2-Rectifications are Enough for Straight Flows: A Theoretical Insight into Wasserstein Convergence

Saptarshi Roy, Vansh Bansal, Purnamrita Sarkar, Alessandro Rinaldo

Keywords: generative model, optimal transport, rectified flow, wasserstein distance

TL;DR: We examine the straightness of Rectified Flow and its Wasserstein convergence.

Abstract:

Diffusion models have emerged as a powerful tool for image generation and denoising. Typically, generative models learn a trajectory between the starting noise distribution and the target data distribution. Recently Liu et al. (2023b) designed a novel alternative generative model Rectified Flow(RF), which aims to learn straight flow trajectories from noise to data using a sequence of convex optimization problems with close ties to optimal transport. If the trajectory is curved, one must use many Euler discretization steps or novel strategies, such as exponential integrators, to achieve a satisfactory generation quality. In contrast, RF has been shown to theoretically straighten the trajectory through successive rectifications, reducing the number of function evaluations (NFEs) while sampling. It has also been shown empirically that RF may improve the straightness in two rectifications if one can solve the underlying optimization problem within a sufficiently small error. In this paper, we make two key theoretical contributions: 1) we provide the first theoretical analysis of the
Wasserstein distance between the sampling distribution of RF and the target distribution. Our error rate is characterized by the number of discretization steps and a new formulation of straightness stronger than that in the original work. 2) under a mild regularity assumption, we show that for a rectified flow from a Gaussian to any general target distribution with finite first moment (e.g. mixture of Gaussians), two rectifications are sufficient to achieve a straight flow, which is in line with the previous empirical findings. Additionally, we also present empirical results on both simulated and real datasets to validate our theoretical findings. The codes are available at https://github.com/bansal-vansh/rectified-flow.

Primary Area: generative models

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Submission Number: 13240