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：我们考虑从$p$个联合分布的希尔伯特随机元素的$n$次实现中恢复条件独立关系的问题。我们在稀疏高维场景下开展工作，其中$n \ll p$且每个元素至多与$d \ll p$个其他元素相关。在此背景下，我们提出图拉索的无穷维推广。我们在自然假设下证明模型选择一致性，并将许多经典结果推广至无穷维情形。特别地，我们不需要有限截断或额外结构限制。我们的方法基于插件性质，可适用于任何观测场景（无论是稀疏还是密集），且对序列依赖性不敏感。重要的是，我们的方法可理解为源自一致最大似然哲学的自然产物。