Algebraic Algorithm Of Fourth-order Tensor Decomposition

Tensor is multidimensional array.A single array is a first-order tensor,also a vector.A binary array is a second-order tensor,also a matrix.A triple array is a third-order tensor.A tensor whose order is greater than two is called a high-order tensor.The decomposition of tensors has great application in data compression,image processing,quantum information and so on.The CP decomposition algorithm of tensors is usually divided into two categories,the numerical algorithm and the algebraic algorithm.Numerical algorithms are mainly applied to the low-rank approximation of tensors,while algebraic algorithms are applied to the exact CP decomposition of tensors.This paper studies the canonical polyadic decomposition of fourth-order tensors.The CP decomposition of tensors is to represent the tensor as a sum of some rank-1 tensor,and the number is the least.Assuming that the CP rank of the CP decomposition for tensor T is R,the factor matrices are A,B,C,D.This paper discuss the algebra algorithm for CP Decomposition only in the case that the rank of the Khatri-rao product of the factor matrix C and D is less than or equal to R:(1)the case that the rank of the Khatri-rao product of the factor matrix C and D is equal to R.In this paper,we present a special algorithm for CP decomposition of fourth-order tensors,and prove that we can get a CP decomposition of tensor T by the algorithm under certain conditions,and the decomposition is unique.The uniqueness here means the uniqueness in the sense of equivalence class.(2)the case that the rank of the Khatri-rao product of the factor matrix C and D is less than or equal to R.we present a general algorithm for the CP decomposition of fourth-order tensors,and prove that the decomposition obtained by the algorithm is the CP decomposition of tensor T,and the CP decomposition of T is unique.