Flow Straight and Fast in Hilbert Space: Functional Rectified Flow

Jianxin Zhang, Clayton Scott

Keywords: Hilbert space, superposition principle

TL;DR: We generalize rectified flow to a separable Hilbert space and unify it with other functional generative model under a common framework.

Abstract: Many generative models originally developed in finite-dimensional Euclidean space have functional generalizations in infinite-dimensional settings. However, the extension of rectified flow to infinite-dimensional spaces remains unexplored. In this work, we establish a rigorous functional formulation of rectified flow in an infinite-dimensional Hilbert space. Our approach builds upon the superposition principle for continuity equations in an infinite-dimensional space. We further show that this framework extends naturally to functional flow matching and functional probability flow ODEs, interpreting them as nonlinear generalizations of rectified flow. Notably, our extension to functional flow matching removes the restrictive measure-theoretic assumptions in the existing theory of \citet{kerrigan2024functional}. Furthermore, we demonstrate experimentally that our method achieves superior performance compared to existing functional generative models.

Primary Area: generative models

Submission Number: 7774