On Recovery of Sparse Signals Via ₁ Minimization

This paper considers constrained lscr1minimization methods in a unified framework for the recovery of high-dimensional sparse signals in three settings: noiseless, bounded error, and Gaussian noise. Both lscr1minimization with an lscrinfinconstraint (Dantzig selector) and lscr1minimization under anllscr2constraint 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 lscr1minimization with anllscr2constraint and lscr1minimization with an lscrinfinconstraint. 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.