  Generating accurate extremes from an observational data set is crucial when
seeking to estimate risks associated with the occurrence of future extremes
which could be larger than those already observed. Applications range from the
occurrence of natural disasters to financial crashes. Generative approaches
from the machine learning community do not apply to extreme samples without
careful adaptation. Besides, asymptotic results from extreme value theory (EVT)
give a theoretical framework to model multivariate extreme events, especially
through the notion of multivariate regular variation. Bridging these two
fields, this paper details a variational autoencoder (VAE) approach for
sampling multivariate heavy-tailed distributions, i.e., distributions likely to
have extremes of particularly large intensities. We illustrate the relevance of
our approach on a synthetic data set and on a real data set of discharge
measurements along the Danube river network. The latter shows the potential of
our approach for flood risks' assessment. In addition to outperforming the
standard VAE for the tested data sets, we also provide a comparison with a
competing EVT-based generative approach. On the tested cases, our approach
improves the learning of the dependency structure between extremes.
