Kantorovich-Rubenstein Duality

Definition

$W(P_r,P_{\theta })={\sup }_{||f|{|}_L\le 1}\left[E_{x\sim P_r}[f(x)]-E_{x\sim P_{\theta }}[f(x)]\right]={\sup }_{||f|{|}_L\le 1}\left[\int f(x)d(P_r,P_{\theta })(x)\right]$ where $W(\cdot ,\cdot )$ is Wasserstein Distance, and $||f|{|}_L\le 1$ denotes the 1-Lipschitz Continuity of the function $f$.