发表机构
Institute of Science and Technology Austria (ISTA)（奥地利科学技术学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出从尺度空间动力学视角研究扩散模型，通过临界尺度σ_c量化样本记忆持久性，实验验证其能识别重复、过拟合及离群点导致的记忆，并提供可解释的度量。
AI 中文摘要
扩散模型通常被视为将噪声转化为数据的随机过程。我们采取一个互补的视角：扩散模型定义了一个由噪声尺度索引的确定性动力系统族。在每个固定尺度σ下，我们将去噪器视为一个自映射并研究其动力学。对于精确的去噪器，不动点对应于平滑数据密度的临界点，而吸引子对应于其模态；随着σ增加，样本级模态合并为逐渐更粗糙的模态。这提示了一种关于记忆的几何观点：由于重复或过拟合而获得过多概率质量的样本，以及离群点，在更强的平滑下应比普通样本保持更易区分。我们通过临界尺度σ_c量化这种持久性，即样本被固定尺度动力学保留的最大噪声尺度。在条件模型中，相同的构造自然扩展到图像-标题对。在受控设置和大规模模型上的实验表明，σ_c追踪由重复、过拟合和离群点引起的记忆，并识别Stable Diffusion中记忆化和部分记忆化的样本。此外，σ_c提供了关于记忆的图像空间分布和标题依赖性的可解释度量。
英文摘要
Diffusion models are typically viewed as stochastic processes that transform noise into data. We take a complementary perspective: a diffusion model defines a family of deterministic dynamical systems indexed by noise scale. At each fixed scale $σ$, we treat the denoiser as a self-map and study its dynamics. For an exact denoiser, fixed points correspond to critical points of the smoothed data density, while attractors correspond to its modes; as $σ$ increases, sample-level modes merge into progressively coarser ones. This suggests a geometric view of memorization: examples that receive excess probability mass due to duplication or overfitting, as well as outliers, should remain distinguishable under stronger smoothing than ordinary examples. We quantify this persistence by the critical scale $σ_c$, the largest noise scale at which an example is retained by the fixed-scale dynamics. In conditional models, the same construction extends naturally to image--caption pairs. Experiments in controlled settings and on large-scale models show that $σ_c$ tracks memorization arising from duplication, overfitting, and outliers, and identifies both memorized and partially memorized examples in Stable Diffusion. Moreover, $σ_c$ yields interpretable measures of the image spatial distribution and caption dependence of memorization.
