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Uniform Concentration for $α$-subexponential Random Operators

Tiankun Diao, Xuanang Hu, Vladimir V. Ulyanov, Hanchao Wang

TL;DR

This work studies random matrices A whose rows (or columns) have $\alpha$-subexponential tail distributions with $\alpha \in (0,2]$, and establishes concentration type inequality for $Ax$, showing that their geometric distortion is governed by Talagrand's functional of the set and depends on the tail parameter $\alpha$.

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 $α$-subexponential tail distributions with $α\in (0,2]$. 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 $α$. Our results extend the known optimal inequalities in the subgaussian regime ($α=2$), 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.

Uniform Concentration for $α$-subexponential Random Operators

TL;DR

This work studies random matrices A whose rows (or columns) have -subexponential tail distributions with , and establishes concentration type inequality for , showing that their geometric distortion is governed by Talagrand's functional of the set and depends on the tail parameter .

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 -subexponential tail distributions with . So subgaussian and sub-exponential models are included in as special cases. We establish concentration type inequality for , where x belongs to the bounded subsets of , showing that their geometric distortion is governed by Talagrand's functional of the set and depends on the tail parameter . Our results extend the known optimal inequalities in the subgaussian regime (), 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.
Paper Structure (13 sections, 17 theorems, 131 equations)

This paper contains 13 sections, 17 theorems, 131 equations.

Table of Contents

  1. Introduction
  2. Row-wise $\alpha$-subexponential model
  3. Column-wise $\alpha$-subexponential model
  4. Methodological remarks
  5. Preliminaries
  6. $\alpha$-subexponential Random Variables and Isotropic Vectors
  7. Generic chaining and Talagrand's $\gamma$-functionals
  8. Proof of Theorem \ref{['thm:row']}
  9. Proof of Theorem \ref{['thm:column']}
  10. Applications
  11. Johnson-Lindenstrauss Lemma
  12. R.I.P. of $\alpha$-subexponential random matrices
  13. Normalizing the columns of an $\alpha$-subexponential matrix

Key Result

Theorem 1.1

Let $B \in \mathbb{R}^{l \times m}$ be a fixed matrix and $A \in \mathbb{R}^{m \times n}$ be a random matrix with zero mean. Assume $\alpha\in(0,2]$. Let rows of $A$ be independent isotropic and have $\psi_\alpha$-norm (or quasi-norm when $\alpha<1$) bounded by $K$ uniformly. Let $T \subset \mathbb{ Moreover, for any $u \geq 0$, with probability at least $1 - C \exp(-u^\alpha)$, where $C(\alpha)$

Theorems & Definitions (28)

  • Theorem 1.1: Row-wise model
  • Corollary 1.1
  • Theorem 1.2: Column-wise model
  • Corollary 1.2
  • Remark 1.1: On the necessity of column normalization
  • Definition 2.1: alpha-subexponential random variables
  • Proposition 2.1
  • Definition 2.2: Admissible sequence
  • Theorem 2.1: See Talagrand Talagrand2014
  • Theorem 2.2
  • ...and 18 more