About exchanging expectation and supremum for conditional Wasserstein GANs
This provides a theoretical foundation for more flexible discriminator designs in conditional Wasserstein GANs, but it is incremental as it focuses on clarifying a specific mathematical operation.
The paper addresses the mathematical justification for exchanging supremum and expectation in conditional Wasserstein GANs, enabling the use of discriminators that are Lipschitz-1 in only one argument, which fills a gap in existing literature.
In cases where a Wasserstein GAN depends on a condition the latter is usually handled via an expectation within the loss function. Depending on the way this is motivated, the discriminator is either required to be Lipschitz-1 in both or in only one of its arguments. For the weaker requirement to become usable one needs to exchange a supremum and an expectation. This is a mathematically perilous operation, which is, so far, only partially justified in the literature. This short mathematical note intends to fill this gap and provides the mathematical rationale for discriminators that are only partially Lipschitz-1 for cases where this approach is more appropriate or successful.