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Robust Model Selection of Gaussian Graphical Models

Abrar Zahin, Rajasekhar Anguluri, Lalitha Sankar, Oliver Kosut, Gautam Dasarathy

TL;DR

This work tackles robust model selection for Gaussian graphical models under unknown additive noise, showing that exact graph recovery is impossible while identifiability can be achieved up to an equivalence class captured by an articulated set tree (AST). It introduces NoMAD, an ancestor-discovery algorithm that reconstructs the AST and recovers the equivalence class [G], with rigorous population and finite-sample guarantees. The approach leverages a Test Identical Ancestor (TIA), distance extensions to latent ancestors, and clustering of leaf/internal blocks to reveal local and global graph structure in noisy settings. Experimental results on synthetic and IEEE 33-bus networks demonstrate NoMAD’s robustness to noise and its advantage over traditional methods like GLASSO in preserving the correct equivalence class. The findings enable meaningful partial structure recovery useful for applications in power grids, neuroscience, and complex networks where measurement noise is inevitable.

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.

Robust Model Selection of Gaussian Graphical Models

TL;DR

This work tackles robust model selection for Gaussian graphical models under unknown additive noise, showing that exact graph recovery is impossible while identifiability can be achieved up to an equivalence class captured by an articulated set tree (AST). It introduces NoMAD, an ancestor-discovery algorithm that reconstructs the AST and recovers the equivalence class [G], with rigorous population and finite-sample guarantees. The approach leverages a Test Identical Ancestor (TIA), distance extensions to latent ancestors, and clustering of leaf/internal blocks to reveal local and global graph structure in noisy settings. Experimental results on synthetic and IEEE 33-bus networks demonstrate NoMAD’s robustness to noise and its advantage over traditional methods like GLASSO in preserving the correct equivalence class. The findings enable meaningful partial structure recovery useful for applications in power grids, neuroscience, and complex networks where measurement noise is inevitable.

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.
Paper Structure (17 sections, 20 theorems, 27 equations, 7 figures, 5 algorithms)

This paper contains 17 sections, 20 theorems, 27 equations, 7 figures, 5 algorithms.

Table of Contents

  1. Introduction
  2. Related Work
  3. Preliminaries and Problem Statement
  4. The Robust Model Selection Problem
  5. An Identifiability Result
  6. The Robust Model Selection Algorithm
  7. The NoMAD Algorithm
  8. Performance Analysis of NoMAD in the Population Setting
  9. Performance Analysis of NoMAD in Finite Sample Setting
  10. Experiments
  11. Conclusion and Future Directions
  12. Algorithmic Details
  13. Learning $\mathcal{P}_{\rm algo}{}$ and $A_{\rm algo}{}$ for $\mathcal{T}_{\rm algo}{}$
  14. Learning $E_{\rm algo}{}$ for $\mathcal{T}_{\rm algo}{}$
  15. Theory: Guaranteeing the Correctness of the NoMAD
  16. ...and 2 more sections

Key Result

Theorem 2.1

Fix a covariance matrix $\Sigma^*$ whose conditional independence structure is given by the graph $G$. Fix $D$, where $D=\textrm{diag} (D_{11},\ldots,D_{pp})\geq 0$. Let $\Sigma^{o}=\Sigma^*+D$. Then, there exists at least one $H\in [G]$ such that $\Sigma^{o}$ can be written as $\Sigma^{q}+D^{q}$, w

Figures (7)

  • Figure 1: (a) a true underlying graph where both $B_1 \cup \{6,8\}$ ($B_2 \cup \{7,9\}$) are non-trivial blocks where $B_1$ ($B_2$) is an arbitrary set of vertices such that the subgraph on $B_1 \cup \{6,8\}$ ($B_2 \cup \{7,9\}$) is a biconnected component, (b) joint graph $G^{\textsc{j}}{}$; noisy vertices associated with the non-trivial blocks containing $B_1$ and $B_2$, and some other vertices are not numbered to reduce the clutter, and (c) the articulated set tree $\mathcal{T}_{\textsc{bc}}{(G{})}$.
  • Figure 2: An illustration of three graphs from the same equivalence class of $G$ in Fig. \ref{['fig:true_graph (zero_corrupted)']}. For all three graphs, $B_1 \cup \{6,8\}$ ($B_2 \cup \{7,9\}$) can have any induced subgraph as long as subgraphs on $B_1 \cup \{6,8\}$ ($B_2 \cup \{7,9\}$) is a biconnected component.
  • Figure 3: Graph with multiple minimal mutual separators.
  • Figure 4: (a) The joint graph $G^{\textsc{j}}{}$, (b) the leaf clusters and internal clusters of $G^{\textsc{j}}{}$; $B_1^e$ ($B_2^e$) denote the set of noisy vertices associated with the vertices in $B_1$ ($B_2$); Also, recall that clusters are a set of vertices; grey vertices are identified but unlabeled articulation points associated with the clusters, (c) non-trivial blocks and trivial blocks along with the identified and labeled articulation points, and (d) the edges between different articulation points.
  • Figure 5: Synthetic graph and IEEE-33 Bus system considered for our simulation
  • ...and 2 more figures

Theorems & Definitions (57)

  • Definition 2.1: Articulated Set Tree
  • Definition 2.2: Equivalence Relation, $\sim$
  • Theorem 2.1: Identifiability
  • Definition 2.3: Information distances
  • Definition 2.4: Minimal mutual separators, Star triplets, Ancestors
  • Definition 2.5: TIA Test
  • Theorem 3.2
  • Theorem 4.3
  • Remark 4.1
  • Theorem B.1
  • ...and 47 more