Random Double Tensors Integrals

TL;DR

This paper develops a theoretical framework for random double tensor integrals, establishing tail bounds for unitarily invariant norms, and explores their applications in tensor mean, perturbation analysis, and derivatives.

Contribution

It introduces the concept of random double tensor integrals and derives new tail bounds and continuity properties for tensor functions involving randomness.

Findings

Established tail bounds for unitarily invariant norms of random DTI

Derived tail bounds for tensor means like arithmetic, geometric, harmonic, and general means

Proved continuity properties and tail bounds for derivatives of tensor-valued functions

Abstract

In this work, we try to build a theory for random double tensor integrals (DTI). We begin with the definition of DTI and discuss how randomness structure is built upon DTI. Then, the tail bound of the unitarily invariant norm for the random DTI is established and this bound can help us to derive tail bounds of the unitarily invariant norm for various types of two tensors means, e.g., arithmetic mean, geometric mean, harmonic mean, and general mean. By associating DTI with perturbation formula, i.e., a formula to relate the tensor-valued function difference with respect the difference of the function input tensors, the tail bounds of the unitarily invariant norm for the Lipschitz estimate of tensor-valued function with random tensors as arguments are derived for vanilla case and quasi-commutator case, respectively. We also establish the continuity property for random DTI in the sense of convergence in the random tensor mean, and we apply this continuity property to obtain the tail bound of the unitarily invariant norm for the derivative of the tensor-valued function.