  In this paper, we focus on learning the underlying product graph structure
from multidomain training data. We assume that the product graph is formed from
a Cartesian graph product of two smaller factor graphs. We then pose the
product graph learning problem as the factor graph Laplacian matrix estimation
problem. To estimate the factor graph Laplacian matrices, we assume that the
data is smooth with respect to the underlying product graph. When the training
data is noise free or complete, learning factor graphs can be formulated as a
convex optimization problem, which has an explicit solution based on the
water-filling algorithm. The developed framework is illustrated using numerical
experiments on synthetic data as well as real data related to air quality
monitoring in India.
