非负元素随机矩阵的 $r$-到-$p$ 范数：渐近正态性与最大值点的 $\ell_\infty$ 界
On $r$-to-$p$ norms of random matrices with nonnegative entries: Asymptotic normality and $\ell_\infty$-bounds for the maximizer
AI总结：
本文研究非负元素对称随机矩阵的 $r\to p$ 范数，建立了其渐近正态性及最大化向量的 $\ell_\infty$ 稳定性界，推广了 Füredi-Komlós 关于最大奇异值的结果，并提出了新的非线性幂法分析。
摘要：
对于 $n\times n$ 矩阵 $A_n$，其 $r\to p$ 算子范数定义为 $$\\|A_n\\|_{r\to p}:= \sup_{\mathbf{x}\in\mathbb{R}^n:\\|\mathbf{x} \\|_r\leq 1 } \\|A_n\mathbf{x} \\|_p\quad\text{对于}\quad r,p\geq 1.$$ 对于 $r$ 和 $p$ 的不同取值，该范数对应于多种应用中出现的关键量，包括矩阵条件数估计、数据聚类以及交通网络中 oblivious 路由方案的构造。本文考虑具有非负元素的对称随机矩阵的 $r\to p$ 范数，包括 Erdős-Rényi 随机图的邻接矩阵、具有正 sub-Gaussian 元素的矩阵以及某些稀疏矩阵。对于 $1<p\leq r<\infty$，当 $n\to\infty$ 时，建立了适当中心化和缩放后的范数 $\\|A_n\\|_{r\to p}$ 的渐近正态性。当 $p \geq 2$ 时，这被证明蕴含 $\ell_p$ 二次最大化问题（也称为 $\ell_p$ Grothendieck 问题）解的渐近正态性。此外，对于 $\\|A_n\\|_{r\to p}$ 定义中的唯一最大化向量，获得了尖锐的 $\ell_\infty$ 逼近界，这可视为在矩阵均值元素随机扰动下最大化向量的 $\ell_\infty$ 稳定性结果。事实上，该结果被证明对具有某些渐近展开性质的广泛确定性矩阵序列类也成立。所获得的结果可视为 Füredi 和 Komlós (1981) 关于一类对称随机矩阵最大奇异值渐近正态性的开创性结果的推广。在一般情形 $1<p\leq r< \infty$ 下，谱方法不再适用，因此开发了一种新方法，涉及非线性幂法的精细收敛分析以及最大化向量的扰动界，这可能具有独立的研究兴趣。
英文摘要：
For an $n\times n$ matrix $A_n$, the $r\to p$ operator norm is defined as $$\|A_n\|_{r\to p}:= \sup_{\mathbf{x}\in\mathbb{R}^n:\|\mathbf{x} \|_r\leq 1 } \|A_n\mathbf{x} \|_p\quad\text{for}\quad r,p\geq 1.$$ For different choices of $r$ and $p$, this norm corresponds to key quantities that arise in diverse applications including matrix condition number estimation, clustering of data, and construction of oblivious routing schemes in transportation networks. This article considers $r\to p$ norms of symmetric random matrices with nonnegative entries, including adjacency matrices of Erdős-Rényi random graphs, matrices with positive sub-Gaussian entries, and certain sparse matrices. For $1<p\leq r<\infty$, the asymptotic normality, as $n\to\infty$, of the appropriately centered and scaled norm $\|A_n\|_{r\to p}$ is established. When $p \geq 2$, this is shown to imply asymptotic normality of the solution to the $\ell_p$ quadratic maximization problem, also known as the $\ell_p$ Grothendieck problem. Furthermore, a sharp $\ell_\infty$-approximation bound for the unique maximizing vector in the definition of $\|A_n\|_{r\to p}$ is obtained, and may be viewed as an $\ell_\infty$-stability result of the maximizer under random perturbations of the matrix with mean entries. This result is in fact shown to hold for a broad class of deterministic sequences of matrices having certain asymptotic expansion properties. The results obtained can be viewed as a generalization of the seminal results of Füredi and Komlós (1981) on asymptotic normality of the largest singular value of a class of symmetric random matrices. In the general case with $1<p\leq r< \infty$, spectral methods are no longer applicable, and so a new approach is developed involving a refined convergence analysis of a nonlinear power method and a perturbation bound on the maximizing vector, which may be of independent interest.
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