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中国工业与应用数学学会会刊
主管：中华人民共和国教育部
主办：西安交通大学
ISSN 1005-3085  CN 61-1269/O1
工程数学学报

工程数学学报 ›› 2016, Vol. 33 ›› Issue (3): 243-258.doi: 10.3969/j.issn.1005-3085.2016.03.003

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基于SCAD的压缩感知阈值迭代算法的收敛性分析

张  会1,   张  海1,2,   勾  明1   

  1. 1- 西北大学数学学院，西安 710069
    2- 中国科学院数学与系统科学研究院应用数学研究所，北京 100190
  • 收稿日期:2014-05-28 接受日期:2014-11-07 出版日期:2016-06-15 发布日期:2016-08-15
  • 基金资助:
    国家自然科学基金 (11171272; 11571011)；陕西省自然科学基金 (2011JM1008)；陕西省教育厅专项科研计划 (JC11217).

Convergence Analysis of Compressive Sensing Based on SCAD Iterative Thresholding Algorithm

ZHANG Hui1,  ZHANG Hai1,2,  GOU Ming1   

  1. 1- School of Mathematics, Northwest University, Xi'an 710069
    2- Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190
  • Received:2014-05-28 Accepted:2014-11-07 Online:2016-06-15 Published:2016-08-15
  • Supported by:
    The National Natural Science Foundation of China (11171272; 11571011); the Natural Science Foundation of Shaanxi Province (2011JM1008); the Specialized Research Plan in Shaanxi Province Department of Education (JC11217).

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摘要： 基于SCAD罚函数的压缩感知在有噪声稀疏信号重建中具有优良的理论及应用效果，开展其快速重建算法研究有着重要的意义，阈值迭代算法是解决压缩传感问题最有效的算法之一．本文研究了基于SCAD罚函数的压缩感知阈值迭代算法的收敛性问题，给出了算法收敛到稀疏解的充分条件，并证明了迭代估计值以指数阶速率收敛于最优值．进一步，本文给出了基于AMP改进的SCAD阈值迭代算法的收敛性分析．

关键词: 压缩感知, SCAD, 阈值迭代算法, 稀疏性

Abstract:

Compressive sensing based on SCAD has good theoretical properties for sparse signal reconstruction with noise. It is vital to study this kind of algorithms. The iterative thresholding algorithm is one of the most efficient algorithms to solve the problem of compressed sensing. In this paper, we study the convergence of the iterative thresholding algorithm for compressive sensing based on SCAD. We give some sufficient conditions on the convergence of the iterative thresholding algorithm. We prove that the algorithm is convergent with exponentially decaying error. Furthermore, we study the convergence of an improved iterative thresholding SCAD algorithm based on an approximate message passing algorithm.

Key words: compressive sensing, SCAD, iterative thresholding algorithm, sparsity

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