Gaussian random projections for Euclidean membership problems

TL;DR

This paper explores how Gaussian random projections can efficiently determine point membership in sets within high-dimensional Euclidean spaces, preserving problem feasibility with high probability.

Contribution

It demonstrates that random projections maintain the feasibility or infeasibility of Euclidean membership problems across various assumptions, aiding high-dimensional algorithmic solutions.

Findings

Feasibility and infeasibility are preserved under random projections.

Applicable to any algorithmic setting involving high-dimensional Euclidean membership.

Results hold under multiple assumptions about the data.

Abstract

We discuss the application of random projections to the fundamental problem of deciding whether a given point in a Euclidean space belongs to a given set. We show that, under a number of different assumptions, the feasibility and infeasibility of this problem are preserved with high probability when the problem data is projected to a lower dimensional space. Our results are applicable to any algorithmic setting which needs to solve Euclidean membership problems in a high-dimensional space.