  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
may 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 data and real
data.
