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 - <p style="text-align: justify;">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 $ℓ_p \ (0 &lt; p ≤ 1)$ minimization method. The restricted isometry constant (RIC) condition $δ_{tk} &lt;\frac{1}{pη^{ \frac{2}{p}−1} +1}$ 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.</p>