基于（非）凸极小化的高维数据分离与重构研究

发布时间：2018-03-12 11:25

本文选题：压缩数据分离　切入点：扰动　出处：《西南大学》2017年硕士论文　论文类型：学位论文

【摘要】：随着信息时代的来临,在生产与生活中我们常常会面对各种各样的复杂且富有价值的高维数据,如何有效地挖掘和处理这些高维数据一直是学术界与工业界研究的热点.压缩感知是一种新颖且有效的高维数据处理理论,它利用信号数据的稀疏性和可压缩性,能够以高概率实现对信号的精确重构,目前已在压缩成像,医学成像,模式识别,图像处理等领域得到了广泛应用.本文基于压缩感知理论并结合应用背景研究了不同类型的高维数据处理,主要内容如下:第一章,概述了压缩感知理论产生的背景与研究意义,并简要地介绍了压缩感知的最新研究进展以及实际应用成果.第二章,介绍了压缩感知的三个主要方面:信号的稀疏表示,测量矩阵的设计和信号的重构理论与重构算法.第三章,针对多模态数据,首先引入了压缩数据分离模型,然后基于冗余紧框架并利用非凸的D-?q-极小化方法研究了扰动数据分离问题.当冗余紧框架和测量矩阵满足互相关性,零空间性质,限制性等容条件时,建立了稀疏信号的重构条件并获得了局部最优解与原始信号的误差上界.研究表明了D-?q-极小化方法对冗余紧框架下的稀疏信号恢复是鲁棒的和稳定的.第四章,采用凸的?2/?1极小化方法和Block D-RIP理论研究了在冗余紧框架下的块稀疏信号,所获结果表明,当Block D-RIP常数δ2k|τ满足0δ2k|τ0.2时,?2/?1极小化方法能够鲁棒重构原始信号,同时改进了已有的重构条件和误差上限.基于离散傅里叶变换(DFT)字典,我们执行了一系列仿真实验充分地证实了理论结果.第五章,研究了低秩张量修补问题,基于目标秩之前的奇异值不会影响张量秩的极小化这一事实,本文提出了奇异值的部分和极小化的低秩张量修补算法(PSSV-LRTC).针对模拟数据和真实数据执行了一系列实验,结果表明我们的算法比已有的算法具有更高的精度和收敛率.第六章,总结了全文的主要工作,并对扰动数据分离,块稀疏压缩感知以及张量修补中有进一步研究价值的内容作了分析与展望.
[Abstract]:With the advent of the information age, we often face a variety of complex and valuable high-dimensional data in production and life. How to effectively mine and process these high-dimensional data has been a hot topic in academia and industry. Compression perception is a novel and effective theory of high-dimensional data processing, which makes use of the sparsity and compressibility of signal data. It is possible to reconstruct the signal accurately with high probability, and it has been used in compression imaging, medical imaging, pattern recognition and so on. Image processing and other fields have been widely used. In this paper, different types of high-dimensional data processing are studied based on compressed sensing theory and combined with application background. The main contents are as follows: chapter one, This paper summarizes the background and significance of the theory of compressed sensing, and briefly introduces the latest research progress and practical application of compressed sensing. Chapter two introduces three main aspects of compressed sensing: sparse representation of signals. The design of measurement matrix and the theory and algorithm of signal reconstruction. Chapter 3, for multi-modal data, the compression data separation model is introduced first, then based on redundant compact frame and using non-convex D-? The q-minimization method is used to study the problem of perturbed data separation. When the redundant compact frame and the measurement matrix satisfy the mutual correlation, the property of zero space and the restricted equal volume condition, The reconstruction condition of sparse signal is established and the error upper bound between the local optimal solution and the original signal is obtained. The q-minimization method is robust and stable for sparse signal recovery under redundant compact frame. Chapter 4th, convex? 2/? 1 minimization method and Block D-RIP theory are used to study block sparse signals under redundant compact frame. The results show that when the Block D-RIP constant 未 2k 蟿 satisfies 0 未 2k 蟿 0.2? 2/? 1 minimization method can reconstruct the original signal robustly, at the same time improve the existing reconstruction condition and error upper limit. Based on the DFT dictionary of discrete Fourier transform, we have carried out a series of simulation experiments to fully verify the theoretical results. Chapter 5th, Based on the fact that the singular value before the target rank does not affect the minimization of Zhang Liang rank, the problem of low rank Zhang Liang repair is studied. In this paper, a partial and minimized low rank Zhang Liang repair algorithm for singular values is proposed. A series of experiments are performed on simulated and real data. The results show that our algorithm has higher accuracy and convergence rate than the existing algorithms. Chapter 6th. The main work of this paper is summarized, and the contents of disturbance data separation, block sparse compression perception and Zhang Liang repair are analyzed and prospected.
【学位授予单位】：西南大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：TN911.7

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