Determining And Applications Of Strong H-tensors

Tensor theory has widespread application background.Many problems in image processing,date mining,computer vision,neural network,nonlinear optimization,high order Markov chains,quantum entanglement,chemomet-rics,psychometrics and so on all have a close relation with tensor theory.As an important class of tensors,strong H-tensors have an essential impact on practical problems.Particularly,the criteria for strong H-tensors play an important role in determining the positive definiteness of an even-degree homogeneous multivariate polynomial form.However,it is difficult to deter-mine whether a tensor,especially for a large scale tensor,is a strong H-tensor or not.Therefore,the research of the criteria for identifying strong H-tensors has theory value and practical significance.In this dissertation,we mainly research the criteria for identifying strong H-tensors and obtain the following results.Firstly,some sufficient and nec-essary conditions for a tensor to be a strong H-tensor are given,and by these conditions we propose five sufficient conditions for strong H-tensors,which only refer to the elements of tensors and are easy to be verified.Mean-while,we present a direct algorithm for identifying strong H-tensors with these sufficient conditions.Secondly,we construct an iterative algorithm for identifying weakly irreducible strong H-tensors by choosing positive diag-onal matrix progressively,which is obtained by using sum of the absolute values of all the off-diagonal entries in each row of a tensor.It is proved that this algorithm always stops in finite iterative steps and its output is cor-rect.Then,two different techniques are used to transform arbitrary strong H-tensors identification problem into weakly irreducible strong H-tensors i-dentification problem,and we give two iterative algorithms for identifying strong H-tensors.Finally,some numerical examples are given to verify the feasibility and effectiveness of these algorithms.As applications of strong H-tensors,we also study the localization for tensor eigenvalues and the identification of positive definiteness(semi-definiteness)for an even-degree homogeneous multivariate polynomial form.Firstly,we present p-norm strictly diagonally dominant tensors,which is a subclass of strong H-tensors,and then a new eigenvalue inclusion set is obtained.The numerical example shows that the proposed set is contained in the Brauer-type eigenvalue inclusion set[C.Q.Li,J.J.Zhou,Y.T.Li.A new Brauer-type eigenvalue localization set for tensors.Linear and Multilinear Algebra,2016,64:727-736].Secondly,based on the sufficient conditions of strong H-tensors,some sufficient conditions for identifying the positive definiteness of an even-degree homogeneous multivariate polynomial form are given.Then,we introduce two new classes of tensors(p-norm B-tensor and p-norm Bo-tensor)and prove that an even-order symetric p-norm B-tensor(p-norm B0-tensor)is positive definite(semi-definite).Lastly,we design an iterative algorithm for identifying the positive definiteness of an even-degree homoge-neous multivariate polynomial form.