7
$\begingroup$

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?

$\endgroup$
  • 3
    $\begingroup$ 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 '13 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 '13 at 4:21
  • 1
    $\begingroup$ Related question: mathoverflow.net/questions/120291/… $\endgroup$ – Dirk Feb 12 '13 at 7:15
4
$\begingroup$

(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 $\delta(x_2)$.

$\endgroup$
  • $\begingroup$ 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 '15 at 17:55
1
$\begingroup$

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

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.