A central limit theorem for projections of the cube

TL;DR

This paper establishes a central limit theorem describing the distribution of the volume of projections of a high-dimensional cube onto random subspaces as the dimension of the cube grows infinitely large.

Contribution

It provides the first CLT for the volume of cube projections onto random subspaces in the high-dimensional limit, with respect to Haar measure.

Findings

Volume of cube projections converges to a normal distribution as N tends to infinity.

The CLT holds for fixed projection dimension n and increasing cube dimension N.

Results apply to random subspaces chosen uniformly from the Grassmannian.

Abstract

We prove a central limit theorem for the volume of projections of the N-cube onto a random subspace of dimension n, when n is fixed and N tends to infinity. Randomness in this case is with respect to the Haar measure on the Grassmannian manifold.