Integral Probability Metrics (IPMs)

Let $\mathcal{F}$ be a class of measurable functions $\mathcal{X}\to \mathbb{R}$. An IPM measures the largest gap between the expected value of $f$ under the two distributions:

It is 0 exactly when the distributions match. Different classes $\mathcal{F}$ give different metrics: classifiers (basic GANs like GAN, CGAN, DCGAN); 1-Lipschitz functions $\{f:\Vert f{\Vert }_L\le 1\}$ give Wasserstein-1 (WGAN, WGAN-GP, RD-GAN); $\{f:\Vert f{\Vert }_{\infty }\le 1\}$ gives total variation (EBGANs); autoencoder mean error (BEGANs); and the unit ball of an RKHS $\{f:\Vert f{\Vert }_{\mathcal{H}}\le 1\}$ gives MMD (MMDNets).