We consider the problem of recovering conditional independence relationships between $p$ jointly distributed Hilbertian random elements given $n$ realizations thereof. We operate in the sparse high-dimensional regime, where $n \ll p$ and no element is related to more than $d \ll p$ other elements. In this context, we propose an infinite-dimensional generalization of the graphical lasso. We prove model selection consistency under natural assumptions and extend many classical results to infinite dimensions. In particular, we do not require finite truncation or additional structural restrictions. The plug-in nature of our method makes it applicable to any observational regime, whether sparse or dense, and indifferent to serial dependence. Importantly, our method can be understood as naturally arising from a coherent maximum likelihood philosophy.

翻译：我们考虑基于 $n$ 次观测恢复 $p$ 个联合分布的希尔伯特随机元之间条件独立关系的问题。研究场景设定于稀疏高维情形，即 $n \ll p$ 且每个元素至多与 $d \ll p$ 个其他元素存在关联。在此背景下，我们提出图套索的无限维推广。在自然假设下，我们证明了模型选择的一致性，并将众多经典结论推广至无限维空间。特别地，该方法无需有限截断或附加结构性约束。其即插即用特性使其适用于任意观测模式（无论是稀疏还是密集），且对序列相关性不敏感。重要地，该方法可被理解为源于一致的极大似然估计哲学。