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.

Wasserstein Generative Adversarial Networks (WGANs) are the popular generative models built on the theory of Optimal Transport (OT) and the Kantorovich duality. Despite the success of WGANs, it is still unclear how well the underlying OT dual solvers approximate the OT cost (Wasserstein-1 distance, W1) and the OT gradient needed to update the generator. In this paper, we address these questions. We construct 1-Lipschitz functions and use them to build ray monotone transport plans. This strategy yields pairs of continuous benchmark distributions with the analytically known OT plan, OT cost and OT gradient in high-dimensional spaces such as spaces of images. We thoroughly evaluate popular WGAN dual form solvers (gradient penalty, spectral normalization, entropic regularization, etc.) using these benchmark pairs. Even though these solvers perform well in WGANs, none of them faithfully compute W1 in high dimensions. Nevertheless, many provide a meaningful approximation of the OT gradient. These observations suggest that these solvers should not be treated as good estimators of W1 but to some extent they indeed can be used in variational problems requiring the minimization of W1.

Generative Adversarial Networks (GANs) are powerful generative models, but suffer from training instability. The recently proposed Wasserstein GAN (WGAN) makes progress toward stable training of GANs, but sometimes can still generate only poor samples or fail to converge. We find that these problems are often due to the use of weight clipping in WGAN to enforce a Lipschitz constraint on the critic, which can lead to undesired behavior. We propose an alternative to clipping weights: penalize the norm of gradient of the critic with respect to its input. Our proposed method performs better than standard WGAN and enables stable training of a wide variety of GAN architectures with almost no hyperparameter tuning, including 101-layer ResNets and language models with continuous generators. We also achieve high quality generations on CIFAR-10 and LSUN bedrooms.

Neural Information Processing Systems http://nips.cc/

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.

We describe how BWGAN can be efficiently implemented.

Neural Information Processing Systems http://nips.cc/

The'Held-out Digit' is the digit In practice, 50,000 generated samples are used to measure the FID of a GAN.

We thank the reviewers for their valuable feedback. We'll modify to include LS-GAN and R-LS-GAN. W-GAN is difficult to analyze. R1&R4-Experiments to validate proposed GAN losses. R1&R4-Experiments to validate proposed GAN losses.

Limited visibility of power distribution network power flows at the low voltage level presents challenges to both distribution network operators from a planning perspective and distribution system operators from a congestion management perspective. Forestalling these challenges through scenario analysis is confounded by the lack of realistic and coherent load data across representative distribution feeders. Load profiling approaches often rely on summarising demand through typical profiles, which oversimplifies the complexity of substation-level operations and limits their applicability in specific power system studies. Sampling methods, and more recently generative models, have attempted to address this through synthesising representative loads from historical exemplars; however, while these approaches can approximate load shapes to a convincing degree of fidelity, the co-behaviour between substations, which ultimately impacts higher voltage level network operation, is often overlooked. This limitation will become even more pronounced with the increasing integration of low-carbon technologies, as estimates of base loads fail to capture load diversity. To address this gap, a Conditional Diffusion model for synthesising daily active and reactive power profiles at the low voltage distribution substation level is proposed. The evaluation of fidelity is demonstrated through conventional metrics capturing temporal and statistical realism, as well as power flow modelling. The results show synthesised load profiles are plausible both independently and as a cohort in a wider power systems context. The Conditional Diffusion model is benchmarked against both naive and state-of-the-art models to demonstrate its effectiveness in producing realistic scenarios on which to base sub-regional power distribution network planning and operations.