# Decompositions of Dependence for High-Dimensional Extremes

arXiv: https://arxiv.org/abs/1612.07190 · HTML: https://mlsanity.com/paper/1612.07190/

## Key facts

- Cited by 13 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 20 Dec 2016.

## Built on by (most cited first, 13 of 13)

- [Regression modelling of spatiotemporal extreme U.S. wildfires via partially-interpretable neural networks](https://mlsanity.com/paper/2208.07581/) (2022)
- [Concentration bounds for the empirical angular measure with statistical learning applications](https://mlsanity.com/paper/2104.03966/) (2021)
- [On Regression in Extreme Regions](https://mlsanity.com/paper/2303.03084/) (2023)
- [Spectral learning of multivariate extremes](https://mlsanity.com/paper/2111.07799/) (2021)
- [Regular Variation in Hilbert Spaces and Principal Component Analysis for Functional Extremes](https://mlsanity.com/paper/2308.01023/) (2023)
- [Kernel PCA for multivariate extremes](https://mlsanity.com/paper/2211.13172/) (2022)
- [High-dimensional variable clustering based on maxima of a weakly dependent random process](https://mlsanity.com/paper/2302.00934/) (2023)
- [Simulation of Multivariate Extremes: a Wasserstein-Aitchison GAN approach](https://mlsanity.com/paper/2504.21438/) (2025)
- [Multi-site modelling and reconstruction of past extreme skew surges along the French Atlantic coast](https://mlsanity.com/paper/2505.00835/) (2025)
- [Extrapolation in Statistical Learning with Extreme Value Theory](https://mlsanity.com/paper/2605.01909/) (2026)
- [Canonical Tail Dependence for Soft Extremal Clustering of Multichannel Brain Signals](https://mlsanity.com/paper/2512.06435/) (2025)
- [Principal Component Analysis for Multivariate Extremes](https://mlsanity.com/paper/2606.07213/) (2026)
- [Low-Rank Dependence Decomposition via Accelerated Symmetric Non-negative Matrix Factorization](https://mlsanity.com/paper/2607.24518/) (2026)

## Abstract (arXiv)

Employing the framework of regular variation, we propose two decompositions which help to summarize and describel high-dimensional tail dependence. Via transformation, we define a vector space on the positive orthant, yielding the notion of basis. With a suitably-chosen transformation, we show that transformed-linear operations applied to regularly varying random vectors preserve regular variation. Rather than model regular-variation's angular measure, we summarize tail dependence via a matrix of pairwise tail dependence metrics. This matrix is positive semidefinite, and eigendecomposition allows one to interpret tail dependence via the resulting eigenbasis. Additionally this matrix is completely positive, and a resulting decomposition allows one to easily construct regularly varying random vectors which share the same pairwise tail dependencies. We illustrate our methods with Swiss rainfall data and financial return data.