Multi-projection of unequal dimension optimal transport theory for Generative Adversary Networks

As a major step forward in machine learning, generative adversarial networks (GANs) employ the Wasserstein distance as a metric between the generative distribution and target data distribution, and thus can be viewed as optimal transport (OT) problems to reflect the underlying geometry of the probability distribution. However, the unequal dimensions between the source random distribution and the target data, result in often instability in the training processes, and lack of diversity in the generative images. To resolve the challenges, we propose here a multiple-projection approach, to project the source and target probability measures into multiple different low-dimensional subspaces. Moreover, we show that the original problem can be transformed into a variant multi-marginal OT problem, and we provide the explicit properties of the solutions. In addition, we employ parameterized approximation for the objective, and study the corresponding differentiability and convergence properties, ensuring that the problem can indeed be computed.