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Concentration Inequalities for Sub-Weibull Random Tensors

Yunfan Zhao

Abstract

We extend the theory of concentration inequalities to simple random tensors with heavy-tailed coefficients. Specifically, we consider the class of sub-Weibull distributions $\mathcal{S}_α$ for $α\in [1, 2]$. We establish concentration bounds for Euclidean functions of such tensors, exhibiting a phase transition between sub-gaussian and heavy-tailed regimes. Our results rely on a new Generalized Maximal Inequality for products of heavy-tailed random variables and a martingale analysis using Nagaev-type inequalities.

Concentration Inequalities for Sub-Weibull Random Tensors

Abstract

We extend the theory of concentration inequalities to simple random tensors with heavy-tailed coefficients. Specifically, we consider the class of sub-Weibull distributions for . We establish concentration bounds for Euclidean functions of such tensors, exhibiting a phase transition between sub-gaussian and heavy-tailed regimes. Our results rely on a new Generalized Maximal Inequality for products of heavy-tailed random variables and a martingale analysis using Nagaev-type inequalities.

Paper Structure

This paper contains 14 sections, 7 theorems, 51 equations.

Table of Contents

  1. Introduction
  2. Heavy tails
  3. New results
  4. Related results
  5. Our approach
  6. Organization
  7. The Probabilistic Setting
  8. Concentration of Measure for Quadratic Forms
  9. Geometry of High-Dimensional Tensors
  10. Martingale Analysis of Euclidean Functions
  11. The Main Theorem
  12. Conclusion
  13. Summary of contributions
  14. Future directions

Key Result

Theorem 3.1

Let $(\Omega, \mathcal{F}, \mathbb{P})$ be a probability space. Let $X = (X_1, \dots, X_n)$ be a random vector with independent components such that $\mathbb{E}X_i = 0$ and $\|X_i\|_{\psi_\alpha} \le K$ for some $\alpha \in (0, 2]$ and $K>0$. Let $A \in \mathbb{R}^{n \times n}$ be a deterministic ma

Theorems & Definitions (30)

  • Definition 2.1: The Class $\mathcal{S}_\alpha$ for $\alpha \geq 1$
  • Remark
  • Definition 2.2: Simple Random Tensor Space
  • Remark
  • Definition 2.3: Euclidean Functions
  • Remark
  • Definition 2.4
  • Remark
  • Theorem 3.1
  • proof
  • ...and 20 more