Conditional Sampling With Monotone GANs

We present a new approach for sampling conditional probability measures, enabling consistent uncertainty quantification in supervised learning tasks. We construct a mapping that transforms a reference measure to the measure of the output conditioned on new inputs. The mapping is trained via a modification of generative adversarial networks (GANs), called monotone GANs, that imposes monotonicity and a block triangular structure. We present theoretical guarantees for the consistency of our proposed method, as well as numerical experiments demonstrating the ability of our method to accurately sample conditional measures in applications ranging from inverse problems to image in-painting.