  We consider the problem of optimizing the discriminator in generative
adversarial networks (GANs) subject to higher-order gradient regularization. We
show analytically, via the least-squares (LSGAN) and Wasserstein (WGAN) GAN
variants, that the discriminator optimization problem is one of interpolation
in $n$-dimensions. The optimal discriminator, derived using variational
Calculus, turns out to be the solution to a partial differential equation
involving the iterated Laplacian or the polyharmonic operator. The solution is
implementable in closed-form via polyharmonic radial basis function (RBF)
interpolation. In view of the polyharmonic connection, we refer to the
corresponding GANs as Poly-LSGAN and Poly-WGAN. Through experimental validation
on multivariate Gaussians, we show that implementing the optimal RBF
discriminator in closed-form, with penalty orders $m \approx\lceil \frac{n}{2}
\rceil $, results in superior performance, compared to training GAN with
arbitrarily chosen discriminator architectures. We employ the Poly-WGAN
discriminator to model the latent space distribution of the data with
encoder-decoder-based GAN flavors such as Wasserstein autoencoders.
