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.

翻译：在逆问题中，许多条件生成模型通过最小化联合分布与其学习近似之间的散度来逼近后验测度。虽然该方法在使用Kullback-Leibler散度时也能控制后验测度间的距离，但在Wasserstein距离下并不成立。本文将引入一类具有受限耦合的条件Wasserstein距离，其等于后验期望的Wasserstein距离。通过推导其对偶形式，我们为条件Wasserstein GAN的损失函数提供了严谨的理论依据。我们论述了普通Wasserstein距离与条件Wasserstein距离相一致的充分条件。此外，数值实验表明，基于条件Wasserstein距离的训练能为后验采样带来更优的性质。