张欢^{1, 2,,},
雷宏¹
1.中国科学院电子学研究所 北京 100190
2.中国科学院大学 北京 100049

详细信息

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

¹⁾ 意思为ε依次取以下值进行仿真实验：0.001, 0.01, 0.05, 0.10, 0.15, 0.20, 0.25, 0.30, 0.35, 0.40, 0.45, 0.50, 0.55, 0.60, 0.65, 0.70, 0.75, 0.80, 0.85, 0.90, 0.95, 1.00。
中图分类号:TN911.72

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出版历程

收稿日期:2018-12-06
修回日期:2019-04-01
网络出版日期:2019-05-28
刊出日期:2019-12-01

An Error Bound of Signal Recovery for Penalized Programs in Linear Inverse Problems

Huan ZHANG^{1, 2,,},
Hong LEI¹
1. Institute of Electronics, Chinese Academy of Sciences, Beijing 100190, China
2. University of Chinese Academy of Sciences, Beijing 100049, China


摘要
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。
关键词:线性逆问题/
压缩感知/
稳定恢复/
惩罚优化问题/
权重选择
Abstract:Penalized programs are widely used to solve linear inverse problems in the presence of noise. For now, the study of the performance of panelized programs has two disadvantages. First, the results have some limitations on the tradeoff parameters. Second, the effect of the direction of the noise is not clear. This paper studies the performance of penalized programs when bounded noise is presented. A geometry condition which is used to study the noise-free problems and constrained problems is provided. Under this condition, an explicit error bound which guarantees stable recovery (i.e., the recovery error is bounded by the observation noise up to some constant factor) is proposed. The results are different from many previous studies in two folds. First, the results provide an explicit bound for all positive tradeoff parameters, while many previous studies require that the tradeoff parameter is sufficiently large. Second, the results clear the role of the direction of the observation noise playing in the recovery error, and reveal the relationship between the optimal tradeoff parameters and the noise direction. Furthermore, if the sensing matrix has independent standard normal entries, the above geometry condition can be studied using Gaussian process theory, and the measurement number needed to guarantee stable recovery with high probability is obtained. Simulations are provided to verify the theoretical results.
Key words:Linear inverse problem/
Compressed sensing/
Stable recovery/
Penalized program/
Tradeoff
注释:
1) ¹⁾ 意思为ε依次取以下值进行仿真实验：0.001, 0.01, 0.05, 0.10, 0.15, 0.20, 0.25, 0.30, 0.35, 0.40, 0.45, 0.50, 0.55, 0.60, 0.65, 0.70, 0.75, 0.80, 0.85, 0.90, 0.95, 1.00。



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