Sharp parameter dependence for structured Gaussian operator norms
Let $m, n \ge 1$, $1 \le p \le 2 \le q \le \infty$ and $A = (a_{ij}) \in \mathbb{R}^{m \times n}$, and let $p^*$ be the conjugate exponent of $p$, with $p^* = \infty$ when $p = 1$. Let $G_A = (a_{ij} g_{ij})$ with independent standard real Gaussian variables $g_{ij}$, let $\|B\|_{p \to q} = \sup_{\|x\|_p \le 1} \|Bx\|_q$, and put $$D_1 = \max_{1 \le i \le m} \|(a_{ij})_{j=1}^n\|_{p^*}, \qquad D_2 = \max_{1 \le j \le n} \|(a_{ij})_{i=1}^m\|_q, \qquad L(k) = \max\{1, \ln k\},$$ with $\min\{\infty, L(k)\} = L(k)$. Latała and Strzelecka proved, confirming a conjecture of Guédon, Hinrichs, Litvak and Prochno, that $\mathbb{E}\|G_A\|_{p \to q}$ is comparable, up to constants depending only on $p$ and $q$, to $D_1 + D_2 + \mathbb{E}\max_{i,j} |a_{ij} g_{ij}|$. When all $a_{ij} = 1$, $\mathbb{E}\|G_A\|_{p \to q}$ is of the order of the first two terms of the bound below, up to absolute constants, and for $p = q = 2$ the bound below holds (Latała, van Handel and Youssef).
Problem. Is there an absolute constant $C > 0$ such that $$\mathbb{E}\|G_A\|_{p \to q} \le C \Bigl[\sqrt{\min\{p^*, L(n)\}}\, D_1 + \sqrt{\min\{q, L(m)\}}\, D_2 + \mathbb{E}\max_{i,j} |a_{ij} g_{ij}|\Bigr]$$ for all $m, n \ge 1$, all $1 \le p \le 2 \le q \le \infty$ and every $A \in \mathbb{R}^{m \times n}$?
Source: Latała and Strzelecka (2026), Conjecture 5 in arXiv:2502.02186v3 (Adv. Math. 501); posed without a number in the first arXiv version (2025).
Imported from github.com/ajt60gaibb/OpenProblemsInNLA.
Classification
- Primary
- PR — Probability
- Secondary
- FA, NA
- Keywords
- OpenProblemsInNLA, operator norms, random matrices
- Literature checked
- Oct. 8, 2026 by clemens-admin
References
@article{latala2026,
author = {Lata{\l}a, Rafa{\l} and Strzelecka, Marta},
title = {Operator $\ell_p \to \ell_q$ norms of {G}aussian matrices},
journal = {Adv. Math.},
volume = {501},
year = {2026},
pages = {Paper No. 111097},
doi = {10.1016/j.aim.2026.111097},
eprint = {2502.02186},
archivePrefix = {arXiv},
}
@incollection{guedon2017,
author = {Gu{\'e}don, Olivier and Hinrichs, Aicke and Litvak, Alexander E. and Prochno, Joscha},
title = {On the expectation of operator norms of random matrices},
booktitle = {Geometric Aspects of Functional Analysis},
series = {Lecture Notes in Math.},
volume = {2169},
publisher = {Springer},
address = {Cham},
year = {2017},
pages = {151--162},
doi = {10.1007/978-3-319-45282-1_10},
eprint = {1603.02401},
archivePrefix = {arXiv},
}
@article{latala2018,
author = {Lata{\l}a, Rafa{\l} and van Handel, Ramon and Youssef, Pierre},
title = {The dimension-free structure of nonhomogeneous random matrices},
journal = {Invent. Math.},
volume = {214},
number = {3},
year = {2018},
pages = {1031--1080},
doi = {10.1007/s00222-018-0817-x},
eprint = {1711.00807},
archivePrefix = {arXiv},
}
@misc{openproblemsinnla-mi31,
author = {Townsend, Alex},
title = {{MI-31}: Sharp parameter dependence for structured Gaussian operator norms},
howpublished = {Open Problems in Numerical Linear Algebra, GitHub repository},
year = {2026},
url = {https://github.com/ajt60gaibb/OpenProblemsInNLA/blob/main/matrix-inequalities-and-norms/MI-31/README.md},
note = {Read at commit eb7c9d6, 2026-10-07},
}
Resolution claims
We give a candidate proof of Conjecture 5 in the third arXiv version of Latala and Strzelecka Operator lp lq norms of Gaussian matrices.
Status history
| When | Change | By | Reason |
|---|---|---|---|
| 2026-10-08 | Needs verification → Solution claimed | clemens-admin | Listed with the record of its solution, unreviewed. |
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00251, "Sharp parameter dependence for structured Gaussian operator norms", clopen.solutions.