General Tail Bounds for Random Tensors Summation: Majorization Approach

TL;DR

This paper develops new tail bounds for the sum of random tensors focusing on the top k-largest singular values, using majorization techniques and extending classical inequalities to tensors.

Contribution

It introduces a novel approach to tensor tail bounds based on majorization, extending Chernoff and Bernstein inequalities to tensor sums.

Findings

Derived tail bounds for the Ky Fan k-norm of tensor sums.

Extended classical Chernoff and Bernstein inequalities to tensors.

Provided inequalities for unitarily norms of multivariate tensors.

Abstract

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 work 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 Shih Yu's work: Convenient tail bounds for sums of random tensors. Majorization and antisymmetric tensor product tools are main techniques utilized to establish inequalities for unitarily norms of multivariate tensors. The Laplace transform method is integrated with these inequalities for unitarily norms of multivariate tensors to give us tail bounds estimation for 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.