On the Injective Norm of Sums of Random Tensors and the Moments of Gaussian Chaoses
Ishaq Aden-Ali
TL;DR
This work bounds the expected $\ar{\ell}_p$ injective norm of sums of subgaussian random tensors using a simple PAC-Bayesian argument, avoiding geometric chaining. The bound is expressed in terms of variance parameters capturing tensor contractions, and it sharpens prior results, including that of Bandeira et al. In the Euclidean case, the bound aligns with and sharpens Latała's moment estimates for Gaussian chaoses, yielding an elementary proof of these fundamental results. The approach offers a dimension-aware, non-asymptotic tool for analyzing high-order random tensors and their moments, with potential applications to tensor concentration phenomena and chaos analysis.
Abstract
We prove an upper bound on the expected $\ell_p$ 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 Latała 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.