Uniform Concentration for $\alpha$-subexponential Random Operators

TL;DR

This paper extends concentration inequalities for random matrices with heavy-tailed, $ ext{alpha}$-subexponential distributions, broadening the scope of high-dimensional geometric analysis beyond subgaussian models.

Contribution

It introduces new concentration inequalities for $ ext{alpha}$-subexponential random matrices, covering heavier tails and providing guarantees for dimension reduction and robust inference.

Findings

Extends optimal inequalities to $ ext{alpha}$-subexponential tails

Provides near-isometric embedding results for heavy-tailed matrices

Enables robust high-dimensional inference with non-Gaussian data

Abstract

Random matrices acting on structured sets play a fundamental role in high-dimensional geometry, compressed sensing, and randomized algorithms. Existing results primarily focus on subgaussian models, when random matrices act as near-isometries on sets with optimal tail behavior. Nevertheless, very often in applications we deal with distributions with heavy tails that are not subgaussian but have at least exponential-type tails. In this work, we study random matrices A whose rows (or columns) have $\alpha$-subexponential tail distributions with $\alpha \in (0,2]$. So subgaussian and sub-exponential models are included in as special cases. We establish concentration type inequality for $Ax$, where x belongs to the bounded subsets of $\mathbb{R}^n$, showing that their geometric distortion is governed by Talagrand's functional of the set and depends on the tail parameter $\alpha$. Our results extend the known optimal inequalities in the subgaussian regime ($\alpha=2$), and provide new guarantees for heavier-tailed, yet exponentially integrable, random matrices. These findings extend the theory of random matrices beyond the subgaussian framework. Moreover, they yield near-isometric embedding results applicable to dimension reduction and allow us to make robust high-dimensional inference under non-Gaussian measurements.