Norms of structured random matrices

TL;DR

This paper establishes optimal bounds for the expected operator norm of structured random matrices with various entry distributions, expressed through Hadamard product norms, advancing understanding of their spectral properties.

Contribution

It provides the first sharp bounds for the expected operator norm of structured random matrices with diverse entry distributions, including Gaussian and bounded entries.

Findings

Optimal bounds up to logarithmic factors for Gaussian, bounded, and $\psi_r$ entries.

Precise order of expected norms determined in certain cases.

Results expressed via operator norms of Hadamard products.

Abstract

For $m,n\in\mathbb{N}$ let $X=(X_{ij})_{i\leq m,j\leq n}$ be a random matrix, $A=(a_{ij})_{i\leq m,j\leq n}$ a real deterministic matrix, and $X_A=(a_{ij}X_{ij})_{i\leq m,j\leq n}$ the corresponding structured random matrix. We study the expected operator norm of $X_A$ considered as a random operator between $\ell_p^n$ and $\ell_q^m$ for $1\leq p,q \leq \infty$. We prove optimal bounds up to logarithmic terms when the underlying random matrix $X$ has i.i.d. Gaussian entries, independent mean-zero bounded entries, or independent mean-zero $\psi_r$ ($r\in(0,2]$) entries. In certain cases, we determine the precise order of the expected norm up to constants. Our results are expressed through a sum of operator norms of Hadamard products $A\circ A$ and $(A\circ A)^T$.