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.

翻译：极值图模型编码了多元极值的条件独立结构，为量化罕见事件风险提供了有力工具。先前从数据中学习这些图模型的研究主要集中于所有相关变量均可观测的情境。针对流行的Hüsler-Reiss模型类，我们提出了\texttt{eglatent}方法——一种可处理的凸优化程序，用于在存在潜变量的情况下学习极值图模型。我们的方法将Hüsler-Reiss精度矩阵分解为两个部分：一个稀疏分量，编码在给定潜变量条件下观测变量间的图结构；一个低秩分量，编码少数潜变量对观测变量的影响。我们为\texttt{eglatent}提供了有限样本的理论保证，证明其能够一致地恢复条件图结构及潜变量数量。我们通过合成数据与真实数据展示了该方法性能的提升。