On ${\ell }_p$-Gaussian-Grothendieck problem

Abstract

For $p\ge 1$ and $(g_{ij}{)}_{1\le i,j\le n}$ 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 $1\le p<2$ 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.

arXiv comment63 pages

References in corpus (13)

• Level-Spacing Distributions and the Airy Kernel
• Broken Replica Symmetry Bounds in the Mean Field Spin Glass Model
• The Parisi formula has a unique minimizer
• Equilibrium statistical mechanics of bipartite spin systems
• A Dynamic Programming Approach to the Parisi Functional
• Disorder chaos and multiple valleys in spin glasses
• Free energy in the generalized Sherrington-Kirkpatrick mean field model
• Random version of Dvoretzky's theorem in ${\ell }_p^n$
• A connection between MAX $\kappa$-CUT and the inhomogeneous Potts spin glass in the large degree limit
• The Legendre structure of the Parisi formula
• Following the ground-states of full-RSB spherical spin glasses
• The Crisanti-Sommers Formula for Spherical Spin Glasses with Vector Spins
• On properties of the spherical mixed vector p-spin model

Cited by in corpus (1)

• The ${\ell }^p$-Gaussian-Grothendieck problem with vector spins