Gaussian fluctuations for high-dimensional random projections of $\ell_p^n$-balls

TL;DR

This paper investigates the Gaussian fluctuations of projected points in high-dimensional $\, ext{l}_p^n$-balls, establishing a central limit theorem and Berry-Esseen bounds for the Euclidean norm of these projections.

Contribution

It provides the first detailed description of the Gaussian fluctuations and convergence rates for random projections of high-dimensional $\, ext{l}_p^n$-balls, including large deviations analysis.

Findings

Central limit theorem for Euclidean norms of projections

Berry-Esseen bounds on convergence rates

Discussion of large deviations for the projections

Abstract

In this paper, we study high-dimensional random projections of $\ell_p^n$-balls. More precisely, for any $n\in\mathbb N$ let $E_n$ be a random subspace of dimension $k_n\in\{1,\ldots,n\}$ and $X_n$ be a random point in the unit ball of $\ell_p^n$. Our work provides a description of the Gaussian fluctuations of the Euclidean norm $\|P_{E_n}X_n\|_2$ of random orthogonal projections of $X_n$ onto $E_n$. In particular, under the condition that $k_n\to\infty$ it is shown that these random variables satisfy a central limit theorem, as the space dimension $n$ tends to infinity. Moreover, if $k_n\to\infty$ fast enough, we provide a Berry-Esseen bound on the rate of convergence in the central limit theorem. At the end we provide a discussion of the large deviations counterpart to our central limit theorem.