A Seginer theorem for arbitrary independent identically distributed tensor entries
For integers $r \ge 3$ and $n_1, \dots, n_r \ge 2$, let $T \in \mathbb{R}^{n_1 \times \cdots \times n_r}$ have independent identically distributed integrable real entries of mean zero. The distribution may depend on the dimensions, and no moment beyond the first is assumed. Let $$\|T\|_{\mathrm{inj}} = \sup_{\|x_1\|_2 = \cdots = \|x_r\|_2 = 1} |\langle T, x_1 \otimes \cdots \otimes x_r \rangle|$$ be its injective norm, and let $$F(T) = \max_{1 \le k \le r} \mathbb{E} \max_{i_j,\ j \ne k} \Big( \sum_{a=1}^{n_k} T_{i_1, \dots, i_{k-1}, a, i_{k+1}, \dots, i_r}^2 \Big)^{1/2}$$ be the largest expected Euclidean fiber norm, the maximum over the modes $k$ being taken outside the expectation. For matrices ($r = 2$) the comparison below is Seginer's theorem. Lucca and Pesenti proved it for entries distributed as a random variable $Z$ with $\mathbb{E}|Z|^{2+\varepsilon} \le K (\mathbb{E} Z^2)^{1 + \varepsilon/2}$ for some $\varepsilon \in (0, 1)$ and $K \ge 1$, with constants depending on $r$, $\varepsilon$ and $K$.
Problem. For each $r \ge 3$, are there constants $0 < c_r \le C_r < \infty$, depending only on $r$, such that $$c_r F(T) \le \mathbb{E} \|T\|_{\mathrm{inj}} \le C_r F(T)$$ for all $n_1, \dots, n_r \ge 2$ and every such entry distribution?
Source: Lucca and Pesenti (2026), Conjecture 1.8, for every order r; stated for r ≥ 3 and n_j ≥ 2 by OpenProblemsInNLA, the case r = 2 being Seginer's theorem.
Imported from github.com/ajt60gaibb/OpenProblemsInNLA.
Classification
- Primary
- PR — Probability
- Secondary
- FA, NA
- Keywords
- injective norms, OpenProblemsInNLA, random tensors
- Literature checked
- Oct. 8, 2026 by clemens-admin
References
@misc{lucca2026,
author = {Lucca, Kevin and Pesenti, Lucas},
title = {Norm Bounds for Sparse Random Tensors and Spectral Gap of Random Hypergraphs},
year = {2026},
eprint = {2607.07308},
archivePrefix = {arXiv},
}
@article{seginer2000,
author = {Seginer, Yoav},
title = {The Expected Norm of Random Matrices},
journal = {Combin. Probab. Comput.},
volume = {9},
number = {2},
year = {2000},
pages = {149--166},
doi = {10.1017/S096354830000420X},
}
@misc{openproblemsinnla-tr21,
author = {Townsend, Alex},
title = {{TR-21}: A Seginer theorem for arbitrary independent identically distributed tensor entries},
howpublished = {Open Problems in Numerical Linear Algebra, GitHub repository},
year = {2026},
url = {https://github.com/ajt60gaibb/OpenProblemsInNLA/blob/main/tensor-computations/TR-21/README.md},
note = {Read at commit eb7c9d6, 2026-10-07},
}
Resolution claims
For every fixed order $k \ge 3$, we prove that the expected injective norm of a tensor with iid mean-zero integrable entries is comparable to its largest expected Euclidean fiber norm, uniformly over all rectangular dimensions and entry laws. Precisely, writing $F(T)$ for that fiber norm, $F(T) \le \mathbb{E}\|T\|_{\mathrm{inj}} \le C_k F(T)$, where $C_k$ depends only on $k$; no finite-variance …
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-00214, "A Seginer theorem for arbitrary independent identically distributed tensor entries", clopen.solutions.