# Sparse regular variation

arXiv: https://arxiv.org/abs/1907.00686 · HTML: https://mlsanity.com/paper/1907.00686/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 1 Jul 2019.

## Built on by (most cited first, 8 of 8)

- [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)
- [Multi-site modelling and reconstruction of past extreme skew surges along the French Atlantic coast](https://mlsanity.com/paper/2505.00835/) (2025)
- [Principal Component Analysis for Multivariate Extremes](https://mlsanity.com/paper/2606.07213/) (2026)

## Abstract (arXiv)

Regular variation provides a convenient theoretical framework to study large events. In the multivariate setting, the dependence structure of the positive extremes is characterized by a measure - the spectral measure - defined on the positive orthant of the unit sphere. This measure gathers information on the localization of extreme events and has often a sparse support since severe events do not simultaneously occur in all directions. However, it is defined through weak convergence which does not provide a natural way to capture this sparsity structure.In this paper, we introduce the notion of sparse regular variation which allows to better learn the dependence structure of extreme events. This concept is based on the Euclidean projection onto the simplex for which efficient algorithms are known. We prove that under mild assumptions sparse regular variation and regular variation are two equivalent notions and we establish several results for sparsely regularly varying random vectors. Finally, we illustrate on numerical examples how this new concept allows one to detect extremal directions.