A new look at random projections of the cube and general product measures

TL;DR

This paper establishes a strong law of large numbers for random projections of high-dimensional cubes, showing convergence to a Euclidean ball, and studies large deviations for projections of general product measures.

Contribution

It provides new asymptotic results for random projections of the cube and general product measures, including convergence and large deviations principles.

Findings

Random projections of the cube converge to a Euclidean ball of radius √(2/π).

Asymptotic counts of vertices and volume near points inside the limit ball are derived.

Large deviations principles are established for projections of product measures, with explicit rate functions.

Abstract

A strong law of large numbers for $d$-dimensional random projections of the $n$-dimensional cube is derived. It shows that with respect to the Hausdorff distance a properly normalized random projection of $[-1,1]^n$ onto $\mathbb{R}^d$ almost surely converges to a centered $d$-dimensional Euclidean ball of radius $\sqrt{2/\pi}$, as $n\to\infty$. For every point inside this ball we determine the asymptotic number of vertices and the volume of the part of the cube projected `close' to this point. Moreover, large deviations for random projections of general product measures are studied. Let $\nu^{\otimes n}$ be the $n$-fold product measure of a Borel probability measure $\nu$ on $\mathbb{R}$, and let $I$ be uniformly distributed on the Stiefel manifold of orthogonal $d$-frames in $\mathbb{R}^n$. It is shown that the sequence of random measures $\nu^{\otimes n}\circ(n^{-1/2}I^*)^{-1}$, $n\in\mathbb{N}$, satisfies a large deviations principle with probability $1$. The rate function is explicitly identified in terms of the moment generating function of $\nu$. At the heart of the proofs lies a transition trick which allows to replace the uniform projection by the Gaussian one. A number of concrete examples are discussed as well, including the uniform distributions on the cube $[-1,1]^n$ and the discrete cube $\{-1,1\}^n$ as a special cases.