full
search.png

Search

close.png

Cancel

arrow
Volume 41, Issue 6
Stable and Robust Recovery of Approximately $k$-Sparse Signals with Partial Support Information in Noise Settings via Weighted $ℓ_p\ (0 < p ≤ 1)$ Minimization

Biao Du & Anhua Wan

DOI: 10.4208/jcm.2207-m2022-0058

J. Comp. Math., 41 (2023), pp. 1137-1170.

Published online: 2023-11

Preview Full PDF 1525 18402

Cited by

google scholar semantic scholar
Export citation
  • Abstract

In the existing work, the recovery of strictly $k$-sparse signals with partial support information was derived in the $ℓ_2$ bounded noise setting. In this paper, the recovery of approximately $k$-sparse signals with partial support information in two noise settings is investigated via weighted $ℓ_p \ (0 < p ≤ 1)$ minimization method. The restricted isometry constant (RIC) condition $δ_{tk} <\frac{1}{pη^{ \frac{2}{p}−1} +1}$ on the measurement matrix for some $t ∈ [1+\frac{ 2−p}{ 2+p} σ, 2]$ is proved to be sufficient to guarantee the stable and robust recovery of signals under sparsity defect in noisy cases. Herein, $σ ∈ [0, 1]$ is a parameter related to the prior support information of the original signal, and $η ≥ 0$ is determined by $p,$ $t$ and $σ.$ The new results not only improve the recent work in [17], but also include the optimal results by weighted $ℓ_1$ minimization or by standard $ℓ_p$ minimization as special cases.

  • Keywords

Signal recovery, weighted $ℓ_p$ minimization, Approximately $k$-sparse signal, Noise setting, Reconstruction error bound, Restricted isometry property.

  • AMS Subject Headings

94A12, 94A15

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address
  • BibTex
  • RIS
  • TXT
@Article{JCM-41-1137, author = {Du , Biao and Wan , Anhua}, title = {Stable and Robust Recovery of Approximately $k$-Sparse Signals with Partial Support Information in Noise Settings via Weighted $ℓ_p\ (0 < p ≤ 1)$ Minimization}, journal = {Journal of Computational Mathematics}, year = {2023}, volume = {41}, number = {6}, pages = {1137--1170}, abstract = {

In the existing work, the recovery of strictly $k$-sparse signals with partial support information was derived in the $ℓ_2$ bounded noise setting. In this paper, the recovery of approximately $k$-sparse signals with partial support information in two noise settings is investigated via weighted $ℓ_p \ (0 < p ≤ 1)$ minimization method. The restricted isometry constant (RIC) condition $δ_{tk} <\frac{1}{pη^{ \frac{2}{p}−1} +1}$ on the measurement matrix for some $t ∈ [1+\frac{ 2−p}{ 2+p} σ, 2]$ is proved to be sufficient to guarantee the stable and robust recovery of signals under sparsity defect in noisy cases. Herein, $σ ∈ [0, 1]$ is a parameter related to the prior support information of the original signal, and $η ≥ 0$ is determined by $p,$ $t$ and $σ.$ The new results not only improve the recent work in [17], but also include the optimal results by weighted $ℓ_1$ minimization or by standard $ℓ_p$ minimization as special cases.

}, issn = {1991-7139}, doi = {https://doi.org/10.4208/jcm.2207-m2022-0058}, url = {http://global-sci.org/intro/article_detail/jcm/22107.html} }
TY - JOUR T1 - Stable and Robust Recovery of Approximately $k$-Sparse Signals with Partial Support Information in Noise Settings via Weighted $ℓ_p\ (0 < p ≤ 1)$ Minimization AU - Du , Biao AU - Wan , Anhua JO - Journal of Computational Mathematics VL - 6 SP - 1137 EP - 1170 PY - 2023 DA - 2023/11 SN - 41 DO - http://doi.org/10.4208/jcm.2207-m2022-0058 UR - https://global-sci.org/intro/article_detail/jcm/22107.html KW - Signal recovery, weighted $ℓ_p$ minimization, Approximately $k$-sparse signal, Noise setting, Reconstruction error bound, Restricted isometry property. AB -

In the existing work, the recovery of strictly $k$-sparse signals with partial support information was derived in the $ℓ_2$ bounded noise setting. In this paper, the recovery of approximately $k$-sparse signals with partial support information in two noise settings is investigated via weighted $ℓ_p \ (0 < p ≤ 1)$ minimization method. The restricted isometry constant (RIC) condition $δ_{tk} <\frac{1}{pη^{ \frac{2}{p}−1} +1}$ on the measurement matrix for some $t ∈ [1+\frac{ 2−p}{ 2+p} σ, 2]$ is proved to be sufficient to guarantee the stable and robust recovery of signals under sparsity defect in noisy cases. Herein, $σ ∈ [0, 1]$ is a parameter related to the prior support information of the original signal, and $η ≥ 0$ is determined by $p,$ $t$ and $σ.$ The new results not only improve the recent work in [17], but also include the optimal results by weighted $ℓ_1$ minimization or by standard $ℓ_p$ minimization as special cases.

Du , Biao and Wan , Anhua. (2023). Stable and Robust Recovery of Approximately $k$-Sparse Signals with Partial Support Information in Noise Settings via Weighted $ℓ_p\ (0 < p ≤ 1)$ Minimization. Journal of Computational Mathematics. 41 (6). 1137-1170. doi:10.4208/jcm.2207-m2022-0058
Copy to clipboard
BibteX RIS TXT
The citation has been copied to your clipboard