The $\ell_r$-Levy-Grothendieck problem and $r\rightarrow p$ norms of Levy matrices

TL;DR

This paper investigates the asymptotic behavior of the $ ext{ell}_r$-Grothendieck problem and $r ightarrow p$ norms of Levy matrices with heavy-tailed entries, revealing convergence to Fréchet or stable distributions depending on parameters.

Contribution

It provides the first detailed analysis of high-dimensional limits of these quantities for Levy matrices with heavy tails, including phase transitions based on tail index and matrix entry centering.

Findings

Convergence to Fréchet distribution for certain parameter ranges.

Convergence to stable distribution powers for other parameter ranges.

Different limiting behaviors when matrix entries are centered or have non-zero mean.

Abstract

Given an $n\times n$ matrix $A_n$ and $1\leq r, p \leq\infty$, consider the following quadratic optimization problem referred to as the $\ell_r$-Grothendieck problem: \begin{align}M_r(A_n)\coloneqq\max_{\boldsymbol{x}\in\mathbb{R}^n:\|\boldsymbol{x}\|_r\leq1}\boldsymbol{x}^{\top} A_n \boldsymbol{x},\end{align} as well as the $r\rightarrow p$ operator norm of the matrix $A_n$, defined as \begin{align}\|A_n\|_{r \rightarrow p}\coloneqq \sup _{\boldsymbol{x}\in\mathbb{R}^n:\|\boldsymbol{x}\|_r \leq 1}\|A_n \boldsymbol{x}\|_p,\end{align} where $\|\boldsymbol{x}\|_r$ denotes the $\ell_r$-norm of the vector $\boldsymbol{x}$. This work analyzes high-dimensional asymptotics of these quantities when $A_n$ are symmetric random matrices with independent and identically distributed heavy-tailed upper-triangular entries with index $\alpha$. When $1\leq r\leq 2$ (respectively, $1\leq r\leq p$) and $\alpha\in(0,2)$, suitably scaled versions of $M_r(A_n)$ and $\|A_n\|_{r\rightarrow p}$ are shown to converge to a Fr\'echet distribution as $n\rightarrow\infty$. In contrast, when $2< r<\infty$ (respectively, $1\leq p< r$), it is shown that there exists $\alpha_*\in(1,2)$ such that for every $\alpha\in(0,\alpha_*)$, suitably scaled versions of $M_r(A_n)$ and $\|A_n\|_{r\rightarrow p}$ converge to the power of a stable distribution. Furthermore, it is shown that there exists $\bar\alpha_*>\alpha_*$ such that when $\alpha\in(\alpha_*,\bar\alpha_*)$, the latter convergence result holds only when the matrix entries are centered; when the entries have non-zero mean, a different limit arises after additional centering and scaling. As a corollary, these results yield a characterization of the limiting ground state of the Levy spin glass when $\alpha \in (0,1)$. The analysis uses a combination of tools from the theory of heavy-tailed distributions, the nonlinear power method and concentration inequalities.