High-order sum-of-squares structured tensors:theory and applications  

作　　者：Haibin CHEN Yiju WANG Guanglu ZHOU 

机构地区：[1]School of Management Science,Qufu Normal University,Rizhao 276826,China [2]Department of Mathematics and Statistics,Curtin University,Perth,WA,Australia

出　　处：《Frontiers of Mathematics in China》2020年第2期255-284,共30页中国高等学校学术文摘·数学（英文）

基　　金：This work was supported in part by the National Natural Science Foundation of China(Grant Nos.11601261,11671228);the Natural Science Foundation of Shandong Province(No.ZR2019MA022).

摘　　要：Tensor decomposition is an important research area with numerous applications in data mining and computational neuroscience.An important class of tensor decomposition is sum-of-squares(SOS)tensor decomposition.SOS tensor decomposition has a close connection with SOS polynomials,and SOS polynomials are very important in polynomial theory and polynomial optimization.In this paper,we give a detailed survey on recent advances of high-order SOS tensors and their applications.It first shows that several classes of symmetric structured tensors available in the literature have SOS decomposition in the even order symmetric case.Then,the SOS-rank for tensors with SOS decomposition and the SOS-width for SOS tensor cones are established.Further,a sharper explicit upper bound of the SOS-rank for tensors with bounded exponent is provided,and the exact SOS-width for the cone consists of all such tensors with SOS decomposition is identified.Some potential research directions in the future are also listed in this paper.

关 键 词：Sum-of-squares(SOS)tensor positive semi-definite(PSD)tensor H-eigenvalue structured tensor 

分 类 号：O183[理学—数学]