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Refining Concentration for Gaussian Quadratic Chaos

Kamyar Moshksar

TL;DR

The paper advances concentration results for Gaussian quadratic chaos by tightening the Hanson-Wright constant to at least 0.1457 and sharpening Laurent-Massart bounds in the PSD case to about 0.1524. It introduces a versatile m-bound framework indexed by m that leverages Schatten norms, revealing a phase-transition-like behavior where the optimal bound switches between m=1 (HWI) and m→∞. It also uncovers a twin inequality that outperforms HWI for both small and large tail parameters and presents five A-dependent PSD-bound candidates, identifying the m∞-bound and χ^2-bound as the prevailing contenders. A detailed even-n analysis shows nuanced dominance relationships among these bounds, with asymptotics tightening as dimension grows. Collectively, these results provide sharper, more flexible tools for Gaussian quadratic chaos concentration and PSD-specific regimes.

Abstract

We slightly modify the proof of Hanson-Wright inequality (HWI) for concentration of Gaussian quadratic chaos where we tighten the bound by increasing the absolute constant in its formulation from the largest known value of 0.125 to at least 0.145 in the symmetric case. We also present a sharper version of an inequality due to Laurent and Massart (LMI) through which we increase the absolute constant in HWI from the largest available value of approximately $0.134$ due to LMI itself to at least $0.152$ in the positive-semidefinite case. A new sequence of concentration bounds indexed by $m=1,2,3,\cdots, \infty$ is developed that involves Schatten norms of the underlying matrix. The case $m=1$ recovers HWI. These bounds undergo a phase transition in the sense that if the tail parameter is smaller than a critical threshold $τ_c$, then $m=1$ is the tightest and if it is larger than $τ_c$, then $m=\infty$ is the tightest. This leads to a novel bound called the~$m_\infty$-bound. A separate concentration bound named twin to HWI is also developed that is tighter than HWI for both sufficiently small and large tail parameter. Finally, we explore concentration bounds when the underlying matrix is positive-semidefinite and only the dimension~$n$ and its largest eigenvalue are known. Five candidates are examined, namely, the $m_\infty$-bound, relaxed versions of HWI and LMI, the $χ^2$-bound and the large deviations bound. The sharpest among these is always either the $m_\infty$-bound or the $χ^2$-bound. The case of even dimension is given special attention. If $n=2,4,6$, the $χ^2$-bound is tighter than the $m_\infty$-bound. If $n$ is an even integer greater than or equal to 8, the $m_\infty$-bound is sharper than the $χ^2$-bound if and only if the ratio of the tail parameter over the largest eigenvalue lies inside a finite open interval which expands indefinitely as $n$ grows.

Refining Concentration for Gaussian Quadratic Chaos

TL;DR

The paper advances concentration results for Gaussian quadratic chaos by tightening the Hanson-Wright constant to at least 0.1457 and sharpening Laurent-Massart bounds in the PSD case to about 0.1524. It introduces a versatile m-bound framework indexed by m that leverages Schatten norms, revealing a phase-transition-like behavior where the optimal bound switches between m=1 (HWI) and m→∞. It also uncovers a twin inequality that outperforms HWI for both small and large tail parameters and presents five A-dependent PSD-bound candidates, identifying the m∞-bound and χ^2-bound as the prevailing contenders. A detailed even-n analysis shows nuanced dominance relationships among these bounds, with asymptotics tightening as dimension grows. Collectively, these results provide sharper, more flexible tools for Gaussian quadratic chaos concentration and PSD-specific regimes.

Abstract

We slightly modify the proof of Hanson-Wright inequality (HWI) for concentration of Gaussian quadratic chaos where we tighten the bound by increasing the absolute constant in its formulation from the largest known value of 0.125 to at least 0.145 in the symmetric case. We also present a sharper version of an inequality due to Laurent and Massart (LMI) through which we increase the absolute constant in HWI from the largest available value of approximately due to LMI itself to at least in the positive-semidefinite case. A new sequence of concentration bounds indexed by is developed that involves Schatten norms of the underlying matrix. The case recovers HWI. These bounds undergo a phase transition in the sense that if the tail parameter is smaller than a critical threshold , then is the tightest and if it is larger than , then is the tightest. This leads to a novel bound called the~-bound. A separate concentration bound named twin to HWI is also developed that is tighter than HWI for both sufficiently small and large tail parameter. Finally, we explore concentration bounds when the underlying matrix is positive-semidefinite and only the dimension~ and its largest eigenvalue are known. Five candidates are examined, namely, the -bound, relaxed versions of HWI and LMI, the -bound and the large deviations bound. The sharpest among these is always either the -bound or the -bound. The case of even dimension is given special attention. If , the -bound is tighter than the -bound. If is an even integer greater than or equal to 8, the -bound is sharper than the -bound if and only if the ratio of the tail parameter over the largest eigenvalue lies inside a finite open interval which expands indefinitely as grows.

Paper Structure

This paper contains 17 sections, 5 theorems, 156 equations, 11 figures.

Table of Contents

  1. Introduction
  2. Summary of prior art
  3. Contributions and main results
  4. Notations
  5. Analysis of the results in Subsection \ref{['1B']}
  6. Improving LMI
  7. Beyond HWI
  8. Improvement over HWI for both all sufficiently small $t$ and large $t$
  9. Upper bounds on $\Pr(\boldsymbol{\Delta}>t)$ that depend on $A$ only through $n$ and $\|A\|$
  10. Proof of Proposition \ref{['prop_1']}
  11. Proof of Proposition \ref{['prop_2']}
  12. Proof of Proposition \ref{['prop_3']}
  13. Proof of Theorem \ref{['prop_4']}
  14. Proof of Theorem \ref{['prop_5']}
  15. Proof of Proposition \ref{['prop_6']}
  16. ...and 2 more sections

Key Result

Corollary 1

For a symmetric matrix $A$,

Figures (11)

  • Figure 1: The plot for $\hat{a}_{opt}-a_{opt}$ in terms of $\rho$. Both $a_{opt}$ and $\hat{a}_{opt}$ depend on the matrix $A$ and the tail parameter $t$ through the ratio $\rho$. We observe that $\hat{a}_{opt}-a_{opt}<0.035$ regardless of the value of $\rho$.
  • Figure 2: Panel (a) presents plots of the upper bounds in (\ref{['augmented_LM']}), (\ref{['optimal_LM']}) and (\ref{['LM_itself']}) in terms of $0\leq t\leq 50$ for the positive-semidefinite matrix $A$ given (\ref{['matrix']}). The difference between $e^{\Lambda(t,a_{opt})}$ and $e^{\Lambda(t,\hat{a}_{opt})}$ is not noticeable in this plot. Panel (b) skectches the difference $e^{\Lambda(t, \hat{a}_{opt})}-e^{\Lambda(t, a_{opt})}$.
  • Figure 4: Panel (a) presents plots of the upper bound $e^{\Lambda_m(t,b)}$ in terms of $b$ for $t=5$ and $m=1,2,3,4,5$. We see that the infimum $\inf_{0<b<1}e^{\Lambda_m(5,b)}$ is the smallest for $m=1$, i.e., HWI offers the tightest upper bound. Panel (b) presents similar plots for $t=10$ and $m=1,5,9,13,17$. We see that the achieved infimum value decreases as $m$ grows.
  • Figure 5: Panel (a) presents plots of $\inf _{0<b<1}e^{\Lambda_m(t,b)}$ in terms of $1\leq m\leq 20$ for $5\leq t\leq 10$ in steps of $0.2$. Panel (b) shows the optimum $m$ denoted by $m_{opt}$ over the domain $1\leq m\leq 20$ at which $\inf _{0<b<1}e^{\Lambda_m(t,b)}$ is the smallest. If Conjecture 1 is true, then these plots imply that $\tau_c\le 7.2$.
  • Figure 6: Inequality (\ref{['Ext_3']}) in Theorem \ref{['prop_5']} is tighter than HWI for both sufficiently small and large values of $t$. For the symmetric matrix $A$ given in (\ref{['ex_sym']}), Panel (a) and Panel (b) compare the two bounds for $0\leq t\leq 2$ and $5\leq t\leq 14$, respectively. We have also included the $m_\infty$-bound in Corollary \ref{['coro_2']} for comparison.
  • ...and 6 more figures

Theorems & Definitions (10)

  • Corollary 1
  • proof
  • Corollary 2
  • proof
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 4
  • proof