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线性逆问题中惩罚优化方法信号重建误差界研究

张欢 雷宏

引用本文: 张欢, 雷宏. 线性逆问题中惩罚优化方法信号重建误差界研究[J]. 电子与信息学报, 2019, 41(12): 2939-2944. doi: 10.11999/JEIT181125 shu
Citation:  Huan ZHANG, Hong LEI. An Error Bound of Signal Recovery for Penalized Programs in Linear Inverse Problems[J]. Journal of Electronics and Information Technology, 2019, 41(12): 2939-2944. doi: 10.11999/JEIT181125 shu

线性逆问题中惩罚优化方法信号重建误差界研究

    作者简介: 张欢: 男,1991年生,博士生,研究方向为低维结构信号恢复理论及应用;
    雷宏: 男,1963年生,研究员,博士生导师,研究方向为电磁场与微波技术、信号处理理论与技术方面的研究
    通讯作者: 张欢,zhanghuan13@mails.ucas.ac.cn
摘要: 惩罚优化问题常常用于在有噪声的条件下用较少的观测个数来求解线性逆问题。目前,对惩罚优化问题恢复误差的研究主要存在以下两点不足:一是对权重参数往往有要求;二是噪声的方向对误差的影响未知。针对这两个问题,该文研究了当存在有界噪声时,惩罚优化问题恢复的误差界。首先,该文从问题的几何出发,给定了一个几何条件。当这一条件满足时,就能够推导出惩罚优化问题恢复的一个明确的误差界。这个误差界保证了恢复的解是稳定的,也就是说,恢复误差不会超过观测误差的常数倍。同时,这一误差界对于任意的正权重参数都成立,并且揭示了恢复误差以及最优的权重选择与观测噪声的方向之间的联系。进一步地,当观测矩阵是一个高斯矩阵时,依据这一几何条件可以得到高概率稳定恢复所需的观测次数。仿真实验证明了理论结果的正确性。

English

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      席博, 洪涛, 张更新. 卫星物联网场景下基于节点选择的协作波束成形技术研究. 电子与信息学报, 2020, 42(0): 1-9.

  • 图 1  数值仿真结果

    表 1  本文符号说明

    符号说明
    ${\text{y}}$观测信号
    ${\text{A}}$观测矩阵
    ${{\text{x}}^ * }$真实信号
    ${\text{z}}$观测噪声
    $\varepsilon $观测噪声${\text{z}}$的幅度
    $m$观测个数(观测信号${\text{y}}$的维度)
    $n$真实信号${{\text{x}}^ * }$的维度
    $f$真实信号${{\text{x}}^ * }$的结构函数
    $\lambda $惩罚优化问题中的权重调节参数
    $\partial f({{\text{x}}^ * })$函数$f$在点${{\text{x}}^ * }$的次梯度
    $D(f,{{\text{x}}^ * })$函数$f$在点${{\text{x}}^ * }$的下降锥
    $N(f,{{\text{x}}^ * })$函数$f$在点${{\text{x}}^ * }$的法锥
    $w(D(f,{{\text{x}}^ * }) \cap {{\text{S}}^{n - 1}})$下降锥$D(f,{{\text{x}}^ * })$的球面高斯宽度
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文章相关
  • 通讯作者:  张欢, zhanghuan13@mails.ucas.ac.cn
  • 收稿日期:  2018-12-06
  • 录用日期:  2019-04-01
  • 网络出版日期:  2019-05-28
  • 刊出日期:  2019-12-01
通讯作者: 陈斌, bchen63@163.com
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