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Statistical Limits in Random Tensors with Multiple Correlated Spikes

Yang Qi, Alexis Decurninge

TL;DR

The paper analyzes statistical limits in the multi-spiked, symmetric tensor model by focusing on critical points of the high-dimensional ML objective under KKT conditions. It introduces the flat$$(\mathcal{T})$$ operator and derives its asymptotic spectrum via random matrix theory, revealing BBP-like outlier behavior and a detailed phase-transition landscape for the existence of detectable critical points. The authors characterize limiting alignments between ground-truth signals and estimated critical-point signals through explicit equations involving correlation matrices and Stieltjes-transform–based functions, and they propose an asymptotically unbiased estimator for tensor weights by solving a polynomial system. Together, these results illuminate information-theoretic and algorithmic thresholds in high-dimensional tensor inference and offer practical, data-driven methods for improved weight estimation and summary-statistics inference.

Abstract

We use tools from random matrix theory to study the multi-spiked tensor model, i.e., a rank-$r$ deformation of a symmetric random Gaussian tensor. In particular, thanks to the nature of local optimization methods used to find the maximum likelihood estimator of this model, we propose to study the phase transition phenomenon for finding critical points of the corresponding optimization problem, i.e., those points defined by the Karush-Kuhn-Tucker (KKT) conditions. Moreover, we characterize the limiting alignments between the estimated signals corresponding to a critical point of the likelihood and the ground truth signals. With the help of these results, we propose a new estimator of the rank-$r$ tensor weights by solving a system of polynomial equations, which is asymptotically unbiased contrary the maximum likelihood estimator.

Statistical Limits in Random Tensors with Multiple Correlated Spikes

TL;DR

The paper analyzes statistical limits in the multi-spiked, symmetric tensor model by focusing on critical points of the high-dimensional ML objective under KKT conditions. It introduces the flat operator and derives its asymptotic spectrum via random matrix theory, revealing BBP-like outlier behavior and a detailed phase-transition landscape for the existence of detectable critical points. The authors characterize limiting alignments between ground-truth signals and estimated critical-point signals through explicit equations involving correlation matrices and Stieltjes-transform–based functions, and they propose an asymptotically unbiased estimator for tensor weights by solving a polynomial system. Together, these results illuminate information-theoretic and algorithmic thresholds in high-dimensional tensor inference and offer practical, data-driven methods for improved weight estimation and summary-statistics inference.

Abstract

We use tools from random matrix theory to study the multi-spiked tensor model, i.e., a rank- deformation of a symmetric random Gaussian tensor. In particular, thanks to the nature of local optimization methods used to find the maximum likelihood estimator of this model, we propose to study the phase transition phenomenon for finding critical points of the corresponding optimization problem, i.e., those points defined by the Karush-Kuhn-Tucker (KKT) conditions. Moreover, we characterize the limiting alignments between the estimated signals corresponding to a critical point of the likelihood and the ground truth signals. With the help of these results, we propose a new estimator of the rank- tensor weights by solving a system of polynomial equations, which is asymptotically unbiased contrary the maximum likelihood estimator.

Paper Structure

This paper contains 33 sections, 18 theorems, 189 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Related work
  3. Contributions
  4. Organization
  5. Preliminaries
  6. Notations
  7. Random matrix theory tools
  8. Gaussian calculations
  9. Symmetric Gaussian distribution
  10. General setting
  11. Asymptotic spectrum of $\flat(\mathcal{T})$
  12. Sequence of critical points with fully convergent summary statistics
  13. Generalization to sequence with partially convergent summary statistics
  14. Limiting alignments
  15. Asymptotic results
  16. ...and 18 more sections

Key Result

Theorem 2.1

For two points $a < b$ where the cumulative distribution function of $\mu$ is continuous, If $\mu$ admits a density function at $x$, then the density at $x$ is given by

Figures (4)

  • Figure 1: Density functions of the spectral measure of Theorem \ref{['thm:generalized_measure']} (in black) for $d = 3$ and $d = 4$ and histograms of eigenvalues for one realization of $\flat (\mathcal{T})$.
  • Figure 2: Solutions with $\rho=0$
  • Figure 3: Solutions with $\rho=0.7$
  • Figure 4: Solutions with $\beta_1=2\beta_2$ and $\rho=0.7$

Theorems & Definitions (32)

  • Theorem 2.1: Inverse formula of Stieltjes-Perron
  • Theorem 2.2
  • Lemma 2.3: Stein's lemma
  • Lemma 2.4
  • Proposition 3.1
  • proof
  • Proposition 3.2
  • proof
  • Lemma 3.3
  • proof
  • ...and 22 more