From Global to Tail Convergence: Dynamics of $\alpha$-Stable Score-Based Generative Models
Score-based generative models with heavy-tailed $\alpha$-stable Lévy noise have shown empirical promise for extreme-event modeling, yet theoretical guarantees remain scarce. This paper develops a non-asymptotic $W_p$ convergence framework for scale-preserving $\alpha$-stable dynamics and proves that global Wasserstein convergence alone does not separate stable from Brownian models on heavy-tailed targets. A sharp distinction emerges instead in the tails: stable dynamics inherit power-law tails with index $\alpha$, while Brownian outputs stay light-tailed, and asymptotically correct extreme-quantile scaling requires matching the target tail index $\nu$ to the noise index $\alpha$. These results establish tail-index matching as a principled design mechanism for capturing heavy-tailed phenomena.
The paper delivers a technically sophisticated convergence analysis, but its impact is constrained by several theoretical gaps. Its central thesis that global $W_p$ convergence does not fundamentally distinguish the two dynamics is rigorously shown for exact-score dynamics when $\nu \ge 1 + \alpha$, yet the accompanying tail separation is both elegant and decisive: "At any positive generation time, stable dynamics produces power-law tails with index $\alpha$, whereas Brownian dynamics remains light-tailed." The conclusion that tail-index matching is the unique regime allowing asymptotically correct extreme-quantile scaling is well-supported by asymptotic theory and synthetic experiments, though the reliance on exact scores and one-dimensional Student-$t$ targets limits practical applicability.
The theoretical decomposition is clean and the tail-regime analysis is genuinely novel. Theorem 4 supplies an explicit non-asymptotic $W_p$ bound with five interpretable error terms, including a discretization contribution that scales as $h^{1/\alpha}$ reflecting stable fluctuations over an exponential-integrator step: "The last discretization term, of order $h^{1/\alpha}$, reflects stable fluctuations over an EI step, where the nonlinear drift in (3.3) is frozen." The proof technique leveraging the fractional Tweedie formula to link score regularity to the denoiser modulus $C_t$ is a principled extension of Gaussian diffusion theory. The extreme-value-theoretic diagnostics and the exact extreme-quantile ratio in Lemma 13 yield a crisp, testable index-matching condition.
There is a notable disconnect between the global convergence and tail analyses: Theorem 7 requires $\nu \ge 1 + \alpha$ for power-law targets, whereas Section 5.2 holds for any $\nu > 1$. Moreover, Proposition 6 shows that in the intermediate regime $\alpha \le \nu < 1 + \alpha$ the key regularity condition can fail catastrophically: "If $\alpha \le \nu < 1 + \alpha$, Assumption A5 alone does not ensure $C_t < \infty$: for each such $\nu$, there is a smooth, positive density in dimension one satisfying this assumption for which $C_t = \infty$ for every $t > 0$."
The Brownian comparison is also asymmetrical: Theorem 10 assumes $\nu > 2$ while the stable Corollary 9 only needs $\nu \ge 1 + \alpha$, so the common target class is narrower than the text suggests. Finally, the experiments are one-dimensional and training-free; the authors themselves warn that "extending the tail analysis to learned scores is essential: estimation errors in low-density regions may interact strongly with the jump structure precisely where the stable dynamics is meant to improve tail coverage."
The asymptotic tail claims are supported convincingly, but the global comparison is incomplete. Proposition 14 and Lemma 13 are derived cleanly, and Figure 3's quantile ratios follow the "predicted scaling $(1 - \kappa)^{1/\nu - 1/\alpha}$." Yet the authors concede that "the available sample size does not reliably resolve their asymptotic exponents" away from index match, and Figure 4 shows Hill-estimate drift across thresholds. The global rate comparison with Brownian dynamics further relies on discretization rates imported from Strasman et al. (2025), which were obtained under "global score regularity and strong log-concavity and without the early-stopping error," making the comparison "suggestive rather than conclusive."
The numerical protocols are documented with admirable precision. Appendix F states exact parameters: "$\beta(t) \equiv 1$, $\alpha = 1.5$ and $\eta = 0.5$ for the stable backward runs," together with tabled step sizes, horizons, and sample counts (Table 3). The authors disclose software versions and provide code at https://github.com/ainsi333/stable-sbgm-experiments. Because every experiment is training-free and evaluates exact scores via Fourier inversion, the synthetic results are fully reproducible. The limitation is that this exact-score protocol deliberately avoids the learned-score regime, leaving open the very question flagged in the conclusion.
Score-based generative models driven by $\alpha$-stable L{\'e}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 $\alpha$-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 $\alpha$, whereas Brownian dynamics remains light-tailed. For a target with tail index $\nu$, asymptotically correct extreme-quantile scaling is possible precisely when $\alpha$=$\nu$. These results identify tail-index matching as a principled mechanism behind the ability of stable generative models to capture extreme events.
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