Y-Diagonal Couplings: Approximating Posteriors with Conditional Wasserstein Distances

• URL: http://arxiv.org/abs/2310.13433v1
• Date: Fri, 20 Oct 2023 11:46:05 GMT
• Title: Y-Diagonal Couplings: Approximating Posteriors with Conditional Wasserstein Distances
• Authors: Jannis Chemseddine, Paul Hagemann, Christian Wald
• Abstract summary: In inverse problems, many conditional generative models approximate the posterior measure by minimizing a distance between the joint measure and its learned approximation. We will introduce a conditional Wasserstein distance with a set of restricted couplings that equals the expected Wasserstein distance of the posteriors.
• Score: 0.4419843514606336
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: 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.

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