In this paper, we focus on learning product graphs from multi-domain data. We assume that the product graph is formed by the Cartesian product of two smaller graphs, which we refer to as graph factors. We pose the product graph learning problem as the problem of estimating the graph factor Laplacian matrices. To capture local interactions in data, we seek sparse graph factors and assume a smoothness model for data. We propose an efficient iterative solver for learning sparse product graphs from data. We then extend this solver to infer multi-component graph factors with applications to product graph clustering by imposing rank constraints on the graph Laplacian matrices. Although working with smaller graph factors is computationally more attractive, not all graphs readily admit an exact Cartesian product factorization. To this end, we propose efficient algorithms to approximate a graph by a nearest Cartesian product of two smaller graphs. The efficacy of the developed framework is demonstrated using several numerical experiments on synthetic and real data.

中文翻译：

---

从具有稀疏性和秩约束的多域数据中学习产品图

在本文中，我们专注于从多领域数据中学习产品图。我们假设乘积图是由两个较小图的笛卡尔积形成的，我们将其称为图因子。我们将乘积图学习问题视为估计图因子拉普拉斯矩阵的问题。为了捕捉数据中的局部交互，我们寻求稀疏图因子并假设数据的平滑模型。我们提出了一种有效的迭代求解器，用于从数据中学习稀疏产品图。然后我们扩展这个求解器，通过对图拉普拉斯矩阵施加秩约束来推断多分量图因子，并应用于产品图聚类。尽管使用较小的图因子在计算上更具吸引力，但并非所有图都容易承认精确的笛卡尔积因式分解。为此，我们提出了有效的算法，通过两个较小图的最近笛卡尔积来近似图。所开发框架的有效性通过对合成数据和真实数据的若干数值实验得到证明。