Graph Laplacian learning, also known as network topology inference, is a problem of great interest to multiple communities. In Gaussian graphical models (GM), graph learning amounts to endowing covariance selection with the Laplacian structure. In graph signal processing (GSP), it is essential to infer the unobserved graph from the outputs of a filtering system. In this paper, we study the problem of learning Cartesian product graphs under Laplacian constraints. The Cartesian graph product is a natural way for modeling higher-order conditional dependencies and is also the key for generalizing GSP to multi-way tensors. We establish statistical consistency for the penalized maximum likelihood estimation (MLE) of a Cartesian product Laplacian, and propose an efficient algorithm to solve the problem. We also extend our method for efficient joint graph learning and imputation in the presence of structural missing values. Experiments on synthetic and real-world datasets demonstrate that our method is superior to previous GSP and GM methods.

翻译：图拉普拉斯学习，也称为网络拓扑推断，是多个领域高度关注的问题。在高斯图模型（GM）中，图学习相当于将协方差选择赋予拉普拉斯结构。在图信号处理（GSP）中，从滤波系统的输出推断未观测图至关重要。本文研究在拉普拉斯约束下学习笛卡尔积图的问题。笛卡尔图积是建模高阶条件依赖关系的自然方式，也是将GSP推广到多路张量的关键。我们建立了笛卡尔积拉普拉斯惩罚最大似然估计（MLE）的统计一致性，并提出高效算法求解该问题。我们还扩展了方法，可在存在结构性缺失值时实现高效的联合图学习与插补。在合成和真实数据集上的实验表明，我们的方法优于以往的GSP和GM方法。