Operator $\ell_p\to\ell_q$ norms of random matrices with iid entries
Rafał Latała, Marta Strzelecka
TL;DR
The paper derives sharp, two-sided bounds for the expected operator norms of iid random matrices from $\ell_p^n$ to $\ell_q^m$ under a mild $\alpha$-regularity condition that captures many common distributions (Gaussian, Rademacher, log-concave, Weibull, etc.). The core result is a Chevet-type formula: $\mathbb E\|X\|_{\ell_p^n\to\ell_q^m} \sim_\alpha m^{1/q}\sup_{t\in B_p^n}\|\sum_{j=1}^n t_j X_{1j}\|_{q\wedge\log m} + n^{1/p^*}\sup_{s\in B_{q^*}^m}\|\sum_{i=1}^m s_i X_{i1}\|_{p^*\wedge\log n}$, with explicit asymptotics provided for Gaussian, Weibull, log-concave, and log-convex tails. The authors develop lower bounds via Paley–Zygmund-type arguments and upper bounds through a comprehensive, case-by-case analysis relying on tools like Chevet’s inequality, contractions, symmetrization, and detailed tail-decomposition techniques. The results unify and extend classical Seginer-type bounds beyond the square/orthogonal setting, yielding precise scaling laws in a broad range of $(p,q)$ and tail behaviors, thereby enriching the understanding of random matrix norms in high-dimensional geometry. Overall, the work offers a versatile framework for predicting operator-norm behavior of rectangular iid matrices across diverse distributional regimes, with potential applications in probability, statistics, and data science where random linear maps arise.
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 $\|X_{i,j}\|_{2ρ} \le α\|X_{i,j}\|_ρ$ for every $ρ\ge 1$, 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 $α$, 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 $(\mathbb{E} X_{1,1}^4)^{1/4}\leq α(\mathbb{E} X_{1,1}^2)^{1/2}$.