kernel MMD

Let $\mathcal{H}$ be a RKHS with Mercer kernel $K$. The MMD (maximum mean discrepancy) is the integral probability metric over the unit ball $\{f\in \mathcal{H}:\Vert f{\Vert }_{\mathcal{H}}\le 1\}$.

If the data come in pairs $(X_t,Y_t)$, the MMD has the following plug-in estimator

$$\text{MMD}(P_t,Q_t)=\sqrt{\frac{1}{t^2}\sum _{i,j}(K(X_i,X_j)+K(Y_i,Y_j)-2K(X_i,Y_j))},$$

which is a second-order v-statistic. Because the MMD is a convex distributional distance, we can develop confidence sequences for convex functionals for it. The Kernel MMD also plays a big role in two-sample testing.