Sparse Recovery Conditions and Performance Bounds for $\ell _p$-Minimization

Abstract

In sparse recovery, a sparse signal ${\mathbf x}\in \mathbb {R}^N$ with $K$ nonzero entries is to be reconstructed from a compressed measurement $\mathbf y=Ax$ with ${\mathbf A}\in \mathbb {R}^{M\times N}$ ( $M ). The $\ell _p$ $(0\leq p pseudonorm has been found to be a sparsity inducing function superior to the $\ell _1$ norm, and the null space constant (NSC) and restricted isometry constant (RIC) have been used as key notions in the performance analyses of the corresponding $\ell _p$ -minimization. In this paper, we study sparse recovery conditions and performance bounds for the $\ell _p$ -minimization. We devise a new NSC upper bound that outperforms the state-of-the-art result. Based on the improved NSC upper bound, we provide a new RIC upper bound dependent on the sparsity level $K$ as a sufficient condition for precise recovery, and it is tighter than the existing bound for small $K$ . Then, we study the largest choice of $p$ for the $\ell _p$ -minimization problem to recover any $K$ -sparse signal, and the largest recoverable $K$ for a fixed $p$ . Numerical experiments demonstrate the improvement of the proposed bounds in the recovery conditions over the up-to-date counterparts.