We propose a unified fractional regularization framework for sparse signal recovery based on the $\ell_1/\ell_p^q$ model. Our main theoretical contribution is the characterization of the equivalence between the first-order stationary points of the $\ell_1/\ell_p^q$ formulation and the subtractive $\ell_1 - α\ell_p$ model, providing a unified perspective on these nonconvex regularizers. In addition, we establish a new sufficient recovery condition under the Restricted Isometry Property (RIP), showing that the framework's robustness even under high-coherence sensing matrices. To solve the resulting problem, we develop a majorization-minimization (MM) algorithm and prove its convergence via the Kurdyka-Lojasiewicz (KL) property. Numerical experiments on different sensing matrices and MRI reconstruction demonstrate that the proposed approach consistently outperforms existing methods.

翻译：我们提出了一种基于$\ell_1/\ell_p^q$模型的统一分数阶正则化框架，用于稀疏信号恢复。本文的主要理论贡献在于刻画了$\ell_1/\ell_p^q$公式的一阶稳定点与减法$\ell_1 - α\ell_p$模型之间的等价性，从而为这些非凸正则化器提供了统一视角。此外，我们基于受限等距性质（RIP）建立了新的充分恢复条件，表明该框架即使在高度相关的感知矩阵下仍具有鲁棒性。为解决由此产生的问题，我们开发了一种主化-最小化（MM）算法，并利用Kurdyka-Lojasiewicz（KL）性质证明了其收敛性。在不同感知矩阵与MRI重建上的数值实验表明，所提方法始终优于现有方法。