# Concentration bounds for the empirical angular measure with statistical learning applications

arXiv: https://arxiv.org/abs/2104.03966 · HTML: https://mlsanity.com/paper/2104.03966/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- Builds on 8 of its 61 references that are in the map.
- First posted on arXiv 7 Apr 2021.

## Builds on (most cited first, 8 of 8)

- [Decompositions of Dependence for High-Dimensional Extremes](https://mlsanity.com/paper/1612.07190/) (2016)
- [Sparse Structures for Multivariate Extremes](https://mlsanity.com/paper/2004.12182/) (2020)
- [k-means clustering of extremes](https://mlsanity.com/paper/1904.02970/) (2019)
- [Principal Component Analysis for Multivariate Extremes](https://mlsanity.com/paper/1906.11043/) (2019)
- [Sparsity in Multivariate Extremes with Applications to Anomaly Detection](https://mlsanity.com/paper/1507.05899/) (2015)
- [Sparse regular variation](https://mlsanity.com/paper/1907.00686/) (2019)
- [Modelling across extremal dependence classes](https://mlsanity.com/paper/1408.5060/) (2014)
- [Semi-parametric modeling of excesses above high multivariate thresholds with censored data](https://mlsanity.com/paper/1412.0838/) (2014)

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

- [A VAE Approach to Sample Multivariate Extremes](https://mlsanity.com/paper/2306.10987/) (2023)
- [On Regression in Extreme Regions](https://mlsanity.com/paper/2303.03084/) (2023)
- [Spectral learning of multivariate extremes](https://mlsanity.com/paper/2111.07799/) (2021)
- [Kernel PCA for multivariate extremes](https://mlsanity.com/paper/2211.13172/) (2022)
- [Simulation of Multivariate Extremes: a Wasserstein-Aitchison GAN approach](https://mlsanity.com/paper/2504.21438/) (2025)
- [Extrapolation in Statistical Learning with Extreme Value Theory](https://mlsanity.com/paper/2605.01909/) (2026)
- [Out-of-Distribution generalization of quantile regression with heavy tailed inputs: an SVM approach](https://mlsanity.com/paper/2606.00265/) (2026)
- [Principal Component Analysis for Multivariate Extremes](https://mlsanity.com/paper/2606.07213/) (2026)

## Abstract (arXiv)

The angular measure on the unit sphere characterizes the first-order dependence structure of the components of a random vector in extreme regions and is defined in terms of standardized margins. Its statistical recovery is an important step in learning problems involving observations far away from the center. In the common situation that the components of the vector have different distributions, the rank transformation offers a convenient and robust way of standardizing data in order to build an empirical version of the angular measure based on the most extreme observations. However, the study of the sampling distribution of the resulting empirical angular measure is challenging. It is the purpose of the paper to establish finite-sample bounds for the maximal deviations between the empirical and true angular measures, uniformly over classes of Borel sets of controlled combinatorial complexity. The bounds are valid with high probability and, up to logarithmic factors, scale as the square root of the effective sample size. The bounds are applied to provide performance guarantees for two statistical learning procedures tailored to extreme regions of the input space and built upon the empirical angular measure: binary classification in extreme regions through empirical risk minimization and unsupervised anomaly detection through minimum-volume sets of the sphere.