Low-Dimensional Adaptation of Rectified Flow: A New Perspective through the Lens of Diffusion and Stochastic Localization
This work addresses sampling efficiency for generative models like RF and DDPM, offering incremental improvements through low-dimensional adaptation and new discretization schemes.
The paper tackles the problem of accelerating sampling in Rectified Flow (RF) by showing it adapts to the low intrinsic dimensionality of target distributions, achieving an iteration complexity of O(k/ε) with precise drift estimates, and proposes a stochastic RF sampler with improved performance in simulations.
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.