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数据的条件独立结构，为量化罕见事件风险提供了有力工具。现有从数据中学习此类图结构的研究主要聚焦于所有相关变量均可观测的场景。针对广泛使用的Hüsler-Reiss模型族，我们提出\texttt{eglatent}方法——一种用于在含隐变量条件下学习极值图模型的可处理凸优化算法。该方法将Hüsler-Reiss精度矩阵分解为两个组成部分：稀疏分量编码在隐变量条件约束下观测变量间的图结构，以及低秩分量编码少量隐变量对观测变量的影响。我们提供了\texttt{eglatent}的有限样本保证，证明其能一致地恢复条件图结构及隐变量个数。在合成数据与真实数据上的实验进一步凸显了本方法的性能优势。