Numerical approximation to the Wasserstein metric? 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? 
 A: *

*If you're just looking to approximate the Wasserstein metric then Sinkhorn's algorithm / entropy regularization  is a nice option since it doesn't really restrict you to any particular case. Have a look at the book Computational Optimal Transport by Gabriel Peyré and Marco Cuturi, in particular Section 4 on entropic regularization.


*Now with respect to the densities $f,g$ : Lets assume $f,g$ are densities on $\mathbb{R}^d$ with some compact support. Since Sinkhorn's algorithm works for measures on a finite state space we need to discretise the model - that is define a finite grid over the support of $f,g$ and approximate $f,g$ on this finite grid. Then use Sinkhorn algorithm for this approximation. Note there will be two sources of error, one from the discretisation of the model and the other from the entropic blurring.
A: $\sqrt{\int_{0}^{1}|F^{-1}(t)-G^{-1}(t)|^{2}dt} = \inf E(ξ-η)^{2}$ 
$\int_{0}^{1}|F^{-1}(t)-G^{-1}(t)|dt = \inf E|ξ-η|$
Where infium is taken over all pairs (ξ,η) of random variables with distribution functions F and G. 
