Large deviations for random matrices in the orthogonal group and Stiefel manifold with applications to random projections of product distributions

TL;DR

This paper establishes large deviation principles for random matrices in the orthogonal group and Stiefel manifold, and applies these results to analyze the behavior of high-dimensional random projections of product distributions, including uniform distributions on $ ext{ell}_p$-balls.

Contribution

It provides explicit LDPs for matrices in the orthogonal group and Stiefel manifold, and applies these to characterize large deviations of high-dimensional projections of product measures.

Findings

Explicit large deviation principles for random matrices in orthogonal group and Stiefel manifold.

Precise characterization of large deviations for high-dimensional projections of product distributions.

Extension of previous work on projections of uniform distributions on $ ext{ell}_p$-balls.

Abstract

We prove large deviation principles (LDPs) for random matrices in the orthogonal group and Stiefel manifold, determining both the speed and good convex rate functions that are explicitly given in terms of certain log-determinants of trace-class operators and are finite on the set of Hilbert-Schmidt operators $M$ satisfying $\|MM^*\|<1$. As an application of those LDPs, we determine the precise large deviation behavior of $k$-dimensional random projections of high-dimensional product distributions using an appropriate interpretation in terms of point processes, also characterizing the space of all possible deviations. The case of uniform distributions on $\ell_p$-balls, $1\leq p \leq \infty$, is then considered and reduced to appropriate product measures. Those applications generalize considerably the recent work [Johnston, Kabluchko, Prochno: Projections of the uniform distribution on the cube - a large deviation perspective, Studia Mathematica 264 (2022), 103-119].