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

TL;DR

This paper confirms a conjecture about the expected operator norms of Gaussian matrices between ll_p and ll_q spaces, establishing bounds that depend only on the matrix entries and the parameters p and q.

Contribution

It proves the conjecture for a broader range of p and q, including cases previously known only in special instances, and provides a new proof for the spectral case without spectral theory.

Findings

Expected ll_p to ll_q operator norms are comparable to a sum involving row and column norms and the maximum entry.

The result extends known cases to a wider range of p and q, confirming the conjecture.

A new proof for the spectral case p=2=q is provided without spectral theory.

Abstract

We confirm the conjecture posed by Gu\'edon, 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).