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 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 $\ell_0$, whose significance also lies in revealing the functional relationship of the parameters involved to keep a high performance of a multi-parameter minimization model. (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：本文引入了基于非凸变换ℓₚ（TLp）惩罚函数（含两个参数a∈(0,∞)与p∈(0,1]）的极小化模型，其中p=1的情形已知且由S. Zhang与J. Xin建立。利用稀疏凸组合技术，我们基于限定等距性质建立了精确与稳定的稀疏信号恢复理论。采用修正迭代重加权最小二乘方法与凸函数差分算法构造了用于无约束TLp极小化的IRLSTLp算法，并证明了若干收敛性结果。最后，通过数值实验验证了IRLSTLp算法的鲁棒性及TLp极小化模型的灵活性。本文的创新性体现在三个方面：(i) 引入可分离惩罚函数P的松弛度RD_P概念，定量刻画P逼近ℓ₀的程度，其重要意义在于揭示多参数极小化模型中保持高性能的参数函数关系；(ii) 提出包含两个可调参数的TLp惩罚函数，相较于ℓₚ与TL1极小化模型，TLp极小化模型具有更强的灵活性与稀疏促进能力；(iii) 通过TLp极小化进行信号恢复所得的RIP上界，当p∈(0,1]且a→∞时可退化为R. Zhang与S. Li获得的紧致RIP界，特别地，当p=1时能够恢复经典紧致界δ_{2s}<√2/2。