Successive Rank-one Decomposition Of Higher-order Complex Tensors

High-dimensional model is becoming important in the sciences and engineering be-cause of development in sensor technology and storage technology.Lots of problems in the matrix space are becoming more and more difficult.The researches about tensor is becom-ing more and more popular.In recent years,researches on tensor are mainly concentrated in the tensor eigenvalue problem,the best rank-one approximation problem,the tensor decom-position problem and the tensor rank problem.It has shown that the problem of computing largest Z-eigenvalue is related to the best symmetric real rank-one approximation problem,and the problems of computing largest unitary symmetric eigenvalue(US-eigenvalue)and unitary eigenvalue(U-eigenvalue)are related to the best symmetric complex rank-one ap-proximation problem to a symmetric tensor and the best complex rank-one approximation problem to a general tensor,respectively.Inspired by the successive rank-one decompo-sition method of symmetric real tensor over the real filed,we try to use the successive rank-one decomposition method to decompose the higher-order complex tensor.The main contributions of this paper are as follows:1,We extend the successive rank-one decomposition method to the complex field,and propose the successive symmetric rank-one decomposition method of a higher-order symmetric complex tensor.We show that the successive decomposition of real tensors may actually be different over the real field and the complex field by an example.For some symmetric real tensors,we can get a CANDECOMP/PARAFAC(CP)decomposition with the iteration steps less than the dimension of a tensor over the complex field,but it needs infinite iteration steps to get one over the real field,namely,it just can get an approximation.Furthermore,we prove that the CP decomposition could be obtained when the successive symmetric rank-one complex decomposition method is applied to a unitary diagonalizable symmetric tensor.2,We extend the successive rank-one decomposition method to decompose a non-symmetric tensor.According to the tensor symmetric embedding,we demonstrate the ef-fectiveness constructively of the successive rank-one decomposition method.Similar to the case of symmetric tensor,we only get an approximation of a general tensor with the successive rank-one decomposition method,but we prove that the CP decomposition could be obtained when the successive rank-one decomposition method is applied to a unitary diagonalizable tensor.Through the numerical experiment,we find that the successive de-composition of real tensors may actually be different over the real field and the complex field.For some real tensors,we can get a CP decomposition with the iteration steps less than the smallest dimension of a tensor over the complex field,but it needs infinite iteration steps to get one over the real field,namely,it just can get an approximation.