  Several works in implicit and explicit generative modeling empirically
observed that feature-learning discriminators outperform fixed-kernel
discriminators in terms of the sample quality of the models. We provide
separation results between probability metrics with fixed-kernel and
feature-learning discriminators using the function classes $\mathcal{F}_2$ and
$\mathcal{F}_1$ respectively, which were developed to study overparametrized
two-layer neural networks. In particular, we construct pairs of distributions
over hyper-spheres that can not be discriminated by fixed kernel
$(\mathcal{F}_2)$ integral probability metric (IPM) and Stein discrepancy (SD)
in high dimensions, but that can be discriminated by their feature learning
($\mathcal{F}_1$) counterparts. To further study the separation we provide
links between the $\mathcal{F}_1$ and $\mathcal{F}_2$ IPMs with sliced
Wasserstein distances. Our work suggests that fixed-kernel discriminators
perform worse than their feature learning counterparts because their
corresponding metrics are weaker.
