Operator $\ell_p\to\ell_q$ norms of Gaussian matrices

TL;DR

The paper resolves the conjecture that the expected operator norm $\mathbb{E}\|G_A\|_{p\to q}$ of a centered Gaussian matrix with variance profile $A\circ A$ is comparable, up to constants depending only on $p$ and $q$, to $\max_i\|(a_{ij})_j\|_{p^*} + \max_j\|(a_{ij})_i\|_{q} + \mathbb{E}\max_{i,j}|a_{ij}g_{ij}|$ for $1\le p\le 2\le q\le \infty$. The authors reduce logarithmic losses present in prior bounds by developing two key dimension-dependent estimates, an exponent/log reduction, and a decomposition into tractable blocks, with Gaussian concentration and hypercontractivity guiding the control of each piece. Their results yield two-sided bounds for higher moments and tails, a deterministic criterion for the boundedness of Gaussian operators between $\ell_p$ and $\ell_q$, and extensions to mixtures and Weibull-type entries. In particular, the conjecture is confirmed in the general parameter range, and a new spectral-free proof is provided for the $p=q=2$ case. This work advances understanding of structured Gaussian matrices with variance profiles and offers tools applicable to broader random-matrix models with independent entries.

Abstract

We confirm the conjecture posed by Guédon, Hinrichs, Litvak, and Prochno in 2017 that $\mathbb{E}\|(a_{ij}g_{ij})_{i\le m, j\le n}\colon \ell_p^n \to \ell_q^m\|$ is comparable, up to constants depending only on $p$ and $q$, to \[ \max_i \|(a_{ij})_j\|_{p^*} +\max_j \|(a_{ij})_i\|_{q} +\mathbb{E} \max_{i,j} |a_{ij}g_{ij}| \] provided that $1\le p \le 2\le q \le \infty$. This was known before only in the case $p=1$ or $q=\infty$, and in the spectral case $p=2=q$. We also reprove the conjecture in the case $p=2=q$ without using spectral theory (which was employed in the previously known proof).

Theorems & Definitions (78)

• Conjecture 1
• Theorem 2
• Remark 3
• Remark 4
• Corollary 5
• Remark 6
• Corollary 7
• Corollary 8
• Remark 9
• Corollary 10