A type-2 inequality for injective norms of Gaussian sums of tensors
Let $T_1, \dots, T_n \in (\mathbb{R}^d)^{\otimes r}$ be symmetric tensors, $g_1, \dots, g_n$ i.i.d. standard Gaussians, and $\|T\|_{\mathcal{I}_p} = \max_{\|x\|_p \le 1} |\langle T, x^{\otimes r}\rangle|$ the symmetric injective $\ell_p$ norm. For $r = p = 2$ the inequality below is the matrix bound of Tomczak-Jaegermann and of Ahlswede and Winter. Bandeira, Gopi, Jiang, Lucca and Rothvoss proved it for $p \ge 2r$; a volumetric barrier stops their method for $p < 2r$.
Problem. Prove or disprove that for every $p \ge 2$, $$\mathbb{E}\,\Bigl\|\sum_{i=1}^n g_i T_i\Bigr\|_{\mathcal{I}_p} \le \tilde{O}_{r,p}\Bigl(d^{\frac12 - \frac1p}\sqrt{\textstyle\sum_{i=1}^n \|T_i\|_{\mathcal{I}_p}^2}\Bigr),$$ where $\tilde{O}_{r,p}$ hides constants depending on $r$ and $p$ and factors polylogarithmic in $d$ and $n$.
Source: Lucca, Conjecture 16 of Randomstrasse101, in “Tensor Concentration Inequalities” (2025); Entry 8 of arXiv:2603.29571.
The statement is adapted from randomstrasse101.math.ethz.ch, whose text is available under CC BY 4.0.
Classification
- Primary
- PR — Probability
- Secondary
- FA
- Keywords
- injective norm, matrix concentration inequalities, Randomstrasse101, random tensors
- Literature checked
- Oct. 2, 2026 by clemens-admin
References
@misc{randomstrasse2025,
author = {Bandeira, Afonso S. and Dmitriev, Daniil and Lucca, Kevin and Nizi{\'c}-Nikolac, Petar and R{\"o}dder, Almut},
title = {{Randomstrasse101}: Open Problems of 2025},
year = {2026},
eprint = {2603.29571},
archivePrefix = {arXiv}
}
@misc{bandeira2024tensors,
author = {Bandeira, Afonso S. and Gopi, Sivakanth and Jiang, Haotian and Lucca, Kevin and Rothvoss, Thomas},
title = {A geometric perspective on the injective norm of sums of random tensors},
year = {2024},
eprint = {2411.10633},
archivePrefix = {arXiv}
}
@misc{adenali2025injective,
author = {Aden-Ali, Ishaq},
title = {On the injective norm of sums of random tensors and the moments of {G}aussian chaoses},
year = {2025},
eprint = {2503.10580},
archivePrefix = {arXiv}
}
@article{tomczak1974moduli,
author = {Tomczak-Jaegermann, Nicole},
title = {The moduli of smoothness and convexity and the {R}ademacher averages of trace classes {$S_p$} $(1 \le p < \infty)$},
journal = {Studia Mathematica},
volume = {50},
pages = {163--182},
year = {1974}
}
@article{ahlswede2002strong,
author = {Ahlswede, Rudolf and Winter, Andreas},
title = {Strong converse for identification via quantum channels},
journal = {IEEE Transactions on Information Theory},
volume = {48},
number = {3},
pages = {569--579},
year = {2002},
doi = {10.1109/18.985947}
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00034, "A type-2 inequality for injective norms of Gaussian sums of tensors", clopen.solutions.