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 can be 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 inappropriate in this setting and may lead to a complete graph, we employ the nonconvex minimax concave penalty (MCP), which promotes sparse solutions with lower estimation bias. Second, as opposed to existing first-order methods for this problem, we develop a second-order proximal Newton approach to obtain an efficient solver, utilizing several algorithmic features, such as using Conjugate Gradients, preconditioning, and splitting to active/free sets. Numerical experiments demonstrate the advantages of the proposed method in terms of 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）是一种常见的多变量统计模型，用于从给定数据中学习加权稀疏依赖图。该图学习问题可表述为精度矩阵的极大似然估计（MLE），需满足拉普拉斯结构约束并加入稀疏诱导惩罚项。本文旨在准确高效地求解该学习问题。首先，由于常用的$\ell_1$范数惩罚在此场景下不适用且可能导致全连接图，我们采用非凸极小极大凹惩罚（MCP），该惩罚能促进低估计偏差的稀疏解。其次，不同于该问题的现有的一阶方法，我们开发了一种二阶近端牛顿方法以获得高效求解器，该方法融合了共轭梯度法、预处理及活跃/自由集分割等多种算法特性。数值实验表明，与现有方法相比，本方法在计算复杂度和图学习精度方面均具有优势。