Solving Sylvester Tensor Equation Based On Tensor Train Decomposition

A tensor is a high-dimensional array,which can be regarded as a high-order generalization of a matrix.It has a wide range of applications in the fields of signal processing,nonlinear optimization,image processing,model reduction,and data mining.This paper mainly aims at solving the problem of Sylvester tensor equation,and proposes a fast and effective optimization algorithm and conducts theoretical analysis.The full text is divided into five chapters:Chapter 1.First,the relevant symbols used in this article are given,and then the research background and current situation of tensor decomposition and Sylvester tensor equation are introduced.Chapter 2,Preliminary Knowledge.First introduce the definitions of CP decomposition,Tucker decomposition and Tensor Train(TT)decomposition,and then give tensor Kronecker product,left orthogonal and right orthogonal definitions,and tensor-related properties to prepare for the following content.Chapter 3 mainly introduces two methods of solving Sylvester tensor equation.First,the tensor projection method is introduced to solve the linear equation in the form of Ax=c,which is solved by the tensor-based GMRES method,and the preprocessing method is also given.The second method is based on the alternate direction implicit iteration(ADI)method,which is extended to the structure tensor,and the ADI iteration based on the structure tensor equation is proposed.Chapter 4 mainly studies the problem of solving the Sylveter tensor equation based on the form of Tensor Train(TT).At the same time,in order to reduce the calculation amount of the coefficient matrix,the coefficient matrix is randomly sampled,and a random method based on TT decomposition is proposed.In addition,for the sampled sub-problems,two methods are given here,namely the gradient method and the Newton method.Under certain assumptions,this paper presents the convergence analysis of the stochastic gradient method based on TT decomposition.Chapter 5,this part mainly gives some numerical experiments to verify the effectiveness of the proposed method.Experiment 1 gives numerical experiments of stochastic gradient descent method and stochastic Newton method based on TT decomposition.The scale of the calculation example is 10¹³.Numerical experiments show that both methods can handle large-scale problems and each has advantages.The calculation example of experiment 2 is used to compare the stochastic gradient descent method and the projection method based on tensor format.The experiment is small in scale,and the experiment shows that the effects of the two algorithms are not much different.but the projection method cannot handle large-scale tensors.Chapter 6,Summarize the full text and determine the future research direction.