  Identifying directions where extreme events occur is a major challenge in
multivariate extreme value analysis. In this paper, we use the concept of
sparse regular variation introduced by Meyer and Wintenberger (2021)} to infer
the tail dependence of a random vector X. This approach relies on the Euclidean
projection onto the simplex which better exhibits the sparsity structure of the
tail of X than the standard methods. Our procedure based on a rigorous
methodology aims at capturing clusters of extremal coordinates of X. It also
includes the identification of the threshold above which the values taken by X
are considered as extreme. We provide an efficient and scalable algorithm
called MUSCLE and apply it on numerical examples to highlight the relevance of
our findings. Finally we illustrate our approach with financial return data.
