非线性压缩感知问题研究

Compressed sensing is one of the popular research direction，which has made rich achivements and has a great influence in many areas , such as image processing , magnetic resonance imaging and radar data analysis. Traditional compressed sensing theory mainly discusses the sparse signal recovery under the linear observation system. However, in many practical problems, the observation system is often nonlinear, for example, the phase retrieval problem and 1 bit compressed sensing problem. Hence，recovering sparse signal from nonlinear measurements concerned by many scholars. The project will mainly study on nonlinear compressed sensing problem. Firstly, combined with the theory of linear compressed sensing, we will analyze the problem of structured sparse signal recovery from nonlinear measurements. Members will also construct suitable structured random measurements which satisfy certain properties . Secondly, this team will generalize ADM algorithm and Proximity algorithm to nonlinear compressed sensing to construct fast and efficient algorithms. The convergence and robustness of these algorithms will be analyzed. Furthermore, the application of non-convex algorithms to nonlinear compressed sensing will be discussed. We believe these research results will further promote the progress of the nonlinear compressed sensing theory and provide some new algorithms for solving practical problems in science and Engineering.

压缩感知是目前非常热门的研究方向之一，已经取得了丰富的成果，在核磁共振成像，图像处理，雷达数据分析等很多领域都有了很大影响。传统的压缩感知理论主要分析线性观测系统下的稀疏信号的恢复，但是在很多实际问题中，观测系统往往是非线性的，例如相位恢复问题和1比特压缩感知问题。因此，研究在非线性观测系统下稀疏信号的恢复问题受到了很多学者的关注。本项目将主要研究非线性压缩感知问题，结合线性压缩感知的相关理论，分析非线性观测系统下结构性稀疏信号的恢复问题，构造合适的结构随机测量，用于非线性压缩感知问题的恢复；将ADM算法，Proximity算法等推广到非线性压缩感知问题，构造快速，高效的恢复算法，并分析算法的收敛性和鲁棒性；对非凸优化算法在非线性压缩感知问题中的应用进行分析。相信这些研究结果必将进一步推动非线性压缩感知理论的进展，并为科学和工程中实际问题的解决提供一些新的算法。

压缩感知是一个非常热门的研究领域，已经取得了非常丰富的研究成果，在物理，电信，医学，生物等很多领域都有非常重要的应用。压缩感知主要是利用信号的稀疏性，可以通过有效的算法从少数的线性测量中恢复信号。但是在实际问题中，也会遇到很多测量是非线性的，从非线性的测量中恢复稀疏信号的问题称为非线性压缩感知问题，比如相位恢复问题，一比特压缩感知问题等等。针对非线性压缩感知问题，已经取得了非常丰富的研究成果。本项目对于稀疏相位恢复问题和一比特压缩感知问题和一般非线性压缩感知问题进行了研究。首先对非凸的Shatten_p 范数最小化恢复算法进行了分析，利用限制等距条件得到了恢复成功的条件。应用秩1模型分析了稀疏相位恢复所需的测量数。结构性稀疏信号的恢复问题也是很受关注的问题，本项目分析了非负稀疏信号和组稀疏信号的相位恢复问题，将两段式相位恢复算法推广了非负稀疏和组稀疏相位恢复的情形，进行了数值模拟。最后对利用结构性随机测量进行1比特压缩感知恢复的问题进行了一些初步探索。