Core Concepts
Conditional Wasserstein distances play a crucial role in Bayesian OT flow matching, offering insights into posterior sampling algorithms.
Abstract
The content discusses the introduction of conditional Wasserstein distances in the context of Bayesian OT flow matching. It covers theoretical properties, geodesics, velocity fields, and the application of these concepts in solving Bayesian inverse problems. The paper proposes a new OT Bayesian flow matching algorithm and presents numerical experiments showcasing its advantages in various scenarios.
Stats
In [31], the authors investigated the relation between the joint measures D(PY,Z, PY,X) and its relation to the expected error between the posteriors EY W1(PZ|Y=y, PX|Y=y).
For the Wasserstein-1 distance, it is shown that W1(PY,X, PY,Z) ≤ Ey∼PY W1(PX|Y=y, PZ|Y=y).
The loss function in conditional Wasserstein GAN literature arises naturally in the dual formulation of the conditional Wasserstein-1 distance.
Quotes
"Inverse problems, many conditional generative models approximate the posterior measure by minimizing a distance between the joint measure and its learned approximation."