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Universality of General Spiked Tensor Models

Yanjin Xiang, Zhihua Zhang

TL;DR

This work proves a universality principle for general asymmetric rank-one spiked tensor models in high dimensions: when the noise has independent, zero-mean, unit-variance entries with finite fourth moment, the asymptotic spectral behavior and the ML-estimated spike alignments coincide with the Gaussian-noise case. It develops a resolvent-based analysis combined with a cumulant expansion to control cross terms arising from dependence between the spike and the noise, and it shows that the limiting spectrum of block-wise tensor contractions, as well as explicit formulas for the leading singular value and alignments, match those obtained under Gaussian noise. The results extend to arbitrary tensor order $d\ge3$ and to rank-$r$ models with orthogonal components, yielding a decoupled, componentwise universality in the high-dimensional limit. The findings justify Gaussian-based predictions in practical tensor inference settings and highlight robust spectral and statistical phenomena that persist beyond Gaussian assumptions. The work also outlines open problems on the optimization landscape, the exact critical thresholds, and potential extensions to weaker moment conditions or algorithmic estimators beyond maximum likelihood.

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.

Universality of General Spiked Tensor Models

TL;DR

This work proves a universality principle for general asymmetric rank-one spiked tensor models in high dimensions: when the noise has independent, zero-mean, unit-variance entries with finite fourth moment, the asymptotic spectral behavior and the ML-estimated spike alignments coincide with the Gaussian-noise case. It develops a resolvent-based analysis combined with a cumulant expansion to control cross terms arising from dependence between the spike and the noise, and it shows that the limiting spectrum of block-wise tensor contractions, as well as explicit formulas for the leading singular value and alignments, match those obtained under Gaussian noise. The results extend to arbitrary tensor order and to rank- models with orthogonal components, yielding a decoupled, componentwise universality in the high-dimensional limit. The findings justify Gaussian-based predictions in practical tensor inference settings and highlight robust spectral and statistical phenomena that persist beyond Gaussian assumptions. The work also outlines open problems on the optimization landscape, the exact critical thresholds, and potential extensions to weaker moment conditions or algorithmic estimators beyond maximum likelihood.

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 (), 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.
Paper Structure (86 sections, 14 theorems, 567 equations)

This paper contains 86 sections, 14 theorems, 567 equations.

Table of Contents

  1. Introduction
  2. Main Contributions
  3. Related work
  4. Organization
  5. Preliminaries
  6. Notation and Notions
  7. Tensor contraction operator $\Phi_d$
  8. Cumulant expansion
  9. Spectral norm control
  10. The order-three asymmetric spiked tensor model with a rank-one perturbation
  11. Tensor singular value and singular vectors
  12. The limiting spectral measure associated with block-wise contractions of third-order tensors
  13. Concentration of the singular value and the alignments
  14. Asymptotic singular value and alignments.
  15. Cubic third-order tensors: the balanced case $c_1=c_2=c_3=\tfrac{1}{3}$
  16. ...and 71 more sections

Key Result

Lemma 2.1

Khorunzhy_1996 Let $\xi$ be a real-valued random variable such that $\mathbb{E}|\xi|^{p+2}<\infty$ for some integer $p\ge 0$, and let $f\colon \mathbb{R}\to\mathbb{C}$ be a function whose first $p+1$ derivatives are continuous and bounded. Then where $\kappa_a$ denotes the $a$-th cumulant of $\xi$, and the remaining term satisfies with a constant $C$ depending only on $p$.

Theorems & Definitions (28)

  • Remark 1
  • Definition 1: Stieltjes transform
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 3.1
  • Theorem 3.1
  • proof
  • Corollary 3.1
  • proof
  • Theorem 3.2
  • ...and 18 more