About the regularity of the discriminator in conditional WGANs

Training of conditional WGANs is usually done by averaging the underlying loss over the condition. Depending on the way this is motivated different constraints on the Lipschitz continuity of the discriminator arise. For the weaker requirement on the regularity there is however so far no mathematically complete justification for the used loss function. 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.