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, this is in general not hold true for the Wasserstein distance. In this paper, we introduce a conditional Wasserstein distance via a set of restricted couplings that equals the expected Wasserstein distance of the posteriors. Interestingly, the dual formulation of the conditional Wasserstein-1 flow resembles losses in the conditional Wasserstein GAN literature in a quite natural way. We derive theoretical properties of the conditional Wasserstein distance, characterize the corresponding geodesics and velocity fields as well as the flow ODEs. Subsequently, we propose to approximate the velocity fields by relaxing the conditional Wasserstein distance. Based on this, we propose an extension of OT Flow Matching for solving Bayesian inverse problems and demonstrate its numerical advantages on an inverse problem and class-conditional image generation.

翻译：在逆问题中，许多条件生成模型通过最小化联合测度与其学习近似之间的距离来逼近后验测度。虽然这种方法在使用Kullback-Leibler散度时也能控制后验测度之间的距离，但对于Wasserstein距离而言，这一结论通常不成立。本文通过一组受限耦合引入条件Wasserstein距离，该距离等于后验期望Wasserstein距离。有趣的是，条件Wasserstein-1流的对偶形式以非常自然的方式类似于条件Wasserstein生成对抗网络文献中的损失函数。我们推导了条件Wasserstein距离的理论性质，刻画了相应的测地线、速度场以及流常微分方程。随后，我们通过松弛条件Wasserstein距离来逼近速度场。基于此，我们提出了一种扩展的最优传输流匹配方法用于求解贝叶斯逆问题，并在一个逆问题及类别条件图像生成任务中展示了其数值优势。