Recovery of Signals with Low Density

Abstract

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 report, we study a signal-density measure, the ratio between the ${\ell }_1$-norm and the ${\ell }_{\infty }$-norm of a vector, which 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 kernel result and an uncertainty relation that are more general and less restrictive than those based on the ${\ell }_0$-pseudonorm. Furthermore, we use this density measure to analyze orthogonal matching pursuit (OMP). We show that OMP provably (i) recovers sparse signals with decaying magnitudes using up to 2$\times$ more non-zero coefficients than guaranteed by standard, sparsity-based results and (ii) identifies the largest entries of arbitrary signals under a suitable magnitude-decay condition.

Revised from a manuscript first posted to arXiv on July 10, 2015, and later submitted to a journal, where it remained in review limbo. This version corrects an error in the original OMP recovery proof, restores previously omitted results, and updates the discussion and references. It is released solely as a technical report; no further journal submission is planned

References in corpus (2)

• Performance Analysis of Sparse Recovery Based on Constrained Minimal Singular Values
• Uncertainty Relations and Sparse Signal Recovery for Pairs of General Signal Sets