Particle Dynamics of Flow Matching and Classifier-Free Guidance from a Stagewise Geometry Perspective

2026-09-07

AI Summary

研究从逐阶段几何角度出发，通过建立统一的吸引和吸收理论，解决了流匹配及无分类器引导在生成建模中的轨迹交互问题。

Abstract

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.

Problem

• Flow Matching
• Classifier-Free Guidance
• Particle Dynamics
• Data Geometry
• Trajectory

Innovation

• stagewise geometric theory
• flow matching
• classifier-free guidance
• particle dynamics
• continuous and discrete sampling