# Relation Between Wasserstein Distance and Relative Entropy

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?

• 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 – Coca Sep 21 '19 at 17:25
• 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. – Di.Ma Jan 23 at 1:53