# 1-Wasserstein metric and generalizations

Let ${\displaystyle {\mathcal {M}}}$ denote the space of all Radon measures on ${\displaystyle \mathbb {R} ^{d}}$ with finite mass. Moreover, let ${\displaystyle {\mathcal {M}}^{p}}$ denote the space of Radon measures with finite ${\displaystyle p^{th}}$ moment, that is, ${\displaystyle \int _{\mathbb {R} ^{d}}|x|^{p}d\mu (x)<\infty }$. Then, for ${\displaystyle \mu ,\nu \in {\mathcal {M}}^{p}}$ with ${\displaystyle |\mu |=|\nu |}$ the ${\displaystyle p}$-Wasserstein distance is defined as[1]

${\displaystyle W_{p}(\mu ,\nu )=(|\mu |{\underset {\pi \in \Gamma (\mu ,\nu )}{\inf }}\int _{\mathbb {R} ^{d}\times \mathbb {R} ^{d}}|x-y|^{p}d\pi (x,y))^{1/p}}$

where ${\displaystyle \Gamma (\mu ,\nu )}$ denotes the set of all transport plans from ${\displaystyle \mu }$ to ${\displaystyle \nu }$. Note that, if we restrict ${\displaystyle \mu }$ and ${\displaystyle \nu }$ to be probability measures, then the ${\displaystyle p}$-Wasserstein distance can be seen as a special case of the Kantorovich Problem, where ${\displaystyle c(x_{1},x_{2})=|x_{1}-x_{2}|^{p}}$. Furthermore, if we let ${\displaystyle {\mathcal {P}}^{p}}$ denote the space of probability measures on ${\displaystyle \mathbb {R} ^{d}}$ with finite ${\displaystyle p^{th}}$ moment, then ${\displaystyle W_{p}}$ defines a metric on ${\displaystyle {\mathcal {P}}^{p}}$.

# 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 ${\displaystyle \mu }$ and ${\displaystyle \nu }$.[1] In this case, there is unit cost ${\displaystyle a>0}$ for the addition or removal of mass from both ${\displaystyle \mu }$ and ${\displaystyle \nu }$, whereas the transport cost of mass between ${\displaystyle \mu }$ and ${\displaystyle \nu }$ stays the same with the classical Kantorovich Problem; multiplied with some rate ${\displaystyle b>0}$ .

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

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

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

Note that, under this definition ${\displaystyle {W}_{p}^{a,b}}$ defines a metric on ${\displaystyle {\mathcal {M}}}$, and ${\displaystyle ({\mathcal {M}},{W}_{p}^{a,b})}$ is a complete metric space.[1]. Furthermore, this generalization allows one to extend the ${\displaystyle 1}$-Wasserstein distance to the signed Radon measures as well. Let us denote the space of all signed Radon measures on ${\displaystyle \mathbb {R} ^{d}}$ with finite mass with ${\displaystyle {\mathcal {M}}^{s}(\mathbb {R} ^{d})}$.

Definition.[2] For some ${\displaystyle a,b\in \mathbb {R} _{++}}$, the generalized Wasserstein distance for signed measures ${\displaystyle \mathbb {W} _{1}^{a,b}}$ is given by

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

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

Moreover, if we let ${\displaystyle ||\mu ||^{a,b}=\mathbb {W} _{1}^{a,b}(\mu ,0)=W_{1}^{a,b}(\mu _{+},\mu _{-})}$, then ${\displaystyle ({\mathcal {M}}^{s},||\mu ||^{a,b})}$ is a normed vector space. However, as opposed to the completeness of ${\displaystyle ({\mathcal {M}},{W}_{p}^{a,b})}$ , ${\displaystyle ({\mathcal {M}}^{s},||\mu ||^{a,b})}$ fails to be a Banach space.[2]

# Duality

As a special case of the Kantorovich Dual Problem when ${\displaystyle c(x_{1},x_{2})=|x_{1}-x_{2}|}$, ${\displaystyle 1}$-Wasserstein metric has the following dual characterization.

Theorem.[3] Let Lip${\displaystyle (\mathbb {R} ^{d})}$ denote the space of all Lipschitz functions on ${\displaystyle \mathbb {R} ^{d}}$, and let ${\displaystyle ||\varphi ||_{\text{Lip}}={\underset {x\neq y}{\sup }}{\frac {\varphi (x)-\varphi (y)}{|x-y|}}}$. Then,

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

In a similar spirit, this duality result can be extended for generalized ${\displaystyle 1}$-Wasserstein distances ${\displaystyle W_{1}^{1,1}}$ and ${\displaystyle \mathbb {W} _{1}^{1,1}}$ as well, where ${\displaystyle a}$ and ${\displaystyle b}$ are taken as 1. For measures with unequal mass, when the additional constraint ${\displaystyle ||\varphi ||_{\infty }\leq 1}$ is imposed on the test functions, it holds that[4]

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

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.[2]

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