Gaussian random projections for Euclidean membership problems

We discuss the application of Gaussian random projections to the fundamental problem of deciding whether a given point in a Euclidean space belongs to a given set. In particular, we consider the two cases when the target set is either at most countable or of low doubling dimension. We show that, under a number of different assumptions, the feasibility (or infeasibility) of this problem is preserved almost surely when the problem data is projected to a lower dimensional space. We also consider the threshold version of this problem, in which we require that the projected point and the projected set are separated by a certain distance error. As a consequence of these results, we are able to improve the bound of Indyk–Naor on the Nearest Neigbour preserving embeddings. Our results are applicable to any algorithmic setting which needs to solve Euclidean membership problems in a high-dimensional space.