α-stable 生成模型收敛性分析：尾部指数匹配是捕捉极端事件的关键
From Global to Tail Convergence: Dynamics of $α$-Stable Score-Based Generative Models
理论证明来了：α-stable 扩散为何比 Brownian 更会抓重尾——α 对上数据尾部指数，极端分位数才准。
一篇 arXiv 论文为基于 α-stable Lévy 过程的评分生成模型建立了非渐近收敛分析，覆盖紧支撑与幂律分布两类目标。在 p-Wasserstein 距离下，论文量化了早停、初始化、评分近似和时间离散化各自的影响，并证明初始化误差随前向时长指数衰减。与 Brownian 扩散的对比显示，仅看全局 Wasserstein 收敛分不出两种模型的优劣，两者的初始化误差在同一类重尾目标上都指数衰减。真正的分野在尾部：任意正的生成时刻，stable 动力学生成指数为 α 的幂律尾部，Brownian 动力学仍是轻尾。当 α 等于目标尾部指数 ν 时，极端分位数的渐近标度才可能正确，这解释了 stable 模型擅长刻画极端事件的原因。
From Global to Tail Convergence: Dynamics of $α$-Stable Score-Based Generative Models
Score-based generative models driven by $α$-stable L{é}vy processes have been proposed as an alternative to Brownian diffusion models, with empirical evidence suggesting improved sampling and better modeling of heavy-tailed data. Yet theoretical guarantees for these models remain limited. We develop a non-asymptotic convergence analysis for scale-preserving $α$-stable generative dynamics, covering both compactly supported and power-law targets. In p-Wasserstein distance, we quantify the effects of early stopping, initialization, score approximation, and time discretization, and show exponential decay of the initialization error. A comparison with Brownian dynamics reveals that global Wasserstein convergence alone does not fundamentally distinguish the two models: both initialization errors decay exponentially with the forward horizon on the common class of heavy-tailed targets. A sharp separation emerges instead in the tails. At any positive generation time, stable dynamics produces power-law tails with index $α$, whereas Brownian dynamics remains light-tailed. For a target with tail index $ν$, asymptotically correct extreme-quantile scaling is possible precisely when $α$=$ν$. These results identify tail-index matching as a principled mechanism behind the ability of stable generative models to capture extreme events.
