Non-parametric assessment of extreme dependence structures between an arbitrary number of variables, though quite well-established in dimension $2$ and recently extended to moderate dimensions such as $5$, still represents a statistical challenge in larger dimensions. Here, we propose a novel approach that combines clustering techniques with angular/spectral measure analysis to find groups of variables (not necessarily disjoint) exhibiting asymptotic dependence, thereby reducing the dimension of the initial problem. A heuristic criterion is proposed to choose the threshold over which it is acceptable to consider observations as extreme and the appropriate number of clusters. When empirically evaluated through numerical experiments, the approach we promote here is found to be very efficient under some regularity constraints, even in dimension $20$. For illustration purpose, we also carry out a case study in dietary risk assessment.
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