The Symmetric Decomposition Of Symmetric Tensor And The Best Low Rank Approximation

A tensor is a higher order generalization of a matrix, because of the limitationof traditional matrix theory in processing data, tensor analysis become a veryimportant tool in science and engineering field, tensor decomposition and the bestlow rank approximation is one of the hot spots in the recent research, it plays animportant role in many fields such as the signal processing, data communication,computer vision, and image processing, symmetric tensor is a very special class oftensor, it can be thought as a higher order generalization of a symmetric matrix,recently it also become a hot topic.This article can be divided into two parts, the first part is the third chapter, wemain research the uniqueness of symmetric decomposition of symmetric tensor,first we derived the uniqueness sufficient condition of symmetric decomposition ofthree order symmetric tensors, and thus deduces to any order, part two of this articleis mainly about the best symmetric low rank approximation of symmetrical tensors,first we proved that for any symmetric tensor of arbitrary order, they all have a bestsymmetric rank1approximation, secondly, we discuss that when the symmetricrank of a symmetric tensor has rank s(A)> r, and its order is more than2,then Adoes not have the best rank-1approximation, we also prove that some special korder symmetric tensor dosent have the best rank-r approximation, and at the end,when the condition is limited to nonnegative, we prove the nonnegative symmetrictensors have best nonnegative symmetric rank-r approximation.