矩阵联合(块)对角化算法研究及在盲信号分离中的应用

In this project, we mainly study the matrix joint (block) diagonalization algorithms and their applications in blind source separation and other areas. This project is problem-motivated, involving cross of multiple disciplines. We intend to introduce some constraint conditions in cost function, treat the joint diagonalization problem as the two-objective optimization problem, and establish some new algorithms which can avoid trivial and degenerate solution; To improve the computational efficiency, some new algorithms proposed are based on the matrix decomposition, the convergence and stopping criterion of the joint approximate diagonalization of eigen-matrices, altering row diagonalization and other algorithms; Using linear algebraic theory, we analyze and reconstruct the joint block diagonalization algorithm which is based on matrix *-algebra theory, and mainly consider the computational demanding and precision; We also intend to treat the set of target matrices as third-order Tensor, and establish some new joint (block) diagonalization algorithms by Tensor decomposition technique, and the software application platform will be established.

本项目主要研究矩阵联合(块)对角化算法，以及此类算法在盲信号分离等领域的应用，属于问题驱动型课题，涉及到多个学科的交叉。本项目拟在代价函数中引入约束条件，把联合对角问题转化为一个双目标的优化问题，从而建立可以避免平凡解和退化解的算法；通过研究特征矩阵联合近似对角化、交替行对角化等算法的收敛性和终止标准，以矩阵分解理论为基础，构造新算法，提高计算效率；基于线性代数理论，对以矩阵*-代数理论为基础的联合块对角化算法进行分析和重构，并着重考虑新算法的计算量和计算精度；将目标矩阵组转化为三阶Tensor，利用Tensor分解技术建立联合（块）对角化算法，并搭建算法应用的软件平台。

多个矩阵联合角化/块对角化方法是解决盲源分离（BSS）和主成分分析（ICA）等问题的一类重要的方法，同时也是矩阵计算研究的重要内容之一。利用目标矩阵集合具有的对角化/块对角化的内在共同分解结构特点，结合Hermitian矩阵求解特征值的经典Jacobi算法、矩阵LU分解和矩阵特征值及特征向量关系等，构造正交和非正交联合对角化算法、研究经典算法的收敛性及其算法在BSS上的应用是本项目的研究重点。基于单个Hermitian矩阵求解特征值的Jacobi算法、矩阵的LU分解理论和矩阵的特征值和特征向量关系等，建立了新型非正交矩阵联合对角化Jacobi-like算法、基于LU分解的非正交矩阵联合对角化算法和基于Hermitan矩阵分解的正交联合对角化算法，同时也给出部分算法的收敛性分析，已发表或录用论文7篇。