  Extremal graphical models encode the conditional independence structure of
multivariate extremes and provide a powerful tool for quantifying the risk of
rare events. Prior work on learning these graphs from data has focused on the
setting where all relevant variables are observed. For the popular class of
H\"usler-Reiss models, we propose the \texttt{eglatent} method, a tractable
convex program for learning extremal graphical models in the presence of latent
variables. Our approach decomposes the H\"usler-Reiss precision matrix into a
sparse component encoding the graphical structure among the observed variables
after conditioning on the latent variables, and a low-rank component encoding
the effect of a few latent variables on the observed variables. We provide
finite-sample guarantees of \texttt{eglatent} and show that it consistently
recovers the conditional graph as well as the number of latent variables. We
highlight the improved performances of our approach on synthetic and real data.
