When the signal does not have a sparse structure but has sparsity under a certain transformation domain, Nam et al. \cite{NS} introduced the cosparse analysis model, which provides a dual perspective on the sparse representation model. This paper mainly discusses the error estimation of non-convex $\ell_p(0<p<1)$ relaxation cosparse optimization model with noise condition. Compared with the existing literature, under the same conditions, the value range of the $\Omega$-RIP constant $\delta_{7s}$ given in this paper is wider. When $p=0.5$ and $\delta_{7s}=0.5$, the error constants $C_0$ and $C_1$ in this paper are better than those corresponding results in the literature \cite{Cand,LiSong1}. Moreover, when $0<p<1$, the error results of the non-convex relaxation method are significantly smaller than those of the convex relaxation method. The experimental results verify the correctness of the theoretical analysis and illustrate that the $\ell_p(0<p<1)$ method can provide robust reconstruction for cosparse optimization problems.

翻译：当信号不具有稀疏结构但属于某一变换域下的稀疏时，Nam等人\cite{NS}引入了共稀疏分析模型，该模型为稀疏表示模型提供了对偶视角。本文主要讨论含噪条件下非凸$\ell_p(0<p<1)$松弛共稀疏优化模型的误差估计。与现有文献相比，在相同条件下，本文给出的$\Omega$-RIP常数$\delta_{7s}$的取值范围更广。当$p=0.5$且$\delta_{7s}=0.5$时，本文的误差常数$C_0$和$C_1$优于文献\cite{Cand,LiSong1}中的相应结果。此外，当$0<p<1$时，非凸松弛方法的误差结果显著小于凸松弛方法。实验结果验证了理论分析的正确性，并说明$\ell_p(0<p<1)$方法能为共稀疏优化问题提供稳健的重构。