T-product tensors—part II: tail bounds for sums of random T-product tensors

Abstract

This paper is the part II of a serious work about T-product tensors focusing at establishing new probability bounds for sums of random, independent, T-product tensors. These probability bounds characterize large-deviation behavior of the extreme eigenvalue of the sums of random T-product tensors. We apply Laplace transform method and Lieb’s concavity theorem for T-product tensors obtained from our part I paper, and apply these tools to generalize the classical bounds associated with the names Chernoff, and Bernstein from the scalar to the T-product tensor setting. Tail bounds for the norm of a sum of random rectangular T-product tensors are also derived from corollaries of random Hermitian T-product tensors cases. The proof mechanism is also applied to T-product tensor-valued martingales and T-product tensor-based Azuma, Hoeffding and McDiarmid inequalities are derived.