# Regular Variation in Hilbert Spaces and Principal Component Analysis for Functional Extremes

arXiv: https://arxiv.org/abs/2308.01023 · HTML: https://mlsanity.com/paper/2308.01023/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 11 of its 60 references that are in the map.
- First posted on arXiv 2 Aug 2023.

## Builds on (most cited first, 11 of 11)

- [Causal discovery in heavy-tailed models](https://mlsanity.com/paper/1908.05097/) (2019)
- [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)
- [Advances in Statistical Modeling of Spatial Extremes](https://mlsanity.com/paper/2007.00774/) (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)
- [Determining the Dependence Structure of Multivariate Extremes](https://mlsanity.com/paper/1809.01606/) (2018)
- [A Multivariate Extreme Value Theory Approach to Anomaly Clustering and Visualization](https://mlsanity.com/paper/1907.07523/) (2019)
- [Spectral learning of multivariate extremes](https://mlsanity.com/paper/2111.07799/) (2021)

## Built on by (most cited first, 3 of 3)

- [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)
- [Principal Component Analysis for Multivariate Extremes](https://mlsanity.com/paper/2606.07213/) (2026)

## Abstract (arXiv)

Motivated by the increasing availability of data of functional nature, we develop a general probabilistic and statistical framework for extremes of regularly varying random elements X in L2[0,1]. We place ourselves in a Peaks-Over-Threshold framework where a functional extreme is defined as an observation X whose L2-norm |X| is comparatively large. Our goal is to propose a dimension reduction framework resulting into finite dimensional projections for such extreme observations. Our contribution is double. First, we investigate the notion of Regular Variation for random quantities valued in a general separable Hilbert space, for which we propose a novel concrete characterization involving solely stochastic convergence of real-valued random variables. Second, we propose a notion of functional Principal Component Analysis (PCA) accounting for the principal `directions' of functional extremes. We investigate the statistical properties of the empirical covariance operator of the angular component of extreme functions, by upper-bounding the Hilbert-Schmidt norm of the estimation error for finite sample sizes. Numerical experiments with simulated and real data illustrate this work.