Recovery of Signals with Low Density

Sparse signals (i.e., vectors with a small number of non-zero entries) build the foundation of most kernel (or nullspace) results, uncertainty relations, and recovery guarantees in the sparse signal processing and compressive sensing literature. In this paper, we introduce a novel signal-density measure that extends the common notion of sparsity to non-sparse signals whose entries' magnitudes decay rapidly. By taking into account such magnitude information, we derive a more general and less restrictive kernel result and uncertainty relation. Furthermore, we demonstrate the effectiveness of the proposed signal-density measure by showing that orthogonal matching pursuit (OMP) provably recovers low-density signals with up to 2$\times$ more non-zero coefficients compared to that guaranteed by standard results for sparse signals.