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On The Absolute Constant in Hanson-Wright Inequality

Kamyar Moshksar

TL;DR

This short report investigates the following concentration of measure inequality which is a special case of the Hanson-Wright inequality, and presents a value for κ in the special case where the matrix A in (1) is a real symmetric matrix.

Abstract

We revisit and slightly modify the proof of the Gaussian Hanson-Wright inequality where we keep track of the absolute constant in its formulation.

On The Absolute Constant in Hanson-Wright Inequality

TL;DR

This short report investigates the following concentration of measure inequality which is a special case of the Hanson-Wright inequality, and presents a value for κ in the special case where the matrix A in (1) is a real symmetric matrix.

Abstract

We revisit and slightly modify the proof of the Gaussian Hanson-Wright inequality where we keep track of the absolute constant in its formulation.

Paper Structure

This paper contains 1 section, 2 theorems, 13 equations, 1 figure.

Table of Contents

  1. Appendix

Key Result

Theorem 1

Let $\underline{\boldsymbol{x}}\sim \mathrm{N}(\underline{0}_n,I_n)$. If $A$ is a nonzero $n\times n$ matrix, then for every $a>0$ where $C$ is an absolute constant that does not depend on $n$, $A$ and $a$.

Figures (1)

  • Figure 1: Plots of $\frac{r}{4}$ and $\frac{1}{8f(r)}$ in terms of $0<r<1$. We see that $\min\{\frac{r}{4},\frac{1}{8f(r)}\}$ is maximized for $r\approx 0.583$ and its maximum value is approximately $0.1457$.

Theorems & Definitions (3)

  • Theorem 1: Hanson-Wright Inequality
  • Lemma 1
  • proof