New Results On Eigenvalues And Rank-one Approximation Of Tensors

Higher-order tensors play an increasingly important role in many fields such as signal process-ing,computer vision,data mining,neuroscience,and etc..Extracting useful information from higher-order tensors is an important task.Tensor decomposition and low-rank approximation are important means to accomplish this task.In this thesis,from a new perspective,tensor eigenvalues,tensor decompositions,and algorithms for tensor rank-one approximation which are different from the existing definitions are given.In addition,we also study the H-eigenvalue of a special structure tensor-M tensor.The main research ideas and achievements of this thesis are as follows:1.Matrix eigenvalues play an important role in matrix analysis.As a high-order generaliza-tion of matrix,the definition of eigenvalues of tensors is also of great significance in the study of tensors.In this thesis,based on the idea that tensor can be regarded as mapping from one space to another,we define an eigenvalue to ensure the stability of eigenspace.By using the optimality condition of the extremum problem of "P-degree-type",the unified form of tensor eigenvalues is given,that makes it possible to define more tensor eigenvalues.The practical significance and properties of some eigenvalues are also analyzed.In addition,the unified form of tensor singular value is given.A new tensor multiplication is defined based on the relationship between tensor and mapping.Furthermore,the definitions of tensor QR decomposition,LU decomposition,eigendecomposition and singular value decomposition(SVD)are proposed.2.The rank-one approximation plays an important role in the study of higher-order ten-sors,and the existent finite methods for computing the rank-one approximation are generally based on the matricization of the given tensor.In this thesis,we give two methods to calculate the rank-one approximation of tensors.First,exploiting the segmentation of indices of the ten-sor,we present a dynamical strategy for the matricization of the tensor,and design a numerical method to compute the rank-one approximation of the given tensor.Second,different from the common techniques of regarding a tensor as a matrix and using the matrix SVD as the approx-imation tool,we transform the given tensor into an order-3 tensor,and introduce the projection technique to the calculation of the rank-one approximation of the given tensor.The complexity analysis shows that the computational cost of our algorithm is no more than that of the exist-ing algorithms.Numerical results show that,compared with the existent finite algorithms,our algorithm can provide a more accurate rank-one approximation for more than 93%examples.3.In recent years,people have paid more and more attention to the tensor with special structure,and the M-tensor as the representative of the special structure tensor has also become an important research object for scholars at home and abroad.In this thesis,we propose an alter-native least squares algorithm that calculates the minimum H-eigenvalue of M-tensor.We regard the calculation of the minimum eigenvalue as an optimization problem and solve it by alternately solving two sub-problems.Of these two sub-problems,one only needs to be solved by a simple derivative operation,and the other is a multi-linear system problem.While using existing al-gorithms to solve multi-linear systems,we also improve them.Numerical experimental results show that the improved algorithm for solving multi-linear systems requires fewer iteration step-s than existing ones.And the alternative least squares algorithm that calculates the minimum H-eigenvalue of the M-tensor is far superior to the existing methods in some examples.