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 is 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 less appropriate in this setting, we employ the nonconvex minimax concave penalty (MCP), which promotes sparse solutions with lower estimation bias. Second, as opposed to most existing first-order methods for this problem, we base our method on the second-order proximal Newton approach to obtain an efficient solver for large-scale networks. This approach is considered the most efficient for the related graphical LASSO problem and allows for several algorithmic features we exploit, such as using Conjugate Gradients, preconditioning, and splitting to active/free sets. Numerical experiments demonstrate the advantages of the proposed method in terms of \emph{both} computational complexity and graph learning accuracy compared to existing methods.

翻译：拉普拉斯约束高斯马尔可夫随机场（LGMRF）是一种常见的多元统计模型，用于从给定数据中学习加权稀疏依赖图。该图学习问题被表述为精度矩阵在拉普拉斯结构约束下的最大似然估计（MLE），并附加了稀疏诱导惩罚项。本文旨在精确且高效地解决此学习问题。首先，针对常用$\ell_1$范数惩罚在此场景下适用性不足的问题，我们采用非凸的最小最大凹惩罚（MCP），该惩罚能够以更低估计偏差促进稀疏解。其次，与现有大多数基于一阶方法的研究不同，我们以二阶邻近牛顿法为基础构建求解器，以高效处理大规模网络。该方法被认为是解决相关图LASSO问题的最有效方案，并支持多项算法特性，包括共轭梯度法、预处理技术以及主动/自由集分裂策略。数值实验表明，与现有方法相比，所提方法在计算复杂度与图学习精度两个方面均具有显著优势。