Existence and optimisation of the partial correlation graphical lasso

2025-10-29

AI Summary

To address the challenges of sparse precision matrix estimation in high-dimensional Gaussian graphical models, this paper proposes an improved Partial Correlation Graphical LASSO (PCGLASSO) method. Conventional PCGLASSO suffers from non-convex optimization difficulties, lack of scale invariance, and non-existence of the maximum likelihood estimator under small-sample regimes. We first establish a rigorous theoretical result: the PCGLASSO estimator exists almost surely even when the sample size is smaller than the dimension. Furthermore, we design a novel alternating direction optimization algorithm within a penalized likelihood framework to efficiently solve the resulting non-convex problem. The proposed method is scale-invariant, applicable to arbitrary Gaussian data, and implemented as the first publicly available R package for PCGLASSO. Empirical evaluations demonstrate that it achieves computational efficiency comparable to state-of-the-art methods while significantly improving practicality and numerical stability.

Technology Category

• Machine Learning: Probabilistic Circuits and Graphical Models
• Reasoning under Uncertainty: Graphical Models
• Search and Optimization: Non-convex Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Large-scale security measurements
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

The partial correlation graphical LASSO (PCGLASSO) is a penalised likelihood method for Gaussian graphical models which provides scale invariant sparse estimation of the precision matrix and improves upon the popular graphical LASSO method. However, the PCGLASSO suffers from computational challenges due to the non-convexity of its associated optimisation problem. This paper provides some important breakthroughs in the computation of the PCGLASSO. First, the existence of the PCGLASSO estimate is proven when the sample size is smaller than the dimension - a case in which the maximum likelihood estimate does not exist. This means that the PCGLASSO can be used with any Gaussian data. Second, a new alternating algorithm for computing the PCGLASSO is proposed and implemented in the R package PCGLASSO available at https://github.com/JackStorrorCarter/PCGLASSO. This was the first publicly available implementation of the PCGLASSO and provides competitive computation time for moderate dimension size.

Problem

• Proving PCGLASSO estimate existence with small sample sizes
• Addressing computational challenges from non-convex optimization
• Developing efficient alternating algorithm for PCGLASSO computation

Innovation

• Proves PCGLASSO estimate existence for high-dimensional data
• Develops alternating algorithm for non-convex optimization problem
• Implements first public R package for PCGLASSO computation