简单随机张量的无维度鲁棒估计：重尾与对抗污染下的最优保证
Robust dimension-free estimation of simple random tensors: optimal guarantees under heavy tails and adversarial contamination
AI总结：
本文针对任意阶简单随机张量，提出基于方向截尾均值与极小极大聚合的鲁棒估计器，在重尾与对抗污染下实现近最优无维度统计速率，其建立的高阶过程集中不等式具独立应用价值。
AI中文摘要：
我们研究了任意阶数 $q\in\mathbb{N}$ 的简单随机张量在有限矩假设与对抗污染下的鲁棒估计问题。我们提出了该场景下首个达到近最优无维度统计速率的鲁棒估计器：当 $p\ge2q$ 阶矩有限时，该估计器可达到近最优污染率；在弱矩区间 $q\le p\le2q$ 内，它仍能提供非平凡保证。该估计器基于方向截尾均值与极小极大聚合，无需区间交并过程即可适配 $p$ 及超压缩常数的上界。我们的分析将近期鲁棒均值与协方差估计进展所依托的截尾均值框架拓展至任意阶张量，尤其建立了高阶计数与截断经验多向量积过程的集中不等式，这些不等式或在当前应用之外（包括算法鲁棒估计）具有独立价值。
英文摘要：
We study robust estimation of simple random tensors of arbitrary order $q\in\mathbb{N}$ under finite-moment assumptions and adversarial contamination. We propose the first robust estimator achieving near-optimal dimension-free statistical rates in this setting. The estimator attains the near-optimal corruption rate whenever $p\ge2q$ moments are finite and continues to provide nontrivial guarantees throughout the weak-moment regime $q\le p\le2q$. Being based on directional trimmed means and minimax aggregation, our estimator is adaptive to $p$ and upper bounds on hypercontractive constants without resorting to interval-intersection procedures. Our analysis extends the trimmed-mean framework underlying recent advances in robust mean and covariance estimation to arbitrary tensor order. In particular, we establish concentration inequalities for higher-order counting and truncated empirical multi-vector product processes. We believe these inequalities could be of independent interest beyond the present application, including algorithmic robust estimation.
