# From Global to Tail Convergence: Dynamics of α-Stable Score-Based Generative Models

arXiv: https://arxiv.org/abs/2610.19773 · HTML: https://mlsanity.com/paper/2610.19773/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Placed in Diffusion foundations and sampling, one of 14 directions in Diffusion models.
- Builds on 13 of its 42 references that are in the map.
- First posted on arXiv 14 Oct 2026.

## Placed in

- Diffusion models / [Diffusion foundations and sampling](https://mlsanity.com/diffusion/diffusion-foundations-and-sampling/)

## Builds on (most cited first, 13 of 13)

- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [KL Convergence Guarantees for Score diffusion models under minimal data assumptions](https://mlsanity.com/paper/2308.12240/) (2023)
- [Wasserstein Convergence Guarantees for a General Class of Score-Based Generative Models](https://mlsanity.com/paper/2311.11003/) (2023)
- [Non Gaussian Denoising Diffusion Models](https://mlsanity.com/paper/2106.07582/) (2021)
- [Heavy-Tailed Diffusion Models](https://mlsanity.com/paper/2410.14171/) (2024)
- [An analysis of the noise schedule for score-based generative models](https://mlsanity.com/paper/2402.04650/) (2024)
- [Heavy-tailed denoising score matching](https://mlsanity.com/paper/2112.09788/) (2021)
- [Heavy-Tailed Diffusion with Denoising L'evy Probabilistic Models](https://mlsanity.com/paper/2407.18609/) (2024)
- [A Unified Approach to Analysis and Design of Denoising Markov Models](https://mlsanity.com/paper/2504.01938/) (2025)
- [A General Derivative Identity for the Conditional Mean Estimator in Gaussian Noise and Some Applications](https://mlsanity.com/paper/2104.01883/) (2021)
- [Generalizing Score-based generative models for Heavy-tailed Distributions](https://mlsanity.com/paper/2603.00772/) (2026)
- [Do Heavy Tails Help Diffusion? On the Subtle Trade-off Between Initialization and Training](https://mlsanity.com/paper/2605.13175/) (2026)
- [Generator-Guided Inverse Sampling for L'evy-Driven Generative Models](https://mlsanity.com/paper/2608.10384/) (2026)

## Abstract (arXiv)

Score-based generative models driven by α-stable L'evy 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 nu, asymptotically correct extreme-quantile scaling is possible precisely when α=nu. These results identify tail-index matching as a principled mechanism behind the ability of stable generative models to capture extreme events.