TY - JOUR T1 - Generalized T-Product Tensor Bernstein Bounds AU - Chang , Shih Yu AU - Wei , Yimin JO - Annals of Applied Mathematics VL - 1 SP - 25 EP - 61 PY - 2022 DA - 2022/01 SN - 38 DO - http://doi.org/10.4208/aam.OA-2021-0012 UR - https://global-sci.org/intro/article_detail/aam/20172.html KW - T-product tensors, T-eigenvalues, T-singular values, Bernstein bound, Courant-Fischer theorem for T-product tensors. AB -

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.