Random Parametrization Double Tensors Integrals and Their Applications

TL;DR

This paper introduces parametrization double tensors integrals (PDTI), extending previous double tensor integrals, and develops new bounds and perturbation formulas to analyze random tensors and their applications.

Contribution

It extends double tensor integrals to PDTI, establishes new bounds and perturbation formulas, and investigates convergence properties for random tensors.

Findings

Derived new tail bounds for random tensors.

Established a perturbation formula for PDTI.

Analyzed convergence properties of random PDTI.

Abstract

In this work, we extend double tensor integrals (DTI) from our previous work to parametrization double tensors integrals (PDTI) by applying integral kernel transform bounds to upper bound PDTI norm and establishing a new perturbation formula. Besides, the convergence property of random PDTI is investigated and this property is utilized to characterize the relation between the original derivative tensor and the action result of PDTI to the original derivative tensor. These tools help us to derive new tail bounds for random tensors according to more general operator inequalities, e.g., Heinz inequality and Birman-Koplienko-Solomyak inequality. Moreover, new tail bounds about random tensors are also obtained according to our new derived perturbation formula and integral kernel transform bounds.