The ${\ell }_r$-Levy-Grothendieck problem and $r\to p$ norms of Levy matrices

Abstract

Given an $n\times n$ matrix $A_n$ and $1\le r,p\le \infty$, 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 $r\to p$ 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 $\Vert \mathbf{x}{\Vert }_r$ denotes the ${\ell }_r$-norm of the vector $\mathbf{x}$. 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 $1\le r\le 2$ (respectively, $1\le r\le p$) and $\alpha \in (0,2)$, suitably scaled versions of $M_r(A_n)$ and $\Vert A_n{\Vert }_{r\to p}$ are shown to converge to a Fréchet distribution as $n\to \infty$. In contrast, when $2<r<\infty$ (respectively, $1\le p<r$), it is shown that there exists ${\alpha }_{\ast }\in (1,2)$ such that for every $\alpha \in (0,{\alpha }_{\ast })$, suitably scaled versions of $M_r(A_n)$ and $\Vert A_n{\Vert }_{r\to p}$ converge to the power of a stable distribution. Furthermore, it is shown that there exists ${\bar{\alpha }}_{\ast }>{\alpha }_{\ast }$ such that when $\alpha \in ({\alpha }_{\ast },{\bar{\alpha }}_{\ast })$, 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.

38 pages, 3 figures