generalized Wasserstein distance ${W}_{p}^{a,b}$ is given by

${W}_{p}^{a,b}(\mu ,\nu )={\underset {\underset {|{\bar {\mu }}|=|{\bar {\nu }}|}{{\bar {\mu }},{\bar {\nu }}\in {\mathcal {M}}^{p}}}{\inf }}(a(|\mu -{\bar {\mu }}|)+a(|\nu -{\bar {\nu }}|)+b{W}_{p}({\bar {\mu }},{\bar {\nu }}))$

where ${W}_{p}({\bar {\mu }},{\bar {\nu }})$ is the classical $p$-Wasserstein distance for measures with equal mass.

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 ${\mathcal {M}}^{s}(\mathbb {R} ^{d})$.

Definition. For some $a,b\in \mathbb {R} _{++}$, the generalized Wasserstein distance for signed measures $\mathbb {W} _{1}^{a,b}$ is given by

$\mathbb {W} _{1}^{a,b}(\mu ,\nu )=W_{1}^{a,b}(\mu _{+}+\nu _{-},\mu _{-}+\nu _{+})$

where $\mu _{+},\nu _{-},\mu _{-},\nu _{+}$ are any measures in ${\mathcal {M}}(\mathbb {R} ^{d})$ such that $\mu =\mu _{+}-\mu _{-}$ and $\nu =\nu _{+}-\nu _{-}$.

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,

$W_{1}(\mu ,\nu )=\sup\{\int _{\mathbb {R} ^{d}}\varphi d(\mu -\nu ):\varphi \in {\mathcal {C}}_{c}^{0},{\text{ }}||\varphi ||_{\text{Lip}}\leq 1\}$.

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

$W_{1}^{1,1}(\mu ,\nu )=\sup\{\int _{\mathbb {R} ^{d}}\varphi d(\mu -\nu ):\varphi \in {\mathcal {C}}_{c}^{0},{\text{ }}||\varphi ||_{\infty }\leq 1,{\text{ }}||\varphi ||_{\text{Lip}}\leq 1\}$.

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.

$\mathbb {W} _{1}^{1,1}(\mu ,\nu )=\sup\{\int _{\mathbb {R} ^{d}}\varphi d(\mu -\nu ):\varphi \in {\mathcal {C}}_{b}^{0},{\text{ }}||\varphi ||_{\infty }\leq 1,{\text{ }}||\varphi ||_{\text{Lip}}\leq 1\}$.