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Learning the hub graphical Lasso model with the structured sparsity via an efficient algorithm

Chengjing Wang, Peipei Tang, Wenling He, Meixia Lin

TL;DR

This work tackles learning discriminated hub graphical models (DHGL) in high dimensions, where conventional pADMM methods struggle with accuracy and speed. It introduces a two-phase optimization framework: Phase I uses a dual ADMM (dADMM) to generate a strong initialization, and Phase II applies an augmented Lagrangian method (ALM) with a semismooth Newton (SSN) inner solver to obtain highly accurate solutions by exploiting hub-induced sparsity via a surrogate generalized Jacobian. The authors provide convergence analyses for the dADMM, ALM, and SSN procedures and validate the approach through extensive synthetic and real-world experiments, showing up to 70% reductions in runtime while maintaining high-quality estimates. The results demonstrate scalable, accurate inference for hub-structured Gaussian graphical models with broad applicability to biological networks, social graphs, and financial portfolios.

Abstract

Graphical models have exhibited their performance in numerous tasks ranging from biological analysis to recommender systems. However, graphical models with hub nodes are computationally difficult to fit, particularly when the dimension of the data is large. To efficiently estimate the hub graphical models, we introduce a two-phase algorithm. The proposed algorithm first generates a good initial point via a dual alternating direction method of multipliers (ADMM), and then warm starts a semismooth Newton (SSN) based augmented Lagrangian method (ALM) to compute a solution that is accurate enough for practical tasks. We fully excavate the sparsity structure of the generalized Jacobian arising from the hubs in the graphical models, which ensures that the algorithm can obtain a nice solution very efficiently. Comprehensive experiments on both synthetic data and real data show that it obviously outperforms the existing state-of-the-art algorithms. In particular, in some high dimensional tasks, it can save more than 70\% of the execution time, meanwhile still achieves a high-quality estimation.

Learning the hub graphical Lasso model with the structured sparsity via an efficient algorithm

TL;DR

This work tackles learning discriminated hub graphical models (DHGL) in high dimensions, where conventional pADMM methods struggle with accuracy and speed. It introduces a two-phase optimization framework: Phase I uses a dual ADMM (dADMM) to generate a strong initialization, and Phase II applies an augmented Lagrangian method (ALM) with a semismooth Newton (SSN) inner solver to obtain highly accurate solutions by exploiting hub-induced sparsity via a surrogate generalized Jacobian. The authors provide convergence analyses for the dADMM, ALM, and SSN procedures and validate the approach through extensive synthetic and real-world experiments, showing up to 70% reductions in runtime while maintaining high-quality estimates. The results demonstrate scalable, accurate inference for hub-structured Gaussian graphical models with broad applicability to biological networks, social graphs, and financial portfolios.

Abstract

Graphical models have exhibited their performance in numerous tasks ranging from biological analysis to recommender systems. However, graphical models with hub nodes are computationally difficult to fit, particularly when the dimension of the data is large. To efficiently estimate the hub graphical models, we introduce a two-phase algorithm. The proposed algorithm first generates a good initial point via a dual alternating direction method of multipliers (ADMM), and then warm starts a semismooth Newton (SSN) based augmented Lagrangian method (ALM) to compute a solution that is accurate enough for practical tasks. We fully excavate the sparsity structure of the generalized Jacobian arising from the hubs in the graphical models, which ensures that the algorithm can obtain a nice solution very efficiently. Comprehensive experiments on both synthetic data and real data show that it obviously outperforms the existing state-of-the-art algorithms. In particular, in some high dimensional tasks, it can save more than 70\% of the execution time, meanwhile still achieves a high-quality estimation.
Paper Structure (28 sections, 8 theorems, 89 equations, 6 figures, 8 tables, 5 algorithms)

This paper contains 28 sections, 8 theorems, 89 equations, 6 figures, 8 tables, 5 algorithms.

Table of Contents

  1. Introduction
  2. Additional notations
  3. Preliminaries
  4. Duality and the optimality condition
  5. The proximal mapping and the Moreau envelope
  6. Properties of the negative log-determinant function
  7. Properties of the regularization term $Q(\cdot)$
  8. Properties of the regularization term $R(\cdot)$
  9. A two-phase algorithm
  10. Phase I: dADMM
  11. Phase I I: ALM
  12. An implementable stopping criteria for the ALM subproblems
  13. Convergence results of the ALM
  14. The SSN for solving the ALM subproblem
  15. Numerical issues for solving the subproblem \ref{['linear systems for CG']}
  16. ...and 13 more sections

Key Result

proposition thmcounterproposition

For any $X \in \mathbb{S}^p$, let its eigendecomposition be $X = P \textrm{Diag}(d)P^T$, where $d$ is the vector of eigenvalues in the descending order and $P$ is an orthonormal matrix whose columns are the corresponding eigenvectors. Given $\gamma > 0$, we define two scalar functions We also define their vector counterparts $\phi_{\gamma}^+ (\cdot): \mathbb{R}^p \to \mathbb{R}^p$ and $\phi_{\gam

Figures (6)

  • Figure 1: (a) The inverse covariance matrix in a toy example of a Gaussian graphical model featuring five hub nodes, the inverse covariance matrix displays white elements as zeros and colored elements as non-zeros. Consequently, the colored elements represent the edges in the graph. (b) Estimate from the hub graphical lasso. (c) Graphical lasso estimate
  • Figure 2: Examples of the inverse covariance matrix in three set-ups.
  • Figure 3: The adjacency matrix of graph estimation by using three algorithms. The four subfigures are: (a) the true adjacency matrix (i.e., ground truth), (b) the estimated result by the pADMM, (c) the estimated result by the dADMM, and (d) the estimated result by the two-phase algorithm, respectively.
  • Figure 4: Results of synthetic data efficacy measures
  • Figure 5: The resulting network of student webpages data. The nodes represent 50 words. Detected hub nodes are marked in red.
  • ...and 1 more figures

Theorems & Definitions (16)

  • proposition thmcounterproposition
  • proposition thmcounterproposition
  • proposition thmcounterproposition
  • proposition thmcounterproposition
  • proof
  • definition thmcounterdefinition
  • proposition thmcounterproposition
  • proof
  • remark thmcounterremark
  • theorem 1
  • ...and 6 more