Concentration bounds for the empirical angular measure with statistical learning applications

Stephan Clémençon (1, 2) , Hamid Jalalzai (1, 2) , Stéphane Lhaut (3) , Anne Sabourin (4) , Johan Segers (3)

1 IDS - Département Images, Données, Signal
2 S2A - Signal, Statistique et Apprentissage
3 Institut de Statistique, Biostatistique et Sciences Actuarielles (ISBA)
4 MAP5 - UMR 8145 - Mathématiques Appliquées Paris 5

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 minimumvolume sets of the sphere.

Keywords: MSC2020 subject classifications: Primary 62G05 62G30 62G32 secondary 62H30; angular measure; classification; concentration inequality; extreme value analysis; minimum-volume sets; ranks

Domains: Statistics [math.ST]; Applications [stat.AP]; Methodology [stat.ME]; Statistics Theory [stat.TH]; Machine Learning [stat.ML]

Stephan Clémençon, Hamid Jalalzai, Stéphane Lhaut, Anne Sabourin, Johan Segers. Concentration bounds for the empirical angular measure with statistical learning applications. 2023. ⟨hal-03920536⟩