  This paper introduces Polynomial Graphical Lasso (PGL), a new approach to
learning graph structures from nodal signals. Our key contribution lies in
modeling the signals as Gaussian and stationary on the graph, enabling the
development of a graph-learning formulation that combines the strengths of
graphical lasso with a more encompassing model. Specifically, we assume that
the precision matrix can take any polynomial form of the sought graph, allowing
for increased flexibility in modeling nodal relationships. Given the resulting
complexity and nonconvexity of the resulting optimization problem, we (i)
propose a low-complexity algorithm that alternates between estimating the graph
and precision matrices, and (ii) characterize its convergence. We evaluate the
performance of PGL through comprehensive numerical simulations using both
synthetic and real data, demonstrating its superiority over several
alternatives. Overall, this approach presents a significant advancement in
graph learning and holds promise for various applications in graph-aware signal
analysis and beyond.
