Spectral learning of multivariate extremes

TL;DR

This paper introduces a spectral clustering method tailored for analyzing the dependence structure of multivariate extremes, providing theoretical guarantees and practical estimation strategies for the angular measure in extreme value theory.

Contribution

It develops a spectral clustering approach for multivariate extremes, with proven consistency and a new estimation method for the angular measure based on extremal samples.

Findings

Spectral clustering can consistently identify extremal clusters in multivariate data.

The proposed estimation strategy accurately learns the angular measure in finite samples.

Numerical experiments demonstrate the effectiveness of the method in practical scenarios.

Abstract

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.