Kernel PCA for multivariate extremes

TL;DR

This paper introduces kernel PCA as a novel approach for analyzing the dependence structure of multivariate extremes, providing theoretical insights and practical guarantees for clustering and dimension reduction in extreme value analysis.

Contribution

It offers a new theoretical understanding of kernel PCA preimages for multivariate extremes and characterizes its performance under asymptotic dependence models.

Findings

Kernel PCA effectively identifies clusters in extreme value data.

Theoretical guarantees are established for kernel PCA preimages in extreme value analysis.

Numerical experiments demonstrate finite sample performance of the proposed method.

Abstract

We propose kernel PCA as a method for analyzing the dependence structure of multivariate extremes and demonstrate that it can be a powerful tool for clustering and dimension reduction. Our work provides some theoretical insight into the preimages obtained by kernel PCA, demonstrating that under certain conditions they can effectively identify clusters in the data. We build on these new insights to characterize rigorously the performance of kernel PCA based on an extremal sample, i.e., the angular part of random vectors for which the radius exceeds a large threshold. More specifically, we focus on the asymptotic dependence of multivariate extremes characterized by the angular or spectral measure in extreme value theory and provide a careful analysis in the case where the extremes are generated from a linear factor model. We give theoretical guarantees on the performance of kernel PCA preimages of such extremes by leveraging their asymptotic distribution together with Davis-Kahan perturbation bounds. Our theoretical findings are complemented with numerical experiments illustrating the finite sample performance of our methods.