In inverse problems, many conditional generative models approximate the posterior measure by minimizing a distance between the joint measure and its learned approximation. While this approach also controls the distance between the posterior measures in the case of the Kullback Leibler divergence, it does not hold true for the Wasserstein distance. We will introduce a conditional Wasserstein distance with a set of restricted couplings that equals the expected Wasserstein distance of the posteriors. By deriving its dual, we find a rigorous way to motivate the loss of conditional Wasserstein GANs. We outline conditions under which the vanilla and the conditional Wasserstein distance coincide. Furthermore, we will show numerical examples where training with the conditional Wasserstein distance yields favorable properties for posterior sampling.

逆问题中，通过最小化联合度量与其学习近似度量之间的距离，许多条件生成模型近似后验测度。尽管这种方法对于KL散度的情况也控制了后验测度之间的距离，但对于Wasserstein距离则不成立。我们引入一种带有一组受限耦合的条件Wasserstein距离，它等于后验分布的期望Wasserstein距离。通过推导其对偶性，我们找到了正式推动条件Wasserstein GAN损失的方法。我们概述了条件Wasserstein距离和普通Wasserstein距离重合的条件，并且还展示了使用条件Wasserstein距离训练时获得有利的后验抽样性质的数值示例。

Y-对角耦合：用条件瓦砾均衡距离近似后验