Maximum Mean Discrepancy (MMD)

The maximum mean discrepancy (MMD) maps each distribution to its mean in a rich feature space defined by a kernel $k$ (a reproducing kernel Hilbert space), and measures the distance between those means:

with $x,x^{\prime }\sim P$ and $y,y^{\prime }\sim Q$. It is the IPM whose function class is the unit ball of the RKHS. With a characteristic kernel such as the Gaussian $k(x,y)=\exp (-\Vert x-y{\Vert }^2/2{\sigma }^2)$, MMD is zero only when $P=Q$.

MMD is easy to estimate from samples — just average kernel values within and between the two sample sets — and needs no density estimates or adversarial training. It is used in two-sample tests, domain adaptation and MMD-GANs.