AI 中文总结
本文针对样本矩张量建立无维度Sharp Frobenius范数集中不等式，发现奇数阶张量的高斯与次高斯尺度重合、偶数阶不同的奇偶效应，通过结合结构化鞅耦合定理等完成矩界证明，为相关领域提供关键理论支撑。
AI 中文摘要
本文建立了样本矩张量的无维度Sharp Frobenius范数集中不等式。我们的界在次高斯类上是最优的，对于高斯数据，得到了匹配的双侧估计。我们还确定了奇偶效应：中间的高斯和次高斯尺度在奇数阶张量时重合，但在偶数阶时不同。第二个主要结果为这些界提供了弱矩估计，证明了高斯凸支配下希尔伯特值多项式的无维度矩界，该界的证明结合了近期提出的结构化鞅耦合定理与均匀对数凹分布矩张量的新估计。
英文摘要
This paper establishes sharp dimension-free Frobenius-norm concentration inequalities for sample moment tensors. Our bounds are optimal over the sub-Gaussian class, while for Gaussian data we obtain matching two-sided estimates. We also identify a parity effect: the intermediate Gaussian and sub-Gaussian scales coincide at odd tensor orders but differ at even orders. Our second main result, which supplies the weak-moment estimate behind these bounds, proves a dimension-free moment bound for Hilbert-valued polynomials under Gaussian convex domination. The proof of this bound combines a recently developed structured martingale coupling theorem with new estimates for moment tensors of uniformly log-concave distributions.
Comments35 pages
