On the Trajectory Regularity of ODE-based Diffusion Sampling

Abstract: Diffusion-based generative models use stochastic differential equations (SDEs) and their equivalent ordinary differential equations (ODEs) to establish a smooth connection between a complex data distribution and a tractable prior distribution. In this paper, we identify several intriguing trajectory properties in the ODE-based sampling process of diffusion models. We characterize an implicit denoising trajectory and discuss its vital role in forming the coupled sampling trajectory with a strong shape regularity, regardless of the generated content. We also describe a dynamic programming-based scheme to make the time schedule in sampling better fit the underlying trajectory structure. This simple strategy requires minimal modification to any given ODE-based numerical solvers and incurs negligible computational cost, while delivering superior performance in image generation, especially in $5\sim 10$ function evaluations.

• The paper demonstrates that ODE-based diffusion sampling exhibits a boomerang-shaped regularity in its trajectories, providing a unified synthesis schedule.
• It introduces a dynamic programming approach to align ODE solver steps with this inherent trajectory structure, thereby enhancing sampling efficiency.
• Empirical results show improved image quality and reduced computational costs, validated through metrics like the Fréchet Inception Distance.

Detailed Summary and Analysis of "On the Trajectory Regularity of ODE-based Diffusion Sampling"

Overview of Diffusion-Based Generative Models

Diffusion-based generative models leverage stochastic differential equations (SDEs) and their equivalent ordinary differential equations (ODEs) to effectively map a complex data distribution to a tractable prior distribution. This formulation is central to the model's capabilities in tasks like image synthesis, audio, and video generation. The key element of these models is the score function, which is defined as the gradient of the log data density with respect to the input. Recently, research has indicated that the backward SDE can be substituted by an equivalent probability flow ODE (PF-ODE) while maintaining identical marginal distributions. This deterministic approach simplifies the generative process by only introducing stochasticity in the initial sample selection.

Significance of Trajectory Regularity

Despite the successful application of diffusion models, the intricate mathematical structure of SDEs and the high-dimensional data involved leave several aspects unexplored. An empirical observation is the consistent shape regularity formed by the PF-ODE sampling trajectories—commonly displaying a linear-nonlinear-linear "boomerang" pattern. This trajectory structure is intriguing as it appears regardless of the initial random samples or the content generated. The inability of 1-D projections to fully capture this pattern suggests a multi-dimensional geometric organization of trajectories, offering insight into a unified time schedule for sample synthesis. The trajectory's intrinsic regularity enables effective use of large sampling steps without introducing significant truncation errors.

Implicit Denoising Trajectory

A critical construct in this study is the implicit denoising trajectory, which corresponds to a rotation of each sample point along the trajectory and significantly influences its curvature. This rotation can be computationally aligned with a kernel density estimation (KDE) of the original data distribution with time-varying widths, drawing an analogy to the classical mean-shift algorithm. Though not directly feasible for practical sampling, this KDE-derived interpretation offers a theoretical framework illustrating the expected regularity of sampling trajectories. The denoising trajectory's closeness to a closed-form under KDE makes it a powerful analytical tool.

Practical Implications and Accelerated Sampling

The explicit identification of the trajectory regularity has direct implications for sampling efficiency in diffusion models. By using dynamic programming techniques to align the sampling steps with the inherent structure of the trajectory, an accelerated sampling strategy can be developed. This approach optimally reallocates time in the sampling schedule, allowing for significant performance improvements with minimal computational cost—particularly noticeable in scenarios with limited function evaluations.

Technical Contributions and Experimental Validation

The paper contributes several key insights and methodologies:

• Demonstration of a shape regularity in the trajectories of ODE-based diffusion sampling, which naturally arise from the exchange between the implicit denoising trajectory and the explicit trajectory path.
• Proposal of an easy-to-implement dynamic programming approach to align ODE solvers' time schedules with the embedded trajectory structure, yielding superior image quality and reduced computational overhead.

Empirical validations across various datasets underscore the effectiveness of these advancements. This includes quantitative enhancements in image synthesis performance with fewer function evaluations, substantiated by metrics like Fréchet Inception Distance (FID).

Conclusion

The paper "On the Trajectory Regularity of ODE-based Diffusion Sampling" provides not only a novel theoretical understanding of sampling trajectories in diffusion models but also an application framework that significantly boosts sampling efficacy. The insights into the geometric structures underlying sampling trajectories hope to inspire further developments in both theoretical explorations and practical implementations of AI-driven generative models. Future work may explore deeper regulatory structures within trajectories or leverage these insights for novel applications in AI technologies.