  Graph Learning (GL) is at the core of inference and analysis of connections
in data mining and machine learning (ML). By observing a dataset of graph
signals, and considering specific assumptions, Graph Signal Processing (GSP)
tools can provide practical constraints in the GL approach. One applicable
constraint can infer a graph with desired frequency signatures, i.e., spectral
templates. However, a severe computational burden is a challenging barrier,
especially for inference from high-dimensional graph signals. To address this
issue and in the case of the underlying graph having graph product structure,
we propose learning product (high dimensional) graphs from product spectral
templates with significantly reduced complexity rather than learning them
directly from high-dimensional graph signals, which, to the best of our
knowledge, has not been addressed in the related areas. In contrast to the rare
current approaches, our approach can learn all types of product graphs (with
more than two graphs) without knowing the type of graph products and has fewer
parameters. Experimental results on both the synthetic and real-world data,
i.e., brain signal analysis and multi-view object images, illustrate
explainable and meaningful factor graphs supported by expert-related research,
as well as outperforming the rare current restricted approaches.
