Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Efficient Graph Laplacian Estimation by Proximal Newton

Yakov Medvedovsky, Eran Treister, Tirza Routtenberg

TL;DR

This work tackles the problem of learning sparse graph Laplacians under the LGMRF model, where the precision matrix must be a Laplacian and standard $\ell_1$ regularization is biased for this structure.The authors propose NewGLE, a proximal Newton method that preserves Laplacian constraints and applies a nonconvex MCP penalty to promote sparsity, using a second-order quadratic approximation of the smooth objective and an inner nonlinear projected CG solver with diagonal preconditioning.Key innovations include a Laplacian parameterization to reduce variables, a free-set strategy to limit updates, and a robust convergence analysis showing that the method converges to a stationary point with monotone objective decrease.Numerical experiments on synthetic graphs and real datasets show that NewGLE achieves higher accuracy (better RE and F-score) and significantly faster runtimes than competing approaches, especially in high-dimensional, low-sample regimes.

Abstract

The Laplacian-constrained Gaussian Markov Random Field (LGMRF) is a common multivariate statistical model for learning a weighted sparse dependency graph from given data. This graph learning problem can be formulated as a maximum likelihood estimation (MLE) of the precision matrix, subject to Laplacian structural constraints, with a sparsity-inducing penalty term. This paper aims to solve this learning problem accurately and efficiently. First, since the commonly used $\ell_1$-norm penalty is inappropriate in this setting and may lead to a complete graph, we employ the nonconvex minimax concave penalty (MCP), which promotes sparse solutions with lower estimation bias. Second, as opposed to existing first-order methods for this problem, we develop a second-order proximal Newton approach to obtain an efficient solver, utilizing several algorithmic features, such as using Conjugate Gradients, preconditioning, and splitting to active/free sets. Numerical experiments demonstrate the advantages of the proposed method in terms of both computational complexity and graph learning accuracy compared to existing methods.

Efficient Graph Laplacian Estimation by Proximal Newton

TL;DR

This work tackles the problem of learning sparse graph Laplacians under the LGMRF model, where the precision matrix must be a Laplacian and standard $\ell_1$ regularization is biased for this structure.The authors propose NewGLE, a proximal Newton method that preserves Laplacian constraints and applies a nonconvex MCP penalty to promote sparsity, using a second-order quadratic approximation of the smooth objective and an inner nonlinear projected CG solver with diagonal preconditioning.Key innovations include a Laplacian parameterization to reduce variables, a free-set strategy to limit updates, and a robust convergence analysis showing that the method converges to a stationary point with monotone objective decrease.Numerical experiments on synthetic graphs and real datasets show that NewGLE achieves higher accuracy (better RE and F-score) and significantly faster runtimes than competing approaches, especially in high-dimensional, low-sample regimes.

Abstract

The Laplacian-constrained Gaussian Markov Random Field (LGMRF) is a common multivariate statistical model for learning a weighted sparse dependency graph from given data. This graph learning problem can be formulated as a maximum likelihood estimation (MLE) of the precision matrix, subject to Laplacian structural constraints, with a sparsity-inducing penalty term. This paper aims to solve this learning problem accurately and efficiently. First, since the commonly used -norm penalty is inappropriate in this setting and may lead to a complete graph, we employ the nonconvex minimax concave penalty (MCP), which promotes sparse solutions with lower estimation bias. Second, as opposed to existing first-order methods for this problem, we develop a second-order proximal Newton approach to obtain an efficient solver, utilizing several algorithmic features, such as using Conjugate Gradients, preconditioning, and splitting to active/free sets. Numerical experiments demonstrate the advantages of the proposed method in terms of both computational complexity and graph learning accuracy compared to existing methods.
Paper Structure (28 sections, 6 theorems, 60 equations, 7 figures, 2 algorithms)

This paper contains 28 sections, 6 theorems, 60 equations, 7 figures, 2 algorithms.

Table of Contents

  1. INTRODUCTION
  2. PROBLEM FORMULATION
  3. GMRF and Constrained MLE
  4. Sparsity Promoting Penalty Functions
  5. METHOD
  6. Proximal Newton for Graph Laplacian Estimation
  7. Laplacian parameterization
  8. Restricting the updates to free sets
  9. Solution of the Newton problem
  10. MORE RELATED WORKS
  11. EXPERIMENTAL RESULTS
  12. Experimental setup
  13. Results - graph learning performance
  14. Results - Computational Costs
  15. Ablation Study
  16. ...and 13 more sections

Key Result

Theorem 1

The optimization problem where the feasible set is given by is equivalent to the optimization problem stated in minimization_problem_3.

Figures (7)

  • Figure 1: Performance comparisons under (a) RE, (b) F-score with different sample size ratios $n/p$ for planar graphs with 1,000 nodes.
  • Figure 2: Performance comparisons in terms of (a) RE, and (b) F-score with different sample size ratios $n/p$ for Barabasi-Albert graph of degree 2 with 100 nodes.
  • Figure 3: Run-time performance comparisons under RE with a sample size ratio (a) $n/p = 0.5$ and (b) $n/p = 15$ for random planar graphs with 1,000 nodes.
  • Figure 4: Run-time performance comparisons under F-score with a sample size ratio (a) $n/p = 0.5$ and (b) $n/p = 15$ for random planar graphs with 1,000 nodes.
  • Figure 5: Average convergence times over sample size ratio $n/p$ for random planar graphs with 1,000 nodes.
  • ...and 2 more figures

Theorems & Definitions (12)

  • Theorem 1
  • Theorem 2
  • Definition 1
  • proof
  • Lemma 1
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Lemma 4
  • ...and 2 more