# 2-Wasserstein (optimal transport) and extension to the set of all signed measures

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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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 – Mark Meckes Feb 12 '13 at 3:40
Thanks for the reference. I will think more about the dual version. A more direct approach is also welcome. – passerby51 Feb 12 '13 at 4:21
Related question: mathoverflow.net/questions/120291/… – Dirk Feb 12 '13 at 7:15
@Dirk: Thanks for the link. – passerby51 Feb 12 '13 at 16:37

(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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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? – passerby51 Oct 25 '15 at 17:55

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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Thanks for the pointer. – passerby51 Feb 2 at 5:36