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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, 2022, 48(7): 1782−1793 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, 2022, 48(7): 1782−1793 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, 2022, 48(7): 1782−1793 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, 2022, 48(7): 1782−1793 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, 12031003)资助
详细信息
    作者简介:

    周洁容:广州大学机器生命与智能研究中心硕士研究生. 广州大学数学与信息科学学院硕士研究生. 2014 年获得岭南师范学院数学学士学位. 主要研究方向为稀疏信息处理. E-mail: jrzhouzoe@163.com

    李海洋:广州大学机器生命与智能研究中心教授. 广州大学数学与信息科学学院教授. 2008 年获得陕西师范大学基础数学博士学位. 主要研究方向为量子逻辑理论, 机器学习理论和稀疏信息处理. E-mail: fplihaiyang@126.com; fplihaiyang@126.com

    凌军:广州大学机器生命与智能研究中心博士研究生. 广州大学数学与信息科学学院博士研究生. 2019 年获得南昌大学概率论与数理统计硕士学位. 主要研究方向为小目标运动检测, 算子半群理论和人工智能. E-mail: gdalingjun@163.com

    陈浩:广州大学机器生命与智能研究中心硕士研究生. 广州大学数学与信息科学学院硕士研究生. 2015 年获得广州大学数学学士学位. 主要研究方向为小目标运动检测, 人工智能和计算神经学. E-mail: gdchenhao@126.com

    彭济根:广州大学机器生命与智能研究中心教授. 广州大学数学与信息科学学院教授. 1998 年获得西安交通大学计算数学博士学位. 主要研究方向为非线性泛函分析, 机器学习理论, 稀疏优化和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 China (11771347, 12031003)
More Information
    Author Bio:

    ZHOU Jie-Rong Master student at the Research Center for Machine Life and Intelligence, Guangzhou University. Master student at the Mathematics and Information Science College, Guangzhou University. She received her bachelor degree in mathematics from the School of Lingnan Normal University in 2014. Her main research interest is sparse information processing

    LI Hai-Yang Professor at the Research Center for Machine Life and Intelligence, Guangzhou University. Professor at the Mathematics and Information Science College, Guangzhou University. He received his Ph.D. degree in fundamental mathematics from Shaanxi Normal University in 2008. His research interest covers quantum logic theory, machine learning theory, and sparse information processing

    LING Jun Ph.D. candidate at the Research Center for Machine Life and Intelligence, Guangzhou University. Ph.D. candidate at the Mathematics and Information Science College, Guangzhou University. He received his master degree in probability theory and mathematical statistics from Nanchang University in 2019. His research interest covers small target detection, operator semigroup theory, and artificial intelligence

    CHEN Hao Master student at the Research Center for Machine Life and Intelligence, Guangzhou University. Master student at the Mathematics and Information Science College, Guangzhou University. He received his bachelor degree in mathematics from Guangzhou University in 2015. His research interest covers small target motion detection, artificial intelligence, and computational neuroscience

    PENG Ji-Gen Professor at the Research Center for Machine Life and Intelligence, Guangzhou University. Professor at the Mathematics and Information Science College, Guangzhou University. He received his Ph.D. degree in applied mathematics from Xi'an Jiaotong University in 1998. His research interest covers nonlinear functional analysis, machine learning theory, sparse optimization, and mathematical theory and methods for collision detection. Corresponding author of this paper

  • 摘要: 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 (Majorize minimization)技术、外点罚函数法和共轭梯度法, 提出一种求解该模型的算法, 称为NCCS (Non-convex composite sparse)算法. 为降低重构信号陷入局部极值的可能性, 提出在算法的每步迭代中以BP (Basis pursuit)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, 进行了多项数值实验. 实验结果表明, 相较于SL0 (Smoothed $L_0$)算法、IRLS (Iterative reweighed least squares)算法、SCSA (Successive concave sparsity approximation)算法以及BP 算法等经典算法, 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  4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 }({\boldsymbol{x}})$${h_\sigma }({\boldsymbol{x}})$${g_\sigma }({\boldsymbol{x}})$和函数${f_\sigma }({\boldsymbol{x}})$$\sigma = 0.1$时的二元函数分布

    Fig.  2  The bivariate distribution of ${p_\sigma }({\boldsymbol{x}}),$ ${h_\sigma }({\boldsymbol{x}}),$ ${g_\sigma }({\boldsymbol{x}})$ and the function ${f_\sigma }({\boldsymbol{x}})$ 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 5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 5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 5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 5种算法的支撑集恢复成功率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 5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  5种算法的归一化均方差的数值记录

    Table  1  Numerical records 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
    SL04.11×10−102.86×10−102.06×10−101.55×10−101.16×10−109.38×10−111.46×10−41.28×10−39.22×10−33.13×10−2
    IRLS2.02×10−44.08×10−41.15×10−34.23×10−31.07×10−22.79×10−25.04×10−27.20×10−21.14×10−11.38×10−1
    BP1.65×10−224.49×10−225.68×10−224.95×10−224.19×10−224.38×10−151.18×10−59.52×10−41.03×10−23.57×10−2
    SCSA2.05×10−231.94×10−221.94×10−222.45×10−222.50×10−222.96×10−223.18×10−221.27×10−161.27×10−133.57×10−12
    NCCS5.36×10−272.10×10−264.81×10−244.98×10−245.87×10−151.85×10−173.35×10−162.62×10−173.58×10−161.00×10−16
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  • 收稿日期:  2020-08-18
  • 录用日期:  2020-12-14
  • 网络出版日期:  2021-01-06
  • 刊出日期:  2022-07-01

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