From Static Geometry to Dynamical Singularity: Detecting Memorization in Diffusion Models via Score Evolution

Detecting training data memorization in diffusion models is important for copyright protection and privacy auditing. We view memorization as an abnormally local concentration process: during reverse-time generation, a memorized instance acts as a point-like attractor whose probability basin is sharper and more position-sensitive than that of a generalized concept. Starting from the Fokker–Planck equation, we derive an exact evolution law for the score field ${\partial }_t{\mathbf{s}}_t$ and show that its leading geometric contribution in score-dominated local regimes is ${\mathbf{H}}_t{\mathbf{s}}_t$, linking temporal concentration dynamics to spatial curvature. We then show that standard sampling along a shared guided trajectory already exposes this curvature information through a complementary transport-curvature response, without requiring explicit Hessian computation. Motivated by this analysis, we propose the **Dynamical Singularity Metric (DSM)**, an on-trajectory detector that measures the pathwise score-evolution discrepancy between conditional and unconditional branches. DSM requires no additional backpropagation or network evaluations beyond sampling itself. Experiments on Stable Diffusion show that DSM matches or exceeds curvature-based baselines while being substantially cheaper and effective at the earliest reverse step.