Computational Methods for Symmetric Tensor Problems

对称张量问题的计算方法

• 批准号: 1619973
• 负责人: Jiawang Nie
• 金额: $ 15万
• 依托单位: University of California-San Diego
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2016
• 资助国家: 美国
• 起止时间: 2016-07-01 至 2020-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1619973&HistoricalAwards=false
• 关键词: Computational; Methods; Symmetric; Tensor; Problems

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.

该项目针对对称计算问题。张量是矩阵的概括。对称张量的条目遵守对称模式。关于它们的计算问题在大数据时间内变得越来越重要。像矩阵的情况一样，关于对称张量的基本问题正在计算它们在真实和复杂的字段上的分解，确定其等级，计算低级别近似值并将其应用于相关应用程序。该项目致力于对对称张量的计算问题的研究。TENSOR是计算数学中的强大工具。对称张量具有美丽的代数和几何特性。一个基本重要性的问题是将张量作为一台张量的总和，最小长度。这是所谓的张量分解问题。张量分解可以在真实或复杂场上。尽管它们是相关的，但实际场上的分解与复杂场的情况大不相同。在应用程序中，张量可能很大，但是它们的排名可能很小。人们通常需要尽可能接近一个低等级的对称张量。生成多项式是解决对称张量计算问题的有效工具。它优雅地使用代数属性。 对称张量可以看作是对称的多线性功能，可以用多元多项式表示。该项目使用计算代数，多项式系统，矩阵计算，复杂和真实代数几何形状和优化的数学知识。该项目产生的结果在多线性代数，信号处理，盲源分离，数值分析，高阶马尔可夫链中具有潜在的应用。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。产生的结果将迅速传播给科学界。