A Unified Fractional Regularization Framework for Sparse Recovery

TL;DR

This paper introduces a unified fractional regularization framework for sparse signal recovery, providing theoretical insights, a new recovery condition, and an effective algorithm validated by numerical experiments.

Contribution

It characterizes the equivalence of stationary points in fractional regularization models, establishes a new RIP-based recovery condition, and develops a convergent MM algorithm.

Findings

The framework outperforms existing methods in numerical experiments.

A new sufficient recovery condition under RIP is established.

The proposed algorithm converges reliably across various sensing matrices.

Abstract

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 $\ell_1 - \alpha \ell_p$ 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.