Computational Methods for Symmetric Tensor Problems

这个项目以对称计算问题为目标。张量是矩阵的推广。对称张量的条目服从对称模式。在大数据时代，与之相关的计算问题变得越来越重要。与矩阵的情况一样，关于对称张量的基本问题是计算它们在实域和复域上的分解，确定它们的秩数，计算低阶近似，并将它们应用于相关的应用。该项目致力于研究对称张量的计算问题。张量是计算数学中的一个强有力的工具。对称张量具有优美的代数和几何性质。一个基本的重要问题是将张量写成具有最小长度的一阶张量的和。这就是所谓的张量分解问题。张量分解可以在实数域上进行，也可以在复数域上进行。虽然它们是相关的，但实数域上的分解与复数域上的分解有很大的不同。在应用中，张量可以很大，但它们的级别可能很小。人们经常需要用一个低阶张量来近似一个对称张量，尽可能地接近。生成多项式是解决对称张量计算问题的有效工具。它巧妙地使用了代数性质。对称张量可以看作是对称的多线性泛函，可以用多元多项式来表示。这个项目使用的数学知识来自计算代数、多项式系统、矩阵计算、复代数几何和实代数几何以及最优化。本课题的研究成果在多线性代数、信号处理、盲源分离、数值分析、高阶马尔可夫链等领域具有潜在的应用价值。该项目将为对这一主题感兴趣的学生和年轻研究人员提供培训。所产生的成果将迅速传播给科学界。

This project targets at symmetric computational problems. Tensors are generalizations of matrices. The entries of symmetric tensors obey symmetric patterns. Computational problems about them become more and more important in big data time. Like the case of matrices, basic problems about symmetric tensors are computing their decompositions over the real and complex fields, determining their ranks, computing low-rank approximations, and applying them in relevant applications. This project devotes to the research of computational problems about symmetric tensors.Tensor is a powerful tool in computational mathematics. Symmetric tensors have beautiful algebraic and geometric properties. A problem of fundamental importance is to write a tensor as a sum of rank one tensors, with minimum length. This is the so-called tensor decomposition problem. Tensor decompositions can be over either the real or complex field. Although they are related, the decomposition over the real field is very different from the case of complex field. In applications, tensors can be very large, but their ranks may be small. People often need to approximate a symmetric tensor by a low rank one, as close as possible. Generating polynomial is an efficient tool for solving symmetric tensor computational problems. It uses the algebraic properties elegantly. A symmetric tensor can be viewed as a symmetric multi-linear functional, which can be expressed by a multivariate polynomial. This project uses mathematical knowledge from computational algebra, polynomial systems, matrix computations, complex and real algebraic geometry, and optimization. The results produced by this project have potential applications in multilinear algebra, signal processing, blind source separation, numerical analysis, higher order Markov chains. The project is going to provide training for students and young researchers who are interested in the subject. Produced results will be promptly disseminated to the scientific community.