基于两类算法求解实对称张量的正交逼近问题

In the field of engineering, there are many problems to be solved, such as undetermined blind source separation and implicit variable identification. The non-uniqueness of implicit mixed matrices, which can not be solved by traditional matrix method, can be solved theoretically effectively by using the tensor method of orthogonal approximations and decompositions based on symmetric tensors. Naturally, orthogonal approximations and decompositions of real symmetric tensors are the key computational problems in many practical applications, such as blind source signal separation, implicit variable recognition and so on. In view of the complexity of the structure of tensor itself, many theories and algorithms related to orthogonal approximation of real symmetric tensors need to be explored urgently. This project will focus on the theory and algorithms of orthogonal approximation of real symmetric tensor. Firstly, explore the uniqueness and perturbation analysis of the solution of the orthogonal approximation problem of real symmetric tensors; secondly, establish an augmented Lagrangian function method and a feasible method on the matrix manifold, and implement the numerical experiments; thirdly, apply the obtained theory and algorithm to the blind source signal separation problem. This research will provide theoretical basis and corresponding numerical software for the problems mentioned above, and promote the further study of tensor approximation problems.

在工程应用中，需要求解大量的欠定盲源信号分离、隐式变量识别等问题。基于实对称张量的正交逼近和分解的张量方法来求解这些问题在理论上能有效地解决传统矩阵方法无法解决的隐式混合矩阵的不唯一性。自然地，实对称张量正交逼近和分解是诸如此类众多实际问题的关键科学计算问题。鉴于张量数据本身的复杂性，与实对称张量正交逼近相关的许多关键理论和算法亟需探索。本项目将聚焦研究实对称张量正交逼近问题的理论与算法:（1）探索实对称张量正交逼近问题解的唯一性及扰动分析等理论基础；（2）建立求解该问题的增广拉格朗日函数法和矩阵流形上的算法等与问题结构相匹配的计算方法并进行数值软件实现；（3）将得到的理论与算法应用到盲源信号分离问题。本项目的研究将为上述实际问题的准确、高效求解提供理论依据和相应数值软件，为张量逼近优化问题的进一步研究提供推动作用。

本项目围绕实对称张量的正交逼近问题进行了相关的研究。主要成果包括：研究了实对称张量正交逼近问题解的性质，建立了求解算法的全局收敛性和局部收敛率；刻画了对称和非对称完全正交可分解张量的Von Neumann型不等式和谱函数性质；研究了Stiefel流形约束下的基于实对称张量表达的四次型极小化问题的高效求解；此外，项目还研究了张量低秩逼近问题的其它相关基础性质，设计了一种基于谱梯度法和非线性共轭梯度法的优化方法高效求解张量CP分解问题；通过张量分解技术，建立了一类多项式优化问题解集的更好的误差界结果；探讨了低秩张量在图像处理中的应用。