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基于非凸复合函数的稀疏信号恢复算法

周洁容 李海洋 凌军 陈浩 彭济根

周洁容, 李海洋, 凌军, 陈浩, 彭济根. 基于非凸复合函数的稀疏信号恢复算法. 自动化学报, 2021, 47(x): 1−12 doi: 10.16383/j.aas.c200666
引用本文: 周洁容, 李海洋, 凌军, 陈浩, 彭济根. 基于非凸复合函数的稀疏信号恢复算法. 自动化学报, 2021, 47(x): 1−12 doi: 10.16383/j.aas.c200666
Zhou Jie-Rong, Li Hai-Yang, Ling Jun, Chen Hao, Peng Ji-Gen. Sparse signal reconstruction algorithm based on non-convex composite function. Acta Automatica Sinica, 2021, 47(x): 1−12 doi: 10.16383/j.aas.c200666
Citation: Zhou Jie-Rong, Li Hai-Yang, Ling Jun, Chen Hao, Peng Ji-Gen. Sparse signal reconstruction algorithm based on non-convex composite function. Acta Automatica Sinica, 2021, 47(x): 1−12 doi: 10.16383/j.aas.c200666

基于非凸复合函数的稀疏信号恢复算法

doi: 10.16383/j.aas.c200666
基金项目: 国家自然科学基金(11771347), 国家自然科学基金(12031003)资助
详细信息
    作者简介:

    周洁容:2014年在岭南师范学院获得学士学位, 目前在广州大学数学与信息科学学院攻读硕士研究生. 主要研究方向为稀疏信息处理. E-mail: jrzhouzoe@163.com

    李海洋:1996年在延安大学获得数学学士学位, 2005年和2008年在陕西师范大学获得基础数学硕士和博士学位. 目前是广州大学数学与信息科学学院教授. 他目前主要研究方向包括量子逻辑理论, 机器学习理论, 和稀疏信息处理. E-mail: fplihaiyang@126.com; fplihaiyang@126.com

    凌军:2016年于上饶师范学院获得数学学士学位, 2019年在南昌大学获得概率论于数理统计硕士学位, 现于广州大学数学与信息科学学院攻读博士学位, 主要研究方向包括小目标运动检测, 算子半群理论以及人工智能. E-mail: gdalingjun@163.com

    陈浩:2015年于广州大学数学与信息科学学院获得学士学位, 现于广州大学数学与信息科学学院攻读硕士研究生. 主要研究包括小目标运动检测, 人工智能和计算神经学. E-mail: gdchenhao@126.com

    彭济根:1989年在江西大学获得数学学士学位, 1992年和1998年在西安交通大学分别获得应用数学硕士学位和计算数学博士学位. 曾于1992年至2017年在西安交通大学工作, 2017年入职广州大学. 现为广州大学数学与信息科学学院教授. 他目前的研究方向包括非线性泛函分析及其应用, 机器学习理论, 稀疏优化, 碰撞检测的数学理论与方法. 本文通信作者. E-mail: jgpeng@gzhu.edu.cn

Sparse Signal Reconstruction Algorithm Based on Non-Convex Composite Function

Funds: Supported by National Natural Science Foundation of P. R. China (11771347), National Natural Science Foundation of P. R. China (12031003)
  • 摘要: 本文基于泛函深度作用的思想, 通过将两种非凸稀疏泛函进行复合, 构造了一种新的稀疏信号重构模型, 实现了对0范数的深度逼近. 综合运用MM技术、外点罚函数法和共轭梯度法, 提出了一种求解该模型的算法, 称为NCCS算法. 为降低重构信号陷入局部极值的可能性, 提出了在算法的每步迭代中以BP模型的解作为初始迭代值. 为验证所建模型和所提算法的有效性, 本文进行了多项数值实验. 实验结果表明: 相较于SL0算法、IRLS算法、SCSA算法以及BP算法等经典算法, 本文提出的算法在重构误差、信噪比、归一化均方差、支撑集恢复成功率等方面都有更优的表现.
  • 图  1  四种函数在$\sigma = 0.1$时的一元函数分布

    Fig.  1  The unary distribution of the four functions at $\sigma = 0.1$

    图  2  ${p_\sigma }(x)$${h_\sigma }(x)$${g_\sigma }(x)$和函数${f_\sigma }(x)$$\sigma = 0.1$时的二元函数分布

    Fig.  2  The bivariate distribution of ${p_\sigma }(x)$, ${h_\sigma }(x)$, ${g_\sigma }(x)$ and the function ${f_\sigma }(x)$ at $\sigma = 0.1$

    图  3  待定数$\alpha $对NCCS算法运行时间的影响

    Fig.  3  The influence of undetermined number $\alpha $ on the running time of NCCS algorithm

    图  4  NCCS算法的一维信号重构仿真图, 信号大小为: $500 \times 1$, 稀疏度为$65$

    Fig.  4  One-dimensional signal reconstruction simulation diagram of NCCS algorithm, the signal size is: 500×1, the sparsity is 65

    图  5  SL0、IRLS、BP、SCSA、NCCS五种算法的重构误差和稀疏度的变化关系

    Fig.  5  The relationship between the reconstruction error and sparsity of the five algorithms of SL0, IRLS, BP, SCSA, and NCCS

    图  6  SL0、IRLS、BP、SCSA、NCCS五种算法的重构信噪比和稀疏度的变化关系

    Fig.  6  The relationship between the reconstructed signal-to-noise ratio and sparsity of the five algorithms of SL0, IRLS, BP, SCSA, and NCCS

    图  7  SL0、IRLS、BP、SCSA、NCCS五种算法的运行时间和稀疏度的变化关系

    Fig.  7  The relationship between the running time and sparsity of the five algorithms of SL0, IRLS, BP, SCSA, and NCCS

    图  8  SL0、IRLS、BP、SCSA、NCCS五种算法的支撑集恢复成功和稀疏度的变化关系

    Fig.  8  The relationship between the recovery success rate of the support set and sparsity of the five algorithms of SL0, IRLS, BP, SCSA, and NCCS

    图  9  SL0、IRLS、BP、SCSA、NCCS五种算法的归一化均方差和稀疏度的变化关系

    Fig.  9  The relationship between the normalized mean square error and sparsity of the five algorithms of SL0, IRLS, BP, SCSA, and NCCS

    表  1  五种算法的归一化均方差的数值记录

    Table  1  Numerical record of the normalized mean square error of five algorithms

    算法归一化均方差$NMSE$
    k=20k=30k=40k=50k=60k=70k=80k=90k=100k=110
    SL04.11E-102.86E-102.06E-101.55E-101.16E-109.38E-111.46E-041.28E-039.22E-033.13E-02
    IRLS2.02E-044.08E-041.15E-034.23E-031.07E-022.79E-025.04E-027.20E-021.14E-011.38E-01
    BP1.65E-224.49E-225.68E-224.95E-224.19E-224.38E-151.18E-059.52E-041.03E-023.57E-02
    SCSA2.05E-231.94E-221.94E-222.45E-222.50E-222.96E-223.18E-221.27E-161.27E-133.57E-12
    NCCS5.36E-272.10E-264.81E-244.98E-245.87E-151.85E-173.35E-162.62E-173.58E-161.00E-16
    下载: 导出CSV
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  • 收稿日期:  2020-12-01
  • 录用日期:  2020-12-14
  • 网络出版日期:  2021-01-06

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