In recent years, Rectified flow (RF) has gained considerable popularity largely due to its generation efficiency and state-of-the-art performance. In this paper, we investigate the degree to which RF automatically adapts to the intrinsic low dimensionality of the support of the target distribution to accelerate sampling. We show that, using a carefully designed choice of the time-discretization scheme and with sufficiently accurate drift estimates, the RF sampler enjoys an iteration complexity of order $O(k/\varepsilon)$ (up to log factors), where $\varepsilon$ is the precision in total variation distance and $k$ is the intrinsic dimension of the target distribution. In addition, we show that the denoising diffusion probabilistic model (DDPM) procedure is equivalent to a stochastic version of RF by establishing a novel connection between these processes and stochastic localization. Building on this connection, we further design a stochastic RF sampler that also adapts to the low-dimensionality of the target distribution under milder requirements on the accuracy of the drift estimates, and also with a specific time schedule. We illustrate with simulations on the synthetic data and text-to-image data experiments the improved performance of the proposed samplers implementing the newly designed time-discretization schedules.

翻译：近年来，整流流因其生成效率与最先进的性能而广受关注。本文研究了整流流在何种程度上能自动适应目标分布支撑集的本质低维特性以加速采样。我们证明，通过精心设计的时间离散化方案与足够精确的漂移估计，整流流采样器可实现阶数为 $O(k/\varepsilon)$（忽略对数因子）的迭代复杂度，其中 $\varepsilon$ 为总变差距离精度，$k$ 为目标分布的本质维度。此外，通过建立去噪扩散概率模型过程与随机局部化之间的新联系，我们证明该过程等价于整流流的随机化版本。基于此关联，我们进一步设计了一种随机整流流采样器，该采样器在漂移估计精度要求更宽松的条件下，配合特定的时间调度方案，同样能适应目标分布的低维特性。通过合成数据与文生图数据实验的模拟，我们验证了采用新设计的时间离散化方案的采样器在性能上的提升。