Computational Methods for Symmetric Tensor Problems

这个项目的目标是对称计算问题。张量是矩阵的推广。对称张量的项服从对称模式。在大数据时代，关于它们的计算问题变得越来越重要。像矩阵一样，对称张量的基本问题是计算它们在真实的和复数域上的分解，确定它们的秩，计算低秩近似，并将它们应用于相关应用。本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。一个非常重要的问题是把一个张量写成一个秩为一的张量的和，并且长度最小。这就是所谓的张量分解问题。张量分解可以在真实的域上进行，也可以在复域上进行。虽然它们是相关的，但真实的域上的分解与复域上的分解有很大的不同。在应用中，张量可以非常大，但它们的秩可能很小。人们经常需要用一个低秩张量来近似一个对称张量，尽可能地接近。生成多项式是求解对称张量计算问题的有效工具。它巧妙地运用了代数性质。 对称张量可以看作是一个对称的多线性泛函，它可以用多元多项式表示。这个项目使用的数学知识，从计算代数，多项式系统，矩阵计算，复杂和真实的代数几何，优化。本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。该项目将为对该主题感兴趣的学生和年轻研究人员提供培训。所产生的结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.