基于宽度理论的稀疏测量矩阵的构造研究

Compressed sensing is a popular research direction in recent years. Its fundamental idea is to recover a high-dimensional sparse signal from remarkably small number of measurements using the sparsity of signals. One of the main problems in this field is the construction of measurement matrices, which is mainly based on the Restricted Isometry Property (RIP) or the column coherence of matrices. However, the current construction methods of the deterministic measurement matrices and structured random matrices have difficulties in making.them arrive at the optimal measurements, due to insufficient characterization of the measurement matrices. The project is to study the construction of measurement matrices based on discovering the relevance between width theory and optimal conditions of reconstruction of sparse signals. Our main work is summarized as follows..(1) We will weaken the RIP condition by improving its constants and giving the optimal upper bound of some RIP constant, and investigate more accurate and elaborate characterization of the optimal measurement matrices. (2) We will study the optimal approximation order of l_q-minimization decoder and the conditions of constructing measurement matrices under l_q-minimization for 0<q<1, and analyze the the relevance between this order and the optimal sampling number. (3) With respect to the above new characterization conditions, we will construct the deterministic measurement matrices and structured random matrices with optimal orders by overcoming the barrier of the matrices with small column coherence, thus, the signals under these encoders can be reconstructed accurately or with high probability via l_1-minimization or l_q-minimization. The completion of this project will provide theoretical guarantee for the further applications of compressed sensing.

压缩感知是近来国际上热门的研究方向。其主要思想是利用信号的稀疏性，通过尽量少的观测信息恢复信号。测量矩阵的构造是压缩感知研究的核心问题之一，其构造主要基于有限等距性质(RIP)或矩阵的列相干性。遗憾的是现有的确定性测量矩阵和结构随机矩阵的构造方法很难使其达到最优测量数。究其原因是对测量矩阵的特有性质还没有完全揭示和精确刻画。本项目主要在揭示宽度理论和稀疏信号重构最优性条件的关联性基础上，研究测量矩阵的构造问题。（1）通过给出某一RIP常数的最优上界进一步松弛RIP条件，研究刻画最优测量矩阵的更加精细的条件。（2）研究非凸lq(0<q<1)极小化稀疏解码的最优逼近阶，分析该阶和最优采样数的联系。（3）基于以上新的刻画条件，通过克服矩阵列相干性带来的瓶颈，构造阶最优的确定性矩阵和结构随机矩阵，使得信号在其编码下用l1或lq解码能够准确或高概率地重构。从而为压缩感知的进一步应用提供理论保证。

压缩感知是近来国际上热门的研究方向，其主要思想为：利用信号稀疏性的特征，通过尽量少的观测信息恢复信号。压缩感知所研究的核心问题有三个，分别是信号稀疏表示的字典设计、测量矩阵的设计、信号快速恢复的算法设计。测量矩阵的构造又是一个最为关键的问题。事实上，快速算法的设计与测量矩阵紧密相关，优良的测量矩阵的设计对快速恢复的算法设计奠定了基础。本项目探讨的是测量矩阵的构造问题，为此，建立了保证非凸稀疏优化模型恢复稀疏信号的本质弱于有限等距性质(RIP)的一系列新条件，由此刻画了非凸稀疏优化模型在块稀疏信号、字典意义下稀疏信号、低秩矩阵恢复方面的显著优势；探讨了一类比Gauss随机矩阵更宽泛的 Weibull随机矩阵，证实其作为测量矩阵在稀疏信号恢复方面具有类似于Gauss随机矩阵的优良性能，为结构随机测量矩阵的构造这一困难问题奠定理论基础并提供了全新尝试；针对结构稀疏信号和测量含噪但噪声水平难以获得的信号恢复问题，提出并发展了一系列新的预处理和恢复条件，给出了一类加权稀疏意义下的信号恢复的条件刻画，建立了一类加权非凸稀疏优化模型解的稳定性和鲁棒性；针对多模态数据分离问题，建立了稀疏表示的非凸优化模型，利用字典意义下的零空间性质刻画了测量矩阵的特征，进而估计了模型解的误差上界，为数据分离提供了新方法。该项目的研究完善了压缩感知的理论，并对该理论在医学图像、遥感图像和量子信息处理方面的进一步应用提供理论保障和方法支持。

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