Spectra of contractions of the Gaussian Orthogonal Tensor Ensemble

TL;DR

This paper investigates the spectral properties of matrix contractions derived from the Gaussian Orthogonal Tensor Ensemble, revealing semi-circle laws, phase transitions, and eigenvector insights across different regimes of tensor order and dimension.

Contribution

It generalizes spectral limit results for tensor contractions to large-r and n regimes, including phase transitions and eigenvector behavior, extending prior fixed-r analyses.

Findings

Semi-circle bulk limits in all regimes

Baik-Ben Arous-Péché phase transition at r=4

Eigenvectors encode contraction direction information

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-P\'{e}ch\'{e} 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 $\langle \mathbf{u}, \mathbf{v} \rangle = o(1)$.