发现扩散模型采样轨迹具有低维几何规律，可提升生成效率。
Geometric Regularity in Deterministic Sampling Dynamics of Diffusion-based Generative Models
- 采样路径在低维子空间内呈几乎一致的回旋形状。
- 仅需5-10次函数求值即可实现高质量图像生成。
- 无需改动现有求解器，轻量级优化显著提升性能。
基于扩散的生成模型通过随机微分方程（SDE）及其等价的概率流常微分方程（ODE）建立复杂高维数据分布与易处理先验分布之间的平滑变换。本文揭示了扩散生成模型确定性采样动力学中存在显著的几何规律：沿梯度场模拟的每条采样轨迹均位于极低维子空间内，且所有轨迹呈现几乎相同的回旋形状，与模型架构、应用条件或生成内容无关。我们对这些轨迹的若干有趣特性进行了刻画，尤其在基于核估计数据建模的闭式解下。此外，我们提出一种基于动态规划的采样时间调度优化方案，以更好地匹配底层轨迹结构。该策略仅需对现有确定性数值求解器进行微小修改，计算开销可忽略，并在仅5-10次函数求值的区域实现更优的图像生成效果。
原文摘要 · Abstract (English)
Diffusion-based generative models employ stochastic differential equations (SDEs) and their equivalent probability flow ordinary differential equations (ODEs) to establish a smooth transformation between complex high-dimensional data distributions and tractable prior distributions. In this paper, we reveal a striking geometric regularity in the deterministic sampling dynamics of diffusion generative models: each simulated sampling trajectory along the gradient field lies within an extremely low-dimensional subspace, and all trajectories exhibit an almost identical boomerang shape, regardless of the model architecture, applied conditions, or generated content. We characterize several intriguing properties of these trajectories, particularly under closed-form solutions based on kernel-estimated data modeling. We also demonstrate a practical application of the discovered trajectory regularity by proposing a dynamic programming-based scheme to better align the sampling time schedule with the underlying trajectory structure. This simple strategy requires minimal modification to existing deterministic numerical solvers, incurs negligible computational overhead, and achieves superior image generation performance, especially in regions with only 5 - 10 function evaluations.
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