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依赖结构。具体而言，我们聚焦于极值理论中由角度或谱测度表征的多元极值渐近依赖性。本研究基于从极值样本（即半径超过大阈值的随机向量的角度部分）构建的随机$k$-近邻图，探讨了谱聚类的理论性能。特别地，我们推导了线性因子模型中极值的渐近分布，并证明在特定条件下，谱聚类能够一致地识别该模型中极值的聚类结构。基于这一结果，我们提出了一种简单且一致的角度测度估计策略。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。