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.