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Classical results in sparse recovery guarantee the exact reconstruction of s-sparse signals under assumptions on the dictionary that are either too strong or NP-hard to check. Moreover, such results may be pessimistic in practice since they are based on a worst-case analysis. In this paper, we consider the sparse recovery of signals defined over a graph, for which the dictionary takes the form of an incidence matrix. We derive necessary and sufficient conditions for sparse recovery, which depend on properties of the cycles of the graph that can be checked in polynomial time. We also derive support-dependent conditions for sparse recovery that depend only on the intersection of the cycles of the graph with the support of the signal. Finally, we exploit sparsity properties on the measurements and the structure of incidence matrices to propose a specialized sub-graph-based recovery algorithm that outperforms the standard l1 -minimization approach. 

Classical results in sparse representation guarantee the exact recovery of sparse signals under assumptions on the dictionary that are either too strong or NP hard to check. Moreover, such results may be too pessimistic in practice since they are based on a worst-case analysis. In this paper, we consider the sparse recovery of signals defined over a graph, for which the dictionary takes the form of an incidence matrix. We show that in this case necessary and sufficient conditions can be derived in terms of properties of the cycles of the graph, which can be checked in polynomial time. Our analysis further allows us to derive location dependent conditions for recovery that only depend on the cycles of the graph that intersect this support. Finally, we exploit sparsity properties on the measurements to a specialized sub-graph-based recovery algorithm that outperforms the standard $$l_1$$-minimization.