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Consider the 2-Wasserstein distance between probability measures $\mu$ and $\nu$ (on $\mathbb{R}^d$), defined as $$ d_{W_2}(\mu,\nu) = \inf_{\gamma} \Big[\int \|x-y\|^2 d\gamma(x,y)\Big]^{1/2} $$ where the $\inf$ is over all couplings $\gamma$ of $\mu$ and $\nu$. Can we define a norm (or something norm-like) on the space of signed measures (or a linear subspace of it containing the cone of probability measures) which gives rise to $W_2$ for probability measures. (I suppose not, but why?)

If not, can we approximate $d_{W_2}$ by a norm?

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    $\begingroup$ There is a dual formulation of Wasserstein distance which makes perfect sense for signed measures, although I don't know what pathological behaviors it might have in that generality. For comparison, this paper discusses the fact that if bounded-Lipschitz distance is extended in the obvious way to signed measures then it fails to be a complete metric: worldscientific.com/doi/abs/10.1142/S0219493712003584 $\endgroup$
    – Mark Meckes
    Feb 12, 2013 at 3:40
  • $\begingroup$ Thanks for the reference. I will think more about the dual version. A more direct approach is also welcome. $\endgroup$
    – passerby51
    Feb 12, 2013 at 4:21
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    $\begingroup$ Related question: mathoverflow.net/questions/120291/… $\endgroup$
    – Dirk
    Feb 12, 2013 at 7:15

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(I guess you missed a square in your definition.)

2-Wasserstein distance doesn't respect the convex structure on measures. Consider two points $x_1 \ne x_2$ and Dirac measures $\delta(x_1), \delta(x_2)$. The measure $\frac{\delta(x_1)+\delta(x_2)}{2}$ is not a midpoint between $\delta(x_1)$ and $\delta(x_2)$.

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  • $\begingroup$ You are right, I missed a square. I guess you are arguing why it can't be approximated by a norm? If I remember correctly, 2-Wasserstein is geodesically convex. So maybe it is possible to approximate it locally by a norm? $\endgroup$
    – passerby51
    Oct 25, 2015 at 17:55
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This paper has several links to relevant literature by Kantorovich & Rubinstein who define an OT inspired norm for signed measures.

https://hal.inria.fr/inria-00072186/en

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For the Earth-mover $W_1$ distance (based upon the cost function $c(x,y)=\|x-y\|$) this is exactly the purpose of this paper. Note however that their construction does not work for the quadratic cost $c(x,y)=\|x-y\|^2$ as you seem to be hoping for.

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    $\begingroup$ Maybe I'm misunderstanding something, but I think the fact that the $W_1$ distance extends to a norm on the space of all signed measures with finite first moment is actually an immediate consequence of the classical Kantorovich-Rubinstein theorem: namely, this theorem shows that the $W_1$-distance is just a restriction of the (metric induced by the) norm on the dual space of the space of Lipschitz continuous functions that vanish at infinity. I guess the paper you linked is up to something more involved. $\endgroup$
    – Jochen Glueck
    May 24, 2021 at 10:56

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