Kenji Fukumizu: Analysis of OT-Flow Matching under Manifold Hypothesis

To understand why generative models such as flow matching and diffusion models succeed on high-dimensional data, it is essential to analyze their behavior under the manifold hypothesis, which posits that the data distribution is supported on a low-dimensional submanifold. In this talk, we focus on Optimal Transport Conditional Flow Matching and show an exact proximal form via an extended Brenier potential under the manifold hypothesis. More precisely, the mapping to recover the target point, or the "decoder", is expressed by a proximal operator, which yields an explicit expression of the vector field. Using mathematical tools from convex analysis, we analyze the local behavior of the vector field around the manifold and show the stability of the manifold structure against perturbation to the dynamics: the dynamics does not expand or shrink exponentially along manifold directions.