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Simultaneous Identification of Sparse Structures and Communities in Heterogeneous Graphical Models

Dapeng Shi, Tiandong Wang, Zhiliang Ying

TL;DR

The paper introduces a sparse plus low-rank diagonal-block decomposition of the residual precision matrix in Gaussian graphical models to simultaneously identify sparse edges and non-overlapped communities. It proposes a three-stage estimation procedure—LS-based regression, adaptive-$\ell_1$ penalized estimation for $S$ and $L$, and K-means clustering on the latent-community rows—along with an ADMM algorithm for efficient computation and data-driven tuning. Theoretical contributions include identifiability via tangent-space analysis and an adaptive irrepresentability condition ensuring model-selection consistency, plus a clustering error bound for the final stage. Empirical results on synthetic data and stock market data demonstrate superior performance in recovering community structure and edges, with practical implications for genetics, neuroscience, finance, and beyond.

Abstract

Exploring and detecting community structures hold significant importance in genetics, social sciences, neuroscience, and finance. Especially in graphical models, community detection can encourage the exploration of sets of variables with group-like properties. In this paper, within the framework of Gaussian graphical models, we introduce a novel decomposition of the underlying graphical structure into a sparse part and low-rank diagonal blocks (non-overlapped communities). We illustrate the significance of this decomposition through two modeling perspectives and propose a three-stage estimation procedure with a fast and efficient algorithm for the identification of the sparse structure and communities. Also on the theoretical front, we establish conditions for local identifiability and extend the traditional irrepresentability condition to an adaptive form by constructing an effective norm, which ensures the consistency of model selection for the adaptive $\ell_1$ penalized estimator in the second stage. Moreover, we also provide the clustering error bound for the K-means procedure in the third stage. Extensive numerical experiments are conducted to demonstrate the superiority of the proposed method over existing approaches in estimating graph structures. Furthermore, we apply our method to the stock return data, revealing its capability to accurately identify non-overlapped community structures.

Simultaneous Identification of Sparse Structures and Communities in Heterogeneous Graphical Models

TL;DR

The paper introduces a sparse plus low-rank diagonal-block decomposition of the residual precision matrix in Gaussian graphical models to simultaneously identify sparse edges and non-overlapped communities. It proposes a three-stage estimation procedure—LS-based regression, adaptive- penalized estimation for and , and K-means clustering on the latent-community rows—along with an ADMM algorithm for efficient computation and data-driven tuning. Theoretical contributions include identifiability via tangent-space analysis and an adaptive irrepresentability condition ensuring model-selection consistency, plus a clustering error bound for the final stage. Empirical results on synthetic data and stock market data demonstrate superior performance in recovering community structure and edges, with practical implications for genetics, neuroscience, finance, and beyond.

Abstract

Exploring and detecting community structures hold significant importance in genetics, social sciences, neuroscience, and finance. Especially in graphical models, community detection can encourage the exploration of sets of variables with group-like properties. In this paper, within the framework of Gaussian graphical models, we introduce a novel decomposition of the underlying graphical structure into a sparse part and low-rank diagonal blocks (non-overlapped communities). We illustrate the significance of this decomposition through two modeling perspectives and propose a three-stage estimation procedure with a fast and efficient algorithm for the identification of the sparse structure and communities. Also on the theoretical front, we establish conditions for local identifiability and extend the traditional irrepresentability condition to an adaptive form by constructing an effective norm, which ensures the consistency of model selection for the adaptive penalized estimator in the second stage. Moreover, we also provide the clustering error bound for the K-means procedure in the third stage. Extensive numerical experiments are conducted to demonstrate the superiority of the proposed method over existing approaches in estimating graph structures. Furthermore, we apply our method to the stock return data, revealing its capability to accurately identify non-overlapped community structures.
Paper Structure (46 sections, 18 theorems, 199 equations, 6 figures, 4 tables, 1 algorithm)

This paper contains 46 sections, 18 theorems, 199 equations, 6 figures, 4 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Notation
  3. Graphical models with community-based decomposition
  4. Latent community graphical models
  5. Grouped latent variable graphical models
  6. Proposed method
  7. A three-stage estimation procedure
  8. Algorithm
  9. Weight assignment and choices of tuning parameters
  10. Theoretical properties
  11. Identifiability
  12. Asymptotic analysis
  13. Simulation study
  14. Second stage: adaptive $\ell_1$ penalized estimation
  15. Third stage: $K$-means clustering
  16. ...and 31 more sections

Key Result

Proposition 1

The tangent space at any smooth point $L= U_1\Sigma U_1^T\in\mathcal{LS}(m, r)$ is given by where $\mathcal{T}_1(L) = \left\{U_1Y^T + YU_1^T: Y\in\mathbb{R}^{p\times r} \right\}, \mathcal{T}_2(L^*) = \{ N = \operatorname{diag}(N_1,\cdots, N_m): \operatorname{Supp}(N_i)\subseteq \operatorname{Supp}(L_i) \}$. Moreover, with $\mathcal{V}(L) = \left\{ N = \operatorname{diag}(N_1,\cdots, N_m): N_i\

Figures (6)

  • Figure 1: Comparison of Hamming error rates under different values of $a$ (representing the strengths of eigenvalues of latent community part $L$). Left panel: clustering based on $\widehat{L}$. Right panel: clustering based on $\operatorname{Cor}(\operatorname{abs}(\widehat{L}))$. $x$-axis: different values of a. $y$-axis: Hamming error rates.
  • Figure 2: Comparison of Hamming error rates under different sample sizes. Left panel: clustering based on $\widehat{L}$. Right panel: clustering based on $\operatorname{Cor}(\operatorname{abs}(\widehat{L}))$. $x$-axis: different values of sample size. $y$-axis: Hamming error rates.
  • Figure 3: Heatmaps of latent community graph estimated by LVGGM method and proposed method respectively. The top 15 companies on the vertical axis are from the energy sector, the middle 15 are from the financial sector, and the bottom 15 are from the healthcare sector.
  • Figure 4: Latent community graphs estimated by LVGGM method and proposed method respectively. Green represents the healthcare sector, pink represents the financial sector, and blue represents the energy sector.
  • Figure 5: Heatmaps of sparse graph estimated by LVGGM method and proposed method respectively. The top 15 companies on the vertical axis are from the energy sector, the middle 15 are from the financial sector, and the bottom 15 are from the healthcare sector.
  • ...and 1 more figures

Theorems & Definitions (32)

  • Proposition 1
  • Theorem 1
  • Remark 1
  • Remark 2
  • Remark 3
  • Theorem 2
  • Theorem 3
  • Remark 4
  • Corollary 1
  • Theorem 4
  • ...and 22 more