Spectra of contractions of the Gaussian Orthogonal Tensor Ensemble

TL;DR

This work analyzes the spectra of matrix-valued contractions of the Gaussian Orthogonal Tensor Ensemble (GOTE), focusing on pure contractions M_n = \mathcal{G} \cdot \mathbf{w}^{\otimes (r-2)} and mixed contractions. It proves a semicircular limiting spectral distribution under appropriate scaling for pure contractions across growing r and n, and identifies Baik–Ben Arous–Péché-type phase transitions at the edge for r \ge 4, including explicit edge limits in several regimes and information-carrying edge eigenvectors. For mixed contractions at r=4, the paper shows the LSD depends on the contraction overlap and establishes contiguity results between mixed and pure cases, with no outliers when the overlap vanishes. The proofs hinge on representing contractions as low-rank perturbations of GOE matrices and employing an isotropic local semicircle law to derive directional spectral measures, contributing to tensor-PCA analyses and random-hypergraph adjacency theory.

Abstract

In this article, we study the spectra of matrix-valued contractions of the Gaussian Orthogonal Tensor Ensemble (GOTE). Let $\mathcal{G}$ denote a random tensor of order $r$ and dimension $n$ drawn from the density \[ f(\mathcal{G}) \propto \exp\bigg(-\frac{1}{2r}\|\mathcal{G}\|^2_{\mathrm{F}}\bigg). \] For $\mathbf{w} \in \mathbb{S}^{n - 1}$, the unit-sphere in $\mathbb{R}^n$, we consider the matrix-valued contraction $\mathcal{G} \cdot \mathbf{w}^{\otimes (r - 2)}$ when both $r$ and $n$ go to infinity such that $r / n \to c \in [0, \infty]$. We obtain semi-circle bulk-limits in all regimes, generalising the works of Goulart et al. (2022); Au and Garza-Vargas (2023); Bonnin (2024) in the fixed-$r$ setting. We also study the edge-spectrum. We obtain a Baik-Ben Arous-Péché phase-transition for the largest and the smallest eigenvalues at $r = 4$, generalising a result of Mukherjee et al. (2024) in the context of adjacency matrices of random hypergraphs. We also show that the extreme eigenvectors of $\mathcal{G} \cdot \mathbf{w}^{\otimes (r - 2)}$ contain non-trivial information about the contraction direction $\mathbf{w}$. Finally, we report some results, in the case $r = 4$, on mixed contractions $\mathcal{G} \cdot \mathbf{u} \otimes \mathbf{v}$, $\mathbf{u}, \mathbf{v} \in \mathbb{S}^{n - 1}$. While the total variation distance between the joint distribution of the entries of $\mathcal{G} \cdot \mathbf{u} \otimes \mathbf{v}$ and that of $\mathcal{G} \cdot \mathbf{u} \otimes \mathbf{u}$ goes to $0$ when $\|\mathbf{u} - \mathbf{v}\| = o(n^{-1})$, the bulk and the largest eigenvalues of these two matrices have the same limit profile as long as $\|\mathbf{u} - \mathbf{v}\| = o(1)$. Furthermore, it turns out that there are no outlier eigenvalues in the spectrum of $\mathcal{G} \cdot \mathbf{u} \otimes \mathbf{v}$ when $\langle \mathbf{u}, \mathbf{v} \rangle = o(1)$.

Figures (1)

• Figure 1: Histograms of the eigenvalues of $n^{-1/2}\mathcal{G} \cdot \mathbf{e}_1 \otimes \mathbf{e}_1$ (left) and $n^{-1/2}\mathcal{G} \cdot~\mathbf{e}_1 \otimes \mathbf{e}_2$ (right) for $n = 2500$, based on $100$ replications. The solid (blue) curves depict the corresponding semi-circle densities. The dotted vertical lines mark the locations $\pm \varpi_4 = \pm \frac{2}{\sqrt{3}}$. Note that there are two outlier eigenvalues in the spectrum of $n^{-1/2}\mathcal{G} \cdot \mathbf{e}_1 \otimes \mathbf{e}_1$ and none in that of $n^{-1/2}\mathcal{G} \cdot \mathbf{e}_1 \otimes \mathbf{e}_2$.

Theorems & Definitions (63)

• Proposition 2.1
• Theorem 2.1: Edge eigenvalues -- first order results
• Theorem 2.2: Edge eigenvalues -- second order results
• Theorem 2.3
• Definition 2.1: Directional Spectral Measure
• Theorem 2.4: Spectral measure in the direction $\mathbf{x}$
• Proposition 2.2
• Theorem 2.5
• Remark 2.1
• Proposition 2.3