提出统一分数阶正则化框架，提升稀疏信号恢复精度与鲁棒性。
A Unified Fractional Regularization Framework for Sparse Recovery
- 基于ℓ₁/ℓₚ^q模型构建统一正则化框架，参数灵活可调。
- 在RIP条件下实现更宽松的恢复保证，理论性能更优。
- 适用于信号恢复、MRI重建等场景，适合追求高精度的科研人员。
我们提出一种基于ℓ₁/ℓₚ^q模型的统一分数阶正则化框架，用于稀疏信号恢复。该模型推广了多种常用稀疏性促进正则项，并通过参数p和q提供额外灵活性。主要理论贡献在于揭示了ℓ₁/ℓₚ^q公式与减法型ℓ₁−αℓₚ模型的一阶驻点等价性，为非凸正则项提供了统一视角。此外，我们在受限等距性质（RIP）下建立了新的充分恢复条件，表明该框架能提供更宽松的恢复保证并增强鲁棒性。为求解由此产生的非凸问题，我们设计了一种极大化-极小化（MM）算法，并利用Kurdyka–Łojasiewicz（KL）性质证明其收敛性。在不同传感矩阵下的稀疏恢复问题及MRI重建的数值实验表明，所提方法在恢复精度上优于现有方法。
原文摘要 · Abstract (English)
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.
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