A Unified Fractional Regularization Framework for Sparse Recovery

2026-04-25

AI Summary

This work addresses the challenge of sparse signal recovery under highly coherent sensing matrices by proposing a unified fractional regularization framework based on the ℓ₁/ℓₚ^q model. It establishes, for the first time, the equivalence between this model and the subtractive ℓ₁−αℓₚ formulation at first-order stationary points, offering a novel perspective on nonconvex regularization. Theoretical recovery guarantees are provided under the restricted isometry property, specifically tailored to high-coherence settings. Algorithmically, a majorization–minimization (MM) approach is employed, with convergence ensured via the Kurdyka–Łojasiewicz property. Extensive experiments demonstrate that the proposed method significantly outperforms existing sparse recovery techniques across various sensing matrices and in magnetic resonance imaging (MRI) reconstruction tasks.

Technology Category

• Search and Optimization: Non-convex Optimization
• Machine Learning: Matrix & Tensor Methods
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• User Modeling, Personalization and Recommendation: User privacy protection in personalized systems
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

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 $\ell_1 - α\ell_p$ 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.

Problem

• sparse recovery
• fractional regularization
• nonconvex regularizers
• Restricted Isometry Property
• high-coherence sensing matrices

Innovation

• fractional regularization
• sparse recovery
• nonconvex regularizers
• majorization-minimization
• restricted isometry property