In this paper,we extend the notion of the T-Schur decomposition to the weighted T-core-EP decomposition.Next,the weighted T-core-EP inverse of rectangular tensors is defined by a system,and its existence and uniquenes...In this paper,we extend the notion of the T-Schur decomposition to the weighted T-core-EP decomposition.Next,the weighted T-core-EP inverse of rectangular tensors is defined by a system,and its existence and uniqueness are obtained.Furthermore,the perturbation of the weighted T-core-EP inverse is studied under several conditions,and the relevant examples are provided to verify the perturbation bounds of the weighted T-core-EP inverse of tensors.展开更多
Since Kilmer et al.introduced the new multiplication method between two third-order tensors around 2008 and third-order tensors with such multiplication structure are also called as T-product tensors,T-product tensors...Since Kilmer et al.introduced the new multiplication method between two third-order tensors around 2008 and third-order tensors with such multiplication structure are also called as T-product tensors,T-product tensors have been applied to many fields in science and engineering,such as low-rank tensor approximation,signal processing,image feature extraction,machine learning,computer vision,and the multi-view clustering problem,etc.However,there are very few works dedicated to exploring the behavior of random T-product tensors.This work considers the problem about the tail behavior of the unitarily invariant norm for the summation of random symmetric T-product tensors.Majorization and antisymmetric Kronecker product tools are main techniques utilized to establish inequalities for unitarily norms of multivariate T-product tensors.The Laplace transform method is integrated with these inequalities for unitarily norms of multivariate T-product tensors to provide us with Bernstein bound estimation of Ky Fan k-norm for functions of the symmetric random T-product tensors summation.Finally,we also apply T-product Bernstein inequality to bound Ky Fan norm of covariance T-product tensor induced by hypergraph signal processing.展开更多
基金Yuhang Liu is supported by the innovative research project for post-graduates of Harbin Normal University(Grant HSDSSCX2020-32)Haifeng Ma is supported by the National Natural Science Foundation of China(Grant 11971136).
文摘In this paper,we extend the notion of the T-Schur decomposition to the weighted T-core-EP decomposition.Next,the weighted T-core-EP inverse of rectangular tensors is defined by a system,and its existence and uniqueness are obtained.Furthermore,the perturbation of the weighted T-core-EP inverse is studied under several conditions,and the relevant examples are provided to verify the perturbation bounds of the weighted T-core-EP inverse of tensors.
基金supported by Innovation Program of Shanghai Municipal Education Commissionthe National Natural Science Foundation of China under grant No.11771099
文摘Since Kilmer et al.introduced the new multiplication method between two third-order tensors around 2008 and third-order tensors with such multiplication structure are also called as T-product tensors,T-product tensors have been applied to many fields in science and engineering,such as low-rank tensor approximation,signal processing,image feature extraction,machine learning,computer vision,and the multi-view clustering problem,etc.However,there are very few works dedicated to exploring the behavior of random T-product tensors.This work considers the problem about the tail behavior of the unitarily invariant norm for the summation of random symmetric T-product tensors.Majorization and antisymmetric Kronecker product tools are main techniques utilized to establish inequalities for unitarily norms of multivariate T-product tensors.The Laplace transform method is integrated with these inequalities for unitarily norms of multivariate T-product tensors to provide us with Bernstein bound estimation of Ky Fan k-norm for functions of the symmetric random T-product tensors summation.Finally,we also apply T-product Bernstein inequality to bound Ky Fan norm of covariance T-product tensor induced by hypergraph signal processing.
