On Recovery of Sparse Signals via $\ell_1$ Minimization

TL;DR

This paper provides a unified analysis of $ ext{l}_1$ minimization methods for recovering high-dimensional sparse signals across noiseless, bounded error, and Gaussian noise settings, improving conditions and error bounds.

Contribution

It offers a simplified, unified treatment of Dantzig selector and $ ext{l}_1$ minimization, weakening recovery conditions and tightening error bounds, extending key existing results.

Findings

Improved recovery conditions for sparse signals.

Tighter error bounds in various noise settings.

Established connections between RIP and mutual incoherence.

Abstract

This article considers constrained $\ell_1$ minimization methods for the recovery of high dimensional sparse signals in three settings: noiseless, bounded error and Gaussian noise. A unified and elementary treatment is given in these noise settings for two $\ell_1$ minimization methods: the Dantzig selector and $\ell_1$ minimization with an $\ell_2$ constraint. The results of this paper improve the existing results in the literature by weakening the conditions and tightening the error bounds. The improvement on the conditions shows that signals with larger support can be recovered accurately. This paper also establishes connections between restricted isometry property and the mutual incoherence property. Some results of Candes, Romberg and Tao (2006) and Donoho, Elad, and Temlyakov (2006) are extended.