Banach Wasserstein GAN (1806.06621v2)

Abstract: Wasserstein Generative Adversarial Networks (WGANs) can be used to generate realistic samples from complicated image distributions. The Wasserstein metric used in WGANs is based on a notion of distance between individual images, which induces a notion of distance between probability distributions of images. So far the community has considered $\ell^2$ as the underlying distance. We generalize the theory of WGAN with gradient penalty to Banach spaces, allowing practitioners to select the features to emphasize in the generator. We further discuss the effect of some particular choices of underlying norms, focusing on Sobolev norms. Finally, we demonstrate a boost in performance for an appropriate choice of norm on CIFAR-10 and CelebA.

• The paper introduces Banach Wasserstein GAN by extending traditional WGANs to arbitrary separable Banach spaces using non-standard distance metrics.
• It replaces standard ℓ2 norms with dual norms and employs gradient penalties to maintain the Lipschitz condition for stable training.
• Experimental results on CIFAR-10 and CelebA demonstrate enhanced inception and FID scores, highlighting the efficacy of selecting targeted image features.

An Overview of Banach Wasserstein GAN

The paper "Banach Wasserstein GAN" introduces a novel approach to the implementation of Wasserstein Generative Adversarial Networks (WGANs) by extending their theoretical framework to separable Banach spaces. This novel conceptualization allows the practitioner to select non-standard distance metrics, potentially optimizing the image generation process by emphasizing specific image features, such as edges or textures, that are more indicative of visually realistic results.

Core Contributions and Methodology

The authors first provide an essential theoretical expansion from the standard $\ell^2$ norm WGANs to WGANs utilizing arbitrary underlying norms defined within Banach spaces. Notably, they frame the procedure in a manner that retains the use of a gradient penalty (GP), a vital component due to its role in enforcing the Lipschitz condition necessary for stable training of GANs.

Key contributions within this work include:

• Introduction of Banach Wasserstein GAN (BWGAN): The extension of WGAN with GP to encompass any separable complete normed space, allowing for the incorporation of alternative norms beyond the typical $\ell^2$. This essentially broadens the functional applicability of WGANs by allowing specific features to dictate the necessary norm.
• Implementation Details and Enhanced Performance: The authors delineate the procedural adaptations necessary for implementing BWGANs and highlight that the primary modification compared to traditional methods involves substituting the $\ell^2$ norm with a dual norm. They provide heuristically derived suggestions for the selection of regularization parameters fostering improved training stability and convergence.
• Performance Validation: Strong numerical results were demonstrated on datasets such as CIFAR-10 and CelebA. Significant performance improvements were observed using spaces like $L^{10}$, achieving an unsupervised inception score of 8.31 on CIFAR-10—a notable milestone for non-progressive growing methods.

Experimental Results and Implications

The experimental evaluation underlines the impact of norm selection on GAN performance. By utilizing various Sobolev and $L^p$ spaces, the researchers observed enhancements in generated image quality metrics—Inception and FID scores—suggesting that emphasizing specific image characteristics can indeed beneficially alter the learning dynamics. This effectively opens a new dimension in GAN design space, encouraging further exploration into domain-specific objective formulations.

The application of BWGAN on image datasets not only demonstrates improvements in quantitative evaluations but also showcases a potential pathway toward more interpretable and semantically aware generative models. As model performance showed notable variation across different norms, this work provides a basis for other researchers to investigate applications outside the domain of image synthesis, potentially extending these concepts to domains where data can be represented in non-standard metric spaces.

Theoretical Considerations and Future Directions

The theoretical groundwork laid out provides a robust foundation for future explorations into the generalization of GANs in metric spaces beyond separable Banach spaces. Moreover, the authors demonstrate that while their intricately constructed regularizer remains effective in Banach spaces, alternative Lipschitz enforcement methods could unlock potential for GAN training in broader metric contexts.

Given the implications for both practical advancements and theoretical insights into the GAN framework, this paper sets the stage for future work to explore advanced metric formulations across a wide spectrum of generative tasks. Particular attention might be given to understanding GAN behavior and structure through the prisms of Sobolev and other advanced function spaces, potentially intertwining this work with advances in variational formulations and optimal transport theory.

In summary, this paper not only expands the theoretical boundaries around Wasserstein GANs but also provides actionable insights into the use of flexible and targeted distance metrics for superior generative modeling in complex data domains.