In this article, we introduce a minimization model via a non-convex transformed $\ell_p$ (TLp) penalty function with two parameters $a\in(0,\infty)$ and $p\in(0,1]$, where the case $p=1$ is known and was established by S. Zhang and J. Xin. Using the sparse convex-combination technique, we establish the exact and the stable sparse signal recovery based on the restricted isometry property (RIP). We apply a modified iteratively re-weighted least squares method and the difference of convex functions algorithm (DCA) to give the IRLSTLp algorithm for unconstrained TLp minimization and prove some related convergence results. Finally, we conduct some numerical experiments to show the robustness of the IRLSTLp and the flexibility of the TLp minimization model. The novelty of these results lies in three aspects: (i) We introduce the concept of the relaxation degree RD$_P$ of a separable penalty function $P$ to quantitatively measure how closely $P$ approaches $\ell_0$. (ii) We introduce the TLp penalty, which includes two aforementioned adjustable parameters, offering more flexibility and stronger sparsity-promotion capability of the TLp minimization model, compared with the $\ell_p$ and the TL1 minimization models. (iii) The obtained RIP upper bound for signal recovery via TLp minimization can reduce, when $p\in(0,1]$ and as $a\to \infty$, to the sharp RIP bound obtained by R. Zhang and S. Li and, especially, can recover, when $p=1$, the well-known sharp bound $δ_{2s}<\frac{\sqrt{2}}{2}$.

翻译：本文提出了一种基于非凸变换$\ell_p$（TLp）罚函数的最小化模型，该函数包含两个参数$a\in(0,\infty)$和$p\in(0,1]$，其中$p=1$的情形已由张帅与辛洁建立。利用稀疏凸组合技术，我们基于限制等距性质（RIP）建立了精确与稳定的稀疏信号恢复理论。通过改进的迭代重加权最小二乘法与凸差函数算法（DCA），我们提出了用于无约束TLp最小化问题的IRLSTLp算法，并证明了相关的收敛性结果。最后，我们通过数值实验展示了IRLSTLp算法的鲁棒性以及TLp最小化模型的灵活性。本文结果的创新性体现在三个方面：（i）引入可分离罚函数$P$的松弛度RD$_P$概念，以定量度量$P$逼近$\ell_0$范数的程度；（ii）提出的TLp罚函数包含两个可调参数，相较于$\ell_p$与TL1最小化模型，TLp最小化模型具有更强的灵活性及稀疏促进能力；（iii）通过TLp最小化进行信号恢复所获得的RIP上界，当$p\in(0,1]$且$a\to \infty$时可退化为张睿与李松获得的尖锐RIP界，特别当$p=1$时，可恢复经典的尖锐界$δ_{2s}<\frac{\sqrt{2}}{2}$。