In Gaussian graphical model selection, noise-corrupted samples present significant challenges. It is known that even minimal amounts of noise can obscure the underlying structure, leading to fundamental identifiability issues. A recent line of work addressing this "robust model selection" problem narrows its focus to tree-structured graphical models. Even within this specific class of models, exact structure recovery is shown to be impossible. However, several algorithms have been developed that are known to provably recover the underlying tree-structure up to an (unavoidable) equivalence class. In this paper, we extend these results beyond tree-structured graphs. We first characterize the equivalence class up to which general graphs can be recovered in the presence of noise. Despite the inherent ambiguity (which we prove is unavoidable), the structure that can be recovered reveals local clustering information and global connectivity patterns in the underlying model. Such information is useful in a range of real-world problems, including power grids, social networks, protein-protein interactions, and neural structures. We then propose an algorithm which provably recovers the underlying graph up to the identified ambiguity. We further provide finite sample guarantees in the high-dimensional regime for our algorithm and validate our results through numerical simulations.

翻译：在高斯图模型的选择问题中，噪声污染的样本带来了显著挑战。已知即使微量噪声也可能掩盖潜在结构，导致根本性的可辨识性问题。近期针对“鲁棒模型选择”问题的研究将焦点限定于树结构图模型。即使在此类特定模型内，精确结构恢复被证明是不可能的。然而，已有若干算法被证明能够（在不可避免的等价类内）可证明地恢复潜在树结构。本文将这些结果推广至非树结构图。我们首先刻画了噪声环境下一般图可被恢复的等价类特征。尽管存在内在歧义性（我们证明其不可避免），但可恢复的结构揭示了潜在模型中的局部聚类信息与全局连通性模式。这类信息在电力网络、社交网络、蛋白质相互作用及神经结构等实际场景中具有重要价值。随后我们提出一种算法，该算法可证明地恢复至所识别歧义性范围内的潜在图结构。进一步地，我们在高维统计框架下给出了算法的有限样本保证，并通过数值仿真验证了理论结果。