Recovery of sparsest signals via $\ell^q$-minimization

TL;DR

This paper proves that all s-sparse signals can be exactly recovered from measurements using ll^q-minimization for 0<q, provided the signals are uniquely determined by their measurements, advancing sparse signal recovery methods.

Contribution

It establishes that ll^q-minimization guarantees exact recovery of s-sparse signals under conditions of unique measurement determination.

Findings

Exact recovery of s-sparse signals via ll^q-minimization for 0<q.

Recovery is guaranteed when each s-sparse vector is uniquely determined by measurements.

Theoretical proof of recovery conditions for ll^q-minimization.

Abstract

In this paper, it is proved that every $s$-sparse vector ${\bf x}\in {\mathbb R}^n$ can be exactly recovered from the measurement vector ${\bf z}={\bf A} {\bf x}\in {\mathbb R}^m$ via some $\ell^q$-minimization with $0< q\le 1$, as soon as each $s$-sparse vector ${\bf x}\in {\mathbb R}^n$ is uniquely determined by the measurement ${\bf z}$.