快速稀疏恢复算法研究

发布时间：2018-04-12 19:06

本文选题：线性Bregman算法 + 快速软阈值算法　；参考：《国防科学技术大学》2014年硕士论文

【摘要】：压缩感知,即从一组线性观测中恢复出原来的稀疏信号,在许多领域如信号处理、图像处理、机器学习中发挥了相当重要的作用。由于压缩感知的巨大的理论与应用价值,各种各样的稀疏恢复算法不断涌现。根据方法来分,这些算法大致可以分为三类:凸松弛、贪婪算法和硬阈值方法。但是随着实际问题规模的不断增长,这三大类算法的效率都受到很大的影响。本文使用了一些先进的加速技术对其中的部分算法进行加速。主要的工作可以归纳如下:一:提出了预条件的线性Bregman算法,并从理论上说明了该算法的优势,数值实验也验证了该方法的优势。然后将此方法推广到不相容的增广的1极小化问题,并提出了相应的算法。二:针对1-1极小化问题提出了重加权的快速软阈值算法。在本算法中,主要是对光滑化后的1-1极小化问题使用快速软阈值算法进行计算,再用重启技术加速。实验结果验证了该算法的优势。三:提出了一类加速的硬阈值追踪算法,并证明了当感知矩阵满足合适的限制自同构条件时,该类算法依然具有线性收敛率。数值实验结果表明了该类算法要比已有的快速硬阈值追踪算法好。四:针对非负稀疏信号恢复问题,提出了投影硬阈值追踪算法,并证明了当感知矩阵满足合适的限制自同构条件时,该算法依然具有线性收敛率。数值实验结果表明了该算法要比硬阈值追踪算法效果好。
[Abstract]:Compressed sensing, that is, recovering the original sparse signals from a set of linear observations, plays an important role in many fields such as signal processing, image processing, and machine learning.Due to the great theoretical and practical value of compressed sensing, a variety of sparse restoration algorithms are emerging.According to the method, these algorithms can be divided into three categories: convex relaxation, greedy algorithm and hard threshold method.However, with the increasing scale of practical problems, the efficiency of these three algorithms is greatly affected.This paper uses some advanced acceleration techniques to accelerate some of the algorithms.The main work can be summarized as follows: first, the preconditioned linear Bregman algorithm is proposed, and the advantages of the algorithm are explained theoretically, and the advantages of the method are verified by numerical experiments.Then the method is extended to the incompatible augmented minimization problem and the corresponding algorithm is proposed.Second, a fast soft threshold algorithm with reweighting for 1-1 minimization problem is proposed.In this algorithm, the fast soft threshold algorithm is used to calculate the 1-1 minimization problem after smoothing, and the restart technique is used to accelerate the algorithm.Experimental results verify the advantages of the algorithm.Numerical results show that this algorithm is better than the existing fast hard threshold tracking algorithm.Numerical results show that the algorithm is more effective than the hard threshold tracking algorithm.
【学位授予单位】：国防科学技术大学
【学位级别】：硕士
【学位授予年份】：2014
【分类号】：TN911.7

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