Research On Theory And Algorithm Of Tensor Rank 1 Approximation Problem

Tensor,because of its high load carrying information,has become an effective way to express complex data.It has been widely used in signal and image processing,machine learning,neuroscience and other fields.Feature extraction from tensor representation of massive data is an important application of tensor low-rank approximation.In order to simplify the calculation,people often transform the tensor low-rank approximation problem into a series of tensor rank-1 approximation problems.Furthermore,because tensor rank-1approximation in the general sense cannot extract important information hidden behind the data,people introduced sparse decomposition of tensor.Based on this,this paper considers the theoretical and algorithm research of the tensor sparse rank-1 approximation problem and partial symmetric tensor rank-1 approximation problem.This paper aims at the sparse rank-1 approximation of high-order tensors and the rank-1 approximation of high-order partial symmetric tensors.The numerical algorithms of these two types of optimization problems are established by analyzing their properties,and the convergence analysis of the algorithms is given.The effectiveness of the algorithms is verified by numerical experiments.The structure of this paper is as follows:In Chapter 1,mainly introduces the research background and development status of tensor,tensor rank-1 approximation and tensor sparse rank-1 approximation,as well as the main research contents of this paper.In Chapter 2,for the sparse rank-1 approximation problem of high-order tensors,we introduce a threshold operator to sparsely control the decomposition factors,and then we establish an optimization model for the best sparse rank-1 approximation problem.An alternative algorithm for this problem is given.Not only the calculation amount of each iteration step of the algorithm is small,but also the global convergence of the algorithm is established without any assumptions.The effectiveness of the algorithm are verified by some numerical examples.In Chapter 3,for the rank-1 approximation problem of partial symmetric tensors,we established an alternating linear minimization method for the problem,and the convergence of the algorithm is proved under certain assumptions.Finally,the validity of the algorithm are verified by some numerical examples.In chapter 4,a brief summary of the research contents of this article is given,and on the outlook for future research topics.