Flow Straight and Fast: Learning to Generate and Transfer Data with Rectified Flow (2209.03003v1)

Abstract: We present rectified flow, a surprisingly simple approach to learning (neural) ordinary differential equation (ODE) models to transport between two empirically observed distributions \pi_0 and \pi_1, hence providing a unified solution to generative modeling and domain transfer, among various other tasks involving distribution transport. The idea of rectified flow is to learn the ODE to follow the straight paths connecting the points drawn from \pi_0 and \pi_1 as much as possible. This is achieved by solving a straightforward nonlinear least squares optimization problem, which can be easily scaled to large models without introducing extra parameters beyond standard supervised learning. The straight paths are special and preferred because they are the shortest paths between two points, and can be simulated exactly without time discretization and hence yield computationally efficient models. We show that the procedure of learning a rectified flow from data, called rectification, turns an arbitrary coupling of \pi_0 and \pi_1 to a new deterministic coupling with provably non-increasing convex transport costs. In addition, recursively applying rectification allows us to obtain a sequence of flows with increasingly straight paths, which can be simulated accurately with coarse time discretization in the inference phase. In empirical studies, we show that rectified flow performs superbly on image generation, image-to-image translation, and domain adaptation. In particular, on image generation and translation, our method yields nearly straight flows that give high quality results even with a single Euler discretization step.

• The paper introduces rectified flow, an innovative method that learns ODE models to transport data in straight-line paths between distributions.
• It employs a nonlinear least squares approach to achieve deterministic couplings with provably non-increasing convex transport costs.
• Empirical results show that rectified flow outperforms traditional diffusion methods in image generation and domain adaptation with minimal discretization steps.

Overview of Rectified Flow for Distribution Transport

The paper introduces a new method termed "rectified flow" as a novel approach for learning ordinary differential equation (ODE) models to facilitate data transport between two empirical distributions, denoted as $\tg_0$ and $\tg_1$. This method is positioned as a solution that unifies generative modeling and domain transfer, two tasks commonly encountered in machine learning and data science.

Rectified flow offers a unique perspective by solving the ODE models to follow direct, straight-line paths between corresponding data points in the distributions $\tg_0$ and $\tg_1$. The significance of this approach lies in its computational efficiency, as it allows for the simulation of data paths without time discretization. This characteristic is particularly beneficial for generating computational models that are scalable to large datasets and complex models without the need for introducing additional parameters.

The paper posits an optimization-based approach, in which the rectified flow procedure minimizes a nonlinear least squares problem without the instability issues associated with alternative methods such as GANs. By transforming the pairing between $\tg_0$ and $\tg_1$, rectified flow produces deterministic couplings with provably non-increasing convex transport costs. Recursive application of this rectification process results in flows with paths that become increasingly straight, thereby enhancing the accuracy and efficiency of simulation using coarse time discretization.

Empirically, rectified flow demonstrates impressive capabilities in tasks such as image generation and domain transfer. Notably, it achieves high-quality results even with a minimal Euler discretization step, outperforming traditional diffusion methods and other generative models in specific benchmarks.

Implications and Future Directions

The paper implies that adopting rectified flow could significantly impact both theoretical and practical applications in AI, particularly in fields where efficient data transport and generative modeling are critical. By reducing transport costs and enabling efficient one-step simulations, this approach opens avenues for fast and scalable modeling across various domains.

The rectified flow paradigm can potentially serve as a foundation for future developments in AI, especially in designing hybrid models that combine the generalized functionalities of ODEs with the structural flexibility of neural networks. Additionally, the framework provides a theoretical basis for understanding and improving upon existing methods that involve distribution transport, such as optimal transport algorithms, by focusing on the straightening and stabilization characteristics of rectified flows.

Moving forward, researchers may explore more sophisticated applications of rectified flow beyond image generation and domain adaptation, including its role in understanding the embeddings in latent space for more complex, multimodal datasets. This could lead to advancements in unsupervised learning frameworks and other domains where transport between distributions is a fundamental challenge.

In summary, rectified flow presents a significant stride towards efficient and scalable solutions for generative modeling and domain transfer, holding promise for enhancing both the computational sophistication and theoretical understanding of distribution transport in machine learning.