On Recovery of Sparse Signals Via $\ell _{1}$ Minimization

This paper considers constrained lscr₁ minimization methods in a unified framework for the recovery of high-dimensional sparse signals in three settings: noiseless, bounded error, and Gaussian noise. Both lscr₁ minimization with an lscr_infin constraint (Dantzig selector) and lscr₁ minimization under an llscr₂ constraint are considered. 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. In particular, our results illustrate the relationship between lscr₁ minimization with an llscr₂ constraint and lscr₁ minimization with an lscr_infin constraint. This paper also establishes connections between restricted isometry property and the mutual incoherence property. Some results of Candes, Romberg, and Tao (2006), Candes and Tao (2007), and Donoho, Elad, and Temlyakov (2006) are extended.