Verifying the Smoothness of Graph Signals: A Graph Signal Processing Approach

Graph signal processing (GSP) deals with the representation, analysis, and processing of structured data, i.e.
graph signals
that are defined on the vertex set of a generic graph. A crucial prerequisite for applying various GSP and graph neural network (GNN) approaches is that the examined signals are
smooth graph signals
with respect to the underlying graph, or, equivalently, have low graph total variation (TV). In this paper, we develop GSP-based approaches to verify the validity of the smoothness assumption of given signals (data) and an associated graph. The proposed approaches are based on the representation of network data as the output of a graph filter with a given graph topology. In particular, we develop two smoothness detectors for the graph-filter-output model: 1) the likelihood ratio test (LRT) for known model parameters; and 2) a semi-parametric detector that estimates the graph filter and then validates its smoothness. The properties of the proposed GSP-based detectors are investigated, and some special cases are discussed. The performance of the GSP-based detectors is evaluated using synthetic data, data from the IEEE 14-bus power system, and measurements from a network of light intensity sensors, under different setups. The results demonstrate the good detection performance of the proposed approach and its robustness to different generating models, noise levels, topology changes, and the number of samples.