Sparse Recovery over Graph Incidence Matrices

TL;DR

This paper studies sparse recovery of signals over graphs using incidence matrices, providing new polynomial-time verifiable conditions and a specialized algorithm that outperforms standard methods.

Contribution

It introduces necessary and sufficient cycle-based conditions for sparse recovery over graphs and proposes a novel sub-graph-based algorithm tailored for incidence matrices.

Findings

Cycle-based conditions enable efficient sparse recovery verification.

The proposed algorithm outperforms standard -minimization in experiments.

Support-dependent conditions improve recovery accuracy.

Abstract

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 $\ell_1$-minimization approach.