TITLE: Numerical approximation to the Wasserstein metric?
QUESTION [9 upvotes]: Are there numerical methods for approximating/calculating the Wasserstein metric in particular cases?
Suppose that $f$ and $g$ are two density functions with the same support. How can I calculate the Wasserstein metric for these two models?

REPLY [5 votes]: Yes, there are. First note that the Wasserstein metric is, after discretization, the solution of a linear program (LP) that can be fed into any LP solver.
Moreover, there are specialized algorithms, try googling for "Earth-Movers-Distance" (mostly Wasserstein-1).
Then is the Benamou-Brennier framework which puts optimal transport for Wasserstein-2 into the framework of fluid-mechanics, see here.
Also you may want to look at the numerical method here for the Wasserstein-1 distance.
In the end, it really depends, what your goals are, and how you discretize your density functions.