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.

翻译：我们提出了一种新的条件概率测度采样框架，该框架基于分块三角传输映射。在巴拿赫空间框架下，我们发展了分块三角传输的理论基础，建立了实现条件采样的通用条件，并揭示了单调分块三角映射与最优传输之间的内在联系。基于该理论，我们进一步提出了一种名为单调生成对抗网络（M-GAN）的计算方法，用于学习合适的分块三角映射。我们的算法仅利用底层联合概率测度的样本，因此属于无似然方法。数值实验表明，M-GAN在合成示例、涉及常微分方程和偏微分方程的贝叶斯反问题以及概率图像修复中，均能实现对条件测度的精确采样。