On $\ell_p$-Gaussian-Grothendieck problem

TL;DR

This paper investigates the asymptotic behavior of the $ ext{ell}_p$-Gaussian-Grothendieck problem for large matrices, providing explicit limits for $1 \\leq p < 2$ and a variational formula for $p > 2$, revealing structural properties of near optimizers.

Contribution

It computes the limit of the $ ext{ell}_p$-Gaussian-Grothendieck problem for $1 \\leq p < 2$ and establishes a Parisi-type variational representation for $p > 2$, extending understanding of these optimization problems.

Findings

Explicit limit for $1 \\leq p < 2$ cases.

Variational formula for $p > 2$ case.

Near optimizers are stable and weakly delocalized.

Abstract

For $p\geq 1$ and $(g_{ij})_{1\leq i,j\leq n}$ being a matrix of i.i.d. standard Gaussian entries, we study the $n$-limit of the $\ell_p$-Gaussian-Grothendieck problem defined as \begin{align*}\max\Bigl\{\sum_{i,j=1}^n g_{ij}x_ix_j: x\in \mathbb{R}^n,\sum_{i=1}^n |x_i|^p=1\Bigr\}.\end{align*} The case $p=2$ corresponds to the top eigenvalue of the Gaussian Orthogonal Ensemble; when $p=\infty$, the maximum value is essentially the ground state energy of the Sherrington-Kirkpatrick mean-field spin glass model and its limit can be expressed by the famous Parisi formula. In the present work, we focus on the cases $1\leq p<2$ and $2<p<\infty.$ For the former, we compute the limit of the $\ell_p$-Gaussian-Grothendieck problem and investigate the structure of the set of all near optimizers along with stability estimates. In the latter case, we show that this problem admits a Parisi-type variational representation and the corresponding optimizer is weakly delocalized in the sense that its entries vanish uniformly in a polynomial order.