The square of energy Distance between CDFs $F$ and $G$, of $X$ and $Y$ resp., is defined here as $$d^2(F, G) = E||X-Y|| - E||X-X'|| - E||Y-Y'||$$ where $(X, X')$ and $(Y, Y')$ are IID pairs.
I am looking for an explicit form for multivariate Gaussians $F = N(0, C_F)$ and $G = N(0, C_G)$, in terms of $C_F$ and $C_G$
I found something similar (this paper's Theorem 2.2), which gives a result for the Wasserstein metric under some other distance function called the Riemannian metric(in the above formulation we used Euclidean distance). However, the paper's theory was too complicated for me to follow.
Any help in this direction would be really appreciated.
Thanks!
