Spectral asymptotics for contracted tensor ensembles

TL;DR

This paper analyzes the spectral distribution of contracted random symmetric tensors, showing it approximates a semicircular family with covariance determined by overlaps of tensor contractions, extending graphical calculus methods.

Contribution

It introduces a tensorial extension of graphical calculus for spectral analysis of random tensors, providing new insights into their joint spectral distribution and covariance structure.

Findings

Spectral distribution approximates a semicircular family.

Covariance is given by overlaps of tensor contractions.

Characterization of extreme variance cases.

Abstract

Let $\mathbf{T}_{d, N}: \Omega \to \mathbb{R}^{N^d}$ be a random real symmetric Wigner-type tensor. For unit vectors $(u_N^{(i, j)})_{i \in I, j \in [d-2]} \subset \mathbb{S}^{N-1}$, we study the contracted tensor ensemble \[ \left(\frac{1}{\sqrt{N}}\mathbf{T}_{d, N}\left[u_N^{(i, 1)} \otimes \cdots \otimes u_N^{(i, d-2)}\right]\right)_{i \in I}. \] For large $N$, we show that the joint spectral distribution of this ensemble is well-approximated by a semicircular family $(s_i)_{i \in I}$ whose covariance $(\mathbf{K}_{i, i'}^{(N)})_{i, i'\in I}$ is given by the rescaled overlaps of the corresponding symmetrized contractions \[ \mathbf{K}_{i, i'}^{(N)} = \frac{1}{d(d-1)}\langle u_N^{(i, 1)} \odot \cdots \odot u_N^{(i, d-2)}, u_N^{(i', 1)} \odot \cdots \odot u_N^{(i', d-2)} \rangle, \] which is the true covariance of the ensemble up to a $O_d(N^{-1})$ correction. We further characterize the extreme cases of the variance $\mathbf{K}_{i, i}^{(N)} \in [\frac{1}{d!}, \frac{1}{d(d-1)}]$. Our analysis relies on a tensorial extension of the usual graphical calculus for moment method calculations in random matrix theory, allowing us to access the independence in our random tensor ensemble.