Theories And Numerical Methods For Several Approximation Problems Of Constrained Tensors

The constrained tensors approximation problems are important topics of numerical algebra,which are widely applied in the blind source separation,higher-order statistics,machine learning and harmonic retrieval problems.In this paper,we study systematically the theories and numerical methods for the following constrained tensors approximation problems.In Chapter 2,the best low multilinear rank approximation of symmetric tensors(?)is studied.We first reformulate this problem as the maximation problem on the Riemannian manifold,and then we design the Riemannian spectral conjugate gradient(RSCG)method for solving the equivalent maximation problem.Finally,the numerical results show that our algorithm is feasible and effective.In Chapter 3,we focus on the Hankel tensor approximation problems(?)The strong Hankel tensor approximation problem is transformed into a smooth unconstrained optimization problem by using the Vandermonde decomposition of positive semidefinite Hankel matrix,and the nonlinear conjugate gradient method with Armijo linear search is designed to solve this problem.We construct Dykstra's algorithm and its acceleration algorithms to solve the Hankel tensor approximations problem with structure constraint.Alternating direction method is used for solving the best low multilinear rank approximation of Hankel tensor.Numerical experiments are performed to illustrate the feasibility of the proposed methods.In Chapter 4,we study the low rank approximation solution of the following second order tensor equation(?)The low rank approximation solution problem is transformed into an unconstrained optimization problem base on Gramian expression,and then the nonlinear conjugate gradient method is designed to solve the equivalent unconstrained optimization problem.Some numerical examples are presented to illustrate that the new method has faster convergence speed than the LR-ADI method and Krylov subspace method.