Fourth Order Tensor Decomposition Approach And Its Application

With the development of science and technology,the scale of the data we got becomes larger and larger,and type of the data also becames more and more complex.The data have multi-storage format,the grayscale photos are often stored in matrix form,the color photos are stored in the form of third-order tensors,and the video data are the fourth-order tensor.Therefore,this paper regards practical application with video compression as the application background to research the fourth-order tensor decomposition.First,the relation between the special third-order symmetric tensor decomposition and tensor approximation problem is considered.We present the rank-1 decomposition method of the third-order symmetric tensor,based on the tensor best rank approximation problem.This problem is an important part in the field of tensor research.Based on the above new method,the best rank-1 approximation of the third-order symmetric tensor is given.Meanwhile,the upper bound of the conditional error is estimate.Secondly,we propose three third-order tensor decomposition methods based on different fourth-order tensor multiplications.(1)based on the matrix multiplication,the F-product of the fourth-order tensor is proposed.Using the definitions of the circulant matrix,the Discrete Fourier Transform and the Singular Value Decomposition,the fourth-order tensor decomposition is obtained based on the F-product,record as F-TD;(2)the TH-product of the fourth-order tensor is proposed,based on the t-product definition of third-order tensor.Using the t-svd of the third-order tensor and the TH-product,the fourth-order tensor decomposition is obtained,record as FT-SVD;(3)based on the Face-wise multipliers of third-order tensors,we define a new fourth-order tensor B-product.The fourth-order tensor decomposition are obtain,which can be degenerated into matrix singular value decomposition,denoted as B-TD.Based on the above theoretical work,the resolution accuracy and running time of the two methods of F-TD and FT-SVD are compared by calculating artificial data and large scale random data.The results show that the above two decomposition methods have high decomposition accuracy.For the B-TD method,an accurate formula for the compression ratio of fourth-order tensors and the video compression algorithm are given.At last,the compressed videos of monochrome videos and color videos are presented.The experiments show that the B-TD method is feasible and has good compression effect.