An Extreme Value Perspective on Learning Stress Laws

2026-07-12

AI Summary

This work addresses the challenge of simulating multivariate extreme events and estimating rare-event probabilities under both heavy-tailed and light-tailed distributions by proposing Self-Similar Generative Estimation (SS-GEN). The method uniquely integrates the asymptotic tail structure from extreme value theory into a deep generative modeling framework, leveraging self-similar decomposition to decouple the tail distribution into an explicit radial component and a nonparametric angular component. This transformation recasts tail modeling as a standard generative task on a compact domain, eliminating the need for specialized architectures or parametric tail assumptions. Theoretical analysis shows that SS-GEN achieves vanishing relative error for regularly varying distributions and vanishing log-relative error for Weibull-type tails. Empirical results confirm its ability to accurately generate representative extreme samples and reliably estimate rare-event probabilities.

Abstract

We introduce Self-Similar Generative Estimation (SS-GEN), a method for simulating multivariate tail events and estimating rare-event probabilities in both heavy and light-tailed settings. SS-GEN exploits asymptotic tail structure to decompose the tail distribution into an explicit radial component and a nonparametric angular component, reducing tail learning to a compact-domain problem that can be handled by off-the-shelf deep generative models. The resulting sampler generates representative extreme scenarios and supports probability estimation far beyond the observed data. Under mild nonparametric tail assumptions, we show that the SS-GEN density is asymptotically exact in the tail, with vanishing uniform relative error for regularly varying distributions and vanishing uniform log-relative error for Weibull-type distributions. Unlike existing approaches that rely on specialized architectures or parametric tail specifications, SS-GEN leverages asymptotic tail structure to enable standard generative models to generate representative extreme samples and estimate rare-event probabilities beyond the observed data.

Problem

• extreme value theory
• rare-event probability
• tail estimation
• multivariate extremes
• generative modeling

Innovation

• Self-Similar Generative Estimation
• tail dependence
• rare-event simulation
• extreme value theory
• deep generative models