Large deviation principles induced by the Stiefel manifold, and random multidimensional projections

Abstract

For fixed positive integers $k<n$, given an n-dimensional random vector $X^{(n)}$, consider its k-dimensional projection ${\mathbf{a}}_{n,k}^{⊺}{X}^{(n)}$, where ${\mathbf{a}}_{n,k}$ is an $n\times k$-dimensional matrix belonging to the Stiefel manifold ${\mathbb{V}}_{n,k}$ of orthonormal k-frames in ${\mathbb{R}}^n$. For a class of sequences ${\{X^{(n)}\}}_{n\in \mathbb{N}}$ that includes uniform distributions on suitably scaled ${\ell }_p^n$ balls, $p\in (1,\mathrm{\infty }]$, and product measures with sufficiently light tails, it is shown that the sequence of projected vectors ${\{{\mathbf{a}}_{n,k}^{⊺}{X}^{(n)}\}}_{n\in \mathbb{N}}$ satisfies a large deviation principle whenever the empirical measures of the rows of $\sqrt{n}{\mathbf{a}}_{n,k}$ converge, as $n\to \mathrm{\infty }$, to a probability measure on ${\mathbb{R}}^k$. In particular, this implies a (quenched) large deviation principle for the sequence ${\{{\mathbf{a}}_{n,k}^{⊺}{X}^{(n)}\}}_{n\in \mathbb{N}}$ for almost every realization ${\{{\mathbf{a}}_{n,k}\}}_{n\in \mathbb{N}}$ of ${\{{\mathbf{A}}_{n,k}\}}_{n\in \mathbb{N}}$, where each ${\mathbf{A}}_{n,k}$ is a random matrix, independent of ${\{X^{(n)}\}}_{n\in \mathbb{N}}$, that is distributed according to the normalized Haar measure on ${\mathbb{V}}_{n,k}$. Moreover, a variational formula is obtained for the rate function of the large deviation principle for the annealed projections ${\{{\mathbf{A}}_{n,k}^{⊺}{X}^{(n)}\}}_{n\in \mathbb{N}}$, in terms of a family of quenched rate functions and a modified entropy term. A key step in this analysis is a large deviation principle for the sequence of empirical measures of rows of the random matrices $\sqrt{n}{\mathbf{A}}_{n,k}$, $n\ge k$, which may be of independent interest. The study of multidimensional random projections of high-dimensional measures is of interest in asymptotic functional analysis, convex geometry and statistics. Prior results on quenched large deviations for random projections of ${\ell }_p^n$ balls have been essentially restricted to the one-dimensional setting.