Y-Diagonal Couplings: Approximating Posteriors with Conditional Wasserstein Distances
This work addresses a theoretical gap in generative modeling for inverse problems, though it appears incremental in improving Wasserstein-based methods.
The paper tackles the problem of approximating posterior measures in inverse problems using conditional generative models, showing that training with a newly introduced conditional Wasserstein distance yields favorable properties for posterior sampling.
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.