H-Eigenvalue Inclusion Set Of Sparse Tensors And Its Application

The eigenvalue problem of sparse tensors is widely used in large-scale data analysis,image processing,statistics and other fields,so it is of great significance to estimate the eigenvalues of tensors.On the basis of previous studies,this thesis studies H-eigenvalue inclusion sets of sparse tensors and their applications,and obtains some valuable results.The main content and results of this thesis are as follows:In Chapter 1,we introduce the research background and current situation of the tensor eigenvalue problem at home and abroad,and gives the main research content.In Chapter 2,by using the representation matrix’s digraph and majorization matrix’s digraph,we establish H-eigenvalue inclusion sets of sparse tensors under appropriate conditions,and give the H-eigenvalue inclusion sets of sparse tensors in general condition.Furthermore,this thesis theoretically proves that the obtained results can more accurately characterize the region of the H-eigenvalue of a tensor than some existing results.Numerical examples also verify that the obtained results are more accurate in estimating the H-eigenvalue and require less computation.As applications,we provide some checkable sufficient conditions for the positive definiteness of sparse tensors,and propose bounds estimate on the H-spectral radius of nonnegative sparse tensors.In Chapter 3,we establish bounds of the minimum H-eigenvalue for a Z-tensor by its representation matrix’s digraph and majorization matrix’s digraph.By analyzing tensor elements,we propose lower and upper bounds under weak and general conditions.Based on the lower bound estimations for the minimum H-eigenvalue,we provide some sufficient conditions for judging nonsingular M-tensors and the positive definiteness of Z-tensors.In Chapter 4,we summarize the content of this thesis and propose research topics that can be further explored in the future.Based on directed graphs,this paper establishes H-eigenvalue inclusion sets,which require more accurate and less computations.This research not only enriches the theory and research results of H-eigenvalue,but also provides a research foundation for large-scale sparse tensor optimization problems.