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J. Comp. Math., 41 (2023), pp. 1137-1170.
Published online: 2023-11
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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} }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.