We study estimation and detection of high-order moment and cumulant tensors from $n$ i.i.d.\ observations of a $p$-dimensional random vector, with performance measured in tensor spectral norm. Under sub-Gaussianity, we show that the minimax rate for estimating the order-$d$ moment and cumulant tensors is $\sqrt{p/n}\wedge 1$. In contrast to covariance estimation, the sample moment tensor is generally not rate-optimal for $d\ge 3$, and we construct an estimator that attains the minimax rate up to logarithmic factors. On the computational side, we study testing whether the $d$-th order cumulant tensor vanishes after whitening. Using the low-degree polynomial framework, we provide evidence that detection is computationally hard when $n\ll p^{d/2}$. At the same time, we identify a regime in which an efficiently computable estimator achieves error smaller than the separation at which low-degree tests can reliably distinguish the null from the alternative. This reveals an unusual reverse detection--estimation gap: computationally efficient detection can be harder than computationally efficient estimation. The underlying reason is that the relevant loss, tensor spectral norm, is itself NP-hard to compute, creating a new form of computational--statistical gap.

翻译：我们从$n$个$p$维随机向量的独立同分布观测出发，研究高阶矩张量与累积量张量的估计与检测问题，其性能通过张量谱范数衡量。在次高斯性假设下，我们证明$d$阶矩张量与累积量张量的极小化极大估计率为$\sqrt{p/n}\wedge 1$。与协方差估计不同，样本矩张量在$d\ge 3$时通常不是率最优的，我们构造了一个估计量，在对数因子范围内达到了极小化极大估计率。在计算层面，我们研究了白化后$d$阶累积量张量是否为零的检验问题。利用低次多项式框架，我们提供了证据表明当$n\ll p^{d/2}$时，检测问题是计算困难的。同时，我们识别出一个区域，在该区域内存在高效可计算的估计量，其误差小于低次检验能够可靠区分原假设与备择假设所需的间隔。这揭示了一种反常的逆向检测—估计差距：高效可计算的检测可能比高效可计算的估计更加困难。其根本原因在于相关损失函数（张量谱范数）本身是NP难计算的，从而产生了一种新形式的计算—统计差距。