Technical report: Two observations on probability distribution symmetries for randomly-projected data

TL;DR

This technical report investigates symmetries in probability distributions of high-dimensional data after random projection, highlighting invariance under reflection and rotation about the original data vector.

Contribution

It presents two novel observations about symmetry properties of projected data distributions that were previously unnoted.

Findings

Distributions are invariant under reflection across the original data vector.

Distributions are invariant under rotation about the original data vector.

These symmetries hold for projections onto lower-dimensional subspaces.

Abstract

In this technical report, we will make two observations concerning symmetries of the probability distribution resulting from projection of a piece of p-dimensional data onto a random m-dimensional subspace of $\mathbb{R}^p$, where m < p. In particular, we shall observe that such distributions are unchanged by reflection across the original data vector and by rotation about the original data vector