Universality of General Spiked Tensor Models

2026-02-04

AI Summary

This study investigates statistical inference and spectral universality in rank-one spiked tensor models under high-dimensional, non-Gaussian noise. Focusing on rank-one signals embedded in higher-order asymmetric tensors, the authors analyze the spectral distribution of specific tensor contractions. By leveraging resolvent methods from random matrix theory, cumulant expansions under a finite fourth-moment condition, and Efron–Stein-type variance bounds, they establish that—provided the noise possesses a finite fourth moment—the empirical spectral distribution converges almost surely to a deterministic limit. Moreover, the leading singular value and its associated singular vector exhibit asymptotic alignment with the underlying signal direction, matching precisely the behavior observed in the Gaussian setting. This result removes the restrictive Gaussian assumption traditionally imposed in such analyses and establishes universality for spiked tensor models under general non-Gaussian noise.

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Reasoning under Uncertainty: Graphical Models
• Cognitive Modeling & Cognitive Systems: Neural Spike Coding

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Large-scale security measurements
• Web Mining and Content Analysis: Models for Web evolution

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.

Problem

• spiked tensor models
• universality
• high-dimensional statistics
• non-Gaussian noise
• spectral behavior

Innovation

• spiked tensor models
• universality
• random matrix theory
• cumulant expansion
• high-dimensional statistics