Research PaperNo Access

The null space property of the weighted $ℓ_{r} - ℓ_{1}$ minimization

Jianwen Huang

School of Mathematics, Chongqing Normal University, Chongqing 400047, P. R. China

E-mail Address: hjw1303987297@126.com

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Xinling Liu

College of Mathematics and Information, China West Normal University, Nanchong 637009, P. R. China

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, and

Jinping Jia

School of Mathematics and Statistics, Tianshui Normal University, Tianshui 741001, P. R. China

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https://doi.org/10.1142/S0219691323500212Cited by:1 (Source: Crossref)

Abstract

The null space property (NSP), which relies merely on the null space of the sensing matrix column space, has drawn numerous interests in sparse signal recovery. This paper studies NSP of the weighted $ℓ_{r} - ℓ_{1}$ ( $r \in (0, 1]$ ) minimization. Several versions of NSP of the weighted $ℓ_{r} - ℓ_{1}$ minimization including the weighted $ℓ_{r} - ℓ_{1}$ NSP, the weighted $ℓ_{r} - ℓ_{1}$ stable NSP, the weighted $ℓ_{r} - ℓ_{1}$ robust NSP and the $ℓ_{q}$ weighted $ℓ_{r} - ℓ_{1}$ robust NSP for $1 \leq q \leq 2$ , are proposed, as well as the associating considerable results are derived. Under these NSPs, sufficient conditions for the recovery of (sparse) signals with the weighted $ℓ_{r} - ℓ_{1}$ minimization are established. Furthermore, we show that to some extent, the weighted $ℓ_{r} - ℓ_{1}$ stable NSP is weaker than the restricted isometric property (RIP). And the RIP condition we obtained is better than that of Zhou (2022).

Keywords:

AMSC: 60E05, 91A28, 94A12, 94A20

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