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| Generalized T-Product Tensor Bernstein Bounds |
| Shih Yu Chang,Yimin Wei |
| (Department of Applied Data Science, San Jose State University, San Jose, CA 95192, USA;School of Mathematical Sciences and Shanghai Key Laboratory of Contemporary Applied Mathematics, Fudan University, Shanghai 200433) |
| DOI： |
| Abstract: |
| Since Kilmer et al. introduced the new multiplication method be tween 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 ten sor approximation, signal processing, image feature extraction, machine learn ing, 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 unitar ily 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 in equality to bound Ky Fan norm of covariance T-product tensor induced by hypergraph signal processing. |
| Key words: T-product tensors, T-eigenvalues, T-singular values, Bernstein bound, Courant-Fischer theorem for T-product tensors. |