Convenient tail bounds for sums of random tensors

TL;DR

This paper develops new probability bounds for sums of random Hermitian tensors, extending classical scalar inequalities to the tensor setting and providing tools for analyzing large deviations of tensor eigenvalues.

Contribution

It generalizes Laplace transform and Lieb's concavity theorem from matrices to tensors, enabling tensor-specific tail bounds and inequalities.

Findings

Derived tail bounds for eigenvalues of sums of random Hermitian tensors

Extended classical scalar inequalities like Chernoff, Bennett, Bernstein to tensors

Established tensor-valued martingale inequalities such as Azuma, Hoeffding, McDiarmid

Abstract

This work prepares new probability bounds for sums of random, independent, Hermitian tensors. These probability bounds characterize large-deviation behavior of the extreme eigenvalue of the sums of random tensors. We extend Lapalace transform method and Lieb's concavity theorem from matrices to tensors, and apply these tools to generalize the classical bounds associated with the names Chernoff, Bennett, and Bernstein from the scalar to the tensor setting. Tail bounds for the norm of a sum of random rectangular tensors are also derived from corollaries of random Hermitian tensors cases. The proof mechanism can also be applied to tensor-valued martingales and tensor-based Azuma, Hoeffding and McDiarmid inequalities are established.