A new upper bound of p for lp-minimization in compressed sensing

Abstract

Signal recovery through l1-minimization may fail for restricted isometry constant δ2s>2/2, therefore lp-minimization is an alternative method. In this paper, we consider the use of lp-minimization with 0 < p ≤ 1 to recover sparse signals from an under-determined linear measurement Ax=y. We prove that if A satisfies a restricted isometry property with δ2s>2/2, an upper bound p¯=(2+2)(1−δ2s) exists such that any s-sparse signal can be recovered through a nonconvex lp-minimization with any p≤p¯. To the best of our knowledge, this upper bound p¯ is a significant improvement compared to the best existing results proposed by Wen et al. (2015) [19], i.e.,p¯={5031(1−δ2s),δ2s∈[2/2,0.7183),0.4541,δ2s∈[0.7183,0.7729),2(1−δ2s),δ2s∈[0.7729,1).Numerical experiment indicates that lp-minimization with a larger p has better recovery effect than with a smaller p. Our result expands the upper bound of p.