流匹配与无分类器引导的粒子动力学：从分段几何视角
Particle Dynamics of Flow Matching and Classifier-Free Guidance from a Stagewise Geometry Perspective
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中文总结 AI 辅助
本文从分段几何视角统一分析了流匹配与无分类器引导的粒子动力学，证明轨迹逐步吸引至均值、凸包及局部簇，并给出距离衰减估计，为连续与离散采样提供理论解释。
中文摘要 AI 辅助
流匹配结合无分类器引导（CFG）在生成建模中被广泛使用，然而其理论理解大多停留在分布层面。由于实际采样遵循个体轨迹，仅凭分布层面的保证无法完全捕捉轨迹如何与数据几何相互作用，以及引导如何重塑该几何。为克服这一局限，我们为连续动力学和显式欧拉离散化建立了统一的吸引与吸收的分段几何理论。具体而言，在$t\in[0,1]$从噪声运行到数据的过程中，我们证明无条件流轨迹依次被吸引到全局均值附近、数据凸包附近以及可能非凸的局部簇附近。在这些阶段中，相应的距离满足一个共同的收缩估计，最终阶段距离衰减为${O}(1-t)$。对于CFG，相同的结构在推断均值、膨胀的条件凸包以及目标簇附近恢复条件流匹配的局部几何中得以保持。我们进一步证明，一般时间调度$a(t)$将$O(1-t)$衰减替换为$O(1-a(t))$。综合来看，这些结果为流匹配和CFG在连续与离散采样中提供了统一的粒子级几何解释。
英文摘要
Flow matching, together with classifier-free guidance (CFG), is widely used in generative modeling, yet much of the theoretical understanding remains distribution-wise. Since practical sampling follows individual trajectories, distribution-level guarantees alone do not fully capture how trajectories interact with the data geometry or how guidance reshapes it. To overcome this limitation, we establish a unified stagewise geometric theory of attraction and absorption for both continuous dynamics and explicit Euler discretization. Specifically, with $t\in[0,1]$ running from noise to data, we show that unconditional flow trajectories are successively attracted toward a neighborhood of the global mean, the data convex hull, and a neighborhood of a possibly nonconvex local cluster. Across these stages, the corresponding distance satisfies a common contraction estimate, yielding an ${O}(1-t)$ decay of the distance in the final stage. For CFG, the same structure persists with an extrapolated mean, an inflated conditional convex hull, and, near the target cluster, the restored local geometry of conditional flow matching. We further show that a general time schedule $a(t)$ replaces the $O(1-t)$ decay by $O(1-a(t))$. Together, these results provide a unified particle-level geometric account of flow matching and CFG across continuous and discrete sampling.
发表机构
- The Hong Kong University of Science and Technology（香港科技大学）
- Southern University of Science and Technology（南方科技大学）
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