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?


Your Answer

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

Browse other questions tagged or ask your own question.