Lower Bounds for Local Versions of Dimension Reductions

Abstract

We consider the problem of embedding vectors from an arbitrary Euclidean space into a low-dimensional Euclidean space while preserving, up to a small distortion, a subset of the distances. In particular, preserving only the distance of each vector to a small number of its nearest neighbors. We show that even when the subset of distances we wish to preserve is very small, the problem does not become easier than when one is required to preserve all the distances.