  This paper investigates the recovery of a node-domain sparse graph signal
from the output of a graph filter. This problem, often referred to as the
identification of the source of a diffused sparse graph signal, is seminal in
the field of graph signal processing (GSP). Sparse graph signals can be used in
the modeling of a variety of real-world applications in networks, such as
social, biological, and power systems, and enable various GSP tasks, such as
graph signal reconstruction, blind deconvolution, and sampling. In this paper,
we assume double sparsity of both the graph signal and the graph topology, as
well as a low-order graph filter. We propose three algorithms to reconstruct
the support set of the input sparse graph signal from the graph filter output
samples, leveraging these assumptions and the generalized information criterion
(GIC). First, we describe the graph multiple GIC (GM-GIC) method, which is
based on partitioning the dictionary elements (graph filter matrix columns)
that capture information on the signal into smaller subsets. Then, the local
GICs are computed for each subset and aggregated to make a global decision.
Second, inspired by the well-known branch and bound (BNB) approach, we develop
the graph-based branch and bound GIC (graph-BNB-GIC), and incorporate a new
tractable heuristic bound tailored to the graph and graph filter
characteristics. Finally, we propose the graph-based first order correction
(GFOC) method, which improves existing sparse recovery methods by iteratively
examining potential improvements to the GIC cost function through replacing
elements from the estimated support set with elements from their one-hop
neighborhood. We conduct simulations that demonstrate that the proposed sparse
recovery methods outperform existing methods in terms of support set recovery
accuracy, and without a significant computational overhead.
