Universality of General Spiked Tensor Models

Abstract: 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.