  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.
