Identifying directions where severe events occur is a major challenge in multivariate Extreme Value Analysis. The support of the spectral measure of regularly varying vectors brings out which coordinates contribute to the extremes. This measure is defined via weak convergence which fails at providing an estimator of its support, especially in high dimension. The estimation of the support is all the more challenging since it relies on a threshold above which the data are considered to be extreme and the choice of such a threshold is still an open problem. In this paper we extend the framework of sparse regular variation introduced by Meyer and Wintenberger (2020) to infer tail dependence. This approach relies on the Euclidean projection onto the simplex which exhibits sparsity and reduces the dimension of the extremes' analysis. We provide an algorithmic approach based on model selection to tackle both the choice of an optimal threshold and the learning of relevant directions on which extreme events appear. We apply our method on numerical experiments to highlight the relevance of our findings. Finally we illustrate our approach with financial return data.