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Study on \(L_1\) over \(L_2\) Minimization for Nonnegative Signal Recovery

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Abstract

In this paper, we carry out a comprehensive study for the unconstrained \(L_1\) over \(L_2\) sparsity promoting model, widely used in the regime of coherent dictionaries for recovering nonnegative sparse signals. First, we prove the existence of global solutions. Second, we analyze the sparse property of any local minimizer of the \(L_{1}/L_{2}\) model. This property serves as a certificate to rule out the nonlocal minimizers. Third, we focus on algorithmic development on the unconstrained model with nonnegative constraint. We derive an analytical solution for the proximal operator of the \(L_{1} / L_{2}\) with nonnegative constraint. Then, we apply the alternating direction method of multipliers in a particular splitting way, referred to as ADMM\(_p^+\). We establish its global convergence to a d-stationary solution (sharpest stationary) by verifying the Lyapunov function with the Kurdyka-Łojasiewicz property instead of imposing. Extensive numerical simulations confirm the superiority of ADMM\(_p^+\) over the existing state-of-the-art methods in nonnegative sparse recovery.

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Data Availibility

The datasets generated during and/or analysed during the current study are available from the corresponding author on reasonable request.

Notes

  1. The original SOOT algorithm proposed in [29] aims to solve blind deconvolution, i.e., finding the unknown signal and blur simultaneously.

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Acknowledgements

We appreciate Dr. Penghang Yin from the Department of Mathematics and Statistics of University at Albany providing us the codes of generating DOAS problems, Algorithm 3 in [36], and the SGPM.

Funding

Min Tao was partially supported by the Natural Science Foundation of China (No. 11971228) and Jiangsu University QingLan Project. The work of Xiao-Ping Zhang is supported by the Natural Sciences and Engineering Research Council of Canada (NSERC), Grant No. RGPIN-2020-04661.

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Authors and Affiliations

  1. Department of Mathematics, National Key Laboratory for Novel Software Technology, Nanjing University, Nanjing, 210093, China

    Min Tao

  2. Department of Tsinghua-Berkeley Shenzhen Institute, Tsinghua University, Beijing, China

    Xiao-Ping Zhang

  3. Department of Electrical, Computer and Biomedical Engineering, Toronto Metropolitan University, Toronto, ON M5B 2K3, Canada

    Xiao-Ping Zhang

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  1. Min Tao
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Correspondence to Min Tao.

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Tao, M., Zhang, XP. Study on \(L_1\) over \(L_2\) Minimization for Nonnegative Signal Recovery. J Sci Comput 95, 94 (2023). https://doi.org/10.1007/s10915-023-02225-2

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