Recovery of Signals with Low Density

TL;DR

This paper introduces a new signal-density measure that generalizes sparsity to signals with rapidly decaying entries, enabling improved recovery guarantees for algorithms like OMP in compressive sensing.

Contribution

It proposes a novel density measure extending sparsity concepts, leading to less restrictive recovery guarantees and demonstrating enhanced performance of OMP for low-density signals.

Findings

OMP recovers signals with up to 2× more non-zero entries than standard sparse guarantees.

The new measure provides a broader framework for signal recovery beyond strict sparsity.

Derived kernel and uncertainty relations are less restrictive and more general.

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 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.