  We present a novel framework for conditional sampling of probability
measures, using block triangular transport maps. We develop the theoretical
foundations of block triangular transport in a Banach space setting,
establishing general conditions under which conditional sampling can be
achieved and drawing connections between monotone block triangular maps and
optimal transport. Based on this theory, we then introduce a computational
approach, called monotone generative adversarial networks (M-GANs), to learn
suitable block triangular maps. Our algorithm uses only samples from the
underlying joint probability measure and is hence likelihood-free. Numerical
experiments with M-GAN demonstrate accurate sampling of conditional measures in
synthetic examples, Bayesian inverse problems involving ordinary and partial
differential equations, and probabilistic image in-painting.
