Tail Bounds for Tensor-valued Random Process

TL;DR

This paper introduces tail bounds for tensor-valued random processes using generic chaining, providing new probabilistic tools for high-dimensional data analysis and applications in compressed sensing.

Contribution

It develops a novel framework for tail bounds of tensor-valued processes, extending existing methods to high-dimensional and exponential tail scenarios.

Findings

Derived tail bounds for supremum of tensor-valued processes

Extended bounds to processes with multiple exponential tails

Applied bounds to high-dimensional compressed sensing

Abstract

To consider a high-dimensional random process, we propose a notion about stochastic tensor-valued random process (TRP). In this work, we first attempt to apply a generic chaining method to derive tail bounds for all p-th moments of the supremum of TRPs. We first establish tail bounds for suprema of processes with an exponential tail, and further derive tail bounds for suprema of processes with arbitrary number of exponential tails. We apply these bounds to high-dimensional compressed sensing and empirical process characterizations.