Robust Model Selection of Gaussian Graphical Models

TL;DR

This paper advances robust model selection for Gaussian graphical models by extending recovery guarantees beyond trees, characterizing the recoverable structure under noise, and proposing an algorithm with finite sample guarantees.

Contribution

It generalizes robust structure recovery from trees to arbitrary graphs, characterizes the inherent ambiguity, and introduces an algorithm with provable guarantees.

Findings

Recovery of graph structure up to an equivalence class in noisy settings

Algorithm provably recovers the underlying graph within the identified ambiguity

Finite sample guarantees demonstrated through numerical simulations

Abstract

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.