Several Gradient Algorithms For Tensor Eigenvalue Problems

With the development of science and technology and the advent of big data era,more and more practical ap-plications need to use tensor to describe some data problems,such as computer visualization,signal processing,quantum entanglement,automatic control,statistical data analysis,high order Markov chain,hypergraph theory,machine learning,medical imaging and so on.In order to explore the inherent characteristics of these data,it is often necessary to decompose or compute the eigenvalues of the involved tensors.Tensor eigenvalue problem has become an important topic of multiple linear algebra.In this thesis,several fast algorithms are proposed and analyzed in the framework of generalized tensor eigenpairs.The first one is the adaptive gradient method(AG)which calculates the generalized eigenpair of tensors.The second is the spectral gradient method(SPG)for calculating the generalized tensor eigenvalue complementarity problem.The third one is the SPA with an adaptive shift.The thesis mainly studies the algorithms of generalized tensor eigenvalues to related problems.The specific contents are as follows:The first chapter is the introduction part,briefly introduced tensor definition and its operations,as well as some applications of tensor.The second chapter gives some necessary concepts and conclusions about tensor eigenvalue problem.We use the inexact gradient method to improve the SSPM proposed in the literature[23]and use to solve the generalized eigenpairs problem of tensor.And establish its global convergence and linear convergence under some suitable assumptions.Finally,our numerical experiments show that our method is efficient and superior.In the fourth chapter,we present two spectral projection gradient algorithms of TEiCP and establish the global convergence results under some suitable assumptions.One of the main features of SPG is to take BB steps in the search direction.This method is better than the steepest descent gradient method or projection gradient method in practice.We also propose SSPA,which is a great improvement to the original SPA approach.The final numerical experiments also show that our proposed methods are effective.