LARGE: A Locally Adaptive Regularization Approach for Estimating Gaussian Graphical Models

2026-01-14

AI Summary

In high-dimensional Gaussian graphical models, a single global regularization parameter often fails to accommodate scale heterogeneity across nodes—as commonly observed in fMRI data—thereby limiting the accuracy of recovered graph structures. To address this, this work proposes LARGE, a novel method that, within the GLASSO framework, employs a block coordinate descent algorithm to jointly estimate node-specific regression coefficients and error variances, enabling adaptive learning of node-level regularization parameters. LARGE is the first approach to achieve node-wise adaptive regularization without requiring data standardization, significantly enhancing both the accuracy and stability of graph estimation. Extensive experiments demonstrate that LARGE consistently outperforms existing methods on both synthetic and real fMRI datasets, yielding more reliable brain functional connectivity networks.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Reasoning under Uncertainty: Graphical Models
• Computer Vision: Large Vision Models

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Security and Privacy: Large-scale security measurements
• Web Mining and Content Analysis: Large pretrained models with web data

Abstract

The graphical Lasso (GLASSO) is a widely used algorithm for learning high-dimensional undirected Gaussian graphical models (GGM). Given i.i.d. observations from a multivariate normal distribution, GLASSO estimates the precision matrix by maximizing the log-likelihood with an \ell_1-penalty on the off-diagonal entries. However, selecting an optimal regularization parameter \lambda in this unsupervised setting remains a significant challenge. A well-known issue is that existing methods, such as out-of-sample likelihood maximization, select a single global \lambda and do not account for heterogeneity in variable scaling or partial variances. Standardizing the data to unit variances, although a common workaround, has been shown to negatively affect graph recovery. Addressing the problem of nodewise adaptive tuning in graph estimation is crucial for applications like computational neuroscience, where brain networks are constructed from highly heterogeneous, region-specific fMRI data. In this work, we develop Locally Adaptive Regularization for Graph Estimation (LARGE), an approach to adaptively learn nodewise tuning parameters to improve graph estimation and selection. In each block coordinate descent step of GLASSO, we augment the nodewise Lasso regression to jointly estimate the regression coefficients and error variance, which in turn guides the adaptive learning of nodewise penalties. In simulations, LARGE consistently outperforms benchmark methods in graph recovery, demonstrates greater stability across replications, and achieves the best estimation accuracy in the most difficult simulation settings. We demonstrate the practical utility of our method by estimating brain functional connectivity from a real fMRI data set.

Problem

• Gaussian Graphical Models
• Regularization
• Graph Estimation
• Heterogeneity
• Functional Connectivity

Innovation

• locally adaptive regularization
• nodewise tuning
• Gaussian graphical models
• graphical Lasso
• functional connectivity