ON r-TO-p NORMS OF RANDOM MATRICES WITH NONNEGATIVE ENTRIES: ASYMPTOTIC NORMALITY AND l∞-BOUNDS FOR THE MAXIMIZER

• 成果类型: Article
• 署名作者: Dhara, Souvik; Mukherjee, Debankur; Ramanan, Kavita
• 署名单位: University System of Georgia; Georgia Institute of Technology; Brown University
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/24-AAP2061
• 发表日期: 2024
• 页码: 5076-5115
• 关键词: eigenvectors; eigenvalues; statistics

摘要

For an nxn matrix A(n), the r -> p operator norm is defined as parallel to An parallel to(r -> p):= sup(x is an element of R)(n):parallel to x parallel to r(<= 1)(n)(parallel to A)x parallel to(p) for r,p >= 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 -> p norms of symmetric random matrices with nonnegative entries, including adjacency matrices of Erdos-R & eacute;nyi random graphs, matrices with positive sub-Gaussian entries, and certain sparse matrices. For 1 < p <= r < infinity, the asymptotic normality, as n ->infinity, of the appropriately centered and scaled norm parallel to An parallel to(r -> p) is established. When p >= 2, this is shown to imply, as a corollary, asymptotic normality of the solution to the l(p) quadratic maximization problem, also known as the l(p) Grothendieck problem. Furthermore, a sharp l(infinity)-approximation bound for the unique maximizing vector in the definition of parallel to An parallel to(r -> p) is obtained, and may be viewed as an l(infinity)-stability result of the maximizer under random perturbations of the matrix with mean entries. This result, which may be of independent interest, 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 F & uuml;redi and Koml & oacute;s (1981) on asymptotic normality of the largest singular value of a class of symmetric random matrices, which corresponds to the special case r = p = 2 considered here. In the general case with 1 < p <= r < infinity, spectral methods are no longer applicable, and so a new approach is developed involving a refined convergence analysis of a nonlinear power method and a perturbation bound on the maximizing vector,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.