Anticoncentration and Berry--Esseen bounds for random tensors

TL;DR

This paper develops bounds on the distance to normality for linear functions of symmetric, exchangeable random tensors, extending classical CLT results to high-dimensional tensor settings with optimal estimates.

Contribution

It introduces a combinatorial CLT for high-dimensional tensors with permutation-based statistics, providing optimal non-asymptotic normality bounds in multiple regimes.

Findings

Derived Kolmogorov distance estimates for Gaussian approximation of tensor linear functions.

Established a combinatorial CLT for permutation-based tensor statistics.

Extended classical CLT results to arbitrary tensor dimensions.

Abstract

We obtain estimates for the Kolmogorov distance to appropriately chosen gaussians, of linear functions \[ \sum_{i\in [n]^d} \theta_i X_i \] of random tensors $\boldsymbol{X}=\langle X_i:i\in [n]^d\rangle$ which are symmetric and exchangeable, and whose entries have bounded third moment and vanish on diagonal indices. These estimates are expressed in terms of intrinsic (and easily computable) parameters associated with the random tensor $\boldsymbol{X}$ and the given coefficients $\langle \theta_i:i\in [n]^d\rangle$, and they are optimal in various regimes. The key ingredient -- which is of independent interest -- is a combinatorial CLT for high-dimensional tensors which provides quantitative non-asymptotic normality under suitable conditions, of statistics of the form \[ \sum_{(i_1,\dots,i_d)\in [n]^d} \boldsymbol{\zeta}\big(i_1,\dots,i_d,\pi(i_1),\dots,\pi(i_d)\big) \] where $\boldsymbol{\zeta}\colon [n]^d\times [n]^d\to\mathbb{R}$ is a deterministic real tensor, and $\pi$ is a random permutation uniformly distributed on the symmetric group $\mathbb{S}_n$. Our results extend, in any dimension $d$, classical work of Bolthausen who covered the one-dimensional case, and more recent work of Barbour/Chen who treated the two-dimensional case.