Stable recovery of sparse signals via lplp-minimization

In this paper, we show that, under the assumption that ‖e‖2≤ϵ‖e‖2≤ϵ, every k  -sparse signal x∈Rnx∈Rn can be stably (ϵ≠0ϵ≠0) or exactly recovered (ϵ=0ϵ=0) from y=Ax+ey=Ax+e via lplp-minimization with p∈(0,p¯], wherep¯={5031(1−δ2k),δ2k∈[22,0.7183)0.4541,δ2k∈[0.7183,0.7729)2(1−δ2k),δ2k∈[0.7729,1), even if the restricted isometry constant of A   satisfies δ2k∈[22,1). Furthermore, under the assumption that n≤4kn≤4k, we show that the range of p   can be further improved to p∈(0,3+222(1−δ2k)]. This not only extends some discussions of only the noiseless recovery (Lai et al. (2011) [16] and Wu et al. (2013) [17]) to the noise recovery, but also greatly improves the best existing results where p∈(0,min⁡{1,1.0873(1−δ2k)})p∈(0,min⁡{1,1.0873(1−δ2k)}) (Wu et al. (2013) [17]).