| As a form of high-dimensional array,the tensor plays a more and more important role in the era of big data and artificial intelligence.With the advent of the age of 5G communication,efficient data transmission puts forward higher requirements for data processing capacity.As the important problems of data processing and signal processing,data mining,data compression,data dimension reduction,data clustering and data recovery have been redefined in the form of tensors.In the process,the low rank approximation of the tensor becomes the core step of data processing.This is because the elements in high-dimensional data are not necessarily the key support of the whole data,the redundant data often contains the low rank characteristic,which is the fundamental basis of data processing.Therefore,the low rank approximation of the tensor,as the representation of low rank and low dimension of the data,has important research value.In this dissertation,based on the structural characteristics of the tensors,the low rank approximation algorithms of the tensors and their applications are studied.The main research content of this dissertation includes the following aspects.Firstly,based on the partial symmetries of partially symmetric tensors such as the paired symmetric tensors and the k-mode symmetric tensors,models and algorithms of the structure preserving best rank one approximations(partially symmetric best rank one approximations)are proposed,and the convergence of the algorithms is discussed.Furthermore,numerical examples are given to verify the effectiveness of the algorithms,and applications of the algorithms in solving tensor eigenvalues and tensor sample clustering are also given.Secondly,for the tensor data sets with manifold structure characteristics,the manifold regularization nonnegative triple decomposition model is proposed by applying the manifold learning method to represent the correlation between the tensor objects.Then an algorithm for solving the manifold regularization nonnegative triple decomposition is proposed,and the convergence of the algorithm is discussed.Numerical experiments of the algorithm are also given to show the convergence rate and applications for the compression and the low rank representation of the video data.Thirdly,according to the characteristics of the matrices obtained by slicing the tensor,a new tensor completion model and the corresponding algorithm are proposed.Then the algorithm is applied to the color image completion,MRI recovery and video recovery,and the comparative experiments are used to show the advantages of the algorithm in data recovery. |
