Network Topology Inference with Sparsity and Laplacian Constraints

TL;DR

This paper introduces a novel method for network topology inference using Laplacian constrained Gaussian graphical models, addressing limitations of traditional sparsity-inducing norms by incorporating an $ ext{l}_0$-norm constraint and demonstrating its effectiveness through experiments.

Contribution

It proposes an $ ext{l}_0$-norm constrained Laplacian estimation method with an efficient gradient projection algorithm for improved network inference.

Findings

The method effectively infers sparse network topologies.

It outperforms $ ext{l}_1$-norm based approaches in experiments.

Demonstrates success on synthetic and financial datasets.

Abstract

We tackle the network topology inference problem by utilizing Laplacian constrained Gaussian graphical models, which recast the task as estimating a precision matrix in the form of a graph Laplacian. Recent research \cite{ying2020nonconvex} has uncovered the limitations of the widely used $\ell_1$-norm in learning sparse graphs under this model: empirically, the number of nonzero entries in the solution grows with the regularization parameter of the $\ell_1$-norm; theoretically, a large regularization parameter leads to a fully connected (densest) graph. To overcome these challenges, we propose a graph Laplacian estimation method incorporating the $\ell_0$-norm constraint. An efficient gradient projection algorithm is developed to solve the resulting optimization problem, characterized by sparsity and Laplacian constraints. Through numerical experiments with synthetic and financial time-series datasets, we demonstrate the effectiveness of the proposed method in network topology inference.