揭示修正流方法的统计特性，证明其比传统非参数估计更快收敛。
Statistical Properties of Rectified Flow
- 基于回归与密度估计构建修正流的可计算版本
- 在有界和无界情况下均实现比常规估计更快的收敛速度
- 为修正流提供理论支撑，适合关注生成模型理论的研究者
修正流（Rectified flow）是一种定义两个分布间传输映射的方法，在机器学习中广受欢迎，但其理论支持仍较匮乏。修正流可视为最优传输的近似，区别于需在函数空间优化的其他传输方法，其计算仅依赖标准统计工具如回归或密度估计，我们据此发展了传输映射的实证版本。本文研究修正流的结构性质，包括存在性、唯一性和正则性，以及部分估计器的统计性质，如收敛速率和中心极限定理。针对有界与无界情形分别分析，两者均面临独特挑战。在两种情况下，均证明收敛速率优于常规非参数回归与密度估计。
原文摘要 · Abstract (English)
Rectified flow (Liu et al., 2022; Liu, 2022; Wu et al., 2023) is a method for defining a transport map between two distributions, and enjoys popularity in machine learning, although theoretical results supporting the validity of these methods are scant. The rectified flow can be regarded as an approximation to optimal transport, but in contrast to other transport methods that require optimization over a function space, computing the rectified flow only requires standard statistical tools such as regression or density estimation, which we leverage to develop empirical versions of transport maps. We study some structural properties of the rectified flow, including existence, uniqueness, and regularity, as well as the related statistical properties, such as rates of convergence and central limit theorems, for some selected estimators. To do so, we analyze the bounded and unbounded cases separately as each presents unique challenges. In both cases, we are able to establish convergence at faster rates than those for the usual nonparametric regression and density estimation.
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