Y-Diagonal Couplings: Approximating Posteriors with Conditional
  Wasserstein Distances
Jannis Chemseddine Paul Hagemann Christian Wald
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.