将矩形流扩展到希尔伯特空间，实现更优的函数生成模型。
Flow Straight and Fast in Hilbert Space: Functional Rectified Flow
- 基于无限维连续性方程的叠加原理，构建希尔伯特空间中的功能矩形流框架。
- 实验表明该方法在函数生成任务上优于现有模型。
- 无需严格测度论假设，适用于更广泛的生成任务，适合生成模型研究者。
许多原本在有限维欧氏空间中开发的生成模型已在无限维设置中实现功能化。然而，矩形流向无限维空间的扩展尚未被探索。本文在无限维希尔伯特空间中建立了矩形流的严格功能形式。该方法基于无限维空间中连续性方程的叠加原理，并进一步证明该框架可自然推广至功能流匹配与功能概率流微分方程，将其解释为矩形流的非线性推广。值得注意的是，我们的功能流匹配扩展消除了现有理论中 extcite{kerrigan2024functional} 的限制性测度论假设。此外，实验结果表明，该方法在性能上显著优于现有的功能生成模型。
原文摘要 · Abstract (English)
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.
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