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 ${\ma...Show MoreMetadata
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< N). The \ell _p (0\leq p < 1) 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.
Published in: IEEE Transactions on Signal Processing ( Volume: 66, Issue: 19, 01 October 2018)