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.
