An asymptotic thin shell condition and large deviations for random multidimensional projections

TL;DR

This paper establishes large deviation principles for high-dimensional random projections under an asymptotic thin shell condition across various regimes, extending previous results to broader classes of measures and multidimensional settings.

Contribution

It introduces a general asymptotic thin shell condition and derives LDPs for random projections and their norms in multiple regimes, broadening the scope of large deviation results in high-dimensional geometry.

Findings

LDPs are proven for projections in constant, sublinear, and linear regimes.

Assumptions verified for Gibbs measures, $ ext{l}_p^n$ balls, and Orlicz balls.

Results extend existing univariate LDPs to multidimensional projections.

Abstract

Consider the projection of an $n$-dimensional random vector onto a random $k_n$-dimensional basis, $k_n \leq n$, drawn uniformly from the Haar measure on the Stiefel manifold of orthonormal $k_n$-frames in $\mathbb{R}^n$, in three different asymptotic regimes as $n \rightarrow \infty$: "constant" ($k_n=k$), "sublinear" ($k_n \rightarrow \infty$ but $k_n/n \rightarrow 0$) and "linear" $k_n/n \rightarrow \lambda$ with $0 < \lambda \le 1$). When the sequence of random vectors satisfies a certain "asymptotic thin shell condition", we establish annealed large deviation principles (LDPs) for the corresponding sequence of random projections in the constant regime, and for the sequence of empirical measures of the coordinates of the random projections in the sublinear and linear regimes. We also establish LDPs for certain scaled $\ell_q$ norms of the random projections in these different regimes. Moreover, we verify our assumptions for various sequences of random vectors of interest, including those distributed according to Gibbs measures with superquadratic interaction potential, or the uniform measure on suitably scaled $\ell_p^n$ balls, for $p \in [1,\infty)$, and generalized Orlicz balls defined via a superquadratic function. Our results complement the central limit theorem for convex sets and related results which are known to hold under a "thin shell" condition. These results also substantially extend existing large deviation results for random projections, which are first, restricted to the setting of measures on $\ell_p^n$ balls, and secondly, limited to univariate LDPs (i.e., in $\mathbb{R}$) involving either the norm of a $k_n$-dimensional projection or the projection of $X^{(n)}$ onto a random one-dimensional subspace. Random projections of high-dimensional random vectors are of interest in a range of fields including asymptotic convex geometry and high-dimensional statistics.