论文标题
在$ \ ell_p $ -gaussian-grothendieck问题上
On $\ell_p$-Gaussian-Grothendieck problem
论文作者
论文摘要
对于$ p \ geq 1 $和$（g_ {ij}）_ {1 \ leq i，j \ leq n} $是I.I.D.的矩阵。标准高斯条目，我们研究$ \ ell_p $ -gaussian-grothendieck问题的$ n $限制，定义为\ begin {align*} \ max \ max \ bigl \ {\ sum_ {\ sum_ {i，j = 1} | x_i |^p = 1 \ bigr \}。\ end {align*} case $ p = 2 $对应于高斯正交集合的顶部特征值；当$ p = \ infty $时，最大值本质上是Sherrington-kirkpatrick平均田间旋转玻璃模型的基态能量及其极限可以用著名的巴黎公式表示。在目前的工作中，我们专注于$ 1 \ leq p <2 $和$ 2 <p <\ infty。$对于前者，我们计算了$ \ ell_p $ -Gaussian-grothendieck问题的限制，并调查了所有接近优化者的结构以及稳定性估计。在后一种情况下，我们表明这个问题承认了parisi型变分表示，并且相应的优化器被薄弱地定位，因为其条目以多项式顺序均匀消失。
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.
