Flow Straight and Fast in Hilbert Space: Functional Rectified Flow

2025-09-12

AI Summary

This work addresses generative modeling in infinite-dimensional Hilbert spaces and introduces, for the first time, a functional rectified flow framework by generalizing rectified flow to function spaces. Methodologically, it leverages the superposition principle of the continuity equation to jointly model functional flow matching and nonlinear probability flow ODEs—without requiring stringent measure-theoretic assumptions—enabling differentiable and invertible manifold optimization over function spaces. Theoretically, it establishes a rigorous infinite-dimensional functional formulation, substantially extending the applicability boundary of rectified flow. Empirically, the proposed method outperforms existing functional generative models across diverse function-generation tasks—including stochastic process synthesis and functional time-series generation—demonstrating superior effectiveness and generalization capability.

Technology Category

• Machine Learning: Deep Generative Models & Autoencoders
• Natural Language Processing: Generation
• Reasoning under Uncertainty: Relational Probabilistic Models

Application Category

• Economics, Online Markets and Human Computation: Economic ramifications for generative AI infrastructure and applications
• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs
• User Modeling, Personalization and Recommendation: On-Device user modeling, personalization, and recommendation

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.

Problem

• Extends rectified flow to infinite-dimensional Hilbert spaces
• Removes restrictive assumptions in functional flow matching
• Achieves superior performance over existing functional generative models

Innovation

• Functional rectified flow in Hilbert space
• Nonlinear generalization of flow matching
• Superior performance in functional generative models