Conditional Wasserstein Distances with Applications in Bayesian OT Flow Matching

TL;DR

This paper introduces a new conditional Wasserstein distance tailored for Bayesian inverse problems, providing theoretical insights and practical algorithms that improve posterior approximation and class-conditional image generation.

Contribution

It defines a novel conditional Wasserstein distance, characterizes its properties, and extends OT Flow Matching to enhance Bayesian inverse problem solutions.

Findings

The proposed method improves posterior measure approximation.

It demonstrates numerical advantages in inverse problems.

It enhances class-conditional image generation.

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, 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.