Conditional Wasserstein Distances with Applications in Bayesian OT Flow Matching
This addresses a theoretical gap in conditional generative models for inverse problems, with incremental improvements for Bayesian inference and image generation.
The paper tackles the problem that minimizing Wasserstein distance between joint measures does not control the posterior distance in inverse problems, by introducing a conditional Wasserstein distance that equals the expected posterior distance and applying it to Bayesian OT flow matching. The result demonstrates numerical advantages on an inverse problem and class-conditional image generation.
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.