Gaussian random projections for Euclidean membership problems
Résumé
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.
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