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.

翻译：我们研究了在生成对抗网络（GAN）中优化受高阶梯度正则化约束的判别器问题。通过最小二乘GAN（LSGAN）和Wasserstein GAN（WGAN）变体，我们分析性地证明了该判别器优化问题本质上是$n$维空间中的插值问题。利用变分法推导出的最优判别器，实际上是涉及迭代拉普拉斯算子或多调和算子的偏微分方程的解。该解可通过多调和径向基函数（RBF）插值以闭式形式实现。基于多调和关联，我们将对应的GAN分别命名为Poly-LSGAN和Poly-WGAN。通过对多元高斯分布的实验验证，我们表明：采用惩罚阶数$m \approx\lceil \frac{n}{2} \rceil$的闭式最优RBF判别器，其性能优于使用任意选择判别器架构训练的GAN。我们进一步将Poly-WGAN判别器应用于编码器-解码器型GAN（如Wasserstein自编码器）中，以对数据的隐空间分布进行建模。