Efficient Graph Laplacian Estimation by Proximal Newton

TL;DR

This paper introduces a second-order proximal Newton method for efficient and accurate estimation of Laplacian-constrained Gaussian Markov Random Fields, improving computational speed and graph sparsity accuracy.

Contribution

It develops a novel second-order optimization algorithm using MCP penalty for better sparsity and lower bias in Laplacian graph learning.

Findings

Outperforms existing methods in computational efficiency.

Achieves more accurate sparse graph structures.

Demonstrates robustness on numerical experiments.

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.