Near optimal dimensionality reductions that preserve volumes

Abstract

Let P be a set of n points in Euclidean space and let 0 < ε < 1. A well-known result of Johnson and Lindenstrauss states that there is a projection of P onto a subspace of dimension O(ε-2 log n) such that distances change by at most a factor of 1 + ε. We consider an extension of this result. Our goal is to find an analogous dimension reduction where not only pairs but all subsets of at most k points maintain their volume approximately. More precisely, we require that sets of size s ≤ k preserve their volumes within a factor of (1 + ε)s-1. We show that this can be achieved using O(max{k/ε, ε-2 log n}) dimensions. This in particular means that for k = O (log n/ε) we require no more dimensions (asymptotically) than the special case k = 2, handled by Johnson and Lindenstrauss. Our work improves on a result of Magen (that required as many as O(kε-2 log n)dimensions) and is tight up to a factor of O(1/ε). Another outcome of our work is an alternative and greatly simplified proof of the result of Magen showing that all distances between points and affine subspaces spanned by a small number of points are approximately preserved when projecting onto O(kε-2 log n) dimensions. © 2008 Springer-Verlag Berlin Heidelberg.