Separation Results between Fixed-Kernel and Feature-Learning Probability MetricsDownload PDF

May 21, 2021 (edited Oct 26, 2021)NeurIPS 2021 OralReaders: Everyone
  • Keywords: integral probability metric, kernel, feature, separation, distribution, harmonics, Legendre
  • TL;DR: We provide separation results between probability metrics with fixed-kernel and feature-learning discriminators.
  • Abstract: 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.
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  • Code: https://github.com/CDEnrich/separation_ipms
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