Large deviations for high-dimensional random projections of $\ell_p^n$-balls

Abstract

The paper provides a description of the large deviation behavior for the Euclidean norm of projections of $\ell_p^n$-balls to high-dimensional random subspaces. More precisely, for each integer $n\geq 1$, let $k_n\in\{1,\ldots,n-1\}$, $E^{(n)}$ be a uniform random $k_n$-dimensional subspace of $\mathbb R^n$ and $X^{(n)}$ be a random point that is uniformly distributed in the $\ell_p^n$-ball of $\mathbb R^n$ for some $p\in[1,\infty]$. Then the Euclidean norms $\|P_{E^{(n)}}X^{(n)}\|_2$ of the orthogonal projections are shown to satisfy a large deviation principle as the space dimension $n$ tends to infinity. Its speed and rate function are identified, making thereby visible how they depend on $p$ and the growth of the sequence of subspace dimensions $k_n$. As a key tool we prove a probabilistic representation of $\|P_{E^{(n)}}X^{(n)}\|_2$ which allows us to separate the influence of the parameter $p$ and the subspace dimension $k_n$.