Operator $\ell_p\to\ell_q$ norms of random matrices with iid entries

TL;DR

This paper derives bounds and explicit formulas for the expected operator norms of random matrices with iid entries from \,\ell_p^n\ to \,\ell_q^m\, spaces, covering various distributions and regularity conditions.

Contribution

It provides a unified framework for estimating operator norms of iid random matrices across different \,p,q\, norms and distribution types, with explicit asymptotic formulas.

Findings

Expected operator norms are comparable to specific supremum-based expressions.

Explicit formulas are given for Gaussian, Weibullian, log-concave, and log-convex distributions.

Two-sided bounds are established under weaker regularity assumptions in certain ranges.

Abstract

We prove that for every $p,q\in[1,\infty]$ and every random matrix $X=(X_{i,j})_{i\le m, j\le n}$ with iid centered entries satisfying the regularity assumption $\|X_{i,j}\|_{2\rho} \le \alpha \|X_{i,j}\|_{\rho}$ for every $\rho \ge 1$, the expectation of the operator norm of $X$ from $\ell_p^n$ to $\ell_q^m$ is comparable, up to a constant depending only on $\alpha$, to \[ m^{1/q}\sup_{t\in B_p^n}\Bigl\|\sum_{j=1}^nt_jX_{1,j}\Bigr\|_{ q\wedge \operatorname{Log} m} +n^{1/p^*}\sup_{s\in B_{q^*}^m}\Bigl\|\sum_{i=1}^{m} s_iX_{i,1}\Bigr\|_{ p^*\wedge \operatorname{Log} n}. \] We give more explicit formulas, expressed as exact functions of $p$, $q$, $m$, and $n$, for the asymptotic operator norms in the case when the entries $X_{i,j}$ are: Gaussian, Weibullian, log-concave tailed, and log-convex tailed. In the range $1\le q\le 2\le p$ we provide two-sided bounds under a weaker regularity assumption $(\mathbb{E} X_{1,1}^4)^{1/4}\leq \alpha (\mathbb{E} X_{1,1}^2)^{1/2}$.