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

2025-03-13

AI Summary

This work addresses the problem of deriving expectation upper bounds for the ℓₚ-injective norm of sums of sub-Gaussian random tensors—a fundamental estimation challenge at the intersection of high-dimensional probability and tensor analysis. Methodologically, we introduce the PAC-Bayesian lemma to control random tensor norms, marking the first application of this technique in sub-Gaussian tensor analysis and overcoming key limitations of classical moment-based approaches. Our main contribution is a universal expectation upper bound that strictly improves upon the seminal results of Bandeira et al. and Latała. In particular, when *p* = 2, our bound sharpens Latała’s theorem and yields a concise, elementary derivation of all moments of Gaussian chaos. Moreover, the result unifies treatment across all *p* ≥ 1, substantially broadening the scope and applicability of sub-Gaussian tensor analysis.

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Reasoning under Uncertainty: Stochastic Optimization
• Natural Language Processing: Learning & Optimization for NLP

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Social Networks and Social Media: Social media analysis through the lenses of networks
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

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 Lata{l}a 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.

Problem

• Upper bound on expected injective norm of random tensors
• Improvement over recent results in tensor analysis
• Elementary proof of Gaussian chaos moment estimates

Innovation

• Uses PAC-Bayesian lemma for bounds
• Improves Bandeira et al.'s recent result
• Sharpens Latała's Gaussian chaos estimates