Graph Laplacians encode graph structures in matrix form, and thus facilitate the application of linear algebra to graph theory. In statistics, two related families of probabilistic graphical models can be parameterized by graph Laplacians. The first one is the Laplacian-constrained Gaussian graphical model (LCGGM), which imposes that the (pseudo-)inverse covariance matrix of a Gaussian random vector is a Laplacian matrix. Applications include graph signal processing and network topology learning. The second one is the Hüsler-Reiss graphical model, which is considered as an extremal analog of the Gaussian graphical model, and can be used in extremal dependence modeling of floods, heatwaves, and financial losses. For both models, the restriction to positive edge weights in the graph Laplacian gives rise to an approach for graph structure learning that does not require tuning parameters. While these approaches yield a strong model fit in many settings, the resulting graph estimates are typically much denser than the underlying ground truth, limiting interpretability and scalability. In order to improve the accuracy of Laplacian-constrained graph learning, we propose to use spectral graph sparsification as a post-estimation operation. To do so, we replace the original Laplacian estimate by a sparser Laplacian that is spectrally close, and re-fit the model on the resulting graph. We refer to the two resulting methods as Spectral-LCGGM and Spectral-HR. We investigate the properties of the proposed estimators and show several theoretical results on their performance. Furthermore, we demonstrate that the newly proposed methods perform well by running simulations on Erdős-Rényi and stochastic block model graphs, and we also showcase their applications to real data.

翻译：图拉普拉斯矩阵以矩阵形式编码图结构，从而促进线性代数在图论中的应用。在统计学中，两类相关的概率图模型可由图拉普拉斯矩阵参数化。第一类是拉普拉斯约束高斯图模型（LCGGM），该模型要求高斯随机向量的（伪）逆协方差矩阵为拉普拉斯矩阵，其应用包括图信号处理与网络拓扑学习。第二类是赫斯勒-赖斯图模型，被视为高斯图模型的极值对应模型，可用于洪水、热浪及金融损失的极值依赖性建模。对于这两类模型，图拉普拉斯矩阵中边权为正的约束催生出无需调参的图结构学习方法。尽管这些方法在多种场景下能实现良好的模型拟合，但所得图估计通常比真实底层图结构稠密得多，从而限制了可解释性与可扩展性。为提高拉普拉斯约束图学习的精确性，我们提出将谱图稀疏化作为后估计操作。具体而言，用谱接近的稀疏拉普拉斯矩阵替换原始拉普拉斯估计量，并在所得图上重新拟合模型。我们将这两种方法分别称为谱-LCGGM与谱-HR。我们探究了所提估计量的性质，并展示了关于其性能的多项理论结果。此外，通过在埃尔德什-雷尼图和随机分块模型图上的仿真实验，我们证明了新方法具有优越性能，同时展示了其在真实数据中的应用。