A Unified Fractional Regularization Framework for Sparse Recovery

arXiv:2604.23184v3 Announce Type: replace-cross Abstract: 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 $\ell_1-\alpha\ell_p$ 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--{\L}ojasiewicz (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.

Abstract: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 $\ell_1-\alpha\ell_p$ 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.

Information Theory (cs.IT); Machine Learning (cs.LG)

arXiv:2604.23184 [cs.IT]

(or arXiv:2604.23184v3 [cs.IT] for this version)

From: Hao Wang Dr. [view email] [v1] Sat, 25 Apr 2026 07:32:39 UTC (594 KB) [v2] Wed, 27 May 2026 07:10:41 UTC (586 KB) [v3] Mon, 12 Oct 2026 08:25:39 UTC (565 KB)

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A Unified Fractional Regularization Framework for Sparse Recovery

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