[beagle3]

Actually, “apply a random matrix” is often the solution to a large dimensional space involving near neighbours.

The Johnson-Lindenstrauss lemma asserts that a multiplying by a random matrix (some conditions apply, but iirc rotation matrices satisfy them) keeps, in many senses, the distances between points even if the dimension drops very significantly (some conditions apply but usually satisfied by real world data)

This is, in fact, the theoretical underpinning of compressed sensing.