We propose a unified fractional regularization framework for sparse signal recovery based on the $\ell_1/\ell_p^q$ model. This model generalizes several widely used sparsity-promoting regularizers and provides additional flexibility through the parameters $p$ and $q$. 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, thereby offering a unified perspective on these nonconvex regularizers. In addition, we establish a new sufficient recovery condition under the Restricted Isometry Property (RIP), which shows that the proposed framework can provide relaxed recovery guarantees and improved robustness. To solve the resulting nonconvex problem, we develop a majorization--minimization (MM) algorithm and prove its convergence by using the Kurdyka--Łojasiewicz (KL) property. Numerical experiments on sparse recovery problems with different sensing matrices and MRI reconstruction demonstrate that the proposed approach outperforms existing methods in recovery accuracy.

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。该模型推广了几种广泛使用的稀疏促进正则化器，并通过参数$p$和$q$提供了额外的灵活性。我们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，我们建立了受限等距性质（RIP）下的一个新的充分恢复条件，表明所提出的框架能够提供更宽松的恢复保证和更强的鲁棒性。为了求解由此产生的非凸问题，我们开发了一种最大-最小化（MM）算法，并利用Kurdyka–Łojasiewicz（KL）性质证明了其收敛性。针对不同感知矩阵下的稀疏恢复问题和MRI重建的数值实验表明，所提出的方法在恢复精度上优于现有方法。