# Earth Movers Distance and Maximum Mean Discrepency

By Kantorovich-Rubinstein duality the Earth Movers Distance (EMD)/Wasserstein Metric is equivalent to Maximum Mean Discrepancy (MMD) correct? See here for a more thorough explanation. Why then does the original Kernel MMD paper compare their method to EMD (see the first paragraph in section 7.2) and corresponding lectures by people like Alex Smola seem to imply that they are different metrics. I feel like I'm going crazy because I always assumed that everyone knew that MMD and EMD were the same via the Wasserstein GAN paper (granted Gretton et. al. published way before WGAN was introduced, but I can't find anyone referencing the equality even with a google search now), but now reading these papers I feel like up is down and down is up.

• You mean "equivalent" in the sense that these metrics are integral probability metrics? Then they are equivalent. But Gretton et.al. discuss MMD for Reproducing Kernel Hilbert Spaces (RKHS). In case of EMD the unit ball is made of Lipschitz functions with $L\leq 1$. This is not a RKHS, So it makes sense to compare. Or do you mean "equivalent metrics" in the topological sense? This would not follow from duality and I don't think it is true in general. – g g Jun 14 at 21:21
• So I mean equivalent in the topological sense, although I also mean theoretically as I understand MMD with a RKHS will not give the same answer as the Wasserstein distance with at Lipschitz 1 neural network. I was under the impression that the Wasserstein GAN proof showed that, theoretically (if you could compare every possible Lipschitz <= 1 function), this was equivalent to the EMD. Is my understanding off? If you have a good explanation of these you can make it an answer and I will mark your answer as answered. – www3 Jun 20 at 17:02