We propose a spectral clustering algorithm for analyzing the dependence structure of multivariate extremes. More specifically, we focus on the asymptotic dependence of multivariate extremes characterized by the angular or spectral measure in extreme value theory. Our work studies the theoretical performance of spectral clustering based on a random $k$-nearest neighbor graph constructed from an extremal sample, i.e., the angular part of random vectors for which the radius exceeds a large threshold. In particular, we derive the asymptotic distribution of extremes arising from a linear factor model and prove that, under certain conditions, spectral clustering can consistently identify the clusters of extremes arising in this model. Leveraging this result we propose a simple consistent estimation strategy for learning the angular measure. Our theoretical findings are complemented with numerical experiments illustrating the finite sample performance of our methods.

翻译：我们提出了一种谱聚类算法，用于分析多元极值的相依结构。具体而言，我们聚焦于极值理论中由角度测度或谱测度表征的多元极值渐近相依性。本研究探讨了基于极值样本构建的随机$k$最近邻图的谱聚类的理论性能，其中极值样本指半径超过大阈值的随机向量的角度分量。特别地，我们推导了线性因子模型下极值的渐近分布，并证明在特定条件下，谱聚类能够一致识别该模型中极值形成的聚类。基于此结果，我们提出了一种用于学习角度测度的简单一致估计策略。理论发现辅以数值实验，展示了所提方法在有限样本下的性能。