Theoretical Guarantees for Sparse Graph Signal Recovery

Sparse graph signals have recently been utilized in graph signal processing (GSP) for tasks such as graph signal reconstruction, blind deconvolution, and sampling. In addition, sparse graph signals can be used to model real-world network applications across various domains, such as social, biological, and power systems. Despite the extensive use of sparse graph signals, limited attention has been paid to the derivation of theoretical guarantees on their recovery. In this paper, we present a novel theoretical analysis of the problem of recovering a node-domain sparse graph signal from the output of a first-order graph filter. The graph filter we study is the Laplacian matrix, and we derive upper and lower bounds on its mutual coherence. Our results establish a connection between the recovery performance and the minimal graph nodal degree. The proposed bounds are evaluated via simulations on the Erdős-Rényi graph.