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

Publication Date

4-1-2022

Document Type

Article

Publication Title

Computational and Applied Mathematics

Volume

41

Issue

3

DOI

10.1007/s40314-022-01811-8

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.

Funding Number

11771099

Funding Sponsor

National Natural Science Foundation of China

Keywords

Random T-product tensors, T-product tensor Azuma inequality, T-product tensor Bernstein bound, T-product tensor Chernoff bound, T-product tensor McDiarmid inequality, T-product tensor-valued martingale

Department

Applied Data Science

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