Computational Methods for Symmetric Tensor Problems

University Of California-San Diego, La Jolla CA

Investigators

Abstract

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.