Saptarshi Roy

Vansh Bansal, Saptarshi Roy, Purnamrita Sarkar et al.

Flow Matching has become a cornerstone of modern generative models like Stable Diffusion 3, largely due to the efficiency of its Rectified Flow (RF) variant. The success of RF hinges on iteratively learning straight trajectories, pushing generation towards fewer sampling steps. However, the theoretical link between path geometry and sampling efficiency has been underexplored. This paper fills this gap by introducing a novel \textit{Piecewise Straightness} parameter, $γ_{2,T}$. We establish the first Wasserstein convergence bound that explicitly links the discretization error of \textit{any} general flow-model to $γ_{2,T}$, proving that minimizing curvature is the key to achieving high-fidelity, one-step sampling. Building on this theory, we establish the first theoretical framework to analyze the straightness of RF. We begin by offering intuitive geometric arguments for simple cases before identifying sufficient conditions under which a single rectification step (1-RF) yields a perfectly straight or even a Monge optimal coupling. While whether these sufficient conditions are met depends on the problem geometry, they enable the first concrete proofs in this area. Critically, fulfilling these conditions makes the subsequent flow (2-RF) perfectly straight ($γ_{2,T}=0$). This eliminates the discretization error in our bound and makes flawless, single-step sampling possible.

Vansh Bansal, Saptarshi Roy, Purnamrita Sarkar et al.

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) proposed Rectified Flow (RF), a generative model that 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 contributions. First, we provide a 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 novel formulation of straightness stronger than that in the original work. Secondly, we present general conditions guaranteeing uniqueness and straightness of 1-RF, which is in line with previous empirical findings. As a byproduct of our analysis, we show that, in one dimension, RF started at the standard Gaussian distribution yields the Monge map. Additionally, we also present empirical results on both simulated and real datasets to validate our theoretical findings. The code is available at https://github.com/bansal-vansh/rectified-flow.