Spectral learning of multivariate extremes
This work addresses the challenge of understanding extreme events in multivariate data for fields like risk management or environmental science, representing an incremental advance by applying spectral clustering to a specific theoretical model.
The authors tackled the problem of analyzing dependence in multivariate extremes by proposing a spectral clustering algorithm based on a random k-nearest neighbor graph from extremal samples, and they proved that under certain conditions, it can consistently identify clusters and estimate the angular measure, with numerical experiments validating finite-sample performance.
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.