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We consider the problem of optimizing the discriminator in generativeadversarial networks (GANs) subject to higher-order gradient regularization. Weshow analytically, via the least-squares (LSGAN) and Wasserstein (WGAN) GANvariants, that the discriminator optimization problem is one of interpolationin $n$-dimensions. The optimal discriminator, derived using variationalCalculus, turns out to be the solution to a partial differential equationinvolving the iterated Laplacian or the polyharmonic operator. The solution isimplementable in closed-form via polyharmonic radial basis function (RBF)interpolation. In view of the polyharmonic connection, we refer to thecorresponding GANs as Poly-LSGAN and Poly-WGAN. Through experimental validationon multivariate Gaussians, we show that implementing the optimal RBFdiscriminator in closed-form, with penalty orders $m \approx\lceil \frac{n}{2}\rceil $, results in superior performance, compared to training GAN witharbitrarily chosen discriminator architectures. We employ the Poly-WGANdiscriminator to model the latent space distribution of the data withencoder-decoder-based GAN flavors such as Wasserstein autoencoders.

Data Interpolants -- That's What Discriminators in Higher-order Gradient-regularized GANs Are