Research On Structured Tensors And Their Related Computing Problems

Tensor computation is one of the hot topics in the field of numerical algebra.It is widely applied in hyperspectral image,computer vision,multi-line image analysis of face recognition,seismic signal,electroencephalogram,medicine,neuroscience and so on.This paper studies the structured tensors and their related computing problems as following.In Chapter 2,we consider the symmetric and nonnegative tensor completion problem.We first reformulate this problem as the minimization problem of the nuclear norm,and then design the alternating direction method(ADM)to solve this problem.The X-subproblem is treated by the singular value truncation method,and the y-subproblem is solved by the nonmonotone spectral projected gradient method.The convergence of ADM method is given.Numerical examples illustrate that the new method is feasible.In Chapter 3,we consider the Toeplitz tensor approximation problem with some element constraints.We study the structure of Toeplitz tensor and obtain the projection of any tensor on Toeplitz tensor.Then we construct the nonmonotone spectral projected gradient method to solve the problem.Numerical experiments show the feasibility of the new method.In Chapter 4,we consider the Vandermonde decomposition of Hankel tensor.We first reformulate the Hankel tensor decomposition problem as the systems of nonlinear equations.Then we design a new method to solve the nonlinear equations by a series of matrix changes and subproblems.Numerical examples illustrate that the new method is feasible.