This article considers constrained
l1 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
minimization methods: the Dantzig selector and
minimization with an l2
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.