Novel Iterative Reweighted ℓ 1 Minimization for Sparse Recovery

Qi An, Li Wang and Nana Zhang ()
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Qi An: Department of Computer Science, The Open University of China, 75 Fuxing Road, Beijing 100039, China
Li Wang: Department of Computer Science, The Open University of China, 75 Fuxing Road, Beijing 100039, China
Nana Zhang: College of Economics & Management, Zhejiang University of Water Resources and Electric Power, 583 Xuelin Road, Hangzhou 310018, China

Mathematics, 2025, vol. 13, issue 8, 1-13

Abstract: Data acquisition and high-dimensional signal processing often require the recovery of sparse representations of signals to minimize the resources needed for data collection. ℓ p quasi-norm minimization excels in exactly reconstructing sparse signals from fewer measurements, but it is NP-hard and challenging to solve. In this paper, we propose two distinct Iteratively Re-weighted ℓ 1 Minimization (IR ℓ 1 ) formulations for solving this non-convex sparse recovery problem by introducing two novel reweighting strategies. These strategies ensure that the ϵ -regularizations adjust dynamically based on the magnitudes of the solution components, leading to more effective approximations of the non-convex sparsity penalty. The resulting IR ℓ 1 formulations provide first-order approximations of tighter surrogates for the original ℓ p quasi-norm objective. We prove that both algorithms converge to the true sparse solution under appropriate conditions on the sensing matrix. Our numerical experiments demonstrate that the proposed IR ℓ 1 algorithms outperform the conventional approach in enhancing recovery success rate and computational efficiency, especially in cases with small values of p .

Keywords: non-convex sparse recovery; ℓ p quasi-norm; iteratively reweighted ℓ 1 minimization (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
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