  Generative adversarial networks (GANs) have shown great success in
applications such as image generation and inpainting. However, they typically
require large datasets, which are often not available, especially in the
context of prediction tasks such as image segmentation that require labels.
Therefore, methods such as the CycleGAN use more easily available unlabelled
data, but do not offer a way to leverage additional labelled data for improved
performance. To address this shortcoming, we show how to factorise the joint
data distribution into a set of lower-dimensional distributions along with
their dependencies. This allows splitting the discriminator in a GAN into
multiple "sub-discriminators" that can be independently trained from incomplete
observations. Their outputs can be combined to estimate the density ratio
between the joint real and the generator distribution, which enables training
generators as in the original GAN framework. We apply our method to image
generation, image segmentation and audio source separation, and obtain improved
performance over a standard GAN when additional incomplete training examples
are available. For the Cityscapes segmentation task in particular, our method
also improves accuracy by an absolute 14.9% over CycleGAN while using only 25
additional paired examples.
