用简单方法改进随机张量范数上界，推动高斯混沌矩研究
On the Injective Norm of Sums of Random Tensors and the Moments of Gaussian Chaoses
- 基于PAC-Bayesian引理，不依赖几何或链式论证
- 对p=2情形改进拉塔拉结果，提升高斯混沌矩估计精度
- 提供高斯混沌矩的初等证明，适合概率与统计研究者
我们证明了子高斯随机张量和的ℓ_p注入范数的期望上界。证明方法简洁，不依赖任何显式的几何或链式论证，仅通过近年在控制某些'光滑'经验过程上确界时表现出色的PAC-Bayesian引理即可完成。该上界严格优于近期Bandeira、Gopi、Jiang、Lucca和Rothvoss的结果。在欧几里得情形（p=2）下，我们的上界改进了拉塔拉（Latała）的关键结果，而后者曾是其关于高斯混沌矩估计的核心基础。因此，我们得到了这一基本结果的一个初等证明。
原文摘要 · Abstract (English)
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.
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