Minimizing Trajectory Curvature of ODE-based Generative Models (2301.12003v3)

Abstract: Recent ODE/SDE-based generative models, such as diffusion models, rectified flows, and flow matching, define a generative process as a time reversal of a fixed forward process. Even though these models show impressive performance on large-scale datasets, numerical simulation requires multiple evaluations of a neural network, leading to a slow sampling speed. We attribute the reason to the high curvature of the learned generative trajectories, as it is directly related to the truncation error of a numerical solver. Based on the relationship between the forward process and the curvature, here we present an efficient method of training the forward process to minimize the curvature of generative trajectories without any ODE/SDE simulation. Experiments show that our method achieves a lower curvature than previous models and, therefore, decreased sampling costs while maintaining competitive performance. Code is available at https://github.com/sangyun884/fast-ode.

• The paper introduces a forward process training method that minimizes trajectory curvature, reducing sampling costs while preserving model quality.
• It reveals that high curvature leads to increased solver truncation errors, emphasizing the importance of optimized forward coupling.
• Empirical tests on datasets like MNIST, CIFAR-10, and CelebAHQ confirm accelerated sampling and improved efficiency with lower FID scores.

Minimizing Trajectory Curvature of ODE-based Generative Models: An Overview

The paper "Minimizing Trajectory Curvature of ODE-based Generative Models" by Sangyun Lee, Beomsu Kim, and Jong Chul Ye introduces a method to improve the efficiency of ODE-based generative models by addressing the high curvature of learned generative trajectories. This research explores the hypothesis that minimizing curvature leads to reduced sampling costs while maintaining competitive performance.

Summary of Contributions

The authors contribute a novel technique that trains the forward process to minimize trajectory curvature, thereby enhancing sampling speed without losing model quality. This approach diverges from existing methods that often rely on multiple neural network evaluations, resulting in slower generation times.

Key Observations

The paper highlights the direct relationship between curvature and numerical solver truncation error. High curvature indicates greater truncation errors, which necessitates extensive evaluations for accuracy. Consequently, minimizing curvature is pivotal for efficient sampling.

Key Findings:

1. Relationship Between Forward Process and Curvature: The authors investigate this relationship from a rectified flow perspective, demonstrating that intersections between forward trajectories correlate with increased curvature of generative processes.
2. Optimizing Forward Coupling: By redefining forward coupling with a learned process, the paper proposes methods to untangle trajectory intersections, effectively reducing curvature. This is achieved by parameterizing the coupling as a neural network.
3. Empirical Validation: The experimental results show substantial improvement in reducing curvature, leading to accelerated sampling. The models demonstrate improved efficiency across various datasets, including MNIST, CIFAR-10, and CelebAHQ.

Experimental Results

The paper's experiments elucidate notable improvement in sampling efficiency, showcasing a marked reduction in FID scores with limited function evaluations. This is observed across different curvature setups, affirming the hypothesis that lower curvature enhances performance without compromising fidelity.

Implications and Future Directions

Practical Implications:

• Sampling Efficiency: The practical reduction in computational costs during sampling makes this method a valuable asset in large-scale data applications where speed and performance are critical.
• Distillation Benefits: The research indicates minimized distillation error, which implies better performance in student models when distilled from teacher trajectories using this method.

Theoretical Implications:

• Model Design: By addressing the curvature during training, the paper introduces a framework that potentially informs future designs of ODE-based models and highlights the importance of forward process design.

Future Research:

The paper opens the door for further exploration in enhancing encoder distributions beyond simple Gaussian models and refining the trade-off between sample quality and computational efficiency through advanced parameter tuning.

Conclusion

This research contributes significantly to the understanding of trajectory curvature in generative modeling. By providing an efficient method to minimize curvature, it lays the groundwork for developing ODE-based models that are both scalable and rapid in sampling, thus holding promise for widespread applicability in machine learning and AI-driven tasks.