Optimal Transport, CycleGAN, and Penalized LS for Unsupervised Learning in Inverse Problems

The penalized least squares (PLS) is a classic method to inverse problems, where a regularization term is added to stabilize the solution. Optimal transport (OT) is another mathematical framework for computer vision tasks that provides means to transport one measure to another at minimal cost. The cycle-consistent generative adversarial network (cycleGAN) is a recent extension of GAN to learn target distributions with less mode collapsing behavior. Although similar in that no supervised training is required, the algorithms look different, so the mathematical relationship between these approaches is not clear. In this article, we provide an important advance to unveil the missing link. Specifically, we reveal that a cycleGAN architecture is originated from formulating a dual OT problem, by using the consistency constraint of PLS as a regularization term in the primal OT problem. This suggests that cycleGAN can be considered stochastic generalization of classical PLS approaches. Our derivation is so general that various types of cycleGAN architectures can be easily derived by merely changing the transport cost. As proofs of concept, we provide three distinct cycleGAN architecture for various biomedical imaging problems, such as accelerated magnetic resonance imaging (MRI), super-resolution microscopy, and low-dose x-ray computed tomography (CT). Experimental results confirm the efficacy and the flexibility of the theory.