  Real-world data is often times associated with irregular structures that can
analytically be represented as graphs. Having access to this graph, which is
sometimes trivially evident from domain knowledge, provides a better
representation of the data and facilitates various information processing
tasks. However, in cases where the underlying graph is unavailable, it needs to
be learned from the data itself for data representation, data processing and
inference purposes. Existing literature on learning graphs from data has mostly
considered arbitrary graphs, whereas the graphs generating real-world data tend
to have additional structure that can be incorporated in the graph learning
procedure. Structure-aware graph learning methods require learning fewer
parameters and have the potential to reduce computational, memory and sample
complexities. In light of this, the focus of this paper is to devise a method
to learn structured graphs from data that are given in the form of product
graphs. Product graphs arise naturally in many real-world datasets and provide
an efficient and compact representation of large-scale graphs through several
smaller factor graphs. To this end, first the graph learning problem is posed
as a linear program, which (on average) outperforms the state-of-the-art graph
learning algorithms. This formulation is of independent interest itself as it
shows that graph learning is possible through a simple linear program.
Afterwards, an alternating minimization-based algorithm aimed at learning
various types of product graphs is proposed, and local convergence guarantees
to the true solution are established for this algorithm. Finally the
performance gains, reduced sample complexity, and inference capabilities of the
proposed algorithm over existing methods are also validated through numerical
simulations on synthetic and real datasets.
