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.

翻译：图套索（GLASSO）是学习高维无向高斯图模型（GGM）的常用算法。给定来自多元正态分布的独立同分布观测值，GLASSO通过对非对角线元素施加ℓ_1惩罚来最大化对数似然，从而估计精度矩阵。然而，在这种无监督设置中选择最优的正则化参数λ仍然是一个重大挑战。一个众所周知的问题是，现有方法（如样本外似然最大化）选择单一的全局λ，未能考虑变量尺度或局部方差的异质性。将数据标准化为单位方差虽是一种常见的变通方法，但已被证明会对图恢复产生负面影响。解决图估计中的节点自适应调参问题对于计算神经科学等应用至关重要，例如从高度异质、区域特异的功能磁共振成像（fMRI）数据构建脑网络。本研究提出了图估计的局部自适应正则化方法（LARGE），通过自适应学习节点级调优参数来改进图估计与选择。在GLASSO的每个块坐标下降步骤中，我们扩展节点级Lasso回归以联合估计回归系数与误差方差，进而指导节点级惩罚的自适应学习。在模拟实验中，LARGE在图像恢复方面持续优于基准方法，在不同重复实验中表现出更高的稳定性，并在最困难的模拟设置中实现了最佳的估计精度。我们通过从真实fMRI数据集中估计脑功能连接，证明了该方法的实用价值。