General tail bounds for random tensors summation: Majorization approach
Journal of Computational and Applied Mathematics
In recent years, tensors have been applied to different applications in science and engineering fields. In order to establish theory about tail bounds of the tensors summation behavior, this research extends previous work by considering the tensors summation tail behavior of the top k-largest singular values of a function of the tensors summation, instead of the largest/smallest singular value of the tensors summation directly (identity function) explored in Chang (2020). Majorization and antisymmetric tensor product tools are main techniques utilized to establish inequalities for unitarily invariant norms of multivariate tensors. The Laplace transform method is integrated with these inequalities for unitarily invariant norms of multivariate tensors to give us tail bounds estimation for the Ky Fan k-norm for a function of the tensors summation. By restricting different random tensor conditions, we obtain generalized tensor Chernoff and Bernstein inequalities.
National Natural Science Foundation of China
Bernstein inequality, Chernoff inequality, Log-majorization, Random tensors, Unitarily invariant norm
Applied Data Science
Shih Yu Chang and Yimin Wei. "General tail bounds for random tensors summation: Majorization approach" Journal of Computational and Applied Mathematics (2022). https://doi.org/10.1016/j.cam.2022.114533