Learning structures in extreme values in high dimension.

Summary We present and study methods for unsupervised learning of multivariate extreme phenomena in high dimension. In the case where each of the marginal distributions of a random vector is heavy-tailed, the study of its behavior in extreme regions (i.e. far from the origin) can no longer be done via the usual methods which assume a finite mean and variance. Extreme value theory then offers a framework adapted to this study, by giving in particular a theoretical basis to the reduction of dimension through the angular measurement. The thesis is structured around two main steps: - Reducing the dimension of the problem by finding a summary of the dependence structure in the extreme regions. In particular, this step aims at finding the subgroups of components that are likely to exceed a high threshold simultaneously. - Modeling the angular measurement by a mixing density that follows a predetermined dependency structure. These two steps allow the development of unsupervised classification methods through the construction of a similarity matrix for the extreme points.