ELeGANt: An Euler-Lagrange Analysis of Wasserstein Generative Adversarial Networks
Keywords: Generative adversarial networks, Poisson PDE, Euler-Lagrange condition, Fourier series, Optimal GAN discriminator
TL;DR: Leveraging the Euler-Lagrange condition, we show that the optimal gradient-regularized WGAN discriminator, given the generator, solves a Poisson PDE, and can be implemented via a Fourier-series approximation.
Abstract: We consider Wasserstein generative adversarial networks (WGAN) with a gradient-norm penalty and analyze the underlying functional optimization problem within a variational setting. The optimal discriminator in this setting is the solution to a Poisson differential equation, and can be obtained in closed form without having to train a neural network. We illustrate this by employing a Fourier-series approximation to solve the Poisson differential equation. Experimental results based on synthesized low-dimensional Gaussian data demonstrate superior convergence behavior of the proposed approach in comparison with the baseline WGAN variants that employ weight-clipping, gradient or Lipschitz penalties on the discriminator. Further, within this setting, the optimal Lagrange multiplier can be computed in closed-form, and serves as a proxy for measuring GAN generator convergence. This work is an extended abstract, summarizing Asokan & Seelamantula (2023).
Submission Number: 17
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