揭示生成模型中突变现象的几何根源，定位关键控制点。
The Geometry of Phase Transitions in Generative Dynamics via Projection Caustics
- 将去噪视为自由能梯度下降，发现相变源于投影奇点。
- 提出CBD检测器，精准定位模式锁定与敏感窗口。
- 适用于扩散模型的可控生成，尤其适合几何敏感场景。
连续状态生成采样器（如扩散模型、流匹配模型）在逆时序动态中演化，但其生成样本常出现突变：轨迹迅速锁定模式、语义替代崩溃，微小扰动在狭窄时间窗内可引发显著下游影响。本文从几何角度解析此类相变行为：将去噪视为自由能梯度下降，证明尖锐突变发生在投影奇点附近，即数据支撑集的最近点投影不再唯一。基于此，提出临界边界检测器（CBD），用于诊断得分方向不稳定性。在玩具模型、标准扩散模型及潜在文本到图像扩散模型中，CBD均能有效定位模式锁定位置，预测干预敏感窗口，并支持对几何敏感区域的定向控制。结果连接了数据几何与扩散生成动力学。
原文摘要 · Abstract (English)
Continuous-state generative samplers, including diffusion and flow-matching models, evolve through continuous reverse-time dynamics, yet their samples often undergo abrupt qualitative changes: trajectories commit to modes, semantic alternatives collapse, and small perturbations in narrow time windows can produce large downstream effects. This paper develops a geometric account of such phase-transition-like behaviour. We view denoising as gradient descent on a free energy landscape and show that sharp transitions arise near projection caustics, where the nearest-point projection onto the data support ceases to be unique. Motivated by this perspective, we introduce the Critical Boundary Detector (CBD), as practical diagnostics for score-direction instability. Across toy models, standard diffusion models, and latent text-to-image diffusion models, CBD localises mode commitment, predicts intervention-sensitive windows, and supports targeted control in geometrically sensitive regions. Our results connect geometry of data and dynamics of diffusion generation.
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