Consider the Wasserstein metric of order one $W_1$ (aka the Earth Movers Distance). I would like to know whether it is possible to link $W_1$ and relative entropy and what this would mean intuitively. I can't find it anymore, but if I am not mistaken the following holds true for some constant $C$ $$ W_1(\mu, \nu)\le \sqrt{C\cdot \text{KL}(\nu ||\mu)}, $$

where KL is the Kullback–Leibler divergence. My first question would be: Is the above-mentioned inequality true? Secondly, how should one interpret this estimation?

  • $\begingroup$ I am looking for inequalities relating W_p to KL (ideally in the direction opposite to yours), so if you find references please answer your own question and include them here. Thanks $\endgroup$ – Coca Sep 21 '19 at 17:25
  • $\begingroup$ You might want to have a look at this paper >>> arxiv.org/abs/1709.10219 Didn't read more than the abstract but seems relevant. $\endgroup$ – Di.Ma Jan 23 at 1:53

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