Frechet Distance

The Frechet distance $D_{FR}$ between two distributions $p(x)$ and $q(y)$ is given by:

$$D_{FR}[p(x)||q(y)]=\sqrt{\underset{\pi (x,y)}{\min }\left[\iint \pi (x,y){\left|x-y\right|}^{2}dxdy\right]}$$

where $\pi (x,y)$ represents the set of joint distributions that are compatible with the marginal distributions $p(x)$ and $q(y)$.

The Frechet distance can also be formulated as a measure of the maximum distance between the cumulative probability curves.