The distribution-level view: a two-sample test

The objective is to approximate an unknown data distribution through the generator function. Given i.i.d. observations $x=\{x_1,\dots ,x_m\}$ from $\mathbb{X}$ and $\tilde{x}=\{{\tilde{x}}_1,\dots ,{\tilde{x}}_n\}$ from ${\mathcal{G}}_{\theta }(\mathcal{K})$, how do we decide whether $\mathbb{X}={\mathcal{G}}_{\theta }(\mathcal{K})$?

We use a two-sample test with a discrepancy $\mathcal{E}$: null hypothesis ${\mathcal{H}}_0:\mathbb{X}={\mathcal{G}}_{\theta }(\mathcal{K})$, alternative ${\mathcal{H}}_1:\mathbb{X}\ne {\mathcal{G}}_{\theta }(\mathcal{K})$. Ideally $\mathcal{E}=0$ when they are equal. We design an estimator $\hat{\mathcal{E}}$ with a rejection threshold $r_{\alpha }$: if $\hat{\mathcal{E}}>r_{\alpha }$, conclude the distributions differ.

Seen this way, the discriminator is a learned two-sample test and the generator tries to make the test fail.