# Wasserstein distance for categorical data? Relationship to TVD?

Is the Wasserstein distance applicable for categorical data? e.g. if we have the distribution of different coloured balls in two bags,

bag_1 = {"red": 0.33, "blue": 0.33, "green": 0.33}
bag_2 = {"red": 0.45, "blue": 0.20, "green": 0.35}


we can use measures like TVD to compare the two bags. From my understanding, TVD makes more sense here as TVD works on non-metric spaces while Wasserstein is not (e.g. according to this paper).

The paper does describe a relationship between TVD and Wasserstein but I'm a little confused... and wondering what the benefit of the Wasserstein distance is, especially for non-metric data like these. Would love your comments, thanks!

• What does "TVD" stand for? – Saleh Jan 13 at 7:52
• Total variation distance :) – Anonymous Scientist Jan 15 at 9:50
• Indeed, the Wasserstein distance makes more sense for metric spaces, and this is amongst others simply because it uses the metric between symbols in its definition (contrary to the TVD that only uses the probabilities of the symbols). However, this also means that when you are working on a non metric space like here, it's not even possible to define the Wasserstein distance between those two distrbutions. – William de Vazelhes Mar 2 at 14:13