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.
Categories: math.PR