Recovery of High-Dimensional Sparse Signals via ℓ1-Minimization

Abstract

We consider the recovery of high-dimensional sparse signals via ${\ell }_1$-minimization under mutual incoherence condition, which is shown to be sufficient for sparse signals recovery in the noiseless and noise cases. We study both ${\ell }_1$-minimization under the ${\ell }_2$ constraint and the Dantzig selector. Using the two ${\ell }_1$-minimization methods and a technical inequality, some results are obtained. They improve the results of the error bounds in the literature and are extended to the general case of reconstructing an arbitrary signal.