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

TL;DR

This work tackles the problem of learning sparse precision matrices for high-dimensional GGMs when node variances are heterogeneous. It introduces LARGE, which replaces a single global penalty in GLASSO with nodewise penalties learned through an inner Autotune Lasso that estimates per-node noise and guides edge selection via sequential F-tests at level $\alpha$. The approach improves graph recovery and stability, especially in challenging regimes where $n$ is not large relative to $p$, and demonstrates practical utility by estimating brain functional connectivity from real fMRI data. The method offers an interpretable sparsity control mechanism and opens directions for theoretical guarantees and further methodological refinements.

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 λin 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 λand 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.

Figures (5)

• Figure 1: Boxplot of noise variance estimates $\hat{\sigma}_j^2$ for a randomly selected node under a tridiagonal precision matrix. Data are generated from $X \stackrel{\text{iid}}{\sim} \mathcal{N}(0, \Theta^{-1})$ with $(n, p) = (200, 50)$, aggregated over 20 replications. Estimates from the guiding procedure more closely align with those from an oracle linear model.
• Figure 2: ROC curves from a single replication across all simulation settings where $(n, p) = (500, 300)$. In this high-dimensional regime, LARGE consistently achieves the highest AUROC across the various graph structures.
• Figure 3: Estimated precision matrices $\hat{\Theta}$ for a band-2 graph model, plotted in absolute correlation scale. Data are generated from $X \stackrel{\text{iid}}{\sim} \mathcal{N}(0, \Theta^{-1})$ with $(n, p) = (500, 100)$. In the second row, methods relying on a global tuning parameter (EBIC, RIC, StARS, and CV) tend to either under- or over-estimate different regions of the network in the presence of heterogeneous local noise levels.
• Figure 4: Distribution of sample partial variances of across 86 brain regions with $n = 1200$ time points from one fMRI scan.
• Figure 5: FC networks estimated by LARGE and benchmark methods for a single individual, visualized on the correlation scale. All methods recover known biological patterns of brain connectivity.