基于对称锥规划的压缩感知信号重建模型与算法研究

Compressed sensing is a recently introduced novel signal processing theory. It has become a hot international research subject in mathematics and engineering applications. Based on Jordan algebraic technique, this topic will set up new algorithms for compressed sensing signal reconstruction in the context of symmetric cone programming. Firstly, we will establish and optimize some complex compressed sensing signal reconstruction models. By Jordan algebraic tool, we will develop some equivalent relationships between these models and the symmetric cone programming. The optimal conditions of the symmetric cone programming will also be studied. Secondly, by combing the inexact algorithm and the smoothing method, and then introducing the predictor-corrector technique, we will propose an inexact predictor-corrector smoothing method based on a class of symmetric cone complementarity functions with two parameters. This inexact predictor-corrector smoothing method is then applied to deal with large scale sparse signal reconstruction problems with noise. Lastly, we will design a gradient-based neural network and a projection neural network to solve the symmetric cone programming. Based on these two neural networks, we can perform the compressed sensing signal reconstruction in real-time. The scientific significance of the research is to find a new breakthrough point between the symmetric cone programming and compressed sensing, open up new research ideas for compressed sensing signal reconstruction, especially offer new algorithms for large scale signal processing and real-time signal processing, and also provide a theoretical basis and technical support for the actual environmental applications of compressed sensing.

压缩感知是近年来兴起的一种新型信号处理理论，是国际上数学领域和工程应用领域的研究热点。本课题将利用若当代数工具，在对称锥规划框架下，提出压缩感知信号重建新方法。首先，建立并优化一些复杂的压缩感知信号重建模型，借助若当代数工具，研究该模型与对称锥规划的等价转化，给出对称锥规划的最优性条件；其次，基于含两个参数的对称锥互补函数，把非精确算法与光滑算法相结合，引入预估矫正技术，建立求解对称锥规划的非精确预估矫正光滑算法，并利用该方法处理含噪声的大尺度信号重建问题；最后，在对称锥规划框架下，设计基于梯度的神经网络和投影神经网络，分析网络的稳定性，并利用这两种神经网络方法给出压缩感知信号的实时重建。本项目的研究成果将为对称锥规划与压缩感知找到新的切入点，为压缩感知信号重建开辟新的研究思路，尤其为大规模信号处理和实时信号处理提供新的研究方法；能够进一步把压缩感知应用到实际环境中提供理论依据和技术支持。

在若当代数中引入了非线性变换的伪单调性质，通过研究非线性变换的静态性质，解决了非线性对称锥规划问题的解的存在性问题；建立了压缩感知信号重建模型与对称锥规划的关系，设计了求解对称锥规划的神经网络方法，并在此基础上提出了压缩感知信号重建的神经网络方法和延时神经网络方法，分析了网络的收敛性和稳定性；针对压缩感知信号重建的原始L0-范数问题，通过光滑化技巧，借助拉格朗日函数及其偏导数，提出了一种新的迭代算法进行压缩感知信号重建；针对含噪信号，将该算法与正交匹配追踪算法相结合，提出了改进的正交匹配追踪算法，并将加权迭代最小二乘法和正交匹配追踪算法进行结合，形成一个更具有优势的联合算法。这些研究成果将为对称锥规划与压缩感知找到了新的切入点，为压缩感知信号重建开辟新的研究思路，尤其为大规模信号处理和实时信号处理提供新的思路和方法。