# On the Injective Norm of Sums of Random Tensors and the Moments of Gaussian Chaoses

arXiv: https://arxiv.org/abs/2503.10580 · HTML: https://mlsanity.com/paper/2503.10580/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 3 of its 24 references that are in the map.
- First posted on arXiv 13 Mar 2025.

## Builds on (most cited first, 3 of 3)

- [An Introduction to Matrix Concentration Inequalities](https://mlsanity.com/paper/1501.01571/) (2015)
- [Generalization Bounds: Perspectives from Information Theory and PAC-Bayes](https://mlsanity.com/paper/2309.04381/) (2023)
- [Exact minimax risk for linear least squares, and the lower tail of sample covariance matrices](https://mlsanity.com/paper/1912.10754/) (2019)

## Built on by (most cited first, 1 of 1)

- [Variational Tail Bounds for Norms of Random Vectors and Matrices](https://mlsanity.com/paper/2503.17300/) (2025)

## Abstract (arXiv)

We prove an upper bound on the expected ℓp injective norm of sums of subgaussian random tensors. Our proof is simple and does not rely on any explicit geometric or chaining arguments. Instead, it follows from a simple application of the PAC-Bayesian lemma, a tool that has proven effective at controlling the suprema of certain ``smooth'' empirical processes in recent years. Our bound strictly improves a very recent result of Bandeira, Gopi, Jiang, Lucca, and Rothvoss. In the Euclidean case (p=2), our bound sharpens a result of Latala that was central to proving his estimates on the moments of Gaussian chaoses. As a consequence, we obtain an elementary proof of this fundamental result.