QUESTION
Given
How to pass from the general definition of the Wasserstein distance (let's call it Equation (1)):
to the closed forms with d=1, here below (let's call them Equation (2) and Equation(3))?
Where can I find the details to pass from Equation (1) to Equations (2) and (3)?
Notes:
- It looks like you need to know notions from "Optimal transport" (for example from Villani, C., (2008) Optimal transport, old and new), "measure theory" (i.e. Lebesgue measure and Lebesgue integration) and, maybe, from "multivariate statistics" (?).
- It looks like that from Equation (1), where there are a "transport plan" a.k.a. joint distributions "J", with the corresponding marginal distributions "P" and "Q", and a Lp-norm among "x" and "y", you jump to Equation (2) and (3) where you have, respectively, inverse-CDFs, i.e. "F^{-1}" and "G^{-1}" and the Lp-norm among "X" and "Y". How?