Random embeddings with an almost Gaussian distortion

TL;DR

This paper demonstrates that under certain conditions, random matrices with columns derived from symmetric, isotropic vectors exhibit Gaussian-like distortions, leading to near-optimal bounds on their extremal singular values.

Contribution

The authors establish conditions under which random matrices with non-Gaussian columns mimic Gaussian behavior in terms of distortion and singular value bounds.

Findings

Random matrices with isotropic, symmetric vectors show Gaussian-like distortion.

Extremal singular values are tightly bounded for certain dimensions and distributions.

Results apply to log-concave vectors with high probability.

Abstract

Let $X$ be a symmetric, isotropic random vector in $\mathbb{R}^m$ and let $X_1...,X_n$ be independent copies of $X$. We show that under mild assumptions on $\|X\|_2$ (a suitable thin-shell bound) and on the tail-decay of the marginals $\langle X,u\rangle$, the random matrix $A$, whose columns are $X_i/\sqrt{m}$ exhibits a Gaussian-like behaviour in the following sense: for an arbitrary subset of $T\subset \mathbb{R}^n$, the distortion $\sup_{t \in T} | \|At\|_2^2 - \|t\|_2^2 |$ is almost the same as if $A$ were a Gaussian matrix. A simple outcome of our result is that if $X$ is a symmetric, isotropic, log-concave random vector and $n \leq m \leq c_1(\alpha)n^\alpha$ for some $\alpha>1$, then with high probability, the extremal singular values of $A$ satisfy the optimal estimate: $1-c_2(\alpha) \sqrt{n/m} \leq \lambda_{\rm min} \leq \lambda_{\rm max} \leq 1+c_2(\alpha) \sqrt{n/m}$.