低秩模型重构的理论与应用

发布时间：2018-03-30 14:19

本文选题：压缩感知　切入点：矩阵补全　出处：《电子科技大学》2017年博士论文

【摘要】：在电子工程中存在一类非常重要的数学问题,那就是:在欠采样情况下满足观测数据的可行向量不是唯一的,而是可行解组成的一个线性子空间,在此类情况下怎样寻找符合现实问题的唯一可行解。要在一个线性子空间中确定唯一正确的可行解,则这个唯一正确的可行解必须具有其他特殊性质,在电子工程中特殊性质研究较多的是稀疏性(非零元素个数比较少)。原始问题可以描述为:在欠采样情况下,已知观测数据求解具备稀疏性的可行向量。随着问题研究的深入,提出压缩感知的这一崭新理论,同时针对压缩感知问题学者们提出各种各样求解压缩感知问题的算法,以及利用压缩感知理论和相应算法求解电子工程中很多棘手的现实问题。压缩感知理论的研究也拉开了低维度结构化成份重构问题的研究,比如:矩阵作为高维向量的稀疏性以及矩阵奇异值向量组的稀疏性(也称为矩阵的低秩性)。本文研究的重点放在低秩结构矩阵的重构问题以及低维度结构化成份重构的一般模型稳定性研究,主要研究内容分为四部分:第一部分重点研究矩阵低秩稀疏重构模型——观测矩阵是由一个稀疏矩阵和一个低秩矩阵叠加组成时,怎样重构观测矩阵的稀疏分量以及低秩分量。首先在全空间采样下情况考虑此类问题,并利用凸优化理论证明此类问题的强凸优化模型在一定条件下能准确且唯一重构低秩分量和稀疏分量,并在此基础上给出算法设计中具体参数的选择标准。第二部分重点研究矩阵的低秩稀疏重构模型比全空间采样更一般的情况,即研究矩阵低秩稀疏重构模型在随机子空间采样下的重构问题。利用凸优化理论证明在随机采样下满足一定条件时强凸优化模型的最优解与凸优化模型的最优解一致性,并给出算法设计中具体参数选择标准。第三部分重点研究矩阵低秩稀疏重构模型强凸优化模型中向随机子空间投影矩阵算子。证明在一定条件下此随机矩阵算子满足严格等距性质,并利用矩阵算子的严格等距性质优化算法设计中参数选择标准。第四部分重点研究矩阵低秩稀疏重构模型的推广形式,也就是低维度结构化重构问题。证明在有界噪声干扰下低维度结构化重构问题的凸优化模型最优解具有稳定性。
[Abstract]:There is a very important mathematical problem in electronic engineering, that is, the feasible vector satisfying the observed data in the case of under-sampling is not unique, but a linear subspace composed of feasible solutions. In this case, how to find the only feasible solution of the problem in accordance with the reality, in order to determine the only correct feasible solution in a linear subspace, the only correct feasible solution must have other special properties. In electronic engineering, sparseness (the number of non-zero elements is relatively small) is more studied in electronic engineering. The original problem can be described as: in the case of under-sampling, the known observation data can be used to solve the feasible vector with sparsity. This new theory of compressed perception is put forward, and a variety of algorithms to solve the problem of compressed perception are put forward for the problem of compressed perception. And using the theory of compressed perception and corresponding algorithms to solve many thorny practical problems in electronic engineering. The research of compressed perception theory also opens the research of low-dimensional structural component reconstruction. For example, the sparsity of matrix as high dimensional vector and the sparsity of singular value vector system of matrix (also called low rank of matrix). In this paper, we focus on the reconstruction of low rank structure matrix and the structural composition of low dimension. Research on the stability of the general model of reconfiguration, The main research contents are divided into four parts: the first part focuses on the low-rank sparse reconstruction model of matrix, when the observation matrix is composed of a sparse matrix and a low-rank matrix. How to reconstruct the sparse component and the low rank component of the observation matrix. The convex optimization theory is used to prove that the strong convex optimization model of this kind of problems can reconstruct the low rank and sparse components accurately and uniquely under certain conditions. On this basis, the selection criteria of the specific parameters in the algorithm design are given. In the second part, the low rank sparse reconstruction model of matrix is studied more generally than the full space sampling. By using convex optimization theory, it is proved that the optimal solution of strongly convex optimization model is consistent with that of convex optimization model when it satisfies certain conditions under random sampling. In the third part, we focus on studying the projection matrix operator of the directed random subspace in the strong convex optimization model of the sparse reconstruction model with low rank matrix, and prove the random matrix operator under certain conditions. Satisfying the strict isometric property, In the fourth part, we focus on the generalized form of matrix low rank sparse reconstruction model. It is proved that the optimal solution of the convex optimization model for the low dimensional structured reconstruction problem under bounded noise disturbance is stable.
【学位授予单位】：电子科技大学
【学位级别】：博士
【学位授予年份】：2017
【分类号】：O151.21

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