An asymptotic thin shell condition and large deviations for multidimensional projections
Abstract
Random projections of highdimensional random vectors are of interest in a range of fields including asymptotic convex geometry and highdimensional statistics. We 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 $\ell_q^n$ norms of the random projections in these different regimes. Moreover, we verify the aforementioned asymptotic thin shell condition for various sequences of random vectors of interest, including those distributed according to a Gibbs measure with superquadratic interaction potential. Our results serve to 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 $\ell_p^n$ 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 onedimensional subspace.
 Publication:

arXiv eprints
 Pub Date:
 December 2019
 arXiv:
 arXiv:1912.13447
 Bibcode:
 2019arXiv191213447S
 Keywords:

 Mathematics  Probability;
 Mathematics  Functional Analysis;
 60F10;
 52A23;
 46B06
 EPrint:
 24 pages