We study the rank-one spiked tensor model in the high-dimensional regime, where the noise entries are independent and identically distributed with zero mean, unit variance, and finite fourth moment.This setting extends the classical Gaussian framework to a substantially broader class of noise distributions.Focusing on asymmetric tensors of order $d$ ($\ge 3$), we analyze the maximum likelihood estimator of the best rank-one approximation.Under a mild assumption isolating informative critical points of the associated optimization landscape, we show that the empirical spectral distribution of a suitably defined block-wise tensor contraction converges almost surely to a deterministic limit that coincides with the Gaussian case.As a consequence, the asymptotic singular value and the alignments between the estimated and true spike directions admit explicit characterizations identical to those obtained under Gaussian noise. These results establish a universality principle for spiked tensor models, demonstrating that their high-dimensional spectral behavior and statistical limits are robust to non-Gaussian noise. Our analysis relies on resolvent methods from random matrix theory, cumulant expansions valid under finite moment assumptions, and variance bounds based on Efron-Stein-type arguments. A key challenge in the proof is how to handle the statistical dependence between the signal term and the noise term.

翻译：我们研究高维情形下的秩一尖峰张量模型，其中噪声项为零均值、单位方差且具有有限四阶矩的独立同分布随机变量。该设定将经典高斯框架扩展至更广泛的噪声分布类别。针对$d$阶（$\ge 3$）非对称张量，我们分析了最佳秩一逼近的最大似然估计量。在通过温和假设分离出相关优化景观中信息性临界点的前提下，我们证明：经适当定义的块状张量收缩算子的经验谱分布几乎必然收敛于确定性极限，且该极限与高斯情形完全一致。由此可得，渐近奇异值以及估计尖峰方向与真实尖峰方向之间的对齐关系具有显式表征，且这些表征与高斯噪声下获得的结果完全相同。这些结论确立了尖峰张量模型的普适性原理，表明其高维谱行为与统计极限对非高斯噪声具有鲁棒性。我们的分析基于随机矩阵理论中的预解式方法、有限矩假设下有效的累积量展开，以及基于Efron-Stein型论证的方差界。证明中的关键挑战在于如何处理信号项与噪声项之间的统计依赖性。