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Polynomial Graphical Lasso: Learning Edges from Gaussian Graph-Stationary Signals

Andrei Buciulea, Jiaxi Ying, Antonio G. Marques, Daniel P. Palomar

TL;DR

The paper tackles learning graph topology from nodal signals when the data are Gaussian and graph-stationary, by introducing Polynomial Graphical Lasso (PGL) that jointly estimates the graph shift operator ${\mathbf S}$ and a precision matrix ${\boldsymbol{\Theta}}$ that can be a polynomial in ${\mathbf S}$. It formulates a biconvex constrained optimization that generalizes graphical Lasso and graph-stationarity, and develops a low-complexity alternating algorithm with convergence guarantees to a coordinatewise minimum. Through extensive synthetic and real-data experiments, PGL shows improved graph recovery and improved downstream tasks (clustering, investing) over GL and GSR, while maintaining scalable computation. The work provides a flexible, statistically principled framework for graph learning from stationary Gaussian graph signals with practical convergence and implementational benefits for large-scale graphs.

Abstract

This paper introduces Polynomial Graphical Lasso (PGL), a new approach to learning graph structures from nodal signals. Our key contribution lies in modeling the signals as Gaussian and stationary on the graph, enabling the development of a graph-learning formulation that combines the strengths of graphical lasso with a more encompassing model. Specifically, we assume that the precision matrix can take any polynomial form of the sought graph, allowing for increased flexibility in modeling nodal relationships. Given the resulting complexity and nonconvexity of the resulting optimization problem, we (i) propose a low-complexity algorithm that alternates between estimating the graph and precision matrices, and (ii) characterize its convergence. We evaluate the performance of PGL through comprehensive numerical simulations using both synthetic and real data, demonstrating its superiority over several alternatives. Overall, this approach presents a significant advancement in graph learning and holds promise for various applications in graph-aware signal analysis and beyond.

Polynomial Graphical Lasso: Learning Edges from Gaussian Graph-Stationary Signals

TL;DR

The paper tackles learning graph topology from nodal signals when the data are Gaussian and graph-stationary, by introducing Polynomial Graphical Lasso (PGL) that jointly estimates the graph shift operator and a precision matrix that can be a polynomial in . It formulates a biconvex constrained optimization that generalizes graphical Lasso and graph-stationarity, and develops a low-complexity alternating algorithm with convergence guarantees to a coordinatewise minimum. Through extensive synthetic and real-data experiments, PGL shows improved graph recovery and improved downstream tasks (clustering, investing) over GL and GSR, while maintaining scalable computation. The work provides a flexible, statistically principled framework for graph learning from stationary Gaussian graph signals with practical convergence and implementational benefits for large-scale graphs.

Abstract

This paper introduces Polynomial Graphical Lasso (PGL), a new approach to learning graph structures from nodal signals. Our key contribution lies in modeling the signals as Gaussian and stationary on the graph, enabling the development of a graph-learning formulation that combines the strengths of graphical lasso with a more encompassing model. Specifically, we assume that the precision matrix can take any polynomial form of the sought graph, allowing for increased flexibility in modeling nodal relationships. Given the resulting complexity and nonconvexity of the resulting optimization problem, we (i) propose a low-complexity algorithm that alternates between estimating the graph and precision matrices, and (ii) characterize its convergence. We evaluate the performance of PGL through comprehensive numerical simulations using both synthetic and real data, demonstrating its superiority over several alternatives. Overall, this approach presents a significant advancement in graph learning and holds promise for various applications in graph-aware signal analysis and beyond.
Paper Structure (17 sections, 2 theorems, 47 equations, 4 figures, 1 table, 3 algorithms)

This paper contains 17 sections, 2 theorems, 47 equations, 4 figures, 1 table, 3 algorithms.

Table of Contents

  1. Introduction
  2. Graphs and Random Graph Signals
  3. Graph learning problem formulation
  4. Biconvex relation and algorithm design
  5. Biconvex relaxation
  6. Solving subproblem for ${\mathbf S}$
  7. Solving subproblem for $\boldsymbol{\Theta}$
  8. Graph-learning algorithm and convergence analysis
  9. Numerical experiments
  10. Test case 1: Estimation error vs. number of samples for multiple synthetic scenarios.
  11. Test case 2: Noisy GMRF graph signals.
  12. Test case 3: Computational complexity.
  13. Test case 4: Real data scenarios.
  14. Conclusions
  15. Computations of projections
  16. ...and 2 more sections

Key Result

Theorem 1

Let $\{(\hat{\boldsymbol{\Theta}}^{(k)}, {\hat{\mathbf S} }^{(k)} ) \}_{k \in \mathbb{N}}$ be a sequence generated by Algorithm A:BSUM. Under Assumptions assumption1 and assumption2, every limit point of $\{(\hat{\boldsymbol{\Theta}}^{(k)}, {\hat{\mathbf S} }^{(k)} ) \}_{k \in \mathbb{N}}$ is a bloc

Figures (4)

  • Figure 1: Graph estimation error $\mathrm{nme}({\mathbf S}^*,{\hat{\mathbf S} })$ vs. number of samples $R$ for different graph learning approaches (PGL, GL, and GSR) and simulations setups. Specifically, the six lines reported in each subplot correspond to the combination of (a) 3 different graph learning methods and 2 different covariance setups (SSEM and Poly) for a noise-free scenario; (b) 3 different graph learning schemes and 2 noise levels $\sigma\in\{0.05,0.2\}$ for a Poly setup; and (c) 3 graph generation models and 2 covariance models (SSEM and Poly) for the PGL algorithm in a noise-free scenario.
  • Figure 2: Graph estimation error $\mathrm{nme}({\mathbf S}^*,{\hat{\mathbf S} })$ vs. number of samples $R$ for data generated according to a GMRF. We consider three different graph learning schemes (PGL, GL, and GSR) and two levels of additive white noise ($sigma\in\{0.05,0.2\}$), giving rise to the six lines in the figure.
  • Figure 3: The experiment considers 40 companies of the S&P 500 from 4 different sectors, using as nodal signals the daily returns in the period 2010-2015. Four different graph-learning schemes are considered: correlation networks (Corr), PGL, GL and GSR. After learning the graph, a spectral clustering method is implemented. The figure shows the normalized node clustering error (fraction of wrongly clustering nodes) as the percentage of available signals to learn the graph increases.
  • Figure 4: This experiment learns the graph connecting the 7 FAAMUNG stocks using as signals the closing price from July 2019 to May 2020. Three different graph learning methods are used (PGL, GL, and GSR) and a different graph is learned for every day (using the signals of the previous 30 days). Subplot (a) shows the value of the algebraic connectivity ($\lambda_2$) associated with each one of the $200\times 3$ estimated graphs. Subplot (b) shows the value of the portfolio for 4 different investment strategies, 3 of which are based on the algebraic connectivity estimated in the subplot (a).

Theorems & Definitions (4)

  • Theorem 1
  • Proposition 1
  • proof
  • proof