  Extreme value statistics provides accurate estimates for the small occurrence
probabilities of rare events. While theory and statistical tools for univariate
extremes are well-developed, methods for high-dimensional and complex data sets
are still scarce. Appropriate notions of sparsity and connections to other
fields such as machine learning, graphical models and high-dimensional
statistics have only recently been established. This article reviews the new
domain of research concerned with the detection and modeling of sparse patterns
in rare events. We first describe the different forms of extremal dependence
that can arise between the largest observations of a multivariate random
vector. We then discuss the current research topics including clustering,
principal component analysis and graphical modeling for extremes.
Identification of groups of variables which can be concomitantly extreme is
also addressed. The methods are illustrated with an application to flood risk
assessment.
